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🔥 Content Introduction
In the field of heat conduction physics and numerical computation, the 3D heat conduction equation is the core model that describes the temperature changes of three-dimensional objects over time — from food processing (such as egg cooking) to material heat treatment (such as metal quenching), its numerical solutions hold significant engineering value. This article focuses on the specific scenario of “temperature evolution during egg cooking,” implementing the solution of the 3D heat conduction equation based on MATLAB. It calculates the temperature changes at each point in a 3D grid using the Euler Method, while also analyzing the code logic, parameter settings, and result visualization. Additionally, it will relate to the commonalities of 1D/2D diffusion equations, providing a reference for simulating similar heat conduction problems.
1. Core Background: The Physical Connection Between the 3D Heat Conduction Equation and Egg Cooking
To understand the implementation logic of the code, it is essential to clarify the physical significance of the heat conduction equation and how it describes the temperature changes during the egg cooking process.
⛳️ Results
📣 Sample Code
%parameters%
L = 1; %X-Y grid (LxL)
N = 60; %Number of space-steps in each X and Y direction
dx = L/N; %space-step size
D = 0.4; %diffusive coefficient
dt = 0.4*(dx*dx)/2; %time-step size (so that method is stable) Q: D dependence? see iteration->no D at the end.
U0 = 0.867; %initial temperature
i = 0; %index for loop over time
tStop = 1; %
%%initialize%%
U = zeros(N);
%loop%
while i < tStop
U(1,:) = U0; %initial condition. Resets every time.
delta_2D = laplacian_2D(U,dx); %computes the Laplacian of a spatial mesh (matrix U) for given dx.
Uc = U(2:end-1,2:end-1); %removes the edges of the grid.
U(2:end-1,2:end-1) = Uc + dt*D*delta_2D; %updates the matrix adding dt*D*delta every time.
U(:,1) = U(:,2); %Neumann BC at the edges of the XY-grid
U(:,end) = U(:,end-1); %why does the order of equality matter?
U(1,:) = U(2,:);
U(end,:) = U(end-1,:);
i = i + dt;
%include plotting in loop to graph U against X,Y at different
%times, for example at the start, halfway, and at the end.
% #time-steps =tStop/dt.
%Neumann BC ensures that heat is not lost. Remember that the derivative with
%the normal direction to the surface must be nullified at the walls.
end
figure(1)
set(gca,’Fontsize’,15);
mesh(linspace(0,L,N-2),linspace(0,L,N-2),Uc);
colormap(hot);
xlabel(‘distance[cm]’,’Fontsize’,15);
ylabel(‘distance[cm]’,’Fontsize’,15);
title(‘Solution of the diffusion equation in 2D’, ‘Fontsize’,15);
🔗 References
🎈 Some theoretical references are from online literature; if there is any infringement, please contact the author for removal.
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