The continuity equation is a physical law that states that the amount of mass or fluid entering a closed system is equal to the amount of mass or fluid leaving the system in the same period of time.
The continuity equation is valid for any type of fluid, as long as the fluid is incompressible and the flow is stationary, that is, the speed and properties of the fluid at any point in the system do not vary with time.
An incompressible fluid is one that has a constant density and does not change its volume in response to the application of pressure.
Examples of the continuity equation
Below are some examples of its application in everyday life:
Liquid flow in a tube
A classic example of the application of the continuity equation is the flow of liquid in a pipe.
Suppose a liquid flows through a pipe of cross section A₁ with a velocity v₁ and then enters a pipe of cross section A₂ with a velocity v₂.
Using this equation we can size the sections of the pipe to alter the flow velocity.
Flow of water in a river
The continuity equation also applies to the flow of water in a river.
This equation is used to calculate the velocity of water at different points in the river. Thus, the behaviour of the river under different conditions can be predicted, such as when dams are built or engineering works are carried out for flood control.
Mathematical formula
In mathematical terms, the continuity equation is expressed by the following formula:
A₁ * v₁ = A₂ * v₂
Where:
- A₁ and A₂ are the cross-sectional areas of the duct or pipe at points 1 and 2 respectively.
- v₁ and v₂ are the velocities of the fluid at points 1 and 2 respectively.
According to the continuity equation, if the cross-sectional area of the conduit or pipe through which the fluid flows remains constant, then the fluid velocity and the flow rate are inversely related. In other words, if the fluid velocity increases, the flow rate decreases and vice versa.
Relationship with the principle of continuity
The continuity equation is closely related to the continuity principle which states that in a steady flow system the amount of fluid entering must be equal to the amount leaving, provided there are no losses or accumulations.
This principle is based on the conservation of mass and applies to incompressible fluids (those whose density does not change significantly, such as water) and in certain cases to compressible fluids (such as gases).
In practical terms, the principle of continuity implies that if a fluid moves through a conduit with different cross-sectional areas, its velocity will change to keep the volumetric or mass flow rate constant. For example, if the conduit narrows, the fluid must increase its velocity to compensate for the reduction in area, and vice versa if the conduit widens.
Uses and practical applications
The continuity equation has many applications in physics and engineering, particularly in fluid mechanics. Some of its main applications are presented below:
- Piping system design: Used to calculate the flow rate and velocity of fluid at different points in the piping system, allowing the diameter and length of pipes to be dimensioned to ensure a constant and uniform flow.
- Flow analysis in ducts and channels: it is used to analyse the flow of liquids in ducts and channels, allowing the speed and flow rate to be determined at different points in the system.
- Optimizing the efficiency of hydraulic systems: It is used to optimize the efficiency of hydraulic systems, such as turbines and pumps, since it allows calculating the flow rate and speed of the fluid at different points in the system and determining the optimal geometry of the system components.
Solved exercises
Exercise 1
A pipe with a cross section of 0.02 m² transports water at a speed of 2 m/s. If the diameter of the pipe is reduced to half its original value, what is the velocity of the water in the narrow pipe?
Solution:
The continuity equation states that the volumetric flow rate of the fluid flowing through the pipe is constant throughout the flow. Therefore, we can write:
A₁·v₁ = A₂·v₂
where A1 is the original cross section of the pipe, v1 is the original velocity of water, A₂ is the cross section of the narrow pipe, and v₂ is the velocity of water in the narrow pipe.
We have A₂ = A₁/4, since the diameter of the tube is reduced to half its original value, therefore A₂ = π(0.01 m)² = 0.000314 m².
Substituting the known values into the continuity equation, we get:
0.02 m² × 2 m/s = 0.000314 m² × v₂
v₂ = (0.02 m² × 2 m/s) / 0.000314 m² = 127.39 m/s
Therefore, the velocity of water in the narrow pipe is 127.39 m/s.
Exercise 2
A pipe of 0.1 m diameter transports water at a speed of 2 m/s. If two pipes of 0.05 m diameter are added, what is the speed of the water in each of the smaller pipes?
Solution:
The cross section of a 0.1 m diameter pipe is A₁ = π(0.05 m)² = 0.00785 m². Therefore, the volumetric flow rate of water flowing through the 0.1 m pipe is:
Q = A₁v₁ = 0.00785 m² × 2 m/s = 0.0157 m³/s
The cross section of a 0.05 m diameter pipe is A₂ = π(0.025 m)² = 0.0001963 m². Since there are two 0.05 m diameter pipes, the total area is A₃ = 2A₂ = 0.0003926 m². Therefore, the volumetric flow rate of water flowing through the two 0.05 m pipes is:
Q = A₃·v₃
v3 = Q / A3 = 0.0157 m³/s / 0.0003926 m² = 40.11 m/s
Therefore, the velocity of water in each of the 0.05 m diameter pipes is 40.11 m/s.