Chapter 5, Problem 5.19P

(a)

Interpretation Introduction

Interpretation:

The maximum flow rate through a fire hose for a pump is to be determined.

Concept Introduction:

The flow rate through the fire hose is given as,

  $\begin{array}{l}\text{Q}\text{=}\text{A}\text{×}\text{<mover accent="true"><mo>V</mo><mo>¯</mo></mover>}\\ \text{Q}\text{=}\frac{\text{π}}{\text{4}}{\text{D}}^{\text{2}}\text{<mover accent="true"><mo>V</mo><mo>¯</mo></mover>}\mathrm{.......}\text{ (1)}\end{array}$

The notations used here are,

D = Diameter of the fire hose

  $\text{<mover accent="true"><mo>V</mo><mo>¯</mo></mover>}$ = Average velocity of the fluid

The average velocity can be calculated from the pressure drop of the fluid,

  $\text{ΔP}\text{=}\frac{{\text{2fLρ<mover accent="true"><mo>V</mo><mo>¯</mo></mover>}}^{\text{2}}}{\text{Dg}}$ ....... (2)

The notations used here are,

  $\text{ΔP}$ = Pressure drop in the fluid

g = Acceleration due to gravity

f = Fanning friction factor

  $\text{ρ}$ = Density of the fluid

L = Length of the duct

The Reynolds number is given as,

  $\text{Re = }\frac{\text{ρ<mover accent="true"><mo>V</mo><mo>¯</mo></mover><mo>D</mo>}}{\text{μ}}$ ....... (3)

The notations used here are,

  $\text{<mover accent="true"><mo>V</mo><mo>¯</mo></mover>}$ = Average velocity of the fluid

D = Diameter of duct

  $\text{μ}$ = Viscosity of the fluid

  $\text{ρ}$ = Density of fluid

The roughness parameter is given as, $\frac{\text{k}}{\text{D}}$ where k is roughness of the pipe. It shows how much friction is present in the pipe and with the help of the roughness parameter, f can be calculated from the friction factor chart.

(b)

Interpretation Introduction

Interpretation:

The increase in flow is to be determined due to contamination by ethylene oxide.

Concept Introduction:

The change in friction factor determines the change in flow rate as it depends on the average velocity keeping other factors constant.

Thus,

The flow rate through the fire hose is given as,

  $\begin{array}{l}\text{Q}\text{=}\text{A}\text{×}\text{<mover accent="true"><mo>V</mo><mo>¯</mo></mover>}\\ \text{Q}\text{=}\frac{\text{π}}{\text{4}}{\text{D}}^{\text{2}}\text{<mover accent="true"><mo>V</mo><mo>¯</mo></mover>}\mathrm{.......}\text{ (1)}\end{array}$

The notations used here are,

D = Diameter of the fire hose

  $\text{<mover accent="true"><mo>V</mo><mo>¯</mo></mover>}$ = Average velocity of the fluid

The average velocity can be calculated from the pressure drop of the fluid,

  $\text{ΔP}\text{=}\frac{{\text{2fLρ<mover accent="true"><mo>V</mo><mo>¯</mo></mover>}}^{\text{2}}}{\text{Dg}}$ ....... (2)

The notations used here are,

  $\text{ΔP}$ = Pressure drop in the fluid

g = Acceleration due to gravity

f = Fanning friction factor

  $\text{ρ}$ = Density of the fluid

L = Length of the duct

Combining equation (1) and (2), it can be deduced that,

  $\text{Q}\text{α}\frac{\text{1}}{\sqrt{\text{f}}}$ ....... (3)