# Venturi Wet Gas Flow Modeling Based on Homogeneous and Separated Flow Theory

Authors: Fang Lide; Zhang Tao; Xu Ying
Journal: Mathematical Problems in Engineering
 (2009)

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## Abstract

Wet gas metering is becoming increasingly important to petroleum and chemical engineering. The Venturi meter is a favored device for the metering of the unprocessed wet natural gas production flows. When Venturi meters are used in wet gas, the measured differential pressure is higher than it would be in gas phases flowing alone. This phenomenon is called over reading. Eight famous over-reading correlations have been studied by many researchers under low- and high-pressure conditions, the conclusion is separate flow model and homogeneous flow model performing well both under high and low pressure. In this study, a new metering method is presented based on homogeneous and separate flow theory; the acceleration pressure drop and the friction pressure drop of Venturi under two-phase flow conditions are considered in new correlation, and its validity is verified through experiment. For low pressure, a new test program has been implemented in Tianjin University's low-pressure wet gas loop. Three 50 mm Venturi meters withβ values of 0.4048, 0.55, and 0.70 have been tested at the working operation pressure of 0.15, 0.20, and 0.25 MPa, the gas densimetric Froude number from 0.6 to 2.5, the modified Lockhart-Martinelli parameter from 0.0022 to 0.3, and the ratio of the gas liquid mass flow rate from 0.5 to 0.99. For high pressure, the National Engineering Laboratory offered their reports on the web  and  their test range with β values of 0.4, 0.60, and 0.75, pressure of 1.5, 3.0, and 6.0 Mpa, gas Froude number from 0.5 to 5.5, and the modified Lockhart-Martinelli parameter from 0.0035 to 0.3. The coefficients of the new proposed correlation are fitted with all independent data both under high and low pressure. Finally, the applicability and errors of new correlation are analyzed.


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## Body

## 1. Introduction

Wet gas  metering  has  been  described as a subset of multiphase
flow measurement, where the volume of gas at actual measuring conditions is very high when compared to the volume of liquid in the flow stream. High-gas volume
fraction has been defined in the range of 90–98% by different
technical papers; more details are shown by Agar and Farchy [1]. Normally, these conditions
need wet gas metering; for instance, some small or remote gas fields are
processed together in common platform facilities, the
individual unprocessed streams must be metered before mixing. In other
circumstances, some gas meters may also be subjected to small amounts of liquid
in the gas. This can happen to the gas output of a separator as a result of
unexpected well conditions or liquid slugging.Two ways are employed to meter wet gas: one
approach is to use a multiphase flow meter in wet gas, and the other approach
is to use a standard dry gas meter applying corrections to the measurements
based on knowledge of how this type of meter is affected by the presence of
liquid in the gas stream. This method requires prior knowledge of the liquid
flow, which has to be obtained through another means; more details were shown by Lupeau et al. [2].As a mature single-phase flow measurement device, the
Venturi meter has been successfully applied in a variety of industrial fields
and scientific research. Just owing to its successful applications in
single-phase flows, the Venturi meter can easily be considered for two-phase
flow measurement. When Venturi meters are used in
wet gas, the measured differential pressure is higher than it would be
with the gas phase flowing alone. If uncorrected, this additional pressure drop
will result in an over reading of the gas mass flow rate. More details were shown by Geng et al. [3].Eight famous over-reading correlations
have been studied in low- and high-pressure conditions
[4–10]. In Steven’s paper [10], an ISA Controls standard North
Sea specification 6” Venturi meters with a 0.55 diameter ratio (or “beta”)
of 6 mm pressure tappings was the meter installed in National Engineering Laboratory (NEL) with pressure from 2 to 6 MPa and
LM parameter from
0 to 0.3. NEL’s engineer tested
three 4-inch meters with different beta values (0.4, 0.60, 0.75) and tested
over a range of pressures (1.5–6.0 MPa) gas
densimetric Froude number (Frg),
0.5–5.5, and
Lockhart-Martinelli parameter, X, 0–0.4 [11–13]. The results show that the liquid
existence causes the meters to “over-read” the gas flow rate. This over reading
is affected by the liquid fraction, gas velocity, pressure, and Venturi beta
value. They predicted that some of the data seem to tend to a value slightly
above unity, particularly at low X values. Furthermore, in 2002, Britton et al.
did some tests in Colorado Engineering Experiment Station, Inc., Colo, USA, [14, 15] with pressure between 1.4–7.6 MPa and X values between 0–0.25. Their study
also confirmed the over-reading existence in Venturi meters.The result of high-pressure
comparison is shown in Table1 [10].Table 1
Result
of high-pressure comparison.
ModelsRoot mean square
errorRankDe Leeuw0.02111Homogeneous0.02372Lin0.04623Murdork 1.50.04824Murdork 1.260.06505Chisholm0.07106Smith and Leang0.12607Under low pressure, eight correlations are compared with
Tianjin University's low-pressure wet gas test
facilities [16] (see Table 2).Table 2
Result of low-pressure comparison.
ModelsRMSERankHomogenous0.110211Steven0.147872De Leeuw0.148543Smith and Leang0.188214Chisholm0.195975Murdock1.50.206586Lin0.207427Murdock0.210788The method of comparing the
seven correlations performances was chosen to be by comparison of the root mean
square error (defined asδ):(1.1)δ=1N∑1N(ORp(i)−ORe(i)ORe(i))2, where ORp(i) is prediction over reading; ORe(i) is experimentation over reading; N is data numbers.Tables1 and 2 show the models performance in low and high pressure. By De Leeuw model
being based on separated flow assumption, more parameters have been considered
so it performs well. Although the assumptions of homogeneous models are
simple, it performs well at both low pressure and high pressure (see Steven’s
results), for wet gas, homogeneous models may be true to some extent. This
means that wet gas flow structure holds homogeneous character and separation
character. Therefore, a new correlation considering homogeneous and separation
flow theory together could be better than the previous ones.This paper proposed a new Venturi wet gas correlation based on homogenous
and separate assumption. The acceleration
pressure drop and the friction
pressure drop of Venturi under two-phase flow conditions are considered in new correlation,
and its validity is verified through experiment. Finally, the performance of
the new proposed correlations is compared with the old eight correlations both
under low and high pressure.


## 2. New Model Based on Homogeneous and Separated Flow Theory

### 2.1. Over-Reading Theory of Venturi Wet Gas Metering

When Venturi meters are used in
wet gas the measured differential pressure is higher than it would be for the
gas phase flowing alone. If uncorrected, this additional pressure drop will
result in an over reading of the gas mass flow rate:(2.1)OR=mg′mg, where mg is the correct gas
mass flow rate, mg′ is the apparent gas mass flow rate determined from the two-phase measured
differential pressure ΔPtp, ΔPtp is the actual two-phase differential pressure between the
upstream and throat tappings, and ΔPg is the gas differential pressure between the upstream and
throat tappings:(2.2)mg=CεAT2ρgΔPg1−β4,(2.3)mg′=CεAT2ρgΔPtp1−β4. In (2.2) and (2.3), C is
discharge coefficient, AT is the area of the Venturi throat, ε is expansibility factor, ρg is gas density, and β is diameter ratio. In
fact, the discharge coefficient C is variable under different flow conditions.
Here, given that the discharge coefficient C is constant, and take into account
the fact that different flow conditions only have effect on over reading, but
not have effect on the discharge coefficient given C.The real gas mass flow rate can been obtained by(2.4)mg=mg′OR.The homogeneous flow theory treats the two-phase flow as if
it was a single-phase flow by using a homogeneous density expressionρtp which averages the phase densities so that
the single-phase differential pressure meter equation can be used(2.5)1ρtp=xρg+1−xρl, where x is the mass quality, ρtp is the homogeneous density, and subscripts “l” and “g” are for liquid and gas, respectively.With
this models the gas mass flow rate of the two phase flow can be written as(2.6)mg=xCεAT2ρtpΔPtp1−β4.Let (2.3) divide (2.6),
then the homogeneous model gives(2.7)ORh=mg′mg=CεAT2ρgΔPtp/1−β4x(CεAT2ρtpΔPtp/1−β4),ORh=1xρgρl+(1−ρgρl)x.However, (2.6) is also an
estimation function about gas mass flow rate; the real gas mass flow rate
should be (2.2) and then (2.6) as the apparent gas mass flow rate will be more rational. So let (2.6) divide (2.2), the real over reading under the
homogeneous flow theory is shown in the following form:(2.8)ORh=mg′mg=x⁢⋅(CεAT2ρtpΔPtp/1−β4)CεAT2ρgΔPg/1−β4=x⁢⋅ρtpρg⁢⋅ΔPtpΔPg.Equation (2.8) derived from
homogeneous flow theory, if ΔPtp/ΔPg derived from separation flow theory, the
combination of homogeneous and separation flow theory is implemented.Separated flow theory takes
into account the fact that the two phases can have differing properties and
different velocities. Separate equations of continuity, momentum, and energy
are written for each phase, and these six equations are solved simultaneously,
together with rate equations which describe how the phases interact with each
other and with the walls of duct. In the simplest version, only one parameter,
such as velocity, is allowed to differ for the two phases while conservation
equations are only written for the combined flow.Equation (2.9) shows the momentum
function of one dimension two-phase flow based on separated flow assumption.
The pressure drop of fluids in the pipe come from three parts, the first is
friction; the second is gravitation; the third is acceleration [17–21]:(2.9)−dPdz=τ0UA+[ρgα+ρl(1−α)]g⁢sinθ+1Addz{AG2[(1−x)2ρl(1−α)+x2ρgα]}(2.10)−dPdz=dPfdz+dPgdz+dPadz, where τ0 is friction force, U is perimeter of pipe,  α is void fraction,  G is mass velocity of mixture, dPf/dz is pressure drop caused by friction, dPg/dz is pressure drop caused by gravitation,  dPa/dz is pressure drop caused by acceleration.


### 2.2. The Friction Pressure Ddrop of Venturi Under Two-Phase
Flow Condition

For single-phase flow in straight pipe, the friction pressure drop can be calculated with(2.11)dPfdz=λd⁢⋅ρu22, where λ is the friction factor; d is the pipe diameter, u is the velocity.Givenλ is constant in conical convergent of Venturi,
the fluid velocity in the straight pipe keep unchanged, d0 is diameter of straight pipe, A0 is cross-section of straight pipe, d1 is diameter of Venturi throat, l0 is the length of conical convergent, θ is convergent angle. The schematic of Venturi conical
convergent part is shown in Figure 1.Figure 1
Sketch of Venturi conical convergent region.Analyzing an infinitesimaldl given d is diameter of the analyzing part, A is cross-section, l is the distance from Venturi inlet to
infinitesimal dl make integral  to (2.11):(2.12)ΔPf=∫0l0λd⁢⋅ρu22⁢dz,(2.13)ΔPf=λρ2∫0l01d⁢⋅u2⁢dz.Multiplyd0 to (2.13) in two sides:(2.14)ΔPf=λρ2d0∫0l0d0d⁢⋅u2⁢dz.From continuity equation,(2.15)u=A0Au0,A0A=(d0d)2.Substitute (2.15) into (2.14):(2.16)ΔPf=λρu022d0∫0l0(d0d)5dz.According to geometrical relationship showed
in Figure1,(2.17)dz=dlcosθ,(2.18)ll0=d0−dd0−d1,(2.19)⟹d0d=l0l0−l(1−d1/d0).Letβ be diameter ratio of Venturi, then(2.20)β=d1d0.Substitute (2.17), (2.19), and (2.20) into (2.16):(2.21)ΔPf=λρu02l052d0⁢cosθ∫0l0(1l0−l(1−β))5⁢dl,(2.22)ΔPf=λρu02l058d0⁢cos⁢θ(1−β)(1l0−l(1−β))4|0l0,(2.23a)ΔPf=(1+β)⋅(1+β2)β4⁢⋅14⁢cosθ⁢⋅λd0⁢⋅ρu022l0. Equation
(2.23a) shows that the friction pressure drop is affected by diameter ratio,
convergent angle, convergent length, inlet diameter, and inlet velocity.In a constant section pipe withl0 length, the friction pressure drop is(2.23b)ΔPfl0=λl0d0⁢⋅ρu022.Equation (2.23a) that is divided by (2.23b)
is(2.24)Kf=(1+β)⋅(1+β2)β4⁢⋅14⁢cosθ. Equation
(2.24) shows that the ratio Kf is a function of
diameter ratio and convergent angle. For a definite Venturi, Kf is constant.As for gas liquid two-phase flow, (2.23a) and (2.23b)
changes into (2.25)ΔPf=Kf⁢⋅λl0d0⁢⋅αρgug2+(1−α)ρlul22.When the pipe is full of gas (α = 1) or liquid (α = 0), (2.25) changes to (2.23a).From gas liquid two-phase flow continuity
equation,(2.26)xGA=Agugρg,(1−x)GA=Alulρl.Consider the definition of void fraction,(2.27)xαG=ugρg,(1−x)(1−α)G=ulρl,(2.28)G=mA=αρgug+(1−α)ρlul which defined S as slip ratio, that is, gas and liquid real velocity ratio combine (2.26) and
(2.27):(2.29)1α=1+s1−xx⁢⋅ρgρl.Substitute (2.26) and (2.27) into (2.25):(2.30)ΔPf=Kf⁢⋅λl0d0⁢⋅G22⁢⋅1ρl[x2α⁢⋅ρlρg+(1−x)21−α].When the pipe is full of  gas,(2.31)ΔPfg=Kf⁢⋅λgl0d0⁢⋅G22⁢⋅x2ρg.When the pipe is full of liquid,(2.32)ΔPfl=Kf⁢⋅λll0d0⁢⋅G22⁢⋅(1−x)2ρl.Letλ=λg=λl,
define Xf as(2.33)Xf=ΔPflΔPfg=(1−xx)ρgρl.Equation (2.30) divided by (2.31) is(2.34)ΔPfΔPfg=1α+(1−x)2x2ρgρl⁢⋅11−α.Substitute (2.29) into (2.34):(2.35)ΔPfΔPfg=1+CfXf+Xf2, where(2.36)Cf=1sρlρg+sρgρl.


### 2.3. The Aacceleration Pressure Drop of Venturi Under
Two-Phase Flow Condition

According to (2.9), the acceleration pressure
drop is(2.37)dPadz=1Addz{AG2[(1−x)2ρl(1−α)+x2ρgα]},(2.38)ΔPa=∫dPa=∫ATA01A⁢d{AG2[(1−x)2ρl(1−α)+x2ρgα]}.Given the fluid is incompressible, the void fractionα is constant in the Venturi throat. Integrate (2.38):(2.39)ΔPa=G2[(1−x)2ρl(1−α)+x2ρgα]⁢⋅lnA0AT.When the gas was flowing alone in the pipe, the
pressure drop can be expressed as(2.40)ΔPag=G2x2ρg⁢⋅lnA0AT.The similar equation for the liquid phase is(2.41)ΔPal=G2(1−x)2ρl⁢⋅lnA0AT.DefineXa:(2.42)Xa=ΔPalΔPag=(1−x)x⁢⋅ρgρl.Equation (2.39) divided by (2.40) is(2.43)ΔPaΔPag=(1−x)2x2⁢⋅ρgρl⁢⋅11−α+1α.Substitute (2.29) and (2.42) into (2.34):(2.44)ΔPaΔPag=1+Ca⁢⋅Xa+Xa2, where Ca is expressed as(2.45)Ca=1s⁢⋅ρlρg+s⁢⋅ρgρl.Compare (2.33) with (2.42), it is obvious that Xf is the same as Xa.Also, compared (2.33) with (2.42), Cf is equal to Ca.And then, (2.44) is equal to(2.46)ΔPfΔPfg=ΔPaΔPag=1+Cg⁢⋅X+X2, where(2.47)Cg=Ca=Cf=1s⁢⋅ρlρg+s⁢⋅ρgρl,(2.48)X=Xa=Xf=(1−x)x⁢⋅ρgρl.Equation (2.46) notes that the ratio of two-phase
and single-phase friction pressure drop is equal to the ratio of two- phase and
single-phase acceleration pressure drop.


### 2.4. The Total Pressure Ddrop of Venturi Under Two-Phase
Flow Condition

For a horizontal mounted Venturi,
gravitation pressure drop can be ignored. The total pressure drop is(2.49)ΔPtp=ΔPf+ΔPa.The total pressure drop of Venturi under
single-phase flow condition is(2.50)ΔPg=ΔPfg+ΔPag.Divide (2.49) by (2.50):(2.51)ΔPtpΔPg=ΔPf+ΔPaΔPfg+ΔPag.According to (2.46) and geometric axiom,(2.52)ΔPfΔPfg=ΔPaΔPag=ΔPf+ΔPaΔPfg+ΔPag.Combine (2.51) and (2.52):(2.53)ΔPtpΔPg=ΔPfΔPfg=ΔPaΔPag=1+Cg⁢⋅X+X2.So the model combined homogeneous and
separation flow theory can be expressed as (2.55). Call this correlation as H-S model:(2.54)ORH-S=x⁢⋅ρtpρg⁢⋅ΔPtpΔPg=x⁢⋅ρlρlx+ρg(1−x)⁢⋅1+Cg⋅X+X2(2.55)=x⁢⋅1+Cg⁢⋅X+X21+ρg/ρl⁢X.Equation (2.55) shows that Cg is an effect
factor to OR, it must be known first when (2.55) is used. However, slip ratio S is
contained in Cg equation, and
slip ratio is hard to be determined accurately, so it needs to fit a
correlation with experiment.


## 2.1. Over-Reading Theory of Venturi Wet Gas Metering

When Venturi meters are used in
wet gas the measured differential pressure is higher than it would be for the
gas phase flowing alone. If uncorrected, this additional pressure drop will
result in an over reading of the gas mass flow rate:(2.1)OR=mg′mg, where mg is the correct gas
mass flow rate, mg′ is the apparent gas mass flow rate determined from the two-phase measured
differential pressure ΔPtp, ΔPtp is the actual two-phase differential pressure between the
upstream and throat tappings, and ΔPg is the gas differential pressure between the upstream and
throat tappings:(2.2)mg=CεAT2ρgΔPg1−β4,(2.3)mg′=CεAT2ρgΔPtp1−β4. In (2.2) and (2.3), C is
discharge coefficient, AT is the area of the Venturi throat, ε is expansibility factor, ρg is gas density, and β is diameter ratio. In
fact, the discharge coefficient C is variable under different flow conditions.
Here, given that the discharge coefficient C is constant, and take into account
the fact that different flow conditions only have effect on over reading, but
not have effect on the discharge coefficient given C.The real gas mass flow rate can been obtained by(2.4)mg=mg′OR.The homogeneous flow theory treats the two-phase flow as if
it was a single-phase flow by using a homogeneous density expressionρtp which averages the phase densities so that
the single-phase differential pressure meter equation can be used(2.5)1ρtp=xρg+1−xρl, where x is the mass quality, ρtp is the homogeneous density, and subscripts “l” and “g” are for liquid and gas, respectively.With
this models the gas mass flow rate of the two phase flow can be written as(2.6)mg=xCεAT2ρtpΔPtp1−β4.Let (2.3) divide (2.6),
then the homogeneous model gives(2.7)ORh=mg′mg=CεAT2ρgΔPtp/1−β4x(CεAT2ρtpΔPtp/1−β4),ORh=1xρgρl+(1−ρgρl)x.However, (2.6) is also an
estimation function about gas mass flow rate; the real gas mass flow rate
should be (2.2) and then (2.6) as the apparent gas mass flow rate will be more rational. So let (2.6) divide (2.2), the real over reading under the
homogeneous flow theory is shown in the following form:(2.8)ORh=mg′mg=x⁢⋅(CεAT2ρtpΔPtp/1−β4)CεAT2ρgΔPg/1−β4=x⁢⋅ρtpρg⁢⋅ΔPtpΔPg.Equation (2.8) derived from
homogeneous flow theory, if ΔPtp/ΔPg derived from separation flow theory, the
combination of homogeneous and separation flow theory is implemented.Separated flow theory takes
into account the fact that the two phases can have differing properties and
different velocities. Separate equations of continuity, momentum, and energy
are written for each phase, and these six equations are solved simultaneously,
together with rate equations which describe how the phases interact with each
other and with the walls of duct. In the simplest version, only one parameter,
such as velocity, is allowed to differ for the two phases while conservation
equations are only written for the combined flow.Equation (2.9) shows the momentum
function of one dimension two-phase flow based on separated flow assumption.
The pressure drop of fluids in the pipe come from three parts, the first is
friction; the second is gravitation; the third is acceleration [17–21]:(2.9)−dPdz=τ0UA+[ρgα+ρl(1−α)]g⁢sinθ+1Addz{AG2[(1−x)2ρl(1−α)+x2ρgα]}(2.10)−dPdz=dPfdz+dPgdz+dPadz, where τ0 is friction force, U is perimeter of pipe,  α is void fraction,  G is mass velocity of mixture, dPf/dz is pressure drop caused by friction, dPg/dz is pressure drop caused by gravitation,  dPa/dz is pressure drop caused by acceleration.


## 2.2. The Friction Pressure Ddrop of Venturi Under Two-Phase
Flow Condition

For single-phase flow in straight pipe, the friction pressure drop can be calculated with(2.11)dPfdz=λd⁢⋅ρu22, where λ is the friction factor; d is the pipe diameter, u is the velocity.Givenλ is constant in conical convergent of Venturi,
the fluid velocity in the straight pipe keep unchanged, d0 is diameter of straight pipe, A0 is cross-section of straight pipe, d1 is diameter of Venturi throat, l0 is the length of conical convergent, θ is convergent angle. The schematic of Venturi conical
convergent part is shown in Figure 1.Figure 1
Sketch of Venturi conical convergent region.Analyzing an infinitesimaldl given d is diameter of the analyzing part, A is cross-section, l is the distance from Venturi inlet to
infinitesimal dl make integral  to (2.11):(2.12)ΔPf=∫0l0λd⁢⋅ρu22⁢dz,(2.13)ΔPf=λρ2∫0l01d⁢⋅u2⁢dz.Multiplyd0 to (2.13) in two sides:(2.14)ΔPf=λρ2d0∫0l0d0d⁢⋅u2⁢dz.From continuity equation,(2.15)u=A0Au0,A0A=(d0d)2.Substitute (2.15) into (2.14):(2.16)ΔPf=λρu022d0∫0l0(d0d)5dz.According to geometrical relationship showed
in Figure1,(2.17)dz=dlcosθ,(2.18)ll0=d0−dd0−d1,(2.19)⟹d0d=l0l0−l(1−d1/d0).Letβ be diameter ratio of Venturi, then(2.20)β=d1d0.Substitute (2.17), (2.19), and (2.20) into (2.16):(2.21)ΔPf=λρu02l052d0⁢cosθ∫0l0(1l0−l(1−β))5⁢dl,(2.22)ΔPf=λρu02l058d0⁢cos⁢θ(1−β)(1l0−l(1−β))4|0l0,(2.23a)ΔPf=(1+β)⋅(1+β2)β4⁢⋅14⁢cosθ⁢⋅λd0⁢⋅ρu022l0. Equation
(2.23a) shows that the friction pressure drop is affected by diameter ratio,
convergent angle, convergent length, inlet diameter, and inlet velocity.In a constant section pipe withl0 length, the friction pressure drop is(2.23b)ΔPfl0=λl0d0⁢⋅ρu022.Equation (2.23a) that is divided by (2.23b)
is(2.24)Kf=(1+β)⋅(1+β2)β4⁢⋅14⁢cosθ. Equation
(2.24) shows that the ratio Kf is a function of
diameter ratio and convergent angle. For a definite Venturi, Kf is constant.As for gas liquid two-phase flow, (2.23a) and (2.23b)
changes into (2.25)ΔPf=Kf⁢⋅λl0d0⁢⋅αρgug2+(1−α)ρlul22.When the pipe is full of gas (α = 1) or liquid (α = 0), (2.25) changes to (2.23a).From gas liquid two-phase flow continuity
equation,(2.26)xGA=Agugρg,(1−x)GA=Alulρl.Consider the definition of void fraction,(2.27)xαG=ugρg,(1−x)(1−α)G=ulρl,(2.28)G=mA=αρgug+(1−α)ρlul which defined S as slip ratio, that is, gas and liquid real velocity ratio combine (2.26) and
(2.27):(2.29)1α=1+s1−xx⁢⋅ρgρl.Substitute (2.26) and (2.27) into (2.25):(2.30)ΔPf=Kf⁢⋅λl0d0⁢⋅G22⁢⋅1ρl[x2α⁢⋅ρlρg+(1−x)21−α].When the pipe is full of  gas,(2.31)ΔPfg=Kf⁢⋅λgl0d0⁢⋅G22⁢⋅x2ρg.When the pipe is full of liquid,(2.32)ΔPfl=Kf⁢⋅λll0d0⁢⋅G22⁢⋅(1−x)2ρl.Letλ=λg=λl,
define Xf as(2.33)Xf=ΔPflΔPfg=(1−xx)ρgρl.Equation (2.30) divided by (2.31) is(2.34)ΔPfΔPfg=1α+(1−x)2x2ρgρl⁢⋅11−α.Substitute (2.29) into (2.34):(2.35)ΔPfΔPfg=1+CfXf+Xf2, where(2.36)Cf=1sρlρg+sρgρl.


## 2.3. The Aacceleration Pressure Drop of Venturi Under
Two-Phase Flow Condition

According to (2.9), the acceleration pressure
drop is(2.37)dPadz=1Addz{AG2[(1−x)2ρl(1−α)+x2ρgα]},(2.38)ΔPa=∫dPa=∫ATA01A⁢d{AG2[(1−x)2ρl(1−α)+x2ρgα]}.Given the fluid is incompressible, the void fractionα is constant in the Venturi throat. Integrate (2.38):(2.39)ΔPa=G2[(1−x)2ρl(1−α)+x2ρgα]⁢⋅lnA0AT.When the gas was flowing alone in the pipe, the
pressure drop can be expressed as(2.40)ΔPag=G2x2ρg⁢⋅lnA0AT.The similar equation for the liquid phase is(2.41)ΔPal=G2(1−x)2ρl⁢⋅lnA0AT.DefineXa:(2.42)Xa=ΔPalΔPag=(1−x)x⁢⋅ρgρl.Equation (2.39) divided by (2.40) is(2.43)ΔPaΔPag=(1−x)2x2⁢⋅ρgρl⁢⋅11−α+1α.Substitute (2.29) and (2.42) into (2.34):(2.44)ΔPaΔPag=1+Ca⁢⋅Xa+Xa2, where Ca is expressed as(2.45)Ca=1s⁢⋅ρlρg+s⁢⋅ρgρl.Compare (2.33) with (2.42), it is obvious that Xf is the same as Xa.Also, compared (2.33) with (2.42), Cf is equal to Ca.And then, (2.44) is equal to(2.46)ΔPfΔPfg=ΔPaΔPag=1+Cg⁢⋅X+X2, where(2.47)Cg=Ca=Cf=1s⁢⋅ρlρg+s⁢⋅ρgρl,(2.48)X=Xa=Xf=(1−x)x⁢⋅ρgρl.Equation (2.46) notes that the ratio of two-phase
and single-phase friction pressure drop is equal to the ratio of two- phase and
single-phase acceleration pressure drop.


## 2.4. The Total Pressure Ddrop of Venturi Under Two-Phase
Flow Condition

For a horizontal mounted Venturi,
gravitation pressure drop can be ignored. The total pressure drop is(2.49)ΔPtp=ΔPf+ΔPa.The total pressure drop of Venturi under
single-phase flow condition is(2.50)ΔPg=ΔPfg+ΔPag.Divide (2.49) by (2.50):(2.51)ΔPtpΔPg=ΔPf+ΔPaΔPfg+ΔPag.According to (2.46) and geometric axiom,(2.52)ΔPfΔPfg=ΔPaΔPag=ΔPf+ΔPaΔPfg+ΔPag.Combine (2.51) and (2.52):(2.53)ΔPtpΔPg=ΔPfΔPfg=ΔPaΔPag=1+Cg⁢⋅X+X2.So the model combined homogeneous and
separation flow theory can be expressed as (2.55). Call this correlation as H-S model:(2.54)ORH-S=x⁢⋅ρtpρg⁢⋅ΔPtpΔPg=x⁢⋅ρlρlx+ρg(1−x)⁢⋅1+Cg⋅X+X2(2.55)=x⁢⋅1+Cg⁢⋅X+X21+ρg/ρl⁢X.Equation (2.55) shows that Cg is an effect
factor to OR, it must be known first when (2.55) is used. However, slip ratio S is
contained in Cg equation, and
slip ratio is hard to be determined accurately, so it needs to fit a
correlation with experiment.


## 3. Dry Gas Calibration and Wet Gas Tests

### 3.1. Dry gas calibration

Three
venture meters are calibrated in TJU critical sonic nozzle flow calibration
facility; see Figure2.Figure 2
Schematic
diagram of TJU  critical sonic nozzle flow calibration facility.The facility has eleven sonic nozzles of different
discharge coefficient, and the calibration range varies from 2.50 to 660 m3/h with a step of 2 m3/h. The maximum calibrated flow rate is about 380 m3/h due to the beta ratio and pipe diameter. At the
same time, the TJU multiphase flow loop also has the
calibration function. So the dry gas calibration for three Venturis was done in
both. The test data from the two facilities show the same results. Figure 3
shows the calibration coefficient C with different diameter ratio. When the
Reynolds number is higher than 1 × 105, the value of coefficient is in
accord with the standard discharge coefficient for flows with Reynolds
numbers less than  one  million [22].Figure 3
Discharge coefficient of Venturi tube in single phase flow.Fit  the coefficientC in different diameter ratio, the parameters listed in table 3:Table 3
Parameters value.
0.40480.550.7P117.88251−0.588676.06974P2−0.000740.00006−0.00016P31.283E−8−9.7022E−101.88E−9P4−1.0944E−138.4168E−15−1.0733E−14P54.5672E−19−3.5094E−202.9549E−20P6−7.4476E−255.5411E−26−3.1419E−26(3.1)C=P1+P2⁢⋅Re+P3⁢⋅Re⁡2+P4⁢⋅Re⁡3+P5⁢⋅Re⁡4+P6⁢⋅Re⁡5.


### 3.2. Test System and Experimental
Procedures

The tests were  conducted on TJU multiphase flow loop at pressures from
0.15 MPa to 0.25 MPa across a range of gas velocities and liquid fractions.
TJU’s low-pressure wet gas test facilities are a fully automatic control and
functional complete system, which is not only a multiphase flow experiment
system, but also a multiphase flow meter calibration system. As an experiment
system, the test can be conducted in a horizontal pipe, vertical pipe and 0–90°lean
pipe; as a calibration system, the test meter can be calibrated in standard
meter method. Figure 4 shows schematic diagram of
TJU multiphase flow loop.Figure 4
Schematic
diagram of TJU multiphase flow loop.Thess facilities have six components, named as medium source, measurement
pipe, horizontal pipe, vertical pipe, 0–90°lean pipe
and computer control system.Gas medium is compression air, and two compressors provide dynamic force,
the compressor air is passing through cooling and drying unit which access to
two 12 m3 accumulator tanks; the accumulator tanks and pressure
maintaining valve can hold a stable pressure 0–0.8 MPa for
the test. The liquids used in test are water (oil or oil and water mixture also
can be used) and a water pump pushes the water to a 30-meter-high water tower,
which can hold a stable pressure for liquid.In standard meter calibration system, gas calibration system has five
paths; three of them are low-flow channels metering with three mass flow
controllers made in America by Alicat scientific company, Ariz, USA, the lowest
flux is 10 l/min, the other two paths
are middle and large flow channels metering with a Roots type flow meter and a
vortex flow meter. All temperature and pressure measurements use traceable
calibrated instrumentation for gas temperature and pressure compensation.Liquid calibration system has six paths: four of them are low-flow
channels metering with an electrical flow meter made in Germany
combined by four magnet
valves, the lowest flux is 0.01 m3/h,
the other two paths are middle and large flow channels metering with a
electrical flow meter and a vortex flow meter. See parameters of the standard
meter in Table 4.Table 4
Parameters of the standard sensor.
PhasesRange (m3/h)AccuracyWater0.01 ~ 3.0±0.2%0.75 ~ 19±1.0%1.7 ~ 43±0.5%Air0 ~ 6.0±0.8%0.15 ~ 17±3.0%6.5 ~ 130±1.5%Oil0.02 ~ 2.51.0%0.75 ~ 191.0%Gas and liquid calibrate through standard meter access to mixer, and then
go through the experimental pipe. There are two paths in experimental pipes,
one is made in rustless steel, the other is made in organic glass, their diameter
is 50 mm, and a cutoff valve which can adjust the pressure is installed at
outlet of the pipe.Figure5 shows horizontal experiment pipe, which includes mixer,
temperature sensor, straight lengths, pressure sensor, and Venturi tube.Figure 5
Horizontal
experiment pipe.According to ISO 5167-1, 4:2003 [23, 24], a classical Venturi tube
with a machined convergent section, straight lengths and diameter ratio must
accord with Table 5.Table 5
Required straight lengths for classical Venturi tubes with a
machined convergent section.
Diameter ratioStraight length (D)0.4080.5080.60100.70100.7518In this test, three Venturi tubes withβ values of 0.4048, 0.55, and
0.70 have been produced, the length of Venturi tubes is 388 mm, diameter is 50 mm,
the length of cylindrical throat is 20 mm, conical convergent angle is 21°,
conical divergent angle is 15°, diameter of pressure tappings is 4 mm, the pipe wall
roughness is 0.06 mm, and stainless
steel flange is used in connecting. 1151 differential pressure transducers were
made in Rosemont company, Colo,
USA, the
uncertainty of whole equipment is 2.5‰.The test data are collected and saved as Microsoft Excel file
automatically (see
experimental parameters in Table6).Table 6
Experimental parameters.
βP (MPa)FrgX0.40480.150.8 ~ 1.50.0022 ~ 0.03380.201.0 ~ 1.880.0022 ~ 0.04720.250.67 ~ 1.810.0022 ~ 0.04950.550.151.04 ~ 1.780.0022 ~ 0.25720.201.09 ~ 1.850.0022 ~ 0.34310.250.92 ~ 1.730.0022 ~ 0.35140.70.151.04 ~ 2.00.0024 ~ 0.04800.201.08 ~ 2.00.0025 ~ 0.05250.250.87 ~ 1.660.0027 ~ 0.0576The flow pattern of the test included annular and drop-annular, where Frg is gas Froude number:(3.2)Frg=vggDρgρl−ρg.vg is superficial velocity of the gas phase: vg=mg/(ρgA).


## 3.1. Dry gas calibration

Three
venture meters are calibrated in TJU critical sonic nozzle flow calibration
facility; see Figure2.Figure 2
Schematic
diagram of TJU  critical sonic nozzle flow calibration facility.The facility has eleven sonic nozzles of different
discharge coefficient, and the calibration range varies from 2.50 to 660 m3/h with a step of 2 m3/h. The maximum calibrated flow rate is about 380 m3/h due to the beta ratio and pipe diameter. At the
same time, the TJU multiphase flow loop also has the
calibration function. So the dry gas calibration for three Venturis was done in
both. The test data from the two facilities show the same results. Figure 3
shows the calibration coefficient C with different diameter ratio. When the
Reynolds number is higher than 1 × 105, the value of coefficient is in
accord with the standard discharge coefficient for flows with Reynolds
numbers less than  one  million [22].Figure 3
Discharge coefficient of Venturi tube in single phase flow.Fit  the coefficientC in different diameter ratio, the parameters listed in table 3:Table 3
Parameters value.
0.40480.550.7P117.88251−0.588676.06974P2−0.000740.00006−0.00016P31.283E−8−9.7022E−101.88E−9P4−1.0944E−138.4168E−15−1.0733E−14P54.5672E−19−3.5094E−202.9549E−20P6−7.4476E−255.5411E−26−3.1419E−26(3.1)C=P1+P2⁢⋅Re+P3⁢⋅Re⁡2+P4⁢⋅Re⁡3+P5⁢⋅Re⁡4+P6⁢⋅Re⁡5.


## 3.2. Test System and Experimental
Procedures

The tests were  conducted on TJU multiphase flow loop at pressures from
0.15 MPa to 0.25 MPa across a range of gas velocities and liquid fractions.
TJU’s low-pressure wet gas test facilities are a fully automatic control and
functional complete system, which is not only a multiphase flow experiment
system, but also a multiphase flow meter calibration system. As an experiment
system, the test can be conducted in a horizontal pipe, vertical pipe and 0–90°lean
pipe; as a calibration system, the test meter can be calibrated in standard
meter method. Figure 4 shows schematic diagram of
TJU multiphase flow loop.Figure 4
Schematic
diagram of TJU multiphase flow loop.Thess facilities have six components, named as medium source, measurement
pipe, horizontal pipe, vertical pipe, 0–90°lean pipe
and computer control system.Gas medium is compression air, and two compressors provide dynamic force,
the compressor air is passing through cooling and drying unit which access to
two 12 m3 accumulator tanks; the accumulator tanks and pressure
maintaining valve can hold a stable pressure 0–0.8 MPa for
the test. The liquids used in test are water (oil or oil and water mixture also
can be used) and a water pump pushes the water to a 30-meter-high water tower,
which can hold a stable pressure for liquid.In standard meter calibration system, gas calibration system has five
paths; three of them are low-flow channels metering with three mass flow
controllers made in America by Alicat scientific company, Ariz, USA, the lowest
flux is 10 l/min, the other two paths
are middle and large flow channels metering with a Roots type flow meter and a
vortex flow meter. All temperature and pressure measurements use traceable
calibrated instrumentation for gas temperature and pressure compensation.Liquid calibration system has six paths: four of them are low-flow
channels metering with an electrical flow meter made in Germany
combined by four magnet
valves, the lowest flux is 0.01 m3/h,
the other two paths are middle and large flow channels metering with a
electrical flow meter and a vortex flow meter. See parameters of the standard
meter in Table 4.Table 4
Parameters of the standard sensor.
PhasesRange (m3/h)AccuracyWater0.01 ~ 3.0±0.2%0.75 ~ 19±1.0%1.7 ~ 43±0.5%Air0 ~ 6.0±0.8%0.15 ~ 17±3.0%6.5 ~ 130±1.5%Oil0.02 ~ 2.51.0%0.75 ~ 191.0%Gas and liquid calibrate through standard meter access to mixer, and then
go through the experimental pipe. There are two paths in experimental pipes,
one is made in rustless steel, the other is made in organic glass, their diameter
is 50 mm, and a cutoff valve which can adjust the pressure is installed at
outlet of the pipe.Figure5 shows horizontal experiment pipe, which includes mixer,
temperature sensor, straight lengths, pressure sensor, and Venturi tube.Figure 5
Horizontal
experiment pipe.According to ISO 5167-1, 4:2003 [23, 24], a classical Venturi tube
with a machined convergent section, straight lengths and diameter ratio must
accord with Table 5.Table 5
Required straight lengths for classical Venturi tubes with a
machined convergent section.
Diameter ratioStraight length (D)0.4080.5080.60100.70100.7518In this test, three Venturi tubes withβ values of 0.4048, 0.55, and
0.70 have been produced, the length of Venturi tubes is 388 mm, diameter is 50 mm,
the length of cylindrical throat is 20 mm, conical convergent angle is 21°,
conical divergent angle is 15°, diameter of pressure tappings is 4 mm, the pipe wall
roughness is 0.06 mm, and stainless
steel flange is used in connecting. 1151 differential pressure transducers were
made in Rosemont company, Colo,
USA, the
uncertainty of whole equipment is 2.5‰.The test data are collected and saved as Microsoft Excel file
automatically (see
experimental parameters in Table6).Table 6
Experimental parameters.
βP (MPa)FrgX0.40480.150.8 ~ 1.50.0022 ~ 0.03380.201.0 ~ 1.880.0022 ~ 0.04720.250.67 ~ 1.810.0022 ~ 0.04950.550.151.04 ~ 1.780.0022 ~ 0.25720.201.09 ~ 1.850.0022 ~ 0.34310.250.92 ~ 1.730.0022 ~ 0.35140.70.151.04 ~ 2.00.0024 ~ 0.04800.201.08 ~ 2.00.0025 ~ 0.05250.250.87 ~ 1.660.0027 ~ 0.0576The flow pattern of the test included annular and drop-annular, where Frg is gas Froude number:(3.2)Frg=vggDρgρl−ρg.vg is superficial velocity of the gas phase: vg=mg/(ρgA).


## 4. Model Parameters Determining and Error Analyzing

The coefficientCg can be calculated by experimental data. On TJU
multiphase flow system, the real gas, liquid mass flow rateand gas, liquid density can be determined by
standard sensor. The gas mass fraction is known parameter. The Lockhart-Martinelli parameter can be obtained by (2.33). The
over reading can be calculated with (2.1) and (2.3). Therefore, the coefficient Cg can be calculated by (2.55) (H-S model). The study shows that coefficient Cg decreases with increasing Lockhart-Martinelli parameter X, decreases with increasing pressure P, decreases with increasing diameter
ratio β, increases with increasing Gas Froude number Frg,and increases with increasing gas liquid quality ratio x/(1 − x).Equation (2.47) can be expressed as(4.1)Cg=f(s,ρgρl).Equation (4.1) shows that
the gas-liquid quality ratio x/(1 − x)
contains the same parameter with coefficient Cg:(4.2)mgml=x1−x=S⁢⋅ρgρl⁢⋅α1−α.Combining
(3.2) and (4.1)
can gain(4.3)Cg=f(x1−x,α1−α,ρgρl).De Leeuw model considers the coefficientCg as a function of gas-liquid density ratio and
gas Froude number:(4.4)CDe Leeuw=(ρlρg)n+(ρgρl)n,n={0.410.5≤Frg≤1.5,0.606(1−e−0.746⁢Frg)Frg≥1.5.However,
inherited the form of the coefficientCg of De Leeuw model’s, and using gas liquid
density ratio as base of exponential function, the exponent n is a
severe nonlinear curve with other parameters such as Lockhart-Martinelli
parameter X, or gas Froude number (see Figure 6).Figure 6
Lockhart-Martinelli parameterX   effect on n of De Leeuw model.Research found that using gas
liquid volume ratio (gas liquid mass ratio
divided by gas liquid density ratio) as a base of exponential functionCg in H-S model,
the exponent n almost linear increases with increasing
Lockhart-Martinelli parameter X, it can be seen as Figure 7, so defined the coefficient Cg of the H-S model asFigure 7
Lockhart-Martinelli parameterX effect to n.(4.5)CH-S=(x/(1−x)ρg/ρl)n+(ρg/ρlx/(1−x))n.In fact, gas  liquid  mass ratio divided by gas liquid
density ratio is equal to gas liquid volume ratio:(4.6)CH-S=(φ1−φ)n+(1−φφ)n,n=f(β,P(ORρgρl),Frg,X,…), where φ is gas volume fraction.Next, a correlation of
exponentn with other parameters  will be approached.


### 4.1. Effect of Parameters to Exponentn of H-S Model

Figure7 shows the effect of Lockhart-Martinelli parameter X to n, and exponent n almost linearly increases
with increasing Lockhart-Martinelli parameter X. Figure 8 shows the effect of pressure to n, apparently,
exponent n decreases with the increasing pressure. Figure 9 shows the effect of gas Froude number to n, seemingly, n increases
with the increasing of gas Froude number. Figure
10 shows  the  effect of diameter ratio to n, and n decreases with the increasing diameter ratio.Figure 8
PressureP effect on coefficient  n   under same gas  Froude number.Figure 9
Gas
Froude number  Frg   effect on coefficient n under same pressure.Figure 10
Diameter ratio  effect on
coefficientn
under
6 MPa.


### 4.2. Fitting Exponentn of H-S Model

According
to the results of these figures,n varied
linearly with Lockhart-Martinelli parameter X, and with the rate of curves effect by diameter ratio,
pressure, and Gas Froude number.
So the experiment correlation of coefficient n should take the Lockhart-Martinelli
parameter X as
a key independent variable, and pressure P (or gas liquid density ratio),
diameter ratio β, Gas Froude number Frg as
auxiliary variable. Exponent n can be defined as(4.7)n=A+B⁢⋅Xk, where(4.8)A=a1⁢⋅(β)a2⋅(Frg)a3⋅(ρgρl)a4,B=a5⁢⋅(β)a6⋅(Frg)a7⋅(ρgρl)a8, where K is constant, a1, a2, a3, a4, a5, a6, a7, a8 are undetermined coefficient,
which will be determined through experimental data. The fit coefficient showed
in Table 7.Table 7
Fit exponentn with all data.
a1a2a3a4a5a6a7a8K1.29203−0.171610.12618−0.018840.301960.05205−0.071220.02590.78105Table7 is the  coefficient n fit by independent data
from TJU low-pressure wet gas loop and National Engineering Laboratory high-pressure wet gas loop. Using exponent n and
coefficient Cg for H-S over-reading
model, over 98% data set will express the prediction error within ±5%, and the maximum error within ±6.5%. See Figures 11 and 12.Figure 11
Comparison of H-S prediction OR and
experimental OR.Figure 12
The prediction error of H-S model.


### 4.3. Comparison of H-S OR
Model and the Eeight Previous OR Models

Compare new
model to 8 old models with the condition of pressureP varied from 0.15 to
6.0 MPa, beta ratio varied from 0.4 to 0.75, gas
densimetric Froude number Frg varied from
0.5 to 5.5, the modified Lockhart-Martinelli
parameter X varied from
0.002 to 0.3, the ratio of the gas to total mass flow rate x varied from 0.5 to 0.99. The data used for comparison is
independent data different from training data. A Part of independent data was obtained from NEL’s report. Figures 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, and 24 are a part of
the compared results.Figure 13
Comparison of models
underβ = 0.4, P = 1.5 MPa, and Frg = 2.Figure 14
Comparison errors of models underβ = 0.4, P = 1.5 MPa, and Frg = 2.Figure 15
Comparison of models underβ = 0.4048, P = 0.20 MPa, and
Frg = 1.5.Figure 16
Comparison errors of models underβ = 0.4048, P = 0.20 MPa, and
Frg = 1.5.Figure 17
Comparison of models underβ = 0.55, P = 0.15 MPa, and
Frg = 2.Figure 18
Comparison errors of models underβ = 0.55, P = 0.15 MPa, and
Frg = 2.Figure 19
Comparison of models underβ = 0.60, P = 3 MPa, and Frg = 1.5.Figure 20
Comparison errors of models underβ = 0.60, P = 3 MPa, and Frg = 1.5.Figure 21
Comparison of models underβ = 0.70, P = 0.20 MPa, and
Frg = 1.7.Figure 22
Comparison errors of models underβ =0.70, P = 0.20 MPa, and
Frg = 1.7.Figure 23
Comparison of models underβ = 0.75, P = 6 MPa, and Frg = 3.5.Figure 24
Comparison errors of models underβ = 0.75, P = 6 MPa, and Frg = 3.5.These figures
show that H-S model can accurately predict Venturi OR in all kinds of flow
conditions, the error of H-S wet gas model is stable with OR increasing, and
within 5%. It again proved that new wet gas model has good adaptability and
wide application range. Particularly, as the wet gas flow fluctuate intensively
under low pressure, all old OR models cannot predict OR accurately, the absolute
of maximum error almost reached 40%. However, the new wet gas model reflects
this change perfectly, the prediction OR has the same distribution with real
OR. This is mainly because the homogenous model can well reflect the
fluctuation of real OR, and the H-S model has inherited this ability. NEL’ data
have evidence trends because it is obtained in middle and high pressure. Even
though, old correlations predicted errors are also large than H-S correlation,
they varied from 10% to −35%.


## 4.1. Effect of Parameters to Exponentn of H-S Model

Figure7 shows the effect of Lockhart-Martinelli parameter X to n, and exponent n almost linearly increases
with increasing Lockhart-Martinelli parameter X. Figure 8 shows the effect of pressure to n, apparently,
exponent n decreases with the increasing pressure. Figure 9 shows the effect of gas Froude number to n, seemingly, n increases
with the increasing of gas Froude number. Figure
10 shows  the  effect of diameter ratio to n, and n decreases with the increasing diameter ratio.Figure 8
PressureP effect on coefficient  n   under same gas  Froude number.Figure 9
Gas
Froude number  Frg   effect on coefficient n under same pressure.Figure 10
Diameter ratio  effect on
coefficientn
under
6 MPa.


## 4.2. Fitting Exponentn of H-S Model

According
to the results of these figures,n varied
linearly with Lockhart-Martinelli parameter X, and with the rate of curves effect by diameter ratio,
pressure, and Gas Froude number.
So the experiment correlation of coefficient n should take the Lockhart-Martinelli
parameter X as
a key independent variable, and pressure P (or gas liquid density ratio),
diameter ratio β, Gas Froude number Frg as
auxiliary variable. Exponent n can be defined as(4.7)n=A+B⁢⋅Xk, where(4.8)A=a1⁢⋅(β)a2⋅(Frg)a3⋅(ρgρl)a4,B=a5⁢⋅(β)a6⋅(Frg)a7⋅(ρgρl)a8, where K is constant, a1, a2, a3, a4, a5, a6, a7, a8 are undetermined coefficient,
which will be determined through experimental data. The fit coefficient showed
in Table 7.Table 7
Fit exponentn with all data.
a1a2a3a4a5a6a7a8K1.29203−0.171610.12618−0.018840.301960.05205−0.071220.02590.78105Table7 is the  coefficient n fit by independent data
from TJU low-pressure wet gas loop and National Engineering Laboratory high-pressure wet gas loop. Using exponent n and
coefficient Cg for H-S over-reading
model, over 98% data set will express the prediction error within ±5%, and the maximum error within ±6.5%. See Figures 11 and 12.Figure 11
Comparison of H-S prediction OR and
experimental OR.Figure 12
The prediction error of H-S model.


## 4.3. Comparison of H-S OR
Model and the Eeight Previous OR Models

Compare new
model to 8 old models with the condition of pressureP varied from 0.15 to
6.0 MPa, beta ratio varied from 0.4 to 0.75, gas
densimetric Froude number Frg varied from
0.5 to 5.5, the modified Lockhart-Martinelli
parameter X varied from
0.002 to 0.3, the ratio of the gas to total mass flow rate x varied from 0.5 to 0.99. The data used for comparison is
independent data different from training data. A Part of independent data was obtained from NEL’s report. Figures 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, and 24 are a part of
the compared results.Figure 13
Comparison of models
underβ = 0.4, P = 1.5 MPa, and Frg = 2.Figure 14
Comparison errors of models underβ = 0.4, P = 1.5 MPa, and Frg = 2.Figure 15
Comparison of models underβ = 0.4048, P = 0.20 MPa, and
Frg = 1.5.Figure 16
Comparison errors of models underβ = 0.4048, P = 0.20 MPa, and
Frg = 1.5.Figure 17
Comparison of models underβ = 0.55, P = 0.15 MPa, and
Frg = 2.Figure 18
Comparison errors of models underβ = 0.55, P = 0.15 MPa, and
Frg = 2.Figure 19
Comparison of models underβ = 0.60, P = 3 MPa, and Frg = 1.5.Figure 20
Comparison errors of models underβ = 0.60, P = 3 MPa, and Frg = 1.5.Figure 21
Comparison of models underβ = 0.70, P = 0.20 MPa, and
Frg = 1.7.Figure 22
Comparison errors of models underβ =0.70, P = 0.20 MPa, and
Frg = 1.7.Figure 23
Comparison of models underβ = 0.75, P = 6 MPa, and Frg = 3.5.Figure 24
Comparison errors of models underβ = 0.75, P = 6 MPa, and Frg = 3.5.These figures
show that H-S model can accurately predict Venturi OR in all kinds of flow
conditions, the error of H-S wet gas model is stable with OR increasing, and
within 5%. It again proved that new wet gas model has good adaptability and
wide application range. Particularly, as the wet gas flow fluctuate intensively
under low pressure, all old OR models cannot predict OR accurately, the absolute
of maximum error almost reached 40%. However, the new wet gas model reflects
this change perfectly, the prediction OR has the same distribution with real
OR. This is mainly because the homogenous model can well reflect the
fluctuation of real OR, and the H-S model has inherited this ability. NEL’ data
have evidence trends because it is obtained in middle and high pressure. Even
though, old correlations predicted errors are also large than H-S correlation,
they varied from 10% to −35%.


## 5. Conclusions

Separation and
homogeneous assumptions reflect the wet gas flow character, so a correlation
combining these two assumptions performed well than each single one. The
H-S model has inherited merits of homogeneous correlation and separation
correlation, and can predict Venturi over
reading accurately with the conditions of pressure varied from 0.15 to 6 MPa, beta ratio varied from 0.4 to 0.75, gas
densimetric Froude number varied from 1 to 5.5, the modified Lockhart-Martinelli
parameter varied from 0.002 to 0.3, the ratio of the gas to total mass flow rate
varied from 0.5 to 0.99. The prediction error of H-S model is within±6.5%.



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*Source: 807287-2008-10-20.xml*