### 1. Introduction

*R*

^{2}) were used to indicate the most appropriate order.

### 2. Methodology

### 2.1. Kinetic Study

*g*/

*L*, under different pressures. Experimental data has been summarised in Table 1.

#### 2.1.1. Adsorption kinetics modelling

*n*

*order using a true hypothesis based on real conditions:*

^{th}Solid surface is not uniform;

Adsorption heat depends on the recovery rate of the solid surface;

Adsorption is done using two layers;

No balance exists between the molecules of the two phases;

Filtration can be addressed by adsorption;

Morphological meshes of the membrane structure are equivalent to the sites;

Fouling begins after filling the meshes;

Physico-chemical parameters are independent of temperature and initial concentration;

*n*∈

*R*

^{+}),

*q*(

*t*) is the adsorbed amount of protein per unit of membrane mass(

*mg/g*),

*q*

_{max}is th–e maximum adsorption capacity of the membrane (

*mg/g*) and

*k*

*the constant of the reaction rate [(*

_{n}*mg/g*)

^{1 − n}

*s*

^{−1}].

*M*is the mass of the membrane,

*V*is the cumulative filtrate volume,

*C*

*is the mass concentration of the chemical substances in feed flow, and*

_{in}*δ*is the retention rate of the membrane.

*δ*< 1), if

*δ*= 0 : no adsorption,

*δ*= 1 : total absorption

##### (3)

$$\frac{d(\delta .V.{C}_{in}/M)}{dt}={k}_{n}{\left[{\left(\frac{\delta .V.{C}_{in}}{M}\right)}_{\text{max}}-{\left(\frac{\delta .V.{C}_{in}}{M}\right)}_{t}\right]}^{n}$$##### (4)

$$\frac{d{V}_{t}}{dt}={k}_{n}{\left(\frac{\delta .V.{C}_{in}}{M}\right)}^{n-1}\xb7{({V}_{m}-{V}_{t})}^{n}$$##### (5)

$$\{\begin{array}{l}\frac{dV(t)}{dt}={K}_{n}{({V}_{m}-{V}_{t})}^{n}\hfill \\ {V}_{t=0}=V(0)=0\hfill \end{array}$$### 2.1.1.1. Pseudo-first order equation

*min*

^{−1}) and

*V*

*is the filtrate volume. The constant*

_{m}*k*

_{1}model can be determined using

*V*versus

*t*plot. Fig. 2 shows a schematic representation of this variation for different concentration suspensions.

### 2.1.1.2. Pseudo second order equation

*k*

_{2}is the pseudo-second order rate constant (

*kg/g min*). The filtrate volume

*V*and the constant

*k*

_{2}can be determined experimentally from plot

*V*versus

*t*(Fig. 3).

#### 2.1.2. Adsorption model

*n*

*pseudo-order, the integration of Eq. (4) between the initial state (*

^{th}*t*= 0,

*V*= 0) and the instantaneous state (

*t*=

*t*,

*V*=

*V*) is given as follows:

##### (9)

$$V={V}_{m}-{\left[\frac{{V}_{m}^{n-1}}{1+t(n-1){K}_{n}\hspace{0.17em}{V}_{m}^{n-1}}\right]}^{{\scriptstyle \frac{1}{n-1}}}$$*K*

*and the maximum filtrate volume, according to the pseudo-order*

_{n}*n*and concentration

*C*. The relationship allows us to write the final equation of the model.

##### (10)

$$V(t)=(1.68276-\mathrm{0.58581.}n)-{\left[\frac{{(1.68276-\mathrm{0.58581.}n)}^{n-1}}{1+t(n-1)({\mathrm{3.67.10}}^{-8}{e}^{\left({\scriptstyle \frac{n}{0.17911}}\right)}+{\mathrm{8.6468.10}}^{-4}){(1.68276-\mathrm{0.58581.}n)}^{n-1}}\right]}^{{\scriptstyle \frac{1}{n-1}}}$$##### (11)

$$\begin{array}{l}V(t)={V}_{m}-{\left[\frac{{V}_{m}^{(b0+b1*c+b2*{C}^{2})-1}}{1+t*(d0+d1*C)*((b0+b1*c+b2*{C}^{2})-1)*{V}_{m}^{(b0+b1*C+b2*{C}^{2})-1}}\right]}^{{\scriptstyle \frac{1}{((b0+b1*C+b2*{C}^{2})-1)}}}\\ n={b}_{0}+{b}_{1}*C+{b}_{2}*{C}^{2}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}k={d}_{0}+{d}_{1}*C\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}{b}_{0}^{*}={b}_{0}-1\end{array}$$### 2.2. Mathematical Method

*V*

*,*

_{m}*K*

*and pseudo order*

_{n}*n*, respectively by nonlinear regression and give very good adjustments to the experimental data as most of their respective regression correlation coefficients (

*R*

^{2}) are close to unity [31].

### 2.3. Error Function

#### 2.3.1. The mean absolute error (MAE)

*Y*

_{i}_{, exp}and

*Y*

_{i, cal}are referring, respectively to the experimentally and calculated values obtained from the proposed model.

#### 2.3.2. RMSE

*n*is the number of data points;

*Y*

_{i}_{, exp}are referring to the experimental value and the

*Y*

_{i, cal}calculated values obtained from the proposed model.

### 3. Results and Discussion

### 3.1. Kinetic Model Validity

*K*

*and to optimize the pseudo-order n from Eq. (12), when the value of filtrate volume experimental data*

_{n}*V*and time

*t*is plotted. The different values of

*K*

*are obtained from Eq. (6), (7) and (12), respectively. For fitting the proposed model, the results of*

_{n}*R*

^{2}= 0.993 are obtained for concentration

*C*= 1.4

*gL*

^{−1}, optimal pseudo order equation

*n*= 1.115 and constant

*K*

*= 1.11 × 10*

_{n}^{−3}

*mgg*

^{− (1 − n )}

*S*

^{−1}, thus a reasonable small value of RSME and MAE also gave a good correlation between the predicted filtrate volume and experimental values. This result confirms the performance and the accuracy of the new model, which is able to describe the adsorption of BSA on PCTE membrane.

*R*

^{2}values for PSO equation are higher than those of PFO equation but it is the best result of the new model for all the different concentrations.

### 3.2. Models Comparison

*R*

^{2}of 0.255–0.54 for different concentration of BSA, showing relatively large RMSE values of 0.074–0.28. Consequently, the experimental results did not follow the first-order kinetic model.

*V*

_{i, cal}values did not agree with the experimental

*V*

_{i}_{, exp}value, which also gives a relatively large deviation of RMSE 0.0494–0.0171.

*V*

_{i, cal}values were consistent with the experimental

*V*

_{i}_{, exp}values, resulting in a relatively small deviation of RMSE 0.0171–0.0364, in addition to linear regression coefficients, close to 0.997.

*R*

^{2}and to the least RMSE are almost identical. The value of

*R*

^{2}obtained from the nonlinear regression of the MATLAB software can therefore be used to perfectly judge the best-fit order.

### 3.3. Validation of the Model for Other Experimental Data in the Literature

### 4. Conclusions

*n*

*order was developed based on a real assumption, to correlate the kinetic data of BSA adsorption on PCTE membrane for different concentrations. MATLAB software was used to optimize the pseudo-order and to describe the kinetic parameters, thus a comparison of PFO, PSO and the proposed model equations for the description of adsorption kinetics has been examined.*

^{th}*R*

^{2}up to 0.997 and a reasonable small value of RMSE and MAE, respectively up to 0.0171 and 0.2141 The optimal pseudo order of BSA adsorption on PCTE membrane

*n*= 1.115 was between 1 and 2; the best-fit curves were plotted for the proposed model.