Heat Transfer

The movement of thermal energy due to temperature difference. Heat flows from higher temperature to lower temperature.

Three modes: conduction, convection, radiation. Heat transfer mode depends on the medium (solid, fluid, or vacuum).

Conduction

Heat transfer due to random motion of atoms or molecules. Occurs in solids and stationary fluids.

Fourier’s Law

$$\dot{q} = -k \frac{dT}{dx}$$

• $\dot{q}$: heat flux
• $k$: thermal conductivity
• $\frac{dT}{dx}$: temperature gradient

Larger $k$ → faster heat transfer.

Heat Flux

Rate of heat transfer per unit area. Denoted by $\dot{q}$. Unit: $\text{W}\,\text{m}^{-2}$.

Thermal Conductivity

Measure of a material’s ability to conduct heat. Denoted by $k$. Unit: $\text{W}\,\text{m}^{-1}\,\text{K}^{-1}$.

Convection

Heat transfer from a solid surface to a moving fluid next to the surface (or vice versa).

Can be caused by either:

• Advection: bulk fluid motion.
• Diffusion: random fluid movement.

2 types:

• Natural convection: buoyancy-driven flow.
• Forced convection: external forcing (fan, pump).

Newton’s Law of Cooling

$$\dot{q} = h (T_s - T_\infty)$$

• $\dot{q}$: convective heat flux
• $h$: heat transfer coefficient
• $T_s$: surface temperature
• $T_\infty$: fluid temperature

Heat Transfer Coefficient

Measure of convective heat transfer between surface and fluid. Denoted by $h$. Unit: $\text{W}\,\text{m}^{-2}\,\text{K}^{-1}$.

Varies with flow conditions.

Radiation

Heat transfer through electromagnetic waves. No medium required. Wavelength range: about 0.1–100 μm. Depends on surface temperature and emissivity. Dominant at high temperatures or between large surfaces.

$$\dot{q} = \sigma \varepsilon \sigma f (T_1^4 - T_2^4)$$

Here:

• $\dot{q}$: net radiative heat flux
• $\sigma$: Stefan-Boltzmann constant
• $\varepsilon$: emissivity
• $f$: geometrical factor

Stefan-Boltzmann Constant

Denoted by $\sigma$. Value: $5.67 \times 10^{-8} \;\text{W}\,\text{m}^{-2}\,\text{K}^{-4}$.

Emissivity

Measure of a surface’s ability to emit thermal radiation. Denoted by $\varepsilon \in [0,1]$.

• $\varepsilon=0$: perfect reflector
• $\varepsilon=1$: perfect emitter, blackbody

Geometrical Factor

Depends on the orientation and shape of the surfaces exchanging radiation.

One-Dimensional Conduction

Used for walls, rods, slabs where temperature varies in only one direction.

$$\frac{dQ}{dx} = \frac{d}{dx} \left(k \frac{dT}{dx}\right) = 0$$ $$\frac{d^2 T}{dx^2} + \left( \frac{1}{A} \frac{\text{d}A}{\text{d}x} \right) \frac{\text{d}T}{\text{d}x} = 0$$

For constant $A$:

$$\frac{d^2 T}{dx^2} = 0$$

Temperature Distribution for Plane Slab

For a plane slab, W$A$ is constant w.r.t $x$. Integrating the above equation twice gives:

$$T(x) = a x + b$$

Temperature varies linearly throughout the conduction direction.

$$T(x) = T_1 + \frac{T_2 - T_1}{L}x$$

Note

$$\frac{\partial T}{\partial t} = \frac{k}{\rho c} \frac{\text{d}T^2}{\text{d}x^2} + \frac{\dot{q_g}}{\rho c}$$

Here:

• $k$ - thermal conductivity
• $\rho$ - density

Thermal Resistance

Conduction–convection systems can be modeled using electrical analogy.

$$R = \frac{\Delta T}{Q}$$

For a composite wall series and parallel thermal resistances are combined similar to electrical resistances.

Conduction Resistance

$$R_\text{cond} = \frac{L}{kA}$$

Convection Resistance

$$R_\text{conv} = \frac{1}{hA}$$

Note

Real problems often involve a wall with convection on both sides. In that case, conduction resistance and convection resistance are combined as usual.