Continuity Equation

Definition of Flux

A continuity equation becomes useful if a flux can be defined. To explain flux, first, there must be a quantity q which can flow or move, such as energy, mass, electric charge, momentum, number of molecules, etc. Let us assume ρ be the volume density of this quantity (q) that is, the amount of q per unit volume.

The way by which this quantity q is flowing is described by its flux.

What is the Continuity Equation?

The continuity equation is an equation that describes the transport of some quantities like fluid or gas. The continuity equation is very simple and powerful when it is applied to a conserved quantity. When it is applied to an extensive quantity it can be generalized. Physical phenomena are conserved using continuity equations like energy, mass, momentum, natural quantities, and electric charge.

Continuity equations are a local and stronger form of conservation laws. For example, a weak version of the law of conservation of energy states that energy can neither be created nor destroyed which means that the total amount of energy in the universe is fixed. It means energy can neither be created nor destroyed nor can it teleport from one place to another—it can only move by continuous flow. A continuity equation is nothing but a mathematical way to explain this kind of statement.

Integral Form

The integral form of the continuity equation says that-

• When additional q flows inward through the surface of the region, the amount of q in a region increases and decreases when it flows outward;

• When new q is created inside a region the number of q increases and decreases

• When q is destroyed;

• Apart from these two methods, there is no other way for the amount of q in a region to change.

In terms of mathematics, the integral form of the continuity equation expressing the rate of increase of q within a volume V is:

$\frac{dq}{dt}$ + $\ _{S} j \cdot dS = \sum$

• Here, S  denotes an imaginary closed surface, that encloses a volume V,

• S dS  is a surface integral over that closed surface,

• q denotes the total amount of the quantity in volume V,

• J is the flux of q,

• t denotes time.

• And Σ is the net rate that q is being produced inside the volume V.

Flow Rate Formula

This equation gives very useful information about the flow of liquids and their behavior when it flows in a pipe or hose. The hose, a flexible tube, whose diameter decreases along its length has a direct consequence. The volume of water flowing through the hose must be equal to the flow rate on the other end. The flow rate of a liquid means how much a liquid passes through an area in a given time.

The formula for the flow rate is given below-

The Equation of Continuity can be written as:

m=ρi1 vi1 Ai1+ρi2 vi2 Ai2+…..+ρin vin Aim

m=ρo1 vo1 Ao1+ρo2 vo2 Ao2+…..+ρon von Aom……….. (1)

Where,

m = Mass flow rate

ρ = Density

v = Speed

A = Area

With uniform density equation (1) it can be modified further -

q=vi1 Ai1+vi2 Ai2+…. + vin Aim

q=vo1 Ao1+vo2 Ao2+…. + von Aom……….. (2)

Where,

q = Flow rate

ρi1=ρi2.. = ρin = ρo1 = ρo2= …. = ρom

Fluid Dynamics

The continuity equation in fluid dynamics says that in any steady-state process, the rate at which mass leaves the system is equal to the rate at which mass enters a system including the accumulation of mass within the system.

The differential form of the continuity equation is:

∂ρ∂t + ▽⋅(ρu)=0

Where,

t = Time

ρ = Fluid density

u = Flow velocity vector field.

The derivative time can be understood as the loss of mass in accumulation inside the system, while the divergence term means the difference in flow in and flow out. The above mention equation is also one of the (fluid dynamics) Euler equations. The equations of Navier–Stokes form a vector continuity equation expressing the conservation of linear momentum.

Continuity Equation Example

Question: If 10 m³/h of water flows through a 100 mm inside diameter pipe. If the inside diameter of the pipe is reduced to 80 mm. Calculate the velocities.

Solution

Velocity of 100 mm pipe

Putting the equation (2), to calculate the velocity of 100 mm pipe

(10 m³/h)(1/3600 h/s)=v100 (3.14(0.1 m) 2/4)

Or

v100= (10 m³/h) (1/3600 h/s) (3.14(0.1)2/4)

=0.35 m/s

Velocity of 80 mm pipe

Again applying equation (2), to calculate the velocity of 80 mm pipe

(10 m³/h)(1/3600 h/s)= v80 (3.14(0.08 m) 2/4)

Or

v80= (10 m³/h) (1/3600 h/s) (3.14(0.08 m)2/4)

=0.55 m/s.

Q1. What is the Differential Form of the Continuity Equation?

Solution) According to the divergence theorem, a general continuity equation can also be written in a "differential form."

The differential form of the continuity equation can be written as given below-

∂⍴/∂t  + ∇・j = σ

Where

∇⋅ denotes divergence,

ρ = the amount of the q per unit volume,

j means the flux of q,

t is the time,

σ = q per unit volume per unit produced.

Terms that generate q (i.e. σ > 0) or remove q (i.e. σ < 0) are known as a "sources" and "sinks" respectively.

This general continuity equation can be used to derive any continuity equation, from simple to complicate.

Q2.  How to Derive the Continuity Equation?

Solution) Derivation of the continuity equation is regarded as one of the most important derivations in fluid dynamics.

The continuity equation can also be defined as the product of the cross-sectional area of the pipe and the velocity of the fluid at any given point when the pipe is always constant and this product is equal to the volume flow per second.

The continuity equation is given as:

R=Av=Constant

Where,

• R = the volume flow rate

• A = the flow area

• v = the flow velocity

With considering the following assumptions-

• The tube is having a single entry and exit.

• The fluid flowing is non-viscous.

• The flow is incompressible.

• The fluid flow is steady.