Frictional pressure drop of two-phase flow in a horizontal tube under low void fraction

The frictional pressure drop of air-water two-phase flow is a crucial parameter in hydrodynamic calculations. Solving for precisely calculating frictional pressure drop in it remains a significant challenge due to the various factors affecting two-phase flow characteristic. Currently, various scholars have proposed multiple frictional pressure drop prediction models of two-phase flow. Typically, these models exhibit great errors under low void fraction. To more accurately predict the frictional pressure drop of two-phase flow under low void fraction, this paper experimentally investigated the flow characteristics of air-water two-phase flow in a horizontal tube. The flow rates for water and air are 3-6 kg/s and 0.0004-0.003 kg/s, respectively. The void fraction ranged from 0 to 0.07%. The experimental data and flow patterns of two-phase flow were analyzed, and these data was compared with commonly used frictional pressure drop prediction model of two-phase flow. The results indicated that: (1) Within the scope of this paper, two-phase flow pattern observed include bubbly flow and slug flow. With changes in the void fraction, the number of bubbles and the length of slugs have undergo alterations; (2) All three commonly used prediction models underestimated the frictional pressure drop under low void fraction. Therefore, Considering more influencing factors to improve the accuracy of prediction model is the next research direction. This research has showed the limitations of existing prediction models and emphasized the importance of research on frictional pressure drop of two-phase flow.


Introduction
Air-water two-phase flow finds wide applications across various industries such as energy, hydraulic engineering and transportation.Precisely calculating the frictional pressure drop of two-phase flow is crucial for process design in industrial production [1].Many researchers have conducted studies on the two-phase flow characteristic and have proposed various equations to predict its frictional pressure drop [2][3][4][5].However, there are large errors between the predicted values of these equations and the actual values [6][7][8].Most of these studies have been carried out under conditions of medium to high void fraction, therefor the conclusions are often not applicable to low void fraction.With the advancement in the study and application of two-phase flow in tubes, there is an increasing need for a better understanding of the two-phase flow characteristics under low void fractions.Under low void fraction conditions, there is a significant difference in flow rates between air and water.Therefore, the interactions between difference phases and between the fluids and the wall are more intricate, greatly affecting frictional pressure drop.Chen [9] collected 30 models for calculating frictional pressure drop of two-phase flow within tubes.He also collected and organized a database comprising 2922 sets of data from 23 articles, and used this database to evaluate and analyze these models.The results indicate that these existing models fail to provide exactly calculations for adiabatic two-phase flow frictional pressure drop, with most models having errors exceeding 30%.This is due to the significant variation in flow patterns of two-phase flow under different operating conditions, the range of experimental condition on which these models are based affects their adaptability.Also, the void fraction has a significant influence on the accuracy of predicted values of these models.
Therefore, this paper investigates the frictional pressure drop and flow pattern of two-phase flow in circular horizontal tube using air and water as the working fluids.A comparison is made between the experimental data and commonly used frictional pressure drop predication models of two-phase flow (including the Homogeneous model, Chisholm model, and Friedel model), and the accuracy of these models under low void fractions was evaluated.Based on the experimental results, the variation trend of friction pressure drop under low void fraction are analyzed, and a better calculation formula for frictional pressure drop is provided.

Frictional pressure drop prediction models for gas-liquid two-phase flow
To calculate the frictional pressure drop of two-phase flow, researchers have proposed numerous prediction models, which can be classified into two main categories: homogeneous flow models and separated flow models.

Homogeneous flow model
In the homogeneous flow model, the gas-liquid two-phase fluid is considered as a uniform medium with homogeneous flow parameters.This model is based on the following two assumptions: (1) the actual velocities of the gas and liquid phases are equal, and (2) the interactions at the gas-liquid interface can be neglected.
In the homogeneous model, the calculation formula for the frictional pressure drop of two-phase flow is as follows: Here, / is the frictional pressure drop gradient of two-phase flow, in Pa/m; is the frictional resistance coefficient of two-phase flow; is the density of two-phase flow, in kg/m3; is the hydraulic diameter of the channel, in m; and is the mass flow velocity of two-phase flow, in kg/(m2•s).
The calculation formula for the density of two-phase flow is as follows: Where and are the density of liquid phase and gas phase, respectively, in kg/m3.represents the void fraction mentioned earlier.The void fraction is the fraction of gas phase in the overall two-phase flow and is an important parameter for analyzing two-phase flow situations.The calculation formula for void fraction is as follows: and are the mass flow rate of liquid and gas phases, respectively, in kg/s.The frictional resistance coefficient of two-phase flow is calculated as follows Where is the average Reynolds number of two-phase flow, and is the dynamic viscosity of two-phase flow in Pa • s.In this paper, the McAdams equivalent flow density calculation method is used to calculate the dynamic viscosity [2], which is given by:

Separated flow models
The separated flow model considers the gas and liquid phases in two-phase flow as independent flows, each with its own real average velocity and physical properties.From a modeling perspective, the separated flow model can better reflect the actual conditions of two-phase flow compared to the homogeneous flow model.In the separated flow model, the frictional pressure drop of two-phase flow is obtained by multiplying the frictional pressure drop of single-phase flow by a frictional pressure drop multiplier.This idea was first proposed by Lockhart and Martinelli and has been increasingly accepted by scholars, making it a classic method for calculating the frictional pressure drop of gasliquid two-phase flow and organizing related data [10].
The calculation of frictional pressure drop of two-phase flow using the separated flow model is mainly based on the following four equations: Where and are the frictional pressure drop gradients of liquid phase and gas phase, respectively.They represent the frictional pressure drop gradients when the liquid phase (or gas phase) flows alone through the same tube segment.
2 and 2 are the frictional pressure drop multipliers of liquid phase and gas phase, respectively.and are the frictional pressure drop gradients of total liquid phase and total gas phase, respectively.They represent the frictional pressure drop gradients when total two-phase are converted into the liquid phase (or gas phase) based on mass, and these liquid phases (or gas phase) flow through the same tube segment.
2 and 2 are the frictional pressure drop multipliers of total liquid phase and gas phase, respectively.In other words, in the separated flow model, the key to calculating the frictional pressure drop of two-phase flow lies in the calculation of these four multipliers.By fitting data from different experimental conditions, various correlation equations have been derived to calculate the frictional pressure drop multipliers.The most commonly used methods are the Chisholm model and Friedel model.

Chisholm model
To facilitate data processing, Lockhart and Martinelli proposed a dimensionless number , also known as the Lockhart-Martinelli parameter, which is defined as Where and are the frictional resistance coefficients of liquid phase and gas phase, respectively.Based on experimental data, they plotted the relationship between 2 − .However, graphical representation of the relationship is not convenient for calculations.Therefore, Chisholm correlated the frictional pressure drop multiplier of liquid phase 2 and the Lockhart-Martinelli parameter using the coefficient in the following expression [3]: The values of coefficient recommended by Chisholm are based on the flow regimes of gas and liquid phases (laminar or turbulent), as shown in Table 1.

Friedel model
The Friedel model is used to calculate the frictional pressure drop of two-phase flow by using the frictional pressure drop multiplier of total liquid phase 2 .Based on 25,000 experimental data, Friedel proposed the following formula: Where, and are the Froude number and Weber number of two-phase flow respectively, which are calculated using density of two-phase flow .The formula for calculating the coefficient is

Experimental equipment and research methods
To investigate the pressure drop variation of air-water two-phase flow at low void fractions in a horizontal tube with different flow rates, a visualization two-phase flow experimental system was designed.The experimental system consists of four main components: the water supply system, the air supply system, the test section, and the data acquisition system, as shown in Figure 1.Water stored in a water pool is pumped into the main tube through a vortex pump, and the water flow rate is controlled by adjusting the pump rotational speed and the butterfly valve opening.Air is discharged from an air compressor, and its flow rate is regulated by a flow control valve before entering the main tube, where it mixes thoroughly with the water.The air-water two-phase flow passes through the test section and is then discharged.The discharged water phase is collected in a water storage tank and weighed to accurately calculate the water flow rate.
The main tube and the test section tube in the experimental system are both constructed with circular tube with an inner diameter of 52.9 mm.The water flow rate range in this study is 4-6 kg/s, the air flow rate range is 0.0004-0.003kg/s, and the void fraction range is x < 0.07%.The pressure in the test section is measured by drilling small holes in the tube and connecting pressure sensors (KYOWA PGM-5KH, Rated capacity: 500 kPa, Nonlinear error:0.37%), with a distance of 700 mm between the two pressure measurement points.The pressure sensors are connected to a data acquisition device, which is controlled and recorded by a computer.The data is collected at a frequency of 1024 Hz, with a sampling time of one minute.To visualize the flow, the test section tube is made of transparent organic glass.A high-speed camera is used to observe the air-water two-phase flow patterns.
Before starting the experiment, the pressure sensors and high-speed camera are calibrated.During the experiment, the water flow rate is fixed first, then the air flow rate is adjusted to the expected value.After each flow rate adjustment, wait approximately 5 minutes for pressure drop fluctuations to stabilize, and then record the flow patterns and collect the data.Once a set of operating conditions is completed, the air flow rate is changed, and the same procedure is followed for the next set of operating conditions.

Frictional resistance coefficient of single-phase flow
To verify the validity and accuracy of the experimental system and measurement system, a water resistance characteristic experiment was conducted before the two-phase flow experiment.The experiment measured the pressure drop of water in the test section under turbulent flow conditions, and the calculated values of the frictional resistance coefficient from the experimental data were compared with theoretical values.The theoretical values were obtained from the Blasius equation [11]: As shown in Figure 2, both the frictional resistance coefficient obtained from experiment and the one from the Blasius equation decreases with the increase of water velocity, and the relative error between the experimental value and the theoretical value is not more than 20%.It can be seen that the experimental system and the measurement system can be used to investigate the pressure drop of fluid in the tube.

Two-phase flow pattern
Figures 3 and 4 show the observed flow pattern of two-phase flow under different flow conditions.In this study, two type flow patterns were observed: bubbly flow and slug flow.It can be observed that the majority of the observed flow patterns are slug flow.Only when the water flow rate is relatively high and the air flow rate is low (e.g., water flow rate Gw=5kg/s and air flow rate Ga=0.00043kg/s),bubbly flow can be observed, where the air is dispersed in a bubble form within the continuous water phase.The bubbles are not uniformly distributed across the tube cross-section, and due to the influence of gravity, the bubbles tend to flow in the upper part of tube.As the air flow rate increases or the water flow rate decreases, the bubbly flow transitions to slug flow.This is because the void fraction increases, the bubbles to coalesce and form elongated air slugs.In slug flow, there are still numerous small bubbles distributed within the water phase.From Figures 3 and 4, the following two rules can be observed: (1) With increasing air flow rate, the length of air slug increases, and the number of surrounding bubbles also increases.( 2) With increasing water flow rate, the length of air slug decreases, and the number of surrounding bubbles increases.This indicates that the total flow rate of two-phase flow affects the number of bubbles, while the void fraction affects the length of air plugs.Additionally, at a water flow rate of 4 kg/s, the bubbles tend to be circular in shape under the effect of surface tension.As the water flow rate increases to 6 kg/s, larger non-circular bubbles appear inside the tube.This is because the air and water velocities differ significantly, causing the bubbles to deviate from a circular shape by the shear force exerted from the surrounding water flow.When the water flow rate is 5 kg/s, the bubbles remain circular at different air flow rates, and there is no significant increase in the number or volume of bubbles.This indicates that the interaction between the bubbles and the surrounding water is predominantly governed by surface tension.

Friction pressure drop of two-phase flow
The total pressure drop of two-phase flow through a tube includes frictional pressure drop ∆ , gravitational pressure drop ∆ , accelerating pressure drop ∆ and local pressure drop ∆ , that is In this paper, due to the use of a horizontal tube with a uniform diameter for the test section, the gravity pressure drop and local pressure drop are not considered.Since there is no heat transfer, the mass flow rates of the two phases remain relatively constant, therefore the acceleration pressure drop is also neglected.Hence, the measured pressure drop can be considered as the frictional pressure drop of two-phase flow, that is As shown in Figure 5, the graph illustrates the variation of frictional pressure drop of two-phase flow.It is evident that, at the same water flow rate, the frictional pressure drop of two-phase flow always increases with an increase in air flow rate (void fraction).When the water flow rate is higher, the growth rate of frictional pressure drop is greater compared to that in other two water flow rates.Considering the analysis of flow patterns mentioned earlier, it can be inferred that even in the same flow pattern (slug flow), when the flow rate is different, the dominant force at the interface between two phases may be different.Therefore, the total force has different effects on frictional pressure drop, and the growth rate of frictional pressure drop will also change.Under the experimental conditions of this study, at higher water flow rate, the influence of surface tension on frictional pressure drop is weaker, while at lower water flow rate, this influence becomes pronounced.

Comparison between the predicted results and experimental results
Based on the experimental conditions, predictions of frictional pressure drop of two-phase flow were made using the Homogeneous model, Chisholm model and Friedel model.Figure 6-8 present the comparison between the predicted data from these three models and the experimental data.It can be observed that all three prediction models underestimated the frictional pressure drop of two-phase flow at low void fraction.The predicted values from the Chisholm model were also around 45% lower than the experimental values.In Chisholm model, the C coefficient is taken as a constant value based on the flow regime of gas and liquid phases (laminar or turbulent).However, in reality, the C coefficient is a measure of interaction between the gas and liquid phases, and its value varies with the flow conditions.This leads to a significant deviation between the predicted frictional pressure drop based on the suggested C coefficient by Chisholm and the experimental values.
The predicted values from the Friedel model were approximately 25% lower than the experimental values.As shown in the equations (12) provided earlier, the Friedel model incorporates the Froude number and Weber number to calculate the frictional pressure drop, taking into account the influence of gravity and surface tension.Considering the flow pattern analysis in this paper, the effects of surface tension at the interface between air and water vary significantly under different flow conditions.Additionally, bubbles tend to distribute predominantly in the upper region of tube due to the influence of gravity.Therefore, the predicted values from the Friedel model are closer to experimental values compared to previous two models.However, the 25% error is still not negligible.
In conclusion, due to the different analytical approaches adopted by researchers and limitations in experimental conditions, the proposed frictional pressure drop prediction models are not well-suited for two-phase flow with low void fraction.

Correction of C coefficient
Since the Chisholm model is widely used and has a simple form that allows for easy incorporation of other parameters, this paper aims to modify the C coefficient in the Chisholm model based on experimental data.The values of the C coefficient obtained from experimental data is processed by Chisholm equation are shown in Figure 9, depicting the relationship between the C coefficient and the Lockhart-Martinelli parameter X.The range of C coefficient is between 50 and 200, significantly higher than the recommended C coefficient by Chisholm.This could be attributed to the large difference in flow velocities between water and air at low void fraction, leading to more intense interactions between the two phases and an increase in C coefficient.The C coefficient increases with the increase in the X coefficient, and their relationship can be approximated by a power function.Based on this observed trend, a relationship equation between the C coefficient and the X coefficient is obtained through fitting the experimental data, as follow: = 5.83 0.66  (17) This equation can provide reference for the prediction of frictional pressure drop of two-phase flow.However, due to the limitation of the experimental scope, the application range of this equation is limited, and it is more suitable for the slug flow of air-water two-phase flow with low void fraction.

Conclusion
To investigate the flow characteristics of air-water two-phase flow with low void fraction, frictional pressure drop and flow visualization experiments were conducted in a two-phase flow experimental system.The following conclusions were drawn based on the analysis of experimental results: (1) In this paper, the observed flow patterns include bubble flow and slug flow.As the flow rate changes, the number of bubbles and the length of slug also change.In slug flow, as the total flow rate increases, the number of bubbles dispersed in the continuous water also increases.As the void fraction increases, the length of slug becomes longer.
(2) At low water flow, bubbles exhibit a circular shape under the influence of surface tension.At high water flow, bubbles deviate from circular due to the shear forces exerted by surrounding water.
(3) At the same water flow, the frictional pressure drop of two-phase flow always increases with an increase in air flow rate.The growth rate of frictional pressure drop is higher at high water flow rates.
(4) Based on the experimental conditions, predictions of frictional pressure drop of two-phase flow were made using the homogeneous model, Chisholm model, and Friedel model.All three prediction models underestimated the frictional pressure drop of two-phase flow under low void fractions.The predicted values from the Chisholm and homogeneous model were approximately 45% lower than the experimental values, while the Friedel model were approximately 25% lower.
(5) Based on the experimental data and the Chisholm model, the key coefficient C was modified using the Lockhart-Martinelli parameter X to make it suitable for low void fraction conditions.The

Figure 5 .
Figure 5.The variation of frictional pressure drop of two-phase flow.

Figure 6 .
Figure 6.The comparison between Homogeneous model and experimental data.

Figure 7 .
Figure 7.The comparison between Chisholm model and experimental data.

Figure 8 .
Figure 8.The comparison between Friedel model and experimental data.The predicted values from the homogeneous model were approximately 45% lower than the experimental values.Clearly, since the majority of flow patterns observed in this paper were slug flow, which significantly deviates from the "homogeneous medium assumption" of the homogeneous model, there is a significant discrepancy between the predicted values of homogeneous model and experimental values.The predicted values from the Chisholm model were also around 45% lower than the experimental values.In Chisholm model, the C coefficient is taken as a constant value based on the flow regime of gas and liquid phases (laminar or turbulent).However, in reality, the C coefficient is a measure of interaction between the gas and liquid phases, and its value varies with the flow conditions.This leads

Table 1 .
The values of coefficient recommended by Chisholm