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Synthesis of Reactive Mass-Exchange Networks
CHAPTER EIGHT
Syn thesis of Rea ctive Mass-Exchange Networks
Chapters Three, Five and Six have covered the synthesis of physical mass-exchange networks. In these systems, the targeted species were transferred from the rich phase to the lean phase in an intact molecular form. In some cases, it may be advantageous to convert the transferred species into other compounds using reactive MSAs. Typically, reactive MSAs have a greater capacity and selectivity to remove an undesirable component than physical MSAs. Furthermore, since they react with the undesirable species, it may be possible to convert pollutants into other species that may either be reused within the plant itself or sold. The synthesis of a network of reactive mass exchangers involves the same challenges described in synthesizing physical MENs. The problem is further compounded by virtue of the reactivity of the MSAs. Driven by the need to address this important problem, Srinivas and E1-Halwagi introduced the problem of synthesizing reactive mass-exchange networks REAMENs and developed systematic techniques for its solution (Srinivas and E1-Halwagi, 1994; E1-Halwagi and Srinivas, 1992). This chapter provides the basic principles of synthesizing REAMENs. The necessary thermodynamic concepts are covered. Chemical equilibrium is then tackled in a manner that renders the REAMEN synthesis task close to the MEN problem. Finally, an optimization-based approach is presented and illustrated by a case study.
8.1 Objectives of REAMEN Synthesis The problem of synthesizing REAMENs can be stated as follows (E1-Halwagi and Srinivas, 1992): 191
192
CHAPTER EIGHT Synthesis of reactive mass-exchange networks
Physical I Reactive MSAa In
11111 REACTIVE MASS-EXCHANGE NETWORK
Waste (Rich) Streams (Sources) In
v
.~ ~
Waste (Rich) Streams (Sources) Out
,
11111
Physical/Reactive MSAa Out F i g u r e 8 . 1 Schematic representation of the REAMEN synthesis problem.
Given a number NR of waste (rich) streams and a number Ns of lean streams (physical and reactive MSAs), it is desired to synthesize a cost-effective network of physical and/or reactive mass exchangers which can preferentially transfer a certain undesirable species, A, from the waste streams to the MSAs whereby it may be reacted into other species. Given also are the flowrate of each waste stream, G i, its supply (inlet) composition, y~, and target (outlet) composition, y~, where t are i = 1, 2, .. ., NR. In addition, the supply and target compositions, xjs and x j, given for each MSA, where j = 1, 2 . . . . . Ns. The flowrate of any lean stream, L j, is unknown but is bounded by a given maximum available flowrate of that stream, i.e., L j < L jc.
(8.1)
Figure 8.1 is a schematic illustration of the REAMEN synthesis problem. As has been previously discussed in the synthesis of physical MENs, several design decisions are to be made: 9 Which mass-exchange operations should be used (e.g., absorption, adsorption, etc.)? 9 Which MSAs should be selected (e.g., physical/reactive transfer, which solvents, adsorbents, etc.)? 9 What is the optimal flowrate of each MSA? 9 How should these MSAs be matched with the waste streams (i.e., stream pairings)? 9 What is the optimal system configuration (e.g., how should these mass exchangers be arranged?, is there any stream splitting and mixing?, etc.)? The first step in synthesizing a REAMEN is to establish the conditions for which the reactive mass exchange is thermodynamically feasible. This issue is covered by the next section.
8.2 Corresponding composition scales for reactive mass exchange
193
8.2 Corresponding Composition Scales for Reactive Mass Exchange The fundamentals of reactive mass exchange, design of individual units, chemical equilibrium and kinetics are covered in the literature (e.g., Fogler, 1992; Friedly, 1991; E1-Halwagi, 1990; Kohl and Reisenfeld, 1985; Westerterp et al., 1984; Astarita et al., 1983; Smith and Missen, 1982; Espenson, 1981; Levenspiel, 1972; E1-Halwagi, 1971). This section presents the salient basics of these systems. In order to establish the conditions for thermodynamic feasibility of reactive mass exchange, it is necessary to invoke the basic principles of mass transfer with chemical reactions. Consider a lean phase j that contains a set Bj = {Bz,j I z = 1. . . . . NZj } of reactive species (i.e., the set Bj contains NZj reactive species, each denoted by Bz, j, where the index z assumes values from 1 to NZj). These species react with the transferable key solute, A, or among themselves via Qj independent chemical reactions which may be represented by NZj
A + Z
(8.2)
1)l'z'J Bz'J - - 0
z=l
and NZj
~-'~ l)qj,z,jBz, j -- 0
qj : 2, 3 . . . . . Q j,
(8.3)
Z--1
where the stoichiometric coefficients 1)q,z, j a r e negative for products and positive for the reactants. It is worth noting that stoichiometric equations can be mathematically handled as algebraic equations. Therefore, although component A may be involved in more than the first reaction, one can always manipulate the stoichiometric equations algebraically to keep A in the first reaction and eliminate it from the other stoichiometric equations. Compositions of the different species can be tracked by relating them to the extents of the reactions through the following expression Qj
bz,j
b~z , j - Z
l)qj ,z, j~qj
(z=l
'
2'
"""'
NZj) '
(8.4)
qj --2
where bz, j is the composition of species Bz, j in the jth lean phase, b~ is the admissible composition of species Bz,j in the jth lean phase and ~qj is the extent of the qjth reaction (or the qjth reaction coordinate). The extent of reaction is defined for reactions two through Qj. The reason for not defining it for the first reaction is that the variable uj plays indirectly the role of the extent of reaction for
194
CHAPTER EIGHT Synthesis of reactive mass-exchange networks
the first reaction. 1 The admissible compositions may be selected as the lean-phase composition at some particular instant of time, or any other situation which is compatible with stoichiometry and mass-balance bounds. The equilibrium constant of a reaction is the product of compositions of reactants and products each raised to its stoichiometric coefficient. Hence, for the reaction described by Eq. (8.2) one may write
1 NZj ( Qj ) gl,J = ~j ~ bz,J - Z Pqj,z,j~qj z-1 ~k qj=2
Pl,z,j
i.e.,
aj
o
bz,j
__ ~ 1
-
z-1
and for the reactions given by Eq. (8.3):
Nzj( oj
gqj,j -- 1-I z=l
1)qj
,z,j~qj
)
Vl,z,j (8.5)
qj=2
)vqjzj
bz, J - Z VqJ'z'J~qJ qj=2
qj = 2, 3 . . . . . Qj,
(8.6)
where Kqj,j is the equilibrium constant for the qjth reaction and aj is the composition of the physically dissolved A in lean phase j. It is now useful to recall the concepts of molarity and fractional saturation (Astarita et al., 1983). The molarity, m j, of a reactive MSA is the total equivalent concentration of species that may react with component A. On the other hand, the fractional saturation, u j, is a variable that represents the degree of saturation of chemically combined A in the j th lean phase. Therefore, uj mj is the total concentration of chemically combined A in the jth MSA. Hence, the total concentration of A in MSA j can be expressed as
xj = aj + u j m j ,
j ~ S,
(8.7)
where the physically-dissolved concentration of A, a j, equilibrates with the richphase composition through a distribution function Fj, i.e.,
y* = Fj (aj ),
j r S.
(8.8)
Equations (8.4)-(8.8) represent a complete mathematical description of the chemical equilibrium between a rich phase and the jth MSA. The simultaneous solution 1For this reason, whenever there is a single reaction taking place in the jth MSA, no extent of reaction is defined. Insteadthe fractional saturation, uj, is employed.
Example 8.1. Absorption of H2S in aqueous sodium hydroxide
195
of these nonlinear equations (for instance by using the software LINGO) yields the equilibrium compositions of both phase in the form y;=fj(x;)
jrS.
(8.9)
For any mass-exchange operation to be thermodynamically feasible, the following conditions must be satisfied: xj < x j ,
j ~ S,
(8.10a)
Yi > Y*,
i e R,
(8.10b)
and/or
i.e., y=
f (xj + e ~ ) ,
j r S,
(8.11a)
where Y --- Yi -- eiR, i ~ R, (8.1 lb) s where e]~ and e j are positive quantities called, respectively, the rich and the lean minimum allowable composition differences. The parameters e/R and e js are optimizable quantities that can be used for trading off capital versus operating costs (see Chapters Two and Three). Equations (8.1 l a) and (8.1 l b) provides a correspondence among the rich and the lean composition scales for which mass exchange is practically feasible. This is the reactive equivalent to Eq. (3.5) used for establishing the corresponding composition scales for physical MENs.
Example 8.1. Absorption of HzS in Aqueous Sodium Hydroxide Consider an aqueous caustic soda solution whose molarity m l - - 5 . 0 k m o l / m 3 (20 wt.% NaOH). This solution is to be used in absorbing HES from a gaseous waste. The operating range of interest is 0.0 _< Xl (kmol/m 3) < 5.0. Derive an equilibrium relation for this chemical absorption over the operating range of interest.
Solution The absorption of H2S in this solution is accompanied by the following chemical reactions (Astarita and Gioia, 1964): H2S + NaOH = NariS + H20
(8.12)
H2S -t- 2NaOH = Na2S + 2H20.
(8.13)
196
CHAPTER EIGHT Synthesis of reactive mass-exchange networks
As has been described earlier, the stoichiometric reactions should be manipulated algebraically to retain the transferable species (H2S) only in the first equation. Therefore, H2S can be eliminated from Eq. (8.13) by subtracting (8.12) from (8.13) to get NariS + NaOH = Na2S + H20.
(8.14)
Equations (8.12) and (8.14) can be written in ionic terms as follows: H2S q- OH- = HS- + H20
(8.15)
HS- + OH- = S 2- + H20.
(8.16)
Let us denote the three ionic species as follows: OH- = BI,1 HS- = B2,1 S 2- -- B3,1,
with aqueous-phase concentrations referred to as bl,1, b2,1 and b3,1, respectively. Also, let us denote the composition of the physically dissolved H2S in the aqueous solution as al. For cases when the concentration of water remains almost constant with respect to the other species, one can define the following reaction equilibrium constants for Eqs. (8.15) and (8.16), respectively: g l , 1 "-
b2,1 a l "bl,1
(8.17)
and b3,1 K2,1 -- ~ , b2,1 9 bl,1
(8.18)
where KI,1 = 9.0 • 106 m3/kmol and K2,1 = 0.12 m3/kmol. The distribution coefficient for the physically dissolved H2S is given by yl = 0.368,
(8.19)
al
where yl is the composition of H2S in the gaseous stream. It is useful to relate the molarity of the aqueous caustic soda (ml = 5.0 kmol NaOH/m 3) to the other reactive species. Once the reactions start, the composition of NaOH will decrease. However, it is possible to relate the molarity of the solution to the concentration of the reactive species at any reaction coordinate. Suppose that after a certain extent of reaction (8.15) and (8.16) an analyzer is placed in the solution to measure the compositions of OH-, HS- and S 2- with the measured
Example 8.1: Absorption of H2S in aqueous sodium hydroxide
197
concentrations denoted by bl,1, b2,1 and b3,1, respectively. These measured concentrations are related to the molarity as follows. According to Eq. (8.15), b2,1 kmoles of OH- must have reacted to yield b2,1 kmoles of HS-. Similarly, according to Eq. (8.16), b3,1 kmoles of OH- must have reacted with b3,1 kmoles of H S to yield b3,1 kmoles of S 2-. But the b3,1 kmoles of HS- must have resulted from the reaction of b3,1 kmoles of OH- (according to Eq. (8.15)). Hence, 2b3,1 kmoles of O H - are consumed in producing b3,1 kmoles of S 2-. Therefore, the molarity of the aqueous caustic soda can be related to the concentrations of the reactive ions as follows: 5 -- bl,1 -k- b2,1 q'- 2b3,1.
(8.20)
There are two forms ofreacted HzS in the solution: HS- and S 2-. By recalling that U lml is the total concentration of chemically combined HzS in the aqueous caustic soda and conducting an atomic balance on S over Eqs. (8.15) and (8.16), we get 5Ul -- b2,1 "+" b3,1.
(8.21)
As discussed earlier, the admissible compositions may be selected as the leanphase composition at some particular instant of time, or any other situation which is compatible with stoichiometry and mass-balance bounds such as Eqs. (8.20) and (8.21). Let us arbitrarily select the admissible composition of S 2- to be zero, i.e., b~ = 0. This selection automatically fixes the corresponding values of b~ and b~ cording to Eqs. (8.20) and (8.22), we get b 2,1 ~ = 5u 1.
(8.22) Ac-
(8.23)
Similarly, according to Eqs. (8.21)-(8.23), we obtain: b~ = 5(1 - Ul).
(8.24)
As mentioned earlier, the extent of reaction is defined for all reactions except the first one. Hence, we define ~2 as the extent of reaction for Eq. (8.16). Equation (8.4) can now be used to describe the compositions of the reactive species as a function of ~2 and the admissible compositions, i.e., bl,1 "- 5(1 - ul) - ~2
(8.25)
b2,1 -" 5Ul - ~2
(8.26)
b3,1 -" ~2-
(8.27)
198
CHAPTER EIGHT Synthesis of reactive mass-exchange networks
Substituting from Eqs. (8.19), (8.25), (8.26) and (8.27) into Eq. (8.17), we get 9 . 0 x 106--
5Ul -- ~2
Yl ~[5(1 0.368
-- Ul) -- ~2]
or Yl - - 4 . 0 9 X 10 -8
5Ul - ~2 . [5(1 -- Ul) -- ~2]
(8.28)
Substituting from Eqs. (8.25)-(8.27) into Eq. (8.18), we obtain 0.12=
~2
[5Ul - ~2][5(1 - ul) - ~z]'
which can be rearranged to ~2 _ 13.33 ~2 "!- 25Ul (1 -- Ul) = 0.
(8.29)
According to Eq. (8.7), the total concentration of H2S in the aqueous caustic soda (physically dissolved and chemically reacted) can be expressed as follows Xl = al + 5Ul.
(8.30a)
In this system, the physically dissolved H2S is much less than the chemically combined, 2 i.e., al << 5Ul,
(8.30b)
X1 -- 5Ul.
(8.30c)
which simplifies Eq. (8.30a) to
Equations (8.28), (8.29), and (8.30c) can be used to develop an expression for reactive mass exchange of H2S. First, a value of fractional saturation is selected (where xS/5 < ul _< x]/5). Then, Eq. (8.30c) is used to calculate the corresponding xl. Next, Eq. (8.29) is solved to determine the value of ~2. Finally, Eq. (8.28) is solved to evaluate Yl. The pair (yl, Xl) are in equilibrium. The same procedure is repeated for several values of u 1 (between x s/5 and x]/5) to yield pairs of (yl, x i) which are in equilibrium. Nonlinear regression can be employed to derive an equilibrium expression for these pairs. To illustrate the above-mentioned procedure, let us start with a fractional saturation oful = 0.1. According to Eq. (8.30c), xl = 0.5 kmol/m 3. By substituting for Ul = 0.1 in Eq. (8.29), we obtain: ~2 _ 13.33~2 + 2.25 = 0, 2This assumption can be numerically verified by comparing values of aj with ujmj after the equilibrium equation is generated.
8.3 Synthesis approach
199
Figure 8.2 Equilibrium data for the example of absorbing H2S in caustic soda.
which can be solved to get ~2---~
13.33 - ~/13.33 • 13.33 - 4 2
• 2.25
= 0.171 kmol/m 3 (the other root is rejected)
(8.31)
The values o f u l and ~2 are plugged into Eq. (8.28) to get yl - 3.1 x 10 -9 kmol]m 3. Hence, the pair (3.1 x 10 -9, 0.5) are in equilibrium. 3 The same procedure is repeated for values of u l between 0.0 and 0.1. The results are plotted as shown in Fig. 8.2. The plotted data are slightly convex and can be fitted to the following quadratic function (all compositions in kmol/m3): Yl = 2.364 X 10-9x 2 + 5.022 x 10-9xl
(8.32)
with a correlation coefficient, r 2, of 0.9999.
8.3 Synthesis Approach Now that a procedure for establishing the corresponding composition scales for the rich-lean pairs of stream has been outlined, it is possible to develop the CID. The CID is constructed in a manner similar to that described in Chapter Five. However, it should be noted that the conversion among the corresponding composition scales may be more laborious due to the nonlinearity of equilibrium relations. Furthermore, a lean scale, x j , represents all forms (physically dissolved and chemically combined) of the pollutant. First, a composition scale, y, for component A in 3We can now test the validity of Eq. (8.30b). According to Eq. (8.19), al = 8.45 x 10-9 kmol/m3 which is indeed much less than u l m l = 0.5.
200
CHAPTER EIGHT Synthesis of reactive mass-exchange networks
any rich stream is created. This scale is in a one-to-one correspondence with any composition scale of component A in the ith rich stream, yi, via Eq. (8.1 lb). Then, Eq. (8.1 la) is used to generate Ns composition scales for component A in the lean streams. Next, each stream is represented by an arrow whose tail and head correspond to the supply and target compositions, respectively, of the stream. The partitions corresponding to these heads and tails establish the composition intervals. Similar to the CID for physical MENs, within any composition interval, it is thermodynamically feasible to transfer component A from the rich streams to the lean streams. Also, according to the second law of thermodynamics it is spontaneously possible to transfer component A from the rich streams in a given composition interval to any lean stream within a lower composition interval. It is worth noting that the foregoing composition partitioning procedure ensures thermodynamic feasibility only when all the equilibrium relations described by Eq. (8.9) are convex. In this case by merely satisfying Eq. (8.11) at both ends of a composition interval, Eqs. (8.10) are automatically satisfied throughout that interval (Fig. 8.3a). On the other hand, when at least one of the equilibrium relations expressed by Eq. (8.9) is non-convex, the satisfaction of Eq. (8.11) at both ends of an interval does not necessarily imply the realization of inequalities (Eqs. 8.10) throughout that interval (Fig. 8.3b). In such case, additional composition partitioning is needed. This can be achieved by discretizing the non-convex portions of the equilibrium curves through linear Yi
Y i,k
Y i,k
~E s I I ] i I t I
operating line
J
,/ I equilibrium line Y i,k+!
Y i~§
[
-
I Es t I I X j~,~
J
. !
t I I!
X j,k+}
I I I I I I I I I I i X j~,
Xj~,
Xj
8.3a A reactive massexchangerwith convexequilibrium(EI-Halwagiand Srinivas, Synthesis of reactive mass-exchangenetworks, Chem. Eng. Sci., 47(8), p. 2115, Copyright9 1992,withkind permission from Elsevier Science Ltd., The Boulevard, LangfordLane, Kidlington0X5 1GB,UK). Figure
8.3 Synthesis approach
201
Yi
Yi,k
v~k
~ - - /
'
equilibrium line
Yi,k+l Yi,k+ t
F i g u r e 8 . 3 b A reactive mass exchanger with non-convex equilibrium (E1-Halwagi and Sfinivas, Synthesis of reactive mass-exchange networks, Chem. Eng. Sci., 47(8), p. 2115, Copyright 9 1992, with kind permission from Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington 0X5 1GB, UK). Yi
Yi,a Yi,b
2k
__--~---__------. equilibrium line
vi.c
Yi,k+1 i
i~ _ J
~
I I
Xj~+ l
Ej
s
I I I
Xj,a
I I I I
X j,b
,I I
Xjs
I I
Xj~
Xj
Figure 8.3c A reactive mass exchanger with discretization of non-convex equilibrium (E1-Halwagi and Srinivas, Synthesis of reactive mass-exchange networks, Chem. Eng. Sci., 47(8), p. 2115, Copyfight 9 1992, with kind permission from Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington 0X5 1GB, UK).
202
CHAPTER EIGHT Synthesis of reactive m a s s - e x c h a n g e networks
overestimators. 4 Clearly, the number of these linear segments depends upon the desired degree of accuracy. For instance, Fig. 8.3b may be convexified by introducing two linear parts at the concavity inflection point (Fig. 8.3c). Consequently, additional intervals will be created within the CID by partitioning the composition scales at the locations corresponding to points a, b and c. The end result of this partitioning procedure is that the entire composition range is divided into nint composition intervals, with k = 1 being the highest and k =/'tint being the lowest. Having established a one-to-one thermodynamically-feasible correspondence among all the composition scales, we can now solve the REAMEN problem via a transshipment formulation similar to that described in Chapter Five for the synthesis of physical MENs. Example 8.2. Removal Pulping Process
o f H2S f r o m a K r a f t
Kraft pulping is a common process in the paper industry. Figure 8.4 shows a simplified flowsheet of the process. In this process, wood chips are reacted (cooked) with white liquor in a digester. White liquor (which contains primarily NaOH, Na2S, Na2CO3 and water) is employed to dissolve lignin from the wood chips. The cooked pulp and liquor are passed to a blow tank where the pulp is separated from the spent liquor "weak black liquor" which is fed to a recovery system for ToAtmosphere *
Aq.NaOH, $1
I
REAMEN it
El Activated Carbon,S3
~ I
I I
! /
S2, ToRegenerationand Recycle Wood C;ips
~ Gaseous Waste, Ri
/
LW ~ 9~Z
Steam
Digester
v I
r
Steam
Pulpto Further
,
eak BlackLiquor
~
Gases
Blow Tank
] I
"-1 "-I
Washers
I~
Slaking and Caustiizing
Condensate Flue Gas Recovery Furnace
Lime
Smelt
Kiln
GOa~
;Water
Lime
Calcium Carbonate
o
Settling and Filtration
Figure 8.4 Kraftpulping process. 4A more rigorous method of tackling nonconvex equilibrium without discretization has been developed by Srinivas and E1-Halwagi (1994) and is beyond the scope of this book.
Example 8.2: Removal of H2S from a Kraft pulping process
203
Table 8.1 Data for the Gaseous Emission of the Kraft Pulping Process
Stream R1
Flowrate Gi, m3/s
Description Gaseous waste from evaporators
Target composition (10-1~ kmol/m3) y[
Supply composition (10-1~ kmol/m3) yS
16.2
1,600
3.0
conversion to white liquor. The first step in recovery is concentration of the weak liquor via multiple effect evaporators. The concentrated solution is sprayed in a furnace. The smelt from the furnace is dissolved in water to form green liquor which is reacted with lime (CaO) to produce white liquor and calcium carbonate "mud". The recovered white liquor is mixed with make-up materials and recycled to the digester. The calcium carbonate mud is thermally decomposed in a kiln to produce lime which is used in the causticizing reaction. There are several gaseous wastes emitted from the process (see Dunn and E1-Halwagi, 1993, and Homework Problem 8.5). In this example, we focus on the gaseous waste leaving the multiple effect evaporators, R1, whose primary pollutant is HzS. Stream data for this waste stream are given in Table 8.1. A rich-phase minimum allowable composition difference, e/R, of 1.5 x 10 -1~ kmol/m 3 is used. A process lean stream and an external MSA are considered for removing H2S. The process lean stream, S 1, is a caustic soda solution which can be used as a solvent for the reactive separation of HES. An added bonus for using the process MSA is the conversion of a portion of the absorbed HES into Na2S, which is needed for whiteliquor makeup. In other words, HES "the pollutant" is converted into a valuable chemical which is needed in the process. The external MSA, $2, is a polymeric adsorbent. The data for the candidate MSAs are given in Table 8.2. The equilibrium
Table 8.2 Data for the MSAs of the Kraft Pulping Example
Stream
Upper bound on flowrate LjC m3/s
Supply composition (kmol/m3) xs.J
,,,, ,,
Target composition (kmol/m3) xt.j .
Sl $2
0.01 oo
0.000 0.010
.
0.500 0.025
.
. es. (kmol/m3) J
.
.
.
0.10 0.01
Cj $/m3 MSA
.
2,900
204
CHAPTER EIGHT Synthesis of reactive mass-exchange networks
data for the transfer of H2S from the waste stream to the adsorbent is given by Yl = 2 x 10-9x2,
(8.33)
where Yl and x2 are given in kmol/m 3. For the given data, determine the minimum operating cost of the REAMEN and construct a network with the minimum number of exchangers.
Solution The following expression for the equilibrium data of H2S in caustic soda has been derived in Eq. (8.32) of Example 8.1 (all compositions are in kmol/m3): yl = 2.364 x 10 -9 + 5.022 • 10-9Xl.
(8.32)
By invoking Eq. (8.11), we get the following equation for the practical-feasibility curve: yl - 1.5 x 10 -1~ = 2.364 x 10-9(Xl + 0.1) 2 + 5.022 • 10-9(Xl + 0.1).
(8.34)
Similarly, for activated carbon Yl -- 1.5 • 10 -10 = 2 x 10-9(x2 d-0.01).
(8.35)
The CID for the problem is shown in Fig. 8.5. The table of exchangeable loads for the waste streams and the MSAs are shown by Tables 8.3 and 8.4, respectively. According to the two-stage targeting procedure, we first minimize the annual operating cost of the MSAs (given by 3600 x 8760 • 2100 x L2 ---- 6.62256x 101~ L2 when we assume 8760 operating hours per year). By applying the linear-
Figure 8.5 CID for Kraft pulping example.
Example 8.2: Removal of HES from a Kraft pulping p r o c e s s
205
Table 8.3 The TEL for the Gaseous Waste
Interval
Load of R1 (10 -10 kmol H2S/s)
1 2 3 4 5
25,270 559 62 0 0
p r o g r a m m i n g f o r m u l a t i o n d e s c r i b e d in C h a p t e r Six (P6.1), o n e can write the following optimization program: m i n 6 . 6 2 2 5 6 • 1010 L2, subject to 31 - - 2 5 , 2 7 0 x 10 -1~
32-
31 4- 0.5L1 -- 559 x 10 -10 33 - 3 2 34-
= 62 x 10 -1~
33 -- 0.0
- 3 4 4- 0 . 0 1 5 L 2 = 0.0 6k > 0
Lj>O
k--1,2,3,4 j=l,2
L1 < 0.01 x 10 -1~ .
Table 8.4 The TEL for the Lean S t r e a m s Capacity of lean streams per m 3 of MSA (kmol H2S/m 3 MSA) Interval
S1
$2
2 3 4 5
0.5 -
0.015
(P8.1)
206
CHAPTER EIGHT Synthesis of reactive mass-exchange networks
It is worth pointing out that the wide range of coefficients may cause computational problems for the optimization software. This is commonly referred to as the "scaling" problem. One way of circumventing this problem is to define scaled flowrates of MSAs in units of 10 -1~ m3/s and scaled residual loads in units of 10 -1~ kmol/s, i.e., let
Lscaled j --
1010L j,
skcaled-"
10108k,
j = 1, 2 k = 1, 2, 3, 4
(8.36) (8.37)
With the new units, the scaled program becomes: min 6.62256
LS2caled
(P8.2)
subject to ~caled = 25,270 ~caled _ ~caled .+_ 0.5L~caled
__ 559
S~ caled -- S~ caled --" 62 ~caled _ ~caled = 0.0 _ ~caled _~_ 0.015 L ~caled = 0.0
~caled>0
k=1,2,3,4
LjScaled __> 0
j = 1, 2
L~caled < 0.01 In terms of LINGO input, the scaled program can be written as follows (with the S in each variable indicating that it is a scaled variable): model: min
=
6.62256-LS2;
deltaSl
=
25270;
deltaS2
-
deltaSl
+
0.5.LSI
deltaS3
-
deltaS2
=
62;
deltaS4
-
deltaS3
=
0.0;
-
deltaS4
+
0.015-LS2
deltaSl
>=
0.0;
deltaS2
>=
0.0;
deltaS3
>=
0.0;
deltaS4
>=
0.0;
LSI
>=
0.0;
LS2
>=
0.0;
end
=
0.0;
=
559;
Problems
207
The solution report from LINGO gives the following results: Objective
value-
Variable LS2 DELTAS1 DELTAS2 LSI DELTAS3 DELTAS4
27373 Value 4133.333 25270.00 0.0000000E+00 51658.00 62.00000 62.00000
Therefore, the minimum operating cost is approximately $27,000/year and the pinch location is between the second and third composition intervals. The REAMEN involves two exchangers; one above the pinch matching R1 with S1 and one below the pinch matching R1 with $2.
Problems 8.1 Derive an equilibrium expression for the reactive absorption of H2S in diethanolamine "DEA". The molarity of DEA is 2 kmol/m3. The following reaction takes place: H2S + (C2Hs)2NH = HS- + (C2Hs)2NH+, whose equilibrium constant is given by (Lal et al., 1985): K = 234.48 =
[HS-][(C2Hs)2[NH~-] [H2S][(C2Hs)2NH]
(8.38)
with all concentrations in kmol/m3. The physical distribution coefficient is: 0.363 =
Composition of H2S in gas (kmol/m3) Composition of physically-dissolved H2S in DEA (kmol]m 3)
(8.39)
8.2 Develop equilibrium equations for the reactive absorption of CO2 into: (a) Aqueous potassium carbonate (b) Monoethanolamine
(Hint: See pp.
68-79 of Astarita et al., 1983.)
8.3 Coal may be catalytically hydrogenated to yield liquid transportation fuels. A simplified process flow diagram of a coal-liquefaction process is shown in Fig. 8.6. Coal is mixed with organic solvents to form a slurry which is reacted with hydrogen. The reaction products are fractionated into several transportation fuels. Hydrogen sulfide is among the primary gaseous pollutants of the process (Warren et al., 1995). Hence, it is desired to design a cost-effective H2S recovery system. Two major sources of H2S emissions from the process are the acid gas stream evolving from hydrogen manufacture, R1, and the gaseous waste emitted from the separation
208
CHAPTER EIGHT Synthesis of reactive mass-exchange networks
Table 8.5
Data for the Waste Streams of the Coal.Liquefaction Problem (All Compositions Are in kmoi/m 3)
Stream
Flowrate LCm3/hr
Supply composition yjs kmol/m3
Target composition yjt kmol/m3
R1 R2
121.1 28.9
3.98 x 10-4 71.6 x 10-4
2.1 x 10-7 2.1 x 10-7
section of the process, R2, as shown in Fig. 8.6. Stream data for these acid gas streams are summarized in Table 8.5. Six potential MSAs should be simultaneously screened. These include absorption in water, S1, adsorption onto activated carbon, $2, absorption in chilled methanol, $3, and the use of the following reactive solvents; diethanolamine (DEA), $4, hot potassium carbonate, $5, and diisopropanolamine (DIPA), $6. Equilibrium relations governing the transfer of hydrogen sulfide from the gaseous waste streams to the various separating agents can be approximated over the range of operating
F i g u r e 8 . 6 Coal-liquefaction process (Warren et al., 1995. Courtesy of ASCE).
Problems
209
Table 8.6
Data for the MSAs of the Coal-Liquefaction Problem (All Compositions Are in kmol/m 3)
Stream
Lc m3/s
Supply composition x~ 10-6 kmol/m3
$1 $2 $3 S4 $5 S6
c~ c~ oo (x) o~ ~
1000 2.00 2.15 3.40 3.20 1.30
Target composition x~ kmol/m3 0.0150 0.4773 0.2652 0.1310 0.0412 0.7160
compositions (kmol/m 3) as follows: y = 0.398xl
(8.40)
y = 0.015x2
(8.41)
y = 0.027x3
(8.42)
y -- 0.079x426
(8.43)
y = 0.013x5
(8.44)
y -- 0.010x6
(8.45)
The unit operating costs of water, $1, activated carbon, $2, chilled methanol, $3, diethanolamine, $4, hot potassium carbonate, $5, and diisopropanolamine, $6, are 0.001, 8.34, 2.46, 5.94, 3.97, and 4.82 in $/m 3, respectively. These costs include the cost of regeneration and make-up. Stream data for the MSA's are given in Table 8.6. The values of eft a n d e jS are taken to be 0.0 and 2 x 10 -7 kmol/m 3 respectively. Synthesize a REAMEN which features the minimum number of units that realize the minimum operating cost. 8.4 Most of the world's rayon is produced through the viscose process (E1-Halwagi and Srinivas, 1992). Figure 8.7 provides a schematic representation of the process. In this process cellulose pulp is treated with caustic soda, then reacted with carbon disulfide to produce cellulose xanthate. This compound is dissolved in dilute caustic soda to give a viscose syrup, which is fed to vacuum-flash boiling deaerator to remove air. The gaseous stream leaving the deaerator, RI, should be treated for the removal of H2S prior to its atmospheric discharge. In spinning, a viscose solution is extruded through fine holes submerged in an acid bath to produce the rayon fibers. The acid-bath solution contains sulfuric acid, which neutralizes caustic soda and decomposes xanthate and various sulfurcontaining species, thus producing H2S as the major hazardous compound in the exhaust gas stream, R2.
210
CHAPTER EIGHT Synthesis of reactive mass-exchange networks
Caustic Soda
Cellulosle Pulp Steeping Press
I
Xanthating Churn
R 1,to
Dissolver Reactive
Viscose
Syrup Deaerator
Deaerator Exhaust,R
Mass
1
R 2, to atmosphere
Exchange
Gaseous Waste, R
Network v
Spinning
i
i
"
Rayon Fibers (to finishing)
~
F i g u r e 8.7 Simplified flowsheet for viscose rayon production (from E1-Halwagi and Srinivas, Synthesis of reactive mass-exchange networks, Chem. Eng. Sci., 47(8), p. 2116, Copyright 9 1992, with kind permission from Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington 0X5 1GB, UK).
It is desired to synthesize a REAMEN for treating the gaseous wastes (R1 and RE) of a viscose rayon plant. Three MSAs are available to select from. These MSAs are caustic soda, $1, (a process stream already existing in the plant with ml = 5.0 kmol/m3), diethanolamine, $2, (with m2 = 2.0 kmol/m 3) and activated carbon, $3. The unit costs for $2 and $3 including stream makeup and subsequent regeneration are 64.9 $/m 3 and 169.4 $/m 3, respectively. Stream data are given in Table 8.7. The chemical absorption of H2S into caustic-soda solution (Astarita and Gioia, 1964) involves the following two reactions: H2S + OH- ~ HS- + H20
(8.46)
HS- + OH- ~
(8.47)
S 2- + H20.
Since the concentration of water remains approximately constant, one can define the following two reaction equilibrium constants KI.1 -"
[HS-] [H2S][OH-]
(8.48)
Problems
211
Table 8.7
Stream Data for the Viscose-Rayon Example Rich streams
Lean streams
xjS
xjt
Stream
Gi
(kmol/m 3)
(kmol/m 3)
Stream
(m3/s)
(kmol/m 3)
(kmol/m 3)
R1 R2
0.87 0.10
1.3 x 10-5 0.9 x 10-5
2.2 x 10-7 2.2 x 10-7
S1
2.0 x 10-4
S2
(x~
S3
OO
0.0 2.0 x 10-6 1.0 x 10-6
0.1 1.0 x 10-3 3.0 x 10-6
y~
y~
LjC
and [S 2- ] K2.1 "-
[HS-][OH-]
(8.49)
where KI,1 = 9.0 x 106 m3/kmol and K2,1 = 0.12 m3/kmol. The distribution coefficient for the physically dissolved portion of H2S between a rich phase and caustic-soda solution is given by
yi/al = 0.368.
(8.50)
The overall reaction of hydrogen sulfide with diethanol amine (Lal et al., 1985) is given by H2S + (C2H4OH)2NH ~
H S - + (C2H4OH)2NH +,
(8.51)
for which the equilibrium constant is given by K1.2 = 234.48 =
[HS-][R2NH +] [H2S][R2NH]
(8.52)
and the physical distribution coefficient is given by (Kent and Eisenberg, 1976):
Yi/a2 -- 0.363.
(8.53)
The adsorption isotherm for H2S on activated carbon is represented by (Valenzuela and Myers, 1989)
yi/a3
"- 0.015.
(8.54)
The following values for the minimum allowable composition differences are selected: e s = 3.50, e s = 0.014, e s = 10 -6, e R "- 10 -7, e R = 1.5 • 10-7kmol/m 3
8.5 Consider the Kraft pulping process shown in Fig. 8.8 (Dunn and E1-Halwagi, 1993). The first step in the process is digestion in which wood chips, containing primarily lignin, cellulose, and hemicellulose, are "cooked" in white liquor (NaOH, Na2S, Na2CO3 and
212
CHAPTER EIGHT Synthesis of reactive mass-exchange networks
WhiteLiquor
WoodChips;
i Pulpto PaperMachine
Digester [ Canstieizer
]
I
I
Lime
[
Weak
Eor,,'or,,
e~ Lim
~
t
~'~ I~e
Dissolving
r - f - - Tank ---
~
VicthanolAmiue~EA), S4 ~
ActivatedCarbon,S 5 Hot K2CO3Solution,S 6
Wastewater to Additional Treatment
~ ~
IL
' BlackLiquor, S 3
Air Stripping
_Recovery Furnace
Water ud
Contaminated Condensate
Blar "
Green Liquor.
LimeKiln
.L_
~
~'~
Reactive
I
Mass-Exchange Network
i'~"
Sl, to digestor $2, to slaker "S3 to evaporators $4 to regeneration SS to regeneration -'-$6 to regeneration
To atmosphere Figure 8.8 A simplified flowsheet of the Kraft pulping process (Dunn and E1-Halwagi, 1993).
water) to solubilize the lignin. The off gases leaving the digester contain substantial quantifies of H2S. The dissolved lignin leaves the digester in a spent solution referred to as the "weak black liquor". This liquor is processed through a set of multiple-effect evaporators designed to increase the solid content of this stream from approximately 15% to approximately 65 %. At the higher concentration, this stream is referred to as strong black liquor. The contaminated condensate removed through the evaporation process can be processed through an air stripper to transfer sulfur compounds (primarily H2S) to an air stream prior to further treatment and discharge of the condensate stream. The strong black liquor is burned in a furnace to supply energy for the pulping processes and to allow the recovery of chemicals needed for subsequent pulp production. The burning of black liquor yields an inorganic smelt (NaaCO3 and Na2S) that is dissolved in water to produce green liquor (NaOH, Na2S, Na2CO3 and water), which is reacted with quick lime (CaO) to convert the Na2CO3 into NaOH. The conversion of the Na2CO3 into NaOH is referred to as the causticizing reaction and involves two reactions. The first reaction is the conversion of calcium oxide to calcium hydroxide in the presence of water in an agitated slaker. The calcium hydroxide subsequently reacts with NaECO3 to form NaOH and a calcium carbonate precipitant. The calcium carbonate is then heated in the lime kiln to regenerate the calcium oxide and release
Problems
213
Table 8.8 Data for the Gaseous Emission of the Kraft Pulping Process
Flowrate Gi, m3/s
Supply composition (10-5 kmol/m3) yS
Target composition (10-7 kmol/m3) y[
3.08
2.1
8.2
2.1
1.19
2.1
Stream
Description
R1
Gaseous waste from recovery furnace
117.00
R2
Gaseous waste from evaporator Gaseous waste from air stripper
0.43
R3
465.80
carbon dioxide. These reactions result in the formation of the original white liquor for reuse in the digesting process. Three major sources in the kraft process are responsible for the majority of the H2S emissions. These involve the gaseous waste streams leaving the recovery furnace, the evaporator and the air stripper, respectively denoted by R1, RE and R3. Stream data for the gaseous wastes are summarized in Table 8.8. Several candidate MSAs are screened. These include three process MSAs and three external MSAs. The process MSAs are the white, the green and the black liquors (referred to as S1, $2 and $3, respectively). The external MSAs include diethanolamine (DEA), $4, activated carbon, $5, and 30 wt% hot potassium carbonate solution, $6. Stream data for the MSAs is summarized in Table 8.9. Synthesize a MOC REAMEN that can accomplish the desulfurization task for the three waste streams.
Table 8.9 Data for the MSAs of the Kraft Pulping Example
Stream
Description
$1 $2 $3
White liquor Green liquor Black liquor DEA Activated carbon Hot Potassium carbonate
S4
$5 $6
Upper bound on flowrate L jc m 3/s
Supply composition (kmol/m3) xjs
Target composition (kmol/m3) xjt
0.040 0.049 0.100 cxz oo
0.320 0.290 0.020 2 x 10, 6 1 x 10-6
3.100 1.290 0.100 0.020 0.002
o~
0.003
0.280
214
CHAPTER EIGHT Synthesis of reactive mass-exchange networks
Symbols aj A bz,j bz~
Bj Bz,j
cj
fj Pj ai
i
J k
Iqj,j Lj mj np
NR Ns NZj qj
Qj r 2
R
Ri S
sj uj xj xj$ xjt Xj,k
xj Y Yi
y7 y* Z
physically dissolved concentration of A in lean stream j, kmol/m3 key transferable component composition of reactive species Bz,j in lean phase j, kmol/m3 admissible composition of reactive species Bzj in lean phase j, kmol/m3 set of reactive species in lean stream j index of reactive species in lean stream j unit cost of the jth MSA j, ($/kg) equilibrium distribution function between rich phase composition and total content of A in lean phase j, defined in Eq. (8.9) equilibrium distribution function between rich phase composition and physically dissolved A in lean phase j, defined in Eq. (8.8) flow rate of rich stream i, kmol/s index for rich streams index for lean streams index for composition intervals equilibrium constant for the qjth reaction in lean phase j flow rate of lean stream j, kmol/s upper bound on flow rate of lean stream j, kmol/s index for subnetworks molarity of lean stream j, kmol/m3 number of mass-exchange pinch points in the problem number of rich streams number of lean streams number of reactive species in lean stream j index for the independent reactions in lean stream j number of independent reactions in lean stream j correlation coefficient set of rich streams the ith rich stream set of lean streams the j th lean stream fractional saturation of chemically combined A in the jth MSA composition of key component in lean stream j, kmol/m3 supply composition of key component in lean stream j, kmol/m3 target composition of key component in lean stream j, kmol/m3 the upper bound composition for interval k on the scale Sj, kmol/m3 equilibrium composition of key component in lean stream j, kmol/m3 composition of key component in any rich stream, kmol/m3 composition of key component in rich stream i, kmol/m3 supply composition of key component in rich stream i, kmol/m3 target composition of key component in rich stream i, kmol/m3 equilibrium composition of key component in any rich stream, kmol/m3 index for the reactive species
References
215
Greek letters
t~k t~i,k ,s I)qj ,z,j
~qj
residual load leaving interval k, kmol/s residual load leaving interval k for rich stream i, kmol/s rich-phase minimum allowable composition difference, kmol/m 3 lean-phase minimum allowable composition difference, kmol/m 3 stoichiometric coefficient of reactive species z in reaction qj in lean phase j extent of reaction qj in the jth MSA
Special symbol []
concentration, kmol/m 3
References Astarita, G., and Gioia, E (1964). Hydrogen sulfide chemical absorption. Chem. Eng. Sci., 19, 963-971. Astarita, G., Savage, D. W., and Bisio, A. (1983). "Gas Treating with Chemical Solvents," John Wiley and Sons, New York. Dunn, R. E, and E1-Halwagi, M. M. (1993). Optimal recycle/reuse policies for minimizing the wastes of pulp and paper plants. J. Environ. Sci. Health, A28(1), 217-234. E1-Halwagi, M. M. f1971). An engineering concept of reaction rate. Chem. Eng., May, 75-78. E1-Halwagi, M. M. (1990). Optimization of bubble column slurry reactors via natural delayed feed addition. Chem. Eng. Commun., 92, 103-119. E1-Halwagi, M. M., and Srinivas, B. K. (1992), Synthesis of reactive mass-exchange networks. Chem. Eng. Sci., 47(8), 2113-2119. Espenson, J. H. (1981). "Chemical Kinetics and Reaction Mechanisms," McGraw-Hill Book Company, New York. Fogler, S. H. (1992). "Elements of Chemical Reaction Engineering," Prentice Hall, Englewood Cliffs, NJ. Friedly, J. C. (1991). Extent of reaction in open systems with multiple heterogeneous reactions. AIChE J., 37(5), 687-693. Kent, R. L., and Eisenberg, B. (1976). Better data for amine treating. Hydrocarbon Processing, February, pp. 87-90. Kohl, A., and Reisenfeld, F. (1985). "Gas Purification," 4th ed., Gulf Publishing Co., Houston, TX. Lal, D., Otto, E D., and Mather, A. E. (1985). The solubility of H2S and CO2 in a diethanolamine solution at low partial pressures. Can. J. Chem. Eng., 63, 681-685. Levenspiel, O. (1972). "Chemical Reaction Engineering" 2nd ed., John Wiley and Sons, New York. Smith, W. R., and Missen, R. W. (1982). "Chemical Reaction Equilibrium Analysis: Theory and Algorithms" John Wiley and Sons, New York.
216
CHAPTER EIGHT Synthesis of reactive mass-exchange networks
Srinivas, B. K., and E1-Halwagi, M. M. (1994). Synthesis of reactive mass-exchange networks with general nonlinear equilibrium functions. AIChE J., 40(3), 463-472. Valenzuela, D. P., and Myers, A. (1989). "Adsorption Equilibrium Data Handbook," Prentice Hall, Englewood Cliffs, NJ. Warren, A., Srinivas, B. K., and E1-Halwagi, M. M. (1995). Design of cost-effective wastereduction systems for synthetic fuel plants. J. Environ. Eng., 121(10), 742-747. Westerterp, K. R., Van Swaaij, W. P. M., and Beenackers, A. A. C. M. (1984). "Chemical Reactor Design and Operation," John Wiley and Sons, New York.
Syn thesis of Rea ctive Mass-Exchange Networks
Chapters Three, Five and Six have covered the synthesis of physical mass-exchange networks. In these systems, the targeted species were transferred from the rich phase to the lean phase in an intact molecular form. In some cases, it may be advantageous to convert the transferred species into other compounds using reactive MSAs. Typically, reactive MSAs have a greater capacity and selectivity to remove an undesirable component than physical MSAs. Furthermore, since they react with the undesirable species, it may be possible to convert pollutants into other species that may either be reused within the plant itself or sold. The synthesis of a network of reactive mass exchangers involves the same challenges described in synthesizing physical MENs. The problem is further compounded by virtue of the reactivity of the MSAs. Driven by the need to address this important problem, Srinivas and E1-Halwagi introduced the problem of synthesizing reactive mass-exchange networks REAMENs and developed systematic techniques for its solution (Srinivas and E1-Halwagi, 1994; E1-Halwagi and Srinivas, 1992). This chapter provides the basic principles of synthesizing REAMENs. The necessary thermodynamic concepts are covered. Chemical equilibrium is then tackled in a manner that renders the REAMEN synthesis task close to the MEN problem. Finally, an optimization-based approach is presented and illustrated by a case study.
8.1 Objectives of REAMEN Synthesis The problem of synthesizing REAMENs can be stated as follows (E1-Halwagi and Srinivas, 1992): 191
192
CHAPTER EIGHT Synthesis of reactive mass-exchange networks
Physical I Reactive MSAa In
11111 REACTIVE MASS-EXCHANGE NETWORK
Waste (Rich) Streams (Sources) In
v
.~ ~
Waste (Rich) Streams (Sources) Out
,
11111
Physical/Reactive MSAa Out F i g u r e 8 . 1 Schematic representation of the REAMEN synthesis problem.
Given a number NR of waste (rich) streams and a number Ns of lean streams (physical and reactive MSAs), it is desired to synthesize a cost-effective network of physical and/or reactive mass exchangers which can preferentially transfer a certain undesirable species, A, from the waste streams to the MSAs whereby it may be reacted into other species. Given also are the flowrate of each waste stream, G i, its supply (inlet) composition, y~, and target (outlet) composition, y~, where t are i = 1, 2, .. ., NR. In addition, the supply and target compositions, xjs and x j, given for each MSA, where j = 1, 2 . . . . . Ns. The flowrate of any lean stream, L j, is unknown but is bounded by a given maximum available flowrate of that stream, i.e., L j < L jc.
(8.1)
Figure 8.1 is a schematic illustration of the REAMEN synthesis problem. As has been previously discussed in the synthesis of physical MENs, several design decisions are to be made: 9 Which mass-exchange operations should be used (e.g., absorption, adsorption, etc.)? 9 Which MSAs should be selected (e.g., physical/reactive transfer, which solvents, adsorbents, etc.)? 9 What is the optimal flowrate of each MSA? 9 How should these MSAs be matched with the waste streams (i.e., stream pairings)? 9 What is the optimal system configuration (e.g., how should these mass exchangers be arranged?, is there any stream splitting and mixing?, etc.)? The first step in synthesizing a REAMEN is to establish the conditions for which the reactive mass exchange is thermodynamically feasible. This issue is covered by the next section.
8.2 Corresponding composition scales for reactive mass exchange
193
8.2 Corresponding Composition Scales for Reactive Mass Exchange The fundamentals of reactive mass exchange, design of individual units, chemical equilibrium and kinetics are covered in the literature (e.g., Fogler, 1992; Friedly, 1991; E1-Halwagi, 1990; Kohl and Reisenfeld, 1985; Westerterp et al., 1984; Astarita et al., 1983; Smith and Missen, 1982; Espenson, 1981; Levenspiel, 1972; E1-Halwagi, 1971). This section presents the salient basics of these systems. In order to establish the conditions for thermodynamic feasibility of reactive mass exchange, it is necessary to invoke the basic principles of mass transfer with chemical reactions. Consider a lean phase j that contains a set Bj = {Bz,j I z = 1. . . . . NZj } of reactive species (i.e., the set Bj contains NZj reactive species, each denoted by Bz, j, where the index z assumes values from 1 to NZj). These species react with the transferable key solute, A, or among themselves via Qj independent chemical reactions which may be represented by NZj
A + Z
(8.2)
1)l'z'J Bz'J - - 0
z=l
and NZj
~-'~ l)qj,z,jBz, j -- 0
qj : 2, 3 . . . . . Q j,
(8.3)
Z--1
where the stoichiometric coefficients 1)q,z, j a r e negative for products and positive for the reactants. It is worth noting that stoichiometric equations can be mathematically handled as algebraic equations. Therefore, although component A may be involved in more than the first reaction, one can always manipulate the stoichiometric equations algebraically to keep A in the first reaction and eliminate it from the other stoichiometric equations. Compositions of the different species can be tracked by relating them to the extents of the reactions through the following expression Qj
bz,j
b~z , j - Z
l)qj ,z, j~qj
(z=l
'
2'
"""'
NZj) '
(8.4)
qj --2
where bz, j is the composition of species Bz, j in the jth lean phase, b~ is the admissible composition of species Bz,j in the jth lean phase and ~qj is the extent of the qjth reaction (or the qjth reaction coordinate). The extent of reaction is defined for reactions two through Qj. The reason for not defining it for the first reaction is that the variable uj plays indirectly the role of the extent of reaction for
194
CHAPTER EIGHT Synthesis of reactive mass-exchange networks
the first reaction. 1 The admissible compositions may be selected as the lean-phase composition at some particular instant of time, or any other situation which is compatible with stoichiometry and mass-balance bounds. The equilibrium constant of a reaction is the product of compositions of reactants and products each raised to its stoichiometric coefficient. Hence, for the reaction described by Eq. (8.2) one may write
1 NZj ( Qj ) gl,J = ~j ~ bz,J - Z Pqj,z,j~qj z-1 ~k qj=2
Pl,z,j
i.e.,
aj
o
bz,j
__ ~ 1
-
z-1
and for the reactions given by Eq. (8.3):
Nzj( oj
gqj,j -- 1-I z=l
1)qj
,z,j~qj
)
Vl,z,j (8.5)
qj=2
)vqjzj
bz, J - Z VqJ'z'J~qJ qj=2
qj = 2, 3 . . . . . Qj,
(8.6)
where Kqj,j is the equilibrium constant for the qjth reaction and aj is the composition of the physically dissolved A in lean phase j. It is now useful to recall the concepts of molarity and fractional saturation (Astarita et al., 1983). The molarity, m j, of a reactive MSA is the total equivalent concentration of species that may react with component A. On the other hand, the fractional saturation, u j, is a variable that represents the degree of saturation of chemically combined A in the j th lean phase. Therefore, uj mj is the total concentration of chemically combined A in the jth MSA. Hence, the total concentration of A in MSA j can be expressed as
xj = aj + u j m j ,
j ~ S,
(8.7)
where the physically-dissolved concentration of A, a j, equilibrates with the richphase composition through a distribution function Fj, i.e.,
y* = Fj (aj ),
j r S.
(8.8)
Equations (8.4)-(8.8) represent a complete mathematical description of the chemical equilibrium between a rich phase and the jth MSA. The simultaneous solution 1For this reason, whenever there is a single reaction taking place in the jth MSA, no extent of reaction is defined. Insteadthe fractional saturation, uj, is employed.
Example 8.1. Absorption of H2S in aqueous sodium hydroxide
195
of these nonlinear equations (for instance by using the software LINGO) yields the equilibrium compositions of both phase in the form y;=fj(x;)
jrS.
(8.9)
For any mass-exchange operation to be thermodynamically feasible, the following conditions must be satisfied: xj < x j ,
j ~ S,
(8.10a)
Yi > Y*,
i e R,
(8.10b)
and/or
i.e., y=
f (xj + e ~ ) ,
j r S,
(8.11a)
where Y --- Yi -- eiR, i ~ R, (8.1 lb) s where e]~ and e j are positive quantities called, respectively, the rich and the lean minimum allowable composition differences. The parameters e/R and e js are optimizable quantities that can be used for trading off capital versus operating costs (see Chapters Two and Three). Equations (8.1 l a) and (8.1 l b) provides a correspondence among the rich and the lean composition scales for which mass exchange is practically feasible. This is the reactive equivalent to Eq. (3.5) used for establishing the corresponding composition scales for physical MENs.
Example 8.1. Absorption of HzS in Aqueous Sodium Hydroxide Consider an aqueous caustic soda solution whose molarity m l - - 5 . 0 k m o l / m 3 (20 wt.% NaOH). This solution is to be used in absorbing HES from a gaseous waste. The operating range of interest is 0.0 _< Xl (kmol/m 3) < 5.0. Derive an equilibrium relation for this chemical absorption over the operating range of interest.
Solution The absorption of H2S in this solution is accompanied by the following chemical reactions (Astarita and Gioia, 1964): H2S + NaOH = NariS + H20
(8.12)
H2S -t- 2NaOH = Na2S + 2H20.
(8.13)
196
CHAPTER EIGHT Synthesis of reactive mass-exchange networks
As has been described earlier, the stoichiometric reactions should be manipulated algebraically to retain the transferable species (H2S) only in the first equation. Therefore, H2S can be eliminated from Eq. (8.13) by subtracting (8.12) from (8.13) to get NariS + NaOH = Na2S + H20.
(8.14)
Equations (8.12) and (8.14) can be written in ionic terms as follows: H2S q- OH- = HS- + H20
(8.15)
HS- + OH- = S 2- + H20.
(8.16)
Let us denote the three ionic species as follows: OH- = BI,1 HS- = B2,1 S 2- -- B3,1,
with aqueous-phase concentrations referred to as bl,1, b2,1 and b3,1, respectively. Also, let us denote the composition of the physically dissolved H2S in the aqueous solution as al. For cases when the concentration of water remains almost constant with respect to the other species, one can define the following reaction equilibrium constants for Eqs. (8.15) and (8.16), respectively: g l , 1 "-
b2,1 a l "bl,1
(8.17)
and b3,1 K2,1 -- ~ , b2,1 9 bl,1
(8.18)
where KI,1 = 9.0 • 106 m3/kmol and K2,1 = 0.12 m3/kmol. The distribution coefficient for the physically dissolved H2S is given by yl = 0.368,
(8.19)
al
where yl is the composition of H2S in the gaseous stream. It is useful to relate the molarity of the aqueous caustic soda (ml = 5.0 kmol NaOH/m 3) to the other reactive species. Once the reactions start, the composition of NaOH will decrease. However, it is possible to relate the molarity of the solution to the concentration of the reactive species at any reaction coordinate. Suppose that after a certain extent of reaction (8.15) and (8.16) an analyzer is placed in the solution to measure the compositions of OH-, HS- and S 2- with the measured
Example 8.1: Absorption of H2S in aqueous sodium hydroxide
197
concentrations denoted by bl,1, b2,1 and b3,1, respectively. These measured concentrations are related to the molarity as follows. According to Eq. (8.15), b2,1 kmoles of OH- must have reacted to yield b2,1 kmoles of HS-. Similarly, according to Eq. (8.16), b3,1 kmoles of OH- must have reacted with b3,1 kmoles of H S to yield b3,1 kmoles of S 2-. But the b3,1 kmoles of HS- must have resulted from the reaction of b3,1 kmoles of OH- (according to Eq. (8.15)). Hence, 2b3,1 kmoles of O H - are consumed in producing b3,1 kmoles of S 2-. Therefore, the molarity of the aqueous caustic soda can be related to the concentrations of the reactive ions as follows: 5 -- bl,1 -k- b2,1 q'- 2b3,1.
(8.20)
There are two forms ofreacted HzS in the solution: HS- and S 2-. By recalling that U lml is the total concentration of chemically combined HzS in the aqueous caustic soda and conducting an atomic balance on S over Eqs. (8.15) and (8.16), we get 5Ul -- b2,1 "+" b3,1.
(8.21)
As discussed earlier, the admissible compositions may be selected as the leanphase composition at some particular instant of time, or any other situation which is compatible with stoichiometry and mass-balance bounds such as Eqs. (8.20) and (8.21). Let us arbitrarily select the admissible composition of S 2- to be zero, i.e., b~ = 0. This selection automatically fixes the corresponding values of b~ and b~ cording to Eqs. (8.20) and (8.22), we get b 2,1 ~ = 5u 1.
(8.22) Ac-
(8.23)
Similarly, according to Eqs. (8.21)-(8.23), we obtain: b~ = 5(1 - Ul).
(8.24)
As mentioned earlier, the extent of reaction is defined for all reactions except the first one. Hence, we define ~2 as the extent of reaction for Eq. (8.16). Equation (8.4) can now be used to describe the compositions of the reactive species as a function of ~2 and the admissible compositions, i.e., bl,1 "- 5(1 - ul) - ~2
(8.25)
b2,1 -" 5Ul - ~2
(8.26)
b3,1 -" ~2-
(8.27)
198
CHAPTER EIGHT Synthesis of reactive mass-exchange networks
Substituting from Eqs. (8.19), (8.25), (8.26) and (8.27) into Eq. (8.17), we get 9 . 0 x 106--
5Ul -- ~2
Yl ~[5(1 0.368
-- Ul) -- ~2]
or Yl - - 4 . 0 9 X 10 -8
5Ul - ~2 . [5(1 -- Ul) -- ~2]
(8.28)
Substituting from Eqs. (8.25)-(8.27) into Eq. (8.18), we obtain 0.12=
~2
[5Ul - ~2][5(1 - ul) - ~z]'
which can be rearranged to ~2 _ 13.33 ~2 "!- 25Ul (1 -- Ul) = 0.
(8.29)
According to Eq. (8.7), the total concentration of H2S in the aqueous caustic soda (physically dissolved and chemically reacted) can be expressed as follows Xl = al + 5Ul.
(8.30a)
In this system, the physically dissolved H2S is much less than the chemically combined, 2 i.e., al << 5Ul,
(8.30b)
X1 -- 5Ul.
(8.30c)
which simplifies Eq. (8.30a) to
Equations (8.28), (8.29), and (8.30c) can be used to develop an expression for reactive mass exchange of H2S. First, a value of fractional saturation is selected (where xS/5 < ul _< x]/5). Then, Eq. (8.30c) is used to calculate the corresponding xl. Next, Eq. (8.29) is solved to determine the value of ~2. Finally, Eq. (8.28) is solved to evaluate Yl. The pair (yl, Xl) are in equilibrium. The same procedure is repeated for several values of u 1 (between x s/5 and x]/5) to yield pairs of (yl, x i) which are in equilibrium. Nonlinear regression can be employed to derive an equilibrium expression for these pairs. To illustrate the above-mentioned procedure, let us start with a fractional saturation oful = 0.1. According to Eq. (8.30c), xl = 0.5 kmol/m 3. By substituting for Ul = 0.1 in Eq. (8.29), we obtain: ~2 _ 13.33~2 + 2.25 = 0, 2This assumption can be numerically verified by comparing values of aj with ujmj after the equilibrium equation is generated.
8.3 Synthesis approach
199
Figure 8.2 Equilibrium data for the example of absorbing H2S in caustic soda.
which can be solved to get ~2---~
13.33 - ~/13.33 • 13.33 - 4 2
• 2.25
= 0.171 kmol/m 3 (the other root is rejected)
(8.31)
The values o f u l and ~2 are plugged into Eq. (8.28) to get yl - 3.1 x 10 -9 kmol]m 3. Hence, the pair (3.1 x 10 -9, 0.5) are in equilibrium. 3 The same procedure is repeated for values of u l between 0.0 and 0.1. The results are plotted as shown in Fig. 8.2. The plotted data are slightly convex and can be fitted to the following quadratic function (all compositions in kmol/m3): Yl = 2.364 X 10-9x 2 + 5.022 x 10-9xl
(8.32)
with a correlation coefficient, r 2, of 0.9999.
8.3 Synthesis Approach Now that a procedure for establishing the corresponding composition scales for the rich-lean pairs of stream has been outlined, it is possible to develop the CID. The CID is constructed in a manner similar to that described in Chapter Five. However, it should be noted that the conversion among the corresponding composition scales may be more laborious due to the nonlinearity of equilibrium relations. Furthermore, a lean scale, x j , represents all forms (physically dissolved and chemically combined) of the pollutant. First, a composition scale, y, for component A in 3We can now test the validity of Eq. (8.30b). According to Eq. (8.19), al = 8.45 x 10-9 kmol/m3 which is indeed much less than u l m l = 0.5.
200
CHAPTER EIGHT Synthesis of reactive mass-exchange networks
any rich stream is created. This scale is in a one-to-one correspondence with any composition scale of component A in the ith rich stream, yi, via Eq. (8.1 lb). Then, Eq. (8.1 la) is used to generate Ns composition scales for component A in the lean streams. Next, each stream is represented by an arrow whose tail and head correspond to the supply and target compositions, respectively, of the stream. The partitions corresponding to these heads and tails establish the composition intervals. Similar to the CID for physical MENs, within any composition interval, it is thermodynamically feasible to transfer component A from the rich streams to the lean streams. Also, according to the second law of thermodynamics it is spontaneously possible to transfer component A from the rich streams in a given composition interval to any lean stream within a lower composition interval. It is worth noting that the foregoing composition partitioning procedure ensures thermodynamic feasibility only when all the equilibrium relations described by Eq. (8.9) are convex. In this case by merely satisfying Eq. (8.11) at both ends of a composition interval, Eqs. (8.10) are automatically satisfied throughout that interval (Fig. 8.3a). On the other hand, when at least one of the equilibrium relations expressed by Eq. (8.9) is non-convex, the satisfaction of Eq. (8.11) at both ends of an interval does not necessarily imply the realization of inequalities (Eqs. 8.10) throughout that interval (Fig. 8.3b). In such case, additional composition partitioning is needed. This can be achieved by discretizing the non-convex portions of the equilibrium curves through linear Yi
Y i,k
Y i,k
~E s I I ] i I t I
operating line
J
,/ I equilibrium line Y i,k+!
Y i~§
[
-
I Es t I I X j~,~
J
. !
t I I!
X j,k+}
I I I I I I I I I I i X j~,
Xj~,
Xj
8.3a A reactive massexchangerwith convexequilibrium(EI-Halwagiand Srinivas, Synthesis of reactive mass-exchangenetworks, Chem. Eng. Sci., 47(8), p. 2115, Copyright9 1992,withkind permission from Elsevier Science Ltd., The Boulevard, LangfordLane, Kidlington0X5 1GB,UK). Figure
8.3 Synthesis approach
201
Yi
Yi,k
v~k
~ - - /
'
equilibrium line
Yi,k+l Yi,k+ t
F i g u r e 8 . 3 b A reactive mass exchanger with non-convex equilibrium (E1-Halwagi and Sfinivas, Synthesis of reactive mass-exchange networks, Chem. Eng. Sci., 47(8), p. 2115, Copyright 9 1992, with kind permission from Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington 0X5 1GB, UK). Yi
Yi,a Yi,b
2k
__--~---__------. equilibrium line
vi.c
Yi,k+1 i
i~ _ J
~
I I
Xj~+ l
Ej
s
I I I
Xj,a
I I I I
X j,b
,I I
Xjs
I I
Xj~
Xj
Figure 8.3c A reactive mass exchanger with discretization of non-convex equilibrium (E1-Halwagi and Srinivas, Synthesis of reactive mass-exchange networks, Chem. Eng. Sci., 47(8), p. 2115, Copyfight 9 1992, with kind permission from Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington 0X5 1GB, UK).
202
CHAPTER EIGHT Synthesis of reactive m a s s - e x c h a n g e networks
overestimators. 4 Clearly, the number of these linear segments depends upon the desired degree of accuracy. For instance, Fig. 8.3b may be convexified by introducing two linear parts at the concavity inflection point (Fig. 8.3c). Consequently, additional intervals will be created within the CID by partitioning the composition scales at the locations corresponding to points a, b and c. The end result of this partitioning procedure is that the entire composition range is divided into nint composition intervals, with k = 1 being the highest and k =/'tint being the lowest. Having established a one-to-one thermodynamically-feasible correspondence among all the composition scales, we can now solve the REAMEN problem via a transshipment formulation similar to that described in Chapter Five for the synthesis of physical MENs. Example 8.2. Removal Pulping Process
o f H2S f r o m a K r a f t
Kraft pulping is a common process in the paper industry. Figure 8.4 shows a simplified flowsheet of the process. In this process, wood chips are reacted (cooked) with white liquor in a digester. White liquor (which contains primarily NaOH, Na2S, Na2CO3 and water) is employed to dissolve lignin from the wood chips. The cooked pulp and liquor are passed to a blow tank where the pulp is separated from the spent liquor "weak black liquor" which is fed to a recovery system for ToAtmosphere *
Aq.NaOH, $1
I
REAMEN it
El Activated Carbon,S3
~ I
I I
! /
S2, ToRegenerationand Recycle Wood C;ips
~ Gaseous Waste, Ri
/
LW ~ 9~Z
Steam
Digester
v I
r
Steam
Pulpto Further
,
eak BlackLiquor
~
Gases
Blow Tank
] I
"-1 "-I
Washers
I~
Slaking and Caustiizing
Condensate Flue Gas Recovery Furnace
Lime
Smelt
Kiln
GOa~
;Water
Lime
Calcium Carbonate
o
Settling and Filtration
Figure 8.4 Kraftpulping process. 4A more rigorous method of tackling nonconvex equilibrium without discretization has been developed by Srinivas and E1-Halwagi (1994) and is beyond the scope of this book.
Example 8.2: Removal of H2S from a Kraft pulping process
203
Table 8.1 Data for the Gaseous Emission of the Kraft Pulping Process
Stream R1
Flowrate Gi, m3/s
Description Gaseous waste from evaporators
Target composition (10-1~ kmol/m3) y[
Supply composition (10-1~ kmol/m3) yS
16.2
1,600
3.0
conversion to white liquor. The first step in recovery is concentration of the weak liquor via multiple effect evaporators. The concentrated solution is sprayed in a furnace. The smelt from the furnace is dissolved in water to form green liquor which is reacted with lime (CaO) to produce white liquor and calcium carbonate "mud". The recovered white liquor is mixed with make-up materials and recycled to the digester. The calcium carbonate mud is thermally decomposed in a kiln to produce lime which is used in the causticizing reaction. There are several gaseous wastes emitted from the process (see Dunn and E1-Halwagi, 1993, and Homework Problem 8.5). In this example, we focus on the gaseous waste leaving the multiple effect evaporators, R1, whose primary pollutant is HzS. Stream data for this waste stream are given in Table 8.1. A rich-phase minimum allowable composition difference, e/R, of 1.5 x 10 -1~ kmol/m 3 is used. A process lean stream and an external MSA are considered for removing H2S. The process lean stream, S 1, is a caustic soda solution which can be used as a solvent for the reactive separation of HES. An added bonus for using the process MSA is the conversion of a portion of the absorbed HES into Na2S, which is needed for whiteliquor makeup. In other words, HES "the pollutant" is converted into a valuable chemical which is needed in the process. The external MSA, $2, is a polymeric adsorbent. The data for the candidate MSAs are given in Table 8.2. The equilibrium
Table 8.2 Data for the MSAs of the Kraft Pulping Example
Stream
Upper bound on flowrate LjC m3/s
Supply composition (kmol/m3) xs.J
,,,, ,,
Target composition (kmol/m3) xt.j .
Sl $2
0.01 oo
0.000 0.010
.
0.500 0.025
.
. es. (kmol/m3) J
.
.
.
0.10 0.01
Cj $/m3 MSA
.
2,900
204
CHAPTER EIGHT Synthesis of reactive mass-exchange networks
data for the transfer of H2S from the waste stream to the adsorbent is given by Yl = 2 x 10-9x2,
(8.33)
where Yl and x2 are given in kmol/m 3. For the given data, determine the minimum operating cost of the REAMEN and construct a network with the minimum number of exchangers.
Solution The following expression for the equilibrium data of H2S in caustic soda has been derived in Eq. (8.32) of Example 8.1 (all compositions are in kmol/m3): yl = 2.364 x 10 -9 + 5.022 • 10-9Xl.
(8.32)
By invoking Eq. (8.11), we get the following equation for the practical-feasibility curve: yl - 1.5 x 10 -1~ = 2.364 x 10-9(Xl + 0.1) 2 + 5.022 • 10-9(Xl + 0.1).
(8.34)
Similarly, for activated carbon Yl -- 1.5 • 10 -10 = 2 x 10-9(x2 d-0.01).
(8.35)
The CID for the problem is shown in Fig. 8.5. The table of exchangeable loads for the waste streams and the MSAs are shown by Tables 8.3 and 8.4, respectively. According to the two-stage targeting procedure, we first minimize the annual operating cost of the MSAs (given by 3600 x 8760 • 2100 x L2 ---- 6.62256x 101~ L2 when we assume 8760 operating hours per year). By applying the linear-
Figure 8.5 CID for Kraft pulping example.
Example 8.2: Removal of HES from a Kraft pulping p r o c e s s
205
Table 8.3 The TEL for the Gaseous Waste
Interval
Load of R1 (10 -10 kmol H2S/s)
1 2 3 4 5
25,270 559 62 0 0
p r o g r a m m i n g f o r m u l a t i o n d e s c r i b e d in C h a p t e r Six (P6.1), o n e can write the following optimization program: m i n 6 . 6 2 2 5 6 • 1010 L2, subject to 31 - - 2 5 , 2 7 0 x 10 -1~
32-
31 4- 0.5L1 -- 559 x 10 -10 33 - 3 2 34-
= 62 x 10 -1~
33 -- 0.0
- 3 4 4- 0 . 0 1 5 L 2 = 0.0 6k > 0
Lj>O
k--1,2,3,4 j=l,2
L1 < 0.01 x 10 -1~ .
Table 8.4 The TEL for the Lean S t r e a m s Capacity of lean streams per m 3 of MSA (kmol H2S/m 3 MSA) Interval
S1
$2
2 3 4 5
0.5 -
0.015
(P8.1)
206
CHAPTER EIGHT Synthesis of reactive mass-exchange networks
It is worth pointing out that the wide range of coefficients may cause computational problems for the optimization software. This is commonly referred to as the "scaling" problem. One way of circumventing this problem is to define scaled flowrates of MSAs in units of 10 -1~ m3/s and scaled residual loads in units of 10 -1~ kmol/s, i.e., let
Lscaled j --
1010L j,
skcaled-"
10108k,
j = 1, 2 k = 1, 2, 3, 4
(8.36) (8.37)
With the new units, the scaled program becomes: min 6.62256
LS2caled
(P8.2)
subject to ~caled = 25,270 ~caled _ ~caled .+_ 0.5L~caled
__ 559
S~ caled -- S~ caled --" 62 ~caled _ ~caled = 0.0 _ ~caled _~_ 0.015 L ~caled = 0.0
~caled>0
k=1,2,3,4
LjScaled __> 0
j = 1, 2
L~caled < 0.01 In terms of LINGO input, the scaled program can be written as follows (with the S in each variable indicating that it is a scaled variable): model: min
=
6.62256-LS2;
deltaSl
=
25270;
deltaS2
-
deltaSl
+
0.5.LSI
deltaS3
-
deltaS2
=
62;
deltaS4
-
deltaS3
=
0.0;
-
deltaS4
+
0.015-LS2
deltaSl
>=
0.0;
deltaS2
>=
0.0;
deltaS3
>=
0.0;
deltaS4
>=
0.0;
LSI
>=
0.0;
LS2
>=
0.0;
end
=
0.0;
=
559;
Problems
207
The solution report from LINGO gives the following results: Objective
value-
Variable LS2 DELTAS1 DELTAS2 LSI DELTAS3 DELTAS4
27373 Value 4133.333 25270.00 0.0000000E+00 51658.00 62.00000 62.00000
Therefore, the minimum operating cost is approximately $27,000/year and the pinch location is between the second and third composition intervals. The REAMEN involves two exchangers; one above the pinch matching R1 with S1 and one below the pinch matching R1 with $2.
Problems 8.1 Derive an equilibrium expression for the reactive absorption of H2S in diethanolamine "DEA". The molarity of DEA is 2 kmol/m3. The following reaction takes place: H2S + (C2Hs)2NH = HS- + (C2Hs)2NH+, whose equilibrium constant is given by (Lal et al., 1985): K = 234.48 =
[HS-][(C2Hs)2[NH~-] [H2S][(C2Hs)2NH]
(8.38)
with all concentrations in kmol/m3. The physical distribution coefficient is: 0.363 =
Composition of H2S in gas (kmol/m3) Composition of physically-dissolved H2S in DEA (kmol]m 3)
(8.39)
8.2 Develop equilibrium equations for the reactive absorption of CO2 into: (a) Aqueous potassium carbonate (b) Monoethanolamine
(Hint: See pp.
68-79 of Astarita et al., 1983.)
8.3 Coal may be catalytically hydrogenated to yield liquid transportation fuels. A simplified process flow diagram of a coal-liquefaction process is shown in Fig. 8.6. Coal is mixed with organic solvents to form a slurry which is reacted with hydrogen. The reaction products are fractionated into several transportation fuels. Hydrogen sulfide is among the primary gaseous pollutants of the process (Warren et al., 1995). Hence, it is desired to design a cost-effective H2S recovery system. Two major sources of H2S emissions from the process are the acid gas stream evolving from hydrogen manufacture, R1, and the gaseous waste emitted from the separation
208
CHAPTER EIGHT Synthesis of reactive mass-exchange networks
Table 8.5
Data for the Waste Streams of the Coal.Liquefaction Problem (All Compositions Are in kmoi/m 3)
Stream
Flowrate LCm3/hr
Supply composition yjs kmol/m3
Target composition yjt kmol/m3
R1 R2
121.1 28.9
3.98 x 10-4 71.6 x 10-4
2.1 x 10-7 2.1 x 10-7
section of the process, R2, as shown in Fig. 8.6. Stream data for these acid gas streams are summarized in Table 8.5. Six potential MSAs should be simultaneously screened. These include absorption in water, S1, adsorption onto activated carbon, $2, absorption in chilled methanol, $3, and the use of the following reactive solvents; diethanolamine (DEA), $4, hot potassium carbonate, $5, and diisopropanolamine (DIPA), $6. Equilibrium relations governing the transfer of hydrogen sulfide from the gaseous waste streams to the various separating agents can be approximated over the range of operating
F i g u r e 8 . 6 Coal-liquefaction process (Warren et al., 1995. Courtesy of ASCE).
Problems
209
Table 8.6
Data for the MSAs of the Coal-Liquefaction Problem (All Compositions Are in kmol/m 3)
Stream
Lc m3/s
Supply composition x~ 10-6 kmol/m3
$1 $2 $3 S4 $5 S6
c~ c~ oo (x) o~ ~
1000 2.00 2.15 3.40 3.20 1.30
Target composition x~ kmol/m3 0.0150 0.4773 0.2652 0.1310 0.0412 0.7160
compositions (kmol/m 3) as follows: y = 0.398xl
(8.40)
y = 0.015x2
(8.41)
y = 0.027x3
(8.42)
y -- 0.079x426
(8.43)
y = 0.013x5
(8.44)
y -- 0.010x6
(8.45)
The unit operating costs of water, $1, activated carbon, $2, chilled methanol, $3, diethanolamine, $4, hot potassium carbonate, $5, and diisopropanolamine, $6, are 0.001, 8.34, 2.46, 5.94, 3.97, and 4.82 in $/m 3, respectively. These costs include the cost of regeneration and make-up. Stream data for the MSA's are given in Table 8.6. The values of eft a n d e jS are taken to be 0.0 and 2 x 10 -7 kmol/m 3 respectively. Synthesize a REAMEN which features the minimum number of units that realize the minimum operating cost. 8.4 Most of the world's rayon is produced through the viscose process (E1-Halwagi and Srinivas, 1992). Figure 8.7 provides a schematic representation of the process. In this process cellulose pulp is treated with caustic soda, then reacted with carbon disulfide to produce cellulose xanthate. This compound is dissolved in dilute caustic soda to give a viscose syrup, which is fed to vacuum-flash boiling deaerator to remove air. The gaseous stream leaving the deaerator, RI, should be treated for the removal of H2S prior to its atmospheric discharge. In spinning, a viscose solution is extruded through fine holes submerged in an acid bath to produce the rayon fibers. The acid-bath solution contains sulfuric acid, which neutralizes caustic soda and decomposes xanthate and various sulfurcontaining species, thus producing H2S as the major hazardous compound in the exhaust gas stream, R2.
210
CHAPTER EIGHT Synthesis of reactive mass-exchange networks
Caustic Soda
Cellulosle Pulp Steeping Press
I
Xanthating Churn
R 1,to
Dissolver Reactive
Viscose
Syrup Deaerator
Deaerator Exhaust,R
Mass
1
R 2, to atmosphere
Exchange
Gaseous Waste, R
Network v
Spinning
i
i
"
Rayon Fibers (to finishing)
~
F i g u r e 8.7 Simplified flowsheet for viscose rayon production (from E1-Halwagi and Srinivas, Synthesis of reactive mass-exchange networks, Chem. Eng. Sci., 47(8), p. 2116, Copyright 9 1992, with kind permission from Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington 0X5 1GB, UK).
It is desired to synthesize a REAMEN for treating the gaseous wastes (R1 and RE) of a viscose rayon plant. Three MSAs are available to select from. These MSAs are caustic soda, $1, (a process stream already existing in the plant with ml = 5.0 kmol/m3), diethanolamine, $2, (with m2 = 2.0 kmol/m 3) and activated carbon, $3. The unit costs for $2 and $3 including stream makeup and subsequent regeneration are 64.9 $/m 3 and 169.4 $/m 3, respectively. Stream data are given in Table 8.7. The chemical absorption of H2S into caustic-soda solution (Astarita and Gioia, 1964) involves the following two reactions: H2S + OH- ~ HS- + H20
(8.46)
HS- + OH- ~
(8.47)
S 2- + H20.
Since the concentration of water remains approximately constant, one can define the following two reaction equilibrium constants KI.1 -"
[HS-] [H2S][OH-]
(8.48)
Problems
211
Table 8.7
Stream Data for the Viscose-Rayon Example Rich streams
Lean streams
xjS
xjt
Stream
Gi
(kmol/m 3)
(kmol/m 3)
Stream
(m3/s)
(kmol/m 3)
(kmol/m 3)
R1 R2
0.87 0.10
1.3 x 10-5 0.9 x 10-5
2.2 x 10-7 2.2 x 10-7
S1
2.0 x 10-4
S2
(x~
S3
OO
0.0 2.0 x 10-6 1.0 x 10-6
0.1 1.0 x 10-3 3.0 x 10-6
y~
y~
LjC
and [S 2- ] K2.1 "-
[HS-][OH-]
(8.49)
where KI,1 = 9.0 x 106 m3/kmol and K2,1 = 0.12 m3/kmol. The distribution coefficient for the physically dissolved portion of H2S between a rich phase and caustic-soda solution is given by
yi/al = 0.368.
(8.50)
The overall reaction of hydrogen sulfide with diethanol amine (Lal et al., 1985) is given by H2S + (C2H4OH)2NH ~
H S - + (C2H4OH)2NH +,
(8.51)
for which the equilibrium constant is given by K1.2 = 234.48 =
[HS-][R2NH +] [H2S][R2NH]
(8.52)
and the physical distribution coefficient is given by (Kent and Eisenberg, 1976):
Yi/a2 -- 0.363.
(8.53)
The adsorption isotherm for H2S on activated carbon is represented by (Valenzuela and Myers, 1989)
yi/a3
"- 0.015.
(8.54)
The following values for the minimum allowable composition differences are selected: e s = 3.50, e s = 0.014, e s = 10 -6, e R "- 10 -7, e R = 1.5 • 10-7kmol/m 3
8.5 Consider the Kraft pulping process shown in Fig. 8.8 (Dunn and E1-Halwagi, 1993). The first step in the process is digestion in which wood chips, containing primarily lignin, cellulose, and hemicellulose, are "cooked" in white liquor (NaOH, Na2S, Na2CO3 and
212
CHAPTER EIGHT Synthesis of reactive mass-exchange networks
WhiteLiquor
WoodChips;
i Pulpto PaperMachine
Digester [ Canstieizer
]
I
I
Lime
[
Weak
Eor,,'or,,
e~ Lim
~
t
~'~ I~e
Dissolving
r - f - - Tank ---
~
VicthanolAmiue~EA), S4 ~
ActivatedCarbon,S 5 Hot K2CO3Solution,S 6
Wastewater to Additional Treatment
~ ~
IL
' BlackLiquor, S 3
Air Stripping
_Recovery Furnace
Water ud
Contaminated Condensate
Blar "
Green Liquor.
LimeKiln
.L_
~
~'~
Reactive
I
Mass-Exchange Network
i'~"
Sl, to digestor $2, to slaker "S3 to evaporators $4 to regeneration SS to regeneration -'-$6 to regeneration
To atmosphere Figure 8.8 A simplified flowsheet of the Kraft pulping process (Dunn and E1-Halwagi, 1993).
water) to solubilize the lignin. The off gases leaving the digester contain substantial quantifies of H2S. The dissolved lignin leaves the digester in a spent solution referred to as the "weak black liquor". This liquor is processed through a set of multiple-effect evaporators designed to increase the solid content of this stream from approximately 15% to approximately 65 %. At the higher concentration, this stream is referred to as strong black liquor. The contaminated condensate removed through the evaporation process can be processed through an air stripper to transfer sulfur compounds (primarily H2S) to an air stream prior to further treatment and discharge of the condensate stream. The strong black liquor is burned in a furnace to supply energy for the pulping processes and to allow the recovery of chemicals needed for subsequent pulp production. The burning of black liquor yields an inorganic smelt (NaaCO3 and Na2S) that is dissolved in water to produce green liquor (NaOH, Na2S, Na2CO3 and water), which is reacted with quick lime (CaO) to convert the Na2CO3 into NaOH. The conversion of the Na2CO3 into NaOH is referred to as the causticizing reaction and involves two reactions. The first reaction is the conversion of calcium oxide to calcium hydroxide in the presence of water in an agitated slaker. The calcium hydroxide subsequently reacts with NaECO3 to form NaOH and a calcium carbonate precipitant. The calcium carbonate is then heated in the lime kiln to regenerate the calcium oxide and release
Problems
213
Table 8.8 Data for the Gaseous Emission of the Kraft Pulping Process
Flowrate Gi, m3/s
Supply composition (10-5 kmol/m3) yS
Target composition (10-7 kmol/m3) y[
3.08
2.1
8.2
2.1
1.19
2.1
Stream
Description
R1
Gaseous waste from recovery furnace
117.00
R2
Gaseous waste from evaporator Gaseous waste from air stripper
0.43
R3
465.80
carbon dioxide. These reactions result in the formation of the original white liquor for reuse in the digesting process. Three major sources in the kraft process are responsible for the majority of the H2S emissions. These involve the gaseous waste streams leaving the recovery furnace, the evaporator and the air stripper, respectively denoted by R1, RE and R3. Stream data for the gaseous wastes are summarized in Table 8.8. Several candidate MSAs are screened. These include three process MSAs and three external MSAs. The process MSAs are the white, the green and the black liquors (referred to as S1, $2 and $3, respectively). The external MSAs include diethanolamine (DEA), $4, activated carbon, $5, and 30 wt% hot potassium carbonate solution, $6. Stream data for the MSAs is summarized in Table 8.9. Synthesize a MOC REAMEN that can accomplish the desulfurization task for the three waste streams.
Table 8.9 Data for the MSAs of the Kraft Pulping Example
Stream
Description
$1 $2 $3
White liquor Green liquor Black liquor DEA Activated carbon Hot Potassium carbonate
S4
$5 $6
Upper bound on flowrate L jc m 3/s
Supply composition (kmol/m3) xjs
Target composition (kmol/m3) xjt
0.040 0.049 0.100 cxz oo
0.320 0.290 0.020 2 x 10, 6 1 x 10-6
3.100 1.290 0.100 0.020 0.002
o~
0.003
0.280
214
CHAPTER EIGHT Synthesis of reactive mass-exchange networks
Symbols aj A bz,j bz~
Bj Bz,j
cj
fj Pj ai
i
J k
Iqj,j Lj mj np
NR Ns NZj qj
Qj r 2
R
Ri S
sj uj xj xj$ xjt Xj,k
xj Y Yi
y7 y* Z
physically dissolved concentration of A in lean stream j, kmol/m3 key transferable component composition of reactive species Bz,j in lean phase j, kmol/m3 admissible composition of reactive species Bzj in lean phase j, kmol/m3 set of reactive species in lean stream j index of reactive species in lean stream j unit cost of the jth MSA j, ($/kg) equilibrium distribution function between rich phase composition and total content of A in lean phase j, defined in Eq. (8.9) equilibrium distribution function between rich phase composition and physically dissolved A in lean phase j, defined in Eq. (8.8) flow rate of rich stream i, kmol/s index for rich streams index for lean streams index for composition intervals equilibrium constant for the qjth reaction in lean phase j flow rate of lean stream j, kmol/s upper bound on flow rate of lean stream j, kmol/s index for subnetworks molarity of lean stream j, kmol/m3 number of mass-exchange pinch points in the problem number of rich streams number of lean streams number of reactive species in lean stream j index for the independent reactions in lean stream j number of independent reactions in lean stream j correlation coefficient set of rich streams the ith rich stream set of lean streams the j th lean stream fractional saturation of chemically combined A in the jth MSA composition of key component in lean stream j, kmol/m3 supply composition of key component in lean stream j, kmol/m3 target composition of key component in lean stream j, kmol/m3 the upper bound composition for interval k on the scale Sj, kmol/m3 equilibrium composition of key component in lean stream j, kmol/m3 composition of key component in any rich stream, kmol/m3 composition of key component in rich stream i, kmol/m3 supply composition of key component in rich stream i, kmol/m3 target composition of key component in rich stream i, kmol/m3 equilibrium composition of key component in any rich stream, kmol/m3 index for the reactive species
References
215
Greek letters
t~k t~i,k ,s I)qj ,z,j
~qj
residual load leaving interval k, kmol/s residual load leaving interval k for rich stream i, kmol/s rich-phase minimum allowable composition difference, kmol/m 3 lean-phase minimum allowable composition difference, kmol/m 3 stoichiometric coefficient of reactive species z in reaction qj in lean phase j extent of reaction qj in the jth MSA
Special symbol []
concentration, kmol/m 3
References Astarita, G., and Gioia, E (1964). Hydrogen sulfide chemical absorption. Chem. Eng. Sci., 19, 963-971. Astarita, G., Savage, D. W., and Bisio, A. (1983). "Gas Treating with Chemical Solvents," John Wiley and Sons, New York. Dunn, R. E, and E1-Halwagi, M. M. (1993). Optimal recycle/reuse policies for minimizing the wastes of pulp and paper plants. J. Environ. Sci. Health, A28(1), 217-234. E1-Halwagi, M. M. f1971). An engineering concept of reaction rate. Chem. Eng., May, 75-78. E1-Halwagi, M. M. (1990). Optimization of bubble column slurry reactors via natural delayed feed addition. Chem. Eng. Commun., 92, 103-119. E1-Halwagi, M. M., and Srinivas, B. K. (1992), Synthesis of reactive mass-exchange networks. Chem. Eng. Sci., 47(8), 2113-2119. Espenson, J. H. (1981). "Chemical Kinetics and Reaction Mechanisms," McGraw-Hill Book Company, New York. Fogler, S. H. (1992). "Elements of Chemical Reaction Engineering," Prentice Hall, Englewood Cliffs, NJ. Friedly, J. C. (1991). Extent of reaction in open systems with multiple heterogeneous reactions. AIChE J., 37(5), 687-693. Kent, R. L., and Eisenberg, B. (1976). Better data for amine treating. Hydrocarbon Processing, February, pp. 87-90. Kohl, A., and Reisenfeld, F. (1985). "Gas Purification," 4th ed., Gulf Publishing Co., Houston, TX. Lal, D., Otto, E D., and Mather, A. E. (1985). The solubility of H2S and CO2 in a diethanolamine solution at low partial pressures. Can. J. Chem. Eng., 63, 681-685. Levenspiel, O. (1972). "Chemical Reaction Engineering" 2nd ed., John Wiley and Sons, New York. Smith, W. R., and Missen, R. W. (1982). "Chemical Reaction Equilibrium Analysis: Theory and Algorithms" John Wiley and Sons, New York.
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CHAPTER EIGHT Synthesis of reactive mass-exchange networks
Srinivas, B. K., and E1-Halwagi, M. M. (1994). Synthesis of reactive mass-exchange networks with general nonlinear equilibrium functions. AIChE J., 40(3), 463-472. Valenzuela, D. P., and Myers, A. (1989). "Adsorption Equilibrium Data Handbook," Prentice Hall, Englewood Cliffs, NJ. Warren, A., Srinivas, B. K., and E1-Halwagi, M. M. (1995). Design of cost-effective wastereduction systems for synthetic fuel plants. J. Environ. Eng., 121(10), 742-747. Westerterp, K. R., Van Swaaij, W. P. M., and Beenackers, A. A. C. M. (1984). "Chemical Reactor Design and Operation," John Wiley and Sons, New York.