Reaction and diffusion

Processes in which diffusion is accompanied by a chemical reaction arise frequently and in a variety of different contexts. All catalytic reactions, in which the catalyst resides within a porous matrix, are necessarily accompanied by diffusional transport of the reactants and products into and out of this catalyst particle. In noncatalytic gas-solid and liquid-solid reactions, diffusion occurs not so much within the solid particle but rather tiirough a gas or liquid film, or through ash layers surroxmding the reacting core. Here again diffusion is coupled with reaction. 

Reactions accompanied by diffusion also occur in fluid systems. The atmosphere is one vast reacting reservoir in which gaseous pollutants such as the nitrogen oxides (the famous Nox) or sulfur oxides (the equally famous Sox) diffuse into the air and undergo reactions with atmospheric oxygen. In gas-liquid systems, the liquid phase often contains a reacting component that interacts with the gas diffusing into it. Here, too, reaction is linked to diffusion. 

Evidently, in each of the systems mentioned, a host of different reachons are possible. A reacting solid particle, for example, may involve the combustion of a fuel or the calcining of calcium carbonate (limestone) to calcium oxide and carbon dioxide. Reactants involved in catalytic reactions are almost infinite in their variety, as are those participating in atmospheric reactions. 

To convey a flavor of these events without overwhelming the reader with a mass of details, we limit ourselves to the following cases. 

Most cellular functions depend on enzymatic reactions. Turnover rates for intracellular enzymes are typically several hundred reactions per second, although the variation in catalytic power is high (Table 4.10). A typical intracellular compartment (L 1 fim) will be well mixed by diffusion if D 10 cm /s, with a diffusion time of 10 ms. In this micrometer-scale compartment, any two molecules will collide every 1 s [131]. Clearly, this rate of collision will have important implications for the overall rate of reaction for enzymes with high turnover number. 

The turnover number, which is equal to the maximal reaction rate Vmax divided by the concentration of active sites on the enzyme, is indicated for a variety of enzymes (from [58], p. 191). The reaction time was determined as l/(tumover number), which estimates a characteristic time for a round of catalysis by the enzyme. 

A mass balance equation for the reaction kinetics can be written for each of the species A, B, A—B, and A-B. The assumption that the intermediate species (A—B) reaches a steady-state concentration rapidly compared to the other species yields  

In this formulation, the overall apparent rates of the forward and reverse reaction kf and k ) can be written in terms of rates of encounter k and k ) and the intrinsic forward and reverse reaction rates (/c+i and k ). 

The encounter step involves the simultaneous diffusion of two separated solute species that collide during random motion. The kinetics of this event ( + and kj) can be determined by analysis of the diffusion equation. The relevant diffusion equation was provided in the previous chapter, in the context of the diffusion of solutes to the surface of a cell. If the cell is replaced with one of the solute molecules. A, which is assumed to be fixed in space and surrounded by 

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Figure C2.7.13. Schematic representation of diffusion and reaction in pores of HZSM-5 zeolite-catalysed toluene disproportionation the numbers are approximate relative diffusion coefficients in the pores 1131.

Rabinowitch E 1937 Collision, coordination, diffusion and reaction velocity in condensed systems Trans. Faraday See. 33 1225-33... 

The equations of combiaed diffusion and reaction, and their solutions, are analogous to those for gas absorption (qv) (47). It has been shown how the concentration profiles and rate-controlling steps change as the rate constant iacreases (48). When the reaction is very slow and the B-rich phase is essentially saturated with C, the mass-transfer rate is governed by the kinetics within the bulk of the B-rich phase. This is defined as regime 1. 

Aris, R. The Mathematical Theoiy of Diffusion and Reaction in Feimeahle Catalysis, vols. 1 and 2, Oxford University Press, Oxford (1975). 

Example Consider the equation for convection, diffusion, and reaction in a tiihiilar reactor. 

In each form of attack, solute concentration differences arise primarily by diffusion-related processes. As a consequence, stagnant conditions may promote attack, since concentration gradients near affected areas are reduced by flow and these concentration gradients supply the energy that drives diffusion. Similarly, high concentrations of dissolved species increase attack. Elevated temperature usually stimulates attack by increasing both diffusion and reaction rates. 

Many authors contributed to the field of diffusion and chemical reaction. Crank (1975) dealt with the mathematics of diffusion, as did Frank-Kamenetskii (1961), and Aris (1975). The book of Sherwood and Satterfield (1963) and later Satterfield (1970) discussed the theme in detail. Most of the published papers deal with a single reaction case, but this has limited practical significance. In the 1960s, when the subject was in vogue, hundreds of papers were presented on this subject. A fraction of the presented papers dealt with the selectivity problem as influenced by diffitsion. This field was reviewed by Carberry (1976). Mears (1971) developed criteria for important practical cases. Most books on reaction engineering give a good summary of the literature and the important aspects of the interaction of diffusion and reaction. 

Aris, R., 1975, The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts, Vols. I and II, Oxford University Press, London. 

An example of a journal hovering between broad and narrow spectrum is Journal of Alloys and Compounds, subtitled an interdiciplinary journal of materials science and solid-state chemistry and physics. One which is more restrictively focused is Journal of Nuclear Materials (which I edited for its first 25 years). Ceramics has a range of journals, of which the most substantial is Journal of the American Ceramic Society. Ceramics International is an example of an international journal in the field, while Journal of the European Ceramic Society is a rather unusual instance of a periodical with a continental remit. More specialised journals include Solid State Ionics Diffusion and Reactions, and a new Journal of Electroceramics, started in 1997. 

M. Tammaro, J. W. Evans. Reactive removal of unstable mixed NO -I- CO adlayers Chemical diffusion and reaction front propagation. J Chem Phys 705 7795-7806, 1998. 

G. Schulz, M. Martin. Computer simulations of pattern formation in ionconducting systems. Solid State Ionics, Diffusion and Reactions 101-103AM,... 

Phase transition occurs at a state of thermodynamic equilibrium, inducing a change in the microstructure of atoms. However, corrosion is a typical nonequilibrium phenomenon accompanied by diffusion and reaction processes. We can also observe that this phenomenon is characterized by much larger scales of length than an atomic order (i.e., masses of a lot of atoms), which is obvious if we can see the morphological change in the pitted surface. 

A detailed description of AA, BB, CC step-growth copolymerization with phase separation is an involved task. Generally, the system we are attempting to model is a polymerization which proceeds homogeneously until some critical point when phase separation occurs into what we will call hard and soft domains. Each chemical species present is assumed to distribute itself between the two phases at the instant of phase separation as dictated by equilibrium thermodynamics. The polymerization proceeds now in the separate domains, perhaps at differen-rates. The monomers continue to distribute themselves between the phases, according to thermodynamic dictates, insofar as the time scales of diffusion and reaction will allow. Newly-formed polymer goes to one or the other phase, also dictated by the thermodynamic preference of its built-in chain micro — architecture. 

In laminar flow stirred tanks, the packet diffusion model is replaced by a slab-diffusion model. The diffusion and reaction calculations are similar to those for the turbulent flow case. Again, the conclusion is that perfect mixing is almost always a good approximation. 

Figure 2.1 Dependence of the effectiveness factor on the Thiele modulus for a first-order irreversible reaction. Steady-state diffusion and reaction, slab model, and isothermal conditions are assumed.

The reactions are still most often carried out in batch and semi-batch reactors, which implies that time-dependent, dynamic models are required to obtain a realistic description of the process. Diffusion and reaction in porous catalyst layers play a central role. The ultimate goal of the modehng based on the principles of chemical reaction engineering is the intensification of the process by maximizing the yields and selectivities of the desired products and optimizing the conditions for mass transfer. 

Ochoa, JA Stroeve, P Whitaker, S, Diffusion and Reaction in Cellular Media, Chemical Engineering Science 41, 2999, 1986. 

Ryan, DJ Carbonell, RG Whitaker, S, A Theory of Diffusion and Reaction in Porous Media, AIChE Symposium Series 71, 46, 1981. 

Whitaker, S, The Method of Volume Averaging An Application to Diffusion and Reaction in Porous Catalysts. In Proceedings of the National Science Council, Part A Physical Science and Engineering National Science Council Taipei, Taiwan, Repubhc of China, 1991 Vol. 15, p 465. 

We first explain the setting of reactors for all CFD simulations. We used Fluent 6.2 as a CFD code. Each reactant fluid is split into laminated fluid segments at the reactor inlet. The flow in reactors was assumed to be laminar flow. Thus, the reactants mix only by molecular diffusion, and reactions take place fi om the interface between each reactant fluid. The reaction formulas and the rate equations of multiple reactions proceeding in reactors were as follows A + B R, ri = A iCaCb B + R S, t2 = CbCr, where R was the desired product and S was the by-product. The other assumptions were as follows the diffusion coefficient of every component was 10" m /s the reactants reacted isothermally, that is, k was fixed at... 

ENZSPLIT - Diffusion and Reaction Split Boundary Solution... 

DIFFUSION AND REACTION IN A POROUS ENZYME CARRIER STEADY-STATE SPLIT BOUNDARY SOLUTION... 

Diffusion and reaction takes place within a bead of volume 7t Rp- and area 4 71 Rp2. It is of interest to find the penetration distance of oxygen for given specific activities and aggregate diameters. The system is modelled by taking small spherical shell increments of volumes 4/3 (rn -rn 3). The outside area... 

Diffusion and reaction within a spherical bead Constant D=3.0E-06 Diffusion coefficient (m2/h) Constant R=0.1 rRadius of aggregate (m)... 

Diffusion and reaction with the target (e.g., complex with Al or Fe )... 

We focus on the effects of crowding on small molecule reactive dynamics and consider again the irreversible catalytic reaction A + C B + C asin the previous subsection, except now a volume fraction < )0 of the total volume is occupied by obstacles (see Fig. 20). The A and B particles diffuse in this crowded environment before encountering the catalytic sphere where reaction takes place. Crowding influences both the diffusion and reaction dynamics, leading to nontrivial volume fraction dependence of the rate coefficient fy (4>) for a single catalytic sphere. This dependence is shown in Fig. 21a. The rate constant has the form discussed earlier,... 

The above model was solved numerically by writing finite difference approximations for each term. The equations were decoupled by writing the reaction terms on the previous time steps where the concentrations are known. Similarly the equations were linearized by writing the diffusivities on the previous time step also. The model was solved numerically using a linear matrix inversion routine, updating the solution matrix between iterations to include the proper concentration dependent diffusivities and reactions. 

The steady-state continuity equations which describe mass balance over a fluid volume element for the species in the stagnant film which are subject to uniaxial diffusion and reaction in the z direction are... 

DP McNamara, GL Amidon. Dissolution of acidic and basic compounds from the rotating disk Influence of convective diffusion and reaction. J Pharm Sci 75 858-868, 1986. 

The interpretation of FCS measurements of chemical kinetics is complicated by the coupling of diffusion and reaction in the open reaction system. Consider the simple bimolecular reaction ... 

In a conventional relaxation kinetics experiment in a closed reaction system, because of mass conservation, the system can be described in a single equation, e.g., SCc(t) = SCc(0)e Rt where R = ((Ca) + ( C b)) + kh- The forward and reverse rate constants are k and k t, respectively. In an open system A, B, and C, can change independently and so three equations, one each for A, B, and C, are required, each equation having contributions from both diffusion and reaction. Consequently, three normal modes rather than one will be required to describe the fluctuation dynamics. Despite this complexity, some general comments about FCS measurements of reaction kinetics are useful. 

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