Edited by: Jorge Norberto Beltramini, Queensland University of Technology, Australia
Reviewed by: Arne Reinsdorf, Evonik Industries, Germany; Heather Trajano, University of British Columbia, Canada
This article was submitted to Chemical Reaction Engineering, a section of the journal Frontiers in Chemical Engineering
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Throughout the decades, the solid–fluid processes have been modeled with the hypothesis of ideal, non-porous particles. Numerous relevant industrial reactions with reactive solids do not involve ideal slabs, spheres, or cylinders, but instead the particles have rough surfaces comparable of the Moon with cracks and craters. The theory of non-ideal solid particles is briefly reviewed and the effect of the non-ideality is illustrated for batch processes. In general, surface roughness results in the increase of the apparent reaction order with respect to the solid reactant.
A few years ago, as the 2nd Edition of the textbook
However, reactive solids appear in numerous chemical processes—not only in the pure chemical industry, but in process industry in general. Let us think about the reactions which are applied in fine chemical and pharmaceutical industries in one corner, and reactions with minerals, leaching of ores, and combustion processes on the other side. The key issue is that the solid material undergoes dramatic changes during the progress of the process—not only chemically, but physically as well. Reactants are changed chemically to products, but structural changes of the solid particles take place simultaneously, which makes this sector of chemical reaction engineering particularly challenging, probably the most demanding field in contemporary reaction engineering.
The author's first contact to the field of reactive solids was in the marvelous reaction engineering course for undergraduates given at Åbo Akademi University by professors Leif Hummelstedt and Lars-Eric Lindfors—
Product layer model. The shrinking particle model is obtained by removing the product layer from the figure.
The equations landed on the pages of the compendium without any derivation, but luckily, a good study circle of undergraduates provided a way out. We were often spending time in the undergraduate library of our university, in a beautiful historic Reuter's building. After screening the shelves in the library, some textbooks became a source of great inspiration. The first one was the classical, very condensed treatment of K.G. Denbigh and J.C.R. Turner:
Later on, one of the authors of this article got contact to the classical textbook written by Octave Levenspiel—
The model equations which are the basis of the tests are collected in
Standard relationships between particle radius and reaction time.
Chemical reaction control | (T1) | |
Film diffusion control (Stokes' regime) | (T2) | |
Chemical reaction control | (T3) | |
Product layer diffusion control | (T4) | |
Film diffusion control | (T5) | |
The plots |
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The radius ( |
The classical theory of Yagi and Kunii (
The generalized model equations are presented [e.g., by Salmi et al. (
Generalized models for first order kinetics.
Chemical reaction control: ϕ″ → 0, Equation (T1) ( |
|
Film diffusion control: ϕ″ large, Equation (T2) | |
Chemical reaction control: ϕ′ → 0, |
|
Product layer diffusion control: ϕ′ is large, |
|
Film diffusion control: |
For the general case that no one of steps alone is the rate limiting one, Equations (T6), (T7) can be used to get information about the rate controlling steps. The parameters Φ′, Φ″ and
The models discussed above were restricted to a unimodal particle size distribution, to one particle size, but particle size distributions can be incorporated in a rather straightforward way: each particle is assumed to have a similar behavior, and the particle size distribution is shifted downwards during the progress of the reaction. The treatment of particle size distributions in batch reactors is described for instance by Grénman et al. (
Is there anything still to be developed further? The issues which have been discussed in this text so far, are based on the idea of completely ideal particles, spheres, long cylinders, and slabs. For instance, if we have a cylindrical particle like a very long pencil, then we need to calculate the lateral surface, but the ends of the cylinder can be neglected because they are so tiny in the proportion—or on the other hand, the reactive particles might be perfect non-porous spheres, as presented in classical texts. It is of course intuitively understood that this theory is a simplification, but the reality is sometimes different. Sohn and Szekely (
How do the real solid particles applied in industrial processes look? Microscopy provides the answer; there has been a tremendous leap in the technology of microscopy in the last decades. Particularly useful in this sense is Scanning Electron Microscopy (SEM), which provides a helicopter view of solid particle surfaces down to few micrometer resolution. What has been assumed to be an ideal particle with shape factors one, two, and three, for slab, cylinder, and sphere, is not the reality, but instead, it might resemble the moon landscape. It brings our memory to the landing of Apollo 11 on the moon surface in July 1969. The surface of our satellite has a lot of defects, such as cracks and craters. This implies that the actual surface area-to-volume ratio can be much higher than that predicted by ideal geometry as indicated by the samples displayed in
SEM images of some real reactive solid particles.
Another very useful experimental technique to be kept in mind is physisorption (i.e., the measurement of the solid surface area by nitrogen adsorption—a very well-established tool in the field of heterogeneous catalysis for the measurement of relatively large specific surface areas). Nowadays it is possible with the most modern gas adsorption technique to determine the surface areas of particles with a very low porosity, possessing specific surface areas of only a few square meters per gram. These materials could be characterized as non-porous or semi-porous. They have a low surface area and a very low amount of nitrogen adsorption, but modern equipment can still handle the small surface areas. In fact, solid particles of this kind deviate strongly from ideal geometry (
Classical chemical kinetics is a strong tool in the determination of the mechanisms of solid–fluid processes. In experimental devices, where a high degree of turbulence prevails around the solid particles, the resistance of film diffusion is suppressed, and if no product layer is formed since the product is dissolved or cracked away from the reactive surface, then the rate controlling step is the chemical process itself.
When we derive the rate equations for ideal solid geometries with respect to the amount of substance, or the concentration of the solid material measured in moles per liter, the reaction order is 1/2 for a long cylinder and 2/3 for a sphere. Reaction orders <1 imply that a complete conversion of the solid reactant is achieved within a finite reaction time (if the reaction order is zero with respect to the fluid component; Salmi et al.,
A former experimental project in the authors' laboratory brought the dilemma of ideal particles in daylight, as the leaching of zinc was measured, and it was observed that the reaction order was completely different from 2/3 predicted by ideal spherical particles (Salmi et al.,
How to obtain a simple but realistic model for non-ideal solids so that the experimental observations could be described by a molecular-based but still simple mathematical model? The equations for the new approach were written down on a napkin: the basic idea was simply that the integer values for the shape factor in the reactivity equations were replaced with non-integer values. A non-integer shape factor can be related to the experimentally measured surface area of a solid particle. This model had the potential to explain many of our previously obtained experimental results. The first article combining the experimental facts and the theoretical approach was published (Salmi et al.,
The basic concepts of the theory of non-ideal surfaces is summarized below. For the sake of illustration, a well-stirred batch reactor with equal particle sized solids in the liquid phase is presumed. For fluid–solid processes, the overall rate is assumed to be proportional to the accessible surface area (
The specific rates (
The total area of accessible surface are of solid particles in the reactor is
while the volume of an individual particle is
where
from which the number of particles can be solved and inserted in Equation (2), which becomes
The area-to-volume ratio
The ratio
For a constant number of particles, the particle volumes and amounts of substance are related by
which is inserted in Equation (6):
The ratio
which is expressed with the original specific surface area (σ) of the particle
We have now
which is inserted in the expression of
If the mole fraction of the solid material is constant (
For a long cylinder,
However, as the real particles deviate from the ideal ones, the accessible surface area for the chemical reactions easily become higher. Parameter
The effect of particle non-ideality on the reaction order.
5 | 3 | 1/3 | 2/3 |
10 | 6 | 1/6 | 5/6 |
100 | 60 | 1/60 | 59/60 |
1,000 | 600 | 1/600 | 599/600 |
Two observations are important: the effective reaction order (1 –
A simplified case with a single chemical reaction and chemical reaction control is considered here, as well as the reaction order (
For a well-stirred batch reactor, the mass balance of a solid component (
For a liquid-phase component (
Very often the component concentrations are used in the treatment of the mass balances.
For the liquid-phase components, the use of concentrations is evident. The amount of substance (
The term
After inserting the rate expression (16) in the mass balances (19) and (20) and introducing a dimensionless Damköhler number (
the balance equations obtain the forms displayed below,
where λ = νi/νj. The dimensionless concentrations (
The differential equation is solved by separation of variables and integration. The initial condition
for
for
Two special cases are of particular interest: the reaction is of zero order or first order with respect to the fluid-phase component. After inserting the values
For
For
a ≠ 1
For
which de facto implies that first order kinetics is observed. On the other hand, for
which corresponds to second order kinetics.
Numerical simulations of model Equations (26), (27) are provided in
Numerical simulations of models (26) (upper) and (27) (lower) in Damköhler space (
The approach presented in this article enables to describe the kinetics of non-ideal solids with surface defects and, in the ultimate case, the model approaches the model for completely porous particles. The main benefit is that the reaction kinetics close to first order with respect to the solid material can be predicted by the model. Only one parameter, the shape factor is needed to take into account the non-ideality. Furthermore, the value of the shape factor can be obtained from measurable quantities (particle density, surface area, and characteristic thickness) as summarized in
New perspectives should be discussed—where shall we go in future? A multitude of fluid-solid processes are conducted not only in batch reactors, but they appear in continuous reactors, where particles enter the system, they react and leave the reactor vessel. There, we approach the theories of crystallization, but in a reversed manner. We do not only have a stagnant particle size distribution, but the population balances will be included, because particles which exceed a specific size, or go below it, are not only created by the solid-liquid or solid-gas reaction, but they are also influenced by the inflow and outflow. Theories of non-ideal particles, particle size distributions in dynamic forms, population balances, and flow models are needed in future. The dream is to have kinetic models, particle size distributions, and population balances coupled to the most modern approach to computational fluid dynamics. With such a sophisticated multiscale approach, the behavior of sophisticated gas-liquid-solid processes could be predicted and optimized, from the treatment of minerals to fine chemistry.
The main benefit of including non-ideality is to get better kinetic models which do not have systematic deviations from the experimental data. The improved modeling approach presented by us might be beneficial in process scale-up.
The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.
TS and PT wrote the 1st version of the manuscript. JW made the computations. All authors took part in the final editing of the manuscript.
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
The economic support is gratefully acknowledged.
Surface area (m2) | |
Shape factor (–) | |
Biot number for mass transfer (–) | |
Concentration (mol/m3) | |
Diffusion coefficient (m2/s) | |
Damköhler number (–) | |
Rate constant (mol/m2/s)(mol/m3)n, |
|
Molar mass (kg/mol) | |
Mass (kg) | |
Amount of substance (mol) | |
Number of particles (–) | |
Initial radius of particle (m) | |
Reaction rate of reaction step |
|
Particle radius (m) | |
Reaction rate (mol/m2/s) | |
S | Number of reactions (–) |
Time (s) | |
Conversion (–) | |
Reciprocal shape factor |
|
Mole fraction of the reactive solid in solid material (–) | |
y | Dimensionless concentration (–) |
λ | Ratio between stoichiometric coefficients (–) |
ν | Stoichiometric coefficient (–) |
ρ | Density (kg/m3) |
σ | Specific surface area (m2/kg) |
τ | Reaction time for complete conversion (t) |
Thiele moduli (–) | |
i | Fluid-phase component index |
j | Solid reactant index |
k | Reaction index |
n | Reaction order |
P | Particle |
0 | Initial state |