1. Introduction
The assessment of bacterial deposition becomes a prerequisite for the correct prediction of bacterial transport for bioaugmentation methods or the prevention of microbial pathogens from contaminating drinking water supplies in alluivial aquifers. The deposition refers to the complex process of removing bacteria from a suspension in a porous medium, such as straining, attachment, sorption, etc. To date, numerous studies on the process of bacterial deposition have been conducted to verify the effects of chemical factors such as solution chemistry [
1,
2], physical factors such as grain size and bacterial surface protein [
3], and saturated and unsaturated porous media [
4], grain size and pore water velocity [
5], and physico-chemical factors such as saturated and unsaturated media and ionic strength [
6].
Deposition coefficients are typically determined by fitting transport models to the observed bacterial breakthrough curves (BTCs) eluted from the porous medium. Several studies have been conducted to use easily determined measures in order to determine the deposition coefficient [
7,
8]. A method for the determination of sticking efficiencies has been proposed using the filtration model of Yao et al. [
7]. This method requires time-consuming treatment such as the measurement of complete BTCs or the dissection of a column and enumeration the deposited bacteria.
Bolster et al. proposed another method for calculating the bacterial deposition coefficent using a simple measure—the fraction of bacteria recovered from a laboratory column breakthrough experiment [
8]. This method was derived from the analytical solution of the bacteria transport equation for pulse input. The dimensionless form of the bacterial transport equation for advection, disperstion, and deposition processes can be given as [
8]:
where peclet number
, Damkohler number representing time-scale ratios for deposition to flow velocity
, dimensionless length
, dimensionless time
, normalized concentration of bacteria in aqueous phase
, and normalized concentration of bacteria on the solid phase
(
v: porewater velocity or advection coefficient,
L: column length,
D: dispersion coefficient,
c: bacterial concentration in aqueous phase,
s: bacterial concentration in solid phase,
: injection concentration,
: deposition coefficient). A solution to Equation (1) for the bacterial flux concentration at the end of the column (
X = 1) for a step input was given as [
8]:
The bacterial flux concentration at the end of the column for a pulse input was generated by using the principle of superposition.
where
is the solution for a pulse input and
is the length of the pulse input.
The cummulative recovery of bacteria is obtained by integrating the equation for the bacterial flux concentration. The fractional mass recovery (fr) for a pulse input is obtained by normalizing the cummulative recovery with the mass input.
Rearranging Equation (5) yields the folllowing dimensionless deposition coefficient (Bolster et al. 1998 [
8]):
Equation (6) is independent of pulse length, and can be conveniently used for a porous media which undergoes bacterial dispersion and deposition. The deposition coefficient () is a function of the fractional mass recovery (fr) and the peclet number (Pe), and for large Pe the relationship between and becomes linear.
Retrospection of Equation (3) enables us to gain some insights on the relationship between
C(1,
T) and
for the step input experiment. The Equation (3) can be simply expressed as:
where:
As time goes to infinity and
, the complement error function (erfc) terms approach a constant value, as follows:
Under these conditions, the effluent concentration reaches the steady-state concentration
and is simplified so that:
and at that condition,
. Therefore, we can obtain the following relation:
Rearranging this equation gives the following relationship for
:
This relationship between the steady-state concentration (plateau concentration) and the deposition rate can be used to obtain an accurate bacterial deposition coefficient for a step input.
Alternatively, the deposition coefficient can also be estimated using the approach of Yao et al. [
7]. The effluent bacterial concentration of a packed bed with length
L is related to the efficiency of a single spherical collector:
where
is the bed porosity,
is a collision efficiency factor,
is the collector efficiency, and
d is the grain diameter. The integration of Equation (19) yields the following relationship between the deposition
and the effluent concentration in steady-state
, so that:
where:
The relationship between
and
given by Equation 18 implies that the deposition coefficient is a function of
and
Pe, and for high
Pe it becomes only a function of
. A similarity can be found between Equations (6) and (18) in that both of them have an identical form, as is shown in
Table 1, and they differ in that the deposition coefficient is a function of
for Equation (18) while it is a function of
fr for Equation (6). Another similarity also can be found between Equations (18) and (20) in that both of them are functions of
except for the presence of the additional term
in Equation (18). These similarities suggest that both simple measures,
fr as well
, are applicable for the determination of the bacterial deposition coefficient. However, it is necessary to compare the accuracy of the two methods using
. The objective of this study is to propose a simple method for the estimation of the deposition coefficient from breakthrough curves of pulse and step input. The validity of the new method was tested by comparing the accuracy of the estimated deposition coefficient with the other two methods. Furthermore, it is shown how they deviate from each other according to
Pe and the injection duration of the bacterial solution (pulse or step input).
2. Theoretical Analysis
The validity of the simple method for the determination of was tested by comparing the value of given for simulating bacterial transport and the values estimated using Equations (6), (18) and (20). For the simulation of bacterial transport, a clean bed system with step and pulse injection for fixed were postulated. The bacterial flux concentration at the end of the column for step and pulse input were calculated using Equations (3) and (4), respectively.
Sensitivity analysis for
was performed to compare the accuracy of the methods for the estimation of
. The bacterial BTC was simulated for short (
) and long pulse inputs (
) with
and various
. The deposition coefficient (
) was estimated using Equations (6), (18) and (20) based on the
and
of the simulated BTCs. From the BTC of long pulse and step inputs, the steady-state concentration
could be determined from the concetration of the plateau part. From the BTC of short pulse input, the relative peak concentration
was used instead of
. The fractional mass recovery was calculated by the following:
Coincidence of the given and estimated values could indicate the validity of the equation for the estimation of .
The sensitivity analysis was also conducted for
and
to confirm the applicable range of parameters for Equations (6), (18) and (20). The range of parameters used for the sensitivity analysis are shown in
Table 2. The applicability was also determined by comparing the given
values with those previously estimated.