Znd. Eng. Chem. Res. 1993,32,293-301 dures where water repellency is a primary performance requirement. It is common knowledge that the surfactants are usually mixed with pesticides to reduce the surface energy of the formulation and to enhance the ease of spreading the pesticide on the substzates. Since this study indicated that surfactant build-up on the surface of the fabrics reduced their water repellency, additional work is recommended to determine the effect of the application of different typea of pesticide formulations on the water repellency of fabrics.
Acknowledgment
293
Bikerman, J. J. Theory of Interfacial Tension. Phys. Chem. 1970, 20, 225.
Evans, W. P. Cationic Fabric Softeners. Chem. Ind. 1969,893-903. Ginn, M. E.; Kinney, F. B.; Harris, J. C. Effect of Cotton Substrate Characteristics upon Surfactant Adsorption. J. Am. Oil Chem. SOC.1961,38, 139-143. Ginn, M. E.; Schenach, T. A.; Jungermann, E. Performance Evaluation of Selected Fabric Softeners. J. Am. Oil Chem. SOC.1965, 42,1084-1091.
Hughes, G . K.; Koch, S. D. Evaluation of Fabric Softeners. Soap Chem. Spec. 1965,109-112. Linfield, W. M.; Sherrill, J. C.; Davis, G. A,; Raschke, R. M. Fabric Treatment with Cationic Softeners. J. Am. Oil Chem. SOC.1958, 35, 590-593.
The authors wish to express sincere gratitude to the members of the Southern Regional Project (S208) Committee for their help and stimulating discussions. Funding for research was provided by the USDA, Southern Regional Hatch Project S 208. This contribution is indeed appreciated.
Literature Cited AATCC Technical Manual; American Association of Textile Chemists and Colorists: Research Triangle Park, NC, 1988. Auerbach, M. E. Germicidal Quaternary Ammonium Salts in Dilute Solution, a Colorimetric Assay Method. Ind. Eng. Chem., Anal. Ed. 1943,16,492-493.
Miller, B.; Young, R. A. Methodology for Studying the Wettability of Filaments. Text. Res. J. 1975,45, 359-365. Miller, B.; Tyomkin, I. Spontaneous Transplanar Uptake of Liquids by Fabrics. Text. Res. J. 1984,54, 702-706. Sarmadi, A. M.; Kwon, Y. A.; Young, R. A. Wettability of Nonwoven Fabrics. 1. Effect of Fluorochemical Finishes on Water Repellency. Znd. Eng. Chem. Res. 1993, preceding paper in this issue. Sexsmith, F. H.; White, H. J., Jr. The Absorption of Cationic Surfactants by Cellulosic Materials, I, The Uptake of Cation and Anion by a Variety of Substrates. J. Colloid Sci. 1959, 14, 508-618.
Received for review March 23, 1992 Revised manuscript received October 1, 1992 Accepted October 23, 1992
PROCESS DESIGN AND CONTROL Design and Operating Targets for Nonideal Multicomponent Batch Distillation Christine Bernot,+Michael F. Doherty, and Michael F. Malone* Department of Chemical Engineering, University of Massachusetts, Amherst, Massachusetts 01003-0011
A method is reported to estimate the batch sizes, operating times, equipment sizes, utility loads, and costa for the batch distillation of nonideal multicomponent mixtures. The method provides an estimate for both the design and the operating policy for the reflux or reboil ratio without the need to integrate the full column model numerically. This operating policy approximates the constant product composition policy for high purities and high fractional recoveries and shows a substantial cost savings over the constant reflux or reboil policies, primarily on account of a substantial reduction in the vapor rate. The approach provides a good approximation to the results of a detailed optimization in an example for a binary mixture. A more interesting application is illustrated for the separation of a quaternary azeotropic mixture arising from the transesterification of ethyl acetate with methanol to produce ethanol and methyl acetate.
Introduction Batch processes are used extensively in the specialty chemical industry because they are well-suited to lowproduction and to changing markets' studies Of the Optimal design and Operation Of batch were reviewed by Rippin (1983), who pointed out that the batch distillation of nonideal mixtures had not received much attention.
* T o whom correspondence should be addressed.
'Current address: Solvay & Cie, Rue de Ransbeek 310,1120 Bruxelles, Belgium.
Recent work on design and optimal operation of batch distillation has focused mainly on relatively ideal mixtures where volatilities can be used to describe the vapor-liquid equilibrium. For example, computational to solve the problem have been developed for binary mixtures in tray or packed columns (Hansen and Jorgensen, 1986) with a recycled intermediate cut (Christensen and Jorgensen, 198,) and in multicomponent systems (Diwekar et al., 1987). Optimal design and operating variables were also studied by Luyben (1988) for ternary mixtures including recycled intermediate cuts; Al-"'Uwaim and Luyben (1991) described a shortcut design method for well-balanced feeds. Diwekar et al. (1989)
oa8s-5885/93/2632-Q293$04.00/00 1993 American Chemical Society
294 Ind. Eng. Chem. Res., Vol. 32, No. 2, 1993
studied the optimal design for ternary mixtures using nonlinear programming techniques, and a shortcut design method was developed by Diwekar and Madhavan (1991). The interaction between design variables and the operating policy in the optimal design for binary mixtures was analyzed by Kim (1985) and Logsdon et al. (1989). Wu and Chiou (1989) described a design for a quaternary mixture and found a variable reflux policy that was significantly more economical than a constant reflux operation. Performance simulations for the batch distillation of ternary azeotropic mixtures were described by Kolber and Anderson (1987) for a ternary azeotropic mixture, but neither the design nor choice of the operating variables was addressed. In fact, to the best of our knowledge, no literature is available concerning the optimal design for batch distillation of nonideal and azeotropic mixtures. A feasible separation sequence for batch distillation can be found using the synthesis procedure described by Bernot, Doherty, and Malone (1991). In this paper we describe a method to estimate the batch sizes, operating times, equipment sizes, utility loads, and costs for any single batch rectifier or stripper in the sequence. Two operating policies are studied: one maintains the reflux or reboil ratio constant over each cut, primarily for comparison with the second strategy which approximates the constant product composition policy. After a description of the basic model and the design procedure, this approach is compared to the results of a detailed optimization in an example for a binary mixture. The full power of the approach is demonstrated by consideration of the design of a batch column for the separation of a quaternary azeotropic mixture containing methyl acetate, ethyl acetate, ethanol, and methanol produced in the transesterification reaction. Basic Model Our goal is to present a method for designing batch distillation columns in which a given amount of a nonideal multicomponent mixture is separated into some or all of its pure components, during a specified period of time, and at a cost close to the minimum. Existing simulators such as BATCHFRAC are useful for rating calculations with specified operating policies and equipment sizes. The operating policy, e.g., reflux ratio vs time, to attain certain product purity goals can also be approximated by the use of "controllers" for the dynamic models. However, such calculations may be quite time-consuming for nonideal mixtures and introduce additional requirements of tuning the controller parameters. We desire a more rapid method that is generally applicable to ideal, nonideal, and azeotropic mixtures and that provides good targets as a basis for more detailed simulation or pilot studies. Overall Balance Constraints. The scheduling constraints on the number of batches per year, nb, and the amount of feed per batch, H,,must be met regardless of the details of a design or the particular operating policies for vapor boil-up, reflux ratio, or reboil ratio. The number of batches, the time available tw (e.g., the number of working hours per year), the operating time, t p , and the "dead" time, td, are related by
The dead time includes the startup time and the product removal and recharging times. The number of moles of fresh feed per batch is then Ho = F/nb (2) where F is the total number of moles of fresh feed to be
N
n I-1
1
Figure 1. Schematic representation of (a) the batch rectifier or "conventional" column and (b) the batch stripper or 'inverted" column.
processed in time tw. Equations 1and 2 contain six variables, nb, Ho, tp,tw, td,and F,and we consider caeeawhere values are prescribed for three of these: F,tw, and td. One degree of freedom remains, which we will take as the operating time per batch; this can eventually be chosen by an optimization. Evolution of Compositions. We describe a model for the variation of the compositions in both the conventional and the inverted configurations. In a "conventional" column or batch rectifier, shown schematically in Figure la, the batch is held in the reboiler and products are withdrawn at the top of the column, while in an "inverted" column or batch stripper, Figure lb, the batch tank acts as the reflux drum and producta are withdrawn as bottoms from the reboiler. The latter is an uncommon arrangement but offers some important advantages for the separation of azeotropic mixtures (see Bernot et al., 1990, 1991). We consider a column with a total condenser and account only for the hold up of the batch wing a quasi-steady description of the column. Constant molar overflow is also assumed, although heat effects can be incorporated in a straightforward manner when enthalpy data are available. In this model, we decouple the variation of compositions from the variation of flows and the batch size using a dimensionless "warped" time. This is defined so that d t = (P/H) dt (3) where P is the instantaneous production rate (moles per hour) and H is the instantaneous liquid holdup in the batch (moles). For each warped time, the differential ma88 balances can be integrated for the batch composition, x , dx/dt = x - xp (4) together with the algebraic mass balancea and vapopliquid equilibrium (VLE) for each stage. If the product has mole fractions xp,the (quasi-steady) mass balance on any stage j in a batch rectifier is r+l 1 1 Ii IN (5) xj = y y j b l - ;xP or, in a batch stripper, l I j I N These are combined with the vapor-liquid equilibrium relationships for each stage and the reboiler, e.g., wing the
Ind. Eng. Chem. Res., Vol. 32, No. 2,1993 295 Wilson equation for the liquid-phase activity coefficients, along with the mass balance for a total condenser. xN+1 = YN (7) Here, yo is the vapor composition in equilibrium with the reboiler composition xR,and xN+1is the condenser composition vector. In a batch rectifier, xR = x and xN+1 = xp,while in a batch stripper xR = xp and xN+1 = x. At each value of the warped time, there are two degrees of freedom in eqs 4, with 5 or 6, and 7, and the VLE. These are taken to be the number of theoretical stages and the reflux ratio, r([), in a batch rectifier or the reboil ratio, s([), in a batch stripper. (Since the flows and compositions are decoupled, the reflux or reboil ratio policy can be expressed as a function of only the warped time.) Cuts, Operating Times,and Vapor Rate. The variables tp, N, and r([) or s([) can be optimized by formulating a suitable mixed calculus-variational problem (e.g., Kim, 1985) and the results generally show a time-varying reflux or reboil policy with discontinuities between the “cuts”. However, the exact solution of this optimization problem is quite difficult and cannot normally be justified at the design stage where the main focus is on choosing a good structure and estimating targets. Instead, we choose a reboil or reflux ratio policy for each cut, and formulate an optimization problem using the variables tp, N , and [ k along with any parameters in the reflux or reboil policy. The cut number is k, and [ k is the warped time from the start of the batch to the end of cut k. The total number of cuts is n, of which n, - 1are collected by withdrawing product streams while the last cut consists of the remainder of the batch. We wish to estimate the batch size, cycle time, compositions, equipment sizes, costs, and operating policy for the column. For a given reflux or reboil ratio policy in each cut and a given number of stages, the system of differential and algebraic equations in eqs 4 , or 5 or 6, and 7, and the VLE can be integrated to obtain the variation of the compositions as a function of the warped time. We used Gear’s method as implemented in the Livermore Solver DLSODI. An overall mass balance for each cut can be written k = 1, ..., n, - 1 Hk + wk = Hk-1 (8) where wk is the total amount of material collected as a product during cut k, and Hk is the size of the batch at the end of cut k. From the quasi-steady material balances, dHx/dt = -Pxp, along with eq 3, we find 1=1
Zi,k
- fi,kni,k-l
Once the time-averaged values of the flows and compositions are obtained, the vapor rate necessary to perform the separation can be determined and this is sufficient to set the column diameter, the heat exchanger duties, and the utility loads. The relationship of the vapor rate during cut k to the other flows is given by a maas balance around the condenser (or around the reboiler in a batch stripper). Integrating over the duration of cut k for a batch rectifier, we have
or for a batch stripper
We choose a constant vapor rate policy and vk = v for each cut throughout the separation; this is appropriate because the column will eventually have a fixed diameter, permitting little or no variation of the vapor rate. Both the design and the operation are simplified under this policy, with little or no economic penalty (Kim, 1985). The real time for the duration of a cut, tk, can be determined by rewriting eq 16 and using the definition of the warped time in eq 3 Vtk = Ldh(rk+ 1)Hd[
(18)
which leads to
k
Hk = Hk-1 eXp(-bk) = Ho eXp(-CbL)
to the specification for the average product purity of one of the components (say component i) Xi,k-l - %i,k exp(-bk) = xi,k - Zi,k It may be convenient for certain cuts to make a specification (or, eventually, an optimization) of the fractional recovery for one of the components. The fractional recovery of component i in cut k is Zi,kWk - %i,k(xi,k-l- x i , k ) f. I (14) I ,k ICi,k-lHk-l ni.k-l(Zi,k - xi,k) and the batch composition at the end of the cut can be obtained by rearranging eq 14: (l - fi,k)Xi,k-lZi,k xi,k = (15)
(9)
and k-1
wk
= Ho exp(-CbJ[l - exp(-bk)l 1=1
(10)
where the duration of cut k (measured in warped time) is b k = [ k - [k-1. The average product composition for cut k, zk, can be found by time-averagingthe instantaneous distillate composition, but it is simpler to use the overall mass balance for each component over the cut HkXk + wkzk = Hk-1Xk-1 (11) which gives (with eqs 9 and 10) xk exp(-bk) + zk[l - exp(-bk)] xk-1 (12) Knowing the initial composition, so,we can follow the batch composition as a function of the warped time and compute zk using eq 12. The duration of each cut is related
for a batch rectifier (or to a similar expression for the batch stripper). The operating time is the sum of the cut times, excluding the final cut which is taken from the batch tank, i.e., n.-1
Equations 19 and 20 can be solved for the vapor rate in terms of the operating time for the rectifier
or for a stripper
296 Ind. Eng. Chem. Res., Vol. 32, No. 2, 1993
Substituting for the vapor rate in eq 19 gives the actual cut times. Equation 21 or 22 relates the vapor rate and the operating time, and the vapor rate could be used as an optimization variable in place of tp. For a constant reflux or reboil ratio policy in each cut, the integrals in eqs 21 and 22 can be evaluated explicitly. For the conventional column,
Honc-1 k-1 C ((rk + 1 ) exp(-C 6 J [ 1 - exp(-dk)lI
v = -tp
k=l
1=1
(234
or simply 1
4-1
v=tp k = l
(rk +1)wk
(23b)
In the special (and generally suboptimal) case where the same constant reflux ratio is used in each cut, eq 23b simplifies to V = (r + l)(Ckwk/tp),giving a relationship between the vapor rate and the average distillate rate that is similar to the expression describing a continuous rectifier. For a batch stripper, a similar approach gives
or 1
n.-1
Design Procedure Specifications. The design model is formulated using the variables tp, N,rk([) Or S k ( [ ) , and b k , k = 1,..., n, - 1. Because we decouple the compositions from the flows using the warped time, it is not necessary to know the operating time in order to determine the composition variationsalthough this information is eventually required for a full design. For a given number of stages there are two degrees of freedom for each of the cuts; the reflux or reboil ratio and the duration of the cut, bk. For each cut there are also two specifications such as the product purity and/or a fractional recovery that could be made. We generally choose the purity and the fractional recovery of the major component in each of the cuts and determine the corresponding values of bk and either rk([) or S k ( [ ) , as appropriate. Constant Reflux or Reboil Ratio Policy. In the case of a constant reflux or reboil ratio policy, bk is determined using a specification of the average product purity as a 'stopping criterion" while the reflux or reboil ratio is computed to meet the desired fractional recovery. For azeotropic mixtures, we distinguish "product" cuts from "azeotropic" cuts; the latter do not meet the desired purities but are unavoidable on account of the nature of the mixture (Bernot et al., 1990,1991). Other 'intermediate" or 'slop" cute may be useful to reduce costa, e.g., Quintero-Marmol and Luyben (1990) or Farhat et al. (1991). These intermediate cute can be avoided in the design problem by choosing a sufficiently large number of stages. A purity specification is made for zj,k if j is the major component obtained during a product cut k. For an aZeOtrOpiC cut, the specification is zj,k = @x.- where @ is a number either slightly smaller or slightiy larger than unity depending upon the results of the feasibility study, cf. Bernot et al. (1991). The fractional recovery f j , k may be constrained by the amount of a component that must be removed during a cut to ensure that the purity requirements on subsequent cuts can be met (there is an obvious corresponding case
in squences of continuous columns). We can compute a lower bound for the fractional recovery of each cut as discussed in the example problems. The fractional recovery may also be an optimization variable, but we do not consider theae cases here. Equations 14 and 15 show how the amount of a key component remaining in the batch at the end of a cut can be used instead of a fractional recovery. For a given purity and fractional recovery and a known number of stages, the reflux or reboil ratio required to perform the separation is found by iteration for each cut. In a batch rectifier, we guess a value for the reflux ratio, integrate eqs 4,5, and 7 until the specified average distillate purity is obtained, and check the corresponding fractional recovery. The reflux ratio is varied until the fractional recovery is equal to the specified value. Often, the logarithm of the reflux ratio displays a linear dependence on the logarithm of the final still cornposition of the major component being removed; this provides a way of converging the problem in a few iterations (see Bernot, 1990). Variable Reflux o r Reboil Ratio Policy. The optimum reflux ratio policy does not always show large savings over policies that are simpler to compute, such as the constant product composition policy. In fact, studies of several binary and ternary mixturea &own that a constant distillate composition policy gives results close to the optimal policy provided there are an adequate number of stages in the column (Robinson, 1969,1971; Kim, 1985). For this policy, eq 4 can be integratad exactly for binary mixtures to give the batch composition % ( E ) = x p + [x(O) - x p l exp(O (25) where x p is constant. Generally, an iterative calculation is required at each value of the warped time in order to find the reflux or reboil ratio that gives the specified product purity. We have found that the constant distillate composition policy is often closely approximated by a policy where the reflux ratio is changed in proportion to its instantaneous mininum value at each warped time. The actual reflux ratio policy is taken as r ( [ ) = l.5rh([). For each cut the minimum reflux is computed at each warped time in the integration of eqs 4,5, and 7 using the desired distillate purity and the instantaneousliquid compoeition in the atill. The minimum reflux can be calculated efficiently using the method of Underwood (1946a,b, 1948) and King (1980) for ideal mixtures or using the method of Julka and Doherty (1990) for nonideal or azeotropic mixtures. This procedure can be greatly simplified for multicomponent mixtures by recognizing that it producea a distillate compoeition for some component(s)that is almoet constant during a given cut. With this approximation, eq 4 is a set of decoupled first-order linear differential equations that can be integrated analytically to give an estimate for the variation of each mole fraction in the still as a function of the warped time x(5) = xP,k + [x((k-l) - xP&I exp(C - &) (26) where xp = zk is the (constant) product composition for cut k. Otcourse, there are too few degrees of freedom for all of the product compositions to be constant, and eq 26 will be a good approximation only for the major components. Values for the other product cornpositions can be estimated if (i) trace amounts of the lighter components already removed as major components in earlier cuts are ignored, (ii) the purity requirement of the key component during cut k is given, and (iii) the balance of material in the distillate consists only of the next heaver component.
Ind. Eng. Chem. Res., Vol. 32,No. 2, 1993 297 Table I. Procedure for Calculating the Number of Stages for a Batch Rectifier 1. Find the mininum number of stanes for each cut. 1.1. From the specified producipurity and the fractional recovery, calculate the batch composition at the end of the cut (eq 15). 1.2. Compute the compositions in the column stage by stage at total reflux or reboil, beginning at the composition found in step 1.1 and stopping at the stage number that first meets or exceeds the product purity specification. 2. Find the controlling cut. 2.1. Compare the number of stages found for each cut and determine which cut has the largest Nmin.For example, see Diwekar et al. (1989). 2.2. If the product purity constraint for this cut is hard (e.g., the cut is a final product or it is a recycle cut with a high purity specification which cannot be relaxed), then this is the 'controlling" cut for choosing N. If the cut is recycled or a byproduct, we might attempt to relax the purity specification and return to step 1, leading to an optimization. 3. Find the number of theoretical stages. 3.1. Compute the minimum reflux at the end of the controlling out. 3.2. Compute the number of stages that gives the desired purity at the end of the controlling cut, using a reflux ratio r = 1.5rmin. (The factor of 1.5 may be adjusted to optimize or to account for uncertainty but should be the same value used to integrate for the batch composition.) Table 11. Summary of the Design Algorithm I. Composition changes 1. Specify the initial batch composition, x,,, and the pressure, p. 2. Identify the sequence of cuts for this initial composition, using the results of Bernot, Doherty, and Malone (1990, 1991). 3. Specify the average purities for the key component j in the product cuts k, 2 j . k in each of the cuts. 4. Specify a fractional recovery for each key component in the first n, - 1 cuts. 5. For each cut in sequence, calculate a. the average product composition, zk,as described in the text b. the batch composition of the key component at the end of the cut (eq 15) c. the duration of the cut (eq 13) d. the batch composition of the remaining components at the end of the cut (eq 12) e. the number of stages as described in Table I 6. Calculate the reflux or reboil ratio a. constant policy: calculate rk or sk for each of the n, - 1 cuts ~~" each of the n, - 1 cuts b. variable policy: calculate rk = l.5rminor sk = 1 . 5 ~ throughout 11. Batch size, time, and costs 7. Specify a. the amount of the initial feed, F b. the number of working hours, tw c. the dead time, t d 8. Choose an operating time, tp (possibly for later optimization) 9. Calculate a. the number of batches per year, nb (eq 1) b. the moles of feed per batch, Ho(eq 2) c. the vapor rate, V (eq 21 or 22) d. the cut times, t k (eq 19) 10. Estimate the column size, utility loads, condenser and reboiler sizes. and associated costa
The minimum reflux is relatively easy to compute if the composition in the still is estimated from eq 26 at each warped time. The actual reflux must be chosen somewhat above the minimum, e.g., r = 1.5rmin,and the procedure described here and below could be used to determine the optimal value. The great advantage of this reflux ratio policy is that it is possible to obtain a good estimate of the column design and reflux policy without numerical integration of the complete column model. A similar procedure applies to batch stripping, e.g., where the reboil policy is s ( U = 1*5smi,([)* Number of Stages. The number of theoretical stages, N, is chosen so that the required average purity can be obtained for each cut. Table I shows the procedure for a batch rectifier; the inverted sequence can be treated with the obvious minor changes. Normally, the algorithm can be simplified by eliminating step 3 and adding a few stages (e.g., 10%)to the number obtained in step 2. This is due to the fact that, at the end of each cut, the reflux ratio is large so that the corresponding number of stages is near Nmin.
If the minimumreflux computed in step 3.1 is very large, an intermediate cut should be considered. Sizes and Costs. Using the method described above, part of the design problem can be uncoupled from the rest and solved in terms of the compositions and warped time. The integral necessary to compute the vapor rate can be evaluated for each cut, and once a value is chosen for the
operating time, the vapor rate can be calculated from eq 21 or 22. The real cut times are found from eq 19 for the conventional configuration, or the analogous equation for the inverted configuration. The dead time is estimated from where td,o is the minimum dead time and t p / 8 represents a setup time which increases with the amount of material to process. To complete the design we minimize a total annual cost (TAC) similar to that described by Douglas (1988). A capital charge factor of 0.75 is used, leading to a TAC approximately equal to the operating cost plus twice the capital cost; this is functionallty equivalent to a 6-month payout time. The capital cost correlations (stainless steel construction) and the operating cost expressions are given in more detail by Bernot (1990); the labor costa are assumed equal for each of the different alternatives and are thus neglected for the purposes of relative comparisons. The design algorithm is summarized in Table II. Valuea for the vapor rate, flows, and costa can be obtained for any specifications of F, tw, and td, as well as for any value of the batch time, t p , without repeating the integration.
Example 1: A Simple Binary Mixture We consider an example of Kim (1985),who found the optimum design and reflux policy to minimize the TAC
298 Ind. Eng. Chem. Res., Vol. 32, No. 2, 1993 Optimal
1
Batch Composition, mol Z
Figure 2. Reflux ratio as a function of the batch composition for example 1.
using a detailed and timeconsuming integration of the full column model. The feed is a binary mixture for which a constant volatility of 2.0 adequately describes the vapor-liquid equilibrium. The feed contains 30 mol % of the light component. The distillate is to contain 95% of the light component originally present in the feed at a purity of 99 mol %. By material balance, this fractional recovery will be reached at a warped time of 0.34. If all of the light component could be removed, the warped time would be 0.36 but, of course, this would be indefinitely expensive. From eq 25, the constant distillate composition would correspond to a batch composition that obeys x = 0.99 - 0.69 exp(t) (28) For this simple mixture, the reflux can be found analytically as r = 1.5( 0.99 - y Y-x
-)
with y = 2x/(1 + x ) from the vapor-liquid equilibrium. The approximation in eq 29 is compared in Figure 2 with the results of Kim (1985), who reported the optimum reflux policy as a function of the batch composition. The number of stages can also be found easily for this example, using the well-known expressions of Fenske and Smoker. The procedure suggested here gives 14 theoretical stages; Kim reports an optimal value of 18 but with a reflux ratio that is substantially closer to the minimum. Even for this very simple mixture, there is a substantial savings in engineering time and the results are in good agreement with the detailed modeling. The treatment of nonideal binary mixtures is straightforward and requires only an efficient method for computing pinch compositions. This can sometimes be done analytically, even for azeotropic mixtures, using the coordinate transforms of Anderson and Doherty (1984) to describe the vapor-liquid equilibrium. Diwekar (1991) used the vapor-liquid equilibrium description to study the design of batch distillation columns for binary azeotropic mixtures. Extension of the ideas developed in this paper to multicomponent, ideal mixtures is described in detail by Bernot (1990). Since alternative design and optimization methods are already available for multicomponent ideal mixtures, we focus next on a four-component, azeotropic mixture which cannot be easily treated with traditional methods. Example 2: A Quaternary Aaeotropic Mixture We consider the distillation of a mixture produced via the reversible transesterification reaction
ethyl acetate (EA) + methanol (M) methyl acetate (MA)+ ethanol (E) Bernot et al. (1991) discuss the feasibility and synthesis of a batch distillation system for the separation, and we describe the design of a single column in one of the alternatives (column C1 in Figure 13a of Bernot et al., (1991)). This mixture has five batch distillation regions, and for this alternative, the feed composition should lie in the batch region that leads to the sequence of cut MA-M (azeo), EA-M (azeo), M, and E. To ensure this, the reactor feed must have a composition richer in methanol than the binary EA-M azeotrope. For a ratio of M/EA in the reactor feed of 2.5 and a 50% conversion of the ethyl acetate in the reactor, the feed to be separated contains 14.3 mol % MA, 57.1 mol % M, 14.3 mol % EA, and 14.3 mol % E. The column is operated at a pressure of 1atm and is designed for a feed of loo00 kmol/yr (4.8 x lo5 kg/yr = 1.1 X lo6 lb/yr), with tw = 2500 h/yr and td," = 0.5 h. The first cut produces a mixture of MA and M which must have a composition of MA slightly below that of the binary azeotrope;we set the methyl acetate product purity ~ ~ and ~ ~the~ final still mole to Z Y A , ~= 0 . 9 9 5 =~ 0.673 fraction of MA to 0.005 which corresponds to a fractional recovery of 0.972 for MA in the first cut. The first cut is processed further to recover the MA in high purity; see Bernot et al., (1990). The second cut must have a composition near the binary EA-M azeotrope; the product = 0 . 9 8 =~0.291 ~ with ~ a fractional purity is set to recovery of 95% EA; this cut is recycled to the reactor. During the last distillate cut, methanol is recovered in relatively high purity along with small amounts of ethyl acetate and ethanol; this cut is recycled to the reactor. Small amounts of the by-product ethanol are permissible in the recycle, and we set the average distillate purity of ethanol to 0.5 mol % for the third cut. Assuming that all of the ethyl acetate appears in the product and removing sufficient methanol to achieve 99 mol % purity of ethanol in the fourth cut, we find an average distillate composition of 94.5 mol % methanol in the third cut. Neglecting the small amounts of ethyl acetate and ethanol in the first cut, the average methanol composition is 0.327 and the corresponding still compositions can be calculated for each component. The key component for the second cut is ethyl acetate, and the mole fraction of EA in the still at the end of this cut is 0.022 for the average distillate purity and fractional recovery given above. The methyl acetate in the still at the end of the first cut is taken overhead as an impurity and Z M ,=~0.009. The average ~ 0.050,and if composition of EA in the third cut is z W , = the distillate contains 0.5 mol % ethanol, 99 mol % of the ethanol is left in the still at the end of the cut. Neglecting the ethanol collected during this cut, the mole fraction of methanol remaining in the still is 0.700 and most of this is removed during the third cut. Table 111summarizes the composibions and warped times required for the cuts. Using these estimated still compositions, we calculate the minimum number of theoretical stages a t the end of each cut required to reach the specified average distillate purity; these are 18 for the fmt cut, 10 for the second cut, and 14 for the third. The first cut is controlling; at the end of this cut the m i n i u m reflux ratio is 119 and taking r = 1,5r-, 20 theoretical stages are required for the design. For the constant reflux ratio policy, we found the minimum reflux at the beginning and end of each cut with the results shown in Table IV. For the first cut, a constant reflux ratio of 76 meets the purity specifications but r1 = 90 was chosen to take into account uncertainties (corre-
Ind. Eng. Chem. Res., Vol. 32,No. 2,1993 299 Table 111. Estimated Still and Average Distillate Mole Fractions for Example 2 MA M EA E Cut 1: Duration, bl = 0.23 0.143 0.571 0.143 still begin 0.143 0.005 0.635 0.180 still end 0.180 product 0.673 0.327 0 0 still begin still end produt
Cut 2: Duration, 8, = 0.89 0.005 0.635 0.180 0 0.542 0.022 0.009 0.700 0.291
0.180 0.436 0
still begin still end product
Cut 3 Duration, h3 = 0.82 0.542 0.022 0.010 0 0.945 0.050
0.436 0.990 0.005
Table IV. Minimum Reflux Ratio at the Beginning and End of Eech Cut for Example 2" rmln MA M EA E cut 1 still begin 0.143 0.143 4.6 0.143 0.571 still end 0.180 0.180 119 0.005 0.635 10-3 10-16 0.673 0.327 distillate 1.5 12.0
atill begin still end distillate
2.5 133
0.638 0.542 0.709
0.181 0.022 0.291
1000
8
800j
u Constant reflux policy --t Variable reflux policy
0.0
2.0
6.0
4.0
0.554 0.010 0.995
0.446 0.990 0.005
"The still compositions and distillate mole fractions are estimated from the problem specifications or mass balances as discussed in the text. Table V. Computed Still and Average Distillate Compositions for Example 2 MA M EA E Cut 1: r = 90,Duration, a1 = 0.231,and W,/Ho = 0.206 still begin 0.143 0.571 0.143 0.143 still end 0.005 0.635 0.180 0.180 product 0.675 0.325 0.002 0 Cut 2 r = 12,Duration, h2 = 0.878,and W2/Ho = 0.463 still begin 0.005 0.635 0.180 0.180 still end 0 0.543 0.021 0.436 product 0.009 0.699 0.292 0 = 0.824,and W3/Ho= 0.184 0.533 0.021 0.436 0.010 0 0.990 0.961 0.036 0.003
sponding to a specification of zm,l = O.998XMm = 0.675). For the second and third cuts, we choose a reflux of 12 and 68,respectively. The corresponding durations and sizes of the product cuts, and the average distillate and still compositions, are summarized in Table V. The warped times obtained from simulation are close to those obtained by estimation of the cut compositions and these can be used to recalculate the flows without another integration of the model in order to find a design at higher reflux ratios. The variable reflux policy yields substantially different results for the design. For each cut, we calculate the vapor rate from eq 21 using an instantaneous reflux ratio of F = l.Srmi,,. The minimum reflux ratio is calculated as described above using the given distillate specifications and compositions in Table IV. The vapor rate for this policy at tp = 4h ie 44 kmol/h is contrast to a value of 188 kmol/h for the constant reflux policy. This leads to a large re-
8.0
Operating Time (hr)
Figure 3. Cost comparison for the constant and variable reflux policies for example 2. 200
160
I
-1
1
120
f
40
-1
il
- full model .... simplified model
i
0.181 0.436
cut 3
Cut 3 r = 68,Duration, still begin 0 0 still end 0 product
s
2
cut 2 still begin still end distillate
I200
0
1
-I
0
0 05
0 IO
0 IS
020
0 2s
Warped Time
Figure 4. Reflux ratio as a function of the warped time for the f i i t cut in example 2. In the full model, the reflux ratio is computed to maintain the distillate mole fraction of methyl acetate precisely constant. In the simplified model, eq 26 is used to approximate the distillate composition and the still composition is found by material balance.
duction in both the capital and operating costs for the variable reflux policy as shown in Figure 3. This result is not sensitive to the cost models or the uncertainty in parameters. To compare the simplified approach to the full set of differential and algebraic equations (eqs 4,5, and 7),we examined the variation of the reflux ratio as a function of the warped time predicted from the two models. The results for the reflux ratio during the first cut are shown in Figure 4; the two curves are nearly identical and this accuracy is typical. Figure 5 shows the distillate and bottoms temperatures along with the reflux policy as a function of the warped time. Each cut is clearly visible in the transitions of the reflux ratio or the distillate temperature, while bottoms temperature changes more smoothly. Figure 6 show the distillate compositions corresponding to the reflux policy in Figure 5. The first two distillate cuts have composition very close to the desired values. The first is near the MA-M azeotrope (which is separated in a different batch column with the addition of methyl formate as an entrainer); the second is near the EA-M azeotrope which is recycled to the reactor. The third cut has a relatively low average purity of 96.1 mol 3' % M, 3.6 mol 9i EA,and the balance ethanol; the methanol purity is somewhat higher than the specification on account of the overdesign.
Conclusions We have developed a simple design procedure for batch distillation of nonideal multicomponent mixtures. An
300 Ind. Eng. Chem. Res., Vol. 32, No. 2, 1993
,
80
product purities and compares favorably with more de-
/
Bottoms
'-t
tailed optimizations, but is substantially easier to compute. As illustrated in the examples, the approach is suitable for
both ideal and nonideal multicomponent mixtures, including those with distillation boundaries and azeotropes. The method gives a reflux policy with a substantial cost savings compared to the constant reflux program and also provides a simple technique for obtaining the column design rapidly without the need of integrating the full differential-algebraic model for the column.
70-
iI
I
Distillate
c
60-
Acknowledgment We are grateful to Vivek Julka for helpful discussions. Parts of this work were presented a t the AIChE Annual Meeting, Los Angeles CA, 1991, paper 133e. Financial support was provided by the sponsors of the University of Massachusetts Process Design and Control Center.
l
l
l
I
I
l
I
I
I
I
I
l
l
I
I
l
l
l
I
l
l
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 Warped Time
Figure 5. Distillate and bottoms temperatures (a) and reflux ratios (b) for the variable reflux policy in example 2. The corresponding distillate compositions are shown in Figure 6.
0.8
-2
0.6
c
3
.1
0.4
e
eY
E
Nomenclature Vedor quantities have length equal to 1 less than the number of components since the summation constraint on mole fractions is assumed implicitly. Ho= initial batch size (mol) H = instantaneous batch size (mol) Hk= batch size at the end of cut k (mol) P = instantaneous production rate (mol/h) F = amount of feed to be processed in time tw (mol) f i , k = fractional recovery of component i in cut k nb = number of batches in time tw n, = number of cuts per batch N = number of theoretical stages r = reflux ratio s = reboil ratio rk = reflux ratio during cut k sk = reboil ratio during cut k rmin= minimum reflux ratio smin= minimum reboil ratio t = time (h) tp = operating time per batch (h) td = dead time per batch (h) tk = time for cut k (h) t w = total working time to process amount F (2500,5000, or 7500 h/yr) V = vapor boil-up rate (mol/h) wk = amount withdrawn as product in cut k, (mol) vk = vapor boil-up rate during cut k (mol/h) x, = vector of initial mole fractions for the batch x = vector of instantaneous mole fractions for the batch xk = vector of mole fractions in the batch during cut k xJ0) = vector of mole fractions in the batch at the beginning of cut k (and the end of cut k - 1; ~0 is the initial feed composition) x , , k = mole fraction of component i in the batch during cut k xBZm = vector of mole fractions giving the cornposition of an azeotrope x p = vector of instantaneous mole fractions in product P x, = vector of instantaneous liquid mole fractions on stage
i :J -
0.2 -
in
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 Warped Time
Figure 6. Distillate compositions of methanol (a) and ethyl acetate, methyl acetate, and ethanol (b) with the variable reflux policy in example 2. The key component in each cut is nearly constant: methyl acetate or methanol in the first (azeotropic)cut, ethyl acetate or methanol in the second (azeotropic) cut, and methanol in the third cut. Figure 5 shows the reflux ratios and the distillate and bottoms temperatures.
economical reflux or reboil ratio policy, proportional to the instantaneous minimum value, is relatively easy to find with good accuracy. This approximates certain features of the constant product composition policy for high
1
y, = vector of instantaneous vapor mole fractions on stage i X N + ~=
vector of instantaneous liquid mole fractions in the reflux drum XR = vector of instantaneous liquid mole fractions in the reboiler zk = vector of average mole fractions at the end of cut k in product P z,b = average mole fraction for component i in product P at the end of cut k t = warped time, defined by eq 3 ( k = warped time for cut k
Znd. Eng. Chem. Res. 1993,32, 301-314 6 = warped time measured from the beginning of a cut bk = duration of cut k
Literature Cited Al-Tuwaim, M. S.; Luyben, W. L. Multicomponent Batch Distillation. 3. Shortcut Design of Batch Distillation Columns. Ind. Eng. Chem. Res. 1991,30,507-516. Anderson, N. J.; Doherty, M. F. An Approximate Model for Binary, Azeotropic Distillation Design. Chem. Eng. Sci. 1984,39,11-16. Bernot, C. Design and Synthesis of Multicomponent Batch Distillation. Ph.D. Dissertation, Chemical Engineering Department, University of Massachusetts at Amherst, 1990. Bernot, C.; Doherty, M. F.; Malone, M. F. Patterns of Composition Change in Multicomponent Batch Distillation. Chem. Eng. Sci. 1990,45,1207-1221. Bernot, C.; Doherty, M. F.; Malone, M. F. Feasibility and Separation Sequencing in Multicomponent Batch Distillation. Chem. Eng. Sci. 1991,46,1311-1326. Christensen, F. M.; Jorgensen, S. B. Optimal Control of Binary Batch Distillation With Recycled Waste Cut. Chem. Eng. J. 1987, 34,57434. Diwekar, U. M. An Efficient Design Method for Binary, Azeotropic Batch Distillation Columns. AIChE J. 1991,37, 1571-1578. Diwekar, U. M.; Madhavan, K. P. Multicomponent Batch Distillation Column Design. Ind. Eng. Chem. Res. 1991,30, 713-721. Diwekar, U. M.; Malik, R. K.; Madhavan, K. P. Optimal Reflux Rate Policy Determination for Multicomponent Batch Distillation Columns. Comput. Chem. Eng. 1987,11,629-637. Diwekar, U. M.; Madhavan, K. P.; Swaney, R. E. Optimization of Multicomponent Batch Distillation Columns. Ind. Eng. Chern. Res. 1989,28,1011-1017. Douglas, J. M. Conceptual Design of Chemical Processes; McGrawHill: New York, 1988;Chapter 2. Farhat, S.; Czernicki, M.; Pibouleau, L.; Domenech, S. Optimization of Multiple-Fraction Batch Distillation by Nonlinear Programming. AIChE J . 1990,36,1349-1360. Hansen, T.T.; Jorgensen, S. B. Optimal Control of Binary Batch Distillation in Tray or Packed Columns, Chem. Eng. J. 1986,33, 151-155.
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Julka, V.; Doherty, M. F. Geometric Behavior and Minimum Flows for Nonideal Multicomponent Distillation. Chem. Ena. - Sci. 1990. 45, 1801-1822. Kim, Y. S. Optimal Control of Time-Dependent Processes. Ph.D. Thesis. Chemical Eneineering-DeDartment. Universitv of Maasa. chusetts at Amherst: 1985. King, C. J. Separation Processes; McGraw-Hill: New York, 1980; pp 417-423. Kolber, M. J.; Anderson, T. F. Design of Batch Distillation by Interactive Simulation on Microcomputer. Presented at the AIChE Annual Meeting, New York, NY, 1987;paper 92c. Logsdon, J. S.;Diwekar, U. M.; Biegler, L. T. On the Simultaneous Optimal Design and Operation of Batch Distillation Columns. Presented at the AIChE Annual Meeting, San Francisco, CA, 1989;paper 27b. Luyben, W. L. Multicomponent Batch Distillation. 1. Ternary Systems with Slop Recycle. Ind. Eng. Chem. Res. 1988, 27, 642-647. Quintero-Marmol, E.; Luyben, W. L. Multicomponent Batch Distillation. 2. Comparison of Alternative Slop Handling and Operating Strategies. Ind. Eng. Chem. Res. 1990,29,1915-1921. Rippin, D. W. T. Simulation of Single- and Multiproduct Batch Chemical Plants for Optimal Design and Operation. Comput. Chem. Eng. 1983,7, 137-156. Robinson, E. R. The Optimisation of Batch Distillation Operations. Chem. Eng. Sci. 1969,24,1661-1668. Robinson, E. R. Optimal Reflux Policies for Batch Distillation. Chem. Process Eng. 1971,52,47-49and 55. Underwood, A. J. V. Fractional Distillation of Ternary Mixtures. Part 11. J. Znst. Pet. 19468,32,598-613. Underwood, A. J. V. Fractional Distillation of Multicomponent Mixtures-Calculation of Minimum Reflux Ratio. J. Znst. Pet. 1946b,32,614426. Underwood, A. J. V. Fractional Distillation of Multicomponent Mixtures. Chem. Eng. Prog. 1948,44,603-614. Wu, W.-H.; Chiou, T.-N. Determination of Feasible Reflux Ratios and Minimum Number of Plates Required in Multicomponent Batch Distillation. Ind. Eng. Chem. Res. 1989,28, 1862-1867.
Received for reuiew October 30, 1992 Accepted November 12, 1992
Monitoring and Diagnosing Process Control Performance: The Single-Loop Case Nives Stanfelj, Thomas
E. Marlin,* a n d J o h n F. MacGregor
Chemical Engineering Department, McMaster University, Hamilton, Ontario, Canada L8S 4L7
This paper presents a hierarchical method for monitoring and diagnosing the performance of single-loop control systems based primarily on typical operating plant data. It (1)identifies significant deviations from control objectives, (2) determines the best achievable performance with the current control structure, and (3) identifies steps to improve the current performance. Within the last point, the method can isolate whether poor performance is due to the feedforward loop or the feedback loop. If in the feedback loop, it is sometimes possible to determine whether the cause of poor performance is plant/model mismatch or poor tuning. The methods are based on simple but rigorous statistical analysis of plant time series data using autocorrelation and cross correlation functions. The theoretical basis of the method is developed, and it is applied to simulation studies which clarify the principles. Then, results of studies on two industrial processes are reported. The first is a heat exchanger feedback temperature controller, and the second is a feedforward-feedback tray temperature controller in a 50-tray distillation column. The initial diagnosis and subsequent control performance improvements are reported for both cases. The process industries make wide use of automatic process control to achieve objectives from safety to optimization. Every plant has many control loops operating in automatic; the number of control loops varies from the low lo's in small, simple processes to over 1000 in large integrated planta. Due to this high degree of automation, succedul plant operation depends on the proper operation of the control strategies.
Currently, only overall measures of process and control performance are monitored. The most commonly used measure of performance is the variance on standard deviation of key process variables. If the control strategies do not work well, the standard deviations can be very large. The reason that the standard deviation is used for monitoring is ita direct relationship to process performance and profit (Bozenhardt and Dybeck, 1986;Marlin et al., 1987;
Oaaa-5sa5/93/2632-03Q1~04.00/00 1993 American Chemical Society
