2018 Convective Heat Transfer Midterm Exam
Monday April 23rd, 2018
17:00 to 19:00

NO NOTES OR BOOKS; USE CONVECTIVE HEAT TRANSFER TABLES THAT WERE DISTRIBUTED; ANSWER ALL 4 QUESTIONS; ALL QUESTIONS HAVE EQUAL VALUE; INDICATE CLEARLY YOUR ASSUMPTIONS WHEN APPLICABLE.
 04.17.18
 Question #1
Derive Fourier's law of heat conduction in a gas: $$q^{\prime \prime}_x=-k \frac{\partial T}{\partial x}$$ with $$k=\frac{5 k_{\rm B}}{4 \sigma}\sqrt{\frac{3 RT}{2}}$$ with $k$ the thermal conductivity, $\sigma$ the collision cross-section, $k_{\rm B}$ the Boltzmann constant and $R$ the gas constant.

 Question #2
Consider the following piston-cylinder assembly:
In the latter, the cylinder is fixed while the piston is allowed to move and is subject to a gravity force $mg$. Knowing that the gravitational acceleration is of $g=9.8$ m/s$^2$, that the radius of the piston and of the cylinder are of $R_{\rm p}=10$ cm and $R_{\rm c}=10.3$ cm, respectively, that the height of the piston is of $H=5$ cm, that the density of the piston is of $\rho_{\rm p}=2000$ kg/m$^3$, and that the oil viscosity and density are of $\mu_{\rm oil}=0.5$ kg/ms and $\rho_{\rm oil}=800$ kg/m$^3$, do the following:
 (a) Find the force acting on the piston in the positive $y$ direction due to viscous effects as a function of the piston speed $q$. For simplicity, you can assume that $R_{\rm c}-R_{\rm p}\ll R_{\rm c}$. (b) Using the expression derived in (a), find the maximum speed $q$ that the piston would get if it is allowed to fall freely assuming negligible drag on its top and bottom surfaces.

 Question #3
Consider air flowing around a cone as follows:
Knowing that the air has a density of 1 kg/m$^3$, a viscosity of $10^{-5}$ kg/ms, a pressure of 90 kPa, and a velocity of 20 m/s, and that the cone has dimensions of $L=0.954$ m and $R=0.3$ m, calculate the drag force acting on the cone due to friction within the boundary layer.
 Question #4
Consider a flow of air forming a boundary layer above and below a flat plate as follows:
Knowing that the air freestream properties are of $\mu=10^{-5}$ kg/ms, $k=0.03~$W/mK, $\rho=1$ kg/m$^3$, $T=300$ K, ${\rm Pr}=0.7$, and $u_\infty=3$ m/s, and knowing that the flat plate temperature is kept constant at $350$ K, calculate the average Nusselt number $\overline{\rm Nu}$ between $x=1$ m and $x=3$ m.
 $\pi$