Permeate flux decrease due to concentration polarization in a closed roto-dynamic reverse osmosis filtration system

Desalination 402 (2017) 152–161

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Permeate flux decrease due to concentration polarization in a closed roto-dynamic reverse osmosis filtration system Abhijit Chaudhuri ⁎, Anoop Jogdand Department of Applied Mechanics, Indian Institute of Technology Madras, Chennai 600036, Tamil Nadu, India

H I G H L I G H T S • • • • •

Numerical modeling of concentration polarization in closed roto-dynamic RO system More mixing and less concentration polarization for higher rotor speed Permeate flux decreases with time but at slower rate for larger aspect ratio. Energy required for rotating the disk and flow through membrane were derived. Specific energy consumption increases when Reynolds number is very large.

a r t i c l e

i n f o

Article history: Received 4 August 2016 Received in revised form 8 October 2016 Accepted 8 October 2016 Available online xxxx

a b s t r a c t The efficiency of water purification system is largely affected by the concentration polarization. This can be reduced by increasing the mixing, mass transfer above the membrane and shear rate over the membrane surface. It is easy to increase shear rate and mixing in a roto-dynamic system than spiral wound system. A closed roto-dynamic filtration system without retentate outlet can be used when the supply of feed water is limited and reuse of water should be maximized. In a closed system, concentration inside the rotor cavity increases with time. We simulated unsteady flow and transport processes along with reverse osmosis through the membrane using ANSYS-Fluent for different values of rotor speed, feed pressure and aspect ratio. We compared the effects of these parameters on concentration polarization and permeate flux reduction. We also estimated the energy input for rotating the rotor and pressure driven flow. The specific energy consumptions for all cases were compared. The permeate flux increases with rotor speed but specific energy consumption also increases rapidly with rotor speed. So high rotor speed may not be always very effective. The present study and computational framework would be useful for optimum design of roto-dynamics cross-flow RO system. © 2016 Elsevier B.V. All rights reserved.

1. Introduction Membrane separation is a common industrial filtration process, which involves selective separation of components from a solution by passing it through a membrane. Based on the pore size, chemical characteristics of the membrane and the compound, certain components can pass through the membrane. Cross flow filtration system where the feed flow is tangential to the membrane surface is superior in performance than dead end filtration [1]. For desalination and deionization of water, the applied feed pressure should be sufficient to overcome the resistance offered by the porous membrane and osmotic pressure in case of Reverse Osmosis (RO) filtration system. The total energy requirement in cross flow reverse osmosis (CFRO) system is mainly

⁎ Corresponding author. E-mail address: [email protected] (A. Chaudhuri).

http://dx.doi.org/10.1016/j.desal.2016.10.005 0011-9164/© 2016 Elsevier B.V. All rights reserved.

governed by the generating the cross flow over the membrane and permeate flow though the membrane overcoming the osmotic pressure. The osmotic pressure increases with time due to concentration polarization on the membrane in feed flow side. Most of the researches in desalination and similar separation applications are driven by two main objectives: (i) reduction of the cost and energy per unit volume of the water filtered and (ii) reducing the environmental footprint of these processes. The spiral wound CFRO system is at the heart of the whole system of any commercial desalination plant. Johnson et.al [2] describes in detail the components of this module, its development over the years, various key aspects in its design and optimization. Various attempts on a macroscopic, microscopic scale were made to understand and describe the complex transport phenomenon such as concentration polarization (CP) in this module either theoretically [3–15] or experimentally [16–18]. In past several studies also discussed the issue of fouling and scaling on the membrane [4,19–23]. A significant learning from all these studies is that the increase of shear

A. Chaudhuri, A. Jogdand / Desalination 402 (2017) 152–161

rotating disk to impart motion of the fluid in the system. In this paper we have formulated the net energy requirement based on first law of thermodynamic for a control volume.

rate on the membrane surface, mixing and increase of mass transfer etc. help to reduce this concentration polarization layer thickness and hence the scaling and fouling [24]. Hence it is important to enhance the circulation of water in the feed flow side such that the shear rate on the membrane can reduce these adverse effects. Roto-dynamic RO system, which involves a rotating disk over a membrane in an enclosed chamber can do this more effectively than spiral wound RO system [25,26]. Jaffrin [25] discussed about more energy saving in dynamic filtration systems than in tubular systems for the ultra and micro filtration. Previous experimental and/or computational studies [27–30] on dynamic filtration systems related to ultra and nano-filtration revealed that permeate flux is controlled by angular speed of the rotating plate, height of rotor-cavity chamber, feed flow and feed pressure. Jogdand and Chaudhuri [31] developed a computational framework and demonstrated how CP layer on the membrane surface affects the permeate flux for an open roto-dynamic RO system with a feed inlet and retentate outlet. They showed that the rotation of rotor could generate high shear rate on the surface of the stationary membrane reduce the concentration polarization and enhance permeate flux. In this paper we presented our simulation results as obtained using ANSYS-Fluent V14.5 for a closed roto-dynamic RO filtration system i.e. without a retentate outlet. In a rotary system, the flow regime (laminar or turbulent) is controlled by Reynolds number (based on angular speed of the rotor) as well as aspect ratio. Daily and Nece [32] prepared a graph which distinguished different regimes for flow inside a closed and confined rotor cavity. In a closed system the feed rate at the inlet is very small and it is equal to the rate of permeate discharge. So in a closed system, the feed flow magnitude has an insignificant role on the characteristic of flow field inside the rotor-cavity. In an open rotodynamic cross flow RO filtration system, which consists of a feed inlet and retentate outlet the concentration polarization reaches a steady state [31]. Hence steady state simulation of flow and transport process along with salt retention by semipermeable membrane is enough to numerically investigate the effect of parameters on the concentration polarization and permeate flux. However for closed roto-dynamic cross flow RO system, due to continuous accumulation of salt inside the rotor-cavity, the system never reaches steady state. So it is important to study the time dependent behavior of concentration polarization and permeate flux numerically. Time dependent behavior can also be possible when membrane fouling and scaling cause reduction of membrane permeability with time. This computation framework for unsteady simulation of closed system can be used to characterize the membrane permeability reduction due to scaling and fouling if supplemented by experimental data. The cost optimization and sustainability are the driving factors for the research on desalination by reverse osmosis and membrane technology. The running costs of desalination process in a treatment plant and household device are directly proportional to the energy loss. The possible sources of energy loss are (i) viscous flow inside rotor cavity, (ii) viscous flow through semi–permeable membrane and (iii) permeate flow against the osmotic pressure. In a roto-dynamic CFRO system, the energy is supplied while feed is injected at high pressure and

2. Mathematical modeling The schematic diagram of a closed roto-dynamic RO filtration system is shown in Fig. 1. The components of the RO system such as rotor, membrane, stationary sidewall and inlet are clearly indicated in Fig. 1a. The axes for the polar coordinates are also indicated in the same figure. Since the geometry and the physical boundary conditions for flow and transport are independent of θ, the axisymmetric model of flow and transport processes were considered. The geometric variables such as radius of the membrane (R), radius of the inlet (R1) and gap between the rotor and membrane (h) are also shown in Fig. 1b. The flow regime inside the rotor cavity depends on Reynolds number and aspect ratio which are defined as Re = R2Ωρ/μ and h⁎ = h/R. Here angular velocity, density and fluid viscosity are denoted as Ω, ρ and μ. In the present study, the flow for 10000 ≤ Re ≤ 75000 and 0.08b h⁎ b 0.2 belong to II regime, which is laminar but the boundary layers are disjoint [32]. The numerical simulation of the feed flow, concentration polarization and permeate flux through the membrane for laminar flow regime requires the solution of the following coupled PDEs which are based on the conservation of mass, momentum and species transport in an axisymmetric set up. ∂ρ 1 ∂ðρrvr Þ ∂ðρvz Þ þ þ ¼ 0: ∂t r ∂r ∂z


 ∂vr ∂vr v2θ ∂vz ∂p ¼− þ vr − þ vz ρ r ∂r
∂t ∂r ∂z
  4∂ μ ∂ 2∂ ∂vz ðrvr Þ − þ μ 3 ∂r
r ∂r
3 ∂r ∂z ∂ ∂vz ∂ ∂vr þ ð2Þ þ μ μ ∂z ∂z ∂r ∂z ρ

ρ



 ∂vz ∂vz ∂vz ∂p 1 ∂ ∂vz ¼− þ μr þ vr þ vz ∂z
r ∂r ∂t ∂r ∂z ∂r
 1∂ ∂vr 2 ∂ ∂ þ ðrvr Þ − μ μ ðrvr Þ r ∂r
∂z  3r ∂z ∂r 4∂ ∂vz ð3Þ μ þ 3 ∂z ∂z


 ∂vθ ∂vθ ∂vθ vr vθ 1 ∂ ∂ vθ  ∂ ∂vθ ; ð4Þ ¼ 2 þ μr 3 μ þ vr þ vz þ r r ∂r ∂r r ∂z ∂t ∂r ∂z ∂z



 ∂ðρC Þ 1 ∂ðρrvr C Þ ∂ðρvz C Þ 1 ∂ ∂ðρC Þ ∂ ∂ðρC Þ þ þ þ ¼ rD D r r ∂r ∂t ∂r ∂z ∂r ∂z ∂z

ð5Þ

Here vr, vz and vθ are respectively radial, axial and swirl velocity components. Fluid pressure and salt concentration are denoted by p and C

Feed inlet z

Rotor Side wall

R1 h

r

Axis

Membrane

r

R Permeate outlet

Side wall (a)

ð1Þ

z

Feed flow inlet Membrane

153

(b)

Fig. 1. Schematic diagram to show a closed roto-dynamic cross flow reverse osmosis filtration system where the feed inlet is located at the center and permeate outflows through the membrane. (a) Three dimensional view and (b) axisymmetric representation of the domain.

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respectively. In the equations the notations for salt diffusivity are D. The dependence of fluid properties on the concentration makes the flow and transport as coupled processes and augments further nonlinearity into the system of equations. The following are the relevant relations as taken from [7,18] for modeling concentration dependent fluid and transport properties:

(Δp) and osmotic pressure difference (ΔΠ) between both sides of the membrane. The expression for permeate mass flux (J) is given as, J¼

ρKΔp−ΔΠ ; μ

ð13Þ

D ¼ 1:16  10−9 −3:9  10−12 C

ð6Þ

The osmotic pressure at both sides of membrane is calculated for sodium sulfate using the following relation [18]:

ρ ¼ 997:1 þ 0:909C

ð7Þ

Π ¼ 0:523C

μ ¼ 8:9  10−4 þ 3:133  10−6 C

ð8Þ

Here the units for C, and Π are g/kg, and bar respectively. Flow and transport inside the rotor-cavity was solved using ANSYSFluent. The computational grid to solve the axisymmetric flow and transport equations is shown in Fig. 2. Near the boundaries (impermeable walls and semipermeable membrane) a finer mesh is used to properly model the momentum and solute boundary layer. Since ANSYSFluent was not equipped with the implementation of Robin BC for solute transport (Eq. (12)). So we converted Robin BC into Dirichlet BC using finite difference approach. The expression of the salt concentration at the membrane surface was derived based on 2nd order backward finite difference formula. In terms of the concentration in two numerical cells adjacent to the membrane, it was obtained as,

The units of C, D, ρ, and μ in empirical relations (Eqs. (8)–(10)) are /kg, m2/s, kg/m3, and Pa·s respectively. We imposed no slip and impermeable boundary conditions for velocities on the surfaces of the rotating disk and stationary walls. Along the surface of the membrane tangential velocity was set to zero but normal flow was taken same as the permeate flux. The mathematical equations representing the flow and transport boundary conditions along the rotor, sidewall and inlet are given below, vθ ¼ rΩ; vr ¼ 0; vz ¼ 0;

∂C ¼ 0; @z ¼ h; R1 ≤r ≤R; ∀θ; ∂z

ð9Þ

∂C ¼ 0 @r ¼ R; ∀z; ∀θ: vθ ¼ 0; vr ¼ 0; vz ¼ 0; ∂r

ð10Þ

p ¼ pfeed ; C ¼ C in

ð11Þ

@z ¼ h; 0 ≤r ≤R1 ; ∀θ

Along the surface of semipermeable porous membrane, the rejection of salt ion when water permeates through the membrane is modeled as the following set of coupled flow and transport boundary conditions. ∂ðρC Þ vθ ¼ 0; vr ¼ 0; ρvz ¼ −J; JC þ D ¼ JC p ∂z

@ z ¼ 0; ∀r; ∀θ

ð12Þ

where J denotes the mass flux of permeate (kg/m2 /s) and Cp is the concentration in permeate side. It is expected to be zero but usually very small. In the present setup the positive value of Jcorresponds to water leaving out through the membrane at z = 0. For semipermeable membranes permeate flux is determined from the fluid pressure difference

CX ¼

ð14Þ

Dρð4C A −C B Þ ; 3Dρ−2JRC Δz

ð15Þ

where Cx, CA, and CB denote respectively the concentration at the center of RO zone and adjacent shells above the membrane as illustrated in Fig. 2c. For numerical simulation we took CX as the value of concentration at membrane surface (Cm). Here Δz is the grid spacing near the membrane. C

The salt rejection coefficient (RC) is given as RC ¼ 1− C mp . In this study we considered RC = 0.9. During numerical simulations of CFRO process the value of Cm, which is obtained from Eq. (15) was specified as Dirichlet BC at z = 0 in each iteration loop. The same value of Cm was also used to calculate the permeate flux using Eqs. (13) and (14). The initial guess of Cmcould be either taken as the converged value from the previous time step or feed inlet concentration. With the values of Cm and J the governing equations and boundary conditions (Eqs. (1)–(12)) were solved and then Cm was recalculated using Eq. (15). Such iterative process continued for each time step until the convergence criteria for

Fig. 2. Computational mesh for simulating flows and transport in roto-dynamic system: (a) entire domain, (b) enlarged view at the membrane interface and (c) illustration of RO zone modeling.

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all state variables were satisfied. Thus the overall solution procedure became implicit in nature. The computational framework for steady state simulation and implementation Robin BC for semipermeable membrane was validated by Jogdand and Chaudhuri [31] against the experimental study by Diaz et al. [18] of concentration polarization during cross flow filtration in a channel. Jogdand and Chaudhuri [31] also performed transient simulation for an open roto-dynamic cross flow RO system. They observed that the concentration field reaches steady state within tD/R2 b 10−5 for different values of Reynolds numbers. We noticed that the initial transient behavior could be very well captured if ΔtD/R2 = 2 × 10−9. So all the transient simulations for closed system were performed using Δ tD/R2 = 2× 10−9. 3. Results and discussion The results of closed roto-dynamic cross flow RO filtration system are presented to discuss the evolution of flow field and concentration field with time. To study the effects of angular speed of the rotor, height of the roto-dynamic chamber and feed pressure on the temporal variation of permeate flux in details we simulated several cases with varying Reynolds number (Re = ΩR2ρ/μ), aspect ratio (h*=h/R) and feed pressure ratio (pfeed⁎ =(pfeed −Πin)/Πin). We considered osmotic pressure for inlet concentration, Πin =Π(Cin) to nondimensionalize the fluid pressure. Since the salt concentration inside the rotor-cavity increases with time, the fluid properties such as density, viscosity, and permeate flux (J) vary with time. In the present study the maximum change of density and viscosity are within 1% and 3% of initial values respectively. But the change in permeate flux reduction is more than 50% in many cases. To visualize the temporal variation of flow field, the velocity profiles along z-axis at r/R = 0.5 are plotted in Fig. 3 for different values of dimensionless times, tν/R2 = 0.05 , 0.2 , and 1. However Fig. 3 does not show any significant change of radial and swirl velocity components with time. The variation of swirl, radial and axial velocity profiles with radius are shown in Fig. 4 for Reynolds number, Re = 50000 aspect ratio h/ R = 0.16, and feed pressure ratio, pfeed ⁎ = 0.91. Fig. 4 shows that the boundary layers at the rotor and stationary membrane are disjoint. This is in accordance with the flow regime as discussed by Daily and Nece [32]. In Fig. 4 it can also be seen that at any given radial distance the maximum value of radial velocity (vr) is approximately one order smaller than that of swirl velocity (vθ). But the axial velocity component (vz) is a few orders smaller than vθ. From Fig. 4b we find that the flow is radially inward near the membrane and outward near the rotor. Swirl velocity increases with radius. Radial velocity also increases with radius but near the edge it decreases. However axial velocity follow different

155

trend. It is the highest near the axis and decreases with radius. The vertical concentration profiles tν/R2 = 1 at different radial locations are shown in Fig. 4d. Sharp concentration gradients near semipermeable membrane and impermeable walls are also evident in Fig. 5b–d. For better understanding of the flow field and evolution of concentration field, the projection of the pathlines on r-z plane and concentration contours in r-z plane at different dimensionless times, t D/R2 = 2 × 10−5 , 3.6 × 10−4 and 1 × 10− 3 are shown in Fig. 5. The pathlines indicate that the injected fluid flow outward and inward ear the impermeable rotor and semipermeable membrane respectively. The increase of concentration with time is clearly observed from this figure. For obvious reason concentration is the highest at the surface of the membrane. Near the surface of rotor and sidewall, concentration is also higher than the central zone. This implies that the near membrane and impermeable boundaries the normal solute flux is dominated by diffusion since normal velocity is either negligibly small or zero. Below the inlet the concentration is equal to injection concentration up to 60% of the height. But below that the concentration is observed to be the highest. This is expected, as pathlines below the inlet appear to be closed recirculation lines like corkscrew due to the swirl velocity (see Fig. 5a). Fig. 5c and d also show that the highest concentration at this location in the later stage, tD/R2 = 3.6× 10−4 and 1 × 10−3. Even though the concentration increases with time, the concentration contour lines closely follow the pathlines. The effects of angular speed or Reynolds number on the concentration polarization are discussed using Figs. 6 and 7. The concentration distributions along r-z plane for different Reynolds number (Re = 10000 , 25000 , 50000 , and 75000) are shown in Fig. 6. These concentration contours are plotted after t D/R2 = 3.6 × 10−4 for h⁎ = 0.16 and pfeed* = 0.91. The effects of angular speed of the rotor on the mixing and concentration polarization are observed in this figure. For smaller Reynolds number radially outward spreading of salt is less. Consequently the accumulation of salt and concentration polarization below the inlet is more. This is because radial velocity increases with Reynolds number. The shear rate distribution on the membrane surface along the radius is shown in Fig. 7a for four different values of Reynolds _ number. The dimensionless shear rate (γ=Ω) increases with radius but has a peak at r ≈ 0.95R.. At r = R solid sidewall causes the pathlines to bend upward (see Fig. 5a) and as a consequence shear rate approaches to zero. Since rotation of the rotor generates the shear rate on membrane, we nondimensionalized shear rate with respect to angular speed, Ω. This dimensionless form is useful to know if the shear rate increases linearly with Reynolds number, which is proportional to Ω. Fig. _ 7a shows that the dimension shear rate, γ=Ω, is not same for all Re but it increases with Re. This implies that resultant shear rate on the

Fig. 3. To show that the hydrodynamics steady state reaches very early, (a) swirl and (b) radial velocity profiles at r/R=0.5 are plotted for three different dimensionless time instances.

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Fig. 4. For the closed roto-dynamic system the steady state profiles of dimensionless (a) swirl velocity, (b) radial velocity, (c) axial velocity and (d) concentration are shown for different radial locations.

membrane surface due to rotation of the rotor increases more than a linear function of angular speed. The radial distribution of concentration on the membrane surface and permeate flux after tD/R2 = 3.6 × 10−4 are shown in Fig. 7b and c respectively for same set of Reynolds numbers. To eliminate the effect of membrane constant (K) and feed pressure (pfeed) we choose to nondimensionalize the permeate flux as J/ (Kpfeed). The increase of permeate flux with radius and Reynolds number validate the fact that shear rate and mass flux on the membrane enhance the permeate flux for cross flow RO filtration system. The effect of Reynolds number on the time variation of averaged permeate flux (averaged over the membrane area) is shown in Fig. 7d. The average permeate flux (Javg) decreases rapidly with time in the early stage of

Fig. 5. For the closed roto-dynamic system and Re = 50000, h⁎ = 0.16, pfeed* = 0.91 the steady state velocity field and unsteady concentrations fields are shown: (a) pathlines as projected on r-z plane and (b,c,d) concentration distribution in r-z plane at different time instances.

filtration in a closed roto-dynamic system. But at the later stage it decreases slowly. The effect of Re on Javg is prominent in the intermediate stage but at the later stage Javg become very close for all Reynolds numbers. To study the effect of height of the rotor cavity chamber and aspect ratio (h⁎ = h/R) on the concentration polarization and permeate flux we performed simulations for four different values of h⁎. Fig. 8 shows the concentration field inside the rotor cavity at an intermediate time t D/R2 = 3.6 × 10−4 for h⁎ = 0.08 , 0.12 , 0.16 , and 0.20. Since the volume of rotor cavity chamber increases with height, the concentration near the wall is low for larger value of h⁎. The effect of h⁎on the shear rate is shown in Fig. 9a. The influence of h⁎ on the shear rate is not significant. Near r = 0.95R the shear rate increases slightly with the

Fig. 6. The effects of angular speed of the rotor on concentration polarization and mixing are shown by plotting the concentration field distribution in r-z plane at t D/R2 = 3.6× 10−4 for (a) Re=10000, (b) Re=25000, (c) Re=50000, and (d) Re=75000.

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157

Fig. 7. The effects of angular speed of the rotor on radial distribution of (a) resultant shear rate, (b) concentration along the membrane interface at t D/R2 =3.6×10−4 (c) permeate flux at t D/R2 =3.6×10−4 and (d) temporal variation of average permeate flux.

decrease of h⁎. Since the boundary layers are separated for all of these values of h⁎ and Reynolds numbers [32], the effect of height on the shear rate is small. Fig. 9b and c show the effect of h⁎on the radial distribution of concentration on membrane and dimensionless permeate flux after t D/R2 = 3.6 × 10−4. The shapes of curves in Fig. 9c are very similar and there is approximately 25% reduction in total permeate discharge rate when aspect ratio is reduced to 0.08 from 0.16. Thus higher concentration polarization and lower permeate flux for smaller aspect ratio implies that slight increase of shear rate is insufficient to remove salt from membrane surface and to increase the permeate flux. Fig. 9d shows the effect of aspect ratio on the long term temporal variations of average permeate flux. For smaller value of h⁎, Javg decreases faster with time.

Hence the advantage of larger volume of rotor cavity chamber is observed at the later stage of cross flow filtration. Figs. 10 and 11 show the effect of feed pressure (pfeed) on the concentration polarization and permeate flux when other dimensionless parameters are fixed (Re = 50000 and h⁎ = 0.16). Since feed pressure is the driving force for permeate to flow though the porous membrane against the viscous and osmotic pressure force, the increase of permeate flux with pfeed is obvious. However it is not known how pfeedcan affect concentration polarization, which in turn controls the nature of decay of permeate flux with time. Fig. 10 shows that significant increase in the concentration inside the rotor cavity for lager value of feed pressure. A minor difference in the pattern of the concentration contour lines near the inlet for higher feed pressure is observed in Fig. 10. Based on the discussion on the similarity between concentration contour line and pathlines (see Fig. 5), we can claim that the flow field near the inlet is slightly modified for higher value of feed pressure. This is possibly because of very high inlet pressure. However Fig. 11a shows that shear rate is almost independent of the feed pressure. In Fig. 11b it is seen that the concentration on the membrane is quite high for higher feed pressure. Interestingly the value of concentration at r = 0 is proportional to the value of pfeed⁎. The radial variations of dimensionless permeate flux for different pfeed is shown in Fig. 11c. The radial distributions of permeate flux for pfeed⁎ = 2.83 and 4.75 are close but are from that for pfeed⁎ = 0.91. Fig. 11d shows the temporal variation of average permeate flux with feed pressure. It is observed that average permeate flux for pfeed⁎ = 2.83 and 4.75 are almost same when normalized with respect to Kpfeed. 4. Energy balance equation for a control volume

Fig. 8. The effect of aspect ratio of the rotor-cavity on concentration polarization and mixing are shown by plotting the concentration field distribution in r-z plane at t D/ R2 =3.6×10−4 for (a) h⁎ =0.08, (b) h⁎ =0.12, (c) h⁎ =0.16, and (d) h⁎ =0.2.

The power requirement for filtration by the closed roto-dynamic system can be derived based on first law of thermodynamics [33]. The complete roto-dynamic filtration system including the porous membrane is considered as a “Control Volume” (CV). In absence of any heat

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Fig. 9. The effects of aspect ratio on radial distribution of (a) resultant shear rate, (b) concentration along the membrane interface at t D/R2 =3.6×10−4 (c) permeate flux at t D/R2 =3.6× 10−4 and (d) temporal variation of average permeate flux.

source, the energy balance equation for a control volume is given as,

Based on the above assumption we simplify Eq. (16) and rearrange the terms as follows:

ð16Þ ð17Þ where e = U + V2/2 + gz and U is internal energy per unit mass. Along the “Control Surface” (CS) the normal unit vector and the vector ^ and ! representing the surface stress are denoted by n τ respectively. The increase of heat energy inside the roto-cavity due to viscous dissipation and the work done against viscous force and osmotic force while flowing through the porous membrane are considered as energy loss. It is because the transformation of mechanical energy into the heat energy during these processes is irreversible. Here we assume that the heat energy flux at the permeate outlet is not useful. The gravitational potential energy can also be ignored due to small size of the system. The results in previous section have shown steady state velocity field.

_ In Eq. (17) the LHS corresponds to the net mechanical power input (W). The rate of energy loss due to flow against viscous and osmotic forces is bal_ The expression of Wcan _ anced by the power input W. be derived as, ! ! ! Z R ρV 2in ρV 2out pout þ vz;in r dr− vz;out r dr 2 2 R1 0 0 ! ! Z R α 1 ρQ 3in α 2 ρQ 3out 2 − pout Q out þ ¼ 2π τ zθ Ωr dr þ pin Q in þ 2A2in 2A2out R1

_ ¼ 2π W

Z

R

Z

τ zθ Ωr 2 dr þ

R1

pin þ

ð18Þ For a closed roto-dynamic filtration system the inlet and outlet discharge rates are same i.e. Q =Qin = Qout. Since Q is very small, the kinetic energy terms are dropped from Eq. (18). When the fluid in permeate outlet is under atmospheric condition and feed pressure is taken as the gauge pressure, Eq. (18) can be simply written as, _ ¼ 2π W

Z

R R1

τzθ Ωr 2 dr þ pfeed Q

ð19Þ

_ R and W _P Let us denote first and second terms in RHS of Eq. (19) as W

_ R, represents the rate of work done by the respectively. The first terms, W _ P , corresponds to the power required rotating disk. The second term, W for pressure driven flow. Based on the physical understanding of the en-

Fig. 10. The effect of feed pressure on concentration are shown by plotting the concentration field distribution in r-z plane at t D/R2 =3.6×10−4 for (a) pfeed⁎ =0.91, (b) pfeed⁎ =2.83 and (c) pfeed⁎ =4.75.

_ R is the source ergy loss and mechanical work done, one can think that W of the energy for creating cross flow over the membrane and it compensates the energy loss due to viscous dissipation inside the roto-cavity. _ P , balances the rate of energy The power for pressure driven flow, W

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159

Fig. 11. The effects of feed pressure on radial distribution of (a) resultant shear rate, (b) concentration along the membrane interface at t D/R2 =3.6×10−4 (c) permeate flux at t D/R2 = 3.6×10−4 and (d) temporal variation of average permeate flux.

loss due to permeate flow through porous membrane against the osmotic pressure and viscous force. Since total permeate discharge rate _ P is not constant in time. However W _ R can (Q) changes with time, W

by numerical time integration of the permeate discharge (Q), inputs _ R and W _ P ) over the duration of Q. For comparison purpose, powers (W we have normalized the total volume of permeate as,

be considered as constant since it depends on the velocity field that _ R , tangential shear reaches steady state very quickly. To determineW stress on the rotor (τzθ) is obtained from the numerical solution of flow field. Like the shear rate on the membrane, the tangential shear stress on the rotor (τ zθ ¼ μ γ_ zθ ) also depend on the Reynolds number and aspect ratio. Fig. 12 shows the radial distribution of γ_ zθ for different values of Re and h⁎. Similar to total shear rate on the membrane, the dimensionless tangential shear rate (γ_ zθ =Ω) also increases with Reynolds number. But the effect of aspect ratio on γ_ zθ =Ω is insignificant. So rotor _ R , is insensitive to the aspect ratio. The Table 1 presents the power, W total volume of permeate ( Vp ) and total energy supplied (WRand WP) for the duration of [0, td], where td = 0.001R2/D. These are calculated

ð20Þ

The denominator in Eq. (20) corresponds to the volume of water that passes through the entire porous membrane for a duration of [0, td] in absence of osmotic pressure. Thus the effect of feed pressure on the dimensionless volume of permeate is eliminated. The values of dimensionless permeate volume for different cases are given in Table 1. The permeate volume increases with Reynolds number. However for larger range of Reynolds number, permeate volume does not increase much. Table 1 shows that the permeate volume increases linearly

Fig. 12. The variations of tangential shear rate on the surface of rotor are shown for different values of (a) Reynolds numbers and (b) aspect ratios.

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Table 1 Dimensionless output results for different values of dimensionless input parameters. Values of dimensionless input parameters Reynolds number, Re

Aspect ratio, h⁎

Feed pressure ratio, pfeeed⁎

Vp* 1000 10,000 25,000 50,000 75,000 0.08 0.12 0.16 0.20 0.91 2.83 4.75

0.083 0.119 0.140 0.154 0.161 0.128 0.143 0.154 0.175 0.154 0.221 0.245

with aspect ratio. The permeate flow also increases linearly with the feed pressure excess to the inlet osmotic pressure. The energy consumption due to pressure driven flow and rotating the disk are compared for different values of parameters. The pumping energy calculated based on first law of thermodynamics (see Eq. (19)) is proportional to permeate volume. The rotor energy increases with Reynolds number. It increases more rapidly than Ω2because normalized shear rate on the surface of rotor (γ_ zθ =Ω) increases with angular velocity. However the rotor energy changes marginally with the aspect ratio and feed pressure. The total energy is normalized as, ð21Þ The denominator in Eq. (21) corresponds to the energy for flow through porous membrane when feed pressure is equal to pfeed. The values of normalized rotor energy, WR⁎, are listed in Table 1 for different sets of parameters. The specific energy consumption (SEC) is an important measure used in designing a desalination system. It is defined as, ð22Þ From the definition of pumping power in Eq. (19), it is clear that specific energy consumption due to pressure driven flow is equal to the feed pressure. The normalized SEC with respect to feed pressure becomes, ð23Þ

W⁎

SEC*

0.083 0.120 0.148 0.191 0.253 0.162 0.179 0.191 0.210 0.191 0.229 0.248

1.00 1.001 1.054 1.238 1.568 1.263 1.245 1.238 1.199 1.238 1.037 1.010

fluid properties and hence on the flow field. The simulations for the duration of 0 ≤t D/R2 ≤ 1× 10−3, showed that the concentration below the inlet was always the highest even in the early stage of filtration. In rest of the domain it increases gradually. Except the zone very near the boundaries, the concentration contour lines were very similar to the pathlines. This inferred that the solute transport away from the membrane and impermeable boundaries is dominated by advection. Results for different Reynolds number showed that for higher angular speed of the rotor, the mixing was more and concentration polarization was less. Hence the permeate flux increases with angular speed. In the initial stage the permeate discharge rate is almost same for all aspect ratios but it decreases slowly for higher aspect ratio. We noted that the permeate flux increases with feed pressure. A system level energy analysis was carried out. It showed that for small Reynolds number the energy required to rotate the disk is very small compared to the energy supplied for permeate flow through the porous membrane. The rotor-energy increased very rapidly with angular speed but the permeate flux did not increase so much. We found that specific energy consumption were increasing when larger angular speed and feed pressure were used to enhance permeate volume. So the flow at large Reynolds number may not be always effective to optimize the permeate flux. However there may be other advantage of rotating the rotor at higher angular speed. Membrane fouling and scaling should be less for higher angular speed since shear rate is more and concentration polarization is less. This should be experimentally investigated. The present computational framework and results for closed rotodynamic CFRO, can be used to characterize the membrane fouling and scaling from any experimental studies. Abbreviations

But the specific energy consumption due to the rotation of the rotor increases with Reynolds number but decreases with aspect ratio and feed pressure. But it decreases with aspect ratio because of larger volume of roto-cavity system is available for better mixing. The dimensionless specific energy consumption decreases with feed pressure because rotating energy is almost constant for different values of feed pressure.

CP RO CFRO UDF CV CS SEC

Concentration polarization Reverse Osmosis Cross Flow Reverse Osmosis User Defined Function Control volume Control surface Specific energy consumption

5. Summary and conclusions Notations The continuous growth of concentration polarization and reduction of permeate flux was modeled for a closed roto-dynamic cross flow RO system, without a retentate outlet. The numerical simulations were performed using ANSYS-Fluent and several UDFs were written to accurately model all flow and transport boundary conditions at the semipermeable membrane interface. The unsteady flow and transport simulation results showed that the flow reached steady state. This implies that the effect of concentration increase with time is insignificant on the concentration dependent

R h R1 r θ z

Radius (m) Gap between the stator and rotor (m) Radius of feed inlet (m) Radial Co-ordinate Azimuthal Co-ordinate Axial Co-ordinate

A. Chaudhuri, A. Jogdand / Desalination 402 (2017) 152–161

Ω ρ μ ν Re h⁎ vθ vr vz p C t D Cin pfeed J Π K CX CA CB RC Cm Cp pfeed⁎ Javg γ_ γ_ zθ ! n ! v V U ! τ τzθ Q _ W _R W _P W Vp V⁎p W⁎ SEC SEC*

Angular speed of the rotor (rad/s) Density of fluid (kg/m3) Dynamic viscosity of fluid (Pa·s) Kinematic viscosity of fluid (Pa·s) Reynolds number Aspect ratio Swirl velocity (m/s) Radial velocity (m/s) Axial velocity (m/s) Fluid pressure (Bar) Salt concentration (g/kg of solution) Time (s) Salt diffusivity (m2/s) Feed inlet salt concentration (g/kg of solution) Fluid pressure at feed inlet (Bar) Mass flux of permeate (kg/m2 −s) Osmotic pressure (Bar) Permeability of the porous membrane, (m2/m) Salt concentration at membrane, (g/kg of solution) Salt concentration at a cell center in Zone A, (g/kg of solution) Salt concentration at a cell center in Zone B, (g/kg of solution) Rejection coefficient Salt concentration at the membrane surface (g/kg of solution) Salt concentration in the permeate side (g/kg of solution) Dimensionless feed pressure Average permeate flux (kg/m2 −s) Shear rate on the surface of membrane (s−1) Tangential shear rate on the surface of rotor (s−1) Unit vector Velocity vector (m/s) Resultant velocity (m/s) Internal energy per unit mass (joule/kg) Shear stress vector on the surface of rotor (MPa) Tangential shear stress on the surface of rotor (MPa) Feed flow (m3/s) Mechanical power (Watt) Power required for rotating the disk (Watt) Power required for pressure driven flow (Watt) Permeate volume (m3) Dimensionless permeate volume Dimensionless work Specific energy consumption (joule/m3) Dimensionless specific energy consumption

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