heat transfer for spherical coordinates using 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.