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PACKED COLUMNS
Chapter 11
PACKED COLUMNS
11.1. INTRODUCTION Packed columns are the units most often used in absorption operations (Fig. 11-1). Usually, they are cylindrical columns up to several metres in diameter and over 10 metres high. The packing is placed on a support whose free cross section should be at least equal to the packing porosity. One of the various supports is shown schematically in Fig. 11-2. Liquid is fed in at the top of the column and distributed over the packing through which it flows downwards. To guarantee a uniform liquid distribution over the GAS OUT
Fig. 11-1. Packed column
Fig. 11-2. Packing supports
PACKED COLUMNS
411
cross-section of the column, a liquid distributor is employed. There are many types of distributor. Figure 11-3 presents schematically a weir-trough liquid distributor. Gas flows upwards countercurrently to the falling liquid, which absorbs soluble species from the gas. The gas which is not absorbed flows away from the top of the column, usually through a mist eliminator. The mist eliminator separates liquid drops entrained by the gas from the packing. The separator may be a layer of the packing, mesh or it may be specially designed.
Fig. 11-3. Weir-trough liquid distributor
Historically, the first type of packing which was widely used in industry, not only in absorption but also in distillation processes, were Raschig rings. Introduced at the beginning of the 20th century they are still used in industry. Raschig rings (Fig. 11-4) have a large specific packing surface and high porosity. They are hollow cylinders with an external diameter equal to the ring height. They can be made of ceramic, metal, graphite or plastic. In the 1930s Berl saddles were used and after the Second World War many other very efficient packings were manufactured. The main virtues of a good packing are a large specific surface area, high void fraction (porosity),chemical resistance to liquid and gas, high mass transfer rates in both phases, a small pressure drop, and low cost. Two of the features mentioned above characterize the packing particularly well. These are the specific surface of the packing and the void fraction. The specific surface
Fig. 11-4. Raschig ring and its modifications a) Raschig ring, b) Lessig ring, c) partition ring, d) Pall ring, e) Bialecki ring, f) Κ ring, g) Hy-Pak packing
412
INTRODUCTION
of the packing is the total surface area of all packing elements contained in a unit volume. Usually, the specific surface area of 2
3
industrial packings ranges from 100 to 1000 m /m . However, there are packings with smaller or larger surface areas. The void fraction is the ratio of the free volume to the bed volume. For ceramic packings the void fraction is usually about 0.7, while for metal packing it is about 0.95 and more. Packings may be divided into two main groups: random and structural. Structural packings are manufactured for a given column diameter and column height. Usually the same packing cannot be used in columns of different diameter. Random packings may be relocated to various columns. The most popular random packings are rings. Since the time when Raschig introduced his rings in industry, they have been continuously modified. Figure 11-4 shows a Raschig ring and its main modified forms. The objective in modifying the rings is to increase the mass transfer area, the porosity of the packing, and to guarantee a low pressure drop and good conditions for mixing the gas and liquid phases (in the column cross-section but not along its height), which leads to high mass transfer coefficients in both phases. From experimental investigations it follows that the most efficient of all the types of ring are Pall and Κ rings. Table 11-1 gives the basic characteristics of these rings together with data for Raschig rings. Figure 11-5 shows the comparison between these three packings [1], Table
11-1. Basic characteristics of
Type of
Dimensions
packing
mm
Raschig, Pall and Κ
Specific a
rings
surface 2
3
[m /m ]
area
Void c
fraction ["]
Raschig
25x25x3
190
rings
38x38x6
115
0.68
(ceramic)
50x50x6
92
0.74
Pall
25x25x1
206
0.90
rings
38x38x1
128
0.91
(plastic)
50x50x1
103
0.92
0.73
Κ
25x0.5
210
0.950
rings
35x0.6
155
0.953
(metal)
50x0.8
100
0.960
PACKED COLUMNS
413
The liquid mass transfer coefficient in the form of the Sherwood number and the pressure drop as a function of the gas and liquid flow rates are presented there. Apart from rings, other shapes characterized by a large specific surface area and high void fraction are used as packing. Figure 11-6 illustrates such packing shapes.
4 6 61
2 4 - UqN*]
200 400 600 — Re,
Fig. 11-5. Liquid mass transfer coefficient and pressure drop for different kinds of packings
a)
Fig. 11-6.
Some random packing: a) Berl saddle, b) Intalox saddle,
c) Interpack, d) Tellerette, e) IMTP
Recently, the application of structural packing in absorption has increased rapidly owing to the fact that it has a larger mass transfer area than random packing, all other features being similar. Another advantage is that by using this packing it is possible to obtain the same values of mass transfer coefficient in the entire column, in contrast to random packing, for which this coefficient has different values at various points of the column and depends on the location of the particular packing elements. It is also easier and cheaper to install columns already with the structural packing inside rather than assemble a column and then fill it with a random packing. One of the first structural packings which is particularly recommended, is the Sulzer BX packing. A
414
HYDRODYNAMICS OF PACKED COLUMNS
classical form of the Sulzer BX packing is grid strips bent alternately at an angle of about 30° to the column axis. Bent strips form bunches located in the column one on top of the other. The liquid and the gas flow in channels which change their flow direction while passing through the bunches. Figure 11-7 presents
Fig. 11-7. Sulzer BX packing
schematically the Sulzer BX packing. Several modifications have been recently introduced to this packing. In Table 11-2 the fundamental data concerning the Sulzer BX packing are given. Table 11-2. Characteristics of Sulzer BX packing specific
surface
area
[m /m ]
a
500
2
3
channel dimension
base
angles
bQ
r
Ψ
78
30
0.0102
[m]
Above a short description of a packed column was given. Further information can be found in the excellent books [2, 3, 4]. 11.2· HYDRODYNAMICS OF PACKED COLUMNS When considering packed columns the following problems should be discussed: a) the distribution of the liquid over the packing, b) the axial mixing of the liquid and the gas (residence time distribution), c) liquid hold-up,
PACKED COLUMNS
415
d) pressure drop and flooding of the column, e) the wetted area of the packing and mass transfer area in columns.
11.2.1. Distribution of Liquid over the Packing and Residence Time Distribution of Liquids in a Packed Column The liquid fed to the top of the packing has a natural tendency to distribute over the packing in a radial direction [5-11]. This phenomenon occurs in all random packings and in some structural packings as well. At some distance from the column top the liquid reaches the column wall and thereafter flows down it. At a lower level more liquid accumulates on the column wall, although part of the liquid returns back to the packing. At some distance from the packing top, dynamic equilibrium between the liquid flowing through the packing and flowing down the column wall is established. This ratio depends on the packing size, the column diameter, the kind of packing, the physicochemical properties of the liquid, the contact angle of the materials of which the column and packing are made, and the liquid flow rate. Fig. 11-8 presents schematically the
o)
liquid out
liquid ouf
Fig. 11-8. Distribution of liquid over the packing
phenomenon of liquid flow over packing which is uniformly (Fig. 11-8a) and nonuniformly wetted (Fig. 11-8b). The figures show deviations from an ideal flow pattern in particular annular sections for four different packing heights. If maldistribution does not occur and the liquid does not accumulate on the walls then in each annular section the ratio (W ) f / W T should be equal to 1. L loc
L
Actually, the liquid accumulates on the wall. It is shown in the figure that at a particular height (the lowest level) dynamic equilibrium, independent of the original type of column feeding, is
416
HYDRODYNAMICS OF PACKED COLUMNS
attained. In columns of small diameter the liquid flowing down the column wall can constitute up to 50% of the total liquid. In columns of larger diameter the amount of liquid flowing along the wall compared to that flowing over the packing is small. It is evident that the mass transfer coefficients for the liquid flowing over the packing and down the wall are different. This is why liquid accumulation on the column walls is, apart from maldistribution, the main reason why small packed columns cannot be expected to fully represent the mass transfer conditions when applied to large industrial columns. Sometimes, due to maldistribution of the packing elements or poor wetting of the packing, some regions of the column remain unwetted or slightly wetted. This is a highly undesirable condition which results in the column dramatically losing its absorption abilities. To avoid these undesirable conditions different types of redistributors of liquid are installed in the column. Apart from the large-scale maldistibution described above nonuniform flow on this scale, equivalent to the size of the packing, is observed [11]. It is called a "natural flow" maldistribution. Even at a high liquid velocity some packing elements are not wetted, while over other elements much more liquid flows than would result from the calculated mean flow velocity. The "natural flow" maldistribution may seriously hinder the absorption process. The liquid distribution over the packing is connected with the problem of the residence time distribution in a packed column. Investigations concerning this problem are incomplete and most of the experimental studies refer to columns of small diameter. It follows from the performed investigations that in columns of larger diameter plug flow of the liquid may be assumed. This refers particularly to tall columns. For small columns deviations from plug flow occur. Gas flow in a column with a structural packing is near to plug flow. The stream of gas flowing through the packing splits into smaller parts which combine afterwards. Continuous changes in the flow direction mean that even for columns of small diameter the residence time of the gas for particular molecules is close to a mean residence time. For random packing fairly high deviations from plug flow sometimes appear. This is connected, amongst other things, with the influence of the column walls on the packing structure. Near the wall, the packing elements are arranged in a different way than in the column centre. With an increase in the distance from the wall to the column axis the void fraction of the packing decreases to
PACKED COLUMNS
attain a constant value for a distance from the several diameters of the packing elements (Fig. 11-9).
wall
equal
417
to
Distance from the wall
Fig. 11-9. Porosity as a function of distance from the wall
Therefore, for random packings the gas velocity distribution is as follows: at the column wall the velocity is equal to zero, then it increases to reach a maximum at a small distance from the wall. Next, the velocity decreases to attain a constant value near the column axis. For equipment with a large diameter the effect of the wall regions can be negligible and thus a flat profile for the gas velocity may be assumed for such columns. The above statement is also valid for two-phase flow, except for the case where a small amount of gas is in contact with a large quantity of liquid (absorption of sparingly soluble gases). It was observed that for such conditions the liquid entrains gas particles downwards, mixing the gas along the column height, and instead of gas plug flow intermediate flow occurs, which combines the features of plug flow and flow with ideal mixing. Example l l - A . Absorption is carried out in two identical columns with structural packing. For the process conditions the equilibrium line is given by the equation Y* = m X A
A
The concentration of the soluble component in the gas at the inlet is Y kmol A/ kmol in, and the solvent is free of solute. AZ
In the first column the solute concentration in the gas drops to Y kmol A/ kmol in. Determine the final concentration of solute AO
species in the gas in the second column if, due to improper wetting, c% of the packing remains unwetted. The packing is constructed so that the gas and the liquid are not radially translocated. The gas and liquid superficial molar velocities in both columns are
418
HYDRODYNAMICS OF PACKED COLUMNS 2
identical. They are W _
2
kmol S/(m s),
kmol in/(m s) and W T
Gl η
LS
respectively. In a narrow range of changing velocities the volumetric overall mass transfer coefficient is proportional to the gas superficial molar velocity Ψ Λ 1 and to the liquid Gin
superficial molar velocity raised to the power n. Κ
a GY
~
W
m
W Gin
n
LS
A gas mixture may be treated as a dilute system. Solution The operating line described by the equation W Y
A
=
X
+
A
" W ^ Gin
for
the
first
column
(Fig.ll-Al)
Y
A0
is
<"-
A 1
>
The packing height is given by the equation Λ
ζ d Y
Ζ =
G ni K
a
G Y
m
Y
J
γ
A
A -
(11-A2)
Y A r
AO Using the equilibrium dependence (11-A2) can be written as W Z = B W
f.
°
i
n
.
and
Y A Z Γ
W ,„N" JI
Γ, r
Y
ι η ί
equation
dY. -
w
OINU mW _. W
MW
I
(11-A1),
π mGIN W.
(11-A3)
MW
~w
A
is J
equation
V
is
and upon integration
z
[
J^y" I i
i
y s w - r
1UÎÏ
l
G i n·>
l
1- W A schematic diagram of the in Fig. 11-A2.
n
w
j ^
Az
4
w —
<
U
-
A
4
>
0 A
G pl
LJ
flows in the second
column is illustrated
PACKED COLUMNS
AI)
A2)
w LS
Wo
- w r„ n
Gin
100
^AO
AO
419
,ΥΔ
T
Y' A0
YA
L n
100
G ,
Y AZ .
Fig. 11-A. Mass balance for the first and the second column
The operating line for the wetted part of the column is 100 W Y
A
=
S
L
(100-C ) W .
γ
Gin
A
ι γ »
(11-A5)
AO
The packing height in the second column equal height in the first column, is given by the relation
fl 0 0-cl
W Η
1
Η [
AZ
1 00 J
f °- 1 1Q
0 0
c
b f I V w J Y V [l00-cj L S G i n [ 100 J Yj^ o Deleting Y
ζ =
Y A
χ ln-
A
to the
packing
dY (11-A6)
1 Y -
Y
A
A
from equation (11-A6) and, upon integration we obtain
ι F
100-C
[
100 loo
Γι Z L
mW W
I
e
Η Γ
J LS L
O i n f 1l O 0 0O - c l l
LS
mW
L JJ
1 -
,
Y '
Gin
w
Ls
F
Ί"
100-C [
mW. G i n f l O O - c 1 Y, Y W L s[ l 0 0 j A 0
100
J.
(11-A7)
AO
Comparing equations (11-A4) and (11-A7) and after rearrangement, we have
HYDRODYNAMICS OF PACKED COLUMNS
420
mW. A Z
Y
L
A0
w Β
e -
JJ
[ 100
l s
EW^ JinflOO-c] Gi W
< - > U
A8
« I 100 J
LS
where mW m Y V Ginfl00-c1'
f 100 Ί " Γ
tioo- J L -
W
1
Β =
c
mW
j
W mW.
>
γ
J
I 100 J.
S
L
.
Gin LS
+
mW
AZ
2L£ γ
AO
W
X In — γ (11-A9) After mixing the gas at the top of the column the final concentration of a solute gas 0at the column outlet is obtained
A
Y ' (100-c)+ Y AO Y
c A7
AK »
100
—
< n
A 1 0
>
For instance, for the values m = 0.9
γ AZ = o . i the value of Y
c = 20%
? ! m o
η = 0.5
γ AO = o.oi
A
kmol i n is 0.0235
k
m
°!
A
k m o 1
kmol
A
iη
.
AK kmol ι η Hence, due to the maldistribution of the liquid over the packing a large decrease in absorption efficiency occurs. 11.2.2.
Liquid Hold-op on the Packing
It is particularly important to know the amount of liquid held on the packing when absorption with a chemical reaction occurs. The total hold-up is the sum of a dynamic and static hold-up. The static hold-up is the volume of liquid suspended on the packing when gas and liquid flow is interrupted in the column. The difference between the total hold-up and static hold-up is called the dynamic hold-up. From experimental investigations it follows
PACKED COLUMNS 421 that the volume of liquid hold-up on the packing depends on the kind and size of packing, the physicochemical properties of the liquid and the liquid flow velocity. With an increase in liquid flow velocity, the total hold-up rises. In a countercurrent gas and liquid flow two characteristic regions can be distinguished. In the first one, the total liquid hold-up is practically independent of the gas flow velocity. After exceeding a certain flow rate the liquid hold-up also depends on the gas flow rate (Fig. 11-10). Such flow rates form a so-called loading line.
Fig. 11-10. Hold-up in packed column
The static hold-up was found [15] to be dependent mainly on the packing size. It can be determined by the following equation ,
1.53X10·«
)
,1 . 2
The dynamic hold-up was determined by many authors [12-15]. One of the correlations [15] which appears to fit fairly well over a broad range of operational conditions is β
=2.9xl0- *Re;
where d
5
0
66
f ^ l
0
'
7
5
. ! -
1
-
2
(Π-2)
H0
Ρ
is the nominal packing size, 2 m; R e ' is the Reynolds number L
for the liquid phase, defined as ww d 1 d L Ρ R e L " S//T ε S is the cross-sectional area of the column, m ; 1 is the liquid flow rate, kg/s. This equation may be applied in flow regions below the loading line of the column.
a
M
422
HYDRODYNAMICS OF PACKED COLUMNS
There are several other equations for which liquid hold-up can be calculated* One, which is valid over very wide flow conditions, is given by Billet [16].
11.2.3. Flooding of the Column, Pressure Drop in Packed Columns
Fig. 11-11. Pressure drop as a function of gas flow velocity
Figure 11-11 illustrates a schematic diagram of the pressure drop as a function of the gas flow velocity for constant liquid velocities in random packings. For dry packing and one phase gas flow the pressure drop may be determined from a modified Darcy-Weisbach equation [15] (11-3) where the friction factor A is a function of the Reynolds number for the gas. If the Reynolds number for the gas is less than 200 A = 456 Re whereas for
,-0.742 G
(11-4)
R e ' ^ 200 G
A = 27.4 R e ' ' * 0
2 1 6
G
where the Reynolds number for the gas phase is defined as
(11-5)
PACKED COLUMNS
423
W
"G
€
'G
'
For wetted packings there are many equations and diagrams to determine the pressure drop during gas flow through the packing. These relations are not general in nature and usually relate to one type (and often one size) of packing. The exact data for a given type and size of packing can be obtained from manufacturers. As an example, the pressure drop for 25 X 25 x 2.5mm Raschig rings is shown in Fig. 11-12.
0.1
0.2 0.3 10.5 0.5 1 0.4
Fig. 11-12. Pressure drop for 25 mm Raschig rings
The increase in gas flow velocity at constant liquid velocity causes an increase in flow resistance. In the pressure drop diagrams two characteristic bends are observed as a function of gas flow velocity. The first one occurs at the point of column loading. An increase in gas flow velocity above the loading line of the column causes an abrupt rise in flow resistance. This is basically connected with a very fast increase in the total liquid hold-up on the packing and, as a result, a decrease in the free volume of the packing available for gas phase flow. At the flooding point the pressure drop is very high. Just below this point the liquid hold-up is very large and this means that the real gas flow velocities in the packing channels are very high. This, in turn, results in an abrupt increase in the friction forces between the gas and liquid phases and a further stop in the liquid flow down. At the flooding point the entire column cross-section is filled with liquid at some distance down the column. This liquid layer is maintained by a compressed gas. The liquid flowing down accumulates, the liquid layer grows and, after some time, the
424
HYDRODYNAMICS OF PACKED COLUMNS
weight of the liquid surpasses the gas pressure and the liquid abruptly descends. This permits the gas to flow upwards. After some time the phenomenon is repeated. Sometimes, the gas ejects the liquid from the packing top. At such gas and liquid flow velocities the column cannot operate in a stable way. Thus, flooding conditions represent the admissible flow rates of phases which may exist in the column. Just before flooding, the mass transfer conditions are optimal. Therefore, from the point of view of mass transfer the flow velocities should be chosen in such a manner that the column operates in the vicinity of the flooding point. However, when safe operation of the equipment is considered the designed gas flow rates are usually taken as 0.6 to 0.9 of the velocity at which the column is flooded. In the literature many relations can be found [2,16,17,18] which allow flow velocities of the phases for flooding to be determined. Below, one such correlation [18] which may be used for random packings is given. The column is flooded when the two dimensionless gas and liquid velocities satisfy the relation w
=
(11-6)
G
*
2
10430(w ) j 0.007-w*
l+2430w* + C
c
The two dimensionless velocities are given by the equations wc = d r- ι pz '^ οr gi) f ^1/3 ζ
and 1-β w^ = 2.6 — G 3 ε ι
ρ
£
2
G
l
p.
L
u, τ A
Gmax
-τ g d ζ
.
(11-8)xo
pz
where constant A reflects the effect of the column diameter A =
2 3
1+ and
-j 1 1-e
(11-9) pz
d
kmin
d^ z is the effective size of the packing element d
=6 pz
a
(11-10)
7N
/ Λ
425
PACKED COLUMNS
Example 11-B. A column packed with 25 mm Bialecki steel rings is to be used in absorption. Determine the column diameter at which the gas flow velocity is 0.8 of the flooding rate. The volumetric 3
3
gas and liquid flow rates are 3000 m /h and 30 m /h, respectively. The physicochemical properties of the gas and the liquid in the column are close to those of air and water at 30°C The basis data for 2
3
Bialecki rings are as follows: a = 206 m /m , ε = 0.959 -5 2 3 The gas parameters are υ = 1.661x10 m /s, 0 = 1.12 kg/m . G
= 8.05x10
The liquid parameters are υ
Li
-7
G
2
m /s, ρ
LI
3
= 996 kg/m .
Solution Introducing data to equations (11-7) and (11-8), using equations (11-9) and (11-10), we obtain *= 30x4 ~ 3600x*xd
W c
2
X
206 6 x d - 0.959)
additionally
1- 0 . 9 5 9 0.959
kmin -7-v 1/3 _ _« ^ _ « Λ - 4 %
χ
Γ8.05Χ10-Ί
L 9. 8 l
7.716X10-
J
2
d
2
km
0.959
y0
1
η 1 +
§ ^ . k
ϊ*-πγ x
x
2
2
x
206xd,
L
3000 x4 x206 3600 χπ χά x 9 . 8 1 X6X(1-0.959) 2
( Bn 1 )
.
2 . 6 * i ^ x j j j i x[ y
_
1+
4
kmi η
J
kmin
1.3X 10"
2
.
2
0.0 1 9 l d 4 d1 km k mii 1n JI km i i ηaJ Irm
(11-B2) Substituting equations (11-B1) and (11-B2) into equation (11-6) we have 1.3 XlO" 1 +
0
0
d
t
2
1 9
.
kmin
kmin
0.15
2430χ(7.716Χ 10" ) 4
1
+
— - v " '
d
/
.
km ι η
.
+
10430X (7.716X 10 ' ) 8
,« . L
0
0
7
2
. 7.7 1 6 X 1 0 ^ d k m i.n t
(11-B3)
426
HYDRODYNAMICS OF PACKED COLUMNS
In equation (11-B3) there is one variable, that is the column diameter,diameter, which is calculated by an iterative method. It is d = 0.84 m.This means that at such a diameter flooding kmi η
conditions appear in the column. The superficial under these conditions is 30 00x4 , . u = = 1.5 m/s G m ax 2 3600χπχ0.84 Under the conditions presented above should be 0.8 of the flooding rate. Thus, u
= 0.8 u G
the
gas
gas
velocity
flow
velocity
= 0.8 x 1.5 = 1.2 m/s Gmax
For such a velocity the column diameter is
d
= ν k
u
= ν
4 χ 3600 χ
3000 π x 1.2
= 0.94 m
G
11.2.4. Mass Transfer Area The wetted surface area of the packing and mass transfer area have been determined in many studies using various measuring techniques [19-22]. The following conclusions may be drawn from these works: a) The mass transfer area is a function of the type and dimensions of the packing elements. b) The material from which the packing is made has a very strong influence on the mass transfer area. c) Up to the loading point, the mass transfer area depends only on the liquid flow velocity. Above this point it also depends on the gas velocity. d) When a distributor with about 100 points per 1 m is applied, the mass transfer area is in fact independent of the column height. e) In columns of small diameter the mass transfer area is a function of the column diameter. f) The mass transfer area depends on the physicochemical properties of the liquid flowing down through the packing. Many data referring to the mass transfer area are presented in Fig. 11-13 [23]. These values represent the mass transfer area for various types of packing under operating conditions below the loading point.
PACKED COLUMNS
427
.a
ο
15
20
0
8
Fig. 11-13. Mass transfer area for different packings
There are significant discrepancies between the experimental results concerned with the wetted surface area of the packing, the mass transfer area in physical processes and the mass transfer area in processes of mass transfer with a chemical reaction. Puranik and Vogelpohl [24] attempted to overcome these discrepancies. They assumed that in a column there are two types of area: a static area a , proportional to the static hold-up, and a dynamic a . s
d
proportional to the dynamic hold-up. In the case of physical mass transfer such as liquid evaporation to the gas, both static and dynamic areas constitute a mass transfer area a m
a
m
= a
s
+ a, d
(11-11)
In physical absorption, the liquid suspended on the packing in the form of a static hold-up very quickly becomes saturated by soluble species and therefore the mass transfer area is only the dynamic area. In absorption with an instantaneous chemical reaction, when the concentration of a reacting substance in the liquid is very high, due to gradual exhaustion of the reacting substance in the liquid, the mass transfer area should correspond both to the dynamic and static area. In the case of slow chemical reactions occurring in the liquid phase, the mass transfer area is equal to the sum of the dynamic
428
HYDRODYNAMICS OF PACKED COLUMNS
area and a fraction of the static area. the ratio of the given rate of chemical instantaneous chemical reaction. On the basis of data presented in equations yielding static and dynamic areas r _ a, A A _ Ä d
= 0.229 - 0.091 In a
a
We __ FrL
the
following
° (11-13) v '
a + ( tanh R - 1 ) — î -
m
Re
the literature, are proposed ϊ -0.182
(11-12)
a a — a a where
This fraction follows from reaction to the rate of an
a
(11-14)
w L
a
nL 2
We
Fr and
L
=
L
σ
=
w
(11-15) v '
a pL σ 2 W l a 2
is the critical surface tension.
c
The following range of variables has been taken — 5 - = 0.08 - 0.8 ; w = 0.25 - 12 kg/(m s); a L 3 2 / / l = (0.5 - 13 ) x l 0 " kg/(m s) ; a = (2.5 - 7.5)XlO" N/m; ρ
= 800 - 1900 kg/m ; 3
L
Several values of σ Table 11-3.
d
ρ
= 10 - 37.5 mm
for different materials are given in Table 11-3.
c Critical values Ο
for different materials c
Material
Critical
tension [N/m]
Glass
7,,4xl0"
Ceramic
6..2xl0'
Graphit
6.,4xl0"
Steel
7.,2xl0"
Parafine
2 2 2
2 2
2..OxlO"
PACKED COLUMNS
429
1 1 3 · MASS TRANSFER COEFFICIENTS IN PACKED COLUMNS Mass transfer coefficients in packed columns were determined in numerous experimental investigations and theoretical studies [25-29]. It follows from these studies that mass transfer coefficients in the gas and liquid phases depend on the flow velocities of both phases, the physicochemical properties of the liquid and gas, represented by density, viscosity and diffusion coefficients, and the type and size of packing. In the case of volumetric mass transfer coefficients, their values also depend on the liquid surface tension and contact angle. In the literature there are many equations, diagrams and tables giving the values of mass transfer coefficients as a function of various parameters. Below, those equations typical for a given packing are presented. Mass transfer coefficients in the liquid phase were investigated mostly on the basis of water systems. For such systems and traditional packings (Raschig rings and Berl saddles) the correlation derived by Mohunta et al. is recommended [30]
i M l 'j iV* î0'^i l 'r fi V Vi VV rf fτT 2 /
î
1 / I9
3
Br)
AL m
w
a
9
,
/ / 44
η
1 / 2
fe)
(-7V] z
1v
L
(11-16)
The ranges of particular variables are as follows 2
w L = 0.1 - 42 kg/(m s) η d
Wq = 0.015 - 1.2 kg/(m s) 2
3
Ρ
where d
= (0.7 - 1.5 ) X 1 0 ' kg/(m s) = 6 - 5 0 mm,
S c L = 140 - 1030
is the nominal dimension of the packing element.
This equation determines the volumetric mass transfer coefficients. It follows from the investigations performed by Sherwood and Holloway [25], and from the above equation, that the volumetric mass transfer coefficients in the liquid phase are in fact independent of packing size. In the case of viscous liquids no complete or detailed studies have been devoted to the determination of the effect of viscosity on the mass transfer coefficient. From fragmentary investigations it follows that this effect is very strong. For such system the Norman and Sammak [32] equation can be applied k
d
3
4w > 0 . 6 i r d g ρ τ Ί ΐ / 6 Γ ηΊ f
^1/2
430
MASS TRANSFER COEFFICIENTS IN PACKED COLUMNS
This equation is valid for the following range of parameters S C l = 108 - 43500
R e L = 1.2 - 545
/fL = (0.4 - 20 ) x l 0 " kg/(m s). 3
Mass transfer coefficients in the gas phase for traditional packings may be calculated using the equation given by Onda [33] k ^
a
AG
w_
r
AO
^o.7r
^ Ό
J
^0.33
η_ G
AG
J
V
where the constant C depends on the nominal packing size d^ for
d
for
d
^
ρ
15 mm
C = 5.23
< 15 mm
p
C = 2
For Sulzer packing the equations derived by Bravo [34] are valid - for the gas phase Sh
= 0.0338 R e ' G
0
Sc
8
G
(11-19)
0 , 33
G
where
AG d
+
PM^
Re'
»
t
°»
e<
U
a n G
e ff
d
U
e i
L
e ff
u
v a
e nt
«)
(11.21)
AL
=
2
V
effective gas and liquid velocities,
diameter packing.
<2) u AAL i
,
hîlL
e f f
a re
^eq " l * ^ - for the liquid phase
k
u
ηο
Ο
where
t
L,eff
\ 71 *ο
(
H
-
2
2
>
where S' is the length of the corrugation side. Another equation is given by Spiegel and Meier [35]. Fair and Bravo [36] gave an excellent review of calculation of a column containg structured packing.
PACKED COLUMNS 431
Example 11-C. Ammonia of low concentration is removed from air in a column packed with 25 mm Raschig rings. Determine the values of
the mass transfer coefficients in both phases, if the gas and liquid flow velocities
in the
2
G
column
2
L
are w = 1 kg/(m s)
and w = 5 kg/(m s),
respectively. The physicochemical parameters of the gas and liquid are %
= 2.3XlO"
m /s
5
%
2
= 1.98XlO'
m /s
9
2
AL
AG
pQ
= 1 . 1 5 kg/m
if
= 1.84XlO"
3
pL
= 997 kg/m
η
= 9 X l O ' kg/(m s)
kg/(m s)
5
3
4
a = 190 m Im . Solution The mass transfer coefficient calculated from equation (11-16)
AL
m
0 V| ^ fΓ „ •Î 5I 'Xx9.Xx1l0o' - 'χx1 l9w J
<·
3 ,
4
9 . 8 1 X 9 97 2
the
l ^ f
9.81X997 = 2.5 X l 0 - x f M90X(9X10"V 3
k a
in
4
2
3
4
phase
99 7 x 9 . 8 l V l
99 XXl O1"
Ι / 4
liquid
9X 1 0 "
. " VV ' " .
0
is
/ 9
J
4
.
0 1 2 9
...
*-997x 1 . 9 8 X 10" J 9
-'
For a linear liquid flow velocity \
- ~T~
=
L
- 9 5 Γ ~"
0
0
05
M S/
the interface a r e | determined from Fig. 11-13 for 25 mm Raschig rings is 105 m Im . Hence the mass transfer coefficient in the liquid phase is k a , AL m 0.0129 - ^ - 4 . k = = —ϊ-τ^ = 1.23x10 m/s a 105
AL
w
in
the
gas
phase
is
determined
>0.7
= 5.23x(2.3xl0" )xl90x ί — ] 5 J H1.84xlO' )xl90 5
A G
f 1 Ll.l5x(2. 3 xlO'V 1
χ
A
m
The mass transfer coefficient from Onda's equation (11-18) k
l
8
4 x
1 0
(190X0.0025)"
2
= 0.047 m/s
432
COCURRENT PHASE FLOW
11.4. COCURRENT PHASE FLOW Packed columns may also operate in a system of co-current phase flow. This operation is particularly desirable when gas solubility in the liquid is very high and the driving force of absorption is, in fact, independent of counter- or co-current flow systems. In the case of a cocurrent system no limitations connected with column flooding occur with the gas and liquid velocities. Since it is possible to use much higher linear velocities for both phases, the column diameter in the co-current system is smaller than in the countercurrent system. Packed columns with cocurrent flow of the phases are also employed in absorption accompanied by a chemical reaction when the packing also acts as a catalyst (trickle bed). From the hydrodynamics point of view, the mechanism for cocurrent flow of the phases is more complicated compared to that for countercurrent flow. For high loading of the column with the liquid and at a low gas flow, or with a significant gas flow and a low liquid flow, as well as for intermediate cases, both phases flow downward in the column as continuous phases. The mechanism of flow in this region is similar to that in a countercurrent column. An increase in the velocity of one phase at a constant velocity of the other phase causes pulsating flow of phases in the column or spraying of the liquid. The characteristics of a cocurrent column operation can be represented by Fig. 11-14. In co-current absorption, when the liquid and gas flow as continuous phases, the mass transfer coefficients and mass transfer areas are similar to those in the counter-current flow. This region of operation is the most suitable for absorption processes.
0.01
OOS
0.1
QS
1
Fig. 11-14. Characteristics of co-current gas and liquid flows
PACKED COLUMNS
433
11.5. BALANCE EQUATIONS OF PACKED COLUMNS FOR BINARY SYSTEMS Chapter 9 presents balance equations for the differential volume of any absorber type for various models of phase flow. In packed columns the real flow of both the gas and the liquid phases is usually approximated by a plug flow model or by a plug flow model with longitudinal dispersion. These absorbers can operate with cocurrent and countercurrent flow, and, apart from a short start-up period, under steady-state conditions. Owing to discrepancies encountered in the literature, which concern the effect of longitudinal mixing on the process of absorption, unreliability of correlations for the dispersion coefficient, and the fact that the existing correlations for mass and heat transfer are valid for a plug flow model, we shall present only those balance equations for packed columns which are based on the plug flow model. In addition we shall assume that (i) in the gas phase there is only one active component and a component insoluble in the liquid (inert component), and in the liquid phase an active component and a nonvolatile solvent are present. (ii) the steady-state pressure. w A6
isothermal
w
process
takes
Λ
AL
a
place
W
constant
WAL
A0 XA
*A
under
ΧΑ
Gm
NAG
1
1
s 0)1
ί b)
W
AG*dWA0
V<*A
V
Gin
Fig. 11-15. Scheme of differential element of a packed column binary systems: a) cocurrent phase flow, b) countercurrent flow
For a column element of differential volume dV (Fig. 11-15), under the above simplifying assumptions, and in the case of physical absorption, equations (9-23) and (9-31) using additionally equations (9-50) and (9-51), are reduced to the form
434
BALANCE EQUATIONS OF PACKED COLUMNS
- gas phase dW = + a_ N ? „
A O
Ζ
m
(11-23)
AG
- liquid phase dW - j — ^
λ
= a
Cl Z
N* m
(11-24)
AL
The minus sign on the right hand side of equation (11-23) refers to the cocurrent phase flow, while the plus sign refers to countercurrent flow. The composition of each phase results from the following formulae S
Λ =
A
W
A r
Q
"
2
5)
i η
s w
s X
A
L
=
(11-26)
s
where S is the cross-sectional area of the column. Substituting the above equations into equations (11-24), we obtain after rearrangement d Y
.
a
= +
A
dz
m
G
a
(11-27)
AO
n
t
ι η
i dX
and
f«
N
s
(11-23)
S Ν* S
The boundary conditions for differential equations (11-27) and (11-28) are known, provided that the concentrations in the gas and the liquid components entering the column are known (Fig. 11-16). Thus, - for the cocurrent flow V»
=
Y
A0
A
X( 0 )
»
X
U
2 9
n
3 0
A0
< -
X
< -
>
- for the countercurrent flow Y
Z
A< >
=
Y
AZ
X
A<°> »
A0
>
PACKED COLUMNS
435
ο)
Fig. 11-16. Diagram of a packed column - binary systems: a) cocurrent flow, b) countercurrent flow
Due to the continuity condition of the mass fluxes Ν
- Ν AG
δ
= Ν
AG
*
*
= Ν
AG
= Ν
AL
δ AL
- Ν
(11-31)
AL
the differential equation (11-28) can be solved comparing equations (11-27) and (11-28) we have dY in
dz
analytically.
After
dX = + L s
(11-32)
dz
Upon integration of the above equation in the range from Y (0) to Y A (z)
and
from
X A (0)
A
to X fz),
A
taking into account
A
conditions (11-29) or (11-30) and neglecting to the current column height z, we obtain - for the cocurrent flow G. (Υ
- Y J
δ
in
A
= L (X
AO'
S
V
A
boundary
ö
- X
A
J
the
symbols
referring
(11-33)
AO
- for the countercurrent flow G
iJ Î - V » Y
-
L
S Î ( X
-
X
A0>
(11-34)
Equations (11-33) and (11-34), as was mentioned in chapter 10, are called the operating line equations. They represent a relationship between the actual compositions of the gas and liquid phase in a given packed column cross section.
436
BALANCE EQUATIONS OF PACKED COLUMNS
For binary systems the calculation problem for packed columns is usually formulated in one of two ways: a) For a given absorber height Ζ (or mass transfer area) calculate the concentration (or absorption efficiency) of the active component in the gas phase, which leaves the equip ment (the calculation refers to the existing absorber which is to work under new operating conditions). b) Determine the absorber height Ζ (or mass transfer area) at which the desired concentration (or absorption efficiency) of the active component in the gas leaving the absorber is achieved. The solution of type a) consists of numerical integration of the differential equation (11-27) in the interval < 0 , Z > for the initial condition Y (0) = Y . A
v
7
AO
To solve a problem equation (11-27) to d
Y
A
of type b), it
S Ν
a
m
is convenient
to
transform
δ
AG
since for this form of differential equation the integration interval is known for the independent variable Y . For the assumed A
absorption
fj
efficiency
of
the
active
component,
the
molar
abs
ratio of this component in the gas leaving the column is - for the cocurrent flow Y
AZ
«
=
Y
- ".bs>
U
A0
< -
3 6
>
- for the countercurrent flow Y
A<°>
=
Y
A0
«
<
!
- ".bs>
Y
AZ
Π
< -
3 7
>
The calculation of the column height Ζ (or mass transfer area) at which the desired active component concentration (or η t ) is abs
achieved consists in a numerical solution of differential equation (11-35) in the interval < Y ,Y > for the initial condition z(Y ) = 0. AO
AZ
AO
The column height Ζ can also be calculated using the integrated form of equation (11-27)
PACKED COLUMNS
437
The application of equation (11-38) is particularly convenient when both the gas and liquid phases are dilute solutions and the absorption equilibrium line is a straight line (see Chapter 10). In such a case the integral on the right hand side of equation (11-38), and thus the column height, can be calculated analytically (example 11-D). For concentrated solutions or when the equilibrium line is not a straight line, the integral constituting die right hand side of the equation must be calculated numerically. The gas (liquid) phase, apart from the active component, often encompasses a mixture of several inert (nonvolatile) components. As was mentioned in Chapter 9, from the formal point of view such an absorption system is a multicomponent system and the calculation of packed columns should be made on the basis of equations for multicomponent systems. In some cases, the mass fluxes and N ^ L are calculated on the basis of approximate relations, in which the concept of efficient diffusion coefficient is used (Chapter 4), and then it will be relatively simple to calculate the columns and to solve the balance equations for binary systems. The calculation of packed columns is also relatively simple in the case of a chemical absorption, when in the liquid phase a instantaneous reaction occurs A
+
ν
Β
Β
>
C
(11-39)
and the reagents Β and C are nonvolatile components. In such a case, the mass balance for the active component in the gas phase is given by equation (11-23), and the balance equations (9-16) of the liquid phase reduce to a single differential equation in the form d
*
Z
W T B L
λ
= a m NT T BL
(11-40)
The molar ratio of component Β in the liquid phase results from the following equation c
X°
Β
S wnT = —,—— Lr
(11-41)
S
Substituting this equation into equation (11-40) we obtain .„δ
„
„δ
438
BALANCE EQUATIONS OF PACKED COLUMNS
The assumption of a instantaneous chemical reaction (equation 11-39) is equivalent to the assumption that this reaction takes place only in the liquid film and the concentration of component A in the liquid bulk is, in fact, equal to zero. The differential equation (11-42) can be solved analytically * δ after the relation between mass fluxes Ν and Ν is determined. AL
BL
For the reaction taking place in the liquid film, according to the considerations presented in Chapters 7 and 10 one may write dN dN t . = - — . (11-43) dz ν dz Β
* δ Upon integration of this equation in the range from Ν to Ν and AL AL c * δ from Ν to Ν , we have BL AL
BL
= —
- N*
AL
V
ÎN* \
B ^
BL
- Ν* 1 BL
(11-44) '
J
*
Since it has been assumed that reagent Β is nonvolatile (N
BL
=
0)
and the concentration of component A in the liquid bulk is equal to zero (N Ν*
= 0 ) , equation (11-44) simplifies to the form AL
= -υ
BL
N* Β
(11-45)
AL
'
At the interface there is the condition N
AO = 1
<- >
N
U 46
L
After employing equations (11-45) and equations (11-27) and (11-42), one obtains d Y
G.
in
, dz
f
L
<
= + —? v
d X
Β
(11-46)
f
^—dz
in
differential
(11-47)
In this equation the plus sign refers to co-current phase flow and the minus sign to the counter-current flow. Upon integration of equation (11-47) in the range from Y (0) to δ δ Υ (ζ) and from Χ (0) = X to Χ (ζ), and after neglecting the Α
Β
B0
Β
symbols concerning current column height z, we have
v
PACKED COLUMNS
439
- for the cocurrent flow
in
A
AO
V
Β
Β
7
BO
- for the countercurrent flow G
iJ Î - V » = - "IT Y
-V
1
Β
^- > U 49
The calculation procedure for a instantaneous chemical reaction expressed by equation (11-39) is almost the same as in the case of physical absorption. For problem a) the procedure includes a numerical solution of differential equation (11-27) in the interval < 0 , Z > with the initial condition Y (0) = Y . For problem b) it A
AO
encompasses either a numerical solution of differential (11-35) in the interval < Y .Y > with the condition z(Y J AO
AZ
equation = 0, or
AO
a numerical (sometimes analytical) calculation of the integral in equation (11-38). The difference is that in the case of physical absorption the liquid phase composition is determined by equation (11-33) or (11-34), while in absorption with a chemical reaction (11-39) by equation (11-48) or (11-49). Example 11-D. In a countercurrent packed column ( F i g . l l - D l ) pure water is used as the solvent in removing ammonia from air. Determine the column dimensions at which 95% absorption of NH^ takes place under the following conditions: - mean temperature in the column Τ = 298.15K - mean pressure in the column Ρ = 1.013 χ 10 Pa - volumetric flow rate of the gas phase at the column inlet (at 5
273.15K and 1.013 Χ 10 Pa) 5
'
ν Λ „ = 1 m /s 3
OGZ
- mole fraction of NH^ in the gas phase at the column inlet y = 0.03 7
AZ
- water flow rate at the column inlet equal to 1.1 of the minimum value. The equilibrium line is given by the equation Y* = 0.98 X* A
A
v
440
BALANCE EQUATIONS OF PACKED COLUMNS
The packings are ceramic Raschig rings of size d = 0.025 m. The physicochemical parameters of the gas and liquid phases as follows - density
1.18 kg/in
PG=
viscosity
ρΊ = 997 kg/m Li
= 1.86X 10' kg/(m s) 5
η
η = 8.94X 10" kg/(ms) 4
G
L
surface tension
σ = 0.072 N/m
molecular mass
M = 29 kg/kmol
M = 18 kg/kmol L
G
diffusivity
are
AG
=
2.31xl0" m /s 5
2
AL
=
2.0xl0' m /s 9
2
Assume that the gas phase is a mixture of ideal gases.
Fig. 11-D1. Diagram of a packed column for ammonia absorption in water
Solution Due to the very small concentration of NH^ in the inlet gas, the column calculation will be made using the equations applicable to dilute systems. 1. Mass balances of the gas phase The volumetric flow rate and total gas phase concentration the process conditions are ν
°
Z
= ν
V °°
Z
P
T
o
=
l
x
L 0 l W
298,15 = 1.013 XlO"' 2 7 3 . 1 5 x
where subscript ο refers to normal conditions.
l
W
/
s
.
( 1 1 D 1 )
in
PACKED COLUMNS
C
o -
π
-
lO^kmol/m
=
8 3 Ϊ 4 ^ 9 8 ° 1 5
441
(11-D2)
3
The flow rate of the inert component (air) is calculated from the formula
ο i n,
- ooz ™ 4R T«
0
-
υ Ά ζ· ' .
< "
1
0
1
8314 χ
3
x
1
0 3
273.15
Χ (1 - 0.03) = 4.327Χ10" kmol/s
(11-D3)
2
The molar ratio of NH^ in the gas mixture is given by - at the inlet to the column = J
Y
= T T T T ^ U = 3.093 X 10" kmol A/kmol in
A Z
2
y
" AZ
~
(11-D4)
'
- at the outlet Y A f t = (1 - 0.95) Y AU
= (1 - 0.95) X (3.093 χ 10" ) 2
ΑΖι
= 1.547x10 kmol A/kmol in (11-D5) The transfer rate of ammonia absorbed between the phases in the column is G. ( Y . . - Y . J in
AZ
= (4.327X10" ) χ (3.093X10" - 1.547xl0" ) 2
2
3
AO
= 1.271 XlO" kmol A/s
(11-D6)
3
2. Mass balances of the liquid phase. A minimum ratio (L IG ) . (Fig.ll-D2) is a result of equation S
(10-5) rearranged to form
f "TT" 1 » in
min
in min
YaZ
" °
ΑΖ
^
Y
^
Ya
m AO where the slope of the equilibrium line Since pure water enters the absorber ΧΑΛ = 0
m = 0.98.
AO
Substituting numerical values into equation (11-D7), we obtain -2 -3 r L "\ A
mM
442
BALANCE EQUATIONS OF PACKED COLUMNS
A'
Fig. 11-D2. Minimum ratio L / O .
A
for ammonia absorption in water
The flow rate of the solvent (water) is equal to L
S
= 1.1 G
ί-τΛ]
in
νI
Cj.
= 1.1 Χ (4.327X10" ) χ
0.931
2
J
in ^ min
= 4.431 XlO" kmol S/s
(11-D8)
2
The molar ratio of NH^ in the liquid leaving the column is X
AZ
=X
G +-r^(Y - Y J = 0 + AO Ls AZ AO'
-2 4
3
2
1
x
10
4.431 x l O "
2
x (3.093XlO"
- 1.547X10' ) = 2.869X10' kmol A/kmol S 3
2
(11-D9)
2
3. Column diameter In Example 11-B it has been shown that equation (11-6) through (11-10) can be transformed to a single nonlinear algebraic equation f( d
. ) = 0
(11-D10)
kmin
where the only unknown is the minimum diameter d
,
at which the
kmin
column is flooded. Proceeding as in example 11-B, the column diameter was calculated to be d = 0.8271 m for Raschig rings of d = 0.025 m kmin
(a = 190 m Im
ρ
and ε = 0.73, cf. Table 11-1) under the assumed
PACKED COLUMNS
process conditions. The superficial to d t . is equal to corresponding to
linear
gas
flow
443
velocity
k kmin
4 ν
GZ
=
"
G m ax π
d
x
4
=
π x
2
1 . 0 9 2 = 2.032 m/s
( 0.8271)
(11-D11)
2
km in
It was assumed that u
(11-D12)
= 0.7 X 2.032 = 1.422 m/s
= 0.7 u G
Gmax
The column diameter corresponding to this velocity is
/ GZ 4 V
= ν
π u
4 x 1 092 πΧ 1 422
/ = ν
G
(11-D13)
= 0.989 m
In further calculations the column diameter d = 1 m was assumed. k
4. Mass transfer coefficients The cross-sectional area of the column is equal to S =
n
S
2
=
π
?
= 0.7854 m
1
The superficial mass flow are, respectively _ j A W G= —g L M
s
W
= —g"
L
velocities of the gas and liquid
(4.327xl0" )x29 „ 2N 1 < o f t. ο 7 8 5 4' = 1.598 kg/(m s)
phase
2
=
w
w
(11-D14)
2
(4.431 x l 0 ' ) x 18 o. 7 854 2
L =
The mass transfer coefficient Onda's correlation (11-18)
, ,1
1
=
1
0 1 6
Λ
k
8
2
7
/
/ ( m
s
/ tl (11-D15)
Ν
>
(H-D16)
in the gas phase is calculated
from
,0.7
k
AG
=
5.23x(2.31xl0" )xl90x 5
1 . 598 ( 1 . 8 6 X 10
1 .86x10
0 . 33
-5
1.18X(2. 3 1x10
) x 190j
X(190 x 0.025)" = 6.478 Xl0" m/s 2
2
) (11-D17)
444
BALANCE EQUATIONS OF PACKED COLUMNS
For dilute systems it can be written as k
= k Gy
= k GY
C AG
= (6.478XlO" ) χ 2
(4.087XlO" ) 2
G
= 2.648xlO" kmol/(m s) 3
2
(11-D18)
The volumetric mass transfer coefficient in calculated from Mohunta's correlation (11-16)
the
liquid
phase
2/ 3
k
9 .81x997
a = 2.5XHTX
AL m
1/9
9.81
190χ (8. 9 4X10" )
X997
8.94X 1 0'
4
1/ 4
1 . 0 1 6 x ( 8 . 9 4 x 10" )x 190 3
4
9 .81 x 9 97 2
3
8 .94x 1 0
X
4
is
-1/2
-4
[997 X ( 2 . 0 χ 10"')J
.-3
= 3.940x10'" 1/s
(11-D19)
The mass transfer area is given by the correlation (11-14) 0.04 1
1.01 6
a = 1.045 X190X m
.
r
-» 0 . 1 3 3
1.0 1 6 X 190x997x0.072
1 9 0 x 8 . 9 4 X 10" -0.182
0.072 0.062
= 58.86 m / m 2
3
(11-D20)
The critical surface tension σ = 0.062 N/m has been obtained from c
Table 11-3. The molar mass transfer coefficient is equal to P
^ A L V k
Lx
3.940X10" 58. 8 6
L
M
=
3
997 18
= 3.708XlO" kmol/(m s) 3
2
(11-D21)
The overal mass transfer coefficient Κ
o,-
K
2.648x10"
Lx
= 1.558XlO" kmol/(m s) 3
is equal to
1
m Gy
Gy
2
0.98 3.708x10" (11-D22)
PACKED COLUMNS
445
5. Packing height The
packing
height
sufficient
for
achieving
the
desired
NH^
absorption can be determined analytically since the equilibrium curve is a straight line, and the gas and liquid phases are dilute solutions. The differential mass balance equation for NH^ in the gas phase is given by equation (11-27) dY
a _
S Ν -
A°
o
.
2
7)
i η
At the interface, NH holds
the
condition
of
continuity
of
molar
flux
for
3
N
(11-D23)
=
A G
In a gaseous film, NH^ diffuses through an inert component
(air)
and in a liquid film through a nonvolatile solvent (water). The molar fluxes are calculated from the equations holding for dilute solutions (Table 5.1) N
AG « V A - l> Y
Y
The value Y
(U
-
D24)
will be expressed as a function of Y . A
A
Using equations (11-D24), (11-D25) and the equilibrium line equation Y* = m X* A
(11-D26)
A
in equation (11-D23), after rearrangement, we obtain m k Y* = A m
m k Y
τ-
k
+
A
k
Gy
The value of X X
A
»
X
A0
A +
+
=-
m k
Lx
Gx
xV-
+ k
x A
(11-D27)
A
Lx
is calculated from the operation line equation (11-34) -TT-
( Y
S
Y
A - A0>
Substituting this value into equation (11-D27) we obtain
n
( -
D
2
8
>
(
446
BALANCE EQUATIONS OF PACKED COLUMNS
mk
γ*
m k_ G y + k Lx
A
γ
r
mk mk.
+
X
-ί mkΓG G yy
A
G. G.
r
Γ -LxL A 0
+
-, Υ
Υ
- Τ "S< Α - Α 0 >
J
(11-D29) Application of equations (11-D24) and (11-D29) in equation (11-27), after rearrangement, leads to the following differential equation d
Γ
Y
A
indz
m
G7
yL
f
i
m
k + Gy
k
l Lx
k
m
G. ^ L
L
ΙJ
S
m k m k
+
r - î T - h r »· « Y
Gy
Lx ^
x
< J
S
"-
D30)
J
Upon integration of the above equation in the range Y A = Y A 0 to Y
= Y A
and of ζ = 0 to ζ = Ζ, the relation determining the
AZ
column height is obtained
x In
G.
[
Ζ =
r
Y
mG. s AA Z
Ί J
mG LY - ^
a SK m
Gy
A Z -
Y
Y
A 0 > -
f
f
i
X
A 0
(11-D31)
- mX AO
AO
Introducing numerical values, we have Ζ = 8.341 m In the example considered,
molar
fluxes
Ν ^ and Ν AG
τ
were
AL
discussed on the basis of formulae valid for dilute systems. Moreever, mass gas and liquid velocities were assumed constant along the column.If these fluxes were calculated on the basis of formulae holding for concentrated systems, the packing height which would guarantee 95% NH^ absorption would be equal to Ζ = 8.05 m. Therefore, the difference between the two solutions is insignificant as it does not exceed 4%. Moreover, the column height obtained from the exact solution is lower than that under the assumption that the system is diluted, so the result obtained
PACKED COLUMNS
447
is safer from the practical point of view. Therefore, for dilute systems in which the concentration of the active component in the gas and liquid phase does not exceed 10%, for the sake of simplicity of calculation, the use of equations (11-D24) and (11-D25), which hold for dilute systems, is recommended. Example 11-E. In a packed countercurrent column of diameter d^ = 1.5
m absorption of CO^ from
a gas mixture is carried out by
means of water. Determine the packing height at which 99% absorption of CO^ is achieved. The process parameters are as follows - mean temperature in the column
Τ = 298.15 Κ
- mean pressure in the column Ρ = 2 χ 10 Pa - molar flow rate of the gas phase at the column inlet 6
G
= 4.46XlO" kmol/s ζ - molar flow rate of the liquid phase at the column inlet 2
L
= 4.03 kmol/s ο - molar fraction of CO^ in the liquid entering the column χ
= 3.07XlO" kmol A/kmol 5
Λ
AO
The composition of the gas phase entering the column is given in Table l l - E l . Table l l - E l . Composition of gas phase
Component
c o 2 (A)
Mole
fraction
0. 300
(B)
0 .012
(C)
0. 515
(D)
0. 170
CH4 (E)
0. 003
CO N 2
The equilibrium line is given by Henry's law in the form where
He = 1.65x10
Pa.
The packing consists of ceramic Raschig rings of diameter
d^ =
0.025 m. The physicochemical parameters of the liquid are: /> L =
448
BALANCE EQUATIONS OF PACKED COLUMNS
997 kg/m? η = 8.94 χ 10" Pas, σ = 0.072 N/m, Ni = 18 kg/kmol, SB = 4
-9
Li
L«
2
AL»
2.01 x 10 m /s. The physicochemical parameters of the gas mixture components are tabulated in Table 11-E2. Assume that the gas phase is a mixture of ideal gases. In the calculation of multicomponent mass transfer use an approximate formula for an effective diffusion coefficient for C 0 2 through a mixture of inert gases. Table 11-E2. Physicochemical properties of the gas phase components (T = 273.15K, Ρ = 1.013X10*Pa) ο ο j
component
M
η,χΙΟ
/ H T J
J
J k r
5
DxlO
J
5
AJ
A Β
44
115.5
28
61.4
1.76
1.37
C
2
81.3
0.898
5.5
D
28
59.5
1.82
1.33
Ε
16
55.2
1.15
1.53
1.57
Solution Owing to the very low (as compared to CO^) solubility of the other components of the gas mixture, the presence of these components in the liquid phase has been neglected. Moreover, it has been assumed that the solvent (water) is not volatile. 1. Mass balances of the gas phase The total concentration of the gas phase in the process conditions is C
o - Ί Γ Τ - - 8314X 2 9 8 1 5 = 6
8
0 6 8
X 1 0
"
2
"
( 1 1E 1 )
The total flow rate of inert components is equal to G. = G (1 - i l in L AL = 3.122x10
= (4.46 Χ ΙΟ" ) Χ (1 - 0.3) 2
(11-E2)
kmol/s
The molar flow rates of particular inert components are as follows = 4.46XlO"
G B
2
-2
χ 0.012 = 5.352χ 10" kmol/s 4
-2
GC - 4.46XlO χ 0.515 - 2.297XlO'' kmol/s -2 -3 G d = 4.46x10 χ 0.170 = 7 . 5 8 2 x 1 0 kmol/s 2 4 G - 4.46XlO" χ 0.003 = 1.338XlO" kmol/s Ε
PACKED COLUMNS
449
The molar ratio of C 0 2 in the gas mixture is - at the column inlet y A Y = 1 - ^ = -^j^ A Z l - y l - 0 . 3
= 4.286 X l O ^ k m o l A/kmol in
(11-E3)/
- at the column outlet Y
1
= 0.01 Y AO
= 0.01 Χ (4.286 Χ 10" ) AZ
3
= 4.286xl0" kmol A/kmol in
(11-E4)
The quantity of C 0 2 absorbed in the column per unit time 2 1 3 G. (Y - Y . J = (3.122X10" ) χ (4.286X10" - 4 . 2 8 6 X 1 0 " ) ιη
AZ
AO
= 1.325x10
2
kmol A/s
(11-E5)
The molar flow rate of the gas phase leaving the column 2 3 G 0 = G . n ( l + Y A 0 ) = (3.122X10" ) χ (1 + 4 . 2 8 6 X 1 0 " ) = 3.135X10"
2
kmol/s
(11-E6)
2. Mass balances of the liquid phase The molar flow rate of the solvent 3
L s = L 0 ( l - x A 0 ) = 4.03χ(1-3.07XlO" ) s 4.03 kmol/s
(11-E7)
The molar ratio of CO^ in the liquid - entering the column X
= .
A
3
° = AO
0
7 X
1 0
5
= 3.07 XlO" kmol A/kmol S
1-3.07x10" (11-E8)
- leaving the column X
AZ
=X
+ - ^ - ( Y - Y ) = 3.07 x 10" + — — - j — T T T ~ — L AZ AO 4.03 5
AO
S
3
3
X ( 4 . 2 8 6 X l O ' U ^ e ö X 10" )= 3.318x 10" kmol A/kmol S
(11-E9)
The molar flow rate of the liquid phase leaving the column 3
L=L(1+X Ζ
S
) = 4.03 χ (1+3.318 x l O " ) = 4.043 kmol/s AZ
(11-E10)
v
450 BALANCE EQUATO INS OF PACKED COLUMNS 3. Mass transfer coefficients A comparison of G
with ϋΛ
Z
(equation 11-E6), and Y
AZ
0
(equation
11-E3) with Y ^ o (equation 11-E4) shows that the composition and molar flow rate of the gas (and, therefore, the viscosity and density) are subject to appreciable changes along the column height. These variations, flow rate in particular, significantly affect the mass transfer coefficient k , causing it to vary along
the
column
height.
The
AG
value
of
the
coefficient
k
AG
is
therefore calculated from equation (11-18) at each integration step of the mass balance equation for CO^ In the calculation, changes in the viscosity, density and gas phase flow rate along the column are taken into account. The diffusivity occurring in equation (11-18) is calculated from the approximate equation (4-137) transformed to the equivalent form D
Io
G
G
B r _
%
G
C +
_
«
AB
—
G —
D +
AC
_
0
4.
AD
E
-
( 1 1E U
_
»
AE
The binary diffusion coefficients ® for normal conditions presented in Table 11-E2. Substituting numerical values into above equation, we obtain
(AG) D
= 301X10
"
5 M2/S
are the
(11-E12)
The subscript ο refers to the normal conditions. For process conditions, the diffusivity the equation given by Fuller et al. (3-70) ,
175
Χ [ 273 15 ]
>
= 1-78XlO"
6
is ,
m /s 2
The viscosity of the gas mixture is determined the equation [37]
calculated Λ
, --10
using
5
(11-Ε13) on the basis of
PACKED COLUMNS 4 5 1
Σ y.
j=i
-
ηο
*t.
J
7
V
J
Μ.Τ J kr
J
-V
(H-E14)
Σ y. v
7
M.Τ J Jkr
j = i, J
transformed into G
inV
"c G
/
A
M
A
ιnY
A
T
k
+
r
, / A
Vi
i
/ I
M
T A
Ak r
+
.
Σ
j=2
jν
G
,
(- > U
M
j T ι kr
E15
Substituting numerical values (Table 1 1 - E 2 ) , we obtain 5.661 Χ 10" "G
5
Y
3.606Υ
A
+ 2 . 5 6 4 χ
ΙΟ"
5
+ 2 . 3 5 9
"
Ε
1
6)
A
The mean molar mass of the gas mixture is calculated from the equation M Y + M ηι MG = * * (11-E17) A
where
5
Μ
= in
j
Σ M y = 8.82 kg/kmol j Zj =2
(11-E18)
The gas mixture density, molar flow velocity are calculated from the formulae
superficial
mass
=
G
= G (1 + Y ) in A
(11-E20)
w
=
(11-E21)
G
O
C
and
Pg
where
M
rate
(11-E19)
O
G M /S
G 2
π d S = — = 4 The values of η , ρ
n
π
-v2
4 and w
GG
= 1.767 m
2
(11-E22)
depend on the concentration Y , and
G
so they are implicit functions of the current column height.
A
(
1
1
452
BALANCE EQUATIONS OF PACKED COLUMNS
The molar mass transfer coefficient is given by the formula k
= k Gy
C
(11-E23)
AG G
A comparison
of L
(equation
(11-E10)) with L
TL
reveals that 0
the molar flow rate of the liquid phase does not in practice change along the column. Besides, the values of X (equation (11-E9))
and
X
(equation
(11-E8)) prove that
the
AZ
liquid
phase
AO
can be treated as a diluted solution. For these reasons the volumetric mass transfer coefficient in the liquid phase and the specific dynamic mass transfer area are calculated assuming that these values are constant along the entire column. Upon substitution of numerical values into equation (11-16) and (11-14) one has 2 1 k a = 6.344 χ 10 s" AL m
a
2
m
= 182.7 m / m
3
The molar mass transfer coefficient in the liquid phase is ( k
.
a
AL m
)
a
Lx
P
_ 6.344X10
L
M m
2
18 2 . 7
997
18
L
2
2
= 1.923 χ 1 0 " kmol/(m s) 4. Packing height At the interface CO holds
(11-Ε24)
the condition
of continuity of molar
2
NAo =
flux
of
(11-E25)
It is assumed that in a gas film the process of masstransfer is described by the equation for one-component diffusion ( C 0 2) through an inert mixture of other gases 1 + Ν
= k
In
Y* — 1 + Y
(11-E26)
A
Similarly,
in a liquid
an absorbent (water)
film,
the solute
CO^ diffuses
through
PACKED COLUMNS
453
1 + X AL
(11-E27)
Lx
1 + X
The equation has the form Y
A
For
of absorption "5
^ - 5
He X
He
=
given
equilibrium
expressed
in molar
(11-E28)
- 1
values of Y
and
ratios
X
A
(Fig.ll-El)
equations
(11-E25)
A
through (11-E28) form a nonlinear algebraic equation (1 + Y f ) [ P (1 + X * ) - H e X J A * — Ρ (1 + x A )
f(X A ) = k Q y in 1 + X - k
Lx
ln
(11-E29)
= 0 1 + X
with one unknown X . A solution of this equation and
equation
A
(11-E28)
allow
molar
ratios
Χ ,
Y
at
the
in the liquid bulk follows from
the
A
interface (Fig. 11-E1).
A
to
be
determined
operating line
eq(11-E29) YA
'
*A
|\equilibrium curve
*A
Fig. 11-E1. Equilibrium for a given state of phase bulks
The molar ratio of C 0 2 operation line equation (11-34) G X* A
= X
+ AO
μ . L
(γ*. γ A
) AO
(11-E30)
454
BALANCE EQUATIONS OF PACKED COLUMNS
The height of the column can be calculated from equation (11-35) ,
G. Ζ" ( Υ AO = 0 α λ )
I N
dY
a A
S
Ν
m
(11-35)
v
AG
Numerical integration of this differential equation in the range from Y 4 = Υ 4 Λ to Y = Y allows us to obtain a value for the A
AO
A
AZ
packing height at which the desired CO^ absorption level is achieved. Each call for the function on the right hand side of the differential equation in the integration procedure involves the following calculations: - molar ratio of X
A
-
equation (11-E30)
- gas mixture viscosity, η
-
equation (11-E16)
- mean molar mass of the gas mixture, M
-
equation (11-E17)
- gas mixture density, p^ - equation (11-E19) - molar flow rate and superficial mass velocity equations (11-E20) and (11-E21) - mass transfer coefficient, k - equation (11-18)
of
the
gas
AG
- molar mass transfer coefficient, k Q ^ - equation (11-E23) - molar ratio X as a result of a numerical solution
of
A
the
nonlinear algebraic equation in the form (11-E29) * - molar ratio, Y - equation (11-E28) - molar flux, Ν
A
AG
-
equation (11-E26)
The differential equation (11-35) was integrated using the classical fourth-order Runge-Kutta method with a variable integration step. The final result was Ζ = 8.106 m. Typical results, calculated for several values of Υ , are displayed in A
Table
11-E3.
From
this table
it
follows
that
the
value
of
-3
mass transfer
coefficient
k
varies from
1.393x10 -4
the 2
kmol/(m
s)
2
in the lower part of the column, to 8.678x10 kmol/(m s) in the upper part, thus it decreases by over 37%. This change is caused by the variables w . ρ and η . G
G
G
The computer program for this example is called E l IE.PAS.
455
PACKED COLUMNS Table 11-E3. Result of calculations Ya
[kmol A / k m o l
in]
Parameters
4.286χ1θ"
3
0.0
3.136χ10" 1.592x10' c A
4.729xl0'
xa
3
5
xa
3.070xl0"
\
1.092xl0"
M
2
1
3.917xl0"
Y*
3
5
2.6xl0
2. 025x10'
2
Pc
7.237
k
8.678xl0'
8.106
3.580X10'
2
6..445x10"*
4..627xl0"
2.014xl0"
5
9..537xl0'
4.890x10"
1
3
3.615xl0"
3
3
3.321xl0~
S
1.280χ10"
16.08
19.37
12.97
15.63
1.209χ10"
2
1
4.228x10"
3
1.226xl0"
4
4.461xl0~
1
2.442xl0"
3
2
1
2.515xl0"
8 .723 4
4.286x10'
3.934xl0"
1
5,,612xl0'
1..13xl0'
_1
5.808
3.,310xl0"
10 .81
8.970
c
2
2 .889
ζ G w
6xl0"
3
5
1.393xl0'
3
Gy
Example 11-F. In a countercurrent packed column of diameter d f c
=
1.1 m the absorption of C 0 2 from a gas mixture using a 2.5 molar aqueous solution of monoethanolamine was performed. Determine the packing height at which 99.995% of C 0 2 absorption is guaranteed. The
ratio
of
CO^
concentration
in
concentration at the column inlet is α outlet it is α 1.5x10
-3
Z
0
the
liquid
the
amine
= 0.15, and at the column
= 0 . 4 . The viscosity of the liquid
kg/(m s), and the amine diffusivity
-10
to
phase is η
=
in water is SH = BL
2
7.7x10 m / s . The other process parameters (except for the molar flow rate of the absorbent and CO^ content in the liquid entering the column) are the same as in Example 11-E. Solution 1. Reaction kinetics From the considerations presented in Chapter 7 it follows that the overall chemical reaction rate in the liquid phase in which CO and monoethanolamine (MEA) are involved CO
+ 2
2RNH
RNHCOO" + 2
RNH* 3
(11-F1)
456
BALANCE EQUATIONS OF PACKED COLUMNS
can be reaction
described
R = kC
by
the
kinetics
of
a
second-order
irreversible
C
(11-F2)
1 A Β
In
the
liquid
phase
the
subscript
A
refers
to
CO^y
and
the
subscript Β to ME A. The dependence of the reaction rate constant on the temperature is expressed by the formula l o g i o fti = 10.99 -
(11-F3) 3
where the constant
k^ is given in
m /(kmol s).
For process conditions, i.e. for
= 5917.7
Τ = 298.15 Κ
m /(kmol s)
(11-F4)
3
2. Absorption equilibrium and CO^ diffusivity in MEA solution Due to a chemical reaction taking place in the solution, solubility and diffusivity of C 0 2 are determined indirectly the basis of the data on inactive nitrogen monoxide (N). To formulate an absorption equilibrium equation for
the
the on CO^
aqueous MEA solution system and to determine CO^ diffusivity
in
the liquid phase, we shall use the correlations mentioned in Chapter 7. %
=
ni VSTW A
m Ν
1
43
%
= 3.61x10
U
^NL -5
< -
-1.87x10
-7
AL
C
-8.73x10
-7
C
BL
1
F 5
>
(11-F6)
BL
o1 r
"
( 1 1 F 7 )
where m = J
P. i R Τ C
(j = A, N)
(11-F8)
J
The mean value of nitrogen monoxide diffusivity in the aqueous MEA solution was determined as %
NL
=1.48x10
-9
2
m/s.
PACKED COLUMNS
457
The diffusivity of C 0 2 in the MEA solution is I
I
Γ
4L Λ
0 AL
= 1.61 x 1 0 " m /s 9
(11-F9)
2
Substituting equation (11-F8) advantage of the relation Ρ*
= Ρ y*
A
A
into
equation
(11-F6)
and
taking
(11-FlO)
the absorption equilibrium equation for the system considered is obtained 3 2 2 y* = (1.134 - 5.873X10" C - 2 . 7 4 2 x l 0 " C ) C* (11-F11) A
BL
where molar concentrations C
BL
and C
* AL
BL
AL
3
are given in kmol/m.
3. Mass balances for the gas phase The values of C , G. , G , G , G , G , Y G
in example 11-Ε.
in
B
C
D
E
AZ
have been calculated
The molar ratio of C 0 2 in the gas mixture leaving the column Y
= (1 - 0.99995) Y AU
= (1 - 0.99995 ) X ( 4 . 2 8 6 x l 0 ) _1
Pî.Lj
= 2.143X10"
5
kmol A/kmol in
(11-F12)
The quantity of C 0 2 absorbed in the column per unit time G. (Y ιη
- Y J AZ
-
(3.122X10" ) χ (4.286X10" - 2 . 1 4 3 X 1 0 " ) 2
1
5
AO
= 1.338X10" kmol/s (11-F13) The molar flow rate of the gas phase leaving the column 2 5 G = G (1 + Y J = (3.122 χ ΙΟ" ) χ (1 + 2.143 XlO' ) 0
2
in
= 3.122Χ10"
AO 2
kmol/s
4. Mass balances for the liquid phase The free amine concentration in a liquid - entering the column
(11-F14)
458
BALANCE EQUATIONS OF PACKED COLUMNS
= 2.5 (1-t) a)
C BLO
= 2.5 X (1-2x0.15) = 1.75 kmol/m
(11-F15)
3
Β 0
- leaving the column = 2.5 (l-o a ) = 2.5 X (1-2x0.40) = 0.50 kmol/m
C BLZ
(11-F16)
3
Β Ζ
ν = 2 is the stoichiometric coefficient of ME A in reaction (11-F1). Β
The reaction between CO^ and MEA is a fast one; it takes place only in the liquid film. The volumetric flow rate of the liquid is therefore calculated as V
G
Y
B
V
L
in^ AZ
C.f
=
BLO
_ Y
-C
2
A0^
^
2X ( 1 . 3 3 8 X ΙΟ" )
1.75
=
BLZ
-
0.5
-
2
1
-3
Λ ΑΛ
4
1
3.
1 Λ
X
10
m
/S
(11-F17)
5. Mass transfer coefficient
The coefficient of mass transfer in the gas phase is calculated from the equations given in example 11-E. The volumetric mass transfer coefficient in the liquid phase and the dynamic surface area are calculated from equation (11-16) and (11-14), respectively. After substitution of numerical values into these correlations, we have k a
AL
a
= 2.118X10"
m
= 148.7 m / m 2
m
s"
2
1
3
hence,
( k
k
a
AL m
a
AL
2.188X10" "Ί'^Τ 14o . 7
)
m
2
, -A 4 = 1.424X IO""* m/s
(11-F18)
6. Packing height At the interface molar flux continuity holds
AG » 1L
<- >
N
N
U F19
The molar fluxes Ν
and Ν AG
are determined by the formulae AL
A
1 + Y
Ν
A G
= k
Gy
ln
-— 1 + Y*
(11-F20)
A
AL = A AL
N
E
K
(- > N F21
PACKED COLUMNS
where Ε
is the second-order reaction
A
enhancement factor.
459
It is
calculated from Wellek's equation
• * (EAA - Ι )
- 1
·
ι « = Œ00 -x I ) '
3 3
r ^ 7 •+ fà
1 3 5
ν · 1
3 5
ι
tgh V M A
(7.50)
where g>
k
AL
M
k
1
c
AL
c*
BL
+1
00
BL
2
k E
s
o k Β
BL
C
AL
^ AL
For given values of the molar ratio of CO
in the bulk of the gas δ and MEA molar concentration in the liquid bulk C , 2
phase Y A
BL
equations (11-F19) and (11-F11) with (11-F20), (11-F21), (7-28), (7-48), (7-50) and y
Y
A
allow
auxiliary
relations
A
(11-F22)
- ya to determine 1
us
concentration, i.e. Y
the
molar
and C A
computational algorithm is: * a) assume C τ
ratio
of
CO
2
and
its
molar
at the interface. The corresponding AL
AL *
b) calculate y
from equation (11-F11) A *
c) calculate Y d) calculate Ε
A
from equation (11-F22) , M
oo
e) calculate N f)
check
if
a
and Ε A
q
for
from equations (7-28), (7-48) and (7-50) A
and N ^ L from equations (11-F20) and (11-F21) the
(11-F19) is fulfilled.
required If
accuracy
ΙΝ 1
of
calculation
€
equation
- Ν* I < ε, the calculation can AG
AL
be terminated, otherwise, a new value of C
^ τ
AL
should be taken
and the calculation procedure repeated starting from point b).
460
BALANCE EQUATIONS OF PACKED COLUMNS
The
concentration
of
amine
C
in
the
liquid
bulk
can
be
BL
calculated on the basis of equation (11-49). Assuming that the absorbed CO^ does not in practice change the volumetric flow rate of the liquid phase, one can write L
X
S< B - V
»
\
C
<
* BL0>
Substituting the above relation into equation (11-49) one obtains
C
BL
»
C
BL0 -
A -
( YY
A0>
Similarly, as in example 11-E, the packing height is calculated using the numerical solution of differential equation (11-35). ΊΪΥ- -
a
A
G
S N n
m
Y
* A0>
»
°
<»-
3 5
>
AG
Each call for the function on the right hand differential equation requires the calculation of - molar concentration of C
BL
-
side of the
above
equation (11-F24)
- gas mixture viscosity η > mean molar mass M
and gas mixture
density ρ - equations (11-E16),(11-E17) and (11-E19) - molar flow rate and superficial mass flow velocity of the gas equations (11-E20) and (11-E21) - mass transfer coefficients k^&nd k Q - equations (11-18) and (11-E23) - values o f C
T
, Y , E
AL
A
00
, Μ , Ε , Ν A
A
, Ν , - using the algorithm AG
AL
presented on page 454. After integration of the differential equation (11-35) by classical fourth-order Runge-Kutta method in the range from Y Υ
= 2.143 χ 10" kmol A/kmol to Y = Y
= 4.286 X 10" kmol A/kmol
5
AO
A
A
the =
1
AZ
in, we obtain the packing height Ζ = 3.222 m. The most important results of the calculations are shown in Figs. 11-Fl, 11-F2 and 11-F3. From Fig. 11-F2 it follows that in the upper part of the column, the reaction (11-F1) in the liquid phase takes place in the region of fast pseudo-first-order reactions, while in the lower part of the column it takes place in the fast reaction region. Over a very short segment in the lower part of the column
PACKED COLUMNS
461
(starting from the gas inlet) the reaction takes place in the instantaneous reaction region. The mass transfer coefficient k Gy
(Fig.ll-F3) changes from 2.15XlO"
kmol/(m
3
-3
2
s) in the lower part
2
of the column to 1.33x10 kmol/(m s) in the upper part, thus decreasing by over 38%. The computer program for this example is called E l IF.PAS. 10° r-
10-
ι
J
1
1
ui— 2
z
3
m
( )
Fig. 11-F2. Distribution of characteristic chemical reaction factors _
ι
ι
Fig. 11-F1. Concentration distribution of particular components
ζ
( m )
Fig. 11-F3. Distribution of mass transfer coefficients in a gas phase
462
BALANCE EQUATIONS OF PACKED COLUMNS
11.6. BALANCE EQUATIONS OF PACKED COLUMNS FOR MULTICOMPONENT SYSTEMS The balance equations presented so far for packed columns and the methods of their solution referred to the simplest cases of absorption: - isothermal physical absorption of one component from an inert component (Example 11-D) - isothermal physical absorption of one component from a mixture of inert components (Example 11-E) - isothermal chemical absorption of one component with a simple very fast chemical reaction in the liquid phase (Example 11-F). The two latter cases, although referring to multicomponent systems from the formal point of view, can be calculated in the same way as e f
for binary systems, if the notion of efficient diffusivity DAG is employed. In this section balance equations for packed columns and multicomponent systems will be presented. Examples will include isothermal and adiabatic absorption. The balance equations for packed columns for multicomponent systems are formulated (in a similar way to binary systems) for the plug flow model of gas and liquid phase. Assuming additionally steady-state conditions of the absorption process, for the element of the column with differential volume dV (Fig. 11-17), the balance (b)
(a) W
A
iG Σα
ΣD «V H W
iO iG
Λ .H W
W
w
iO n «\»H W IO iO
iL iL
i=l
iL η «ν, H
Σ
Σ
i=l
i=l
- j -
Ni
ε,
G
SE Ο c
Gas bulk
W i 0+dW iG π
«ν.
Liquid film Liquid bulk
η
W
iL iL
Gas bulk
Ρ
Ε 3
ε,
^ 'Liquid film 5
Liquid bulk WjL+dWjL
«ν.
W
Σ H i GW i 0+ d Œ H i GW 1 0)J £ » i L i L >i=l i=l i=l
+ d
< £ H i LW 1 L) i=l
ζ
Σ
H i 0W | 04.d
- z+dz
SîïiLWu.+diLHjLWjL)
Fig. 11-17. Scheme of differential element of a packed column a) cocurrent flow, b) countercurrent flow
equations (9-7), (9-16),(9-36) and (9-40) are reduced to the form - gas phase dW.
iG
dz
a
m
Ν
iG
(i = l,...,n)
(11-50)
PACKED COLUMNS
—
= T a d ζ
qn
Ta
m G
Σ
m
Νιρ
(11-51)
1>·,η)
(11-52)
8.
^ i= l
463
iG
iG
- liquid phase dW d ζ ^ - = a m NfiTL + β
Σ υ . · R.
. , j =1
-a
dz
qf m
iiJ
+a
L
(i -
ιJ
β.,
£ m i. = ^l
(11-53)
iL
iL
The minus sign on the right hand side of equation (11-50) and (11-51) refers to the cocurrent flow of phases and the plus sign to the countercurrent flow. The assumption of a plug flow model for each phase in an absorber allows us to determine their compositions using the following simple relations *i
=
W
i O'
W
o
<* -
1
—
;
n )
o
W
-
Σ W.G
(11-54)
i=1
x. = 1
W . f / Wf 1
Li
(i = Ι,.,.,η)
Σ W.f
; W. -
L»
Li 1
The temperatures T Q and T L are determined of molar enthalpy.
=1
directly from the definition
(a)
(b) (Σ H i LW i L) Q
W i G( 0 ) (Σ H j , W | 0) ( 0 ) | i=l
Ο
W
iGZ
à Hi i=l
W
L iL)z
Fig. 11-18. Diagram of a packed column a) cocurrent flow, b) countercurrent flow
(11-55)
1 LI
W
<5 " i G i G > Z i=l
464
BALANCE EQUATIONS OF PACKED COLUMNS
For the general case, the calculations of packed columns depend on solving the differential equations (11-50) through (11-53) with the algebraic equations describing mass and heat transfer between phases and a chemical reaction taking place in a liquid film (cf. Chapters 6 and 7) with appropriate boundary conditions. The boundary conditions are a result of the assumption of a given mode of absorption process. Usually, all the flux parameters entering the column are given or easily determined. These flux input parameters are the boundary conditions for the differential equations considered. For cocurrent columns (Fig. ll-18a) W
i G
(
0)
W
»
ί
iG0
(
1
»
D )
(11-56)
(£ft
1
w
0
l
0
]
4 =1 W
i L
[ £ f i
(
0)
( £ ö
W
»
i
L
w
iL0
i
L
w
0
i
]
0
4 =1 1
Β
·-'
( k
J
J
0
<» »
] ( 0 ) =
4 =1
i
(11-57)
>
w
L
i
L
l
4 =1
o
J
and the countercurrent columns (Fig. ll-18b) W
Z
iG< >
W
=
iGZ
1
<* »
» -
D
>
(11-58) ( £ ö
i
G
w
i
G
] ( Z ) = J
4 =1 W
i L( ° >
[ £ ö 4 =1
[ È f f
i
= L
w
W
i
L
]
w
1
0
]
(
1
» i =
" -
L
l
J
Z
(0)= [ È f f
J
0
4 =1 ί
iL0
|
n
>
1
L
(11-59)
w
i
L
] o
J
As with binary systems, for known parameters of the gas and liquid entering the column, the calculation procedure for packed columns is formulated in two ways a) for a known absorber height Ζ all parameters of the components leaving the column should be calculated. (Usually, the calculation is made for existing equipment which is to work under new operating conditions of the absorption process.)
PACKED COLUMNS
465
b)
one should calculate the absorber height Ζ (equivalently, mass transfer surface area) at which the required value of a chosen parameter of the gas or liquid leaving the equipment is achieved. (Usually, this parameter is the absorption efficiency of a given gas component.). The calculation for packed columns operating under cocurrent flow conditions is known in the theory of ordinary differential equations as an initial-value problem since at one end of the column (at the inlet) all the parameters of the gas and liquid are known - cf. equations (11-56) and (11-57). The solution of the problem does not pose, therefore, any significant numerical difficulty. For problem a) the solution consists of the integration of the differential balance equations of the absorber (11-50) through (11-53) with initial conditions (11-56) and (11-57) in the interval < 0 , Z > . For problem b) the integration of the differential equations considered should be continued until the additional condition formulated in the problem is fulfilled. The calculation for packed columns operating under countercurrent flow conditions is, in turn, called a two-point boundary value problem since the inlet gas and liquid parameters are given at two opposite column ends - cf. equations (11-58) and (11-59). Such a problem can be solved numerically using, for instance, iterative methods that involve a multiple integration of the initial problem. Generally, these calculations are laborious and time-consuming. Problem a) is now a so-called two-point boundary value problem with a fixed boundary (the column height is given). Its solution comprises an iterative choice of unknown values of an arbitrarily selected flux at the column outlet. The parameters should be varied until the known values of this flux are obtained at the other end of the column, i.e. at the inlet. For case b) the problem is called a two-point boundary problem with a free boundary (the absorber height is unknown). The problem can be solved by appropriate transformation of differential equations, which leads to a boundary value problem with a fixed boundary. There are many iterative methods for solving a two-point boundary value problem with fixed and free boundaries [38-41], One of the simplest methods is a shooting method whose basic elements are discussed below. In problem a) such a vector u(0) = [ w ( 0 ) , W ( 0 ) , . . , W ( 0 ) , [ Σ 0 iG
2 o
n G
i G
W
i O
)(0)]
T
(11-60)
466
BALANCE EQUATIONS OF PACKED COLUMNS
is to be found which will satisfy n + 1 boundary conditions written in the form of the algebraic equation F[U (0)j
«
u (Z) - u z
(11-58)
= 0
(11-61)
where
τ
u ( Z ) = [ w i G( Z ) , W 2 G( Z ) , . . , W n G( Z ) , [ ^ f t i oW i G] ( Z ) ]
»Z
=
W
[ 1CZ '
W
W
2GZ —
nGZ'
(
Σ
f
&*) ιο
(11-62)
lQ
Vector u (Z) is a function of vector u ( 0 ) since it results from the solution of equations ( 1 1 - 5 0 ) through ( 1 1 - 5 3 ) for the initial conditions u ( 0 ) and v Q in the interval < 0 , Z > , where \
-
W
[ 1L0'
W
W
2L0 · · - nL0 ' ( Σ «
iL
*
i
4 =1
L
)
(H-64)
] ' 0
J
J
Vector u ( 0 ) can be searched using an arbitrary method for solution of nonlinear systems of algebraic equations [38,42-48]. One of the most efficient method is the gradient Newton-Raphson type method. The Jacobi matrix of function ( 1 1 - 6 1 ) required in these methods should^ be determined numerically. Its particular elements in point u ( 0 ) , which is the k-th approximation of the solution u ( 0 ) being searched, are substituted by the finite differences , r 3F. -, r άη.(Ζ). ^ u: (Z)- u!(Z)
I L
du.(0)1 /f\\ k v / J j u(0)=u (0) /
r
i
= N
I
L
du.(0)1 /f\\ v / J j u(0)=u (0)
δ u . v( 07 ) j
(i, j = l , . . . , n + l)
(11-65)
The elements of vector u"(Z) in finite differences are obtained upon integration of the ^system of equations ( 1 1 - 5 0 ) through ( 1 1 - 5 3 ) for the initial conditions u ( 0 ) + iu(O) and • , where *o(0) = | W
L
O
( 0 ) ^
Ö . GW
The elements of v|ctor u'(Z) are initial conditions u ( 0 ) and ν .
o
i G
j(0)]
T
(11-66)
calculated in the same way for
In problem b) the column height Ζ is not known (the boundary value problem with a free boundary), but an additional condition must be satisfied. It can be written in the form of a single equation
PACKED COLUMNS
G [ U ( 0 ) , VO,
Uz,
v(Z)j
= 0
467
(11-67)
re
[Ê^wJ©]
v(Z) = [ w i L ( Z ) , W 2 L ( Z ) , . . . , W n L ( Z ) ,
(11-68)
Introduction of a new independent variable ξ = z/Z leads to the following boundary value problem with a fixed boundary dW = Ζ a
ς
Ν
m
(i = l...,n)
iG
z
a v
q
+
» o
a
(11-69)
Σ
»
1
dW iL .
= Ζ fa
dζ
\l
d
iL
( 1
=1
-
^
(i = l,...,n)
(11-71)
ö.w.l
>
=
W
ί
iGZ
(
*
Σ β.,
qf+ a
^ m
d£ iG
( 1 17 0 )
]J
ij
= Z a
W
N
.o
+ Α Σ v. . R.l
Ν*
l m
S
L
1
—
mim .=_i
u
n
<
iL
(11-72) îLJ
)
(11-73)
[Σ L
fi w ] i0
(D - (È ö w ]
10
i G
J
i =l
i G
M=l
J
Z
(11-74)
[ Σ ö w ] (0) = ( ς i L
i L
J
M =l
M= l
ft w ) iL
iL
0
J
where
ί e
<0,
1>
Problem b) consists, now, in finding such a vector
'-[z
< 0 )
]
(11-75)
BALANCE EQUATIONS OF PACKED COLUMNS
468
which would satisfy the system of η + 2 algebraic equations in the form u (1) - u F (r)
(11-76)
G [u ( 0 ) , v , u , ν ( 1 ) ] 0
z
where u (1) -
u (Z)
(11-77)
ν (1) -
ν (Z)
(11-78)
Vectors u ( l ) and v(l) are the functions of vector r because they result from equations (11-69) through (11-72) for the initial conditions u(0) and v in the interval < 0 , 1 > . The root r in Q
equation (11-76) is calculated in the same way as for problem a). In many practical calculations equation (11-67) has a particularly simple form. For example, when the absorption efficiency η of a chosen component of a gas mixture (e.g. abs
component no. 1) is needed, this equation has the form W
ο
- W (0)
-
•«
w
.
f
i
t
<
.,
( 1 1
.
7 9 )
IG ζ Equation (11-79) allows us to calculate directly one of the unknown boundary conditions ^ \ q ^ ^* * ^ ^ l of vector u (0)). For such a case another method of solution of the two-point boundary value problem with a free boundary is proposed. It starts with finding a vector u (0) = [ w ( 0 ) , . . . , W ( 0 ) , [ £ f i W ] (0)] (11-80) e
e
r s t
e
e m e nt
T
2o
nG
i
o
i Q
which would satisfy η boundary conditions (11-58), expressed in the form of the equation F p (0)j = u (Z) - u
= 0
z
(11-81)
e u (Z) = [ w ( Z ) , . . . , W ( Z ) , [ £ f t 2G
»z=
KGZ
nG
—
W
n G Z ' (Σ
L
^1
Ö
=1
i o
W
W
i o
i o
]
i o
] Z
] (Z)]
T
J J
T
(11-82)
(11-83)
PACKED COLUMNS
469
The elements of vector u (Z) are a function of vector u (0), since they follow from equations (11-50) through (11-53) for the initial conditions u (0) and Vq. In this case integration of the differential equations is continued until a value of the independent variable is obtained such that the following condition is satisfied
i[ i
F
u
( 0 )
]
" V
Z
" iz
)
u
=
"
0
( 1 1
8 4 )
The procedure for obtaining the root of equation (11-81) is the same as that presented in problem a). In a special case of physical absorption both the initial and boundary value problem can be partially integrated analytically. In this case the following equations hold a a
m m
Ν q
= a
iO
+a
G
m
m η
Ν
(i = Ι,.,.,η)
iL
(11-85) η
[H.
'-'
N.
iO
i=l
=a
iG
m
q, + a L
Ε 1,
m . . i=l
iL
Ν,
(11-86)
iL
or dW
dW = T
dz
(i = l,...,n)
d z J
M= 1
(11-87)
J
M= 1
d ζ
~
(11-88)
d ζ
Integration of equations (11-87) and (11-88) gives relation of state variables in the gas and liquid phase
the
simple
- for co-current flow W
il/
Z
>
(.^Ö ^i = l
-
i L
W
W
iL0
i L
)
W
-
Z
+
iG< >
W
iG0
(Ζ) = [ Ê J
^1
Ö
«
1
»
- "
D
W i L) - ( È
i L
=1
H
>
< -
fiio
J
^i=l
0
W i Q)
J
8 9
>
(Z)
(11-90)
IE,".-*.-] o - for counter-current flow W
i L
(
Z)
»
W
iL0
+
W
iG<
Z )
-
W
i G
(
0)
ί
(
»
1
—
n
)
-
9(
1U)
470
BALANCE EQUATIONS OF PACKED COLUMNS
iL
r
w iL
ίΣ
(Ζ) =
8
iL
W
iLJ
0
M=1
J
n
M= l
iG
W
iG
(11-92)
(0)
The parameters for packed columns for physical absorption are calculated in the same way as presented previously (depending on the type of problem), however the number of differential equations to be integrated numerically is half that of the previous case. When the absorption process is isothermal, in order to calculate the parameters of the packed columns one should solve numerically the differential equations for the mass balances of the components in the gas and liquid phase - cf. equations (11-50) and (11-51). In the special case of isothermal physical absorption, the calculation only requires a numerical solution of the differential equations describing the mass balances of the gas phase components, since the differential equations describing the state of the liquid phase can be solved analytically. The method of solution of these equations is identical to those presented previously (depending on the type of problem).
Example 11-G. In a packed column of diameter d^ = 1.4 m and height Ζ = 4 m means of
(Fig. 11-Gl)
simultaneous
water is carried out
absorption of H S and CO by 2 2 6 under a pressure Ρ = 1.013x10 Pa
Lo « 3.0 kmol/s Xi »0.i»1.Z3A X 5. 1
y«»o.oe
yu·-0.16 y«*o.io
HZ) . xjlZ)
X,Z«0.66
Fig.
11-Gl.
Diagram
of
a
packed
column
PACKED COLUMNS
471
and at a temperature Τ = 303.2 Κ. A gas mixture enters the column at the bottom at a rate G^ = 0.04 kmol/s, and pure water is supplied to the top of the column at a rate L Q = The packing consists of ceramic Raschig rings of d
3.0 kmol/s. =
p
0,025 m.
The composition of the gas mixture at the column inlet, given in mole fractions, is H S (1) - 0,08 ; CO (2) - 0.16 ; C Η (3) 2
2
2
6
0.1; CH^ (4) - 0.66. Calculate the absorption efficiency of H 2 S . Solution The superficial molar gas velocity at the column inlet is Q
\V
= -=^- =
„ = 2.598 X 10" kmol/(m s) (11-G1) 2 ° πχ(1.4) 4 and the molar velocities of particular components are 2 3 2 W , „ = W ν = ( 2 . 5 9 8 X l 0 " ) x 0 . 0 8 = 2 . 0 7 8 X 1 0 " kmol/(m s) Z
0
0 4
2
2
S
lOZ
GZMZ
W„„ = W 2GZ
W„
= W
W
= W
3GZ
4GZ
ν
= (2.598xl0" )x0.16 = 4.157XlO"
y
= (2.598XlO" )X0.1 = 2 . 5 9 8 X l O "
y
= ( 2 . 5 9 8 X l 0 " ) x 0 . 6 6 = 1.715X10"
2
GZ'2Z
2
GZ'3Z
2
GZ'4Z
3
(11-G2) 2 kmol/(m 2 s) >l/(m s) (11-G3) 2 kmol/(m2 s) /(m s)
3
2
(11-G4) 2 kmol/(m2 s) )l/(m s) (11-G5)
The molar water velocity at the column inlet is L W
= L 0
= S
— — = 1.949 kmol/(m s) 2 πχ(1.4) 4 2
(11-G6)
The active components are H 2 S and C 0 2 > The mass balance equations have the form , G
Z
= a
m
Ν
iG
(i = 1,2)
(11-G8)
(i = 1,2)
(11-G8)
dW l L
d
Z
= a
m
Ν
iL
472
BALANCE EQUATIONS OF PACKED
COLUMNS
It is possible to solve the above differential equations directly only if the values of W and W are given at the same level of the absorber cross section (the initial problem). For a counter-current column W is given at the level ζ = Ζ and W.. iO
at the level
ζ = 0 by
W (Z) = W 1Q
W
iL
(Z) = W 2G
= 2.078X10" kmol/(m s)
(11-G9)
= 4.157X10" kmol/(m s)
(11-G10)
= 0 kmol/(m s)
(11-G11)
3
i o z
2
3
2
2GZ
W (0) = W i l
2
l L o
W (0) = W = 0 kmol/(m s) (11-G12) Equations (11-G7) and (11-G8) with boundary conditions (11-G9) to (11-G12) represent a two-point boundary value problem with a fixed boundary. To decrease the number of differential equations to be solved numerically, it is advantageous to express the values of W as functions of W . Following the considerations 2
2L
2 L 0
iL
iG
in Chapter 11.6 and using equations (11-G11) and (11-G12) we have W.fz) = W.fz) - W . iL
(i = 1, 2)
r n
(11-G13)
iGO
iG
Therefore, the boundary value problem now comprises the solution of differential equations (11-G7) and algebraic equations (11-G13) with boundary conditions (11-G9) and (11-G10). The method for calculation of H (i = 1, 2) and a , required in iG
m
differential equations (11-G7), is discussed in detail in example 6B. The boundary value problem formulated in this way is solved using the method presented in Chapter 11.6 (problem a) for countercurrent columns). To calculate the components of the vector τ u (0) = [ w ( 0 ) , W ( 0 ) ] (11-G14) the system of nonlinear algebraic equations (11-66) was solved by Marquardts method [42]. The elements of the Jacobi matrix required in Marquardt's method were calculated using equation (11-70). Differential equations (11-G7) were integrated by Mer son's method [49], After calculations which included five iterations, the components of vector u (0) satisfying equation (11-66) were determined W ( 0 ) = 1.326xl0" kmol/(m s) W ( 0 ) = 2.450 X 10" kmol/(m s) iG
2G
4
1G
2
3
2Q
2
PACKED COLUMNS
The efficiency of H 2 S absorption is
tfab
=
473
93.62%.
The distribution of parameters of particular fluxes along the column corresponding to W l Q ( 0 ) and W 2 Q ( 0 ) are presented in Figs. 11-G2 through 11-G8.
Fig. 11-03. Distribution of
Fig. 11-G5. Operating and
concentrations of components
equilibrium lines
BALANCE EQUATIONS OF PACKED COLUMNS 1(Γ
-
2G
/
0.996
l ο-
ο
t 1 0.994 0.0015 0.0010 1
23G
ίο-
— 33G f
-0.0005
ι 0" 10"
/ —11G
"
ι
1
1
1
— 13G — ^ ,·=·120
1
1
1
1
2
3
1
z (m)
1
_L
J-
2
3
z (m)
Fig. 11-G7. Matrix elements distribution of correction factors
3.6x10"
Fig. 11-G6. Matrix elements distribution of mass transfer coefficients in gas phase
3.4xl0
-4
3.2xl0" t4
3.0X10-
4 1
0
1 IL
1 1
1
1
2
1
3
4
z (m) Fig. 11-G8. Matrix elements distribution of mass transfer coefficients in liquid phase
The results obtained show that in the lower part of the column (Figs.ll-G4 and 11-G5) C 0 2 is desorbed. This phenomenon, surprising at first sight, (part of the column operates like an absorber of C 0 2 and the other part as a desorber of C 0 2 ) , can be easily explained. The
rate of H 2 S absorption, due to the greater solubility of
H S compared to CO ,is much higher than the rate of CO 2
2
absorption. 2
This causes a rapid decrease in H 2 S concentration in the gas phase (Fig. 11-G3). Pure water entering the column becomes slowly saturated with C 0 2 (the saturation takes place in practice at a distance of 2.5 m from the top of the column). Water saturated with carbon dioxide flows down the column and encounters gas in which the mole fraction of C 0 2 is lower than in the upper part of the apparatus. This causes a desorption of COn in this part of the column.
PACKED COLUMNS
475
Nondiagonal elements of the mass transfer coefficients matrix (Fig. 11-G6) are smaller than the diagonal ones by at least one order of magnitude. This is evidence of a negliglibly small cross effect of multicomponent diffusion on mass transfer in the gas phase. The diagonal elements of the correction matrix at the bottom of the mass flux (Fig. 11-G7) are very close to unity. The nondiagonal elements of this matrix are at least four orders of magnitude smaller. The mass transfer coefficients in the liquid phase (Fig. 11-G8) are, in fact, constant along the column. This follows from the fact that the superficial molar velocity of the liquid phase (under the assumption of constant flux, and superficial mass velocity) is actually constant along the column height (Fig. 11-G2). The computer program for this example is called E U G . P A S .
Example 11-H. In a packed column (Fig. 11-H1) the adiabatic process of hydrogen chloride (1) absorption from air (3) in water W G(0) yil0),y 2(0) TGIOI
2
WQ2« 3.89W0* kmot/nfsl
1
y^-o.u y^O.OI Τ 0 2«29β.15Κ
'
I W L( Z I χ , (Ζ) . x 7 [ Z ) T L( Z |
m
Fig. 11-H1. Diagram of a packed column
(2), under a pressure Ρ = l . O x l O Pa takes place. A gas of composition y = 0.14 and y = 0.01 and temperature Τ = 5
1Z
2Z
KJZj
298.15 Κ enters the bottom of the 2 column at a superficial molar 2 2 velocity W = 3.898XlO" kmol/(m s). Pure water at temperature Τ
=
GZ
303.15 Κ is supplied to the column countercurrently
at a
476
BALANCE EQUATIONS OF PACKED COLUMNS -2
2
kmol/(m s). The column superficial molar velocity W L 0 = 3.118x10 is packed with ceramic Raschig rings of diameter d = 0.025 m. Calculate the column absorption occurs.
height
at
which
96%
hydrogen
chloride
Solution The superficial inlet are W_
= W
1GZ
\V
molar
velocities of 2
GZMZ
2GZ
at the
column
= (3.898 X l 0 " ) x 0.14 = 5.457 X 10" kmol/(m s)
ν
= W
components
3
2
7
'
(11-H1) 2 4 2 = (3.898 X l O " ) χ 0.01 = 3.898 X 1 0 " k m o l / ( m s )
ν GZ 2Z
(11-H2) m
w
=
« ^
W
(3.898 X l O ) χ 0.85 = 3.313X 10" kmol/(m s)
=
2
2
2
(11-H3) The superficial molar outlet of the column is W
(0) = \V IG
Λ
velocity
hydrogen
at
the
3
1GZ
kmol/(m s)
4
chloride
= 0.04 x 5 . 4 5 7 X 1 0 "
= (1-0.96) \\r
1G0
= 2.183X10'
of
(11-H4)
2
The soluble components are hydrogen chloride and water vapour. Since the process is no ni sothermal, the balance equations of the column comprise the mass balance equations of soluble components in the gas and liquid phases and the heat balance equations of these phases. The balance equations have the form dW 7
Γ
=
"
a
m
N
iG
i
a
•
(
»
n
^
3
m
5
< - °>
q„ + a Ε 8. N.n Λ3 m . . iO iO
(11-51) v '
dW ΎΓ-
= a
»
N
iL
i
(
=
^
n
( "
5
2
>
v
PACKED COLUMNS
S =l
J
=
dΖ
\
+ IU
Σ Ö
^1
i L
N
=
W
iO
1 W
o
<* -
W
(11-53)
i L
1 11
i=l
The composition of each phase follows and (11-65) *i
477
>
>
W
o -
from
Σ W.0
relations
(11-54)
(11-54)
i=1 X
i -
iL '
W
W
L
» !> >
(I
'
2
W
L
=
Σ W.L
(11-55)
i=1
In the worked example the partial molar enthalpies of the gas components have been defined by the equations Ö
= £
.„(T
iG
piG
v
- T°) + r. G
G
(i = 1, 2)
For such definitions of partial molar the equations presented in Chapter (11-51) can be transformed into C pG
G
where C
3
=
Ε 2
pG
,
7
enthalpies, according to 4, differential equation
dT - = - 5 - = a q^ dz m ^G
\V
(6-C5)
i
(11-H5)
ν,
(11-H6)
piG M
Equation (11-H5) will be used to calculate directly the gas phase temperature. The boundary conditions for differential equations (11-50), (11-H5), (11-52) and (11-53) are as follows W
W
=
I G Z
2GZ
=
5
3
·
, 7 4 X5
8
9
W l o( 0 ) = W l T
GZ
Wl
=
L 0
2 98
= 0
15
8
χ
1
°
L
0
~
3
K
M
O ML 2 /
(
)
kmol/(m s)
4
(11-H8)
2
= 2 . 1 8 3 X 1 0 ' kmol/(m s) 4
Q 0
(11-H7)
S
2
(11-H9) (11-H10)
K
kmol/(m s) 2
(11-H11)
v
478
BALANCE EQUATIONS OF PACKED COLUMNS W
=
2L0
3
i r S
1
1
8
iL
M = l
01
2
1
W
I
x
lLI
2
kmol/(m s)
(11-H12)
= 13.08 kW/m
2
(11-H13)
J
0
The superficial molar velocities of the liquid components and the enthalpy of this phase are calculated using equations (11-91) and (11-92) W
[i iL
(
Z)
W
=
ö
i L
+
iL0
w
i
) <
L
z
W
iG
>
- [σ
J
Ö i G
W
i G
Z)
*
W
i G
(
0)
ί
(
0
' >
(
(Σ
Λ
4 =1
! 2
»
+
[ Σ Λ Ι \ ]
-
4 =1
(
J
ö
i G
U
9
-
1)
w )(z) i o
J
4 =1
)(0)
(11-92) J
4 =1
Equation (11-92) will be used directly in the calculation of liquid temperature T L (z). Taking into account the equations the partial molar enthalpies of the components in phase (Example 6-C) in the equation considered we have
the
the for
liquid
Τ (ζ) = 298.15 Li
(.σ
S=l
fl
iL
W
i L)
J
0
+ (.σ, 4 =1
ff
iG
W
,ο]
( Z ) J
- (Λ 4 =1
0
iG
W
i G)
w
J
pL
WL(z)
(11-H14) The values of a , Ν. . a_. m
iG
n
G
and J H _ iG
pG
iL
pL
were
S
calculated using the procedure given in detail in Example 6-C. Equations (11-54), (11-55), (11-91) and (11-H14), part of which are analytically integrated balance equations for the liquid phase, are treated as auxiliary in the description. The main part of the description is a system of three differential equations (11-50) and (11-H5) with four boundary conditions (11-H7) to (11-H10). The problem formulated in such a way is called a two-point boundary value problem with a free boundary. It is worthwhile noting that in the problem the column height Ζ is not given and, thus, the integration interval for the variable ζ is not determined.
PACKED COLUMNS
479
The method of solution of the above problem consists of multiple integration of differential equations (11-50) and W a nd T (11-H5) at the values assumed initially. These 2 G( ° ) Q( ° ) values should be chosen in an iterative way so that conditions (11-H8) and (11-H10) should be fulfilled at the other end of the column. In each iteration cycle the integration of differential equations is stopped at a value of variable ζ at which condition (11-H7) is satisfied. The formal procedure followed in the case of boundary value problems is described in Chapter 11.6. The differential balance equations for the column were integrated by Runge-Kutta's fourth-order method with a variable integration step. The iterative calculations began with the value of
W 2 G ( 0 ) = 2.6XlO" kmol/(m s) and 3
2
calculations comprising were obtained
eight
iterations
W 2 Q ( 0 ) = 6.399XlO" kmol/(m s) 3
T Q (0) = 314 K. After long the
following
values
T Q (0) = 328.85 Κ
2
Taking into account these initial values of differential equations, conditions (11-H8) and (11-H10) were satisfied to within 0.01% accuracy while condition (11-H7) was satisfied for a value of the variable ζ = 1.464 m. This value can be taken, therefore, as the column height which guarantees 96% absorption of hydrogen chloride. The most important results of the calculations are presented in Figs. 11-H2 through 11-H9. The following conclusions may be drawn from the calculations performed. The liquid superficial molar velocity (starting from the level ζ = 0, cf. Fig. 11-Hl) first increases as a result of hydrogen chloride absorption and water vapour condensation (Fig. 11-H2), -2
and after reaching a maximum (W
=3.25x10
2
kmol/(m s) at ζ =
Lmax
0.3 m ) it decreases due to rapid evaporation of water to the gas stream. The gas superficial molar velocity (starting from the level of ζ = Z, cf. Fig. 11-Hl) also rises, and after attaining a -2
maximum (W
=4.1x10
2
kmol/(m
s) at ζ = 0.3 m) it decreases
Gmax
due to hydrogen chloride absorption and water vapour condensation. The liquid phase temperature rises abruptly at the beginning, then the growth becomes moderate, and after reaching a maximum (T = 339 Κ at ζ = 1.05 m) it falls quickly (Fig. 11-H4). The Lmax
480
BALANCE EQUATIONS OF PACKED COLUMNS
TV
Z= 1.464.
_L 0
J
I
0.2 0.4 0.6 0.8
I
1
1.0 1.2
(
>J 1.4
I
1.6
z (m) 0
0.2
0.4 0.6
0.8
1.0
1.2
1.4
1.6
ζ (m)
Fig. 11-H2. Distribution of
Fig. 11-H4. Distribution of temperature
superficial molar velocities
0
0.2 0.4 0.6 0.8
1.0
1.2 1.4
ζ (m)
Fig. 11-H3. Distribution of concentrations of components
1.6
0
0.2 0.4 0.6 0.8
1.0
1.2
1.4
1.6
ζ (m)
Fig. 11-H5. Distribution of elements for mass transfer
matrix coefficients
liquid phase composition corresponding to the maximum temperature is close to an azeotropic composition of the HCl-H^O system (Figs. 11-H3 and 11-H4). The temperature at the interface is very close to the liquid phase temperature. Only in a short, upper part of the column where hot gas is in contact with a cool liquid, is the difference between the temperature at the interface and the liquid bulk appreciable high, attaining 8 K.
PACKED COLUMNS
481
482
BALANCE EQUATIONS OF PACKED COLUMNS
The gas phase temperature increases along 70% of the column height to reach a maximum (T = 333 Κ at ζ = 0.3 m) and then Gmax
drops due to water vapour condensation as well as heat transfer and conductivity to the liquid phase. Hydrogen chloride is absorbed in the whole column and therefore the molar flux of this component is always positive, i.e. directed from the gas to the liquid phase (Fig. 11-H8). The heat of solution of hydrogen chloride is used to evaporate water and to heat up the gas phase. Hence, the molar flux of water is negative over 70% of the column height. The water vapour condenses only in the upper part of the column where the hot gas is in contact with cool liquid, and the molar flux of the water is positive. The heat of vapour condensation is used to heat up the liquid phase. The total flux of heat transfer between phases comprises the heat flux resulting from conduction and the heat flux resulting from mass flux. The percentage of heat transferred by conduction q is very small along almost the whole column compared with the flux of heat transferred by mass flow, q 2 G (Fig. 11-H9). The same refers to q j L and q
. Only
over a short, upper
part of the
column
where the hot gas is in contact with cool liquid and the hydrogen chloride concentration in the liquid phase is very small, is the percentage of flux q i L much higher than that of flux The corrections for large, mass fluxes and the Ackermann correction factors Ξ and Ξ , respectively, differ only slightly L
HL
from unity along the whole column and therefore can be neglected (Fig. 11-H7). The nondiagonal elements of the matrix S Q are smaller than the diagonal elements by at least two orders of magnitude. Hence, the influence of cross effects is practically negligible. The Ackermann correction in the gas phase Ξ differs HO
from unity by 7% at the most, and therefore can also be neglected in practice. The remark concerning nondiagonal elements of the matrix also refers to nondiagonal elements of the matrix k Q . The computer program for this example is called E11H.PAS. Example 11-1. Raal and Khurana [SO] have investigated the absorption of ammonia (1) from air (3) in water (2) in a column,
PACKED COLUMNS
483
packed with 1/2" ceramic Berl saddles. The column diameter was d k
= 0.15 m, and the packing height Ζ was 0.55 m. In one of the experiments gas composed of y = 0.4414 and y = 0.0069 was fed IZ
2Ζ
to the bottom of the column at a temperature Τ molar flow rate G
= 287.8 Κ and a Z
-4
= 6.194x10"
G
kmol/s. From the top, pure water
at a temperature ^ L 0 = 298.8K was supplied at a molar flow rate L Q = 3.919x10 kmol/s. The process was performed adiabatically 5 under a pressure Ρ = 1.067X10 Pa. The authors found that 96.5% ammonia was absorbed and the liquid leaving the column at a temperature T L (Z) = 326.1 Κ contained x ^ Z ) = 0.0638 mole fraction of ammonia. Assuming the same parameters of the gas and liquid streams entering the column as in Raal and Khurana's experiment, calculate the percentage of NH^ absorbed in the column and compare it with the experimental value. Solution The balance equations determining this process are the same as those presented in Example 11-H. The boundary conditions for differential equations (11-50), (11-H5), (11-52) and (11-53) describing the absorption process considered, are as follows W , „ „ = 1.5473xlO" W _
= 2.4536x10
Τ
= 287.8 Κ
2GZ
GZ
WJ W
L 0
= 0
2
Ä
f Σ
(11-11)
kmol/(m s)
(11-12)
2
(11-13) (11-14)
kmol/(m s) 2
(11-15)
= 2.2183 XlO" kmol/(m s) 1
2L0
kmol/(m s)
ff.
W
1
2
= 10.851 kW/m
2
(11-16)
The above differential equations and boundary conditions are classified as a two-point boundary value problem with a fixed boundary. In this case the column height Ζ is known, so the integration interval for the independent variable ζ is determined.
484
BALANCE EQUATIONS OF PACKED COLUMNS
The method of solution of this problem consists of a multiple integration of differential equations (11-50) and (11-H5) in the w interval < 0 , Z > at the assumed initial values of l Q ( 0 ) , W 2 Q ( 0 ) and T Q (0). These values need to be chosen iteratively in such a way that, finally, the boundary conditions (11-11) to (11-13) are satisfied at the other side of the column. The formal procedure in the case of the boundary problems is described in Chapters 11.6 and 14.2. The values of a , Ν , q , iG
m
ft._.
c ..
iG
ft...
pG
c
iL
and AH pL
XJ
necessary to solve the differential S
equations were calculated as described in detail in Example 6-C. Part of the physicochemical parameters needed to calculate the above values for ΝΗ^-Η,^Ο system are listed in Example 11-C, the other ones are given here. The values of ιι , λ . S, and V Μ Τ ' in equations (6-C 12) 1 v or or ι ι ci to (6-C15) are η
= 0.91 Χ 10" kg/(ms), λ
S, = 626 K, • Μ Τ 1
= 2 . 1 4 x 1 0 ' W/(m K),
5
1
1
2
= 83.04.
Cl
The binary diffusion coefficients in the gas phase determined by the method proposed by Fuller et al. - equation (3-70) - for Τ 5 5 2 = 298.15 Κ and Ρ = 1.067X 10 Pa are fll1 2 = 2 . 6 0 x 10" m / s , fll3 = 2.06x10
-5
2
m / s and 0
23
= 2.32x10
-5
The mean partial molar heat of NH
2
m/s. in the gas phase is Ö
3
o
= plG
r
35.94 kJ/(kmol K). The physicochemical properties of the liquid phase as a function of composition and temperature are described by the following correlation - density PL
(11-17)
log — — = A m P
L0
ρ
= 7.949265X 10
+ 1.637338 Τ - 3.217544x10
T
kg/m (11-18)
A = - 1.644871 XlO" - 5.514286Χ10" T 2
4
(11-19)
1
PACKED COLUMNS
485
- viscosity
>/ = 2.6090X10" - 1.4814X10" Τ + 2.1386xl0" T 2
4
7
2
L
2
5
2
+ (1.5255 X l 0 ' - 6 . 9 0 5 6 x 10" T + 7.3594T )x i kg/(ms)
(11-111)
- thermal conductivity AL =
1.6640X10"
- 1.1044X10"
7
4
+
1.4777X10"
T)Xj
6
Τ + (-5.3512X 1 0 "
4
W/(mK)
(11-112)
- surface tension 1
4
2
a = 1.1952 X l O " -1.6051 X l O " Τ + (-5.4779 + 1.8428 Χ Ι Ο " Ύ)αχ 1
+ (2.0462X10 - 6.9488X10"
2
T)x
2
N/m
(11-113)
- molar heat (at an average temperature) C
pL
= 75.2573 + 7.4622 χ
1
+ 3.146 χ
2
1
kJ/(kmol Κ)
(11-114)
- molar enthalpy of solution at 298.15 Κ = 34558.38 χ
AU S
+ 503.6361 χ 1
- 196784.6 χ*
2
- 63565.2 χ
1
3
+ 183376.8 χ
1
4
1
kJ/kmol
(11-115)
- diffusion coefficient (Τ = 298.15 Κ) for diluted solutions βι
IL
= 2.67 χ 10"
9
2
m /s
The absorption equilibrium of the NH^-H^O system was calculated using the equations and algorithm presented in Example 2-A. Iterative calculations of NH^ absorption in the packed
column
start with the values 4
2
4
W i Q( 0 ) = 5.0xl0" kmol/(m s)
2
W 2 Q ( 0 ) = l.Ox 10" kmol/(m s)
Τ (0) = 305.0 Κ G
After lengthy calculations values have been obtained 4
including 2
W i g ( 0 ) = 4.7966 x 10" kmol/(m s)
six
iterations
the
following
4
2
W 2 Q ( 0 ) = 6.1547X 10" kmol/(m s)
Τ (0) = 303.24 Κ G
Taking
into
account
these
initial
values,
conditions
(11-11)
to
486
BALANCE EQUATIONS OF PACKED COLUMNS
(11-13) were satisfied with an accuracy of 0.01%. The efficiency of the absorption of NH^ in the column is 96.9%. The calculated temperature of the liquid leaving the column was T L ( Z ) = 325.IK. The calculated results are in good agreement with the experimental results of Raal and Khurana. The distribution of particular parameters along the column height is presented in Figs. 11-11 through 11-18. The following conclusions can be drawn from the calculations The mole fraction of ammonia in the liquid phase increases steadily from zero at the inlet of the column to 0.0629 at the outlet (Fig. 11-11); the gas phase mole fraction of this compound decreases from 0.4414 at the inlet to 0.0235 at the outlet. In the column above 0.15 m from the gas inlet, the water vapour condenses and its concentration at the interface on the gas side is lower than in the bulk. In the column there is a cross section at which the vapour behaves like an inert component. Below this cross section water evaporates from the liquid phase. The gas phase temperature, starting from the bottom, increases rapidly and after reaching a maximum (Τ = 317K at ζ = 0.3 m), Gmax
decreases due to condensation of water vapour and heat transfer to the liquid phase (Fig. 11-12). The liquid phase temperature increases in the lower part of the column, and after reaching a weak maximum, just before the outlet of water from the column, it falls to 325.1 K. The interfacial temperature is close to the liquid phase temperature. The maximum difference of these temperatures does not exceed 3 K. The superficial molar velocity of the gas phase decreases with column height, while the superficial molar velocity of the liquid phase grows steadily (Fig. 11-13). In the lower part of the column water evaporates to the gas stream, however, due to ammonia absorption, which is more intensive than evaporation, the superficial molar gas velocity reduces along the column. A great deal of interesting data concerning the process of ammonia absorption in water can be obtained from the mass and heat flux distribution along the column (Figs. 11-14 and 11-15). The nonequimolarity of the mass transfer is confirmed directly by the overall mass flux distribution (Fig. 11-14). The molar flux of ammonia is positive over the total height of the column (no desorption of NH^ from the solution was observed). On the other hand, the molar flux of water vapour allows us to distinguish water vapour condensation and water evaporation regions in the column. In the water vapour condensation region
PACKED COLUMNS
0
0.2
0.4
0
0.
0.2
0.4
487
0.6
z (m)
z (m)
Fig. 11-11. Distribution of
Fig. 11-12. Distribution of
concentrations of components
temperature
3 ι-
0
0.2
0.4
0.6
Ο
Fig. 11-13. Distribution of superficial molar velocities
0.2
0.4
z (m)
z (m)
Fig. 11-14. Distribution of mass flux
0.6
488
BALANCE EQUATIONS OF PACKED COLUMNS 8 ι
0.2
0
0.4
0.6
0
0.2
z (m)
0.4
0.6
z (m)
Fig. 11-15. Distribution of heat flux
Fig. 11-16. Distribution of matrix elements for mass transfer coefficients
the overall heat flux is positive; the sign changes in the evaporation region (Fig. 11-15). The total molar flux of heat transferred between phases comprises the molar heat flux due to conduction and the molar heat flux due to mass transfer. The percentage of heat transferred by conduction, q^, is very small along almost the whole column compared with the molar flux of heat carried with mass, q 2 Q (in Example 11-H). The case is different in the liquid phase. Over almost the whole column length, the molar heat flux transferred by conduction, q i L , is of the same order as the molar heat flux as a result of the mass flow q^ Only over a short distances in the lower part of the column
is q_
IL
much
lower
than
q
2L
. All
mass
fluxes
and
the
overall heat flux have characteristic extremes which are a result of changes in the mass and heat driving forces (Figs. 11-11 and 11-12) and changes in the mass and heat transfer along the column (Figs. 11-16 and 11-17). The active specific mass transfer area is constant throughout the whole column (Fig. 11-17).
PACKED COLUMNS
489
The values of the nondiagonal elements of the mass transfer coefficient matrix in a multicomponent mixture are at most several percent of the values of the diagonal elements (Fig. 11-16). A strong predominance of diagonal elements in the mass transfer coefficient matrix over nondiagonal elements is evidence of the negligible influence of the so-called cross effects of multicomponent diffusion in mass transfer in the gas phase. The diagonal elements of the mass transfer coefficient matrix change with the column height by over 50% and the mass transfer coefficient in the liquid phase by over 100%. The rate of change of these values is a result of appreciable changes in the superficial molar velocities of the phases and physicochemical parameters along the column. The correction for large mass fluxes in the liquid phase Λ deviates only slightly from nondiagonal
unity (Fig.
elements of the correction
11-18). The values of the *
matrix
Ε > are only of
490
SIMULATION OF PACKED COLUMNS
the order of several percent of the diagonal elements. On the other hand, the diagonal elements of this matrix differ appreciably from unity only in the region of transition from water evaporation to water vapour condensation. The Ackermann correction factor in the liquid phase Ξ
*
along the entire column
HL
is, in fact, equal to unity. The maximum deviation of the Ackermann correction factor in the gas phase Ξ does not exceed 10%. HG
The computer program for this example is called E11LPAS. 11.7. SIMULATION O F INDUSTRIAL PACKED COLUMNS From the considerations presented in Chapters 6 and 10 it follows that in order to design an absorber correctly it is necessary to know many constants, parameters and physicochemical properties of a system; the most important are: (i)
the mass transfer coefficients in both phases k r
and k , iG
iL
(ii) the gas-liquid equilibrium data (the Henry constant, dependence of the gas solubility on temperature, pressure concentrations of the particular components), (iii) the kinetic data for the chemical reactions taking place the liquid phase, (iv) the hydrodynamic data (mass transfer area, gas and in an absorber, e t c ) .
the and in
liquid hold-up
All of these quantities most often depend on the concentrations of particular components and the temperature, and, therefore, they vary along the height of the equipment. The determination of these values from various empirical and semiempirical relations and correlations can be subject to serious error. Some of these quantities, for instance, the chemical reaction kinetics, cannot be calculated and their experimental determination often requires equipment of a special design. This is why in some cases instead of calculating the performance of the columns, which involves lengthy auxiliary laboratory investigations, another procedure is adopted, namely the simulation of an industrial packed column using laboratory equipment. Sometimes, despite the availability of all the data and the possibility of designing an absorber (or desorber), it is still necessary to verify the correctness of the calculations and to simulate an industrial absorber operation in a laboratory rig.
PACKED COLUMNS
491
Two types of simulation should be considered [51-54]: (i) simulation of the mass transfer process occurring in a differential cross section of an industrial column (differential simulation), (ii) the simulation of the column (integral simulation).
performance
of
the
entire
industrial
In both cases the simulation consists of performing in a laboratory absorber a measurement or a series of measurements under strictly defined and programmed conditions, so that the results obtained could correspond to those conditions found in industrial columns. 11.7.1. Mass Transfer Simulation in Absorber Cross Sections The mass transfer simulation in a given absorber cross sections is employed in the case where concentrations in the gas and liquid phases may be steadily interrelated in any cross section of an industrial apparatus. Such conditions appear during physical absorption (or desorption) of one component and during absorption with an instantaneous or fast chemical reaction that takes place in a film. Additionally, it should be confirmed that in an industrial column the process takes place under isothermal or close-toisothermal conditions. The respective mass balance equations for the gas and liquid phases in a countercurrent regime have the form dW
AG
= a
Z
dW .
m
= a
A L
α Z
Ν
(11-23)
AG
Ν m
(11-24) AL
For dilute systems the above equations can be written in the form dYΛ A
a _
S Ν m
"dz
G
dX dz
a A
AG
(11-27)
AL
(11-28)
ι η
S Ν m
Ls
In the case of absorption with a fast or instantaneous reaction, instead of equation (11-28) the mass balance liquid phase is described by equation (11-42)
chemical for the
492
SIMULATION
dX„
OF PACKED
a
ST-
-
COLUMNS
S Ν " L
^ S
The fluxes of components and (11-46)
< N
=-% A G
N
A and
Β are related by equations (11-45)
1L
^
(11-46)
=
The idea of simulation is as follows. In a laboratory absorber flow and mixing conditions for both phases should be arranged so that the mass transfer coefficients k and k are the same as in AG
AL
an industrial absorber. Gas and liquid concentrations as well as temperature and pressure in the laboratory absorber should also be the same as in the various cross sections of the industrial absorber. If these conditions are fulfilled, the experimentally determined mass transfer rates in the laboratory absorber should be equal to the mass transfer rates in the respective cross sections of the industrial absorber. The packing height in the industrial absorber can be determined from an integrated form of equation (11-27), (11-28) or (11-42); for example, when equation (11-27) is integrated, we obtain Y A Z G dY i n A Ζ = (11-93) a S Ν
Λ
Y
m
„
AG
A0
N ^ G is the flux of component A determined in the laboratory absorber. The example presented below illustrates the proposed method [52]. Example 11-J. In order to neutralize alkaline sewage a column packed with 2 5 x 2 5 x 3 mm Raschig rings was used. The initial concentration of NaOH in the sewage was C
= 0 . 4 kmol/m
BLO
the final concentration should decrease to C
BLZ
=0.04
3
and 3
kmol/m .
3
= 50 m /h and, the
Sewage enters the column at a flow rate of ν LU
excess gas should be 2/3 in relation to that calculated from the reaction. The process is performed at Τ = 303 Κ under atmospheric
PACKED COLUMNS
493
3
pressure. The sewage density is ρ L viscosity coefficient
is t\
=
=
.3
0.9x10
1020 kg/m , the
dynamic
kg/(m s). The composition
of the exhaust gases at the inlet to the column is CO^ - 8%, Ο
-
11%, N 2 - 8 1 % . Determine the required column diameter and packing height. Due to possible changes in the intensity of sewage flow rate to the column, in the calculations one should assume a gas flow velocity equal to half of the maximum velocity at which the column is flooded. Solution The flow rate of soda lye, which reacts in the column, is L = v (C - C ) = ÔÎSÂX(0.4 - 0.04) = 5 x l 0 ' k m o l / s V v 7 Β L 0 BLZ BL0 3600 3
The flow rate of carbon dioxide entering the column L
G
AZ
=
B
L
χ 1.67 = 4 . 1 7 x 1 0
3
kmol/s
(ii-Ji) (11-J2)
The flow rate of exhaust gases G
ζ
= —
^AZ — yJ
-2 4 17 X l O " Λο = 5 . 2 x 10 kmol/s 0.0 8 3
=
AZ
(11-J3)
The mass flow rate of gas at the inlet G
= G M Ζ
= (5.2χ Ι Ο " ) χ ( 0 . 0 8 x 4 4 + 0.11x32 + 0.81x28) 2
G
= 1.55 kg/s
(11-J4)
The mass flow rate of the liquid 1
=^6ϋδ- =
5
° 6θΓ° 3
-
1 4 1 6 7
k
*
/s
( U
-
J 5 )
The diameter at which the column would be flooded is determined from equations (11-6) to (11-10). It is equal to d = 1.285 m. The calculation is performed in the same way as kmin
in Example 11-B. The maximum gas flow velocity is
SIMULATION
494
OF PACKED
COLUMNS
^
u />
G m ax
4 1X
π d
G
= 1.0 m/s
55
1.19ΧπΧΐ.285
.
2
kmin
(11-J6)
2
The gas density was determined from the equation of state for ideal gases. The gas velocity in the column and the column diameter are u
= 0.5 u G
= 0.5 m/s
(11-J7)
Gmax 4
d=V
/ 4x1.55 = V 1.1 9 Χ π χ 0 . 5
8
p n u
k
G
For this calculated value liquid flow velocities are w
=
= π d
G
=
V
' L
L
=
of
X
π
k
πα,
L
4
2
4 W
= 1.82m
(11-J8)
G
1
Χ
,
the
column
diameter,
the
= 0.6 kg/(m s)
55
and
(11-J9)
2
1.82
gas
2
4 X1020X50
. 54
5
k
g
/
(
2m s
.(
)
Jn
1
0)
πχ 1 .82 x3600
2
2
k
The mass transfer area at a linear liquid velocity in the column
\
-
-77
=
-TÖ4Ö-
determined from Fig. 11-13, is
0
-
0
0
5
4
MS
/
2
a
-
J 1 (1) 1 1
3
= 105 m /m . m
The physical mass transfer coefficients in both phases, determined from equations analogous to those in Example 11-C are k = 0.0264 m/s AG
k
= 1.32X10" m/s 4
AL
In the calculations the following physicochemical of the gas and liquid were taken into account - for the gas 0
AG
= 1.69 XlO"
Pg
=1.19
nQ
= 1.86XlO"
5
m /s 2
kg/m 5
Pa s
properties
PACKED COLUMNS
495
- for the liquid AL
= 2 XlO"
m /s
9
= 1020
2
kg/m
= 0.9XlO"
3
3
Pa s
For the case being considered, the chemical reaction between NaOH and CO^ is fast enough for it to take place completely in a liquid film. Therefore, while preparing a mass balance for the lower or upper part of the column (Fig. 11-J1), a dependence may | v L0
&o
Κ
BIO
G
Vi
yA
CBL
11-J1. Schematic diagram for mass balance of the column
Fig.
be found between the molar fraction
of CO^ in the gas and the
NaOH concentration in the liquid phase ν „ G
C LZ
Z*AZ
+
ν BZ
2
„ =
C (11-J12)
L B G
* A
+
— Τ "
The value of the volumetric flow rate V
l
can be assumed to be
constant along the column height ν
= ν L
(11-J13)
= ν LZ
LO
while the molar gas flow correlated by the equation G (1 - y A ) - G z (1 - y A Z ) Thus
rates
in
both
cross
sections
are
(11-J14)
4%
SIMULATION OF PACKED COLUMNS
G =
(U-J15)
1 - y.
Introducing equation formula is obtained G y
A
=
y
+
Z
into
(11-J12)
^(C - C ) BLZ BL 2 v
Z 'AZ G
(11-J15)
+
^
C
B L Z -
C
the
following
(11-J16)
B L >
This relation shows the changes in CO^ concentration in the gas and NaOH concentration in the liquid phase along the column height of the industrial equipment. Several values of the soda lye concentration and the corresponding mole fraction of carbon dioxide y are given in Table 11-J1. A
Table 11-J1. Concentration of
ΒL
*A
NaOH (C
δ BL
) and mole fraction of CO (y ) 2 A
0.4
0.3
0.2
0.1
0.034
0.047
0.060
0.073
0.07
0 04
0.076
0 08
Next, in a laboratory absorber simulation of several cross sections of an industrial absorber was performed. The operation was carried out in a contactor with a flat contact area. First, a series of measurements were carried out for CO - water and NH2
3
air-sulfuric acid systems which were used for the determination of the mass transfer coefficients in both phases, as a function of impeller rotational speed. These data are presented in Figs. 11-J2 and 11-J3. 1
Ks* .
V $"
Fig. 11-J2. Mass transfer coefficient k Q versus impeller rotational
PACKED COLUMNS
497
10
± Fig. 11-J3. Mass transfer coefficient k
Lt
versus impeller rotational speed 1
Next, at impeller revolution of η = 5.2 s" , η = L
1
18 s" , which
Ο
correspond to physical coefficients for a given system, measurements were made at various NaOH/Na 2 C0 3 concentration ratios in the liquid and at corresponding molar fractions of CO^ in the gas, which were determined from equation (11-J16). The absorption experimentally, are presented in Table 11-J2 .
rates,
obtained
Table 11-J2. Experimental absorption rates C*L
3
[mol/m ]
N* 1 0
6
2
[mol/ (m s)]
0.4
0.3
0.2
0.1
0.07
0.04
6.4
4.2
2.0
0.7
0.4
0.2
The packing height in an industrial column can be defined from the integrated form of equation (11-42). Since carbon dioxide does not accumulate in the liquid, equation (11-42), using in addition eqsuation (11-45) and (11-46), can be rearranged to give
(11-J17) BLZ
Using data from Table 11-J2, the integral in equation (11-J17) can be calculated and then the packing height may be determined Ζ =
0. 0 054 5 X ( 3 . 0 5 X l O ) = 7.84 m 2 χ 105
498
SIMULATION OF PACKED COLUMNS
11.7.2. Simulation of the Industrial Absorber The simulation procedure discussed above is not valid if many chemical reactions occur in the liquid phase or if one or more reactions take place in the liquid bulk. If the chemical reaction kinetics are not known, it is impossible to relate the compositions of gas and liquid phases in particular cross sections of the industrial absorber. The same restrictions are valid for multicomponent physical absorption in dilute and concentrated systems. In these cases simulation of the entire industrial absorber is necessary. The respective mass balance equations have the form dW
dW ^
=
a
i L
N m
+
r ^ ^
i
j
R
j
(11-52)
j=l The analysis of the above equations reveals that for the general case, that is multicomponent absorption with reactions that take place partly in the film and partly in the liquid bulk, the simulation of an industrial absorber operation is only possible if a laboratory absorber is used. In the laboratory absorber the ratio of gas to liquid flow velocity, liquid hold-up, interfacial area in a unit volume, and mass transfer coefficients in both phases should be equal to those values which occur in the industrial absorber. When the experiments are carried out in a laboratory absorber in which the height can be varied, a height can be determined at which a given absorption level is obtained. This height is also equal to the height which must be used in the industrial column. Some more specific conclusions concerning the simulation of industrial absorbers can be drawn for particular cases of multicomponent physical absorption with inert material and multicomponent absorption with a slow chemical reaction which occurs entirely in the bulk of the liquid phase. For multicomponent physical absorption the mass balance equation for the liquid phase reduces to the form dW "dT^
-
a m
N
iL
ί
(
«
1
' -
n
>
Upon integration of equations (11-50) and (11-94) we obtain
U
< -
9 4
>
PACKED COLUMNS
Ζ
a W.
„ iz dY I -jj-*-
499
Y
m
=
Y
in
(i = l,...,n)
(11-95)
(i = l,...,n)
(11-96)
i G i 0
Ζ
a
X
„ iZ
m
=
w
S
dX
-jq——
j γ
iL iO
If the mass transfer coefficients in the model of the industrial absorber are the same as in the actual industrial absorber, and the ratio of gas to liquid flow rate in both is equal, the left hand side will also be equal Z a - N
Z
a
M , -hH r
v
in
•'lab
t^r-}, L
S
r
in
-
(,| w>
Ί
^ ind
-i ^-}
-
1
J
lab
L
S
J
ind
The simulation of an industrial column can be carried out in the following steps: a) For known systems, the mass transfer coefficients in both phases should be determined as a function of gas and liquid flow velocities, and as a function of the geometrical parameters of the laboratory apparatus. As the packing model either a string-of-disks or a string-of-spheres column or an inclinedwall column can be used. For instance, for a string-of-spheres column the mass transfer coefficients are determined as functions of gas and liquid velocity and sphere diameter, tube diameter, hollow size, etc. b) For known physical mass transfer coefficients in an industrial column, the laboratory apparatus design should be chosen in such a way as to obtain the same values of mass transfer coefficients in both apparatus at the same liquid to gas ratio L / G. . S
c)
in
In the model column the process of industrial absorption is studied. The investigations should be carried out in such a way as to achieve the same absorption efficiency in the laboratory column as in the industrial column. d) The experimentally determined laboratory column height is used to calculate from equation (11-97) or (11-98) the required packing height in the industrial absorber.
500
SIMULATION OF PACKED COLUMNS
The modelling of absorption with a slow single chemical reaction is described in detail in the literature [55, 56]. Experiments described in these works prove that the method presented above has been fully verified.
PROBLEMS Ρ 11-A.
Using the data of Example
11-E calculate the
absorption
efficiency of CO^ in a packed column of height a) b)
Ζ = 5 m, Z = 10 m.
Ρ 11-Β.
Repeat
the
calculation
in
Example
11-E using
the
exact
solution of the Stefan-Maxwell equation. Ρ 11-C. For the data in Example 11-G calculate the height of the packing for 99% removal of H^S. Ρ 11-D. For the data in Example 11-H estimate the packed height for 99% of HCl. Ρ 11-E. A gas consisting of a) 0.1 mole fraction of HCl, 0.01 mole fraction of vapour and 0.89 air, b) 0.6 mole fraction of HCl, 0.01 mole fraction of vapour and 0.39 air, c) 0.99 mole fraction of HCl and 0.01 mole fraction of vapour is to be absorbed in an adiabatic packed column. Calculate the absorption efficiency in a 3 m packed column. Other data are given in Example 11-H. Ρ 11-F. A gas consisting of 0.1 mole fraction of S 0 2 >
0.02 mole
fraction of vapour and 0.88 mole fraction of air is introduced to a column of diameter 1 m filled with 0.025 m Raschig rings. Gas enters the column at a rate of 20 mol/s at 293.15 Κ and a pressure of 1.013x10 Pa. Pure water at a rate 4 kg/s is flowing into the column. Calculate the height of the column for 99% removal of SO^. Equilibrium data for the system
SO^- H^O can be found in [57].
PACKED COLUMNS
501
REFERENCES 1. Jaroszynski M., private communication, 1990. 2. Stringle R.F. Jr., Random Packings and Packed Towers. Design and Applications. Gulf, Houston 1987. 3. Treybal R.E., Mass Transfer Operations. McGraw-Hill, New York 1980. 4. Kohl A.L., F.C. Riesenfeld, Gas Purification. Gulf, Houston 1985. 5. Cihla Z., O. Schmidt, Czech. Chem. Comm., 22, 896, 1957. 6. Jameston G.J., Trans. Inst. Chem. Engrs., 44, 198, 1966. 7. Porter K.E., J.J. Templeman, Trans. Inst. Chem. Engrs., 46, 86, 1968. 8. Dutkai Ε., E. Ruckenstein, Chem. Eng. Sei., 23, 1365, 1968. 9. Porter K.E., Trans. Inst. Chem. Engrs., 46, 69, 1968. 10. Hoek P.J., J.A. Wesselingh, F.J. Zuiderweg, Chem. Eng. Res. Dev., 64, 431, 1986. 11. Distillation and Absorption, Institution of Chemical Engineers, Symposium Series, No. 104, Brighton 1987. 12. Shulman H.L., C F . Ullrich, N. Wells, AIChE J., 1, 247, 1955. 13. Shulman H.L., C F . Ullrich, N. Wells, A.N. Proulx, AIChE J., 1, 259, 1955. 14. Stanek V., V. Kolar, Chem. Eng. J., 5, 5 1 , 1973. 15. Takahashi T., Y. Akagi, Κ. Ueyama, J. Chem. Eng. Japan., 12, 341, 1979. 16. Billet R., M. Schuhes, Distillation and Absorption, Institution of Chemical Engineers, Symposium Series, No 104, Brighton 1987. 17. Eckert J . S . , Chem. Eng. Progr., 57 (9), 54, 1961. 18. Mersmann Α., Chem. Ing. Techn., 3, 212, 1965. 19. Mayo F . , T.G. Hunter, A.W. Nash, J. Soc. Chem. Ind., 54, 376, 1935. 20. Yoshida F . , T. Koyanagi, AIChE J., 8, 309, 1962. 21. Onda Κ., H. Takeuchi, Koyama, Chem. Eng. Japan, 3 1 , 127, 1967. 22. Javari A . S . , M.M. Sharma, Chem. Eng. Sei., 23, 1, 1968. 23. Charpentier J . C , B.I. Morsi, NATO ASI on Mass Transfer with Chemical Reaction, Izmir 1981. 24. Puranik S.S., A. Vogelpohl, Chem. Eng. Sei., 29, 501, 1974. 25. Sherwood T.K., F.A.L. Hollo way, Trans. Am. Inst. Chem. Engrs., 36, 39, 1940. 26. Van Krevelen D.W., P.J. Hoftijzer, Ree. Trav. Chim. des Pays-Bas, 66, 49, 1947. 27. Ramm W,M., A.Ju. Zakgeim, N.M. Gurowa, Zagig, Z.W. Alina, Trud. Chim. Tech., 1, 913, 1961. 28. Shulman H.L., C G . Savini, P.V. Edvin, AIChE J, 9, 479, 1963. 29. Danckwerts P.V.,Gas-Liquid Reaction, McGraw-Hill, New York 1970.
502
SIMULATION OF PACKED COLUMNS
30. Mohunta D., A.S. Vaidyanathan, G.S. Laddha, Ind. Chem. Eng., 11, 39, 1969. 31. Au-Yeung P.H., A.B. Ponter, Can. J. Chem. Eng., 61, 481, 1983. 32. Norman W.S., F.Y.Y. Sammak, Trans. Inst. Chem. Engrs, 4 1 , 109, 1963. 33. Onda K., H. Takeuchi, Y. Okumoto, J. Chem. Eng. Japan., 1, 56, 1968. 34. Bravo J.L., J.A. Rocha, J.R. Fair, Hydrocarb. Proc., 64, 9 1 , Jan. 1985. 35. Spiegel L., W, Meier, I. Chem. E. Symp. Ser., 104, A203, 1987. 36. Fair J.R., J.L. Bravo, Chem. Eng. Prog., 86, 91, Jan. 1990. 37. Herning F., L. Zipperer, Gas u. Wasserfach, 5, 1936. 38. Jacobs D.A.H. (ed.), The State of the Art in Numerical Analysis, Academic Press, New York 1977. 39. Stoer J., R. Burlisch, Einfuhrung in die numerische Mathematik, Vol. II, Springer-Verlag, Berlin 1973. 40. Constantinides Α., Applied Numerical Methods with Personal Computers, McGraw-Hill, New York 1987. 41. Hlavacek V. (ed.), Dynamics of Nonlinear Systems, Gordon and Breach Science Publ., New York 1986. 42. Marquardt D.W., J. Soc. Ind. Appl. Math., 11, 431, 1963. 43. Broyden C G . , Math. Comp., 19, 577, 1965. 44. Broyden C.G., Computer Journal, 12, 25, 1965. 45. Avila J., Univ. of Maryland Comp. Sei. Ctr. Tech. Rep. TR-142, 1971. 46. Brent R.P., SIAM J. Numer. Anal., 10, 327, 1973. 47. Cosnard M., M.Sc.Thesis, Cornell Univ. Comp. Sei., TR75-248, 1975. 48. Duflhard P., Numer. Math., 5, 289, 1974. 49. Christiansen J., Numer. Math., 14, 317, 1970. 50. Raal J.D., M.K. Khurana, Can. J. Chem. Eng., 5 1 , 162, 1973. 51. Danckwerts P.V., A.J. Gillham, Trans. Instn Chem. Engrs., 44, 42, 1966. 52. Danckwerts P.V., E. Alper, Trans. Instn Chem. Engrs, 53, 34, 1975. 53. Alper E., P.V. Danckwerts, Chem. Eng. Sei., 3 1 , 599, 1976. 54. Shah Y.T., W.-D. Deckwer, Fluid-Fluid Reactors, Chap. 6 in A. Bisio, R.L. Kabel, Scaleup of Chemical Processes, Wiley, New York 1985. 55. Alper E., Chem. Ing. Tech., 51, 1136, 1979. 56. Laurent Α., C. Fonteix, J.C. Charpentier, AIChE J, 26, 282, 1980. 57. Edwards T.J., G. Maurer, J. Newman, J.M. Prausnitz, AIChE J., 24, 966, 1978.
PACKED COLUMNS
11.1. INTRODUCTION Packed columns are the units most often used in absorption operations (Fig. 11-1). Usually, they are cylindrical columns up to several metres in diameter and over 10 metres high. The packing is placed on a support whose free cross section should be at least equal to the packing porosity. One of the various supports is shown schematically in Fig. 11-2. Liquid is fed in at the top of the column and distributed over the packing through which it flows downwards. To guarantee a uniform liquid distribution over the GAS OUT
Fig. 11-1. Packed column
Fig. 11-2. Packing supports
PACKED COLUMNS
411
cross-section of the column, a liquid distributor is employed. There are many types of distributor. Figure 11-3 presents schematically a weir-trough liquid distributor. Gas flows upwards countercurrently to the falling liquid, which absorbs soluble species from the gas. The gas which is not absorbed flows away from the top of the column, usually through a mist eliminator. The mist eliminator separates liquid drops entrained by the gas from the packing. The separator may be a layer of the packing, mesh or it may be specially designed.
Fig. 11-3. Weir-trough liquid distributor
Historically, the first type of packing which was widely used in industry, not only in absorption but also in distillation processes, were Raschig rings. Introduced at the beginning of the 20th century they are still used in industry. Raschig rings (Fig. 11-4) have a large specific packing surface and high porosity. They are hollow cylinders with an external diameter equal to the ring height. They can be made of ceramic, metal, graphite or plastic. In the 1930s Berl saddles were used and after the Second World War many other very efficient packings were manufactured. The main virtues of a good packing are a large specific surface area, high void fraction (porosity),chemical resistance to liquid and gas, high mass transfer rates in both phases, a small pressure drop, and low cost. Two of the features mentioned above characterize the packing particularly well. These are the specific surface of the packing and the void fraction. The specific surface
Fig. 11-4. Raschig ring and its modifications a) Raschig ring, b) Lessig ring, c) partition ring, d) Pall ring, e) Bialecki ring, f) Κ ring, g) Hy-Pak packing
412
INTRODUCTION
of the packing is the total surface area of all packing elements contained in a unit volume. Usually, the specific surface area of 2
3
industrial packings ranges from 100 to 1000 m /m . However, there are packings with smaller or larger surface areas. The void fraction is the ratio of the free volume to the bed volume. For ceramic packings the void fraction is usually about 0.7, while for metal packing it is about 0.95 and more. Packings may be divided into two main groups: random and structural. Structural packings are manufactured for a given column diameter and column height. Usually the same packing cannot be used in columns of different diameter. Random packings may be relocated to various columns. The most popular random packings are rings. Since the time when Raschig introduced his rings in industry, they have been continuously modified. Figure 11-4 shows a Raschig ring and its main modified forms. The objective in modifying the rings is to increase the mass transfer area, the porosity of the packing, and to guarantee a low pressure drop and good conditions for mixing the gas and liquid phases (in the column cross-section but not along its height), which leads to high mass transfer coefficients in both phases. From experimental investigations it follows that the most efficient of all the types of ring are Pall and Κ rings. Table 11-1 gives the basic characteristics of these rings together with data for Raschig rings. Figure 11-5 shows the comparison between these three packings [1], Table
11-1. Basic characteristics of
Type of
Dimensions
packing
mm
Raschig, Pall and Κ
Specific a
rings
surface 2
3
[m /m ]
area
Void c
fraction ["]
Raschig
25x25x3
190
rings
38x38x6
115
0.68
(ceramic)
50x50x6
92
0.74
Pall
25x25x1
206
0.90
rings
38x38x1
128
0.91
(plastic)
50x50x1
103
0.92
0.73
Κ
25x0.5
210
0.950
rings
35x0.6
155
0.953
(metal)
50x0.8
100
0.960
PACKED COLUMNS
413
The liquid mass transfer coefficient in the form of the Sherwood number and the pressure drop as a function of the gas and liquid flow rates are presented there. Apart from rings, other shapes characterized by a large specific surface area and high void fraction are used as packing. Figure 11-6 illustrates such packing shapes.
4 6 61
2 4 - UqN*]
200 400 600 — Re,
Fig. 11-5. Liquid mass transfer coefficient and pressure drop for different kinds of packings
a)
Fig. 11-6.
Some random packing: a) Berl saddle, b) Intalox saddle,
c) Interpack, d) Tellerette, e) IMTP
Recently, the application of structural packing in absorption has increased rapidly owing to the fact that it has a larger mass transfer area than random packing, all other features being similar. Another advantage is that by using this packing it is possible to obtain the same values of mass transfer coefficient in the entire column, in contrast to random packing, for which this coefficient has different values at various points of the column and depends on the location of the particular packing elements. It is also easier and cheaper to install columns already with the structural packing inside rather than assemble a column and then fill it with a random packing. One of the first structural packings which is particularly recommended, is the Sulzer BX packing. A
414
HYDRODYNAMICS OF PACKED COLUMNS
classical form of the Sulzer BX packing is grid strips bent alternately at an angle of about 30° to the column axis. Bent strips form bunches located in the column one on top of the other. The liquid and the gas flow in channels which change their flow direction while passing through the bunches. Figure 11-7 presents
Fig. 11-7. Sulzer BX packing
schematically the Sulzer BX packing. Several modifications have been recently introduced to this packing. In Table 11-2 the fundamental data concerning the Sulzer BX packing are given. Table 11-2. Characteristics of Sulzer BX packing specific
surface
area
[m /m ]
a
500
2
3
channel dimension
base
angles
bQ
r
Ψ
78
30
0.0102
[m]
Above a short description of a packed column was given. Further information can be found in the excellent books [2, 3, 4]. 11.2· HYDRODYNAMICS OF PACKED COLUMNS When considering packed columns the following problems should be discussed: a) the distribution of the liquid over the packing, b) the axial mixing of the liquid and the gas (residence time distribution), c) liquid hold-up,
PACKED COLUMNS
415
d) pressure drop and flooding of the column, e) the wetted area of the packing and mass transfer area in columns.
11.2.1. Distribution of Liquid over the Packing and Residence Time Distribution of Liquids in a Packed Column The liquid fed to the top of the packing has a natural tendency to distribute over the packing in a radial direction [5-11]. This phenomenon occurs in all random packings and in some structural packings as well. At some distance from the column top the liquid reaches the column wall and thereafter flows down it. At a lower level more liquid accumulates on the column wall, although part of the liquid returns back to the packing. At some distance from the packing top, dynamic equilibrium between the liquid flowing through the packing and flowing down the column wall is established. This ratio depends on the packing size, the column diameter, the kind of packing, the physicochemical properties of the liquid, the contact angle of the materials of which the column and packing are made, and the liquid flow rate. Fig. 11-8 presents schematically the
o)
liquid out
liquid ouf
Fig. 11-8. Distribution of liquid over the packing
phenomenon of liquid flow over packing which is uniformly (Fig. 11-8a) and nonuniformly wetted (Fig. 11-8b). The figures show deviations from an ideal flow pattern in particular annular sections for four different packing heights. If maldistribution does not occur and the liquid does not accumulate on the walls then in each annular section the ratio (W ) f / W T should be equal to 1. L loc
L
Actually, the liquid accumulates on the wall. It is shown in the figure that at a particular height (the lowest level) dynamic equilibrium, independent of the original type of column feeding, is
416
HYDRODYNAMICS OF PACKED COLUMNS
attained. In columns of small diameter the liquid flowing down the column wall can constitute up to 50% of the total liquid. In columns of larger diameter the amount of liquid flowing along the wall compared to that flowing over the packing is small. It is evident that the mass transfer coefficients for the liquid flowing over the packing and down the wall are different. This is why liquid accumulation on the column walls is, apart from maldistribution, the main reason why small packed columns cannot be expected to fully represent the mass transfer conditions when applied to large industrial columns. Sometimes, due to maldistribution of the packing elements or poor wetting of the packing, some regions of the column remain unwetted or slightly wetted. This is a highly undesirable condition which results in the column dramatically losing its absorption abilities. To avoid these undesirable conditions different types of redistributors of liquid are installed in the column. Apart from the large-scale maldistibution described above nonuniform flow on this scale, equivalent to the size of the packing, is observed [11]. It is called a "natural flow" maldistribution. Even at a high liquid velocity some packing elements are not wetted, while over other elements much more liquid flows than would result from the calculated mean flow velocity. The "natural flow" maldistribution may seriously hinder the absorption process. The liquid distribution over the packing is connected with the problem of the residence time distribution in a packed column. Investigations concerning this problem are incomplete and most of the experimental studies refer to columns of small diameter. It follows from the performed investigations that in columns of larger diameter plug flow of the liquid may be assumed. This refers particularly to tall columns. For small columns deviations from plug flow occur. Gas flow in a column with a structural packing is near to plug flow. The stream of gas flowing through the packing splits into smaller parts which combine afterwards. Continuous changes in the flow direction mean that even for columns of small diameter the residence time of the gas for particular molecules is close to a mean residence time. For random packing fairly high deviations from plug flow sometimes appear. This is connected, amongst other things, with the influence of the column walls on the packing structure. Near the wall, the packing elements are arranged in a different way than in the column centre. With an increase in the distance from the wall to the column axis the void fraction of the packing decreases to
PACKED COLUMNS
attain a constant value for a distance from the several diameters of the packing elements (Fig. 11-9).
wall
equal
417
to
Distance from the wall
Fig. 11-9. Porosity as a function of distance from the wall
Therefore, for random packings the gas velocity distribution is as follows: at the column wall the velocity is equal to zero, then it increases to reach a maximum at a small distance from the wall. Next, the velocity decreases to attain a constant value near the column axis. For equipment with a large diameter the effect of the wall regions can be negligible and thus a flat profile for the gas velocity may be assumed for such columns. The above statement is also valid for two-phase flow, except for the case where a small amount of gas is in contact with a large quantity of liquid (absorption of sparingly soluble gases). It was observed that for such conditions the liquid entrains gas particles downwards, mixing the gas along the column height, and instead of gas plug flow intermediate flow occurs, which combines the features of plug flow and flow with ideal mixing. Example l l - A . Absorption is carried out in two identical columns with structural packing. For the process conditions the equilibrium line is given by the equation Y* = m X A
A
The concentration of the soluble component in the gas at the inlet is Y kmol A/ kmol in, and the solvent is free of solute. AZ
In the first column the solute concentration in the gas drops to Y kmol A/ kmol in. Determine the final concentration of solute AO
species in the gas in the second column if, due to improper wetting, c% of the packing remains unwetted. The packing is constructed so that the gas and the liquid are not radially translocated. The gas and liquid superficial molar velocities in both columns are
418
HYDRODYNAMICS OF PACKED COLUMNS 2
identical. They are W _
2
kmol S/(m s),
kmol in/(m s) and W T
Gl η
LS
respectively. In a narrow range of changing velocities the volumetric overall mass transfer coefficient is proportional to the gas superficial molar velocity Ψ Λ 1 and to the liquid Gin
superficial molar velocity raised to the power n. Κ
a GY
~
W
m
W Gin
n
LS
A gas mixture may be treated as a dilute system. Solution The operating line described by the equation W Y
A
=
X
+
A
" W ^ Gin
for
the
first
column
(Fig.ll-Al)
Y
A0
is
<"-
A 1
>
The packing height is given by the equation Λ
ζ d Y
Ζ =
G ni K
a
G Y
m
Y
J
γ
A
A -
(11-A2)
Y A r
AO Using the equilibrium dependence (11-A2) can be written as W Z = B W
f.
°
i
n
.
and
Y A Z Γ
W ,„N" JI
Γ, r
Y
ι η ί
equation
dY. -
w
OINU mW _. W
MW
I
(11-A1),
π mGIN W.
(11-A3)
MW
~w
A
is J
equation
V
is
and upon integration
z
[
J^y" I i
i
y s w - r
1UÎÏ
l
G i n·>
l
1- W A schematic diagram of the in Fig. 11-A2.
n
w
j ^
Az
4
w —
<
U
-
A
4
>
0 A
G pl
LJ
flows in the second
column is illustrated
PACKED COLUMNS
AI)
A2)
w LS
Wo
- w r„ n
Gin
100
^AO
AO
419
,ΥΔ
T
Y' A0
YA
L n
100
G ,
Y AZ .
Fig. 11-A. Mass balance for the first and the second column
The operating line for the wetted part of the column is 100 W Y
A
=
S
L
(100-C ) W .
γ
Gin
A
ι γ »
(11-A5)
AO
The packing height in the second column equal height in the first column, is given by the relation
fl 0 0-cl
W Η
1
Η [
AZ
1 00 J
f °- 1 1Q
0 0
c
b f I V w J Y V [l00-cj L S G i n [ 100 J Yj^ o Deleting Y
ζ =
Y A
χ ln-
A
to the
packing
dY (11-A6)
1 Y -
Y
A
A
from equation (11-A6) and, upon integration we obtain
ι F
100-C
[
100 loo
Γι Z L
mW W
I
e
Η Γ
J LS L
O i n f 1l O 0 0O - c l l
LS
mW
L JJ
1 -
,
Y '
Gin
w
Ls
F
Ί"
100-C [
mW. G i n f l O O - c 1 Y, Y W L s[ l 0 0 j A 0
100
J.
(11-A7)
AO
Comparing equations (11-A4) and (11-A7) and after rearrangement, we have
HYDRODYNAMICS OF PACKED COLUMNS
420
mW. A Z
Y
L
A0
w Β
e -
JJ
[ 100
l s
EW^ JinflOO-c] Gi W
< - > U
A8
« I 100 J
LS
where mW m Y V Ginfl00-c1'
f 100 Ί " Γ
tioo- J L -
W
1
Β =
c
mW
j
W mW.
>
γ
J
I 100 J.
S
L
.
Gin LS
+
mW
AZ
2L£ γ
AO
W
X In — γ (11-A9) After mixing the gas at the top of the column the final concentration of a solute gas 0at the column outlet is obtained
A
Y ' (100-c)+ Y AO Y
c A7
AK »
100
—
< n
A 1 0
>
For instance, for the values m = 0.9
γ AZ = o . i the value of Y
c = 20%
? ! m o
η = 0.5
γ AO = o.oi
A
kmol i n is 0.0235
k
m
°!
A
k m o 1
kmol
A
iη
.
AK kmol ι η Hence, due to the maldistribution of the liquid over the packing a large decrease in absorption efficiency occurs. 11.2.2.
Liquid Hold-op on the Packing
It is particularly important to know the amount of liquid held on the packing when absorption with a chemical reaction occurs. The total hold-up is the sum of a dynamic and static hold-up. The static hold-up is the volume of liquid suspended on the packing when gas and liquid flow is interrupted in the column. The difference between the total hold-up and static hold-up is called the dynamic hold-up. From experimental investigations it follows
PACKED COLUMNS 421 that the volume of liquid hold-up on the packing depends on the kind and size of packing, the physicochemical properties of the liquid and the liquid flow velocity. With an increase in liquid flow velocity, the total hold-up rises. In a countercurrent gas and liquid flow two characteristic regions can be distinguished. In the first one, the total liquid hold-up is practically independent of the gas flow velocity. After exceeding a certain flow rate the liquid hold-up also depends on the gas flow rate (Fig. 11-10). Such flow rates form a so-called loading line.
Fig. 11-10. Hold-up in packed column
The static hold-up was found [15] to be dependent mainly on the packing size. It can be determined by the following equation ,
1.53X10·«
)
,1 . 2
The dynamic hold-up was determined by many authors [12-15]. One of the correlations [15] which appears to fit fairly well over a broad range of operational conditions is β
=2.9xl0- *Re;
where d
5
0
66
f ^ l
0
'
7
5
. ! -
1
-
2
(Π-2)
H0
Ρ
is the nominal packing size, 2 m; R e ' is the Reynolds number L
for the liquid phase, defined as ww d 1 d L Ρ R e L " S//T ε S is the cross-sectional area of the column, m ; 1 is the liquid flow rate, kg/s. This equation may be applied in flow regions below the loading line of the column.
a
M
422
HYDRODYNAMICS OF PACKED COLUMNS
There are several other equations for which liquid hold-up can be calculated* One, which is valid over very wide flow conditions, is given by Billet [16].
11.2.3. Flooding of the Column, Pressure Drop in Packed Columns
Fig. 11-11. Pressure drop as a function of gas flow velocity
Figure 11-11 illustrates a schematic diagram of the pressure drop as a function of the gas flow velocity for constant liquid velocities in random packings. For dry packing and one phase gas flow the pressure drop may be determined from a modified Darcy-Weisbach equation [15] (11-3) where the friction factor A is a function of the Reynolds number for the gas. If the Reynolds number for the gas is less than 200 A = 456 Re whereas for
,-0.742 G
(11-4)
R e ' ^ 200 G
A = 27.4 R e ' ' * 0
2 1 6
G
where the Reynolds number for the gas phase is defined as
(11-5)
PACKED COLUMNS
423
W
"G
€
'G
'
For wetted packings there are many equations and diagrams to determine the pressure drop during gas flow through the packing. These relations are not general in nature and usually relate to one type (and often one size) of packing. The exact data for a given type and size of packing can be obtained from manufacturers. As an example, the pressure drop for 25 X 25 x 2.5mm Raschig rings is shown in Fig. 11-12.
0.1
0.2 0.3 10.5 0.5 1 0.4
Fig. 11-12. Pressure drop for 25 mm Raschig rings
The increase in gas flow velocity at constant liquid velocity causes an increase in flow resistance. In the pressure drop diagrams two characteristic bends are observed as a function of gas flow velocity. The first one occurs at the point of column loading. An increase in gas flow velocity above the loading line of the column causes an abrupt rise in flow resistance. This is basically connected with a very fast increase in the total liquid hold-up on the packing and, as a result, a decrease in the free volume of the packing available for gas phase flow. At the flooding point the pressure drop is very high. Just below this point the liquid hold-up is very large and this means that the real gas flow velocities in the packing channels are very high. This, in turn, results in an abrupt increase in the friction forces between the gas and liquid phases and a further stop in the liquid flow down. At the flooding point the entire column cross-section is filled with liquid at some distance down the column. This liquid layer is maintained by a compressed gas. The liquid flowing down accumulates, the liquid layer grows and, after some time, the
424
HYDRODYNAMICS OF PACKED COLUMNS
weight of the liquid surpasses the gas pressure and the liquid abruptly descends. This permits the gas to flow upwards. After some time the phenomenon is repeated. Sometimes, the gas ejects the liquid from the packing top. At such gas and liquid flow velocities the column cannot operate in a stable way. Thus, flooding conditions represent the admissible flow rates of phases which may exist in the column. Just before flooding, the mass transfer conditions are optimal. Therefore, from the point of view of mass transfer the flow velocities should be chosen in such a manner that the column operates in the vicinity of the flooding point. However, when safe operation of the equipment is considered the designed gas flow rates are usually taken as 0.6 to 0.9 of the velocity at which the column is flooded. In the literature many relations can be found [2,16,17,18] which allow flow velocities of the phases for flooding to be determined. Below, one such correlation [18] which may be used for random packings is given. The column is flooded when the two dimensionless gas and liquid velocities satisfy the relation w
=
(11-6)
G
*
2
10430(w ) j 0.007-w*
l+2430w* + C
c
The two dimensionless velocities are given by the equations wc = d r- ι pz '^ οr gi) f ^1/3 ζ
and 1-β w^ = 2.6 — G 3 ε ι
ρ
£
2
G
l
p.
L
u, τ A
Gmax
-τ g d ζ
.
(11-8)xo
pz
where constant A reflects the effect of the column diameter A =
2 3
1+ and
-j 1 1-e
(11-9) pz
d
kmin
d^ z is the effective size of the packing element d
=6 pz
a
(11-10)
7N
/ Λ
425
PACKED COLUMNS
Example 11-B. A column packed with 25 mm Bialecki steel rings is to be used in absorption. Determine the column diameter at which the gas flow velocity is 0.8 of the flooding rate. The volumetric 3
3
gas and liquid flow rates are 3000 m /h and 30 m /h, respectively. The physicochemical properties of the gas and the liquid in the column are close to those of air and water at 30°C The basis data for 2
3
Bialecki rings are as follows: a = 206 m /m , ε = 0.959 -5 2 3 The gas parameters are υ = 1.661x10 m /s, 0 = 1.12 kg/m . G
= 8.05x10
The liquid parameters are υ
Li
-7
G
2
m /s, ρ
LI
3
= 996 kg/m .
Solution Introducing data to equations (11-7) and (11-8), using equations (11-9) and (11-10), we obtain *= 30x4 ~ 3600x*xd
W c
2
X
206 6 x d - 0.959)
additionally
1- 0 . 9 5 9 0.959
kmin -7-v 1/3 _ _« ^ _ « Λ - 4 %
χ
Γ8.05Χ10-Ί
L 9. 8 l
7.716X10-
J
2
d
2
km
0.959
y0
1
η 1 +
§ ^ . k
ϊ*-πγ x
x
2
2
x
206xd,
L
3000 x4 x206 3600 χπ χά x 9 . 8 1 X6X(1-0.959) 2
( Bn 1 )
.
2 . 6 * i ^ x j j j i x[ y
_
1+
4
kmi η
J
kmin
1.3X 10"
2
.
2
0.0 1 9 l d 4 d1 km k mii 1n JI km i i ηaJ Irm
(11-B2) Substituting equations (11-B1) and (11-B2) into equation (11-6) we have 1.3 XlO" 1 +
0
0
d
t
2
1 9
.
kmin
kmin
0.15
2430χ(7.716Χ 10" ) 4
1
+
— - v " '
d
/
.
km ι η
.
+
10430X (7.716X 10 ' ) 8
,« . L
0
0
7
2
. 7.7 1 6 X 1 0 ^ d k m i.n t
(11-B3)
426
HYDRODYNAMICS OF PACKED COLUMNS
In equation (11-B3) there is one variable, that is the column diameter,diameter, which is calculated by an iterative method. It is d = 0.84 m.This means that at such a diameter flooding kmi η
conditions appear in the column. The superficial under these conditions is 30 00x4 , . u = = 1.5 m/s G m ax 2 3600χπχ0.84 Under the conditions presented above should be 0.8 of the flooding rate. Thus, u
= 0.8 u G
the
gas
gas
velocity
flow
velocity
= 0.8 x 1.5 = 1.2 m/s Gmax
For such a velocity the column diameter is
d
= ν k
u
= ν
4 χ 3600 χ
3000 π x 1.2
= 0.94 m
G
11.2.4. Mass Transfer Area The wetted surface area of the packing and mass transfer area have been determined in many studies using various measuring techniques [19-22]. The following conclusions may be drawn from these works: a) The mass transfer area is a function of the type and dimensions of the packing elements. b) The material from which the packing is made has a very strong influence on the mass transfer area. c) Up to the loading point, the mass transfer area depends only on the liquid flow velocity. Above this point it also depends on the gas velocity. d) When a distributor with about 100 points per 1 m is applied, the mass transfer area is in fact independent of the column height. e) In columns of small diameter the mass transfer area is a function of the column diameter. f) The mass transfer area depends on the physicochemical properties of the liquid flowing down through the packing. Many data referring to the mass transfer area are presented in Fig. 11-13 [23]. These values represent the mass transfer area for various types of packing under operating conditions below the loading point.
PACKED COLUMNS
427
.a
ο
15
20
0
8
Fig. 11-13. Mass transfer area for different packings
There are significant discrepancies between the experimental results concerned with the wetted surface area of the packing, the mass transfer area in physical processes and the mass transfer area in processes of mass transfer with a chemical reaction. Puranik and Vogelpohl [24] attempted to overcome these discrepancies. They assumed that in a column there are two types of area: a static area a , proportional to the static hold-up, and a dynamic a . s
d
proportional to the dynamic hold-up. In the case of physical mass transfer such as liquid evaporation to the gas, both static and dynamic areas constitute a mass transfer area a m
a
m
= a
s
+ a, d
(11-11)
In physical absorption, the liquid suspended on the packing in the form of a static hold-up very quickly becomes saturated by soluble species and therefore the mass transfer area is only the dynamic area. In absorption with an instantaneous chemical reaction, when the concentration of a reacting substance in the liquid is very high, due to gradual exhaustion of the reacting substance in the liquid, the mass transfer area should correspond both to the dynamic and static area. In the case of slow chemical reactions occurring in the liquid phase, the mass transfer area is equal to the sum of the dynamic
428
HYDRODYNAMICS OF PACKED COLUMNS
area and a fraction of the static area. the ratio of the given rate of chemical instantaneous chemical reaction. On the basis of data presented in equations yielding static and dynamic areas r _ a, A A _ Ä d
= 0.229 - 0.091 In a
a
We __ FrL
the
following
° (11-13) v '
a + ( tanh R - 1 ) — î -
m
Re
the literature, are proposed ϊ -0.182
(11-12)
a a — a a where
This fraction follows from reaction to the rate of an
a
(11-14)
w L
a
nL 2
We
Fr and
L
=
L
σ
=
w
(11-15) v '
a pL σ 2 W l a 2
is the critical surface tension.
c
The following range of variables has been taken — 5 - = 0.08 - 0.8 ; w = 0.25 - 12 kg/(m s); a L 3 2 / / l = (0.5 - 13 ) x l 0 " kg/(m s) ; a = (2.5 - 7.5)XlO" N/m; ρ
= 800 - 1900 kg/m ; 3
L
Several values of σ Table 11-3.
d
ρ
= 10 - 37.5 mm
for different materials are given in Table 11-3.
c Critical values Ο
for different materials c
Material
Critical
tension [N/m]
Glass
7,,4xl0"
Ceramic
6..2xl0'
Graphit
6.,4xl0"
Steel
7.,2xl0"
Parafine
2 2 2
2 2
2..OxlO"
PACKED COLUMNS
429
1 1 3 · MASS TRANSFER COEFFICIENTS IN PACKED COLUMNS Mass transfer coefficients in packed columns were determined in numerous experimental investigations and theoretical studies [25-29]. It follows from these studies that mass transfer coefficients in the gas and liquid phases depend on the flow velocities of both phases, the physicochemical properties of the liquid and gas, represented by density, viscosity and diffusion coefficients, and the type and size of packing. In the case of volumetric mass transfer coefficients, their values also depend on the liquid surface tension and contact angle. In the literature there are many equations, diagrams and tables giving the values of mass transfer coefficients as a function of various parameters. Below, those equations typical for a given packing are presented. Mass transfer coefficients in the liquid phase were investigated mostly on the basis of water systems. For such systems and traditional packings (Raschig rings and Berl saddles) the correlation derived by Mohunta et al. is recommended [30]
i M l 'j iV* î0'^i l 'r fi V Vi VV rf fτT 2 /
î
1 / I9
3
Br)
AL m
w
a
9
,
/ / 44
η
1 / 2
fe)
(-7V] z
1v
L
(11-16)
The ranges of particular variables are as follows 2
w L = 0.1 - 42 kg/(m s) η d
Wq = 0.015 - 1.2 kg/(m s) 2
3
Ρ
where d
= (0.7 - 1.5 ) X 1 0 ' kg/(m s) = 6 - 5 0 mm,
S c L = 140 - 1030
is the nominal dimension of the packing element.
This equation determines the volumetric mass transfer coefficients. It follows from the investigations performed by Sherwood and Holloway [25], and from the above equation, that the volumetric mass transfer coefficients in the liquid phase are in fact independent of packing size. In the case of viscous liquids no complete or detailed studies have been devoted to the determination of the effect of viscosity on the mass transfer coefficient. From fragmentary investigations it follows that this effect is very strong. For such system the Norman and Sammak [32] equation can be applied k
d
3
4w > 0 . 6 i r d g ρ τ Ί ΐ / 6 Γ ηΊ f
^1/2
430
MASS TRANSFER COEFFICIENTS IN PACKED COLUMNS
This equation is valid for the following range of parameters S C l = 108 - 43500
R e L = 1.2 - 545
/fL = (0.4 - 20 ) x l 0 " kg/(m s). 3
Mass transfer coefficients in the gas phase for traditional packings may be calculated using the equation given by Onda [33] k ^
a
AG
w_
r
AO
^o.7r
^ Ό
J
^0.33
η_ G
AG
J
V
where the constant C depends on the nominal packing size d^ for
d
for
d
^
ρ
15 mm
C = 5.23
< 15 mm
p
C = 2
For Sulzer packing the equations derived by Bravo [34] are valid - for the gas phase Sh
= 0.0338 R e ' G
0
Sc
8
G
(11-19)
0 , 33
G
where
AG d
+
PM^
Re'
»
t
°»
e<
U
a n G
e ff
d
U
e i
L
e ff
u
v a
e nt
«)
(11.21)
AL
=
2
V
effective gas and liquid velocities,
diameter packing.
<2) u AAL i
,
hîlL
e f f
a re
^eq " l * ^ - for the liquid phase
k
u
ηο
Ο
where
t
L,eff
\ 71 *ο
(
H
-
2
2
>
where S' is the length of the corrugation side. Another equation is given by Spiegel and Meier [35]. Fair and Bravo [36] gave an excellent review of calculation of a column containg structured packing.
PACKED COLUMNS 431
Example 11-C. Ammonia of low concentration is removed from air in a column packed with 25 mm Raschig rings. Determine the values of
the mass transfer coefficients in both phases, if the gas and liquid flow velocities
in the
2
G
column
2
L
are w = 1 kg/(m s)
and w = 5 kg/(m s),
respectively. The physicochemical parameters of the gas and liquid are %
= 2.3XlO"
m /s
5
%
2
= 1.98XlO'
m /s
9
2
AL
AG
pQ
= 1 . 1 5 kg/m
if
= 1.84XlO"
3
pL
= 997 kg/m
η
= 9 X l O ' kg/(m s)
kg/(m s)
5
3
4
a = 190 m Im . Solution The mass transfer coefficient calculated from equation (11-16)
AL
m
0 V| ^ fΓ „ •Î 5I 'Xx9.Xx1l0o' - 'χx1 l9w J
<·
3 ,
4
9 . 8 1 X 9 97 2
the
l ^ f
9.81X997 = 2.5 X l 0 - x f M90X(9X10"V 3
k a
in
4
2
3
4
phase
99 7 x 9 . 8 l V l
99 XXl O1"
Ι / 4
liquid
9X 1 0 "
. " VV ' " .
0
is
/ 9
J
4
.
0 1 2 9
...
*-997x 1 . 9 8 X 10" J 9
-'
For a linear liquid flow velocity \
- ~T~
=
L
- 9 5 Γ ~"
0
0
05
M S/
the interface a r e | determined from Fig. 11-13 for 25 mm Raschig rings is 105 m Im . Hence the mass transfer coefficient in the liquid phase is k a , AL m 0.0129 - ^ - 4 . k = = —ϊ-τ^ = 1.23x10 m/s a 105
AL
w
in
the
gas
phase
is
determined
>0.7
= 5.23x(2.3xl0" )xl90x ί — ] 5 J H1.84xlO' )xl90 5
A G
f 1 Ll.l5x(2. 3 xlO'V 1
χ
A
m
The mass transfer coefficient from Onda's equation (11-18) k
l
8
4 x
1 0
(190X0.0025)"
2
= 0.047 m/s
432
COCURRENT PHASE FLOW
11.4. COCURRENT PHASE FLOW Packed columns may also operate in a system of co-current phase flow. This operation is particularly desirable when gas solubility in the liquid is very high and the driving force of absorption is, in fact, independent of counter- or co-current flow systems. In the case of a cocurrent system no limitations connected with column flooding occur with the gas and liquid velocities. Since it is possible to use much higher linear velocities for both phases, the column diameter in the co-current system is smaller than in the countercurrent system. Packed columns with cocurrent flow of the phases are also employed in absorption accompanied by a chemical reaction when the packing also acts as a catalyst (trickle bed). From the hydrodynamics point of view, the mechanism for cocurrent flow of the phases is more complicated compared to that for countercurrent flow. For high loading of the column with the liquid and at a low gas flow, or with a significant gas flow and a low liquid flow, as well as for intermediate cases, both phases flow downward in the column as continuous phases. The mechanism of flow in this region is similar to that in a countercurrent column. An increase in the velocity of one phase at a constant velocity of the other phase causes pulsating flow of phases in the column or spraying of the liquid. The characteristics of a cocurrent column operation can be represented by Fig. 11-14. In co-current absorption, when the liquid and gas flow as continuous phases, the mass transfer coefficients and mass transfer areas are similar to those in the counter-current flow. This region of operation is the most suitable for absorption processes.
0.01
OOS
0.1
QS
1
Fig. 11-14. Characteristics of co-current gas and liquid flows
PACKED COLUMNS
433
11.5. BALANCE EQUATIONS OF PACKED COLUMNS FOR BINARY SYSTEMS Chapter 9 presents balance equations for the differential volume of any absorber type for various models of phase flow. In packed columns the real flow of both the gas and the liquid phases is usually approximated by a plug flow model or by a plug flow model with longitudinal dispersion. These absorbers can operate with cocurrent and countercurrent flow, and, apart from a short start-up period, under steady-state conditions. Owing to discrepancies encountered in the literature, which concern the effect of longitudinal mixing on the process of absorption, unreliability of correlations for the dispersion coefficient, and the fact that the existing correlations for mass and heat transfer are valid for a plug flow model, we shall present only those balance equations for packed columns which are based on the plug flow model. In addition we shall assume that (i) in the gas phase there is only one active component and a component insoluble in the liquid (inert component), and in the liquid phase an active component and a nonvolatile solvent are present. (ii) the steady-state pressure. w A6
isothermal
w
process
takes
Λ
AL
a
place
W
constant
WAL
A0 XA
*A
under
ΧΑ
Gm
NAG
1
1
s 0)1
ί b)
W
AG*dWA0
V<*A
V
Gin
Fig. 11-15. Scheme of differential element of a packed column binary systems: a) cocurrent phase flow, b) countercurrent flow
For a column element of differential volume dV (Fig. 11-15), under the above simplifying assumptions, and in the case of physical absorption, equations (9-23) and (9-31) using additionally equations (9-50) and (9-51), are reduced to the form
434
BALANCE EQUATIONS OF PACKED COLUMNS
- gas phase dW = + a_ N ? „
A O
Ζ
m
(11-23)
AG
- liquid phase dW - j — ^
λ
= a
Cl Z
N* m
(11-24)
AL
The minus sign on the right hand side of equation (11-23) refers to the cocurrent phase flow, while the plus sign refers to countercurrent flow. The composition of each phase results from the following formulae S
Λ =
A
W
A r
Q
"
2
5)
i η
s w
s X
A
L
=
(11-26)
s
where S is the cross-sectional area of the column. Substituting the above equations into equations (11-24), we obtain after rearrangement d Y
.
a
= +
A
dz
m
G
a
(11-27)
AO
n
t
ι η
i dX
and
f«
N
s
(11-23)
S Ν* S
The boundary conditions for differential equations (11-27) and (11-28) are known, provided that the concentrations in the gas and the liquid components entering the column are known (Fig. 11-16). Thus, - for the cocurrent flow V»
=
Y
A0
A
X( 0 )
»
X
U
2 9
n
3 0
A0
< -
X
< -
>
- for the countercurrent flow Y
Z
A< >
=
Y
AZ
X
A<°> »
A0
>
PACKED COLUMNS
435
ο)
Fig. 11-16. Diagram of a packed column - binary systems: a) cocurrent flow, b) countercurrent flow
Due to the continuity condition of the mass fluxes Ν
- Ν AG
δ
= Ν
AG
*
*
= Ν
AG
= Ν
AL
δ AL
- Ν
(11-31)
AL
the differential equation (11-28) can be solved comparing equations (11-27) and (11-28) we have dY in
dz
analytically.
After
dX = + L s
(11-32)
dz
Upon integration of the above equation in the range from Y (0) to Y A (z)
and
from
X A (0)
A
to X fz),
A
taking into account
A
conditions (11-29) or (11-30) and neglecting to the current column height z, we obtain - for the cocurrent flow G. (Υ
- Y J
δ
in
A
= L (X
AO'
S
V
A
boundary
ö
- X
A
J
the
symbols
referring
(11-33)
AO
- for the countercurrent flow G
iJ Î - V » Y
-
L
S Î ( X
-
X
A0>
(11-34)
Equations (11-33) and (11-34), as was mentioned in chapter 10, are called the operating line equations. They represent a relationship between the actual compositions of the gas and liquid phase in a given packed column cross section.
436
BALANCE EQUATIONS OF PACKED COLUMNS
For binary systems the calculation problem for packed columns is usually formulated in one of two ways: a) For a given absorber height Ζ (or mass transfer area) calculate the concentration (or absorption efficiency) of the active component in the gas phase, which leaves the equip ment (the calculation refers to the existing absorber which is to work under new operating conditions). b) Determine the absorber height Ζ (or mass transfer area) at which the desired concentration (or absorption efficiency) of the active component in the gas leaving the absorber is achieved. The solution of type a) consists of numerical integration of the differential equation (11-27) in the interval < 0 , Z > for the initial condition Y (0) = Y . A
v
7
AO
To solve a problem equation (11-27) to d
Y
A
of type b), it
S Ν
a
m
is convenient
to
transform
δ
AG
since for this form of differential equation the integration interval is known for the independent variable Y . For the assumed A
absorption
fj
efficiency
of
the
active
component,
the
molar
abs
ratio of this component in the gas leaving the column is - for the cocurrent flow Y
AZ
«
=
Y
- ".bs>
U
A0
< -
3 6
>
- for the countercurrent flow Y
A<°>
=
Y
A0
«
<
!
- ".bs>
Y
AZ
Π
< -
3 7
>
The calculation of the column height Ζ (or mass transfer area) at which the desired active component concentration (or η t ) is abs
achieved consists in a numerical solution of differential equation (11-35) in the interval < Y ,Y > for the initial condition z(Y ) = 0. AO
AZ
AO
The column height Ζ can also be calculated using the integrated form of equation (11-27)
PACKED COLUMNS
437
The application of equation (11-38) is particularly convenient when both the gas and liquid phases are dilute solutions and the absorption equilibrium line is a straight line (see Chapter 10). In such a case the integral on the right hand side of equation (11-38), and thus the column height, can be calculated analytically (example 11-D). For concentrated solutions or when the equilibrium line is not a straight line, the integral constituting die right hand side of the equation must be calculated numerically. The gas (liquid) phase, apart from the active component, often encompasses a mixture of several inert (nonvolatile) components. As was mentioned in Chapter 9, from the formal point of view such an absorption system is a multicomponent system and the calculation of packed columns should be made on the basis of equations for multicomponent systems. In some cases, the mass fluxes and N ^ L are calculated on the basis of approximate relations, in which the concept of efficient diffusion coefficient is used (Chapter 4), and then it will be relatively simple to calculate the columns and to solve the balance equations for binary systems. The calculation of packed columns is also relatively simple in the case of a chemical absorption, when in the liquid phase a instantaneous reaction occurs A
+
ν
Β
Β
>
C
(11-39)
and the reagents Β and C are nonvolatile components. In such a case, the mass balance for the active component in the gas phase is given by equation (11-23), and the balance equations (9-16) of the liquid phase reduce to a single differential equation in the form d
*
Z
W T B L
λ
= a m NT T BL
(11-40)
The molar ratio of component Β in the liquid phase results from the following equation c
X°
Β
S wnT = —,—— Lr
(11-41)
S
Substituting this equation into equation (11-40) we obtain .„δ
„
„δ
438
BALANCE EQUATIONS OF PACKED COLUMNS
The assumption of a instantaneous chemical reaction (equation 11-39) is equivalent to the assumption that this reaction takes place only in the liquid film and the concentration of component A in the liquid bulk is, in fact, equal to zero. The differential equation (11-42) can be solved analytically * δ after the relation between mass fluxes Ν and Ν is determined. AL
BL
For the reaction taking place in the liquid film, according to the considerations presented in Chapters 7 and 10 one may write dN dN t . = - — . (11-43) dz ν dz Β
* δ Upon integration of this equation in the range from Ν to Ν and AL AL c * δ from Ν to Ν , we have BL AL
BL
= —
- N*
AL
V
ÎN* \
B ^
BL
- Ν* 1 BL
(11-44) '
J
*
Since it has been assumed that reagent Β is nonvolatile (N
BL
=
0)
and the concentration of component A in the liquid bulk is equal to zero (N Ν*
= 0 ) , equation (11-44) simplifies to the form AL
= -υ
BL
N* Β
(11-45)
AL
'
At the interface there is the condition N
AO = 1
<- >
N
U 46
L
After employing equations (11-45) and equations (11-27) and (11-42), one obtains d Y
G.
in
, dz
f
L
<
= + —? v
d X
Β
(11-46)
f
^—dz
in
differential
(11-47)
In this equation the plus sign refers to co-current phase flow and the minus sign to the counter-current flow. Upon integration of equation (11-47) in the range from Y (0) to δ δ Υ (ζ) and from Χ (0) = X to Χ (ζ), and after neglecting the Α
Β
B0
Β
symbols concerning current column height z, we have
v
PACKED COLUMNS
439
- for the cocurrent flow
in
A
AO
V
Β
Β
7
BO
- for the countercurrent flow G
iJ Î - V » = - "IT Y
-V
1
Β
^- > U 49
The calculation procedure for a instantaneous chemical reaction expressed by equation (11-39) is almost the same as in the case of physical absorption. For problem a) the procedure includes a numerical solution of differential equation (11-27) in the interval < 0 , Z > with the initial condition Y (0) = Y . For problem b) it A
AO
encompasses either a numerical solution of differential (11-35) in the interval < Y .Y > with the condition z(Y J AO
AZ
equation = 0, or
AO
a numerical (sometimes analytical) calculation of the integral in equation (11-38). The difference is that in the case of physical absorption the liquid phase composition is determined by equation (11-33) or (11-34), while in absorption with a chemical reaction (11-39) by equation (11-48) or (11-49). Example 11-D. In a countercurrent packed column ( F i g . l l - D l ) pure water is used as the solvent in removing ammonia from air. Determine the column dimensions at which 95% absorption of NH^ takes place under the following conditions: - mean temperature in the column Τ = 298.15K - mean pressure in the column Ρ = 1.013 χ 10 Pa - volumetric flow rate of the gas phase at the column inlet (at 5
273.15K and 1.013 Χ 10 Pa) 5
'
ν Λ „ = 1 m /s 3
OGZ
- mole fraction of NH^ in the gas phase at the column inlet y = 0.03 7
AZ
- water flow rate at the column inlet equal to 1.1 of the minimum value. The equilibrium line is given by the equation Y* = 0.98 X* A
A
v
440
BALANCE EQUATIONS OF PACKED COLUMNS
The packings are ceramic Raschig rings of size d = 0.025 m. The physicochemical parameters of the gas and liquid phases as follows - density
1.18 kg/in
PG=
viscosity
ρΊ = 997 kg/m Li
= 1.86X 10' kg/(m s) 5
η
η = 8.94X 10" kg/(ms) 4
G
L
surface tension
σ = 0.072 N/m
molecular mass
M = 29 kg/kmol
M = 18 kg/kmol L
G
diffusivity
are
AG
=
2.31xl0" m /s 5
2
AL
=
2.0xl0' m /s 9
2
Assume that the gas phase is a mixture of ideal gases.
Fig. 11-D1. Diagram of a packed column for ammonia absorption in water
Solution Due to the very small concentration of NH^ in the inlet gas, the column calculation will be made using the equations applicable to dilute systems. 1. Mass balances of the gas phase The volumetric flow rate and total gas phase concentration the process conditions are ν
°
Z
= ν
V °°
Z
P
T
o
=
l
x
L 0 l W
298,15 = 1.013 XlO"' 2 7 3 . 1 5 x
where subscript ο refers to normal conditions.
l
W
/
s
.
( 1 1 D 1 )
in
PACKED COLUMNS
C
o -
π
-
lO^kmol/m
=
8 3 Ϊ 4 ^ 9 8 ° 1 5
441
(11-D2)
3
The flow rate of the inert component (air) is calculated from the formula
ο i n,
- ooz ™ 4R T«
0
-
υ Ά ζ· ' .
< "
1
0
1
8314 χ
3
x
1
0 3
273.15
Χ (1 - 0.03) = 4.327Χ10" kmol/s
(11-D3)
2
The molar ratio of NH^ in the gas mixture is given by - at the inlet to the column = J
Y
= T T T T ^ U = 3.093 X 10" kmol A/kmol in
A Z
2
y
" AZ
~
(11-D4)
'
- at the outlet Y A f t = (1 - 0.95) Y AU
= (1 - 0.95) X (3.093 χ 10" ) 2
ΑΖι
= 1.547x10 kmol A/kmol in (11-D5) The transfer rate of ammonia absorbed between the phases in the column is G. ( Y . . - Y . J in
AZ
= (4.327X10" ) χ (3.093X10" - 1.547xl0" ) 2
2
3
AO
= 1.271 XlO" kmol A/s
(11-D6)
3
2. Mass balances of the liquid phase. A minimum ratio (L IG ) . (Fig.ll-D2) is a result of equation S
(10-5) rearranged to form
f "TT" 1 » in
min
in min
YaZ
" °
ΑΖ
^
Y
^
Ya
m AO where the slope of the equilibrium line Since pure water enters the absorber ΧΑΛ = 0
m = 0.98.
AO
Substituting numerical values into equation (11-D7), we obtain -2 -3 r L "\ A
mM
442
BALANCE EQUATIONS OF PACKED COLUMNS
A'
Fig. 11-D2. Minimum ratio L / O .
A
for ammonia absorption in water
The flow rate of the solvent (water) is equal to L
S
= 1.1 G
ί-τΛ]
in
νI
Cj.
= 1.1 Χ (4.327X10" ) χ
0.931
2
J
in ^ min
= 4.431 XlO" kmol S/s
(11-D8)
2
The molar ratio of NH^ in the liquid leaving the column is X
AZ
=X
G +-r^(Y - Y J = 0 + AO Ls AZ AO'
-2 4
3
2
1
x
10
4.431 x l O "
2
x (3.093XlO"
- 1.547X10' ) = 2.869X10' kmol A/kmol S 3
2
(11-D9)
2
3. Column diameter In Example 11-B it has been shown that equation (11-6) through (11-10) can be transformed to a single nonlinear algebraic equation f( d
. ) = 0
(11-D10)
kmin
where the only unknown is the minimum diameter d
,
at which the
kmin
column is flooded. Proceeding as in example 11-B, the column diameter was calculated to be d = 0.8271 m for Raschig rings of d = 0.025 m kmin
(a = 190 m Im
ρ
and ε = 0.73, cf. Table 11-1) under the assumed
PACKED COLUMNS
process conditions. The superficial to d t . is equal to corresponding to
linear
gas
flow
443
velocity
k kmin
4 ν
GZ
=
"
G m ax π
d
x
4
=
π x
2
1 . 0 9 2 = 2.032 m/s
( 0.8271)
(11-D11)
2
km in
It was assumed that u
(11-D12)
= 0.7 X 2.032 = 1.422 m/s
= 0.7 u G
Gmax
The column diameter corresponding to this velocity is
/ GZ 4 V
= ν
π u
4 x 1 092 πΧ 1 422
/ = ν
G
(11-D13)
= 0.989 m
In further calculations the column diameter d = 1 m was assumed. k
4. Mass transfer coefficients The cross-sectional area of the column is equal to S =
n
S
2
=
π
?
= 0.7854 m
1
The superficial mass flow are, respectively _ j A W G= —g L M
s
W
= —g"
L
velocities of the gas and liquid
(4.327xl0" )x29 „ 2N 1 < o f t. ο 7 8 5 4' = 1.598 kg/(m s)
phase
2
=
w
w
(11-D14)
2
(4.431 x l 0 ' ) x 18 o. 7 854 2
L =
The mass transfer coefficient Onda's correlation (11-18)
, ,1
1
=
1
0 1 6
Λ
k
8
2
7
/
/ ( m
s
/ tl (11-D15)
Ν
>
(H-D16)
in the gas phase is calculated
from
,0.7
k
AG
=
5.23x(2.31xl0" )xl90x 5
1 . 598 ( 1 . 8 6 X 10
1 .86x10
0 . 33
-5
1.18X(2. 3 1x10
) x 190j
X(190 x 0.025)" = 6.478 Xl0" m/s 2
2
) (11-D17)
444
BALANCE EQUATIONS OF PACKED COLUMNS
For dilute systems it can be written as k
= k Gy
= k GY
C AG
= (6.478XlO" ) χ 2
(4.087XlO" ) 2
G
= 2.648xlO" kmol/(m s) 3
2
(11-D18)
The volumetric mass transfer coefficient in calculated from Mohunta's correlation (11-16)
the
liquid
phase
2/ 3
k
9 .81x997
a = 2.5XHTX
AL m
1/9
9.81
190χ (8. 9 4X10" )
X997
8.94X 1 0'
4
1/ 4
1 . 0 1 6 x ( 8 . 9 4 x 10" )x 190 3
4
9 .81 x 9 97 2
3
8 .94x 1 0
X
4
is
-1/2
-4
[997 X ( 2 . 0 χ 10"')J
.-3
= 3.940x10'" 1/s
(11-D19)
The mass transfer area is given by the correlation (11-14) 0.04 1
1.01 6
a = 1.045 X190X m
.
r
-» 0 . 1 3 3
1.0 1 6 X 190x997x0.072
1 9 0 x 8 . 9 4 X 10" -0.182
0.072 0.062
= 58.86 m / m 2
3
(11-D20)
The critical surface tension σ = 0.062 N/m has been obtained from c
Table 11-3. The molar mass transfer coefficient is equal to P
^ A L V k
Lx
3.940X10" 58. 8 6
L
M
=
3
997 18
= 3.708XlO" kmol/(m s) 3
2
(11-D21)
The overal mass transfer coefficient Κ
o,-
K
2.648x10"
Lx
= 1.558XlO" kmol/(m s) 3
is equal to
1
m Gy
Gy
2
0.98 3.708x10" (11-D22)
PACKED COLUMNS
445
5. Packing height The
packing
height
sufficient
for
achieving
the
desired
NH^
absorption can be determined analytically since the equilibrium curve is a straight line, and the gas and liquid phases are dilute solutions. The differential mass balance equation for NH^ in the gas phase is given by equation (11-27) dY
a _
S Ν -
A°
o
.
2
7)
i η
At the interface, NH holds
the
condition
of
continuity
of
molar
flux
for
3
N
(11-D23)
=
A G
In a gaseous film, NH^ diffuses through an inert component
(air)
and in a liquid film through a nonvolatile solvent (water). The molar fluxes are calculated from the equations holding for dilute solutions (Table 5.1) N
AG « V A - l> Y
Y
The value Y
(U
-
D24)
will be expressed as a function of Y . A
A
Using equations (11-D24), (11-D25) and the equilibrium line equation Y* = m X* A
(11-D26)
A
in equation (11-D23), after rearrangement, we obtain m k Y* = A m
m k Y
τ-
k
+
A
k
Gy
The value of X X
A
»
X
A0
A +
+
=-
m k
Lx
Gx
xV-
+ k
x A
(11-D27)
A
Lx
is calculated from the operation line equation (11-34) -TT-
( Y
S
Y
A - A0>
Substituting this value into equation (11-D27) we obtain
n
( -
D
2
8
>
(
446
BALANCE EQUATIONS OF PACKED COLUMNS
mk
γ*
m k_ G y + k Lx
A
γ
r
mk mk.
+
X
-ί mkΓG G yy
A
G. G.
r
Γ -LxL A 0
+
-, Υ
Υ
- Τ "S< Α - Α 0 >
J
(11-D29) Application of equations (11-D24) and (11-D29) in equation (11-27), after rearrangement, leads to the following differential equation d
Γ
Y
A
indz
m
G7
yL
f
i
m
k + Gy
k
l Lx
k
m
G. ^ L
L
ΙJ
S
m k m k
+
r - î T - h r »· « Y
Gy
Lx ^
x
< J
S
"-
D30)
J
Upon integration of the above equation in the range Y A = Y A 0 to Y
= Y A
and of ζ = 0 to ζ = Ζ, the relation determining the
AZ
column height is obtained
x In
G.
[
Ζ =
r
Y
mG. s AA Z
Ί J
mG LY - ^
a SK m
Gy
A Z -
Y
Y
A 0 > -
f
f
i
X
A 0
(11-D31)
- mX AO
AO
Introducing numerical values, we have Ζ = 8.341 m In the example considered,
molar
fluxes
Ν ^ and Ν AG
τ
were
AL
discussed on the basis of formulae valid for dilute systems. Moreever, mass gas and liquid velocities were assumed constant along the column.If these fluxes were calculated on the basis of formulae holding for concentrated systems, the packing height which would guarantee 95% NH^ absorption would be equal to Ζ = 8.05 m. Therefore, the difference between the two solutions is insignificant as it does not exceed 4%. Moreover, the column height obtained from the exact solution is lower than that under the assumption that the system is diluted, so the result obtained
PACKED COLUMNS
447
is safer from the practical point of view. Therefore, for dilute systems in which the concentration of the active component in the gas and liquid phase does not exceed 10%, for the sake of simplicity of calculation, the use of equations (11-D24) and (11-D25), which hold for dilute systems, is recommended. Example 11-E. In a packed countercurrent column of diameter d^ = 1.5
m absorption of CO^ from
a gas mixture is carried out by
means of water. Determine the packing height at which 99% absorption of CO^ is achieved. The process parameters are as follows - mean temperature in the column
Τ = 298.15 Κ
- mean pressure in the column Ρ = 2 χ 10 Pa - molar flow rate of the gas phase at the column inlet 6
G
= 4.46XlO" kmol/s ζ - molar flow rate of the liquid phase at the column inlet 2
L
= 4.03 kmol/s ο - molar fraction of CO^ in the liquid entering the column χ
= 3.07XlO" kmol A/kmol 5
Λ
AO
The composition of the gas phase entering the column is given in Table l l - E l . Table l l - E l . Composition of gas phase
Component
c o 2 (A)
Mole
fraction
0. 300
(B)
0 .012
(C)
0. 515
(D)
0. 170
CH4 (E)
0. 003
CO N 2
The equilibrium line is given by Henry's law in the form where
He = 1.65x10
Pa.
The packing consists of ceramic Raschig rings of diameter
d^ =
0.025 m. The physicochemical parameters of the liquid are: /> L =
448
BALANCE EQUATIONS OF PACKED COLUMNS
997 kg/m? η = 8.94 χ 10" Pas, σ = 0.072 N/m, Ni = 18 kg/kmol, SB = 4
-9
Li
L«
2
AL»
2.01 x 10 m /s. The physicochemical parameters of the gas mixture components are tabulated in Table 11-E2. Assume that the gas phase is a mixture of ideal gases. In the calculation of multicomponent mass transfer use an approximate formula for an effective diffusion coefficient for C 0 2 through a mixture of inert gases. Table 11-E2. Physicochemical properties of the gas phase components (T = 273.15K, Ρ = 1.013X10*Pa) ο ο j
component
M
η,χΙΟ
/ H T J
J
J k r
5
DxlO
J
5
AJ
A Β
44
115.5
28
61.4
1.76
1.37
C
2
81.3
0.898
5.5
D
28
59.5
1.82
1.33
Ε
16
55.2
1.15
1.53
1.57
Solution Owing to the very low (as compared to CO^) solubility of the other components of the gas mixture, the presence of these components in the liquid phase has been neglected. Moreover, it has been assumed that the solvent (water) is not volatile. 1. Mass balances of the gas phase The total concentration of the gas phase in the process conditions is C
o - Ί Γ Τ - - 8314X 2 9 8 1 5 = 6
8
0 6 8
X 1 0
"
2
"
( 1 1E 1 )
The total flow rate of inert components is equal to G. = G (1 - i l in L AL = 3.122x10
= (4.46 Χ ΙΟ" ) Χ (1 - 0.3) 2
(11-E2)
kmol/s
The molar flow rates of particular inert components are as follows = 4.46XlO"
G B
2
-2
χ 0.012 = 5.352χ 10" kmol/s 4
-2
GC - 4.46XlO χ 0.515 - 2.297XlO'' kmol/s -2 -3 G d = 4.46x10 χ 0.170 = 7 . 5 8 2 x 1 0 kmol/s 2 4 G - 4.46XlO" χ 0.003 = 1.338XlO" kmol/s Ε
PACKED COLUMNS
449
The molar ratio of C 0 2 in the gas mixture is - at the column inlet y A Y = 1 - ^ = -^j^ A Z l - y l - 0 . 3
= 4.286 X l O ^ k m o l A/kmol in
(11-E3)/
- at the column outlet Y
1
= 0.01 Y AO
= 0.01 Χ (4.286 Χ 10" ) AZ
3
= 4.286xl0" kmol A/kmol in
(11-E4)
The quantity of C 0 2 absorbed in the column per unit time 2 1 3 G. (Y - Y . J = (3.122X10" ) χ (4.286X10" - 4 . 2 8 6 X 1 0 " ) ιη
AZ
AO
= 1.325x10
2
kmol A/s
(11-E5)
The molar flow rate of the gas phase leaving the column 2 3 G 0 = G . n ( l + Y A 0 ) = (3.122X10" ) χ (1 + 4 . 2 8 6 X 1 0 " ) = 3.135X10"
2
kmol/s
(11-E6)
2. Mass balances of the liquid phase The molar flow rate of the solvent 3
L s = L 0 ( l - x A 0 ) = 4.03χ(1-3.07XlO" ) s 4.03 kmol/s
(11-E7)
The molar ratio of CO^ in the liquid - entering the column X
= .
A
3
° = AO
0
7 X
1 0
5
= 3.07 XlO" kmol A/kmol S
1-3.07x10" (11-E8)
- leaving the column X
AZ
=X
+ - ^ - ( Y - Y ) = 3.07 x 10" + — — - j — T T T ~ — L AZ AO 4.03 5
AO
S
3
3
X ( 4 . 2 8 6 X l O ' U ^ e ö X 10" )= 3.318x 10" kmol A/kmol S
(11-E9)
The molar flow rate of the liquid phase leaving the column 3
L=L(1+X Ζ
S
) = 4.03 χ (1+3.318 x l O " ) = 4.043 kmol/s AZ
(11-E10)
v
450 BALANCE EQUATO INS OF PACKED COLUMNS 3. Mass transfer coefficients A comparison of G
with ϋΛ
Z
(equation 11-E6), and Y
AZ
0
(equation
11-E3) with Y ^ o (equation 11-E4) shows that the composition and molar flow rate of the gas (and, therefore, the viscosity and density) are subject to appreciable changes along the column height. These variations, flow rate in particular, significantly affect the mass transfer coefficient k , causing it to vary along
the
column
height.
The
AG
value
of
the
coefficient
k
AG
is
therefore calculated from equation (11-18) at each integration step of the mass balance equation for CO^ In the calculation, changes in the viscosity, density and gas phase flow rate along the column are taken into account. The diffusivity occurring in equation (11-18) is calculated from the approximate equation (4-137) transformed to the equivalent form D
Io
G
G
B r _
%
G
C +
_
«
AB
—
G —
D +
AC
_
0
4.
AD
E
-
( 1 1E U
_
»
AE
The binary diffusion coefficients ® for normal conditions presented in Table 11-E2. Substituting numerical values into above equation, we obtain
(AG) D
= 301X10
"
5 M2/S
are the
(11-E12)
The subscript ο refers to the normal conditions. For process conditions, the diffusivity the equation given by Fuller et al. (3-70) ,
175
Χ [ 273 15 ]
>
= 1-78XlO"
6
is ,
m /s 2
The viscosity of the gas mixture is determined the equation [37]
calculated Λ
, --10
using
5
(11-Ε13) on the basis of
PACKED COLUMNS 4 5 1
Σ y.
j=i
-
ηο
*t.
J
7
V
J
Μ.Τ J kr
J
-V
(H-E14)
Σ y. v
7
M.Τ J Jkr
j = i, J
transformed into G
inV
"c G
/
A
M
A
ιnY
A
T
k
+
r
, / A
Vi
i
/ I
M
T A
Ak r
+
.
Σ
j=2
jν
G
,
(- > U
M
j T ι kr
E15
Substituting numerical values (Table 1 1 - E 2 ) , we obtain 5.661 Χ 10" "G
5
Y
3.606Υ
A
+ 2 . 5 6 4 χ
ΙΟ"
5
+ 2 . 3 5 9
"
Ε
1
6)
A
The mean molar mass of the gas mixture is calculated from the equation M Y + M ηι MG = * * (11-E17) A
where
5
Μ
= in
j
Σ M y = 8.82 kg/kmol j Zj =2
(11-E18)
The gas mixture density, molar flow velocity are calculated from the formulae
superficial
mass
=
G
= G (1 + Y ) in A
(11-E20)
w
=
(11-E21)
G
O
C
and
Pg
where
M
rate
(11-E19)
O
G M /S
G 2
π d S = — = 4 The values of η , ρ
n
π
-v2
4 and w
GG
= 1.767 m
2
(11-E22)
depend on the concentration Y , and
G
so they are implicit functions of the current column height.
A
(
1
1
452
BALANCE EQUATIONS OF PACKED COLUMNS
The molar mass transfer coefficient is given by the formula k
= k Gy
C
(11-E23)
AG G
A comparison
of L
(equation
(11-E10)) with L
TL
reveals that 0
the molar flow rate of the liquid phase does not in practice change along the column. Besides, the values of X (equation (11-E9))
and
X
(equation
(11-E8)) prove that
the
AZ
liquid
phase
AO
can be treated as a diluted solution. For these reasons the volumetric mass transfer coefficient in the liquid phase and the specific dynamic mass transfer area are calculated assuming that these values are constant along the entire column. Upon substitution of numerical values into equation (11-16) and (11-14) one has 2 1 k a = 6.344 χ 10 s" AL m
a
2
m
= 182.7 m / m
3
The molar mass transfer coefficient in the liquid phase is ( k
.
a
AL m
)
a
Lx
P
_ 6.344X10
L
M m
2
18 2 . 7
997
18
L
2
2
= 1.923 χ 1 0 " kmol/(m s) 4. Packing height At the interface CO holds
(11-Ε24)
the condition
of continuity of molar
2
NAo =
flux
of
(11-E25)
It is assumed that in a gas film the process of masstransfer is described by the equation for one-component diffusion ( C 0 2) through an inert mixture of other gases 1 + Ν
= k
In
Y* — 1 + Y
(11-E26)
A
Similarly,
in a liquid
an absorbent (water)
film,
the solute
CO^ diffuses
through
PACKED COLUMNS
453
1 + X AL
(11-E27)
Lx
1 + X
The equation has the form Y
A
For
of absorption "5
^ - 5
He X
He
=
given
equilibrium
expressed
in molar
(11-E28)
- 1
values of Y
and
ratios
X
A
(Fig.ll-El)
equations
(11-E25)
A
through (11-E28) form a nonlinear algebraic equation (1 + Y f ) [ P (1 + X * ) - H e X J A * — Ρ (1 + x A )
f(X A ) = k Q y in 1 + X - k
Lx
ln
(11-E29)
= 0 1 + X
with one unknown X . A solution of this equation and
equation
A
(11-E28)
allow
molar
ratios
Χ ,
Y
at
the
in the liquid bulk follows from
the
A
interface (Fig. 11-E1).
A
to
be
determined
operating line
eq(11-E29) YA
'
*A
|\equilibrium curve
*A
Fig. 11-E1. Equilibrium for a given state of phase bulks
The molar ratio of C 0 2 operation line equation (11-34) G X* A
= X
+ AO
μ . L
(γ*. γ A
) AO
(11-E30)
454
BALANCE EQUATIONS OF PACKED COLUMNS
The height of the column can be calculated from equation (11-35) ,
G. Ζ" ( Υ AO = 0 α λ )
I N
dY
a A
S
Ν
m
(11-35)
v
AG
Numerical integration of this differential equation in the range from Y 4 = Υ 4 Λ to Y = Y allows us to obtain a value for the A
AO
A
AZ
packing height at which the desired CO^ absorption level is achieved. Each call for the function on the right hand side of the differential equation in the integration procedure involves the following calculations: - molar ratio of X
A
-
equation (11-E30)
- gas mixture viscosity, η
-
equation (11-E16)
- mean molar mass of the gas mixture, M
-
equation (11-E17)
- gas mixture density, p^ - equation (11-E19) - molar flow rate and superficial mass velocity equations (11-E20) and (11-E21) - mass transfer coefficient, k - equation (11-18)
of
the
gas
AG
- molar mass transfer coefficient, k Q ^ - equation (11-E23) - molar ratio X as a result of a numerical solution
of
A
the
nonlinear algebraic equation in the form (11-E29) * - molar ratio, Y - equation (11-E28) - molar flux, Ν
A
AG
-
equation (11-E26)
The differential equation (11-35) was integrated using the classical fourth-order Runge-Kutta method with a variable integration step. The final result was Ζ = 8.106 m. Typical results, calculated for several values of Υ , are displayed in A
Table
11-E3.
From
this table
it
follows
that
the
value
of
-3
mass transfer
coefficient
k
varies from
1.393x10 -4
the 2
kmol/(m
s)
2
in the lower part of the column, to 8.678x10 kmol/(m s) in the upper part, thus it decreases by over 37%. This change is caused by the variables w . ρ and η . G
G
G
The computer program for this example is called E l IE.PAS.
455
PACKED COLUMNS Table 11-E3. Result of calculations Ya
[kmol A / k m o l
in]
Parameters
4.286χ1θ"
3
0.0
3.136χ10" 1.592x10' c A
4.729xl0'
xa
3
5
xa
3.070xl0"
\
1.092xl0"
M
2
1
3.917xl0"
Y*
3
5
2.6xl0
2. 025x10'
2
Pc
7.237
k
8.678xl0'
8.106
3.580X10'
2
6..445x10"*
4..627xl0"
2.014xl0"
5
9..537xl0'
4.890x10"
1
3
3.615xl0"
3
3
3.321xl0~
S
1.280χ10"
16.08
19.37
12.97
15.63
1.209χ10"
2
1
4.228x10"
3
1.226xl0"
4
4.461xl0~
1
2.442xl0"
3
2
1
2.515xl0"
8 .723 4
4.286x10'
3.934xl0"
1
5,,612xl0'
1..13xl0'
_1
5.808
3.,310xl0"
10 .81
8.970
c
2
2 .889
ζ G w
6xl0"
3
5
1.393xl0'
3
Gy
Example 11-F. In a countercurrent packed column of diameter d f c
=
1.1 m the absorption of C 0 2 from a gas mixture using a 2.5 molar aqueous solution of monoethanolamine was performed. Determine the packing height at which 99.995% of C 0 2 absorption is guaranteed. The
ratio
of
CO^
concentration
in
concentration at the column inlet is α outlet it is α 1.5x10
-3
Z
0
the
liquid
the
amine
= 0.15, and at the column
= 0 . 4 . The viscosity of the liquid
kg/(m s), and the amine diffusivity
-10
to
phase is η
=
in water is SH = BL
2
7.7x10 m / s . The other process parameters (except for the molar flow rate of the absorbent and CO^ content in the liquid entering the column) are the same as in Example 11-E. Solution 1. Reaction kinetics From the considerations presented in Chapter 7 it follows that the overall chemical reaction rate in the liquid phase in which CO and monoethanolamine (MEA) are involved CO
+ 2
2RNH
RNHCOO" + 2
RNH* 3
(11-F1)
456
BALANCE EQUATIONS OF PACKED COLUMNS
can be reaction
described
R = kC
by
the
kinetics
of
a
second-order
irreversible
C
(11-F2)
1 A Β
In
the
liquid
phase
the
subscript
A
refers
to
CO^y
and
the
subscript Β to ME A. The dependence of the reaction rate constant on the temperature is expressed by the formula l o g i o fti = 10.99 -
(11-F3) 3
where the constant
k^ is given in
m /(kmol s).
For process conditions, i.e. for
= 5917.7
Τ = 298.15 Κ
m /(kmol s)
(11-F4)
3
2. Absorption equilibrium and CO^ diffusivity in MEA solution Due to a chemical reaction taking place in the solution, solubility and diffusivity of C 0 2 are determined indirectly the basis of the data on inactive nitrogen monoxide (N). To formulate an absorption equilibrium equation for
the
the on CO^
aqueous MEA solution system and to determine CO^ diffusivity
in
the liquid phase, we shall use the correlations mentioned in Chapter 7. %
=
ni VSTW A
m Ν
1
43
%
= 3.61x10
U
^NL -5
< -
-1.87x10
-7
AL
C
-8.73x10
-7
C
BL
1
F 5
>
(11-F6)
BL
o1 r
"
( 1 1 F 7 )
where m = J
P. i R Τ C
(j = A, N)
(11-F8)
J
The mean value of nitrogen monoxide diffusivity in the aqueous MEA solution was determined as %
NL
=1.48x10
-9
2
m/s.
PACKED COLUMNS
457
The diffusivity of C 0 2 in the MEA solution is I
I
Γ
4L Λ
0 AL
= 1.61 x 1 0 " m /s 9
(11-F9)
2
Substituting equation (11-F8) advantage of the relation Ρ*
= Ρ y*
A
A
into
equation
(11-F6)
and
taking
(11-FlO)
the absorption equilibrium equation for the system considered is obtained 3 2 2 y* = (1.134 - 5.873X10" C - 2 . 7 4 2 x l 0 " C ) C* (11-F11) A
BL
where molar concentrations C
BL
and C
* AL
BL
AL
3
are given in kmol/m.
3. Mass balances for the gas phase The values of C , G. , G , G , G , G , Y G
in example 11-Ε.
in
B
C
D
E
AZ
have been calculated
The molar ratio of C 0 2 in the gas mixture leaving the column Y
= (1 - 0.99995) Y AU
= (1 - 0.99995 ) X ( 4 . 2 8 6 x l 0 ) _1
Pî.Lj
= 2.143X10"
5
kmol A/kmol in
(11-F12)
The quantity of C 0 2 absorbed in the column per unit time G. (Y ιη
- Y J AZ
-
(3.122X10" ) χ (4.286X10" - 2 . 1 4 3 X 1 0 " ) 2
1
5
AO
= 1.338X10" kmol/s (11-F13) The molar flow rate of the gas phase leaving the column 2 5 G = G (1 + Y J = (3.122 χ ΙΟ" ) χ (1 + 2.143 XlO' ) 0
2
in
= 3.122Χ10"
AO 2
kmol/s
4. Mass balances for the liquid phase The free amine concentration in a liquid - entering the column
(11-F14)
458
BALANCE EQUATIONS OF PACKED COLUMNS
= 2.5 (1-t) a)
C BLO
= 2.5 X (1-2x0.15) = 1.75 kmol/m
(11-F15)
3
Β 0
- leaving the column = 2.5 (l-o a ) = 2.5 X (1-2x0.40) = 0.50 kmol/m
C BLZ
(11-F16)
3
Β Ζ
ν = 2 is the stoichiometric coefficient of ME A in reaction (11-F1). Β
The reaction between CO^ and MEA is a fast one; it takes place only in the liquid film. The volumetric flow rate of the liquid is therefore calculated as V
G
Y
B
V
L
in^ AZ
C.f
=
BLO
_ Y
-C
2
A0^
^
2X ( 1 . 3 3 8 X ΙΟ" )
1.75
=
BLZ
-
0.5
-
2
1
-3
Λ ΑΛ
4
1
3.
1 Λ
X
10
m
/S
(11-F17)
5. Mass transfer coefficient
The coefficient of mass transfer in the gas phase is calculated from the equations given in example 11-E. The volumetric mass transfer coefficient in the liquid phase and the dynamic surface area are calculated from equation (11-16) and (11-14), respectively. After substitution of numerical values into these correlations, we have k a
AL
a
= 2.118X10"
m
= 148.7 m / m 2
m
s"
2
1
3
hence,
( k
k
a
AL m
a
AL
2.188X10" "Ί'^Τ 14o . 7
)
m
2
, -A 4 = 1.424X IO""* m/s
(11-F18)
6. Packing height At the interface molar flux continuity holds
AG » 1L
<- >
N
N
U F19
The molar fluxes Ν
and Ν AG
are determined by the formulae AL
A
1 + Y
Ν
A G
= k
Gy
ln
-— 1 + Y*
(11-F20)
A
AL = A AL
N
E
K
(- > N F21
PACKED COLUMNS
where Ε
is the second-order reaction
A
enhancement factor.
459
It is
calculated from Wellek's equation
• * (EAA - Ι )
- 1
·
ι « = Œ00 -x I ) '
3 3
r ^ 7 •+ fà
1 3 5
ν · 1
3 5
ι
tgh V M A
(7.50)
where g>
k
AL
M
k
1
c
AL
c*
BL
+1
00
BL
2
k E
s
o k Β
BL
C
AL
^ AL
For given values of the molar ratio of CO
in the bulk of the gas δ and MEA molar concentration in the liquid bulk C , 2
phase Y A
BL
equations (11-F19) and (11-F11) with (11-F20), (11-F21), (7-28), (7-48), (7-50) and y
Y
A
allow
auxiliary
relations
A
(11-F22)
- ya to determine 1
us
concentration, i.e. Y
the
molar
and C A
computational algorithm is: * a) assume C τ
ratio
of
CO
2
and
its
molar
at the interface. The corresponding AL
AL *
b) calculate y
from equation (11-F11) A *
c) calculate Y d) calculate Ε
A
from equation (11-F22) , M
oo
e) calculate N f)
check
if
a
and Ε A
q
for
from equations (7-28), (7-48) and (7-50) A
and N ^ L from equations (11-F20) and (11-F21) the
(11-F19) is fulfilled.
required If
accuracy
ΙΝ 1
of
calculation
€
equation
- Ν* I < ε, the calculation can AG
AL
be terminated, otherwise, a new value of C
^ τ
AL
should be taken
and the calculation procedure repeated starting from point b).
460
BALANCE EQUATIONS OF PACKED COLUMNS
The
concentration
of
amine
C
in
the
liquid
bulk
can
be
BL
calculated on the basis of equation (11-49). Assuming that the absorbed CO^ does not in practice change the volumetric flow rate of the liquid phase, one can write L
X
S< B - V
»
\
C
<
* BL0>
Substituting the above relation into equation (11-49) one obtains
C
BL
»
C
BL0 -
A -
( YY
A0>
Similarly, as in example 11-E, the packing height is calculated using the numerical solution of differential equation (11-35). ΊΪΥ- -
a
A
G
S N n
m
Y
* A0>
»
°
<»-
3 5
>
AG
Each call for the function on the right hand differential equation requires the calculation of - molar concentration of C
BL
-
side of the
above
equation (11-F24)
- gas mixture viscosity η > mean molar mass M
and gas mixture
density ρ - equations (11-E16),(11-E17) and (11-E19) - molar flow rate and superficial mass flow velocity of the gas equations (11-E20) and (11-E21) - mass transfer coefficients k^&nd k Q - equations (11-18) and (11-E23) - values o f C
T
, Y , E
AL
A
00
, Μ , Ε , Ν A
A
, Ν , - using the algorithm AG
AL
presented on page 454. After integration of the differential equation (11-35) by classical fourth-order Runge-Kutta method in the range from Y Υ
= 2.143 χ 10" kmol A/kmol to Y = Y
= 4.286 X 10" kmol A/kmol
5
AO
A
A
the =
1
AZ
in, we obtain the packing height Ζ = 3.222 m. The most important results of the calculations are shown in Figs. 11-Fl, 11-F2 and 11-F3. From Fig. 11-F2 it follows that in the upper part of the column, the reaction (11-F1) in the liquid phase takes place in the region of fast pseudo-first-order reactions, while in the lower part of the column it takes place in the fast reaction region. Over a very short segment in the lower part of the column
PACKED COLUMNS
461
(starting from the gas inlet) the reaction takes place in the instantaneous reaction region. The mass transfer coefficient k Gy
(Fig.ll-F3) changes from 2.15XlO"
kmol/(m
3
-3
2
s) in the lower part
2
of the column to 1.33x10 kmol/(m s) in the upper part, thus decreasing by over 38%. The computer program for this example is called E l IF.PAS. 10° r-
10-
ι
J
1
1
ui— 2
z
3
m
( )
Fig. 11-F2. Distribution of characteristic chemical reaction factors _
ι
ι
Fig. 11-F1. Concentration distribution of particular components
ζ
( m )
Fig. 11-F3. Distribution of mass transfer coefficients in a gas phase
462
BALANCE EQUATIONS OF PACKED COLUMNS
11.6. BALANCE EQUATIONS OF PACKED COLUMNS FOR MULTICOMPONENT SYSTEMS The balance equations presented so far for packed columns and the methods of their solution referred to the simplest cases of absorption: - isothermal physical absorption of one component from an inert component (Example 11-D) - isothermal physical absorption of one component from a mixture of inert components (Example 11-E) - isothermal chemical absorption of one component with a simple very fast chemical reaction in the liquid phase (Example 11-F). The two latter cases, although referring to multicomponent systems from the formal point of view, can be calculated in the same way as e f
for binary systems, if the notion of efficient diffusivity DAG is employed. In this section balance equations for packed columns and multicomponent systems will be presented. Examples will include isothermal and adiabatic absorption. The balance equations for packed columns for multicomponent systems are formulated (in a similar way to binary systems) for the plug flow model of gas and liquid phase. Assuming additionally steady-state conditions of the absorption process, for the element of the column with differential volume dV (Fig. 11-17), the balance (b)
(a) W
A
iG Σα
ΣD «V H W
iO iG
Λ .H W
W
w
iO n «\»H W IO iO
iL iL
i=l
iL η «ν, H
Σ
Σ
i=l
i=l
- j -
Ni
ε,
G
SE Ο c
Gas bulk
W i 0+dW iG π
«ν.
Liquid film Liquid bulk
η
W
iL iL
Gas bulk
Ρ
Ε 3
ε,
^ 'Liquid film 5
Liquid bulk WjL+dWjL
«ν.
W
Σ H i GW i 0+ d Œ H i GW 1 0)J £ » i L i L >i=l i=l i=l
+ d
< £ H i LW 1 L) i=l
ζ
Σ
H i 0W | 04.d
- z+dz
SîïiLWu.+diLHjLWjL)
Fig. 11-17. Scheme of differential element of a packed column a) cocurrent flow, b) countercurrent flow
equations (9-7), (9-16),(9-36) and (9-40) are reduced to the form - gas phase dW.
iG
dz
a
m
Ν
iG
(i = l,...,n)
(11-50)
PACKED COLUMNS
—
= T a d ζ
qn
Ta
m G
Σ
m
Νιρ
(11-51)
1>·,η)
(11-52)
8.
^ i= l
463
iG
iG
- liquid phase dW d ζ ^ - = a m NfiTL + β
Σ υ . · R.
. , j =1
-a
dz
qf m
iiJ
+a
L
(i -
ιJ
β.,
£ m i. = ^l
(11-53)
iL
iL
The minus sign on the right hand side of equation (11-50) and (11-51) refers to the cocurrent flow of phases and the plus sign to the countercurrent flow. The assumption of a plug flow model for each phase in an absorber allows us to determine their compositions using the following simple relations *i
=
W
i O'
W
o
<* -
1
—
;
n )
o
W
-
Σ W.G
(11-54)
i=1
x. = 1
W . f / Wf 1
Li
(i = Ι,.,.,η)
Σ W.f
; W. -
L»
Li 1
The temperatures T Q and T L are determined of molar enthalpy.
=1
directly from the definition
(a)
(b) (Σ H i LW i L) Q
W i G( 0 ) (Σ H j , W | 0) ( 0 ) | i=l
Ο
W
iGZ
à Hi i=l
W
L iL)z
Fig. 11-18. Diagram of a packed column a) cocurrent flow, b) countercurrent flow
(11-55)
1 LI
W
<5 " i G i G > Z i=l
464
BALANCE EQUATIONS OF PACKED COLUMNS
For the general case, the calculations of packed columns depend on solving the differential equations (11-50) through (11-53) with the algebraic equations describing mass and heat transfer between phases and a chemical reaction taking place in a liquid film (cf. Chapters 6 and 7) with appropriate boundary conditions. The boundary conditions are a result of the assumption of a given mode of absorption process. Usually, all the flux parameters entering the column are given or easily determined. These flux input parameters are the boundary conditions for the differential equations considered. For cocurrent columns (Fig. ll-18a) W
i G
(
0)
W
»
ί
iG0
(
1
»
D )
(11-56)
(£ft
1
w
0
l
0
]
4 =1 W
i L
[ £ f i
(
0)
( £ ö
W
»
i
L
w
iL0
i
L
w
0
i
]
0
4 =1 1
Β
·-'
( k
J
J
0
<» »
] ( 0 ) =
4 =1
i
(11-57)
>
w
L
i
L
l
4 =1
o
J
and the countercurrent columns (Fig. ll-18b) W
Z
iG< >
W
=
iGZ
1
<* »
» -
D
>
(11-58) ( £ ö
i
G
w
i
G
] ( Z ) = J
4 =1 W
i L( ° >
[ £ ö 4 =1
[ È f f
i
= L
w
W
i
L
]
w
1
0
]
(
1
» i =
" -
L
l
J
Z
(0)= [ È f f
J
0
4 =1 ί
iL0
|
n
>
1
L
(11-59)
w
i
L
] o
J
As with binary systems, for known parameters of the gas and liquid entering the column, the calculation procedure for packed columns is formulated in two ways a) for a known absorber height Ζ all parameters of the components leaving the column should be calculated. (Usually, the calculation is made for existing equipment which is to work under new operating conditions of the absorption process.)
PACKED COLUMNS
465
b)
one should calculate the absorber height Ζ (equivalently, mass transfer surface area) at which the required value of a chosen parameter of the gas or liquid leaving the equipment is achieved. (Usually, this parameter is the absorption efficiency of a given gas component.). The calculation for packed columns operating under cocurrent flow conditions is known in the theory of ordinary differential equations as an initial-value problem since at one end of the column (at the inlet) all the parameters of the gas and liquid are known - cf. equations (11-56) and (11-57). The solution of the problem does not pose, therefore, any significant numerical difficulty. For problem a) the solution consists of the integration of the differential balance equations of the absorber (11-50) through (11-53) with initial conditions (11-56) and (11-57) in the interval < 0 , Z > . For problem b) the integration of the differential equations considered should be continued until the additional condition formulated in the problem is fulfilled. The calculation for packed columns operating under countercurrent flow conditions is, in turn, called a two-point boundary value problem since the inlet gas and liquid parameters are given at two opposite column ends - cf. equations (11-58) and (11-59). Such a problem can be solved numerically using, for instance, iterative methods that involve a multiple integration of the initial problem. Generally, these calculations are laborious and time-consuming. Problem a) is now a so-called two-point boundary value problem with a fixed boundary (the column height is given). Its solution comprises an iterative choice of unknown values of an arbitrarily selected flux at the column outlet. The parameters should be varied until the known values of this flux are obtained at the other end of the column, i.e. at the inlet. For case b) the problem is called a two-point boundary problem with a free boundary (the absorber height is unknown). The problem can be solved by appropriate transformation of differential equations, which leads to a boundary value problem with a fixed boundary. There are many iterative methods for solving a two-point boundary value problem with fixed and free boundaries [38-41], One of the simplest methods is a shooting method whose basic elements are discussed below. In problem a) such a vector u(0) = [ w ( 0 ) , W ( 0 ) , . . , W ( 0 ) , [ Σ 0 iG
2 o
n G
i G
W
i O
)(0)]
T
(11-60)
466
BALANCE EQUATIONS OF PACKED COLUMNS
is to be found which will satisfy n + 1 boundary conditions written in the form of the algebraic equation F[U (0)j
«
u (Z) - u z
(11-58)
= 0
(11-61)
where
τ
u ( Z ) = [ w i G( Z ) , W 2 G( Z ) , . . , W n G( Z ) , [ ^ f t i oW i G] ( Z ) ]
»Z
=
W
[ 1CZ '
W
W
2GZ —
nGZ'
(
Σ
f
&*) ιο
(11-62)
lQ
Vector u (Z) is a function of vector u ( 0 ) since it results from the solution of equations ( 1 1 - 5 0 ) through ( 1 1 - 5 3 ) for the initial conditions u ( 0 ) and v Q in the interval < 0 , Z > , where \
-
W
[ 1L0'
W
W
2L0 · · - nL0 ' ( Σ «
iL
*
i
4 =1
L
)
(H-64)
] ' 0
J
J
Vector u ( 0 ) can be searched using an arbitrary method for solution of nonlinear systems of algebraic equations [38,42-48]. One of the most efficient method is the gradient Newton-Raphson type method. The Jacobi matrix of function ( 1 1 - 6 1 ) required in these methods should^ be determined numerically. Its particular elements in point u ( 0 ) , which is the k-th approximation of the solution u ( 0 ) being searched, are substituted by the finite differences , r 3F. -, r άη.(Ζ). ^ u: (Z)- u!(Z)
I L
du.(0)1 /f\\ k v / J j u(0)=u (0) /
r
i
= N
I
L
du.(0)1 /f\\ v / J j u(0)=u (0)
δ u . v( 07 ) j
(i, j = l , . . . , n + l)
(11-65)
The elements of vector u"(Z) in finite differences are obtained upon integration of the ^system of equations ( 1 1 - 5 0 ) through ( 1 1 - 5 3 ) for the initial conditions u ( 0 ) + iu(O) and • , where *o(0) = | W
L
O
( 0 ) ^
Ö . GW
The elements of v|ctor u'(Z) are initial conditions u ( 0 ) and ν .
o
i G
j(0)]
T
(11-66)
calculated in the same way for
In problem b) the column height Ζ is not known (the boundary value problem with a free boundary), but an additional condition must be satisfied. It can be written in the form of a single equation
PACKED COLUMNS
G [ U ( 0 ) , VO,
Uz,
v(Z)j
= 0
467
(11-67)
re
[Ê^wJ©]
v(Z) = [ w i L ( Z ) , W 2 L ( Z ) , . . . , W n L ( Z ) ,
(11-68)
Introduction of a new independent variable ξ = z/Z leads to the following boundary value problem with a fixed boundary dW = Ζ a
ς
Ν
m
(i = l...,n)
iG
z
a v
q
+
» o
a
(11-69)
Σ
»
1
dW iL .
= Ζ fa
dζ
\l
d
iL
( 1
=1
-
^
(i = l,...,n)
(11-71)
ö.w.l
>
=
W
ί
iGZ
(
*
Σ β.,
qf+ a
^ m
d£ iG
( 1 17 0 )
]J
ij
= Z a
W
N
.o
+ Α Σ v. . R.l
Ν*
l m
S
L
1
—
mim .=_i
u
n
<
iL
(11-72) îLJ
)
(11-73)
[Σ L
fi w ] i0
(D - (È ö w ]
10
i G
J
i =l
i G
M=l
J
Z
(11-74)
[ Σ ö w ] (0) = ( ς i L
i L
J
M =l
M= l
ft w ) iL
iL
0
J
where
ί e
<0,
1>
Problem b) consists, now, in finding such a vector
'-[z
< 0 )
]
(11-75)
BALANCE EQUATIONS OF PACKED COLUMNS
468
which would satisfy the system of η + 2 algebraic equations in the form u (1) - u F (r)
(11-76)
G [u ( 0 ) , v , u , ν ( 1 ) ] 0
z
where u (1) -
u (Z)
(11-77)
ν (1) -
ν (Z)
(11-78)
Vectors u ( l ) and v(l) are the functions of vector r because they result from equations (11-69) through (11-72) for the initial conditions u(0) and v in the interval < 0 , 1 > . The root r in Q
equation (11-76) is calculated in the same way as for problem a). In many practical calculations equation (11-67) has a particularly simple form. For example, when the absorption efficiency η of a chosen component of a gas mixture (e.g. abs
component no. 1) is needed, this equation has the form W
ο
- W (0)
-
•«
w
.
f
i
t
<
.,
( 1 1
.
7 9 )
IG ζ Equation (11-79) allows us to calculate directly one of the unknown boundary conditions ^ \ q ^ ^* * ^ ^ l of vector u (0)). For such a case another method of solution of the two-point boundary value problem with a free boundary is proposed. It starts with finding a vector u (0) = [ w ( 0 ) , . . . , W ( 0 ) , [ £ f i W ] (0)] (11-80) e
e
r s t
e
e m e nt
T
2o
nG
i
o
i Q
which would satisfy η boundary conditions (11-58), expressed in the form of the equation F p (0)j = u (Z) - u
= 0
z
(11-81)
e u (Z) = [ w ( Z ) , . . . , W ( Z ) , [ £ f t 2G
»z=
KGZ
nG
—
W
n G Z ' (Σ
L
^1
Ö
=1
i o
W
W
i o
i o
]
i o
] Z
] (Z)]
T
J J
T
(11-82)
(11-83)
PACKED COLUMNS
469
The elements of vector u (Z) are a function of vector u (0), since they follow from equations (11-50) through (11-53) for the initial conditions u (0) and Vq. In this case integration of the differential equations is continued until a value of the independent variable is obtained such that the following condition is satisfied
i[ i
F
u
( 0 )
]
" V
Z
" iz
)
u
=
"
0
( 1 1
8 4 )
The procedure for obtaining the root of equation (11-81) is the same as that presented in problem a). In a special case of physical absorption both the initial and boundary value problem can be partially integrated analytically. In this case the following equations hold a a
m m
Ν q
= a
iO
+a
G
m
m η
Ν
(i = Ι,.,.,η)
iL
(11-85) η
[H.
'-'
N.
iO
i=l
=a
iG
m
q, + a L
Ε 1,
m . . i=l
iL
Ν,
(11-86)
iL
or dW
dW = T
dz
(i = l,...,n)
d z J
M= 1
(11-87)
J
M= 1
d ζ
~
(11-88)
d ζ
Integration of equations (11-87) and (11-88) gives relation of state variables in the gas and liquid phase
the
simple
- for co-current flow W
il/
Z
>
(.^Ö ^i = l
-
i L
W
W
iL0
i L
)
W
-
Z
+
iG< >
W
iG0
(Ζ) = [ Ê J
^1
Ö
«
1
»
- "
D
W i L) - ( È
i L
=1
H
>
< -
fiio
J
^i=l
0
W i Q)
J
8 9
>
(Z)
(11-90)
IE,".-*.-] o - for counter-current flow W
i L
(
Z)
»
W
iL0
+
W
iG<
Z )
-
W
i G
(
0)
ί
(
»
1
—
n
)
-
9(
1U)
470
BALANCE EQUATIONS OF PACKED COLUMNS
iL
r
w iL
ίΣ
(Ζ) =
8
iL
W
iLJ
0
M=1
J
n
M= l
iG
W
iG
(11-92)
(0)
The parameters for packed columns for physical absorption are calculated in the same way as presented previously (depending on the type of problem), however the number of differential equations to be integrated numerically is half that of the previous case. When the absorption process is isothermal, in order to calculate the parameters of the packed columns one should solve numerically the differential equations for the mass balances of the components in the gas and liquid phase - cf. equations (11-50) and (11-51). In the special case of isothermal physical absorption, the calculation only requires a numerical solution of the differential equations describing the mass balances of the gas phase components, since the differential equations describing the state of the liquid phase can be solved analytically. The method of solution of these equations is identical to those presented previously (depending on the type of problem).
Example 11-G. In a packed column of diameter d^ = 1.4 m and height Ζ = 4 m means of
(Fig. 11-Gl)
simultaneous
water is carried out
absorption of H S and CO by 2 2 6 under a pressure Ρ = 1.013x10 Pa
Lo « 3.0 kmol/s Xi »0.i»1.Z3A X 5. 1
y«»o.oe
yu·-0.16 y«*o.io
HZ) . xjlZ)
X,Z«0.66
Fig.
11-Gl.
Diagram
of
a
packed
column
PACKED COLUMNS
471
and at a temperature Τ = 303.2 Κ. A gas mixture enters the column at the bottom at a rate G^ = 0.04 kmol/s, and pure water is supplied to the top of the column at a rate L Q = The packing consists of ceramic Raschig rings of d
3.0 kmol/s. =
p
0,025 m.
The composition of the gas mixture at the column inlet, given in mole fractions, is H S (1) - 0,08 ; CO (2) - 0.16 ; C Η (3) 2
2
2
6
0.1; CH^ (4) - 0.66. Calculate the absorption efficiency of H 2 S . Solution The superficial molar gas velocity at the column inlet is Q
\V
= -=^- =
„ = 2.598 X 10" kmol/(m s) (11-G1) 2 ° πχ(1.4) 4 and the molar velocities of particular components are 2 3 2 W , „ = W ν = ( 2 . 5 9 8 X l 0 " ) x 0 . 0 8 = 2 . 0 7 8 X 1 0 " kmol/(m s) Z
0
0 4
2
2
S
lOZ
GZMZ
W„„ = W 2GZ
W„
= W
W
= W
3GZ
4GZ
ν
= (2.598xl0" )x0.16 = 4.157XlO"
y
= (2.598XlO" )X0.1 = 2 . 5 9 8 X l O "
y
= ( 2 . 5 9 8 X l 0 " ) x 0 . 6 6 = 1.715X10"
2
GZ'2Z
2
GZ'3Z
2
GZ'4Z
3
(11-G2) 2 kmol/(m 2 s) >l/(m s) (11-G3) 2 kmol/(m2 s) /(m s)
3
2
(11-G4) 2 kmol/(m2 s) )l/(m s) (11-G5)
The molar water velocity at the column inlet is L W
= L 0
= S
— — = 1.949 kmol/(m s) 2 πχ(1.4) 4 2
(11-G6)
The active components are H 2 S and C 0 2 > The mass balance equations have the form , G
Z
= a
m
Ν
iG
(i = 1,2)
(11-G8)
(i = 1,2)
(11-G8)
dW l L
d
Z
= a
m
Ν
iL
472
BALANCE EQUATIONS OF PACKED
COLUMNS
It is possible to solve the above differential equations directly only if the values of W and W are given at the same level of the absorber cross section (the initial problem). For a counter-current column W is given at the level ζ = Ζ and W.. iO
at the level
ζ = 0 by
W (Z) = W 1Q
W
iL
(Z) = W 2G
= 2.078X10" kmol/(m s)
(11-G9)
= 4.157X10" kmol/(m s)
(11-G10)
= 0 kmol/(m s)
(11-G11)
3
i o z
2
3
2
2GZ
W (0) = W i l
2
l L o
W (0) = W = 0 kmol/(m s) (11-G12) Equations (11-G7) and (11-G8) with boundary conditions (11-G9) to (11-G12) represent a two-point boundary value problem with a fixed boundary. To decrease the number of differential equations to be solved numerically, it is advantageous to express the values of W as functions of W . Following the considerations 2
2L
2 L 0
iL
iG
in Chapter 11.6 and using equations (11-G11) and (11-G12) we have W.fz) = W.fz) - W . iL
(i = 1, 2)
r n
(11-G13)
iGO
iG
Therefore, the boundary value problem now comprises the solution of differential equations (11-G7) and algebraic equations (11-G13) with boundary conditions (11-G9) and (11-G10). The method for calculation of H (i = 1, 2) and a , required in iG
m
differential equations (11-G7), is discussed in detail in example 6B. The boundary value problem formulated in this way is solved using the method presented in Chapter 11.6 (problem a) for countercurrent columns). To calculate the components of the vector τ u (0) = [ w ( 0 ) , W ( 0 ) ] (11-G14) the system of nonlinear algebraic equations (11-66) was solved by Marquardts method [42]. The elements of the Jacobi matrix required in Marquardt's method were calculated using equation (11-70). Differential equations (11-G7) were integrated by Mer son's method [49], After calculations which included five iterations, the components of vector u (0) satisfying equation (11-66) were determined W ( 0 ) = 1.326xl0" kmol/(m s) W ( 0 ) = 2.450 X 10" kmol/(m s) iG
2G
4
1G
2
3
2Q
2
PACKED COLUMNS
The efficiency of H 2 S absorption is
tfab
=
473
93.62%.
The distribution of parameters of particular fluxes along the column corresponding to W l Q ( 0 ) and W 2 Q ( 0 ) are presented in Figs. 11-G2 through 11-G8.
Fig. 11-03. Distribution of
Fig. 11-G5. Operating and
concentrations of components
equilibrium lines
BALANCE EQUATIONS OF PACKED COLUMNS 1(Γ
-
2G
/
0.996
l ο-
ο
t 1 0.994 0.0015 0.0010 1
23G
ίο-
— 33G f
-0.0005
ι 0" 10"
/ —11G
"
ι
1
1
1
— 13G — ^ ,·=·120
1
1
1
1
2
3
1
z (m)
1
_L
J-
2
3
z (m)
Fig. 11-G7. Matrix elements distribution of correction factors
3.6x10"
Fig. 11-G6. Matrix elements distribution of mass transfer coefficients in gas phase
3.4xl0
-4
3.2xl0" t4
3.0X10-
4 1
0
1 IL
1 1
1
1
2
1
3
4
z (m) Fig. 11-G8. Matrix elements distribution of mass transfer coefficients in liquid phase
The results obtained show that in the lower part of the column (Figs.ll-G4 and 11-G5) C 0 2 is desorbed. This phenomenon, surprising at first sight, (part of the column operates like an absorber of C 0 2 and the other part as a desorber of C 0 2 ) , can be easily explained. The
rate of H 2 S absorption, due to the greater solubility of
H S compared to CO ,is much higher than the rate of CO 2
2
absorption. 2
This causes a rapid decrease in H 2 S concentration in the gas phase (Fig. 11-G3). Pure water entering the column becomes slowly saturated with C 0 2 (the saturation takes place in practice at a distance of 2.5 m from the top of the column). Water saturated with carbon dioxide flows down the column and encounters gas in which the mole fraction of C 0 2 is lower than in the upper part of the apparatus. This causes a desorption of COn in this part of the column.
PACKED COLUMNS
475
Nondiagonal elements of the mass transfer coefficients matrix (Fig. 11-G6) are smaller than the diagonal ones by at least one order of magnitude. This is evidence of a negliglibly small cross effect of multicomponent diffusion on mass transfer in the gas phase. The diagonal elements of the correction matrix at the bottom of the mass flux (Fig. 11-G7) are very close to unity. The nondiagonal elements of this matrix are at least four orders of magnitude smaller. The mass transfer coefficients in the liquid phase (Fig. 11-G8) are, in fact, constant along the column. This follows from the fact that the superficial molar velocity of the liquid phase (under the assumption of constant flux, and superficial mass velocity) is actually constant along the column height (Fig. 11-G2). The computer program for this example is called E U G . P A S .
Example 11-H. In a packed column (Fig. 11-H1) the adiabatic process of hydrogen chloride (1) absorption from air (3) in water W G(0) yil0),y 2(0) TGIOI
2
WQ2« 3.89W0* kmot/nfsl
1
y^-o.u y^O.OI Τ 0 2«29β.15Κ
'
I W L( Z I χ , (Ζ) . x 7 [ Z ) T L( Z |
m
Fig. 11-H1. Diagram of a packed column
(2), under a pressure Ρ = l . O x l O Pa takes place. A gas of composition y = 0.14 and y = 0.01 and temperature Τ = 5
1Z
2Z
KJZj
298.15 Κ enters the bottom of the 2 column at a superficial molar 2 2 velocity W = 3.898XlO" kmol/(m s). Pure water at temperature Τ
=
GZ
303.15 Κ is supplied to the column countercurrently
at a
476
BALANCE EQUATIONS OF PACKED COLUMNS -2
2
kmol/(m s). The column superficial molar velocity W L 0 = 3.118x10 is packed with ceramic Raschig rings of diameter d = 0.025 m. Calculate the column absorption occurs.
height
at
which
96%
hydrogen
chloride
Solution The superficial inlet are W_
= W
1GZ
\V
molar
velocities of 2
GZMZ
2GZ
at the
column
= (3.898 X l 0 " ) x 0.14 = 5.457 X 10" kmol/(m s)
ν
= W
components
3
2
7
'
(11-H1) 2 4 2 = (3.898 X l O " ) χ 0.01 = 3.898 X 1 0 " k m o l / ( m s )
ν GZ 2Z
(11-H2) m
w
=
« ^
W
(3.898 X l O ) χ 0.85 = 3.313X 10" kmol/(m s)
=
2
2
2
(11-H3) The superficial molar outlet of the column is W
(0) = \V IG
Λ
velocity
hydrogen
at
the
3
1GZ
kmol/(m s)
4
chloride
= 0.04 x 5 . 4 5 7 X 1 0 "
= (1-0.96) \\r
1G0
= 2.183X10'
of
(11-H4)
2
The soluble components are hydrogen chloride and water vapour. Since the process is no ni sothermal, the balance equations of the column comprise the mass balance equations of soluble components in the gas and liquid phases and the heat balance equations of these phases. The balance equations have the form dW 7
Γ
=
"
a
m
N
iG
i
a
•
(
»
n
^
3
m
5
< - °>
q„ + a Ε 8. N.n Λ3 m . . iO iO
(11-51) v '
dW ΎΓ-
= a
»
N
iL
i
(
=
^
n
( "
5
2
>
v
PACKED COLUMNS
S =l
J
=
dΖ
\
+ IU
Σ Ö
^1
i L
N
=
W
iO
1 W
o
<* -
W
(11-53)
i L
1 11
i=l
The composition of each phase follows and (11-65) *i
477
>
>
W
o -
from
Σ W.0
relations
(11-54)
(11-54)
i=1 X
i -
iL '
W
W
L
» !> >
(I
'
2
W
L
=
Σ W.L
(11-55)
i=1
In the worked example the partial molar enthalpies of the gas components have been defined by the equations Ö
= £
.„(T
iG
piG
v
- T°) + r. G
G
(i = 1, 2)
For such definitions of partial molar the equations presented in Chapter (11-51) can be transformed into C pG
G
where C
3
=
Ε 2
pG
,
7
enthalpies, according to 4, differential equation
dT - = - 5 - = a q^ dz m ^G
\V
(6-C5)
i
(11-H5)
ν,
(11-H6)
piG M
Equation (11-H5) will be used to calculate directly the gas phase temperature. The boundary conditions for differential equations (11-50), (11-H5), (11-52) and (11-53) are as follows W
W
=
I G Z
2GZ
=
5
3
·
, 7 4 X5
8
9
W l o( 0 ) = W l T
GZ
Wl
=
L 0
2 98
= 0
15
8
χ
1
°
L
0
~
3
K
M
O ML 2 /
(
)
kmol/(m s)
4
(11-H8)
2
= 2 . 1 8 3 X 1 0 ' kmol/(m s) 4
Q 0
(11-H7)
S
2
(11-H9) (11-H10)
K
kmol/(m s) 2
(11-H11)
v
478
BALANCE EQUATIONS OF PACKED COLUMNS W
=
2L0
3
i r S
1
1
8
iL
M = l
01
2
1
W
I
x
lLI
2
kmol/(m s)
(11-H12)
= 13.08 kW/m
2
(11-H13)
J
0
The superficial molar velocities of the liquid components and the enthalpy of this phase are calculated using equations (11-91) and (11-92) W
[i iL
(
Z)
W
=
ö
i L
+
iL0
w
i
) <
L
z
W
iG
>
- [σ
J
Ö i G
W
i G
Z)
*
W
i G
(
0)
ί
(
0
' >
(
(Σ
Λ
4 =1
! 2
»
+
[ Σ Λ Ι \ ]
-
4 =1
(
J
ö
i G
U
9
-
1)
w )(z) i o
J
4 =1
)(0)
(11-92) J
4 =1
Equation (11-92) will be used directly in the calculation of liquid temperature T L (z). Taking into account the equations the partial molar enthalpies of the components in phase (Example 6-C) in the equation considered we have
the
the for
liquid
Τ (ζ) = 298.15 Li
(.σ
S=l
fl
iL
W
i L)
J
0
+ (.σ, 4 =1
ff
iG
W
,ο]
( Z ) J
- (Λ 4 =1
0
iG
W
i G)
w
J
pL
WL(z)
(11-H14) The values of a , Ν. . a_. m
iG
n
G
and J H _ iG
pG
iL
pL
were
S
calculated using the procedure given in detail in Example 6-C. Equations (11-54), (11-55), (11-91) and (11-H14), part of which are analytically integrated balance equations for the liquid phase, are treated as auxiliary in the description. The main part of the description is a system of three differential equations (11-50) and (11-H5) with four boundary conditions (11-H7) to (11-H10). The problem formulated in such a way is called a two-point boundary value problem with a free boundary. It is worthwhile noting that in the problem the column height Ζ is not given and, thus, the integration interval for the variable ζ is not determined.
PACKED COLUMNS
479
The method of solution of the above problem consists of multiple integration of differential equations (11-50) and W a nd T (11-H5) at the values assumed initially. These 2 G( ° ) Q( ° ) values should be chosen in an iterative way so that conditions (11-H8) and (11-H10) should be fulfilled at the other end of the column. In each iteration cycle the integration of differential equations is stopped at a value of variable ζ at which condition (11-H7) is satisfied. The formal procedure followed in the case of boundary value problems is described in Chapter 11.6. The differential balance equations for the column were integrated by Runge-Kutta's fourth-order method with a variable integration step. The iterative calculations began with the value of
W 2 G ( 0 ) = 2.6XlO" kmol/(m s) and 3
2
calculations comprising were obtained
eight
iterations
W 2 Q ( 0 ) = 6.399XlO" kmol/(m s) 3
T Q (0) = 314 K. After long the
following
values
T Q (0) = 328.85 Κ
2
Taking into account these initial values of differential equations, conditions (11-H8) and (11-H10) were satisfied to within 0.01% accuracy while condition (11-H7) was satisfied for a value of the variable ζ = 1.464 m. This value can be taken, therefore, as the column height which guarantees 96% absorption of hydrogen chloride. The most important results of the calculations are presented in Figs. 11-H2 through 11-H9. The following conclusions may be drawn from the calculations performed. The liquid superficial molar velocity (starting from the level ζ = 0, cf. Fig. 11-Hl) first increases as a result of hydrogen chloride absorption and water vapour condensation (Fig. 11-H2), -2
and after reaching a maximum (W
=3.25x10
2
kmol/(m s) at ζ =
Lmax
0.3 m ) it decreases due to rapid evaporation of water to the gas stream. The gas superficial molar velocity (starting from the level of ζ = Z, cf. Fig. 11-Hl) also rises, and after attaining a -2
maximum (W
=4.1x10
2
kmol/(m
s) at ζ = 0.3 m) it decreases
Gmax
due to hydrogen chloride absorption and water vapour condensation. The liquid phase temperature rises abruptly at the beginning, then the growth becomes moderate, and after reaching a maximum (T = 339 Κ at ζ = 1.05 m) it falls quickly (Fig. 11-H4). The Lmax
480
BALANCE EQUATIONS OF PACKED COLUMNS
TV
Z= 1.464.
_L 0
J
I
0.2 0.4 0.6 0.8
I
1
1.0 1.2
(
>J 1.4
I
1.6
z (m) 0
0.2
0.4 0.6
0.8
1.0
1.2
1.4
1.6
ζ (m)
Fig. 11-H2. Distribution of
Fig. 11-H4. Distribution of temperature
superficial molar velocities
0
0.2 0.4 0.6 0.8
1.0
1.2 1.4
ζ (m)
Fig. 11-H3. Distribution of concentrations of components
1.6
0
0.2 0.4 0.6 0.8
1.0
1.2
1.4
1.6
ζ (m)
Fig. 11-H5. Distribution of elements for mass transfer
matrix coefficients
liquid phase composition corresponding to the maximum temperature is close to an azeotropic composition of the HCl-H^O system (Figs. 11-H3 and 11-H4). The temperature at the interface is very close to the liquid phase temperature. Only in a short, upper part of the column where hot gas is in contact with a cool liquid, is the difference between the temperature at the interface and the liquid bulk appreciable high, attaining 8 K.
PACKED COLUMNS
481
482
BALANCE EQUATIONS OF PACKED COLUMNS
The gas phase temperature increases along 70% of the column height to reach a maximum (T = 333 Κ at ζ = 0.3 m) and then Gmax
drops due to water vapour condensation as well as heat transfer and conductivity to the liquid phase. Hydrogen chloride is absorbed in the whole column and therefore the molar flux of this component is always positive, i.e. directed from the gas to the liquid phase (Fig. 11-H8). The heat of solution of hydrogen chloride is used to evaporate water and to heat up the gas phase. Hence, the molar flux of water is negative over 70% of the column height. The water vapour condenses only in the upper part of the column where the hot gas is in contact with cool liquid, and the molar flux of the water is positive. The heat of vapour condensation is used to heat up the liquid phase. The total flux of heat transfer between phases comprises the heat flux resulting from conduction and the heat flux resulting from mass flux. The percentage of heat transferred by conduction q is very small along almost the whole column compared with the flux of heat transferred by mass flow, q 2 G (Fig. 11-H9). The same refers to q j L and q
. Only
over a short, upper
part of the
column
where the hot gas is in contact with cool liquid and the hydrogen chloride concentration in the liquid phase is very small, is the percentage of flux q i L much higher than that of flux The corrections for large, mass fluxes and the Ackermann correction factors Ξ and Ξ , respectively, differ only slightly L
HL
from unity along the whole column and therefore can be neglected (Fig. 11-H7). The nondiagonal elements of the matrix S Q are smaller than the diagonal elements by at least two orders of magnitude. Hence, the influence of cross effects is practically negligible. The Ackermann correction in the gas phase Ξ differs HO
from unity by 7% at the most, and therefore can also be neglected in practice. The remark concerning nondiagonal elements of the matrix also refers to nondiagonal elements of the matrix k Q . The computer program for this example is called E11H.PAS. Example 11-1. Raal and Khurana [SO] have investigated the absorption of ammonia (1) from air (3) in water (2) in a column,
PACKED COLUMNS
483
packed with 1/2" ceramic Berl saddles. The column diameter was d k
= 0.15 m, and the packing height Ζ was 0.55 m. In one of the experiments gas composed of y = 0.4414 and y = 0.0069 was fed IZ
2Ζ
to the bottom of the column at a temperature Τ molar flow rate G
= 287.8 Κ and a Z
-4
= 6.194x10"
G
kmol/s. From the top, pure water
at a temperature ^ L 0 = 298.8K was supplied at a molar flow rate L Q = 3.919x10 kmol/s. The process was performed adiabatically 5 under a pressure Ρ = 1.067X10 Pa. The authors found that 96.5% ammonia was absorbed and the liquid leaving the column at a temperature T L (Z) = 326.1 Κ contained x ^ Z ) = 0.0638 mole fraction of ammonia. Assuming the same parameters of the gas and liquid streams entering the column as in Raal and Khurana's experiment, calculate the percentage of NH^ absorbed in the column and compare it with the experimental value. Solution The balance equations determining this process are the same as those presented in Example 11-H. The boundary conditions for differential equations (11-50), (11-H5), (11-52) and (11-53) describing the absorption process considered, are as follows W , „ „ = 1.5473xlO" W _
= 2.4536x10
Τ
= 287.8 Κ
2GZ
GZ
WJ W
L 0
= 0
2
Ä
f Σ
(11-11)
kmol/(m s)
(11-12)
2
(11-13) (11-14)
kmol/(m s) 2
(11-15)
= 2.2183 XlO" kmol/(m s) 1
2L0
kmol/(m s)
ff.
W
1
2
= 10.851 kW/m
2
(11-16)
The above differential equations and boundary conditions are classified as a two-point boundary value problem with a fixed boundary. In this case the column height Ζ is known, so the integration interval for the independent variable ζ is determined.
484
BALANCE EQUATIONS OF PACKED COLUMNS
The method of solution of this problem consists of a multiple integration of differential equations (11-50) and (11-H5) in the w interval < 0 , Z > at the assumed initial values of l Q ( 0 ) , W 2 Q ( 0 ) and T Q (0). These values need to be chosen iteratively in such a way that, finally, the boundary conditions (11-11) to (11-13) are satisfied at the other side of the column. The formal procedure in the case of the boundary problems is described in Chapters 11.6 and 14.2. The values of a , Ν , q , iG
m
ft._.
c ..
iG
ft...
pG
c
iL
and AH pL
XJ
necessary to solve the differential S
equations were calculated as described in detail in Example 6-C. Part of the physicochemical parameters needed to calculate the above values for ΝΗ^-Η,^Ο system are listed in Example 11-C, the other ones are given here. The values of ιι , λ . S, and V Μ Τ ' in equations (6-C 12) 1 v or or ι ι ci to (6-C15) are η
= 0.91 Χ 10" kg/(ms), λ
S, = 626 K, • Μ Τ 1
= 2 . 1 4 x 1 0 ' W/(m K),
5
1
1
2
= 83.04.
Cl
The binary diffusion coefficients in the gas phase determined by the method proposed by Fuller et al. - equation (3-70) - for Τ 5 5 2 = 298.15 Κ and Ρ = 1.067X 10 Pa are fll1 2 = 2 . 6 0 x 10" m / s , fll3 = 2.06x10
-5
2
m / s and 0
23
= 2.32x10
-5
The mean partial molar heat of NH
2
m/s. in the gas phase is Ö
3
o
= plG
r
35.94 kJ/(kmol K). The physicochemical properties of the liquid phase as a function of composition and temperature are described by the following correlation - density PL
(11-17)
log — — = A m P
L0
ρ
= 7.949265X 10
+ 1.637338 Τ - 3.217544x10
T
kg/m (11-18)
A = - 1.644871 XlO" - 5.514286Χ10" T 2
4
(11-19)
1
PACKED COLUMNS
485
- viscosity
>/ = 2.6090X10" - 1.4814X10" Τ + 2.1386xl0" T 2
4
7
2
L
2
5
2
+ (1.5255 X l 0 ' - 6 . 9 0 5 6 x 10" T + 7.3594T )x i kg/(ms)
(11-111)
- thermal conductivity AL =
1.6640X10"
- 1.1044X10"
7
4
+
1.4777X10"
T)Xj
6
Τ + (-5.3512X 1 0 "
4
W/(mK)
(11-112)
- surface tension 1
4
2
a = 1.1952 X l O " -1.6051 X l O " Τ + (-5.4779 + 1.8428 Χ Ι Ο " Ύ)αχ 1
+ (2.0462X10 - 6.9488X10"
2
T)x
2
N/m
(11-113)
- molar heat (at an average temperature) C
pL
= 75.2573 + 7.4622 χ
1
+ 3.146 χ
2
1
kJ/(kmol Κ)
(11-114)
- molar enthalpy of solution at 298.15 Κ = 34558.38 χ
AU S
+ 503.6361 χ 1
- 196784.6 χ*
2
- 63565.2 χ
1
3
+ 183376.8 χ
1
4
1
kJ/kmol
(11-115)
- diffusion coefficient (Τ = 298.15 Κ) for diluted solutions βι
IL
= 2.67 χ 10"
9
2
m /s
The absorption equilibrium of the NH^-H^O system was calculated using the equations and algorithm presented in Example 2-A. Iterative calculations of NH^ absorption in the packed
column
start with the values 4
2
4
W i Q( 0 ) = 5.0xl0" kmol/(m s)
2
W 2 Q ( 0 ) = l.Ox 10" kmol/(m s)
Τ (0) = 305.0 Κ G
After lengthy calculations values have been obtained 4
including 2
W i g ( 0 ) = 4.7966 x 10" kmol/(m s)
six
iterations
the
following
4
2
W 2 Q ( 0 ) = 6.1547X 10" kmol/(m s)
Τ (0) = 303.24 Κ G
Taking
into
account
these
initial
values,
conditions
(11-11)
to
486
BALANCE EQUATIONS OF PACKED COLUMNS
(11-13) were satisfied with an accuracy of 0.01%. The efficiency of the absorption of NH^ in the column is 96.9%. The calculated temperature of the liquid leaving the column was T L ( Z ) = 325.IK. The calculated results are in good agreement with the experimental results of Raal and Khurana. The distribution of particular parameters along the column height is presented in Figs. 11-11 through 11-18. The following conclusions can be drawn from the calculations The mole fraction of ammonia in the liquid phase increases steadily from zero at the inlet of the column to 0.0629 at the outlet (Fig. 11-11); the gas phase mole fraction of this compound decreases from 0.4414 at the inlet to 0.0235 at the outlet. In the column above 0.15 m from the gas inlet, the water vapour condenses and its concentration at the interface on the gas side is lower than in the bulk. In the column there is a cross section at which the vapour behaves like an inert component. Below this cross section water evaporates from the liquid phase. The gas phase temperature, starting from the bottom, increases rapidly and after reaching a maximum (Τ = 317K at ζ = 0.3 m), Gmax
decreases due to condensation of water vapour and heat transfer to the liquid phase (Fig. 11-12). The liquid phase temperature increases in the lower part of the column, and after reaching a weak maximum, just before the outlet of water from the column, it falls to 325.1 K. The interfacial temperature is close to the liquid phase temperature. The maximum difference of these temperatures does not exceed 3 K. The superficial molar velocity of the gas phase decreases with column height, while the superficial molar velocity of the liquid phase grows steadily (Fig. 11-13). In the lower part of the column water evaporates to the gas stream, however, due to ammonia absorption, which is more intensive than evaporation, the superficial molar gas velocity reduces along the column. A great deal of interesting data concerning the process of ammonia absorption in water can be obtained from the mass and heat flux distribution along the column (Figs. 11-14 and 11-15). The nonequimolarity of the mass transfer is confirmed directly by the overall mass flux distribution (Fig. 11-14). The molar flux of ammonia is positive over the total height of the column (no desorption of NH^ from the solution was observed). On the other hand, the molar flux of water vapour allows us to distinguish water vapour condensation and water evaporation regions in the column. In the water vapour condensation region
PACKED COLUMNS
0
0.2
0.4
0
0.
0.2
0.4
487
0.6
z (m)
z (m)
Fig. 11-11. Distribution of
Fig. 11-12. Distribution of
concentrations of components
temperature
3 ι-
0
0.2
0.4
0.6
Ο
Fig. 11-13. Distribution of superficial molar velocities
0.2
0.4
z (m)
z (m)
Fig. 11-14. Distribution of mass flux
0.6
488
BALANCE EQUATIONS OF PACKED COLUMNS 8 ι
0.2
0
0.4
0.6
0
0.2
z (m)
0.4
0.6
z (m)
Fig. 11-15. Distribution of heat flux
Fig. 11-16. Distribution of matrix elements for mass transfer coefficients
the overall heat flux is positive; the sign changes in the evaporation region (Fig. 11-15). The total molar flux of heat transferred between phases comprises the molar heat flux due to conduction and the molar heat flux due to mass transfer. The percentage of heat transferred by conduction, q^, is very small along almost the whole column compared with the molar flux of heat carried with mass, q 2 Q (in Example 11-H). The case is different in the liquid phase. Over almost the whole column length, the molar heat flux transferred by conduction, q i L , is of the same order as the molar heat flux as a result of the mass flow q^ Only over a short distances in the lower part of the column
is q_
IL
much
lower
than
q
2L
. All
mass
fluxes
and
the
overall heat flux have characteristic extremes which are a result of changes in the mass and heat driving forces (Figs. 11-11 and 11-12) and changes in the mass and heat transfer along the column (Figs. 11-16 and 11-17). The active specific mass transfer area is constant throughout the whole column (Fig. 11-17).
PACKED COLUMNS
489
The values of the nondiagonal elements of the mass transfer coefficient matrix in a multicomponent mixture are at most several percent of the values of the diagonal elements (Fig. 11-16). A strong predominance of diagonal elements in the mass transfer coefficient matrix over nondiagonal elements is evidence of the negligible influence of the so-called cross effects of multicomponent diffusion in mass transfer in the gas phase. The diagonal elements of the mass transfer coefficient matrix change with the column height by over 50% and the mass transfer coefficient in the liquid phase by over 100%. The rate of change of these values is a result of appreciable changes in the superficial molar velocities of the phases and physicochemical parameters along the column. The correction for large mass fluxes in the liquid phase Λ deviates only slightly from nondiagonal
unity (Fig.
elements of the correction
11-18). The values of the *
matrix
Ε > are only of
490
SIMULATION OF PACKED COLUMNS
the order of several percent of the diagonal elements. On the other hand, the diagonal elements of this matrix differ appreciably from unity only in the region of transition from water evaporation to water vapour condensation. The Ackermann correction factor in the liquid phase Ξ
*
along the entire column
HL
is, in fact, equal to unity. The maximum deviation of the Ackermann correction factor in the gas phase Ξ does not exceed 10%. HG
The computer program for this example is called E11LPAS. 11.7. SIMULATION O F INDUSTRIAL PACKED COLUMNS From the considerations presented in Chapters 6 and 10 it follows that in order to design an absorber correctly it is necessary to know many constants, parameters and physicochemical properties of a system; the most important are: (i)
the mass transfer coefficients in both phases k r
and k , iG
iL
(ii) the gas-liquid equilibrium data (the Henry constant, dependence of the gas solubility on temperature, pressure concentrations of the particular components), (iii) the kinetic data for the chemical reactions taking place the liquid phase, (iv) the hydrodynamic data (mass transfer area, gas and in an absorber, e t c ) .
the and in
liquid hold-up
All of these quantities most often depend on the concentrations of particular components and the temperature, and, therefore, they vary along the height of the equipment. The determination of these values from various empirical and semiempirical relations and correlations can be subject to serious error. Some of these quantities, for instance, the chemical reaction kinetics, cannot be calculated and their experimental determination often requires equipment of a special design. This is why in some cases instead of calculating the performance of the columns, which involves lengthy auxiliary laboratory investigations, another procedure is adopted, namely the simulation of an industrial packed column using laboratory equipment. Sometimes, despite the availability of all the data and the possibility of designing an absorber (or desorber), it is still necessary to verify the correctness of the calculations and to simulate an industrial absorber operation in a laboratory rig.
PACKED COLUMNS
491
Two types of simulation should be considered [51-54]: (i) simulation of the mass transfer process occurring in a differential cross section of an industrial column (differential simulation), (ii) the simulation of the column (integral simulation).
performance
of
the
entire
industrial
In both cases the simulation consists of performing in a laboratory absorber a measurement or a series of measurements under strictly defined and programmed conditions, so that the results obtained could correspond to those conditions found in industrial columns. 11.7.1. Mass Transfer Simulation in Absorber Cross Sections The mass transfer simulation in a given absorber cross sections is employed in the case where concentrations in the gas and liquid phases may be steadily interrelated in any cross section of an industrial apparatus. Such conditions appear during physical absorption (or desorption) of one component and during absorption with an instantaneous or fast chemical reaction that takes place in a film. Additionally, it should be confirmed that in an industrial column the process takes place under isothermal or close-toisothermal conditions. The respective mass balance equations for the gas and liquid phases in a countercurrent regime have the form dW
AG
= a
Z
dW .
m
= a
A L
α Z
Ν
(11-23)
AG
Ν m
(11-24) AL
For dilute systems the above equations can be written in the form dYΛ A
a _
S Ν m
"dz
G
dX dz
a A
AG
(11-27)
AL
(11-28)
ι η
S Ν m
Ls
In the case of absorption with a fast or instantaneous reaction, instead of equation (11-28) the mass balance liquid phase is described by equation (11-42)
chemical for the
492
SIMULATION
dX„
OF PACKED
a
ST-
-
COLUMNS
S Ν " L
^ S
The fluxes of components and (11-46)
< N
=-% A G
N
A and
Β are related by equations (11-45)
1L
^
(11-46)
=
The idea of simulation is as follows. In a laboratory absorber flow and mixing conditions for both phases should be arranged so that the mass transfer coefficients k and k are the same as in AG
AL
an industrial absorber. Gas and liquid concentrations as well as temperature and pressure in the laboratory absorber should also be the same as in the various cross sections of the industrial absorber. If these conditions are fulfilled, the experimentally determined mass transfer rates in the laboratory absorber should be equal to the mass transfer rates in the respective cross sections of the industrial absorber. The packing height in the industrial absorber can be determined from an integrated form of equation (11-27), (11-28) or (11-42); for example, when equation (11-27) is integrated, we obtain Y A Z G dY i n A Ζ = (11-93) a S Ν
Λ
Y
m
„
AG
A0
N ^ G is the flux of component A determined in the laboratory absorber. The example presented below illustrates the proposed method [52]. Example 11-J. In order to neutralize alkaline sewage a column packed with 2 5 x 2 5 x 3 mm Raschig rings was used. The initial concentration of NaOH in the sewage was C
= 0 . 4 kmol/m
BLO
the final concentration should decrease to C
BLZ
=0.04
3
and 3
kmol/m .
3
= 50 m /h and, the
Sewage enters the column at a flow rate of ν LU
excess gas should be 2/3 in relation to that calculated from the reaction. The process is performed at Τ = 303 Κ under atmospheric
PACKED COLUMNS
493
3
pressure. The sewage density is ρ L viscosity coefficient
is t\
=
=
.3
0.9x10
1020 kg/m , the
dynamic
kg/(m s). The composition
of the exhaust gases at the inlet to the column is CO^ - 8%, Ο
-
11%, N 2 - 8 1 % . Determine the required column diameter and packing height. Due to possible changes in the intensity of sewage flow rate to the column, in the calculations one should assume a gas flow velocity equal to half of the maximum velocity at which the column is flooded. Solution The flow rate of soda lye, which reacts in the column, is L = v (C - C ) = ÔÎSÂX(0.4 - 0.04) = 5 x l 0 ' k m o l / s V v 7 Β L 0 BLZ BL0 3600 3
The flow rate of carbon dioxide entering the column L
G
AZ
=
B
L
χ 1.67 = 4 . 1 7 x 1 0
3
kmol/s
(ii-Ji) (11-J2)
The flow rate of exhaust gases G
ζ
= —
^AZ — yJ
-2 4 17 X l O " Λο = 5 . 2 x 10 kmol/s 0.0 8 3
=
AZ
(11-J3)
The mass flow rate of gas at the inlet G
= G M Ζ
= (5.2χ Ι Ο " ) χ ( 0 . 0 8 x 4 4 + 0.11x32 + 0.81x28) 2
G
= 1.55 kg/s
(11-J4)
The mass flow rate of the liquid 1
=^6ϋδ- =
5
° 6θΓ° 3
-
1 4 1 6 7
k
*
/s
( U
-
J 5 )
The diameter at which the column would be flooded is determined from equations (11-6) to (11-10). It is equal to d = 1.285 m. The calculation is performed in the same way as kmin
in Example 11-B. The maximum gas flow velocity is
SIMULATION
494
OF PACKED
COLUMNS
^
u />
G m ax
4 1X
π d
G
= 1.0 m/s
55
1.19ΧπΧΐ.285
.
2
kmin
(11-J6)
2
The gas density was determined from the equation of state for ideal gases. The gas velocity in the column and the column diameter are u
= 0.5 u G
= 0.5 m/s
(11-J7)
Gmax 4
d=V
/ 4x1.55 = V 1.1 9 Χ π χ 0 . 5
8
p n u
k
G
For this calculated value liquid flow velocities are w
=
= π d
G
=
V
' L
L
=
of
X
π
k
πα,
L
4
2
4 W
= 1.82m
(11-J8)
G
1
Χ
,
the
column
diameter,
the
= 0.6 kg/(m s)
55
and
(11-J9)
2
1.82
gas
2
4 X1020X50
. 54
5
k
g
/
(
2m s
.(
)
Jn
1
0)
πχ 1 .82 x3600
2
2
k
The mass transfer area at a linear liquid velocity in the column
\
-
-77
=
-TÖ4Ö-
determined from Fig. 11-13, is
0
-
0
0
5
4
MS
/
2
a
-
J 1 (1) 1 1
3
= 105 m /m . m
The physical mass transfer coefficients in both phases, determined from equations analogous to those in Example 11-C are k = 0.0264 m/s AG
k
= 1.32X10" m/s 4
AL
In the calculations the following physicochemical of the gas and liquid were taken into account - for the gas 0
AG
= 1.69 XlO"
Pg
=1.19
nQ
= 1.86XlO"
5
m /s 2
kg/m 5
Pa s
properties
PACKED COLUMNS
495
- for the liquid AL
= 2 XlO"
m /s
9
= 1020
2
kg/m
= 0.9XlO"
3
3
Pa s
For the case being considered, the chemical reaction between NaOH and CO^ is fast enough for it to take place completely in a liquid film. Therefore, while preparing a mass balance for the lower or upper part of the column (Fig. 11-J1), a dependence may | v L0
&o
Κ
BIO
G
Vi
yA
CBL
11-J1. Schematic diagram for mass balance of the column
Fig.
be found between the molar fraction
of CO^ in the gas and the
NaOH concentration in the liquid phase ν „ G
C LZ
Z*AZ
+
ν BZ
2
„ =
C (11-J12)
L B G
* A
+
— Τ "
The value of the volumetric flow rate V
l
can be assumed to be
constant along the column height ν
= ν L
(11-J13)
= ν LZ
LO
while the molar gas flow correlated by the equation G (1 - y A ) - G z (1 - y A Z ) Thus
rates
in
both
cross
sections
are
(11-J14)
4%
SIMULATION OF PACKED COLUMNS
G =
(U-J15)
1 - y.
Introducing equation formula is obtained G y
A
=
y
+
Z
into
(11-J12)
^(C - C ) BLZ BL 2 v
Z 'AZ G
(11-J15)
+
^
C
B L Z -
C
the
following
(11-J16)
B L >
This relation shows the changes in CO^ concentration in the gas and NaOH concentration in the liquid phase along the column height of the industrial equipment. Several values of the soda lye concentration and the corresponding mole fraction of carbon dioxide y are given in Table 11-J1. A
Table 11-J1. Concentration of
ΒL
*A
NaOH (C
δ BL
) and mole fraction of CO (y ) 2 A
0.4
0.3
0.2
0.1
0.034
0.047
0.060
0.073
0.07
0 04
0.076
0 08
Next, in a laboratory absorber simulation of several cross sections of an industrial absorber was performed. The operation was carried out in a contactor with a flat contact area. First, a series of measurements were carried out for CO - water and NH2
3
air-sulfuric acid systems which were used for the determination of the mass transfer coefficients in both phases, as a function of impeller rotational speed. These data are presented in Figs. 11-J2 and 11-J3. 1
Ks* .
V $"
Fig. 11-J2. Mass transfer coefficient k Q versus impeller rotational
PACKED COLUMNS
497
10
± Fig. 11-J3. Mass transfer coefficient k
Lt
versus impeller rotational speed 1
Next, at impeller revolution of η = 5.2 s" , η = L
1
18 s" , which
Ο
correspond to physical coefficients for a given system, measurements were made at various NaOH/Na 2 C0 3 concentration ratios in the liquid and at corresponding molar fractions of CO^ in the gas, which were determined from equation (11-J16). The absorption experimentally, are presented in Table 11-J2 .
rates,
obtained
Table 11-J2. Experimental absorption rates C*L
3
[mol/m ]
N* 1 0
6
2
[mol/ (m s)]
0.4
0.3
0.2
0.1
0.07
0.04
6.4
4.2
2.0
0.7
0.4
0.2
The packing height in an industrial column can be defined from the integrated form of equation (11-42). Since carbon dioxide does not accumulate in the liquid, equation (11-42), using in addition eqsuation (11-45) and (11-46), can be rearranged to give
(11-J17) BLZ
Using data from Table 11-J2, the integral in equation (11-J17) can be calculated and then the packing height may be determined Ζ =
0. 0 054 5 X ( 3 . 0 5 X l O ) = 7.84 m 2 χ 105
498
SIMULATION OF PACKED COLUMNS
11.7.2. Simulation of the Industrial Absorber The simulation procedure discussed above is not valid if many chemical reactions occur in the liquid phase or if one or more reactions take place in the liquid bulk. If the chemical reaction kinetics are not known, it is impossible to relate the compositions of gas and liquid phases in particular cross sections of the industrial absorber. The same restrictions are valid for multicomponent physical absorption in dilute and concentrated systems. In these cases simulation of the entire industrial absorber is necessary. The respective mass balance equations have the form dW
dW ^
=
a
i L
N m
+
r ^ ^
i
j
R
j
(11-52)
j=l The analysis of the above equations reveals that for the general case, that is multicomponent absorption with reactions that take place partly in the film and partly in the liquid bulk, the simulation of an industrial absorber operation is only possible if a laboratory absorber is used. In the laboratory absorber the ratio of gas to liquid flow velocity, liquid hold-up, interfacial area in a unit volume, and mass transfer coefficients in both phases should be equal to those values which occur in the industrial absorber. When the experiments are carried out in a laboratory absorber in which the height can be varied, a height can be determined at which a given absorption level is obtained. This height is also equal to the height which must be used in the industrial column. Some more specific conclusions concerning the simulation of industrial absorbers can be drawn for particular cases of multicomponent physical absorption with inert material and multicomponent absorption with a slow chemical reaction which occurs entirely in the bulk of the liquid phase. For multicomponent physical absorption the mass balance equation for the liquid phase reduces to the form dW "dT^
-
a m
N
iL
ί
(
«
1
' -
n
>
Upon integration of equations (11-50) and (11-94) we obtain
U
< -
9 4
>
PACKED COLUMNS
Ζ
a W.
„ iz dY I -jj-*-
499
Y
m
=
Y
in
(i = l,...,n)
(11-95)
(i = l,...,n)
(11-96)
i G i 0
Ζ
a
X
„ iZ
m
=
w
S
dX
-jq——
j γ
iL iO
If the mass transfer coefficients in the model of the industrial absorber are the same as in the actual industrial absorber, and the ratio of gas to liquid flow rate in both is equal, the left hand side will also be equal Z a - N
Z
a
M , -hH r
v
in
•'lab
t^r-}, L
S
r
in
-
(,| w>
Ί
^ ind
-i ^-}
-
1
J
lab
L
S
J
ind
The simulation of an industrial column can be carried out in the following steps: a) For known systems, the mass transfer coefficients in both phases should be determined as a function of gas and liquid flow velocities, and as a function of the geometrical parameters of the laboratory apparatus. As the packing model either a string-of-disks or a string-of-spheres column or an inclinedwall column can be used. For instance, for a string-of-spheres column the mass transfer coefficients are determined as functions of gas and liquid velocity and sphere diameter, tube diameter, hollow size, etc. b) For known physical mass transfer coefficients in an industrial column, the laboratory apparatus design should be chosen in such a way as to obtain the same values of mass transfer coefficients in both apparatus at the same liquid to gas ratio L / G. . S
c)
in
In the model column the process of industrial absorption is studied. The investigations should be carried out in such a way as to achieve the same absorption efficiency in the laboratory column as in the industrial column. d) The experimentally determined laboratory column height is used to calculate from equation (11-97) or (11-98) the required packing height in the industrial absorber.
500
SIMULATION OF PACKED COLUMNS
The modelling of absorption with a slow single chemical reaction is described in detail in the literature [55, 56]. Experiments described in these works prove that the method presented above has been fully verified.
PROBLEMS Ρ 11-A.
Using the data of Example
11-E calculate the
absorption
efficiency of CO^ in a packed column of height a) b)
Ζ = 5 m, Z = 10 m.
Ρ 11-Β.
Repeat
the
calculation
in
Example
11-E using
the
exact
solution of the Stefan-Maxwell equation. Ρ 11-C. For the data in Example 11-G calculate the height of the packing for 99% removal of H^S. Ρ 11-D. For the data in Example 11-H estimate the packed height for 99% of HCl. Ρ 11-E. A gas consisting of a) 0.1 mole fraction of HCl, 0.01 mole fraction of vapour and 0.89 air, b) 0.6 mole fraction of HCl, 0.01 mole fraction of vapour and 0.39 air, c) 0.99 mole fraction of HCl and 0.01 mole fraction of vapour is to be absorbed in an adiabatic packed column. Calculate the absorption efficiency in a 3 m packed column. Other data are given in Example 11-H. Ρ 11-F. A gas consisting of 0.1 mole fraction of S 0 2 >
0.02 mole
fraction of vapour and 0.88 mole fraction of air is introduced to a column of diameter 1 m filled with 0.025 m Raschig rings. Gas enters the column at a rate of 20 mol/s at 293.15 Κ and a pressure of 1.013x10 Pa. Pure water at a rate 4 kg/s is flowing into the column. Calculate the height of the column for 99% removal of SO^. Equilibrium data for the system
SO^- H^O can be found in [57].
PACKED COLUMNS
501
REFERENCES 1. Jaroszynski M., private communication, 1990. 2. Stringle R.F. Jr., Random Packings and Packed Towers. Design and Applications. Gulf, Houston 1987. 3. Treybal R.E., Mass Transfer Operations. McGraw-Hill, New York 1980. 4. Kohl A.L., F.C. Riesenfeld, Gas Purification. Gulf, Houston 1985. 5. Cihla Z., O. Schmidt, Czech. Chem. Comm., 22, 896, 1957. 6. Jameston G.J., Trans. Inst. Chem. Engrs., 44, 198, 1966. 7. Porter K.E., J.J. Templeman, Trans. Inst. Chem. Engrs., 46, 86, 1968. 8. Dutkai Ε., E. Ruckenstein, Chem. Eng. Sei., 23, 1365, 1968. 9. Porter K.E., Trans. Inst. Chem. Engrs., 46, 69, 1968. 10. Hoek P.J., J.A. Wesselingh, F.J. Zuiderweg, Chem. Eng. Res. Dev., 64, 431, 1986. 11. Distillation and Absorption, Institution of Chemical Engineers, Symposium Series, No. 104, Brighton 1987. 12. Shulman H.L., C F . Ullrich, N. Wells, AIChE J., 1, 247, 1955. 13. Shulman H.L., C F . Ullrich, N. Wells, A.N. Proulx, AIChE J., 1, 259, 1955. 14. Stanek V., V. Kolar, Chem. Eng. J., 5, 5 1 , 1973. 15. Takahashi T., Y. Akagi, Κ. Ueyama, J. Chem. Eng. Japan., 12, 341, 1979. 16. Billet R., M. Schuhes, Distillation and Absorption, Institution of Chemical Engineers, Symposium Series, No 104, Brighton 1987. 17. Eckert J . S . , Chem. Eng. Progr., 57 (9), 54, 1961. 18. Mersmann Α., Chem. Ing. Techn., 3, 212, 1965. 19. Mayo F . , T.G. Hunter, A.W. Nash, J. Soc. Chem. Ind., 54, 376, 1935. 20. Yoshida F . , T. Koyanagi, AIChE J., 8, 309, 1962. 21. Onda Κ., H. Takeuchi, Koyama, Chem. Eng. Japan, 3 1 , 127, 1967. 22. Javari A . S . , M.M. Sharma, Chem. Eng. Sei., 23, 1, 1968. 23. Charpentier J . C , B.I. Morsi, NATO ASI on Mass Transfer with Chemical Reaction, Izmir 1981. 24. Puranik S.S., A. Vogelpohl, Chem. Eng. Sei., 29, 501, 1974. 25. Sherwood T.K., F.A.L. Hollo way, Trans. Am. Inst. Chem. Engrs., 36, 39, 1940. 26. Van Krevelen D.W., P.J. Hoftijzer, Ree. Trav. Chim. des Pays-Bas, 66, 49, 1947. 27. Ramm W,M., A.Ju. Zakgeim, N.M. Gurowa, Zagig, Z.W. Alina, Trud. Chim. Tech., 1, 913, 1961. 28. Shulman H.L., C G . Savini, P.V. Edvin, AIChE J, 9, 479, 1963. 29. Danckwerts P.V.,Gas-Liquid Reaction, McGraw-Hill, New York 1970.
502
SIMULATION OF PACKED COLUMNS
30. Mohunta D., A.S. Vaidyanathan, G.S. Laddha, Ind. Chem. Eng., 11, 39, 1969. 31. Au-Yeung P.H., A.B. Ponter, Can. J. Chem. Eng., 61, 481, 1983. 32. Norman W.S., F.Y.Y. Sammak, Trans. Inst. Chem. Engrs, 4 1 , 109, 1963. 33. Onda K., H. Takeuchi, Y. Okumoto, J. Chem. Eng. Japan., 1, 56, 1968. 34. Bravo J.L., J.A. Rocha, J.R. Fair, Hydrocarb. Proc., 64, 9 1 , Jan. 1985. 35. Spiegel L., W, Meier, I. Chem. E. Symp. Ser., 104, A203, 1987. 36. Fair J.R., J.L. Bravo, Chem. Eng. Prog., 86, 91, Jan. 1990. 37. Herning F., L. Zipperer, Gas u. Wasserfach, 5, 1936. 38. Jacobs D.A.H. (ed.), The State of the Art in Numerical Analysis, Academic Press, New York 1977. 39. Stoer J., R. Burlisch, Einfuhrung in die numerische Mathematik, Vol. II, Springer-Verlag, Berlin 1973. 40. Constantinides Α., Applied Numerical Methods with Personal Computers, McGraw-Hill, New York 1987. 41. Hlavacek V. (ed.), Dynamics of Nonlinear Systems, Gordon and Breach Science Publ., New York 1986. 42. Marquardt D.W., J. Soc. Ind. Appl. Math., 11, 431, 1963. 43. Broyden C G . , Math. Comp., 19, 577, 1965. 44. Broyden C.G., Computer Journal, 12, 25, 1965. 45. Avila J., Univ. of Maryland Comp. Sei. Ctr. Tech. Rep. TR-142, 1971. 46. Brent R.P., SIAM J. Numer. Anal., 10, 327, 1973. 47. Cosnard M., M.Sc.Thesis, Cornell Univ. Comp. Sei., TR75-248, 1975. 48. Duflhard P., Numer. Math., 5, 289, 1974. 49. Christiansen J., Numer. Math., 14, 317, 1970. 50. Raal J.D., M.K. Khurana, Can. J. Chem. Eng., 5 1 , 162, 1973. 51. Danckwerts P.V., A.J. Gillham, Trans. Instn Chem. Engrs., 44, 42, 1966. 52. Danckwerts P.V., E. Alper, Trans. Instn Chem. Engrs, 53, 34, 1975. 53. Alper E., P.V. Danckwerts, Chem. Eng. Sei., 3 1 , 599, 1976. 54. Shah Y.T., W.-D. Deckwer, Fluid-Fluid Reactors, Chap. 6 in A. Bisio, R.L. Kabel, Scaleup of Chemical Processes, Wiley, New York 1985. 55. Alper E., Chem. Ing. Tech., 51, 1136, 1979. 56. Laurent Α., C. Fonteix, J.C. Charpentier, AIChE J, 26, 282, 1980. 57. Edwards T.J., G. Maurer, J. Newman, J.M. Prausnitz, AIChE J., 24, 966, 1978.