Ind. Eng. Chem. Res. 1993,32, 511-518
511
Shortcut Procedure for Simulating Batch Distillation Operations Suresh Sundaram and Lawrence B. Evans’ Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
A fast, accurate model for batch distillation simulation will be useful both in batch distillation synthesis and in batch process design. A shortcut model using the Fenske-Underwood-Gilliland (FUG) equations of continuous distillation design was developed. The model consists of stepping forward in time using a first-order explicit integration scheme with a variable time step and solving the FUG equations a t each time step. A large number of example problems were used for model testing and validation. Agreement between the shortcut model and rigorous simulation was excellent. The model is a powerful and computationally fast tool that can be used both in batch process design and in synthesis of batch distillation systems. Introduction Specialty chemicals represent about 15% ’ of worldwide chemicals production with approximately $150 billion produced annually. Batch distillation is one of the most widely used separation processes in the manufacture of specialty chemicals. In contrast to continuousdistillation, until recently there have been few computer tools available for computer simulation and optimization of batch distillation. Batch distillations are performed based on trial and error experience, and the operations can be far from the economical optimum. There is a real need for systematic methodologies for the design and operation of batch separations. This paper is part of an overall research program aimed at developing improved design methods for batch processing. In this paper, we present a simplified model for batch distillation that is based on the Fenske-UnderwoodGilliland (FUG) shortcut equations of continuous distillation design. The basic assumption is that, at any instant in time, the batch column is identical to the rectifying section of a continuouscolumn. This permits the shortcut design methods which have been successfully used in simulation of continuous distillation columns to be used at every step in batch distillation. The model has shown excellent agreementwith rigorous simulation results when used for distilling multicomponent feed streams under the assumptions of constant molal overflow,and zero vapor, and liquid holdup. A shortcut model using the FUG equations was also developed independently by Diwekar et al. (1991) for design of batch columns. This paper differs from Diwekar et al. by formulating the model in a form that can be used conveniently to simulate an existing column. This paper also demonstratesthe validity of the model by an extensive comparison with the results of rigorous simulation.
ColburnandStearns (1941),RobinsonandGilliland (19501, and Pigford, Tepe, and Graham (1951). Huckaba and Danly successfullysolved batch distillation with enthalpy balances in 1960 for a constant holdup, adiabatic case. In 1963,Meadows presented the first model of multicomponent batch distillation, and used finite differencesto solve the set of equations. Distefano (1968) extended Meadows’ model and conducted a study on the degree of stiffness of the differential equations. An efficient method to solve the equations was developed in 1981 by Boston et al. (1981). Their solution technique, which used an “inside-out”algorithm (Boston, 1980),was demonstrated to be a robust and efficient method which could cope with the stiff nonlinear equations produced by the integration formula. Their work has been incorporated into a commerical batch distillation simulation package called BATCHFRAC. Cuille and Reklaitis in 1986 extended Meadows’ model to account for chemical reactions. Ruiz (1988) described the application of the generalized dynamic distillation model of Gani, Ruiz and Cameron (1986) to multicomponent distillation. Diwekar et al. (1991) presented a shortcut model based on the FUG method. The model was developed for two operating scenarios: constant overhead composition and constant reflux ratio. The constant overhead composition case was further classified into two cases, when all components have constant overhead compositions and when only one component has a constant overhead composition. When all components have constant overhead compositions, the FUG equations are used in a noniterative procedure to determinethe reflux ratio profile to maintain the specified compositions. A systematic comparison of their results with rigorous simulation was not made.
Fenske-Underwood-Gilliland Method for Design of Continuous Distillation Columns
Batch Distillation Simulation: Literature Review Seader (1988) recently presented an excellent review of modeling of batch distillation. Lord Rayleigh was the first to develop a mathematical relationship between the initial charge to a still pot in simple batch distillation, the liquid left in the pot at any time t , the liquid composition in the pot, and the vapor composition. Bogart, in 1937,developed a design method for the case of constant overhead composition for distillation with a mounted column. The procedure for constant reflux ration was treated by Smoker and Rose in 1940. Important investigations on the effect of holdup were made by Rose, Welshans, and Long (19401,
Before describing the FUG method for designing continuous columns, some important definitions are presented. The light key component (lk) is defined as one which is present in the residue in important amounts. All components lighter than the light key are present only in small amounts in the residue. The heavy key component (hk) is present in the distillate in important amounts. All components heavier than the heavy key are present only in small amounts in the distillate. At the start of the design, the feed flow rate, composition, temperature, and pressure are known. Two key components and their splits between the distillate and bottoms
0888-588519312632-0511$04.00/0 0 1993 American Chemical Society
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512 Ind. Eng. Chem. Res., Vol. 32, No. 3, 1993 Specified feed. vapor rite. number of stages. reflux ratio of column
-c.
c
Ne~ton,s
Solve F W quatiom:
- Guess minimumreflux ratio - Compute minimumnumber of stages - Compute
~
W,i,
a
0.4 0.6 0 , : m 0.4
(Fcarke’r quation)
/amin (Gillihd’s correlation)
Solve for minimum reflux ratio
(b): Reflux ratio = 25.0
(s): Reflux ratio = 1 0
0.2
( U d e m d ’ s equation)
0.2
0
+
0
50
100
0
0 0
50
’ 100
0
50
100
50
100
Compute splits of all components
1
Repeat if stop criterion i s unsatisfied
Take a small time slep
I
I
CIllcuLPte new fced mount m d compositions I
Figure 1. Shortcut model algorithm.
are also specified. A preliminary estimation of the distribution of the nonkey components is made. Column pressure and type of condenser are determined from bubble and dew point calculations of the distillate and bottoms. The feed-phase condition can be calculated once the column pressure is known. The next step in the procedure is the calculation of the minimum number of stages for the specified splits of the two key components using Fenske’s equation:
%distilled
% distilled
Figure 2. Comparison of shortcut model with rigorous simulation using BATCHFRAC. Binary separation: XF = 0.8 0.2; UAB 1.1; Nstg= 20 (+, component A (BATCHFRAC); X, component B (BATCHFRAC);solid line is shortcut model). (R):
(b): Reflux rntio = 25.0
Reflux rntio = 1.0
Ij
I I
--===-l ‘ 7
ll++-
‘hk,B \I --
1r.n (’lk,D
0.200
A;
The minimum number of stages corresponds to the state of total reflux. It is influenced by the presence of the nonkey components only if they have any effect on the relative volatility between the key components. Fenske’s equation is used again in the next step to determine the splits of the nonkey components, by replacing lk and hk by any two nonkey components. If the calculated splits are considerably different from the ones estimated earlier, an iterative procedure is used. Once the estimated and calculated splits of the nonkey components are close, Underwood’s equations are used to determine the minimum reflux ratio required to achieve the specified separation of the two keys. The minimum reflux ratio corresponds to a column of infinite stages, and the point at which this occurs is referred to as a pinch point. When all components in the feed distribute to both the distillate and bottoms, a single pinch point is obtained, and the separation is classified as a class 1 separation (Shiras et al., 1950). If one or more of the components occur in either the distillate or the bottoms but not in both, two pinch points occur in the column, and the separation is classified as a class 2 separation. For the rectifying section pinch point of a continuous column, the following equation can be written from mass balances and equilibrium relationships:
For a class 1separation, X, can be directly replaced by the feed composition ZF.If the above equation is used for
50
100
I
w
0.6-
- w
0.4 -
0.6 0.4 .
o 00
50
100
O 0
.
50 :
L 100
% distilled
% distilled
Figure 3. Comparison of shortcut model with rigorous simulation using BATCHFRAC. Binary separation: XF = 0.1 0.9; U A B = 10.0; Nstg= 3 (+, component A (BATCHFRAC); X, component B (BATCHFRAC);solid line is shortcut model).
a class 2 separation, the minimum reflux ratio obtained will be an upper bound to the true minimum. This is because the presence of the nonkey components in the pinch zone increases the difficulty of the separation, thus increasing the reflux requirement. For a class 2 separation, x , is not simply related to the feed composition. The two equations devised by Underwood in this case are
“ij’i 1J
D
(4)
:m:m
(a): Reflux ratio
a
(b): Reflux ratio = 25.0
= 1.0
a
0.4
SO
0.200
Y
t/
w
0.4
SO
0
50
100
I o,6k w
J100
0.4
i -
0
so
0
%distilled
100
(b): o*c=6.0 o,gc=l.l
o ~ c = 2 . 0osc=l.5
c distilled
RC distilled
% distilled
Figure 4. Comparison of shortcut model with rigorous simulation using BATCHFRAC. Binary separation: XF = 0.4 0.6;a A B = 10.0; Nsts = 3 (+, component A (BATCHFRAC); X, component B (BATCHFRAC); solid line is shortcut model). (a):
a
-7
o,6K
o.8
(b): zr-0.4 0.5 0.1
(s): z p - 0 . 3 3 0.33 0.34
0.4
0.200
100
Ind. Eng. Chem. Res., Vol. 32,No.3, 1993 513
Figure 6. Comparison of shortcut model with rigorous simulation using BATCHFRAC. Ternary separation: (XAC = 2.0; a g c = 1.5; N8e = 20; reflux ratio = 10.0 (+, component A (BATCHFRAC); X, component B (BATCHFRAC); 0,component C (BATCHFRAC); solid line is shortcut model). (b): %c=6.0
( a ) : O ~ c = 2 . 0Oec=1.5
1
0.8
a
0.8
0.8
a
o.6
0.4
0.6
' I
1
0.6
a
0.4
0.4
0.2
0.2 0
0
0sc=1.2
0.2
so
0 0
100
50
I00
n
"0
50
""f o'8M 1
:::M :::w ::M
0.8
-:.0
R
0.200
50
100
I
0.6
0.4
R
0.2
0
0
%distilled
0.6
0.6
50
100
%dictilled
Figure 5. Comparison of shortcut model with rigorous simulation using BATCHFRAC. Ternary separation: XF = 0.33 0.33 0.34;N 8 ~ = 3; reflux ratio = 10.0 (+, component A (BATCHFRAC); X, component B (BATCHFRAC); 0,component C (BATCHFRAC); solid line is shortcut model).
Equation 3 is first solved for m roots of 8 where m is one lees than the number of distributing Components. Quation 4 is then written for each value of 8, and the m equations are solved simultaneously to yield R m i n and the splits of the distributing components. At this point in the design, the minimum reflux ratio, the minimum number of stages, and the splits of all the componentsat totaland minimum reflux are known. The actual reflux ratio is generally established at some multiple of the minimum reflux. The value of R I R m i n usually lies between 1.1 and 1.5. The number of theoretical stages for the specified separation is then determinedfrom Gilliland's empirical correlation relating Rmin, N m i n , R,and N. One
0 0
00
50 cdblilkd
50
100
100
%distilled
Figure 7. Comparison of shortcut model with rigorous simulation using BATCHFRAC. Ternary separation: ZF = 0.4 0.5 0.1;N.Q = 3;reflux ratio = 10.0 (+,component A (BATCHRRAC);X, compOnent B (BATCHFRAC); 0 , component C (BATCHFRAC); solid line 1 shortcut model).
equation form of Gilliland's correlation is (Eduljee, 1975)
The feed stage location can be located using the Kirkbridge equation. The distribution of the nonkey components at actual reflux is approximated to be close to that estimated by Fenske's equation at total reflux.
The Shortcut Model Algorithm The shortcut model is made up of two nested loops, as shown in Figure 1. For the outer loop, the differential
614 Ind. Eng. Chem. Res., Vol. 32, No. 3, 1993 Table I. Problem Description: Quaternary Feed SeDaration column specification number of internal stages 5 initial charge 100 feed comp (mole fraction) A 0.25 B 0.25 C 0.25 D 0.25 distill until 99% of operation step specifications init charge distilled
0.8l
Q
I
I
0.6
0.4
0.2 0 0
50
100
50
100
Physical Property Models component re1 volatility A 2.0 B 1.5 C 1.2 D 1.0
balances for total mass in the pot and for each component can be written as follows
dM
-v
0
dt R + l where Vis the vapor rate of the column and R is the reflux ratio, and (7)
% distilled
Figure 8. Comparison of shortcut model with rigorous simulation using BATCHFRAC. Quaternary separation: XF = 0.25 0.25 0.25; N,, = 6; reflux ratio = 10.0 (+, component A (BATCHFRAC); 0 , component B (BATCHFRAC); X, component C (BATCHFRAC);1, component D (BATCHFRAC); solid line is shortcut model).
the actual reflux ratio. The final equation is that the sum of the overhead compositions should be equal to unity. An analysis of the degrees of freedom of the inner loop system of equations was conducted. The total number of equations are Equation 9 can be integrated numercially after substituting for dM from eq 6. For example, if first-order integration with at as the time step is used, dXi,B can be approximated as (xy- x $ ) and dM can be approximated as (Mnew Mold). Equation 9 would then become
with
At the beginning of the distillation, the feed composition Zi,F) and the total mass of the feed (Mold)are known. Assuming a small positive at, the new pot compositions and the new maas in the pot can be calculated from eqs 10and 11,assuming the vapor rate of the column, the reflux ratio, and the overhead compositions corresponding to the initial feed composition are known. After each iteration of the outer loop, the pot compositions and amount, the vapor rate of the column, the number of stages, and the reflux ratio are known. The inner loop uses the FUG equations to compute the overhead vapor compositions from these given values. Fenske's equation (eq 1) is used to determine the minimum number of stages for the system. For batch distillation, the pinch has to occur in the rectifying section. It will be assumed that all components distribute, making it a class 1separation. Hence the applicable Underwood's equations are eq 2 with xi,.. replaced by x~,B. Gilliland's correlation is used to relate the minimum number of stages, minimum reflux ratio, the actual number of stages, and (xfi =
Fenske's equation Underwood's equations Gilliland's correlation sum of X~,D'S
1 N-1 1 1
The unknowns in this system of equations are overhead compositions x ~ , ~ minimum reflux ratio minimum number of stages
N 1 1
The number of equations and number of unknowns both total N + 2. The system is thus a well-determined system which can be solved. The equations can be simplified to a single nonlinear equation in one unknown as shown in the Appendix. It is solved using Newton's method for which the required derivative is derived in the Appendix.
Model Testing and Validation The model was extensively tested on a number of examples. The examples presented here fall into three classes: (1)binary feed separations where feed composition, relative volatility, number of stages, and reflux ratio were systematically varied; (2) ternary feed separations with the same variations as the bindary examples; (3) specific multicomponent examples with varying reflux ratio policies. In each case, the separation was simulated using BATCHFRAC, and the results from the shortcut simulation are compared to those from BATCHFRAC. Binary Examples. These were the first set of comparisons to be made. Twenty-four separations were simulated using the shortcut model with three values of the feed composition and two values each of the relative
Ind. Eng. Chem. Res,, Vol. 32, No. 3, 1993 515 Table 11. Test Problems Used for Comparison of Shortcut Model with Rigorous Simulation Binary Comparisons feed composition, XA QAB N8-e reflux ratio 1.1 3 1.0 0.1 3 1.1 25.0 0.1 20 20 3 3 20 20 3 3 20 20 3 3 20 20 3 3 20 20 3 3 20 20
1.1 1.1 10.0 10.0 10.0 10.0 1.1 1.1 1.1 1.1 10.0 10.0 10.0 10.0 1.1 1.1 1.1 1.1 10.0 10.0 10.0 10.0
0.1 0.1 0.1 0.1 0.1 0.1
0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8
remarks
1.0 25.0 1.0 25.0 1.0 25.0 1.0 25.0 1.0 25.0 1.0 25.0 1.0 25.0 1.0 25.0 1.0 25.0 1.0 25.0 1.0 25.0
Figure 4 Figure 4
Figure 5 Figure 5
Figure 3 Figure 3
Ternary Comparisons feed composition
Q
XA
X B
QAC
QBC
Natane
0.33 0.33 0.33 0.4 0.4 0.4
0.33 0.33 0.33 0.5 0.5 0.5
2.0 6.0 2.0 2.0 2.0 6.0
1.5 1.2 1.5 1.5 1.5 1.2
3 3 20 20 3 3
reflux ratio 10.0 10.0 10.0 10.0 10.0 10.0
remarks Figure 6 Figure 6 Figure I Figure I Figure 8 Figure 8
Quaternary Comparisons ~~
feed composition
ff
XA
X B
xc
ffAB
ffBD
W D
Nstage
0.1 0.25
0.3 0.25
0.1 0.25
2.0 2.0
1.5 1.5
1.2 1.2
3 10
volatility, the number of stages, and the reflux ratio. For each separation, distillation was simulated until 99 % of the initial charge had been distilled. Pot liquid composition and instantaneous overhead vapor composition profiles were compared to those obtained using BATCHFRAC to simulate the same separation. When using BATCHFRAC, constant molal overflow and constant relative volatility were assumed. Some comparisons are shown in Figures 2-4. The results indicate that the shortcut model does a good job of predicting batch distillation composition profiles, at least for binary separations. Ternary Examples. The shortcut model was used to simulate three-component separations for various values of feed compositions, relative volatilities between components, number of stages, and reflux ratios. Some of the results are shown in Figures 5-7. Again, the shortcut model accurately predicts composition profiles for three-component separations. Quaternary Feed Separation. The problem description is given in Table I. The feed consists of four components and distillation is conducted in a column with six stages until 99% of the initial charge is distilled. The column is operated at a reflux ratio of 10.0. Instantaneous vapor and pot liquid compositions from the shortcut model are compared to those obtained from rigorous simulation in Figure 8.
reflux ratio 10.0 10.0
remarks Figure 9
A list of all the comparisons made is shown in Table 11. Comparisons with rigorous simulations for all these test problems are provided in Sundaram (1991). Separation of Multicomponent Organic Feed. This example is identical to the one used by Boston et al. (1981) in their paper on BATCHFRAC. The problem description is given in Table 111. The feed consists of four components, propane, butane, pentane, and hexane. The column consists of eight internal stages. There are five operating steps, with stopping criteria given for each operating step. The objectives are to produce butane at 99.0% purity and hexane at 99.98% purity. The purpose of the first two steps is to remove the most volatile component, propane. In actual operation, the second propane removal step is terminated when the mole fraction of butane reaches 0.985 in the distillate. Then the main accumulator is dumped, and material containing 40% butane and 60% hexane is added to the pot, before beginning the third step. This step is terminated when the mole fraction of butane in the accumulator reaches 0.9900. The accumulator is dumped again before the first of two pentane removal steps is started. At the end of the last operation step, the mole fraction of hexane in the step is 0.9998, and this may be drained off as the final hexane product. The problem was simulated using BATCHFRAC assuming zero holdups and constant relative volatilities. Material balances for each operating step were obtained from the simulation, and the percent distilled at the end
516 Ind. Eng. Chem. Res., Vol. 32, No. 3, 1993
Potential Shortcomings of the Model
Table 111. Problem Description: Multicomponent Organic Feed Sesaration column specification number of internal stages 8 pressure, psia 14.7 initial charge, lb-mol 100 feed composition (mole fraction) propane 0.1 n-butane 0.3 n-pentane 0.1 n-hexane 0.5 OPSTEP 1 2 3 4 5
Operation Step Specifications purpose reflux ratio propane removal 5 propane removal 20 butane production 25 pentane removal 15 pentane removal 25
distillate rate 2 2 2 2 2
Physical Property Modele: Zero Holdup Case component re1 volatility 4.58 propane 1.00 n-butane 0.258 n-pentane n-hexane 0.067
of each operating step was computed. This was used as the stopping criteria for the shortcut model. The comparison of results from the shortcut model and from BATCHFRAC are shown in Table IV. From Table IV, it can be seen that the material balances agree well with those obtained from rigorous simulation. Since constant molal overflow and constant distillate rate were assumed for both the shortcut model and BATCHFRAC, the distillation times were identical for both simulations.
The shortcut model did an excellent job of matching simulation results for all of the test cases studied. However, the model is an approximate one and cannot be expected to perform well in all possible situations. To determine possible causes of model deficiency, the effects of all the major assumptions made during the development of the model equations were examined. The first major assumption is that, during each time step, the batch column can be represented using the Fenske-Underwood-Gilliland equations. The implicit assumption is that, at every time step, the vapor and liquid at each stage in the column are in equilibrium and a quasisteady-state profile is established equivalent to the profile for a continuous column. This assumption effectively neglects the effect of holdup on the dynamic behavior of the column. A second set of assumptions is related to the FUG method itself, and ita implementation. In the model, it was assumed that all components present in the column distribute between the bottoms and the distillate. The lightest component was assumed to be the light key component, and the heaviest component was assumed to be the heavy key component. This means that at minimum reflux, the zone of constant composition (or the pinch) lies adjacent to the feed stage (Shiras et al., 1950). This allowed the use of a simplified version of Underwood's equation, thereby considerably simplifying the solution procedure. There are two exceptionsto the assumed behavior. The first is when all components distribute but the zone of constant composition is not adjacent to the feed stage. This happens when t t z system is sufficiently nonideal so as to give the equivalent of a tangent pinch point. Such
Table IV. Mass Balances: Shortcut vs BATCHFRAC: Zero Holdup Propane Removal Steps ace mol OPSTEP 1 component shortcut rigorous 8.1370 8.0348 propane 0.0003 0.1028 n-butane n-pentane 0.0 0.0 0.0 0.0 n-hexane
ace mol OPSTEP 2 shortcut rigorous 9.7261 9.9933 1.5904 1.7734 0.0 0.0 0.0 0.0
Butane Production Steps ace mol OPSTEP 3 ~~
comDonent propane n-butane n-pentane n-hexane
component propane n-butane n-pentane n-hexane
ace mol OPSTEP 4 shortcut rigorous 0.0 0.0 0.0 0.0583 8.4812 8.5032 0.1353 0.0554
mol n-butane recovered % n-butane recovered purity of n-butane mol n-hexane recovered % ' n-hexane recovered purity of n-hexane
shortcut 0.0 35.467 0.1626 0.0045 Pentane Removal Steps acc mol OPSTEP 5 shortcut rigorous 0.0 0.0 0.0 0.0583 9.4656 9.6312 2.7162 2.4926 Performance Parameters shortcut 35.47 93.33 97.10 59.11 95.34 99.32
riaorous 0.0087 36.171 0.3567 0 reboiler mol shortcut rigorous 0.0 0.0 0.0 0.0 0.4047 0.0119 59.1096 59.505
BATCHFRAC 36.17 95.19 99.00 59.51 95.98 99.98
casesare rare, however, and is not considered to be a serious issue. Second, some or all of the nonkey components may be nondistributing a t minimum reflux. Here, again, the zone of constant composition cannot be presumed to be adjacent to the feed stage. Some of the light nonkey components will be absent from the bottoms, and some of the heavy key componentswill be absent from the distillate. In this case, it is necessary to first determine which components are distributing, and then determine the distribution of all the components. This can be done using a procedure outlined by Shiras et. al. (1950) and also described by King (1980). The method involves an iterative procedure in which Underwood's equations (eq 3 and 4) are solved repeatedly to determine the distributing components first and the actual distribution of all the components next. Since this more complicated procedure is not used, the model is likely to fail in situations when the nonkey components are nondistributing. Practically, this is most likely to happen when there is a very small amount of the intermediate component in the column and when the number of stages in the column is very high. However, in the test cases, numbers of stages as high as 30 were used without numerical difficulties. If the method were used to model nonideal systems, it would be necessary to incorporate a thermodynamic package to calculate the change in relative volatilities with changing composition.
Summary
A shortcut model for batch distillation simulation using the FenskeUnderwood-Gilliland equations of continuous distillation design was developed. The model consists of stepping forward in time using a first-order explicit integration scheme with a variable time step, and solving the FUG equations a t each time step. At each snapshot in time, the batch column was assumed to resemble the rectifying section of a continuous column. A large number of example problems were used for model testing and validation. The problems were rigorously simulated using BATCHFRAC, and the results of the shortcut model were compared to these results. Comparisons were made by plotting the instantaneous vapor and liquid compositions as a function of initial charge that is distilled. Agreement between the shortcut model and rigorous simulation was excellent. Assumptions that were made either explicitly or implicitly during model development were identified, and their impact on simulation results was discussed. The model is a powerful and computationally fast tool that can be used both in batch process design and a synthesis of batch distillation systems. Nomenclature B = boilup rate, kmol/h D = distillate rate, kmol/h
Ind. Eng. Chem. Res., Vol. 32, No. 3, 1993 517 x = composition Zi,F feed composition of component i 6 = unknown in Underwood's equation a = relative volatility Subscripts B = bottoms D = distillate hk = heavy key component i = component i j = component j lk = light key component N = component N = pinch point
Appendix The equations that are used in the shortcut model with the accompanying simplifications are presented. Fenske's equation gives us the minimum number of stages:
The equation can be rearranged to give NXhk,D
Xlk,D
= CYlk.hk min
For any component i,
Since the individual overhead compositions have to add up to unity
or
Rearranging eq 15 1.0
=
'hk,D
(17)
1.0
-
"i,hkNminXi,B
xhk,B ~1
Underwood's equation for batch distillation is eq 2 with component 1being the light key component and component N being the heavy key component. This is because in batch distillation all components are expected to distribute between the pot and overheads. Also, the pitch compositions are the same as the pot liquid compositions. Underwood's equation is thus xl,D
_.-
hk = heavy key component
L, = reflux rate, kmol/h lk = light key component M = amount of material in reboiler, kmol N = number of stages in column N m i n = minimum number of stages q = variable in Underwood's equation R = reflux ratio Rmin = minimum reflux ratio V = vapor rate, kmol/h
XIk,B Xhk,B
Rmin =
XN,D
a -
'"XN,B
'1,B a1,N
-1
(18)
The equation solved using Newton's method is xl,D -%in
Substituting
=
xl,B
C Y -
XN D
's\'rN'B
ala - 1
= 0 = Res(Rmin)
(19)
518 Ind. Eng. Chem. Res., Vol. 32, No. 3, 1993
from Fenske’s equation, and subsequent use of eq 17, yields
N m i n can be substituted from Eduljee’s equation for Gilliland’s correlation as
( (R;fy)0.5m8) (21)
Nmin= N - 0.75(N + 1) 1 -
Equation 19 is an implicit nonlinear equation in Rmin for which the Newton’s scheme employed is
The required derivative, Res’(RLin), can be obtained by differentiating eq 20 with respect to Rmin. After some simplification Res’(Riin) =
with
and N
s = x“i.NNm%i,B
(25)
i=l
Literature Cited Bogart, M. J. P. The Design of Equipment for Fractional Batch Distillation. Trans. Am. Znst. Chem. Eng. 1937,33, 139.
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