b 

b 

RESEARCH TRIANGLE INSTITUTE 

Center for Environmental Analysis 

July 1, 1996 

MEMORANDUM 

TO: Rebecca Daiss 

FROM: Jeff Coburn 

SUBJECT: Emission Flux Equations for Showering 

Purpose 

The purpose of this memorandum is to document the equations used for 

estimating the emissions during showering in a closed bathroom. 

BackProuncUTheorv 

Little (1992a) presented an approach for applying the two-resistance theory to 

model emissions in showers. However, the approach developed by Little only provided a 

means to estimate the gas and liquid phase mass transfer coefficients from experimentally 

observed overall mass transfer coefficients for compounds of with differing Henry's law 

constants. The approach developed here attempts to estimate the gas and liquid phase 

mass transfer coefficients using Nusselt numbers correlations for falling drops. A 

comparison of the "falling drop model" and the models proposed by McKone (1987) and 

Little (1992a) is provided in the "Model Results" section. 

The basic solution algorithm is to estimate the velocity of the water leaving the 

nozzle, calculate the average droplet size by assuming the nozzle velocity is the terminal 

velocity,calculate the gas, liquid, and overall mass transfer coefficients for the falling 

drop, then calculate the fraction of contaminant emitted based on the time required for 

the drop to fall assuming the contaminant concentration in the air is constant over the 

time frame of the droplet fall. Finally, the shower and bathroom air concentrations are 

calculated using mass balance considerations. 

As only volumetric flow rates are reported in the shower experiment literature, the 

3040 Cornwallis Road Post Office Box 12194 Research Triangle Park, North Carolina 27709-2194 USA 

Telephone 919 541-5995 Fax 919 541-7155 

6% 
nozzle velocity becomes the primary assumption. Options for estimating a nozzle 

velocity include: 

use a typical shower flow rate and estimate the orifice crosssectional area of the shower; 

1 0 use Equation 5-12 from Perry and Chilton (1973) for flow 

through an orifice meter and estimate a water line pressure in 

shower head (simplifies to ~[cdsec] = 238(~p)'.~where AP = 

water line pressure - atmospheric pressure, in psi); 

select one that correlates the existing data well. 

A nozzle velocity was selected based on the ability to correlate the existing data. 

As many of the studies yielded an overall mass transfer coefficient (GLA)of 

approximately 10 to 20 L/min, as reported by Little (1992a), a nozzle velocity of 

400 cdsec (a AP of approximately 3 psi) provided a reasonable correlation of the data. 

From the data reported by Little, it also appears that the &,A increases slightly with 

increasing water flow rate. Using a set nozzle velocity will increase KOLAlinearly with 

shower flow rate. It may be possible to develop a nozzle velocity correlation that is 

weakly dependent on flow rate, but selecting a set nozzle velocity (or shower head 

pressure) appears to provide a reasonable correlation with the reported data. 

The droplet diameter was estimated from the approximate equations used to 

describe the experimental data correlation between the friction factor (drag coefficient) 

and the Reynolds from Bird et al. (1960, Figure 6.3-1, p. 192). For water droplets falling 

in air at 35"C, the "intermediate law" (Bird et al., 1960, Equation 6.3-21, p. 194) prevails 

between droplet velocities of 40 to 600 cdsec. Using the properties of water and air at 

35 "Cyand rearranging the equation to solve for the droplet diameter as a function of 

terminal velocity, the "intermediate law" equation simplifies to: 

d, = (~,0.*~~)/1,935[40 i v, I 600 cdsec] 

where 

d, = droplet diameter (cm) 

v, = terminal velocity (cdsec2),for 40 I v, i 600 cdsec. 

(Eqn. la) 

Above a Reynold's number of 500 (v, 2 600 cdsec for water droplets falling in air at 

35"C), "Newton's law" applies. Therefore, for nozzle velocities set above 600 cdsec, 

the following equation should be used to calculate the droplet diameter. 

2 

I 
x 

d, = (v?)/(2.6E+6) [v, 2 600 cdsec] (Eqn. lb) 

For forced convection around a sphere, Bird et al. (1960) provides a correlation of 

the Nusselt number to the Reynolds and Schmidt numbers (Bird et al., 1960, Equation 

21.2-25, p. 647) as follows: 

Nu, = (k, dp)/(Dab,,)= (2 + 0.6 Re:5 SC~.~~) (Eqn. 2) 

where 

Nu, = Nusselt number for phase f (dimensionless) 

k, = mass transfer coefficient for phase f (cm/sec) 

d, = dropJet diameter (cm) 

Dab,,= diffusivity of constituent in phase f (cm2/sec). 

Re, = d, v, pf /pf = Reynolds number for phase f (dimensionless) 

Sc, = p, /(pf Dab,f)= Schmidt number for phase f (dimensionless) 

v, = terminal velocity (cm/sec2) 

pf = density for phase f, (g/cm3) 

p, = viscosity for phase f, (g/cm-sec). 

Equation 2 is used to estimate both the gas and liquid phase mass transfer 

coefficients. The overall mass transfer coefficient is then calculated assuming Henry's 

law partitioning as follows: 

where 

KO,= overall mass transfer coefficient (cm/sec 

k, = mass transfer coefficient for liquid phase (cm/sec) 

k, = mass transfer coefficient for gas phase (cm/sec) 

H '= WRT = 40 H = dimensionless Henry's law constant (cdsec) 

H = Henry's law constant (atm-m3/mol) 

R = Ideal gas law constant = 8.205E-5 atm-m3/mol-"K 

T = temperature (OK) 

As there are several different aidliquid interfaces (falling drops, falling film on 

walls, and puddle at shower floor), McKone (1987) used only the Nusselt numbers 

functionality to the Schmidt number, and developed an equation for the proportionality 

of the overall mass transfer coefficient in terms of the air and liquid diffusivities. The 
"McKone equation" for the overall mass transfer coefficient is: 

where, 

p = proportionality constant (cm-set)-". 

In a correspondence to the editor, Little (1992b) suggested that another 

proportionality constant should be included in the McKone equation as follows: 

where, 

a = proportionality constant (cm-sec)-'h1/3. 

The main difference, therefore, in the current model compared to the McKone equation 

or the "Little equation" provided in Equation 5 is the value of a. The McKone equation 

assumes a = 1;Little suggests a = 70 (Little, 1992b); and the current model yields a = 4. 

Ramifications of these differences will be discussed in the model results. 

The contaminant loss rate from the water droplet can be expressed as follows: 

where 

C= liquid phase contaminant concentration (pg/cm3or mg/L) 

t= time (sec) 

.A= droplet surface area (cm2) 

V= droplet volume (cm3) 

kg= mass transfer coefficient for gas phase (cm/sec) 

Ys = gas phase contaminant concentration in the shower (pg/cm3or mg/L) 

The time required for the droplet to fall equals the nozzle height divided by the 

nozzle velocity. The ratio of the surface area to volume for the droplet is calculated as 

6/d, (Le., by assuming a spherical shape). Therefore, a dimensionless overall mass 

transfer coefficient, N, can be defined as: 

pu' = Kol(6/d,) (h/v,) 

4 
where 

N = dimensionless overall mass transfer coefficient (dimensionless) 

h = nozzle height (cm) 

Equation 6 can be integrated assuming the constituent concentration in the gas 

phase is constant over the time frame of the droplet fall. The fraction of contaminant 

emitted from a water droplet at any given time can therefore be calculated as: 

where 

fern= fraction of contaminant emitted from the droplet (dimensionless) 

tout = droplet contaminant concentration at shower floor/drain (mg/L) 

cin= droplet contaminant concentration entering the shower (mg/L) 

fsat= y,/(H’cin) = fraction of gas phase saturation (dimensionless) 

Note: Equation 8 is equivalent to the equation presented in Little (1992a, Equation 5) for 

liquid phase concentration change. Using a fraction emitted allows subsequent equations 

to be written in different, but self-consistent units. 

The change in the shower air Concentrationis influenced by the mass of 

contaminant lost by the water and the exchange of air between the shower and the rest of 

the bathroom. It is assumed that the bathroom is unventilated during the showering and 

bathroom activities. Consequently, the following equations apply: 

(Eqn. 9) 

(Eqn. 10) 

(Eqn. 11) 

where 

t= time (min) 

V, = volume of the shower (L) 

Qi= volumetric liquid flow rate (L/min) 

Qgs = volumetric gas exchange rate between shower/bathroom (Llmin) 

Yb = gas phase contaminant concentration in the bathroom (mg/L) 

5 
V, = volume of the bathroom excluding the shower (L) 

Qgb = volumetric gas exchange rate between bathroomhouse (L/min) 

yh = gas phase contaminant concentration in the house (mg/L) 

Vh= volume of the house excluding the bathroom and shower (L) 

Qgh= volumetric gas exchange rate between house/outdoors (L/min) 

yam= gas phase contaminant concentration in the atmosphere (mg/L). 

Note : Equations 9 and 10 are equivalent to the equations presented in Little (1992a, 

Equations 9 and 10) with yh = Ybin. 

Equations 8,9, 10, and 11 can be simultaneously solved using a simple finite 

difference numerical integration. The various gas phase concentrations can be calculated 

for any point in time, and an average concentrations can be calculated for any time 

period. 

Model Results 

Little (1992a) summarized the results of several shower studies and calculated the 

experimental overall mass transfer coefficients (i.e., K,,A). Most of the experimental 

overall mass transfer coefficients are based on the contaminant loss from the water, 

although some are based on air concentration data. Table 1 presents the overall mass 

transfer coefficients as reported by Little (1992) with those calculated using the falling 

drop model equations and the McKone equation. The McKone equation parameter p 

was selected so that the droplet model and the McKone equation provided the same 

overall mass transfer coefficients for benzene. 

Minimal effort has been taken at this time to verify Little's calculation of the 

overall mass transfer coefficients from the original sources. As the overall mass transfer 

coefficients for Jo et al. provided the worst correlation (overestimate the 10-minute 

concentrations reported by Jo et al. by a factor of 3 to 4 as Table l),the Jo et al. study (Jo 

et al., 1990) was reviewed. A study of the Jo et al. paper reveals that the reported 

concentrations are actually 10-minute average shower concentrations (one tenax trap 

sample over a 10-minute period). As the average shower air concentration is 

approximately one-half the shower concentration at the 10th minute, the recommended 

model parameters (v, = 400 cdsec) actually only overestimates the average 10-minute 

concentrations reported by Jo et al. by a factor of 1.5 to 2. This factor is within the range 

of differences in the overall mass transfer coefficients as determined by water data 

compared to air data [as seen in Study T reported in Little (1992a)l. Furthermore, Jo et 

6 
al. reported a recovery for "the entire breath sampling system" to be 63 percent for 

chloroform. From the information reported in Jo et al. it is difficult to determine if (or 

how) this recovery impacts the reported air concentrations, but if this is a mass balance 

recovery factor, the estimated average 10-min. concentrations are only overestimated by 

10 to 20 percent, not by a factor of 4 as expected from the data as reported in Table 1. 

Although a complete review of all of the referenced studies would be beneficial, it is 

concluded from the data in Table 1 and the above discussion, that a fixed terminal 

velocity or a fixed value for p provides correlation of the volatile organic emission data 

[withina factor of 2. 

Although the overall mass transfer coefficient will directly influence the initial 

rate of emissions, the Henry's Jaw constant will also affect the effective concentration 

driving force, which may limit the predicted emissions. To compare different model 

predictions for contaminants exhibiting a wide range of Henry's law constants, the total , 

mass of contaminant emitted over a 10-minute shower was calculated using the droplet I 

model, the hcKone equation, the McKone equation as modified by Little for 10 different 

contaminants that exhibit a range of Henry's law constant values. The constants for p in 

the Little equation was selected to yield the same overall mass transfer coefficient as the 1 

other models for benzene. The 10 contaminants were selected by sorting the current 

hemical properties data base on Henry's law values, determining the percent ranking, 

nd selecting the 5, 15,25, 35,45, 55, 65,75, 85, and 95th percentile constituents. The 

sults are presented in Table 2. 

\References 

Bird, R.B., W.E. Stewart and E.N. Lightfoot. 1960. Transport Phenomena. John Wiley 

& Sons. New York, New York. 

Jo, W.C., P.W. Clifford, and P.J Lioy. 1990. Routes of chloroform exposure and body 

burden from showering with chlorinated tap water. Risk Analysis. 10(4):575-580. 

Little, J.C. 1992a. Applying the two-resistance theory to contaminant volatilizatin in ,

showers. Environ. Sci. Technol. 26(7):1341-1349. 

Little, J.C. 1992b. Correspondence. Comment on "Human exposure to volatile organic 

compounds in household tap water: the indoor inhalation pathway." Environ. 

Sci. Technol. 26(4):836-837. 

McKone, T.E. 1987. Human exposure to volatile organic compounds in household tap 

water: the indoor inhalation pathway. Environ. Sci. Technol. 21(12):1 194-1201. 

7 
TABLE I. COMPARISON OF OVERALL MA 5 TRANSFER COEFFICENTS 

1 I I 

Reported Referenced Study*/ 

rate* height* KO,A* Compound (Wmin) 

11 StudyT I 13.5 I 1.7 

carbon tetrachloride 

tetrachloroethene 

I,1,1-trichloroethane 

chloroform - ~~ I

' 1,2,3-trichloropropane 

Study G 5 1.8 

trichloroethene 

choroform 

1,2-dibromo-3­

chloropropane 

Study H 13.7 1.6 

trichlorofluoromethane 

1,1,l-tricholorethane I 

tetrachloroethene 

trichloroethene 

Studv M 

trichloroethene I ~~ 

Study J 8.I 1.8 

chloroform ~ ~~ ~ ~~ 

*Data as reported in Little (1992a). 

9 - 15 

8 - 15 

2-6 

9 

Calc'd Calc'd 

Ko,A** Ko,A*** 

droplet McKone 

(L/min) (Wmin) 

13 13 

14 14 

10 11 

, 

5.2 5.2 

4-7 I 5.5 I 5.5 

I I 

25 -34 13 13 

21 -29 12 12 

26 12 12 

30 13 13 

2

8% - 9.7

I I I I 

2.5 I 9.5 9.5 

**Calculated assuming a terminal velocity of 400 cdsec using the "droplet model." 

***Calculated assuming p = 216 using the "McKoneequation." 

8 

11 
TABLE 2. MODEL PREDICTIONS FOR TOTAL MASS EMITTED 

*Calculated assuming p = 216 using the "McKoneequation." 

**Calculated assuming a terminal velocity of 400 cdsec using the "droplet model." 

***Calculated assuming p = 276 and a = 70 using the "Little equation." 

9