# A Novel Mass Transfer Model to Describe the Separation Process in Reverse Osmosis of Glucose

Authors: Chenghan Chen; Yanwei Wang; Furong Tan; Qili Zhu
Journal: International Journal of Chemical Engineering
 (2021)

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## Abstract

Basic and theoretical research on processes such as reverse osmosis (RO) is essential in the fermentation industry to improve production efficiency and reduce cost. Here, we focus on the RO concentration of glucose solutions. We constructed a mathematic model that incorporates various membrane and experimental parameters to characterize the mass transfer process of RO membrane and validated the model output with experimental data. Calculation results were highly consistent with the experimental data, demonstrating that this model can be useful for predicting the RO concentration process.


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## Body

## 1. Introduction

The field of fermentation technology has rapidly advanced over recent years [1]. In the microbial fermentation process, the concentration of fermentable sugar is a key component that directly impacts the ultimate concentration of target products [2]. Low concentrations of fermentable sugar in the fermentation broth lead to low concentrations of the target product, and the subsequent purification process is expensive [3]. As such, it is important to increase the concentrations of fermentable sugars prior to fermentation.Many methods to concentrate fermentable sugars have been developed, including vacuum distillation [4], pervaporation [5], and freezing [6]. In general, these techniques are useful [7], but typically usually involve complicated operations performed by expensive equipment with high energy demands. This industry urgently needs to develop a convenient and environment-friendly in situ concentration method. Here, we present membrane technology as a feasible alternative.Widespread interest has developed in the use of membrane technology as a sugar concentration method. In recent years, this technique has become widely adopted in industrial applications and research laboratories as it is energy saving, environment friendly, and exhibits permselectivity capability [8–10]. At present, biorefining uses pressure-, chemical-, or thermal-driven membrane systems such as RO, microfiltration (MF), ultrafiltration (UF), nanofiltration (NF), pervaporation (PV), and membrane distillation (MD) to concentrate sugars [11]. Of these, NF and RO have been successfully employed to concentrate glucose and xylose [12]; however, RO membranes are more efficient as they possess pore sizes that more effectively prevent sugar loss [13].Increased focus on RO technology has resulted in a high demand for RO membranes, elevating the cost of RO systems. Optimizing the RO membrane can reduce these costs. One method of making the system more efficient is to develop a mathematical model that adequately describes the performance of the RO process [14]. On the other hand, the concentration of sugar in the concentrate is an important parameter in the RO concentration process. Therefore, further research needs to be carried out to reveal the impact of various parameters in the RO process on the sugar concentration process, including membrane parameters and experimental parameters.Here, we investigated the mass transfer glucose during the RO process by carrying out a series of concentration experiments using glucose in a laboratory-scale RO system. With the results from these experiments, we developed a novel mathematical model that comprehensively reflects the effect of various parameters to characterize the mass transfer process of the RO membrane. Studies on validation of the model with experimental results were carried out. This model will provide valuable information that can be used to predict the RO concentration process.


## 2. Experimental

### 2.1. Membrane and Module

RO membranes are comprised primarily of polyamide and cellulose acetate (CA). Nowadays, polyamide membranes dominate RO market sales with a 91% share, with asymmetric CA hollow fiber membranes holding a distant second spot. Although the latter has superior chlorine resistance, the former has higher salt rejection and net pressure driving force [15]. A polyamide composite membrane (PA2-4040, HYDECANME, USA) was selected for the experiments (see Table 1 for characteristics of the membrane).Table 1
Main characteristics of the used membrane.
TypePA2-4040Membrane propertiesCompositionPolyamidePermeable capacity (average, m3·d−1); membrane area effectively (m2)7.2; 7.9Recovery rate (single, %)15Usage conditionsMaximum pressure (Mpa)4.14Temperature (°C)5–45Maximum flow (m3·h−1)3.6The experimental setup (Figure1) consisted of a cooling and heating circulation tank to control the temperature of the model solution, a pump for feeding glucose solution into the membrane module, a check valve to control the solution flux, the RO membrane module which included pressure gauges and a flow meter, and a device for sampling the solution.Figure 1
Schematic illustration of the RO system.


### 2.2. Model Solution Preparation

In this study, glucose anhydrous (AR) was purchased from Kelong Chemical Reagent Factory, Chengdu, China. Model solutions (24.8–166.8 mol•m−3) were prepared using ultrapure water, adding suitable amounts of the analytes (calculated).


### 2.3. Concentration Experiment

The glucose model solutions were selected to carry out RO concentration experiments (Figure1). In this experiment, the RO membrane remained unchanged. Prior to running the experiment, the system was flushed with deionized water. Model solutions (15 L) were transferred into the membrane module using a feed pump. Next, the solution was concentrated and permeate outflow from the RO membrane module was recirculated into the circulation tank. A series of operation parameters were examined during this procedure, including glucose solution inlet flux (Qf, 28 × 10−6 m3 s−1 < Qf < 250 × 10−6 m3 s−1, obtained using a flowmeter), inlet concentration of glucose solution (Cf, 35 mol m−3 < Cf < 140 mol m−3, calculated), transmembrane pressure (△p, 300 kPa <△p < 620 kPa, obtained using a pressure gauge), and temperature (T, T = 306 K, obtained using a temperature gauge). Samples were collected three times to obtain an average value under each experimental condition.In this study, membrane performance was evaluated by the observed retention. The observed retention is displayed asRo and characterizes the ability of the Ro membrane to retain a component, where the greater the Ro value, the greater the difficulty to pass through the membrane. Ro can be expressed by the following equation [16]:(1)Ro=Cf−CpCf,where Cf (g·L−1) and Cp (g·L−1) are the feed concentration and permeate concentration of the solute.


### 2.4. Assay

The concentration of glucose solutions (amount of sample for each test: 20 µL) was quantified by high-performance liquid chromatography (HPLC, Agilent LC1200, USA) equipped with a differential refraction detector (RID), and an Aminex HPX-87H column (Bio-Rad Co., Hercules, CA, USA). The mobile phase was 5 mmol L−1 H2SO4 at a flow rate of 0.6 mL·min−1, and the column temperature was maintained at 35°C.


## 2.1. Membrane and Module

RO membranes are comprised primarily of polyamide and cellulose acetate (CA). Nowadays, polyamide membranes dominate RO market sales with a 91% share, with asymmetric CA hollow fiber membranes holding a distant second spot. Although the latter has superior chlorine resistance, the former has higher salt rejection and net pressure driving force [15]. A polyamide composite membrane (PA2-4040, HYDECANME, USA) was selected for the experiments (see Table 1 for characteristics of the membrane).Table 1
Main characteristics of the used membrane.
TypePA2-4040Membrane propertiesCompositionPolyamidePermeable capacity (average, m3·d−1); membrane area effectively (m2)7.2; 7.9Recovery rate (single, %)15Usage conditionsMaximum pressure (Mpa)4.14Temperature (°C)5–45Maximum flow (m3·h−1)3.6The experimental setup (Figure1) consisted of a cooling and heating circulation tank to control the temperature of the model solution, a pump for feeding glucose solution into the membrane module, a check valve to control the solution flux, the RO membrane module which included pressure gauges and a flow meter, and a device for sampling the solution.Figure 1
Schematic illustration of the RO system.


## 2.2. Model Solution Preparation

In this study, glucose anhydrous (AR) was purchased from Kelong Chemical Reagent Factory, Chengdu, China. Model solutions (24.8–166.8 mol•m−3) were prepared using ultrapure water, adding suitable amounts of the analytes (calculated).


## 2.3. Concentration Experiment

The glucose model solutions were selected to carry out RO concentration experiments (Figure1). In this experiment, the RO membrane remained unchanged. Prior to running the experiment, the system was flushed with deionized water. Model solutions (15 L) were transferred into the membrane module using a feed pump. Next, the solution was concentrated and permeate outflow from the RO membrane module was recirculated into the circulation tank. A series of operation parameters were examined during this procedure, including glucose solution inlet flux (Qf, 28 × 10−6 m3 s−1 < Qf < 250 × 10−6 m3 s−1, obtained using a flowmeter), inlet concentration of glucose solution (Cf, 35 mol m−3 < Cf < 140 mol m−3, calculated), transmembrane pressure (△p, 300 kPa <△p < 620 kPa, obtained using a pressure gauge), and temperature (T, T = 306 K, obtained using a temperature gauge). Samples were collected three times to obtain an average value under each experimental condition.In this study, membrane performance was evaluated by the observed retention. The observed retention is displayed asRo and characterizes the ability of the Ro membrane to retain a component, where the greater the Ro value, the greater the difficulty to pass through the membrane. Ro can be expressed by the following equation [16]:(1)Ro=Cf−CpCf,where Cf (g·L−1) and Cp (g·L−1) are the feed concentration and permeate concentration of the solute.


## 2.4. Assay

The concentration of glucose solutions (amount of sample for each test: 20 µL) was quantified by high-performance liquid chromatography (HPLC, Agilent LC1200, USA) equipped with a differential refraction detector (RID), and an Aminex HPX-87H column (Bio-Rad Co., Hercules, CA, USA). The mobile phase was 5 mmol L−1 H2SO4 at a flow rate of 0.6 mL·min−1, and the column temperature was maintained at 35°C.


## 3. Results and Discussion

### 3.1. RO Experiments

To obtain the membrane performance in the process of RO concentration experiments, some operation parameters were examined. Transmembrane pressure and feed flow were found to interact with each other, with flow rate decreasing as transmembrane pressure increased (Figure2). The feed flow was changed by adjusting valve 1, and transmembrane pressure in the system was changed by adjusting valve 2 and valve 3.Figure 2
Relation between transmembrane pressure and feed solution flow (Cf = 70.17 mol m−3, PH = 6, and T = 306 K).When other experimental conditions were held constant (feed solution concentrationCf = 70.17 mol m−3, PH = 6, and feed solution temperature T = 306 K), the total flux of solvent and solute Jv, Ro, and concentrate solution concentration Cb decreased with feed solution flow (Figure 3). Ro slightly decreased with the increase of Qf (Figure 3(a)). △p also decreased with Qf, which mean that a unit volume of feed solution can be divided into less pressure to facilitate the RO process, and the macroscopic performance of this process was the decrease in Ro and Jv (Figure 2). As Qf increased, the rate of Cb reduction changed from fast to slow (Figure 3(b)). Overall, Jv decreased with Qf, which means the concentrate flow Qb increased as Qf increased, and the Qb was close to Qf; then, Cb also approached Cf (Figure 3(a)). If Qf continues to increase, two consequences may result: (1) Qf exceeds the processing capacity of the membrane, resulting in membrane damage; (2) Cb becomes approximately the same as Cf such that the RO process substantially loses the ability to separate.Figure 3
Influence of feed solution flow on the solvent flux, solute rejection, and concentrate solution concentration. (Cf = 70.17 mol m−3, PH = 6, and T = 306 K).
(a)(b)An increase inCf led to increases in Cb and decreases in Jv (Figure 4). RO did not detectably vary with Cf. In the RO process, concentration polarization (CP) is a phenomenon where the solute accumulates on the surface of the membrane causing substantial reduction in the rejection coefficient of the membrane. When Qf is constant, the increase in Cf led to severe CP, decreasing Jv. In addition, high levels of CP led to a slightly lower trend of Ro. Cb also increased with Cf, as predicted.Figure 4
Influence of feed solution concentration on the solvent flux, solute rejection, and concentrate solution concentration (Qf = 1.56 × 10−4 m3 s−1, PH = 6, and T = 306 K).
(a)(b)Overall, we found that different operating parameters have different effects onCb and are difficult to quantify (Figures 3 and 4). Therefore, this paper aims to derive a mathematical model that can synthesize various parameters to better express Cb.


### 3.2. Membrane Transport Equations

Mass transfer in RO separation processes is mainly governed by two aspects, (1) inside-membrane mass transfer and (2) outside-membrane mass transfer. In outside-membrane mass transfer, the main object of investigation is the mass transfer process near the membrane, which can be significantly hindered by CP (Figure5). The relationship of solute concentration at the membrane surface, feed solution concentration, and permeate solution concentration can be expressed by the following equation [17]:(2)Cδ,1−CpCf−Cp=eJwδc,p/D=eJw/k,where Jw is the solvent (pure water) flux, Cp is the permeate solution concentration, D is the diffusion coefficient, Cδ,1 is the solute concentration at the membrane surface (feed side, Figure 5), δcp is the CP layer thickness, and k is the mass transfer coefficient in the CP layer.Figure 5
Schematic diagram of the mass transfer inRo membrane separation process.For the RO process, the relationship between the solute fluxJs, Jw , and Jv is as follows:(3)Jv=Jw+Js≈Jw.To describe the mass transfer process of inside-membrane, the irreversible thermodynamic Spiegler–Kedem model [18, 19] is assumed to be appropriate to explain the separation performance of solute through the membrane in this work. As much, Jv can be expressed by the following equation:(4)Jv=Lp+Δp−σΔπ,where Lp is the hydraulic permeability constant, △π is the difference in the osmotic pressure across the membrane, and σ is the reflection coefficient. The reflection coefficient represents the solute separation capability of a membrane, which, for permeable membranes, is bounded by 0<puncsp> </puncsp><<puncsp> </puncsp> σ < 1. △π is calculated using the following equation:(5)Δπ=RTCδ,1−Cδ,2=RTCδ,1−Cp,where R is the gas law constant, T is the temperature, and Cδ,2 is the solute concentration at the membrane surface (permeate side, Figure 5).Substituting equation (5) in equation (4), we get(6)Jv=LpΔp−σΔπ=LpΔp−σRTCδ,1−Cp.


### 3.3. Conservation Equations

According to the principle of mass conservation, we can get the following equations:(7)QfCf=QbCb+QpCp,(8)Qf=Qb+Qp,where Qp is the permeate flow. Also, the relationship between Qp and Jv is shown in the following equation:(9)Jv=Qps,where S is the Ro membrane area effectively.Substituting equations (8) and (9) in equation (7), we can get an equation for Cb(L) at the exit:(10)CbL=QfCfQf−JvS−JvSQf−JvSCp.Substituting equation (6) in equation (10), we get(11)CbL=QfCf−CpLpΔp−σRTCδ,1−CpSQf−CpΔp−σRTCδ,1−CpS.SinceCδ,1 is difficult to determine, the key to solving Cb is to express Cδ,1−Cp.


### 3.4. Flow Equations

According to equations (7) and (8), we can get the following equations:(12)QfCf=QbxCbx+QpxCpx,(13)Qf=Qbx+Qpx.As the concentration of the permeate solution concentration in the RO process is small,Cp is assumed to be uniform in the permeate channel and can, therefore, be expressed as a constant. Thus, equation (12) can be rewritten as(14)QfCf=QbxCbx+QpxCp.According to Figure6, the total mass balance and the solute mass balance at any point along the feed channel are given as follows [20]:(15)dubxdx=−Jvhf,(16)dCbxdx=−JvubhfCbx−Cδ,1x,where ub and hf are the feed solution flow rate and feed channel thickness.Figure 6
Schematic diagram of macroseparation process in theRo membrane.According to equations (13) and (15), we get(17)dQbxdx=−dQpxdx,(18)dQbxdx=dubxdxA=−Jvhfwhf=−wJv,where A and w are the feed channel cross-sectional area and feed channel width.Also, according to equation (16), we get(19)0=dQbxdxCbx+dCbxdxQbx+dQpxdxCp.Substituting equations (16), (17), and (18) in equation (19), we get(20)2Cbx=Cδ,1x+Cp.We defineφ = Cδ,1(L)−Cp; then, equation (20) can be rewritten as follows:(21)CbL=φ+2Cp2.Combining equations (11) and (21), we can get the following expression:(22)φ+12Qf−LpΔpSLpSσRT2=2QfCf−CpLpSσRT+14Qf−LpΔpSLpSσRT2.Then, we can derive the expression ofφ:(23)φ=θ−Qf−LpΔpS2LpSσRT.We defineθ as follows:(24)θ=8QfCf−CpLpSσRT+Qf−LpΔpS2.According to equations (21) and (23), we can get the concentrate solution concentration at the exit as the following expression:(25)CbL=2QfCf−QfCp−LpΔpSCp+CpθQf−LpΔpS+θ.According to equation (25), the concentrate solution concentration Cb at the exit is related to Cf and Cp, membrane parameters Lp, σ, and S, and operating parameters Qf, T, and △p. When these parameters are known, Cb can be calculated.


### 3.5. Model Validation

In this section, model validation needs to be carried out to verify whether the calculated values of the mathematical formulaCb is consistent with experimental data. First, we need to determine the parameter values used in the model from our experimental system. The nonlinear parameter estimation technique with Levenberg–Marquardt with Gauss–Newton algorithm [21, 22] was used to determine these parameters. The calculated value of σ is 0.9981. Lp, values were also calculated and found to remain constant across within the range of experimental parameters (Table 2).Table 2
Calculated value ofLp (Cf = 70.17 mol m−3, PH = 6, T = 306 K, and R = 8.314 J mol−1 K−1).
Qf (m3 s−1, × 10−6)△p (Pa, × 103)Cp (mol m−3)Lp (m·s−1Pa−1 × 10−12)248.88893113.91421.3923218.6111346.53.89761.3611175.5556421.53.39721.3423152.2222488.53.27481.3238123.3333505.53.24711.304875.55565303.08581.388959.72225532.96351.370948.33335782.96351.353836.1111598.52.80781.35329.7222616.52.74111.3529The concentrate solution concentrationCb from our experiment was obtained using HPLC. The theoretical value of Cb was also calculated based on equation (20). The comparison of the two Cb value is shown in Figure 7. The relative error between treatment and theoretical Cb values was calculated, and the maximum value of the relative error was found to be 5.62%.Figure 7
Comparison of the twoCb.


## 3.1. RO Experiments

To obtain the membrane performance in the process of RO concentration experiments, some operation parameters were examined. Transmembrane pressure and feed flow were found to interact with each other, with flow rate decreasing as transmembrane pressure increased (Figure2). The feed flow was changed by adjusting valve 1, and transmembrane pressure in the system was changed by adjusting valve 2 and valve 3.Figure 2
Relation between transmembrane pressure and feed solution flow (Cf = 70.17 mol m−3, PH = 6, and T = 306 K).When other experimental conditions were held constant (feed solution concentrationCf = 70.17 mol m−3, PH = 6, and feed solution temperature T = 306 K), the total flux of solvent and solute Jv, Ro, and concentrate solution concentration Cb decreased with feed solution flow (Figure 3). Ro slightly decreased with the increase of Qf (Figure 3(a)). △p also decreased with Qf, which mean that a unit volume of feed solution can be divided into less pressure to facilitate the RO process, and the macroscopic performance of this process was the decrease in Ro and Jv (Figure 2). As Qf increased, the rate of Cb reduction changed from fast to slow (Figure 3(b)). Overall, Jv decreased with Qf, which means the concentrate flow Qb increased as Qf increased, and the Qb was close to Qf; then, Cb also approached Cf (Figure 3(a)). If Qf continues to increase, two consequences may result: (1) Qf exceeds the processing capacity of the membrane, resulting in membrane damage; (2) Cb becomes approximately the same as Cf such that the RO process substantially loses the ability to separate.Figure 3
Influence of feed solution flow on the solvent flux, solute rejection, and concentrate solution concentration. (Cf = 70.17 mol m−3, PH = 6, and T = 306 K).
(a)(b)An increase inCf led to increases in Cb and decreases in Jv (Figure 4). RO did not detectably vary with Cf. In the RO process, concentration polarization (CP) is a phenomenon where the solute accumulates on the surface of the membrane causing substantial reduction in the rejection coefficient of the membrane. When Qf is constant, the increase in Cf led to severe CP, decreasing Jv. In addition, high levels of CP led to a slightly lower trend of Ro. Cb also increased with Cf, as predicted.Figure 4
Influence of feed solution concentration on the solvent flux, solute rejection, and concentrate solution concentration (Qf = 1.56 × 10−4 m3 s−1, PH = 6, and T = 306 K).
(a)(b)Overall, we found that different operating parameters have different effects onCb and are difficult to quantify (Figures 3 and 4). Therefore, this paper aims to derive a mathematical model that can synthesize various parameters to better express Cb.


## 3.2. Membrane Transport Equations

Mass transfer in RO separation processes is mainly governed by two aspects, (1) inside-membrane mass transfer and (2) outside-membrane mass transfer. In outside-membrane mass transfer, the main object of investigation is the mass transfer process near the membrane, which can be significantly hindered by CP (Figure5). The relationship of solute concentration at the membrane surface, feed solution concentration, and permeate solution concentration can be expressed by the following equation [17]:(2)Cδ,1−CpCf−Cp=eJwδc,p/D=eJw/k,where Jw is the solvent (pure water) flux, Cp is the permeate solution concentration, D is the diffusion coefficient, Cδ,1 is the solute concentration at the membrane surface (feed side, Figure 5), δcp is the CP layer thickness, and k is the mass transfer coefficient in the CP layer.Figure 5
Schematic diagram of the mass transfer inRo membrane separation process.For the RO process, the relationship between the solute fluxJs, Jw , and Jv is as follows:(3)Jv=Jw+Js≈Jw.To describe the mass transfer process of inside-membrane, the irreversible thermodynamic Spiegler–Kedem model [18, 19] is assumed to be appropriate to explain the separation performance of solute through the membrane in this work. As much, Jv can be expressed by the following equation:(4)Jv=Lp+Δp−σΔπ,where Lp is the hydraulic permeability constant, △π is the difference in the osmotic pressure across the membrane, and σ is the reflection coefficient. The reflection coefficient represents the solute separation capability of a membrane, which, for permeable membranes, is bounded by 0<puncsp> </puncsp><<puncsp> </puncsp> σ < 1. △π is calculated using the following equation:(5)Δπ=RTCδ,1−Cδ,2=RTCδ,1−Cp,where R is the gas law constant, T is the temperature, and Cδ,2 is the solute concentration at the membrane surface (permeate side, Figure 5).Substituting equation (5) in equation (4), we get(6)Jv=LpΔp−σΔπ=LpΔp−σRTCδ,1−Cp.


## 3.3. Conservation Equations

According to the principle of mass conservation, we can get the following equations:(7)QfCf=QbCb+QpCp,(8)Qf=Qb+Qp,where Qp is the permeate flow. Also, the relationship between Qp and Jv is shown in the following equation:(9)Jv=Qps,where S is the Ro membrane area effectively.Substituting equations (8) and (9) in equation (7), we can get an equation for Cb(L) at the exit:(10)CbL=QfCfQf−JvS−JvSQf−JvSCp.Substituting equation (6) in equation (10), we get(11)CbL=QfCf−CpLpΔp−σRTCδ,1−CpSQf−CpΔp−σRTCδ,1−CpS.SinceCδ,1 is difficult to determine, the key to solving Cb is to express Cδ,1−Cp.


## 3.4. Flow Equations

According to equations (7) and (8), we can get the following equations:(12)QfCf=QbxCbx+QpxCpx,(13)Qf=Qbx+Qpx.As the concentration of the permeate solution concentration in the RO process is small,Cp is assumed to be uniform in the permeate channel and can, therefore, be expressed as a constant. Thus, equation (12) can be rewritten as(14)QfCf=QbxCbx+QpxCp.According to Figure6, the total mass balance and the solute mass balance at any point along the feed channel are given as follows [20]:(15)dubxdx=−Jvhf,(16)dCbxdx=−JvubhfCbx−Cδ,1x,where ub and hf are the feed solution flow rate and feed channel thickness.Figure 6
Schematic diagram of macroseparation process in theRo membrane.According to equations (13) and (15), we get(17)dQbxdx=−dQpxdx,(18)dQbxdx=dubxdxA=−Jvhfwhf=−wJv,where A and w are the feed channel cross-sectional area and feed channel width.Also, according to equation (16), we get(19)0=dQbxdxCbx+dCbxdxQbx+dQpxdxCp.Substituting equations (16), (17), and (18) in equation (19), we get(20)2Cbx=Cδ,1x+Cp.We defineφ = Cδ,1(L)−Cp; then, equation (20) can be rewritten as follows:(21)CbL=φ+2Cp2.Combining equations (11) and (21), we can get the following expression:(22)φ+12Qf−LpΔpSLpSσRT2=2QfCf−CpLpSσRT+14Qf−LpΔpSLpSσRT2.Then, we can derive the expression ofφ:(23)φ=θ−Qf−LpΔpS2LpSσRT.We defineθ as follows:(24)θ=8QfCf−CpLpSσRT+Qf−LpΔpS2.According to equations (21) and (23), we can get the concentrate solution concentration at the exit as the following expression:(25)CbL=2QfCf−QfCp−LpΔpSCp+CpθQf−LpΔpS+θ.According to equation (25), the concentrate solution concentration Cb at the exit is related to Cf and Cp, membrane parameters Lp, σ, and S, and operating parameters Qf, T, and △p. When these parameters are known, Cb can be calculated.


## 3.5. Model Validation

In this section, model validation needs to be carried out to verify whether the calculated values of the mathematical formulaCb is consistent with experimental data. First, we need to determine the parameter values used in the model from our experimental system. The nonlinear parameter estimation technique with Levenberg–Marquardt with Gauss–Newton algorithm [21, 22] was used to determine these parameters. The calculated value of σ is 0.9981. Lp, values were also calculated and found to remain constant across within the range of experimental parameters (Table 2).Table 2
Calculated value ofLp (Cf = 70.17 mol m−3, PH = 6, T = 306 K, and R = 8.314 J mol−1 K−1).
Qf (m3 s−1, × 10−6)△p (Pa, × 103)Cp (mol m−3)Lp (m·s−1Pa−1 × 10−12)248.88893113.91421.3923218.6111346.53.89761.3611175.5556421.53.39721.3423152.2222488.53.27481.3238123.3333505.53.24711.304875.55565303.08581.388959.72225532.96351.370948.33335782.96351.353836.1111598.52.80781.35329.7222616.52.74111.3529The concentrate solution concentrationCb from our experiment was obtained using HPLC. The theoretical value of Cb was also calculated based on equation (20). The comparison of the two Cb value is shown in Figure 7. The relative error between treatment and theoretical Cb values was calculated, and the maximum value of the relative error was found to be 5.62%.Figure 7
Comparison of the twoCb.


## 4. Conclusions

Reverse osmosis (RO) concentration of glucose solution was studied to provide information that can improve the fermentation efficiency and reduce cost. To describe the concentration process, a mathematical model that accurately incorporates the effects of various parameters was developed to characterize the mass transfer process of the RO membrane. Comparison between experimental findings and model results revealed that the calculated concentrate solution concentration was consistent with the experimental data within 5.62%. This model can be used to predict the RO concentration process.



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