Convective Heat Transfer Assignment 6 — 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
The first design project given to you after you join a water distribution company is to prevent water flowing in an underground pipe from freezing. Consider a long 100 m pipe with a 0.15 m radius buried 2 m under ground (the center of the pipe is 2 m below the earth surface). Water flows in the pipe with the following properties: $$ \rho=1000~{\rm kg/m}^3,~~~c_p=~4000~{\rm J/kgK},~~~k=0.6~{\rm W/m^\circ C},~~~\mu=10^{-3}~{\rm kg/ms} $$ On a cold winter day, the surface of the ground is measured to be $-10^\circ$C. Water enters the pipe at a bulk temperature of $20^\circ$C. To prevent freezing (with a safety margin), the water temperature should not drop below $3.3^\circ$C at any location. The ground conductivity can be taken as 1.5 W/m$^\circ$C, and the pipe walls can be assumed smooth and to oppose negligible resistance to heat flow. Do the following:
(a)  Determine the minimum water mass flow rate through the pipe that prevents the water temperature to fall below 3.3$^\circ$C anywhere within the pipe; make your design safe by taking into consideration that the ground surface temperature varies by as much as $\pm 2.4^\circ$C and that the ground conductivity varies by as much as $\rm \pm 0.5~W/m^\circ$C.
(b)  Determine the wall temperature of the pipe for the mass flow rate found in (a)
(c)  Determine the bulk temperature of the water exiting the pipe for the mass flow rate found in (a)
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.  0.12 kg/s, $3.3^\circ$C, $7^\circ$C.
3.  $0.001~{\rm kg/ms}$, $8.88$, $60^\circ{\rm C}$.
Due on Wednesday June 7th at 16:30. Do all 3 problems.
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