Heat Transfer | Sahithyan's Notes

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

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}