Assessment of effectiveness factor in porous catalysts under non-symmetric external conditions of concentration

3.1 System settings

The model settings of the catalytic layer are the same as those reported in ref (Caravella et al. 2012), whose sketch is recalled in Figure 2 along with the relative boundary conditions. Computational meshing, simulations and post-processing analysis were performed in the environment of the commercial software ANSYS CFX (CFX-Pre, CFX-Solver and CFD-Post), after designing the actual physical models in AutoCAD^®.

Figure 2:

Sketch of the catalytic layer investigated (Figure 1d, e) along with the corresponding boundary conditions for the two calculation approaches considered in this paper: (a) Heterogeneous approach and (b) pseudo-homogeneous one.

As for the effectiveness factor η _Int, its definition is reported in Eq. (3) (see, for example (Aris 1969; Froment and Bishoff 1979; Fogler 2011):

where V _Part is the particle volume, R _A is the reaction rate, C _A,Int is the (variable) concentration of the limiting reactant inside particle and C _A,Surf is the concentration of the limiting reactant over the catalyst external surface.

Since an ensemble of particles with a reactive stream flowing along the axial direction is considered, the real conditions inside particles could be quite far from the ideal case of concentration spherical symmetry. This implies that the concentration – and, thus, the reaction rate – is generally non-uniform over the particle surface. Based on this fact, the quantity C _A,Surf in Eq. 3 is calculated here as the average value of concentration over the surface (Eq. 4):

Observing Eqs. 3 and 4, it can be noted that the particle volume and the surface considered for the calculation of the averages can be chosen in several ways. In this investigation, two different approaches are used. As for the former, which will be referred to as “Sphere-by-Sphere” (SbS), the definition of the internal effectiveness factor η _Int (Eq. 3) is applied by calculating the average values of Eqs. 3 and 4 over the volume and the surface of each entire particle, as indicated in Figure 3a.

Figure 3:

Sketch of the locations chosen for calculation of the internal effectiveness factor η _Int using the “Sphere-by-Sphere” approach (a), and the “Equisized-Volume” approach (b).

Differently, for the latter approach – here referred to as “Equisized-Volume” (EV) –, volume and surface are chosen to belong to two consecutive half particles (Figure 3b). The term “equisized” indicates that all volumes along the ensemble have the same length. Obviously, according to this choice, surface areas and volumes in the two methods are equivalent. The only difference is that they are shifted by the distance of a layer.

The choice for the EV approach is made basing on the consideration that, since the concentration profile due to reaction mostly develops along the axial direction, the transport phenomena of interest could be better caught within locations whose boundaries are normal to the axis. Let us remark that, in the ideal case – i.e., with spherical symmetry valid for both geometry and boundary conditions – the following expression of internal effectiveness factor holds for first-order kinetics (Eq. 5) (Aris 1969; Fogler 2011; Froment and Bishoff 1979):

where the subscript “Th” indicates “Theoretical” and the parameter ϕ is the Thiele modulus generalized with respect to geometry, defined in Eq. 6 (Aris 1969; Fogler 2011; Froment and Bishoff 1979), where r _Part is the particle radius, k _c,1 is the first-order kinetic constant expressed in s⁻¹ and D _A,Eff is the effective diffusivity inside particle. For the purpose of this study and without loss of generality, the diffusivity inside particle is considered to be of Knudsen-type.

In principle, there are no theoretical indications establishing that Eq. 5 can be applied also to particles where the concentration boundary conditions at the surface are non-symmetric. Therefore, a possible indication of conditions under which CFD results and analytical solution are equivalent represents new information that is very useful to simplify the simulation of complex systems of particles without loss of precision due to the non-uniformity of boundary conditions around the particles. This issue will be addressed in Section 4.2.

3.2 Simulation settings for the pseudo-homogeneous approximation

In parallel with the analysis of the intra-particle mass transport limitations, simulation should also indicate a way of simplifying the original system in order to achieve the same results in a simpler and faster way without loss of significant information. This is important, for example, in a scale-up procedure aiming at reproducing the behaviour of the original system in the frame of a more complex and multi-zone structure.

Concerning this specific case, it is not computationally feasible to consider the particle system with its heterogeneity in simulations of the overall catalytic converter. To do that, a pseudo-homogeneous approach is needed. However, the more complex the original system, the more strict the conditions under which such an approach is effectively usable can be.

Therefore, the choice of the average transport and kinetic parameters by means of which original system (heterogeneous) and pseudo-homogeneous one have to be matched can be quite hard and, in some case, simply not possible unless accepting a significant discrepancy.

With these remarks, the same simulation settings as those used in Caravella et al. (2012) are chosen here to model the particle ensembles by a pseudo-homogeneous approach, which we anticipate have shown to be effective in providing a satisfactory match.

4.1 Profile analysis

In Figure 4, the local decane concentration profile in each particle is visualized. In this type of plot, our interest is to show the concentration distribution inside each particle and, thus, a local scale is used for each particle. The zones inside particles with a lower and higher concentration are respectively indicated by the blue and red colour, which a slower and a faster reaction rate respectively correspond to.

Figure 4:

Visualization of the symmetry degree inside each particle in terms of molar concentration profiles of decane for tortuosity values of (a) 3 and (b) 10. T = 260 °C, d _Pore = 5 nm.

As depicted in the plots, as the reaction rate increases with respect to the Knudsen diffusion, the concentration inside particles tends to become lower and its profiles more similar to have spherical symmetry. However, this is not true for the superficial molar concentration, which is significantly non-uniform.

As for the influence of particle tortuosity, it affects the diffusion coefficient: the higher the tortuosity, the lower the diffusion coefficient inside particle. At higher tortuosity, it is more difficult for the reactant to penetrate inside particle and, thus, the reaction tends to occur closer to the external surface.

However, in this type of plots, it is not possible to understand in what conditions the concentration symmetry inside particles is favoured, as we observe a substantial transverse concentration gradient along the axis of the considered particle ensemble. In practise, we would like to evaluate the effectiveness factor of the entire ensemble of particles, which will be assessed by evaluating the effectiveness factor of each particle of the ensemble. Nevertheless, we have a non-uniform concentration profile along the ensemble and, thus, a question arises concerning the way of evaluating the effectiveness factor of particles in our system, to which it is not possible to say a priori whether the analytical expression can be applied or not. This issue is justly dealt with in the incoming sections.

4.2 Internal effectiveness factor of particles

As shown previously, the concentration profiles inside and outside the porous catalytic particles of the here-considered ensemble are not provided with spherical symmetry. Actually, generally speaking, it is very difficult for real spherical particles to work under conditions of perfectly symmetry. Nevertheless, the catalyst effectiveness factors for spherical particles are usually calculated using the theory that considers a perfect symmetry (Alvarez-Ramirez, Ochoa-Tapia, and Valdes-Parada 2005; Burghardt and Kubaczka 1996; Mariani et al. 2009; Smith, Zahradnik, and Carberry 1975).

The goal of this section is to check if and to what extent this can be done for systems for which spherical symmetry does not hold. To do that, the internal effectiveness factor will be evaluated by CFD simulations, comparing the obtained values to those corresponding to the analytical expression (Eq. 5).

As explained in Section 3.1, in the present paper two different methods are used to evaluate the effectiveness factor inside particles η _Int, hare-named as “Sphere-by-Sphere” (SbS) and “Equisized-Volume” (EV) method, respectively.

Figure 5 shows the obtained results in terms of internal effectiveness factor profiles (indicated by the respective symbols) evaluated by means of the SbS approach for a d_Pore  = 5 nm at several temperature values and two particle tortuosity factors. The continuous lines are the values corresponding to the theory valid for spherical symmetry. It has to be remarked that, in principle, each particle has different value of η _Int, and, thus, a profile is expected, whose abscissas are chosen to be located in correspondence of the centre of each particle.

Figure 5:

Profiles of the internal effectiveness factor at several temperatures for different values of tortuosity calculated with the “sphere-by-sphere” (SbS) method for tortuosity values of (a) 3 and (b) 10. d _Pore = 5 nm.

Moreover, it is necessary to indicate that the first particle is excluded from the profile evaluation, because its “special” role of particle with the highest concentration set as boundary condition does not allow a fair comparison with the other particles.

Examining the obtained results, the first thing to note is that the calculated values are practically constant at all temperatures and tortuosities considered, the differences among each other being negligible with respect to their values (<0.1%). Actually, such a similarity is quite surprising, especially if considering that the concentration conditions along the ensemble change more or less rapidly with temperature. Especially at 300 °C, where the reaction rate is considerably faster than at lower temperature values, it could be expectable a more significant scattering of calculated values. Nevertheless, no significant irregularity is observed.

As for τ _Part = 10, all values of internal effectiveness factor are lower than those obtained for τ _Part = 3 at the same temperature. This happens because of a larger relative difference between reaction and diffusion rate inside particles.

In fact, since Knudsen diffusion – and, thus, flux – decreases with tortuosity, less amount of reacting species passes through particles, preferring passing by. This causes a higher concentration on the surface, whose consequence is a higher reaction rate. Therefore, a smaller zone inside the catalyst particles is exploited for reaction, causing the internal effectiveness factor to be lower.

As a further remarkable result, it was found that, except for the case at 300 °C and τ _Part = 10, where a discrepancy of 4.5% is observed, the internal effectiveness factor is almost coincident with that calculated from the theory exploiting symmetry. Actually, this result is quite surprising, too, because there is no apparent reason to expect that the values should be the same.

In fact, if looking back at Figure 4, the concentration is not uniform over the particle surface and there is no spherical symmetry inside it. The only symmetry that holds for particles is the one established with respect to the plane passing for the ensemble axis and normal to the open boundary.

Nevertheless, the overall result of integration provides a substantial equivalence with the analytical solution. This occurs even in zones of ensemble where decane concentration and reaction rate are extremely low. If the internal effectiveness factor is calculated according to the second method (EV), the results of Figure 6 are obtained.

Figure 6:

Profiles of the internal effectiveness factor at several temperatures for different values of tortuosity calculated with the “Equisized-Volume” (EV) method for tortuosity values of (a) 3 and (b) 10. d _Pore = 5 nm.

As shown in figure, the EV method provides results almost equivalent to those from the theory in spherical symmetry even in the case at 300 °C and τ _Part = 10 (about 1.0% of discrepancy). This indicates that the differences between the SbS and EV methods could be not neglected when reaction is significantly faster than Knudsen diffusion.

To check these expectations and provide a more general view of the situation, the internal effectiveness factor (averaged along the ensemble profiles) is shown in Figure 7 as a function of the Thiele modulus (ϕ), which includes the dependence of η _Int on the different factors investigated in the present paper (i.e., temperature, particle tortuosity and mean pore diameter).

Figure 7:

Average internal effectiveness factor calculated from theory (continuous line) and for both SbS and EV method (symbols) as a function of Thiele modulus. The labels indicate the percent difference of each method from the theory in spherical symmetry.

In the figure, the values calculated by means of SbS and EV method, respectively, are shown and compared with the analytical solution (“Theory” in the plot) obtained in spherical symmetry. The labels refer to the percent difference between methods and theory.

As for the EV method, it is possible to see that the value of effectiveness factor is very close to the theoretical ones in symmetric conditions within the whole range of Thiele modulus investigated, with a maximum discrepancy of 2.1% evaluated at the highest ϕ.

The situation is different for the SbS method. In fact, below a ϕ of about 1.3, the three values (SbS, EV and Theory) are still almost coincident, with a maximum discrepancy below 1%. However, at a higher ϕ, the difference between SbS, EV and Theory becomes much more significant, reaching maximum values of 21% (SbS) and 9.7% (EV) at a ϕ of about 5.4.

An explanation for the reason why the results of the EV method are much closer to the ideal case (spherical symmetry) than those of the SbS one can be found analysing the difference between the geometries considered for calculation. When internal effectiveness factor is calculated by SbS method, the axial range involved in the evaluation is equal to the particle diameter (1 μm). However, when EV method is used, the axial range is reduced to about 0.74 μm, which is the length of each equisized volume.

Therefore, under the same conditions, the variation of concentration along the ensemble axial direction is found to be lower in the equisized volume than in the entire sphere. In other words, the concentration is “more uniform” in the second case. This difference is not evident when reaction rate is slower, because the concentration gradient is lower as well.

For the same reason, the particle volume considered for the SbS method has a concentration meanly lower than the particle volume considered for EV. Consequently, the internal effectiveness factor calculated by SbS is always lower than that calculated by the EV one under the same conditions. It is also interesting that the theoretical behaviour of the symmetric system is comprised between those of the two methods proposed.

Another important result is that, although the geometries chosen for the EV and SbS methods are different from each other, the internal effectiveness factor does not depend on the particular geometry chosen for the calculation at a low Thiele modulus (<1.3). Consequently, there is a strong indication that under these conditions Eq. 5 is applicable to any other structure (SC, BCC, random, etc). Differently, at a higher Thiele modulus (>2 ca.), these considerations cannot be generally confirmed, because the differences between EV, Theory and SbS are significant. Therefore, in this last case, the most appropriate calculation method for the effectiveness factor has to be chosen and checked carefully.

Concerning this aspect, until this point we demonstrated the existence of operating ranges where the particular calculation method chosen (Theory, EV and SbS) affect the value of the internal effectiveness factor and no indications on which “the best method” could be have been provided.

In order to provide an adequate answer to such a pending question, let us recall that the internal effectiveness factor is useful because it allows the complexity of heterogeneous catalytic particles to be by-passed by means of a homogeneous approach instead. Therefore, the choice of the appropriate effectiveness factor has to be done by using it in the pseudo-homogeneous approach and checking the difference of the obtained results from those of the heterogeneous approach. This issue will be dealt with in next section.

4.3 Pseudo-homogeneous approximation

This section is focused on the feasibility of using a pseudo-homogeneous approach that could effectively match the heterogeneous results. The simulation settings used for this analysis are those described in Section 3.2.

The comparison between the two approaches is made considering the molar concentration profile of the key-reactant (decane) for all conditions investigated (temperatures, particle tortuosity and pore diameter). This variable is chosen because it is directly proportional to the reaction rate at a fixed temperature. Therefore, by looking at this variable, it is possible to visualize the qualitative behaviour of reaction, too.

Moreover, it has to be specified that, in all comparisons, the concentration values for the heterogeneous cases are calculated as the averages over several cross sections normal to the ensemble axis including the fluid domain only, i.e., excluding the porous particles. This choice is made to make the comparison fair, because the “infinitesimal” control volume in the homogeneous case is supposed to be sufficiently larger than particles and the corresponding solution regards just the concentration in the fluid.

In this situation, the profiles inside particles are indirectly taken into account by the internal effectiveness factor only, and no further information about them can be obtained from simulations in homogeneous mode.

The first data set introduced regards conditions corresponding to low values of Thiele modulus (<0.23, Figure 8a–c), evaluated at 260 °C and τ _Part = 3 by changing the particle mean pore diameter from 5 to 50 nm. This figure is divided into three sub-plots to visualize some details of heterogeneous (Figure 8b) and homogeneous (Figure 8c) concentration profiles, which cannot be analysed in the full scale figure because of being very close to each other. The profiles in homogeneous are obtained using the internal effectiveness factor calculated by means of the SbS method.

Figure 8:

Comparison between heterogeneous (symbols with lines) and pseudo-homogeneous (continuous line) approach in terms of decane molar concentration profiles at 260 °C for a particle tortuosity of 3 (a) full-scale profiles; (b) detail of heterogeneous profiles; (c) detail of homogeneous profiles.

However, under the considered operating conditions, this choice is completely arbitrary, since all the three methods provide the same values according to the previously described Figure 7, the situation being different for a higher Thiele modulus, as will be shown later.

Considering Figure 8a, because of the low ϕ considered, the reaction is very slow and, thus, the sensitivity of the profiles on the mean pore diameter is so low that they can be hardly distinguished. Nevertheless an interesting and peculiar behaviour can be observed for a mean pore diameter of 50 nm (ϕ = 0.07).

In fact, the profile in homogeneous (Figure 8c) is lower than that at 20 nm, whereas the profile in heterogeneous (Figure 8b) is above that at 5 nm. In homogeneous case, if the mean pore diameter of particle is augmented, the only effect that is observed is the increase of Knudsen diffusivity, with a consequent larger amount of reactant consumed by reaction (lower profiles). This occurs because the pseudo-homogeneous approach is based on the spherical symmetry for particles.

However, in the heterogeneous approach, another effect is taken into account: the net mass flux of reactant passing through the particles is not null, because there is no constraint of the symmetric boundary conditions imposed on the surface. Therefore, if the Knudsen diffusion is sufficiently fast and, at the same time, the reaction is slow enough, less amount of reactant is consumed by reaction, with a consequent increase of its profile.

This is confirmed by the plot at 300 °C (Figure 9), where both heterogeneous and homogeneous profiles show the same trend. In fact, under these conditions, the reaction rate is much higher than that at 260 °C and, thus, the effect of the higher diffusion flux due to the increased diameter is completely balanced and overcome by the reaction velocity, causing the heterogeneous profile for 50 nm to be below that for 20 nm, as in homogeneous case.

Figure 9:

Comparison between heterogeneous (symbols with lines) and pseudo-homogeneous (continuous line) approach in terms of decane molar concentration profiles at 300 °C and τ _Part = 3. The data related to a d _pore of 20 nm are comprised between 5 and 50 nm, very close to the latter.

The difference between the two situations can be observed better in Figure 10, where the normalised amount of decane consumed by reaction defined in Eq. 7 – actually a kind of conversion – is plotted as a function of the mean pore diameter under the same conditions as those of Figures 8 and 9.

Figure 10:

Comparison between heterogeneous (“Het.”, empty symbols) and pseudo-homogeneous (“Hom.”, filled symbols) approach in terms of normalized decane molar concentration drop (defined in Eq. 7) at 260 and 300 °C for a particle tortuosity of 3.

Although such a difference is very small, nevertheless an important information arises from this figure. In fact, it is indicated that, under certain operating conditions, the residence time of the reactants inside catalyst can be so short that the performance of a single catalytic layer can decrease, with the overall effect for the entire catalytic converter of a productivity decrease.

This can be avoided by increasing the reaction rate, which means acting on operating conditions like temperature, or on geometrical factors like pore size and/or the catalyst distribution inside the catalytic layer. However, as anticipated before, the situation is quite different for higher values of Thiele modulus.

To investigate this range of conditions, the concentration profiles of decane are shown in Figure 11 (left-hand side) together with the profiles of the difference from the heterogeneous results (right-hand side) for increasing ϕ up to a value of 5.4. Plotting such differences is necessary in order to highlight discrepancies that otherwise would remain hidden because of the large scale used in the profile plots. Simulations in homogeneous mode are carried out by considering the three different methods used to evaluate the internal effectiveness factor of particles described above, i.e., EV, Theory and SbS. From comparing these data, we will be able to:

1. decide if and to what extent the pseudo-homogeneous approach is effective to replace the heterogeneous one with an acceptable approximation;

2. check which the best calculation method for the effectiveness factor is.

Figure 11:

Comparison between heterogeneous (symbols) and pseudo-homogeneous (continuous lines) approach in terms of decane molar concentration profiles – left-hand side, a (1) ϕ = 1.32, b (1) ϕ = 2.96 and c (1) ϕ = 5.40 – and in terms of percent difference from the data calculated in heterogeneous case – right-hand side, a 2) ϕ = 1.32, b (2) ϕ = 2.96 and c (2) ϕ = 5.40) – at different values of Thiele modulus.

Examining the obtained results for a ϕ of 1.32 (Figure 11a.1), it is possible to observe that the differences between heterogeneous and homogeneous solutions are very small. Furthermore, the different homogeneous profiles are so close to each other that they cannot be distinguished within the whole range of the ensemble positions.

The quantification of this behaviour is provided in Figure 11a.2, where it is shown that all the three calculation methods generate concentration profiles with an error below 1.6%. In particular, the SbS method provides a maximum discrepancy (0.2%) significantly smaller than the other two. However, no clear conclusions can be drawn from this, because such differences are too small.

For this reason, also in force of the results previously shown in Figure 7 – reporting a progressive increase of the difference among the internal effectiveness factors with the Thiele modulus – the sensitivity of profiles on ϕ has been checked in Figure 11b.1, b.2, c.1 and c.2 in order to establish the most suitable calculation method. As a result, the three homogeneous profiles become more distinguishable for a progressively higher ϕ, allowing us to recognize that the SbS method provides in all cases concentration profiles much closer to the heterogeneous ones.

Even for a ϕ of about 3 (Figure 11b) – where profiles almost overlap – the maximum errors generated by the EV and Theory method are significant (11.0 and 8.6%, respectively). For the highest ϕ considered, this fact is put in evidence much more, with the errors provided by the EV and Theory method reaching 19 and 26%, respectively.

Therefore, in force of these results, we can say that, generally speaking, the best method to calculate the internal effectiveness factor for a high Thiele modulus is the SbS one, whereas the theoretical expression obtained by considering spherical symmetry can be applied only for a low Thiele modulus.

Once the best method has been identified, the question concerning the practical calculation of the internal effectiveness factor arises. In fact, the pseudo-homogeneous approach would become useless if every time a heterogeneous system should be used to evaluate the internal effectiveness factor. For this reason, we want to propose an alternative calculation method, starting from the results of Figure 7. Such a method is explained as follows.

First, let us note that the shape of the behaviour of the internal effectiveness factor calculated by the SbS method – henceforth referred to as η _Int,SbS  – reported in Figure 7 is qualitatively the same as that of the internal effectiveness factor calculated by the analytical expression (η _Int,Th). Actually, we can say that the difference between the two profiles consists in a horizontal gap that tends to be zero for a small ϕ, progressively increasing for a higher ϕ.

From these considerations, our idea is to keep using the analytical expression represented by Eq. 5, correcting, however, the Thiele modulus in order to match the results obtained for η _Int,SbS . In other words, we want to calculate η_Int,SbS by using the analytical expression of the theory, which is useful because of being simple to implement in simulation software. The procedure to do that is briefly described in Figure 12. For each ϕ, the corresponding value of η _Int,SbS coming from simulations in heterogeneous is put into Eq. 8, which is solved to find the corresponding value of the modified Thiele modulus ϕ _Mod.

Figure 12:

Qualitative scheme representing the correction procedure by which to calculate the modified Thiele modulus ϕ _Mod from the original one ϕ.

By performing this procedure for all values of η _Int,SbS , data of ϕ _Mod versus ϕ is obtained. Then, a fitting of this data set is performed with the following empirical expression (Eq. 9):

This type of polynomial form is chosen not only for its simplicity, but also because the simulation data indicate that, for a low ϕ, the values of ϕ and ϕ _Mod tend to have gradually the same behaviour. This implies an expression passing through the origin and a derivative in the origin itself equal to the unity.

Moreover, the absence of the square term in Eq. 9 is because this expression with just one parameter is fully able to describe the discrepancy between the two Thiele moduli, as shown in Figure 13 and reported in Table 2.

Figure 13:

Fitting of the modified Thiele modulus ϕ _Mod (Eq. 8) versus the theoretical one, ϕ (Eq. 6).

By looking at the value of the fitting parameter a, it is possible to see that its value is very close to 0.01. Therefore, without significant error, we suggest the following final expression to be used for the calculation of the modified Thiele modulus, which is valid for values of Thiele modulus lower than 5.4 (Eq. 10):

Obviously, because of the empirical nature of Eq. 10, extrapolations at values of Thiele modulus higher than those considered in the present paper has to be carried out very carefully in order to avoid significant errors in pseudo-homogeneous simulations.

However, Eq. 10 can be easily used in simulation software to correct the Thiele modulus within the range of values investigated, calculating the internal effectiveness factor in conditions of non-spherical symmetry. This allows users to avoid carrying out simulations in heterogeneous mode by performing, on the other hand, simulations in simpler pseudo-homogeneous mode without significant errors, which matches the target of the present paper.

List of symbols

C

molar concentration [mol m⁻³]

D

diffusion coefficient [m² s⁻¹]

d _Part

particle diameter [m]

d _Pore

mean pore diameter of particles [m]

J

diffusive molar flux [mol m⁻² s⁻¹]

k _c

kinetic constant [m³ s⁻¹ kg⁻¹]

k _c,1

kinetic constant [s⁻¹]

M

molar mass [kg mol⁻¹]

P

absolute pressure [Pa]

r

generic radial position from the center of the porous particles [m]

r _Part

radius of particle [m]

S _P

external surface area of a porous particles [m²]

R _i

volumetric reaction rate of the generic i species [mol m⁻³ s⁻¹]

T

temperature [K]

V _Part

particle volume [m³]

x

molar fraction [−]

Greek Symbols

δ

inter-particle distance [m]

ε

porosity [−]

η

effectiveness factor [−]

φ

thiele modulus [−]

ν

stoichiometric coefficient [−]

ρ

density [kg m⁻³]

τ

tortuosity [−]

Subscripts

A

key-reacting species (decane)

Eff

effective (for diffusivity)

Ens

(particle) ensemble (considering dense particles)

Het

heterogeneous

Hom

(pseudo) homogeneous

i,j

generic species in the diffusivity definition

Int

internal (for effectiveness factor, inside the porous particles)

Part

particle (for density, tortuosity and porosity)

Surf

surface (of the particles)

T

total (for molar concentration)

Superscripts

Kn

Knudsen (for flux and diffusivity)

MS

Maxwell-Stefan (for free diffusivity)

Acronyms

EV

equisized-volume (approach)

SbS

sphere-by-sphere (approach)