Two-phase Pressure Drop

Two-phase Pressure Drop

In the practical analysis of piping systems the quantity of most importance is the pressure loss due to viscous effects along the length of the system, as well as additional pressure losses arising from other technological equipments like, valves, elbows, piping entrances, fittings and tees.

In contrast to single-phase pressure drops, calculation and prediction of two-phase pressure drops is much more sophisticated problem and leading methods differ significantly. Experimental data indicates that the frictional pressure drop in the two-phase flow (e.g. in a boiling channel) is substantially higher than that for a single-phase flow with the same length and mass flow rate. Explanations for this include an apparent increased surface roughness due to bubble formation on the heated surface and increased flow velocities.

Pressure Drop – Homogeneous Flow Model

The simplest approach to the prediction of two-phase flows is to treat the entire two-phase flow as if it were all liquid, except flowing at the two-phase mixture velocity. The two-phase pressure drops for flows inside pipes and channels are the sum of three contributions:

The total pressure drop of the two-phase flow is then:

∆ptotal = ∆pstatic + ∆pmom + ∆pfrict

The static and momentum pressure drops can be calculated similarly as in case of single-phase flow and using the homogeneous mixture density:

mixture density - definition

The most problematic term is the frictional pressure drop ∆pfrict, which is based on the single-phase pressure drop that is multiplied by the two-phase correction factor (homogeneous friction multiplier – Φlo2). By this approach the frictional component of the two-phase pressure drop is:

two-phase pressure drop - equation

where (dP/dz)2f is frictional pressure gradient of two-phase flow and (dP/dz)1f is frictional pressure gradient if entire flow (of total mass flow rate G) flows as liquid in the channel (standard single-phase pressure drop). The term Φlois the homogeneous friction multiplier, that can be derived according to various methods. One of possible multipliers is equal to Φlo2 = (1+xglg – 1)) and therefore:
two-phase pressure drop - equation2

As can be seen this simple model suggests that the two-phase frictional losses are in any event higher than the single-phase frictional losses. The homogeneous friction multiplier increases rapidly with flow quality.

Typical flow qualities in steam generators and BWR cores are on the order of 10 to 20 %. The corresponding two phase frictional loss would then be 2 – 4 times that in an equivalent single-phase system.

 
Martinelli-Nelson Friction Multiplicator
Martinelli-Nelson frictional pressure drop
The Martinelli-Nelson frictional pressure drop function for water as a function of the prevailing pressure level and the exit mass quality.
Source: http://authors.library.caltech.edu/25021/1/chap8.pdf
Separated-phases flow model - Lockhart-Martinelli correlation
An alternate approach to calculate two-phase pressure drop is the separated-phases model.

In this model, the phases are considered to be flowing separately in the channel, each occupying a given fraction of the channel cross section and each with a given velocity. It is obvious the predicting of the void fraction is very important for these methods. Numerous methods are available for predicting the void fraction.

The method of Lockhart and Martinelli is the original method that predicted the two-phase frictional pressure drop based on a friction multiplicator for the liquid-phase, or the vapor-phase:

∆pfrict = Φltt2 . ∆pl (liquid-phase ∆p)

∆pfrict = Φgtt2 . ∆pg (vapor-phase ∆p)

The single-phase friction factors of the liquid fl and the vapor fg are based on the single phase flowing alone in the channel, in either viscous laminar (v) or turbulent (t) regimes.

∆pl can be calculated classically, but with application of (1-x)2 in the expression and ∆pg with application of vapor quality x2 respectively.

The two-phase multipliers Φltt2 and Φgtt2 are equal to:

Lockhart-Martinelli - multipliers

where Xtt is the Martinelli’s parameter defined as:

Martinelli parameter

Lockhart-Martinelli - tableand the value of C in these equations depends on the flow regimes of the liquid and vapor. These values are in the following table.

The Lockhart-Martinelli correlation has been found to be adequate for two-phase flows at low and moderate pressures. For applications at higher pressures, the revised models of Martinelli and Nelson (1948) and Thom (1964) are recommended.

separated flow model - Lockhart-Martinelli correlation
Correlations for void fraction and frictional pressure drop (Lockhart and Martinelli, 1949)
Example: Pressure Drop - BWR
Pressure Drop - BWR
Source: http://www.nrc.gov/docs/ML1214/ML12142A157.pdf

Two-phase Minor Loss

In industry any pipe system contains different technological elements as bends, fittings, valves or heated channels. These additional components add to the overall head loss of the system. Such losses are generally termed minor losses, although they often account for a major portion of the head loss. For relatively short pipe systems, with a relatively large number of bends and fittings, minor losses can easily exceed major losses (especially with a partially closed valve that can cause a greater pressure loss than a long pipe, in fact when a valve is closed or nearly closed, the minor loss is infinite).

Single-phase minor losses are commonly measured experimentally. The data, especially for valves, are somewhat dependent upon the particular manufacturer’s design. The two-phase pressure loss due to local flow obstructions is treated in a manner similar to the single-phase frictional lossesvia local loss multiplier.

See more: TWO-PHASE FRICTIONAL PRESSURE LOSS IN HORIZONTAL BUBBLY FLOW WITH 90-DEGREE BEND 

 
References:
Reactor Physics and Thermal Hydraulics:
  1. J. R. Lamarsh, Introduction to Nuclear Reactor Theory, 2nd ed., Addison-Wesley, Reading, MA (1983).
  2. J. R. Lamarsh, A. J. Baratta, Introduction to Nuclear Engineering, 3d ed., Prentice-Hall, 2001, ISBN: 0-201-82498-1.
  3. W. M. Stacey, Nuclear Reactor Physics, John Wiley & Sons, 2001, ISBN: 0- 471-39127-1.
  4. Glasstone, Sesonske. Nuclear Reactor Engineering: Reactor Systems Engineering, Springer; 4th edition, 1994, ISBN: 978-0412985317
  5. Todreas Neil E., Kazimi Mujid S. Nuclear Systems Volume I: Thermal Hydraulic Fundamentals, Second Edition. CRC Press; 2 edition, 2012, ISBN: 978-0415802871
  6. Zohuri B., McDaniel P. Thermodynamics in Nuclear Power Plant Systems. Springer; 2015, ISBN: 978-3-319-13419-2
  7. Moran Michal J., Shapiro Howard N. Fundamentals of Engineering Thermodynamics, Fifth Edition, John Wiley & Sons, 2006, ISBN: 978-0-470-03037-0
  8. Kleinstreuer C. Modern Fluid Dynamics. Springer, 2010, ISBN 978-1-4020-8670-0.
  9. U.S. Department of Energy, THERMODYNAMICS, HEAT TRANSFER, AND FLUID FLOW. DOE Fundamentals Handbook, Volume 1, 2 and 3. June 1992.
  10. White Frank M., Fluid Mechanics, McGraw-Hill Education, 7th edition, February, 2010, ISBN: 978-0077422417

See above:

Two-phase Flow