Instantaneous reaction enhancement factor

For a semi-batch operation, the liquid-solid mass-transfer coefficient can also be obtained by monitoring a reaction between the dissolving solid B and a liquid reactant C. If this reaction is instantaneous, the enhancement factor for the reaction is... 

The parameter p (= 7(5 ) in gas-liquid sy.stems plays the same role as V/Aex in catalytic reactions. This parameter amounts to 10-40 for a gas and liquid in film contact, and increases to lO -lO" for gas bubbles dispersed in a liquid. If the Hatta number (see section 5.4.3) is low (below I) this indicates a slow reaction, and high values of p (e.g. bubble columns) should be chosen. For instantaneous reactions Ha > 100, enhancement factor E = 10-50) a low p should be selected with a high degree of gas-phase turbulence. The sulphonation of aromatics with gaseous SO3 is an instantaneous reaction and is controlled by gas-phase mass transfer. In commercial thin-film sulphonators, the liquid reactant flows down as a thin film (low p) in contact with a highly turbulent gas stream (high ka). A thin-film reactor was chosen instead of a liquid droplet system due to the desire to remove heat generated in the liquid phase as a result of the exothermic reaction. Similar considerations are valid for liquid-liquid systems. Sometimes, practical considerations prevail over the decisions dictated from a transport-reaction analysis. Corrosive liquids should always be in the dispersed phase to reduce contact with the reactor walls. Hazardous liquids are usually dispensed to reduce their hold-up, i.e. their inventory inside the reactor. 

Enhancement factor E. For reaction occurring only in the liquid film, whether instantaneous or fast, the rate law may be put in an alternative form by means of a factor that measures the enhancement of the rate relative to the rate of physical absorption of A in the liquid without reaction. Reaction occurring only in the liquid film is characterized by cA - 0 somewhere in the liquid film, and the enhancement factor E is defined by... 

Note that the enhancement factor E is relevant only for reaction occurring in the liquid film. For an instantaneous reaction, the expressions may or may not involve E, except that for liquid-film control, it is convenient, and for gas-film control, its use is not practicable (see problem 9-12(a)). The Hatta number Ha, on the other hand, is not relevant for the extremes of slow reaction (occurring in bulk liquid only) and instantaneous reaction. The two quantities are both involved in rate expressions for fast reactions (occurring in the liquid film only). 

Situation 4 very fast reaction, Ha> 3 and Eboundary layer. The hydrogen concentration in the bulk of the liquid falls to zero. Thus, all the catalyst in the bulk is useless. For instantaneous reactions, Ha 3, E=EX and the reaction takes place in a narrow plane located somewhere in the boundary layer the larger Ea0 the closer to the interface the reaction plane. If the limiting enhancement factor E is very high, it is said that the reaction takes place at the gas-liquid interface. Such a case is referred to as surface reaction . 

Figures 7(a)-(c) show a comparison between the numerically computed absorption flux and the absorption flux obtained from expression (31), using eqs (24), (30) and (34)-(37). From these figures it can be concluded that for both equal and different binary mass transfer coefficients absorption without reaction can be described well with eq. (24), whereas absorption with instantaneous reaction can be described well with eq. (30). If the Maxwell-Stefan theory is used to describe the mass transfer process, the enhancement factor obeys the same expression as the one obtained on the basis of Fick s law [eq. (35)]. Finally, from Figs 7(b) and 7(c) it appears that the use of an effective mass transfer coefficient m the Hatta number again produces satisfactory results.

Regime 5 - instantaneous reactions at an reaction plane developing inside the film For very high reaction rates and/or (very) low mass transfer rates, ozone reacts immediately at the surface of the bubbles. The reaction is no longer dependent on ozone transfer through the liquid film kL or the reaction constant kD, but rather on the specific interfacial surface area a and the gas phase concentration. Here the resistance in the gas phase may be important. For lower c(M) the reaction plane is within the liquid film and both film transfer coefficients as well as a can play a role. The enhancement factor can increase to a high value E > > 3. 

The enhancement factors are either obtained by fitting experimental results or are derived theoretically on the grounds of simplified model assumptions. They depend on reaction character (reversible or irreversible) and order, as well as on the assumptions of the particular mass transfer model chosen [19, 26]. For very simple cases, analytical solutions are obtained, for example, for a reaction of the first or pseudo-first order or for an instantaneous reaction of the first and second order. Frequently, the enhancement factors are expressed via Hatta-numbers [26, 28]. They can be used in combination with the HTU/NTU-method or with a more advanced mass transfer description method. However, it is generally not possible to derive the enhancement factors properly from binary experiments, and a theoretical description of reversible, parallel or consecutive reactions is based on rough simplifications. Thus, for many reactive absorption processes, this approach appears questionable. 

Certain fast or instantaneous reactions (for example, proton transfer reactions) are always mass-transfer controlled. The enhancement factor, I, acconnts for the effect of chemical reactions on mass transfer and is dehned as the ratio of the process rate over the mass-transfer rate in absence of a chemical reaction. 

So far no attention has been given in this chapter on the effect of the diffusivities. Often instantaneous reactions involve ionic species. Care has to be taken in such case to account for the influence of ionic strength on the rate coefficient, but also on the mobility of the ions. For example, the absorption of HCI into NaOH, which can be represented by H + OH HjO. This is an instantaneous irreversible reaction. When the ionic diffusivities arc equal the diffusivities may be calculated from Pick s law. But, H and OH have much greater mobilities than the other ionic species and the results may be greatly in error if based solely on molecular diffusivities. This is illustrated by Fig. 6.3c-2, adapted from Sherwood and Wei s [4] work on the absorption of HCI and NaOH by Danckwerts. The enhancement factor may be low by a factor of 2 if only molecular diffusion is accounted for in the mobility of the species. Important differences would also occur in the system HAc-NaOH. When CO2 is absorbed in dilute aqueous NaOH the effective diffusivity of OH is about twkx that of COj. 

For intermediate reaction rates the use of the enhancement factor is not consistent with the standard approach of diffusional limitations in reactor design and may be somewhat confusing. Furthermore, there are cases where there simply is no purely physical mass transfer process to refer to. For example, the chlorination of decane, which is dealt with in the coming Sec. 6.3.f on complex reactions or the oxidation of o-xylene in the liquid phase. Since those processes do not involve a diluent there is no corresponding mass transfer process to be referred to. This contrasts with gas-absorption processes like COj-absorption in aqueous alkaline solutions for which a comparison with C02-absorption in water is possible. The utilization factor approach for pseudo-first-order reactions leads to = tfikC i and, for these cases, refers to known concentrations C., and C . For very fast reactions, however, the utilization factor approach is less convenient, since the reaction rate coefficient frequently is not accurately known. The enhancement factor is based on the readily determined and in this case there is no problem with the driving force, since Cm = 0- Note also that both factors and Fji are closely related. Indeed, from Eqs. 6.3.C-5 and 6.3.C-10 for instantaneous reactions ... 

So far, only pseudo-first-order and instantaneous second-order reactions were discussed. In between there is the range of truly second-order behavior for which the continuity equations for A (Eq. 6.3.a-l) or B (Eq. 6.3.a-2), cannot be solved analytically, only numerically. To obtain an approximate analytical solution. Van Krevelen and Hoftijzer [3] dealt with this situation in a way analogous to that apfdied to pseudo-first-order kinetics, namely by assuming that the concentration of B remains approximately constant close to the interface. They mainly considered very fast reactions encountered in gas absorption so that they could set Cm - 0, that is, the reaction is completed in the film. Their development is in terms of the enhancement factor, F. The approximate equation for is entirely analogous with that obtained for a pseudo-first-order reaction Eq. 

This approximate solution is valid to within 10 percent of the numerical solution. Obviously when Cgi, > C, then y = y and the enhancement factor equals that for pseudo-first-order. When this is not the case f is now obtained from an implicit equation. Van Krevelen and Hoftijzer solved Eq. 6.3.e-l and plotted F versus y in the diagram of Fig. 6.3.C-2, given in Sec. 6.3.c connecting the results for pseudo-first-order and instantaneous second-order reactions. 

It can be seen from this equation that the reversibility of the reaction can have an important effect on the enhancement factor compared to the corresponding irreversible case with the same y-value. Instantaneous reversible reactions were studied by Olander [13],... 

Unlike the enhancement factor of Equation 14.19 or 14.20, that of the above equation refers to a terminal value r] that corresponds to the enhancement caused by an instantaneous reaction. Therefore, it is referred to as the asymptotic enhancement factor. Because this regime involves a reaction plane, it is reasonable to expect that the reaction plane will move with time. It is possible to allow for this transient situation by invoking the penetration theory (Karlsson and Bjerle, 1980). A comparison of the expressions based on the film and penetration theories shows that... 

In Chapter 14 we saw that the enhancement factor for this regime is an asymptotic value because it corresponds to the extreme case of enhancement due to an instantaneous reaction. The enhancement for no reaction is obviously one and that corresponding to any other regime will lie between these two asymptotes. 

We have qualitatively established so far that, when ( ) <<1, the rate enhancement factor is unity, while when 6 its value is very large and is given by Eq.8. Clearly, an intermediate region exists where ())>>1, and therefore I is appreciably larger than unity, yet the reaction is not so fast as to be instantaneous, and therefore I is appreciably less than This intermediate region... 

While the issue of chemical kinetics can be avoided in the slow reaction regime (since the rate of reaction is so slow that its actual value needs not be known), and in the instantaneous reaction regime (since the rate of reaction is so fast that, again, its value needs not be known), it cannot be avoided in the case of the fast reaction regime. However, considerable simplifications arise also in this limiting case, and the following simple equation is obtained for the enhancement factor I ... 

Fig. 6 indicates an initial 22% reduction in the enhancement factor as the temperature and solubility adjust instantaneously at t = o. Thereafter, with k = 10 s" as the time of exposure increases, the enhancement factor continues to decline to a value of 0.65 after 100 milliseconds. The chemical reaction is therefore seen to significantly hinder the absorption process, rather than to enhance it. To some extent an enhancement factor of less than unity is a misnomer. In the complete absence of heat effects the enhancement factor would have become 1.25. The hindrance of the chemical reaction intensifies as the rate constant is made larger. In Fig. 6, the limit of reasonable applicability of the linear solubility approximation is shown by reverting to a broken line. 

In the instantaneous regime the enhancement factor and reaction factor are identical and Eqn (53) applies. The authors (18) used the following generalised expression... 

Since the earlier treatments of this problem by Ramachandran and Sharma(4) and Uchida et.al.(7).several experimental studies and verifications of predictions of enhancement factors have been reported(7,15,16) several detailed models based on film concept have also been proposed(7-12).Recently a penetration model for an instantaneous irreversible chemical reaction has also been presented.which however differs numerically only negligibly than the film model(13).The most important modification of Ramachandran and Sharma s treatment is due to Uchida et. al.(7-9) who consider that the rate of solid dissolution may be accelerated by the absorption of gas as discussed above.They have also considered the case where the concentration of solid component in the bulk liquid phase may not be maintained at the saturation solubility(that is,"finite" slurry) which occurs of course when the rate of solid dissolution is relatively slow compared with gas absorption rate(8).The case where the solid dissolution is finite was further considered by Sada et.al.(12) both theoretically and experimentally.Uchida et.al.(8) could also explain the data of Takeda et.al.(14) by their modified model.Analytical solutions presented above are for instantaneous reactions ... 

Ej = enhancement factor for instantaneous reaction b = number of moles of B reacting with 1 mol of A... 

If the rate is slow, Hatta modulus is small. If the rate is fast, the Hatta modulus is large. For very large Hatta modulus, the enhancement factor is equal to Hatta modulus, since tanh oo = i. Finally, for instantaneous reactions, the enhancement factor has a limiting value given by Equation 6.80. A qualitative plot of enhancement factor versus Hatta modulus is given in Figure 6.12. The merits of the figure and how it can be used for equipment selection will be discussed in Chapter 11. 

A classical description of the selection of the equipment is given by Krishna and Sie (1994). The qnalitadve enhancement factor versus Hatta number diagram shown in the sidebar figure is classified for the reaction rate (slow, fast, or instantaneous) and a dispersion ratio J3, the ratio of the liquid phase volume to the volume of the diffusion layer. 

If the reaction in the liquid film leads to an increase in the concentration gradient of the reactant at the interface, the mass transfer of the absorbed gas from the interface into the liquid phase is enhanced compared to the absence of reaction. This effect is considered by the enhancement factor , which depends on Ha. For a slow reaction (small Ha), the mass transfer rate is not enhanced. For a fast reaction, the conversion mostly takes place in the liquid film. For an instantaneous reaction, the absorbed reactant A and the liquid reactant B do not coexist, and conversion takes place at a reaction plane. 

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