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34-SM-03-01
Page 4
equation:
ρ
vD
Re =
υ
where,
ν = velocity
D = inside pipe diameter
ρ = fluid density
µ = fluid viscosity
The SMV 3000 can be configured
to dynamically compensate for
discharge coefficient.
This method follows the standard
Stoltz equation for orifice, Venturi
and nozzle primary elements to
predict discharge coefficient for
flowrate in the turbulent regime -
Re > 4000.
b
C = C
+
∞
Re
Where,
C
= Discharge coefficient at
∞
infinite Re #
b = function of primary element
Re = Reynolds number
n = depends on the primary
element
Dynamically compensating for
discharge coefficient allows the
SMV 3000 to obtain better flow
accuracy at higher turndowns for
orifice, Venturi and nozzles.
Thermal Expansion Factor
The material of the process pipe
and primary flow element expands
or contracts with changes in
temperature of the fluid being
measured. When a primary flow
element, such as an orifice, is
sized, the flowrate is calculated
based on the Beta ratio (d/D) at 68
degrees F. The SMV 3000, using
the thermal expansion coefficients
which are dependent of the
material of the pipe and flow
element, calculates the change in
Beta ratio per the following
equations:
β = d/D
D = 1 + α
d = 1 + α
where,
β = beta ratio
D = pipe diameter
d = bore diameter
D
= pipe diameter at design
ref
d
= bore diameter at design
ref
α
= Thermal Expansion Coef.
p
α
= Thermal Expansion Coef.
pe
T
= flowing temperature
f
As an example, a fluid at 600
degrees F could cause as much
as 1% error in flow measurement
using 300 series stainless steel
n
materials.
Gas Expansion Factor
The gas expansion factor corrects
for density differences between
pressure taps due to expansion of
compressible fluids. It does not
apply for liquids which are
essentially non-compressible and
approaches unity when there are
small differential pressures for gas
and steam measurements. The
gas expansion factor is dependent
on the Beta ratio, the Isentropic
exponent, the differential pressure
and the static pressure of the fluid
per the following equation:
Υ
= 1 - (0.41 + 0.35β
1
where,
β = beta ratio
X
= h
1
k = isentropic exp. (ratio of
The SMV 3000 dynamically
compensates for gas expansion
effects and provides better mass
flow accuracy, especially for low
static pressure applications.
(T
- 68)D
p
f
ref
(T
- 68)d
pe
f
ref
temperature
temperature
of pipe
of bore
4
)X
/k
1
/P
w
specific heats)
Velocity of Approach Factor
E
is dependent on the Beta ratio
v
as defined by the following
equation:
Β
1-
E
= 1/
v
In turn, Beta ratio is dependent on
the bore diameter and pipe
diameter which are functions of
temperature. The SMV 3000
compensates dynamically for
velocity of approach factor by
calculating the true Beta ratio at
flowing temperature. This ensures
high flowrate accuracy at low and
high temperature applications.
Density and Viscosity of Fluids
Density directly effects the flowrate
calculation as well as the
discharge coefficient due to
changes in the Reynolds number.
The SMV 3000 can be configured
to compensate for density of fluids
due to changes in the temperature
and/or pressure per the following:
•
Gases as a function of P and
T per the Gas Law Equations.
•
Steam as function of P and T
based on the ASME Tables.
•
Liquids as a function of T per a
5th Order Polynomial.
2
ρ = d
+ d
T
+ d
T
+ d
1
2
F
3
F
Changes in the viscosity of a fluid
due to changes in temperature can
also effect the Reynolds number
and therefore discharge
coefficient. The SMV 3000 can
compensate the viscosity of liquids
based on the following 5th order
polynomial equation:
2
µ = v
+ v
T
+ v
T
+ v
1
2
F
3
F
4
3
4
T
+ d
T
4
F
5
F
3
4
T
+ v
T
4
F
5
F
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