Chapter 2, Problem 2.52P

A chemically reacting mixture is stored in a thin-walled spherical container of radius $r_1=200\text{mm,}$ and the exothermic reaction generates heat at a uniform, but temperature-dependent volumetric rate of $\stackrel{.}{q}={\stackrel{.}{q}}_o\exp \left(-A/T_o\right),$ where ${\stackrel{.}{q}}_o=5000{\text{W/m}}^{\text{3}},A=75\text{K,}$ and $T_o$ is the mixture temperature in kelvins. vessel is enclosed by an insulating material of outer radius $r_2,$ thermal conductivity k, and emissivity $\epsilon .$ The outer surface of the insulation experiences convection heat transfer and net radiation exchange with the adjoining air and large surroundings, respectively.

1. Write the steady-state form of the heat diffusion equation for the insulation. Verify that this equation is satisfied by the temperature distribution
   $T\left(r\right)=T_{s,1}-\left(T_{s,1}-T_{s,2}\right)\left[\frac{1-\left(r_1/r\right)}{1-\left(r_1/r_2\right)}\right]$
   Sketch the temperature distribution, $T\left(r\right),$ labeling key features.
2. Applying Fourier's law. show that the rate of heat transfer by conduction through the insulation may be expressed as

$q_r=\frac{4\pi k\left(T_{s,1}-T_{s,2}\right)}{\left(1/r_1\right)-\left(1/r_2\right)}$
Applying an energy balance to a control surface about the container. obtain an alternative expression for $q_r,$ expressing your result in terms of $\stackrel{.}{q}$ and $r_1.$

3. Applying an energy balance to a control surface placed around the outer surface of the insulation, obtain an expression from which may be $T_{s,2}$ determined as a function of $\stackrel{.}{q},r_1,h,T_{\infty },\epsilon ,$ and $T_{\text{sur}}.$
4. The process engineer wishes to maintain a reactor temperature of $T_o=T\left(r_1\right)=95°\text{C}$ under conditions for which $k=0.05\text{W/m}\cdot \text{K,}{r}_2=208\text{mm,}h=5{\text{W/m}}^{\text{2}}\cdot \text{K,}\epsilon =0.9,T_{\infty }=25°\text{C,}$ and $T_{sur}=35°\text{C}.$ What is the actual reactor temperature and the outer surface temperature $T_{s,2}$ of the insulation?
5. Compute and plot the variation of $T_{s,2}$ with $r_2$ for $201\le r_2\le 210\text{mm}\text{.}$ The engineer is concerned about potential burn injuries to personnel who may come into contact with the exposed surface of the insulation. Is increasing the insulation thickness a practical solution to maintaining $T_{s,2}\le 45°\text{C?}$ What other parameter could be varied to reduce $T_{s,2}?$