MECHANICS of FLUIDS LABORATORY - Mechanical Engineering

Incompressible Flow Through a Meter
Incompressible flow through a venturi and an
orifice meter was discussed in Experiment 9. For
our purposes here, we merely re-state the
equations for convenience. For an air over liquid
manometer, the theoretical equation for both
meters is
Q th = A √⎺⎺⎺⎺⎺
2g∆h
2
(1 - D 4 2 /D 4 1 )
Now for any pressure drop ∆h i , there are two
corresponding flow rates: Q ac and Q th . The ratio of
these flow rates is the venturi discharge
coefficient C v , defined as
C v = Q ac
Q th
= 0.985
for turbulent flow. The orifice discharge
coefficient can be expressed in terms of the Stolz
equation:
C o = 0.595 9 + 0.031 2β 2.1 - 0.184β 8 +
+ 0.002 9β 2.5 10
⎛
6 0.75
⎞
⎝ Re β⎠
where Re = ρV oD o
µ
L 1 = 0
L 1 = 1/D 1
L 1 = 1
+ 0.09L 1
⎝ ⎛ β 4
1 - β ⎠ ⎞ 4 - L 2 (0.003 37β 3 )
= 4ρQ ac
πD o µ
for corner taps
for flange taps
for 1D & 1 2 D taps
β = D o
D 1
β 4
and if L 1 ≥ 0.433 3, the coefficient of the ⎛ ⎞
⎝ 1 - β 4 ⎠
term becomes 0.039.
L 2 = 0 for corner taps
L 2 = 1/D 1 for flange taps
L 2 = 0.5 - E/D 1 for 1D & 1 2 D taps
E = orifice plate thickness
Compressible Flow Through a Meter
When a compressible fluid (vapor or gas)
flows through a meter, compressibility effects
must be accounted for. This is done by introduction
of a compressibility factor which can be
determined analytically for some meters
(venturi). For an orifice meter, on the other hand,
the compressibility factor must be measured.
The equations and formulation developed
thus far were for incompressible flow through a
meter. For compressible flows, the derivation is
somewhat different. When the fluid flows
through a meter and encounters a change in area,
the velocity changes as does the pressure. When
pressure changes, the density of the fluid changes
and this effect must be accounted for in order to
obtain accurate results. To account for
compressibility, we will rewrite the descriptive
equations.
Venturi Meter
Consider isentropic, subsonic, steady flow of
an ideal gas through a venturi meter. The
continuity equation is
ρ 1 A 1 V 1 = ρ 2 A 2 V 2 = ·m isentropic = ·m s
where section 1 is upstream of the meter, and
section 2 is at the throat. Neglecting changes in
potential energy (negligible compared to changes
in enthalpy), the energy equation is
h 1 + V 1 2
2 = h 2 + V 2 2
2
The enthalpy change can be found by assuming
that the compressible fluid is ideal:
h 1 - h 2 = C p (T 1 - T 2 )
Combining these equations and rearranging gives
or
C p T 1 +
·
m s
2
2ρ 1 2 A 1
2 = C pT 2 +
m
· 2 s
2ρ 2 2 A
2 2
m
· 1
2 s
⎛
⎞
⎝ ρ 2 2 A
2 - 1
2 ρ 2 1 A
2 1 ⎠
= 2C p(T 1 - T 2 )
= 2C p T 1
⎝ ⎛ 1 - T 2
T ⎠ ⎞
1
If we assume an isentropic compression process
through the meter, then we can write
p 2 T
= ⎛ 2
⎞
p 1 ⎝ T 1 ⎠
γ
γ - 1
where γ is the ratio of specific heats (γ = C p /C v ).
Also, recall that for an ideal gas,
C p = R γ
γ - 1
Substituting, rearranging and simplifying, we get
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