Vapor-liquid equilibria calculations for components of natural gas using Huron-Vidal mixing rules

Abstract

Fossil fuels, such as oil or natural gas are composed of a complex mixture of hydrocarbons and impurities such as water, carbon dioxide, or nitrogen. The oil and gas companies have great interest in removing these impurities since they impact negatively natural gas production. In this regard, natural gas is commonly dehydrated with an absorbent agent, such as triethylene glycol (TEG). Due to its low volatility, there is little research concerning its partition throughout vaporization. Therefore, simulations of the natural gas industry processes need accurate calculations of thermodynamic properties and phase equilibria for the optimization of operations and design of new installations. For that, the Peng-Robinson equation of state is the most frequent model used in gas applications, refineries, and petrochemistry in a wide range of simulators. In this work, three modeling approaches are proposed to improve thermodynamics calculations: The models are based on the Peng-Robinson equation of state associated with non-random two-liquid model through the original Huron-Vidal (HV), first order modified Huron-Vidal (PR-MHV1-NRTL) and Van der Waals (PR-VdW) mixing rules. Moreover, the Peng-Robinson was coupled with the Almeida-Aznar-Telles modification in the attractive term. Parameters for the binary systems composed of components of natural gas as well as triethylene glycol were estimated. The most important results are that vapor-liquid equilibria data from 15 binary systems were correlated with the proposed modeling approaches, and a phase diagram for the ternary system methane/CO₂/TEG was predicted. The main conclusions are that the PR-VdW and PR-MHV1-NRTL models are adequate to represent the available binary VLE data for natural gas components and mixtures of TEG with methane and CO₂, but more experimental data regarding the molar fraction of TEG in the vapor phase are important; and that the TEG loss through vaporization in a flash vent could be around 204g per day.

Graphic abstract

Appendix A Demonstration of the matrix form of the NRTL equation

Models of $G^E$ usually contain variables with one or two indexes that engage in operations as $\sum _i{a_i}$, $\sum _j{A_{ij}b_j}$ or $\sum _k{A_{ik}B_{kj}}$ . The NRTL model Renon and Prausnitz (1968), has two binary interaction parameters ($A_{ij}$ and $A_{ji}$) and a non-randomness parameter ($\alpha _{ij}$ = $\alpha _{ji}$) for each pair of components ($A_{ii}$ = 0 and $\alpha _{ii}$ = 0 for all i). For the NRTL model, the molar Gibbs energy in excess of a mixture is given by Eq. A.1:

$$\begin{aligned} \frac{\bar{G}^E}{RT} = \sum _{i=1}^{n} \frac{x_i\sum _{j=1}^{n} \Lambda _{ji}x_j}{ \sum _{j=1}^{n}G_{ji} x_j} \end{aligned}$$

(A.1)

where

$$\begin{aligned} G_{ij} = \exp {\left( \frac{-\alpha _{ij}A_{ij}}{T}\right) } \end{aligned}$$

(A.2)

$$\begin{aligned} \Lambda _{ij} = {\left( \frac{A_{ij}G_{ij}}{T}\right) } \end{aligned}$$

(A.3)

To express this model in matrix notation, first we define the matrices in Eqs. A.4 and A.5:

$$\begin{aligned} \underline{\underline{G}} = \exp {(-T^{-1}\underline{\underline{\alpha }} \circ \underline{\underline{A}})} \end{aligned}$$

(A.4)

$$\begin{aligned} \underline{\underline{\Lambda }} = {T^{-1}\underline{\underline{A}} \circ \underline{\underline{G}})} \end{aligned}$$

(A.5)

where the double underlined letters represent matrices and the symbol “$\circ$” refers to the matrices term-by-term multiplication (Hadamard’sproduct). Translating Eq. A.1 requires a sequence of indices elimination. For example, we eliminate j by defining Eqs. A.6 and A.7:

$$\begin{aligned} a_i= & {} \sum _j G_{ij}^tx_j \end{aligned}$$

(A.6)

$$\begin{aligned} b_i= & {} \sum _j \Lambda _{ji}x_j \end{aligned}$$

(A.7)

In which the superscript “t” indicates transpose operation. Eqs. A.6 and A.7 are equivalent to the Eqs. A.8 and A.9:

$$\begin{aligned} \underline{a} = \underline{\underline{G}}^t \underline{x} \end{aligned}$$

(A.8)

$$\begin{aligned} \underline{b} = \underline{\underline{\Lambda }}^t \underline{x} \end{aligned}$$

(A.9)

where the underline letters represent vector. Next, i is eliminated by making Eqs. A.10 and A.11:

$$\begin{aligned} c_i = \frac{b_i}{a_i} \end{aligned}$$

(A.10)

$$\begin{aligned} d_i = {c_i}{x_i} \end{aligned}$$

(A.11)

Which allows us to write Eq. A.12:

$$\begin{aligned} \frac{\bar{G}^E}{RT} = \sum _i d_i = \underline{1}^t\underline{d} \end{aligned}$$

(A.12)

where:

$$\begin{aligned} \underline{c} = \mathscr {D}^{-1}(\underline{a})\underline{b} \end{aligned}$$

(A.13)

$$\begin{aligned} \underline{d} = \underline{\mathscr {D}}(\underline{c})\underline{x} \end{aligned}$$

(A.14)

Then substitutions lead to Eq. A.8:

$$\begin{aligned} \frac{\bar{G}^E}{RT} = \underline{1}^{t}{\mathscr {D}}[{\mathscr {D}}^{-1}(\underline{\underline{G}}^t \underline{x})\underline{\underline{\Lambda }}^t \underline{x}]\underline{x} \end{aligned}$$

(A.15)

Finally, we can use algebraic rules in order to obtain the Eq. A.16, a matrix expression for the molar excess Gibbs energy via NRTL:

$$\begin{aligned} \frac{\bar{G}^E}{RT} = \underline{x}^t[ {\mathscr {D}}^{-1}(\underline{\underline{G}}^t \underline{x})\underline{\underline{\Lambda }}^t \underline{x}] \end{aligned}$$

(A.16)

The next step is to obtain a similar expression for the components activity coefficients ($\gamma _i$ for all i) by applying differentiation rules to the Eq. A.16. For that, let us consider the Eq. A.17:

$$\begin{aligned} RT\ln (\gamma _i) = \frac{\partial N\bar{G}^E}{\partial n_i} \end{aligned}$$

(A.17)

In which N is the number of moles of the mixture, $n_i$ is the number of moles of component i. The Eq. A.18 expresses the activity coefficients in matrix notation:

$$\begin{aligned} \ln (\underline{\gamma }) = {J}_n^t \left( \frac{ N\bar{G}^E}{RT}\right) \end{aligned}$$

(A.18)

One can then apply the product rules and form the chain to get the Eq. A.19:

$$\begin{aligned} \ln (\underline{\gamma }) = {J}_n^t (\underline{x}) {J}_x^t \left( \frac{ N\bar{G}^E}{RT}\right) N + \underline{1}\frac{ \bar{G}^E}{RT} \end{aligned}$$

(A.19)

Since $N = \underline{1}^t \underline{n}$ and, therefore, ${J}_n^t N = \underline{1}$, the vector of mole fraction is given by $\underline{x} = N^{-1}\underline{n}$, which leads to Eq. A.20:

$$\begin{aligned} {J}_n\underline{x} = N^{-1}\left( \underline{\underline{I}} - \underline{x}\;\underline{1}^t\right) \end{aligned}$$

(A.20)

By applying the jacobian operator and using the product rule, we obtain an expression for $J_x\left( \frac{\bar{G}^E}{RT}\right)$ and finally, substituting it in Eq. A.19, we have the Eq. A.21, a matrix expression for the activity coefficients via NRTL:

$$\begin{aligned} \begin{aligned} \ln (\underline{{\gamma }}) = (((\underline{\underline{E}} +\underline{\underline{E}}^t )&-((\underline{\underline{G}}\;\underline{\underline{D}})\underline{x}^t))\circ \underline{\underline{E}}^t)\circ \underline{x} \\ \underline{\underline{E}} =&( \underline{\underline{ \tau }}\;\underline{\underline{G}}) \underline{\underline{D}} \\ \underline{\underline{ \tau }} =&{\underline{\underline{A}}}/{T} \\ \underline{\underline{D}} =&\left( \frac{1}{ \underline{\underline{G}}^t \circ \underline{ x} }\right) ^t\\ \underline{\underline{G}} =&\exp {\left( - \underline{\underline{\alpha }}\; \underline{\underline{\tau }} \right) } \end{aligned} \end{aligned}$$

(A.21)

The two working equations (or sets) are Eqs. A.16 and A.21, where the $\bar{G}^E$ is used directly in the expression for the mixing rule to obtain the EoS parameter $a(\underline{x})$ and the expression for the vector $\ln (\underline{\gamma })$ is used directly in the vector $\ln (\underline{\phi })$ expression derived for that mixing rule.