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RESEARCH NOTES Predicton of Pressure Loss and Holdup in Two-Phase Horizontal Stratified Roll-Wave Pipe Flow J. K. Watterson, R. K. Cooper, and P. L. Spedding* School of Aeronautical Engineering, Queen’s University Belfast, Stranmillis Road, Belfast BT9 5AG, U.K.

A new method is presented for the calculation of two-phase parameters for horizontal cocurrent stratified plus roll-wave pipe flow. An iterative momentum balance was performed based on the actual flow conditions and the system geometry. An inflection point in the predicted pressure drop relation was shown to correspond with the balancing of the actual forces in the pipe. Using the technique, satisfactory agreement was formed between calculated two-phase parameters and reliable data. Introduction Spedding and Cooper1 have recently emphasized the current difficulties experienced in attempting to predict both pressure drop and holdup in horizontal cocurrent two-phase pipe flow. Detailed analyses of existing momentum balance models2-5 have demonstrated that they all experience some difficulties with the predicting of both pressure drop and holdup particularly for the intractable stratified plus roll-wave flows. This is unfortunate because these regimes are important elements found in most horizontal multiphase lines. All models, with the exception of the MARS method,4 assumed a straight horizontal liquid/gas interface, whereas the reality would be a concave shape of the form shown in Figure 1. This work outlines a momentum balance model that permits predictions to be made of pressure loss and holdup for two-phase horizontal pipe flow with the stratified roll-wave regimes. Momentum Balance Model The relevant equations are

-(∆P/∆x)LAL ) τWLSL - τiSi

(1)

-(∆P/∆x)GAG ) τWGSG + τiSi

(2)

-(∆P/∆x)TAT ) τWGSG + τWLSL

(3)

τWG ) 0.5fGVG2FG

(4)

τWL ) 0.5fLVL2FL

(5)

VG ) VSG/RG; VL ) VSL/RL

(6)

DG ) 4AG/(SG + Si); DL ) 4AL/(SL + Si)

(7)

ReG ) DGVGFG/µG; ReL ) DLVLFL/µL

(8)

fG ) CGReG-n; fL ) CLReL-m

(9)

Re < 2100 Re > 2100

[

C ) 16 C ) 0.046

FL(VSL/RL)2

]

m, n ) 1

(10)

m, n ) 0.2

(11)

0.5

FG(VSG/RG)2

) 3.65VSL0.28

(12)

log(Si/SL) ) [0.2626VSL0.3501] log VSG - 0.4927VSL0.4059 (13) There are several differences between this model and other momentum balance theories. These are as follows: (i) The holdup can be predicted within (10% using eq 12 independently of the pipe diameter. (ii) This allows Si/SL to be found within (5% using eq 13. (iii) The actual phase velocities (cf. eq 6) are used in the calculations. (iv) The equivalent diameters (cf. eq 7) include the interfacial length Si. Subsequently, it is found that τL ∼ τi, which gives justification to this assumption.

Figure 1. Pipe cross section.

where

Calculation Method

* Author to whom correspondence should be addressed. Tel: +44 028 90335417. Fax: +44 028 90382701. E-mail: [email protected].

The calculation involves the following: (a) Determine RL and RG by eq 12, Si/SL by eq 13, and VL and VG by eq 6.

10.1021/ie010749y CCC: $22.00 © 2002 American Chemical Society Published on Web 11/14/2002

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Ind. Eng. Chem. Res., Vol. 41, No. 25, 2002

Nomenclature A ) area, m2 C ) coefficient equations 9 D ) equivalent diameter, m d ) diameter, m f ) friction factor m ) exponent in eq 9 n ) exponent in eq 9 P ) pressure, kg m-1 s-2 R ) holdup S ) phase parameter or interface, m V ) velocity, m s-1 x ) length, m R ) angle suspended at the pipe center line by the wetted perimeter, deg µ ) viscosity, kg m-1 s-1 F ) density, kg m-3 τ ) shear stress, kg m-1 s-2 Figure 2. Determination of the inflection in the predicted pressure drop for various assumed liquid holdup values. Data: VSG ) 9.152 m s-1, VSL ) 0.1135 m s-1, d ) 0.0454 m, and RL ) 0.149.

(b) Assume an angle R above the minimum dictated by RL. (c) Find SL, SG, and Si from the pipe diameter. (d) Calculate D by eq 7, Re by eq 8, and f by eq 9. (e) Find the wall shear stresses from eqs 4 and 5. (f) Calculate (∆P/∆x)T by eq 3. Iterative calculation is performed to determine the inflection in the (∆P/∆x)T versus R curve. An example is shown in Figure 2, where the inflection in the curve gives the correct pressure loss for the holdup value found from eq 12. This latter value will be at a maximum within (10% of the true value of liquid holdup. (g) Assume new values of RL of (2.5%, (5%, (7.5%, and (10% around the value given by eq 12. (The first assumption is given in Figure 2.) (h) Repeat steps a-f for each new assumed value of RL. (i) The correct RL and (∆P/∆x)T values will be at the inflection of this curve, giving values of the shear stresses. Discussion and Conclusions The method relies on the holdup being independent of the diameter1 and the shear stresses and other twophase parameters forming into a balance in the pipe. Measurement of two-phase data can exhibit a wide variation unless particular care is exercised in the experimental procedure. It was found that the results of calculations using this method of approach sat neatly within the experimental data obtained from a number of sources,2-9 suggesting that the method correctly determined the actual force balance point within the physical pipe. The method shows that, if the holdup can be reliably predicted, there is a possibility of determining other two-phase parameters for the given geometrical conditions.

Subscripts G ) gas i ) interface L ) liquid S ) superficial SG ) superficial gas SL ) superficial liquid T ) total

Literature Cited (1) Spedding, P. L.; Cooper, R. K. A note on the prediction of liquid holdup with the stratified roll wave regime for gas/liquid co-current flow in horizontal pipes. Int. J. Heat Mass Transfer 2002, 45, 219-222. (2) Hand, N. P. Gas-liquid co-current flow in a horizontal pipeline. Ph.D. Thesis, Queen’s University Belfast, Belfast, U.K., 1991. (3) Ferguson, M. E. G. An investigation of horizontal and inclined two-phase flow. Ph.D. Thesis, Queen’s University Belfast, Belfast, U.K., 1993. (4) Grolman, E. Gas-liquid flow with low liquid loading in slightly inclined pipes. Ph.D. Dissertation, University of Amsterdam, Amsterdam, Holland, 1992. (5) Donnelly, G. F. An analytical evaluation of horizontal multiphase flow Ph.D. Thesis, Queen’s University Belfast, Belfast, U.K., 1997. (6) Nguyen, V. T. Two phase gas-liquid co-current flow. An investigation of holdup, pressure drop and flow pattern in a pipe at various inclinations. Ph.D. Thesis, University of Auckland, Auckland, New Zealand, 1975. (7) Chen, J. J. J. Two phase gas-liquid flow with particular emphasis on holdup measurement and prediction. Ph.D. Thesis, University of Auckland, Auckland, New Zealand, 1979. (8) Andritsos, N. Effect of pipe diameter and liquid viscosity on horizontal stratified flow. Ph.D. Dissertation, University of Illinois, Urbana, IL, 1986. (9) McBride, W. J. Division of a multiphase flow in a horizontal bifuriation. Ph.D. Thesis, Queen’s University Belfast, Belfast, U.K., 1995.

Received for review September 10, 2001 Revised manuscript received August 22, 2002 Accepted August 28, 2002 IE010749Y