Any analytical expression would depend entirely on the shape of the breakthrough curve. Let's assume that C=0 initially and that we have corrected t for the dead volume of the system: 1. In the ideal case, C increases instantaneously from C=0 to C=C0 and thus t_star is when this increase happens. 2, For a symmetric breakthrough curve, t_star is the time when C=C_0/2. 3. In any other case, t_star is given by the solution to an equation involving integrals int_0^tstar F C dt = int_tstar^infinity F (C_0 - C) dt but that's exactly what is done graphically in this video from 2:27 onwards int_0^tstar F C dt = light orange surface int_tstar^infinity F (C_0 - C) dt = dark purple area The problem is that for any real system, the breakthrough curve likely does not follow a simple, easily predictable equation. Please also note that we are here assuming that we have constant pattern and that might be an incorrect assumption. (But we do need to start with simplified examples in teaching adsorption, otherwise there are much complications that make it difficult to learn the underlying principles)
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Is there an analytical expression for t_star available for constant pattern adsorption? Thanks!
Any analytical expression would depend entirely on the shape of the breakthrough curve. Let's assume that C=0 initially and that we have corrected t for the dead volume of the system:
1. In the ideal case, C increases instantaneously from C=0 to C=C0 and thus t_star is when this increase happens.
2, For a symmetric breakthrough curve, t_star is the time when C=C_0/2.
3. In any other case, t_star is given by the solution to an equation involving integrals
int_0^tstar F C dt = int_tstar^infinity F (C_0 - C) dt
but that's exactly what is done graphically in this video from 2:27 onwards
int_0^tstar F C dt = light orange surface
int_tstar^infinity F (C_0 - C) dt = dark purple area
The problem is that for any real system, the breakthrough curve likely does not follow a simple, easily predictable equation. Please also note that we are here assuming that we have constant pattern and that might be an incorrect assumption. (But we do need to start with simplified examples in teaching adsorption, otherwise there are much complications that make it difficult to learn the underlying principles)
@@PLE_LU nice response, thanks a lot!!