Create a model to solve an energy balance

Hi, I am trying to use MATLAB to create a model that can solve a set of equations. I have attached an image of the equations I am trying to solve. Essentially, there are 3 unknowns, Tout, Tp & Tg.

I wanted to ask which would be the best way to solve this on MATLAB, i have tried to use matrices but the temperatures I am getting are very high. Is there any other method that could be used to solve these equations?

Kind regards

[darova]

Can you attach the data? What values are known and what are uknown?

[kasim]

Hi darova

I have attached the data I have, those highlighted in yellow need to be determined iteratively which is where I keep going wrong.

Regards

[darova]

Did you try fsolve?

[kasim]

i haven't used fsolve in MATLAB before, I tried using matrices however that didn't work for me.

[darova]

You can't use matrices because you have nonlinear equations

[kasim]

Yes I am aware that was one of the mistakes I was making. Is there any tutorial or example I could follow that would show me how to use the fsolve in a case similar to this?

[darova]

MATLAB has good help page

Or just solve example

syms Tm Tp Tg hg hrpg
eq1 = ag*I + hg*(Tg-Tm) + hrpg*(Tp-Tg) - ...
eq2 = hg*A*(Tm-Tg) + ...
eqs = [eq1 eq2 .. eq6];
vars = symvar(eqs);
S = solve(eqs,vars);

[kasim]

Thank you for your response, I have tried that but it doesn't output any values? Or am i missing something?

clc

syms Tm Tp Tg Tout hrgs hrpg

I = 750; %Solar radiation

Ta = 330; %ambient temperature

Tin = Ta; %inlet temperature

m = 0.02; %mass flowrate

cp = 1; %specific heat capacity of air

ag = 0.01; %glass absorption

taug = 0.91; %glass transmissivity

ap = 0.95; %plate absorption

Eg = 0.96; %glass emittance

Ep = 0.3; %plate emittance

Sigma = 5.67E-8; %Boltzman constant

L = 0.2; %Collector Length

W = 1.2; %Collector Width

A=L*W; %Collector Area

hw = 5;

hp = 10;

hg = 10;

eq1 = ag*I + hg*(Tg-Tm) + hrpg*(Tp-Tg) - (hw+hrgs)*(Tg-Ta)

eq2 = hg*A*(Tm-Tg)+ m*cp*(Tout-Tin)-A*hp*(Tp-Tm)

eq3 = hp*(Tp-Tm)+hrpg*(Tp-Tg)--ap*taug*I

eq4 = 2*Tm-Tin-Tout

eq5 = hrpg - (Sigma*((Tp^2)+(Tg^2))*(Tp+Tg))/((1/Ep)+(1/Eg)-1)

eq6 = hrgs - Sigma*Eg*((Tg^4)-(Ta^4))/(Tg-Ta)

eqs = [eq1 eq2 eq3 eq4 eq5 eq6];

vars = symvar(eqs);

S = solve(eqs,vars);

[darova]

Try fsolve

[kasim]

Again, thank you!

with regards to this line: [Tg Tout Tp Tm hrpg hrgs] = deal(x(1),x(2),x(3),x(4),x(5),x(6));

is this the order in which the answers are outputted? i.e. Tg = Value 1, Tout = Value 2

[darova]

Yes

Experiment with x0 (initial guess) for different outputs

Answer 1

[Alex Sha]

Hi, I get two solutions:

No.	1	2
tg	-328.861557610457	-549.471818197512
tm	1.72830356360143	-541.712011528394
hrpg	-0.00727597589137702	-11.1180945451184
tp	326.846969797053	-548.48073838475
hrgs	0.0134500837438822	-4.90776624779464
tout	-326.543392872797	-1413.42402305679

[kasim]

Hi Alex, thank you for your response.

Unfortunately those solutions are not accurate for this model. For the model to be correct, the output temperatures should be in the range of 300-350 and not negative.

Kind regards

[Alex Sha]

if parameter tm is in the range of 300-350, then the result should be:

tm: 330.8818199199
tg: 330.390690546287
hrpg: 2.43454558766769
tp: 331.387646292175
hrgs: 7.83839735511287
tout: 331.7636398398

Feval:
3.02957658959713E-12
7.26019244723375E-12
1.32972655109143E-9
0
7.17292891749821E-12
-2.30926389122033E-14

[kasim]

Hi, I have found a mistake in my model, which is perhaps why you're getting different results.

eq1 should read as ag*I + hg*(Tg-Tm) + hrpg*(Tp-Tg) + (hw+hrgs)*(Tg-Ta)

Perhaps this may provide some accurate values?

[Alex Sha]

then the result will be:

tm: 330.112356511329
tg: 329.614806294714
hrpg: 2.41746002404232
tp: 330.611779336469
hrgs: 7.81080204482525
tout: 330.224713022658

Feval:
9.35251875944232E-12
-6.95421498164706E-12
-1.83834281131112E-10
-5.6843418860808E-14
-3.1827873669954E-12
-3.30402372128447E-13

[kasim]

Hi, Could you show me how you coded this?

Answer 2

[Afaf]

% Consts mcp_g2 = 1 * 800; alpha_g = 0.35; tau_g = 0.35; A = 1.20; G = 910; hc2 = 4.4; Tab = 30; hc1_g1_g2 = 3.8; hcl = 13.8; Ag2 = 1; sigma = 5.669e-6; S = 0.18; Tam = 30; Tg1 = 300; Eg1 = 0.35; Eg2 = 0.35;

% Time span and initial condition (chat gbtعشان تتوجد حلول المعادلة الحل دا % اقترحو(مكن تكون اي قيمة ما فارقة م t_span = [0 200]; % Simulation time in seconds Tg2_0 = 300; % Initial guess for Tg2

% the differential equation dTg2_dt = @(t, Tg2) (1 / mcp_g2) * (alpha_g * tau_g * A * G + hc2 * A * (Tab - Tg2) + hc1_g1_g2 * A * (Tg1 - Tg2) + Ag2 * sigma * ((Tg1^4 - Tg2^4)) / (1 / Eg1 + 1 / Eg2 - 1) + (Tab^2 + Tab^2) * (Tg1 + Tab) * (Tab - Tg2) * sigma * S / (1 / Eg1 + 1 / Eg2 - 1));

% Solve the ODE [t, Tg2] = ode45(dTg2_dt, t_span, Tg2_0);

% Plot of results figure; plot(t, Tg2, 'LineWidth', 2); xlabel('Time (s)'); ylabel('Temperature (K)'); title('Temperature vs Time'); grid on;

[Afaf]

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