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Numerical simulation of the flow in a rotating disk filtration module
Desalination 235 (2009) 122–138
Numerical simulation of the flow in a rotating disk filtration module C. Torrasa, J. Pallaresb*, R. Garcia-Vallsa, M.Y. Jaffrinc Structured Systems Engineering for Energy, Materials and Chemistry, Chemical Engineering Department, Experiments, Computation and Modelization in Fluid Mechanics and Turbulence Group, Mechanical Engineering Department, Universitat Rovira i Virgili. Av. Països Catalans, 26, 43007 Tarragona, Catalunya, Spain Tel. +34 (977) 55-96-82; Fax +34 (977) 55-96-91; email: [email protected] c Biomechanics and Biomedical Engineering laboratory, UMR CNRS 6600, Technological University of Compiegne, Compiegne, France a
b
Received 18 April 2007; accepted revised 3 February 2008
Abstract The characterization of the flow inside an experimental flat membrane module with a smooth rotating disk was performed. The module consists of a disk rotating at speeds up to 3000 rpm inside a cylindrical housing equipped with a stationary circular flat membrane. The characterization was carried out by using a finite volume CFD software with the κ-omega turbulence model and results of the range of rotation speeds 300 ≤ Ω ≤ 20000 rpm were compared with experimental and theoretical data reported in previous studies. The simulations suggest high permeate fluxes for the device due to large average shear stresses on the membrane and the absence of stagnant zones inside the module, which are desirable features to avoid membrane fouling processes. The simulations show an overall good agreement with theoretical results based on the main assumption that the wall shear stress on the membrane and on the disk can be predicted using modified correlations for rotating flow over a stationary wall and for flow induced by a rotating disk, respectively and with experimental pressure measurements. It has been found that the flow rate imposed at the inlet of the module has an important effect on the pressure distribution. At the membrane some discrepancies were found between the results obtained with the simulations and with the theoretical approach because of the limitations of the assumptions, especially at low rotating speeds for which the effect of the flow through the module becomes important. The correlations relating the disk rotation rate with the surface averaged pressure and the shear stress on the membrane were determined. Keywords: Ultrafiltration; Microfiltration; Fouling; Membrane module; Rotating disk; Shear stress; Computational fluid dynamics
*Corresponding author. 0011-9164/09/$– See front matter © 2008 Elsevier B.V. All rights reserved doi:10.1016/j.desal.2008.02.006
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1. Introduction In filtration experiments (mainly micro- and ultrafiltration) carried out with porous membranes, not only the separation capability of the membrane is important but also its permeability. Large efforts are invested in order to maximize the membrane effective area but also to prevent flux decrement during experiment because of usual phenomena such as the fouling or the concentration polarization. The last one is caused by the imbalance in transport between the membrane surface and the core [1,2]. Fouling phenomena is caused by the solids that are in the feed solution, which block the pores or create a cake over the membrane, and therefore, reduce its effective area [2]. To overcome this problem, two strategies are available: prevention and cure. Usually, both are needed. In terms of a cure, backflushing is an often used process to remove the solids that are blocking the pores [3]. This process implies that normal operation is stopped periodically or alternatively, the need of coupled membrane systems, and thus, an increment of the cost of the process. Therefore, prevention is an important issue. One prevention method consists of a proper design of the membrane module and thus, the flow over the membrane [4]. Rotating disk filtration modules have been shown to yield higher permeate fluxes and better solute transmission than conventional cross flow filtration due to the high shear rates they generate at the membrane which prevent or limit cake formation in microfiltration [5–7]. In addition these shear rates are obtained at relatively low inlet flow rates and low pressure drops inside the module, resulting in a relatively uniform transmembrane pressure (TMP). The combination of high shear rates and low TMP facilitates macromolecule transmission through the membrane. Conversely, when used in ultrafiltration (UF) and nanofiltration (NF), these systems reduce concentration polarisation on the membrane, and their permeate fluxes continue to increase with pressure to higher levels than in classical tubular systems [8]. In addition since rejected solute con-
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centration at the membrane is reduced by the high shear rate, their diffusive transmission through the membrane is smaller and their apparent rejection is higher, which is very advantageous in the treatment of waste waters by NF [9]. For a correct analysis of the performance of these systems, it is therefore important to evaluate accurately the membrane shear rate and pressure distribution at the membrane. Since disk rotation speeds are generally high, these systems operate in the turbulent regime, for which no analytic solution is available. Thus it is logical to use a Computational Fluid Dynamics (CFD) approach for this purpose. CFD has been demonstrated to be a powerful tool to predict the flow inside housings and to resolve properly boundary layers and flow distribution effects as several works show [10–13]. This paper, which is the extension of a published abstract [14], illustrates the detailed description of the flow inside the module from CFD analysis considering the main variables and some results are contrasted with experimental ones. 2. Materials and methods 2.1. Description of the membrane module The module consists of a metal disk rotating inside a cylindrical housing near a circular stationary organic membrane. Fig. 1 shows two pictures of the module. The simulations were performed on the physical model of the module shown in Fig. 2, which considers the annular inlet, the fluid layer between the disk and the membrane confined laterally by the housing and the exit tube with a length equal to 2 diameters in order to avoid the influence of the exit boundary condition on the flow near the membrane. The dimensions of the physical device are also shown in Fig. 2. The flow through the module was considered to be 3 l/min, which corresponds to an axial velocity component at the module annular inlet of 0.017 m/s. It should be noted that the flow at the
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Fig. 1. Rotating disk membrane module device A.
Fig. 2. Simulated 3D physical model from module A.
inlet is considered to be rotating at the same rotating speed as the disk. The flow rate across the membrane depends on the type of membrane, the
transmembrane pressure, the presence of fouling or polarization phenomena, but it rarely exceeds 2% of inlet flow rate and consequently, has been
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ignored because of its small effect in the overall flow on the module, which is dominated by the disk rotation. The module analyzed in this study (module A: disk radius = 72.3 mm, inner radius of housing = 78.5 mm and gap between the disk and the membrane = 10.7 mm), shown in Figs. 1 and 2, is the result of several modifications introduced in an earlier module (B) used by Bouzerar et al. [15]. The difference of the modules is the change of the position of the retentate outlet that previously was located in the hollow shaft of the rotating disk, and the fluid exited in the opposite direction of the membrane. Since many results were obtained with the old module, it was also simulated for comparison with the new one. Flow calculations were carried out at six rotation speeds of the disk (Ω): 300, 1000, 2000, 3000, 10000 and 20000 rpm. Considering the Reynolds number based on the maximum linear velocity and the inlet width (Re = 1.4·105), which characterizes the Couette type flow generated at the inlet of the module, the κ-omega turbulence model [16] was used in the simulations. This turbulence model has been previously used successfully in simulations of turbulent rotating disk flows [16]. 2.2. Brief description of the software used for the simulations The commercial software used in the study was Gambit & Fluent 6.2 [17]. The geometry and the mesh were created with Gambit using the following boundary conditions: a) Prescribed velocity inlet at the inlet annular space b) Outflow at the retentate exit c) Constant rotating velocity on the disk surface d) Zero velocity on the housing wall and on the membrane The mesh used in the simulations contains 1,750,000 hexahedral elements. The first grid node near the membrane was located at 10-6 m from the wall to properly resolve the thin boundary
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layer and, thus, the wall shear stress distribution on the membrane, produced by the effect of the rotation of the disk. 2.3. Comparison of simulation and experiments in an enclosed rotor-stator cavity In order to check the consistency of the numerical results of the flow in the module that are presented in the next section, we performed a simulation in an enclosed rotor-stator cavity of aspect ratio h/R = 0.127 to reproduce the experimental conditions of Cheah et al. [18] who measured profiles of the angular velocity component and the angular component of the wall shear stress at r/R = 0.8 on the stator and on the disk at Re = 1.6·106. Fig. 3 shows a good agreement between the numerical predictions of the time averaged profiles of vθ at r/R = 0.4 (Fig. 3a) and r/R = 0.8 (Fig. 3b) and the corresponding measurements. The wall shear stress obtained from the simulations of the flow in the membrane module, as well as those predicted by the approximation used by Bouzerar et al [19], were compared with measurements carried out by Cheah et al. [18] and Itoh et al. [20]. These experimental studies report the friction coefficient [Eq. (1)] based on the local azimuthal component of the wall shear stress at r = 0.8·R in the range 3·105 ≤ Re ≤ 1.6·106 in enclosed rotor-stator cavities with aspect ratios 0.127 and 0.08, similar to that of the module considered in this study (0.15). CT
WT 1 U VT2,ref 2
(1)
Note that the definition of Cθ, given in Eq. (1), is based on the difference in the local tangential velocity between the solid surface and the rotating fluid core, vθ,core (i.e. for the static membrane, vθ,ref = vθ,core and for the rotating disk vθ,ref = ω·r – vθ,core). Fig. 4 shows the measurements of Cheah et al. [18] and Itoh et al. [20] together with the predic-
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Fig. 3. Comparison of mean velocity profiles along the gap between the disk and the membrane at Re = 1.6·106. (a) r/R = 0.4 and (b) r/R = 0.8 measured by Cheah et al. and calculated by CFD.
tions of the model used by Bouzerar et al. [19] and the present numerical simulations of the flow in the membrane module. The results of the simulation of the enclosed rotor-stator cavity are also included in this figure and show good agreement with the measurements of Cheah et al. [18]. It can be seen in Fig. 4 that the model of Bouzerar et al. [19] and the numerical simulations
predict values of the friction coefficients of the disk not too far from the experimental results of Cheah et al. [18] and Itoh et al. [20]. For the membrane, both the model and the simulations underpredict the measurements of the friction coefficient but the predictions of the simulations are closer to the experimental values than those of the model.
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Fig. 4. Comparison of the circumferential skin friction coefficient at r/R = 0.8 as a function of the Reynolds number.
3. Theoretical models In the case of laminar boundary layers (Re = ω·r2/ν < 105) and neglecting the end effects of the disk and the membrane and the interaction between the two boundary layers developed near both walls, a prediction of the wall shear stress can be obtained by solving the simplified continuity and Navier–Stokes Eqs. (2)–(5) [21,22], where the variation of the velocities and pressure with the azimuthal direction has been eliminated. vr
wvr vT2 wv vz r wr r wz ª w 2v w §v 1 wp Q « 2r ¨ r U wr wr © r ¬ wr
2 · w vr º ¸ 2 » ¹ wz ¼
(2)
vr
vr
wvT vr vT wv vz T wr r wz 2 ªw v w § v · w 2v º Q « 2T ¨ T ¸ 2T » wr © r ¹ wz ¼ ¬ wr wvz wv vz z wr wz ª w 2 v 1 wvz w 2 vz º 1 wp Q « 2z 2 » U wz r wr wz ¼ ¬ wr
1 w rvr wvz r wr wz
0
(3)
(4)
(5)
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This set of partial differential equations can be transformed into a system of four ordinary differential equations using the following dimensionless variables:
z*
z G
z
vr , r Z
, vr* ( z * )
Q/Z vT vz , v*z ( z * ) vT* ( z * ) GZ r Z p p p* ( z * ) PZ UQZ
vz QZ
,
3
1
0.94 U Q 2 k Z 2 r
W m , Tl
0.77 U Q 1
1
2
3
(7b)
3
(7c)
k Z 2 r
1.21 U Q
Wd , r
0.51 U Q 2 k Z 2 r
W d , Tl
0.62 U Q
W dt
2
3
1
0.81 U Q
1
2
1
2
3
(8a)
k Z 2 r
(8b)
3
(8c)
k Z 2 r
0.057 U Q
Wm T B
0.0296 U Q
5
9
k Z 5 r 1
5
8
9
(9)
5
k Z 5 r
8
(10)
5
If Eqs. (8) and (9) are integrated in order to obtain the area-weighted average of the azimuthal component of the wall shear stress, Eqs. (11) and (12) are obtained. Wdisk
0.057 U Q
0.0317 U Q
Wm
1
5
1
5
k Z
9
R
5
1 d 8 ³ r 5 dA A0 8
9
1
9
0.0296 U Q 5 k Z 5 1
(11)
k Z 5 Rd 5
9
0.0164 U Q 5 k Z 5
1 A
Rmem
³
r
8
dA
Rin
1 R Rin2 2 m
5
(12)
Rm18 / 5 Rin18 / 5
(7a)
k Z 2 r
Wmt
1
W d TB
(6)
The integration of this system, with the appropriate boundary conditions, can be performed by a conventional Runge–Kutta method and the expressions for the radial (subscript r), azimuthal (subscript θ) and total (subscript t) wall shear stresses can be obtained: for the membrane [subscript m, Eqs. (7)] and for the disk [subscript d, Eqs. (8)] [23]. In these equations kω represents the angular velocity of the inviscid region between membrane and disk when the disk rotates at angular velocity ω. The coefficient k is smaller than 1 and is assumed to be dependent only upon disk geometry. It can be seen, that, according to Eqs. (7) and (8) the shear stress on the membrane is about 50% larger than on the disk Wm,r
Carlsson [24] and the Blasius correlation for turbulent boundary layers to predict the azimuthal component of the disk and membrane shear stress, respectively. The resulting local azimuthal wall shear stresses are indicated in Eqs. (9) and (10), for the disk and the membrane.
For turbulent boundary layers, Bouzerar et al. [19] used the correlation reported by Murkes and
where Rd is the disk radius, A is the area and Rm is the membrane outer radius and Rin the inner one. Eqs. (13a) and (13b) relate the azimuthal component of the wall shear stress for the disk and the membrane, respectively, for the fluid properties and the dimensions considered in this study. In these equations the value of the velocity factor k, that models the fact that the fluid outside the boundary layers is rotating at a smaller rotation rate than the disk, is set to 0.45 according to Bouzerar et al. [15]. For skim milk at 45°C, with the following constant physical properties:
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a) Viscosity: ν = 1.06·10–6 m2/s (μ = 9.8·10–4 Pa·s) b) Density: ρ = 924 kg/m3 we obtain
Wdisk
0.0059 Z1.8 1.02 104 :1.8
Wmembrane
0.0037 Z1.8
6.37 105 :1.8
(13a) (13b)
The pressure distribution on the membrane is of importance because it directly affects the transmembrane pressure and, thus, the permeate flux. The pressure on the membrane can be estimated by assuming that the flow is potential outside the boundary layers (i.e. the viscous and turbulent stresses are negligible) developed near the disk and near the membrane. Under this assumption Eq. (2) can be simplified and Eq. (14) is obtained.
vT2 r
r k Z2
1 wp U wr
(14)
The integration of Eq. (14) along the radial direction leads to the pressure distribution outside the boundary layers given in Eq. (15), which can be used to estimate the pressure at the disk and the membrane according to the conventional boundary layer approximation: 2
p 0.5 U k Zr p0
(15)
where p0 is the pressure at the symmetry axis (r = 0).
4. Results and discussion We first present results obtained at 2000 rpm. At this rotation speed the flow is dominated by the disk rotation. For higher rotational speeds, the profiles are similar and the results will be shown and discussed in the next subsection. Near the disk, the fluid is moved outward, along the radial direction, by the centrifugal acceleration with a positive radial velocity, while
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near the membrane the fluid flows with a negative radial velocity towards the outlet, located at the membrane centre of the through a thin boundary layer. Due to the disk rotation, the azimuthal fluid velocity is important at the edge of the disk reaching 16.4 m/s at 2000 rpm. In order to investigate the thin boundary layer developed near the membrane, the grid nodes were stretched towards the wall. Fig. 5 shows the velocity vectors within this boundary layer. Note that the vectors indicate the position of the grid nodes and, consequently, the boundary layer can be considered as well resolved by the grid used. The velocity vectors in the boundary layer near the rotating disk have almost the same velocity as the disk. On the other hand, it can be seen in Fig. 5 that the boundary layers at the rotating disk and at the membrane do not interact because of the large space between them in comparison with their thickness. Fig. 6 shows the radial pressure distribution at the membrane and at the disk, including the ringshaped inlet, located at r > 0.072 m. The values of pressure are relative to the pressure at the exit of the outlet tube (i.e. pressure at the outlet is 0). In both cases, the pressure increases with radius and differences between the two pressures are negligible, indicating that the pressure distribution on the walls is imposed by the rotating inviscid flow in the core of the module. Fig. 7 shows the radial distribution of the wall shear stress modulus at the membrane at Ω = 2000 rpm. The shear stress tends to increase with the radius. The mean shear stress at the membrane is slightly larger than at the disk, as suggested by the wall shear stress distributions predicted by Eqs. (7c) and (8c). It can be seen in Fig. 7 that the end effects near the outlet (r = 0.01 m) and near the lateral housing (r = 0.0785 m) are important and that the theoretical model given by Eq. (7c) underestimates the wall shear stress modulus, especially near the lateral housing, due to the effect of the inlet flow.
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Fig. 5. Velocity vectors in the r,z plane for module A.
Fig. 6. CFD calculations of distribution of the pressure along the radius at the membrane and at the disk for module A at 2000 rpm and at 3 l/min.
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Fig. 7. Distribution of the shear stress along the radius at the membrane (simulated and theoretical) of module A, at 2000 rpm and at 3 l/min.
4.1. Variation of the shear stress as a function of the rotating speed Fig. 8 compares the modulus of the surface averaged wall shear stresses predicted by the simulations and predictions using the model of Bouzerar et al. [19] at the membrane (8a) and at the disk (8b) for modules A and B, as a function of the disk rotation speed. It can be seen that the wall shear stress increases as the rotation speed is increased with a power law with an exponent for the membrane about 10% larger than for the disk. These exponents are less than 2 and range between 1.5 and 1.7. It can be seen in Fig. 8 that the shear stress modulus on the membrane and the disk in the two modules are not significantly different because the flow is dominated by the effect of disk rotation, and the location of the retentate outlet is not important. The model proposed by Bouzerar et al. [19] predicts wall shear stresses on the disk in good agreement with the simulations as shown in Fig. 8b. However, for the membrane, the model, given in Eq. (13b) and derived from the Blasius correlation for the shear stress on a flat plate, underes-
timates the wall shear stress by about a factor of 4 at large rotation rates, as shown in Fig. 8a. This deviation can be partly attributed to the fact that the model does not take into account the end effects of the inlet flow, of the lateral confining wall and the outlet, as suggested by Fig. 7a. Also, the theoretical model neglects the radial shear stress contribution to the overall wall shear stress. The simulations show that both components have an effect on the shear stress modulus. The radial distributions of the wall shear stress at different rotation speeds and on the membrane are shown in Figs. 9 and 10. At low rotation rates (Fig. 9) both components contribute to the modulus of the wall shear stress and the end effects of the outlet, located at r < 0.01 m, have an important effect in the shear stress distribution. It can be seen that the wall shear stress increases as the radial flow near the membrane approaches the outlet. As seen when comparing Fig. 9 and Fig. 10, the increase of the rotation rate from 300 rpm to 1000 rpm produces an overall increase of the values of the local wall shear stress. At 1000 rpm and at moderate values of the radius, (0.04 m ≤ r ≤ 0.07 m) the modulus of the wall shear stress
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Fig. 8. Variation of the shear stress with the rotating speed at (a) the membrane and at (b) the disk for modules A and B at 3 l/min. Comparison between simulations and Bouzerar’s approximation.
increases with radius, indicating the progressive increase in importance of the rotation imposed by the disk on the membrane boundary layer. From this point to the positive radial direction, the force applied to the fluid due to the rotation is the dominant one. It can be seen in Figs. 9 and 10 that the extension of this recirculation bubble is reduced as the rotation rate is increased from
300 rpm to 1000 rpm and is not observed at 2000 rpm, as shown in Fig. 7. 4.2. Variation of the radial pressure distribution as a function of the rotating speed As shown in Fig. 6, the pressure field within the module depends mainly on the radial coordi-
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Fig. 9. Shear stress distribution along the radius at the membrane, at 300 rpm, for the module A and at 3 l/min.
Fig. 10. Shear stress distribution along the radius at the membrane, at 1000 rpm, for the module A and at 3 l/min.
nate. We investigated the influence of the flow rate through the module and the rotation rate on the radial pressure distribution and also we com-
pared experimental pressure measurements with results obtained with the simulations. Fig. 11 shows the pressure on the membrane
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Fig. 11. Distribution of pressure on the membrane at several inlet velocities for module A at Ω = 2000 rpm. Comparison with the inviscid model.
at Ω = 2000 rpm for three different imposed flow rates through the module, 3.0, 1.6 and 0.2 l/min. The corresponding inlet velocities are 0.017, 0.009 and 0.0011 m/s, respectively, as indicated in Table 1. The theoretical pressure distributions predicted by Eq. (15) using k = 0.45 and k = 1 are also plotted in Fig. 11 for comparison purpose. It can be seen that the increase of the imposed inlet flow rate, introduced at 0.0723 < r < 0.0785 m, produces the progressive increase of the value of the maximum pressure, located on the lateral housing of the module. This fact has an important implication if the factor k, used in the theoretical predictions of the wall shear stresses [Eqs. (7)– (12)] is determined from pressure measurements Table 1 Values of k as a function of the inlet velocity at Ω = 2000 rpm
Flow rate (l/min)
Inlet velocity (m/s) k
0.2 1.6 3.0
0.0011 0.0090 0.0170
0.47 0.53 0.61
using Eq. (15), as reported by Bouzerar et al. [19]. These authors measured k = 0.45 in a similar filtration module, using the readings of pressure transducers for a flow rate of 0.2 l/min. Fig. 11 and Table 1 show that the factor k, computed using the value of the maximum pressure and Eq. (15), at Ω = 2000 rpm strongly depends on the inlet flow rate and that the value at the lowest flow rate (0.2 l/min) agrees with the theoretical curve for k = 0.42 reported by Bouzerar et al. [19]. Two experimental pressure measurements at vin = 0.0011 m/s and vin = 0.017 m/s performed at the housing by using the pressure transducer are also plotted in the figure, showing the good agreement between them and those obtained with the simulations. Fig. 12 shows the pressure profiles along the radius at different rotating speeds for the flow rate considered in this study (3.0 l/min). It can be seen that as the rotating speed increases, the profile becomes more parabolic indicating better agreement with the theoretical model given in Eq. (15), using, for these conditions, k = 0.67. Fig. 13 shows the influence of the rotating
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Fig. 12. Distribution of pressure on the membrane calculated by CFD at several rotating speeds for module A and at 3 l/min.
speed on the surface averaged pressure at the membrane and at the disk for the two modules. As shown in Fig. 12, the local pressure on the membrane is not proportional to Ω2 for an inlet flow rate of 3 l/min and for Ω < 20000 rpm. It can be seen in Fig. 13 that, on average, the pressure at membrane is slightly higher than at the disk with differences ranging from 5% to 20%. 4.3. Energy considerations The modules considered require two energetic inputs to operate: that of the pump, which is related to the pressure drop of the flow across the module, and that of the motor to overcome the viscous drag on the disk. The target benefit in this case (output) is the prevention of the fouling, which is achieved by increasing the membrane shear stress. We analyzed the influence of the flow rate and the rotating speed on the pressure drop in the modules and the torque needed to overcome the viscous drag on the disk. To study the influence of the flow rate, we performed several simulations by changing the
inlet velocity at zero rotating speed and we examined the changes on the membrane shear stress and the pressure drop in the module cavity. At zero rotation rate, the membrane shear stress is very small for the cross-flow velocities considered. For example at Ω = 0 and at a flow rate of 3 l/min the membrane shear stress of module A is τmem = 0.03 Pa, while at Ω = 2000 rpm this value increases up to τmem = 112.50 Pa. The comparison of the two modules without rotation shows that the membrane shear stress in the module A (τmem = 0.033 Pa) is nearly one order of magnitude higher than that of module B (τmem = 0.006 Pa) at 3 l/min. However when rotation is applied, there are no significant differences between them. A similar tendency is observed for the pressure drop, but in all cases, the absolute loss is insignificant. For example at Ω = 0 and at a flow rate of 3 l/min the pressure drop of module A is only ΔP = 20.6 Pa. To study the influence of the rotating speed on the energy requirements, we performed several simulations by keeping constant the velocity inlet (0.017 m/s) and changing the rotating speed.
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Fig. 13. Variation of the logarithm of pressure with the logarithm of the rotating speed and at 3 l/min for the (a) module A and (b) module B. Comparison with the inviscid model.
We examined the changes on the membrane shear stress and the power required for the motor (calculated from the torque needed to overcome the viscous drag on the disk) and the power required for the pump (calculated from the pressure drop between the inlet and the outlet of the module).
At 2000 rpm, the power needed to operate the disk is about 24 W and there are no significant differences between the two modules (also at the other rotating speeds considered). The pressure drop at 2000 rpm is 45 kPa and there are no significant differences between the two modules. If these re-
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sults are compared with the ones obtained at zero rotation, it can be observed that when rotation is applied, the main contribution to the pressure drop is the rotation but not the intensity of the crossflow. The pressure drop at 2000 rpm in terms of power is about 2.3 W. Therefore, the energy requirement for the motor is about 10 times the energy requirement for the pump. Results as function of the rotating speed, shows that the energy for the motor increases with an exponent of 2.8, while that corresponding to energy for the pump is 1.6. These results show that when rotation is applied, the input energy requirements depend mainly on the rotation and neither on the geometry nor the cross-flow rate. Also, the membrane shear stress depends mainly on the rotating speed and consequently, module A is more appropriate considering fluid temperature increment due to friction as observed experimentally. Thus, when future designs of rotating modules with smooth disks have to be performed, parameters such as geometry or pressure loss can be considered secondary in comparison with the rotation speed, which gives more degrees of freedom for the design. 5. Discussion and conclusions The simulations of the turbulent flow in the membrane module equipped with a rotating disk show that the velocity and shear stress profiles are dominated by centrifugal forces at rotating velocities greater than 2000 rpm. At lower rotation rates the end effects of the outlet produce recirculating bubbles near the membrane that affect considerably the membrane wall shear stress distribution. At high rotation rates, Ω ≥ 2000 rpm, the flow distribution in the module does not present recirculating streamlines or dead zones especially near the membrane, where a thin boundary layer occurs producing large values of the shear stress over the membrane and consequently a reduction of the fouling. A rotating velocity
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around 2000 rpm seems to be a good compromise between energy consumption and permeate flux. The model proposed by Bouzerar et al. [15,19] to predict wall shear stresses partially agrees with the simulations. Results for shear stress at the disk coincide, but the total shear stress at the membrane of the simulations is two times larger than the azimuthal component estimated by the model of Bouzerar et al. [15,19]. Measurements of the wall shear stress could clarify the accuracy of the predictions of the model and those of the numerical simulations. For the disk rotation rates and the flow rates considered, the pressure distribution within the module depends mainly on the radial coordinate. The pressure level of the flow near the lateral housing is strongly affected by the rotation rate and the inlet flow rate imposed. The validity of the theoretical radial pressure distribution obtained from Bernoulli’s equation assuming an inviscid fluid core in solid body rotation [Eq. (15)] improves as the rotation rate is increased and deteriorates as the imposed flow rates through the module increases. The simulations indicate that the input energy requirements depend mainly on the rotation rate of the disk and not on the geometry nor the crossflow rate. At the usual operating rotating rates of the disk both modules give similar performance in terms of membrane shear stress and pressure drop. Acknowledgements Authors acknowledge Omar Al-Akoum and Matthieu Frappart for their contribution on the experimental work. This work has been supported by the Spanish Ministry of Science and Technology; project PPQ2002-04201-C02. Symbols A C
— Area, m2 — Friction coefficient, adim
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h k P r R v z
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— Gap between the disk and the membrane, m — Velocity factor vθ,core/ω r, adim. — Pressure, Pa — Radial distance, m/radial component — Radius, m — Velocity, m/s — Normal component to the membrane, m
Greek δ μ ν θ ρ τ ω Ω
— — — — — — — —
Boundary layer thickness, m Dynamic viscosity, kg/m/s Kinematic viscosity, m2/s Tangential component Fluid density, kg/m3 Shear stress, Pa Angular velocity, rad/s Rotating speed, rpm
References [1] S. Sablani, M.F.A. Goosen, R. Al-Belushi and M. Wilf, Desalination, 141 (2001) 269–298. [2] M. Mulder, Basic Principles of Membrane Technology. Kluwer, 2nd ed., 1996. [3] M.Y. Jaffrin, B.B. Gupta and P. Blanpain, Membrane fouling control by backflushing in microfiltration with mineral membranes. Proc. Fifth World Filtration Congress, Nice, France, 1990, pp. 479–483. [4] J. Schwinge, D.E. Wiley and A.G. Fane, J. Membr. Sci., 229 (2004) 53–61. [5] U. Frenander and A.S. Jönsson, Biotech. Bioeng., 52 (1996) 397–403. [6] S.S. Lee, A. Burt, G. Russotti and B. Buckland, Biotech. Bioeng., 48 (1995) 386–400. [7] A. Pessoa and M. Vitolo, Process Biochem., 33 (1998) 39–45. [8] A. Brou, L. Ding, P. Boulnois and M.Y. Jaffrin, J.
Membr. Sci., 197 (2002) 269–282. [9] O. Al-Akoum, M.Y. Jaffrin, L.H. Ding and M. Frappart, J. Membr. Sci., 235 (2004) 111–122. [10] G. Brans, C.G.P.H. Schroën, R.G.M. van der Sman and R.M. Boom, J. Membr. Sci., 243 (2004) 263– 272. [11] J. Schwinge, D.E. Wiley and D.F. Fletcher, Desalination, 146 (2002) 195–201. [12] S.K. Karode and A. Kumar, J. Membr. Sci., 193 (2001) 69–84. [13] S.X. Liu, M. Peng and L. Vane, Chem. Eng. Sci., 59 (2004) 5853–5857. [14] C. Torras, J. Pallarés, R. Garcia-Valls and M.Y. Jaffrin, Desalination, 200 (2006) 453–455. [15] R. Bouzerar, L. Ding and M.Y. Jaffrin, J. Membr. Sci., 170 (2000) 127–141. [16] S.Z. Liu and H.M. Tsai, Simulation of boundary layer transition with a modified k-ω model. AIAAPaper 1998-340, Aerospace Sciences Meeting and Exhibit, 36th, Reno, NV, January 12–15, 1998. [17] Fluent software. http://www.fluent.com. [18] S.C. Cheah, H. Jacovides, D.C. Jackson, H. Ji, and B.E. Launder, Experim. Thermal Fluid Sci., 9 (1994) 445–455. [19] R. Bouzerar, M.Y. Jaffrin, L. Ding and P. Paullier, AIChE J., 46 (2000) 257–265. [20] M. Itoh, Y. Yamada, S. Imao and M. Gouda, Experiments on turbulent flow due to an enclosed rotating disk. Proc. 1st symposium on Engineering Turbulence Modelling and Measurements, W. Rodi and E.N. Ganic, eds., Elsevier, New York, 1990, pp. 659–668. [21] U.T. Bödewadt, Die Drehströmung über festem Grunde, Z. Angew. Math. Mech., 20 (1940) 241– 253. [22] T. Von Kármán, Uber laminare und turbulente Reibung, Z. Angew. Math. Mech., 1 (1921) 233– 252. [23] J.P. Vanyo, Rotating Fluids in Engineering and Science. Butterworth-Heinemann, 1993. [24] J. Murkes and C.G. Carlsson, Crossflow Filtration, Wiley, New York, 1988.
Numerical simulation of the flow in a rotating disk filtration module C. Torrasa, J. Pallaresb*, R. Garcia-Vallsa, M.Y. Jaffrinc Structured Systems Engineering for Energy, Materials and Chemistry, Chemical Engineering Department, Experiments, Computation and Modelization in Fluid Mechanics and Turbulence Group, Mechanical Engineering Department, Universitat Rovira i Virgili. Av. Països Catalans, 26, 43007 Tarragona, Catalunya, Spain Tel. +34 (977) 55-96-82; Fax +34 (977) 55-96-91; email: [email protected] c Biomechanics and Biomedical Engineering laboratory, UMR CNRS 6600, Technological University of Compiegne, Compiegne, France a
b
Received 18 April 2007; accepted revised 3 February 2008
Abstract The characterization of the flow inside an experimental flat membrane module with a smooth rotating disk was performed. The module consists of a disk rotating at speeds up to 3000 rpm inside a cylindrical housing equipped with a stationary circular flat membrane. The characterization was carried out by using a finite volume CFD software with the κ-omega turbulence model and results of the range of rotation speeds 300 ≤ Ω ≤ 20000 rpm were compared with experimental and theoretical data reported in previous studies. The simulations suggest high permeate fluxes for the device due to large average shear stresses on the membrane and the absence of stagnant zones inside the module, which are desirable features to avoid membrane fouling processes. The simulations show an overall good agreement with theoretical results based on the main assumption that the wall shear stress on the membrane and on the disk can be predicted using modified correlations for rotating flow over a stationary wall and for flow induced by a rotating disk, respectively and with experimental pressure measurements. It has been found that the flow rate imposed at the inlet of the module has an important effect on the pressure distribution. At the membrane some discrepancies were found between the results obtained with the simulations and with the theoretical approach because of the limitations of the assumptions, especially at low rotating speeds for which the effect of the flow through the module becomes important. The correlations relating the disk rotation rate with the surface averaged pressure and the shear stress on the membrane were determined. Keywords: Ultrafiltration; Microfiltration; Fouling; Membrane module; Rotating disk; Shear stress; Computational fluid dynamics
*Corresponding author. 0011-9164/09/$– See front matter © 2008 Elsevier B.V. All rights reserved doi:10.1016/j.desal.2008.02.006
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1. Introduction In filtration experiments (mainly micro- and ultrafiltration) carried out with porous membranes, not only the separation capability of the membrane is important but also its permeability. Large efforts are invested in order to maximize the membrane effective area but also to prevent flux decrement during experiment because of usual phenomena such as the fouling or the concentration polarization. The last one is caused by the imbalance in transport between the membrane surface and the core [1,2]. Fouling phenomena is caused by the solids that are in the feed solution, which block the pores or create a cake over the membrane, and therefore, reduce its effective area [2]. To overcome this problem, two strategies are available: prevention and cure. Usually, both are needed. In terms of a cure, backflushing is an often used process to remove the solids that are blocking the pores [3]. This process implies that normal operation is stopped periodically or alternatively, the need of coupled membrane systems, and thus, an increment of the cost of the process. Therefore, prevention is an important issue. One prevention method consists of a proper design of the membrane module and thus, the flow over the membrane [4]. Rotating disk filtration modules have been shown to yield higher permeate fluxes and better solute transmission than conventional cross flow filtration due to the high shear rates they generate at the membrane which prevent or limit cake formation in microfiltration [5–7]. In addition these shear rates are obtained at relatively low inlet flow rates and low pressure drops inside the module, resulting in a relatively uniform transmembrane pressure (TMP). The combination of high shear rates and low TMP facilitates macromolecule transmission through the membrane. Conversely, when used in ultrafiltration (UF) and nanofiltration (NF), these systems reduce concentration polarisation on the membrane, and their permeate fluxes continue to increase with pressure to higher levels than in classical tubular systems [8]. In addition since rejected solute con-
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centration at the membrane is reduced by the high shear rate, their diffusive transmission through the membrane is smaller and their apparent rejection is higher, which is very advantageous in the treatment of waste waters by NF [9]. For a correct analysis of the performance of these systems, it is therefore important to evaluate accurately the membrane shear rate and pressure distribution at the membrane. Since disk rotation speeds are generally high, these systems operate in the turbulent regime, for which no analytic solution is available. Thus it is logical to use a Computational Fluid Dynamics (CFD) approach for this purpose. CFD has been demonstrated to be a powerful tool to predict the flow inside housings and to resolve properly boundary layers and flow distribution effects as several works show [10–13]. This paper, which is the extension of a published abstract [14], illustrates the detailed description of the flow inside the module from CFD analysis considering the main variables and some results are contrasted with experimental ones. 2. Materials and methods 2.1. Description of the membrane module The module consists of a metal disk rotating inside a cylindrical housing near a circular stationary organic membrane. Fig. 1 shows two pictures of the module. The simulations were performed on the physical model of the module shown in Fig. 2, which considers the annular inlet, the fluid layer between the disk and the membrane confined laterally by the housing and the exit tube with a length equal to 2 diameters in order to avoid the influence of the exit boundary condition on the flow near the membrane. The dimensions of the physical device are also shown in Fig. 2. The flow through the module was considered to be 3 l/min, which corresponds to an axial velocity component at the module annular inlet of 0.017 m/s. It should be noted that the flow at the
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Fig. 1. Rotating disk membrane module device A.
Fig. 2. Simulated 3D physical model from module A.
inlet is considered to be rotating at the same rotating speed as the disk. The flow rate across the membrane depends on the type of membrane, the
transmembrane pressure, the presence of fouling or polarization phenomena, but it rarely exceeds 2% of inlet flow rate and consequently, has been
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ignored because of its small effect in the overall flow on the module, which is dominated by the disk rotation. The module analyzed in this study (module A: disk radius = 72.3 mm, inner radius of housing = 78.5 mm and gap between the disk and the membrane = 10.7 mm), shown in Figs. 1 and 2, is the result of several modifications introduced in an earlier module (B) used by Bouzerar et al. [15]. The difference of the modules is the change of the position of the retentate outlet that previously was located in the hollow shaft of the rotating disk, and the fluid exited in the opposite direction of the membrane. Since many results were obtained with the old module, it was also simulated for comparison with the new one. Flow calculations were carried out at six rotation speeds of the disk (Ω): 300, 1000, 2000, 3000, 10000 and 20000 rpm. Considering the Reynolds number based on the maximum linear velocity and the inlet width (Re = 1.4·105), which characterizes the Couette type flow generated at the inlet of the module, the κ-omega turbulence model [16] was used in the simulations. This turbulence model has been previously used successfully in simulations of turbulent rotating disk flows [16]. 2.2. Brief description of the software used for the simulations The commercial software used in the study was Gambit & Fluent 6.2 [17]. The geometry and the mesh were created with Gambit using the following boundary conditions: a) Prescribed velocity inlet at the inlet annular space b) Outflow at the retentate exit c) Constant rotating velocity on the disk surface d) Zero velocity on the housing wall and on the membrane The mesh used in the simulations contains 1,750,000 hexahedral elements. The first grid node near the membrane was located at 10-6 m from the wall to properly resolve the thin boundary
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layer and, thus, the wall shear stress distribution on the membrane, produced by the effect of the rotation of the disk. 2.3. Comparison of simulation and experiments in an enclosed rotor-stator cavity In order to check the consistency of the numerical results of the flow in the module that are presented in the next section, we performed a simulation in an enclosed rotor-stator cavity of aspect ratio h/R = 0.127 to reproduce the experimental conditions of Cheah et al. [18] who measured profiles of the angular velocity component and the angular component of the wall shear stress at r/R = 0.8 on the stator and on the disk at Re = 1.6·106. Fig. 3 shows a good agreement between the numerical predictions of the time averaged profiles of vθ at r/R = 0.4 (Fig. 3a) and r/R = 0.8 (Fig. 3b) and the corresponding measurements. The wall shear stress obtained from the simulations of the flow in the membrane module, as well as those predicted by the approximation used by Bouzerar et al [19], were compared with measurements carried out by Cheah et al. [18] and Itoh et al. [20]. These experimental studies report the friction coefficient [Eq. (1)] based on the local azimuthal component of the wall shear stress at r = 0.8·R in the range 3·105 ≤ Re ≤ 1.6·106 in enclosed rotor-stator cavities with aspect ratios 0.127 and 0.08, similar to that of the module considered in this study (0.15). CT
WT 1 U VT2,ref 2
(1)
Note that the definition of Cθ, given in Eq. (1), is based on the difference in the local tangential velocity between the solid surface and the rotating fluid core, vθ,core (i.e. for the static membrane, vθ,ref = vθ,core and for the rotating disk vθ,ref = ω·r – vθ,core). Fig. 4 shows the measurements of Cheah et al. [18] and Itoh et al. [20] together with the predic-
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Fig. 3. Comparison of mean velocity profiles along the gap between the disk and the membrane at Re = 1.6·106. (a) r/R = 0.4 and (b) r/R = 0.8 measured by Cheah et al. and calculated by CFD.
tions of the model used by Bouzerar et al. [19] and the present numerical simulations of the flow in the membrane module. The results of the simulation of the enclosed rotor-stator cavity are also included in this figure and show good agreement with the measurements of Cheah et al. [18]. It can be seen in Fig. 4 that the model of Bouzerar et al. [19] and the numerical simulations
predict values of the friction coefficients of the disk not too far from the experimental results of Cheah et al. [18] and Itoh et al. [20]. For the membrane, both the model and the simulations underpredict the measurements of the friction coefficient but the predictions of the simulations are closer to the experimental values than those of the model.
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Fig. 4. Comparison of the circumferential skin friction coefficient at r/R = 0.8 as a function of the Reynolds number.
3. Theoretical models In the case of laminar boundary layers (Re = ω·r2/ν < 105) and neglecting the end effects of the disk and the membrane and the interaction between the two boundary layers developed near both walls, a prediction of the wall shear stress can be obtained by solving the simplified continuity and Navier–Stokes Eqs. (2)–(5) [21,22], where the variation of the velocities and pressure with the azimuthal direction has been eliminated. vr
wvr vT2 wv vz r wr r wz ª w 2v w §v 1 wp Q « 2r ¨ r U wr wr © r ¬ wr
2 · w vr º ¸ 2 » ¹ wz ¼
(2)
vr
vr
wvT vr vT wv vz T wr r wz 2 ªw v w § v · w 2v º Q « 2T ¨ T ¸ 2T » wr © r ¹ wz ¼ ¬ wr wvz wv vz z wr wz ª w 2 v 1 wvz w 2 vz º 1 wp Q « 2z 2 » U wz r wr wz ¼ ¬ wr
1 w rvr wvz r wr wz
0
(3)
(4)
(5)
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This set of partial differential equations can be transformed into a system of four ordinary differential equations using the following dimensionless variables:
z*
z G
z
vr , r Z
, vr* ( z * )
Q/Z vT vz , v*z ( z * ) vT* ( z * ) GZ r Z p p p* ( z * ) PZ UQZ
vz QZ
,
3
1
0.94 U Q 2 k Z 2 r
W m , Tl
0.77 U Q 1
1
2
3
(7b)
3
(7c)
k Z 2 r
1.21 U Q
Wd , r
0.51 U Q 2 k Z 2 r
W d , Tl
0.62 U Q
W dt
2
3
1
0.81 U Q
1
2
1
2
3
(8a)
k Z 2 r
(8b)
3
(8c)
k Z 2 r
0.057 U Q
Wm T B
0.0296 U Q
5
9
k Z 5 r 1
5
8
9
(9)
5
k Z 5 r
8
(10)
5
If Eqs. (8) and (9) are integrated in order to obtain the area-weighted average of the azimuthal component of the wall shear stress, Eqs. (11) and (12) are obtained. Wdisk
0.057 U Q
0.0317 U Q
Wm
1
5
1
5
k Z
9
R
5
1 d 8 ³ r 5 dA A0 8
9
1
9
0.0296 U Q 5 k Z 5 1
(11)
k Z 5 Rd 5
9
0.0164 U Q 5 k Z 5
1 A
Rmem
³
r
8
dA
Rin
1 R Rin2 2 m
5
(12)
Rm18 / 5 Rin18 / 5
(7a)
k Z 2 r
Wmt
1
W d TB
(6)
The integration of this system, with the appropriate boundary conditions, can be performed by a conventional Runge–Kutta method and the expressions for the radial (subscript r), azimuthal (subscript θ) and total (subscript t) wall shear stresses can be obtained: for the membrane [subscript m, Eqs. (7)] and for the disk [subscript d, Eqs. (8)] [23]. In these equations kω represents the angular velocity of the inviscid region between membrane and disk when the disk rotates at angular velocity ω. The coefficient k is smaller than 1 and is assumed to be dependent only upon disk geometry. It can be seen, that, according to Eqs. (7) and (8) the shear stress on the membrane is about 50% larger than on the disk Wm,r
Carlsson [24] and the Blasius correlation for turbulent boundary layers to predict the azimuthal component of the disk and membrane shear stress, respectively. The resulting local azimuthal wall shear stresses are indicated in Eqs. (9) and (10), for the disk and the membrane.
For turbulent boundary layers, Bouzerar et al. [19] used the correlation reported by Murkes and
where Rd is the disk radius, A is the area and Rm is the membrane outer radius and Rin the inner one. Eqs. (13a) and (13b) relate the azimuthal component of the wall shear stress for the disk and the membrane, respectively, for the fluid properties and the dimensions considered in this study. In these equations the value of the velocity factor k, that models the fact that the fluid outside the boundary layers is rotating at a smaller rotation rate than the disk, is set to 0.45 according to Bouzerar et al. [15]. For skim milk at 45°C, with the following constant physical properties:
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a) Viscosity: ν = 1.06·10–6 m2/s (μ = 9.8·10–4 Pa·s) b) Density: ρ = 924 kg/m3 we obtain
Wdisk
0.0059 Z1.8 1.02 104 :1.8
Wmembrane
0.0037 Z1.8
6.37 105 :1.8
(13a) (13b)
The pressure distribution on the membrane is of importance because it directly affects the transmembrane pressure and, thus, the permeate flux. The pressure on the membrane can be estimated by assuming that the flow is potential outside the boundary layers (i.e. the viscous and turbulent stresses are negligible) developed near the disk and near the membrane. Under this assumption Eq. (2) can be simplified and Eq. (14) is obtained.
vT2 r
r k Z2
1 wp U wr
(14)
The integration of Eq. (14) along the radial direction leads to the pressure distribution outside the boundary layers given in Eq. (15), which can be used to estimate the pressure at the disk and the membrane according to the conventional boundary layer approximation: 2
p 0.5 U k Zr p0
(15)
where p0 is the pressure at the symmetry axis (r = 0).
4. Results and discussion We first present results obtained at 2000 rpm. At this rotation speed the flow is dominated by the disk rotation. For higher rotational speeds, the profiles are similar and the results will be shown and discussed in the next subsection. Near the disk, the fluid is moved outward, along the radial direction, by the centrifugal acceleration with a positive radial velocity, while
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near the membrane the fluid flows with a negative radial velocity towards the outlet, located at the membrane centre of the through a thin boundary layer. Due to the disk rotation, the azimuthal fluid velocity is important at the edge of the disk reaching 16.4 m/s at 2000 rpm. In order to investigate the thin boundary layer developed near the membrane, the grid nodes were stretched towards the wall. Fig. 5 shows the velocity vectors within this boundary layer. Note that the vectors indicate the position of the grid nodes and, consequently, the boundary layer can be considered as well resolved by the grid used. The velocity vectors in the boundary layer near the rotating disk have almost the same velocity as the disk. On the other hand, it can be seen in Fig. 5 that the boundary layers at the rotating disk and at the membrane do not interact because of the large space between them in comparison with their thickness. Fig. 6 shows the radial pressure distribution at the membrane and at the disk, including the ringshaped inlet, located at r > 0.072 m. The values of pressure are relative to the pressure at the exit of the outlet tube (i.e. pressure at the outlet is 0). In both cases, the pressure increases with radius and differences between the two pressures are negligible, indicating that the pressure distribution on the walls is imposed by the rotating inviscid flow in the core of the module. Fig. 7 shows the radial distribution of the wall shear stress modulus at the membrane at Ω = 2000 rpm. The shear stress tends to increase with the radius. The mean shear stress at the membrane is slightly larger than at the disk, as suggested by the wall shear stress distributions predicted by Eqs. (7c) and (8c). It can be seen in Fig. 7 that the end effects near the outlet (r = 0.01 m) and near the lateral housing (r = 0.0785 m) are important and that the theoretical model given by Eq. (7c) underestimates the wall shear stress modulus, especially near the lateral housing, due to the effect of the inlet flow.
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Fig. 5. Velocity vectors in the r,z plane for module A.
Fig. 6. CFD calculations of distribution of the pressure along the radius at the membrane and at the disk for module A at 2000 rpm and at 3 l/min.
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Fig. 7. Distribution of the shear stress along the radius at the membrane (simulated and theoretical) of module A, at 2000 rpm and at 3 l/min.
4.1. Variation of the shear stress as a function of the rotating speed Fig. 8 compares the modulus of the surface averaged wall shear stresses predicted by the simulations and predictions using the model of Bouzerar et al. [19] at the membrane (8a) and at the disk (8b) for modules A and B, as a function of the disk rotation speed. It can be seen that the wall shear stress increases as the rotation speed is increased with a power law with an exponent for the membrane about 10% larger than for the disk. These exponents are less than 2 and range between 1.5 and 1.7. It can be seen in Fig. 8 that the shear stress modulus on the membrane and the disk in the two modules are not significantly different because the flow is dominated by the effect of disk rotation, and the location of the retentate outlet is not important. The model proposed by Bouzerar et al. [19] predicts wall shear stresses on the disk in good agreement with the simulations as shown in Fig. 8b. However, for the membrane, the model, given in Eq. (13b) and derived from the Blasius correlation for the shear stress on a flat plate, underes-
timates the wall shear stress by about a factor of 4 at large rotation rates, as shown in Fig. 8a. This deviation can be partly attributed to the fact that the model does not take into account the end effects of the inlet flow, of the lateral confining wall and the outlet, as suggested by Fig. 7a. Also, the theoretical model neglects the radial shear stress contribution to the overall wall shear stress. The simulations show that both components have an effect on the shear stress modulus. The radial distributions of the wall shear stress at different rotation speeds and on the membrane are shown in Figs. 9 and 10. At low rotation rates (Fig. 9) both components contribute to the modulus of the wall shear stress and the end effects of the outlet, located at r < 0.01 m, have an important effect in the shear stress distribution. It can be seen that the wall shear stress increases as the radial flow near the membrane approaches the outlet. As seen when comparing Fig. 9 and Fig. 10, the increase of the rotation rate from 300 rpm to 1000 rpm produces an overall increase of the values of the local wall shear stress. At 1000 rpm and at moderate values of the radius, (0.04 m ≤ r ≤ 0.07 m) the modulus of the wall shear stress
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Fig. 8. Variation of the shear stress with the rotating speed at (a) the membrane and at (b) the disk for modules A and B at 3 l/min. Comparison between simulations and Bouzerar’s approximation.
increases with radius, indicating the progressive increase in importance of the rotation imposed by the disk on the membrane boundary layer. From this point to the positive radial direction, the force applied to the fluid due to the rotation is the dominant one. It can be seen in Figs. 9 and 10 that the extension of this recirculation bubble is reduced as the rotation rate is increased from
300 rpm to 1000 rpm and is not observed at 2000 rpm, as shown in Fig. 7. 4.2. Variation of the radial pressure distribution as a function of the rotating speed As shown in Fig. 6, the pressure field within the module depends mainly on the radial coordi-
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Fig. 9. Shear stress distribution along the radius at the membrane, at 300 rpm, for the module A and at 3 l/min.
Fig. 10. Shear stress distribution along the radius at the membrane, at 1000 rpm, for the module A and at 3 l/min.
nate. We investigated the influence of the flow rate through the module and the rotation rate on the radial pressure distribution and also we com-
pared experimental pressure measurements with results obtained with the simulations. Fig. 11 shows the pressure on the membrane
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Fig. 11. Distribution of pressure on the membrane at several inlet velocities for module A at Ω = 2000 rpm. Comparison with the inviscid model.
at Ω = 2000 rpm for three different imposed flow rates through the module, 3.0, 1.6 and 0.2 l/min. The corresponding inlet velocities are 0.017, 0.009 and 0.0011 m/s, respectively, as indicated in Table 1. The theoretical pressure distributions predicted by Eq. (15) using k = 0.45 and k = 1 are also plotted in Fig. 11 for comparison purpose. It can be seen that the increase of the imposed inlet flow rate, introduced at 0.0723 < r < 0.0785 m, produces the progressive increase of the value of the maximum pressure, located on the lateral housing of the module. This fact has an important implication if the factor k, used in the theoretical predictions of the wall shear stresses [Eqs. (7)– (12)] is determined from pressure measurements Table 1 Values of k as a function of the inlet velocity at Ω = 2000 rpm
Flow rate (l/min)
Inlet velocity (m/s) k
0.2 1.6 3.0
0.0011 0.0090 0.0170
0.47 0.53 0.61
using Eq. (15), as reported by Bouzerar et al. [19]. These authors measured k = 0.45 in a similar filtration module, using the readings of pressure transducers for a flow rate of 0.2 l/min. Fig. 11 and Table 1 show that the factor k, computed using the value of the maximum pressure and Eq. (15), at Ω = 2000 rpm strongly depends on the inlet flow rate and that the value at the lowest flow rate (0.2 l/min) agrees with the theoretical curve for k = 0.42 reported by Bouzerar et al. [19]. Two experimental pressure measurements at vin = 0.0011 m/s and vin = 0.017 m/s performed at the housing by using the pressure transducer are also plotted in the figure, showing the good agreement between them and those obtained with the simulations. Fig. 12 shows the pressure profiles along the radius at different rotating speeds for the flow rate considered in this study (3.0 l/min). It can be seen that as the rotating speed increases, the profile becomes more parabolic indicating better agreement with the theoretical model given in Eq. (15), using, for these conditions, k = 0.67. Fig. 13 shows the influence of the rotating
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Fig. 12. Distribution of pressure on the membrane calculated by CFD at several rotating speeds for module A and at 3 l/min.
speed on the surface averaged pressure at the membrane and at the disk for the two modules. As shown in Fig. 12, the local pressure on the membrane is not proportional to Ω2 for an inlet flow rate of 3 l/min and for Ω < 20000 rpm. It can be seen in Fig. 13 that, on average, the pressure at membrane is slightly higher than at the disk with differences ranging from 5% to 20%. 4.3. Energy considerations The modules considered require two energetic inputs to operate: that of the pump, which is related to the pressure drop of the flow across the module, and that of the motor to overcome the viscous drag on the disk. The target benefit in this case (output) is the prevention of the fouling, which is achieved by increasing the membrane shear stress. We analyzed the influence of the flow rate and the rotating speed on the pressure drop in the modules and the torque needed to overcome the viscous drag on the disk. To study the influence of the flow rate, we performed several simulations by changing the
inlet velocity at zero rotating speed and we examined the changes on the membrane shear stress and the pressure drop in the module cavity. At zero rotation rate, the membrane shear stress is very small for the cross-flow velocities considered. For example at Ω = 0 and at a flow rate of 3 l/min the membrane shear stress of module A is τmem = 0.03 Pa, while at Ω = 2000 rpm this value increases up to τmem = 112.50 Pa. The comparison of the two modules without rotation shows that the membrane shear stress in the module A (τmem = 0.033 Pa) is nearly one order of magnitude higher than that of module B (τmem = 0.006 Pa) at 3 l/min. However when rotation is applied, there are no significant differences between them. A similar tendency is observed for the pressure drop, but in all cases, the absolute loss is insignificant. For example at Ω = 0 and at a flow rate of 3 l/min the pressure drop of module A is only ΔP = 20.6 Pa. To study the influence of the rotating speed on the energy requirements, we performed several simulations by keeping constant the velocity inlet (0.017 m/s) and changing the rotating speed.
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Fig. 13. Variation of the logarithm of pressure with the logarithm of the rotating speed and at 3 l/min for the (a) module A and (b) module B. Comparison with the inviscid model.
We examined the changes on the membrane shear stress and the power required for the motor (calculated from the torque needed to overcome the viscous drag on the disk) and the power required for the pump (calculated from the pressure drop between the inlet and the outlet of the module).
At 2000 rpm, the power needed to operate the disk is about 24 W and there are no significant differences between the two modules (also at the other rotating speeds considered). The pressure drop at 2000 rpm is 45 kPa and there are no significant differences between the two modules. If these re-
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sults are compared with the ones obtained at zero rotation, it can be observed that when rotation is applied, the main contribution to the pressure drop is the rotation but not the intensity of the crossflow. The pressure drop at 2000 rpm in terms of power is about 2.3 W. Therefore, the energy requirement for the motor is about 10 times the energy requirement for the pump. Results as function of the rotating speed, shows that the energy for the motor increases with an exponent of 2.8, while that corresponding to energy for the pump is 1.6. These results show that when rotation is applied, the input energy requirements depend mainly on the rotation and neither on the geometry nor the cross-flow rate. Also, the membrane shear stress depends mainly on the rotating speed and consequently, module A is more appropriate considering fluid temperature increment due to friction as observed experimentally. Thus, when future designs of rotating modules with smooth disks have to be performed, parameters such as geometry or pressure loss can be considered secondary in comparison with the rotation speed, which gives more degrees of freedom for the design. 5. Discussion and conclusions The simulations of the turbulent flow in the membrane module equipped with a rotating disk show that the velocity and shear stress profiles are dominated by centrifugal forces at rotating velocities greater than 2000 rpm. At lower rotation rates the end effects of the outlet produce recirculating bubbles near the membrane that affect considerably the membrane wall shear stress distribution. At high rotation rates, Ω ≥ 2000 rpm, the flow distribution in the module does not present recirculating streamlines or dead zones especially near the membrane, where a thin boundary layer occurs producing large values of the shear stress over the membrane and consequently a reduction of the fouling. A rotating velocity
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around 2000 rpm seems to be a good compromise between energy consumption and permeate flux. The model proposed by Bouzerar et al. [15,19] to predict wall shear stresses partially agrees with the simulations. Results for shear stress at the disk coincide, but the total shear stress at the membrane of the simulations is two times larger than the azimuthal component estimated by the model of Bouzerar et al. [15,19]. Measurements of the wall shear stress could clarify the accuracy of the predictions of the model and those of the numerical simulations. For the disk rotation rates and the flow rates considered, the pressure distribution within the module depends mainly on the radial coordinate. The pressure level of the flow near the lateral housing is strongly affected by the rotation rate and the inlet flow rate imposed. The validity of the theoretical radial pressure distribution obtained from Bernoulli’s equation assuming an inviscid fluid core in solid body rotation [Eq. (15)] improves as the rotation rate is increased and deteriorates as the imposed flow rates through the module increases. The simulations indicate that the input energy requirements depend mainly on the rotation rate of the disk and not on the geometry nor the crossflow rate. At the usual operating rotating rates of the disk both modules give similar performance in terms of membrane shear stress and pressure drop. Acknowledgements Authors acknowledge Omar Al-Akoum and Matthieu Frappart for their contribution on the experimental work. This work has been supported by the Spanish Ministry of Science and Technology; project PPQ2002-04201-C02. Symbols A C
— Area, m2 — Friction coefficient, adim
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h k P r R v z
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— Gap between the disk and the membrane, m — Velocity factor vθ,core/ω r, adim. — Pressure, Pa — Radial distance, m/radial component — Radius, m — Velocity, m/s — Normal component to the membrane, m
Greek δ μ ν θ ρ τ ω Ω
— — — — — — — —
Boundary layer thickness, m Dynamic viscosity, kg/m/s Kinematic viscosity, m2/s Tangential component Fluid density, kg/m3 Shear stress, Pa Angular velocity, rad/s Rotating speed, rpm
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