model temperature as a function of position in a plane wall undergoing transient heat conduction with cyclic heat flux applied to one side and negligible h in matlab

To model temperature as a function of position in a plane wall undergoing transient heat conduction with cyclic heat flux applied to one side and negligible h in MATLAB, we can use the finite difference method.

The equation for transient heat conduction in one dimension is given as:

where T(x, t) is the temperature at position x and time t, k is the thermal conductivity, ρ is the density, c is the specific heat, and Q is the heat source.

We can discretize the above equation using the central difference method and obtain the following expression:

where i is the position index and n is the time index, Δx and Δt are the spatial and time steps, respectively.

To solve the above equation numerically using MATLAB, we can use a for loop to iterate through all the spatial and time steps and calculate the temperature at each point using the above equation. Here is a sample code that demonstrates how to implement the finite difference method for transient heat conduction:

main.m
% Parameters
L = 1; % Length of wall
nx = 100; % Number of grid points
dx = L/(nx-1); % Spatial step
k = 1; % Thermal conductivity
rho = 1; % Density
c = 1; % Specific heat
alpha = k/(rho*c); % Thermal diffusivity
h = 0; % Heat transfer coefficient
q0 = 1; % Cyclic heat flux
t_end = 100; % End time
nt = 10000; % Number of time steps
dt = t_end/nt; % Time step

% Initialize temperature matrix
T = zeros(nx, nt);

% Boundary conditions
T(1,:) = 0; % Left boundary: T = 0
T(nx,:) = 0; % Right boundary: T = 0

% Initial condition
T(:,1) = 0; % T = 0 everywhere at t = 0

% Main loop
for n = 1:nt-1
    Q = q0*sin(2*pi*n*dt/t_end); % Cyclic heat flux
    for i = 2:nx-1
        T(i,n+1) = T(i,n) + alpha*dt/dx^2*(T(i+1,n)-2*T(i,n)+T(i-1,n)) + dt/(rho*c*dx)*Q;
    end
end

% Plot temperature profile
x = linspace(0,L,nx);
t = linspace(0,t_end,nt);
[X,T] = meshgrid(x,t);
surf(X,T,T');
xlabel('Position');
ylabel('Time');
zlabel('Temperature');
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In this code, we first define the parameters such as the length of the wall, number of grid points, thermal conductivity, density, specific heat, thermal diffusivity, heat transfer coefficient, cyclic heat flux, end time, and number of time steps.

Next, we initialize the temperature matrix as a nx by nt matrix and set the boundary and initial conditions.

In the main loop, we first calculate the cyclic heat flux at the current time step using the sin function. Then, we use a for loop to update the temperature at each point using the finite difference method and the discretized equation.

Finally, we plot the temperature profile as a function of position and time using the surf function.

Note that since the heat transfer coefficient is negligible (h = 0), there is no heat loss from the wall and hence we only need to consider the heat flux from the cyclic heat source.

I hope this helps!