Collocation Methods for Distillation Design. 1. Model Description and

Fast and accurate distillation design requires a model that significantly reduces the problem size while accurately approximating a full-order distill...
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Ind. Eng. Chem. Res. 1996, 35, 1603-1610

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Collocation Methods for Distillation Design. 1. Model Description and Testing Robert S. Huss and Arthur W. Westerberg* Department of Chemical Engineering, Carnegie-Mellon University, Pittsburgh, Pennsylvania 15213

Fast and accurate distillation design requires a model that significantly reduces the problem size while accurately approximating a full-order distillation column model. This collocation model builds on the concepts of past collocation models for design of complex real-world separation systems. Two variable transformations make this method unique. Polynomials cannot accurately fit trajectories which flatten out. In columns, flat sections occur in the middle of large column sections or where concentrations go to 0 or 1. With an exponential transformation of the tray number which maps zero to an infinite number of trays onto the range 0-1, four collocation trays can accurately simulate a large column section. With a hyperbolic tangent transformation of the mole fractions, the model can simulate columns which reach high purities. Furthermore, this model uses multiple collocation elements for a column section, which is more accurate than a single high-order collocation section. Introduction Several researchers have explored and developed collocation for distillation column modeling. In this paper, we present a collocation model which expands on prior models, addressing the problems specific to steady-state, continuous columns. We describe the formulation of the model in detail and compare it to prior models, using rigorous column simulations as a benchmark. We had two primary reasons for examining collocation models. First, we are interested in designing columns and not just simulating them, and, second, we discovered a need when carrying out general minimum reflux computations. For the design of a column, one must determine its diameter, number of trays, and feed tray location. The latter two are discrete variables for tray-by-tray column models. For collocation models, they are continuous variables, allowing continuous variable optimization approaches for their selection. Also, as we shall observe, our collocation model has a number of equations equivalent to a column with about 18 trays, no matter what the actual number of trays in the column is. Thus, such models can be much smaller for very large columns. If one examines the minimum reflux problem where the products are not sharply split, one needs to include equations that state that two points in a section of the column must be joined by stage-by-stage computations. The exact number of stages cannot be known except by computation. A collocation model for this column section provides these equations where the number of stages is computed, not given. While we were interested in using collocation models, we also were advised that it was difficult to make such models accurate especially for difficult and sharp separations. We thus wished to assess the validity of this advice and, if it proved valid, to find ways to improve their accuracy. Background of Collocation Applied to Distillation Cho and Joseph [1983] developed a reduced-order method for modeling staged separation processes. They used orthogonal collocation, placing their collocation points at the roots of Jacobi polynomials to obtain more

accurate solutions. Their model had a single collocation section for the column section above the feed and another for the section below the feed. They tested by modeling a simple absorber system, and binary and three component distillation, and used the Antoine equation for the equilibrium relationship. In later papers Srivastava and Joseph [1984, 1987a] developed methods for handling multiple feeds and side draws. They also developed a complex method for handling steep and flat composition profiles by fitting the composition profiles with different polynomials for each component. They developed a complicated approach using two sets of collocation points, global and local, to fit both the key components and non key components [Srivastava and Joseph, 1987b]. They tested these later ideas using constant relative volatility systems. Stewart, Levien, and Morari [1984] developed an orthogonal collocation method that stresses selecting gridpoints based on the roots of Hahn polynomials. These polynomials are defined over the set of integers, making them more suitable for models based on difference equations such as for plate distillation columns. Their method becomes a stage-by-stage model when the number of collocation points equals the number of trays in a column section. Their results showed errors at least an order of magnitude smaller than Cho and Joseph’s default choice of collocation points. They tested for binary and six-component distillation columns with constant relative volatility and for a ternary system using UNIQUAC for equilibrium. Swartz and Stewart [1986] applied the method to design, iteratively passing from the model to an SQP optimization algorithm. Swartz and Stewart [1987] also developed a finiteelement method for handling multiphase distillation problems. Recently, Seferlis and Hrymak [1994a,b] developed a finite element model to simulate existing distillation columns. Each element used the collocation model of Stewart et al. [1984]. They obtained higher accuracy with multiple collocation sections of lower order than with a single collocation section of higher order. Table 1 lists the characteristics of each of these collocation methods as well as the characteristics of the model presented in this paper.

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1604 Ind. Eng. Chem. Res., Vol. 35, No. 5, 1996 Table 1. Comparison of Collocation Method

Cho and Joseph, 1983 Srivastava and Joseph, 1987 Stewart et al., 1984 Swartz and Stewart, 1986 Swartz and Stewart, 1987 Seferlis and Hrymak, 1994 this work

variable transformations

elements/section

placement of points

thermodynamics tested

single single (global), multiple (local) single single multiple (break points at phase changes) multiple multiple

continuous orthogonal (Jacobi) continuous orthogonal (Jacobi)

Antoine constant relative volatility

none none

discrete orthogonal (Hahn) discrete orthogonal (Hahn) discrete orthogonal (Hahn)

UNIQUAC ideal nonideal three phase

none none none

discrete orthogonal (Hahn) continuous (orthogonal for 2 or 3 points)

regression of data UNIFAC/Pitzer

none transform tray number and mole fractions

Figure 2. Diagram of a tray. Figure 1. Collocation of a differential equation.

Collocation Collocation [Finlayson, 1980; p 61] is generally thought of as a method for numerically solving differential equations. The use of collocation for simulation of a distillation column is an extension of this technique. Given a differential equation

dy/dx ) f(x,y),

y(0) ) y0

(1)

we want to find y as a function of x. We can approximate y as a polynomial in x,

yˆ ) y0 + a1x + x2x2 + ... + anxn

(2)

We can then approximate y and f as functions of x. At any point in x, we can define an error for this approximation,

error(x) ) f(x,yˆ ) - (a1 + 2a2x2 + ... + nanxn-1)

(3)

At n collocation points, x, ..., xn, we require the error to be zero and get n equations to solve for n coefficients to the polynomial. See Figure 1. If we place the collocation points strategically at the roots of orthogonal polynomials, then the error decreases much faster as the number of terms increases [Finlayson, 1980; p 73]. Collocation of a distillation column uses the same idea. The error equation is formed by requiring an output, yout, from a tray to lie on the same polynomial as the corresponding input, yin. The tray number is the independent variable x and differs by 1 for yin and yout. A Collocation Model for a Tray Section In this section we develop our collocation model. We shall examine the model using a degree of freedom analysis. This analysis will expose some interesting issues about earlier published models and also about our model. We start first by considering a stream, then we consider a tray, and finally we consider a column section. We shall see that all collocation models seem to ignore some of the equations we can write. Using Duhem’s (other) theorem [Prygogine and Defay, 1954; Westerberg et al., 1989], we note that a stream at thermodynamic equilibrium requires precisely nc + 2 specifications to completely characterize it, where nc is the number of components in it. This theorem is true even if the stream breaks up into multiple phases and if there are reactions among the components in it. A

typical specification is to state the original flows of the nc components (e.g., 1 mol/s of ethylene, 1.5 mol/s of water, and 0 mol/s of ethanol and diethyl ether) and the temperature and pressure for the stream. Ethylene and water react to form ethanol, while two molecules of ethanol will decompose into water and diethyl ether. If we say the stream is at thermodynamic equilibrium at the given temperature and pressure, Duhem’s theorem says we can, in principle, compute the flows for all the phases into which it partitions and the compositions of those phases. The net number of new variables a stream will introduce to a model must be equal to the number of specifications we can make for it, nc + 2. This theorem guarantees we can write equations to compute all the other variables we might use in defining an equilibrium stream model. The discussion for a conventional equilibrium tray (see Figure 2) is very similar to that for a stream. If we know the two streams entering it (nc + 2 specifications each), we can, by material balance, know the material flows entering it. Two added specificationsse.g., the temperature and pressure of the traysand the fact that it is at equilibrium allow us to compute everything else about it and its output streams. If we require the tray to be adiabatic, we add an equation stating this. We write this equation without introducing new variables, thus reducing the allowable specifications by one. For example, we could remove our specification of the tray temperature. For an adiabatic tray, we thus have specified 2(nc + 2) + 1 variables. By using the same argument as above for a stream, an adiabatic equilibrium tray is equivalent to introducing 2(nc + 2) + 1 net new variables into a model. We are guaranteed that we can, in principle, compute any remaining variables in the tray model. A tray exists inside a column section, as we illustrate in Figure 3. To model a column section, we introduce the four streams that enter and leave the column section. Invoking Duhem’s other theorem, each introduces nc + 2 net new variables. In a tray-by-tray model, we tie the streams entering and leaving a tray to the input and output streams of the section by connecting each tray to its adjoining trays. However, in a collocation model, a tray not located at the end of the section is disjoint from the other trays and the ends of the column. It is tied to the other trays and the section input and output streams only because its variables will satisfy the same polynomials we have written to characterize the trajectories of these variables through the section. We can, therefore, write nc material balances and one heat balance that tie a tray to the conditions

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Figure 3. Diagram of a collocation section.

at one end of the column section, as indicated by the dashed envelopes. We also write nc material balances and one energy balance around the entire column section, relating the streams at the ends to each other. Our next decision is to select the variables whose trajectories we would like to represent using polynomials against tray position. Previous papers on collocation pick nc - 1 of the compositions and the molar enthalpy for each stream. We would, therefore, write nc “vapor” polynomials whose coefficients we wish to find for the vapor streams and, similarly, nc “liquid” polynomials for the liquid streams. To say that a stream must “lie on” one of these sets of polynomials says we have to write the appropriate nc polynomials once for that stream. We would write each polynomial in terms of the appropriate stream composition or enthalpy value at the tray number used to position the stream in the section. We show an accounting of the equations we have written so far in Table 2. We also add the specifications we need for a column section. These include specifications for the input streams and for the pressures throughout. For a well-posed collocation model, the number of equations must equal the number of variables; the net number of equations will be zero. Solving for the order of the polynomial, using the net number of equations set to zero in Table 2, gives

3 3 1 n o ) ns + 2 2 2nc

right. Intuitively we want the order to be related in a simple manner to the number of stages and for it always to be an integer. Previous papers set the order of the polynomial to the number of stages, i.e., no ) ns. If we do that here, then we find from Table 2 that there are (ns + 1)(nc - 1) too many equations. Cho and Joseph [1983] very cleverly proved that, (1) if the section model satisfied the assumption of constant molar overflow (i.e., CMO, which is allowed if one drops the energy balance) and (2) if the trays satisfied the collocation polynomials for compositions, then the component material balances we proposed to write relating the trays to the streams at the end of the section are exactly satisfied. This is not an obvious result. Thus, for cases approximating CMO, these equations are not useful for providing independent information from which to fit polynomial coefficients. We chose to drop these nsnc equations from the model, leaving us with nc - 1 - ns extra equations. We next chose to enforce only the overall material balance tying each tray to the end of the section, giving us ns additional equations, and not to require the compositions at the end of the column for one output stream to satisfy the polynomial equations, removing nc - 1 equations. Our model is now square. Previous works did not make it clear just what they had to assume to get their models to be square. There are other ways to deal with the excess equations: (1) solve all the equations in a least-squares sense and (2) monitor the residuals of these excess equations to detect problems with the accuracy of the collocation fit and, if detected, add more collocation trays to the section. Our own computational experience has shown that the dropped equations are very closely satisfied (errors of 10-5 while we detected convergence by errors of size 10-8) even when one does not assume constant molar overflow. Thus monitoring them has proved to be rather uninteresting. This is not to say that there will not be problems where they could prove interesting. However, we have not seen any yet. Collocation Polynomials We elected to use the Lagrange form for our collocation polynomials. In this form, the coefficients are the values of the fitted variables at the collocation points so we can know readily how to scale them and how to provide initial estimates for them. Using a Lagrange polynomial, the following equations:

(4)

If we set the number of stages, ns, to 2, we get the polynomial order, no, to be 3 for three components and 3.125 for four components. Generally the polynomial order will not be an integer so something cannot be

n+1

xi(w) )

∑ Wk(w) xi(wk)

i ) 1, ..., nc - 1

(5)

k)0

are the polynomial approximations of order n for variable x at position w. The kth term of a Lagrange

Table 2. Degrees of Freedom for Section new variables ns trays 4 streams into and out of section nc polynomials (for nc - 1 compositions and 1 molar enthalpy) of order no for each stream (introduces no+ 1 polynomial coefficients per polynomial, of which there are nc for vapor streams and nc for liquid streams) nc component material balances and 1 energy balance tying trays to end of column section and around the entire section pressure specifications for each tray (for each input stream and for tray) and for 2 streams leaving the section specifications for input streams to the section totals net net if polynomial order, no, equals number of stages, ns

ns(2(nc + 2) + 1) 4(nc + 2) 2nc(no + 1)

new equations

nc(4ns + 4) (nc + 1)(ns + 1) 3ns + 2

2(nc + 2) 2nsnc + 5ns + 6nc + 2ncno + 8 5ncns + 4ns + 7nc + 7 3ncns + nc - ns - 2ncno - 1 ncns + nc - ns - 1 ) (ns + 1)(nc - 1)

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Figure 4. Column trajectories over a range of reflux ratios, r, for components c1, c2, and c3 with relative volatilities of 1.5, 1.2, and 1.0, respectively.

polynomial, Wk, is defined by the equation:

Wk(w) )

n+1 w

- wj

k

- wj

∏ j)0 w j*k

(6)

Point Placement An interesting decision in collocation is the placement of the collocation points. Cho and Joseph [1983] placed their points at the zeros of Jacobi polynomials defined by,

∫01zβ(1 - z)aRPn(R,β)(z) dz ) 0

(7)

j ) 0, 1, ..., n - 1 where R and β are parameters. Their default choice of the parameters for the Jacobi polynomial (R ) 1, β ) 1) resulted in symmetrically spaced points. They could have moved the points toward either end of the collocation section by increasing both parameters while keeping them equal. Stewart et al. [1984] showed that placing the collocation points at the zeros of the Hahn polynomial created smaller errors than placing the points by the Jacobi polynomial. They argued that the Hahn polynomial was a better choice because it maintained the discrete nature of the column. For full order (number of collocation points equals the number of actual trays in the column), the collocation points would be placed exactly at the tray locations, an exact fit which the Jacobi polynomial placement cannot match. We shall be introducing variable transformations to remap tray number which can go from zero to infinity to the range 0-1. When active, we use Jacobi point placement as our trays are no longer evenly spaced vs the variable, wk, that corresponds to kth point location in a column section. The parameters R and β for the Jacobi (and for Hahn polynomials) allow us to move the point locations while still carrying out an orthogonal collocation. We wondered if the nature of column profiles might suggest where this placement should be. Figure 4 shows the form for column trajectories for a column with 15 trays above and 15 trays below the feed, separating a three component constant relative volatility mixture. We also assumed constant molar overflow. We show the trajectories for three different reflux ratios. For small reflux ratios, the trajectories are fairly flat for major portions of the column. Large reflux ratios, on the other hand, led to steep trajectories.

Figure 5. Effect of point placement on error for 2- and 3-tray collocation sections. The scale of each plot is different, noted in the upper left corner of each plot.

For Jacobi placement with two or three collocation points, we can move the points symmetrically away from the middle of a column section smoothly by increasing the parameters R and β while keeping them equal. We elected, therefore, to think of the point location by the fraction of the distance the points were between the midpoint and the end of the section, calling this their spread. A fractional distance of one-third would evenly distribute two points in the collocation section. Figure 5 gives the error vs point placement for different values of the reflux ratio when using collocation models. We used two collocation sections to model the column, one above and one below the feed. We used sections containing two and three collocation trays. Figure 5 certainly suggests that points more to the ends of the sections, about 70% of the distance from the middle to the end, give more accurate answers, where we measured accuracy by comparing exit compositions for a section to those obtained using tray-by-tray models. Many experiments corroborated this observation. For three collocation points, placing one in the middle and the other two symmetrically away from the middle corresponds to a Jacobi placement. From these experiments, we set our default placement so the points are about 70% of the way from the midpoint to the end of the section. In the next section, we shall show how variable transformations appear to be more important than point placement for increasing accuracy. Variable Transformations We use two variable transformations in this model to alleviate the problem of flattened trajectories. When the mole fraction of a component is not changing over part of a tray section, we call that a flattened trajectory. Flattened trajectories cannot be fit well with polynomials, so we perform variable transformations to alter the shape. The first is a transformation of the tray number. To simulate a large number of trays, or an infinite number of trays, we want an index that goes to a finite value as the tray number goes to infinity. Even for finite columns with a large number of trays, the trajectories flatten out as the number of trays increases. Some possible transformations are:

z ) 1 - e-as z)

s s+a

(8) (9)

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Figure 6. Comparison of variable transformations on s. The curves show the best fit given by a third-order polynomial for s/smax, a quadratic for s/s + 1.5, and a linear polynomial for 1 exp(-0.59s).

Figure 8. Column trajectories for a large column over a range of reflux ratios with three components with relative volatilities of 1.5, 1.2, and 1.0.

Figure 7. Effect of choice of a on the exponential transformation. For the correct choice of a, the trajectory is exactly straightened out.

Figure 9. Error curves for s- and z-based collocation.

In both these equations, s is the stage number, z is the transform variable, and a is a parameter. In both cases, z ) 0 when s ) 0, and z tends to 1 as s tends to infinity. To discover the better form of the transformation, we first investigated fitting results from the Kremser approximation:

ys )

1 - As A - As y1 yˆ 1-A 1-A 0

A)

L , KV

yˆ 0 ) Kx0

(10) (11) Figure 10. Effect of transformation on mole fraction, x.

Figure 6 shows that the variable transformation given by eq 8 did the best at straightening out the trajectory. In the appendix, we show that, with the exponential variable transformation, the Kremser approximation can be exactly straightened out for the correct choice of a. Figure 7 shows how the choice of a affects the shape of the trajectories. At the proper selection of a, the data can be fit with a linear function. Therefore, we use the variable transformation in eq 8. Figure 8 shows column trajectories for a three component, constant relative volatility, constant molar overflow column with 50 trays above and below the feed, for three different reflux ratios, generated by a trayby-tray model. Over the range of reflux ratios, r, we tested the collocation model, comparing the composition of the distillate product to the tray-by-tray calculation. The collocation model used had two collocation sections per column section, with two trays in each collocation section. We compared the s-based and z-based collocations over a range of point placements, using the point placements described in the last section. Figure 9 shows the average errors of both cases over the same range of

point spreads for different reflux ratios. The z-based collocation had lower average errors for every reflux ratio. For all but the lowest reflux ratio, the best solution was achieved with the z-based collocation. The second variable transformation is one on the mole fractions. When distilling to high purity, mole fractions go to 1 or zero, again flattening out the trajectory. We need to transform the asymptotic approach to zero and 1 into a decreasing and increasing function that can be fitted by the polynomial used in the collocation. We use the following transformation:

2xi - 1 ) tanh(xˆ i)

(12)

As the mole fraction goes to 1 or zero, the transformation variable, xˆ i, goes to negative infinity and plus infinity. Figure 10 shows the effect of this transformation. For an exponential approach to 1 and zero, the transformation straightens the trajectory out, so the slope never goes to zero. Without this transformation, modeling sharp splits is very difficult. As the mole fractions of some of the

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Figure 11. Effect of x transformation in a column simulation.

Figure 14. Column configuration.

the bottom half of the column. This also demonstrates why it is beneficial in an s-based collocation for the collocation points to be spread toward the ends of the collocation section to keep the curvature to a minimum. Figure 12. Blowup of the lower left corner of Figure 11.

Formulation of the Collocation Column Model

Figure 13. Effect of s transformation on a column with many trays.

components approach zero or 1, the polynomial will create a curved trajectory, “bouncing” off the boundary. It becomes impossible to model a column with a component going to a mole fraction of 10-6 or smaller. However, the above transformation will give the polynomial room to move and will allow an asymptotic approach to the boundary. Figure 11 shows the four possible models of a 63-tray column which is removing all of the heavy components from the distillate. Figure 12 is a blowup of the trajectories for the heavy component near the top of the column. The two simulations without the transformation on mole fraction are curved and “bounce” up. The two solutions using the transformation smoothly approach the top of the column. Figure 13 shows the combined benefits of the two variable transformation, showing two collocation models of a 103-tray column with high purities. Both models used the transformation on mole fraction, since this problem did not converge without it. For one, the polynomial is based on the stage number, and, for the other, the polynomial is based on the transformed stage number. The s-based solution has a high curvature in

Our standard formulation of the collocation column model is shown by Figure 14. Each column section is divided into two collocation sections with two trays each. Sererflis and Hrymak [1994a] used multiple collocation sections so they could use more collocation points in specific areas of the column where the temperature and composition profiles changed rapidly. We observe that the areas of activity are at the ends of the column sections. By breaking each column section into two parts and using the transformation on the stage number, we deemphasize the center of the column section by numbering the top collocation section downward and the bottom collocation section upward. With this transformation, the points at the beginning of a collocation section are stressed, and those at the end are less important. Therefore, for large columns with relatively low reflux ratios, the collocation points will be located where the compositions are changing, and the area of no activity will join the two collocation sections, but no collocation points need to be located there. This standard formulation is sufficient for modeling large columns with reasonable accuracy and is small enough to model small columns without overkill. The model has four collocation sections with two trays each, a feed tray, a condenser, and a reboiler. These are 11tray calculations. Since the collocation trays are not connected the way they would be for a tray-by-tray model and since there are polynomial equations, there are more equations than there would be for an 11-tray column model. For a three-component system, the collocation model has 1811 equations and variables, including all thermodynamic equations. A tray-by-tray model with 19 trays has 1856 equations and variables. For a four-component system, the collocation model has 2215 equations and variables compared to 2205 for a tray-by-tray model with 18 trays. For a set number of trays below 18, a tray-by-tray model might be more

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Figure 15. Comparison of collocation to a rigorous model for a methanol/water column. Curves show a tray-by-tray model; points show a collocation model.

Figure 16. Comparison of collocation to a rigorous model of the separation range over a D:F ratio for acetone/benzene/chloroform system.

efficient, or a nonstandard collocation model can be used with fewer collocation sections. Testing the Collocation Model We have performed several tests of the collocation model using nonideal thermodynamics. In each of the following examples, the standard collocation model was used, with four collocation sections of two trays each. We used UNIFAC liquid mixture and Pitzer vapor mixture models for the thermodynamics and equilibrium and assumed constant molar overflow. We would like to note that performing tests like these is a nontrivial task. The process of obtaining a full thermodynamic model is complex, but once a tray-by-tray collocation model has been successfully refined and converged, it is relatively easy to perform many sequential incremental changes to obtain a wealth of data. The two examples below, where we performed a series of calculations to determine the binary separations over a range of operations, required 50 solutions of the trayby-tray and collocation models. Most of the work was done in getting that first useful solution. Then the models could be re-solved repeatedly as the distillate to feed ratio was increased incrementally. The collocation has many parameters that can be adjusted, but it is much more robust than a tray-by-tray model. The process of solving these models will be discussed further in a companion paper. The first example is the separation of a 50/50 mixture of methanol and water. The column has 46 trays and a reflux of 1.0. The purity of each product is 99.6%. Figure 15 shows a comparison between the tray-by-tray solution and the collocation model. The curves are the tray-by-tray, and the points are the collocation. The figure shows an excellent fit. Including all the thermodynamic calculations, the tray-by-tray and collocation models had 3255 and 1407 equations, respectively. The error in the distillate composition is 0.02%. Using the acetone/chloroform/benzene system, we performed many tests of the collocation model. The feed was a 36/24/40 mixture of acetone, chloroform, and benzene. A 33 tray column with a reflux ratio of 4.0 was used. We performed a search over the range of distillate to feed ratios to find the maximum binary separations, as done by Wahnschafft [1992]. Figure 16 shows the comparison of the binary separation range plots with those for a tray-by-tray calculation. The chloroform/benzene binary separation factor is not meaningful before a D:F of 0.3, since practically nothing of either component is coming out of the distillate at low D:F. For the acetone/benzene binary separation factor curve the average error was 1%, and for the

Figure 17. Comparison of collocation to a rigorous model for acetone/benzene/chloroform column, for three different D:F ratios.

Figure 18. Comparison of collocation to a rigorous model of the separation range over a D:F ratio for acetone/ethanol/propanol/ isobutyl alcohol/butanol system.

acetone-chloroform separation factor curve the average error was 3%. The collocation shows very good agreement with tray-by-tray calculations. The error in the acetone concentration in the distillate was less than 2% over the range of D:F ratios, with an average error of 1%. Figure 17 shows comparisons of three different column simulations on a ternary diagram. Including all the thermodynamic calculations, the tray-by-tray and collocation models had 3144 and 1811 equations, respectively. Finally, we performed a set of tests on an equimolar mixture of acetone, ethanol, propanol, isobutyl alcohol, and normal butanol, using a 23-tray column with a reflux ratio of 0.8. Figure 18 shows the comparison of the binary separation range plots for the components which are adjacent in the order of relative volatility. The other binary separation ranges compare equally well but would clutter the figure. Conclusions In this paper, we have demonstrated that this new collocation method can accurately reduce the order of

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column models. The two variable transformations greatly expand the capabilities of standard collocation methods. We have found that the degree of freedom selection is important and demonstrated what equations can be ignored. The choice of point placement is nontrivial, and no particular polynomial will give optimal point placement. Variable transformations more significantly reduced errors than point placement at the roots of orthogonal polynomials. In a companion paper, we will discuss how collocation provides the missing link for simulation of minimum reflux conditions. We will also discuss a design algorithm for designing arbitrary columns using the collocation model.

the following equation.

y(z) )

y1 - Ayˆ 0 1-A

-

y1 - yˆ 0 ln((1-z)-a) A 1-A

(16)

If we define A ) exp(B), then we can take the following steps:

Aln((1-z)

-a)

) (exp(B))ln((1-z)

-a)

) exp(ln((exp(B))ln((1-z) ))) -a

) exp(B(ln((1 - z)-a))) ) exp(ln((1 - z)-aB)) ) (1 - z)-aB

Acknowledgment Support from Eastman Chemicals and DOE Contract No. DE FG02-85ER13396 provide support for this research. Facilities support is from NSF Grant No. EEC-8942146 which supports the EDRC, an NSF funded Engineering Research Center.

(17)

Now eq 16 becomes the following.

y(z) )

y1 - Ayˆ 0 1-A

-

y1 - yˆ 0 (1 - z)-aB 1-A

(18)

Therefore, when -aB ) 1, the equation is linear in z. Nomenclature Literature Cited A ) simplification variable for Kremser approximation (L/ KV) a ) parameter for exponential transformation of stage location R ) parameter of Jacobi polynomial β ) parameter of Jacobi polynomial K ) equilibrium constant used in Kremser approximation L ) liquid molar flow rate no ) order of polynomial nc ) number of components ns ) number of collocation points Pno ) Jacobi polynomial of order no s ) stage location V ) vapor molar flow rate Wk ) kth term of Lagrange polynomial wk ) position of the bottom of tray k xi ) liquid mole fraction of component i xˆ i ) transformed mole fraction yi ) vapor mole fraction of component i z ) transformed stage location

Appendix Rewriting the Kremser approximation to gather terms with s produces the following equation.

ys )

y1 - Ayˆ 0 1-A

-

y1 - yˆ 0 s A 1-A

(13)

Rewriting the variable transformation

z ) 1 - e-as

(14)

for s in terms of z,

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Received for review January 10, 1996 Accepted February 5, 1996X IE9503499

s ) ln((1 - z)-a)

(15)

and placing it into the Kremser approximation produces

X Abstract published in Advance ACS Abstracts, April 15, 1996.