Equation of continuity
Try to open the water tap slowly while paying attention to the speed of the water coming out of the mouth of the faucet. Now compare, where is the rate of water flow faster, when the mouth of the faucet is closed or not? Why is that? To understand this, please learn the continuity equation.
In a steady stream, the speed of each fluid particle at a point, say point A is always the same. When passing point B, the velocity of the fluid particles may change. However, when arriving at point B, the fluid particles that follow from behind flow at the same speed as the fluid particles that precede it. Likewise when coming at point C and so on. The Flow Line is a curve that connects points A, B, and C.
We can describe each streamlines through each point in the fluid flow. If we perceive a steady fluid flow, some streamlines that pass a particular angle on the imaginary surface area form a flow tube. No fluid particles intersect but are always parallel, and the flow tube will resemble a pipe whose shape is still the same. Fluid entering at one end of the tube will come out of the tube at the other end.
The debit states the volume of a fluid flowing through a certain cross section at a certain time interval. Mathematically, it can be stated:
Debit = Volume of Fluid / time interval
Q = V / t
To increase your understanding, we use an example. For example, fluid flows through a pipe. Pipes are usually cylindrical and have a certain cross-sectional area. The pipe also has a length.
When the fluid flows in the pipe as far as L, the volume of fluid in the pipe is V = A L (V = volume of fluid, A = cross-sectional area and L = length of pipe). Because as long as it flows in a pipe along L, the fluid need a certain time interval, then we can say that the amount of fluid is:
Because v = s / t = L / t —> L = v t, the equation above is changed to:
Thus, when the fluid flows through a pipe that has a certain cross-sectional area and length, for a certain period, the amount of fluid discharge (Q) is equal to the cross-sectional area (A) multiplied by the fluid flow velocity (v).
The equation of continuity
Review the fluid flow in a pipe that has a different diameter, as shown in the figure below.
This image show fluid flow from left to right (fluid flows from a large diameter pipe to a small diameter). The dashed line is streamlined.
A1 = large-diameter pipe cross section area, A2 = small-diameter cross section area, v1 = velocity of fluid flow in a large-diameter pipe, v2 = velocity of fluid flow in a small-diameter pipe, L = distance traveled by fluid.
In a steady stream, the velocity of the flow of fluid particles at a point equals the speed of the flow of other fluid particles that pass through that point. Fluid flow also does not intersect (the line is parallel). Therefore the mass of fluid that enters one end of the pipe must be equal to the mass of the fluid that exits at the other end. If the fluid has a certain mass coming to the tube with a large diameter, then the fluid will exit on a small diameter pipe with the same mass.
Now consider the figure of the pipe above. Review the large diameter pipe and the small diameter pipe.
During a certain time interval, some fluids flow through the section of the pipe with a large diameter (A1) as far as L1 (L1 = v1 t). The volume of fluid flowing is V1 = A1 L1 = A1 v1 t. During the same time interval, some other fluids flow through the small diameter pipe section (A2) as far as L2 (L2 = v2 t). The volume of fluid flowing is V2 = A2 L2 = A2 v2 t.
The equation of continuity for the incompressible fluids
In incompressible fluids, the fluid density is always the same at every point it passes. The fluid mass flowing in a pipe that has a cross-sectional area of A1 (large pipe diameter) during a certain time interval is:
Likewise, the mass of fluid flowing in a pipe that has a cross-sectional area of A2 (small pipe diameter) during a certain time interval is:
In a steady stream, the mass of the incoming fluid is the same as the mass of the fluid coming out, so:
m1 = m2
ρ A1 v1 t = ρ A2 v2 t
A1 v1 = A2 v2
So, the equation of continuity equation in incompressible fluids:
A1 v1 = A2 v2 — Equation 1
A1 = cross-sectional area 1, A2 = cross-sectional area 2, v1 = velocity of fluid flow in section 1, v2 = velocity of fluid flow in section 2, A v = flow rate of volume (V / t) = debit.
Equation 1 shows that the debit is always the same at each point along the pipe or flow tube. When the pipe cross-section decreases, the fluid flow rate increases, whereas when the pipe section becomes large, the fluid flow rate becomes small.
When part of the mouth of the faucet is closed, the flow of water becomes heavier than when part of the mouth of the faucet is not closed. That is because the cross-sectional area of the faucet becomes small when part of the mouth of the faucet is closed so that the rate of flow of water increases. But it is essential to know that the flow rate or debit of fluids is always the same at each point along the flow of water, whether part of the mouth of our faucet is closed or not. So what changes is the flow rate of the fluid?
Then what about the flow of water in the river? The deep river section has a larger cross-section than the shallow part of the river, so the rate of water flow in the deep river is smaller than the speed of water flow in the shallow part of the river.
The equation of continuity for the compressible fluids
For compressed fluid, the density of the fluid is not always the same. In other words, fluid density changes when compressed. If the fluid is not compressed the density of the fluid is removed from the equation, so in this case, the fluid density included in the equation. Referring to the equation that has been derived before, let us derive the equation for the compressed fluid.
The mass of fluid that enters is equal to the mass of the fluid that exit, so :
m1 = m2
ρ A1 v1 t = ρ A2 v2 t
The interval of fluid flow is the same so that it can be eliminated. Equations change to:
ρ A1 v1 = ρ A2 v2 → Equation 2
This is the equation for the compressed fluid. The difference lies in the density of the fluid. If the fluid is compressed, the density changed. Conversely, if the fluid is not compressed, the density is always the same so that it can be removed from the equation.
Example problem 1:
Water flows through a pipe with a diameter of 10 cm at a speed of 2 m / s. What is the water debit?
Diameter of pipe = 10 cm
Radius of pipe (r) = 5 cm = 0.05 m
Speed of water (v) = 2 m/s
Wanted : Debit
Q = A v
Q = ( π r2) (2 m/s)
Q = (3.14)(0.05 m)2 (2 m/s)
Q = (3.14)(0.0025 m2)(2 m/s)
Q = 0.0157 m3/s
Example problem 2:
A pipe with a diameter of 20 cm is connected to another pipe with a diameter of 10 cm. If the speed of water in the pipe with a diameter of 20 cm is 4 m/s, what is the speed of water in a pipe with a diameter of 10 cm?
Diameter 1 = 20 cm (r1 = 10 cm = 0.1 m)
v1 = 4 m/s
Diameter 2 = 10 cm (r2 = 5 cm = 0.05 m)
Wanted : v1
Q1 = Q2
A1 v1 = A2 v2
( π r12) (4 m/s) = (π r22) (v2)
(0.1 m)2 (4 m/s) = (0.05 m)2 (v2)
(0.01 m2)(4 m/s) = (0.0025 m2)(v2)
(0.04 m3 /s) = (0.0025 m2)(v2)
v2 = 16 m/s