## Conduction
Thermal heat transfer through contact between objects.
### Fourier's Law
$q''_{x}=\frac{q_{x}}{A} = -k \frac{ \partial T }{ \partial x }$
$k$ = thermal conductivity $\left[ \mathrm{\frac{W}{mK}} \right]$
$q''$ = heat flux $\left[ \mathrm{\frac{W}{m^2}} \right]$
or more generally,
$\vec{q}'' = -k\nabla T$
## Convection
Mode of heat transfer due to conduction / thermal diffusion plus bulk motion of fluid.
$\mathrm{Nu} = \frac{hD}{k}=f(\mathrm{Re}, \mathrm{\mathrm{Pr}})$ $\mathrm{Pr} = \frac{\nu}{\alpha}=\frac{\text{kinematic viscosity}}{\text{thermal diffusivity}}$
1. Calculate the Reynolds number ($\frac{\rho vL}{v}$) and Prandtl number ($\frac{\nu}{\alpha}=\frac{c_{p}\mu}{k}$)
2. Choose the right correlation and calculate Nusselt number
3. Calculate $h$ from Nusselt number: $h= \frac{\mathrm{Nu}\cdot k}{d}$
4. $\dot{q}''=h\Delta T$ or $\dot{Q}=hA\Delta T$
## Newton's Law of Cooling
$\dot{Q} = hA\Delta T$
$q''_{s} = h(T_{s} - T_{\infty})$
$q =$ heat transfer $\mathrm{W}$
$q =$ heat flux $\mathrm{W/m^{2}}$
$h =$ heat transfer coefficient $\mathrm{\frac{W}{m^{2}K}}$
## External Flow
### Thermal Boundary Layer
Region in which you have thermal gradients between $T_{s}$ and $T_{\infty}$
![[Pasted image 20250316225128.png|350]]
### Total Heat Transfer Rate
$q = \bar{h} A_{s} (T_{s} - T_{\infty})$
where average convection coefficient $\bar{h}$ is defined as
$\bar{h} = \frac{1}{A_{s}} \int _{A_{s}} h \, dA_{s}$
which for a flat plate reduces to
$\bar{h} = \frac{1}{L} \int_{0}^{L} h \, dx$
### Empirical Correlation
With dimensional analysis and experimentation the following relationship has been shown:
$\overline{Nu_{L}} = C \mathrm{Re}_{L}^{m}\mathrm{Pr}^{n}$
$C$, $m$, and $n$, are typically independent of the nature of the fluid and instead vary with surface geometry and type of flow.
#### Flow over Isothermal Plate
Assuming steady, incompressible, laminar flow with constant fluid properties and negigible viscous dissipation and $\frac{dp}{dx}=0$
| Flow Type | Condition | Correlation |
| --------- | -------------------------------------------------------------------------------------- | ------------------------------------------------------------------------- |
| Laminar | $\mathrm{Pr} \geq 0.6$ | $\overline{Nu}_{x} = 0.664 \mathrm{Re}_{x}^{1/2} \mathrm{Pr}^{1/3}$ |
| Turbulent | $\mathrm{Pr} \geq 0.6$ | $\overline{Nu}_{x} = 0.0296 \mathrm{Re}_{x}^{4/5} \mathrm{Pr}^{1/3}$ |
| Mixed | $0.6 \leq \mathrm{Pr} \leq 60
lt;br>$\mathrm{Re}_{x,c} \leq \mathrm{Re}_{L} \leq 10^{8}$ | $\overline{Nu}_{x} = (0.037 \mathrm{Re}_{x}^{4/5} - A) \mathrm{Pr}^{1/3}$ |
## Natural Convection
No forced / defined bulk flow velocity, so we cannot use Reynold's number
Instead use Rayleigh number:
$\mathrm{Ra} = \mathrm{Gr}\times \mathrm{Pr}=\frac{\text{Buoyancy}}{\text{viscosity}}\times\frac{\text{momentum diffusivity}}{\text{thermal diffusivity}}$
$Nu = f(\mathrm{Ra},\mathrm{Pr})$
## Internal Flows
Laminar, fully developed flow → $\mathrm{Nu}= \begin{cases}\text{constant heat flux } \mathrm{ q''}: & 4.36\\ \text{constant wall }T_{s}: & 3.66\end{cases}$
$\Delta T = T_{s}-T_{b}$, where $T_{b}$ is a chosen "bulk mean temperature" analogous to $v_{\text{avg}}$
$\dot{Q} = hA\Delta T_{mean}$
where $\Delta T_{mean}$ is the mean of the temperature difference which logarithmically decays across the length of the pipe
$\Delta T_{mean} = \frac{\Delta T_{1}-\Delta T_{2}}{\ln(\Delta T_{1})-\ln\Delta (T_{2})}$
![[Pasted image 20250516235942.png]]
![[Pasted image 20250517000532.png]]
#### Fully developed flow through a circular tube
| Flow Type | Condition | Correlation |
| ----------------------- | ----------------------------------------------------------------------------------- | ----------------------------------------------------------------------------------------------------------- |
| Turbulent (Internal) | $0.7 \leq \mathrm{Pr} \leq 160lt;br>$\mathrm{Re}_D > 10,\!000lt;br>$L/D \geq 10$ | $\mathrm{Nu}_D = 0.023 \mathrm{Re}_D^{4/5} \mathrm{Pr}^{n}lt;br>$n = 0.3$ for cooling, $n = 0.4$ for heating |
| Turbulent (Sieder-Tate) | $0.7 \leq \mathrm{Pr} \leq 16,\!700lt;br>$\mathrm{Re}_D > 10,\!000lt;br>$L/D \geq 10$ | $\mathrm{Nu}_D = 0.027 \mathrm{Re}_D^{4/5} \mathrm{Pr}^{1/3} \left( \frac{\mu}{\mu_s} \right)^{0.14}$ |
| Turbulent (Gnielinski) | $0.5 \leq \mathrm{Pr} \leq 2000lt;br>$3,\!000 \leq \mathrm{Re}_D \leq 5 \times 10^6$ | $\mathrm{Nu}_D = \frac{(f/8)(\mathrm{Re}_D - 1000)\mathrm{Pr}}{1 + 12.7(f/8)^{1/2}(\mathrm{Pr}^{2/3} - 1)}$ |