1024
Ind. Eng. Chem. Res. 1999, 38, 1024-1031
SEPARATIONS Shortcurt Procedure for Multicomponent Batch Distillation with Distillate Receiver Jeung Kun Kim and Dong Pyo Ju* Division of Chemical Engineering & Biotechnology, Ajou University, 5 Woncheon-dong, Suwon 442-749, Korea
A shortcut calculation procedure applicable to a multicomponent batch distillation with a distillate receiver under total reflux condition was developed. Based on the assumptions of negligible vapor and liquid holdups in stages and the condenser, constant molar overflows, constant vapor boilup rates, constant relative volatilities, and adibatic equilibrium stages, the proposed model provides the estimates of the volume of the distillate receiver, changes in compositions in the receiver and still, concentration profiles of the column, and total distillation time required to complete the separation, provided that the desired product purities of the most volatile component are specified for the receiver and still. Experimental data for the separation of BTX mixture using a 10 cm i.d. column having six theoretical stages were compared with the simulation results for the two cases when the distillate receiver is initially empty and initially full. 1. Introduction Batch distillation is becoming more important as a result of the recent increase in the production of highvalue-added, low-volume specialty chemicals and biochemicals. The flexibility in operation and the lower costs for separating relatively pure components are the advantages offered by batch distillation over continuous distillation. In many cases, the objective of the batch distillation is to recover the most volatile component of a feed mixture at a high degree of purity, leaving relatively heavy components in the still. There are two basic modes of operating a batch distillation column: (1) constant reflux and variable product composition and (2) variable reflux and constant product composition. The unsteady-state behavior of these columns can be analyzed either by rigorous stageby-stage calculation methods or by shortcut methods assuming negligible vapor and liquid holdups and constant molar overflows. Meadows1 developed the first rigorous multicomponent batch distillation model, based on the assumptions of adiabatic equilibrium stages and constant molar liquid holdups for stages and the condenser. Distefano2 extended the model and developed a computer-based method for solving a set of differential mass and energy balance equations. Distefano’s work forms the basis for almost all of the later work on rigorous modeling of batch distillation columns. Seader3 presented an excellent review of modeling of batch distillation columns. Recently Diwekar and Madhavan4 presented shortcut methods for the multicomponent batch distillation operation for the two cases of constant reflux and constant distillate composition. Sundaram and Evans5 also simulated the multicomponent batch distillation operations under constant reflux. Adopting same assumption of constant molal overflow and negligible * To whom correspondence should be addressed. E-mail:
[email protected].
vapor and liquid holdups, these methods employed the Fenske-Underwood-Gilliland shortcut procedure for continuous distillation to calculate the vapor and liquid compositions at successive time steps. They treated batch distillation as a sequence of continuous distillation and were able to circumvent stage-by-stage calculations. Another mode of operating batch distillation columns, suitable for pilot-plant or small-scale distillation, was proposed by Treybal.6 Applying this technique for separating a binary mixture, the author reported a detailed analysis for the dynamics of a binary column. The apparatus used is conventional as depicted in Figure 1. A heated still, surmounted by a distillation column, condenser, and distillate receiver, has provision for total reflux. The batch charge is introduced into the still, and the level of the overflow reflux line is adjusted so that the distillate receiver will contain the correct amount of distillate. Heat is then applied, and the distillation proceeds without supervision until the liquid in the distillate receiver, which becomes progressively richer in the light component, comes to the desired composition. The operation can be performed with the distillate receiver either initially empty or full with the original batch charge. The advantages of this operating mode are evident: the operation is exceptionally convenient because it requires no reflux control. Neither the yield nor the quality of products is influenced by the variation in the heating rate or interruption of the distillation. Moreover, since the column operates under total reflux condition, it operates at its maximum distillation capacity and less stages are required in comparison to the other modes of separation under a finite reflux ratio. Sørensen and Skogestad7 found the total reflux operation to be better for separations with a small amount of light component. The objective of this article is to extend Treybal’s mode of operation for a binary mixture to a multicomponent system with constant relative volatilities. Under
10.1021/ie9805584 CCC: $18.00 © 1999 American Chemical Society Published on Web 02/05/1999
Ind. Eng. Chem. Res., Vol. 38, No. 3, 1999 1025
[( )( )]
log N+1)
xDi xWn xWi xDn log Rin
(2)
Equation 2 gives the concentration in the distillate receiver for any component as
( )
xDi ) RinN+1xWi
xDn xWn
(3)
n xDi ) 1, one obtains Since ∑ i)1
xDn
1
)
xWn
n
(4)
RinN+1xWi ∑ i)1
Substitution of eq 4 into 3 gives
xDi )
RinN+1xWi n
(5)
RinN+1xWi ∑ i)1
Figure 1. Batch distillation column with a distillate receiver under total reflux conditions: (a) total condenser; (b) distillate receiver; (c) still; (d) column; (e) reflux line.
the specification of the desired product concentrations of the lightest component in the distillate receiver and the still, the calculation procedure for determining the characteristics of the operation such as the volume of the distillate receiver, the variation of compositions in the distillate receiver and the still, the stagewise concentration profiles, and the distillation time necessary to accomplish the separation is to be presented for both cases of startup policies of the distillate receiver. The validity of the proposed method is demonstrated by comparing the theoretical concentration profiles with experimental results. 2. Analysis of the System 2.1. Assumptions. W0 mol of a multicomponent mixture, composed of n components 1, 2, ..., n in the order of decreasing volatility, are to be distilled in a batch column of N + 1 theoretical stages including the still. Initial feed compositions are xWi0. The dynamic behavior of the column with a distillate receiver which operates under total reflux mode can be analyzed using the following assumptions: (i) Holdup in the column and condenser is negligible relative to that in the still and distillate receiver. (ii) There is constant molal overflow. (iii) There is a constant vapor boilup rate (V, mol/h). (iv) There are constant relative volatilities of the mixture (Rin ) constant; i ) 1, 2, ..., n). (v) There are adiabatic equilibrium stages. (vi) There is constant molal holdup in the distillate receiver (D mol). (vii) Contents of the still and distillate receiver are well mixed. 2.2. Maximum Distillate Concentration of the Lightest Component (xD1max). The relation between the distillate and the still compositions under total reflux is given by Fenske equation:
Introducting xWi0 for xWi in eq 5, one can estimate the maximum attainable distillate concentration of the lightest component (xD1max). But if one desires xD1max as a product concentration, the volume of the distillate accumulated in the receiver will be null. So the desired product concentration (xD1F) must be specified somewhat lower value than xD1max. 2.3. Minimum Still Concentration of the Lighest Component (xW1min). As distillation continues, the concentration of the lightest component in the still (xW1) decreases continuously. But, under given operating conditions, there exists always a lower limit of this concentration (xW1min) which differs according to xD1F given. When the distillate receiver comes to the desired concentration of xD1F, the system can be described by the following equations. Specification of product concentration (1 equation):
xD1 ) xD1F
(6)
Component material balances (n - 1 equations):
DxDi + (W0 - D)xWi ) W0xWi0 i ) 1, 2, ..., n - 1 (7) Mole fraction summations (2 equations): n
xDi ) 1 ∑ i)1
(8)
n
xWi ) 1 ∑ i)1
(9)
Total reflux restrictions (n - 1 equations):
( )
xDi ) RinN+1xWi
xDn xWn
i ) 1, 2, ..., n - 1
(3)
There are 2n + 1 equations in 2n + 1 variables (D, xDi, xWi). Thus the value of xW1 will be determined solving eqs 6-9 and 3 simultaneously, which is lower
1026
Ind. Eng. Chem. Res., Vol. 38, No. 3, 1999
limit of the concentration of the lightest component in the still (xW1min) corresponding to xD1F. But if this composition is used as a product purity for the still, the distillation time will be infinite. So it is desirable that the actual specification of the product purity for the still (xW1F) is somewhat superior to xW1min. 2.4. Volume of the Distillate Receiver Corresponding to xD1F and xW1F. The volume of the distillate receiver (D mol) estimated in section 2-3 is the maximum amount of distillate (Dmax) when the product purities are specified to xD1F and xW1min. On the other hand, the volume of the receiver corresponding to the product purities of xD1F and xW1F is easily determined from the material balance for the lightest component:
D)
W0(xW10 - xW1F)
Since xW1F is in practice larger than xW1min, the actual amount of D will be less than Dmax. 2.5. Operation with a Distillate Receiver Initially Empty. 2.5.1. Analysis of the Regime for Rayleigh Distillation. Suppose initially the distillate receiver is empty. Then a time θ1 will be required to condense D mol of vapor to fill the receiver. If the vapor boilup rate is V mol/h, then
D V
(11)
During this time, there will be no reflux and the contents of the still undergo a Rayleigh distillation. The compositions of the distillate receiver and the still vary continuously during this period. Since the theoretical stages in the column play no role during Rayleigh distillation, the composition of the first drop of distillate deposited in the distillate receiver can be estimated from eq 5 by substituting N ) 0: 0
xDi0 )
φi ) 1 - (1 - φ1)1/R1i i ) 1, 2, ..., n Then n
D)
∑ i)1
n
Di′ )
∑ i)1
n
Wi0φi )
RinxWi
(12)
n
xDi′ )
φi )
Wi0
)
Wi0
W10
)
( ) W2′
R12
W20
)1-
) ‚‚‚ )
( ) W i′
Wi0
(17)
Wi0 Wi0(1 - φi) ) (1 - φ1)1/R1i W0 - D W0 - D
(18)
xWi′ )
During Rayleigh distillation the composition of the distillate receiver varies from xDi0 to xDi′ while that of the still changes from xWi0 to xWi′. The evolution of the compositions in the receiver and the still can be estimated as follows: Let the amount of each component accumulated in the receiver by Rayleigh distillation up to the time m∆t from startup be Di(m) mol, where m is the time-increment index. When the distillation starts, no distillate exists in the receiver. Then Di(0) ) 0 and Di(m) is estimated by the relation
Di(m) ) Di(m-1) + xDi(m-1) × V × ∆t m g 1 (19) from which the total distillate accumulated in the receiver and the amount of residue remaining in the still become respectively
( ) ( ) Wi
0
)
W1′
W1
0
1/R1i
(20)
W(m) ) W0 - D(m)
(21)
Then the compositions of the still and the receiver are calculated by (m)
xWi
)
W0xWi0 - Di(m) W(m)
(22)
and
( ) W i′
Di(m) ∑ i)1
and
(13)
R1i
(m)
xDi
Wi0
or
W i′
(16)
n
Noting that for a Rayleigh distillation
W1 ′
1i
Di′ Wi0φi Wi0 ) ) [1 - (1 - φ1)1/R1i] D D D
D(m) )
The compositions of the receiver and the still when the receiver is first full can be estimated by the following procedure. Let the amount of each component in the distillate receiver and the still when the receiver just becomes full be Di′ and Wi′ mol, respectively. Then the fractional vaporization of each component (φi) at this instant is
Wi0 - Wi′
Wi0[1 - (1 - φ1)1/R ] ∑ i)1
Since Wi0, R1i, and D are known, φ1 will be obtained from eq 16. Using φ1, the distillate receiver and the still compositions at the end of the Rayleigh distillation are calculated as
RinxWi0 ∑ i)1
D i′
(15)
(10)
xD1F - xW1F
θ1 )
one obtains, combining eqs 13 and 14, the fractional vaporization of each component (φi) in terms of that of the lightest component (φ1) as
i ) 1, 2, ..., n
(14)
)
Di(m) D(m)
(23)
xD1(m) is then compared to xD1′, and if xD1′ is smaller than xD1(m), calculation for the next time step is undertaken by substituting the newly obtained xDi(m) for xDi(m-1) in eq 19. The calculation is repeated until xD1(m) becomes xD1′.
Ind. Eng. Chem. Res., Vol. 38, No. 3, 1999 1027
2.5.2. Analysis of the Operation under Total Reflux. As distillation proceeds from this point, reflux runs down the column at the same molar rate as vapor boils up. As distillation continues, the compositions of the distillate receiver and the still vary continuously. The composition of the lightest component in the still and distillate receiver become leaner and richer, respectively, until ultimately at the end of distillation, xD1 becomes xD1F and xW1 becomes xW1F. This will require a time θ2. The analysis of this process, sketched in Figure 1, is as follows: A component material balance for the still can be written as
dxWi V ) (x - yWi) dt W0 - D 1i
(25)
RinxWi(k)
n
inxWi
xWi(k+1) ) xWi(k) +
V [x (k) - yWi(k)]∆t (26) W0 - D 1i
xDi(k+1) ) xDi(k) +
V [y (k) - xDi(k)]∆t D Ni
(27)
Here k is the time-increment index. For a given ∆t time increment, xWi(k+1) and xDi(k+1) are computed for each component from eqs 26 and 27, respectively. Calculations are initiated at k ) 0. At this instant, xWi(k) ) xWi′, xDi(k) ) xDi′, and yWi(k) are calculated from the relation
yWi(k) )
RinxWi′ n
(28)
RinxWi′ ∑ i)1
inxNi
stage 1:
x2i
(k)
) x1i
+
Rinx1i(k)
RinxWi(k)
-
n
∑
n
∑R
(k)
Rinx1i
inxWi
i)1
stage 2:
x3i
(k)
) x2i
+
Rinx2i
∑R
stage N - 1: xNi
(k)
) xN-1i
+
(30)
n
(k)
inx2i
i)1
(k)
Rinx1i(k)
-
n
‚ ‚ ‚
(k)
i)1
(k)
(k)
(k)
i)1
∑
Rinx1i(k)
i)1
‚ ‚ ‚
‚ ‚ ‚
RinxN-1i(k)
RinxN-2i(k)
n
∑
-
n
(k)
RinxN-1i
i)1
Equations 24 and 25 can be expressed in terms of finitedifference form, using Euler’s method:
∑R
(k)
i)1
(k)
RinxNi(k)
-
n
∑R
(24)
A component material balance for the distillate receiver gives
dxDi V ) (yNi - xDi) dt D
x1i(k) ) xDi(k) +
∑R
(k)
inxN-2i
i)1
For N theoretical stage columns with n component system, there are nN equations in nN unknowns (xji(k), where j ) 1, 2, ..., N and i ) 1, 2, ..., n). Thus all values of stage compositions (xji(k)) are obtained by solving eq 30 simultaneously. This may be readily done on a digital computer. The x1i(k) and yNi(k) calculated from eqs 30 and 29 are substituted into eqs 26 and 27 along with xWi(k), yWi(k), xDi(k), and ∆t to estimate xWi(k+1) and xDi(k+1). xD1(k+1) is then compared to xD1F, and if xD1(k+1) is smaller than xD1F, a new calculation for the next time step is undertaken by substituting the values of xWi(k+1) and xDi(k+1) in the place of xWi(k) and xDi(k) in eqs 26 and 27, respectively. The calculation procedure is repeated until xD1(k+1) is greater than or equal to the specified concentration of xD1F. Counting the number of the timeincrement index, k, the distillation time required for total reflux operation is calculated by
θ2 ) k∆t
(31)
from which the total distillation time is found to be
θ ) θ1 + θ2
However, eqs 26 and 27 require values for x1i(k) and xNi(k), since yNi(k) are related to xNi(k) by
(32)
entire column: Lx1i(k) ) LxDi(k) + VyWi(k) - VyNi(k)
An arbitrary value of 0.001 h for ∆t was assigned in this article, but it may be changed if desired. The calculation procedure of the proposed method is shown in Figure 2. 2.6. Operation Using the Distillate Receiver Initially Full. An alternate procedure is to start the operation with the distillate receiver full of the same liquid as the initial charge. In this procedure, there exists no region for Rayleigh distillation (θ1 ) 0) and the operation is carried out entirely under the condition of total reflux (θ ) θ2). Analysis of this mode of operation is analogous to that of operation with the receiver initially empty. Equations 24-30 are applied once again with initial values of xWi(0) ) xWi0 and xDi(0) ) xWi0. The calculation procedure is summarized also in Figure 2.
If we note that V ) L for total reflux operation and substituting eqs 28 and 29 for yWi(k) and yNi(k), respectively, the above material balance becomes
3. Separation of a BTX Mixture In order to test the validity of the shortcut method proposed in this study, distillation experiments of a
(k)
yNi
)
RinxNi(k) n
(29)
RinxNi(k) ∑ i)1 These values can be determined by solving simultaneously the following material balance equations.
1028
Ind. Eng. Chem. Res., Vol. 38, No. 3, 1999
each stage, still, and distillate receiver during experiments for the two cases of operating mode and analyzed by gas chromatography (Hewlett Packard 5890 series II plus). Experimental data are compared with the calculations based on the proposed model in section 4. The BTX system may be considered nearly ideal, and the relative volatilities may be assumed to be the ratio of the vapor pressures of two components at the system temperature. In our experiments, the temperatures at the top and bottom of the column were estimated by the bubble point calculations. Peng-Robinson equation of state and the experimental compositions at the end of the distillation were used in these calculations. Since the difference of the temperatures between the two ends of the column was found to exceed more than 40 °C, the relative volatilities at each end of the column were estimated, respectively. Then the average relative volatilities were determined by
Rij ) (RijtopRijbottom)1/2 Values of the average relative volatilities of BTX mixture, estimated in this manner, are presented in Table 1. During the simulation, these values were used without any modifications.
Figure 2. Flowchart for batch distillation calculations with a distillate receiver.
ternary system, a mixture of benzene, toluene, and o-xylene, were undertaken using a batch column depicted in Figure 1. The details of the experimental apparatus and operating conditions are summarized in Table 1. In order to follow up the variation of the concentrations in the column, samples were taken from
4. Results and Discussion By means of the shortcut method proposed herein, one can estimate the variation of compositions in the distillate receiver and still, batch distillation time required to complete the separation, and the stagewise concentration profiles at any instant during the distillation. Prior to the initiation of the simulation, one must examine the ability of the equipment to deliver the specified product purities. If the specifications are attainable by the apparatus at hand, then the volume of the distillate receiver (D mol) corresponding to these purities are determined. The calculation procedures are described in sections 2.2-2.4 and summarized in Figure 2. In case of the separation specified in Table 1, the maximum concentration of the component 1 (benzene) in the distillate receiver (xD1max) is 0.9879, which is greater than the desired purity of xD1F ) 0.95. Minimum concentration of the component 1 in the still (xW1min) corresponding to xD1F ) 0.95 is 0.0444 which is smaller than the specified purity of xW1F ) 0.05. Thus the separation we specified is feasible. Figure 3 shows the variation of xW1min corresponding to xD1F. The minimum concentration xW1min decreases gradually as the specified final distillate concentration xD1F decreases from its maximum value of xD1max ) 0.9879. Using the specified purities of xD1F ) 0.95 and xW1F ) 0.05, the volume of the distillate receiver is calculated to be D ) 15.2 mol. Figure 4 represents the relation
Table 1. Description of the Batch Column and Operating Conditions Distillation Column column: i.d. 10 cm, Pyrex glass, thermally insulated, 7 actual plates equivalent to 5 theoretical stages (N ) 5) plate: cross-flow, sieve-plate type, no. of holes/plate ) 251, hole diameter ) 0.2 cm, weir length ) 8.4 cm, weir height ) 0.5 cm, spacing ) 15 cm still: 20 L, Pyrex glass, heated by electric mantle Operating Conditions system: benzene (1)-toluene (2)-o-xylene (3) relative volatility: R13 ) 6.29, R23 ) 2.72, R12 ) 2.31 initial charge: 115.2 mol feed composition: xW10 ) 0.1687, xW20 ) 0.3140, xW30 ) 0.5173 vapor boilup rate: 106.7 mol/h product purity: xD1F ) 0.95 (receiver), xW1F ) 0.05 (still)
Ind. Eng. Chem. Res., Vol. 38, No. 3, 1999 1029
Figure 3. Variation of the minimum concentration of the lightest component in the still corresponding to the specified distillate composition.
Figure 5. Changes in compositions in the distillate receiver for the product purities of xD1F ) 0.95 and xW1F ) 0.05. The receiver is initially empty.
Figure 4. Volume of the distillate receiver corresponding to the specified product purities in the distillate receiver and the still.
Figure 6. Changes in compositions in the still for the product purities of xD1F ) 0.95 and xW1F ) 0.05. The receiver is initially empty.
between the volume of the receiver and the specified product purities. For a fixed value of xW1F, the volume of the receiver increases as xD1F decreases. In all cases of xD1F, the volume D decreases gradually as xW1F increases giving a final value of zero when xW1F is specified at 0.1687, which is the feed concentration of component 1 given. The variations of compositions in the receiver and still are different due to the initial conditions of the receiver. Assuming the receiver is initially empty, the contents in the still undergo a Rayleigh distillation during θ1 ) D/V ) 15.2/106.7 ) 0.142 h. After this time, the distillation continues under total reflux condition and xD1 becomes richer continuously, until ultimately at the end of the distillation xD1 becomes xD1F. The theoretical time required for total reflux operation (θ2) was found to be 1.509 h. Summation of θ1 and θ2 gives the total distillation time as 1.651 h. The changes of the compositions in the receiver and the still during this mode of operation, obtained by simulation, are shown in Figures 5 and 6, respectively. Experimental concentration variations are also indicated in these figures. Experimental compositions during Rayleigh distillation in Figure 5 show that the degree of enrichment of benzene in the distillate receiver for the first few minutes exceeds the estimation while that of toluene and o-xylene fall short of the expected value. The discrepancy is probably due to the relatively low temperature conditions inside the column during the start up period. In this case, condensation of some vapor can take place before its arrival to the condenser which causes the enrichment of benzene over the other components. Taking into account
Figure 7. Changes in compositions in the distillate receiver for the product purities of xD1F ) 0.95 and xW1F ) 0.05. The receiver is initially full with original batch charge.
such experimental constraints and the fact that the average relative volatilities were used without any modifications throughout the simulation, the conformity of the theoretical profiles with experimental results in Figures 5 and 6 are considered satisfactory. As mentioned in section 2.6, the operation can be performed with the distillate receiver initially full with the original batch charge. Figures 7 and 8 show the trends of the composition change in the receiver and the still for the operating conditions described in Table 1. Since the period of Rayleigh distillation does not exist in this case, xD1 increases directly from xW10 to xD1F and the theoretical batch time required is equal to θ2, which
1030
Ind. Eng. Chem. Res., Vol. 38, No. 3, 1999
Figure 8. Changes in compositions in the still for the product purities of xD1F ) 0.95 and xW1F ) 0.05. The receiver is initially full with batch charge.
for the two cases of the initial state of the distillate receiver. It shows that in all cases of xD1F considered, the batch time θ increases as xW1F is lowered with an increase in D as expected. Table 2 also shows that the batch time for the operation with the distillate receiver initially empty is always longer than that for the operation with the receiver initially full. It appears that when the volume of the receiver is relatively small, the batch times are nearly equal regardless of the mode of operation. The difference of batch time between the two cases grows as the volume of the receiver increases. For the separations examined in Table 2, the time differences are negligible and the initial condition of the receiver is not critical for the batch distillation time. However one can expect that, if the relative volatilities are similar and/or the volume of the distillate receiver is important relative to the volume of batch charge, then the differences in batch times, empty and full, will become significant. 5. Concluding Remarks
Figure 9. Concentration profiles of the liquid phase at the end of distillation. Stage locations: 0 refers to the still, and 6, the distillate receiver. Table 2. Comparison of the Batch Distillation Times initial state of the distillate receiver empty xD1F
xW1min
xW1F
D (mol)
θ1 (h)
θ2 (h)
θ (h)
full θ (h)
0.98
0.1081
0.95
0.0444
0.90
0.0211
0.85
0.0131
0.80
0.0090
0.14 0.11 0.14 0.11 0.08 0.05 0.14 0.11 0.08 0.05 0.03 0.14 0.11 0.08 0.05 0.02 0.14 0.11 0.08 0.05 0.02
3.94 7.77 4.08 8.05 11.7 15.2 4.35 8.56 12.5 16.1 18.4 4.66 9.14 13.3 17.1 20.6 5.01 9.80 14.2 18.2 22.0
0.037 0.073 0.038 0.075 0.110 0.142 0.041 0.080 0.117 0.151 0.172 0.044 0.086 0.124 0.160 0.193 0.047 0.092 0.133 0.171 0.206
0.342 1.066 0.215 0.463 0.777 1.509 0.162 0.341 0.551 0.859 1.330 0.134 0.284 0.456 0.695 1.328 0.116 0.245 0.394 0.599 1.062
0.379 1.139 0.253 0.538 0.887 1.651 0.203 0.421 0.668 1.010 1.502 0.178 0.370 0.580 0.855 1.521 0.163 0.337 0.527 0.770 1.268
0.376 1.120 0.251 0.532 0.877 1.635 0.201 0.417 0.659 0.996 1.483 0.177 0.365 0.572 0.842 1.501 0.162 0.333 0.520 0.757 1.248
is 1.635 h. Figure 9 represents the stagewise composition profiles of the liquid phase at the end of the distillation. Figures 7-9 show that the agreement between the theoretical and experimental profiles is excellent. The batch distillation times required for several specifications of xD1F and xW1F are compared in Table 2
On the basis of the assumptions of negligible holdups in the column and condenser, constant molal overflows, constant relative volatilities, constant vapor boilup rates, and adiabatic equilibrium stages, a shortcut model was developed. This is to simulate a multicomponent batch distillation column equipped with a distillate receiver which operates under total reflux condition. To initiate the simulation, values of some parameters such as the number of equilibrium stages in the column, amount of the batch charge, initial feed compositions, relative volatilities of the system, and the vapor boilup rate are required. First of all, the model estimates the highest composition of the lightest component in the distillate receiver theoretically attainable under given operating conditions. In consideration of this maximum composition, actual distillate purity desired is specified as an input. Then the model calculates the lower limit of the composition of the lightest component in the still which will be different according to the desired distillate purity. Once the product purities in the distillate receiver and the still are specified for the lightest component, the proposed model estimates the volume of the receiver, the variation of the compositions in the receiver and the still, composition profiles of the column, and the batch time required to complete the separation in both cases of the receiver, initially empty and initially filled with the original batch charge. The validity of the proposed method was demonstrated by comparing the simulation results with the experimental data, taking the separation of a ternary BTX mixture as an example. The primary advantage of the batch distillation with distillate receiver over conventional constant reflux or variable reflux operation is its expediency. In principle, this mode of operation is applicable to any multicomponent batch distillations to separate the lightest component from the feed mixture at a high purity. But it would be more suitable for small-scale distillation operations when the required number of theoretical stages is not important. For those situations, one may find the shortcut procedure developed herein convenient and useful. Nomenclature D ) volume of distillate receiver, mol Di ) amount of component i in distillate receiver during Rayleigh distillation, mol
Ind. Eng. Chem. Res., Vol. 38, No. 3, 1999 1031 Di′ ) amount of component i in distillate receiver when the receiver is first full by Rayleigh distillation, mol i ) component k ) time-increment index for the distillation under total reflux condition L ) flow rate of liquid phase, mol/h N ) number of equilibrium stages in the column n ) number of components in feed mixture, or heaviest component t ) time, h V ) vapor boilup rate, mol/h W ) amount of liquid remaining in the still, mol Wi′ ) amount of component i in the still when the receiver is first full by Rayleigh distillation, mol Wi0 ) amount of component i in the original batch charge, mol W0 ) amount of the original batch charge, mol xD1F ) specified product purity of the lightest component in the distillate receiver xD1max ) maximum concentration of the lightest component in the distillate receiver xDi ) mole fraction of component i in the distillate receiver xDi′ ) mole fraction of component i in the distillate receiver when the receiver just becomes full by Rayleigh distillation xji ) mole fraction of component i in liquid phase from stage j xW1F ) specified product purity of the lightest component in the still xW1min ) minimum concentration of the lightest component in the still xWi ) mole fraction of component i in the still xWi′ ) mole fraction of component i in the still when the distillate receiver just becomes full by Rayleigh distillation xWi0 ) initial concentration of the batch charge yji ) mole fraction of component i in vapor phase from stage j
Superscripts k ) time-increment index for the distillation under total reflux condition m ) time-increment index for Rayleigh distillation Greek Symbols Rij ) relative volatility between components i and j ∆t ) time increment with a default value of 0.001, h θ ) total distillation time, h θ1 ) time required for Rayleigh distillation, h θ2 ) time required for total reflux operation, h φi ) fractional vaporization of component i when the receiver is first full
Literature Cited (1) Meadows, E. L. Multicomponent Batch Distillation Column Design. Chem. Eng. Prog. Symp. Ser. 1963, 59, 48-55. (2) Distefano, G. P. Mathematical Modelling and Numerical Integration of Multicomponent Batch Distillation Equations. AIChE J. 1968, 14, 190. (3) Seader, J. D. Essential Features of a Model for Batch Distillation. Presented at ASPENWORLD 88, Amsterdam, The Netherlands, 1988. (4) Diwekar, U. M.; Madhavan, K. P. Multicomponent Batch Distillation Column Design. Ind. Eng. Chem. Res. 1991, 30, 713721. (5) Sundaram, S.; Evans, L. B. Shortcut Procedure for Simulating Batch Distillation Operations. Ind. Eng. Chem. Res. 1993, 32, 511-518. (6) Treybal, R. E. A Simple Method for Batch Distillation. Chem. Eng. 1970, Oct. 5, 95. (7) Sørensen, E.; Skogestad, S. Optimal Operating Policies of Batch Distillation. In Proceedings of Symposium PSE ‘94; KIChE: Kyungju, Korea, 1994; p 449.
Received for review August 25, 1998 Revised manuscript received November 23, 1998 Accepted November 25, 1998 IE9805584
