Transport properties dilute gases

If these assumptions are satisfied then the ideas developed earlier about the mean free path can be used to provide qualitative but useful estimates of the transport properties of a dilute gas. While many varied and complicated processes can take place in fluid systems, such as turbulent flow, pattern fonnation, and so on, the principles on which these flows are analysed are remarkably simple. The description of both simple and complicated flows m fluids is based on five hydrodynamic equations, die Navier-Stokes equations. These equations, in trim, are based upon the mechanical laws of conservation of particles, momentum and energy in a fluid, together with a set of phenomenological equations, such as Fourier s law of themial conduction and Newton s law of fluid friction. When these phenomenological laws are used in combination with the conservation equations, one obtains the Navier-Stokes equations. Our goal here is to derive the phenomenological laws from elementary mean free path considerations, and to obtain estimates of the associated transport coefficients. Flere we will consider themial conduction and viscous flow as examples. 

Assume that all species are highly dilute in ammonia. For the purpose of this problem, consider the following transport properties as constants n = 2.7 x 10-4 g/cm-s, k = 1.1 x 104 erg/cni S-K, and Dtmg-nh3 = 3 cm2/s. The specific heat for ammonia may be taken as cp = 2600 J/kg-K. Assume that the density may be determined from a perfect-gas equation of state for ammonia alone. 

Heat and mass transfer constitute fundamentally important transport properties for design of a fluidized catalyst bed. Intense mixing of emulsion phase with a large heat capacity results in uniform temperature at a level determined by the balance between the rates of heat generation from reaction and heat removal through wall heat transfer, and by the heat capacity of feed gas. However, thermal stability of the dilute phase depends also on the heat-diffusive power of the phase (Section IX). The mechanism by which a reactant gas is transferred from the bubble phase to the emulsion phase is part of the basic information needed to formulate the design equation for the bed (Sections VII-IX). These properties are closely related to the flow behavior of the bed (Sections II-V) and to the bubble dynamics. 

It is more usual with closed-shell atoms to consider potential models such as the exchange-Coulomb and Hartree-Fock dispersion potentials and to determine the parameters from dilute gas properties such as the second virial coefficient and the transport properties, viscosity and thermal conductivity, together with... 

For the mathematical description of the component transport through a porous membrane there are two modeling approaches common. The first is the so-called extended Pick model (EFM), which can be applied to describe the transport of diluted, nonadsorbable gases in mesoporous membrane materials at low pressure (Veldsink et al., 1995 Papavassiliou et al., 1997 Al-Juaied et al., 2001). The second, the more general dusty gas model, is based on the Stefan-Maxwell approach for multicomponent diffusion (Mason et al., 1967 Krishna and WesseUngh, 2000). Both models require knowledge of the above-mentioned membrane properties (e.g.. Bo, c/t). Because for a specific membrane material these parameters are a priori not predictable, they have to be determined experimentally. Typical membrane materials and characteristic transport parameters used in this work are Hsted in Table 5.1. 

The kinetic theory of gases has had a long and rich history.t In its modem form, kinetic theory begins with the work of D. Bernoulli, Qausius, and most importantly Maxwell, who first used statistical methods to compute the properties of gases, recognizing that the random motions of the gas molecules could be best described by a distribution function. Besides giving the form of this distribution function for a gas at equilibrium, Maxwell derived equations for the transport of mass, momentum, and energy in a dilute gas. For a gas composed of so-called Maxwell molecules, which interact with repulsive forces... 

Quinones-Cisneros et al. proposed the friction theory (the so called f-the-ory) to predict viscosity using an equation of state. According to f-theory, the viscosity of dense fluids is a mechanical property rather than a transport property. Consequently, the total viscosity of a dense fluid can be written as the sum of a dilute-gas term y q and a friction term t]f through ... 

Determining the numerical potential of Fig. 2.10 is a formidable task, even with modern computers. However, once the potential is known, it is relatively simple to use it to calculate thermodynamic and transport properties. It is therefore natural to hope that the argon potential could be expressed in a parametric form and used to compute properties of other gaseous systems. This would be particularly important for determining gas properties in regions of high temperature not easily reached in the laboratory. While the available data are insufficient to provide a completely reliable test, it appears that in a two-parameter form the potential of Fig. 2.10 can be used to correlate the properties of dilute rare gases. ... 

Substantial progress has been made recently in applying the kinetic theory dilute gas expressions, both for the transport and equilibrium properties. For example, we now have a good grasp on how a model intermolecular potential can be used to relate theory to data. Also simple nonspherlcal molecules can be considered systematically. 

This present volume, which is complementary to the previous publication, discusses the present state of theory with regard to the dilute-gas state, the initial density dependence, the critical region and the very dense gas and liquid states for pure components and mixtures. In all cases, the intention is to present the theory in usable form and examples are given of its application to nonelectrolyte systems. This will be of particular use to chemical and mechanical engineers. The subtitle of this volume Their correlation, prediction and estimation reflects the preferred order of rqrplication to obtain accurate values of transport properties. Careful correlation of accurate experimental data gives reliable values at interpolated temperatures and pressures (densities), and at different compositions when the measurements are for mixtures. Unfortunately, there are only a limited number of systems where data of such accuracy are available. In other cases, sound theoretical methods are necessary to predict the required values. Where information is lacking - for intermolecular forces, for example - estimation methcxls have to be used. These are of lower accuracy, but usually have more general tq)plicability. 

Here, X is the transport property associated with the particular process under consideration. It follows that the transport coefficient, which itself may be a function of the temperature and density of the fluid, will reflect the interactions between the molecules of the dilute gas. For that reason there has been, for tq)proximately 150 years, a purely scientific interest in the transport properties of fluids as a means of probing the forces between pairs of molecules. Within the last twenty years, at least for the interactions of the monatomic, spherically symmetric inert gases, the transport prq>erties have played a significant role in the elucidation of these forces. 

However, it should be recognized that although the theory of the transport properties of fluids is not completely developed, it can provide some guidance in the process of correlation. For example, all kinetic theories of transport reveal that it is the temperature and density that are the fundamental state variables and that pressure is of no direct significance. Since most measurements are carried out at specified pressures and not specified densities, this automatically means that a single, uniform equation of state must be used to convert any experimental data to p, T) space from the experimental p, T) space. Furthermore, the dilute-gas kinetic theory reveals a number of relationships between different properties of a gas that are exact or nearly exact so that these relationships provide consistency tests for experimental data as well as constraints that must be satisfied by the final correlation of the properties. 

It should be clear from the preceding classiflcation of methods of correlation, prediction and estimation of the transport properties of fluids that the list has been presented in the preferred order of application. That is, whenever a correlation of critically evaluated data is available it should be used. Examples of the development of some of these correlations are given in later chapters for different classes of fluids. Wide-ranging correlations of this type are available for only a small subset of the fluids of interest, and the next best means of obtaining the properties is either directly from theory (in rare cases) or from a representation of the results of an exact theory supported by experimental data. This would, in fact, always be the preferred choice of method for the evaluation of the properties of mixtures where wide-ranging correlations in temperature, density and composition are not practicable. This approach is viable at present only for the dilute state of gases and gas mixtures. 

It is worth noting that in this context the terms dilute or low-density gas represent a real physical situation, whereas the frequently used expression zero-density limit is related to results of a mathematical extrapolation of a density series of a particular transport property at constant temperature to zero density. The derived value is assumed to be identical with the true value for the dilute-gas state, a statement that in most cases turns out to be correct. 

The ease of the practical evaluation of the transport properties of a dilute gas by means of these relationships decreases as the complexity of the molecules increases. Thus for a pure monatomic gas, with no internal degrees of freedom, the calculations are now trivial, consuming minutes on a personal computer. For systems involving atoms and rigid rotors the computations are now almost routine and take hours on a work station (see, for instance, Dickinson Heck 1990). For two rigid rotors the calculations are only just beginning to be carried out and take days on a work station (Heck Dickinson 1994). For systems that involve molecules other than rigid rotors the theory is still approximate and calculations are heuristic. 

Section 4.2 considered the kinetic theory of the transport properties of dilute, pure fluids in some detail in this section the theory is extended to gas mixtures. Naturally, the theory of mixtures shares many features with that of pure species so that the same pattern will be adopted for the presentation, although duplication will be avoided whenever possible. For these reasons the semiclassical kinetic theory description is immediately adopted here, and similar consequences for the description of some phenomena as they pertain to pure gases are accepted. 

The formulation set out above does, however, provide a hierarchy of procedures for the prediction of the transport properties of dilute-gas mixtures in the sense set out in Chapter 3. In the following the formulas given above are detailed in a manner which will enable each level of the hierarchy to be discussed in turn. 

Recent advances in the theoretical description of the initial density dependence of the transport properties justify a separate treatment. If moderately dense gases are considered, only the linearized equations (5.1) are needed that is, the virial form of the density expansion can be truncated after the term linear in density. This means that the deviation from the dilute-gas behavior can be represented by the second transport virial coefficients Bx or alternatively by the initial-density coefficients which are... 

For calculation of transport properties on the basis of hard-sphere theory, the hard-sphere expressions for the dilute-gas transport properties are required. These are given... 

Transport property measurements are normally reported in terms of the measured state variables of temperature, pressure and composition. Density is not usually measured at each state point but is instead obtained from an equation of state formulation. It is crucial to consider the uncertainty in the density which is calculated by the equation of state. As in the case for the transport property correlations, the uncertainty in the fluid density is not uniform over the entire PVT x) surface (Younglove 1982). The uncertainty in the fluid density in the critical region is almost certainly larger than it is in the dilute-gas limit or near the phase boundaries far from the critical point Chapter 8 discusses equations of state and their importance in the analysis of transport properties. 

If the terms of equation (7.13) are truly independent, then it may be appropriate to fit the dilute-gas transport properties to primary data at low density the dilute-gas contribution calculated from the resultant correlation can then be subtracted from the experimental data at higher densities to study the remaining excess and critical enhancement terms. Since the dilute-gas term is a function of temperature alone (or a reduced temperature such as T = T/Tc or T = k T/i) and both the excess and critical enhancement terms are zero in the limit of zero density, the dilute-gas term is expected to be mathematically independent of the excess and critical enhancement terms. However, if the dilute-gas term has some cross correlation with respect to density (because of its determination from experimental data at nonzero density), then it may not be proper to fit the dilute gas data independently. Any cross correlation which is present in the data is not necessarily due to the behavior of the transport property itself, but may be due to systematic effects. This uncertainty may be a function of the other independent variables such as density or composition. 

Developments in kinetic theory have led to theoretically based correlations of the transport properties of gases both in the dilute-gas limit and, more recently, at increasing densities. The forms of these limited surface correlations are given by... 

At the present time, the most successful correlations of dense-fluid transport properties are based upon consideration of the hard-sphere model. One reason for this is that, as discussed in Chapter 5, it is possible to calculate values from theory for this model at densities from the dilute-gas state up to solidification. Second, this is physically a reasonably realistic molecular model because the van der Waals model, which has been successfully applied to equilibrium properties of dense fluids, becomes equivalent to the hard-sphere model for transport properties. 

The correlation earlier developed by Hanley (1974) for the dilute-gas thermal conductivity was calculated for the 11-6-8 interatomic potential with parameters a and /k from fitting both transport and equilibrium properties. The correlation of Kestin et al. (1984) is a universal correlation based on the extended principle of corresponding states (see Chapter 11) for transport and equilibrium properties. The correlation of... 

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