## 3 Adsorption of Model Compounds by Eggshell Biocomposite Adsorbents
### Kinetic Models  
Aside from the isotherm models that account for the adsorption profiles at equilibrium conditions, the adsorption profile under non-equilibrium (dynamic) conditions provides insight on the adsorption kinetics and rate parameters of the adsorption process. In simple terms, adsorption kinetics describe the time dependent adsorption of an adsorbate onto the adsorbent versus time. The rate of an adsorption processes is influenced by the contact time, adsorbent surface structure, and initial adsorbate concentration.  
Adsorption mass transfer kinetics usually involves some basic steps, which are the transportation of the adsorbate from the bulk of the solution to the adsorbent surface (bulk diffusion), diffusion of the adsorbate into the liquid film (film diffusion), diffusion into the internal pores of the adsorbent (intra-particle diffusion), and the adsorption and desorption of the adsorbates from the adsorbent (surface reaction) [92; 93]. An evaluation of the adsorption profile versus time provides insight on the factors that control the rate of the process. Kinetic models can give details about adsorption rates, the performance of the adsorbent, and the probable reaction mechanism to provide a better understanding of the adsorption process, for system design and scale up [94]. The net rate of an adsorption process can be controlled by each step, or a combination of the basic steps involved in the mass transfer, where the rate-determining step may change in the course of the adsorption profile [93].  
Some of the methods where contact is achieved between adsorbate and adsorbent in adsorption systems involve batch mode, continuous fixed bed, continuous moving bed, continuous fluidized bed, and pulsed bed. For this review, the batch method and continuous fixed bed method are emphasized. Both methods are low cost, facile, and commonly deployed in research studies. In the batch method, the adsorbate and adsorbent are thoroughly mixed in constant volume of diluted solution while for the continuous fixed bed, the adsorbate is prepared as a solution and allowed to continuously pass through a bed or column packed with adsorbent. The batch adsorption technique requires less volume of dilute adsorbate solution while the fixed bed usually needs more volume and usually of higher concentration of adsorbate. By comparison, the fixed bed method is utilized by industry or other large-scale applications [95] while the batch method analysed by models such as pseudo-first order, pseudo-second order [96] and Elovich models [97] is often applicable for laboratory studies. The fixed bed method is analysed by the [PERSON] [98; 99]; [PERSON] [100]; bed depth service time models [101]. Some of the kinetic models used to shed light on the mechanisms involved in the adsorptive uptake of pollutants by ESP are briefly outlined in the Supplementary Materials. The first four kinetic models outlined in Equations [12; 13; 14; 15; 16; 17] are mainly used in the batch adsorption process, while other kinetic models shown in Equations [18; 19; 20; 21; 22; 23; 24] are applied to fixed bed systems.  
Adsorption can occur either via physisorption (weak interactions) or chemisorption (strong interactions). Examples for physisorption are electrostatic interactions (reversible), while chemisorption can occur via coordination through ligand exchange. For specific contaminants, such as phosphate, physisorption via outer sphere coordination or chemisorption via inner sphere coordination (ligand exchange with -OH groups on the adsorbent sites, typically for metal oxides) [102]. The threshold is typically set to 80 kJ/mol for physisorption while chemisorption involves higher enthalpy values above 80 kJ/mol. Section S2 of the Supplementary Materials outline various types of models used in both batch and continuous fixed bed column, along with related parameters that are listed in Table S2.