Higbie equation

The resulting ktjdb) plot in Fig. 4.11 proves that bubbles with db < 0.15 mm follow the Frdsling equation i.e. are rigid. For larger diameters the bubbles begin to rotate, whereby kt also increases, kt attains its maximum at db = 2 mm. The Higby equation applies to db > 2 mm. 

To make a valid comparison between these two cases, one needs to express transient diffusion in ferms of an equivalent mass transfer coefficient Icq. That is provided by the Higbie equation that we derive in Chapter 4 and reproduce below ... 

Illustration 4.5 Cumulative Uptake by Diffusion for the Semi-Infinite Domain The Higbie Equation... 

A notable exception occurs in the case of mass transfer from rising gas bubbles to a surrounding liquid. Here one sees a shift of the resistance in its entirety to the external medium, even if it is in a state of turbulence. For single bubbles, the Higbie equation (Equation 4.11g) provides the appropriate kz value. Mass transfer from swarms of bubbles is taken up in Illustration 5.5. 

Assuming that all the surface elements are exposed for the same time te (Higbie s assumption), from equation 10.113, the moles of A (nA) transferred at an area A in time... 

Equations 10.142 and 10.143 give the point value of NA at time t. The average values Na can then be obtained by applying the age distribution functions obtained by Higbie and by Danckwerts, respectively, as discussed Section 10.5.2. 

The penetration theory is attributed to Higbie (1935). In this theory, the fluid in the diffusive boundary layer is periodically removed by eddies. The penetration theory also assumes that the viscous sublayer, for transport of momentum, is thick, relative to the concentration boundary layer, and that each renewal event is complete or extends right down to the interface. The diffusion process is then continually unsteady because of this periodic renewal. This process can be described by a generalization of equation (E8.5.6) ... 

Instead of determining 4, in equation (8.33), we must determine r in equation (8.34). Although the difference between Higbie s penetration theory and Danckwerts surface renewal theory is not great, the fact that a statistical renewal period would have a similar result to a fixed renewal period brought much credibility to Higbie s penetration theory. Equation (8.34) is probably the most used to date, where r is a quantity that must be determined from the analysis of experimental data. 

This provides an equation which has the same form as that obtained on the basis of the Higbie penetration theory ... 

While Higbie proposed the model only for the liquid phase, it has been extended to include the gas phase. To satisfy Pick s theory, Higbie assumed that, at the time of exposure, the transfer process involved only molecular diffusion or, alternatively, the packets assumed a laminar character as they approached the interface. The net result of Higbie s model provides the flux equation ... 

Higbie was the first to apply this equation to gas absorption in a liquid, showing that diffusing molecules will not reach the other side of a thin layer if the contact time is short. The depth of penetration, defined as the distance at which the concentration change is 1 percent of the final value, is 3.6 /d. For... 

The transfer efficiency in the drop-formation region varies approximately as the square root of the drop-formation time and inversely as the drop diameter. Since the drop acceleration interval is quite short, acceleration effects are normally combined with drop-formation effects. Heertjes (HI3), basing his analysis on Higbie s penetration theory, suggested equations for the drop-formation and coalescence regions. For the drop-formation region. 

Astarita (1967) has combined the Higbie and Danckwerts approaches to give the following general equation for the mass transfer coefficient ... 

A relatively simple model to describe the gas-liquid mass transfer in circular channels with slug flow pattern was proposed by van Eaten and Krishna [47]. For their fundamental model the authors considered an idealized geometry of the Taylor bubbles as shown in Figure 7.12. The bubbles consist of two hemispherical caps and a cylindrical body. The Higbie penetration model was applied to describe the mass transfer process of a compound from the gas phase to the liquid (Equation 7.8). For a rising bubble, the liquid will flow along the bubble surface of the cap. The average distance... 

For the case of absorption accompanied by a chemical reaction, from the Higbie s theory, 0(0) is deduced from the solution of the equation,... 

It is now justifiable to solve this equation using the boundary conditions of the simplified physical absorption models (1,2). Solution of Eqn (2) then enables us to calculate the absorption rate at a particular "point" in the absorber. These results are usually expressed in terms of an enhancement factor, i.e. a factor by which the rate of absorption is increased by the chemical reaction. It is well known that this enhancement factor differs little in value whether film or the Higbie or Danckwerts surface renewal models are used as the basis of calculation. Figure (2) shows the typical representation of the effect of chemical reaction for a second order (r=k2A B) irreversible reaction. With the exception of Region IV, all regions are amenable to analytical solutions. In fact, the enhancement factor predic-... 

The enhancement of mass transfer due to chemical reaction depends on the order of the reaction as well as its rate. Order is defined as the sum of all the exponents to which the concentrations in the rate equation are raised. In elementary reactions, this number is equal to the number of molecules involved in the reaction however, this is only true if the correct reaction path has been assumed. Danckwerts presents a review of many cases of importance in gas absorption operations. He compares the results of using the film model and the Higbie and Danckweits surface-renewal models and concludes that, in general, the predictions based on the three models are quite similar. Mass transfer rate equations for a few of the cases encountered in a gas absorption operation are summarized in the following paragraphs, which are based primarily on discussions presented by Danckweits. ... 

Enhancement fectors for second-order reaction for quiescent liquid or agitated liquid (film or Higbie models). Based on equation... 

The basics of charge transfer may also be presented in the form of two analogies. One involves using equations that describe the collision mechanics between particles and the wall, as presented by Timoshenko (1951) and developed by Soo (1967). This is quite similar to the basic heat-transfer analysis. The second approach is to use the penetration theory as given by Higbie (1935) and Danckwerts (1951) for heat, mass, and momentum transfer for the analysis of charge transfer. 

If u in the foregoing equation is replaced by the average velocity uo, it becomes Higbie penetration model. Here, we considered a disturbance velocity Su is added to uq as follows ... 

This equation, due to Higbie, was originally derived to describe mass transfer between rising gas bubbles and a surrounding liquid Tran. AIChE, 31,368 [1935]). It applies quite generally to situations where the contact time between the phases is short and the penetration (or depletion) depth is so small that transfer may be viewed as taking place from a plant to a semiinfinite domain. In Section 4.1.2.3 we will provide a quantitative criterion for this approach, which is also referred to as the Penetration Theory. It also describes both the short- and long-term behavior in diffusion between a plane and a semi-infinite space, and we used this property in Chapter 1, Table 1.4, to help us set upper and lower bounds to mass transfer coefficients and "film" thickness Zp j. 

What Higbie s solution does not do is provide the detailed concentration profiles that result in these cases. To obtain that information, one must return to the fundamental solution (Equation 4.9a). 

Furthermore, van Baten and Krishna [55] formulated an equation for the description of kia, which is based on the fundamental approach. The authors combined the Higbie penetration model to describe mass transfer from the two hemispherical caps and the Pigford model to describe the mass transfer to the liquid film [56]. Furthermore, they used CFD simulations for studying the separate influence of different parameters on the volumetric mass transfer coefficient and compared these results with the predictions of Equation 12.29. 

Because of veiy small difference between the results obtained after the models of Higbie [18] and Danckwerts [19], it is offered [20 p.20] for both of them to use equivalent diftiision time 0 calculated by the equation ... 

According to the Higbie model and equation (243), the equation for the liquid-side controlled mass transfer is as follows ... 

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