notes/uni/mmme/2047_thermodynamics_and_fluid_dynamics/heat_transfer.md

---
author: Akbar Rahman
date: \today
title: MMME2047 // Heat Transfer
tags: [ heat_transfer ]
uuid: d3ba66c2-e486-464a-a4df-f23f2155ee6d
lecture_slides: [ ./lecture_slides/6ConvHeatTransfer-without-written-comments.pptx ]
lecture_notes: [ ./lecture_notes/ConvectHeatTrans2022-2023.pdf ]
exercise_sheets: [ ./exercise_sheets/ExamplesConvectionHeatTransfer.pdf ]
---

# Errata

## Grashof Number (formula booklet, lecture slides p. 24, lecture recording 2, 1:19:30)

The Grashof number formula should be written with $\nu$, not $\mu$,
It is correctly written in lecture notes (p. 8), [Wikipedia](https://en.wikipedia.org/wiki/Grashof_number),
and on this page.

Somehow it is also incorrect in the formula booklet.

You can check which one is correct by checking which results in a dimensionless Grashof number.

# Convection

- conduction and radiation heat transfer can be estimated by calculations and properties
- convection is dependent on fluid properties, flow type, and flow characteristics

The rate of convective heat transfer, $\dot Q$, is given by Newton's law of cooling:

\begin{equation}
\dot Q = hA(T_f-T_w)
\label{eqn:convectionheattransfer}
\end{equation}

where $T_w$ is the wall temperature, $T_f$ is the fluid temperature, $A$ is the area of heat flow,
and $h$ is the heat transfer coefficient.

## Thermal Resistance

Equation \ref{eqn:convectionheattransfer} can also be expressed in terms of thermal resistance, $R_\text{thermal}$:

$$\dot Q = \frac{T_f-T_w}{\sum R_\text{thermal}}$$

where $R_\text{thermal} = \frac{1}{hA}$.

In a way this analogous to Ohm's law, specifically with resistors in series ($I = \frac{\Delta V}{\sum R_\text{electrical}}$).

## Analysis of how Convection Works

At a hot wall, the velocity of the fluid touching it will be zero.
Therefore the heat transfer into the fluid must happen by conduction.
This means that the local heat flux per unit area, $\dot Q''$ (dot for rate, double dash for per
unit area) is given by:

$$\dot Q'' = -k \frac{\delta T}{\delta y}|_{\text{wall}}$$

where $k$ is the conductivity of the wall

![](./images/convection.png)

The heated fluid is carried away by convection, therefore at steady state we can say that:

$$\dot Q'' = -k \frac{\delta T}{\delta y}|_\text{wall} = -h(T_\infty - T_\text{wall})$$

Rearranging allows $h$ to be found.

# Nusselt Number - Relation Between Fluid Conductivity and Convection

Nusselt number is a dimensionless number:

$$\text{Nu} = \frac{hL}{k_f}$$

where $k_f$ is conductivity of the fluid, $L$ is the representative length (e.g. diameter, length,
internal width, etc.), and $h$ is heat transfer coefficient.

Since $h$ is unknown a lot of the time, sometimes Nusselt number must be found through approximating
by other dimensionless numbers: Prandtl, Reynolds, and Grashof.


Nusselt number for a laminar forced flow is around 3.66.
For a turbulent forced flow it is estimated to be:

$$\text{Nu}_x = 0.023\text{Re}_x^{0.8}\text{Pr}^{0.4}$$

For a laminar forced flow over a flat plate:

$$\text{Nu}_x = 0.332\text{Re}_x^{0.5}\text{Pr}^{0.33}$$

For natural convection of a vertical wall:

\begin{align*}
\text{Nu}_x = 0.59(\text{Gr}_x\text{Pr})^{0.25} &\text{ for }10^3 < \text{GrPr} < 10^9 \\
\text{Nu}_x = 0.13(\text{Gr}_x\text{Pr})^{0.25} &\text{ for }10^9 < \text{GrPr} < 10^{12}
\end{align*}

[removing annoying html syntax highlighti]: >

# Prandtl Number

This number relates thickness of velocity boundary layer to thickness of thermal boundary layer:

$$\text{Pr} = \frac{c_p\mu}{k_f} = \frac{\nu}{\alpha}$$

where $\nu$ is the kinematic viscosity and $\alpha$ is the thermal diffusivity (equations given in
lecture notes p. 5).

# Grashof Number

Grashof number compares the buoyancy of the fluid (due to compressibility, $\beta = T^{-1}$,
where $T$ is the film temperature, or average temperature between fluid and wall, in kelvin)
and the viscous resistance to buoyant motion.

$$\text{Gr} = \frac{g\beta L^3\rho^2\Delta T}{\nu^2}$$

where $g$ is acceleration due to gravity, $L$ is the height or length of the tube, $\rho$ is density
of the fluid, $\Delta T = T_\text{wall} - T_\infty$, and $\nu$ is the kinematic viscosity.

# Axisymmetric Shenanigans

Axisymmetric shapes are symmetric about an axis.

$$\dot Q' = -kA\frac{\mathrm dT}{\mathrm dr} = -k2\pi r\frac{\mathrm dT}{\mathrm dr}$$

Note that the single dash on $\dot Q'$ implies *per unit length*.
For any length, $L$, $A = 2\pi rL$.

In this case, the temperature profile is no longer linear, even if $k$ is constant:

$$R_\text{th} = \frac{\ln r_o - \ln r_i}{2\pi kL}$$

# Definitions

- *lagging* - insulation