Gradient flux law

Two Basic Descriptions of Transport by Random Motion Mass Transfer Model and Gradient-Flux Law... 

The second model, the so-called gradient-flux law, is considered to be more fundamental, although it is based on a more restrictive physical picture. In contrast to the mass transfer model, in which no assumption is made regarding the spatial separation of subsystems A and B, in the gradient-flux law it is assumed that the subsystems and the distance between them, Axa/b, become infinitely small. For very small subsystems the term occupation number loses its meaning and must be replaced by occupation density or concentration. Obviously, the difference in occupation density tends toward zero, as well. Yet the ratio of the two differences, Aoccupa-tion density Axa/b, is equal to the spatial gradient of the occupation density and usually different from zero ... 

One well-known example of the gradient-flux law is Fick s first law, which relates the diffusive flux of a chemical to its concentration gradient and to the molecular diffusion coefficient ... 

Manifestations of the gradient-flux law are not due to a random process but due the equilibrium between external force and internal friction. The positive sign results from the special sign convention used for electric currents and fields. 

The relation between length and time scales of diffusion, calculated from the Einstein-Smoluchowski law (Eq. 18-8), are shown in Fig. 18.11 for diffusivities between 10 10 cm2s 1 (helium in solid KC1) and 108 cm2s (horizontal turbulent diffusion in the atmosphere). Note that the relevant time scales extend from less than a millisecond to more than a million years while the spatial scales vary between 1 micrometer and a hundred kilometers. The fact that all these situations can be described by the same gradient-flux law (Eq. 18-6) demonstrates the great power of this concept. 

In fight of this analogy, we anticipate that the effect of turbulence may be dealt with in a similar manner like the random motion of molecules for which die gradient-flux law of diffusion (Eq. 18-6) has been developed. In addition, the mass transfer model (Eq. 18-4) may provide an alternative tool for describing the effect of turbulence on transport... 

The connection between the direct coefficients in Eq. 2.21 and the empirical force-flux laws discussed in Section 2.1.2 can be illustrated for heat flow. If a bar of pure material that is an electrical insulator has a constant thermal gradient imposed along it, and no other fields are present and no fluxes but heat exist, then according to Eq. 2.21 and Table 2.1,... 

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 ... 

Tick s laws Classical laws of diffusion relating the flux and the concentration gradient (first law, Eq 5.4), and the rate of variation of the local concentration to the gradient of the concentration gradient (second law, Eq. 5.5). 

Fick s First Law This law relates flux of a component to its composition gradient, employing a constant of proportionahty called a... 

Mutual Diffusivity, Mass Diffusivity, Interdiffusion Coefficient Diffusivity is denoted by D g and is defined by Tick s first law as the ratio of the flux to the concentration gradient, as in Eq. (5-181). It is analogous to the thermal diffusivity in Fourier s law and to the kinematic viscosity in Newton s law. These analogies are flawed because both heat and momentum are conveniently defined with respec t to fixed coordinates, irrespective of the direction of transfer or its magnitude, while mass diffusivity most commonly requires information about bulk motion of the medium in which diffusion occurs. For hquids, it is common to refer to the hmit of infinite dilution of A in B using the symbol, D°g. 

Physically, diffusion occurs because atoms, even in a solid, are able to move - to jump from one atomic site to another. Figure 18.4 shows a solid in which there is a concentration gradient of black atoms there are more to the left of the broken line than there are to the right. If atoms jump across the broken line at random, then there will be a net flux of black atoms to the right (simply because there are more on the left to jump), and, of course, a net flux of white atoms to the left. Pick s Law describes this. It is derived in the following way. 

Either the and the two e s diffuse outward through the film to meet the 0 at the outer surface, or the oxygen diffuses inwards (with two electron holes) to meet the at the inner surface. The concentration gradient of oxygen is simply the concentration in the gas, c, divided by the film thickness, x and the rate of growth of the film dx/dt is obviously proportional to the flux of atoms diffusing through the film. So, from Pick s Law (eqn. (18.1)) ... 

The Fourier law gives the rate at which heat is transferred by conduction through a substance without mass transfer. This states that the heat flow rate per unit area, or heat flux, is proportional to the temperature gradient in the direction of heat flow. The relationship between heat flux and temperature gradient is characterized by the thermal conductivity which is a property of the substance. It is temperature dependent and is determined experimentally. 

According to Fick s first law, the rate of diffusion (i.e., the flux) is directly proportional to the slope of the concentration gradient ... 

Hence, the current (at any time) is proportional to the concentration gradient of the electroactive species. As indicated by the above equations, the dififusional flux is time dependent. Such dependence is described by Fick s second law (for linear diffusion) ... 

The quantity of solute B crossing a plane of area A in unit time defines the flux. It is symbolized by J, and is a vector with units of molecules per second. Fick s first law of diffusion states that the flux is directly proportional to the distance gradient of the concentration. The flux is negative because the flow occurs in a direction so as to offset the gradient ... 

The kinetics of transport depends on the nature and concentration of the penetrant and on whether the plastic is in the glassy or rubbery state. The simplest situation is found when the penetrant is a gas and the polymer is above its glass transition. Under these conditions Fick s law, with a concentration independent diffusion coefficient, D, and Henry s law are obeyed. Differences in concentration, C, are related to the flux of matter passing through the unit area in unit time, Jx, and to the concentration gradient by,... 

The relationship between the diffusional flux, i.e., the molar flow rate per unit area, and concentration gradient was first postulated by Pick [116], based upon analogy to heat conduction Fourier [121] and electrical conduction (Ohm), and later extended using a number of different approaches, including irreversible thermodynamics [92] and kinetic theory [162], Pick s law states that the diffusion flux is proportional to the concentration gradient through... 

It is empirically known that a linear relation exists between a potential gradient or the force X and the conjugate flux J, and the laws of Ohm, Fourier, and Pick s first law for electrical conduction, thermal conduction, and diffusion, respectively, within a range of suitably small gradients ... 

Pick s first law, 1855). The proportionality factor Dj is called the diffusion coefficient of the substance concerned (units cm%). In the diffusion of ions in solutions, Eq. (4.1) is obeyed only at low concentrations of these ions. At higher concentrations the proportionality between flux and concentration gradient is lost (i.e., coefficient D, ceases to be constant). 

The development of the theory of solute diffusion in soils was largely due to the work of Nye and his coworkers in the late sixties and early seventies, culminating in their essential reference work (5). They adapted the Fickian diffusion equations to describe diffusion in a heterogeneous porous medium. Pick s law describes the relationship between the flux of a solute (mass per unit surface area per unit time, Ji) and the concentration gradient driving the flux. In vector terms. 

The basic flux across the membranes may be related to that across a thin film [83]. Pick s First Law of Diffusion indicates that the total flux of diffusant across a homogeneous membrane, /, is proportional to the concentration gradient of the diffusant ... 

The constant of proportionality k is known as the thermal conductivity of the material and the above relationship is known as Fourier s law for conduction in one dimension. The thermal conductivity k is the heat flux which results from unit temperature gradient in unit distance. In s.i. units the thermal conductivity, k, is expressed in Wm"1 K. Integration of Fourier s law yields... 

This result shows that the most likely rate of change of the moment due to internal processes is linearly proportional to the imposed temperature gradient. This is a particular form of the linear transport law, Eq. (54), with the imposed temperature gradient providing the thermodynamic driving force for the flux. Note that for driven transport x is taken to be positive because it is assumed that the system has been in a steady state for some time already (i.e., the system is not time reversible). 

Fourier s law for thermal conduction An equation describing the relationship between the rate of heat flux and the temperature gradient. See Eq. (23). 

Fick first recognized the analogy among diffusion, heat conduction, and electrical conduction and described diffusion on a quantitative basis by adopting the mathematical equations of Fourier s law for heat conduction or Ohm s law for electrical conduction [1], Fick s first law relates flux of a solute to its concentration gradient, employing a constant of proportionality called a diffusion coefficient or diffu-sivity ... 

The first step in the process is to relate heat flow to a temperature gradient, just as a diffusive flux can be related to a concentration gradient. The fundamental law of heat conduction was proposed by Jean Fourier in 1807 and relates the heat flux (q) to the temperature gradient ... 

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