Finite-Difference Heat Propagation Through the Sheath of a Temperature Probe

Project info

• Field of studyInstrumentation and Sensors
• LevelDoctoral dissertation (S3)
• Method typeNumerical Simulation
• Estimated build time± 7 working days
• Toolsloops/vektorisasi MATLAB inti; plot
• PriceUS$38 - US$228

Research goal

Simulate probe tip temperature with Finite-Difference Heat Equation and examine its sensitivity to sheath thermal conductivity and sheath thickness.

Variables

Example result

• 0.4Fourier number r
• 43Grid nodes
• 54.014Peak value at end
• 0.016Max abs. error
• 0.33Remaining heat fraction
• 417Time steps

The explicit finite-difference scheme (r=0.40, 43 nodes) decays the peak of Probe tip temperature from 105.8 to 54.0 with a maximum error of 0.016 against the analytical solution; 33% of the heat remains.

Computed from illustrative synthetic data, not from a real study.

About Finite-Difference Heat Equation

The one-dimensional heat equation solved by finite differences computes temperature over position and time by discretising spatial and time derivatives. It is used to simulate simple conduction heat transfer in engineering theses.

Steps

1. 1Set thermal diffusivity, rod length, and initial and boundary conditions
2. 2Discretise space (dx) and time (dt)
3. 3Ensure the explicit stability condition r = alpha dt/dx^2 <= 0.5
4. 4Update temperature at each time step
5. 5Compare with the analytic solution
6. 6Plot temperature profiles and error

Assumptions to check

• The explicit scheme is stable only if r is at most 0.5
• Material properties are constant and heat flow is one-dimensional
• Boundary and initial conditions are clearly defined

Reading and reporting the results

Report discretisation parameters, the stability number r, and the maximum error against the analytic solution. Show temperature profiles at several times and explain the decay behaviour.

Example MATLAB code

A short starting script with this project's variable names. The delivered version is adapted to your data and fully commented.

alpha = 0.01;  L = 1;
nx = 51;  sheath_thermal_conductivity = linspace(0, L, nx)';           % spatial grid
dx = sheath_thermal_conductivity(2) - sheath_thermal_conductivity(1);
dt = 0.4 * dx^2 / alpha;                           % r = 0.4 (stable)
r = alpha * dt / dx^2;
tend = 20;  nt = round(tend / dt);

probe_tip_temperature = sin(pi * sheath_thermal_conductivity);                          % initial temperature
u = probe_tip_temperature;
snap = zeros(nx, 4);  snapT = [0 5 10 20];  ks = 1;
for n = 1:nt
    u(2:end-1) = u(2:end-1) + r * (u(3:end) - 2*u(2:end-1) + u(1:end-2));
    u(1) = 0;  u(end) = 0;                         % fixed-temperature ends
    if ks <= 4 && n*dt >= snapT(ks) - dt/2
        snap(:, ks) = u;  ks = ks + 1;
    end
end
exact = exp(-alpha * pi^2 * nt*dt) * sin(pi * sheath_thermal_conductivity);
fprintf('r = %.2f, steps = %d\n', r, nt);
fprintf('Max error at t = %.1f: %.2e\n', nt*dt, max(abs(u - exact)));

figure;
plot(sheath_thermal_conductivity, snap(:, 1:ks-1), 'LineWidth', 1.2);
xlabel('Position x'); ylabel('Temperature');
legend(arrayfun(@(s) sprintf('t = %g', s), snapT(1:ks-1), 'UniformOutput', false));

Packages and prices

Choose how much help you need. Each higher package includes everything in the lower ones. Prices exclude 11% VAT.

• Payment: 50% down payment before work starts, balance after the deliverables are handed over
• Tax: Prices exclude 11% VAT
• Data and academic integrity: Results are computed from your own research data; you write and stand behind your manuscript, we help with the analysis and your understanding of it
• Revisions: Up to 2 revisions are included in the price (Package 3: 3 rounds)

Frequently asked questions