temperature of a plane wall originally at 20 degrees subject to a large heat flux applied for 0.2 seconds in matlab

Assuming that the plane wall has a constant thermal conductivity and constant density, we can use the following one-dimensional heat conduction equation to calculate the temperature distribution:

$\frac{\partial T}{\partial t} = \frac{\alpha}{\rho c_p} \frac{\partial^2 T}{\partial x^2}$

where $T$ is the temperature as a function of time $t$ and position $x$ along the plane wall. $\alpha$ is the thermal diffusivity given by $\alpha = \frac{k}{\rho c_p}$, where $k$ is the thermal conductivity, $\rho$ is the density, and $c_p$ is the specific heat capacity.

Since the heat flux is already known, we can apply the following boundary condition at the surface of the wall:

$-k\frac{\partial T}{\partial x}\bigg|_{x=0} = q''$

where $q''$ is the heat flux applied and negative sign indicates direction toward the wall surface. During the period of heating, $q''$ is assumed to stay constant.

To solve this equation numerically, we need to discretize the wall into small segments and approximate the second derivative using finite difference method:

$\frac{\partial^2 T}{\partial x^2} \approx \frac{T(x+\Delta x)-2T(x)+T(x-\Delta x)}{(\Delta x)^2}$

where $\Delta x$ is the size of each segment. Then, we can use the following forward difference scheme to move forward in time:

$T_i^{n+1} = T_i^n + \frac{\Delta t \alpha}{(\Delta x)^2 \rho c_p}(T_{i+1}^n - 2T_i^n + T_{i-1}^n)$

where $i$ is the index for each segment, $n$ is the index for each time step, and $\Delta t$ is the time step size. The initial condition is $T_i^0 = 20$ for all $i$.

We can implement this scheme in Matlab as follows:

main.m
% Define parameters
k = 1;          % thermal conductivity
rho = 1;        % density
c_p = 1;        % specific heat capacity
alpha = k/(rho*c_p);    % thermal diffusivity
T0 = 20;        % initial temperature
q = 100000;     % heat flux
t_end = 0.2;    % end time
dx = 0.01;      % segment size
dt = 0.001;     % time step size

% Discretize space and time
x = 0:dx:1;     % wall segment
t = 0:dt:t_end; % time steps

% Set initial condition
T = ones(length(x),length(t)) * T0;

% Solve using forward difference scheme
for n = 1:length(t)-1
    for i = 2:length(x)-1
        T(i,n+1) = T(i,n) + alpha*dt/((dx)^2*rho*c_p)*(T(i+1,n) - 2*T(i,n) + T(i-1,n));
    end
    % Apply boundary condition at surface
    T(1,n+1) = T(2,n+1) - dx*q/k;
end

% Plot temperature distribution
figure
plot(x,T(:,end),'LineWidth',2)
xlabel('Distance (m)')
ylabel('Temperature (C)')
title('Temperature Distribution after 0.2 seconds')
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This will give us the temperature distribution across the wall at the end of the heating period, which can be plotted using plot(x,T(:,end)).