Acentric Factor and Normal Boiling Point

Definitions and Relationships

Normal Boiling Point

The mathematical formulation of the normal boiling point is given by

$$\begin{equation}\label{eq:normal_boiling_point_definition} T_b = \{T_b \;\; \textrm{such that} \;\; p_v(T_b) = 1 \textrm{atm}\} \end{equation}$$

Acentric Factor

The mathematical definition of the acentric factor (ω) is given by

$$\begin{equation}\label{eq:acentric_factor_definition} \omega ≝ - \log \left[ \frac{p_v(T^*)}{p_c} \right] - 1 \end{equation}$$

where \(T^* = 0.7\cdot T_c\) (temperatures in absolute units), \(p_v\) is the vapor pressure, \(p_c\) is the critical pressure, and \(T_c\) is the critical temperature.

Relationship

Both the acentric factor and normal boiling points are tied to a point on the vapor pressure curve at conditions below the critical point. There is a one-to-one relationship between the acentric factor and normal boiling point, meaning that there is only one acentric factor that honers the critical point and a given normal boiling point, and vice versa. This relationship is shown in Figure 1 below, and shows what acentric factors result in a ± 10C shift in the normal boiling point.

Since the normal boiling point is the defining bound for single carbon number components, it is a physical constraint that should be honored. This is the reason why, in whitson^PVT, there is no regression on the acentric facotrs, but rathre on the normal boiling points.

Figure 1: Figure showing the vapor pressure line (solid lines), critical point (circle), normal boiling points (square symbols), and the vapor pressure at T_r = T^* / T_c = 0.7 for three EOS models where (ω, T_b) = (0.27056, 93.197C) (black), (ω, T_b) = (0.17378, 83.197C) (blue), and (ω, T_b) = (0.38288, 103.197C) (red). The component is a SCN C₇ component and the EOS is a PR78 model with volume shifts.

Acentric Factor from the Normal Boiling Point

The only accurate method honor both the normal boiling point and critical point for a given component is to solve for the acentric factor iteratively until the vapor pressure at the normal boiling point temperature is equal to atmospheric pressure! This is shown in the animation below.

Acentric Factor from Correlations

Approximate values for the acentric factor can be estimated from correlations. The Lee-Kesler and Edmister models utilize the component normal boiling points to estimate the acentric factor, making it a bit more accurate, but the Pedersen model does not!

Acentric factor correlations do NOT honor the normal boiling points!

It is important to note that these values do not honor the normal boiling point exactly and will impact the EOS calculated normal boiling points!

Lee-Kesler Model

The Lee-Kesler correlation for acentric factors aims to convert the normal boiling point to a reasonable estimate for the acentric factor. The correlation is given by

$$\begin{equation}\label{eq:lee_kesler_acentric_factor_tbr_less_than_0.8} \omega = \frac{ -\ln(p_c/p_{sc}) + A_1 + A_2 / T_{br} + A_3 \ln(T_{br}) + A_4 T_{br}^6 }{A_5 + A_6 / T_{br} + A_7 \ln(T_{br}) + A_8 T_{br}^6}, \; T_{br} < 0.8 \end{equation}$$

and

$$\begin{equation}\label{eq:lee_kesler_acentric_factor_tbr_more_than_0.8} \omega = -7.904 + 0.1352 K_w - 0.007465 K_w^2 + 8.359 T_{br} + \frac{1.408 - 0.01063 K_w}{T_{br}} , \; T_{br} \geq 0.8 \end{equation}$$

where \(T_{br} = T_b / T_c\) is the reduced normal boiling point, \(K_w\) is the component Watson factor, and the coefficients are given below. The temperature units are in absolute values.

Table 1: Coefficients for the Lee-Kesler acentric factor model.

Coefficient	Value
A1	-5.92714
A2	6.09648
A3	1.28862
A4	-0.169347
A5	15.2518
A6	-15.6875
A7	-13.4721
A8	0.43577

Edmister Model

$$\begin{equation}\label{eq:edmister_acentric_factor} \omega = \frac{3}{7} \frac{\log(p_c/p_{sc})}{(T_c/T_b)-1} - 1 \end{equation}$$

where \(p_c\) is the critical pressure, \(T_c\) is the critical temperature, \(p_{sc}\) is the surface pressure, \(T_b\) is the normal boiling point. The temperature units are in absolute values.

Pedersen Model

$$\begin{equation}\label{eq:pedersen_acentric_factor} m(\omega) = e_1 + e_2 M + e_3 \rho +e_4 M^2 \end{equation}$$

where \(m(\omega)\) is the slope in Soave's correction term (\(\alpha(T)\)) to the EOS a parameter, \(\omega\) is the component acnetric factor, \(M\) is the component molecular weight, \(\rho\) is the component liquid viscosity at surface conditions. The equations for \(m(\omega)\) for the PR EOS can be found here, and for the SRK EOS can be found here.

Non-physical component acentric factors

The acentric factors using the Pedersen model are non-monotonic for heavier SCN components, resulting in non-physical thermodynamic behavior! This is shown in the figure below using the Pedersen model with the PR EOS model.

Parameters for SRK and PR EOS models

EOS Type	e1	e2	e3	e4
SRK	0.74310	4.8122e-3	9.6707e-3	-3.7184e-6
PR	0.373765	5.49269e-3	1.17934e-2	-4.93049e-6

Excel Test File

We at whitson have made an Excel file with all the correlations above, that you can download for free here.