Internal Mass-Transfer Resistance

If enzymes are immobilized by copolymerization or microencapsulation, the intraparticle mass-transfer resistance can affect the rate of enzyme reaction. In order to derive an equation that shows how the mass-transfer resistance affects the effectiveness of an immobilized enzyme, let s make a series of assumptions as follows  

The reaction occurs at every position within the immobilized enzyme, and the kinetics of the reaction are of the same form as observed for free enzyme. 

Mass transfer through the immobilized enzyme occurs via molecular diffusion. 

There is no mass-transfer limitation at the outside surface of the immobilized enzyme. 

For a steady-state condition, the change of substrate concentration, dCs/dt, is equal to zero. After opening up the brackets and simplifying by eliminating all terms containing dr2 or dr3, we obtain the second order differential equation  

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Figure 8.8 Internal mass transfer resistance and catalyst deactivation concentration profiles inside a catalyst particle-lactose hydrogenation to lactitol and by-products (sponge Ni).

Very precise kinetic experiments were performed with sponge Ni and Ru/C catalysts in a laboratory-scale pressurized slurry reactor (autoclave) by using small catalyst particles to suppress internal mass transfer resistance. The temperature and pressure domains of the experiments were 20-70 bar and 110-130°C, respectively. Lactitol was the absolutely dominating main product in all of the experiments, but minor amounts of lactulose, lactulitol, lactobionic acid, sorbitol and galactitol were observed as by-products on both Ni and Ru catalysts. The selectivity of the main product, lactitol typically exceeded 96%. 

The signal generated by the complex is governed by several physical phenomena associated with the matrix thickness. As soon as the probe is placed in contact with the analyte, external mass transfer controls the movement of the analyte toward the surface of the optical probe.(S4) The osmotic pressure and Gibbs free energy dictate the permeation of the analyte into the matrix. Once the analyte has penetrated the matrix, internal mass transfer resistance controls the movement of the analyte in the matrix. Eventually, the probe reaches a steady state of equilibrium with molecules continuously moving in and out of the matrix. 

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. 

In this section we will develop the internal effectiveness factor for spherical catalyst pellets. The development of models that treat individual pores and pellets of different shapes is undertaken in the problems at the end of this chapter. We will first look at the internal mass transfer resistance to either the products or reactants that occurs between the external pellet surface and the interior of the pellet. To illustrate the salient principles of this model, we consider the irreversible isomerization... 

The sources of band broadening of kinetic origin include molecular diffusion, eddy diffusion, mass transfer resistances, and the finite rate of the kinetics of ad-sorption/desorption. In turn, the mass transfer resistances can be sorted out into several different contributions. First, the film mass transfer resistance takes place at the interface separating the stream of mobile phase percolating through the column bed and the mobile phase stagnant inside the pores of the particles. Second, the internal mass transfer resistance controls the rate of mass transfer between this interface and the adsorbent surface. It is composed of two contributions, the pore diffusion, which is molecular diffusion taking place in the tortuous, constricted network of pores, and surface diffusion, which takes place in the electric field at the liquid-solid interface [60]. All these mass transfer resistances, except the kinetics of adsorption-desorption, depend on the molecular diffusivity. Thus, it is important to study diffusion in bulk liquids and in porous media. 

The differences between the profiles calculated with the two models, the GM and FOR models, are too small to be seen and, thus, the GR and the FOR models are interchangeable at St/Bi > 5. In the case illustrated in Figure 16.20, the St/Bi ratio is of the order of 100. The results of this work can be used to observe the relative intensity of the external and internal mass transfer resistances. The value of was equal to kgxt = 3.8 cm/min based on a Dm of 0.0017 cm /min. This value is a bulk property and can not be applied to the stagnant solution inside the pore to measure the internal mass transfer resistance. Assuming a Dm = 0.0001 cm /min an internal mass transfer coefficient, kj t = 0.076 cm/min is obtained. Thus the ratio of the external and internal mass transfer is of the order 50 meaning the external resistance can be neglected compared to the internal one. 

Lee outlines three different physical methods that are commonly utilized for enzyme immobilization. Enzymes can be adsorbed physically onto a surface-active adsorbent, and adsorption is the simplest and easiest method. They can also be entrapped within a cross-linked polymer matrix. Even though the enzyme is not chemically modified during such entrapment, the enzyme can become deactivated during gel formation and enzyme leakage can be problematic. The microencapsulation technique immobilizes the enzyme within semipermeable membrane microcapsules by interfacial polymerization. All of these methods for immobilization facilitate the reuse of high-value enzymes, but they can also introduce external and internal mass-transfer resistances that must be accounted for in design and economic considerations. 

This example also shows the effects of mass- and enei y-transfer resistances within the catalyst pellet. The temperature increases toward the center of the pellet and increases the rate, but the oxygen concentration goes down, tending to reduce the rate. The global value of 49.8 x 10" is the resultant balance of both factors. Hence the net error in using the bulk conditions to evaluate the rate would be [(49.8 — 43.6)/49.8] (100), 12.5%. In this case the rate increase due to external and internal thermal effects more than balances the adverse effect of internal mass-transfer resistance. The procedure for calculating the effects of internal gradients on the rate is presented in Chap. 11. 

Equation (F) is for the intrinsic rate at a catalyst site, and so C is the p-Wj o i-centration at a site within the catalyst pellet. To convert this to a rate for the pellet, Tp, that is, to account for internal mass-transfer resistance, we use Eq. (11-43) to obtain... 

If the form of the intrinsic rate equation is satisfactory, K should be constant at a given pressure. The values of I/K in Table 12-2 are about constant for a given catalyst and pressure. Results for different catalyst sizes may be different, because K includes the effects of external and internal mass-transfer resistances. 

While r k was the same for the two small particle sizes. Table 12-2 shows that rjk is significantly less for the -in. pellets. The comparison must be made at the same pressure because k and r both vary with pressure. Therefore, considerable internal mass-transfer resistance exists. We can evaluate rj from the rjk data using Eq. (C). For example, at 40 psig... 

For -in. pellets both external and internal mass transport retard the rate. The internal mass-transfer resistance is sizable, as indicated by r values of the order of 0.5. The external resistance seldom exceeded 10% of the total. 

Walter et al. [84] discussed several common experimental methods to estimate the influence of internal mass transfer resistances on the observed rates of heterogeneous catalytic reactions. For example, when the reaction temperature is varied, because the intrinsic reaction rate increases more strongly with temperature than the rate of diffusion, the influence of mass transfer becomes more important and the observed apparent activation energy decreases as the temperature increases effects of temperature on selectivity may be more complex. 

Another approach to varying the influence of internal mass transfer resistances consists of changing the thickness of the catalytic layer. The application of this method is limited because of the experimental difficulty of depositing layers of various thicknesses with identical pore structures and catalytic activities. 

If the internal mass transfer resistance is negligible, what is the concentration of the substrate at the surface of the particle What is the effectiveness factor for this immobilized enzyme ... 

The use of small affinity adsorbent particles immobilized in hydrogel beads has been proposed to circumvent some of these problems (1). The hydrogel matrix can be provided by Ca-Alginate, K-Carrageenan or any other reversible gel. Previous research in our laboratory has indicated that significantly higher adsorption rates and overall adsorption capacities can be achieved by using immobilized affinity adsorbent beads in the whole broth. These beads provide low overall internal mass transfer resistance due to the... 

External mass transfer, such as diffusion to particles or to the outside of pipes or cylinders, requires different correlations from those for internal mass transfer, because there is boundary-layer flow over part of the surface, and boundary-layer separation is common. The mass-transfer coefficients can be determined by studying evaporation of liquid from porous wet solids. However, it is not easy to ensure that there is no effect of internal mass-transfer resistance. Complications from diffusion in the solid are eliminated if the solid is made from a slightly soluble substance that dissolves in the liquid or sublimes into a gas. This method also permits measurement of local mass-transfer coefficients for different points on the solid particle or cylinder. 

Equation (21.50) has been used to predict the internal mass-transfer resistance for separation processes using hollow-fiber membranes. The recommended equation for heat transfer, Eq. (12,23), has an empirical coefficient of 2.0, and this higher... 

Prior to the systematic kinetic e q)eriments a screening of the mass transfer effect was carried out. A conparison of erqjeriments performed widi a catalyst particle mean diameter of 760-960 pm and of 250-500 pm showed quite similar results which indicate the absence of internal mass transfer resistances. Moreover, ejq)eriments performed with different agitation velocities (800 and 200 rpm) gave similar results, suggesting the absence of macro-mixing effects and external mass transfer resistances around the catalyst particles. 

Using artificial neural networks (ANN) the reaction system, including intrinsic reaction kinetics but also internal mass transfer resistances, is considered as a black-box and only input-output signals are analysed. With this approach the conversion rate of the i-th reactant into the j-th species can be expressed in a general form as a complex function, being a mathematical superposition of all above mentioned functional dependencies. This function includes also a contribution of the internal diffusion resistances. So each of the rate equations of Eq. 5 can be described with the following function based on the vanables which uniquely define the state of the system ... 

Fig. 2.1-11 Concentration profile ofthe educt A during mass transfer through a laminar bounda layer of thickness (5 (external mass-transfer resistance) and a cylindrical catalyst pore of length L (internal mass-transfer resistance).

Figure 11.5 Effective yield of the intermediate product as a function of conversion for first-order consecutive reactions (plug flow reactor), (a) Influence of internal mass transfer resistance and (b) influence of external mass transfer resistance.

To avoid internal mass transfer resistances in the porous catalytic layer, its thickness, must be limited. To ensure an effectiveness factor of > 0.95 in an isothermal catalyst layer, the following criterion must be fulfilled [14] ... 

To enhance the adsorptive and catalytic performance, both tailoring the pore structure and tuning the pore chemistry should be considered. The former is related to the internal mass-transfer resistance, while the latter contributes to selective adsorption and nature of the active center. High-efficiency porous heterogeneous catalysts could be produced on the basis of the synergistic effect of the high surface area, uniform pore channel, and the reactivity. 

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