Conductive flux transport

For physical process simulation, as well as for mathematical model development, we can use the isomorphism principle. This is based on the formal analogy of the mathematical and physical descriptions of different phenomena. We can detail this principle by considering the conductive flux transport of various properties, which can be written as follows ... 

We now repeat the derivation of the steady-state heat transport limited moisture uptake model for the system described by VanCampen et al. [17], The experimental geometry is shown in Figure 9, and the coordinate system of choice is spherical. It will be assumed that only conduction and radiation contribute significantly to heat transport (convective heat transport is negligible), and since radiative flux is assumed to be independent of position, the steady-state solution for the temperature profile is derived as if it were a pure conductive heat transport problem. We have already solved this problem in Section m.B, and the derivation is summarized below. At steady state we have already shown (in spherical coordinates) that... 

As can be seen in the different boundary conditions, the main effects of having ribs are electronic conductivity and transport of oxygen and water, especially in the liquid phase. In terms of electronic conductivity, the diffusion media are mainly carbon, a material that is fairly conductive. However, for very hydro-phobic or porous gas-diffusion layers that have a small volume fraction of carbon, electronic conductivity can become important. Because the electrons leave the fuel cell through the ribs, hot spots can develop with large gradients in electron flux density next to the channel. " Furthermore, if the conductivity of the gas-diffusion layer becomes too small, a... 

The thermoelectric effect is due to the gradient in electrochemical potential caused by a temperature gradient in a conducting material. The Seebeck coefficient a is the constant of proportionality between the voltage and the temperature gradient which causes it when there is no current flow, and is defined as (A F/A7) as AT- 0 where A Fis the thermo-emf caused by the temperature gradient AT it is related to the entropy transported per charge carrier (a = — S /e). The Peltier coefficient n is the proportionality constant between the heat flux transported by electrons and the current density a and n are related as a = Tr/T. 

Hence, thermal conductivity, X, is the heat flux transported through a material due to a... 

Since conduction (i.e., migration) and diffusion are the two possible modes of transport for an ionic species, the total flux J. must be the sum of the conduction flux (y ),- and the diffusion flux (7 ),.. Thus,... 

It should be emphasized therefore that the transport number only pertains to the conduction flux (i.e., to that portion of the flux produced by an electric field) and any flux of an ionic species arising from a chemical potential gradient (i.e., any diffusion flux) is not counted in its transport number. From this definition, the transport number of a particular species can tend to zero, f, 0, and at the same time its diffusion flux can be finite. 

The observed conductivity is always found to be less than that calculated from the sum of the diffusion coefficients (Table 5.27), i.e., from the Nemst-Einstein relation [Eq. (5.61)]. Conductive transport depends only on the charged species because it is only charged particles that respond to an external field. If therefore two species of opposite charge unite, either permanently or temporarily, to give an uncharged entity, they will not contribute to the conduction flux (Fig. 5.34). They will, however, contribute to the diffusion flux. There will therefore be a certain amount of currentless diffusion, and the conductivity calculated from the sum of the diffusion coefficients will exceed the observed value. Currentless diffusion will lead to a deviationfrom the Nernst-Einstein relation. 

Equations (20 and 21) describe ionic transport as a combination of electrolyte diffusion and ionic conduction, and hence the name diffusion-conduction flux equation is suggested. Its comparison with the diffusion-migration flux equation, Eq. (7), evidences the difference between migration and conduction. Migration is due to electric fields, either external or internal, and does not require a nonzero current density. Conduction is the ionic motion associated to the part of the electric field that is controlled externally. Note that the electric current density is a measurable magnitude but not the local electric field. Conduction requires a nonzero electric current density. 

Momentum can be transported by convection and conduction. Convection of momentum is due to the bulk flow of the fluid across the surface associated with it is a momentum flux. Conduction of momentum is due to intermolecular forces on each side of the surface. The momentum flux associated with conductive momentum transport is the stress tensor. The general momentum balance equation is also referred to as Cauchy s equation. The Navier-Stokes equations are a special case of the general equation of motion for which the density and viscosity are constant. The well-known Euler equation is again a special case of the general equation of motion it applies to flow systems in which the viscous effects are negligible. 

Here j is the current density, or equivalently, the charge flux. The electric field E, which is the gradient of the potential p, provides the driving force for conduction. The transport coefficient a is the specific conductivity, expressed in ohm m"k While (3.1) may appear unfamiliar, it reduces to the common expression I — VjR in the special case of conduction in a wire of uniform cross section (see Problem 3.1). 

Although Equation 4.282 does not contain the convective term, the right-hand side of this equation contains all the sources of heat and, hence, liquid heat flux is included in the conductive heat flux, on the left-hand side of Equation 4.298. The heat flux transported, with the liquid water produced in the ORR, is... 

On quiescent planets or satellites without substantial atmospheres, the surface temperature is determined by a balance between incident solar flux, thermally emitted radiation, and conductive heat transport into or out of the opaque surface. By measuring the surface temperature and the bolometric albedo the absorbed and emitted radiation can be found and the conductive flux into the solid body derived. After sunset or dining a solar eclipse the cooling rate of the surface depends on the thermal inertia of the subsurface layers. A study of such cooling rates provides a sensitive means of discriminating between powdery, sandy, or solid rock surfaces. We now review the theory behind such an analysis, and discuss examples of thermal inertia measurements. 

Neurotransmitter transport can be electrogenic if it results in the net translocation of electrical charge (e.g. if more cations than anions are transferred into the cell interior). Moreover, some transporters may direction-ally conduct ions in a manner akin to ligand-gated ion channels this ion flux is not coupled to substrate transport and requires a separate permeation pathway associated with the transporter molecule. In the case of the monoamine transporters (DAT, NET, SERT) the sodium current triggered by amphetamine, a monoamine and psychostimulant (see Fig. 4) is considered responsible for a high internal sodium concentration... 

The Chemkin package deals with problems that can be stated in terms of equation of state, thermodynamic properties, and chemical kinetics, but it does not consider the effects of fluid transport. Once fluid transport is introduced it is usually necessary to model diffusive fluxes of mass, momentum, and energy, which requires knowledge of transport coefficients such as viscosity, thermal conductivity, species diffusion coefficients, and thermal diffusion coefficients. Therefore, in a software package analogous to Chemkin, we provide the capabilities for evaluating these coefficients. ... 

A combination of continuum transport theory and the Poisson distribution of solution charges has been popular in interpreting transport of ions or conductivity of electrolytes. Assuming zero gradient in pressure and concentration of other species, the flux of an ion depends on the concentration gradient, the electrical potential gradient, and a convection... 

Although the transport properties, conductivity, and viscosity can be obtained quantitatively from fluctuations in a system at equilibrium in the absence of any driving forces, it is most common to determine the values from experiments in which a flux is induced by an external stress. In the case of viscous flow, the shear viscosity r is the proportionality constant connecting the magnitude of shear stress S to the flux of matter relative to a stationary surface. If the flux is measured as a velocity gradient, then... 

The flux vector accounts for mass transport by both convection (i.e., blood flow, interstitial fluid flow) and conduction (i.e., molecular diffusion), whereas S describes membrane transport between adjacent compartments and irreversible elimination processes. For the three-subcompartment organ model presented in Figure 2, with concentration both space- and time-dependent, the conservation equations are... 

If initial solute uptake rate is determined from intestinal tissue incubated in drug solution, uptake must be normalized for intestinal tissue weight. Alternative capacity normalizations are required for vesicular or cellular uptake of solute (see Section VII). Cellular transport parameters can be defined either in terms of kinetic rate-time constants or in terms of concentration normalized flux [Eq. (5)]. Relationships between kinetic and transport descriptions can be made on the basis of information on solute transport distances. Note that division of Eq. (11) or (12) by transport distance defines a transport resistance of reciprocal permeability (conductance). 

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