Adsorption wave shape and length

The shape and length of the wave that propagates in the bed is related to the mathematical form of the uptake rate expression that is substituted into the last term in Eq. (9.10). Equation (9.15) is the pde describing diffusion of an adsorbate from the gas phase into the adsorbent This is but one of many diffusion based uptake models that might be substituted into the uptake rate term of Eq. (9.9) or (9.10). 

Equation 9.15, when solved for the case of macro-pore diffusion gives us, in the low loading Hmit, the famihar relationship that mass uptake is proportional to the square root of time. The same relationship can be derived for micro-pore diffusion as well. The solution to this equation can be used with the appropriate particle sizes to estimate diffusivity from uptake rate measurements. 

A second and vitally important conclusion that came out of the literature in the 1970s was that linear driving force models also produced solutions that were indistinguishable from diffusion based models. 

That this was known in industry at least 15 years earlier is one of the unfortunate discrepancies between academic research and commercial industrial research and development. Not all that is known is necessarily published. This realization subsequently lead to the development of both solid phase and gas phase linear driving force models that each provide very good representations of measured data without the excess labor involved with the diffusion-based models. For trace systems there are quite a few analytical solutions that are available and quite tractable for both design work and the analysis of adsorption column performance. 

Linear Driving Force Approximation and Resistance Modeiing 

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