# Modeling Improved Performance of Reduced-Height Biosand Water Filter Designs

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Biosand Filter Overview

#### 2.2. Design Simplification

#### 2.3. Experimental Design and Conditions

^{2}), reservoir volume (12 L), and biolayer depth (5 cm) were conserved in R1 and R2 to match the control BSF, and all three filters (Control, R1, R2) were simplified to have square bases and an outlet located at the bottom and 6 cm from the right edge of the filter. The measurements of the control, R1, and R2 are displayed in Figure 1.

#### 2.4. Finite Element Approximation of Darcy’s Law

^{3}], ${V}_{i}$ [l/t], and ${P}_{top,i}$ [l] represent the volume discharged, filter velocity, and height of the water in the reservoir at the current time step, respectively,${Q}_{i+1}$ [l

^{3}], ${V}_{i+1}$ [l/t], and ${P}_{top,i+1}$ [l] represent the volume discharged, filter velocity, and height of the water in the reservoir at the next time step, ${A}_{top}$ represents the filter reservoir area [l

^{2}], and $dt$ represents the time interval [t]. Using the finite element approximation of Darcy’s law at the current time step yielded velocity, which was then used to calculate the new driving pressure, ${P}_{top,i+1}$ and the new volume discharged, ${Q}_{i+1}$. For this study, we used 25 s time steps, ending at 5 h. Due to practical considerations for a 12 L reservoir volume, the model was capped at 5 h. To quantify the average velocity throughout the filter over the entire time interval, we assumed each configuration exhibited a behavior similar to exponential decay. The mean lifetime of an exponential decay function was calculated where the value of the function is reduced to 1/$e$ of the function’s initial value. The average velocity was then approximated using Equation (10):

^{3}/t]. The results for the average velocity for each filter design and media type were then used to analyze the percent removal of E. coli in the contaminant modeling experiments described in Section 2.5.

#### 2.5. Contaminant Removal Modeling

^{−1}], and $a$ was the age of the biolayer [t] [20]. The single collector efficiency (η) was calculated using the colloid filtration theory equations developed by Tufenkji and Elimelech (2004) [38]. The fluid viscosity ($\mu $) of water [kg/m°C] was assumed to be temperature dependent and was calculated using Equation (14) [39].

## 3. Results

#### 3.1. Fluid Velocity and Discharge

#### 3.2. Contaminant Removal

## 4. Discussion

#### 4.1. Comparison Between Designs

#### 4.2. Biolayer Age and Media Depth

#### 4.3. Literature Agreement

#### 4.4. Comparison to Ceramic Filtration

#### 4.5. Model Limitations and Experimental Error

#### 4.6. Practical Design Applications

## 5. Conclusions

- Slower fluid velocities through the filter require less effective area depth, as residence times inside the filter increase. Increased residence times allow for longer contact time with the both the biolayer and effective media which leads to greater bacteria removal and virus deactivation. For the BSF, slow velocities are directly related to the hydraulic conductivity of the effective media, where fine sands have the greatest reduction in fluid velocity. Thus, BSF designs with finer-grained media can be designed with shorter filter bodies relative to the traditional BSF design size. Reduced velocities can also be achieved through decreased head pressure, which can be obtained by a shorter standing height on top of the filter media with each use. To maintain the total volume of discharged water, reduced standing heights require a wider filter body than the traditional design.
- Increased biolayer area leads to greater contaminant removal. Particularly for bacteria, contact with the biolayer is the most notable filtering mechanism in the BSF. Designs which increase the biolayer area relative to the traditional BSF design will have greater contaminant removal rates, assuming other conditions are consistent between filters. This can be accomplished through a wider design, which also enables a reduced standing water height above the filter media and slower fluid velocity as outlined in Conclusion #1. Under the proper conditions, BSF technology can remove nearly all bacteria contaminants through just the biolayer.
- Viruses, unlike bacteria, are less impacted by the biolayer and are more effectively removed with longer residence times inside the BSF. Longer residence times can be achieved by decreased media grain size (i.e., hydraulic conductivity), taller effective areas (i.e., taller filter bodies), or slower fluid velocity (i.e., slower water flow). With a modified design, BSF technology can remove 100% of virus contaminants.
- The R1 and R2 designs outperformed the traditional BSF in contaminant removal at all media grain sizes, but their total discharge was notably less. While not outside of other common HWT solutions at fine grain sizes, the discharge rates of R1 and R2 can be improved by a larger filter surface area or larger media grain sizes. With sand characteristics commonly used in the traditional BSF, both the R1 and R2 designs outperformed the traditional design while also maintaining practical discharge rates.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Pressure Distribution

## Appendix B. Collector and Sticking Efficiency

**Figure A2.**Single collector efficiency versus particle size for the control design with varying media.

## Appendix C. Model Constants

Constant | Constant Description | Value | Units |
---|---|---|---|

${\rho}_{p}$ | Particle Density (E. coli) ^{1} | 1160 | kg/m^{3} |

${\rho}_{p}$ | Particle Density (MS2) * | 1000 | kg/m^{3} |

${d}_{p}$ | Particle Diameter (E. coli) | 1 | μm |

${d}_{p}$ | Particle Diameter (MS2) | 27.5 | nm |

$g$ | Gravitational Constant | 9.80 | m/s^{2} |

$k$ | Boltzmann Constant | 1.38 × 10^{−23} | J/K |

$A$ | Hamaker Constant ^{2} | 2.15 × 10^{−20} | J |

$f$ | Porosity ^{3} | 0.42 | - |

${\rho}_{l}$ | Fluid Density | 1000 | kg/m^{3} |

$p$ | Power (E. coli) | 0.2 | - |

$p$ | Power (MS2) | 0.1 | - |

a | Biolayer Age | 14 | days |

${f}_{0}$ | Scale Factor ^{4,5} | 1.9 × 10^{−4} | m°C |

${f}_{1}$ | Rate Coefficient ^{4,5} | 0.072 | day^{−1} |

${f}_{2}$ | Sticking Factor (E. coli) | 0.0029 | - |

${f}_{2}$ | Sticking Factor (MS2) | 0.00075 | - |

**Table A2.**Results for the sticking and the single collector efficiency for a traditional BSF design (C), a 40% reduced-height design (R1), and a 70% reduced-height design (R2).

Type of Sand | Filter Design | Sticking Efficiency (α) | Single Collector Efficiency (η) | ||
---|---|---|---|---|---|

E. coli | MS2 | E. coli | MS2 | ||

Coarse | Control | 0.0401 | 0.0028 | 0.0006 | 0.0087 |

R1 | 0.0563 | 0.0033 | 0.0022 | 0.0301 | |

R2 | 0.0837 | 0.0041 | 0.0112 | 0.1302 | |

Medium–Coarse | Control | 0.0744 | 0.0039 | 0.0056 | 0.0794 |

R1 | 0.1037 | 0.0046 | 0.0211 | 0.2753 | |

R2 | 0.1521 | 0.0056 | 0.1176 | 1.0 | |

Medium | Control | 0.1012 | 0.0045 | 0.0163 | 0.2373 |

R1 | 0.1402 | 0.0054 | 0.0601 | 0.8239 | |

R2 | 0.2036 | 0.0066 | 0.3200 | 1.0 | |

Fine | Control | 0.1708 | 0.0060 | 0.1199 | 1.0 |

R1 | 0.2330 | 0.0071 | 0.4759 | 1.0 | |

R2 | 0.3295 | 0.0088 | 1.0 | 1.0 |

## Appendix D. Model Error

**Figure A4.**Error bars for MS2 based on the 0.6 log error as reported by Schijven et al. (2013) [20] for a traditional BSF design (C), a 40% reduced-height design (R1), and a 70% reduced-height design (R2).

## Appendix E. Heavy Metals

## References

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**Figure 1.**Visualizations and volumetric properties of (

**a**) a traditional biosand water filter (BSF) design (control); (

**b**) a 40% reduced-height design (R1); and (

**c**) a 70% reduced-height design (R2).

**Figure 2.**Visualization of all boundary conditions used in the fluid modeling experiments, where q denotes flow between adjacent cells.

**Figure 3.**Modeled filter velocity vs. time at varying sand media for (

**a**) a traditional BSF design (control); (

**b**) a 40% reduced-height filter design (R1); and (

**c**) a 70% reduced-height filter design (R2).

**Figure 4.**Modeled volume discharged vs. time at varying sand media for (

**a**) a traditional BSF design (control); (

**b**) a 40% reduced-height design (R1); and (

**c**) a 70% reduced-height design (R2). Color fills are used to represent a range between given media such as that between coarse and medium–coarse sand.

**Figure 5.**Modeled average volumetric flow rate by filter design (a traditional BSF design (C), a 40% reduced-height design (R1), and a 70% reduced-height design (R2)) for varying sand media.

**Figure 6.**Modeled percent removal of E. coli by depth at varying sand media for (

**a**) a traditional BSF design (control); (

**b**) a 40% reduced-height design (R1); and (

**c**) a 70% reduced-height design (R2). Color fills are used to represent a range between given media such as that between fine and medium sand. Note the x-axis scales are different.

**Figure 7.**Modeled percent removal of E. coli at the biolayer and at the full filter length for a traditional BSF design (C), a 40% reduced-height design (R1), and a 70% reduced-height design (R2).

**Figure 8.**Modeled percent removal of MS2 by depth at varying sand media for (

**a**) a traditional BSF design (control); (

**b**) a 40% reduced-height design (R1); and (

**c**) a 70% reduced-height design (R2). Color fills are used to represent a range between given media such as that between fine and medium sand. Note the x-axis scales are different.

**Figure 9.**Modeled percent removal of MS2 at the biolayer and at the full filter length for a traditional BSF design (C), a 40% reduced-height design (R1), and a 70% reduced-height design (R2).

**Figure 10.**Error bars for the E. coli results based on the 0.6 log error as reported by Schijven et al. (2013) [20] for a traditional BSF design (C), a 40% reduced-height design (R1), and a 70% reduced-height design (R2).

**Table 1.**Literature values for percent removal of Escherichia coli (E. coli) vs. depth in slow-sand/biosand filters derived from physical experiments.

Sand Bed Depth (cm) | Percent Removal (%) |
---|---|

<1 | 94.380 ^{1} |

5 | 99.370 ^{1} |

10 | 99.980 ^{2} |

15 | 99.984 ^{2} |

40 | 99.700 ^{3} |

54 | 98.500 ^{4} |

55 | 99.987 ^{2} |

**Table 2.**Selected hydraulic conductivity (K) and grain size values per media type chosen from physical experiments.

Media | K ^{1} (cm/s) | Grain Size ^{2} (mm) |
---|---|---|

Coarse | 0.6 | 1.0 |

Medium–Coarse | 0.05 | 0.5 |

Medium | 0.02 | 0.25 |

Fine | 0.002 | 0.15 |

**Table 3.**Modeled average filter velocity for each filter configuration (a traditional BSF design (control), a 40% reduced-height design (R1), and a 70% reduced-height design (R2)) and media type.

Media | Design | Average Velocity (m/h) |
---|---|---|

Coarse | Control | 2.6933 |

R1 | 0.4723 | |

R2 | 0.0608 | |

Medium–Coarse | Control | 0.2244 |

R1 | 0.0394 | |

R2 | 0.0051 | |

Medium | Control | 0.0898 |

R1 | 0.0157 | |

R2 | 0.0020 | |

Fine | Control | 0.0090 |

R1 | 0.0016 | |

R2 | 0.0002 |

Sand Bed Depth (cm) | Percent Removal of E. coli (%) | |||
---|---|---|---|---|

Literature Values | Fine Sand ^{†} | Medium Sand ^{†} | Percent Difference | |

0 | 94.38 ^{1,5} | 98.72 | N/A ^{‡} | 4.60% |

5 | 99.37 ^{1,5} | 99.99 | N/A ^{‡} | 0.62% |

10 | 99.98 ^{2,5} | ~100.00 | N/A ^{‡} | 0.02% |

15 | 99.984 ^{2,5} | ~100.00 | N/A ^{‡} | 0.02% |

40 | 99.70 ^{3,6} | N/A ^{‡} | 97.97 | 1.73% |

54 | 98.50 ^{4,6} | N/A ^{‡} | 99.09 | 0.60% |

55 | 99.987 ^{2,5} | ~100.00 | N/A ^{‡} | 0.01% |

^{1}[17]

^{2}[16]

^{3}[33]

^{4}[6].

^{5}Fine sand.

^{6}Medium sand.

^{†}Modeled in this study.

^{‡}N/A represents a missing literature value at the corresponding media size to compare with our modeled results (i.e., we have a modeled value, but no matching value can be found in the literature at the specific media conditions).

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Phillips, J.A.; Smidt, S.J.
Modeling Improved Performance of Reduced-Height Biosand Water Filter Designs. *Water* **2020**, *12*, 1337.
https://doi.org/10.3390/w12051337

**AMA Style**

Phillips JA, Smidt SJ.
Modeling Improved Performance of Reduced-Height Biosand Water Filter Designs. *Water*. 2020; 12(5):1337.
https://doi.org/10.3390/w12051337

**Chicago/Turabian Style**

Phillips, James A., and Samuel J. Smidt.
2020. "Modeling Improved Performance of Reduced-Height Biosand Water Filter Designs" *Water* 12, no. 5: 1337.
https://doi.org/10.3390/w12051337