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Fluid Dynamics and Flow Measurements
Fluid measurement is the chapter in Fluid mechanics which deals with the application of Bernoulli equation applicable to many situations, not just the pipe flow. It consists of various measurement apparatus such as Venturimeter, Orifice meter, Pitot Tube, etc. and its application to flow measurement from tanks, within pipes as well as in open channels.
Flow measurement is the quantification of bulk fluid movement. Flow can be measured in a variety of ways. Positive displacement flow meters accumulate of a fixed volume of fluid and then count the number of times the volume is filled to measure flow.
Some flow measuring instruments are given as:
- The Pitot tube is a simple velocity measuring device.
- Uniform velocity flow hitting a solid blunt body, has streamlines similar to this:
- Some move to the left and some to the right. The centre one hits the blunt body and stops.
- At this point (2) velocity is zero The fluid does not move at this one point. This point is known as the stagnation point.
- Using the Bernoulli equation we can calculate the pressure at this point.
- Along the central streamline at 1: velocity u1, pressure p1
- At the stagnation point of: u2 = 0. (Also z1 = z2)
The blunt body does not have to be a solid. It could be a static column of fluid, this feature is used measure the flow velocity.
- Two piezometers, one as normal and one as a Pitot tube within the pipe can be used such as
- The above expression is for velocity from two pressure measurements and the application of the Bernoulli equation.
Pitot Static Tube
- The Pitot static tube combines the tubes and they can then be easily connected to a manometer:
- In reality, its diameter is very small and can be ignored i.e. points 1 and 2 are considered to be at the same level.
- The holes on the side connected to one side of a manometer, while the central hole connects to the other side of the manometer.
- Using the theory of the manometer
We know that
- Advantages of Pitot tube:
- Simple to use (and analyse)
- Gives velocities (not discharge)
- May block easily as the holes are small
- The Venturi meter is a device for measuring discharge in a pipe.
- It is a rapidly converging section which increases the velocity of flow and hence reduces the pressure.It then returns to the original dimensions of the pipe by a gently diverging ‘diffuser’ section.
Using Bernoulli along the streamline from point 1 to point 2,
Substituting and rearranging gives
- Actual discharge takes into account the losses due to friction, we include a coefficient of discharge (Cd=0.9)
- In terms of the manometer readings,
- This expression does not include any elevation terms. (z1 or z2)
- When used with a manometer The Venturimeter can be used without knowing its angle
Venturimeter Design Note Point:
- The diffuser assures a gradual and steady deceleration after the throat. So that pressure rises to something near that before the meter.
- The angle of the diffuser is usually between 6 and 8 degrees.
- Wider and the flow might separate from the walls increasing energy loss.
- If the angle is less the meter becomes very long and pressure losses again become significant.
- The efficiency of the diffuser of increasing pressure back to the original is rarely greater than 80%.
- Care must be taken when connecting the manometer so that no burrs are present.
- Orifice meter is a device used for measuring the rate of flow of a fluid flowing through a pipe which works on the same principle as that of venturimeter.
- Setup for Orifice meter consists of a flat circular plate which has a circular hole, in concentric with the pipe which is called orifice.
- The diameter of orifice is generally 0.5 times the diameter of the pipe (D), although it may vary from 0.4 to 0.8 times the pipe diameter.
Applying Bernoulli’s equations at sections 1 and 2, we get
where h is the differential head.
Let A0 is the area of the orifice, Coefficient of contraction, Cc = A2 /A0
By continuity equation, we have
If Cd is the co-efficient of discharge for orifice meter, which is defined as
So, the final discharge of Orifice meter ,
Flow Over Notches and Weirs
- A notch is an opening in the side of a tank or reservoir.
- It is a device for measuring discharge.
- A weir is a notch on a larger scale - usually found in rivers.
- It is used as both a discharge measuring device and a device to raise water levels.
- Velocity of the fluid approaching the weir is small so we can ignore kinetic energy.
- The velocity in the flow depends only on the depth below the free surface.
A General Weir Equation
Consider a horizontal strip of width b, depth h below the free surface ,
Integrating from the free surface, h=0, to the weir crest,h=H, gives the total theoretical discharge
This is different for every differently shaped weir or notch.
Some common types of Weirs
- Rectangular Weir
The width does not change with depth so, b= Constant= B
Substituting this into the general weir equation gives
- To get the actual discharge we introduce a coefficient of discharge, Cd, to account for losses at the edges of the weir and contractions in the area of flow,
- ‘V’ Notch Weir
The relationship between width and depth is dependent on the angle of the “V”.
The width, b, a depth h from the free surface is
So the discharge is
The actual discharge is obtained by introducing a coefficient of discharge
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