differential equations for heat transfer on a wire in matlab

To solve differential equations for heat transfer on a wire in MATLAB, you can use the built-in function pdepe. Here are the steps to follow:

1. Define the problem: Write down the partial differential equation (PDE) that describes heat transfer on the wire. For example, if we assume the wire is one-dimensional and has constant thermal conductivity, the PDE can be written as:

   ρ * c * ∂T/∂t = ∂/∂x(k * ∂T/∂x)

   where:

   • ρ is the density of the wire,
   • c is the specific heat capacity of the wire,
   • T is the temperature of the wire,
   • t is time, and
   • k is the thermal conductivity of the wire.

2. Set the boundary conditions: Specify the temperature at the boundaries of the wire. For example, you may have fixed temperature values at both ends or a combination of fixed temperature and heat flux conditions.

3. Define the initial condition: Specify the initial temperature distribution along the wire.

4. Set up the MATLAB code: Write a MATLAB script or function that sets up the problem using pdepe function. Here's an example code snippet:

main.m
function temperature = solveHeatingWire()
    % Parameters
    rho = 7800; % kg/m^3
    c = 500; % J/(kg*K)
    k = 30; % W/(m*K)
    L = 1; % length of the wire (m)
    tspan = [0 100]; % time span for the simulation

    % Solve the PDE
    m = 0;
    sol = pdepe(m, @heatPDE, @heatIC, @heatBC, @heatX, tspan);

    % Extract the temperature profile
    temperature = sol(:,:,1);

    % Plot the results
    x = heatX();
    t = sol(end,:,2);
    surf(x, t, temperature)
    xlabel('Distance (m)')
    ylabel('Time (s)')
    zlabel('Temperature (K)')
end

function [c, f, s] = heatPDE(x, t, u, DuDx)
    % PDE function
    rho = 7800; % kg/m^3
    c = 500; % J/(kg*K)
    k = 30; % W/(m*K)
    c = rho * c;
    f = k * DuDx;
    s = 0;
end

function u0 = heatIC(x)
    % Initial condition
    u0 = 273 + 10 * sin(pi * x);
end

function [pl, ql, pr, qr] = heatBC(xl, ul, xr, ur, t)
    % Boundary conditions
    pl = 0; % insulated boundary
    ql = 1;
    pr = 0; % insulated boundary
    qr = 1;
end

function x = heatX()
    % Spatial discretization
    L = 1; % length of the wire (m)
    Nx = 100; % number of points on the wire
    x = linspace(0, L, Nx);
end
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5. Run the code: Call the solveHeatingWire function to solve the heat transfer problem and visualize the results.

Note: Adjust the parameters and boundary conditions according to your specific problem. Also, make sure to check the MATLAB documentation of pdepe for more detailed information and examples.

This code sets up a one-dimensional heat transfer problem on a wire with specific properties (density, specific heat capacity, and thermal conductivity). It uses the pdepe function to solve the PDE numerically. The initial condition is a sinusoidal temperature distribution, and the boundaries are assumed to be insulated (zero heat flux). Finally, the code plots the temperature evolution over time and space using a 3D surface plot.