First-order deposition reaction

A combined analytical and numerical method is employed to optimize process conditions for composites fiber coating by chemical vapor infiltration (CVI). For a first-order deposition reaction, the optimum pressure yielding the maximum deposition rate at a preform center is obtained in closed form and is found to depend only on the activation energy of the deposition reaction, the characteristic pore size, and properties of the reactant and product gases. It does not depend on the preform specific surface area, effective diffusivity or preform thickness, nor on the gas-phase yield of the deposition reaction. Further, this optimum pressure is unaltered by the additional constraint of prescribed deposition uniformity. Optimum temperatures are obtained using an analytical expression for the optimum value along with numerical... 

To solve generally for the optimum conditions and to plot resulting deposition rates, it is useful to rewrite the pressure, temperature, density and deposition rate in terms of normalized variables. For this purpose, the normalized values are defined as p = p/p, T = T/T, and p = p/p, . Similarly, the normalized deposition rate and molecular speed are S = S/S and, 15 t5/i5 . The reference temperature, density and speed for a first-order deposition reaction are taken as... 

Figure 16. Examples of deposition shapes in pyrolytic-laser-assisted CVD (a) first-order surface reaction, no mass-transfer effects (b) first-order surface reaction, depletion effects and (c) Langmuir-Hinshelwood surface kinetics, no mass transfer effects. The ratio r/w is the radial position relative to the beam...

The rate of deposition of Brownian particles is predicted by taking into account the effects of diffusion and convection of single particles and interaction forces between particles and collector [2.1] -[2.6]. It is demonstrated that the interaction forces can be incorporated into a boundary condition that has the form of a first order chemical reaction which takes place on the collector [2.1], and an expression is derived for the rate constant The rate of deposition is obtained by solving the convective diffusion equation subject to that boundary condition. The procedure developed for deposition is extended to the case when both deposition and desorption occur. In the latter case, the interaction potential contains the Bom repulsion, in addition to the London and double-layer interactions [2.2]-[2.7]. Paper [2.7] differs from [2.2] because it considers the deposition at both primary and secondary minima. Papers [2.8], [2.9] and [2.10] treat the deposition of cancer cells or platelets on surfaces. 

The objective of the present research is to predict the rate of deposition of Brownian particles by considering the effects of diffusion, convection, and interaction forces between particle and collector. It will be shown that, when the repulsion due to the double-layer is sufficiently large, the interaction forces can be incorporated into a boundary condition for the convective-diffusion equation. This boundary condition takes the form of a virtual first-order chemical reaction which occurs on the surface of the collector. 

In this paper it is shown that the rate of deposition of Brownian particles on the collector can be calculated by solving the convective diffusion equation subject to a virtual first order chemical reaction as a boundary condition at the surface. The boundary condition concentrates the surface-particle interaction forces. When the interaction potential between the particle and the collector experiences a sufficiently high maximum (see f ig. 2) the apparent rate constant of the boundary condition has the Arrhenius form. Equations for the apparent activation energy and the apparent frequency factor are established for this case as functions of Hamaker s constant, dielectric constant, ionic strength, surface potentials and particle radius. The rate... 

When van der Waals and double-layer forces are effective over a distance which is short compared to the diffusion boundary-layer thickness, the rate of deposition may be calculated by lumping the effect of the particle-collector interactions into a boundary condition on the usual convective-diffusion equation. This condition takes the form of a first-order irreversible reaction (10, 11). Using this boundary condition to eliminate the solute concentration next to the disk from Levich s (12) boundaiy-kyersolution of the convective-diffusion equation for a rotating disk, one obtains... 

The second chapter examines the deposition of Brownian particles on surfaces when the interaction forces between particles and collector play a role. When the range of interactions between the two (which can be called the interaction force boundary layer) is small compared to the thickness of the diffusion boundary layer of the particles, the interactions can be replaced by a boundary condition. This has the form of a first order chemical reaction, and an expression is derived for the reaction rate constant. Although cells are larger than the usual Brownian particles, the deposition of cancer cells or platelets on surfaces is treated similarly but on the basis of a Fokker-Plank equation. 

Oxidation kinetics over platinum proceeds at a negative first order at high concentrations of CO, and reverts to a first-order dependency at very low concentrations. As the CO concentration falls towards the center of a porous catalyst, the rate of reaction increases in a reciprocal fashion, so that the effectiveness factor may be greater than one. This effectiveness factor has been discussed by Roberts and Satterfield (106), and in a paper to be published by Wei and Becker. A reversal of the conventional wisdom is sometimes warranted. When the reaction kinetics has a negative order, and when the catalyst poisons are deposited in a thin layer near the surface, the optimum distribution of active catalytic material is away from the surface to form an egg yolk catalyst. 

Ni3C decomposition is included in this class on the basis of Doremieux s conclusion [669] that the slow step is the combination of carbon atoms on reactant surfaces. The reaction (543—613 K) obeyed first-order [eqn. (15)] kinetics. The rate was not significantly different in nitrogen and, unlike the hydrides and nitrides, the mobile lattice constituent was not volatilized but deposited as amorphous carbon. The mechanism suggested is that carbon diffuses from within the structure to a surface where combination occurs. When carbon concentration within the crystal has been decreased sufficiently, nuclei of nickel metal are formed and thereafter reaction proceeds through boundary displacement. 

Catalysts include oxides, mixed oxides (perovskites) and zeolites [3]. The latter, transition metal ion-exchanged systems, have been shown to exhibit high activities for the decomposition reaction [4-9]. Most studies deal with Fe-zeolites [5-8,10,11], but also Co- and Cu-systems exhibit high activities [4,5]. Especially ZSM-5 catalysts are quite active [3]. Detailed kinetic studies, and those accounting for the influence of other components that may be present, like O2, H2O, NO and SO2, have hardly been reported. For Fe-zeolites mainly a first order in N2O and a zero order in O2 is reported [7,8], although also a positive influence of O2 has been found [11]. Mechanistic studies mainly concern Fe-systems, too [5,7,8,10]. Generally, the reaction can be described by an oxidation of active sites, followed by a removal of the deposited oxygen, either by N2O itself or by recombination, eqs. (2)-(4). 

There are several factors that may be invoked to explain the discrepancy between predicted and measured results, but the discrepancy highlights the necessity for good pilot plant scale data to properly design these types of reactors. Obviously, the reaction does not involve simple first-order kinetics or equimolal counterdiffusion. The fact that the catalyst activity varies significantly with time on-stream and some carbon deposition is observed indicates that perhaps the coke residues within the catalyst may have effects like those to be discussed in Section 12.3.3. Consult the original article for further discussion of the nonisothermal catalyst pellet problem. 

Moffat [80] reported the electrodeposition of Ni-Al alloy from solutions of Ni(II) in the 66.7 m/o AlCl3-NaCl melt at 150 °C. The results obtained in this melt system are very similar to those found in the AlCh-EtMcImCI melt. For example, Ni deposits at the mass-transport-limited rate during the co-deposition of Al, and the co-deposition of Al commences several hundred millivolts positive of the thermodynamic potential for the A1(III)/A1 couple. A significant difference between the voltammetric-derived compositions from the AlCl3-NaCl melt and AlCl3-EtMeImCl melt is that alloy composition is independent of Ni(II) concentration at the elevated temperature. Similar to what has been observed for room-temperature Cu-Al, the rate of the aluminum partial reaction is first order in the Ni(II) concentration. Moffat s... 

Palmer et al have studied the pyrolysis of C302 at temperatures in the range 900-1100 °K by following the rate of carbon deposition from a He stream containing 0.1-0.5 mole % C302. The reaction was first order in C302 and was inhibited by the addition of CO a substance other than C302 or CO was responsible for carbon deposition at the wall. Reaction (1) and its reverse... 

The thermal decomposition and photolysis of this alkyl have been studied by Buchanan and Creutzberg112. The pyrolysis mechanism is not fully understood. The overall process is first-order and is unaffected by an 8.5-fold increase in surface-to-volume ratio. Based on measurements of pressure increase, the reaction exhibits an induction period ranging from 2-3 minutes at 513 °C to 40 minutes at 466 °C. Short chains are apparently involved. A polymer initially of empirical formula (BCH2) but slowly losing hydrogen to form (BCH) is deposited on the surface. The mechanism probably involves the reactions... 

ECALE is then ALE where the elements are deposited by controlling the substrate s electrochemical potential, so that atomic layers are formed at underpotentials [Eq. (1)]. The underpotentials are used in order to obtain surface-limited deposition reactions. Compounds are deposited using a cycle where a first solution containing a precursor to one of the elements is introduced to the substrate and an atomic layer is electrodeposited at its underpotential. The cell is then rinsed, a solution containing a precursor to... 

With this information in mind, we can construct a model for the deposition rate. In the simplest case, the rate of flux of reactants to the surface (step 2) is equal to the rate at which the reactants are consumed at steady state (step 5). All other processes (decomposition, adsorption, surface diffusion, desorption, and transport away from the substrate) are assumed to be rapid. It is generally assumed that most CVD reactions are heterogeneous and first order with respect to the major reactant species, such that a general rate expression of the form of Eq. (3.2) would reduce to... 

In the course of the reaction, all the catalysts deactivated by deposition of coke. Several deactivation laws were fitted with the experimental data. The Voohries law, r = r t n, often claimed to represent ageing of acid catalysts did not apply. The best fit for the rate law of the deactivation process was obtained with -dr/dt = kd>r°<, with o(= 1+0.2 depending on the catalyst. Therefore a first order deactivation rate applies which takes the integral form r = rQ expt-k. t). In most cases, correlation coefficients better than 0.9 were obtained when determining kd (refs.5,15). 

This model is equivalent to the model for a first-order reaction in an infinite cylinder of catalysts (214). Analogous to the solution of the catalyst particle problem, the notation of an effectiveness factor can be introduced as the ratio of reaction on each pair of wafers (back and front) to the deposition rate at the wafer edge, that is,... 

Thus, if T < 1, the deposition proceeds more rapidly at the wafer edge than at the interior of the wafer, and a nonuniform film thickness results. This result is the so-called bull s-eye effect. Figure 15a illustrates the variation in i] with < ) for a first-order reaction and a large aspect ratio, A = RJA. For the first-order reaction, the film thickness has the well-known form... 

The theory of coupled multicomponent first-order reaction and diffusion given by Wei (1962) has been used to successfully model the metal deposition profiles (Agrawal and Wei, 1984 Ware and Wei, 1985a). The... 

Newson (1975) was among the first to develop a pore plugging model of demetallation to predict catalyst life. By using the pore structure model of Wheeler (1951), the pellet was assumed to have N pores of identical length but with a specified distribution of pore radii. Metal deposition was assumed to be a first-order reaction over an outer fraction of the pore length and to have a uniform thickness. This model showed that the broadness of the size distribution had little effect on the catalyst life for the same average radii, but that increasing the radii from 45 to 65 A more than doubled the catalyst life. The restricted form of the diffusivity (see Section IV,B,5) was not employed in this model. 

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