Mass transfer dimensionless groups

For heat transfer the Fourier number is cct/a. The heat transfer analogs of the mass transfer dimensionless groups can be found by making the substitutions described in Chapter 1. 

Average transport coefficients between the bulk stream and the particle surface in a fixed-bed reactor can be correlated in terms of dimensionless groups which describe the flow conditions For mass transfer the group kj p(G is a function of the Reynolds number dpGIfi and the Schmidt number pIp. Chilton and Colburn suggested plotting 7 vs dpGfp., where... 

Note that the group on the left side of Eq. (14-182) is dimensionless. When turbulence promoters are used at the inlet-gas seclion, an improvement in gas mass-transfer coefficient for absorption of water vapor by sulfuric acid was obsei ved by Greenewalt [Ind. Eng. Chem., 18, 1291 (1926)]. A falhug off of the rate of mass transfer below that indicated in Eq. (14-182) was obsei ved by Cogan and Cogan (thesis, Massachusetts Institute of Technology, 1932) when a cauTiiug zone preceded the gas inlet in ammonia absorption (Fig. 14-76). 

Not only is the type of flow related to the impeller Reynolds number, but also such process performance characteristics as mixing time, impeller pumping rate, impeller power consumption, and heat- and mass-transfer coefficients can be correlated with this dimensionless group. 

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 ... 

The reaction (Eqn. 5.4-65) takes place in the liquid phase. The molecules are transferred away from the interface to the bulk of the liquid, while reaction takes place simultaneously. Two limiting cases can be envisaged (1) reaction is very fast compared to mass transfer, which means that reaction only takes place in the film, and (2) reaction is very slow compared to mass transfer, and reaction only takes place in the liquid bulk. A convenient dimensionless group, the Hatta number, has been defined, which characterizes the situation compared to the limiting cases. For a reaction that is first order in the gaseous reactant and zero order in the liquid reactant (cm = 1, as = 0), Hatta is ... 

Some of the important dimensionless groups pertinent to heat and mass transfer problems are listed in Table 3.5. 

The solution of Eqs. (9) is straightforward if the six parameters are known and the boundary conditions are specified. Two boundary conditions are necessary for each equation. Pavlica and Olson (PI) have discussed the applicability of the Wehner-Wilhelm boundary conditions (W3) to two-phase mass-transfer model equations, and have described a numerical method for solving these equations. In many cases this is not necessary, for the second-order differentials can be neglected. Methods for evaluating the dimensionless groups in Eqs. (9) are given in Section II,B,1. 

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. 

On the submicron scale, the current distribution is determined by the diffusive transport of metal ion and additives under the influence of local conditions at the interface. Transport of additives in solution may be non-locally controlled if they are consumed at a mass-transfer limited rate at the deposit surface. The diffusion of additives in solution must then be solved simultaneously with the flux of reactive ion. Diffusive transport of inhibitors forms the basis for leveling [144-147] where a diffusion-limited inhibitor reduces the current density on protrusions. West has treated the theory of filling based on leveling alone [148], In his model, the controlling dimensionless groups are equivalent to and D divided by the trench aspect ratio. They determine the ranges of concentration within which filling can be achieved. 

Dimensionless blend time method, 16 688 Dimensionless groups, 15 685, 686t, 687t Dimensionless mass transfer equation,... 

The effect of agitation, as produced by a rotary stirrer, for example, on mass transfer rates has been investigated by Hixson and Baum 2-1 who measured the rate of dissolution of pure salts in water. The degree of agitation is expressed by means of a dimensionless group (Nd2p/fx) in which ... 

The decrease in the exit concentration with decreases in the extraction pressure seen in Figs. 17 and 18 is a consequence of the fact that the driving force for mass transfer is directly related to the partial pressure of the volatile component in the vapor phase, which, in this case, is constant and equal to the extraction pressure. In fact, reasonably good agreement between the data in Fig. 17 and the predictions of Eq. (38) can be obtained provided it is assumed that the dimensionless group (ki ATlk y p/L) is independent of pressure. This point is illustrated in Fig. 19, which is a plot of Eq. (38) for Pe =. The value used for (ki Aj/k v(kp/L) was chosen so as to obtain the asymptotic value of wi in Fig. 17. 

Many of the results and correlations in heat and mass transfer are expressed in terms of dimensionless groups such as the Nusselt, Reynolds and Prandtl numbers. The definitions of those dimensionless groups referred to in this chapter are given in Appendix 2. 

The fundamental physical laws governing motion of and transfer to particles immersed in fluids are Newton s second law, the principle of conservation of mass, and the first law of thermodynamics. Application of these laws to an infinitesimal element of material or to an infinitesimal control volume leads to the Navier-Stokes, continuity, and energy equations. Exact analytical solutions to these equations have been derived only under restricted conditions. More usually, it is necessary to solve the equations numerically or to resort to approximate techniques where certain terms are omitted or modified in favor of those which are known to be more important. In other cases, the governing equations can do no more than suggest relevant dimensionless groups with which to correlate experimental data. Boundary conditions must also be specified carefully to solve the equations and these conditions are discussed below together with the equations themselves. 

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... 

Consider the mass transfer across a flat plate. In this case, the important variables are (dimensions in parentheses) the mass transfer coefficient k (L/T), the bulk fluid velocity u (L/T), the kinematic viscosity of the fluid v (L2/T), the solute diffusion coefficient I) (L2/T), and the plate length / (L). The number of independent variables n = 5 and the number of the involved dimensions m = 2. Hence, die number of dimensionless groups Pi = n — m = 3. 

As a result, many correlations are available for heat and mass transfer at moderate pressures that have been developed over time. Perry and Green [6] give a fairly complete amount of data with regards to correlations for different arrangements [7], On the other hand, very few data and correlations are available in the field of high pressure heat and mass transfer, as will be reviewed later. Correlations are in terms of the individual coefficients, ki and h, included in dimensionless groups such as those given before in Eqns. (3.4-10). 

The Hatta number Ha is a dimensionless group used to characterize the speed of reaction in relation to the diffu-sional resistance to mass transfer,... 

Figure 4.19 illustrates the effect of liquid phase mass transfer, represented by the dimensionless group Kuq (see Eqs. (55) and (57)). If the evaporation velocity is in the same order of magnitude as the liquid phase mass transfer coefficient, then the selectivity of the evaporation process vanishes though the relative volatility as well as the gas phase mass transfer coefficients remain unchanged. 

The rate constant K3 which appears in the dimensionless group A5 is also unknown. It corresponds to the combustion of the unstable polymeric residue which is assumed to be very fast, i.e., mass transfer controlled. There are two ways to account mathematically for the destruction of the polymeric residue by gaseous oxygen when it becomes unstable. The first is to use equation (14) with a larger but finite rate constant K3 (or A5) together with the parameter o defined above. If this approach is taken there exists a minimum integration step of order I/A5 that can be used in order to account for the finite mixing time in the reactor and also to account for the assumption that the combustion of the polymer is mass transfer controlled. 

The mass-transfer performance of trays often is expressed by way of a dimensionless group called the number of transfer units [see Lockett (op. cit.), Kister (op. cit.), and Sec. 14 for additional background]. These dimensionless numbers are defined by... 

A number of experimental and theoretical studies of mass transfer in solution processes have been published. Since this literature is fairly well known, it will be mentioned briefly, but not analyzed in detail. Most of the earlier work in agitation employed dissolution rates as performance criteria (H6, H8, W5). Experimental studies of dissolution itself have employed suspended solute plates (B7, Wl), single crystals (M12, P5), revolving crystals (D2), and packed beds (Gl, L3, M5, V4). Recently, several theoretical analyses of literature data have appeared (El, HI, R3). A number of Russian investigators have also studied dissolution (N6, Zl) they prefer to correlate data in terms of individual variables rather than the dimensionless groups customary in English and American literature. 

---