Journal of Environmental Treatment Techniques  
2020, Volume 8, Issue 3, Pages: 900-907  
J. Environ. Treat. Tech.  
ISSN: 2309-1185  
Journal web link: http://www.jett.dormaj.com  
Mathematical Modelling for Predicting Rejection of  
Trace Organic Contaminants by the Nanofiltration  
Membrane NF270  
Hai Quang Dang*  
Faculty of Environment, the University of Danang, University of Science and Technology, Danang, Vietnam  
Received: 21/01/2020  
Accepted: 13/04/2020  
Published: 20/09/2020  
Abstract  
This study implemented multiple linear regression model to predict rejection of trace organic contaminants (TrOCs) by the nanofiltration  
NF) membrane NF270. Multiple regression analysis by the Statgraphics Centurion software were used to find an optimal mathematical  
(
modeling that combines interactions between molecular width, molecular height, molecular length, molecular weight and log D of TrOCs for  
predicting rejection. The result shows a relatively good agreement between the predicted rejection and the observed rejection and an  
2
acceptable R-squared correlation coefficient were found (R = 91.42 %) for the best model. In conclusion, a unified general multiple linear  
regression equation was able to predict rejections of TrOCs during nanofiltration with the explanatory variables of molecular width, molecular  
height, molecular length and molecular weight. Moreover, the present approach is a basis to continue investigation using multiple regression  
analysis techniques for understanding rejection of TrOCs by the NF membranes.  
Keywords: Trace organic contaminants (TrOCs), Nanofiltration, Multiple linear regression, Mathematical modeling, Rejection  
Introduction1  
health. It has been found that the physicochemical properties of  
1
TrOCs, such as molecular size, molecular weight (MW),  
hydrophobicity and charge caused by the functional groups, have  
significant effects on their rejection by NF membranes [10,11].  
According to Kimura et al. [12], the molecular size of the TrOCs  
could be considered one of the most important factors influencing  
their rejection by the NF membranes. The molecular weight of the  
solutes is often used as an indication of size while the molecular  
size parameters such as molecular width, molecular length and  
molecular height have been confirmed to more appropriate  
predicators for describing size exclusion effects on the rejection  
of TrOCs by NF membranes [13-16].  
The demand for fresh water worldwide is increasing  
dramatically caused by continued population growth posing  
challenges in the last few decades by growing water stress, both  
in terms of water scarcity and quality deterioration. Some of most  
important problems in water supply are the necessity of fresh  
water production for drinking, domestic, agricultural, landscape  
or industrial uses, the requirement of higher performance methods  
for wastewater reclamation and reusing applications, as well as  
lower maximum levels of contaminants [1]. However, the  
potential presence of trace organic contaminants (TrOCs) such as  
endocrine disrupting chemicals (EDCs), pharmaceutically active  
compounds (PhACs) and disinfection by products (DBPs) in  
treated wastewater and other water sources has become a public  
concern because of their potential risks on ecological and human  
health in recent decades [2-4].  
Recognizing these problems, the rejection of TrOCs in water  
treatment processes, which are associated with potentially  
adverse human health effects, is of increasing interest for  
membrane applications. Nanofiltration (NF) has been  
demonstrated to be appropriate technologies for removing most  
TrOCs [5,6]. An important driving force for the widespread  
implementation of NF membranes is their high removal efficiency  
for a large number of inorganic salts and TrOCs amongst the  
membrane processes [7-9]. This will have special significance  
because satisfactory elimination of micropollutants in water  
sources is of paramount importance for the protection of public  
A
number of articles have proposed a mechanistic  
understanding of the rejection of TrOCs, others have tried to  
apply fitting parameter models to model rejection [17-19].  
However, there have been few models to predict the rejection of  
TrOCs by NF membranes. It would therefore be of interest to have  
a statistical model that can use for predicting rejection of TrOCs.  
Among modeling approaches, multiple linear regression analysis  
is a relatively simple statistical method used to examine the  
correlation among variables. The present study is aimed at  
developing multiple regression model that can usefully estimate  
the rejection of TrOCs by NF membrane based on an integral  
approach that considers physicochemical properties of the  
compounds.  
Corresponding author: Hai Quang Dang, Faculty of Environment, the University of Danang, University of Science and Technology, Danang,  
Vietnam. E-mail: dqhai@dut.udn.vn.  
900  
Journal of Environmental Treatment Techniques  
2020, Volume 8, Issue 3, Pages: 900-907  
2
.2 Nanofiltration membrane  
The NF270 membrane (Dow-Filmtec, Minneapolis, MN) was  
2
Materials and methods  
2
.1 Background of statistical analysis  
selected for this study. According to the manufacturer, it is a thin-  
film composite polyamide membrane that is widely used for water  
and wastewater treatment application. This is a loose NF  
membrane with a relatively high permeability (of approximately  
1
Statistical analysis is useful for exploring and examining the  
basic features of the data prior to applying statistical tests and  
fitting statistical models. Because the fact that many factors  
influence some phenomena, it is necessary to calculate the  
interaction among phenomena. In order to attain this, multiple  
regression methods can be used. Multiple regression has taken a  
very significant place in statistical science. It is a method of  
analysis for assessing the strength of the relationship between a  
set of explanatory variables known as independent variables, and  
a single response or dependent variable. Applying multiple  
regression analysis to a set of data results in what are known as  
regression coefficients, one for each explanatory variable [20,21].  
Regression analysis is a mathematical measure of the average  
relationship between two or more variables in terms of the  
original units of the data. The concept of regression analysis  
involves finding the best relationship between variables.  
2
1 L/bar m h). At pH 4 and above, this membrane is negatively  
charged [23]. The flat sheet membrane samples were stored dry  
before use.  
2
.3 Trace organic contaminants, analytical chemicals and  
reagents  
The target TrOCs for this research have been chosen from the  
major classes of EDCs, PhACs and DBPs. They have diverse  
physicochemical properties such as hydrophobicity, charge,  
solubility, and molecular size. A stock solution was prepared at a  
concentration of 1 mg/mL in pure methanol. A working solution  
of these TrOCs was also prepared in pure methanol. Both these  
solutions were stored in a freezer at -18 ºC prior to use.  
Chemical solutions and feed waters were prepared with Milli-  
Q water. Both the solvents used for solid phase extraction and  
analysis of samples including methanol and dichloromethane,  
purchased from Sigma-Aldrich (Sydney, Australia). Internal  
standard of bisphenol A-d16 and N,O-bis (trimethylsilyl)  
In case of researching relationship between two phenomena  
and in case of prediction of the value of dependent variable, first  
to identify variables and then to find out random sample n size for  
the chosen values of dependent variables. Suppose that k  
phenomenon is identified as independent variable (predictor), or  
x
i
, i = 1, 2,…, k, and Y as dependent random variable. The whole  
multiple linear model can be presented as one equation for  
dependent variable Y  
trifluoroacetamide  
(BSTFA)  
containing  
1
%
of  
trimethylchlorosilane (TMCS). Pyridine used in the derivatization  
process. All reagents and chemicals were purchased from Sigma-  
Aldrich (Sydney, Australia).  
i
:
Y
i
= β  
0
+ β  
1
x
1
+ β  
2
x
2
+ …+ β  
k
x
k
+ ε  
i
(1)  
2
.4 Experimental protocol  
A laboratory scale cross flow NF/RO system consisted of a  
where Y  
of independent variable, β  
regression coefficient), and ε  
random error which has normal distribution, zero mean and  
constant variance. The whole regression model can be estimated  
by the sample regression model using least squares fitted  
i
is dependent random variable, x  
1
, x  
2
, …, x  
k
are values  
0
, β  
1
, …, β  
k
are model parameters  
stainless steel NF/RO membrane cell with an effective surface  
area of 40 cm and channel height of 2 mm, a stainless steel feed  
(
i
is a supporting element, or a  
2
reservoir, and a high pressure pump (Hydra-Cell, Wanner  
Engineering Inc., Minneapolis, MN) was used. The temperature  
of the feed solution was controlled by a chiller/heater (Neslab  
RTE 7, Thermo Scientific Inc., Waltham, MA, USA) equipped  
with a stainless steel heat exchanger coil which was submerged  
directly into the feed reservoir. A digital flow meter (Optiflow  
(prediction) equation (obtained by minimizing Error Sum of  
Squares, SSE):  
y
i
= b  
0
+ b  
1
x
1
+ b  
2
x
2
+ …+ b  
k
x
k
(2)  
1
000, Agilent Technologies, Palo Alto, CA) connected to a PC  
was utilized to measure permeate flow, and the cross flow was  
monitored with a rotameter.  
where y  
i
is adjustable or foreseen value of dependent variable Y  
, …, x are values of independent variables, and b , b , …,  
are estimations of unknown parameters β , β , …, β . Once a  
i
,
x
b
1
, x  
k
2
k
0
1
The rejection of TrOCs was performed in a background  
0
1
k
electrolyte solution containing 10 mM of NaCl, 1 mM of CaCl  
2
,
regression model has been constructed, it may be important to  
confirm the goodness of fit of the model and the statistical  
significance of the estimated parameters. Commonly used  
techniques to verify the goodness of fit include the R square,  
Adjusted R square, Multiple R, and hypothesis testing. Statistical  
significance can be verified by a Fisher distribution (F-test) of the  
overall model, followed by tests of the individual parameters  
using Student's t-distribution (t-test) [22].  
The fit of a multiple regression model can be judged with  
calculation of the multiple correlation coefficient, Multiple R,  
defined as the correlation between the observed values of the  
response variable and the values predicted by the model. The  
and 1 mM of NaH PO (pH 7) and conducted over 24 hours. Prior  
2
4
to each experiment, the NF270 membrane samples were gently  
washed with copious Milli-Q water to remove any preservatives.  
They were then compacted using Milli-Q water at 1,000 kPa for  
at least one hour until a stable permeate flux had been obtained.  
The background electrolyte solution was then added to the feed  
reservoir, and made up to the total feed volume of 10 litres.  
During the experiment, the feed reservoir temperature and cross  
flow velocity were kept constant at 20 ± 0.1 °C and 42 cm/s,  
respectively. The permeate flux was set to the manufacturer’s  
2
quoted nominal membrane flux of 42 L/m h throughout the  
2
experiment. Both permeate and retentate were recirculated to the  
feed reservoir. A mixture of 25 target TrOCs was then added to  
the feed reservoir to obtain a concentration of 25 μg/L of each.  
The feed solution pH was kept constant during the experiments  
by periodically adding a small amount of 1 M of NaOH or 1 M  
HCl. Approximately 100 mL of feed and permeate samples were  
taken at specific times. Samples were stored in clean glass bottles,  
squared value of R (R ) gives the proportion of the variability of  
the response variable accounted for by the explanatory variables.  
Adjusted R square used to compare models with different sets of  
independent variables in terms of predictive capabilities. Analysis  
of variance (ANOVA) will provide an F-test of the null  
hypothesis that each of b  
words that R is zero [21].  
0
1 k  
, b , b2, b , is equal to zero, or in other  
2
901  
Journal of Environmental Treatment Techniques  
2020, Volume 8, Issue 3, Pages: 900-907  
wrapped in aluminium foil, stored in the fridge for subsequent  
extraction and GC/MS analysis. The effective rejection was  
defined as R (%):  
than 0.99. The detection limits and quantification limits for  
analytes were estimated with the signal to noise (s/n) ratio higher  
than 3 and higher than 10, respectively.  
A Metrohm model 744 pH Meter was calibrated before  
beginning of an experiment and utilized to measure the feed  
solution pH for the duration of the experiment.  
C
p
R 100(1  
)
(3)  
C
f
3
Results and discussion  
where C  
respectively.  
f
and C  
p
were the feed and the permeate concentrations,  
3.1 Properties of trace organic contaminants and their rejection  
efficiency  
The major physicochemical properties of the target TrOCs  
and their rejection efficiency are shown in Table 1. The standard  
deviation of data obtained from two independent experiments.  
The compounds selected for this investigation exhibited  
considerably difference in their physicochemical properties.  
These compounds have low molecular weight, ranging between  
138.12 and 361.82 g/mol for salicylic acid and bezafibrate,  
respectively. However, they are markedly different in their  
a
2
.5 Analytical methods  
The Oasis HLB SPE cartridges (6 mL, 200 mg, Waters,  
Milford, MA, USA) for extraction of the TrOCs in feed and  
permeate samples were used in this investigation. The feed and  
permeate samples of 100 mL were allowed to reach room  
temperature and adjusted by 4 M sulphuric acid to pH range  
between 2 and 3. Before the samples were extracted, the SPE  
cartridges were conditioned sequentially by  
7
mL  
dissociation constants (pK ), molecular dimension (width, height  
dichloromethane and methanol (1:1, v/v), 7 mL methanol, and  
about 2 x 7 mL reagent water on a vacuum manifold at a flow rate  
of 2 mL/min. Subsequently, the samples were passed through the  
cartridges with a flow rate of 2 mL/min. The loaded cartridges  
were washed with 6 x 7 mL of Milli-Q water and dried under  
vacuum for 30 minutes along with a stream of nitrogen. The SPE  
columns containing the TrOCs were eluted with 7 mL methanol  
followed by 7 mL dichloromethane and methanol (1:1, v/v) at a  
flow rate of 1 - 5 mL/min. The elution volume was then  
evaporated to dryness under a gentle stream of nitrogen in a water  
bath at 40 °C. An amount of 200 µL methanol solution containing  
and length) and hydrophobicity properties. Most TrOCs are weak  
acids and will dissociate into an ionic form at pH above the pKa.  
The molecular widths, heights and lengths of these TrOCs are  
from 0.354 to 0.435, 0.505 to 1.313 and 0.615 to 1.179 nm,  
respectively. The difference in molecular weight and dimension  
can play a major role in the rejection of TrOCs.  
On the other hand, it is striking to note that the intrinsic  
hydrophobicity of TrOCs was an important factor in determining  
their rejection by a NF process [27,28]. The logarithm of the  
effective octanol-water distribution coefficient, log D, is a good  
parameter which can be used to evaluate the hydrophobicity of  
TrOCs at any pH value [29,30]. According to Wells [31] and  
Alturki et al. [32], organic compounds with log D equal to 3 or  
higher are generally referred to as hydrophobic. By contrast,  
organic compounds with log D below 3 are referred to as  
hydrophilic.  
It can be observed that rejection efficiency of TrOCs varied  
considerably depending on the different physicochemical  
characteristics of the compounds, ranging from 49.27 to 98.30 %.  
In general, larger MW and molecular dimension compounds  
showed higher rejections than small MW and molecular  
dimension compounds on size exclusion grounds. Because of the  
large MW and molecular dimension, TrOCs do not significantly  
penetrate into the membrane pores, resulting in their adsorption  
occurring mainly at the membrane surface. Consequently, the  
diffusion of these compounds across the membrane is very  
limited, leading to the high rejection efficiencies observed. These  
results are consistent with the observations of Yangali-Quintanilla  
5
µg bisphenol A-d16 was utilized to dissolve the extracted  
residues, and was transferred into 1.5 mL vials before further  
evaporation to dryness under a gentle nitrogen stream. Finally, the  
derivatization of the dried residues in the vials was performed by  
adding 100 µL of BSTFA (N,O-bis (trimethylsilyl)  
trifluoroacetamide) (1 % TMCS (trimethylchlorosilane)) and 100  
µL of pyridine (dried with KOH solid). The conditions of the  
derivatization reaction were 30 min at 60 - 70 °C. The derivatives  
were allowed to cool to room temperature before analysis by GC-  
MS [24].  
A
Shimadzu GCMS-QP5000 system consisting of  
Shimadzu AOC 20i autosampler and a Phenomenex Zebron ZB-  
(5 % diphenyl - 95 % dimethylpolysiloxane) capillary column  
30 m × 0.25 mm ID, df = 0.25 µm) was used to determine the  
a
5
(
concentrations of the organic compounds. Helium was used as the  
carrier gas at a constant flow rate of 1.3 mL/min. The GC oven  
temperature program was conducted as follows: 100 °C for 1 min,  
first ramp 10 °C/min to 175 °C, 3 min at 175 °C, second ramp 30  
and coworkers [33,34], who also demonstrated  
a strong  
°C to 210 °C, third ramp 2 °C/min to 228 °C, fourth ramp 30 °C  
correlation between molecular weight, width, length and the  
rejection for hydrophilic compounds (such as acetaminophen,  
phenacetine, caffeine, metronidazole, phenazone and  
sulfamethoxazole) by NF200 and NF90 membranes, and that  
rejection of these compounds may be attributed to the domination  
of the size exclusion effect. Agenson et al. [13] demonstrated that  
size exclusion effect represented by molecular weight and  
molecular width of solutes played a major influence on the  
rejection efficiency in membrane separation. Additionally, Van  
der Bruggen and Vandecasteele [35] suggested that the rejection  
of neutral TrOCs can be predicted using the molecular weight of  
the compound, a higher rejection for the compounds with larger  
molecular weight was obtained.  
to 260 °C, fifth ramp 3 °C/min to 290 °C, 3 min at 290 °C. The  
injector port and the temperature of the GCMS interface were set  
at 280 °C. A sample volume of 1 µL was injected in splitless  
mode.  
The MS was obtained by electron impact ionisation in full  
scan mode from 50 to 600 of m/z, and later on in selected ion  
monitoring (SIM) mode for qualitative determinations. The most  
abundant ions of each organic compound were selected from its  
spectrum for quantitation, in accordance with previous studies  
[25,26]. A series of standard TrOCs at 1, 10, 50, 100, 500, and  
1
000 ng/mL and a bisphenol A-d16 internal standard were  
prepared for the instrument calibration. The calibration curves  
obtained for each compound had correlation coefficients greater  
902  
Journal of Environmental Treatment Techniques  
2020, Volume 8, Issue 3, Pages: 900-907  
Table 1: Physicochemical properties and rejection efficiency of the selected trace organic contaminants  
Trace organic compounds Log Kow a  
pK  
a
Molecular Molecular Molecular Molecular Log D at pH  
Rejection ±  
STDEV (%)  
a
b
b
c
7
Width (nm) Height (nm)  
Length weight (MW)  
b
(nm)  
(g/mol)  
138.12  
206.28  
250.33  
296.15  
236.27  
266.34  
150.22  
206.32  
220.35  
289.54  
228.29  
270.37  
272.38  
288.38  
296.40  
314.42  
194.19  
218.25  
290.32  
253.28  
277.40  
361.82  
249.09  
268.26  
270.24  
Salicylic acid  
Ibuprofen  
Gemfibrozil  
Diclofenac  
Carbamazepine  
Pentachlorophenol  
2.011  
3.502  
4.302  
4.548  
1.895  
5.115  
3.397  
5.180  
6.142  
5.343  
3.641  
3.624  
4.146  
2.527  
4.106  
5.111  
-0.628  
0.829  
0.594  
0.659  
4.410  
2.504  
3.125  
2.860  
3.114  
3.01  
4.41  
4.75  
4.18  
13.94  
4.68  
10.13  
10.15  
10.15  
7.80  
10.29  
10.25  
10.27  
10.25  
10.24  
10.26  
0.52  
12.26  
7.04  
5.81  
9.18  
3.29  
12.13  
6.99  
6.51  
0.354  
0.354  
0.354  
0.354  
0.354  
0.354  
0.354  
0.354  
0.354  
0.354  
0.354  
0.340  
0.340  
0.340  
0.356  
0.354  
0.412  
0.426  
0.420  
0.412  
0.435  
0.420  
0.412  
0.412  
0.354  
TM  
0.577  
0.561  
0.670  
0.767  
0.676  
0.640  
0.505  
0.595  
0.519  
0.602  
0.570  
0.693  
0.693  
0.693  
0.693  
0.842  
0.676  
0.740  
0.766  
0.595  
0.933  
1.313  
0.668  
0.760  
0.706  
c
0.615  
0.900  
0.972  
0.829  
0.818  
0.659  
0.735  
0.822  
1.179  
0.926  
0.876  
0.697  
0.697  
0.751  
0.788  
0.947  
0.750  
0.734  
1.047  
1.032  
0.871  
0.773  
0.902  
1.015  
1.033  
-1.130  
0.940  
2.070  
1.770  
1.890  
2.850  
3.400  
5.180  
6.140  
5.280  
3.640  
3.620  
4.150  
2.530  
4.110  
5.110  
-7.110  
0.830  
0.310  
-0.560  
4.410  
-1.210  
3.120  
2.550  
2.500  
49.27 ± 3.21  
80.61 ± 2.13  
93.08 ± 2.45  
98.03 ± 1.05  
81.32 ± 2.88  
92.74 ± 3.96  
51.28 ± 5.99  
80.77 ± 2.73  
88.95 ± 2.76  
92.28 ± 1.49  
78.61 ± 4.68  
82.40 ± 4.83  
84.38 ± 1.10  
95.72 ± 0.40  
95.36 ± 1.94  
98.30 ± 1.03  
89.52 ± 3.75  
92.07 ± 1.86  
93.64 ± 3.27  
88.15 ± 1.75  
91.05 ± 0.84  
98.23 ± 0.05  
81.02 ± 5.36  
85.88 ± 5.27  
87.15 ± 5.52  
(pH-pKa)  
4
4
4
-tert-butylphenol  
-tert-octylphenol  
-n-nonylphenol  
Triclosan  
Bisphenol A  
Estrone  
1
-estradiol  
Estriol  
1
1
-ethinylestradiol  
-estradiol acetate  
Caffeine  
Primidone  
Trimethoprim  
Sulfamethoxazole  
Amitriptyline  
Bezafibrate  
Linuron  
Formononetin  
Genistein  
a
b
Scifinder Scholar, calculated using Molecular Modeling Pro Plus software, calculated by the equation: log D(pH) = Log Kow  Log (1+10  
).  
It is however noteworthy that there was no correlation  
statistics included R-squared = 91.42 percent, R-squared  
between rejection and log D of these TrOCs. These observations  
can be attributed to the fact that in addition to the effect of the log  
D, there are a number of other factors which may influence TrOCs  
rejection such as MW, molecular dimension, charge, and so on.  
Correlation between the physicochemical properties and the  
rejection for the TrOCs will be discussed in detail in the following  
section.  
(adjusted for d.f.) = 86.27 percent and Standard error of Est. =  
4.56.  
The output shows the results of fitting a multiple linear  
regression model to describe the relationship between Y and  
independent variables. The equation of the fitted model is as  
follows:  
Y = -1886.76 + 5020.79X1 + 2267.82X2 + 2092.90X3 +  
0.267792X4 - 0.837608X5 - 5891.10X1X2 - 5464.52X1X3 -  
2505.03X2X3 + 6448.92X1X2X3  
3
.2 Multiple linear regression model for trace organic  
contaminants rejection  
Physicochemical properties of the TrOCs and their rejection  
efficiency were used as indicators in the model development.  
Database using for multiple regression analysis are presented in  
the Table 2. Statistical analysis of multiple regression by  
Statgraphics Centurion software was used to construct the best  
optimal mathematical modeling for trace organic contaminants  
rejection by the NF270 membrane. The whole multiple regression  
model can be presented as one equation for dependent variable Y:  
In determining whether the model can be simplified, the  
highest P-value on the independent variables is 0.1038, belonging  
to X5 (Table 3). Since the P-value is greater or equal to 0.05, that  
term is not statistically significant at the 95.0 % or higher  
confidence level. Consequently, it should consider removing X5  
from the model. Whereas, the regression coefficients (include X1,  
X2, X3, X4, X1X2, X1X3, X2X3, and X1X2X3) contribute  
significantly to the model (exist in multiple regression equation)  
due to their P-values are less than 0.05. Since the P-value in the  
ANOVA table is less than 0.05, there is a statistically significant  
relationship between the variables at the 95.0 % confidence level.  
The R-squared statistic indicates that the model as fitted