Simultaneous solution of systems of interlinked multistaged separators using the Cray X-MP supercomputer

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UNIVERSITY OF ILLINOIS

... tik4*m*i**«»*

THIS IS TO CERTIFY THAT THE THESIS PREPARED UNDER MY SUPERVISION BY

KARL ROBERT KRAUSE

ENTITLED SIMULTANE0US s o l u t i o n o r s y s t e m s o f i n t e r l i n k e d m u l t i s t a g e d

SEPARATORS USING THE CRAY X-MP SUPERCOMPUTER

IS APPROVED BY ME AS FULFILLING THIS PART OF THE REQUIREMENTS FOR THE

DEGREE OF BACHEL0R 0 F SCIENCE IN CHEMICAL ENGINEERING

Instructor in Chtrgt

A m o v tn : ../JJrfSSfcSz

UVATS A f f n iT P A O T U P V T A V CHEMICAL ENGINEERING

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Simultaneous Solution of Systems o f Interlinked Multistaged Separators Using the Cray X-M P Supercomputer

By

Karl Robert Krause

Thesis

for the

Degree o f Bachelor of Science in

Chemical Engineering

College of Liberal Arts and Sciences University o f Illinois

Urbana, Illinois

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6 m₁ T 8 10 10 10 14 14 15 15 15 17 17 18 1 8 21

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31 5 16

List of Figures

1 Generic Distillation Column Plate . . . .

2 Revision of Code to Improve Vectorization

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List of Tables

1 Problem Specifications... . 7

2 Comparison of Average Solution Time ... 9

3 Comparison o f Nonlinear Equation Solvers... \[ 4 Comparison of Linear Equation Solvers ... 12

5 Comparison of Vectorizability of Linear Equation Solvers ... 12

6 Comparison of Time for Jacobian Evaluations and Function Evaluations . 13 T Breakdown of Execution Time by S u b ro u tin e ... 14

8 Total Solution Time for Each Hun for Problem I ... 23

9 Total Solution Time for Each Run for Problem 2 ... 24

10 Total Solution Time for Each Hun for Problem 3 ... 24

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1

Introduction

Distillation is one o f the most widely used industrial chemical processes and the modelling o f simple distillation systems has received much attention. More complicated systems with interlinking sidestreams are frequently the most cost effective, however, hut are also more difficult to simulate. The traditional method for simple systems has been to simulate one column, use the product streams from this column as the feed streams to the second column, simulate the second column, and so on until all of the columns have been solved. This sequential modular approach works well for simple systems but is not very efficient for interlinked systems because a sidestream from one column may affect a previously solved column. As a result, a large number of iterations over all id' the columns is required to solve the system.

A more efficient solution method is to solve the entire system simultaneously. This technique works well with interlinked systems, but requires the solution o f a large number o f nonlinear equations. Each plate is modelled bv 2C ♦ I equations, where C is the number o f components. Therefore, over 1,000 nonlinear equations must be solved simultaneously to simulate a typical system o f 5 components and 100 plates. While advances in computer technology have allowed this technique to be used on computers such as the CYBER 175, the development of supercomputers is important because l) their increased memory allows larger problems to be solved and 2) their vector processing capabilities can be used to improve the efficiency o f existing programs.

The second aspect has been investigated in this report. The performance of existing programs1 on the CYBER and the Cray X-MP supercomputer has been compared. Both computers were located a* the University o f Illinois-Urhana campus. The total solution time on the Cray was less than on the CYBER because of the faster clock speed of the Cray. The programs were then run on the Cray with the vectorizAtion both on and off. Any difference in solution time was due only to the effect o f vectorization. Finally, the CPU time spent in each DO loop And subroutine was determined to identify the locations in the code where vectorization would he the most beneficial.

The efficient solution o f distillation problems is important as computer resources he come more valuable. Also, flowsheeting programs are finding increased use in industry. The distillation columns frequently require the most time to simulate accurately. Signif icant reductions in the solution time for the distillation system, therefore, will improve the overall performance o f the flowsheeting program.

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2

Simultaneous Solution of Interlinked Distillation Columns

The equations that must he solved are nonlinear and successive linearization was used to solve the system. This involves approximating the nonlinear system as a linear system, solving the linear system, and using this new solution to relinearize the nonlinear system. The solution o f the linear system converges to the solution o f the nonlinear system. 2.1 Equation Formulation

The Naphtali*Samlholm equation formulation was used for most of the problems2.

In this formulation, each stage is represented bv I energy balance, (' mass balances, and

C equilibrium equations; 1 energy balance; En ( I * '^n )//n t ( I t S», )/l»» If n * \ fin t ft f n V mass balances: m ( l ) I'»i ,»r» * ( I * ‘^n)f*n,tr\ ^ n ► I M l , w In.in V equilibrium equations: Q n ,m ( ft E n ,w ^ n f*m , m ) > f* n I ( I 0 ) V n f l , »n h i / 1 n • 1 N number of plates, n I. 2. . . . N, m b 2. . . . ( ’

Figure l shows a generic plate ami delines the variables.

The equilibrium K-values ami the enthalpies are functions o f temperature. The

independent variables are NC vapor How rates, N(! liquid flow rates, an plate tem

peratures. The En, Mn>m, and Q n m are residuals which will equal zero wm;i the system

has been solved.

An alternative formula! ioi»4 involves summing the (' equilibrium equations;

dn l*n j En,inhn,tn

and using this to replace one of the equilibrium equations. This formulation was used with the BBTF linear equation solver and is indicated by BBTF*.

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Figure I: Generic Distillation Column Plate including Nomenclature*

p l o t * n * I

n • place number a • component nuaber

Lq • liquid flow race from place n

■ vapor flow race from place • liquid aidescream from plate n Wn • vapor sidesereaa from place n Fn - feed rare to plate n

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The term ‘solver* is actually a misnomer because the nonlinear equation solver is used to approximate the actual system with a linear system. This linear system can then he directly solved. Four different linear equation solvers were used.'*

The Newton-Raphson (\ H) method is the classic method. The nonlinear sytein is

expanded in a Taylor series about the current approximation to the solution. j\ and all

partial derivatives higher than first order are dropped. The resulting linear system can he

represented by i f ik i where .1 is the Jacobian matrix of partial derivatives,

p is the vector of variable corrections, ami / is the set of nonlinear equations. The

superscript (F) refers to the iteration number. I lie new estimate of the solution is

jr'kt n p{k’ t r u’\ The N R method converges faster than the other methods hut the Jacobian must he reevaluated at each iteration.

The Jacobian evaluation frequently requires more CPI time than solving the linear system. This is due to calls to time consuming thermodynamic property evaluation routines. The other methods, known as quasi-Newton methods, approximate or hold the Jacobian constant for several iterations. This reduces the time for each iteration but results in slower convergence and more iterations. The simplified Newton-Raphson (simp N-R) method simply holds the Jacobian constant for several iterations. The modified Broyden (M B) method updates the Jacobian without reevaluating the thermodynamic partial derivatives. Also, the linear system doesn't have to he resolved at each iteration. Schubert's (Schubert) method approximates the Jacobian in a way that preserves the sparsity o f the matrix, but the linear system must lie resolved at each iteration. The method o f Dennis and Marwil (MARWIL) method essentially updates the Jacobian in a way that makes the solution o f the linear system trivial. The drawback is that more information about the linear system is required. MARWIL is most efficient when used with the GENSOL linear equation solver.

2.3 Linear Equation Solvers

The methods of solving the linear system differ in the data structure used and in the matrix block structure they take advantage of.h The solvers used were continuous back substition (CBS), bordered-block-triangular form (BBTF). bordered-block-tridiagonal form (BBTDF1), LU factorisation (LUIP9), Harwell subroutine library method MA28 (M A28A), sparse Gaussian elimination with partial pivoting (NSPIV), and implicit back substitution (RANKI). B B TD Fl stores each block o f nonzero elements as a full matrix and is referred to as having a l*tau storage scheme, where tau is the number of nonzero elements. NSPIV, LU1P9, RANKI, and CBS have 2*tau data structures, where each nonzero element and its column number are stored along with a small array of row locator indices. A 3*tau data structure is used by MA28A. Each nonzero element is stored along with its row and column numbers.

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3 Comparison of Performance on the CYBER and the Cray

The program* wet® tested on 9 problem* with 5 nonlinear equation solvers, 8 lin* ear equation solvers, and 3 thermodynamic property evaluation routines/The problem specifications are summarized in Table l and detailed in reference I.

Table I: Problem Specifications

p r o b le m r t -c o m p o n e n t * s ta g e s ♦i) n a t io n s c o lu m n s fe e d s s id e s t r e a m s . . „„„ . -■ ■ ’[...3 ... . 22 154 " 2 .... 1 3 i 2 4 45 4 0 5 2 3 l 3 4 90 8 1 0 2 3 4 10 37 m m mi 11 2 l 3 5 _IS 25 77 5 2 1 l 6 6 95 1235 3 1 3 7 ! _i 6 120 1560 2 3 l _: 8 _! ₃ 3 0 0 2 1 0 0 2 2 4 !_i 9 ! 3 33 0 2 3 1 0 3 1 2 :

3.1 Differences Between CYBER and Cray FORTRAN

Originally, the programs were written in FORTRAN 4 on the CYBER. The Cray uses CFT, an extended version of FORTRAN 77, which has some different syntaxes than FORTRAN 4.7 Minor changes in the program statement were necessary, Also, the syntax of the RETURNS statement was different. RETURNS allows a subroutine to return to different locations in the calling routine depending on results in the subroutine.

The CYBER does not distinguish between upper and lower case letters and the programs and data files were all upper case. The Cray uses both cases. The programs initially did not execute correctly on the Cray. The problem was found to be in the way character data was being read from the data file. The terms %LIQ’ and ‘ VAP* were used to designate liquid and vapor feed streams, respectively, but the Cray was reading *L!q’ and *VAp\ Since these are not equivalent on the ( ’ray, the programs were not calculating the stream properties correctly. The programs ran property after changing all letters to lower case in both the data and FORTRAN files. The reason that the last letter being read as lower case has not been determined.

Another surprise occured when comparing runs on the Cray with the vectorization on (Cray-on) and off (Cray-off). In most cases the solution was reached slightly faster, as expected, by Cray-on than by Cray-off. Several runs, however, took l or 2 more iterations

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with the vertorization on resulting in ( ’ray on taking more CIM' time to converge. Kv«*n when both methods converged in the same number of iterations the final sums of squares were slightly (up to 10/7) different. While differences in the internal precision of the v'YBER and die Cray are expected to produce slightly different results after several hundred thousand operations, calculations on the ( ’rav should produce exactly the same result regardless of whether the vectori/ation is on or off. Slight tmmeric.il differences between matrix elements could result in different pivot selections, which could lead to a different number of iterations.

it.2 Total Solution Time

The average solution time for each problem is summarized in Table 2 for both mi merical and analytical derivatives. The total solution times for each run on the CYBER. Cray on, anti Cray-off are listed in fables 8 to Hi. The Cray offers two advantages that allow programs to execute faster. The first is a clock speed that is about 5 times faster than conventional computers. This means that each numerical operation can be performed five times faster m, the Cray. The second advantage is the ability to process large vector loops simultaneously. This is refered to as vectorizatinn. Instead of iterat ing through a DO loop to update each element o f an array or matrix, the Cray can update the entire matrix simultaneously, ( ’ode that is written to take advantage of vectorization can show a speedup of up to ten.1* Speedup is defined, in this case, as the ratio of solution times Cray-off/Crav-on. Highly vectorized programs on the Cray can show a speedup of up 50 when compared to the CYBER due to both the faster clock speed and vectorization.

The column labeled ‘ runs' in Table 2 indicates the number of combinations of equation solvers anti thermodynamic property derivatives used to solve a particular problem. Onlv runs where results were available from both the CYBER and the ( ’rav were included in the tables to make the comparison between machines valid.

Comparison between solution times for CYBER anti Crav off indicate an av rage speedup of only 2.5, varying between l.fi and 3.7 from problem to problem. This speedup is due to the faster clock speed only. One possible explanation for the less than expected increase is the memory structure of the Cray. The Cray memory is organized into 32 ‘ banks.’ Reading from a memory location in a given bank makes that bank unavailable for four clock cycles.1* A ‘ bank conflict* occurs when an unavailable memory location is accessed and the number o f bank conflicts depends on the dimensions o f the array variables. Severe bank conflicts can reduce the efficiency by up to a factor o f four. The variation in average solution times from problem to problem support the possibility of bank conflicts because each problem involves a different number o f components, stages, and variables. Therefore, the arrays will have different dimensions.

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Table 2: Comparison of Average Solution Time

numerical thermodynamic property derivatives

average solution time ( s e e )

problem run CYBER ( ray off ( rav cm speedup H

1 7 G. 12 2.533 2.172 1.02 1; 2 7 23.9 8.649 7.675 1.13 | 3 7 70.4 18.907 18.788 l.oi : 1 \ HO.9 49.601 49.331 1.01 jj 5 4 105. 47.167 43.347 1.09 jj 6 2 53.6 23.016 22.494 1.02 m t 3 125. 47.410 46.084 1.03 j 8 1 51.9 24.954 24.769 1.01 9 2 68.7 32.922 31.104 1.06

average speedup from CYBER to ( ray: 2.4

analytical thermodynamic property derivatives

average solution time ( sec) ]

speedup

jjproblem run , CYBER Cray off Crayon

i 1 8 1.667 1.611 1.03 4 2 38.8 15.610 16.033 r 5 4 49.2 31.385 28.829 1.09 6 ₂ 25.5 11.548 11.442 1.01 1 7 3 61.0 21.042 20.570 1.02 ! 8 l 21.3 7.976 7.922 i.oi i 1 L _ ® _ 2 | 37.7 14.185 13.987 1.01 !

average speedup from CYBER to Cray: 2.4

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Very little speedup or cured when the vert orient ion was turned on. This was expected because the code was not written to take ad vantage o f this feature. Problem five showed the greatest improvement in solution time with the vectorization on and had the largest number of components. While the improvement was not very great, it could indicate that more loops involving the number of components were vectorized than loops that

iterate on the number of plates.

3.3 Nonlinear Equation Solvers

The nonlinear equation solvers are compared in fable 3. Again, verv little reduction

in CPE time v noticed with the vectorization on. Problems I and 9 were run using

each nonlinear equation solver with the same linear equation solver. Onlv deferences in the nonlinear equation solvers accounted for the differences in solution times. All of the runs executed faster when analytical derivatives where used but the Newton-Raphson method, which reevaluates the Jacobian at each iteration, improv'd the most. Significant improvements in the vectorization of the Jacobian and physical property evaluation subroutines could greatly improve the efficiency of this method. Schubert’s method and the Newton-Raphson method solve the problems faster than the other methods.

3.4 Linear Equation Solvers

The linear equation solvers are compared in Tables 4 and 5. The earlier results from the CYBER are not included for clarity but may be found in Table 17. The execution time for all of the linear equation solvers except CBS decreased by a factor of 4 or greater when run on the ( ray. This was significantly larger than the average decrease in total solution time. There were no significant improvements in solution time when the vectorization was turned on except for BBTDPl with a speedup o f 1.14 This routine also executed 10 to 15 times faster on the Cray without vectorization than on the CYBER. This could indicate that the data was structured to avoid bank conflicts. BBTDFl and LU1P9 showed the best overall performance.

3.5 Jacobian and Function Evaluations

Finally, neither the function nor the Jacobian evaluation routines were greatly af fected by vectorization. Table 6 compares the times for the Cray and the CYBER. The speedup was the same for both derivative methods indicating that neither method had a large degree o f vectorization. Numerical derivative calculations require about 5 times more CPU time than analytical calculations but may be easier to vectorize. These results further show that significant improvement in efficiency could be achieved by rewriting the programs to take advantage o f vectorization.

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Table 3: Comparison of Nonlinear Kquation Solvers

Problem 1: 154 equations* 22 stages, 3 components CBS linear equation solver. ( 'hao-Seader correlation

nonlinear thertnodymanic _{total solutir, n time (sec i}

equation solver

proper! v

derivatives CYBKR Cray off t ray-on speedup

N R numerical j 3. TO 1.854 1.841 1.01 M B numerical i 2.123 2.099 1.01 Simp. N R numerical i 2.275 2.245 1.01 Schubert numerical j 1.470 1.445 1.02 N-R analytical j 0.K52 0.044 1.01 M B analytical , 1.103 1.140 1.02 Simp. N-R analytical | 1.310 1.298 1.01 Schubert analytical ] 0.995 0.974 1.02

Problem 9: 2310 equations* 330 stages, 3 components LU1P9 linear equation solver,J^hao-Seader correlation

nonlinear thertnodymanic ] total solution time (sec)

equation property |

solver derivatives JJC'YBKR

’ | 05.7..

Cray-off Cray on speedup

N-R numerical 32.078 31.951 1.00 M B numerical _ii 44.150 43.555 1.01 Simp. N-R numerical i _ 42.069 41.530 1.01 Schubert numerical | 71.0 33.765 30.250 1.12 N-R analytical | 25.9 9.860 9.776 1.01 M-B analytical _i _- 21.940 21.533 1.02 Simp. N-R analytical 19.789 19.513 1.01 Schubert analytical 1 49.4 18.509 18.197 1.02 11

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Table I: Comparison of Linear Equation Solvers

! linear time to so Ive linear system for Crav-on (sec)

I equation problem number

! solver t 2 3 4 5 0 7 8 9 [ " T bs _{. 1:1} _.52 _2.9 _3.1 3.0 1.9 LI'IPD .020 .048 .ioa .00 .71 .38 .32 .31 .25 RANKI ! .028 .19 .73 1.5 1.3 2.7 NSPIV : .021 .051 .74 .45 .37 .32 BBTDF1 ! .017 .057 .117 23 .33 MA28A .020 .9 2.8 BBTF* .028 .052 -GENSOL : .029

-Table 5: Comparison of Vectorizability of Linear Equation Solvers

ir _linear _number _{average CPU time to solve linear system (sec)}

equation of

solver problems CYBER Cray-off Cray-on speedup

CBS | 7 2.63 1.644 1.648 LUIP9 9 1.26 0.315 0.311 1.01 RANKI 6 3.42 1.069 1.082 — NSPIV 2 0.19 0.036 0.036 1.00 BBTDF1 5 2.50 0.171 0.150 1.14 MA28A 3 4.18 1.274 1.240 1.03 BBTF* 2 3.12 0.040 0.040 1.00

Use to compare vectorizability of each linear equ.it ion solver-can’ t compare equation solvers because different

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Table 6: Comparison of Time for Jacobian Evaluations and Function Evaluations

thermodynamic property derivatives

average CPU time for evaluation (sec)

CYBER Cray-off Cray-on speedup

Jacobian numerical 6.01 2.296 2.241 1.02 1 evaluation analytical 1.49 0.515 0.502 1.03 function evaluation both 0.30 0.120 0.117 1.03 1 13

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4 Opportunities for Improved Performance

The results clearly indicate that the programs are not written for efficient execution o f the Cray. The execution time was broken done at the subroutine and the DO loop level by using the TIM ER/TALLY utility. The results are promising in that a large portion of the execution time is spent in a small area of the code. While it is impossible to vectorize every DO loop, vectorization of small portions o f the code could result in large speedups.

4.1 Breakdown of CTO Time by Subroutine

Problem 4 was run using the TIMER/TALLY utility and the results are summarized in Table 7. The highest percentage of time was spent in the linear equation solver subroutines regardless o f the nonlinear equation solver used. The percentage increased, up to 91% for CBS, when analytical thermodynamic derivatives were used because less tune was spent evaluating the Jacobian. Further analysis at the DO loop level indicated that over 80% o f the total solution time was being spent in one DO loop within the subroutine CBSLUl. If this loop can be vectorized to achieve a speedup of 10 within the loop, the overall program will show a speedup of up to 8.

Table 7: Breakdown of Execution Time by Subroutine

Problem 4 run with Cray-on

j nonlinear linear | equation equation | solver solver percentage o f t thermodynamic numerical ime in subroutine >roperty derivatives analytical | N-R NSPIV | Schubert LUIP9 1 NSPIV1 28% VLElO 4% LU1P2 64% SCHUB 2% NSPIV 1 69% JACOB2 6% LUIP2 74% SCHUB 3% ! | N-R CBS 1 CBSLUl 60% VLElO 2% CBSLUl 91% [ JACOB2 1% | Schubert RANKI 1 RANKI 81% VLElO 1% RANKI 90% ; SPK2 1% | M-B BBTDFl l 1 SOLVE2 20% INVERT 5% SOLVE2 23% ’ INVERT 9% .

less than 2% o f all operations were vectorized j|

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number that were scalar. Less than 2% of the operations were vectorised. This is not surprising considering the above results.

1.2 Vectorization o f Linear Kquation Solvers

The possibilitv of vectorizing the CBSLCl subroutine was examined. The portion of code where the majority of the time was being spent is shown in the top of f igure 2, Only the innermost loop can be vectorized and a loop with an IF statement in it can not be vectorized. Also, a loop can not .alter an array value that would be used in a latter iteration if the loop was not vectorized. The innermost loop, |)() .V20 . . . . in the original code was not vectorized for these and other reasons.

The total solution time was reduced bv 2 seconds by rewriting the rode as three separate loops ami using a temporary array. The loop with label 527 in the bottom

portion of Figure 2 is vectorized, hut the loop with label f»>5 still is not. \ more

advanced understanding of the intricacies of vectorization would probably show that the whole loop can he vectorized, but this small modification confirms that the programs can he more efficient.

4.3 Vectorization o f Other Subroutines

The thermodynamic property subroutines are called many times during execution, particularly when the N-R nonlinear equation solver is used. These would be likely candidates for further study. A point of diminishing returns is reached quickly, however, because very little time is spent in most o f the subroutines. Completely vectorizing a

subroutine that only takes \% of the total time will only result in a speedup of about

l.l.

4.4 Alternative Solution Strategies

Perhaps the most promising approach would be an entirely different solution strategy. A more tfllcent algorithm written with vectorization in mind might give better results than patching up programs that were designed for scalar operations. Also, the virtually unlimited memory capacity o f the Cray could be utilized in a way that allows for an efficient vectorized algorithm.

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Figure 2: Revision of Code to Improve Vectorization

er

Subroutine CBSLCl whs revised to improve the vectorization of one 1)0 loop. Ov

80% of the total solution time was spent in this subroutine when problem t

with the Newton-Raphson nonlinear and the CBS linear equation solvers a

derivatives.

* . r <1^)5 sef

Original Code: 1% of all operations were vectorized, solution time I '* -'''

IF (NSPK .EQ. 0) G OTO 540 DO 530 NS = 1. NSPK JJP CLi(JSPK(NS)) DO 520 JJ - JJF, JJL IF (A (JP + JJ) .EQ. 0.) GOTO 520 NOP NOP + 1

IF( A(JJP+JJ) .EQ. 0.) NFILL - NFILL r 1

A(JJP f JJ) A(JJP -J J ) • A(JJP + II)*A(JP -JJ )

520 CONTINUE

530 CONTINUE

n(*7

Modified Code: 4% o f all operations were vectorized, solution time -DO 523 JJ = JJF, JJL 523 ATEMP(JJ) = 0. IF (NSPK .EQ. 0) G OTO 540 DO 530 NS 1. NSPK JJP ^ CLI(JSPK(NS)) IF (A (J J P t ll) .EQ. 0.) GOTO 530 DO 525 JJ JJF, JJL IF (A (JP +JJ) .EQ. 0.) GOTO 525 NOP ^ NOP t- 1

IF (A (JJP+JJ) .EQ. 0.) NFILL = NFILL + l ATEMP(JJ) = A (J J P + Il)*A (J P + J J ) 525 CONTINUE DO 527 JJ = JJF, JJL A(JJP+JJ) -r A (J J P fJ J ) • ATEMP(JJ) 527 ATEMP(JJ) ^ 0. 530 CONTINUE

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5

Acknowledgements

The guidance and patience of Or. M.A. Stadtherr was essential throughout this project. The members of Professor Stadtherr’* research group were also very helpful. Finally, the inspiration and motivational support provided bv Julie Staley was great I v appreciated.

6 Bibliography

The following references were used in this report:

1. Malachowski, Mike., ’ Simultaneous Solution of Multistaged Interlinked

Separators,' Ph.D. Thesis, I niv. of Illinois. Frbana. Illinois (1983)

2. Ibid, pp. 37-41. 3. Ibid, p. 28.

I. Ibid, p. 43. 5. Ibid, pp. 86-95. 6. Ibid, pp. 59-86.

7. ('ray Research Inc., ’ Cray 1 and Cray X-MP Computer Systems. | ()R- TRAN ((-F T ) Reference Manual,' revision J 02 (1984), p. I I.

8. Natio i il Center for Supercornputing Applications, T s e r’s Ciuide,' version 1.0, (January, 1986), p. 10-7.

9. Ibid, p. 10 9.

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r«* C » t Ci l* — . w l tM ' £ R C O fl C 'C tt X* 4 t~,

A

Appendices

A .I Sample Program O utput - 1 •: J ■,

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A.3 Subroutine Structure of the Programs

The program itself is SIM25 and is stored in the file »im290. The program performs some initialization and then calls the subroutine MAIN located in the file lcsim28i. Main then calls the following subroutines:

routine: tile: routine: file: routine: file:

DATA Ic28x VSCALK lc28x PRINTF Ic28x

SC'HUBO Ic28x Sf'HFB lc28x BH Ic28x

BHA lr28x SIMP lc28x MARWIL Ic28g5

JACOBS Ic28b JACOB3 Ic28b JACOBI Ir28b

PRINT1 le28x STORE3 Ic28 x ST0RE2 Ic28x

TALLY1 lcoriler TALLY3 Ic28x RSf'ALE Ic28x

FSC’ALE Ic28x JACOB lc28b BBTFO Ic28gfi

CiENSOL Ic28g5 RANK! tcrank CBSLU Iccbslu

BBTF Ic28g4 MA28AA Ic28x M28*BB lc28c

LUIP9 lclu 1 p9 NSPIVC lcnspivl SOLVEO Ic28g6

STORE4 Ic28x BBTFP Ic28x S0LVE3 Ic28g4

MA28CC Ic28x SOLVECi Ic28g5 STEP Ir28x

PRINTR Ic28x RESIDi lc28i RESIDO l<-28i

The thermodynamic function routines for C’hao-Seader correlation with numerical deriva tives are stored in Ic21d4 and with analytical derivatives in !c21d. The Uniquac routines are stored in lcuniq. Most o f the subroutines listed above call other subroutines located in the same hie. The modified version of the subroutine CBSLUl is stored in vcbs. Data files are stored as lcdXXX where XXX denotes the problem and linear equation solver used.

A.4 Total Solution Time o f Each Run

The following tables list the total solution time for each run on the Cray. Data from the CYBER is included where available.

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Table 8: Total Solution Time for Each Run for Problem l

Chao-Sender correlation with numerical derivatives

I nonlinear linear f total OPT time (sec)

equat ion equation _!

solver solver CYBER

4.00 Crayon ('ray-off J Schubert BHTDFl 3.686 N-R GENSOL 10.5 3.568 3.577 i Schuli*; ? MA28A 8.38 1.436 1.474 | N R RANK1 3.79 1.808 1.816 | N R CBS 3.70 1.841 1.854 j N R LCIP9 3.76 1.763 1.771 j N-R NSPIV — 1.769 1.783 ! MARWIL CiENSOL 8.70 3.202 3.220 M B RANKI — 2.069 2.096 M B LUIP9 — 2.013 2.036 M B CBS — 2.245 2.275 Simp N R CBS - 2.245 2.275 Schubert CBS 2.96 1.445 1.476

Chao-Sender correlation with analytical derivatives

nonlinear linear total CPU time (sec)

equation equation

solver solver CYBER Cray-on Cray-off

Schubert BBTDFl — 4.665 5.050 N-R GENSOL — 2.379 2.377 Schubert MA28A — 0.960 0.994 N-R RANKI — 0.614 0.618 N-R CBS — 0.644 0.652 N-R LUIP9 — 0.570 0.575 N-R NSPIV — 0.576 0.581 MARWIL GENSOL — 2.482 2.490 M B RANKI ...— 1.116 1.140 M-B LUIP9 — _1.039 _1.073 M-B CBS — * 1.140 1.163 Simp N-R CBS — 1.298 1.316 Schubert CBS — 0.974 0.995

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Table 9: Total Solution Time for Each Run for Problem 2

Uniquac correlation with numerical derivatives

! nonlinear linear total CPU time (sec)

| equation equation | _i

■ solver solver ! CYBER Cray on Crav-off

| Schubert BBTDFl f 18.4 13.291 16.150 ! | Schubert MA28A 53.7 16.695 18.381 Simp N-R BBTF* 1 13.2 4.220 4.115 N R CBS 13.8 3.077 2.970 ! Schubert LIUP9 13.7 2.828 2.825 j N R NSP1V !₁ 2.716 2.605 MARWII. (JENSOL 34.9 8.971 8.859 | | Schubert RANK! ; 19.3 4.640 4.641 j

Uniquac correlation with numerical derivatives nonlinear equation solver linear equation solver

total CPU time (sec)

CYBER Crayon Cray-off

Schubert LU1P9 40.4 8.704 9.389 N R CBS 47.6 12.101 11.736 N-R RANKI 60.0 13.829 13.713 MARWIL _GENSOL _197.2 _50.866 _51.030 Simp N-R BBTF* 48.4 16.867 16.412 Simp N-R BBTDFl 37.2 14.219 15.986 Sit° r N-R MA28A 61.7 14.927 14.984

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Table 11: Total Solution lime for Each Run for Problem 4

Chao*Header correlation with numerical derivatives

nonlinear linear total CPC time (sec)

equation equation

solver solver ^ CYBER

63.5 Cray-on Cray-off ) Schubert L I I P 9 18.741 17.980 1 N-R CBS 55.1 23.877 24.408 ;i Schubert RANKI 103.4 31.433 29.473 jl Schubert BBTDFl 101.5 123.273 126.633 |

Chao-Sender correlation with analytical derivatives nonlinear equation solver linear equation solver

total CPU time (sec)

CYBER C rayon Cray-off

Schubert LU1P9 109.7 42.171 43.888

N-R CBS 29.3 15.995 15.865

Schubert RANKI — 23.737 24.992

(28)

Chao-Seader correlation with numerical derivatives

equation solver

equation

solver i CYBER Cray-on

! i f'rav-off N-R CBS ' 1 0 9 . 7 12.171 43.888 N R NSPIV 14.290 15.381 N R LU1P9 89.3 22.832 2-1.525 J N-R RANKI 107.1 27.366 29.161 N R BBTDF1 115.2 81.019 91.092 j

Chao-Sender correlation with analytical derivatives nonlinear equation solver linear equation solver

total CPU time (see)

CYBER Cray on Cray-off |

N-R CBS 53.4 27.768 28.095 !

N-R NSPIV - _5.303 _5.478 _!

N-R LUIP9 33.2 8.338 8.619

N-R RANKI 50.9 13.019 13.580

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Table 13: lota! Solution Time for Each Hun for Problem b

Chao-Sender correlation with numerical derivatives

equation equation

solver solver CYBER Cray-on Cray-off

N R NSP!V 15.881 16.402

N R CBS 62.8 29.459 29.952

N R LU1P9 44.4 15.528 16.079

( hao-Seader correlation with analytical derivatives

nonlinear linear total CPU time i(sec)

equal ion solver

equation

solver CYBER ( ’ray on (>ay-of

N R NSPIV 4.829 5.003

N-R CBS 34.5 18.379 18.428

N-R LC1P9 16.4 4.508 4.670

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Fablf 11: I'otal Solution Time for Each Hun for Problem

Chao Header correlation with numerical derivatives

nonlinear linear ! total CPC time ( sec)

equation equation

solver solver j CYBER ( ’ray-on Cray-off ^

N R NSPIV 1I 34.952 35,957 i

N R CBS |! 112.5 47.909 48.997

N R o 'i p ? ) ! 95.6 34.469 35.724

N-R RANK! 1 167.1 55.874 57.510 |

Chao-Header correlation with analytical derivatives

equat ion equation

solver solver CYBER Crayon Cray-off j.

N R NSPIV - 9.327 9.642 “

N R C’BS 47.9 22.345 22.721 jj

N-R LC1P9 31.7 8.929 9.279 jj

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Tablt* 15: Total Solution Time for Each Run for Problem 8

Chao-Header correlation with numerical derivatives

equation equation

j solver solver CYBER Cray on Cray off i

f N R ' SPIV ' 24.836 24.97'! \

J N R M ' IPO 5L9 24.769 24.954 |

Chao-Seader correlation with analytical derivatives

equation equation i

solver solver CYBER Cray-on Crav-off

N R NSPIV 7.1)82 8.070 1

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Table 16: Total Solution Time for Bach Run for Problem 0

Chao-Header correlation with numerical derivatives

| nonlinear linear total CPU time (sec)

equation

1 solver

equation

solver CYBER Cray-on ( ’ray-off

f N R LU1P9 65.7 31.951 32.078

| Simp N- R LU1P9 - - 11.530 42.069

| M B MUP9 - 43.555 44.150

| Schubert LIMP*) 71.6 30.256 33.765

( ’ hao* Seader c or relation with analytical der i vat i ves

equation equation

solver solver CYBER Crayon Cray-off

N R LU1P9 25.9 9.776 9.860

Simp N-R H U P9 — 19.513 19.789

M-B LU1P9 21.553 21.940

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A.5 Performance of Linear Lt|uation Solvers on the i ' \ B L R

The results from reference I are aiven in the I able below.

Table 17: Solution Times for Linear Reflation Solvers on the t ’Y B K R

LINEAR SOLUTION LINEAR SOLUTION

EQUATION TIME EQUATION TIME

SOLVER (SECONDS) SOLVER (SECONDS)

PROBLEM 1 CBS .07 PROBLEM 2 LU1P9 .21

EQUATIONS LU1P9 .09 A05 EQUATIONS CBS .21

22 STAGES RANK I . 10 A5 STAGES NSPIV .27

3 COMPONENTS NSPIV .11 A COMPONENTS BBTDF2 .56

BBTDF1 .16 BBTDF1 .59 BBTDF2 . 17 RANKI .73 BBTF* .18 THOMAS .8* THOMAS .26 BBTF* .97 BBTF .36 BBTF 1.A7 MA28A .67 MA28A 3.08

PROBLEM 3 LU1P9 . A6 PROBLEM A LU1P9 2 . A A

810 EQUATIONS CBS .71 777 EQUATIONS BBTDF2 3.18

90 STAGES BBTDF2 1.13 37 STAGES BBTDF1 A.A2

A COMPONENTS BBT0F1 1.21 10 COMPONENTS RANKI A. A]

THOMAS 1.6A CBS A.73

RANKI 2.25 THOMAS 10.23

BBTF* 5.26

BBTF 6.07

MA28A 8.78

PROBLEM 5 LU1P9 2.81 PROBLEM 6 LU1P9 1.A8

775 EQUATIONS BBTDF2 3*62 1235 EQUATIONS BBTDF2 A.23

25 STAGES RANKI A.86 95 STAGES CBS A.55

15 COMPONENTS BBT0F1 6.10 6 COMPONENTS BBTBF1 5.01

CBS 6.11 THOMAS 6.12

RANKI 6.91

PROBLEM 7 LU1P9 1.3* PROBLEM 8 LU1P9 1.3*

1560 EQUATIONS CBS 2.0A 2100 EQUATIONS B8TDF1 2.19

120 STAGES BBTDF2 3.A5 300 STAGES BBTBF2 2.35

6 COMPONENTS BBTDF1 A.03 3 COMPONENTS

RANKI 8.1A PROBLEM 9 LU1P9 1.13 2310 EQUATIONS BBTDF1 2.A2 330 STAGES THOMAS 3.01 3 COMPONENTS

SI

Simultaneous solution of systems of interlinked multistaged separators using the Cray X-MP supercomputer

UNIVERSITY OF ILLINOIS

THIS IS TO CERTIFY THAT THE THESIS PREPARED UNDER MY SUPERVISION BY

Contents

List of Figures

Introduction

Simultaneous Solution of Interlinked Distillation Columns

n • place number a • component nuaber

3 Comparison of Performance on the CYBER and the Cray

CBS | 7 2.63 1.644 1.648 LUIP9 9 1.26 0.315 0.311 1.01 RANKI 6 3.42 1.069 1.082 — NSPIV 2 0.19 0.036 0.036 1.00 BBTDF1 5 2.50 0.171 0.150 1.14 MA28A 3 4.18 1.274 1.240 1.03 BBTF* 2 3.12 0.040 0.040 1.00

4 Opportunities for Improved Performance

* . r <1^)5 sef

Acknowledgements

Appendices

N-R CBS 29.3 15.995 15.865

N-R CBS 53.4 27.768 28.095 !

N R NSPIV 4.829 5.003

N R NSPIV - 9.327 9.642 “

N R NSPIV 7.1)82 8.070 1

A.5 Performance of Linear Lt|uation Solvers on the i ' \ B L R

BBTDF1 .16 BBTDF1 .59 BBTDF2 . 17 RANKI .73 BBTF* .18 THOMAS .8* THOMAS .26 BBTF* .97 BBTF .36 BBTF 1.A7 MA28A .67 MA28A 3.08

120 STAGES BBTDF2 3.A5 300 STAGES BBTBF2 2.35