Viscosity kinetic theory expression

The ordinary multicomponent diffusion coefficients D j and the viscosity and thermal conductivity are computed from appropriate kinetic theory expressions. First, pure species properties are computed from the standard kinetic theory expressions. For example, the binary diffusion coefficients are given in terms of pressure and temperature as... 

At high pressure, Lid effective viscosity becomes independent of pressure and equal to the kinetic theory expression developed earlier. At low pressure, Lid is no longer negligible compared to unity, the slip distance becomes a significant fraction of the separation between the surfaces, and the effective viscosity decreases. 

For gases in the low-density limit, a kinetic-theory expression similar to that for viscosity can be used to evaluate single-component thermal conductivity. For a monatomic gas, meaning a gas with no rotational of vibrational degrees of freedom, the thermal conductivity is expressed as... 

It turns out to be a surprisingly difficult task to determine accurately the thermal conductivity of polyatomic gases from the viscosity. Many of the approaches are motivated by the ideas of Eucken. The so-called Eucken factor is a nondimensional group determined by dividing the kinetic-theory expression for a monatomic gas by that for viscosity, yielding... 

Using appropriate kinetic theory expressions, evaluate and plot the pure-species viscosities and thermal conductivities in the range 1000 K < T < 2000 K. 

Pure species viscosities are given by the standard kinetic theory expression [178]... 

APPENDIX 5.3. THE KINETIC THEORY EXPRESSION FOR THE VISCOSITY OF A FLUID... 

Appendix 5.3. The Kinetic Theory Expression for the Viscosity of a Eluid. 756... 

If the diameter of HI (obtained from viscosity) is cr = 435 pm, estimate the rate of decomposition of HI at 700 K using the kinetic theory expression for the number of collisions between like molecules and the values of the activation energy obtained in Problem 33.2. 

Note that the pure-component viscosities t] are calculated with the correction factors frj, but the interaction viscosities jj/y omit any /. This apparently slightly inconsistent procedure compensates for the error involved in the use of only a first-order kinetic-theory expression for /jmix (Najafi et al. 1983). 

The Molecular Origins of Mass Diffusivity. In a manner directly analogous to the derivations of Eq. (4.6) for viscosity and Eq. (4.34) for thermal conductivity, the diffusion coefficient, or mass diffusivity, D, in units of m /s, can be derived from the kinetic theory of gases for rigid-sphere molecules. By means of summary, we present all three expressions for transport coefficients here to further illustrate their similarities. 

As illustrated in the low-density limit of Fig. 3.3, the viscosity of gases increases with increasing temperature. Moreover, for pressures well below the critical pressure, there is very little pressure dependence. The kinetic theory of dilute gases provides the theoretical basis for the temperature dependence. The Chapman-Enskog theory provides an expression for dilute pure-species viscosities as... 

Comparing with Eq. 12.1 gives a kinetic gas theory expression for the viscosity ... 

At low pressure, the viscosity and thermal conductivity are independent of the pressure. This is observed experimentally, in confirmation of the kinetic theory. Therefore, these quantities can be expressed as a function of the temperature alone. Most of these correlations will also be based on the square root of the temperature, although the exact expressions tend to be more complicated. 

The first term on the right-hand side represents momentum exchange between solid phases I and s and Kis is the solid-solid exchange coefficient. The last term represent additional shear stresses, which appear in granular flows (due to particle translation and collisions). Expressions for solids pressure, solids viscosity (shear and bulk) and solid-solid exchange coefficients are derived from the kinetic theory of granular flows. 

Several different expressions have been derived for solids pressure, solids shear viscosity and solids bulk viscosity, employing different approximations and assumptions while applying the kinetic theory of granular flows. Some of the commonly used equations are described below (see Gidaspow, 1994 and a review given by Peirano, 1998) Solids pressure ... 

Boltzmann, Ludwig. (1844-1906). Bom in Vienna, Boltzmann was interested primarily in physical chemistry and thermodynamics. His work has importance for chemistry because of his development of the kinetic theory of gases and the rules governing their viscosity and diffusion. The mathematical expression of his most important generalizations is known as Boltzmann s law, still regarded as one of the cornerstones of physical science. 

In the experimental arrangement shown in Figure 7.1 at steady state, the net flux is equal to the mean molecular velocity multiplied by the total concentration. According to the kinetic theory, the viscosity of a gas is independent of pressure, while it is expected to vary with the gas composition. Substitution of Eq. (7.14) into Eqs. (7.10) and (7.11), with Vd from Eq. (7.12) yields the expression for the fluxes caused by gradients in both composition and total pressure... 

Note that the viscosity parameter p has been introduced as a prefactor in front of the tensor functions by substitution of the kinetic theory transport coefficient expression after comparing the kinetic theory result with the definition of the viscous stress tensor o, (2.69). In other words, this model inter-comparison defines the viscosity parameter in accordance with the Enskog theory. 

Many mathematical expressions of varying complexity and form have been proposed in the literature to model shear-thinning characteristics some of these are straightforward attempts at cmve fitting, giving empirical relationships for the shear stress (or apparent viscosity)-shear rate curves for example, while others have some theoretical basis in statistical mechanics - as an extension of the application of the kinetic theory to the liquid state or the theory of rate processes, etc. Only a selection of the more widely used viscosity models is given here more complete descriptions of such models are available in many books [Bird et al., 1987 Carreau et al., 1997] and in a review paper [Bird, 1976],... 

The kinetic theory of gases attempts to explain the macroscopic nonequilibrium properties of gases in terms of the microscopic properties of the individual gas molecules and the forces between them. A central aim of this theory is to provide a microscopic explanation for the fact that a wide variety of gas flows can be described by the Navier-Stokes hydrodynamic equations and to provide expressions for the transport coefficients appearing in these equations, such as the coefficients of shear viscosity and thermal conductivity, in terms of the microscopic prop>erties of the molecules. We devote most of our attention in this article to this problem. 

The kinetic theory of gases yields theoretical expressions for the thermal conductivity and other transport properties of gases. For ideal gases around atmospheric pressure, where the mean free path is much less than the smallest dimension of the container, the ratio of the thermal conductivities of the isotopic molecules is inversely proportional to the ratio of the square-roots of their molecular masses. At lower pressures, where the mean free path becomes comparable to, or larger than the dimensions of the container, the thermal conductivity will be strongly pressure dependent. The isotope analysis of isotopic gas mixtures by using a catharometer is based on the fact that, to a first approximation, the relationship between the thermal conductivity and isotopic composition of the mixture is linear (Muller et al. 1969). The isotope ratio of the viscosities of gases is equal, to a first approximation, to the square-root of the molecular mass ratio. 

The dynamic viscosity or absolute viscosity of gases is denoted p or ti and is expressed in Pa.s. The kinetic theory of gases allows accurate prediction of the viscosity of ideal gases, and states that viscosity is independent of pressure and increases as temperature increases because atomic or molecular collisions increase. 

If gaseous systems have high densities, both the kinetic theory of gases and the Chapman-Enskog theory fail to properly describe the transport coefficients behavior. Furthermore, the previously derived expression for viscosity and... 

As the development of the kinetic theory outlined above has shown, the transport coefficients are obtained in different orders of approximations according to the number of terms included in the basis vectors of equation (4.7). Fortunately, the lowest-order approximations, at least for viscosity and thermal conductivity, are remarkably accurate and adequate for many purposes. However, for the most accurate work it is necessary to take account of higher-order kinetic theory corrections. These corrections may be expressed in the form... 

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