Analyzing how temperature changes over time, often using the Finite Difference Method (FDM) or Finite Element Analysis (FEA). 2. Convection
Fourier's Law of Heat Conduction in one dimension is expressed as:
While the "RapidShare" link is likely dead and the "patched" software obsolete, the methodology contained within—solving partial differential equations numerically for thermal analysis—remains a cornerstone of modern engineering education.
Using the Central Finite Difference approximation for an interior node Analyzing how temperature changes over time, often using
The 2D steady-state heat conduction equation (Laplace equation) is . Using central differences, the temperature at node is approximately the average of its neighbors:
% Define the parameters alpha = 0.1; L = 1; T = 1; Nx = 100; Nt = 100;
𝜕T𝜕t=α𝜕2T𝜕x2the fraction with numerator partial cap T and denominator partial t end-fraction equals alpha the fraction with numerator partial squared cap T and denominator partial x squared end-fraction is the thermal diffusivity. Step 1: Discretize Time Using the Central Finite Difference approximation for an
% MATLAB Script: Transient Cooling using Lumped Capacitance Method clear; clc; % Material and Geometric Properties (Copper Sphere) rho = 8933; % Density (kg/m^3) Cp = 385; % Specific heat (J/kg*K) k_solid = 401; % Thermal conductivity of copper (W/m*K) D = 0.05; % Diameter (m) R = D / 2; % Radius (m) V = (4/3) * pi * R^3; % Volume of sphere A = 4 * pi * R^2; % Surface area of sphere % Environmental Conditions h = 200; % Convection coefficient (W/m^2*K) T_inf = 20; % Ambient fluid temp (C) T_i = 350; % Initial solid temp (C) % Verify Biot Number Criterion (Bi = h * L_c / k) Lc = V / A; % Characteristic length Bi = (h * Lc) / k_solid; fprintf('Calculated Biot Number: %.4f\n', Bi); if Bi > 0.1 warning('Biot number exceeds 0.1. Lumped capacitance may be inaccurate.'); else disp('Biot number is valid (< 0.1). Proceeding with lumped model.'); end % Time Vector Setup t_final = 500; % Total time simulation (seconds) t = linspace(0, t_final, 1000); % Analytical Solution Formula thermal_time_constant = (rho * V * Cp) / (h * A); T = T_inf + (T_i - T_inf) * exp(-t / thermal_time_constant); % Plot Temperature History figure; plot(t, T, 'b-', 'LineWidth', 2); grid on; xlabel('Time (seconds)'); ylabel('Temperature (^\circC)'); title('Transient Cooling Response of a Copper Sphere'); Use code with caution. 4. Radiation: Multi-Surface Enclosure Exchange
This example is a gold standard for engineers because it models a real-world scenario (heat dissipation from electronics) where nonlinearity comes from the ( T^4 ) radiation term.
Solving Heat Transfer Problems Using MATLAB: Comprehensive Practical Examples Lumped capacitance may be inaccurate
% Plot the results surf(x, t, T); xlabel('Distance'); ylabel('Time'); zlabel('Temperature');
% Step 3: Substitute and separate eqMain = subs(eqMain, lhs(eqTheta), rhs(eqTheta)); eqMain = eqMain / g(tau) / f(eta); % Separate the variables