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I need some assistance or feedback on a Matlab program that I working on by applying the method of finite difference to calculate and output the steady state space distribution of temperature and resulting rate of hear flow in an I Beam made of Duralumin. I have written a partial code but I am not sure if I am on the right path in solving this problem.

First: I divided the I Beam into a nodal grid, which would include 289 interior nodes (not including boundaries). The beam measures 2.4cm(.024m) x 3cm (.03m) and dx=dy=.001m.

Basically in my code I inputted all the variables given, which includes the upper/lower surface temperatures, thermoconductivity (K). Next, I created a outline of the beam by inputing the boundaries/ initial conditions. Then I applied an interior node equation that I looked up in my Heat Transfer Text:

T(m,n+1) + T(m,n-1) + T(m+1,n) +T(m-1,n) -4*T(m,n)=0

I applied this equation in 3 FOR loops for the interior nodes: Upper, center and lower sections of the I Beam. Next, I used a heat flux equation to calculate the rate of heat flow by applying: qx=k*((T1-T2)/L)

I used the same method as above to calculate this equation by stating the boundaries/ initial conditions and then apply 3 FOR loops for the three sections of the I Beam.

At this point I am not sure if I am applying the correct equations or methods. I get Temp. outputs for each interior nodes but at some pts. it seems that it is not correct. For the Heat Flux I obtain section that are 0 and neg., so that it wrong.

Any help or tips would be appreciated. Below is my code and I have attached the problem (NMProb.doc) if you would like to take a look at it. Thank you for your time.

clc

clear all;

T1 = input('Fixed Temperature at the upper surface Tt[60 deg C] : '); % deg C fixed Temperature at the upper surface.

T2 = input('Fixed Temperature at the lower surface. Tl[15 deg C] : '); % deg C fixed Temperature at the lower surface.

k = input('Enter the thermal conductivity k[Duralumin 164 W/m deg C] : '); % W/m deg C (thermal conductivity).

%Intilializing Constant Variables

k=164; %Thermoconductivity

Dn=2787; %Dn is defined as Density

Qi=0;

Qw=0;

%Outline dimensions of I beam

Wt=0.024; %Width

Ht=0.03; %Height

%Temperature Conditions

Tt=60; %Upper face Temperature (333.15K)

Tb=15; %Lower face Temperature (288.15K)

Ti=20; %Initial Temperature of interior nodes (293.15K)

%Conditions of nodal points

dx=.001; %Delta x - distance between each node

dy=.001; %Delta y - distance between each node

M=31; %Number of nodes in the Y direction -i

N=25; %Number of nodes in the X direction -j

%Applying Boundaries/ Initial Conditions to I Beam(Temperature)

T=zeros(31,25);

T(1,:)=T1; %Top Surface

T(31,:)=T2; %Bottom Surface

%Left

T(1:6,1)=0; %Upper Top Left

T(6,1:10)=0; %Bottom Top Left

T(6:26,10)=0; %Inner Center Left

T(26,1:10)=0; %Lower Bottom Left

T(26:31,1)=0; %Lower Left

%Right

T(1:6,25)=0; %Upper Top Right

T(6,16:25)=0; %Bottom Top Right

T(6:26,16)=0; %Inner Center Right

T(26,16:25)=0; %Lower Bottom Right

T(26:31,25)=0; %Lower Right

T(2:5,2:24)=Ti; %Upper Section

T(6:26,11:15)=Ti; %Center Section

T(27:30,2:24)=Ti; %Lower Section

T;

%Temperature Distribution of Interior Nodes (289 Nodal Points)

for(i=2:5)

for(j=2:24)

%Temperature of Nodal Points for Upper Portion

T(i,j)=(T(i,j+1)+T(i,j-1)+T(i+1,j)+T(i-1,j))/4;

end

end

for(i=6:26)

for(j=11:15)

%Temperature of Nodal Points for Center Portion

T(i,j)=(T(i,j+1)+T(i,j-1)+T(i+1,j)+T(i-1,j))/4;

end

end

for(i=27:30)

for(j=2:24)

%Temperature of Nodal Points for Bottom Portion

T(i,j)=(T(i,j+1)+T(i,j-1)+T(i+1,j)+T(i-1,j))/4;

end

end

Q=zeros(31,25);

Q(1,:)=Qw; %Top Surface

Q(31,:)=Qw; %Bottom Surface

%Left

Q(1:6,1)=Qw; %Upper Top Left

Q(6,1:10)=Qw; %Bottom Top Left

Q(6:26,10)=Qw; %Inner Center Left

Q(26,1:10)=Qw; %Lower Bottom Left

Q(26:31,1)=Qw; %Lower Left

%Right

Q(1:6,25)=Qw; %Upper Top Right

Q(6,16:25)=Qw; %Bottom Top Right

Q(6:26,16)=Qw; %Inner Center Right

Q(26,16:25)=Qw; %Lower Bottom Right

Q(26:31,25)=Qw; %Lower Right

Q(2:5,2:24)=Qi; %Upper Section

Q(6:26,11:15)=Qi; %Center Section

Q(27:30,2:24)=Qi; %Lower Section

%Rate of Heat Flow of Interior Nodes (289 Nodal Points

% for(i=2:5)

% for(j=2:24)

% %Temperature of Nodal Points for Upper Portion

% Q(i,j)=k*((T(i,j+1)-T(i,j))/dx);

% end

% end

for(i=6:26)

for(j=11:15)

%Temperature of Nodal Points for Center Portion

T(i,j)=(T(i,j+1)+T(i,j-1)+T(i+1,j)+T(i-1,j))/4;

end

end

%

% for(i=27:30)

% for(j=2:24)

% %Temperature of Nodal Points for Bottom Portion

% T(i,j)=(T(i,j+1)+T(i,j-1)+T(i+1,j)+T(i-1,j))/4;

% end

% end

Q;

fprintf(1,'Press enter and type T for Temp. Distribution');pause;

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# Matlab in applying Finite Difference for Temp. distribution/ rate of heat flow

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