Viscous Flow Assignment 6 — Pipe Systems  
Question #1
Starting from the first law of thermodynamics in integral form: $$ \frac{{\rm d}}{{\rm d}t} \int_{V} \rho E {\rm d}V + \int_{S} (\rho E + P)(\vec{v}\cdot\vec{n}){\rm d}S=\dot{Q}-\dot{W} $$ with: $$ E\equiv e+\frac{1}{2}q^2 + g y $$ Show that conservation of energy within a pipe system can be expressed as: $$ \left( \frac{P_1}{\rho} + g y_1 + \frac{\alpha_1}{2} (u_{\rm b})_1^2\right) -\left( \frac{P_2}{\rho} + g y_2 + \frac{\alpha_2}{2} (u_{\rm b})_2^2\right)=h_{\rm L} $$ Outline all assumptions and provide the definitions for the kinetic energy coefficient $\alpha$ and the head loss $h_{\rm L}$.
Question #2
Consider water with a viscosity $\mu=10^{-3}$ kg/ms and a density $\rho=1000$ kg/m$^3$ flowing from a large reservoir through two pipes as follows:
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The first pipe has a length $L_1=60$ km and a diameter $D_1=0.1$ m while the second pipe has a length $L_2=1.2$ m. For a height difference between the water in the reservoir and the water exiting the second pipe $\Delta H=26.24$ m, find the diameter of the second pipe $D_2$ that will yield a mass flow rate $\dot{m}=0.79$ kg/s. Also find the Reynolds number within both pipes and indicate whether the flow is turbulent or laminar within pipe 1 and 2. You can assume that both pipes are smooth.
Question #3
Consider a large pump with the exit pipe being $\Delta H=20$ meters higher than the entrance pipe as follows:
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Water enters the pump with a mass flow rate of 100 kg/s, a pressure of 1 atm and a temperature of 300 K. It is observed that the water entering the pump has a uniform velocity profile (constant $q$) while the water exiting the pump has a velocity profile that is fully-developed. The pipe diameter at the pump entrance is of $D_1=8$ cm and the pipe diameter at the pump exit is $D_2=12$ cm. Given a pump power of 140 kW and a pump efficiency of 50%, and knowing that the pump is well insulated and does not lose heat to the environment, do the following:
(a)  Calculate the pressure of the water exiting the pump
(b)  Calculate the temperature of the water exiting the pump
Use the following data for liquid water: density $\rho=1000$ kg/m$^3$, viscosity $\mu=10^{-3}$ kg/ms, heat capacity $c=c_p=c_v=4200~$J/kgK, conductivity $k=0.6~$W/mK.
Question #4
A 8.5 m long plastic tube with the inner diameter of 8.0 mm is used to empty an aquarium. The tube is led from the aquarium to a floor drain in a bathroom. The water surface of the aquarium is located 0.90 m above the outlet of the tube. The roughness of the tube is $e=0.001$ mm. Determine the volumetric flow rate in the tube (in m$^3$/s). The density of water is 10$^3$ kg/m$^3$ and the kinematic viscosity is $10^{-6}$ m$^2$/s. Hint: in this problem you will have to iterate and also find out if the flow is laminar or turbulent.
Question #5
Oil flows in a pipeline at a rate of 200 million liters per day. The pipe inside diameter is 100 cm, and its roughness is 0.1 cm. The maximum pressure that the pipeline can sustain is 80 atmospheres above atmospheric pressure. The minimum pressure necessary to keep dissolved gases within the oil is 3 atmospheres (absolute pressure). The oil has a density of 920 kg/m$^3$ and a kinematic viscosity of $5$ CentiStokes (1 CentiStoke = $10^{-6}$ $\rm m^2$/s). For these conditions, determine the maximum possible spacing between pumping stations. If the pump efficiency is 90 percent, determine the power that must be supplied at each pumping station.
Question #6
A water tap is fed through a 370 m long pipe from a container. The water surface of the container is $21$ m above the ground, and the opening of the tap is located 3.4 m above the ground. The friction factor of the tube is $f=0.020$. Five bends give a pressure loss of $0.8\cdot \frac{1}{2} \rho u_{\rm b}^2$ each. In addition, there are two valves present in the system that each introduce a pressure loss of $1.5\cdot \frac{1}{2} \rho u_{\rm b}^2$. How much time will it take to fill up a 32 m$^3$ cistern?
2.  10000, 0.01 m, $1.01\times10^5$.
3.  764 kPa, 300.167 K.
4.  32 cm$^3$/s.
5.  98.9 km, 20.3 MW.
6.  4 hours 44 mins.
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