An Introduction of 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.
FAQs on Continuity Equation
1. What is the principle of continuity in the context of fluid dynamics?
The principle of continuity states that for an incompressible fluid undergoing a steady, streamlined flow, the volume of fluid entering a pipe in a given time interval must be equal to the volume of fluid leaving the pipe in the same time interval. This is a direct consequence of the law of conservation of mass, implying that the volume flow rate (the product of cross-sectional area and fluid velocity) remains constant at all points along the pipe.
2. On what fundamental conservation law is the equation of continuity based?
The equation of continuity is fundamentally based on the Law of Conservation of Mass. This law asserts that for any closed system, mass can neither be created nor destroyed. In the context of fluid flow through a pipe, it means that the rate at which mass enters one end of the pipe must equal the rate at which it leaves the other end, assuming there are no sources or sinks of fluid within the pipe.
3. What does the common form of the continuity equation, A₁V₁ = A₂V₂, represent?
The equation A₁V₁ = A₂V₂ is the mathematical representation of the principle of continuity for an incompressible fluid. In this formula:
- A₁ and A₂ are the cross-sectional areas of the pipe at two different points.
- V₁ and V₂ are the corresponding velocities of the fluid at those points.
The equation shows that the product of area and velocity, known as the volume flow rate, is constant. Therefore, where the pipe is narrower (smaller A), the fluid velocity (V) must be higher, and vice-versa.
4. Can you provide a real-world example of the continuity equation in action?
A classic real-world example is using a garden hose. When you place your thumb over the end of the hose, you decrease the cross-sectional area (A) of the opening. To maintain a constant flow rate as dictated by the continuity equation, the velocity (V) of the water must increase significantly. This is why the water sprays out much faster and travels a greater distance.
5. What are the key assumptions made for the simple continuity equation (A₁V₁ = A₂V₂) to be valid?
For the simplified equation A₁V₁ = A₂V₂ to accurately describe fluid flow, several conditions must be met. These are the key assumptions:
- The fluid must be incompressible, meaning its density (ρ) remains constant.
- The flow must be steady, meaning the velocity of the fluid at any given point does not change over time.
- The flow must be non-viscous, meaning we ignore the effects of internal friction within the fluid.
- The flow should be streamlined or laminar, where fluid particles move in smooth, parallel paths.
6. If the volume flow rate is constant in a pipe, does that mean the fluid's velocity is also constant everywhere?
No, this is a common misconception. The continuity equation (A₁V₁ = A₂V₂) states that the volume flow rate (Q = A × V) is constant, not the velocity itself. The velocity (V) is inversely proportional to the cross-sectional area (A). Therefore, if the pipe's diameter changes, the velocity of the fluid must change to keep the product A × V constant. The velocity only remains constant if the pipe has a uniform diameter.
7. How does the continuity equation change for compressible fluids like gases?
For compressible fluids, such as gases, the density (ρ) can change. Therefore, the principle of continuity must be applied to the mass flow rate, not the volume flow rate. The equation becomes ρ₁A₁V₁ = ρ₂A₂V₂. This more general form accounts for the fact that if a gas is compressed (density increases), its velocity or the area must adjust to ensure that the mass flowing per second remains constant throughout the system.
8. What is the importance of the continuity equation in fields other than basic fluid mechanics?
The principle of continuity has wide-ranging applications beyond simple pipes. For example:
- In meteorology, it helps model air currents and wind patterns in the atmosphere.
- In aerodynamics, it is crucial for understanding how air flows over an airplane's wings to generate lift.
- In medicine, it is used to model blood flow through arteries, helping to identify constrictions or aneurysms.
- In civil engineering, it's used to design river channels, canals, and urban water supply systems.
