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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 + + diff --git a/uni/mmme/2047_thermodynamics_and_fluid_dynamics/images/convection.png b/uni/mmme/2047_thermodynamics_and_fluid_dynamics/images/convection.png new file mode 100644 index 0000000000000000000000000000000000000000..31a3b5f1893894610133cb5e28ae89053b5f4acb GIT binary patch literal 27180 zcmX`TWmr{R*EYOCKtQ^rTSB_KOF|IoPU(_vkdPEmx*L>kq`ON&*fi2z(rn_JT=)BY z2S3=rT64`c=E!rLgNS#^a_FeUs1OJQUH*;qdk6%c8v=n#K!yjuDJNs22j7rP73HKM zk56AYZN>2r2qi>bT0-43;~>+=SADwaDnQ8NTO0{WGL~cp5jI?+o~m&W76;cQU9rsD z?-%bDdUiMJA`|0J59N(EUHJVn1^!6F$hQ)vbnD6^c+0g{T+H*nBt*ku=#go87kib%)mOg@e z`Vc~IuKe_Ejnai0IfxSPc`7crAeVnI1N?;CC$S|Bu42I8qz6AqevA`9WKK*>bjJLY zf|addD0)Kicq2BjlI%l?K=sUA`IEeh%URLe^bK-E4k}LaJMU#owCe5gArY*gI5^JY zPV<#^KXChm$W_MTGphbrLUK6DvPP|=Pgo&nU}?&+u`$@y@$=3H#cL+juaFS(;>C4J zWFkrzRPoEh1?b`;T250_DxJ!MqLDdm;5+(MTn2VwwPU7iy}S_$r6J>936;qf~4}@twUSAJYFe4Hr7_-=$)GWcY}b 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