In this study a new viscosity model based on the Soave-Redlich-Kwong equation of state (SRK EOS) has been proposed to predict the viscosity of pure and mixed light hydrocarbons in a wide range of pressure and temperature. Genetic algorithm in MATLAB environment is used to obtain parameters of the model. The accuracy of this new model has been compared with Lohrenz et.al correlation (LBC), the PR viscosity model of Guo et al. and the PRµ viscosity model of Fan et al. For having a fair comparison between these viscosity models, the binary interaction coefficients in mixing rules are set to zero. The results show that the proposed model base on the SRK EOS predicts the viscosity of pure light components and their mixtures with best accuracy among the other methods.
Keywords: Equation of state, Hydrocarbon, Genetic algorithm, SRK, Viscosity
1. Introduction
Viscosity of hydrocarbons is an important property which has a wide application in oil industry such as reservoir simulation, petroleum production and processing, pipeline and separation equipment design. Several methods are available for the estimation of viscosities of hydrocarbons which can be classified into three main groups: theoretical, semi-theoretical, and empirical models. The base of the theoretical models are statistical mechanics. The Erying (Glasstone et al., 1941) and the Chapman-Enskog (Chapman and Cowling., 1970) models are considered to be in this category. These models are used for hydrocarbon gases however, provide unsatisfactory results for liquid hydrocarbons (Poling et al., 2000).
Semi theoretical models have a theoretical basis but in these models there are some adjustable parameters that must be determined from experimental data. Examples of semi-theoretical models are the modified Chapman_Enskog model (Chung et al., 1984), corresponding states (Ely and Hanley, 1984; Pedersen and Fredenslund ,1987; Huber and Ely, 1992), Eyring’s reaction rate theory (Eyring,1936; Ewell and Eyring,1937), friction theory (F-Theory)( Quiñones-Cisneros et al. (2000); Quiñones-Cisneros et al. (2001)) and the Expanded Fluid model (Yarranton and Satyro,2009). The base of the empirical models is experimental data. In these models viscosities of liquids or gases are related to pressure, temperature or density of the fluid. Examples of empirical methods are the empirical residuals (Lohrenz et al., 1964; Jossi et al. ,1982), Andrade ( 1934) and Walther(1931). However, most of the empirical models cannot be applied to both liquid and gas phases. The correlation of Lohrenz et al.(1964) known as LBC method and the viscosity model of Pedersen and Fredenslund (1987) are the most popular and commonly used method in petroleum industry(Abhijit,2006). However, one of the main drawbacks of the empirical and semi-empirical model is that these methods generally require density to calculate the fluid viscosity.
The first viscosity model based on equation of state was developed by Little and Kennedy (1966) by using the van der Waals EOS and analogy between the P-v-T and T-μ-P relationship. Lawal (1986) proposed a viscosity model for hydrocarbons based on Lawal-Lake-Silberberg equation of state. However, poor results were obtained when it was applied to oil reservoir fluids. Guo et al. (1997) proposed two viscosity models based on Peng-Robinson(PR) and Patel-Teja (PT) equation of state for calculating hydrocarbons viscosity. In 2001, Guo et la. modified the viscosity model based on the PR EOS to improve the results of calculation (Guo et al., 2001). Fan and Wang (2006) based on the Peng-Robinson EOS, suggested a viscosity model for hydrocarbons. The results showed that this model predicted the viscosity of hydrocarbons more accurate respect to the Lohrenz et al. correlation and Guo et al. model. Also, Wang et al. (2007) applied the Peng-Robinson(PR) viscosity model for some hydrocarbons such as heptane, octane, nonane, and their mixtures. Predicted results showed that this model can predict the viscosity of specified system of hydrocarbons with reasonable errors. The main advantage of viscosity models based on EOS is that the viscosity of gas and liquid phase at high and low pressure can be calculated by a single model.
In this work based on the SRK EOS (Soave,1972) and analogy between P-v-T and T-μ-P, a new viscosity model has been proposed to properly predict viscosity of light hydrocarbons and carbon dioxide over a wide range of temperature and pressure.
2- Viscosity model
The equation of state used in this work is the SRK EOS (Soave,1972):
(1)
Parameters of SRK EOS can be expressed as follows:
Based on the analogy of P-v-T and T-μ-P relationships, the positions of T and P in the SRK EOS are interchanged; v and R, are replaced by μ and r, respectively (defined subsequently). Eq. (1) is then transformed to the following SRK viscosity equation:
In Eqs.(13) and (14) the values of the Q1,Q2 and Q3 have been correlated with the acentric factor. The resulting correlation are given by:
The nine constants in Eqs. 15 to17 were obtained by minimizing the average absolute deviation (AAD) between the calculated and experimental viscosity data:
For the present study, the Genetic Algorithm (GA) in MATLAB environment has been used to minimize the objective function. GA, is a metaheuristic search method that is widely used for solving optimization and approximation problems which is based on the process of natural selection. In genetic algorithm, a population of individual with associated fitness values is maintained at each generation that are represents a point in a search space and a possible solution. The individuals with higher fitness score are assigned and undergo genetic transformation by genetic operators such as crossover and mutation. The crossover operator represents of mating between two individuals (as parents) and the mutation operator introduces random modifications that selects one individuals from the parent population and changes its internal representation and consider it as the child population. Individuals (parents) in the population die and are replaced by the child population. These steps are repeated until a stopping criterion has been achieved. The GA parameters used in this study are shown in Table 1.
Table 1. Parameters used in the genetic algorithm.
Population size 20
Selection function Roulette wheel
Crossover fraction 0.402
Elite count 2
Maximum generation 80
Function tolerance 1e-6
2-1. Extension to mixtures
The proposed viscosity model was extended to mixtures by applying the following mixing rules:
In solving Eq. (7), if three viscosity roots were obtained at the specified temperature and pressure, the correct viscosity root is chosen as follows (Guo et al. (1997)):
1) In the subcritical gas region, the smallest real root greater than bʹ should be taken as gas viscosity.
2) In the liquid region, the maximum real root should be taken as liquid viscosity.
3) In the supercritical region (T> Tc), the maximum real root should be taken.
3. Results and Discussions
3.1. Pure hydrocarbon viscosity
The proposed model has been used to calculate the viscosities of pure hydrocarbons and carbon dioxide in liquid and gas states over wide ranges of pressure and temperature. Table 2 shows the results of the viscosity calculations based on the SRK viscosity model, the PRµ model, the PR viscosity model of Guo et al. and the LBC correlation. The total average absolute deviation of viscosity, predicted by the SRK model, is 8.35% whereas it is 10.29% for the PRµ, 49.01 % for the Guo et al. model and 36.46% for the Lohrenz et al. correlation (LBC), respectively. It can be seen that the SRK model is more accurate than the three other models. Figs.1-4 represents the comparison of the calculated pressure-viscosity curves of methane, ethane, n-butane and i-butane obtained by using the SRK model to the experimental data at different temperatures. As can be seen, there is a good agreement between predicted results and experimental data.
Figs. 5-7 show a graphical comparison between experimental and predicted viscosity by SRK, PRµ and Guo et al models for three ternary mixtures at atmospheric pressure. Mixtures 1 to 3 include the following composition: mixture 1 (Fig 5): Methane (1)-Ethane (2)-Propane (3) (x1=52.03mole%, x2=29.18 mole%), mixture 2 (Fig 6): Methane (1)-Propane (2)-n-Butane (3) (x1=49.28mole%, x2=24.87 mole%) and mixture 3 (Fig 7): n-Butane (1)-Propane (2)-Ethane (3) (x1=23.36 mole%, x2=37.63 mole%).
As can be seen in Figs 5-7, the predicted results by using the SRK model are in good agreement with the experimental data. The average absolute deviations are 1.62%, 4.08% and 4.05% for mixture 1, mixture 2 and mixture 3, respectively. However, large errors are observed for PRµ and Guo et al. model.
In Fig. 8 for a quaternary mixture at 0.1 MPa, the predicted results of the proposed model, PRµ and Guo et al models are compared to the experimental data. The average absolute deviation of 2.49% for the proposed model, 18.17% for the PRµ model and 74.25% for the Guo et al. model, show that the SRK viscosity model is more accurate than the other two models.
A new viscosity model based on SRK EOS has been proposed to predict the viscosity of pure and mixed light hydrocarbons. The proposed model can be applied in a wide range of pressure and temperature. For 5581 experimental data points, the proposed model predicts viscosity of pure components more accurate than the PRµ model, the PR viscosity model of Guo et al. and the LBC correlation. The results of calculation for 38 binary (2772 data points), three ternary and one quaternary mixtures prove that the present model successfully describes the viscosity of these mixtures and has the lowest deviation in comparison to the other viscosity models studied in this work.