Equations of State

Constant Density

ρ = ρ_{0}

(669)

where $ρ_{0}$ is a constant.

Polynomial Density

For gases, liquids, and solids, density can be specified as a function of temperature $T$ in the form of a polynomial. The temperature range can be subdivided into intervals, each with its own polynomial function:

Figure 1. EQUATION_DISPLAY

ρ = \sum_{i = 0}^{n} {a_{i} \cdot T}^{i - 1}

(670)

where $a$ is a polynomial coefficient.

Ideal Gas

The ideal gas law expresses density as a function of temperature and pressure:

Figure 2. EQUATION_DISPLAY

ρ = \frac{p}{R T}

(671)

where:

$p$ is the pressure
$R$ is the specific gas constant

and:

Figure 3. EQUATION_DISPLAY

R = R_{u} / M

(672)

where:

$R_{u}$ is the universal gas constant [8314.4621 J/kmol K]
$M$ is the molecular weight

Real Gas

At high pressure and low temperature, the p-v-T behavior of real gases deviates from that predicted by the ideal gas equation. This change of behavior results from the molecules of the gas taking up a significant portion of the total volume as the gas density increases. In addition, intermolecular attractive forces become increasingly important.

Van der Waals

The van der Waals equation is given by:

Figure 4. EQUATION_DISPLAY

(p + a / v^{2}) (v - b) = R T

(673)

Van der Waals replaced specific volume in the ideal gas relation, $p v = R T$ , with $(v - b)$ to account for the volume that the particles of the gas occupy, while replacing the pressure with the term $(p + a / v^{2})$ . The constant $b$ is the co-volume of the particles, and the constant $a$ is a measure of the attractive forces.

The constants $a$ and $b$ are evaluated based on the experimental observation that, in p-v space, the critical isotherm $T = T_{c} = constant$ has zero slope and an inflection point at the critical state, $p = p_{c}$ and $v = v_{c}$ . Thus the following can be written:

Figure 5. EQUATION_DISPLAY

\frac{d p}{d v} | T_{c} = 0

(674)

and:

Figure 6. EQUATION_DISPLAY

\frac{d^{2} p}{{d v}^{2}} | T_{c} = 0

(675)

which leads to the following expressions for $a$ and $b$ :

Figure 7. EQUATION_DISPLAY

a = \frac{27}{64} \frac{R^{2} T_{c}^{2}}{p_{c}}, b = \frac{1}{8} \frac{R T_{c}}{p_{c}}

(676)

These equations allow $a$ and $b$ to be determined from experimentally measured values of critical pressure and temperature, $p_{c}$ and $T_{c}$ .

Peng Robinson

The Peng-Robinson equation is given by:

Figure 8. EQUATION_DISPLAY

p = \frac{R T}{(v - b)} - \frac{a α (T_{r})}{(v^{2} + 2 b v - b^{2})}

(677)

where $v$ is the specific volume.

The function

α (T_{r})

is given by:

Figure 9. EQUATION_DISPLAY

α (T_{r}) = {[1 + (0.37464 + 1.54226 ω - 0.2699 ω^{2}) (1 - {T_{r}}^{0.5})]}^{2}

(678)

where $T_{r} = T / T_{c}$ is the reduced temperature and $ω$ is the acentric factor of the gas.

The constants are found to be:

Figure 10. EQUATION_DISPLAY

a = \frac{0.4572 R^{2} T_{c}^{2}}{p_{c}}, b = \frac{0.0778 R T_{c}}{p_{c}}

(679)

Redlich-Kwong: The Redlich-Kwong equation is given by:

Figure 11. EQUATION_DISPLAY

$p = \frac{R T}{(v - b)} - \frac{a}{[T^{0.5} v (v + b)]}$

(680)
and
Figure 12. EQUATION_DISPLAY

$a = \frac{0.4275 R^{2} T_{c}^{2}}{p_{c}}, b = \frac{0.0867 R T_{c}}{p_{c}}$

(681)

Soave-Redlich-Kwong: Soave modified the Redlich-Kwong equation, replacing the $T^{0.5}$ power in the denominator of Eqn. (681) with a different temperature-dependent expression [210]. This modification improves predictions for liquid densities and vapor-liquid equilibrium, and is:
Figure 13. EQUATION_DISPLAY

$p = \frac{R T}{v - b} - \frac{a α (T_{r})}{v (v + b)}$

(682)
The function $α (T_{r})$ is:
Figure 14. EQUATION_DISPLAY

$α (T_{r}) = {[1 + (0.48 + 1.574 ω - 0.176 ω^{2}) (1 - T_{r}^{0.5})]}^{2}$

(683)
and
Figure 15. EQUATION_DISPLAY

$a = \frac{0.42748 R^{2} T_{c}^{2}}{p_{c}}, b = \frac{0.08664 R T_{c}}{p_{c}}$

(684)

Modified Soave-Redlich-Kwong: Graboski and Daubert modified the Soave-Redlich-Kwong model to improve its behavior for phase equilibrium calculations that involve non-polar molecules [173] and [174]. All equations remain identical to the Soave-Redlich-Kwong model, except the function $α (T_{r})$ which changes to:
Figure 16. EQUATION_DISPLAY

$α (T_{r}) = {[1 + (0.48503 + 1.55171 ω - 0.15613 ω^{2}) (1 - T_{r}^{0.5})]}^{2}$

(685)

Equilibrium Air

For air at high temperatures, the effects of molecular dissociation, internal energy excitation, and ionization become significant.

For situations where the time scale of these effects is much shorter than the time scale of the flow, these effects can be treated in an equilibrium manner. In this case, the properties of the flow, such as the density, specific heat, and transport properties, become expressions of two thermodynamic variables. These expressions typically take the forms of curve fits that account for the chemical reactions and energy modes present at high temperature. In particular, the curve fits of Gupta and others [176] were used. These curve fits specify the compressibility factor, specific heat, enthalpy, viscosity, and thermal conductivity as a function of temperature and pressure. The curve fits are valid for temperatures less than 30,000 K and for pressures between 10^–4 and 10² atmospheres.

The equilibrium air equation of state provides the compressibility factor as a field function. In the context of this model, the compressibility factor represents the ratio of the undissociated molecular weight to the mean molecular weight of air [176]:

Figure 17. EQUATION_DISPLAY

Z (T, P) = \frac{M_{o}}{\bar{M}}

(686)

where $M_{o}$ represents the molecular weight of undissociated air and $\bar{M}$ represents the mean molecular weight of air at the given temperature and pressure. Because the mean molecular weight is a function of the composition, the compressibility factor can be used as a measure of the degree of dissociation in the air, with higher values signifying a large degree of dissociation and ionization.

Thermal Non-Equilibrium

For a gas, the internal energy is composed of translational, rotational, vibrational, and electronic modes. In equilibrium, these energy modes can be adequately described by a single characteristic temperature. For thermal non-equilibrium, the translational and rotational energy modes are described by one temperature, while the vibrational and electronic energy modes are described by an additional vibrational-electronic temperature. The vibrational-electronic temperature is determined by solving an additional energy equation describing the conservation of vibrational-electronic energy. Gnoffo and others [172] give this equation, simplified to non-ionizing flows by Lockwood [192]):

Figure 18. EQUATION_DISPLAY

\begin{matrix} \frac{\partial ρ e_{v e}}{\partial t} + \nabla \cdot (ρ e_{v e} U) = \nabla \cdot (k_{v e} \nabla T) + \nabla \cdot (ρ \sum_{s}^{} h_{v e, s} D_{s} {\nabla y}_{s}) \\ + \sum_{s = mol}^{} ρ_{s} \frac{(e_{v, s}^{*} - e_{v, s})}{< τ_{s} >} + \sum_{s = mol}^{} {\dot{ω}}_{s} e_{v e, s} \end{matrix}

(687)

The exchange of vibrational-electronic energy and translational-rotational energy is modeled through a relaxation term. Gnoffo and others [172] present this relaxation, and provide a simplification which allows the calculation to include fewer species-dependent parameters:

Figure 19. EQUATION_DISPLAY

e_{v, s}^{*} - e_{v, s} \approx C_{v, v}^{s} (T - T_{v})

(688)

Figure 20. EQUATION_DISPLAY

\sum_{s = mol}^{} \frac{ρ_{s} C_{v, v}^{s}}{〈 τ_{s} 〉} \approx \frac{ρ C_{v, v}^{s}}{{\bar{τ}}_{v}}

(689)

To describe the relationship between vibrational-electronic enthalpy and temperature, the vibrational specific heat must be specified. This vibrational specific heat is specified by simultaneously setting the total specific heat and the translational-rotational specific heat. For this model, the translational-rotational energy modes are assumed fully excited $(F - E)$ , giving a constant translational-rotational specific heat $(C_{p, t r})$ . With these two values specified, the vibrational specific heat and enthalpy are given by [172]:

Figure 21. EQUATION_DISPLAY

C_{p, v e} (T_{v e}) = C_{p, total} (T_{v e}) - C_{p, t r}^{F - E}

(690)

Figure 22. EQUATION_DISPLAY

h_{v e} (T_{v e}) = h_{total} (T_{v e}) - C_{p, t r}^{F - E} (T_{v e} - T_{ref}) - h_{o}

(691)

IAPWS-IF97

The models of IAPWS-IF97 (International Association for the Properties of Water and Steam, Industrial Formulation 1997) allow you to run simulations of liquid water or gaseous steam.

These models include calculations of density and other thermodynamic properties. The IAPWS-IF97 provides fundamental polynomial equations for the specific Gibbs free energy, $g (p, T)$ . Specific volume, internal energy, entropy, enthalpy, heat capacity, and speed of sound are derived from the fundamental equation using appropriate combinations of the dimensionless Gibbs free energy and its derivatives [182].

Based on the Gibbs free energy, the IAPWS-IF97 models in Simcenter STAR-CCM+ are only valid within certain ranges of temperature and pressure. These ranges are represented in a diagram of regions [182] which are essentially the phases of water and steam.

The following table lists the components of the region diagram and how they work with IAPWS-IF97 models in Simcenter STAR-CCM+.


Component	Ranges of Validity		STAR-CCM+ Models
Region 1 (liquid)	273.15 K ≤ T ≤ 623.15 K	p ≤ 100 MPa	IAPWS-IF97 (Water)
Region 2 (steam)	273.15 K ≤ T ≤ 1073.15 K	p ≤ 100 MPa	IAPWS-IF97 (Steam)
Region 3 (saturation)	This phase region is not supported in Simcenter STAR-CCM+.
Boundary 4	This boundary lies between liquid and saturation on one side, and steam on the other.
Region 5 (steam)	1073.15 K < T ≤ 2273.15 K	p ≤ 50 MPa	IAPWS-IF97 (Steam)

The enthalpies $H_{s h}^{L}$ of superheated liquid and $H_{s c}^{V}$ of subcooled vapor are calculated as:

Figure 23. EQUATION_DISPLAY

\begin{matrix} H_{s h}^{L} (T, P) = H_{r 1} (T_{s a t}, P) + C_{p, r 1} (T_{s a t}, P) (T - T_{s a t}) & (T > T_{s a t}) \end{matrix}

(692)

Figure 24. EQUATION_DISPLAY

\begin{matrix} H_{s c}^{V} (T, P) = H_{r 2} (T_{s a t}, P) + C_{p, r 2} (T_{s a t}, P) (T - T_{s a t}) & (T < T_{s a t}) \end{matrix}

(693)

where:

$T_{s a t} (P)$ is the saturation temperature
The subscripts $r 1$ and $r 2$ refer to Region 1 and Region 2, respectively.

Multicomponent Gases

For multicomponent gases using the Peng Robinson, Soave-Redlich-Kwong, or Modified Soave-Redlich-Kwong equations, $a α (T_{r})$ and $b$ are based on geometric and arithmetic summations over the components $i$ :

Figure 25. EQUATION_DISPLAY

{(a α (T_{r}))}_{mix} = {(\sum_{i = 1}^{} y_{i} \sqrt{a_{i} α_{i} (T_{r})})}^{2}, b_{mix} = \sum_{i = 1}^{} y_{i} b_{i}

(694)

where $y_{i}$ is mole fraction.