Chapter 4, Problem 4.10P

Interpretation Introduction

Interpretation:For the given statement, the equation $\frac{\partial M}{\partial t}+{\displaystyle {\iint }_{c.s}d\stackrel{•}{m}}=0$ should be proved.

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]

Equation of integrated mass balance over control volume,

  ${\displaystyle {\iint }_{c,s}\text{ρ}\left(\overrightarrow{v}\cdot \overrightarrow{n}\right)}\text{dA+}\frac{\text{d}}{\text{dt}}{\displaystyle {\iiint }_{c,V}\text{ρdV}}$

Where,

  $\overrightarrow{v}$ is the velocity vector.

  $\overrightarrow{n}$ is the normal vector

  $\text{ρ}$ is the density

dA is the small element of area on the control surface.

dV is the control volume.

The rate of accumulation of mass within control volume is given by equation:

  $\frac{\partial }{\partial t}{\displaystyle {\iiint }_{c.V.}\text{ρdV}}$

Where,

c.Vis the control volume.

  $\text{ρ}$ is the density.

dV is the control volume.