Convective Heat Transfer Assignment 4 — Nusselt and Stanton Number  
Question #1
Starting from the energy equation for a constant-$\rho$ and constant-$\mu$ fluid: $$ \frac{\partial \rho e}{\partial t}+\frac{\partial \rho u e}{\partial x}+\frac{\partial \rho v e}{\partial y} = \frac{\partial }{\partial x}\left( k \frac{\partial T}{\partial x} \right) +\frac{\partial }{\partial y}\left( k \frac{\partial T}{\partial y} \right) + \mu\left(\frac{\partial u}{\partial x} \right)^2 + \mu\left(\frac{\partial u}{\partial y} \right)^2 + \mu\left(\frac{\partial v}{\partial x} \right)^2 + \mu\left(\frac{\partial v}{\partial y} \right)^2 $$ Show that the heat transfer for a laminar flow over a constant-temperature flat plate can be expressed as: $$ {\rm Nu}_x=0.332 \,\,{\rm Re}_x^{1/2}\,\, {\rm Pr}^{1/3} $$ with ${\rm Nu}_x$ the Nusselt number, ${\rm Re}_x$ the Reynolds number and Pr the Prandtl number. Outline all assumptions and also show that the restrictions on the Prandtl number correspond to: $$ 0.6 \le {\rm Pr} \le 50 $$ In doing the derivation, you can use the following boundary layer relationships applicable to laminar flow over a flat plate: $$ \delta\frac{{\rm d}\delta}{{\rm d}x}=\frac{140}{13}\frac{\mu}{\rho u_\infty} $$ $$ \delta^2=\frac{280}{13} \frac{\mu x}{\rho u_\infty} $$ $$ \frac{u}{u_\infty}= \frac{3}{2} \frac{y}{\delta} - \frac{1}{2} \left(\frac{y}{\delta} \right)^3 $$
Question #2
Starting from the local Nusselt number: $$ {\rm Nu}_x=0.332 \, {\rm Re}_x^{1/2}{\rm Pr}^{1/3} $$ Show that for laminar flow flowing over a flat plate of length $L$, the average Nusselt number corresponds to: $$ \overline{{\rm Nu}_L}=0.664 \,{\rm Re}_L^{1/2}{\rm Pr}^{1/3} $$
Question #3
Experiments to determine the local convection heat transfer coefficient for uniform flow normal to a heated circular disk have yielded a radial Nusselt number distribution of the form $$ {\rm Nu}_{D} = \frac{h(r) D}{k} = {\rm Nu}_0 \left[1+a \left(\frac{r}{r_0} \right)^n \right]$$ where both $n$ and $a$ are positive. The Nusselt number at the stagnation point is correlated in terms of the Reynolds number (${\rm Re}_D=\rho v D/\mu$) and Prandtl number (${\rm Pr}=c_p\mu/k$): $$ {\rm Nu}_0 = \frac{h(r=0) D}{k}=0.814 \,{\rm Re}_D^{0.5} \,{\rm Pr}^{0.36}$$
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Obtain an expression for the average Nusselt number, $\overline{{\rm Nu}_D}= \overline{h} D / k$, corresponding to heat transfer from an isothermal disk.
Question #4
A flat plate 10 cm long and 1 m wide is placed in a wind tunnel where the air properties in the test section are $u=2500$ m/s, $P=1/40$ atm, and $T=-40^\circ$C. Assuming the flow is laminar throughout, how much cooling must be used to maintain the plate temperature at $700^\circ$C?
3.  $\overline{{\rm Nu}_D}=2 {\rm Nu}_0 \left(0.5 + a/(n+2) \right) $.
4.  34.8 kW.
Due on Wednesday May 17th at 16:30. Do all 4 problems.
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