Convective Heat Transfer Assignment 7 — Internal Convection  
Question #1
Starting from the energy equation for a constant density fluid in axisymmetric coordinates: $$ \rho \left(\frac{\partial e}{\partial t} + u \frac{\partial e}{\partial x} + v \frac{\partial e}{\partial r} \right) = \frac{\partial }{\partial x}\left(k\frac{\partial T}{\partial x} \right) + \frac{1}{r} \frac{\partial }{\partial r} \left(kr \frac{\partial T}{\partial r} \right) + \mu \left(\frac{\partial u}{\partial x} \right)^2 + \mu \left(\frac{\partial u}{\partial r} \right)^2 + \mu \left(\frac{\partial v}{\partial x} \right)^2 + \mu \left(\frac{\partial v}{\partial r} \right)^2 $$ In the thermally fully-developed region of a pipe of diameter $D$, show that $$ {\rm Nu}_D=\frac{h D}{k}=\frac{48}{11} $$ $$ {h}\equiv \frac{q_{\rm w}^{\prime\prime}}{T_{\rm w}-T_{\rm b}} $$ with $T_{\rm b}$ the bulk temperature. Outline all assumptions. Note: you can make use of the velocity profile in the fully-developed region $u=2 u_{\rm b}(1-r^2/R^2)$ with $u_{\rm b}$ the bulk velocity.
Question #2
Water at $30^\circ$C enters at a rate of 0.25 kg/s a 4-cm-diameter smooth pipe.
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Over its entire length of $L=6$ m, the pipe is heated on its outside surface by a cross flow of air at a pressure of 1 atm, a temperature of $300^\circ$C, and a speed of 100 m/s. For a pipe of negligible wall thickness (resulting in essentially equal internal and external pipe diameters), calculate the bulk water temperature at the pipe exit.
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: $$f=\frac{(-{\rm d}P/{\rm d}x)D}{\rho u_{\rm b}^2/2}$$
2.  71.4$^\circ$C.
3.  $0.001~{\rm kg/ms}$, $8.88$, $60^\circ{\rm C}$.
Due on May 30th at 17:00. Do all questions.
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