Experimental study of liquid holdup of liquid-gas two-phase flow in horizontal and inclined pipes

Since Lockhart and Martinelli (1949) established the first empirical liquid holdup equation [6], many scholars have proposed classical correlations for liquid holdup. For example, Hughmark (1962) tested the mixture of different liquid mediums and the air in vertical pipes (ID: 16mm~63.5mm), calculated liquid holdup in light of the test results, and applied the calculation method to horizontal pipes [7]. Hughmark’s method was modified by Garcia in 2005 [8].

In 1967, Guzhov et al. developed the correlation for liquid holdup based on the data for liquid-gas mixture in pipes with an inclination angle between -9° to 9° [9]. Six year later, Beggs and Brill studied two-phase flow in pipes tilted at -90° ~ +90°, presented the correlation between liquid holdup and pressure drop, and derived the correlation for different pipe inclination angles based on that for horizontal pipes [10].

In the past decades, many other scholars, such as Eaton et al. [11], Abdul-Majeed [12], Minami-Brill [13], Ansari [14] and Xiao [15], have developed liquid holdup correlations for implementation in the petroleum industry. Despite these achievements, these liquid holdup correlations fail to realize desirable accuracy across different experimental or field conditions [16]-[17]. This conclusion is drawn through numerous evaluations.

In 1964, Dukler evaluated the correlations of Hoogeendoom, Hughmark and Lockhart-Martinelli with the AGA/API database, and discovered that even the best-performing Hughmark’s correlation cannot achieve the desired accuracy [16].

In 1975, Vohra tested the correlations of Beggs-Brill, Dukler, Eaton et al., Guzhov, Hughmark and Lockhart-Martinelli. The test data include 58 groups from Beggs’ research on a horizontal pipe and 238 groups from that of Eaton et al. The results show that the correlation of Eaton et al. enjoyed the highest accuracy with an average percent error (APE) of -5.9%, followed by the Beggs-Brill model (18.9%) [17]. This is because most of the data came from Eaton et al.

In the same year, Mandhane et al. assessed the liquid holdup correlation presented by Beggs-Brill, Eaton et al. and Hughmark based on 2,685 measured values of liquid holdup in horizontal gas-liquid two-phase flow, and recommended a calculation method in view of the flow pattern [18] (Table 1).

Table 1. Liquid holdup calculation method recommended by Mandhane et al.

In 1993, Abdul-Majeed made some improvements to the Beggs-Brill correlation, and compared the improved method with 11 related correlations (e.g. Eaton et al, Minami-Brill Ⅰ and Ⅱ) under horizontal, inclined and vertical conditions. The scholar proved that the modified method had the smallest APE of 6.8% [19].

Based on the data from various horizontal pipe experiments, Garcia et al. (2005) created a theoretical model for liquid holdup prediction in a horizontal pipe, and verified that the model is more accurate than 25 existing methods [20]

In 2008, Cheng et al. compared the Hughmark correlation and Garcia’s modified-Hughmark correlation based on the test data from the National Engineering Laboratory for Pipeline Safety in China University of Petroleum (Beijing). They recommended to forecast the liquid holdup in horizontal pipes with Garcia’s modified model [21].

4.1 Evaluation of liquid holdup correlations based on air-water mixture

The six most popular models, including Beggs-Brill, Mukherjee-Brill, Eaton et al., modified-Hughmark, Minami-Brill I and Minami-Brill II, were evaluated against 261 sets of test data on air-water mixture at the inclination angles of 0°, 15° and 30°.

To predict the liquid holdup at the varied inclinations, the Beggs-Brill inclination modifications were applied to the Eaton et al. and the modified-Hughmark correlations.

The accuracy of each method was tested by three statistical parameters: the average relative error ε₁, the absolute relative average error ε₂ and the standard deviation of relative error ε₃:

$e_{i}=\frac{\left(H_{L}\right)_{i p r e}-\left(H_{L}\right)_{i e x p}}{\left(H_{L}\right)_{i e x p}} \times 100 \%$      (1)

$\varepsilon_{1}=\frac{1}{N} \sum_{i=1}^{N}\left(e_{i}\right)$     (2)

$\varepsilon_{2}=\frac{1}{N} \sum_{i=1}^{N}\left|e_{i}\right|$      (3)

$\varepsilon_{3}=\sqrt{\frac{\sum_{i=1}^{N}\left(e_{i}-\varepsilon_{1}\right)^{2}}{N-1}}$       (4)

The average relative error ε₁ reflects the difference between the predicted value and the measured value. A positive value indicates over-prediction and the inverse is also true. However, the true average error might be concealed as the individual errors could offset each other in the summation process. That is why the absolute average relative error ε₂ was introduced. The standard deviation of relative error ε₃ reveals the degree of dispersion between the predicted value and the measured value. The verification results are shown in Table 3.

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Figure 2. Accuracy of the existing models at liquid flow rates of 6.25 m³/h and 8.33 m³/h

From Table 3, it can be seen that the modified-Hughmark model outperformed the other models with the lowest absolute average relative errors at 0°, 15° and 30° for the 60mm-diameter pipe, while Minami-Brill I correlation boasted the best accuracy for the 75mm-diameter pipe. Except for these two models, the other four models underwent significant increases in liquid holdup prediction errors with the expansion of the pipe diameter. When the liquid flow rate grew from 6.25m³/h to 8.33m³/h, Mukherjee-Brill, Minami I, Minami II and modified-Hughmark correlations all enjoyed better accuracy under horizontal and inclined conditions (Figure 2). Therefore, the accuracy of liquid holdup correlation is yet to be improved at low liquid flow rates. It is also observed that the accuracy of Beggs-Brill model and Eaton et al. model degenerated with the increase in liquid flow rate. The error difference was more pronounced in Eaton et al. model, indicating the need for improvement at high liquid flow rates.

Table 3. Verification results of the six models based on air-water mixture

4.2 Evaluation of liquid holdup correlations based on air-oil mixture

The six models were also tested with 170 sets of data on air-white oil two-phase flow. According to the absolute average relative errors, all models other than the Beggs and Brill model exceeded 100% in terms of the error (Figure 3).

As shown in Figure 4, the absolute average relative errors of the modified-Hughmark model rocketed up with the increase in liquid flow rate, while the accuracy of Beggs-Brill correlation was enhanced when the liquid flow rate climbed up from 15m³/h to 30m³/h. Comparing Figure 3 and Figure 4, it is clear that the Beggs-Brill model outshined the other correlations for the white oil-air two phase flow. The statistical errors of liquid holdup by this method are shown in Table 4. It can be seen that the accuracy of Beggs-Brill correlation under the horizontal condition differed greatly from that under the inclined condition. Meanwhile, when the predicted liquid holdups for water-air flow (Table 3) were contrasted with those of white oil-air flow (Table 4), the model was subject to less error for the inclined pipe. Furthermore, the model was less accurate for the 75mm-diameter pipe than the 60mm-diameter pipe at all inclination angles.

To sum up, the modified-Hughmark model performs well in predicting liquid holdup of water-air two-phase flow in the horizontal direction and at small inclination angles (0~30°), while the Beggs-Brill correlation is applicable to the cases that the measurement accuracy is inadequate for the physical properties of the liquid.

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Figure 3. Absolute average relative errors of the existing models

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Figure 4. Absolute average relative errors of the existing models at different liquid flow rates

Table 4. Statistical errors of liquid holdup by Beggs-Brill method

4.3 Effect of pipe diameter

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Figure 5. Measured liquid holdup at different pipe diameters and gas-liquid ratios

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Figure 6. Measured liquid holdup at different pipe diameters and gas-liquid ratios at the inclination angle of 15°

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Figure 7. Measured liquid holdup at different pipe diameters and gas-liquid ratios at the inclination angle of 30°

In this section, three different inner diameters (40mm, 60mm and 75mm) of the pipe were selected to investigate the effect of pipe diameter on liquid holdup. As shown in Figure 5, the liquid holdup is positively correlated with the pipe diameter at the same gas-liquid ratio. This means the liquid flow area increases with the diameter, and occupies a greater portion in the pipe section. In addition, with the increase of gas-liquid ratio, the influence of pipe diameter on liquid holdup was rather small at the same liquid flow rate. As the gas picked up speed, the liquid carrying capacity increased, and the liquid holdup at each pipe diameter gradually moved to zero.

Similar patterns were observed in the tests at the inclination angles of 15° and 30° (Figures 6 and 7). Therefore, a smaller pipe diameter or higher gas-liquid ratio is recommended to reduce liquid holdup in gas transmission pipe and to balance the throughput and flow rate.

4.4 Effect of liquid medium

Before the test on the effect of liquid medium, a theoretical analysis was conducted on the assumption that the physical properties of the liquid medium vary independently. The analysis was made in reference to the modified-Hughmark correlation. According to the correlation, Equations (5)~(10) are true:

$H_{L}=1-K \frac{U_{s g}}{U_{s g}+U_{s l}}$       (5)

$K=0.1746-0.1301(\ln Z)+0.7508(\ln Z)^{2}-0.4308(\ln Z)^{3}$$+0.09553(\ln Z)^{4}-0.007452(\ln Z)^{5}$       (6)

$Z=\frac{R e^{1 / 6} F r^{1 / 8}}{\lambda_{L}^{1 / 4}}$        (7)

$F r=\frac{\left(U_{s g}+U_{s l}\right)^{2}}{g D}$      (8)

$R e=\frac{\left(\rho_{g} U_{s g}+\rho_{l} U_{s l}\right) D}{\lambda_{L} \mu_{L}+\left(1-\lambda_{L}\right) \mu_{g}}$    (9)

$\lambda_{L}=\frac{U_{s g}}{U_{s g}+U_{s l}}$     (10)

Equation (6) increases monotonically when the value of ln(Z) is greater than 1. According to the test data, the values of ln(Z) in this research always exceed 1. Following this equation, any increase in viscosity will lead to the decline in Reynolds number, provided that all the other parameters are constant. In this scenario, the Z in Equation (7) and K in Equation (6) will also decrease. As can be seen from Equation (5), the liquid holdup will grow in magnitude, revealing that viscosity is proportional to liquid holdup. Diffusion is retarded by an increase in viscosity ratio at a fixed fluidity for the dispersed phase[24].  It is also deduced that density is negatively correlated with liquid holdup. Moreover, the liquid holdup in the pipe will increase with the proportion of heavy components [25]. Under the same gas-liquid volume flow, it is expected that liquid holdup will be relatively high if the liquid medium has a small density, high viscosity and high content of heavy components. According to the test results in Figure 8, it is clear that the white oil-air two-phase flow had a higher liquid holdup than the water-air mixture under the same gas-liquid volume flow. Since the former has a smaller density, higher viscosity and higher content of heavy components than the latter, the test results are consistent with the predicted results.

Then, the relationship between the gas-liquid ratio and the liquid holdup at the inclined angle of 15° was compared with that at 30°. The comparison shows that the liquid holdups varied in a similar pattern with those for the horizontal pipe at the same liquid flow conditions (Figure 9 and Figure 10).

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Figure 8. Measured liquid holdups at different gas-liquid ratios for white oil-air two-phase flow and water-air mixture in the horizontal pipe

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Figure 9. Measured liquid holdups at different gas-liquid ratios for white oil-air two-phase flow and water-air mixture in a pipe with the inclined angle of 15°

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Figure 10. Measured liquid holdups at different gas-liquid ratios for white oil-air two-phase flow and water-air mixture in a pipe with the inclined angle of 30°

4.5 Effect of pipe inclination

According to the theoretical liquid holdup calculated by Beggs and Brill, the correlations for horizontal pipe cannot be directly applied to the inclined pipe. This implies the importance of inclination on liquid holdup.

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Figure 11. Measured liquid holdups at different inclined angles and gas-liquid ratios under the liquid flow of 10.42 m³/h

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Figure 12. Measured liquid holdup at different inclined angles and gas-liquid ratios under the liquid flow of 12.5 m³/h

As shown in Figure 11, under the liquid flow rate of 10.42m³/h and the gas-liquid ratio of 40, the liquid hold decreased as the inclined angle shifted from 0° to 15°, and increased as the angle expanded from 15° to 30°. By contrast, when the liquid flow rate increased to 12.5 m³/h in Figure 12, the liquid holdup rose slightly as the inclined angle increased from 0° to 30°. To sum up, the liquid holdup varied insignificantly when the inclined angle fell between 0° and 30°.

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Figure 13. Measured liquid holdups at different gas-liquid ratios and inclined angles at the liquid flow of 10.42 m³/h

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Figure 14. Measured liquid holdups at different gas-liquid ratios and inclined angles at the liquid flow of 12.5 m³/h

As can be seen in Figures 13 and 14, with the increase in the liquid flow rate, the liquid holdup curves of three inclined angles overlapped each other. This phenomenon reveals that the liquid holdup is not heavily affected by small variation in the inclined angle under a low gas-liquid ratio (<100), and the influence will further decrease with the growth of the gas-liquid ratio.

4.6 Effect of gas-liquid ratio

It can be seen from Figure 13 and 14 that the liquid holdup decreased with increase in gas-liquid ratio. The rate of decrease was rapid at low gas-liquid ratios, and slows down as the ratio increased. Eventually, the liquid holdup stabilized at a certain value and grew with the increase in the liquid flow. Hence, the gas-liquid ratio has a great impact on liquid holdup in a certain range.

In this test, it is observed that when the gas-liquid ratio was less than 200, the liquid holdup decreased at a rather fast pace; when the ratio fell between 200 and 300, the decrease rate gradually slowed down; when the ratio was greater than 300, the liquid holdup curves became asymptotic.

Through the simulation of the actual working conditions (gas volume: 4×10⁵ m³/d; liquid volume: 8~200 m³/d), the author discovered that the effect of the gas-liquid ratio on liquid holdup was minimal when the former exceeded 20,000 [26]. The simulation results under other working conditions show that the effect was no longer obvious after the gas-liquid ratio surpassed 1,000 [27]. Therefore, the range of gas-liquid ratio has a significant impact on the liquid holdup, and the exact impact depends on the specific conditions.

Thanks for Luo Wei of the corresponding author for the article. This work is supported by the National Natural Science Found Project (NO. 61572084) and National Key Scientific and Technological Project (2016ZX05056004-002, 2016ZX05046004-003)