An experimental and analytical study to calculate pressure drop in sand filters taking into account the effect of the auxiliary elements

Transcription

1 Ref: C0405 An experimental and analytical study to calculate pressure drop in sand filters taking into account the effect of the auxiliary elements Gerard Arbat, Jaume Puig-Bargués, Miquel Duran-Ros and Francisco Ramírez de Cartagena, Department of Chemical and Agricultural Engineering and Technology, University of Girona, C. de Maria Aurèlia Capmany, 61, Girona, Catalonia, Spain Toni Pujol and Lino Montoro, Department of Mechanical Engineering and Industrial Construction, University of Girona, C. de Maria Aurèlia Capmany, 61, Girona, Catalonia, Spain Javier Barragán, Department of Agricultural and Forestry Engineering, University of Lleida, Av. de l'alcalde Rovira Roure, 191, Lleida, Catalonia, Spain Abstract Sand filters are frequently used in micro-irrigation especially when water contains large amounts of organic contaminants. This type of filter has the advantage of its simplicity and that the main filtration mechanism is based on depth filtration, giving an additional removal capacity in comparison with screen or disc filters, which essentially work by surface filtration. As the prediction of the head losses is of practical interest for the design of the microirrigation systems, the main objective of this paper is to present and validate an analytical model to compute the head losses produced by a sand filter, which takes into account the effect of the filter underdrain. It has been shown that the classical Ergun equation works well for predicting the pressure drop within a sand bed when it is sufficiently far from the underdrain, but it fails in the region immediately next to this element. To overcome this problem, a model based on a set of connected channels of the same diameter and progressive reduction of its number as the flow approaches the nozzles is presented. An experimental study with a scaled commercial sand filter was conducted with two different media bed weights (3.86 and kg) and two sand grain size ranges ( and mm) in order to compare the reliability of the proposed analytical model. The experimental results were compared with those yielded by the proposed analytical model, as well as with the Ergun equation. Statistical indexes show that the proposed model improved the results of the Ergun equation. Thus, the root mean square error (RMSE) comparing the results of the proposed model with experimental data ranged from 2.85 to 7.16 kpa, while when the results of Ergun equation were compared with the experimental data the RMSE ranged from to kpa. It can be concluded that the new analytical model improves the results of the Ergun equation to predict the pressure drop produced by the entire sand bed by taking into account the effect of the underdrain (nozzle-type) and can be applied to accurately predict the pressure drop in sand filters. Keywords: drip irrigation; head losses; sand media filter; packed bed; mathematical modeling Proceedings International Conference of Agricultural Engineering, Zurich, /8

2 1. Introduction Water quality is a major concern in the management of drip irrigation systems as the solids contained in the irrigation water can clog the emitters. Thus, filtration is essential for the successful operation of micro-irrigation systems. Media filters are commonly used for protecting micro-irrigation systems and are specially recommended when large amount of organic contaminants are present (Duran-Ros, Puig-Bargués, Arbat, Barragán, & Ramírez de Cartagena, 2009) Few studies have analyzed the effect of the auxiliary components of media filters on the head loss, which is related with water and energy consumption as well as filter efficiency (Burt, 2010; Mesquita, Testezlaf, & Ramírez, 2012). The Ergun equation has been extensively used to predict head losses in the filter media (Macdonald, El-Sayed, Mow, & Dullien, 1979) since there is a general consensus on its accuracy. However, its application is strictly valid in the limit of an infinitely extended packed bed (Nemec & Levec, 2005). The boundary condition at the sides of the packed bed, known as the wall effect, has been studied by numerous investigations. These studies demonstrate that when the bed diameter is not significantly larger than the particle diameter the friction produced by the wall of the tank is not negligible, invalidating the applicability of Ergun equation (Foumeny, Benyahia, Castro, Moallemi, & Roshani, 1993). The boundary condition near the filter underdrain also violates the assumption of an infinitely extended packed bed made by the Ergun equation. The capillary model based on the hydraulic radius concept, the physical model in which is based the Ergun equation, assumes that porous media consists of a set of parallel identical channels (Nemec & Levec, 2005). This simplified model of the flow within the porous media may fail to represent the actual flow occurring near the underdrain elements since the streamlines must converge near the passing area through the slots of the underdrains. Indeed, different empirical studies have shown that an important part of the total pressure drop in commercial sand filters is due to the underdrain system (Burt, 2010; Mesquita et al., 2012). From a previous computational study Arbat et al. (2011) showed that a great part of the head losses in a commercial sand filter are produced in a tiny area inside the packed bed close to the underdrain. Their results reveal the importance of properly including the underdrain system when developing models for predicting head losses in packed beds. Nevertheless, it must be noticed that the application of computational fluid dynamic (CFD) software to simulate the pressure drop in a sand filter has strong limitations since it requires an enormous computational force to discretizate the complex geometries of the sand filter. Arbat et al. (2013) developed an analytical solution that can be applied to compute the head losses produced by the underdrain of a porous media filter. The main objective of this paper is to quantify the degree of agreement, using statistics of comparison, of the pressure drop predicted with the analytical model presented in Arbat et al. (2013) as well as with Ergun s equation with experimental results obtained in a commercial scaled sand filter in order to analyze the convenience to use the new equation to predict head losses in sand filter with nozzle-type underdrains. 2. Materials and methods 2.1. Experimental procedures A scaled sand filter was constructed based on a commercial sand filter (Regaber, Parets del Vallès, Spain) of 500 mm internal diameter and with a filtration surface of 1963 cm 2. The dimensions of the scaled filter were chosen such that the Reynolds number corresponding to the circular tank of the filter was similar to that of the commercial filter. Since the commercial filter had 12 nozzles and the scaled filter 1 nozzle, the scaling factor for all lengths was approximately 1/12 1/3. Therefore the volume flowing through each nozzle was the same in Proceedings International Conference of Agricultural Engineering, Zurich, /8

3 both filters. A detailed description of the experimental filter and experiment can be found in Arbat et al. (2013). The relationship between superficial velocity (V 0 ) and the pressure difference from the filter inlet and outlet (D p =p in -p out ) was obtained at the filtration regime for different filter media configurations. Each experiment was repeated 3 times to ensure repeatability of the data. Two different sand particle size ranges ( mm and mm) and three different sand bed weights (or, equivalently, heights) were tested at the filtration regime (Table 1). The main characteristics of the sands are shown in Table 2.These tests were the same showed by Arbat et al. (2013). The underdrain configuration was a nozzle of identical characteristics than that of the commercial sand filter (Regaber, Parets del Vallès, Spain). The nozzle had 45 slots 0.45 mm wide and 30 mm long with a total passing area of mm 2. The passing area through the orifice was 415 mm 2. As the scaled filter has only one nozzle and 200 mm internal diameter, each nozzle served an area of 314 cm 2. The experiments with kg of sand represent approximately the 1/12 of the volume of sand in the commercial sand filter that was taken as a reference and, therefore, correspond to the nominal column height of the scaled filter. The tests with kg of sand with the nozzle gave a media bed depth of 20 mm over the mid height of the nozzle, which represents a very short sand column in relationship with the experiments with kg of sand. The experiments with the short sand column were designed to analyze the pressure drop generated in a tiny area over the underdrain, and to compare their results with those obtained in the analytical model. Additionally, experiments with an intermediate media bed depth corresponding to kg of sand were carried out. The difference of the experimental pressure losses obtained in the experiments with and kg of sand would allow verifying the pressure losses calculated with the Ergun equation in a 0.21 m long sand column without the influence of the underdrain system. Similarly, the difference of the pressure losses obtained with and kg of sand would serve to know the pressure losses in a sand column 0.26 m long Nozzle type underdrain analytical model The model presented in Arbat et al. (2013) is based on a series pressure drop representation from point 1 located at the inlet of the filter to point 2 located at the exit filter. The different regions of the filter are shown in Fig. 1. Thus, the measured total pressure loss p 12 follows: p 12 = p s + p ns (4) where p s and p ns are the pressure losses through the porous media and the non-porous media, respectively. p ns can be divided into three main contributions p ns = p wi + p o + p we (5) where p wi is the pressure drop from point 1 to the top of the sand column (water inlet region), p o is the pressure drop through the nozzle orifice and p we is the pressure drop from the exit of the nozzle orifice to point 2 (water exit region). Note that these terms may include losses from auxiliary elements (e.g., backflush valves). On the other hand, the pressure loss through the sand bed p s is divided into two terms p s = p si + p sii (6) where p si,ii correspond to the pressure losses in regions I and II shown in Fig. 1 In region I, the flow is uniform through the sand bed and the pressure drop follows the Ergun equation. In region II, however, the flow is not uniform due to the influence of the nozzle. Since the drain opening area of the nozzle is smaller than the filter cross-sectional area, the fluid velocity within the sand increases as it approaches the nozzle. This leads to large values of pressure losses in a very small zone close to the nozzle. In this region, the Ergun equation does not provide reasonable predictions. This effect was already noted by means of detailed Proceedings International Conference of Agricultural Engineering, Zurich, /8

4 CFD simulations of a commercial filter in Arbat et al. (2011). A new expression for determining the contribution of the p sii term was developed by Arbat et al. (2013) based on a generalization of the idealized representation of the porous media. The porous media is assumed of being formed by inter-connected cylindrical channels of equivalent diameter D eq through which water flows. Analytical expressions for all the terms shown in equations (4-6) can be seen in Arbat et al. (2013). In the present work we compare the results obtained with these equations with the ones obtained in the experimental tests described in section 2.1, and also with the ones obtained with the classical Ergun s equation Statistics of comparison between the head loss determined experimentally and predicted by the models In order to determine the agreement of the analytical model, the root mean square error, RMSE and the Willmott s index of agreement, d (Willmott, 1982) were computed for the pressure drop at the different velocity regimes during the experimental tests. The statistics were computed as follows: RMSE n i1 n P O i n i 2 Pi Oi i1 d 1 (8) n 2 Pi O Oi O i1 being O, the pressure drop obtained from the experimental tests at a particular superficial i velocity, P i the predicted pressure drop at the same regime, O the average pressure drop at the different tested regimes. Therefore, the units of the RMSE are kpa, d has no dimensions and its upper limit is 1, which indicates a perfect agreement between the analytical solution and the experimental tests. 3. Results and Discussion 2 (7) 3.1. Proportion of head losses in the different parts of the filter The pressure drop in a sand bed column 0.21 m long filled with sand size ranging from 0.63 to 0.75 mm was calculated as the difference between the pressure drop measured with kg and that with kg of the indicated sand sizes (Table 1). Similarly, it was calculated the pressure drop for the same sand bed height with mm sand size. The pressure drop in a sand bed column 0.26 m long filled with mm sand size was calculated doing the difference between the pressure drop measured with kg and that with kg. Using this procedure, the pressure losses in the auxiliary elements of the fitler, as well as those produced in the sand bed closest to the nozzle cancelled out. Therefore, the pressure drop can be compared with the one predicted with Ergun equation. Figure 2 shows that the pressure drop predicted with the Ergun equation follows a similar trend than that obtained in the experiments although it slightly over-predicts the pressure drop for the three cases. In both experimental and Ergun's predictions, the lower pressure drop was produced by the sand column of 0.21 m height and the bigger grain size ( mm), followed by the sand column of the same height filled with grain sizes from Proceedings International Conference of Agricultural Engineering, Zurich, /8

5 mm. The greatest pressure drop was produced by the longer sand bed (L=0.26 m) filled with mm sand size. As generally admitted, Ergun equation is suitable to predict pressure drop in infinitely extended packed bed, especially when the porosity range is rather narrow (0.35<ε<0.55), the bed is made up of similar sized particles and the flow rates are moderate (Nemec & Levec, 2005), which was the case of the sand filter experiments presented here. Applying the model developed by Arbat et al. (2013), the sand bed was responsible of most of the pressure losses, roughly 90%. Figure 3 shows that in the experiments with kg of sand approximately the 10% of the total pressure losses were produced in the region ranging from the top of the sand bed to a short distance from the nozzle (Region I) and 70% in a very tiny region at the bottom of the sand column, close to the nozzle slots (Region II). In the experiments with and kg of sand, approximately 35 and 50% of the total pressure drop was produced in regions I and II respectively. The extension of region II, where most of the pressure losses are produced, was initially unknown but the experiment with 1.54 kg of sand, where the sand column was as short as 2.5 mm over the top of the nozzle, would confirm that it is restricted to a very tiny area at the bottom of the sand column. Indeed, the experimental results corresponding to a 0.26 m sand column (Fig. 2), obtained by subtracting the measured values of the kg case with those of the kg of sand, did not show any abrupt increase of the pressure drop compared with the one obtained with a 0.21 m sand column and followed the good agreement with Ergun equation. Therefore it was experimentally confirmed that the length of region II is less than the 2.5 mm over the top of the nozzle Statistics of comparison of the predictions of the different models with the experimental ones The degree of agreement of the analytical model and Ergun s equation compared to the experimental results was evaluated using the Willmott s index of agreement (d) and the RMSE (Table 3). The results show that when the proposed analytical model, that takes the effect of the underdrain in the flow inside the porous medium (Region II), the difference between the model prediction and the experimental results was reduced greatly in all the cases compared with the results obtained with Ergun s equation. It must be noticed that when Ergun s equation was used to compute the pressure drop, the pressure losses produced by the auxiliary elements were taken into account and computed in the same way than in the proposed analytical solution. The results of the Willmott s index of agreement (d) also indicate an improvement of the predictions of the proposed model, being this index greater than 0.90 in all the comparisons with the proposed model while it ranged 0.60 to 0.80 when Ergun s equation was used. Therefore, the proposed model showed a closer prediction of the head losses than that of the Ergun s equation in the entire set of experimental tests. 4. Conclusions The Ergun equation is well suited to predict the pressure drop produced by an infinitely long packed bed. Nevertheless, the idealization of parallel channels, which is the physical model that supports Ergun equation, does not match the hydraulic behavior in the lower part of a filter since there exists an underdrain element (nozzle). The experiments carried out confirmed that the flow through the porous media is not uniform near the nozzle where streamlines converge towards its slots. In the idealized analytical model presented in Arbat et al. (2013), this effect was implemented by a reduction of the number of channels available as well as a reduction of the effective filter diameter. The statistics of comparison showed that the proposed model clearly improves Ergun s predictions, since remarkably fits experimental data. The analytical model presented in this paper could be applied to predict Proceedings International Conference of Agricultural Engineering, Zurich, /8

6 the pressure drop produced by commercial sand filters using nozzle-type underdrains, which are currently used in the filtration units used in micro-irrigation systems. 5. Acknowledgements The authors would like to express their gratitude to the Spanish Ministry of Economy and Competitiveness for its financial support of this study through grants CGL , and FIS , to the Autonomous Government of Catalonia for grant 2009-SGR-374. The authors would also like to thank Sergi Saus, Jordi Vicens, and Regaber for their help in carrying out this investigation. Figure 1: Sand bed regions considered to compute pressure drop in the model of Arbat et al. (2013). Figure 2: Experimental and Ergun's predictions of the pressure drop in the sand bed ( p, kpa) for different superficial velocities (m h -1 ). Proceedings International Conference of Agricultural Engineering, Zurich, /8

7 Figure 3: Percentage of the total pressure drop produced in the different regions considered to compute pressure drop in the model of Arbat et al. (2013) Table 1: Different filter configurations tested in the experiments with the scaled sand filter. Sand size (mm) Sand weight (kg) Media bed depth * (mm) * * Sand bed depths over the mid height of the nozzle. Table 2: Main characteristics of the two silica sand sizes used in the study. Grain size Average sand Bulk density Particle density Porosity ranges diameter (mm) (kg m -3 ) (kg m -3 ) (%) (mm) (16) 2556 (0) (0.00) (39) 2507 (6) (1.44) The number in brackets is the standard deviation of three different samples. Proceedings International Conference of Agricultural Engineering, Zurich, /8

8 Table 3: Statistics of comparison of proposed analytical model with the experimental results, and of Ergun equation with the experimental results. RMSE Willmott s index of agreement Grain sizes (mm) Sand weight (kg) Proposed analytical solution (kpa) Ergun s equation* (d), (adimensional) Proposed Ergun s analytical equation* solution * (*) When Ergun s equation was applied the pressure losses in the auxiliary elements were computed in the same way than in the proposed analytical solution. 6. References Arbat, G., Pujol, T., Puig-Bargués, J., Duran-Ros, M., Barragán, J., Montoro, L., & Ramírez de Cartagena, F. (2011). Using computational fluid dynamics to predict head losses in the auxiliary elements of a microirrigation sand filter. Transactions of the ASABE, 54(4), Arbat, G., Pujol, T., Puig-Bargués, J., Duran-Ros, M., Montoro, L., Barragán, J., & Ramírez de Cartagena, F. (2013). An experimental and analytical study to analyze hydraulic behavior of nozzle-type underdrains in porous media filters. Agricultural Water Management, 126, Burt, C. M. (2010). Hydraulics of commercial sand media filters tanks used for agricultural drip irrigation: criteria for energy efficiency. San Luis Obispo: Irrigation Training and Research Center. Duran-Ros, M., Puig-Bargués, J., Arbat, G., Barragán, J., & Ramírez de Cartagena, F. (2009). Effect of filter, emitter and location on clogging when using effluents. Agricultural Water Management, 96(1), Foumeny, E., Benyahia, F., Castro, J., Moallemi, H., & Roshani, S. (1993). Correlations of pressure drop in packed beds taking into account the effect of confining wall. International Journal of Heat and Mass Transfer, 36(2), Macdonald, I., El-Sayed, M., Mow, K., & Dullien, F. (1979). Flow through porous media-the Ergun equation revisited. Industrial & Engineering Chemistry Fundamentals, 18(3), Mesquita, M., Testezlaf, R., & Ramírez, J. (2012). The effect of media bed characteristics and internal auxiliary elements on sand filter head loss. Agricultural Water Management, 115, Nemec, D., & Levec, J. (2005). Flow through packed bed reactors: 1. Single-phase flow. Chemical Engineering Science, 60(24), Proceedings International Conference of Agricultural Engineering, Zurich, /8