Fundamentals Of Thermal-fluid Sciences In Si Units
Fundamentals Of Thermal-fluid Sciences In Si Units
5th Edition
ISBN: 9789814720953
Author: Yunus Cengel, Robert Turner, John Cimbala
Publisher: McGraw-Hill Education
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Chapter 13, Problem 25P
To determine

The anchoring force needed to hold the elbow in place.

Expert Solution & Answer
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Explanation of Solution

Given:

The elbow angle (θ) is 45°.

The cross sectional area at inlet (A1) is 150cm2.

The cross sectional area at exit (A2) is 25cm2.

The elevation difference (z2z1) is 40cm.

The mass of the elbow and the water in it is (m) is 50kg.

The momentum-flux correction factor is (β) is 1.03.

The mass flow rate (m˙) is 30kg/s.

Calculation:

Calculate the weight density of water by using the relation.

    W=mg=(50kg)(9.81m/s2)=490.5kgm/s2(1N1kgm/s2)=490.5N

Calculate the velocity at the inlet (V1) of the pipe using the relation.

  V1=m˙ρA1=30kg/s(1000kg/m3)(150cm2(1m100cm)2)=2.0m/s

Calculate the velocity at the outlet (V2) of the pipe using the relation.

  V2=m˙ρA2=30kg/s(1000kg/m3)(25cm2(1m100cm)2)=12.0m/s

Calculate the gauge pressure at the center on the inlet of the elbow by using Bernoulli’s

Equation.

    p1ρg+V122g+z1=p2ρg+V222g+z2p1p2=ρg(z2z1+V22V122g)

  p=ρg(z2z1+V22V122g)=[(1000(Kg/m3))(9.81(m/s2))(40cm(1m100cm)+((12m/s)2(2m/s)22(9.81m/s)))×(1kN1000kg.m/s2)]=73.9kN/m2(1kPa1kN/m2)=73.9kPa

Calculate the anchoring force in x direction by using moment equation.

  FRx+p1A1=βm˙V2cosθβm˙V1FRx=βm˙(V2cosθV1)p1A1=[1.03(30kg/s)(12cos45°2)m/s(1kN1000kg.m/s2)(73.9kN/m2)(150cm2(1m100cm)2)]=0.908kN

Calculate the anchoring force in y direction by using moment equation.

    FRz=βm˙V2sinθ+W=1.03(30kg/s)(12sin45°m/s)(1kN1000kg.m/s2)+0.4905kN=0.753kN

Calculate the total anchoring force by using the expression.

    FR=FRx2+FRy2=(0.908kN)2+(0.753kN)2=1.18kN

Calculate the angle of the total anchoring force by using the expression.

    θ=tan1(FRzFRx)=tan1(0.7530.908)=39.7°

Thus, the anchoring force needed to hold the elbow is 1.18kN_ in the direction of negative in the direction of 39.7°.

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