Heat Transfer Assignment 4 — Transient Conduction  
Instructions
$\xi$ 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.
05.05.14
Question #1
On a hot summer day a concrete driveway may reach a temperature of $50^\circ$C. Suppose that a stream of water is directed on the driveway so that the surface temperature is suddenly lowered to $10^\circ$C. How long will it take to cool the concrete to $25^\circ$C at a depth of $5$ cm from the surface?
Question #2
A plate of stainless steel (18% Chrome, 8% Nickel) has a thickness of 3.0 cm and is initially uniform in temperature at $500^\circ$C. The plate is suddenly exposed to a convection environment on both sides at $40^\circ$C with $h=150$ W/m$^2 \cdot ^\circ$C. Calculate the times for the center and face temperatures to reach $100^\circ$C.
Question #3
A stainless steel cylinder (18% Chrome, 8% Nickel) is heated to a uniform temperature of $200^\circ$C and then allowed to cool in an environment where the air temperature is maintained constant at $30^\circ$C. The convection heat-transfer coefficient may be taken as $200$ W/m$^2\cdot^\circ$C. The cylinder has a diameter of 10 cm and a length of 15 cm. Calculate the temperature of the geometric center of the cylinder after a time of $10$ min. Also calculate the heat loss.
Question #4
Show that the general solution of the heat equation for 2D transient problems corresponds to: $$ \frac{T-T_\infty}{T_{\rm i}-T_\infty}= \left(\frac{T_1 -T_\infty}{T_{\rm i}-T_\infty} \right)\left(\frac{T_2-T_\infty}{T_{\rm i}-T_\infty} \right) $$ with: $$ \frac{1}{\alpha}\frac{\partial T_1}{\partial t} = \frac{\partial^2 T_1}{\partial x^2} $$ $$ \frac{1}{\alpha}\frac{\partial T_2}{\partial t} = \frac{\partial^2 T_2}{\partial y^2} $$
Question #5
Consider a 0.0245m-radius sphere made in yellow-pine wood initially at a temperature of $200^\circ$C. The sphere is cooled with cold air at a temperature of $T_\infty=20^\circ$C and a convective heat transfer coefficient $h=3~{\rm W/m^2 K}$. Knowing that after a time $\Delta t$, the sphere loses 13.114 kJ to the environment, do the following:
(a)  Find the time elapsed, $\Delta t$, in seconds.
(b)  At a time of $t=\Delta t$, find the center temperature of the sphere in Celcius.
(c)  At a time of $t=\Delta t$, find the temperature on the surface of the sphere in Celcius.
04.05.18
Answers
1.  7086 s.
2.  788 s, 767 s.
3.  72$^\circ$C, 533 kJ.
5.  5853 s, 92$^\circ$C, 76$^\circ$C.
Due on Monday April 23rd at 15:00. Do Questions #1, #3, and #5 only.
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