Energy Equation Derivation of the Oil-Gas Flowin Pipelines
J.M Duan1, W. Wang1, Y. Zhang2, L.J. Zheng1, H.S. Liu1 and J. Gong1*
1 Beijing Key Laboratory of Urban Oil and Gas Distribution Technology, China University of Petroleum-Beijing, Beijing, P.R.China, 1022492 South East Asia Pipeline Company Limited, China National Petroleum Corporation, Beijing, P.R.China, 100028e-mail: [email protected] - [email protected] - [email protected] - [email protected]
Résumé — Dérivation de l’équation d’énergie de l’écoulement huile-gaz dans des pipelines — Lorsde la simulation d’un écoulement multiphasique huile-gaz dans une conduite, le calcul thermodynamiquereprésente un processus important en interaction avec le calcul hydraulique ; il influence la convergencedu programme et la précision des résultats. La forme de l’équation d’énergie constitue la clef du calculthermodynamique. Basée sur l’équation d’énergie de l’écoulement huile-gaz dans un pipeline, la formulede chute de température explicite (ETDF ; Explicit Temperature Drop Formula) est dérivée pour uncalcul de température d’état stable huile-gaz. Cette nouvelle équation d’énergie prend en compte denombreux facteurs, tels que l’effet Joule-Thomson, le travail de pression, le travail de frottement, ainsique l’incidence des ondulations de terrain et le transfert de chaleur avec le milieu extérieur le long de laligne. Ainsi, il s’agit d’une forme globale de l’équation d’énergie, laquelle pourrait décrire précisément laréalité d’un pipeline à phases multiples. Pour cette raison, un certain nombre de points de vue de lalittérature à propos du calcul de température d’un écoulement diphasique huile-gaz dans des pipelinessont passés en revue. L’élimination de la boucle d’itération de température et l’intégration de l’équationde température explicite, au lieu de l’équation d’énergie d’enthalpie, dans le calcul conjugué hydrauliqueet thermique, se sont avérées améliorer l’efficacité de l’algorithme. Le calcul a été appliqué nonseulement au modèle de composants mais aussi au modèle Black-Oil. Ce modèle est incorporérespectivement dans le modèle de composants ainsi que le modèle Black-Oil et deux simulations sonteffectuées sur deux pipelines en service, les pipelines multiphasiques Yingmai-Yaha et Lufeng ; lesrésultats de température sont comparés à la simulation calculée par OLGA et aux résultats mesurés. Il estmontré que ce modèle a très bien simulé la distribution de températures. Enfin, on a analysé l’influencede la capacité thermique spécifique du pétrole et du gaz sur la température du mélange des fluides etl’influence de l’effet Joule-Thomson sur la répartition de température sur le pipeline. Il est montré que lecoefficient de Joule-Thomson représente un facteur clef pour décrire correctement un écoulementdiphasique huile-gaz.
Abstract — Energy Equation Derivation of the Oil-Gas Flow in Pipelines — In the simulation of oil-gaspipeline multiphase flow, thermodynamic computation is an important process interacting with thehydraulic calculation and it influences the convergence of the program and the accuracy of the results.The form of the energy equation is the key to the thermodynamic computation. Based on the energyequation of oil-gas flow in pipeline, the Explicit Temperature Drop Formula (ETDF) is derived for oil-gas steady state temperature calculation. This new energy equation has considered many factors, such asJoule-Thomson effect, pressure work, friction work and impact of terrain undulation and heat transfer
Oil & Gas Science and Technology – Rev. IFP Energies nouvelles, Vol. 68 (2013), No. 2342
LIST OF SYMBOLS
Q Transferring energy per unit time (J/s)u Internal energy of per unit mass (J/kg)v Fluid velocity (m/s)g Gravity acceleration (m/s2)H Altitude (m)ρ Density (kg/m3)α Cross-sectional void fractionA Cross section area of the pile (m2)W Work per unit time (J/s)P Pressure (Pa)τ Shear stress (Pa)S Wetted perimeter (m)Δx Pipe length of one section (m)v– Mean velocity in pipe (m/s)q Heat of oil-gas mixture transfer with the surroundings (J/s)h Enthalpy of unit mass (J/kg)T Temperature (K)cp Specific heat at constant pressure (J/(K.kg))μJ Coefficient of Joule-Thomson effect (K/Pa)ς Specific volume (m3/kg)D Inner diameter (m)U Overall heat transfer coefficient, (W/(m2.K))β Volume expansion coefficient of liquid (K-1)d4
20 Relative density of the crude oil at 20°C to density of thewater at 4°C
φ1, φ2 and φ3 Intermediate variable
Subscripts
g Gas phasel Oil phasee Environmenti Inleto Outlet
When there are two letters marked in subscript, the first letteron behalf of the phase, the second represents the inlet or outlet.
Abbreviations
ETDF Explicit Temperature Drop FormulaCUPMFP China University of Petroleum Multiphase Flow
ProjectsPR EOS Peng-Robinson Equation of StateRETD Relative Errors of Temperature Drop
INTRODUCTION
In recent years, with the vigorous development of oil and gasresources and constant exploitation of waxy crude oil, multi-phase transportation technology brings about flow-insuranceissues such as wax deposition, hydrate formation and pig-ging, which wax deposition is believed to be a significantinfluence to the security and economical operation of multi-phase transportation system. As wax deposition narrows theeffective flow area of pipe, delivery capacity reduces, deliverypressure increases and if serious, blockage may occur. All ofthe thermo-physical parameters of multiphase mixture arelinked with the temperature of the mixed fluid and the waxdeposition rate is relative to the inner wall temperaturedirectly bound up with that of the fluid [1]. Therefore, it isnecessary to do some research on the temperatures of oil-gasflow and inner wall, which is crucial to security and economicaloperation of the pipeline system.
In the study of fluid dynamics and thermodynamics, theeach form of item in energy equation for describing variouscomplex phenomenon has been very complex. There aremany variables requiring introduced, so there are some diffi-culties in solving governing problems, not benefit for engi-neering application [2]. The various fields relating to oilindustry orienting to engineering need an equation easy to besolved on the premise of keeping enough engineering errorurgently.
with the surroundings along the line. So it is an overall form of energy equation, which could describethe actual fact of multiphase pipeline accurately. Therefore, some standpoints in literatures on thetemperature calculation of oil-gas two-phase flow in pipelines are reviewed. Elimination of temperatureiteration loop and integration of the explicit temperature equation, instead of enthalpy energy equation,into the conjugated hydraulic and thermal computation have been found to improve the efficiency ofalgorithm. The calculation applied to both the component model, also applied to the black-oil model.This model is incorporated into the component model and black-oil model, respectively, and twosimulations are carried out with two practical pipeline Yingmai-Yaha and Lufeng multiphase pipelineand the temperature results are compared with the simulation calculated by the OLGA and themeasured. It is shown that this model has simulated the temperature distribution very well. Finally, weanalyzed the influence of the specific heat capacity of oil and gas on the temperature of the mixture offluids and the influence of the Joule-Thomson effect on the temperature distribution on the pipeline. It isshown that the Joule-Thomson coefficient is a key factor to well describe the oil-gas two-phase flow.
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JM Duan et al. / Energy Equation Derivation of the Oil-Gas Flow in Pipelines 343
As the fluid flows in pipe, heat is constantly transferred tothe surroundings and the temperatures of the fluid andenthalpy value are changed. The temperature drop calcula-tion for single flow usually calculated by the Sukhoi modelthat only takes heat transfer with the surroundings and fric-tion work into account [3]. For oil-gas flow differs from thatfor single-phase liquid or gas in that not only the oil-gas mix-ture transfers heat to the surroundings through wall but thequality and heat exchanges between gas and liquid shouldalso be considered. The calculation should take into consider-ation the Joule-Thomson effect caused by the gas cubicexpansion, due to the gas and the temperature rise as a resultof the heat generated by friction in the liquid flow, due to theoil [4]. So the accurate prediction of the temperature distribu-tion of oil-gas flow is very complicated.
The models calculating temperature drop in oil-gas twophase flow pipeline used by the scholars and business soft-ware such as pipephase often only consider the wall heattransfer and energy conversion caused by elevation changes.Although an accurate prediction of the temperature distribu-tion of the oil-gas flow is very complicated, the temperatureof the mixed fluid can be calculated using the energy equa-tion, that is, the enthalpy equation combining with continuityequation and equation of momentum [5].
Gregory and Aziz [6] proposed a simple relationshipbetween enthalpy of oil-gas mixture and liquid holdup butthey only find that the effect of liquid holdup on enthalpy ofthe mixture is insignificant. According to the previous deriva-tion, it is inappropriate to calculate enthalpy of the mixture onthe basis of liquid holdup. Instead, it should be calculated byuse of mass liquid holdup of the cross section.
In his method, Cawkwell and Charles [7] added thechange of the latent heat of phase to the calculation ofenthalpy increment in the energy equation, but methods forcalculating latent heat of phase change are not stated. In fact,there’s no need to additionally compute the latent heat ofphase change, for the fact that the phase change between gasand liquid is a gradual process and is included in enthalpydifference between the two phases.
Alves et al. [8] proposed a model that applies to calculatingthe temperature drop of single phase fluid and oil-gas fluidswith a full range of contained angles and unified Coulter-Bardon formula [9] and Ramey formula [10]. They neglectedmass transfer between gas and liquid but they took intoaccount pressure gradient, slope of the pipeline, accelerationenergy loss and Joule-Thomson effect in the temperature cal-culation and they applied a new method to adjust the specificheat capacity and the Joule-Thomson effect coefficient. Thismodel is widely used to calculate temperatures of pipe fluidsand is accurate for both compositional model and black-oilmodel.
Dulchovnaya and Adewumi [11] suggested a novel approachin calculating the temperature of oil-gas flow. In their model,
the energy equation of oil-gas mixed fluid doesn’t include thekinetic energy term and the potential energy term. One of themain disadvantages of their model is that it neglects the dif-ference between the internal energy and the enthalpy.Moshfeghian et al. [12] used the energy equation that theirmodel doesn’t contain potential energy term to calculate thetemperature of mixed fluid in undulant pipelines. In practice,gas has a high density under high pressure conditions, so theeffect of the undulation or of the potential energy, on thetemperature drop of mixed fluid should not be ignored. Theliquid holdup in oil-gas pipes effects significantly the temper-ature drop computation and is dependent on the pipelineinclination. For declined pipes, the liquid holdup is relativelylow and the temperature drop of oil-gas mixed fluid increases,while for upward pipes, the liquid holdup is high and tempera-ture drop decreases.
Li et al. [13, 14] propose to use a method deviating fromthe calculating temperature drop of the oil-gas flow in anhorizontal pipe, to directly calculate the temperature drop ofa two-phases flow in an undulant pipe, by replacing mass gascontent with section gas content to calculate specific heat ofmixed fluid. They guess that they would obtain a higher pre-cision on the calculating temperature of the mixed fluid incomparison with the former method. However, based on theformer derivation, slipping between gas and liquid has littleinfluence on the enthalpy of the mixed fluid if the kineticenergy is neglected. Therefore, calculate the temperaturedrop in an undulant pipe, using a section gas content in placeof a mass gas content for the determination of the specificheat to is apparently not well-founded. The feasibility of uti-lizing the temperature drop formula, with no consideration ofthe potential energy, to calculate the temperature drop of oil-gas flow in an undulant pipe is questionable. Furthermore,the temperature drop formula considering the Joule-Thomsoneffect of the gas and the heat generated by friction of theliquid, respectively, is non-uniform in theory. As the Joule-Thomson effect coefficient of the oil is below 0°C and as thegas temperature is over 0°C within a certain range, the oil isheated and the gas is cooled due to friction. The friction-generated heat for the oil and the Joule-Thomson effect of thegas can be uniformly expressed by the Joule-Thomson effectof the mixed fluid.
1 TEMPERATURE DROP MODEL FOR OIL-GAS FLOW
In order to simplify the complexity of an oil-gas pipingsystem, some assumptions on to the oil-gas flow are made asfollows:– the cross section of the pipe is constant;– the mixed flow in pipe is regarded as one-dimensional
steady flow and the temperature, pressure and other para-meters of the mixed fluid are treated as the average ofpipeline section;
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Oil & Gas Science and Technology – Rev. IFP Energies nouvelles, Vol. 68 (2013), No. 2344
– the heat conduction of the mixed fluid is neglectedcompared to the convective heat transfer of inner wall;
– the flow in pipeline is steady, no change of the state para-meters of the fluid as time goes on.When oil-gas fluids flow through the pipe and when the
surrounding temperature is colder than the fluids, the heatwill be lost from the fluids, leading to a decline in tempera-ture. In an oil-gas two phase flow pipeline, some ways ofenergy conversion and transformation of mixing fluid in pipeare proposed:– the heat transfer with the surroundings along the line;– the pressure work in boundaries of inlet and outlet of control
volume to fluid;– the friction work between the fluid and pipe the wall to
fluid;– the Joule-Thomson effect is caused by the gas cubic expan-
sion. Take segment i as the object to study (see Fig. 1).
1.1 Gas Energy Equation
Per unit of time, the gas energy that flows into and pours outof control volume are respectively expressed:
(1)
(2)
Per unit of time, the pressure work in boundaries of inletand outlet of control volume to gas are respectivelyexpressed:
(3)
(4)W P A vgo o o go= ( )α
W P A vgi i i gi= ( )α
Q u v gH v Ago go go o go o go= + +( ) ( )2 2/ ρ α
Q u v gH v Agi gi gi i gi i gi= + +( ) ( )2 2/ ρ α
Per unit of time, the friction work between the fluid andthe pipe wall to gas are respectively expressed:
(5)W S xvwg g g g= τ Δ
where v–g is mean velocity of gas in pipe section, .
Wlg is given by the power to gas between gas and oil phase.According to the law of conservation of energy, the
energy equation becomes:
(6)
Substituting Equations (1-5) into Equation (6), Equation(6) can be rewritten as: see Equation (7) where Δqg is thetotal heat transferred with the surroundings by gas in per unittime and h = u + v2/2 with the assumption, ρgo = ρgi + Δρg, αo= αi + Δα, vgo = vgi + Δvg, hgo = hgi + Δhg. Ignore high orderan infinitesimal, Equation (7) can be expressed by dividingΔx: see Equation (8).
Q Q W W W W qgi go go gi wg lg g− = − − − + Δ
vv v
ggi go=+
2
ρ α ρ αρ α
ρgi i gigi
gi i gii gi i giv A
h
xgv A
H
x
Av
xΔ Δ Δ+ − + ggi i
ggi gi i gi
gAv
xv A
xgv A
xα ρ
αα
ρΔ
ΔΔΔ Δ
+ +Δ⎛
⎝⎜
⎞
⎠⎟(hh h gH
v AP P
xP A
v
xP
gi g
i gio i
o ig
o
+ Δ +
=−( )
+ +
0 )
α αΔ
Δ
ΔAAv
x
S x v v
x
W
x
q
xgig g gi g lg gΔ
−+
− +α τ
Δ
Δ Δ
Δ Δ
Δ
Δ
( )2
2
(8)
(7)
(9)
− − − −ρ α ρ α ρ ρgi i gig
gi i gi gi gi giv Adh
dxgv A
dH
dxv Ah ggi gi i gi o i gi i gi giv AgH v A
dP
dxP A Ah0 = + + +α α ρ α ρ α( ii
g
o gi gi gi gi gi gi
AgHdv
dx
P Av v Ah v AgH
0
0
)
( )+ + +ρ ρdd
dxgv Ah gv AgH
d
dxS vi gi gi i gi
gg g gi
αα α
ρτ+ + − +( )0 (( )gH Av W
x
dq
dxgi i gi lgg
0
1ρ α − +
Δ
u v gH v A u v gHgi gi i gi i gi go go o+ +( ) ( ) − + +(2 22 2/ /ρ α )) ( )= ( ) − ( ) −
ρ α
α α τ
go o go
o o go i i gi g g
v A
P A v P A v S xΔ (vv v W qgi go lg g+ − +) 2 Δ
Pi
Ti
hi
Po
Δxi
Δzθ
To
ho
Figure 1
Energy balance of pipe section i.
ogst110166_Duan 21/06/13 18:15 Page 344
Difference quotient replaced by derivative and neglecthigh order minority, Equation (8) can be rewritten as: seeEquation (9).
Specific heat at constant pressure and coefficient of Joule-Thomson effect can be defined as:
(10)c
h
Tpgg
p
=∂
∂
⎛
⎝⎜
⎞
⎠⎟
(11)
Under the condition that the composition of the mixedfluid is constant, hg is given as a function of the pressure andthe temperature by hg = hg(p, T). Therefore:
(12)
Substituting Equation (12) into Equation (9) and combiningrelationship among thermo-dynamical parameters, we canobtain: see Equation (13).
dhh
TdT
h
pdP c dT cg
g
p
g
T
pg J=∂
∂
⎛
⎝⎜
⎞
⎠⎟ +
∂
∂
⎛
⎝⎜
⎞
⎠⎟ = −μ ppgdP
μJ
hg
T
p=∂∂
⎛
⎝⎜
⎞
⎠⎟
1.2 Oil Energy Equation
Analogously, the oil energy equation can be expressed (eachexpression seen in Appendix): see Equation (14) where Δql isthe total heat transferred with the surroundings by oil per unitof time and h = u + v2/2. With the assumption, ρlo = ρli + Δρ,αo = αi + Δα, vlo = vli + Δvl, hlo = hli + Δhl, Equation (14) can beexpressed by dividing Δx: see Equation (15).
Difference quotient replaced by derivative and neglecthigh order minority, Equation (15) can be rewritten as: seeEquation (16).
The volume expansion coefficient of liquid β can bedefined as:
(17)
and
(18)∂∂
⎛
⎝⎜
⎞
⎠⎟ = −
∂∂
⎛
⎝⎜
⎞
⎠⎟
h
PT
TT p
ςς
βς
ς=
∂∂
⎛
⎝⎜
⎞
⎠⎟ =
∂∂
⎛
⎝⎜
⎞
⎠⎟
1 1
V
V
T Tp p
JM Duan et al. / Energy Equation Derivation of the Oil-Gas Flow in Pipelines 345
− − − − −ρ α ρ α ρli i lil
li i li liv Adh
dxgv A
dH
dxv( ) ( )1 1 lli li li li i li
o li
Ah v AgH v AdP
dx
P A A
− = −
+ +
ρ α
ρ
0 1( )
( hh AgHdv
dxP Av v Ahli li i
lo li li li li+ − + +ρ α ρ0 1)( ) ( ++
+ + −
ρα
α
li li
li li li i
v AgHd
dx
gv Ah gv AgH
0
0 1
)
( )( )dd
dxS v gH Av W
dx
dqll l li li i li gl
ρτ ρ α− + − −( ) +0 1
1( ) ll
dx
(16)
ρ α ρ α
ρ
li i lili
li i lii
li
v Ah
xgv A
H
x( ) ( )
(
1 1
1
− + − −Δ Δ
−−+ −
− + −
αρ α
ρα
i lili i
l
li li
Av
xA
v
x
v Ax
)( )
(
ΔΔΔ
ΔΔ
1
1 ααρ
i lil
li l
gv Ax
h h gH)
( )
(
Δ
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
+ Δ +
= −
Δ
0
1 αα αα
i lio i
i ol
o liv AP P
xP A
v
xP Av
x) ( )
−( )+ − −
ΔΔ
ΔΔ Δ
1 −−+
− +τ l l li l gl lS x v v
x
W
x
q
x
Δ ΔΔ Δ
ΔΔ
( )2
2
(15)
u v gH v A u v gHli li i i li li lo lo+ +( ) − − + +2 22 1 2/ ( ) /α ρ oo o lo lo
o o lo i i li
v A
P Av P Av
( ) −
= −( ) − −
( )
( )
1
1 1
α ρ
α α −− + + +τ l l li lo gl lS x v v W qΔ Δ( ) 2(14)
− −⎛
⎝⎜
⎞
⎠⎟−ρ α μ ρ αgi i gi pg J pg gi i giv A c
dT
dxc
dP
dxgv AA
dH
dxv Ah v AgH v A
dP
dx
P
gi gi gi gi gi i gi
o
− − =
+
ρ ρ α0
( αα ρ α ρ α ρi gi i gi gi ig
o gi giA Ah AgHdv
dxP Av v+ + + +0 ) ( ggi gi gi gi
i gi gi i gi
Ah v AgHd
dx
gv Ah gv A
+
+ +
ρα
α α
0 )
( ggHd
dxS v gH Av W
x
dqgg g gi gi i gi lg0 0
1) ( )ρ
τ ρ α− + − +Δ
gg
dx
(13)
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Oil & Gas Science and Technology – Rev. IFP Energies nouvelles, Vol. 68 (2013), No. 2346
Then:
(19)
So:
(20)
βl is obtained by the relation [15]:
(21)
The fluid in the pipeline is far from a critical state andβ is, in general, relatively small. Equation (16) can beexpressed: see Equation (22).
1.3 Oil-Gas Mixing Fluid Energy Equation
The heat of the oil-gas mixture transfer with the surroundingsper unit of time q is expressed:
dq = dql + dqg (23)
where dq = {[UπDo(T – Te)]/Wm} dx, the power to gasbetween gas and oil phase Wlg, to oil Wgl are equal in quantityand contrary sign. Equation (13) added by Equation (22) canobtain: see Equation (24).
Then:
(25)dT
dx
T Te+ = +φ φ
φ1 2
3
βl d d T=
− + −1
2310 6340 5965420
420
dhh
TdT
h
pdP c dTl
l
p
l
T
pl=∂∂
⎛
⎝⎜
⎞
⎠⎟ +
∂∂
⎛
⎝⎜
⎞
⎠⎟ = +
−1 βll
l
TdP
ρ
∂∂
⎛
⎝⎜
⎞
⎠⎟ =
−h
P
T
T
1 βρ
where φ1, φ2 and φ3 are defined as:
(26)
(27)
See Equation (28).The pipeline section of (i, i + 1), the length is Δxi, the angle
contained by the infinitesimal section and horizontal is θ. Theparameters φ1, φ2 and φ3 are assumed to be constant for a dxlength. With this assumption, Equation (25) can be integratedin the pipeline section of (i, i + 1) as:
(29)
This model is called the Explicit Temperature DropFormula (ETDF). The average temperature of the gas-liquid
mixture in the pipe Δxi is , then:
(30)
T Tx
T Tx
i ei
i ei
+ = + −⎛
⎝⎜
⎞
⎠⎟ − −1
1
2
1 1
2 1
1φφ
φ φφ φΔ
Δexp
⎛⎛
⎝⎜
⎞
⎠⎟
⎡
⎣⎢
⎤
⎦⎥
+ − − −⎛
⎝⎜
⎞
⎠⎟
⎡
⎣⎢
⎤φ φ φ
φ1 3 11
1 1 expΔxi
⎦⎦⎥
⎧⎨⎪
⎩⎪
⎫⎬⎪
⎭⎪
Tx
Tdxi
xi= ∫10Δ
Δ
T T T Tx
i e i ei
+ = + −⎛
⎝⎜
⎞
⎠⎟ −
⎛
⎝⎜
⎞
⎠⎟
+
11
2
1
2 1
φφ
φφ φ
expΔ
φφ φφ1 3
1
1− −⎛
⎝⎜
⎞
⎠⎟
⎛
⎝⎜⎜
⎞
⎠⎟⎟exp
Δxi
φρ α ρ α
2
1=
+ −( )⎡⎣ ⎤⎦W v Ac v Ac
U Dm gi i gi pg l i i li pl
oπ
φρ α ρ α
1
1=
+ −( )⎡⎣ ⎤⎦
+
gi i gi pg l i i li pl
o
m
v Ac v Ac
U D
W
π11−( )α βl liv A
dP
dx
− + −( )⎡⎣ ⎤⎦ −ρ α ρ αgi i gi pg l i i li plov Ac v Ac
dT
dx
U D1
π
WWv A
dP
dxT
U D T
Wv
ml li
o e
mgi i+ −( )
⎡
⎣⎢
⎤
⎦⎥ = − +1 α β ρ α
πggi l i i li
i gi gi i gi
A v A g
v A v
+ −( )⎡⎣ ⎤⎦
+ −
ρ α θ
α ρ α
1 sin
AA c v A v AdP
dxP AJ pg li i li o iμ α α α+ −( ) + −( )⎡⎣ ⎤⎦ +1 1 ( ++ +
+ + +
ρ α ρ α
ρ ρ
gi i gi gi ig
o li li l
Ah AgHdv
dx
P A Ah
0 )
( ii il
o gi gi gi gi gi gAgHdv
dxP Av v Ah v0 1)( ) (− + + +α ρ ρ ii o li li li li li liAgH P Av v Ah v AgH
d
dx0 0+ + +
+
ρ ρα
α
)
( ii gi gi i gig
li li ligv Ah gv AgHd
dxgv Ah gv A+ + +α
ρ0 ) ( ggH
d
dxgH Av gH Ai
lgi i gi li i0 0 01 1)( ) ( )− + + −α
ρρ α ρ α vv
dxS v S v v Ah v
li
g g gi l l li gi gi gi gi gi
( )− − + −
1
τ τ ρ ρ AAgH v Ah v AgHli li li li li0 0− −ρ ρ
(24)
(22)
− − +−⎛
⎝⎜
⎞
⎠⎟−ρ α
βρ
ρli i li pll
lliv A c
dT
dx
T dP
dx( )1
1(( ) (1 10− − − = −α ρ ρ αi li li li li li ligv A
dH
dxv Ah v AgH ii li
o li li li il
v AdP
dx
P A Ah AgHdv
d
)
( )( )+ + + −ρ ρ α0 1xx
P Av v Ah v AgHd
dxgvo li li li li li li li+ + + +( ) (ρ ρ
α0 AAh gv AgH
d
dx
S v gH
li li il
l l li li
+ −
− +
0
0
1
1
)( )
(
αρ
τ ρ −− −( ) +αi li gllAv W
dx
dq
dx)
1
ogst110166_Duan 21/06/13 18:15 Page 346
JM Duan et al. / Energy Equation Derivation of the Oil-Gas Flow in Pipelines 347
Equation (29) is the new energy equation of the oil-gasflow in the pipeline; many factors have been considered, suchas Joule-Thomson effect, pressure work, friction work andimpact of the ground undulation and the heat transfer with thesurroundings along the line. So it is an overall form of energyequation, which could fully describe the actual fact of oil-gastwo-phase pipeline accurately on the thermodynamics.
In the simulation of multiphase flow, the convergence ofequations is crucial and most scholars focus lies in how toimprove the stability of the solution of equations. The heattransfer characteristics in the energy equation proposed byChen et al. [16] is represented by only one variable. Andeven the heat transfer is not included in the model presentedby Munkejord et al. [17]. Cazarez et al. [18] consider theinfluence of the force existed in fluid interior to energy but hehas not considered the heat transfer with the surroundingsand the phase transformation. These practices are more com-mon in multiphase flow simulation. There are some scholarsstudying heat transfer in multiphase flow for further.Andreani [19] takes into account the phase transformationheat and the radiant heat transfer. Collado and Munoz [20]focus on the work considering the pressure work, the gasexpanding work and the entropy is introduced in the energyequation. Deng and Gong [21] and Sagar et al. [22] introducesthe Joule-Thomson effect into an equation using a thermody-namic circular connection. The circular relationship of singlegas is not accordance with the oil. When the liquid holdup is
slightly higher, the assumption that the oil-gas mixture fitswith the relationship of a single gas may not be tenable. So therelationship is applied to the gas and oil phase, respectivelyand independently.
A computer code in C++ language was developed and theprogram flow chart is shown in Figure 2. As shown in theFigure 2, pressure iteration is adopted to couple the hydraulicand thermal-dynamic models, due to the replacement of theenthalpy equation with the explicit equation of the tempera-ture field. Therefore, the temperature loop is avoided and thealgorithm is fast-convergent. The calculation applies to boththe component model and the black-oil model.
The main calculation process is composed of five parts:– input the based data;– estimate the initial pressure of mixing fluid P0, calculate
the flow parameters and the physical properties and thecross-sectional void fraction according to known networkcondition;
– decide flow pattern;– calculate the pressure drop of fluid and obtain the pressure
Pk according to the flow pattern;– repeat the step 2-4 until ⏐Pk –Po⏐< ε, satisfy the precision
requirement;– receive the temperature of the fluid using Equation (29).
The calculations of continuity equation, momentum equa-tion, regime transition criterion and thermal properties is seenin Reference [23].
(28)φρ α ρ α
ρ α
3
1
1=
−+ −( )gi i gi pg l i i li pl
gi i gi
v Ac v Ac
v A++ −( )⎡⎣ ⎤⎦
+−
ρ α θ
α ρ α μ
l i i li
i gi gi i gi J
v A g
v A v A
1 sin
cc
v A v A
dP
dx
P
pg
li i li
o i
+ −( ) + −( )
⎡
⎣⎢⎢
⎤
⎦⎥⎥
+
1 1α α
α( AA Ah AgHdv
dx
P A Ah
gi i gi gi ig
o li li
+ +
+ + +
ρ α ρ α
ρ ρ
0 )
( lli il
o gi gi gi gi gi g
AgHdv
dx
P Av v Ah v
0 1)( )−
++ +
α
ρ ρ ii
o li li li li li li
AgH
P Av v Ah v AgH
d0
0+ + +
⎡
⎣⎢
⎤
⎦⎥
ρ ρ
αddx
gv Ah gv AgHd
dx
gv Ah
i gi gi i gig
li li
+ +
+ +
( )
(
α αρ
0
ggv AgHd
dx
gH Av gH
li il
gi i gi li
0
0 0
1)( )
( (
−
+ +
αρ
ρ α ρ 111
−
− − +
−
α
τ τ ρ
ρ
i li
g g gi l l li gi gi gi
Avdx
S v S v v Ah
) )
ggi gi li li li li liv AgH v Ah v AgH0 0− −
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜⎜⎜
ρ ρ
⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟
ogst110166_Duan 21/06/13 18:15 Page 347
Oil & Gas Science and Technology – Rev. IFP Energies nouvelles, Vol. 68 (2013), No. 2348
2 EXAMPLE OF ANALYSIS AND COMPARISONS
In order to verify the correctness and the validity of theETDC model, the model is prepared for the program mod-ules and embedded into the UPTP multiphase flow simula-tion software (based on the black-oil model) and TPCOMsoftware (based on component model) and both developedby the China University of Petroleum Multiphase FlowProjects (CUPMFP). The TPCOMP software has been appliedin a feasibility analysis of the Yingmai-Yaha multiphasepipeline.
2.1 Example 1: the Black-Oil Model
Taking a oil-gas pipeline of Lufeng 13-2 oilfield as an example,verify the proposed model for calculating the temperatures ofmixed fluid in this paper. The input parameters (altitude, pipeparameters, surrounding temperature and overall heat transfer
Input based data
Initial pressure of mixing fluid, P0
Calculate the physical and flow characteristic parameters
Overall heat2.0 W/(m2.K) Inlet temperature 323.15 K
transfer coefficient
Density of0.710 kg/m3 Outlet pressure 2.4 MPa
natural gas
Density of degassed886.9 kg/m3 Outlet temperature 278.75 K
crude oil (20°C)
ogst110166_Duan 21/06/13 18:15 Page 348
coefficient, throughput of mixture fluid, inlet pressure andtemperature) of UPTP are in accordance with the measuredvalue in order to preferably compare the calculation result.
The gas-liquid two phases pipelines is 50 km long andthey undulate along with topography. Table 1 presents theoperation parameters. Figure 3 presents the vertical sectionalprofile of pipeline.
In Table 2 the temperature drop means the difference ofthe threshold temperature and the endpoint temperature; theRelative Errors of Temperature Drop (RETD) be defined as:
It can be seen from Table 2 that the prediction of ETDFagrees well with the measured value for oil-gas flow. It is anoverall form of the energy equation, which could reflect theactual fact of multiphase pipeline accurately.
2.2 Example 2: the Component Model
Taking an oil-gas pipeline of Yingmai-Yaha oilfield as anexample, verify the proposed model for calculating the tem-peratures of mixed fluid in this paper. The gas-liquid pipeline
RETDUPTP MV
UPTP=
−×100%
is 145 km in length, pipe diameter Φ 610×17.5 and upwardin topography. Table 3 presents the component of pipe-conveying fluid. The inlet pressure and temperature are 50°Cand 5 MPa. The proportion of light component in the pipingfluid is large, so the phase change easily occurs when the tem-perature and pressure fluctuate. The input parameters ofTPCOM are in accordance with the OLGA that is advancedmultiphase flow simulation software in order to preferablycompare the calculation result. The vertical sectional profileof the pipeline is shown in Figure 4.
TABLE 3
Composition data of transported petroleum in Yingmai-Yaha pipeline
Molar Molar
Ordinal Component Mol% mass Ordinal Component Mol% mass
(g/mol) (g/mol)
1 C1 86.37 16.043 7 n-C5 0.18 72.151
2 C2 6.59 30.07 8 C6 0.24 86.178
3 C3 1.34 44.097 9 C7+ 1.9 171.21
4 i-C4 0.32 58.124 10 CO2 0.16 44.01
5 n-C4 0.42 58.124 11 N2 2.3 28.014
6 i-C5 0.18 72.151
Figure 4 shows the temperature distribution of gas-liquidmixing fluid comparisons between TPCOM and OLGA.OLGA never clarifies the model of thermodynamic computa-tion but the result simulated result of OLGA is high inreliability and it has been widely recognized in the processdesign and applied of multiphase flow pipeline. So OLGA isselected for the comparison of calculations.
JM Duan et al. / Energy Equation Derivation of the Oil-Gas Flow in Pipelines 349
45 504035302520151050
Alti
tude
(m
)430
400402404406408410412414416418420422424426428
Distance (km)
Figure 3
Vertical sectional profile of pipeline.
20 40 60 80 100 120 1400
Alti
tude
(m
)
Tem
pera
ture
(K
)
1080
950 270
325
320
315
310
305
300
295
290
285
280
275
1070
1060
1050
1040
1030
1020
1010
1000
990
980
970
960
Distance (km)
Altitude
TPCOM
OLGA
Figure 4
Comparison of temperature distribution of oil-gas mixingfluid.
TABLE 2
Result comparison of temperature drop of a 50.0 km long pipeline
MethodInlet Endpoint Temperature
RETDtemperature (K) temperature (K) drop (K)
UPTP 323.15 277.25 45.9 3.37%
MV* 323.15 278.75 44.4
* MV represented by measured value.
ogst110166_Duan 21/06/13 18:15 Page 349
Oil & Gas Science and Technology – Rev. IFP Energies nouvelles, Vol. 68 (2013), No. 2350
This new energy equation has considered many factors,such as the Joule-Thomson effect, the pressure work, the fric-tion work and the impact of the terrain undulation and the heattransfer with the surroundings along the line. It is an overallform of energy equation, which could reflect the actual fact ofmultiphase pipeline accurately. So the simulation result of theTPCOM is close to that of OLGA. That demonstrates theETDF model is correct at certain extent.
From the previous derivation, gas-oil mixing enthalpy ofthe fluid is the most important factor effecting the tempera-ture distribution and the enthalpy is closely related to the gasand oil specific heat at constant pressure and gas Joule-Thomson effect coefficient. In order to verify the accuracy ofETDF, the contrast distribution of the gas and the oil specificheat at constant pressure between the TPCOM using thePeng-Robinson Equation of State (PR EOS) and OLGA seein Figure 5.
The difference of gas and oil specific heat at constantpressure calculated between TPCOM and OLGA is smallfrom Figure 5.
OLGA cannot output the gas Joule-Thomson effectcoefficient, so the indirect method is applied to analyze theeffect of it on the temperature of the gas-oil mixing. Figure 6provides the simulated result of OLGA considering theJoule-Thomson effect, TPCOM considering Joule-Thomsoneffect and not.
The temperature distribution of TPCOM that does not takeinto account the Joule-Thomson effect is above the one ofOLGA and TPCOM that both consider Joule-Thomson
effect. The difference of outlet temperature between theTPCOM and TPCOM (without Joule-Thomson) is 4 K. It isshown that the Joule-Thomson coefficient is a key factor towell describe the flow.
The Joule-Thomson coefficient is expressed:
(31)μ
ςς
JP
p
TTC
=
∂∂
⎛
⎝⎜
⎞
⎠⎟ −
10 20 30 40 50 60 70 80 90 100 110 120 130 1400
Tem
pera
ture
(K
)
325
270
320
315
310
305
300
295
290
285
280
275
Distance (km)
OLGA
TPCOM
TPCOM (without Joule-Thomson)
Figure 6
Comparison of temperature distribution between TPCOMand OLGA.
Comparison of specific heat at constant pressure distributionof oil-gas mixing fluid.
ogst110166_Duan 21/06/13 18:15 Page 350
JM Duan et al. / Energy Equation Derivation of the Oil-Gas Flow in Pipelines 351
According to the thermodynamics equation, Equation (31)can be rewritten as:
(32)
The Joule-Thomson coefficient in TPCOM is calculatedby using PR EOS. Figure 7 provides the information aboutthe calculation of Joule-Thomson effect parameter.
Figure 7 provides the simulated value of the Joule-Thomsoncoefficient becomes larger and larger along the pipeline.Because the temperature and pressure of the oil-gas fluidbecomes smaller and smaller and the correlation between theJoule-Thomson coefficient and temperature is positive and thepressure is same also. The difference of the temperature calcu-lated by the OLGA and TPCOM without Joule-Thomson ismore and more significant seen in Figure 6, as a result of thevalue of the Joule-Thomson coefficient increasing along thepipeline. On the basis of comprehensive analysis of Table 2and Figure 6, the temperature distribution calculated by ETDFmodel is persuasive.
Main factors affecting on temperature are pressure, liquidholdup and gas and oil flow rate. The calculation of theseparameters is base on the thermodynamic properties of thefluid. The thermodynamic properties such as the Joule-Thomson coefficient, the specific heat at constant pressureand so on are calculated by PR EOS. The PR EOS is suitablefor the oil-gas fluid containing high level of light constituent.On the other hand, the liquid holdup is small when this kindof fluid flows in the pipeline. So that the state parameters ofthe fluid at the calculation node change little that can be con-sidered as the flow in pipeline is steady, no change of thestate parameters of the fluid as time goes on. That is in agood accordance with the main assumption of the model. TheETDF model self can be used to simulate any kind of fluid inthe steady flow.
CONCLUSION
A model named ETDF based on the general energy equationhas been introduced to describe the Explicit TemperatureDrop Formula for an oil-gas steady flow in a pipeline. Themodel considers many factors, such as the Joule-Thomsoneffect, the pressure work, the friction work and the impact ofthe terrain undulation and the heat transfer with the surround-ings along the line. The model is taking into account of theinteraction and mass transfer of both phases for energy. Themodel can be applicable to analyze the component model andthe black-oil model.
In this model, the temperature iteration loop is canceled andthe model has accurately predicted the temperature distribution
μρ
ρ
ρρ
Jp
T
C
T
P
T
P= ×
∂∂
⎛
⎝⎜
⎞
⎠⎟
∂∂
⎛
⎝⎜
⎞
⎠⎟
−
⎡
⎣
⎢⎢⎢⎢
⎤
⎦
1 12
⎥⎥⎥⎥⎥
of the mixed fluid. This model has improved the efficiency ofalgorithm and is applicable to both compositional model andblack-oil model. The model and algorithm can predict thetemperature distribution in an oil-gas flow pipeline presentedin this study accurately, comparing with the measured resultof the actual pipe and OLGA software simulation.
ACKNOWLEDGMENT
The authors thank the financial support from the Key NationalScience and Technology Specific Project (2011ZX05026-004-03), the National Natural Science Foundation of China(51104167) and the China National Petroleum Corporation(CNPC) Innovation Foundation (2010D-5006-0604).
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Final manuscript received in January 2012Published online in September 2012
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