Chapter 4, Problem 4.14P

(a)

Interpretation Introduction

Interpretation:

For the uniform exit velocity, the value of mass flow rate and maximum velocity should be calculated.

Concept Introduction:

According to law of conservation of mass, energy can neither be created nor be destroyed.

The generalized form of equation is written as:

[Rate of mass in from control volume] - [Rate of mass out from control volume] + [Rate of accumulation]

Mass flow rate is defined as the ratio of the mass of fluid passing the point with respect to time.

$\text{m}=\text{ρ×v}$

Law of conservation of mass for control volume is given by,

$\frac{\partial \stackrel{\text{•}}{\text{M}}}{\partial \text{t}}+{\displaystyle \int \text{d}\stackrel{\text{•}}{\text{m}}}=0$

Where,

$\frac{\partial \stackrel{\text{•}}{\text{M}}}{\partial \text{t}}$ is the rate of change of mass in control volume.

$\text{d}\stackrel{\text{•}}{\text{m}}$ is the net mass outflow across the control surface

t is the time.

Mass flow rate in terms of average velocity is,

${\text{(ρv)}}_{\text{average}}\text{A}={\displaystyle {\iint }_A\text{ρvdA}}$

Where,

A is the area.

v is the average velocity.

$\text{ρ}$ is the density.

Maximum velocity is the velocity which exists at the center of circular passage. The relationship between average velocity and maximum velocity is,

${\text{v}}_{\text{average}}=\frac{{\text{v}}_{\text{max}}}{2}$

(b)

Interpretation Introduction

Interpretation:

For the parabolic exit velocity,the value of mass flow rate and maximum velocity should be calculated

Concept Introduction:

According to law of conservation of mass, energy can neither be created nor be destroyed.

The generalized form of equation is written as:

[Rate of mass in from control volume] - [Rate of mass out from control volume] + [Rate of accumulation]

Mass flow rate is defined as the ratio of the mass of fluid passing the point with respect to time.

$\text{m}=\text{ρ×v}$

Law of conservation of mass for control volume is given by,

$\frac{\partial \stackrel{\text{•}}{\text{M}}}{\partial \text{t}}+{\displaystyle \int \text{d}\stackrel{\text{•}}{\text{m}}}=0$

Where,

$\frac{\partial \stackrel{\text{•}}{\text{M}}}{\partial \text{t}}$ is the rate of change of mass in control volume.

$\text{d}\stackrel{\text{•}}{\text{m}}$ is the net mass outflow across the control surface

t is the time.

Mass flow rate in terms of average velocity is,

${\text{(ρv)}}_{\text{average}}\text{A}={\displaystyle {\iint }_A\text{ρvdA}}$

Where,

A is the area.

v is the average velocity.

$\text{ρ}$ is the density.

The equation of parabolic velocity profile is given by,

$\text{v(y)}\text{=}{\text{C}}_{\text{1}}\text{y}+{\text{C}}_{\text{2}}{\text{y}}^{\text{2}}$

Where,

${\text{C}}_{\text{1}}$ and ${\text{C}}_{\text{2}}$ are constant.