2017 Convective Heat Transfer Midterm Exam
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 Mon 17 Apr 16:30 -- 18:30   2 Wed 17 Apr 16:30 -- 18:30   0 Mon 24 Apr 16:30 -- 18:30   4 Wed 24 Apr 16:30 -- 18:30   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: 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: 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 $T_0$.
 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. $T_{\rm tip}=T_\infty+\frac{(T_0-T_\infty)\left({\rm cosh}(mL)-{\rm tanh}(mL) {\rm sinh}(mL) \right)}{1+\frac{h}{km}{\rm tanh}(mL)}$
 $\pi$