Flow measuring devices are the devices that are used to measure the rate of flow (i.e., discharge) or velocity of a flowing fluid. All these devices work on the basis of Bernoulli's equations. Basics of Bernoulli's equation, different flow measuring devices, and formulas for calculating the discharge are discussed further.
Basics of Bernoulli's Equation
Bernoulli's equation is also called the energy equation.
It is based on the law of conservation of energy, for more details on the law of conservation on fluid mechanics click here.
According to Bernoulli's equation, the total energy of a flowing fluid at any section consists of three parts as mentioned below.
Pressure or Static head (P/γ)
Kinetic or Velocity head (v^2/2*g)
Datum or Potential head (Z)
By combining these three parts, we get Bernoulli's equation as,
(P/γ) + (v^2/2*g) + Z = constant, a pictorial representation is given for better understanding,
In the above equation,
(P/γ) + Z, is called the piezometric head
(v^2/2*g), is called the kinematic head
The piezometric head is represented by hydraulic gradient line (HGL) Both piezometric head and kinematic head combined represents the total energy line (TEL). It could be easily understood that HGL is always below the TEL. Also, TEL always decreases along the flow direction except at the pump section.
List of Flow Measuring Devices
Based on Bernoulli's equation there are different flow measuring devices as mentioned below.
Venturimeter
Orifice meter
Pitot tube
Nozzle meter
Rotameter
Elbow meter
Out of the above-mentioned devices, Venturimeter is the most popular flow measuring device. Pitot tube and orifice meter are also used to some extent. Therefore, it's important to know the concept, design, and formulas, of these devices and the same is discussed further.
Venturimeter
Venturimeter is a device to measure the rate of flow of fluid in pipes. Venturimeter consists of three portions i.e., convergent, throat, and divergent portion. Design aspects of these three portions are discussed further.
1. Convergent portion
Diameter at convergent and divergent portions are taken as the diameter of the pipe to which the venturimeter is connected. Therefore, Dinlet = Dpipe = Doutlet
The angle of convergent is maintained as (21°± 1°), mostly 20°
The length of the convergent portion is taken as 2.7*(D - d), where, D - diameter of the pipe, and d - diameter of the throat
2. Throat portion
The diameter of the throat(d) is an important factor for the safety of both the pipe and the Venturimeter. If d is too small, velocity head increases and pressure head decreases to a great extent. This negative pressure creates bubbles that explode on the downstream side (where pressure increases) and causes notching of the material. To avoid this d is taken as half of the diameter of the inlet pipe, i.e., 1/2*D
The length of the throat portion (Lthroat) is taken as the diameter of the throat portion (d)
3. Divergent portion
The divergent portion is provided to protect the Venturimeter from cavitation i.e., pressure is gradually increased
The angle of the divergent portion is taken between 5° to 15°, mostly 6°
Other general considerations include,
To measure the pressure difference between inlet and outlet a Utube manometer is adopted
Head loss in convergent portion is neglected (therefore Venturimeter gives theoretical discharge only)
To get the actual discharge, the co-efficient of discharge (Cd) is multiplied with theoretical discharge. Cd value varies between 0.95 to 0.99
The size of a Venturimeter is specified as the diameter of the main pipe and the diameter of the throat of the venturimeter. For example, venturimeter with size 150 * 75 mm represents main pipe diameter (D) as 150 mm and throat diameter (d) as 75mm
Refer to the below attached pictorial representation of the Venturimeter.
Formula Derivation for Venturimeter
Considering the reference line of U-Tube manometer,
Py = Pz
P1 + γ1*x = P2 + γ1*(x-h) + γ2*h,
P1 - P2 = -γ1*x + γ1*x - γ1*h + γ2*h,
P1 - P2 = -γ1*h + γ2*h,
P1 - P2 = h*(γ2 - γ1),
P1 - P2 = (h*γ1) * [(γ2/γ1) - 1],
(P1 - P2)/γ1 = h*[((S2*γw)/(S1*γw)) - 1],
(P1 - P2)/γ1 = h*[(S2/S1) - 1]
The difference in piezometric head between inlet and outlet (ΔH),
ΔH = [(P1/γ)+Z1] - [(P2/γ)+Z2], (here, Z1 = Z2)
ΔH = h*[(S2/S1) - 1]
ΔH = h*[(Sm/S) - 1], if Sm > S
ΔH = h*[1 - (Sm/S)], if Sm < S
where,
h - the manometric reading as shown in the diagram
Sm - specific gravity of the mercury (i.e., 13.6)
S - specific gravity of the flowing fluid
Continuing derivation, apply Bernoulli's equation between inlet and throat i.e., between section 1 & section 2
(P1/γ) + (v1^2/2*g) + Z1 = (P2/γ) + (v2^2/2*g) + Z2 + hL (head loss - neglected)
[(P1/γ) + Z1] - [(P2/γ) + Z2] = [(V2^2) - (V1^2)]/(2*g)
ΔH = [(V2^2) - (V1^2)] / (2*g) --- 1
Applying continuity equation, Q1 = Q2
A1*V1 = A2*V2 = Q
V1 = Q/A1, V2 = Q/A2, substituimg V1 and V2 in equation 1,
ΔH = [(Q/A2)^2 - (Q/A1)^2] / (2*g)
2*g*ΔH = Q^2[(1/A1^2) - (1/A2^2)]
2*g*ΔH*(A1^2)*(A2^2) = Q^2 * ((A1^2)-(A2^2))
Qth = (A1*A2*√(2*g*ΔH)) / √((A1^2) - (A2^2)),
Here discharge is theoretical (Qth) as the head loss was neglected. Actual discharge is calculated by multiplying the coefficient of discharge (Cd) with theoretical discharge.
Qactual = Cd * Qth,
Venturimeter formulas
The same formulas are represented in a pictorial form below.
where,
A1 - area of the pipe at the inlet (i.e., diameter = D)
A2 - area of the pipe at the throat (i.e., diameter = d)
ΔH - the difference in the piezometric head (as explained earlier)
Cd - coefficient of discharge 0.95 to 0.99
g - acceleration due to gravity (9.8 m/sec^2)
Orifice Meter
An orifice meter is a similar device used to measure the discharge of the fluid. It consists of a flat plate containing a circular orifice which is provided concentrically with the pipe across the flow. This device works on the same principle as that of the venturimeter.
Orifice meter is not most commonly used because of the below-mentioned disadvantages which are not present in a venturimeter.
Low accuracy than a venturimeter
Subjected to wear
Dirt and sediments are collected due to the obstructive design
It has low efficiency than a venturimeter
Though the orifice meter has some disadvantages it can be still easily made as it is simple in design and is cheap.
Pitot Tube
A pitot tube is a simple device used for measuring the velocity of a flowing fluid at any point. It appears as an L-shaped bent pipe in its simplest form and is used to measure the velocity of flow which are open to the atmosphere.
Applying Bernoulli's equation,
(P1/γ) + (v1^2/2*g) + Z1 = (P2/γ) + (v2^2/2*g) + Z2 + hL + ΔH,
here,
(P1/γ) = Z1 = (P2/γ) = v2^2 = Z2 = hL(headloss neglected) = 0, therefore we get,
(v1^2/2*g) = ΔH
Vth = √(2*g*ΔH), here velocity is theoretical (Vth) as the head loss was neglected. To get the actual velocity, the coefficient of velocity (Cv) must be multiplied with theoretical velocity.
Vactual = Cv * Vth,
Vactual = Cv * √(2*g*ΔH),
where,
Cv - the coefficient of velocity
g - acceleration due to gravity (9.8 m/sec^2)
ΔH - obtained from pitot tube
Flow Nozzle
A flow nozzle is a similar device to an orifice meter and venturimeter. It has a smooth converging portion as that of a venturimeter but with no diverging portion. Flow nozzle is also not a commonly used device.
Practise Problem
Check out the complete video lecture on open channel flow below.
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