Solute Movement Theory


A simple derivation of the major results of Solute Movement Theory is presented next. This approach implies several simplifying assumptions:

Consider that the packed column is originally at a solute concentration cA (in the interparticular gas phase). A differencial step in concentration, dCA, is introduced in the column. It moves a distance dz in a time dt (figure 1).

Figure 1 – Movement of a differencial step disturbance in concentration along the column.

 

One does not yet know the velocity (us = dz / dt) at which of this concentration step travels along the column. This can be obtained from a mass balance to a column element of volume Adz, that is crossed by the concentration step in a time dt:
 

solute entering at z during dt =

                                solute leaving at z+dz during dt +

                                 + solute accumulating in interparticular phase during Dt +

                                 + solute adsorbing in intraparticular phase during Dt                             (1)

  Or, in more mathematical terms:                                     e A v dt (cA+ dcA) =

                                    e A v dt cA +

                                    + e A dzdcA +

                                    + r(1-e) A dzdqA                                                                                    (2)

 Where v is the (intersticial) velocity of the inert carrier gas, A is the column’s cross-section, e is the packing porosity, r is the adsorbent’s apparent density and qA is the concentration of A adsorbed in the solid (mol/mass of solid), in equilibrium with cA. From equation 2 one obtains then an expression for the solute velocity in the column, already written in terms of an infinitesimal step:

                             (3)

Equation 3 is the major result of SMT. A solute element of concentration cA will travel in the column at a velocity us, which depends (inversely) on the slope of the adsorption isotherm at cA, dqA/dcA. This result will allow us to understand how compressive and dispersive waves are formed in a column.
 

Consider that a concentration wave with a shape as described in figure 2 enters the column. The wave is discretized in five steps in order to simplify the analysis.


Figure 2 – Inlet concentration wave.

 


Now consider two possible adsorption isotherms over the concentration interval c1 to c6 (figure 3).

Figura 3 - Two adsorption isotherms: a) linear b) Langmuir-type.

 

In the case of the linear isotherm, dqA / dcAis the same in any point of the concentration front. Therefore, acording to equation 3, us is constant. The front travels along the column essentialy unchanged. On the other hand, in case b), the low concentration region of the isotherm has a higher slop, and thus higher value of dqA / dcA. Higher concentrations will therefore move faster than the lower ones. This leads to the formation of a compressive wave, which can then, ideally, become a shock wave as illustrated in figure 4.
 

 

Figure 4 - Shock wave formation.


Solute Movement Theory can be used to quantify the velocity of the shock wave along the column. One has simply to repeat the previous material balance (equations 1 and 2) , but now the infinitesimal concentration step is replaced by the finite concentration step that is the shock wave itself. With reference to the previous inlet conditions, the shock wave material balance would lead to:

                                (4)

 Now consider the concentration front shown in figure 5. It is easy to understand that, in this case, the front will distend as it moves along the column. This is a dispersivewave (figure 5).
 

 
 

Figura 5 - Formation of a dispersive wave.

 

The isotherm in case b) is called a favorable isotherm. An hypothetic isotherm with inverted concavity would be called unfavorable, and it would originate a dispersive wave in response to the inlet front shown figure 2.

Back to Home Page


Last modified December 22, 1999
Send comments and suggestions to - cruz@fe.up.pt