Constant-Size Particle

To develop a kinetics model (i.e., a rate law) for the reaction represented in 9.1-1, we focus on a single particle, initially all substance B, reacting with (an unlimited amount 

The single particle acts as a batch reactor in which conditions change with respect to time, This unsteady-state behavior for a reacting particle differs from the steady-state behavior of a catalyst particle in heterogeneous catalysis (Chapter 8). The treatment of it leads to the development of an integrated rate law in which, say, the fraction of B converted, /B, is a function oft, or the inverse. 

Isothermal spherical particle. Consider the isothermal spherical particle of 

Equations 9.1-5 and -7 are two coupled partial differential equations with initial and 

Equation 9.1-15 equates the rate of heat transfer by conduction at the surface to the rate of heat transfer by conduction/convection across a thermal boundary layer exterior to the particle (corresponding to the gas film for mass transfer), expressed in terms of a film coefficient, h, and the difference in temperature between bulk gas at Tg and particle surface at Ts  

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In Figure 9.1, a gas film for external mass transfer of A is shown in all three cases. A further significance of a constant-size particle is that any effect of external mass transfer is the same in all cases, regardless of the situation within the particle. 

Figure 9.1 Constant-size particle (B) in reaction A(g) + bB(s) - products instantaneous concentration profiles for isothermal spherical particle illustrating general case (b) and two extreme cases (a) and (c) solid product porous arrows indicate direction of movement of profile with respect to time...

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. 

Combination of Resistances. The above conversion-time expressions assume that a single resistance controls throughout reaction of the particle. However, the relative importance of the gas film, ash layer, and reaction steps will vary as particle conversion progresses. For example, for a constant size particle the gas film resistance remains unchanged, the resistance to reaction increases as the surface of unreacted core decreases, while the ash layer resistance is nonexistent at the start because no ash is present, but becomes progressively more and more important as the ash layer builds up. In general, then, it may not be reasonable to consider that just one step controls throughout reaction. 

On considering the whole progression from fresh to completely converted constant size particle, we find on the average that the relative roles of these three resistances is given by... 

As with any system, there are complications in the details. The CO sticking probability is high and constant until a 0 of about 0.5, but then drops rapidly [306a]. Practical catalysts often consist of nanometer size particles supported on an oxide such as alumina or silica. Different crystal facets behave differently and RAIRS spectroscopy reveals that CO may adsorb with various kinds of bonding and on various kinds of sites (three-fold hollow, bridging, linear) [307]. See Ref 309 for a discussion of some debates on the matter. In the case of Pd crystallites on a-Al203, it is proposed that CO impinging on the support... 

During the actual preparation of the GPC/SEC gel, there are several noteworthy items in the procedure. When combining aqueous and organic phases, always pour the organic phase into the water phase as the reverse procedure produces very large particles. This mixture must be held at 40°C to prevent the initiator from starting the reaction before the right size particles are formed. Rotor speed determines the particle size of the spheres the faster the speed the smaller the particles. Constant torque mixers produce the best results with more narrow particle-size distributions. The initial mixture should be stirred at 300-400 rpm to ensure a particle-size distribution from 2 to 20 yam. 

The effects of dispersion of the electrocatalyst and of particle size on the kinetics of electrooxidation of methanol have been the subject of numerous studies because of the utilization of carbon support in DMFC anodes. The main objective is to determine the optimum size of the platinum anode particles in order to increase the effectiveness factor of platinum. Such a size effect, which is widely recognized in the case of the reduction of oxygen, is still a subject of discussion for the oxidation of methanol. According to some investigators, an optimum of 2 nm for the platinum particle size exists, but studying particle sizes up to 1.4 nm, other authors observed no size effect. According to a recent study, the rate of oxidation of methanol remains constant for particles greater than 4.5 nm, but decreases with size for smaller particles (up to 2.2 nm). 

This value is considerably higher than the experimental value (0.17) obtained from rate measurements on different size particles, but several factors may be invoked to explain the inconsistency. There will be a distribution of both pore radii and pore lengths present in the actual catalyst rather than uniquely specified values. Alumina catalysts often have a bimodal pore-size distribution. Our estimate of an apparent first-order rate constant using the method outlined above will be somewhat in error. The catalyst surface may not be equally active throughout if selective deactivation has taken place and the peripheral region is less active than the catalyst core. Other sources of error are the... 

It will be shown below that, for different-size particles, the term 6eff has no significance and that the assumption of constant volume of the adsorbed layer is inappropriate. Table V shows that the value of 6eff calculated according to Garvey s pro-... 

Reaction A(g h bB(s) - product (s),(g)] first order with respect to A at core surface. Particle(B) constant-size (L, r constant) isothermal. 

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 ... 

For example for the preparation of a 15 cm long, 4.6 mm i.d. stainless tube column, 2.5 g of octadecyl-bonded silica gel was suspended in 25 ml of hexanol-methanol mixture, and kept in an ultrasonic bath for a few minutes to remove air. After the reservoir was filled with the slurry, methanol was pumped in at 10 ml min -1 under constant pressure, 45 MPa (450 bar). After the replacement of slurry solvent by methanol, the flow was stopped and the pressure allowed to drop. When 0 MPa was reached the reservoir was removed. Then, 20 ml of water was added and methanol was again pumped in under the same conditions as before. Again, the flow was stopped and the pressure allowed to drop until it reached 0 MPa. The pre-column was removed and the analytical column closed. The maximum pressure that can be applied in the filling stage is based on the pore size, particle shape, and purity of the silica gel. This reproducible packing procedure is performed at constant temperature by using a water bath (60-BO °C). 

Mg particles. The burning rate increases with increasing surface area. In other words, the burning rate increases with either decreasing particle size of Mg at constant ot increasing constant Mg particle size.I => l... 

Although there are algebraic analyses in the literature relating the progress of solid—solid reactions to diffusion constant and particle sizes, there are of little use for either prediction of even for extrapolation. Experiments frequently produce measurements which, equally badly, fit a number of models. Even solid—liquid reactions with a liquid product are likely to be difficult to model as most solids are non-homogeneous in structure. Dissolution of a solid therefore does not proceed uniformly from the outside, but rather attack occurs preferentially leading, on occasions, with larger multi-crystalline particles, to their break up. 

As with particles of constant size, let us see what rate expressions result when one or the other of the resistances controls. 

When chemical reaction controls, the behavior is identical to that of particles of unchanging size therefore. Fig. 25.7 and Eq. 21 or 23 will represent the conversion-time behavior of single particles, both shrinking and of constant size. 

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... 

The identification of the pertinent HETP equation must, therefore, be arrived at from the results of a sequential series of experiments. Firstly, all the equations must be fitted to a series of (H) and (u) data sets and those equations that give positive and real values for the constants of the equations identified. The explicit form of those equations that satisfy the preliminary data, must then be tested against a series of data sets that have been obtained from different chromatographic systems. Such systems might involve columns packed with different size particles or employ mobile phases or solutes having different but known physical properties. 

Powder size two fractions with a particle size of 0.05-0.09 mm and 0.40-0.63 mm were used. The vibromilling was performed for 48 h at 10° C The PVC content of the polymer was 6.57% for the first size particle and only 3.82% for the second, as the constant surface between polymer-monomer is less. 

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