Helmholtz Energy Equations of State

Overview

The accurate prediction of the thermodynamic properties of substances is critical in a broad swath of applications in engineering and scientific research. An example application would be the simulation of pressure vessel failure due to fire exposure. The accurate simulation of such a scenario would require not only a sophisticated multiscale model of the metal components, but also an equation of state that can accurately describe both the caloric properties and phase equilibria of the fluid within the vessel. For a small number of important substances, reference equations have been developed that are capable of representing the best experimental physical property data within their reported uncertainties. The current state of the art is to use an equation of state explicit in the Helmholtz energy to accomplish this goal. For a pure component:

$\alpha(\tau,\delta) = \alpha^0(\tau,\delta) + \alpha^r(\tau,\delta)$

$\tau=\frac{T_r}{T}$

$\delta=\frac{\rho}{\rho_r}$

$\alpha=\frac{a}{RT}$

Where $\alpha$ is the reduced Helmholtz energy, $T$ is the temperature, $T_r$ is often (but not necessarily) the critical temperature, $\rho$ is the density, $\rho_r$ is often (but not necessarily) the critical density, $\alpha^0$ is the ideal gas contribution to the reduced Helmholtz energy, $\alpha^r$ is the residual contribution to the reduced Helmholtz energy, and $R$ is the ideal gas constant. An equation of state explicit in the Helmholtz energy has the advantage that all other thermodynamic properties may be written as a function of its derivatives. Because it is easier to obtain analytic derivatives than analytic integrals, this allows for a larger number of terms that may be used in the optimization of the functional form. This equation of state also allows one to utilize all available thermodynamic data in fitting the equation. This includes data from properties such as saturated vapor pressure, heat of vaporization, liquid density, vapor density, liquid heat capacity, vapor heat capacity, and speed of sound. The $\alpha^0$ term is typically obtained from the integration of an equation fitted to ideal gas heat capacity data. An example optimized functional form for the $\alpha^r$ term is the following:

$\alpha^r(\tau,\delta) = \sum_{m=1}^{I_{pol}} c_m \delta^{i_m} \tau^{j_m} + \sum_{m=I_{pol}+1}^{I_{pol}+I_{exp}} c_m \delta^{i_m} \tau^{j_m} e^{\delta^{k_m}}$

Where $I_{pol}$, $I_{exp}$, $c_m$, $i_m$, $j_m$, and $k_m$ are constants. A typical value for $I_{pol}$ and $I_{exp}$ is six, so an equation of this form would have 42 fitted constants (excluding $I_{pol}$ and $I_{exp}$). Fitting a new substance to an equation of state of this form does not necessarily mean fitting several dozen constants however. If there is an existing reference equation of state for a chemically similar molecule (e.g. a saturated hydrocarbon), then it may only be necessary to fit $c_m$ to obtain an adequate fit to the available experimental data. The $c_m$ constants may also be obtained by a corresponding states correlation. Span proposes a correlation for nonpolar substances where $c_m=f(m,\omega)$, where $\omega$ is the accentric factor. Xiang recommends a correlation that applies to nonpolar and polar substances of the form $c_m=f(m,\omega,Z_c)$, where $Z_c$ is the critical compressibility. Use of one of these correlations will result in poorer accuracy than fitting $c_m$, $i_m$, $j_m$, and $k_m$ or just $c_m$, but the results are superior in accuracy over using a cubic equation of state such as Peng-Robinson or a corresponding states approach such as Lee-Kesler, all of which require a comparable amount of user-input information.

Vapor-Liquid Equilibrium

In implementing a Helmholtz energy equation of state in a code, the ability to distinguish between single-phase vapor or liquid and a two-phase mixture must be made. One approach is to calculate the vapor-liquid phase envelope once and save the results in a look-up table. These values are then used as initial guesses for future subroutine calls in either converging on vapor liquid equilibrium or determining that the system is in a single phase region. Calculation of the vapor-liquid phase envelope entails the identification of all thermodynamic states that satisfy three conditions for phase equilibria. The first requirement for phase equilibria in a single component system is chemical potential equilibrium:

$g_L = g_V$

Where $g_L$ and $g_V$ are the liquid and vapor Gibbs energy. The second requirement is mechanical equilibrium:

$P_L = P_V$

Where $P_L$ and $P_V$ are the liquid and vapor pressures. The third requirement is thermal equilibrium:

$T_L = T_V$

Where $T_L$ and $T_V$ are the liquid and vapor temperatures. One approach in calculating the phase envelope is to start by solving for the liquid and vapor densities that satisfy the Gibbs energy and pressure constraints at the triple point temperature. This solution of the two equations for the two unknowns can be accomplished by Newton-Raphson provided suitable initial guesses are provided. These initial guesses can come from experimental data or simple corresponding states correlations such as those found in Properties of Liquids and Gases. Once the saturated liquid and vapor densities at the triple point are converged on, these can be used as initial guesses for the saturation properties at $T_{tr}+\Delta T$, where $\Delta T$ is some small increment in temperature, perhaps 0.1 Kelvin. Proceeding in this manner to the critical point, the phase envelope is calculated.

Resources

Access to property information calculated by reference equations of state for 75 substances of great industrial and scientific importance is provided by NIST online. Citation information for the original journal or book where each equation was published is also provided by NIST. The book by Span is a must have for anyone interested in learning more about Helmholtz energy equations of state and their implementation in scientific codes.