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Wet gas pressure drop across orifice plate in horizontal pipes in the region of flow pattern transition
Flow Measurement and Instrumentation 71 (2020) 101678
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Flow Measurement and Instrumentation journal homepage: http://www.elsevier.com/locate/flowmeasinst
Wet gas pressure drop across orifice plate in horizontal pipes in the region of flow pattern transition Weiye Liu a, Youfu Ma a, b, *, Junfu Lyu b, Jie Shao a, Shuo Wang a a
Shanghai Key Laboratory of Multiphase Flow and Heat Transfer in Power Engineering, School of Energy and Power Engineering, University of Shanghai for Science and Technology, Shanghai 200093, China b Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Department of Energy and Power Engineering, Tsinghua University, Beijing 100084, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Gas–liquid flow Wet gas Orifice plate Pressure drop Prediction model
As a basis for measuring the mass flow rate of wet gas using differential pressure meters, predicting the pressure drop of a wet gas flowing through orifice plates is important; however, this has not yet been solved satisfactorily, although many studies have reported on that subject. In this study, the pressure drop of wet gas across sharpedged orifice plates was experimentally investigated in the region of flow pattern transition using air and water as the two phases, and the prediction performance of the available pressure drop models was compared based on the experimental data. The results show that the homogenous flow models overestimate the pressure drop, whereas those models based on the separated flow model often present underestimations. The models reported for wet gas are also incapable of predicting the pressure drop in this region with acceptable accuracy. Through an analysis of the prediction deviations, it is found that the Froude number of the liquid phase has a significant influence on the pressure drop of the wet gas, besides the Froude number of the gas phase. Then, three new correlations that are based on the homogeneous flow, Chisholm model, and Murdock model, respectively, were proposed based on the experimental result.
1. Introduction The use of wet gases is widespread in the nuclear power, oil, natural gas, chemical, and other industries. The definition of wet gas is not uniform at present. The American Society of Mechanical Engineers (ASME) uses the Lockhart–Martinelli (L–M) parameter X < 0.3 to define a wet gas, while in the oil and gas industry, well fluids with a gas volume fraction (GVF) greater than 90% are often considered as wet gas [1,2]. To acquire the flow rate of wet gas in pipelines, a throttling unit such as an orifice plate or venturi is generally assembled in the pipeline to take the differential pressure of the throttling unit as the signal of the wet gas flow rate [3–5]. Therefore, establishing an exact relation between the pressure drop and the wet gas flow rate is a fundamental work for measuring the wet gas flow rate in the pipelines of various industrial processes. The pressure drop model of a two-phase flow for throttling units has been advanced continuously since the 1960s, and the homogeneous flow model by James [6] and the separated flow models by Chisholm [7,8],
Murdock [9], and Lin [10] are widely regarded as classical models. These models take the prediction method of the gas phase multiplier ΦG or liquid phase multiplier ΦL as the goal, thus obtaining the pressure drop of a two-phase flow across the throttling unit based on that of a single-phase flow. Accordingly, it is considered that the parameters ΦG and ΦL mainly depend on the quality x (or X) and gas–liquid density ratio ρG/ρL, whereas they are independent of the structural parameters of the throttling unit, such as the diameter ratio β. Relatively speaking, ΦG is often used for wet gas because it is close to a pure gas. In addition, the term “over–reading (OR)” was also used in some studies [11–13] to represent the phenomenon that the pressure drop of a wet gas is larger than that of a single gas under the same mass flow rate. Different from the parameter ΦG, which uses the pressure drop ratio as the definition, the term OR uses the mass flow rate ratio as the definition; thus, OR can be regarded as an altered expression of ΦG. Nevertheless, it was reported that the prediction result by the classical two-phase flow pressure drop models for orifices tends to deviate under the wet gas condition. In other words, more factors need to be considered in the ΦG prediction of a wet
* Corresponding author. Shanghai Key Laboratory of Multiphase Flow and Heat Transfer in Power Engineering, School of Energy and Power Engineering, Uni versity of Shanghai for Science and Technology, Shanghai 200093, China. E-mail address: [email protected] (Y. Ma). https://doi.org/10.1016/j.flowmeasinst.2019.101678 Received 28 September 2019; Accepted 12 December 2019 Available online 14 December 2019 0955-5986/© 2019 Elsevier Ltd. All rights reserved.
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gas, such as the Froude number of the gas phase FrG [14–16]. In a wet gas the volume of the gas phase occupies the main part. Generally, the ideal wet gas flow pattern is expected to be the annu lar–mist flow, in which much of the liquid phase is distributed in the gas phase in the form of droplets. Therefore, some scholars believed that the homogeneous flow model is capable of predicting the wet gas pressure drop. For example, Fang et al. [17] performed an experimental study on the pressure drop of wet natural gas under low pressure (0.15–0.25 MPa) and high quality (x ¼ 0.50–0.99) using a venturi with diameter ratio β ¼ 0.4–0.7, and concluded that the original homogeneous flow model has the best prediction performance among the eight pressure drop models (including the Chisholm and Murdock models) they compared. More over, based on an experiment on slotted orifice plates with diameter ratio β � 0.5, Xing et al. [18] considered that the James homogeneous flow model has the best accuracy for predicting the wet gas pressure drop. Kumar et al. [19] also recommended the original homogeneous flow model based on their numerical simulation study on the wet gas pressure drop across multi-orifice plates. However, separated flow models, which basically involve a correc tion of the Chisholm or Murdock model, were also recommended by some studies. Regarding the correction of the Chisholm model, De Leeuw [20] proposed a correction on the power n (n ¼ 0.25 in the original Chisholm model) considering that n is a variable and is dependent on FrG. Steven and Hall [21] also recommended such a correction method to the wet gas pressure drop across orifice plates based on their experimental result. Besides, Steven and Hall [22] further examined the effect of changing the diameter ratio β (0.24–0.73) on the correction and concluded that the correction of n is not sensitive to a change in β. According to De Leeuw [20] and Steven and Hall [21], the power n increases with the increase in FrG. Xu et al. [19] also recom mended a correction on the Chisholm model based on an experiment on the wet gas pressure drop across a venturi. However, Xu et al. [22] concluded that the power n not only is dependent on FrG, but it is also related to the diameter ratio β and the relative length of the throat. He et al. [23] experimentally studied the wet gas pressure drop through a v-cone flowmeter under the conditions of low pressure 0.1–0.4 MPa, β ¼ 0.55, gas–liquid density ratio ρG/ρL ¼ 0.00247–0.00610, and FrG ¼ 0.3–1.5, with the result that n decreases with the increase in the liq uid–gas mass flow ratio, whereas it is almost independent of ρG/ρL and FrG. Then they correlated n with the liquid–gas mass flow ratio. There fore, regarding the wet gas pressure drop across orifices, although a correction of the Chisholm model has been recommended by many au thors, there is currently no unified conclusion on the correction method or on which factor should be considered for correcting the value of n. Moreover, a correction of the Murdock model was also recom mended by Xing et al. [18]. Based on an experimental study on the pressure drop across slotted multi-orifice plates under the GVF range of 99.49%–99.97%, they concluded that in the linear Murdock model, the slope k is dependent on both FrG and β, and it is positively correlated with FrG and negatively correlated with β, whereas the intercept b is only dependent on β, and it increases first and then decreases with the in crease in β. As a result, they proposed a correction of the Murdock model to predict the wet gas pressure drop across slotted multi-orifice plates by considering the effects of FrG and β on k and b. The relationship between the pressure drop and the flow pattern of a wet gas has rarely been reported. For a wet gas with GVF greater than 90%, its flow pattern can be a stratified, slug, or annular–mist flow. Generally, it is considered that the pressure drop model of a gas–liquid flow, a homogenous flow model or a separated flow model, should be in accordance with its flow pattern. Therefore, it is significant to clarify the pressure drop characteristics of a wet gas in the region of flow pattern transition. In this study, the wet gas pressure drop across orifice plates was experimentally investigated in the region of flow pattern transition, wherein the GVFs were in the range of 93.05%–99.31%. Based on the experimental results, the existing pressure drop models of orifice plates were evaluated, and new correlations were proposed for an accurate
prediction of the pressure drop in this flow region. 2. Prediction models for the wet gas pressure drop across orifice plate 2.1. James homogeneous flow model by 1
ρTP
For a gas–liquid mixture, the mixture density, ρTP, can be determined
¼
x
ρG
þ
1
x
ρL
(1)
:
The homogeneous flow model assumes that the two phases are mixed uniformly. As in the original homogeneous flow model, the pressure drop of a two–phase flow across an orifice plate is obtained by replacing the fluid density in the pressure drop equation for the single-phase flow with ρTP. That is ΔpTP ¼
m2TP ½1 þ ðρL =ρG
β4 Þ :
1Þx�ð1
2ρL ðKAÞ
2
(2)
In Eqs. (1) and (2), ΔpTP, mTP, and x are respectively the pressure drop (Pa), mass flow rate (kg/s), and quality of the two–phase flow; ρG and ρL are the density of the gas and liquid phases, respectively (kg/m3); β is the ratio of the orifice diameter d to the inner pipe diameter D, i.e., β ¼ d/D; and K and A are respectively the discharge coefficient and orifice area (m2) of the orifice plate. James [6] proposed a correction of the original homogeneous flow model by replacing the actual quality x in Eq. (2) with a corrected quality xm based on the experiment of measuring the flow rate of a geothermal steam–water two-phase flow. The corrected quality xm is given by (3)
xm ¼ x1:5 : Thus, the James model can be expressed as follows: ΔpTP ¼
m2TP ½1 þ ðρL =ρG
1Þx1:5 �ð1 2
2ρL ðKAÞ
β4 Þ :
(4)
2.2. Chisholm model and modified Chisholm model for wet gas The separated flow model assumes that the two phases flow through an orifice plate separately and the two phases have the same discharge coefficient. Chisholm [7,8] derived a pressure drop model for the gas–liquid flow through orifices as Eq. (5) based on the momentum conservation equations and considering the influence of the interfacial shear force on each flowing phase. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ΦG ¼ 1 þ CX þ X 2 ; (5) � when X < 1; C ¼
ρL ρG
� when X > 1; C ¼
�
�0:25
ρTP ρG
þ
ρG ρL
�
�0:25 þ
�0:25
ρG ρTP
; �0:25 ;
where X is the well-known Lockhart–Martinelli parameter, which can be determined by Eq. (12). The power n of the density ratio shown in the original Chisholm model is a constant (n ¼ 0.25). Steven and Hall [21] proposed a modified Chisholm model for a wet gas flowing through an orifice plate by considering the influence of FrG on ΦG. They concluded that the influ ence of FrG on ΦG is closely related to the flow pattern upstream of the orifice plate. That is, ΦG is insensitive to FrG and the Chisholm power n is a constant of 0.214 when the flow pattern upstream of the orifice plate is a stratified flow, which is judged by FrG � 1.5. With FrG increasing, the flow pattern could transit into an annular–mist flow when FrG > 1.5; in 2
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this case, the power n increases with an increment in FrG. As a result, they proposed a modified Chisholm model as given by Eq. (6), which is also recommended to use in the standard ISO/TR 11583:2012 (Mea surement of wet gas flow by means of pressure differential devices inserted in circular cross-section conduits) [24]. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ΦG ¼ 1 þ CX þ X 2 (6) � where C ¼
ρG ρL
�
�n þ
ρL ρG
where
� 60:6150
ρG ρL
� �2 � � ρ ρ 9:26541 G þ 44:6954 G �3
ρL
� �4 ρ 5:12966 G
ρL
ρL
� �5 . ρ 26:5743 G
ρL
From an experiment on wet gas flowing through slotted multi–orifice plates with the range of FrG ¼ 0.5–2.0 and β ¼ 0.50–0.75, Xing et al. [18] recommended the Murdock model for the pressure drop prediction with further consideration of the effect of β and FrG on ΦG. The detailed model by Xing et al. [18] is presented in Eq. (9), which indicates that the slope in the Murdock model depends on both β and FrG, whereas the intercept only depends on β. h �pffiffiffiffiffiffiffi . � i ΦG ¼ Cx 1 þ (9) FrG β X ;
�n ;
when FrG � 1:5; n ¼ 0:214; � 1 and when FrG > 1:5; n ¼ pffiffi 2
k ¼ 1:48625
��2 � 0:3 pffiffiffiffiffiffiffi ; FrG
where Cx ¼ 12:60β2
16:01β þ 5:81.
where FrG is the gas phase Froude number, which can be determined by Eq. (13).
3. Experimental system and data processing method
2.3. Murdock model and modified Murdock model for wet gas
3.1. Experimental system
The Murdock model [9], expressed in Eq. (7), is also representative of the separated flow model. Unlike the form of the Chisholm model, the Murdock model uses a linear relationship to relate ΦG and X with the experimental result of orifice pressure drop obtained from the steam– water, air–water, and natural gas–water fluids.
The experimental system is shown in Fig. 1. In the experiment, air and water at ambient temperature were used as the fluids of the gas and liquid phases, respectively. The air and water were introduced into a gas–liquid mixer by a compressor and a pump, respectively, to produce a gas–liquid mixture, where the intake air was taken from the atmosphere and the intake water was drawn from a water tank filled with tap water. The structure of the mixer is illustrated in Fig. 1, in which the water evenly permeated into the air pipe through twenty-five holes with diameter of 10 mm. Those holes were placed in a layout of five lines in circumference, each line having five holes with an axial spacing of 20 mm. Downstream of the mixer, the air–water mixture horizontally flowed through a test section, which consisted of two cir cular plexiglass pipes and an orifice plate fixed between them with a flange connection. Both the pipe for the air flow in the mixer and the pipe used for the air–water mixture flow had an inner diameter of 50 mm. The lengths of the plexiglass pipes upstream and downstream of the orifice plate were 0.5 m and 1.5 m, respectively. Besides, two steel pipes were placed upstream and downstream of the plexiglass pipe sections,
ΦG ¼ kX þ b
(7)
where k ¼ 1:26; b ¼ 1:0. Subsequently, Lin [10] advanced an improved Murdock model, expressed in Eq. (8), by considering the influence of the changing den sity ratio ρG/ρL on the prediction accuracy of the model. He proposed a correlation to the slope k of the original Murdock model, based on his experiment on a steam–water mixture under the ρG/ρL range of 0.00455–0.328. ΦG ¼ kX þ 1;
(8)
Fig. 1. Experimental system for the pressure drop of wet gas across orifice plate. 3
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respectively, which also had an inner diameter of 50 mm and whose lengths were 1.0 m and 2.0 m, respectively. The pressure taps used for the pressure drop across orifice plate were located at 20D upstream and 40D downstream of the orifice plate, respectively. Therefore, the two–phase flow pattern could be observed in the plexiglass pipe section while the pressure drop across the orifice plate could be obtained using a differential pressure transmitter by placing the pressure taps on the steel pipes. Besides, the measuring points for the temperature, pressure, and flow rate of the air and water were arranged in their intake pipes, as shown in Fig. 1. Meanwhile, valves were also equipped in the air and water intake pipes to regulate the flow rate. The measuring instruments used in the experiment are listed in Table 1. For an accurate measurement of the pressure drop across the orifice plate, considering that this pressure drop varies in a wide range with the change in test conditions, three differential pressure meters in different test ranges were used in the experiment to choose the proper meter in use according to the pressure drop. During the test process, the output analog signals exported by the measuring instruments were collected by a data acquisition module and then input in the computer through an A/ D communication module.
Table 2 Test conditions in the experiment. Water flow rate, QL (m3/h) Air flow rate, QG0 (m3/h)
40 60 80 100 120
1
2
3
(40, 1) (60, 1) (80, 1) (100, 1) (120, 1)
(40, 2) (60, 2) (80, 2) (100, 2) (120, 2)
(40, 3) (60, 3) (80, 3) (100, 3) (120, 3)
obtained by experimental measurement in this study, thus eliminating the uncertainty of determining the ΔpGO by calculations. Moreover, the gas quality x can be determined by x¼
mG QG ρG ¼ ; mG þ mL QG ρG þ QL ρL
(11)
where m, Q, and ρ are the mass flow rate (kg/h), volumetric flow rate (m3/h) and density of a phase, and subscripts G and L denote the gas and liquid phase, respectively. In this study, QG and QL are obtained by the flow rate meters, whereas ρG and ρL are determined by calculation ac cording to the temperature and pressure of the air and water used. It should be noted that the air flow rates from the meter (QG0 ) were used to carry out the experiment according to the designed conditions listed in Table 2, but QG in Eq. (11) and JG in Eqs. (13) and (14) were determined by a pressure correction according to the pressure change between the location of the air flow rate meter and the location of the orifice plate upstream. Similarly, ρG in Eq. (11), and in Eqs. (12)–(14), were also determined by the pressure upstream of the orifice plate. Then, the LM parameter X can be determined by rffiffiffiffiffi 1 x ρG : (12) X¼ x ρL
3.2. The test orifice plate and the test conditions In this study, a sharp-edged orifice plate with orifice diameter d ¼ 25 mm and orifice thickness t ¼ 5 mm was used for the experiment. Thus, the orifice plate for the test has a diameter ratio β ¼ 0.50 (D ¼ 50 mm) and relative orifice thickness t/d ¼ 0.2. Fifteen test conditions were designed with ranges of water flow rate QL ¼ 1–3 m3/h and air flow rate QG0 ¼ 40–120 m3/h, as presented in Table 2. As a result, the GVFs of the two-phase flow are in a range of 93.05%–99.31%. Each condition was tested twice to reduce the uncer tainty of the experimental result. In addition, the pressure drops of air flowing alone through the orifice plate were also tested under the five air flow rates listed in Table 2, thereby obtaining the gas phase multiplier ΦG from the experimental result.
The gas phase Froude number FrG and the liquid phase Froude number FrL are also considered as important factors influencing ΦG in a wet gas. They can be determined as follows: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi JG ρG ; (13) FrG ¼ pffiffiffiffiffiffi gD ρL ρG
3.3. Data processing methods In this study, the gas phase multiplier ΦG is used to represent the pressure drop of the wet gas. ΦG is determined by sffiffiffiffiffiffiffiffiffiffi ΔpTP ; (10) ΦG ¼ ΔpGO
JL FrL ¼ pffiffiffiffiffiffi gD
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ρL
ρL
ρG
(14)
;
where g is the gravitational acceleration (9.81 m/s2); D is the inner diameter of the pipe (m) and D ¼ 0.05 m in this study; JG and JL are the superficial velocity of the gas and liquid phase (m/s), which are given by
where ΔpTP and ΔpGO are the orifice pressure drops of the two–phase flow and the gas phase only, respectively (Pa). Both ΔpTP and ΔpGO are
JG ¼
Table 1 Instruments used in the experiment.
QG =3600 Q =3600 � ; JL ¼ L � : π D2 4 π D2 4
(15)
Parameters
Devices
Type
Range
Accuracy
Air flow rate
Vortex flowmeter Pressure transmitter
35–350 m3/h 0–100 kPa
1.5
Air pressure near the flow rate meter Water flow rate
LUGB-MKDN50 EJA430ADAS4A-92DA
In addition, the relative deviation ER is defined as follows to compare the prediction performance of the orifice pressure drop models with the experimental result.
0.2
For a homogeneous flow model: ER ¼ ðΔpTPC
LWGY15A0A3C3 MIK-WZPK20A
0.5–5 m3/h 0–100 � C
0.5
For a separated flow model: ER ¼ ðΦGC
0.5
MIK-P300
0–50 kPa
0.5
EJA120ADES4A-92DA EJA110ADLS4A-92DA MIK-3051
0.1–1.0 kPa 0.5–10 kPa 0–100 kPa
0.2
Air/water temperature Pressure upstream of orifice plates Pressure drop across orifice plates
Turbine flowmeter Platinum resistance thermometer Pressure transmitter Differential pressure transmitter
ΔpTPE Þ = ΔpTPE � 100% (16)
ΦGE Þ = ΦE � 100%;
(17)
where ΔpTPE and ΔpTPC are the pressure drops of the gas–liquid flow obtained by the experiment and the prediction model, respectively (Pa); ΦGE and ΦGC are the gas phase multipliers obtained by the experiment and by the prediction model, respectively.
0.2
3.4. Uncertainty analysis
0.5
In the experiment, ΔpTP and ΔpGO were obtained directly by differ ential pressure transmitters. Therefore, their uncertainties can be 4
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determined by the measuring value and the range, accuracy of trans mitters. Then, the uncertainty of ΦG can be obtained by uncertainty propagation principle with its definition as Eq. (10). As the dimension less parameters, FrG and FrL are often used to indicate the influences of superficial gas and liquid velocity, respectively, on the pressure drop of wet gas flow. Therefore, their uncertainties were also analyzed. Un certainties of FrG and FrL mainly depend on the uncertainties of measuring gas and liquid flow rates, and the uncertainties of ρG that are derived from the measurements of gas temperature and pressure, while the uncertainties of ρL are not considered because it does not change much in the experiment. The uncertainty analysis results are listed in Table 3, in which the uncertainty of a parameter Y is expressed by U(Y). 4. Experimental results and discussion 4.1. Results of two-phase flow patterns and pressure drops
Fig. 2. Experiment results of the flow pattern upstream of the orifice plate.
The flow patterns in a wet gas are generally classified as stratified, slug, and annular–mist flows. Based on the flow pattern map of the Shell Company [1], the test conditions of this study and the corresponding results of flow patterns upstream of the orifice plate are shown in Fig. 2, using FrG as the horizontal axis and FrL as the vertical axis. It can be observed from Fig. 2 that the test conditions of this study are in the region of flow pattern transition according to the flow pattern map of the Shell Company. From the observation of the flow pattern up stream of the orifice plate, the three flow patterns, namely, stratified, slug, and annular–mist, were observed in the experiment, as shown in Fig. 3. The flow pattern results obtained by the visualization are listed in Fig. 2. This indicates that, in principle, the experimental results of flow patterns upstream of the orifice plate agree with the flow pattern map. Fig. 4 shows the experimental result of pressure drop across the test orifice plate. It can be seen from Fig. 4 that the wet gas pressure drop across the orifice plate increases with an increase in both FrG and FrL. This indicates that for a wet gas, the pressure drop across orifice plates can effectively reflect the change in wet gas flow rate. 4.2. Evaluating the existing models with the experimental result It was stated in Section 2 that the existing prediction models for pressure drop of a wet gas across an orifice plate can be classified into three categories, namely, the James, Chisholm, and Murdock models. These models are based on different assumptions such as a homogeneous or separated flow, and differ in relation to the influencing factors used. To compare their predicting performance, the three types of prediction models are evaluated hereinafter with the experiment results of this study.
Fig. 3. The three flow patterns of wet gas upstream of the orifice plate.
4.2.1. Evaluation of homogeneous flow models The relative deviations between the experimental result and the result predicted by the original homogeneous flow model, i.e., Eq. (2) are shown in Fig. 5. It can be observed from Fig. 5 that the original homogeneous flow model largely overestimates the pressure drop. Corresponding to the FrL of the experiment at 0.20, 0.40, and 0.60, ER values are in the ranges of 294%–922%, 252%–555%, and 177%–446%, respectively. The relative deviations between the experimental result and the result predicted by the James homogeneous flow model, i.e., Eq. (4) are shown in Fig. 6. Corresponding to FrL at 0.20, 0.40, and 0.60, ER values are in the ranges of 61%–144%, 11%–31%, and 18% – 4.4%, respec tively. It can be observed from Fig. 6 that the deviations mainly depend Table 3 Uncertainty analysis results of the experimental result. U(FrG)
U(FrL)
U(ΔpTP)
U(ΔpGO)
U(ΦG)
1.7%–8.3%
0.7%–2.3%
0.2%–4.1%
0.2%–1.2%
0.1%–2.2%
Fig. 4. Experimental results of pressure drop across the orifice plate. 5
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Fig. 7. Prediction deviations of the Chisholm model.
Fig. 5. Prediction deviations of the original homogeneous flow model.
the ranges of 16%–7.9%, 28% to 12%, and 41% to 19%, respectively. It can be observed from Fig. 7 that the deviations of the model are mostly negative, meaning that, unlike the overestimation by the homogeneous flow models, the Chisholm model yields a certain underestimation of the pressure drop of wet gas in the region of flow pattern transition. As can be seen from Fig. 7, the deviations are small when FrL ¼ 0.20, and they tend to increase with an increase in FrL. Meanwhile, the deviations are also amplified as FrG increases under the same FrL. Based on the Chisholm model, Steven and Hall [21] proposed a correction of the power n of the density ratio of gas to liquid considering the influence of FrG on n, as expressed by Eq. (6). According to Eq. (6), the influence of FrG on n is relevant only when FrG > 1.5, and it is suggested that n ¼ 0.214 when FrG � 1.5. In the experiment of this study, the FrG values are in the range of 0.24–1.29, which are all below 1.5; thus, n ¼ 0.214 was used in this study for the calculations. The relative deviations between the experimental result and the result predicted by the Steven and Hall model, i.e., Eq. (6) are shown in Fig. 8. Corre sponding to FrL values of the experiment at 0.20, 0.40, and 0.60, ER values are in the ranges of 20%–0.034%, 33% to 18%, and 45% to 25%, respectively. It can be observed from Fig. 8 that the deviations of the Steven and Hall model are similar to those of the Chisholm model, except that in this case, all the deviation points are shifted down by a certain degree.
Fig. 6. Prediction deviations of the James homogeneous flow model.
on the values of FrL, i.e., the deviations are small when FrL ¼ 0.60 and tend to increase with a decrease in FrL. Meanwhile, the deviations also show a relation with FrG, that is, under the same FrL, the points of the deviations show a decline as FrG increases. Such a variation in the deviations shown by Fig. 6 might be attrib uted to the change in wet gas flow patterns in the experiment of this study. The flow patterns in the experiment are mostly stratified flows when FrL ¼ 0.20, while they are mostly slug flows when FrL ¼ 0.40 and FrL ¼ 0.60. Therefore, it might be interpreted from the deviation results that the James model is capable of predicting pressure drops of a wet gas with slug flow, whereas that model is not suitable for a stratified flow because it deviates from a homogeneous flow significantly. In addition, it can be found from Fig. 2 that the flow pattern of a wet gas tends to be an annular–mist flow as FrG reaches a certain value. An annular–mist flow is closer to a homogeneous flow than a slug flow. This might be the reason why, as shown in Fig. 6, the deviations of the conditions with a high FrG are relatively small. Xing et al. [18] approved the use of the James homogeneous flow model for slotted orifice plates based on their experiments, and the reason might be that their test was conducted under large gas velocity conditions (FrG ¼ 1.11–1.95), and therefore, with a fully-developed annular–mist flow according to the flow pattern map of wet gas, i.e., Fig. 2.
4.2.3. Evaluation of the Murdock–type models The Murdock model is another typical separated flow model. The relative deviations between the experimental result and the result pre dicted by the Murdock model, i.e., Eq. (7) are shown in Fig. 9.
4.2.2. Evaluation of Chisholm–type models The Chisholm model is a typical separated flow model. The relative deviations between the experimental result and the result predicted by the Chisholm model, i.e., Eq. (5) are shown in Fig. 7. Corresponding to FrL values of the experiment at 0.20, 0.40, and 0.60, the ER values are in
Fig. 8. Prediction deviations of the Steven and Hall model. 6
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Fig. 9. Prediction deviations of the Murdock model.
Fig. 11. Prediction deviations of the Xing model.
Corresponding to the FrL values of the experiment at 0.20, 0.40, and 0.60, ER values are in the ranges of 28% to 12%, 41% to 25%, and 52% to 29%, respectively. It can be observed from Fig. 9 that the deviations of the model are all negative, meaning that, as in the un derestimation by the Chisholm model, the Murdock model yields an obvious underestimation of the pressure drop of wet gas in the region of flow pattern transition. In addition, the results of the evaluation of the Murdock model are similar to those of the Chisholm model or Steven and Hall model regarding the effect of FrG and FrL on the deviations. That is, the deviations are small when FrL ¼ 0.20 and tend to increase with an increase in FrL; meanwhile, the deviations tend to augment as FrG in creases under the same FrL. Accordingly, the relative deviations between the experimental result and the result predicted by the Lin model, i.e., Eq. (8) are shown in Fig. 10. Corresponding to the FrL values of the experiment at 0.20, 0.40, and 0.60, ER values are in the ranges of 26% to 5.0%, 37% to 17%, and 49% to 20%, respectively. It can be observed from Fig. 10 that the Lin model also yields an underestimation of the pressure drop of wet gas in the region of flow pattern transition. Altogether, the de viations of the Lin model are similar to those of the Murdock model, except that in this case, all the deviation points are shifted up by a certain degree. This indicates that the consideration of the influence of the gas–liquid density ratio could improve the prediction performance of the Murdock model. However, as can be seen in Fig. 10, the deviations are still large. The relative deviations between the experimental result and the result predicted by the Xing model, i.e., Eq. (9) are shown in Fig. 11. Corresponding to the FrL values of the experiment at 0.20, 0.40, and 0.60, ER values are in the ranges of 24% to 14%, 37% to 29%, and
45% to 32%, respectively. It can be observed from Fig. 11 that the Xing model also yields an underestimation of the pressure drop of wet gas in the region of flow pattern transition. Because the effect of FrG on ΦG is considered in the Xing model, under the same FrL, the deviations for different FrG values become stable. This means that the correction proposed by Xing et al. [18] is effective to some extent for the effect of FrG on the pressure drop of wet gas. However, as can be seen in Fig. 11, the prediction deviations of the model are still large, and the deviations tend to increase as FrL increases. 4.3. New correlations for the pressure drop of wet gas across orifice plates The evaluations of the existing models for the two–phase pressure drop across orifice plates show that the deviations of these models are large when they are used for the wet gas pressure drop in the region of flow pattern transition. In summary, the models based on a homogenous flow model overestimate the pressure drop, whereas those based on a separated flow model provide an underestimation. For an accurate prediction of the wet gas pressure drop in the region of flow pattern transition, the abovementioned three types of prediction models are improved hereinafter with the experiment results of this study. 4.3.1. Correlations from a modification of the homogeneous flow model By comparing Eqs. (2) and (4), it can be observed that they differ only in the power of the quality x, with power 1.0 for the original ho mogeneous flow model and power 1.5 for the James homogeneous flow model. This means that a correction of the power of x is an effective method for the modification of the homogeneous flow model. The evaluation results shown by Figs. 5 and 6 indicate that, when using the homogeneous flow model, both FrG and FrL are related to the prediction deviations. Therefore, to make the prediction result equal to the experimental result of this study, the power n in Eq. (4) should vary with both FrG and FrL. With the experimental result of this study, the power n can be obtained by letting the power of x in Eq. (4) as an unknown. According to the calculations, the results of the power n are shown in Fig. 12. It can be seen from Fig. 12 that the power n decreases rapidly with an increase in FrL, and decreases slowly with an increase in FrG. With the surface-fitting method, the effect of FrL and FrG on the power n can be expressed by the new equation for the corrected quality xm. Thus, from the modification of the homogeneous flow model, the new correlation proposed in this study is given by ΔpTP ¼
m2TP ½1 þ ðρL =ρG
1Þxn �ð1
2ρL ðKAÞ
2
β4 Þ ;
(18)
where n ¼ 2:17 0:10FrG 1:76FrL þ 1:26Fr2L . The prediction accuracies of the correlation, Eq. (18), are shown in
Fig. 10. Prediction deviations of the Lin model. 7
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Fig. 12. Effects of FrG and FrL on the power n in the homogeneous flow model.
Fig. 14. Effects of FrG and FrL on the power n in the Chisholm model.
The prediction accuracies of the correlation, Eq. (19), are shown in Fig. 15. As can be seen, the relative deviations of Eq. (19) are in the range of 6%–6%. 4.3.3. Correlations from a modification of the Murdock model The Murdock model was also proposed by Xing et al. [18] as a base model for predicting the wet gas pressure drop across orifice plates, wherein they used FrG to modify the slope in the Murdock model, as expressed by Eq. (9). However, the evaluation results shown in Figs. 9–11 indicate that, when using the Murdock model, both FrG and FrL are related to the prediction deviations. Because there are two un knowns in the Murdock model, i.e., the slope k and the intercept b, the effects of changing FrG and FrL on the Murdock model, specifically the k and b in the model, must be clarified before advancing a new correlation. With the experimental result of this study, the effect of changing FrG on the Murdock model is illustrated by Fig. 16. In Fig. 16, the test results are classified by the different air flow rate QG0 . Because of the variations in the air pressure difference between the flow meter point and the upstream point of the orifice plate for the different test conditions, FrG varies in a certain range under the same QG0 . For the analysis, the test results under the same QG0 are approximately regarded as under the same FrG. It can be observed in Fig. 16 that ΦG and X show a linear relationship under the same FrG, and the slope k of the line increases with the increase in FrG. This agrees with the result obtained by Xing et al. [18], who proposed a correction of k by relating that with FrG. Meanwhile, it is interesting that in Fig. 16 the lines of different FrG have an almost identical b, which varies in the narrow range of 0.68–0.73, i.
Fig. 13. Prediction accuracies of the new correlation based on the homoge neous flow model.
Fig. 13. As can be seen, the relative deviations of Eq. (18) are in the range of 5%–6%. 4.3.2. Correlations from a modification of the Chisholm model The Chisholm model was widely used as the base model for pre dicting the wet gas pressure drop across orifice plates. The specific modification method considers the effect of FrG on the pressure drop by letting the power n of gas–liquid density ratio (ρG/ρL) be related with FrG, as expressed in Eq. (6). Such a modification can be found in the works of De Leeuw [20], Steven and Hall [21], and Xu et al. [22]. However, the evaluation results shown by Figs. 7 and 8 indicate that, when using the Chisholm model, both FrG and FrL are related to the prediction deviations. Therefore, to make the prediction result equal to the experimental result of this study, the power n in Eq. (6) should vary with both FrG and FrL. With the experimental result of this study, the powers can be obtained by letting the n in Eq. (6) as an unknown. Ac cording to the calculations, the results of the power n are shown in Fig. 14. It can be observed from Fig. 14 that the power n increases with the increase in both FrG and FrL. With the surface-fitting method, the effect of FrL and FrG on the power n can be expressed by a new equation. Thus, from the modification of the Chisholm model, the new correlation proposed in this study is expressed by pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ΦG ¼ 1 þ CX þ X 2 ; (19) where C ¼
� �n ρG ρL
� �n þ
ρL ρG
Fig. 15. Prediction accuracies of the new correlation based on the Chis holm model.
0:33 ; n ¼ 0:52Fr0:25 . G FrL
8
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Fig. 16. Effects of changing FrG on the slope of the Murdock model.
Fig. 18. Relationship between intercept b and FrL based on Murdock model.
e., with the average value of 0.71. Meanwhile, with the experimental result of this study, the effect of changing FrL on the Murdock model can also be illustrated by Fig. 17. In Fig. 17, the test results are classified by the different FrL. It can be found from Fig. 17 that, in this case, the effect of changing FrL on the Murdock model also shows a certain regularity. That is, the intercept b of the model increases with the increase in FrL, while the slope k approximately remains unchanged (k ¼ 1.22). This indicates that the parameter FrL can also be used to modify the Murdock model to achieve an accurate pre diction of the wet gas pressure drop across orifice plates in the region of flow pattern transition. It can be found that both the correction of k with FrG and the correction of b with FrL are effective for a modification of the Murdock model. As a stable water flow rate, i.e., a stable FrL, was utilized in this experiment, the modification of the Murdock model is conducted by the correction of b with FrL. In this case, k is equal to 1.22, as shown by Fig. 17. The three results of b can be obtained by extending the line in Fig. 17 to the crossing point with the vertical axis. Then, the relationship between b and FrL is given by a quadratic polynomial expression, as shown in Fig. 18. As a result, from the modification of the Murdock model, the new correlation proposed in this study is given by ΦG ¼ 1:22X þ 1:75Fr2L þ 1:75FrL þ 0:95:
Fig. 19. Deviation diagram of novel correlation by Murdock model.
three models, i.e., the modifications of the homogenous flow model, the Chisholm model, and the Murdock model show approximately the same performance. The correlations proposed in this study, i.e., Eqs. (18)– (20), are validated by the experiment of this work, carried out for a wet gas in the region of flow pattern transition under the conditions of FrG ¼ 0.29–1.23 and FrL ¼ 0.20–0.60.
(20)
The prediction accuracies of the correlation, Eq. (20), are shown in Fig. 19. As can be seen, the relative deviations of Eq. (20) are in the range of 8%–8%. Overall, regarding the prediction accuracy of the wet gas pressure drop across orifice plates in the region of flow pattern transition, all the
5. Conclusions In this study, the pressure drop characteristics of a wet gas flowing through a sharp–edged orifice plate were experimentally studied in the region of flow pattern transition. Based on a summary of the existing prediction models for the wet gas pressure drop across orifice plates, the three typical models, i.e., the homogenous flow, Chisholm, and Murdock models, were evaluated with the experimental result of this study. Then, based on the three models, new correlations were proposed for the wet gas pressure drop across orifice plates in the region of flow pattern transition. The following conclusions can be drawn: The experiment of this study was conducted under the conditions of FrG ¼ 0.29–1.23 and FrL ¼ 0.20–0.60, for a wet gas in the region of flow pattern transition according to the Shell flow pattern map for wet gases. All the flow patterns of wet gases, i.e., stratified, slug, and annular–mist flows, were observed in the experiment of this study, and it is confirmed that the boundaries of the flow pattern transitions judged by the experimental result are in accordance with those of the Shell flow pattern map. Compared with the experimental result, the original homogeneous flow model provides a large overestimation of the pressure drop. The
Fig. 17. Effects of changing FrL on the intercept of the Murdock model. 9
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prediction deviations of the James homogeneous flow model are significantly mitigated in comparison to those of the original homoge neous flow model, but there is an obvious increase in prediction de viations as FrL decreases from 0.60 to 0.20. The prediction deviations indicate that the homogeneous flow model must be corrected by the two factors, FrG and FrL, to improve its prediction accuracy. By modifying the homogeneous flow model, a new correlation was proposed in this study, which shows accuracies within 6% when compared with the experi mental result. Compared to the experimental result, the three typical separated flow models, i.e., the Chisholm, Murdock, and Lin models, present an obvious underestimation on the pressure drop. The evaluation results also indicate that the traditional separated flow models must be cor rected by the two factors, FrG and FrL, to improve their prediction ac curacies. By modifying the Chisholm and Murdock models, two new correlations were proposed in this study, respectively, attaining pre diction accuracies within 6% and 8%. The models reported in the literature for the wet gas pressure drop across orifice plates, such as the Chisholm-type model by Steven and Hall [18] and the Murdock-type model by Xing et al. [15], are also evaluated with the experimental result of this study. The results indicate that they also exhibit an underestimation of the pressure drop, although the magnitude of the deviations is mitigated to some extent. Generally, the factor FrG was considered in those corrected models for wet gas. However, the evaluation results of this study reveal that FrL is also an important factor that cannot be neglected, especially when a homoge neous flow or Chisholm-type model is used. Three new correlations based on the homogenous flow, Chisholm model, and Murdock model, respectively, were proposed in this study, as expressed by Eqs. (18)–(20). They exhibit approximately the same per formance with respect to their prediction accuracy.
[2] R. Steven, Wet Gas Flow Metering with Gas Meter Technologies, Colorado Engineering Experiment Station Inc, USA, 2007, pp. 1–40. [3] S.R.V. Campos, J.L. Balino, I. Slobodcicov, D.F. Filho, E.F. Paz, Orifice plate meter field performance: formulation and validation in multiphase flow conditions, Exp. Therm. Fluid Sci. 58 (2014) 93–104. [4] C. Alimonti, G. Falcone, O. Bello, Two-phase flow characteristics in multiple orifice valves, Exp. Therm. Fluid Sci. 34 (2010) 1324–1333. [5] A.R. Almeida, A model to calculate the theoretical critical flow rate through venturi gas lift valves (includes Addendum), SPE J. 16 (2013) 134–147. [6] R. James, Metering steam-water two-phase by sharp–edged orifices, Proc. Inst. Mech. Eng. 180 (1965) 549–566. [7] D. Chisholm, Flow of incompressible two–phase mixtures through sharp-edged orifices, J. Mech. Eng. Sci. 9 (1967) 72–78. [8] D. Chisholm, Research note: two–phase flow through sharp-edged orifices, J. Mech. Eng. Sci. 19 (1977) 128–130. [9] J.W. Murdock, Two-phase flow measurement with orifices, J. Basic Eng. 84 (1962) 419–433. [10] Z.H. Lin, Two-phase flow measurements with sharp-edged orifice, Int. J. Multiph. Flow 8 (1982) 683–693. [11] L.D. Fang, Y. Zhang, X.J. Wang, The study of wet gas measurement model for new inside and outside tube differential pressure flowmeter, J. Sens. Actuators (China) 8 (2013) 1173–1177. [12] X.B. Zheng, D.F. He, Z.G. Yu, B.F. Bai, Deviation analysis of gas and liquid flow rates metering method based on differential pressure in wet gas, Exp. Therm. Fluid Sci. 79 (2016) 245–253. [13] W.X. Chen, Y. Xu, C. Yuan, H.T. Wu, T. Zhang, An investigation of wet gas overreading in orifice plates under ultra–low liquid fraction conditions using dimensional analysis, J. Nat. Gas Sci. Eng. 32 (2016) 390–394. [14] Y.F. Geng, J.W. Zheng, T.M. Shi, Study on the metering characteristics of a slotted orifice for wet gas flow, Flow Meas. Instrum. 2 (2006) 123–128. [15] L.D. Fang, T. Zhang, Performance of a horizontally mounted venturi in low–pressure wet gas flow, Chin. J. Chem. Eng. 2 (2008) 320–324. [16] C. Yuan, Y. Xu, T. Zhang, J. Li, H.X. Wang, Experimental investigation of wet gas over reading in Venturi, Exp. Therm. Fluid Sci. 66 (2015) 63–71. [17] L.D. Fang, T. Zhang, N.D. Jin, A comparison of correlations used for Venturi wet gas metering in oil and gas industry, J. Pet. Sci. Eng. 57 (2007) 247–256. [18] L.C. Xing, Y.F. Geng, M.M. Sun, A new correlation of slotted orifice for gas-liquid two-phase flow with low liquid fractions, Proc. Chin. Soc. Electr. Eng. 14 (2008) 86–91 (China). [19] P. Kumar, W.M.B. Michael, A CFD study of low pressure wet gas metering using slotted orifice meters, Flow Meas. Instrum. 22 (2011) 33–42. [20] R. De Leeuw, Liquid Correction of Venturi Meter Readings in Wet Gas Flow, North Sea Flow Measurement Workshop 1997, Norway, 1997. [21] R. Steven, A. Hall, Orifice plate meter wet gas flow performance, Flow Meas. Instrum. 20 (2009) 141–151. [22] Y. Xu, Q. Zhang, T. Zhang, X.L. Ba, An overreading model for nonstandard Venturi meters based on H correction factor, Measurement 61 (2015) 100–106. [23] D.F. He, S.L. Chen, B.F. Bai, A V-Cone meter measurement correlation in low pressure wet gas based on Chisholm model, Flow Meas. Instrum. 66 (2019) 12–17. [24] International Organization for Standardization, ISO/TR 11583:2012, Measurement of Wet Gas Flow by Means of Pressure Differential Devices Inserted in Circular Cross-Section Conduits, 2012, pp. 9–12.
Acknowledgments This study was supported by the National Key Research and Devel opment Program of China (2016YFB0600201). References [1] ASME, Wet Gas Flowmetering Guideline, ASME MFC-19G-2008, New York, USA, 2008.
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Contents lists available at ScienceDirect
Flow Measurement and Instrumentation journal homepage: http://www.elsevier.com/locate/flowmeasinst
Wet gas pressure drop across orifice plate in horizontal pipes in the region of flow pattern transition Weiye Liu a, Youfu Ma a, b, *, Junfu Lyu b, Jie Shao a, Shuo Wang a a
Shanghai Key Laboratory of Multiphase Flow and Heat Transfer in Power Engineering, School of Energy and Power Engineering, University of Shanghai for Science and Technology, Shanghai 200093, China b Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Department of Energy and Power Engineering, Tsinghua University, Beijing 100084, China
A R T I C L E I N F O
A B S T R A C T
Keywords: Gas–liquid flow Wet gas Orifice plate Pressure drop Prediction model
As a basis for measuring the mass flow rate of wet gas using differential pressure meters, predicting the pressure drop of a wet gas flowing through orifice plates is important; however, this has not yet been solved satisfactorily, although many studies have reported on that subject. In this study, the pressure drop of wet gas across sharpedged orifice plates was experimentally investigated in the region of flow pattern transition using air and water as the two phases, and the prediction performance of the available pressure drop models was compared based on the experimental data. The results show that the homogenous flow models overestimate the pressure drop, whereas those models based on the separated flow model often present underestimations. The models reported for wet gas are also incapable of predicting the pressure drop in this region with acceptable accuracy. Through an analysis of the prediction deviations, it is found that the Froude number of the liquid phase has a significant influence on the pressure drop of the wet gas, besides the Froude number of the gas phase. Then, three new correlations that are based on the homogeneous flow, Chisholm model, and Murdock model, respectively, were proposed based on the experimental result.
1. Introduction The use of wet gases is widespread in the nuclear power, oil, natural gas, chemical, and other industries. The definition of wet gas is not uniform at present. The American Society of Mechanical Engineers (ASME) uses the Lockhart–Martinelli (L–M) parameter X < 0.3 to define a wet gas, while in the oil and gas industry, well fluids with a gas volume fraction (GVF) greater than 90% are often considered as wet gas [1,2]. To acquire the flow rate of wet gas in pipelines, a throttling unit such as an orifice plate or venturi is generally assembled in the pipeline to take the differential pressure of the throttling unit as the signal of the wet gas flow rate [3–5]. Therefore, establishing an exact relation between the pressure drop and the wet gas flow rate is a fundamental work for measuring the wet gas flow rate in the pipelines of various industrial processes. The pressure drop model of a two-phase flow for throttling units has been advanced continuously since the 1960s, and the homogeneous flow model by James [6] and the separated flow models by Chisholm [7,8],
Murdock [9], and Lin [10] are widely regarded as classical models. These models take the prediction method of the gas phase multiplier ΦG or liquid phase multiplier ΦL as the goal, thus obtaining the pressure drop of a two-phase flow across the throttling unit based on that of a single-phase flow. Accordingly, it is considered that the parameters ΦG and ΦL mainly depend on the quality x (or X) and gas–liquid density ratio ρG/ρL, whereas they are independent of the structural parameters of the throttling unit, such as the diameter ratio β. Relatively speaking, ΦG is often used for wet gas because it is close to a pure gas. In addition, the term “over–reading (OR)” was also used in some studies [11–13] to represent the phenomenon that the pressure drop of a wet gas is larger than that of a single gas under the same mass flow rate. Different from the parameter ΦG, which uses the pressure drop ratio as the definition, the term OR uses the mass flow rate ratio as the definition; thus, OR can be regarded as an altered expression of ΦG. Nevertheless, it was reported that the prediction result by the classical two-phase flow pressure drop models for orifices tends to deviate under the wet gas condition. In other words, more factors need to be considered in the ΦG prediction of a wet
* Corresponding author. Shanghai Key Laboratory of Multiphase Flow and Heat Transfer in Power Engineering, School of Energy and Power Engineering, Uni versity of Shanghai for Science and Technology, Shanghai 200093, China. E-mail address: [email protected] (Y. Ma). https://doi.org/10.1016/j.flowmeasinst.2019.101678 Received 28 September 2019; Accepted 12 December 2019 Available online 14 December 2019 0955-5986/© 2019 Elsevier Ltd. All rights reserved.
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gas, such as the Froude number of the gas phase FrG [14–16]. In a wet gas the volume of the gas phase occupies the main part. Generally, the ideal wet gas flow pattern is expected to be the annu lar–mist flow, in which much of the liquid phase is distributed in the gas phase in the form of droplets. Therefore, some scholars believed that the homogeneous flow model is capable of predicting the wet gas pressure drop. For example, Fang et al. [17] performed an experimental study on the pressure drop of wet natural gas under low pressure (0.15–0.25 MPa) and high quality (x ¼ 0.50–0.99) using a venturi with diameter ratio β ¼ 0.4–0.7, and concluded that the original homogeneous flow model has the best prediction performance among the eight pressure drop models (including the Chisholm and Murdock models) they compared. More over, based on an experiment on slotted orifice plates with diameter ratio β � 0.5, Xing et al. [18] considered that the James homogeneous flow model has the best accuracy for predicting the wet gas pressure drop. Kumar et al. [19] also recommended the original homogeneous flow model based on their numerical simulation study on the wet gas pressure drop across multi-orifice plates. However, separated flow models, which basically involve a correc tion of the Chisholm or Murdock model, were also recommended by some studies. Regarding the correction of the Chisholm model, De Leeuw [20] proposed a correction on the power n (n ¼ 0.25 in the original Chisholm model) considering that n is a variable and is dependent on FrG. Steven and Hall [21] also recommended such a correction method to the wet gas pressure drop across orifice plates based on their experimental result. Besides, Steven and Hall [22] further examined the effect of changing the diameter ratio β (0.24–0.73) on the correction and concluded that the correction of n is not sensitive to a change in β. According to De Leeuw [20] and Steven and Hall [21], the power n increases with the increase in FrG. Xu et al. [19] also recom mended a correction on the Chisholm model based on an experiment on the wet gas pressure drop across a venturi. However, Xu et al. [22] concluded that the power n not only is dependent on FrG, but it is also related to the diameter ratio β and the relative length of the throat. He et al. [23] experimentally studied the wet gas pressure drop through a v-cone flowmeter under the conditions of low pressure 0.1–0.4 MPa, β ¼ 0.55, gas–liquid density ratio ρG/ρL ¼ 0.00247–0.00610, and FrG ¼ 0.3–1.5, with the result that n decreases with the increase in the liq uid–gas mass flow ratio, whereas it is almost independent of ρG/ρL and FrG. Then they correlated n with the liquid–gas mass flow ratio. There fore, regarding the wet gas pressure drop across orifices, although a correction of the Chisholm model has been recommended by many au thors, there is currently no unified conclusion on the correction method or on which factor should be considered for correcting the value of n. Moreover, a correction of the Murdock model was also recom mended by Xing et al. [18]. Based on an experimental study on the pressure drop across slotted multi-orifice plates under the GVF range of 99.49%–99.97%, they concluded that in the linear Murdock model, the slope k is dependent on both FrG and β, and it is positively correlated with FrG and negatively correlated with β, whereas the intercept b is only dependent on β, and it increases first and then decreases with the in crease in β. As a result, they proposed a correction of the Murdock model to predict the wet gas pressure drop across slotted multi-orifice plates by considering the effects of FrG and β on k and b. The relationship between the pressure drop and the flow pattern of a wet gas has rarely been reported. For a wet gas with GVF greater than 90%, its flow pattern can be a stratified, slug, or annular–mist flow. Generally, it is considered that the pressure drop model of a gas–liquid flow, a homogenous flow model or a separated flow model, should be in accordance with its flow pattern. Therefore, it is significant to clarify the pressure drop characteristics of a wet gas in the region of flow pattern transition. In this study, the wet gas pressure drop across orifice plates was experimentally investigated in the region of flow pattern transition, wherein the GVFs were in the range of 93.05%–99.31%. Based on the experimental results, the existing pressure drop models of orifice plates were evaluated, and new correlations were proposed for an accurate
prediction of the pressure drop in this flow region. 2. Prediction models for the wet gas pressure drop across orifice plate 2.1. James homogeneous flow model by 1
ρTP
For a gas–liquid mixture, the mixture density, ρTP, can be determined
¼
x
ρG
þ
1
x
ρL
(1)
:
The homogeneous flow model assumes that the two phases are mixed uniformly. As in the original homogeneous flow model, the pressure drop of a two–phase flow across an orifice plate is obtained by replacing the fluid density in the pressure drop equation for the single-phase flow with ρTP. That is ΔpTP ¼
m2TP ½1 þ ðρL =ρG
β4 Þ :
1Þx�ð1
2ρL ðKAÞ
2
(2)
In Eqs. (1) and (2), ΔpTP, mTP, and x are respectively the pressure drop (Pa), mass flow rate (kg/s), and quality of the two–phase flow; ρG and ρL are the density of the gas and liquid phases, respectively (kg/m3); β is the ratio of the orifice diameter d to the inner pipe diameter D, i.e., β ¼ d/D; and K and A are respectively the discharge coefficient and orifice area (m2) of the orifice plate. James [6] proposed a correction of the original homogeneous flow model by replacing the actual quality x in Eq. (2) with a corrected quality xm based on the experiment of measuring the flow rate of a geothermal steam–water two-phase flow. The corrected quality xm is given by (3)
xm ¼ x1:5 : Thus, the James model can be expressed as follows: ΔpTP ¼
m2TP ½1 þ ðρL =ρG
1Þx1:5 �ð1 2
2ρL ðKAÞ
β4 Þ :
(4)
2.2. Chisholm model and modified Chisholm model for wet gas The separated flow model assumes that the two phases flow through an orifice plate separately and the two phases have the same discharge coefficient. Chisholm [7,8] derived a pressure drop model for the gas–liquid flow through orifices as Eq. (5) based on the momentum conservation equations and considering the influence of the interfacial shear force on each flowing phase. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ΦG ¼ 1 þ CX þ X 2 ; (5) � when X < 1; C ¼
ρL ρG
� when X > 1; C ¼
�
�0:25
ρTP ρG
þ
ρG ρL
�
�0:25 þ
�0:25
ρG ρTP
; �0:25 ;
where X is the well-known Lockhart–Martinelli parameter, which can be determined by Eq. (12). The power n of the density ratio shown in the original Chisholm model is a constant (n ¼ 0.25). Steven and Hall [21] proposed a modified Chisholm model for a wet gas flowing through an orifice plate by considering the influence of FrG on ΦG. They concluded that the influ ence of FrG on ΦG is closely related to the flow pattern upstream of the orifice plate. That is, ΦG is insensitive to FrG and the Chisholm power n is a constant of 0.214 when the flow pattern upstream of the orifice plate is a stratified flow, which is judged by FrG � 1.5. With FrG increasing, the flow pattern could transit into an annular–mist flow when FrG > 1.5; in 2
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this case, the power n increases with an increment in FrG. As a result, they proposed a modified Chisholm model as given by Eq. (6), which is also recommended to use in the standard ISO/TR 11583:2012 (Mea surement of wet gas flow by means of pressure differential devices inserted in circular cross-section conduits) [24]. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ΦG ¼ 1 þ CX þ X 2 (6) � where C ¼
ρG ρL
�
�n þ
ρL ρG
where
� 60:6150
ρG ρL
� �2 � � ρ ρ 9:26541 G þ 44:6954 G �3
ρL
� �4 ρ 5:12966 G
ρL
ρL
� �5 . ρ 26:5743 G
ρL
From an experiment on wet gas flowing through slotted multi–orifice plates with the range of FrG ¼ 0.5–2.0 and β ¼ 0.50–0.75, Xing et al. [18] recommended the Murdock model for the pressure drop prediction with further consideration of the effect of β and FrG on ΦG. The detailed model by Xing et al. [18] is presented in Eq. (9), which indicates that the slope in the Murdock model depends on both β and FrG, whereas the intercept only depends on β. h �pffiffiffiffiffiffiffi . � i ΦG ¼ Cx 1 þ (9) FrG β X ;
�n ;
when FrG � 1:5; n ¼ 0:214; � 1 and when FrG > 1:5; n ¼ pffiffi 2
k ¼ 1:48625
��2 � 0:3 pffiffiffiffiffiffiffi ; FrG
where Cx ¼ 12:60β2
16:01β þ 5:81.
where FrG is the gas phase Froude number, which can be determined by Eq. (13).
3. Experimental system and data processing method
2.3. Murdock model and modified Murdock model for wet gas
3.1. Experimental system
The Murdock model [9], expressed in Eq. (7), is also representative of the separated flow model. Unlike the form of the Chisholm model, the Murdock model uses a linear relationship to relate ΦG and X with the experimental result of orifice pressure drop obtained from the steam– water, air–water, and natural gas–water fluids.
The experimental system is shown in Fig. 1. In the experiment, air and water at ambient temperature were used as the fluids of the gas and liquid phases, respectively. The air and water were introduced into a gas–liquid mixer by a compressor and a pump, respectively, to produce a gas–liquid mixture, where the intake air was taken from the atmosphere and the intake water was drawn from a water tank filled with tap water. The structure of the mixer is illustrated in Fig. 1, in which the water evenly permeated into the air pipe through twenty-five holes with diameter of 10 mm. Those holes were placed in a layout of five lines in circumference, each line having five holes with an axial spacing of 20 mm. Downstream of the mixer, the air–water mixture horizontally flowed through a test section, which consisted of two cir cular plexiglass pipes and an orifice plate fixed between them with a flange connection. Both the pipe for the air flow in the mixer and the pipe used for the air–water mixture flow had an inner diameter of 50 mm. The lengths of the plexiglass pipes upstream and downstream of the orifice plate were 0.5 m and 1.5 m, respectively. Besides, two steel pipes were placed upstream and downstream of the plexiglass pipe sections,
ΦG ¼ kX þ b
(7)
where k ¼ 1:26; b ¼ 1:0. Subsequently, Lin [10] advanced an improved Murdock model, expressed in Eq. (8), by considering the influence of the changing den sity ratio ρG/ρL on the prediction accuracy of the model. He proposed a correlation to the slope k of the original Murdock model, based on his experiment on a steam–water mixture under the ρG/ρL range of 0.00455–0.328. ΦG ¼ kX þ 1;
(8)
Fig. 1. Experimental system for the pressure drop of wet gas across orifice plate. 3
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respectively, which also had an inner diameter of 50 mm and whose lengths were 1.0 m and 2.0 m, respectively. The pressure taps used for the pressure drop across orifice plate were located at 20D upstream and 40D downstream of the orifice plate, respectively. Therefore, the two–phase flow pattern could be observed in the plexiglass pipe section while the pressure drop across the orifice plate could be obtained using a differential pressure transmitter by placing the pressure taps on the steel pipes. Besides, the measuring points for the temperature, pressure, and flow rate of the air and water were arranged in their intake pipes, as shown in Fig. 1. Meanwhile, valves were also equipped in the air and water intake pipes to regulate the flow rate. The measuring instruments used in the experiment are listed in Table 1. For an accurate measurement of the pressure drop across the orifice plate, considering that this pressure drop varies in a wide range with the change in test conditions, three differential pressure meters in different test ranges were used in the experiment to choose the proper meter in use according to the pressure drop. During the test process, the output analog signals exported by the measuring instruments were collected by a data acquisition module and then input in the computer through an A/ D communication module.
Table 2 Test conditions in the experiment. Water flow rate, QL (m3/h) Air flow rate, QG0 (m3/h)
40 60 80 100 120
1
2
3
(40, 1) (60, 1) (80, 1) (100, 1) (120, 1)
(40, 2) (60, 2) (80, 2) (100, 2) (120, 2)
(40, 3) (60, 3) (80, 3) (100, 3) (120, 3)
obtained by experimental measurement in this study, thus eliminating the uncertainty of determining the ΔpGO by calculations. Moreover, the gas quality x can be determined by x¼
mG QG ρG ¼ ; mG þ mL QG ρG þ QL ρL
(11)
where m, Q, and ρ are the mass flow rate (kg/h), volumetric flow rate (m3/h) and density of a phase, and subscripts G and L denote the gas and liquid phase, respectively. In this study, QG and QL are obtained by the flow rate meters, whereas ρG and ρL are determined by calculation ac cording to the temperature and pressure of the air and water used. It should be noted that the air flow rates from the meter (QG0 ) were used to carry out the experiment according to the designed conditions listed in Table 2, but QG in Eq. (11) and JG in Eqs. (13) and (14) were determined by a pressure correction according to the pressure change between the location of the air flow rate meter and the location of the orifice plate upstream. Similarly, ρG in Eq. (11), and in Eqs. (12)–(14), were also determined by the pressure upstream of the orifice plate. Then, the LM parameter X can be determined by rffiffiffiffiffi 1 x ρG : (12) X¼ x ρL
3.2. The test orifice plate and the test conditions In this study, a sharp-edged orifice plate with orifice diameter d ¼ 25 mm and orifice thickness t ¼ 5 mm was used for the experiment. Thus, the orifice plate for the test has a diameter ratio β ¼ 0.50 (D ¼ 50 mm) and relative orifice thickness t/d ¼ 0.2. Fifteen test conditions were designed with ranges of water flow rate QL ¼ 1–3 m3/h and air flow rate QG0 ¼ 40–120 m3/h, as presented in Table 2. As a result, the GVFs of the two-phase flow are in a range of 93.05%–99.31%. Each condition was tested twice to reduce the uncer tainty of the experimental result. In addition, the pressure drops of air flowing alone through the orifice plate were also tested under the five air flow rates listed in Table 2, thereby obtaining the gas phase multiplier ΦG from the experimental result.
The gas phase Froude number FrG and the liquid phase Froude number FrL are also considered as important factors influencing ΦG in a wet gas. They can be determined as follows: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi JG ρG ; (13) FrG ¼ pffiffiffiffiffiffi gD ρL ρG
3.3. Data processing methods In this study, the gas phase multiplier ΦG is used to represent the pressure drop of the wet gas. ΦG is determined by sffiffiffiffiffiffiffiffiffiffi ΔpTP ; (10) ΦG ¼ ΔpGO
JL FrL ¼ pffiffiffiffiffiffi gD
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ρL
ρL
ρG
(14)
;
where g is the gravitational acceleration (9.81 m/s2); D is the inner diameter of the pipe (m) and D ¼ 0.05 m in this study; JG and JL are the superficial velocity of the gas and liquid phase (m/s), which are given by
where ΔpTP and ΔpGO are the orifice pressure drops of the two–phase flow and the gas phase only, respectively (Pa). Both ΔpTP and ΔpGO are
JG ¼
Table 1 Instruments used in the experiment.
QG =3600 Q =3600 � ; JL ¼ L � : π D2 4 π D2 4
(15)
Parameters
Devices
Type
Range
Accuracy
Air flow rate
Vortex flowmeter Pressure transmitter
35–350 m3/h 0–100 kPa
1.5
Air pressure near the flow rate meter Water flow rate
LUGB-MKDN50 EJA430ADAS4A-92DA
In addition, the relative deviation ER is defined as follows to compare the prediction performance of the orifice pressure drop models with the experimental result.
0.2
For a homogeneous flow model: ER ¼ ðΔpTPC
LWGY15A0A3C3 MIK-WZPK20A
0.5–5 m3/h 0–100 � C
0.5
For a separated flow model: ER ¼ ðΦGC
0.5
MIK-P300
0–50 kPa
0.5
EJA120ADES4A-92DA EJA110ADLS4A-92DA MIK-3051
0.1–1.0 kPa 0.5–10 kPa 0–100 kPa
0.2
Air/water temperature Pressure upstream of orifice plates Pressure drop across orifice plates
Turbine flowmeter Platinum resistance thermometer Pressure transmitter Differential pressure transmitter
ΔpTPE Þ = ΔpTPE � 100% (16)
ΦGE Þ = ΦE � 100%;
(17)
where ΔpTPE and ΔpTPC are the pressure drops of the gas–liquid flow obtained by the experiment and the prediction model, respectively (Pa); ΦGE and ΦGC are the gas phase multipliers obtained by the experiment and by the prediction model, respectively.
0.2
3.4. Uncertainty analysis
0.5
In the experiment, ΔpTP and ΔpGO were obtained directly by differ ential pressure transmitters. Therefore, their uncertainties can be 4
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determined by the measuring value and the range, accuracy of trans mitters. Then, the uncertainty of ΦG can be obtained by uncertainty propagation principle with its definition as Eq. (10). As the dimension less parameters, FrG and FrL are often used to indicate the influences of superficial gas and liquid velocity, respectively, on the pressure drop of wet gas flow. Therefore, their uncertainties were also analyzed. Un certainties of FrG and FrL mainly depend on the uncertainties of measuring gas and liquid flow rates, and the uncertainties of ρG that are derived from the measurements of gas temperature and pressure, while the uncertainties of ρL are not considered because it does not change much in the experiment. The uncertainty analysis results are listed in Table 3, in which the uncertainty of a parameter Y is expressed by U(Y). 4. Experimental results and discussion 4.1. Results of two-phase flow patterns and pressure drops
Fig. 2. Experiment results of the flow pattern upstream of the orifice plate.
The flow patterns in a wet gas are generally classified as stratified, slug, and annular–mist flows. Based on the flow pattern map of the Shell Company [1], the test conditions of this study and the corresponding results of flow patterns upstream of the orifice plate are shown in Fig. 2, using FrG as the horizontal axis and FrL as the vertical axis. It can be observed from Fig. 2 that the test conditions of this study are in the region of flow pattern transition according to the flow pattern map of the Shell Company. From the observation of the flow pattern up stream of the orifice plate, the three flow patterns, namely, stratified, slug, and annular–mist, were observed in the experiment, as shown in Fig. 3. The flow pattern results obtained by the visualization are listed in Fig. 2. This indicates that, in principle, the experimental results of flow patterns upstream of the orifice plate agree with the flow pattern map. Fig. 4 shows the experimental result of pressure drop across the test orifice plate. It can be seen from Fig. 4 that the wet gas pressure drop across the orifice plate increases with an increase in both FrG and FrL. This indicates that for a wet gas, the pressure drop across orifice plates can effectively reflect the change in wet gas flow rate. 4.2. Evaluating the existing models with the experimental result It was stated in Section 2 that the existing prediction models for pressure drop of a wet gas across an orifice plate can be classified into three categories, namely, the James, Chisholm, and Murdock models. These models are based on different assumptions such as a homogeneous or separated flow, and differ in relation to the influencing factors used. To compare their predicting performance, the three types of prediction models are evaluated hereinafter with the experiment results of this study.
Fig. 3. The three flow patterns of wet gas upstream of the orifice plate.
4.2.1. Evaluation of homogeneous flow models The relative deviations between the experimental result and the result predicted by the original homogeneous flow model, i.e., Eq. (2) are shown in Fig. 5. It can be observed from Fig. 5 that the original homogeneous flow model largely overestimates the pressure drop. Corresponding to the FrL of the experiment at 0.20, 0.40, and 0.60, ER values are in the ranges of 294%–922%, 252%–555%, and 177%–446%, respectively. The relative deviations between the experimental result and the result predicted by the James homogeneous flow model, i.e., Eq. (4) are shown in Fig. 6. Corresponding to FrL at 0.20, 0.40, and 0.60, ER values are in the ranges of 61%–144%, 11%–31%, and 18% – 4.4%, respec tively. It can be observed from Fig. 6 that the deviations mainly depend Table 3 Uncertainty analysis results of the experimental result. U(FrG)
U(FrL)
U(ΔpTP)
U(ΔpGO)
U(ΦG)
1.7%–8.3%
0.7%–2.3%
0.2%–4.1%
0.2%–1.2%
0.1%–2.2%
Fig. 4. Experimental results of pressure drop across the orifice plate. 5
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Fig. 7. Prediction deviations of the Chisholm model.
Fig. 5. Prediction deviations of the original homogeneous flow model.
the ranges of 16%–7.9%, 28% to 12%, and 41% to 19%, respectively. It can be observed from Fig. 7 that the deviations of the model are mostly negative, meaning that, unlike the overestimation by the homogeneous flow models, the Chisholm model yields a certain underestimation of the pressure drop of wet gas in the region of flow pattern transition. As can be seen from Fig. 7, the deviations are small when FrL ¼ 0.20, and they tend to increase with an increase in FrL. Meanwhile, the deviations are also amplified as FrG increases under the same FrL. Based on the Chisholm model, Steven and Hall [21] proposed a correction of the power n of the density ratio of gas to liquid considering the influence of FrG on n, as expressed by Eq. (6). According to Eq. (6), the influence of FrG on n is relevant only when FrG > 1.5, and it is suggested that n ¼ 0.214 when FrG � 1.5. In the experiment of this study, the FrG values are in the range of 0.24–1.29, which are all below 1.5; thus, n ¼ 0.214 was used in this study for the calculations. The relative deviations between the experimental result and the result predicted by the Steven and Hall model, i.e., Eq. (6) are shown in Fig. 8. Corre sponding to FrL values of the experiment at 0.20, 0.40, and 0.60, ER values are in the ranges of 20%–0.034%, 33% to 18%, and 45% to 25%, respectively. It can be observed from Fig. 8 that the deviations of the Steven and Hall model are similar to those of the Chisholm model, except that in this case, all the deviation points are shifted down by a certain degree.
Fig. 6. Prediction deviations of the James homogeneous flow model.
on the values of FrL, i.e., the deviations are small when FrL ¼ 0.60 and tend to increase with a decrease in FrL. Meanwhile, the deviations also show a relation with FrG, that is, under the same FrL, the points of the deviations show a decline as FrG increases. Such a variation in the deviations shown by Fig. 6 might be attrib uted to the change in wet gas flow patterns in the experiment of this study. The flow patterns in the experiment are mostly stratified flows when FrL ¼ 0.20, while they are mostly slug flows when FrL ¼ 0.40 and FrL ¼ 0.60. Therefore, it might be interpreted from the deviation results that the James model is capable of predicting pressure drops of a wet gas with slug flow, whereas that model is not suitable for a stratified flow because it deviates from a homogeneous flow significantly. In addition, it can be found from Fig. 2 that the flow pattern of a wet gas tends to be an annular–mist flow as FrG reaches a certain value. An annular–mist flow is closer to a homogeneous flow than a slug flow. This might be the reason why, as shown in Fig. 6, the deviations of the conditions with a high FrG are relatively small. Xing et al. [18] approved the use of the James homogeneous flow model for slotted orifice plates based on their experiments, and the reason might be that their test was conducted under large gas velocity conditions (FrG ¼ 1.11–1.95), and therefore, with a fully-developed annular–mist flow according to the flow pattern map of wet gas, i.e., Fig. 2.
4.2.3. Evaluation of the Murdock–type models The Murdock model is another typical separated flow model. The relative deviations between the experimental result and the result pre dicted by the Murdock model, i.e., Eq. (7) are shown in Fig. 9.
4.2.2. Evaluation of Chisholm–type models The Chisholm model is a typical separated flow model. The relative deviations between the experimental result and the result predicted by the Chisholm model, i.e., Eq. (5) are shown in Fig. 7. Corresponding to FrL values of the experiment at 0.20, 0.40, and 0.60, the ER values are in
Fig. 8. Prediction deviations of the Steven and Hall model. 6
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Fig. 9. Prediction deviations of the Murdock model.
Fig. 11. Prediction deviations of the Xing model.
Corresponding to the FrL values of the experiment at 0.20, 0.40, and 0.60, ER values are in the ranges of 28% to 12%, 41% to 25%, and 52% to 29%, respectively. It can be observed from Fig. 9 that the deviations of the model are all negative, meaning that, as in the un derestimation by the Chisholm model, the Murdock model yields an obvious underestimation of the pressure drop of wet gas in the region of flow pattern transition. In addition, the results of the evaluation of the Murdock model are similar to those of the Chisholm model or Steven and Hall model regarding the effect of FrG and FrL on the deviations. That is, the deviations are small when FrL ¼ 0.20 and tend to increase with an increase in FrL; meanwhile, the deviations tend to augment as FrG in creases under the same FrL. Accordingly, the relative deviations between the experimental result and the result predicted by the Lin model, i.e., Eq. (8) are shown in Fig. 10. Corresponding to the FrL values of the experiment at 0.20, 0.40, and 0.60, ER values are in the ranges of 26% to 5.0%, 37% to 17%, and 49% to 20%, respectively. It can be observed from Fig. 10 that the Lin model also yields an underestimation of the pressure drop of wet gas in the region of flow pattern transition. Altogether, the de viations of the Lin model are similar to those of the Murdock model, except that in this case, all the deviation points are shifted up by a certain degree. This indicates that the consideration of the influence of the gas–liquid density ratio could improve the prediction performance of the Murdock model. However, as can be seen in Fig. 10, the deviations are still large. The relative deviations between the experimental result and the result predicted by the Xing model, i.e., Eq. (9) are shown in Fig. 11. Corresponding to the FrL values of the experiment at 0.20, 0.40, and 0.60, ER values are in the ranges of 24% to 14%, 37% to 29%, and
45% to 32%, respectively. It can be observed from Fig. 11 that the Xing model also yields an underestimation of the pressure drop of wet gas in the region of flow pattern transition. Because the effect of FrG on ΦG is considered in the Xing model, under the same FrL, the deviations for different FrG values become stable. This means that the correction proposed by Xing et al. [18] is effective to some extent for the effect of FrG on the pressure drop of wet gas. However, as can be seen in Fig. 11, the prediction deviations of the model are still large, and the deviations tend to increase as FrL increases. 4.3. New correlations for the pressure drop of wet gas across orifice plates The evaluations of the existing models for the two–phase pressure drop across orifice plates show that the deviations of these models are large when they are used for the wet gas pressure drop in the region of flow pattern transition. In summary, the models based on a homogenous flow model overestimate the pressure drop, whereas those based on a separated flow model provide an underestimation. For an accurate prediction of the wet gas pressure drop in the region of flow pattern transition, the abovementioned three types of prediction models are improved hereinafter with the experiment results of this study. 4.3.1. Correlations from a modification of the homogeneous flow model By comparing Eqs. (2) and (4), it can be observed that they differ only in the power of the quality x, with power 1.0 for the original ho mogeneous flow model and power 1.5 for the James homogeneous flow model. This means that a correction of the power of x is an effective method for the modification of the homogeneous flow model. The evaluation results shown by Figs. 5 and 6 indicate that, when using the homogeneous flow model, both FrG and FrL are related to the prediction deviations. Therefore, to make the prediction result equal to the experimental result of this study, the power n in Eq. (4) should vary with both FrG and FrL. With the experimental result of this study, the power n can be obtained by letting the power of x in Eq. (4) as an unknown. According to the calculations, the results of the power n are shown in Fig. 12. It can be seen from Fig. 12 that the power n decreases rapidly with an increase in FrL, and decreases slowly with an increase in FrG. With the surface-fitting method, the effect of FrL and FrG on the power n can be expressed by the new equation for the corrected quality xm. Thus, from the modification of the homogeneous flow model, the new correlation proposed in this study is given by ΔpTP ¼
m2TP ½1 þ ðρL =ρG
1Þxn �ð1
2ρL ðKAÞ
2
β4 Þ ;
(18)
where n ¼ 2:17 0:10FrG 1:76FrL þ 1:26Fr2L . The prediction accuracies of the correlation, Eq. (18), are shown in
Fig. 10. Prediction deviations of the Lin model. 7
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Fig. 12. Effects of FrG and FrL on the power n in the homogeneous flow model.
Fig. 14. Effects of FrG and FrL on the power n in the Chisholm model.
The prediction accuracies of the correlation, Eq. (19), are shown in Fig. 15. As can be seen, the relative deviations of Eq. (19) are in the range of 6%–6%. 4.3.3. Correlations from a modification of the Murdock model The Murdock model was also proposed by Xing et al. [18] as a base model for predicting the wet gas pressure drop across orifice plates, wherein they used FrG to modify the slope in the Murdock model, as expressed by Eq. (9). However, the evaluation results shown in Figs. 9–11 indicate that, when using the Murdock model, both FrG and FrL are related to the prediction deviations. Because there are two un knowns in the Murdock model, i.e., the slope k and the intercept b, the effects of changing FrG and FrL on the Murdock model, specifically the k and b in the model, must be clarified before advancing a new correlation. With the experimental result of this study, the effect of changing FrG on the Murdock model is illustrated by Fig. 16. In Fig. 16, the test results are classified by the different air flow rate QG0 . Because of the variations in the air pressure difference between the flow meter point and the upstream point of the orifice plate for the different test conditions, FrG varies in a certain range under the same QG0 . For the analysis, the test results under the same QG0 are approximately regarded as under the same FrG. It can be observed in Fig. 16 that ΦG and X show a linear relationship under the same FrG, and the slope k of the line increases with the increase in FrG. This agrees with the result obtained by Xing et al. [18], who proposed a correction of k by relating that with FrG. Meanwhile, it is interesting that in Fig. 16 the lines of different FrG have an almost identical b, which varies in the narrow range of 0.68–0.73, i.
Fig. 13. Prediction accuracies of the new correlation based on the homoge neous flow model.
Fig. 13. As can be seen, the relative deviations of Eq. (18) are in the range of 5%–6%. 4.3.2. Correlations from a modification of the Chisholm model The Chisholm model was widely used as the base model for pre dicting the wet gas pressure drop across orifice plates. The specific modification method considers the effect of FrG on the pressure drop by letting the power n of gas–liquid density ratio (ρG/ρL) be related with FrG, as expressed in Eq. (6). Such a modification can be found in the works of De Leeuw [20], Steven and Hall [21], and Xu et al. [22]. However, the evaluation results shown by Figs. 7 and 8 indicate that, when using the Chisholm model, both FrG and FrL are related to the prediction deviations. Therefore, to make the prediction result equal to the experimental result of this study, the power n in Eq. (6) should vary with both FrG and FrL. With the experimental result of this study, the powers can be obtained by letting the n in Eq. (6) as an unknown. Ac cording to the calculations, the results of the power n are shown in Fig. 14. It can be observed from Fig. 14 that the power n increases with the increase in both FrG and FrL. With the surface-fitting method, the effect of FrL and FrG on the power n can be expressed by a new equation. Thus, from the modification of the Chisholm model, the new correlation proposed in this study is expressed by pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ΦG ¼ 1 þ CX þ X 2 ; (19) where C ¼
� �n ρG ρL
� �n þ
ρL ρG
Fig. 15. Prediction accuracies of the new correlation based on the Chis holm model.
0:33 ; n ¼ 0:52Fr0:25 . G FrL
8
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Fig. 16. Effects of changing FrG on the slope of the Murdock model.
Fig. 18. Relationship between intercept b and FrL based on Murdock model.
e., with the average value of 0.71. Meanwhile, with the experimental result of this study, the effect of changing FrL on the Murdock model can also be illustrated by Fig. 17. In Fig. 17, the test results are classified by the different FrL. It can be found from Fig. 17 that, in this case, the effect of changing FrL on the Murdock model also shows a certain regularity. That is, the intercept b of the model increases with the increase in FrL, while the slope k approximately remains unchanged (k ¼ 1.22). This indicates that the parameter FrL can also be used to modify the Murdock model to achieve an accurate pre diction of the wet gas pressure drop across orifice plates in the region of flow pattern transition. It can be found that both the correction of k with FrG and the correction of b with FrL are effective for a modification of the Murdock model. As a stable water flow rate, i.e., a stable FrL, was utilized in this experiment, the modification of the Murdock model is conducted by the correction of b with FrL. In this case, k is equal to 1.22, as shown by Fig. 17. The three results of b can be obtained by extending the line in Fig. 17 to the crossing point with the vertical axis. Then, the relationship between b and FrL is given by a quadratic polynomial expression, as shown in Fig. 18. As a result, from the modification of the Murdock model, the new correlation proposed in this study is given by ΦG ¼ 1:22X þ 1:75Fr2L þ 1:75FrL þ 0:95:
Fig. 19. Deviation diagram of novel correlation by Murdock model.
three models, i.e., the modifications of the homogenous flow model, the Chisholm model, and the Murdock model show approximately the same performance. The correlations proposed in this study, i.e., Eqs. (18)– (20), are validated by the experiment of this work, carried out for a wet gas in the region of flow pattern transition under the conditions of FrG ¼ 0.29–1.23 and FrL ¼ 0.20–0.60.
(20)
The prediction accuracies of the correlation, Eq. (20), are shown in Fig. 19. As can be seen, the relative deviations of Eq. (20) are in the range of 8%–8%. Overall, regarding the prediction accuracy of the wet gas pressure drop across orifice plates in the region of flow pattern transition, all the
5. Conclusions In this study, the pressure drop characteristics of a wet gas flowing through a sharp–edged orifice plate were experimentally studied in the region of flow pattern transition. Based on a summary of the existing prediction models for the wet gas pressure drop across orifice plates, the three typical models, i.e., the homogenous flow, Chisholm, and Murdock models, were evaluated with the experimental result of this study. Then, based on the three models, new correlations were proposed for the wet gas pressure drop across orifice plates in the region of flow pattern transition. The following conclusions can be drawn: The experiment of this study was conducted under the conditions of FrG ¼ 0.29–1.23 and FrL ¼ 0.20–0.60, for a wet gas in the region of flow pattern transition according to the Shell flow pattern map for wet gases. All the flow patterns of wet gases, i.e., stratified, slug, and annular–mist flows, were observed in the experiment of this study, and it is confirmed that the boundaries of the flow pattern transitions judged by the experimental result are in accordance with those of the Shell flow pattern map. Compared with the experimental result, the original homogeneous flow model provides a large overestimation of the pressure drop. The
Fig. 17. Effects of changing FrL on the intercept of the Murdock model. 9
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prediction deviations of the James homogeneous flow model are significantly mitigated in comparison to those of the original homoge neous flow model, but there is an obvious increase in prediction de viations as FrL decreases from 0.60 to 0.20. The prediction deviations indicate that the homogeneous flow model must be corrected by the two factors, FrG and FrL, to improve its prediction accuracy. By modifying the homogeneous flow model, a new correlation was proposed in this study, which shows accuracies within 6% when compared with the experi mental result. Compared to the experimental result, the three typical separated flow models, i.e., the Chisholm, Murdock, and Lin models, present an obvious underestimation on the pressure drop. The evaluation results also indicate that the traditional separated flow models must be cor rected by the two factors, FrG and FrL, to improve their prediction ac curacies. By modifying the Chisholm and Murdock models, two new correlations were proposed in this study, respectively, attaining pre diction accuracies within 6% and 8%. The models reported in the literature for the wet gas pressure drop across orifice plates, such as the Chisholm-type model by Steven and Hall [18] and the Murdock-type model by Xing et al. [15], are also evaluated with the experimental result of this study. The results indicate that they also exhibit an underestimation of the pressure drop, although the magnitude of the deviations is mitigated to some extent. Generally, the factor FrG was considered in those corrected models for wet gas. However, the evaluation results of this study reveal that FrL is also an important factor that cannot be neglected, especially when a homoge neous flow or Chisholm-type model is used. Three new correlations based on the homogenous flow, Chisholm model, and Murdock model, respectively, were proposed in this study, as expressed by Eqs. (18)–(20). They exhibit approximately the same per formance with respect to their prediction accuracy.
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Acknowledgments This study was supported by the National Key Research and Devel opment Program of China (2016YFB0600201). References [1] ASME, Wet Gas Flowmetering Guideline, ASME MFC-19G-2008, New York, USA, 2008.
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