Convection film theory

A simplified model usiag a stagnant boundary layer assumption and the one-dimension diffusion—convection equation has been used to calculate wall concentration ia an RO module. The iategrated form of this equation, the widely appHed film theory (41), is given ia equation 8. 

Consider a vessel containing an agitated liquid. Heat transfer occurs mainly through forced convection in the liquid, conduction through the vessel wall, and forced convection in the jacket media. The heat flow may be based on the basic film theory equation and can be expressed by... 

Another difficulty is related to the mass transfer by convection, as, by definition, the films are stagnant and hence, there should be no mass transport mechanism, except for molecular diffusion in the direction normal to the interface (Kenig, 2000). Nevertheless, convection in films is directly accounted for in correlations. Moreover, in case of reactive systems, the film thickness should depend on the reaction rate, which is beyond the two-film theory consideration. 

Mass-Transfer Coefficient Denoted by kc, kx, Kx, 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 theory, the surface renewal theory and the penetration theory, all of which pertain to idealized cases. For many situations of practical interest like investigating the flow inside tubes and over flat surfaces as well as measuring external flow through 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 right-hand side of Eq. (2-4) takes into account the time variation of the concentration of the reacting species in the liquid phase. Just as in the case of film theory, classical penetration theory neglects the convective mass transfer at... 

As the film thickness 8 is not normally known, the mass transfer coefficient / , cannot be calculated from this equation. However the values for the cases used most often in practice can be found from the relevant literature (i.e. [1.23] to [1.26]) which then allows the film thickness to be approximated using (1.189). In film theory the mass transfer coefficient / for vanishing convection flux h( — 0 is proportional to the diffusion coefficient D. 

Putting in the mass transfer coefficient [3 = D/5 for negligible convection from (1.189), and using the principles of film theory the following relationship between the mass transfer coefficients / and li, as shown in Fig. 1.49, can be found ... 

The factor is known as the Stefan correction factor , [1.28]. In order to calculate the mass transferred using film theory, the mass transfer coefficient (3 has to be found. In cases where convection is negligible the mass transferred is calculated from equation (1.181), whilst where convection is significant the mass transferred is given by (1.183). 

Equations (1.198) and (1.199) are also known as Lewis equations. The mass transfer coefficients f3m calculated using this equation are only valid, according to the definition, for insignificant convective currents, fn the event of convection being important they must be corrected. The correction factors C, = /3 n/l3m for transverse flow over a plate, under the boundary layer theory assumptions are shown in Fig. 1.50. They are larger than those in film theory for a convective flow out of the phase, but smaller for a convective flow into the phase. 

The value of a varies with the system under consideration. For example, in equimolar counter diffusion, Na and Nb are of the same magnitude, but in opposite direction. As a result, a is equal to 1 and hence, Eq. (2) reduces to Eq. (1), where is equal to Convective mass transfer coefficients are used in the design of mass transfer equipment. However, in most cases, these coefficients are extracted from empirical correlations that are determined from experimental data. The theories, which are often used to describe the mechanism of convective mass transfer, are the film theory, the penetration theory, and the surface renewal theory. 

The thickness of the fictitious film in the film theory can never be measured. The film theory predicts that the convective mass transfer coefficient k is directly proportional to the diffusivity whereas experimental data from various studies show that k is proportional to the two-third exponent of the diffusivity. In addition, the concept of a stagnant film is unrealistic for a fluid-fluid interface that tends to be unstable. Therefore, the penetration theory was proposed by Higbie to better describe the mass transfer in the liquid phase... 

Convective effects in the reactor space external to the feature are included by making use of the film theory. The effects of the bath hydrodynamics external to the wafer are included by assuming a thin concentration boundary layer adjacent to the wafer. 

Concentration Polarisation is the accumulation of solute due to solvent convection through the membrane and was first documented by Sherwood (1965). It appears in every pressure dri en membrane process, but depending on the rejected species, to a very different extent. It reduces permeate flux, either via an increased osmotic pressure on the feed side, or the formation of a cake or gel layer on the membrane surface. Concentration polarisation creates a high solute concentration at the membrane surface compared to the bulk solution. This creates a back diffusion of solute from the membrane which is assumed to be in equilibrium with the convective transport. At the membrane, a laminar boundary layer exists (Nernst type layer), with mass conservation through this layer described by the Film Theory Model in equation (3.7) (Staude (1992)). cf is the feed concentration, Ds the solute diffusivity, cbj, the solute concentration in the boundary layer and x die distance from the membrane. 

Now, it is necessary to discuss the mass transfer coefficient for component j in the boundary layer on the vapor side of the gas-liquid interface, fc ,gas, with units of mol/(area-time). The final expression for gas is based on results from the steady-state film theory of interphase mass transfer across a flat interface. The only mass transfer mechanism accounted for in this extremely simple derivation is one-dimensional diffusion perpendicular to the gas-liquid interface. There is essentially no chemical reaction in the gas-phase boundary layer, and convection normal to the interface is neglected. This problem corresponds to a Sherwood number (i.e., Sh) of 1 or 2, depending on characteristic length scale that is used to define Sh. Remember that the Sherwood number is a dimensionless mass transfer coefficient for interphase transport. In other words, Sh is a ratio of the actual mass transfer coefficient divided by the simplest mass transfer coefficient when the only important mass transfer mechanism is one-dimensional diffusion normal to the interface. For each component j in the gas mixture. 

Several elementary aspects of mass diffusion, heat transfer and fluid flow are considered in the context of the separation and control of mixtures of liquid metals and semiconductors by crystallization and float-zone refining. First, the effect of convection on mass transfer in several configurations is considered from the viewpoint of film theory. Then a nonlinear, simplified, model of a low Prandtl number floating zone in microgravity is discussed. It is shown that the nonlinear inertia terms of the momentum equations play an important role in determining surface deflection in thermocapillary flow, and that the deflection is small in the case considered, but it is intimately related to the pressure distribution which may exist in the zone. However, thermocapillary flows may be vigorous and can affect temperature and solute distributions profoundly in zone refining, and thus they affect the quality of the crystals produced. 

The preceding discussion assumes that no convection exists in the melt, and this is rarely, if ever, the case. Next we shall consider two approaches which account for convection in the melt, a transport mechanism which is especially important in mass transfer because Dl is small and even weak convection markedly alters solute concentration profiles and may cause macrosegregation. First we shall discuss film theory which is a very simple approach that gives qualitative information and often provides considerable physical insight into the mechanisms involved. Second, we shall discuss a simplified model of zone refining. 

Blasco and Alvarez [28] and Alvarez and Blasco [29] considered the application of flash drying to moisture removal of fish and soya meals. Heat, momentum, aud mass balauce equations were formulated. The model was solved numerically with appropriate coefficients of convective heat and mass transfer. Dilute phase transport of homogeneous radial mono-size particle distribution was considered. The conveying superheated steam was assumed to be an ideal gas. The initial period for heating the particles, during which condensation takes place, was neglected. Using the film theory [30], the effect of the mass transfer on the heat transfer coefficient... 

The CP model, which is based on the film theory, was developed to desaibe the back diffusion phenomenon during filtration of macromolecules. In this model, the rejection of particles gives rise to a thin fouling layer on the membrane surface, overlaid by a CP layer in which particles diffuse away from the membrane surface, where solute concentration is high, to the bulk phase, where the solute concentration is low [162], At steady state, convection of particles toward the membrane surface is balanced by diffusion away from the membrane. Thus, integrating the ID convection-diffusion equation across the CP layer gives... 

One of the simplest and widely used theories for modeling flux in pressure-independent mass transfer controlled systems is the film theory. As solution is ultrafiltered, solute is brought onto the membrane surface by convective transport at a rate, J defined as follows ... 

Mass transfer coefficient (fe) A measure of the solute s mobility due to forced or natural convection in the system. Analogous to a heat transfer coefficient, it is measured as the ratio of the mass flux to the driving force. In membrane processes the driving force is the difference in solute concentration at the membrane surface and at some arbitrarily defined point in the bulk fluid. When lasing the film theory to model mass transfer, k is also defined as D/S, where D is solute diffusivity and d is the thickness of the concentration boundary layer. 

Sherwood number (Sh) A dimensionless measure of the ratio of convective mass transfer to molecular mass transfer. If the mass transfer coefficient k is defined in terms of the film theory, then Sh is a measure of the ratio of hydraulic diameter to the thickness of the boundary layer. See Section 6.5. 

With considering concentration polarization effects, the performance of osmosis process can be predicted. There are generally two dominant theoretical models for osmosis phenomena, the thin-film theory and the convection-diffusion theory, which are separately elaborated as follow. 

Film Theory ModeL There will be convective mass transfer of solute to the wall of the membrane, vsbich is balanced by divisional transport away from the membrane owing to a concentration gradient. At steady state, assuming the diOudon coefficient D to be constant ... 

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. 

The two film theory [0.4] describes the mass transfer between two adjacent phases. The main resistance occurs in the two boundary layers at either side of the interface. In these laminar boundary layers, matter is only transported by molecular diffusion. With a phase equilibrium at the interface, the interface itself offers no resistance to the mass transfer. Mass transfer is very fast in the bulk of the phase, due to turbulent convection. The concentrations c i and are uniform throughout the bulk phase (see Fig. 1-45). 

In the film theory any convective heat transfer is accounted for if we assume that all the liquid contents mix thoroughly—except the thin film of condensed liquid on the interface. The proportionality constant allows for the fact that the theory is only approximate and must be modified to fit existing data. T e proportionality constant r is essentially constant at 4 for all systems studied. The film theory can allow for variations in fluid properties, pressurization levels and time. The proportionality constant r although a function of these same parameters is, however, not strongly affected by changes in them. 

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