Van Krevelen and Hoftijzer

Pag), where y o mole fraction of A in bulk gas phase can be determined iteratively, yAi = mole fiaction of A in gas inlet. Equations (1) to (6) were solved using fourth order Runge-Kutta method [1, 8]. The value of enhancement factor, E, was predicted using equation of Van Krevelen and Hoftijzer [2]. 

In this case, an exact analytical solution of the continuity equations for A and B does not exist. An approximate solution has been developed by Van Krevelen and Hoftijzer (1948) in terms of E. Results of a numerical solution could be fitted approximately by the implicit relation... 

An approximate implicit solution (van Krevelen and Hoftijzer, 1948) for the enhancement... 

An approximate analytical solution to this system has been proposed by van Krevelen and Hoftijzer (Eq. (27) and Fig. 45.3)) ... 

Van Krevelen and Hoftijzer developed an empirical correlation in which the coefficient, on an ionic strength basis, is considered to be the sum of contributions from the cation, the anion, and the gas(24) ... 

This equation closely resembles the empirical expression of Van Krevelen and Hoftijzer (VI) for the bubble formation in inviscid liquids, provided that the gas density is negligible compared to the liquid density. Their relationship... 

This simplified equation is of the same form as the equations of Van Krevelen and Hoftijzer (VI) for air-water system and the theoretical equation of Davidson and Schuler (D9) for inviscid liquids. It is interesting to observe that the various equations differ only in the value of the constant although they are based on different mechanisms. 

The simplest method of representing data for gas-film coefficients is to relate the Sherwood number [(hod/Dv)(PBm/Z3)] to the Reynolds number (Re) and the Schmidt number (p,/pDv). The indices used vary between investigators though van Krevelen and Hoftijzer(28) have given the following expression, which is claimed to be valid over a wide range of Reynolds numbers ... 

The difference between a physical absorption, and one in which a chemical reaction occurs, can also be shown by Figures 12.11a and 12.11 b, taken from a paper by van Krevelen and Hoftijzer(28 Figure 12.11a shows the normal concentration profile for... 

Figure 23.4 The enhancement factor for fluid-fluid reactions as a function of Mf and modified from the numerical solution of van Krevelens and Hoftijzer (1954).

Comment. In this example we see that two distinct zones are present. Situations may be encountered where even another zone may be present. For example, if the entering liquid contains insufficient reactant, a point is reached in the tower where all this reactant is consumed. Below this point physical absorption alone takes place in reactant-free liquid. The methods of these examples, when used together, deal in a straightforward manner with this three-zone situation and van Krevelens and Hoftijzer (1948) discuss actual situations where these three distinct zones are present. 

The enhancement factor can be evaluated from equations originally developed by Van Krevelen and Hoftijzer (1948). A convenient chart based on the equations is shown in Fig. 6. The parameter for the curves is — 1, where 0V is the enhancement factor as Ha approaches... 

The data have, however, been obtained for only a few systems by Onda et al. (J), Joosten and Danckwerts (2), and Hikita et al. (3). These authors also proposed methods for estimating the solubility of gases in aqueous mixed-salt solutions from the corresponding data for each salt in the systems. These studies are extensions of the empirical method proposed by van Krevelen and Hoftijzer (4) on the basis of the following modified Setschenow Equation. 

The range of applicability of the Setschenow Equation on the salt concentration in aqueous single-salt solutions varies with the system (gas plus an electrolyte) and is never confirmed clearly. Van Krevelen and Hoftijzer (4) showed the range to be up to 2 mol/L of ionic strength in all the systems, while Onda et al. (5) showed that the equation could be applied to the more concentrated solutions for some systems, such as up to 15 mol/L of ionic strength for carbon dioxide systems at the maximum. 

The estimating methods by van Krevelen and Hoftijzer (4) and Onda et al. (I) are based on the linear relationship between log(L0/L) and salt concentration. When these methods are used to estimate the solubility of gases in aqueous salt solutions over a wide range of salt concentration, however, the estimates are sometimes in serious error, as shown in Figures 3 and 4. 

Equations (14-206) and (14-207) result from a balance of bubble buoyancy against interfacial tension. They include no inertia or viscosity effects. At low bubbling rates (carbon tetrachloride for vertically oriented orifices with 0.004 < D < 0.95 cm. If the orifice diameter becomes too large, the bubble diameter will be smaller than the orifice diameter, as predicted by Eq. (14-206), and instability results consequently, stable, stationary bubbles cannot be producecl. 

Van Krevelen and Hoftijzer (Vl) studied the time of passage of three different materials in a small drier 10 cm in diameter and 76 cm long. The materials studied were nitrochalk fertilizer granules, sand, and marl powder. 

This expression, which was first derived by Van Krevelen and Hoftijzer (35), cannot be solved explicitly for /cl- It is shown graphically in Figure 5. 

As explained in Section II,B,4, Van Krevelen and Hoftijzer (V4) computed approximated solutions for the film model, expressed by Eq. (26). It is interesting to note that other correlations giving implicit or explicit dependence between the enhancement factor E and the parameters Ha and El have been developed. Hikita and Asai (H7) have proposed an implicit expression for the Higbie model ... 

Van Krevelen and Hoftijzer (V4), Barrett (B7), and Ondaet al. (08) have evaluated h for various species. The values presented in Tables I and II, which extend to ionic strengths of 2 A/ and greater, are characteristic of industrial systems. The effect of pressure on ha values is small up to 200 atm (08). 

The frequency of gas bubbles which are formed steadily through an orifice in a fluidized bed has been studied by Harrison and Leung (H6). Their results show that the mechanism of chainlike bubble formation is the same as that in an inviscid liquid. If all of the excess gas above the minimum fluidization velocity passes through in the form of gas bubbles, the diameter of a sphere having the same volume as the originated bubble is represented by two equations (in units of cm/sec). Van Krevelen and Hoftijzer (V6) found that... 

This differential equation was solved numerically by van Krevelen and Hoftijzer [300] with the simplifying assumption of an idealized concentration profile, as can be seen in Fig. 4.13. The following three ranges could be distinguished ... 

The more general case is illustrated in Figure 7.28a. The reactions are of finite velocity and it is possible for A and B to coexist in a zone of reaction as shown. This was first analyzed by Van Krevelen and Hoftijzer [D.W. Van Krevelen and P.J. Hoftijzer, Rec. Trav. Chim., 67, 563 (1948)], who showed that the film theory solution with concentration boundary conditions specified as Ca = C. at the interface and Ca = Ca. Cq = at L (in effect, a bulk phase concentration) was well approximated by... 

Using the analysis of van Krevelen and Hoftijzer, determine the rate of absorption of CO2 into a NaOH solution under the following conditions. 

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. 

Porter [7] and also Kishinevskii et al. [8] derived approximate equations for the enhancement factor that were found by Alper [9] to be in excellent agreement with the Van Krevelen and Hoftijzer equation (for Porter s equation when y 2) and which are explicit. Porter s equation is ... 

For an irreversible reaction of global order m + n(m with respect to A, n with respect to S), the approach followed by Hikita and Asai [10] was very similar to that of Van Krevelen and Hoftijzer. The rate of reaction was written as ... 

Derivations were given for reversible, consecutive and parallel reactions with any order by Onda et al. [19, 20, 21,22], Onda et al. assumed that the concentrations at y = yj, are the equilibrium values corresponding to the reversible reaction in the bulk. The development was analogous to that of Van Krevelen and Hoftijzer [3] and Hikita and Asai [10]. This led to approximate expressions for the enhancement factor giving values in close agreement with those obtained by numerical integration. 

The parameter curves in the Van Krevelen and Hoftijzer diagram were originally, according to the film theory, (a/h)( >B/D )(CB/C i). To account for the results of the penetration theory, indicating that the mass transfer coefficients are proportional to the square roots of the difiusivities, Danckwerts and co-workers have used (alb)(CB/CAi) /Db/Da as the parameter group in the Van Krevelen and Hoftijzer diagram. The value of the tatter group amounts to... 

It is obvious from this previous equation that the solubility values of some salts could be represented by a straight line relationship. According to the model of Van Krevelen and Hoftijzer (50), the ionic strength I is introduced as a better measure of electrolyte activity and the salting-out constant is considered to be the result of contributions from the various gas species (h ), and ions (h, h ) present ... 

Onda et al (21) then used the linearisation method of Hikita and Asai (22) (which is analogous to the Van Krevelen and Hoftijzer treatment (17)), so that Eqn (70) becomes... 

A graphical representation of these equations is given in Fig. 6.4-14. van Krevelen and Hoftijzer originally developed their correlation only for irreversible second-order reactions (first-order in each reactant) and for equal difiusivities of the two reactants. Danckwerts pointed out that the results also are applicable to the case where />a is not equal to Of. Decoursey developed an approximate solution for absoiption with irreversible second-order reaction based on the Danckwerts surface-renewal model. The resulting expression, which is somewhat easier to use than the van Krevelen-Hoftijzer approach, is... 

At this point a diagram can be constructed showing Fa as a function of y, as first given by Van Krevelen and Hoftijzer [1948], but only for the case of no reaction in the bulk (Fig. 6.3.2-1). The other curves in the diagram pertain to... 

Equation (6.3.3-10) is also represented in Fig. 6.3.2-1. Since Fa is independent of y in the present case, a set of horizontal lines with a/b Db/D Cbi/Ca as a parameter is obtained. The curves in the central part that connect the lines for infinitely fast reactions to the curve for a pseudo-first-order reaction correspond to moderately fast second-order reactions. They were calculated by Van Krevelen and Hoftijzer [1948] under the assumption that B is only weakly depleted near the interface. For moderately fast reactions, this assumption was reasonably confirmed by more rigorous computations. 

---