The Mass Transfer Number

From Equation 3.39, the FIRR per unit area is given by the following expression  

As diffusion flames are expected to bum at stoichiometric conditions then 

And substitution into Equation 3.50 leads to a phenomenological definition of the mass transfer number  

As can be noted, the mass transfer number represents the mass of fuel generated per unit mass of fuel burnt, and is the additional physical parameter that controls the burning rate. 

Although there are many definitions of the mass transfer number, they are mostly generated for specific conditions such as droplet burning or boundary layer burning. All retain the same physical concept, which is the capability of the flame to self-sustain by generating more fuel. If B 1, then the flame will produce more fuel than that necessary to sustain burning. 

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The mass transfer number B represents the ratio of the energy available for vaporization to the energy required for vaporization, and may be thought of as a driving force for mass transfer. It can be expressed as... 

Under conditions of convective heat transfer, the mass transfer number B may be written as... 

For the very small particles and the very dilute two-phase system considered, the mass transfer number can be estimated by Sh=kgdp/D=2 (Ranz and Marshall, 1952). The diffiision coefficient, D, can be estimated by Fuller et al, (9) The solids hold-up equals G,/psU, (10), with G. as the solids flux in kg/(m, s), p, the particle density with u, approximating the gas velocity Ug. Values for these parameters are listed in Table 3. 

Consequently the mass transfer rate during diffusion combustion of polymers is determined by the ratio of the heat of combustion to the effective enthalpy of polymer gasification. The lower the combustion heat and the higher the polymer gasification enthalpy or, in other words the more heat resistant the polymer, the lower is the B value. For polymer combustion in air the B value of e.g., PMMA varies between 1.3-1.4, that of polyethylene between 0.5-0.6, of phenolic resins between 0.14-0.4 74 75). An increase of the oxygen concentration in the oxidative medium and of the oxidant temperature causes a rise of the mass transfer number B. Lower B values have been observed in thermally stable polymers of the carbonizable types. 

Two approaches have usually been taken based on (1) correlating the mass transfer number Sh with Re and Sc, and the heat transfer number Nu with Re and Pr and (2) defining a composite dimensionless factor for mass and Jh for heat transfer) and correlating it with Re ... 

Other correlations based partially on theoretical considerations but made to fit existing data also exist (71—75). A number of researchers have also attempted to separate from a by measuring the latter, sometimes in terms of the wetted area (76—78). Finally, a number of correlations for the mass transfer coefficient itself exist. These ate based on a mote fundamental theory of mass transfer in packed columns (79—82). Although certain predictions were verified by experimental evidence, these models often cannot serve as design basis because the equations contain the interfacial area as an independent variable. 

For hquid systems v is approximately independent of velocity, so that a plot of JT versus v provides a convenient method of determining both the axial dispersion and mass transfer resistance. For vapor-phase systems at low Reynolds numbers is approximately constant since dispersion is determined mainly by molecular diffusion. It is therefore more convenient to plot H./v versus 1/, which yields as the slope and the mass transfer resistance as the intercept. Examples of such plots are shown in Figure 16. 

Values of the mass-transfer coefficient k have been obtained for single drops rising (or falling) through a continuous immiscible Hquid phase. Extensive Hterature data have been summarized (40,42). The mass-transfer coefficient is often expressed in dimensionless form as the Sherwood number ... 

Pressure. Within limits, pressure may have Htfle effect in air-sparged LPO reactors. Consider the case where the pressure is high enough to supply oxygen to the Hquid at a reasonable rate and to maintain the gas holdup relatively low. If pressure is doubled, the concentration of oxygen in the bubbles is approximately doubled and the rate of oxygen deHvery from each bubble is also approximately doubled in the mass-transfer rate-limited zone. The total number of bubbles, however, is approximately halved. The overall effect, therefore, can be small. The optimum pressure is likely to be determined by the permissible maximum gas holdup and/or the desirable maximum vapor load in the vent gas. 

The constant depends on the hydraulic diameter of the static mixer. The mass-transfer coefficient expressed as a Sherwood number Sh = df /D is related to the pipe Reynolds number Re = D vp/p and Schmidt number Sc = p/pD by Sh = 0.0062Re Sc R. ... 

Chemistry. Chemical separation is achieved by countercurrent Hquid— Hquid extraction and involves the mass transfer of solutes between an aqueous phase and an immiscible organic phase. In the PUREX process, the organic phase is typically a mixture of 30% by volume tri- -butyl phosphate (solvent) and a normal paraffin hydrocarbon (diluent). The latter is typically dodecane or a high grade kerosene (20). A number of other solvent or diluent systems have been investigated, but none has proved to be a substantial improvement (21). 

Correlations for the mass-transfer coefficient, as the Sherwood number for various membrane geometries have been reviewed (39). 

Computation of Tower Height The required height of a gas-absorption or stripping tower depends on (1) the phase equilibria involved, (2) the specified degree of removal of the solute from the gas, and (3) the mass-transfer efficiency of the apparatus. These same considerations apply both to plate towers and to packed towers. Items 1 and 2 dictate the required number of theoretic stages (plate tower) or transfer units (packed tower). Item 3 is derived from the tray efficiency and spacing (plate tower) or from the height of one transfer unit (packed tower). Solute-removal specifications normally are derived from economic considerations. 

There are a number of different types of experimental laboratory units that could be used to develop design data for chemically reacting systems. Charpentier [ACS Symp. Sen, 72, 223-261 (1978)] has summarized the state of the art with respect to methods of scaUng up lab-oratoiy data and tabulated typical values of the mass-transfer coefficients, interfacial areas, and contact times to be found in various commercial gas absorbers as well as in currently available laboratoiy units. 

The mass-transfer coefficient, /c, is contained in the Sherwood number ... 

When the two liquid phases are in relative motion, the mass transfer coefficients in eidrer phase must be related to die dynamical properties of the liquids. The boundary layer thicknesses are related to the Reynolds number, and the diffusive Uansfer to the Schmidt number. Another complication is that such a boundaty cannot in many circumstances be regarded as a simple planar interface, but eddies of material are U ansported to the interface from the bulk of each liquid which change the concenuation profile normal to the interface. In the simple isothermal model there is no need to take account of this fact, but in most indusuial chcumstances the two liquids are not in an isothermal system, but in one in which there is a temperature gradient. The simple stationary mass U ansfer model must therefore be replaced by an eddy mass U ansfer which takes account of this surface replenishment. 

When bodr phases are producing eddies a more complicated equation due to Mayers (1962) gives the value of the mass transfer coefficient in terms of the Reynolds and Schmidt numbers which shows that die coefficient is proportional to... 

For conditions in industrial production reactors and in corresponding recycle reactors, the mass transfer coefficients of Gamson et al (1943) will be used. These are approximately correct and simple to use. There may be better correlations for specific cases and especially for larger molecules, where diffiisivity is low and Schmidt number is high. In such cases literature referring to given conditions should be consulted. 

In their analysis, however, they neglected the surface tension and the diffusivity. As has already been pointed out, the volumetric mass-transfer coefficient is a function of the interfacial area, which will be strongly affected by the surface tension. The mass-transfer coefficient per unit area will be a function of the diffusivity. The omission of these two important factors, surface tension and diffusivity, even though they were held constant in Pavlu-shenko s work, can result in changes in the values of the exponents in Eq. (48). For example, the omission of the surface tension would eliminate the Weber number, and the omission of the diffusivity eliminates the Schmidt number. Since these numbers include variables that already appear in Eq. (48), the groups in this equation that also contain these same variables could end up with different values for the exponents. 

Some of the exponents in Eqs. (50) and (52) can be evaluated from experimental data. For example, Calderbank and Moo-Young (C4) investigated several chemical systems and found that the mass-transfer coefficient per unit area was a function of the Schmidt number to the power of from 0.50 to 0.67 this would also be the value of B6. In addition, they found that agitation had no effect on KL therefore, s is equal to zero. 

An attempt has been made by Johnson and co-workers to relate such theoretical results with experimental data for the absorption of a single carbon dioxide bubble into aqueous solutions of monoethanolamine, determined under forced convection conditions over a Reynolds number range from 30 to 220. The numerical results were found to be much higher than the measured values for noncirculating bubbles. The numerical solutions indicate that the mass-transfer rate should be independent of Peclet number, whereas the experimentally measured rates increase gradually with increasing Peclet number. The discrepancy is attributed to the experimental technique, where-... 

Experimental results for fixed packed beds are very sensitive to the structure of the bed which may be strongly influenced by its method of formation. GUPTA and Thodos157 have studied both heat transfer and mass transfer in fixed beds and have shown that the results for both processes may be correlated by similar equations based on. / -factors (see Section 10.8.1). Re-arrangement of the terms in the mass transfer equation, permits the results for the Sherwood number (Sh1) to be expressed as a function of the Reynolds (Re,) and Schmidt numbers (Sc) ... 

Water flows at 0.50 m/s through a 20 mm tube lined with 0-naphthol. What is the mass transfer coefficient if the Sehtnidl number is 2330 ... 

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