1. Introduction
World annual freshwater extraction for municipal, industrial, and agricultural needs is approximately 3928 km
3 y
−1 [
1]. Some 44% of this water (1716 km
3 y
−1) is consumed, mainly for irrigating farmland (38% of freshwater extraction), and the remainder (56%, 2212 km
3 y
−1) is primarily released to environment in the form of wastewater, industrial effluents or agricultural drainage water [
1]. The expected change in future rainfall scenarios as well as the increase of water demand for socio-economical needs require the implementation of adaptation measures in some areas, as in the Mediterranean Basin [
2].
Water reuse from wastewater treatment plants appears as a suitable strategy to mitigate the stress on natural resources, with a water quality level depending on the end-user types [
2]. Reclaimed water for crop irrigation must meet a minimum of quality standards to avoid any health hazard, but the cost of additional treatments and wastewater discharge taxes may compromise its financial viability [
3]. However, drip irrigation methods are very efficient in water consumption terms [
4] and avoid the contact between plant leaves or fruits and wastewater, which is advisable to further reduce contamination risks in some crops [
5]. Therefore, drip irrigation with reclaimed water might be an appropriate method to cope with forthcoming water scarcity issues.
One of the main problems of the drip irrigation technique is the emitter clogging that raises maintenance costs and reduces crop productivity [
5]. The filtration process of treated water may effectively prevent drippers to clog, thereby becoming essential for sustainable irrigation with non-conventional water resources [
6]. The removal of particles and microorganisms and the decrease in turbidity may be satisfactorily achieved with pressurized porous media filters working under deep bed filtration conditions [
7]. These porous media filters use a given amount of granular material like sand or crushed glass so as to create a bed inside a pressurized tank [
8]. Underdrain elements with slots smaller in width than the grain size are located at the base of the packed bed to avoid loss of granular material [
9]. In filtration mode, pumping work is needed to achieve a nominal flow rate, being proportional to the pressure drop through the filter [
10]. Total filter pressure drop increases as particle retention in the packed bed progresses, eventually reaching a threshold value beyond which the backwashing regime begins [
11]. In the latter mode, pumping work more intense than that of filtration is required to develop fluidization of the granular bed assuring particle removal [
9]. As a consequence, an entire filtration cycle involves energy consumption that becomes a setback to drip irrigation implementation. The reduction of filter head losses would imply longer filtration mode runs and, hence, improve energy efficiency.
Filter pressure drop is the addition of purely hydraulic head losses (caused by major flow friction losses and minor losses from auxiliary elements such as diffuser, underdrains, etc.) plus the energy losses through the porous media [
12]. Redesigns of current filters to enhance energy efficiency require models that correctly capture the flow behavior in all regions and, especially, in the packed bed. Erdim et al. [
13] carried out a comprehensive study of the prediction capacity of 38 pressure drop-flow rate correlations found in the literature that applied to porous media formed by spheres of single diameter. The range of applicability of those expressions varied depending on the modified particle Reynolds number
, with many of them sharing
intervals. Erdim et al. [
13] experimentally found that the well-known Ergun equation failed to predict the energy losses at
500, and proposed a new correlation that better fitted data.
A relevant point related with many of those previous expressions was that the pressure drop per unit length was expected to be valid for a region with uniform flow within the packed bed. However, this is not always accomplished in commercial filters since the effect of having underdrain slots with an open area lower than that of the cross-sectional filter implies a flow concentration towards the drainage [
14]. Arbat et al. [
12] and Pujol et al. [
15] aimed to solve this problem by including non-uniform flow corrections to analytical pressure drop equations within the granular packed bed. The modifications of the equations took into account the variation in the number of water channels available when approaching the underdrain element. However, those expressions were not validated locally but, once added to the rest of the momentum sinks, compared globally with measured data of the total filter pressure drop.
Besides analytical expressions, several authors have applied numerical methods to simulate the flow behavior of pressurized porous media filters. The computational fluid dynamics (CFD) technique based on the finite volume method uses a momentum sink to determine the pressure drop per unit length within the packed bed [
16]. This methodology has allowed the analysis of both laboratory and commercial filter type designs. Bové et al. [
17] employed the CFD technique to corroborate the superior performance of a modified underdrain that reduced the pressure drop mainly by increasing the zone with uniform flow within the granular bed. The same authors used simulations to discuss the reasons of having different data values of the total filter pressure drop of two underdrain designs [
18]. Mesquita et al. [
19] confirmed by means of CFD analyses of three different types of underdrains that the best hydraulic behavior corresponded to the design with the highest flow uniformity within the porous media. More recently, Mesquita et al. [
20] modeled different designs of a diffuser plate, finally proposing one that increased the flow uniformity above the sand surface. The validity of this design was experimentally confirmed [
20]. Also based on conclusions from CFD analysis, Pujol et al. [
21] proposed a redistribution of underdrains in a commercial sand media filter that reduced the total filter pressure drop by 5.8%.
However, a drawback of previous CFD studies is that they relied on expressions to estimate the momentum sink within the packed bed extracted from experimentally adjusted data that, in some cases, were not totally independent. For example, [
17,
18] assumed a pressure drop per unit length in the porous region that followed a quadratic function in terms of superficial velocity and whose coefficients were determined from pressure measurements in a slab within the granular bed under those conditions simulated. Analyses found in [
14,
21] also calibrated the momentum sink coefficients of the pressure drop equation employed in the CFD for the porous media by minimizing the error with respect to experimental data. On the other hand, though numerical analyses in [
14,
17,
18,
19,
20,
21,
22] were correctly set up for the objectives proposed, the level of discretization applied in the numerical models was somehow limited to include the recommended growth of layers of prismatic elements at certain walls only, ignoring those small walls of the underdrain element.
From the above, several questions may arise: can common pressure drop-flow rate correlations for packed beds correctly predict the total filter pressure drop even though there exist regions near the underdrain with high values? To obtain a successful CFD simulation, is the common procedure of fitting coefficients of empirical pressure drop equations for the porous media a consequence of not using a massive mesh? Is there a physically based equation with coefficients a function of flow and granular media properties that can accurately describe the pressure drop within the entire packed bed? Can we derive a simple analytical (i.e., closed-form) expression to determine the pressure drop in the non-uniform flow region of the packed bed? The present work was intended to answer the previous questions by analyzing the results of a numerical model of a pressurized packed bed filter that used a commercial type underdrain and comparing them with experimental data.
4. Discussion
Figure 8 and
Figure 10 confirmed the departure from uniform conditions as the flow approached the underdrain element. Arbat et al. [
12] and Pujol et al. [
15] proposed a simplified pressure drop model to take this effect into account. In their analytical model, the granular bed was divided into two regions: region 1 far from the underdrain in which the flow was essentially uniform, and region 2 close to the underdrain with a non-uniform flow behavior. The pressure drop formulation applied to region 1 followed Ergun Equation (10) being understood as the sum of the Poiseuille friction loss term (linear with
) as fluid flowed through the water channels in the packed bed plus a minor loss term (proportional to
) due to sudden expansions and contractions as channels were not straight (see [
27]). In region 2, both major and minor loss terms were modified according to a reduction in the number of channels available. In region 2, [
12,
15] assumed a linear decrease in the available diameter for the water flow from
(=200 mm; filter diameter) to
= 27.2 mm; equivalent diameter of the slots open area). The extent of region 2 was calculated by matching the virtual surface area of a scaled underdrain to the filter cross-sectional area
. The perpendicular distance from this virtual surface to the real underdrain was 18 mm for the design here employed.
However, the information from velocity vectors and streamlines nearby the pod observed in
Figure 11 suggested that the flow was initially directed towards the region occupied by the entire pod surface area and not specifically towards the region occupied by the slots only. Flow trajectories deviated to reach the slots open area at the very end of their paths. Thus, we assumed that in the non-uniform zone, the fluid did not flow through a region with cross-sectional areas linearly decreasing from
to
but linearly decreasing from
to
along a distance
and linearly decreasing from
to
along a distance
, with
(=94.2 mm) the equivalent diameter of the lateral surface of the pod-type underdrain (see
Figure 12).
Therefore, the effect of the non-uniform flow zone in the packed bed was calculated by expressing the pressure drop per unit length
in terms of the volumetric flow rate
, since it was a magnitude conserved along any cross-sectional area of
Figure 12. Here, the length
corresponded to the distance traveled along the streamline starting at the slots and ending at the top of the sand region. Thus, from Equation (9):
where
was the diameter of the available cross-section, being calculated as:
with
,
and
, and
m,
and
(see
Figure 12). Therefore, Equation (13) for
and
applied to regions A and B in
Figure 12, respectively.
By substituting Equation (13) into Equation (12) we found:
whose solution was:
with
the pressure at
,
(
the magnitude of the superficial velocity at
(or, equivalently, at
) and
the diameter ratio defined as:
Note that in the limit = 1, Equation (15) reduced to Equation (9) valid for uniform flow, as it should.
Results of the analytical expression (15) applied to evaluate the pressure drop through the path of a streamline compared with those extracted from the simulation and with the prediction of the analytical model proposed in [
12,
15] are shown in
Figure 13. Results referred to streamlines #4 and #8 in
Figure 11 with the parameters corresponding to the Ergun model with
0.2 L s
−1 or
0.8 L s
−1 for SS1 and MS media and two different heights
of the granular bed. The
value was chosen as half the minimum lateral distance between two consecutive slots, being
1.5 mm. The
value was calculated in a similar way as the non-uniform length
in Pujol et al. [
15] but scaling the pod in the radial direction only since the commercial underdrain here used did not have slots in its top surface. This procedure assumed that flow non-uniformity was essentially caused by the effect of the pod openings, reaching a distance of influence equal to
30 mm in our case. However, for streamline #4 in
Figure 11, we added an additional 10 mm to the previous value of
to take into account that its path was not perpendicular to the slots (and, therefore, it did not travel a distance
30 mm within the non-uniform region) but clearly inclined (
30 mm).
The analytical results from Equation (15) remarkably reproduced the pressure data along the entire path of the streamlines for all packed bed heights, flow rates and porous media types. In contrast, the analytical model in [
15] only explained the behavior found in the uniform flow zone. Thus, simulations were almost matched by the analytical model (Equation (15)) that constrained the fluid to flow through a porous domain consisted of a cylindrical region (uniform zone) plus two consecutive contracted cones till reaching the slots (non-uniform zones; see
Figure 12). The analytical formulation saved all computational costs related to CFD modeling (time and economical). However, its main shortcoming was the requirement of properly estimating the critical distances
and
. Pressure drop values from other underdrain designs would be needed to evaluate these distances in order to confirm the validity of the criterion applied here, which was based on geometrical considerations only.
On the other hand, from
Figure 5,
Figure 6 and
Figure 7, we observed that filter pressure drop
using the Ergun equation for porous media correctly reproduced experimental data for the whole range of superficial velocities tested when using silica sand but not microspheres. In the latter case, simulations substantially overpredicted
at high flow rates. The comprehensive study carried out in [
13] about the pressure drop-flow rate correlations for packed beds of spheres concluded that Ergun equation overestimated the pressure drop when
. For the microspheres used in the present study, this condition was attained when
m s
−1. At
L s
−1, that value of superficial velocity was not reached in the uniform zone (C in
Figure 12), but it was in the non-uniform zone (A in
Figure 12). Therefore, we simulated the microsphere case with a momentum sink term that followed the pressure drop correlation proposed by Erdim et al. [
13]:
This expression was found to better represent the head loss of flows through packed spheres at high particle Reynolds number than the Ergun equation [
13].
Simulations carried out at
L s
−1 and
mm using Equation (17) instead of Equation (10) obtained a reduction in
of 0.50 kPa, though closer to observations than Ergun model, still higher than the measured value (see
Figure 7).
Figure 8, and the streamlines analysis above, confirmed the strong contribution of the non-uniform flow in the packed bed to the filter pressure drop. In this region, flow velocities were higher than superficial velocities employed in
Figure 2. Thus, values of pressure drop per unit length in the non-uniform region would correspond to extrapolated values (at higher velocities) of the regression fits shown in
Figure 2. A qualitative observation of the predicted curves in
Figure 2 for the quadratic expressions (Ergun and Darcy-Forchheimer) indicated that they almost exactly followed measured data for silica sand media (SS1 and SS2) but had a trend with a higher slope than the experimental data (though still within the error bars) for microspheres (MS). Therefore, the deviation when extrapolating the data trend would be expected to increase, and so the head loss predicted in the non-uniform zone.
A careful analysis of microspheres data obtained in the laboratory revealed some discrepancies between pressure drop per slab of height equal to 100 mm within the sand region (case
317 mm). As it was shown in
Figure 1a, the laboratory filter had pressure sensors vertically separated 100 mm between them (i.e., sensor p1 at height above the base plate
= 100 mm, sensor p2 at
= 200 mm and sensor p3 at
= 300 mm).
Figure 2 took into account pressure drop data between sensors p3 and p1.
Figure 14 show data of pressure drop between sensors p3 and p2 and between sensors p2 and p1. We expected similar pressure drop data for both slabs since, from
Figure 10, the level at
= 100 mm was lying in the uniform flow zone. However,
Figure 14 indicated lower pressure drop values in the slab from 200 mm to 100 mm than in the slab from 300 mm to 200 mm at high flow rates. This could be a consequence of an incipient particle retention in the uppermost packed bed layer. Though the test was carried out with tap water and porous media were washed prior to use, it might have occurred that some small particles were remaining in the system. This phenomenon was not observed in SS1 and SS2 cases, in which pressure drop data series for both slabs (
and
were almost indistinguishable. The above point might be the reason that Darcy-Forchheimer equations overestimated the filter pressure drop for microspheres. However, Ergun equation for microspheres did not rely on any fitting with experimental data since we assumed a coefficient of sphericity equal to one. From
Figure 2, the Ergun prediction for the momentum sink tended to overestimate the observed slope at higher velocities, which magnified the contribution of the non-uniform flow zone to the total pressure drop. An assumption of a sphericity value less than one for microspheres would not necessarily improve the prediction from the Ergun Equation. However, the use of the Ergun equation for the porous media correctly predicted the total filter pressure drop for the range of superficial velocities (
50 m h
−1) in which the measured pressure difference at the two 100 mm vertical slabs (
Figure 14) were equal (overlapped error bars).
Finally,
Table 6 summarizes the main advantages and disadvantages found when applying the momentum sink equations to evaluate the pressure drop in porous media. All comments are exclusively related to the findings of the present study. Threshold values of the superficial velocity beyond which either Kozeny-Carman or Darcy equations underpredicted the pressure drop is not included in
Table 6 since it was a function of porous media type and packed bed height (see
Section 3.1).
5. Conclusions
We carried out CFD simulations of a pressurized granular bed filter with a commercial underdrain unit with the finest discretization found in the literature (16 × 106 elements required to mesh half of the laboratory filter). A minimum number of 12 elements across the slots (0.45 mm width) of the pod-type underdrain was used. Five types of equations to evaluate the pressure drop in the porous media were investigated: linear in (Darcy and Kozeny-Carman expressions), quadratic in (Darcy-Forchheimer and Ergun expressions) and a power function model in . Results were compared with experimental data obtained at different superficial velocities for three porous media (microspheres and two types of silica sand) and two different heights of the granular bed.
Total pressure drop results when applying all the previous porous media equations were very similar and lied within the experimental uncertainty range at least up to 38.3 m h−1 for both silica sand media with 117 mm and 317 mm. This superficial velocity corresponded to values ranging from 6.9. to 982 depending on the media used, being figures well above the recommended values for applying the linear Equations (Darcy and Kozeny-Carman).
Filtration modes with higher superficial velocities increased the discrepancies between models. The Ergun equation, based on flow and granular media properties, better followed the observed trend for both silica sand media and packed bed heights (NSE coefficient > 0.978 and RMSE < 1.3 kPa with respect to measured data) than using momentum sink expressions with terms fitted from experimental data. However, microsphere datasets were not accurately described by adopting pressure drop models with quadratic terms at high superficial velocities, likely due to some data inconsistencies for that range.
A detailed analysis of simulated data revealed important head losses inside the underdrain, mainly due to the sharp entrance into its exit pipe and the high velocity reached. For packed beds with low height 117 mm, these minor and major hydraulics losses can represent more than 40% of the total filter pressure drop, so the inner design of the underdrain element is of paramount importance to reduce filter energy requirements.
Simulations also reported the influence of the non-uniform flow zone within the porous media. In quadratic models (Darcy-Forchheimer and Ergun), this region accounted for more than 40% of the total filter pressure drop at low granular bed heights. This confirmed the relevance to correctly design the water drainage so as the maximize the flow uniformity within the porous media.
An analytical model that took into account this non-uniform region within the packed bed was developed. The model divided the granular media intro three regions. The upper part with uniform flow, a second non-uniform zone where the flow was directed to the underdrain and a third non-uniform zone at very short distance from the slots where the flow finally moved towards the underdrain openings. Pressure drop values along streamlines obtained from the analytical model almost exactly matched CFD results, substantially improving previously proposed equations. The extent of non-uniform regions was derived from geometrical considerations of the underdrain element. However, further analyses with different underdrain designs would be needed to generalize the validity of the tree-zone analytical model.