Chapter 1, Problem 1.57P

A furnace tor processing semiconductor materials isformed by a silicon carbide chamber that is zone-heatedon the top section and cooled on the lower section. Withthe elevator in the lowest position, a robot arm inserts thesilicon water on the mounting pins. In a production operation, the wafer is rapidly moved toward the hot zone toachieve the temperature-time history required tor theprocess recipe. In this position, the top and bottom surfaces of the wafer exchange radiation with the hot andcool zones, respectively, of the chamber. The zonetemperatures are $T_h=1500\text{K}$ and $T_c=330\text{K}$ , and theemissivity and thickness of the wafer are $\in =0.65$ and $d=0.78\text{mm}$ . respectively. With the ambient gas at $T_{\infty }=700\text{K}$ , convection coefficients at the upper andlower surfaces of the wafer are 8 and $4{\text{W/m}}^{\text{2}}\cdot \text{K}$ , respectively. The silicon wafer has a density of $2700{\text{kg/m}}^3$ anda specific heat of $875\text{J/kg}\cdot \text{K}$ .

(a) For an initial condition corresponding to a watertemperature of $T_{w,i}=300\text{K}$ and the position of thewater shown schematically, determine the corresponding lime rate of change of the wafer temperature. ${\left(d{T}_w/dt\right)}_i$ .
(b) Determine the steady-slate temperature reachedby the wafer if it remains in this position. Howsignificant is convection heat transfer for thissituation? Sketch how you would expect thewafer temperature to vary as a function of vertical distance.