Pore volume fraction modeling

In the model described in this work every effort has been made to ensure that it embodies physically meaningful parameters. It is inevitcible, however, that some simplistic idealizations of the physical processes involved in GPC must be made in order to arrive at a system of equations which lends itself to mathematical solution. The parameters considered are, the axial dispersion, interstitial volume fraction, flow rate, gel particle size, column length, intra-particle diffusivity, accessible pore volume fraction and mass transfer between the bulk solution eind the gel particles. 

FIG U RE 4.10 Stability diagram showing the stable state of operation of the CCL as a function of the pore volume fraction of secondary pores, Xm, and jq. (Reprinted from Electmchim. Acta, 53(13), Liu, J., and Eikerling, M. Model of cathode catalyst layers for polymer electrolyte fuel cells The role of porous structure and water accumulation. 4435 1446. Copyright (2008), Elsevier. With permission.)... 

In the classical model of the size exclusion mechanism this difference stands for the effective pore volume of the separating model. Any elution of samples or fractions outside this interval always means a perturbation by a different mechanism. Such conditions have to be avoided. It is not possible to expand this elution difference A significantly for a given column. For this reason, GPC column sets are considerably longer than LG columns for other mechanisms. 

Numerous models have been proposed to interpret pore diffusion through polymer networks. The most successful and most widely used model has been that of Yasuda and coworkers [191,192], This theory has its roots in the free volume theory of Cohen and Turnbull [193] for the diffusion of hard spheres in a liquid. According to Yasuda and coworkers, the diffusion coefficient is proportional to exp(-Vj/Vf), where Vs is the characteristic volume of the solute and Vf is the free volume within the gel. Since Vf is assumed to be linearly related to the volume fraction of solvent inside the gel, the following expression is derived ... 

This model, when applied to Nation as a function of water content, indicated a so-called quasi-percolation effect, which was verified by electrical impedance measurements. Quasi-percolation refers to the fact that the percolation threshold calculated using the single bond effective medium approximation (namely, x = 0.58, or 58% blue pore content) is quite larger than that issuing from a more accurate computer simulation. This number does not compare well with the threshold volume fraction calculated by Barkely and Meakin using their percolation approach, namely 0.10, which is less than the value for... 

Water Transport. The dependence of the diffusive permeability of tritiated water Pq., (HTO) and hydraulic permeability LpAX on the lEC are presented in Figures 4 and 5. Both Pj and LpAX can be seen to increase exponentially as a function of lEC. Since the volume fraction of water, (J>, is also a linear function of the lEC, a similar exponential relation is obtained for these parameters vs. (J>. In terms of the pore model, the increase in either diffusion permeability or the hydraulic permeability may be caused by one or both of the following possibilities (a) an increase in the number of passageways, or (b) by increase in the radius of the pores. This question may be resolved by examining the g factor, defined as a ratio of two permeabilities (15) ... 

The coefficient of proportionality K is called the permeability of the reinforcement. According to theory [5] K is only dependent on the geometry between the fibers in the reinforcement (the pore space ). Several models for the dependence of K on the fiber volume fraction Vf has been proposed. The most-cited model is the so-called Kozeny-Carman model [16,17], which predicts a quadratic dependence on the fiber radius R in addition to the dependence on Vf... 

The model can be made somewhat more precise by considering the silica skeleton as a loosely packed system of spheres of equal radii that is derived from a close packing by leaving out a fraction 1 — 0 of the possible sites. If the sphere radius is Rs (A.), it can easily be shown, assuming complete accessibility of all surfaces, that the pore volume F/ is... 

S0l., So2 and SHlo refer to the respective source terms owing to the ORR, e is the electrolyte phase potential, cGl is the oxygen concentration and cHlo is the water vapor concentration, Ke is the proton conductivity duly modified w.r.t. to the actual electrolyte volume fraction, Dsa is the oxygen diffusivity and is the vapor diffusivity. The details about the DNS model for pore-scale description of species and charge transport in the CL microstructure along with its capability of discerning the compositional influence on the CL performance as well as local overpotential and reaction current distributions are furnished in our work.25 27,67... 

Figure 5. Comparison of the experimental results with the new pore model. A) Ethanol concentrations in volume fractions between 0 and 1 in the receiver compartment and saline in the donor compartment. B) Ethanol concentrations on donor and receiver side. C) Ethanol concentrations in volume fraction between 0 and 1 in the donor compartment and saline in the receiver. Lines depict calculated permeabilities, employing Equation 6. Points are experimental permeabilities with respective standard deviations.

Rieckmann and Keil (1997) introduced a model of a 3D network of interconnected cylindrical pores with predefined distribution of pore radii and connectivity and with a volume fraction of pores equal to the porosity. The pore size distribution can be estimated from experimental characteristics obtained, e.g., from nitrogen sorption or mercury porosimetry measurements. Local heterogeneities, e.g., spatial variation in the mean pore size, or the non-uniform distribution of catalytic active centers may be taken into account in pore-network models. In each individual pore of a cylindrical or general shape, the spatially ID reaction-transport model is formulated, and the continuity equations are formulated at the nodes (i.e., connections of cylindrical capillaries) of the pore space. The transport in each individual pore is governed by the Max-well-Stefan multicomponent diffusion and convection model. Any common type of reaction kinetics taking place at the pore wall can be implemented. 

Another approach is to scale the A-term contribution to a so-called effective particle diameter as has been proposed by Vallano and Remcho [19], They defined an effective particle diameter from the perfusive EOF velocity within each volume fraction of the pore size distribution of the particles. Their model allows inclusion of both the intraparticle EOF and a pore size distribution. However, both this approach as well as Eq. (10) are more or less empirical. 

Equation 1 is applicable (i.e., no pore shape assumption) for pores with radius greater than about 5 nm. However, the model has been extended to pores as small as 0.5 nm [16] by assuming a pore shape. In addition, the fraction of pore volume with pore sizes less than 0.5 nm may be obtained (assuming that the concept of pore size in this size range has physical significance) although distribution information in that region can not be determined. 

The content of non-evaporable water, relative to that in a fully hydrated paste of the same cement, was used as a measure of the degree of hydration. Portland cement paste takes up additional water during wet curing, so that its total water content in a saturated, surface dry condition exceeds the initial w/c ratio. Evidence from water vapour sorption isotherms indicated that the properties of the hydration product that were treated by the model were substantially independent of w/c and degree of hydration, and only slightly dependent on the characteristics of the individual cement. The hydration product was thus considered to have a fixed content of non-evaporable water and a fixed volume fraction, around 0.28, of gel pores. 

Parrott and co-workers (P30,P32,P35,P33) described a more sophisticated method for modelling the hydration process. The fraction of the total water porosity that was below 4nm was calculated by multiplying the volume fraction of C-S- H by an appropriate factor, which depended on whether the C-S-H was formed from alite or belite, the temperature and the amount of space available. The constants assumed were based on experimental data obtained using a procedure based on methanol sorption (Section 8.3.4). The effect of drying was allowed for (P35) by introducing a factor of 0.7 - -1.2(RH — 0.5) for 0.5 < RH < 1, or of 0.7 for RH 0.5. These refinements allow some deviation from the Powers-Brownyard postulate of a fixed volume ratio of gel porosity to product. Typical results for the volume fractions of pores larger than 4 nm in mature pastes of a cement with an alite content of 56% were approximately 0.26, 0.16 and 0.07 for w/c ratios of 0.65, 0.50 and 0.35, respectively (P32). For the two higher w/c ratios, these results are near the capillary porosities of Powers and Brownyard, but for w/c 0.35 the latter value is zero. 

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