Center for Turbulence ResearchAnnual Research Briefs 2003

49

Simulation of inviscid compressible multiphaseflow with condensation

Philip Kelleners

Condensation of vapours in rapid expansions of compressible gases is investigated. Inthe case of high temperature gradients the condensation will start at conditions well awayfrom thermodynamic equilibrium of the fluid. In those cases homogeneous condensationis dominant over heterogeneous condensation. The present work is concerned with de-velopment of a simulation tool for computation of high speed compressible flows withhomogeneous condensation. The resulting flow solver should preferably be accurate androbust to be used for simulation of industrial flows in general geometries.

1. Introduction

A substance below its critical temperature can be present in either gaseous or liquidphase, depending on the pressure, and is referred to as a vapour. Vapours present ina mixture of gases and vapours, when subjected to expansion can condensate and formliquid droplets. This phenomenon is observed in e.g. aircraft tip vortices and in industrialflows like steam turbines. Condensation in flows of gas mixtures at high speed has beeninvestigated by, among others; Wegener (1969), Hill (1966), Campbell (1989), Schnerr etal. Schnerr (1996), Dohrmann (1989), Mundinger (1994), Adam (1996) and van Dongenet al. Luijten (1999), Luijten (1998), Prast (1997) and Lamanna (2000). Expansion innozzles of gases to supersonic speeds has often been used to investigate the physics ofcondensation. Condensation in the flow around airfoil sections and in steam turbines hasbeen investigated to a large extent. At the University of Twente a numerical tool hasbeen developed to simulate transonic flows with condensation in confined geometries.The solver operates on the basis of a finite volume method using unstructured meshes.It has been observed that results obtained with the solver are very sensitive to accurateshock prediction, and fine shock resolution in the flow field, especially in cases of stronginteraction between the gasdynamic shock and the condensation process. The focus of thepresent work is to improve the accuracy and robustness of the flow solver by improvingsolid wall boundary treatment and spatial reconstruction for simulations with secondorder spatial accuracy.

2. Physics of Condensation during Rapid Expansion

Below its critical temperature, a fluid can be in gaseous or liquid phase. The thermo-dynamical region of coexistence of vapour and liquid in equilibrium in bulk substancesis given by the Clausius-Clapeyron relation. For rapid expansions of vapour this coexis-tence region is passed without the fluid attaining equilibrium. The vapour is saturatedexpressed by the saturation ratio;

Address: Engineering Fluid Dynamics, Mechanical Engineering - University of Twente,P. O. Box 217, 7500 AE Enschede, The Netherlands

50 P. H. Kelleners

T [K]

p v[P

a]

s

220 240 260 280 3000

500

1000

1500

0

10

20

30

40

50

pv,eq (T)

pv(T)

s(T)

Figure 1. Expansion of airwater mixture in nozzleS2, partial vapour pressures andsaturation ratio s (dashed)

s =nv

nv,eq(2.1)

where nv and nv,eq are the number of moles of the vapour in the actual mixture andthe mixture in equilibrium, respectively. For pressures as high as atmospheric pressureequation 2.1 can be replaced by the more conventional expression:

s =pv

pv,eq(2.2)

where pv is the actual partial vapour pressure in the mixture and pv,eq is the equilibriumpartial vapour pressure at the same thermodynamic conditions. s, pv and pv,eq are plottedin figure 1 for the expansion of an airwater mixture, as functions of temperature.

The curve for the coexistence of liquid and vapour (pv,eq) divides the plane given bypressure and temperature into two regions. For pressures higher than the coexistencepressure the substance under consideration (water) is present in liquid form while inequilibrium state. For lower pressures the substance will be present as watervapourwhile in equilibrium. The expansion in the nozzle, as given by the partial vapour pressurepv, is so quick that the watervapour will not immediately condense under equilibriumconditions. This is the case as the characteristic time of the gasdynamic flow is muchsmaller than the time needed to form the first onsets of the new liquid phase. For thenozzle flow at hand this is due to the highcooling rate in the nozzle. So the watervapourexpands further, driving the airwatervapour mixture well away from equilibrium, asindicated by the saturation ratio which attains values as high as 40. In case s > 1 the fluidis said to be supersaturated. Formation of small liquid clusters, nuclei, at high supersaturation is the first stage of the condensation process that starts in order to reestablishequilibrium. On these newly formed nuclei, the supersaturated vapour condenses as asecond step until equilibrium is reached, the process of droplet growth. This can be seenin figure 1 by the decrease in partial vapour pressure and the increase in temperature.With so much liquid already present, the remainder of the condensation process dueto droplet growth takes place very close to thermodynamic equilibrium of the water,indicated by the saturation close to one, for the remainder of the expansion.

Compressible multiphase flow with condensation 51

The process described above is the process of homogeneous condensation. In contrast,the process of heterogeneous condensation is droplet growth on foreign, already presentparticles. However, for high cooling speeds and consequentially high supersaturationthe number of new nuclei formed exceeds any realistic number of foreign particles byseveral orders of magnitude. For the present applications only homogeneous condensationprocesses are of importance.

An effect resulting from condensation is the release of latent heat during the con-densation process, indicated by the increase in temperature during condensation in theprevious example of nozzle flow. The latent heat L is defined as

L = hv hl (2.3)with hv and hl the enthalpy of the gaseous and liquid phase, respectively. The latentheat is the enthalpy needed to evaporate a unit mass of liquid. It is a material property.

3. Nucleation and Droplet Growth models

From the previous treatment of condensation during rapid expansion, it is observedthat the condensation process consists of two consecutive stages. The first one is theformation of liquid nuclei, nucleation, the second one is the condensation of vapourmolecules on the already present nuclei, making these grow in the process of dropletgrowth. In the following, a brief treatment is given of the physical background of bothnucleation and droplet growth together with the presentation of the models used tosimulate these processes.

3.1. Nucleation

Under supersaturation vapour molecules can condense on liquid already present or onforeign objects. In the absence of both, the vapour molecules can form small clusters. Asa result of the clustering, an additional phase needs to be formed, the interface betweenthe liquid inside the cluster and the gas outside of the cluster. The interface can be re-garded as infinitely thin. At the interface there is surface tension. Thus the creation of aninterface requires energy. For very small clusters, the surface effects are dominant overvolume related effects. As a result the formation of the interface represents an energybarrier in the formation process of the nucleus. If the energy involved in the clusteringof the vapour molecules is less than required in the formation of the interface surface,the cluster will disintegrate immediately following its formation. Therefore at near equi-librium conditions, supersaturations close to one, it is highly unlikely, that a realisticnumber of stable nuclei in a volume at macro scale will be formed although clustersare constantly formed and falling apart. The previous notion results in formulation offormation enthalpy of critically sized stable nuclei, and the number densities in whichthese stable nuclei are likely to come into existence. The creation enthalpy of one liquiddroplet under equilibrium conditions is given by Dohrmann (1989):

G = 4r2 nkT ln(s) (3.1)where, G is the change in Gibbs enthalpy, r is the radius of the droplet, is the dropletsurface tension in a plane with no curvature, n is the number of molecules in the droplet,k is the Boltzmann constant, and T is the temperature. In accordance with classicalnucleation theory, we note that a stable nucleus will be created when the function forG attains a local maximum. G at this maximum is:

52 P. H. Kelleners

G =4

3r2 (3.2)

with r, the radius of this critical sized droplet being given by:

r =2

lRvT ln(s)(3.3)

where l is the density of the liquid in the droplet, and Rv is the gas constant of thevapour. The number of droplets created is given by the nucleation rate. For the nucleationrate several models are available, Classical Nucleation Theory CNTmodel (Wegener1969) and the Internally Consistent Classical Theory, ICCTmodel (Luijten 1998). Inthe present work, the CNTmodel is applied:

J =2vl

2

m3exp

(

163

3

m2l R3vT

3 ln2(s)

)

(3.4)

where v is the density of the vapour and m is the mass of one vapour molecule.

3.2. Droplet Growth

The droplet growth model to be applied depends on the regime in which droplet growthtakes place. For pressures of the order of atmospheric pressure droplet growth is basedon a balance between condensation of vapour molecules onto droplets, and evaporationof vapour molecules from the droplet. For pressures 1 to 2 orders of magni