Heat Transfer — CAE Glossary
Heat Transfer — What It Is
Heat transfer is basically "heat moving," right? What are the specific mechanisms?
Heat transfer has three main modes. Conduction (conduction) is when heat is transmitted through a material by molecular vibration. Convection (convection) is when heat is carried away by the movement of a fluid. Radiation (radiation) is when heat is transmitted as electromagnetic waves and can occur even in a vacuum. In real engineering problems, all three modes typically occur simultaneously.
All three at once? What kind of situations have all three happening?
Consider a laptop's cooling system. CPU heat is transmitted to the heatsink via conduction, a fan draws air that removes heat via convection, and the casing also dissipates heat via radiation. The same applies to car engine blocks — all three modes are involved.
Conduction (Conduction) — Fourier's Law
Fourier's Law is the fundamental law of conduction, right? What is the equation?
Fourier's Law in one dimension is:
$q$ is heat flux (W/m²), $k$ is thermal conductivity (W/(m·K)), and $dT/dx$ is the temperature gradient. The negative sign appears because heat always flows from high to low temperature. In 3D, the vector form is $\mathbf{q} = -k \nabla T$.
How much does the $k$ value vary between materials?
It varies by orders of magnitude. Copper is about 400 W/(m·K) and conducts heat exceptionally well, but air is around 0.026. Insulation materials are typically 0.02–0.04. If you don't input the correct $k$ value for your material in CAE thermal analysis, the results will be completely wrong. Composite materials and gaskets especially require careful attention.
Fourier's Law leads to the heat conduction equation, right?
Yes. Combining Fourier's Law with energy conservation gives the heat conduction equation:
$\rho$ is density, $c_p$ is specific heat, and $\dot{q}_v$ is volumetric heat generation. FEM thermal solvers essentially discretize and solve this equation.
Convection (Convection) — Newton's Law of Cooling
The fundamental equation for convection is Newton's Law of Cooling, right?
That's correct. The equation is very simple:
$T_s$ is the solid surface temperature, $T_\infty$ is the bulk fluid temperature far from the surface, and $h$ is the convection coefficient (W/(m²·K)). But behind this simplicity lies a major challenge: determining $h$.
How do you determine $h$? It's not in a property table, is it?
$h$ varies dramatically depending on fluid type, flow velocity, geometry, and flow regime. Here are typical ranges:
- Natural convection in air: 5–25 W/(m²·K)
- Forced convection in air: 25–250 W/(m²·K)
- Forced convection in water: 100–20,000 W/(m²·K)
- Boiling water: 2,500–100,000 W/(m²·K)
In practice, engineers use empirical Nusselt number correlations or CFD to compute $h$ directly.
Natural and forced convection differ that much. I understand forced convection is more efficient, but what is the Nusselt number?
The Nusselt number is a dimensionless number indicating the effectiveness of convective heat transfer: $\mathrm{Nu} = hL/k_f$, where $k_f$ is the fluid's thermal conductivity. In essence, it tells you how many times more effective convection is compared to pure conduction. Nu = 1 means convection has no effect; Nu = 100 means convection moves 100 times more heat than conduction alone would.
Radiation (Radiation) — Stefan-Boltzmann Law
Radiation transmits heat via electromagnetic waves, right? What is Stefan-Boltzmann Law?
The total radiant energy emitted by a blackbody (ideal radiator) is:
where $\sigma = 5.67 \times 10^{-8}$ W/(m²·K⁴) is the Stefan-Boltzmann constant. For real surfaces, we include emissivity $\varepsilon$ (0 to 1):
$$E = \varepsilon \sigma T^4$$The key observation is the fourth-power dependence on temperature. When temperature doubles, radiation increases 16-fold. This is why radiation becomes dominant at high temperatures.
Fourth power is incredible… How do you write the radiative heat exchange between two surfaces?
The net radiative heat exchange between two surfaces is:
$F_{12}$ is the view factor (form factor), representing what fraction of radiation from surface 1 reaches surface 2. In CAE, computing view factors for complex geometries is tedious — typically done with Monte Carlo methods. Accurate radiation modeling is essential for investment casting mold cooling and spacecraft thermal design.
In space, there's no conduction or convection because it's a vacuum. Radiation is the only control mechanism?
Exactly. Satellite thermal design is radiation-dominated. The sun-facing side gets extremely hot while the opposite side is bitterly cold. Heat pipes and radiator panels maintain uniform temperature. This is a classic radiation-dominated heat transfer problem.
Conjugate Heat Transfer (CHT)
How do you account for conduction, convection, and radiation all together in CAE?
That is called Conjugate Heat Transfer (CHT). Solve the heat conduction equation in the solid region and the Navier-Stokes plus energy equations in the fluid region simultaneously, enforcing temperature and heat flux continuity at the interface:
Because you don't assume $h$, CHT can be highly accurate for complex flow patterns.
CHT sounds computationally expensive. Should every problem use it?
True, it is costly. So you pick the right tool for the job. A simple flat-plate cooling problem can be handled by assuming $h$ and running thermal analysis alone. But for tightly coupled problems like gas turbine blade cooling or engine exhaust manifold design, CHT is essential. Ansys Fluent, STAR-CCM+, and OpenFOAM's chtMultiRegionFoam are the standard CHT solvers.
Heat Transfer Analysis in CAE
What is the first thing to consider when running a CAE thermal analysis?
First, identify "which heat transfer mechanism dominates this problem?" Dimensionless numbers help:
- Biot number $\mathrm{Bi} = hL/k_s$ → Can internal temperature gradients be neglected?
- Fourier number $\mathrm{Fo} = \alpha t / L^2$ → Characterize the time scale of unsteady problems
- Nusselt number $\mathrm{Nu}$ → Quantify convection strength
If Bi < 0.1, temperature within the solid is nearly uniform, so you can use the lumped-capacitance method. For large Bi, you must solve the temperature distribution fully with FEM.
There's also thermal stress analysis, right? That couples heat and structure?
Yes. Temperature change causes material expansion/contraction. If there are constraints, thermal stresses develop. The typical approach is weak coupling: run thermal analysis first to get the temperature field, then map that to structural analysis. But if deformation changes the thermal solution (e.g., contact gaps changing resistance), you need strong coupling. Cylinder head gasket design is a classic strongly coupled problem.
Related Terms
- Fourier's Law — The fundamental law of heat conduction
- Newton's Law of Cooling — The fundamental equation for convective heat transfer
- Stefan-Boltzmann Law — The fundamental law of radiation
- Thermal Conductivity — A material's ability to conduct heat
- Convection Coefficient — A measure of convection strength
- Nusselt Number — Dimensionless number for convective heat transfer
- Biot Number — Ratio of internal to external thermal resistance
- Conjugate Heat Transfer (CHT) — Coupled solid-fluid analysis
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