This provides a useful tool for assessing the performance of a pool in terms of the likely peak turbidity, and which could also be used to inform those responsible for developing guidelines for pool operation.

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

The principle stated by Gage and Bidwell in 1926 [

9] using the term ‘dirt’, but applied here to turbidity, is that if a constant input of turbidity is maintained indefinitely, then the pool water turbidity will rise until the rate of removal of turbidity by filtration (which rises as the turbidity of water being delivered to the filter increases) matches the rate of input.

Turbidity is measured by nephelometry [

6], based on the measurement of scattered light by particles in a sample, and expressed in units of nephelometric turbidity unit (NTU). The intensity of the scattered radiation is related to the intensity of the incident radiation and the concentration of particles that are causing the scattering [

19]. In this analysis, we shall consider the turbidity of water expressed in NTU as a concentration resulting from the quantity of turbidity-forming particles introduced by bathers. Therefore, the rate at which turbidity is removed is equal to the product of the rate of delivery of turbidity-forming particles to the filter (i.e., the pool water NTU multiplied by the circulation rate, Q, in m

^{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).

The hourly input of turbidity will be the product of the number of bathers entering the pool per hour (B) and the quantity of turbidity-forming particles added on average by each bather (K

_{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):

In the analysis presented here, the values of B and Q are unequivocal, and the assumption is that they are kept constant. However, the values of K_{p} and E are more ambiguous and require further discussion.

The value of E depends on a number of factors, including the particle size [

7], and would be expected to have a lower value if being used in the context of

Cryptosporidium oocyst removal than for the removal of turbidity [

7]. The value of E may also change with time, due to fluctuations during the course of a day (as the dirtiness of the water changes), and possible changes in performance of the filter media over periods of several days during the backwash cycle [

20]. However, in the context of establishing the equilibrium turbidity during a period of constant bathing load the value of E for a filter would be expected to be relatively stable during the period that the equilibrium is being approached, assuming that other factors that affect the efficiency (e.g., coagulant dosing rate and the filtration velocity) remain constant.

In the context of turbidity, filtration efficiencies of 0.9 have been reported [

13] for a pool with dual media anthracite/sand filters and coagulant (PAC) dosing optimised to minimise the measured filtrate turbidity. Where coagulation is poor or absent, filtration efficiencies of 0.2 (or less) are likely [

4,

18]. We shall examine the predicted equilibrium turbidity in scenarios used in

Figure 1, where the filter efficiencies for turbidity removal during periods of protracted heavy bathing load will be E = 0.9, 0.5 or 0.2. This covers the range that most swimming pools are likely to be operating in.

There is little information on the likely values for the average amount of turbidity-forming material introduced per bather into pool water. Two approaches have been used to obtain this information. The first is to measure the rise in turbidity in a small body of water (e.g., a spa) following entry of by a known number of bathers, where the input per bather is calculated by dividing the rise in NTU by the number of contributing bathers per m

^{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.

The application of Equation (12) as a guide for pool operators is illustrated by

Figure 2, which shows values for the equilibrium (i.e., the maximum possible) turbidity for several pool scenarios. To facilitate a comparison between very different pools, the x-axis shows the ratio of the number of bathers entering the pool to the volume (m

^{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.

The possible scenarios in

Figure 2 also cover a range of filtration efficiency (E = 0.9, 0.5 or 0.2) [

7]. These are shown in combination with relatively dirty or relatively clean bathers using K

_{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.

The concept of the number of bathers per m

^{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):

This value of 1.7 m

^{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.

Note also that the model predicts that an upper limit guideline of 0.58 bathers m

^{−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

The preceding analysis considered the turbidity reached in swimming pool water when in a state of equilibrium achieved in the case where bathers continue to enter the pool indefinitely at a constant rate. This leads to the question whether bathing loads are ever sustained for long enough for the equilibrium turbidity to be achieved. For example, the measured diurnal courses of turbidity for the heavily used 690 m

^{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.

Modelling using the Gage–Bidwell principles described above involves essentially the same problem as modelling the removal of

Cryptosporidium oocysts following an AFR using Equation (10). The latter describes the transition from some initial concentration (C

_{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:

where the left side of Equation (14) represents the concentration of particles (or the NTU) expressed as a fraction of the step change from the original concentration (C

_{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.

Hence, the progress towards the equilibrium turbidity under conditions of constant bathing load is related to the number of particle-turnovers, irrespective of pool size. The implications are illustrated in

Figure 3, which shows, for three filtration efficiencies, how rapidly the turbidity changes towards a new equilibrium value following a change in bathing load. For example, with relatively good filtration (E = 0.9), 90% of the change towards the new equilibrium turbidity occurs after 2.6 water-turnovers. Hence, for a spa with a 10 min water-turnover time, 90% of the transition towards the equilibrium NTU is predicted to be achieved in 26 min. This suggests that a spa is quite likely to approach the equilibrium NTU predicted for the maximum allowable bathing load. However, for a leisure pool with 1.5 m average-depth and 3 h water-turnover time, it would take 7.8 h of continuous maximum bathing load for the turbidity to reach 90% of the change from C

_{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.

If the filters were only removing 50% of the turbidity from water passing through the filters, the equilibrium turbidity would be higher, but the time taken to reach 90% of the change from C

_{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.

It should be noted that

Figure 3 can also be applied to predict of the rate of reduction in turbidity during a recovery period when bathers are absent from the pool, and where the turbidity of the pool water will fall from its value at the start of the recovery period towards near zero. For example, using Equation (14), a heavily used water park pool with a 2 h particle-turnover would expect a 40% reduction in turbidity after just 1 h of recovery time, increasing to 63% and 78% removal of turbidity after 2 and 3 h, respectively. The implication is that for a pool with good filtration there is little benefit in terms of particle removal of recovery periods longer than a couple of turnovers.