Shrinking Particles

When a solid particle of species B reacts with a gaseous species A to form only gaseous products, the solid can disappear by developing internal porosity, while maintaining its macroscopic shape. An example is the reaction of carbon with water vapor to produce activated carbon the intrinsic rate depends upon the development of sites for the reaction (see Section 9.3). Alternatively, the solid can disappear only from the surface so that the particle progressively shrinks as it reacts and eventually disappears on complete reaction (/B =1). An example is the combustion of carbon in air or oxygen (reaction (E) in Section 9.1.1). In this section, we consider this case, and use reaction 9.1-2 to represent the stoichiometry of a general reaction of this type. 

An important difference between a shrinking particle reacting to form only gaseous product(s) and a constant-size particle reacting so that a product layer surrounds a shrinking core is that, in the former case, there is no product or ash layer, and hence no ash-layer diffusion resistance for A. Thus, only two rate processes, gas-film mass transfer of A, and reaction of A and B, need to be taken into account. 

We can develop a simple shrinking-particle kinetics model by taking the two rate-processes involved as steps in series, in a treatment that is simpler than that used for the SCM, although some of the assumptions are the same  

In the following example, the treatment is illustrated for a spherical particle. 

For a reaction represented by A(g) + bB(s) — produc1(g), derive the relation between time (t) of reaction and fraction of B converted (/B), if the particle is spherical with an initial radius R0, and the Ranz-Marshall correlation for kAg(R) is valid, where R is the radius at t. Other assumptions are given above. 

Earlier, Equation 8.96 was derived for a first-order reaction, assuming the ratio a/ao. For a constant gas concentration (c ), the ratio a/ao = t/to and the reaction time are obtained as 

Depending on the rate-determining step, either diffusion through the gas film or chemical reaction as the rate-determining step, Equation 8.142 is transformed to various different forms. Some limiting cases were considered in Section 8.2.3. 

Let us now consider a stagnant packed bed with a continuous gas flow. The reactor operates, in this case, in a semibatch mode solid particles form the continuous phase, whereas gas is the discontinuous phase. The mass balances must be derived for the transient state for the 

FIGURE 8.5 Particle radius (r) versus the reaction time (f) for a particle containing a product layer. Case shrinking particle. (Data from Levenspiel, O., Chemical Reaction Engineering, 3rd Edition,Wiley, New York, 1999.) 

For a gas component i reacting with the solid component B, the mass balance is given by 

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Two models developed in Chapter 9 to describe the kinetics of such reactions are the shrinking-core model (SCM) and the shrinking-particle model (SPM). The SCM applies to particles of constant size during reaction, and we use it for illustrative purposes in this chapter. The results for three shapes of single solid particle are summarized in Table 9.1 in the form of the integrated time (t conversion (/B) relation, where B is the solid reactant in model reaction 9.1-1 ... 

Figure 56 above describes the phenomenology of the char combustion regime (III). The concept of the shrinking core or shrinking particle model is usually applied in mathematical modelling of char combustion in regime (III). 

This relationship of size versus time for shrinking particles in the Stokes regime is shown in Figs. 25.9 and 25.10, pp. 582 and 583, and it well represents small burning solid particles and small burning liquid droplets. 

Particles of constant size Gas film diffusion controls, Eq. 11 Chemical reaction controls, Eq. 23 Ash layer diffusion controls, Eq. 18 Shrinking particles Stokes regime, Eq. 30 Large, turbulent regime, Eq. 31 Reaction controls, Eq. 23... 

Since no ash is formed at any time during reaction, we have here a case of kinetics of shrinking particles for which two resistances at most, surface reaction and gas film, may play a role. In terms of these, the overall rate constant at any instant from Eq. 35 is... 

The limitation of such a model to first-order reaction rates is not as restricting as it seems. In fact, many reactions might at least be considered as of pseudo -first order, which means that they behave macroscopically like first-order reactions. This is the case for diluted fluids and for non-catalytic gas/solid reactions such as the so-called shrinking core or shrinking particle model. Other examples are electrochemical reactions [106],... 

The solution of liquid extract out of solid particles is described according to a shrinking particle model by convective mass transfer and diffusion in the particle structure. 

Fluid-solid reactions include thermal decomposition of minerals, roasting (oxidation) of sulfide ores, reduction of metal oxides with hydrogen, nitridation of metals, and carburization of metals. Each t3 e of reaction will be discussed finm the thermodynamic point of view. Then reaction kinetics for all of the various rate determining steps in fluid-sohd reactions will be discussed for two general models shrinking core and shrinking particle. 

When the product layer flakes ofT as fast as it is formed, the reaction may be considered to be occurring at the surface of a shrinking particle (see Figure 5.2). This tj pe of reaction is described by the following steps ... 

It may be noted that for growing particles (d /dt) >0 and for shrinking particles (d g/dt) <0. Comparing Eqs. 50 and 51, which represent the mass balances for particles of size g for shrinking and growing particles respectively, it is seen that they both differ from each other only in the sign for the fourth term. In order to represent both of them by a single equation, we may substitute the absolute value of d g/dt in Eq. 51, so that... 

The shrinking-particle model only presumes reactions on the outer surface ... 

VV — theory shrinking particle — theory constant particle size ... 

Thijssen and Coumans, Short-cut Calculation of Non-isothermal Drying Rates of Shrinking and Non-shrinking Particles Containing an Expanding Gas Phase, Proc. 4th Int. Drying Symp., IDS 84, Kyoto, Japan, 1 22-30 (1984). Van der Lijn, doctoral thesis, Wageningen, 1976. 

A moving coordinate accounting for the movement of shrinking particles was used to handle the problem of structural changes during drying (Ratti, 1991). 

The rate of change of particle size is also indicated as Gp and can be positive, in the case of growing particles, or negative, in the case of shrinking particles. This is probably the most popular way to indicate the rate of phase-space advection due to mass exchange, perhaps because it is quite easy to measure the change in particle size at different instants, for example by simple imaging techniques. If the internal coordinate is instead particle volume (i.e. = Vp), the definition becomes... 

Another possible scenario is that as a consequence of mass transfer only the number of primary particles changes, whereas their size remains more or less constant. This hypothesis seems to be realistic in the case of negative molar flux, J <0, or, in other words, in the case of shrinking particles. In fact, in this case it is more likely that the external particles will be consumed before the internal ones. The resulting expressions for the continuous rate of change of the two internal coordinates therefore read as... 

Shrinking Particle This occurs when the particle consists entirely of the sohd reactant, and the reaction does not generate any solid products. The reaction takes place on the surface of the particle, and as it proceeds, the particle shrinks, until it is consumed completely. 

At the lower temperature (783 K open symbols in Fig. 70) a substantially different behavior is observed. The imide band (A in Fig. 69 bottom) decreases quasi-linearly with the elapsed time (see Eq. 24). The aromatic band (V in Fig. 70 top) is complex, revealing two distinct decomposition patterns. At the beginning (first half) of the normalized time a slow linear decrease is observed, followed by a fast decrease. The decrease of the imide band and the change of the aromatic band in the second part of the curve are typical for a film diffusion-controlled reaction of shrinking particles in a gas flow in the Stokes regime. To confirm this observation a new mathematical model is used to fit the curves [321]. Starting from Eq. 20, the reaction velocity ks is substituted with kg=D Rf1 [321]. D is the diffusion velocity and kg the mass transfer coefficient between fluid and particle. The differential equation is solved and the time necessary to reduce a particle from a starting radius R0 to Rt is obtained [see Eq. (22)] [321],... 

The data points (open symbols) in Fig. 70 were fitted using Eq. 24 with the exponent as variable. The obtained values, around 1.5 0.1, confirm the hypothesis of the film diffusion-controlled reaction of shrinking particles in the Stokes regime. 

No satisfactory results were obtained with this model. This is probably due to the fact that this model is not really valid for shrinking particles [321], The ash diffusion model was also applied for the high-temperature ex-... 

Unlike ordinary suspensions with particle sizes down to a few xm, nanosuspensions deserve attention not only for the enormous increase of the specific surface area, which brings about a high dissolution rate upon dilution. Moreover, the extremely small particles have a solubility exceeding the common solubility Cs in equilibrium with macroscopic particles. As the diameter of particles falls below 1 fji,m, the solubility increases with shrinking particle size (although normally perceived as a substance characteristic, for a given solvent composition being dependent only on temperature) as described by the Kelvin equation ... 

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