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Example 2: Equilibrium Flash solved with different nonlinear solvers and SMath Studio installed on an Arch Linux ditstribution Source: Chemical Engineering, July 23, 1984, Nonlinear Models. Example 2. Equilibrium Flash. Given the feedrate and equilibrium pressure and temperature, and applying the criterion that: Ln ( sum ( y) / sum (x) )= 0, the equilibrium flash equation can be expressed as: ( King, C . J . Separation Process, McGraw Hill Book Co. NY, 1980.).

The x and K values for a mixture of methane ethane and propane at 213.7 K and 2,758 kPa are presented below. VF = V / F is the fraction vaporized in the flash destillatio, where V is the vapor flux and F is the liquid flux in the flash unit.

VF f K i el x i el * K i el 1 - (VF * 1 + / i 1 n sum x i el K i el 1 - (VF * 1 + / i 1 n sum / ln :

K 2.3283 0.2446 0.0527 31 mat :

x 0.6199 0.3231 0.057 31 mat :

n x length :

epsilon 106 -^:

iterations 400 :

Bound_P1 0 :

Bound_P2 1 :

The convergence is found for g(VF) = 0 and a good initial value of VF is equal to 1

res VF f VF Bound_P1 Bound_P2 solve :

res 0.5

res f 9.0474 109 -^*

res1 VF f Bound_P1 Bound_P2 epsilon epsilon iterations 110 Secant :

res1 0.5 511 mat i x1 x2 X3 f(x1) f(x2) f(X3) 001 0.3008 0.4222 0.9816 - 0.1902 11 0.3008 0.4142 0.9816 - 0.1902 0.0865 2 0.3008 0.4142 0.5089 0.1902 0.0865 0.0094 - 3 0.4142 0.5089 0.4996 0.0865 0.0094 - 0.0004 4 0.5089 0.4996 0.5 0.0094 - 0.0004 2.2148 106 -^* 5 0.4996 0.5 0.5 0.0004 2.2148 106 -^* 5.2245 1010 -^* - 77 mat 31 mat

Bisection Direct Quasi Regula Falsi

res2 VF f Bound_P1 Bound_P2 epsilon epsilon iterations 110 BDQRF :

res2 0.5 011 mat i Xdn Xup X f(Xdn) f(Xup) f(X) 010 0.5 0.9816 - 0.4222 7.9541 107 -^* 27 mat 31 mat

res3 VF f Bound_P1 epsilon epsilon iterations 110 Broyden :

res3 0.5 711 mat i x B(x) f(x) 00111 mat 0.4222 1 0.4222 - 1.6409 - 11 mat 0.6795 2 0.6928 1.225 - 11 mat 0.2307 - 3 0.4102 0.8802 - 11 mat 0.0904 4 0.4897 0.9995 - 11 mat 0.0108 5 0.5005 0.9525 - 11 mat 0.0005 - 6 0.5 0.947 - 11 mat 3.0735 106 -^* 7 0.5 0.9473 - 11 mat 8.737 1010 -^* 94 mat 31 mat

res4 VF f Bound_P2 epsilon epsilon iterations 110 NewtonRaphson :

res4 0.5 511 mat i x J(x) f(x) 01 7.0879 - 0.9816 - 1 0.8615 2.0719 - 0.5096 - 2 0.6156 1.2196 - 0.1309 - 3 0.5082 1.0656 - 0.0087 - 4 0.5 1.0557 - 4.0514 105 -^* - 5 0.5 1.0557 - 8.7631 1010 -^* - 74 mat 31 mat