Description of Case
The general two-phase heat transfer correlation (see the article on the Knovel website by Ghajar and Tang, Estimations of Heat Transfer in Nonboiling Two-Phase Flow with a General Correlation) is applicable for estimating heat transfer coefficients for nonboiling two-phase, two-component (liquid and permanent gas) flow in pipes. This article presents four practical illustrations of the use of the general two-phase heat transfer correlation. The first illustration involves the flow of air and silicone in a vertical pipe. Liquid silicone such as Dow Corning 200 Fluid 5 CS is used primarily as an ingredient in cosmetic and personal care products. The second illustration involves the application of the correlation to air and gas-oil flow in vertical pipes. The third and fourth illustrations deal with the applications of the correlation to the air and water flow in horizontal and inclined pipes, respectively.

Description of Solution

The estimation of the nonboiling two-phase flow heat transfer coefficient, regardless of flow pattern, gas-liquid combination, and pipe inclination angle, is accomplished by using the following general heat transfer correlation (see the article on the Knovel website by Ghajar and Tang, Estimations of Heat Transfer in Nonboiling Two-Phase Flow with a General Correlation):

The flow pattern factor (F_p) and inclination factor (I*) are given as follows:

The liquid phase heat transfer coefficient (hl) is calculated by using the Sieder and Tate¹ correlation:

The value of the void fraction (α) is calculated by using general void correlation (see the article on the Knovel website by Ghajar and Tang, A General Void Fraction Correlation in Two-Phase Flow for Various Pipe Orientations) proposed by Woldesemayat and Ghajar²:

The two-phase distribution coefficient (C0) and gas drift velocity (ugm) in meters per second (m/s) are given as follows:

The general void correlation is applicable for horizontal to upward vertical pipe orientations. However, in cases in which the two-phase flow is in upward vertical pipe, the use of the void fraction correlation of Rouhani and Axelsson³ is expected to provide more accurate results (see the article on the Knovel website by Ghajar and Tang, Void Fraction Correlations for Vertical Upward Two-Phase Flow in Pipes). When the correlation by Rouhani and Axelsson³ is used, the two-phase distribution coefficient (C0) and gas drift velocity (ugm) are calculated as follows:

Application in Air and Silicone Flow in a Vertical Pipe
A two-phase bubbly flow of air and silicone is transported in an 11.7-mm-diameter vertical pipe. Liquid silicone such as Dow Corning 200 Fluid 5 CS is used primarily as an ingredient in cosmetic and personal care products because of its low surface tension (σ = 19.7 × 10-3 N/m), high spreadability, nongreasy soft feel, and subtle skin lubricity characteristics. The two-phase flow has a gas mass flow rate (ṁg) of 1.89 × 10-5 kg/s and a liquid mass flow rate (ṁl) of 0.907 kg/s, and the system pressure (Psys) is 250 kPa. The gas phase consists of air with density (ρg) of 1.19 kg/m³, dynamic viscosity (μg) of 18.4 × 10-6 kg/m•s, and Prandtl number (Prg) of 0.71. The liquid phase consists of liquid silicone with thermal conductivity (kl) of 0.117 W/m•K, density (ρl) of 913 kg/m3, dynamic viscosity (μl) of 45.7 × 10-4 kg/m•s, and Prandtl number (Prl) of 64. The dynamic viscosity (μs) of liquid silicone evaluated at the pipe surface temperature is 39.8 × 10-4 kg/m•s. The following example calculation illustrates the use of the general two-phase heat transfer correlation to predict the two-phase heat transfer coefficient (htp) for this flow.

From the measured gas and liquid mass flow rates, the quality (x) and the superficial gas (Vsg) and liquid (Vsl) velocities can be calculated as x = 2.084 × 10-5, Vsg = 0.1477 m/s, and Vsl = 9.24 m/s. The void fraction (α) is calculated by using the correlation of Rouhani and Axelsson³ (Equations 5, 8, and 9): α = 0.01295 where C0 = 1.2 and ugm = 0.1423 m/s. From the superficial gas and liquid velocities and void fraction, the gas (Vg) and liquid (Vl) velocities can be calculated as Vg = Vsg / α = 11.41 m/s and Vl = Vsl / (1 – α) = 9.361 m/s.

The flow pattern factor (Fp) and inclination factor (I*) for the vertical pipe (θ = 90°) are calculated by using Equations 2 and 3 to be Fp = 0.9873 and I* = 63.16. The liquid-phase heat transfer coefficient (hl) is calculated by using Equation 4 to be hl = 3248 W/m²•K where Rel = 21739. Using the general two-phase heat transfer correlation (Equation 1), the two-phase heat transfer coefficient is estimated to be htp = 3589 W/m²•K. Compared with the measured two-phase heat transfer coefficient of 3480 W/m²•K of Rezkallah4 in similar flow conditions, the general two-phase heat transfer correlation overpredicted the measured value by only 3.1%.

Application in Air and Gas-Oil Flow in a Vertical Pipe
A two-phase flow of air and gas-oil is being transported through a 70-mm-diameter vertical pipe. The two-phase flow has a gas mass flow rate (ṁg) of 0.077 kg/s and a liquid mass flow rate (ṁl) of 10.2 kg/s, and the system pressure (Psys) is 200 kPa. The liquid phase consists of domestic-grade gas-oil with dynamic viscosity (μl) of 39.2 × 10-4 kg/m•s, density (ρl) of 835 kg/m3, surface tension (σ) of 25 × 10-3 N/m, and Prandtl number (Prl) of 60. The gas phase consists of air with dynamic viscosity (μg) of 18.2 × 10-6 kg/m•s, density (ρg) of 2.5 kg/m³, and Prandtl number (Prg) of 0.71. The following example calculation illustrates the use of the general two-phase heat transfer correlation to predict the two-phase heat transfer coefficient to the liquid phase heat transfer coefficient ratio (htp/hl).

From the measured gas and liquid mass flow rates, the quality (x) and the superficial gas (Vsg) and liquid (Vsl) velocities can be calculated as x = 0.00749, Vsg = 8.0 m/s, and Vsl = 3.17 m/s. The void fraction (α) is calculated by using the correlation of Rouhani and Axelsson³ (Equations 5, 8, and 9): α = 0.5906 where C0 = 1.199 and ugm = 0.1532 m/s. From the superficial gas and liquid velocities and void fraction, the gas (Vg) and liquid (Vl) velocities can be calculated as Vg = Vsg/α = 13.55 m/s and Vl = Vsl / (1 – α) = 7.743 m/s.

The flow pattern factor (Fp) and inclination factor (I*) for the vertical pipe (θ = 90°) are calculated by using Equations 2 and 3 to be Fp = 0.4416 and I* = 1602. Using the general two-phase heat transfer correlation (Equation 1), the value for htp/hl can be calculated to be htp/hl = 1.75. Compared with the measured value of htp/hl = 1.65 by Dorresteijn5 in similar flow conditions, the general two-phase heat transfer correlation overpredicted the measured value by 6.1%.

Application in Air and Water Flow in a Horizontal Pipe
A two-phase slug flow of air and water is transported through an 18.6-mm-diameter horizontal pipe. The liquid phase is water with thermal conductivity (kl) of 0.6046 W/m•K, dynamic viscosity (μl) of 9.33 × 10-4 kg/m•s, density (ρl) of 999 kg/m³, surface tension (σ) of 0.0723 N/m, and Prandtl number (Prl) of 6.52. The dynamic viscosity (μs) of water evaluated at the pipe surface temperature is 8.40 × 10-4 kg/m•s. The gas phase is air with dynamic viscosity (μg) of 18.3 × 10-6 kg/m•s, density (ρg) of 1.19 kg/m3, and Prandtl number (Prg) of 0.71. The flow has a gas mass flow rate (ṁg) of 9.51 × 10-4 kg/s and a liquid mass flow rate (ṁl) of 0.661 kg/s, and the system pressure (Psys) is 200 kPa. The following example calculation illustrates the use of the general two-phase heat transfer correlation to predict the two-phase heat transfer coefficient (htp) for this flow.

From the measured gas and liquid mass flow rates, the quality (x) and the superficial gas (Vsg) and liquid (Vsl) velocities can be calculated as x = 0.00144, Vsg = 2.941 m/s, and Vsl = 2.435 m/s. The void fraction (α) is calculated by using the correlation of Woldesemayat and Ghajar² (Equations 5, 6, and 7): α = 0.5034 where C0 = 1.044 and ugm = 0.2298 m/s. From the superficial gas and liquid velocities and void fraction, the gas (Vg) and liquid (Vl) velocities can be calculated as Vg = Vsg/α = 5.842 m/s and Vl = Vsl/(1 – α) = 4.903 m/s.

The flow pattern factor (Fp) is calculated by using Equation 2 to be Fp = 0.4978. The liquid-phase heat transfer coefficient (hl) is calculated by using Equation 4 to be hl = 12340 W/m²•K where Rel = 68820. Using the general two-phase heat transfer correlation (Equation 1), the two-phase heat transfer coefficient is estimated to be htp = 8848 W/m²•K. Note that for the horizontal pipe (θ = 0°), the inclination factor is I* = 1. Compared with the measured two-phase heat transfer coefficient of 9079 W/m²•K of Franca and associates6 in similar flow conditions, the general two-phase heat transfer correlation underpredicted the measured value by only 2.5%.

Application in Air and Water Flow in a 5° Inclined Pipe
A two-phase flow of air and water is transported through a 27.9-mm-diameter 5° inclined pipe. The liquid phase is water with thermal conductivity (kl) of 0.5787 W/m•K, dynamic viscosity (μl) of 13.4 × -4 kg/m•s, density (ρl) of 999.6 kg/m³, surface tension (σ) of 0.0744 N/m, and Prandtl number (Prl) of 9.74. The dynamic viscosity (μs) of water evaluated at the pipe surface temperature is 12.6 × 10-4 kg/m•s. The gas phase is air with dynamic viscosity (μg) of 17.7 × 10-6 kg/m•s, density (ρg) of 2.38 kg/m3, and Prandtl number (Prg) of 0.71. The flow has a gas mass flow rate (ṁg) of 7.73 × 10-3 kg/s and a liquid mass flow rate (ṁl) of 0.494 kg/s, and the system pressure (Psys) is 250 kPa. The following example calculation illustrates the use of the general two-phase heat transfer correlation to predict the two-phase heat transfer coefficient (htp) for this flow.

From the measured gas and liquid mass flow rates, the quality (x) and the superficial gas (Vsg) and liquid (Vsl) velocities can be calculated as x = 0.01541, Vsg = 5.313 m/s, and Vsl = 0.8084 m/s. The void fraction (α) is calculated by using the correlation of Woldesemayat and Ghajar² (Equations 5, 6, and 7): α = 0.7112 where C0 = 1.178 and ugm = 0.2594 m/s. From the superficial gas and liquid velocities and void fraction, the gas (Vg) and liquid (Vl) velocities can be calculated as Vg = Vsg/α = 7.470 m/s and Vl = Vsl/(1 – α) = 2.799 m/s.

The flow pattern factor (Fl) and inclination factor (I*) for the inclined pipe (θ = 5°) are calculated by using Equations 2 and 3 to be Fp = 0.3375 and I* = 9.921. The liquid-phase heat transfer coefficient (hl) is calculated by using Equation 4 to be hl = 4764 W/m²•K where Rel = 31306. Using the general two-phase heat transfer correlation (Equation 1), the two-phase heat transfer coefficient is estimated to be htp = 3687 W/m²•K. Compared with the measured two-phase heat transfer coefficient of 3700 W/m²•K by Ghajar and Tang7 in similar flow conditions, the general two-phase heat transfer correlation underpredicted the measured value by only 3.5%.

Summary and Wider Applicability of Results
The practical illustrations presented in this article involved the estimation of heat transfer coefficients in nonboiling two-phase flows in pipes with diameters ranging from 11.7 to 70 mm for horizontal, inclined, and vertical pipe orientations. In addition, these practical illustrations included two-phase flow with different gas-liquid combinations. In these practical illustrations, the use of the void fraction correlations to estimate void fractions in two-phase flow was shown. Since void fraction is a hydrodynamic parameter, the correlations are applicable to both boiling and nonboiling flows. These illustrations show the versatility of the general heat transfer correlation, and the calculated results were verified with measured values. In addition to its applicability for a wide range of nonboiling two-phase flows, the general two-phase heat transfer correlation can be utilized as a tool to compare results from computational fluid dynamics (CFD) and computational heat transfer analysis.

References
1.    Sieder, E. N., and Tate, G. E. Heat Transfer and Pressure Drop of Liquids in Tubes. Industrial & Engineering Chemistry, vol. 28, no. 12, pp. 1429–1435, 1936.

2.    Woldesemayat, M. A., and Ghajar, A. J. Comparison of Void Fraction Correlations for Different Flow Patterns in Horizontal and Upward Inclined Pipes. International Journal of Multiphase Flow, vol. 33, no. 4, pp. 347–370, 2007.

3.    Rouhani, S. Z., and Axelsson, E. Calculation of Void Volume Fraction in the Subcooled and Quality Boiling Regions. International Journal of Heat and Mass Transfer, vol. 13, no. 2, pp. 383–393, 1970.

4.    Rezkallah, K. S. Heat Transfer and Hydrodynamics in Two-Phase Two-Component Flow in a Vertical Tube. Ph.D. thesis, University of Manitoba, Winnipeg, Canada, 1987.

5.    Dorresteijn, W. R. Experimental Study of Heat Transfer in Upward and Downward Two-Phase Flow of Air and Oil through 70-mm Tubes. Proceedings of the 4th International Heat Transfer Conference, Paris and Versailles, France, vol. 5, B5.9, 1970.

6.    Franca, F. A., Bannwart, A. C., Camargo, R. M. T., and Gonçalves, M. A. L. Mechanistic Modeling of the Convective Heat Transfer Coefficient in Gas-Liquid Intermittent Flows. Heat Transfer Engineering, vol. 29, no. 12, pp. 984–998, 2008.

7.    Ghajar, A. J., and Tang, C. C. Importance of Non-Boiling Two-Phase Flow Heat Transfer in Pipes for Industrial Applications. Heat Transfer Engineering, vol. 31, no. 9, pp. 711–732, 2010.

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