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The frequency response of a plate gas absorption column
Chemical Enginering Science, 1966, Vol. 21, pp. 77-86. Pergamon Press Ltd., Oxford. Printed in Great Britain.
The frequency response of a plate gas absorption column A. J. HAAGENSENtand F. P. LEES~ Imperial Chemical Industries Ltd., Central Instrument Laboratory, Bozedown House, Reading, Berkshire (Received 2 February 1965; in revisedform 4 March 1965)
Abstract A theoretical model is given for the unsteady-state behaviour of a plate gas absorption column and the theoretical frequencyresponses are derived. Experimental frequencyresponses obtained on a 4-in. i.d. six-plate absorption column operating with the system air-ammonia-water are presented. methylaniline in a sieve-plate column using a step response technique. Their equations are the full differential equations of the system which are integrated numerically and for which the assumptions of isothermal conditions and linear equilibrium are not necessary. Liquid flow change is taken into account, but it is assumed that the flow change is transmitted instantaneously throughout the column. The present paper describes the theoretical and experimental investigation of the unsteady-state absorption of ammonia from air into water in a bubble-cap plate gas absorption column using a frequency response technique. The work was done in 1960-61.
INTRODUCTION
A KNOWLEDGEof the dynamic characteristics of chemical plant is frequently desirable. A number of investigations have been published of the unsteady-state behaviour of distillation columns, but there is little in the literature on that of gas absorption columns. Theoretical investigations of the unsteady-state behaviour of multistage mass transfer operations in general have been made by MARSHALL and PIGFORD [1] and of multistage gas absorption and extraction operations by LAPIDUS and AMUNDSON [2]. A theoretical study of the dynamic behaviour and control of a plate gas absorption column has been made by CEAGLSKE [3]. None of these papers give any experimental work. THEORETICAL WORK Two recent papers do report both theoretical and experimental work. GRAY and PRADOS [4] investiThe theoretical model of the gas absorption progated the response of the outlet gas concentration cess is based on the isothermal absorption of a to the inlet gas concentration for the absorption of solute gas from an inert gas under conditions where carbon dioxide from air into water in a packed both the gas and the liquid flow rates are subcolumn using a frequency response technique. Their stantially constant and where the concentration of theoretical equations are analytical solutions of the the solute in both phases is small. The following are frequency response of gas concentration in a packed the basic assumptions: bed with countercurrent constant gas and liquid 1. Absorption is isothermal flows using plug-flow, mixed-stages and eddy 2. Equilibrium line is straight diffusion models. Isothermal conditions and a 3. Plate efficiency is constant linear equilibrium relation are assumed and flow 4. Liquid and gas flows are constant (at steadychanges are not taken into account. state). 5. There is complete mixing of gas between plates CALVERT and COULMAN [5] investigated the and of liquid on plates. response of the outlet gas concentration to the inlet It is appreciated that there are many industrial gas concentration and to the inlet liquid flow for the absorption of sulphur dioxide from air into di- gas absorption systems in which some of these t Now at Imperial Chemical Industries Ltd., Agricultural Division, Billingham, Co. Durham. $ Now at Imperial Chemical Industries Ltd., Mond Division, P.O. Box 7, Winnington, Norwich, Cheshire. 77
A. J. HAAGENSENand F. P. LEES
assumptions do not apply, but the attempt to take account of such deviations renders the model very much more complex. The equations have been written in terms of mole fractions rather than mole ratios. The use of mole ratios renders the expressions for hold-up more complicated. The basic equations for the absorption system are as follows. The mass balances are
At steady state equations (7) and (8) become Y* = r X
(11)
Y.-1- Y. eg = y . _ l _ y.,
(12)
The steady state mass balance between plate n and the top of the column is E(7,N+ ~ -- X . ) + G(Y.- x - YN) = 0
aw. Ot = L.+I -- L .
(1)
OU.
Ot = G._ l - G. . (~'VX)n
,
. (U~')
°--FF-v°
(14)
From equations (11) and (12)
(2)
Y. - (1 + E,)Y._x + EaflX. (15) Substituting for )?. from equation (14) in equation (15)
n
cgt
= (LX).+I + (GY).-x - ( L X ) . - ( G Y ) .
r. = ~Y.-1 + bX'. + ey.
(3)
(16)
where
The liquid and gas hold-up relations are
G
IV. = Kx + K2 L21a
(4)
U. = K a P .
(5)
~i = 1 - Eg + Earl -~
(17a)
b = Earl
(17b)
In equation (4) the second term represents liquid hold-up associated with the liquid head over the weir.
G
= --Egfl £
The pressure drop relation is -
"1- £G( y.n- ~ -- Yn)
Xn = XN+I
(17c)
It can readily be shown that
8p
d--'n= K , + Ks G2/3
~ =:•o + h~N+,
(6)
(18)
(19a)
= ~N-1 "/t- ~N-2 ... 1
In equation (6) the second term represents pressure drop through bubble cap slots. The index of this term is based on the work of ROGERs and THIELE [6] and of CROSS and RYDER [7].
Y* =
Y.-i -- Y. Y.-x -- Y* X . - X.+I X* - X.+ 1
(8)
Eg E o + [(I--Eg)
h= ~
(19c)
~(fiG~L)]
~
w. °X. + u ° r .
~t
(9)
= L.+l(X.+x
The relation between the plate efficiencies ([8] p. 158) is El =
(19b)
The steady-state gas and liquid concentrations, which are required for the solution of the unsteadystate equations, can be calculated from equations (18), (16) and (14). Combining equations (1), (2) and (3)
(7)
rX
The plate efficiency relations are
Ez =
~n f = ~ 1- ~ 1 -
The equilibrium relation is
Eg =
(13)
- X.) + G._I(Y._,
-
Y.)
(20)
The frequency response of the column may be obtained by expressing the time-dependent variables as the sum of their steady and unsteady-state values, by subtracting the steady-state equations from the
(10) 78
The frequency response of a plate gas absorption column total equations and by then taking Laplace transforms S ~ n ~ - 7n+ 1 - - 7n (21) s ft. = O.-1 - O.
(22)
(36)
D1 = K a K 6 s
Integrating equation (35) with the boundary conditions
W s ~ . + U s y . = ( X . + l - X.)7.+1 + ( Y . _ , - Y.)
n=0,-~n
=K60°
9 . - 1 + E(~,. + 1 - ~.)
x
n = N,p. = 0
+ t~(y.- 1 - .~,)
(23)
2 Ks w. = ~ ~ i ~ 7.
(24)
un = K a P .
(25)
-- ap. On = K6~.
(26)
gives K6
&=
x/(D0(1 + 02) x {expl'-x/(Dx)n ] - D 2 exp[x/(D1)n]}~o (37) where D2 = e x p [ - 2~/(D1)N]
where 2 Ks K6 = ~ GU---x
(27)
(38)
Hence ~___~.= exp[-x/(D1)n] + 0 5 exp[x/(D1)n ]
y* = p~
(28)
Yn- 1 -- Yn
Y . - 1 - Y*~ E t --
~n ~
Xn + 1
~, X n
~ Xn+
--
(29)
(3o)
7n+ 1
1
1 + zs
2
K2
1 -I- D 2
9___~= exp[--x/(Dl)N] + Dz exp[x/(O0N] 1 +D 2
9o
(4o)
= sech[x/(Dt)N ]
(31)
where 3E1/3
(32)
1 (1 + zs) u+l-"
(33)
(39)
Atn =N
1
The response of the outlet liquid flow to the inlet liquid flow is as follows: 7.
9o
The responses of the outlet liquid concentration to the inlet liquid flow, the inlet gas flow, the inlet liquid concentration and the inlet gas concentration are obtained as follows. Combining equations (28) and (30) y.=-~--t~ ( l _ E t ) £ . + 1 + E--~ fl Xn -
(41a)
Hence 7. iv+, Atn=
Y.- 1 = ~
1 11
1
The response of the outlet gas flow to the inlet gas flow is as follows. Differentiating equation (26) and combining it with equations (22) and (25)
alyo + bl£t + c1£2
aep. D1/~. = 0
P
(41b)
Combining equations (23), (41a) and (41b) for all plates except the bottom plate and equations (23) and (41a) for the bottom plate only the following set of equations is obtained
(34)
7N+1 = (1 + zs) N
On 2
_
(1 - E3 x. + ~ - £.-1
(35)
where
el~o
(42a)
a.£.-x + b.£. + c.£.+1 = dflN+l + e.~o
(42b)
a~£s_ 1 + bN£N + CN£N+I = duTN+1 + eN~o
(42c)
where 79
= dllN+l +
A. J. HAAGENSENand F. P. LEES flY0 -Jr glYl + hlY2 =jaTN+x + k19o
(43a)
a,, = - C, ~
El
=j.TN+I + k.0o
(46b)
fNYN-1 + gUYN + h~2N+l = Jflu+l +kN0o
(46c)
fnYn-1
U s ) ~fl+
b,=(G+
E,) + L + Ws
~fl Ll (1
"{- OnYn "dr h,,Y.+t
(46a)
where (43b)
f~ = - ( E + Ws) (1 -
)
G
(47a)
(43c)
c. = - ( G + Us) -~ (1 - E~) - E
g . = (L - + - Ws) +
-- Eg) E (1 - + G + Us
1
dn = (X,+I - X.) (1 + zs) N-"
-E hn =
e. = (Yn-1 - Yn)
(47c)
Eo[3
{exp[-x/(D0(n - 1)] + D2 exp[~(D,)(n - 1)]}
1
J, = (X.+I - X.) (1 + zs) N-"
1 +D 2
(43e) al = - G
k. = ( L - I
(43f) ×
bl = (G + Us) ~ + E, + Ws
X = 7N+tMI-aD + ~oM~- XE + ~oMx-lA + gN+ 1M~-1C where X = {xl .-. XN}
MI=
cl 0 b2 C2 a 3 b3
c3
0
0
a,
0
?
{exp[-x/(Oa)(n -- 1)] + D 2 exp[~/(D0(n - 1)3} 1 +D 2
(47e) (E + Ws) an--
b.
Cn
0
-
-
(47f)
-]- G "[- U S
(47g)
hN = - L (44)
The solution of these equations is the matrix equation Y = ~N+lM21J + ~oM21K + YoM21F
(45a)
0
(47d)
- Y.)
(43g)
The solution of these equations is the matrix equation
'bl a2 0
(47b)
(43d)
0 0 0 0 0 0 aN bN,
+ 2N+aM21H
(48)
where ¥ = {Y~ ... YN}
(49a)
gt fz
0 0 0 0 0 0 9~ (49b)
M2 = l i
th 0 g2 h2 0 .f3
g3
(45b)
ha o f.
v.
h.
dN}
(45c)
E = { e x ... eN}
(45d)
A={-at0
(45e)
J = {Jl ...JN}
(49c)
(45f)
K = {k1-.-kN}
(49d)
F = { - - f l 0 ... 0}
(49e)
H={O...O-hN}
(49f)
D=
{dl...
... 0}
C = {0 ... 0 - cN}
The responses of the outlet gas concentration to the four input variables are obtained in a similar manner. The following set of equations is obtained. 80
The frequency response of a plate gas absorption column The frequency responses are now obtained by making the substitution
Gas
~ffluent
Liquid
~ffluent
s = io2 2rr T
The response is obtained as a complex number The amplitude, and phase lag of the response are
(q + Jr).
Amplitude 5 = x/(q 2 q- r 2) Phase Lag~b = t a n - l ( ~ ) The normalized attenuation is then Normalized attention 0~ = &oo/5 EXPERIMENTAL WORK
o @-@-@
The experimental apparatus used was the plate gas absorption column shown schematically in Fig. 1. The column was constructed of standard QVF glass column sections 4 in. i.d. x 12 in. high. The plates were made of stainless steel and consisted of a disc ~ in. o.d. x -~ in. thick welded inside a thicker annulus 5¼ in. o.d. x ½ in. thick. A circular weir, a circular downcomer and two bubble-caps were located symmetrically on a 2¼ in. p.c.d. The weir and downcomer were 1 in. i.d. and the weir was 2 in. high. The riser was ½ in. i.d. x 1~ in. high and the caps were 1 in. o.d. x 13 in. high with 12 slots 1 in. high x 3/32 in. wide. Six identical plates were used, except that the downcomer of the bottom plate was luted to prevent passage of gas. The experimental absorption system was airammonia-water. The water used was towns water. There were on the inlet water line a hand control valve, a rotameter, a thermometer, an orifice plate and an automatic control valve. The gas used was a mixture of air and ammonia. There were on the air and ammonia lines needle valves and rotameters and on the inlet mixed gas line a thermometer, an orifice plate, an automatic control valve and an analysis point. There were on the outlet water line an electrical conductivity cell and an analysis point and on the outlet gas line an analysis point, an orifice plate and a thermometer. There were liquid
Water
Air Ammonia
FIG. 1. Experimental gas absorption column. FI 1-3 Rotameters FR 1-3 Orifice plates TI 1-3 Mercury-in-glass thermometers An 1-3 Sample points An 4 Conductivity cell and gas sample points on each plate. A small pool of liquid was maintained in the column base so that the conductivity cell ran full and the level of this liquid was maintained by a luted offtake. Sinusoidal signals were impressed on the inlet liquid and mixed gas control valves by a pneumatic sine-wave generator. Pressure transducers were used to convert the signals from the orifice plates and from the sine-wave generator to electrical signals. Signals from the orifice plates, sine-wave generator and conductivity cell were recorded on a multi-channel recorder which used ultra-violet sensitive paper. The column was operated in all experiments at the following mean conditions: Liquid flow 446 lb mole/ft2hr 81
A. J. HAAGENSEN and F. P. LEEs 0.¢
t O.E
/ ×
.a
x
0.7
× G = 4'061b © =6"14
E
= 7.5r
I
.J O'E
200
300
400
Liquid flow L Ib
FIG. 2.
mole / f t Z h r
6OO
5(~)0
m o l e / f t z hr
Liquid hold-up.
Gas flow 6.4 lb mole/ft2hr Gas composition 10 ~o v/v NHa Under these conditions the concentration of ammonia in both gas and liquid is low and absorption is substantially isothermal. The equilibrium relation is given by equation (7) with the following values of the constants: fl = 0.76 This value is taken from PERRY [9] and corresponds to 20°C, which is the temperature at which the experimental work was done. Ammonia is highly soluble in water and the equilibrium partial pressure of the gas above the liquid was small compared with the actual partial pressures of ammonia in the gas. Preliminary experiments were carried out at steady state to determine the liquid hold-up in the column, the pressure drop across the column and the plate efficiency. A graph of liquid hold-up versus liquid flow at the mean gas flow is given in Fig. 2. The liquid hold-up relation is given by equation (4) with the following values of the constants:
K4 = 0-0025 K 5 = 0"001 The plate efficiency, as defined by equation (8), was measured by analysis of the gas at the inlet and outlet of the column and also between plates. The mean value over the whole column was 64 per cent, although values up to 85 per cent were found on the bottom plate. The following frequency responses were obtained: Input
Output
Inlet liquid flow Inlet liquid flow Inlet gas flow
Outlet liquid flow Outlet liquid concentration Outlet liquid concentration
Table 1. Response of outlet liquid flow to inlet liquid flow Run A1 2 3 4 5 6 7 8 9 10
KI = 0.365 K2 = 0.0072 A graph of pressure drop versus gas flow at the mean liquid flow is given in Fig. 3. The pressure drop relation is given by equation (6) with the following values of constants: 82
Period of oscillation
Measured response
(hr)
a
¢
0"0655 0"0342 0"0340 0"0325 0"0292 0'0231 0-0200 0-0183 0"0162 0"0150
1"0 1"06 1"02 1'03 1"01 1"11 1-26 1"23 1"32 1.40
7 21 44 44 43 66 68 81 80 97
The frequency response of a plate gas absorption column 0.007
l.,,.o.oOXGzl~
o.
O.OOG
I ~X I
I
X /
f~"
I
0.005 ×
L ' 4 4 6 Ib mole/ftZhr
0"004
8
Gos flow 0 Ib m o l e / f t 2 hr
FIG. 3.
Pressure drop.
The amplitude of the sine wave impressed on the inlet liquid and gas flows was + 10 per cent of the mean value. The experimental results are expressed as a frequency response made up of a normalized attenuation and a phase lag. At steady state a unit change in the input signal results in a change in the output signal which is known as the potential value. When the input varies sinusoidally the amplitude of the output signal for an input signal of unit amplitude is attenuated. The ratio of the potential value to this attenuated amplitude is the normalized attenuation, which is therefore dimensionless and greater than unity. The response of the outlet liquid flow to the inlet
liquid flow is given in Table 1 and in Fig. 4. It was obtained by replacing the lute shown in Fig. 1 by a graduated measuring cylinder and by timing the rate of rise and fall of liquid in the cylinder with a constant offtake. The response of the outlet liquid concentration to inlet liquid and gas flows is given in Tables 2 and 3 and in Figs. 5 and 6. It was obtained using the conductivity cell to measure the outlet liquid concentration. It is necessary to correct the measured response for the lags associated with the pool of liquid in the base of the column. The transfer characteristics of this pool and of the pipe between the bottom of the
Table 2. Response of outlet liquid concentration to inlet liquid flow
Run B1 2 3 4 5 6 7 8 9 10
Period o f oscillation (hr)
Measured response ~¢
~
0"0433 0"0375 0"0283 0"0230 0"0206 0"0181 0"0144 0"00945 0"0086 0"0075
1"23 1"27 1"50 1"67 1'81 1'95 2"74 3"55 4-8 5"7
64 77 97 107 135 145 177 225 256 272
Correction ~ ~ 1"0 1"0 1"0 1"01 1"01 1"01 1"02 1"04 1"05 1"07
83
9 10 13 16 18 21 26 38 43 48
Corrected response ~ 1"23 1"27 1"50 1"66 1'80 1"93 2"68 3"40 4"56 5"30
55 67 84 91 117 124 151 187 213 224
A. J. HAAGENSEN and F. P. LEES
Table 3. Reponse o f outlet liquid concentration to inlet 9as flow Period of oscillation
Run
C1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Measured response
Corrected response
Correction
(hr)
~
~
~
0"0628 0-0358 0.0340 0"0220 0-0183 0"0153 0"012 0.011 0'0092 0-0083 0.0081 0-0072 0-0064 0"0064 0"0058 0-0050
1"04 1.25 1-25 1"34 1"48 1"45 1"90 1"54 2"00 2.57 2.38 2.52 3'00 3'60 3"26 3"42
5 23 29 47 64 63 83 93 97 85 114 118 132 129 142 141
1"0 1"0 1"0 1"01 1'01 1 "02 1-03 1"03 1 '05 1"06 1"06 1 "07 1 '09 1"09 1"11 1"15
5 10 11 17 21 24 31 33 40 44 45 50 57 57 62 71
360,
,
1-04 1.25 1"25 1"33 1"47 1"42 1"85 1"50 1"90 2"43 2"25 2.36 2"75 3"30 2.94 2'97
0 13 18 30 43 39 52 60 57 41 69 68 75 72 80 70
,
300
240
"--~
180
~
120
G.
6O
i
oI 2~
t
i 2c --•.i-
e 7
g
~
o
Z Z I.
Period
of
oscillation,
hr
0.001
0-01 Period
FXG. 4. Response of outlet liquid flow to inlet liquid flow. Crosses are experimental and full lines theoretical values.
of
oscillation,
0.1 hr
~.
FIG. 5. Response of outlet liquid concentration to inlet liquid flow. Crosses are experimental and full lines theoretical values. 84
The frequency response of a plate gas absorption column
Table 4. Constants for calculation of theoretical responses N E d K1 //2 Ka /(4 K5 fl Eg
6 446 6"4 0-365 0.0072 0"00218 0'0025 0"001 0"76 0"64 0.10
?o
W Xz
Xz X3 X4 X5 X'o
0-785 0"00143 0"000523 0"000190 0.0000674 0-0000225 0.00000605
OYz
?a 73 ?4 ?5 ?8
0.00218 0-0367 0-0135 0"00494 0"00181 0"000663 0"000242
X~+I 0 column and the conductivity cell were obtained by injecting an impulse of 5 per cent sodium hydroxide into the liquid entering the pool from the bottom plate and analysing the response of the conductivity cell. The lags were found to correspond to a transfer lag of 1.6 see and a distance-velocity lag of 2.1 sec. The measured responses have been corrected for these lags as shown in Tables 2 and 3.
The corresponding theoretical responses were calculated from the equations given in the previous Section using the data given in Table 4. The first column of this table gives the basic numerical constants for the experiments. The second and third columns give the calculated liquid and gas hold-ups and the liquid and gas concentrations on each plate, calculated from equations (4), (5) and (11-19c).
360 l
CONCLUSION
00
There is fair agreement between the theoretical and experimental results for the three responses investigated. It was originally intended to carry out experimental work to check the theory at different gas-liquid ratios and with different solute gases, but the work was interrupted before this could be done. The effect of plate efficiency for the experimental absorption system was investigated using the theoretical model. The normalized attenuation and phase lag decrease with increase in plate efficiency. In the six-plate column, at a period 0.01 hr, the normalized attenuation and the phase lag are 3.19 and 162° respectively for a plate efficiency 64 per cent and 2.52 and 159° respectively for a plate efficiency 85 per cent. The application of the model is limited by the assumptions made in its derivation, of which the most important are that the absorption is isothermal, that the equilibrium relation is linear and that the concentration of solute in the inlet gas is small, but these assumptions are applicable in many absorption systems.
"0- 24C
180
120
60
0
t g
o B 7
"6
"t. I
E 7£_ I"
0.001
0-OI
Period of oscillation,
0.1
hr
=~
FIG. 6. Response of outlet liquid concentration to inlet gas flow. Crosses are experimental and full lines theoretical values.
85
A. J. HAAGENSENand F. P. LEES Aclmowledgements--Thanks are due to Mr. A. J. YOUNG, head of the Central Instrument Laboratory, Imperial Chemical Industries Ltd., to Mr. J. McMmLAN for his advice throughout the work and to Mr. R. Rowe, who carried out many of the experiments and calculations, and to other members of the laboratory staff who helped in many ways.
NOTATION a-j a-h Dl-a Eg Et G i Kl-e L n
Constants Constants Complex constants Plate efficiency (gas phase) Plate efficiency (liquid phase) Gas flow lb mole/ft~hr =a/--1 Constants Liquid flow lb mole/ft~hr Number of plates from bottom of column N Number of plates in column P Pressure atm s Laplace operator t Time hr T Period of sine wave hr U Gas hold-up lb mole/ft 2 plate I4/ Liquid hold-up lb mole/ft 2 plate
X Liquid solute concentration Y Gas solute concentration Y* Equilibrium gas solute concentration corresponding to liquid solute concentration X ct Normalized attenuation (defined in tex0 Constant Amplitude of sine wave 7 Time constant for transmission of liquid flow across a plate Phase lag of sine wave to = 2¢r/T
mole fraction mole fraction mole fraction
hr degrees hr -1
Subscripts 0-6, n and N are used to denote plate number. The numbering is from the bottom of the column. Total values of time-dependent variables are represented by a capital letter, steady-state values by a capital letter with a bar above, small tmsteady-state variations about the steadystate value by a small letter and Laplace transforms of the unsteady-state variations by a small letter with a tilde above it: G=~+g .~q) = The Laplace transform is defined as follows: c~o
~ [ f ( t ) ] = S exp(--st)f(t)dt 0
REFERENCES
[1 ] MARSHALLW. R. and I~GFORDR. L., Applications of Differential Equations to Chemical Engineering Problems. University of Delaware Press, 1947. [2] LAPmus L. and AMUr,a3SONN. R., lndustr. Engn# Chem. 1950 42 1071. [3] CEA~LSKEN. H., Proc. 1st Int. Congr. lnt. Federat. Automatic Control Moscow 1960. Vol. IV, p. 288. Butterworths, London, 1961. [4] GRAYR. I. and PRADOSJ. W., A.LC.E. Jl. 1963 9 211. [5] CALVERTS. and COUL~iANG. A., Chem. En#n# Pro#r. Symp. Ser. 46. S p.9. [6] ROGERSM. C. and Trilra_~ E. W., Industr. En#ng Chem. 1934 26 524. [7] CROSSG. A. and RYDER H., J. AppL Chem. 1952 2 51. [8] SHERWOODT. K. and PXGFORDR. L., Absorption and Extraction. McGraw-Hill, New York, 1952. [9] PERRYJ. H. (Ed.), Chemical Engineers Handbook (3rd Ed.). McGraw-Hill, New York, 1950. R~sum6---Les auteurs pr6sentent un mod~le th6orique du comportement transitoire d'une colonne d'absorption ~t plateaux et 6tablissent la r6ponse fr6quentielle th6orique. L'article contient encore des r6ponses fr6quentielles exp6rimentales obtenues avec le syst~me air-ammoniaque sur des colonnes d'absorption d'un diam6tre de 4 in., comprenant 6 plateaux. Zusammenfassung--Es wird ein theoretisches Modell far das nichtstation~ire Verhalten einer Absorptionskolonne mit B6den aufgesteUt und die theoretische Antwortfunktion fiar ein eingegebenes sinusf6rmiges Signal (frequency response-Methode) aufgestellt. Diese Funktionen werden mit experimentellen Werten aus einer 4 inch Absorptionskolonne mit 6 B6den ffir das System Ammoniak-Luft und Wasser verglichen.
86
The frequency response of a plate gas absorption column A. J. HAAGENSENtand F. P. LEES~ Imperial Chemical Industries Ltd., Central Instrument Laboratory, Bozedown House, Reading, Berkshire (Received 2 February 1965; in revisedform 4 March 1965)
Abstract A theoretical model is given for the unsteady-state behaviour of a plate gas absorption column and the theoretical frequencyresponses are derived. Experimental frequencyresponses obtained on a 4-in. i.d. six-plate absorption column operating with the system air-ammonia-water are presented. methylaniline in a sieve-plate column using a step response technique. Their equations are the full differential equations of the system which are integrated numerically and for which the assumptions of isothermal conditions and linear equilibrium are not necessary. Liquid flow change is taken into account, but it is assumed that the flow change is transmitted instantaneously throughout the column. The present paper describes the theoretical and experimental investigation of the unsteady-state absorption of ammonia from air into water in a bubble-cap plate gas absorption column using a frequency response technique. The work was done in 1960-61.
INTRODUCTION
A KNOWLEDGEof the dynamic characteristics of chemical plant is frequently desirable. A number of investigations have been published of the unsteady-state behaviour of distillation columns, but there is little in the literature on that of gas absorption columns. Theoretical investigations of the unsteady-state behaviour of multistage mass transfer operations in general have been made by MARSHALL and PIGFORD [1] and of multistage gas absorption and extraction operations by LAPIDUS and AMUNDSON [2]. A theoretical study of the dynamic behaviour and control of a plate gas absorption column has been made by CEAGLSKE [3]. None of these papers give any experimental work. THEORETICAL WORK Two recent papers do report both theoretical and experimental work. GRAY and PRADOS [4] investiThe theoretical model of the gas absorption progated the response of the outlet gas concentration cess is based on the isothermal absorption of a to the inlet gas concentration for the absorption of solute gas from an inert gas under conditions where carbon dioxide from air into water in a packed both the gas and the liquid flow rates are subcolumn using a frequency response technique. Their stantially constant and where the concentration of theoretical equations are analytical solutions of the the solute in both phases is small. The following are frequency response of gas concentration in a packed the basic assumptions: bed with countercurrent constant gas and liquid 1. Absorption is isothermal flows using plug-flow, mixed-stages and eddy 2. Equilibrium line is straight diffusion models. Isothermal conditions and a 3. Plate efficiency is constant linear equilibrium relation are assumed and flow 4. Liquid and gas flows are constant (at steadychanges are not taken into account. state). 5. There is complete mixing of gas between plates CALVERT and COULMAN [5] investigated the and of liquid on plates. response of the outlet gas concentration to the inlet It is appreciated that there are many industrial gas concentration and to the inlet liquid flow for the absorption of sulphur dioxide from air into di- gas absorption systems in which some of these t Now at Imperial Chemical Industries Ltd., Agricultural Division, Billingham, Co. Durham. $ Now at Imperial Chemical Industries Ltd., Mond Division, P.O. Box 7, Winnington, Norwich, Cheshire. 77
A. J. HAAGENSENand F. P. LEES
assumptions do not apply, but the attempt to take account of such deviations renders the model very much more complex. The equations have been written in terms of mole fractions rather than mole ratios. The use of mole ratios renders the expressions for hold-up more complicated. The basic equations for the absorption system are as follows. The mass balances are
At steady state equations (7) and (8) become Y* = r X
(11)
Y.-1- Y. eg = y . _ l _ y.,
(12)
The steady state mass balance between plate n and the top of the column is E(7,N+ ~ -- X . ) + G(Y.- x - YN) = 0
aw. Ot = L.+I -- L .
(1)
OU.
Ot = G._ l - G. . (~'VX)n
,
. (U~')
°--FF-v°
(14)
From equations (11) and (12)
(2)
Y. - (1 + E,)Y._x + EaflX. (15) Substituting for )?. from equation (14) in equation (15)
n
cgt
= (LX).+I + (GY).-x - ( L X ) . - ( G Y ) .
r. = ~Y.-1 + bX'. + ey.
(3)
(16)
where
The liquid and gas hold-up relations are
G
IV. = Kx + K2 L21a
(4)
U. = K a P .
(5)
~i = 1 - Eg + Earl -~
(17a)
b = Earl
(17b)
In equation (4) the second term represents liquid hold-up associated with the liquid head over the weir.
G
= --Egfl £
The pressure drop relation is -
"1- £G( y.n- ~ -- Yn)
Xn = XN+I
(17c)
It can readily be shown that
8p
d--'n= K , + Ks G2/3
~ =:•o + h~N+,
(6)
(18)
(19a)
= ~N-1 "/t- ~N-2 ... 1
In equation (6) the second term represents pressure drop through bubble cap slots. The index of this term is based on the work of ROGERs and THIELE [6] and of CROSS and RYDER [7].
Y* =
Y.-i -- Y. Y.-x -- Y* X . - X.+I X* - X.+ 1
(8)
Eg E o + [(I--Eg)
h= ~
(19c)
~(fiG~L)]
~
w. °X. + u ° r .
~t
(9)
= L.+l(X.+x
The relation between the plate efficiencies ([8] p. 158) is El =
(19b)
The steady-state gas and liquid concentrations, which are required for the solution of the unsteadystate equations, can be calculated from equations (18), (16) and (14). Combining equations (1), (2) and (3)
(7)
rX
The plate efficiency relations are
Ez =
~n f = ~ 1- ~ 1 -
The equilibrium relation is
Eg =
(13)
- X.) + G._I(Y._,
-
Y.)
(20)
The frequency response of the column may be obtained by expressing the time-dependent variables as the sum of their steady and unsteady-state values, by subtracting the steady-state equations from the
(10) 78
The frequency response of a plate gas absorption column total equations and by then taking Laplace transforms S ~ n ~ - 7n+ 1 - - 7n (21) s ft. = O.-1 - O.
(22)
(36)
D1 = K a K 6 s
Integrating equation (35) with the boundary conditions
W s ~ . + U s y . = ( X . + l - X.)7.+1 + ( Y . _ , - Y.)
n=0,-~n
=K60°
9 . - 1 + E(~,. + 1 - ~.)
x
n = N,p. = 0
+ t~(y.- 1 - .~,)
(23)
2 Ks w. = ~ ~ i ~ 7.
(24)
un = K a P .
(25)
-- ap. On = K6~.
(26)
gives K6
&=
x/(D0(1 + 02) x {expl'-x/(Dx)n ] - D 2 exp[x/(D1)n]}~o (37) where D2 = e x p [ - 2~/(D1)N]
where 2 Ks K6 = ~ GU---x
(27)
(38)
Hence ~___~.= exp[-x/(D1)n] + 0 5 exp[x/(D1)n ]
y* = p~
(28)
Yn- 1 -- Yn
Y . - 1 - Y*~ E t --
~n ~
Xn + 1
~, X n
~ Xn+
--
(29)
(3o)
7n+ 1
1
1 + zs
2
K2
1 -I- D 2
9___~= exp[--x/(Dl)N] + Dz exp[x/(O0N] 1 +D 2
9o
(4o)
= sech[x/(Dt)N ]
(31)
where 3E1/3
(32)
1 (1 + zs) u+l-"
(33)
(39)
Atn =N
1
The response of the outlet liquid flow to the inlet liquid flow is as follows: 7.
9o
The responses of the outlet liquid concentration to the inlet liquid flow, the inlet gas flow, the inlet liquid concentration and the inlet gas concentration are obtained as follows. Combining equations (28) and (30) y.=-~--t~ ( l _ E t ) £ . + 1 + E--~ fl Xn -
(41a)
Hence 7. iv+, Atn=
Y.- 1 = ~
1 11
1
The response of the outlet gas flow to the inlet gas flow is as follows. Differentiating equation (26) and combining it with equations (22) and (25)
alyo + bl£t + c1£2
aep. D1/~. = 0
P
(41b)
Combining equations (23), (41a) and (41b) for all plates except the bottom plate and equations (23) and (41a) for the bottom plate only the following set of equations is obtained
(34)
7N+1 = (1 + zs) N
On 2
_
(1 - E3 x. + ~ - £.-1
(35)
where
el~o
(42a)
a.£.-x + b.£. + c.£.+1 = dflN+l + e.~o
(42b)
a~£s_ 1 + bN£N + CN£N+I = duTN+1 + eN~o
(42c)
where 79
= dllN+l +
A. J. HAAGENSENand F. P. LEES flY0 -Jr glYl + hlY2 =jaTN+x + k19o
(43a)
a,, = - C, ~
El
=j.TN+I + k.0o
(46b)
fNYN-1 + gUYN + h~2N+l = Jflu+l +kN0o
(46c)
fnYn-1
U s ) ~fl+
b,=(G+
E,) + L + Ws
~fl Ll (1
"{- OnYn "dr h,,Y.+t
(46a)
where (43b)
f~ = - ( E + Ws) (1 -
)
G
(47a)
(43c)
c. = - ( G + Us) -~ (1 - E~) - E
g . = (L - + - Ws) +
-- Eg) E (1 - + G + Us
1
dn = (X,+I - X.) (1 + zs) N-"
-E hn =
e. = (Yn-1 - Yn)
(47c)
Eo[3
{exp[-x/(D0(n - 1)] + D2 exp[~(D,)(n - 1)]}
1
J, = (X.+I - X.) (1 + zs) N-"
1 +D 2
(43e) al = - G
k. = ( L - I
(43f) ×
bl = (G + Us) ~ + E, + Ws
X = 7N+tMI-aD + ~oM~- XE + ~oMx-lA + gN+ 1M~-1C where X = {xl .-. XN}
MI=
cl 0 b2 C2 a 3 b3
c3
0
0
a,
0
?
{exp[-x/(Oa)(n -- 1)] + D 2 exp[~/(D0(n - 1)3} 1 +D 2
(47e) (E + Ws) an--
b.
Cn
0
-
-
(47f)
-]- G "[- U S
(47g)
hN = - L (44)
The solution of these equations is the matrix equation Y = ~N+lM21J + ~oM21K + YoM21F
(45a)
0
(47d)
- Y.)
(43g)
The solution of these equations is the matrix equation
'bl a2 0
(47b)
(43d)
0 0 0 0 0 0 aN bN,
+ 2N+aM21H
(48)
where ¥ = {Y~ ... YN}
(49a)
gt fz
0 0 0 0 0 0 9~ (49b)
M2 = l i
th 0 g2 h2 0 .f3
g3
(45b)
ha o f.
v.
h.
dN}
(45c)
E = { e x ... eN}
(45d)
A={-at0
(45e)
J = {Jl ...JN}
(49c)
(45f)
K = {k1-.-kN}
(49d)
F = { - - f l 0 ... 0}
(49e)
H={O...O-hN}
(49f)
D=
{dl...
... 0}
C = {0 ... 0 - cN}
The responses of the outlet gas concentration to the four input variables are obtained in a similar manner. The following set of equations is obtained. 80
The frequency response of a plate gas absorption column The frequency responses are now obtained by making the substitution
Gas
~ffluent
Liquid
~ffluent
s = io2 2rr T
The response is obtained as a complex number The amplitude, and phase lag of the response are
(q + Jr).
Amplitude 5 = x/(q 2 q- r 2) Phase Lag~b = t a n - l ( ~ ) The normalized attenuation is then Normalized attention 0~ = &oo/5 EXPERIMENTAL WORK
o @-@-@
The experimental apparatus used was the plate gas absorption column shown schematically in Fig. 1. The column was constructed of standard QVF glass column sections 4 in. i.d. x 12 in. high. The plates were made of stainless steel and consisted of a disc ~ in. o.d. x -~ in. thick welded inside a thicker annulus 5¼ in. o.d. x ½ in. thick. A circular weir, a circular downcomer and two bubble-caps were located symmetrically on a 2¼ in. p.c.d. The weir and downcomer were 1 in. i.d. and the weir was 2 in. high. The riser was ½ in. i.d. x 1~ in. high and the caps were 1 in. o.d. x 13 in. high with 12 slots 1 in. high x 3/32 in. wide. Six identical plates were used, except that the downcomer of the bottom plate was luted to prevent passage of gas. The experimental absorption system was airammonia-water. The water used was towns water. There were on the inlet water line a hand control valve, a rotameter, a thermometer, an orifice plate and an automatic control valve. The gas used was a mixture of air and ammonia. There were on the air and ammonia lines needle valves and rotameters and on the inlet mixed gas line a thermometer, an orifice plate, an automatic control valve and an analysis point. There were on the outlet water line an electrical conductivity cell and an analysis point and on the outlet gas line an analysis point, an orifice plate and a thermometer. There were liquid
Water
Air Ammonia
FIG. 1. Experimental gas absorption column. FI 1-3 Rotameters FR 1-3 Orifice plates TI 1-3 Mercury-in-glass thermometers An 1-3 Sample points An 4 Conductivity cell and gas sample points on each plate. A small pool of liquid was maintained in the column base so that the conductivity cell ran full and the level of this liquid was maintained by a luted offtake. Sinusoidal signals were impressed on the inlet liquid and mixed gas control valves by a pneumatic sine-wave generator. Pressure transducers were used to convert the signals from the orifice plates and from the sine-wave generator to electrical signals. Signals from the orifice plates, sine-wave generator and conductivity cell were recorded on a multi-channel recorder which used ultra-violet sensitive paper. The column was operated in all experiments at the following mean conditions: Liquid flow 446 lb mole/ft2hr 81
A. J. HAAGENSEN and F. P. LEEs 0.¢
t O.E
/ ×
.a
x
0.7
× G = 4'061b © =6"14
E
= 7.5r
I
.J O'E
200
300
400
Liquid flow L Ib
FIG. 2.
mole / f t Z h r
6OO
5(~)0
m o l e / f t z hr
Liquid hold-up.
Gas flow 6.4 lb mole/ft2hr Gas composition 10 ~o v/v NHa Under these conditions the concentration of ammonia in both gas and liquid is low and absorption is substantially isothermal. The equilibrium relation is given by equation (7) with the following values of the constants: fl = 0.76 This value is taken from PERRY [9] and corresponds to 20°C, which is the temperature at which the experimental work was done. Ammonia is highly soluble in water and the equilibrium partial pressure of the gas above the liquid was small compared with the actual partial pressures of ammonia in the gas. Preliminary experiments were carried out at steady state to determine the liquid hold-up in the column, the pressure drop across the column and the plate efficiency. A graph of liquid hold-up versus liquid flow at the mean gas flow is given in Fig. 2. The liquid hold-up relation is given by equation (4) with the following values of the constants:
K4 = 0-0025 K 5 = 0"001 The plate efficiency, as defined by equation (8), was measured by analysis of the gas at the inlet and outlet of the column and also between plates. The mean value over the whole column was 64 per cent, although values up to 85 per cent were found on the bottom plate. The following frequency responses were obtained: Input
Output
Inlet liquid flow Inlet liquid flow Inlet gas flow
Outlet liquid flow Outlet liquid concentration Outlet liquid concentration
Table 1. Response of outlet liquid flow to inlet liquid flow Run A1 2 3 4 5 6 7 8 9 10
KI = 0.365 K2 = 0.0072 A graph of pressure drop versus gas flow at the mean liquid flow is given in Fig. 3. The pressure drop relation is given by equation (6) with the following values of constants: 82
Period of oscillation
Measured response
(hr)
a
¢
0"0655 0"0342 0"0340 0"0325 0"0292 0'0231 0-0200 0-0183 0"0162 0"0150
1"0 1"06 1"02 1'03 1"01 1"11 1-26 1"23 1"32 1.40
7 21 44 44 43 66 68 81 80 97
The frequency response of a plate gas absorption column 0.007
l.,,.o.oOXGzl~
o.
O.OOG
I ~X I
I
X /
f~"
I
0.005 ×
L ' 4 4 6 Ib mole/ftZhr
0"004
8
Gos flow 0 Ib m o l e / f t 2 hr
FIG. 3.
Pressure drop.
The amplitude of the sine wave impressed on the inlet liquid and gas flows was + 10 per cent of the mean value. The experimental results are expressed as a frequency response made up of a normalized attenuation and a phase lag. At steady state a unit change in the input signal results in a change in the output signal which is known as the potential value. When the input varies sinusoidally the amplitude of the output signal for an input signal of unit amplitude is attenuated. The ratio of the potential value to this attenuated amplitude is the normalized attenuation, which is therefore dimensionless and greater than unity. The response of the outlet liquid flow to the inlet
liquid flow is given in Table 1 and in Fig. 4. It was obtained by replacing the lute shown in Fig. 1 by a graduated measuring cylinder and by timing the rate of rise and fall of liquid in the cylinder with a constant offtake. The response of the outlet liquid concentration to inlet liquid and gas flows is given in Tables 2 and 3 and in Figs. 5 and 6. It was obtained using the conductivity cell to measure the outlet liquid concentration. It is necessary to correct the measured response for the lags associated with the pool of liquid in the base of the column. The transfer characteristics of this pool and of the pipe between the bottom of the
Table 2. Response of outlet liquid concentration to inlet liquid flow
Run B1 2 3 4 5 6 7 8 9 10
Period o f oscillation (hr)
Measured response ~¢
~
0"0433 0"0375 0"0283 0"0230 0"0206 0"0181 0"0144 0"00945 0"0086 0"0075
1"23 1"27 1"50 1"67 1'81 1'95 2"74 3"55 4-8 5"7
64 77 97 107 135 145 177 225 256 272
Correction ~ ~ 1"0 1"0 1"0 1"01 1"01 1"01 1"02 1"04 1"05 1"07
83
9 10 13 16 18 21 26 38 43 48
Corrected response ~ 1"23 1"27 1"50 1"66 1'80 1"93 2"68 3"40 4"56 5"30
55 67 84 91 117 124 151 187 213 224
A. J. HAAGENSEN and F. P. LEES
Table 3. Reponse o f outlet liquid concentration to inlet 9as flow Period of oscillation
Run
C1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Measured response
Corrected response
Correction
(hr)
~
~
~
0"0628 0-0358 0.0340 0"0220 0-0183 0"0153 0"012 0.011 0'0092 0-0083 0.0081 0-0072 0-0064 0"0064 0"0058 0-0050
1"04 1.25 1-25 1"34 1"48 1"45 1"90 1"54 2"00 2.57 2.38 2.52 3'00 3'60 3"26 3"42
5 23 29 47 64 63 83 93 97 85 114 118 132 129 142 141
1"0 1"0 1"0 1"01 1'01 1 "02 1-03 1"03 1 '05 1"06 1"06 1 "07 1 '09 1"09 1"11 1"15
5 10 11 17 21 24 31 33 40 44 45 50 57 57 62 71
360,
,
1-04 1.25 1"25 1"33 1"47 1"42 1"85 1"50 1"90 2"43 2"25 2.36 2"75 3"30 2.94 2'97
0 13 18 30 43 39 52 60 57 41 69 68 75 72 80 70
,
300
240
"--~
180
~
120
G.
6O
i
oI 2~
t
i 2c --•.i-
e 7
g
~
o
Z Z I.
Period
of
oscillation,
hr
0.001
0-01 Period
FXG. 4. Response of outlet liquid flow to inlet liquid flow. Crosses are experimental and full lines theoretical values.
of
oscillation,
0.1 hr
~.
FIG. 5. Response of outlet liquid concentration to inlet liquid flow. Crosses are experimental and full lines theoretical values. 84
The frequency response of a plate gas absorption column
Table 4. Constants for calculation of theoretical responses N E d K1 //2 Ka /(4 K5 fl Eg
6 446 6"4 0-365 0.0072 0"00218 0'0025 0"001 0"76 0"64 0.10
?o
W Xz
Xz X3 X4 X5 X'o
0-785 0"00143 0"000523 0"000190 0.0000674 0-0000225 0.00000605
OYz
?a 73 ?4 ?5 ?8
0.00218 0-0367 0-0135 0"00494 0"00181 0"000663 0"000242
X~+I 0 column and the conductivity cell were obtained by injecting an impulse of 5 per cent sodium hydroxide into the liquid entering the pool from the bottom plate and analysing the response of the conductivity cell. The lags were found to correspond to a transfer lag of 1.6 see and a distance-velocity lag of 2.1 sec. The measured responses have been corrected for these lags as shown in Tables 2 and 3.
The corresponding theoretical responses were calculated from the equations given in the previous Section using the data given in Table 4. The first column of this table gives the basic numerical constants for the experiments. The second and third columns give the calculated liquid and gas hold-ups and the liquid and gas concentrations on each plate, calculated from equations (4), (5) and (11-19c).
360 l
CONCLUSION
00
There is fair agreement between the theoretical and experimental results for the three responses investigated. It was originally intended to carry out experimental work to check the theory at different gas-liquid ratios and with different solute gases, but the work was interrupted before this could be done. The effect of plate efficiency for the experimental absorption system was investigated using the theoretical model. The normalized attenuation and phase lag decrease with increase in plate efficiency. In the six-plate column, at a period 0.01 hr, the normalized attenuation and the phase lag are 3.19 and 162° respectively for a plate efficiency 64 per cent and 2.52 and 159° respectively for a plate efficiency 85 per cent. The application of the model is limited by the assumptions made in its derivation, of which the most important are that the absorption is isothermal, that the equilibrium relation is linear and that the concentration of solute in the inlet gas is small, but these assumptions are applicable in many absorption systems.
"0- 24C
180
120
60
0
t g
o B 7
"6
"t. I
E 7£_ I"
0.001
0-OI
Period of oscillation,
0.1
hr
=~
FIG. 6. Response of outlet liquid concentration to inlet gas flow. Crosses are experimental and full lines theoretical values.
85
A. J. HAAGENSENand F. P. LEES Aclmowledgements--Thanks are due to Mr. A. J. YOUNG, head of the Central Instrument Laboratory, Imperial Chemical Industries Ltd., to Mr. J. McMmLAN for his advice throughout the work and to Mr. R. Rowe, who carried out many of the experiments and calculations, and to other members of the laboratory staff who helped in many ways.
NOTATION a-j a-h Dl-a Eg Et G i Kl-e L n
Constants Constants Complex constants Plate efficiency (gas phase) Plate efficiency (liquid phase) Gas flow lb mole/ft~hr =a/--1 Constants Liquid flow lb mole/ft~hr Number of plates from bottom of column N Number of plates in column P Pressure atm s Laplace operator t Time hr T Period of sine wave hr U Gas hold-up lb mole/ft 2 plate I4/ Liquid hold-up lb mole/ft 2 plate
X Liquid solute concentration Y Gas solute concentration Y* Equilibrium gas solute concentration corresponding to liquid solute concentration X ct Normalized attenuation (defined in tex0 Constant Amplitude of sine wave 7 Time constant for transmission of liquid flow across a plate Phase lag of sine wave to = 2¢r/T
mole fraction mole fraction mole fraction
hr degrees hr -1
Subscripts 0-6, n and N are used to denote plate number. The numbering is from the bottom of the column. Total values of time-dependent variables are represented by a capital letter, steady-state values by a capital letter with a bar above, small tmsteady-state variations about the steadystate value by a small letter and Laplace transforms of the unsteady-state variations by a small letter with a tilde above it: G=~+g .~q) = The Laplace transform is defined as follows: c~o
~ [ f ( t ) ] = S exp(--st)f(t)dt 0
REFERENCES
[1 ] MARSHALLW. R. and I~GFORDR. L., Applications of Differential Equations to Chemical Engineering Problems. University of Delaware Press, 1947. [2] LAPmus L. and AMUr,a3SONN. R., lndustr. Engn# Chem. 1950 42 1071. [3] CEA~LSKEN. H., Proc. 1st Int. Congr. lnt. Federat. Automatic Control Moscow 1960. Vol. IV, p. 288. Butterworths, London, 1961. [4] GRAYR. I. and PRADOSJ. W., A.LC.E. Jl. 1963 9 211. [5] CALVERTS. and COUL~iANG. A., Chem. En#n# Pro#r. Symp. Ser. 46. S p.9. [6] ROGERSM. C. and Trilra_~ E. W., Industr. En#ng Chem. 1934 26 524. [7] CROSSG. A. and RYDER H., J. AppL Chem. 1952 2 51. [8] SHERWOODT. K. and PXGFORDR. L., Absorption and Extraction. McGraw-Hill, New York, 1952. [9] PERRYJ. H. (Ed.), Chemical Engineers Handbook (3rd Ed.). McGraw-Hill, New York, 1950. R~sum6---Les auteurs pr6sentent un mod~le th6orique du comportement transitoire d'une colonne d'absorption ~t plateaux et 6tablissent la r6ponse fr6quentielle th6orique. L'article contient encore des r6ponses fr6quentielles exp6rimentales obtenues avec le syst~me air-ammoniaque sur des colonnes d'absorption d'un diam6tre de 4 in., comprenant 6 plateaux. Zusammenfassung--Es wird ein theoretisches Modell far das nichtstation~ire Verhalten einer Absorptionskolonne mit B6den aufgesteUt und die theoretische Antwortfunktion fiar ein eingegebenes sinusf6rmiges Signal (frequency response-Methode) aufgestellt. Diese Funktionen werden mit experimentellen Werten aus einer 4 inch Absorptionskolonne mit 6 B6den ffir das System Ammoniak-Luft und Wasser verglichen.
86