Drying kinetics as Tool for the Assessment of Dynamic Porosity of Catalyst-Support Materials

Critical drying fraction data (xc) were obtained for a series of porous materials, frequently used as supports for heterogeneous catalysts. This kinetic parameter correlates well with the secondorder rate constants for water diffusion in the pores (p < 0,01). It is concluded that xc is by itself a sufficient parameter for comparison of mass-transfer resistance between different candidate supports.


Introduction
Catalysis implies changes in rates of reaction that seek kinetic optimisation to produce some desired product, or the minimisation of side reactions. Kinetic acceleration is the result of changing the molecular environment where the reaction occurs, and transition-state complexes are stabilised relative to cases where catalysts are absent.
Heterogeneous catalysis is of fundamental importance in the transformation of raw materials into useful products such as fuels, either derived from fossil feedstocks or biofuels from renewable sources. Heterogeneous catalysts are constituted by the catalyst itself, a support that offers mechanical and thermal stability and sometimes promoters are also included, that is, substances that improve the kinetic properties of the whole catalytic material.
Besides mechanical and thermal stability, support materials for heterogeneous catalysts must allow convenient masstransfer dynamics for both entering reactant molecules and exiting product molecules. The geometrical features of pores in supports strongly affect the selectivity of chemical reactions, a phenomenon known as "shape selectivity" (Chen, et al. 2012, Smith & Maesen, 2016. Thus, supports must have high specific surfaces, high porosity (fraction of total volume that equals empty space), low constrictivity (ratio of mean diameter of flowing molecules to mean diameter of pores), and low tortuosity (average lengths of pores relative to macroscopic geometrical dimension of solid sample Theoretical considerations on porosity and tortuosity have been put forward by Matyka and collaborators (Matyka, et al. 2008). There is no general relationship between porosity and tortuosity, but many trial relations have been proposed (Duda, Koza & Matyka, 2011) of various forms such as exponential, logarithmic, linear or polynomial. Nevertheless, porosity and tortuosity are intuitively interdependent variables. This issue makes the overall dynamic porosity concept a more easily handled quantity, for the relative comparison of the mass-transfer features of catalyst-support materials and other porous materials.
Besides heterogeneous catalysis, dynamic porosity is a convenient parameter for the relative assessment of processes such as combustion, drying, fluid storage in porous materials, retention of dangerous spills, permeability of soils and construction materials, or any case where it is necessary to assess transport processes in different porous media.
This work gives dynamic porosity data for the drying operation of different inorganic porous materials, some commonly used as supports in heterogeneous catalysis.

Statement of the Problem
Isothermal drying curves offer useful information on the dynamics of diffusion of fluids within the confined space of pores in solid materials.
Macro as well as microscopic details during drying processes is obtained from so-called Krischer curves (Kemp, et al., 2001) at specific temperature and barometric pressure. If x is the extent of the drying process that has occurred at time t, the graphical representation of dx/dt vs. (1-x) is called a Krischer curve.
Such plots show an initial constant drying rate, such as shown in figure 1 for the case of Molecular sieve 5A at 50 °C and 87 kPa. A critical point is reached (xc), afterwards dx/dt decreases as the process proceeds. During the constant-rate period, the solid particles surfaces are entirely covered by molecules of the soaking liquid and they behave analogously to surface molecules in pure liquids.  The rate of evaporation of liquids against a stagnant gaseous environment is given by the Hertz-Knudsen equation (Rahimi & Ward, 2005): where pv is the vapour pressure of the liquid at temperature T, and M is molecular mass. A correction factor κ is included, that gives the fraction of liquid molecules that remain in the surrounding gas phase. Zeroth-order kinetics is obeyed as long as the rate of evaporation counterbalances the rate of arrival of liquid at the surface.
When xc is reached, dx/dt decreases steadily. This experimental fact indicates that kinetics is now determined by factors other than the intrinsic volatility of the soaking liquid (pv), such as internal transport limitations. More complex kinetics is observed at very high drying extents.
Microscopic approaches to the kinetics of drying have been proposed based on capillary transport in porous matrices (Coussot, 2000 Disconnection of particle superficial water and internal water causes cessation of the capillary transport that maintains the constant evaporative flux. This explains the kinetic transition from the constant-rate stage to the decreasing-rate stage (Shokri & Or, 2011). Once this disruption occurs, the water thermodynamic activity inside pores decreases caused by the formation of menisci (Hu, et al., 2018), as predicted by the Kelvin equation. The kinetic outcome is a slower drying rate.
From a mechanistic point of view, at xc there is no longer a continuous liquid coverage of the surfaces. The surface area covered decreases as the drying process occurs, and it is now observed that dx/dt decreases as (1-x) decreases as well:  [2] and the kinetics are the first order in the remaining moisture content and also proportional to the virtual fluxional area through which the liquid evaporates.
Thus, it is easily understood that fluids diffuse more easily in solids with high xc values (high dynamic porosity). xc is then an indirect measure of diffusibility of fluids and can be used to assess their relative mobility in a group of porous materials.

Drying Curves
All measurements were done at 50 °C and 87 kPa barometric pressure. The laboratory environment was kept at 60% -65% relative humidity.
The solid materials were soaked with tap water for periods of 14 h -16 h. The samples were gently pressed in paper towels prior to kinetic measurements, to obtain free-flowing solids.
10-gramme samples were placed on aluminium foil sample pans of 74 cm 2 surface area. The raw mass-time kinetic data were obtained by monitoring mass loss by using Ohaus MB35 Halogen Moisture Analysers.

Data Treatment
The extent of drying was obtained from the mass balance The "surface" drying regime (-dm/dt = constant) was determined from the masstime data pairs for which a linear correlation is valid with Pearson's rp ≥ 0,9990.
The apparent rate constant for the second drying stage (kap = k × Fluxional area) was obtained as the slope of dx/dt vs. (1-x) that obeys a linear relationship (pseudo firstorder kinetics). The dx/dt values were calculated from cubic polynomials x = f(t). Fluxional areas were calculated by using the previously determined evaporation rate of pure water at 50 °C and 87 kPa of 11,37 mmol s -1 m -2 , as follows:

Results and Discussion
The kinetic data are shown in table 1. No correlation between dynamic porosity and chemical nature is sought in this work, but only the relation between the kinetic parameters xc and k for the different porous materials.

Sigma-Aldrich products (types 3 A and 5 A)
show similar k values, but the zeolite type 4 A from Caledon Laboratories has a lower value (0,060 ± 0,006 vs. 0,14 ± 0,02). The supplier declares this 4 A material to be a mixture of more than 70% aluminosilicate, 5% SiO2 and a mineral binder. XRD patterns indeed show the presence of SiO2 in 4 A (conspicuous signal for quartz at 2θ = 26,7 °), as shown in figure 3. It is plausible to think of the presence of SiO2 causing an obstructive effect on the dynamic porosity for the diffusivity of water in the pores of zeolite type 4 A, relative to 3 A and 5 A analogues.
At present, we have no data from experiments such as BET isotherms that give information on the specific areas of neither these materials nor their pore-size distribution. Nevertheless, a comment is due on the similarity of k values for Molecular sieve type 5A (0,5 nm pore size), Molecular sieve type 3A (0,3 nm pore size), and diatomite (1 µm -5 µm pore size).
Mata-Segreda (2014) studied the drying kinetics of filter paper samples of different pore sizes at 22 °C and 87 kPa. The author found an empirical regression equation that relates xc with pore size: log xc = 0,039 log (pore size/µm) -0,077 [6] The equation is valid for pore sizes up to 11 µm. It was observed that xc tends asymptotically to the expected theoretical value of 1, for larger pore sizes.
A shape-selective heterogeneous catalyst (Masel, 2001, Smit & Maesen, 2006, Chen, et al., 2012 selects product molecules according to their cavity geometry and pore dimensions and how easily they diffuse. So, the desired product can travel out of the solid matrix and if side products are formed, they could be converted into the desired product. Thus, it is of practical importance to assess dynamic porosities amongst a group of possible support candidates.
Both k and xc can be used to assess diffusional barriers of support materials, allowing their comparison amongst a group of possibilities. Since xc is easier to measure, one concludes that it is by itself a sufficient parameter for comparison of mass-transfer resistance between different candidate supports.