Heat Transfer Assignment 8 — Pipe & Duct 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
Water at $43^\circ$C enters at a rate of $6$ kg/s a 5-cm-internal-diameter pipe having a relative roughness $e/D$ of 0.002. If the pipe is 2 m long and the pipe walls are maintained at $71^\circ$C, calculate the exit bulk temperature of the water and the total heat transfer.
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 pipeline used to transport oil over a distance of 100 kilometers. The pipeline is buried 3 m in the earth and is composed of a tube of internal diameter of 1.2 m surrounded by a 15-cm-thick isolator which has a thermal conductivity of 0.05 W/m$\cdot^\circ$C. Oil with a density of $900$ kg/m$^3$, a viscosity of 0.765 kg/ms, a heat capacity of 2000 J/kg$\cdot^\circ$C and a Prandtl number of $10^4$ enters the pipeline with a mass flow rate of 500 kg/s and a bulk temperature of $120^\circ$C. What is the bulk temperature of the oil after it has travelled 100 kilometers if the surface temperature of the earth is $-40^\circ$C and its thermal conductivity 0.5 W/m$\cdot^\circ$C?
Question #4
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)
1.  63.5$^\circ$C, 516 kW.
2.  71.4$^\circ$C.
3.  111.9$^\circ$C.
4.  0.12 kg/s, $3.3^\circ$C, $7^\circ$C.
Due on May 30th at 15:00. Do Questions #2, #3, and #4 only.
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