update notes on heat transfer

uni/mmme/2047_thermodynamics_and_fluid_dynamics/heat_transfer.md

@@ -9,12 +9,24 @@ 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
+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)
@@ -33,3 +45,91 @@ $$\dot Q = \frac{T_f-T_w}{\sum R_\text{thermal}}$$
 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.).
+
+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
+
+