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i^IEli ORNL/TM-7490

A Mathematical Model for Liquid Flow Transients in a

Rotary Dissolver

B. E. Lewis F. E. Weber

DISCLAIMER

This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency Thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.

DISCLAIMER

Portions of this document may be illegible in electronic image products. Images are produced from the best available original document.

Printed in the United States of America. Available from the Department of Energy

Technical Information Center P.O. Box 62, Oak Ridge, Tennessee 37830

This report was prepared as an account of work sponsored by an agency of the UnitedStatesGovernment. NeithertheUnitedStatesGovernment nor any agency thereof. nor any of their ernptoyecw, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or othemise, does not necessarily constitute or imply it8 endorsement, recommendation, or favoring by the United StatesGovernment or any agency thereof. The views and opinions of authors expreseed herein do not necessarily state or refleet those of theunited StatesGwernment or any agency thereof.

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ORNL/TM-7490 Dist. Category UC-86

Contract No. W-7405-«ng-26

CONSOLIDATED FUEL REPROCESSING PROGRAM

A MATHEMATICAL MODEL FOR LIQUID FLOW TRANSIENTS IN A ROTARY DISSOLVER

B. E. Lewis and P. E. Weber Chemical Technology Division

Date Publ ished - October 1980

NOTICE- This document contams information of a preliminary nature. It is subject to revision or correction and therefore does not represent a final report.

OAK RIDGE NATIONAL LABORATORY Oak Ridge, Tennessee 37830

operated by UNION CARBIDE CORPORATION

for the DEPARTMENT OF ENERGY

f:

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»

CONTENTS

HIGHLIGHTS v

1. INTRODUCTION 1

2. MATHEMATICAL TREATMENT 3

3. EXPERIMENTAL REQUIREMENTS 7

4. RESULTS 9

5. DISCUSSION OF RESULTS 11

6. CONCLUSIONS 13

7. ACKNOWLEDGMENTS 15

8. REFERENCES 17

APPENDIX - FORTRAN Program Listing A-1

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HIGHLIGHTS

A model describing the liquid outlet response to perturbations in the flow t o a compart- mented rotary dissolver has been developed. The model incorporates stagewise differential material balances coupled with the general equation for flow over a weir to calculate acid con- centrations and liquid volumes in each stage. Data were taken from step-change flow experi- ments conducted on a 0.5-t/d rotary dissolver. The predicted response of the model was in good agreement with the data from the dissolver experiments. All constantsin the model were obtained by independent tests. The model appears to be applicable over a wide range of dissolver operating conditions; however, temperature fluctuations and the presence of solids were not addressed.

1 . INTRODUCTION

The prediction of the effects of perturbations in the flow and concentration of liquid feed to a continuous rotary dissolver is important in several aspects of dissolver process design and control. The initial startup of a continuous dissolver requires that a certain flow and concen- tration of acid be established before any testing or dissolving can be accomplished effectively. Initiation of the dissolution reactions before the proper acid concentration or flow has been established can lead to acid deficient situations. Also, during normal operations, acid flow and concentration perturbations may cause problems leading to the production of off-specification dissolver products. These problems could be minimized or possibly avoided with an early indication of the effect of flow changes on the acid concentration in each stage of the dissolver. Therefore, a computer code has been developed to model the acid flow and concentration in each stage of the compartmented rotary dissolver for the nitric acid-water system. The code also gives insight into a solution for the more complex unsteady-state material balance model with chemical reaction. This report is concerned with the development and presentation of the results from the compute; simulation of the flow and concentration of the nitric acid-water system within the present 0.5-t/d continuous rotary dissolver.

The dissolver is a compartmented device consisting of a 0.75-m-dim drum enclosed in a rectangular shroud. An isometric view of the dissolver drum is shown in Fig. 1. The dnim is - 2.4-m long and has nine separate stages or compartments. Liquid moves through the dissolver by flowing through the slots in the walls (shown in the cutaway in Fig. 1) separating each stage. Eight of the stages contain nitric acid and one contains only a rinse water-acid solution. The rinse stage is not considered in this analysis. Each stage is - 26-cm long and maintains a nominal liquid inventory of - 5 Q. Each stage contains a single baffle to provide agitation and solids transfer as the drum is rotated; therefore, complete mixing is assumed.

I t has been shown in previous experiments on a single-dissolving stage rotary dissolver that the acid concentration in the outlet stream can be approximated by a simple stirred tanks-in- series model.' Several variations of stirred tank models were used (with varying degrees of success) in an effort to model the 0.5 t/d dissolver. Initially, the simple single-parameter, tanks-in-series model with constant flow rate and tank volume was used.2 This model, which is similar to the single-stage model, predicted concentrations that preceded the actual data by as much as - 10 min. A variation of this model involving the introduction of a time-lag term into the concentration equation was also used. This alternate model predicted the experi- mental data very well during the first half of the run but lagged during the latter half. The two models appeared to have deviated from the actual data because of the assumptions of constant flow rates and stage volumes. In order to take into account the variations in flow and volume, it was necessary t o start with the differential material balance equation. A schematic diagram showing the flow through the dissolver is presented in Fig. 2. The overflow from the ith tank becomes the feed to the (i-1)th tank. The fluid dynamics of the liquid were modeled as flow over a weir, since liquid flows through the slots in the stage walls without completely filling the slots.

I

O R N L DWG. 79-18915

\DRIP L IP Fk. 1. Cutaway view of Ll~e 0.5-t/d crlmpsnmenlen toraly diedirer drum.

ORNL-DWG 80-6863

St.:ic~ n T f! . 1

/ A ?

Stage i

Volume = Vi

C u r ~ c c r ~ t r a l i u r ~ ; Ci

Flow out = Fi St.a~;e 1

Density = pi

Fig. 2. Schematic representations of model parameters.

2. MATHEMATICAL TREATMENT

Basically, there are four variables in this problem: the volumetric flow rate out of stage i,(Fi); the concentration of acid in stage i,(Ci); the liquid volume of stage i,( Vi); and the liquid density in stage i,(pi). Thus, four equations are necessary to solve for unknowns.

Since we are dealing with a water-acid system, two material balances can be written:

d Acid: Fi+ 1Ci+ - F.C. = - (CiVi)

d t (1)

d Total: pi+lFi+l - piFi = ; (piVi) (2)

A third equation that could be applied was an empirical relation between concentration and density for nitric acid:.

The final equation was from the fluid mechanics of the problem. The general equation describing flow over a weir is given by

where F is the volumetric flow rate, H is the height of fluid over the weir, and K and P are empirical constants. If Vo is the volume of stage i up to the bottom of the weir, and As is the horizontal surface area of stage i, then the following relation between the liquid volume and 111e l~eiglit over thc wcir holds:

Vi = Vo + ASH . ( 5 )

combining Eqs. (4) and (9,

Equations ( I ) , (2), (3), and (6) comprise a closed system of simultaneous nonlinear differential equations, with initial conditions at t = 0: Ci = Ci,, and pi = pi,, for i = 1,8.

Before the above set of equations could be solved, the functional relationship between pi and Ci was required. For nitric acid solutions, the density is approximately linear with concentration up to 44% acid. Thus, Eq. (3) could be written as:

where the constants a and b were determined by linear regression.

Equations ( 1) and (2) were solved by a fully implicit finite difference technique. Earlier attempts with an explicit technique led to stability problems. The advantage of unconditional stability with a fully implicit technique was felt to outweigh the increase in computation time.

After application of the finite difference scheme with a time step of At and extensive algebraic manipulation, Eqs. ( I ) , (2 ) , (3) , and (6) were written as:

c . = "Hi

Ji - bHi '

where Ei and G depend only on the.geometric properties of the dissolver and are given by:

and

Hi and Ji depend on the state of the fluid both in the upstream stage and at the previous time step and can be written:

and

where the superscripts indicate the previous time. step. Equations (8) through ( 1 1) and the associated relationships in Eqs. ( 12) through ( 15)

were required t o solve the material balance for each of the eight stages in the 0.5-t/d dissolver. The solution order (Fig. 3) was from stage 8 (the acid-entry stage) to stage. 1 (the liquid-outlet stage). Therefore, for stage i, all the information t o solve for Ci, pi, Fi, and Vi is known either from previous calculations or from the initial conditions. Because of the nonlinearity of Eq. ( l o ) , an iterative solution for Fi was required. The secant method was used since simpler methods, such as successive substitutions, diverge.

A listing of the FORTRAN computer program for the model is given in the Appendix.

ORNL DWG. 80-6862

,-HNO.

PRODUCT

Fig. 3. Liquid flow diagram for the compartmented dissolver.

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3. EXPERIMENTAL REQUIREMENTS

Basically, three different types of data were required to solve the model. The first type was the geometric volume and area of each stage. The volume of each stage was found by measuring the amount of water necessary to fill a stage t o the bottom of the weir. The area was calculated from the known dimensions of the dissolver.

The data of the second type were the constants in the density-concentration correlation. The data for density vs weight fraction of HNO, and a linear fit of this data are presented in Fig. 4. For weight fractions less than 0.44, the constants a and b in Eq. (7) were determined by linear regression to be 1001.2 and 0.489 16 g/Q respectively.

The final required data were the weir equation constants K and P b f ~ ~ . (4). By measuring the total mass collected for various step changes in flow, the constants K and P were calculated to be 0.9888 cm and 2.83 respectively. The value of 2.83 for the exponent P compares reason- ably well to the reported values of 1.8 t o 2.5 for various shaped w e k 3

ORNL OWG. 80- 10396

~ ' ~ " I ' ~ " ~ " " I " ~ ~ I ~ ~ ~ - I ~ ~ ~ '

Solid Line - Data Fro111 ICT

Dashed Line - Least Squares Fit

// / ;

-

- ,/'

,/'

/" -

/' /'>

: /' 1 . . , . 1 . . . . 1 . . . . , . . . . 1 . . . . , . . . .

0.0, 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Weight Fraction

Fig. 4. Densityconcentration correlation for the nitric acid-water system.

Acid concentration data were taken during an experiment in which a series of step changes in the acid and water feed rates were made. An outline of the operating conditions for this test is presented in Fig. 5. This experiment was the most severe test of the final model, covering most of the step changes in liquid flows which would be anticipated in normal operation of the dissolver (i.e., startup, small perturbations in acid flow, and shutdown).

ORNL DWG. 80-6861Rl

TIME (ii~in)

Fig. 5 , Expcrirncntml tnrt pr~e'edom for model verification.

4. RESULTS

The predictions of the model, along with the experimental results from the step change tests, are presented in Fig. 6. Also shown is a summary of actual run conditions. The predicted response and the experimentally determined response agree very well for all cases. The devia- tion at 160 mi11 into the test can be explained by a known failure in the titration equipment used to collect the data. The important feature of the model is that no adjustable parameters were necessary to achieve this close agreement between experimental and predicted values. All constants in the model were chosen by separate independent tests.

H N O TI ME F L O W CON^ (mid (kg/h) (M)

initial 9.84 0.0

0 36.79 7.2

180 46.83 7.9

325 41.67 7.6

415 9.84 0.0

- exper hr~arx La1 data

I I 1 I I I I I L I I 1 I 1 1 I I I I 1 1 1 1 1 1 1 1 1 1 1 ~ 1 1 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ 1

0.0 50.0 100.0 150.0 200.0 250.0 300.0 350.0 400.0 450.0 500.0 550.0 600.0 650.0 Elapsed Time (min)

Fig. 6. Comparison of model concentration predictions with experimental data.

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5 . DISCUSSION OF RESULTS

Several limiting assumptions have gone into the development of the model. A linear relationship between the nitric acid concentration and solution density was used and is only applicable up to an acid concentration of 44 wt %. Caution should be exercised in applying the model outside this concentration range, since deviations from linearity are assured with ' the nitric acid-water system and can be seen in Fig. 4. The success of the model is highly dependent on the accuracy of the density correlation.

The experimentally determined dissolver responses to step changes of 5 to 35 kg/h were compared to the predicted responses. Agreement was good except for the largest step change of 35 kglh. In this case, the predicted response led the measured response by 3 to 4 min. This deviation was well within acceptable limits.

Temperature fluctuations have not been addressed in this model. The model was developed assuming a constant temperature of 25°C. Including a correlation for the nitric acid-water density, temperature, and acid concentration would allow the inclusion of temperature changes, which could easily be incorporated into the model if needed.

The presence of sheared material in the dissolver significantly complicates the modeling of the process. Complications arise primarily because of backmixing during solids transfer, deviations from the assumption of ideal mixing, and depletion of acid during reaction. The effects from sheared materials were not considered; therefore, it is not advisable to use this model to describe a system with sheared material. Only a very rough prediction of performance can be expected from the model for such a system.

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6. CONCLUSIONS

A model has been developed which accurately predicts the outlet response t o perturbations in the feed acid flow and concentration for a 0.5-t/d rotary dissolver. This model will be useful in predicting start-up times and will provide a better understanding of how fluids flow through the dissolver. The model is limited t,o use in the two-component nitric acid-water system (at 25OC) with no solids present.

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7. ACKNOWLEDGMENTS

The authors are grateful t o M. E. Baldwin and R. J . Shannon of the dissolver operating staff for conducting the experiments and taking the requested data.

We also apprcciate the helpful suggestions of J. Q. Kirkman and others associated with the dissolution effort.

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8. REFERENCES

1. W. D. Burch et al, Advanced Fuel Recycle Program Progress Report for Period April 1 to June 30, 1977,ORNL/TM-5993 (September 1977), p. 3-3.

2. 0 . Levenspiel, Chemical Reaction Engineering, 2nd ed., Wiley, New York ( 1972), pp. 290-9 1.

3. Perry 5 Chemical Engineers Handbook, 4th ed., ~c~raw- ill, New York ( 1969), p. 18-9.

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APPENDIX

FORTRAN Program Listing

T H I S PR3GRAM C A L C U L A T E S T H E E P P E C T ON T H E D I S S O L V E R O F A S T E P CHAHG?? I N E I T H E F . THE FLDW, C O N C E N T R A T I O N , OR BOTH.

T H E VARIADLES: A , R , ' C ( T ) , E , ? ( I ) , G, H , J, R H O ( I ) , AND V ( 1 ) ARE AS D E F I N E D I N T H E D E R I V A T I O N O F THE O R I G I N A L EQUATIONS.

I I S TIiE NUMBER 3 P THE S T A G E , WHERE F E E D E N T E R S I N STAGE d RND E X I T S I N S T A G E 1.

COLD ( I ) I S T H E P R E V I 3 U S T I R E S T E P CONCENTRATION. R H O O L D ( 1 ) I S THE P R E V I O U S T I M E S T E P D E N S I T Y . VOLD (I) I S T H E P R E V I n U S T I M E S T E P VOLrJYE. PT.LG1, P C A G Z . AYD I P L A S ARE YRROR P R O C E S S I N G FLAGS. E P S I S THE CONVERGENCE cRTTERTDN. IMAX T S THC !lAXIM!I* NUMBER OF T I H E S T E P S . D E L T A T I S T H E T I N E 1 NCAERENT, AND RDT I S I T S R E C I P R O C A L . P I S THE E X Q n H E N T I N THE WEIR BQrJATION, AND R P I S I T S R E C I P R O C A L . CUAX I S TIIE N\XI90!! CHANGE I N C 2 N C E N T R R r I D N BETWEEN T I R E S T E P S . KO, X I , X 2 , PO, F 1 , A N 3 F 2 A n y SCRATCH V A R I A B L E S USED I A C A L C U L A T I N G P .

U N I T S : C ( 1 ) (=) G/L F ( I ) (=) L/HE RHO ( I ) (=) S/L V ( I ) (=) L COLD ( I ) (=) G / L RIiOOLD ( I ) (= ) G / L VOLD ( I ) (=) L D E L T A T ( = ) HR P (= ) C/L B ( = ) NO U N I T S ":=I L c. (= ) L ? (=) 80 ITNITS

CAUTION: DO N n T ATTENPI ' 'YO USE T H I 3 P R 3 G R A l 4 AS WRITTEN 8 3 R N I T 9 I C A C I D C n N C R N T P A T I n N S ABOVE 44%!

9 Y A L J INT!?GER F L A G 1 , F L A G 2 nTMEWSIClN P H n (9) ,COLD (9) ,Vc?LD (9 ) , R H 0 3 L D (9) ,

,BX ( 3 n q ,Y ( 3 0 0 ) , XI 1 (330) , Y I I (300) COMMON E , R D T , H ,C (9) ,%, RP.1 , P (91 , E P S , V ( 9 ) D R G I N I N I T I A L I Z A T I O N OF THE "ROBLELM S E T THE CONVERGFNCE C R I T E R I O H E P S = 1. E- 3 FLAGZ=I)

K P T = O TCOIlNT=O. 0

S E T THI? U A X T Y U M NLIYBBR O F I T S R A T I O N S IMAX=2r )OO

C ZERO ARRAYS DO 1 6 0 K = 1 , 3 0 0 X (K) =O. r) Y (K) = 0 . 0 X 1 1 (K) =O. 0 Y l l ( K ) =O.O

1 6 0 C O N T I N I J E C READ I N EXPEaTMENTAL DATA

K 1 = 1 2 5 R E A D ( 5 , 3 1 0 ) ( ~ 1 1 ( I ) , Y 1 1 ( I ) , I = l , K l )

3 1 0 FORYAT ( P 1 0 . 2 , P l O . 3 ) C S E T THE T I N E INCREMENT

9 R L T A T = 1 . 5 - 2 RDT= I . /DELTA?

C S E T THE: PI!YSIC AL PAPAMETERS DF I'HE . D I S S 3 L V E R A = 1 0 0 1 . 2 R = . 4 8 9 1 b g=u, 9 3 ~ 0 . 2 0 1 9 4 1 P = 2 . 7 R P = l . / P T I H F = . 3

C S E T UP ! IEADIN3S FOR DATA 3 U T P U T U P I T E ( 6 , 1 5 )

1 5 F O R M . 4 T ( 1 X 8 ' T I ! l Y ' , 5 X 8 ' C ( 1 ) ' . 6 X , ' 2 ( 2 ) ' , 6 X n ' C ( 3 ) ' , 6 X , ' C ( U ) ' , 6 X , % ' C ( 5 ) ' , 6 X 8 ' C ( 6 ) ' , 6 X , ' C ( 7 ) ' , 6 X 8 ' Z ( F 1 ) ' , 6 K , ' C ( 9 ) ' , 6 X , ' F ( 1 ) ' )

C SET THE S T A T E 3 P THE D I S S O L V F R B E P 3 R E T H E S T E P CHANGE DO 1 I 1 = 1 , 8 I = R - T I + 1 c o L n (I) =0. o RHOOLD ( I ) = A+'D*C3LD ( I )

C F'(1) I S TN KG/ER F ( T ) = 9 . 8 4 F ( 1 ) = F ( I ) * 1. 9 3 / R H r ) O L n ( I )

1 VOLD ( I ) =F:*.G*F (I) **RP C SET TYE VALg'S OF T H E S T E P CHANGE

C ( 9 ) = U 5 3 . 6 9 RIIO ( 9 ) = A + B*C (9)

C P ( 9 ) I S I N KG/HP F ( Q ) = 3 6 , 7 9 F (9) = F ( 9 ) * 1. E3 /RHO (Q)

C CALCIILATF T H E NEW C ( I ) , P ( I ) . P H O ( I ) 4ND V (I) FOR C EACH STAGE U S I l C r A E P R F V I O I J S T I Y E S T E P VALUTS.

5 DO 1 0 I 1 = 1 , 9 I = R - I l + l

H = P ( I + 1 ) * C ( I + 1 ) *CDLD ( I ) * V O L D ( I ) *ROT J = R H n ( I + 1 ) * P ( 1 4 1 ) + R H 0 3 L D ( I ) * V o L D ( I ) *RDT D E N O N = l . / ( J - S * H ) C ( I ) =.4*H*DENC# RHO ( I ) =A*J*DENOIY

C CALCULAiE NEW P ( 1 ) AND V ( I ) ONLY IF T H E D I S S O L V E R I S NOT C F L U I D HYDRAU LTCALLY AT STEADY S T A T E

I F ( I B S ( P ( q ) - E ( 1 ) ) . S T . E P S ) CALL SECANT 10 CONTTNTJE

C AFTER N E W VALUPS APE CALCIIEATBU, 1NCKENENT T I M E A N D C O N T I N U E T I ! l E = T I V E t D E L T A T

C S E T UP S T E P CHANGE VALUES TN I N = T I n E * 6 0 . I P (TNIN.GE. 1 SO. AND.TMIN.LT.325) GO T 3 1 1 0 I F (TNIN-SE. 325 .AND.TnIN.LT.415) GO T 3 1 2 0 IP (TNIN.GE. 4 1 5 ) G 3 TO 1 4 0 G O T O 1 3 0

1 4 0 I F (THIN.ST. h 3 0 ) GO TO 1 5 0 C ( 9 ) =O.O F (9 ) =9.91r RHO (9) =A+B*C (9) GO TO 1 3 0

1 1 0 F (9 ) = 4 6 . 8 3 C ( 9 ) = 4 9 7 . 7 RHO ( 9 ) =A+B*C (9) GO TO 1 3 0

1 2 0 P ( 9 ) = 4 1 . 6 7 C (9) =U78. A RHO(9) =A+D*C (9 )

1 3 0 COHTTNIJE C CALCULATE HAXINnM CHANGE TN CONCENTRATI3NS BETWEEN T I H E C S T E P S . I P CONCENTRATIONS HAVE CONVERGED, STOP CALCULATIONS. C I F NOT, CONTINUE UNTIL THE NAXIflUfl NnMBER OF T I N E S T E P S C HAS PASSED

TCOUNT=TCO!JNT+ DELr AT CRAX=O.O FLAG2=FLAG2+ 1

DO h I 1 = 1 , . 8 I=F \ -114 1

6 CYAX=A!lAXl (C!4AX,4BS (C ( I ) -COLD ( I ) ) 1 I F (CHAX. LT. E P S ) C 3 TO 1 5 0 I F (PLAG2.EQ. IH.4X) CALL ERROR(PLAG2, I Y A X I

C RESET THE PREVIOIIS T I Y E S T E P VALUES F3 PRESENT VALUES D O 7 I 1 = 1 , 8

1 = 9 - 7 1 4 1 COLD ( I ) =C ( I ) VOLD(1) = V ( I )

7 RHOOLD (I) = R ! i O ( I ) C P R I N T O U T T#P. CAI.CULkTElJ VALUES.

r B I T E ( 6 , 1 0 0 ) T I M E , ( C ( 1 ) , 1 = 1 , 9 ) , F ( l ) 1 0 0 PORRAT(1 X , F 8 . 0 , 9 (2X8F8 .U) , 2 X , P 9 . U)

I F (TCOONT. LT .0 .033 ) 6 0 TO 5 TCO!JyT=r), 0 KPT=KPT+ 1 X (KPT) = T n I N Y (KPT) =C ( 1 ) / 6 3 . 0 1 2 S 7

c ~ n n ~ R A C K P O R N E X T TIME S T E P GOTO 5

1 5 0 CALL PLOTID(X,Y,KPT,Xll,Y11,Kl) S T O P

E N D

T H I S S U H H O l l r I N E P R O C E S S E S ANY S R R O R S WHICH RAY OCCUR DURING THE C A L C U L A T I O N S AND P R I N T S A N . A F P R O P R I ATE ERROR MESSAGE. SUP,ROUTINE ERRDP ( I FLAG* I V R X ) I F ( 1 F L A G . EQ. 1 5 0 0 ) GO TO 1 0 I F ( I P L A G . E Q . - 1 0 ) G O T O 2 0 r F ( I P L A G . E Q . - 1 1 ) GO TO 30 I F ( I P L A G . E Q . I N A X ) GO TO 4 0

GO TO 60 U!?ITE ( 6 . 1 5 ) PORHAT ( ' S E C A N T PIETIIOD T S 'YOT C O N V E R G I N G ' ) GO TO SO W R T T F . ( h , 2 5 ) FORVAT ( I S E C A N T FETHOD CANNOT F I N D Sl'L R T I N G V A L U E S 1 ) GO T O 50 U R I T 9 ( 6 , 3 5 ) FORMAT( ' P HAS NO ROOT' ) GO TO 50 U Q I T E ( 6 , & 5 ) F O R U A T ( 1 N O C O Y V E R S E N C E O V E R T I M E ' ) V R I T E ( 6 , 5 5 ) P O R Y A T ( ' E R R 9 9 I N ERROR P R O C E S S I N G * ) CONTrNrJE

S T 3 P END T 9 I S SI JRROII l ' INE CALCULATYS THE NEW F ( 1 ) AND V ( 1 ) U S I N G THE P R E V I O U S T I M E S T E P VALUES S U B R 0 9 T I V E S E C A N T COMfiON E , R D l ' , H . C ( 9 ) , C , ? P , I , P ( 9 I , E P S ~ V ( ~ ) I N T E G E R ,FLAG1 F L A G l = O S T T G[I!?SS V h L U E S T O L E F T AND R I S H l ' O F ZERO (KO HAS N E G I T I V E VALUE AND XI H I S P O S I T I V E VALUE) XO=F (I) X 1 = 0 . 0 CON=E* RDT-H/C (I) TEMP=G*RDT

MAKE S U R E XO I S NEBA'PIVE DO 1 K 1 = 1 , 5 O O K = 5 0 0 - K 1 * 1 XO=Xrl-0.1 I F (X '3. LT. 0 . 0 ) G O T 0 1 FO=CON+XO+TEflP*XO**9P I F ( F O . L ' P . O . 0 ) G O T 0 5 CONTTNUE

IF XO I S P 3 S I T I V E . PRIYl' A P P R O P R T A T R ERROR HESSACE V L A G l = - 1 0 C A L L E R R O R ( F L A E 1 , I M A X )

F A K E STIRE X I I S P O S I T I V E DO 1 0 K = 1 . 5 0 0 X 1 = X I+O. 1 F l = C O N + X l + T E Y P * X l * * R P I F (F 1. GT. 0 . 0 ) G O T 0 1 5

C 3 N T T WIJE I F X1 TS N!?;ATIVB, P R I N T A P P R O P R I A T E ERROR HESSAGE P L A G l = - 1 1

CALL ERROR (FLAG1 , IFAT) C I F SECPNT HETHOD HAS CONVERGED, PREPARE TO RETURN

15 I F (ABS (PO) . LT. EPS.OR. ABS ( P l ) . LT. EPS) GOT0 2 5 FLAG I= PLAZ 1 4 1

C I F SECANT HETROD I S NOT CONVERGING, PRINT C APPROPRIATE ERROR UESSAGL

I F (FLAGl. EQ. I U AX) CALL ERROR (FLRG1,IUAX) C CALCULATE N E W END POINT USING THE SECANT UETHOD

ALPHA=(Xl-10) / (F l -PO) X2=X 1-ALPHA*Pl P2=CON*X2*TEqP*XZ**RP

C REPLlCE ONE EYD POINT WIrH NEW END POINT C UPERR THE OLD AND NEW VALUE HAVE THE SANE SIGN

I F (ABS ( F 2 ) .LT.YPS) GOT0 2 5 I F (ARS ( P l / A B S ( P I ) -P2/ABS ( F 2 ) ). GT.EPS) GOT0 2 0 Xl=X2 P1 =P? QOTO 15

2 0 XO = X2 FO = F2 GOT0 1 5

C PREPARE TO RETURN VALVE OF P(I) AND V ( 1 ) 2 5 I F (ABS (PO). LT. EPS) P (I) = X O

I F (ABS (P l ) .LT .EPS) P ( I ) =XI I F (ABS (P2). LT. CPS. A N D . FLAG1 .GT. 0 ) F ( I ) =X2 V ( I ) = E + G * P ( I ) **RP RETURN END

C C C

SUBROIITINE PLOTlD(X,Y,K,X1l~Y11,Kl) DIUENSInN X ( 3 0 0 ) ,Y ( 3 0 0 ) , X I 1 ( 3 0 0 ) , Y l l ( 3 0 0 ) , IPAK ( 1 5 0 ) CALL CALCPP C4 1.1. COMPLX CALL MXlALF ( 'ST4NDAP9' ,') ' ) CALL 3X2ALP ( 'L/CSTD' , ' (I)

C9LL UX3ALP ( ' INSTRLl ' , '+ ' ) CALL T I T L E ( ' $',

~ - I ~ O , ' E ( L A P ~ E U ) 'P ( I H E ( f l X N ) ) % ' . l o o , 2'0 (UTLYT ) HNO*LI1.5) 3+LXIIX) C (ONCENTRATI3N ( 1 H) $', 1 0 0 , 8 . 2 5 , 6 . 0 )

CALL BLNK3 (-0. 015 , - -12 . - .1 ,6 .5 ,0 ) CALL BT.NK4 ( - . I ,8 .95, - . 015,-. 12,O) CALL GRAF (0. ,50. , 6 5 0 . , 0., 1. ,9.) CALL B L N K 1 ( . 1 2 5 , . 2 5 , 0 . 0 , 6 . 5 , 0 ) CALL nLNK2(0.0,q.q5,.1%5,,25,0) CALL Y C R A X 5 ( n - 0 , . 2 , 9 , . 6 . 0 , ' $ ' , -100 ,O. ,0.) CALL TGRAXS (0 .0 ,10 . , 6 5 0 . , 8 . 2 5 , ' $ ' , - 1 3 0 , U . U , 3 , 0 ) CALL RESET ( 'BLNKS') CALL ?i 9RKER ( 2 ) CALL DOT CALL SPLINE CALL CUSVF (X .Y, K, 0 ) CALL HESET ( ' D O T ' ) CALL CUSVE ( X I 1 , V l l , K l , O ) CALL RLVEr ( 1 1 0 . , I . 0 8 , 6 0 . 0 , 1 . 0 9 , 3 0 1 )

CILL RLYESS ( ' (NODEL DATI) $ I . 1 0 0 , 115. .3-74) CALL RLVSC (110.,3.74,Q3.,3.74,301~ CALL RLNESS('(EXPER1YENTAL D A ~ A ) $ ' ,133,115. , 1 - 0 0 ) CALL RLNESS ('HNO+LA.S) ~ + E X I . X ) $ 1 , 1 0 0 , s 2 0 , 5 - 25) CALL RLYESS ( ' TINE PLOU C3NC$', 100,310.0,U.90) CALL RLNESS ( I ( (HIN) ) ( ( K G / H ) 1 ( () H) s r , 100 ,310 .0 .U-55) C4LL RLHESS ( I (INITIAL) 9.84 0.0 $l ,100.310.0,U.O) CALL RLMESS ( ' 0 3 6 - 7 9 7.2 $Q,100 ,310 .0 ,3 .5 ) CALL RLMESS ( ' 180 46.83 7.9 $* , .100, 310.0,3.0) CALL RLNESS ( ' 325 41.67 7.6 $ * , 100, 310.0,2.5) CNLL RLHESS ( ' 4 1 5 '3.94 0.3 $ ' ,100,310.0,2.0) CALL ENDPL ( I ) CALL DONEPL STOP EN D

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