Transport effects temperature gradients

Mass Transport. An expression for the diffusive transport of the light component of a binary gas mixture in the radial direction in the gas centrifuge can be obtained directly from the general diffusion equation and an expression for the radial pressure gradient in the centrifuge. For diffusion in a binary system in the absence of temperature gradients and external forces, the general diffusion equation retains only the pressure diffusion and ordinary diffusion effects and takes the form... 

The gas motion near a disk spinning in an unconfined space in the absence of buoyancy, can be described in terms of a similar solution. Of course, the disk in a real reactor is confined, and since the disk is heated buoyancy can play a large role. However, it is possible to operate the reactor in ways that minimize the effects of buoyancy and confinement. In these regimes the species and temperature gradients normal to the surface are the same everywhere on the disk. From a physical point of view, this property leads to uniform deposition - an important objective in CVD reactors. From a mathematical point of view, this property leads to the similarity transformation that reduces a complex three-dimensional swirling flow to a relatively simple two-point boundary value problem. Once in boundary-value problem form, the computational models can readily incorporate complex chemical kinetics and molecular transport models. 

Mass transfer may be influenced by gradients in variables other than concentration and pressure. In the pharmaceutical sciences, gradients in electrical potential and in temperature are two important examples of these other driving forces. Section IV.B.l describes the effect of electrical potential gradients on the transport of ions, and Section IV.B.2 discusses mass transport in the presence of temperature gradients, known as combined heat and mass transfer. 

The transport of heat between latitude bands is assumed to be diffusive and is proportional to the temperature difference divided by the distance between the midpoints of each latitude band. This is the temperature gradient. In this simulation all these distances are equal, so the distance need not appear explicitly. The temperature gradient is multiplied by a transport coefficient here called diffc, the effective diffusion coefficient. The product of the diffusion coefficient and the temperature gradient gives the energy flux between latitude zones. To find the total energy transport, we must multiply by the length of the boundary between the latitude zones. In... 

The starting points for the continuity and energy equations are again 21.5-1 and 21.5-6 (adiabatic operation), respectively, but the rate quantity7 (—rA) must be properly interpreted. In 21.5-1 and 21.5-6, the implication is that the rate is the intrinsic surface reaction rate, ( rA)int. For a heterogeneous model, we interpret it as an overall observed rate, (—rA)obs, incorporating the transport effects responsible for the gradients in concentration and temperature. As developed in Section 8.5, these effects are lumped into a particle effectiveness factor, 77, or an overall effectiveness factor, r]0. Thus, equations 21.5-1 and 21.5-6 are rewritten as... 

On an international scale, the Alps are a middle-sized chain of mountains which, due to their situation in the central latitude of Europe, are influenced by maritime as well as continental factors. Humidity is generally transported by the west and south winds flowing from the Atlantic or the Mediterranean towards the mountain chain. With altitudes of up to 4,500 m ASL, the Alps present an enormous barrier to the air masses being transported in this way, and this barrier effect reinforces European meridional temperature gradients [10]. 

So far, the effect of temperature gradients within the particle has been ignored. Stringly exothermic reactions generate a considerable amount of heat which, if conditions are to remain stable, must be transported through the particle to the exterior surface where it may then be dissipated. Similarly, an endothermic reaction requires a source of heat and in this case the heat must permeate the particle from the exterior to the interior. In either case, a temperature gradient within the particle may be established and the chemical reaction rate will then vary with position within the particle. 

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. 

Eq. 2.18) and Ohm s law for electrical conduction (where the electrical conductivity is also always positive, according to Exercise 2.1 s solution). However, this is not necessarily the case for the cross-effect tensors /3 and 7. For example, in mass diffusion in a thermal gradient, the heat of transport can be either positive or negative the direction of the atom flux in a temperature gradient can then be in either direction. The anisotropic equivalent to the heat of transport relates the direction of the mass diffusion to the direction of the temperature gradient. There is no physical requirement that these quantities could not be in reversed directions, and indeed, sometimes they are. 

The elution and migration effects are also found in PDC, but not the compression effect, since no precipitant and temperature gradients are applied. Unlike BWF, no simple transport Equation is applied in PDC because the replacement of the linear Eq. (3b) by the non-linear (17c) would lead to an integrodifferential equation similar to (41 a-b), but more complicated, if some explicit formulation were used instead of the implicit one, based on a flow-equilibrium and on a perturbation calculus, applied to an integrated transport Equation... 

Thermal diffusion, also known as the Ludwig-Soret effect [1, 2], is the occurrence of mass transport driven by a temperature gradient in a multicomponent system. While the effect has been known since the last century, the investigation of the Ludwig-Soret effect in polymeric systems dates back to only the middle of this century, where Debye and Bueche employed a Clusius-Dickel thermogravi-tational column for polymer fractionation [3]. Langhammer [4] and recently Ecenarro [5, 6] utilized the same experimental technique, in which separation results from the interplay between thermal diffusion and convection. This results in a rather complicated experimental situation, which has been analyzed in detail by Tyrrell [7]. 

Finally, at this point it should be recalled that when applying theoretical criteria one has to be aware of the fact that if concentration and temperature gradients occur simultaneously inside the catalyst pellet, both effects may compensate each other, and thus a combined heat and mass transport criterion alone may not be able to recognize that, in this case, the observed kinetics are actually influenced by transport effects. 

The movement of macromolecules in a temperature gradient is always in the direction from the hot to the cold region [43,197]. This movement is caused by thermal diffusion, exploited as the driving force in Th-FFF, and called the Soret effect, known already for over 50 years [201-203]. The transport (Eq. (1)) has to be extended by a term taking the thermal diffusion into account. Thus the flux density Jx can be expressed by [34,194] ... 

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