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Hydrodynamics and mass transfer simulation of wastewater treatment in membrane reactors
Desalination 286 (2012) 290–295
Contents lists available at SciVerse ScienceDirect
Desalination journal homepage: www.elsevier.com/locate/desal
Hydrodynamics and mass transfer simulation of wastewater treatment in membrane reactors Saeed Shirazian, Mashallah Rezakazemi, Azam Marjani ⁎, Sadegh Moradi Islamic Azad University, Arak Branch, Department of Chemistry, Arak, Iran
a r t i c l e
i n f o
Article history: Received 8 September 2011 Received in revised form 16 November 2011 Accepted 17 November 2011 Available online 15 December 2011 Keywords: Mass transfer Membrane reactor CFD Wastewater treatment Simulation
a b s t r a c t This work presents CFD simulation of hollow-fiber membrane reactors (HFMRs) for separation of ammonia from aqueous solutions. Simulations are carried out to predict ammonia transport in a membrane reactor. Mass transfer and hydrodynamics of the system were investigated through solving conservation equations. The conservation equations solved here were the Navier–Stokes and continuity equations. Finite element analysis was applied for numerical solution of the system equations. UMFPACK was chosen as linear solver and showed good convergence. The influence of effective parameters on the mass transfer and hydrodynamics of the ammonia removal was investigated. Simulation results showed that total flux of ammonia decreases sharply in the region near the membrane inlet. Hydrodynamic investigations also revealed that velocity reached fully developed after a short distance from the reactor entrance. The results of this work confirmed that feed and solvent velocities are the most important parameters in the removal of ammonia. It is also indicated that the numerical procedure is useful and capable to predict the performance of hollow-fiber membrane reactors for separation of ammonia from aqueous solutions. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Nowadays, releasing industrial wastewater containing dissolved gases is a major environmental problem. The problem could be more serious if the dissolved gases are toxic. One of the toxic dissolved gases which are present in wastewater is ammonia (NH3). For example, wastewater of fertilizer factories contains dissolved ammonia in aqueous solutions. The concentration of dissolved ammonia in their wastewater varies between 500 and 2000 ppm. Although the ammonia is present in low concentrations, it can cause serious environmental and health problems [1–3]. NH3 in solutions is produced from industries such as catalyst factories, coking, chemical fertilizer, coal gasification, petroleum refining, and pharmaceutical. Dissolved ammonia in water and wastewater has adverse effects on human life and fishes. Ammonia (NH3) is highly toxic and limits fish production in intensive systems [4,5]. When dissolved ammonia enters the aquatic system, equilibrium is established between NH3 and ammonium ion (NH4+). Of the two forms of ammonia, NH3 is far more toxic to fish, and its formation is favored by high pH and water temperature. When pH of system exceeds ~ 8.5, any NH3 present can be dangerous. In general, a normally functioning aquatic system should contain no measurable NH3
⁎ Corresponding author. Tel.: + 98 861 3663041; fax: + 98 861 3663055. E-mail address: [email protected] (A. Marjani). 0011-9164/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.desal.2011.11.039
because as soon as it enters the system, it should be removed by bacteria [4–6]. Therefore, separation of ammonia from water and wastewater is very important to prevent the adverse effects of ammonia on human life and environment. There are some separation processes which are currently used for ammonia removal. Separation of dissolved ammonia using conventional processes has some disadvantages. The main disadvantage of separation processes for ammonia removal is that these processes consume high energy. Gas striping is one of the conventional processes which are suitable for ammonia separation. Other disadvantages of these processes are: high operational costs, difficult design and scale up, flooding and foaming. Designing and developing effective separation processes for ammonia separation from solutions have been a great subject of researches. Recently, hollow-fiber membrane reactors (HFMRs) have attracted many attentions as an alternative for separation of ammonia. Both separation and reaction can be carried out in HFMRs. The main advantage of HFMRs upon conventional separation processes is that these reactors can provide a dispersion free contact between two phases. In addition, the flow rates of phases in reactor can vary independently, while neither flooding nor unloading problems may occur. Therefore, HF membrane reactors can be considered as a new technology for separation of ammonia from wastewater [7–10]. Simulation of ammonia separation from water in HF membrane reactors is based on solving conservation equations for ammonia in the feed and membrane phases. Conservation equations including momentum and mass equations are obtained and solved numerically by numerical methods to determine concentration and velocity distribution in the
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occurs in the shell side of membrane reactor. Fig. 2 shows the model domain for numerical simulation [11–13]. 3.1. Equations of lumen side
Fig. 1. Mechanism of ammonia separation in membrane reactors.
membrane reactor. The Navier–Stokes equations are considered as the momentum equation. The computational fluid dynamics (CFD) are usually employed to solve the differential equations. The main purpose of this work is to simulate ammonia separation from water in an HF membrane reactor. The governing equations are solved by numerical method using CFD techniques. Mass transfer and the Navier–Stokes equations are solved simultaneously for the ammonia in the membrane reactor to obtain the concentration distribution. 2. Theory Hollow-fiber membrane reactors (HFMRs) bring two phases including feed and solvent in contact for the purpose of reaction and separation. For wastewater treatment, feed solution containing dissolved ammonia is circulated in the tube (lumen) side of the reactor. Meanwhile, solvent solution which is mostly sulfuric acid flows in the shell side of the membrane reactor. The operational mode can be cross flow or parallel. In the membrane reactor, ammonia is transferred from the bulk of feed solution to the shell side and reacts with the sulfuric acid. The chemical reaction between ammonia and sulfuric acid is instantaneous so that ammonia concentration in the shell side of the membrane reactor is assumed to be zero. Since the membrane is hydrophobic, non-wetted mode is considered for numerical simulation. It means that the aqueous phase cannot penetrate the membrane pores. The basic principle of ammonia separation in membrane reactors is shown in Fig. 1 [1–6]. 3. Equations of model In this study, separation of ammonia from water in membrane reactors is simulated. Two sets of equations are derived and solved numerically to predict distribution of concentration and velocity in the membrane reactor. The velocity distribution is obtained in the lumen side of the reactor while the concentration distribution is obtained in the lumen as well as the membrane side. Navier–Stokes equations are applied for determining velocity in the lumen side. Chemical reaction between ammonia and sulfuric acid solution
Concentration distribution of ammonia in the lumen side is obtained by solving continuity equation for ammonia in this subdomain. The continuity equation is derived considering steady state and also axial and radial diffusions. Fick's law of diffusion is applied for estimation of diffusive flux. The continuity equation may be written as [14]:
DNH3 −lumen
" 2 ∂ C NH3 −lumen
¼ V z−lumen
∂r 2 ∂C NH3 −lumen
2
þ
1 ∂C NH3 −lumen ∂ C NH3 −lumen þ r ∂r ∂z2
#
ð1Þ
∂z
The term Vz − lumen denotes velocity in the lumen side of the membrane reactor. Velocity distribution in the lumen side is calculated by solving the Navier–Stokes equations. Therefore, the Navier–Stokes and continuity equations should be coupled and solved simultaneously to determine concentration distribution of ammonia in the lumen side of the reactor. The Navier–Stokes equations may be written as follows [14]: −∇⋅η ∇V z−lumen þ ð∇V z−lumen ÞT þ ρðV z−lumen :∇ÞV z−lumen þ ∇p ¼ F ∇:V z−lumen ¼ 0 ð2Þ where η,V, and ρ denote fluid dynamic viscosity (kg/m s), velocity vector (m/s), and density (kg/m 3), respectively; p is pressure (Pa) and F is a body force term (N). Boundary conditions for the lumen side: @z ¼ 0;
C NH3 −lumen ¼ C inlet
V z−lumen ¼ V 0
ð3Þ
@z ¼ L; Convective flux; p ¼ patm
ð4Þ
It should be pointed out that the boundary condition for the convective flux assumes that the mass flux passing through the boundary is convection-dominated. In other words, it assumes that the mass flux due to diffusion across this boundary is zero. @r ¼ 0; @r ¼ r t ;
∂C NH3 −lumen ∂r
¼0
ð5Þ
Axial symmetry
C NH3 −lumen ¼ pNH3 −membrane =H;
ð6Þ
No slip condition
where rt, H, and pNH3 − membraneare inner radius of fiber, Henry's law constant, and partial pressure of ammonia in the gas phase of membrane pores in equilibrium with the concentration of ammonia in the aqueous solution, i.e. CNH3 − lumen. Table 1 Parameters used in simulation.
Fig. 2. Model domain for numerical simulation [8].
Parameter
Symbol
Value
Unit
Fiber inner radius Fiber outer radius Fiber porosity Fiber length Number of fibers Temperature Inlet ammonia concentration
rin rout ε L n T Co
110 150 40 25 7000 293 10
μm μm % cm – K mol/m3
292
S. Shirazian et al. / Desalination 286 (2012) 290–295
Fig. 3. Contours of ammonia concentration in the membrane reactor.
3.2. Equations of membrane The continuity equation for the transport of ammonia inside the membrane pores, which is considered to be due to diffusion, may be written as [14]: DNH3 −membrane
" 2 ∂ C NH3 −membrane ∂r
2
þ
# 2 1 ∂C NH3 −membrane ∂ C NH3 −membrane þ ¼0 r ∂r ∂z2
ð7Þ Boundary conditions for the membrane: @r ¼ r t ;
pNH3 −membrane ¼ C NH3 −lumen H ðHenry lawÞ
ð8Þ
@r ¼ r s ;
C NH3 −lumen ¼ 0 ðInstantaneous chemical reactionÞ
ð9Þ
The reaction rate between ammonia and acid sulfuric solution is postulated to be instantaneous; therefore the ammonia concentration in the shell side of the membrane reactor was neglected. It is also
Fig. 4. Total flux distribution in the feed side.
assumed that there is no mass transfer at both edges of the hollow fibers, i.e.: @z ¼ 0&L; Insulation
3.3. Numerical solution of the equations The main objective of this work is to simulate a membrane reactor for removal of ammonia using the CFD method. The equations in the membrane reactor with the boundary conditions were solved
S. Shirazian et al. / Desalination 286 (2012) 290–295
293
Fig. 5. Radial distribution of ammonia mass transfer flux in the lumen side.
numerically using COMSOL Multiphysics 3.5 software. COMSOL uses finite element method (FEM) for numerical solution of the equations. The finite element method is combined with adaptive meshing and error control using numerical solver of UMFPACK version 4.2. The applicability, robustness and accuracy of this numerical method for the membranes have been proved by some researches [15–18]. There is a large difference between the dimensions of membrane reactor which makes the convergence of solving equations quite difficult. A scaling factor was applied for the membrane reactor in the z direction. Application of scaling factor makes the diffusivities anisotropic. It also reduces the solution time and accuracy. An IBM-PC-Pentium 4 (CPU speed is 2800 MHz) was used to solve the sets of equations. Parameters used for numerical simulations are listed in Table 1. 4. Results and discussion 4.1. Concentration distribution of ammonia in the reactor Fig. 3 shows contours of ammonia concentration in the membrane reactor. The contours are used to show the ammonia concentration changes quantitatively and qualitatively and obtained in the lumen side of the membrane reactor where the feed solution flows. The maximum of ammonia concentration can be observed at the inlet of lumen side near the feed–membrane interface. It is also indicated that concentration changes is sharp at the regions near the membrane wall due to high contribution of diffusional mass transfer in the radial direction. Fig. 6. Velocity distribution in the lumen side of the membrane reactor.
4.2. Total flux distribution in the lumen side Variations of the mass transfer flux of ammonia in the lumen side of the membrane reactor are indicated in Fig. 4. The mass transfer fluxes are illustrated by arrows in Fig. 4. The mass transfer flux is the summation of diffusional and convection flux of ammonia in the feed side. It is clearly shown that in the regions near the axis of the feed i.e., r = 0 the total flux is higher than other regions. This is due to high convection mass transfer in these regions. In fact, velocity which causes convection mass transfer is more significant in the z direction and also at the regions near the fiber center. 4.3. Radial distribution of total flux in the feed side Distribution of mass transfer flux of ammonia in radial direction is shown in Fig. 5. As it can be seen from Fig. 5, total mass transfer flux of ammonia in the feed side decreases as the radial distance increases.
The total mass transfer flux is due to convection and diffusion. Since velocity in radial direction is negligible, the contribution of convective mass transfer flux is weak in the radial direction. Furthermore, due to viscous force effects, the velocity of fluid decreases in the radial direction which in turn decreases convective mass transfer flux in the r direction. Fig. 5 also shows total flux distribution at different axial positions. 4.4. Velocity field in the reactor The velocity field in the shell side of HFMC is shown in Fig. 6. Fig. 7 also shows the velocity profile in the shell side, in which the liquid solvent flows. The velocity profile in the shell side of the HFMC was simulated by solving the Navier–Stokes equations. The velocity profile is almost parabolic with a mean velocity which increases with the membrane length because of continuous fluid permeation.
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S. Shirazian et al. / Desalination 286 (2012) 290–295
Fig. 7. Axial velocity profile in the lumen side of the membrane reactor.
Fig. 8. Profile of pressure in the lumen side of the membrane reactor.
Figs. 6 and 7 also reveal that at the inlet regions in the shell side, the velocity is not developed. After a distance from the inlet, the velocity profile becomes fully developed (see Fig. 7). Fig. 7 shows the velocity profile of solvent phase at the center of the shell side (center line velocity). As observed, the model considers the inlet effects on the hydrodynamics of fluid flow in the shell side.
T u V Vz-tube z
4.5. Pressure distribution
Greek symbols ε membrane porosity η dynamic viscosity (kg/m s) ρ density (kg/m 3)
Pressure distribution in the lumen side of the membrane reactor is shown in Fig. 8. As it can be seen from the figure, the pressure drop along the feed side of membrane reactor is low. This is one of the advantages of membrane reactors for ammonia removal. The pressure drop along the membrane reactors is not appreciable. This advantage can reduce the operating costs of the process. 5. Conclusions A CFD simulation was conducted to investigate the separation of ammonia from aqueous solutions using membrane reactors. The type of membrane reactor considered here was hollow-fiber membrane reactor (HFMR). Conservation equations were solved for ammonia in the feed and membrane sides. Distributions of concentration, mass transfer flux, velocity, and pressure were obtained using the model. Simulation results showed that decreasing feed velocity is favorable in ammonia removal in membrane reactors. Furthermore, simulation results revealed that the pressure drop is not significant in the membrane reactor. It was also shown that feed and solvent velocities are the most important parameters in the removal of ammonia. Nomenclature C concentration, mol/m 3 Coutlet outlet concentration of ammonia in the lumen side, mol/m 3 Cintlet inlet concentration of ammonia in the lumen side, mol/m 3 D diffusion coefficient, m 2/s H Henry's law constant, mol/m 3 kPa Ji diffusive flux of species i, mol/m 2 s L length of the fiber, m n number of fibers p pressure, Pa r radial coordinate, m rt inner radius of fibers, m rs outer radius of fibers, m t time, s
temperature, K average velocity, m/s velocity in the module, m/s z-velocity in the tube, m/s axial coordinate, m
Abbreviations FEM finite element method HF hollow fiber HFMR hollow-fiber membrane reactor 2D two dimensional CFD computational fluid dynamics
References [1] A. Mandowara, P.K. Bhattacharya, Simulation studies of ammonia removal from water in a membrane contactor under liquid–liquid extraction mode, J. Environ. Manage. 92 (2011) 121–130. [2] A. Hasanoglu, J. Romero, B. Pérez, A. Plaza, Ammonia removal from wastewater streams through membrane contactors: experimental and theoretical analysis of operation parameters and configuration, Chem. Eng. J. 160 (2010) 530–537. [3] T.C. Jorgensen, L.R. Weatherley, Ammonia removal from wastewater by ion exchange in the presence of organic contaminants, Water Res. 37 (2003) 1723–1728. [4] M.J. Semmens, D.M. Foster, E.L. Cussler, Ammonia removal from water using microporous hollow fibers, J. Membr. Sci. 51 (1990) 127–140. [5] B. Norddahl, V.G. Horn, M. Larsson, J.H. du Preez, K. Christensen, A membrane contactor for ammonia stripping, pilot scale experience and modeling, Desalination 199 (2006) 172–174. [6] M. Sarioglu, Removal of ammonium from municipal wastewater using natural Turkish (Dogantepe) zeolite, Sep. Purif. Technol. 41 (2005) 1–11. [7] A. Gabelman, S.-T. Hwang, Hollow fiber membrane contactors, J. Membr. Sci. 159 (1999) 61–106. [8] J.J. Guo, C.D. Ho, Theoretical study on membrane extraction of Cu2 + with D2EHPA in laminar flow circular tube modules, Desalination 233 (2008) 247–257. [9] H. Ye, D.B. Das, J.T. Triffitt, Z. Cui, Modelling nutrient transport in hollow fibre membrane bioreactors for growing three-dimensional bone tissue, J. Membr. Sci. 272 (2006) 169–178. [10] A. Bottino, G. Capannelli, A. Comite, R. Di Felice, R. Firpo, CO2 removal from a gas stream by membrane contactor, Sep. Purif. Technol. 59 (2008) 85–90. [11] I.S. Chang, C.M. Chung, Pollution prevention for manufacturing of ammonium chloride an experimental study of wastewater recycling, Desalination 127 (2000) 145–153.
S. Shirazian et al. / Desalination 286 (2012) 290–295 [12] H. Mahmud, A. Kumar, R.M. Narbaitz, T. Matsuura, A study of mass transfer in the membrane air-stripping process using microporous polypropylene hollow fibers, J. Membr. Sci. 179 (2000) 29–41. [13] C.I. McDermott, S.A. Tarafder, C. Schüth, Vacuum assisted removal of volatile to semi volatile organic contaminants from water using hollow fibre membrane contactors: II: a hybrid numerical-analytical modelling approach, J. Membr. Sci. 292 (1–2) (2007) 17–28. [14] R.B. Bird, W.E. Stewart, E.N. Lightfoot, Transport Phenomena, 2nd ed. John Wiley & Sons, New York, 1960. [15] S. Shirazian, A. Moghadassi, S. Moradi, Numerical simulation of mass transfer in gas–liquid hollow fiber membrane contactors for laminar flow conditions, Simul. Modell. Pract. Theory 17 (2009) 708–718.
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[16] M.R. Sohrabi, A. Marjani, S. Moradi, M. Davallo, S. Shirazian, Mathematical modeling and numerical simulation of CO2 transport through hollow-fiber membranes, Appl. Math. Modell. 35 (2011) 174–188. [17] S. Shirazian, A. Marjani, F. Fadaei, Supercritical extraction of organic solutes from aqueous solutions by means of membrane contactors: CFD simulation, Desalination 277 (2011) 135–140. [18] N.S. Abdullah, D.B. Das, Modelling nutrient transport in hollow fibre membrane bioreactor for growing bone tissue with consideration of multi-component interactions, Chem. Eng. Sci. 62 (2007) 5821–5839.
Contents lists available at SciVerse ScienceDirect
Desalination journal homepage: www.elsevier.com/locate/desal
Hydrodynamics and mass transfer simulation of wastewater treatment in membrane reactors Saeed Shirazian, Mashallah Rezakazemi, Azam Marjani ⁎, Sadegh Moradi Islamic Azad University, Arak Branch, Department of Chemistry, Arak, Iran
a r t i c l e
i n f o
Article history: Received 8 September 2011 Received in revised form 16 November 2011 Accepted 17 November 2011 Available online 15 December 2011 Keywords: Mass transfer Membrane reactor CFD Wastewater treatment Simulation
a b s t r a c t This work presents CFD simulation of hollow-fiber membrane reactors (HFMRs) for separation of ammonia from aqueous solutions. Simulations are carried out to predict ammonia transport in a membrane reactor. Mass transfer and hydrodynamics of the system were investigated through solving conservation equations. The conservation equations solved here were the Navier–Stokes and continuity equations. Finite element analysis was applied for numerical solution of the system equations. UMFPACK was chosen as linear solver and showed good convergence. The influence of effective parameters on the mass transfer and hydrodynamics of the ammonia removal was investigated. Simulation results showed that total flux of ammonia decreases sharply in the region near the membrane inlet. Hydrodynamic investigations also revealed that velocity reached fully developed after a short distance from the reactor entrance. The results of this work confirmed that feed and solvent velocities are the most important parameters in the removal of ammonia. It is also indicated that the numerical procedure is useful and capable to predict the performance of hollow-fiber membrane reactors for separation of ammonia from aqueous solutions. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Nowadays, releasing industrial wastewater containing dissolved gases is a major environmental problem. The problem could be more serious if the dissolved gases are toxic. One of the toxic dissolved gases which are present in wastewater is ammonia (NH3). For example, wastewater of fertilizer factories contains dissolved ammonia in aqueous solutions. The concentration of dissolved ammonia in their wastewater varies between 500 and 2000 ppm. Although the ammonia is present in low concentrations, it can cause serious environmental and health problems [1–3]. NH3 in solutions is produced from industries such as catalyst factories, coking, chemical fertilizer, coal gasification, petroleum refining, and pharmaceutical. Dissolved ammonia in water and wastewater has adverse effects on human life and fishes. Ammonia (NH3) is highly toxic and limits fish production in intensive systems [4,5]. When dissolved ammonia enters the aquatic system, equilibrium is established between NH3 and ammonium ion (NH4+). Of the two forms of ammonia, NH3 is far more toxic to fish, and its formation is favored by high pH and water temperature. When pH of system exceeds ~ 8.5, any NH3 present can be dangerous. In general, a normally functioning aquatic system should contain no measurable NH3
⁎ Corresponding author. Tel.: + 98 861 3663041; fax: + 98 861 3663055. E-mail address: [email protected] (A. Marjani). 0011-9164/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.desal.2011.11.039
because as soon as it enters the system, it should be removed by bacteria [4–6]. Therefore, separation of ammonia from water and wastewater is very important to prevent the adverse effects of ammonia on human life and environment. There are some separation processes which are currently used for ammonia removal. Separation of dissolved ammonia using conventional processes has some disadvantages. The main disadvantage of separation processes for ammonia removal is that these processes consume high energy. Gas striping is one of the conventional processes which are suitable for ammonia separation. Other disadvantages of these processes are: high operational costs, difficult design and scale up, flooding and foaming. Designing and developing effective separation processes for ammonia separation from solutions have been a great subject of researches. Recently, hollow-fiber membrane reactors (HFMRs) have attracted many attentions as an alternative for separation of ammonia. Both separation and reaction can be carried out in HFMRs. The main advantage of HFMRs upon conventional separation processes is that these reactors can provide a dispersion free contact between two phases. In addition, the flow rates of phases in reactor can vary independently, while neither flooding nor unloading problems may occur. Therefore, HF membrane reactors can be considered as a new technology for separation of ammonia from wastewater [7–10]. Simulation of ammonia separation from water in HF membrane reactors is based on solving conservation equations for ammonia in the feed and membrane phases. Conservation equations including momentum and mass equations are obtained and solved numerically by numerical methods to determine concentration and velocity distribution in the
S. Shirazian et al. / Desalination 286 (2012) 290–295
291
occurs in the shell side of membrane reactor. Fig. 2 shows the model domain for numerical simulation [11–13]. 3.1. Equations of lumen side
Fig. 1. Mechanism of ammonia separation in membrane reactors.
membrane reactor. The Navier–Stokes equations are considered as the momentum equation. The computational fluid dynamics (CFD) are usually employed to solve the differential equations. The main purpose of this work is to simulate ammonia separation from water in an HF membrane reactor. The governing equations are solved by numerical method using CFD techniques. Mass transfer and the Navier–Stokes equations are solved simultaneously for the ammonia in the membrane reactor to obtain the concentration distribution. 2. Theory Hollow-fiber membrane reactors (HFMRs) bring two phases including feed and solvent in contact for the purpose of reaction and separation. For wastewater treatment, feed solution containing dissolved ammonia is circulated in the tube (lumen) side of the reactor. Meanwhile, solvent solution which is mostly sulfuric acid flows in the shell side of the membrane reactor. The operational mode can be cross flow or parallel. In the membrane reactor, ammonia is transferred from the bulk of feed solution to the shell side and reacts with the sulfuric acid. The chemical reaction between ammonia and sulfuric acid is instantaneous so that ammonia concentration in the shell side of the membrane reactor is assumed to be zero. Since the membrane is hydrophobic, non-wetted mode is considered for numerical simulation. It means that the aqueous phase cannot penetrate the membrane pores. The basic principle of ammonia separation in membrane reactors is shown in Fig. 1 [1–6]. 3. Equations of model In this study, separation of ammonia from water in membrane reactors is simulated. Two sets of equations are derived and solved numerically to predict distribution of concentration and velocity in the membrane reactor. The velocity distribution is obtained in the lumen side of the reactor while the concentration distribution is obtained in the lumen as well as the membrane side. Navier–Stokes equations are applied for determining velocity in the lumen side. Chemical reaction between ammonia and sulfuric acid solution
Concentration distribution of ammonia in the lumen side is obtained by solving continuity equation for ammonia in this subdomain. The continuity equation is derived considering steady state and also axial and radial diffusions. Fick's law of diffusion is applied for estimation of diffusive flux. The continuity equation may be written as [14]:
DNH3 −lumen
" 2 ∂ C NH3 −lumen
¼ V z−lumen
∂r 2 ∂C NH3 −lumen
2
þ
1 ∂C NH3 −lumen ∂ C NH3 −lumen þ r ∂r ∂z2
#
ð1Þ
∂z
The term Vz − lumen denotes velocity in the lumen side of the membrane reactor. Velocity distribution in the lumen side is calculated by solving the Navier–Stokes equations. Therefore, the Navier–Stokes and continuity equations should be coupled and solved simultaneously to determine concentration distribution of ammonia in the lumen side of the reactor. The Navier–Stokes equations may be written as follows [14]: −∇⋅η ∇V z−lumen þ ð∇V z−lumen ÞT þ ρðV z−lumen :∇ÞV z−lumen þ ∇p ¼ F ∇:V z−lumen ¼ 0 ð2Þ where η,V, and ρ denote fluid dynamic viscosity (kg/m s), velocity vector (m/s), and density (kg/m 3), respectively; p is pressure (Pa) and F is a body force term (N). Boundary conditions for the lumen side: @z ¼ 0;
C NH3 −lumen ¼ C inlet
V z−lumen ¼ V 0
ð3Þ
@z ¼ L; Convective flux; p ¼ patm
ð4Þ
It should be pointed out that the boundary condition for the convective flux assumes that the mass flux passing through the boundary is convection-dominated. In other words, it assumes that the mass flux due to diffusion across this boundary is zero. @r ¼ 0; @r ¼ r t ;
∂C NH3 −lumen ∂r
¼0
ð5Þ
Axial symmetry
C NH3 −lumen ¼ pNH3 −membrane =H;
ð6Þ
No slip condition
where rt, H, and pNH3 − membraneare inner radius of fiber, Henry's law constant, and partial pressure of ammonia in the gas phase of membrane pores in equilibrium with the concentration of ammonia in the aqueous solution, i.e. CNH3 − lumen. Table 1 Parameters used in simulation.
Fig. 2. Model domain for numerical simulation [8].
Parameter
Symbol
Value
Unit
Fiber inner radius Fiber outer radius Fiber porosity Fiber length Number of fibers Temperature Inlet ammonia concentration
rin rout ε L n T Co
110 150 40 25 7000 293 10
μm μm % cm – K mol/m3
292
S. Shirazian et al. / Desalination 286 (2012) 290–295
Fig. 3. Contours of ammonia concentration in the membrane reactor.
3.2. Equations of membrane The continuity equation for the transport of ammonia inside the membrane pores, which is considered to be due to diffusion, may be written as [14]: DNH3 −membrane
" 2 ∂ C NH3 −membrane ∂r
2
þ
# 2 1 ∂C NH3 −membrane ∂ C NH3 −membrane þ ¼0 r ∂r ∂z2
ð7Þ Boundary conditions for the membrane: @r ¼ r t ;
pNH3 −membrane ¼ C NH3 −lumen H ðHenry lawÞ
ð8Þ
@r ¼ r s ;
C NH3 −lumen ¼ 0 ðInstantaneous chemical reactionÞ
ð9Þ
The reaction rate between ammonia and acid sulfuric solution is postulated to be instantaneous; therefore the ammonia concentration in the shell side of the membrane reactor was neglected. It is also
Fig. 4. Total flux distribution in the feed side.
assumed that there is no mass transfer at both edges of the hollow fibers, i.e.: @z ¼ 0&L; Insulation
3.3. Numerical solution of the equations The main objective of this work is to simulate a membrane reactor for removal of ammonia using the CFD method. The equations in the membrane reactor with the boundary conditions were solved
S. Shirazian et al. / Desalination 286 (2012) 290–295
293
Fig. 5. Radial distribution of ammonia mass transfer flux in the lumen side.
numerically using COMSOL Multiphysics 3.5 software. COMSOL uses finite element method (FEM) for numerical solution of the equations. The finite element method is combined with adaptive meshing and error control using numerical solver of UMFPACK version 4.2. The applicability, robustness and accuracy of this numerical method for the membranes have been proved by some researches [15–18]. There is a large difference between the dimensions of membrane reactor which makes the convergence of solving equations quite difficult. A scaling factor was applied for the membrane reactor in the z direction. Application of scaling factor makes the diffusivities anisotropic. It also reduces the solution time and accuracy. An IBM-PC-Pentium 4 (CPU speed is 2800 MHz) was used to solve the sets of equations. Parameters used for numerical simulations are listed in Table 1. 4. Results and discussion 4.1. Concentration distribution of ammonia in the reactor Fig. 3 shows contours of ammonia concentration in the membrane reactor. The contours are used to show the ammonia concentration changes quantitatively and qualitatively and obtained in the lumen side of the membrane reactor where the feed solution flows. The maximum of ammonia concentration can be observed at the inlet of lumen side near the feed–membrane interface. It is also indicated that concentration changes is sharp at the regions near the membrane wall due to high contribution of diffusional mass transfer in the radial direction. Fig. 6. Velocity distribution in the lumen side of the membrane reactor.
4.2. Total flux distribution in the lumen side Variations of the mass transfer flux of ammonia in the lumen side of the membrane reactor are indicated in Fig. 4. The mass transfer fluxes are illustrated by arrows in Fig. 4. The mass transfer flux is the summation of diffusional and convection flux of ammonia in the feed side. It is clearly shown that in the regions near the axis of the feed i.e., r = 0 the total flux is higher than other regions. This is due to high convection mass transfer in these regions. In fact, velocity which causes convection mass transfer is more significant in the z direction and also at the regions near the fiber center. 4.3. Radial distribution of total flux in the feed side Distribution of mass transfer flux of ammonia in radial direction is shown in Fig. 5. As it can be seen from Fig. 5, total mass transfer flux of ammonia in the feed side decreases as the radial distance increases.
The total mass transfer flux is due to convection and diffusion. Since velocity in radial direction is negligible, the contribution of convective mass transfer flux is weak in the radial direction. Furthermore, due to viscous force effects, the velocity of fluid decreases in the radial direction which in turn decreases convective mass transfer flux in the r direction. Fig. 5 also shows total flux distribution at different axial positions. 4.4. Velocity field in the reactor The velocity field in the shell side of HFMC is shown in Fig. 6. Fig. 7 also shows the velocity profile in the shell side, in which the liquid solvent flows. The velocity profile in the shell side of the HFMC was simulated by solving the Navier–Stokes equations. The velocity profile is almost parabolic with a mean velocity which increases with the membrane length because of continuous fluid permeation.
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Fig. 7. Axial velocity profile in the lumen side of the membrane reactor.
Fig. 8. Profile of pressure in the lumen side of the membrane reactor.
Figs. 6 and 7 also reveal that at the inlet regions in the shell side, the velocity is not developed. After a distance from the inlet, the velocity profile becomes fully developed (see Fig. 7). Fig. 7 shows the velocity profile of solvent phase at the center of the shell side (center line velocity). As observed, the model considers the inlet effects on the hydrodynamics of fluid flow in the shell side.
T u V Vz-tube z
4.5. Pressure distribution
Greek symbols ε membrane porosity η dynamic viscosity (kg/m s) ρ density (kg/m 3)
Pressure distribution in the lumen side of the membrane reactor is shown in Fig. 8. As it can be seen from the figure, the pressure drop along the feed side of membrane reactor is low. This is one of the advantages of membrane reactors for ammonia removal. The pressure drop along the membrane reactors is not appreciable. This advantage can reduce the operating costs of the process. 5. Conclusions A CFD simulation was conducted to investigate the separation of ammonia from aqueous solutions using membrane reactors. The type of membrane reactor considered here was hollow-fiber membrane reactor (HFMR). Conservation equations were solved for ammonia in the feed and membrane sides. Distributions of concentration, mass transfer flux, velocity, and pressure were obtained using the model. Simulation results showed that decreasing feed velocity is favorable in ammonia removal in membrane reactors. Furthermore, simulation results revealed that the pressure drop is not significant in the membrane reactor. It was also shown that feed and solvent velocities are the most important parameters in the removal of ammonia. Nomenclature C concentration, mol/m 3 Coutlet outlet concentration of ammonia in the lumen side, mol/m 3 Cintlet inlet concentration of ammonia in the lumen side, mol/m 3 D diffusion coefficient, m 2/s H Henry's law constant, mol/m 3 kPa Ji diffusive flux of species i, mol/m 2 s L length of the fiber, m n number of fibers p pressure, Pa r radial coordinate, m rt inner radius of fibers, m rs outer radius of fibers, m t time, s
temperature, K average velocity, m/s velocity in the module, m/s z-velocity in the tube, m/s axial coordinate, m
Abbreviations FEM finite element method HF hollow fiber HFMR hollow-fiber membrane reactor 2D two dimensional CFD computational fluid dynamics
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