Ind. Eng. Chem. Res. 1989, 28, 1397-1405
Glinos, K.; Malone, M. F. Optimality Regions for Complex Column Alternatives in Distillation Systems. Chem. Eng. Res. Des. 1988, 66, 229. Petlyuk, F. B.; Platonov, V. M.; Slavinskii, D. M. Thermodynamically Optimal Method for Separating Multicomponent Mixtures. Int. Chem. Eng. 1965, 5(3), 555. Shiras, R. N.; Hanson, D. N.; Gibson, C. H. Calculation of Minimum Reflux in Distillation Columns. Ind. Eng. Chem. 1950,42(5), 871. Terranova, B. E.; Westerberg, A. W. Temperature-Heat Diagrams for Complex Columns. 1. Intercooled/Interheated Distillation Columns. Ind. Eng. Chem. Res. 1989, first of three in a series in this issue. Underwood, A. J. V. Fractional Distillation of Multicomponent Mixtures-Calculation of Minimum Reflux Ratio. J. Inst. Petrol. 1946, 32, 614. Underwood, A. J. V. Fractional Distillation of Multicomponent Mixtures. Chem. E m . Prop. 1948. 44. 603.
Region IV is similar to region I, although $2 is the active root. Equation 3 written with this root can be rearranged to obtain
V=
aMC f M C
ca i t if i
aMC - 4’2
EMC+
i # M C a i - $2
1397
(41)
Similar linear expressions may be found for cases with more than one middle component.
Literature Cited Carlberg, N. A.; Westerberg, A. W. Temperature-Heat Diagrams for Complex Cohmns. 2. U ~ d ~ r w o o dM&hod ’s for Side Strippers and Enrichers. Ind. Eng. Chem. Res. 1989, second of a series of three in this issue. Fidkowski, Z.; Krolikowski, L. Thermally Coupled System of Distillation Columns: Ootimization Procedure. AIChE J . 1986. 32(4), 537.
Received f o r review November 28, 1988 Revised manuscript received May 19, 1989 Accepted June 15, 1989
Dynamic Simulation of Packed- and Tray-Type Absorbers Chittur Chandrasekharan Lakshmanan BHP Melbourne Research Laboratories, 245 Wellington Road, Mulgrave 31 70, Victoria, Australia
Owen Edward Potter* Department of Chemical Engineering, Monash University, Clayton 3168, Victoria, Australia
T h e “cinematic” model is applied t o simulate the dynamic behavior of gas absorbers (packed and tray type) subjected to disturbances in the composition of the entering gas. The simulations include linear and nonlinear equilibrium relations. For the linear absorption case, the results of the cinematic model are compared with the analytical solution. For the nonlinear case, the comparisons are made with the numerical solution obtained by using a n IMSL package based on the GEAR algorithm. In both cases, very good agreement is achieved. T h e effect of the number of trays on the approach to the steady state is also shown for both linear and nonlinear cases. Particularly in the nonlinear case, a significant delay in the approach t o steady state is observed. T h e effect of using two approaches of the cinematic model, namely, forward and reverse processing, on the results of the simulation are discussed. The earliest work on the dynamic simulation of gas absorbers giving time domain solutions is that of Jaswon and Smith (1954). This is a general solution involving evaluation of a complicated series. This approach has been criticized by Tommasi and Rice (1970) as a solution for an unusual set of boundary conditions. Liapis and McAvoy (1981) have developed a method that involves linearization and perturbation to overcome the slow rate of convergence of the solution proposed by Jaswon and Smith (1954). Gray and Prados (1963) have proposed three models, namely, the slug flow model, the mixing cell model, and the axial diffusion model, and reported frequency response studies. Bradley and Andre (1972) have published a dynamic analysis of a packed gas absorber for a system involving absorption followed by chemical reaction using the method of characteristics. Srivastava and Joseph (1984) have used orthogonal collocation to solve the dynamic equations describing binary-component and multicomponent distillation in packed beds to overcome the large dimensionality that is unavoidable for a finite difference method. Tan and Spinner (1984) have also used the method of characteristics to transform the model equations to a set of ordinary differential equations along the characteristic paths and solved the resulting system of equations by modified Euler’s method to obtain the solution along the characteristic paths. After this step, the 0888-5885/89/2628-1397$01.50/0
results are converted to the time domain. Adaptive grid techniques using the method of lines together with biased upwind finite difference representation of the derivatives are used by Schiesser in the package DSS (1977) to simulate a variety of heat- and mass-transfer problems including a packed gas absorber. Lakshmanan and Potter (1984,1987) have developed a new numerical model, namely, the “cinematic” model, to simulate the dynamic behavior of countercurrent systems in the time domain. This model has been applied successfully to a number of dynamic simulation problems including double-pipe heat exchangers, plate heat exchangers, packed gas absorbers, and gas-solid fluidized beds. Based on these numerical experiments, it has been found that this model is computationally fast and accurate. Also, problems usually encountered by other numerical schemes, such as convergence and inversion of results from the transformed domain to the time domain, etc., are not encountered in using the cinematic model. The problem of obtaining the dynamic response becomes increasingly difficult when the system under consideration has severe nonlinearities. Therefore, in this paper, the cinematic model is applied to gas absorption in packed- and tray-type columns. Firstly, the application will be for a tray-type absorber. Secondly, the application will be extended to a packed g a s absorber. Linear and nonlinear equilibrium
0 1989 American Chemical Society
1398 Ind. Eng. Chem. Res., Vol. 28, No. 9, 1989
cases will be considered for the tray absorber problem, and the application to a packed absorber will be illustrated for the case of linear equilibrium. The results from the cinematic model will be compared with those obtained by analytical solution in the case of linear equilibrium. As there exists no analytical solution for the nonlinear problem considered here, the numerical solution obtained by using the DGEAR algorithm from the IMSL package (1982) will be used to perform the comparison.
Application of the Cinematic Model to Gas Absorption Tray Absorber. Prior to the disturbance, solute-free gas and liquid flow through the column. At t = 0, a step increase in the solute concentration of the gas entering the absorber is effected. The cinematic model is applied to simulate the dynamic response of the absorber until steady state is attained. The calculations involved in using the cinematic model consist of two major steps, namely, the mixing step and the mass-transfer step. The sequence in which these two steps are applied to a particular problem differentiates between the two approaches of processing, namely, forward and reverse processing. Note that gas enters the absorber at tray N and the liquid enters a t 1. In applying the cinematic model to the gas absorption problem, the following assumptions are made: (1)The absorber is assumed to be operating under isothermal conditions. (2) Perfect mixing is assumed in both gas and liquid phases. (3) Each tray is assumed to be an ideal stage. (4) On each tray, the exchange of solute between the two phases is assumed to be taking place at equilibrium. ( 5 ) The gas and liquid holdups in each tray are assumed to be constant. (6) A linear equilibrium relationship is considered. Forward Processing. In this mode of processing, the mixing step takes place first followed by the mass-transfer step. The mixing step is considered to be infinitely fast, thereby allowing the whole time for the mass-transfer step which is considered to occur at equilibrium. A quantum of liquid is defined as a fraction of the liquid held on a tray. This quantum, QHL, of the liquid entering the absorber is added to the top tray, and the added quantum trickles through all trays and exits at the liquid outlet. As a result of this mixing step, the mole fraction of the liquid phase in each tray changes. As perfect mixing is assumed in the liquid phase, the new set of liquid-phase mole fractions in every tray can be calculated by using the equation given by Mason and Piret (1950). If x j o is the initial mole fraction of solute in the liquid on tray j before the addition of the quantum and xo is the mole fraction of the solute in the liquid quantum added, then the new set of mole fractions can be given by
x , is the mole fraction of the solute in the nth tray after the mixing step. The mixing step occurs for a time interval At, which is equal to QHLILM, where LMis the molal rate of flow of liquid entering the absorber. Therefore, eq 1 gives the solute mole fraction leaving the nth tray at the end of At. It may be noted that the molar density of the contents of the tray is assumed to be constant in the derivation of eq l. The liquid leaving the absorber is given by the time-averaged value calculated as
By the use of eq 1, the integral in eq 2 can be evaluated and the average mole fraction, can be given as
g, N-1
= xo
+e-pt Iktkdt At SA‘ 0 k=O
(3)
where P = LM/HL and I N - j = [ ( x j 0 - x o ) / ( N- j)!]PN-’for j = 1, 2, ..., N . The integration in eq 3 is performed analytically. The mixing step is now complete, and the mass-transfer step begins. The mass-transfer step is considered to be taking place during all of the time interval At. In this step, a quantum of gas appropriate to the liquid feed quantum is introduced at the gas inlet. In other words, this quantum of gas is equal to AtGM. Also, QHL equals LMAt. Let be the time-averaged mole fraction of solute in the gas leaving the (n + 1)th tray. A differential mass balance can be written for the transfer of solute between the two phases as
HLdx,/dt
= G M [ G- ~
n l
(4)
and the mole fraction y n of the solute in the gas leaving tray n can be given by integrating eq 4 as
where yn,ois the mole fraction of the solute in the gas on tray n at the beginning of the mass-transfer step. Note that this ignores the holdup in the gas phase dispersed in the liquid. Also, the mass-transfer step occurs at equilibrium, implying x , = y,/m. The time-averaged mole fraction of the solute in the gas leaving tray n can be given bY ,, =
Jn
cAtd t
-
JO
Substituting At = QHL/LM after the evaluation of the can be obtained as integral in the above equality,
Substituting eq 5 in the above equality, it can be seen that the time-averaged value of the mole fraction of the solute in the gas leaving tray n can be given by
~
It must be noted that yN+lequals yo, the inlet gas composition. Equations 4-6 are valid only when the absorption equilibrium is linear (y* = mx). Nonlinear equilibrium will be considered later. By the use of eq 1-6, the sequence
Ind. Eng. Chem. Res., Vol. 28, No. 9, 1989 1399 of calculations that needs to be coded into a computer program is given next. Algorithm for Forward Processing. (a) Input number of trays, gas- and liquid-phase flow rates, the equilibrium relation, and the mole fraction of solute in the feed gas. (b) Initialize the gas- and liquid-phase compositions in all trays. (c) Choose a quantum size. (d) Begin the mixing step. (e) Calculate the new set of liquid-phase compositions for all trays using eq 1 and 2. (f) Calculate the time-averaged composition of the liquid leaving the absorber using eq 4. (g) Introduce an appropriate quantum of gas at the gas inlet. (h) Calculate the gas- and liquid-phase compositions after the transfer step for all trays. (i) Calculate the composition of gas leaving the absorber. 6) Reinitialize the gas- and liquid-phase compositions for all trays. (k) Repeat steps c-j until steady state is attained. Algorithm for Reverse Processing. In this approach, a quantum of gas is introduced first and the calculations are performed for the mass-transfer step. Then an appropriate quantum of liquid is introduced at the top of the absorber, and the mixing step is commenced. The sequence in which the calculations are to be performed is given next. (a) Initialize the gas- and liquid-phase compositions in every tray. (b) Introduce a quantum of gas a t the gas inlet. (c) Calculate the gas- and liquid-phase compositions after the transfer step for all trays by using eq 5 and 6. (d) Calculate the composition of gas leaving the absorber. (e) Begin the mixing step by introducing an appropriate quantum of liquid at the liquid inlet. (f) Calculate new set of liquid-phase compositions on trays after the mixing step using eq 1. (g) Calculate the composition of liquid leaving the absorber by using eq 3. (h) Reinitialize the gas- and liquid-phase compositions on each tray. (i) Repeat steps b-h until steady state is established. Example. Consider a tray absorber through which solute-free liquid and gas flow initially. At t = O+, the concentration of the incoming gas is stepped up to a particular value. The required data are given as follows: number of trays = 6; flow rate of liquid = 40.8 mol/min; flow rate of gas = 66.7 mol/min; y* = mx = 0.72~;HL= 75 mol/plate; H G = 1 mol/plate; liquid entering is solute free; at t = 0+, the mole fraction of solute in the feed gas is increased to 0.3. The calculations are performed by using both forward and reverse approaches of the cinematic model until steady state is achieved. The results are also compared with the results obtained using the analytical solution given by Douglas (1972). In the forward processing scheme, the disturbance gets recorded in the liquid leaving the absorber one At later than predicted by the analytical solution. This is due to the fact that the mixing step occurs prior to the transfer step in the case of the forward-processing scheme. In the problem considered here, the concentration of the solute in the absorber is zero prior to the start-up. Therefore, the quantum trickled through the trays at the beginning of the forward processing has no chance to meet the solute until it exits the absorber, which takes At time units. This
I
y
x xx
x-x-x-XX
X-X-XX-X-X-
20 0
0
TRAYt6
60 0
0
LO
TIME (MINI
x x x-x-x-
x-X
X-X-X-X
X-X-X
TRAY* 6
Y
0.3
0.2
0.1
0
0
20 0
LO TIME ( M I N I X X X
0
60 0 A n a l y t i c a l solution
- Cinematic mode Cluantdv size = 0 05
Figure 2. Dynamic response of a tray absorber (linear equilibrium-liquid phase) for a quantum size of 0.05. The symbol X represents the analytical solution, and the continuous line denotes the result of the cinematic model. The tray numbers are shown.
explains the time delay introduced by the forward-processing scheme. Except for this time delay, the profiles agree well with the corresponding ones from the analytical solution. This time delay amounts to 0.3676 which, equals QHL/LM. It is clear that the time delay is directly proportional to the quantum size used. However, the gasphase profiles are not affected by this time delay. Hence, the reverse approach is preferred when the disturbance to the system enters only through the entering gas. Simu-
1400 Ind. Eng. Chem. Res., Vol. 28, No. 9, 1989
lations performed with various quantum sizes (0.05, 0.2, 0.5, 1.0) indicated that quantum sizes 0.05, 0.2, and 0.5 produced identical results. The results shown in Figures 1 and 2 correspond to a quantum size of 0.05. The profiles for quantum sizes 0.2 and 0.5 overlap with these profiles exactly. Figure 1 also can be considered as obtained by the forward-processing scheme. Figure 2 requires a shift in the time values before it can represent the liquid-phase profiles for the forward-processing scheme. This means that t = 0 corresponds to 0.3676 for the forward-processing scheme. Because of this minor difference, plots are not shown for each scheme separately. The example problem is also solved by using the DGEAR subroutine of IMSL (1982), and the cpu times are compared. The cpu time taken by DGEAR is 2.68 s, while the cinematic model with a quantum size of 0.2 requires only 1.27 s for the same buffer i/o (inputfoutput), directory ifo, and page faults. The results agreed well from the time the solute concentration is stepped up until the steady state is established. These computations are performed on a Vax 8700 using double-precision floating-point arithmetic. This clearly indicates that the cinematic model is fast in providing an accurate dynamic solution. Nonlinear Equilibrium. Consider absorption of sulfur dioxide in water in a tray absorber with the following operating conditions: carrier gas is air; N = 13; LSM = 1891.39 mol/h; GMB = 30 mol/h; HL = 5.56 mol of absorbent/plate; gas holdup is neglected. Prior to the disturbance, solute-free air and water flow through the absorber, and at t = 0+, the solute concentration is stepped up to 0.5 mol of solute/mol of carrier gas in the feed. The cinematic model is then used to simulate the approach to steady state of this system. The following equilibrium relations are fitted:
+ a2x2 + a3x3+ a4x4 y = bo + b,x + b2x2+ b3x3
y = a,n
for 0 5 x 5 0.000422 for x
> 0.000422
It must be noted that these equilibrium relations use mole fractions of the solute in gas and liquid. The constants are a, = 14.37; a2 = -65454.04; a3 = 0.3696 X lo9; a4 = -0.4591 X bo = -0.4091 X bl = 27.689; b2 = 425.496; b3 = -6936.947. Due to the absorption of concentrated gas, the temperature of the liquid increases in the column. For the example considered here, this increase due to the heat of absorption alone amounts to about 8 "C. This nonisothermal situation can be considered by including a heat balance for each tray in the absorber. However, for the purpose of illustration of a new model to simulate the dynamics of the absorber, heat effects are not considered in this work. It may be recalled from the results obtained for the linear equilibrium case that the reverse approach produces the least error for a start-up problem such as the one considered here. Hence, for the nonlinear equilibrium case, calculations are shown only for reverse processing. Firstly, the mixing step is performed as described for the linear equilibrium case. The new set of concentrations are given as n C,o - Co C, = Co + e-QZ------@-j ( n- j ) !
;=,
LSM
where Q = -At
(7)
HL
Note that, due to the absorption of concentrated gas, there is a change in density of the liquid phase through the absorber. Because of this, the mixing equation (eq 7) uses concentrations instead of mole fractions. This calculation is repeated for all n from 1 to N. The concentration of
liquid leaving the absorber,
G,is calculated as
where P = L S M / H Land IN-j = [(cj0 - Co)/(N-j)!]PN-jfor j = l , 2 ,..., N . From the data given, it can be inferred that the equilibrium and the operating line are curved. Since the molar density of the liquid changes through the column, the mole fraction of the solute in the liquid over trays after the mixing step is calculated from the molar concentrations using the following relation: x, =
18C, 18C, + 1
For the transfer of solute that takes place for a differential time At, the following mass balance can be written:
Since the equilibrium relation is highly nonlinear, the result of the transfer step cannot be given in closed form. However, the equilibrium can be represented by a linear or a quadratic function between the limits of mole fractions of solute in the liquid before and after the transfer step which occurs for a differential time At. This will allow the solution of the transfer step mass balance analytically. This approach will also save the cpu time considerably. In this work, no such approximation is made in order to assess the ability of the cinematic model to handle the complex nonlinear equilibrium considered. It may also be noted that eq 10 uses Gm instead of G M , as the gas velocity changes significantly between the inlet and the outlet for the problem chosen. Equation 10 together with the nonlinear equilibrium sets up an initial value problem which is highly nonlinear. The solution is obtained using the DGEAR routine of the IMSL (1982) package. The numerical solution is carried out until t = At to give the mole fraction of the solute in the liquid phase after the transfer step. The corresponding gas-phase mole fraction is obtained by using the equilibrium relation. The time-averaged mole fraction of the solute leaving tray n is obtained from an overall mass balance around tray n given by r
E/(l E).
and can be calculated as + x , , ~and x , , ~are the mole fractions of the solute in the liquid phase before and after the transfer step, respectively. Algorithm. (a) Input GMB,L S M ,H L , HG, and N. (b) Choose quantum size Q. (c) Initialize the mole ratios of the solute in the gas and liquid phases on all trays. (d) Commence the mixing step by introducing the quantum of liquid a t the top of the absorber. (e) Calculate the new set of compositions in the liquid phase by using eq 7 and 8. (f) Calculate the time-averaged composition of the liquid leaving the absorber by using eq 9. (9) Begin the mass-transfer step by introducing an appropriate quantum of gas at the gas inlet. (h) Calculate the compositions after the transfer step by using eq 10 and 11. (i) Calculate the time-averaged composition of the gas leaving the absorber by using eq 11.
Ind. Eng. Chem. Res., Vol. 28, No. 9, 1989 1401
0012 TRAY# 6
Z
0.3
TRAY#5
TRAY# L -
0
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-
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80
TIME (MIN 1
XXX
-
GEAR C r n e m a t ~ cmodel
Ouantum ssze
X XX
-
GEAR
Cinematic model
Quantum s l z e = 0 2
Figure 3. Dynamic response of a tray absorber (nonlinear equilibrium-gas phase) for a quantum size of 0.2. The symbol X represents the numerical solution from GEAR,and the continuous line denotes the result of the cinematic model. The tray numbers are shown.
6 ) Reinitialize
*
the compositions of the gas and liquid
phases. (k)Repeat steps d-j until steady state is established. Numerical Solution Using GEAR. Assuming perfect mixing in both gas and liquid phases, the dynamic mass balance for the nth tray can be written as
+
0
06
O’I 0
K(Q)
LIOUID
LL
l n
This equation can be integrated for n = 1to N from t = 0 until steady state is established. yo is the gas composition at the gas inlet, and XNfl is the composition of the liquid entering the column. The integration is performed by using the DGEAR subroutine of the IMSL (1982), and the results are plotted in Figures 3 and 4 along with the results obtained from the cinematic model. It can be seen that the two results agree well. The results shown in Figures 3 and 4 correspond to a quantum size Q = 0.2. Figure 5 shows the effect of quantum size on the error introduced in steady-state recovery. The error in recovery is defined as shown: 70 error in recovery for the gas phase= YI(Q = 0.2) loo 1 -K(Q= 0.2) % error in recovery for the liquid phase = X d Q ) - X,(Q= 0.2) -ol X,(Q = 0.2)
02
A
GAS
w
t-
If the gas holdup is neglected as being insignificant, the mass balance can be rearranged to give
:
Figure 4. Dynamic response of a tray absorber (nonlinear equilibrium-liquid phase) for a quantum size of 0.2. The symbol X represents the numerical solution from GEAR, and the continuous line denotes the result of the cinematic model. The tray numbers are shown.
I
0
OL
08 12 QUANTUM SIZE 10)
I
16
I 20
Figure 5. Effect of quantum size on the recovery (steady statenonlinear equilibrium).
In calculating these errors, Q = 0.2 is chosen as the reference since the simulated results with this quantum size agreed perfectly well with the numerical solution produced by using DGEAR. The mole ratios used in the recovery calculations are the outlet mole fractions of the two phases. It can be inferred from Figure 5 that even with a quantum size of 2, the “percent” error in recovery (both gas and liquid phases) is less than 1% . This indicates that for a reasonable quantum size of 0.2, the solution will be very accurate. A quantum size of 0.2 indicates that a fifth of the liquid holdup on a tray will flow through the column during the mixing step. Figure 5 provides conclusive evidence that the method converges rapidly as the quantum size is reduced. In the case of nonlinear equilibrium, the number of trays in the absorber is varied and the dynamic simulations are performed by using the cinematic model and DGEAR.The
1402 Ind. Eng. Chem. Res., Vol. 28, No. 9, 1989
0'016
5 ~
I
I
N = 11.13.16 x~x~-x-x-x-x-x-x-x-x-xx x-x-x-x-x-x-
4
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tQ
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0
i
LiOCllC O U T L E -
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001L-
-
CINEMATIC
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r
N= 6
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,
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.
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120 180 2LO TIME ( M I N I Figure 8. Effect of the number of trays on the approach to steady state (linear equilibrium-composition of gas flowing out). The symbol X represents the analytical solution, and the continuous line denotes the result of the cinematic model. Quantum size is 0.2.
0
N=11
0.002-
X X X CINEMATIC O U A N T U M SIZEIOI'O;
120 180 2LO TIME ( M I N I Figure 7. Effect of number of trays on the approach to steady state (nonlinear equilibrium-composition of gas flowing out). The continuous line represents the numerical solution from GEAR, and the symbol X denotes the result of the cinematic model. Quantum size is 0.2.
0
60
dynamic profiles are shown to be in good agreement with each other as shown in Figures 6 and 7 . It is interesting that a change in the number of trays has a significant influence on the rate of approach to steady state of the gas phase but has not affected the rate of approach of the liquid phase. Increasing the number of trays increases the residence time of the gas phase in the absorber. But the observed delay in approaching steady state for the gas phase is more than the increase in residence time that can be accounted as due to the additional trays. To find out if this behavior is a characteristic of the nonlinear equilibrium, similar calculations are performed for the linear equilibrium as well. The results shown in Figure 8 indicate that increasing the number of trays increases the time required to attain steady state even for the linear equilibrium. From the analytical solution for a tray-type gas absorber (with linear equilibrium), the time required to attain equilibrium can be calculated as a function of the number of trays. Marshall and Pigford (1952) have defined the degree of approach to equilibrium as the difference be-
60
tween the driving force a t the top of the column at equilibrium and the instantaneous driving force and plotted this rate of approach versus the number of theoretical stages using the absorption factor as a parameter. From their graph, it can be seen that a delayed approach to steady state is possible. Jackson and Pigford (1956) carried out a similar study. It may be noted that the problems solved by them are either linear or a linearized nonlinear problem. As obtaining an analytical solution for a nonlinear problem such as the one considered in this work is not possible, a plot similar to the one given by Marshall and Pigford can be constructed only from a variety of accurate numerical solutions, which further justifies the need for an accurate numerical scheme. Packed Gas Absorber Problem. In the following analysis, the dynamic simulation of a packed gas absorber is considered. The following assumptions are made. (1) Plug flow is assumed in the gas and liquid phases. (2) Isothermal conditions are assumed. ( 3 ) Linear equilibrium relation is assumed. (4) Physical absorption is considered. ( 5 ) Radial dispersion in both the gas and liquid is considered to be absent. The dynamic mass balances representing the transfer of solute gas between the gas and liquid streams can be written as
-r - + - = NTUoG[y - ( m x ) ] "J
"I'
ar
az
The liquid enters at z = 0.
Cinematic Model as Applied to a Packed Gas Absorber The cinematic model is applied to the gas absorber problem defined in eq 13 and 14. The absorber is divided into N cells. Each cell consists of gas and liquid compartments. A quantum of liquid is flown in to the top cell and the contents of the liquid compartments in other cells are appropriately displaced down the column. Liquid is allowed to remain in each cell for a differential time At which is equal to ( l / N ) t h of 7. During this time, gas is flown in at the bottom of the column, and the exchange is allowed to take place in each cell for the differential
Ind. Eng. Chem. Res., Vol. 28, No. 9, 1989 1403 contact time. This exchange is given by
RESPONSE OF THE LlOUlD (2.1) v)
a
a In these equations, ylo is the composition of gas flowing past a liquid-filled compartment. These equations are specifically given for the first cell (near gas inlet). To solve this initial value problem, one needs the following initial conditions. A t the beginning of the exchange ( t = O), y1 = yl* and x = xg. The result of the exchange is given by the solution of these equations as
+
y1 = PleRIAr P2eRzAr+ ylo x =
P1KleRiA7 + P2K2eRzAT + Kola
0
z 2
P i
I 0
(17) (18)
The constants appearing in these equations are defined in the Nomenclature section. After the exchange, the composition of the gas in a cell is set at the value in equilibrium with the liquid in the same cell. The gas flowing out of the cell is trapped and mixed before it is flown in to the next cell above. The composition of the gas thus flown in to the cell above is given by
(19)
Equations 17-19 set up an algorithm that can be easily programmed to simulate the dynamics of a packed gas absorber. Algorithm. (a) Choose the number of cells N . (b) Initialize the gas and liquid compositions in each cell. (c) Admit a quantum of gas at the average composition for At. (Note that the average composition of gas entering the first cell is the same as the gas composition at the inlet.) (d) Compute the composition of gas and liquid after the exchange using eq 17 and 18. (e) Compute the average composition of gas leaving a cell and flow this gas in to the cell above for At, and continue this for all cells up to the top of the absorber by computing step d each time gas is flown. (f) Calculate the gas-phase composition in equilibrium with the liquid-phase composition left after the exchange. (8) Flow the contents of the liquid-phase compartment from each cell into the adjacent one, and reinitialize the compositions in each cell. (h) Repeat steps b-g until steady state is attained. An example of the application of the cinematic model to a start-up problem of a packed gas absorber will be considered next. Start-up Followed by a Step Increase in Gas Composition in a Packed Gas Absorber. Absorption of sulfur dioxide in water from a mixture of sulfur dioxide and air in a packed tower is considered but with a simplified equilibrium, i.e., linear equilibrium. Liquid holdup and the overall mass-transfer coefficient are obtained from Morris and Jackson (1953). Linear equilibrium is assumed to elucidate the idea in a simple manner. However, for a rigorous design, an exact equilibrium relation must be used. The data are as follows: y* = 32x; H = 8 m; cross-sectional
0
100
200 TIME ISECSI
300
coo
Figure 9. Approach to steady state of a packed gas absorber (gas and liquid compositions at the respective outlets are shown) for a start-up followed by a further step increase in gas composition at the inlet. N = 100.
area of the tower = 1.25 m2; a = 95 m2/m3packed volume; G = 0.02785 kmol/(s.m2); mG/L = 1.0; hG= 34.723 mol/m3; hL = 3333.33 mol/m3; KO, = 0.1694 mol/ [s.m24y - y*)]; packing = 2-in. stoneware rings. Initially, solute-free water and soluble-free air enter the absorber. Then, the mole fraction of sulfur dioxide in the inlet gas stream is increased to 0.0125 by a step input. The compartment of the gas stream in the cell adjacent to the gas inlet is filled with gas of this composition. The cinematic model is applied, and the computations are performed as described earlier until steady state (state 1) is established. This steady state is again disturbed by a second step increase in the inlet gas stream, resulting in a composition of 0.0150. Computations are continued with the model until a second steady state (state 2) is established. The responses at the outlets of the gas and the liquid streams are shown in Figure 9. The steady-state profiles are found to be in good agreement with the corresponding values obtained from the analytical solution of the steady-state mass balances.
Summary Based on the computations performed, it can be seen that the cinematic model is easy to use and provides an accurate account of the dynamic behavior of packed- and tray-type countercurrent systems. The application of this new model to other countercurrent systems will be reported elsewhere. It should not be misinterpreted that the cinematic model works well for a two-phase system only when the capacitance of one phase is significantly greater than that of the other phase. To demonstrate that this is not the case, Lakshmanan and Potter (1984) have shown that the model works well for a two-phase system in which the capacitance of one phase is almost comparable to that of the other phase (liquid-liquid heat exchanger). The proposed cinematic model saves cpu time in providing the same accuracy as standard numerical techniques for the following three major reasons. Firstly, the computations using the cinematic model do not involve any infinite series (method of characteristics and methods using Laplace transforms almost always require the evaluation of at least one infinite series), and therefore the problems associated with the evaluation of infinite series such as numerical convergence do not occur. Secondly, the cinematic model provides the solution in the time domain. It may be recalled that the solution using
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Ind. Eng. Chem. Res., Vol. 28, No. 9, 1989
method of characteristics or Laplace transforms generally involves the inversion of results from the transformed domain to the time domain. Often this happens to be a major effort, and numerical inversion methods have their own limitations. Finally, the separation of the mixing step and the transfer step in the manner handled by the cinematic model reduces the computation to repeated evaluation of the same equations followed by updating of the state vector a t every time step, which is performed by a set of nested “DO loops”. In order to handle the time variable inputs, the disturbance entering must be divided into segments of time span equal to A t , which is fixed by the quantum size chosen and the flow rates of the streams. During this At, a mean value for the input can be obtained, and the procedures outlined in the earlier sections can be adopted. In this process, care must be given to the nature of the time-varying function in the inputs (e.g., periodic behavior etc.). By use of this methodology, a frequency response study has been performed by Lakshmanan and Potter (1984) for a packed gas absorber problem. Further work on extending the application of the cinematic model to more complex situations including chemical reactions is in progress.
Nomenclature a = transfer area, m2/m3packed volume A1 = -[N+ N T U ~ G ] / T G A2 = mNTUoG/TG A3 = N / T G A, = N T U o L / ( m r d Co = concentration of the solute in entering liquid, mol/L C,, = concentration of solute in the liquid on tray j before the mixing step, mol/L C, = concentration of the solute in liquid on tray n, mol/L CN = time-averaged concentration of the solute in the liquid leaving the absorber, mol/L G = superficial gas velocity, kmol/(s.m2 tower) GM = flow rate of gas, mol of solute plus carrier gas/h GMB= flow rate of gas, mol of solute free gas/h H = height of the packed column, m hG = holdup in gas, mol/m3 packed volume HG = holdup of gas per plate, mol for linear equilibrium case only hL = holdup in liquid, mol/m3 packed volume H L = holdup of liquid per plate, mol for linear equilibrium case only H L = holdup of liquid per plate, mol on a solute-free basis for nonlinear equilibrium HTUoG = height of overall transfer unit on gas-phase basis = G / ( K o c a ) ,m Zk = summation variable defined in text K , = (Ri - AJ/A2 K2 = (R2 - Ai)/A2 K, = lJm K O G = overall mass-transfer coefficient on gas-phase basis, mol/ [s.m2.(y - y*)] L = superficial liquid velocity, mol/ (s.m2 column) LM = rate of flow of liquid, kmol/h LSM= rate of flow of liquid, mol/h solute-free basis m = equilibrium constant, y* = mx M1 = mA, - AI M 2 = mA3A, N = number of trays; also represents number of cells for packed gas absorber NTUoG = number of overall transfer units on gas-phase basis = HKoGa/G NTUoL = number of overall transfer units on liquid-phase basis = (mG/L)NTUoG
K2)14’10
Q = quantum size, fraction r = uL/uG
R1 = 0.5{-M1 + [ ( h f l 2- 4kf2)]1/z] R2 = 0.5(-M1 - [(MI’- 4M2)]1’2) At = time step uG = linear velocity of the gas phase = G / h c , m/h uL = linear velocity of the liquid phase = L / h L ,m/h x = mole fraction of solute in the liquid phase x o = mole fraction of solute in the liquid in a cell for packed absorber; also represents the mole fraction of solute in the liquid quantum added at the top of the absorber xjo = mole fraction of solute on tray j before the mixing step x, = mole fraction of solute on tray n after the mixing step x , , ~= mole fraction of the solute in the liquid on tray n before the transfer step x,,~ = mole fraction of the solute in the liquid on tray n after the transfer step x , + ~ = moles of solute per mole of solvent in the liquid phase on the (n + 1)th tray = time-averaged mole fraction of the liquid leaving the absorber y = mole fraction of solute in gas ylo = mole fraction of solute in gas flowing past a liquid-filled compartment Y , , ~= mole fraction of the solute in gas on tray n at the start of the mass-transfer step E = time-averaged value of the mole fraction of solute in gas leaving tray n yl* = mole fraction of solute in gas that is in equilibrium with the liquid of composition xo in a cell y* = mole fraction of solute in the gas phase in equilibrium with x, =mx YN+I = average mole fraction of solute in the gas leaving the absorber Y = mole ratio, moles of solute/moles of carrier gas Y , = time-averaged value of the mole ratio of solute to carrier gas in gas leaving tray n
5
Greek Symbols = residence time of gas phase = H / u G ,h = residence time of liquid phase = H / u L , h
iG T~
Literature Cited Bradley, K. J.; Andre, H. A Dynamic Analysis of a Packed Gas Absorber. Can. J. Chem. Eng. 1972,50, 528-533. Douglas, J . M. Process Dynamics and Control; Prentice-Hall: Englewood Cliffs, NJ, 1972; Vol. 1. Gray, R. I.; Prados, J. W. The Dynamics of a Packed Gas Absorber by Frequency Response Analysis. AZChE J. 1963,9(2), 211-216. IMSL, International Mathematical and Statistical Library, Houston, TX, 1982. Jaswon, M. A.; Smith, W. Countercurrent Transfer Processes in the Non-Steady State. Proc. R. SOC.1954,2254, 226-244. Jackson, R. F.; Pigford, R. L. Rate of Approach to Steady State by Distillation Column. Ind. Eng. Chem. 1956, 48(6), 1020-1026. Lakshmanan, C.; Potter, 0. E. Dynamics of Heat Exchangers and Fluidized Beds. Proc. 12th. Aust. Chem. Eng. Coni., Chemeca 84 1984, 2, 871-878. Lakshmanan, C.; Potter, 0. E. Cinematic Modeling of Dynamics of Solids Mixing in Fluidized Beds. Ind. Eng. Chem. Res. 1987, 26(2), 292-296. Liapis, A. I.; McAvoy, T. J. Transient Solutions for a Class of Hyperbolic Counter-current Distributed Heat and Mass Transfer Systems. Trans. Inst. Chem. Eng. 1981, 59, 89-94. Marshall, W. R.; Pigford, R. L. In The Application of Differential Equations to Chemical Engineering Problems. Internal Report, University of Delaware, 1952, pp 159-162. Mason, D. R.; Piret, E. L. Continuous Flow Stirred Tank Reactor Systems. Ind. Eng. Chem. 1950, 42(5), 817-825.
I n d . Eng. Chem. Res. 1989,28, 1405-1410 Morris, G. A.; Jackson, J. Absorption Towers; Butterworths Scientific Publication: Great Britain, 1953. Schiesser, W. E. DSSIP-Release 3. Internal Report, Lehigh University, Bethlehem, PA, 1977. Srivastava, R. K.; Joseph, B. Simulation of Packed-bed Separation Processes Using Orthogonal Collocation. Ind. Eng. Chem. 1984, 8(1),43-50. Tan, K. S.; Spinner, I. H. Numerical Methods of Solution for Con-
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tinuous Countercurrent Processes in the Nonsteady State. AIChE
J. 1984, 30(5), 770-779. Tommasi, G.; Rice, P. Dynamics of Packed Tower Distillation. Ind. Eng. Chem. Process. Des. Deu. 1970, 9, 234-243. Received for review November 12, 1987 Revised manuscript received May 1, 1989 Accepted May 16, 1989
Characteristics of a Cocurrent Multistage Bubble Column B. H. Chen and N. S. Yang* Chemical Engineering Department, Technical University of Nova Scotia, P.O. Box 1000, Halifax, Nova Scotia, Canada
Hydrodynamic and mass-transfer characteristics of a cocurrent multistage bubble column containing 37 plates have been determined as a function of column diameter, liquid viscosity, and flow rate of the two phases. T h e plates are of 6-mesh screen, having a fractional free area of 0.64. T h e use of screen plates has produced a homogeneous dispersion of nearly identical bubbles in the liquid in the column. It also has practically eliminated the diameter effect on the gas holdup, the bubble size, and the volumetric mass-transfer coefficient. In comparison with other gas-liquid contacting devices, the screen plate column has very small backmixing of the liquid phase, a large gas holdup, and a large mass-transfer coefficient. T h e coefficient is affected adversely by viscosity but positively by both gas and liquid flow rates. These favorable characteristics may manifest the high potential of the screen-plate column as an effective and economical gas-liquid contactor. The bubble column is generally an efficient and economical device for bringing about an intimate interfacial contact necessary for numerous gas-liquid or gas-liquidsolid operations in the chemical or biochemical industry. However, some of the hydrodynamic properties of the column, such as the bubble coalescence a t high gas flow rates and the severe backmixing of the liquid phase, may have limited its application at the present level. Detailed reviews of column characteristics were given recently by Shah et al. (1982) and Chen (1986). One method proposed to minimize these undesirable effects is to stage a bubble column. Voigt and Schugerl (1979) and Voigt et al. (1980) studied the absorption of oxygen in a countercurrent multistage bubble column containing perforated plates with downcomers and found a large increase in both a and kL over that reported for single-staged columns. The intensity of the longitudinal dispersion in the liquid was also lowered as a result of staging (Sekizawa and Kubota, 1974). Plates without downcomers but with up to 64% free area have also been tested, with similar findings (Yang et al., 1986a; Schugerl et al., 1977; Nishikawa et al., 1985). The level of accomplishment expected of a modified bubble column is obviously dependent on the nature of the internal solid. An ideal column plate must at least possess the following properties: (1) simple construction and operation, (2) a free area sufficient to produce a large gas holdup at low pressure drop, (3) uniform resistance to two-phase flow over the entire cross section, and (4) small volume relative to the column. It appears that plates made from wire-mesh screens could reasonably satisfy all of these criteria. This is well supported by the successful applications of wire screens in the aerodynamic industry (Schubauer et al., 1950; Laws and Livesey, 1978) and by their proven capability to improve the operation of many two-phase contacting devices (Chen, 1971). *Visiting Scholar from Dalian Institute of Technology, Dalian, People’s Republic of China.
0888-5885/ 8912628-1405$01.5010
Among the various types of screens that have been tested in previous studies (Chen and Vallabh, 1970; Voyer and Miller, 1968), the 6-mesh, 0.58-mm wire diameter woven screen has proven itself to be the most promising as a internal solid. This is mainly due to its particular geometry, which provides the least resistance to rising drops or bubbles while still being able to keep the bubble growth under control. The primary purpose of this study is to examine the operating characteristics of cocurrent multistage bubble columns fitted with screen plates and to compare the results with those previously published for other types of modified bubble columns. Some conclusions can then be drawn regarding its potential as a gas-liquid contacting device.
Experimental Section Apparatus. Figure 1 shows schematically the experimental apparatus used. Each of the three columns employed was constructed from two sections of 1.2-m-long Plexiglas tubing, with diameters of 0.05, 0.075, and 0.15 m, respectively. All three columns have an overall height of about 3 m. Only the 0.075-m-diameter column was insulated for dispersion study. The circular plates installed in each column were made from 6-mesh stainless steel wire-screen sheets having a fractional free area of 0.64. Thirty-seven such plates were mounted 0.05 m apart, forming a stack on a central shaft of 5-mm diameter. The plates were about 1.5 mm smaller in diameter than the column diameter for easy removal or installation. Air, after passing through a filter, a pressure regulator, and a calibrated rotameter, was admitted to a 0.2-m-long humidifying section at the bottom of the column. A distributor consisting of four 1.5-mm nozzles for the two smaller columns and eight such nozzles for the 0.15-m column was used to disperse the air through another 0.15-m-long humidifying section before entering the test section. The air rose in the form of bubbles through the series of 37 screen plates and left the column freely at the 0 1989 American Chemical Society
