2018 Convective Heat Transfer Final Exam  
Monday June 18th 2018
17:00 — 20:00


Question #1
Starting from the energy equation $$ \rho\frac{\partial E}{\partial t} + \rho u\frac{\partial H}{\partial x} + \rho v\frac{\partial H}{\partial y} = \frac{\partial }{\partial x}\left( k \frac{\partial T}{\partial x} \right) +\frac{\partial }{\partial y}\left( k \frac{\partial T}{\partial y} \right) + \frac{\partial u \tau_{xx}}{\partial x} + \frac{\partial u \tau_{yx}}{\partial y} + \frac{\partial v \tau_{xy}}{\partial x} + \frac{\partial v \tau_{yy}}{\partial y} $$ the $x$ momentum equation $$ \rho \frac{\partial u}{\partial t} + \rho u \frac{\partial u}{\partial x} + \rho v \frac{\partial u}{\partial y}=-\frac{\partial P}{\partial x}+ \frac{\partial \tau_{xx}}{\partial x}+\frac{\partial \tau_{yx}}{\partial y} $$ and the $y$ momentum equation $$ \rho \frac{\partial v}{\partial t} + \rho u \frac{\partial v}{\partial x} + \rho v \frac{\partial v}{\partial y}=-\frac{\partial P}{\partial y}+ \frac{\partial \tau_{xy}}{\partial x}+\frac{\partial \tau_{yy}}{\partial y} $$ Show that the energy equation for a constant-$\rho$ and constant-$\mu$ fluid corresponds to: $$ \rho\frac{\partial e}{\partial t}+\rho u\frac{\partial e}{\partial x}+\rho v\frac{\partial e}{\partial y} = \frac{\partial }{\partial x}\left( k \frac{\partial T}{\partial x} \right) +\frac{\partial }{\partial y}\left( k \frac{\partial T}{\partial y} \right) + \phi $$ with $\phi$ the viscous dissipation per unit volume defined as: $$ \phi\equiv\mu\left(\frac{\partial u}{\partial x} \right)^2 + \mu\left(\frac{\partial u}{\partial y} \right)^2 + \mu\left(\frac{\partial v}{\partial x} \right)^2 + \mu\left(\frac{\partial v}{\partial y} \right)^2 $$
Question #2
An air stream with a speed of $50$ m/s and density of $\rho=1.0$ kg/m$^3$ flows parallel to a flat plate with a length of 45 cm and a width of 100 cm. Determine the total drag force on the flat plate and calculate the boundary layer thickness 10 and 45 cm from the leading edge. Take the kinematic viscosity as $15\times 10^{-6}$ m$^2$/s.
Question #3
Consider a 30 m long pipe with a diameter of 1 cm and with a smooth interior wall surface. The pipe wall temperature is kept constant at 60$^\circ$C.
(a)  Some liquid enters the pipe with a temperature of 20$^\circ$C and exits the pipe with a mixing cup (bulk) temperature of 57$^\circ$C. Knowing that the mass flow rate of the liquid is of $0.015$ kg/s, that the liquid density is of 1000 kg/m$^3$, that the friction force exerted on the pipe due to the motion of fluid is equal to 0.144 N, determine the viscosity and the Prandtl number of the liquid.
(b)  Using the Prandtl number and viscosity found in part (a), estimate the bulk temperature at the exit of the pipe for the same inflow temperature as in (a) but with the mass flow rate increased to 0.15 kg/s.
Hint: When the flow in a pipe is fully-developed, the friction factor is equal to: $$f=\frac{(-{\rm d}P/{\rm d}x)D}{\rho u_{\rm b}^2/2}$$
Question #4
A flow of hot water vapor interacts with a 1-meter-long and 1-meter-wide flat plate and forms a boundary layer. In order to keep the plate temperature below 100$^\circ$C, you decide to cool the plate through film cooling. Film cooling consists of injecting some liquid water through the plate so that it evaporates when in contact with the hot vapor flow. The flow of water vapor has the following range of properties: $$10~\frac{\rm m}{\rm s}\le U_\infty\le 100~\frac{\rm m}{\rm s}$$ $$0.2~\frac{\rm kg}{{\rm m}^3}\le \rho_\infty \le 0.6~\frac{\rm kg}{{\rm m}^3}$$ $$400~{\rm K}\le T_\infty \le 800~{\rm K}$$ Knowing that $\Delta H_{\rm vap}=2260$ kJ/kg, $T_{\rm sat}=100^\circ$C, and that the vapor has a viscosity of $\mu_{\rm v}=2\cdot 10^{-5}$ kg/ms, a specific heat at constant pressure of $(c_p)_{\rm v}=2000$ J/kgK, and a thermal conductivity of $k_{\rm v}=0.04$ W/mK, do the following:
(a)  Determine the vapor freestream conditions (within the range specified) that will result in the maximum amount of heat transfer to the plate.
(b)  For the freestream conditions determined in (a), find the minimum amount of injected liquid water in kg/s that prevents the plate temperature to exceed 100$^\circ$C.
(c)  For the freestream conditions determined in (a) and the mass flow rate of injected liquid water found in (b), find the heat flux at the trailing edge of the flat plate in W/m$^2$K.
Question #5
Consider an electrical cable made of copper with a length of 2 meters and a diameter of 2 mm. The cable stands horizontally and is surrounded by air at a temperature of 300 K and a density of 1.2 kg/m$^3$. Knowing that the electrical resistivity of copper is of $15.4\times 10^{-9}~\Omega$m, find the maximum allowed current (in amps) that keeps the surface temperature of the cable below 400 K. Make your design safe by taking into consideration that the convective heat transfer coefficient obtained through the correlations can be off by as much as 30%.
Question #6
Consider a long vertical pipe with smooth walls and with an inner radius of 1 cm. Water fills up the pipe and is pulled downwards through the gravitational acceleration (g=9.81 m/s$^2$). Knowing that the pressure gradient is zero within the pipe, do the following:
(a)  Find the wall shear stress within the pipe in N/m$^2$.
(b)  Find the bulk velocity of the water in m/s within the pipe.
(c)  Find the mass flow rate of the water in kg/s within the pipe.
(d)  Determine whether the flow in the pipe is laminar or turbulent.
Note: for water, $c=4000$ J/kgK, $\rho=1000$ kg/m$^3$, $\mu=10^{-3}$ kg/ms, $k=0.6$ W/mK.
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