The
shear viscosity
The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.
Viscosity quantifies the inter ...
(or viscosity, in short) of a fluid is a material property that describes the friction between internal neighboring fluid surfaces (or sheets) flowing with different fluid velocities. This friction is the effect of (linear) momentum exchange caused by molecules with sufficient energy to move (or "to jump") between these fluid sheets due to fluctuations in their motion. The viscosity is not a material constant, but a material property that depends on temperature, pressure, fluid mixture composition, local velocity variations. This functional relationship is described by a mathematical viscosity model called a
constitutive equation
In physics and engineering, a constitutive equation or constitutive relation is a relation between two physical quantities (especially kinetic quantities as related to kinematic quantities) that is specific to a material or substance, and app ...
which is usually far more complex than the defining equation of shear viscosity. One such complicating feature is the relation between the viscosity model for a pure fluid and the model for a fluid mixture which is called mixing rules. When scientists and engineers use new arguments or theories to develop a new viscosity model, instead of improving the reigning model, it may lead to the first model in a new class of models. This article will display one or two representative models for different classes of viscosity models, and these classes are:
* Elementary kinetic theory and simple empirical models
- viscosity for dilute gas with nearly spherical molecules
*
Power series
In mathematics, a power series (in one variable) is an infinite series of the form
\sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots
where ''an'' represents the coefficient of the ''n''th term and ''c'' is a con ...
- simplest approach after dilute gas
*
Equation of state
In physics, chemistry, and thermodynamics, an equation of state is a thermodynamic equation relating state variables, which describe the state of matter under a given set of physical conditions, such as pressure, volume, temperature, or intern ...
analogy
between PVT and T
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
In chemistry, the transition state of a chemical reaction is a particular configuration along the reaction coordinate. It is defined as the state corresponding to the highest potential energy along this reaction coordinate. It is often marked ...
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 sections. This means that some known contributions of 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.
The microscopic or
molecular origin of fluids means that
transport coefficient A transport coefficient \gamma measures how rapidly a perturbed system returns to equilibrium.
The transport coefficients occur in transport phenomenon with transport laws
: \mathbf_k = \gamma_k \mathbf_k
where:
: \mathbf_k is a flux of the prop ...
s 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
The Boltzmann equation or Boltzmann transport equation (BTE) describes the statistical behaviour of a thermodynamic system not in a state of equilibrium, devised by Ludwig Boltzmann in 1872.Encyclopaedia of Physics (2nd Edition), R. G. Lerne ...
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 analytical derive (analytical) mathematical models for properties characteristic to fluids such as viscosity,
thermal conductivity
The thermal conductivity of a material is a measure of its ability to conduct heat. It is commonly denoted by k, \lambda, or \kappa.
Heat transfer occurs at a lower rate in materials of low thermal conductivity than in materials of high thermal ...
, and
electrical conductivity (by treating the charge carriers in a material as a gas). See also
convection–diffusion equation
The convection–diffusion equation is a combination of the diffusion equation, diffusion and convection (advection equation, advection) equations, and describes physical phenomena where particles, energy, or other physical quantities are transferr ...
. The mathematics is so complicated for polar and non-spherical molecules that it is very difficult to get practical models for viscosity. The purely theoretical approach will therefore be left out for the rest of this article, except for some visits related to dilute gas and significant structure theory.
Use, definition and dependence
The classic
Navier-Stokes equation is the
balance equation for momentum density for an isotropic, compressional and viscous fluid that is used in
fluid mechanics
Fluid mechanics is the branch of physics concerned with the mechanics of fluids ( liquids, gases, and plasmas) and the forces on them.
It has applications in a wide range of disciplines, including mechanical, aerospace, civil, chemical and ...
in general and
fluid dynamics in particular:
:
On the right hand side is (the divergence of) the total stress tensor
which consists of a pressure tensor
and a dissipative (or viscous or deviatoric) stress tensor
. The dissipative stress consists of a compression stress tensor
(term no. 2) and a shear stress tensor
(term no. 3). The rightmost term
is the gravitational force which is the body force contribution, and
is the mass density, and
is the fluid velocity.
:
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
In continuum mechanics, the strain-rate tensor or rate-of-strain tensor is a physical quantity that describes the rate of change of the deformation of a material in the neighborhood of a certain point, at a certain moment of time. It can be def ...
.
Stokes assumptions for a wide class of fluids therefore says that for an isotropic fluid the compression and shear stresses are proportional to their velocity gradients,
and
respectively, and named this class of fluids for
Newtonian fluid
A Newtonian fluid is a fluid in which the viscous stresses arising from its flow are at every point linearly correlated to the local strain rate — the rate of change of its deformation over time. Stresses are proportional to the rate of chang ...
s. The classic defining equation for
volume viscosity Volume viscosity (also called bulk viscosity, or dilatational viscosity) is a material property relevant for characterizing fluid flow. Common symbols are \zeta, \mu', \mu_\mathrm, \kappa or \xi. It has dimensions (mass / (length × time)), and the ...
and
shear viscosity
The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.
Viscosity quantifies the inter ...
are respectively:
:
:
The classic compression velocity "gradient" is a diagonal tensor that describes a compressing (alt. expanding) flow or
attenuating sound waves
In physics, sound is a vibration that propagates as an acoustic wave, through a transmission medium such as a gas, liquid or solid.
In human physiology and psychology, sound is the ''reception'' of such waves and their ''perception'' by the ...
:
:
The classic Cauchy shear velocity gradient, is a symmetric and traceless tensor that describes a pure shear flow (where pure means excluding normal outflow which in mathematical terms means a traceless matrix) around e.g. a wing, propeller, ship hull or in e.g. a river, pipe or vein with or without bends and boundary skin:
:
where the symmetric gradient matrix with non-zero trace is
:
How much the volume viscosity contributes to the flow characteristics in e.g. a
choked flow
Choked flow is a compressible flow effect. The parameter that becomes "choked" or "limited" is the fluid velocity.
Choked flow is a fluid dynamic condition associated with the venturi effect. When a flowing fluid at a given pressure and temperatu ...
such as
convergent-divergent nozzle
A de Laval nozzle (or convergent-divergent nozzle, CD nozzle or con-di nozzle) is a tube which is pinched in the middle, making a carefully balanced, asymmetric hourglass shape. It is used to accelerate a compressible fluid to supersonic speeds ...
or
valve
A valve is a device or natural object that regulates, directs or controls the flow of a fluid (gases, liquids, fluidized solids, or slurries) by opening, closing, or partially obstructing various passageways. Valves are technically fitting ...
flow is not well known, but the shear viscosity is by far the most utilized viscosity coefficient. The volume viscosity will now be abandoned, and the rest of the article will focus on the shear viscosity.
Another application of shear viscosity models is
Darcy's law for multiphase flow
Morris Muskat et al.Muskat M. and Meres M.W. 1936. The Flow of Heterogeneous Fluids Through Porous Media. Paper published in J. Appl. Phys. 1936, 7, pp 346–363. https://dx.doi.org/10.1063/1.1745403Muskat M. and Wyckoff R.D. and Botset H.G. and M ...
.
:
where a = water, oil, gas
and
and
are absolute and relative permeability, respectively. These 3 (vector) equations models flow of water, oil and natural gas in subsurface oil and gas reservoirs in porous rocks. Although the pressures changes are big, the fluid phases will flow slowly through the reservoir due to the flow restriction caused by the porous rock.
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 fluid dynamics, Couette flow is the flow of a viscous fluid in the space between two surfaces, one of which is moving tangentially relative to the other. The relative motion of the surfaces imposes a shear stress on the fluid and induces flow ...
. In this experimental layout, the
shear stress
Shear stress, often denoted by (Greek: tau), is the component of stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross section. ''Normal stress'', on the ...
and the shear velocity gradient
(where now
) takes the simple form:
:
Inserting these simplifications gives us a defining equation that can be used to interpret experimental measurements:
:
where
is the area of the moving plate and the stagnant plate,
is the spatial coordinate normal to the plates. In this experimental setup, value for the force
is first selected. Then a maximum velocity
is measured, and finally both values are entered in the equation to calculate viscosity. This gives one value for the viscosity of the selected fluid. If another value of the force is selected, another maximum velocity will be measured. This will result in another viscosity value if the fluid is a
non-Newtonian fluid
A non-Newtonian fluid is a fluid that does not follow Newton's law of viscosity, i.e., constant viscosity independent of stress. In non-Newtonian fluids, viscosity can change when under force to either more liquid or more solid. Ketchup, for ex ...
such as paint, but it will give the same viscosity value for a
Newtonian fluid
A Newtonian fluid is a fluid in which the viscous stresses arising from its flow are at every point linearly correlated to the local strain rate — the rate of change of its deformation over time. Stresses are proportional to the rate of chang ...
such as water, petroleum oil or gas. If another parameter like temperature,
, is changed, and the experiment is repeated with the same force, a new value for viscosity will be the calculated, 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 mathematical 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 mathematical function is capable to do.
For a Newtonian fluid, the
constitutive equation
In physics and engineering, a constitutive equation or constitutive relation is a relation between two physical quantities (especially kinetic quantities as related to kinematic quantities) that is specific to a material or substance, and app ...
for shear viscosity is generally a
function of temperature, pressure,
fluid composition:
:
where
is the liquid phase composition with molfraction
for fluid component i, and
and
are the gas phase and total fluid compositions, respectively. For a non-Newtonian fluid (in the sense of a
generalized Newtonian fluid A generalized Newtonian fluid is an idealized fluid for which the shear stress is a function of shear rate at the particular time, but not dependent upon the history of deformation. Although this type of fluid is non-Newtonian (i.e. non-linear) in ...
), the constitutive equation for shear viscosity is also a function of the shear velocity gradient:
:
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 equation
In physics and engineering, a constitutive equation or constitutive relation is a relation between two physical quantities (especially kinetic quantities as related to kinematic quantities) that is specific to a material or substance, and app ...
s 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 certainly true. Non-Newtonian fluids will therefore be abandoned, and the rest of this article will focus on Newtonian fluids.
Dilute gas limit and scaled variables
Elementary kinetic theory
In textbooks on elementary kinetic
theory
one can find results for dilute gas modeling that have widespread use. Derivation of the kinetic model for shear viscosity usually starts by considering a
Couette flow
In fluid dynamics, Couette flow is the flow of a viscous fluid in the space between two surfaces, one of which is moving tangentially relative to the other. The relative motion of the surfaces imposes a shear stress on the fluid and induces flow ...
where two parallel plates are separated by a gas layer. This non-equilibrium flow is superimposed on a
Maxwell–Boltzmann equilibrium distribution of molecular motions.
Let
be the collision
cross section
Cross section may refer to:
* Cross section (geometry)
** Cross-sectional views in architecture & engineering 3D
*Cross section (geology)
* Cross section (electronics)
* Radar cross section, measure of detectability
* Cross section (physics)
**Abs ...
of one molecule colliding with another. The number density
is defined as the number of molecules per (extensive) volume
. The collision cross section per volume or collision cross section density is
, and it is related to the
mean free path
In physics, mean free path is the average distance over which a moving particle (such as an atom, a molecule, or a photon) travels before substantially changing its direction or energy (or, in a specific context, other properties), typically as a ...
by
:
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:
:
where
:
where
is the
Boltzmann constant
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constant, ...
,
is the
Avogadro constant
The Avogadro constant, commonly denoted or , is the proportionality factor that relates the number of constituent particles (usually molecules, atoms or ions) in a sample with the amount of substance in that sample. It is an SI defining c ...
,
is the
gas constant
The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . It is the molar equivalent to the Boltzmann constant, expressed in units of energy per temperature increment per ...
,
is the
molar mass
In chemistry, the molar mass of a chemical compound is defined as the mass of a sample of that compound divided by the amount of substance which is the number of moles in that sample, measured in moles. The molar mass is a bulk, not molecular, ...
and
is the
molecular mass
The molecular mass (''m'') is the mass of a given molecule: it is measured in daltons (Da or u). Different molecules of the same compound may have different molecular masses because they contain different isotopes of an element. The related quanti ...
. The equation above presupposes that the gas
density
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
is low (i.e. the pressure is low), hence the subscript zero in the variable
. 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
, implies that the collision cross section of one molecule can be estimated by
:
:
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
The Morse potential, named after physicist Philip M. Morse, is a convenient
interatomic interaction model for the potential energy of a diatomic molecule. It is a better approximation for the vibrational structure of the molecule than the qua ...
. 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.
) is therefore called
kinetic radius. The d-parameter (where
) is called
kinetic diameter.
The macroscopic collision cross section
is often associated with the critical molar volume
, and often without further proof or supporting arguments, by
:
where
is molecular shape parameter that is taken as 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
, and use of reduced temperature
, gives
:
:
which implies that the empirical parameter
is dimensionless, and that
and
have the same units. The parameter
is a scaling parameter that involves the gas constant
and the critical molar volume
, and it used to scale the viscosity. In this article the viscosity scaling parameter will frequently be denoted by
which involve one or more of the parameters
,
,
in addition to critical temperature
and molar mass
. Incomplete scaling parameters, such as the parameter
above where the gas constant
is absorbed into the empirical constant, will often be encountered in practice. In this case the viscosity equation becomes
:
where the empirical parameter
is not dimensionless, and a proposed viscosity model for dense fluid will not be dimensionless if
is the common scaling factor. Notice that
:
Inserting the critical temperature in the equation for dilute viscosity gives
:
The default values of the parameters
and
should be fairly universal values, although
depends on the unit system. However, the critical molar volume in the scaling parameters
and
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
:
where the critical compressibility factor
, 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
a universal average value of
(and the gas constant
) into a default value of the tuning parameter
as a practical solution of the difficulties of getting experimental values for
and/or
. The visocity model for a dilute gas is then
:
:
By inserting the critical temperature in the formula above, the critical viscosity is calculated as
:
Based on an average critical compressibility factor of
and measured critical viscosity values of 60 different molecule types, Uyehara and Watson (1944)
determined an average value of
to be
:
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.
Accepting the cubic EOS, the molar hard core volume
can be calculated from the turning point constraint at the critical point. This gives
:
where the constant
is a universal constant that is specific for the selected variant of the cubic EOS. This says that using
, and disregarding fluid component variations of
, 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 there are families of molecule types (i.e. groups of fluid components). The simplest group is the n-alkanes which are long chains of CH
2-elements. The more CH
2-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
:
This says that the scaling parameter
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
that is used extensively throughout the industry and applied science communities as a scaling factor for (shear) viscosity. The literature often reports the reciprocal parameter and denotes it as
.
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
pressure
ar molar mass
/mol low density (low pressure or dilute) gas viscosity
P It is, however, common in the industry to use another unit for liquid and high density gas viscosity
P
Kinetic theory
From
Boltzmann's equation
The Boltzmann equation or Boltzmann transport equation (BTE) describes the statistical behaviour of a thermodynamic system not in a state of equilibrium, devised by Ludwig Boltzmann in 1872.Encyclopaedia of Physics (2nd Edition), R. G. Lerne ...
Chapman and Enskog derived a
viscosity model for a dilute gas.
:
where
is (the absolute value of) the energy-depth of the potential well (see e.g.
Lennard-Jones interaction potential). The term
is called the collision integral, and it is occurs 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: The (analytical) mathematics gets incredible complex for polar and non-spherical molecules making it very difficult to achieve practical models for viscosity based on a statistical approach. The purely statistical approach will therefore be left out in the rest of this article.
Empirical correlation
Zéberg-Mikkelsen (2001)
proposed empirical models for gas viscosity of fairly spherical molecules that is displayed in the section on Friction Force theory and its models for dilute gases and simple light gases. These simple empirical correlations illustrate that empirical methods competes with the statistical approach with respect to gas viscosity models for simple fluids (simple molecules).
Kinetic theory with empirical extension
The gas viscosity model of Chung et alios (1988)
is combination of the
Chapman–Enskog(1964)
kinetic theory of
viscosity for dilute gases and the empirical expression of
Neufeld et alios (1972)
for the reduced collision integral, but expanded empirical to handle polyatomic, polar and hydrogen
bonding fluids over a wide temperature range. This viscosity model illustrates a successful combination of kinetic theory and empiricism, and it is displayed in the section of Significant structure theory and its model for the gas-like contribution to the total fluid viscosity.
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.
:
:
:
Zéberg-Mikkelsen (2001)
proposed an empirical correlation for the
parameter for n-alkanes, which is
:
:
The critical molar volume of component i
is related to the critical mole density
and critical mole concentration
by the equation
. From the above equation for
it follows that
:
where
is the compressibility factor for component i, which is often used as an alternative to
. By establishing a trend function for the parameter
for a
homologous series, groups or families of molecules, parameter values for unknown fluid components in the homologous group can be found by interpolation and extrapolation, and parameter values can easily re-generateat 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
and the classical pressure dominated scaling parameter
:
:
:
Zéberg-Mikkelsen (2001)
proposed an empirical correlation for critical viscosity η
ci parameter for n-alkanes, which is
:
:
The unit equations for the two constitutive equations above by Zéberg-Mikkelsen (2001) are
:
Inserting the critical temperature in the three viscosity equations from elementary kinetic theory gives three parameter equations.
:
:
The three viscosity equations now coalesce to a single viscosity equation
:
because a
nondimensional scaling is used for the entire viscosity equation. The standard nondimensionality reasoning goes like this: Creating nondimensional variables (with subscript D) by scaling gives
:
Claiming nondimensionality gives
:
The collision cross section and the critical molar volume which are both difficult to access experimentally, are avoided or circumvented. On the other hand, the critical viscosity has appeared 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
Classic mixing rules for gas
Wilke (1950)
derived a mixing rule based on kinetic gas theory
:
:
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 containing 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),
is found to be suitable for gases of hydrocarbon mixtures.
Classic mixing rules for liquid
The classic Arrhenius (1887).
mixing rule for liquid mixtures is
:
where
is the viscosity of the liquid mixture,
is the viscosity (equation) for fluid component i when flowing as a pure fluid, and
is the molfraction of component i in the liquid mixture.
The Grunberg-Nissan (1949)
mixing rule extends the Arrhenius rule to
:
where
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
:
where
is the partial molar volume of component i, and
is the molar volume of the liquid phase and comes from the vapor-liquid equilibrium (VLE) calculation or the EOS for single phase liquid.
A modification of the Katti-Chaudhri mixing rule is
:
:
where
is the excess activation energy of the viscous flow, and
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
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
, and subtracts this contribution from the total viscosity which is measured in the laboratory. This gives a residual viscosity term, often denoted
, which represents the contribution of the dense fluid,
.
:
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 (truncated) power series of reduced mole density or pressure.
Jossi et al. (1962)
presented such a model based on reduced mole density, but its most widespread form is the version proposed by
Lohrenz et al. (1964)
which is displayed below.
:
The LBC-function is then expanded in a (truncated) power series with empirical coefficients as displayed below.
:
The final viscosity equation is thus
:
:
:
Local nomenclature list:
*
: mole density
3">ol/cm3*
: reduced mole density
*
:
molar mass
In chemistry, the molar mass of a chemical compound is defined as the mass of a sample of that compound divided by the amount of substance which is the number of moles in that sample, measured in moles. The molar mass is a bulk, not molecular, ...
/mol*
: critical pressure
tm*
: temperature
*
: critical temperature
*
: critical molar volume
3/mol">m3/mol*
: viscosity
P
Mixture
:
:
:
:
The formula for
that was chosen by LBC, is displayed in the section called Dilute gas contribution.
Mixing rules
:
:
:
:
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 ft
3/lb mole, and it is calculated by
:
where
is the specific gravity of the C7+ fraction.
:
:
:
The molar mass
(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 the molar volume of the reservoir fluid (mixture) is calculated.
:
The molar volume
is converted to mole density
(also called mole concentration and denoted
), and then scaled to be reduced mole density
.
:
Dilute gas contribution
The correlation for dilute gas viscosity of a mixture is taken from Herning and Zipperer
(1936)
and is
:
The correlation for dilute gas viscosity of the individual components is taken from
Stiel and Thodos (1961)
and is
:
where
:
:
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 z) 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
:
When the common CS principle above is applied to viscosity, it reads
:
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.
(1984, 1987, 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
as suggested by Tham and Gubbins
(1970).
The reference fluid is methane, and it is given the subscript z.
:
:
:
Mixing rules
The interaction terms for critical temperature and critical volume are
:
:
The parameter
is usually uncertain or not available. One therefore wants to avoid this parameter. Replacing
with the generic average parameter
for all components, gives
:
:
:
The above expression for
is now inserted into the equation for
. This gives the following mixing rule
:
Mixing rule for the critical pressure of the mixture is established in a similar way.
:
:
:
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
and mole weighting
.
:
:
The rotational coupling parameter for the mixture is
:
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 Benedict-Webb-Rubin (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 calculated 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
:
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
:
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
:
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. Using the BWRM EOS, the molar volume of the reference fluid is calculated as
:
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
:
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
:
The formulas for
,
,
are taken from Hanley et al.
(1975).
The dilute gas contribution is
:
The temperature dependent factor of the first density contribution is
:
The dense fluid term is
:
where exponential function is written both as
and as
. 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
and
are weight factors. Their correction term is
:
:
:
:
:
:
Equation of state analogy
Phillips
(1912)
plotted temperature
versus viscosity
for different isobars for propane, and observed a similarity between these isobaric curves and the classic isothermal curves of the
surface. Later, Little and Kennedy
(1968)
developed the first viscosity model based on analogy between
and
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.
:
The viscosity equation of Guo (1998) is displayed on the next line.
:
To prepare for the mixing rules, the viscosity equation is re-written for a single fluid component i.
:
Details of how the composite elements of the equation are related to basic parameters and variables, is displayed below.
:
:
:
:
:
:
:
:
:
:
:
Mixture
:
Mixing rules
:
:
:
:
Friction force theory
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. (2000, 2001a, 2001b and Z 2001, 2004, 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 (
), and then establish a model for the dense fluid viscosity
. The FF theory states that for a fluid under shear motion, the shear stress
(i.e. the dragging force) acting between two moving layers can be separated into a term
caused by dilute gas collisions, and a term
caused by friction in the dense fluid.
:
The dilute gas viscosity (i.e. the limiting viscosity behavior as the pressure, normal stress, goes to zero) and the dense fluid viscosity (the residual viscosity) can be calculated by
:
where du/dy
is the local velocity gradient orthogonal to the direction of flow. Thus
:
The basic idea of QZS (2000) is that internal surfaces in a
Couette flow
In fluid dynamics, Couette flow is the flow of a viscous fluid in the space between two surfaces, one of which is moving tangentially relative to the other. The relative motion of the surfaces imposes a shear stress on the fluid and induces 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
and the normal force
is given by
:
where
is known as the kinetic friction coefficient, A is the area of the internal flow surface,
is the shear stress and
is the normal stress (or pressure
) between neighboring layers in the
Couette flow
In fluid dynamics, Couette flow is the flow of a viscous fluid in the space between two surfaces, one of which is moving tangentially relative to the other. The relative motion of the surfaces imposes a shear stress on the fluid and induces flow ...
.
:
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
of the dense fluid can therefore be considered to consist of an attractive friction shear contribution
and a repulsive friction shear contribution
. Inserting this gives us
:
The well known cubic equation of states (
SRK, PR and PRSV EOS), can be written in a general form as
:
The parameter pair (u,w)=(1,0) gives the
SRK EOS, and (u,w)=(2,-1) gives 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