Diffusion efficiency factor

A Back-Pressure Efficiency Factor. Because a gaseous diffusion stage operates with a low-side pressure p which is not negligible with respect to there is also some tendency for the lighter component to effuse preferentiahy back through the barrier. To a first approximation the back-pressure efficiency factor is equal to (1 — r), where ris the pressure ratiopjpj. 

The concentration dependence of z/l vs. c/c0 is plotted in Figure 11.14a. It can be seen that from a Thiele modulus cp > 3 the educt does not reach the internal part of the pore. The inner part of the pore system is useless for catalysis. This is especially relevant if expensive metals serve as active components on a porous carrier, which are then wasted. There are chances to master this diffusion limitation, which will be discussed later in detail. Another important variable is the efficiency factor tj. The efficiency factor r is defined as the quotient of the speed of reaction rs to the maximal possible speed of reaction rsmax. r is related to q> as the quotient of the hyperbolic tangent of the Thiele modulus qy. 

Qualitatively, one can now deduct the apparent activation energy As for the case of a diffusion-limited reaction. If we plot the logarithm of the product of the efficiency factor and the constant for the speed of reaction ln[kij] against 1/T, a typical curve with three regimes can be seen (see Figure 11.15). 

The slope of the lines presented in Figure 5 is defined as k(q/v). The q/v term defines the turnover of the tank contents or what is commonly referred to as the retention time. When q is increased, the liquid contacts the carbon more often and the removal of pesticides should increase, however, the efficiency term, k, can be a function of q. As the waste flow rate is increased, the fluid velocity around each carbon particle increases, thereby increasing system turbulence and compressing the liquid boundary layer. The residence time within the carbon bed is also decreased at higher liquid flow rates, which will reduce the time available for the pesticides to diffuse from the bulk liquid into the liquid boundary layer and into the carbon pores. From inspection of Table II, the pesticide concentration also effects the efficiency factor, k can only be determined experimentally and is valid only for the equipment and conditions tested. 

The effect of diffusion on the rate of a heterogeneous catalytic reaction is characterized by the efficiency factor of a catalyst, tj, defined as the ratio of the actual reaction rate to the rate that would be in the kinetic region under the same conditions. 

In the case of a kinetic equation of an arbitrary form, the well-known differential equation describing a combined progress of a reaction and diffusion cannot be integrated. The first terms of series expansion of the efficiency factor in powers of the grain radius, a, can be found. A rather cumbersome calculation gives (77)... 

In industry large pellets of a catalyst were employed (e.g., 6-8 mm in size), and the rate of the process was essentially affected by the slowness of the diffusion of ammonia in the pores of the catalyst the efficiency factor at this size of pellets is about 0.5. The effect of diffusion retardation of the ammonia synthesis was studied both at high pressures (99), when the free path of molecules is much smaller than the radius of catalyst pores so that the bulk diffusion is operative, and at pressures near to 1 atm (116), where there is a transition from the bulk to the Knudsen diffusion. 

For very small catalyst partides, this equation must itsdf be corrected by an efficiency factor to account for diffusion in industrial catalyst systems, in which the particle diameter reaches 6 to 12 mm. 

In a diffusion-free enzyme reaction the reaction rate increases up to a certain critical value exponentially and is described by the Arrhenius equation [82]. In diffusion-controlled reactions the reaction rate is a matter of the efficiency factor ri [see Eqs. (3 - 5)]. In more detail, the maximum reaction rate is expressed within the root of Eq. (4). Conclusively, the temperature dependence is a function of the square root of the enzyme activity. In practice, immobilized enzymes are much less temperature dependent when their reaction rate is diffusion controlled. 

The proportionality of the efficiency of the reactor to the amount of catalyst,the calculation of the rate of diffusion of dissolved hydrogen to the outside of the catalyst particles and the efficiency factor show that the reaction took place in the kinetic region. 

The characteristic transfer times for gases and liquids as function of the hydraulic diameter are shown in Figure 11.10 for three different geometries of the charmel. If we impose a Damkohler number Dal I <0.1 to get efficiency factors rj > 0.9, gas-phase reactions with characteristic reaction times > 0.1 s can be run in rnicrochannel reactors with d, < 1 mm. Owing to the low diffusion coefficient in liquids, the influence of external mass transfer on the transformation rate and the product selectivity must be considered for fast reactions with S 1 s even for small channels of diameter 50-100(rm. Note that the mass transfer time in sht channels (small H/W ratio) is roughly five times smaller than the corresponding values of circular and square channels. 

The integral product yield as function of conversion for different values of the Thiele modulus is shown in Figure 2.28 for k = k /k = 1/4. It is obvious that internal diffusional resistance leads to a drastic decrease of the target product selectivity and yield. In the domain of practical interest with k <1, the maximum obtainable yield for strong diffusion resistance ( 3, Equation 2.194) drops roughly to 50% of the value reached in the kinetic regime (Equation 2.187). At the same time the efficiency factor in the porous catalyst drops to r p<0.2 as indicated. This demonstrates the dramatic impact of pore diffusion limitation on the overall productivity of the catalytic process. 

An eventual influence of the reactant diffusion in the porous catalytic layer can be respected by adding an efficiency factor r to the rate constant (see Chapter 2), and the observed reaction rate is given by Equation 8.11. 

Finally, the efficiency factor of the catalyst should be dose to unity, implying that there should be no diffusion limitation in the catalyst layer. Here porosity generally enhances the diffusion, which brings the optimization process back to the previous features. 

Use of implies that the supposed process of heterocoalescence between antifoam drops and bubbles is diffusion controlled. In practice, it seems likely, however, that convection will be important under the conditions assumed for this model. The adjustable coagulation efficiency factor presumably accounts for the probability of antifoam drops overcoming colloidal repulsion forces to adhere to bubbles to which... 

Wetting, film and pore diffusion should be included by introduction of, for instance, an efficiency factor ri. ... 

So low efficient factors may be explained in terms of diffusion approach mechanism only if both the reagents are anisotropic. The theory of these reactions developed in /20/ shows that... 

No net mass transfer between the gas and the catalyst particles takes place, but the reactive gas species and the gas products diffuse to and away from the catalyst particle surface where the reactants/products are adsorbed/desorbed on the active sites. In the current pseudo-homogeneous modeling approach such effects might be considered by use of an external efficiency factor. However, as a first approach this mass transfer effect is neglected. 

Catalyst Particle Size Smaller catalyst particles give a higher conversion due to lower diffusion restrictions (higher efficiency factor, see [459]). 

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