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Abellar, Rodgie John Aries, Allen Jerry Borlan, Randolph Yanson, Edsel
Problem:
A well stirred storage vessel contains 10000 kg of solution of a dilute methanol solution ( w A = 0.05 mass fraction alcohol). A constant flow of 500 kg/min. of pure water is suddenly introduced into the tank and a constant rate of withdrawal of 500 kg/min is started. These two flows are continued and remain constant. Assuming that the densities of the solutions are the same and that the total contents of the tank remain constant at 10,000 kg of solution, calculate the time for the alcohol content to drop to 1wt%.
Abellar, Rodgie John Aries, Allen Jerry Borlan, Randolph Yanson, Edsel
The particular solution to the DE is . Numerical Solution (Runge-Kutta Method):
Let
,
For an accurate result, we take With the aid of MS Excel: n
t
wA
k1
k2
0
k3
k4
wA+1
0.0
0.050000
-0.001250
-0.001234
-0.001235
-0.001219
0.048765
1
0.5
0.048765
-0.001219
-0.001204
-0.001204
-0.001189
0.047561
2
1.0
0.047561
-0.001189
-0.001174
-0.001174
-0.001160
0.046387
3
1.5
0.046387
-0.001160
-0.001145
-0.001145
-0.001131
0.045242
4
2.0
0.045242
-0.001131
-0.001117
-0.001117
-0.001103
0.044125
5
2.5
0.044125
-0.001103
-0.001089
-0.001090
-0.001076
0.043035
6
3.0
0.043035
-0.001076
-0.001062
-0.001063
-0.001049
0.041973
7
3.5
0.041973
-0.001049
-0.001036
-0.001036
-0.001023
0.040937
8
4.0
0.040937
-0.001023
-0.001011
-0.001011
-0.000998
0.039926
9
4.5
0.039926
-0.000998
-0.000986
-0.000986
-0.000973
0.038940
10
5.0
0.038940
-0.000974
-0.000961
-0.000961
-0.000949
0.037979
11
5.5
0.037979
-0.000949
-0.000938
-0.000938
-0.000926
0.037041
12
6.0
0.037041
-0.000926
-0.000914
-0.000915
-0.000903
0.036126
13
6.5
0.036126
-0.000903
-0.000892
-0.000892
-0.000881
0.035234
14
7.0
0.035234
-0.000881
-0.000870
-0.000870
-0.000859
0.034364
15
7.5
0.034364
-0.000859
-0.000848
-0.000849
-0.000838
0.033516
16
8.0
0.033516
-0.000838
-0.000827
-0.000828
-0.000817
0.032688
17
8.5
0.032688
-0.000817
-0.000807
-0.000807
-0.000797
0.031881
18
9.0
0.031881
-0.000797
-0.000787
-0.000787
-0.000777
0.031094
19
9.5
0.031094
-0.000777
-0.000768
-0.000768
-0.000758
0.030327
20
10.0
0.030327
-0.000758
-0.000749
-0.000749
-0.000739
0.029578
21
10.5
0.029578
-0.000739
-0.000730
-0.000730
-0.000721
0.028847
22
11.0
0.028847
-0.000721
-0.000712
-0.000712
-0.000703
0.028135
Abellar, Rodgie John Aries, Allen Jerry Borlan, Randolph Yanson, Edsel 23
11.5
0.028135
-0.000703
-0.000695
-0.000695
-0.000686
0.027441
24
12.0
0.027441
-0.000686
-0.000677
-0.000678
-0.000669
0.026763
25
12.5
0.026763
-0.000669
-0.000661
-0.000661
-0.000653
0.026102
26
13.0
0.026102
-0.000653
-0.000644
-0.000645
-0.000636
0.025458
27
13.5
0.025458
-0.000636
-0.000628
-0.000629
-0.000621
0.024829
28
14.0
0.024829
-0.000621
-0.000613
-0.000613
-0.000605
0.024216
29
14.5
0.024216
-0.000605
-0.000598
-0.000598
-0.000590
0.023618
30
15.0
0.023618
-0.000590
-0.000583
-0.000583
-0.000576
0.023035
31
15.5
0.023035
-0.000576
-0.000569
-0.000569
-0.000562
0.022466
32
16.0
0.022466
-0.000562
-0.000555
-0.000555
-0.000548
0.021912
33
16.5
0.021912
-0.000548
-0.000541
-0.000541
-0.000534
0.021371
34
17.0
0.021371
-0.000534
-0.000528
-0.000528
-0.000521
0.020843
35
17.5
0.020843
-0.000521
-0.000515
-0.000515
-0.000508
0.020328
36
18.0
0.020328
-0.000508
-0.000502
-0.000502
-0.000496
0.019827
37
18.5
0.019827
-0.000496
-0.000489
-0.000490
-0.000483
0.019337
38
19.0
0.019337
-0.000483
-0.000477
-0.000477
-0.000471
0.018860
39
19.5
0.018860
-0.000471
-0.000466
-0.000466
-0.000460
0.018394
40
20.0
0.018394
-0.000460
-0.000454
-0.000454
-0.000448
0.017940
41
20.5
0.017940
-0.000448
-0.000443
-0.000443
-0.000437
0.017497
42
21.0
0.017497
-0.000437
-0.000432
-0.000432
-0.000427
0.017065
43
21.5
0.017065
-0.000427
-0.000421
-0.000421
-0.000416
0.016644
44
22.0
0.016644
-0.000416
-0.000411
-0.000411
-0.000406
0.016233
45
22.5
0.016233
-0.000406
-0.000401
-0.000401
-0.000396
0.015832
46
23.0
0.015832
-0.000396
-0.000391
-0.000391
-0.000386
0.015441
47
23.5
0.015441
-0.000386
-0.000381
-0.000381
-0.000376
0.015060
48
24.0
0.015060
-0.000376
-0.000372
-0.000372
-0.000367
0.014688
49
24.5
0.014688
-0.000367
-0.000363
-0.000363
-0.000358
0.014325
50
25.0
0.014325
-0.000358
-0.000354
-0.000354
-0.000349
0.013972
51
25.5
0.013972
-0.000349
-0.000345
-0.000345
-0.000341
0.013627
52
26.0
0.013627
-0.000341
-0.000336
-0.000336
-0.000332
0.013290
53
26.5
0.013290
-0.000332
-0.000328
-0.000328
-0.000324
0.012962
54
27.0
0.012962
-0.000324
-0.000320
-0.000320
-0.000316
0.012642
55
27.5
0.012642
-0.000316
-0.000312
-0.000312
-0.000308
0.012330
56
28.0
0.012330
-0.000308
-0.000304
-0.000304
-0.000301
0.012025
57
28.5
0.012025
-0.000301
-0.000297
-0.000297
-0.000293
0.011729
58
29.0
0.011729
-0.000293
-0.000290
-0.000290
-0.000286
0.011439
59
29.5
0.011439
-0.000286
-0.000282
-0.000282
-0.000279
0.011157
60
30.0
0.011157
-0.000279
-0.000275
-0.000275
-0.000272
0.010881
61
30.5
0.010881
-0.000272
-0.000269
-0.000269
-0.000265
0.010612
62
31.0
0.010612
-0.000265
-0.000262
-0.000262
-0.000259
0.010350
63
31.5
0.010350
-0.000259
-0.000256
-0.000256
-0.000252
0.010095
64
32.0
0.010095
-0.000252
-0.000249
-0.000249
-0.000246
0.009846
Interpolating the 63 rd and 64th iteration for
approximately gives
The time for the ethanol content in the tank to drop to 1wt. % is 32.2 min (by means of analytical solution) or approximately 32.0 min (Runge-Kutta, ).
