A method for calculating the liquid holdup of annular flow and churn flow in high-pressure gas wells based on normal-pressure experimental testing

Abstract

This invention provides a method for calculating the liquid holdup of annular and agitated flows in high-pressure gas wells based on atmospheric pressure experimental testing, relating to the field of gas reservoir drainage and production technology. This method involves conducting atmospheric pressure experiments using a simulation device and calculating parameters such as gas density based on the measured data. The friction coefficient is corrected using the measured data and calculation results. Different pressure values ​​are used to calculate the apparent gas flow velocity corresponding to each liquid holdup rate. The calculation results are plotted as apparent gas flow velocity versus pressure curves for different liquid holdup rate series. The liquid holdup rate is then read from the curves based on the pressure and apparent gas flow velocity under high-pressure conditions. This invention addresses the problem that the two-phase flow patterns under high-pressure conditions and experimental atmospheric pressure differ, making it difficult to predict the liquid holdup of annular and agitated flows in high-pressure wellbores. Based on the flow similarity criterion and considering droplet entrainment, this invention proposes a method for calculating the liquid holdup of annular and agitated flows in high-pressure gas wells based on atmospheric pressure experimental testing, providing a theoretical basis for the optimized design of drainage and production processes.

Description

Technical Field

[0001] This invention belongs to the field of gas reservoir drainage and gas production, and particularly relates to a method for calculating the liquid holdup of annular flow and turbulent flow in high-pressure gas wells based on atmospheric pressure experimental testing. Background Technology

[0002] Water production is inevitable during gas reservoir development. As development progresses, reservoir energy decreases, leading to a reduction in gas production, but water production continues to increase, sometimes even causing water lock-up and severely impacting normal gas reservoir production. Accurately predicting the liquid holdup in the wellbore is beneficial for visually reflecting the water production situation of the gas well, helping to plan production and avoid liquid accumulation. In addition, it plays an important role in accurately calculating production and ensuring flow safety.

[0003] Current research on gas-liquid two-phase flow mainly focuses on atmospheric pressure and the pressure range that experimental equipment can withstand. There is relatively little research on gas-liquid two-phase flow under high pressure because two-phase flow is not a uniform and stable process. Its characteristics change with space and time. Flow under high pressure is more complex, poses greater challenges to equipment, and is more dangerous.

[0004] To address the difficulty in predicting the liquid holdup of annular and agitated flows in high-pressure wells, this invention proposes a method for calculating the liquid holdup of annular and agitated flows in high-pressure gas wells based on atmospheric pressure experimental tests. This method relies on atmospheric pressure experimental and calculation results, using the momentum equation for gas-liquid two-phase flow to calculate the apparent gas flow velocity corresponding to various liquid holdup rates under different pressure conditions, plotting curves for different liquid holdup series, and then reading the liquid holdup under high-pressure conditions from the curves. Summary of the Invention

[0005] This invention aims to address the problem that the different two-phase flow patterns under high pressure and experimental atmospheric pressure cannot meet the requirements for predicting the liquid holdup of annular and agitated flows in high-pressure wells. It proposes a method for calculating the liquid holdup of annular and agitated flows in high-pressure gas wells based on atmospheric pressure experimental tests, providing a theoretical basis for the optimized design of drainage and gas production processes.

[0006] Atmospheric pressure test is conducted through Figure 2 The gas-liquid two-phase flow simulation device shown consists of an outlet, a liquid film collection device, a recovery tank, a quick-closing valve, a pressure gauge, a measuring tube, an air inlet, and a liquid inlet. The pipe section at the liquid film collection device has small holes; as the gas-liquid mixture flows through, the liquid film enters the collection device, and the gas core carries the liquid droplets upwards. The plexiglass tube in the measuring tube is graduated. The air inlet and liquid inlet are located a considerable distance from the lower quick-closing valve, ensuring thorough gas-liquid mixing.

[0007] A certain flow rate of gas and liquid is introduced through the air inlet and liquid inlet, respectively. After mixing, the mixture flows through a quick-closing valve into the test section, continuing upwards to the liquid film collection device. The liquid film separates and flows through the collection device into the recovery tank. Measuring the liquid volume in the recovery tank provides the volume of the liquid film that has flowed through the test section over a period of time. Simultaneously closing both quick-closing valves and stopping the gas-liquid supply allows a portion of the gas-liquid mixture to be retained in the test section, and the liquid height can be read from the scale markings on the section. A pressure gauge is used to read the pressure in the test section.

[0008] A method for calculating the liquid holdup of annular and agitated flow in high-pressure gas wells based on atmospheric pressure experimental tests is described below: Step 1: Conduct gas-liquid two-phase annular flow and agitated flow experiments under normal pressure using a simulation device. The experimental pipe diameter remains constant. For each group, the gas flow rate, liquid flow rate, well inclination angle, liquid density, liquid viscosity, gas viscosity, and gas-liquid interfacial tension are given. Measure and record the temperature, pressure, liquid film volume, collection time, and liquid height. Calculate the gas density, pipe cross-sectional area, apparent gas velocity, liquid holdup, liquid film flow rate, apparent liquid film velocity, liquid film Reynolds number, droplet flow rate, core liquid holdup, proportion of droplets in the gas-liquid mixture, proportion of liquid film in the gas-liquid mixture, droplet entrainment rate, liquid film thickness, core Reynolds number, and pressure gradient.

[0009] The gas density is: (1); In the formula, P Pressure, Pa; M Here is the molar mass of the gas, in kg / mol; T Temperature, K; R Let m be the ideal gas constant, taken as 8.314m. 3 ·Pa / (K·mol); Z The gas deviation factor is dimensionless. The cross-sectional area of ​​the pipe is: (2); The apparent flow rate in the gas phase is: (3); In the formula, D The diameter of the pipe is in meters (m). Q G For gas flow rate, m 3 / s; Liquid holdup: (4); In the formula, L L The height of the liquid is in meters (m). L The length of the test pipe section is in meters (m). The liquid film flow rate is: (5); In the formula, V LF Let m be the volume of the liquid film. 3 ; t For collection time, s; The apparent flow rate of the liquid film is: (6); The liquid film Reynolds number is: (7); In the formula, r L The density of the liquid is kg / m³. 3 ; m L The viscosity of the liquid is Pa·s; The droplet flow rate is: (8); In the formula, Q L For liquid flow rate, m 3 / s; Assuming there is no slippage between the droplet and the gas core, the liquid holdup of the gas core is: (9); The proportion of droplets in the gas-liquid mixture is: (10); The proportion of the liquid film in the gas-liquid mixture is: (11); Solve by simultaneously solving equations (10) and (11). H LD and H LF ; The droplet entrainment rate is: (12); The liquid film thickness is: (13); The Reynolds number of the air core is: (14); In the formula, m G Where is the gas viscosity, Pa·s; The pressure gradient is: (15); In the formula, P 2 and P1 represents the readings of pressure gauges 2 and 1, respectively, in Pa; L p The distance between the pressure gauges is in meters (m).

[0010] Step 2: Calculate the gas core density, the friction coefficient of the liquid film and the tube wall, and the interfacial friction coefficient of the gas phase and the liquid film under the experimental conditions based on the data measured in the atmospheric pressure experiment in Step 1.

[0011] The core density is: (16); The interfacial friction coefficient between the gas phase and the liquid film is: (17); In the formula, t i Let be the interfacial shear stress between the gas phase and the liquid film, in Pa; d̃ The thickness of the dimensionless liquid film is dimensionless. The gas phase momentum equation is: (18); The interfacial shear stress between the gas phase and the liquid film is obtained by deformation as follows: (19); In the formula, A G The cross-sectional area of ​​the pipe occupied by the air core, in meters. 2 ; g The acceleration due to gravity is taken as 9.8 m / s². 2 ; The pipe inclination angle is expressed in degrees (°). S i Let be the circumference of the air core, in meters (m). The dimensionless liquid film thickness is: (20); The cross-sectional area of ​​the pipe occupied by the air core is: (twenty one); The circumference of the air core is: (twenty two); The coefficient of friction between the liquid film and the pipe wall is: (twenty three); In the formula, t L Let be the interfacial shear stress between the liquid film and the pipe wall, in Pa; The momentum equation for the liquid film is: (twenty four); In the formula, A LThe cross-sectional area of ​​the pipe occupied by the liquid film, in meters. 2 ; S i The wetted perimeter of the liquid film is measured in meters (m). The interfacial shear stress between the deformed liquid film and the pipe wall is: (25); The cross-sectional area of ​​the pipe occupied by the liquid film is: (26); The wetted perimeter of the liquid film is: (27).

[0012] Step 3: Correct the droplet entrainment rate, gas-liquid interface friction coefficient, and liquid film-tube wall friction coefficient using experimental data and calculation results.

[0013] Corrected droplet entrainment rate: (28); In the formula, a , b , c The fitting coefficient can be obtained from the experimental results; s The surface tension at the gas-liquid interface is N / m; The corrected coefficient of friction at the gas-liquid film interface is: (29); In the formula, d , e , f , h The fitting coefficient can be obtained from the experimental results; The corrected coefficient of friction between the liquid film and the pipe wall is: (30); In the formula, i , j , k The fitting coefficient can be obtained from the experimental results.

[0014] Step 4: Given a pressure value, express the corresponding gas density and core density. Substitute the pipe diameter, liquid density, liquid viscosity, proportion of liquid film in the gas-liquid mixture, gas viscosity, apparent flow velocity of liquid film, dimensionless liquid film thickness, corrected droplet entrainment rate, corrected gas-liquid interface friction coefficient, and corrected friction coefficient between liquid film and pipe wall for each experimental group into the gas-liquid two-phase momentum equation to obtain the apparent gas flow velocity under that pressure condition.

[0015] Eliminating the pressure gradient and simplifying equations (18) and (24), we obtain the two-phase momentum equations for gas and liquid: (31); Substituting the corrected droplet entrainment rate, the corrected gas-liquid interface friction coefficient, and the corrected liquid film and pipe wall friction coefficients: (32); Given a pressure value P I The gas density under this pressure condition can be expressed according to equations (1), (5), (8), (9) and (16). r GI and core density r GCI The pipe inclination angle in a set of experiments i 1. Pipe diameter D Liquid density r L1 Fit coefficient i Apparent flow rate of liquid film v SLF1 Fit coefficient k Dimensionless liquid film thickness d̃ 1. Liquid viscosity m L1 The proportion of liquid film in the gas-liquid mixture H LF1 Fit coefficient a Gas-liquid interfacial tension s 1. Fit coefficients b Fit coefficient c Fit coefficients j Fit coefficients d Gas viscosity m G1 Fit coefficient e Fit coefficients f and fitting coefficients h Substituting into equation (32), the apparent airflow velocity is solved. v SG1 Under the same pressure P I Under these conditions, each set of experimental data was substituted into the equation to solve for the apparent gas flow rate corresponding to each liquid holdup. v SGn , where n is the number of experimental groups.

[0016] Step 5: Take different pressure values ​​and repeat step 4 to plot the calculation results as apparent gas flow rate versus pressure for different liquid holdup series.

[0017] Step 6: Read the corresponding liquid holdup from the obtained curve based on the pressure and apparent gas flow rate under high pressure conditions.

[0018] Compared with the shortcomings and deficiencies of existing technologies, the present invention has the following beneficial effects: Based on atmospheric pressure experimental data, considering droplet entrainment, and relying on flow similarity, a method for calculating liquid holdup in annular and turbulent flow of high-pressure gas wells is proposed. This method solves the problem of difficulty in predicting liquid holdup under high pressure conditions and provides theoretical support for the production optimization of gas wells.

[0019] Obviously, the above description is only an implementation idea of ​​the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions and improvements made within the spirit and principles of the present invention should be included within the protection scope of the present invention. Attached Figure Description

[0020] Figure 1 This is a technical roadmap for a method to calculate the liquid holdup of annular and turbulent flow in high-pressure gas wells based on atmospheric pressure experimental testing. Figure 2 This is a schematic diagram of a gas-liquid two-phase flow simulation device; Figure 3 This is a schematic diagram of gas-liquid two-phase flow; Figure 4 A flowchart for obtaining the apparent gas flow rate versus pressure curves for different liquid holdup series.

[0021] 1-Outlet, 2-Liquid film collection device, 3-Recovery tank, 4-Quick-closing valve 1, 5-Pressure gauge 1, 6-Measuring pipe section, 7-Pressure gauge 2, 8-Quick-closing valve 2, 9-Air inlet, 10-Liquid inlet. Detailed Implementation

[0022] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings.

[0023] like Figure 1 As shown, Figure 1 This is a technical roadmap for the present invention. The present invention provides a method for calculating the liquid holdup in annular and agitated flows of high-pressure gas wells based on atmospheric pressure experiments. Annular and agitated flow experiments of gas-liquid two-phase flow are conducted under atmospheric pressure conditions, using a gas-liquid two-phase flow simulation device such as… Figure 2 As shown, the device consists of an outlet, a liquid film collection device, a recovery tank, a quick-closing valve, a pressure gauge, a measuring tube, an air inlet, and a liquid inlet. The pipe section at the liquid film collection device has small holes; when the gas-liquid mixture flows through, the liquid film enters the collection device, and the gas core carries the liquid droplets upwards. The plexiglass tube in the measuring tube has graduations. The air inlet and liquid inlet are relatively far from the lower quick-closing valve, ensuring thorough mixing of the gas and liquid.

[0024] A certain flow rate of gas and liquid is introduced through the air inlet and liquid inlet, respectively. After mixing, the mixture flows through a quick-closing valve into the test section, continuing upwards to the liquid film collection device. The liquid film separates and flows through the collection device into the recovery tank. Measuring the liquid volume in the recovery tank provides the volume of the liquid film that flowed through over a period of time. Simultaneously closing both quick-closing valves and stopping the gas-liquid supply allows a portion of the gas-liquid mixture to be retained in the test section, and the liquid height can be read from the scale markings on the section. A pressure gauge is used to read the pressure in the test section.

[0025] Gas-liquid two-phase flow simulation experiments were conducted under normal pressure using a gas-liquid two-phase flow simulation apparatus. The experimental pipe diameter remained constant. For each group, given parameters included gas flow rate, liquid flow rate, well inclination angle, liquid density, liquid viscosity, gas viscosity, and gas-liquid interfacial tension. Temperature, pressure, liquid film volume, collection time, and liquid height were measured and recorded. Gas density, pipe cross-sectional area, apparent gas velocity, liquid holdup, liquid film flow rate, apparent liquid film velocity, liquid film Reynolds number, droplet flow rate, core holdup, droplet proportion of the gas-liquid mixture, liquid film proportion of the gas-liquid mixture, droplet entrainment rate, liquid film thickness, core Reynolds number, and pressure gradient. A schematic diagram of the gas-liquid two-phase flow is shown below. Figure 3 As shown.

[0026] The gas density is: (1); In the formula, P Pressure, Pa; M Here is the molar mass of the gas, in kg / mol; T Temperature, K; R Let m be the ideal gas constant, taken as 8.314m. 3 ·Pa / (K·mol); Z The gas deviation factor is dimensionless. The cross-sectional area of ​​the pipe is: (2); The apparent flow rate in the gas phase is: (3); In the formula, D The diameter of the pipe is in meters (m). Q G and Q L These are gas flow rate and liquid flow rate, respectively, in m. 3 / s; Liquid holdup: (4); In the formula, L L The height of the liquid is in meters (m). L The length of the test pipe section is in meters (m). The liquid film flow rate is: (5); In the formula, V LF Let m be the volume of the liquid film. 3 ; t For collection time, s; The apparent flow rate of the liquid film is: (6); The liquid film Reynolds number is: (7); In the formula, r L The density of the liquid is kg / m³. 3 ; m L The viscosity of the liquid is Pa·s; The droplet flow rate is: (8); In the formula, Q L For liquid flow rate, m 3 / s; Assuming there is no slippage between the droplet and the gas core, the liquid holdup of the gas core is: (9); The proportion of droplets in the gas-liquid mixture is: (10); The proportion of the liquid film in the gas-liquid mixture is: (11); Solve by simultaneously solving equations (10) and (11). H LD and H LF ; The droplet entrainment rate is: (12); The liquid film thickness is: (13); The Reynolds number of the air core is: (14); In the formula, m G Where is the gas viscosity, Pa·s; The pressure gradient is: (15); In the formula, P 2 andP 1 represents the readings of pressure gauges 2 and 1, respectively, in Pa; L p The distance between the pressure gauges is in meters (m).

[0027] The gas core density, the friction coefficients of the liquid film and the tube wall, and the interfacial friction coefficients of the gas phase and the liquid film under experimental conditions were calculated based on the data measured in the atmospheric pressure experiment.

[0028] The core density is: (16); The interfacial friction coefficient between the gas phase and the liquid film is: (17); In the formula, t i Let be the interfacial shear stress between the gas phase and the liquid film, in Pa; d̃ The thickness of the dimensionless liquid film is dimensionless. The gas phase momentum equation is: (18); The interfacial shear stress between the gas phase and the liquid film is obtained by deformation as follows: (19); In the formula, A G The cross-sectional area of ​​the pipe occupied by the air core, in meters. 2 ; g The acceleration due to gravity is taken as 9.8 m / s². 2 ; The pipe inclination angle is expressed in degrees (°). S i Let be the circumference of the air core, in meters (m). The dimensionless liquid film thickness is: (20); The cross-sectional area of ​​the pipe occupied by the air core is: (twenty one); The circumference of the air core is: (twenty two); The coefficient of friction between the liquid film and the pipe wall is: (twenty three); In the formula, t L Let be the interfacial shear stress between the liquid film and the pipe wall, in Pa; The momentum equation for the liquid film is: (twenty four); In the formula, A LThe cross-sectional area of ​​the pipe occupied by the liquid film, in meters. 2 ; S i The wetted perimeter of the liquid film is measured in meters (m). The interfacial shear stress between the deformed liquid film and the pipe wall is: (25); The cross-sectional area of ​​the pipe occupied by the liquid film is: (26); The wetted perimeter of the liquid film is: (27).

[0029] The droplet entrainment rate, gas-liquid interface friction coefficient, and liquid film-tube wall friction coefficient were corrected using experimental data and calculation results.

[0030] Corrected droplet entrainment rate: (28); In the formula, a , b , c The fitting coefficient can be obtained from the experimental results; s The surface tension at the gas-liquid interface is N / m; The corrected coefficient of friction at the gas-liquid film interface is: (29); In the formula, d , e , f , h The fitting coefficient can be obtained from the experimental results; The corrected coefficient of friction between the liquid film and the pipe wall is: (30); In the formula, i , j , k The fitting coefficient can be obtained from the experimental results.

[0031] Given a pressure value, express the corresponding gas density and core density, and substitute the pipe diameter, liquid density, liquid viscosity, gas viscosity, apparent flow velocity of the liquid film, dimensionless liquid film thickness, corrected droplet entrainment ratio, corrected gas-liquid interface friction coefficient, and corrected friction coefficient of the liquid film and pipe wall for each experimental group into the gas-liquid two-phase momentum equation to obtain the apparent gas flow velocity under that pressure condition.

[0032] Eliminating the pressure gradient and simplifying equations (18) and (24), we obtain the two-phase momentum equations for gas and liquid: (31); Substituting the corrected droplet entrainment rate, the corrected gas-liquid interface friction coefficient, and the corrected liquid film and pipe wall friction coefficients: (32); Given a pressure value P I The gas density under this pressure condition can be expressed according to equations (1), (5), (8), (9) and (16). r GI and core density r GCI The pipe inclination angle in a set of experiments i 1. Pipe diameter D Liquid density r L1 Fit coefficients i Apparent fluid flow rate v SL1 Fit coefficients k Dimensionless liquid film thickness d̃ 1. Liquid viscosity m L1 The proportion of liquid film in the gas-liquid mixture H LF1 Fit coefficients a Gas-liquid interfacial tension s 1. Fit coefficients b Fit coefficients c Fit coefficients j Fit coefficients d Gas viscosity m G1 Fit coefficients e Fit coefficients f and fitting coefficients h Substituting into equation (32), the apparent airflow velocity is solved. v SG1 Under the same pressure P I Under these conditions, each set of experimental data was substituted into the equation to solve for the apparent gas flow rate corresponding to each liquid holdup. v SGn , where n is the number of experimental groups.

[0033] By taking different pressure values, the calculation results are plotted as apparent gas flow rate versus pressure for different liquid holdup series.

[0034] The liquid holdup is read from the obtained curve based on the pressure and apparent gas flow rate under high pressure conditions.

Claims

1. A method for calculating the liquid holdup of annular flow and churn flow in high-pressure gas wells based on normal-pressure experimental tests, characterized in that, Includes the following steps: Step 1: Conduct gas-liquid two-phase annular flow and agitated flow experiments under normal pressure using a simulation device. The experimental pipe diameter remains constant. For each group, the gas flow rate, liquid flow rate, well inclination angle, liquid density, liquid viscosity, gas viscosity, and gas-liquid interfacial tension are given. Measure and record the temperature, pressure, liquid film volume, collection time, and liquid height. Calculate the gas density, pipe cross-sectional area, apparent gas velocity, liquid holdup, liquid film flow rate, apparent liquid film velocity, liquid film Reynolds number, droplet flow rate, core liquid holdup, proportion of droplets in the gas-liquid mixture, proportion of liquid film in the gas-liquid mixture, droplet entrainment rate, liquid film thickness, core Reynolds number, and pressure gradient. Step 2: Calculate the gas core density, the friction coefficient of the liquid film and the tube wall, and the interfacial friction coefficient of the gas phase and the liquid film under the experimental conditions based on the data measured in the atmospheric pressure experiment in Step 1. Step 3: Correct the droplet entrainment rate, gas-liquid interface friction coefficient, and liquid film-tube wall friction coefficient using experimental data and calculation results; Step 4: Given a pressure value, express the corresponding gas density and core density. Substitute the pipe diameter, liquid density, liquid viscosity, gas viscosity, apparent liquid velocity, dimensionless liquid film thickness, corrected droplet entrainment rate, corrected gas-liquid interface friction coefficient, and corrected liquid film and pipe wall friction coefficient of each experiment into the gas-liquid two-phase momentum equation to obtain the apparent gas velocity under that pressure condition. Step 5: Take different pressure values ​​and repeat step 4 to plot the calculation results as apparent gas flow rate versus pressure for different liquid holdup series. Step 6: Read the corresponding liquid holdup from the obtained curve based on the pressure and apparent gas flow rate under high pressure conditions.

2. The method for calculating liquid holdup in high-pressure gas wellbore based on atmospheric pressure experiments according to claim 1, characterized in that, Step four proposes substituting the results of the atmospheric pressure experiment into the gas-liquid two-phase momentum equation to solve for the apparent gas flow rate under a given pressure condition. The gas-liquid two-phase momentum equation is as follows: ； In the formula, g The acceleration due to gravity is m / s². 2 ; The pipe inclination angle is expressed in degrees (°). D Pipe diameter, in meters (m); ρ GC The density of the air core is kg / m³. 3 ; ρ L The density of the liquid is kg / m³. 3 ; f L is the coefficient of friction between the liquid film and the pipe wall, dimensionless; v SLF The apparent velocity of the liquid film is m / s; δ̃ The thickness of the dimensionless liquid film is dimensionless. f i is the friction coefficient at the gas-liquid film interface, dimensionless; v SG The apparent airflow velocity is in m / s; The liquid film Reynolds number is: ； In the formula, Q LF For liquid film flow rate, m 3 / s; μ L The viscosity of the liquid is Pa·s; The Reynolds number of the air core is: ； In the formula, μ G Where is the gas viscosity, Pa·s; Corrected droplet entrainment rate: ； In the formula, ρ G The density of the gas is kg / m³. 3 ; σ The surface tension at the gas-liquid interface is N / m. μ L The viscosity of the liquid is Pa·s; a , b , c These are all fitting coefficients, which can be obtained from experimental results; Corrected gas-liquid interfacial friction coefficient: ； In the formula, d , e , f and h These are all fitting coefficients, which can be obtained from experimental results; Correct the coefficient of friction between the liquid film and the pipe wall: ； In the formula, i , j , k These are all fitting coefficients, which can be obtained from experimental results; Substituting the above parameters into the two-phase momentum equation: ； Given a pressure value P I This indicates the gas density under that pressure condition. ρ GI and core density ρ GCI The pipe inclination angle in a set of experiments θ 1. Pipe diameter D Liquid density ρ L1 Fit coefficient i Apparent flow rate of liquid film v SLF1 Fit coefficient k Dimensionless liquid film thickness δ̃ 1. Liquid viscosity μ L1 The proportion of liquid film in the gas-liquid mixture H LF1 Fit coefficient a Gas-liquid interfacial tension σ 1. Fit coefficients b Fit coefficients c Fit coefficient j Fit coefficient d Gas viscosity μ G1 Fit coefficients e Fit coefficient f and fitting coefficients h Substituting into the above equation, we can solve for the apparent airflow velocity. v SG1 Under the same pressure P I Under these conditions, each set of experimental data was substituted into the equation to solve for the apparent gas flow rate corresponding to each liquid holdup. v SGn , where n is the number of experimental groups.

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