Conservation of Mass

Figure 4.1: Differential Control Volume

We derive the equation for mass conservation by considering a differential control volume at P(x,y,z)as shown in Fig.4.1. Let the dimensions of the volume be dx, dyand dz and velocity components at P be u,v and w. Assuming that the mass flow rate is continuous across the volume we can calculate the mass flow rates at the various faces of the cell by a Taylor Series expansion as we had done previously (Eqn. 2.5). Accordingly we have,

$\displaystyle ML =\left[ \rho u - {\partial \over {\partial x}}(\rho u) 
 dx\ri...
...z; MR =\left[ \rho u + {\partial \over {\partial x}}(\rho u) 
 dx \right] dy dz$    
$\displaystyle MB =\left[ \rho v - {\partial \over {\partial y}}(\rho v) 
 dy\ri...
...z; MT =\left[ \rho v + {\partial \over {\partial y}}(\rho v) 
 dy \right] dx dz$    
$\displaystyle MA =\left[ \rho w - {\partial \over {\partial z}}(\rho w) 
 dz\ri...
...y; MF =\left[ \rho w + {\partial \over {\partial z}}(\rho w) 
 dz \right] dx dy$ (4.1)

The net mass flow rate into the control volume as a consequence is given by,
$\displaystyle MNET =  MR - ML + MT - MB + MA - MF$ (4.2)

Applying the Reynolds transport theorem for mass (Eqn. 3.30) will give,

$\displaystyle {\int_{CV} {{\partial \rho} \over {\partial t}} d
 \forall}  +  MNET  =  0$ (4.3)

From Eqn.4.1 and 4.2 we have,

$\displaystyle MNET = \left( {\partial \over {\partial x}}(\rho u) + {\partial ...
...\partial y}}(\rho v)
 + {\partial \over {\partial z}}(\rho w) \right) dx dy dz$ (4.4)

Further in Eqn.4.3 noting that the control volume is tiny, the integral can be approximated as

$\displaystyle {\int_{CV} {{\partial \rho} \over {\partial t}} d
 \forall}  \approx  {{\partial \rho} \over {\partial t}} dx dy dz$ (4.5)

The Reynolds Transport Theorem thus gives,

$\displaystyle {{\partial \rho} \over {\partial t}} dx dy dz + \left( {\partial ...
...tial y}}(\rho v)
 + {\partial \over {\partial z}}(\rho w) \right) dx dy dz = 0$ (4.6)

Cancelling out dx dy dz, we have,

$\displaystyle {{\partial \rho} \over {\partial t}}  + {\partial \over {\partial...
...rtial \over {\partial y}}(\rho v)
 + {\partial \over {\partial z}}(\rho w) = 0$ (4.7)

Eqn. 4.7 is known as the Continuity Equation. Note that it is a very general equation with hardly any assumption except that density and velocities vary continually across the element we have considered.

If we now bring in the gradient operator, namely,

$\displaystyle \nabla = \hat{i}{\partial \over {\partial x}} + \hat{j}{\partial \over {\partial y}}
 + \hat{k}{\partial \over {\partial z}} $ (4.8)

and represent velocity as a vector,

$\displaystyle \vec{V} = \hat{i}u + \hat{j}v + \hat{k}w$ (4.9)

Then the Continuity Equation can be written in a compact manner as
$\displaystyle {{\partial \rho} \over {\partial t}}  + \nabla \cdot (\rho \vec{V}) = 0$ (4.10)

Written in this form it enables one to consider any other system of coordinates with ease.


Subsections (c) Aerospace, Mechanical & Mechatronic Engg. 2005
University of Sydney