SHORTCUT MODELS FOR DISTILLATION COLUMNS - I. STEADY-STATE BEHAVIOR. Sigurd Skogestad. Manfred Morari. California Institute of Technology

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1 SHORTCUT MODELS FOR DISTILLATION COLUMNS - I. STEADY-STATE EHAVIOR Sigurd Skogestad Manfred Morari California Institute of Technology Chemical Engineering, Pasadena, CA 925 è88è Chapter 9 in Ph.D. thesis "Studies on Robust Control of Distillation Columns" by Sigurd Skogestad Caliærnia Instutute of Technology, Pasadena, January 987 Note: Figures are missing in this version UNPULISHED Abstract Jafarey et al. è979è derived a simple analytical expression for the separation factor S. This paper provides a simpler and more instructive derivation of this expression, and also evaluates its validity for estimating steady-state gains and RGA-values. The results show that the expression gives a good estimate of how S changes with internal æows, but describes poorly its variation with external æows. Fortunately, when computing the steady-state gains, this error is often not important.

2 . INTRODUCTION Shortcut models. Simple ëshortcut" models are useful for analyzing and understanding the strongly nonlinear behavior of distillation columns. For robust distillation column control, for example, it is important to know the correlation between the transfer matrix elements as they vary with operating conditions. The essential part of this behavior may be predicted from simple analytical models. This paper presents a derivation and evaluation of analytical expressions for the separation factor S which is useful for steady-state calculations. In Part II èskogestad and Morari, 987aè we discuss the dynamic behavior and present an analytical expression for the dominant time constant. Though our main goal is to develop shortcut models for use in control system design, the results presented in these two papers are useful for obtaining general insight into distillation column behavior. For example, simple analytical models are invaluable as a teaching tool; a basic understanding of distillation columns is much more easily acquired by analyzing simple analytical models than by running tray-by-tray simulation programs. We stress that the shortcut models presented are not intended to replace trayby-tray simulations; for design purposes we recommend that more accurate models are obtained using simulation programs. Steady-state behavior. A simple nonlinear steady-state model is useful for deriving analytical expressions for the gains g ij between the process inputs èe.g., reæux L and boilup V for the LV-conægurationè and the controlled outputs ètop and bottom compositions, y D and x è. dy D = g dl + g 2 dv dx = g 2 dl + g 22 dv èè To derive these gains analytically, two equations relating y D and x to L and V and other operating variables for the column are needed. One is given by the overall 2

3 material balance for the light component Fz F = Dy D + x è2è and the other is conveniently written in terms of the separation factor S èshinskey, 984è S = y D=, y D x =, x x è, y D è èthe approximation holds for high-purity separations with y D and è,x è è. S is in general a complex function of the operating and design variables for the column èæow rates, feed composition, VLE-data for system, number of trays, feed location, etc.è. There are three reasons for choosing to express the column behavior in terms of S: è3è èè Reasonably simple and reliable expressions may be derived for S. This is motivated by Fenske's total reæux equation S = æ N which is exact for mixtures with constant relative volatility æ. è2è Usually S does not change much with operating conditions. The essential part of the variation of the gains g ij with operating conditions may therefore be captured by assuming S is constant. è3è S as deæned by è3è is a function of y D and x only. This makes it simple to derive analytical expressions for the steady-state gains. There are other ways of expressing shortcut models for distillation columns, besides using S. For example, Gilliand's empirical correlation relates R,R min R+ to N,N min N+, and analytical equations have been derived to æt the correlation. However, this form is not convenient for obtaining analytical expressions for the gains, since it does not satisfy è2è and è3è above. The separation factor S has been used extensively by Shinskey è984è for predicting the steady-state value of the RGA. The - element of the RGA is =, g, 2g 2 g g 22 è4è 3

4 For columns with both products of high purity Skogestad and Morari è987bè derived the following approximations for the LV - and DV - conægurations èg LV è Dè, y D è+x è@ ln S=@Lè D è5è èg DV è + x Dè,y D è è6è Consequently, a good model for S is needed to evaluate the RGA for the LVconæguration, but is less important for the DV-conægurations. Also note that for the LV-conæguration may bevery large if both products are of high purity, while the value for the DV- conæguration is always beween 0 and. Shinskey uses the simple analytical model for S developed by Jafarey et al. è979è to estimate the RGA. In their paper, Jafarey et al. only checked the validity of the model for predicting the total number of theoretical trays. However, to estimate steady-state gains èg ij è and RGA-values, derivatives of the model are needed. The objective of our paper is to check the models validity for estimating gains and the RGA. We also feel that the derivation by Jafarey et al. does not provide much insight. They start from Smokers exact analytical solution, derive a simpliæed model which is not very good, delete some terms in this model, and ænally arrive at a better model. Another goal of this paper is therefore to rederive Jafarey's model in a more direct manner which yields more insight. Our derivation yields a slightly diæerent form of the model, but it reduces to Jafarey's expression if the assumption D=F z F is made. Assumptions. All the results in this paper are for a two-product column, binary mixture with constant relative volatility èæè and constant molar æows. The extension to multicomponent mixtures is discussed at the end of the paper. 4

5 2. CASE STUDY EXAMPLES In this section we present steady-state data for seven high-purity distillation columns which may be used as case study examples in this and future work. The columns were selected to include a wide range of operating conditions. Data for the columns are given in Table, together with computed values for R=R min ; L=F and D=F. Steady-state gains. The exact steady-state gain matrices for the seven columns were obtained by linearizing the material balance for each tray. For tray i with no feed Lèx i+, x i è+v èy i,, y i è=0; y i = Linearizing è7è for small perturbations æx i +èæ, èx i è7è èx i+, x i èdl +èy i,, y i èdv + Ldx i+ +è,l, K i V èdx i + K i, Vdx i, = 0 è8è where K i is the linearized VLE-constant oneach tray K i = dy i dx i = æ è+èæ, èx i è 2 è9è In matrix form è8è becomes Ax + u + Ed =0; y = Cx è0è Here x =èdx ::: dx N è T are the tray compositions, u =èdl dv è T are the inputs, d =èdf dz F è T are the disturbances, and y =èdy D dx è T are the outputs. Solving these equations give y =,CA, u, CA, Ed and the steady-state gain matrix and disturbance matrix are G =,CA, and G d =,CA, E èè These matrices are given for the seven columns in Table 2. They are given both for the LV - and the DV -conægurations: 5

6 LV, conæguration : DV, conæguration : dyd dx dyd dx = G LV dl dv df + G dlv dz F dd df = G DV + G dv ddv dz F è2aè è2bè Note that G LV and G DV are not independent. ecause of the assumption of constant molar æows we have dl = dv, dd which yields and we derive dl dv = M dd where M = dv, 0 G DV = G LV M è3è Similar relationships are easily derived for other choices of manipulated inputs èskogestad and Morari, 987bè. The matrix G LV is very sensitive to small relative errors in the elements èas is seen from the large value of èskogestad and Morari, 987cèè, while G DV is insensitive to such errors. The reader should therefore ænd G LV from G LV = G DV M, rather than using the G LV -matrix given in Table 2. The disturbance matrices G dlv and G ddv are in general diæerent, but happen to be equal in this case since the feed is liquid. 3. A SIMPLE EXPRESSION FOR S Previous Work For mixtures with constant relative volatility èæè the following exact expressions for N min and R min hold èe.g., Henley and Seader, 98è. R!: S = æ N min èfenskeè è4è N!: R min = æ, èy D, æ, y D è èy D è x F, x F æ, x F è5è x F is the composition of the æashed feed. Simple extensions to multicomponent mixtures exist èe.g., Henley and Seaderè. Jafarey, Douglas and McAvoy è979è 6

7 derived a simple expression for S which has also been adopted by Shinskey è984è. For liquid feed we have x F = z F and Jafarey's expression becomes q F =: S = æ N +=èrz F è N=2 è6è The expression è6è gives the correct limiting value è4è as R!, but its value of R min N!: R min = èæ 2, èz F è7è is lower than the correct one è5è. Jafarey et al. used è6è for design purposes to ænd the number of theoretical stages N needed to accomplish a given separation. For 24 cases studied the average error in N was 5:3è compared to 2:5è for the Gilliland correlation. Derivation of Expression for S The main reason for including this section is that the derivation given here is much simpler and more instructive than the one given by Jafarey et al. Furthermore, extension to multicomponent mixtures is straightforward. We will initially linearize the vapor-liquid equilibrium èvleè curve in the top and bottom parts of the column. Above feed : è, y i è= æ T è, x i è è8aè elow feed : y i = æ x i è8bè Here æ T column. and æ denote the relative volatility in the top and bottom part of the The assumption of linear VLE will clearly make the separation in the middle of the column simpler as seen from Fig.. Note that for high-purity columns, most of the stages will be at the column ends in the regions of high purity where the linear approximation is good. The operating lines èmaterial balance of light component for each trayè are Above feed : V T y i, = L T x i + Dy D è9aè 7

8 elow feed : L x i+ = y i + x è9bè Since the equilibrium and operating lines are assumed linear it is simple to derive exact relationships between the feed tray composition and y D and x. Consider the top part of the column. The vapor composition of the heavy component è, y j è on the jth tray from the top is found by repeated use of è8aè and è9aè èkremser equations, see McCabe and Smith, 976è. è, y j è=è, y D èèa j T + èaj T, èè, L T =V T è è A T, è20è where A T is the absorption factor A T = L T =V T =æ T è2è Since A T é wehave A j T éé for j large. Using this assumption for the feed tray we derive è, y NF è è, y D èa N T T AT, L T =V T A T, where N T is the number of theoretical stages above the feed. Similarly, we derive for the bottom section of the column an expression for the liquid composition on the feed tray x NF = x è A,N x A,N + èa,n A,!, èè, V =L è A,,, V =L A,, Here N is the number of theoretical trays below the feed and A é is the absorption factor in the bottom part of the column è22è è23è A = L =V æ è24è Multiplying equations è22è and è23è gives an approximate expression for the separation factor S : S x è, y D è = æn æn T T èl=v è NT T è, y NF èx NF 8 èl=v è N æ c è25è

9 c = èæ T, èèæ, è èæ T, V T =L T èèæ, L =V è In most cases c is close to one; in particular, this is the case if the reæux is high. Assuming constant relative volatility æ = æ = æ T and using æ = y NF=,y NF x NF =,x NF N = N + N T + we derive from è25è S = æ N èl=v è N T T y NF è, x NF è èl=v è N æ c and è26è We know that S predicted by è26è is too large because of the linearized VLE. However, è26è may be corrected to satisfy the exact relationship S = æ N at inænite reæux simply by dropping =y NF è, x NF è from è26è. y assuming in addition c =we get S = æ N èl=v èn T T èl=v è N This expression is somewhat misleading since it suggests that the separation may always be improved by transferring stages from the bottom to the top section if èl=v è T é èv=lè. This is clearly not generally true, and to avoid this problem we follow Jafarey et al. è979è and choose N T N N=2 and derive the alternative è27è expression N=2 èl=v S = æ N èt èl=v è è28è If the feed is liquid then V = V T and q F =: N=2 S = æ N LT L = æ N L L + F N=2 è28aè Since L=F = L D D F, this reduces to Jafarey's expression è6è if the additional assumption D F z F is made. Jafarey et al. derived è6è as a design equation and the assumption D F z F is reasonable in this case. However, the assumption is not so easy to justify for control purposes where D is a variable which may take onvalues diæerent from D = z F F. The assumption seems particularly misleading if the model is used to compute gains for disturbances in z F.Table 3 gives the estimated 9

10 number of trays in the seven example columns obtained from è28è and è6è. The two formulas yield almost identical values since the columns have D F z F. The main shortcoming of è28è is obvious from the derivation: The model is poor if a pinch zone appears around the feed plate such that the assumption that most of the trays are in the region of high or low purity which led to è26è is no longer valid. A pinch zone occurs around the feed plate if the reæux is near minimum; this explains why è6è gives a value of R min è7è which istoolow. More importantly, pinch zones appear above orbelow the feed plate if the feed location is not optimal. This is likely to happen during operation of the column. The use of è28è implicitly assumes that the feed location is optimal. Finally, for columns with large diæerences between N T and N, è27è should be used instead of è28è. We will study the validity of the models è6è and è28è for estimating steady-state gains in Section WHY ARE DISTILLATION COLUMNS NONLINEAR? For the simple case of constant molar æows and constant relative volatility considered in this paper, there are two possible sources of nonlinearity èaè Nonlinear VLE y i = æx i +èæ +èx i è29è èè ilinear terms èlx i ;Vy i ; etcè. in the material balance for each tray Lx i+ + Vy i, = Lx i + Vy i è30è None of these nonlinearities seem very strong, and the strongly nonlinear behavior observed for èhigh-purityè distillation columns is therefore somewhat surprising. The main reason for the nonlinear behavior is, as we will show, the nonlinear VLE. Consider changes in product compositions èy D and x è caused by achange in feed composition èz F è, when all æows, including reæux L and boilup V, are constant. In this case the material balance è30è is linear, and the only possible source of nonlinearity is the VLE. 0

11 Linear VLE. If the VLE were assumed to be linear èe.g., by linearizing è29è on each tray; y i = a i x i + b i è, all equations describing the column behavior are linear, and a linear relationship between the product compositions and z F results y D = k z F ; x = k 2 z F Here the gains è@y D =dz F è L;V = k and è@y D =dz F è L;V = k 2 are constant and independent of the feed composition. Nonlinear VLE. A completely diæerent result is found, however, when the VLE is not linearized. The simpliæed model è27è for S was derived using the nonlinear VLE model è29è. èactually, the VLE was ærst linearized, but the ænal expression was corrected to match S = æ N at total reæux.è This model predicts that S is constant for changes in feed composition provided L and V are unchanged. Consider a column which nominally has z o F =0:5 and, yo D = xo =0:0. For S constant the following two equations give y D and x as a function of z F y D è, x è è, y D èx = yo D è, xo è è, y o D èxo = 980 è3è z F = D F y D + F x è32è èthe second equation is the overall material balance for light component.è Note that all æows are constant, so D=F = =F =0:5 is constant. è3è and è32è clearly result in a nonlinear relationship between y D and x, and z F. The steady-state gains found by combining these equations are strongly dependent on the operating point. For example, S constant : F L;V = D F + x è,x è F y D è,y D è è33è At the nominal operating point è@y D =@z F è L;V =. However, for z F =0:55, è3è and è32è give x =0:0 and, y D =0:0092, and the linearized gain at this

12 operating point D F è L;V = 0:0202. This is 50 times smaller than the nominal value. The conclusion is that the nonlinear VLE results in a strongly nonlinear behavior for high-purity columns. The bilinear terms in è30è are of much less importance for the observed nonlinearity of distillation columns. 5. STEADY-STATE GAINS FROM EXPRESSION FOR S Consider any input or disturbance èfor example = L; V; ; F; z F ; etc.è. The steady-state D =@ =d may be found by diæerentiating the material balance è2è and using the deæning expression for the separation factor è3è @ = e è34è Here e is deæned ln D y D è, y x è, x è35è e =,èy D, + +èz F, è36è Solving for the gains D è, y D è, x = I s = I s e + x è, x ln e, Dy D è, y D ln è37aè è37bè where I s is the "sum" of impurities leaving the column I s = x è, x è+dy D è, y D è è38è For high-purity separations I s x + Dè, y D è. There are two contributions to the gains in è37è: The e -term and the contribution from changes in the separation factor S. The e -term physically represents the eæect on the gains of changing the external material balance. This is clear = 0 and e = 0 is obtained 2

13 when we change the internal æows in the column only èchange L and V keeping D and constantè. For changes in the external æows èwhich represent most disturbances and inputsè the e -term in è37è is usually dominating - at least when both products are high-purity.this means that the major contribution to the gains in this case may be obtained by assuming S constant and the exact value ln S=@ is of minor importance in this case. On the other hand, a good model for S is important for obtaining the gains for changes in the internal æows : In this case the gains, for example è@y D =@Lè D, are directly proportional to è@ ln S=@Lè D =è@ ln S=@V è D and a good estimate of this quantity is required for estimating the correct value of the gain. ln S=@ from shortcut models Equations è37è are exact. I s is a given constant and e is trivial to ænd for the case of constant molar æows. The only ëunknown" in è37è ln S=@, which may be estimated from the shortcut models è28è or è6è. Analytic expressions ln S=@ obtained with these models for diæerent choices of are given in Table 4. Estimated numerical values ln S=@, and of the gains and the RGA for the LV-conæguration are compared with exact values for the seven columns in Table 5-7 and Fig.2. The results are discussed below for each shortcut model. Using the model è28è for ln S. From Table 5 we see that, with the exception of è@ ln S=@V è D, the estimates ln S=@ are extremely poor. Note that because of the assumption of constant molar æows we ln ln S ln S V è39è and even though the two terms è@ ln S=@V è L and è@ ln S=@Lè V induvidually are estimated very poorly with è28è, the estimate of the sum è@ ln S=@V è D is in reasonable agreement èfig.2è. Also note from Fig. 2 that è@ ln S=@V è L and è@ ln S=@Lè V 3

14 are very sensitive to the feed point location, while their sum ln è D is nearly constant. The conclusion is that the model è28è is only useful for estimating the eæect of changing the internal æows on ln ln S D è40è Fortunately, as pointed out above, this corresponds exactly to the case when the ln S=@ in è37è is most important. Therefore, in spite of the generally poor estimates ln S=@, the model è28è can be useful for estimating steady-state gains and the RGA èrecall Eq. è5èè. This will in particular be the case for columns with both products of high purity. In other cases the errors may be signiæcant. This is evident by comparing the estimated steady-state gain matrices in Table 7 with the exact values in Table 2. Using Jafarey's model è6è for ln S. As seen from Table 5 and 6, the estimated values for è@ ln S=@V è D are very similar to those found using è28è. The estimated values ln S=@ with respect to changes in the external material balance are again poor. Surprisingly, the values are also very diæerent from what is obtained using è28è. The estimated steady-state gain matrices are also very diæerent for the two models in some cases as seen from Table 7. Shinskey è984è uses the model è6è to estimate the RGA. This estimate can be quite poor as seen from Table 8. The model è28è seems to be somewhat better than è6è for estimating the RGA. Choice of Model for ln S The two seemingly very similar shortcut models è6è and è28è for ln S may result in quite diæerent estimates for the steady-state gain matrices ètable 7è. None of the models give very accurate results, but they may still be useful for obtaining a ærst estimate. ased on the numerical results presented above, there is no reason to give preference to one of the models è6è on è28è. However, the assumption D z F F which led to è6è does not seem to be justiæed from a theoretical point of 4

15 view, and we therefore recommend using è28è, which also gives simpler analytical expressions ln S=@ ètable 4è. 6. A NEW FORMULA FOR THE OPTIMAL FEED LOCATION Equations è22è and è23è are reasonably accurate if most of the trays are located in the region of high and low purity. Divide equation è22è by è23è and assume the feed plate is optimally located such that x NF = x F and y NF = y F, y F x F =, y D æ N T,N èl=v è NT T x èv=lè N æ, èl=v è æ, èv=lè T As a crude approximation neglect the last two terms. This approximation is reasonable if N T and N are not too diæerent since èl=v è T and èv=lè are often reasonably close in magnitude. è4è then gives è4è N T, N = ln,y F x F ln æ x,y D è42è x F and y F are functions of z F ;q F and æ and are obtained by æashing the feed. The optimal feed tray location is then given by N F = N += N +, èn T, N è 2 è43è èn is the total number of theoretical stages.è Estimated and exact values of the optimal feed stage locations are shown for the seven columns in Table 9. The exact value is found using Stoppel è946è èfor the feed as liquid N F = n 2 +, where n 2 is found from Stoppel's paperè. The average error jæn F j in percent ofn=2 for the seven columns using è42è is 7.0è. This compares to 2.2è when using the Fenske ratio èhenley and Seader, 98è N T N = ln yd ln,z F,y D z F,x x è44è z F,z F and 8.0è when using the empirical Kirkbride formula èhenley and Seader, 98è N T N = ", z F z F D 5 x, y D 2 è 0:206 è45è

16 èthe crude estimate N F = N=2 gives an average error of 20.0èè. We also computed the optimal feed point location for the 24 columns given by Jafarey et al. è979è. èthere is a misprint in this paper, and R=R min should be.75 for case bè. The following average errors were found jæn F j N=2 This work è42è 4.4è Fenske ratio è44è 5.7è Kirkbride è45è 2.8è 7. EXTENSIONS TO MULTICOMPONENT MIXTURES Deæne separation in terms of two key components èl and Hè, and deæne the pseudo-binary values of composition and æows èsuperscript 'è by considering only these key components. For example y 0 = y 0 L = y L y L + y H ; V 0 = V èy L + y H è è46è èas usual the subscript L has been dropped for the light componentè. The nonkey components are assumed to be non-distributing; the heavy nonkey èhnè is assumed to have muchlower relative volatility than the heavy key èhè. Also, the light nonkey èlnè is assumed to have much higher relative volatility than the light key èlè. Under these assumptions è27è still holds if a pseudo-binary basis is used èappendixè: where as before S = y0 D =, y0 D x 0 =, x0 = æ N èl0 =V 0 è N T T èl 0 =V 0 è N æ = y L=y H x L =x H = y0 =, y 0 x 0 =, x 0 è47è è48è The material balance may also be written on a pseudo-binary basis F 0 z 0 F = 0 x 0 + D 0 y 0 D è49è 6

17 Consequently, all the gain equations è34è-è38è derived above still apply when a pseudo-binary basis is used. Similarly, Eq. è42è for the optimal feed point location applies if x ;y D ;y F and x F are replaced by the pseudo-binary values x 0 ;y0 D ;y0 F and x0 F. 8. CONCLUSION It is convenient to express the column behavior in terms of the separation factor S = y Dè, x è è, y D èx S is often nearly constant for varying operating conditions. è3è For columns with constant relative volatility and constant molar æows, Fenske's exact equation applies at total reæux S = æ N è4è For ænite reæux the following generalization is useful S = æ N èl=v èn T T èl=v è N è27è This model gives a good description of how S changes with the internal æows, but describes poorly the eæect of changes in the external material balance. Fortunately, it turns out that this is of less importance if the model for S is used to obtain estimates for the steady state gains. Finally, a new formula for estimating the optimal feed stage location is proposed N T, N = ln,y F x F ln æ x,y D è42è From its derivation we know that this formula may give poor results if N T and N are very diæerent, but it gave better results than other proposed methods for the seven columns in Table and for the 24 columns studied by Jafarey et al. è979è. Acknowledgements. Partial support from the National Science Foundation and Norsk Hydro is gratefully acknowledged. 7

18 NOMENCLATURE - bottom product rate D - distillate ètop productè rate F - feed rate L T = L - liquid æow in top section of column L = L + q F F - liquid æow in bottom section of column N -number of theoretical trays below feed èincl. reboilerè N F = N + - feed tray location N T -number of theoretical trays above feed N = N + N T + - total number of theoretical trays in column q F - fraction of liquid in feed R = L=D - reæux ratio S = y Dè,x è è,y D èx - separation factor V = V -vapor æow in bottom section of column V T = V +è, q F èf -vapor æow in top part of column x - mole fraction of light component in bottom product x i - liquid mole fraction of light component on stage i x F ;y F - mole fraction in feed at feed stage pressure x NF ;y NF - mole fractions on feed tray y D = x D - mole fraction of light component in distillate ètop productè y i -vapor mole fraction og light component on stage i Subscripts - bottom part of column, bottom product D - distillate product T - top part of column i - tray no. numbered from bottom èi= for reboiler, i=2 for ærst tray, i=n for top tray, i=n+ for condenserè Superscript 0 - pseudo-binary basis is used 8

19 REFERENCES Gilliland, E. R., Ind. and Eng. Chem., 32, 220 è940è. Henley, E. J. and J. D. Seader, Equilibrium-State Separation Operations in Chemical Engineering, John Wiley and Sons è98è. Jafarey, A., J. M. Douglas and T. J. McAvoy, Short-Cut Techniques for Distillation Column Design and Control.. Column Design, Ind. & Eng. Chemistry Process Des. & Dev., 8, è979è. McCabe, W. L. and J. C. Smith, Unit Operations of Chemical Engineering, 3rd Edition, McGraw-Hill è976è. Shinskey, F. G., Distillation Control, 2nd Edition, McGraw-Hill è984è. Skogestad, S. and M. Morari, Shortcut Models for Distillation Columns - II. Dynamic Composition Response, This issue of Comp. & Chem. Eng. è987aè Skogestad, S. and M. Morari, Understanding the Steady State ehavior of Distillation Columns, in preparation è987bè. Skogestad, S. and M. Morari, Implications of Large RGA-elements on Control Performance, submitted to Ind. & Eng. Chemistry Process Des. & Dev. è987cè Stoppel, A. E., Calculation of Number of Theoretical Plates for Rectifying Column, Ind. & Eng. Chemistry, 38, 2, è946è. 9

20 APPENDIX Extensions to multicomponent mixtures The relative volatility between the key components is deæned as èsubscript i for the tray is droppedè ottom section. From è48è we get æ = y L=x L y H =x H = y0 è, x 0 è x 0 è, y 0 è è48è y L =èæ y H x H èx L =^æ x L èaè In Section 3 we linearized the VLE by assuming y H and x H, but the last approximation does not hold in presence of heavy nonkey-components èhnè. Assuming y HN 0 èi.e., the heavy nonkey has very low relative volatilityè we get x HN z HNF L and we ænd y H and x H, x HN =, z HNF L, i.e., ^æ = æ è, x HNF L è èa2è and we have a linear VLE relationship for the bottom section. This leads to an expression for x LNF similar to è23è but with æ replaced by ^æ. Top section. From è48è we get y H =è æ T y L x L èx H = ^æ T x H èa3è A derivation similar to the one for the bottom section yields ^æ T = æ T, z LN F=V T èa4è This leads to an expression for y HNF as a function of y HD similar to è22è, but with æ T replaced by ^æ T. Note that Fenske's expression for S at inænite inæux is exact also when there are nonkey components. S = y LD=x LD y HD =x HD = y0 D è, x0 è x 0 è, y0 D è = æn èa5è 20

21 Multiplying è22è with è24è as before, and correcting with the exact expression èa5è we derive S = æ N, z N HNF, z,nt LNF L V T Since the nonkey components are non-distributing we have èl=v è N T T èl=v è N èa6è L 0 T = L T ; V 0 T = V T, z LN F L 0 = L, x HN F; V 0 = V Substituting this into èa6è we ænally derive S = æ N èl0 =V 0 è N T T èl 0 =V 0 è N è47è 2

22 Column z F æ N N F y D x D F L F N N min R R min A 0:5 : :99 0:0 0:500 2:706 :76 :388 0: " " " " " 0:092 2:329 " :30 C 0:5 " " " 0:90 0:002 0:555 2:737 :93 :645 D 0:65 : :995 0:0 0:64 :862 :66 :529 E 0: :9999 0:05 0:58 0:226 :99 :44 F 0: :9999 0:000 0:500 0:227 :47 3:83 G 0:5 : :9999 0:000 0:500 2:635 :76 :38 Table. Steady-state data for distillation column examples. All columns have liquid feed èq F = è. Column G LV G DV G d èf; z F è A C D E F G 0:878,0:864,0:878 0:04 0:394 0:88 :082,:096,:082,0:04 0:586 :9 :748,:77,:748 : :0858 :636 :9023,:9054,0:9023,:0032 0:0904 0:936 :604,:602,:604 : :8822 :790 :0865,:0248,:0865,: :097 0:070 0:23,0:2,0:23 :0092 0:056 0:252 2:26,2:29,2:26,:00307 :344 2:395 :02033,:035,:02033 :0079 :00045 :00949 :24,:26,:24,:0035 0:780 :86 :074,:073,:074 : :5362 :073 0:9257,0:9267,0:9257,: :4636 0:9269 0:8649,0:8646,0:8649 : :435 0:8647 :35,:35,:35,: :5683 :35 Table 2. Steady-state gain matrices for examples obtained using èè. G DV = G LV M è3è. Unscaled product compositions èy D and x è are used as deæned in è2è. 22

23 Column Actual N Estimated N This work è28è Jafarey è6è A C D E F G Average Error 6.3è 6.3è Table 3. Estimated number of theoretical trays in columns. This work è28è ln S N ln æ + N 2, ln V, ln L, ln ln F L;V, ln N L;V ln F, N 2 L;V N 2 N 2 L V q F F L+q F F èq F =è: N 2 èl=v èt èl=v è Jafarey è6è èfeed liquid; q F =è N ln æ, N 2 ln + Rz F N 2 è,q F èf L+D, N 2 L L=F + N 2 N 0 2 qf L+q F F +,q F L+D L=F +q F, L=F +D=F, N 2 L R+ Rz F + D L Rz F + Rz F + z F Rz F + 0 Rz, F + Rz F +z F Table 4. Analytical expressions ln S=@ obtained from shortcut models è28è and è6è. è@ ln S=@V è D for q F 6= is equal to è@ ln S=@Lè V +è@ ln S=@V è L è39è. 23

24 Column A Column Exact This work Jafarey Exact Thiswork Jafarey è28è è6è è28è è6è è@ ln S=@Lè V,2: :99 2:8 85:3 2:58 64:0 è@ ln S=@V è L 23:9 0,0:8,8:9 0,6:6 è@ ln S=@V è D 2:76 :99 :99 3:44 2:58 2:42 è@ ln S=@z F è L;V,24:0 0 0:8 70:7 0 56:5 è@ ln S=@F è L;V,9:3,5:4 0,0:5,6:0 0 è@ ln S=@q F è L;V,22:7 0:8 0:8 84:8 2:3 :9 Table 5. Estimated values ln S=@ for columns A and. Column Exact Estimates This work è28è Jafarey è6è A C D E F G Average error 29.2è 32.2è Table 6. Estimates of è@ ln S=@V è D =è@ ln S=@Lè D. This represents the eæect of changing the internal æows on S. 24

25 Column Exact Estimated This work è28è A C Jafarey è6è 0:878,0:864 0:990,0:980 :0434,:0335 :082,:096 0:970,0:980 0:966,0:9265 :748,:77 :0032,0:9800 :5554,:5337 :9023,:9054 0:9777,0:9800 0:928,0:9240 :604,:602 :5940,:5909 :6094,:606 :0865,:0248 :0345,:03528 :0074,:0488 Table 7. Estimates of G LV for columns A, and C obtained using è37è with shortcut models è28è and è6è. Column, Exact, Estimated This work è28è Jafarey è6è A C D E F G Average error 26.2è 36.5è Table 8. Estimated and exact RGA-values for the LV-conæguration. Note that the values of Jafarey are used in the book of Shinskey è984è. 25

26 Column N N F, Exact N F, Estimated Stoppel This work Fenske ratio Kirkbride è42è è44è è45è A C D E F G Average error, jæn F j N=2 : 7.0è 2.2è 8.0è Table 9. Optimal feed point locations. 26

27 Figure captions. Fig.. VLE - curve for relative volatility æ = 2:0 and linear approximation with y = æx and, y =,x æ. Fig. 2. Column A. Eæect of feed point location N F on derivatives of ln S with all other column data æxed èincluding, y D = x =0:0è. Exact values are compared with shortcut model è28è. Note that è@ ln S=@Lè V +è@ ln S=@V è L = è@ ln S=@V è D and that è@ ln S=@V è L = 0 for the shortcut model since q F = èsee Table 4.è. 27