Chapter 9, Problem 9.27P

The vertical rear window of an automobile is of thickness
$L=8\text{mm}$ and height $H=0.5\text{m}$ and contains line-meshed heating wires that can induce nearly uniform volumetric heat, $\stackrel{\dot{}}{q}\left({\text{W/m}}^3\right)$ .
(a) Consider steady-slate conditions for which the interior surface of the window is exposed to quiescent air at 10°C, while the exterior surface is exposed to air at -10°C moving in parallel how over the surface with a velocity of 20 m/s. Determine the volumetric heating rate needed to maintain the interior window surface at $T_{s,i}=15°C$ .
(b) The interior and exterior window temperatures, $T_{s,i}$ and $T_{\infty ,o}$ depend on tile compartment and ambient temperatures, $T_{\infty ,i}$ , and $T_{\infty ,o}$ , as well as on the velocity $u_{\infty }$ , of air flowing over the exterior surface and the volumetric heating rate $\stackrel{\dot{}}{q}$ . Subject to the constraint that $T_{s,i}$ , is to be maintained at 15°C, we wish to develop guidelines for varying the heating rate in response to changes in $T_{\infty ,i},T_{\infty ,o}$ , and/or $u_{\infty }$ . If $T_{\infty ,i}$ , is maintained at 10°C, how will $\stackrel{\dot{}}{q}$ and $T_{\infty ,o}$ , vary with $T_{\infty ,o}$ , for $-25\le T_{\infty ,o}\le 5°C$ and $u_{\infty }=10,20$ , and $30\text{m/s}$ ? If a constant vehicle speed is maintained, such that $u_{\infty }=30\text{m/s}$ , how will $\stackrel{\dot{}}{q}$ and $T_{\infty ,i}$ , vary with $T_{\infty ,o}$ , for $5\le T_{\infty ,i}\le 20°C$ and $T_{\infty ,o}=-25,-10,$ , and 5°C’?