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Muhicomponent Separation CaIcuIationsby Linearization

LEONARD M. NAPHTAL I and DONALD P. SANDHOLMP o l y t e c h n i c I n s t i t u t e of Brook lyn, Brook lyn, N ew Y o r k

A new approac h to separat ion ca lcu l at ions has been developed which has many p rac t ic a ladvantages. The equat ions of conservat ion of mass and energy and o f equ i l ib r ium a re g roupedby s tage and then l inea r ized . Th e resu l t ing se t o f equa t ions has a b lock- t r id iagona l s t ruc tu rewhich perm its so lut ion by a s imple techni que. Thu s a new way to apply the Newto n-Raphsontechn ique to s epa ra t ion ca lcu la t ions i s devised. Th e method has been t r ied on a number o fprob lems chosen to exh ib i t cha rac te r is t i cs wh ich cause o the r so lu t ion techn ique to fa i l . Mos tp rob lems a re so lved w i th ten i t e ra t ions or less.

While there are many methods available for calculatingnmlticomponent distillation, their performances leave muchto be desired in many instances. In this paper a newtechnique is presented which has the following advantages:

1. Component volatility ranges do not affect the con-vergence. Absorbers, strippers, and reboiled absorbers canbe calculated with the same algorithm used for distillationproblems with no modifications.2 . The presence of nonideal solutions is accounted forrigorously and simply with no change in the method.3. Temperature, key component composition or rate,reflux ratio, recovery, or any other function of condenseror reboiler flows can be used as specifications.4. Murphree plate efficiencies are taken into account inii rigorous manner.5 . The allowable number of feeds and sidestreams isunlimited. No difficulties are caused by negative flows orrounding errors in multifeed problems.

6. The method is based on a linearization of the dis-tillation equations so that convergence accelerates as thesolution is approached. This is also true of the Wang-Oleson ( 2 3 ) , Greenstadt-Bard ( 9 ) , and Tiemey-Bruno(26) algorithms. It is not true of methods based upon theThiele-Geddes method ( 5 , 7, 11, 22) or unsteady stateapproach (2, 20) .7. No difficulties or complications arise from componentflows being very small, hence seeding procedures are un-necessary.

The primary disadvantage of the new method is its re-quirement of storage space which necessitates the use ofsecondary storage media for medium to large size prob-lems, unless a rather large immediate access memory isavailable. However, recent trends in the computing in-dustry are toward multiprogramming or time-sharingcomputers, on one hand, and toward small low cost, disk-oriented computers, on the other. Time-sharing would al-low this distillation program to retrieve its data fromsecondary storage while another program was being run.Thus no penalty would need be paid for the relativelyslow access time of secondary storage. Th e small com-puters rely on disk storage for most problems, so that boththese schemes tend to reduce the penalty incurred by re-quiring a large amount of storage. When a large computeris used the penalty is eliminated.The equations which describe continuous, multicom-ponent distillation are well known (8, 11,12, 14, 1, 22) .Consider the case of an n plate column separating c com-

Leonard M. Naphtali is in New York, New York. Donald P. Sandholmis at Monsanto Chemical Company, St. Louis, Missouri.

ponents where plate 1 is a condenser and plate n is areboiler. Furthermore, let sidestreams be specified by theratio of the sidestream to the stream which remains afterthey are withdrawn. Figure 1 shows a schematic repre-sentation of one such plate. Lij, Vij, and Ti are the vari-ables and Li and Vi will represent the total phase flows.We shall assume that the pressure is known and the plateis adiabatic.Let us examine three types of functions which describethe physical processes on plate i.Enthalpy balance:

Ei =(1+Si)Hi +(1+~ i ) h i- H i + 1 - h-1- hfi(1 )Component material balances:

A l i j =(1+Si)vij+(1 +Si ) Lij - vi+ ,j- i-1.j - ij(2)Equilibrium relationships:

Qij is derived from the definitions of the vapor phaseklurphree plate efficiency. That isyij - i+1,j13 v % + l J. . x . .- .i=

0Ti Kij xij - ij + (1- i ) i+l , j=0

N and y represent component mole fractions. In terms ofour variables this equation becomes= o .T i Kij Lij Vij (1- i) V i+ 1.j+-

Li Vi V i + lUy multiplying each term by Vi we arrive at Equation (3) .These functions" (or equations) apply to all interiorplates of the column. They also apply to a partial con-denser or reboiler if the heat duty is known and is treatedas feed enthalpy. This is the case for absorber and strippercolumns where the heat duties are zero (that is, thecondenser and reboiler are ordinary trays). Later we shallsee how to make specifications other than these.

* Functions (l), ( 2 ) , and ( 3 ) are discrepancy functions, that is,they are a quantitative measure of the failure of the values of LAJ, t j ,and T , to satisfy the physical relationships. The value of Ei is the num-ber o f heat unitdtime by which the enthalpy balance is unsatisfied,whereas in Mi j and Qii he discrepancy is measured in moledtime. Asolution to the problem has been obtained when one has values of thevnriahles which make Ei, Mi,, nd Otj zero. (I), ( 2 ) , and ( 3 ) can beconsidered equations if we replaced E i, M i j , and Q i i by zero. We wouldthen say we are looking for the values of the variables which satisfythe equations.

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METHOD OF SOLUT10NThere are 2c +1 equations and variables on each plateor a total of n(2c +1) of each. Most methods divide theequations into groups and solve one set as the overallproblem, with the other sets being solved as inner sub-

problems. Friday and Smith ( 8 ) have analyzed the meth-ods which arise from the different groupings of Equations( l ) , 2 ) ,and ( 3 ) . Briefly, they report that each groupingof these equations enables one to obtain solutions for sometypes of problems (that is, wide as opposed to narrowboiling components), but no one grouping will work forall types.The remaining alternative is to solve all the equations

simultaneously. A modified Newton-Raphson procedurewas chosen to solve these nonlinear equations becausethat method is based on a linearization of the equationswhich becomes more correct as the solution is approached.Thus, a s the values of the unknowns become more correct,the convergence to the final values accelerates. This be-havior contrasts with distillation methods (that is, non-steady state and Thiele-Geddes) in which convergenceactually decelerates as the solution is approached. Naphtali(17) found that if the equations and variables are groupedaccording to plate, the matrix of partial derivatives neededin the Newton-Raphson method assumes a form which iseasy to solve and drastically reduces the amount of storageand number of calculations required. I t is this economywhich makes the method feasible.

where xi s the vector of variables on plate i, and Fi isthe vector of functions on that plate. Thusxi =-. VilVi2

X i Cxi ,c tX i s t

and

Using this notation, the Newton-Raphson method be-comes

where zm+ls the calculated correction, which is addedto ?n to obtain the new values of the variables x r n+ l .fthe functions Fmwere linear this correction would makethe value of each of the functions zero. Since they are

I Ivapor

f ,s ides t ream ff!&t '1 r ; h i didest ream

Y+,, L,,Fig. 1. Schematic rep resentat ion o f one plate.

not G"+ls just an approximation to the needed correc-tion. ( x)F is the matrix of partial derivatives of all thefunctions with respect to all of the variables at the presentvalue of the variables Tm .Hence

--- -

where

dFi,zc+1 dFi,iC+l J...dxi1 fi!xi,~c 1

It will be noted that the functions for plate i [Equations( I ) , ( 2 ) , and ( 3 ) ] involve only the variables on platesi - 1, i, and i +1. Thus the partial derivatives of thefunctions on this plate with respect to the variables on allplates other than these three are zero. Thus from (4)wesee that

_ -B1 El 0 0 . .. ... 0A=, B, E2 0 0 ... ... 00 z 3 E3 E3 0 ... ... 0

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where

The actual elements of these arrays are shown in the Ap-pendix.The form of (z)s called block tridiagonal or quasi-

tridiagonal, and the simple structure greatly simplifies thesolution. The solution can be accomplished by a Gaussianelimination scheme in which only the matrices need bestored. (As can be seen in the Appendix, each of the ele-ments in the last c columns of each C matrix is zero andtherefore need not be stored.)

Experience has shown that used by itself the Newton-Raphson method is sometimes unstable. This statement isespecially true if the present values of the variables arefar from correct. For this reason some method of improv-ing the convergence characteristics of the method must beused. The overall measure of convergence used is the sumof the squares of the discrepancy functions. This value( C =F F ) was chosen for two reasons: (1) it is aquantitative measure of the goodness of the solution, and( 2) the Newton-Raphson method will produce a set ofcorrections ax whose direction will decrease the sum ofthe squares ( 1 6 ) at least initially. Naphtali ( 1 6 ) foundthat if either the calculated corrections are halved until asmaller sum of the squares was obtained or the values ofa from a =0 to OL >1 used in defining x m f l =xm +AX + were searched to find the smallest value of thesum of the squares, the linearization method became more

stable. This is the solution procedure which was reducedto a computer program. In order to use this algorithm, one must supply initial

cstimates of all the variables. This can be done either byguessing all the values or by using some other method togenerate the needed numbers. One such generating pro-cedure (1 requires only that temperature and liquid overvapor ratio profiles be estimated to generate all componentflows. This initialization procedure requires that the va-porization coefficient K be independent of composition, arestriction that the convergence algorithm does not have.Even with this restriction, this procedure is useful becauseit greatly reduces the number of variables that must beguessed. Of course other methods may be used. Since theNewton-Raphson method is based on linearization of theequations, the closer to the solution one starts the betteroff he is. Wild initial guesses can make the Newton-Raphson linearization approximation invalid to such anextent that the method will fail to converge.

-F ---

- -

-

OTHER END SPECIFICATIONSWe have previously shown that if the end specificationsare condenser and reboiler heat duties the functions for aninterior plate can be used for the (partia l) condenser and

(part ial ) reboiler also. If other specifications are made,then the first and last plate functions may be modified.The new specification function replaces the enthalpy bal-ance function. E l or E n no longer applies because the heatload is unknown. In fact the heat load can be calculatedby an enthalpy balance around the condenser or reboiler.

Table 1 gives specification functions which would re-place E l or En. n some situations ( that is, reboiled absorb-ers), it is desirable to control the temperature profile byinserting heat exchangers at various points in the column.

The temperature on a specific tray can be fixed by replac-ing the enthalpy balance on that tray by the equation T =T,,,,,.A program using the new algorithm has been writteni n FORTRAN IV. It has been run successfully with manydifferent types of volatility ranges and end specifications.Experience has shown that the method lives up to expec-tations with regard to convergence on both narrow andwide boiling problems.PROGRAM DETAIL S

Some of the details of the program strategy are pre-sented below. A Fibonacci search was incorporated intothe program to find the best fractional par t of the Newton-Raphson corrections to use. This procedure, which isdescribed clearly by Wilde ( 25 ) ,was chosen because ofit s high efficiency in locating the minimum value of afunction. The values of the liquid and vapor flow ratesare never allowed to become negative because of the ad-ditiou of a negative Newton-Raphson correction. Instead,if a liquid or vapor rate would become negative upon theaddition of the correction, a negative exponential correc-tion, as described by Naphtali ( 1 6 ) , is used to decreasethe value of the rate. This strategy keeps the values of thevariables in a physically meaningful region. The storageof the variables x, unctional values F , correctionsx,ndpartial derivative matrices c, as well as other problemvariables (tha t is, plate efficiencies and pressures), canbe either in the computers memory or on disk or tape, de-pending on the memory available and problem size. Thestorage required for these arrays is n ( ( 2 ~ 1 ) (C +1)+2) = n(2c2+9c +6) words. For example, for a10-component, 20-plate problem the requirement is 5,290words, while for a 20-component, 100-plate problem therccjuirement is 98,600 words.

-

TEST OF PROGRAMThe number of different types of problems which poten-tially can be set up by various combinations of problem

specifications is vast. Therefore, looking for the limitationson the capability of the algorithm we set up critical ex-periments to determine whether it solved problems whichstopped other algorithms. Passing these tests is a limitedkind of circumstantial evidence that the algorithm willwork on a wide variety of problems. (A s of this writingthe algorithm is in use in industry and has solved hundredsof live industrial problems, ranging from reboiled ab-sorbers to alcohol dehydration columns.)

One of the most difficult types of problems to solve formany distillation programs is an absorber problem. Thisis true because this kind of problem contains both multi-ple feeds and wide boiling mixtures. As an example weconsidered a 20-plate column with four hypothetical com-ponents with a very wide boiling range. As can be seen

TABLE .Condenser Reboiler

Specification Ei E nReflux (rebo il) ratio L1 - V1 (b> V n- L, (5 )Temperature Ti - T D 7 , - BOverall product rate V1- D Ln -Key component rate V1, - D Lnj -Lnj - n X Bey component mole V1, - V ~ X Dfraction

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TABLE.K Values Plate Temp., F. L/V

MaterialABCD

ABCD

ABCD

Temperature,100F. 200F.500.0 550.01.50 1.800.90 1 oo1.0 x 10-6 1.5 x 10-6

Molar liquid enthalpies, 103 B.t.u./mole0.01 0.0130.30 0.330.40 0.441.50 1.90

1.00 1.0021.80 1.822.00 2.035.75 5.95

Molar vapor enthalpies, 103 B.t.u./mole

from the data in Table 2 the volatility range is extreme.For the absorber problem, the feeds are listed below asare the initial temperature and L/V profiles.

Liquid feed No . 1 o top plate, temperature =125F.X aterial Moles

ABCD

ABCD

0.00.00.0100.0Vapor feed No . 2 to plate 20, temperature =200F.

75.015.010.00.0Plate temperature, initial estimates: linearly interpolated from 145.0 attop to 200.0 at bottom. L/ V values, initial estimates: linearly interpolatedfrom 1.30 at top to 1.25 at bottom.This problem converged in four iterations to a sum ofthe squares of 2.4 X 10-lo. The final values of the top

and bottom products are shown below.Material Top product Bottom product

ABCD74.884.680.0210.0000899

0.1210.329.979100.0At the other end of the volatility range is the following

distillation problem. The components were n-butane, i-pentane, and n-pentane at 1 atm. pressure. The data weretaken from tables and graphs in Browns Unit Opera-tions ( 6 ) . The problem was specified as having a refluxratio of 1,000 while recovering 80% of the feed in thebottoms product. A sidestream on the first plate (con-denser) withdrew a liquid product equal to 30% of thereflux liquid. Many programs converge with difficulty ifthe sidestream has a significant volume. The column con-tained 20 plates with a liquid feed, T =30F., flow =500 moles of each component, on the tenth plate. Theinitial guesses follow.

12-910-1920

50.051.5 to 53.053.0 to 57.060

1,Ooo.o0.771.920.92The initial value of the sum of the squares was 1.7 x lo5and this was reduced to 2.3 x in six iterations. Thefinal product streams and temperature profiles are shownas listed.

MolesCom- Bottomponent Top product Sidestream productn-C4 0.9966 298.9 200.1i-C5 1.042 x 10-4 0.09425 499.9n-C5 4.371 x 5.935 x 10-3 500.0

Temperature profile1 29.7 6 31.2 11 56.0 16 57.62 29.7 7 33.4 12 56.1 17 59.43 29.8 8 37.9 13 56.2 18 62.84 29.9 9 45.7 14 56.3 19 68.75 30.3 10 56.0 15 56.7 20 77.0

Another type of problem which was tested involvednonideal solutions. The components to be separated weren-hexane and ethanol. For simplicity, we assumed thatthere is no heat of mixing (although the method does not

yiPi*require this). Let Ki =- here yi is the activityPcoefficient, Pi* the vapor pressure, and E the total pres-sure. The activity coefficients were calculated using theVan Laar equation data from Perrys Handbook (18) ndthe enthalpy data were taken from Weber (29) and Reidand Smith (19). The feed was a vapor at 200F. con-sisting of 50 moles of each component. The column con-tained ten plates with the feed entering the sixth. Thereflux ratio was specified as being 5.0 and the columnwas to recover 70% of the feed in the bottoms product.This problem converged in ten iterations to a sum of thesquares of 7.4 x The results are shown below.

Total-Hexane EthanolD 22.59 7.413 30.00B 72.41 42.59 70.00Total products 50.00 50.00 100.00

Plate temperature1. 151.4 6 . 153.42. 151.4 7. 153.43. 151.5 8. 153.54. 151.7 9. 153.95. 152.2 10. 155.4

These three problems represent only a small portion of theproblems run, but they illustrate the flexibility and con-vergence of the method.ACKNOWLEDGMENT

Part of this work was supported by National ScienceFoundation Grant No. GP566.Vol. 17, No. 1 AlChE Journal Page 151

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NOTATION=partial derivative of equations on plate i with=partial derivative of equations on plate i with=partial derivative of equations on plate i with

respect to variables on plate i - 1respect to variables on plate irespect to variables on plate i +1

BiCic =number of componentsD =distillate rateEiF iFFi,jh =liquid enthalpyH =vapor enthalpyK,,j

-

=enthalpy balance on plate i=feed to plate i= vector of all functional values=vector of functional values o n plate i=function value of equation j on plate i

-

=equilibrium vaporization constant on plate i for=molar liquid flow of component i on plate i=total liquid molar flow on plate i=material balance equation for component i on

component iLiM,, j plate in =number of platesQi,jsiSiS STi0i.jVixXi,jxi,jYi , jrEv i

=equilibrium equation for component j on plate i=liquid sidestream on plate i=vapor sidestream on plate i=value of sum of the squares=temperature on pIate i=vapor molar flow of component j on plate i=total vapor flow on plate i=vector of all variables= vector of variables on pla te i= th variable on plate i=liquid mole fraction of component i on plate i=vapor moIe fraction of component i on plate i=vector of generated Newton-Raphson correction=convergence criteria (small num ber )=vapor-based Murphree efficiency on plate i

-

LITERATURE CITED1.2.3.4.5.

6.7.8.9

Amundson, N. R., and A . J. Pontinen, Ind. Eng. Chem.,50,730 ( 1958).Ball, W. E., paper presented at AIChE New Orleansmeeting (1961).Bonner, J. S., paper presented at 21st Midyear MeetingAPI Refining Div. ( May 1956),Boyum, A., Ph.D. thesis, Polytechnic Inst. Brooklyn(1966).Brown, G. G., et al., Unit Operations, Wiley, New York(1950).Edmister, W. C., Trans . AIChE, 42( l ) , 15; ( 2 ) , 403(1946).Friday, J. R. , and B. D. Smith, A l C h E J., 10(5) , 698(1964).Greenstadt, J., Y. Bard, and B. Morse, Ind . Eng. Chem. ,50,1644 ( 1958).

, Petrol Processing, 11(6) , 64 (1956).

10. Hanson, D. N., J. H. Duffin, and G. F. Somerville, Com-putation of Multistage Separation Processes, Reinhold,New York (1962).11. Holland, C. D., Multiconiponent Distillation, Prentice-Hall, Englewood Cliffs, N. J. (1963).12. Lewis, W. K. , and G . C. Matheson, Ind . Eng. Chem. , 24,494 (1932).13. Lyster, W. N., S . L. Sullivan, D. S. Billingley, and C. D.Holland, Petrol. Ref., 38,221 (1959),. I I I I I ,14. Maddox, R. N., Chem. Eng. (Dec. 11, 1961).

15. Murdoch, P. G., Che m. En g. Progr., 44,855 (1948).16. Naphtali, L . M., ibid., 60( 9), 70 (1964).17.-paper presented at AIChE San Francisco meeting,18. Perry, J. H., ed., Chemical Engineers Handbook, 4th19. Reid, R. C., and J. M. Smith, Chem. Eng. Progr., 47,20. Rose, A., R. F. Sweeny, and V . N . Schrodt, Ind . Eng.21. Smith, B. D., Design of Equilibrium Stage Processes,22. Thiele. E. W., and R. L. Geddes, Ind. Eng. Chern., 25,

(19G5).edit., McCraw-Hill, New York (1963).415 (1951).Chern., 50,737 (1958).McCraw-Hill, New York (1963).289 ( 1933). -2 3 . Wang, Y. L., and A. P. Oleson, personal communication.24. Weber, J . H. , AlChE J., 2, 514 (1956).25. Wilde. D., Optimum Seeking Methods, Prentice-Hall,Englewood Cliffs, N. J. (1963).-556 (1967).26. Tierney, John W., and Joseph A. Bruno, AIChE I., 13,

APPENDIX. CONSTRUCTION OF THE PARTIALDERIVATIVE MATRICES.A i Matrixplate i with respect to the variables of plate i-1.This is the matrix of partial derivatives of the functions of

Enthalpy balance =Fi,l

dhi-1=-dFii dFi1---=dTi-1 dxi-1,,+1 dTi-1dFi1 dhi- 1=- i = 1,2, ...Fi1---=dLi-l , j dxi- l ,c+j+ 1 dLi-l , j

In all following work we shall omit the x notation, remember-ing that Ti = cij; Ti; Lij].Material balance functions=Fi,j+1 i =1, 2, . . . c0 k = l , 2 , . . .d F i j +1d 0 i - m

dFi,j+i-=

-=dTi-1dFi,j +IdLi- k- - jk k =1, 2 , . . . C 1 i f g = k

o i f i + kwhere 8 j k =Equilibrium functions=Fi, ,+i+j; j =1, 2, . . .c

dFi,c +1 +j - Fi,c +1+ j - Fi,c+ 1 +j -0 ; k = l , 2 , .. .dui-1,k dTi-1 dLi-1,k

Thus the A matrix has the following structure:1 : :-: ate sizeB i Mat r i xplate i with respect to the variables on that plate.

o x x 0 represents an all-zero submatrixX represents a nonzero submatrixI represents a submatrix which isthe identity matrix of appropri-

This is the matrix of partial derivatives of the functions ofEnthalpy balance =Fi,i

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Material balance functions = Fi , j+l; i = 1,2, ...

k = l , 2 , . . .Equilibrium functions = F.,, + l + j ; j = l , 2 , ...

dF i ,c + l+ j = - ( & j + v i ( s ) )L i j - b j kdoili L i

Thus the Bi matrix has th e following structure:X

C M o t r i xplate i with respect to the variables on the plate below.This is the matrix of partial derivatives of the functions 011Enthalpy balance =Fi,l

d H i + ltioi+ 1,j dui+l,j

dH i+1d T i+1 d T i + l-- - 0Fi1d L i+1.j

-- - -. , j = 1,2, . . .Fi1-- --F i1

Material balance functions =Fi, j+1; i = 1,2, ...k = 1,2, ...

k =1,2 , . . cEquilibrium functions =Fi,c+l+j; j =1, 2, . .

-F i , c+ l+ jd v i+ l , k

dFi,c +1+dTi+ 1

dFi,c+ 1+d L i + 1,k

=o

=0; k =1 ,2 , . ..cThus the .C matrix looks like[ZI 0 0IEQUATIONS USED FOR SPECIFICATIONS OTHFR THAN HEAT DUTIES&placing condenser duty specification. Equ ation F1.1 changedType of equationHeflux rati o L1- v 1(i)pecCondenser temperature TI - d s p e cDistillate rate V1 - specKey component flow VI, key - specKe y component m ole fraction V i , ey - i XrpecReplacing reboiler duty F,i changedTyp e of Eq uatio n / 1 7 \Reboil ratioReboiler temperatureBottoms rate L n - specKey component flowKey component mole fractionThe partial derivatives of these equations must replace partialsof the enthalpy balance equations when they are used. Ofcourse, any other feasible end specification might be usedinstead.The functions which describe the phenomeca occurring in atotal condenser are different from the functions of a n equilib-rium stage. Of course, the materials balance functions muststill hold and E l may be modified to describe the desired endspecification. The equilibrium relationships however no longerare applicable because we no longer have two phases. In placeof these c functions we must use c other functions which de-scribe the process. ( c - 1) of th ese functions must express therestriction that both leaving streams (distillate and reflux)have the same composition. The remaining funztion is a bubblepoint function in which the temperature used to calculate Kis the actual condenser temperature plus a specified numberof degrees of subcooling.If the precision of the numbers used in calciilations were in-finite, it would make no difference which component was notused in the ( c - ) composition functions. With limited pre-cision arithmetic, however, it becomes more important thatthe component omitted is not a trace component.The functions which replace the equilibrium function Q ij

' fbr ' a tdtdl 'condenser are shown below.Bubble point function Fi,c+z =Qi,

T n - B pe cLn,key - n specLn,key - n X s p e c

C

i= 1Composition functions (This assumes that component 1 isleft out of the calculation. Usually, if it is the lightest com-ponent, it will be present in the distillate in nontrace amounts.If it is only a trace component in the feed, it could be switchedwith another nontrace component. )

Xl j = l jor

Manuscript received J u l y 12, 1967; revision recvioed December 4,1969; p a p er accepted December lo, 1969.

AlChE Journal Page 153ol. 17, No. 1