Molecular flux entropy

Step 7. Manipulate the equation of change for specific entropy, via definitions of convective and molecular entropy fluxes, to identify all terms that correspond to entropy generation. These terms appear as products of fluxes and forces. 

Although we shall not directly use these four postulates of irreversible thermodynamics as a foundation to our study of molecular transport in separations, a number of important principles are illuminated here. For instance, postulate 2 permits us to use—and this is in no way obvious— equilibrium parameters such as entropy and temperature in descriptions of systems where no equilibrium exists. The importance of this is evident when we ask ourselves how we would describe a system if these parameters were not available. Postulate 3 demonstrates that in the range of our typical experiences, the fluxes of matter or of heat are proportional to the gradients or forces that drive them. However, there are exceptions nonlinear terms enter if the forces become intense enough. 

A change in entropy will thus have a significant effect on the selectivity when molecular sieving is considered. This is thoroughly discussed by Singh and Koros [9]. The flux may be described as in Equation 4.16 where Eq.ms is the activation energy for diffusion in the molecular sieving media. 

The convective flux of entropy is given by p s, where v is the mass-average velocity of the mixture. The molecular flux of entropy with respect to v is deflned by... 

Step 1. Insert the molecular flux of entropy given by (25-37) into the defining equation for entropy production [i.e., (25-38)] ... 

The multicomponent equation of change for specific internal energy, given by (27-4), which is consistent with the first law of thermodynamics and the definition of the molecular flux of thermal energy via the entropy balance, reduces to ... 

Some heat flows in connection with entropy production are associated with other thermodynamic variables. Typical single fluxes and forces are summarized above. It may be noted that steady fluxes are considered. Kinetic theory provides theoretical justification of some of these flux force relations (J = LX). Here, L is called phenomenological coefficient. But kinetic theory has limitation in the sense that first approximation to distribution function corresponds to local equilibrium hypothesis. It may be noted that non-equilibrium molecular dynamics (model and simulation) provides justification of these laws for a wide range. Nevertheless, justification has to be provided by experiments (Table 2.1). 

The general equations of change given in the previous chapter show that the property flux vectors P, q, and s depend on the nonequi-lihrium behavior of the lower-order distribution functions g(r, R, t), f2(r, rf, p, p, t), and fi(r, P, t). These functions are, in turn, obtained from solutions to the reduced Liouville equation (RLE) given in Chap. 3. Unfortunately, this equation is difficult to solve without a significant number of approximations. On the other hand, these approximate solutions have led to the theoretical basis of the so-called phenomenological laws, such as Newton s law of viscosity, Fourier s law of heat conduction, and Boltzmann s entropy generation, and have consequently provided a firm molecular, theoretical basis for such well-known equations as the Navier-Stokes equation in fluid mechanics, Laplace s equation in heat transfer, and the second law of thermodynamics, respectively. Furthermore, theoretical expressions to quantitatively predict fluid transport properties, such as the coefficient of viscosity and thermal... 

Note D.2 (Entropy increase in the isolated system). In the isolated system dQ = 0 as mentioned above, however, the flux dQ caused by some internal heat source is not always zero. Furthermore, it is not completely obvious whether or not an irreversible process which satisfles the inequality (D.51) exists in an isolated system (as mentioned in AppendixD.3, Boltzmann s interpretation of entropy molecular configuration W). Any experiment to verify the inequality (D.51) for the isolated system poses an interesting physical observation problem, since it is almost impossible to measure the state of the isolated system by using a sensor inserted into the system without introducing some disturbance. 

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