misc fix

proj/proj.m

@@ -0,0 +1,278 @@
+%% Project: 1D Heat Transfer Simulation (Numerical Solution of Heat Equation)
+% This Live Script provides a complete solution to the MATLAB Project focusing 
+% on determining heat transfer through a 1D bar by conduction using the explicit 
+% finite difference method. It addresses all questions from the project document 
+% (Questions 1-7).
+%% Introduction to the Heat Equation
+% The general 3D Heat Equation describes temperature distribution over time 
+% and space:
+% 
+% $$\rho c_p \frac{\partial T}{\partial t} = \lambda \left( \frac{\partial^2 
+% T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} + \frac{\partial^2 T}{\partial 
+% z^2} \right)$$
+% 
+% Where:
+% 
+% - $T$: Temperature ($^{\circ}C$)
+% 
+% - $\rho$: Mass density ($kg/m^3$)
+% 
+% - $c_p$: Specific heat capacity ($J/(kg \cdot ^{\circ}C)$)
+% 
+% - $\lambda$: Thermal conductivity ($W/(m \cdot ^{\circ}C)$)
+% 
+% - $x, y, z$: Spatial dimensions (m) - $t$: Time (s)
+% 
+% For a 1D bar, assuming homogeneous and isotropic material and insulated sides 
+% (except for the ends), the equation simplifies to:
+% 
+% $$\rho c_p \frac{\partial T}{\partial t} = \lambda \frac{\partial^2 T}{\partial 
+% x^2}$$
+% 
+% This can be rewritten using thermal diffusivity $\alpha = \frac{\lambda}{\rho 
+% c_p}$:
+% 
+% $$\frac{\partial T}{\partial t} = \alpha \frac{\partial^2 T}{\partial x^2}$$
+%% Define Physical Parameters and Material Properties
+
+% Bar Dimensions and Discretization
+L = 0.4;        % Length of the bar (m)
+dx = 0.01;      % Spatial step size (m)
+dt = 0.5;       % Time step size (s)
+
+% Thermal Conditions
+Tg = 100;       % Temperature at left end (x=0) in Celsius
+Td = 50;        % Temperature at right end (x=L) in Celsius
+T0 = 0;         % Initial temperature of the bar (internal nodes) in Celsius
+
+% Material Properties (Aluminum properties as hinted in the document)
+rho = 2700;     % Density (kg/m^3)
+Cp = 900;       % Specific heat capacity (J/kg/C)
+lambda = 237;   % Thermal conductivity (W/m/C)
+
+% Simulation End Time (chosen to observe steady-state behavior)
+t_final_s_sim = 2000; % Total simulation time in seconds
+
+% Time points at which to capture and plot full temperature profiles
+% This allows for a denser visualization early on, and sparser later as equilibrium is approached.
+specific_early_points = [0, 5, 10, 20];
+mid_range_points = 50:50:750;
+late_range_points = 750:250:t_final_s_sim;
+plot_time_points = unique([specific_early_points, mid_range_points, late_range_points]);
+%% Question 1: Express the left term of the equation (*) using numerical differentiation.
+% The left term is the first partial derivative of temperature with respect 
+% to time, $\frac{\partial T}{\partial t}$. Using a forward finite difference 
+% approximation for the time derivative at node $i$ and time step $n$:
+% 
+% $$\frac{\partial T}{\partial t} \approx \frac{T_i^{n+1} - T_i^n}{\Delta t}$$
+% 
+% This approximates the rate of temperature change at a specific spatial point 
+% $i$ from time $t_n$ to $t_{n+1}$.
+%% Question 2: Express the right term of the equation (*) using numerical differentiation.
+% The right term is the second partial derivative of temperature with respect 
+% to space, $\frac{\partial^2 T}{\partial x^2}$.
+% 
+% Using a central finite difference approximation for the second spatial derivative 
+% at node $i$ and time step $n$:
+% 
+% $$\frac{\partial^2 T}{\partial x^2} \approx \frac{T_{i+1}^n - 2T_i^n + T_{i-1}^n}{(\Delta 
+% x)^2}$$
+% 
+% This relates the curvature of the temperature profile at node $i$ to the temperatures 
+% of its immediate neighbors at the current time step.
+%% Question 3: Reformulate the equation (*) using the results of Question 1 and 2.
+% Substitute the finite difference approximations into the 1D Heat Equation:
+% 
+% $$\frac{T_i^{n+1} - T_i^n}{\Delta t} = \alpha \frac{T_{i+1}^n - 2T_i^n + T_{i-1}^n}{(\Delta 
+% x)^2}$$
+% 
+% Where $\alpha = \frac{\lambda}{\rho c_p}$ is the thermal diffusivity.
+%% Question 4: From the new formula of the equation, express $T_i^{n+1}$.
+% Rearranging the reformulated equation to explicitly solve for the temperature 
+% at the next time step $T_i^{n+1}$:
+% 
+% $$T_i^{n+1} = T_i^n + \alpha \frac{\Delta t}{(\Delta x)^2} (T_{i+1}^n - 2T_i^n 
+% + T_{i-1}^n)$$
+% 
+% This is the core update equation for each interior node.
+% 
+% For convenience, we define the Fourier number (or numerical stability parameter) 
+% $r$:
+% 
+% $$r = \alpha \frac{\Delta t}{(\Delta x)^2}$$
+% 
+% So the update equation becomes:
+% 
+% $$T_i^{n+1} = T_i^n + r (T_{i+1}^n - 2T_i^n + T_{i-1}^n)$$
+%% Question 5: What are the boundary conditions and initial temperature?
+% a. Temperature on the left $x=0$, $T_g$?
+% 
+% The temperature at the left end of the bar ($x=0$) is a fixed boundary condition: 
+% $T_g = 100^{\circ}C$.
+% 
+% This means $T_1^n = 100^{\circ}C$ for all time steps $n$.
+% 
+% b. Temperature on the right $x=L$, $T_d$?
+% 
+% The temperature at the right end of the bar ($x=L$) is a fixed boundary condition: 
+% $T_d = 50^{\circ}C$.
+% 
+% This means $T_{N_x}^n = 50^{\circ}C$ for all time steps $n$.
+% 
+% c. Initial temperature $T_0$?
+% 
+% The initial temperature of the *internal* nodes of the bar (at time $t=0$) 
+% is $T_0 = 0^{\circ}C$.
+% 
+% This means $T_i^0 = 0^{\circ}C$ for all internal nodes $i$.
+%% Question 6: Write the algorithm to calculate all temperatures at each point of the bar and at any time.
+% This section outlines the step-by-step algorithm implemented in MATLAB.
+% 
+% 1.  **Calculate Derived Constants:** Compute thermal diffusivity (alpha) and 
+% the Fourier number (r). Check the stability criterion: r must be <= 0.5 for 
+% the explicit method to be stable.
+% 
+% 2.  **Spatial Grid Setup:** Create a vector 'x' representing the discrete 
+% spatial nodes along the bar. Determine the total number of spatial nodes (Nx).
+% 
+% 3.  **Initialize Temperature Array:** Create 'T_current' (a 1D array/vector) 
+% to hold temperatures at the current time step. Set all interior nodes to 'T0'. 
+% Apply fixed boundary conditions 'Tg' and 'Td' to the first and last nodes.
+% 
+% 4.  **Time Loop:** Iterate through discrete time steps from t=0 up to 't_final_s_sim'.
+% 
+% a. For each time step, create a 'T_next' array (initialized to 'T_current').
+% 
+% b.  **Apply Finite Difference Equation: For each *interior* node (from index 
+% 2 to Nx-1), calculate its temperature at the next time step using the derived 
+% formula from Question 4.
+% 
+% c.  **Update Boundary Conditions:** The boundary node temperatures (T_next(1) 
+% and T_next(Nx)) remain fixed at Tg and Td, respectively. These are carried over 
+% from T_current or explicitly set to override any updates to ensure they remain 
+% fixed.
+% 
+% d.  **Advance Time Step:** Set 'T_current = T_next' to prepare for the next 
+% iteration.
+% 
+% e.  **Data Storage for Plotting:** Store the full temperature profile (T_current) 
+% at predefined 'plot_time_points'. Also, record the temperature at the midpoint 
+% of the bar over all time steps for a separate time-series plot.
+%% Question 7: Use MATLAB to solve the equation.
+
+% This section provides the MATLAB implementation based on the algorithm and detailed parameters.
+
+fprintf('--- Solving 1D Heat Equation Project ---\n');
+
+% Calculate Derived Constants
+alpha = lambda / (rho * Cp);
+r = alpha * dt / (dx^2);
+fprintf('Thermal Diffusivity (alpha): %.4e m^2/s\n', alpha);
+fprintf('Fourier Number (r): %.4f\n', r);
+
+if r > 0.5
+    warning('The explicit method is unstable for r > 0.5. Consider reducing dt or increasing dx.');
+end
+
+% Spatial Grid Setup
+x = 0:dx:L;         % Spatial grid points
+Nx = length(x);     % Number of spatial nodes
+
+% Find the index for the midpoint of the bar for plotting the right pane
+mid_index = find(abs(x - L/2) < 1e-9, 1); % Find index closest to L/2
+if isempty(mid_index) % Fallback if L/2 isn't an exact node
+    mid_index = round(Nx / 2);
+end
+
+% Initialize Temperature Array
+T_current = zeros(1, Nx); % Initialize temperature array with zeros
+T_current(2:Nx-1) = T0;   % Internal nodes at initial temperature (0 C)
+T_current(1) = Tg;        % Left boundary condition (100 C)
+T_current(Nx) = Td;       % Right boundary condition (50 C)
+
+% Time Grid Setup
+time_vector = 0:dt:t_final_s_sim;
+Nt = length(time_vector); % Number of time steps
+
+% Storage for results
+T_midpoint_over_time = zeros(1, Nt); % Stores temp at midpoint over ALL time steps
+T_midpoint_over_time(1) = T_current(mid_index); % Initial midpoint temp
+
+all_profiles = cell(size(plot_time_points)); % Stores full profiles at specific times
+all_profile_times = cell(size(plot_time_points)); % Stores labels for these profiles
+
+% Capture initial profile (t=0) if it's one of the requested plot_time_points
+[is_t0_plot_point, idx_t0_plot_point] = ismembertol(0, plot_time_points, dt/2, 'DataScale', 1);
+if is_t0_plot_point
+    all_profiles{idx_t0_plot_point} = T_current;
+    all_profile_times{idx_t0_plot_point} = sprintf('t = %.0f s', 0);
+end
+
+% Main Time-Stepping Loop
+fprintf('\nStarting simulation...\n');
+for n = 1:Nt-1 % Loop from t=0 (n=1) to t_final_s (n=Nt)
+    T_next = T_current; % Initialize T_next. Boundary values from T_current are implicitly carried over.
+
+    % Apply finite difference equation for internal nodes (from index 2 to Nx-1)
+    for i = 2:Nx-1
+        T_next(i) = T_current(i) + r * (T_current(i+1) - 2*T_current(i) + T_current(i-1));
+    end
+
+    T_current = T_next; % Update temperature for the next iteration
+
+    % Store midpoint temperature for time series plot
+    T_midpoint_over_time(n+1) = T_current(mid_index);
+
+    % Store full spatial profiles for plotting at specified plot_time_points
+    [is_plot_time, idx_to_store] = ismembertol(time_vector(n+1), plot_time_points, dt/2, 'DataScale', 1);
+    if is_plot_time
+        all_profiles{idx_to_store} = T_current;
+        all_profile_times{idx_to_store} = sprintf('t = %.0f s', time_vector(n+1));
+    end
+end
+fprintf('Simulation finished.\n');
+%% Visualization (Question 7 Continued)
+
+% Create a figure to display the results in two subplots
+figure('Color', 'white', 'Position', [100, 100, 1000, 400]); % Adjust figure size for better viewing
+
+% Left Pane: Temperature Profiles Over Time (T vs x)
+subplot(1, 2, 1);
+hold on;
+grid on;
+box on; % Add a box around the plot
+
+% Plot each captured profile
+colors = lines(length(plot_time_points)); % Generate distinct colors based on number of plot points
+for k = 1:length(plot_time_points)
+    if ~isempty(all_profiles{k}) % Ensure data exists for this plot point
+        plot(x, all_profiles{k}, 'DisplayName', all_profile_times{k}, 'LineWidth', 1.5, 'Color', colors(k,:));
+    end
+end
+
+xlabel('Position along bar, x (m)');
+ylabel('Temperature, T (^oC)');
+title('Temperature Profiles Over Time');
+
+% Conditionally display legend if not too many lines
+if length(plot_time_points) < 8 % Adjust threshold as needed
+    legend('Location', 'southeast');
+end
+hold off;
+
+% Right Pane: Temperature at Midpoint Over Time (T vs t)
+subplot(1, 2, 2);
+plot(time_vector, T_midpoint_over_time, 'LineWidth', 2, 'Color', [0.8500 0.3250 0.0980]);
+grid on;
+box on; % Add a box around the plot
+xlabel('Time, t (s)');
+ylabel('Temperature at Midpoint (^oC)');
+title('Temperature at Midpoint (x=L/2) Over Time');
+ylim([0 100]); % Set y-limits to reasonable temperature range
+xlim([0 t_final_s_sim]); % Set x-limits to the simulation time
+
+% Final Output
+fprintf('\nSimulation and plotting complete. Check the generated figure for results.\n');
+% To save the figure, use:
+% saveas(gcf, 'Figure_1.png');
+% (Uncomment and run the above line after the figure appears to save it.)

proj/runHeatProject.m

@@ -23,4 +23,4 @@ plotProfiles(x_coords, all_temp_profiles, profile_labels);
 subplot(1, 2, 2);
 plotMidpointTemperature(time_vec, midpoint_T_series);
 
-exportgraphics(f, 'Figure_1.pdf', 'ContentType', 'vector','Resolution', 150, 'Units', 'centimeters', Width=1920);
+exportgraphics(f, 'Figure_1.pdf', 'ContentType', 'vector');

proj/solveHeatEquation.m

@@ -23,9 +23,13 @@ function [x, T_final_profile, T_time_series, time_vector, all_profiles, all_prof
     Tg = thermal_cond.Tg; Td = thermal_cond.Td; T0 = thermal_cond.T0;
     dx = steps.dx; dt = steps.dt;
 
-    % Calculate alpha and r (Fourier Number)
+    % Calculate Thermal Diffusivity (alpha)
     alpha = lambda / (rho * Cp);
+    fprintf('Thermal Diffusivity (alpha): %.4e m^2/s\n', alpha);
+
+    % Calculate the Fourier Number (r) and check stability
     r = alpha * dt / (dx^2);
+    fprintf('Fourier Number (r): %.4f\n', r);
 
     if r > 0.5
         warning('The explicit method is unstable for r > 0.5. Consider reducing dt or increasing dx.');
@@ -53,7 +57,7 @@ function [x, T_final_profile, T_time_series, time_vector, all_profiles, all_prof
     all_profiles = cell(length(plot_time_points), 1);
     all_profile_times = cell(length(plot_time_points), 1);
     
-    % Capture initial profile (t=0)
+    % Capture initial profile (t=0) if it's one of the requested plot_time_points
     if ismembertol(0, plot_time_points, dt/2, 'DataScale', 1)
         all_profiles{1} = T_current;
         all_profile_times{1} = sprintf('t = %.0f s', 0); % Using sprintf for consistent formatting