Theory, penetration

Assuming the solid smtace is made of a substance with solubility, C in mg/m, is a convenient notion to understand the mass transfer. The bulk fluid far removed has a lesser concentration. Coo, same as does each fluid eddies that approaches the interface. All eddies that encounter the interface reside in direct contact for a time period, t, in seconds. While in contact for this short time-period, diffusive mass transport occurs from the solid surface into the adjoining fluid driven by the [Q - C x ] concentration differences. The entering chemical mass is viewed to be penetrating into the eddy. 

The first school began with Venczel s dissertation in Zurich in 196190 on the transport of ferric ion to an electrode evolving hydrogen gas from one molar sulfuric acid. Venczel found that mass transfer increased rapidly with the onset of gas evolution. Ibl and Venczel26 reported mass transfer at gas-evolving electrodes as Nemst boundary layer thicknesses that are functions of gas evolution rate 

Ibl and Venczel26 hypothesized that diffusion of the reactant from fresh electrolyte, brought to the surface after a bubble detaches, is the mechanism of mass transfer enhancement. They 

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Penetrating stains Penetration resistance Penetration theory Penetrometers Peng-Robinson equation DL-Pemcillamine Penicillamine [52-67-5]... 

Other Models for Mass Transfer. In contrast to the film theory, other approaches assume that transfer of material does not occur by steady-state diffusion. Rather there are large fluid motions which constantiy bring fresh masses of bulk material into direct contact with the interface. According to the penetration theory (33), diffusion proceeds from the interface into the particular element of fluid in contact with the interface. This is an unsteady state, transient process where the rate decreases with time. After a while, the element is replaced by a fresh one brought to the interface by the relative movements of gas and Uquid, and the process is repeated. In order to evaluate a constant average contact time T for the individual fluid elements is assumed (33). This leads to relations such as... 

The values of d and n are given in Table 3 typical values for can be found in Table 4. The exponent of 0.5 on the Schmidt number (l-L /PiLj) supports the penetration theory. Further examples of empirical correlations provide partial experimental confirmation of equation 78 (3,64—68). The correlation reflecting what is probably the most comprehensive experimental basis, the Monsanto Model, also falls in this category (68,69). It is based on 545 observations from 13 different sources and may be summarized as... 

The penetration theory predicts that should vary by the square root of the molecular difriisivity, as compared with film theoiy, which predicts a first-power dependency on D. Various investigators have reported experimental powers of D ranging from 0.5 to 0.75, and the Chilton-Colburn analogy suggests a 2/3 power. 

U. Single water drop in air, liquid side coefficient / jy l/2 ki = 2 ), short contact times / J 1 lcontact times dp [T] Use arithmetic concentration difference. Penetration theory, t = contact time of drop. Gives plot for k a also. Air-water system. [lll]p.. 389... 

The parameter values for the curves of Fig. 14-14 originally were defineci from film theory as (Dg/D )(B /vCi) but later were refined by the results of penetration theory to the definition (( ) — 1), where... 

The rate of mass transfer in the liquid phase in wetted-waU columns is highly dependent on surface conditions. When laminar-flow conditions prevail without the presence of wave formation, the laminar-penetration theory prevails. When, however, ripples form at the surface, and they may occur at a Reynolds number exceeding 4, a significant rate of surface regeneration develops, resulting in an increase in mass-transfer rate. 

Aeeording to the penetration theory, liquid elements or pareels remain in eontaet with the gas for a limited time and subsequently are mixed with the bulk. This eoneept is partieularly suited to bubble flows beeause it refleets their... 

The reaction engineering model links the penetration theory to a population balance that includes particle formation and growth with the aim of predicting the average particle size. The model was then applied to the precipitation of CaC03 via CO2 absorption into Ca(OH)2aq in a draft tube bubble column and draws insight into the phenomena underlying the crystal size evolution. 

Weekman and Myers (W3) measured wall-to-bed heat-transfer coefficients for downward cocurrent flow of air and water in the column used in the experiments referred to in Section V,A,4. The transition from homogeneous to pulsing flow corresponds to an increase of several hundred percent of the radial heat-transfer rate. The heat-transfer coefficients are much higher than those observed for single-phase liquid flow. Correlations were developed on the basis of a radial-transport model, and the penetration theory could be applied for the pulsing-flow pattern. 

The absorption is assumed to occur into elements of liquid moving around the bubble from front to rear in accordance with the penetration theory (H13). These elements maintain their identity for a distance into the fluid greater than the effective penetration of dissolving gas during the time required for this journey. The differential equation and initial and boundary conditions for the rate of absorption are then... 

According to the penetration theory, which assumes equal contact for each element (HI3), one can write that... 

These equations are equivalent, but the first converges rapidly for short times and the second for long times. The equations clearly show that for short times the penetration theory is approached ... 

The penetration theory holds for the region where t is much less than L2jD, the film theory for the region where t is much greater than L2/D. This comparison is shown in Fig. 8, which clearly shows that the film and penetration theories are asymptotes of the film-penetration model. 

Peclet number, 352 trickle operation, 92-93 gas phase, 94 liquid phase, 103 Penetration theory, 340 Polymers, 38-40 n-Propane, 178 Propellants, solid combustion, 4-50 flameless, 45... 

The work of Higbie laid the basis of the penetration theory in which it is assumed that the eddies in the fluid bring an element of fluid to the interface where it is exposed to the second phase for a definite interval of time, after which the surface element is mixed with the bulk again. Thus, fluid whose initial composition corresponds with that of the bulk fluid remote from the interface is suddenly exposed to the second phase. It is assumed that equilibrium is immediately attained by the surface layers, that a process... 

In an experimental wetted wall column, pure carbon dioxide, is absorbed in water. The mass transfer rate is calculated using the penetration theory, application of which is limited by the fact that the concentration should not teach more than 1 per cent of the saturation value at a depth below the surface at which the velocity is 95 per cent of the surface velocity. What is the maximum length of column to which the theory can be applied if the flowrate of water is 3 cm3/s per cm of perimeter ... 

In a gas-liquid contactor, a pure gas is absorbed in a solvent and the Penetration Theory provides a reasonable model by which to describe the transfer mechanism. As fresh solvent is exposed to the gas, the transfer rate is initially limited by the rate at which the gas molecules can reach the surface. If at 293 K and a pressure of 1 bar the maximum possible rate of transfer of gas is 50 m3/m2s, express this as an equivalent resistance, when the gas solubility is 0.04 kmol/m3. 

The penetration theory has been used to calculate the rate of mass transfer across an interface for conditions where the concentration CAi of solute A in the interfacial layers (y = 0) remained constant throughout the process. When there is no resistance to mass transfer in the other phase, for instance when this consists of pure solute A, there will be no concentration gradient in that phase and the composition at the interface will therefore at all Limes lie the same as the bulk composition. Since the composition of the interfacial layers of the penetration phase is determined by the phase equilibrium relationship, it, too. will remain constant anil the conditions necessary for the penetration theory to apply will hold. If, however, the other phase offers a significant resistance to transfer this condition will not, in general, be fulfilled. 

HARRIOTT 25 suggested that, as a result of the effects of interfaeial tension, the layers of fluid in the immediate vicinity of the interface would frequently be unaffected by the mixing process postulated in the penetration theory. There would then be a thin laminar layer unaffected by the mixing process and offering a constant resistance to mass transfer. The overall resistance may be calculated in a manner similar to that used in the previous section where the total resistance to transfer was made up of two components—a Him resistance in one phase and a penetration model resistance in the other. It is necessary in equation 10.132 to put the Henry s law constant equal to unity and the diffusivity Df in the film equal to that in the remainder of the fluid D. The driving force is then CAi — CAo in place of C Ao — JPCAo, and the mass transfer rate at time t is given for a film thickness L by ... 

Thus either the penetration theory or the film theory (equation 10.144 or 10.145) respectively can be used to describe the mass transfer process. The error will not exceed some 9 per cent provided that the appropriate equation is used, equation 10.144 for L2 jDt > n and equation 10.145 for L2/Dt < n. Equation 10.145 will frequently apply quite closely in a wetted-wall column or in a packed tower with large packings. Equation 10.144 will apply when one of the phases is dispersed in the form of droplets, as in a spray tower, or in a packed tower with small packing elements. 

The penetration and film-penetration theories have been developed for conditions of equimolecular counterdiffusion only the equations are too complex to solve explicitly for transfer through a stationary carrier gas. For gas absorption, therefore, they apply only when the concentration of the material under going mass transfer is low. On the other hand, in the two-fihn theory the additional contribution to the mass transfer which is caused by bulk flow is easily calculated and hp (Section 10.23) is equal to (D/L)(Cr/Cum) instead of D/L. 

These relations between the various coefficients are valid provided that the transfer rate is linearly related to the driving force and that the equilibrium relationship is a straight line. They are therefore applicable for the two-film theory, and for any instant of time for the penetration and film-penetration theories. In general, application to time-averaged coefficients obtained from the penetration and film-penetration theories is not permissible because the condition at the interface will be time-dependent unless all of the resistance lies in one of the phases. 

As noted previously, for equimolecular counterdiffusion, the film transfer coefficients, and hence the corresponding HTUs, may be expressed in terms of the physical properties of the system and the assumed film thickness or exposure time, using the two-film, the penetration, or the film-penetration theories. For conditions where bulk flow is important, however, the transfer rate of constituent A is increased by the factor Cr/Cgm and the diffusion equations can be solved only on the basis of the two-film theory. In the design of equipment it is usual to work in terms of transfer coefficients or HTUs and not to endeavour to evaluate them in terms of properties of the system. 

The solution of this equation has been discussed by DANCKWERTSt28), and here a solution will be obtained using the Laplace transform method for a semi-infinite liquid initially free of solute. On the assumption that the liquid is in contact with pure solute gas, the concentration Cm at the liquid interface will be constant and equal to the saturation value. The boundary conditions will be those applicable to the penetration theory, that is ... 

According to the simple penetration theory the instantaneous mass flux, (AfOt is ... 

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