Transport Properties Equations Estimation

The parameters of a model can be estimated by fitting the model to experimental data [182,183]. Using the model of Section 4.7.3, two regression analysis procedures can be applied [43] transport properties estimation and transport properties equations estimation. 

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. 

A means to find or estimate required constitutive properties that appear in the conservation equations. These can include equations of state, thermodynamic and transport properties, and chemical reaction rates. 

The liquid-phase diffusion coefficients are found with the Nemst-Hartley equation (193), which describes the transport properties in weak electrolyte systems. The gas-phase diffusion coefficients are estimated according to the... 

Molecular Simulations Molecular simulations are useful for predicting properties of bulk fluids and solids. Molecular dynamics (MD) simulations solve Newton s equations of motion for a small number (on the order of 10 ) of molecules to obtain the time evolution of the system. MD methods can be used for equilibrium and transport properties. Monte Carlo (MC) simulations use a model for the potential energy between molecules to simulate configurations of the molecules in proportion to their probability of occurrence. Statistical averages of MC configurations are useful for equilibrium properties, particularly for saturated densities, vapor pressures, etc. Property estimations using molecular simulation techniques are not illustrated in the remainder of this section as commercial software implementations are not generally available at this time. 

Equation (11-77) shows that the maximum temperature rise depends on the heat of reaction, transport properties of the pellet, and the surface concentration of reactant. It permits a simple method of estimating whether intrapellet temperature differences are significant (see Example 11-9). 

In Great Britain, the National Engineering Laboratory (NEL, formerly a government agency but now privatized) has prodnced a database for thermodynamic and transport properties. PPDS contains correlations for properties of a large number of pure components these are based on evaluated experimental data where possible but also include some estimated properties. For mixtures, the database contains binary interaction parameters fitted to data for use with common equation-of-state and liqnid-activity methods for calculating phase eqnilibria. Information is available at their Web site [14]. 

In the previous sections, methods of experimental determination of heat and mass transport properties have been discussed. These methods use special apparatus and are based on the equation of definition of the corresponding property. This section discusses the experimental determination of these properties from drying experiments. Some relevant techniques have been already discussed by Molnar [125]. However, a generalized method based on model-building techniques is presented here. The method uses a drying experimental apparatus and estimates the heat and mass transport properties as parameters of a drying model that incorporates these properties [28,43,177-181]. An outline of the method is described below. 

Several empirical equations describing the dependence of transport properties on various factors are tested using a model disaimination procedure. The constants of the empirical eqnations are estimated as parameters of the total model (process model pins properties model) by fitting it to experimental data. 

Each of the property information systems has an extensive set of subroutines to determine the parameters for vapor pressure equations (e.g., the extended Antoine equation), heat capacity equations, etc., by regression and to estimate the thehnophysical and transport properties. The latter subroutines are called to determine the state of a chemical mixture (phases at equilibrium) and its properties (density, enthalpy, entropy, etc.) When calculating phase equilibria, the fugacities of the species are needed for each of the phases. A review of the phase equilibrium equations, as well as the facilities provided by the process simulators for the calculation of phase equilibria, is provided on the CD-ROM that accompanies this book (see ASPEN- Physical Property Estimation and HYSYS Physical Property Estimation). 

Stream properties required for solving material and energy balance equations and other process calculations are predicted from component properties. The properties of petroleum pseudocomponents can be estimated from their boiling points and specific gravities. The component properties include the molecular weight, critical constants, acentric factor, heat of formation, ideal gas enthalpy, latent heat, vapor pressure, and transport properties. These are predicted mainly by empirical correlations based on experimental data. Many of these correlations are documented in the American Petroleum Institute Technical Data Book (API, 1983). 

It is important to note that Equations (7) and (8), and the equivalent equations for a mixture can fail to predict correctly the transport properties of a nonconformal fluid if the corresponding states parameters are estimated from thermodynamic data. The equations can break down, in particular, if p/p > 1. 

The contrast between the predictions of the only available theory for the intermediate-density range and experiment casts further doubt on the validity of the MET. At the same time, the empirical observation itself, validated on a large number of pure materials, at least for supercritical temperamres, provides a valuable empirical estimation procedure. This is because, if the density dependence of a transport property of a pure fluid is available along just one isotherm, it is possible to evaluate the property as a function of density along any other isotherm merely by assuming that the excess property is temperature independent and by making use of equation (5.27) or (5.28). 

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