| Name: |
13eq1 - Gibbs energy minimization |
| Source: |
Himmelblau, D. M., Applied
nonlinear programming, McGraw- Hill, 1972 |
| |
|
|
|
| Reference/s |
Shacham, M., 1986, Int.
J. numerical Meth. Engng, 23, 1455 |
| |
Bullard, L. G. and Biegler,
L. T., Computers Chem. Engng. 15(4), 239 (1991). |
| |
|
|
|
| Model: |
13 implicit equations, indep.
variables v1 to v13 |
| |
Higher difficulty level |
| |
Constraints: vi are nonnegative
for i=1,2,…10 |
| |
Discontinuities: Undefined
for nonpositive values of v1 to v10 |
| |
Initial estimates: 1.(0.05,
0.2, 0.8, 0.001, 0.5, 0.0007, 0.03, 0.02, 0.1, 0.1, 10, 10, 10)
; |
| |
2. (0.1, 0.1, 0.1, 0.1, 0.1,
0.1, 0.1, 0.1, 0.1, 0.1, 10, 10, 10) |
| |
|
|
|
| Solved by |
Shacham, M., POLYMATH 5.1,
build 19, April 18, 2001 |
| |
|
|
|
| Model Eqs. |
Gibbs energy minimization
|POLVER05_3 |
|
EXCEL
FILE |
f(v1) = -6.089+ln(v1/xs)+v11
# |
|
TEXT
FILE |
f(v2) = -17.164+ln(v2/xs)+2*v11
# |
|
POLYMATH
FILE |
f(v3) = -34.054+ln(v3/xs)+2*v11+v13
# |
| |
f(v4) = -5.914+ln(v4/xs)+v12
# |
| |
f(v5) = -24.721+ln(v5/xs)+2*v12
# |
| |
f(v6) = -14.986+ln(v6/xs)+v11+v12
# |
| |
f(v7)=-24.1+ln(v7/xs)+v12+v13
# |
| |
f(v8)=-10.708+ln(v8/xs)+v13
# |
| |
f(v9)=-26.662+ln(v9/xs)+2*v13
# |
| |
f(v10)=-22.197+ln(v10/xs)+v11+v13
# |
| |
f(v11)=v1+2*v2+2*v3+v6+v10-2
# |
| |
f(v12)=v4+2*v5+v6+v7-1 # |
| |
f(v13)=v3+v7+v8+2*v9+v10-1
# |
| |
xs=v1+v2+v3+v4+v5+v6+v7+v8+v9+v10
# |
| |
v1(0)=0.05 |
| |
v2(0)=0.2 |
| |
v3(0)=0.8 |
| |
v4(0)=0.001 |
| |
v5(0)=0.5 |
| |
v6(0)=0.0007 |
| |
v7(0)=0.03 |
| |
v8(0)=0.02 |
| |
v9(0)=0.1 |
| |
v10(0)=0.1 |
| |
v11(0)=10 |
| |
v12(0)=10 |
| |
v13(0)=10 |
| |
|
|
|
| Variable/function values |
Variable |
Value |
f(x) |
| |
v1 |
0.05 |
0.3265 |
| |
v2 |
0.2 |
0.6378 |
| |
v3 |
0.8 |
-4.8659 |
| |
v4 |
0.001 |
-3.4105 |
| |
v5 |
0.5 |
-6.0029 |
| |
v6 |
0.0007 |
-2.8392 |
| |
v7 |
0.03 |
-8.1953 |
| |
v8 |
0.02 |
-5.2088 |
| |
v9 |
0.1 |
-9.5533 |
| |
v10 |
0.1 |
-5.0883 |
| |
v11 |
10 |
0.1507 |
| |
v12 |
10 |
0.0317 |
| |
v13 |
10 |
0.15 |
| |
xs |
1.8017 |
|
| |
|
|
|
| Solution |
Variable |
Value |
f(x) |
| |
v1 |
0.04070664967202 |
0 |
| |
v2 |
0.14796418434584 |
0 |
| |
v3 |
0.78211670254494 |
-1.78E-15 |
| |
v4 |
0.00141449528575 |
1.78E-15 |
| |
v5 |
0.48528331804737 |
3.55E-15 |
| |
v6 |
0.00069374665473 |
0 |
| |
v7 |
0.02732512196478 |
3.553E-15 |
| |
v8 |
0.01790081773259 |
1.776E-15 |
| |
v9 |
0.03710976393300 |
0 |
| |
v10 |
0.09843782989168 |
0 |
| |
v11 |
9.78442121489321 |
-2.22E-16 |
| |
v12 |
12.9690398976484 |
2.22E-16 |
| |
v13 |
15.2249662814130 |
0 |
| |
xs |
1.63895263007271 |
|
| |
|
|
|
| Additional information |
Only a constrained algorithm
that keeps the values of v1, v2,…v10 |
| |
positive throughout the iterations
converges from initial guess2. |