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Octave Script

OCTAVE is a free program highly compatible to MATLAB. It is a handy tool to solve differential equations numerically with built-in feature or use numerical methods dealing with inversion of matrices. A graphical plot of the results can be generated with ease.

The purpose of this calculation approach is to demonstrate the numerical technique and TDMA (Tri-Diagomal Matrix Algorithm) method to solve transient conduction. Various improvisation is possible in this script such as implementation of Heat Flux and Convective boundaries at the tip, incorporating variable cross-section area such as tapered rod and so on.

% Solve 1D heat equation with Backward Time Centre Space scheme % d^2T 1 dT % ---- = ----- * ---- % dx^2 alpha dt % clear; clc; % L = length of the domain [m] L = 0.1; % NN = number of mesh nodes [number] NN = 21; % nTime = number of steps [number], delta_t = tMax / nTime nTime = 4; % tMax = maximum time for the simulation [s] tMax = 5.0; % alpha = diffusion coefficient [m^2/s] % Thermal conductivity [W/m.K] k = 150; % Density [kg/m^3] rho = 2700; % Specific heat capacity [J/kg.K] Cp = 450; alpha = k/rho/Cp; % Specify initial conditions dx = L / (NN - 1); x = [0 : dx: L]; phi = 25 + 10 * sin(pi .* x / L); % Specify boundary conditions phi0 = 10; phiL = 50; dt = tMax / nTime; % Check for Courant number for numerical stability and bounded solution CoNum = alpha * dt / dx^2; %If (CoNum >= 2.0) % N = L / ((alpha * dt / 2.0)^0.5); % NN = round(N); % dx = L / (NN - 1); %endif % --- Coefficients of the Tri-Diagonal System % T^n(i-1) - 2T^n(i) + T^n(i+1) 1 T^n - T^[n-1](i) % ----------------------------- = ----- * -------------- % dx^2 alpha dt % .-- --. % alpha | 1 2*alpha | alpha T^[n-1](i) % - ----- T^n(i-1) + | --- - ------- |T^n(i) - -----T^n(i+1) = ---------- % dx^2 | dt dx^2 | dx^2 dt % |__ __| % % - C * T^n(i-1) + A * T^n(i) - B * T^n(i+1) = d(i), i = 1 ... NN % % A* T^n(i) = B(i) * T(i+1) + C(i) * T(i-1) + d(i), i = 1 ... NN % subdiagonal C: coefficients of phi(i-1) B = (alpha/dx^2) * ones(NN,1); % superdiagonal B: coefficients of phi(i+1) C = B; % diagonal A: coefficients of phi(i) A = (1/dt) * ones(NN, 1) + 2 * B; % constant term d d = zeros(NN, 1); % Plot temperature profile at t = 0 [s] hold on; plot(x, phi, "linestyle", ":", "linewidth", 2, "marker", "o"); for m = 1:nTime % Tridiagonal Matrix Algorithm [Ref: S. V. Patankar] % If T(1) is specified A(1) = 1; B(1) = 0; % Since T(0) and T(NN+1) have no significance B(end) = 0; C(1) = 0; d(1) = phi0 / dt; P(1) = B(1) / A(1); Q(1) = d(1) / A(1); for i = 2: NN d(i) = phi(i) / dt; P(i) = B(i) / (A(i) - C(i) * P(i-1)); Q(i) = (d(i) + C(i) * Q(i-1)) / (A(i) - C(i) * P(i-1)); end phi(NN) = phiL; % --- Backward substitution for i = NN-1 : -1 : 2 phi(i) = P(i) * phi(i+1) + Q(i); end phi(1) = phi0; plot(x, phi, "linestyle", "--", "linewidth", 1, "marker", "o"); end xlabel('x [m]'); ylabel('Temperature [\deg C]'); %grid on; set (gca, "ygrid", "on"); % Format X-axis ticks xtick = get (gca, "xtick"); xticklabel = strsplit (sprintf ("%.3f\n", xtick), "\n", true); set (gca, "xticklabel", xticklabel) % Format Y-Axis ticks ytick = get (gca, "ytick"); yticklabel = strsplit (sprintf ("%.1f\n", ytick), "\n", true); set (gca, "yticklabel", yticklabel); legengTxt = cell(nTime+1, 1); for m = 0: nTime legendTxt{m+1} = strcat('t = ', num2str(m * dt, '%.2f'), ' [s]'); end legend(legendTxt, "location", 'northwest'); hold off; print -dpng transCondOneD.png -color -landscape;

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