# Inside VMG | September 2014

## Expanded Fluid Viscosity Model in VMGSim

*By Herbert Loria, Marco A. Satyro - VMG Calgary*

*and*

*Hamed Motahhari, Harvey W. Yarranton - University of Calgary*

### Introduction

The calculation of viscosities is an important part of process simulation where the correct calculation of pressure drops and heat transfer coefficients is paramount. When applied to computer simulation problems, viscosity correlations should have a small number of adjustable parameters to ensure a maximum of physical significance and predictable extrapolation behaviour, easily determined parameters from incomplete or estimated data, and speed.

Virtual Materials Group is pleased to announce the addition of a new tool that satisfies all of the criteria previously set for a viscosity model suitable for computer simulation purposes. The Expanded Fluid (EF) viscosity model has been adapted for efficient integration in VMGSim thanks to the right selection of the density models, the automatic determination of fluid specific parameters and the use of generalized rules for the calculation of binary interaction parameters.

The enhanced EF model fit experimental viscosities of pure hydrocarbons, water and polar compounds important for the simulation of oil and natural gas systems with average absolute errors just above 5 %. The implemented EF model was tested against experimental viscosity data that included hydrocarbon and aqueous mixtures with average absolutes errors of 0.7 and 6.2 % respectively. The EF model was also applied to crude oil (bitumen) examples.

### EF Viscosity Model

The general principle behind the EF viscosity model is simple: as a fluid expands there is a greater distance between it molecules and its fluidity increases (and therefore, the viscosity decreases). Yarranton and Satyro1 used this principle to develop the following correlation for the fluid viscosity as a departure from the low pressure gas viscosity:

where μ is the viscosity of the fluid in mPa s, μG is the low pressure gas viscosity in mPa s, c2 is a fluid constant fit parameter, and β is a correlating parameter given by:

where ρ is the density, and ρ*s is the density beyond which the fluid cannot be compressed without incurring a solid-liquid phase transition. Note, when the density approaches to zero, the predicted viscosity approaches the gas-low pressure viscosity. When the density approaches the compressed state density, the predicted viscosity goes to infinity.

This form of correlation, between density and viscosity, was found to be adequate at atmospheric pressures over a moderate range of temperatures. The model was improved by introducing pressure dependence into the compressed state density as follows1:

where ρ0s is the compressed state density in a vacuum, P is the pressure and, c3 is a constant. The c3 constant can be roughly related to the molecular weight of the component.

In general, the model has three temperature independent parameters for each fluid: c2, c3 and ρ0s. The inputs to the EF viscosity model are the measured fluid density, pressure, and low pressure gas viscosity. Note that the effect of the temperature on viscosity is introduced through the fluid density and low pressure gas viscosity, both of which are temperature dependent, while only the density has a pressure dependency.

For fluids with significant hydrogen bonding, such as water or methanol, the c2 parameter becomes a function of the temperature4:

where T is the temperature, and c2∞, Kc2 and γc2 are fitting parameters.

### Implementation of the EF Viscosity Model in VMGSim

### Pure Components

Two viscosity databases, one of hydrocarbons and one of natural gas components were compiled from the NIST database. The databases were screened to remove obvious outliers, after screening both databases included 13,609 liquid or vapor phase viscosity points. For each component in the databases the adjustable parameters from the EF viscosity correlation were calculated.

Figures 1 to 4 are some examples of VMGSim’s EF viscosity model results compared against experimental data for n-heptane, cyclohexane, toluene, water, methanol and carbon dioxide, respectively. Observe the almost negligible deviations of the model fits at higher pressures; it was possible to obtain these small deviations thanks to the c3 parameter and its effect on viscosity at different pressures.

**Figure 1.** EF viscosity model (lines) fitted to experimental n-heptane viscosity data for: (a) saturated liquid and vapor, (b) liquid at high pressure

**Figure 2.** EF viscosity model (lines) fitted to experimental toluene viscosity data for: (a) saturated liquid, (b) liquid at high pressure

**Figure 3.** EF viscosity model (lines) fitted to experimental water viscosity data for: (a) saturated liquid, (b) liquid at high pressure. Note the model performance at extreme
pressures.

**Figure 4.** EF viscosity model (lines) fitted to experimental methanol viscosity data for: (a) saturated liquid and vapor, (b) liquid at high pressure

Viscosity experimental data are needed in order to estimate the fluid specific parameters for the compounds not included in the previous datasets. However; it would not be practical to get experimental information for each one of the almost 20,000 compounds available in VMGSim and use it to obtain the EF model fluid specific parameters. To obtain the fluid specific parameters for each compound in VMGSim a general procedure based on viscosity parameters from Yaws database, which are included inside VMGSim, was designed.

### Mixtures

Mass mixing rules were proposed to extend the EF viscosity correlation to mixtures as follows:

where i, j are pure component indices, nc is the number of components, w is the mass fraction of the pure component, and βij is a binary interaction parameter. The low pressure gas viscosity can be calculated using the Wilke’s method defined as follows:

where x is the mole fraction of the pure component and,

The model was tested on binary hydrocarbon and aqueous mixtures databases. The data for the hydrocarbon mixtures were obtained from Chevalier et al. and from NIST6 in the case of the aqueous mixtures. Viscosities were first predicted with no interaction parameters (βij = 0) and then fitted by adjusting the interaction parameters to minimize the error between the measured and calculated viscosity. The interaction parameters were generalized in simple correlations based on the components’ density and normal boiling point.

Figures 5 to 9 are some examples of the results with generalized interaction parameters for different pairs of hydrocarbon and polar compounds.

**Figure 6.** EF viscosity model (lines) fitted to experimental binary viscosity data (symbols) for the mixtures: (a) n-hexane / n-heptane, (b) n-hexane / n-hexadecane

**Figure 7.** EF viscosity model (lines) fitted to experimental binary viscosity data (symbols) for the mixtures: (a) benzene / n-decane, (b) cyclohexane / n-hexadecane

**Figure 8.** EF viscosity model (lines) fitted to experimental binary viscosity data (symbols) for the mixtures: (a) benzene / cyclohexane, (b) benzene / o-xylene

**Figure 9.** EF viscosity model (lines) fitted to experimental binary viscosity data (symbols) for the mixtures: (a) diethylene glycol / water, (b) methanol / water

### Oil Pseudo Components

In the case of oil pseudo components, the fluid specific EF viscosity parameters are calculated inside the Oil Characterization environment based on selected estimation methods or user input experimental data.

### Validations of the EF Viscosity model on Heavy Oil Systems

The application of the EF viscosity correlation to heavy oil systems is illustrated in the following examples. The first example shows how the correlation predictions match high pressure experimental heavy oil viscosities and the second one shows the comparison of the correlation results with experimental data from diluted heavy oil.

### Heavy Oil Viscosities at High Pressures

A Western Canadian bitumen was characterized based on its gas chromatographic compositional analysis. The bitumen was characterized in the Oil Characterization environment of VMGSim using the Cn Compositional Analysis option and experimental viscosities. The bitumen was represented by 75 pseudo components in order to have a smooth molecular weight distribution through its pseudo components.

Once the characterization was performed, the oil was installed in the VMGSim’s flowsheet environment and the viscosities were calculated with the EF model. The predicted viscosities for this bitumen were compared with experimental viscosities at different temperatures and pressures (Figure 10) and the average absolute deviation (AAD) was 7.5%. Note that this small error was obtained without applying any interaction parameter between pseudo-components; that is, the calculated viscosities come from the EF correlation without any tuning.

**Figure 10.** Measured and calculated (EF model) viscosities of a Western Canadian bitumen at different temperatures and pressures

### Diluted Bitumen Viscosities

In this example the EF viscosity model is applied to a Cold Lake bitumen diluted with toluene. Bitumen, toluene and mixtures (containing 1.61, 4.71 and 9.55 wt% toluene) viscosities were measured from 296.15 to 394.15 K at 1 atm.

The bitumen was characterized as a single component using a molecular weight of 582 g/mol and a density of 995kg/m3 at 298.15 K. Using the experimental data presented by Satyro and Yarranton, the EF model parameters were obtained for the Cold Lake bitumen. Toluene parameters for the EF viscosity model were previously obtained. Figure 11 shows “out of the box” model results compared against experimental data for the pure bitumen, toluene and their mixtures, with an AAD of 9.3%.

**Figure 11.** Measured and calculated viscosities from Cold Lake bitumen, toluene and their mixtures at 1 atm

The predictions can of course be improved by tuning the interaction parameter between bitumen and toluene for the mixing rules used by the EF model. VMGSim can be used to automatically tune the interaction parameter based on the experimental data. By tuning the interaction parameter, the AAD was reduced to 4.88% as seen in Figure 12.

**Figure 12.** Measured and calculated (with tuned EF model interaction parameter) viscosities from Cold Lake bitumen, toluene and their mixtures at 1 atm

### Using the EF Viscosity model in VMGSim

The EF viscosity correlation is fully implemented in VMGSim 8.0 and can be used as the viscosity estimation method in conjunction with any property package. To enable the method, open the
**Thermo Model** form and select a property package, then go to the **Settings** tab and select “Expanded Fluid” from the list of Liquid or Vapor
Viscosity methods. Click **Apply** and the method will be ready to use.

To access the component specific EF correlation parameters, go to the **Thermo Model** form select the compound of interest and open the **Compound** form, the
parameters can be seen under the **Other Properties** tab.

The parameters *A, B, C, D* and *E* refer to the parameters ρ0s, c2, Kc2 and γc2 and c3 from equations 1 -5, respectively.

In addition the interaction parameters from the mixing rules from the EF viscosity correlation are also available in VMGSim, to access them go to the **Thermo Model** form, and
click the **Kij** button, the interaction parameters are under the matrix named “VMGEFViscosity”.

### References

1. Expanded Fluid-Based Viscosity Correlation for Hydrocarbons. Yarranton, Harvey W. and Satyro, Marco A. 2009, Industrial Engineering and Chemistry Research, Vol. 48, pp. 3640-3648.

2. Expanded Fluid-Based Viscosity Correlation for Hydrocarbons using an Equation of State. Satyro, Marco A. and Yarranton, Harvey. W. 2010, Fluid Phase Equilibria, Vol. 298, pp. 1-11.

3. Predicting the Viscosity of Asymetric Hydrocarbon Mixtures with the Expanded Fluid Vuiscosity Correlation. Motahhari, Hamed, Satyro, Marco A. and Yarranton, Harvey W. 2011, Industrial and
Engineering Chemistry Research, Vol. 22, pp. 12831-12843.

4. Viscosity Prediction for Natuural Gas Processing Applications. Motahhari, Hamed, Satyro, Marco A. and Yarranton, Harvey W. 2012, Vols. 322-323, pp. 56-65.

5. Extension of the Expanded Fluid Viscosity Model to Characterized Oils. Motahhari, Hamed, et al., et al. 2013, Energy and Fuels, Vol. 27, pp. 1881-1898.

6. NIST Standard Reference Database. NIST/TRC Source Database. s.l. : WinSource, 2008.

7. Yaws, C. L. Chemical properties Handbook: Physical, Thermodynamic, Environmental, Transport, Safety, and Health Related Properties for Organic and Inorganic Chemicals. New York :
McGraw-Hill, 1999.

8. A Viscosity Equation for Gas Mixtures. Wilke, C.R. 1950, Journal of Chemical Physiscs, Vol. 18, pp. 517-519.

9. Viscosity and Density of Some Aliphatic, Cyclic, and Aromatic Hydrocarbons Binary Liquid Mixtures. Chevalier, J.L.E., Petrino, P.J. and Gaston-Bonhomme, Y.H. 1990, Journal of Chemical
Engineering Data, Vol. 35, pp. 206-212

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