Z-Factor

The Z-factor is a main variable in the generalized equation of state model defined by

$\begin{equation}\label{eq:eos_general} Z = \frac{pv}{RT} \end{equation}$

where $v$ is the molar volume ($v=V/n$).

There is a unique Z-factor for each fluid phase in the system. For real gas the Z-factor can be thought of as the gas correction or deviation factor. Some of the most common Z-factor models are given in the following sections.

Real Gases

Note

The main article on real gases is: Real Gas Law

Real gas behavior is defined by its deviation from the ideal gas law. There are several approaches to describe the non-ideal behavior, but the most common is by estimating the Z-factor correction (sometimes called the gas correction factor for real gases). Other approaches involve applying an EOS to describe the phase behavior of the gas. An Example of this is the Benedict-Webb-Rubin EOS models.

Some of the most common methods for estimating the Z-factor for hydrocarbon real gases behavior are given below.

Standing-Katz Z-factor Chart

Note

For more information see the real gas page section on the Standing-Katz chart

The Standing-Katz charts are a set of empirically derived plots made by Standing and Katz in 1944¹. The general approach of the chart is to first find the gas pseudo-reduced conditions by dividing the current conditions by the pseudo-critical conditions , i.e. yielding the pseudo-reduced pressure ($p_{pr}=p/p_{pc}$) and pseudo-reduced temperature ($T_{pr}=T/T_{pc}$). The pseudo-critical conditions can be calculated by correlation (e.g. the Sutton correlation) or by applying Kay's mixing rule if compositions are available. Once the pseudo-reduced conditions are found, then the Standing-Katz chart can be used to look up the associated Z-factor value.

Hall-Yarbrough Estimation of Gas Z-Factor

Note

For more information see the real gas page section on the Hall-Yarbrough gas Z-factor

The Hall-Yarbrough Z-factor approach is an iterative method to calculate the Z-factor for real gases. This approach uses the Carnahan-Standing hard-sphere EOS to iteratively estimate the Z-factor. Similar to the Standing-Katz chart the Hall-Yarbrough approach uses the pseudo-reduced conditions and the molecular weight ($MW$), critical temperature ($T_c$), critical pressure ($p_c$) and accentric factor ($\omega$).

Two-Phase Z-Factors

The most common approach for two-phase EOS modeling involves the Z-factor as the main variable. The industry standard has become using cubic EOS models to describe the variation in Z-factor as a cubic polynomial. A general description of cubic EOS models as well as some specific models are given in the following sections.

Cubic Equation of States

Note

The main article on cubic EOS models is: Cubic Equation of State Models

A cubic EOS is any fluid model that can be written of the following form

$\begin{equation}\label{eq:ceos} A_0 + A_1\cdot Z +A_2\cdot Z^2 + A_3\cdot Z^3 = 0 \end{equation}$

For the most common cubic EOS models used in the industry (i.e. the Peng-Robinson and Soave-Redlich-Kwong EOS models) the general structure of the coefficients and parameters of the model are the same. The main bulk parameters are given by

$\begin{eqnarray}\label{A_gen} A = a\frac{p}{(RT)^2} \end{eqnarray}$

$\begin{eqnarray}\label{B_gen} B = b\frac{p}{RT} \end{eqnarray}$

$\begin{eqnarray}\label{a_gen} a = \Omega_a \frac{R^2T_c^2}{p_c}\alpha(T) \end{eqnarray}$

$\begin{eqnarray}\label{b_gen} b = \Omega_b \frac{RT_c}{p_c} \end{eqnarray}$

$\begin{eqnarray}\label{eq:alpha} \alpha(T) = [1+m(\omega)(1-\sqrt{\frac{T}{T_c}})]^2 \end{eqnarray}$

where the coefficients $\Omega_a$ and $\Omega_b$ are different for the different EOS models, the function for the $m(\omega)$ term is different for the different EOS models and the coefficients of equation \eqref{eq:ceos} are different for the different EOS models. In the following sections, the specific structure of the coefficients in equation \eqref{eq:ceos} are given for the van der Waals, Peng-Robinson and Soave-Redlich-Kwong EOS models are given.

van der Waals EOS

Note

The main article on the van der Waals EOS models is: van der Waals Equation of State

$\begin{equation}\label{eq:ceos_vdw} Z^3-(B+1)Z^2+AZ-AB=0 \end{equation}$

Note that the $\alpha(T)$ term was not introduced until Soave's modification of the original Redlich-Kwong EOS, so the van der Waals EOS model has $\alpha(T)=1$ for all temperatures.

Peng-Robinson EOS

Note

The main article on the Peng-Robinson EOS models is: Peng-Robinson Equation of State

$\begin{equation}\label{eq:ceos_pr} Z^3-(1-B)Z^2+(A-3B^2-2B)Z-(AB-B^2-B^3)=0 \end{equation}$

Soave-Redlich-Kwong EOS

Note

The main article on the Soave-Redlich-Kwong EOS models is: Soave-Redlich-Kwong Equation of State

$\begin{equation}\label{eq:ceos_srk} Z^3-Z^2+(A-B-B^2)Z-AB=0 \end{equation}$

M.B. Standing and D.L. Katz. Vapor-liquid equilibria of natural gas-crude oil systems. Transactions of the AIME, 155:paper SPE–944232–G, 1944. ↩