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Multiphase Science and Technology, Vol. 16, No. 4, pp. 355-387, 2004

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CLOSURE RELATIONS FOR THE SHEAR STRESS IN TWO-FLUID MODELS FOR CORE-ANNULAR FLOW

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Amos Ullmann and Neima Brauner Faculty of Engineering, Tel-Aviv University, Tel-Aviv 6997, Israel

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Abstract. The exact solution for laminar core-annular flow (CAF) in inclined pipes is used to derive new closure relations for the wall and interfacial shear stresses. These are incorporated in a two-fluid model for CAF. With these closure relations, the two-fluid model yields the exact solution for the hold-up and pressure drop in case of laminar horizontal or inclined CAF. It is shown that multiple solutions can be obtained in upward or downward inclined systems. The multiple solution regions are identified in terms of the controlling dimensionless parameters. The predictions of the two-fluid model for laminar and turbulent CAFs are tested against experimental data available from the literature for oil-water systems. The good comparison suggests the two-fluid model as convenient tool for evaluating the pressure drop and holdup in CAF.

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Keywords: Annular, core flow, shear stresses, two-phase, gas, liquid, two-fluid.

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1. INTRODUCTION

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The core-annular flow (CAF) is one of the most important flow patterns in gas-liquid and liquid-liquid systems. In horizontal and upward inclined gas-liquid concurrent flows, the annular flow is mainly inertia driven and high gas flow rate in the core is required to drag the liquid in the annulus. However, as liquid film flow is the basic flow pattern in downward vertical tubes, the CAF is obtained also at relatively low gas and liquid flow rates in concurrent downward and countercurrent gas-liquid flows.

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A. ULLMANN and N. BRAUNER

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In liquid–liquid systems, the CAF appears most attractive for pressure loss reduction and power saving in heavy oil wells and for the transportation of viscous oils. The viscous oil forms the core phase, which is surrounded and lubricated by an immiscible low viscosity liquid (such as water) as the annular phase. A stable CAF is a fully developed flow pattern, where both the core and annular phases are distinct and continuous. Provided this flow pattern can be established, the pressure drop is almost independent of the oil viscosity, and only slightly higher than that obtained in single phase water flow at the mixture flow rate. The continued interest in oil-water CAF resulted in many experimental and theoretical studies, which has been reviewed by Oliemans (1986), Oliemans and Ooms (1986), Joseph and Renardy (1992) and Brauner (1998). In horizontal pipes, the float-up tendency of light (or heavier) core phase may result in break-up of the top (or bottom) wall film and stratification of the liquids. Therefore, the occurrence of annular flow in liquid-liquid systems is more frequently encountered in oil-water systems of low density differential, ∆ρ and small diameter tubes. These systems are characterized by a small Eotvös number, EoD = ∆ρgD2/8σ < 1, and resemble micro-gravity systems. In such systems, an annulus of the wetting phase (surrounding a core of the non-wetting phase) is a natural configuration which complies with surface tension forces and wall-adhesion forces (Gorelik and Brauner, 1999). Indeed, CAF with water in the annulus is promoted by using hydrophilc (rather than hydrophobic) pipe material. In case of viscous oil, the flow in the core is laminar, while the flow of the aqueous phase in the annulus can be laminar or turbulent, depending on its flow rate and tube diameter. Exact, simple solutions are available for the configuration of concentric CAF for the case of laminar flow in both phases in horizontal and inclined tubes (e.g. Russel and Charles, 1959, Bai et al.., 1992). These solutions provide the radial velocity profiles, holdup and pressure drop. However, similarly to gas-liquid systems, also in liquid-liquid systems, the 1-D two-fluid model is a simple and effective tool for calculations of steady or transient CAF for laminar or turbulent flows in the phases (e.g. Barnea and Taitel, 1989, Brauner, 1991). When applied to inclined CAF, predictions of holdup, pressure drop, flow reversal and flooding phenomena require reliable closure relations for the wall and interfacial shear stresses, which correctly represent the balance between frictional and gravitational forces. Similarly to stratified flows, also in annular flows, multiple solutions can be obtained for specified operational conditions in co-current and countercurrent inclined flows. Identifying and mapping the operational conditions associated with multiple solutions is of practical importance. Moreover, for a particular application, all possible solutions and their physical relevance are to be examined in determining the holdup, pressure drop, the stability of the CAF and its transition to other flow patterns.

2. TWO-FLUID MODELS FOR CAF

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The configuration of concentric core annular flow is described in Figure 1. The flow rates are both positive in concurrent down-flow, both negative in concurrent up-flow, whereas in countercurrent flows, the heavier phase flows downward (Qa>0, Qc ρc, otherwise, Qa0 with ρa > ρc). Equating the pressure drop in the core and

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CLOSURE RELATIONS FOR SHEAR STRESS

Figure 1 Schematic description of the CAF configuration.

annular phases, a force balance in steady and fully developed annular flow reads:  1 Sa 1  − τ i Si  +  + ( ρ a − ρc ) g sin β = 0 Aa  Ac Aa 

(1.1)

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with

Si = π Dc

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Sa = π D

Ac = π Dc2 / 4 (1.2) Aa =

π ( D 2 − Dc2 ) 4

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Closure relations are required for the wall and interfacial shear stresses, τa, τi. The vast majority of studies on the modeling of these closure relations relate to annular flow in vertical gas-liquid systems, with the gas flowing in the core and the liquid in the annulus. The following closure relations are commonly adopted: 1 τ a = − ρa fa U a U a 2

(2.1)

1 τ i = − ρ c fi U c U c ; 2

(2.2)

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where Ua,c are the average velocities of the core and annular phases. The wall friction factor, fa is evaluated based on the Reynolds number of the annular phase and by adopting single-phase flow models/correlations (e.g. fa = 16/Rea for laminar flow). The correlations used for the interfacial friction factor, fi, are largely empirical. For example, Wallis (1969) suggested correlations in term of the film thickness, h, of the form fi = A + B (h/D)n. Different values for the constants (A,B,n) were recommended for concurrent and countercurrent flows, which depend also on the tube diameter and flow conditions. In case the annular phase velocity is of the order of the core phase velocity, as is the case in liquid-liquid flows, the velocity head used for evaluating τi in Eq. (2.2) is based on UcUa, rather than on Uc (Brauner, 1991). In the following section, the exact solution for the general case of inclined (concurrent or countercurrent) CAF is obtained in terms of the same dimensionless parameters used in the case of stratified flow. This solution is used to re-examine the structure of the closure laws for the wall and interfacial shear stresses. New closure relations are established, which provide the exact solution for the holdup and pressure drop for the case of laminar concurrent or countercurrent inclined CAF’s. These closure relations are extended to cases of turbulent flow in either one or both of the phases. The application of the so obtained two-fluid model is demonstrated for the prediction of pressure drop and hold-up in oil-water CAF.

3. EXACT SOLUTION FOR INCLINED CAF

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3.1 Velocity profiles, holdup and pressure drop For the case of steady and fully developed laminar flow, the axial velocity profiles in the core (c) and annular (a) phases are obtained by integration of the N.S. equations: 1 ∂  ∂u  dp − ρc g sin β = Pc r  = r ∂r  ∂r  dz

(3.1)

µa

1 ∂  ∂u  dp − ρ a g sin β = Pa r  = r ∂r  ∂r  dz

(3.2)

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The B.C are ua = 0 at r=R, ∂uc/∂r = 0 at r=0, and the continuity of velocities and shear stresses across the phases’ interface. In terms of the (unknown) interfacial velocity, the velocity profiles are given by: uc 2 = − Pc ( Rc2 − r 2 ) + ui µ U as

(4.1)

ua = −2 Pa (1 − r 2 ) + 4YRc2 ln r U as

(4.2)

uc =

ua =

where Uas (= 4Qa/πD2) is the superficial velocity of the annular phase,


µ = µc / µa

Y=

F

Rc = Rc / R

;

( ρ a − ρc ) g sin β

( −dp f / dz )as

(4.3)

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r = r / R

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dp / dz − ρc g sin β dp / dz − ρ a g sin β Pc = ; Pa = − dp f / dz − dp f / dz

)as

(

)as

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(

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and (-dpf /dz)as = 32 µaUas /D2 is the frictional pressure drop in single-phase flow of the annular phase. The interfacial velocity ui = ui / uas (used in Eq. (4.1)) is obtained by substituting r = Rc in Eq. (4.2):

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ui = −2 Pa (1 − Rc2 ) + 4YRc2 ln Rc

(5)

Integration of the velocity profiles over each fluid flow cross sectional area yields the feed flow rates, whereby:

(

)

(6.1)

− 2YRc2 [1 + Rc2 (2 ln Rc − 1)]

(6.2)

U cs 1 = − Pc Rc4 − 2 Pa 1 − Rc2 Rc2 + 4YRc4 ln Rc µ U as

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q=

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2

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1 = − Pa 1 − Rc2

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These two equations can be solved for the unknown driving forces in the core and annular phases Pc and Pa , in terms of µ , Y , q and Rc : (7.1)

−q µ (1 − Rc2 ) + 2 µ Rc2 + 4Y µ Rc4 [1 − Rc2 + (1 + Rc2 ) ln Rc ] Pc = (1 − R 2 ) R 4

(7.2)

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1 + 2YRc2 [1 + Rc2 (2 ln Rc − 1)] Pa = − (1 − Rc2 ) 2

c

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Since Pc − Pa = Y , the following implicit equation is obtained for the core phase

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holdup, ε c = Rc2 :

q µ =

A. ULLMANN and N. BRAUNER εc 1 − εc

 εc  2µ + 1 εc − 

   2 2  ( µ (1 − ε c2 ) + ε c2 ) ln ε c + ε c (1 − ε c )    + Y ε c 4 µ − 1 +  (1 − ε c )2     (8)

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fa

=

( dpg / dz )

(9.1)

 ρ  = 1 − 1 − c  ε c ρ a g sin β  ρa 

(9.2)

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Π ga =

( −dp f / dz ) = − ( Pa + ε cY ) ( −dp f / dz )as

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Π

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Solving Eq. (8) for ε c (or Rc ) , either of Eqs. (7) can be used to obtain the pressure drop. The pressure drop, dp/dz is decomposed into the frictional pressure drop, dpf /dz and the gravitational pressure drop, dpg /dz. In terms of the dimensionless variables these are given by:

Simple explicit expressions can be obtained for some particular cases. For instance, the well-known solution for the case of horizontal CAF, can be easily obtained from Eq. (8) by introducing Y = 0: εc = µ  −1 + 1 + q / µ  1 − εc

=

R (−dp f / dz )

( −dp f / dz )cs

=

1

µ q(1 − ε c )

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fc

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and by Eq. (7.1): Π

(10)

2

; Π

fa

=

(−dp f / dz )

( −dp f / dz )as

=

1 (1 − ε c ) 2

(11)

For highly viscous core liquid ( µ → ∞ ), or rigid core (as in the case of transportation of cylindrical capsules) Eqs.(10) and (11) yield: q ; Π 2+q

fc

=

(2 + q )2 4µ q

(12)

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εc =

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For the case of inclined CAF with a highly viscous core phase, Eq. (8) reduces to: q=

2ε c + 2Y ε c2 (2 + ln ε c ) 1 − εc

(13)

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3.2 Interfacial and wall shear stress

du dr

r = Rc

=

Pc Rc 2

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τ i = µc

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The interfacial shear stress can be easily obtained from the velocity profile of the core phase:

or

τi R  − dp f  = − Rc Pc ; τ as = −   2  dz as τ as

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τi =

(14)

(15)

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Substitution of the solution for Pc (Eq. (7.2)) provides the interfacial shear stress in terms of the system parameters and the flow geometry ( R obtained via solution of Eq. c

τ a = µa

P

(8)). Similarly the wall shear stress, τa, can be derived from the velocity profile of the annular phase: dua R = ( Pa + ε c ∆ρ g sin β ) dr r = R 2

(16.1)

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or: τa = −( Pa + ε cY ) = Π τ as

fa

(16.2)

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τa =

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Hence, τa can be calculated by introducing Eq. (7.1) and the solution for the holdup. The expressions for τi and τa (Eq. (15) and (16)) are not suitable as closure relations for a two-fluid model, as they include the unknown pressure drop. However, using Eqs. (4.1) and (5), it can be easily shown that: τi = −

4 µc (U c − ui ) Rc

(17)

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where ui is the interfacial velocity and Uc is the average velocity of the core phase (Uc = Qc /πRc2):

Equation (17) is equivalent to:

Uc = −

Pc Rc2 + ui 8µc

(18)

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A. ULLMANN and N. BRAUNER 1 fi ρc U c (U c − ciU a ) 2

(19)

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τi = −

with ci = ui /Ua and fi is the friction factor for the case of laminar core: ρ U D 16 ; Rec = c c c ; Rec µc

(20)

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fi =

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The structure of Eq.(19) is evidently similar to the expression obtained for τi in the stratified flow geometry (Brauner et al., 2003). The interfacial shear stress is proportional to the velocity of the core phase times the velocity difference, and not the square of the core velocity (or the velocity difference) as commonly assumed in closure relations for τi. The form of Eq.(19) is obviously suitable as a closure for two-fluid models. However, its exact implementation requires the value of ci. The latter can be obtained from the exact solution. Using Eq. (5) for u and U = U /(1 − R 2 ) as i

ci =

a

P

obtained from Eq. (6.2) and Eq. (7.1)) yields:

ui = 2 + YFci (ε c ) Ua

as

c

(21.1)

with

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  (1 + ε c ) Fci (ε c ) = 4ε c (1 − ε c ) 1 + ln ε c   2(1 − ε c ) 

(21.2)

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Equation (21) indicates that for horizontal flow ci = 2, independently of the viscosity ratio and holdup. For inclined flows, the second term represents a correction that introduces the effect of gravity on the interfacial velocity. While the full expression for ci can be introduced in the closure relation for τi (Eq. (19)), it is worth noting that the effect of the gravity term is expected to be of a minor significance in practical applications where εa 1, χ 2 q / cio ρc, or to concurrent down-flow for ρa < ρc. For large q, a single solution of a thin film is obtained. However, in some range of flow rates (40 < q < 105), three different solutions for the film thickness result for the same q.

Figure 2 Variation of the dimensionless film thickness with the flow rates ratio for u = 10 : a) Y = -2000; b) Y = 20.

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Figure 2b is for positive Y (= 20). In this case a thick film generally corresponds to counter current flow, with the heavier phase flowing downwards (except for very thick films, for small values of q > 0). The counter-current flow is feasible in a limited range of rates, qF < q < 0. Point F represents flooding conditions, beyond which no solution is obtained for the CAF configuration. However, in the flow rates range where countercurrent flow is feasible, there are always two solutions for the film thickness. The concurrent region, q > 0, corresponds to up-flow for ρa < ρc, or down-flow for ρa > ρc. Also in this case, multiple (3) solutions for the film thickness are obtained in the range of low q. The above characteristics of the holdup curve are similar to those obtained in stratified inclined flows. These have been recently theoretically and experimentally explored in Ullmann et al. (2003a,b) and Brauner et al., (2003). In those studies it has been shown that also in stratified flow, there are always two feasible solutions for the holdup in countercurrent flow, and either one or three solutions in concurrent up-flow or down-flow. The commonly used two-fluid models may miss the multiple holdups predicted by the exact solution. This is demonstrated in Figure 3. If the effects of inclination on the closure relations used for the wall friction and interfacial density are ignored (Fa = 0 and ci = 2 in Eqs. (27)), only a single solution is predicted for concurrent up-flow and downflow for all flow rate ratios. Moreover, the counter-current flow became infeasible for the parameters used in Figure 2b. However, a significantly better prediction is obtained when the thin film approximation of Fa and ci (Eqs. (22) and (25) are used. Indeed, as shown in Figure 4, the customarily used Fa = 0 in closure relations for the wall shear stress and a constant ci are invalid as they miss the pronounced effect of the inclination on the wall friction and interfacial velocity as the film becomes thicker. The effect of the inclination parameter and viscosity ratio on the counter-current holdup curve is shown in Figure 5. Obviously, the range of flow rates ratio where counter-current flow is feasible increases with Y (Figure 5a). For large values of Y, the flooding flow rates ratio, qF is proportional to Y (qF /Y is constant). There is, however, a minimal value of Y > 0(~9) for which counter-current flow is feasible. For example, given a two-fluid system, this minimal value puts the limit on the flow rate of the annular phase that can be transported counter-currently to the core phase. Figures 5b,c show that the counter-current flow rates range decreases with µ . For highly viscous core ( µ → ∞) and Y = 100, qF / Y → −0.034 (Figure 5b), however, for Y → ∞ this limit would extend to qF /Y → -0.0444. In the opposite case of viscous annular phase µ 0 and YMS > 0 indicate the limits of the MS region in concurrent down-flow for ρa > ρc (or concurrent up-flow for ρa < ρc). The countercurrent limits are represented by the dashed segment of the curve, where qMS/YMS < 0 with YMS > 0 and qMS < 0. It represents the variation of qF with Y, whereby at the T point qF ≡ qMS → -∞ as Y → ∞. The rest of the curve at negative qMS/YMS represents the limits of the MS region in concurrent up-flow ρa > ρc (or concurrent down-flow for ρa < ρc). It corresponds to qMS > 0 and YMS < 0, starting from point T, where YMS → -∞ and qMS → ∞. A more detailed characterization of the multiple solution boundaries is included in Figures 7-9. The effect of the viscosity ratio on the locus of flooding conditions is depicted in Figure 7. Note that qF /Y represents the maximal core phase flow rate for a given two-fluid system in counter-current flow. The qF is viscosity weighted by µa in case of µ >> 1 and by µc in case of µ 1

0.80

~ µ=1

5 2

T

0.00 -0.15

1.00

R

-0.10 -0.05 F Flooding Conditions, qF /Y = 8µaUcs / (∆ρgsinβD2)

~ b) µ100

0

P

0.20

10

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0.40

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0.60

0.40

1

T

0.1

∼ µ ρc (or down-flow for ρa < ρc) is shown in Figure 8. For instance, for µ = 0.1 , the thickest film on the MS boundaries is h / R ≈ 0.2, hence all the three holdups are of relatively thin films, in particular for µ < 1 (see Figure 8a). The corresponding YMS vs qMS is depicted in Figure (8b). On the other hand, as shown in Figure 9, the multiple (three) holdups regions in concurrent down-flow for ρa > ρc (or up-flow for ρa < ρc) are associated with thick films and slim cores. Such configurations are susceptible to wave bridging phenomena affecting transition to other flow patterns.

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Figure 8 Characterization of the multiple solutions regions boundaries in inclined concurrent flows with Y < 0: up-flow for ρa > ρc, down-flow for ρa < ρc: a) range of film thickness, b) range of flow-rates ratio.

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It is worth noting that in stratified flow, a complete similarity where shown to exist between the multiple solution regions in concurrent up-flow and down-flow. Such a similarity is not obtained in the case in CAF. The relevance of the multiple holdups of the annular configuration in practical applications is demonstrated in Figure 10 with reference to the flow pattern map reported by Govier and Aziz (1972) for upward oil-water flow in a vertical tube. The sequence of flow patterns observed while increasing the oil flow rate at a constant water flow rate resembles that obtained in air-water upward flow: oil bubbles, oil slugs, then transitional flow with churning, and eventually a dispersion of water in oil (Dw/o). The


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Figure 9 Characterization of the multiple solutions regions boundaries in inclined concurrent flows with Y > 0: up-flow for ρa < ρc, down-flow for ρa > ρc: a) range of film thickness, b) range of inclination parameter.

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10

ρo/ρw=0.85, µo/µw=20.1, D=2.64cm

a)

F

1

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Exp. boundary

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100

10-1

TF Churm

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3-s

a

Dw/o

b

b)

0.60 0.40

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0.80

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1.00

b

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0.20 0

0.01

a

0.10 1.0 Superficial Oil Velocity, Uos[ft/s]

10.0

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Figure 10 a) Multiple solution region on the flow patterns map for vertical upward oil-water system reported by Govier and Aziz (1972), b) the holdup curve for a constant water flow rate.

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region where multiple holdups are predicted is also indicated on this map. The associated variation of the film thickness predicted for CAF is depicted in Figure 10b. Left to the triple solution (3-s) region, the annular flow model yields a single unstable solution of a very thick film and a slim oil core, where wave bridging leads to formation of oil bubbles or slugs. Close to the 3-s boundary churn flow was observed. As the 3-s boundary is reached, two additional solutions appear of relatively thin films, whereby

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the CAF becomes a possible stable configuration. At higher oil flow rates (right to the triple solution region), only a single mode of a thin film is predicted. In this region CAF is not necessarily obtained in oil-water systems. Once the oil core becomes turbulent, the wavy water annulus is dispersed into the turbulent oil core to form a dispersion of water in oil. However, Figure 10 demonstrates that stability of the three possible solutions should be analyzed for studying the phenomena related to growth of interfacial waves, drop entrainment and flow pattern transition. The effects of the oil core viscosity and tube diameter on the location of the triple solutions (3-s) region in upward oil-water flows are studied in Figure 11. The 3-s region migrates to lower oil flow rates as the oil core viscosity increases and/or the tube diameter is reduced. For µo > 10 cp, the oil viscosity has a mild effect on the location of the 3-s region. Note that with low oil viscosities and/or large tube diameters, the laminar model is no longer valid. A CAF model with a turbulent oil core should be applied.

Figure 11 The effects of the oil viscosity a) and tube diameter b) on the location of the multiple solution regions in oil-water vertical upward flow.

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5.2 Kinematic wave propagation in the multiple solutions regions

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It is of interest to examine the relation between the multiple solution boundaries and the propagation velocity of kinematic (void fraction) waves, CK. For incompressible fluids, the transient continuity equations for the core and annular phases are (38.1)

∂ε a ∂ ∂ε ∂ε + ( ε aU a ) = a + CK a = 0 ∂t ∂z ∂t ∂z

(38.2)

∂U cs ∂ε c

Um

=

∂U as ∂ε a

(38.3)

R

CK =

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∂ε c ∂ ∂ε ∂ε + ( ε cU c ) = c + CK c = 0 ∂t ∂z ∂t ∂z

Um

P

where Um = Ucs + Uas is the (constant) mixture velocity. In case of negligible inertia effects, the local force balance is represented by the steady-state momentum equations, hence in case of laminar flow, by Eqs. (6) and (7). Putting the latter in the form of F (q, µ , Y , ε a ) = 0, the Ck is obtained from the following relations:

)

(

(39)

)

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whereby:

(

 ∂F   ∂ε  a  (q + 1) Y + ∂F ∂Y ∂q

R

C K

C = K = U as ∂F

with

(40.1)

TH

C K

 µ  ∂q ∂ε a   =− (1 − ε a )  1− εa  µ +  2µ +  εa  εa 

{

U

∂q −2 (3ε + 2) − 2ε a2 − ε a + 2] ε a = ε a µ + 1 − ε a + Y (1 − ε a ) [ µε  a a ∂ε a ε a3

}

(40.2)

a + (1 − ε a )(ε a2 ( µ − 1) + 1)] +2 ln(1 − ε a )[ µε 

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It is worth noting that the kinematic wave velocity is in fact identical to the velocity of the long waves as obtained from a full stability analysis of the transient continuity and momentum equations (e.g. Brauner and Moalem Maron, 1992). In view of Eq. (40), CK vanishes on the multiple solution boundaries. This is demonstrated in Figure 12, showing

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A. ULLMANN and N. BRAUNER ~ = µ /µ = 10, D = 2.5cm, ∆ρ/ρ = 0.8, Y = -1000 a) µ w ο w

F

200 150

O

~ CK

100

1

Ucs

u~ i

2

Ucs

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50

Thin Film

0 10-2

10-1

~ = µ /µ = 10, D = 2.5cm, ∆ρ/ρ = 0.8 b) µ w ο w

P

101

101

100

R

10-3 102

0

~ (CK)max ~ (CK)min

100

10-3 10-3

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10-2

R

τa=0 10-1

10-2

3-s 10-1

100

101

TH

Superficial Oil Velocity, Ucs Uos[m/s]

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Figure 12 a) The variation of the kinematic wave velocity and interfacial velocity with increasing the oil-flow rate at a constant water flow rate, b) Locus of extreme values of the kinematic wave velocity and locus of zero wall shear stress in oil-water vertical upward CAF.

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the variation of CK with the oil flow rate at a constant water flow rate corresponding to Y = -1000. Starting with low Ucs, the C K is positive and increases with Ucs to a maximal value. Then, as the triple-holdup region is approached, the C decreases K

reaching a zero value on the 3-s boundary. Within the 3-s region (between U1cs and U2cs), the C K is positive for two of the three possible modes: the thickest film and the

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thinnest film, and negative for the middle mode (marked with dashed line). A negative C K means that kinematic waves propagate opposite to the flow direction, which may be incompatible with the boundary conditions at the fluids entrance (at z=0). . Note that on the left boundary of the 3-s region, C K = 0 for the thin and middle film thickness modes, while on the right 3-s boundary, C = 0 for the thick and middle film thickness

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modes. For oil flow rates beyond the 3-s region, the C K (of the single thin film mode) monotonously increases with Ucs. The interfacial velocity ui = ui / U as is also indicated on Figure 12a. As expected, it is positive for all three modes, as the heavier water in the annulus are dragged upward by the lighter phase. The locus of the extreme value of C K for this oil-water system is mapped in Figure 12b (see the Appendix). Also shown is the locus of zero wall shear stress. The latter is obtained by equating 1 + YFa in Eq.(24.1) to zero. The locus of τa = 0 indicates the onset of flow reversal in the near wall region. Hence, water flow rates above the τa = 0 boundary are associated with backflow (downward) near the tube walls, τa > 0. Note that the left boundary of the 3-s region coincide with the locus of τa = 0. Backward flow is not obtained with the thin film mode which appears on the 3-s boundaries. From the point of view of kinematical consideration, favorable conditions for establishment of CAF are obtained right to the locus of τa = 0 and the locus of minimal C K values.

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In counter-current CAF, the kinematical wave velocity attains a zero value at the flooding point. Figure 13 demonstrates the variation of C K with increasing the core phase flow rate at constant Uas (constant Y). For the of thin annulus mode (i.e. lower part of the holdup curve in Figure 2b), C K > 0 , indicating that the kinematical waves propagate downwards, namely in the direction of the heavy phase flow. Along the thick film mode C K < 0, namely, kinematical waves move upward in the direction of the light phase. Hence, in counter-current flows, kinematical considerations do not rule out the possibility of obtaining modes of negative C K values.

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5.3 Application of the MTF model to CAF in oil-water systems

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Figure 14a shows a comparison of the MTF model predictions with holdup data for horizontal oil-water CAF obtained by Arney et al. (1993). Water flows in the annulus and oil in the core. With the viscous oils (6 and 27 poise) and tube diameter of D=0.95cm used in the experiments, the flow in the viscous oil core is always laminar. Hence, the results of the MTF model correspond to laminar core (Lc) with either laminar or turbulent flow in the annulus (La or Ta). With highly viscous core, the holdup is determined by the flow rates ratio, and is given by Eq. (12) in case of Lc-La, and by Eq. (35) for Lc-Ta. Figure 14 shows that most of the data are confined by the predictions of these two equations. In fact, the experimental correlation suggested by Bai et al. (1992) is identical to Eq. (35) with cio = 1.39. This value is in between the theoretical value of cio = 2 for laminar flow and cio ≈ 1.15 for turbulent water annulus (see Eq. (33)). In any case, the water holdup exceeds the input water cut, Uws/Um by only a few percents, implying a small slip between the two phases.


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Figure 13 The kinematic wave velocity and interfacial velocity in counter-current oil-water CAF.

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The pressure drop predictions in horizontal oil-water CAF are demonstrated in Figure 14b in comparison with the data reported by Guevara et al.. (1988). The oil viscosities used in the experiments ranged from 50-to-2300 poise. The MTF results are for Lc-Ta, and in this viscosity range are practically independent of the oil viscosity. The model predicts that the pressure loss in the system is higher than the pressure loss obtained in single- phase water flow at the mixture velocity, however of the same order of magnitude. The ratio between the two is typically 1.1 to 3. Both the theory and data indicate that for each oil superficial velocity, there exists an optimal water-cut corresponding to a minimal pressure drop. For the case of Lc-Ta, the optimal water-cut is in the range of Uws/Um =0.08-0.12, compared to the optimal value of Uws/Um =1/3 in the case of Lc-La (e.g. Russel and Charles,1959, Rovinsky et al., 1997). Bai et al. (1992) obtained data for the holdup in concurrent vertical up-flow and down-flow of viscous oil (6.01 poise) lubricated by water flow in the annulus. In the range of flow rates tested, they found gravity effect in up-flow is apparently mild. Similarly to horizontal CAF, the holdup in up-flow depends on the flow rates ratio, rather than on the flow rate of each of the phases. Indeed, as shown in Figure 15a, the Lc-Ta holdup predicted in up-flow is similar to that obtained in horizontal CAF (Figure 14), with a rather mild effect of the water flow rate. However in down-flow, the experimental data indicate that the oil tends to accumulate, in particular, in the low- tointermediate range of the tested water flow rates. As shown in Figure 15b, the MTF model predicts these trends. Moreover, at low water flow rates, εo does not vanish as Uos → 0. With no (fresh) oil fed to the pipe, the solution corresponds to a suspended oil core (or a long oil slug), where the buoyancy force is balanced by the interfacial drag due to the water flow in the annulus. This situation was actually observed by Bai et al. (1992) in down-flow. Obviously, for high water and oil flow rates, gravity effects are suppressed, and the holdup is practically independent of the tube inclination. It worth

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Figure 14 Comparison of the MTF model predictions with experimental data obtained for horizontal-water CAF with µ >> 1. a) Holdup; b) Pressure drop.


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Figure 15 Comparison of the MTF model predictions with experimental data obtained for CAF of 6.01 poise oil and water in vertical tube. a) Concurrent up-flow; b) Concurrent down-flow.

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mentioning, that multiple holdups are not predicted in the range of flow rates where CAF was experimentally established in this oil-water system.

5.4 Implications to the modeling of interfacial shear

Fi =

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The neglect of the gravity effect in closure relations that have been customarily employed in two-fluid models (see Eqs. 2), results in miss prediction of the holdup and pressure drop. In gas-liquid annular flow, poor predictions of two-fluid models have been commonly attributed to the incorrect representation of the interfacial friction factor, fi , that has to account for the wave phenomena at the interface. Various empirical correlations for fi (or Fi in Eq.(28.1)) have been suggested in the literature to match holdup and pressure drop data. However, as gravity effects on the closure relations have been ignored, it is possible that part of the empirical corrections are not related to the wave phenomena. It is therefore of interest here to evaluate the fictitious corrections required on fi to compensate for the neglect of the gravity terms in τi and τa. To this aim the two-fluid combined momentum equation (30.1) can be written in the following form: χ 2ε c(5− nc ) / 2 [1 + Y ( Fa − ε a3 )] ci ε a (ε a − ε c ) q

(41)

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Upon substituting the holdup obtained by the MTF model equations (with the ci and Fa given by Eqs. 28.2 and 29 and Fi = 1) in the RHS of Eq. (41), the resulting Fi value (LHS of Eq.(41)) is in fact identically 1. However, in case this holdup is substituted with other values of ci and Fa, the resulting Fi represents the artificial correction required on τi to match the holdup prediction. The resulting Fi obtained by imposing ci = 0 and Fa = 0 (as assumed in Eqs 2.1 and 2.2) is demonstrated in Figure 16 vs. the film thickness for air-water counter-current flow in a 2.5 cm vertical tube. In this case, the gas phase is turbulent. The increase of the film thickness corresponds to increasing the liquid downward flow rate at a constant upward gas flow rate. The Fi > 1 implies an augmented fi is required to compensate for ci = 0 and Fa = 0 values, the latter being of the main effect. The Fi correction factor is compared with the corrections suggested by the widely used Wallis correlation, Fiw = 0.005(1 + 150h ) / f c and that of Bharathan et al. (1979) for countercurrent air-water flow in 2.5 cm vertical tube, FiB = (0.005 + 280(h / 2)2.13 ] / f . Figure 16 implies that at least part of the apparent fi augmentation, c

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as suggested by these correlations, may be an artifact that results from the misrepresentation of τa.

6. CONCLUSIONS

Accurate prediction of the holdup and pressure drop in core-annular flow (CAF) via 1-D two-fluid models requires reliable closure relations for the wall and interfacial shear


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Figure 16 Apparent augmentation of the interfacial shear required to compensate for using ci = 0 and Fa = 0: Demonstration for counter-current air-water flow in vertical 2.5 cm pipe.

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stresses. The structure of the closure relations that are commonly used is based on knowledge gained in single phase flows with empirical corrections introduced to match with two-phase flow data of a particular application. The approach taken in this study is to re-examine the exact solution for laminar CAF in inclined pipes. The exact solution is used to derive new closure relations for the wall and interfacial shear stresses that correctly represent the effect of the inclination and the interaction between the flows of the two phases. These are incorporated in a two-fluid model for CAF. With the new closure relations, the two-fluid model yields the exact solution for the hold-up and pressure drop in case of laminar horizontal or inclined CAF. The same structures of closure relations are applicable also in turbulent flows in either or both of the phases. It is shown that multiple solutions can be obtained in upward or downward inclined systems. Similarly to stratified flows (Ullmann et al., 2003b, 2004), also in CAF the counter-current region is associated with double-solution for the whole range of its feasibility. In concurrent up-flows and down-flows, up to three solutions are obtained in some limited range of operational conditions. The multiple solution regions are identified in terms of the controlling dimensionless parameters. The introduction of the multi-holdup regions on flow pattern maps of gasliquid and liquid-liquid systems indicates that these regions correspond to operational conditions where the annular or churn flows were experimentally observed. As computational codes usually provide only one solution for specified operational conditions, it is important to make sure that the solution obtained indeed corresponds to the relevant physical configuration. To that end, a-priori mapping of the multiple holdups region can be helpful.

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It is worth noting that the commonly used closure relations may fail in indicating the possibility multiple holdups. This is a result of the neglect of the gravity effect in closure relations used for the wall (and interfacial) shear stresses. It has been demonstrated that at least part of the apparent augmentation of the interfacial shear stress, as suggested by empirical correlations, may be an artifact that results from the misrepresentation of the wall shear stress. The predictions of the two-fluid model for laminar and turbulent CAFs are tested against experimental data available from the literature for oil-water systems. The good comparison suggests the two-fluid model as convenient tool for evaluating the pressure drop and holdup in CAF.

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The extremum of C K corresponds to dCK /dεa = 0. This condition is given by:

where: 4 ε a3

 µ 3 3 1 2 + 4Y [ + µ − + − 1] 4 2 2 2 2 ε a ε a3 εa  ε a 1 1 1 [( µ − 1)(ε a − 2) + ( µ − 2) + ] + 2 (1 − ε a ) εa ε a3

( µ − 1) +

6

(A.2)

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∂ε a2

=

R

∂2 F

(A.1)

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 ∂2 F 1  dC K 1 =  − 2 + 3 C K (2ε a ( µ − 1) + 2)  = 0 1− εa 1 − ε a  ∂ε dεa εa  a [ µ + (2µ + )]  εa εa

+ ln(1 − ε a )[

2

ε a3

( µ − 2) +

 + 1 − µ ] ε a4  3

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Substituting Eq. (40) for C K , Eqs. (A.1) with (A.2) can be solved for Y (εa), which is then substituted into Eq. (8) to obtain the corresponding q. Given a two-fluid system, this Y vs. q relation corresponds to Uas vs. Ucs for which C K attains maximal or minimal values.

REFERENCES

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Arney, M., Bai, R., Guevara, E., Joseph, D.D. and Liu, K. (1993). Friction factor and holdup studies for lubricated pipelining: I. Experiments and correlations. International Journal of Multiphase Flow, 19:1061-1076. Bai, R., Chen, K. and Joseph, D.D. (1992). Lubricated pipelining: Stability of coreannular flow, Part V. experiments and comparison with theory, Journal of Fluid Mechanics, 240:97-132. Bharathan, D., Walis, G.B. and Richter, H.J. (1979) Air-water counter-current annular flow. Electric Power research Institute Rep. EPRI NP-1165. Barnea, D., and Taitel, Y. (1989). Transient formulation modes and stability of steadystate annular flow, Chem. Eng. Sci., 44, 325-332. Brauner, N. (1991). Two-phase liquid-liquid annular flow. International Journal Multiphase Flow, 17(1):59-76. Brauner, N., and Moalem Maron, D. (1992b). Flow pattern transitions in two phase liquid-liquid horizontal tubes International Journal of Multiphase Flow 18:123-140.

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Brauner, N., and Moalem Maron, D. (1992). Analysis of stratified/non-stratified transitional boundaries in inclined gas-liquid flows. International Journal of Multiphase Flow 18(4):541-557 Brauner, N., Ullmann, A., Goldstein, A and Zamir, M. (2003). Closure relations for the shear stresses in two-fluid models for stratified flow, presented at the 41th European Two-Phase Flow Group Meeting, Trondhiem, May 11-14. Brauner, N. (1998). Liquid-Liquid two-phase flow, Chap. 2.3.5 in HEDU - Heat Exchanger Design Update, edited by G.F. Hewitt 1:40. Goldstein, A. (2002). Analytical Solution of Two-Phase Laminar Stratified Flow in Inclined Tubes, M.Sc. Thesis, School of Engineering, Tel-Aviv University, Israel. Gorelik, D. and Brauner, N. (1999). The interface configuration in two-phase stratified flow, International Journal Multiphase Flow. 25:877-1007. Govier, G.W., and Aziz, K. (1972). The Flow of Complex Mixtures in Pipes, Robert E. Krieger Publishing Company, 1st ed., 326-327, New York. Guevara, E., Zagustin, K., Zubillaga, V., and Trallero, J.L. (1988). Core-Annular Flow (CAF): The most economical method for the transportation of viscous hydrocarbons, 4th UNITAR/U.N. Dev. Program AOSTRA-Petro-Can-Pet. Venez., S.A.-DOE Heavy Crude Tar Sands. International Conference Edmonton. 5:194. Joseph, D.D., and Renardy, Y.Y. (1992). Fundamentals of two fluids dynamics: Part I and II (edited by F. John, et al.), Springer-Verlag. Oliemans, R.V.A. (1986). The Lubricating Film Model for Core-Annular Flow. Ph.D. Dissertation, Delft University Press. Oliemans, R.V.A., Ooms, G. (1986). Core-Annular Flow of oil and water through a pipeline. Multiphase Science and Technology. vol. 2, eds. G.F. Hewitt, J.M. Delhaye, and N. Zuber, Hemisphere Publishing Corporation, Washington. Rovinsky, J., Brauner, N. and Moalem Maron, D. (1997). Analytical solution for laminar two-phase flow in a fully eccentric core-annular configuration, International Journal Multiphase Flow 23:523-542. Russell, T.W.F. and Charles, M.E. (1959). The effect of the less viscous liquid in the laminar flow of two immiscible liquids. Canadian Journal Chemical Engineering 37:18-34. Taitel, Y., and Barnea, D. (1983) Counter-current gas-liquid vertical flow, model for flow pattern and pressure drop, Int. J. Multiphase Flow, 9: 637-647. Ullmann, A., Zamir, M., Ludmer, Z. and Brauner, N. (2003a). Stratified laminar counter-current flow of two liquid phases in inclined tubes. Int. J. Multiphase Flow 29: 1583-1604. Ullmann, A., Zamir, M., Gat, S. and Brauner, N. (2003b). Multi-holdups in co-current stratified flow in inclined tubes. Int. J. Multiphase Flow 29: 1565-1581. Ullmann, A., Goldsein, A., Zamir, M. and Brauner, N. 2004. Closure relations for the shear stresses in two-fluid models for laminar stratified flow. International Journal Multiphase Flow, 30 (7-8): 877-900. Wallis, G.B., (1969) One-dimensional two-phase flow. Mcgraw-Hill, N.Y.