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Name: |
Threeq2 - Two phase flash of a nonideal binary
mixture (isobutanol-water) |
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Source: |
Henley, E. J. and Rosen, E. M., Material
and Energy Balance |
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Computations, Wiley:
New York, 1969 |
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Reference/s |
Cutlip, M. B. and Shacham, M, Problem Solving
in Chemical |
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Engineering with Numerical Methods (2nd
Ed.), Prentice Hall Inc., |
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Model: |
3 implicit equations, indep. variables alpha,
x1 and x2 |
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Higher difficulty level |
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Constraints: 0<=alpha<=1, 0<=x1<=1,
0<=x2<=1 |
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Discontinuities: Undefined for (x1+B*x2/A)=0
and (A*x1/B+x2)=0 |
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Initial estimates: 1. alpha=0.5, x1=0, x2=1,
2.alpha=0.9, x1=0.5, x2=0.5; |
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3. alpha=0.9, x1=0.4, x2=0.6 4.alpha=0.5,
x1=0.1, x2=0.9 |
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Solved by |
Shacham, M., POLYMATH 5.1, build 19, April
4, 2001 |
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Model Eqs. |
Two phase flash of a nonideal binary mixture
(isobutanol-water)|POLVER05_0 |
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EXCEL
FILE |
f(x1)=x1-z1/(1+alpha*(k1-1)) # |
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TEXT
FILE |
f(x2)=x2-z2/(1+alpha*(k2-1)) # |
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POLYMATH
FILE |
f(alpha)=x1+x2-(y1+y2) # |
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p1=10^(7.62231-1417.9/(191.15+t)) # |
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p2=10^(8.10765-1750.29/(235+t)) # |
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gamma2=10^(B*x1*x1/((x1+B*x2/A)^2)) # |
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gamma1=10^(A*x2*x2/((A*x1/B+x2)^2)) # |
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k1=gamma1*p1/760 # |
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k2=gamma2*p2/760 # |
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y1=k1*x1 # |
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y2=k2*x2 # |
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t=88.538 # |
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B=0.7 # |
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A=1.7 # |
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z1=0.2 # |
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z2=0.8 # |
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x1(0)=0 |
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x2(0)=1 |
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alpha(0)=0.5 |
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Variable/function values |
Variable |
Value |
f(x) |
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x1 |
0 |
-1.6296E-02 |
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x2 |
1 |
3.3895E-02 |
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alpha |
0.5 |
3.4387E-01 |
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t |
88.538 |
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p1 |
357.053 |
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B |
0.70 |
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A |
1.70 |
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gamma2 |
1.000 |
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p2 |
498.662 |
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gamma1 |
50.119 |
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k2 |
0.656 |
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y2 |
0.656 |
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k1 |
23.546 |
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y1 |
0.00 |
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z1 |
0.20 |
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z2 |
0.80 |
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Solution |
Variable |
Value |
f(x) |
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x1 |
0.0226974766367 |
-3.4694E-18 |
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x2 |
0.9773025233633 |
0.0000E+00 |
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alpha |
0.5322677863643 |
0.00E+00 |
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t |
88.538 |
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p1 |
357.05282269240 |
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B |
0.7 |
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A |
1.7 |
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gamma2 |
1.0046052345120 |
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p2 |
498.66206831320 |
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gamma1 |
33.36690033259 |
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k2 |
0.6591559527368 |
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y2 |
0.6441947758996 |
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k1 |
15.67598151085 |
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y1 |
0.3558052241004 |
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z1 |
0.2 |
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z2 |
0.8 |
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Infeasible Solution |
Variable |
Value |
f(x) |
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x1 |
0.6867568052506 |
-1.3212E-14 |
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x2 |
0.3132431762583 |
-1.1102E-16 |
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alpha |
1.4708209249600 |
-5.55E-16 |
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t |
88.538 |
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p1 |
357.05282269240 |
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B |
0.7 |
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A |
1.7 |
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gamma2 |
3.1342710428540 |
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p2 |
498.66206831320 |
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gamma1 |
1.10281288233 |
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k2 |
2.0565027380050 |
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y2 |
0.6441854496365 |
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k1 |
0.51810849018 |
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y1 |
0.3558145314887 |
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z1 |
0.2 |
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z2 |
0.8 |
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False solution |
Variable |
Value |
f(x) |
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x1 |
-7.97906491100E-13 |
1.1309E-13 |
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x2 |
2.54516136300E-10 |
3.8950E-14 |
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alpha |
-9.14388809400E+09 |
1.07E-10 |
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t |
88.538 |
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p1 |
357.05282269240 |
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B |
0.7 |
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A |
1.7 |
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gamma2 |
1.0000948741280 |
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p2 |
498.66206831320 |
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gamma1 |
53.23346661201 |
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k2 |
0.6561965505819 |
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y2 |
1.67012610700E-10 |
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k1 |
25.00942041516 |
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y1 |
-1.99551788900E-11 |
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z1 |
0.2 |
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z2 |
0.8 |
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Additional information |
Most programs do not converge, converge to
an infeasible |
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solution or to a false solution from initial
guesses 2,3, and 4. |
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Using a constrained method which keeps alpha,x1
and x2 |
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positive throughout the iterations enables
convergence |
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from initial guesses 3 and 4. |