Mass transfer equation dimensional scaling factors

Introduce dimensionless variables (i.e., C, = Cps fi, etc.) and write each mass transfer rate process in terms of these dimensionless variables and the corresponding dimensional scaling factor. This scaling factor contains all the dimensions of, as well as an order-of-magnitude estimate for, the particular mass transfer rate process. For example, the left-hand side of equation (10-4) is written as follows, where all the variables are dimensionless ... 

Divide the entire mass transfer equation by the scahng factor for diffusion (i.e ,mixCAo/i )- This is an arbitrary but convenient choice. Any of the r + 2 dimensional scaling factors can be chosen for this purpose. When the scaling factor for the diffusion term in the dimensional mass transfer equation is divided by ,mixCAo/T, the Laplacian of the molar density contains a coefficient of unity. When the remaining r - -1 scaling factors in the dimensional mass transfer equation are divided by i,mixCAo/T, the dimensionless mass transfer equation is obtained. Most important, r -h 1 dimensionless transport numbers appear in this equation as coefficients of each of the dimensionless mass transfer rate processes, except diffusion. Remember that the same dimensionless number appears as a coefficient for the accumulation and convective mass transfer rate processes on the left-hand side of the equation. 

As mentioned above, the dimensionless transport numbers in the mass transfer equation are generated from ratios of dimensional scaling factors. If one divides the scaling factor for convective mass transfer by the scaling factor for diffusion, the result is... 

For unsteady-state diffusion into a quiescent medium with no chemical reaction, the mass transfer Peclet number does not appear in the dimensionless mass transfer equation for species i because it is not appropriate to make variable time t dimensionless via division by L/ v) if there is no bulk fluid flow (i.e., (d) = 0). In this case, the first term on each side of equation (10-11) survives, which corresponds to the unsteady-state diffusion equation. However, the characteristic time for diffusion of species i over a length scale L, given by L /50i,mix. replaces L/ v) to make variable time t dimensionless. Now, the accumulation and diffusional rate processes scale as CAo i.mix/A, with dimensions of moles per volume per time. Since both surviving mass transfer rate processes exhibit the same dimensional scaling factor, there are no dimensionless numbers in the mass transfer equation which describes unsteady-state diffusion for species i in nonreactive systems. 

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