solving iterative equations in matlab

For a heat exchanger modelling, I would like to give different equations into the Matlab, and solve the unknowns with respect to equations' relationship. For example these are different equations, and parameters:

. . .

rhogi=rhogo*Pin1g*Tout1g/(Pout1g*Tin1g);
  rhog=0.5*(rhogo+rhogi);
  Gg=m1g/**Aflow**;
  Dh=4***Aflow**/n*pi*Do;
  Redh=Gg*Dh/nug;
  fo=(delpag-(Gg^2*(1/rhogo-1/rhogi)))*rhog*0.5*Dh/(L*Gg^2);

    ratio=Redh/Red;
    fo=f*ratio^-0.2;   %f=0.046*Redh^-0.2

    h=0.95*Ga*0.5*f*(power(Pra,-2/3));
    ho=cpg*Gg*0.5*fo*(power(Prg,-2/3));

    X=Do/D;

    UAh=UA/(h*n*pi*D*L);UAho=UA/(ho.n.pi.Do*L);

    UAho=1-Uah;
    Adim=**Aflow**/n*D^2;
...

, and among them for example "Aflow" shall exist in three different equations; but in the first line, Matlab stops due to "undefined variable Aflow". I need Matlab to go over all the equations, and find Aflow itself. How can I handle this?

Answer 1

[Walter Roberson]

There are three approaches:

1. Use the Symbolic Toolbox with symbolic equations and use dsolve() or like routines to find a symbolic solution if possible. This would involve defining Aflow as a symbolic variable and solving for it
2. Use iterative equations. Initialize Aflow to something (0 perhaps) and run a calculation using that value and use the results of the calculation to refine the Aflow value, repeating this process until Aflow stabilizes, indicating consistent equations
3. Use an ODE routine such as ode45() or ode15s() or possibly bvp4c() depending what you are trying to do.

[fert]

syms D L n Aflow X
Ga=m1a/(pi*D^2/4*n);
Red=Ga*D/nua;
F1=(delpaa-(Ga^2*(1/rhoao-1/rhoai)))*rhoa*0.5*D/(L*Ga^2)-f;  %%Equation 1

%%Pressure Drop in Gas Side
delpag=Pin1g-Pout1g;
rhogo=(Pout1g/287*Tout1g)
rhogi=rhogo*Pin1g*Tout1g/(Pout1g*Tin1g);
rhog=0.5*(rhogo+rhogi);

Gg=m1g/(Aflow);
X=Do/D;
Dh=4*Aflow/n*pi*Do;
Redh=Gg*Dh/nug;
F2=(delpag-(Gg^2*(1/rhogo-1/rhogi)))*rhog*0.5*Dh/(L*Gg^2)-fo;

ratio=Redh/Red;
fo=f*ratio^-0.2;   %f=0.046*Redh^-0.2

F3=0.95*Ga*0.5*f*(power(Pra,-2/3))-h;
F4=cpg*Gg*0.5*fo*(power(Prg,-2/3))-ho;

F5=(UA/(h*n*pi*D*L))+(UA/(ho.n.pi.Do*L))-1;

% UAho=1-Uah;
Adim=Aflow/n*D^2;

S = solve(F1,F2,F3,F4,F5,D, L, n, Aflow, X);
S.D
S.L
S.n
S.Aflow
S.X

when I execute above code I get the error of "SYM objects do not allow nested indexing. Assign intermediate values to variables instead." in the line of "F5"? What I am doing wrong, and how can it be handled?

[Walter Roberson]

My guess is that you want to replace the "." with "*" to indicate multiplication.

[fert]

So sorry. Thank you very much!!! However, now it gives the:

"Undefined function or variable 'Do'. X=Do/D;"

But, I already put the D and X to the syms so, it should guess also the Do, shouldn't it?

[Walter Roberson]

Maybe. Probably. Is it an input to be given later, or is it a quantity to be solved for, or does it represent some particular operation on D, or ... ?

[fert]

%%%Recuperator 1
Tin1g=197.17+273; Tin1a=29.71+273;
Tout1g=90+273;Tout1a=167.21+273;
Pin1g=1.023;Pin1a=1.176;
Pout1g=1.013;Pout1a=1.166;
m1g=13.53;m1a=11.16;  
%%Heat Exchanger Design

Cair1=0.95*m1a;  %Tairaverage=371.46 [K]
Cgas1=Cair1*(Tout1a-Tin1a)/(Tin1g-Tout1g); cpg=Cgas1/m1g;
%Cmin=Cair
E=Cair1*(Tout1a-Tin1a)/(Cair1*(Tin1g-Tin1a));
NTU=log((1-(E*Cair1/Cgas1))/(1-E))/(1-(Cair1/Cgas1));
UA=Cair1*NTU;

%%Pressure Drop in Air Side
delpaa=Pin1a-Pout1a;
rhoai=(Pin1a/287*Tin1a);
rhoao=rhoai*Pout1a*Tin1a/(Pin1a*Tout1a);
rhoa=0.5*(rhoao+rhoai);
nua=3.407;
nug=3.600;
Pra=0.685;
Prg=0.686;

syms D L n Aflow Do 
Ga=m1a/(pi*D^2/4*n);
Red=Ga*D/nua;
f=(delpaa-(Ga^2*(1/rhoao-1/rhoai)))*rhoa*0.5*D/(L*Ga^2);  %%Equation 1

%%Pressure Drop in Gas Side
delpag=Pin1g-Pout1g;
rhogo=(Pout1g/287*Tout1g)
rhogi=rhogo*Pin1g*Tout1g/(Pout1g*Tin1g);
rhog=0.5*(rhogo+rhogi);

Gg=m1g/(Aflow);
Do=1.2*D;
Dh=4*Aflow/n*pi*Do;
Redh=Gg*Dh/nug;
fo=(delpag-(Gg^2*(1/rhogo-1/rhogi)))*rhog*0.5*Dh/(L*Gg^2);

ratio=Redh/Red;
fo=f*ratio^-0.2;   %f=0.046*Redh^-0.2

h=0.95*Ga*0.5*f*(power(Pra,-2/3));
ho=cpg*Gg*0.5*fo*(power(Prg,-2/3));

F5=(UA/(h*n*pi*D*L))+(UA/(ho*n*pi*Do*L))-1;

% UAho=1-Uah;
Adim=Aflow/n*D^2;

S = solve(f,fo,h,ho,F5,D, L, n, Aflow,Do);
S.D
S.L
S.n
S.Aflow
S.Do

This is my whole code. Now I am getting the error of: "Reference to non-existent field 'Do'.

Error in EnergySystem (line 69) S.Do "

[fert]

Lets forget about the "solve" function. How can I solve this with "for" loops?

[Walter Roberson]

After you defined Do as being a symbol, you used

Do=1.2*D;

which redefined it in terms of the symbol D. It is now an expression rather than a symbol by itself, so it cannot be solved for.

Caution: you define

fo=(delpag-(Gg^2*(1/rhogo-1/rhogi)))*rhog*0.5*Dh/(L*Gg^2);

but a couple of lines later you have

fo=f*ratio^-0.2;   %f=0.046*Redh^-0.2

which redefines fo as being completely different. There is no point in calculating the first expression if you are going to change the variable before even using it ?