Mass transfer dimensional analysis

Static mixing of immiscible Hquids can provide exceUent enhancement of the interphase area for increasing mass-transfer rate. The drop size distribution is relatively narrow compared to agitated tanks. Three forces are known to influence the formation of drops in a static mixer shear stress, surface tension, and viscous stress in the dispersed phase. Dimensional analysis shows that the drop size of the dispersed phase is controUed by the Weber number. The average drop size, in a Kenics mixer is a function of Weber number We = df /a, and the ratio of dispersed to continuous-phase viscosities (Eig. 32). 

Although the absorption of a gas in a gas-liquid disperser is governed by basic mass-transfer phenomena, our knowledge of bubble dynamics and of the fluid dynamic conditions in the vessel are insufficient to permit the calculation of mass-transfer rates from first principles. One approach that is sometimes fruitful under conditions where our knowledge is insufficient to completely define the system is that of dimensional analysis. 

Pavlushenko et al. (P4) in their dimensional analysis considered Ks, the volumetric mass transfer coefficient, to be a function of pc, pc, L, Dr, N, Vs, and g. They determined the following relationship for the dimensionless groupings ... 

Clearly, the maximum degree of simplification of the problem is achieved by using the greatest possible number of fundamentals since each yields a simultaneous equation of its own. In certain problems, force may be used as a fundamental in addition to mass, length, and time, provided that at no stage in the problem is force defined in terms of mass and acceleration. In heat transfer problems, temperature is usually an additional fundamental, and heat can also be used as a fundamental provided it is not defined in terms of mass and temperature and provided that the equivalence of mechanical and thermal energy is not utilised. Considerable experience is needed in the proper use of dimensional analysis, and its application in a number of areas of fluid flow and heat transfer is seen in the relevant chapters of this Volume. 

Kroeker CJ, Soliman HM, Ormiston SJ (2004) Three-dimensional thermal analysis of heat sinks with circular cooling micro-channels. Int J Heat Mass Transfer 47 4733 744 Lee PS, Garimella SV, Liu D (2005) Investigation of heat transfer in rectangular micro-channels. Int J Heat Mass Transfer 48 1688-1704... 

Lelea D, Nishio S, Takano K (2004) The experimental research on micro-tube heat transfer and fluid flow of distilled water. Int J Heat Mass Transfer 47 2817-2830 Li J, Peterson GP, Cheng P (2004) Three-dimensional analysis of heat transfer in a micro-heat sink with single phase flow. Int J Heat Mass Transfer 47 4215-4231 Lin TY, Yang CY (2007) An experimental investigation by method of fluid crystal thermography. Int. J. Heat Mass Transfer 50(23-24) 4736-4742... 

Qu W, Mudawar I (2002) Analysis of three-dimensional heat transfer in micro-channel heat sinks. Int J Heat Mass Transfer 45 3973-3985... 

Wallis GB (1969) One-dimensional two-phase flow. McGraw-HUl, New York Wayner PC, Kao YK, LaCroix LV (1976) The interline heat transfer coefiicient of an evaporating wetting film. Int 1 Heat Mass Transfer 19 487-492 Weisberg A, Bau HH, Zemel IN (1992) Analysis of micro-channels for integrated cooling. Int 1 Heat Mass Transfer 35 2465-2472... 

Dimensional analysis of the variables characteristic of mass transfer under flow conditions suggests that the following dimensionless groups are appropriate for correlating mass transfer data. 

Conventional dimensional analysis employs single length and time scales. Correlations are thus obtained for the mass or heat transfer coefficients in terms of the minimum number of independent dimensionless groups these can generally be represented by power functions such as... 

The point of view based on a physical model started with the 1935 paper of Higbie [30], While the main problem treated by Higbie was that of the mass transfer from a bubble to a liquid, it appears that he had recognized the utility of his representation for both packed beds and turbulent motion. The basic idea is that an element of liquid remains in contact with the other phase for a time A and during this time, absorption takes place in that element as in the unsteady diffusion in a semiinfinite solid. The mass transfer coefficient k should therefore depend on the diffusion coefficient D and on the time A. Dimensional analysis leads in this case to the expression... 

One may note that the preceding two-step dimensional analysis provides an equation for the mass transfer coefficient, whereas the conventional dimensional analysis, which involves the quantities k, D, U, and R, can tell us only that... 

The second approach consists in deliberately abandoning certain similarity criteria and checking the effect on the entire process. This technique was used by Gerhard Damkbhler (1908-1944) in his trials to treat a chemical reaction in a catalytic fixed-bed reactor by means of dimensional analysis. Here the problem of a simultaneous mass and heat transfer arises—they are two processes that obey completely different fundamental principles ... 

There are some good chemical-vapor-deposition reactors that deliberately starve the rotating disk. However, the similarity is broken by the recirculation, and the one-dimensional analysis techniques described herein lose their validity. If the chemical reaction on the surface is sufficiently slow, compared to mass transfer through the boundary layer, then the deposition uniformity will not be much affected by the boundary-layer similarity. In these... 

Correlations based on dimensional analysis with the above variables in equation 3-10 would allow mass transfer rates to be easily predicted, e. g. in scaling-up lab results to full-scale or for changes in the liquid properties. However, no correlations have been developed with this complexity. 

Yuan, J., Rokni, M. and Sunden, B. (2003) Three-dimensional computational analysis of gas and heat transport phenomena in ducts relevant for anode-supported solid oxide fuel cells, International Journal of Heat and Mass Transfer 46, 809-821. 

After introducing Fick s law and dimensional analysis using mass transfer coefficients, the most useful solutions will be presented. The solutions will be... 

Mass transfer coefficients, dimensional analysis, and dimensionless numbers... 

Heat transfer processes are described by physical properties and process-related parameters, the dimensions of which not only include the base dimensions of Mass, Length and Time but also Temperature, , as the fourth one. In the discussion of the heat transfer characteristic of a mixing vessel (Example 20) it was shown that, in the dimensional analysis of thermal problems, it is advantageous to expand the dimensional system to include the amount of heat, H [kcal], as the fifth base dimension. Joule s mechanical equivalent of heat, J, must then be introduced as the corresponding dimensional constant in the relevance list. Although this procedure does not change the pi-space, a dimensionless number is formed which contains J and, as such, frequently proves to be irrelevant. As a result, the pi-set is finally reduced by one dimensionless number. 

In fact, the advantage of these combinations of numbers obtained by making differential equations dimensionless, over those combinations delivered by dimensional analysis, is that they characterize certain types of mass and heat transfer, respec-... 

Example 42 Description of the mass and heat transfer in solid-catalyzed gas reactions by dimensional analysis... 

Numerous empirical correlations for the prediction of residual NAPL dissolution have been presented in the literature and have been compiled by Khachikian and Harmon [68]. On the other hand, just a few correlations for the rate of interface mass transfer from single-component NAPL pools in saturated, homogeneous porous media have been established, and they are based on numerically determined mass transfer coefficients [69, 70]. These correlations relate a dimensionless mass transfer coefficient, i.e., Sherwood number, to appropriate Peclet numbers, as dictated by dimensional analysis with application of the Buckingham Pi theorem [71,72], and they have been developed under the assumption that the thickness of the concentration boundary layer originating from a dissolving NAPL pool is mainly controlled by the contact time of groundwater with the NAPL-water interface that is directly affected by the interstitial groundwater velocity, hydrodynamic dispersion, and pool size. For uniform... 

An extensive analysis of the behaviour of different types of non-adiabatic fixed bed reactor models is carried out and the importance of the heterogeneous one and two-dimensional models III-0 and III-T is stressed. Although in these models the heat and mass transfer phenomena are correctly taken into account, they... 

For a typical chemical engineering problem, the dimensions considered are generally M, L, T and 0. The dimensions F, L, T, 9 can also be used but, in this case, especially for heat transfer problems and for coupled heat and mass transfer processes, complicated dimensional formulae are derived. To establish a dimensional formula for a variable, it is necessary to have a relationship containing this variable. This relationship can be independent of the process to which the dimensional analysis is applied. The use of tables containing dimensional formulae for physical variables can also be effective. 

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