b

b 

## RESEARCHTRIANGLE INSTITUTE
Center for Environmental Analysis July 

### 1, 1996 

### MEMORANDUM

TO: Rebecca Daiss 

### FROM: Jeff Coburn SUBJECT: Emission Flux Equationsfor Showering


### Purpose The purpose of this memorandum is to document the equationsused for estimating the emissions during showeringin a closedbathroom.



### BackProuncUTheorv Little (1992a)presented an approach for applying the two-resistancetheory to model emissions in showers. However, the approach developedby Little onlyprovided 

### a means to estimatethe gas and liquid phase mass transfer coefficientsfrom experimentally observed overallmass transfer coefficients for compoundsof with differing Henry's law constants. The approach developed here attemptsto estimate the gas and liquid phase mass transfer coefficientsusing Nusselt numbers correlationsfor falling drops. 

### A comparison of the "fallingdrop model" and the models proposed by McKone (1987) 

### and Little (1992a)is provided in the "ModelResults" section. The basic solution algorithm is to estimate the velocity of the water leaving the nozzle, calculate the averagedroplet sizeby assuming the nozzle velocityis the terminal velocity,calculate the gas, liquid, and overall mass transfer coefficientsfor the falling drop, then calculate the fraction of contaminant emitted based on the time required for the drop to fall assuming the contaminantconcentrationin 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 arereported 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 orificemeter and estimate a water line pressure in =238(~p)'.~ showerhead (simplifies to ~[cdsec] where AP 

### = water line pressure - atmosphericpressure, in psi); select one that correlates the existing data well. 

### A nozzle velocity was selected based on the ability to correlatethe existing data. As many of the studies yielded an overall mass transfer coefficient 

### (GLA) of approximately 10to 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 KOLA linearly with showerflow 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 showerhead 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 "intermediatelaw" (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"Cy and rearranging the equation to solve for the droplet diameter as a function of terminal velocity, the "intermediatelaw" 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 cdsecfor water droplets falling in air at 35 "C), "Newton'slaw" applies. Therefore, for nozzle velocities set above 600 cdsec, the following equation should be used to calculate the droplet diameter.
2 Ix 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 Schmidtnumbers (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 numberfor phase f (dimensionless) k, = mass transfer coefficientfor phase f (cm/sec) d, = dropJetdiameter (cm) 

### Dab,, = diffusivity of constituentin phase 

### f (cm2/sec). Re, = d, v, 

### pf/pf =Reynoldsnumberforphase f (dimensionless) Sc, = p, 

### /(pfDab,f) = Schmidtnumberforphase f (dimensionless) v, = terminal velocity (cm/sec2) pf= densityforphase f, (g/cm3) p, = viscosity forphase f, (g/cm-sec). Equation 2 isused to estimateboth 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 gasphase (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"McKoneequation" for the overall mass transfer coefficientis:

where, 

## p = proportionality constant 

### (cm-set)-".

In a correspondence to the editor, Little (1992b) suggested that another proportionality constant should be included in the McKoneequation 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 "Littleequation"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 coefficientfor 

### 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 dimensionlessoverall mass transfer coefficient, 

### N, can be defined as: pu' = Kol (6/d,) (h/v,)
4where 

### N = dimensionless overallmass transfer coefficient (dimensionless) h = nozzle height (cm)
Equation 

### 6 can be integrated assumingthe constituent concentration in the gas phase is constant over 

### the time frame of the dropletfall. Thefraction 

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

where fern = fraction of contaminantemitted from the droplet (dimensionless) 

### tout = droplet contaminantconcentration at shower floor/drain (mg/L)

cin = droplet contaminantconcentration entering the shower (mg/L) 

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

Note: Equation 8 is equivalentto the equationpresented in Little (1992a, Equation 

### 5)for liquid phase concentration change. Using a fraction emitted allows subsequentequations to be written in different,but self-consistentunits. 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 exchangerate between bathroomhouse 

### (L/min)

yh = gas phase contaminantconcentration in the house (mg/L) Vh= volume of the house excluding the bathroom and shower 

### (L) Qgh = volumetric gas exchangerate between house/outdoors 

### (L/min) yam= gas phase contaminantconcentration 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 11can be simultaneously solved using a simple finite difference numerical integration. The various gas phase concentrationscan be calculated for any point in time, and an average concentrations can be calculated for any time period.



### Model Results Little (1992a) summarizedthe 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 1presents 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 equationprovided the same overall mass transfer coefficients for benzene. Minimaleffort 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 coefficientsfor 

### 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-minuteperiod). As the average shower 

### air concentration

### is approximately one-half the shower concentration at the 10thminute, the recommended model parameters (v, =400 cdsec) actually only overestimates the average 10-minute concentrations reported by Jo et al. by a factor of 

### 1.5to 2. This factor is within the range of differences in the overall mass transfer coefficients as determinedby water data comparedto air data [as seen in Study T reported in Little (1992a)l. Furthermore, Jo et
6al. reported a recovery for "theentire breath sampling system"

### to be 63 percent for chloroform. From the informationreported 

### in Jo et al. it is difficult to determineif (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 

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

## pprovides correlation of the volatile organic emission data [withina factor of 2. Although the overall mass transfer coefficient will directlyinfluence 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 differentmodel predictions for contaminants exhibiting a wide range 

### of Henry's law constants, the total 

# , mass of contaminant emitted over a 10-minute showerwas calculated using the droplet 

# I model, the hcKoneequation, the McKone equation as modifiedby Little for 10different contaminants that exhibit a range of Henry's law constant values. The constantsfor 

## p in the Little equation was selected to yield the same overall mass transfer coefficient as the 1 other models for benzene. The 10contaminants 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 95thpercentile 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 contaminantvolatilizatin 

### in
, showers. 

### Environ. Sci. Technol. 26(7):1341-1349. Little, J.C. 1992b. Correspondence. Commenton "Humanexposure to volatile organic compoundsin household tap water: the indoor inhalation pathway." 

### Environ. Sci. Technol. 26(4):836-837. McKone,T.E. 1987. Human exposure to volatile organic compoundsin household tap water: the indoor inhalation pathway. 

### Environ. 

### Sci. Technol. 21(12):1 194-1201.
7TABLE 

### I. COMPARISONOF OVERALLMA 

### 5 TRANSFERCOEFFICENTS 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

8% 

### - 9.7 

# 2

I I I I 2.5 

# I 9.5 9.5

**Calculated assuming a terminal velocity of 400 cdsecusing the "dropletmodel." ***Calculated assuming 

## p = 

### 216 using the "McKoneequation."

8 

# 11TABLE 

### 2. MODEL PREDICTIONS

### FOR TOTAL MASS EMITTED

*Calculated assuming 

## p =216 using the "McKoneequation." **Calculated assuming a terminal velocity of 400 cdsecusing the "dropletmodel." ***Calculated assuming 

## p = 276 and a = 70using the "Littleequation."

9