/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 169 A large reservoir supplies water... [FREE SOLUTION] | 91影视

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Chapter 8: Problem 169

A large reservoir supplies water for a community. A portion of the water supply system is shown. Water is pumped from the reservoir to a large storage tank before being sent on to the water treatment facility. The system is designed to provide \(1310 \mathrm{L} / \mathrm{s}\) of water at \(20^{\circ} \mathrm{C}\). From \(B\) to \(C\) the system consists of a square-edged entrance, \(760 \mathrm{m}\) of pipe, three gate valves, four \(45^{\circ}\) elbows, and two \(90^{\circ}\) elbows. Gage pressure at \(C\) is 197 kPa. The system between \(F\) and \(G\) contains \(760 \mathrm{m}\) of pipe, two gate valves, and four \(90^{\circ}\) elbows. All pipe is \(508 \mathrm{mm}\) diameter, cast iron. Calculate the average velocity of water in the pipe, the gage pressure at section \(F,\) the power input to the pump (its efficicncy is 80 percent), and the wall shear stress in section \(F G\)

Short Answer

Expert verified
Calculations from the four steps will give the average velocity of the water in the pipe, gage pressure at section F, power input to the pump, and the wall shear stress in section FG. These depend on specific data provided in the problem, so they would need to be calculated during the solving process.

Step by step solution

01

Calculate the average velocity of water in the pipe

The continuity equation is given by \(Q = A \times V\), where \(Q\) is the flow rate, \(A\) is the cross-sectional area of the pipe, and \(V\) is the average velocity. Rearranging gives \(V = Q / A\). The diameter of the pipe is given, so the area of the pipe can be calculated using \(A = Pi \times (D/2)^2\). Substituting given values provides the average velocity.
02

Calculate the gage pressure at section F

The total head loss (\(h_{L,total}\)) in the system can be calculated using Bernoulli's equation and the head loss due to each component in the system (i.e., the square-edged entrance, pipe, gate valves, elbows). The gage pressure at section F (\(P_F\)) then is \(P_F = P_C - \rho \cdot g \cdot h_{L,total}\), where \(\rho\) is the density of water, \(g\) is the acceleration due to gravity, and \(P_C\) is the gage pressure at section C.
03

Calculate the power input to the pump

The power required by the pump can be determined by \(P_r = \rho \cdot g \cdot Q \cdot h_{L}/\eta\), where \(h_{L}\) is the head loss, \(\eta\) is the pump efficiency. Substituting the efficiency of the pump and the calculated total head loss gives the power input to the pump.
04

Calculate the wall shear stress in section FG

The calculation of the wall shear stress (\(\tau\)) in section FG involves the formula: \(\tau = f \cdot \frac{1}{2} \cdot \rho \cdot V^2\), where \(f\) is the friction factor which can be determined using the given pipe material, diameter and flow velocity, and \(V\) is the velocity at section FG (same as the average velocity calculated in step 1).

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Most popular questions from this chapter

A fire nozzle is supplied through \(100 \mathrm{m}\) of \(3.5-\mathrm{cm}\) diameter, smooth, rubber-lined hose. Water from a hydrant is supplied to a booster pump on board the pumper truck at \(350 \mathrm{kPa}(\text { gage }) .\) At design conditions, the pressure at the nozzle inlet is 700 kPa (gage), and the pressure drop along the hose is 750 kPa per \(100 \mathrm{m}\) of length. Determine (a) the design flow rate, (b) the nozzle exit velocity, assuming no losses in the nozzle, and (c) the power required to drive the booster pump, if its efficiency is 70 percent.

Consider blood flow in an artery. Blood is nonNewtonian; the shear stress versus shear rate is described by the Casson relationship: $$\left\\{\begin{array}{ll} \sqrt{\tau}=\sqrt{\tau_{c}}+\sqrt{\mu \frac{d u}{d r}} & \text { for } \tau \geq \tau_{c} \\ \tau=0 & \text { for } \tau<\tau_{c} \end{array}\right.$$ where \(\tau_{e}\) is the critical shear stress, and \(\mu\) is a constant having the same dimensions as dynamic viscosity. The Casson relationship shows a linear relationship between \(\sqrt{\tau}\) and \(\sqrt{d u / d r},\) with intercept \(\sqrt{\tau_{c}}\) and slope \(\sqrt{\mu} .\) The Casson relationship approaches Newtonian behavior at high values of deformation rate. Derive the velocity profile of steady fully developed blood flow in an artery of radius \(R\). Determine the flow rate in the blood vessel. Calculate the flow rate due to a pressure gradient \(d p / d x=-100 \mathrm{Pa} / \mathrm{m},\) in an artery of radius \(R=1 \mathrm{mm},\) using the following blood data: \(\mu=3.5 \mathrm{cP}\) \(\tau_{c}=0.05\) dynes \(/ \mathrm{cm}^{2}\)

Two immiscible fluids are contained between infinite parallel plates. The plates are separated by distance \(2 h\), and the two fluid layers are of equal thickness \(h=5 \mathrm{mm}\). The dynamic viscosity of the upper fluid is four times that of the lower fluid, which is \(\mu_{10 \text { wer }}=0.1 \mathrm{N} \cdot \mathrm{s} / \mathrm{m}^{2}\). If the plates are stationary and the applied pressure gradient is \(-50 \mathrm{kPa} / \mathrm{m}\) find the velocity at the interface. What is the maximum velocity of the flow? Plot the velocity distribution.

Consider steady, fully developed laminar flow of a viscous liquid down an inclined surface. The liquid layer is of constant thickness, \(h\), Use a suitably chosen differential control volume to obtain the velocity profile. Develop an expression for the volume flow rate.

Water discharges to atmosphere from a large reservoir through a moderately rounded horizontal nozzle of \(25 \mathrm{mm}\) diameter. The free surface is \(2.5 \mathrm{m}\) above the nozzle exit plane. Calculate the change in thow rate when a short section of 50 -mm-diameter pipe is attached to the end of the nozzle to form a sudden expansion. Determine the location and estimate the magnitude of the minimum pressure with the sudden expansion in place. If the flow were frictionless (with the sudden expansion in place), would the minimum pressure be higher, lower, or the same? Would the flow rate be higher, lower, or the same?
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