Continuity Equation

This equation is a fundamental principle in physics and liquid elements or fluid dynamics that describes the transport of some quantity. It tells the preservation of mass within that system. It is expressed as a mass that is neither created nor destroyed but is conserved for a liquid flowing through a given area.

What is Continuity Equation?

Continuity Equation is an equation that tells about the conservation of mass within the system. This equation used many fields like liquid or fluid mechanics, power and magnetism, and even in the analysis of traffic flow or stream. It is expressed as a mass that is neither created nor destroyed but is conserved for a liquid flowing through a given region or area. The equation applies to many fields, including liquid or fluid mechanics, power and magnetism, and even in the analysis of traffic flow or stream.

Principle Of Continuity

Here are some general principles of continuity are given :

• Conservation or Preservation Of Mass: The basic thought behind the principle of continuity is the conservation of mass that mass is neither created nor destroyed but is conserved for a liquid flowing through a given region or area. In a closed system, mass is neither created nor destroyed.
• Incompressibility: Incompressibility means impossible or difficult to compress like oil and water. For incompressible liquids or fluids, the density is remains constant so the equation of continuity simplifies that ∇⋅v=0. This suggests that the liquid or fluid is incompressible, and changes in speed are remunerated by changes in cross-sectional region.
• Conservation of different amounts or quantities: This equation can be adjusted the preservation or conservation of different quantities, like electric charge in electromagnetism.

Derivation Of Continuity Equation

Continuity Equation

Now, consider the pipe with fluid of density density (ρ) flows for a short interval of time in the pipe, and assume that short interval of time as Δt. In this time Δt, the fluid will cover a distance of Δx1 with a speed or velocity V1 at the lower end of the pipe to the upper end of pipe V2.

The distance covered by fluid in Δt will be

Δx1 = V1Δt

Now, the volume of fluid of at lower end of pipe will be

V = A1 Δx1 = A1 V1 Δt

We know that mass (m) = Density (ρ) × Volume (V), so the mass of liquid in Δx1 will

Δm1= Density × Volume

Δm1 = ρ1A1V1Δt ——–(Equ.1)

Now, the mass flux(the mass of the fluid per unit time passing through any cross-sectional area) for lower end with area A1 will be

Δm1/Δt = ρ1A1V1 ——–(Equ2)

And similarly, the mass flux for upper end with area A2 will be

Δm2/Δt = ρ2A2V2 ——–(Equ3)

Now, the fluid is steady so the density of fluid remains constant with time and mass flux at lower end is equal to mass flux at upper end.

(Equ2) = (Equ3)

Thus,

ρ1A1V1 = ρ2A2V2 ——–(Equ4)

ρ A V = constant

This proves the law of conservation of mass.

For steady flow, the density remains constant ρ1 = ρ2

Thus,

A1 V1 = A2 V2

A V = Constant

This is the derivation of continuity derivation….

Continuity Equation in Different Fields

• In Thermodynamics: This equation is used in thermodynamics to depicts the conservation or preservation of mass and energy in the systems. It assists us to learn how energy and mass passes or flows through various cycles and systems.
• In liquid dynamics or fluid mechanics: This equation is used in liquid dynamics or mechanics to determines or depicts the rate of flow of fluid or liquid and speed of fluid while flowing. It describes that the rate of flow of liquid entering a region must be equal to the rate of flow of liquid leaving that region.
• In Network or Graph theory: This equation used in network hypothesis or graph hypothesis to depicts the flow or progression of important data, information, assets and resources in an organization as well as in network. It assists us to understand the management or distribution inside in organization as well as in network.
• In Electricity: This equation is used in electricity to depicts or determines the flow of electric current with in the system. It describes or tells us that the current entering a region must be equal to the current leaving that region.
• In Transferring of Heat: This equation is used in transferring of heat to determine or depicts the conservation or preservation of energy while heat transfer. It tells or expresses that the flow of heat entering a area must be equal to the heat flow leaving that area.

Flow Rate Formula

This equation expresses that the flow rate of fluid or quantities at one point in the system is equal to the flow rate of fluid or quantities at another point.It works on the principle of continuity.

Consider a fluid with flow rate (Q), the cross-sectional area (A) and the velocity (V).

The flow rate of liquid represents the volume of liquid passing through a given cross-sectional region per unit time.

The continuity equation for flow can be expressed as:

A1V1 = A2V2

where, A1​ and A2 are the cross-sectional areas at two different end in the fluid flow and V1​ and V2​ are the fluid velocities at those ends.

The product of cross-sectional area (A1 and A2) and fluid velocity (V1 and V2) is the flow rate.

Q = A1​V1​ = A2​V2​

Continuity Equation in Integral Form

The integral form of the continuity equation is given below:

∫V ​( ∂t/∂ρ​ + ∇ ⋅ (ρv) ) dV = −∫S​ (ρv) ⋅ dS

Continuity Equation in Differential Form

The differential form of the continuity equation is given below:

∂t/∂ρ ​+ ∇ ⋅ (ρv) = 0

Fluid Dynamics

Fluid dynamics is a liquid mechanics that deals with the study of the movements of fluids (liquids and gases) and the forces acting on each other. It is a part of physics and engineering that has applications in many fields, including aerospace engineering, civil engineering, chemical engineering, natural science, and meteorology, among others.

Incompressibility

It defines as the property of liquid or substance where the density of substance remains constant with changes in pressure. In simple words, the substances which cannot be easily compressed in the smaller volume, the substances are called incompressible substances and the property shows by this substances is known as incompressibility. This concept generally used in fluid dynamics or mechanics.

Streamlines and Stream-Tubes

Streamlines: These are the imaginary lines that represent the direction of liquid or fluid flow at any random point in a liquid or fluid field. These lines show the way that a liquid or fluid would follow at specific moment. They never cross each other. They give us a idea of how the liquid is moving at a specific moment. They never cross each other and help us to determine the velocity of the liquid at various points.

Streamlines

Stream-Tubes: These are the groups or bundles of streamlines. In other words, they are the collections of streamlines. These are the 3d structures and forming a tube like structure or shape. They are used to determine the flow within a specific volume of space. They enclose the flow of a liquid and help us analyze all flow pattern within a specific region. They can different size and shape, and they provide a more comprehensive understanding of how the liquid moves within a particular area or region.

Streamtubes

Assumptions of Continuity Equation

• The main assumption is that the total mass is conserved or preserved. the meaning of this line is the total mass of the system remains constant over time.
• The another assumption is that the flow of liquid or fluid is constant, continuous, regular, and difference in density are primarily because of changes in volume and than changes in the actual quantity of fluid. This assumption is frequently for compressible flows.
• This assumes that inside the system there is no sources of mass and sinks of mass. It means that mass neither created nor destroyed.
• This equation depends on the general principle of conservation of mass. It considers changes in mass of the system at each subtly little volume in the system.
• This equation depends on the general principle of conservation of mass so there is no chemical reactions and phase or chemical changes.

These above given assumptions make this equation very valuable for describing the fluid flow of various system in different scientific and engineering application.

Applications of Continuity Equation

• This equation is used in fluid dynamics to determine the liquid flow in pipes, tubes, channels, and other systems.
• This equation helps to determine or analyze variables like flow rate, velocity, and pressure distribution.
• This equation is used in aerodynamics to determine the flow of air around objects, such as airplanes, jets, car and understand the lift and drag system.
• This equation is used to find applications in electrical circuits system, where it is used to determine the electric charge flow through conductors.
• This equation is used in industrial areas to determine the flow of liquids in industrial process.
• This equation is also used to determine the flow rate of water in streams, rivers, groundwater systems.

Solve Examples on Continuity Equation

Example 1: Imagine a pipe with water flowing through it. The pipe has a cross-sectional area of 0.1 square meters. Find the mass flow rate into pipe.

Solution

According to the continuity equation, mass flow rate into the pipe must be equal to the mass flow rate out of the pipe.

Use the equation:

Mass flow rate = Density * Velocity * Area

So, the mass flow rate into the pipe would be:

Mass flow rate = 1000 kg/m^3 * 2 m/s * 0.1 m^2 = 200 kg/s

Hence, the mast flow rate is 200 kg/s.

Example 2: A tube has a cross-sectional area of 5 m². Water flows through the tube with the velocity of 50 m/s. Calculate the volume flow rate of water.

Solution

The continuity equation tells that the volume flow rate (Q) is equal to the product of the cross-sectional area (A) and the velocity (v). So, we can use the formula Q = A × V.

Given:

A = 5 m2

V = 50 m/s

Now, the values in the formula,

we get:

Q = A × V

Q = 5 m2 × 50 m/s

Q = 250 m3/s

Therefore, the volume flow rate of water is 250 m3/s.

Example 3: A river has a width of 20 m and a depth of 15 m. The water is flowing with a velocity of 4 m/s. If the river is 200 m long, what is the volume flow rate of water?

Solution

Here, we need to use the continuity equation, which tells that the volume flow rate (Q) is equal to the product of the cross-sectional area (A) and the velocity (v).

Given:

Width (w) = 20 m

Depth (d) = 15 m

Velocity (v) = 4 m/s

Length (L) = 200 m

Now, first find area of cross-sectional (A) of river

A = Width × Depth

A = 20 m × 15 m

A = 300 m2

Now, we can use the formula of flow rate Q = A × V:

Q = 300 m2 × 4 m/s

Q = 1200 m3/s

Therefore, the volume flow rate of water is 1200 m3/s.

Example 4: A cylindrical tube with a diameter of 0.5 m carries a fluid with a velocity of 10 m/s. The pipe gradually narrows down to a diameter of 0.2 m. If the volume flow rate at the wider end is 20 m/s, what is the volume flow rate at the narrower end?

Solution

Here, we need to use the continuity equation, which tells that the volume flow rate (Q) is remains same throughout a pipe of varying diameter. So, the volume flow rate at the wider end is equal to the volume flow rate at the narrower end.

Given:

Diameter at wider end (D1) = 0.5 m

Diameter at narrower end (D2) = 0.2 m

Velocity (v) = 10 m

Volume flow rate at wider end (Q1) = 20 m/s

Now, first find area of cross-sectional (A) at each end using the formula A = πr2:

Area at wider end (A1) = π × (D1/2)²

Area at narrower end (A2) = π × (D2/2)²

Now, we can use the continuity equation to find the volume flow rate at the narrower end:

Q1 = A1 × V

Q2 = A1 × V

Since, Q1 = Q2 then,

A1 × V = A1 × V

Now, substitute the formulas for A1 and A2:

π × (D1/2)² × V = π × (D2/2)² × V

Simplify the equation, we get:

(D1/2)² = (D2/2)²

Taking the square root of both sides, we have:

D1/2 = D2/2

Since D1 = 0.5 m and D2 = 0.2 m, we can solve for the volume flow rate at the narrower end:

Q2 = Q1 × (D2/D1)

Q2 = 10 m3/s × (0.2 m / 0.5 m)

Q2 = 4 m3/s

Therefore, the volume flow rate at the narrower end is 4 m3/s.

Conclusion

Well, the continuity equation is a basic principle in liquid dynamics that deals with the conservation of mass. It basically defines that the rate of mass flow into a control volume must equal the rate of mass flow out of that volume, assuming there are no sources or sinks of mass within the volume of system.

This equation is used in various fields, such as fluid mechanics, aerodynamics, and hydrodynamics. It helps us understand and determine the behavior of fluids in different scenarios.

In last, the equation is a powerful tool that helps us to determine the conservation of mass in liquid flow and has wide-ranging applications in various fields. It’s fascinating how this simple equation can have such a huge impact on our understanding of fluid mechanics or dynamics. I hope this conclusion gives you a nice over view of the continuity equation and its importance.

FAQs on Continuity Equation

What is Continuity Equation?

Continuity Equation is a equation that tells about conservation of mass with in the system. This equation used many fields like liquid or fluid mechanics, power and magnetism, and even in the analysis of traffic flow or stream. It is expressed as a mass is neither created nor destroyed but is conserved for a liquid flowing through a given region or area.

Who discovered Continuity Equation?

This equation describes the conservation of mass and conservation of fluid momentum. This equation first published by L. Euler in 1752 and N.E. Zhukovsky in 1876 by precise calculations.

What is the formula of continuity equation and flow rate?

Continuity equation – A1 V1 = A2 V2

A V = Constant

Flow rate – Q = A1​V1​ = A2​V2​

where, Q is referred to as flow rate, A1​ and A2 are the cross-sectional areas at two different end in the fluid flow and V1​ and V2​ are the fluid velocities at those ends.