Fundamentals of Thermal-Fluid Sciences
Fundamentals of Thermal-Fluid Sciences
5th Edition
ISBN: 9780078027680
Author: Yunus A. Cengel Dr., Robert H. Turner, John M. Cimbala
Publisher: McGraw-Hill Education
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Chapter 12, Problem 22P

(a)

To determine

The maximum height the water will reach in the tank.

(a)

Expert Solution
Check Mark

Explanation of Solution

Given:

The diameter of the tank is DT.

The mass flow rate is m˙in..

The diameter of the orifice is Do.

Neglect the frictional losses, since the orifice has rounded entrance.

Calculation:

Consider the point 1 as the free surface of the tank and the point 2 as the exit of the orifice.

Consider the reference line is at the orifice. Therefore, z2=0.

The pressure at the points 1 and 2 is atmospheric. Therefore, P1=P2=Patm.

The velocity at the tank is very small compared to the orifice. Therefore, V10

Write the Bernoulli’s equation between the points 1 and 2 as follows;

  P1ρg+V122g+z1=P2ρg+V222g+z2Patmρg+022g+z1=Patmρg+V222g+0z1=V222gV2=2gz1        (I)

Write the equation for the mass flow rate through the orifice as follows;

  m˙out=ρV˙out=ρAorificeV2=ρ(πDo24)2gzz=12g(4m˙outρπDo2)2        (II)

When the flow rate is incoming, m˙out=m˙in;z=hmax;

Substitute m˙out=m˙in and z=hmax in Equation (II).

hmax=12g(4m˙inρπDo2)2

Thus, the maximum height the water will reach in the tank is hmax=12g(4m˙inρπDo2)2_.

(b)

To determine

The relationship for the water height as a function of time.

(b)

Expert Solution
Check Mark

Explanation of Solution

The amount of water flows through the orifice at a differential time is;

  dmout=m˙outdt=ρ×πDo24×2gzdt

The increase in water in the tank at the differential time is;

  dmtank=ρAtankdz=ρ×πDT24×dz

The amount of water enters during the differential time is;

dmin=m˙indt

Write the conservation of mass equation;

  dmtank=m˙indtm˙outdtρ×πDT24×dz=m˙indtρ×πDo24×2gzdtρ×πDT24×dz=(m˙inρ×πDo24×2gz)dtρ×πDT24×dzm˙inρ×πDo24×2gz=dt

Integrate the above equation and apply the boundary conditions (t=0tot;z=0toz).

  z=0z=zρ×πDT24×dzm˙inρ×πDo24×2gz=t=0t=tdtρπDT22(ρπDo22g4)2(ρπDo22gz4m˙inlnm˙inρπDo22gz4m˙in)=t

Thus, the relationship for the water height as a function of time is ρπDT22(ρπDo22g4)2(ρπDo22gz4m˙inlnm˙inρπDo22gz4m˙in)=t_.

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Chapter 12 Solutions

Fundamentals of Thermal-Fluid Sciences

Ch. 12 - A glass manometer with oil as the working fluid is...Ch. 12 - The velocity of a fluid flowing in a pipe is to be...Ch. 12 - The water level of a tank on a building roof is 20...Ch. 12 - Prob. 14PCh. 12 - Prob. 15PCh. 12 - Prob. 16PCh. 12 - Prob. 17PCh. 12 - Prob. 18PCh. 12 - A piezometer and a Pitot tube are tapped into a...Ch. 12 - The diameter of a cylindrical water tank is Do and...Ch. 12 - Prob. 21PCh. 12 - Prob. 22PCh. 12 - Prob. 23PCh. 12 - An airplane is flying at an altitude of 12,000 m....Ch. 12 - While traveling on a dirt road, the bottom of a...Ch. 12 - Prob. 26PCh. 12 - Prob. 27PCh. 12 - Air at 105 kPa and 37°C flows upward through a...Ch. 12 - A handheld bicycle pump can be used as an atomizer...Ch. 12 - Prob. 31PCh. 12 - The water pressure in the mains of a city at a...Ch. 12 - Prob. 33PCh. 12 - Air is flowing through a venturi meter whose...Ch. 12 - The water level in a tank is 15 m above the...Ch. 12 - What is useful pump head? How is it related to the...Ch. 12 - Prob. 38PCh. 12 - What is irreversible head loss? How is it related...Ch. 12 - Consider the steady adiabatic flow of an...Ch. 12 - Prob. 41PCh. 12 - Prob. 42PCh. 12 - Prob. 43PCh. 12 - Prob. 44PCh. 12 - In a hydroelectric power plant, water flows from...Ch. 12 - Reconsider Prob. 12–45E. Determine the flow rate...Ch. 12 - Prob. 47PCh. 12 - Water is being pumped from a large lake to a...Ch. 12 - A 15-hp (shaft) pump is used to raise water to a...Ch. 12 - Prob. 51PCh. 12 - The water level in a tank is 20 m above the...Ch. 12 - Prob. 53PCh. 12 - Prob. 54PCh. 12 - Water flows at a rate of 20 L/s through a...Ch. 12 - Prob. 56PCh. 12 - Prob. 57PCh. 12 - Prob. 58PCh. 12 - Prob. 59PCh. 12 - Prob. 60PCh. 12 - Prob. 61PCh. 12 - Prob. 62PCh. 12 - Prob. 63PCh. 12 - A 73-percent efficient 12-hp pump is pumping water...Ch. 12 - Prob. 65PCh. 12 - Air flows through a pipe at a rate of 120 L/s. The...Ch. 12 - Air at 100 kPa and 25°C flows in a horizontal duct...Ch. 12 - Prob. 68RQCh. 12 - Prob. 69RQCh. 12 - Prob. 70RQCh. 12 - A 3-m-high large tank is initially filled with...Ch. 12 - Prob. 73RQCh. 12 - Prob. 74RQ
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