Ministry of Higher Education &scientific Research Mustansiriyah University

1. Ministry of Higher Education &scientific Research Mustansiriyah University College of Engineering Mechanical Engineering Department Separated Two-Phase Flow (Section 5) Asst. Prof. Dr. Hayder Mohammad Jaffal

2. Separated Two-Phase Flow The separated flow model considers that the two phases travelling in the channel to be artificially segregated into two distinct streams (much like Stratified flow, one for liquid and the other for vapor. Assumptions and derivation of the model The separated flow model is based on the following premises • Constant but not necessarily equal velocities for the vapor and liquid phases • The attainment of thermodynamic equilibrium between phases • The use of empirical correlations and simplified concepts to relate the two phase friction multiplier and the void fraction to the independent variables of flow. flowing separately with different velocities and sharing a definite interface between them. Analytical solution to the separated flow model requires in all six equations; mass, momentum and energy conservation equations for each phase. Additional information such as velocity and temperature profile and other hydrodynamic parameters are required to solve these equations thus making it complex and difficult. An easy and a quick approach is to use empirical methods based on extensive experimental data.

3. One- Dimensional Steady Separated Equilibrium Flow The basic equations for steady one-dimensional separated flow along the tube shown in figure (1) are: Figure (1): Forced applied on a tube differential

4. Continuity: The mass flux of each stream is:

5. By using the definition of x, the above two equations can be used to give two alternative expressions for the overall mass flux as follows: Momentum:

6. Energy:

7. The energy equation in conveniently written in terms of the quality. Thus The total pressure gradient is then the sum of the components, as follows:

12. The void fraction correlation is often expressed, for vapor liquid mixture of a given substance, in the form Sub in Eq. 30

13. Sub Eqs.18,19, and 38 into Eq.25, the total pressure gradient for variable tube area is :

14. The empirical model based on the concept of separated flow was first conceived by Lockhart and Martinelli and since then several investigators have proposed different correlations by modifying the correlation of. The separated flow multiplier that accounts for the pressure drop due to the flow of single phase liquid or gas. The subscripts ‘lo’ and ‘go’ correspond to frictional pressure drop when the single phase liquid or gas flow rate is assumed to be equivalent to the entire two phase mixture flow rate (G). Whereas, the subscripts ‘l’ and ‘g’ indicate the frictional pressure drop when single phase liquid or gas is flowing at a rate of G(1-x) and Gx, respectively. The understanding of these four approaches that can be used in separated flow model becomes more apparent from their mathematical definitions presented by Eqs. (40) to (43).

15. Literature provides several correlations to determine the two phase frictional multipliers as shown in the above equations to predict two phase frictional pressure drop.

16. Correlations of void fraction: Most void fraction correlations are actually correlations of the slip ratio (S). Experimentally , it is found that the slip ratio depends on ( in decreasing order of importance): a- Physical void fraction correlations (usually expressed as (ρL/ρG); b- Quality (x); C-Mass flux (G); and then d- Relatively minor variables such as tube diameter, inclination of tube, length, heat flux, and flow pattern. Some of the verbal correlations are listed below with full description of each: 1-Zivi correlation (1964) Zivi effectively assumes that the total kinetic energy flow is a minimum. The kinetic energy flow of each phase is 0.5 ρiui2Qi where qi is the volume flow rate of the phase (m3/s) then using the indications:

18. With the relation mentioned above for velocities , the kinetic energy flow: Simplifying the above expression yields: Where:

19. Now differentiating (y) by (50) to find the minimum kinetic energy flow: The minimum therefore occurs when

20. Comparing with Eq. (52) gives The slip ratio is therefore assumed to depend only on the phase density ratio. 2-Chishlom correlation (1972) Chishlom produced a simple correlation which is expressed in terms of the quality and the densities of both gas and liquid phases as:

21. Both the Zivi and the Chishlom correlations reveal that S→1 as (ρL/ρG)→1, that is, as the critical point is approached ( where the phase densities are equal) the flow becomes homogenous in character. The Chishlom correlation also gives: S→1 as x→0 S→( ρL/ρG )0.5 as x→1 The latter limit for (S) is actually the condition that the momentum flow is a minimum. It should be noted that the Chishlom correlation provides a very simple, reasonably accurate results. 3-CISE correlation (1970) The correlation of Permoli et al. (1970) usually known as the CISE correlation is a correlation in terms of the slip ratio (S). the void fraction (α) is then given by:

22. The slip ratio is then given by:

24. Here, (x) is the quality ; (ρG) is the gas density (kg/m3); (ρL) is the liquid density (kg/m3); (d) is the tube diameter (m); (μL) is the liquid viscosity (N/m2s); (G) is the total (liquid +gas) mass flux (kg/ m2s); and (σ) is the surface tension (N/m).

25. Frictional pressure gradient: The fractional pressure gradient can be correlated by various values of (фL2 ), (фLO2 ), (фG2 ), and (фGO2 ), as for homogeneous flow, the actual values of (ф2 ) multipliers usually being determined experimentally. 1-Lockhart-Martinelli correlation The experimental work on which the correlation is based was done for horizontal flow of air-liquid mixtures at near-atmospheric pressures and with no change of phase. It is inadvisable to use the correlation for other conditions. For the conditions employed, the accelerative component of component vanishes. Consequently the frictional pressure gradients in the two phases were assumed to be equal. Martinelli (1949) used ‘only liquid’ and ‘only gas’ reference flows and, having derived equations for the frictional pressure gradient in the two-phase flow in terms of shape factors and equivalent diameters of the portions of the pipe through which the phases are assumed to flow, argued that the two-phase multipliers 4; and & could be uniquely correlated against the ratio X z of the pressure gradients of the two reference flows:

26. Different curves were suggested, depending on whether the phase-alone flows were laminar (“viscous”) or turbulent, and the multipliers subscripted accordingly. The graphical correlation of Lockhart and Martinelli is shown in Figure 1.

27. Figure (1)Two-phase pressure drop correlation of Lockhart and Martinelli [1949]

28. The curves of Figure (1) can be well represented [Collier (1972)] by equations of the form: and

29. where, for air-water mixtures, the values of C are as follows:

30. 2- Friedel correlation The Friedel (1979) correlation is probably more accurate generally available correlation for frictional two-phase pressure gradient. It was written in terms of two-phase multiplier: where (-dp/dz)F is the frictional pressure gradient (N/m2) in two-phase flow , and (-dp/dz)LO is the frictional pressure gradient (N/m2) in single-phase liquid flow with the same mass flow rate as the total two-phase flow rate. Then:

32. and The correlation is applicable to a vertical upflow and horizontal flow.

33. Example (1): A boiling channel receives a thermal power of 6000 kW. If subcooled water at 275 °C enters the channel at flow rate of 15 kg/s, what is the void fraction at the channel outlet, where the slip ratio is 1.687 and pressure is 68 bar. Solution : At 68 bar : hL=1256 kJ/kg hLG=1518 kJ/kg ρL=744.04762 kg/m3 ρG=35.3732 kg/m3 At 275 °C : hL=1209.9 kJ/kg h1=hL at 275 °C = 1209.9 kJ/kg h2=1256+x2x1518

34. X2=0.233

36. Example (2): For vertical tube of 16 mm internal diameter , determine the total pressure drop from the inlet to outlet if the tube is 2 m long and the flow is upwards with vapor quality changing from 0.0 to 0.2 (assume Zivi void fraction) using Friedel model with these properties: Mass flow rate=0.4002 kg/s; surface tension=0.015 N/m; Liquid density=1300 kg/m3; vapor density=20 kg/m3; liquid viscosity =0.0002 Ns/m2; vapor viscosity =0.00001 Ns/m2. Solution: xin=0 xout=0.2 Assume xmean=0.1 as necessary. Use the Zivi equation to calculate the void fraction:

38. To calculate frictional pressure gradient from Friedel model:

43. To calculate gravitational pressure gradient:

44. To calculate acceleration pressure gradient (for adiabatic flow)