Heat Transfer Assignment 1 — Essentials of Conduction, Convection, and Radiation  
$\xi$ is a parameter related to your student ID, with $\xi_1$ corresponding to the last digit, $\xi_2$ to the last two digits, $\xi_3$ to the last three digits, etc. For instance, if your ID is 199225962, then $\xi_1=2$, $\xi_2=62$, $\xi_3=962$, $\xi_4=5962$, etc. Keep a copy of the assignment — the assignment will not be handed back to you. You must be capable of remembering the solutions you hand in.
Question #1
A flat wall is exposed to an environmental temperature of $38^\circ$C. The wall is covered with a layer of insulation $2.5$ cm thick whose thermal conductivity is $1.4~{\rm W/m} \cdot ^\circ$C, and the temperature of the wall on the inside of the insulation is $315^\circ$C. The wall loses heat to the environment by convection. Compute the value of the convection heat-transfer coefficient which must be maintained on the outer surface of the insulation to ensure that the outer-surface temperature does not exceed $41^\circ$C.
Question #2
A black 20-by-20 cm plate has air forced over it at a velocity of 2 m/s and a temperature of $0^\circ$C. The plate is placed in a large room whose walls are at $30^\circ$C. The back side of the plate is perfectly insulated. Calculate the temperature of the plate resulting from the convection-radiation balance. Take the convective heat transfer $h$ as $12$ W/m$^2\cdot^\circ$C. Are you surprised at the result?
Question #3
Two large black plates are separated by a vacuum. On the outside of one plate is a convection environment of $T=80^\circ$C and $h=100~{\rm W/m}^2 \cdot ^\circ$C, while the outside of the other plate is exposed to 20$^\circ$C and $h=15~{\rm W/m}^2 \cdot ^\circ$C. Make an energy balance on the system (i.e. apply the heat equation in control volume form) and determine the plate temperatures.
Question #4
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 #5
Starting from the first law of thermo $$ d (me)+P dV=\delta Q-\delta W $$ and Fourier's law $q^{\prime\prime}_x=-k \partial T/\partial x$ derive the heat equation: $$ \frac{\partial (\rho c T)}{\partial t}=\frac{\partial}{\partial x}\left(k \frac{\partial T}{\partial x} \right)+\frac{\partial}{\partial y}\left(k \frac{\partial T}{\partial y} \right)+\frac{\partial}{\partial z}\left(k \frac{\partial T}{\partial z} \right)+S $$
Question #6
Consider a block of glass surrounded by two silver plates and two glass walls as follows:
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It is known that the distance between the silver and the glass is much less than either $L_1$, $L_2$, $L_3$, or $H$. Knowing that $L_1=1~$m, $L_3=1$ m, $H=1~$m, that the temperature on the right side of the glass block is $T_{\rm G2}=127^\circ$C, that the temperature of the left wall is $T_{\rm W1}=227^\circ$C, that the temperature of the environment is $T_\infty=27^\circ$C, that the convective heat transfer coefficient on the right wall is $h_2=5.669~$W/m$^2$$^\circ$C, do the following:
(a)  Find the temperature of the right wall, $T_{\rm W2}$.
(b)  Find the heat flux in W/m$^2$ due to convective heat transfer on the right wall.
(c)  Find the temperature on the left side of the glass block, $T_{\rm G1}$.
(d)  Find the length of the glass block, $L_2$.
You can use the following properties for glass and silver:
Hints: You can assume that the walls are thin with no temperature gradient within. The temperature on the right face of the glass block is uniform.
1.  $h\ge 5115$ W/m$^2$$^\circ$C
2.  9.6$^\circ$C
3.  $350$ K and $312$ K
6.  485.5 K, 0.174 m.
Due on Wednesday March 20th at 9:00. Do Questions #2, #5, and #6 only.
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