First-order reaction, with diffusion

Illustration 4.11 First-Order Reaction with Diffusion in a Liquid Film Selection of a Reaction Solvent... 

The equations used in these models are primarily those described above. Mainly, the diffusion equation with reaction is used (e.g., eq 56). For the flooded-agglomerate models, diffusion across the electrolyte film is included, along with the use of equilibrium for the dissolved gas concentration in the electrolyte. These models were able to match the experimental findings such as the doubling of the Tafel slope due to mass-transport limitations. The equations are amenable to analytic solution mainly because of the assumption of first-order reaction with Tafel kinetics, which means that eq 13 and not eq 15 must be used for the kinetic expression. The different equations and limiting cases are described in the literature models as well as elsewhere. 

Reaction in the liquid phase (Trxn). Now consider the case where an irreversible, first-order reaction with rate constant k (s-1) takes place, in addition to diffusion and solubilization. Equation (CCC) becomes... 

The absorption of ozone from the gas occurred simultaneously with the reaction of the PAH inside the oil droplets. In order to prove that the mass transfer rates of ozone were not limiting in this case, the mass transfer gas/water was optimized and the influence of the mass transfer water/oil was studied by ozonating various oil/water-emulsions with defined oil droplet size distributions. No influence of the mean droplet diameter (1.2 15 pm) on the reaction rate of PAH was observed, consequently the chemical reaction was not controlled by mass transfer at the water/oil interface or diffusion inside the oil droplets. Therefore, a microkinetic description was possible by a first order reaction with regard to the PAH concentration (Kornmuller et al., 1997 a). The effects of pH variation and addition of scavengers indicated a selective direct reaction mechanism of PAH inside the oil droplets... 

An enzyme is immobilized by copolymerization technique. The diameter of the spherical particle is 2 mm and the number density of the particles in a substrate solution is 10,000/L. Initial concentration of substrate is 0.1 mole/L. A substrate catalyzed by the enzyme can be adequately represented by the first-order reaction with k0 = 0.002 mol/Ls. It has been found that both external and internal mass-transfer resistance are significant for this immobilized enzyme. The mass-transfer coefficient at the stagnant film around the particle is about 0.02 cm/s and the diffusivity of the substrate in the particle is 5 x 10-6 cm2/s. 

For example, consider the short residence time case presented earlier (7 s residence time, 1-m long channel with 0.007-inch diameter). For a first order reaction with = 0.05 and an inlet concentration of 100 mol m, about 30% conversion would be achieved. The right-hand side of the dimensionless inequality above, assuming a molecular diffusivity of 1.16 X 10 m s is 0.007. Thus, the plug flow criterion is easily satisfied. For rate constants lower than the value in this example, the above criterion is even more readily satisfied. Thus, the channel dimensions used in this example will enable a plug flow mass balance to be the basis for all reaction rate parameter calculations, simplifying the data analysis. 

Reaction (l) accounts for 0 release from the oxygen evolving system (OES), assuming a first order reaction with a time constant of 1 ms, or 10 ms. Reaction (2) was assumed to be diffusion-limited, taking a distance of 0.5 pm between the OES and the oxidase. A first order reaction with a time constant of 2 ms was taken for (3). This treatment is obviously a crude approximation, for a number of reasons such as the assumption of a fixed OES-oxidase distance, of a purely diffusion-limited reaction (2), the disregard for other redox centers (CuA and CuB) and further transfer steps (second reduction of a3, turnover of Cyt c, etc..). Nevertheless, this model accounts qualitatively for several features of the experimental data, such as the absence of a detectable lag in the oxidation, or the presence of such a lag for Cyt c. 

EXAMPLE 7J-3. Reaction and Unsteady-State Diffusion Pure CO2 gas at 101.32 kPa pressure is absorbed into a dilute alkaline buffer solution containing a catalyst. The dilute, absorbed solute COg undergoes a first-order reaction with k = 25 s ... 

Based on these equations, the effective reaction rate rmeff (= km.effCoc.iiq) is calculated. The rate constant km,efF takes into account not only the influence of pore diffusion and external diffusion but also the influence of H2 on the chemical reaction. This leads to a simple first-order reaction with respect to octene, as km.eff still includes the influence of H2. 

With the plug flow assumption, the concentration profile in the freeboard (x>H) is then given for a first order reaction with negligible internal pore diffusion control (15,73) by... 

R is the overall resistance for a first order reaction with internal diffusion within the porous catalyst particles For kinetics different from first order, the relations for both R and R will be different and, as a rule, will still contain a concentration term For instance for n-th order kinetics Equation (24) is still valid in good approximation provided is defined according to ... 

The transformation (conversion) and carbonation processes are considered to be diffusion controlled first order reactions with rate constants = 0.48 and = 0.007. One unit of conversion results in a 0.57 fold increase in porosity and a corresponding unit of carbonation, 0.18 times decrease in porosity. Therefore, the change in porosity (AZ ) due to these processes can be expressed as follows ... 

This analysis was carried out for an irreversible, first-order reaction with a very low effectiveness factor. However, for the situation shown in Figure 9-11, the rate per particle (or per unit weight or per unit volume) always depends on both the intrinsic rate constant and the effective diffusivity, no matter what rate equation is obeyed and no matter how steep the concentration gradients might be. 

In other words, diffusion is like a heterogeneous reversible first-order reaction with an equilibrium constant of unity. 

The first-order reaction of hydrogen with Ni3C at 443 K is relatively more rapid than the decomposition [669], indicating facile hydrogenation of the residual carbon at the reactant surface and the possibility of diffusion control is mentioned. 

A pure gas is absorbed into a liquid with which it reacts. The concentration in the liquid is sufficiently low for the mass transfer to be governed by Pick s law and the reaction is first order with respect to the solute gas. It may be assumed that the film theory may be applied to the liquid and that the concentration of solute gas falls from the saturation value to zero across the film. Obtain an expression for the mass transfer rate across the gas-liquid interface in terms of the molecular diffusivity, 1), the first order reaction rate constant k. the film thickness L and the concentration Cas of solute in a saturated solution. The reaction is initially carried our at 293 K. By what factor will the mass transfer rate across the interface change, if the temperature is raised to 313 K7... 

FIGURE 8.1 Fraction unreacted versus dimensionless rate constant for a first-order reaction in various isothermal reactors. The case illustrated with diffusion is for = 0.1. 

Comparing this equation with Equation (8.34) shows that 3At/Y is the flat-plate counterpart of aAII - We thus seek a value for t/T below which diffusion has a negligible effect on the yield of a first-order reaction. 

Suppose that catalyst pellets in the shape of right-circular cylinders have a measured effectiveness factor of r] when used in a packed-bed reactor for a first-order reaction. In an effort to increase catalyst activity, it is proposed to use a pellet with a central hole of radius i /, < Rp. Determine the best value for RhjRp based on an effective diffusivity model similar to Equation (10.33). Assume isothermal operation ignore any diffusion limitations in the central hole, and assume that the ends of the cylinder are sealed to diffusion. You may assume that k, Rp, and eff are known. 

In order to verify that the fixed bed and the micro-channel reactor are equivalent concerning chemical conversion, an irreversible first-order reaction A —) B with kinetic constant was considered. For simplicity, the reaction was assumed to occur at the channel surface or at the surface of the catalyst pellets, respectively. Diffusive mass transfer to the surface of the catalyst pellets was described by a correlation given by Villermaux [115]. 

Consider the case when the equilibrium concentration of substance Red, and hence its limiting CD due to diffusion from the bulk solution, is low. In this case the reactant species Red can be supplied to the reaction zone only as a result of the chemical step. When the electrochemical step is sufficiently fast and activation polarization is low, the overall behavior of the reaction will be determined precisely by the special features of the chemical step concentration polarization will be observed for the reaction at the electrode, not because of slow diffusion of the substance but because of a slow chemical step. We shall assume that the concentrations of substance A and of the reaction components are high enough so that they will remain practically unchanged when the chemical reaction proceeds. We shall assume, moreover, that reaction (13.37) follows first-order kinetics with respect to Red and A. We shall write Cg for the equilibrium (bulk) concentration of substance Red, and we shall write Cg and c for the surface concentration and the instantaneous concentration (to simplify the equations, we shall not use the subscript red ). 

The Effectiveness Factor Analysis in Terms of Effective Diffusivities First-Order Reactions on Spherical Pellets. Useful expressions for catalyst effectiveness factors may also be developed in terms of the concept of effective diffusivities. This approach permits one to write an expression for the mass transfer within the pellet in terms of a form of Fick s first law based on the superficial cross-sectional area of a porous medium. We thereby circumvent the necessity of developing a detailed mathematical model of the pore geometry and size distribution. This subsection is devoted to an analysis of simultaneous mass transfer and chemical reaction in porous catalyst pellets in terms of the effective diffusivity. In order to use the analysis with confidence, the effective diffusivity should be determined experimentally, since it is difficult to obtain accurate estimates of this parameter on an a priori basis. 

Thus a zero-order reaction appears to be 1/2 order and a second-order reaction appears to be 3/2 order when dealing with a fast reaction taking place in porous catalyst pellets. First-order reactions do not appear to undergo a shift in reaction order in going from high to low effectiveness factors. These statements presume that the combined diffusivity lies in the Knudsen range, so that this parameter is pressure independent. 

According to equation 8.5-28, the nth-order surface reaction becomes a reaction for which the observed order is (n + l)/2. Thus, a zero-order surface reaction becomes one of order 1/2, a first-order reaction remains first-order, and second-order becomes order 3/2. This is the result if De is independent of concentration, as would be the case if Knud-sen diffusion predominated. If molecular diffusion predominates, for pure A, DecrcA, and the observed order becomes n/2,with corresponding results for particular orders of surface reaction (e.g., a first-order surface reaction is observed to have order 1/2). 

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