find the inner surface of a wall undergoing transient heat conduction and a heat flux for 0.2 seconds in matlab

To find the inner surface temperature of a wall undergoing transient heat conduction and subjected to a heat flux for 0.2 seconds, you can use the finite difference method in MATLAB.

Assuming a 1D heat transfer through a wall with thickness dx, and an initial uniform temperature Ti, the transient heat conduction equation can be written as:

main.m
rho * cp * A * dT/dt = k * A * (dT/dx)^2 + Q
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where

  • rho is the density of the material,
  • cp is the specific heat capacity of the material,
  • k is the thermal conductivity of the material,
  • Q is the heat generation rate (in this case, the heat flux),
  • A is the cross-sectional area of the wall,
  • T is the temperature of the wall at a given time and position.

Using the finite difference approach, the equation can be discretized in both time and space. For example, the temperature at position i and time n can be represented as Ti,n. The finite difference method can be written as:

Ti,n+1 = Ti,n + (k * dt / (rho * cp * dx^2)) * (Ti+1,n - 2Ti,n + Ti-1,n) + (Q * dt / (rho * cp * A))

where dt is the time step.

To implement this in MATLAB, we can initialize the parameters of the problem and set up a loop to solve for the temperature at each time step. Here's an example code:

main.m
% Parameters
rho = 2700;         % density of the wall material, kg/m^3
cp = 900;           % specific heat capacity of the wall material, J/kg-K
k = 2.5;            % thermal conductivity of the wall material, W/m-K
T0 = 20;            % initial temperature of the wall, degC
L = 0.1;            % thickness of the wall, m
dx = 0.001;         % spatial step size, m
Q = 10000;          % heat flux, W/m^2
A = 1;              % cross-sectional area, m^2

% Time step
tmax = 0.2;         % duration of heating, sec
dt = 0.01;          % time step size, sec
t = 0:dt:tmax;      % time vector

% Initial temperature profile
nx = L/dx + 1;      % number of spatial points
Ti = T0 * ones(nx,1);% temperature at all spatial points
Ti(nargout > 0) = Ti; % output initial temperature profile if requested

% Finite difference solution
for n = 1:length(t)-1
    dTi = (k * dt / (rho * cp * dx^2)) * (Ti(3:end) - 2*Ti(2:end-1) + Ti(1:end-2)) + (Q * dt / (rho * cp * A));
    Ti(2:end-1) = Ti(2:end-1) + dTi;
end

% Plot the temperature profile at final time
x = (0:nx-1) * dx;
plot(x, Ti);
xlabel('Distance (m)');
ylabel('Temperature (degC)');
title(['Temperature profile after ', num2str(tmax), ' sec of heating']);
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This code will output a plot of the temperature profile at the final time. Note that this is a simplified example and does not account for certain effects, such as radiation heat transfer and temperature-dependent material properties.

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