130
Ind. Eng. Chem. Res. 1997, 36, 130-136
Short-Cut Design Procedure for Batch Distillations H. E. Salomone, O. J. Chiotti, and O. A. Iribarren* INGAR (Instituto de Desarrollo y Disen˜ o), Avellaneda 3657, 3000 Sante Fe, Argentina
This paper presents a short-cut procedure for the design of batch distillations which is equivalent to the Fenske-Underwood-Gilliland procedure for continuous distillations. Given a separation task, we propose computing the minimum number of stages and minimum reflux ratio required to perform it by batch distillation. These parameters are quite different from the continuous Nmin and Rmin parameters for the same separation task, because of the difference in the form of the material balances (steady state in continuous but integrated differential mass balances in the batch case). We propose that they should be computed as the number of stages and reflux ratio required by hypothetical batch distillation columns that operate at total reflux or have an infinite number of stages, respectively. Then, it is shown that the performance of batch distillations does correlate in Gilliland coordinates constructed with these batch Nmin and Rmin. This produces a quite simple, straightforward method to do preliminary design of batch distillations. Introduction Batch processes are the alternative of choice for processing specialty and fine chemicals, which are highvalue products to be produced in small volumes, in multiproduct plants. Batch distillation plays an important role in these processes, in the purification of liquid products and in the recovery of solvents. This explains the increased interest of the present literature in the development of process models and design procedures to handle batch distillation problems. Although rigorous dynamic simulators have already reached a commercial stage, their demand in terms of computation time is very large, so they are useful for testing the performance of given distillation columns, but they are hardly useful at the decision-making steps of design or production planning. Thus, many articles deal with the development of simple models and short-cut design procedures, appropriate at these decision-making steps. The simplifying assumptions usually adopted to reduce computation time are constant relative volatilities (this reduces equilibrium computations), constant molal overflow (this avoids energy balance computations), and zero holdup in the column (only the still differential equations are considered, with the column in a pseudosteady state). Furthermore, some present literature reports using the Fenske-Underwood-Gilliland (FUG) method for continuous distillation to describe the instantaneous separation performance of batch columns. This approach was presented by Diwekar and Madhavan (1991) and later on used to solve both design problems, as in Diwekar (1992), and simulations, as in Sundaram and Evans (1993). The approach allows for a considerable reduction in computation demand, which in turn becomes independent of the number of stages. However, the computation effort is still large if used inside nested calculations, as in the design optimization. Thus, there is still room for short-cut design procedures to do preliminary estimates without solving the optimization. One important contribution in this field is made by Al-Tuwain and Luyben (1991), who construct design correlations which allow us to read off an optimum number of trays and reflux ratio for a given separation. The design correlations are parametric on * Author to whom correspondence should be addressed. S0888-5885(95)00458-1 CCC: $14.00
different values for the design specifications: relative volatilities, feed and product purities, and economic data. The short-cut procedure for continuous distillation is much easier: given a separation task, one finds the minimum number of stages by Fenske, the minimum reflux ratio by Underwood, and then uses either the “shortest cut” procedure of adopting R ) 1.2Rmin and N ) 2Nmin, or the Gilliland correlation which relates R with N (given Rmin and Nmin) to incorporate the economic data into a quite simple optimization problem. This problem is the trade-off between capital costs associated with N (which affects the height of the column) and both capital and energy costs associated with R (which affects the diameter of the column and utility consumptions). The short-cut procedure presented in this paper follows the same sequence of steps outlined in the previous paragraph. In the sections that follow, we first define the separation task and next the minimum number of stages and minimum reflux ratio required by a batch separation to perform this task. Then, we use extensive simulation data to construct a correlation in Gilliland coordinates based on these minimum stages and reflux. Finally, we summarize the design procedure and compare it with the cases of continuous distillation and the approaches for batch distillation previously proposed in the literature. Separation Task Specification For each splitting of an amount of feed into a distillate and a residue, this separation task is fully specified by the following data: (1) the amount of feed and the mole fraction of each component; (2) the relative volatilities among the components; (3) the fractional recoveries of any two components. The fractional recovery of a component is defined as the amount of this component in the distillate at the end of the separation, relative to the amount of the same component in the feed before the separation. Even when the design specifications may originally be given as compositions, their transformation into fractional recoveries is straightforward and presents an advantage: they are monotonically increasing quantities during the separations (mole fractions are not) so they are convenient as termination criteria. Two fractional recoveries unambiguously define the separation extent. © 1997 American Chemical Society
Ind. Eng. Chem. Res., Vol. 36, No. 1, 1997 131
Material Balances. The differential material balances for each component are
dni ) -xi,t dD
(1)
Table 1. Procedure for the Computation of Nmin step 1: define the separation task xi,f Ri,h fl* fh* step 2: guess a value for N step 3:
where ni corresponds to the moles of component i in the still, D is the distillate, and xi,t is the mole fraction of i at the top of the column. Dividing eq 1 by the one that corresponds to the lightest specified component l yields the form
dni xi,t ) dnl xl,t
integrate
dni xi,t ) up to fl ) fl* dnl xl,t
tracking
fl )
with
xi,b )
n0l - nl n0l ni C
∑n
(2)
i
i)1
which is convenient for numerical integration because nl is both the integration variable and the variable on which the termination criterion is applied. Minimum Number of Stages. The minimum number of stages to perform a separation task by batch distillation is defined as the number of stages required by a hypothetical distillation column that operates at total reflux. The instantaneous separation performance of such a column is given by Fenske equations:
[
ln N)
]
i ) 1, ..., C - 1
(3)
There are C - 1 independent equations like (3), where C is the number of components, and the extra needed equation is that the summation of xi,t be unity. These equations can be arranged to an explicit form whose derivation has been published in Sundaram and Evans (1993):
xh,b
xh,t )
C
∑ i)1 xh,t xi,t ) Ri,hN xi,b xh,b
step 4: compute the heavy key fractional recovery
n0h
Fenske equation relating instantaneous top and bottom compositions is
xt )
RNxb
(6)
1 + (RN - 1)xb
which is used to integrate the Rayleigh equation
to give
ln
If the differential mass balance equations (2) are integrated using eqs 4 and 5 to predict the instantaneous top compositions, this corresponds to simulating the performance of a hypothetical total reflux column of N stages. The method that we propose in order to get the minimum number of stages is an iterative procedure where an initial guess of N is refined until the specified recovery fractions of the two components is reached in the distillate. The simulations are stopped when the amount of the lightest specified component in the distillate corresponds to fl. Then, the amount of the heaviest specified component in the distillate is used to drive the iteration on N until this amount corresponds to the specified fractional recovery fh. The procedure is outlined in Table 1. In the case of binary mixtures, it is possible to get an analytical expression for the minimum number of stages. In this case, a separation task is univocally determined given the distillate and residual mole fractions of the lightest component, so we use these compositions instead of the fractional recoveries. The
n0h - nh
step 5: if fh 〈 〉 fh*, guess a new value for N and go to step 3 step 6: Nmin ) N
Ri,h xi,b
(5)
xh,t x xh,b i,b
xi,t ) Ri,hN
N
i ) 1, ..., C - 1
N i,h xi,b
i)1
ln (4)
C
∑R
fh )
xi,t xh,b xi,b xh,t ln Ri,h
Xh,b
xh,t )
F ) W
dx
∫xx)x)x xt -bxb b
b
f
(7)
w
((
)) (
xf 1 - xw 1 F ) N ln W R -1 xw 1 - xf
+ ln
)
1 - xw 1 - xf
(8)
From the total and light component mass balances when a feed F is split into a distillate D and a residue W
F xd - xw ) W xd - xf
(9)
replacing in eq 8 and rearranging give a simple expression for Nmin as a function of the feed, distillate, and residue compositions:
[
]
xd - xw xf xd - xf xw RNmin ) xd - xw 1 - xf ln xd - xf 1 - xw ln
[
]
(10)
Equation 10 predicts the same figure as the numerical procedure in Table 1, and simulations with this number of stages tend to perform the required separation only when the reflux ratio tends to be infinite. Minimum Reflux Ratio The minimum reflux to perform a separation task by batch distillation is defined as the reflux required by a
132 Ind. Eng. Chem. Res., Vol. 36, No. 1, 1997
hypothetical batch distillation column having an infinite number of stages, so its instantaneous separation performance is given by Underwood equations. At each step of the integration, one must compute the C - 1 roots θm of C
i,h
- θm
)0
∑ i)1R
Ri,hxi,t
i,h
- θm
|
)1+R
Rj Rj - θi Ax ) b bi ) 1 + R
m ) 1, ..., C - 1
(12)
i ) 1, ..., C - 1
aC,j ) 1
i ) 1, ..., C - 1
bc ) 1
the distillate compositions xi,t so it can be readily solved by a linear systems solver. The resolution of eqs 11 and then 13 should be followed by a verification step to test that the heaviest component also distributes:
xi,t g 0
i)C
with
xi,b )
n0l ni C
∑n
i
[
1 - xt 1 xt -R R - 1 xb 1 - xb
]
(15)
Rearranging eq 15 to get an explicit expression for xt as a function of R, R, and xb, replacing xt into Rayleigh equation (7), and performing the integration yield the expression already derived by Bauerle and Sandall (1987):
[
xi,t calculated in subprocedure A step 4: compute the heavy key fractional recovery
fh )
n0h - nh n0h
step 5: if fh 〈 〉 fh*, guess a new value for R and go to step 3 step 6: Rmin ) R Subprocedure A step 1: given xi,b Ri,h R step 2: find the C - 1 roots θm of C
Ri,hxi,b
∑R
i,h
- θm
)0
step 3: obtain top compositions xi,t solving the linear system
Ax ) b
|
Rj Rj - θi bi ) 1 + R
ai,j )
i ) 1, ..., C - 1
aC,j ) 1
i ) 1...C - 1
bC ) 1
step 4: if xC,t < 0, then set xC,t ) 0, let C ) C - 1, and go to step 3 step 5: return xi,t
(14)
If not, then its composition in the distillate is set to zero and eq 13 are solved again with C ) C - 1 (without considering the previous heaviest component) until verification of condition (14). The differential mass balance equations (2) are integrated with the above sequence of calculations predicting the instantaneous top compositions. Now, the manipulated variable for converging to the desired fractional recoveries is R of eqs 13. This procedure provides the minimum reflux ratio Rmin required to perform the specified separation task. The procedure is outlined in Table 2. In the case of binary mixtures, it is possible to get an analytical expression for the minimum reflux ratio. The Underwood equation relating instantaneous top and bottom compositions is
ln
fl )
i)C
(13)
R)
n0l - nl
tracking
i)1
There are C - 1 independent equations like (12), and the required extra equation is that the summation of xi,t equals unity. This system of equations is not explicit, as in the case of the minimum number of stages. However, it is a simple linear system with the form shown in eq 13, with j ) 1, ..., C, and vector x contains
ai,j )
dni xi,t ) up to fl ) fl* dnl xl,t
(11)
Each of these roots is located between the consecutive values of Ri,h and is readily found by an area elimination method. Then one finds the top compositions by solving C
step 1: define the separation task xi,f Ri,h fl* fh* step 2: guess a value for R step 3:
integrate
Ri,hxi,b
∑ i)1R
Table 2. Procedure for the Computation of Rmin
]
1 - xf xw W 1 ln ) + F (R + 1)(R - 1) 1 - xw xf
(
)
1 - xf 1 ln (16) R+1 1 - xw
These authors derived this expression for columns having an infinite number of stages and proposed using it as an approximation to the performance of real columns having a finite number of stages. Replacing the mass balance in eq 9 into eq 16 and rearranging give a simple expression for Rmin as a function of the feed, distillate, and residue compositions:
[( ) ] ( )
ln (Rmin + 1)(R - 1) )
1 - xf R xw 1 - xw xf xd - xf ln xd - xw
(17)
Equation 17 is appropriate as long as both components distribute. For a given reflux, the limit condition for distribution occurs when xb has an intermediate value xi such that Underwood equation (15) predicts xt ) 1:
Rmin )
1 (R - 1)xi
(18)
For compositions at the still of xb g xi, the heaviest component does not distribute: eq 15 predicts xt > 1, and the heaviest component should not be considered. The top composition is just xt ) 1, and integration of Rayleigh equation (7) yields
ln
(
)
1 - xf W ) ln F 1 - xw
(19)
which does not depend on R and corresponds to the
Ind. Eng. Chem. Res., Vol. 36, No. 1, 1997 133
mass balance after extraction of distillate of composition xd ) 1. Usually, the distillation will go through both zones, so Rayleigh equation (7) becomes
ln
W ) F
dx
∫xx)x)x 1 - bxb + ∫xx)x)x b
b
i
b
f
b
i
w
dxb xt - xb
(20)
with xt given by Underwood equation (15). After integration and replacing the mass balance of eq 9, one gets
[(
) ] )( )]
1 - xi R xw 1 - xw xi (Rmin + 1)(R - 1) ) xd - xf 1 - xi ln xd - xw 1 - xf ln
[(
(21) Figure 1.
which must be solved in conjunction with eq 18 to get xi and Rmin. Equations 18 and 21 predict the same Rmin as the numerical procedure in Table 2, and simulations with this reflux ratio tend to perform the required separation only when the number of stages tends to be infinite. Simulation Model The model used to simulate batch distillations assumes constant volatilities and constant molal overflow and neglects the column holdup. So, the equilibrium relations and mass balances are given by
yi,k )
Ri,hxi,k
(22)
C
Ri,hxi,k ∑ i)1
[
xi,k+1 ) yi,k -
]
xi,t R + 1 R+1 R
(23)
where subscript k corresponds to successive separation stages, starting from the bottom. At each integration step, the instantaneous xi,t compositions are computed climbing from the reboiler up to stage N with eqs 22 and 23. This requires a first guess for the xi,t to be used in eq 23. After reaching stage N, the mean quadratic error between the calculated and guessed values of xi,t is minimized by a Simplex search. The differential mass balance equations (2) are integrated with the instantaneous top compositions computed by the iterative procedure described above. Given a number of stages (larger than Nmin) and a guess for R, the simulations are conducted until the lightest specified component fractional recovery is obtained. Then, the reflux ratio is manipulated to converge the heaviest specified component to the desired fractional recovery. Construction of the Correlations We generated a set of cases covering a range for the variables involved similar to the original article by Gilliland (1940), that is, R ranging from 1.26 to 4.05, Rmin from 0.53 to 7, and Nmin from 1.4 to 42. The procedure used is as follows: 1. Separation tasks were generated by randomly choosing over a range of values for the following parameters: relative volatilities (1.26 to 4.05), feed compositions (0.2 to 0.8 for the first component and 0.01
to the amount not yet assigned for the next components), light component fractional recoveries (0.95 to 0.999), and heavy component fractional recoveries (0.001 to 0.05). 2. The values for Rmin and Nmin were computed for each of these tasks. 3. Those tasks having Rmin values outside the range 0.53 to 7, or Nmin values outside the range 1.4 to 42, were discarded. 4. A simulation was conducted for each of the tasks assigning a random value for N in the range from 1.01 to 6 times Nmin. Each simulation resulted in a value for R such as would satisfy the specified separation task. 5. Gilliland coordinates Y ) (N - Nmin)/(N + 1) and X ) (R - Rmin)/(R + 1) were computed for each simulation. We first tested this construction procedure with continuous distillations reproducing Gilliland’s original work, with the results plotted in Figure 1. Furthermore, the equation form of the Gilliland correlation by Eduljee (1975) was drawn and fits well
Y ) 0.75(1 - X0.57)
(24)
The scattering of Figure 1 is roughly 20%; i.e., the actual value of the ordinate is Y ( 0.2Y, where Y is the ordinate value given by the equation form of the correlation. Scattering is directly related to the range of the parameters. Broadening this range, e.g., with feed compositions and fractional recoveries allowed to take values over a wider range, the points spread out. At the other extreme, for a single separation task, the correlation is obviously a single curve, because the simulation model becomes an implicit function of the two variables N and R. In the middle, for the ranges studied, the Fenske-Underwood-Gilliland method is quite a good approximation, widely popular over the years due to its simplicity. Next, we used the same construction procedure for batch distillations following the calculation procedures for Nmin, Rmin, and simulations as described in the previous sections of the paper and increasing the upper bound for Rmin from 7 to 40 because the Rmin for batch distillations tends to be much larger than for continuous distillations that perform the same separation tasks. The results are plotted in Figure 2. Even if the scattering is larger than for the continuous case (about 35%), the points do correlate. Furthermore, the correlation for batch distillations is clearly below the one for continuous distillations; i.e., its equation form has different coefficients:
134 Ind. Eng. Chem. Res., Vol. 36, No. 1, 1997
Figure 3.
Figure 2.
Y ) 0.62(1 - X0.34)
(25)
With regard to the “shortest cut”, plotting the points on coordinates N/Nmin vs R/Rmin, the scattering is worse than with Gilliland coordinates but allowed us to learn that the point corresponding to R ) 1.2Rmin is about N ) 1.7Nmin. Sloppy Separations Actually, the Gilliland (1940) correlation reproduced in Figure 1 corresponds to complete columns (both rectifying and stripping sections) performing sharp separations. However, when applied to batch distillations, the FUG approach involves a rectifying section that starts operation with mixed bottom-pure top compositions and ends operation with pure bottommixed top compositions. We computed such conditions and plotted the results in Figure 3. They correspond to binary mixtures which in the case of mixed bottompure top correspond to bottom compositions ranging from 0.2 to 0.6 and top compositions from 0.9 to 0.999 and in the case of pure bottom-mixed top correspond to bottom compositions ranging from 0.001 to 0.05 and top compositions from 0.4 to 0.8. As has already been discussed, it is natural that the scattering should increase with respect to the Gilliland plot when broadening the range of the parameters. Also, it seems natural that the points corresponding to pure bottom-mixed top be farthest from the original correlation because these compositions are quite unlikely to be encountered in continuous rectifications. Their location below the Eduljee correlation is due to the large Rmin and Nmin required by the separations in this zone. Since the FUG literature reports discussed in the introduction use the Eduljee correlation, which has also been drawn in Figure 3 and acts as an upper bound, they can be regarded as conservative. We propose the following equation form
Y ) 0.75(1 - X0.34)
(26)
as the appropriate correlation for the FUG approach to batch distillation because it will render a better approximation to the actual performance of continuous rectifying sections performing batch distillations. Finally, we investigated the case of batch distillations performing sloppy separations and plotted the results in Figure 4. In the case of splitting a batch distillation, the initial phase corresponds to the production of a pure
Figure 4.
distillate, while the final phase corresponds to the extraction of a slop cut to be recycled, leaving a pure residue in the still. In this case, the first phase needs no large fractional recoveries of the light component (the intermediate cut is recycled) but requires small recoveries of the heavy component: the squares represent cases with fl ranging from 0.2 to 0.6 and fh ranging from 0.001 to 0.005. Otherwise, the second phase demands large fractional recoveries of the light component (so it is not present in the residue) and intermediate recoveries for the heavy component: the circles represent cases with fl ranging from 0.995 to 0.999 and fh ranging from 0.2 to 0.6. These points are located above the ones for sharp separations (sloppy separations require smaller Rmin and Nmin) and are adequately fitted by the original Eduljee equation, although we understand that this is just a coincidence because the points correlated correspond to a quite different case from continuous columns performing sharp separations. All the correlations of interest to batch distillation (Figures 2-4) show in common a larger scattering than the correlation for continuous distillation performing sharp separations (Figure 1). The scattering of Figure 1 is roughly 20%, while the scattering for both the FUG approach (Figure 3) and our approach (Figures 2 and 4) is roughly 35%. This is a result of the broader range of bottom-top compositions that occur during the batch operation. Summary of the Design Procedure We recommend the use of eq 25 if no recycle is adopted and eq 24 if recycle of sloppy cuts is adopted.
Ind. Eng. Chem. Res., Vol. 36, No. 1, 1997 135
The design decision “recycle: yes or no” is beyond the scope of the present paper and will probably demand a trial and error procedure; for example, first assume no recycle for any of the separations, and then try recycle for those separations that demanded the larger Nmin and Rmin. Unification of both cases using eq 24 for either sloppy or sharp separations would be too conservative in the case of sharp separations that already demand larger Nmin and Rmin; this would bias the decision “recycle: yes or no” toward always recycling. So, the design procedure can be summarized as follows: 1. Define the separation task by specifying the feed compositions, the relative volatilities between components, and recovery factors for any two components. 2. Find Nmin with eq 10 for binary systems or by the numerical integration of the still differential balances using Fenske equations to get the instantaneous top composition. The procedure is outlined in Table 1. 3. Find Rmin by solving eqs 18 and 21 for binary systems or by the numerical integration of the still differential balances using Underwood equations to get the instantaneous top compositions. The procedure is outlined in Table 2. 4. Use the appropriate Y vs X correlation for batch distillations to get the optimal design: For sharp separations (no slop cut), use eq 25 to optimize the trade-off between R and N or use the shortest cut R ) 1.2Rmin and N ) 1.7Nmin. For sloppy separations (a product cut followed by a slop cut to be recycled), use eq 24 or R ) 1.2Rmin and N ) 2Nmin. Comparison with Continuous Distillation The procedure outlined above follows exactly the same sequence of steps as the Fenske-Underwood-Gilliland procedure for continuous distillations. The computations for finding Nmin and Rmin are more laborious in the case of batch distillation because they involve numerical integrations. Also, the points correlated are more laborious in the batch case because they correspond to dynamic simulations, so the extent of simplification introduced by the correlations can be regarded to be equivalent in the batch and continuous cases. For binary mixtures, the procedure for batch is almost as simple as for continuous distillations because the prediction of Rmin and Nmin is analytical. We compared the Rmin and Nmin required by batch vs continuous distillations to perform the same separation tasks. First of all, their dependency on the feed, distillate, and waste compositions is quite different because of the mass balances being different in the batch and continuous cases. While Nmin continuous depends only on the distillate and waste compositions and Rmin continuous depends only on the distillate and feed compositions, both Nmin and Rmin batch depend on the three compositions. With respect to the numerical values, Nmin for continuous distillation is always larger than for batch (1.5 to 3 times in most of the cases). Actually, Nmin continuous is an upper bound for Nmin batch: if the hypothetical total reflux batch column were provided with a number of separation stages Nmin continuous, it would be able to produce distillate in specification up to the last drop when the bottom is also in specification. As in the earlier stages of the batch distillation, the column will be producing a distillate richer in the lighter components, the actual Nmin batch will be smaller.
On the other hand, Rmin for batch distillation is always larger than for continuous (between 2.5 and 10 times as large and even more). Actually, Rmin continuous is a lower bound for Rmin batch: if the hypothetical infinite separation stages batch column were operated at a reflux ratio Rmin continuous, it would be able to produce just 1 drop of distillate in specification, when the bottom has the feed composition. As in the later stages of the batch distillation, the column will be producing a distillate poorer in the lighter components, the actual Rmin batch will be larger. Comparison with Fenske-Underwood-Gilliland Approaches These approaches use the Fenske-Underwood-Gilliland procedure for continuous distillation to predict the instantaneous top compositions of the batch column, as proposed by Diwekar and Madhavan (1991) and further developed for simulations by Sundaram and Evans (1993). At each integration step, the computation demand is larger than the one required by the calculation of Rmin and Nmin because it involves the convergence of the top compositions. Thus, if used to size a column for performing a specified separation task, these approaches would spend at least 1 order of magnitude more computation time than ours, with no gain in precision. For the purpose of design, it is more effective to use the correlation of already integrated batch distillations than to use the continuous distillations correlation to predict the instantaneous separation at each integration step. On the other hand, the Fenske-Underwood-Gilliland approach is appropriate for performance simulation given a number of stages and reflux ratio, which our approach is not, and furthermore, it is more appropriate to handle variable reflux policies. Variable reflux ratio policies could be handled with our approach as piecewise constant reflux by splitting the separation into a discrete number of periods. However, the reductions in vapor generation by adopting an optimal reflux policy have been estimated by King (1980) in less than 10%, so this optimization is hardly justified but as a final refinement. Comparison with Design Correlations These are short-cut design approaches that recommend a number of stages and reflux ratios providing minimal data about the separation and cost items (only about the separation in the simpler ones), without computations involved but just entering in the tables and/or figures. The most comprehensive of these approaches is the one by Al-Tuwain and Luyben (1991) which condenses a large number of optimizations in graphs that are parametric on the principal economic data (construction materials, payback period, energy costs) and separation task characteristics (relative volatilities, feed and product purities). In doing so, many other data must be assumed, such as heats of vaporization, temperature approaches in heat exchangers, production rate to size the still, etc.: characteristic values are used for these quantities. With regard to computation effort, this procedure is obviously much more expeditious as long as the particular separation or economic factors of interest to the user are in the range of those used in the design correlations. For example, the results reported by Al-
136 Ind. Eng. Chem. Res., Vol. 36, No. 1, 1997
Tuwain and Luyben (1991) consider feeds with essentially equal fractions of all components, equal relative volatilities between consecutive components, and equal product purities. More asymmetric specifications would require new design correlations. On the other hand, the procedure presented in this paper provides the lower computation demand approach already specific to the particular separation and economic factors. Relation with Previous Works on Binary Distillation The pioneer work in this area is the one by Bauerle and Sandall (1987), who derived analytical expressions to describe the performance of minimum reflux (infinite stages) columns. For the case of constant reflux operation, they arrived at the expression in eq 16. Working with this equation, one can get an expression for Rmin as an analytical function of feed, distillate, and residue compositions as presented in eq 17. However, this expression may yield misleading results if the heavier component does not distribute, which is likely to occur at the beginning of most distillations. Our first work in this area (Chiotti and Iribarren, 1991) used a constant product composition policy, followed by constant reflux ratio for the withdrawal of the slop cut, so eq 17 could be used safely. At that time, we did not handle the concept of an Nmin for batch separations, so we proposed computing the R (associated to a finite number of stages N) necessary to achieve the same separation as obtained by Rmin (associated to infinite stages) at an arbitrary intermediate point of the distillation, when the distillate was diverted from the light product receiver to the slop cut receiver. In the present paper, we fixed the problem with Rmin by considering the distribution of the heavier component, introduced the Nmin, and found that Gilliland coordinates constructed with these Rmin and Nmin correlated the performance of simulations. Conclusion We have presented a short-cut procedure for the design of batch distillations that is equivalent to the Fenske-Underwood-Gilliland procedure for continuous distillations. The procedure is based on the fact that after computing a minimum number of stages and minimum reflux ratio, as those required by batch distillations whose instantaneous separation performances are given by Fenske and Underwood equations respectively, Gilliland coordinates constructed with these minimum values correlate the performance predicted by simulations. For the purpose of preliminary design, our approach provides considerable saving of computation effort as compared with the previous shortest cut procedure that uses the FUG method for continuous distillations to predict the instantaneous separation performance of batch distillations. Furthermore, we propose a correlation for the FUG approach that improves its accuracy with respect to using Eduljee’s equation. The simplicity of the method proposed here is even more evident in the case of binary separations, for which it remains analytical.
Nomenclature D ) amount of distillate [kmol] fi ) fractional recovery of component i F ) amount of feed [kmol] ni ) amount of component i in the still [kmol] N ) number of separation stages Nmin ) minimum N to perform a separation task R ) reflux ratio Rmin ) minimum R to perform a separation task W ) amount of residue [kmol] x ) composition of liquid [mole fraction] X ) Gilliland coordinate for reflux ratio y ) composition of vapor [mole fraction] Y ) Gilliland coordinate for number of stages Greek Symbols Ri ) relative volatility of component i θm ) root of Underwood equation lying between Rm and Rm+1 Subscripts b ) bottom d ) distillate f ) feed h ) heavy key component i ) component or intermediate composition in binary separation k ) number of stages starting from bottom l ) light key component m ) root of Underwood equation t ) top w ) residue Superscripts 0 ) initial * ) specification
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Received for review July 24, 1995 Accepted May 31, 1996X IE950458N X Abstract published in Advance ACS Abstracts, August 15, 1996.
