## 2 Experimental
### Methodology
#### 2.2.3 Mathematical Modelling of the Results of SMAHS  
The homogeneous surface diffusion model (HSDM) was used to study the adsorption kinetics process which comprises 3 steps: (i) the adsorbate diffuses through a stagnant liquid film layer surrounding the adsorbent particle; (ii) the adsorbate adsorbs from the liquid phase onto the outer surface of the adsorbent; and (iii) the adsorbate diffuses along the inner surface of the adsorbent particles until it reaches its adsorption site. A detailed description of the model was reported elsewhere [17; 18]. The model equations are:  
\[\frac{\partial\mathrm{q_{td}}}{\partial\mathrm{t}}=\mathrm{D_{s}}\left(\frac {\partial^{2}\mathrm{q_{td}}}{\partial\mathrm{r^{2}}}+\frac{2}{\mathrm{r}} \frac{\partial\mathrm{q_{td}}}{\partial\mathrm{r}}\right) \tag{1}\]  
Initial and boundary conditions for the process are given in Equations (2)-(4):  
\[\mathrm{t}=0;\mathrm{q_{td}}=0 \tag{2}\]  
\[\mathrm{r}=0;\frac{\partial\mathrm{q_{td}}}{\partial\mathrm{r}}=0 \tag{3}\]  
\[\mathrm{r}=\mathrm{r_{p}};\mathrm{D_{s}}\mathrm{\rho_{p}}\frac{\partial \mathrm{q_{td}}}{\partial\mathrm{r}}=\mathrm{k_{f}}(\mathrm{C}-\mathrm{C_{s}}) \tag{4}\]  
where, \(\mathrm{D_{s}}=\) surface diffusion coefficient (m\({}^{2}\)/s) which is the rate of diffusion of nitrate along the surface of the adsorbent particle, \(\mathrm{q_{td}}=\) nitrate-N concentration in the solid phase (mg/L), \(\mathrm{C}=\) nitrate-N concentration in the liquid phase (mg/L), \(\mathrm{C_{s}}=\) liquid phase nitrate concentration at the adsorbent surface (mg N/L), \(\mathrm{r}=\) radial distance from the centre of adsorbent particle (m), \(\mathrm{k_{f}}=\) external mass transfer coefficient (m/s), and \(\mathrm{\rho_{p}}=\) particle density (kg/m\({}^{3}\)). The mass balance of the nitrate in bulk solution in the membrane reactor was calculated using Equation (5). This model was developed using the concept of a continuous flow stirred tank reactor (CSTR) and by incorporating membrane packing density (\(\mathrm{A_{M}}\)/\(\mathrm{V_{M}}\)) and membrane correlation coefficient (MCC) into the adsorption model [3; 13]. \[\frac{\mathrm{d}\mathrm{C_{b}}}{\mathrm{dt}}=\frac{\mathrm{Q}}{\mathrm{V}} \left(\mathrm{C_{0}}-\mathrm{C_{b}}\right)-\ \frac{\mathrm{M}}{\mathrm{V}}\ \frac{\mathrm{dq}}{\mathrm{dt}}-\ \frac{\mathrm{A_{M}}}{\mathrm{V_{M}}}\ \mathrm{MCC}\ \mathrm{C_{b}} \tag{5}\]  
where \(\mathrm{C_{b}}\) is the nitrate concentration in the bulk phase in the reactor (mg N/L); \(\mathrm{Q}\) is the flow rate (m\({}^{3}\)/s); \(\mathrm{V}\) is the volume of the bulk solution in the reactor (m\({}^{3}\)); \(\mathrm{C_{0}}\) is the nitrate concentration in the feeding tank (mg N/L); \(\mathrm{M}\) is the mass of the adsorbent used (g); \(\mathrm{A_{M}}\) is the surface area of the membrane (m\({}^{2}\)); \(\mathrm{V_{M}}\) is the volume of membrane (m\({}^{3}\)); MCC is the membrane correlation coefficient. The term [(M/V) (dq/dt)] denotes the adsorption of nitrate onto the adsorbent in suspension, and [(A\({}_{\mathrm{M}}\)/\(\mathrm{V_{M}}\)) MCC \(\mathrm{C_{b}}\)] represents the adsorption onto the adsorbents layer deposited onto membrane surface. The first two terms on the right-hand side of Equation (5) represent adsorption. Once the particles are adsorbed, the adsorbent (together with nitrate) is retained by the membranes which is represented by the third term of the equation on the right-hand side. The retention and possible further adsorption on the membrane are assumed to be proportional to the packing density (A\({}_{\mathrm{m}}\)/\(\mathrm{V_{m}}\)). Here, the packing density is the area of membrane in a unit volume of the membrane module. The proportionality isrepresented by a surrogate parameter of membrane correlation coefficient (MCC). More details are given in the paper of [PERSON] et al. [3]. A higher value of MCC indicates a more efficient removal of adsorbate and adsorbent which contained the adsorbate by the membrane [13].