## Related questions with answers

A furnace for processing semiconductor materials is formed by a silicon carbide chamber that is zone-heated on the top section and cooled on the lower section. With the elevator in the lowest position, a robot arm inserts the silicon wafer on the mounting pins. In a production operation, the wafer is rapidly moved toward the hot zone to achieve the temperature-time history required for the process recipe. In this position, the top and bottom surfaces of the wafer exchange radiation with the hot and cool zones, respectively, of the chamber. The zone temperatures are $T_{h}=1500 \mathrm{K}$ and $T_{c}=330 \mathrm{K}$, and the emissivity and thickness of the wafer are $\varepsilon=0.65$ and d=0.78 mm, respectively. With the ambient gas at $T_{\infty}=700 \mathrm{K}$, convection coefficients at the upper and lower surfaces of the wafer are 8 and $4 \mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}$, respectively. The silicon wafer has a density of $2700 \mathrm{kg} / \mathrm{m}^{3}$ and a specific heat of $875 \mathrm{J} / \mathrm{kg} \cdot \mathrm{K}$. (a) For an initial condition corresponding to a wafer temperature of $T_{w, i}=300 \mathrm{K}$ and the position of the wafer shown schematically, determine the corresponding time rate of change of the wafer temperature, $\left(d T_{w} / d t\right)_{i}$. (b) Determine the steady-state temperature reached by the wafer if it remains in this position. How significant is convection heat transfer for this situation? Sketch how you would expect the wafer temperature to vary as a function of vertical distance.

Solution

Verified- hot zone temperature $T_h = 1500$K
- cold zone temperature $T_c = 330$K
- wafer thickness $\epsilon = 0.65$
- wafer thickness $d=0.78$mm
- ambient gas temperature $T_\infty = 700$K
- upper and lower surface convection factors $h_u = 8 h_l = 4$W/m$^2$K
- silicon wafer density $\rho = 2700$kg/m$^3$
- silicon wafer specific heat $c_p = 875$J/kgK
- initial wafer temperature $T_{w,i}=300$K

Set up energy equation for wafer over time:

$\begin{align*} \dot{E}_{in}'' - \dot{E}_{out}'' &= \dot{E}_{st}''\\ q_{rad,h}''+q_{rad,c}''-&q_{conv,u}''+q_{conv,l}''=mc_p\frac{dT}{dt}\\ \epsilon \sigma[(T_{sur,h}^4-T_w^4&)+(T_{sur,c}^4-T_w^4)-[h_u(T_w-T_\infty)+h_l(T_w-T_\infty)]= \rho c_p d \frac{dT}{dt} \end{align*}$

$\textbf{a)}$Initial condition for this case is $T_w=300$K:

$\begin{align*} &0.65 \cdot 5.67 \cdot 10^{-8}[(1500^4-300^4)+(330^4-300^4)-[8(300-700)+4(300-700)]\\ &=2700 \cdot 875 \cdot 0.78 \cdot 10^{-3} \frac{dT}{dt}\\ \\ \rightarrow &\frac{dT}{dt} = \frac{0.65 \cdot 5.67 \cdot 10^{-8}[(1500^4-300^4)+(330^4-300^4)-[8(300-700)+4(300-700)]}{2700 \cdot 875 \cdot 0.78 \cdot 10^{-3}}\\ &\frac{dT}{dt} = 103.78 \text{K/s} \end{align*}$

$\textbf{b)}$ In this steady-state case it is possible to neglect energy stored:

$\begin{align*} &\epsilon \sigma[(T_{sur,h}^4-T_w^4)+(T_{sur,c}^4-T_w^4)-[h_u(T_w-T_\infty)+h_l(T_w-T_\infty)]=0\\ &0.65 \cdot 5.67 \cdot 10^{-8}[(1500^4-T_{w,b}^4)+(330^4-T_{w,b}^4)-[8(T_{w,b}-700)+4(T_{w,b}-700)] \rightarrow &T_{w,b}= 1250.78\text{K} \end{align*}$

If we ignore convection heat transfer, temperature would only rise by about 11K which is insignificant at this value.

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