- Email: [email protected]
A dimensional analysis of two phase flow through a horizontally installed Venturi flow meter
Flow Measurement and Instrumentation 19 (2008) 342–349
Contents lists available at ScienceDirect
Flow Measurement and Instrumentation journal homepage: www.elsevier.com/locate/flowmeasinst
A dimensional analysis of two phase flow through a horizontally installed Venturi flow meter Richard Steven ∗ Multiphase and Wet Gas Flow Research Facility, Colorado Engineering Experiment Station, Inc., 54043 WCR 37-Nunn, CO 80648, USA
article
info
Article history: Received 4 May 2007 Received in revised form 14 March 2008 Accepted 12 May 2008
a b s t r a c t A dimensional analysis on two phase flow through a horizontally mounted Venturi meter is performed. The derived dimensional groups are compared and related to those used in existing DP meter two phase flow models and correlations. The results give confidence in the existing dimensionless numbers in industrial use and lead to comments on the state of this art and possible future research trends. © 2008 Elsevier Ltd. All rights reserved.
Keywords: Two phase flow Dimensional analysis Venturi Differential pressure meter
1. Introduction Industry has a requirement to meter two-phase (i.e. combined gas and liquid) flows. Part of this requirement is a need to meter wet gas flows, that is, two-phase flows where the liquid flowrate is relatively small compared to the gas flowrate. Two-phase flow test results from various flow meters over many years has led to the Differential Pressure (DP) flow meter being one meter of choice for such applications. One of the most common DP meters on the market is the Venturi flow meter. The Venturi is typically used to meter wet gas/two-phase flows in three general ways: (1) The Venturi is used as a stand alone device along with other independent systems. For example, an independent system predicts the two-phase flows liquid component flowrate (e.g. test separator historical records or a tracer injection technology) and the Venturi meter reading with the liquid flowrate prediction are inputs to some Venturi two-phase flow correlation that gives the gas flowrate as an output. (2) The Venturi is used in series with another gas flow meter that has a different liquid induced gas flowrate error than the Venturi. The difference in the two meters erroneous gas flowrate predictions allow a measurement by difference technique to be used to predict the actual gas and liquid flowrates. (3) The Venturi has phase fraction devices (e.g. microwave systems, dual energy densitometers or capacitance systems etc.) imbedded
∗
Tel.: +1 970 897 2711; fax: +1 970 897 2710. E-mail address: [email protected].
0955-5986/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.flowmeasinst.2008.05.004
in some part of the meter body or neighboring spool pieces and the Venturi reading in conjunction with the phase fraction device readings are used in a wet gas/two-phase flow mathematical model to predict the gas and liquid flowrates. In all three cases understanding the response of the Venturi meters primary reading, that is the differential pressure read, to the wet gas/two-phase flow is of importance to the calculation procedure. This response is usually described by a wet gas/twophase Venturi meter correlation (or flow model). The Venturi meter is just one of several types of DP meter designs investigated for two-phase flow metering over the last fifty years. The DP meter wet gas/two-phase correlations that are in the public domain utilize dimensionless numbers that appear to have been chosen independently by various researchers developing two-phase flow models. It is therefore an interesting exercise to perform a full dimensional analysis on two-phase flow through a generic Venturi meter using the Buckingham Pi theorem. This exercise shows a list of dimensionless groups that could have an influence on a Venturi meters wet gas/two-phase flow response. It will be shown that these dimensionless groups can be re-arranged to give a set of dimensionless numbers that include all the dimensionless numbers currently used by two-phase flow metering researchers. Furthermore, this set of dimensionless numbers can be split into meter geometry dependant and independent sets. This exercise could therefore be repeated for any DP meter geometry and the same dimensionless numbers independent of geometry would be derived. This exercise is important as it fundamentally proves the validity of using several dimensionless numbers that have been in common use for many years.
R. Steven / Flow Measurement and Instrumentation 19 (2008) 342–349
343
Therefore equating the exponents gives: m : a + b + 1 = 0,
l : −3a + c − 1 = 0,
t : −b − 2 = 0.
Therefore, a = 1, b = −2, c = 4 and we have:
Π1 =
n
ρg
1
˙g m
−2
(D)4
ρg D4 ∆Ptp ∆Ptp = 2 . ˙g m
o
However, we know from mass continuity that:
˙ g = ρg Ag Ug = ρg AUsg = ρg m Fig. 1. Sketch of generic Venturi meter.
2. The dimensional analysis procedure set up The following is a dimensional analysis for general two-phase flow (with one gas component and one liquid component) through a Venturi meter. As with all dimensional analysis investigations the first step is to list all possible parameters that may affect the system (whether they in reality do or do not) and state their dimensions (i.e. the relationship of mass ‘‘m’’, length ‘‘l’’ and time ‘‘t’’ that they represent). A sketch of a generic Venturi meter is drawn in Fig. 1. The list that will be analyzed is:
∆Ptp (i.e. the differential pressure from upstream to throat) ˙g &m ˙ l (i.e. the gas and liquid mass flow rates m respectively) ρg (i.e. the gas density) ∆ρ (i.e. ρl − ρg , the difference between phase densities) µg & µl (i.e. the gas and liquid viscosities respectively) σl (i.e. the liquid interfacial tension) d (i.e. the Venturi inside bore throat diameter) D (i.e. the Venturi inlet inside bore diameter) Lc (i.e. the length of the converging section) Lt (i.e. the length of the throat) Ld (i.e. the length of the diffuser section) e (i.e. the roughness of the wetted surface) g (i.e. the gravitational constant) P (i.e. the system pressure)
m/lt
2
(m/t ) m/l3 m/l3 (m/lt ) m/t 2 (l) (l) (l) (l) (l) (l) l/t 2 m/lt 2
Note: The combination of the gas density (ρg ) and the density difference between the phases (∆ρ = ρl − ρg ) will be used here instead of liquid density (ρl ) as this combination contains the liquid density information and the liquid density therefore becomes a superfluous parameter.
3. The dimensional analysis procedure There are sixteen parameters listed, i.e. ‘‘m’’ = 16, that must be investigated. There are three primary parameters, i.e. length (l), mass (m) and time (t), i.e. ‘‘n’’ = 3. Number of independent groups = m − n = 16 − 3 = 13. That is, we have: f (Π1 , Π2 , Π3 , Π4 , Π5 , Π6 , Π7 , Π8 , Π9 , Π10 , Π11 , Π12 , Π13 ) = 0
where f1 is some function different to function f . ˙ g & D the Taking an arbitrary choice of 3 parameters: ρg , m thirteen groups are found in turn.
n
o
a b ˙ g (D)c ∆Ptp ρg m a b m m m c = m0 l0 t 0 . (l) 3 2
Group 1: Π1 =
l
t
lt
D2 Usg
(1) 4 where Ag and Ug are the actual gas cross sectional area and actual average gas velocity respectively and A and Usg are the Venturi inlet area and superficial gas velocity (i.e. the average gas velocity if that gas mass flow rate flowed alone in the pipe at that gas density) respectively. As, when considering dimensional analysis, constants do not influence results (i.e. they can be dropped) we can substitute Eq. (1) into Π1 and re-arranging we get:
Π1 =
ρg D4 ∆Ptp ∆Ptp 2 = ρg Usg2 ˙g m
or as the dynamic pressure of the gas flow if it flowed alone is 1 ρ U2 : 2 g sg
Π1 =
2∆Ptp ρg D4 ∆Ptp . 2 = ρg Usg2 ˙g m
By similar procedures the rest of the groups can be derived:
o ˙l m ˙ l) = (D)0 (m ˙g m n o 0 ∆ρ −1 ˙ g (D)0 (∆ρ) = Group 3: Π3 = ρg m ρg n o µ µg 0 −1 gD ˙g m = Group 4: Π4 = ρg (D)1 µg = ˙g m ρg Usg D n o −1 µl µ D 0 l ˙g = Group 5: Π5 = ρg m (D)1 (µl ) = ˙g m ρg Usg D n o −2 σl σl ρg D3 1 ˙g Group 6: Π6 = ρg m (D)3 (µl ) = 2 = 2 ρ U g sg D ˙g m n o 0 d 0 ˙ g (D)−1 (d) = m Group 7: Π7 = ρg D n o 0 Lc 0 −1 ˙ g (D) Group 8: Π8 = ρg m (Lc ) = D n o 0 Lt 0 −1 ˙ g (D) Group 9: Π9 = ρg m (Lt ) = D n o 0 Ld 0 −1 ˙ g (D) Group 10: Π10 = ρg m ( Ld ) = D n o 0 e 0 ˙ g (D)−1 (e) = Group 11: Π11 = ρg m Π2 =
Group 2:
n
ρg
0
˙g m
−1
D
Group 12: Π12 =
n
Group 13: Π13 =
n
ρg
2
ρg
1
ρg2 D5 g gD 2 = 2 Usg ˙g m
˙g m
(D)5 (g ) =
˙g m
−2
o ρg D4 P P . (D)4 (P ) = 2 = 2 ρ g Usg ˙g m
Therefore we have: Π1
Π9 , Π10 , Π11 , Π12 , Π13 )
o
−2
where f is some function, or in an alternate expression:
Π1 = f1 (Π2 , Π3 , Π4 , Π5 , Π6 , Π7 , Π8 , Π9 , Π10 , Π11 , Π12 , Π13 )
π
= f1 (Π2 , Π3 , Π4 , Π5 , Π6 , Π7 , Π8 ,
That is: 2∆Ptp
ρg Usg2
˙l m
= f1
˙g m
ρ
2 5 gD g
m2g
˙
, ,
∆ρ µg D µl D σl ρg D3 d Lc Lt Ld e , , , , , , , , , ˙g m ˙g ˙ 2g ρg m m D D D D D ! P
ρg Usg2
344
R. Steven / Flow Measurement and Instrumentation 19 (2008) 342–349
which is:
(i.e. a modified Weber number for two phase flow).
ρg Usg2
∆Ptp =
2
f1
2 5 e ρg D g
D
,
m2g
˙
˙ l ∆ρ µg D µl D σl ρg D3 d Lc Lt Ld m , , , , , , , , , ˙ g ρg m ˙g m ˙g ˙ 2g m m D D D D ! ,
˙ 2g m
ρ
2 g Usg
2ρg
˙l m
f1 A2
˙g m
2 5 e ρg D g
D
,
˙ 2g m
,
, P
∆ρ µg D µl D σl ρg D3 d Lc Lt Ld , , , , , , , , ˙g m ˙g ˙ 2g ρg m m D D D D !
.
ρg Usg2
4. Analyse of two-phase flow (with one liquid component) — Re-arranging groups ∆ρ
ρ −ρ
∆ρ
ρ
Examining ρ we see that ρ = l ρ g = ρ l − 1. g g g Therefore, with dimensional analysis theory it can be said that: ρl ∆ρ Π3 = ρ ≡ ρ or instead, if we wish to express this dimensionless g
group as the reciprocal of the above derived expression, it could be ρ said Π30 = ρg (i.e. the gas to liquid density ratio). l When examining the derived groups, we can derive several known wet gas/ two-phase flow parameters. Dimensional analysis allows any dimensionless group to be replaced by a new dimensionless group that is found by combining the group to be replaced with one of the other derived dimensionless groups. Therefore, let Π2 = Π2 ∗ 0
s =
√
Π3 =
˙l m
r
˙g m
ρg ρl
superficial liquid inertia superficial gas inertia
= XLM
(i.e. the Lockhart Martinelli parameter1 ). Let Π12 = √ 0
Usg
1
Π3 ∗ Π12
s =
r
= √
gD
ρg ρl − ρg
superficial gas inertia liquid gravity force
=
1
Π6
=
=
Π4
ρg Usg D µg
superficial gas inertia
ρg Usg2 D2
(i.e. the superficial gas Reynolds number which is defined as the gas Reynolds number at the meter inlet if the gas phase flowed alone). Let:
Π50 =
1
=
Π5
ρg Usg2 D2 ρg Usg D = µl µl Usg D superficial gas inertia
superficial gas inertia
liquid viscous forces if liquid at superficial gas velocity = RetpLiquid i.e. a liquid Reynolds number which is defined as the liquid Reynolds number at the meter inlet if the liquid phase flowed at the superficial gas velocity. The best way to describe the liquid viscous forces in two-phase flow is debatable. Π50 is a rather abstract parameter. It is difficult to see the relevance of this parameter. A parameter that has a clearer physical meaning as a stand alone dimensionless number that could be used as an alternative is created as follows: ˙l ˙g ˙l m m m Π2 00
Π5 =
=
Π5
µl D
∗
˙g m
=
µl D
.
The mass continuity equation for the liquid mass flow rate is:
˙ l = ρl Al Ul = ρl AUsl = ρl m
π
D2 Usl
(2) 4 where Al and Ul are the actual liquid cross sectional area and actual average liquid velocity respectively and A and Usl are the Venturi inlet area and superficial liquid velocity (i.e. the average liquid velocity if that liquid mass flow rate flowed alone in the pipe) respectively. Therefore dropping the π4 constant as it is irrelevant in dimensional analysis we get:
˙l m µl D ρl Usl2 D2 superficial liquid velocity = = = Resl . µl Usl D liquid viscous forces
Π500 =
This is the classical single liquid phase Reynolds number for single phase liquid flows. However, it still has questionable relevance with relation to wet gas/two phase flow — especially wet gas flow where the majority of the mass flow is gas flow. By definition, Π7 = Dd = β (i.e. the Venturi meter ‘‘beta’’ or ‘‘diameter’’ ratio). Π11 = De , is a standard single phase flow dimensionless number and is called the ‘‘relative roughness’’.
Π8 Lc Π8 Lc 00 = and Π10 = = . Π9 Lt Π10 Ld Note that Π1 is a modified Euler Number and Π13 can be re-
arranged like so: 0 Π13 =
σl D
liquid interfacial tension force
= Resg
gas viscous force
Let Π90 =
= Frg
(i.e. the gas densiometric Froude parameter2 ). Let Π60 =
1
=
That is, the differential pressure read by a Venturi flow meter in use with wet gas/two-phase flow can be found by the product of the dynamic pressure of the gas flow if it flowed alone and some unknown function (f1 ) that contains the above set of dimensionless groups. The inclusion of each dimensionless group is not a guarantee that each parameter does in fact have any significant effect on the meter’s differential pressure. It simply states that it may be found to do so after experimental analysis is carried out. No information on the form of this function is offered by the Buckingham Pi theorem. It is required to be found by modeling and/or experimentation and data analysis.
g
=
P
or by substituting in Eq. (1):
∆Ptp =
Let Π40 =
= WetpLiquid
Therefore whereas there was originally the following set of groups:
∆Ptp = 1 There are several different definitions of the Lockhart Martinelli parameter. This definition is the one most favored by metering engineers in the last decade. 2 This terminology is not universally used. It is described in this way by de Leeuw [5] and others but there are cases where this group is otherwise named.
Π1 ∆Ptp = . Π13 P
˙ 2g m
˙l m
f1 2ρg A2 Ld
,
˙g m
e ρ
D D
,
,
2 5 gD g
m2g
˙
,
∆ρ µg D µl D σl ρg D3 d Lc Lt , , , , , , , ˙g m ˙g ˙ 2g ρg m m D D D ! P
ρg Usg2
.
R. Steven / Flow Measurement and Instrumentation 19 (2008) 342–349
The following set of groups can be derived from the original set of groups.
∆Ptp =
˙ 2g m ,
f2
,
p
˙g = m
P
where the function f2 is different to function f1 . This is a statement that the differential pressure read by a Venturi flow meter in use with two-phase flow can be found by the product of the dynamic pressure of the gas flow if it flowed alone and some unknown function (f2 ) that contains the above set of dimensionless groups. No information regarding the form of the function (f2 ) is offered by dimensional analysis. Another way of expressing this result is:
˙ g = A 2ρg ∆Ptp ∗ f3 m p
Ld
,
Ld
,
e ∆Ptp
D D
,
XLM ,
ρg Lc , Frg , Resg , Resl , Wetp , β, , ρl Lt
Resg , β,
Lc Lt
,
Lt Ld
,
Ld
,
e ∆Pg
D D
,
P
where f4 is a different function to f3 . If we consider one particular geometry the geometry terms can be dropped:
˙ g = A 2ρg ∆Pg ∗ f5 m p
Resg ,
∆Pg
P
where f5 is a different function to f4 . This is the form of a single phase DP meter equation (see Ref. [1]). If the analysis was performed on different geometry DP meters the form of function f5 would change accordingly for each geometry. If a single phase flow DP meter, such as a Venturi meter, is used with two-phase flow with no wet gas correction applied the single phase flow equation gives an erroneous gas mass flow rate ˙ g ,Apparent ) and an associated erroneous gas Reynolds prediction (m number prediction (Retp ) as well as an erroneous expansibility factor (due to ∆Ptp being recorded instead of the differential pressure if the gas phase flowed alone ∆Pg ). This means in practical terms that the single phase flow equation, applied when the meter is in use with two-phase flow, gives the following result:
˙ g ,Apparent = A 2ρg ∆Ptp ∗ f4 m p
Retp , β,
Lc Lt
,
Lt Ld
,
Ld
,
e ∆Ptp
D D
,
P
.
The aim of a ‘‘wet gas’’ correction factor (or ‘‘correlation’’) is to correct this two-phase flow error. It has been earlier derived that the factors that affect a DP meters differential pressure when the flow is two-phase are:
˙ g = A 2ρg ∆Ptp ∗ f3 m p
Lt Ld
,
Ld
,
e ∆Ptp
D D
,
XLM ,
ρg Lc , Frg , Resg , Resl , Wetp , β, , ρl Lt
P
we can therefore re-arrange these equations to give the form: ˙ g ,Apparent m
f6 XLM ,
ρg ρl
, LLdt , LDd , De ,
, Frg , Resg , Resl , Wetp , β,
Lc Lt
,
Lt Ld
∆Ptp
,
P Ld D
, De
.
It is important to understand that although the Buckingham Pi theorem is a very powerful tool, it does not indicate the best set of dimensionless groups to use or the form of the functions that relate the chosen dimensionless groups. It only states which dimensionless groups are valid. Amongst the valid groupings there is, for example, no proof that a researcher gets more benefit in using say group Π20 (i.e. the Lockhart–Martinelli parameter) as compared to group Π2 (i.e. the liquid to gas mass flowrate). The best choice of which groupings to use comes from experience and trial and error during data analysis. 5. Dimensional analysis with one gas phase and two liquid components
where f3 is a different function to f2 . Now, if we were to drop the liquid terms (i.e. we consider dry gas) we are left with the following:
p
Lc Lt
P
˙ g = A 2ρg ∆Pg ∗ f4 m
A 2ρg ∆Ptp ∗ f4 Retp , β,
e ∆Ptp
D D
Lt
differential pressure with two-phase flow, we can say that function six may contain these groups, i.e:
2ρg A2 Ld
ρg Lc Lt XLM , , Frg , Resg , Resl , Wetp , β, , , ρl Lt Ld
345
˙g = m where the function f6 is the corrective function. f6 As we know the dimensionless groups that could affect the
Before discussing the gas and one liquid component two phase flow DP meter dimensional analysis results and comparing them to published correlations it is necessary to discuss the case of gas with two liquid components. Some DP meters are now being used to meter flows of natural gas, oil and water. It has often been assumed that the gas type has little to no affect on the response of a DP meter with two phase flow. That is, for any given meter, it was assumed that as long as the gas density of two different gases matched, the gas dynamic pressure on the liquid was equal for a set gas flow rate and the wet gas flow response would therefore be the same. In 2006 Reader Harris et al [2] showed data to back this assumption. However, it was also assumed that for a set gas dynamic pressure, different liquid properties would affect a wet gas response of a DP meter. In the same paper Reader Harris et al [2] also showed data to back this assumption. The majority of the traceable two phase/wet gas flow data through DP meters is for gas with a single liquid component. Hence, most existing papers on the subject deal with one liquid component two phase/wet gas data. Therefore, the comparisons between the dimensional analysis results and the existing DP meter correlations needs to be done with the results of Sections 3 and 4. However, it is necessary here, in Section 5, to address the issue of two liquid components. If two liquids (say water and oil with subscripts ‘‘w’’ and ‘‘o’’ respectively) are treated separately then there are two parameters ˙ w and m ˙ o ), liquid density (ρw and ρo ), for liquid mass flow rate (m liquid viscosity (µw and µo ) and surface tension (σw and σo ). This adds four extra parameters to the parameter list in Section 2. We therefore derive four extra groups. That is, where as in Section 3 we had four dimensionless groups that account for liquid property effects (i.e. Π2 , Π3 , Π5 and Π6 ) we now have eight dimensionless groups that account for liquid property effects. Group 2: Π2A = Group 3: Π3A Group 5: Π5A Group 6: Π6A
˙o m ˙g m
and
Π2B =
˙w m
˙g m ρo ρw = and Π3A = ρg ρg µo µw = and Π5B = ρg Usg D ρg Usg D σo σw = and Π6B = . ρg Usg2 D ρg Usg2 D
346
R. Steven / Flow Measurement and Instrumentation 19 (2008) 342–349
Therefore we have:
∆Ptp = f7
˙ 2g m f 2ρg A2 7
where the function f7 can be written as:
˙w m ˙ o ρw ρo µg D µw D µo D σw ρg D3 σo ρg D3 m
, , , , , ˙g m ˙g m ˙g ρg ρg m ! 2 5 P d Lc Lt Ld e ρg D g , , , , , , . ˙ 2g m D D D D D ρg Usg2 ˙g m
,
˙g m
,
˙ 2g m
,
˙ 2g m
,
Here again, we can combine groups to create new groups. For example:
˙o m Π2A = , ˙w Π2B m µo Π5A = , = Π5B µw
Π3A ρo = , Π3B ρw Π6A σo = = . Π6B σw
Π2C =
Π3C =
Π5C
Π6C
Fig. 2. Murdock plot of two-phase dp meter data.
Note that Cd is the DP meter discharge coefficient and Y is the DP meter expansion factor for the case of dry gas flows. Note:
Therefore, we can re-arrange the expression to:
∆Ptp =
˙ 2g m
f 2ρg A2 8
Cd = f9 Resg
where the function f8 is different to function f7
and can be written as: f8
ρw ρo µg D µw D µo σw ρg D3 σo , , , , , , , ˙g m ˙ w ρg ρw m ˙g m ˙ g µw ˙ 2g m m σw ! 2 5 P d Lc Lt Ld e ρg D g , , , , , , . ˙ 2g m D D D D D ρg Usg2 ˙w m
,
˙o m
,
6. Discussion of the gas and one liquid component results An early detailed analysis of two-phase flow through a DP meter was the two-phase orifice plate meter analysis by Murdock [3]. In this analysis Murdock derived a two-phase flow model through an orifice plate and then took a wide array of data from various sources to fit an orifice plate meter two-phase correlation. The data included different meter diameters (D), different beta ratios (β ), different liquid properties (with the data including water, salt water and hydrocarbon liquids) and different gas velocities. Murdock’s model did not directly account for any two-phase effects of geometry (as he considered orifice plate meters alone and ignored any diameter and beta effects), fluid properties (including densities) or gas velocity. That is, Murdock did not directly consider any pressure or velocity effects. Note that the ‘‘over-reading’’ of a DP gas meter with twophase flow is defined as the ratio of the DP meters erroneous ˙ g Apparent ) to the actual gas mass gas mass flowrate prediction (m ˙ g ). Murdock plotted his orifice plate meter two-phase flowrate (m data on a graph that could be approximately described as the Lockhart Martinelli parameter (group Π10 ) vs. the over-reading ˙ g Apparent /m ˙ g ) as shown in Fig. 2. This form of plot is now (m generally called a ‘‘Murdock Plot’’. The Murdock correlation is:
˙g ≈ m
˙ g ,Apparent m 1 + 1.26XLM
Aβ 2 Ytp C d tp
√
=
√
2ρg ∆Ptp
1−β 4
1 + 1.26XLM
.
(3)
Y = f10 β, P , ∆Pg , κ
and
Ytp = f10 β, P , ∆Ptp , κ .
The numerator of the Murdock correlation is the erroneous ˙ g Apparent . However, two-phase flow gas mass flowrate term, m correction factors are functions that correct the erroneous gas flowrate result. The Venturi two-phase analysis produced the following equation:
A 2ρg ∆Ptp ∗ f4 Retp , β,
p
˙g = m
f6 XLM ,
ρg ρl
Lc Lt
, LLdt , LDd , De ,
, Frg , Resg , Resl , Wetp , β,
Lc Lt
,
Lt Ld
∆Ptp
,
P Ld D
, De
.
Note that we could do the same dimensional analysis on an orifice plate as has been done here for a Venturi. If we ignored geometric terms and concentrated on flow conditions we would get the same result:
A 2ρg ∆Ptp ∗ f5 Retp ,
p
˙g = m
f6 XLM ,
ρg ρl
∆Ptp
P
, Frg , Resg , Resl , Wetp
.
Therefore, as Murdock has not included any geometric terms we can say he has fitted function f6 . However, from modeling of the two-phase flow through an orifice plate Murdock suggested that the DP over-reading was only a function of the Lockhart Martinelli parameter. The Lockhart Martinelli parameter includes the gas to liquid mass flow rate ratio and gas to liquid density ratio terms in one parameter. Murdock did not account for any independent gas to liquid density ratio effect. In fact, with Murdock being one of the early researchers, he did not take account of several of the parameters listed above as having possible affects on DP meter two-phase flow over-readings. He did not account for the gas velocity (i.e. gas densiometric Froude number for any set meter diameter, pressure and liquid properties). He did not account for the meter geometry. He also did not directly consider the effect of liquid properties. Therefore the dimensional analysis result reduces to:
A 2ρg ∆Ptp ∗ f5 Retp ,
p
˙g = m
and
where κ is the gas isentropic exponent (a dimensionless term). When the flow is two-phase flow and the differential pressure read is not that of the dry gas flow functions f9 and f10 give the incorrect values (C dtp and Ytp respectively), i.e.: Cdtp = f9 Retp
The understanding of liquid property and ‘‘water cut’’ (i.e. the ratio of water to total volume flowrate) effects on DP meters in use with two-phase flow is beyond the limit of the published understanding. This topic is therefore at the forefront of twophase/wet gas flow metering research and as yet no particular set of dimensionless groups that combine these liquid property parameters has been selected by any researcher as the most appropriate. Therefore, no more can yet be said on this issue.
f11 (XLM )
∆Ptp P
.
It should be noted the Murdock correlation (Eq. (3)) matches this form. Note that Eq. (3) does include the beta ratio term in
R. Steven / Flow Measurement and Instrumentation 19 (2008) 342–349
Fig. 4. Venturi meter gas densiometric froude number effect for a set gas to liquid density ratio.
Fig. 3. Venturi meter gas to liquid density ratio effect.
the dry gas flow prediction but note we have ignored geometry terms here and therefore removed the beta ratio term. As we are discussing a generic meter geometry this is allowed by dimensional analysis. The effect of the geometry, including the beta ratio, would be accounted for in the form of function, f5 . In the following examples this argument also holds. Note that strictly speaking on the same argument we could choose to also drop the area term, A, as well but the author has chosen not to. Between 1967–77 Chisholm [4,5] developed orifice two-phase flow theory and for cases of XLM ≤ 1 Chisholm gave a two-phase flow correlation:
Aβ 2 Ytp Cdtp
√
√
˙g = m
2ρg ∆Ptp
1−β 4
q
(4)
2 1 + CXLM + XLM
on Murdock’s work by including an independent parameter of the gas to liquid density ratio along with the Lockhart Martinelli parameter. This gave more accurate over-reading predictions indicating the Lockhart Martinelli parameter did not fully account for the gas to liquid density ratio phenomenon. No further work was released on orifice plate meter two-phase flow response until 2007. However, in 1997, de Leeuw [6] released a seminal paper on Venturi meter two-phase flow characteristics. De Leeuw developed Chisholm’s orifice plate meter work to apply to Venturi meters and included the effect the gas velocity has on the system. Fig. 4 shows the gas densiometric Froude number effect de Leeuw discovered. De Leeuw had data recorded from one geometry of Venturi meter (i.e. an ISO 5167 Part 4 [1] compliant, 400 , schedule 80, 0.401 beta ratio). The result was the following correlation:
where
C =
14
+
ρl ρg
14
˙g = m
.
√
A 2ρg ∆Ptp ∗ f5 Retp ,
p
f12 XLM ,
ρg ρl
∆Ptp P
.
It should be noted the Chisholm correlation matches this form. A similar effect has been seen to exist for Venturi meters. Fig. 3 shows data from a 400 , schedule 80, 0.4 Beta Ratio ISO 5167 compliant Venturi meter tested at CEESI with natural gas and decane (C10 hydrocarbon liquid). A clear gas to liquid density ratio effect is visible. As the gas to liquid density ratio value increases for all other parameters held constant the over-reading reduces. Note that with regards to the Chisholm equation, an important point with regards to general dimensional analysis can be made here. There is often a misunderstanding that if a parameter is imbedded in one dimensionless group then the effect is fully accounted for in the system. Buckingham Pi theory does not state this. Here, as an example, we see that Murdock had the gas to liquid density ratio paired with the gas to liquid mass flowrate only (as part of the Lockhart Martinelli parameter). Chisholm improved
2ρg ∆Ptp
1−β 4
(4)
q
2 1 + CXLM + XLM
(4a)
With a similar flow model to Murdock, Chisholm considered slip between separated gas and liquid phases and stated that the slip was purely a function of the gas to liquid density ratio. However, like Murdock, Chisholm did not account for the geometry of the DP meter and did not independently consider any affect related to the gas velocity (e.g. gas densiometric Froude number for any set meter diameter, pressure and liquid properties). He also did not directly consider the effect of liquid properties (other than density). Therefore the dimensional analysis result reduces to:
˙g = m
Aβ 2 Ytp Cdtp
√
ρg ρl
347
where
C =
ρg ρl
n
+
ρl ρg
n (4b)
where n = f Frg .
(5)
De Leeuw did not take into consideration the geometry of the DP meter device and did not independently consider the effect of liquid properties. Therefore the dimensional analysis result reduces to:
A 2ρg ∆Ptp ∗ f5 Retp ,
p
˙g = m
ρ
f13 XLM , ρg , Frg l
∆Ptp P
.
It should be noted the de Leeuw correlation matches this form. It should be noted that Murdock, Chisholm and de Leeuw derived Eqs. (3), (4), (4a) and (5) respectively from flow modeling. No dimensional analysis was carried out in these derivations and yet some of the most important dimensional numbers (i.e. the Lockhart Martinelli parameter, the gas to liquid density ratio and the gas densiometric Froude number) were found by direct consideration of the physical phenomena. This shows that it is possible to derive practical correlations by flow modeling alone without the use of dimensional analysis. However, dimensional analysis is an extremely valuable tool that can aid engineers significantly when researching physical phenomena. Carrying out dimensional analysis in conjunction with modeling can
348
R. Steven / Flow Measurement and Instrumentation 19 (2008) 342–349
Fig. 6. NEL 400 , 0.75 beta ratio Venturi meter fluid property two-phase data.
Fig. 5. Venturi beta ratio effect for a set pressure (31 bara) and gas densiometric froude number (Frg = 1.5).
give significant extra insight and reassurance to the engineer developing a physical model. In 2002–2005 Stewart and Steven [7,8] produced a similar correlation for 400 and 600 cone type DP meters with a 0.75 beta ratio. The dimensionless groups used in this equation was the same as used by de Leeuw. Here again, as with de Leeuw’s Venturi meter work, the beta ratio was set. However, Stewart and Steven [7] did show comparisons of two different beta ratio cone type DP meters and showed that a beta ratio effect existed. For all other parameters held constant the over-reading reduced with increasing beta ratio. In 2003 Stewart [9] subsequently showed this same beta ratio effect to also exist for Venturi meters. Fig. 5 shows this effect. For all other parameters held constant increasing the beta ratio increases the over-reading. In 2007 Steven et al [10] then showed orifice plate meters have a slight beta ratio effect. No new correlations were supplied that accounted for any beta ratio affect. However, such a correlation would be of the form:
A 2ρg ∆Ptp ∗ f5 Retp ,
p
˙g = m
ρ
f14 XLM , ρg , Frg , β l
∆Ptp
P
.
It is unlikely that any practical correlation would be produced along the form of function f14 . It is considerably more practical to keep the correlations as simple as possible by setting a function for a given beta ratio. That is, it is more practical to restrict the choice of DP meter to a few set beta ratios where correlations are available. In 2006 Reader Harris et al [2,11] and Steven [12,13] showed that liquid properties do, under some flow conditions, have an influence on the scale of a DP meters two-phase over-reading. It was stated that water, with higher interfacial tension values compared to light hydrocarbon liquids, had lower over-readings for all other parameters held constant. Fig. 6 shows a NEL result for a 400 , schedule 80, 0.75 beta ratio, a gas to liquid density ratio of 0.046 and gas densiometric Froude number of 2.5 (see Reader Harris [11] for more details). Note that gas properties do not have an effect on this DP meters wet gas over-reading. No updated correlations were proposed by Reader Harris et al or Steven. However, under these findings it is practical to assume such a correlation could have the form (for a set meter geometry):
A 2ρg ∆Ptp ∗ f5 Retp ,
p
˙g = m
f15 XLM ,
ρg ρl
∆Ptp
P
.
, Frg , Resl , Wetp
It remains to be seen which, if either, of these liquid property dimensionless groups will be useful. It can be seen in Section 5 that there are several options for manipulating the liquid property
Fig. 7. A comparison of two different diameter geometrically similar venturi meters under similar two-phase flow dimensionless parameter values.
dimensionless groups and the usefulness of each group could be ascertained by applying it to experimental data. Physical modeling of the flow can also give significant insight and indicate the likely best form for liquid property dimensional groups. As it has been shown by Reader Harris and Steven that a liquid property effect does exist, dimensional analysis can help engineers when investigating this fluid property effect. Here, a liquid Reynolds number dimensionless group effect would imply a significant liquid viscosity effect, and a modified Weber number dimensionless group effect would imply a significant interfacial tension effect. Little is known with regards to the affect meter diameter has on two-phase flow DP meters. Steven [12,14] has indicated that a diameter effect may exist. It was pointed out by Steven [14] that the data set which led to this statement was extremely small and therefore no definite claim was made. However, the available data suggested that for an increasing diameter with the other dimensionless group values held constant the over-reading was seen to decrease. Fig. 7 shows the comparison made by Steven [14] between 400 and 200 0.6 beta ratio Venturi meters at similar gas to liquid density ratio and gas densiometric Froude numbers, with similar fluids. The smaller meter has a smaller over-reading. If a diameter effect is confirmed for two-phase DP meters, and industry desired one universal two-phase/wet gas correlation for a given DP meter design, this effect would be required to be a included. The form of the correlation would therefore be (for otherwise set geometry):
A 2ρg ∆Ptp ∗ f5 Retp ,
p
˙g = m
f16 XLM ,
ρg ρl
∆Ptp
P
, Frg , Resl , Wetp , β, LDd , De
.
R. Steven / Flow Measurement and Instrumentation 19 (2008) 342–349
However, as with the beta ratio effect, the diameter effect is unlikely to be added to any practical DP meter two-phase/wet gas flow correlation as it is far more practical to keep the correlations as simple as possible by setting a function for a set geometry. If this was not a research policy the correlations could become too complicated for practical use. Furthermore, each new affect required to be included in a correlation adds to the correlations over all uncertainty. With the potential for liquid fluid properties requiring to be added to the correlation variables, a policy of producing wet gas DP meter correlations for set geometries looks increasingly desirable. As the correlations require yet more flow condition parameters (i.e. parameters out with the control of the meter designer) an obvious method of reducing the number of overall variables in a correlation is to eliminate the geometry parameters (i.e. parameters controlled by the meter designer). That is, create DP meter wet gas correlations for set geometries only. 7. Conclusions Dimensional analysis has produced a set of dimensional numbers for two phase flow through a DP meter. When this set was reduced to dry gas flow conditions, the set correctly showed similarity with the actual gas Venturi flow equation, as required. When this set was reduced to the particular two phase flow conditions tested by individual researchers, the set correctly showed similarity with the correlation in question. There is a very limited quantity of DP meter wet gas flow data sets available in the public domain, where liquid property effects and geometry effects (including beta ratio and diameter effects) are examined. The few data sets that are discussed in the public domain indicate that there are liquid property and geometry effects. Therefore as the industry develops its understanding of these parameters effects on DP meters, dimensional analysis will aid the research and analysis of the experimental results. From a practical stand point, with more factors being found to influence the magnitude of gas DP meter wet gas flow overreadings, more factors are required to be accounted for in the corrective equations. However, the more factors there are, the greater the uncertainty in the overall correction. For this reason, and to avoid unnecessary complexity, it is likely that new correlations will not include geometry terms. New correlations are likely to be based on one geometry of DP meter. It will not be practical to apply such a policy to the case of liquid properties and have a practical DP meter wet gas correlation for the case
349
of gas flows with more than one liquid component. Whereas, meter users can have some influence on the geometry of a DP meter chosen, in most industrial metering applications the user has little to no choice of the flow conditions the chosen device must meter. For example, in many oil and gas industry twophase/wet gas flow applications there is a mix of water (sometimes salt water) and hydrocarbon liquids and this can change in composition through the useful life of a DP meter. It will therefore be desirable that the installed DP meter (of whatever geometry) has a wet gas correlation that can account for liquid property effects. Industry is still far from having a full set of trustworthy twophase/wet gas DP meter correlations for all possible applications. Much research remains to be conducted in this area. Several parameters are known to affect a DP meters two-phase/wet gas over-reading that as yet are not included in the correlations. The use of suitably chosen dimensionless groups is likely to be very important in understanding the phenomena and in helping developing useable two-phase/wet gas flow DP meter correlations. References [1] International Standards Organisation. International standard 5167-part 4. 2003. [2] Reader-Harris M. Ventuti-tube performance in wet gas using different test fluids. In: North sea flow measurement workshop 2006. [3] Murdock JW. Two-phase flow measurements with orifices. Journal of Basic Engineering 1962;84:419–33. [4] Chisholm D. Flow of incompressible two-phase mixtures through sharp-edged orifices. Journal of Mechanical Engineering Science 1967;9(1). [5] Chisholm D. Research note: Two-phase flow through sharp-edged orifices. Journal of Mechanical Engineering Science 1977;19(3). [6] De Leeuw R. Liquid correction of Venturi meter readings in wet gas flow. In: North sea workshop 1997. [7] Stewart D, Steven R, Peter RJW, Hodges D. Wet gas metering with V-cone meters. In: North sea flow measurement workshop 2002. Paper No.4.2. [8] Steven R, Kegel T, Britton C. An update on V-Cone meter wet gas flow metering research Flomeko 2005, Peebles, Scotland. [9] Stewart DG. Suitability of dry gas metering technology for wet gas metering. In: North sea flow measurement workshop 2003. [10] Steven, Stobie, Ting. A re-evaluation of axioms regarding orifice meter wet gas flow performance. In: 6th international south East Asia hydrocarbon flow measurement workshop. 2007. [11] NEL report 2005/206 Venturi-tube performance in wet gas using different test fluids, 2006. [12] Steven R, Kinney J, Britton C. Liquid property and diameter effects on venturi meters used with wet gas flows. In: 6th international symposium on fluid flow measurement. 2006. [13] Steven R. Horizontally installed differential pressure meter wet gas flow performance review. In: North sea flow measurement workshop. 2006. [14] Steven R. A discussion on wet gas flow parameter definitions. In: Energy institute/CEESI one day wet gas flow metering seminar. 2006.
Contents lists available at ScienceDirect
Flow Measurement and Instrumentation journal homepage: www.elsevier.com/locate/flowmeasinst
A dimensional analysis of two phase flow through a horizontally installed Venturi flow meter Richard Steven ∗ Multiphase and Wet Gas Flow Research Facility, Colorado Engineering Experiment Station, Inc., 54043 WCR 37-Nunn, CO 80648, USA
article
info
Article history: Received 4 May 2007 Received in revised form 14 March 2008 Accepted 12 May 2008
a b s t r a c t A dimensional analysis on two phase flow through a horizontally mounted Venturi meter is performed. The derived dimensional groups are compared and related to those used in existing DP meter two phase flow models and correlations. The results give confidence in the existing dimensionless numbers in industrial use and lead to comments on the state of this art and possible future research trends. © 2008 Elsevier Ltd. All rights reserved.
Keywords: Two phase flow Dimensional analysis Venturi Differential pressure meter
1. Introduction Industry has a requirement to meter two-phase (i.e. combined gas and liquid) flows. Part of this requirement is a need to meter wet gas flows, that is, two-phase flows where the liquid flowrate is relatively small compared to the gas flowrate. Two-phase flow test results from various flow meters over many years has led to the Differential Pressure (DP) flow meter being one meter of choice for such applications. One of the most common DP meters on the market is the Venturi flow meter. The Venturi is typically used to meter wet gas/two-phase flows in three general ways: (1) The Venturi is used as a stand alone device along with other independent systems. For example, an independent system predicts the two-phase flows liquid component flowrate (e.g. test separator historical records or a tracer injection technology) and the Venturi meter reading with the liquid flowrate prediction are inputs to some Venturi two-phase flow correlation that gives the gas flowrate as an output. (2) The Venturi is used in series with another gas flow meter that has a different liquid induced gas flowrate error than the Venturi. The difference in the two meters erroneous gas flowrate predictions allow a measurement by difference technique to be used to predict the actual gas and liquid flowrates. (3) The Venturi has phase fraction devices (e.g. microwave systems, dual energy densitometers or capacitance systems etc.) imbedded
∗
Tel.: +1 970 897 2711; fax: +1 970 897 2710. E-mail address: [email protected].
0955-5986/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.flowmeasinst.2008.05.004
in some part of the meter body or neighboring spool pieces and the Venturi reading in conjunction with the phase fraction device readings are used in a wet gas/two-phase flow mathematical model to predict the gas and liquid flowrates. In all three cases understanding the response of the Venturi meters primary reading, that is the differential pressure read, to the wet gas/two-phase flow is of importance to the calculation procedure. This response is usually described by a wet gas/twophase Venturi meter correlation (or flow model). The Venturi meter is just one of several types of DP meter designs investigated for two-phase flow metering over the last fifty years. The DP meter wet gas/two-phase correlations that are in the public domain utilize dimensionless numbers that appear to have been chosen independently by various researchers developing two-phase flow models. It is therefore an interesting exercise to perform a full dimensional analysis on two-phase flow through a generic Venturi meter using the Buckingham Pi theorem. This exercise shows a list of dimensionless groups that could have an influence on a Venturi meters wet gas/two-phase flow response. It will be shown that these dimensionless groups can be re-arranged to give a set of dimensionless numbers that include all the dimensionless numbers currently used by two-phase flow metering researchers. Furthermore, this set of dimensionless numbers can be split into meter geometry dependant and independent sets. This exercise could therefore be repeated for any DP meter geometry and the same dimensionless numbers independent of geometry would be derived. This exercise is important as it fundamentally proves the validity of using several dimensionless numbers that have been in common use for many years.
R. Steven / Flow Measurement and Instrumentation 19 (2008) 342–349
343
Therefore equating the exponents gives: m : a + b + 1 = 0,
l : −3a + c − 1 = 0,
t : −b − 2 = 0.
Therefore, a = 1, b = −2, c = 4 and we have:
Π1 =
n
ρg
1
˙g m
−2
(D)4
ρg D4 ∆Ptp ∆Ptp = 2 . ˙g m
o
However, we know from mass continuity that:
˙ g = ρg Ag Ug = ρg AUsg = ρg m Fig. 1. Sketch of generic Venturi meter.
2. The dimensional analysis procedure set up The following is a dimensional analysis for general two-phase flow (with one gas component and one liquid component) through a Venturi meter. As with all dimensional analysis investigations the first step is to list all possible parameters that may affect the system (whether they in reality do or do not) and state their dimensions (i.e. the relationship of mass ‘‘m’’, length ‘‘l’’ and time ‘‘t’’ that they represent). A sketch of a generic Venturi meter is drawn in Fig. 1. The list that will be analyzed is:
∆Ptp (i.e. the differential pressure from upstream to throat) ˙g &m ˙ l (i.e. the gas and liquid mass flow rates m respectively) ρg (i.e. the gas density) ∆ρ (i.e. ρl − ρg , the difference between phase densities) µg & µl (i.e. the gas and liquid viscosities respectively) σl (i.e. the liquid interfacial tension) d (i.e. the Venturi inside bore throat diameter) D (i.e. the Venturi inlet inside bore diameter) Lc (i.e. the length of the converging section) Lt (i.e. the length of the throat) Ld (i.e. the length of the diffuser section) e (i.e. the roughness of the wetted surface) g (i.e. the gravitational constant) P (i.e. the system pressure)
m/lt
2
(m/t ) m/l3 m/l3 (m/lt ) m/t 2 (l) (l) (l) (l) (l) (l) l/t 2 m/lt 2
Note: The combination of the gas density (ρg ) and the density difference between the phases (∆ρ = ρl − ρg ) will be used here instead of liquid density (ρl ) as this combination contains the liquid density information and the liquid density therefore becomes a superfluous parameter.
3. The dimensional analysis procedure There are sixteen parameters listed, i.e. ‘‘m’’ = 16, that must be investigated. There are three primary parameters, i.e. length (l), mass (m) and time (t), i.e. ‘‘n’’ = 3. Number of independent groups = m − n = 16 − 3 = 13. That is, we have: f (Π1 , Π2 , Π3 , Π4 , Π5 , Π6 , Π7 , Π8 , Π9 , Π10 , Π11 , Π12 , Π13 ) = 0
where f1 is some function different to function f . ˙ g & D the Taking an arbitrary choice of 3 parameters: ρg , m thirteen groups are found in turn.
n
o
a b ˙ g (D)c ∆Ptp ρg m a b m m m c = m0 l0 t 0 . (l) 3 2
Group 1: Π1 =
l
t
lt
D2 Usg
(1) 4 where Ag and Ug are the actual gas cross sectional area and actual average gas velocity respectively and A and Usg are the Venturi inlet area and superficial gas velocity (i.e. the average gas velocity if that gas mass flow rate flowed alone in the pipe at that gas density) respectively. As, when considering dimensional analysis, constants do not influence results (i.e. they can be dropped) we can substitute Eq. (1) into Π1 and re-arranging we get:
Π1 =
ρg D4 ∆Ptp ∆Ptp 2 = ρg Usg2 ˙g m
or as the dynamic pressure of the gas flow if it flowed alone is 1 ρ U2 : 2 g sg
Π1 =
2∆Ptp ρg D4 ∆Ptp . 2 = ρg Usg2 ˙g m
By similar procedures the rest of the groups can be derived:
o ˙l m ˙ l) = (D)0 (m ˙g m n o 0 ∆ρ −1 ˙ g (D)0 (∆ρ) = Group 3: Π3 = ρg m ρg n o µ µg 0 −1 gD ˙g m = Group 4: Π4 = ρg (D)1 µg = ˙g m ρg Usg D n o −1 µl µ D 0 l ˙g = Group 5: Π5 = ρg m (D)1 (µl ) = ˙g m ρg Usg D n o −2 σl σl ρg D3 1 ˙g Group 6: Π6 = ρg m (D)3 (µl ) = 2 = 2 ρ U g sg D ˙g m n o 0 d 0 ˙ g (D)−1 (d) = m Group 7: Π7 = ρg D n o 0 Lc 0 −1 ˙ g (D) Group 8: Π8 = ρg m (Lc ) = D n o 0 Lt 0 −1 ˙ g (D) Group 9: Π9 = ρg m (Lt ) = D n o 0 Ld 0 −1 ˙ g (D) Group 10: Π10 = ρg m ( Ld ) = D n o 0 e 0 ˙ g (D)−1 (e) = Group 11: Π11 = ρg m Π2 =
Group 2:
n
ρg
0
˙g m
−1
D
Group 12: Π12 =
n
Group 13: Π13 =
n
ρg
2
ρg
1
ρg2 D5 g gD 2 = 2 Usg ˙g m
˙g m
(D)5 (g ) =
˙g m
−2
o ρg D4 P P . (D)4 (P ) = 2 = 2 ρ g Usg ˙g m
Therefore we have: Π1
Π9 , Π10 , Π11 , Π12 , Π13 )
o
−2
where f is some function, or in an alternate expression:
Π1 = f1 (Π2 , Π3 , Π4 , Π5 , Π6 , Π7 , Π8 , Π9 , Π10 , Π11 , Π12 , Π13 )
π
= f1 (Π2 , Π3 , Π4 , Π5 , Π6 , Π7 , Π8 ,
That is: 2∆Ptp
ρg Usg2
˙l m
= f1
˙g m
ρ
2 5 gD g
m2g
˙
, ,
∆ρ µg D µl D σl ρg D3 d Lc Lt Ld e , , , , , , , , , ˙g m ˙g ˙ 2g ρg m m D D D D D ! P
ρg Usg2
344
R. Steven / Flow Measurement and Instrumentation 19 (2008) 342–349
which is:
(i.e. a modified Weber number for two phase flow).
ρg Usg2
∆Ptp =
2
f1
2 5 e ρg D g
D
,
m2g
˙
˙ l ∆ρ µg D µl D σl ρg D3 d Lc Lt Ld m , , , , , , , , , ˙ g ρg m ˙g m ˙g ˙ 2g m m D D D D ! ,
˙ 2g m
ρ
2 g Usg
2ρg
˙l m
f1 A2
˙g m
2 5 e ρg D g
D
,
˙ 2g m
,
, P
∆ρ µg D µl D σl ρg D3 d Lc Lt Ld , , , , , , , , ˙g m ˙g ˙ 2g ρg m m D D D D !
.
ρg Usg2
4. Analyse of two-phase flow (with one liquid component) — Re-arranging groups ∆ρ
ρ −ρ
∆ρ
ρ
Examining ρ we see that ρ = l ρ g = ρ l − 1. g g g Therefore, with dimensional analysis theory it can be said that: ρl ∆ρ Π3 = ρ ≡ ρ or instead, if we wish to express this dimensionless g
group as the reciprocal of the above derived expression, it could be ρ said Π30 = ρg (i.e. the gas to liquid density ratio). l When examining the derived groups, we can derive several known wet gas/ two-phase flow parameters. Dimensional analysis allows any dimensionless group to be replaced by a new dimensionless group that is found by combining the group to be replaced with one of the other derived dimensionless groups. Therefore, let Π2 = Π2 ∗ 0
s =
√
Π3 =
˙l m
r
˙g m
ρg ρl
superficial liquid inertia superficial gas inertia
= XLM
(i.e. the Lockhart Martinelli parameter1 ). Let Π12 = √ 0
Usg
1
Π3 ∗ Π12
s =
r
= √
gD
ρg ρl − ρg
superficial gas inertia liquid gravity force
=
1
Π6
=
=
Π4
ρg Usg D µg
superficial gas inertia
ρg Usg2 D2
(i.e. the superficial gas Reynolds number which is defined as the gas Reynolds number at the meter inlet if the gas phase flowed alone). Let:
Π50 =
1
=
Π5
ρg Usg2 D2 ρg Usg D = µl µl Usg D superficial gas inertia
superficial gas inertia
liquid viscous forces if liquid at superficial gas velocity = RetpLiquid i.e. a liquid Reynolds number which is defined as the liquid Reynolds number at the meter inlet if the liquid phase flowed at the superficial gas velocity. The best way to describe the liquid viscous forces in two-phase flow is debatable. Π50 is a rather abstract parameter. It is difficult to see the relevance of this parameter. A parameter that has a clearer physical meaning as a stand alone dimensionless number that could be used as an alternative is created as follows: ˙l ˙g ˙l m m m Π2 00
Π5 =
=
Π5
µl D
∗
˙g m
=
µl D
.
The mass continuity equation for the liquid mass flow rate is:
˙ l = ρl Al Ul = ρl AUsl = ρl m
π
D2 Usl
(2) 4 where Al and Ul are the actual liquid cross sectional area and actual average liquid velocity respectively and A and Usl are the Venturi inlet area and superficial liquid velocity (i.e. the average liquid velocity if that liquid mass flow rate flowed alone in the pipe) respectively. Therefore dropping the π4 constant as it is irrelevant in dimensional analysis we get:
˙l m µl D ρl Usl2 D2 superficial liquid velocity = = = Resl . µl Usl D liquid viscous forces
Π500 =
This is the classical single liquid phase Reynolds number for single phase liquid flows. However, it still has questionable relevance with relation to wet gas/two phase flow — especially wet gas flow where the majority of the mass flow is gas flow. By definition, Π7 = Dd = β (i.e. the Venturi meter ‘‘beta’’ or ‘‘diameter’’ ratio). Π11 = De , is a standard single phase flow dimensionless number and is called the ‘‘relative roughness’’.
Π8 Lc Π8 Lc 00 = and Π10 = = . Π9 Lt Π10 Ld Note that Π1 is a modified Euler Number and Π13 can be re-
arranged like so: 0 Π13 =
σl D
liquid interfacial tension force
= Resg
gas viscous force
Let Π90 =
= Frg
(i.e. the gas densiometric Froude parameter2 ). Let Π60 =
1
=
That is, the differential pressure read by a Venturi flow meter in use with wet gas/two-phase flow can be found by the product of the dynamic pressure of the gas flow if it flowed alone and some unknown function (f1 ) that contains the above set of dimensionless groups. The inclusion of each dimensionless group is not a guarantee that each parameter does in fact have any significant effect on the meter’s differential pressure. It simply states that it may be found to do so after experimental analysis is carried out. No information on the form of this function is offered by the Buckingham Pi theorem. It is required to be found by modeling and/or experimentation and data analysis.
g
=
P
or by substituting in Eq. (1):
∆Ptp =
Let Π40 =
= WetpLiquid
Therefore whereas there was originally the following set of groups:
∆Ptp = 1 There are several different definitions of the Lockhart Martinelli parameter. This definition is the one most favored by metering engineers in the last decade. 2 This terminology is not universally used. It is described in this way by de Leeuw [5] and others but there are cases where this group is otherwise named.
Π1 ∆Ptp = . Π13 P
˙ 2g m
˙l m
f1 2ρg A2 Ld
,
˙g m
e ρ
D D
,
,
2 5 gD g
m2g
˙
,
∆ρ µg D µl D σl ρg D3 d Lc Lt , , , , , , , ˙g m ˙g ˙ 2g ρg m m D D D ! P
ρg Usg2
.
R. Steven / Flow Measurement and Instrumentation 19 (2008) 342–349
The following set of groups can be derived from the original set of groups.
∆Ptp =
˙ 2g m ,
f2
,
p
˙g = m
P
where the function f2 is different to function f1 . This is a statement that the differential pressure read by a Venturi flow meter in use with two-phase flow can be found by the product of the dynamic pressure of the gas flow if it flowed alone and some unknown function (f2 ) that contains the above set of dimensionless groups. No information regarding the form of the function (f2 ) is offered by dimensional analysis. Another way of expressing this result is:
˙ g = A 2ρg ∆Ptp ∗ f3 m p
Ld
,
Ld
,
e ∆Ptp
D D
,
XLM ,
ρg Lc , Frg , Resg , Resl , Wetp , β, , ρl Lt
Resg , β,
Lc Lt
,
Lt Ld
,
Ld
,
e ∆Pg
D D
,
P
where f4 is a different function to f3 . If we consider one particular geometry the geometry terms can be dropped:
˙ g = A 2ρg ∆Pg ∗ f5 m p
Resg ,
∆Pg
P
where f5 is a different function to f4 . This is the form of a single phase DP meter equation (see Ref. [1]). If the analysis was performed on different geometry DP meters the form of function f5 would change accordingly for each geometry. If a single phase flow DP meter, such as a Venturi meter, is used with two-phase flow with no wet gas correction applied the single phase flow equation gives an erroneous gas mass flow rate ˙ g ,Apparent ) and an associated erroneous gas Reynolds prediction (m number prediction (Retp ) as well as an erroneous expansibility factor (due to ∆Ptp being recorded instead of the differential pressure if the gas phase flowed alone ∆Pg ). This means in practical terms that the single phase flow equation, applied when the meter is in use with two-phase flow, gives the following result:
˙ g ,Apparent = A 2ρg ∆Ptp ∗ f4 m p
Retp , β,
Lc Lt
,
Lt Ld
,
Ld
,
e ∆Ptp
D D
,
P
.
The aim of a ‘‘wet gas’’ correction factor (or ‘‘correlation’’) is to correct this two-phase flow error. It has been earlier derived that the factors that affect a DP meters differential pressure when the flow is two-phase are:
˙ g = A 2ρg ∆Ptp ∗ f3 m p
Lt Ld
,
Ld
,
e ∆Ptp
D D
,
XLM ,
ρg Lc , Frg , Resg , Resl , Wetp , β, , ρl Lt
P
we can therefore re-arrange these equations to give the form: ˙ g ,Apparent m
f6 XLM ,
ρg ρl
, LLdt , LDd , De ,
, Frg , Resg , Resl , Wetp , β,
Lc Lt
,
Lt Ld
∆Ptp
,
P Ld D
, De
.
It is important to understand that although the Buckingham Pi theorem is a very powerful tool, it does not indicate the best set of dimensionless groups to use or the form of the functions that relate the chosen dimensionless groups. It only states which dimensionless groups are valid. Amongst the valid groupings there is, for example, no proof that a researcher gets more benefit in using say group Π20 (i.e. the Lockhart–Martinelli parameter) as compared to group Π2 (i.e. the liquid to gas mass flowrate). The best choice of which groupings to use comes from experience and trial and error during data analysis. 5. Dimensional analysis with one gas phase and two liquid components
where f3 is a different function to f2 . Now, if we were to drop the liquid terms (i.e. we consider dry gas) we are left with the following:
p
Lc Lt
P
˙ g = A 2ρg ∆Pg ∗ f4 m
A 2ρg ∆Ptp ∗ f4 Retp , β,
e ∆Ptp
D D
Lt
differential pressure with two-phase flow, we can say that function six may contain these groups, i.e:
2ρg A2 Ld
ρg Lc Lt XLM , , Frg , Resg , Resl , Wetp , β, , , ρl Lt Ld
345
˙g = m where the function f6 is the corrective function. f6 As we know the dimensionless groups that could affect the
Before discussing the gas and one liquid component two phase flow DP meter dimensional analysis results and comparing them to published correlations it is necessary to discuss the case of gas with two liquid components. Some DP meters are now being used to meter flows of natural gas, oil and water. It has often been assumed that the gas type has little to no affect on the response of a DP meter with two phase flow. That is, for any given meter, it was assumed that as long as the gas density of two different gases matched, the gas dynamic pressure on the liquid was equal for a set gas flow rate and the wet gas flow response would therefore be the same. In 2006 Reader Harris et al [2] showed data to back this assumption. However, it was also assumed that for a set gas dynamic pressure, different liquid properties would affect a wet gas response of a DP meter. In the same paper Reader Harris et al [2] also showed data to back this assumption. The majority of the traceable two phase/wet gas flow data through DP meters is for gas with a single liquid component. Hence, most existing papers on the subject deal with one liquid component two phase/wet gas data. Therefore, the comparisons between the dimensional analysis results and the existing DP meter correlations needs to be done with the results of Sections 3 and 4. However, it is necessary here, in Section 5, to address the issue of two liquid components. If two liquids (say water and oil with subscripts ‘‘w’’ and ‘‘o’’ respectively) are treated separately then there are two parameters ˙ w and m ˙ o ), liquid density (ρw and ρo ), for liquid mass flow rate (m liquid viscosity (µw and µo ) and surface tension (σw and σo ). This adds four extra parameters to the parameter list in Section 2. We therefore derive four extra groups. That is, where as in Section 3 we had four dimensionless groups that account for liquid property effects (i.e. Π2 , Π3 , Π5 and Π6 ) we now have eight dimensionless groups that account for liquid property effects. Group 2: Π2A = Group 3: Π3A Group 5: Π5A Group 6: Π6A
˙o m ˙g m
and
Π2B =
˙w m
˙g m ρo ρw = and Π3A = ρg ρg µo µw = and Π5B = ρg Usg D ρg Usg D σo σw = and Π6B = . ρg Usg2 D ρg Usg2 D
346
R. Steven / Flow Measurement and Instrumentation 19 (2008) 342–349
Therefore we have:
∆Ptp = f7
˙ 2g m f 2ρg A2 7
where the function f7 can be written as:
˙w m ˙ o ρw ρo µg D µw D µo D σw ρg D3 σo ρg D3 m
, , , , , ˙g m ˙g m ˙g ρg ρg m ! 2 5 P d Lc Lt Ld e ρg D g , , , , , , . ˙ 2g m D D D D D ρg Usg2 ˙g m
,
˙g m
,
˙ 2g m
,
˙ 2g m
,
Here again, we can combine groups to create new groups. For example:
˙o m Π2A = , ˙w Π2B m µo Π5A = , = Π5B µw
Π3A ρo = , Π3B ρw Π6A σo = = . Π6B σw
Π2C =
Π3C =
Π5C
Π6C
Fig. 2. Murdock plot of two-phase dp meter data.
Note that Cd is the DP meter discharge coefficient and Y is the DP meter expansion factor for the case of dry gas flows. Note:
Therefore, we can re-arrange the expression to:
∆Ptp =
˙ 2g m
f 2ρg A2 8
Cd = f9 Resg
where the function f8 is different to function f7
and can be written as: f8
ρw ρo µg D µw D µo σw ρg D3 σo , , , , , , , ˙g m ˙ w ρg ρw m ˙g m ˙ g µw ˙ 2g m m σw ! 2 5 P d Lc Lt Ld e ρg D g , , , , , , . ˙ 2g m D D D D D ρg Usg2 ˙w m
,
˙o m
,
6. Discussion of the gas and one liquid component results An early detailed analysis of two-phase flow through a DP meter was the two-phase orifice plate meter analysis by Murdock [3]. In this analysis Murdock derived a two-phase flow model through an orifice plate and then took a wide array of data from various sources to fit an orifice plate meter two-phase correlation. The data included different meter diameters (D), different beta ratios (β ), different liquid properties (with the data including water, salt water and hydrocarbon liquids) and different gas velocities. Murdock’s model did not directly account for any two-phase effects of geometry (as he considered orifice plate meters alone and ignored any diameter and beta effects), fluid properties (including densities) or gas velocity. That is, Murdock did not directly consider any pressure or velocity effects. Note that the ‘‘over-reading’’ of a DP gas meter with twophase flow is defined as the ratio of the DP meters erroneous ˙ g Apparent ) to the actual gas mass gas mass flowrate prediction (m ˙ g ). Murdock plotted his orifice plate meter two-phase flowrate (m data on a graph that could be approximately described as the Lockhart Martinelli parameter (group Π10 ) vs. the over-reading ˙ g Apparent /m ˙ g ) as shown in Fig. 2. This form of plot is now (m generally called a ‘‘Murdock Plot’’. The Murdock correlation is:
˙g ≈ m
˙ g ,Apparent m 1 + 1.26XLM
Aβ 2 Ytp C d tp
√
=
√
2ρg ∆Ptp
1−β 4
1 + 1.26XLM
.
(3)
Y = f10 β, P , ∆Pg , κ
and
Ytp = f10 β, P , ∆Ptp , κ .
The numerator of the Murdock correlation is the erroneous ˙ g Apparent . However, two-phase flow gas mass flowrate term, m correction factors are functions that correct the erroneous gas flowrate result. The Venturi two-phase analysis produced the following equation:
A 2ρg ∆Ptp ∗ f4 Retp , β,
p
˙g = m
f6 XLM ,
ρg ρl
Lc Lt
, LLdt , LDd , De ,
, Frg , Resg , Resl , Wetp , β,
Lc Lt
,
Lt Ld
∆Ptp
,
P Ld D
, De
.
Note that we could do the same dimensional analysis on an orifice plate as has been done here for a Venturi. If we ignored geometric terms and concentrated on flow conditions we would get the same result:
A 2ρg ∆Ptp ∗ f5 Retp ,
p
˙g = m
f6 XLM ,
ρg ρl
∆Ptp
P
, Frg , Resg , Resl , Wetp
.
Therefore, as Murdock has not included any geometric terms we can say he has fitted function f6 . However, from modeling of the two-phase flow through an orifice plate Murdock suggested that the DP over-reading was only a function of the Lockhart Martinelli parameter. The Lockhart Martinelli parameter includes the gas to liquid mass flow rate ratio and gas to liquid density ratio terms in one parameter. Murdock did not account for any independent gas to liquid density ratio effect. In fact, with Murdock being one of the early researchers, he did not take account of several of the parameters listed above as having possible affects on DP meter two-phase flow over-readings. He did not account for the gas velocity (i.e. gas densiometric Froude number for any set meter diameter, pressure and liquid properties). He did not account for the meter geometry. He also did not directly consider the effect of liquid properties. Therefore the dimensional analysis result reduces to:
A 2ρg ∆Ptp ∗ f5 Retp ,
p
˙g = m
and
where κ is the gas isentropic exponent (a dimensionless term). When the flow is two-phase flow and the differential pressure read is not that of the dry gas flow functions f9 and f10 give the incorrect values (C dtp and Ytp respectively), i.e.: Cdtp = f9 Retp
The understanding of liquid property and ‘‘water cut’’ (i.e. the ratio of water to total volume flowrate) effects on DP meters in use with two-phase flow is beyond the limit of the published understanding. This topic is therefore at the forefront of twophase/wet gas flow metering research and as yet no particular set of dimensionless groups that combine these liquid property parameters has been selected by any researcher as the most appropriate. Therefore, no more can yet be said on this issue.
f11 (XLM )
∆Ptp P
.
It should be noted the Murdock correlation (Eq. (3)) matches this form. Note that Eq. (3) does include the beta ratio term in
R. Steven / Flow Measurement and Instrumentation 19 (2008) 342–349
Fig. 4. Venturi meter gas densiometric froude number effect for a set gas to liquid density ratio.
Fig. 3. Venturi meter gas to liquid density ratio effect.
the dry gas flow prediction but note we have ignored geometry terms here and therefore removed the beta ratio term. As we are discussing a generic meter geometry this is allowed by dimensional analysis. The effect of the geometry, including the beta ratio, would be accounted for in the form of function, f5 . In the following examples this argument also holds. Note that strictly speaking on the same argument we could choose to also drop the area term, A, as well but the author has chosen not to. Between 1967–77 Chisholm [4,5] developed orifice two-phase flow theory and for cases of XLM ≤ 1 Chisholm gave a two-phase flow correlation:
Aβ 2 Ytp Cdtp
√
√
˙g = m
2ρg ∆Ptp
1−β 4
q
(4)
2 1 + CXLM + XLM
on Murdock’s work by including an independent parameter of the gas to liquid density ratio along with the Lockhart Martinelli parameter. This gave more accurate over-reading predictions indicating the Lockhart Martinelli parameter did not fully account for the gas to liquid density ratio phenomenon. No further work was released on orifice plate meter two-phase flow response until 2007. However, in 1997, de Leeuw [6] released a seminal paper on Venturi meter two-phase flow characteristics. De Leeuw developed Chisholm’s orifice plate meter work to apply to Venturi meters and included the effect the gas velocity has on the system. Fig. 4 shows the gas densiometric Froude number effect de Leeuw discovered. De Leeuw had data recorded from one geometry of Venturi meter (i.e. an ISO 5167 Part 4 [1] compliant, 400 , schedule 80, 0.401 beta ratio). The result was the following correlation:
where
C =
14
+
ρl ρg
14
˙g = m
.
√
A 2ρg ∆Ptp ∗ f5 Retp ,
p
f12 XLM ,
ρg ρl
∆Ptp P
.
It should be noted the Chisholm correlation matches this form. A similar effect has been seen to exist for Venturi meters. Fig. 3 shows data from a 400 , schedule 80, 0.4 Beta Ratio ISO 5167 compliant Venturi meter tested at CEESI with natural gas and decane (C10 hydrocarbon liquid). A clear gas to liquid density ratio effect is visible. As the gas to liquid density ratio value increases for all other parameters held constant the over-reading reduces. Note that with regards to the Chisholm equation, an important point with regards to general dimensional analysis can be made here. There is often a misunderstanding that if a parameter is imbedded in one dimensionless group then the effect is fully accounted for in the system. Buckingham Pi theory does not state this. Here, as an example, we see that Murdock had the gas to liquid density ratio paired with the gas to liquid mass flowrate only (as part of the Lockhart Martinelli parameter). Chisholm improved
2ρg ∆Ptp
1−β 4
(4)
q
2 1 + CXLM + XLM
(4a)
With a similar flow model to Murdock, Chisholm considered slip between separated gas and liquid phases and stated that the slip was purely a function of the gas to liquid density ratio. However, like Murdock, Chisholm did not account for the geometry of the DP meter and did not independently consider any affect related to the gas velocity (e.g. gas densiometric Froude number for any set meter diameter, pressure and liquid properties). He also did not directly consider the effect of liquid properties (other than density). Therefore the dimensional analysis result reduces to:
˙g = m
Aβ 2 Ytp Cdtp
√
ρg ρl
347
where
C =
ρg ρl
n
+
ρl ρg
n (4b)
where n = f Frg .
(5)
De Leeuw did not take into consideration the geometry of the DP meter device and did not independently consider the effect of liquid properties. Therefore the dimensional analysis result reduces to:
A 2ρg ∆Ptp ∗ f5 Retp ,
p
˙g = m
ρ
f13 XLM , ρg , Frg l
∆Ptp P
.
It should be noted the de Leeuw correlation matches this form. It should be noted that Murdock, Chisholm and de Leeuw derived Eqs. (3), (4), (4a) and (5) respectively from flow modeling. No dimensional analysis was carried out in these derivations and yet some of the most important dimensional numbers (i.e. the Lockhart Martinelli parameter, the gas to liquid density ratio and the gas densiometric Froude number) were found by direct consideration of the physical phenomena. This shows that it is possible to derive practical correlations by flow modeling alone without the use of dimensional analysis. However, dimensional analysis is an extremely valuable tool that can aid engineers significantly when researching physical phenomena. Carrying out dimensional analysis in conjunction with modeling can
348
R. Steven / Flow Measurement and Instrumentation 19 (2008) 342–349
Fig. 6. NEL 400 , 0.75 beta ratio Venturi meter fluid property two-phase data.
Fig. 5. Venturi beta ratio effect for a set pressure (31 bara) and gas densiometric froude number (Frg = 1.5).
give significant extra insight and reassurance to the engineer developing a physical model. In 2002–2005 Stewart and Steven [7,8] produced a similar correlation for 400 and 600 cone type DP meters with a 0.75 beta ratio. The dimensionless groups used in this equation was the same as used by de Leeuw. Here again, as with de Leeuw’s Venturi meter work, the beta ratio was set. However, Stewart and Steven [7] did show comparisons of two different beta ratio cone type DP meters and showed that a beta ratio effect existed. For all other parameters held constant the over-reading reduced with increasing beta ratio. In 2003 Stewart [9] subsequently showed this same beta ratio effect to also exist for Venturi meters. Fig. 5 shows this effect. For all other parameters held constant increasing the beta ratio increases the over-reading. In 2007 Steven et al [10] then showed orifice plate meters have a slight beta ratio effect. No new correlations were supplied that accounted for any beta ratio affect. However, such a correlation would be of the form:
A 2ρg ∆Ptp ∗ f5 Retp ,
p
˙g = m
ρ
f14 XLM , ρg , Frg , β l
∆Ptp
P
.
It is unlikely that any practical correlation would be produced along the form of function f14 . It is considerably more practical to keep the correlations as simple as possible by setting a function for a given beta ratio. That is, it is more practical to restrict the choice of DP meter to a few set beta ratios where correlations are available. In 2006 Reader Harris et al [2,11] and Steven [12,13] showed that liquid properties do, under some flow conditions, have an influence on the scale of a DP meters two-phase over-reading. It was stated that water, with higher interfacial tension values compared to light hydrocarbon liquids, had lower over-readings for all other parameters held constant. Fig. 6 shows a NEL result for a 400 , schedule 80, 0.75 beta ratio, a gas to liquid density ratio of 0.046 and gas densiometric Froude number of 2.5 (see Reader Harris [11] for more details). Note that gas properties do not have an effect on this DP meters wet gas over-reading. No updated correlations were proposed by Reader Harris et al or Steven. However, under these findings it is practical to assume such a correlation could have the form (for a set meter geometry):
A 2ρg ∆Ptp ∗ f5 Retp ,
p
˙g = m
f15 XLM ,
ρg ρl
∆Ptp
P
.
, Frg , Resl , Wetp
It remains to be seen which, if either, of these liquid property dimensionless groups will be useful. It can be seen in Section 5 that there are several options for manipulating the liquid property
Fig. 7. A comparison of two different diameter geometrically similar venturi meters under similar two-phase flow dimensionless parameter values.
dimensionless groups and the usefulness of each group could be ascertained by applying it to experimental data. Physical modeling of the flow can also give significant insight and indicate the likely best form for liquid property dimensional groups. As it has been shown by Reader Harris and Steven that a liquid property effect does exist, dimensional analysis can help engineers when investigating this fluid property effect. Here, a liquid Reynolds number dimensionless group effect would imply a significant liquid viscosity effect, and a modified Weber number dimensionless group effect would imply a significant interfacial tension effect. Little is known with regards to the affect meter diameter has on two-phase flow DP meters. Steven [12,14] has indicated that a diameter effect may exist. It was pointed out by Steven [14] that the data set which led to this statement was extremely small and therefore no definite claim was made. However, the available data suggested that for an increasing diameter with the other dimensionless group values held constant the over-reading was seen to decrease. Fig. 7 shows the comparison made by Steven [14] between 400 and 200 0.6 beta ratio Venturi meters at similar gas to liquid density ratio and gas densiometric Froude numbers, with similar fluids. The smaller meter has a smaller over-reading. If a diameter effect is confirmed for two-phase DP meters, and industry desired one universal two-phase/wet gas correlation for a given DP meter design, this effect would be required to be a included. The form of the correlation would therefore be (for otherwise set geometry):
A 2ρg ∆Ptp ∗ f5 Retp ,
p
˙g = m
f16 XLM ,
ρg ρl
∆Ptp
P
, Frg , Resl , Wetp , β, LDd , De
.
R. Steven / Flow Measurement and Instrumentation 19 (2008) 342–349
However, as with the beta ratio effect, the diameter effect is unlikely to be added to any practical DP meter two-phase/wet gas flow correlation as it is far more practical to keep the correlations as simple as possible by setting a function for a set geometry. If this was not a research policy the correlations could become too complicated for practical use. Furthermore, each new affect required to be included in a correlation adds to the correlations over all uncertainty. With the potential for liquid fluid properties requiring to be added to the correlation variables, a policy of producing wet gas DP meter correlations for set geometries looks increasingly desirable. As the correlations require yet more flow condition parameters (i.e. parameters out with the control of the meter designer) an obvious method of reducing the number of overall variables in a correlation is to eliminate the geometry parameters (i.e. parameters controlled by the meter designer). That is, create DP meter wet gas correlations for set geometries only. 7. Conclusions Dimensional analysis has produced a set of dimensional numbers for two phase flow through a DP meter. When this set was reduced to dry gas flow conditions, the set correctly showed similarity with the actual gas Venturi flow equation, as required. When this set was reduced to the particular two phase flow conditions tested by individual researchers, the set correctly showed similarity with the correlation in question. There is a very limited quantity of DP meter wet gas flow data sets available in the public domain, where liquid property effects and geometry effects (including beta ratio and diameter effects) are examined. The few data sets that are discussed in the public domain indicate that there are liquid property and geometry effects. Therefore as the industry develops its understanding of these parameters effects on DP meters, dimensional analysis will aid the research and analysis of the experimental results. From a practical stand point, with more factors being found to influence the magnitude of gas DP meter wet gas flow overreadings, more factors are required to be accounted for in the corrective equations. However, the more factors there are, the greater the uncertainty in the overall correction. For this reason, and to avoid unnecessary complexity, it is likely that new correlations will not include geometry terms. New correlations are likely to be based on one geometry of DP meter. It will not be practical to apply such a policy to the case of liquid properties and have a practical DP meter wet gas correlation for the case
349
of gas flows with more than one liquid component. Whereas, meter users can have some influence on the geometry of a DP meter chosen, in most industrial metering applications the user has little to no choice of the flow conditions the chosen device must meter. For example, in many oil and gas industry twophase/wet gas flow applications there is a mix of water (sometimes salt water) and hydrocarbon liquids and this can change in composition through the useful life of a DP meter. It will therefore be desirable that the installed DP meter (of whatever geometry) has a wet gas correlation that can account for liquid property effects. Industry is still far from having a full set of trustworthy twophase/wet gas DP meter correlations for all possible applications. Much research remains to be conducted in this area. Several parameters are known to affect a DP meters two-phase/wet gas over-reading that as yet are not included in the correlations. The use of suitably chosen dimensionless groups is likely to be very important in understanding the phenomena and in helping developing useable two-phase/wet gas flow DP meter correlations. References [1] International Standards Organisation. International standard 5167-part 4. 2003. [2] Reader-Harris M. Ventuti-tube performance in wet gas using different test fluids. In: North sea flow measurement workshop 2006. [3] Murdock JW. Two-phase flow measurements with orifices. Journal of Basic Engineering 1962;84:419–33. [4] Chisholm D. Flow of incompressible two-phase mixtures through sharp-edged orifices. Journal of Mechanical Engineering Science 1967;9(1). [5] Chisholm D. Research note: Two-phase flow through sharp-edged orifices. Journal of Mechanical Engineering Science 1977;19(3). [6] De Leeuw R. Liquid correction of Venturi meter readings in wet gas flow. In: North sea workshop 1997. [7] Stewart D, Steven R, Peter RJW, Hodges D. Wet gas metering with V-cone meters. In: North sea flow measurement workshop 2002. Paper No.4.2. [8] Steven R, Kegel T, Britton C. An update on V-Cone meter wet gas flow metering research Flomeko 2005, Peebles, Scotland. [9] Stewart DG. Suitability of dry gas metering technology for wet gas metering. In: North sea flow measurement workshop 2003. [10] Steven, Stobie, Ting. A re-evaluation of axioms regarding orifice meter wet gas flow performance. In: 6th international south East Asia hydrocarbon flow measurement workshop. 2007. [11] NEL report 2005/206 Venturi-tube performance in wet gas using different test fluids, 2006. [12] Steven R, Kinney J, Britton C. Liquid property and diameter effects on venturi meters used with wet gas flows. In: 6th international symposium on fluid flow measurement. 2006. [13] Steven R. Horizontally installed differential pressure meter wet gas flow performance review. In: North sea flow measurement workshop. 2006. [14] Steven R. A discussion on wet gas flow parameter definitions. In: Energy institute/CEESI one day wet gas flow metering seminar. 2006.