The Model Fluid

Similarly to the model of Vortmann et al. [132], it is assumed in this study that the model fluid is an ensemble of clusters of molecules. A cluster is viewed as an elementary subsystem. The size of the clusters should be selected so that subsystems are small enough portions of the system that are characterized by the property to be entirely either in liquid or vapour phase (local bistability) and large enough to still can be characterise with thermodynamic intensive variables. Each cluster consists of the same number of water molecules and, thus it has a constant mass. Changes of pressure or temperature modify the volume occupied by a cluster, and this leads to the changing of specific volume of a cluster. The fluid clusters are characterized by a sharp existence function x∈

{ }

0,1 . If a cluster is in the vapour phase, the existence function is equal to 1, x=1, and if a cluster is in the liquid phase, the existence function is zero, x=0. (This assignment does not agree with the Landau formalism where the high temperature phase is set to 0 and the low temperature phase is 1, but only with this assignment can we obtain the hysteresis function in its common form.) The isibaric phase transition can be studied on the isobar curve in T−v plane. In the ideal case of a phase transition, vapour-liquid phase change occurs if the temperature drops to the local pressure

determined equilibrium saturation temperature and it is valid for the inverse process for the liquid-vapour transition as well. A non-equilibrium phase transformation assumption enables some degree of supersaturation-superheating in the cluster without phase change. If the constant pressure is considered, the supersaturation means that the temperature in the cluster decreases without phase change happens. In this context the saturation temperature is the earliest temperature where the condensation of cluster may occur. Further decrease of temperature the cluster gets into metastable state (see isobar curve in T−v plan in Fig. 43), and at the very latest when the spinodal condition is reached, the vapour phase becomes unstable and the cluster is converted into liquid. On reheating a reverse transformation takes place, at latest when the isobar curve intersects the spinodal [20]. Hence theoretically the supersaturation (or superheating) of a cluster can approximate the appropriate spinodal conditions. Physically, the metastable state becomes short-lived before the spinodal is reached [81], therefore the assumption that the spinodal let be the upper limit of supersaturation is not acceptable.

Let we assume, that the allowable degree for the temperature decreases below the saturation temperature is ΔT. If the upper limit of the superheating can be assumed to be equal to the corresponding supersaturation limit ΔT, than each individual cluster that is in the vapour state can be transferred into the liquid state in the temperature interval defined by [T_s,T_s−ΔT] and the vaporization of a cluster that is in the liquid form can occur in the temperature interval of [T_s,T_s+ΔT]. Consequently this phase transformation is a switching process that shows hysteretic character, which can be seen in Fig. 48. In this way the thermally induced non-equilibrium vapour-liquid phase transformation of a fluid cluster is similar to the local bistability hypothesis assumption which has been developed for magnetic materials [15].

T x

Ts

...

...

Ts+ΔT Ts-ΔT

1

0

(vapour)

(liquid)

Fig. 48. The existence function x of the individual water cluster switching from 0 to 1 and 1 to 0 around the saturation temperature T_s

The upper limit of the acceptable supersaturation i.e. the value of ΔT has to be determined. This is the only parameter that needed to identify for the presented model.

The spinodal of vapour supersaturation is proportional to the relative pressure, p/ p_c where p_c is the critical pressure [4], [81]. This give me the idea to introduce a relative temperature T/ (T_c T_c is the critical temperature) based heuristic supersaturation limit.

This limit is close to the saturation curve and vanishes at the critical state (see Fig. 49), red curve under the vapour coexistent curve. The limit has been obtained by calculating a weighted specific volume ν_w from the liquid v_l

( )

T_s and vapour specific volume

( )

_s

v T

v at the saturation temperature T_s, by a weighing factor of T_s/ T_c

( ) (

_{s s c}

) ( ) (

_{v s s c}

) ( )

_{l s}

w T = T/T v T + 1−T/T v T

ν . (41)

100^-5 10^-4 10^-3 10^-2 10^-1

100 200 300 400 500 600 700

v (m³/mol)

T (K)

p = 10⁶ Pa

van der Waals liquid metastable vapor metastable IAPWS IF95 liquid binodal vapor binodal hysteresis limit 2Δ T

T_c

°

Fig. 49. Isobars of water calculated by the van der Waals equation of state in the metastable regions and with IAPWS IF 95 formulation in the stable phase domains.

The corresponding pressure can be determined by the van der Waals equation of state.

This calculation is valid for 300K<T_s <T_c, since this is the temperature limit for the spinodal approximation as well in [81]. ΔT can be determined from the pressure with the hysteresis limit curve. Graphical interpretation this calculation is shown in Fig. 50.

The projection of the vapour coexistence curve (binodal), the spinodal decomposition curve and the hysteresis limit to the p−T plane are shown in this figure as well. The proposed limit agrees with experimental data that can be found in the literature. For example Hewitt in [53] suggested that superheat temperatures found in practice for water about 10°C and Lienhard and Stephenson [88], in their experiments with water jets, found the water could not withstand superheat temperatures higher than about 27°C.

It can be seen in Fig. 50 that the relation between the log(p) and the reciprocal of temperature

( )

1T can be approximated by a linear fitting in the form of

( )

^{p a T b}

log = + . The values of parameters a and b for the approximation of curves of vapour binodal, spinodal and hysteresis limit are given in Table VI.

500 550 600 650 10⁶

10⁷ 10⁸

T (K)

p (Pa) p=const.

vapour binodal (IAPWS IF95) hysteresis

limit

vapour spinodal

ΔT

Fig. 50. Spinodal decomposition curve, binodal curve and the hysteresis limit in p−T plane

Table VI

Parameters of linear approximation

[ ]

1 K

a b

[ ]

−

Vapour binodal −4.74×10³ 24.3 Vapour

spinodal −1.58×10³ 19.3 Hysteresis limit −4.33×10³ 23.7

Fig. 51. Linear approximation of vapour binodal and the proposed limit of hysteresis

The linear approximation of the hysteresis limit in Table VI is valid for temperature interval of 350K<T_s <600K, therefore (41) has to be limited as well. The value of

( )

T_s

ΔT can be expressed by

(

−68033

) (

−7903

)

=

ΔT T_s T_s . T_s , 600350≤T_s ≤ (K). (42)

In document Nemlineáris hőterjedés és hőképanalízis (Pldal 64-67)