# Dimensional Analysis Model of Head Loss for Sand Media Filters in a Drip Irrigation System Using Reclaimed Water

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

^{−1}), $\mu $ is water viscosity (Pa s), $\Delta H$ is the head loss (Pa), $C$ is the TSS concentration (kg m

^{−3}), ${D}_{p}$ is the filter inlet or the outlet diameter, and $\rho $ is the water density (kg m

^{−3}).

_{e}is the sand’s effective diameter in m, which refers to the sieve diameter that allows 10% of the total filter sand to pass through during the sieving process, and it reflects the particle size of the filter sand. ${d}_{f}$ is the internal diameter of the sand media filter, and $V$ is the volume of water filtered (m

^{3}).

_{s}is the ratio between the ${d}_{e}$ and ${d}_{60}$ which refers to the sieve diameter that allows 60% of the total filter sand to pass through the sieve during the sieving process. The higher the $U{C}_{s}$, the more rational the filtration media gradation and the better the filtration efficiency [13]. However, no parameters of sand-filtration media were considered in Equation (1). Although the effects of particle diameter and porosity of sand-filtration media were considered in Equations (2) and (3), the effect of $U{C}_{s}$ was not considered, and the effect of the concentration of suspended solids was not considered in Equation (3). Other models have used the thickness of the pollutant aggregation layer [26,27] or the mean diameter of suspended solids [18] as parameters. However, because these parameters are difficult to measure, their application in engineering practice is restricted.

## 2. Materials and Methods

^{−1}, with a mean value of 21.4 mg L

^{−1}. The sand media filters (model SS-400 × 50) used were manufactured by Beijing Tongjie Company (Beijing, China) with a tank body diameter of 40 cm, a water inlet diameter of 50 mm, and a designed range of flow rate of 5.0–18 m

^{3}/h. As shown in Figure 1, three sand media filters were installed in the testing system. A pressure sensor (accuracy: 0.001 MPa) was installed before and after each sand media filter. A flow sensor (accuracy: 0.1 m

^{3}) was installed at each filter inlet. The system flow rate and pressure data were collected in real time every 5 min using a computer.

## 3. Model Building

#### Generating Dimensionless Parameters

_{p}are usually used to characterize the special nature of the quality of reclaimed water; of these two parameters, the TSS concentration is used more often because it is easier to measure and usually has higher predictive accuracy [18,23,24]. d

_{p}is rarely used because it is difficult to measure. In the model proposed by Puig-Bargués [18], there was an overlap between the volume of filtered liquid $V$ and the filtered liquid flow rate $Q$. The selected parameters in the dimensional analysis should be closely correlated while avoiding overlap. The removal of suspended particles by the sand media filter mainly depends on the effect of sedimentation and interception [19]. The head loss is affected primarily by the interception of suspended particles and by the media structure. Therefore, an accurate description of the physical characteristics of the sand-filtration media and the accumulation of suspended particles is critical for the accuracy of the head loss calculation model for sand media filters.

^{−3}), acceleration of gravity $g$ (m s

^{−2}), filtration velocity $\nu $ (m s

^{−1}), pollution load $D$ (kg), and the rate of change of head loss $\Delta H$ (Pa), were selected.

^{−3}) in period i in the backwash cycle period $t$. ${Q}_{t}$ is the volume of filtered liquid in period i within backwash cycle $t$ (m

^{−3}).

## 4. Statistical Analysis and Model Validation

_{3}and ln π

_{4}, suggesting a significance level of correlation of p > 0.05. Therefore, $\mathrm{ln}{\pi}_{3}$ and $\mathrm{ln}{\pi}_{4}$ were excluded. The significance level of constants $\mathrm{ln}{\pi}_{5}$ and $\mathrm{ln}{\pi}_{6}$ in Table 6 was p < 0.001, and the significance level of $\mathrm{ln}{\pi}_{1}$ and $\mathrm{ln}{\pi}_{2}$ was p < 0.05, suggesting that the pollution load $D$ had more significant effects on dependent variables than the filtration velocity $\nu $ or particle diameter parameters ${d}_{60}$.

_{7}and the calculated value from Equation (7) (p < 0.01). Figure 3b shows that the residues have a relatively even distribution of approximately 0. Moreover, the significance level of the model reached p < 0.001 (as shown in Table 5), suggesting that the model derived with dimensional regression analysis could be used to calculate the head loss for a sand filter using reclaimed water in irrigation. The model may give satisfactory predictions within the range of operational and filter structure parameters, which is valid for the following conditions:

## 5. Results and Discussion

^{−3}for Equation (1) [23], 3.8–68.6 g m

^{−3}for Equation (2), and 3.5–88 g m

^{−3}for Equation (3), which was used to calculate the head loss of the clean filtration layer [24]. The TSS concentration differences suggest that there are differences in the number of suspended particles and their diameters; these differences further suggest that the differences in the removal efficiency of the filtration system [18] might cause differences in the sedimentation and interception of suspended particles in the pores of the sand media, resulting in head loss differences, which might cause the model to have low accuracy.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The measured value of ${\pi}_{7}$, the calculated value from Equation (6) (

**a**), and residues (

**b**). p < 0.05.

**Figure 3.**The measured value of ${\pi}_{7}$, the calculated value from Equation (6), (

**a**) and residues (

**b**).

Treatment | $\mathbf{Sand}\mathbf{Mass}\phantom{\rule{0ex}{0ex}}\mathit{m},\mathit{kg}$ | $\mathbf{Porosity}\phantom{\rule{0ex}{0ex}}\mathit{\epsilon}$ | $\mathbf{Equivalent}\phantom{\rule{0ex}{0ex}}\mathbf{Diameter}\mathit{d},\mathit{mm}$ | ${\mathit{d}}_{\mathit{e}},\phantom{\rule{0ex}{0ex}}\mathbf{mm}$ | ${\mathit{d}}_{60},\phantom{\rule{0ex}{0ex}}\mathbf{mm}$ | $\mathit{U}{\mathit{C}}_{\mathit{s}}$ |
---|---|---|---|---|---|---|

Treatment 1 | 79.1 | 0.406 | 2.94 | 2.1 | 3.1 | 1.48 |

Treatment 2 | 76.2 | 0.428 | 2.02 | 1.41 | 2.20 | 1.56 |

Treatment 3 | 75.5 | 0.433 | 0.65 | 0.41 | 0.80 | 1.95 |

Treatment 4 | 79.5 | 0.402 | 1.19 | 0.55 | 1.82 | 3.31 |

Treatment 5 | 80.4 | 0.397 | 0.98 | 0.50 | 1.50 | 3.00 |

Treatment 6 | 78.3 | 0.413 | 0.84 | 0.45 | 1.10 | 2.40 |

Variable | Range | Mean and Standard Deviation |
---|---|---|

$\mathrm{TSS}(\mathrm{kg}{m}^{-3}$) | 0.0035–0.088 | 0.0214 ± 0.0152 |

$\mathrm{Filtration}\mathrm{velocity}(m{s}^{-1}$) | 0.0083–0.0398 | 0.03 ± 0.007 |

$\mathrm{Head}\mathrm{loss}(\mathrm{Pa}$) | 8750.00–166,517.85 | 55,584.47 ± 32,777.10 |

$\mathrm{Pollution}\mathrm{load}(\mathrm{Kg}$) | 0.0169–4.2049 | 0.91 ± 0.919 |

References | Tank Body Structure Parameters | Sand Media Parameters | Filtered Liquid Parameters |
---|---|---|---|

Dong, 1997 [25] | Porosity ε_{0}Equivalent diameter d Sphericity coefficient ψ Filtration layer thickness L | Water density ρ Water viscosity μ Acceleration of gravity g Filtration velocity v | |

Puig-Bargués et al., 2005 [22] | Total filtration surface area A Effective diameter d _{e} | Solution density ρ Water viscosity u Volume of water filtered V TSS concentration C Flow rate of filtered liquid Q Mean diameter of particle size distribution, d _{p} * | |

Duran-Ros et al., 2010 [23] | The filter inlet or the outlet diameter, D _{p} | Water density ρ Water viscosity u Filtration velocity V the TSS concentration C | |

Elbana et al., 2013 [17] | The internal diameter of the sand media filter tank d _{f} | Effective diameter d_{e} | Water density ρ Acceleration of gravity g Volume of water filtered V TSS concentration C |

$\mathbf{\Delta}\mathit{H}$ | ${\mathit{d}}_{\mathit{e}}$ | ${\mathit{d}}_{60}$ | $\mathit{l}$ | $\mathit{m}$ | $\mathit{\rho}$ | $\mathit{g}$ | $\mathit{v}$ | $\mathit{\mu}$ | $\mathit{D}$ | |
---|---|---|---|---|---|---|---|---|---|---|

M | 1 | 0 | 0 | 0 | 1 | 1 | 0 | 1 | −1 | 1 |

L | 1 | 1 | 1 | 1 | 0 | −3 | 1 | 0 | 1 | 0 |

T | −2 | 0 | 0 | 0 | 0 | 0 | −2 | −1 | −1 | 0 |

Dependent Variable | Significance Level p | ${\mathit{R}}_{\mathit{a}\mathit{d}\mathit{j}}^{2}$ | Independent Variable | Non-Standardized Coefficients | |
---|---|---|---|---|---|

B | Standard Deviation | ||||

$\mathrm{ln}{\pi}_{7}$ | <0.001 | 0.94 | Constants | 1.019 | 0.292 |

$\mathrm{ln}{\pi}_{1}$ | −0.192 | 0.083 | |||

$\mathrm{ln}{\pi}_{2}$ | −0.160 | 0.123 | |||

$\mathrm{ln}{\pi}_{5}$ | 0.636 | 0.021 | |||

$\mathrm{ln}{\pi}_{6}$ | 0.191 | 0.018 |

Statistical Parameters | Calculated Value by Equation (1) ^{a} | Calculated Value by Equation (2) ^{a} | Calculated Value by Equation (3) ^{a} | Calculated Value by Equation (7) ^{b} | Measured Value ^{b} |
---|---|---|---|---|---|

Maximum | 2.15 × 10^{−24} | 1.83 × 10^{4} | 6.19 × 10^{4} | 1.52 × 10^{5} | 1.67 × 10^{5} |

Minimum | 6.45 × 10^{−292} | 1.24 × 10^{4} | 9.57 × 10^{2} | 9.06 × 10^{3} | 8.75 × 10^{3} |

RMSE | 1.29 × 10^{−26} | 8.04 × 10 | 1.13 × 10^{3} | 2.12 × 10^{3} | 2.54 × 10^{3} |

Mean | 1.29 × 10^{−26} | 1.53 × 10^{4} | 1.93 × 10^{4} | 5.37 × 10^{4} | 5.56 × 10^{4} |

Standard deviation | 1.66 × 10^{−25} | 1.04 × 10^{3} | 1.46 × 10^{4} | 2.74 × 10^{4} | 3.28 × 10^{4} |

Variation coefficient | 12.92 | 0.07 | 0.75 | 0.51 | 0.59 |

^{a}represents values with the same letters that are not significantly different (p < 0.05).

^{b}represents values with the same letters that are significantly different (p < 0.05).

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**MDPI and ACS Style**

Hu, Y.; Wu, W.; Liu, H.; Huang, Y.; Bi, X.; Liao, R.; Yin, S.
Dimensional Analysis Model of Head Loss for Sand Media Filters in a Drip Irrigation System Using Reclaimed Water. *Water* **2022**, *14*, 961.
https://doi.org/10.3390/w14060961

**AMA Style**

Hu Y, Wu W, Liu H, Huang Y, Bi X, Liao R, Yin S.
Dimensional Analysis Model of Head Loss for Sand Media Filters in a Drip Irrigation System Using Reclaimed Water. *Water*. 2022; 14(6):961.
https://doi.org/10.3390/w14060961

**Chicago/Turabian Style**

Hu, Yaqi, Wenyong Wu, Honglu Liu, Yan Huang, Xiangshuai Bi, Renkuan Liao, and Shiyang Yin.
2022. "Dimensional Analysis Model of Head Loss for Sand Media Filters in a Drip Irrigation System Using Reclaimed Water" *Water* 14, no. 6: 961.
https://doi.org/10.3390/w14060961