2.3 N UMERICAL S IMULATIONS OF P HASE T RANSITIONS : THE D IFFUSE I NTERFACE M ETHOD
2.3.2 Phase Non-Equilibrium Models
where Δp is the half-width of transition from vapour to liquid, see Fig. 47. Three types of functions are usually used to ensure the transition between the vapour and the liquid:
integration of analytical formula of speed of sound, polynomial connection and sinusoidal connection [110]. The most frequently used method proposed originally by Delannoy [32] in which the connection from liquid to vapour can be defined with a sinusoidal function is
⎟⎟⎠
where cmin corresponds to the minimal speed of sound in the mixture. The parameter cmin can be approximated by cmin =2cv ρv ρl if ρl >>ρv. For water at standard conditions m/scmin≈25 [32]. The graphical representation of ρ
( )
p for water is shown in Fig. 47 with assumption of ρv =0.5kg m3, ρl =1000kg m3. In this method the order parameter is the density, and the variation of density describes the interface between the two phases. The model parameter is the minimal sound speed in the two phase mixture, which must be determined by experiments [24].-1 -0.5 0 0.5 1
2.3.2 Phase Non-Equilibrium Models
Vapour that is supersaturated above its equilibrium saturation pressure or is sub-cooled
metastable state [21]. Contrary, the liquid phase can be prepared in a metastable state by superheating above its boiling temperature or by stretching below its saturated vapour pressure [31]. Non-equilibrium vapour-liquid transitions can be expected for example in film condensation in heat exchangers, condensation tubes, in condensation type cooling of electronic devices, nuclear power plants [79], in turbines [53]; capillary condensation with hysteresis in porous materials [103]; in cavitation phenomenon [24], [132] and in liquid flashing phenomenon [88], [139], etc.
Kinetic Phase Transition Models: the Ginsburg-Landau Theory
In kinetic relaxation models, the phase transition is considered to be a molecular exchange process that develops during a finite time interval [20]. The derivation of a kinetic model for a particular phase transition in a thermodynamical system consists of identifying the order parameter which characterizes the difference between the two phases. Then the equations of state are determined by constructing the Helmholtz free energy density f or the Gibbs free energy density g as a function of order parameter φ and absolute temperature T. Then all relevant thermodynamic quantities are derived from free energy density functions. The free energy based phase transition models differ from each other in the form of the approximation of the free energy function and in the simplification considerations [19], [37].
The Landau theory and the phase field model
In the Landau theory of phase transition [85], it is assumed that the free energy density g admits a power series expansion of the form
( )
∑
∞=
=
0
2 1 i
i r i T,P, ,N ,N ,...,N g
g φ φ , (33)
where the coefficient functions gi are analytic functions. The kinetics of the phase transition implies that the order parameter has to be a function of space x and time t, that is φ =φ
( )
x t, . If the order parameter φ is a function, then the total free energy g must be a functional acting on a function space to which φ belongs. The variational derivative(
∂g ∂φ)
can be interpreted as a generalized thermodynamic force that tends to decrease the value of the total free energy. A possible kinetic equation is obtained by assuming that the order parameter φ responds to the impact of this thermodynamic force at a rate proportional to its negative value. Therefore, in isothermal conditions, the Ginzburg–Landau equation is obtained as the relaxation lawφ φ
∂
−∂
∂ =
∂ g
b t , (34)
where b is a function of φ and ∂g ∂φ is the variational derivative of a free-energy functional g, which has minimum value at equilibrium states. The kinetic equation of the order parameter is called as Cahn-Allen equation for non-conserving dynamics and Cahn-Hillard equation for conserving dynamics [19]. Both equations have been derived under the premise that the temperature is constant. In non-isothermal conditions the integrand of g usually is modified through a rescaling factor of 1/T. The thermodynamically consistent phase field model were described by Penrose and Fife [108].
It can be concluded that in the kinetic models the dynamics of pure phase transition have been studied very extensively but separately from fluid flow problems [37]. These methods require identifications of coefficient functions of the polynomial in (33). In most cases it is a very difficult task, and several assumptions have to be made to obtain an appropriate solution. For engineering applications the kinetic models are usually highly simplified.
A Simplified Relaxation Model
Downar et al. [35] developed a simplified non-equilibrium relaxation model for one-dimensional flashing liquid flow. This model a homogenous relaxation model taking into account the non-equilibrium evaporation leading to metastable conditions and predicts the mass flow rates, the pressure distributions for one-dimensional flashing water. The vapour generation rate Γ has been approximated by the first term of Taylor series as
τ φ ρφ ρ φ = −
=
Γ e
Dt
D , (35)
where ρ is the mixture density, φe is the equilibrium vapour quality, τ is the relaxation time, which is fitted as a power function of void fraction and pressure decay amount [35].
Statistical Approach
Statistical models are based on the consideration of a local bistability hypothesis. The local bistability hypothesis consists in considering the first-order phase transformation as an ensemble of several independent simple transitions. This hypothesis goes back to the magnetic hysteresis models which consist in subdividing the system for small elements in such a way that each single element has just two configurations in its free energy [1]. The system is then considered as a collection of independent bistable objects. This idea has been used in the last few years to study hysteresis and relaxation effects [15], [100]. The martensitic–austenitic phase transformations of shape memory alloys are the typical case for the thermodynamics of systems with hysteresis, described with the statistical method constructed on the local bistability hypotheses [100].
Basso et al. [15] proposed a hysteresis operator for thermally induced first-order phase transformations. Basso et al. find that the switching rules of phase change turn out to be identical to the rules governing the Preisach model of hysteresis [57]. The switching dynamics from 0 to 1 and 1 to 0 is connected to an intrinsic damping. According to the kinetic phase transition model, locally the switching rate is proportional to a force given by the gradient of the free energy with respect to the independent variable x, x∈
{ }
0,1x g t
x
∂
−∂
∂ =
γ ∂ , (36)
where γ is a damping coefficient. The kinetic depends therefore on the specific shape of
(
∂g ∂x)
. If it is not assumed any specific shape, the typical time for the transition to occur can be estimated as approximation with Δt=γ( )
2gb , where gb is the energy barrier between the two minima of free energy function. The thermal relaxation dynamics are connected to the probability that the transitions occur spontaneously by thermal activation over the energy barrier. The master equation for the transition is1 10 0
1 w01P w P
dt
dP = + , (37)
where P0 and P1 are the probability to find the system state in 0 and 1, respectively. The transition rates w01 and w10 are given by exponential of the energy barrier to overcome the thermal energy kBT, kB is the Boltzmann constant.
The difficulty in this development are that the Gibbs free energy functions of the pure phases have to be known in advance and applying a Preisach-type hysteresis model to fluid systems, the hysteresis memories have to be coupled to small volumes of fluids, which are in motion. These models are very complicated to handle numerically.
Before enter upon the proposed hysteresis phase transition model, a statistical approach of modelling a cavitating flow presented by Vortmann et al. [132] has to be shortly reviewed. Vortmann et al. modified the original concept for shape memory alloys of Achenbach and Müller [1] for constructing a statistical model for water to describe cavitating flow. They designed a model fluid comparable to the metallic lattice. For this reason, the concept of a cluster of molecules has been introduced. Each cluster consists of the same number of water molecules and, thus, has the same constant mass. If a cluster grows, it has to change the volume occupied. This happens in the Vortmann model due to decreasing pressure. The rate equation for vapour quality x (x=mv m,
mv is the mass of vapour, m is the total mass) is formulated as
(
x)
Kl v xKv l dtdx
→
→ −
−
= 1 . (38)
The first part in the right hand side of the (38) is the gain term the last part is the loss term. The gain of vapour clusters depends on the number of liquid clusters expressed by the liquid quality
(
1−x)
, and further on the probability Kl→v of liquid clusters changing their state of aggregation from liquid to vapour. Then the probabilities are( ) ( ) and v∞ are the smallest and greatest possible specific volume occupied by the molecules respectively, and τ is the relaxation time. The probability Kv→l differs from Kl→v only in the integration boundaries that cover the region of the respective phase area. For the solution of (38), it is necessary to declare the specific Gibbs free energy functions for the pure phases and in the metastable and unstable regions as well.
In result, the homogenous type model of 2D cavitating flow consists of mass and momentum conservation equations for a mixture, completed by the evolution equation of fluid properties (38) and an additional transport equation for the vapour volume fraction, ξ. The transport of vapour volume fraction is introduced to overcome problems due to the high density variation between the phases; it turns out the relation between the derivatives of mass and volume fractions.
It can be concluded that the kinetic type of phase transition models demand several estimations for the model parameters that could not be verified by experience or measurement.