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Chapter 8: Problem 110

A 10 -m-long \(, 5.042\) -cm I.D. copper pipe has two fully open gate valves, a swing check valve, and a sudden enlargement to a \(9.919-\mathrm{cm}\) I.D. copper pipe. The \(9.919 \mathrm{cm}\) copper pipe is \(5.0 \mathrm{m}\) Iong and then has a sudden contraction to another 5.042 -cm copper pipe. Find the head loss for a \(20^{\circ} \mathrm{C}\) water flow rate of \(0.05 \mathrm{m}^{3} / \mathrm{s}\)

Short Answer

Expert verified
The total head loss is the sum of the head losses due to friction in the different sections of the pipe, the gate valves, the check valve, the sudden enlargement, and the sudden contraction.

Step by step solution

01

Identify Constants

The first task is to list all known quantities, these include inner diameters of the pipes in cm (5.042 cm and 9.919 cm), lengths of the pipes in meters (10 m and 5 m), flow rate of water in cubic meters per second \((0.05 m^3/s)\), and the temperature of water (\(20^{\circ}C\)). The K-values for the fully open gate valve and swing check valve are 0.19 and 2.00 respectively according to the Crane Technical Paper No. 410. For sudden enlargement and sudden contraction, the K values can be calculated using the equation \(K_{enlargement} = (1 - d1/d2)^2\) and \(K_{contraction} = 0.5 * (1 - d1/d2)^2\) respectively.
02

Calculation of Velocities

The next step is to use the formula \( v = Q / A \) to determine the velocities, where v is the velocity of fluid, Q is the flow rate, and A is the cross sectional area. The cross sectional area can be calculated using the formula \( A = \pi( d/2)^2 \), where d is the inner diameter of the pipe.
03

Calculating Reynolds Number and Friction Factor

Use the velocity of fluid (v) and hydraulic diameter (d) to compute the Reynolds Number (\( Re = \rho v d/ \mu \)), where \(\rho\) is the density and \(\mu\) is the dynamic viscosity of water at given temperature. Consider the properties of water at \(20^{\circ}C\), where \( \mu = 1.003 x 10^{-3}\) Pascal second and \(\rho = 998.2 kg/m^3\). Friction factor (f) is then calculated using the Moody Chart or other approximations based on the known Reynolds Number.
04

Calculating Individual Head losses

Head losses due to friction in the pipe can be calculated using Darcy-Weisbach equation \( h_{f} = f L v^2 / (2 g d) \) for each section of the pipe. Then, calculate head losses due to valves and sudden contraction and enlargement, using the formula \( h_{v} = K v^2 / 2g \), where h represents the head loss, K represents the friction loss factor, v is the velocity of the fluid, and g is the acceleration due to gravity.
05

Calculating Total Head Loss

The final step is to sum all the head losses obtained in the step 4 to get the total head loss.

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Most popular questions from this chapter

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