PolymathPlus Report
📉   Differential Equations 2022-04-01 12:45 

# Example 22 - ODE System
# Reversible Reaction in a PBR
# Verified Final Values: T = 1149.6, x = 0.725004, y = 0.766736
# Ref.: Comput. Appl. Eng. Educ. 6: 176-178, 1998

d(T)/d(W)=(.8*(Ta-T)+rA*delH)/(CPA*FA0)
d(x)/d(W)=-rA/FA0
d(y)/d(W)=-0.015*(1-.5*x)*(T/450)/(2*y)
FA0=5
Ta=500
delH=-40000
CPA=40
k=.5*exp(5032*(1/450-1/T))
CA=.271*(1-x)*(450/T)/(1-.5*x)*y
CC=.271*.5*x*(450/T)/(1-.5*x)*y
Kc=25000*exp(delH/8.314*(1/450-1/T))
rA=-k*(CA^2-CC/Kc)
W(0)=0
T(0)=450
x(0)=0
y(0)=1
W(f)=20


Calculated values

  Variable Initial value Minimal value Maximal value Final value
1 CA 0.271 0.03508562 0.271 0.03508562
2 CC 0 0 0.05106577 0.04625013
3 CPA 40 40 40 40
4 delH -40000 -40000 -40000 -40000
5 FA0 5 5 5 5
6 k 0.5 0.5 478.9746 451.19419
7 Kc 25000 35.27272 25000 37.346432
8 rA -0.0367205 -0.97415801 0.00334228 0.00334228
9 T 450 450 1165.4991 1149.5993
10 Ta 500 500 500 500
11 W 0 0 20 20
12 x 0 0 0.72718001 0.7250039
13 y 1 0.7667365 1 0.7667365

Integration chart

 0 200 400 600 800 1000 1200 0 4 8 12 16 20 W T x y

Formatted equations

  d T ( ) d W ( ) = + . 8 T a T ( ) r A d e l H C P A F A 0
  d x ( ) d W ( ) = r A F A 0
  d y ( ) d W ( ) = 0 . 0 1 5 1 . 5 x ( ) T 4 5 0 2 y
  F A 0 = 5
  T a = 5 0 0
  d e l H = 4 0 0 0 0
  C P A = 4 0
  k = . 5 e x p 5 0 3 2 1 4 5 0 1 T
  C A = . 2 7 1 1 x ( ) 4 5 0 T 1 . 5 x y
  C C = . 2 7 1 . 5 x 4 5 0 T 1 . 5 x y
  K c = 2 5 0 0 0 e x p d e l H 8 . 3 1 4 1 4 5 0 1 T
  r A = k C A ( ) 2 C C K c

Differential equations

1 d(T)/d(W) = (.8*(Ta-T)+rA*delH)/(CPA*FA0)
2 d(x)/d(W) = -rA/FA0
3 d(y)/d(W) = -0.015*(1-.5*x)*(T/450)/(2*y)

Explicit equations

1 FA0 = 5
2 Ta = 500
3 delH = -40000
4 CPA = 40
5 k = .5*exp(5032*(1/450-1/T))
6 CA = .271*(1-x)*(450/T)/(1-.5*x)*y
7 CC = .271*.5*x*(450/T)/(1-.5*x)*y
8 Kc = 25000*exp(delH/8.314*(1/450-1/T))
9 rA = -k*(CA^2-CC/Kc)

General

Total number of equations 12
Number of differential equations 3
Number of explicit equations 9
Reporting digits 10
Elapsed time 0.01 sec
Solution method RKF_45
Step size guess. h 1E-06
Truncation error tolerance. eps 1E-06
Calculated Intermediate data points 50

Calculated data points

    W T x y FA0 Ta delH CPA k CA CC Kc rA
1 0 450 0 1 5 500 -40000 40 0.5 0.271 0 25000 -0.0367205
2 0.97949469 457.87433 0.00769353 0.99257731 5 500 -40000 40 0.58654052 0.26462784 0.00084739 21461.228 -0.04107417
3 1.2994947 460.67999 0.01044705 0.9901177 5 500 -40000 40 0.62646921 0.26204123 0.00119707 20151.533 -0.04301685
4 1.6194947 463.61663 0.01333522 0.98764006 5 500 -40000 40 0.67057759 0.25939378 0.00155842 18882.332 -0.04511985
5 2.2594947 469.92952 0.01956292 0.98262802 5 500 -40000 40 0.77386155 0.25389913 0.00231973 16465.38 -0.0498867
6 2.5794947 473.33289 0.02292996 0.98009217 5 500 -40000 40 0.83460829 0.25104214 0.00272172 15317.669 -0.05259867
7 2.8994947 476.92195 0.02648716 0.97753536 5 500 -40000 40 0.90273593 0.2481046 0.00313947 14210.528 -0.05556852
8 3.2194947 480.71526 0.03025334 0.9749567 5 500 -40000 40 0.97948452 0.24508016 0.00357434 13144.031 -0.05883177
9 3.8594947 489.00262 0.03850172 0.9697299 5 500 -40000 40 1.1652219 0.23874125 0.00450167 11133.38 -0.06641413
10 4.1794947 493.5492 0.0430371 0.96707953 5 500 -40000 40 1.2783481 0.23540981 0.00499778 10189.493 -0.07084259
11 4.4994947 498.40648 0.0478892 0.96440286 5 500 -40000 40 1.4085933 0.23195725 0.00551838 9286.7839 -0.07578735
12 4.8194947 503.61269 0.05309666 0.96169847 5 500 -40000 40 1.5595462 0.22837202 0.00606602 8425.449 -0.0813351
13 5.4594947 515.25997 0.06476772 0.95620008 5 500 -40000 40 1.9431258 0.22074912 0.00725459 6827.814 -0.0946868
14 5.7794947 521.81777 0.07134919 0.95340237 5 500 -40000 40 2.1892407 0.21667911 0.0079029 6092.0308 -0.10278165
15 6.0994947 528.96247 0.07852632 0.95056947 5 500 -40000 40 2.4841568 0.21241096 0.00859319 5398.6522 -0.11207727
16 6.4194947 536.78653 0.08639239 0.94769887 5 500 -40000 40 2.8412863 0.20792147 0.00933099 4747.9872 -0.12282703
17 7.0594947 554.95121 0.10467369 0.94183277 5 500 -40000 40 3.8217967 0.19816553 0.01097656 3576.0822 -0.15006861
18 7.3794947 565.60478 0.11540429 0.93883026 5 500 -40000 40 4.5053899 0.1928301 0.01190168 3055.4772 -0.16750838
19 7.6994947 577.5818 0.1274728 0.93577581 5 500 -40000 40 5.379859 0.18713335 0.01290973 2578.8228 -0.18836976
20 8.0194947 591.15889 0.14115769 0.93266434 5 500 -40000 40 6.5185766 0.18102347 0.01401488 2146.3423 -0.21356792
21 8.6594947 624.62866 0.17490161 0.92624543 5 500 -40000 40 10.084249 0.1673106 0.01659045 1414.244 -0.28216844
22 8.9794947 645.56083 0.19600635 0.92292273 5 500 -40000 40 12.939185 0.15955528 0.01810938 1114.3261 -0.32919402
23 9.28527 669.04447 0.219682 0.91966575 5 500 -40000 40 16.904969 0.15127899 0.0197843 862.97936 -0.38648828
24 9.8341193 722.40864 0.27347288 0.91358707 5 500 -40000 40 29.407338 0.13419061 0.02343964 508.28965 -0.52818532
25 10.083891 752.69908 0.30400031 0.91070792 5 500 -40000 40 38.910873 0.12552758 0.0254081 388.89585 -0.61058314
26 10.55458 822.0483 0.37388799 0.90506435 5 500 -40000 40 68.411374 0.10789817 0.02970256 226.74136 -0.78748446
27 11.025552 906.6351 0.45915921 0.89911547 5 500 -40000 40 121.10671 0.08958015 0.03466998 131.3338 -0.93986311
28 11.297334 958.95811 0.51195269 0.89555054 5 500 -40000 40 163.08836 0.07971088 0.03760974 98.808502 -0.97415801
29 11.617334 1018.0371 0.57165754 0.89124829 5 500 -40000 40 221.89188 0.06914354 0.04099639 73.611275 -0.93724878
30 12.107612 1090.2671 0.6449789 0.88449588 5 500 -40000 40 334.22426 0.05422235 0.04621907 49.757102 -0.67218157
31 12.435209 1122.064 0.67757152 0.87991287 5 500 -40000 40 394.92231 0.0476838 0.04861763 42.419156 -0.44532225
32 12.880853 1147.2887 0.70393022 0.87361961 5 500 -40000 40 441.99411 0.04295226 0.05026981 38.089343 -0.23209534
33 13.344658 1159.4106 0.71726577 0.86701486 5 500 -40000 40 465.73829 0.04048423 0.05094221 36.230587 -0.10847867
34 13.683497 1163.3259 0.72207786 0.86215859 5 500 -40000 40 473.78242 0.03950219 0.05106577 35.64222 -0.06049754
35 14.058864 1165.1332 0.72488269 0.85674951 5 500 -40000 40 477.77717 0.03882963 0.051012 35.357238 -0.03104632
36 14.483961 1165.4991 0.72638011 0.85058696 5 500 -40000 40 478.9746 0.03834328 0.05082272 35.27272 -0.01406088
37 14.976233 1164.8596 0.72705049 0.84340135 5 500 -40000 40 478.22799 0.03795436 0.05052064 35.325369 -0.00496473
38 15.254916 1164.2483 0.72718001 0.8393097 5 500 -40000 40 477.3019 0.03777501 0.0503296 35.390899 -0.00231296
39 15.880551 1162.5761 0.72716806 0.83006015 5 500 -40000 40 474.58231 0.03741643 0.04986902 35.584781 0.00067644
40 16.200551 1161.6384 0.72707785 0.8252943 5 500 -40000 40 472.96521 0.03723946 0.04961597 35.7011 0.00141094
41 16.520551 1160.6726 0.72695834 0.82050431 5 500 -40000 40 471.28751 0.03706618 0.04935873 35.822602 0.00186806
42 16.840551 1159.6889 0.72681973 0.81568975 5 500 -40000 40 469.57356 0.03689462 0.04909818 35.947606 0.00216607
43 17.480551 1157.6891 0.72650618 0.80598521 5 500 -40000 40 466.08695 0.03655254 0.04856887 36.204673 0.00252646
44 17.800551 1156.6783 0.72633655 0.8010943 5 500 -40000 40 464.32698 0.03638093 0.04830054 36.335869 0.00265018
45 18.120551 1155.662 0.72616009 0.79617701 5 500 -40000 40 462.5601 0.03620858 0.04802987 36.468561 0.00275683
46 18.440551 1154.6407 0.7259774 0.79123282 5 500 -40000 40 460.78784 0.03603532 0.04775689 36.602658 0.00285407
47 19.080551 1152.5845 0.72559443 0.78126175 5 500 -40000 40 457.23014 0.03568571 0.04720401 36.874917 0.00303661
48 19.400551 1151.5497 0.72539433 0.7762338 5 500 -40000 40 455.44533 0.03550924 0.04692406 37.013069 0.00312624
49 19.720551 1150.5105 0.72518845 0.77117682 5 500 -40000 40 453.6567 0.03533158 0.04664175 37.152585 0.00321633
50 20 1149.5993 0.7250039 0.7667365 5 500 -40000 40 451.19419 0.03508562 0.04625013 37.346432 0.00334228