New non-stationary gradient model of heat-mass-electric charge transfer in thin porous media

The well-known complicated system of non-equilibrium balance equations for a continuous fluid (f) medium needs the new non-Gibbsian model of f-phase to be applicable for description of the heterogeneous porous media (PMs). It should be supplemented by the respective coupled thermal and caloric equations of state (EOS) developed specially for PMs to become adequate and solvable for the irreversible transport f-processes. The set of standard assumptions adopted by the linear (or quasi-linear) non-equilibrium thermodynamics are based on the empirical gradient-caused correlations between flows and forces. It leads, in particular, to the oversimplified stationary solutions for PMs. The most questionable but typical modeling suppositions of the stationary gradient (SG) theory are: 1) the assumption of incompressibility accepted, as a rule, for f-flows; 2) the ignorance of distinctions between the hydrophilic and hydrophobic influence of a porous matrix on the properties; 3) the omission of effects arising due to the concomitant phase intra-porous transitions between the neighboring f-fragments with the sharp differences in densities; 4) the use of exclusively Gibbsian (i.e. homogeneous and everywhere differentiable) description of any f-phase in PM; 5) the very restrictive reduction of the mechanical velocity field to its specific potential form in the balance equation of f-motion as well as of the heat velocity field in the balance equation of internal energy; 6) the neglect of the new specific peculiarities arising due to the study of any non-equilibrium PM in the mesoand nano-scales of a finite-size macroscopic (N,V)-system of discrete particles. This work is an attempt to develop the alternative non-stationary gradient (NSG) model of real irreversible processes in PM. Another aim is to apply it without the above restrictions 1)6) to the description of f-flows through the obviously non-Gibbsian thin porous medium (TPM). We will suppose that it is composed by two inter-penetrable fractal sf-structures of f-phase (formed by the “mixture” of gand l-phases termed, in total, interphase) and solid (s) porous matrix termed below s-phase. The permanent influence of humidity and the respective increase of the moisture content in TPM including the unavoidable phenomenon of capillary condensation are the main factors to occur the non-stationary transport f-flows through its texture.

It was recently shown [1] that all above-mentioned characteristics of a realistic PM can manifest their cumulative effect by the rather simple experimental observation, at least, for the particular case of TPMs (see below).The specific feature of latters is the very small actual magnitude of thickness ( L   ) which becomes much less than two other linear TPM-sizes.From a formal viewpoint, all described trends of a supposed NSG-model should be pronounced in such actually one-dimensional transport.Indeed, the thermodynamic forces-gradients become augmented: in comparison with a usual PM.Moreover, it is naturally to admit the failure for TPM of the non-equilibrium linear SG-model of thermodynamics [2,3].Formally, the gradients of fields (

 
of electrostatic potential) through TPM become great while the respective vector convection of mass ( u  ) and/or diffusion flow (of momentum mu j , heat Q j , mass m j and free charge q j ) cannot be too great to provide its expected proportionality to the above gradients.Oppositely, linear non-equilibrium SG-model postulates that gradients and flows should be small and linearly-dependent.The well-established only in the framework of a bilinear entropy production [2,3] crosseffects of thermo-and electro-diffusion as well as the thermoelectric phenomena must be the non-linearly interdependent in a TPM and the determinative factors for a common transport flow.In other words, the problem of NSG-model becomes so complex in the case of TPM that its any simplified (and, even, rather approximate) solution seems to be very useful for applications.
Additionally, the appearance in the recent years of a "smart-texture's" (ST-) concept applied initially just to the textile fabrics makes the theoretical investigation of TPMs especially actual.The construction for them of an adequate NSG-model confirmed by the relatively scarce and restricted experimental data can lead to the novel insight into ST-problem.The main aim of such investigations is the search for the appropriate controlling parameters and factors.

Reference ideal-liquid and perturbation idealliquid with thermal conductivity regimes proposed for NSG-model of heterophase phenomena in any PM.
We refer now the readers of present work to our previous results reported not only in [1] but also in [4][5][6] where the foundation of fluctuational thermodynamics model (FT-model) has been in detail represented.The main idea of latter was the extension of macroscopic nonequilibrium thermodynamics [2,3] on the spatial mesoscopic, nano-and, even, microscopic (i.e.compatible with the effective sizes of atoms, ions and molecules) scales.The implied methodology of such extrapolation maintains the hypothesis of LE-states but formulated, exclusively, in terms of the independent fields   where s S / m  is the specific (per unit of mass) entropy and q e q / m  is the specific charge (e denotes below the specific internal energy E / m ).
To avoid the misunderstandings, let us note that the used also denotation .Its dimensionality prompts to many authors the questionable idea at the formulation of LE-hypothesis in terms of the co-called modified fluidpressure: where the difference between the gradient applied to the The further FT-transformation of two main Eulerian regimes for the above hierarchy has the following meaning and value illustrated schematically by the reported below sequence of steps.The first reference regime is introduced as an alternative to the widely usable in the phenomenology of heat and mass transfer EOS-models of ideal gas and of its mixtures.An appearance of time t [s] in Eq.( 1) is the realistic feature at the study of actual non-equilibrium processes in the finite volumes V [m 3 ] of any locally heterogeneous (i.e.non-Gibbsian) N-systems [4][5][6].
Assumptions: 1a) the common density of external forces in Eq.(4) tends to zero 0 ext f / V  .1b) the adopted approximation of isotropicity in Eq.( 4) for the tensor of deformation:

Consequences and results
1A) the introduction by FT-model of the dimensionless "thermodynamic time" for PM (and for any other inhomogeneous media) is postulated by the equality including the inverse volume: Its aim is the adequate account for the elasticity of medium by the isothermal or isoentropic bulk modulus 1D) Its usage in a combination with the LE-hypothesis of Eq.( 1) leads immediately to the generalized Bernoulli's integral: which is (again formally) identified by FT-model with the total "mechanical" energy (i.e. with the "hamiltonian" of mechanics) for a conservative "potential" enthalpy field 1E) The IL-regime admits that any arbitrary path of the possible perturbation non-equilibrium processes (see below regime ILT) can be adequately expressed in terms of two experimentally controllable caloric and thermal EOSs: where Namely the alternative to entropy from Eq.( 10) value of temperature in the system of Eqs.(11,12) stands it on the second place (after the pressure) in the above thermodynamic hierarchy of physical fields at the description of real irreversibility.To corroborate such a special role of T and of its conjugated variable of entropy s for any non-equilibrium process, let us remind that just this pair of isolines was chosen by Carnot to form the wellknown reversible cycle.It was supposedly realized by the extremely (infinitely) quick processes at s const  and by the extremely (infinitely) slow processes at

T const 
. Strictly speaking, a literal recognition of such extremes should lead to the certain inconsistency between the differential forms of First and Second Laws for the reversible processes (all realistic processes are occurred, of course, during the finite time intervals): Another questionable extreme of Carnot's ingenious cycle is, of course, the choice of an ideal gas (ig) as one-phase working medium.Its density and pressure To avoid such oversimplifications, the strategy proposed long ago by FT-model [4] seems to be the most appropriate.Indeed, due to its realization one does not omit on the ad hoc basis the divergences of the vector velocity field   u x,T  in IL-regime and of the directed heat flow  

Q j x,T 
in ILT-regime below induced by a gradient of temperature T  .Instead of this, it is naturally to eliminate their finite (i.e.unknown but realistic) values from the explicit sequence of thermodynamically-consistent transformation steps by the following scheme.
Assumptions: 1a) the system of balance equations for entropy and internal energy leads to the irreversible production of entropy   s Zt which becomes the changeable and x,t-dependent in comparison with Eq.( 8) assumed for IL-regime: 1b) FT-model proposes to eliminate the itself divergence Q j  from the above system of balance equations instead of an attempt to obtain the implied complex solution for the respective parabolic equation of thermal conductivity: where the subscript P,v for P,v C is recognizable; 1c) the fundamental exact result [4] of such elimination introduces the characteristic relaxation time-scale τ [s] for any PM and/or TPM: which is the main parameter of ILT-non-stationary changes; 1d) the necessary estimate of initial can be obtained by solution of the much more simple elliptic (stationary) variant of parabolic Eq.( 16) in which the entropy density and the relaxation parameter τ should be preliminarily found: Consequences and results 1A) the discussed Poison's-type Eq.( 18) for the T-field may be, of course, supplemented by the similar equations for the μ-field and φ-field following from the LE-condition assumed by Eq.( 1): where coefficient of the mass conductivity and the electric conductivity supplemented by two measurable linear SG-correlations for thermal conductivity (Fourier's law) and barodiffusion (D'Arcy's law): 1C) the rejection from the combined definition of an electrochemical potential ( q e   ) in Eq.( 1) and the remarkable "flexibility" of adopted by FT-model LEcondition which can be represented in terms of the Gibbs-Duhem's finite differences: Leads immediately to the following SG-description of thermal-electric diffusion for any f-and/or s-phase: It seems that the well-known Videmann-Frantz's law obtained for the proposed by Zommerfeld explanation of the Lorentz's coefficient can be considered as the limiting discrete form of the more general FT-Eq.(23) derived here for the thermal-electric phenomena; 1D) the similar FT-correlation for the electric diffusion has the especially simple form expressed in terms of the specific charge q e q / m  : It is interesting and informative to compare this result with the well-known Nernst-Einstein's law expressed in terms of the particle concentration n N / V  , elementary electric charge 0 q and the self-diffusion coefficient m D from the Fick's law: Such comparison provides immediately the following explicit T-dependence of Eqs. (20,25): as well as the generalized FT-correlation for electric diffusion:

ILT-regime of NSG-model for a compressible ideal liquid with the perturbation contribution of thermal conductivity but without the viscous damping implied by the D'Arcy's law
Its physic sense is obvious: the more is temperature at any density of electric charges, the worse becomes the electric conductivity of a medium; 1E) the combined test of the previous 1C)-and 1D)-results leads to the following FT-correlations for the thermal diffusion: For comparison, the formal elimination of q  from the Videmann-Frantz's and Nernst-Einstein's laws leads to the much more restrictive description of the thermodiffusion ratio from Eq.(29b): because it ignores, in fact, the physical reason of thermal conductivity (i.e.transport of heat which is related just to the entropy density s  ).
The physical adequacy, simplicity and the experimental testability of the reported FT-correlations are the main advantages used in this work to construct the solvable TPM-model in Sects.3,4.Let us discuss below, for comparison, the conventional theoretical and simulation approaches to the same or similar PM-problems.The interested reader can find the more detailed description in the cited references [7][8][9][22][23][24][25][26][27][28][29].Our aim here is to emphasize the distinctions between the conventional PMmodels and the proposed NSG-model without the detailed additional comments.We have changed some denotations of the original references to make the comparison more informative.
There are two main concepts in the discussed problem, which can be termed one-medium and two-medium approaches.The former adopts the local thermal equilibrium for the volume-average fields of f-velocity where K [m 2 ] is permeability and η [Pa•s] is viscosity.In accordance with our previous criticism, this approach combined with the further omission of the f-velocity FT-Eqs.(5)(6)(7)).This conclusion is also related to the "old" conventional interpretations of the D'Arcy's law termed, respectively, the drag theory and the hydraulic radius theory.They were described comprehensively by Iberall long ago [30] for usual PMs: The D'Arcy's correlation (developed for the flow of liquid water through sands) resembles, of course, the Poiseuille's law for the laminar flow of a continuous liquid through the cross-section area (see also Eq.(31b)): The conjectural replacement of the ( 28 R/ )-quantity in a medium by the effective quantity termed permeability of PM K [m 2 ] leads, often, to the confusions at its experimental determination and interpretation [1].The additional, rather crude, from our viewpoint, approximations of the supposed convective velocity D u (termed the D'Arcy's velocity) and the conjectural Reynold's number for PM: provide two alternative modeling variants of K/ -ratio from Eq.( 32): The former expression of the drag theory of permeability modified by Iberall [30] contains "the dimension characteristic of the medium structure, for instance, it is (either) the diameter (d) of the granule (or) the fiber diameter".In accordance with assumptions of Eq.34 (a,b) this is very complicated and implicit for the input parameters ( ).Such rather formidable drag model is best applicable at high porosities (ε belongs to the range of 0.7 to 0.9).The latter expression of the hydraulic radius (r) theory of permeability developed, mainly, by Kozeny contains the purely adjustable ratio of "an orientation PM-factor O to a shape PM-factor S".It turns out [30] that the hydraulic radius model is applicable exclusively at low porosities in range ε of 0.1 to 0.3.
After such interpretation of the mechanical D'Arcy's flow contribution, the volume-averaged balance energy equation becomes [2][3][4][5] a variant of Eq.( 16): where two last terms in the left-hand-side provide a combination of the convection contribution (?) for the vector T  -field with the "volume-averaged heat source term that can be used to describe the cure kinetics (?) of the resin" [24].The tensors of the effective thermal conductivity e  and of the thermal dispersion d  introduced on the ad hoc basis in the right-hand-side of Eq.(37) make, to our mind, any its solution to be rather arbitrary.
We have not reported here the even more formidable system of two coupled phase-averaged f,s-energy equations proposed for the two-medium treatment of PMs-data [26][27][28][29].It is hardly to discuss the quite complex differential equations in which the vast majority of input parameters either unknown or determinable with the large uncertainties.The concept of two different f T and s T - temperatures itself seems to be rather questionable if the medium approach is realizable.Nevertheless, we will demonstrate below (Section 3) that the similar discontinuities of the pressure P  and temperature T  but observable within the same f-phase are unavoidable in the realistic f-flows through TPM.
This observation returns us to the problem of equilibrium first-order (I) VLE-phase transition arising within any realistic PM due to the appearance of moisture content ω in its porous texture.We consider, however, the conventional approach [7][8][9] based on the attempt to model, separately, both v-and l-components of the common fvelocity field f u as the purely mechanical methodology (proposed long ago by Clebsh): The substitution of such superposition into Eq.( 2) leads, simultaneously, to the following modification of the energy balance equation [7][8][9] in comparison, for example, with the one medium description of Eq.(37): where the latent heat  8).It uses also the following alternative formulation of Second Law proposed long ago by Joule, Horstmann and others [16]: where δV is a variation of volume.It emphasizes here the discrete nature of description accepted for Q  on the molecular and/or atomic levels.Again let us remind (to avoid the misinterpretation) that the same symbol δ is used in the present work for the thickness of TPM and to distinct it from the large thickness L of usual PM.Due to such distinction, we have postulated the principal anisotropy of one-dimensional transport flows in which the above "thermodynamic time" of Eq.( 5) should be replaced for TPM by the one-dimensional ratios: The presence in the right-hand side of Eq.(41) the only parameters of a ther-mal EOS   P,V ,T supports the above-mentioned concepts of FT-model.The me-chanical theory of heat assumes in this formulation (without any appeals to the unrealistic extremes) that the internal energy content and/or the enthalpy content in Eqs.(10-12) of a body during its state change can be calculated through nothing more than the P,V,T-information.Hence, the heat quantity consumed during any such change is irrespective of what are the path (usually, an isoline) and the thermodynamic cause (usually, a gradient) for this consumption.
Particularly, Joule corroborated the consequence of Second Law by the experimental adiabatic compression of different liquids [16]: where    The meaning of the generalized Bernoulli's integral from Eq.( 9) becomes now recognizable for any PM.It is the derived in the present work thermodynamic extension of its traditional mechanical value [3] (see our comments to Eq.( 1)): Pt-components.However, this interconnection of "kinetic" and "potential" parts of "hamiltonian" in the brackets of Eq.( 9) should be additionally controllable by its changeable mass per unit of volume(i.e. by   t  ).Besides, the dynamical nature of such changes in the LE-state should take into account not only the compressibility of IL-flow by Eq.( 7) but also the possibility of internal vapor-liquid ( vl  or lv  ) capillary phase transitions occurring in the pores of PM.It is the remarkable feature of the developed here NSG-model and its interpretation adopted by Eq.( 41) that the latter corresponds congruently to the differential Clausius-Clapeyron's equation for any I-phase transition: Thus, the T-dependent ratio of discontinuities in the enthalpy (i.e. in the specific heat content) and the specific volume (i.e. in the inverse     heat contents) can be experimentally corroborated for any non-Gibbsian i-phases (i denotes g (gas), v (vapor), l (liquid), s (solid) etc.).Their main signs and features are the finite volume V and the finite time t  of observation in which the LE-hypothesis is applied to the heterogeneous two-phase I-phase transition's i,j-state.Both coexistent non-Gibbsian i,j-phases cannot (separately) be completely homogeneous in contrast to their idealized infinite Gibbsian counterparts.The typical example of a realistic heterogeneous i,j-coexistence for a pure fluid is, of course, the LE-states of a moist vapor (i.e. of the mechanical mixture formed by the bubbles of v-phase in l-phase or by the drops of l-phase in v-phase).This type of distribution for a mass content is a typical situation also for PMs.
FT-model [4][5][6] excludes (as an inaccessible result for the finite-size measurements) the reality of the strict Gibbsian equalities between the respective i,j-fields Thus the same constraint is also realizable for the LE-state between two actual non-Gibbsian i,j-phases.However, FT-model admits the easily verifiable by experiment with the moist air (ma) relative stability of such mixed i,j-phase in the saturated density range of a 0-phase transition: ji     (see below).This relatively stable colloid (as a matter of fact) heterogeneous formation has been termed interphase (not interface (!), i.e. not the interfacial layer presumed in the standard van der Waals-Maxwell-Gibbs (WMG) theory of a unified EOS and its equilibrium phase transition [17,18]).FT-model assumes so the metastable LE-states (they exist during the finite t  -interval) but rejects completely the unstable states of a spinodal decomposition.
This natural admission leads to the novel concept of a congruent zero-order (0) phase transition [17][18][19][20][21] Thus Eq.( 44) for a Gibbsian's I-phase transition should be, for example, replaced by the following equation for the 0phase transition: Its congruence with the reference s-path of Eq.( 42) is obvious.Moreover, it is straightforward task [16] to derive the similar equation for the second-order phase transition (II) in which the role of given (initial) density i  becomes pronounced: This variant of FT-model corresponds to the limiting case of an equilibrium state between two ij-phases occurring at their compatible densities and enthalpies.The respective conclusion is that the increase of pressure   0 This assumption is in a complete correspondence with the Second and First Laws combined by the so-called thermodynamic EOS for single-phase i-states and twophase ij-states [18] (the subscripts δρ and δT emphasize the heterogeneous nature of the 0-phase transition): The accepted by FT-model order of priority for fields from Eq.( 1) is here confirmed.Moreover, the appearance of both caloric Δ-discontinuities for entropy and internal energy in Eq.( 48) may become the elucidative factor for explanation of an irreversible 0-phase transition introduced for the non-Gibbsian i,j-phases in Sect.3. We will omit the superscript 0 and the double subscript ij below to avoid the overcomplications in denotations.
where the D'Arcy's law determines by Eq.( 32) the experimental τ-values [1] as the ratio of the hardly measurable permeability K [m 2 ] to the purely theoretical kinematic viscosity ν [m 2 /s].Eq.( 49) corresponds to the usual Joule's density of heat Q  which is supposedly compensated by the thermal conductivity: The derived relatively simple NSG-correlations may form the reliable basis for the creation of smart textures.Despite the widespread belief on contrary, neither P  nor T  can be directly usable as the transport governing variablesforces.Both are dependent on the changeable external conditions of TPM-exploitation.Nevertheless, it follows from our consideration that a combination of two desirable fixed flows of mass and heat (

 
q T  -parameters in Eqs.(32,49,50) become the important indicative factors to control, for example, the given optimal conditions for the comfortable wear of TPM-fabrics.We have argued in our previous work [1] that the interrelation between the volume densities of matrix MM m / V , moist air ma ma m / V and water content (liquid) ll m / V is essential to specify the following factor of difference between the hydrophobic (hb) and hydrophilic (hl) TPMs: where the additivity of the dry air (da) volume ( with the standard "dry" porosity da V / V   have been assumed.Thus the indicative factor of above difference can be introduced by two inequalities for the a priori unknown liquid volume fraction   Since both densities of TPM: V  (at the given level of relative humidity) and   (at the zero moisture content 0   ) are, in principle, measurable quantities, one should, firstly, assume the following experimental estimate: where is the saturated T-dependent liquid density of water.This simple assumption leads, however, to the rather interesting and useful for practice observation.The hbcondition in Eq.(52) can achieve the limit of its applicability with the permanent increase of moisture content   t  .At this t-moment the given hb-texture becomes, formally, the modified hl-texture in which the liquid fragments may exist inside of some capillary pores.So the study of capillary condensation is necessary for both hl-and hb-structures.
The further treatment of the wide set of experimental TPM-data in [1] has completely corroborated the result of such analysis.Indeed, the calculation of standard TPM'ssurface density .In other words, the very small δ-thickness of TPM can become in a combination with the given asymptotic parameter 0   the determinative factors for its changeable properties and, first of all, permeability in accordance with the proposed in [1] alternate basis l/b-model of the moist permeable media (see, in particular, Eq.( 50)).
The general f-and T-dependent FT-EOS was introduced by one of us (V.B.R.) [17] in the van der Waals (vdW)-like 3-coefficient's form: where ρ [m -3 ] denotes here the concentration of particles . Its further investigation and application to the wide set of problems [5,6,[18][19][20][21] has corroborated its universality and high level of accuracy.The latter is achievable at the description of any i-phase and/or ij-phase states if the coexistence curve (CXC) data of I-phase transition are known from the reliable experiment .More accurately, such information is necessary and enough to evaluate the T- a ,b ,c of f-phase without any adjustable parameters.In this work we propose to extend the FT-EOS' methodology (which rejects the vdW-concept of a unified (i.e. common for both coexistent f-phases) EOS [21]) on the description of s-phase (i.e.TPM's-matrix) too.Our arguments can be discussed below for the convenience of reader in the simplified vdW-form with two constant coefficients   00 a ,b : But in this framework any structural aggregate properties of an excluded volume are omitted.We have used this simple observation to extend its notion on two hband hl-variants of s-phase, respectively: To the best of our knowledge, such postulated on the base of Eq.( 56 f , f -mixtures.We plan to represent the respective foundation of such idea in the next publications for f,s-type non-Gibbsian phases.Some results of the discussed FT-methodology seem to be especially informative [31,32]  ZZ  lead in accordance with Eq.(57) to the basic well-established vdW-behavior of f-phase [31].Two values of free volume in l-phase   l eT  are compared in Fig. 2 with that in g-phase   g eT to confirm the obvious heterogeneous nature of former.This is a typical macroscopic-interphase: Its substitution in the universal form of thermodynamic two-phase EOS (48) gives the following system of the coupled f-porosities:         

ig t t ig t l t h T T s T h T 
, where   0 We propose to take into account that the experimental value of the consumed vaporization heat: is the main measurable parameter at the adjustable determination of the above initial value [33] for water:  

Conclusion
The proposed NSG-model opens the new wide field of the relevant investigations in the non-equilibrium thermodynamics due to the formal absence of an xintegration and of its usual complexities.We consider that only t-dependence of indicative parameter are essential at the description of transport processes through TPMs.

11 T ,P and external   22 T 1 T and 2 T as well as between 1 P and 2 P
,P sides-planes.Second Law determines, of course, the direction of a resulting forced heat-mass flow in dependence on the given relationship between  and the certain implied spatial structure of s-phase (porous matrix M) are the determinative factors for any PM at the traditional study of transport processes.The unavoidable moisture content of f-phase f da m / m   is ignorable, as a rule, in such onephase investigations.

mD
is related, mainly, to the isothermal-isobaric mass diffusion coefficient [m 2 /s] in the Fick's law for the density gradient   [kg/m 4 ].This "force" as well as the similar, widely usable moisture content gradient     1 /m (see Eqs.(2,3)) exist in the interfacial layers finite thickness of the first-order vaporliquid (v,l) VLE-equilibrium phase transition.Hence, both ones correspond to the equality of chemical potential obligatory lead to the thermodynamic irreversibility of such heterophase self-diffusion.The real cause for latter may be only gradient   [m/s 2 ] fields determine the relatively small external force-field influence per unit of volume ext f / V on the moving fluid: provides the LE-interpretation of P; 1c) the implied identification of the resulting Eulerian Eq.(4) with the vector equation of motion in the Newton's mechanics   D / Dt d / dt  for the conservative field of a potential pressure-field   Px per unit of volume.
be, in fact, negligible) in accordance with the respective ig-EOS, while the specific ig-internal energy from Eq.(13)

q[
Cl 2 /m•s•J] have been introduced.Their subordinated role in relation to the T-field and the use of common τ-value correspond to the abovementioned the respective cross-effects leading to the known transport equations of thermal diffusion (Soret effect), thermal-electric diffusion (Peltier effect) and electric diffusion (Nernst effect) may be straightforwardly analyzed for the respective set of quasi-linear T-dependent SGcoefficients: f u and both main thermodynamic fields T , P .The abovementioned modified f-pressure determines the socalled D'Arcy's f-velocity D u namely by the generalized D'Arcy's law: ) equation of the stationary modeling D'Arcy's convective flow ( 0 D u  related to the capillary condensation and plays, formally, the role of a heat source term.Any solution of the coupled Eqs.(2,39) depends completely on the chosen (supposedly experimental) magnitudes of ill-founded coefficients: and PM  related to the "dry" porosity ε [m 3 /m 3 ] of PM.In total, the described methodology of I-phase transition in PMs' bears the strong resemblance with the known Clebsh's potential of the convection velocity fields [4].One may introduce them by Eq.(1) without the above complexities:     , [m/s].(40) 3 Second law and zero-order (0) VLE-transition in non-Gibbsian f-phases To go beyond the Carnot's cycle artificial extremes, FT-model introduces not only the described reference ILregime with the zero entropy production From the viewpoint of FTmodel[4][5][6] both variations in Eq.(41) should be considered as the quantisized finite values (i.e. the quantities composed by the great but finite number of the microscopic portions of energy B kT[J] and volume of particles 3 b ~d   [m 3 ]).
time" introduced by FT-model.All processes are here realizable during the finite physical time-intervals t  .The main conclusion from the Second Law is that the adiabatic compression of a liquid: as it is in water at temperatures below than: 4 t  °C, for example).

2 2
or, even, gas) in the external gravitation field g ; b) the absence of the internal energy and heat content h (enthalpy) contributions; c) the implied absence of the permeability for the fluid particles in its imaginary walls.One may conclude that the more is a convection contribution in the brackets, the less becomes a molecular momentum flux (pressure) in such rather restrictive and purely mechanical integral of a fluid motion.The NSG-model rejects in its IL-regime all above assumptions a), b), c) and proposes the much more flexible description of IL-flow through any PM.In this case, the more is a convection non-stationary contribution   u t / , the less becomes a heat content   ht composed by the interchangeable internal energy   et -and pressure   a matter of fact, the thermodynamic irreversibility of a real phase transition (!).The natural conclusion is that a spontaneous process of condensation (including its local capillary form) concomitant process of vaporization consuming the external heat at the same temperature   Q / T  from Eq.(44).The supposed hysteresis of the "latent" heat h  (i.e. its thermodynamically well-grounded by FT-model irreversibility between the consuming lv h   and emitting vl h

T   and 0 PC
. The standard positive characteristics for I-phase transitions 0  should become sometimes great but still finite ones for 0-phase transitions.The magnitudes of these derivatives correspond to the supposed by FT-model very small disbalances between ij T -fields ji T T T   at the forcedly adopted Gibbsian equality for ij-pressures ij i j P P P .Vice versa, the same is true for the small disbalance:

/
 ).This result can be of great importance (Sect. 4)for the development of a flexible predictive NSG-model for TPMs too.In this type of PM the sharp "jumps" (discontinuities) of second thermal ( ) derivatives in the different pores are not only possible but unavoidable.4Fluctuational EOS for hydrophobic and hydrophilic TPMsWe have assumed in Sects.2,3 that the thermal conductivity λ [W/m K] and the relaxation period τ [s] in Eqs.(21a,b) for two main measurable thermodynamic forces of medium ( T  and P  ) maintain their constant values while two other transport coefficient of the particle selfdiffusion dependent in the proposed model of NSG.

.
mQ j , j ) could be provided through TPM just by the appropriate choice of P   -and T   -magnitudes.They should be co-ordinated with the controllable variable namely of internal pressure We intend to demonstrate below that both are determinable by the Tdependent cohesive vdW-coefficient   f aT introduced in FT-EOS.In this case, the obtained explicit correlations for

A
[g/m 2 ] (where the TPM-volume is VA  ) has shown the existence of two quite different types of its behavior .It was revealed for the different sets of hb-and hl-fabrics, respectively: trivial linear δ-correlation with the homogeneous TPM's-volume density AV This form defines the "dry" porosity ε by means of the ratio of free volume   0 molecular volume, to the representative total volume V. by the specific repulsive interparticle potential of f-phase.
) PM-definitions are the first attempt to incorporate the physically plausible molecular-based parameters   00 a ,b of matrix M for the description of its interaction with the molecular structure of f-phase.An absence of cohesion contribution 0 0 hb a  in hb-textures of Eq.(59) reflects, in particular, their inability to wetting by the moist f-flow.The remarkable consequence of the proposed approach is a possibility to combine the given properties of f-and s-phases by the well-known vdWconcept of one-fluid approximation.It is most usable at the description of binary 12 mol] but the constant vdW-estimate of the cohesion coefficient 0 474 a  [J dm 3 /mol 2 ] seems to be rather crude for our aims.The more appropriate T-dependent estimate of   0 aT can be obtained for the low-temperature variant of FT-EOS (56) at the I-phase transition constraints: here the very interesting feature of the proposed f  -estimates.It is coherent, of course, with the concept of 0-phase transition, which implies the interaction of coexistent f-phases.Such FT-correlation expressed in terms of the non-dimensional f Z -and f A -factors by Eq.(63) cannot be described by the conventional WMG-theory of the I-phase transition.The latter adopts, at best, the thermodynamic reversibility of gl  and lg  VLE-transient phenomena followingfrom the unified EOS'-concept and leading to the common f-coefficients of the type those from Eq.(62).slopes in the derived here Eq.(63) corresponds, namely, to the 0phase transition from Eq.(45) but not to the I-phase transition from should be strictly fulfilled).The fundamental distinction of the congruent CVLdiagram[5,6,[18][19][20][21] from the traditional VLE-diagram requires the account for irreversibility indicated by the difference in the direct ( ) two-phase changes described by the caloric EOSs from Eq.(11,12).

FIG. 1
FIG.1(a) -CXC of water[33] as an example of the strongly curvilinear diameter.The formal application of Zeno-line's methodology may lead, in principle, to the serious errors at the prediction of critical parameters   c c c T ,P 

Fig. 2 -
Fig. 2 -Volume fraction of liquid water   l T  s T -ratio at the triple point corresponds to the slope of reference isoenthalpy ma hq  in the   T, - plane at the construction of standard   h, -diagram.

.
Fig. 3 -FT-calculated caloric functions of discontinuities in the enthalpy and entropy compared with the classical WMG-estimates in PMs.
are the standard heat capacities.Hence, to construct the thermodynamic description of medium one should also know both f-and stypes of EOS for any non-Gibbsian (i.e.fractal by its nature) complex phase.It is composed by the fractal solid (s) matrix of PM and by the fractal fluid (liquid and/or gas) flows moving inside of it.FT-model imitates both ones by the molecular-based concept of an excluded volume introduced long ago by van der Waals.
Bothones should be dependent on the discrete by nature variables (i.e. the numbers of particles and free electric charges).They have to be determined as the complementary parameters of a thermodynamic medium with the following exceptional role.They should compensate and relax any external irreversible changes of two main thermodynamic contributions arising due to the forces    (see Eqs. (44-46)) leads to the unambiguous conclusion.It concerns the so-called internal (energy) pressure term   T e / v   from Eq.(48).
at the application to PMs in the low-temperature range of water (the main component of a moisture air) located between its triple g Z  ) which becomes

Table 1 -
[33]input CXC-data[33](P s , Z l , Z g ) and FT-predicted (A l , A v , ε l P hl )-parameters (see text) for the low-temperature range [T t , T b ] of water.