Question 30.1: A horizontal chemical vapor deposition (CVD) reactor for gro...

At sufficiently high flowrate within the reactor, the physical system represents convective mass transfer over a flat plate, where the hydrodynamic boundary layer develops before the concentration boundary layer. Trimethylgallium diluted in \mathrm{H}_{2} gas serves as the source for mass transfer, and the reaction at the boundary surface is the sink for trimethylgallium gas.

(a) First, the local Reynolds number is evaluated at x=9 \mathrm{~cm}, the distance from the edge of the susceptor (4 \mathrm{~cm}) to the middle of the wafer (5 \mathrm{~cm}). The kinematic viscosity of the feed gas is approximated by the properties of the \mathrm{H}_{2} gas, as trimethylgallium is present in only dilute concentration; from Appendix I, ν=5.686 \mathrm{~cm}^{2} / \mathrm{s} at 800 \mathrm{~K} and 1 \mathrm{~atm} for \mathrm{H}_{2} gas. Therefore

\operatorname{Re}_{x}=\frac{v_{\infty} x}{ν}=\frac{(100 \mathrm{~cm} / \mathrm{s})(9 \mathrm{~cm})}{5.686 \mathrm{~cm}^{2} / \mathrm{s}}=158.3

and so the flow is laminar. The Schmidt number is

\mathrm{Sc}=\frac{ν}{D_{A B}}=\frac{5.686 \mathrm{~cm}^{2} / \mathrm{s}}{1.55 \mathrm{~cm}^{2} / \mathrm{s}}=3.67

For the present system where the hydrodynamic boundary layer starts to develop before the concentration boundary layer, the local Sherwood number for laminar flow is

\operatorname{Sh}_{x}=0.332 \operatorname{Re}_{x}^{1 / 2}\left(\frac{\mathrm{Sc}}{1-\left(\frac{X}{x}\right)^{3 / 4}}\right)^{1 / 3}=0.332(158.3)^{1 / 2}\left(\frac{3.67}{1-\left(\frac{4 \mathrm{~cm}}{9 \mathrm{~cm}}\right)^{3 / 4}}\right)^{1 / 3}=8.375

Finally, the local mass-transfer coefficient predicted by boundary-layer theory is

k_{c}=\frac{\mathrm{Sh}_{x}}{x} D_{A B}=\left(\frac{8.375}{9 \mathrm{~cm}}\right)\left(1.55 \frac{\mathrm{cm}^{2}}{\mathrm{~s}}\right)=1.44 \frac{\mathrm{cm}}{\mathrm{s}}

(b) For film theory, recall that the local mass-transfer coefficient is defined as

k_{c}=\frac{D_{A B}}{\delta_{c}}

For laminar flow, the hydrodynamic boundary-layer thickness is

\delta=\frac{5 x}{\sqrt{\operatorname{Re}_{x}}}=\frac{5(9 \mathrm{~cm})}{\sqrt{158.3}}=3.58 \mathrm{~cm}

Now recall that the relationship between the hydrodynamic boundary-layer thickness and the concentration boundary-layer thickness is

\frac{\delta}{\delta_{c}}=\mathrm{Sc}^{1 / 3}   (28-18)

for laminar flow over a flat plate where the hydrodynamic and concentration boundary layers have the same starting point. For laminar flow over a flat plate where the hydrodynamic boundary layer is initiated before the concentration boundary layer, the combination of equations (28-21), (28-26), and (30-5)

\frac{\overline{{{k}}}_{c}x}{D_{A B}}=\mathrm{Sh}_{L}=0.664\,\mathrm{Re}_{L}^{1/2}\mathrm{Sc}^{1/3}       (28-21)

\begin{aligned} & \bar{k}_c=\frac{0.332 D_{A B}\left(\frac{v}{ν}\right)^{1 / 2}(\mathrm{Sc})^{1 / 3} \int_0^{L_t} x^{-1 / 2} d x+0.0292 D_{A B}\left(\frac{v}{ν}\right)^{4 / 5}(\mathrm{Sc})^{1 / 3} \int_{L_t}^L x^{-1 / 5} d x}{L} \\ & \bar{k}_c=\frac{0.664 D_{A B}\left(\frac{v}{ν}\right)^{1 / 2}(\mathrm{Sc})^{1 / 3} L_t^{1 / 2}+0.0365 D_{A B}\left(\frac{v}{ν}\right)^{4 / 5}(\mathrm{Sc})^{1 / 3}\left[(L)^{4 / 5}-\left(L_t\right)^{4 / 5}\right]}{L} \\ & \bar{k}_c=\frac{0.664 D_{A B}\left(\operatorname{Re}_t\right)^{1 / 2}(\mathrm{Sc})^{1 / 3}+0.0365 D_{A B}(\mathrm{Sc})^{1 / 3}\left[\left(\operatorname{Re}_L\right)^{4 / 5}-\left(\mathrm{Re}_t\right)^{4 / 5}\right]}{L} \end{aligned}        (28-26)

\mathrm{Sh}_{x}=0.332\,\mathrm{Re}_{x}^{1/2}\Bigg(\frac{\mathrm{Sc}}{1-\left(\frac{X}{x}\right)^{3/4}}\Bigg)^{1/3}         (30-5)

yields

\frac{\delta}{\delta_{c}}=\left(\frac{\mathrm{Sc}}{1-\left(\frac{X}{x}\right)^{3 / 4}}\right)^{1 / 3}     (30-6)

Substituting in the appropriate values, we obtain

\frac{\delta}{\delta_{c}}=\left(\frac{\mathrm{Sc}}{1-\left(\frac{X}{x}\right)^{3 / 4}}\right)^{1 / 3}=\left(\frac{3.67}{1-\left(\frac{4 \mathrm{~cm}}{9 \mathrm{~cm}}\right)^{3 / 4}}\right)^{1 / 3}=2.004

and so

\delta_{c}=\frac{3.58 \mathrm{~cm}}{2.004}=1.79 \mathrm{~cm}

Finally

k_{c}=\frac{D_{A B}}{\delta_{c}}=\frac{1.55 \mathrm{~cm}^{2} / \mathrm{s}}{1.79 \mathrm{~cm}}=0.868 \frac{\mathrm{cm}}{\mathrm{s}}

We note that k_{c} predicted by film theory is 37 \% lower than k_{c} predicted by boundary-layer theory.

The preceding analysis is valid only in the limiting case where the thickness of the hydrodynamic boundary layer is significantly less than the height of the enclosure housing the susceptor plate. However, if the Reynolds number is sufficiently low, then the hydrodynamic boundary-layer thickness approaches the height of the enclosure. In this situation, the CVD reactor housing itself becomes the conduit for flow, and the analyses provided by Middleman and Hochberg { }^{2} and Middleman ^{3} are recommended.