Heat Transfer Assignment 6 — Flat Plate Flow  
$\xi$ is a parameter related to your student ID, with $\xi_1$ corresponding to the last digit, $\xi_2$ to the last two digits, $\xi_3$ to the last three digits, etc. For instance, if your ID is 199225962, then $\xi_1=2$, $\xi_2=62$, $\xi_3=962$, $\xi_4=5962$, etc. Keep a copy of the assignment — the assignment will not be handed back to you. You must be capable of remembering the solutions you hand in.
Question #1
Consider the wing of an aircraft as a flat plate of 2.5 m length in the flow direction. The plane is moving at $100$ m/s in air that is at a pressure of 0.7 bar and a temperature of $-10^\circ$C. If the top surface of the wing absorbs solar radiation at a rate of $800$ $\rm W/m^2$, estimate its steady-state temperature with and without the effect of viscous dissipation. Assume the wing to be of solid construction and to have a single, uniform temperature. Ignore incident radiation on the bottom surface and take $\epsilon=0.4$ on the top and bottom surfaces of the wing.
Question #2
A thin, flat plate of length $L=1$ m separates two airstreams that are in parallel flow over opposite surfaces of the plate. One airstream has a temperature of $T_{\infty,1}=200^\circ$C and a velocity of $u_{\infty,1}=60$ m/s, while the other airstream has a temperature of $T_{\infty,2}=25^\circ$C and a velocity of $u_{\infty,2}=10$ m/s. The pressure in both streams corresponds to 1 atm. What is the temperature at the midpoint of the plate?
Question #3
Consider liquid water flowing over a flat plate of length $L=1$ m. The water has the following properties: $$ \rho=1000~{\rm kg/m^3},~~~c_p=4000~{\rm J/kgK},~~~\mu=10^{-3}~{\rm kg/ms},~~~k=0.6~{\rm W/m\cdot^\circ C} $$ Midway through the plate at $x=0.5~$m, you measure a heat flux to the surface of: $$ q^"_{x=0.5~{\rm m}}=3181 ~{\rm W/{m^2}} $$ You also measure an average heat flux to the surface over the length of the plate of: $$ \overline{q^"}=4500 ~{\rm W/{m^2}} $$ Knowing the latter, and knowing that the plate temperature is equal to $20^\circ$C do the following:
(a)  Is the flow laminar or turbulent, or a mix of both? You must provide proof of this using the data provided.
(b)  What is the possible range of the freestream velocity $U_{\infty}$?
(c)  Find a relationship between $T_\infty$ and $U_\infty$
1.  $-3.66^\circ$C, $-8.13^\circ$C.
2.  460.8 K.
3.  $T_\infty=20^\circ {\rm C}+6^\circ {\rm C m^{0.5}s^{-0.5}} U_\infty^{-0.5}$.
Due on Thursday May 18th at 9:00. Do all 3 problems.
Note: I made a modification to Question #3 May 8th at 22:05. Make sure you have the latest version. For Question #3, the freestream velocity is not known and its possible range must be proven somehow.
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