General molecular transport equation

GENERAL MOLECULAR TRANSPORT EQUATION FOR MOMENTUM, HEAT, AND MASS TRANSFER... 

A General Molecular Transport Equation and General Property Balance... 

General molecular transport equation. All three of the molecular transport processes of momentum, heat or thermal energy, and mass are characterized in the elementary sense by the same general type of transport equation. First we start by noting the following ... 

As discussed in Section 2.3 for the general molecular transport equation, all three main types of rate-transfer processes—momentum transfer, heat transfer, and mass transfer— are characterized by the same general type of equation. The transfer of electric current can also be included in this category. This basic equation is as follows ... 

A good agreement is generally obtained between the models based on transport equations and the SDE for mass and heat molecular transport. However, as explained above, the SDE can only be applied when convective flow does not take place. This restrictive condition limits the application of SDE to the transport in a porous solid medium where there is no convective flow by a concentration gradient. The starting point for the transformation of a molecular transport equation into a SDE system is Eq. (4.108). Indeed, we can consider the absence of convective flow in a non-steady state one-directional transport, together with a diffusion coefficient depending on the concentration of the transported property... 

In this case, if the boundary and initial conditions allow it, either ej or c can be used to define the mixture fraction. The number of conserved scalar transport equations that must be solved then reduces to one. In general, depending on the initial conditions, it may be possible to reduce the number of conserved scalar transport equations that must be solved to min(Mi, M2) where M = K - Nr and M2 = number of feed streams - 1. In many practical applications of turbulent reacting flows, M =E and M2 = 1, and one can assume that the molecular-diffusion coefficients are equal thus, only one conserved scalar transport equation (i.e., the mixture fraction) is required to describe the flow. 

In the case of pure advection (no molecular transport), the diffusion term in the general transport equation (8.2.5) is made equal to zero and time-dependent mass balance is expressed as... 

It is worth noting at this point that the various scientific theories that quantitatively and mathematically formulate natural phenomena are in fact mathematical models of nature. Such, for example, are the kinetic theory of gases and rubber elasticity, Bohr s atomic model, molecular theories of polymer solutions, and even the equations of transport phenomena cited earlier in this chapter. Not unlike the engineering mathematical models, they contain simplifying assumptions. For example, the transport equations involve the assumption that matter can be viewed as a continuum and that even in fast, irreversible processes, local equilibrium can be achieved. The paramount difference between a mathematical model of a natural process and that of an engineering system is the required level of accuracy and, of course, the generality of the phenomena involved. 

We cannot finish this short introduction on the property transport problems without some observations and commentaries about the content of Figs. 3.2 and 3.3. First, we have to note that, for the generalization of the equations, only vectorial expressions can be accepted. Indeed, considering the equations given in the figures above, some particular situations have been omitted. For example, we show the case of the vector of molecular transport of the momentum that in Fig. 3.3 has been used in a simplified form by eliminating the viscous dissipation. So, in order to generalize this vector, we must complete the Tjj expression with consideration of the difference between the molecular and volume viscosities q — ri ... 

The dimensionless form of the equation contains one dimensionless parameter as a multiplier of the first term of the right-hand side and maybe some additional dimensionless parameters, which may appear within the dimensionless source term, S. Depending on the general variable, 0, the effective diffusion coefficient, F, appearing in this dimensionless number will be different, leading to different dimensionless numbers. For the species mass fraction, momentum and enthalpy transport equations, the effective diffusion coefficient will be molecular diffusion coefficient, the kinematic viscosity of the fluid and the thermal diffusivity of the fluid respectively. The corresponding dimensionless numbers are, therefore, defined as follows. 

The moment method can then be employed to derive a generalized equation of change for a mean particle property < ip > in the same manner as described in chap 2 for molecular systems. In particular, the generalized transport equation for < ip r, t) > is derived multiplying (4.1) by a microscopic quantity ip r, c, t) and integrating the resulting relation over the whole velocity space. 

Here the situation is very similar to that encountered in connection with the need for continuum (constitutive) models for the molecular transport processes in that a derivation of appropriate boundary conditions from the more fundamental, molecular description has not been accomphshed to date. In both cases, the knowledge that we have of constitutive models and boundary conditions that are appropriate for the continuum-level description is largely empirical in nature. In effect, we make an educated guess for both constitutive equations and boundary conditions and then normally judge the success of our choices by the resulting comparison between predicted and experimentally measured continuum velocity or temperature fields. Models derived from molecular theories, with the exception of kinetic theory for gases, are generally not available for comparison with the empirically proposed models. We discuss some of these matters in more detail later in this chapter, where specific choices will be proposed for both the constitutive equations and boundary conditions. 

The section begins with the random walk, a useful model from statistical physics that provides insight into the kinetics of molecular diffusion. From this starting point, the fundamental relationship between diffusive flux and solute concentration. Pick s law, is described and used to develop general mass-conservation equations. These conservation equations are essential for analysis of rates of solute transport in tissues. 

Once again, mass diffusivity J0a, mix and thermal conductivity tc in these expressions represent molecular transport properties via Pick s and Fourier s law, respectively. However, the fluid properties that appear in Sc and St should be interpreted as diffusivities, not molecular transport properties. In terms of the analogies between heat and mass transfer, sometimes 30A,mix represents a diffusivity, and other times it represents a molecular transport property. This ambiguity does not exist in the corresponding expressions for heat transfer. In general, 30a, mix represents a diffusivity in the mass transfer equation and in expressions for the boundary layer thickness Sc. 

Equations (2.3-11) and (2.3-12) are general equations for the conservation of momentum, thermal energy, or mass and will be used in many sections of this text. The equations consider here only molecular transport occurring and the equations do not consider other transport mechanisms such as convection, and so on, which will be considered when the specific conservation equations are derived in later sections of this text for momentum, energy, or mass. 

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