Constitutive relations mass flux

To yield a constitutive relation for the mass flux of particles, it is convenient to begin with the equation of motion of a single particle, which is expressed by... 

The generic equations of balance are statements of truth, which is a priori self-evident and which must apply to all continuum materials regardless of their individual characteristics. Constitutive relations relate diffusive flux vectors to concentration gradients through phenomenological parameters called transport coefficients. They describe the detailed response characteristics of specific materials. There are seven generic principles (1) conservation of mass, (2) balance of linear momentum, (3) balance of ro-... 

Equations (10) are generally valid for both liquid and gas phases if reactions take place there. They represent nothing but a differential mass balance for the film region with the account of the source term due to the reaction. To link this balance to the process variables like component concentrations, some additional relationships - often called constitutive relations (see Ref. [16]) - are necessary. For the component fluxes Ni, these constitutive relations result from the multicomponent diffusion description (Eqs. (1), (2)) for the source terms, from the reaction kinetics description. The latter strongly depends on the specific reaction mechanism [27]. The reaction rate expressions lli usually represent nonlinear dependencies on the mixture composition and temperature of the corresponding phase. 

As we have already seen, the analysis of a diffusion problem proceeds by solving the conservation equations together with appropriate constitutive relations for the diffusion fluxes. Use of Eq. 6.1.1 for the diffusion flux with Eq. 1.3.9 for the conservation of mass leads to... 

For diffusion in isothermal multicomponent systems the generalized driving force was written as a linear function of the relative velocities (m/ — My). In the general case, we must allow for coupling between the processes of heat and mass transfer and write constitutive relations for and q in terms of the (m — My) and V(l/r). With this allowance, the complete expression for the conductive heat flux is... 

Chapter 1 serves to remind readers of the basic continuity relations for mass, momentum, and energy. Mass transfer fluxes and reference velocity frames are discussed here. Chapter 2 introduces the Maxwell-Stefan relations and, in many ways, is the cornerstone of the theoretical developments in this book. Chapter 2 includes (in Section 2.4) an introductory treatment of diffusion in electrolyte systems. The reader is referred to a dedicated text (e.g., Newman, 1991) for further reading. Chapter 3 introduces the familiar Fick s law for binary mixtures and generalizes it for multicomponent systems. The short section on transformations between fluxes in Section 1.2.1 is needed only to accompany the material in Section 3.2.2. Chapter 2 (The Maxwell-Stefan relations) and Chapter 3 (Fick s laws) can be presented in reverse order if this suits the tastes of the instructor. The material on irreversible thermodynamics in Section 2.3 could be omitted from a short introductory course or postponed until it is required for the treatment of diffusion in electrolyte systems (Section 2.4) and for the development of constitutive relations for simultaneous heat and mass transfer (Section 11.2). The section on irreversible thermodynamics in Chapter 3 should be studied in conjunction with the application of multicomponent diffusion theory in Section 5.6. 

A statement of the constitutive relation analogous to those for mass, heat, and momentum is that the flux due to migration in an electric field is proportional to the force acting on the particle multiplied by the particle concentration. The molar flux in stationary coordinates is then... 

The important relation used in the mass balance equation is the constitutive flux equation, which relates the flux and the concentration gradient of the adsorbed species. For the diffusion of the adsorbed species inside a micro-particle, the flux can be written in terms of the chemical potential gradient as follows ... 

The most common types of models in chemical engineering are those related to the transport of mass, heat, and momentum. In addition to the balance equation, a constitutive equation that relates the flux of interest to the dependent variable (e.g. mass flux to concentration) is needed. These relations (in simple ID form) for the microscopic level and for flow at the porous media level are given in Table 3.1. It should be noted that all these relations have the general form... 

The mass conservation equation only relates concentration variation with flux, and hence cannot be used to solve for the concentration. To describe how the concentrations evolve with time in a nonuniform system, in addition to the mass balance equations, another equation describing how the flux is related to concentration is necessary. This equation is called the constitutive equation. In a binary system, if the phase (diffusion medium) is stable and isotropic, the diffusion equation is based on the constitutive equation of Pick s law ... 

The mass transfer flux law is analogous to the laws for heat and momentum transport. The constitutive equation for Ja, the diffusional flux of A resulting from a concentration difference, is related to the concentration gradient by Pick s first law ... 

It is valid for each continuum independent of the individual material properties and is therefore one of the fundamental equations in fluid mechanics and subsequently also in heat and mass transfer. The movement of a particular substance can only be described by introducing a so-called constitutive equation which links the stress tensor with the movement of a substance. Generally speaking, constitutive equations relate stresses, heat fluxes and diffusion velocities to macroscopic variables such as density, velocity and temperature. These equations also depend on the properties of the substances under consideration. For example, Fourier s law of heat conduction is invoked to relate the heat flux to the temperature gradient using the thermal conductivity. An understanding of the strain tensor is useful for the derivation of the consitutive law for the shear stress. This strain tensor is introduced in the next section. 

Setting the mass balance equation around a very thin element in the capillary between the two reservoir and making use of the constitutive flux relation (8.3-7), we have ... 

A system may contain energy when it possesses the ability to store it, as already stated. This ability takes the form of two constitutive properties, one for each subvariety of energy inductance for the inductive subvariety, and capacitance for the capacitive one. These names are borrowed from electrodynamics and generalized to all energy varieties. So, there is a translational mechanical inductance (inertial mass), a rotational mechanical inductance (inertia), a hydrodynamical capacitance (compressibility integrated over the volume), a thermal capacitance (which depends on the specific heat), and so on. In electrodynamics, inductance and capacitance feature components called inductor (or self-inductance) and capacitor, respectively. Electric inductance relates current to the quantity of induction (induction flux) and electric capacitance relates potential to charge. 

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