Penetration theoiy

The stagnant-film model discussed previously assumes a steady state in which the local flux across each element of area is constant i.e., there is no accumulation of the diffusing species within the film. Higbie [Trans. Am. Jn.st. Chem. Eng., 31,365 (1935)] pointed out that industrial contactors often operate with repeated brief contacts between phases in which the contact times are too short for the steady state to be achieved. For example, Higbie advanced the theory that in a packed tower the liquid flows across each packing piece in laminar flow and is remixed at the points of discontinuity between the packing elements. Thus, a fresh liquid surface is formed at the top of each piece, and as it moves downward, it absorbs gas at a decreasing rate until it is mixed at the next discontinuity. This is the basis of penetration theoiy. 

If the velocity of the flowing stream is uniform over a veiy deep region of hquid (total thickness, 6r V ), the time-averaged mass-transfer coefficient according to penetration theoiy is given oy... 

Penetration theoiy 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 veiy complex, however, penetration theoiy is more difficult to use than film theory, and the latter method normally is preferred. 

Danckwerts [Jnd. Eng. Chem., 42, 1460(1951)] proposed an extension of the penetration theoiy, called the surface renewal theoiy, which allows for the eddy motion in the liquid to bring masses of fresh liquid continually from the interior to the surface, where they are exposed to the gas for finite lengths of time before being replaced. In his development, Danckwerts assumed that every element of fluid has an equal chance of being replaced regardless of its age. The Danck-werts model gives... 

Equations (13-115) to (13-117) contain terms, for rates of heat transfer from the vapor phase to the hquid phase. These rates are estimated from convective and bulk-flow contributions, where the former are based on interfacial area, average-temperature driving forces, and convective heat-transfer coefficients, which are determined from the Chilton-Colburn analogy for the vapor phase and from the penetration theoiy for the liquid phase. 

The Chan and Fair correlation uses Eqs. (7.13) and (7.16) to calculate the point efficiency E0g- Values of Na and Nj, in Eq. (7.13) are obtained using Eq. (7.14). Chan and Fair derived the following equation for ktfij, based on Highie s penetration theoiy, observations fay Calderbank et al. (136,137), and data regression. 

One of the most significant contributions of the penetration theory is the prediction that the mass transfer coefficient varies as D. As will be seen in Section 2.4-3, experimental mass transfer coefficients generally are correlated with an exponent on Pab ranging from to. The penetration theory model has also been successfully used to predict the effect of simultaneous chemical reaction on mass transfer in gas absorption and in carrier-facilitated membranes. Stewart solved the penetration theoiy model by taking into account bulk flow at the interface. The results, as in the film theory case, are conveniently expressed as the dependence of the ratio kj/k on the dimensionless total flux, (Nf, + Na)k°. This curve is also shown in Fig. 2.4-2 and generally predicts greater effects of convection than does the film theoiy. 

For the liquid phase, the penetration theoiy is used, with one exposure being the residence time for liquid flow between corrugation changes ... 

Harriott" presents a model that incorporates the penetration mechanism, but which includes thin, laminar, interfacial films that are stabilized by interfacial tension and are therefore excluded from intermittent mixture with the bulk fluid. The penetration theoiy has been extended by Angelo et al. to allow for stretching surfaces, such as those that occur, for exanq>Ie, in large, oscilladiig droplets. Ruckenstein has further refined the penetration theoiy to accommodate the effects of velocity distributions within the eddies during solute penetration. 

In its simplest form, the penetration theoiy assumes that a fluid of initial composition xX is brought into contact with an interface at a fixed composition Xa. I forahmer. For short contact times the composition far from the interface (z oo) remains at xX- If bulk flow is neglected (dilute solution or low transfer rates), solution of the unsteady-state diflusion equation provides an expression for the average mass transfer flux and coefficient for a contact time 6. 

Further relaxation of the assumptions of the film and penetration theories was suggested by Danckwerts who viewed the process as one of transient one-dimensional diffusion to packets or elements of fluid that reside at the phaM interface for vaiying periods of time. Therefore, the model is that of the penetration theoiy with a distribution of contact times. The surface age distribution d(0 i defined such that Mffdt is the fraction of surface that has resided at the interface for a time between t and t + dt. The mass transfer flux for the entire suiface is obtained by integration of the instantaneous flux over all exposure times ... 

Mass-Transfer Coefficient Denoted by /c, K, and so on, the mass-transfer coefficient is the ratio of the flux to a concentration (or composition) difference. These coefficients generally represent rates of transfer that are much greater than those that occur by diffusion alone, as a result of convection or turbulence at the interface where mass transfer occurs. There exist several principles that relate that coefficient to the diffusivity and other fluid properties and to the intensity of motion and geometry. Examples that are outlined later are the film theoiy, the surface renewal theoiy, and the penetration the-oiy, all of which pertain to ideahzed cases. For many situations of practical interest like investigating the flow inside tubes and over flat surfaces as well as measuring external flowthrough banks of tubes, in fixed beds of particles, and the like, correlations have been developed that follow the same forms as the above theories. Examples of these are provided in the subsequent section on mass-transfer coefficient correlations. 

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. 

This result suggests that the mass transfer coeflicient varies as DfjH, a result intermediate to those of the film theoiy and the penetration theory. In fact, the power dependence of the Sherwood number on the Schmidt number is observed in a number of correlations for mass transfer in turbulent flow in conduits and for flow about submerged objects. (See Section 2.4-3.)... 

Skelland (Ref. 1, p. 106) gives the results of a theoiy of penetration with surface renewal in which it is assumed that the fractional rate of surface renewal s is related to the age of the surfoce throu the constants and n as r = where (, n + 1) > 0. A theoiy of penetration with periodically vaiying rates of surface renewal has been developed by Skelland and Lee for two-phase dispersions in turbine-agitated vessels. This attempts to allow for the varying degrees of turbulence encountered by bubbles or drops as they circulate through all regions of the vessel. 

H. L. Frisch, D. Klempner, and T. Kwei, Modified Free Volume Theoiy of Penetrant Diffusion... 

Several versions of the fiee-vohnne theoiy of diffusion have been pronniigated foiiowing its introduction for polyineric systems in die woifc of Fujita. " Odier more fundamentally based molecular approaches have been attemiNed at times based on actual molecular properties of the polymer and penetrant however, their greater complexity has discounted widespread application. The free-volume theoiy is sinqde and has evidved dumigh die woik of Vrentas and Duda" Stem, Frisch, and coworkeis to a readily useful... 

The diffusion coefficient appearing in Eq. (20.4-3) is a true measure of the molecular mobility of the penetrant in question. Intuitively, the free-v ume theoiy proponents argue that a penetrant can execute a diffusive jump when a free-volume element greater than or equal to a critical size presmits itseif to a penetrant. The native poiymer, totaiiy devoid of penetrant, still possesses a certain amoum of fiee-volume packets of distributed size which wander spontaneously and randomly through the lubbeiy matrix. In fact, when a packet of sufficient size presents itself to a polymer segment, the polymer may execute a self-diffusive motion, and this is the action that causes slow interdiffusion of polymer chains. 

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