A previously developed and validated two-dimensional computational fluid dynamics (CFD) model to study the hydrodynamics in a desalination membrane filled with spacers in zig-zag arrangements has been further developed to include the effects of a pulsating flow with the profile of a heartbeat. Numerical solutions were obtained with Fluent for pulsating laminar flows in channels filled with four different spacers and four lengths of cells. Hydrodynamics was investigated for unsteady state, using a characteristic function of a heartbeat, in order to study the influence of temporal variation in the hydrodynamic behavior. The results show the velocities distribution, streamlines, pressure drop and the wall shear stress on the impermeable wall of the membrane, for Reynolds numbers up to 100. The reduction in the distance between the filaments of the spacers, leads to the appearance of more active recirculation zones that can promote mass transfer and decreasing concentrations layers. On the other hand, this reduction increases the pressure drop and consequently the energy expended in the process. Further, the characteristic function of heartbeat demonstrates promising results, with regard to the energy consumption in the process and optimization of the recirculation zones.
- reverse osmosis
- zig-zag spacers
- heartbeat flow profile
The planet has not only freshwater, but from the enormous volume of water available on our planet, only 3% is not salty. This resource is limited and finite, vital for the existence of life on earth and for the economic and social development .
The exploitation of natural water resources combined with the increase in the world population, the changes in the life style, the inefficient use of water and its contamination, among other factors, leads to scarcity of natural freshwater to respond the necessities of the population . Due to these factors, the demand for potable water, has led to an increasing need to find new alternative sources of drinking water.
Desalination provides a good alternative to reduce the problem of the scarcity of drinking water, once it provides clean water which otherwise would not be accessible for agricultural, industrial and services supply [3, 4].
Membrane systems are widely used in water treatment processes. Depending on the pore size, membrane processes can be classified as microfiltration (MF), ultrafiltration (UF), nanofiltration (NF) and reverse osmosis (RO). The MF and UF are used to remove small colloidal particles and bacteria. For removal of viruses and large molecules such as proteins, the UF is only used. The NF and RO are capable of removing the smallest particles and components, like salts dissolved in water. In the case of RO, it is possible to remove almost all the components dissolved in the water .
The RO is the most used membrane separation method. This is applied at chemical, textile, petrochemical, electrochemical and paper industry, as well as in the treatment of municipal wastewaters . But, the major application of RO process is the desalination of brackish and salty water to obtain potable water . This is due to its versatility to operate over a wide range of feed water salinities, lower intake and pumping costs, minimal space requirements and operation at ambient temperatures . Spiral wound modules (SWMs) have been largely used in industrial applications to obtain potable water, due to their low cost and high membrane area to volume ratio . The performance of a SWM is affected by many factors such as leaf geometry (number and dimensions), spacers, fouling propensity, cleaning ability and operating conditions . Despite several benefits of membrane technology over other separation techniques, the phenomena inherent to any membrane separation method are concentration polarization and fouling. As the filtration process occurs though the semipermeable membrane, the rejected impurities begin to accumulate on the membrane surface. After a certain period of time, a concentration layer is formed on the membrane surface due to the solute. This phenomenon, in which, the concentration of solute is higher in the vicinity of the membrane as compared with the observed in the fluid, is known as concentration polarization [10, 11]. This phenomenon can significantly affect the performance of the membrane, for example, the permeate water quality and productively are affected by fouling deposited on the membrane surface. On the other hand, from the literature it is well known that spacers are essential to separate the sheets of membranes in both spiral wound modules based thin rectangular channels (slits). Spacers are also important in the optimization of the mass transfer enhancement and pressure loss minimization [12, 13].
One important way by which the efficiency of module can be enhanced is to modify the hydrodynamics in membrane channel, which can be done by improving feed channel spacers design and using the unsteady state to induce a temporal variation in the feed velocity. This can prevent the accumulation of solutes near the membrane surface, thereby lowering the concentration gradient between the bulk fluid and the membrane surface, and reducing fouling phenomenon .
The hydrodynamics inside the feed channel using computational fluid dynamics (CFD) has been investigated awhile. Cao et al.  investigated, trough CFD, the effect of three different cylindrical spacers arrangements: spacers touching the same channel wall, spacers touching opposite channel walls and spacers suspended in the channel. The flow simulations, in the spacer-filled channels, showed the distribution of velocity and turbulent kinetic energy as well as the detailed flow patterns. It was found that the presence of the spacer in the membrane is responsible for the local shear stress distribution on the membrane surface and improves the production of the eddy activities. It is also found that spacer configuration, geometry and position in the channel are crucial in the distribution of eddies and high shear stress on the membrane surface. The investigation also suggests that transverse spacer cylinders suspended in the channel are more desirable. Furthermore, decrease in the distance between transverse spacer cylinders can produce more active eddies and reduce the distance between shear stress peaks and consequently improve the mass transfer at the membrane surfaces. Besides that, it was demonstrated that the pressure drop is significantly increased by reduction in the distance between transverse cylinders which increase the operating costs. Schwinger et al.  perform CFD studies on how the use of spacers in membrane systems can affect its hydrodynamics. The flow fluid was studied in three different spacer configurations, the cavity, zig-zag, and submerged spacers. Were considered, in a cell, a single filament adjacent to the wall and centered in the channel for Reynolds numbers from 90 to 768 for different values of the length of the mesh and filament diameter. The results obtained show that the flow around the filament increases the wall shear stress and promotes the formation of large recirculation zones behind the filaments. Furthermore, under the same conditions, identical Reynolds number and filament diameter, a single filament adjacent to a membrane wall produced a larger recirculation zone than a single filament in the center of the channel. According to the authors, the utilization of the spacer in a SWM should enhance the mass transfer and reduce the accumulation of solute in the membrane surface while maintaining a low pressure loss along the channel. Saeed et al.  used a CFD tool to investigate the flow patterns associated with fluids within the membrane module. The effects on flow patterns through a spacer filled RO membrane with the secondary structure of the membranes (feed spacer filaments) at various angles with the inlet flow are analyzed. The results revealed that the alignment of the filaments in the direction of flow has a great influence on the generation of recirculation streams in channels filled with spacers. The optimization of the orientation of the filaments can lead to desirable recirculation patterns inside the modulus, and therefore, may increase the performance of the membrane. Sousa et al.  studied the hydrodynamics in a desalination membrane. CFD techniques were used to study the hydrodynamics of feed channels of a desalination membrane filled with elliptical spacers in zig-zag arrangements and transverse in relation to the flow, for four spacers and four cell lengths. They used
In the present investigation, a previously developed and validated two-dimensional CFD model to study the hydrodynamics in a desalination membrane filled with spacers in zig-zag arrangements , is further developed to include a characteristic function of a heartbeat within a cell of a semipermeable desalination membrane, in order to study the global and local impact of hydrodynamic behavior on the velocity, pressure drop, and wall shear stress for different spacer arrangements. The numerical approach used allows uncoupling of several phenomena and contributes to a better understanding of the mechanisms through which the shape of the flow profiles can affect membrane performance.
2. Governing equations and boundary conditions
An incompressible Newtonian fluid was used to simulate the bi-dimensional unsteady state flow in a feed channel of a SWM with zig-zag spacers (Figure 1). The governing equations are the continuity and Navier-Stokes equations. For a bi-dimensional flow, the governing equations can be written as:
In the present study, the Reynolds number is defined as
2.1. Inlet velocity profiles
In order to obtain a better numerical approach taking into account the flow periodicity, it was necessary to determine the velocity profile at the inlet and outlet of the cell . As far as our knowledge in previous investigations, for example Refs. [8, 17, 22], it was considered a fully developed velocity profile at inlet. However, the velocity profiles between the filaments are not fully developed because there is no enough distance between filaments to allow the full development of the fluid flow. Therefore, for a given average velocity, the inlet velocity profile was obtained using the following procedure: for the value of
For the case of impermeable walls, when the solution is achieved, the boundary condition at the inlet is replaced by the solution obtained for the velocity profile at the outlet boundary. This process was repeated until the differences between the inlet and outlet profiles were less than 0.15%. Similarly, in order to verify the permeability influence on the flow, a study was also done for a semipermeable membrane. To this, the bottom wall was considered permeable, with a constant flow rate of permeate. The value of permeate velocity (
where the second term on the right-hand side of Eq. (4b) is the velocity profile in the outlet boundary obtained for the case of impermeable walls, when the imposed inlet flow rate is equal to the permeate flow rate.
The time-dependent velocity at the inlet was obtained from Doppler ultrasound images for a heartbeat, in the abdominal aorta, with a period of
Then the normalized velocity
The no-slip condition was applied at the wall boundaries.
Note that in the presence of permeability, it is needed to differentiate inlet and outlet profiles because some fluid is permeated through the permeable wall, see Eq. (4b).
3. Results and discussion
In this study, two-dimensional simulations using different feed channel geometries filled with spacers (different cells lengths and different shape of the elliptical filaments) were performed. We studied the impact on the velocity field, streamlines, average wall shear stress (
3.1. Inlet velocity profiles
Since for normal operation of the SWM, the velocity profiles inside the cell cannot be fully developed due to the filaments periodicity, so that the inlet profiles used in this study were determinate by the aforementioned procedure. As example, Figure 4 shows the profiles determined for the four elliptical filaments with
The width profiles decrease from
3.2. Velocity field
Figure 5 illustrates the velocity field for
From Figure 5, it can be seen that the well-known velocity behavior, that is, for each instant, the maximum velocity inside the membrane is located in the zones between the filament and the adjacent wall, for example Refs. [17, 23, 24]. This behavior is explained by the channel narrowing due to the spacers, with the consequent velocity increase to ensure the mass conservation. From general analysis of Figure 5, we can see that the maximum of velocity occurs at T/8. From this instant, velocity decreases until reaches approximately half of the period (T/2) and after that remains almost constant until the start of a new cycle. A more detailed analysis reveals that the permeability has a very little impact in the velocity. For instance, at 5T/8, velocity achieves the minimum value of 0.213 m/s for
With increasing distance between the spacers, it is verified that for each instant, the average velocity within the cell decreases with increases of
3.3. Average velocity inside the cell
Figure 7 shows the average velocity inside the cell for both cases, impermeable and semipermeable membrane, for a cell with length
A comparison between average velocities for impermeable and semipermeable cases also shows that the biggest difference is of 4.49% and occurs at 3T/8, for
Figures 9 and 10 show the streamlines for
The recirculation zones are visible in all instants analyzed for all studied cell geometries. These zones are generally located downstream of the filaments, however, under specific conditions, small upstream recirculation zones can occur as shown in Figure 9 (a) at
We can infer that the length between filaments has no influence on the eddies formation. Therefore, for all conditions studied,
The variation of the length eddies during a pulsatile flow modeled by a heartbeat shape combined with spacer arrangement may lead to better performance of the desalination membrane. Thus, the formation of desirable flow patterns within the membrane can lead to its better performance, which are in line with literature, for example, Saeed et al. .
3.5. Pressure drop per length unit
Figure 11 shows the pressure drop per unit of length (Δ
The maximum value of Δ
On the other hand,
Figure 12 shows the pressure drop per unit of length (Δ
The behaviors and conclusions seen in Figure 11, on the pressure drop per unit length, are also observed for this case, where
The results also show that
In general, an energy gain is expected when an inlet heartbeat function is used, compared to sinusoidal functions because in heartbeat function the flow keeps small fluctuations, at least, during half cycle, so in this case the pressure gradients developed during a cycle are smaller than that for a sinusoidal function.
3.6. Wall shear stresses per unit length
Figure 13 presents, for
For studied conditions,
Figure 14 presents, for
Comparing the results obtained by Cao et. al.  and Sousa et al.  with the present results, one can infer that the use of inlet flux with heartbeat profiles can benefit the membrane performance because the concentration polarization may be reduced, improving the mass transfer, once it dynamically changes the boundary layer concentrations and prevents clogging of the membrane surface.
In this chapter, CFD techniques were used to study the hydrodynamics inside the feeding channel of a semipermeable membrane in spiral wound configuration. The main goal of this research was to investigate the effects of a pulsating flow with a profile of a heartbeat on the hydrodynamics of feed channels of a desalination membrane filled with spacers in zig-zag arrangements and transverse to the flow. Both membrane impermeable and semipermeable cases were analyzed. For the semipermeable case, we considered the membrane bottom wall with a typical permeation rate of 0.1 mm/s. The comparison between the two cases shows that the permeability has no significant influence on hydrodynamics. On the other hand, the permeability implies that there is a slight decrease in average velocity, Δ
Due to the existence of the filaments in the cell, the velocity profiles are not fully developed. Thus, depending on spacer type (
In summary, this study suggests that the inter-filaments length combined with the flow variation characteristic of a heartbeat can control the development of concentration polarization, and thus reduce the probability of fouling and energy consumption in the process, through an optimization of the recirculation zones.
This work was supported by the Portuguese Foundation for Science and Technology under UID/SEM/04252/2013
|h||Cell height (m)|
|hf||Filament height (m)|
|g||Gravitational acceleration (m s−2)|
|L||Cell length (m)|
|p||Static pressure (Pa)|
|Δp||Pressure drop (Pa)|
|v¯||Average inlet velocity (m s−1)|
|vp||Permeate velocity (m s−1)|
|vh(t)||Normalized inlet velocity|
|vx||Velocity component in x-direction (m s−1)|
|vy||Velocity component in y-direction (m s−1)|
|x||Longitudinal rectangular coordinate (m)|
|y||Transverse rectangular coordinate (m)|
|ρ||Fluid density (kg m−3)|
|µ||Dynamic viscosity (Pa s)|