Electro-driven membrane separation technologies have been widely used in various industrial fields such as seawater desalination, wastewater treatment, and resource utilization of salt-lakes brine and seawater due to its efficient ion separation performance[1-5]. Among the electro-driven membrane separation technologies, the electrochemically switched ion permselective (ESIP)[6-11]technology has attracted extensive attention due to its ability to achieve continuous and highly selective separation and recovery of target ions from dilute solutions. The ESIP combines electrochemically switched ion exchange (ESIX)[12-13]and electrodialysis (ED) techniques to achieve continuous and highly selective separation of target ions by applying a double pulse potential over the permselective membrane electrode under effect of an external electric field. The ESIP technology achieves a breakthrough from intermittent operation of ESIX technology to continuous operation, which has realized the transformation from in-situ adsorption/desorption to continuous ion separation and recovery. In recent years, ESIP membrane separation technology has made progress in the fields of energy and environment, such as water softening (Ca2+[10], Mg2+[10]), alkali metal separation (K+[9]), heavy metal ion separation (Pb2+[14], Cu2+[8], Zn2+[5]) and anion separation (F-[11]). However, there is still a lack of a complete theoretical system of ion mass transfer to guide the design and optimization of membrane materials and modules in ESIP system.
There are several main categories of mass transfer theories describing electrically driven membrane systems: Maxwell-Stefan equation[16], Nernst-Planck equation[17], Kedem-Katchalsky equation, Artificial Neural network models, Fuzzy Logic and Semi-empirical models[18]. Among them, the Nernst-Planck model is generally used to describe the ion transfer process because of its simplicity and accuracy. In the Nernst-Planck model, the ion transfer within the electric field is attributed to three main aspects: concentration difference diffusion, electromigration and convective mass transfer. In the ESIP system, the ion migration in the solution conforms to the Nernst-Planck theory. However, the oscillation permeation effect on the target ions is caused by adjusting the redox state of the ESIP membrane material. Thus, the ion transfer in the ESIP membrane is disobedient to the Nernst-Planck theory. GAO et al. found that the membrane thickness and ionic active site concentration are the key factors affecting the mass transfer of the target ions[8]. Therefore, modifying the Nernst-Planck model by introducing the effect of ionic active site concentration and membrane thickness on the ion transport is the key to establish the ion mass transfer model for ESIP.
In this work, a copper nitrate electrolyte was chosen as the simulated solution and the ESIP system was divided into four regions: bulk solution phase, diffusion layer, Donnan interface layer and membrane phase. Based on the modified Nernst-Planck model, Donnan equilibrium and electroneutrality assumptions, the effects of current density, membrane thickness and ionic active site concentration on the distribution of ion concentration and resistance in different regions of the system were investigated. The main factors affecting the separation performance of ESIP membrane system were obtained and optimized by simulating the ion concentration and resistance distribution in the system.
1 Simulation and experiment
1.1 ESIP system with multiple pairs of membranes
As shown in Fig.1, the ESIP system with multiple pairs of membranes can be divided into three parts: the ESIP membrane stack, the circuit control system, and the fluid delivery system. A series of electroactive cation and anion permselective membranes are installed in parallel between a pair of cathode and anode to form an ESIP membrane stack. Spacers are introduced between the individual electroactive membrane sheets both to transport fluid. The external circuit system consists of a DC regulated power supply for providing a stable electric field on both sides of the membrane stack and a pulsed power supply for modulating redox state of membrane material. In this paper, the chamber with increasing salt concentration is referred to as the receiving and the chamber with decreasing salt concentration is called the source.
Fig.1 Configuration diagram of membrane
separation system of ESIP
1.2 Model construction
Based on the modular symmetry principle, half of the source, half of the receiving and half of one electroactive cation permselective membrane were selected as representative calculation units. As shown in Fig.2, the model was divided into seven regions: Ⅰ/Ⅶ is the bulk solution of the source and the receiving; Ⅱ/Ⅵ is the diffusion layer regions of the feed solution and the receiving solution; Ⅲ/Ⅴ is the Donnan interface layer on both sides of the membrane material; Ⅳ is the membrane phase. Copper nitrate solution was selected as the simulated salt solution. The calculation procedure was steady state, neglecting the effects of water transport, concentration polarization, electric convection and fluid convection flow on the ion mass transfer. And it can be assumed that each region meets the conditions of electric neutrality and thermodynamic equilibrium, and the concentration of the bulk solution does not change.
Fig.2 Model mechanism diagram of ESIP
In this work, a modified Nernst-Planck model was established to describe the ion mass transfer process in ESIPM by introducing the ionic active site concentration and membrane thickness. The modified model is as follows:
WhereJiis the flux of ioni;
ziandcidenote the diffusion coefficient, valence, and concentration of ioniin the membrane, respectively;φ,R,T, andFare the potential, gas constant, absolute temperature, and Faraday’s constant; andσ,W, andLare the Electron amount of gained by or lost from the ionic active sites, ionic active site concentration, and membrane thickness, respectively. In the equation, the first term represents the concentration diffusion and electromigration, and the second term is the pulse-driven term[19].
The Nernst-Planck theory can be used to express the mass transfer of ions in the process of adsorption, diffusion and desorption:
(2)
In this paper, the effect of convective parallel to the membrane direction was neglected. The mass transfer direction of ions is mainly horizontal direction, the formula (2) can be converted into:
(3)
The mass conservation equation (continuity equation) for the whole system is:
(4)
Since the system operates in the steady state, the formula (4) is evolved as:
(5)
To solve equation (3), the boundary conditions are set to:
1) Within each region, the assumption of charge neutrality is satisfied:
(6)
In the diffusion layer
2) The anion and cation fluxes are related to current density by Faraday’s law.
(7)
Wherejis the current density.
3) From the formula (5), it is known that the ion flux is not a function of the positionx. Thus, the ion flux remains unchanged in the horizontal direction. In addition, the ion flux can be calculated based on the current density and the ion transference number:
(8)
1.2.1The ion mass transfer model in the diffusion layer
Combining formula (3) and formulas (4)-(8), the nitrate ion concentration distribution in the diffusion layer is
(9)
(10)
In the above equation, the ion concentration is assumed to be linearly distributed. In that formula (10),
is the copper ion concentration in the bulk solution.
Then the copper ion concentration distribution is:
(11)
(12)
The diffusion layer resistance can be calculated from the ion equivalent conductance and the ion concentration:
(13)
Λis the equivalent conductance, which can be calculated by the Deybe-Hückel-Onsager formula:
(14)
Λ0is the equivalent conductance of dilute solution, m2·S/mol, andAandBare constants.
The potential distribution of the diffusion layer can be expressed by the following equation:
(15)
1.2.2The ion mass transfer model in the Donnan interface
In the Donnan interface layer, the ion concentration can be calculated by Donnan equilibrium.
(16)
Where
is the concentration of ions in the membrane immediately at the electric double layer interface.
Equation (16) combined with equation (6) can be expressed as:
(17)
(18)
The ion concentration of membrane phase at the right membrane-electric double layer interface is solved by the model in 1.2.3 below. The ion concentration of receiving solution on the right electric double layer is similar to the algorithm on the left side above.
The Donnan potential can be calculated by the following equation:
(19)
whereais the ionic activity, the subscript s and m represents the solution phase and the membrane phase. The ionic activity coefficientγican be calculated by the Debye-Hückel equation as follows:
(20)
In the above equation,λis a constant, 0.509 (kg/mol)1/2, andIis the ionic strength, mol/kg, which can be calculated by the following equation:
(21)
wherebiis the mass molar concentration of ioni.
The resistance of the Donnan interface layer can be calculated by Ohm's law as follows:
(22)
1.2.3The ion mass transfer model in the membrane phase
A modified Nernst-Planck equation was used to represent the ion transfer process in the membrane phase by introducing ionic active site concentration:
(23)
(24)
Combining the above equations (6), (23) and (24), it follows that
(25)
The active site concentration in the cation permselective membrane is much greater than the cation concentration in the membrane (regardless of the ion concentration distribution in the membrane). The active site concentration is constant in the horizontal direction of the membrane for simplicity of calculation. Therefore, according to the formula (25), the ion concentration in the membrane is as follows:
(26)
(27)
The membrane potential can be calculated by Ohm's law as follows:
φ=jRm.
(28)
1.2.4The parameters of model
The parameters of model is in table 1.
Table 1 The parameters of model
ParametersValueSolution propertiesΛ1Equivalent conductance of Cu2+,m2·S/mol107.2×10-4Λ2Equivalent conductance of NO-3,m2·S/mol71.42×10-4D1Diffusion coefficient of Cu2+ in the compartment,cm2/s1.428×10-9D2Diffusion coefficient of NO-3 in the compartment,cm2/s1.902×10-9δThickness of diffusion layer,μm200Membrane characteristicsD1Diffusion coefficient of Cu2+ in the membrane phase,cm2/s1.428×10-10D2Diffusion coefficient of NO-3 in the membrane phase,cm2/s1.902×10-10RmResistance of membrane,Ω/cm29×10-8σElectron amount of gained by or lost from the ionic active sites1Other parametersAConstant,-60.02BConstant,-0.229λConstant,(kg/mol)1/20.509
1.3 Experimental
The self-made BPEI-CQD/PPy/PSS membrane with an effective area of 4 cm×4 cm was assembled in the ESIP membrane module. 200 mL of copper nitrate solution with the concentration of 15.625 mol/m3and 200 mL of nitric acid solution (pH=3) were poured into two tanks as the source and receiving, respectively. Then peristaltic pump was turned on to circulate continuous liquid. The coupled circuit system described in Section 1 was applied to the ESIP membrane system.
During the operation of the system with various membrane thicknesses and different active site concentrations, aliquots of 1 mL were sampled from the source or receiving chambers to detect the concentration of Cu2+by AAS-990. The flux of Cu2+across the composite membrane was calculated. The active site concentration was defined with reference to the calculation of the concentration of the immobilized functional group of the ion exchange membrane[20]:
WhereCIEandDSare the target ion exchange capacity and the swelling degree of the membrane material, respectively.
2 Results and discussion
2.1 Effect of current density on the ion mass transfer
The diffusion layer thickness was determined to 200 μm, and the diffusion coefficient in the membrane was assumed to be one order of magnitude lower than that in the solution. When the electrolyte mass concentration is 2 937.5 g/m3(Cu2+mass concentration: 1 000 g/m3) , i.e., 15.625 mol/m3, the active site mass concentration is 10 000 mg/L, i.e., 156 mol/m3, and the membrane thickness is 7 μm, the distribution of Cu2+concentration and resistance in different regions at various current densities is shown in Fig.3. Fig.3(a) shows the concentration distribution of Cu2+in different regions. The concentration gradient of Cu2+decreases with the increase of current density in the diffusion layer of the feed solution (0≤x≤200 μm), while the concentration gradient of Cu2+in the diffusion layer of the receiving solution is basically unchanged with the increase of current density. It is attributed to the driving force provided by the membrane in the redox state is much greater than the effect of current density on the ion mass transfer. In the membrane phase, the ion concentration in the right electric double layer varied less at different current densities, and the concentration gradient of Cu2+in the membrane is significantly larger than that in the dilute solution diffusion layer. This further indicates that the effect of membrane is greater than the effect of current density on the ion mass transfer in ESIP systems. In addition, the effects of ESIP membrane on the ion mass transfer are discussed in detail in Section 2.2 and 2.3.
Fig.3 Distribution of (a) Cu2+concentration and (b) resistance
under different current densities
Fig.3(b) reflects the variation of resistance in different regions at different current densities. With the increase of current density, the resistance of the left and right Donnan layers decreases, the resistances of the diffusion layer of the feed solution increase, while the resistances of membrane phase and the diffusion layers of the receiving solution is fluctuate slightly unchanged, and the total resistance decreases. At low current density, the Donnan layer resistance is the main resistance. The contribution of diffusion layer resistance to the total resistance increases with the increase of current density. For example, when the current density is 5 A/m2, the Donnan layer resistance accounts for 90.43% of the total resistance, while when the current density is 60 A/m2, the diffusion layer resistance accounts for 81.02% of the total resistance. Therefore, it is important to understand the resistance conditions of each region under different current densities and determine the main contribution resistance to improve the ion separation performance of ESIP. For instance, at low current density, Donnan layer resistance can be reduced by adjusting the membrane thickness and the active site concentration. At high current density, diffusion layer resistance can be reduced by increasing the fluid flow rate and decreasing the chamber thickness.
2.2 Effect of membrane thickness on the ion mass transfer
Since the ion transference number in the membrane is influenced by the membrane thickness, it is another important factor affecting ion mass transfer in ESIP system. The empirical correlation between ion transference number and the membrane thickness was obtained by experimental determination as follows:
When the Cu2+concentration is 1 000 g/m3, i.e., 15.625 mol/m3and the active site concentration is 10 000 mg/L, i.e., 156 mol/m3, the distribution of Cu2+concentration and resistance in different regions at various membrane thickness are shown in Fig.4. The variation of Cu2+concentration distribution in different regions under different membrane thickness are shown in Fig.4(a). The concentration gradient of Cu2+increases with the increase of membrane thickness in the diffusion layer on the source, while the concentration gradient of Cu2+in the diffusion layer of the receiving solution is fluctuate slightly. It is attributed to the effect of ion concentration at the membrane interface. Fig.4(a) shows that increasing the film thickness results in an increase in the ion concentration gradient in the membrane. Fig.4(b) shows that the resistance changes in each region at different membrane thicknesses. It can be found that the diffusion layer resistance on the source dominates the overall resistance at thin membrane. With the increase of membrane thickness, the diffusion layer resistance on the source decreases continuously and the left Donnan layer resistance gradually dominates the total resistance. Therefore, it is beneficial to understand the resistance distribution of each region at different membrane thickness conditions to optimize the process parameters to improve the ion separation efficiency of ESIP. For example, in the case of a thin membrane, the mass transfer rate of ion can be improved by increasing the flow rate to enhance the turbulence condition of the fluid in the source.
Fig.4 Distribution of (a) Cu2+concentration and (b) resistance
under different membrane thicknesses
2.3 Effect of ion active site concentration on the ion mass transfer
The driving force of ion migration in ESIP system is provided by the redox (gain or loss of electrons) of the ionic active site. Therefore, the active site concentration in membrane is a crucial factor affecting ion mass transfer. The empirical correlation between the ion transference number and the active site concentration was obtained by experimental determination as follows:
When the Cu2+concentration is 1 000 g/m3, i.e., 15.625 mol/m3, and the membrane thickness is 7 μm, the distribution of Cu2+concentration and resistance in different regions at various active site concentration is shown in Fig.5. Fig.5(a) shows the concentration distribution of Cu2+in different regions at various active site concentrations. In the diffusion layer of the feed solution, the concentration gradient of Cu2+decreases with the increase of active site concentration. In the membrane phase, the concentration gradient of Cu2+increases significantly with the increase of active site concentration. Fig.5(b) shows the concentration distribution of Cu2+in the diffusion layer of the receiving solution and the results show that the ion concentration gradient increases as the ionic active site concentration increases. The variation of resistance in different regions at various active site concentration is shown in Fig.5(c). With the increase of the active site concentration, the resistance of the diffusion layer of the feed solution gradually increases and occupies a dominant position, and the resistance of the left Donnan layer increases to a certain extent. Therefore, it is necessary to increase the flow rate of the feed solution to reduce the resistance in the diffusion layer under high active site concentration.
Fig.5 Distribution of (a) Cu2+concentration, (b) Cu2+concentration of right diffusion zone and
(c) resistance under different ionic active site concentration
3 Conclusions
In this work, a modified Nernst-Planck equa-tion was proposed to study the ion transport behavior in membrane phase based on the ion transport characteristics in ESIP membrane separation system. The distribution of the ion concentration and resistance was simulated under different current densities, membrane thicknesses and active site concentration in the membrane. The main conclusions from the calculations were as follows:
1) In the diffusion layer of feed solution, the membrane thickness and the ionic active site concentration have a great influence on the ion concentration distribution. In the membrane phase, the order of the factors that affects the ion concentration gradient is ionic active site concentration>membrane thickness>current density. In the diffusion layer of the receiving solution, the ion concentration distribution is less influenced by the three factors. Therefore, reducing the membrane thickness and increasing the ionic active site concentration are the main ways to improve the ion mass transfer in ESIP membrane separation system.
2) The resistance distribution in each region is influenced by current density, membrane thickness, and ionic active site concentration. The Donnan resistance plays a dominant role in the overall resistance at low current density, while diffusion layer resistance is dominant at high current density. The diffusion layer resistance plays a dominant role in the overall resistance at thin membrane, while Donnan layer resistance is dominant at thick membrane. The diffusion layer resistance of feed solution dominates at different ionic active site concentration, and the dominanting role of diffusion layer resistance increases with the increas of ionic active site concentration. Therefore, under the influence of high current density, thin membrane, and high ionic active site concentration, the separation performance in ESIP system can be optimized by reducing resistance in Diffusion layer (increasing the flow rate or decreasing the chamber thickness).
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