two-phase blowdown from pipelines

34
Pergamon Chemical Enoineering Science, Vol. 50, No. 4, pp. 695 713, 1995 Copyright © 1995 Elsevier Science Lid Printed in Great Britain. All rights reserved 0009 2509/95 $9.50 + 0.00 0009-2509(94)00246-0 MODELLING OF TWO-PHASE BLOWDOWN FROM PIPELINES--I. A HYPERBOLIC MODEL BASED ON VARIATIONAL PRINCIPLES J. R. CHEN,* S. M. RICHARDSON and G. SAVILLE Department of Chemical Engineering, Imperial College, London SW7 2BY, U.K. (Received 11 October 1993; accepted in revised form 18 August 1994) Abstract--In this paper, Geurst's variational principle for bubbly flow is extended to generalised multi- component two-phase dispersions. The present variational principle allows both phases to be compressible in deriving the momentum equations. A mixture energy equation is obtained using Noether's invariant theorem and is shown to be comparable with the averaging formulation. The hyperbolicity of the equations is achieved by forcing the flow to be marginally stable. Under the marginally stable condition, all the information related to the structure of the flow is found to be embedded in an inertial coupling constant and an expression for this constant is obtained based on critical flow data. The marginally stability model gives correct sonic characteristics up to void fractions of 0.8. The clearly defined sonic characteristics make possible the rigorous determination of the critical flow condition for rapid depressurisation of pipelines. 1. INTRODUCTION Most accidents in chemical plants, nuclear power plants and offshore oil and gas platforms usually result in the spillage of toxic, radioactive, flammable or explosive materials. Accurate prediction of the re- leasing process is important in determining the conse- quences of an accident. The predicted information including the rate of material release, the total quant- ity released and the physical state of the material is valuable for evaluating new process designs, process improvements and the safety of existing processes. The blowdown phenomenon, amongst other transient release processes, is a subject of particular interest to the chemical, oil/gas and power industries. In the chemical industries, blowdown of pressure vessels and pipelines can be a hazardous operation due to the very low temperature generated within the fluid dur- ing depressurisation (Haque et al., 1990). For the offshore oil and gas industries, controlled blowdown of sub-sea transportation lines is frequently required in order to perform maintenance on the lines. The low temperature generated during the blowdown opera- tion may lead to the formation of hydrates and block the pipeline if free water is present. A detailed invest- igation into the transient behaviour of pipeline blow- down is necessary for defining operational limits (Ellul et al., 1991). In the case of emergency blowdown or accidental rupture of sub-sea pipelines, the hazard arises not only because of the low temperature that can arise in the pipe wall but also because of the large total efflux and high efflux rates that arise from the large inventory of the long pipelines (Richardson and Saville, 1991). For example, during the tragic loss of *Author to whom correspondence should be addressed. Present address: Institute of Applied Mechanics, National Taiwan University, Taipei, Taiwan. Piper Alpha platform on the night of 6th July 1988, the rupture of three large gas lines connected the Piper Alpha platform, although not the primary cause of the accident, were actually the major source for escalation of the incident (Sylvester-Evans, 1991). The prediction of blowdown of long sub-sea pipelines is therefore of prime importance in the risk assessment of offshore oil and gas installations. In the nuclear power industries, the hypothetical loss-of-coolant ac- cident (LOCA) is one of the most significant aspects of the design and test of the emergency cooling system of a reactor. Therefore, a fundamental study of the blow- down process is crucial in the assessment of safety practices and procedures to prevent or minimise the consequences of controlled or uncontrolled releases. Blowdown from pipes involves fast transient behav- iour and choking or critical flow phenomena and is not a trivial modelling task. In the simplest case of modelling, the two phases are considered as a homo- geneous (pseudo-one-phase) mixture. The result is the homogeneous model that resembles the Euler equa- tions of gas dynamics. The homogeneous model indi- cates infinite coupling between the two phases. How- ever, the two-phase critical mass flow rate, one of the most important factors in risk assessment, is usually underestimated by the homogeneous model (Ardron and Furness, 1976). Rigorous approaches based on various averaging methods [e.g. Ishii (1975), Drew (1983) and Lahey and Drew (1992)] have been applied to the local field equations for each phase. This method of phase averaging gives two sets of averaged conservation equations for each phases, usually called the two-fluid model. However, the local information lost in the averaging process must be supplied as closure relations in order to close the model which in most cases is unknown. The simplest two-fluid model, usually called the separated or the Wallis model 695

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Page 1: Two-phase Blowdown From Pipelines

Pergamon Chemical Enoineering Science, Vol. 50, No. 4, pp. 695 713, 1995 Copyright © 1995 Elsevier Science Lid

Printed in Great Britain. All rights reserved 0009 2509/95 $9.50 + 0.00

0009-2509(94)00246-0

MODELLING OF TWO-PHASE BLOWDOWN FROM PIPELINES--I. A HYPERBOLIC MODEL

BASED ON VARIATIONAL PRINCIPLES

J. R. CHEN,* S. M. RICHARDSON and G. SAVILLE Department of Chemical Engineering, Imperial College, London SW7 2BY, U.K.

(Received 11 October 1993; accepted in revised form 18 August 1994)

Abstract--In this paper, Geurst's variational principle for bubbly flow is extended to generalised multi- component two-phase dispersions. The present variational principle allows both phases to be compressible in deriving the momentum equations. A mixture energy equation is obtained using Noether's invariant theorem and is shown to be comparable with the averaging formulation. The hyperbolicity of the equations is achieved by forcing the flow to be marginally stable. Under the marginally stable condition, all the information related to the structure of the flow is found to be embedded in an inertial coupling constant and an expression for this constant is obtained based on critical flow data. The marginally stability model gives correct sonic characteristics up to void fractions of 0.8. The clearly defined sonic characteristics make possible the rigorous determination of the critical flow condition for rapid depressurisation of pipelines.

1. INTRODUCTION

Most accidents in chemical plants, nuclear power plants and offshore oil and gas platforms usually result in the spillage of toxic, radioactive, flammable or explosive materials. Accurate prediction of the re- leasing process is important in determining the conse- quences of an accident. The predicted information including the rate of material release, the total quant- ity released and the physical state of the material is valuable for evaluating new process designs, process improvements and the safety of existing processes. The blowdown phenomenon, amongst other transient release processes, is a subject of particular interest to the chemical, oil/gas and power industries. In the chemical industries, blowdown of pressure vessels and pipelines can be a hazardous operation due to the very low temperature generated within the fluid dur- ing depressurisation (Haque et al., 1990). For the offshore oil and gas industries, controlled blowdown of sub-sea transportation lines is frequently required in order to perform maintenance on the lines. The low temperature generated during the blowdown opera- tion may lead to the formation of hydrates and block the pipeline if free water is present. A detailed invest- igation into the transient behaviour of pipeline blow- down is necessary for defining operational limits (Ellul et al., 1991). In the case of emergency blowdown or accidental rupture of sub-sea pipelines, the hazard arises not only because of the low temperature that can arise in the pipe wall but also because of the large total efflux and high efflux rates that arise from the large inventory of the long pipelines (Richardson and Saville, 1991). For example, during the tragic loss of

*Author to whom correspondence should be addressed. Present address: Institute of Applied Mechanics, National Taiwan University, Taipei, Taiwan.

Piper Alpha platform on the night of 6th July 1988, the rupture of three large gas lines connected the Piper Alpha platform, although not the primary cause of the accident, were actually the major source for escalation of the incident (Sylvester-Evans, 1991). The prediction of blowdown of long sub-sea pipelines is therefore of prime importance in the risk assessment of offshore oil and gas installations. In the nuclear power industries, the hypothetical loss-of-coolant ac- cident (LOCA) is one of the most significant aspects of the design and test of the emergency cooling system of a reactor. Therefore, a fundamental study of the blow- down process is crucial in the assessment of safety practices and procedures to prevent or minimise the consequences of controlled or uncontrolled releases.

Blowdown from pipes involves fast transient behav- iour and choking or critical flow phenomena and is not a trivial modelling task. In the simplest case of modelling, the two phases are considered as a homo- geneous (pseudo-one-phase) mixture. The result is the homogeneous model that resembles the Euler equa- tions of gas dynamics. The homogeneous model indi- cates infinite coupling between the two phases. How- ever, the two-phase critical mass flow rate, one of the most important factors in risk assessment, is usually underestimated by the homogeneous model (Ardron and Furness, 1976). Rigorous approaches based on various averaging methods [e.g. Ishii (1975), Drew (1983) and Lahey and Drew (1992)] have been applied to the local field equations for each phase. This method of phase averaging gives two sets of averaged conservation equations for each phases, usually called the two-fluid model. However, the local information lost in the averaging process must be supplied as closure relations in order to close the model which in most cases is unknown. The simplest two-fluid model, usually called the separated or the Wallis model

695

Page 2: Two-phase Blowdown From Pipelines

696 J. R. CHEN et al.

(Wallis, 1969), neglects all the interactions between the two phases and possesses complex characteristics, i.e. the equations are elliptic rather than hyperbolic, and the model is ill-posed in the sense of Hadamard to initial value problems (Gidaspow, 1974). Apparently, the modelling of the interactions between the two phases is not only important to the physical problems concerned but also crucial for a proper mathematical formulation. Despite the fact that an ill-posed model of an initial value problem violates the causality law (Sursock, 1982) and its solution is unstable (Drew and Flaherty, 1992), stable numerical solutions of these models are possible provided that the numerical diffu- sion inherent in the numerical method, usually the finite difference method, is sufficiently large to damp the instability of the model. The accuracy of the nu- merical solution in this case is uncertain because mesh refinement is not always possible. Furthermore, the complex characteristics render the rigorous deter- mination of boundary conditions of the flow, in par- ticular the choking condition, difficult if not im- possible. The existing work on transient two-phase blowdown is therefore confined to two catego- ries: either the homogeneous model is used, e.g. Lyczkowski et al. (1978), Richardson and Saville (1991) or an ill-posed two-fluid model is used with unknown accuracy of finite difference approximation and empirical or physically unrealistic critical flow conditions, e.g. Solbrig et al. (1976) and Hall et al. (1993), among others.

Well-posed models are still possible but are usually limited to a small range of two-phase flow, e.g. bubbly flow. For example, Lahey et al. (1992) has developed a well-posed model for bubbly flow by taking into account all the two-phase interactions through a cell model ensemble-averaging method. However, the ex- tension of this model to other flow patterns or high void fraction regime is still far from complete. Re- cently, an entirely different approach has appeared which uses an extended form of Hamilton's principle to model the two-phase interaction. Hamilton's prin- ciple simply states that the motion of a body is a result of minimisation of the time integral of the Lagrangian defined by the difference between kinetic energy and potential energy. Based on the variation of the Lagrangian, the Euler-Lagrange equations can be derived which can be combined to give the equations of motion of the system.

The application of Hamilton's principle in two- phase flow problems was initiated in the work of Bedford and Drumheller (1978) for developing a theory of mixtures. The full potential of the varia- tional principle in studying the interactions or the inertial coupling in two-phase flow in a systematic way was first realised by Geurst (1985a, b, 1986) using an explicit expression of averaged kinetic energy den- sity containing an extra term related to the square of relative velocity and an unknown function which he called the virtual mass coefficient. Subsequent studies by Wallis (I 988, 1990a) show that Geurst's variational model is the only model to satisfy all the basic tests for

a proper macroscopic theory of two-phase flow. Later, Pauchon and Smereka (1992) show that Geurst's model of the equations of motion can indeed be recast into a form which is compatible with the conventional averaging formulations except that all the closure laws are determined by the unknown func- tion. Clearly, the variational principle can provide an effective way for developing and testing specific closure laws required by the averaging models.

In this paper, we will concentrate on the modelling of inertial coupling in two-phase flow using a modi- fied variational principle based on Geurst (1985a, b, 1986). In particular, we focus on the development of the energy conservation equation and equations of motion for compressible single- or multi-component vapour-liquid mixtures using a thermodynamic equi- librium assumption. The characteristics of the model are analysed. Constraints of well-posedness and stable wave propagation lead us to propose a class of model which is marginally stable by using an adjusted struc- ture parameter. The physical significance of this para- meter is discussed and an expression is also proposed based on a wide range of two-phase critical flow and pressure pulse propagation data. The result is a com- plete set of hyperbolic equations, numerical solutions of which for blowdown problems will be presented in a second paper (Chen et al., 1993a).

2. DEVELOPMENT OF TWO-PHASE FLOW MODEL

2.1. Basic definitions and assumptions o f vapour-liquid mixtures

Consider a multi-component vapour-liquid mix- ture in which both phases are allowed to be compress- ible. The two phases can depart from the equilibrium state depending on their thermal and flow states. We shall assume here that the two phases are in thermo- dynamic equilibrium when the flow is homogeneous but retain non-equilibrium effects from non-homo- geneity of the flow. The thermodynamic equilibrium assumption will ensure the maximum possible mass transfer rate during any phase change process and significantly simplify the requirement of modelling the interracial heat/mass transfer processes into a simple phase equilibrium calculation. Although the neglected thermodynamic non-equilibrium effects such as de- layed bubble nucleation in a superheated liquid can also be considered under the framework considered here, additional relations must be supplied in the form of interfacial transfer closure laws. However, there is little information about these closure laws except for the simplest cases such as a steam-water mixture. The thermodynamic equilibrium assumption is therefore a useful and probably inevitable assumption for studying complicated two-phase systems such as hy- drocarbon mixtures. The same assumption has also been used in other dynamic two-phase flow simula- tion for multi-component hydrocarbon mixtures, e.g. Bendiksen et al. (1991).

When the flow is non-homogeneous, the averaged thermodynamic states for each phase may differ from

Page 3: Two-phase Blowdown From Pipelines

Modelling of two-phase blowdown from pipelines 691

each other. The non-homogeneity can be caused by two types of dynamic effects, namely non-uniform phase distributions and local velocity slip between the

two phases. In a one-dimensional macroscopic flow field, the non-homogeneity is represented by the aver- aged relative motion uo - uL, where uG and uL are the cross-sectionally averaged velocities of the vapour and liquid phases, respectively. By including an ex- plicit form of flow non-homogeneity or Reynolds stress in the kinetic energy of the two-phase system,

we shall show later that the departure from equilib- rium can be expressed explicitly in terms of the flow non-homogeneity or uG - uL. Regard the averaged state of vapour as the equilibrium state. The averaged mass densities of the vapour and liquid denoted by pG and pL are both assumed to be functions of the aver-

aged thermodynamic pressure pc, averaged thermo- dynamic temperature To, and individual mass frac- tions yoi,i=r, ,N and yLi,i= ,, ,N. Any departure of the averaged liquid state from equilibrium with the vapour state is only caused by dynamic effects, i.e. by the relative motion between the two phases. The

liquid is therefore attributed a dynamic temperature TL and a dynamic pressure pL in which T, = TG and pL = pG when uL = uo. The same equilibrium assump- tions are also applied to other thermodynamic vari- ables such as chemical potential pki, etc.

Let tl denote the volume fraction of the vapour

phase, or the so-called void fraction. We define the following reduced densities of vapour and liquid phases:

Pl = (1 - dPL> P2 = WC;. (1)

Similar expressions can be given for the entropy dens- ities and reduced densities of component i. We have

s1 = (1 - COPLSL, s2 = WG~G, (2)

Yli = (1 - E)~LYL~, Y2i = NpGYGi, i= l,...,N,

(3)

where sir is the averaged specific entropy of phase k. The total internal energy density U of the mixture is

given by

u = PlhL + P2hc - EPG - (1 - ~PL

=SITL+S~TG +CPL~YI~ I

•CPG~Y~~-~PG-(~ -~)PL. (4) I

Note that the validity of the above expression requires that the fluctuation of internal energy and other fluc- tuations of thermodynamic properties are effectively zero. Additional closure relations must be supplied if these fluctuation terms are not zero and in most cases they are not known. We therefore confine ourselves to the above simple expression and neglect all the fluctu- ations of thermodynamic properties. By using the

Gibbs-Duhem equation given by

1 1 yki dpki = - st dT, + m dp,, k = G or L (5)

CES 50:4-J

one gets

dU =CpLidYli + CpoidYZi + TLdS1 1 I

+ TcdS2 -(PC - PL)~U. (6)

This expression is important in performing the varia- tional analysis. Note that Geurst (1985a, b, 1986) used a single pressure pG for both phases and the last term in the right-hand side of eq. (6) does not appear. This term, however, is necessary to the formulation of

a compressible liquid phase. The interfacial mass, entropy and species transfer

rate are defined as

^ + -(%?+F) (7)

!!!$= -(2+%) (8)

8 Y2i 8 Y2fuG ry;‘dt+t= -

C?Z

i= l,...,N. (9)

Combining each equation for the two phases gives the conservation equation of total mass, entropy and spe- cies for non-dissipative flow:

$1 + p2) + ;(pI"L +pZUG) = o (10)

;@I + SZ) +&,uL + sZ"G) = 0 (11)

i( Yli $ Y2i) + f ( YliUL f YziUG) = 0,

i = 1, . , N. (12)

For dissipative flow, eq. (11) becomes an inequality. Note that mass and thermal diffusions are insignifi- cant compared to convection during the transient process and are therefore neglected in all the dis- cussion here. The flow will be assumed to be inviscid: viscous or other dissipative effects will be taken into account later by using quasi-steady-state correlations,

e.g. algebraic viscous drag laws. When the fluid contains only one-component,

eq. (12) is no longer necessary and the Gibbs-Duhem equation reduces to

dpk= -sxdTx+Idp,, k = G or L. (13) pk

Equation (6) also reduces to

dU = (1 - a)pLdpL + LY~G~PG + TLdS1 + TodS2

+ (PGPG -PLPL-PC +PL)~@. (14)

2.2. Kinetic energy of two-phase flow Following Geurst, we assume that the averaged

kinetic energy density K can be written as

K=:p,ut+:p2U~+:m(a)p1(UG-_L)‘. (15)

Page 4: Two-phase Blowdown From Pipelines

698 J.R. CHEN et al.

The first two terms represent the averaged kinetic energy for the two phases at their centres of mass in a volume element. The last term takes into account the kinetic energy associated with the possible velo- city fluctuation in the liquid phase due to the pres- ence of the vapour phase. The last term includes the effects such as virtual mass acceleration of vapour bubbles having a drift velocity relative to the liquid. For dilute non-interacting spherical particles, it is well known that m(ct)= ½~t [e.g. Smereka and Milton (1991)]. Clearly, this form may not necessarily repres- ent the correct kinetic energy due to relative motion in other flow regimes such as annular or stratified flow where the interactions in the two phases are less significant. However, under fast transient conditions, the two phases are strongly coupled and such flow patterns usually do not occur. A similar form of kin- etic energy is also used by Lhuillier (1985) in a differ- ent approach to two-phase flow modelling. However, Lhuillier's approach ends up very similar to that of Geurst.

The coefficient (1 - ct)m(ct) is called the virtual mass coefficient by Geurst. The analysis of Smereka and Milton (1991) suggested the name Reynolds stress coefficient for m(~) based on the analogy of single- phase turbulent flow. Wallis (1990a) called the func- tion m(ct) exertia. In this work, the suggestion of Smereka and Milton (1991) will be adopted. One should note that the above expression also neglects the kinetic energy associated with bubble pulsation which is only important near the bubble resonance frequency. The kinetic energy associated with the ve- locity fluctuation of the vapour phase due to the presence of the liquid is also neglected. This is justified because Pc is always much smaller than PL except at very high pressure and, in particular, in the critical region.

2.3. Hamilton's principle and equations of motion for one-component systems

First of all, consider a one-component system. Hamilton's principle for a one-component system has been studied by Geurst (1985b). The present formula- tion is different from that of Geurst since the liquid is allowed to be compressible. Nevertheless, the equa- tions of motion turn out to be the same. The deriva- tion here will be limited to one-dimensional flow only. The derivation of a three-dimensional form is similar to that given in Geurst (1986).

The Lagrangian density of the system is written as

L = K -- U. (16)

According to the extended Hamilton's principle (Geurst, 1985a, b), time and space (in the present treat- ment the axial or z coordinate) integration of Lagran- gian of the system is a minimum. In the variations, the constraints such as mass and entropy conservations must also be satisfied. The constraints of mass and entropy conservations are introduced through the use of Lagrange multipliers. One has therefore the follow-

ing variational principle:

6 dt dz £ = 0 (17) o 0

where

£ = L + ~o ~ t (p , + p~) + ~z(p,uL + p~uc)

+ ~l -~(S~ + S2) + (S,uL + S2uc) . (18)

Euler-Lagrange equations may be obtained by per- forming independent variations of the dependent vari- ables. There are two possible choices of these depen- dent variables. One can simply choose the thermo- dynamic variables Pc, Tc and the flow variables uc and UL. Alternatively, one can use the combined vari- ables, Pl, P2, S1, $2, UL and uc, as done by Geurst. The former choice inevitably involves complicated thermodynamic derivatives of density and entropy. Therefore, the latter approach is used. However, as the liquid is assumed to be compressible, ~, Pc and PL, will be used as independent variation variables in- stead of pl and P2 to account for the additional degree of freedom. Also, no variation of pressure, temper- ature and chemical potential is necessary since the Gibbs-Duhem equation has already been used. Per- forming independent variation of ~t, PL, PC, $1, $2, UL and uc, gives the following Euler-Lagrange equa- tions:

6~: - ½pLu~ 1 2 _ + :pcuo ½[re(a) - (1 - ct)m'(g)]pL(u~ - UL) 2 -4- ItLPL

-- Itc, Pc + Pc -- PI. + PL ( ~--t + UI. ~--Z )

-pc(~+uc~)=O (19a)

6p:: ½u~ + ½m(~)(u~ - u:) ~

- # L - - ~ - + u ~ - z = 0 (19b)

2 u G - # o - ~ - + u ~ = 0 (19c)

C 8Sl: TL + ~+UL~-~Z = 0 (19d)

c%/ uo dt/'] = 0 (19e) 6S2: TG + -~ + dz /

dqJ ~tl bUL: plUL -- plm(ot)(uo -- UL) -- Pl ~Z -- $1 ~z = 0

(19f)

~uc: p2uc +plm(~t)(uc -- uL) -- P2 - $ 2 ~ Z = 0

(19g)

Page 5: Two-phase Blowdown From Pipelines

Modelling of two-phase blowdown from pipelines

where the prime denotes a derivative with respect to ~c 1 + - - ( F , - F,,sD Substituting eqs (19b) and (19c) into (19a) gives Px

1 pL - Po = ~ plm'(u)(ua - uL) 2 (20)

which relates the pressure difference between the two phases to the velocity difference. Also, from eqs (19b)-(19g), it is readily shown that

To-TL = - [ l , +m(~ )p , (~ + l--'lIF(u°-uL)21p,/lL so-so. _I

(21)

and

~ o - - # L = --2p2m(e)

+ c'° • .,so _ s . . . (22)

Equations (20)-(22) show that any departure from thermodynamic equilibrium is a result of dynamic non-equilibrium. These are consistent with our pre- vious definitions. The jumps in the averaged thermo- dynamic states such as pressure or temperature do not necessarily violate the local equilibrium states. For example, the averaged temperatures for the two phases can be different when the phase distributions are different. This is also true when the local slip between the two phases is zero. Even when m(~) is zero, i.e the two phases are uncoupled, a difference in p, T, etc. can still arise from non-uniform phase distri- butions or slip velocity of the two phases. The only case where the differences vanish is when the flow is homogeneous. In this case, uo = uz and m(ct) --* oo. Note that Geurst (1985b) arrives at a slightly different form of eq. (22) due to a different definition of StL: the where difference can be found to be (Po - Pz)/P~ through the Gibbs-Duhem equation.

The equations of motion can be derived from ma- nipulation of the Euler-Lagrange equations. Elimin- ating the spatial derivatives in eqs (19b) and (19c) using eqs (19d)-(19g) gives the generalised Bernoulli theorem

(pl + p2)~t + (S~ + Sz)~

[ (' +llleo-Uq=o x l+p lm(c t ) ~ p z / d k S a - S L /

+ K + U + ~ p a + ( 1 - ~ ) p L = 0 . (23)

Using the pressure difference relation, eq. (20), the above equation reduces to the same generalised Bernoulli theorem as obtained by Geurst (1985a, b). Further manipulations yield the following equations of motion in non-conservative form:

[uL -m(~)(u~ - u , . ) ] + ~ ~ 2 u~ -m(~)tuG -u~)uL

(1 - ~ ) OpL - ½m(~)(u~ - u ~ ) 2 } + - -

Pl Oz

699

(24a)

~ [ u a +p,m(Ct)p= ( u o - u D ]

a f l z mm(~)(u~-uDu~}

ot Opo 1 + (F, - F.,so)

P2 ~z P2

[ x l + p , m ( ~ t ) -~ ~ #so-sL /

(24b)

By using eq. (7), eqs (24a) and (24b) can be recast into a form which is equivalent to the averaged equa- tions of Pauchon and Smereka (1992):

~ [(1 - ot)pLuL] + Oz [(1 -- ¢t)pLu~

+ (1 - - OOpLm(Ot)(U o -- UL) 2]

+ (1 - ~ )~z z + ½(1 - ~t)ptm'(~t)(u ~ - uL) 2

6(1 - ~) x - - + FuL -- Mi a = 0 (25a)

0z

~ 2 Opo a-t{~poud + ~[~pou~] + • ~ + r ~ + Mf = 0

(25b)

Mi a = ~ [(1 - .)pLm(~)(uo - uL)]

+ ~z [(1 - ot)pLm(OO(U o -- UL)Uo]

+ (1 -- ~)pLm(a)(uo -- uD ~ - ; (25c)

FML = Fm[UL -- m(o t ) (Uo -- uL)l + (rs -- FmSL)

x [1 + Ol in (a ) (1 + 1 ) ] ( ~ )

--- -- FUL. (25e)

No assumption of incompressible liquid is required in

Page 6: Two-phase Blowdown From Pipelines

700

recasting the above equations owing to the introduc- tion of a separate pL. Equations (25a)-(25e) are the generalised equations of motion for a one-component two-phase flow with phase change. They are identical to those derived by Geurst (1985b). It is therefore concluded that compressibility of the liquid has no effect on the form of the equations of motion. The effect of interfacial mass transfer on the equations of motion is uniquely determined by the above model in eqs (25d) and (25e).

It is interesting to compare the present model with the conventional averaged two-fluid model. For example, the three-dimensional ensemble-averaged momentum equation can be written as (Lahey and Drew, 1992):

~ p i u i - - + V . OtkPkUkU k -~- - - OtkVPk + (Pki - - pk)VO~k

Ot

+ FkUki + V" ~k(Zk + ~ )

+ M~ (26)

with the momentum jump condition given by

(puVOtk + FgUkl + M'~) = M'[ k=C,L

J. R. CHEN et al.

Pauchon and Smereka (1992) except that the term M~ is called the symmetric virtual mass acceleration. Note that Wallis (1990) has shown that M~ contains more than just the virtual mass acceleration. Finally, the equation of motion of the mixture can obtained by adding up eqs (25a) and (25b) and replacing PL by eq. (20):

0 2 -t[_otpauc + (1 - OOpLUL] --}- -~z [OtpGuG at- (l -- O~)pLU 2

+ (1 - :z)pLm(ot)(ua -- Uz) 2]

+ ~ + 1 ~? ~z [pL(1 -- o'~)2m'(~)(ua -- uL) 2] = O.

(29)

2.4. Equat ions o f mot ion for mul t i -component sys tems The extension of the above analysis to a system

containing multi-component mixtures is straightfor- ward. The only difference between a one- and a multi-component system is that the latter has more degrees of freedom and requires extra variations in performing the analysis. Extra constraints of conser-

(27) vation of each species, eq. (9) must also be satisfied in performing the variations. The Euler-Lagrange equa- tions are obtained by performing independent vari- ations of ~, PL, Pc, Sa, $2, Y l i , i = 1 ..... N, Y2i , i = 1 . . . . . N, UI.. and uc:

3ct: -- • ~ u 2 1 2 2PL L + 2 p a u c -- ½[m(~)

-- (1 -- ct)m'(~)]pL(Uc -- UL) 2 + PC -- PL

+pL -~+UL~z - p c ~ + u ~ =0

(30a)

bpL: i u 2 (a~O ~ ) 2 L W ½ m ( o O ( t t c - - U L ) 2 - " f f ~ + U L = 0

where ~k, Tk Re, Pki, M'~, FkUki and M 7 are the shear stress, fluctuating momentum flux or the Reynolds stress, interfacial pressure, interfacial force density, interfacial momentum flux and surface tension force density, respectively. For convenience, all the aver- aging notations have been neglected. Comparison of eq. (26) with (25a) and (25b) reveals that

Pal = PLi = Pc = PL -- ½(1 -- ~)pLm'(OO(Uc -- UL) 2

¢ ~ = -- m(oOpL(U G -- UL) 2

~ e = o 128)

"I' c = "r L = 0

M~ = - M t = M~

F c u c i = - - FLULi = F M C = - - F M L .

All the closure relationships are uniquely determined and depend on known flow variables and one un- known Reynolds stress coefficient m(~). Equation (28) also shows that the interfacial pressure difference, the velocity fluctuations and the interfacial force all have the same origin of stress from the relative motion between the two phases. The zero gas-phase Reynolds stress is consistent with the assumption made in the expression for the averaged kinetic energy. The van- ishing shear stresses are also consistent with the in- viscid assumption made earlier. The momentum jump condition, eq. (27), is also embedded in eq. (28) with M~' = 0. This is expected since surface tension force is not considered in the present framework. The above analysis shows that the Euler-Lagrange equations provide not only the equations of motion of the sys- tem but also the closure relations that completely close the system. Similar comparisons with volume- averaged momentum equation are also shown by

6p6: j. 2 (a tp Otp) 2 u c - -~-+ucff- z =0

3Sl: TL+ +uL~z = 0

~$2: To+ -~-+uc~ =0

6Y,,: ~L,+ \ ~ + uL~/=o,

6Y2i: #a~ + + u~ c~z ] = O,

(30b)

(30c)

(30d)

(30e)

i = 1 . . . . . N

(30f)

i = 1, ... ,N

(30g)

t3cp 6UL: p l U L - - p l m ( O O ( U a - uL) - 1 0 1

Oq O~i - - S l - ~ , E l i = 0 7z "7

(30h)

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Modelling of two-phase blowdown from pipelines

&o bU~: p2Ua + plm(~t)(u~ -- UL) -- P20~

Orl - ~ Y2i O~ ̀= 0. (30i) - S2 ~ az

Substituting eqs (30b) and (30c) into (30a) gives the same pressure difference relation, eq. (20). Also, from eqs (30d) and (30e), it is readily shown that

T~ -- T L = (u~ - UL)~Z. (31)

Similarly, from eqs (30f) and (30g) one gets

# O i - - # L i = ( U ~ - - U L ) ~ z , i = 1 . . . . . N. (32)

Again, eqs (31) and (32) show that any thermodyn- amic non-equilibrium is a result of dynamic non-equi- librium. These relations are consistent with the pre- vious development of a one-component mixture. However, it is no longer possible to express the tem- perature and chemical potential differences explicitly in terms of velocity difference. Combining eqs (30f)-(30i) gives the following N + 1 relationships:

-- SL( Ta -- T L ) , - 2 yLi(I.t¢,i -- #Li) i

= ½[1 + m(~)] (u~ -- UL) 2

s ~ ( T ~ - TL) + ~ y ~ ( ~ -- ~ ) i

(y~j - y~j)[½(u~ - ug) - ½ m ( ~ ) ( u ~ - u~) ~]

-- (yLjUo -- yajUL)(U6 -- UL)

-- (Y~j + YLJ ~2)m(a)(U~ -- UL)2

+ (yLjsa -- Y~iSL)(Ta -- TL)

+ ~ (YLjYoi -- Yo~YLi)(IZa~ -- #U) = O, i ~ j

j = l . . . . . N - 1 . (33c)

The generalised Bernoulli theorem for a multi-com- ponent two-phase mixture is

(Px + P2)-~ + (Sl +S2) + ~(Yai + Y2i) i

+ K + U + ~ t p ~ + ( t - - a ) p L = 0 .

The equations of motion can be shown to be the same as the one-component counterparts except that the interfacial momentum flux terms become

r u L = F . , [ U L - - m ( ~ ) ( u ~ - - UL)] + ( r , -- r . s ~ )

x [1 + Pa m(a' ( 1 + -~2/dl'~](u°--UL'~\ so - s---~/

701

- - ~ [ F y , - F m y L , - ( F ~ - - F m S L )

×(r°,- (35a) \ s o SL / d \ UG -- UL /

F M ~ = - - F , . [ u a + P l m ( a ) ( u a - u L ) - ] - ( F ~ -

(35b) \ s~ SL / A k UG UL /

The extra effect of different volatilities of the fluids, YG~- YL~, on the interracial momentum transfer of a multi-component mixture appears in the last term of FMk. Note that chemical potentials cannot be ex- pressed explicitly in terms of relative velocity but must be solved simultaneously for all the components from eqs (33a)-(33c).

2.5. Energy conservation equation of mixtures The energy conservation equation of two-phase

mixtures can be derived from Noether's invariant (33a) theorem as [see Geurst (1985b) and Appendix]

~ [ p l hL + p2hf; - otp(r, - (1 - a)pL + ½PlU 2

(33b) + z 2 2P2U6 + ½m(~)Px(Ua -- UL) 2]

1 2 ½m(~) - u L ) : ] + ~ z { U L [ p l h L + 2PlUL + p t ( u a

1 2 + u6[p2h6 + 2p2u6]}

+ Ozz [m(a)pl (uo - UL) z U~] = 0. (36)

Note that eq. (36) is valid for both one- and multi- component systems. Here we are interested in the equivalence between the present and conventional averaging formulations. For example, the ensemble- averaged energy conservation equation of mixtures is (Lahey and Drew, 1992; Drew and Wallis, 1992):

1 2 eRe~ [ap6(h~ + ½u 2 + e~ e) + (1 - ct)pL(hL + 2UL + L J

- ~ P G - ( 1 - ~ ) P L ]

(34) a Zu2 + e~ e) + ~z[aPouG(h~ + 2 G

& 2 + (1 -- OOpLUL(hL + 2UL + e[e)]

- - - - [gz~ 'uG + (1 -- g) z~eUz -- q~ Oz

--qL g - q ~ - q [ ] = 0 (37)

where one-dimensional flow is assumed as well as that

Page 8: Two-phase Blowdown From Pipelines

702

there is no gravity, shear stress, internal heat source, external heat flux and interfacial energy source from surface tension, e~ e, qk x and q~ are the fluctuation kinetic energy, fluctuation kinetic energy flux and fluctuation internal energy flux, respectively. The equivalence of these two formulations becomes more clear if the last term on the RHS of eq. (36) is rear- ranged as

~z [m(a) Pt (ua - uL) z ua] = ~z [m(~)pl (uo - u t ) 2 UL]

d + ~Z [m(~)pl (uo -- UL)3]. (38)

One immediately infers the following closure relations for eq. (37):

e~ e = 0

e~ e = ½ m(z )p l ( ua - Uz) 2

q~ = q~ = q~ = 0

q~ = m(oOpl (u~ -- UL) 3 (39)

$Re z = -- m(g)pL(uc; - - UL) 2

~ge = 0 .

These relations are consistent with the closures of the momentum equations. These relations also reveal that the last term ofeq. (36) is the summation of two effects: the flux of fluctuating kinetic energy of the liquid qL x and the product of liquid phase Reynolds stress and mean velocity (1 - g)T~euL. Both terms originate from the fluctuation of kinetic energy flux:

½(u~ukuD = ½ ( ( ( u D + u ; , p )

t ¢ t t 1 t ~ t = (Uk)(½(Uk) 2 + ½(ukuD) + ( u D (ukuD + 2(UkU~Uk)

~ Tr Re Re q~. related to e k U k ' ~ k

The only surprise is that the factor ½ in the definition K ofqr. disappears in the closure relation in eq. (39). This

is however correct and can be verified from the kinetic energy equation.

To derive the conservation equation of kinetic en- ergy of the mixture, we take the sum of the product of UL with the terms in eq. (69) and the product of u~ with the terms in eq. (70). After lengthy manipulation, one arrives at:

i 2 1 2 -~t [2ptUL + 2P2UG + ½m(o~)Pl(U~ -- Ul.) 2]

c3 ri. U3 _~_ ½ p2U3"] (~ + ~z,~,.,~ L + ~z(½m(~)m(u~ - uL)~u~

+ m(oOp~(ua -- UL)2Uo} + ½m'(oOp~(ua -- UL)2~tt

dPo dPL + ~ u ~ - ~ z + (1 - ~)u~-fz + (u~ - UL)rU~

+ F,[½u~ - ½u~ -- ½m(~)(ua - u~.)Zl = 0. (41)

J. R. CHEN et al.

One notes that the last term in eq. (36) also appears in the kinetic energy equation of the mixtures. Subtract- ing eq. (36) from eq. (41) and using the relation for pressure difference gives

~-~[p2h6 ~3P6- ( 1 - ct) ~tL + plhL] -- ot--~-

c~ gp~ + 7 - [ p 2 u ~ h ~ + plULht] --

O2

OPL - (1 - ~)Uz ~ - z -- (U~ -- u L ) F ~

F r l u 2 I 2 - mL2 ~--2UL--½m(0t)(U~--UL) 2 ] = 0 . (42)

Equation (42) is similar in form to the energy conser- vation equation commonly used in the literature, both from averaging and phenomenological models; e.g. Ishii (1975) and Yadigaroglu and Lahey (1976). Equa- tion (42) is applicable to both one- and multi-compon- ent systems provided the correct relation is used for FMa. Using the definition of entropy and some lengthy algebraic manipulations, it is possible to show that eq. (42) and the entropy conservation equation, eq. (8), are equivalent. The equivalence of entropy and energy equations is a result of the non-dissipative flow assumption. However, the selection of an energy equation rather than an entropy equation renders the conventional extension of dissipative flow using steady viscous boundary layer approximation straightforward.

One should note that the mixture energy conserva- tion equation alone is enough to close the present

(40)

model where thermodynamic equilibrium between the two phases is assumed and the degree of non-equilib- rium resulted from flow non-homogeneity can be ex- plicitly calculated from eqs (20)-(22) or eqs (31)-(33). If, however, other thermodynamic non-equilibrium is present, extra information such as another energy equation relating to the non-equilibrium states must be supplied in order to close the model. In this work, we will neglect all thermodynamic non-equilibrium such as delayed nucleation in superheated liquid or delayed condensation in subcooled vapour. During the change of state of the system, e.g. heating, the equilibrium assumption will ensure the maximum possible mass transfer rate while non-equilibrium theories, provided that they exist, predict finite rate of transfer. Therefore, provided that the relaxation time of thermodynamic equilibrium is shorter than the relaxation time of the overall system, thermodynamic equilibrium assumption is justified, i.e. the irreversible transfer processes can be well approximated by revers- ible transfer processes. As will be discussed in Part II

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Modelling of two-phase

of this paper (Chen et al., 1993a), these non-equilib- rium effects are only important in the very early stages of the blowdown process and, once phase change occurs, non-equilibrium states quickly approach equi- librium ones. We also would like to stress that, once the thermodynamic equilibrium assumption is made, there is no need to calculate F,,, Fs or Fr~ as only mixture species, mass and energy equations are enough to close the system completely. Instead, F 's can be calculated from eqs (7)-(9) if necessary. Details of the calculation schemes are given in Chen et al. (1993a).

In practical calculations, the phase enthalpies and other thermodynamic properties are determined by equilibrium flash calculations. However, the liquid enthalpy and pressure in momentum and energy equations may deviate from equilibrium with the va- pour state as a result of dynamic non-equilibrium. It is desirable to convert these variables into equilibrium variables explicitly in the conservation equations. For liquid pressure, the pressure difference relation, eq. (20) can be readily used. For liquid enthalpy, some manipulations are necessary.

Gibbs equation for enthalpy density can be written a s

d(plhL) = Tzd(plsz ) + u~.dPx + (1 - ~)dpL.

(43)

At equilibrium conditions, we have

d(plh~.) = Tcd(pls~.) + pcdPl + (1 - ~t)dpc (44)

where the superscript e denotes equilibrium state. The above equation also assumes that the variation of liquid density is negligible. Subtracting the two equa- tions gives

d(plhL) = d(plh~) + TLd(plSL) -- Tcd(plseL)

+ (PL -- Pc)dPl + (1 - ct)d(pL -- PC)-

(45)

Similarly, one can also derive a Gibbs equation for the enthalpy flux:

d(plULhL) = d(plULh[) + TL d(plULSL)

-- Tcd(pluLSeL) + (PL -- IIc)d(pxUL)

+ (1 -- Ct)uLd(pL -- PC)" (46)

Therefore, the mixture energy equation, eq. (42), can be recast as

ff_~[pEhc 63Pc . 63 + plheL] -- ~-~ + ~Z [-Pmuchc + paULheL]

Opt -- [~uc + (1 -- OOUL] C3Z

= r ~ - ~ ( p , ( s ~ - s [ ) ) + N ( p ~ u ~ ( s L -- ~))

- - ( T o - - TL)F~ + (~L -- ~C) r . , - - ( u c -- UL)FMC 1 2 1 2

- F,,[mUo - 2UL -- ½m(ot)(uc -- uz) z] (47)

blowdown from pipelines 703

where F[ is the equilibrium entropy source term de- fined by

0 - ~ (Pl uLs~). (48) r ~ = - ~ (p~ s,.)

The only term unknown in eq. (47) is the non-equilib- rium liquid entropy. The departure from equilibrium of the liquid entropy can be determined by the follow- ing thermodynamic relation:

/ \ CpL / d (1 /pL) / dpL+ dTL. (49) dSL =

The first term on the RHS ofeq. (49) can be neglected since the liquid is nearly incompressible and PL is approximately constant. Therefore, the entropy de- parture from equilibrium can be approximated by

SL -- S~ = ASL ~ -~L ( TL -- TG). (50)

The model is therefore completely closed in the frame- work of equilibrium thermodynamics. By defining various mass and entropy source terms, the energy equation, eq. (47), possesses a similar structure of derivatives as the simple one-pressure Wallis model (Wallis, 1969) and can be solved directly using the simplified numerical method proposed by Chen et al. (1993b).

3. CHARACTERISTICS AND WAVE PROPAGATION

ANALYSIS

The number of characteristics associated with the model is the same as the number of independent differential equations. Since the present model as- sumes isentropic flow, the energy equations can be decoupled from the mass and momentum equations. The effect of interfacial mass transfer on the mo- mentum equation for the present model is given by eqs (25d), (25e) and eqs (35a), (35b) for single- and multi-component systems, respectively. None of these terms contains derivatives and thus have no effect on the characteristics of the system. Complete character- istics analysis of the mass and momentum equations is still very complicated because of the unknown Reynolds stress coefficient. The characteristics asso- ciated with the mass conservation equations are the sonic characteristics which are related to the com- pressibility of the fluid. The characteristics associated with the momentum equations are the void/concentra- tion wave or simply the void wave characteristics. In the following, the sonic and void/concentration wave characteristics will be analysed separately. Note that Geurst (1985a) has carried out an elaborate character° istics analysis using linear stability analysis.

3.1. Analysis of pressure wave propagation For linear hyperbolic systems, any discontinuity or

shock propagates only along the characteristics of the system. For non-linear hyperbolic systems, the dis- continuity does not necessarily propagate along the

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704 J. R. CHEN et al.

characteristics. Instead, any propagating discontinu- ity is characterised by the Rankine-Hugoniot jump condition where the propagation speeds usually differ from the characteristics of the system. When the mag- nitude of the discontinuity becomes infinitely small, the speeds of propagating discontinuities approach the characteristic speeds of the non-linear system. Therefore, it is possible to study the characteristics of a non-linear system by just looking at the propaga- tion speed of an infinitesimal perturbation without the need of solving the complicated eigenvalue problem. Sergeev and Wallis (1991) were the first to utilise this idea to investigate small pressure pulse propagation in a uniform two-phase media. Although their analysis cannot be extended to the case of non-uniform flow, the analysis provides important information regard- ing the upper and lower bounds of inertial coupling between the two phases which is crucial in determin- ing the Reynolds stress coefficient.

Based on the assumption of incompressible liquid, Sergeev and Wallis (1991) obtained the following ex- pression for the propagation speed D:

(D--u____~°) 2 = P6 PL(O~2 +m(oO)+ct(1--0OP6

\ a~ / ~PL ctp~(m(oO + 1) + (1 - ct)pLra(ct )

(51)

where a6 is the speed of sound of vapour and u ° is the homogeneous velocity in front of the shock. Since PL >> Pa for general cases, one gets

( D - u°~ 2 pG(cd + m(ct)) (52)

a - - T - / -

Clearly, in order to have real and finite propagation speed, we require that re(e) must be positive.

Figure 1 compares the experimental data on pres- sure pulse propagation of Ruggles et al. (1988) and Semenov and Kosterin (1964) with the propagation speed predicted by eq. (51) using two different func- tion forms for m(ct): the Maxwellian exertia, m(ct) = ½~, which is the virtual mass coefficient of spherical particles in the dilute limit, and homogene- ous flow, m(ct) = oo. For completeness, the speed of sound data of Karplus (1958) are also included. The speed of sound in two-phase media is frequency-de- pendent and will approach the speed of the pressure pulse when the sound frequency approaches infinity. Figure 1 shows that the exact degree of inertial coup- ling in dispersed two-phase flow lies between those given by Maxwellian exertia and homogeneous flow, i.e.

½c~ < m(~) < oo (53)

with the preference given to the Maxwellian exertia. In particular, the data of Ruggles et al. (1988) fit quite closely to the prediction using Maxwellian exertia. Figure 1 also indicates that decreasing two-phase coupling or interaction leads to an increase in the wave speed. It is also interesting to note that in an- other extreme case where there is no interaction at all between the two phases, e.g. in smooth separated flow, the pressure wave speed will bifurcate into the speeds of sound in the gas and liquid phases (Nguyen et al., 1981). When measuring the wave speed of such sys- tems, only the slower sonic speed of gas is measured

0.5

0.45

0.4

K 0.35 c-

o

0.3

0

~. 0.25

..1

-u 0.2

0.15

0.1

005

• Semenov, P=1.25 bar

o Karplus, P=lbar

o Ruggles, P=-lbar

- - Maxwellian Exertia

. . . . . . Homogeneous

A

I I t I

0 0.2 0.4 0,6 0.8 1

Void fraction

Fig. 1. Effect of Reynolds stress coefficient on pressure pulse propagation. The predictions are based on a gasfliquid density ratio of 0.00139 for an air-water system.

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Modelling of two-phase blowdown from pipelines

[e.g. Evans and Gouse (1970)]. In two-phase blow- down from pipelines, non-interacting separated flow is rarely encountered and is therefore not considered here.

3.2. Analysis of void wave propaoation The characteristics associated with the void/con-

centration wave can be studied separately by assum- ing both phases to be incompressible and using the concept of drift flux (Pauchon and Smereka, 1992). The drift flux is given by

J = 0t(ua --J0) = ~t(1 - ct)u, (54)

where jo is the volumetric flux given by ctua + (1 - a)UL and u, is relative velocity given by ua - UL. Using the following coordinate transformations:

z * = z - j o t , t * = t (55)

both mass conservation equations are reduced to

c3~ 0J &--~ + ~z* = 0. (56)

The kinetic energy can therefore be expressed in terms o f :t a n d J

PL p6 + pLm(Gt) K = ½F(a)J2; r ( a ) (1 - a ) ~- a ( ~ - ~ - 2 "

(57)

The variational principle now reads

. dt d z L = O ; L = K - d p ~ + ~ z * " d t o d Z~

(58)

The Euler-Lagrange equations for variations of • and J are

0¢ 6a: ½F' (a ) J 2 + ~-~ = 0 (59a)

6,I: J F + az ~ = 0. (59b)

Eliminating q~ in eq. (59a) by using eq. (59b) gives

0-~, J F - - ~ , (½ r ' J 2 ) = 0. (60)

Equations (56) and (60) form a reduced system of the variational model based on an incompressible as- sumption for both phases. The characteristics of the reduced system are given by

J F ' _ x± = ~ [ - +(r '~-½rr') ' /q. (61)

The condition for the reduced system to be hyperbolic is

2F '2 - FF" / > 0. (62)

When the equality is satisfied, the second-order differ-

705

ential equation has the following solution:

F = - e l / ( ¢ 2 + 0~) (63)

where cl and e2 are the integration constants. By using the definition of F we get

c1(1 - a)ct 2 (1 - ~t)0~p~ rn(~) :t 2. (64)

(c2 + ~)PL pL

In the dilute limit of spherical particles one knows that m(a)/~ --* l/2 as ~ ~ 0. Therefore, we choose

ci = - ½thpL-- Pc (65)

where th ~ 1 when ~ --* 0. rh will be called the inertial coupling constant. The constant c2 can now be deter- mined by eq. (62). It turns out that the only possibility to satisfy the inequality of eq. (62) is cz = 0. One can therefore obtain the condition

m(~) = ½~t(rh - (d~ + 2)~) (66)

in which the equality in eq. (62) will be satisfied. Equality of eq. (62) also implies that the two complex conjugate characteristics will collapse into one real root. The flow is therefore in a neutral or marginally stable state. The same marginal stability condition is also obtained by Geurst (1985a) using linear stability analysis of the complete variational model and by Lhuillier (1985) using a different thermodynamic for- mulation.

When ~h = 1, eq. (66) reduces to

m(00 = ½~(1 - c3o0 (67)

in which c3 >/3 for stable two-phase flow according to eq. (62). Clearly, the expression for the Reynolds stress coefficient for dilute spherical particles or the Max- wellian exertia, rn(ct)= ½~, is unstable for any finite void fraction. If the Reynolds stress coefficient is set to zero, the model reduces to the Wallis model and the void wave characteristics are complex for any void fraction according to eq. (62).

Another important feature of the marginal stability condition is that the characteristics of the reduced system in the Eulerian coordinate become

;~ ± = uG (68)

which indicates that the void wave propagates at the constant velocity of the gas phase and is independent of void fraction. It has been observed from experi- ments that void waves in bubbly flow propagate at a velocity lying between the gas and liquid velocities but with preference for the gas velocity [e.g. Matuszkiewicz et al. (1987)]. Lahey et al. (1992) also showed that instability of a void wave in bubbly flow is actually related to bubble-to-slug flow regime transitions. Clearly, the exact modelling of a void wave will be flow-regime dependent and is beyond the current state-of-the-art.

3.3. Analysis of non-linear void wave propagation Equation (68) indicates the possible existence of

a non-linear void wave characterised by the gas velo-

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706

city when the flow is marginally stable. This observa- tion led Geurst and Vreenegoor (1988) to propose the following analysis of non-linear void wave propaga- tion based on constant gas velocity. This analysis turns out to be a convenient way of analysing the marginal stability condition of different forms of Reynolds stress coefficient without the need to resort to the characteristics analysis of the complete model. We now recast their analysis in terms of the present variational principle in which the two phases are both compressible.

The propagation equation of the non-linear void wave characterised by constant gas velocity and gas density can be written as

0~ 0~ g t + ua ~zz = 0 (69)

which is reduced from the gas mass conservation equation. The solution for a has the functional form

= f ( x - uat). Combining with the liquid mass con- servation equation gives

C uL = uG - - - (70)

1 - ~

where c is a constant velocity. For simplicity, it is assumed that the flow is isother-

mal and there is no mass transfer between the two phases. The latter assumption relaxes the connection between the chemical potentials of the two phases. The additional degree of freedom is introduced through the introduction of two separate Lagrange multipliers on the constraints of continuity of the two phases. The Lagrangian becomes

FOP1 OpIUL~ FOp2 ..] OP21UG 1 £=K-v+ 'LOt - 0 z j"

(71) Performing independent variations of ~, PL, Pc, u~ and u~ gives the following Euler-Lagrange equations:

2vt L + 2 p ~ u ~ - ½[rn(~) -- (1 - ~)m'(ot)]p~(uo -- uz) 2 + ,uLpz

{ O~o2 o,~2 ~ -- p ~ , - f f ~ + u~--~-z ) = O (72a)

3PL: •2 UL2 + ½rn(~)(u~ -- UL) 2

--/.tz -- k, 0t + U L ~ Z ) = 0 (728)

2 u a - # a \ 0t + u a = 0 (72c)

3ul: Pl uL - plrn(oO(ua - UL)

Oqh -- 01 -~z = 0 (72d)

J. R. CHEN et al.

Ot#2 3u~: p2u~ + ptm(o~)(uG - uL) - p 2 ~ - z = O. (72e)

Substituting eqs (72b) and (72c) into (72a) gives the same pressure difference equation as eq. (20). The same equations of motion as those of Section 2.3 excluding the Fu6 and FML terms can be obtained from combinations of eqs (72b)-(72e). The contribu- tion of/~L from the liquid pressure in eq. (72b) can be separated by using the Gibbs-Duhem equation:

• 2uL2 + ½ [-rn(00 + (1 - ~)m'(~)] (u~ - uD 2

(0q,~ + UL~z ~) =0 (73)

where the modified chemical potential kt* is a function of gas density and gas pressure. We therefore recover the same Euler-Lagrange equation as the formulation of Geurst and Vreenegoor (1988) using an incom- pressible liquid assumption. Since Pc is constant and the flow is isothermal, Pc and therefore/~* must also be constant. Geurst and Vreenegoor (1988) have shown eq. (73) can only be satisfied provided that the terms containing void fraction are constant. With the help of eq. (70), the condition of existence of a non- linear void wave characterised by eq. (69) can be written as

1 ½ [ l + m ( ~ ) + ( 1 - ~ ) r n ' ( ~ ) ] ( l _ g ) 2 ¼(r~+2)

(74)

where ~ is a constant. Equation (74) constitutes a dif- ferential equation for the Reynolds stress coefficient. Solving the differential equation with the condition, m(0) = 0, gives

re(g) = ½~(~fi - (rh + 2)g) (75)

which is identical to eq. (66) and implies ~ = fit. The analysis shows that, by describing the void wave propagation a priori, one can also derive the corres- ponding marginal stability condition of the governing equation without actually going through the detailed stability analysis such as those of Section 3.2. This is particularly useful in analysing the functional depend- ence of the Reynolds stress coefficient as shown below.

In analysing the effect of bubble deformation, Geurst and Vreenegoor (1987) showed an interesting result. By assuming that the Reynolds stress coeffi- cient is also a function of the Weber number defined by

pL(u~ - uL) 2 We (76)

(a/Db)

they found using linear stability analysis that the marginal stability condition becomes

m(~, We) = ½~(rh - (rh + 2)cz) (77)

where rh is not a constant but a function of = We(l - ~ t ) 2 = ObpLC2/ff. Vreenegoor (1990) per-

formed a non-linear void wave analysis based on eq.

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Modelling of two-phase blowdown from pipelines

(69) and found that the Reynolds stress coefficient also satisfies the same condition as eq. (77).

The Weber number is of course a plausible para- meter that will affect bubble deformation. Another possible effect that will also affect the s tructure of the flow is the viscosity and the characterising parameter is the two-phase Reynolds number defined by

DhpL(Uo -- uL) Re = (78)

where Dh is the hydraulic diameter of the flow channel and # is the viscosity of the liquid. Therefore, the Reynolds stress coefficient is now defined as m = m(ot, Re). Viscous dissipation at the bubble sur- face can be assumed to be negligible compared with the kinetic energy of the bulk liquid at high flow velocity. Yet the shear-induced bubble deformation has a significant effect on the fluctuation kinetic en- ergy. The inclusion of the two-phase Reynolds num- ber in the Reynolds stress coefficient is a preliminary way of taking into account the effect of viscous shear on the bubble deformation. To show the effect of Reynolds number on the marginal stability condition, the non-linear analysis of Geurst and Vreenegoor (1988) is used. Performing independent variation of ct, pL and PG, and introducing the modified chemical potential of liquid gives

I 2 2uL + ½[m + (1 -- ~)m~R emR e] (U6 -- uL) 2

- /~* - ( ~ / + UL~zl) = 0 (79)

where

0m(ct, Re) Om( ct, Re ) (80)

m~ = Oot ' mR~ = ORe

The condition for the existence of a non-linear void wave becomes

½[1 + m + (1 - ct)m~ + R e m R e ] - - (1 - ~ )2

- ¼(~ + 2).

(81)

Under the prescribed void wave propagation, eqs (69) and (70), the Reynolds stress coefficient can be recast as a function of void fraction and a constant G0:

m ( ~ , R e ) = m*(ct,(o/(1 - ~t)), ~o = DhpLC/ld. (82)

Furthermore,

d Go ~m*(ct , (o / ( l -- ct)) = m~ + ~ m R e

R e = m~ + mRe. (83)

1 - ~ t

Therefore, eq. (81) can be rewritten as an ordinary differential equation:

½[1 + m* + ( 1 - ~ ) d m * ] ( 1 - - 1 Gt)2 - ~(rh + 2)

(84)

707

which is similar to eq. (74). The solution of above equation satisfying m*(0) = 0 is

m* = ½ct(n~ -(n~ + 2)~). (85)

Note that now fit can be an arbitrary function of Go = Re(1 - ct) which is a function of viscosity that is not considered in the inviscid flow analysis. The above analysis suggests that any functional dependence of the Reynolds stress coefficient other than void frac- tion only appears in the constant rh under the mar- ginal stability condition eq. (66), i.e., when the flow is marginally stable; all the information relating to the s tructure of the flow which is not considered under the non-dissipative, inviscid flow assumption is embed- ded in the inertial coupling constant rh. This im- portant feature leads us to propose the following marginal stability model.

3.4. Marg ina l s tabil i ty model

For the purpose of modelling transient two-phase flow in practical problems relevant to risk assessment, the void wave characteristics are important but not really essential. The most important ones are the sonic characteristics which determine the critical flow rate and therefore must be modelled accurately. Yet any complex characteristics in the model will render the model ill-posed. Therefore, it is assumed that the void wave propagates at the gas velocity whatever the flow regime. The flow regime transitions are specified em- pirically by using a flow regime map. This approach avoids the problem of complex characteristics and arrives at a flow which is marginally stable for the void wave characteristics. The Reynolds stress coeffi- cient is therefore given by the marginal stability con- dition, eq. (66).

The marginal stability condition for the case of spherical particles, rh = 1, gives a negative value of m(~t) when ct > 1/3. The negative value of Reynolds stress gives complex values of sonic characteristics even though the void wave characteristics are real. Geurst (1985a, b) suggests this breakdown of model is associated with the breakdown of bubbly flow, i.e. the flow regime is changed. This is however in contradic- tion to the observations of Lahey et al. (1992) that flow regime transition is related to the instability of void wave characteristics rather than the instability of sonic characteristics. Inspection of the marginal stab- ility condition, eq. (66), suggests that a larger value of rh can shift the breakdown to a larger value of void fraction. The analyses in previous section also show that, when the flow is marginally stable according to the condition eq. (66), all the information relating to the s tructure of the flow that is not considered under the non-dissipative, inviscid flow assumption is em- bedded in the constant rh. It is therefore possible to find appropriate values of th for all the void fractions so that the condition of real sonic characteristics, eq. (53), is not only satisfied but also gives the correct degree of inertial coupling and realistic sonic charac- teristics. An accurate prediction of the latter is essen- tial for accurate prediction of critical flow rate, one of

Page 14: Two-phase Blowdown From Pipelines

708 J. R. CHEN et al.

the most important factors in the risk assessment of pressurised components.

Under the framework of inviscid flow, the only variable characterising the flow structure is the void fraction. Therefore, a dependence of th on c< is ex- pected in order to have real sonic characteristics for all void fractions. Combining the conditions, eq. (53) with eq. (66), gives

or

th -- (rh + 2)ct t> 1 (86)

1 + 2 ~ th > / - - (87)

1--c t

A possible choice of th is

1 + k~oc rh = - - ( 8 8 )

where k~ is a constant to be determined from experi- ments. Note that eq. (88) also fulfils the requirement that rh --, I when c~ ~ 0 for the dilute limit of spberical particles.

The four characteristics of the marginal stability model have the following forms:

2 = uG, uG, uc +_ ac. (89)

The first two and the last two eigenvalues correspond to void wave characteristics and sonic characteristics, respectively. For convenience, denote the left-running sonic characteristic velocity by ,IZ = uc - ac. Under critical flow or choking conditions, any disturbance downstream cannot propagate upstream and 2Z must be zero. Therefore, provided that the flow conditions at the choking plane are known from experiments, the constant kc can be determined so that the correspond- ing 2Z of the model is zero. Although critical flow data are abundant in the literature, critical flow data supplied with flow conditions, in particular the void fraction, at the choking plane are rare. Fauske (1965) reports the first of these data for low-pressure air-water systems. More comprehensive data are pro- vided by Deichsel (1988) and Deichsel and Winter (1990) for air-water systems at various pressures be- tween 1.3 and 5.9 bar.

20

-20

0

-6o ?

-80

-tO0

-120

i i i I ' I i f i i i i i

. . . . T l T . . . .

I i i I ~_x-X-X-X-X-X-X-X-X-X-X_Xx I

~ x ~ X - , I I --X-x-x-x-x X ~ x I I I I - X - X - x -

I I I I X-X-X-x

) I I )

i

Void fractions

• 0.325

0.4

• 0.472

0.49

- - . t 0.513

0.56

• 0.61

0.692

- - x - - 0.76

• 0.81

[] 0.865

0.91

0.936

0.964

0.99

2 2.5 3 3.5 4 4.5 5

kc

Fig. 2. Effect of kc on the left-running sonic characteristic under critical flow conditions. Data from Deichsel and Winter (1990) for an air-water system with Pexi t = 1.7 bara.

Page 15: Two-phase Blowdown From Pipelines

Modelling of two-phase

Figure 2 shows the calculated 2~- for one series of the data of Deichsel and Winter (1990) using different values of kc. It shows that kc -~ 3 gives the best fit of zero 2~- for 0 ~< ct ~< 0.8. Figure 3 also shows that it is impossible to fit the data to the zero 2~- when ~t > 0.8 no matter what the value of kc is. Figure 3 shows the calculated 2~ for the data of Fauske (1965) and Deichsel and Winter (1990). Again, all predicted char- acteristics are close to zero except when ~t > 0.8. More critical flow data are given in Deichsel (1988) and the resulted left-running sonic characteristic velocities are plotted in Fig. 4. Other than the data from the pipe of 0.0011 m diameter, the overall agreement with zero characteristics remains the same as Fig. 3. Deichsel (1988) attributed the peculiar results from the 0.0011 m diameter pipe as a non-negligible wall effect. It is suspected that liquid is no longer the continuous phase due to the small pipe diameter. It is therefore suggested that the present marginal stability model with

1 +3ct rh = (90)

1 - 0 ~

can be used for pipes with diameter > 0.002 m up to = 0.8 and with less accuracy for • > 0.8. Geurst 's

blowdown from pipelines 709

model of dilute dispersion is therefore successfully generalised to non-dilute two-phase dispersions.

One must be aware that the marginally stable con- dition is associated with the assumption of a constant void wave velocity propagating at the drift velocity of the vapour. This is a reasonable approximation when the vapour phase is the discrete phase and constitute the characterising structure of the flow. When the vapour phase is no longer the discrete phase and the liquid becomes the discrete phase, e.g. liquid droplets, the characterising structure of the flow is the discrete liquid phase and the void-induced structure wave should propagate at a velocity close to that of the liquid phase. Therefore, the effect of this assumption is most significant when the slip between the two phases is very large and the liquid is the discrete phase. This is probably the reason why it is impossible to fit the critical flow data for ct > 0.8 no matter how large the inertial coupling constant is.

Calculations have also been made to check the flow regime of the data of Deichsel and Winter (1990). It is found that all the data satisfy the intermittent to dispersed bubble transition criterion of Taitel and Dukler (1976), i.e. the flow regime is dispersed bubble flow. For high void fraction, bubble flow is certainly

r.) O

O ~

&

20

10

0

-10

-20

-30

-40

-50

-60

-70

-80

-90

-100

i I i i i i i I I

I I I I I I I . . . . . . . . . t - . . . . . . . ~ . . . . . . . . - t . . . . . . . . t - . . . . . . . . I - . . . . . . . ~ . . . . . . . . - t . . . . . . .

t I I I I ^ L ,-~ O I O I I

i m J • e l f t f f i i I - r 0 • 4 1 , I ~li - t I , 0

. . . . . . . . V - - - ~ . . . . -4 . . . . . . ~ t - - ¢ . . . . . . . 4- . . . . . . . . l - . . . . . . . . . . . . . . -~ . . . . . . . .

t 7 I { ~ I I A t O ^ I O I

i i I I I I I & A I [ 3 I I I OI . . . . . . . . . ~ . . . . . . . . . . . . . . . . ~ . . . . . . . . . . . . . . . . . . . . . . . . . ~, . . . . . . . . ~ . . . . . . .

f I . . . . . . . . . ~ . . . . . . . . . . . . . . . . ~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 . . . . . I I I z~

I It 11 . . . . . . . . I . . . . . . . . i ~ - . . . . . . . . . . . . . . . . ,~ . . . . . . . . . . . . . . . ~, . . . . . . . . . . . . . . . . . .

I I ',

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . t I

. . . . . . . . ,F . . . . . . . . . . . . . . . . . . . . . . . . . I-, . . . . . . . . . . . . . . . . . . . . . . . . . t . . . . . . . . t I I I

. . . . . . . . • . . . . . . . . 1 I I I

', I I I t I t I

. . . . . . . . r . . . . . . . . . . . . . . . . - t . . . . . . . . *" . . . . . . . . r - . . . . . . . -1 . . . . . . . . - t • . . . . . . . i

, 1 I . . . . . . . . ~ . . . . . . . ~ . . . . . . . . ~ . . . . . . . . ~ . . . . . . . . ~ . . . . . . . ] . . . . . . . . ~ l -- r . . . .

I f I

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Void fraction

• D & W , * D & W , P e r • D & W , P c = Pe= 1.Tbar 1.3-2.0bar 1.8-3.5bar

a D&W, Pc= o D&W, Pe= zx Fauske, 2.3-5.2bar 2.4-5.9bar Pc= 1.156bar

Fig. 3. Left-running sonic characteristic of the marginal stability model with kc = 3 for the critical flow data of Fauske (1965) and Deichsel and Winter (1990).

Page 16: Two-phase Blowdown From Pipelines

710

70

60

50

40

30

20

10

o

.,~ -10

- 2 0

O • ~ -30 O

-40

• ~ -50 e., ,~ -60

~ -70

-80

-90

-100

-110

-120

-130

-140

J. R. CHEN et al.

i i i i i i d I I r I I I

......... L ....... J ........ ~ ................ J ........ ~ ....... -L ....... I I I I I I I i I I I I I I I I I

I I I I I I I I I I I I

I I I I I I I I I I I I I I • I / / / I r . . . . . . . 7 . . . . . . . . r . . . . . . . . r . . . . . . . 1 . . . . . . . . r - - - - - - - - r . . . . . . I I I I I I I I I I I I I I

I I I I A I I I I I I v I I

J . . . . . . . . . ~- . . . . . . . ~ . . . . . . . . . - - , - - - - ~ . . . . . . . , - - - ,m,, , ,~. l - . . . . -~. . . . . I I I I I "¢ @I N I l 0 , , , • , . ' ~ , . ~ l " l , ' . o . , . ~,

i I i I 1 7 0 U I O ~ I~ •

....... ~ ........ ~ ........ ~ ....... ~ ....... _,_-..~,._-~ .... o-

. . . . . . . . . . . . ~ I l i ~ . . . . . . . . . . ( ~ I I ~ . . . . . . . . . . . ~ I I I * - - ~ I ~ I ~ . . . . 4

, , ~ n , t9 , , o ~ I~,, QQ

r 7 ~ r ~ r o ~ f i m e I i i 13 t ~ l n I t I 0 I I I - I - ! I •

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ~ . . . . . . . ~ -- ~ I I I ~ .... " I l l ~ I II ~ I I I I L I I I / - - I I , , a , , u , • ,,-,~,~, ~ * :

. . . . . . . . . L . . . . . . J . . . . . . . . I . . . . . . . . ~ . . . . . . . J . . . . . . . . I . . . . [

I I I I I i # I l l l l l Z~ m~ ~

I I I I I r l I -- ~ ~- - . . . . . . . . L . . . . . . . J . . . . . . . . l . . . . . . . . L . . . . . . . • . . . . . . 1 . . . . .~ --

I l I I I I r

I i I I i i - - l u • I I I I I I I •

. . . . . . . . ,~ . . . . . . . ~ . . . . . . . . t . . . . . . . . ,~ . . . . . . . fi . . . . . . . . f . . . . a - - - h T r . . . . . I I I I I I I • • I I I I I I I

. . . . . . . . r . . . . . . . 7 . . . . . . . . T . . . . . . . . f " . . . . . . . 7 . . . . . . . . T . . . . . . . . F ' I - - 1 ~ . . . . I I I I I I I I I I I I I

I I I I I I I

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Void fraction

a D=0.0011 m o D=0.0019 m * D=0.003 m • D=0.005 m

Fig. 4. Left-running sonic characteristic of the marginal stability model with kc = 3 for the critical flow data of Deichsel (1988).

no t possible because the liquid level is not enough to suppor t bubble dispersion and bubbles should co- alesce into a cont inuous gas phase and form a differ- ent flow pat tern, i.e. the annular-dispersed flow. The Taitel and Dukler (1976) cri terion for t ransi t ion be- tween annular-dispersed and dispersed-bubble flow depends only on the equil ibrium liquid level. If the equi l ibr ium liquid level falls below half of the pipe diameter, annular-dispersed flow is expected; if the equi l ibr ium liquid level is greater than half of the pipe diameter, dispersed-bubble flow is expected. In terms of void fraction, this cri terion corresponds to the condi t ion ~ < 0.5 for dispersed-bubble flow and

> 0.5 for annular-dispersed flow. Therefore, the present model of two-phase dispersion is strictly ap- plicable to ~ ~< 0.5 only. The fact tha t sonic character- istics are predicted well up to ~ = 0.8 suggests tha t large dis tor ted bubble flow or slug flow might still persist at void fractions greater than 0.5 under critical flow condit ion. Fur the r studies on the dispersed- bubble to annular-dispersed t rans i t ion might help to clarify such a possibility.

I t is also apparen t tha t the pressure difference term given by eq. (20) becomes negative when • > 0.5 under the marginal stability condition. This unphysical re- sult results from the assumpt ion of void wave propa- gat ion at a cons tan t gas-phase velocity a l though the

negative pressure difference does not lead to a cata- s t rophic failure of the model when ~ > 0.5. This also coincides with the ideal Ta i te l -Dukle r transi- t ion bounda ry of d ispersed-bubble /annular-dis- persed flow. Clearly, a p roper descript ion of void wave p ropaga t ion is essential to formulate the flow- regime dependent two-fluid model. However, to match void wave p ropaga t ion at the two extremes of void fraction s imultaneously is not a trivial task and does not necessarily yield consis tent results with the single-particle limit. For these reasons, this work will be conten t with the present simple assumpt ions of void wave propagat ion.

4. C O N C L U S I O N

The inertially coupled equat ions of mot ion of bubbly flow of Geurs t (1985a, b) are extended to gen- eralised two-phase dispersions. The extension can be summarised as follows.

• The modified var ia t ional principle allows both phases to be compressible.

• An energy equa t ion of the mixture is obta ined by using Noether ' s invar ian t theorem and is shown to be comparab le with the averaging formula- tions.

Page 17: Two-phase Blowdown From Pipelines

Modelling of two-phase blowdown from

• The closure of interfacial mass and energy trans- Fm fer is established by using a thermodynamic FMk equilibrium assumption. F~

• Hyperbolicity of the equations is achieved by Fri forcing the flow to be marginally stable. Under 2 the marginal stability condition, all the informa- # tion related to the structure of the flow that is not #k considered under the framework of non-dissi- P~.j=I.2 pative, inviscid flow is found to be embedded in an inertial coupling constant. ,Ok

• An expression for the inertial coupling constant ¢r is obtained based on critical flow data. The mar- Zk

Re ginal stability model gives correct sonic charac- r~ teristics up to ct = 0.8.

The well-posedness and clearly defined character- istics of the marginal stability model make the numer- ical simulation of physical problems possible and meaningful. In the second part of this paper, the simplified finite difference method of Chen et al.

(1993b) for equilibrium two-phase flow models will be extended and applied to the marginal stability model for the cases of blowdown from pipes containing multi-component mixtures.

Acknowledgements--The authors would like to thank Prof. J. A. Geurst who kindly read and commented on a prelimi- nary version of this paper. Financial support from British Gas plc for JRC through the award of research scholarship is also gratefully acknowledged.

NOTATION

a~ speed of.sound of vapour c integration constant D propagation speed of pressure pulse Se fluctuation kinetic energy of phase k e k

hk specific enthaipy of phase k J drift flux K kinetic energy density L Lagrangian density m Reynolds stress coefficient th inertial coupling constant M~ interfacial force density M~' surface tension force density N number of component of the fluid Pk pressure of phase k qk r fluctuation kinetic energy flux of phase k qk r fluctuation internal energy flux of phase k sk specific entropy of phase k t time Tk temperature of phase k Uk velocity of phase k U total internal energy density Yk, mass fraction of component i in phase k Yo.j=I.2 reduced densities of liquid and vapour

phases of component i z axial coordinate

Greek letters ct void fraction

pipelines 711

interfacial mass transfer rate interfacial momentum flux of phase k interracial equilibrium entropy transfer rate interfacial species transfer rate characteristics or eigenvalues of the model viscosity of liquid chemical potential of phase k reduced densities of liquid and vapour phases density of phase k surface tension shear stress of phase k fluctuating momentum flux of phase k

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APPENDIX

When the equations of motion are derivable from a varia- tional principle, a general and systematic way for establish- ing the conservative theorems can be developed from a direct study of the invariance of the variational integral. This is called Noether's invariant theorem (Logan, 1977; Geurst, 1985a, b).

Consider a Lagrangian L = L(~ ,c~ /c~ t ~) that depends only on the argument functions @i and their first derivatives of the independent variables t ~, i = 1 . . . . . M, j = 1 . . . . . N. The R-parameter family of transformations on the variables t ~ and tP j can be written as

r i = dpi(ti,~J,g), t~ j = goJ(ti,~J,g), r = 1 . . . . . R (A1)

where g denote a set of R independent parameters. The zero value of g gives the identity transformation

i- '= $'(t ' ,$J,0) = t i, (~J = tpJ(ti,$J,O) = $J. (A2)

Equation (A1) can be expanded about g = 0 to obtain

~i = t i + ~( t i , ~ j ) s r "4- 0 (8" ) , ~J = ~J + ~J(ti, l[lJ)F, " + O(sr)

(A3)

where the Jacobians or the generators of the transformations are given by

¢9 i • a ~ ° J i " ~(ti,~d) = ~, ' ( t i ,¢J,0), (i(ti ,~ ') = ~7( t ,g , ' , 0 ) . (A4)

For simplicity, the convention that repeated indices repres- ent summation over the index is used.

Noether's invariant theorem states that a necessary condi- tion for the fundamental integral SL dt 1, . . . , dt u to be abso- lutely invariant under the R-parameter family of transforma- tions (A1) is that the following R identities hold true:

~ k - - + - - J

(A5)

where g~ is the Kronecker delta function and ~f, = O~J/63t i. Assume that the fundamental integral S L d f , . . . , dt u is

invariant under the M-parameter transformation

~i=t,+g, (i,r=l.. . M), ~J=~b j (A6)

which represents a translation of (t 1, . . . , tU)-space. The Jacobians of this family are given by

~=,~, ~=o , ( i , r = 1 . . . . . N) . (A7)

In the present work, there are only two independent vari- ables, i.e time and space, and t I = t and t 2 = z. Equation (A5)

Page 19: Two-phase Blowdown From Pipelines

becomes, for r = 1,

Modelling of two-phase blowdown from pipelines

OIL sOLq 01" .OL'~ -+ ,~J-o~t+; o+~) :o

o~L 0; ~+--TJ + ~1_ ~ - 0'=~ J : o.

and for r = 2,

(A8)

(A9)

The Lag~ngian is chosen to be the equivalent Lagrangian density of L obtained from integration by parts of, e.g., eq.

713

(18) for a one-component system:

--S,(~+.L~)~--S2(~+uG~s)~. (AiO)

The argument functions are given by ~, pG, PL, S,, $2, u~, uL, ~o and r/. Equation (A8) becomes the energy conservation equation of mixtures, eq. (36), while eq. (Ag) becomes the momentum conservation equation of mixtures, eq. (29).

CES 50:4-K

Page 20: Two-phase Blowdown From Pipelines

P e r g a m o n Chemical Enoineerimj Science, Vol. 50. No. 13, pp. 2173-2187, 1995 Copyright ,~(~; 1995 Elsevier Science Ltd

Pnnted in Great Britain. All rights t,eservod 0009-2509/95 $9.50 + 0.00

0 0 0 9 - 2 5 0 9 / 9 5 ) t ) 0 0 0 9 - 7

M O D E L L I N G O F T W O - P H A S E B L O W D O W N F R O M

P I P E L I N E S - - I I . A S I M P L I F I E D N U M E R I C A L M E T H O D F O R

M U L T I - C O M P O N E N T M I X T U R E S

J. R. CHEN ~, S. M. RICHARDSON and G. SAVILLE Department of Chemical Engineering, Imperial College, London SW7 2BY, U.K.

(Received 11 October 1993; accepted in revised form 20 December 1994)

Abstract--A simplified numerical method is proposed to solve general two-phase flow equations for multi-component mixtures. The method is applied to solve the marginal stability model proposed in the first part of this paper. Case studies are performed and validated against experimental data for the blowdown of pipelines containing one- or two-component mixtures. The results show that the marginal stability model performs better than the simple homogeneous model for blowdown from short pipes. For blowdown from long pipes, the results of both models are quite similar. Concentration stratification is found to be insignificant in the overall blowdown predictions.

I. INTRODUCTION

In the first part of this paper [Chen et al. (1995), hereafter referred to as Part I], a marginal stability two-phase flow model is proposed which possesses real and clearly defined sonic characteristics. Critical flow or choking, the most difficult aspect of the mod- elling of depressurisation processes, can therefore be studied systematically through the consideration of characteristics. In the second part of this paper, the simplified numerical method of Chen et al. (1993) is extended and applied to the marginal stability model for mult i-component mixtures. In particular, a de- coupled method is developed to solve the conserva- tion equations of each species efficiently. The effect of concentrat ion stratification, the change of overall composition of the two-phase mixtures due to slip between the two phases, during the rapid de- pressurisation process is also studied.

2. TIlE SIMPLIFIED NUMERICAL METHOD

2.1. Generalised equations of t wo-phase phase flow The generalised equations of two-phase flow in

thermodynamic equilibrium can be written as follows.

Mass balance of mixtures:

0 =-[~p~ + (1 -- ~)PL] (Tt

1 O + a ~z A[~PclUG + (l -- ot)pC, uL] = 0. (1)

~'Author to whom correspondence should be addressed. Present address: Industrial Technology Research Institute, Center of Industrial Safety and Health Technology, Bldg. 32, 195 Chung Hsing Rd., Sect. 4, Chutung, Hsinchu, Taiwan.

Momentum balance of liquid and vapour phases:

- I LuL AI1- +tl

= rwL + ~ + ~ - (1 - ~)pLgCOsO. (2)

? I OA~p~u~ + Op

= rw~ - ri - r~ - ctp~ 9 cos 0. (3)

Energy balance of mixtures:

?P ~ [~tpGht; + (1 - ~t)pLht] dt

1 t? + ~ ~ A [otp~uc, htl + (! - ~t)ptuLhL]

- [ ~ u ~ + (1 - ~ ) u L ] Op ~ - ~ - - UL'~wL - - WG'(wG OZ

+ (u~ - UL)~i + Qw. (4)

Mass balance of component i:

~~(Pl Yt.i -t- P2.Vri) + ~ ~ A(pl ULyLi + P2uaYGi) = 0,

i = 1 . . . . . N - 1. (5)

where ~: denotes two-phase interaction terms not considered in the algebraic drag z~ and p = p~, hL = h[. zw~ is the wall drag of phase k and Qw is the external heat source term. For completeness, the area variation of the flow channel is also included. When ~ contains derivatives, the differential equations and their difference approximations can be modified ac- cordingly without much difficulty.

2.2. Finite difference method of one-component systems The finite difference method used is semi-implicit

and is similar to the ICE scheme of Hariow and

2173

Page 21: Two-phase Blowdown From Pipelines

2174

Amsden (1971) for compressible flow. The scheme for the homogeneous equilibrium model (HEM) is based on the following features (Chen et al., 1993):

• The mixture mass equation is solved in conserva- tive form to maintain mass conservation.

• The momentum convection term can be treated explicitly without affecting the stability of the scheme according to the von Neumann stability analysis of Chen (1993).

• All the other terms including the mass convec- tion, energy convection and pressure gradient terms must be treated implicitly to remove the Courant number restriction on the integration time-step.

• The dependent variables are pressure and mix- ture enthalpy; therefore no special treatment is required for transition from single to two-phase flows.

We now extend the simplified method to the general- ised two-fluid model. Consider the simplest case of a one-component system. It is more convenient to define the following mixture properties based on the mass centre average of the two phases:

p,,, = c~pG + (1 - : t )pL (6)

g p G U 6 + (1 - - O:)pL U L u m - (7)

P,.

~tpc, h . + (1 - ~)pLhL hm = (8)

Pm

Equations (1) and (4) become:

~- Apm um= 0 (9) ~P"+~,.,z

?.t

- [ ~ u ~ + (1 - ~ ) u L ]

= Q~ - u,~w6 - ULr~L + (U~ - UL)Z~. (10)

The momentum equations for each phase are added to give the mixture momentum equation:

(; 1 ~ Op dt p"u" + A ~z A[~peu~. + (1 - =)pLU~] + ,3S

= Z w ~ + r , .L - -p , .OcosO. (11)

When u~ = UL, the above equations reduce to the simple HEM. H E M is simpler to solve than a two- fluid model because the convection terms can be readily expressed in terms of mixture variables. For two-fluid models, some special treatment is required. Since momentum convection can be treated explicitly, only the energy convection term requires special treat- ment, as shown below. The energy convection terms are written as

~tpeueh 6 + (1 -- ~t)pLULh L -= pmu,,hm

+ ~tpe(l -- a)pL (Ue -- UL}(he -- hL). (12) P,,

J. R. CHEN et al.

The mixture energy equation becomes

?' h 1 0 dt p" " + A -~z Ap,.u. ,h, .

l t7 [ a p ~ ( l - - a t ) P L ( u o u L ) ( h t ~ - h L ) ] +~NA

- ~--P - [~u~ + (I - ~)u~] ~--P Ot ?~z

= Qw - u ~ , - ULr,L + (u~ -- UL)q. (13)

The second and third terms of LHS of eq. (13) are called the homogeneous and mixin9 energy convection terms, respectively.

Equations (9), (11) and (13) are the governing equa- tions for mixture properties. They are discretised similar to. the scheme of H E M (Chen et al., 1993) except for the mixing energy convection term:

• mixture mass equation at mass cell centre z /

Vj n + l n n + l ~t (p , , j - p . j ) + Aj+ t/2(p,.u,,b+ i/2

- A t - l:2(pmu,~)~+-~/2 = 0 (14)

• mixture momentum equation at momentum cell centre zi+ 1/2 :

Vj+ 1/2 n - I n At [(p,.u,,)j+ 1/2 - - ( P m U m ) j + 1 / 2 ]

+ A~+l[~p6u~ + (1 2 . - - 9 0 p L U L ] j + 1

n * l n + l + A~+x,z(pj+l - Pj )

gV~+ .+1/2 (15) - - l l 2 P m j + 1/2 COS 01+ 1t2

• mixture energy equation at mass cell centre z~:

Vj [(p,.h.)7+ 1 At - (pmhm)j]

+ A j + t /2 (p ,numhm)~ .+~i2

- Aj-l/2(pmu,~hm)~'+-~/2 + At+ 1:2

X ~tPo( ~)PL(ut~ - - UL)(/,It; - - h=)j,+,/2

"] n+ 1/2

x(u~ - uD(h~ - hL) J~_ 1/2

vj [ ~ . 1 _ p)'-I - A j r ~ u ~ + (1 - ,~ )uL] ; At

n + ! n + l x (pj+ 1:2 - P;- 1:2)

= E[Qw - u~T,,~ - UL TwL

-~n+ 1/2 -t- (U G - - ULJZiJ J (16)

Page 22: Two-phase Blowdown From Pipelines

Modelling of two-phase blowdown from pipelines--ll

where the half time-step properties, such as (p,.)~+ t/2 , + , . + t/2 PmJ -- Pmj

etc., denote the iteration values of the n + 1 new time-step flow variables with the old time-step values as initial values. The n + i new time-step variables in where the difference equations are considered as the new iteration values. An additional outer iteration loop is used in each time-step to ensure the convergence of the new time-step value. All the source terms in the mixture equations and the mixing energy convection term involve individual phase properties and dis- cretised using such half-time-step iteration properties to avoid the difficulty of evaluating the individual phase properties. An outer iteration loop is used to achieve the convergence of these iterative properties. Therefore, the discretised equations are the same as HEM except that an extra mixing convection term evaluated at the half time-step appears in the energy equation and an outer iteration loop is also required. The phase velocities are solved individually from the discretised momentum equations of the two phases with the new time-step pressure obtained from the above mixture equations. The introduction of an outer iteration loop certainly increases the computa- tional load. But, since the choking or critical flow boundary condition usually requires iterations as well, this iterative semi-implicit method actually be- comes advantageous over fully implicit methods in terms of simplicity and efficiency for the present blow- down problems. Note that the cell length, inclination and volume are defined at the mass cell centre while the cell cross-sectional area is defined at the edge of mass cell. Therefore

1 Vj = ~(A~+ ,,2 + A s - I :2)Azj • (17)

Figure I shows the basic definitions of the flow chan- nel.

The mass flux .+ I (P,~Um)i* ,:2 in eq. (15) can be ex- pressed in terms of pressure owing to the explicitly treated momentum convection term. Therefore, the mass flux in eqs (14) and (16) can be recast in terms of pressure. Choosing the mixture enthalpy and pressure as the dependent variables, the new time-step densities can be eliminated by the following linearised equation of state for the two-phase mixture:

g

Fig. 1. Definitions of flow channel.

- 2 . n + 1/2. n+, n+,/2x = ~ap,, )j {pj - pj ! I -2xn+l /2{hn+' n+,/2)

+ ~ahm )j w,mj -- hrnj

2175

(18)

a , , = \ ap - -~ ) , . ' aim = \ t ~ , p = ] j (19)

Equations (14)-(16) therefore reduce to a set of two non-linear algebraic equations. The nonlinearity only arises from the product of two mixture variables and can be eliminated by using the following linearisation:

f •f.+,..2 \ . f " ~ ' O " ' l = , * t , ' 2+ ~{ a ; )

n* 1.'2 X (.qn+ 1,2 ~r" c q A ~ )

= f . ~ , .q.+ ,:2 + f . + ,:2.q.* 1

_ f . + ,;2On+ l/2 + O(A~2) . (20)

Note that the linearisation must be made along the iteration-step not the time-step to ensure that the higher-order truncation terms are negligible,

Finally, the linearised difference equations may be written for cells 2 to M - 1 and solved simultaneously with the boundary cells, cells 1 and M. to obtain the pressure and mixture enthalpy for each cell. Details of the difference equations are given in Chen (1993). The solution matrix has a block tri-diagonal structure and typical algorithms such as the Thomas algorithm can be used to solve the algebraic equations efficiently. An equilibrium pressure-enthalpy flash calculation is then performed for each cell to obtain the void frac- tion. phase densities, phase enthalpies and temper- ature. The new-time step velocities are obtained from the discretised momentum equation for each phase when the new time-step pressure is known.

2.3. Fini te di f ference me thod o f mul t i - componen t sys- tems

The overall mass fraction of component i can be defined as

~PGYt;i + (1 -- :t)Pt.YLi Y . i = (21 )

P=

Similar to eq. (12), the convection term in the mixture species equation may also be separated into a homo- geneous term and a mixing term:

~tpc;u~; Yc, i + (1 -- ~:)pL ut.Yt.i = flmumYmi

+ ap~(l - ~t)PZ(u ~ _ UL)(y~i -- Yt.i). (22) X)m

Therefore the mixture species equation becomes

-~ p.y., + ~ ~ Ap.u,.y.,

+ -A ~z t_ p . (uc; - UL)(YG, -- Yl.i) 1 = O.

(23)

Page 23: Two-phase Blowdown From Pipelines

2176 J.R. CHEN et al.

B y using eq. (9), the above equation can be recast into the following non-conservative form:

d c~

+ ~ A ( u G - - U L ) ( Y G i - - YLi) = O.

(24)

The above equation shows that the change of com- position is totally dependent on three factors, namely, convective transport characterised by the mean con- vective velocity urn, the slip velocity uG - ur, and the volatility of the fluids characterised by the mass frac- tion difference between the two phases. The first factor will offset the separation of components resulting from the latter two factors. It is therefore expected that the maximum separation occurs in countercur- rent flow where the mean flow is small, e.g. the case of a distillation column. In cocurrent pipe flow, separ- ation is expected to be diminished due to the large mean flow. The exact extent of separation will be dependent on the slip and volatility of the fluids. When the two phases move at the same velocity, eq. (24) reduces to

d ~t y'~ + u,, -;--ez Y,i = 0 (25)

a + 1 n + 1/2 P,.j -- Pmj

which shows that concentration stratification will not occur provided that the initial concentration is uni- form throughout the flow channel.

Solving eq. (23) exactly by the methodology of the previous section is very tedious and time consum- ing since the linearised EOS becomes

, - 2 ~ n + 1 / 2 . n + l n + l / 2 1 = l a ~ )~ [p~ - - p~

n+ 1,/2) + (a~217 + '/2(h:+j' - - h,~

N - I l - 2 - n + 1 / 2 , n + l n + l / 2 ~

-~ E "a~mi)j ~Ymij --Ymij 1 i = 1

where

(26)

a,mi \ ~P,. / . , h. (27)

and therefore dependent variables must increase as the number of components increases. This results in a larger coupled system and therefore larger computa- tion time. The variable number of components also makes the programming more difficult. Therefore, a decoupled method is developed which solves eq. (23) independently from other equations.

A completely decoupled species equation will im- pose a convective Courant condition on the integra- tion time-step as shown by Chen (1993). Since the scheme proposed in the previous section involves an outer iteration loop, the following decoupled dis- cretisation for eq. (24) is used in which the iterative half time-step is used in the convective terms instead

of the old time-step:

n-# l n At un+112, n + l / 2 n + l i 2 Ymij = Ymlj - - AZ ral l Ymij - - Ymij 1 1

A t { [ ~ t p o ( l - - o t ) p L Aj4 .1 /2

Vjp~,~ pm ~ n ~- 1/2

X (UG -- UL)(YGi -- Y L i ) [ -l j + 1/2

- A j_ i/2 ( u ~ - UL)

In + 1/2 X(YGi--YLi)Jj_l/2}. (28)

The above equation can be solved explicitly inside the iteration loop for each component. It has been tested and works well provided that the Courant number, defined by Um,,At/Az where urn,, is the maximum convection speed, is not much greater than one. This decoupled treatment therefore significantly reduces the complexity and difficulty associated with multi- component mixtures. The results of concentration stratification will be presented in the next section.

2.4. Application to the marginal stability model Applying the simplified finite difference method to

the marginal stability model of Part I is not a difficult task but caution must be taken to avoid possible difficulties in the iteration. The mixture momentum equation can be written as

l t~ ~t (pmu') + A ~z A[~tpau~ + (1 - oQpLu 2

+ (1 - a)PLm(:t)(uG -- UL) 2 ]

&p I + ~ + ~ ~ [p,.(l - z)~m'(~)(u~ - uL) 2 ]

= zw~ + zwt, - ,ap,. cos 0 (29)

where mixture variables are used where applicable and p -- PG- Comparing with eq. {11), one notes that two extra terms exist: the velocity fluctuations in the liquid phase and the interfacial pressure difference. Both terms contain a Reynolds stress coefficient func- tion or its derivative which are given by the following marginal stability condition:

m ( ~ ) = ~ l ' ~ - (rh + 2 ) ~ )

I m'(ct) = ~m - (rh + 2)or (30)

m " ( ~ ) = - ( ~ + 2).

Substituting this equation into eq. (29) gives

c) 1 c~ cgp

a(~ - (~ + 21~t) d "4- A [ ( l - - Ot),OL(U G - - UL) 2 ]

2A az

1 ] a~ + ~ - ( ~ + 2 1 ~ t (I -- 0t)PL(U~ -- UL) 2

Page 24: Two-phase Blowdown From Pipelines

+ ~ - (~ + 2)a

x [(1 - ~)2p~(u~ - uD ~ ]

1 2 O~ 2(~ + 2)(1 - ct)2pL(uo -- UL) ~,Z

= z,,~ + r~L -- gp., COS 0. (31)

Therefore, under the marginal stability condition, every term containing the derivatives of the Reynolds stress coefficient becomes two separate terms contain- ing the inertial coupling constant, ~ which is given by eq. (90) of Part I. Equation (31) is the final equation to be discretised. The first three terms of equation (31) are the same as the HEM diseretisation. The fourth and fifth terms originate from the flux of velocity fluctuations and can be treated explicitly. The sixth and seventh terms originate from the inteffacial pres- sure difference and can be treated semi-implicitly, i.¢. by using the half time-step variables and iterations. The finite difference discretisation can then be written down without much difficulty.

Next, consider the momentum equations of indi- vidual phase. Both equations have been used and it is found that the gas phase momentum equation is easier to solve both in terms of numerical stability and complexity than the liquid phase one. Therefore, only the finite difference discretisation of the gas mo- mentum equation is discussed here. The gas phase momentum equation, eq. (25b) of Part I, shows that there are two time-dependent terms with one coming from the inteffacial force term M~. Substituting the marginal stability condition into M~ gives:

1 0 M, ~ = ~,(r~ - (~ + 2 ) ~ ) ~ [ ( 1 - ~ ) p ~ ( u ~ - u , ) ]

],o + :t(rh - ( ~ + 2)~t) A Oz

x A[(1 - ~t)pL(U 6 - - UL)UG]

+ rh - (~ + 2)~t (1 - ~)p~(uo - u~)uo Oz

+~O~[r f i - - ( t f i+2)~] ( I - -~ )pL(U~- -UL)~Z .

(32)

Note that the derivative in the first term contains unknown uo and UL which can be further reduced to one unknown only by using the following identity:

(I - ~)pL(u~ - UL)

= (! - =)p~.Iu~ p.u. - (1 - ~e)m.~P°UGq_J

= p .ua - pmu. (33)

Modelling of two-phase blowdown from pipelines--II 2177

where u,. is known since the mixture momentum equation is solved before the individual momentum equation. All other state variables are also known variables. Therefore the final momentum equation of the gas phase under the marginal stability condition becomes

~ ( ~ - ( ~ + 2 ) ~ )

0 x [ ~ ( p . u G ) - ~ ( p . u , . ) ] + 1 0 2 Op ~z A["PGuo] + ~

a(~- (~ + 2)a) a + 2A ~ A[(I - ~)pL(uG -- UL)Uo]

+ • - ( ~ + 2 ) ~ (1 - e)OL(UG -- UL)~

+ ~,[*h - (*h + 2 ) a ] ( l - ~ ' )#L(UG--UL)~

+ FUG = ~,o -- q. (34)

The discretisation is as follows: other than mo- mentum convection, which is treated explicitly, all the other terms can be evaluated using new iterative values for u,. and all the state variables, and old iterative values for uG and uL. The discretised equa- tion can be solved to give a new ue and subsequently a new uL can also be obtained by using the definition of u. . Again, this lengthy difference equation can be written down without difficulty.

Now, consider the energy equation, eq. (47) of Part I. The derivative structure of this equation is exactly the same as eq. (4) of the generalised model and requires no special treatment. The final things to be given in detail are the various source terms in the momentum and the energy equations. In this work, all the source terms are discretised semi-implicitly. For example, the equilibrium entropy source term, eq. (48) of Part I, is determined by

1 ,, . + , n { [ ( 1 ~~ , , . . , . + . 2 = - - u~ IPLSL j j ( r , ) s At

- [ ( l - ~ ) p L s ? . ] ~ }

1 V / { A j + 1/z[(1 . . . . . ~/2 - - ~ ] P L S L U L J J + I/2

• n + l / 2 - A~_ ~n [(I - ~)PLSLUL]s- ~12 }" (35)

In a similar manner, the interfacial mass transfer rate is discretised by

n+ 1 n+1/2

1 r ~ / ~ n + 1 / 2 + ~ LAIj+ I / 2 ~ B o U G ) / + I/2

, , n + 1/2- I - - A j - 1/2[OZpGUG J.I- 112 J (36)

Page 25: Two-phase Blowdown From Pipelines

2178 J. R. CHEN et al.

while the F, - F,,sG term in the source term FM~ is determined by the following discretised equation:

( r , . . . . , , 2 1 . . . - l , . s c , ~ j = ~(cwc, b(s~'~ ''~ - s~,)

1 n n ~ l n + l + ~ z J ~ p c , U ~ ) i ( s c , , . , , - so, . . . . ) .

i

(37)

Since it is usually tedious to determine the chemical potential difference in the multi-component mixture, the interracial momentum flux term, FM~, is deter- mined by the simpler equation of a one-component system, i.e. eqs (25d) and (25e) of Part I. Note that this simple treatment is also compatible with our de- coupled approach in solving the mass equations for each species where the system is solved as a frozen mixture without composition changes within each iteration step. If however, a fully coupled approach is used for the species equations, then the multi-com- ponent version of FMG, eqs (35a) and (35b) of Part I, should be used. For the present purpose of studying transient blowdown processes for risk assessment, the decoupled approach is considered as sufficient and no attempt is made to pursue a fully-coupled approach. As we will show in the next section, the effect of concentration stratification on the whole biowdown process is usually insignificant.

Finally, we consider the critical flow boundary con- dition for the marginal stability model. Once u~ and UL are obtained, the characteristics or eigenvalues of the model at the open end are determined numer- ically. There are four eigenvalues [see eq. (89) of Part I]. Two of them are the gas phase velocity and must be positive. The other two values are the sonic characteristics with one being negative (the left-run- ning) and another positive (the right-running) for sub-critical flow. Critical flow is expected to occur when the left-running sonic characteristic equals or exceeds zero, i.e. the downstream disturbance can no longer propagate upstream. In this case, the velocity of the two phases are reduced in accordance with the magnitude of the left-running sonic characteristic. It- eration on the half-time step variables is repeated until the left-running sonic characteristic converges to zero.

3. RESULTS AND DISCUSSION

3.1. Hydrodynamic constitutive relations The hydrodynamic constitutive relations are the

relations that describe the momentum interactions or exchanges at the fluid-fluid and fluid-wall interfaces. The momentum interactions are usually expressed in terms of drag forces. The generalised drag force at the fluid-fluid interfaces is usually modelled as a linear combination of three forces: steady viscous drag, tran- sient viscous drag or the Basset force and transient non-viscous drag such as the virtual mass force. By taking into account the back-flow explicitly in the variational formulation of two-phase flow, we have

derived an inviscid two-fluid model in Part I which includes the effect of transient non-viscous drag. On the other hand, the transient viscous drag, usually appearing as a time integral (lshii and Mishima, 1984), is very difficult to incorporate as all the history of the flow must be stored and is generally neglected. Liang and Michaelides (1992) have studied the magnitude of unsteady viscous drag for small bubbles for finite Reynolds number flow. They found that unsteady viscous drag term accounts for approximately 25% of the total drag for bubble size smaller than 10-4m. They also found that the unsteady viscous drag de- creases rapidly for bubbles of larger size and larger Reynolds number. This work will consider only the steady part of interfacial viscous drag and neglect all the contribution from the unsteady part.

At the fluid -wall interface, the viscous drag force is also modelled as a linear combination of unsteady and steady wall drag or friction. The unsteady wall friction, usually also modelled as a time integral in single phase flow (Zielke, 1968), can become dominant in laminar flow but diminishes quickly with time and in turbulent flow. Chen (1993) performed an approx- imate study of the unsteady wall friction using single phase expression of Zielke (1968) with the single phase density and viscosity replaced by the two-phase mix- ture density and viscosity of Beattie and Whalley (1982) and found that the unsteady wall friction is negligible compared with the steady wall friction un- der critical flow conditions. Therefore, the transient blowdown process will be studied by using transient inviscid one-dimensional models with viscous bound- ary layers at the fluid--fluid and fluid-wall interfaces described by algebraic drag correlations to represent viscous effects.

The interfacial viscous drag is flow-regime depend- ent and the following simple flow-pattern map is used:

~ ~< 0.3,

0.3 < cc ~< 0.8,

~ > 0.8,

dispersed-bubble flow

intermittent flow

annular-dispersed flow.

(381

Although more mechanistic descriptions of the flow regimes exist, e.g. the Taitel-Dukler (1976) flow-pat- tern map, the large uncertainty associated with the drag correlations usually prevents any gain in accu- racy from using these flow-pattern maps. This simple flow-pattern map is also widely used in modelling steady state critical flow, e.g. Richter (1983), Dobran (1987), and Schwellnus and Shoukri (1991).

A large amount of literature exists for the hy- drodynamic constitutive relations. The following se- lection are based on suggestions from the sensitivity tests of Chen (1993). The generalised interfacial vis- cous drag for any flow pattern is written as follows:

zi = 2-~i~p~u, lu, I (39)

where u, is the slip velocity and Dh is hydraulic dia- meter of the flow channel. The interfacial drag coeffi-

Page 26: Two-phase Blowdown From Pipelines

Modelling of two-phase blowdown from pipelines--ll

cient is flow pattern dependent and is given as follows:

3 D, PG Cs.i=-~Co-~rb-~L for 0 < ot ~< 0.3 (40)

CIi = {0.005[1 + 75(1 - ~t)](l - E)ct- l/2

3 Dh ! - c t E } + t for 0.8 ~< e < I (41)

Co, the drag coefficient of distorted particles (bubbles and droplets), is given by (lshii and Zuber, 1979)

= 4 FO(PL-- {.1 + 17.76[f(ot)]6/7~ 2 Co 5 g , [ a P~)] 1/2 ~ j

where

f(ct) = (l - ~)L5 for bubbles, r o = rb (42)

f(ct) = ct 3 for droplets, rp = rd.

The entrainment fraction E in annular-dispersed flow is given by Ishii and Misbima (1984):

E = tanh(7.25 x 10- 7 We,.2s ReO.25)

w e = P~ t2u2 Dh ( p L -- PGY/3 (43)

(1 - =)uLpLD, Ret. =

In intermittent flow, the value of C[~ varies by several orders of magnitude and is interpolated exponentially with void fraction according to Schwellnus and Shoukri (1991):

C[~ = x~exp(z2~) for 0.3 < a < 0.8

z2 = ln ( C f " 2 ~ - 1- (44) \Csi . l / ct~ ctt

~ = exp (In CI~ ' ~ - ;~2 ¢t ~ )

where the subscripts 1 and 2 denote the values at void fraction of 0.3 and 0.8, respectively.

The inteffacial viscous drag discussed above deter- mines the slip between the two phases at a continuum point in a general three-dimensional flow field. In bubbly or droplet flow, such local slip is actually relatively small compared to the mean flow velocity. In one-dimensional flow, the non-uniform distribu- tion of the two phases in the transverse direction of the flow channel also affects the overall one-dimen- sional slip between the two phases. The effect of phase distribution still exists even when the local slip is zero and is usually larger than the local slip (Ishii and Mishima, 1984). Therefore, only the averaged local relative velocity rather than the difference between averaged velocities should be used in the above rela- tions for averaged interfacial drag in one-dimensional flow.

In bubble and intermittent flow, the averaged local relative velocity, denoted by ti,, is related to averaged velocities by (Ishii, 1979)

2179

I - Coot u, ~- uG - CouL (45)

1--~t

where Co is the distribution parameter. For bubble flow, the following empirical correlation of Co for a round tube is used (Ishii, 1979):

Co = 1.2 - 0.2 P f~t6. (46)

For annular or stratified flow, the distribution para- meter is generally very close to one and the phase distribution effect can be neglected (Ishii, 1977). For intermittent flow, a linear interpolation between Co = I at ~t > 0.8 and Co given by eq. (46) at x < 0.3 is used:

= _ _ _ ~g ~< ct ~< ~t2. \ct2 - ~ / '

(47)

The friction resulting from the viscous boundary at the wall for two-phase flow is similar to the single phase one. It is usually represented by the single phase wall friction multiplied by a two-phase multiplier. Since gas is rarely in contact with the wall, the wall friction is assumed to arise from the liquid phase only and

rwL 4)2°2 CfLo 62 = - (48) D, PL

where the two-phase multiplier 4)20 of Friedel (1979) is used.

Finally, the wall to fluid heat transfer rate per unit volume, Qw, is given by

4 Qw = ~ hw(Tamb -- T6) (49)

where hw is overall heat transfer coefficient, T~mb is the ambient temperature and the fluid temperature is chosen to be T6.

3.2. Model validation: one-component system A computer program called META (Multi-com-

ponent Equilibrium Two-phase Analyser) has been developed based on the present simplified numerical method and incorporating both the HEM and the marginal stability two-fluid model. The thermodyn- amic constitutive relations, namely the phase equilib- rium relations and thermophysical properties of the fluids, are provided by a computer program called PREPROP which is a general thermodynamic pack- age developed in the Fluids group in the Chemical Engineering Department of Imperial College. PREP- ROP incorporates two different methods for predic- ting phase equilibrium and thermodynamic proper- ties. The first method is a corresponding states principle (CSP) based on an accurate equation of state for methane (Saville and Szczepanski, 1982), and the second method is the Peng-Robinson (Peng and Robinson, 1976) equation of state (PR). PR is gener- ally more efficient than CSP while CSP predict more

Page 27: Two-phase Blowdown From Pipelines

2180 J. R. CrtEt~ et al.

accurate mixture properties if the mixtures are all similar to methane, Both methods has been used in the validations.

To validate the finite difference method indepen- dently from the interference of real fluid effect, an extra option of perfect gas properties is built into META. The perfect gas option of META is used to simulate the blowdown of a 1000 m long pipe and compared to the numerical benchmark solution of wave-tracing method-of-characteristics (Chen et al.,

1992). The comparison shows that the simplified finite difference method is about as efficient and accurate as the wave-tracing algorithm of method-of-character- istics (Chen, 1993). Another validation is also made using HEM against the numerical benchmark solu- tion of Hancox et al. (1976) for the blowdown of a subcooled water line that emulates the blowdown experiment of Edwards and O'Brien (1970). Again, the present numerical method agrees well with the numer- ical benchmark as show in Chen (1993). Nevertheless, the agreement between the predictions and the experi- ment is not completely satisfactory; in particular, the predicted pressure always drops more quickly than the measurements. Since the HEM usually predicts the lower limit of critical flow data (Ardron and Fur- ness, 1976), the correct predicted pressure should drop more slowly rather than more quickly than the measurements. Solbrig et al. (1976) suggested that the discrepancy arises from the uncertainty in the pipe rupture mechanism, e.g. some obstacle remains at the circumstance of the pipe. As the exact pipe bore size is unknown, the experiments of Edwards and O'Brien (1970) will not be used for model validation. Alterna- tive blowdown experiments performed by Necmi and Hancox (1978) are used for model validation instead.

The experiment is described briefly here. The pipe is horizontal, 4 m in length, 0.032 m in diameter and was made from transparent PVC. The pipe is filled with dichlorotetrafluoroethane (Freon ! 14 or R 114) and pressurised to 15 bar. The temperature is maintained at room temperature around 21.4'JC. The saturation pressure of R 114 at this temperature is about 2 bar. One end of the pipe is closed and the other end is fitted with a combination of a 0.0005 m thick glass disc and a diaphragm. The glass disc and the dia- phragm are ruptured almost simultaneously by a plunger in less than 0.5 ms. Necmi and Hancox (1978) reported that this arrangement gave a clear break to the full pipe bore.

The calculations are made by assuming a constant wall heat transfer coefficient of 5 W/m2K and a con- stant roughness length scale of 0.0015 mm. Varying these parameters does not have any significant effect on the results as shown in Chen (1993). 25 uniform meshes of 0.16 m and time-step of 0.0004- 0.04 s were used. Further mesh refinement does not have any significant effect on the result. The calculation is made with the Peng-Robinson equation of state. Since R- 114 is significantly different from methane, the methane-based CSP is not used. Each calculation takes about 40 min on a DEC 5000/240 station. Fig-

ure 2(a)-(d) shows the results of pressure at 0.86 m from the closed end, pressure at 0.04m from the ruptured end, void fraction at 0.86 m from the closed end and void fraction at 1.06 m from the ruptured end, respectively. Both the results of HEM and the two-fluid marginal stability model are shown.

The results of HEM show pressure dropping more slowly than the measurements of Necmi and Hancox. This is in line with the observation that HEM predicts a lower critical flow rate than the actual flow rate out of the line. The measured pressure drops quickly be- low the saturation pressure at the point when the rarefaction wave reaches the closed end of the line but never recovers to the saturation pressure. It is quite clear that thermodynamic non-equilibrium, i.e. de- layed bubble nucleation and slower vapour genera- tion rate, is dominant in this stage and the observed flow pattern is bubbly flow. After 0.4 s, the pressure starts to drop again until the end of blowdown. The observed flow patterns less than and greater than I m from the ruptured end are dispersed droplet flow and entrained stratified flow, respectively. The calculated void fraction history at 0.86 m from the closed end shows that around 1 s the calculated void fraction is underestimated compared with the experiments. Since thermodynamic equilibrium ensures the maximum vapour generation rate, this underestimation suggests the existence and the importance of mechanical non- equilibrium or non-homogeneity at this stage and the outflow rate is expected to be underestimated con- siderably by the HEM prior to 1 s of blowdown. The observed flow patterns, void fraction and pressure histories suggest both thermal and mechanical non- equilibria are important at this stage.

The pressure at the closed end predicted by the marginal stability model (MSM) remains at the satu- ration pressure until about 1 s after which the pres- sure drops more quickly than the measurements and predictions of HEM. Previously observed over-es- timation of void fraction around 1 s at 0.86 m from the closed end for HEM is found to have better agreement with the prediction of MSM. This confirms the existence and importance of mechanical non-equi- librium. The predicted pressure of MSM at the rup- tured end drops to 1.1 bar within 0.1 s and then de- creases slowly until the end of the blowdown. This result is actually a combination of the assumption of thermodynamic equilibrium and the inaccuracy of the MSM at high void fraction. We mentioned in Part I that the marginal stability model predicts the sonic characteristics well up to a void fraction of about 0.8. At a higher void fraction, the model requires a higher flow velocity to reach the choking condition. This is because the model still predicts negative character- istics as shown in Fig. 3 of Part I while the real flow should already have choked. The consequence of this deficiency is that the predicted critical flow rate will be higher than the actual flow rate and therefore results in a quicker pressure drop. Thermodynamic equilib- rium further enforces the maximum possible vapour generation rate and brings the flow quickly into a high

Page 28: Two-phase Blowdown From Pipelines

_g

"6 >

Modelling of two-phase blowdown from pipelines--II

(a) Pressure histories at 0.86 m from the closed end.

2.4.

2.2

2.0-

,, " . . , , 1.8" " x .

1.6- ". ~ '"'4

1 . 4 , ~ ",

1"2 I 1.0

0.8 I I ~ ~ I 0 0.5 1.0 1.5 2.0 2.5

Time (s)

2.4

2.2.

2.0

(b) Pressure histories at 0.04 m from the raptured end.

Measurement . . . . . . . META-HEM . . . . META-MSM

. . . . . . . . . . . . • . . . . . ,

1.8

! . 6 ~ "',,

1 . 4 ""'"'.-,.

t ~° , ".

0.8 • t i 0 0.5 1.0 1.5 2.0

Time (s) 2.5

(c) Void fraction histories at 0.86 m from the closed end.

| . 0 j • ~-- ." ."

o9 T / 0"8 / "" / I

I

o.7Jf

0.6 I 0.5

0.4

0.3 /

0.2 xx x

0.1 ~//× ×

0 0/5

/l x" × x l x . '

i / X ." I ..

x × o. I .'"

t I I /0 1.5 2.0

Time (s)

1.0

0.9 ×

0.8

0.7

0.6

0.5

~ 0.4

0.3

0.2

0.!

2.5 0

(d) Void fraction histories at 0.04 m from the ruptured end.

I i /

l I I

I x

I × I I I x I I

I x I I

I. o.'5

. _ _ - - - / ; .....

×

× Measuremen .. . . . . META-HEM

. . . . META-MSM

1.0 1.5 210 Time (s)

2.5

Fig. 2. Results of one-component systems.

2181

void fraction regime as shown in Fig. 2(d). When thermodynamic non-equilibrium is incorporated into MSM, it is expected that delayed vapour generation will result in a lower void fraction near the rupture plane and choking is expected to occur at a lower void fraction and therefore make pressure drop more slowly. Another reason why the predicted pressure in this particular case seems to deviate considerably from the measurements is that the saturation pressure is very close to the ambient pressure. In high pressure systems as we shall see in next validation, the magni- tude of deviation is about the same as for this system which is actually within 1 bar [see Fig. 4(a) and (b)]. The calculated void fraction histories at 1.06 m from the ruptured end shown in Fig. 2(d) show that the actual void fraction is lying between that predicted by MSM and HEM with preference to that of MSM. The HEM, once again, underestimates the void fraction and this suggests the importance of mechanical non- equilibrium.

Finally, we compare the mass flow rate predicted by MSM and HEM in Fig. 3. The maximum mass flow rate predicted by MSM is about 4-5 times larger than that of HEM in the first 0.1 s. The true mass flow rate is expected to lie between the two. However, the exact mass flow rate in this experiment can only be accu- rately predicted by relaxing the thermal equilibrium assumption. Although thermodynamic non-equilib- rium can be incorporated in this work for this particu- lar case, it will not be significant in the problems we are mainly interested, namely, multi-component two- phase flow in long pipes (/> 100m). In next section, the comparison of predictions and measurements for blowdown from a loom pipe will confirm this argu- ment.

3.3. Model validation: two-component system Very few experimental or field data exist for transi-

ent multi-component two-phase flow. The most com- prehensive are the full scale experiments performed by

Page 29: Two-phase Blowdown From Pipelines

. . . . . . . META-HEM

. . . . META-MSM

2182

1o

~ s

h , 6 o

4 m

2 . - . . . . . . . . . . . . . _ . ' : . - . . . . . .

- - - + _ _ - - - - ¢ - - - . . ~ ~ - - - - __q 0.5 1.0 1.5 2.0 2.5

Time (s)

Fig. 3. Comparison of calculated mass flow rate histories at the rupture plane for HEM and MSM.

British Petroleum and Shell Oil [see Tam and Cowley (1988)] on the Isle of Grain, England.

The experiments are very extensive and contain not only transient measurements inside the pipe but also

J. R. CHEN et al.

jet dispersion outside the pipe, Only the information relevant to this work is described here. The pipe is horizontal, 100 m in length, 0.15 m in diameter and made from commercial steel. The pipe is filled with pressurised LPG containing 95 mole% propane and 5 mole% butane. The test pressure varies from 8 to 21 bar and the temperature is ambient temperature varying between 15 and 2ff~C. The saturation pressure of the mixture around this temperature is about 8 bar. One end of the pipe is closed and the other end is fitted with a diaphragm to the desired bore size. The bore size is controlled by a circular orifice plate and varies from full-bore to 0.05 m in diameter. Tam and Cowley did not mention the mechanism used to rup- ture the diaphragm but they reported that the rupture is rapid and produces an unobstructed full-bore aper- ture. Tam and Cowley also gave the roughness of the pipe which is characterised by a length scale of 0.05 mm. Pressure and temperature are measured at ten different locations along the pipe. The pipe is also weighed by 20 uniformly spaced Ioadcells. To further characterise the flow, eight neutron back-scattering sensors are installed to measure the void fraction.

12 ! (a) Pressure histories at closed end.

~ . . . . . .

• ~ 6

t

o 5 ,o , , 25 Time (s)

(c) Temperature histories at closed end. 3o]

2o i

'° I -I0

E o -20 F-

-30

-40

-50 0

~ ~ ~ ' ~ ~

I 0 20 25 Time (s)

12

I0

8

(b) Pressure histories at open end.

c

t-.

Measurement . . . . . . . META-HEM . . . . MSM-CS . . . . . MSM-no CS

6 ' - , . . . . .

4 ' \ , "'""" ......

I 0 5 I0 15 20 25

Time (s)

(d) Temperature histories at open end.

20 L Measurement lOg N.~ . . . . . . . META-HEM

]~ ~ . . . . MSM-CS MSM-noCS

I0 "",

0 5 10 15 20 25 Time (s)

Fig. 4(a)-(d). Results of two-component systems.

Page 30: Two-phase Blowdown From Pipelines

Modelling of two-phase blowdown from pipelines--II 2183

1.2

1.0

0.8

o "~. 0.6 g

N o.4 o

0.2

O"

-0.2 0

I000-

900-

800-

700"

o SO0-

400-

300"

200" I00"

(e) Void fraction histories at closed end.

I .'"

)

5 I0 15 20 25

Time (s)

1.0.

0"8 i i 0.6"1:

(f) Void fraction histories at open end.

~ J

. . . . MSM-CS

0, I 0.2

o

. . . . . MSM-no

) ) i

I0 15 20 25

Time (s)

(g) Total inventory of line.

"'~t~ , Load cell "~'.'.: .~ t ..... Holdup

~,~"~; ....... MET^-HEM "~"~:'. - - - - MSM-CS

~N~'~'~,j ~ . . . . . MSM-no "x~--~: .~ .

"., "l 'v':r L

" ;.. ' ','.,.: , . ,

- ~ ' , , :.:.,,.,,

10 15 20 25 Time (s)

Fig. 4(e)-(g). Results of two-component systems.

The measurements chosen here for validation are for a full-bore rupture and the initial pressure and temperature are 11.25 bar and 19.9°C, respectively. The calculation is made by assuming a constant wall heat transfer coefficient of I00 W/m2K and a constant roughness length scale of 0.05 ram. 25 uniform meshes of 4 m and time-step of 0.005-0.02 s were used. Fur- ther mesh refinement does not have significant effect on the result. The Peng-Robinson equation of state is used for all calculations of thermodynamic and phase behaviour. The computation time is about 20 h for the marginal stability model and 8 h for the homogeneous model of which 95% of the CPU time is spent on the thermodynamic and phase equilibrium calculations. Varying the above parameters does not have a signifi- cant effect on the predictions except the roughness length scale, which we will discuss later. More valida- tions are given in Chen (1993).

Figure 4(a)-(g) shows the results for pressure at closed and open ends, temperature at closed and open ends, void fraction at the closed and open ends, and inventory of the pipe, respectively. By closed and open

ends we mean the gauges or sensors that are closest to the closed or open ends. For pressure and temper- ature gauges, they are 1.5 m from closed end and 0.14 m from open end. For void fraction sensors, they are 1.74 m from the closed end and 0.18 m from the open end. The calculated results shown are the values at cells closest to open and closed ends. Two different measurements of inventory are shown. The first one is based on the direct weighing by the load cell (legend "load cell"). The second method is based on the liquid holdup measurement from the neutron back-scatter- ing (legend "holdup").

The measured pressure at the closed end shows a rapid drop to close to the saturation pressure and then stays around the saturation pressure until 12 s after which the pressure drops gradually to ambient pressure. This result shows that the effect of thermo- dynamic non-equilibrium in long and large diameter pipe is not so significant compared to that in small pipes. The most significant non-equilibrium effect, the delayed bubble nucleation shown by the pressure undershoot below the saturation pressure, is probably

Page 31: Two-phase Blowdown From Pipelines

2184

dominant only in the first 0.5 s. Within 1 s the pres- sure recovers to above the saturation line and the remaining results do not seem to be affected by the delayed nucleation.

The comparison of measurements and HEM pre- dictions are surprisingly good. All the predictions of pressure, temperature, void fraction and inventory histories almost coincide with those of measurements. The only exception is the void fraction at the closed end which is over-predicted after 7 s. The HEM pre- dicts a rapid increase of void fraction after 7 s while the measured void fraction does not show a rapid increase until 10 s. This discrepancy is attributed to another effect of thermal non-equilibrium in which the vapour generated is less than the equilibrium value and therefore the measured increase in void fraction is slower than the prediction. Nevertheless, the good agreement between the measurement and HEM predictions of pressure and temperature sug- gests that the state of the two-phase mixture is not far from equilibrium. One other point to note is that the predicted pressure drops slightly more quickly (rather than slowly) than that measured. However, as the discrepancy is relatively small, it is difficult to judge whether the quicker pressure drop is a result of uncertainty in the rupture mechanism or in other sources. Another possibility of quicker predicted pres- sure drop could be the smaller roughness length scale. Figure 5 shows the predicted pressure at the closed and open ends for roughness length scales varying from 0.0 to 0.02 mm. If the pipe is assumed to be smooth, the pressure can be under-predicted signifi- cantly at the closed end. We take this opportunity to correct an argument in Chen et al. (1993). In that paper, it was mistakenly said that the HEM over- predicts the discharge rate because the pipe is as- sumed smooth. This is incorrect as we discussed be- fore. By increasing the roughness to larger than 0.1 mm, it is possible to get the predicted closed end pressure higher than the measurement. Nevertheless,

J. R. CHEN et al.

the pressure at the open end is still under-predicted before I I s which is the time when pressure/void waves reflect back to the open end. During this per- iod, an increase of roughness will result in a decrease of pressure at the open end as one can see from Fig. 5. As we already showed in Fig. 2(b) that HEM should predict a higher pressure even at the open end, we therefore suggest that the discrepancy in the pressure drop prediction does not result from the wall friction but perhaps results from an obstacle that remains at the pipe end after the pipe is ruptured. Uncertainty in the roughness length scale is still possible but it is considered less significant than the uncertainty of the rupture mechanism.

Figure 4(a)-(g) also shows the predicted results of MSM with and without concentration stratification. As there could be some uncertainty associated with the measurements, good agreement between MSM and the measurements is not expected. The first thing to note in these results is that the results of MSM with and without concentration stratification (CS) are al- most indistinguishable. This is not unexpected as we mentioned in Section 2.3 that the mean flow velocity will offset the change of composition due to the slip. In the present case, the mean flow velocity is greater than 100 m/s while the slip is only about 20 m/s as shown in Fig. 6. The change of overall composition as shown in Fig. 7 is found to be less than 5% before choking ends and increases thereafter to about 50% at the closed end and 15% at the open end. The change of composition is larger at the closed end than the open end because the mean flow velocity for the former is smaller than for the latter. Nevertheless, the large change of composition does not have any significance on the predicted pressure, temperature or inventory histories because the inventory after choking ceases is quite small.

Returning to the results in Fig. 4(a)-(gL both the predicted pressure and temperature at both ends of the pipe are lower than the HEM predictions and

12

10 ~ . . ~ C!osed / end

~ $ ~ _ ~ " ~ . . ~ . ~ . , 0.2mm ~.̂ . ~ ~ O. lmm

. o ",~.:~Smooth" \~"..'.~. ";.-. x "', "\ ,'

Az " ~ ' ; ~ ' ~ ' ~ ' - ' ' ~ 0.05 mm

2 Open end ""~i\x~,: i ~

, i 0 5 10 15 20 25

Time (s)

Fig. 5. Effect of roughness on predicted pressure histories.

120 t

.,,'t V - ' ~ ~. ak" apour ~ ' ~ " . . . . " ~ 17 velocity

.~ 80

> 60-

40-

. . . . MSM-C$ ',', \ 20- . . . . . . . MSM-no CS "~,,.~._....

0 5 10 15 20 25 Time (s)

Fig. 6. Comparison of predicted flow velocities at rupture plane of HEM, MSM and MSM-CS.

Page 32: Two-phase Blowdown From Pipelines

0.075

Modelling of two-phase blowdown from pipelines--ll

250.

2185

0.070

r~

0.065 Q .~,

o 0.060 o E

0.055

200 I '

z" / ' ..'

4 m from closed end / .'

• Closed end~ ./'Y'

4 m from open end\~/:':\\ Open end! ~" "~. / . ; ~

5 10 15 20 Time (s)

150

_~ 100

50

- - META-HEM . . . . MSM-CS . . . . . . . MSM-no CS

0.05 0 25 0 25 5 10 15 20

Time (s)

Fig. 7. Predicted variations of total mole fraction of butane Fig. 8. Comparison of predicted release rate of HEM, MSM in different locations of pipe. without CS and MSM with CS.

measurements while the void fraction is higher at both ends. This suggests the out flow rate is over-predicted by MSM. The over-predicted release rate is con- sidered a result of the uncertainty of the experiment and inaccuracy of MSM itself. However, the good agreement of inventory history of Fig. 4(g) shows that the release rate of both models might not differ signifi- cantly. The comparison of the release rate of both models is shown in Fig. 8. Except for the first second of blowdown, the predicted release rate of MSM (both with and without CS) and HEM are quite close. This result confirms that HEM is a good approximation for strongly coupled two-phase flow in long and large diameter pipes.

Finally we compare the present predictions with the other work. The Isle of Grain experiment has also been compared with predictions from the BLOW- DOWN code of Richardson and Saville (1991) and the PLAC code (Hall et al., 1993). The BLOWDOWN code for pipelines is developed based on quasi-steady state, equilibrium and homogeneous two-phase flow assumptions. It is expected that, provided the time step used in BLOWDOWN is small enough, the re- sults will approach those of HEM. The thermodyn- amic package used by BLOWDOWN is the same as this work. PLAC (Phiibin and Govan, 1990), on the other hand, is a shortened version of the nuclear safety code TRAC (LANL, 1986) with a different built-in thermodynamic package for handling the phase beha- viour of multi-component mixtures. The conservation equations solved by TRAC and PLAC are similar to the ill-posed Wallis model except that thermodynamic non-equilibrium is allowed. This poses the question of which temperature should be chosen for the equilib- rium phase behaviour as there is no other way except a thermodynamic equilibrium condition to determine the phase behaviour of multi-component mixtures. This is the problem of allowing thermal non-equilib- rium in the two phases for multi-component mixtures which is noticed and avoided in this work by using the

consistent equilibrium assumption. Also, the original critical flow boundary condition in TRAC, the Trapp and Ransom (1982) criterion which consider the real part of the characteristics to be zero if the character- istics are complex, is replaced by a homogeneous frozen flow model which is not consistent with the flow model in PLAC.

The results are shown in Fig. 9(a) and (b) for pres- sure and inventory histories, respectively. The predic- tion of BLOWDOWN is found to be quite close to that of HEM at the open end except for the initial rapid pressure drop which the quasi-steady assump- tion cannot resolve properly. The predicted pressure at the open end is found to be close to the result of MSM rather than the HEM. It is not clear why this difference arises. The predicted inventory history is slightly lower than the HEM but is still within the uncertainty of the experiment. The predicted initial inventory of the line is lower because a methane-based CSP is used. Overall, the BLOWDOWN code does a good job judged on its simplifications. PLAC, on the other hand, performs surprisingly poorly. Firstly, the sharp drop of pressure at the closed end is not re- solved but shows smearing similar to the quasi-steady assumption of BLOWDOWN. Secondly, the pressure at both ends drop almost simultaneously by over 2 bar about 5 s after the start of the blowdown. This behviour is not observed in the measurements or the present predictions and the reason is not clear. Most of all the predicted inventory, shown in Fig. 9(b), shows a surprisingly low initial inventory. Hall et al. (1993) mentioned that the initial state of the fluid is 80% liquid. Apparently, this results from incorrect thermodynamic predictions. Also the predicted inven- tory drops to less than 20% of its initial value within 5 s. Clearly, the release rate is considerably over-es- timated by the homogeneous frozen critical flow boundary condition. In summary, the poor perfor- mance of PLAC suggests the importance of accurate predictions of thermodynamic and phase behaviour

Page 33: Two-phase Blowdown From Pipelines

2186

(a) Pressure histories at closed and open ends of line. 1 2 , -

Measurement . . . . . . . META-HEM . . . . MSM-CS 10- . . . . . BLOWDOWN

\ - . . . . PLAC 8- ~ Closed end

,~ [ ' - - . : ~ .

, . . . . . . . . . " : b ~ . - - . ? , k

/ Openend-" .-"~.:.', .N,

o ; ,o 15 20 25 Time (s)

(b) Total inventory of line.

1900~ t 0 0 0 ~ - - . . . . Load cell

"\ --. Holdup s0o~, - M~TA-.EM

. . w °004- . . . . . . LAC

\

! -

• \ h . k I

t

I 0

J. R. CHEN et al.

a perfect gas and the homogeneous two-phase flow model.

• In full-bore blowdown from short pipes (< 10m), both thermal and mechanical non- equilibrium are significant. Validation against Necmi and Hancox's measurements shows that the release rate is significantly under-predicted by the HEM but is slightly over-estimated by the marginal stability two-fluid model which allows mechanical non-equilibrium. To model the blowdown behaviour more accurately, relax- ation of thermal equilibrium assumption is sug- gested.

• In blowdown from long pipes ( I> 100 m), thermal non-equilibrium is found to be insignificant ex- cept for the very early stages of rarefaction wave propagation. The results from HEM and MSM show good agreement with the Isle of Grain experiments. In particular, the release rates pre- dicted by HEM and MSM are very close to each other.

• The effect of concentration stratification in the Isle of Grain experiments is found to be insigni- ficant. It is suggested that, in transient blowdown processes in simple geometry pipelines contain- ing multi-component mixtures, this effect can be generally neglected.

• Thermodynamic and phase equilibrium calcu- lations are found to occupy at least 95% of the CPU time of the total calculations. Accurate predictions of the thermodynamic and phase equilibrium behaviour are essential to transient flow calculations. But more efficient thermodyn- amic calculation methods are clearly desirable.

5 10 15 20 25 Time (s)

Fig. 9. Comparison of results of META, BLOWDOWN and PLAC.

and the consistency of the two-phase model and criti- cal flow boundary condition in transient blowdown calculations.

4. CONCLUSIONS

A simplified numerical method is developed for solving general equilibrium multi-component two- phase flow models. Extension of this method to the proposed marginal stability model is demonstrated. Case studies have been made with the homogeneous equilibrium model and the marginal stability model and these have been compared with experimental measurements for two-phase blowdown from pipes containing one- and two-component mixtures. The most important conclusions are as follows.

• The simplified numerical method is found to be satisfactory for blowdown problems through validations against the wave-tracing solution for

Acknowledyements Financial support from British Gas pie for JRC through the award of research scholarship is grate- fully acknowledged.

NOTATION

A cross-sectional area of flow channel Co distribution parameter Cs~ interracial drag coefficient Dh hydraulic diameter of flow channel E entrainment fraction hk Specific enthalpy of phase k h,, mixture specific enthalpy g acceleration of gravity G mass flux m Reynolds stress coefficient

inertial coupling constant M~ interfacial force density N number of components of fluid p pressure Q~ external heat flux rp radius of particles t time T~ fluid temperature T~mh ambient temperature uk velocity of phase k u,, mixture velocity

Page 34: Two-phase Blowdown From Pipelines

Modelling of two-phase blowdown from pipelines--II

~, averaged local relative velocity V~ volume of computat ion cell j yk~ mass fraction of component i in phase k y , ,~ mixture mass fraction of component i Y~j.j=I.2 reduced densities of liquid and vapour

phases of component i z spatial coordinate

Greek letters ~t void fraction At time-step Az space-step I'm interfacial mass transfer rate Fuk interfacial momentum flux of phase k F~ interfacial equilibrium entropy transfer rate /~L viscosity of liquid pj, j : l . 2 reduced densities of liquid and vapour

phases pk density of phase k pm mixture density a surface tension r~ interfacial viscous drag rwk wall drag of phase k

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