%2D Finite-difference transient conduction calculator%Greg Teichert & Kyle Halgren%This program calculates the transient conduction of a rectangular cross%section.%%% h(2), T_inf(2)% _________________% | |% h(1), T_inf(1) | | a h(3), T_inf(3)% |_________________|% b% h(4), T_inf(4)

%%Inputs%Convective heat transfer coefficients%For a side to be completely insulated, set h = 0.%For a side to be a constant temperature, set h to be very high%The program assumes that the convective coefficients of the corners are%the average of the two adjecent sidesh = zeros(4,1);%this is to set up thevector of h values. DO NO CHANGE.

%Userdefined h valuesh(1) = 10;h(2) = .1;h(3) = 10;h(4) = .1;

%Boundary conditions%Userdefined T infinity values in kelvinT_inf(1) = 293;T_inf(2) = 293;T_inf(3) = 353;T_inf(4) = 353;

%Initial condition (assume uniform initial temperature)%Userdefined initial temperature valueT_0 = 573;

%Material properties%Userdefined material valuesk = .08;rho = 7480;c_p = .460;

%Userdefined physical variablesa = 1; %height of cross sectionb = 1.3; %width of cross sectiont = 3600; %time at which results are given

%%Calculations%alpha (thermal diffusivity)alpha = k/(rho*c_p);

%delta_x (space increment, equal to delta_y)delta_x = gcd(a*100000,b*100000)/100000;while delta_x > min(a,b)/20 delta_x = delta_x/2;end

%Biot numbersBi = zeros(4,1);Bi = h*delta_x/k;

%delta_t (time increment)Bi_max = max(Bi);delta_t1 = 0.25*delta_x^2/alpha;delta_t2 = delta_x^2/(2*alpha*(2 + Bi_max));delta_t3 = delta_x^2/(4*alpha*(1 + Bi_max));delta_t = min([delta_t1 delta_t2 delta_t3]);

%Fourier numberFo = alpha*delta_t/delta_x^2;

%%Create matrices for solution%Creation of two constants that are frequently used in the code; they are%the number of nodes in a column and row of nodes, respectively.a_step = a/delta_x + 1;b_step = b/delta_x + 1;totalnodes = a_step*b_step;

%Master temperature matrix: Each column contains the nodal temperatures of%the cross section for one time step. The columns are made made going down%the column of nodes in the cross section, from left to right.T = zeros(totalnodes,round(t/delta_t) + 1);

%Initial conditionT(1:totalnodes) = T_0*ones(totalnodes,1);

for k = 2:(round(t/delta_t) + 1) %Setting up the coefficient matrix and constant matrix coeff = zeros(totalnodes,totalnodes); const = zeros(totalnodes,1); for i = 1:totalnodes %Top left corner if i == 1 coeff(i,i + a_step) = 2*Fo; coeff(i,i + 1) = 2*Fo; const(i,1) = 4*Fo*(Bi(1)*T_inf(1) + Bi(2)*T_inf(2))/2; coeff(i,i) = 1 - 4*Fo - 4*Fo*(Bi(1) + Bi(2))/2; %Top right corner elseif i == (totalnodes - a/delta_x) coeff(i,i - a_step) = 2*Fo; coeff(i,i + 1) = 2*Fo; const(i,1) = 4*Fo*(Bi(2)*T_inf(2) + Bi(3)*T_inf(3))/2; coeff(i,i) = 1 - 4*Fo - 4*Fo*(Bi(2) + Bi(3))/2; %Bottom left corner elseif i == a_step coeff(i,i + a_step) = 2*Fo; coeff(i,i - 1) = 2*Fo; const(i,1) = 4*Fo*(Bi(4)*T_inf(4) + Bi(1)*T_inf(1))/2; coeff(i,i) = 1 - 4*Fo - 4*Fo*(Bi(4) + Bi(1))/2; %Bottom right corner elseif i == totalnodes coeff(i,i - a_step) = 2*Fo; coeff(i,i - 1) = 2*Fo; const(i,1) = 4*Fo*(Bi(3)*T_inf(3) + Bi(4)*T_inf(4))/2;

coeff(i,i) = 1 - 4*Fo - 4*Fo*(Bi(3) + Bi(4))/2; %Left side elseif 1 < i && i < a_step coeff(i,i + a_step) = 2*Fo; coeff(i,i - 1) = Fo; coeff(i,i + 1) = Fo; const(i,1) = 2*Fo*Bi(1)*T_inf(1); coeff(i,i) = 1 - 4*Fo - 2*Bi(1)*Fo; %Right side elseif (totalnodes - a/delta_x) < i && i < totalnodes coeff(i,i - a_step) = 2*Fo; coeff(i,i - 1) = Fo; coeff(i,i + 1) = Fo; const(i,1) = 2*Fo*Bi(3)*T_inf(3); coeff(i,i) = 1 - 4*Fo - 2*Bi(3)*Fo; %Bottom elseif rem(i/a_step,2) == 0 || rem(i/a_step+1,2) == 0 coeff(i,i - 1) = 2*Fo; coeff(i,i + a_step) = Fo; coeff(i,i - a_step) = Fo; const(i,1) = 2*Fo*Bi(4)*T_inf(4); coeff(i,i) = 1 - 4*Fo - 2*Bi(4)*Fo; %Top elseif rem((i - 1)/a_step,2) == 0 || rem((i - 1)/a_step+1,2) == 0 coeff(i,i + 1) = 2*Fo; coeff(i,i + a_step) = Fo; coeff(i,i - a_step) = Fo; const(i,1) = 2*Fo*Bi(2)*T_inf(2); coeff(i,i) = 1 - 4*Fo - 2*Bi(2)*Fo; %Inside else coeff(i,i + a_step) = Fo; coeff(i,i - a_step) = Fo; coeff(i,i + 1) = Fo; coeff(i,i - 1) = Fo; coeff(i,i) = 1 - 4*Fo; end end T(:,k) = const + coeff*T(:,k-1);end

%%Create the final temperature distribution matrix of the cross section

T_distribution = zeros(a_step,b_step);

for j = 1:b_step T_distribution(:,j) = T(((j - 1)*a_step + 1):j*a_step,round(t/delta_t) + 1);end

%Plot T_distributionx = 0:delta_x:b;y = 0:delta_x:a;

figure(1)clfsurf(y,x,T_distribution')

xlabel('a (m)')ylabel('b (m)')

zlabel('Temperature (K)')title('Transient conduction (the origin of the plot is the top left corner of the cross section)')