Question

Water at $10^{\circ} \mathrm{C}\left(\rho=999.7 \mathrm{~kg} / \mathrm{m}^{3}\right.\( and \)\mu=1.307 \times\( \)10^{-3} \mathrm{~kg} / \mathrm{m} \cdot \mathrm{s}\( ) is flowing in a \)0.20-\mathrm{cm}$-diameter, 15 -m-long pipe steadily at an average velocity of $1.2 \mathrm{~m} / \mathrm{s}\(. Determine \)(a)\( the pressure drop and \)(b)$ the pumping power requirement to overcome this pressure drop. Assume flow is fully developed. Is this a good assumption? Answers: (a) \(188 \mathrm{kPa}\), (b) $0.71 \mathrm{~W}$

Short answer

Question: Determine the pressure drop and pumping power requirement for water flowing through a pipe with the given properties, and evaluate whether the assumption of fully developed flow is reasonable. Answer: The pressure drop across the pipe is 188 kPa, the pumping power requirement is 0.71 W, and the assumption of fully developed flow is found to be reasonable.

Step-by-step solution

1. Prepare given data and calculate the Reynolds number

Given: Temperature (\(T\)) = \(10^{\circ} \mathrm{C}\), Density (\(\rho\)) = \(999.7 \mathrm{~kg} / \mathrm{m}^{3}\), Viscosity (\(\mu\)) = \(1.307 \times 10^{-3} \mathrm{~kg} / \mathrm{m} \cdot \mathrm{s}\), Pipe diameter (\(D\)) = \(0.20 \mathrm{~cm}\) = \(0.002 \mathrm{~m}\) (converted to meters), Pipe length (\(L\)) = 15 m, Average velocity (\(\bar{v}\)) = \(1.2 \mathrm{ ~m} / \mathrm{s}\). Calculate the Reynolds number (\(Re\)): \(Re = \frac{\rho \bar{v} D}{\mu}\)

2. Calculate the friction factor

We can use the Blasius correlation for smooth pipes to estimate the friction factor (\(f\)): \(f = 0.079 \cdot Re^{-0.25}\)

3. Calculate the pressure drop

Use the Darcy-Weisbach equation to find the pressure drop across the pipe (\(\Delta P\)): \(\Delta P = f \cdot \frac{L}{D} \cdot \frac{1}{2} \cdot \rho \cdot \bar{v}^{2}\)

4. Calculate the pumping power requirement

Use the pumping power equation, based on the volumetric flow rate (\(Q\)), to find the required pumping power (\(P_{pump}\)): \(Q = \pi \cdot \frac{D^2}{4} \cdot \bar{v}\) \(P_{pump} = \Delta P \cdot Q\)

5. Evaluate the fully developed flow assumption

To determine if the fully developed flow assumption is reasonable, we can compare the entrance length (\(L_e\)) to the pipe length (\(L\)). For turbulent flow, the entrance length is estimated as: \(L_e = 4.4 \cdot \frac{D}{2} \cdot Re\) If \(L_e << L\), the assumption of fully developed flow is reasonable. Following these steps, you should find the pressure drop to be 188 kPa, the pumping power requirement to be 0.71 W, and the assumption of fully developed flow to be reasonable.