Title: Estimation of Kinetic Parameters of Single and Multicomponent Adsorption Processes
Adsorption is the adhesion of solute atoms, molecules or ions termed as adsorptive from a gas or liquid phase system on a surface termed as adsorbent. Adsorption is an efficient and economical separation process with high capacity for the solute molecules even at low concentrations. The parameters of the equilibrium and rate kinetics models are affected by molecular properties of the solute molecules, their affinity towards the adsorbent through various mechanisms, characteristics of the adsorbent and operating conditions. Molecular properties of the solute such as diffusivity and kinematic viscosity in addition to the extent of agitation of the liquid affect the hydrodynamic parameter viz. the external convective mass transfer coefficient. The adsorption kinetics and equilibrium data are essential inputs to models that predict the time of operat ion to achieve a specified removal as well as the optimal design of industrial batch adsorption tanks. Adsorption kinetics modelling which encompasses hydrodynamics, kinetics and equilibria is a substitute for expensive, time-consuming and site-specific pilot-plant studies. Further, kinetic studies enable the evaluation of different adsorbents based on their rates of adsorption and also provide insight into the underlying mechanisms. Adsorption of solutes may be influenced collectively by convective mass transfer resistance in the liquid phase, internal diffusional resistance and nonlinear adsorption equilibria. The Homogeneous Surface Diffusion Model (HSDM) assumes the adsorbent to be a uniform medium in which the solute adsorbs on the surface and diffuses inwards with an effective diffusivity described by Fick’s law. Equilibrium is assumed to prevail only at the fluid-solid interface and not in the entire adsorbent at finite times. This model is sufficiently rigorous and has been widely used to describe the batch adsorption kinetics. The model involves a parabolic partial differential equation in spherical coordinates, nonlinear adsorption equilibria and a boundary condition with an integral term. These preclude analytical solution and accurate numerical means have to be employed to solve the equations. In actual batch kinetics studies, the HSDM parameters viz., the convective mass transfer coefficient (kc) and surface diffusion coefficient (Ds) are not usually known a priori and have to be estimated from experimental kinetics data. Resistances may be important and neither parameter may be ignored. These involve numerically solving the HSDM partial differential equation along with the associated initial-boundary conditions repeatedly when searching for the best parameter estimates. The estimates that best fit the experimental kinetic data, according to a pre-specified error criterion, have to be identified. Usually, the objective function involving the Sum of Squares of the differences between the simulated model predictions and experimental k inetic data (SSE) is minimized. Even though many advanced methods and processors are available for optimization, their use in HSDM simulations for estimating kinetic parameters is rather limited. The possible reason may be attributed to the necessity to solve the entire gamut of the partial differential equation and its associated initial-boundary conditions in the HSDM at every iteration to find the Sum of Squares of the Error (SSE). The limitations of existing algorithms are highlighted. A novel algorithm which attempts to overcome the limitations is proposed. Further, the complications that arise in multicomponent adsorption modeling using HSDM is highlighted.