Heat Transfer Assignment 4 — Transient Conduction  
$\xi$ 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
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.
Question #6
Consider a block of concrete initially at a temperature of 100$^\circ$C cooled by some air flow as follows:
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Knowing that the dimensions and properties of the concrete can be taken as:

$L=0.4$ m,   $H=0.2$ m,  $D=0.2$ m

$c=900$ J/kg$^\circ$C,   $\rho=2000$ kg/m$^3$,   $k=1.4$ W/m$^\circ$C

and that $h$ can be taken as 14 W/m$^2$$^\circ$C and that $T_\infty$ corresponds to 20$^\circ$C, find the following temperatures at a time 3 hours after the concrete starts to be cooled by the air flow:
(a)  The temperature at the point ($x=0$, $y=H$, $z=L$)
(b)  The average temperature within the concrete
Question #7
Consider a cylinder made of concrete with a length of 20 cm and a diameter of 10 cm. The cylinder is initially at a temperature of 500$^\circ$C and is cooled by a fluid with a temperature $T_\infty$ of 20$^\circ$C and a convective heat transfer coefficient $h$ of 10 W/m$^2$$^\circ$C. Noting that the thermal conductivity, density and heat capacity of concrete can be taken as 1.37 W/m$^\circ$C, 1900 kg/m$^3$, and 880 J/kg$^\circ$C, do the following tasks:
(a)  Find the time needed for the minimum temperature within the cylinder to reach 100$^\circ$C
(b)  At the time found in part (a), find the average temperature within the cylinder
1.  7086 s.
2.  788 s, 767 s.
3.  72$^\circ$C, 533 kJ.
5.  5853 s, 92$^\circ$C, 76$^\circ$C.
6.  $\rm 24.1^\circ C$, $\rm39.2^\circ C$.
7.  $5400~{\rm s}$, $130^\circ{\rm C}$.
Due on Wednesday April 10th at 9:00. Do Questions #1, #6, and #7 only.
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