CL: Film Theory

Again assuming that their is a "film" near the solid boundary where both velocity and concentration change fromt he bulk value to the surface value, we get something like this:

Again, the shear stresses on the solid are constant, so that we might use the friction factor to write the size of the fluid film as:

$\displaystyle{\frac{\delta}{L} = \frac{2}{fRe}}$

We can then write the mass flux due to convection as a product of the mass transfer coefficient and the driving force. For reasons that will be discussed in the next section, if we assume a linear concentration profile within a mass film (see picture) we can write:

$\displaystyle{N_A = D\frac{\Delta C_A}{\delta '} = k_c\Delta C_A}$

Making an analogy to the friction factor and the Nu, we can define a dimensionless mass flux, Sherwood number, as:

$\displaystyle{Sh_L = \frac{N_A}{D\left (\Delta C_A/L \right )}}$

which can be combined with our convection expression to give:

$\displaystyle{Sh_L = \frac{k_c L}{D}}$

Using the film theory definition of kc (i.e., $k_c = D/\delta '$), we can get:

$\displaystyle{Sh_L = \frac{L}{\delta '} = \frac{L}{\delta}\left (\frac{\delta}{\delta '}\right ) = \frac{fRe_L}{2}\left ( \frac{\delta}{\delta '}\right )}$

As we will see later in the course, the ratio of film thicknesses can be related to the material properties for mass transfer, through the Schmidt number, Sc, to give:

$\displaystyle{Sh_L = \frac{fRe}{2}\left ( \frac{\nu}{D}\right )^{1/3} = \frac{f}{2}Re Sc^{1/3}}$

where the factor of 1/3 will also be seen explained later in the course.

This expression is a useful generic formula for the convective mass transfer coefficient that can be used when no better correlation is available (provided you have a decent correlation for f). One further aid to correlations was devised by Chilton and Colburn, where a "j-factor" has been defined as:

$\displaystyle{j_D = \frac{Sh}{Re Sc^{1/3}}}$

so that a direct relation to the friction factor is given as:

$\displaystyle{j_D = \frac{f}{2}}$

NOTE:

Using the above analysis, one can use correlations for Sh directly, j-factor correlation directly, or film theory along with correlations for f to yield kc values!

HIST NOTE:

The precusor of the Chilton-Colburn analogy, the Reynolds analogy, assumed that the two films where the same size, and thus works for Sc=1

OUTCOME:

Use film theory and other correlations to obtain kc