Description of Case

In many industrial applications, such as the flow of oil and natural gas in flow lines and wellbores, knowledge of two-phase pressure drop and nonboiling two-phase heat transfer is required. As a result of the complex nature of two-phase gas-liquid flow, heat transfer data and applicable correlations for nonboiling two-phase flow in various pipe inclinations are not readily available. The hydrodynamic and thermal conditions of nonboiling two-phase flow are dependent on the interaction between the two phases. In most situations encountered by practicing engineers, direct heat transfer measurements for two-phase flow are extremely difficult to perform. It is for this reason that correlating heat transfer and pressure drop can provide useful information.

Description of Solution

To correlate heat transfer and pressure drop for nonboiling two-phase flow, the Reynolds analogy is utilized. The Reynolds analogy relates important parameters of momentum and thermal boundary layers in a simplified form. When it is adopted for use in nonboiling two-phase pipe flow, the ratio of the heat transfer coefficient for two-phase flow (h_tp) to the heat transfer coefficient for liquid single-phase flow (h_l) becomes:

The flow pattern factor (F_p) is given as

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

where Re_sl = 4ṁ_l(πμ_lD)

The two-phase density ρ_tp is defined as

The value of the void fraction (α) is calculated by using the general void correlation (see http://engineeringcases.knovelblogs.com/2010/04/14/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 (u_gm) in m/s, respectively, are given as

The expression for the pressure drop multiplier (Φ_l) is given as

The single-phase liquid frictional pressure gradient, (dp/dz)_f,l, is calculated by using c_f = 16/Re_sl for Re_sl ≤2000 and the Blasius³ equation, c_f = 0.079/Re_sl^0.25 , for Re_sl >2000. The values for the two-phase frictional pressure gradient, (dp/dz)_f,tp, are determined by experimental measurements. In the equations above, the subscripts g, l, and tp refer to the gas phase, liquid phase, and two-phase; sg and sl refer to superficial gas and superficial liquid, and
C₀ = distribution coefficient
c_f = Fanning friction coefficient
D = pipe diameter, m
(dp/dz)_f = frictional pressure gradient, = −2c_fV²ρ/D, Pa/m
F_p = flow pattern factor
g = gravitational acceleration, m/s²
h = heat transfer coefficient, W/(m²·K)
k = thermal conductivity, W/(m·K)
ṁ = sum of the gas phase (ṁ_g) and liquid phase (ṁ_l) mass flow rates, kg/s
Pr = Prandtl number
p_atm = atmosphere pressure, N/m²
p_sys = system pressure, N/m²
Re = Reynolds number
u_gm = gas drift velocity, m/s
V = mean velocity, m/s
α = void fraction
θ = inclination angle, rad.
μ= dynamic viscosity (μ_s is the viscosity evaluated at surface temperature), kg/(m·s)
ρ = density, kg/m³
σ = surface tension, N/m
Φ_l = pressure drop multiplier

Description of Results

To evaluate the correlation, Equation 1, which is based on the analogy between friction factor and heat transfer, experimentally measured heat transfer and pressure drop data were used to confirm the feasibility of the correlation. Experimental data for horizontal and slightly inclined (2-, 5-, and 7-degree) pipes with a diameter of 27.9 mm were measured by using the experimental setup described by Ghajar and Tang.⁴ Experimental data for vertical pipe with a diameter of 11.7 mm were measured by Sujumnong.⁵ In total, 637 experimental data points were used for comparison with the correlation given in Equation 1. The two-phase heat transfer data points were measured for different gas-liquid combinations.

Overall, the correlation successfully predicted 87% of the 637 experimental data points within ±30% agreement. Figure 1 shows the comparison of the calculated h_tp values from the correlation, Equation 1, with experimental data for nonboiling two-phase flow in horizontal, slightly inclined, and vertical pipes. The predicted results for the entire database of 637 data points have an average error range of −16 to 19% and a root mean square error of 23%.

Figure 1: Comparison of the predictions by Equation 1 with all 637 experimental data points for different pipe inclination angles and gas-liquid combinations.

Wider Applicability of Results

The result from this effort of relating heat transfer and pressure drop for nonboiling two-phase flow is in the form of the correlation given in Equation 1. Correlating heat transfer and pressure drop allows engineers to estimate the nonboiling two-phase heat transfer coefficient from a known two-phase pressure drop. As a corollary, it also suggests that one can estimate two-phase pressure drop from a known heat transfer coefficient, and the heat transfer coefficient can be estimated from the general two-phase heat transfer correlation proposed by Ghajar and Tang⁴ (see http://engineeringcases.knovelblogs.com/2010/03/24/estimations-of-heat-transfer-in-nonboiling-two-phase-flow-with-a-general-correlation/). In essence, this effort of correlating heat transfer and pressure drop along with the general two-phase heat transfer correlation provides a closure relation for the two important parameters in two-phase flow.

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.    Blasius, H. Das Ähnlichkeitsgesetz bei Reibungsvorgängen in Flüssigkeiten. Forsch. Arb. Ing.-Wes., no. 134, 1913.
4.    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.
5.    Sujumnong, M. Heat Transfer, Pressure Drop and Void Fraction in Two-Phase, Two-Component Flow in a Vertical Tube. Ph.D. thesis, University of Manitoba, Winnipeg, Canada, 1998.

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