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The continuity equation, or the transport equation, explains the transport of quantities such as fluid or gas. For instance, the continuity equation shows how a fluid conserves mass within its motion. In fact, many physical phenomena including energy, momentum, mass, and electric charge can be conserved by means of the continuity equation. The continuity equation can be expressed as:
R = Av = Constant
Here,
- R = Volume flow rate
- A = Flow area
- v = Flow velocity
The equation of continuity shows the flow of fluids and their behavior during their flow in a pipe or hose. The continuity Equation can be applied to tubes, pipes, rivers, and ducts that possess flowing fluids or gases, among others. Generally, the law of conservation states that energy can neither be created nor destroyed.
Read More: Kinetic Theory of Gases
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Key Terms: Continuity Equation, Rate of Flow, Fluid, Volume Flow, Law of Conservation, Principle of Continuity
Continuity Equation
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The Continuity equation denotes that the product of the cross-sectional area of the pipe and the velocity of the fluid at any point along the pipe remains constant. This product is equivalent to the volume flow per second or the rate of flow. The continuity equation can be represented as:
| R = Av = constant |
Here,
- R = Volume flow rate
- A = Flow area
- v = Flow velocity’
Hence,
Volume flow in over A1 = A1V2\(\Delta\)t
Volume flow in over A2 = A2V2\(\Delta\)t
Therefore,
mass in over A = \(\rho\)A1V1\(\Delta\)t
mass out over A = \(\rho\)A2V2\(\Delta\)t
Thus, \(\rho\)A1V1\(\Delta\) = \(\rho\)A2V2
The law of conservation states that energy can neither be created nor destroyed, meaning the total amount of energy needs to be conserved. The continuity equation is said to be the mathematical statement for the law of conservation of energy. Science has different physical phenomena conserved in energy, momentum, electric charges, mass, and other natural quantities.
The continuity equation contains of several other transport equations, including the convection-diffusion equation, Navier–Stokes equations, and the Boltzmann transport equation. Thus,
- Convection–Diffusion Equation: A combination of Convection and Diffusion equations. It can further explain the physical phenomena wherein the particles, energy, and other physical quantities are transported by means of 'diffusion and convection' inside a physical system.
- Boltzmann Transport Equation: Boltzmann transport equation helps explain the thermodynamic system behavior, which cannot be found in the state of rest or equilibrium.
Assumption of Continuity Equation
There are several assumptions of continuity equation, including:
- The tube has a single entry and a single exit
- The flowing fluid in the tube is apparently non-viscous
- Incompressible flow
- Steady fluid flow
Continuity Equation Principle
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Continuity is the major prospect in describing the transport of fluids or gases. It explains the principle of mass conservation in fluids or gases.
- The principle of continuity states that, in a steady state, the rate at which the mass enters the system is equal to the rate at which the mass leaves the system.
- The continuity equation is used to prove the law of conservation of mass in fluid dynamics.
- Continuity equation explains the flow and behavior of fluids and gases in the pipe or a hose. This equation is applied to fluids and gases flowing through pipes, ducts, rivers, and hoses.
- One of the most important aspects of the continuity equation in terms of fluids is that it is applied to all types of fluids with compressible and non-compressible flows, Newtonian and non-Newtonian fluids.
Flux, in the Continuity Equation, can be divided into two segments:
-
Volumetric Flux: Across a unit area, The rate of volume flow, across a unit area, is called Volumetric flux. It can be calculated by the formula Volumetric flux =liters/(second*area). The SI unit of voumetric flux is m3s−1m−2.
-
Mass Flux: It can be defined as the rate of mass flow with an SI unit of (kgm−2s−1). It can be further represented by the symbols j, J, Q, q.
Uses of Continuity Equation
The continuity equation is commonly used in pipes, tubes, and ducts.
- These structures typically possess a flowing fluid or gas that requires a specific flow to be moved.
- Continuity equation is also found in huge water sources like rivers, lakes, and more.
- It can also be applied in power plants, road logistics, and others.
- Modern technologies, such as semiconductor technologies, use the application of continuity equations.
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| Hydrostatic Paradox | Difference Between Molar mass and Molecular mass | Hydraulic Lift |
Continuity Equation Derivation
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To derive the continuity equation, let us take M as the mass flow rate of the fluid through a pipe or a hose with a density ρ, and with a speed taken to be v. let a be the area of cross-section of that pipe or a hose. So,
M = ρi1 vi1 Ai1+ρi2 vi2 Ai2+…………+ρin vin Aim
M = ρo1 vo1 Ao1+ρo2 vo2 Ao2+……….+ρon von Aom … (1)
Here,
- M = Mass flow rate
- ρ= Densit
- v = Speed
- a = Area
With uniform density equation (1), the above equation can further be modified to:
Q = vi1 Ai1+vi2 Ai2+…,,,,,.+vin Aim
Q = vo1 Ao1+vo2 Ao2+….+von Aom …(2)
Here, Q is taken to be the flow rate of the fluid.
ρi1 = ρi2..= ρin = ρo1 = ρo2 = … =ρom
the explicit continuity equation becomes as follows,
⇒ ρ1A1v1 = ρ2A2v2 … (3)
Thus, it can also be expressed in a more general form:
⇒ ρ A v = constant
The equation claims and proves that the law of conservation of mass in fluid dynamics.
Assuming that the fluid is incompressible, the density here is going to be constant for steady flow. So, ρ1 = ρ2.
Hence, Equation 3 can also be shown as:
⇒ A1 v1 = A2 v2
In a general form:
⇒ A v = constant
Thus, considering that R is the volume flow rate, the above equation can be represented as:
R = A v = constant
Hence, this is the derivation of Continuity Equation.
Integral Form
The integral form of the continuity equation claims that:
- When an additional q is seen to flow inward via the surface of the region, the amount of q in the given region further increases. It, however, decreases when flowing outward
- When a new q is formed inside the region, thus the number of q increases and decreases
- In the case q is destroyed
- Except the two methods, there can be no other way for the amount of q in a region to vary.
The integral form of the continuity equation claims that the rate of increase of q within a volume V. Thus,
| \(\frac{dq}{dt} + ∯ SJ.dS = \sum\) |
Here,
- S = imaginary closed surface, that encloses a volume V
- ∯S dS = surface integral over that closed surface
- q = total amount of the quantity in volume V
- J = flux of q
- t = time
- Σ = net rate that q is being produced inside the volume V
Continuity Equation Fluid Dynamics
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The Continuity Equation in Fluid Dynamics can be expressed as:
- The continuity equation has wide utility in the principle of fluid dynamics. This equation is used to conserve the flow rate of the fluid in a steady state.
- In a steady-state of the fluid, the rate at which the mass leaves the system is always equal to the rate at which it enters the system.
- To derive the differential form of the continuity equation, let us take, t is the period for the rate of flow, ρ as the density of the fluid, and v as the vector field flow velocity. So, the differential form of the continuity equation in fluid dynamics is,
| \(\frac{\partial \rho}{\partial t} + \bigtriangledown \cdot \left (\rho u \right) = 0\) |
Continuity Equation in Cylindrical Coordinates
The continuity equation in cylindrical coordinates can be demonstrated as:
| \(\frac{\partial \rho }{\partial t}+\frac{1}{r}\frac{\partial r\rho u}{\partial r}+\frac{1}{r}\frac{\partial \rho v}{\partial \theta }+\frac{\partial \rho w}{\partial z}=0\) |
Incompressible Flow Continuity Equation
The continuity equation for incompressible flow wherein the density, ρ is constant and also independent of space and time, thus:
| ∇.v = 0 |
Steady Flow Continuity Equation
The continuity equation in cylindrical coordinates can be shown as:
| \(\frac{\partial }{\partial x}(\rho u)+\frac{\partial}{\partial y}(\rho v)+\frac{\partial }{\partial z}(\rho w)=0\) |
Fluid Dynamics
Things to Remember
- The law of conservation states that energy can neither be created nor destroyed, meaning the total amount of energy needs to be conserved. The continuity equation is said to be the mathematical statement for the law of conservation of energy.
- There are different physical phenomena conserved in science like energy, momentum, electric charges, mass, and other natural quantities.
- One of the most important aspects of the continuity equation in terms of fluids is that it is applied to all types of fluids with compressible and non-compressible flows, Newtonian and non-Newtonian fluids.
- The continuity equation has wide utility in the principle of fluid dynamics. This equation is used to conserve the flow rate of the fluid in a steady state.
- In a steady state of the fluid, the rate at which the mass leaves the system is always equal to the rate at which it enters the system.
Read Also:
Sample Questions
Ques. What is Continuity Equation? (1 mark)
Ans. The Continuity equation represented that the product of the cross-sectional area of the pipe and the velocity of the fluid at any given point along the pipe is going to remain constant.
Ques. What are the two types of flux in Continuity Equation? (1 mark)
Ans. The two types of flux in continuity equation are:
- Volumetric Flux
- Mass Flux
Ques. Find out the velocity of fluid when 10 m3/h of fluid flows through a pipe with an inside diameter of 100mm. However, later the pipe is reduced to a diameter of 80mm. (3 Marks)
Ans: First, let us find the velocity of fluid flowing through a pipe with a 100mm diameter,
Using equation 2 from the above derivation,
Q=vo1 Ao1+vo2 Ao2+….+von Aom………..(2)
(10 m3/h)(1/3600 h/s)=v100(3.14(0.1 m)2/4)
v100=(10 m3/h)(1/3600 h/s)/(3.14(0.1)2/4)
=0.35 m/s
Now, for the pipe with a diameter of 80mm
Again, using equation 2 here,
Q=
vo1 Ao1+vo2 Ao2+….+von Aom………..(2)
(10 m3/h)(1/3600 h/s)=v80(3.14(0.08 m)2/4)
v80=(10 m3/h)(1/3600 h/s)/(3.14(0.08 m)2/4)
=0.55 m/s
Ques. Find the new flow rate of the water, if it is flowing through a pipe of diameter 1cm with a flow velocity of 2m/s and the diameter of the pipe is expanded to 3 cm. (3 Marks)
Ans: According to the continuity equation,
ρ1A1v1=ρ2A2v2
π(d1/2)2v1=π(d2/2)2v2
simplifying the equation,
v2=d12v1/d22
= (1)2 * 2/ (3)2
= 0.22 m/s
Ques. A compressible gas is flowing through a pipe of a cross-section area of 0.02m2, with a flow rate of 4m/s and a density of 2 kg/m3. Find out the new density of gas if it flows through the different regions with the same pipe having a cross-section area of 0.03 m2 at a velocity of 1 m/s. (3 Marks)
Ans: Using the continuity equation,
ρ1A1v1=ρ2A2v2
ρ2= ρ1A2v2/A1v1
= 2 * (0.02*4)/(0.03*1)
= 5.33 kg/m3
Ques. Find the new flow rate of the water, if it is flowing through a pipe of diameter 10cm with a flow velocity of 9 m/s and the diameter of the pipe is reduced to 6 cm. (3 Marks)
Ans: According to the continuity equation,
A1v1=A2v2
π(d1/2)2v1=π(d2/2)2v2
simplifying the equation,
v2=d12v1/d22
= (5)2 * 9/ (3)2
= 25cm/s
Ques. A pipe with a diameter of 4 centimeters is attached to a garden hose with a nozzle. If the velocity of flow in the pipe is 2m/s, what is the velocity of the flow at the nozzle when it is adjusted to have a diameter of 8 millimeters? (3 Marks)
Ans: According to the continuity equation,
A1v1=A2v2
A1=π(0.02m)2=0.0004πm2
A2=π(0.004m)2=0.000016πm2
v2=d12v1/d22
(0.0004πm2)(2ms)=(0.000016πm2)v2
v2=(0.0004πm2)(2ms)0.000016πm2
v2=50ms
Ques. What would be the relation of flow speed at the two points, If a pipe with flowing water has a cross-sectional area nine times greater at point R than at point S? (3 Marks)
Ans: According to the continuity equation,
A1v1=A2v2
A1= 1 and A2 = 9
1V1=9V2
V1/V2=9/1
So, the flow speed at point R is nine times that of point S.
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