Fluxes and Driving Forces

The modeling of mass transport from the bulk fluid to the interface in capillary flow typically applies an empirical mass transfer coefficient approach. The mass transfer coefficient is defined in terms of the flux and driving force J = kc(cbuik-c). For non-reactive steady state laminar flow in a square conduit with constant molecular diffusion D, the mass balance in the fluid takes the form... 

Even then, the formulation of the system of fluxes and driving forces is not yet satisfactory, because of the quantity Acw. Thus, a new definition of the fluxes will be introduced, i.e. the volume flux of the solution, defined by the equation... 

This chapter establishes a direct relation between lost work and the fluxes and driving forces of a process. The Carnot cycle is revisited to investigate how the Carnot efficiency is affected by the irreversibilities in the process. We show to what extent the constraints of finite size and finite time reduce the efficiency of the process, but we also show that these constraints still allow a most favorable operation mode, the thermodynamic optimum, where the entropy generation and thus the lost work are at a minimum. Attention is given to the equipartitioning principle, which seems to be a universal characteristic of optimal operation in both animate and inanimate dynamic systems. 

Local heat flux and driving force x = A(l/T) in a heat exchanger. The temperature increase of the cold stream is AT. 

Table 2 presents the measured water fluxes and driving forces at the two salt concentrations. The water flux components Jh and Je are defined from (1) Jv = Jh + Je, with. //, = k hV(-h) the water flux in short-circuited situation and Je = keV —E) the water flux driven by the streaming potential. 

Table 2. Water fluxes and driving forces derived from permeameter experiments with water of two different salt concentrations...

Diagram for the non-equilibrium behavior in solids close to equilibrium. Conventional chemical reaction (x = x ) and particle transport (A s B) are described in a general manner. Particle transport also includes the limiting cases of pure diffusion (zFA =0) and pure electrical conduction (A// = 0).21 Note slight deviations to the notation in the text (e.g. P as pro-portionality factor between flux and driving force). 

The first postulate of irreversible thermodynamics is that the fluxes (or dependent variables) are directly proportional to the driving forces (or independent variables). [Actually, it may be shown that the assumption of local equilibrium follows from the assumption of a linear relation between the fluxes and driving forces (Truesdell, 1969).] If we take the di as dependent variables and the (m, — Wy) as independent variables we may, therefore, write... 

We now postulate a linear relationship between independent fluxes and driving forces... 

There are different approaches that can be taken to estimate the OMTC (or resistance). The first is to directly measure the flux and driving force and calculate the coefficient. The second uses correlations that are available to estimate the value based on the particular process and operating conditions. Examples of this approach will be included in later chapters that deal with a particular separation technology. The third is to determine each mass transfer resistance and combine the terms to calculate the total resistance. This approach is analogous to the calculation of an equivalent resistance for an electrical circuit. 

Fluxes and Driving Forces in Membrane Separation Processes... 

As we noted below, the equation (4.489) the expressions (4.493) show that dependence of reaction rate on affinity is not so simple [158, 159] as it is assumed in classical non-equilibrium thermodynamics [1, 3, 4, 130] based on entropy production (by chemical reactions), i.e. as a product of fluxes and driving forces (4.178). Projection B of chemical potential vector p, to the subspace W also plays a role in expression for reaction rates J as (4.493) in our example the affinity A is projection of p into orthogonal reaction subspace V only, cf. (4.174). Cf. detailed discussion and criticism in review [108] and references [159, 160]. 

These three simple e qiressions plus momentum, energy, and mass balances can be used to derive general transport equations that can, in principle, represent solvent dynamic behavior in all possible uses or applications. The difference between one application and another depends on the initial and boundary conditions and other assumptions that are imposed on the equations. There is extensive literature (id) in this area. The important point is that, in these three simple equations as well as in the more general transport expressions, the fluxes and driving forces do not depend on the individual components present the form of these equations is independent of the particular chemical components present. The identity of the components appears only in the proportionality factors n, X, and Dab- In fact, because all the possible dynamic behavior of a solvent is embodied in the three modes of transport, if two fluids have the same values for these three quantities, then all of the dynamic behavior as represented by these expressions will be the same for both fluids. 

This expression is not a way to calculate the permeating flux (known from experiments), but is the simple definition of permeance, which can be calculated by the ratio between flux and driving force (eqn (14.4)) ... 

Since we assume that the gas (species 2) is insoluble in the liquid (species 1), then N2 = 0. Hence, eq. (8.2-68) can be rewritten in terms of flux and driving force of the component 1 ... 

Extended Stefan-Maxwell constitutive laws for diffusion Eq. 4 resolve a number of fundamental problems presented by the Nemst-Planck transport formulation Eq. 1. A thermodynamically proper pair of fluxes and driving forces is used, guaranteeing that all the entropy generated by transport is taken into account. The symmetric formulation of Eq. 4 makes it unnecessary to identify a particular species as a solvent - every species in a solution is a solute on equal footing. Use of velocity differences reflects the physical criterion that the forces driving diffusion of species i relative to species j be invariant with respect to the convective velocity. Finally, all possible binary solute/solute interactions are quantified by distinct transport coefficients each species i in the solution has a diffusivity or mobility relative to every other species j, Djj or up, respectively. 

The cross-effects, described by the off-diagonal elements, are especially important for systems in which multi-component mass transfer takes place. Cross effects between scalar and vectorial effects are impossible. The entropy production in the system is given as a function of the fluxes and driving forces ... 

Fig. 4.31 Mass transfer flux and driving force of isopropanol in three-component system versus liquid height...

First of all we have obtained an overview of the relationship between fluxes and driving forces, and have particularly investigated the validity range of linear relationships. 

This just expresses the linear relation between flux and driving force. Chemical reactions between freshly mixed reactands do not usually fall within the range of applicabihty of linear, irreversible thermodynamics. However, in this case we only disturb an already established equilibrium (i.e. AG < RT). 

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