Phase Equilibrium Models

Numerical simulations of phase transformations are frequently performed by applying simple laws of phase changes. Here, the phase transition criterion is usually defined by assuming that phase transformation occurs, if the temperature reaches the pressure defined equilibrium (saturation) temperature (isobaric processes) or in isothermal conditions the pressure reaches the equilibrium. These types of phase transitions are most commonly described by a so called equation-of-state type models. In these models the order parameter can be directly determined from the physical quantities appearing in the general conservation laws. However, these formulations could suffer from a singularity problem for a phase change that occurs at a fixed temperature/pressure. This difficulty can be circumvented by assuming that the region where phase transition takes place covers a certain interval.

Equation-of-Sate Type Models for Thermally-Induced Phase Transformations

If the phase transition takes place instantaneously at a fixed temperature, then for example the volumetric fraction of the high temperature phase can be the order parameter φ and it can be described by a mathematical function in form of φ=1

(

T −T_s

)

as shown in Fig. 44a, where 1

( )

x is a step function whose value is zero when T <T_s but equals one otherwise [27]. Its derivative, i.e., the variation of the phase fraction with temperature shown in Fig. 44d and has the form

(

T T_s

)

dT

dφ =δ −

, (25)

in which δ

(

T −T_s

)

is the Dirac delta function whose value is infinity at the transition temperature T_s, but is zero at all other temperatures. To alleviate this singularity the Dirac delta function can be approximated by a uniform distribution over a finite temperature interval, see in Fig. 44e. Accordingly, (25) can be replaced by

T dT

d

= Δ 2 φ 1

. (26)

Alternatively, the normal distribution function like in Fig. 44f can be used as a smooth variation [27]

(

¹²

)

^{e 2(T Ts)2} phase change interval and δ is a sufficiently small positive number. Consequently, the integrals of (26) and (27) yield the linear and the error function approximations for the high temperature phase fraction, respectively, which can be seen in Fig. 44b and Fig. 44c.

Fig. 44. Volumetric fraction of gaseous phase and its variation with temperature, phase change occurs in a) and d) at a fixed temperature; in b) and e) smoothed approximation with uniform distribution; in c) and

f) smoothed approximation with normal distribution, over a 2ΔT temperature interval

Apparent Heat Capacity Method

The so called apparent heat capacity method is one of the most preferred methods for modelling problems in heat conduction with phase change [27]. The model is commonly used in simulation of freeze-thaw in soil [51]. Latent heat effects due to phase change are incorporated in the apparent heat capacity. The advantage of this approach is the same as the method presented above, i.e. that the temperature is the primary dependent variable that derives directly from the solution of the heat equation.

The heat conduction in the two-phase system can be described by

( )

0

where C_p denotes the volumetric heat capacity C_p =c_p1ρ₁ξ+c_p2ρ₂

(

1−ξ

)

, c_p is the specific heat capacity, ξ is the volume fraction of phase 1, L the specific latent heat for change from phase 1 to phase 2 and λ is the effective thermal conductivity of the mixture. The first term in (28) represents the change in energy storage with respect to time. The second term represents the rate of the latent heat released during the change of phases. The third term represents the net energy flow by conduction. The latent heat can be incorporated into the heat capacity and (28) can be rewritten as

(

^T

)

The smoothing of latent heat over a temperature interval depends on the approximation of derivative function, dξ dT. Applying this method to the water vapour-liquid phase transformation, by assuming constant heat capacity of the pure phases and constant latent heat with parameters ^c_pv ^=2.08×103kJ

(

^kgK

)

for water vapour, capacity functions have the shape similar those shown in Fig. 45.

-10 -5 0 5 10

The effective heat capacity can be obtained from the apparent heat capacity divided by the average density. The effective heat capacities with respect to the temperature at various ΔT are shown in Fig. 46. The volumetric heat capacities of pure phases are

numerical difficulties, but too large ΔT smoothes the latent heat over a broad temperature interval, causes release/absorption of heat at temperatures where phase transformations are still unexpected. Therefore the accuracy of the apparent heat capacity method is sensitive to the magnitude of the assumed temperature interval that is arbitrarily selected to approximate the Dirac-delta function. It appears from the numerical solutions presented in [27] that the results obtained by approximating the phase change at a fixed temperature with a gradual change over a small temperature interval should be acceptable if 2ΔT T_initial −T_end <0.1.

-10 -5 0 5 10

2 4 6 8 10 12 14 16 18 20 22

T - T_s (K) Ca /ρ (kJ/kgK)

ΔT=1 ΔT=2 ΔT=4

c_pv c_pl

Fig. 46. Effective heat capacity at vapour-liquid phase transition of water at Ts =373K

From the figure it can be seen that the maximum locations of the specific heat capacity functions are shifted from the location of transition temperature in the direction of higher temperature regions, i.e. the latent heat is mainly released before the transition temperature is reached. This displacement can be observed all of the models, in which the volume fraction is the order parameter of the phase transition, and there is a large difference between the densities of the pure phases. This problem can be solved by the selection of the vapour mass fraction as an order parameter.

Barotropic State Law for Isothermal Phase Transformations

Delannoy and Kueny [32] proposed a formulation that strongly links the mixture density to the static pressure: they use a barotropic law ρ

( )

p , which describes the mixture density both in the pure phases and in the transition zone. This model has been used mainly in cavitating flow models with consideration of an isothermal homogeneous equilibrium mixture model. The barotropic state law ρ

( )

p can be formulated as

⎪

( )

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 c_min corresponds to the minimal speed of sound in the mixture. The parameter cmin can be approximated by c_min =2c_v ρ_v ρ_l if ρ_l >>ρ_v. For water at standard conditions m/sc_min≈25 [32]. The graphical representation of ρ

( )

p for water is shown in Fig. 47 with assumption of ρ_v =0.5kg m³, ρ_l =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

In document Nemlineáris hőterjedés és hőképanalízis (Pldal 55-59)