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[WestwoodChemE]

Hello,

I would appreciate any assistance; when I try to solve this nonlinear system of equations I find that the balances (as I have written them) are underspecified. I am not sure whether the problem is ill-posed (this is an exploratory problem as part of an open-ended design project) or if I need to revise my equations.

It is a classic reaction loop problem with a flash separator & a purge. I would like to solve with matlab (newton's method) but before I do that I need to make sure that I have set the system up correctly.

reaction: a + 3 b --> 2 c
inert: d

streams:
1=makeup; 2 = reactor feed; 3 = reactor effluent; 4 = separator bottoms; 5 = separator overhead (goes to purge intersection); 6 = purge; 7 = recycle gas (from purge intersection to feed-makeup intersection)

Givens:
Flow rate of all 4 species in stream 1 is given: n1a,n1b,n1c,n1d.
One pass fractional conversion of species A in reactor is X1p.
Split fraction in flash separator of all 4 species is given: phia,phib,phic,phid
Overall conversion of a in the system = 96 %

Unknowns:
flowrates of all species in streams 2-7 & recycle fraction R. (6*4 + 1 = 25 unknowns)

Balances:
-4 @ makeup-recycle intersection
-4 @ reactor
-8 @ flash separator
-8 @ purge point
-1 @ overall inert balance

Code:

phia = 0.9;
phib = 0.9;
phic = (1-0.7653);
phid = 1;

n1a = 5110;
n1b = 15330;
n1c = 0;
n1d = 72;

X1p = .2;

%Mixer (4 equations @ makeup-recycle intersection)
F1 = n2a - n7a - n1a;
F2 = n2b - n7b - n1b;
F3 = n2c - n7c - n1c;
F4 = n2d - n7d - n1d;
%Reactor (4 equations)
F5 = n3a - n2a*(1-X1p);
F6 = n3b - (n2b-3*X1p*n2a);
F7 = n3c - 2*X1p*n2a;
F8 = n3d - n2d;%Flash/Separations (8 equations)
F9 = n5a - phia*n3a;
F10 = n4a - (1-phia)*n3a;
F11 = n5b - phib*n3b;
F12 = n4b - (1-phib)*n3b;
F13 = n5c - phic*n3c;
F14 = n4c - (1-phic)*n3c;
F15 = n5d - phid*n3d;
F16 = n4d - (1-phid)*n3d;
%Purge Intersection (8 equations)
F17 = n5a - n6a - n7a;
F18 = n5b - n6b - n7b;
F19 = n5c - n6c - n7c;
F20 = n5d - n6d - n7d;
F21 = n5a*R - n7a;
F22 = n5b*R - n7a;
F23 = n5c*R - n7c;
F24 = n5d*R - n7d;
%Overall Inert Balance
F25 = n1d - n6d;

I am especially not sure as to whether my 25th equation should be an overall inert balance or if I somehow need to work the 96% overall conversion in. Either equation still results in an underspecified system. Do I have too many equations somewhere else?

Again, thanks for your time.

Edited by Ankur Amlani, 23 April 2012 - 12:59 AM.