Question

Water at \(15^{\circ} \mathrm{C}\) is flowing through a 200 -m-long standard 1-in Schedule 40 cast iron pipe with a mass flow rate of $0.5 \mathrm{~kg} / \mathrm{s}$. If accuracy is an important issue, use the appropriate equation to determine \((a)\) the pressure loss and \((b)\) the pumping power required to overcome the pressure loss. Assume flow is fully developed. Is this a good assumption?

Short answer

Answer: The pressure loss in the pipe is 16654 Pa, and the pumping power required to overcome the pressure loss is 10.4 W.

Step-by-step solution

Gather given information

Here's a summary of the information given in the problem statement: - Water temperature: \(15^{\circ} \mathrm{C}\) - Length of pipe: \(L = 200 \mathrm{~m}\) - Pipe material: Standard Schedule 40 cast-iron pipe with a diameter of 1-in - Mass flow rate: \(\dot{m} = 0.5 \mathrm{~kg/s}\)

Calculate the pipe's diameter and cross-sectional area

Convert the 1-inch diameter pipe to meters to have consistent units: \(D = 1 ~\mathrm{in} \times \frac{0.0254 ~\mathrm{m}}{1 ~\mathrm{in}} = 0.0254 ~\mathrm{m}\) Now, find the cross-sectional area of the pipe: \(A = \frac{\pi D^2}{4} = \frac{\pi (0.0254)^2}{4} = 5.067 × 10^{-4} ~\mathrm{m^2}\)

Calculate the volumetric flow rate

Using the mass flow rate given and the density of water at \(15^{\circ} \mathrm{C}\) (\(\rho = 999 \mathrm{~kg/m^3}\)), calculate the volumetric flow rate, Q: \(Q = \frac{\dot{m}}{\rho} = \frac{0.5 ~\mathrm{kg/s}}{999 ~\mathrm{kg/m^3}} = 5.006 × 10^{-4} ~\mathrm{m^3/s}\)

Calculate average velocity of the flow

Calculate the average velocity, \(V\), in the pipe: \(V = \frac{Q}{A} = \frac{5.006 × 10^{-4} ~\mathrm{m^3/s}}{5.067 × 10^{-4} ~\mathrm{m^2}} = 0.988 ~\mathrm{m/s}\)

Determine the Reynolds number and friction factor

To find the friction factor (\(f\)) in the Darcy-Weisbach equation, Reynolds number needs to be calculated first. The kinematic viscosity of water at \(15^{\circ} \mathrm{C}\) is \(\nu = 1.14 × 10^{-6} ~\mathrm{m^2/s}\). Calculate the Reynolds number (Re): \(\mathrm{Re} = \frac{VD}{\nu} = \frac{0.988 ~\mathrm{m/s} \times 0.0254 ~\mathrm{m}}{1.14 × 10^{-6} ~\mathrm{m^2/s}} = 22175\) Since \(22175 < 4000\), we can assume that the flow is laminar, and the friction factor could be found using the simplification: \(f = \frac{64}{\mathrm{Re}} = \frac{64}{22175} = 0.00288\)

Calculate the pressure loss

Using the Darcy-Weisbach equation, calculate the pressure loss, \(\Delta P\): \(\Delta P = f \frac{L}{D}\frac{1}{2} \rho V^2 = 0.00288 \times \frac{200}{0.0254} \times \frac{1}{2} \times 999 ~\mathrm{kg/m^3} \times (0.988 ~\mathrm{m/s})^2 = 16654 ~\mathrm{Pa}\) The pressure loss in the pipe is \(16654 ~\mathrm{Pa}\).

Calculate the pumping power required

To find the pumping power required to overcome the pressure loss, use the formula: \(P = \frac{\Delta P \times Q}{\eta}\) Assuming a pump efficiency of 80% (\(\eta = 0.80\)), we get: \(P = \frac{16654 ~\mathrm{Pa} \times 5.006 × 10^{-4} ~\mathrm{m^3/s}}{0.80} = 10.4 ~\mathrm{W}\) The pumping power required to overcome the pressure loss is \(10.4 ~\mathrm{W}\).

Evaluate the fully-developed flow assumption

The problem asked if our assumption of fully developed flow is good. Since the flow is laminar, the entrance length is given by \(Le = 0.05 \times \mathrm{Re} \times D\). Calculating the entrance length: \(Le = 0.05 \times 22175 \times 0.0254 ~\mathrm{m} = 28.3 ~\mathrm{m}\) Since the entrance length (\(28.3 ~\mathrm{m}\)) is much smaller than the total length of the pipe (\(200 ~\mathrm{m}\)), the assumption of fully developed flow is reasonable for this problem. In conclusion, the pressure loss in the pipe is \(16654 ~\mathrm{Pa}\), and the pumping power required to overcome the pressure loss is \(10.4 ~\mathrm{W}\). Our assumption of fully developed flow is reasonable.