**Greek symbols**


*Microwave Digestion of Hazardous Waste Sludge in Geothermal Hot Waters by Char/Fly Ash… DOI: http://dx.doi.org/10.5772/intechopen.97656*

*B* reactant solid defined *Bim* Biot number for mass transfer *Ci* concentration of manganese in the bulk external phase of stage i (mg L<sup>1</sup> ) *C0* feed concentration of manganese in the column (mg L<sup>1</sup> ) *Def* effective diffusion coefficient (m<sup>2</sup> s 1 ) *F* objective function *h* fixed bed height (m) *ke* mass transfer coefficient in the bulk external phase (m s<sup>1</sup> ) *kr* reaction rate constant for heterogeneous systems (m s<sup>1</sup> ) *N* number of stages *Q* volumetric flowrate (m3 s 1 ) *qi* concentration of immobilized manganese within the adsorbent particle at stage i (mg g<sup>1</sup> ) *qm* theoretical maximum adsorption capacity of the Langmuir isotherm (mg g<sup>1</sup> ) *r* radial distance from the center of the particle, *0 < r < Rp*(m) *R* radius of column (m) *Rp* radius of adsorbent particle (m) *<sup>R</sup><sup>2</sup>* determination coefficient () *rc,i* unreacted core radius at stage i (m) *t* time (s) *Vi* volume of stage i (L) α backmixing coefficient () φ column hold-up () ρ density of adsorbent particle (g m<sup>3</sup> ) τ mean residence time of fluid in the column (s)
