Isothermal overall effectiveness

Equations 2.40 and 2.57a. At steady state, the rate of physical transport of reactants from the bulk fluid to the outer surface of the particles is equal to the total rate of chemical reaction on and within the catalyst particles  

and a represent the external and internal surface areas per unit weight of catalyst, respectively. The internal surface area as can safely be equated to the specific surface area, Sg, for all practical purposes. Solving for Qs and substituting back, 

After rearrangement of terms, the overall effectiveness factor Q for linear kinetics can be written as follows  

External concentration gradients are dependent on the internal mass transfer process. The Biot number for mass (B()m is defined as a measure of the ratio of internal to external mass transport resistances and is expressed in terms of the size factor L of Equation 2.65 [13, 19]  

Experimental results show that the mass Biot number (B()m which is sometimes called the modified Sherwood number [17], is much larger than unity, indicating that the major resistance to mass transfer resides in the internal pore diffusion process. 

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Consider a catalyst particle located in a fluid medium where both external and internal mass transfer limitations affect the global rate. The reactant concentration profile for this situation is represented by curve III in Figure 2.3. The isothermal overall effectiveness factor, Q, expressed in terms of bulk fluid conditions, is derived primarily for first-order reactions using... 

It is possible to combine the resistances of internal and external mass transfer through an overall effectiveness factor, for isothermal particles and first-order reaction. Two approaches can be applied. The general idea is that the catalyst can be divided into two parts its exterior surface and its interior surface. Therefore, the global reaction rates used here are per unit surface area of catalyst. 

Find an expression for the overall effectiveness factor of a first-order isothermal reaction in a flat plate catalyst pellet. 

Thus far, the overall effectiveness factor has been used in the mass and energy balances. Since 77 is a function of the local conditions, it must be computed along the length of the reactor. If there is an analytical expression for 17, for example for an isothermal, first-order reaction rate ... 

Another interesting phenomenon can emerge under non-isothermal conditions for strongly exothermic reactions there will be multiple solutions to the coupled system of energy and mass balances even for the simplest first-order reaction. Such steady-state multiplicity results in the existance of several possible solutions for the steady state overall effectiveness factor, usually up to three with the middle point usually unstable. One should, however, note that the phenomenon is, in practice, rather rarely encountered, as can be understood from a comparison of real parameter values (Table 9.2). 

The observed effects of heat transfer on the flow in micro-nozzles are readily explained as follows. From compressible Rayleigh flow, it is known that removing heat from a supersonic flow acts to accelerate the flow. At steady-state, the bulk of the flow in the micro-nozzle expander is supersonic, and thus, heat transfer acts to further accelerate the supersonic flow. Concurrently, as the flow is cooled, the exit density p increases. The overall effect is an increase in thrust. Heat extraction from the flow into the substrate increases performance from the subsonic layer point of view as well. For low nozzle wall temperatures, the local sonic velocity is diminished and the near-wall Mach number increases. This phenomenon is the force driving the reduction in subsonic layer size for micro-nozzle flows with heat removal. In fact, with sufficient heat extraction from the flow, the subsonic layer can be reduced to the point where the competing effects of viscous forces and nozzle geometry cause the optimum expander half angle to be shifted from 30° to a more traditional expander half angle of 15°. This is demonstrated in Fig. 7 for isothermal wall temperatures less than 700 K. 

Figure 2.30 Overall effectiveness factor as a function of the Weisz modulus for different mass Biot numbers (isothermal, irreversible first order reaction in a porous slab). (Adapted from Ref. [16], Figure 4.17 Copyright 2012, Wiley-VCH GmbH Co. KGaA.)...

Since the reactant concentrations along the pores and within the particles are lower than the external surface concentrations, the overall effect of internal mass transfer resistances is to reduce the actually observed global rate below that measured at exterior surface conditions. It can be stated for isothermal effectiveness factors that r]concentration profile showing the pore diffusion-affected surface reaction is labeled as II in Figure 2.3. 

The general problem of diffusion-reaction for the overall effectiveness factor D is rather complicated. However, the physical and chemical rate processes prevailing under practical conditions promote isothermal particles and negligible external mass transfer limitations. In other words, the key transport limitations are external heat transfer and internal mass transfer. External temperature gradients can be significant even when external mass transfer resistances are negligibly small. 

Charles and Thomas (1963) in their paper considered firstly a spherical catalyst pellet of radius R and focus attention on a spherical shell of thickness dr and radius, as shown in Fig. 9. He assumes isothermal condition and that the complicated diffusion phenomena within the porous structure can be represented by a single overall effective diffusion coefficient, D ff. In the catalyst pellet, reactants are transported to the shell by diffusion and consumed... 

Based on the equations derived in Sections 4.5.3 and 4.5.4, we now define an overall effectiveness factor tjoveraih vvhich includes external and internal diffusion resistances. Here we only consider irreversible and reversible first-order isothermal reactions for more complex cases see Baerns et al. (2006) or Westerterp, van Swaaij, and Beenackers (1998). 

The value of 6 may then be determined without the need to assume a particular rate equation, by measuring the conversion in an isothermal reactor without diffusion limitations. The quantity 1-6 then arises as the ratio of the flow necessary to yield the same exit concentration of ammonia in the poisoned and unpoisoned cases. In practice, diffusional limitations may be greatly reduced by using preconverted synthesis gas in order to reduce the reaction rates. Low reaction rates would make it easier to control the temperature increase associated with an exothermic reaction. Diffusional limitations may also be reduced by using small-size catalyst particles. The overall effect is that the effectiveness factor comes very close to 1. See Chapter 7 of this book and Chapter 5 in the book by Nielsen. ... 

It is clear from the foregoing discussion that the general problem of diffusion-reaction for the overall effectiveness factor is quite involved. Fortunately, however, the physical and chemical processes at work under realistic conditions favor isothermal pellets and negligible external mass transfer resistances. A more detailed examination of this is in order. Combining Eqs. 4.32 and 4.33 results in ... 

Future work in this area will involve the extension of these techniques to other temperatures in an effort to better characterize the overall reaction kinetics of these two processes. In addition, degree of cure obtained through isothermal DSC measurements will be compared with the fraction of acetylene consumed as measured by isothermal FTIR experiments for the same temperature and time. Also, the effect of the incorporation of metal fillers on the isomerization and crosslinking reactions will be addressed. 

Figure 1 has shown that the maximum chemisorption of oxygen on chars from untreated wood occurs at HTT 450°-500°C. However, in order to understand better the effect of metal ions on the total process consisting of pyrolysis and subsequent chemisorption and oxidation of wood char, it was necessary to carry out pyrolysis, isothermal chemisorption and oxidation reactions in a single experiment. A typical overall pyrolysis, isothermal chemisorption (140°C) and oxidation curve is shown in Figure 2. The temperature program is (1) heat from 25° to 500°C at 5°C/min, (2) cool at...

If, however, both reactions were influenced by intraparticle diffusion effects, the rate of reaction of a particular component would be given by the product of the intrinsic reaction rate, fecg, and the effectiveness factor, Tj. Substituting eqn. (6) for the effectiveness factor gives (for a first-order isothermal reaction) the overall rate as 0tanh< >. As is often the case, the molecular weights of the diffusing reactants are similar and can be... 

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