# Applications of Bernoulli’s Equation

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 m3 /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

P1 / p g + v1 / 2 g + z1 = p2 / p g + v2 / 2 g + z2

Let us assume the venturimeter is horizontal.

So                                            z1 = z2

P1 / p g + v1 /2 g = p2 p g + v2 2 g

P1 –p2 / p g = v2 / p g + v2 / 2g

h  = v2 – v2 / 2 g

From continuity equation

a1 v1 = a2 v2

v2 = a1 / a2 v1.

From(5.4)

= ( 1a22a2 / 2a2 ) 1v2 / 2 g

1V2 = 2a2 / 1a22 a2

Discharge                    Q th = a1 v1

Q th  = a1 v2 / 1a2 = 2a2 = 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 Cd is called the coefficient of discharge for venturimeter. Cd < 1.

If we use differential manometer

Then                            h = x [ S h / So – 1]

For                               x = difference of heavier liquid in u-tube

S h = sp. gravity of the heavier liquid

S0 = sp. gravity of the liquid flowing through pipe, here s0 < s h

If               So > St then h = x [ 1 – St / So]

S1 = sp. gravity of lighter liquid.

x = difference of lighter liquid in u-tube.

For inclined venturimeter

Z1 = z2

h  = (p1 / p g + z1) – (p2 / p g + z2) = x [ S h / So – 1] or [ 1 – St / So]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

P1 / p g + 1v2 / 2 g + z1 = p2 /p g + 2v2 / 2 g + z2

Now                            Z1 = Z2, v2 = 0(at nose the kinetic head is converted to pressure head)

P1 /p g + 1v2 / 2g = p2 / p g

or                                 H + 1v2 / 2 g = H + h

v1 = 2 g h

v act = C v v1 = 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,

P1 / p g + 1v2 / 2 g + z1 = p2 / p g + 2v2 / 2 g + z2

or                     ( p1 / pg + z1) – ( p2 / p g + z2) = 2v2 /2 g – 1v2 / 2g

h = 2v2 / 2 g – 1v2 / 2 g

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

a2 v2 = a1 v1 or v2 = a1 / Cc a o v1

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·