User:Frode54/Viscosity

Introduction
The classical Navier-Stokes equation is probably the most famous balance equation for momentum density for an isotropic, compressional and viscous fluid that is used in fluid mechanics in general and fluid dynamics in particular:

$$ \quad \rho \left[\frac{\partial \mathbf{u}}{\partial t} + \mathbf{u} \cdot \nabla \mathbf{u}\right] = -\nabla P + \nabla[\zeta(\nabla\cdot \mathbf{u})] + \nabla\cdot\left[\eta\left(\nabla\mathbf{u} + \left(\nabla\mathbf{u}\right)^T - \frac{2}{3} (\nabla\cdot\mathbf{u})\mathbf{I}\right) \right] + \rho \mathbf{g}$$

On the right hand side we have (the divergence of) the total stress tensor $$ \boldsymbol{\sigma}$$ which consists of a pressure tensor $$ \left(-P\mathbf{I}\right) $$ and a dissipative (or viscous or deviatoric) stress tensor $$ \boldsymbol{\tau}_{d} $$. The dissipative stress consists of a compression stress tensor $$ \boldsymbol{\tau}_{c} $$ (term no. 2) and a shear stress tensor $$ \boldsymbol{\tau}_{s} $$ (term no. 3). The rightmost term is the gravitational force which is the body force contribution.

$$ \quad \boldsymbol{\sigma} = -P \mathbf{I} +\boldsymbol{\tau}_{d} = -P \mathbf{I} +\boldsymbol{\tau}_{c} +\boldsymbol{\tau}_{s} $$

For fluids, the spatial or Eularian form of the governing equations is preferred to the material or Lagrangian form, and the concept of velocity gradient is preferred to the equivalent concept of strain rate tensor. Stokes assumption for a wide class of fluids therefore says that the compression and shear stresses are proportional to their velocity gradients, $$ \mathbf{C} $$ and $$ \mathbf{S} $$ respectively, and named this class of fluids for Newtonian fluids. The classic defining equation for volume viscosity $$ \zeta $$ and shear viscosity $$ \eta $$ are respectively:

$$ \quad \boldsymbol{\tau}_{c} = 3 \zeta \mathbf{C} $$

$$ \quad \boldsymbol{\tau}_{s} = 2 \eta \mathbf{S} $$

The classic compression velocity "gradient" is a diagonal tensor that describes a compressing (alt. expanding) flow or attenuating sound waves:

$$ \quad \mathbf{C} = \frac{1}{3} \left( \nabla\!\cdot\!\mathbf{u} \right) \mathbf{I} $$

The classic Cauchy shear velocity gradient, is a symmetric and traceless tensor that describes a pure shear flow (where pure means excluding normal outflow) around e.g. a wing, propellar, ship hull or in e.g. a river, pipe or vein with or without bends and boundary skin:

$$ \quad \mathbf{S} = \frac{1}{2} \left[ \nabla \mathbf{u} + \left( \nabla \mathbf{u} \right)^{\mathrm{T}} \right] - \frac{1}{3} \left( \nabla\!\cdot\!\mathbf{u} \right) \mathbf{I} $$

How much the volume viscosity contributes to the flow characteristics in e.g. a choked flow such as convergent-divergent nozzle or valve flow is not well known, but the shear viscosity is by far the most utilized viscosity coefficient. We will now leave the volume viscosity and focus on the shear viscosity in the rest of this article.

The above definition is based on a shear-driven fluid motion that in its most general form is modelled by a shear stress tensor and a velocity gradient tensor. The fluid dynamics of a shear flow is, however, very well illustrated by the simple Couette flow. In this experimental layout, the shear stress $$ \tau $$ and the shear velocity gradient $$ S $$ takes the simple forms:

$$ \quad \tau = \eta S \quad \text{where} \quad \tau = \frac {F} {A} \quad \text{and} \quad S = {d u_{} \over dy} = {u_{max} \over y_{max} }$$

Inserting these simplifications gives us a defining equation that can be used to interpret experimental measurements:

$$ \quad \frac {F} {A} = \eta {d u_{} \over dy} = \eta { u_{max} \over y_{max}} $$

In this experimental setup, we select a value for the force $$ F $$ and measure maximum velocity $$ u_{max} $$, and enter them into the equation to calculate viscosity. This gives one value for the viscosity of the selected fluid. We can select another value of the force, and measure maximum velocity. This will result in another viscosity value if we have a non-Newtonian fluid such as paint, but we will get the same viscosity value for a Newtonian fluid such as water, petroleum oil or gas. If we change another parameter like temperature, $$ T $$, and repeat the experiment with the same force, we will get a new value for viscosity, for both non-Newtonian and Newtonian fluids. The great majority of material properties varies as a function of temperature, and this goes for viscosity also. The viscosity is also a function of pressure and, of course, the material itself. For a fluid mixture, this means that the shear viscosity will also vary according to the fluid composition. To map the viscosity as a function of all these variables require a large sequence of experiments that generates an even larger set of numbers called measured data, observed data or observations. Prior to, or at the same time as, the experiments is a material property model (or short material model) proposed to describe or explain the observations. This matemathical model is called the constitutive equation for shear viscosity. It is usually an explicit function that contains some empirical parameters that is adjusted in order to match the observations as good as the matemathical function is capable to do.

For a Newtonian fluid, the constitutive equation for shear viscosity is generally a function of temperature, pressure, fluid composition:

$$ \quad \eta = f(T,P,\mathbf{w}) \quad \text{where} \quad \mathbf{w} =\mathbf{x}, \mathbf{y},\mathbf{z},1_{purefluid} $$

For a non-Newtonian fluid (in the sense of a generalized Newtonian fluid), the constitutive equation for shear viscosity is also a function of the shear velocity gradient:

$$ \quad \eta = f(T,P,\mathbf{w},\mathbf{S}) \quad \text{where} \quad \mathbf{w} =\mathbf{x}, \mathbf{y},\mathbf{z},1_{purefluid} $$

The existence of the velocity gradient in the functional relationship for non-Newtonian fluids says that viscosity is generally not an equation of state, so the term constitutional equation will in general be used for viscosity equations (or functions). The free variables in the two equations above, also indicates that specific constitutive equations for shear viscosity will be quite different from the simple defining equation for shear viscosity that is shown further up. The rest of this article will show that this is certaintly true. We will now leave non-Newtonian fluids and focus on Newtonian fluids in the rest of this article.

Local nomenclature list:


 * $$W_{xy} $$ : temperature
 * $$c \ \ \ \ \ \, $$ : molar concentration or molar density [mol/cm3]
 * $$A \ \ \ \ \, $$ : area of each (upper and lower) boundary plate
 * $$F \ \ \ \ \, $$ : force pulling the upper boundary plate in the Couette flow experiment
 * $$N_{c} \ \ \ $$ : number of fluid components
 * $$P \ \ \ \ \, $$ : pressure
 * $$T \ \ \ \ \,$$ : temperature


 * $$W_{xy} $$ : temperature


 * $$u_{} \ \ \ \ \,$$ : velocity in x-direction
 * $$\mathbf{x} \ \ \ \ \,$$ : liquid phase composition; molfractions $$x_{i} \ i=1,...,N_{c}$$
 * $$\mathbf{y} \ \ \ \ \,$$ : gas phase composition; molfractions $$y_{i} \ i=1,...,N_{c}$$
 * $$\mathbf{z} \ \ \ \ \,$$ : total fluid composition; molfractions $$z_{i} \ i=1,...,N_{c}$$
 * $$\mathbf{\dot{e}} \ \ \ \ \,$$ : strain rate tensor is equivalent to velocity gradient
 * $$\eta \ \ \ \ \,$$ : shear viscosity or viscosity
 * $$\tau \ \ \ \ \,$$ : shear stress
 * $$\zeta \ \ \ \ \,$$ : volume viscosity
 * $$W_{xy} $$ : temperature
 * $$K_{row} $$ : temperature

The above discussion is based on macroscopic arguments. The viscosity has, however, a microscopic or molecular origin, and  transport coefficients like viscosity can be calculated by  time correlations which are valid for both gases and liquids, but it is computer intensive calculations. Another approach is the Boltzmann equation which describes the statistical behaviour of a thermodynamic system not in a state of equilibrium. It can be used to determine how physical quantities change, such as heat energy and momentum, when a fluid is in transport, but it is computer intensive simulations.

From Boltzmann's equation one may also derive other properties characteristic to fluids such as viscosity, thermal conductivity, and electrical conductivity (by treating the charge carriers in a material as a gas). See also convection-diffusion equation. Chapman and Enskog derived a viscosity model for a dilute gas.

$$ \quad \eta_{0} \times 10^{6} = 2.6693 \frac {\sqrt{MT}} {\sigma^{2} \Omega \left( T^{*} \right)} \quad \text{where} \quad T^{*} = k_{B}T / \varepsilon $$

The term dilute gas is a concept that refers to the boundary state of zero pressure which, of course, means that the gas also has zero density, hence the subscript zero on this viscosity variable. The term $$ \Omega ( T^{*} ) $$ is called the collision integral, and we notice that it is inserted as a general function of temperature that the user must specify, and that is not a simple task. This illustrates the situation for the molecular or statistical approach: Create a sophisticated (and often complicated) model for the behavior of large complicated molecules in a fluid mixture and their non-equilibrium partition/distribution-function, and then integrate this model (i.e. time correlation function or Boltzmann equation) analytically. No wonder that the statistical approach is still mostly working with spherical particles (i.e. spherical molecules), and even then the mathematics is so complicated that it is very difficult to get a practical model for viscosity. We will therefore leave the purely theoretical approach for the rest of this article, except for some visits related to dilute gas contributions.

Development of constitutive equations for viscosity is still dominated by empirical, semi-empirical and semi-macroscopic approaches. The target of this article is to cover a variety of viscosity models based on such approaches. The first chapter deals with dilute gas viscosity developed using elementary kinetic theory. This approach provides a classic (or common) viscosity model for dilute gas, and a common estimate of critical viscosity (i.e. viscosity at the critical point). This last contribution is perhaps the most important contribution from the elementary kinetic theory. From dilute gas viscosity the research moved on to more complex and demanding cases as dense gas (i.e. higher pressure and temperature), fluid mixtures and two-phase fluid systems where calculation of the non-equilibrium transport property is (normally) combined with equilibrium calculations of fluid phase components. Besides equilibrium calculations, the normal modeling approach for two-phase systems has been to calculate mass density, via molar volume, from an equation of state, and use mass density in the viscosity equation (formula). The relation between pressure and liquid molar volume is, however, very sensitive as a small change in liquid molar volume is related to a large change in pressure. This approach therefore requires a very accurate equation of state in order to get accurate liquid viscosity calculations. In the time after year 2000 some researchers have looked for ways to mitigate this calculation problem. One group went back to the friction force concept in Newtonian mechanics and combined it with Van der Waals idea that the repulsive and attractive parts of the molecular energy potential corresponds to repulsive term and attractive term in the eqation of state. The group transfeerred this idea to the viscosity equation, but they had to use more terms than Van der Waals in his equation of state. Other groups of researchers switched from the concepts of force and pressure to the concept of energy, which has been used successfully in chemical reaction theory, and how much is needed to jump into neighboring holes in a densely packed liquid.

The list below shows important develoment directions when the research and development moved from dilute gas to more complex and demanding cases:

Dense fluid viscosity:
 * Power series - simplest approach after dilute gas
 * Equation of state analogy between PVT and T$$\eta$$P
 * Corresponding state model - scaling a variable with its value at the critical point
 * Friction force theory - internal sliding surface analogy to a sliding box on an inclined surface
 * Multi- and one-parameter version of friction force theory
 * Transition state analogy - molecular energy needed to squeeze into a vacancy analogous to molecules locking into each other in a chemical reaction
 * Free volume theory - molecular energy needed to jump into a vacant position in the neighboring surface
 * Significant structure theory - based on Eyring's concept of liquid as a blend of solid-like and gas-like behavior / features

Selected contributions from these development directions is displayed in the following chapters. This means that some known contributions and research and development directions are not included. For example is the group contribution method applied to a shear viscosity model not displayed. Even though it is an important method, it is thought to be a method for parameterization of a selected viscosity model, rather than a viscosity model in itself.

Elementary kinetic theory
In books on elementary kinetic theory one can find results for dilute gas modeling that has widespread use. Derivation of the kinetic model for shear viscosity usually starts by considering a Couette flow where two parallell plates are separated by a gas layer. This non-equilibrium flow is superimposed on a Maxwell-Boltzmann equilibrium distribution of molecular motions.

Let $$ \sigma $$ be the microscopic collision cross section of one molecule colliding with another. The macroscopic collision cross section of $$ C \left( = N/V \right)$$ molecules per volume is $$ C \sigma $$, and it is related to the mean free path $$l$$ by

$$ \quad l = \frac {1} {\sqrt{2} C \sigma} $$

Combining the kinetic equations for molecular motion with the defining equation of shear viscosity gives the well known equation for shear viscosity for dilute gases:

$$ \quad \eta_{0} = \frac {2} {3 \sqrt{\pi} } \cdot \frac {\sqrt{m k_{B}T}} {\sigma} = \frac {2} {3 \sqrt{\pi} } \cdot \frac {\sqrt{MRT}} {\sigma N_{A}} $$

where

$$ \quad k_{B} \cdot N_{A} = R \quad \text{and} \quad M = m \cdot N_{A} $$

The equation above presupposes that the gas density is low (i.e. the pressure is low), hence the subscript zero in the variable $$ \eta_{0} $$. This implies that the kinetic translational energy dominates over rotational and vibrational molecule energies. The viscosity equation displayed above further presupposes that there is only one type of gas molecules, and that the gas molecules are perfect elastic hard core particles of spherical shape. This assumption of particles being like billiard balls with radius $$r$$, implies that the microscopic collision cross section of one molecule can be estimated by

$$ \quad \sigma = \pi \left( 2 r_{} \right)^2 = \pi d^2 \qquad \qquad \qquad \, \quad \text{for monomolecular gases and monoparticle beam experiments } $$

$$ \quad \sigma_{ij} = \pi \left( r_{i} + r_{j} \right)^2 = \frac{\pi}{4} \left( d_{i} + d_{j} \right)^2 \quad \text{for binary collision in gas mixtures and dissimilar bullet / target particles} $$

But molecules are not hard particles. For a reasonably spherical molecule the interaction potential is more like the Lennard-Jones potential or even more like the Morse potential. Both have a negative part that attracts the other molecule from distances much longer than the hard core radius, and thus models the van der Waals forces. The positive part models the repulsive forces as the electron clouds of the two molecules overlap. The radius for zero interaction potential is therefore appropriate for estimating (or defining) the collision cross section in kinetic gas theory, and the r-parameter (conf. $$r, r_{i}$$) is therefore called kinetic radius. The d-parameter (where $$d=2r, d_{i}=2r_{i}$$) is called kinetic diameter.

The macroscopic collision cross section $$ \sigma \cdot N_{A}$$ is often associated with the critical molar volume $$ V_{c} $$ by $$ \quad \sigma N_{A} \propto V_{c}^{2/3} \quad \text{or} \quad \sigma N_{A} = \frac {2} {3 \sqrt{\pi} } \cdot K_{rv}^{-1} V_{c}^{2/3} $$

where $$K_{rv}$$ is an empirical tuning parameter, and the pure numerical part is included in order to make the final viscosity formula more suitably for practical use. Inserting this interpretation of $$ \sigma N_{A} $$, and use of reduced temperature $$ T_{r}$$, gives

$$ \quad \eta_{0} = \sqrt{T_{r}} K_{rv} D_{rv} \quad \text{where} \quad T_{r} = T / T_{c} \quad \text{and} $$

$$ \quad D_{rv} = \left( MRT_{c} \right)^{1/2} V_{c}^{-2/3} = R^{1/2} D_{v} $$

which implies that the empirical parameter $$K_{rv}$$ is dimensionless, and that $$D_{rv}$$ and $$\eta_{0}$$ have the same units. The parameter $$D_{rv}$$ is a scaling parameter that involves the gas constant R and the critical molar volume $$V_{c}$$, and it used to scale the viscosity. In this article we will frequently denote the viscosity scaling parameter by Dxyz which involve one or more of the parameters R, Vc, Pc in addition to critical temperature Tc and molar mass M. In practice we will often meet incomplete scaling parameters, such as the parameter $$D_{v}$$ above, where the gas constant R is absorbed into the empirical constant. In the case the viscosity equation becomes

$$ \quad \eta_{0} = \sqrt{T_{r}} K_{v} D_{v} $$

where the empirical parameter $$K_{v}$$ is not dimensionless, and a proposed viscosity model for dense fluid will not be dimensionless if $$D_{v}$$ is the common scaling factor. We notice that

$$ \quad \eta_{0} = \sqrt{T_{r}} K_{rv} D_{rv} = \sqrt{T_{r}} K_{v} D_{v} \implies K_{v} = R^{1/2}K_{rv} $$

If we insert the critical temperature in the equation for dilute viscosity, we get

$$ \quad \eta_{0c} = K_{rv} D_{rv} = K_{v} D_{v} $$

The default values of the parameters $$ K_{rv} $$ and $$ K_{v} $$ should be fairly universal values, although $$ K_{v} $$ depends on the unit system. However, the critical molar volume in the scaling parameters $$ D_{rv} $$ and $$ D_{v} $$ is not easily accessible from experimental measurements, and that is a significant disadvantage. The general equation of state for a real gas is usually written as

$$ PV = ZRT \implies P_{c}V_{c} = Z_{c}RT_{c}$$

where the critical compressibility factor $$ Z_{c} $$, which reflects the volumetric deviation of the real gases from the ideal gas, is also not easily accessible from laboratory experiments. However, critical pressure and critical temperature are more accessible from measurements. It should be added that critical viscosity is also not readily available from experiments.

Uyehara and Watson (1944) proposed to absorb an universal average value of $$ Z_{c} $$ (and the gas constant $$ R $$) into a default value of the tuning parameter $$ K_{p} $$ as a practical solution of the difficulties of getting experimental values for $$ V_{c} $$ and/or $$ Z_{c} $$. The visocity model for a dilute gas is then

$$ \quad \eta_{0} = \sqrt{T_{r}} K_{p} D_{p} \quad \text{where} \quad T_{r} = T / T_{c} \quad \text{and} $$

$$ \quad D_{p} = T_{c}^{-1/6} P_{c}^{2/3} M^{1/2} $$

By inserting the critical temperature in the formula above, the critical viscosity is calculated as

$$ \quad \eta_{0c} = K_{p} D_{p} $$

Based on an average critical compressibility factor of $$ \bar Z_{c} = 0.275 $$ and measured critical viscosity values of 60 different molecule types, Uyehara and Watson (1944) determined an average value of $$ K_{p} $$ to be

$$ \quad \bar K_{p} = 7.7 \cdot 1.01325^{2/3}  \approx 7.77 \quad \text{for} \quad \left[ \eta_{0} \right] = \mu P \quad \text{and} \quad \left[ P_{c} \right] = bar $$

The cubic equation of state (EOS) are very popular equations that are sufficiently accurate for most industrial computations both in vapor-liquid equilibrium and molar volume. Their weakest points are perhaps molar volum in the liquid region and in the critical region. If we accept the cubic EOS, we can calculate the molar hard core volume $$ b $$ from the turning point constraint at the critical point. This gives

$$ b = \Omega_{b} \frac{RT_{c}}{P_{c}} \quad \text{which is similar to} \quad V_{c} = \bar Z_{c} \frac{RT_{c}}{P_{c}} $$

where the constant $$ \Omega_{b} $$ is a universal constant that is specific for the selected variant of the cubic EOS. This says that using $$ D_{p} $$, and disregarding fluid component variations of $$ Z_{c} $$, is in practice equivalent to say that the macroscopic collision cross section is proportional to the hard core molar volume rather than the critical molar volume.

In a fluid mixture like a petroleum gas or oil there are lots of molecule types, and within this mixture we have families of molecule types (i.e. groups of fluid components). The simplest group is the n-alkanes which are long chains of CH2-elements. The more CH2-elements, or carbon atoms, the longer molecule. Critical viscosity and critical thermodynamic properties of n-alkanes therefore show a trend, or functional behaviour, when plotted against molecular mass or number of carbon atoms in the molecule (i.e. carbon number). Parameters in equations for properties like viscosity usually also show such trend behaviour. This means that

$$ \quad \eta_{0cj} = K_{pj} D_{pj} \neq \bar K_{p} D_{pj} \quad \text{for many or most fluid components j }$$

This says that the scaling parameter $$ D_{p} $$ alone is not a true or complete scaling factor unless all fluid components have a fairly similar (and preferably spherical) shape.

The most important result of this kinetic derivation is perhaps not the viscosity formula, but the semi-empirical parameter $$ D_{p} $$ that is used extensively throughout the industry and applied science communities as a scaling factor for (shear) viscosity. The litterature often reports the reciprocal parameter and denotes it as $$ \xi $$.

It should be noted that the dilute gas viscosity contribution to the total viscosity of a fluid will only be important when predicting the viscosity of vapors at low pressures or the viscosity of dense fluids at high temperatures. The viscosity model for dilute gas, that is shown above, is widely used throughout the industry and applied science communities. Therefore, many researchers do not specify a dilute gas viscosity model when they propose a total viscosity model, but leave it to the user to select and include the dilute gas contribution. Some researchers do not include a separate dilute gas model term, but propose an overall gas viscosity model that cover the entire pressure and temperature range they investigated.

In this section our central macroscopic variables and parameters and their units are temperature $$T$$ [K], pressure $$P$$ [bar], molar mass $$M$$ [g/mol], viscosity $$\eta_{0}$$ [cP].

Local nomenclature list:


 * $$c \ \ \,$$ : molar concentration or molar density [mol/cm3]
 * $$c_{r} \ $$ : reduced molar concentration or reduced molar density [1]
 * $$M$$ : molar mass [g/mol]
 * $$P_c$$ : critical pressure [atm]
 * $$T \ $$ : temperature [K]
 * $$T_c $$ : critical temperature [K]
 * $$V_c$$ : critical molar volume [cm3/mol]
 * $$\eta \ \ $$ : viscosity [cP]

Empirical correlation
Zéberg-Mikkelsen (2001) proposed an empirical model for dilute gas viscosity of fairly spherical molecules as follows

$$ \quad \eta_{0} = d_{g1} \sqrt{T} + d_{g2} T^{d_{g3}}$$

or

$$ \quad \eta_{0} = D_{g1} \sqrt{T_{r}} + D_{g2} T_{r}^{D_{g3}}$$

$$ \quad D_{g1} = d_{g1} \cdot \sqrt{T_{c}} \quad \text{and} \quad D_{g2} = d_{g2} \cdot T_{c}^{d_{g3}} \quad \text{and} \quad D_{g3} = d_{g3} $$

where the unit equations for viscosity and temperature are:

$$ \quad \left[ \eta_{0} \right] = \mu P \quad \text{and} \quad \left[ T \right]  = K $$

The second term is a correction term for high temperatures, and we note that most $$ d_{g2} $$ parameters are negative. A table with the empirical d-parameters for some near spherical molecules can be found in the section on Friction Force theory and its model for dilute gases.

Kinetic theory with empirical extension
The gas-like viscosity contribution is taken from the viscosity model of Chung et al.(1984, 1988), which is based on the Chapman-Enskog(1964) kinetic theory of viscosity for dilute gases and the empirical expression of Neufeld et al.(1972) for the reduced collision integral, but expanded empirical to handle polyatomic, polar and hydrogen bonding fluids over a wide temperature range. The viscosity model of Chung et al.(1988) is

$$ \quad \eta_{gl} = 40.785 \frac{\sqrt{MT^{*}}}{V_{c}^{2/3} \Omega^{*}} * F_{c} \quad \text{with unit equation} \quad [ \eta_{gl} ] = \mu P $$

$$ \quad \Omega^{*} = \frac{1.16145}{(T^{*})^{0.14874}} + \frac{0.52487}{exp(0.7732T^{*})} + \frac{2.16178}{exp(2.43787T^{*})} - 6.435 \times 10^{-4} (T^{*})^{0.14874} * sin \left[ 18.0323 (T^{*})^{0.7683} - 7.27371 \right] $$

where

$$ \quad T^{*} = 1.2593 * T / T_{c} \quad \text{and} \quad F_{c} = 1 - 0.2756 \omega + 0.059035 \mu_{r}^{4} + \kappa $$

Local nomenclature list:


 * $$F_{c} \, $$ : factor for molecular shape and polarities of dilute gases [1]
 * $$M \ $$ : molar mass, conf. molecular weight [g/mol]
 * $$T \ \ $$ : temperature [K]
 * $$T_{c} \ $$ : critical temperature [K]
 * $$V_{c} \ $$ : molar critical volume [cm3/mol]
 * $$\eta_{gl} \, $$ : gas-like viscosity contribution [μP]
 * $$\kappa \ \ \, $$ : correction factor for hydrogen bonding effects [1]
 * $$\mu_{r} \, $$ : reduced dipole moment [1]
 * $$\Omega^{*}$$ : reduced collision integral [1]
 * $$\omega \ \ $$ : acentric factor [1]

Trend functions and scaling
In the section with models based on elementary kinetic theory, several variants of scaling the viscosity equation was discussed, and they are displayed below for fluid component i, as a service to the reader.

$$ \quad \eta_{0i} = \sqrt{T_{r}} K_{rvi} D_{rvi} \quad \text{where} \quad D_{rvi} = \sqrt{ M_{i} R T_{ci}} \cdot V_{ci}^{-2/3} $$

$$ \quad \eta_{0i} = \sqrt{T_{r}} K_{vi} D_{vi} \ \ \, \quad \text{where} \quad D_{vi} \ \ = \sqrt{ M_{i} T_{ci}} \cdot V_{ci}^{-2/3} $$

$$ \quad \eta_{0i} = \sqrt{T_{r}} K_{pi} D_{pi} \ \ \, \quad \text{where} \quad D_{pi} \ = M_{i}^{1/2}P_{ci}^{2/3} \cdot T_{ci}^{-1/6} $$

Zéberg-Mikkelsen (2001) proposed an empirical correlation for the Vci parameter for n-alkanes, which is

$$ \quad V _{ci}^{-1} = A + B \cdot \frac {P_{ci}}{RT_{ci}} \iff V _{ci} = \frac {RT_{ci}}{ART_{ci} + BP_{ci}} $$

$$ \quad A = 0.000235751 \ mol/cm^{3} \quad \text{and} \quad B = 3.42770 $$

The critical molar volume of component i $$ V_{ci} $$ is related to the critical amount (or mole) density $$ \rho_{nci} $$ and critical amount (or mole, but often molar) concentration $$c_{ci} $$ by the equation $$ V _{ci}^{-1} = \rho_{nci} = c_{ci} $$. From the above equation for $$ V _{ci}^{-1} $$ we easily see that

$$ \quad Z _{ci} = \frac {P_{ci}}{ART_{ci} + BP_{ci}} \iff \frac{Z_{ci}RT_{ci}}{P_{ci}V_{ci}} = 1 $$

where $$ Z_{ci} $$ is the compressibility factor for component i, which is often used as an alternative to $$ V_{ci} $$. By establishing a trend function for the parameter $$ V_{ci} $$ for a homologous series, groups or families of molecules, we can interpolate and extrapolate parameter values for unknown fluid components in the homologous group, and we can easily re-generate parameter values at later need. Use of trend functions for parameters of homologous groups of molecules have greatly enhanced the usefulness of viscosity equations (and thermodynamic EOSs) for fluid mixtures such as petroleum gas and oil.

Uyehara and Watson (1944) proposed a correlation for critical viscosity (for fluid component i) for n-alkanes using their average parameter $$ \bar K_{p} $$ and the classical pressure dominated scaling parameter $$ D_{pi} $$ :

$$ \quad \eta_{ci} = \bar K_{p} D_{pi} $$

$$ \ \ \bar K_{p} \, = 7.7 \cdot 1.01325^{2/3} \approx 7.77 \quad \text{for} \quad \left[ \eta_{0} \right] = \mu P \quad \text{and} \quad \left[ P_{c} \right] = bar $$

Zéberg-Mikkelsen (2001) proposed an empirical correlation for critical viscosity ηci parameter for n-alkanes, which is

$$ \quad \eta_{ci} = C \cdot P_{ci} M_{i}^{D} $$

$$ \quad \ C = 0.597556 \ \mu P /bar \cdot (g/mol)^{-D} \quad \text{and} \quad D = 0.601652 $$

The unit equations for the two constitutive equations above by Zéberg-Mikkelsen (2001) are

$$ \quad [P_{c}] = bar \quad \text{and} \quad [V_{c}] = [RT_{c}/P_{c}] = cm^{3}/mol \quad \text{and} \quad  [T] = K \quad \text{and} \quad [Z_{c}] = 1 \quad \text{and} \quad [\eta_{c}] = \mu P $$

Inserting the critical temperature in the three viscosity equations from elementary kinetic theory gives three parameter equations.

$$ \quad \eta_{ci} = K_{rvi} D_{rvi} = K_{vi} D_{vi} = K_{pi} D_{pi} \quad \text{or} \quad$$

$$ \quad K_{rvi} = \frac {\eta_{ci}} {D_{rvi}} \quad \text{and} \quad K_{vi} = \frac {\eta_{ci}} {D_{vi}} \quad \text{and} \quad K_{pi} = \frac {\eta_{ci}} {D_{pi}} $$

The three viscosity equations now coalesce to a single viscosity equation

$$ \quad \eta_{0i} = \sqrt{T_{r}} \eta_{ci} = \sqrt{T} \frac{\eta_{ci}}{\sqrt{T_{ci}}} $$

because we now use a proper scaling of viscosity. The collision cross section and the critical molar volume which are both difficult to access experimentally, are avoided or circumvented. On the other hand, we have got the critical viscosity as a new parameter, and critical viscosity is just as difficult to access experimentally as the other two parameters. Fortunately, the best viscosity equations have become so accurate that they justify calculation in the critical point, especially if the equation is matched to surrounding experimental data points.

Classic mixing rules for gas
Wilke (1950) derived a mixing rule based on kinetic gas theory

$$ \quad \eta_{gmix} = \sum_{i=1}^{N} \frac{y_{i} \eta_{gi}} {\sum_{j=1}^{N} y_{j} \varphi_{ij} } $$

$$ \quad \varphi_{ij} =  \frac{\left[ 1+ \sqrt[2]{\frac{\eta_{0i}}{\eta_{0j}}}\cdot\sqrt[4]{\frac{M_{j}}{M_{i}}} \right]^2} { \frac{4}{\sqrt[2]{\pi}} \sqrt[2]{ 1+ \frac{M_{i}}{M_{j}} }  } $$

The Wilke mixing rule is capable of describing the correct viscosity behavior of gas mixtures showing a nonlinear and non-monotonical behavior, or showing a characteristic bump shape, when the viscosity is plotted versus mass density at critical temperature, for mixtures containnig molecules of very different sizes. Due to its complexity, it has not gained widespread use. Instead, the slightly simpler mixing rule proposed by Herning and Zipperer (1936), has been found more suitable for gases of hydrocarbon mixtures.

Classic mixing rules for liquid
The classic Grunberg-Nissan (1949) mixing rule for liquid mixture is

$$ \quad \ln \eta_{lmix} = \sum_{i=1}^{N} x_{i} \ln \eta_{li} $$

where $$ \eta_{lmix} $$ is the viscosity of the liquid mixture, $$ \eta_{li} $$ is the viscosity (equation) for fluid component i when flowing as a pure fluid, and $$ x_{i} $$ is the molfraction of component i in the liquid mixture. The Grunberg-Nissan mixing rule is equivalent to a mixing rule derived by Arrhenius (1887).

A natural modification of the Grunberg-Nissan Mixing rule is

$$ \quad \ln \eta_{lmix} = \sum_{i=1}^{N} x_{i} \ln \eta_{li} + \sum_{i=1}^{N}\sum_{j=1}^{N} x_{i}x_{j}d_{ij}$$

where $$ d_{ij}$$ are empiric binary interaction coefficients that are special for the Grunberg-Nissan theory. Binary interaction coefficients are widely used in cubic EOS where they often are used as tuning parameters, especially if component j is an uncertain component (i.e. have uncertain parameter values).

Katti-Chaudhri (1964) mixing rule is

$$ \quad \ln \left( \eta_{lmix} V_{lmix} \right) = \sum_{i=1}^{N} x_{i} \ln \left( \eta_{li} V_{li} \right)$$

where $$ V_{li} $$ is the partial molar volume of component i, and $$ V_{lmix} $$ is the molar volume of the liquid phase and comes from the vapor-liquid equilium (VLE) calculation or the EOS for single phase liquid.

A modification of the Katti-Chaudhri mixing rule is

$$ \quad \ln \left( \eta_{lmix} V \right) = \sum_{i=1}^{N} z_{i} \ln \left( \eta_{li} V_{i} \right) + \frac {\Delta G^{E}} {RT} $$

$$ \quad \Delta G^{E} = \sum_{i=1}^{N}\sum_{j=1}^{N} z_{i}z_{j}E_{ij} $$

where $$ G^{E} $$ is the excess activation energy of the viscous flow, and $$ E_{ij} $$ is the energy that is characteristic of intermolecular interactions between component i and component j, and therefore is responsible for the excess energy of activation for viscous flow. This mixing rule is theoretically justified by Eyring's representation of the viscosity of a pure fluid according to Glasstone et alios (1941). The quantity $$ \eta_{li} V_{li} $$ has been obtained from the time-correlation expression for shear viscosity by Zwanzig (1965).

Power series
Very often one simply selects a known correlation for the dilute gas viscosity $$ \eta_{0} $$, and substracts this contribution from the total viscosity which is measured in the laboratory. We then get a residual viscosity term, often denoted $$ \Delta\eta $$, which represents the contribution of the dense fluid, $$ \eta_{df} $$.

$$ \quad \eta_{df} = \eta - \eta_{0} \quad \iff \quad \eta = \eta_{0} + \eta_{df} $$

The dense fluid viscosity is thus defined as the viscosity in excess of the dilute gas viscosity. This technique is often used in developing mathematical models for both purely empirical correlations and models with a theoretical support. The dilute gas viscosity contribution becomes important when the zero density limit (i.e. zero pressure limit) is approached. It is also very common to scale the dense fluid viscosity by the critical viscosity, or by an estimate of the critical viscosity, which is a characteristic point far into the dense fluid region. The simplest model of the dense fluid viscosity is a (trunkated) power series of reduced (mass or molar) density or pressure. Jossi et al. (JST 1962) presented such a model based on reduced molar density, but its most widespread form is the version proposed by Lohrenz et al. (LBC 1964) which is displayed below.

$$ \quad \left[ \frac{\eta_{df}} {D_{p}} + 10^{-4} \right]^{1/4} = L_{} $$

The L-function is then expanded in a (truncated) power series with empirical coefficients as displayed below.

$$ \quad L_{} = L_{} \left( c_{r} \right) = \sum_{i=1}^{5} a_{i} c_{r}^{i-1} $$

The final viscosity equation is thus

$$ \quad \eta = \eta_{0} -10^{-4} D_{p} + D_{p} L_{}^{4} $$

$$ \quad \eta_{0} = \eta_{0} \left( T \right) $$

$$ \quad D_{p} = T_{c}^{-1/6} P_{c}^{2/3} M_{n}^{1/2} $$

Local nomenclature list:


 * $$c \ \ \,$$ : molar concentration or molar density [mol/cm3]
 * $$c_{r} \ $$ : reduced molar concentration or reduced molar density [1]
 * $$M$$ : molar mass [g/mol]
 * $$P_c$$ : critical pressure [atm]
 * $$T \ $$ : temperature [K]
 * $$T_c $$ : critical temperature [K]
 * $$V_c$$ : critical molar volume [cm3/mol]
 * $$\eta \ \ $$ : viscosity [cP]

Local nomenclature table:

Mixture
$$ \quad \eta_{mix} = \eta_{0mix} -10^{-4} D_{pmix} + D_{pmix} L_{mix}^{4} $$

$$ \quad L_{mix} = L_{mix} \left( c_{rmix} \right) = \sum_{i=1}^{5} a_{i} c_{rmix}^{i-1} $$

$$ \quad \eta_{0mix} = \eta_{0mix} \left( T \right) $$

$$ \quad D_{pmix} = T_{cmix}^{-1/6} P_{cmix}^{2/3} M_{mix}^{1/2} $$

Mixing rules
$$ \quad T_{cmix} = \sum_{i} z_{i} T_{ci} $$

$$ \quad M_{mix} = M_{n} = \sum_{i} z_{i} M_{i} $$

$$ \quad P_{cmix} = \sum_{i} z_{i} P_{ci} $$

$$ \quad \rho_{ncmix}^{-1} = V_{cmix} = \sum_{i} z_{i} V_{ci} + z_{C7+} \cdot V_{cC7+} \quad i < C7+ $$

The subscript C7+ refers to the collection of hydrocarbon molecules in a reservoir fluid with oil and/or gas that have 7 or more carbon atoms in the molecule. The critical volume of C7+ fraction has unit ft3/lb mole, and it is calculated by

$$ \quad V_{cC7+} = 21.573 + 0.015122 \cdot M_{C7+} - 27.656 \cdot SG_{C7+} + 0.070615 \cdot M_{C7+} SG_{C7+} $$

where $$ SG_{C7+} $$ is the specific gravity of the C7+ fraction.

$$ \quad T_{ci} \quad \text{for} \quad i \geq C7+ \quad \text{or} \quad T_{cC7+} \quad \text{is taken from EOS characterization}$$

$$ \quad M_{i} \quad \text{for} \quad i \geq C7+ \quad \text{or} \quad M_{C7+} \quad \text{is taken from EOS characterization}$$

$$ \quad P_{ci} \quad \text{for} \quad i \geq C7+ \quad \text{or} \quad P_{cC7+} \quad \text{is taken from EOS characterization}$$

The molar mass $$M_{i}$$ (or molecular mass) is normally not included in the EOS formula, but it usually enters the characterization of the EOS parameters.

EOS
From the equation of state we calculate the molar volume of the reservoir fluid (mixture).

$$ \quad V_{mix} = V_{mix}(T,P) \quad \text{for 1 mole fluid} $$

The molar volume is converted to molar concentration, which is called molar density, and then scaled to be reduced molar density.

$$ \quad \rho_{nmix} = 1/V_{mix} \quad and \quad \rho_{ncmix} = 1/V_{cmix} \quad and \quad \rho_{nrmix} = V_{cmix}/V_{mix} = \rho_{nmix}/\rho_{ncmix} $$

Dilute gas contribution
The correlation for dilute gas viscosity of a mixture is taken from Herning and Zipperer (HZ 1936) and is

$$ \quad \eta_{0mix} \left( T \right) = \frac {\sum_{i} z_{i} \eta_{0i} \left( T_{ri} \right) M_{i}^{1/2}} {\sum_{j} z_{j} M_{j}^{1/2}} \quad i,j < C7+ $$

The correlation for dilute gas viscosity of the individual components is taken from Stiel and Thodos (ST 1961) and is

$$ \quad \eta_{0i} \left( T_{ri} \right) = \begin{cases} 34 \times 10^{-5} \cdot D_{pi} T_{ri}^{0.94} & \text{if} \quad T_{ri} \leqslant 1.5 \\ 17.78 \times 10^{-5} \cdot D_{pi} \left( 4.58 \cdot T_{ri} - 1.67 \right)^{5/8} & \text{if} \quad T_{ri} > 1.5 \end{cases} $$

where

$$ \quad D_{pi} = T_{ci}^{-1/6} P_{ci}^{2/3} M_{i}^{1/2} \quad i < C7+ $$

$$ \quad T_{ri} = \frac {T}{T_{ci}} \quad i < C7+ $$

Corresponding state principle
The principle of corresponding states (CS principle or CSP) was first formulated by van der Waals, and it says that two fluids (subscript a and o) of a group (e.g. fluids of non-polar molecules) have approximately the same reduced molar volume (or reduced compressibility factor) when compared at the same reduced temperature and reduced pressure. In mathematical terms this is

$$ \quad \frac { V_{a} \left( P_{ra}, T_{ra} \right)} {V_{ca}} = \frac { V_{o} \left( P_{ro}, T_{ro} \right)} {V_{co}} \iff V_{a} \left( P_{a}, T_{a} \right) = \frac {V_{ca}} {V_{co}} \cdot V_{o} \left( P_{o} = \frac {P_{a}P_{co}}{P_{ca}},T_{o}=\frac{T_{a}T_{co}}{T_{ca}} \right) $$

When the common CS principle above is applied to viscosity, it reads

$$ \eta \left( P, T \right) = \frac {\eta_{c}} {\eta_{co}} \cdot \eta_{o} \left( P_{o},T_{o}\right) \approx \frac {K_{p}D_{p}} {K_{po}D_{po}} \cdot \eta_{o} \left( P_{o},T_{o}\right) $$

We note that the CS principle was originally formulated for equilibrium states, but it is now applied on a transport property - viscosity, and this tells us that another CS formula may be needed for viscosity.

In order to increase the calculation speed for viscosity calculations based on CS theory, which is important in e.g. compositional reservoir simulations, while keeping the accuracy of the CS method, Pedersen et al. (PFT 1984, PF 1987, PFT 1989)  proposed a CS method that uses a simple (or conventional) CS formula when calculating the reduced mass density that is used in the rotational coupling constants (displayed in the sections below), and a more complex CS formula, involving the rotational coupling constants, elsewhere.

Mixture
The simple corresponding state principle is extended by including a rotational coupling coefficient $$ \alpha $$ as suggested by Tham and Gubbins (1970). The reference fluid is methane, and it is given the subscript o.

$$ \quad \eta_{mix} \left( P,T \right) = \left( \frac {T_{cmix}} {T_{co}} \right)^{-1/6} \cdot \left( \frac {P_{cmix}} {P_{co}} \right)^{2/3} \cdot \left( \frac {M_{mix}} {M_{o}} \right)^{1/2} \cdot \frac {\alpha_{cmix}} {\alpha_{co}} \cdot \eta_{o} \left( P_{o},T_{o} \right) $$

$$ \quad P_{o} = \frac {P \cdot P_{co} \alpha_{o}} {P_{cmix} \alpha_{mix}} $$

$$ \quad T_{o} = \frac {T \cdot T_{co} \alpha_{o}} {T_{cmix} \alpha_{mix}} $$

Mixing rules
The interaction terms for critical temperature and critical volume are

$$ \quad T_{cij} = \left( T_{ci} T_{cj} \right)^{1/2} $$

$$ \quad V_{cij} = \frac {1}{8} \left( V_{ci}^{1/3} + V_{cj}^{1/3} \right)^{3} $$

The parameter $$ V_{ci} $$ is usually uncertain or not available. One therefore wants to avoid this parameter. Assuming that $$ Z_{ci} = Z_{c} $$ is the same for all components, we have

$$ \quad V_{ci} = R Z_{ci} T_{ci} / P_{ci} = R_{zc} T_{ci} / P_{ci} \quad \text{where} \quad R_{zc} = R Z_{c} = R Z_{ci} $$

$$ \quad V_{cij} = \frac {1}{8} R_{zc} \left( \left( \frac {T_{ci}} {P_{ci}} \right)^{1/3} + \left( \frac {T_{cj}} {P_{cj}} \right)^{1/3} \right)^{3} $$

$$ \quad T_{cmix} = \frac {\sum_{i} \sum_{j} z_{i} z_{j} V_{cij} T_{cij}} {\sum_{i} \sum_{j} z_{i} z_{j} V_{cij}} $$

The above expression for $$ V_{cij} $$ is now inserted into the equation for $$ T_{cmix}  $$. This gives the following mixing rule

$$ \quad T_{cmix} = \frac {\sum_{i} \sum_{j} z_{i} z_{j} \left( \left( \frac {T_{ci}} {P_{ci}} \right)^{1/3} + \left( \frac {T_{cj}} {P_{cj}} \right)^{1/3} \right)^{3} \left( T_{ci} T_{cj} \right)^{1/2}}

{\sum_{i} \sum_{j} z_{i} z_{j} \left( \left( \frac {T_{ci}} {P_{ci}} \right)^{1/3} + \left( \frac {T_{cj}} {P_{cj}} \right)^{1/3} \right)^{3}} $$

In a similar way we establish the mixing rule for the critical pressure of the mixture.

$$ \quad P_{cmix} = R_{zc} T_{cmix} / V_{cmix} $$

$$ \quad V_{cmix} = \sum_{i} \sum_{j} z_{i} z_{j} V_{cij} $$

$$ \quad P_{cmix} = \frac {8 \sum_{i} \sum_{j} z_{i} z_{j} \left( \left( \frac {T_{ci}} {P_{ci}} \right)^{1/3} + \left( \frac {T_{cj}} {P_{cj}} \right)^{1/3} \right)^{3} \left( T_{ci} T_{cj} \right)^{1/2}}

{\left( \sum_{i} \sum_{j} z_{i} z_{j} \left( \left( \frac {T_{ci}} {P_{ci}} \right)^{1/3} + \left( \frac {T_{cj}} {P_{cj}} \right)^{1/3} \right)^{3} \right)^{2}} $$

The mixing rule for molecular weight is much simpler, but it is not entirely intuitive. It is an empirical combination of the more intuitive formulas with mass weighting $$ \overline{M}_{w} $$ and mole weighting $$ \overline{M}_{n} $$.

$$ \quad M_{mix} = 1.304 \times 10^{-4} \left( \overline{M}_{w}^{2.303} - \overline{M}_{n}^{2.303} \right) + \overline{M}_{n} $$

$$ \quad \overline{M}_{w} = \frac {\sum_{i} z_{i} M_{i}^{2}} {\sum_{j} z_{j} M_{j}} \quad and \quad \overline{M}_{n} = \sum_{i} z_{i} M_{i} $$

The rotational coupling parameter for the mixture is

$$ \quad \alpha_{mix} = 1+7.378 \times 10^{-3} \rho_{ro \alpha}^{1.847} M_{mix}^{0.5173} $$

Reference fluid
The accuracy of the final viscosity of the CS method needs a very accurate density prediction of the reference fluid. The molar volume of the reference fluid methane is therefore calculated by a special EOS, and the Bendict-Webb-Rubin (BWR 1940) equation of state variant suggested by McCarty (1974), and abbreviated BWRM, is recommended by Pedersen et al. (1987) for this purpose. This means that the fluid mass density in a grid cell of the reservoir model may be calulated via e.g. a cubic EOS or by an input table with unknown establishment. In order to avoid iterative calculations, the reference (mass) density used in the rotational coupling parameters is therefore calculated using a simpler corresponding state principle which says that

$$ \quad P_{o \alpha} = \frac {P \cdot P_{co}} {P_{cmix}} \quad \text{and} \quad T_{o \alpha} = \frac {T \cdot T_{co}} {T_{cmix}} \quad \Rightarrow \quad V_{o \alpha} = V(T_{o \alpha},P_{o \alpha}) \quad \text{for 1 mole methane} $$

The molar volume is used to calculate the mass concentration, which is called (mass) density, and then scaled to be reduced density which is equal to reciprocal of reduced molar volume because there is only on component (molecule type). In mathematical terms this is

$$ \quad \rho_{o \alpha} = M_{o}/V_{o \alpha} \quad and \quad \rho_{co} = M_{o}/V_{co} \quad \Rightarrow \quad \rho_{ro \alpha} = \rho_{o \alpha}/\rho_{co} = V_{co}/V_{o \alpha} $$

The formula for the rotational coupling parameter of the mixture is shown further up, and the rotational coupling parameter for the reference fluid (methane) is

$$ \quad \alpha_{o} = 1+0.031 \rho_{ro \alpha}^{1.847} $$

The methane mass density used in viscosity formulas is based on the extended corresponding state, shown at the beginning of this chapter on CS-methods, and using the BWRM EOS we calculate the molar volume of the reference fluid as

$$ \quad V_{o} = V(T_{o},P_{o}) \quad \text{for 1 mole methane} $$

Once again, the molar volume is used to calculate the mass concentration, or mass density, but the reference fluid is a single component fluid, and the reduced density is independent of the relative molar mass. In mathematical terms this is

$$ \quad \rho_{o} = M_{o}/V_{o} \quad and \quad \rho_{co} = M_{o}/V_{co} \quad \Rightarrow \quad \rho_{ro} = \rho_{o}/\rho_{co} = V_{co}/V_{o} $$

The effect of a changing composition of e.g. the liquid phase is related to the scaling factors for viscosity, temperature and pressure, and that is the corresponding state principle.

The reference viscosity correlation of Pedersen et al. (1987) is

$$ \quad \eta_{o} \left( \rho_{o}, T_{o} \right) = \eta_{0}(T_{o}) + \hat{\eta}_{1}(T_{o}) \rho_{o} + F_{1} \Delta \eta' (\rho_{o} , T_{o}) +  F_{2} \Delta \eta'' (\rho_{o} , T_{o}) $$

The formulas for $$ \eta_{0}(T_{o}) $$, $$ \hat{\eta}_{1}(T_{o}) $$, $$ \Delta \eta' (\rho_{o}, T_{o}) $$ are taken from Hanley et al. (HMH 1975).

The dilute gas contribution is

$$ \quad \eta_{0} \left( T_{o} \right) = \textstyle \sum_{i=1}^{9} g_i T_{o}^{\frac{i-3}{4}}$$

The temperature dependent factor of the first density contribution is

$$ \quad \hat{\eta}_{1}\left( T_{o} \right) = h_1 - h_2 \left\lbrack h_3 - ln \left( \frac{T_{o}}{h_4}\right)\right\rbrack^2 $$

The dense fluid term is

$$ \quad \Delta \eta' \left( \rho_{o}, T_{o} \right) = e^{j_1 + j_4 / T_{o}}  \times  \lbrack exp{\lbrack \rho_{o}^{0.1} ( j_2+j_3/T_{o}^{3/2} )  + \theta_{ro} \rho_{o}^{0.5} \left( j_5+j_6/T_{o}+j_7/T_{o}^2  \right) \rbrack } - 1 \rbrack $$

where exponential function is written both as $$ e^{x} $$ and as $$ exp{ \lbrack x \rbrack } $$. The molar volume of the reference fluid methane, which is used to calculate the mass density in the viscosity formulas above, is calculated at a reduced temperature that is proportional to the reduced temperature of the mixture. Due to the high critical temperatures of heavier hydrocarbon molecules, the reduced temperature of heavier reservoir oils (i.e. mixtures) can give a transferred reduced methane temperature that is in the neighborhood of the freezing temperature of methane. This is illustrated using two fairly heavy hydrocarbon molecules, in the table below.The selected temperatures are a typical oil or gas reservoir temperature, the reference temperature of the International Standard Metric Conditions for Natural Gas (and similar fluids) and the freezing temperature of methane.

Pedersen et al. (1987) added a fourth term, that is correcting the reference viscosity formula at low reduced temperatures. The temperature functions $$ F_1 $$ and $$ F_2 $$ are weight factors. Their correction term is

$$ \quad \Delta \eta'' \left( \rho_{o}, T_{o} \right) = e^{k_1 + k_4 / T_{o}}  \times  \lbrack exp{\lbrack \rho_{o}^{0.1} ( k_2+k_3/T_{o}^{3/2} )  + \theta_{ro} \rho_{o}^{0.5} \left( k_5+k_6/T_{o}+k_7/T_{o}^2  \right) \rbrack } - 1 \rbrack $$

$$ \quad \theta_{ro} = \left( \rho_{o} - \rho_ {co} \right) / \rho_ {co} = \rho_{ro} - 1$$

$$ \quad F_{1}  = \frac {HTAN+1}{2} $$

$$ \quad F_{2}  = \frac {1-HTAN}{2} $$

$$ \quad HTAN  = tanh \left( \Delta T_{o} \right) = \frac {e^{\left( \Delta T_{o} \right)} - e^{\left( -\Delta T_{o} \right)}}   {e^{\left( \Delta T_{o} \right)} + e^{\left( -\Delta T_{o} \right)}} $$

$$ \quad \Delta T_{o}  = T_{o} - T_{fo} $$

Equation of state analogy
Phillips (1912) plotted temperature $$T$$ versus viscosity $$\eta$$ for different isobars for propane, and observed a similarity between these isobaric curves and the classic isothermal curves of the $$PVT$$ surface. Later, Little and Kennedy (1968) developed the first viscosity model based on analogy between $$T \eta P$$ and $$PVT$$ using van der Waals EOS. Van der Waals EOS was the first cubic EOS, but the cubic EOS has over the years been improved and now make up a widely used class of EOS. Therefore Guo et al. (1997) developed two new analogy models for viscosity based on PR EOS (Peng and Robinson 1976) and PRPT EOS (Patel and Teja 1982) respectively. The following year T.-M. Guo (1998) modified the PR based viscosity model slightly, and it is this version that will be presented below as a representative of EOS analogy models for viscosity.

PR EOS is displayed on the next line.

$$ \quad P = \frac {RT}{V-b_{eos}} - \frac {a_{eos}} {V(V+b_{eos})+b_{eos}(V-b_{eos})} $$

The viscosity equation of Guo (1998) is displayed on the next line.

$$ \quad T = \frac{rP} {\eta-d} - \frac {a} {\eta \left( \eta+b \right)+b \left( \eta-b \right) } $$

To prepare for the mixing rules we re-write the viscosity equation for a single fluid component i.

$$ \quad T = \frac{r_{i} P} {\eta_{i}-d_{i}} - \frac {a_{i}} {\eta_{i} \left( \eta_{i}+b_{i} \right)+b_{i} \left( \eta_{i}-b_{i} \right) } $$

Details of how the composite elements of the equation are related to basic parameters and variables, is displayed below.

$$ \quad a_{i} = 0.45724 \frac {r_{ci}^{2} P_{ci}^{2} } {T_{ci}} $$

$$ \quad b_{i} = 0.07780 \frac {r_{ci} P_{ci} } {T_{ci}} $$

$$ \quad r_{i} = r_{ci} \tau_{i} \left( T_{ri}, P_{ri} \right)$$

$$ \quad d_{i} = b_{i} \phi_{i} \left( T_{ri}, P_{ri} \right)$$

$$ \quad r_{ci} = \frac {\eta_{ci} T_{ci}} {P_{ci} Z_{ci}} $$

$$ \quad \eta_{ci} = K_{p} D_{pi} \quad \text{where} \quad K_{p} = 7.7\cdot 10^{4} \quad \text{and} \quad D_{pi} = T_{ci}^{-1/6} M_{i}^{1/2} P_{ci}^{2/3} $$

$$ \quad \tau_{i} = \tau_{i} \left( T_{ri}, P_{ri} \right) = \left( 1 + Q_{1i} \left( \sqrt{T_{ri}P_{ri}} -1 \right) \right)^{-2} $$

$$ \quad \phi_{i} = \phi_{i} \left( T_{ri}, P_{ri} \right) = \exp \left[ Q_{2i} \left( \sqrt{T_{ri}} -1 \right) \right] + Q_{3i} \left( \sqrt{P_{ri}} -1 \right)^{2}$$

$$ \quad Q_{1i} = \begin{cases} 0.829599 + 0.350857 \, \omega_{i} - 0.747680 \, \omega_{i}^{2}, & \text{if } & \omega_{i} < 0.3 \\ 0.956763 + 0.192829 \, \omega_{i} - 0.303189 \, \omega_{i}^{2}, & \text{if } & \omega_{i} \ge 0.3 \end{cases} $$

$$ \quad Q_{2i} = \begin{cases} \;\;\; 1.94546 \;\, - 3.19777 \, \omega_{i} + 2.80193 \, \omega_{i}^{2}, & \; \text{if } & \omega_{i} < 0.3 \\ - 0.258789 - 37.1071 \, \omega_{i} + 20.5510 \, \omega_{i}^{2}, & \; \text{if } & \omega_{i} \ge 0.3 \end{cases} $$

$$ \quad Q_{3i} = \begin{cases} 0.299757 + 2.20855 \, \omega_{i} - 6.64959 \, \omega_{i}^{2}, & & \text{if } & \omega_{i} < 0.3 \\ 5.16307 \;\; - 12.8207 \, \omega_{i} + 11.0109 \, \omega_{i}^{2}, & & \text{if } & \omega_{i} \ge 0.3 \end{cases} $$

Mixture
$$ \quad T = \frac{r_{mix}P} {\eta_{mix}-d_{mix}} - \frac{a_{mix}} {\eta_{mix}\left(\eta_{mix}+b_{mix} \right)+b_{mix} \left( \eta_{mix}-b_{mix} \right) } $$

Mixing rules
$$ \quad a_{mix} = \sum_{i=1} z_{i}a_{i} $$

$$ \quad b_{mix} = \sum_{i=1} z_{i}b_{i} $$

$$ \quad d_{mix} = \sum_{i=1} \sum_{i=1} z_{i}z_{i}\sqrt{ d_{i}d_{i} } \left( 1- k_{ij} \right) $$

$$ \quad r_{mix} = \sum_{i=1} z_{i}r_{i} $$

Multi-parameter friction force theory
The multi-parameter version of the friction force theory (short FF theory and FF model), also called friction theory (short F-theory), was developed by Quiñones-Cisneros et al. (QZS 2000, 2001a, 2001b and Z 2001, QDS 2004, QD 2006), and its basic elements, using some well known cubic EOSs, are displayed below.

It is a common modeling technique to accept a viscosity model for dilute gas ($$\eta_{0}$$), and then establish a model for the dense fluid viscosity $$\eta_{df}$$. The FF theory states that for a fluid under shear motion, the shear stress $$\tau$$ (i.e. the dragging force) acting between two moving layers can be separated into a term $$\tau_{0}$$ caused by dilute gas collisions, and a term $$\tau_{df}$$ caused by friction in the dense fluid.

$$ \quad \eta_{} = \eta_{0} + \eta_{df} \quad \text{and} \quad \tau_{} = \tau_{0} + \tau_{df} $$

The dilute gas viscosity (i.e. the limiting viscosity behavior as the pressure, normal stress, goes to zero) and the dense fluid viscsoity (the residual viscosity) can be calculated by

$$ \quad \tau_{0} = \eta_{0} \frac{du}{dy} \quad \text{and} \quad \tau_{df} = \eta_{df}  \frac{du}{dy}$$

where du/dy $$du/dy$$ is the local velocity gradient orthogonal to the direction of flow. Thus

$$ \quad \eta_{0} = \frac{\tau_{0}}{du/dy} \quad \text{and} \quad \eta_{df} = \frac{\tau_{df}}{du/dy} $$

The basic idea of QZS (2000) is that internal surfaces in a Couette flow acts like (or is analogue to) mechanical slabs with friction forces acting on each surface as they slide past each other. According to the Amontons-Coulomb friction law in classical mechanics, the ratio between the kinetic friction force $$F$$ and the normal force $$N$$ is given by

$$ \quad \zeta = \frac{F}{N} = \frac{A\tau_{df}}{A\sigma} = \frac{\tau_{df}}{\sigma} $$

where $$\zeta$$ is known as the kinetic friction coefficient, A is the area of the internal flow surface, $$\tau$$ is the shear stress and $$\sigma$$ is the normal stress (or pressure $$P$$) between neighboring layers in the Couette flow.

$$ \quad \eta_{df} = \frac{\tau_{df}}{du/dy} = \frac{\zeta\sigma}{du/dy} $$

The FF theory of QZS says that when a fluid is brought to have shear motion, the attractive and repulsive intermolecular forces will contribute to amplify or diminish the mechanical properties of the fluid. The friction shear stress term $$\tau_{df}$$ of the dense fluid can therefore be considered to consist of an attractive friction shear contribution $$\tau_{dfatt}$$ and a repulsive friction shear contribution $$\tau_{dfrep}$$. Inserting this gives us

$$ \quad \eta_{df} = \frac{\tau_{dfrep}+\tau_{dfatt}}{du/dy} = \frac{\zeta P}{du/dy} $$

The well known cubic equation of states (SRK, PR and PRSV EOS), can be written in a general form as

$$ \quad P = \frac {RT}{V-b} - \frac{a}{V^2+ubV+wb^2} $$

For (u,w)=(1,0) we get the SRK EOS, and for (u,w)=(2,-1) we get both the PR EOS and the PRSV EOS because they differ only in the temperature and composition dependent parameter / function a. Input variables are, in our case, pressure (P), temperature (T) and for mixtures also fluid composition which can be single phase (or total) composition $$ \mathbf{z} = \left[z_{1}, \cdots, z_{N}\right]$$, vapor (gas) composition $$ \mathbf{y} = \left[y_{1}, \cdots , y_{N}\right]$$ or liquid (in our example oil) composition $$ \mathbf{x} = \left[x_{1}, \cdots , x_{N}\right]$$. Output is the molar volume of the phase (V). Since the cubic EOS is not perfect, the molar volume is more uncertain than the pressure and temperature values.

The EOS consists of two parts that are related to van der Waals forces, or interactions, that originates in the static electric fields of the colliding parts /spots of the two (or more) colliding molecules. The repulsive part of the EOS is usually modeled as a hard core behavior of molecules, hence the symbol (Ph), and the attractive part (Pa) is based on the attractive interaction between molecules (conf. van der Waals force). We write this as

$$ \quad P = P_{h} - P_{a} $$

Assuming that the molar volume (V) is known from EOS calculations, and prior vapor-liquid equilibrium (VLE) calculations for mixtures, we can define two functions that we expect to be a more accurate and robust than the molar volume (V) itself. These functions are

$$ \quad P_{h} = P_{h} (V,T,\mathbf{w} ) = \frac {RT}{V-b} \quad \text{where} \quad \mathbf{w} = \mathbf{x},\mathbf{y}, \mathbf{z}, 1_{pure fluid} $$

$$ \quad P_{a} = P_{a} (V,T,\mathbf{w} ) = \frac{a}{V^2+ubV+wb^2} \quad \text{where} \quad \mathbf{w} = \mathbf{x},\mathbf{y}, \mathbf{z}, 1_{pure fluid} $$

The friction theory therefore assumes that the residual attractive stress $$ \tau_{fatt}$$ and the residual repulsive stress $$ \tau_{frep}$$ are functions of the attractive pressure term $$ P_{a}$$ and the repulsive pressure term $$ P_{h}$$, respectively.

$$ \quad \tau_{dfatt} = F(T,P_{a},\mathbf{w}) \quad \text{and} \quad \tau_{dfrep} = F(T,P_{h},\mathbf{w}) \quad \text{and} \quad \mathbf{w} = \mathbf{x},\mathbf{y}, \mathbf{z}, 1_{pure fluid} $$

The first attempt is, of course, to try a linear function in the pressure terms / functions. $$ \quad \eta_{df} = K_{a} P_{a} + K_{h} P_{h} $$

All $$ K $$ coefficients are in general functions of temperature and composition, and they are called friction functions. In order to achieve high accurcy over a wide pressure and temperature ranges, it turned out that a second order term was needed even for non-polar molecules types such as hydrocarbon fluids in oil and gas reservoirs, in order to achieve a high accurracy at very high pressures. A test with a presumably difficult 3-component mixture of non-polar molecule types needed a third order power to achieve high accuracy at the most extreme super-critial pressures.

$$ \quad \eta = \eta_{0} + K_{a} P_{a} + K_{h} P_{h} + K_{h2} P_{h}^{2} + K_{h3} P_{h}^{3} $$

This article will concentrate on the second order version, but the third order term will be included whenever possible in order to show the total set of formulas. As an introduction to mixture notation, the above equation is repeated for component i in a mixture.

$$ \quad \eta_{i} = \eta_{0i} + K_{ai} P_{ai} + K_{hi} P_{hi} + K_{h2i} P_{hi}^{2} + K_{h3i} P_{hi}^{3} $$

Local nomenclature list: draft / kladd


 * $$c \ \ \,$$ : molar concentration or molar density [mol/cm3]
 * $$c_{r} \ $$ : reduced molar concentration or reduced molar density [1]
 * $$M$$ : molar mass [g/mol]
 * $$P_c$$ : critical pressure [atm]
 * $$T \ $$ : temperature [K]
 * $$T_c $$ : critical temperature [K]
 * $$V_c$$ : critical molar volume [cm3/mol]
 * $$\eta \ \ $$ : viscosity [cP]

Friction functions
Friction functions for fluid component i in the 5 parameter model for pure n-alkane molecules are presented below.

$$ \quad K_{ai} = B_{a1i} \exp\left( \Gamma_{i} -1 \right) + B_{a2i} \left[ \exp\left( 2 \Gamma_{i} - 2 \right) - 1 \right] $$

$$ \quad K_{hi} = B_{h1i} \exp\left( \Gamma_{i} -1 \right) + B_{h2i} \left[ \exp\left( 2 \Gamma_{i} - 2 \right) - 1 \right] $$

$$ \quad K_{h2i} = B_{h22i} \left[ \exp\left( 2 \Gamma_{i} \right) - 1 \right] $$

$$ \quad \Gamma_{i} = T_{ci} / T $$

Friction functions for fluid component i in the 7- and 8-parameter models are presented below.

$$ \quad K_{ai} = B_{a0i} + B_{a1i} \left[ \exp\left( \Gamma_{i} -1 \right) - 1 \right] + B_{a2i} \left[ \exp\left( 2 \Gamma_{i} - 2 \right) - 1 \right] $$

$$ \quad K_{hi} = B_{h0i} + B_{h1i} \left[ \exp\left( \Gamma_{i} -1 \right) - 1 \right] + B_{h2i} \left[ \exp\left( 2 \Gamma_{i} - 2 \right) - 1 \right] $$

$$ \quad K_{h2i} = B_{h22i} \left[ \exp\left( 2 \Gamma_{i} \right) - 1 \right] $$

$$ \quad K_{h3i} = B_{h32i} \left[ \exp \left( 2 \Gamma_{i} \right) - 1 \right] \left( \Gamma_{i} - 1 \right)^{3}$$

$$ \quad \Gamma_{i} = T_{ci} / T $$

The empirical constants in the friction functios are called friction constants. Friction constants for some n-alkanes in the 5 parameter model using SRK and PRSV EOS (and thus PR EOS) is presented in tables below. Friction constants for some n-alkanes in the 7 parameter model using PRSV EOS are also presented in a table below. The constant $$d_{2}$$ for three fluid components are presented below in the last table of this table-series.

Mixture
$$ \quad P_{dyn} = P = \frac {RT}{V_{eos}-b_{eos}} - \frac{a_{eos}}{V_{eos}^2+ub_{eos}V_{eos}+wb_{eos}^2} $$

In the single phase regions, the mole volume of the fluid mixture is determined by the input variables are pressure (P), temperature (T) and (total) fluid composition $$ \mathbf{z} $$. In the two-phase gas-liquid region a vapor-liquid equilibrium (VLE) calculation splits the fluid into a vapor (gas) phase with composition $$ \mathbf{y} $$ and phase mixture molfraction ng and a liquid phase (in our example oil) with composition $$ \mathbf{x} $$ and phase mixture molfraction no. For liquid phase, vapor phase and single phase fluid we get

$$ \quad P_{hmix} = P_{heos} \left( V_{eos},T, \mathbf{w} \right) = \frac {RT}{V_{eos}-b_{eos}} \quad \text{where} \quad \mathbf{w} =\mathbf{x}, \mathbf{y},\mathbf{z} $$

$$ \quad P_{amix} = P_{aeos} \left( V_{eos},T, \mathbf{w} \right) = \frac{a_{eos}}{V_{eos}^2+ub_{eos}V_{eos}+wb_{eos}^2} \quad \text{where} \quad \mathbf{w} =\mathbf{x}, \mathbf{y},\mathbf{z} $$

In a compositional reservoir simulator the pressure is calculated dynamically for each grid cell and each timestep. This gives dynamic pressures for vapor and liquid (oil) or single phase fluid. Assuming zero capillary pressure between hydrocarbon liquid (oil) and gas, the simulator software code will give a single dynamic pressure $$ P_{dyn} $$ which applies to both the vapor mixture and the liquid (oil) mixture. In this case the reservoir simulator software code may use

$$ \quad P_{amix} = P_{hmix} - P_{dyn} \quad \text{and} \quad P_{hmix} = P_{heos} (V_{eos},T,\mathbf{w}) = \frac {RT}{V_{eos}-b_{eos}} \quad \text{where} \quad \mathbf{w} =\mathbf{x}, \mathbf{y},\mathbf{z} $$

or

$$ \quad P_{hmix} = P_{dyn} + P_{amix} \quad \text{and} \quad P_{amix} = P_{aeos} (V_{eos},T,\mathbf{w}) = \frac{a_{eos}}{V_{eos}^2+ub_{eos}V_{eos}+wb_{eos}^2} \quad \text{where} \quad \mathbf{w} =\mathbf{x}, \mathbf{y},\mathbf{z} $$

The friction model for viscosity of a mixture is

$$ \quad \eta_{mix} = \eta_{0mix} + \eta_{dfmix} $$

$$ \quad \eta_{mix} = \eta_{0mix} + K_{amix} P_{amix} +K_{hmix} P_{hmix} + K_{h2mix} P_{hmix}^{2} + K_{h3mix} P_{hmix}^{3} $$

The cubic power term is only needed when molecules with a fairly rigid 2-D structure are included in the mixture, or the user requires a very high accuracy at exemely high pressures. The standard model includes only linear and quadratic terms in the pressure functions.

Mixing rules
$$ \quad \ln \left( \eta_{0mix} \right) = \sum_{i=1}^{N} z_{i} \ln ( \eta_{0i} ) \quad \text{or} \quad \eta_{0mix} = \prod_{i=1}^{N} \eta_{0i}^{z_{i}} $$

$$ \quad K_{qmix} = \sum_{i=1}^{N} W_{i} K_{qi} \quad \text{where} \quad q = a,h,h2 $$

$$ \quad \ln \left( K_{h3mix} \right) = \sum_{i=1}^{N} z_{i} \ln \left( K_{h3i} \right) \quad \text{or} \quad K_{h3mix} = \prod_{i=1}^{N} K_{h3i}^{z_{i}} $$

where the empirical weight fraction is

$$ \quad W_{i} = \frac {z_{i}} {M_{i}^{\varepsilon} \cdot MM} \quad \text{where} \quad MM = \sum_{j=1}^{N} \frac {z_{j}} {M_{j}^{\varepsilon} } $$

The recommended values for $$\varepsilon$$ are These values are established from binary mixtures of n-alkanes using a 5-parameter viscosity model, and they seems to be used for 7- and 8-parameter models also. The motivation for this weight parameter $$W_{i}$$, and thus the $$\varepsilon$$-parameter, is that in asymmetric mixtures like C1H4 - C10H12, the lightest component tends to decrease the viscosity of the mixture more than linearly when plotted versus molfraction of the light component (or the heavy component).
 * $$ \quad \varepsilon = 0.15 \quad \;\; $$ gave best performance for SRK EOS
 * $$ \quad \varepsilon = 0.075 \quad $$ gave best performance for PRSV EOS

The friction coefficients of some selected fluid components is presented in the tables below for the 5,7 and 8-parameter models. For convenience are critical viscositites also included in the tables.

....

One-parameter friction force theory
The one-parameter version of the friction force theory (FF1 theory and FF1 model) was developed by Quiñones-Cisneros et al. (QZS 2000, 2001a, 2001b and Z 2001, QDS 2004), and its basic elements, using some well known cubic EOSs, are displayed below.

The first step is to define the reduced dense fluid (or frictional) viscosity for a pure (i.e. single component) fluid by dividing by the critical viscosity. The same goes for the dilute gas viscosity.

$$ \quad \eta_{dfr} = \frac {\eta_{df}}{\eta_{c}} \quad \text{and} \quad  \eta_{0r}= \frac {\eta_{0}}{\eta_{c}} $$

The second step is to replace the attractive and repulsive pressure functions by reduced pressure functions. This will of course, affect the friction functions also. We therefore introduce new friction functions that is called reduced friction functions, and that will be of a more universal nature. The reduced frictional viscosity is

$$ \quad \eta_{dfr} = K_{ar} \left( \frac {P_{a}}{P_{c}} \right) + K_{hr} \left( \frac {P_{h}}{P_{c}} \right) + K_{h2r} \left( \frac {P_{h}}{P_{c}} \right)^{2} $$

Next we return to the unreduced frictional viscosity and rephrase the formula as

$$ \quad \eta_{df} = \frac{\eta_{c} K_{ar}}{P_{c}} P_{a} + \frac{\eta_{c} K_{hr}}{P_{c}} P_{h} + \frac{\eta_{c} K_{h2r}}{P_{c}^{2}} P_{h}^{2} $$

Critical viscosity is seldom measured and attempts to predict it by formulas are few. For a pure fluid, or component i in a fluid mixture, we have a formula from kinetic theory.

$$ \quad \eta_{ci} =K_{vi} D_{vi} \quad \text{where} \quad   D_{vi} = M_{i}^{1/2}T_{ci}^{1/2}V_{ci}^{-2/3} $$

where $$K_{vi}$$ is a constant, and critical molar volume Vci is assumed to be proportional to the collision cross section. The critical molar volume Vci is  significantly more uncertain than the parameters Pci and Tci. To get rid of Vci, we assume that the critial compressibility factor Zci can be replaced by a universal average value. Then we have $$ \quad \eta_{ci} = K_{p} D_{pi} \quad \text{where} \quad   D_{pi} = M_{i}^{1/2}P_{ci}^{2/3}T_{ci}^{-1/6} $$

where $$K_{p}$$ is a constant. Based on an average critical compressibility factor of Zc = 0.275 and measured critical viscosity values of 60 different molecule types, Uyehara and Watson (1944) determined an average value of Kp to be

$$ \quad K_{p} = 7.7 \cdot 1.01325^{2/3} \approx 7.77  $$

Zéberg-Mikkelsen (2001) proposed an empirical correlation for Vci, with parameters for n-alkanes, which is

$$ \quad V _{ci}^{-1} = A + B \cdot \frac {P_{ci}}{RT_{ci}} \iff V _{ci} = \frac {RT_{ci}}{ART_{ci} + BP_{ci}} $$

where $$ V _{ci}^{-1} = \rho_{nci} = c_{ci} $$. From the above equation we easily see that

$$ \quad Z _{ci} = \frac {P_{ci}}{ART_{ci} + BP_{ci}} \iff \frac{Z_{ci}RT_{ci}}{P_{ci}V_{ci}} = 1 $$

Zéberg-Mikkelsen (2001) also proposed an empirical correlation for ηci, with parameters for n-alkanes, which is

$$ \quad \eta_{ci} = C \cdot P_{ci} M_{i}^{D} $$

The unit equations for the two constitutive equations above by Zéberg-Mikkelsen (2001) are

$$ \quad [P_{c}] = bar \quad \text{and} \quad [V_{c}] = [RT_{c}/P_{c}] = cm^{3}/mol \quad \text{and} \quad  [T] = K \quad \text{and} \quad [\eta_{c}] = \mu P $$

The next step is to split the formulas into formulas for well defined components (designated by subscript d) with respect critical viscosity and formulas for uncertain components (designated by subscript u) where critical viscosity is estimated using $$D_{pi}$$ and the universal constant $$K_{p}$$ which will be treated as a tuning parameter for the current mixture. The dense fluid viscosity (for fluid component i in a mixture) is then written as

$$ \quad \eta_{dfi} = \eta_{dfdi} + \eta_{dfui} = \eta_{dfdi} + K_{pu}F_{ui} $$

The formulas from friction theory is then related to well defined and uncertain fluid components. The result is

$$ \quad \eta_{dfdi} = \frac{\eta_{ci} K_{ari}}{P_{ci}} P_{ai} + \frac{\eta_{ci} K_{hri}}{P_{ci}} P_{hi} + \frac{\eta_{ci} K_{h2ri}}{P_{ci}^{2}} P_{hi}^{2} \quad \text{for} \quad i=1,\ldots,m $$

$$ \quad F_{ui} = \frac{D_{pi} K_{ari}}{P_{ci}} P_{ai} + \frac{D_{pi} K_{hri}}{P_{ci}} P_{hi} + \frac{D_{pi} K_{h2ri}}{P_{ci}^{2}} P_{hi}^{2} \quad \text{for} \quad i=m+1,\ldots,N $$

$$ \quad D_{pi} = M_{i}^{1/2} P_{ci}^{2/3}T_{ci}^{-1/6} $$

However, in order to obtain the characteristic critical viscosity of the heavy pseudocomponents, the following modification of the Uyehara and Watson (1944) expression for the critical viscosity can be used. The frictional (or residual) viscosity is then written as

$$ \quad \eta_{ci} = K_{p} D_{pi} \quad \text{where} \quad K_{p} = 7.9483 $$

where the unit equations are $$ \left[ \eta \right] = \mu P $$ and $$ \left[ P_{c} \right] = bar $$ and $$ \left[  T_{c}  \right] =  K $$.

Local nomenclature list: draft


 * $$c \ \ \,$$ : molar concentration or molar density [mol/cm3]
 * $$c_{r} \ $$ : reduced molar concentration or reduced molar density [1]
 * $$M$$ : molar mass [g/mol]
 * $$P_c$$ : critical pressure [atm]
 * $$T \ $$ : temperature [K]
 * $$T_c $$ : critical temperature [K]
 * $$V_c$$ : critical molar volume [cm3/mol]
 * $$\eta \ \ $$ : viscosity [cP]

Reduced friction functions
$$ \quad K_{qri} = B_{qrc} +B_{qr00} \left( \Gamma_{i} -1 \right) + \sum_{m=1}^{2} \sum_{n=0}^{m} B_{qrmn} \psi_{i}^{n} \left[ \exp ( m \Gamma_{i} - m ) -1 \right] \quad \text{where} \quad q=a,h$$

$$ \quad K_{h2ri} = B_{h2rc} + B_{h2r21} \psi_{i} \left[ \exp ( 2 \Gamma_{i} ) -1 \right] \left( \Gamma_{i} - 1 \right)^{2}$$

$$ \quad \psi_{i} = \frac {R T_{ci}}{P_{ci}} \quad \text{and} \quad \Gamma_{i} = \frac {T_{ci}}{T} $$

The unit equation of $$ \psi_{i} $$ is $$ \left[ \psi_{i} \right] = cm^{3}/mol $$.

The 1-parameter model have been developed based on single component fluids in the series from methane to n-octadecane (C1H4 to C18H38). The empirical parameters in the reduced friction functions above are treated as universal constants, and they are listed in the following table. For convenience are critical viscosities included in the tables for models with 5- and 7-parameters that was presented further up.

Mixture
The mixture viscosity is given by

$$ \quad \eta_{mix} = \eta_{dmix} + \eta_{umix} = \eta_{dmix} + K_{pu}F_{umix}$$

The mixture viscosity of well defined components is given by

$$ \quad \eta_{dmix} = \eta_{0dmix} + K_{admix}P_{amix}  + K_{hdmix}P_{hmix} + K_{h2dmix}P_{hmix}^{2} + K_{h3dmix}P_{hmix}^{3} $$

The mixture viscosity function of uncertain components is given by

$$ \quad F_{umix} = \eta_{0umix} + K_{aumix}P_{amix}  + K_{humix}P_{hmix} + K_{h2umix}P_{hmix}^{2} + K_{h3umix}P_{hmix}^{3} $$

The mixture viscosity can be tuned to measured viscosity data by optimizing (regressing) the parameter $$ K_{pu} $$.

where the mixture friction coefficients are obtained by eq(I.7.45) through eq(I.7.47) and $$ P_{a} $$ and $$ P_{h} $$ are the attractive and repulsive pressure term of the mixture.

Mixing rules
The mixing rules for the well defined components are

$$ \quad \ln \left( \eta_{0dmix} \right) = \sum_{i=1}^{m} z_{i} \ln ( \eta_{0i} ) \quad \text{or} \quad \eta_{0mix} = \prod_{i=1}^{m} \eta_{0i}^{z_{i}} $$

$$ \quad K_{qrdmix} =\sum_{i=1}^{m}W_{i} \frac{\eta_{ci}K_{qri}}{P_{ci}} \quad \text{where} \quad q=a,h $$

$$ \quad K_{qprdmix} =\sum_{i=1}^{m}W_{i} \frac{\eta_{ci}K_{qrpi}}{P_{ci}^{p}} \quad \text{where} \quad q=a,h  \quad \text{and} \quad p=2,3 $$

QZS recommends to drop the dilute gas term for the uncertain fluid components which are usually the heavier (hydrocarbon) components. The formula is kept here for consistency. The mixing rules for the uncertain components are

$$ \quad \ln \left( \eta_{0umix} \right) = \sum_{i=m+1}^{N} z_{i} \ln ( \eta_{0i} ) \quad \text{or} \quad \eta_{0mix} = \prod_{i=m+1}^{N} \eta_{0i}^{z_{i}} $$

$$ \quad K_{qrumix} =\sum_{i=m+1}^{N}W_{i} \frac{D_{pi}K_{qri}}{P_{ci}} \quad \text{where} \quad q=a,h $$

$$ \quad K_{qprumix} =\sum_{i=m+1}^{N}W_{i} \frac{D_{pi}K_{qpri}}{P_{ci}^{p}} \quad \text{where} \quad q=a,h  \quad \text{and} \quad p=2,3 $$

$$ \quad \varepsilon = 0.30 \quad \text{when SRK, PR or PRSV EOS is used} $$

Dilute gas limit
Zéberg-Mikkelsen (2001) proposed an empirical model for dilute gas viscosity of fairly spherical molecules as follows

$$ \quad \eta_{0} = d_{g1} \sqrt{T} + d_{g2} T^{d_{g3}}$$

or

$$ \quad \eta_{0} = D_{g1} \sqrt{T_{r}} + D_{g2} T_{r}^{D_{g3}}$$

$$ \quad D_{g1} = d_{g1} \cdot \sqrt{T_{c}} \quad \text{and} \quad D_{g2} = d_{g2} \cdot T_{c}^{d_{g3}} \quad \text{and} \quad D_{g3} = d_{g3} $$

The unit equations for viscosity and temperature are

$$ \quad \left[ \eta_{0} \right] = \mu P \quad \text{and} \quad \left[ T \right]  = K $$

The second term is a correction term for high temperatures, and we note that most $$ d_{g2} $$ parameters are negative.

...

Light gases
Zéberg-Mikkelsen (2001) proposed a FF-model for light gas viscosity as follows

$$ \eta_{lg} = \eta_{0} + K_{a} P_{a} + K_{h} P_{h} + K_{h2} P_{h}^{2}$$

The friction functions for light gases are simple

$$ \quad K_{a} = B_{a0} $$

$$ \quad K_{h} = B_{h0} $$

$$ \quad K_{h2} = \frac { B_{h20} } {T_{r}^{2}}$$

The FF-model for light gas is valid for low, normal, critical and super critical conditions for these gases. Although the FF-model for viscosity of dilute gas is recommended, any accurate viscosity model for dilute gas can also be used with good results.

The unit equations for viscosity and temperature are

$$ \quad \left[ \eta_{lg} \right] = \mu P \quad \text{and} \quad \left[ T \right]  = K $$

.....

Transition state analogy
We started this article on viscosity for mixtures by displaying equations for dilute gas based on elementary kinetic theory, hard core (kinetic) theory and proceeded to selected theories (and models) that aimed at modeling viscosity for dense gases, dense fluids and supercritical fluids. Many or most of these theories where based on a philosophy of how gases behaves with molecules flying around, colliding with other molecules and exchanging (linear) momentum and thus creating viscosity. When the fluid became liquid, the models started to deviate from measurements because a small error in the calculated molar volume from the EOS is related to a large change in pressure and vica versa, and thus also in viscosity. We have now come to the other end where theories (or models) are based on a philosophy of how a liquid behaves and give rise to viscosity. Since molecules in a liquid are much closer to each other, one may wonder how often a molecule in one sliding fluid surface finds a free volume in the neighboring sliding surface that is big enough for the molecule to jump into it. This may be rephrased as: we may wonder when a molecule have enough energy in its fluctuating movements to squeeze into a small open volume in the neighboring sliding surface, similar to a molecule that collides with another molecule and locks into it in a chemical reaction, and thus creates a new compound, as modeled in the transition state theory (TS theory and TS model).

Free volume theory
The free volume theory (short FV theory and FV model) originates from Doolittle (1951) who proposed that viscosity is related to the free volume fraction $$ f_{\nu} $$ in a way that is analogous to the Arrhenius equation. The viscosity model of Doolittle (1951) is

$$ \quad \eta = A \exp \left[ \frac{B}{f_{\nu}} \right] \quad \text{where} \quad f_{\nu}  = \frac{V - b} {b} $$

where $$V$$ is the molar volume and $$b$$ is the molar hard core volume.

There where, however, little activity on the FV theory until Allal et al. (AMM 1996, AMB 2001a) proposed a relation between the free volume fraction and parameters (and/or variables) at the molecular level of the fluid (also called the microstructure of the fluid). The 1996-model became the start of a period with high research activity where different models were put forward. The surviving model was presented by Allal et al. (ABB 2001b), and this model will be displayed below.

The viscosity model is composed of a dilute gas contribution $$ \eta_{0} $$ (or $$ \eta_{dg} $$ ) and a dense-fluid contribution $$ \eta_{df} $$ (or dense-state contribution $$ \eta_{ds} $$ or $$ \Delta \eta$$).

$$ \quad \eta = \eta_{0} + \eta_{df} $$

Allal et al. (2001b) showed that the dense-fluid contribution to viscosity can be related to the friction coefficient $$ \zeta $$ of the sliding fluid surface, and Dulliens (1963) has shown that the self-diffusion coefficient $$ D $$ is related to the friction coefficient of an internal fluid surface. These two relations are shown here:

$$ \quad \eta_{df} = \frac{\rho N_{A}L_{p}^{2}\zeta}{M} \quad \text{and} \quad D = \frac{k_{B}T}{\zeta}$$

By eliminating the friction coefficient $$ \zeta $$, Boned et al. (BABZBQ 2004) expressed the characteristic length $$ L_{p} $$ as

$$ \quad L_{p}^{2} = \frac{DM \eta_{df}}{\rho N_{A} k_{B}T} = \frac{DM \eta_{df}}{\rho RT} $$

The right hand side corresponds to the so-called Dullien invariant which was derived by Dullien (1963, 1972). A result from this is that the characteristic length $$ L_{p}$$ is interpreted as the average momentum transfer distance to a molecule that will enter a free volume site and collide with a neigboring molecule.

The friction coefficient $$ \zeta $$ is modeled by Allal et al. (ABB 2001b) as

$$ \quad\zeta = \zeta_{0} \exp \left[ \frac{B}{f_{\nu}} \right] \quad \text{and} \quad \zeta_{0} = \frac{E}{N_{A}L_{d}} \left( \frac{M}{3RT}\right)^{1/2} $$

The free volume fraction is now related to the energy E by

$$ \quad f_{\nu} = \left( \frac {RT}{E} \right)^{3/2} \quad \text{and} \quad E = E_{0} + PV  \quad \text{and} \quad E_{0} = \alpha \rho$$ $$ \quad \text{where} \quad \rho = \frac{M} {V} $$

where $$E $$ is the total energy a the molecule must use in order to diffuse into a vacant volume, and $$PV $$ is connected to the work (or energy) necessary to form or expand a vacant volume available for diffusion of a molecule.The energy $$E_{0} $$ is the barrier energy that the molecule must overcome in order to diffuse, and it is modeled to be proportional to mass density in order to improve match of measured viscosity data. We note that the sensitive term $$V-b $$ in the denominator of Doolittle's (1951) model has disappeared, making the viscosity model of Allal et al. (ABB 2001b) more robust to numerical calculations of liquid molar volume by an imperfect EOS. The pre-exponential factor A is now a function and becomes

$$ \quad A = \frac{L_{c} \rho (\alpha\rho+PV)}{\sqrt{3MRT}} \quad \text{where} \quad L_{c} = \frac{L_{p}^{2}}{L_{d}}$$

The viscosity model proposed by Allal et al.(ABB 2001b) is thus

$$ \quad \eta =\eta_{0} + A  \exp \left[ B \left( \frac{\alpha\rho+PV}{RT} \right)^{3/2} \right] $$

As a digression we can mention that the self-diffusion coefficient of Boned et al. (BABZBQ 2004) becomes

$$ \quad D =\frac{RT L_{d}}{\alpha\rho+PV} \sqrt{\frac{3RT}{M}}  \exp \left[ - B \left( \frac{\alpha\rho+PV}{RT} \right)^{3/2} \right] $$

Local nomenclature list:


 * $$B $$ parameter that characterizes the free volume overlap or empirical tuning parameter [1]
 * $$b $$ molar hard core volume [m3/mol]
 * $$E $$ total energy which the molecule must use in order to diffuse [J/mole]
 * $$E_{0} $$ barrier energy which the molecule must overcome in order to diffuse [J/mole]
 * $$L_{p} $$ average momentum transfer distance for a molecular that transfer linear momentum (conf. hard core radius) and/or angular momentum (conf. radius of gyration) [Å]
 * $$L_{d} $$ dissipation length to the energy E [Å]
 * $$L_{c} $$ composite parameter that is characteristic for viscosity [Å]
 * $$M $$ molar mass, conf. molecular weight [kg/mol]
 * $$N_{A} $$ Avogadros constant
 * $$P $$ pressure [MPa]
 * $$R $$ gas constant R = 8.31451 [K·J/mol]
 * $$V $$ molar volume [m3/mol]
 * $$\alpha $$ characteristic parameter or empirical tuning parameter [1]
 * $$\eta $$ viscosity [Pas]
 * $$\rho $$ mass density [kg/m3]
 * $$\zeta$$ friction coefficient of a molecule related to the mobility of the molecule
 * $$\zeta_{0}$$ friction coefficient for zero mass density i.e. for a dilute system / low pressure limit

Trend functions
The three characteristic viscosity parameters $$ \, \alpha_{}, \, B_{}, L_{c} \, $$ are usually established by optimizing the viscosity formula against measured viscosity data for pure fluids (i.e. single component fluids). Data for these parameters can then be stored in databases together with data for other chemical and physical material properties and information. This happens more often if use of the equation becomes widespread. Hydrocarbon molecules is a huge group of molecules that has several subgroups which itself contains molecules of the same basic structure, but with different lengths. The alkanes is the simplest of these groups. A material property of molecules in such a group normally shows up as a function when plotted against another material property. A mathematical function is then selected based physical/chemical knowledge, experience and intuition, and the empirical parameters (i.e. constants) in the function are determined by curve fitting. Such a function is called a trend or trend function, and the group of molecule types is called a homologous series. Llovell et al. (2013a, 2013b) proposed trend functions for the three FV parameters $$ \alpha, \, B, \, L_{c} \, $$ for alkanes. Oliveira et al. (2014) proposed trend functions for the FV parameters for fatty acid methyl esters (FAME) and fatty acid ethyl esters (FAEE), both including compounds with up to three unsaturated bonds, which are displayed below.

$$ \quad \alpha = a_{0} + a_{1}M $$

$$ \quad B = b_{0} + b_{1}M + b_{2}M^{2} $$

$$ \quad L_{c} = c_{0} + c_{1}M $$

The molecular weights (or molar mass) associated with the parameters used in curve fitting process, correspond to carbon numbers in the range 8-24 and 8-20 for FAME and FAEE respectively.

....

Mixture
The mixture viscosity is

$$ \quad \eta_{mix} = \eta_{0mix} + \eta_{dfmix} $$

The dilute gas viscosity $$\eta_{0} $$ is taken from Chung et al.(1988) which is displayed in the section on SS theory. The dense fluid contribution to viscosity in FV theory is

$$ \quad \eta_{dfmix} = \frac {L_{cmix} \rho_{eos}(\alpha_{mix} \rho_{eos} + PV_{eos})}{\sqrt{3RTM_{mix}}}

\exp {\left[ B_{mix} \left( \frac{ \alpha_{mix} \rho_{eos} + PV_{eos}}{RT} \right)^{3/2} \right] }$$

where $$ \alpha \text{,} \, B \text{,} \, L_{c} \,$$ are three characteristic parameters of the fluid w.r.t. viscosity calculations. For fluid mixtures are these three parameters calculated using mixing rules. If the self-diffusion coefficient is included in the governing equations, probably via the diffusion equation, use of four characteristic parameters (i.e. use of Lp and Ld instead of Lc) will give a consistent flow model, but flow studies that involves the diffusion equation belongs a small class of special studies.

The unit for the viscosity is [Pas], when all other units are kept in SI units.

Mixing rules
At the end of the intensive research period Allal et al. (ABD 2001c) and Canet (2001) proposed two different set of mixing rules, and according to Almasi (2015) there has been no agreement in the literature about which are the best mixing rules. Almasi (2015) therefore recommended the classic linear mole weighted mixing rules which are displayed below for a mixture of N fluid components.

$$ \quad M_{mix} = M_{n} = \sum_{i=1}^{N} z_{i} M_{i} $$

$$ \quad \alpha_{mix} = \sum_{i=1}^{N} z_{i} \alpha_{i} $$

$$ \quad B_{mix} = \sum_{i=1}^{N} z_{i} B_{i} $$

$$\quad L_{cmix} = \sum_{i=1}^{N} z_{i} L_{ci} $$

The three characteristic viscosity parameters $$ \, \alpha_{i}, \, B_{i}, L_{ci} \, $$ are usually established by optimizing the viscosity formula against measured viscosity data for pure fluids (i.e. single component fluids).

Significant structure theory
Viscosity models based on significant structure theory, a designation originating from Eyring, (short SS theory and SS model) has in the first two decades of the 2000s evolved in a development relay. It starting with Macías-Salinas et al.(2003), continued with a significant contribution from Cruz-Reyes et al.(2005), followed by a third stage of development by Macías-Salinas et al.(2013), who's model is displayed here. The SS theories have three basic assumptions:


 * A liquid behaves similar to a solid in many aspects, e.g. a sensitive relation between molar volume (or mass density) and pressure; position and distance between molecules is like a quasi-lattice with "fluidized vacancies" of molecular size distributed randomly throughout the quazi-lattice. The vacancies are assumed to have molecular size and move freely throughout the quasi-lattise structure.
 * The fluid viscosity is calculated from two components which is a gas-like and a solid-like contribution, and both contributions contain all molecule types occurring in the fluid phase. A molecule that jumps from one sliding surface to a vacant site in the neighboring surface, is said to display gas-like behavior. A molecule that remains on its site in the sliding surface for some time, is said to display solid-like behavior.
 * Collisions between molecules from neighboring layers are equivalent to molecules jumping to vacant sites, and these events within viscosity modeling are analogous to chemical reactions between colliding molecules within TS theory.

The fraction of gas-like molecules $$ X_{gl}$$ and solid-like molecules $$ X_{sl}$$ are

$$ \quad X_{gl} = (V - V_{s})/V \quad \text{and} \quad X_{sl} = V_{s}/V \quad \text{and} \quad V_{s} \approx b$$

where $$V$$ is the molar volume of the phase in question, $$V_{s}$$ is the molar volume of solid-like molecules and $$b$$ is the molar hard core volume. The viscosity of the fluid is a mixture of these two classes of molecules

$$ \quad \eta = X_{gl}\eta_{gl} + X_{sl} \eta_{sl}$$

Gas-like contribution
The gas-like viscosity contribution is taken from the viscosity model of Chung et al.(1984, 1988), which is based on the Chapman-Enskog(1964) kinetic theory of viscosity for dilute gases and the empirical expression of Neufeld et al.(1972) for the reduced collision integral, but expanded empirical to handle polyatomic, polar and hydrogen bonding fluids over a wide temperature range. The viscosity model of Chung et al.(1988) is

$$ \quad \eta_{gl} = 40.785 \frac{\sqrt{MT^{*}}}{V_{c}^{2/3} \Omega^{*}} * F_{c} \quad \text{with unit equation} \quad [ \eta_{gl} ] = \mu P $$

$$ \quad \Omega^{*} = \frac{1.16145}{(T^{*})^{0.14874}} + \frac{0.52487}{exp(0.7732T^{*})} + \frac{2.16178}{exp(2.43787T^{*})} - 6.435 \times 10^{-4} (T^{*})^{0.14874} * sin \left[ 18.0323 (T^{*})^{0.7683} - 7.27371 \right] $$

where

$$ \quad T^{*} = 1.2593 * T / T_{c} \quad \text{and} \quad F_{c} = 1 - 0.2756 \omega + 0.059035 \mu_{r}^{4} + \kappa $$

Local nomenclature list:


 * $$F_{c} \, $$ : factor for molecular shape and polarities of dilute gases [1]
 * $$M \ $$ : molar mass, conf. molecular weight [g/mol]
 * $$T \ \ $$ : temperature [K]
 * $$T_{c} \ $$ : critical temperature [K]
 * $$V_{c} \ $$ : molar critical volume [cm3/mol]
 * $$\eta_{gl} \, $$ : gas-like viscosity contribution [μP]
 * $$\kappa \ \ \, $$ : correction factor for hydrogen bonding effects [1]
 * $$\mu_{r} \, $$ : reduced dipole moment [1]
 * $$\Omega^{*}$$ : reduced collision integral [1]
 * $$\omega \ \ $$ : acentric factor [1]

Solid-like contribution
In the 2000s, the development of the solid-like viscosity contribution started with Macías-Salinas et al.(2003) who used the Eyring equation in TS theory as an analogue to the solid-like viscosity contribution, and as a generalization of the first exponential liquid viscosity model proposed by Reynolds(1886). The Eyring equation models irreverible chemical reactions at constant pressure, and the equation therefore uses Gibbs activation energy, $$\Delta G^{\ddagger} $$, to model the transition state energy that the system uses to move matter (i.e. separate molecules) from the initial state to the final state (i.e. the new compound). In the Couette flow, the system moves matter from one sliding surface to another, due to fluctuating internal energy, and probably also due to pressure and the pressure gradient. Besides, the pressure effect on viscosity is somewhat different for systems in a medium pressure range than it is for systems in a very high pressure range. Cruz-Reyes et al.(2005) uses Helmholtz energy (F = U-TS = G-PV) as potential in the exponential function. This gives

$$ \quad \quad \eta_{sl} = A * exp{\left[ -\frac{\Delta G^{\ddagger} - PV}{RT} \right]}$$

Cruz-Reyes et al.(2005) states that the Gibbs activation energy is negative proportional to the internal energy of vaporization (and thus calculated at a point on the freezing curve), but Macías-Salinas et al.(2013) changes that to be the residual internal energy, $$\Delta U^{r} $$, at the general pressure and temperature of the system. One could alternatively use the grand potential ($$\Omega$$ = U-TS-G = -PV, sometimes called Landau energy or potential) in the exponential function and argue that the Couette flow is not a homogeneous system, such that a term with the residual internal energy must be added. Both arguments gives the proposed solid-like contribution which is

$$ \quad \eta_{sl} = A*exp \left[ -\frac{\alpha \Delta U^{r}-PV}{RT} \right] = A*exp \left[ -\frac{ \alpha \Delta U^{r}}{RT} + Z \right]$$

The pre-exponential factor $$A $$ is taken as

$$ \quad A = \frac{RT}{V-b} * \frac{1}{\nu}$$

The jumping frequency of a molecule that jumps from its initial position to a vacant site, $$\nu $$, is made dependent on the number of vacancies, $$X_{gl} $$, and pressure in order to extend the applicability of $$\eta_{sl} $$ to much wider ranges of temperature and pressure than a constant jumping frequency would do. The final jumping frequency model is

$$ \quad \nu = X_{gl}^{-1} * 10^{12} \left( \nu_{0}+\nu_{1}P \right) = \frac {V}{V - b} * 10^{12} \left( \nu_{0}+\nu_{1}P \right)$$

A recurrent problem for viscosity models is the calculation of liquid molar volume for a given pressure using an EOS that is not perfect. This calls for introduction of some emprical parameters. Use of adjustable proportionality factors for both the residual internal energy and the Z-factor is a natural choice. The sensitivity of P versus V-b values for liquids makes it natural to introduce an empirical exponent (power) to the dimensionless Z-factor. The empirical power turns out to be very effective in the high pressure (high Z-factor) region. The solid-like viscosity contribution proposed by Macías-Salinas et al.(2013) is then

$$ \quad \eta_{sl} = \frac{RT}{V}* \frac{1}{10^{12} \left( \nu_{0}+\nu_{1}P \right)}* exp \left[ - \alpha \frac{\Delta U^{r}}{RT} \right]* exp \left[ \beta_{0}Z^{\beta_{1}} \right]$$

Local nomenclature list:


 * $$b \ \ \ \ \ $$ : molar hard core volume of the fluid phase [cm3/mol]
 * $$P \ \ \ \,$$ : pressure [bar]
 * $$T \ \ \ \,$$ : temperature [K]
 * $$V \ \ \ \,$$ : molar volume of the fluid phase [cm3/mol]
 * $$X_{jl} \ $$ : volume fraction of the j-like contribution j=g,s [1]
 * $$Z \ \ \ \,$$ : compressibility factor (Z-factor) [1]
 * $$\alpha \ \ \ \ $$ : proportionality factor [1]
 * $$\beta_{i} \ \ \ $$ : adjustable parameters i=0,1 [1]
 * $$\eta \ \ \ \ \, $$ : viscosity of the fluid phase [μPa·s]
 * $$\eta_{sl} \ \ $$ : solid-like viscosity contribution [μPa·s]
 * $$\nu_{i} \ \ \ $$ : adjustable parameters i=0,1 [s-1] and [bar-1s-1]
 * $$\Delta G^{\ne}$$ : activation energy of the fluid [J/mol]
 * $$\Delta U^{r} \,$$ : residual internal energy of the fluid [J/mol]

Mixture
$$ \quad \eta_{mix} = \frac{V_{mix}-b_{mix}}{V_{mix}} * \eta_{gl}^{mix} + \frac{b_{mix}}{V_{mix}} * \eta_{sl}^{mix}$$

$$ \quad \eta_{gl}^{mix} = F ( T_{cmix},M_{cmix},V_{cmix},\omega_{mix},\mu_{rmix};T)$$

$$ \quad \eta_{sl}^{mix} = F ( V_{mix},\Delta U_{mix}^{r},Z_{mix};P,T)$$

In order to clarify the mathematical statements above, the solid-like contribution for a fluid mixture is displayed in more details below.

$$ \quad \eta_{sl}^{mix} = \frac{RT}{V_{mix}}* \frac{1}{10^{12} \left( \nu_{0}+\nu_{1}P \right)}* exp \left[ - \alpha \frac{\Delta U_{mix}^{r}}{RT} \right]* exp \left[ \beta_{0}Z_{mix}^{\beta_{1}} \right]$$

Mixing rules
The variables $$V_{mix},\Delta U_{mix}^{r},Z_{mix}$$ and all EOS parameters for a fluid mixture are taken from the EOS (conf. W) and the mixing rules used by the EOS (conf. Q). More details on this is displayed below.

A fluid of n mole in the single phase region where the total fluid composition is $$ \mathbf{z} $$ [molefractions]:

$$ \quad Q_{mix} = Q_{eos}(\mathbf{z}) \quad \text{and} \quad W_{mix} = W_{eos}(P,T,\mathbf{z})$$

Gas phase of ng mole in two-phase region where the gas composition is $$ \mathbf{y} $$ [molefractions]:

$$ \quad Q_{mix} = Q_{eos}(\mathbf{y}) \quad \text{and} \quad W_{mix} = W_{eos}(P,T,\mathbf{y})$$

Liquid phase of nl mole in two-phase region where the liquid composition is $$ \mathbf{x} $$ [molefractions]:

$$ \quad Q_{mix} = Q_{eos}(\mathbf{x}) \quad \text{and} \quad W_{mix} = W_{eos}(P,T,\mathbf{x})$$

where

$$ \quad n = n_{l} + n_{g} \quad \text{and} \quad n z_{i} = n_{l}x_{i} + n_{g}y_{i} \quad \text{and} \quad i = 1,\ldots,N$$

$$ \quad Q = T_{c}, M, V_{c}, \omega, b \quad \text{and} \quad W = V, \Delta U^{r}, Z $$

Since nearly all input to this viscosity model is provided by the EOS and the equilibrium calculcations, this SS model (or TS model) for viscosity should be very simple to use for fluid mixtures. The viscosity model also have some empirical parameters that can be used as tuning parameters to compensate for imperfect EOS models and secure high accuracy also for fluid mixtures.