## Lesson-17: CONTINUITY EQUATION

17.1 Principle of conservation of mass

Let us consider a material volume V with bounding surface S. The principle of conservation of mass imposes that: the material derivative of the mass of fluid in V is equal to zero.

The mass of the fluid in V is given by

since the volume V is arbitrary the following differential equation holds

This equation is known in fluid mechanics as continuity equation.

In the particular case in which the fluid is incompressible, i.e. the density ρ is constant, the above equation reduces to

This implies that the velocity field of an incompressible fluid is divergence free.

## 17.2 CONTINUITY EQUATION

Rate of flow or discharge (Q) is the volume of fluid flowing per second. For incompressible fluids flowing across a section,

Volume flow rate,

Q= AV m3/s

where

A=cross sectional area and

V= average velocity.

For compressible fluids, rate of flow is expressed as mass of fluid flowing across a section per second.

Mass flow rate (m) =(ρAV) kg/s where ρ = density.

Continuity equation is based on Law of Conservation of Mass. For a fluid flowing through a pipe, in a steady flow, the quantity of fluid flowing per second at all cross-sections is a constant.

Let v1=average velocity at section [1],

r1=density of fluid at [1], A1=area of flow at [1];

Let v2, r2, A2 be corresponding values at section [2].

Rate of flow at section [1]= r1 A1 v1

Rate of flow at section [2]= r2 A2 v2

r1 A1 v1= r2 A2 v2

This equation is applicable to steady compressible or incompressible fluid flows and is called Continuity Equation.

If the fluid is incompressible, r1 = r2 and the continuity equation reduces to A1 v1= A2 v2

For steady, one dimensional flow with one inlet and one outlet:

r1 A1 v1 - r2 A2v2=0

For control volume with N inlets and outlets

where inflows are positive and outflows are negative .

Velocities are normal to the areas. This is the continuity equation for steady one dimensional flow through a fixed control volume

When density is constant,

17.3 Momentum equation in integral form

Let us consider a material volume V with bounding surface S. Newton’s first principle states that: the material derivative of the momentum of the fluid in V is equal to the resultant of all external forces acting on the volume.

The momentum of the fluid in V is given by:

Therefore we have (in index notation):

This is the integral form of the momentum equation and is often written in compact form as:

I +W = F + ∑,

with I named local inertia and W being the flux of momentum across S.

Last modified: Monday, 2 December 2013, 8:40 AM