Flux-force equations

Does that not imply that this orientation of the molecules has to be built into the flux-forces equations ... 

Thus, the flux-force equations linearly related n thermodynamics fluxes Jy. Ji,.. Ao n conjugate external thermodynamic forces Fi. Fi-- - with the proportionality constants Lij being the n transport coefficients for the process. Note that Eq. (A.9) is an example of Eq. (A. 10) for n = 1 with Ji=J,Lii — a, and Fi—E. Thermoelectric conduction and thermal diffusion are examples of processes that obey Eq. (A. 10) for n > 1. 

We shall make an attempt here to present a coordinated and unified picture for both mass transport (diffusion and sedimentation) and electrical transport (electrophoresis). The general approach of the thermodynamics of irreversible processes which yields correct expressions for the flux-force equations for electrically neutral components will be used in the derivation of transport parameters. 

The so-called coupled kinetic phenomena deserve special consideration in that a certain flux can be caused by different driving forces (that cannot be combined as above). Matter transport or charge transport can be caused not only by chemical or electrical gradients but also by temperature gradients. The inverse is true for the heat flux. This reciprocity is reflected in cross-terms that appear in the flux-force equations and obey the Onsager relations [330,508] (see footnote 6 on page 271). Let us consider a more specific example, namely the thermoelectric effect. It occurs in the chain A B A consisting of two different electrical conductors, A... 

Equating die net momentum flux out of the element to the net retarding force (equations 11.6 and 11.7) and simplifying gives ... 

Rate of Heat Transfer. Fourier s Law may be integrated and solved for a number of geometries to relate the rate of heat transfer by conduction to the temperature driving force. Equations are given below that allow the calculation of steady-state heat flux and temperature profiles for a number of geometries. 

Table G Definitions of the Electric Field E, the (Di)electric Polarization P, the Electric Displacement D, the Magnetic Field H, the Magnetization M, the Magnetic induction or flux density B, statement of the Maxwell equations, and of the Lorentz Force Equation in Various Systems of Units rat. = rationalized (no 477-), unrat. = the explicit factor 477- is used in the definition of dielectric polarization and magnetization c = speed of light) (using SI values for e, me, h, c) [J.D. Jackson, Classical Electrodynamics, 3rd edition, Wiley, New York, 1999.]. For Hartree atomic u nits of mag netism, two conventions exist (1) the "Gauss" or wave convention, which requires that E and H have the same magnitude for electromagnetic waves in vacuo (2) the Lorentz convention, which derives the magnetic field from the Lorentz force equation the ratio between these two sets of units is the Sommerfeld fine-structure constant a = 1/137.0359895...

According to the scheme of MNET we can now list the processes that can take place in the bacteriorhodopsin liposomes and write down the flux-force relation for each of the elemental processes. By adding the fluxes of each of the chemical species, we arrive at a set of equations, represented in matrix form ... 

Three new points are introduced at this time (i) Since fluxes may occur in two orthogonal directions, the conjugate flux-force pairs now are J, VxT), (//, VyT), Jx, Vx( /e), Jy, S/y l /e)). The appropriate geometry is depicted in Fig. 6.10.1. (ii) For later convenience we shall select as independent variables from this particular set the quantities xT, yT, Jx, Jy, so that the phenomenological equations appear in partially inverted form as follows ... 

The usefulness of NET in describing industrial problems has been questioned, because these problems are frequently non-linear. It is then important to know that the flux-force relations in Equation (3) also describe non-linear phenomena, The phenomenological coefficients fj can, for instance, be functions of the state variables. By... 

Mori s [5] generalized Langevin equation is rooted in a specific macroscopic phenomenology namely, the flux-force Eq. (A. 10). Consequently, despite its formal exactness, Mori s equation has a limited... 

Onsager s thermodynamic equation of motion rests on the linear flux-force relations of transport theory [38]. Thus, to start, we beiefly review these phenomenological equations. 

Eq. (A.15) is a basic equation of Onsager s theory. Notice that so far it rests solely on macroscopic phenomenology, since both the flux-force relations and the Second Law derive from purely macroscopic experiments. 

In the limit of vanishingly small Reynolds numbers, forces due to convective momentum flux are negligible relative to viscous, pressure, and gravity forces. Equation (12-4) is simplified considerably by neglecting the left-hand side in the creeping flow regime. For fluids with constant /r and p, the dimensionless constitutive relation between viscons stress and symmetric linear combinations of velocity gradients is... 

Equation (a) identifies the independent fluxes and forces. These forces cause motion and ATP consumption characterized by fluxes (currents) that are average velocity v(/ext,AM) and average rate of ATP hydrolysis J(/ext> A/r)- Molecular motors mostly operate far from equilibrium (Am lOi uT) and the fluxes are not linearly dependent on the forces. Within the linear regime (Am << pT), however, a linear flux—force relationships hold... 

For single component transport, simple transport equations can be derived from linear flux-force relationships ... 

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