heat transfer for spherical coordinates using matlab - matlab

Question

I want to apply heat transfer ( heat conduction and convection) for a hemisphere. It is a transient homogeneous heat transfer in spherical coordinates. There is no heat generation. Boundary conditions of hemisphere is in the beginning at Tinitial= 20 degree room temperature. External-enviromental temperature is -30 degree. You can imagine that hemisphere is a solid material. Also, it is a non-linear model, because thermal conductivity is changing after material is frozen, and this going to change the temperature profile.
I want to find the temperature profile of this solid during a certain time until center temperature reach to -30degree.
In this case, Temperature depends on 3 parameters : T(r,theta,t). radius, angle, and time.
1/α(∂T(r,θ,t))/∂t =1/r^2*∂/∂r(r^2(∂T(r,θ,t))/∂r)+ 1/(r^2*sinθ )∂/∂θ(sinθ(∂T(r,θ,t))/∂θ)
I applied finite difference method using matlab, however program does not calculate anything for inner nodes of the hemisphere, and just giving me initial temperatures values (Which is Told in here) . You can see some scripts which i used for inner nodes.
% initial conditions
Tair = -30.0; % Temperature of air
Tin = 21;
% setting initial values for grid
for i=1:(nodes)
for j=1:(nodes)
Told(i,j) = Tin;
Tnew(i,j) = Tin;
frozen(i) = 0;
latent(i) = Qs*mass(i)*Water/dt;
k(i) = 0.5;
cp(i) = cw;
W(i) = Water;
l(i) = 0;
S(i) = 1-Water;
end
end
%Simulation conditions
J = 9; % No. of space steps
nodes = J+1; % Number of nodes along radius or theta direction
dt =0.1;
t = 0; % time index on start
tmax = 7000; % Time simmulation is supposed to run
R = d/2;
dr = (d/2)/J; % space steps in r direction
y = pi/2; % (theta) for hemisphere
dy = (pi/2)/J; % space steps in Theta direction
% Top surface condition for hemisphere
i=nodes;
for j=1:1:(nodes-1)
Qcd_ot(i,j) = ((k(i)+ k(i-1))/2)*A(i-1)*(( Told(i,j)-Told(i-1,j))/(dr)); % heat conduction out of nod
Qcv(i,j) = h*(Tair-Told(i,j))*A(i); % heat transfer through convectioin on surface
Tnew(i,j) = ((Qcv(i,j)-Qcd_ot(i,j))/(mass(i)*cp(i))/2)*dt + Told(i,j);
end %end of for loop
% Temperature profile for inner nodes
for i=2:1:(nodes-1)
for j=2:1:(nodes-1)
Qcd_in(i,j)= ((k(i)+ k(i+1))/2)*A(i) *((2/R)*(( Told(i+1,j)-Told(i,j))/(2*dr)) + ((Told(i+1,j)-2*Told(i,j)+Told(i-1,j))/(dr^2)) + ((cot(y)/(R^2))*((Told(i,j+1)-Told(i,j))/(2*dy))) + (1/(R^2))*(Told(i,j+1)-2*Told(i,j)+ Told(i,j-1))/(dy^2));
Qcd_out(i,j)= ((k(i)+ k(i-1))/2)*A(i-1)*((2/R)*(( Told(i,j)-Told(i-1,j))/(2*dr)) +((Told(i+1,j)-2*Told(i,j)+Told(i-1,j))/(dr^2)) + ((cot(y)/(R^2))*((Told(i,j)-Told(i,j-1))/(2*dy))) + (1/(R^2))*(Told(i,j+1)-2*Told(i,j)+ Told(i,j-1))/(dy^2));
Tnew(i,j) = (Qcd_in(i,j)-Qcd_out(i,j))/(mass(i)*cp(i)))*dt + Told(i,j);
end
end
%bottom of the hemisphere solid
Tnew(:,nodes)=-30;
Told=Tnew;
t=t+dt;
EDIT *Thanks, now the scripts are working and calculating. And i can see temperature profile for model system.
However, i want to plot in a 2D or 3D plot for this hemisphere temperature profile. Also if it is possible i would like to run animation for temperature change during certain time. The codes what i am using for simulation and to write a file is
t=0;
tmax=7000;
...................
.....................
ss=0; % index for printouts
%start simulation
while t<tmax
ss=ss+1;
.............
.................
................
if ss==2000
dlmwrite('d:\Results_for_model.txt',Tnew,'-append');
ss=0;
end
end % end of while loop
Do you have any suggestion for it ? Because in text file, for Tnew(i,j) values, after every 10 rows, model calculates for next dt value. Therefore, results data looks like a mess, after every 10 rows, it gives for next time values results.
Is there any way to coordinate to write this results according to specific rows and columns ( because otherwise huge amount of data are needed to be organized again and again) ?
and i want to plot in 3d plot for this temperature profile which is hemisphere in this case, i have Tnew(r,theta,t), but i am confused to how to represent this temperature profile to show in a hemisphere graph. I would like to hear your suggestions about it. Thanks in advance !!

Answer 1

To be precise, the proper syntax as defined by MATLAB for the for loop construct is
for index = values
program statements
...
end
where values has one of the following forms:
initval:endval
initval:step:endval
valArray
your code is parsed to initval:endval = 9:2, which means the loop runs 0 times, resulting in no calculations.