4.6.1. Crank (1975) Model

The Crank Model [86] was developed from Fick's Second Law, which considers the diffusion of a single particle in the form of a sphere. This model describes the behavior of the bed, as a whole, from the simplified mass transfer process in a single particle. The model shows that the sphere is initially at a uniform concentration and that the surface concentration is maintained as a constant. The total amount of the diffusing substance entering or leaving the sphere can be written as [86]:

$$\frac{M\_{\rm f}}{M\_{\infty}} = 1 - \frac{6}{\pi^2} \sum\_{n=1}^{\infty} \frac{1}{n^2} \exp\left(-\frac{Dn^2 \pi^2 t}{r^2}\right) \tag{3}$$

where *Mt* and *M*∞ are the mass in a determined time and an infinite time (maximum mass obtained in the extraction), respectively, *D* is the diffusivity of the solute inside the particle (m2·s−1), *<sup>t</sup>* is the extraction time (s), *<sup>r</sup>* is the particle radius (m) and *<sup>n</sup>* is the number of the series expansion.
