## Modeling and simulation

Chronic daily intake (CDI) through inhalation of THMs in the shower room is predicted for the THM concentration in the shower air (*Cair**i*). The calculation of *Cair**i* generally follows Little’s theory which assesses *Cair**i* for a given bathing duration as the average of initial (before the showering event, *C**0.i*) and final (at duration = t, *C**t.i*) concentration (Little, 1992).

\({Cair}_{i}=\frac{{C}_{0,i} + {C}_{t,i}}{2}\) Eq. 1

where, *i* represents individual THMs (*i* = 1 to 4 for TCM, BDCM, DBCM, and TBM). For the first shower, *C**0.i* was taken as zero, and for the successive events, the *Cair**i* of the previous bath was considered as the initial concentration. This was adopted by assuming that back-to-back showering events happen in the shower rooms, which is true in the case of student halls. Therefore, for the second bath of a given duration, *Cair**i* will be equal to ¾ th of *C**t.i*. Hence, for *n*th showering event,

\({Cair}_{i}=\frac{{2}^{n} - 1}{{2}^{n}} \times {C}_{t,i}\) Eq. 2

The term *C**t.i* represents the THM concentration (µg m− 3) in the air at a shower duration, which was determined using the following formula (Ahmed et al., 2019; Kujlu et al., 2020);

\({C}_{t,i}=(1-{e}^{-b\times t}) \times \frac{a}{b}\) Eq. 3

where *t* is the showering duration (min),

\(b=\frac{1}{Vs}\times \left\{\right(\frac{{Q}_{w}}{H}\left) \right(1-{e}^{-N})+{Q}_{g}\}\) , Eq. 4

and

\(a=\frac{1}{Vs}\times \left\{\right({Q}_{w} \times {Cw}_{i } \times (1-{e}^{-N})\}\) , Eq. 5

where *Vs* is the volume of the bathroom (L), *Qw* is the water flow rate in the bathroom (L min− 1), *H* is unitless Henry’s constant at 40℃ for each THM, *Qg* is the airflow rate in the shower (L min− 1), *Cw**i* is the concentration of *i*th THM in the shower water (µg L− 1), and *N* is the non-dimensional overall mass transfer coefficient. *N* can be calculated as;

\(N=\frac{KoLA}{{Q}_{w}}\) Eq. 6

where *KoLA* is the overall mass coefficient of each THM (L min− 1) (Table 2).

The CDI of THMs by an individual through inhalation was determined using the US EPA guidelines of risk assessment, which is shown below (USEPA, 1980);

\({CDI}_{inh,i}=\frac{{Cair}_{i} \times Er \times t \times R \times F \times EF \times ED \times CF}{BW \times AT}\) Eq. 7

where, *CDI**inh,i* is the inhalation chronic daily intake of *i**th* THM (mg kg− 1 day− 1), *Er* is the absorption efficiency of THMs through the respiratory system, *R* is the breathing rate (m3 min− 1), *F* is the showering frequency (events day− 1), *EF* is the exposure frequency (day year− 1), *ED* is the exposure duration (years), *CF* is the conversion from µg to mg (0.001), *BW* is the bodyweight of the individual (kg), and *AT* is the averaging time (days).

Similarly, the CDI through oral as well as dermal pathways were determined using the following equations (Chowdhury, 2013; Téllez Tovar and Rodríguez Susa, 2020);

\({CDI}_{ing,i}=\frac{{Cw}_{i }\times IR \times EF \times ED \times CF}{BW \times AT}\) Eq. 8

\({CDI}_{der,i}=\frac{{Cw}_{i }\times SA \times Pd \times t \times F\times EF\times ED}{BW \times AT}\) Eq. 9

where, *CDI**ing,i* and *CDI**der,i* are respectively the chronic daily intake of *i*th THM through oral and dermal routes, *IR* is the drinking water ingestion rate (L day− 1), *SA* is the skin surface area (m2), *Pd* is the permeability of THMs through human skin (m min− 1), *t* is showering duration (min events− 1), and *F* is showering frequency (events day− 1). The CDI was found by summing up respective individual CDIs, ie,.

\(Total CDI=\sum _{i=1}^{4}\sum _{j=1}^{3}{CDI}_{i,j}\) Eq. 10

Where *i* represents 4 THM species (TCM-TBM) and *j* represents 3 exposure routes (oral, dermal, and inhalation). Hazard quotient (HQ) and hazard index (HI) were assessed using the following equations (Mosaferi et al., 2021).

\({HQ}_{i}=\frac{{CDI}_{i}}{RfD}\) Eq. 11

Where, *RfD* is the chronic reference dose of THMs (mg kg− 1 day− 1), which were taken from USEPA IRIS and RAIS data directories (Table 2) (USEPA IRIS, 2021; USEPA RAIS, 2021). The inhalation reference dose of THMs is under study (USEPA, 2001) and is not yet updated in the IRIS database. When the RfD or cancer slope factors were not available for dermal or inhalation routes, the same for oral exposure was considered as many previous studies (Chowdhury et al., 2020). Total HI through any particular exposure route *j* was found by adding HQ of each THM through that route.

\({HI}_{j}={\sum }_{i=1}^{4}{HQ}_{i,j}\) Eq. 12

similarly, the overall hazard index (*HI**T*) was calculated as follows:

\({HI}_{T}=\sum _{i=1}^{4}\sum _{j=1}^{3}{HQ}_{i,j}\) Eq. 13

In the case of successive showers, total CDI and HI were calculated for both females and males using the respective inhalation CDI and the final result was correlated against the health criterion as hazard index of unity.

## Adjustment for water temperature

Generally, warm water of 35–45 ℃ is preferred for showers and baths, and for that cold (room temperature) and hot waters are mixed (Chowdhury et al., 2020). The term *Cw**i* in Equations 5, 8, and 9 is the THM concentration in cold tap water, which needs to be modified for inhalation and dermal HI to include the result of mixing with heated water. Ingestion HI can be left as it is since generally cold or room temperature water is preferred over hot water for consumption. This modification was admitted as the THMs formation continues in the tap water since water purification systems leave residual free chlorine in the supply water to protect the future contamination and natural organic matters are not completely removed by conventional treatment units. THM formation follows complex mechanisms, and it varies depending on water chemistry, residual chlorine, water temperature, and several other parameters (Padhi et al., 2019b). Therefore, it is difficult to incorporate all the reaction parameters, however, the effect of water temperature can be considered if the THM growth rates in both hot and cold water are determinable. To do that, the *Cw**i* in the current study was altered using a model suggested by Chowdhury et al (2020). Using the THM growth rate of hot water (*k**h*) and that of cold water (*k**w*), THMs in the mixed water can be predicted as;

\({Chw}_{i}={Cw}_{i}\times {e}^{({k}_{h}-{k}_{w})t}\) Eq. 14

where *Chw**i* is the THM concentration in mixed water (µg L− 1), *CW**i* is the THM concentration in cold tap water, *k**h* and *k**w* (S− 1) were determined using the following equation.

\(k=0.0011{e}^{0.0407T}\) Eq. 15

where, T is the water temperature (℃), which was taken as 10 to 20℃ with a median of 15℃ for cold tap water since the sampling was carried out in the winter season. The parameters in all of the above equations, their values, and units are given in Table 2. Since these parameters are prone to uncertainty, triangular distribution was assumed for all of them, which is explained below.