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

# Example 11 - ODE System. Heat Exchange
# Verified Final Values: T1 = 30.9524, T2 = 41.3832 , T3 = 51.3174
# Ref.: Comput. Appl. Eng. Educ. 6: 175, 1998

d(T1)/d(t)=(W*Cp*(T0-T1)+UA*(Tsteam-T1))/(M*Cp)
d(T2)/d(t)=(W*Cp*(T1-T2)+UA*(Tsteam-T2))/(M*Cp)
d(T3)/d(t)=(W*Cp*(T2-T3)+UA*(Tsteam-T3))/(M*Cp)

W=100
Cp=2.0
T0=20
UA=10.
Tsteam=250
M=1000
t(0)=0
T1(0)=20
T2(0)=20
T3(0)=20
t(f)=200


Calculated values

  Variable Initial value Minimal value Maximal value Final value
1 Cp 2 2 2 2
2 M 1000 1000 1000 1000
3 t 0 0 200 200
4 T0 20 20 20 20
5 T1 20 20 30.952381 30.952381
6 T2 20 20 41.38322 41.38322
7 T3 20 20 51.31735 51.31735
8 Tsteam 250 250 250 250
9 UA 10 10 10 10
10 W 100 100 100 100

Integration chart

 15 20 25 30 35 40 45 50 55 60 65 0 40 80 120 160 200 t T1 T2 T3

Formatted equations

  d T 1 d t ( ) = + W C p T 0 T 1 ( ) U A T s t e a m T 1 ( ) M C p
  d T 2 d t ( ) = + W C p T 1 T 2 ( ) U A T s t e a m T 2 ( ) M C p
  d T 3 d t ( ) = + W C p T 2 T 3 ( ) U A T s t e a m T 3 ( ) M C p
  W = 1 0 0
  C p = 2 . 0
  T 0 = 2 0
  U A = 1 0 .
  T s t e a m = 2 5 0
  M = 1 0 0 0

Differential equations

1 d(T1)/d(t) = (W*Cp*(T0-T1)+UA*(Tsteam-T1))/(M*Cp)
2 d(T2)/d(t) = (W*Cp*(T1-T2)+UA*(Tsteam-T2))/(M*Cp)
3 d(T3)/d(t) = (W*Cp*(T2-T3)+UA*(Tsteam-T3))/(M*Cp)

Explicit equations

1 W = 100
2 Cp = 2.0
3 T0 = 20
4 UA = 10.
5 Tsteam = 250
6 M = 1000

General

Total number of equations 9
Number of differential equations 3
Number of explicit equations 6
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

    t T1 T2 T3 W Cp T0 UA Tsteam M
1 0 20 20 20 100 2 20 10 250 1000
2 9.0287351 26.708239 29.265114 29.970363 100 2 20 10 250 1000
3 12.152421 27.895019 31.698647 33.063575 100 2 20 10 250 1000
4 17.550008 29.217731 34.952207 37.742229 100 2 20 10 250 1000
5 21.76496 29.838071 36.782367 40.756664 100 2 20 10 250 1000
6 24.033033 30.074208 37.558174 42.149658 100 2 20 10 250 1000
7 28.902034 30.425699 38.832713 44.639631 100 2 20 10 250 1000
8 34.284975 30.653099 39.772814 46.699282 100 2 20 10 250 1000
9 37.202051 30.732058 40.133416 47.562459 100 2 20 10 250 1000
10 40.291504 30.793095 40.430443 48.315936 100 2 20 10 250 1000
11 46.691504 30.871036 40.844589 49.456506 100 2 20 10 250 1000
12 49.891504 30.89425 40.9797 49.861398 100 2 20 10 250 1000
13 53.091504 30.910839 41.081562 50.18249 100 2 20 10 250 1000
14 56.291504 30.922694 41.158149 50.435851 100 2 20 10 250 1000
15 62.691504 30.937221 41.258577 50.790515 100 2 20 10 250 1000
16 65.891504 30.941547 41.29068 50.911804 100 2 20 10 250 1000
17 69.091504 30.944639 41.314611 51.00598 100 2 20 10 250 1000
18 72.291504 30.946848 41.33242 51.078866 100 2 20 10 250 1000
19 78.691504 30.949555 41.355469 51.178379 100 2 20 10 250 1000
20 81.891504 30.950362 41.362742 51.211589 100 2 20 10 250 1000
21 85.091504 30.950938 41.368124 51.237015 100 2 20 10 250 1000
22 88.291504 30.95135 41.372102 51.256436 100 2 20 10 250 1000
23 94.691504 30.951854 41.377205 51.282502 100 2 20 10 250 1000
24 97.891504 30.952005 41.378801 51.291053 100 2 20 10 250 1000
25 101.0915 30.952112 41.379976 51.297534 100 2 20 10 250 1000
26 104.2915 30.952189 41.38084 51.302438 100 2 20 10 250 1000
27 110.6915 30.952283 41.381942 51.308938 100 2 20 10 250 1000
28 113.8915 30.952311 41.382284 51.311043 100 2 20 10 250 1000
29 117.0915 30.952331 41.382535 51.312627 100 2 20 10 250 1000
30 120.2915 30.952345 41.382719 51.313817 100 2 20 10 250 1000
31 126.6915 30.952363 41.382953 51.31538 100 2 20 10 250 1000
32 129.8915 30.952368 41.383025 51.315881 100 2 20 10 250 1000
33 133.0915 30.952372 41.383077 51.316256 100 2 20 10 250 1000
34 136.2915 30.952374 41.383116 51.316536 100 2 20 10 250 1000
35 142.6915 30.952378 41.383165 51.316901 100 2 20 10 250 1000
36 145.8915 30.952379 41.38318 51.317017 100 2 20 10 250 1000
37 149.0915 30.952379 41.383191 51.317103 100 2 20 10 250 1000
38 152.2915 30.95238 41.383199 51.317168 100 2 20 10 250 1000
39 158.6915 30.95238 41.383209 51.317251 100 2 20 10 250 1000
40 161.8915 30.95238 41.383212 51.317277 100 2 20 10 250 1000
41 165.0915 30.952381 41.383214 51.317297 100 2 20 10 250 1000
42 168.2915 30.952381 41.383216 51.317311 100 2 20 10 250 1000
43 174.6915 30.952381 41.383218 51.31733 100 2 20 10 250 1000
44 177.8915 30.952381 41.383218 51.317336 100 2 20 10 250 1000
45 181.0915 30.952381 41.383219 51.31734 100 2 20 10 250 1000
46 184.2915 30.952381 41.383219 51.317343 100 2 20 10 250 1000
47 190.6915 30.952381 41.383219 51.317347 100 2 20 10 250 1000
48 193.8915 30.952381 41.38322 51.317349 100 2 20 10 250 1000
49 197.0915 30.952381 41.38322 51.31735 100 2 20 10 250 1000
50 200 30.952381 41.38322 51.31735 100 2 20 10 250 1000