Property flux expressions

The solution to the linear integral equations, Eqs. (6.94) and (6.95), are given in detail in various references, which enables an accurate determination of the functions Aj and B/. Since the solution to linear integral equations is a fairly standard mathematical exercise, we instead turn our attention to the specific forms of the property flux expressions and the final closed forms of the conservation equations. 

The Flux Expressions. We begin with the relations between the fluxes and gradients, which serve to define the transport properties. For viscosity the earliest definition was that of Newton (I) in 1687 however about a century and a half elapsed before the most general linear expression for the stress tensor of a Newtonian fluid was developed as a result of the researches by Navier (2), Cauchy (3), Poisson (4), de St. Venant (5), and Stokes (6). For the thermal conductivity of a pure, isotropic material, the linear relationship between heat flux and temperature gradient was proposed by Fourier (7) in 1822. For the difiiisivity in a binary mixture at constant temperature and pressure, the linear relationship between mass flux and concentration gradient was suggested by Pick (8) in 1855, by analogy with thermal conduction. Thus by the mid 1800 s the transport properties in simple systems had been defined. 

Thus, the solvent flux was expressed as a sum of contributions from diffusive and osmotic pressure terms. The osmotic pressure in the solvent flux expression depends upon the viscoelastic properties of the polymer. The relationship between the osmotic pressure and the stresses within the polymer was derived by writing momentum balances as... 

To preview the results somewhat, it will be shown that the general form of the transport equations contains expressions for the property flux variables (momentum flux P, energy flux q, and entropy flux s) involving integrals over lower-order density functions. In this form, the transport equations are referred to as general equations of change since virtually no assumptions are made in their derivation. In order to finally resolve the transport equations, expressions for specific lower-ordered distribution functions must be determined. These are, in turn, obtained from solutions to reduced forms of the Liouville equation, and this is where critical approximations are usually made. For example, the Euler and Navier-Stokes equations of motion derived in the next chapter have flux expressions based on certain approximate solutions to reduced forms of the Liouville equation. Let s first look, however, at the most general forms of the transport equations. 

The general equations of change given in the previous chapter show that the property flux vectors P, q, and s depend on the nonequi-lihrium behavior of the lower-order distribution functions g(r, R, t), f2(r, rf, p, p, t), and fi(r, P, t). These functions are, in turn, obtained from solutions to the reduced Liouville equation (RLE) given in Chap. 3. Unfortunately, this equation is difficult to solve without a significant number of approximations. On the other hand, these approximate solutions have led to the theoretical basis of the so-called phenomenological laws, such as Newton s law of viscosity, Fourier s law of heat conduction, and Boltzmann s entropy generation, and have consequently provided a firm molecular, theoretical basis for such well-known equations as the Navier-Stokes equation in fluid mechanics, Laplace s equation in heat transfer, and the second law of thermodynamics, respectively. Furthermore, theoretical expressions to quantitatively predict fluid transport properties, such as the coefficient of viscosity and thermal... 

As noted by Bom and Green, a superposition approximation for the three-body distribution function, similar to Kirkwood s form in equilibrium systems, Eq. (4.57), is needed for further analysis of Eq. (6.48). Using the solution form Eq. (6.47), we can, however, immediately write down expressions for the so-called auxiliary conditions and the property flux vectors. First, we have to O(e )... 

The permeation flux expressions (3.4.76) and (3.4.81a) are valid for membranes whose properties do not vary across the thickness. Most practical gets separation membranes have an asymmetric or composite structure, in which the properties vary across the thickness in particular ways. Asymmetric membranes are made from a given material therefore the properties varying across Sm are pore sizes, porosity and pore tortuosity. Composite membranes are made from at least two different materials, each present in a separate layer. Not only does the intrinsic Qim of the material vary from layer to layer, but also the pore sizes, porosity and pore tortuosity vary across Sm- At least one layer (in composite membranes) or one section of the membrane (in asymmetric membranes) must be nonporous for efficient gas separation by gas permeation. The flux expressions for such structures can be developed only when the transport through porous membranes has been studied. 

It has been observed that particle flux expressions developed based on shear-induced particle diffusivity describe the observed solvent flux through a microflltration membrane much better than those based on Brownian diffusivity of a particle. For the following system properties, determine the ratio of the solvent fluxes based on shear-induced particle diffusivity and Brownian diffusivity. You should employ the particle volume fraction based solvent flux expression based on the gel polarization model used in ultrafiltration (equation (7.2.72)) ... 

Since an interatomic surface is defined by a set of trajectories of Vp that terminate at a cp and since trajectories never cross, an interatomic surface S(r) is one of local zero flux in the gradient vector field of the electron density that is, it is not traversed by any trajectories of Vp. The zero-flux property is expressed in equation (1) in terms of //(r), the unit vector... 

Dunlop [18] proposed a model for sub-lytic effects in plant cells, based on the same principles, but including four properties postulated to be of particular importance in these systems, namely calcium ion flux, osmo-regulation, cell-cell contact/aggregation and stress protein expression. Of these factors, osmo-regulation (and its inter-relationship with the cell wall) and aggregation patterns, in particular, distinguish plant cells from mammalian cell systems. 

In order to express the importance of the ions to the growth process quantitatively, two related quantities can be defined the fraction of arriving ions per deposited atom, / , and the kinetic energy transferred by ions per deposited atom, Emd, - These quantities are used in ion-beam-assisted deposition in order to relate material properties to ion flux and energy [421]. Their definition is... 

For alkali metals with small cavities at low pressures, the value of / for a given heat flux may not be achievable. Since tw can be expressed as a function of average heat transfer rate per unit area, <7", and liquid properties, Eq. (2-112) can be rearranged and solve for the heat flux ... 

Uniform fluxes and properties. For the case of discrete fluxes of mass and heat at the surface of the control volume with uniform boundary temperatures at these regions, the equation can be expressed as... 

The membrane performance for separations is characterized by the flux of a feed component across the membrane. This flux can be expressed as a quantity called the permeability (P), which is a pressure- and thickness-normalized flux of a given component. The separation of a feed mixture is achieved by a membrane material that permits a faster permeation rate for one component (i.e., higher permeability) over that of another component. The efficiency of the membrane in enriching a component over another component in the permeate stream can be expressed as a quantity called selectivity or separation factor. Selectivity (0 can be defined as the ratio of the permeabilities of the feed components across the membrane (i.e., a/b = Ta/Tb, where A and B are the two components). The permeability and selectivity of a membrane are material properties of the membrane material itself, and thus these properties are ideally constant with feed pressure, flow rate and other process conditions. However, permeability and selectivity are both temperature-dependent... 

---