Thermodynamic Vectors and Derivatives

Let us consider the simplest case of a homogeneous, single-component fluid of fixed mass, with two degrees of freedom. We seek to evaluate a general partial derivative V of the form 

In this two-dimensional space, we can also describe the chosen basis intensities as complementary, and let the prime symbol on R denote the complement of Rh so that 

With these notational conventions, we can uniquely identify the state variables Z (complement), Z (conjugate) and Z (conjugate complement) for any chosen variable Z. 

For example, when temperature T and negative pressure —P are the chosen Rh 

The standard scalar products among vectors T), — P), S), V) are gathered for convenience in Table 12.2, expressed in terms of standard response functions CP, Cv, /3s, oip, 

The standard scalar products among vectors T), — P), S), V) are gathered for convenience in Table 12.2, expressed in terms of standard response functions CP, Cv, /3r, /3S, aP, Ty [see (11.27)—(11.30)]. As described previously, scalar products for S) and V) (i.e., involving properties CP, /3y, aP) are obtained by matrix inversion from those for T) and — P) (i.e., involving Cv, fis, Ty). The vector-algebraic procedure to be described will automatically express any desired derivative in terms of the six properties in Table 12.2, and these expressions may subsequently be reduced (if desired) to involve only three independent properties by identities previously introduced [cf. (11.39)—(11.42)], consistent with the /(/+ l)/2 rule.  

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Technically, COSMO-RS meets all requirements for a thermodynamic model in a process simulation. It is able to evaluate the activity coefficients of the components at a given mixture composition vector, x, and temperature, T. As shown in Appendix C of [Cl 7], even the analytic derivatives of the activity coefficients with respect to temperature and composition, which Eire required in many process simulation programs for most efficient process optimization, can be evaluated within the COSMO-RS framework. Within the COSMOt/ierra program these analytic derivatives Eire available at negligible additionEd expense. COSMOt/ierra can Eilso be csdled as a subroutine, Euid hence a simulator program can request the activity coefficients and derivatives whenever it needs such input. 

It is believed that ASPEN provides a state-of-the-art capability for thermodynamic properties of conventional components. A number of equation-of-state (EOS) models are supplied to handle virtually any mixture over a wide range of temperatures and pressures. The equation-of-state models are programmed to give any subset of the properties of molar density, residual enthalpy, residual free energy, and the fugacity coefficient vector (and temperature derivatives) for a liquid or vapor mixture. The EOS models (named in tribute to the authors of such work) made available in ASPEN are the following ... 

Extended nonequilibrium thermodynamics is not based on the local equilibrium hypothesis, and uses the conserved variables and nonconserved dissipative fluxes as the independent variables to establish evolution equations for the dissipative fluxes satisfying the second law of thermodynamics. For conservation laws in hydrodynamic systems, the independent variables are the mass density, p, velocity, v, and specific internal energy, u, while the nonconserved variables are the heat flux, shear and bulk viscous pressure, diffusion flux, and electrical flux. For the generalized entropy with the properties of additivity and convex function considered, extended nonequilibrium thermodynamics formulations provide a more complete formulation of transport and rate processes beyond local equilibrium. The formulations can relate microscopic phenomena to a macroscopic thermodynamic interpretation by deriving the generalized transport laws expressed in terms of the generalized frequency and wave-vector-dependent transport coefficients. 

Let us now derive phenomenological equations of the kind (5.193) corresponding to the expression (5.205). As has been mentioned before, each flux is a linear function of all thermodynamic forces. However the fluxes and thermodynamic forces that are included in the expression (5.205) for the dissipative function, have different tensor properties. Some fluxes are scalars, others are vectors, and the third one represents a second rank tensor. This means that their components transform in different ways under the coordinate transformations. As a result, it can be proven that if a given material possesses some symmetry, the flux components cannot depend on all components of thermodynamic forces. This fact is known as Curie s symmetry principle. The most widespread and simple medium is isotropic medium, that is, a medium, whose properties in the equilibrium conditions are identical for all directions. For such a medium the fluxes and thermodynamic forces represented by tensors of different ranks, cannot be linearly related to each other. Rather, a vector flux should be linearly expressed only through vectors of thermodynamic forces, a tensor flux can be a liner function only of tensor forces, and a scalar flux - only a scalar function of thermodynamic forces. The said allows us to write phenomenological equations in general form... 

Let us consider a system having N vector-processes with j and VF, thermodynamic fluxes and forces. Study these in the frame of the generalized Onsager constitutive theory. Consequently the flux of the i-th vector process can be derived from the strictly convex... 

Note that differentials (dz) have fundamentally different mathematical character than do functions (such as z, z , z77)- The former are inherently infinitesimal (microscopic) in scale and carry multivariate dependence on all possible directions of change, whereas the latter carry only macroscopic numerical values. Thus, it is mathematically inconsistent to write equations of the form differential = function (or differential = derivative ), just as it would be inconsistent to write equations of the form vector = scalar or apples = oranges. Careful attention to proper balance of thermodynamic equations with respect to differential or functional character will avert many logical errors. 

Schirmer et al. (7.) indicate that the constants and E j may be derived from physical or statistical thermodynamic considerations but do not advise this procedure since theoretical calculations of molecules occluded in zeolites are, at present, at least only approximate, and it is in practice generally more convenient to determine the constants by matching the theoretical equations to experimental isotherms. We have determined the constants in the model by a method of parameter determination using the measured equilibrium data. Defining the entropy constants and energy constants as vectors... 

Herzfeld and Langmuir-Hinshelwood-Hougen-Watson cycles, could be formulated and solved in terms of analytical rate expressions (19,53). These rate expressions, which were derived from mechanistic cycles, are phrased, however, in terms of the formation and destruction of molecular species without the need for computing the composition of reactive intermediates. Thus, these expressions are the relevant kinetics required for molecular models and are rooted to the mechanistic cycles only implicitly by the mechanistic rate constants. The molecular model, in turn, transforms a vector of reactant molecules into a vector of product molecules, either of which is susceptible to thermodynamic analysis. This thermodynamic analysis helps to organize these components into relevant boiling point or solubility product classes. Thus the sequence of mechanistic to molecular to global models is intact. 

It has been emphasized repeatedly that continuum mechanics provides no guidance in the choice of a general constitutive hypothesis for either the heat flux vector q or the stress tensor T. On the other hand, it was noted earlier that (2 41) and (2-56), derived respectively from the law of conservation of angular momentum and the second law of thermodynamics, must be satisfied by the resulting constitutive equations. It thus behooves us to see whether... 

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