SOCIETY OF PETROLEUM ENGINEERS OF AIME Fidelity Union Building Dallas, Texas

PAPER NUMBER 924-G

THIS IS A PREPRINT --- SUBJECT TO CORRECTION

TIO PHASE FLOW IN WELL-FLOWLINES

by

B. T. Yocum, Arabian American Oil Co., New York

Publication Rights Reserved

This paper is to be presented at the 32nd Annual Fall Meeting of the SOCiety of Petroleum Engineers of American Institute of Mining, Metallurgical and Petroleum Engineers in Dallas, October 6-9, 1957. Through agreement with the author it is considered the property of the Society of Petroleum Engineers. Permission to publish is hereby restricted to an abstract of not more than 300 words, with no illustra­tions, unless the paper is specifically released to the press by the Executive Secretary of the Society or the Editor of Journal of Petroleum Technology. Any abstract should contain appropriate, conspicuous acknowledgment of the original presentation. Publication elsewhere after publication in the Journal of Petroleum Technology is granted upon request, providing proper credit is given that publication, the Society, and the authors. When warranted, a notice of copyright ownership by the individual or company which presented the papers must be given.

Discussion of this paper at the meeting is invited. Written discussions for the publication should be sent to the SOCiety of Petroleum Engineers office.

INTRODUCTION

Effective oilfield planning in Arabia re­quired approximate SOlU4ion of the,two phase flow problem in oil wells and'flowlines. Stated in broad terms, the problem resolved to determina­tion of well-flowline potential (BID) as a func­tion of major system variables:

1. Flowing bottom-hole pressure, reservoir pressure and temperature, productivity index.

2. Crude characteristics; pressure - vol­ume - temperature (PVT) relationships, viscosi~ ties, and composition.

3. Depth of oil well, diameter of casing and tubing, length and diameter of flowline.

4. Trap pressure (back pressure at end of flowline), ambient temperature, ground temper­ature.

Relationships developed for well-flowlines have maximum usefulness if they can be extra­polated to low reservoir pressures, representing future conditions, with confidence. It was therefore decided to develop the physical rela­tionships from basic fluid dynamic and thermo­dynamic principles whenever possible rather than relying on existing empirical equations. The co­efficients of the theoretically sound equations; friction factors, heat transfer coeffiCients, thermal gradients, etc., were then correlated for flOwing mjxtures of gas and oil in actual well­flowlines.

For convenience, the oil well will be taken up first and the flowline afterwards. Then, the two will be combined to give the overall well­flowline relationships.

OIL WELL

Fundamental Equations

The controlling physical relationships de­rived for the oil well are:

Pressure Profile:

J dP - jdE. (1 -+ 8f(MW)2 \ y - 144 1( 2f2gc (Di-dO)(ni - d~)V

Eq. 1

Temperature Profile

T = TR - KZ - K' (3 - ZBP)+:K [1 _ e-lfB] r Eq. 2

where r = 3600 MWCll UD

DenSity - Pressure - Temperature Relationship

F = ~(P,T) Eq. 3

This relationship is determined empirically for each crude. It can be derived directly from the PVT data if the assumption of local physical equilibrium between gas and oil at each point in the flo';, is made.

2 TWO PHASE FLOW IN WELL-FLOWLINES 924-G

Simultaneous solution of the three equations is sufficient to define the flow at any point in the oil well.

Development of Equations

Pressure Profile

The mechanical energy balance, neglecting surface tension forces, is:

dB + 144 ~ + vdv = dW - dWF J gc 0

Eq. 4

vdv The change in kinetic energy ( gc) was cal-

culated to be a maximum of 3 ft of head for the oil wells studied. It is negligible compared wi th the other energy terms. There is no shaft work (dwo ) since natural reservoir pressure is supplying the energy for flow.

Most of the flow in the wells tested is up the annulus between the 7 in. casing and the 2-3/8 in. tubing. When tubing and casing are both flowing, tubing flow is never more than 7 per cent of the total and usually much less. Friction losses in annular flow are described by:

Eq. 5 = 8(MW)2fdZ

-rr 2r 2gc(Di-dO)(Df _ d~)2 vdv

Dropping out -g- and dwo ' substituting (5) for dWF and integrating we obtain Eq. 1.

j dP _ jdB ~ + 8f(MW)2 ) ,f - - ill \-'- Tf2 2gc (Di-dO)(nI _ d~)2

Eq. 1

Temperature Profile

The density determined in Eq. 3 and used in Eq. 1 is the mixture density. Since the mixture density depends on both pressure and temperature for a fixed crude composition, the temperature profile of the well must be developed.

Cooling due to heat transfer from the flow­ing oil to the ground is expressed by:

dQ = CpdT - UzrD (T - Tg)<lli - 3600MW Eq. 6

The temperature of the ground increases with well depth. It is assumed that the drop in tem­perature is linear with well depth changing at the rate K:

Eq. 7

There is a temperature drop in the oil flow­ing up the well because of flashing to a gaS-Oil mixture above the bubble point. This effect is small and the resulting Eq. 2 is simplified if it is assumed linear:

6Tf :: K' (B - BBP) Eq. 8

~TF is added to the heat loss to the ground at each point in the oil flow path. By substi­tuting Eq. 7 in Eq. 6, integrating and subtract­ingL1TF, we obtain Eq. 2.

where r K/ (B - ~p)

= 3600 MWCp UD

rK -71f; IT l-e r

Development of Coefficients

Eq. 2

Before Eq. 1 - 3 can be used, K and K' must be empirically determined, U and f must be cor­related, and accurate pvr data developed.

Ground Temperature Gradient, K, ReserVOir to Surface

Measurements made 3 - 4 ft below the ground surface indicate an average temperature of 820F, with a range of 800 - 900 F; at the depth of the reserVOir, the average is 2130F with a range of 2040 - 2200F. The average gradient K is:

K = 213 - 82 = 0 01880 F/ft 6790 . Eq. 7

Rate of Temperature Drop Due to Flashing, K/

Flash calculations were performed ~sing a rough estimate of bubble point pressure and tem­perature of 1,865 psig and 2100 F, and average. casing head conditions of 750 psig and 1850F. These calculations show that the drop in temper­ature due to flashing is only 60F, and this does not vary more than 10F either way for a casing head pressure range of 600 - 950 psig. On the average, flashing will occur over the last 4,000 ft of travel to the surface.

We therefore can estimate an average K/:

K' = 60F - 0 001 0 / 4000' - . 5 F ft Eq. 10

K' is strongly dependent on crude character­istiCS, especially gas-oil ratio, reservoir pres­sure, and temperature. It becomes a more impor-· tant factor in high GOR and low pressure fields. For the crudes tested in our work it was not a controlling factor.

Heat Transfer Coefficient, U

Test data was available on eight oil wells. Reservoir pressure and temperature, and casing pressure and temperature had been measured over a wide range of potential (1,000 - 15,000 B/D). Using K and K' developed above sufficient data was available to determine U, the overall heat transfer coefficient from reservoir to gr01'nd from Eq. 2. The correlation obtained is exhib­ited in Fig. 1. The best average value of U ap­pears to be 2.2 BTU/hr. ft2 OF. This estimate is based on an average specific heat of 0.57 BTU/# OF.

924-G B. T. YOCUM

With U determined, Eq. 2 can be solved to give the temperature profile for the well. This is exhibited in Fig. 2, which presents the temper­ature at any well depth for several values of ~ .

The temperature correlation resulting from use of 2.2 BTU/hr.ft2 OF for U, .0188°F/ft for K, and .00150 F/ft for K.' is exhibited in Fig. 3. This chart shows an average deviation of + 80 F be­tween calculated and observed casing temperature.

Well Friction Factor, f

As a first approximation, the friction factor f in Eq. 1 was determined from the conventional Reynolds number correlation using liquid phase viscosi ty and mixture density. Somewhat surpris­ingly, the correlations obtained with experimental data were good. It had been thought that gravity forces as well as viscous forces might have an ef­fect on the two phase flow friction factor.

With the above friction factors, Eq. 1 was integrated up the well, and the resulting casing pressure values compared with experimental data.

In Fig. 4, the correlation obtained between calculated and observed casing pressures is exhib­ited. The average deviation is + 15 psi which is within the accuracy of the baSic-data, such as PVT analyses, used in the calculation. At least for the data correlated, the friction factor can be obtained from Reynolds number charts.

The mixture density - pressure - temperature chart used in integrating up the well is exhibited in Fig. 5. This stepwise integration while simul­taneously solving Eq. 1, 2 and 3 is very laborious by hand, since only small density increments (10 per cent change) can be used if accurate casing pressure prediction is desired. Computer programs have been written for Eq. 1, 2 and 3 in recent months which greatly reduce the time involved in solving the equations for new fields.

Development of Well Potential Curves

Satisfied that our equations and coefficients conform reasonably with the experimental data, the well potentials are expressed in their practical form of casing pressure versus potential with flowing bottom-hole pressure as parameter. This is demonstrated in Fig. 6.

The casing pressure for annular flow up the well exhibits a maximum in Fig. 6.

This inversion demonstrates the importance of temperature, which has often been neglected in the past by assuming isothermal flow, in well calcula­tions. At low potentials, the cooling of the oil in flOwing up the well is considerable; as the po­tential increases, cooling decreases and the col­umn is lightened because of hotter oil, thereby increasing the wellhead pressure. However, as

potential increases, the friction energy loss in­creases tending to decrease the wellhead pressure. Therefore, a maximum wellhead pressure must be passed through as potential increases representihg a balance between the opposing effects of increas­ing oil temperature and increasing friction loss.

The low potential data, 1,000 - 5,000 BID, was obtained under steady state conditions. How­ever, the wells were not held at low potentials for long periods of time, say, several weeks or months. Under these conditions, the effect of gravi ty forces might begin to show up.

FLOWLINE

Fundamental Equations

Pressure Profile

j 44 dP j _ f 2. 52xlO-3f(MW)2dL 1 f' + dh - f 2D5

Eq. 11

Temperature Profile (Above Ground Lines)

( ) To - Ta Ti = Ta +L'>.TR + 103.26xlO-4 Lj)Cl.2 Eq. 12

(MW)0.6

Density - Pressure - Temperature Relationship

Same as Eq. 3.

Development of Equations

Pressure Profile

As in the well, the velocity head vdv was de­termined to be negligible compared with gthe other heads and there is no shaft work (dwo ). Flowline elevation changes are expressed as.1h to distin­guish from well depth L'>.B. Eq. 11 is readily de­rived from Eq. 4 substituting these considerations

Temperature Profile

Temperature in flowlines is affected by all three heat transfer mechanisms; conduction, con­vection, and radiation. The conduction and con­vection transfer can be determined by a suitably correlated overall heat transfer coefficient using the basic cooling law equation.

e Tl - Ta -UllDL lneo = ln To - Ta = MWCp

Eq. 13

The overall heat transfer coefficient, U, is made up of individual film and steel coefficients:

1 ~+ 1 +~ Eq. 14 U = hi h ho s

Colburn has demonstrated that individual film coefficients can be correlated with Reynolds and Prandtls numbers.

4 TWO PHASE FLOW IN WELL-FLOWLINES 924-G

h _ ~:OO)0.2 __ A_ CpG M

Eg. 15

Using Eg. 15 as a basis, U was correlated from tv.e test data. This ex:pression was substi­tuted in Eq. 13 and integrated giving

To - Ta -4 LDO. 2 Tl = Ta + 3 26 10 10· x (MW)0.6

Eg. 16

The effect of sun radiation on flowline tem­perature was determined from the ex:perimental data. Possibl-= average values forLlTR, the tem­perature added due to radiation, are developed in the correlation section. AddingdTR to Eg. 16 re­sults in Eg. 12.

Development of Coefficients

The friction factor f, heat transfer coeffi­cient U, and radiation effect~TR must be corre­lated.

Flowline Friction Factor

Elementary fluid dynamics texts point out that any flow system with free surface present, like our gas-oil mixture contained in a pipe, theoretically will reguire flow work energy to overcome gravity and interfacial tension forces as well as the usual viscosity forces. From the viewpoint of dimensional analysis, a two phase flow friction factor should correlate with a func­tion lncluding Froude number (gravity), Weber num­ber (interfacial tension) as well as Reynolds num­ber (viscous). To complicate the dynamic picture even more, large pressure drops at low average line pressures may cause compressible forces to arise.

As a first approximation, interfacial tension is neglected and the flow energy is assumed to overcome gravity and vi scous forces only. The theoretical two· phase friction factor is developed below for this case:

Viscous Force

~NR _ ,. _ Fv = ~ 8" - J~ - rrDj v2dL

Gravity Force

Fg = /fn2K"j'gdL

(Reynolds Number Roughness Factor)

Eg. 17

Eg. 18

The flow work energy must overcome both grav­ity and viscous forces which are assumed to add vectorially.

~ --7 -7 J<' = Fv + F g Eq. 19

Fv analogy with Eg. 17 the two phase flow fri~~l.-,,'. factor is related to the net force F:

Eg. 20

Solving Eg. 17 through 20 simultaneously, we find:

Eq. 21

(Froude Number squared)

The ex:perimental data collected on flowline is plotted in Fig. 7. The two phase flow friction factor is plotted against the sguare of the Froude number which Eg. 21 indicates to be the best theo­retical parameter. The correlation behaves in the general manner predicted by Eg. 21. At low Froude numbers, the second term in Eq. 21 is large; much larger than the Reynolds number friction factor. The data show this trend 'very clearly. At large Froude numbers, the second term in Eg. 21 dimin­ishes rapidly since it depends on the inverse square of the Froude number, and the first term, the Reynolds number friction factor becomes con­trolling. The actual data correlate well with Reynolds nWllber alone in the high Froude number region. The leveling off of the friction factor in Fig. 7 indicates that the Froude number does not exert influence in its _igh range. Since the data follovrs the theoretical analysis guite well, neglecting interfacial tension forces appears to be justified for the crudes correlated.

A distinct minimum is indicated by the 6 in. and 8 in. pipe data in· Fig. 7. It is surmised that a transition zone exists. As linear velocity (and Froude number) decrease, the gravity forces show their effect gradually probably through a partial stratification which becomes more pro­nounced as the linear velocity becomes smaller.

In actual calculations, Fig. 7 is not de­pended on fOr friction factor calculations in the Reynolds number controlled region. Instead, if the square of the Froude number is greater than 4.0, the friction factor is esti~ed from the conventional Reynolds number chart just as in the oil well. Below Froude numbers squared of 4.0, the gravity forces become important, so the fric­tion factor is read directly from Fig. 7, since it is the only available correlation in the low Froude number range.

, . Data is being collected at the present time on a high viscosity, low GOR, foamy crude. It is hoped that some information will be developed on the effect of interfacial tension on the friction factor. Also, the effect of large diameters on the friction factor are being studied.

FLOWLINE TEMPERATURE CORRELATIONS

Heat Transfer Coefficients (Above Ground), U

During much of the testing, temperature data were recorded continuously around the c~.8c:k. It

, ,;as i'8und t.hat, at nighttime sufficiently long

4-G B. T. YOCUM

~lowlines, the oil temperature levels off at the 'ambient temperature. This is the behavior ex­pected with conduction and convection heat trans­fer so the ef~ect of radiation ~rom the oil in the ~lowlin-= to the sky must be negligible at night. There~ore, by using the nighttime temperature data, we should be able to correlate U using Eq. 15 as a basis.

For the crudes tested, the Prandtls' number does not vary much, so U was correlated directly with the mass flow rate G(#/~t2-sec). The result­ing correlation

~U (MW)0.4 Cp = 0.0813 if Eq. 22

is then substituted in the cooling Eq. 13 to give

To - Ta T = Ta + 1-03.26xlO-4 LD'2

(MW) '6 Eq. 23

This correlation is exhibited in Fig. 8. Ap­parently, the inside ~ilm hi is almost completely controlling, since the correlation is quite good without consideration o~ wind velocity. Perhaps the outside film, which is controlled by wind velocity, contributed to the scatter of the data.

The accuracy of the correlation, based on study o~ three ~lowlines ~lowing the same crude type, is ± 4 per cent with 100 per cent confidence.

It is doubt~ if the correlation in Fig. 8 would hold fOT radically different crude types. Large di~~erences in viscosity, specific heat, and thermal conductivity exist among crude oils all of which affect the correlating equation. However, ~or any given crude, the technique described should give satisfactory heat transfer coefficient correlations.

Some work has. been done on buried flowline heat transfer coefficients. These are somewhat simpler to handle since the ground temperature does not vary to the same extent as the ambient and there is no appreciable radiation effect. An average value of 0.51 BTU/hr.ft. 2 OF was developed from experimental data for ~lowlines buried in sand. The variation o~ U with ~lowline oil rate is known to be considerable, but a correlation has not been developed yet.

Radiation E~~ects, 6TH

As mentioned above, in su~~iciently long lines, the oil temperatures in the ~lowline leveled o~~ at the ambient temperature at night. The day­time radiation e~~ect can there~ore be measured by

!Subtracting oil temperature from ambient temper­ature. I I

i Temperatures mo:asured 59,000 ft f!:"-'I!j,·,.)lln"cl'~:

Oil Potential = 14,338 spheroid BPD

Oil Ambient Temperature Temperature Radiation Time o~

OF OF E~~ect Day

97 83 14 10:00 AM 113 92 2l 12:00 Noon 125 97 28 2:00 PM 132 97 35 4:00 PM 106 92 14 6:00 PM

88 87 1 8:00 PM 80 80 0 12:00 Midnight 77 79 (0) 4:00 AM

-12 79 0 8:00 AM

100 87 13 Daily Average

In this case, the average oil temperature at the end o~ a 59,000 ft line was 130F higher be­cause o~ radiation effect. On a yearly baSiS, it was found that 100F was a good average value to add ~or radiation e~fect in practical calculations. Although the wellhead temperature is not affected appreciably by sun radiation, a few thousand ~eet down the flowline the radiation e~~ect is as fully developed as it is at the end of the flowline. We are not badly in error, then, in adding 6TH to the ~lowline temperature profile (Eq. 23) at each point in the flowline. Therefore, ~TF is consid­ered to be 100F in Eq. 12 under average yearly condi tions.

Since the radiation effect 6 TH, is dependent on the emissivity of the flowline surface, making the flowlines perfect reflectors by painting them white (or silver) will practically eliminate..6.TR. The 100 cooler oil retains more light ends and the oil potential is increased 2 - 2~ per cent. Painting flowlines white may well become economi­cally attractive some time in the f~ture.

DenSity - Pressure - Temperature Relationships

As in the well, the density term in the hy­draulic equation is equated to the mixture density obtained from the PVT data plotted in Fig. 5. It should be emphasized that the assumption of local thermodynamic equilibrium has been made at every point in the flowline. If we did not make this assumption we could not identify the effective density at any point in the system with the values in Fig. 5 and the equations could not be solved simultaneously.

Development of Well-Flowline Potential Curves

The wellhead pressure vs potential curves de­veloped for the well (Fig. 6) are solved simulta­neously with the flowline pressure drop vs poten­tial as exhibited in Fig. 9. The resulting curve of well-flowline potential vs flowing bottom-hole

., 'Xc (FL;. 1'.,') is, for many applications, the p:·'ie~~;·'tL "'H've. These curves have been de-

----_._--------------

6 TWO PHASE FLOW IN WELL-FLOWLINES 924-G

veloped to cover a wide variety of conditions; 4 in., 6 in., 8 in. fu~d 10 in. diameter flowlines, ° to 100,000 ft lengths, and six types of crude.

We can go one step farther and solve Fig. 10 simultaneously with reservoir flow resistance as characteri zed by Productivity Index. Thi sis shown in Fig. 11 and the resulting relationship well-flowline potential vs reservoir pressure is exhibited in Fig. 12. Fig. 12 has been developed over a wide range of productivity indexes; 10 -200. Although not shown in Fig. 11, Fig. 12 in­cludes the decreasing productivity index found at reservoir pressures below the bubble point. These curves do not apply for the situation where gas­oil ratios in excess of the solution ratio are be­ing produced.

On all charts, the well depth and trap pres­sure are constant and only one PVT analysi s (crude type) is considered. In order to increase the utility of the basic charts, correction charts, Fig. 13 and 14, were developed to correct the po­tential for different well depths and trap pres­sures. Also, the well-flowline potential curves have been extended to six crudes. It is hoped that a correction chart can be developed in the near future for crude type. Thi s would involve a reference well-flowline potential set, perhaps the one illustrated in Fig. 10, and a curve of multi­plying factors applied to the basic set for esti­mation of potential in any field. This develop­ment, however, is still in the 'future.

CONCLUSIONS

1. When annU.Lar casing flow is under consid­eration, in the 1,000 - 7,000 B/D range espe­Cially, temperature profiles are necessary for adeq,uate prediction of well potential. The as­sumption 0: ~othermal flow introduces appreCiable errors in the critical mixture density term thus leading to incorrect oil potential estimates.

2. For the well potential ranges considered in this paper (1,000 - 20,000 B/D), the friction factor predicted from the classical Reynolds num­ber correlation is satisfactory if the Reynolds number is calculated using liq,uid phase viscosity and mixture density.

3. A tentative theory, backed up by consid­erable data, of the flowline two phase flow fric­tion factor is presented. Gravity forces, corre­lated by the Froude number, appear to control the magnitude of the friction factor at low linear velocities. At high velocities, the Reynolds num~ ber appears to be controlling. This dual control theory of the two phase flow friction factor neg­lects interfacial tension forces that may be im­portant in certain crudes. It is clear that much work remains to be done before a generalized two phase flow friction factor correlation is devel­oped. However, the dimensional analysis approach including other dynamic ratios than Reynolds num­ber appears to be fruitful.

4. A flowline temperature correlation is I

presented. Good flowline temperatures are needed to adeq,uately predict mixture density which has a large effect on the pressure drop - oil potential relationship in the flowline.

5. Radiation from the sun is shown to have a defini te effect on oi.l potential. The magnitude of the radiation effect is estimated for Arabian condi tions.

6. A graphical method for determining the oil potential flowing through the three major re­sistances, reservoir, well, and flowline, is given.

ACKNOWLEDGMENTS

The work on two phase flow in well-flowlines summarized in this paper was participated in by many others besides the author. Some of these were: W. M. Compton, New York Engineering, Aramco; F. Caponegro, The Future Oil Development Program, Aramco; R. F. McNamara, New York Engi­neering, Aramco.

Mr. M. Ludwig, Engineering Department, Standard Oil of California, suggested applying di­mensional analysis, including the Froude number, to the flowline friction factor problem.

LIsr OF SYMBOLS

~ Dt do -% e F Fg Fv f fNR G g gc

h hi he hs

K k k' k" L M Nf2 NR P Pm L:,P

PEP Pf PH

- Specific heat, BTU/Lb-OF - Inside diameter of :'1. .. ;vline, ft - Inside casing diameter, ft - Outside tubing diameter, ft - Hydraulic diameter of annulus = D· - do' ft - Base of natural logarithms = 2.7183 - Net force, lbs Force - Gravity force, lbs Force - Viscous force, lbs Force - Friction factor for two phase flow - Friction factor from Reynolds number - Mass flow rate, Lb/Ft2-Hr - Acceleration due to gravity = 32.2 Ft/sec2 - Dimensional factor = 32.2 Lbs Mass-

Ft/Lbs Force-sec2 - Elevation, ft - Inside film coefficient, Ft2-Hr/BTU - Outside film coeffiCient, Ft2-Hr/BTU - Eq,uivalent film coefficient for steel,

Ft2-Rr/BTU - Thermal conductivity, BTU/sec-Ft2- oF/Ft - Ground temperature gradient, of/Ft - Flashing temperature drop gradient, of/Ft - Constant in gravity force eq,uation - Length of flowline, ft - Flow rate, spheroid BPD - Sq,uare of the Froude number - Reynolds number - Pressure, psi - Mean pressure, psi - Pressure drop, psi - Pressure at bubble pOint, psia - Flowing bottom hole pressure, psia - Static reservoir pressure, psia

924-G

Pl

Tm Ta TBP Tg TR ,6 Tf ..6TR

U

v W

B. T. YOCUM

M - Productivity index = P P

R - f Heat loss due to conduction, BTU/Lb

- Temperature, OF - Oil temperature at beginning of flowline

section, OF - Oil temperature at end of flowline section,

OF - Mean oil temperature, OF - Ambient air temperature, OF - Temperature at bubble point, OF - Temperature of ground, OF - Reservoir temperature, OF - Temperature drop due to flashing, OF - Temperature increase due to sun radiation,

OF - Overall heat transfer coefficient,

BTU/hr-ft2- OF - Velocity, ft/sec - Conversion factor, lbs/sec per spheroid

BPD = 4.01 X 10-3 for Ghawar - Work due to friction, ft of head

Shaft work, ft of head - Elevation above reservoir datum plane, ft - Elevation of bubble point, ft - Cooling constant

- 3600 WM Cy = parameter on temperature TID

profiles

- Absolute viscosity, centipoises = s - Kinematic viscosity, centistokes - Initial oil temperature difference,

To - Ta , of e, - Final oil temperature difference,

/' f>BP

<P T

Tl - Ta , of - Mixture density, Lbs/ft 3 - Density at bubble paint, Lbs/ft3 - Function sign - Shear stress, psi

REFERENCES

1.

2.

3·

4.

5·

Hunsaker and Rightmire: Engineering Applica­tions of Fluid Mechanics, McGraw-Hill, 1947

u8 Colburn: "Transaction American Institute of Chemical Engineers", (1933) 29, 174. Lockhart and Martinelli: "Proposed Correla­tion of Data on Isothermal Two Phase, Two Com­ponent Flow in Pipes", Chem. Engr. Prog., 45, No.1, 39. G:" E. Alves: "Cocurrent Liquid-Gas Flow in a Pipeline Contactor" Chem. Engr. Prog., 50, 9, 449. E. Buckingham: "Model Experiment s and the Form of Empirical Equations", ~. ASME (1915) 37, 263.

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