Penetration theory assumptions

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 ... 

Ammonia is absorbed in a falling film of water in an absorption apparatus and the film is disrupted and mixed at regular intervals as it flows down the column. The mass transfer rate is calculated from the penetration theory on the assumption that all the relevant conditions apply. It is found from measurements that the muss transfer rate immediately before mixing is only 16 pet cent of that calculated from the theory anil the difference has been attributed to the existence of a surface film which remains intact and unaffected by the mixing process. If the liquid mixing process lakes place every second, what thickness of surface film would account for the discrepancy, ... 

Explain the basis of the penetration theory for mass transfer across a phase boundary. What arc the assumptions in the theory which lead to the result that the mass transfer rate is inversely proportional to the square root of the time for which a surface element has been expressed (Do not present a solution of the differential equal ion.) Obtain the age distribution function for the surface ... 

On the. assumptions involved in the penetration theory of mass transfer across a phase boundary, the concentration Ca of a solute A at a depth v below the interface at a time l after the formation of the interlace is given by ... 

State the assumptions made in the penetration theory for the absorption of a pure gas inlo a liquid. The surface of an initially solute-free liquid is suddenly exposed to a soluble gas and the liquid is sufficiently deep for no solute to have time to reach the bottom of the liquid. Starting with Hick s second law ol diffusion obtain an expression for (i) the concentration, and (ii) the muss transfer rate at a time t and a depth v below the surface. 

The use of penetration theory requires that the following assumptions be made ... 

The theories vary in the assumptions and boundary conditions used to integrate Fick s law, but all predict the film mass transfer coefficient is proportional to some power of the molecular diffusion coefficient D", with n varying from 0.5 to 1. In the film theory, the concentration gradient is assumed to be at steady state and linear, (Figure 3-2) (Nernst, 1904 Lewis and Whitman, 1924). However, the time of exposure of a fluid to mass transfer may be so short that the steady state gradient of the film theory does not have time to develop. The penetration theory was proposed to account for a limited, but constant time that fluid elements are exposed to mass transfer at the surface (Higbie, 1935). The surface renewal theory brings in a modification to allow the time of exposure to vary (Danckwerts, 1951). 

What is the penetration theory for mass transfer across a phase boundary Give details of the underlying assumptions. 

Penetration theory often is used in analyzing absorption with chemical reaction because it makes no assumption about the depths of penetration of the various reacting species, and it gives a more accurate result when the diffusion coefficients of the reacting species are not equal. When the reaction process is very complex, however, penetration theory is more difficult to use than film theory, and tne latter method normally is preferred. 

The instantaneous mass transfer rate is expressed as a function of time. In order to calculate an average mass transfer coefEcient we need to average the instantaneous coefEcient over the total exposure time period. To do this we need to know the age distribution function, which represents the fraction of elements having ages between t and t + dt t the surface. In the penetration theory, it is assumed that all the elements reside at the interface for a time period of the same length. As a consequence of this assumption the age distribution function is [6] ... 

The classical Danckwerts surface-renewal model is analogous to the penetration theory. The improvement is in the view of the eddy replacement process. Instead of Higbies assumption that all elements have the same recidence time at the interface, Danckwerts [29] proposed to use an averaged exposure time determined from a postulated time distribution. The recidence time distribution of the surface elements is described by a statistical distribution function E(t), defined so that E(t)d,t is the fraction of the interface elements with age between t and t + dt. The rest of the formulation procedure is similar to that of the penetration model. 

Further retaxation of the assumptions of the film aed penetration theories was suggested by Danckwerta1 who viewed the process as one of transient one-dimensioual diffusion to packets or elements of fluid that reside at the phase interface for varying periods of time. Therefore, the model is that of the penetration theory with a distribution of conract times, The surface age distribution p(t) is defined such that 4(i)tb is the fraction of surface that has seslded at the interface for a time between I and t + di. The mass transfer flua for the entire surface is obtained by integration or the instantaneous flux over all exposure times ... 

Objectives and Assumptions. In an effort to complement developments from earlier sections, mass transfer across a gas-liquid interface into a quiescent liquid is considered here. This problem differs slightly from the classic unsteady-state penetration theory analysis because the bubble radius R(t) depends on time, for example, due to the fact that the bubble contracts or expands. The overall objectives are to predict the following quantities ... 

Lekhal et al. [6] proposed a pseudo-homogeneous gas-liquid-liquid model based on the Higbie penetration theory to account for simultaneous absorption of two gases into the liquid phases. Because of the assumption of rapid liquid-liquid mass transfer of reactants leading to the equilibrium between two liquid phases, the model was simplified greatly and the detail of phase dispersion and distribution and multiphase flow was avoided. Reasonable success was achieved and the results of analysis suggested that the only limitation to the conversion of hydroformylation of 1-octene was the gas-liquid mass transfer of CO and H2. 

The absorption of a gas during condensation of water vapor on a cold water droplet Is a complex process characterized by unsteady state mass and heat transfer [6]. In the classical development of absorption of a gas In a liquids three theoretical models have ensued The film theory, the penetration theory, and the boundary layer theory. Each model Invokes different assumptions which result in different conclusions. 

Current research falls into one of two schools of thought Calderbank s slip velocity model and Lamont and Scott s eddy turbulence model (Linek et al., 2004 Linek et al., 2005b). Even though both models are penetration-type models, they make very different assumptions. The slip velocity model assumes different behavior for small and large bubbles. It also assumes a significant difference between average velocities for the two phases. The slip velocity and the surface mobility control mass transfer and, in terms of penetration theory, surface renewal. 

An analysis of chemical desorption has recently been published (Chem.Eng.Sci., 21 0980)), which is based on a number of simplifying assumptions the film theory model is assumed, the diffusivities of all species are taken to be equal to each other, and in the solution of the differential equations an approximation which is second order with respect to distance from the gas-liquid interface is used this approximation was introduced as early as 1948 by Van Krevelen and Hoftizer. However, the assumptions listed above are not at all drastic, and two crucial elements are kept in the analysis reversibility of the chemical reactions and arbitrary chemical mechanisms and stoichiometry.The result is a methodology for developing, for any given chemical mechanism, a highly nonlinear, implicit, but algebraic equation for the calculation of the rate enhancement factor as a function of temperature, bulk-liquid composition, interface gas partial pressure and physical mass transfer coefficient The method of solution is easily gene ralized to the case of unequal diffusivities and corrections for differences between the film theory and the penetration theory models can be calculated. 

The penetration theory in its simplest form represents the case of transient molecular diffusion into a semi-infinite medium It can be applied to real situations if hydrodynamic conditions exist for which such an assumption is approximately valid This would be the case if flow close to the interface is laminar, concentration profiles there are practically nonaal to the interface and time of contact of the phases is reasonably short ... 

Penetration theory can also be applied to turbulent conditions by assuming the turbulence spectrum to consist of large eddies, capable of surface renewal, and small eddies responsible for the presence of eddy diffusivity The small eddies are damped when an element of liquid reaches the interface so that, during its residence time there, mass transfer occurs in accordance with the assumptions of the penetration theory If all the eddies stay at the interface for the same interval of time we talk about penetration theory with regular surface renewal or the Higbie model If there is random distribution of residence times with an age-independent fractional rate of surface renewal, s, the term penetration theory with random surface renewal, or the Danckwerts nK)del, is employed In the case of the Higbie model, the mass transfer coefficient is the same as that given by eqn (18). For the Danckwerts model it takes the form... 

Single-screw systems use mass-transfer penetration theory [10], together with certain geometric assumptions, to develop an expression for a film efficiency Ep -... 

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