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.
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.
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.
) takes the simple form: Inserting these simplifications gives us a defining equation that can be used to interpret experimental measurements: where
In textbooks on elementary kinetic theory[1] 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 where two parallel plates are separated by a gas layer.
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 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.
Notice that Inserting the critical temperature in the equation for dilute viscosity gives The default values of the parameters
The viscosity model for dilute gas, that is shown above, is widely used throughout the industry and applied science communities.
The gas viscosity model of Chung et alios (1988)[5] is combination of the Chapman–Enskog(1964) kinetic theory of viscosity for dilute gases and the empirical expression of Neufeld et alios (1972)[6] for the reduced collision integral, but expanded empirical to handle polyatomic, polar and hydrogen bonding fluids over a wide temperature range.
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.
[2] Uyehara and Watson (1944)[4] proposed a correlation for critical viscosity (for fluid component i) for n-alkanes using their average parameter
Wilke (1950)[7] 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.
Instead, the slightly simpler mixing rule proposed by Herning and Zipperer (1936),[8] is found to be suitable for gases of hydrocarbon mixtures.
The simplest model of the dense fluid viscosity is a (truncated) power series of reduced mole density or pressure.
The rotational coupling parameter for the mixture is 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)[20] equation of state variant suggested by McCarty (1974),[21] 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 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.
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.
Pedersen et al. (1987) added a fourth term, that is correcting the reference viscosity formula at low reduced temperatures.
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-critical pressures.
The unit equations for the central variables in the multi-parameter FF-model is Friction functions for fluid component i in the 5 parameter model for pure n-alkane molecules are presented below.
In the two-phase gas-liquid region a vapor-liquid equilibrium (VLE) calculation splits the fluid into a vapor (gas) phase with composition
In this case the reservoir simulator software code may use or The friction model for viscosity of a mixture is 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.
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.
The article has 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.
Almasi (2015) therefore recommended the classic linear mole weighted mixing rules which are displayed below for a mixture of N fluid components.
The sensitivity of P versus V-b values for liquids makes it natural to introduce an empirical exponent (power) to the dimensionless Z-factor.