Page 21 - Instrumentation Reference Book 3E

6 Measurement of flow
Now consider liquids only. These can be dividing by (1 - AiIA:) equation (1.17) becomes
regarded as being incompressible and their dens-
ity and specific volume will remain constant (1.18)
along the channel and

and taking the square root of both sides
and equation (1.10) may be rewritten as v2 = 4- rn (1.19)
v2
z1 .g+’+-=z PI 2 g+-+- (1.12)
v; P2
’
2 P 2 P Now A2IAl is the ratio (area of section 2)/(area of
Dividing by g, this becomes, section 1) and is often represented by the symbol
m. Therefore

Referring back to Figure 1.4 it is obvious that
there is a height differential between the upstream and
and downstream vertical connections represent- 4~
ing sections 1 and 2 of the fluid. Considering first 1 may be written as 1
the conditions at the upstream tapping, the fluid [l - (A;lA;)] dn=3
will rise in the tube to a height p1Ip . g above the
tapping orpllp . g + 21 above the horizontal level This is termed the velocity of approach factor
taken as the reference plane. Similarly the fluid often represented by E. Equation (1.19) may be
will rise to a height p2Ip. g or p2Ip g + Zz in the written
vertical tube at the downstream tapping.
The differential head will be given by V2 = E2/2gh (1.20)
and
h= ( -+z 1) - (E+-%) (1.14)
P.g P.g Q= A2. V2 = A2 .E2/2ghm3/s (1.21)
but from equation (1.13) we have Mass of liquid flowing per second= W =
(;+zl) +z= (k+Z2) +? p Q = A2 . p . E m kg also since Ap = hp,
2

(1.22)

Therefore (1.23)
(1.15)
1.2.4 Practical realization of equations
and The foregoing equations apply only to stream-
lined (or laminar) flow. To determine actual flow
v; - v; = 2gh (1.16) it is necessary to take into account various other
Now the volume of liquid flowing along the chan- parameters. In practice flow is rarely streamlined,
nel per second will be given by Qm3 where but is turbulent. However, the velocities of par-
ticles across the stream will be entirely random
Q = A1 . Vi = A2. V2 and will not affect the rate of flow very much.
In developing the equations, effects of viscosity
A2. V2
or VI =- have also been neglected. In an actual fluid the
A1 loss of head between sections will be greater than
Now substituting this value in equation (1.16): that which would take place in a fluid free from
viscosity.
In order to correct for these and other effects
another factor is introduced into the equations
for flow. This factor is the discharge coefficient
or V,”(l -&A:) = 2gh (1.17) C and is given by the equation