Force-flux relationship

If the material is also electronically conducting, the general force-flux relationships are... 

In our case jo is naturally identical to the flux density of the neutral component 0 (Jo). Evidently the harmonically averaged conductivity expression in Eq. (6.51) corresponds to an effective, ambipolar conductivity Oq, expressing the fact that both ionic and electronic charge carriers are necessary and the respective resistors so-to-speak connected in series . The expression in brackets obviously represents the chemical potential gradient of the component O fio = IMy/ + 2y h ) The result is a force-flux relationship of the expected form, viz. 

Of course, more complicated situations and conditions will require more sophisticated mathematical treatment, especially for the driving force, but the basic flux relationships are similar for any liquid and gas migration through the subsurface. If the hydraulic conductivities and diffusion coefficients are known for the materials and each migrating fluid of interest, then predictive computer models can often handle the difficult calculations associated with multiple fluids, multiple pressures, and multiple types of materials. 

The concept of equilibrium is quite commonly used in macro- and micro-economics. In fact, in natural systems, one can only expect unstable dynamic equilibrium. It may be noted that one can have stable non-equilibrium steady states. Approach to the so-called equilibrium or to such steady states would depend on a number of variables, which have to be carefully identified. Cause Effect relationship leads to the concept of Force Flux discussed in Part One in this book which is quite relevant for real systems. In the steady state, balance of forces occurs leading to balancing of effects. Typical example in economics is demand-supply relationship in the context of time variations in price and production. 

You are likely already familiar with many of the simple direct force/flux pair relationships that are used to describe mass, charge, and heat transport—they include Pick s first law (diffusion). Ohm s law (electrical conduction), Fourier s law (heat conduction), and Poiseuille s law (convection). These transport processes are summarized in Table 4.1 using molar flux quantities. As this table demonstrates. Pick s first law of diffusion is really nothing more than a simplification of Equation 4.7 for... 

Whatever the relationship used, the particular units for each symbol or entity must be consistent with the units for all other entities, which may or may not involve a conversion of units. Therefore, the units of, say, permeability are defined within the context of the rate or flux relationship used and the units of the driving force for example, partial pressure or pressure times mole fraction. 

The flux of matter from the leading surface to the trailing surface of the pore can be analyzed to derive a equation for the pore mobility Mp. By analogy with the case of a moving boundary, the pore velocity Vp is defined by a force-mobility relationship,... 

To characterize the relationship between the buoyancy forces and momentum flux in different cross-sections of a nonisothermal jet at some distance x, Grimitlyn proposed a local Archimedes number ... 

Irreversible thermodynamics has also been used sometimes to explain reverse osmosis [14,15]. If it can be assumed that the thermodynamic forces responsible for reverse osmosis are sufficiently small, then a linear relationship will exist between the forces and the fluxes in the system, with the coefficients of proportionality then referred to as the phenomenological coefficients. These coefficients are generally notoriously difficult to obtain, although some progress has been made recently using approaches such as cell models [15]. 

Some of the elements of thermodynamics of irreversible processes were described in Sections 2.1 and 2.3. Consider the system represented by n fluxes of thermodynamic quantities and n driving forces it follows from Eqs (2.1.3) and (2.1.4) that n(n +1) independent experiments are needed for determination of all phenomenological coefficients (e.g. by gradual elimination of all the driving forces except one, by gradual elimination of all the fluxes except one, etc.). Suitable selection of the driving forces restricted by relationship (2.3.4) leads to considerable simplification in the determination of the phenomenological coefficients and thus to a complete description of the transport process. 

The conjugate driving force is the pressure gradient Ap multiplied by — 1. Further, the relative flux of the dissolved substance compared to the flux of the solvent, i.e. the exchange flux /D, is defined by the relationship... 

It is assumed that the chloride ion is transported passively across the membrane. Using an approach similar to the formulation of Eqs (2.1.2), (2.3.26) and (2.5.23), relationships can be written for the material fluxes of sodium and chloride ions, /Na+ and Jcr (the driving force is considered to be the electrochemical potential difference), and for the flux of the chemical reaction, Jch ... 

Here Jv is the volumetric flow rate of fluid per unit surface area (the volume flux), and Js is the mass flux for a dissolved solute of interest. The driving forces for mass transfer are expressed in terms of the pressure gradient (AP) and the osmotic pressure gradient (All). The osmotic pressure (n) is related to the concentration of dissolved solutes (c) for dilute ideal solutions, this relationship is given by... 

The power (work by the system per unit time) is thus W = —Fx = —JiXiT. The work is performed under the influence of a heat flux Q leaving the hot reservoir at temperature Ti. The cold reservoir is at temperature T2 (where T > T2). The corresponding thermodynamic force is X2 = I/T2 — 1/Ti, and the flux is J2 = Q. The temperature difference Ti —T2 = AT is assumed to be small compared to T2 T kT, so one can also write X2 = AT/T. Linear irreversible thermodynamics is based on the assumption of local equilibrium with the following linear relationship between the fluxes and forces ... 

The density (or flux) of the lines of force in a solid placed in a magnetic field (H) is termed the magnetic induction, B, and is given by the relationship. 

Yabannavar et al. [81] proposed a proportionality relationship valid for spin-filters based on an analogy to Eq. (15). They defined the Reynolds number based on the tangential velocity at the screen surface. Since in spin-filters the permeation velocity, or perfusion flux, is given by Eq. (16), and it can be assumed that the screen porosity e will be maintained constant throughout the scale-up process, it is possible to write a proportionaHty relationship for the ratio from drag to lift force in spin-filters as given by Eq. (17). 

Studies on multicomponent systems have been mainly restricted to relatively simple ternary systems containing a solvent as component 1 and-solutes as components 2 and 3. For such a system, under zero-volume flow conditions (Eq. (3)), exact expressions for the fluxes of the components 2 and 3 may be written as independent quantities of the forces involved so that the linear laws to which the Onsager reciprocal relationship applies may be written as follows341 ... 

In order to evaluate this expression, we need to know the force v / that is responsible for producing the molecular flux. It could be an external force such as an electric field acting on ions. Then evaluation of Eq. 18-48 would lead to the relationship between electric conductivity, viscosity, and diflusivity known as the Nernst-Einstein relation. 

The transport coefficients lik (i,k = 1,2,.. ., 9) define the linear relationships between the forces and the fluxes... 

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