Chapter 8, Problem 8.98P

It is common practice (o recover waste heat from an oil-or gas-fired furnace by using the exhaust gases to preheatthe combustion air. A device commonly used for thispurpose consists of a concentric pipe arrangement forwhich the exhaust gases are passed through the innerpipe, while the cooler combustion air flows through anannular passage around the pipe.

Consider conditions for which there b a uniformheattransferrateperunitlength, $q{\text{'}}_1=1.25\times {10}^5\text{W/m},$ .from the exhaust gases to the pipe inner surface, while air flows through the annular passage at a rateof $m_a=2.1$ kg/s. The thin-walled inner pipe is ofdiameter $D_i=2$ m, while the outer pipe, which is well insulated from the surroundings, is of diameter $D_0=2.05$ m. The air properties may be taken to be $c_p=1030\text{J/kg}\cdot \text{K},\mu =270\times {10}^{-7}\text{N}\cdot {\text{s/m}}^2,\text{k}=0.041\text{W/m}\cdot \text{K},$ and Pr= 0.68.

(a) Ifairentersat $T_{a,1}=300\text{K}$andL=7m, what isthe air outlet temperature $T_{a,2}$ ?

(b) If the airflow b fully developed throughout theannular region, what b the temperature of the innerpipe at the Inlet $\left(T_{s,i,1}\right)$ and owlet $\left(T_{s,i,2}\right)$ sections ofthe device? What b the outer surface temperature $T_{s,0,1}$ at the inlet?