2017 Convective Heat Transfer Midterm Exam  
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Mon 17 Apr 16:30 -- 18:30  20% of the voters. 14% of the votes.    2
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Wed 24 Apr 16:30 -- 18:30  80% of the voters. 57% of the votes.    8
Poll ended at 5:59 pm on Monday April 10th 2017. Total votes: 14. Total voters: 10.
Wednesday April 26th 2017
16:30 — 18:30


NO NOTES OR BOOKS;
USE HEAT TRANSFER TABLES THAT WERE DISTRIBUTED;
ANSWER ALL 4 QUESTIONS; ALL QUESTIONS HAVE EQUAL VALUE.
04.14.17
Question #1
Starting from the energy equation $$ \rho\frac{\partial E}{\partial t} + \rho u\frac{\partial H}{\partial x} + \rho v\frac{\partial H}{\partial y} = \frac{\partial }{\partial x}\left( k \frac{\partial T}{\partial x} \right) +\frac{\partial }{\partial y}\left( k \frac{\partial T}{\partial y} \right) + \frac{\partial u \tau_{xx}}{\partial x} + \frac{\partial u \tau_{yx}}{\partial y} + \frac{\partial v \tau_{xy}}{\partial x} + \frac{\partial v \tau_{yy}}{\partial y} $$ the $x$ momentum equation $$ \rho \frac{\partial u}{\partial t} + \rho u \frac{\partial u}{\partial x} + \rho v \frac{\partial u}{\partial y}=-\frac{\partial P}{\partial x}+ \frac{\partial \tau_{xx}}{\partial x}+\frac{\partial \tau_{yx}}{\partial y} $$ and the $y$ momentum equation $$ \rho \frac{\partial v}{\partial t} + \rho u \frac{\partial v}{\partial x} + \rho v \frac{\partial v}{\partial y}=-\frac{\partial P}{\partial y}+ \frac{\partial \tau_{xy}}{\partial x}+\frac{\partial \tau_{yy}}{\partial y} $$ Show that the energy equation for a constant-$\rho$ and constant-$\mu$ fluid corresponds to: $$ \rho\frac{\partial e}{\partial t}+\rho u\frac{\partial e}{\partial x}+\rho v\frac{\partial e}{\partial y} = \frac{\partial }{\partial x}\left( k \frac{\partial T}{\partial x} \right) +\frac{\partial }{\partial y}\left( k \frac{\partial T}{\partial y} \right) + \phi $$ with $\phi$ the viscous dissipation per unit volume defined as: $$ \phi\equiv\mu\left(\frac{\partial u}{\partial x} \right)^2 + \mu\left(\frac{\partial u}{\partial y} \right)^2 + \mu\left(\frac{\partial v}{\partial x} \right)^2 + \mu\left(\frac{\partial v}{\partial y} \right)^2 $$

Question #2
Derive Fourier's law of heat conduction in a gas: $$ q^{\prime \prime}_x=-k \frac{\partial T}{\partial x} $$ with $$ k=\frac{5 k_{\rm B}}{4 \sigma}\sqrt{\frac{3 RT}{2}}$$ with $k$ the thermal conductivity, $\sigma$ the collision cross-section, $k_{\rm B}$ the Boltzmann constant and $R$ the gas constant.
Question #3
Consider a journal bearing with a shaft diameter $D_i$ and a casing diameter $D_o$ as follows:
Q3.png  ./download/file.php?id=3448&sid=636f460a9287fa4581a28b04baa96c1f  ./download/file.php?id=3448&t=1&sid=636f460a9287fa4581a28b04baa96c1f
The shaft rotates at a speed $\omega$ (in rad/s), and the oil has a density $\rho$ (in kg/m$^3$), a viscosity $\mu$ (in kg/ms), and a thermal conductivity $k$ in (W/mK). Knowing that there is heat generation inside the shaft of $S$ (in W/m$^3$) and that the temperature of the casing is of ${T_o}$ (in $^\circ$C), do the following:
(a)  From the momentum equation, derive the velocity distribution within the oil as a function of $D_i$, $D_o$, $\omega$ and the distance from the casing, $y$.
(b)  From the energy equation, derive the temperature distribution within the oil as a function of $D_i$, $D_o$, $\omega$, $y$, $T_o$, $S$, $\mu$, and $k$.

Question #4
Consider the following rectangular fin attached to a wall:
Q4.png  ./download/file.php?id=3462&sid=636f460a9287fa4581a28b04baa96c1f  ./download/file.php?id=3462&t=1&sid=636f460a9287fa4581a28b04baa96c1f
Knowing that the tip is not insulated, find the temperature of the fin tip $T_{\rm tip}$ as a function of $h$, $T_\infty$, $L$, $t$, $W$, $\rho$, $c$, $k$, and $T_0$.
Answers
3.  $u=\frac{\omega D_i y}{D_o-D_i}$, $T=T_0+\frac{S D_i y}{4 k}+\frac{\mu}{k}\left(\frac{\omega D_i}{D_o-D_i} \right)^2\left(\frac{(D_o-D_i)y}{2}-\frac{y^2}{2} \right)$
4.  $T_{\rm tip}=T_\infty+\frac{(T_0-T_\infty)\left({\rm cosh}(mL)-{\rm tanh}(mL) {\rm sinh}(mL) \right)}{1+\frac{h}{km}{\rm tanh}(mL)}$
05.01.17
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