Bernoulli’s equation is widely used for the measurement of flow rate with the help of following flow rate measuring devices.

**5.3.1 Venturimeter**

Venturimeter is used to measure flow rate Q in m^{3} /s .

Fig. 5.2 shows a pipe fitted with venturimeter which has three parts (i) converging

part (ii) throat and (iii) diverging part.

Applying Bernoulli’s equation in .sections (1) and (2) we get

P_{1} / p g + v_{1 }/ 2 g + z_{1} = p_{2} / p g + v_{2} / 2 g + z_{2
}

Let us assume the venturimeter is horizontal.

So z_{1} = z_{2}

P_{1} / p g + v_{1} /2 g = p_{2} p g + v_{2} 2 g

P_{1} –p_{2} / p g = v_{2} / p g + v_{2} / 2g

h = v_{2} – v_{2} / 2 g

From continuity equation

a_{1} v_{1} = a_{2} v_{2}

v_{2} = a_{1} / a_{2} v_{1}.

From(5.4)

= ( _{1}a^{2} – _{2}a^{2} / _{2}a^{2} ) _{1}v^{2} / 2 g

_{1}V^{2} = _{2}a^{2 }/ _{1}a^{2} –_{2} a^{2}

Discharge Q _{th} = a_{1} v_{1}

Q _{th} = a_{1} v_{2} / _{1}a^{2} = _{2}a^{2} = 2 g h

Equation (5.5) is the expression for theoretical discharge. In actual discharge, there

are some losses, Q _{act} = Cd x Q _{th}·

where C_{d }is called the coefficient of discharge for venturimeter. C_{d} < 1.

If we use differential manometer

Then h = x [ S _{h} / S_{o} – 1]

For x = difference of heavier liquid in u-tube

S _{h} = sp. gravity of the heavier liquid

S_{0} = sp. gravity of the liquid flowing through pipe, here s_{0} < s _{h}

If S_{o} > S_{t} then h = x [ 1 – S_{t} / S_{o}]

S_{1} = sp. gravity of lighter liquid.

x = difference of lighter liquid in u-tube.

For inclined venturimeter

Z_{1} = z_{2}

h = (p_{1} / p g + z_{1}) – (p_{2} / p g + z_{2}) = x [ S _{h} / S_{o} – 1] or [ 1 – S_{t} / S_{o}]x.

**5.8.2 Pitot-tube**

Pi tot tube is used for measuring flow rate from velocity in a flowing fluid. It is L shaped One end is open to atmosphere where as other end is kept immersed at about mid depth point where velocity is to be measured

Let us consider two points (1) & (2) in the fluid flow as shown in the figure Applying Bernoulli’s equation at sections (1) & (2) we get

P_{1} / p g + _{1}v^{2 }/ 2 g + z_{1} = p_{2} /p g + _{2}v^{2} / 2 g + z_{2}

Now Z_{1} = Z_{2}, v_{2 }= 0(at nose the kinetic head is converted to pressure head)

P_{1} /p g + _{1}v^{2} / 2g = p_{2} / p g

or H + _{1}v^{2} / 2 g = H + h

v_{1} = 2 g h

v act = C v v_{1} = cv 2 g h where

C _{v} = coefficient of velocity for pi tot tube.

a = cross sectional area of pi tot tube, then O _{act} is given by

Q _{act} = C _{v} a 2 g h

**5.8.3 Orificemeter or Orifice Plate**

Orificemeter is used to measure the rate of flow. Generally circular sharp edge plate is

used in orificemeter.

Applying Bernoulli’s equation in sections (1) & (2), we get,

P_{1} / p g + _{1}v2 / 2 g + z_{1} = p_{2} / p g + _{2}v^{2} / 2 g + z_{2}

or ( p_{1} / pg + z_{1}) – ( p_{2} / p g + z_{2}) = _{2}v^{2} /2 g – _{1}v^{2} / 2g

h = _{2}v^{2 }/ 2 g –_{ 1}v^{2} / 2 g

now C, a o where C, = coefficience of contraction

a_{2} v_{2} = a_{1} v_{1} or v_{2} = a_{1} / C_{c} a _{o} v_{1}

from (5.8) =>

Let us simplify by using the following relation between C, & Cd .

where Cd = coefficient of discharge for orificemeter

(Cd)_{orifice} << (Cd)_{venturimeter}·