CHEMICAL ENGINEERING TRANSACTIONS  
 

VOL. 52, 2016 

A publication of 

 

The Italian Association 
of Chemical Engineering 
Online at www.aidic.it/cet 

Guest Editors: Petar Sabev Varbanov, Peng-Yen Liew, Jun-Yow Yong, Jiří Jaromír Klemeš, Hon Loong Lam 
Copyright © 2016, AIDIC Servizi S.r.l., 

ISBN 978-88-95608-42-6; ISSN 2283-9216 
 

Development and Validation of N-nitrosamine Rejection 

Mathematical Model Using a Spiral-wound  

Reverse Osmosis Process 

Mudhar. A. Al-Obaidi, Chakib Kara-Zaïtri, Iqbal. M. Mujtaba* 

Chemical Engineering Division, School of Engineering, University of Bradford, Bradford, West Yorkshire BD7 1DP, UK  

I.M.Mujtaba@bradford.ac.uk 

In this paper, a one-dimensional mathematical model based on coupled differential and algebraic equations 

has been developed for analysing the separation mechanism of a N-nitrosamine in a spiral-wound reverse 

osmosis process. The model is based on Spiegler and Kedem’s work  on mass transport and Darcy’s law and 

concentration polarization to analyse the pressure drop and mass transfer coefficient in the module feed 

channel respectively.  The model is built using the gPROMS software suite and validated using N-nitrosamine 

rejection experimental data from the literature, obtained by using a pilot-scale cross-flow reverse osmosis 

filtration system. Analysis results derived from the model corroborate experimental data.  

1. Introduction 

Reverse Osmosis (RO) is a water purification process, used in in water desalination and waste treatment that 

uses a semipermeable membrane to remove undesirable particles. RO has many immediate advantages 

including minimum thermal damage, high packing density and lower energy consumption (Reverberi et al., 

2014)  . Furthermore, spiral-wound RO modules are less susceptible to membrane fouling and are easier to 

clean (Song, 2002). With the increasing application of RO in the removal of organic compounds, the modelling 

of membrane separation and mass transport mechanisms constitutes a significant factor in the development of 

cost effective and optimum design strategies. Various types of models have been proposed to describe the 

transport phenomena of solvent and solute through the membrane. These models are based on the three-

parameter irreversible thermodynamics model (Spiegler and Kedem, 1966), the solution-diffusion model 

(Lonsdale et al., 1965) and the pore flow models (Jonsson and Boesen, 1975). Successful examples of 

membrane transport mechanisms based on the Spiegler and Kedem model have been well described in the 

literature and validated for both wastewater treatment and seawater desalination processes. For example, 

Ahmad et al. (2007) have developed a model suitable for the multiple solutes system for unsteady-state 

simulation and prediction of membrane filtration dynamics in terms of permeate flux and concentration of each 

solute. This model was validated with experimental data derived from pre-treated palm oil mill effluent as a 

feed using a PVDF hollow fibre membrane module in a pilot plant scale RO system. Whereas, Mane et al. 

(2009) have studied boron rejection using two commercially spiral-wound modules in pilot-scale and full-scale 

RO processes by varying pH and pressures. Patroklou et al. (2013) developed a mathematical model of RO 

for boron rejection and validated the model using experimental data of Mane et al. (2009). 

To the best of authors’ knowledge, the development of a spiral-wound RO model based on the Spiegler and 

Kedem concept and its validation of N-nitrosamine compounds have only been conducted by Fujioka et al. 

(2014). Their model has conidered the variation of operating parameters in the x- axis with assuming zero 

permeate pressure. The aim of this paper is to develop a new distributed model with relaxing the proposed 

assumption of zero pressure at the permeate side. The new idea here is to develop a mathematical steady 

state model based on the work by Spiegler and Kedem to simulate the rejection of N-nitrosamine compounds 

using a spiral-wound RO filtration process. The model will account for the spatial variation of all operating 

parameters along the length of the feed and permeate channels of the module (distributed model). The 

                                

 
 

 

 
   

                                                  
DOI: 10.3303/CET1652189 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

Please cite this article as: Al-Obaidi M. A., Kara-Zaïtri C., Mujtaba I. M., 2016, Development and validation of n-nitrosamine rejection 
mathematical model using a spiral-wound reverse osmosis process, Chemical Engineering Transactions, 52, 1129-1134  
DOI:10.3303/CET1652189   

1129



validation of this model has been corroborated against experimental data derived from the literature. The data 

includes uncharged N-nitrosamine compounds rejection based on a pilot-scale plant of three cylindrical 

pressure vessels with spiral-wound elements. The process model developed can be used later to study the 

variation of solute concentration, pressure, flow rate, solvent and solute fluxes along the length of the feed 

channel and to investigate the impact of variation in operating conditions on the unit performance.  

2. Model Rationale and Development 

2.1 The Assumptions 
The following assumptions are made to develop the proposed process model: 

1. Spiegler and Kedem’s model is used for mass transport through the module. 

2. The membrane characteristics and the channel geometries are assumed constant. 

3. Validity of Darcy’s law where the friction parameter is used to characterize the pressure drop in both 

the feed and permeate channels. 

4. Constant solute concentration in the permeate channel and the average value will be calculated from 

the inlet and outlet calculated concentrations. 

5. The underlying process is assumed to be isothermal. 

2.2 Governing Equations 
The water and solute fluxes (Jw(x), Js(x)) (m/s, kmol/m² s) can be calculated using Spiegler and Kedem’s (1966) 

model: 

𝐽𝑤(𝑥) = 𝐿𝑝 (∆𝑃𝑏 (𝑥)  −  𝜎 ∆𝜋𝑠(𝑥) 
)  (1) 

𝐽𝑠(𝑥) = 𝐽𝑤(𝑥) (1 −  𝜎) 𝐶𝑏(𝑥)
~  +  𝜔 ∆𝜋𝑠(𝑥)  (2) 

Where, 𝐿𝑝, ∆𝑃𝑏 (𝑥), 𝜎, ∆𝜋𝑠(𝑥) and 𝜔 are solvent transport coefficient (m/atm s), trans-membrane pressure (𝑎𝑡𝑚), 

the reflection coefficient (dimensionless), the osmotic pressure difference (atm) and the solute permeability 

coefficients of the membrane (kmol/m²s atm) respectively. While, (𝐶𝑏(𝑥)
~ ) (kmol/m³) is the mean solute 

concentration in the feed side along the x-axis and calculated by. 

𝐶𝑏(𝑥)
~ =

𝐶𝑏(𝑥) − 𝐶𝑝(𝑎𝑣)

𝑙𝑛 (
𝐶𝑏(𝑥)

𝐶𝑝(𝑎𝑣)
)

  
(3) 

Where, Cb(x) and Cp(av) are feed solute concentration and the average permeate concentration (Kmol/m³) 

respectively. The following two equations work well for solute flux and the difference of osmotic pressure: 

𝐽𝑠(𝑥) =  𝐽𝑤(𝑥) 𝐶𝑝(𝑎𝑣)  (4) 

∆𝜋𝑠(𝑥) =  𝑅 𝑇𝑏 (𝐶𝑤(𝑥) −  𝐶𝑝(𝑎𝑣))  (5) 

Where,  𝑅, 𝑇𝑏  𝑎𝑛𝑑 𝐶𝑤(𝑥) (
𝑎𝑡𝑚 𝑚³

°𝐾 𝐾𝑚𝑜𝑙
 , °C, 

𝐾𝑚𝑜𝑙

𝑚3
) are the gas constant, the brine temperature and the molar solute 

concentration on the membrane surface respectively. Substitute Eq(5) in Eqs(1) and (2) gives: 

𝐽𝑤(𝑥) =  𝐿𝑝 (∆𝑃𝑏 (𝑥)  −  𝜎 𝑅 𝑇𝑏 (𝐶𝑤(𝑥)  −  𝐶𝑝(𝑎𝑣))  )  (6) 

𝐽𝑠(𝑥) =  𝐽𝑤(𝑥) (1 −  𝜎) 𝐶𝑏(𝑥)
~  +  𝜔 𝑅 𝑇𝑏 (𝐶𝑤(𝑥)  −  𝐶𝑝(𝑎𝑣))    (7) 

Re-arrangement Eq(7) yields. 

𝐶𝑤(𝑥)  −  𝐶𝑝(𝑎𝑣) =  
𝐽𝑤(𝑥) 𝐶𝑝(𝑎𝑣)

𝜔 𝑅 𝑇𝑏 
 −  

𝐶𝑏(𝑥)
~  ( 1 − 𝜎) 𝐽𝑤(𝑥)

𝜔 𝑅 𝑇𝑏
  (8) 

Then, substituting (𝐶𝑤(𝑥)  −  𝐶𝑝(𝑎𝑣)) in Eq(6) with re-arrangements gives: 

𝐽𝑤(𝑥) =
𝐿𝑝 ∆𝑃𝑏(𝑥)

1+
𝜎 𝐶𝑝(𝑎𝑣) 𝐿𝑝

𝜔
 − 

𝐶
𝑏(𝑥)
~ (1−𝜎) 𝜎 𝐿𝑝

𝜔

  (9) 

Eq(9) can be simplified to: 

𝐽𝑤(𝑥) =  
𝐿𝑝 ∆𝑃𝑏(𝑥)

∅(𝑥)
            where,              ∅(𝑥) =  1 +

𝜎 𝐶𝑝(𝑎𝑣) 𝐿𝑝

𝜔
 −  

𝐶𝑏(𝑥)
~ (1−𝜎) 𝜎 𝐿𝑝

𝜔
 (10) 

The trans-membrane pressure (ΔPb(x)) (atm) is related the pressure in both the feed and permeate channels. 

∆𝑃𝑏(𝑥) =  𝑃𝑏(𝑥)  −  𝑃𝑝(𝑥)  (11) 

Where, Pb(x) and Pp(x) are feed and permeate pressures (atm) respectively. The variation of trans-membrane 

pressure along x-axis can be calculated from: 

𝑑 ∆𝑃𝑏(𝑥)

𝑑𝑥
=  

𝑑 (𝑃𝑏(𝑥) − 𝑃𝑝(𝑥))

𝑑𝑥
  (12) 

1130



Specifically, the mass balance in both channels can be written as: 

𝐹𝑏(0) =  𝐹𝑏(𝑥) +  𝐹𝑝(𝑥)  (13) 

Fb(0), Fb(x) and Fp(x) (m³/s) are feed flow rate at x = 0 and at any point along the x-axis and permeate flow rate 

respectively. Also, Darcy’s law can be used to describe the drop of pressure in both channels (Assumption 3). 

𝑑𝑃𝑏(𝑥)

𝑑𝑥
=  − 𝑏 𝐹𝑏(𝑥)  (14) 

𝑑𝑃𝑝(𝑥)

𝑑𝑥
=  − 𝑏 𝐹𝑝(𝑥)  (15) 

Where, b (atm s/m4) is Feed and permeate channels friction parameter. The substitution of Eqs(14) and (15) 

in Eq(12), gives: 

𝑑 ∆𝑃𝑏(𝑥)

𝑑𝑥
=  −𝑏 (𝐹𝑏(𝑥)  −  𝐹𝑝(𝑥))  (16) 

While, the derivation of Eq(13) with the x-axis, gives: 

𝑑𝐹𝑏(𝑥) 
𝑑𝑥

=  −  
𝑑𝐹𝑝(𝑥) 

𝑑𝑥
=  − 𝑊 𝐽𝑤(𝑥)  (17) 

Where, W (m) is the width of membrane. Then, dividing Eqs(16) and (17), gives: 

𝑑 ∆𝑃𝑏(𝑥)

𝑑𝐹𝑏(𝑥) 
=  

 𝑏(𝐹𝑏(𝑥) − 𝐹𝑝(𝑥))

𝑊 𝐽𝑤(𝑥)
  (18) 

The proposed correlations of Koroneos (2007) to calculate the physical properties of seawater (density, 

viscosity and diffusion coefficient) are considered identical to the analysis of dilute aqueous solutions of 

wastewater. Finally, the re-arrangement and integration yields the specific equations of the model used for 

simulation as follows:  

The feed flow rate (Fb(x)) (m³/s) at any point along the x-axis is calculated by: 

𝐹𝑏(𝑥)
2 =  𝐹𝑏(0)

2 +
𝑊 𝐿𝑝

𝑏 ∅(𝑥)
(∆𝑃𝑏(𝑥)

2 − ∆𝑃𝑏(0)
2 ) + 2 (𝐹𝑏(0) − 𝐹𝑏(𝑥)

 
) (𝐹𝑏(𝑥) − 𝐹𝑏(0))  (19) 

The feed pressure (Pb(x)) (atm) and the trans-membrane pressure (ΔPb(x)) (atm) are calculated by Eqs(20, 21): 

𝑃𝑏(𝑥)
2 = [𝑃𝑏(0) − (𝑏 𝑥 𝐹𝑏(0)) − (𝑏 𝑥 (

𝑊 𝐿𝑝

𝑏 ∅(𝑥)
)

0.5

(∆𝑃𝑏(𝑥) − ∆𝑃𝑏(0)))]

2

−  [𝑏2 𝑥2 (2 (𝐹𝑏(0) − 𝐹𝑏(𝑥)
 

) (𝐹𝑏(𝑥) −

𝐹𝑏(0))) ]  

(20) 

∆𝑃𝑏(𝑥)
2 = [∆𝑃𝑏(0) − (𝑏 𝑥 𝐹𝑏(0)) − (𝑏 𝑥 (

𝑊 𝐿𝑝

𝑏 ∅(𝑥)
)

0.5

(∆𝑃𝑏(𝑥) − ∆𝑃𝑏(0)))]

2

− [𝑏2 𝑥2 (2 (𝐹𝑏(0) − 𝐹𝑏(𝑥)
 

) (𝐹𝑏(𝑥) −

𝐹𝑏(0))) ] + [𝑏
2 𝑥2 (𝐹𝑏(0) − 𝐹𝑏(𝑥)

 
)

2
 ]  

(21) 

The water flux (Jw(x)) (m/sec) and solute flux (Js(x)) (kmol/m² s) depict in the counter of Eqs(22 and 23). 

𝐽𝑤(𝑥) =
𝐿𝑝

∅(𝑥)
 {∆𝑃𝑏(0) − (𝑏 𝑥 𝐹𝑏(0)) − (𝑏 𝑥 (

𝑊 𝐿𝑝

𝑏 ∅(𝑥)
)

0.5

(∆𝑃𝑏(𝑥) − ∆𝑃𝑏(0))) − [𝑏 𝑥 (2 (𝐹𝑏(0) − 𝐹𝑏(𝑥)
 

) (𝐹𝑏(𝑥) −

𝐹𝑏(0)))
0.5

 ] + [𝑏 𝑥 (𝐹𝑏(0)  −  𝐹𝑏(𝑥)
 

) ]}      

(22) 

𝐽𝑠(𝑥) = (𝐽𝑤(𝑥) (1 − 𝜎) 𝐶𝑏(𝑥)
~ ) + (𝜔 𝑅 𝑇𝑏  (𝐶𝑏(𝑥) − 𝐶𝑝(𝑎𝑣)) 𝑒

𝐽𝑤(𝑥)

𝑘(𝑥) )  (23) 

Eq(24) is used to calculate the feed solute concentration along the x-axis (Cb(x)) (kmol/m³) (Lee et al., 2010). 

𝑑
(𝐶𝑏(𝑥) 𝐹𝑏(𝑥))

𝑡𝑓 𝑊

𝑑𝑥
= −

𝐽𝑤(𝑥) 𝐶𝑝(𝑎𝑣)

𝑡𝑓
+

𝐽𝑤(𝑥) 𝐶𝑏(𝑥)

𝑡𝑓
+

𝑑

𝑑𝑥
(𝐷𝑏(𝑥)

𝑑𝐶𝑏(𝑥)

𝑑𝑥
)  

(24) 

Db is the diffusivity parameter of feed (m²/s). Then, the permeate flow rate (Fp(x)) (m³/s) is calculated by: 

𝐹𝑝(𝑥)
2 = 𝐹𝑝(0)

2 + (
𝑊 𝐿𝑝 𝑥

 ∅(𝑥)
∆𝑃𝑏(0) )

2

− (
𝑊 𝐿𝑝 𝑏

 ∅(𝑥)
 𝐹𝑏(0) (

𝑥2

2
))

2

−  {(
𝑊 𝐿𝑝 

 ∅(𝑥)
)

1.5

𝑏0.5  (
𝑥2

2
)  (∆𝑃𝑏(𝑥) − ∆𝑃𝑏(0))}

2

−

{(
𝑊 𝐿𝑝 𝑏

 ∅(𝑥)
) (

𝑥2

2
)  (2 (𝐹𝑏(0) − 𝐹𝑏(𝑥)

 
) (𝐹𝑏(𝑥) − 𝐹𝑏(0) ))

2

} + ((
𝑊 𝐿𝑝 𝑏

 ∅(𝑥)
) (

𝑥2

2
)  (𝐹𝑏(0) − 𝐹𝑏(𝑥)

 
))

2

  

(25) 

The variation of permeate pressure along the x-axis is calculated according to Eq(26). 

1131



𝑃𝑝(𝑥)
2 = 𝑃𝑝(0)

2 − (𝑏 𝑥 𝐹𝑏(0) )
2 

−  ((
𝑊 𝐿𝑝 𝑏

 ∅(𝑥)
) (

𝑥2

2
)  ∆𝑃𝑏(0) )

2

 + ((
𝑊 𝐿𝑝 

 ∅(𝑥)
)  (

𝑥3

6
) 𝑏2 𝐹𝑏(0) )

2

+

 [𝑏1.5 (
𝑥3

6
)  (

𝑊 𝐿𝑝 

 ∅(𝑥)
)

1.5

(∆𝑃𝑏(𝑥) − ∆𝑃𝑏(0) )]

2

+  (𝑏2  (
𝑊 𝐿𝑝 

 ∅(𝑥)
)  (

𝑥3

6
)  (2 (𝐹𝑏(0) − 𝐹𝑏(𝑥)

 
) (𝐹𝑏(𝑥) − 𝐹𝑏(0) )) )

2

−

 ((
𝑥3

6
)  (

𝑊 𝐿𝑝 𝑏

 ∅(𝑥)
)  (𝐹𝑏(0) − 𝐹𝑏(𝑥)

 
))

2

        

(26) 

The mass transfer coefficient (𝑘(𝑥)) (𝑚 𝑠⁄ ) and feed velocity are calculated using Eqs(27 and 28). 

𝑘(𝑥) = 0.753 (
𝐾

2−𝐾
)

0.5
(

𝐷𝑏(𝑥)

 𝑡𝑓
)  (

𝜇𝑏(𝑥) 𝜌𝑏(𝑥)

𝐷𝑏(𝑥)
)

0.1666

(
2 𝑡𝑓

2 𝑈𝑏(𝑥)

𝐷𝑏(𝑥) ∆𝐿
)

0.5

  (27) 

𝑈𝑏(𝑥) = 𝐹𝑏(𝑥) /  (𝑡𝑓 𝑊)  (28) 

Where, 𝐾, 𝜇𝑏(𝑥) , 𝜌𝑏(𝑥)  and Ub(x) are the efficiency of mixing (K = 0.5) (dimensionless), feed viscosity (Kg/m sec), 

density (kg/m³) and feed velocity (m/sec) respectively (Mane et al., 2009). The concentration at the wall 

membrane (Cw(x)) (kmol/m³) is calculated using Eq(29). 

𝐶𝑤(𝑥)−𝐶𝑝(𝑎𝑣)

𝐶𝑏(𝑥)−𝐶𝑝(𝑎𝑣)
=  𝑒𝑥𝑝 (

𝐽𝑤(𝑥)

𝑘(𝑥)
)  (29) 

Eqs(30, 31 and 32) have been used to calculate the permeate solute concentrations (Cp(x)) (kmol/m³) at both  

x = 0 and x = L and the average permeate concentration (Cp(av)) (kmol/m³) respectively. 

𝐶𝑝(0) =
𝐵𝑠 𝑅 𝑇𝑏 𝐶𝑏(0) 𝑒

𝐽𝑤(0)
𝑘(0)

𝐽𝑤(0) + 𝐵𝑠 𝑅 𝑇𝑏   𝑒

𝐽𝑤(0)
𝑘(0)

  (30) 

𝐶𝑝(𝐿) =
𝐵𝑠 𝑅 𝑇𝑏 𝐶𝑏(𝐿) 𝑒

𝐽𝑤(𝐿)
𝑘(𝐿)

𝐽𝑤(𝐿) + 𝐵𝑠 𝑅 𝑇𝑏   𝑒

𝐽𝑤(𝐿)
𝑘(𝐿)

  (31) 

𝐶𝑝(𝑎𝑣) =
𝐶𝑝(0 )+ 𝐶𝑝(𝐿)

2
  (32) 

Bs is solute transport coefficient (m/s) used for the case of assuming (𝜎 = 1) as can be found in the solution-

diffusion model. Finally, Eq(33 and 34) are used to calculate the solute rejection and total water recovery 

(dimensionless). 

𝑅𝑒𝑗 =
𝐶𝑏(𝐿)−𝐶𝑝(𝑎𝑣)

𝐶𝑏(𝐿)
× 100  (33) 

𝑅𝑒𝑐(𝑇𝑜𝑡𝑎𝑙) =
𝐹𝑝(𝑇𝑜𝑡𝑎𝑙)

𝐹𝑏(0)
×  100  (34) 

The model equations have been solved using the gPROMS software where the filtration channel is divided 

into a number of segments of equal intervals (Δx). For a given inlet feed flow rate, pressure, solute 

concentration and temperature with a guesstimate of the initial value of permeate pressure (close to 1 atm), 

the proposed model can predict the longitudinal variation of all parameters in the feed and permeate channels 

in the x-axis by using the estimated values of membrane transport parameters Lp, Bs, 𝜔, 𝜎 and b.  

3. Materials and methods 

A pilot-scale RO filtration system of three 4 inch glass-fiber pressure vessels was used by Fujioka et al. (2014) 

in the experiments (described below) of eight N-nitrosamine solutes rejection with a molecular weight in the 

range of (74 – 158 g/mol) as summarized in Table 1. The N-nitrosamine stock solution contains 10 mg/L of 

each uncharged N-nitrosamine solutes in the tested solution (pH 8) [N-nitrosodimethylamine-D6 (NDMA), N-

nitrosomethylethylamine-D3 (NMEA), N-nitrosopyrrolidine-D8 (NPYR), N-nitrosodiethylamine-D10 (NDEA), N-

nitrosopiperidine-D10 (NPIP), N-nitrosomorpholine-D8 (NMOR), N-nitrosodipropylamine-D14 (NDPA) and N-

nitrosodi-n-butylamine-D9 (NDBA)] were prepared in pure methanol.  Also, an aqueous feed of stock solutions 

of NaCl, CaCl2 and NaHCO3 were also prepared in Milli-Q water at 2 M (NaCl) and 0.1 M (CaCl2 and 

NaHCO3) to simulate the electrolyte composition typically found in treated wastewater. Then, the filtration 

experiments were carried out by introducing the stock solution of N-nitrosamine compounds in the feed to 

obtain approximately (250 ng/L) of each target Nitrosamine compound. Each pressure vessel holds only one 

spiral-wound element. The pressure vessels are connected in series where the concentrated feed solution of 

the first vessel was transferred to the second vessel followed by the third one. The permeate is then collected 

from the stages and recycled back with the retentate solution into the feed tank to maintain constant feed 

concentration.  

1132



The feed was pumped at constant volumetric flow rate of 2.43 × 10-3 (m³/s), while the average permeate flux 

was adjusted at 2.78 × 10-6, 5.56 × 10-6 and 8.33 × 10-6 (m/s) during the experiments by increasing the 

operating feed pressure from 4, 6.5 and 10.1 atm respectively. The feed temperature was controlled at 20 ± 

0.1 °C along the experiments. 

Table 1: Physical and transport parameters of the selected N-nitrosamines (Fujioka et al., 2014) 

Name  

Molecular 

weight 

(g/mol) 

Permeability  

coefficient, Bs (m/s) 

Water permeability coefficient, (Lp) 

(m/s atm) at 20 °C 

Reflection coefficient, 

𝜎 (dimensionless) 

NDMA 

NEMA 

NPYR 

74.05 

88.06 

100.06 

4.75 × 10-6 - 5.35 × 10-6 

1.15 × 10-6 - 1.23 × 10-6 

4.18 × 10-7 - 4.39 × 10-7 

1.06 × 10-6 – 1.27 × 10-6  

1.05 × 10-6 – 1.21 × 10-6  

1.05 × 10-6 – 1.23 × 10-6  

0.951 - 0.985 

0.936 – 0.989 

0.983 – 0.998 

4. Determination of Transport Parameters   

Unknown parameters of the membrane elements and the operating conditions should be determined before 

solving the model equations. In the simulation study, experimental data will be used to predict the best values 

of unknown parameters, which are then used with the known parameters to check the behavior of the unit with 

the variance of operating variables. The friction and the model transport parameters for each run was 

calculated using gEST parameter estimation in gPROMS software using the experiments conducted by 

Fujioka et al. (2014). The registered values of friction parameter are (62, 194 and 395 (atm s/m⁴)) for the 

average permeate fluxes 2.78 × 10-6, 5.56 × 10-6 and 8.33 × 10-6 (m/s) respectively.  The membrane transport 

parameters of the three selected N-nitrosamine solutes (NDMA, NEMA and NPYR) were obtained from the 

experiments conducted by Fujioka et al. (2014) and listed in Table 1. Also, the specifications of the spiral-

wound membrane element can be shown in Table 2. 

Table 2: Specifications of the spiral-wound membrane element 

Make Hydranautics, Oceanside, CA., USA 

Membrane type and configuration ESPA2-4040, Spiral-wound, Composite Polyamide 

Feed and permeate spacer thickness (tf) and (tp), (m)                           6.6 × 10-4 

Membrane sheet area, (m2) 7.9 

Membrane sheet length (L) and width (W) (m)                         0.9 and 8.7778 

5. Model validation 

The model described in section 2 has been validated by comparing the model predictions results with those 

obtained from actual experimentation for the specific RO membrane. Figure 1 shows the comparison of model 

rejections of the three selected N-nitrosamines at three different overall permeate fluxes against experimental 

results. The clear corroboration with experimental data readily shows the suitability of the model to measure 

the observed rejection data with an accepted error within high operating pressures. Furthermore, Figure 1 

shows that the model can be used to simulate the observed data at low operating pressure (low average 

permeate flux) but with some minor deviation. This might be attributed to the inaccurate estimation of the 

transport membrane parameters at such pressure.  

 

 

Figure 1: Experimental and model rejections of NDMA, NEMA and NDEA with average permeate flux 

0

20

40

60

80

100

8.00E-06 6.00E-06 3.00E-06

%
R

e
je

c
ti
o

n

Average Permeate Flux, m³/s

NDMA EXP.

NDMA Model

NEMA EXP.

NEMA Model

NPYR EXP.

NPYR Model

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In addition, Figures 2 and 3 show the linear fittings with a regression coefficient R² (close to 1) for both the 

experimental and model prediction of the outlet feed flow rate and pressure respectively. 
  

Figure 2: Linear fitting of experimental and model 

prediction of outlet feed flow rate 

 

Figure 3: Linear fitting of experimental and model 

prediction of outlet feed pressure 

6. Conclusions 

A steady state one-dimensional mathematical model based on the work of the Spiegler and Kedem has been 

developed for a spiral-wound reverse osmosis process. The model has been validated using a simulation 

study with experimental data derived from N-nitrosamine compounds rejection. Analysis results readily show 

that the proposed model can be used to simulate wastewater rejection during scale-plant design with an 

accepted convergence for most operating parameters. Further work is planned to examine the impact of the 

operating variables on N-nitrosamine rejection. 

References 

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osmosis process in palm oil mill effluent (POME) treatment. Chemical Engineering Journal, 132, 183-193. 

Fujioka T., Khan S. J., Mcdonald J. A., Roux A., Poussade Y., Drewes J. E. and Nghiem L. D., 2014, 

Modelling the rejection of N-nitrosamines by a spiral-wound reverse osmosis system: Mathematical model 

development and validation. Journal of Membrane Science, 454, 212-219. 

Jonsson G. and Boesen C. E., 1975, Water and solute transport through cellulose acetate reverse osmosis 

membranes. Desalination, 17, 145-165. 

Koroneos C., Dompros A. and Roumbas G., 2007, Renewable energy driven desalination systems modelling. 

Journal of Cleaner Production, 15, 449-464. 

Lee C., Chen Y. and Wang G., 2010, A dynamic simulation model of reverse osmosis desalination systems. 

The 5th International Symposium on Design, Operation and Control of Chemical Processes, PSE ASIA, 

Singapore. 

Lonsdale H. K., Merten U. and Riley R. L., 1965, Transport properties of cellulose acetate osmotic 

membranes. Journal of Applied Polymer Science, 9, 1341-1362. 

Mane P. P., Park P.-K., Hyung H., Brown J. C. and Kim J.-H., 2009, Modelling boron rejection in pilot- and full-

scale reverse osmosis desalination processes. Journal of Membrane Science, 338, 119-127. 

Patroklou G., Sassi, K.M. and Mujtaba, I.M., 2013, Simulation of boron rejection by seawater reverse osmosis 

desalination. Chemical Engineering Transaction, 32, 1873-1878. 

Reverberi A., Fabiano B., Cerrato C. and Dovì V., 2014, Concentration polarization in reverse osmposis 

membranes: Effect of membrane splitting. Chemical Engineering Transaction, 39, 763-768. 

Song L., Hong S., Hu J., Ong S. and Ng W., 2002, Simulations of Full-Scale Reverse Osmosis Membrane 

Process. Journal Of Environmental Engineering, 128, 10, 960. 

Spiegler K. S. and Kedem O., 1966, Thermodynamics of hyperfiltration (reverse osmosis): criteria for efficient 

membranes. Desalination, 1, 311-326. 

 

 

R² = 0.9996

0.00213

0.00218

0.00223

0.00228

0.00233

0.00238

0.0022 0.00225 0.0023 0.00235 0.0024

F
b

(L
) 
E

x
p

 (
m

³/
s
)

Fb(L) Model (m³/s)

R² = 1

3

4

5

6

7

8

3 5 7 9

P
b

(L
) 

E
x
p

. 
(a

tm
)

Pb(L) The. (atm)

1134