Mass-transfer coefficients in turbulent flow

Sherwood et al. (19Z5) report that transition from laminar to turbulent occurs in the Reynolds number range from 250 to 500. Although they do not report any correlations for the liquid mass-transfer coefficient in turbulent flow, Treybal (1980) reports the following correlation for a liquid film with constant surface concentration at a somewhat higher range of Reynolds numbers. 

Correlations for Mass Transfer Coefficients in Turbulent Flow... 

When two or more phases are present, it is rarely possible to design a reactor on a strictly first-principles basis. Rather than starting with the mass, energy, and momentum transport equations, as was done for the laminar flow systems in Chapter 8, we tend to use simplified flow models with empirical correlations for mass transfer coefficients and interfacial areas. The approach is conceptually similar to that used for friction factors and heat transfer coefficients in turbulent flow systems. It usually provides an adequate basis for design and scaleup, although extra care must be taken that the correlations are appropriate. 

Heat delivery. Convection and conduction from hot gas sweeping by is the leading mode of heat transfer to a drying coating to supply the latent heat of vaporization of solvent. Except when solvent evaporation is so very rapid as to produce an appreciable convective velocity away from the surface, in turbulent gas flow the mechanisms of heat transfer to and solvent transfer away from the evaporating surface are virtually identical combinations of convective action with thermal conduction on the one hand and molecular diffusion on the other. This is reflected in useful correlations, like Colburn s, of the mass transfer coefficient with the more easily measured heat transfer coefficient in turbulent flow. It is also the reason that the now fairly extensive literature on the performance and design of driers focuses on heat transfer coefficients and heat delivery rates. 

If the concentration of the liquid surface is not constant, there will be mass transfer and a mass-transfer resistance on the gas side also. In separation processes, the gas-phase resistance often controls (the gas-phase mass-transfer coefficient is often significandy smaller than the liquid-phase mass-transfer coefficient). For turbulent flow of the gas in a wetted wall tube, the following correlation was originally reported by Gilliland (see Sherwood et al.. 1975 or Wankat and Knaebel. 2008T... 

There are several reports in the literature of the critical Reynolds number at which turbulent film flow commences. These values of NRe t are usually determined from the breaks which appear in the curves of film thickness, surface velocity of the film, heat or mass transfer coefficients in the film, etc., when plotted against NHe. Some of the numerical values proposed by various investigators are listed in Table I. 

For laminar and turbulent flows, we need appropriate correlation equations for the friction coefficient, heat transfer coefficient, and mass transfer coefficient. For laminar flow in the ranges of 5 X 106 > Re > iO3, and Pr and Sc > 0.5, we have the following relations for the coefficients ... 

Most practically useful mass-transfer situations involve turbulent flow, and for these it is generally not possible to compute mass-transfer coefficients from theoretical considerations. Instead, we must rely principally on experimental data. The data are limited in scope, however, with respect to circumstances and situations as well as to range of fluid properties. Therefore, it is important to be able to extend their applicability to conditions not covered experimentally and to draw upon knowledge of other transport processes (of heat, particularly) for help. A very useful procedure toward this end is dimensional analysis. 

The diversity in correlations reported in the literatme can therefore be attributed to differences in the flow patterns, mean flow strengths and direction, turbulence intensity variations, etc. Table 7A.2 lists some important literature studies and the resulting correlations for the volumetric gas-liquid mass transfer coefficient in stirred vessels. 

Vandu CO, Krishna R. (2004) Volumetric mass transfer coefficients in slurry bubble columns operating in the churn-turbulent flow regime. Chem. Eng. Process., 43 987-995. 

Davis, Ouwerkerk and Venkatesh developed a mathematical model to predict the conversion and temperature distribution in the reactor as a function of the gas and liquid flow rates, physical properties, the feed composition of the reactive gas and carrier gas and other parameters of the system. Transverse and axial temperature profiles are calculated for the laminar flow of the liquid phase with co-current flow of a turbulent gas to establish the peak temperatures in the reactor as a function of the numerous parameters of the system. Also in this model, the reaction rate in the liquid film is considered to be controlled by the rate of transport of reactive gas from the turbulent gas mixture to the gas - liquid interface. The predicted reactor characteristics are shown to agree with large-scale reactor performance. For the calculations of the mass transfer coefficient in the gas phase, kg, Davis et al. used the same correlation as Johnson and Crynes, but multiplied the calculated values arbitrarily by a factor 2 to include the effect of ripples on the organic liquid film caused by the high SOj/air velocities in the core of the reactor. 

We have confined our argument to the flow of liquids. In many situations (usually in the laboratory rather than in industrial practice) turbulence is introduced into the electrolyte by stirring. The argument advanced above applies in principle to a stirred liquid. The problem is that it is difficult to characterize mass transfer coefficients in a small stirred cell, so it is not easy to obtain results which can be applied in practice to a flow system. 

THE PROBLEM An electrolytic reaction takes place in a parallel plate cell of length L, width w, and interelectrode gap d, with w d. The reaction takes place under mass transport control with a mass transfer coefficient kj. The flow regime is turbulent and it can be assumed that the electrolyte volume V= w x d x L. Use dimensional analysis to obtain a scale-up criterion that will keep conversion constant. 

In principle, at least, we do not need mass-transfer coefficients for laminar flow, since molecular diffusion prevails, and the relationships of Chap. 2 can be used to compute mass-transfer rates. A uniform method of dealing with both laminar and turbulent flow is nevertheless desirable. 

Mass-transfer coefficients for laminar flow should be capable of computation. To the extent that the flow conditions are capable of description and the mathematics remains tractable, this is so. These are, however, severe requirements, and frequently the simplification required to permit mathematical manipulation is such that the results fall somewhat short of reality. It is not our purpose to develop these methods in detail, since they are dealt with extensively elsewhere [6, 7]. We shall choose one relatively simple situation to illustrate the general technique and to provide some basis for considering turbulent flow. 

It is possible to predict theoretically the mass transfer rate (or flux AT,) across any surface located in a fluid having laminar flow in many situations by solving the differential equation (or equations) for mass balance (Bird et al, 1960, 2002 SkeUand, 1974 Sherwood et al, 1975). Our capacity to predict the mass transfer rates a priori in turbulent flow from first principles is, however, virtually nil. In practice, we follow the form of the integrated flux expressions in molecular diffusion. Thus, the flux of species i is expressed as the product of a mass-transfer coefficient in phase y and a concentration difference in the forms shown below ... 

Film Theory. Many theories have been put forth to explain and correlate experimentally measured mass transfer coefficients. The classical model has been the film theory (13,26) that proposes to approximate the real situation at the interface by hypothetical "effective" gas and Hquid films. The fluid is assumed to be essentially stagnant within these effective films making a sharp change to totally turbulent flow where the film is in contact with the bulk of the fluid. As a result, mass is transferred through the effective films only by steady-state molecular diffusion and it is possible to compute the concentration profile through the films by integrating Fick s law ... 

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 mass-transfer coefficients depend on complex functions of diffii-sivity, viscosity, density, interfacial tension, and turbulence. Similarly, the mass-transfer area of the droplets depends on complex functions of viscosity, interfacial tension, density difference, extractor geometry, agitation intensity, agitator design, flow rates, and interfacial rag deposits. Only limited success has been achieved in correlating extractor performance with these basic principles. The lumped parameter deals directly with the ultimate design criterion, which is the height of an extraction tower. 

In addition to momentum, both heat and mass can be transferred either by molecular diffusion alone or by molecular diffusion combined with eddy diffusion. Because the effects of eddy diffusion are generally far greater than those of the molecular diffusion, the main resistance to transfer will lie in the regions where only molecular diffusion is occurring. Thus the main resistance to the flow of heat or mass to a surface lies within the laminar sub-layer. It is shown in Chapter 11 that the thickness of the laminar sub-layer is almost inversely proportional to the Reynolds number for fully developed turbulent flow in a pipe. Thus the heat and mass transfer coefficients are much higher at high Reynolds numbers. 

For flow in a smooth pipe, the friction factor for turbulent flow is given approximately by the Blasius equation and is proportional to the Reynolds number (and hence the velocity) raised to a power of -2. From equations 12.102 and 12.103, therefore, the heat and mass transfer coefficients are both proportional to w 75. 

Numerous turbulent mass-transfer relationships are given in Eqs. (39)-(50), Table VII. Although the most important ones in practical applications are those for channels and tubes, several other configurations also have been investigated because of their hydrodynamic interest. Generally, it is not possible to predict mass-transfer rates quantitatively by recourse to turbulent flow theory. An exception to this is for the region of developing mass transfer, where a Leveque-type correlation between the mass-transfer coefficient and friction coefficient/can be established ... 

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