Viscosity models for mixtures

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Viscosity in mixtures: a simple guide

What is viscosity?
- Viscosity is a liquid’s internal friction: how hard it is for layers of fluid moving at different speeds to slide past each other.
- It depends on temperature, pressure, what’s in the mixture, and how the flow changes locally.
- For many common fluids, viscosity changes with the speed of the flow (shear rate). If it stays the same, the fluid is Newtonian; if it changes, it’s non-Newtonian. This guide focuses on Newtonian fluids.

Viscosity in mixtures
- In mixtures, you can’t assume the same viscosity as in a pure substance. You need models (constitutive equations) that describe how viscosity depends on temperature, pressure, and composition.
- To use a pure-fluid model for a mixture, scientists apply mixing rules. These rules tell you how the components’ viscosities combine to give the whole mixture’s viscosity.

Major approaches to viscosity models
1) Kinetic theory for dilute gases
- For very pure, light gases at low density, viscosity can be derived from how molecules collide and exchange momentum.
- This approach uses ideas from the Boltzmann equation and the Chapman–Enskog theory. It works well for simple gases but becomes hard for complex molecules.

2) Real-fluid (dense fluids) and mixture thinking
- In real liquids and dense gases, a common idea is to split viscosity into two parts:
- A dilute-gas contribution (what you’d get if the gas were very thin).
- A dense-fluid, or residual, contribution (what’s left in the liquid-like state).
- This split helps when building models that work over wide ranges of pressure and temperature.

3) Equation of state (EOS) and viscosity
- Many viscosity models rely on an EOS (like cubic EOS) to predict volumes and densities of mixtures.
- The EOS helps link viscosity to pressure, temperature, and composition, and is often combined with mixing rules to handle mixtures.

4) Friction Theory (FF)
- FF treats viscosity as a friction force between sliding layers inside the fluid.
- The total viscosity is the sum of a dilute-gas part and a dense-fluid part, with friction functions that depend on temperature, composition, and pressure.
- There are several versions (5-, 7-, 8-parameter models) that add more detail to fit data for hydrocarbons and other fluids.
- For mixtures, well-defined components use mixing rules, while uncertain components (heavier, less-known parts) have special treatment.

5) Significant Structure Theory (SS)
- SS divides the fluid into gas-like and solid-like groups based on microstructure.
- Gas-like viscosity comes from kinetic theory; solid-like viscosity uses concepts from transition-state theory (energy barriers to molecular motion).
- This approach uses inputs from an EOS and mixes theory with adjustable parameters to fit data.

6) Free Volume (FV) theory
- FV links how easily molecules find space to move (free volume) to viscosity.
- It combines a dilute-gas part with a dense-fluid part, where the dense part depends on free volume and energy barriers.
- For mixtures, FV uses mixing rules and trends (how parameters change along homologous series like alkanes) to predict viscosity across many components.

7) Transition-state theory (TS) and related ideas
- TS theory looks at the energy needed for a molecule to move from one place to another, analogous to passing through a transition state.
- This leads to an exponential form that can be used for the solid-like part of viscosity in some models.

8) Group contributions and trend functions
- For large families of molecules (like alkanes), parameters change predictably with molecular size.
- Trend functions extrapolate viscosity-related parameters from known members of a group to others in the same family.

9) Corresponding states (CS) principle
- CS suggests that reduced (scaled) properties of different fluids behave similarly when compared at the same reduced temperature and pressure.
- It speeds up calculations in reservoir simulations and helps combine with rotational coupling corrections to handle mixtures.

10) Classic mixing rules for viscosity
- Arrhenius rule: viscosity of a liquid mixture is related to the viscosities of its components and their fractions.
- Grunberg-Nissan rule: adds binary interaction terms to account for non-ideal mixing.
- Katti-Chaudhri rule: uses partial molar volumes and other mixture properties.
- In practice, many models subtract the dilute-gas part from the total viscosity to get the dense-fluid contribution, then apply mixing rules to the dense-fluid part.

11) Practical notes
- The choice of model depends on the system: gas, liquid, heavy hydrocarbons, or supercritical fluids all push models in different directions.
- Most robust approaches blend theory (kinetic theory, EOS, TS/SS/FV ideas) with empirical data and tuning parameters. These parameters are fitted to measured viscosities of pure components and mixtures.
- In oil and gas engineering, viscosity modeling often combines EOS-based calculations for volumes, a dilute-gas term, a dense-fluid term, and mixing rules to handle the overall mixture.

Why there isn’t a single best model
- Viscosity is sensitive to microstructure, molecular shape, and interactions that vary across temperature, pressure, and composition.
- Different models excel in different regimes (dilute gases, dense liquids, heavy oils, or very high pressures). Many engineers use a mix: a physical, theory-based foundation plus empirical tuning to match real data.

Bottom line
- Viscosity measures how fluids resist flow, and it changes with temperature, pressure, and what’s in the mixture.
- For mixtures, you use models that combine the behaviors of all components, often by splitting into dilute-gas and dense-fluid parts and applying mixing rules.
- A family of approaches (kinetic theory, EOS-based methods, friction theory, significant structure theory, free volume, and corresponding-states ideas) provides tools to predict viscosity across many fluids and conditions. The best choice depends on the fluid type and the regime of interest, and many practical models blend several ideas with data-driven tuning.