Fourier's Law

Fourier's Law is a mathematical expression of the natural phenomenon that "heat flows where there is a temperature gradient." It was formulated by Joseph Fourier in his 1822 book 'The Analytical Theory of Heat.' All heat conduction analysis starts from here.

Yes, just as Newtonian mechanics begins with F=ma, heat transfer engineering begins with Fourier's Law. In vector form, it is written as follows.

The basic form of Fourier's Law is the relationship between the heat flux vector $\mathbf{q}$ and the temperature gradient.

Here, $k$ is the thermal conductivity [W/(m K)], and $\nabla T$ is the temperature gradient [K/m]. The minus sign reflects the second law of thermodynamics: "heat flows from the high-temperature side to the low-temperature side."

Correct. In the one-dimensional case, the partial derivative becomes an ordinary derivative.

Combining this with the energy conservation law yields the steady-state heat conduction equation.

$\dot{q}_v$ is the internal heat generation per unit volume [W/m³]. If k is constant, it simplifies further to $k \frac{d^2T}{dx^2} + \dot{q}_v = 0$.

Exactly. For a flat plate with surface temperatures $T_1$ and $T_2$, the distribution becomes linear: $T(x) = T_1 + (T_2 - T_1)\frac{x}{L}$. This is a basic case often used for verifying analytical solutions.

For general 3D problems, it is written in tensor form to account for anisotropic materials.

For isotropic materials, $k_{ij} = k \delta_{ij}$, which reduces to Laplace's equation $\nabla^2 T = 0$ (no generation) or Poisson's equation $\nabla^2 T + \frac{\dot{q}_v}{k} = 0$ (with generation).

Correct. For CFRP, the thermal conductivity along the fiber direction is about 5-10 W/(m K), while in the transverse direction it's about 0.5-1 W/(m K). It cannot be handled correctly without the tensor form.

There are three types of boundary conditions.

Correct. In practice, the 3rd kind boundary condition is used most frequently. The estimation of the convection heat transfer coefficient $h$ greatly influences the accuracy of the results.

We start from the weak form (Galerkin method) of the steady-state heat conduction problem. The temperature field $T$ is approximated using shape functions $N_i$.

Applying integration by parts to the weak form yields the element-level equation.

Here, $K^e_{ij} = \int_{\Omega_e} k \nabla N_i \cdot \nabla N_j \, d\Omega$ is the element thermal conductivity matrix, and $f^e_i = \int_{\Omega_e} \dot{q}_v N_i \, d\Omega + \int_{\Gamma_e} q_s N_i \, d\Gamma$ is the element thermal load vector.

Good observation. It has the same mathematical structure as $[K]\{u\}=\{F\}$ in structural analysis. However, in thermal analysis, the unknown is the scalar temperature, so there is only one degree of freedom per node. This makes the problem size much smaller than in structural analysis.

The Finite Difference Method (FDM) is also often used for steady-state heat conduction. Discretizing with central differences yields, in 1D:

The Finite Volume Method (FVM) places temperature at cell centers and conserves heat flux at cell interfaces. CFD solvers (Ansys Fluent, STAR-CCM+) are FVM-based, so in conjugate heat transfer analysis, the solid side is also automatically discretized using FVM.

FEM is strong for complex geometries and can achieve high accuracy with higher-order elements. FVM strictly satisfies conservation laws, making it suitable for fluid-solid coupling. FDM is easy to implement but limited to structured grids. Choosing the right method is important.

The assembled global equation $[K]\{T\}=\{f\}$ is linear for steady-state heat conduction, so it can be solved in a single step.

It becomes a nonlinear problem requiring iterative calculation. The $k(T)$ is updated sequentially using the Newton-Raphson method. The convergence criterion is typically a residual norm around $10^{-6}$, but sometimes a temperature change of $10^{-3}$ K is used as a benchmark.