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*DThe

**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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