Bernoulli’s equation is widely used for the measurement of flow rate with the help of following flow rate measuring devices.
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.
= ( 1a2 – 2a2 / 2a2 ) 1v2 / 2 g
1V2 = 2a2 / 1a2 –2 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.
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·