Continuity Equation

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

In Continuity Equation, Flux is of Two Types:

• Volumetric Flux - Across a unit area, the rate of volume flow is known as Volumetric flux. It is calculated by the formula Volumetric flux =liters/(second*area). Its SI unit is (\[ m^ {3} s^ {-1} m^ {-2} \])

• Mass Flux - It is the rate of mass flow. Its SI unit is (kg \[ m^{-2}s^{-1} \]). It is represented by the symbols j, J, Q, q.

What is the Continuity Equation?

The continuity equation is an equation that describes the transport of some quantities like fluid or gas. It is also known as the transport equation. 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.

According to the continuity equation:

\[ A_{1} V_{1} = A_{2} V_{2} \]

Where,

\[A_{1}\] = cross-sectional area of region 1

\[V_{1}\] = flow velocity in region 1

\[A_{2}\] = cross-sectional area of region 2

\[V_{2}\] = flow velocity in region 2.

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. The continuity equation consists of many other transport equations like the convection-diffusion equation, Navier–Stokes equations, and the Boltzmann transport equation. 

• Convection–Diffusion Equation - It is a combination of convection and diffusion equations. It describes the physical phenomena where particles, energy, and other physical quantities are transferred with the help of 'diffusion and convection' inside a physical system.

• Boltzmann Transport Equation - Boltzmann transport equation describes the behavior (statistical in nature) of the thermodynamic system, which is not in the state of rest or equilibrium. 

Continuity Principle 

Continuity principle refers to the principle of fluid mechanics. The principle of continuity equation is a consequence of the law of conservation of mass. Through the continuity equation, the behavior of fluid is described when it is in motion. Whereas, the second equation is based on Newton's law of motion (which describes the motion of an object and the force acting on its flow) and the third equation is based on 'the law of conservation of energy (which states that mass can be neither created nor destroyed.)

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:

\[ \frac{dq}{dt} + ∯ S j . dS = \sum \]

• 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 = \[ \rho_{i1} v_{i1}A_{i1} + \rho_{i2} v_{i2} A_{i2} +.....+ \rho _{in} v_{in} A_{in} \]

m=\[ \rho_{01} v_{01} A_{01} + \rho_{02} v_{02} A_{02} + ….+ \rho_{0n} v_{0n} A_{0n} \] ……….. (1)

Where,

m = Mass flow rate

\[\rho\] = Density

v = Speed

A = Area

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

q = \[ v_{i1} A_{i1} + v_{i2} A_{i2} +....+ v{_im} A_{im} \]

q = \[ v_{01} A_{01} + v_{02} A_{02} +....+V_{0m} A_{0m} \]

Where,

q = Flow rate

\[ \rho_{i1} =\rho_{i2}.. = \rho_{in} = \rho_{01} = \rho_{02}= …. = \rho_{0m}\]

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

\[\rho\] = 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-mentioned 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.

Uses of the Continuity Equation

The continuity equation is commonly used in pipes, tubes, and ducts. These structures have flowing fluid or gasses etc. which need a specific flow to be moved. Continuity equation can also be applied to huge water sources such as rivers, lakes, etc. This equation can also be applied in diaries, power plants, road logistics, etc. 

Along with this, the modern application of continuity equations includes computer networking and semiconductor technologies, etc. which uses a specific path to move data from one location to another. It is also used in gas pipelines and underground connections to transport gas. 

Continuity Equation Example

1. 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.