# Revisiting the Gage–Bidwell Law of Dilution in Relation to the Effectiveness of Swimming Pool Filtration and the Risk to Swimming Pool Users from Cryptosporidium

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{t}value for a 3 log

_{10}reduction in oocyst viability for free chlorine at pH 7.5 and at 25 °C corresponds to a disinfection time of 10.6 days in pool water containing 1 mg L

^{−1}free chlorine) and can only be controlled adequately in a pool plant room (i.e., external to the pool itself) by filtration, possibly supplemented by non-residual treatments such as UV or ozone [2]. Of particular concern is the large number (potentially > 10

^{8}) of Cryptosporidium oocysts likely to be introduced into the pool water as a result of an accidental faecal release (AFR) by a bather [3,4]. There is a chance of infection from ingestion of just a single oocyst [5] and guidelines have been produced for managing this risk associated with AFRs [6].

^{−}

^{1}). In this context, a turnover cycle is the time taken for a volume of water equivalent to the entire pool volume to pass through the filtration and circulation system once [7]. As we shall demonstrate, this does not mean that all the water in a pool is subject to filtration in a single turnover cycle.

## 2. Materials and Methods

- Disinfection using chlorine gas (0.45 mg L
^{−1}free chlorine in the pool water). - pH adjustment (pH 7.0 in the pool water).
- Flocculant dosing approx. 0.05 mg L
^{−1}Al as poly-aluminium chloride (PAC). - Dual media filter (0.5 m sand depth, 0.7–1.2 mm grain size), (0.5 m anthracite depth, 1.4–2.5 mm grain size).
- Filtration velocity 35 m h
^{−1}.

^{−1}Al coagulant as PAC, based on comparison of turbidity measurements at the filter inlets and outlets.

## 3. Results

#### 3.1. The Gage–Bidwell Law of Dilution: Empirical Approach

_{o}), implying that 30% of the water resident in the pool at the start of the turnover remains in the pool after one turnover. Reworking the example above with two consecutive parcels each containing half the pool volume would have resulted in a corresponding value of 25%; four consecutive parcels each containing a quarter of the pool volume would have resulted in a corresponding value of 32%. The pattern is that as the number of parcels increases (and their size decreases) the percentage of water remaining untreated after one turnover increases towards some maximum value. Furthermore, the only way to ensure none of the water resident in the pool at the start of the turnover remains in the pool after one turnover would be to remove and replace all the water in the pool as a single parcel. This could be achieved by following the ‘empty and fill’ practice used in the early days of municipal pool management [14].

#### 3.2. The Gage–Bidwell Law of Dilution: Computational Approach

_{pv}) is given by Equation (1):

_{pv}/C

_{o}in Equation (1) converges to 0.368 (to three significant figures). In other words, 63.2% of the water present in the pool at the start of the turnover cycle has been treated at the end of the single turnover cycle, with 36.8% remaining untreated.

^{i}). Therefore, we can write Equation (7):

_{o}when T = 0 gives Equation (8):

#### 3.3. The Role of Filter Efficiency in Contaminant Removal

^{3}of untreated water remaining in the case of a 500 m

^{3}pool. As an example of the practical implications, this untreated water might still contain 300,000 oocysts if the pool is well-mixed and if there had been an input of 10

^{8}oocysts prior to the six turnovers [17].

_{p}, as distinct from the water-turnover time, T

_{w}= V/Q, given by Equation (11):

^{3}), Q is the circulation rate (m

^{3}h

^{−1}), and E is the fractional removal of either turbidity (NTU) or particles of a given size class from water in a single pass through the filter.

_{w}) is the time it takes 63.2% of the water in a well-mixed pool to be removed, the particle-turnover time (T

_{p}) is the time it takes 63.2% of particles to be removed. An example of this is illustrated later by diurnal measurements of turbidity (Section 3.5). There is an approximately exponential decrease in turbidity once the pool is closed, where the exponent is the particle-turnover time.

#### 3.4. Application of the Gage–Bidwell Approach to Modelling the Peak Turbidity of Pool Water

- To establish the equilibrium turbidity likely to be achieved if a constant bathing load (in terms of numbers of bathers per hour) is maintained indefinitely.
- To establish the maximum turbidity likely to be achieved if a constant bathing load is sustained for a finite time that is too short for the equilibrium to be achieved.

#### 3.4.1. Modelling the Maximum Turbidity Achievable If the Design Maximum Bathing Load for a Pool Is Sustained Indefinitely

^{3}h

^{−1}) and the filter efficiency (expressed in terms of the fraction of turbidity that is removed in a single pass through the filter, E).

_{p}). If at equilibrium the rates of addition and removal of turbidity are equal, the equilibrium turbidity (C

_{e}) is given as in Equation (12):

_{p}and E are more ambiguous and require further discussion.

^{3}of water. This method was used by Amburgey (personal communication, 2020) who reported an average K

_{p}value of 0.65 NTU (bather m

^{−3})

^{−1}. A variation to this approach might be to collect shower water and measure the recovery of particles from individuals, as done by Keuten at al [21], although the range of values for the sloughing of turbidity-forming material was not reported. An alternative method was used by Stauder and Rodelsperger [13], who used the continuity form of Equation (10) to model the diurnal fluctuations in turbidity from the differences between the rates of input and removal of turbidity, based on the assumption of a well-mixed pool. The parameters affecting the modelled time course of turbidity were the circulation rate (Q), the filter efficiency (E), the known fluctuation in bathing load and the average K

_{p}. As all parameters except K

_{p}were known, values of K

_{p}for each day were obtained by finding the values that gave the best fit between the modelled and measured time course of NTU. This resulted in values ranging from 0.25 to 0.5 NTU (bather m

^{−3})

^{−1}. However, it should be noted that Stauder and Rodelsperger [13] reported the daily visitor number, and it may be that not all the visitors entered the pool; therefore, these values will underestimate K

_{p}. It should also be noted that as this was a paddling pool, not all bathers would be fully immersed, which is likely to reduce the inferred value for K

_{p}. In the scenarios we consider below, we will use values of 0.25 or 0.65 NTU (bather m

^{−3})

^{−1}to represent the range from ‘clean’ to ‘dirty’ bathers.

^{3}) of water being treated (i.e., B/Q from Equation (12)). For example, a pool with 100 bathers h

^{−1}entering the pool with a water circulation rate of 200 m

^{3}h

^{−1}would return a value of 0.5 bathers m

^{−3}circulation, which is the same value as for a spa with 10 bathers h

^{−1}entering the spa with a water circulation rate of 20 m

^{3}h

^{−1}. To put the range of x-axis values into context, a leisure pool with an average depth of 1.5 m operating at maximum bathing load (allowing 4 m

^{2}water area per bather) and a 3 h water-turnover time would have a value of 0.5 bathers m

^{−3}circulation.

_{p}= 0.65 or 0.25 NTU (bather m

^{−3})

^{−1}over the range of values on the x-axis likely to encompass most pools. With relatively good filtration (E = 0.9), the equilibrium turbidity value (achieved after a very long time of bathers entering the pool at a steady rate) will only just reach 0.5 NTU at a value of 1.0 bathers m

^{−3}circulation with dirty bathers. However, pools with less effective filtration (E = 0.5) are at risk of the turbidity exceeding 0.5 NTU at a value of 0.5 bathers m

^{−3}circulation when the bathers are dirty. Pools with relatively poor filtration (E = 0.2) are predicted to have excessive turbidity after prolonged periods of maximum bathing load at a value of 0.4 bathers m

^{−3}circulation even with the cleanest bathers.

^{3}of water treated by filtration (x-axis Figure 2) is already established in pool operation guidelines. For example, the guidelines for pool operation in the UK [10] recommend that where the circulation rate is limited by the design of the pool, the maximum bathing load for the pool should be calculated from Equation (13):

^{3}h

^{−1})/1.7

^{3}circulation/bather is equivalent to an x-axis value in Figure 2 of 0.58 bathers m

^{−3}, shown by the vertical dashed line. Provided the filtration is relatively good (E = 0.9 in this case), this upper limit guideline maintains equilibrium turbidity of the pool water within an acceptable range (no more than 0.3 NTU even with very dirty bathers) in the case where the maximum bathing load is sustained indefinitely.

^{−3}(equivalent to 1.7 m

^{3}water flow through the filtration system per bather) will result in only slight exceedance of the upper acceptable limit of 0.5 NTU, even with dirty bathers, i.e., K

_{p}= 0.65 NTU (bather m

^{−3})

^{−1}, and relatively poor filtration (E = 0.5). In this respect, this guideline [10] is necessarily cautious in that it will maintain acceptable water quality even in pools with relatively dirty bathers and relatively poor filtration performance. Recommendations for water-turnover times for pools may also need some contingency for pools where the water volume behaves as a number of separate compartments and where the ratio of water circulation to bather number within a compartment could be rather less than the overall average for the pool.

#### 3.4.2. Modelling the Maximum Turbidity Achievable If the Design Maximum Bathing Load for a Pool Is Sustained for a Finite Period

^{3}paddling pool studied by Stauder and Rodelsperger [13] showed large fluctuations in turbidity during the day, with the peak values generally appearing as sharp mid-afternoon spikes rather than approaching a plateau. This suggests that equilibrium turbidity values were a long way from being approached in this particular case.

_{o}) to the special case of the final equilibrium concentration being zero. However, as we are now concerned with the accumulation of turbidity-causing particles from some initial starting condition (C

_{o}) to a final non-zero equilibrium turbidity (C

_{e}), Equation (10) can be written in the following more general form:

_{o}) to the final equilibrium concentration C

_{e}. Just as with the removal of Cryptosporidium oocysts, we see that after one particle-turnover time we have reached 63.2% of the final result of the step change and reached 99.7% of the change after six particle-turnovers.

_{o}to C

_{e}. This explains why time courses of turbidity for leisure pools typically show short-term ‘spikes’ at times of peak bathing load, rather than approaching a plateau, because the fluctuations in bathing load are too rapid for equilibrium states to be approached.

_{o}to C

_{e}would increase to 47 min for the spa and 14 h of continuous bathing load for the pool. The implication is that in practice it is only in pools with very short water-turnover times (such as spas and paddling pools) that the turbidity is ever likely to approach the equilibrium value for the maximum instantaneous bathing load. Pools with water-turnover times longer than 2 h would not be expected to approach the equilibrium turbidity for the maximum bathing load that was used as the basis of the nomogram shown in Figure 2.

#### 3.5. Modelling Observed Time Courses of NTU

_{o}was the turbidity value (NTU) at the end of the preceding hour, and C

_{e}was calculated using Equation (12), based on the number of bathers entering the pool that hour and an assumed value for the turbidity input per bather. With a water-turnover time of 1.06 h and a filter efficiency of 0.9, Equation (14) predicts that in 1 h there will be 57% of the transition from C

_{0}to C

_{e}. In this way the diurnal course of turbidity (NTU) was predicted, as shown by the solid line in Figure 4.

^{−3})

^{−1}. Despite these critical assumptions, the modelled time courses showed good agreement with the measured values and predicted the peak daily turbidity values to within 10%.

_{p}, E, T

_{w}and the temporal pattern of bather frequency, but differed only in the daily bather number, which would act to ‘scale’ the peak turbidity value.

#### 3.6. Public Health Implications

- Prediction of the time it takes to achieve satisfactory removal of a contaminant (e.g., Cryptosporidium oocysts) following a single contamination event.
- Prediction of the maximum equilibrium concentration of a contaminant under conditions of a steady input of the contaminant (we considered the maximum turbidity achieved under conditions of a prolonged constant bathing load).
- Prediction of the amount of water that should be circulated per bather to ensure that water clarity remains excellent, even when there is a very prolonged period when bathers are entering the pool.
- Prediction of the peak turbidity likely to be achieved in practice from knowledge of the distribution of bathing load during the day.

_{w}) and the filtration efficiency (E) that provides the best overall key performance indicator of the effectiveness of filtration in swimming pools. We propose a formalisation of this concept in a new combined term, particle-turnover time (T

_{p}= T

_{w}/E), which could provide the basis for assessing the health and safety risks associated with particulate material in pool water. However, this requires the development of a practical methodology for assessing the effectiveness of filtration in operational pools, which is not generally available at present, but which might be based on the use of turbidity measurements or particle counting [7].

^{3}of water (Figure 5).

- Depth ranging from 1–2 m (average depth 1.5 m).
- 4 m
^{2}pool area allowed per bather at maximum bathing load following the UK guidelines [10], i.e., each bather occupies 6 m^{3}of water on average. - 3 h water-turnover time.
- Average bathing time of 0.75 h.

^{3}of water treated per bather. This corresponds to a value of 0.67 bather m

^{−3}for the x-axis of Figure 2, which would imply that with relatively good filtration of E = 0.9 [10] the maximum possible turbidity would be maintained below 0.4 NTU, even with relatively dirty bathers (0.65 NTU (bather m

^{−3})

^{−1}. With poorer filter efficiency (E = 0.5), the turbidity after very prolonged maximum bathing load would just exceed 0.6 NTU (i.e., slightly above the recommended upper limit) with relatively dirty bathers. With any reduction in the period of the maximum bathing load during each day (e.g., only two swim sessions, each of 3 h duration) the resulting maximum turbidity would be expected to be no more than 0.4 NTU.

#### 3.7. Conclusions

^{8}oocysts). If a filter efficiency for Cryptosporidium oocysts of 0.9 is assumed, as for example by PWTAG [10]), then Equation (10) predicts that after six water turnovers the concentration remaining would amount to 9000 m

^{−3}in a 50 m

^{3}pool, and 900 m

^{−3}in a 500 m

^{3}pool. Assuming that the average ingestion of pool water is 37 mL [7], the average ingestion of oocysts from pool water after 6 h of filtration would therefore be 0.3 and 0.03 oocysts in a 50 and 500 m

^{3}pool, respectively. This is below the reported infective dose for Cryptosporidium [2,7]. However, if the filter efficiency is 0.5 or 0.2 (e.g., a sand filter with inadequate coagulation [4,18]), then a similar arithmetic leads to the conclusion that the numbers of oocysts ingested on average following six water turnovers increases in the case of the 50 m

^{3}pool to 3.7 (E = 0.5) and 22.3 (E = 0.2) oocysts. This is within the range of the reported infective dose for Cryptosporidium [2,7] and suggests that in these cases six turnover cycles might be insufficient. This also raises the question of how filtration efficiency can be evaluated in pools [7].

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Lewis, L.; Chew, J.; Woodley, I.; Colbourne, J.; Pond, K. The application of computational fluid dynamics and small-scale physical models to assess the effects of operational practices on the risk to public health within large indoor swimming pools. J. Water Health
**2015**, 13, 939–952. [Google Scholar] [CrossRef] [PubMed][Green Version] - Ryan, U.; Lawler, S.; Reid, S. Limiting swimming pool outbreaks of cryptosporidiosis–the roles of regulations, staff, patrons and research. J. Water Health
**2017**, 15, 1–16. [Google Scholar] [CrossRef] [PubMed] - Chalmers, R.M. Waterborne outbreaks of cryptosporidiosis. Ann. dell’Istituto Super Sanità
**2012**, 48, 429–446. Available online: https://www.iss.it/documents/20126/45616/ANN_12_04_10.pdf (accessed on 26 April 2021). [CrossRef] [PubMed] - Gregory, R. Bench-marking pool water treatment for coping with Cryptosporidium. J. Environ. Health Res.
**2002**, 1, 11–18. [Google Scholar] - Messner, M.J.; Chappell, C.L.; Okhuysen, P.C. Risk assessment for Cryptosporidium: A hierarchical Bayesian analysis of human dose response data. Water Res.
**2001**, 35, 3934–3940. [Google Scholar] [CrossRef] - World Health Organisation. Guidelines for Safe Recreational Water Environments. Volume 2: Swimming Pools and Similar Environments; World Health Organisation: Geneva, Switzerland, 2006; Available online: https://www.who.int/water_sanitation_health/publications/safe-recreational-water-guidelines-2/en/ (accessed on 26 April 2021).
- Wood, M.; Simmonds, L.; MacAdam, J.; Hassard, F.; Jarvis, P.; Chalmers, R.M. Role of filtration in managing the risk from Cryptosporidium in commercial swimming pools—A review. J. Water Health
**2019**, 17, 357–370. [Google Scholar] [CrossRef] [PubMed] - Croll, B. Decontaminating swimming pools and managing Cryptosporidium. Recreation
**2004**, 2004, 32–35. Available online: https://poolsentry.co.uk/s/Croll-2004.pdf (accessed on 26 April 2021). - Gage, S.D.; Ferguson, H.F.; Gillespie, C.G.; Messer, R.; Tisdale, E.S.; Hinman, J.J., Jr.; Green, H.W. Swimming Pools and other Public Bathing Places. Am. J. Public Health
**1926**, 16, 1186–1201. Available online: https://ajph.aphapublications.org/doi/pdf/10.2105/AJPH.16.12.1186 (accessed on 26 April 2021). [CrossRef] [PubMed] - Pool Water Treatment Advisory Group. Code of Practice for Swimming Pool Water; Pool Water Treatment Advisory Group: Loughborough, UK, 2017; Available online: https://www.pwtag.org/code-of-practice/ (accessed on 26 April 2021).
- Kanan, A.; Karanfil, T. Formation of disinfection by-products in indoor swimming pool water: The contribution from filling water natural organic matter and swimmer body fluids. Water Res.
**2011**, 45, 926–932. [Google Scholar] [CrossRef] [PubMed] - Ratajczak, K.; Pobudkowska, A. Pilot Test on Pre-Swim Hygiene as a Factor Limiting Trihalomethane Precursors in Pool Water by Reducing Organic Matter in an Operational Facility. Int. J. Environ. Res. Public Health
**2020**, 17, 7547. [Google Scholar] [CrossRef] [PubMed] - Stauder, S.; Rodelsperger, M. Filtration particles and turbidity in pool water. Removal by in line filtration and modelling of daily courses. In Proceedings of the Fourth International Conference Swimming Pool and Spa, Porto, Portugal, 15–18 March 2011; pp. 71–77. Available online: https://poolsentry.co.uk/s/Stauder-S-and-Rodelsperger-M-2011-filtration-particles-and-turbidity-in-pool-water-removal-by-in-lin.pdf (accessed on 26 April 2021).
- Gordon, I.; Inglis, S. Great Lengths: The Historic Indoor Swimming Pools of Britain; English Heritage: Swindon, UK, 2009. [Google Scholar]
- Alansari, A.; Amburgey, J.; Madding, N. 2018 A quantitative analysis of swimming pool recirculation system efficiency. J. Water Health
**2018**, 16, 449–459. [Google Scholar] [CrossRef] [PubMed][Green Version] - Cloteaux, A.; Gérardin, F.; Midoux, N. Influence of swimming pool design on hydraulic behavior: A numerical and experimental study. Engineering
**2013**, 5, 511–524. [Google Scholar] [CrossRef][Green Version] - Chappell, C.L.; Okhuysen, P.C.; Sterling, C.R.; DuPont, H.L. Cryptosporidium parvum: Intensity of infection and oocyst excretion patterns in healthy volunteers. J. Infect. Dis.
**1996**, 173, 232–236. [Google Scholar] [CrossRef] [PubMed][Green Version] - Lu, P.; Amburgey, J.E. A pilot-scale study of Cryptosporidium-sized microsphere removals from swimming pools via sand filtration. J. Water Health
**2016**, 14, 109–120. [Google Scholar] [CrossRef] [PubMed][Green Version] - Morais, I.P.; Tóth, I.V.; Rangel, A.O. Turbidimetric and nephelometric flow analysis: Concepts and applications. Spectrosc. Lett.
**2006**, 39, 547–579. [Google Scholar] [CrossRef] - Amburgey, J.E. Optimization of the extended terminal subfluidization wash (ETSW) filter backwashing procedure. Water Res.
**2005**, 39, 314–330. [Google Scholar] [CrossRef] [PubMed] - Keuten, M.G.A.; Schets, F.M.; Schijven, J.F.; Verberk, J.Q.J.C.; Van Dijk, J.C. Definition and quantification of initial anthropogenic pollutant release in swimming pools. Water Res.
**2012**, 46, 3682–3692. [Google Scholar] [CrossRef] [PubMed] - Falk, R.A.; Blatchley, E.R.; Kuechler, T.C.; Meyer, E.M.; Pickens, S.R.; Suppes, L.M. Assessing the Impact of Cyanuric Acid on Bather’s Risk of Gastrointestinal Illness at Swimming Pools. Water
**2019**, 11, 1314. [Google Scholar] [CrossRef][Green Version]

**Figure 1.**Effect of filter efficiency (E) on the removal of dirt particles from pool water (expressed as percentage of particles remaining) following successive water-turnover cycles, based on the Gage–Bidwell Law of Dilution.

**Figure 2.**Effect of the number of bathers entering the pool per unit volume of water flow through the filtration system on pool water equilibrium turbidity (NTU), assuming different dirt loadings per bather (solid line K

_{p}= 0.25 NTU (bather m

^{−3})

^{−1}; dashed line K

_{p}= 0.65 NTU (bather m

^{−3})

^{−1}) and different filtration efficiencies (E). The x-axis is the ratio of the number of bathers entering the pool to the volume (m

^{3}) of water flow through the filtration system (i.e., B/Q from Equation (12)) and value of 0.58 is equivalent to 1.7 m

^{3}water flow through the filtration system per bather.

**Figure 3.**Effect of filtration efficiency (E) on the rate at which turbidity approaches equilibrium following a change in bathing load as the number of water-turnovers increases.

**Figure 4.**Comparison of the measured time course of turbidity (NTU) (dashed line) over 7 days with the prediction (solid line) made using Equations (12) and (14) and with values of K

_{p}= 0.35 NTU (bather m

^{−3})

^{−1}, T

_{w}= 1.06 h, and E = 0.9. The number of bathers entering the pool each hour was derived from the recorded daily bather numbers, and an assumed frequency distribution during the opening hours. Based on data of Stauder and Rodelsperger [13].

**Figure 5.**Empirical relationship between daily bathing load and the measured peak turbidity (NTU) over a wide range of daily bather numbers, based on the 21 consecutive days of data presented in Figure 1 of Stauder and Rodelsperger [13]. The solid line shows the comparison with the peak turbidity if the turbidity was at equilibrium with the peak bathing load (Equation (12)).

**Table 1.**Effect on the average concentration of dissolved solids (or particles) by removing water from a pool one container-full at a time and replacing the water removed with the same quantity of pure water, thereby progressively diluting the water in the pool.

State | Cumulative Fraction of Pool Volume Removed | Average Concentration (C) in Pool Water after Mixing |
---|---|---|

Starting state | 0 | C = C_{o} |

After first container | 1/3 | C = (1 − 1/3) C_{o} |

After second container | 2/3 | C = (1 − 1/3) (1 − 1/3) C_{o} |

After third container | 1 | C = (1 − 1/3) (1 − 1/3) (1 − 1/3) C_{o} |

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**MDPI and ACS Style**

Simmonds, L.P.; Simmonds, G.E.; Wood, M.; Marjoribanks, T.I.; Amburgey, J.E. Revisiting the Gage–Bidwell Law of Dilution in Relation to the Effectiveness of Swimming Pool Filtration and the Risk to Swimming Pool Users from *Cryptosporidium*. *Water* **2021**, *13*, 2350.
https://doi.org/10.3390/w13172350

**AMA Style**

Simmonds LP, Simmonds GE, Wood M, Marjoribanks TI, Amburgey JE. Revisiting the Gage–Bidwell Law of Dilution in Relation to the Effectiveness of Swimming Pool Filtration and the Risk to Swimming Pool Users from *Cryptosporidium*. *Water*. 2021; 13(17):2350.
https://doi.org/10.3390/w13172350

**Chicago/Turabian Style**

Simmonds, Lester P., Guy E. Simmonds, Martin Wood, Tim I. Marjoribanks, and James E. Amburgey. 2021. "Revisiting the Gage–Bidwell Law of Dilution in Relation to the Effectiveness of Swimming Pool Filtration and the Risk to Swimming Pool Users from *Cryptosporidium*" *Water* 13, no. 17: 2350.
https://doi.org/10.3390/w13172350