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 54P

a)

To determine

The wattage of the fan-motor unit to be purchased.

a)

Expert Solution
Check Mark

Explanation of Solution

Given:

The dimension of the bathroom is 2m×3m×3m.

The combine efficiency of the fan-motor unit is 50%.

The entire volume of air in the bathroom is replace by the ventilator for every 10 minutes.

Calculation:

Draw the free body diagram of the system as in Figure 1.

Fundamentals of Thermal-Fluid Sciences, Chapter 12, Problem 54P

Assume the frictional losses along the flow is negligible and the flow is steady and incompressible.

The effect of kinetic energy correction factor is negligible α=1.

Take the density of air ρ=1.25kg/m3 .

Consider the acceleration due to gravity (g) as 9.81m/s2.

Calculate the total volume of air (V) in the bathroom;

  V=2m×3m×3m=18m3

Calculate the volume flow rate V˙ using the time it takes to replace the entire volume of air.

  V˙=Vt

Here, t is time.

Substitute 18m3 for V and 10 min for t.

  V˙=18m310min×1min60s=0.03m3/s

Calculate the mass flow rate m˙ using the relation;

  m˙=ρV˙

Substitute 1.25kg/m3 for ρ and 0.03m3/s for V˙.

  m˙=1.25×0.03=0.0375kg/s

Refer to figure 1,

The pressure at the point 1 and point 2 as same as the atmospheric pressure P1=P2=Patm .

The elevation of the point 1 and 2 are same z1=z2 .

Consider the point 1 is far enough from the fan. Thus the flow velocity at the point 1 V1=0 .

Write the energy equation for a steady incompressible flow that relates the pressure (P) , velocity (V) and height (z) for between the point 1 and point 2 as below;

  m˙(P1ρ+α1V122+z1g)+W˙pump=m˙(P2ρ+α2V222+z2g)+W˙turbine+E˙mech,loss

Substitute z1 for z2 , 0 for V1 , P1 for P2 and 0 for W˙pump.

  m˙(P1ρ+0+z1g)+W˙pump=m˙(P1ρ+α2V222+z1g)+0+E˙mech,lossW˙pumpE˙mech,loss=m˙α2V222

Substitute W˙fan,elect for W˙pumpE˙mech,loss  .

  W˙fan,elect=m˙α2V222

Recall that the electric power rating of fan is equal to the effective power divided by the power rating.

  W˙fan,elect=1ηm˙α2V222

Substitute 0.5 for η, 0.0375kg/s for m˙, 1 for α2 and 8 m/s for V2 .

  W˙fan,elect=10.5×0.0375×1×(8)22=2.4W

Thus, the wattage of the fan-motor unit to be purchased is 2.4W_.

b)

To determine

The diameter of fan casing.

b)

Expert Solution
Check Mark

Explanation of Solution

Given:

The dimension of the bathroom is 2m×3m×3m.

The combine efficiency of the fan-motor unit is 50%.

The entire volume of air in the bathroom is replace by the ventilator for every 10 minutes.

Calculation:

Calculate the volume flow rate V˙ using the diameter of fan and velocity of flow.

  V˙=A2V2

Here, A2 is the area of the fan.

Substitute πD24 for A2, 0.03m3/s for V˙ and 8 m/s for V2 .

  0.03=πD24×8D=4(0.03)8π=6.9cm

Thus, the diameter of the casing is 6.9cm_.

c)

To determine

The pressure difference across the fan.

c)

Expert Solution
Check Mark

Explanation of Solution

Given:

The dimension of the bathroom is 2m×3m×3m.

The combine efficiency of the fan-motor unit is 50%.

The entire volume of air in the bathroom is replace by the ventilator for every 10 minutes.

Calculation:

Take that the point 3 and 4 to be on the two sides of the fan on the horizontal line.

The elevation of the point 3 and 4 are same z3=z4 .

The velocity of the point 3 and 4 are same V3=V4 .

Write the energy equation for a steady incompressible flow that relates the pressure (P) , velocity (V) and height (z) for between the point 3 and point 4 as below;

  m˙(P3ρ+α3V322+z3g)+W˙pump=m˙(P4ρ+α4V422+z4g)+W˙turbine+E˙mech,loss

Substitute z3 for z4 , V3  for V4 and 0 for  W˙turbine.

  m˙(P3ρ+αV322+z3g)+W˙pump=m˙(P4ρ+αV322+z3g)+0+E˙mech,lossm˙P3ρ+W˙pumpE˙mech,loss=m˙P4ρW˙pumpE˙mech,loss=m˙P4ρm˙P3ρ

Substitute W˙fan,elect for W˙pumpE˙mech,loss  .

  W˙fan,elect=m˙P4ρm˙P3ρW˙fan,elect=m˙(P4P3)ρP4P3=W˙fan,electm˙/ρ

Substitute V˙ for m˙/ρ.

  P4P3=W˙fan,electV˙

Substitute 1.2 W for W˙fan,elect and 0.03m3/s for V˙.

  P4P3=1.2W0.03m3/s=40Pa

Thus, the pressure difference across the fan is 40Pa_.

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Chapter 12 Solutions

Fundamentals of Thermal-Fluid Sciences

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