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23.6.4 Built-in Thermodynamic Wet Steam Properties

There are many equations that describe the thermodynamic state and properties of steam. While some of these equations are accurate in generating property tables, they are not suitable for fast CFD computations. Therefore, FLUENT uses a simpler form of the thermodynamic state equations [ 410] for efficient CFD calculations that are accurate over a wide range of temperatures and pressures. These equations are described below.



Equation of State


The steam equation of state used in the solver, which relates the pressure to the vapor density and the temperature, is given by [ 410]:


 P = \rho_v R T (1 + B \rho_v + C {\rho_v}^2) (23.6-13)

where $B$, and $C$ are the second and the third virial coefficients given by the following empirical functions:


 B = a_1 (1+ \frac{\tau}{\alpha})^{-1} + a_2 e^{\tau} (1-e^{-\tau})^{\frac{5}{2}} + a_3 \tau (23.6-14)

where $B$ is given in $m^3$/kg, $\tau$ = $\frac{1500}{T}$ with $T$ given in Kelvin, $\alpha$ = 10000.0, $a_1$= 0.0015, $a_2$= -0.000942, and $a_3$= -0.0004882.


 C = a (\tau - \tau_{\rm0})e^{- \alpha \tau} + b (23.6-15)

where $C$ is given in $m^6$/ $kg^2$, $\tau$ = $\frac{T}{647.286}$ with $T$ given in Kelvin, $\tau_o$= 0.8978, $\alpha$=11.16, $a$= 1.772, and $b$= 1.5E-06.

The two empirical functions that define the virial coefficients $B$ and $C$ cover the temperature range from 273 K to 1073 K.

The vapor isobaric specific heat capacity $Cp_v$ is given by:


 Cp_v = Cp_0(T) + R \left [ \left [ (1-\alpha_v T)(B-B_1)-B_2... ... + \alpha_v T C_1 - \frac{C_2}{2} \right ] {\rho_v}^2 \right ] (23.6-16)

The vapor specific enthalpy, $h_v$ is given by:


 h_v = h_{\rm0}(T) + RT \left [ (B-B_1) \rho_v + (C-\frac{C_1}{2}) {\rho_v}^2 \right ] (23.6-17)

The vapor specific entropy, $s_v$ is given by:


 s_v = s_{\rm0}(T) - R \left [ \ln{\rho_{\rm v}} + (B+B_{\rm ... ...ho_{\rm v} + \frac{(C+C_{\rm 1})}{2} {\rho_{\rm v}}^2 \right ] (23.6-18)

The isobaric specific heat at zero pressure is defined by the following empirical equation:


 Cp_0(T) = {\sum_{i=1}}^6 a_i T^{i-2} (23.6-19)

where $Cp_0$ is in KJ/kg K, $a_1$ = 46.0, $a_2$ = 1.47276, $a_3$ = 8.38930E-04, $a_4$ = -2.19989E-07, $a_5$ = 2.46619E-10, and $a_6$ = -9.70466E-14.

and

$B_1$ = $T \frac{dB}{dT}$, $C_1$ = $T \frac{dC}{dT}$, $B_2$ = $T^2 \frac{dB^2}{dT^2}$, and $C_2$ = $T^2 \frac{dC^2}{dT^2}$.

Both $h_0(T)$ and $s_0(T)$ are functions of temperature and they are defined by:


 h_0(T) = \int Cp_0 dT + h_c (23.6-20)


 s_0(T) = \int \frac{Cp_0}{T} dT + s_c (23.6-21)

where $h_c$ and $s_c$ are arbitrary constants.

The vapor dynamic viscosity $\mu_v$ and thermal conductivity $Kt_v$ are also functions of temperature and were obtained from [ 409].



Saturated Vapor Line


The saturation pressure equation as a function of temperature was obtained from [ 302]. The example provided in Section  23.13.5 contains a function called wetst_satP() that represents the formulation for the saturation pressure.



Saturated Liquid Line


At the saturated liquid-line, the liquid density, surface tension, specific heat $Cp$, dynamic viscosity, and thermal conductivity must be defined. The equation for liquid density, $\rho_{\rm l}$, was obtained from [ 302]. The liquid surface tension equation was obtained from [ 409]. While the values of $Cp_l$ , $\mu_l$ and $Kt_l$ were curve fit using published data from [ 93] and then written in polynomial forms. The example provided in Section  23.13.5 contains functions called wetst_cpl(), wetst_mul(), and wetst_ktl() that represent formulations for $Cp_l$ , $\mu_l$ and $Kt_l$.



Mixture Properties


The mixture properties are related to vapor and liquid properties via the wetness factor using the following mixing law:


 \phi_m = \phi_l \beta + (1-\beta) \phi_v (23.6-22)

where $\phi$ represents any of the following thermodynamic properties: $h$, $s$, $Cp$, $Cv$, $\mu$ or $Kt$.


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