1. Asst. Prof. Dr. Hayder Mohammad Jaffal
Ministry of Higher Education &scientific Research
Mustansiriyah University
College of Engineering
Mechanical Engineering Department
Two-Phase Multiplier
(Section 4)
Asst. Prof. Dr. Hayder Mohammad Jaffal

2. Two-Phase Multiplier
The pressure drop in two-phase gas-liquid flow is an important design parameter in many engineering applications. A simple method for calculating two-phase frictional components of the pressure drop involves use the two-phase (ф2) which defined by:
That is, the two-phase frictional pressure drop is calculated from a reference single-phase frictional pressure gradient (dP/dx)R by multiplying by the two-phase multiplier, the value of which is determined from empirical correlations. In equation (1) the two-phase multiplier is written as (ф2) to denote that it corresponds to the reference single-phase flow denoted by k.

3. For a gas-liquid two-phase flow there are four possible reference flows:
1- Whole flow liquid, denoted by subscripts LO
2-Whole flow gas, denoted by subscripts GO
3- Only the liquid in the two-phase flow, denoted by subscript L
4-Only the gas in the two-phase flow, denoted by subscript G.
When the reference flow is the whole of the two-phase flow as liquid, then the two-phase frictional pressure gradient is given by:

4. Here, the sub-scripts L and G indicate the frictional pressure gradient when the single-phase liquid or gas is flowing at a mass flux of GL (or G(1 − χ)) and GG (or Gχ), respectively. Whereas, the sub-scripts Lo and Go indicate the frictional pressure gradient when the single-phase liquid or gas flow rate is assumed to be equivalent to the two-phase flow mixture mass flux Gh.

5. For example, the frictional component of the pressure drop can be written as:
Where Cf,h is friction factor calculated using ρh and μh.

6. The subscript (LO) means;
1- It is single-phase liquid flow.
2- It is calculated at mass flux of G= mass of liquid and gas in two-phase flow.

7. Where the subscript (GO) means that the flow single-phase gas, and that the single pressure drop calculated at a gas mass flux of G, which is the total mass flux in the two-phase flow.

9. Example (1):
For laminar homogenous two-phase flow, prove that:
Solution:

11. Sub Eq.4 in Eq.1

13. Sub Eq.9 and Eq.12 in Eq.5

14. Example (2):
For turbulent homogenous two-phase flow, prove that:
Solution:

16. Sub Eq.9 in Eq.4

17. Sub Eq.10 and Eq.14 in Eq.1

18. Example (3):
Air and water flow at kg/s and 0.4 kg/s upwards in a vertical, smooth-wall tube of internal diameter D = 20 mm and length L = 1.3 m. Using the homogeneous flow model, calculate the pressure drop across the tube (neglecting end effects). The fluids are at a temperature of 20 °C and the expansion of the air may be assumed to be isothermal. The exit pressure is 1 bar. Dynamic viscosity of air and water are: ×10-4 and Pa.s respectively.
Solution:

19. For isothermal expansion, Pv=constant
From thermodynamic tables, ρair= kg/m3 at 20 °C, 1 atm ( bar). Therefore

20. The accelerational pressure drop is estimated from:
There is no change in quality and area

21. The gravitational pressure gradient is estimated with using (θ=90°) from :
The frictional pressure drop calculated using Two-Phase Multiplier
The mixture Reynolds number is:

22. Turbulent flow

23. The total pressure gradient is then the sum of the components, as follows: