Heat Transfer Assignment 7 — Free & Forced Flow  
$\xi$ 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 tube bank uses an in-line arrangement with $S_p=S_n=1.9$ cm and 6.33-mm-diameter tubes. The tube bank is 6 rows deep and $50$ tubes high. The surface temperature of the tubes is constant at $90^\circ$C, and air at a pressure of 1 atmosphere, a temperature of $20^\circ$C, and a speed of $4.5$ m/s is forced across them. Calculate the total heat transfer per unit length for the tube bank as well as the outlet temperature of the air.
Question #2
Repeat the previous problem but with the tubes arranged in the “staggered” configuration with the same values of $S_p$ and $S_n$.
Question #3
A solid sphere made of a radioactive material is cooled by natural convection in an inert gas: $T_\infty=20^\circ$C. The diameter of the sphere is 0.02 m, and there is a constant rate of heat generation per unit volume, $S$, inside it. Under steady-state conditions, measurements indicate that the surface temperature of the sphere is $T_{\rm w}=100^\circ$C. Radiation heat transfer may be considered negligible. The thermal conductivity $k_{\rm s}$ of the radio-active material is of 0.49 W/m$^\circ$C. The thermophysical properties of the inert gas can be taken as $k=0.025$ W/m$^\circ$C, $c_p=1000$ J/kg$^\circ$C, $\mu=2 \times 10^{-5}$ kg/ms, $\rho=1.0$ kg/m$^3$, and $\beta=0.003$ K$^{-1}$. Perform the following tasks:
(a)  Calculate the volumetric rate of heat generation, $S$, inside the sphere.
(b)  Calculate the maximum temperature inside the sphere.
Question #4
You wish to cook some chicken optimally using a convection oven. A convection oven differs from standard ovens by blowing hot air at moderate speeds on the food. This results in the food being heated mostly through convective heat transfer rather than through radiation heat transfer. The chicken you wish to cook can be modeled as a solid sphere with a radius of 1 cm, a thermal conductivity of 0.5 W/m$^\circ$C, a density of 1000 kg/m$^3$, and a heat capacity of 3200 J/kg$^\circ$C. The convection oven blows hot air at atmospheric pressure, a temperature of $130^\circ$C and a speed of 5 m/s towards the chicken. The chicken is initially at a temperature of $5^\circ$C and stands on a grill through which the air can flow freely. You wish to cook the chicken optimally so that it is as tender as possible while being safe to eat. To be safe for eating, the temperature at any location within the chicken must have reached at least $70^\circ$C. To be as tender as possible, the chicken must not be overcooked and must therefore be taken out of the oven as soon as it is safe for eating. Knowing the latter, do the following:
(a)  Find the most accurate possible average convective heat transfer coefficient over the chicken when in the oven.
(b)  Using the average convective heat transfer coefficient found in (a), determine the amount of time the chicken should be left in the oven to be optimally cooked.
(c)  Find the surface temperature of the chicken when it is taken out of the oven.
1.  54.9 kW/m, 30.6$^\circ$C.
2.  62.8 kW/m, 32.1$^\circ$C.
3.  0.24 MW/m$^3$, 108.2$^\circ$C.
4.  61 W/m$^2$$^\circ$C, 256 s, 97$^\circ$ C.
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