2.1. Biosand Filter Overview
The traditional BSF design can be divided into four main components: (1) the filter body, (2) sand and gravel inside the filter (referred to here as the effective area), (3) the active biolayer that occupies the standing water surface above the effective area and may penetrate slightly into the uppermost portion of the effective area, and (4) the reservoir area above the biolayer which serves as storage for contaminated water and establishes head pressure when occupied with water. In sequence, contaminated water is first poured into the top of the filter and fills the reservoir area. This water then passes through a diffuser plate and mixes with the biolayer where microbial activity removes contaminants from the soiled water. Water then infiltrates due to gravity into the effective area and moves downward through the filter. As water continues through the effective area (i.e., sand), particles are further removed by screening, adsorption, and natural death among other processes [
28]. Ultimately, clean water is discharged through an outlet pipe and collected by the user. Flow through the traditional BSF can be characterized as anisotropic, with a heterogeneous media bed (medium to fine sand top layer, coarse sand middle layer, and gravel bottom layer) [
6].
Most notable to the filter is the biolayer which accumulates along the surface of the top sand layer and matures over time as contaminated influent provides new sources of microbes and nutrients. As the microbial ecosystem develops, influent bacteria and viruses undergo a series of chemical and biological degradations while passing through the biolayer including physical straining [
28,
29,
30,
31]; this process is the driving contaminant removal mechanism for the BSF. The biolayer exists because the filter outlet height is slightly above the surface of the effective area to prevent the microbial community from drying out. This is particularly important given that the BSF is inherently an intermittent system as users typically charge the filter one time to a few times per day. Lastly, the BSF is recognized for its easy cleaning, as the surface of the effective layer can be scraped, stirred, or raked as pore spaces become too clogged and the flow rates too slow; the filter can also be emptied and repacked, which further prevents against channeling or preferential flow path issues that may otherwise develop over time.
2.3. Experimental Design and Conditions
We analyzed three different filter designs: (1) the traditional BSF (control), (2) a 40% reduced-height design (R1), and (3) a 70% reduced-height design (R2). We restricted the maximum reduction to 70% to match the 10 to 15 cm sand bed depth recorded by Napotnik et al. (2017), not including the biolayer [
16]. A 40% height reduction was selected as an in-between measure (i.e., between the control and R2) while capturing a sand bed depth of less than 40 cm (common depth in earlier versions of the BSF). The total filter area (~2867 cm
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.
Two modeling experiments were conducted using the properties of each filter. The first modeling experiment was to estimate the average flow rate and pressure distribution throughout the traditional and alternate BSF designs. The second modeling experiment was to estimate the percent removal of
E. coli and MS2 as a function of depth and grain size for each filter. Four different types of media (coarse, medium–coarse, medium, and fine sand) with varying hydraulic conductivities (K) and grain sizes were chosen for the experiments, as outlined in
Table 2. These media were chosen to represent the common types of sand used for filtration systems. Each media type was used for each model, resulting in a total of twelve experimental trials (three designs with four media types) each for fluid modeling and contaminant modeling (twenty-four total).
2.4. Finite Element Approximation of Darcy’s Law
A finite element approximation of Darcy’s law was used to calculate the pressure distribution, velocity gradient, and the average flow rate for the BSF designs at each media condition. Each filter was divided into 1 cm by 1 cm cells, and the finite element approximation was calculated using a combination of constant head, free-field, and impermeable boundary conditions for each respective cell position throughout the filter. These boundary conditions are displayed in
Figure 2 and are defined mathematically by Fox (1996) and Akhter (2006) using the finite difference method [
35,
36]. By splitting the effective area into
M by
N cells, each cell was assigned a unique value (e.g., pressure, velocity, the flow rate), where the entire effective area had
independent equations following the present boundary conditions. The number of cells in the effective area thus depended on the length and width of each cell described in Equations (1) and (2):
where
was the height of the effective area [
l],
was the width of the effective area [
l],
was the length of each cell [
l],
was the width of each cell [
l], and
and
were the height and width count of the cell matrix [unitless]; the total number of cells is then
.
Since the equations representing each cell were independent, we then set them into three matrices—the first with
M ×
N columns and
M ×
N rows, the second with one column and
M ×
N rows, and the third with one column and
M ×
N rows. Multiplying the first and second matrix yields the third matrix, defined in Equation (3):
where
was the first (coefficient) matrix representing the fractional terms in front of the equations outlined by Fox (1996) and Akhter (2006),
represented the second (e.g., pressure or head) matrix of each node throughout the filter, and
represented the third (e.g., constant head) matrix at each point in the filter [
35,
36]. Given the known coefficient matrix,
C, and the known constant matrix,
P, solving for the unknown variable matrix,
h, through a matrix solver yielded a matrix of unique values (e.g., head) at each cell. This relationship is outlined in Equation (4):
where
was the fractional term for the first cell in the first equation,
was the unique value (unknown) for the first cell, and
was the constant value for the first cell (this term was only non-zero for cells with a constant boundary condition—it was zero for every other cell in the matrix). Solving this equation for the unknown matrix
yielded values at each selected cell throughout the filter. A visual representation of the head matrix
and its respective velocity gradient is shown in
Figure A1. The velocity gradient was then obtained using Darcy’s law at each cell, as described in Equations (5) and (6):
where
was the velocity in the x or y direction [
l/t],
was the hydraulic conductivity [
l/t], and
represented the head in the x and y directions, respectively [
l]. The total velocity,
, throughout the BSF was then the sum of each horizontal and vertical velocity at every cell throughout the filter (Equation (7)):
With gravity-driven flow, the driving pressure or head decreases as a function of time as water level in the reservoir declines. This in turn causes the flow rate to decrease exponentially. Using small, discrete time steps with linearization, the flow rate and volume discharged throughout the filter were obtained as a function of time [
37], as outlined in Equations (8) and (9):
where
[
l3],
[
l/t], and
[
l] represent the volume discharged, filter velocity, and height of the water in the reservoir at the current time step, respectively,
[
l3],
[
l/t], and
[
l] represent the volume discharged, filter velocity, and height of the water in the reservoir at the next time step,
represents the filter reservoir area [
l2], and
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,
and the new volume discharged,
. 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/
of the function’s initial value. The average velocity was then approximated using Equation (10):
where
was the average velocity for the entire time interval [
l/t] and
was the initial velocity throughout the filter [
l/t] determined from the initial finite element approximation of Darcy’s Law. The average volumetric flow rate throughout the entire time interval follows Equation (11):
where
is the average volumetric flow rate [
l3/
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
In the BSF, sand particles act as a collector which attracts contaminants (e.g., bacteria and viruses) that act as colloidal particles experiencing interception, diffusion, and sedimentation forces [
25,
38]. As water filters through the sand bed, these particles are further removed by screening, adsorption, and natural death, resulting in an effluent supply of water with reduced bacterial and viral concentrations [
20,
25,
27]. The log removal of microorganisms by attachment can be described using the colloid filtration equation of Yao et al. (1971; Equation (12)):
where
was the concentration of microorganisms at the outlet [CFU/100 mL],
was the concentration of microorganisms in the influent water [CFU/100 mL],
was the porosity of the filtering media [unitless],
was the grain size of the collector [
l],
was the media bed depth [
l],
was the sticking efficiency [unitless], and
was the single collector efficiency [unitless] [
25]. We then included the effect of the biolayer on contaminant removal using Equation (13) developed by Schijven et al. (2013):
where
was the scale factor for the particle (microorganism) [m°C],
was the temperature of the water [°C],
was the rate coefficient [
t−1], and
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 (
) of water [kg/m°C] was assumed to be temperature dependent and was calculated using Equation (14) [
39].
The sticking efficiency (
), introduced by Vissink (2016), was first quantified using Equation (15):
where
was a sticking factor [
],
was the power of the grain size [unitless], and
was the power of the velocity [unitless] [
27]. While this correlation is statistically accurate for filtration systems with velocities greater than 0.3 m/h, this model breaks down for systems at very small velocities. At filter velocities lower than 0.3 m/h, this correlation reports values for the sticking efficiency that are greater than one, violating the restriction that the sticking efficiency must be less than or equal to unity and resulting in an overestimated contaminant removal value. The current model also relates a larger sticking efficiency and percent removal with increase in grain size, which goes against much of the background research on slow-sand and biosand filtration systems [
7,
21,
22].
The contaminant removal model introduced in this paper mirrors that introduced by Vissink but accounts for systems which exhibit very low Darcy velocities, such as the BSF. The model was based on an exponential decay function and agrees with the previous model at a Darcy velocity of 0.4 m/h for the grain size of 0.5 mm using the constants outlined in Vissink (2016) [
27]. The velocity of 0.4 m/h and grain size of 0.5 mm represent average filtration velocities for biosand and slow-sand filtration (0.1–0.4 m/h; [
28]) as well as a common grain size for both filtration techniques (medium–coarse sand). We then quantified the sticking efficiency using Equation (16):
where
was an exponent [unitless] and
was a new sticking factor [
]. Both factors were determined from correlation with the current model developed by Vissink (2016) and from previously reported values from Schijven et al. (2013) [
20,
27]. Data for the sticking and the single collector efficiency in the control filter at varying grain sizes are found in
Figure A2 and
Figure A3. Selected values used to calculate the sticking efficiency, the single collector efficiency, and the log removal of
E. coli and MS2 for each experimental trial are summarized in
Table A1. Values for the particle density, particle diameter, Hamaker constant, porosity, scale factor, and rate coefficients were synthesized from an extensive literature search. For each experimental trial, the fluid temperature was kept constant at 25 °C, porosity at 42%, and the age of the biolayer at 14 days.
E. coli was selected for its pathogenic properties and widespread use as a reference for contaminant removal in slow-sand filters. Although several other strains of pathogenic bacteria cause severe illnesses and waterborne diseases such as cholera, typhoid fever, and gastroenteritis, these bacteria are harder to isolate and culture and therefore the use of indicator bacteria is required [
40];
E. coli is frequently used as a broad bacteria proxy due to its large presence in human and animal intestines [
40]. MS2 was chosen because of its reputation as a conservative model virus [
41].