6188
Ind. Eng. Chem. Res. 2003, 42, 6188-6195
Rate-Based Mixed-Pool Model of a Reactive Distillation Column Antti Pyha1 lahti Neste Engineering, P.O. Box 310, 06101 Porvoo, Finland
Kaj Jakobsson* Helsinki University of Technology, P.O. Box 6100, 02015 Hut, Finland
This paper presents a rate-based mixed-pool model for a reactive distillation column. In the model the tray is divided horizontally into cells where reaction and mass transfer takes place. The basic mass-transfer model applied is the Maxwell-Stefan model, which is implemented in several variants. Various vapor flow patterns can be can approximated by connecting the vapor flows rising from the tray below in different ways. That way it is possible to model the case where the vapor is completely mixed between the trays, the case where vapor is not mixed between the cells and liquid flows always in the same direction with respect to the vapor flow over the trays, and finally the case where vapor is unmixed between the cells and the liquid flow direction is reversed when it flows from a tray to the next tray below. The model is used to test the effect of various vapor flow patterns in a reactive distillation system where the reaction equilibrium is kinetically limited. The model shows a significant effect of the flow pattern on the concentration profiles on the tray. However, the effect on the overall performance of the column depends heavily on the application. When the model is applied to a relatively tall column with dilute feed and moderately exothermal reaction, the effect of the vapor flow pattern is small. On the other hand, the computation cost increases significantly when the number of equations increases. When the model is applied to a reactive distillation system with very high reaction enthalpy, concentrated feed, and a small number of stages, the effect of the lateral concentration profiles is very significant. In the case of a short column, the cost of computation is also not as serious a problem as that with a tall one. Thus, the most likely application for a model of this type in the near future is simulation of absorbers with highly exothermal reactions. Introduction The liquid flow pattern on the tray has an influence on the mass-transfer and reaction rates on a distillation tray. Traditionally, this phenomenon has been lumped together with many other factors affecting the performance of the tray to a quantity called stage efficiency. This has been a practical approach to correct the nonideal behavior of distillation trays. The works of Murphree,1 Hausen,2 and Standart3 are well-known. In recent times, nonequilibrium models or rate-based models4,5 have been introduced. These models are based on calculated mass-transfer fluxes and thus take at least some of the factors included in the efficiency models into account without involving the concept of “efficiency”. The situation becomes even more complicated if a reaction takes place on the trays. Reaction rates depend on concentrations and temperatures, and thus a rigorous model should consider their possible variations. Simulation of a reactive distillation column is a challenging task even without such complications, and thus not many authors have attempted to deal with this problem. Alejski6 has presented a mixed-pool model for reactive distillation, where the mass-transfer model applied to the individual cells was based on a traditional equilibrium stage model and efficiencies. Higler et al.7 have presented a multicell tray model for the reactive distillation column, which is a further elaboration of the mixed-pool model. In that case, the liquid phase on the tray is divided into cells both horizontally and vertically. The mass-transfer model applied in each cell is the nonequilibrium rate-based model. The vertical division
of the liquid phase improves the mass-transfer model of a real tray where vapor rises through the liquid and its concentration is changing continuously. However, the presented solution addresses only the cases where vapor is assumed to be completely mixed between the stages. However, according to Lockett,8 the vapor is practically unmixed between the stages of industrial-scale columns. As part of the Brite Euram III program, a simulation program for reactive distillation applying the rate-based approach has been written. This simulation program contains several hydrodynamic models of the tray; see further work by Kenig et al.9 The most sophisticated models, the mixed-pool model and eddy diffusion model, take the lateral concentration gradients into account. In this paper, the rate-based mixed-pool model is presented. Mixed-Pool Model The principal idea of the mixed-pool model is that the liquid on the tray is assumed to flow through a series of internally completely mixed cells. It can be shown that this kind of system can describe approximately the solution of the eddy diffusion model. It has the advantage that the second-order differential equation group involved in the eddy diffusion model is replaced with a group of algebraic equations. This is an advantage because the computational expense of solving the eddy diffusion problem rigorously in a nonideal, reactive multicomponent system is very high. Of course, if the case is so easy that analytical solution is possible, the eddy diffusion equation is convenient, but this is seldom
10.1021/ie030105l CCC: $25.00 © 2003 American Chemical Society Published on Web 10/31/2003
Ind. Eng. Chem. Res., Vol. 42, No. 24, 2003 6189
Figure 1. Mixed-pool model of a distillation tray when vapor is unmixed and liquid flows in a contrary direction on adjacent trays.
the case with reactive distillation problems, which involve in most cases highly nonideal mixtures with large thermal effects. In Figure 1 is presented the division of the tray into mixed cells. In this case vapor is assumed to flow unmixed from each cell to the corresponding cell on the upper tray and liquid is flowing to opposite directions on the adjacent trays so that the liquid entering the tray is contacted first with vapor from the outlet side cell of the tray below. This is the normal flow pattern with cross-flow trays. With some tray constructions, it is also possible to change the liquid flow pattern so that the liquid entering a tray encounters first the vapor leaving from the liquid inlet side of the tray below. A third flow pattern considered is such that all vapor leaving a tray is assumed to suffer complete mixing between the trays. Lewis10 has treated the theoretical efficiencies in these cases. The basis of the current model is the second-generation rate-based stage model as presented by Taylor and Krishna5 and published in 1993. This model is modified by introducing reaction terms to the balances, but in the basic case of mixed vapor and mixed liquid, the differences are otherwise minor. The model is based on the Maxwell-Stefan multicomponent mass-transfer equations, which take into account the mass-transfer resistances of each phase and also the interactions of the transferring components. An alternative vaporliquid mass-transfer model available is the modification of the previous one as presented by Hung11 and Taylor et al.12 In that model, the mass-transfer resistances of the individual phases are combined to obtain an overall mass-transfer resistance. The main benefit of this model is that it makes it possible to handle vapor plug flow conveniently. On the other hand, lumping the masstransfer resistances of the individual phases poses significant problems. The liquid-film-reaction masstransfer model of Kenig and Go´rak13 is available if fast reaction takes place in the liquid film. The model for reaction and mass transfer in a macroporous catalyst by Sundmacher and Hoffmann14 is implemented as well. Although all of these models are important for the current work, they are taken as given and thus not discussed further here.
the nomenclature based on the cases presented by Lewis:10 Lewis case 1 means total vapor mixing, Lewis case 2 is the case with unidirectional liquid flow on adjacent trays, and Lewis case 3 is that with liquid flow changing direction on each adjacent tray without lateral mixing of vapor. The basic connection schemes are obvious. Case 1 can be simulated by combining all vapor flows rising from one tray, calculating the average composition and enthalpy of the combined stream, and distributing this flow to the pools of the next tray. Case 2 is approximated by connecting the vapor flow from the first cell of the lower tray to the first cell of the tray above, etc., until the last pair of cells on each tray (in the current model, indexing of the pools starts always from the liquid inlet side of the tray). Case 3 is approximated in the same way, but the cells are connected so that the first pool of the lower tray is connected to the last cell of the tray above, etc. Thus, the vapor concentration profiles are simulated with a piecewise constant function. Obviously, a large number of mixed cells would give the best approximation of a continuous vapor distribution. However, because there are no more different liquid conditions available than the number of mixed cells on each tray, there is no advantage of using a larger number of vapor flows. In all cases, it is assumed that the same number of vapor moles enters each liquid cell on each tray. This may not be strictly correct because the vapor throughput of the tray varies slightly from point to point because of the variation of liquid and vapor properties, but because the underlying assumption is that each cell represents an equivalent area on the tray, it is thought to be a reasonably good approximation, at least considering the other simplifications made. The rigorous solution would be to solve the vapor flows through each cell so that the vapor pressure drop of each cell of a tray would be the same. In practice, this would increase the number of the independent variables, and considering the already very large size of the problem, this was not done. When the total vapor mixing is assumed, fulfilling the condition of equal distribution is trivial. However, when vapor is considered unmixed, corrections are necessary to allow the nonuniform change of the vapor flow rate in all cells over the tray. For example, if cold liquid feed enters cell 1 of a tray, the vapor flow is reduced much more in that cell than in the last cell N of the same tray. This effect is especially important in the case of unidirectional liquid flow (Lewis case 2) because if there is a regular change of the vapor flow pattern from tray to tray, this effect is accumulating. This may lead to an anomalous situation with negligible vapor flow on one side of the column and an overly high flow rate on the opposite side. This problem is nonexistent in the complete vapor mixing model and much less profound when liquid changes direction from tray to tray so that the differences have better possibilities to level out. Here the vapor flows are redistributed using special correction factors, which are presented together with other model equations in the next section. Model Equations for the Mixed-Pool Model
Vapor Flow Models Theoretically, the mixed-pool model can approximate the cases of Lewis by connecting the vapor flows rising from the tray below in different ways. Here we apply
Every mixed cell has virtually the same variables and equations as the whole tray in the rate-based model. The pressure is assumed to be the same all over the tray. The liquid is assumed to be homogeneous in the
6190 Ind. Eng. Chem. Res., Vol. 42, No. 24, 2003
direction perpendicular to the direction of the flow. Thus, if the number of mixed cells is p and the number of components is c, there are 5pc + 5c + 1 independent variables on a tray instead of 5c + 6 variables of a corresponding, otherwise identical, rate-based stage without division to mixed cells. The reboiler and condenser are treated as ideal stages, and thus they have only 2c + 4 independent variables each. One benefit of the selected structure of the model is that the simulation of the various vapor flow patterns requires only small changes to the basic equation group. In the actual model, the independent variables of the tray are as follows: leaving liquid flow rates Lk,j (p variables), bulk liquid mole fractions xi,k,j (pc variables), I liquid mole fractions at the interface xi,k,j (pc variL (p variables), liquid temperatures in the bulk Ti,k,j I (p variables), ables), temperatures at the interface Ti,k,j mass-transfer fluxes at the interface Ni,k,j (pc variables), pressure pj (one variable), vapor temperature in the bulk V (p variables), vapor mole fractions at the interface Ti,k,j I yi,k,j (pc variables), vapor mole fractions in the bulk yi,k,j (pc variables), and leaving vapor flow rates Vk,j (p variables). Here subscript i refers to the component, k to the mixed cell, and j to the tray. The tray index j starts from 1 at the reboiler and ends at N at the condenser, corresponding to the actual engineering practice of numbering the trays but differing from the practice in chemical engineering literature. The numbering of the mixed cells of a tray starts always from 1 at the liquid inlet edge, as is shown in Figure 1. The corresponding equation group consists of the following equations. First, p equations are the liquid total mass balances for the mixed cells: c
fk,j ) (Lk-1,j +
∑ i)1
c
L Fi,k,j )+
∑ i)1
c
Ni,k,j +
(Lk,j +
∑ i)1
L Pk,j ),
c
L ri,k,j +
LF ri,k,j ∑ i)1
k ) 1, ..., p (1)
In all liquid-side balances, the feed and product streams (FL and PL) have a nonzero value only for cells located at the edges of the tray because normally liquid product is drawn from the downcomer and liquid feed is introduced either to the downcomer or near the inlet edge of the tray. Moreover, for the cell at the liquid inlet, i.e., cell 1 of tray j, the liquid entering the cell does not come from the adjacent cell of tray j but via the downcomer from the last cell of the next tray above. Formally, this may be handled by defining that in the equation above stream L0,j refers to stream Lp,j+1. The reactions taking place in the liquid phase are presented with two terms: L refers to the rate of reaction in the bulk liquid, ri,k,j LF represents the reaction rate in whereas the term ri,k,j the liquid film if the model of Kenig and Go´rak13 is used. L include the In the case of the solid catalyst, terms ri,k,j possible mass-transfer rate models concerning the catalyst. For example, when the Sundmacher and Hoffmann14 model is used, the mass-transfer resistances of the catalyst particles manifest themselves via this term. The liquid-component mass balances (cp equations) are L L ) + Ni,k,j + ri,k,j + fp+(k-1)c+i,j ) (Lk-1,jxi,k-1,j + Fi,k,j LF L - (Lk,j + Pk,j )xi,k,j, i ) 1, ..., c, k ) 1, ..., p (2) ri,k,j
The same notes as those with liquid total mass balances apply here. The liquid-side mass-transfer equation [p(c - 1) equations] is Lcalc fp(1+c)+(c-1)(k-1)+i,j ) Ni,k,j - Ni,k,j, i ) 1, ..., c - 1, k ) 1, ..., p (3a) Lcalc is the total vapor-side mass-transfer flux Here, Ni,k,j resulting from the mass-transfer correlations with the current values of the independent variables and Ni,k,j is the current value of the independent variable representing the same flux. Obviously, these should be the same when convergence is achieved. When the masstransfer model of Hung,11 based on the overall masstransfer resistances, is applied, eq 3a becomes irrelevant and is replaced with the equation
I - xi,k,j, i ) 1, ..., fp(1+c)+(c-1)(k-1)+i,j ) xi,k,j c - 1, k ) 1, ..., p (3b)
The liquid-side interface concentration summation equation (p equations) is c
f2pc+k,j )
I xi,k,j - 1, ∑ i)1
k ) 1, ..., p
(4)
The liquid energy balance (p equations) is c
∑ i)1
f(2c+1)p+k,j ) [Lk-1,jhk-1,j + (
L Fi,k,j )hFj ]
+
L Ek,j
+
L Qk,j
p L )hk,j, k ) 1, ..., p (5) (Lk,j + Pk,j
It is worth noting that there is no special term for the heat of the reaction. The component enthalpies include the heat of formation, and thus the reaction enthalpy is already included in the enthalpies of the flows. The interface energy balances (p equations) force the heat fluxes to be the same on both sides of the interface: L V - Ek,j , k ) 1, ..., p f(2c+2)p+k,j ) Ek,j
(6)
The interface equilibrium equation (pc equations) is I I - yi,k,j , i ) 1, ..., f(2c+3)p+(k-1)c+i,j ) Ki,k,jxi,k,j c, k ) 1, ..., p (7)
The pressure drop of the tray is represented by one equation:
f(3c+3)p+1,j ) pj - (pj+1 + ∆pj+1)
(8)
Here the pressure drop can be given as a specification or be calculated based on average liquid and vapor compositions on the tray. As discussed earlier, this means simplifying assumptions concerning the vapor flow pattern. The vapor-phase-related equations form the remaining part of the tray equations. In this part the first set is the vapor energy balances (p equations). For the complete vapor mixing case, these become
Ind. Eng. Chem. Res., Vol. 42, No. 24, 2003 6191
1 c V F M M V f(3c+3)p+1+k,j ) Vk,j-1 Hk,j-1 + ( Fi,j )Hj - Ek,j p i)1 QVj PV Vk,j + Hk,j + , k ) 1, ..., p (9a) p p
(
∑
)
Thus, eqs 9a-c can be applied to its evaluation by replacing the enthalpy with the mole fraction. The last equation of the group is the vapor-side total mass balance (p equations): M f(5c+4)p+1+k,j ) Vk,j-1 +
The first term on the right-hand side represents the vapor flow from the tray below to the current cell of the current tray. For cases with complete vapor mixing, the value is the same for all cells of the tray and it can evaluated from equation M M Hk,j-1 ) Vk,j-1
1
p
∑ (Vm,j-1Hm,j-1)
pm)1
(9b)
For the Lewis cases 2 and 3 without vapor mixing, this term becomes M M Vk,j-1 Hk,j-1 ) (Vn,j-1 - Cn-1,j-1)Hn,j-1 + Cn,j-1Hn+1,j-1 (9c)
where k ) 1,..., p, n ) k, for unidirectional liquid flow on adjacent trays and k ) 1, ..., p, n ) p - k + 1, for contrary liquid flow on adjacent trays. The requirement of equimolar vapor flow to each mixed cell is satisfied using the correction factors Cn,j-1, which are determined by applying condition
Vn,j-1 + Cn,j-1 - Cn-1,j-1 )
1
p
M , ∑ (Vm,j-1) ) Vn,j-1 pm)1
n ) 1, ..., p (10)
At the edges of the tray, the correction factors C0,j-1 and Cp,j-1are set to zero. Then follows the vapor-side interface concentration summation equation (p equations): c
f(3c+4)p+k+1,j )
I yi,k,j - 1, ∑ i)1
k ) 1, ..., p
(11)
The next part of the equation group is the vapor-side mass-transfer equation [p(c - 1) equations]: Vcalc - Ni,k,j, i ) 1, ..., f(3c+5)p+(k-1)(c-1)+1+i,j ) Ni,k,j c - 1, k ) 1, ..., p (12) Vcalc is the total vapor-side massAs in eq 3a, here Ni,k,j transfer flux resulting from the mass-transfer correlations with the current values of the independent variables and Ni,k,j is the current value of the independent variable representing the same flux. These should be the same when convergence is achieved. The following equation is the vapor-side component mass balance (pc equations):
V Fi,j - Ni,k,j p PV V Vk,j + y + ri,k,j (13) p i,k,j
M M yk,j-1 + f(4c+4)p+1+(k-1)c+i,j ) Vk,j-1
(
)
The first term on the right-hand side is totally analogous to the one used in the first term on the righthand side of the vapor-side energy balance, eq 9, only the enthalpy is replaced with the vapor mole fraction.
1
c
∑ pi)1
c
V Fi,j +
V ri,k,j - Ni,k,j ∑ i)1
PVj Vk,j - , k ) 1, ..., p (14) p M is the same for all cells of tray j and for all Here Vk,j-1 vapor flow cases because that was just the condition set for it. Thus, there are 5pc + 5c + 1 equations per tray, which matches the number of variables. The reboiler and condenser are treated as equilibrium stages.
Solver The mixed-pool model was solved using Newton’s method. The Thomas algorithm for a block tridiagonal matrix with numerically calculated derivatives was used to solve the linearized subset of the model equations. This is a well-known method that converges well when initial values are good, but it is known as well that without proper initial values the iteration might not converge. As such, the implementation of the solver to this problem posed no special problems, but it was noticed that very heavy damping of the correction steps was necessary in order to keep the solution on the convergence path. Initialization of the Variables Initialization of variables is of crucial importance for finding the solution. One obvious way to provide good initial values is to use a solution with one mixed cell per tray as the basis of the solution. This approach was applied here. The problem is solved first with a ratebased model with each tray as a single mixed cell, and the resulting concentrations, temperatures, etc., are used as the initial values for the multiple-cell mixedpool model. This method gave satisfactory results so that convergence was achieved. On the other hand, using the rate-based model is a rather heavy initialization procedure. This problem is not prohibitive because the mixed-pool model is probably not a tool for every routine simulation. More likely, it will be used for checking the design of some critical devices, and for this case, a tedious initialization process is tolerable. Test Example 1 and Simulations The model has been applied to some test systems. In Figure 2, the column of the test case is presented. It is a small-scale reactive distillation column involving a methyl tert-butyl ether (MTBE) reaction. The simulated column has 20 trays (reboiler tray 1, condenser tray 20). Reactions take place in the reactive section in the middle of the column (trays 8-13). The upper and lower ends of the column consist of inert trays. The simulation involves four components: methanol (MeOH), isobutene, MTBE, and isobutane. The reaction equation is
MeOH + isobutene T MTBE
(15)
Isobutane is an inert component. The system is other-
6192 Ind. Eng. Chem. Res., Vol. 42, No. 24, 2003
Figure 3. Mole fractions of isobutene along the flow path of the reactive trays in example 1, case 3.
Figure 2. Column arrangement of example 1.
wise similar to a normal MTBE reactive distillation column, but the reactive trays are treated as cross-flow trays with catalyst placed as an even layer on the reactive trays. The current industrial catalytic packings do not follow this cross-flow pattern, but here the goal is to test the significance of this flow pattern to the composition profiles and consequently the effect to the reaction rates on the trays, e.g., to find out if developing such devices would be worthwhile. Because there were no results achieved with cross-flow catalyst arrangements available, it was not considered to be feasible to consider catalyst mass-transfer resistances in this study. This test case was simulated using seven different flow models. Case 1 was a mixed liquid-mixed vapor model without division to cells. Case 2 was a mixed liquid-vapor plug-flow model without division to cells. Case 3 was a mixed-pool model otherwise exactly identical with case 1 but with four cells per stage and with complete vapor mixing between the trays (Lewis case 1). Case 4 was otherwise identical with case 2 but with each tray divided into four cells and with complete vapor mixing between the trays. Cases 5-7 applied to this problem were mixed-pool models with five cells per tray and with vapor flow patterns corresponding to the Lewis cases 1-3, respectively. In Figure 3 are presented the calculated molar fractions of isobutene on the reactive trays 8-13 with case 3 (mixed vapor-mixed liquid, four cells per tray). From Figure 3, it can be observed that there is a significant composition gradient of the reactive component along the flow path of the tray. In Figure 4 is presented the formation rate of MTBE on these trays in case 3. In Table 1 are shown the feed and product rates of each component in case 3. Table 2 presents the results of simulating example 1 in case 1 (mixed vapor-mixed liquid, one cell per tray).
Figure 4. Production rates (mmol/s) of MTBE along the flow path of the reactive trays 8-13. Table 1. Feed and Product Flows of the Example 1, Case 3, Mixed Liquid-Mixed Vapor Model with Four Cells per Stage, mmol/s component
feed 1
feed 2
distillate
bottom
methanol isobutene MTBE isobutane
0 12.000 0 68.000
14.000 0 0 0
3.635 1.658 0.322 67.146
0.127 0.103 9.916 0.854
Table 2. Feeds and Products of the Example 1, Case 1, Mixed Liquid-Mixed Vapor Model without Division to Cells, mmol/s component
feed 1
feed 2
distillate
bottom
methanol isobutene MTBE isobutane
14.000 0 0 0
0 12.000 0 68.000
3.634 1.684 0.384 67.090
0.161 0.111 9.820 0.911
Table 3. Feeds and Products of the Example 1, Case 4, Mixed Liquid-Vapor Plug-Flow Model with Four Cells per Stage, mmol/s component
feed 1
feed 2
distillate
bottom
methanol isobutene MTBE isobutane
0 12.000 0 68.000
14.000 0 0 0
3.515 1.501 0.228 67.331
0.60 0.074 10.197 0.669
Table 3 presents the feed and product rates of each component in example 1 in case 4 or the mixed liquidvapor plug-flow model by Hung11 and Taylor et al.12with four cells per tray. Table 4 presents the results of simulating example 1 in case 2, otherwise the same as the previous model but with each tray as a single cell. For brevity, the results of solving this example in cases 5-7 corresponding to Lewis cases 1-3 with five cells per tray are not presented because the difference from the results of cases 1-4 was insignificant. It was interesting to observe that the composition profiles in
Ind. Eng. Chem. Res., Vol. 42, No. 24, 2003 6193 Table 4. Feeds and Products of the Example 1, Case 2, Liquid-Vapor Plug-Flow Model without Division to Cells, mmol/s component
feed 1
feed 2
distillate
bottom
methanol isobutene MTBE isobutane
14.000 0 0 0
0 12.000 0 68.000
3.539 1.506 0.257 67.285
0.58 0.091 10.142 7.19
Table 5. Feeds and Products of the Example 2, Case 1, Mixed Liquid-Mixed Vapor Model without Division to Cells, mol/s feed distillate bottom reaction
isobutene
methanol
MTBE
isobutane
6.000 1.438 3.862 -0.700
5.000 3.957 0.343 -0.700
0.000 0.288 0.411 0.700
5.000 4.616 0.384 0.000
Table 6. Feeds and Products of the Example 2, Case 2, Mixed-Pool Model with Five Cells per Stage and the Mixed Liquid-Mixed Vapor Model, Assuming Complete Vapor Mixing between the Stages (Lewis Case 1), mol/s feed distillate bottom reaction
isobutene
methanol
MTBE
isobutane
6.000 2.049 2.606 -1.345
5.000 2.808 0.848 -1.345
0.000 0.702 0.643 1.345
5.000 4.096 0.904 0.000
the column as well as in the product streams were almost identical with all of the calculated cases. Test Example 2 and Simulation The effect of the tray scale on concentration gradients seemed very small between the various models applied to test example 1. However, one would expect a significant effect at least if the number of trays is small and the heat effects are large so that the lateral variations of the flow rates and concentrations on the individual trays cannot cancel each other out. For testing this hypothesis, another test case was set up. The column should be short. Thus, the test column consisted only of six stages: reboiler, lower feed stage, two reactive stages, upper feed stage, and condenser. The heat of reaction should be high; thus, the MTBE system of example 1 was modified so that the reaction enthalpy was increased by 150 kJ/mol. Thus, the reaction system was changed from reasonably exothermal to highly exothermal. The isobutene-isobutane mixture enters from the lower feed tray and methanol from the upper feed tray, but the feed mixture is much more concentrated, containing 50% of isobutene. The results achieved using the mixed liquid-mixed vapor model with five mixed cells per stage and with complete vapor mixing between the stages are presented in Table 5. The reference case, otherwise identical but without a division to mixed cells, is presented in Table 6. As can be seen, the effect of tray-scale phenomena is very significant. The reaction rate achieved with the reference model in this case is approximately 1.3 mol/ s, whereas according to the mixed-pool model, the reaction rate is only 0.7 mol/s. Also, the distribution of the components between the bottom product and distillate is clearly different. This example, as such, is artificial, but the key issue here is to illustrate under what kinds of conditions the tray-scale effects may be important. Such conditions are more likely to occur in a reactive absorber than in a distillation column.
Figure 5. Liquid streams (mol/s) leaving the stages in example 2 (ref L ) reference calculation with each tray as a single cell; mp L ) calculated with five mixed cells per tray).
Conclusions and Discussion As can be seen in Figure 3, the change of concentrations over a tray is significant in comparison to the difference between adjacent trays. However, Tables 1 and 2 show that the final difference between the mixedpool model and an otherwise identical rate-based model but without division to cells is small. In these tables are compared product flows and compositions of the same problem solved using a rate-based mixed liquidmixed vapor model with each tray as a single cell and a mixed-pool model with four cells per tray. The probable reason is that in a relatively tall distillation column there is rarely such a huge concentration difference over a single tray that the effect of its concentration profile on the overall performance of the column would be significant. MTBE formation is a reaction in which conversion is limited by equilibrium and tends to compensate the changes as well. In the same way, the results involving the vapor plugflow model (Tables 3 and 4) are rather near to each other, but both differ somewhat more from the results achieved with the mixed vapor model. Thus, it seems that, at least in this case, the method used for calculating the local mass-transfer rate is more important than the tray-scale hydrodynamic model. Also, when the effect of the vapor flow pattern from plate to plate (Lewis cases 1-3) was tested, the results differ from the mixed vapor case very little. In a short column with high heat of reaction, the situation is very different. In test example 2 is simulated such a case. The resulting rate of reaction with complete liquid mixing gave an overall reaction rate of 1.3 mol/s, whereas the same problem solved using a mixed-pool model gave an overall reaction rate of only 0.7 mol/s. Figure 5 illustrates the large variation of the internal liquid streams of the column due to the heat of reaction. The example could be relevant to industrial practice for short columns involving high thermal effects. This kind of equipment could be, for example, a reactive absorber. They are often used for highly exothermal systems and, moreover, they have frequently a rather small number of stages. On the other hand, the increase of the computation time is very considerable. When a problem with 4 components and 20 trays in one iteration round with all the necessary hydraulic and thermodynamic calculations was solved using the rate-based model with each plate as a single cell, taking one Newton step lasted on
6194 Ind. Eng. Chem. Res., Vol. 42, No. 24, 2003
average approximately 0.6 s (with a PC equipped with a 1 GHz AMD Athlon processor running Linux and with a program compiled with g77 without any optimization). When the mixed-pool model with five mixed pools was applied, taking one full Newton step lasted on average 16 s; thus, the computation time of a basic iteration step increased by a factor 26. For each tray, there are 26 variables in the rate-based model and 126 variables in the mixed-pool model. Thus, the increase of the time needed for taking one Newton step is approximately proportional to the square of the number of equations of one tray. There was no significant difference between various vapor flow cases. With the current method, this is to be expected because the calculations involved with the vapor distribution are very simple. As an illustration of the increase of the number of equations involved, as mentioned before, there are 5pc + 5c + 1 independent variables on a tray described with a mixed-pool model instead of the 5c + 6 equations of a corresponding normal rate-based model. The resulting amount of equations is formidable even for a moderately sized problem. For example, in a column with reboiler, condenser (treated as ideal stages with 2c + 4 variables each), and 18 trays, 4 components, and 5 mixed pools on each tray, there are 126 equations on each tray with mixed pools and 2292 equations and variables in the whole column. Approximately the same amount of equations results in a normal rate-based model with 23 components present. It is commonplace that engineering problems involve 20-40 components and several dozen trays, and if conditions would require 10-20 mixed pools, the corresponding equation group involves tens of thousands of equations. Applying these models to such problems with current computers is obviously feasible only in very exceptional cases. In the near future, the most likely application is checking the design of some short but large-diameter columns, with significant variation of flow over the trays. Thus, for example, absorbers involving highly exothermal reactions seem to be a more likely application for these types of models than traditional distillation columns. Acknowledgment The support by the European Commission in the form of the BRITE-EURAM program (CEC Project No. BE951335) is greatly acknowledged. The support for K.J. in the form of the PROTEK program of the Academy of Finland is greatly acknowledged. Symbols c ) number of components Ck, j ) vapor flow correction factor for cell k on stage j, defined in eq 10 L Ek, j ) liquid-side energy-transfer rate in cell k on stage j, J/s V Ek, j ) vapor-side energy-transfer rate in cell k on stage j, J/s fl, j ) value of discrepancy function l on tray j, dimension depends on the equation L F i,k,j ) liquid feed of component i to cell k on stage j, mol/s V F i,j ) vapor feed of component i F hj ) liquid feed enthalpy, J/mol
to stage j, mol/s
hk,j ) liquid enthalpy in cell k on tray j, J/mol M Hk,j ) enthalpy from cell k on stage j corrected with the factors Ck,j
HFj ) vapor feed enthalpy, J/mol Hk, j ) vapor enthalpy in cell k on tray j, J/mol Ki,k, j ) distribution factor of component i in cell k on tray j Lk, j ) liquid flow out from cell k on stage j, mol/s Ni,k,j ) mass-transfer rate of component i in cell k on stage j, current value of the variable, mol/s Vcalc Ni,k,j ) vapor-side mass-transfer rate of component i in cell k on stage j, as solved from Maxwell-Stefan equations, mol/s Lcalc Ni,k,j ) liquid-side mass-transfer rate of component i in cell k on stage j, as solved from Maxwell-Stefan equations, mol/s p ) number of mixed cells on each tray P Lj ) liquid product flow rate from tray j, mol/s PVj ) vapor product flow rate from tray j, mol/s QLj ) external energy input to the liquid on stage j, J/s QVj ) external energy input to the vapor on stage j, J/s L ri,k, j ) production rate of component i in cell k on stage j, mol/s I Tk, j ) temperature of the vapor-liquid interface in cell k on stage j, C L Tk,j ) temperature of the liquid in cell k on stage j, C V ) temperature of the vapor in cell k on stage j, C Tk,j Vk, j ) vapor flow out from cell k on stage j, mol/s M ) vapor flow from cell k on stage j corrected with the Vk,j factors Ck,j xi,k,j ) molar fraction of component i in the liquid in cell k on tray j I xi,k,j ) molar fraction of component i in cell k on tray j at the interface xi,k,j ) molar fraction of component i in cell k on tray j I yi,k,j ) molar fraction of component i in the vapor at the interface in cell k on tray j at the interface yi,k,j ) molar fraction of component i in the vapor bulk leaving from cell k on tray j at the interface M yi,k,j ) molar fraction of component i from cell k on stage j corrected with the factors Ck,j p ) number of mixed cells pj ) pressure on tray j, Pa ∆pj ) pressure drop over tray j, Pa
Subscripts i ) ith component k ) kth mixed cell n ) vapor stream flowing from tray j - 1 to the kth mixed cell on tray j j ) jth tray from the column bottom Superscripts I ) interface M ) corrected value L ) liquid phase V ) vapor phase F ) feed
Literature Cited (1) Murphree, E. V. Rectifying column calculations with Particular reference to N component mixtures. Ind. Eng. Chem. 1925, 17, 747. (2) Hausen, H. A definition of exchange efficiency of rectifying plates for binary and ternary mixtures. Chem.-Ing.-Tech. 1953, 25, 595. (3) Standart, G. Studies on distillationsV. Chem. Eng. Sci. 1965, 20, 611.
Ind. Eng. Chem. Res., Vol. 42, No. 24, 2003 6195 (4) Krishnamurthy, R.; Taylor, R. A Nonequilibrium Stage Model of Multicomponent Separation Processes, I: Model Description and Method of Solution. AIChE J. 1985, 31, 449. (5) Taylor, R.; Krishna, R. Multicomponent Mass Transfer; Wiley: New York, 1993. (6) Alejski, K. Computation of the Reacting Distillation Column Using a Liquid Mixing Model on the Plates. Comput. Chem. Eng. 1991, 15, 313. (7) Higler, A.; Krishna, R.; Taylor, R. Nonequilibrium Cell Model for Multicomponent (Reactive) Separation Processes. AIChE J. 1999, 45, 2357. (8) Lockett, M. J. Distillation Tray Fundamentals; Cambridge University Press: Cambridge, U.K., 1986; p 226. (9) Kenig, E.; Jakobsson, K.; Banik, P.; Aittamaa, J.; Go´rak, A.; Koskinen, M.; Wettmann, P. An integrated tool for synthesis and design of reactive distillation. Chem. Eng. Sci. 1999, 54, 1347. (10) Lewis, W. K. Rectification of binary mixtures. Ind. Eng. Chem. 1936, 28, 399.
(11) Hung, J.-S. A second generation nonequilibrium model for computer simulation of multicomponent separation processes. Ph.D. Thesis, Clarkson University, Potsdam, NY, 1991. (12) Taylor, R.; Kooijman, H. A.; Hung, J.-S. A 2nd Generation Nonequilibrium Model for Computer-Simulation of Multicomponent Separation Processes. Comput. Chem. Eng. 1994, 18, 205. (13) Kenig, E. Y.; Go´rak, A. A Film Model Based Approach for Simulation of Multicomponent Reactive Separation. Chem. Eng. Process. 1995, 34, 97. (14) Sundmacher, K.; Hoffmann, U. Development of a New Catalytic Distillation Process for Fuel Ethers via a Detailed Nonequilibrium Model. Chem. Eng. Sci. 1996, 51, 2359.
Received for review February 3, 2003 Revised manuscript received September 15, 2003 Accepted September 22, 2003 IE030105L
