J. Eng. Sci., King Saud Univ., Vol. 14 (2), pp. 423-435 (1408/1988) Improvingthe Lockhart and Martinelli Two-Phase Flow Correlation by SAS Adel Hemeida and Faisal Sumait Petroleum Engineering Department, College of Engineering, King Saud University, Riyadh, Saudi Arabia The correlation of Lockhart and Martinelli in its present form cannot be used to study a large set of data because it requires the use of charts and hence cannot be simulated numerically. A correlation between Lockhart and Martinelli parameters </>and X lor a two phase pressure drop in pipelines was developed using the Statistical Analysis System (SAS). This enabled the development of a computer program for the analysis of data using the Lockhart and Martinelli correlation. Field data from Saudi flowlines were then analysed using the program and the results show that the impoved Lockhart and Martinelli correlation predicts accurately the downstream pressure in flowlines with an average percent difference of 5.1 and standard deviation of 9.6%. Nomenclature Ap d Cross sectional area of pipe, sq. ft Pipe diameter, in The absolute roughness, in Friction factor Pipe length, ft Reynold's number Pressure, psi The reduced pressure Upstream pressure, psi Downstream pressure, psi Flow rate, cu ft/sec Gas flow rate, scf/hr. The multiple correlation coefficient, dimensionless e f L NRe P Ppr PI Pz q Qg R ~.Y'-"~I w~ , dl .i.}5'.(""\ t. II) .)l:J\ ~...w\'?- L~i MI ,~..dl i}J\ ~ f ! I ! i I ! i t ! ! 423
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J. Eng. Sci., King Saud Univ., Vol. 14 (2), pp. 423-435 (1408/1988)
Improvingthe Lockhartand MartinelliTwo-PhaseFlow Correlationby SAS
Adel Hemeida and Faisal Sumait
Petroleum Engineering Department, College of Engineering,King Saud University, Riyadh, Saudi Arabia
The correlation of Lockhart and Martinelli in its present form cannot be used tostudy a large set of data because it requires the use of charts and hence cannot besimulated numerically. A correlation between Lockhart and Martinelli parameters</>and X lor a two phase pressure drop in pipelines was developed using theStatistical Analysis System (SAS). This enabled the development of a computerprogram for the analysis of data using the Lockhart and Martinelli correlation. Fielddata from Saudi flowlines were then analysed using the program and the resultsshow that the impoved Lockhart and Martinelli correlation predicts accurately thedownstream pressure in flowlines with an average percent difference of 5.1 andstandard deviation of 9.6%.
~.Y'-"~I w~ , dl .i.}5'.(""\ t. II) .)l:J\ ~...w\'?- L~i MI ,~..dl i}J\ ~
f!
I
!iI!
i
t!!
423
424 Adel Hemeida and Faisal Sumait
tT
The reciprocal pseudo reduced temperatureAverage temperature, ORSuperficial liquid velocity, cu ftjsec.Superficial gas velocity, cu ftjsec.Lockhart and Martinelli parameterThe reduced densityCompressibility factorSpecific gravityDifferenceDensity, Lbmjcu ftViscosity, cpLockhart and Martinelli parameter
Many of the two-phase flow correlations [1-4J are in use, but the Lockhart andMartinelli correlation [4J, because of its simplicity, has been widely used in industry topredict two-phase flow in horizontal pipes especially for low gas and liquid flow ratesand small pipe sizes. The pressure drop resulting from two-phase flow, is predicted bycalculating the single-phase pressure drop for each phase as ifeach of them was flowingalone through the pipes; and correcting these values for two-phase flow by Lockhartand Martinelliparameter cpo The parameter cp dependsupon whetherthese phases areflowing viscously or turbulently. This coefficient is obtained from standard chartsdeveloped by Lockhart and Martinelli using experimental data obtained at low liquidflow rates and small pipe diameters. Other correlations [1, 2J, can be used in numericalsimulation and in the study oflarge sets offield data. In this work, it was undertaken totackle this deficiencyin the Lockhart and Martinelli correlation and hence use it in theanalysis of large sets of field data. The Statistical Analysis System (SAS) was used todevelopthe equationsto calculatethe Lockhart and Martinelliparameter cp for liquidand gas in their respective flow regimes. A computer program was then developed toanalyse large sets of data using the improved Lockhart and Martinelli correlation.
Development
The computer program flow chart is shown in Fig. (1).One of the subroutines is tocalculate the oil formation volume factor and the gas solubility at operating pressures
Journal of Eng. Sci., Vol. 14. No.2 (1988). College of Eng., King Saud Univ.
Improving the Lockhart and Martinelli Two-Phase Flow Correlation bySAS
ASSUME LIQUID-PHASE
FLOW ONLY, CALCULATE
qL' vsL' PL' (NRe)L' fL' t.PL
ASSUMe GAS-PHASEFLOW ONLY, CALCULATE
qg' Vsg' Pg' (NRe)g' ~Pg
~P CALCULATE
(APL).5,~PTP(9)'~TP(L)9
Fig.(1). Flow chart for improved Lockhart and Martinelli correlation.
and temperatures by the Lagrangian interpolation method [5J, while the other is tocalculate the compressibility factor of gas using the Hall-Yarborough equation [6, 7J,which were developed by using the Starling-Carnahan equation of state:
0.06125 Ppr t e -1.2(1 -c)2z=y
The value of reduced density (y) can be obtained from the solution of the followingequation using the Newton-Raphson iterative technique:
Fig. (2)shows the subroutine for calculating the gas compressibility factor. By assumingthe gas phase flowing alone, downstream pressure is calculated by Weymouth'sequation [7]:
Table (I). Equations for calculating friction factor [7].
Flow mechanism NR< f
laminar (N R<< 2000)
(2000 < N R<< 4000)
64/ N Re
0.5/(N Re)O.3critical
transition [ (200~ 1.16
J4000 < N Re < ;-)
[NRe>C~dr16J
1
,fl = 1.14- 210g~
I
[
e 9.34
J,fl =1.14-2 log d+ NRe,fl
turbulent
Table (2). Reynolds number and flow mechanism
Liquid Gas Flow mechanism
>2000<1000>2000<1000
Liquid turbulent, gas turbulentLiquid viscous, gas viscousLiquid turbulent, gas viscousLiquid viscous, gas viscous
>2000>2000<1000<1000
from the equation [4]:
x = (APL)O'5
APg
Figure (3) is used to obtain the Lockhart and Martinelli parameter </>.The curves for</>Lare used when the pressure gradient for the liquid is used and vice versa. Thetwo-phase pressure drop is then calculated as:
(8)
.(AP
) -</>2 (AP
)AL TP AL LorG
The following equation was derived to calculate the parameter </>using SAS software:
(9)
</>= exp [2.303a + b Ln (X) + 2.~0 (Ln X)2 ]where a,b,and c are constants and they have been selected according to the type of fluidand flow mechanisms (Table 3).
(10)
Table (4) shows a sample of the computer outputs. The validity of the program wastested. The results obtained, showed a percentage error of 1.8 in pressure drop. Thesquare of the multiple correlation coefficients for </>'sare close to unity and hence theequation derived can replace the charts with high accuracy.
Journal of Eng. Sci., Vol. 14, No.2 (1988). College of Eng.. King Saud Univ.
!,iri~
!~""~11e.
it,
t
ii
Improving the Lockhart and Martinelli Two-Phase Flow Correlation by SAS 429
1000.0 1000.0
~"'"
~
5 100.0
~I;;~---..:-'
1
:~
100.0
"&
~ 10.0 10.0..EE/1.
1.00.01 0.1 1.0 10.0
Parameter X =J(:~Y(~:)G=ff!i;
1.0
100.0
Fig. (3). Correlation for multi phase flow (after Lockhart and Martinelli).
Field Data
Tests were done on 101Saudi flowlines of 4,6 and 8 inches diameter [9]. The pipelength varies from approximately 2000 to 35000 ft and flow rate varies fromapproximately 400 to 18000STBjD. Most of the PVT data were taken from theexperimental PVT analysis, such as formation volume factor, gas solubility, gas-liquidratio and average specific gravity of the gas. API gravity and viscosity of oil were takenfrom actual field data. Gas viscosity was taken from reference [10] as a function of gas
Improving the Lockhart and Martinelli Two-Phase Flow Correlation by SAS 431
gra vity. Its average value was estimated as 0.0127 cp after correction for the existance ofN2, CO2, and H2S gases.
Results and Discussion
The computer output using calculated gas compressibility factor, gas solubilityand oil formation factor at operating pressures and temperatures shows a high degreeof accuracy in results with an average percentage difference of 1, 2 and 0.5%respectively.
Table (5) shows samples of test results obtained using improved Lockhart andMartinelli correlation. The statistical results yielded an average percentage error of5.1 %, standard deviation 9.6% and average absolute error 8.4% for downstreampressures.
Table (5). Lockhart and Martinelli correlation - test analysis
Test Diam. Flow rate P2(Measd.) P2(Calc.) %Err press. drop % Err P2
The effects of some important variables such as pipe size, liquid flow rate andliquid Reynolds number, on percentage errors in downstream pressure calculations,were studied. The results are presented in Figs. (4 and 5). The minimum percentageerror in downstream pressure prediction at different liquid flow rates for 4 in, 6 in and8 in pipes, are presented in Fig. (4).As liquid flow rate increases the percentage errorincreases. The rate of increase in percentage error is higher for smaller diameter pipesbecause of the increase in superficial liquid velocities. It can be seen that at low liquidflow rates the absolute value of percentage error decreases to zero and then starts torise. It should be noted that the variation in percentage error gradually fades. Sincea small range in pipe diameters was considered in the Lockhart and Martinellicorrelation (0.0586to 1.017in) it was safe to assume that the accuracy in pressure dropcalculations was a function of flow rate irrespective of the prevailing flow pattern. Forlarge pipe diameters, the flow pattern changes for a constant flow rate which leads tomore erroneous results when using the above Lockhart and Martinelli correlation. Inother words the ratio of the fraction of the space occupied by the liquid decreases withthe increase in pipe diameter. When going to larger diameters, radial static pressuredifference may exist and may produce differencesbetween static pressure drops of theliquid and gas phases which also lead to erroneous results.
Since Reynolds number is used to calculate cp,according to the flow mechanism,Reynolds number for the liquid phase was calculated and plotted against the
30. 4 in
0 6 in
X 8 in
20cr:acr:cr:UJ 10
0
I-ZUJUcr:UJa..
0
_10 LI QUID F LOW RATE, 1000 BB L I DAY
_20
Fig. (4). Effect of liquid flow rate on percentage errors in downstream pressure prediction at various pipediameters.
Journal of Eng. Sci., Vol. ]4, No.2 (1988). College of En!}., King Saud UniL'.
Improving the Lockhart and Martinelli Two-Phase Flow Correlation by SAS 433
30
20a:0a:a:UJ 10
00
.....
zUJua:UJa..
00
0
. .REYNOLDS NUMBER FOR LIQUID PHASE
_10
_20
Fig. (5). Effect of Reynolds number on percentage errors in downstream pressure prediction.
percentage errors in downstream pressure prediction. Fig. (5)shows that the increase inReynolds number causes an increase in percentage errors. This reflects the effect ofliquid superficial velocities on the pressure drop calculations. As the liquid superficialvelocity increases the percentage error increases.
Conclusions
The Lockhart and Martinelli correlation was improved by allowing predictions of<pto be made without the use of charts. The program developed accordingly was usedto study a large set of field data and the following conclusions were drawn:
(1) The equations for parameter <p,developed by the SAS method, are in goodagreement with Lockhart and Martinelli experimental graphs. Therefore, the programthus developed proved its accuracy and capability of analysing large sets of data.
(2) As liquid flow rate increases, the percentage error in downstream pressureprediction increases. The range of increase is higher for smaller diameter pipes.
References
[1] Beggs, H.D. and Brill, J.P. (1973) A study of two-phase flow in inclined pipes. J PT(May), 607.
[2] Dukler, A.E., Wicks, M. and Cleveland, R.G. (1964) Frictional pressure drop in two-phase flow, (A)
~.Y'--".!.ill I "'-'lo:- ," \.J,I.\}5'. (..., i . A) jl:.J1~..wl '? ..I)I -ilihl . "-: GI "\}.JI ~- '-
434 Adel Hemeida and Faisal Sumait
Comparison of existing correlations for pressure loss and holdup, (B) An approach through similarityanalysis", A.ICh.E. Journal (Jan.), pp. 38-51.
[3] Eaton, B.A.; Andrews, D.E.; Knowles, C.R.; Silberberg, I.H. and Brown, K.E. (1967) The prediction offlow patterns. liquid holdup and pressure losses occuring during continuous two-phase flow inhorizontal pipelines, J.PT (June): 815-828; Trans. AIME, 24 (June 1967).
[4] Lockhart, R.W. and Martinelli, R.C. (1949) Proposed correlation of data for isothermal two-phase,two-component flow in pipes, Chemical Engr. Progress, 45, N-l:39--48