### 1. Introduction

### 2. Materials and Methods

### 2.1. Materials

*N*,

*N*’-Dimethyl acetamide (DMAc) was bought from Loba Chemie Pvt. Ltd. India. Polyethylene glycol (PEG) with M.W. 200 (PEG-200), 400 (PEG-400), 600 (PEG-600) Da were procured from High Purity Lab. Pvt. Ltd. India. PEG of M.W. 1500 and 6000 (Da) were obtained from Loba Chemie Pvt. Ltd. India and Sisco Research Lab Pvt. Ltd. India, respectively. Zinc oxide nano powder of 80–100 nm size was bought from Nanoshel LLC, India. HCl was purchased from Merck Ltd., India. Potassium dichromate (K

_{2}Cr

_{2}O

_{7}), and potassium permanganate (KMnO

_{4}) were purchased from Sisco Research Lab. Pvt. Ltd., India. Non-woven polyester backing 3324 was purchased from Ahlstrom Hollytex, Finland.

### 2.2. Membrane Preparation

### 2.3. Membrane Bubble Point, Pore Size, and Number of Pore Analysis

##### (2)

$${N}_{i}=\left({J}_{i}-\frac{{J}_{i-1}\xb7{P}_{i}}{{P}_{i-1}}\right)\xb7\frac{8\eta l}{\pi .{P}_{i}\xb7{\gamma}_{{P}_{i}}^{4}}$$*r*

*is pore radius in m, σ is surface tension (N/m), θ is contact angle (degree),*

_{pi}*Pi*is the applied pressure to open these pores (Pa), N

_{i}is the number of pores per unit area (m

^{2}), η is water viscosity (Pa.s), J

_{i}is water flux at i

^{th}increment when applied pressure is P

_{i}, l is length of pore which is assumed to be equal to skin layer thickness of membrane, similarly J

_{i-1}is water flux at corresponding i

^{th}decremental pressure P

_{i-1}[25, 26].

### 2.4. Rejection and Water Flux Analysis

^{2}·h (LMH), V is volume of water collected, A is the cross-sectional area of the membrane (m

^{2}) and ΔT is the time (s) to collect water of volume V. Though the LMH is not an SI unit, it is largely used to define membrane properties.

### 2.5. Statistical Analysis

_{0}, β

_{1,}… etc. are regression model coefficients and ɛ is random error or supporting element [20]. Y is dependent variable-metal ion rejection efficiency. X

_{1}, X

_{2}, … etc.

_{..}are independent variables and β

_{1,}β

_{2,}… etc. are respective regression model coefficients. Generally, data fitting is validated by R square, adjusted R square, multiple corrélation coefficient R and F-test [16]. R square measures correlation strength on 0 to 100% between dependent variable and linear model. The value of R square ranges from 0 to 1. If R square is zero, it indicates that there is no linear relation between observed and predicted values. If it is one, it means predicted and observed values are identical. While if R square is 0.5, it implies that half of variance is explained by model based upon considered dependent variable The variance measures how each number in the data set is away from average mean. Adjusted R square considers both the number of data points, and independent variables number for predicting data. Adjusted R square is calculated by Eq. (6),

### 3. Results and Discussion

### 3.1. Membrane Transport Properties and Rejection Analysis

*N*,

*N′*-dimethyl formamide (DMF) and N-methyl pyrrolidone (NMP) [18,33]. The polyethylene glycol of different molecular weight and different concentration were added to impart hydrophilicity and smoothness to membrane [22]. It will help to reduce fouling of membrane [8,21,22]. The ZnO nanoparticles from concentration 0.2 to 1.0% were added in dope solution. It helps to improve morphology of membrane and imparts charge to membrane. The use of acid treated ZnO particles shows better performance than nascent [8,23]. Similar enhanced separation of boron and humic acid has been reported [13,14]. The separation principle applied is Donnan exclusion. The different composition membranes used for separation are shown in Table 1.

### 3.2. Multi-Attribute Linear Regression Model for ‘Mn’ and ‘Cr’ Rejection Analysis

##### (9)

$$\begin{array}{l}\text{Y}=-194.8398+2.5578{P}_{1}-0.001797{P}_{2}+10.9039{P}_{3}\\ +248436.8781{P}_{4}+0.0003164{P}_{5}+6118864889.9{P}_{6}\end{array}$$_{1}, P

_{2}, P

_{3}, P

_{4}, P

_{5}, P

_{6}are independent variables

*viz*., PSF concentration, PEG M.W., ZnO concentration, flux in m

^{3}/m

^{2}·s, bubble point pressure in bar, and pore size in nm, respectively; while β

_{1}, β

_{2}, … are respective regression model coefficients.

^{2}= 0.97, R

^{2}

_{adj}= 0.97).

^{2}analysis. R

^{2}value of 0.9712 which shows that the predictors explain 97.1% of the variance of Y variable and excellent data correlation with metal ion rejection. It means data points are scattered less around fitted regression line or there is very small difference between observed and fitted data.

##### (10)

$$\begin{array}{l}\text{Y}=-245.791298+2.694843{Q}_{1}-0.00143342{Q}_{2}+5.324175{Q}_{3}\\ +0.000373141{Q}_{4}+973501404.5{Q}_{5}+1.96113\times {10}^{-11}{Q}_{6}\end{array}$$_{1}, Q

_{2}, Q

_{3}, Q

_{4}, Q

_{5}, Q

_{6}are independent variables viz., PSF concentration, PEG M.W., ZnO concentration, bubble point pressure in bar, pore size in nm and number of pores, respectively, which are affecting Cr. Removal efficiency. Whereas β

_{1}, β

_{2}, … etc. are respective regression model coefficients correlating independent variables with Cr separation efficiency. A variation is observed here compared with Mn based model that the water flux is replaced by pore size and number of pores. These are interdependent parameters, where the water flux is dependent upon combination of number of pores and pore size.

^{2}= 0.98, R

^{2}

_{adj}= 0.98).

### 3.3. Tolerance

### 3.4. Variance Inflation Factor

### 3.5. Model Adequacy Testing

#### 3.5.1. Quantile-Quantile (Q-Q) plot

_{1}(for Mn) and A

_{2}(for Cr) indicates the data is homoscedastic and normally distributed. There are rare outliers and almost all data points are close to reference line shows the data normality.

#### 3.5.2. Residual and predicted versus actual plot

_{1}(for Mn) and 7-B

_{2}(for Cr). Whereas in predicted versus actual graph Fig. 7-C

_{1}(for Mn) and 7-C

_{2}(for Cr)., the data is closer to reference line and no outliers. The predicted Vs actual graph shows that all the points are close to the regressed diagonal line, it suggests good fit of data [38]. This study is satisfactory to correlate the input variable influence on different dependent and independent variables. Though, the ZnO nano particles treatment in model suggests some more experimentation is required to find the dependency of rejection on this parameter.

#### 3.5.3. Standard error and deviation in the analysis

### 4. Conclusions

*viz*., PSF concentration, PEG M.W., ZnO concentration and treatment of ZnO nanomaterial by HCl. These parameters affected membrane properties of bubble point pressure in bar, pore size in nm, number of pores, water flux, which affected the removal properties for Mn and Cr. The anchoring of ZnO nanoparticles suitably modified by treatment with strong acids, vary surface charge to PSF based membranes. These charged membranes showed excellent removal properties for Mn and Cr salts as 97.12 and 98.37%, respectively, which can be optimized to 99% for both materials. The combination of PSF as base material for mechanical, chemical and thermal stability and anchoring such charged nanomaterial provide an excellent method for formation of membranes for industrial applications. This formation parameters were correlated with separation properties using multiple linear regression with the help of R studio software. A model was built and was found to possess excellent correlation with experimental as indicated by R square is 0.97 and 0.98 for Mn and Cr, respectively. Further their smaller (0.01) P-value suggests 99% confidence level for relation between variables, while tolerance of 0.02–0.03 implies good multicollinearity in the data. The nature of residual plots implies the data is homoscedastic and normally distributed.