Am omq>. Hyi., Vol 41. No. 4. pp. 415-435, 1997
British Occupations! Hygiene Society
Publabed by Ebeyicr Science Ltd
Printed in Great Bntam
0003-4878/97 117 00 + 0.00
PII: S0003^878(96)00048-8
DEVELOPMENT OF A MODEL TO PREDICT AIR
CONTAMINANT CONCENTRATIONS FOLLOWING INDOOR
SPILLS OF VOLATILE LIQUIDS
P. H. Reinke and L. M. Brosseau
University of Minnesota, Division of Environmental and Occupational Health, Box 807 U M H C ,
420 Delaware St. SE, Minneapolis, MN 55455, U.S.A.
(Received 30 August 1996)
Abstract—A personal computer spreadsheet model which predicts air contaminant concentrations
following indoor spills of volatile liquids has been developed. Three mass transfer models are
compared for predicting evaporative flux in the model, the flat plate mass transfer theory,
Penetration Theory, and Mackay and Matsugu models. Two methods of predicting spill
temperature during evaporation are presented, the isothermal method, which assumes spill
temperature remains at ambient throughout evaporation, and the spill temperature method, which
predicts spill temperature using equations developed from a heat balance over the spill pool.
Dispersion is approximated in the model as a well-mixed room with short-circuiting. Model
equations are programmed into a Lotus 1,2,3 spreadsheet. Calculations of room concentration,
spill pool surface temperature, and spill area are made at consecutive 20-s intervals following the
spill. Model predictions are compared with concentration measurements made after two test spills
in a laboratory. The model gives a good first estimate of room concentration, performing best
when the spill is assumed to be isothermal and the Penetration Theory or Mackay and Matsugu
methods are used to predict evaporation rate. © 1997 British Occupational Hygiene Society.
Published by Elsevier Science Ltd
NOMENCLATURE
A^n
area of spill at time ( [m2]
^o»piu
area of spill at time zero [m2]
6,pi]i
depth of the spill [m]
b;
thickness of floor [m]
Cp«lr
heat capacity of air [J (kg-K)"']
Cp
heat capacity of spilled liquid [J (kg-K)"']
Z) A .j r
diffusivity moles of contaminant A in air [m2 s~']
GA
molar evaporation rate of A [kg-mol s~']
h
heat transfer coefficient [W (m 2 -K)"']
hmir
heat transfer coefficient of air contacting spill pool [W (m 2 -K)"']
he
heat transfer coefficient on other side of floor [W (m -K)~']
//v, p
latent heat of vaporization [J (kg-mol)"']
k^ r
thermal conductivity of air [W (m-K)~']
kr
thermal conductivity of floor [W ( m - K ) " ' ]
krp
flat
plate mass transfer coefficient [m s ~ ' ]
km
M a c k a y / M a t s u g u mass transfer coefficient [m s ~ ' ]
kp
Penetration mass transfer coefficient [m s " ]
L
flow
path length = length of spill [m]
MW^
molecular weight of A [kg (kg-mol)"']
NA
molar flux of component A [kg-mol A (m 2 -s)"']
Nu
Nussult Number for heat transfer [dimensionless] = hL/k,^
/> A
partial pressure of component A [Pa]
p\.
saturation vapour pressure of component A [Pa]
P
ambient pressure [Pa]
Pr
Prandtl number = (Cpfi/k) [dimensionless]
415
416
Q
R
Re L
S
I
7",jr
7"rer
Tipiu
u
l/ k
V
^room
^.pin
^oipin
yA
yA.
jAroom
yAln
/i nlr
pmv
pAiiq
P. H. Reinke and L. M. Brosseau
volumetric air flow [m 3 s ~ ' ]
universal gas constant [(m3-Pa) (kg-mol K.)"']
Reynolds Number for surface flow [dimensionless]
room short-circuiting factor [dimensionless]
time [s]
air temperature [K]
reference property temperature [K.]
spill pool temperature [K.]
velocity of air flowing across pool [m s~']
overall heat transfer coefficient [W (m 2 - K)" 1 ]
gas phase molar volume [m3 (kg-mol)" 1 ]
room volume [m3]
volume of spill pool [m3]
volume of spill pool at zero time [m3]
mole fraction of A [kg-mole A (kg-mole g a s ) " ' ]
mole fraction A at saturation [kg-mole A (kg-mole gas)" 1 ]
mole fraction of A in well-mixed air [kg-mole A (kg-mole gas)"']
mole fraction A in incoming air [kg-mole A (kg-mole gas)" 1 ]
air viscosity [kg (m-s)" 1 ]
air density [kg m~ 3 ]
density of liquid A [kg m~ 3 ]
INTRODUCTION
The occupational health and safety professional has a number of responsibilities
following an accidental indoor chemical spill. These include assessing exposures of
individuals in the room at the time of the spill and evaluating hazards before
emergency staff enter the area. Such assessments are troublesome because workers
are seldom wearing personal monitoring equipment during an accidental spill, and
measurements of airborne concentration cannot be made until a member of the
emergency staff has entered the contaminated area. A predictive model, in which
exposure can be calculated from estimates of room concentration and data on
contact time with contaminated air, can be a valuable tool for assessing exposures
and evaluating hazards following indoor chemical spills.
A number of works have investigated the evaporation rate of an open pool of
liquid. The model developed in this paper combines evaporation predictions with
contaminant dispersion predictions to estimate air concentration changes following
a small solvent spill in a laboratory room. The model can be combined with data on
contact time to estimate exposure and anticipate health effects.
MODEL DEVELOPMENT
Overview
To predict air contaminant concentrations resulting from evaporation of spilled
liquid, one must consider first the rate at which the contaminant enters the air
(generation) and second the movement of the vapour phase contaminant
throughout, the room (dispersion). Factors which influence generation and
dispersion are shown in Fig. 1.
Evaporation rate, or generation rate (GA), of a spilled liquid depends on the
evaporative molar flux of the contaminant (NA) and the surface area of the spill
(^spiii)- Molar flux is the moles of contaminant evaporating per unit area per unit
time and is a property of the air/liquid system. The surface area of the spill is the
Prediction of air contaminant concentrations
417
Spill Surface Temperature
Two Alternatives
1) Isothermal Method
2) Spill Temperature Method
1
Vapour
Pressure
PA"
1
Room
Mole
Fraction
Mass
Transfer
Coefficient
Saturation
Mole
Fraction
r™
1
*
i
Molar Flux
NA
.
Spill Area
••:•
AspJH
-.
r-..
•
,
Three Alternatives:
1) Flat Plate Mass Transfer
: 2) Penetration Theory
3) Mackay and Matsugu
1
1
Generation, GA
(Evaporation)
.
//
- ,
Dispersion
/
Concentratidn
Fig. 1. Framework for model development.
total area of liquid from which evaporation is occurring, and is influenced by the
liquid spilled, how and where it is spilled, and its evaporation rate. When a spill is
small and the liquid is not restrained, as with a laboratory spill, the surface area
decreases as liquid evaporates. To predict evaporation rate (GA) accurately, changes
in pool surface area must be estimated by the model.
As shown in Fig. 1, molar flux of evaporating contaminant (iVA) is a function of
the saturation mole fraction of the evaporating liquid (FA*)> the mole fraction of
contaminant in the room air (yAroomX and a mass transfer coefficient (k). The mass
transfer coefficient is influenced by the velocity of air across the liquid pool, the path
length of flow across the pool, and the density and viscosity of room air. Several
methods have been developed and validated for predicting a mass transfer coefficient
for an evaporating pool of liquid. These stem from empirical, semi-empirical, and
theoretical analyses, and are described in the Appendix.
Mole fraction of contaminant at the surface of an evaporating liquid (y\t,
saturation mole fraction) is a critical determinant of molar flux. Mole fraction is
calculated from vapour pressure (p^,) which is a function of temperature. Cooling
of a spilled liquid due to evaporation will significantly reduce vapour pressure,
reducing both molar flux and evaporation rate. Pool surface cooling must therefore
be estimated by the model to allow prediction of contaminant evaporation rate.
Two options are presented for predicting spill pool surface temperature. The first
method (isothermal spill) assumes that the surface temperature remains at the
418
P. H. Reinke and L. M. Brosseau
ambient temperature throughout evaporation. The second (spill temperature) uses a
sensible heat balance to predict changes in surface temperature with time.
Figure 1 shows that once the pool surface temperature is estimated, vapour
pressure of the liquid at that temperature can be calculated. Vapour pressure can be
converted to mole fraction and combined with an estimate of room mole fraction
and a mass transfer coefficient to predict molar flux. Finally, the molar flux is used
with an estimate of spill area to determine the generation rate, or evaporation rate,
of the spilled liquid.
When the generation rate is known, a dispersion model is required to estimate
room air concentrations. Dispersion is a function of the air movement patterns in a
room, and can be approximated using a number of available techniques. These
include the well-mixed box method, the near-field/far-field method (Hemeon, 1963),
and turbulent diffusion (Roach, 1981) methods.
In the model developed in this paper, a well-mixed box modified for shortcircuiting (ASHRAE, 1989) is used to describe dispersion. When this method is
used, it is assumed that room air is well-mixed with short-circuiting of a portion of
the air coming into the room. The well-mixed room with short-circuiting model is
used because it accounts somewhat for higher concentrations near the spill pool, and
because it yields relatively simple equations for contaminant concentration. In
addition, the model's only parameter, a short-circuiting factor, can be easily
understood and applied by the model user.
The generation rate predicted in the model can be used with near-field/far-field
and turbulent diffusion techniques to estimate spatial concentration variability more
accurately. Such combinations yield more complex models, however, and require
more detailed information on room air flows.
Contaminant generation, GA
To predict room concentration changes created by an evaporating spill, one must
first estimate the rate at which contaminant is evaporating from the spill (that is,
generation rate). Molar evaporation rate of a liquid pool (GA) can be written as the
product of molar flux of the contaminant (NA) and pool surface area 04,pin).
Evaporation rate = GA = NA(>4spiii) [(kg - mol) s~'j.
(1)
Molar flux is the rate of evaporation per unit surface area of the spill and is
predicted using mass transfer relationships. Pool surface area (Aspm) is the surface
area of the liquid pool from which evaporation is occurring, which decreases as the
spill evaporates. Since the surface area of the spill must be known to predict
evaporation (generation) rate, an equation to predict changes in A^u with time is
developed below.
Predicting area of an unrestrained spill. It is intuitive that the change in area of an
evaporating liquid pool will lie between two extremes: a pool which does not change
in area, only in depth; and a pool which does not change in depth, only in area. A
restrained pool of substantial depth will have constant area for much of its
evaporation time. This occurs when a liquid is contained in a tray, or when a diked
containment area is filled with liquid from a tank spill. An unrestrained spill like a
Prediction of air contaminant concentrations
419
small laboratory spill, however, will decrease in area with time changing the
evaporation rate in Equation (1).
In this analysis, it is assumed that an unrestrained small spill maintains a
constant depth, that is, decreases only in area and not in depth, as it evaporates.
This assumption is probably most accurate for spills on hard flooring (a surface with
high wettability), and is less accurate for spills on carpet or other absorbent surfaces.
To predict changes in pool surface area (^4,pin) with time, a mass balance over an
evaporating pool of liquid A is developed. Changes in liquid density (pAUq) a r e
assumed to be negligible over the range of temperatures encountered.
Rate In + Rate of Generation = Rate Out + Rate of Accumulation.
(2)
Rates are expressed as:
Rate In = 0
(3)
Rate of Generation = 0
(4)
Rate Out = GA{MWK) = NA A^MW^)
Rate of Accumulation = p AUq ^
=
PA]iqd i
(5)
^ E ^ l
(6)
where
^.pjn
GA
NA
^spiii
PAhq
^spiii
MWK
t
area of spill pool at time t [m2]
molar evaporation rate of A [(kg-mol) s~']
molar evaporative flux of A [kg-mol (m 2 - s)" 1 ]
volume of spill pool at time t [m3]
liquid density of A [kg m~ 3 ]
depth of the spill [m]
molecular weight of A [kg (kg-mol)" 1 ]
time [s]
Equations (2-6) are combined to obtain:
N^MWK)Am
\
dt
)
PMiq
Expanding the differential for spill volume yields:
tp
" V dt )
** \
An unrestrained spill is assumed to decrease only in area, so the change in depth
with time is zero.
dt
and
420
P. H. Reinke and L M. Brosseau
,
, . ^ospm Spilled Volume
bspin = constant = — ^ - = \
.
.
v
^oiii
Initial Area
Equation (8) simplifies to:
(10)
where Jospin, ^ospiii. and MWA are constant. Evaporative flux (NA) depends on
length of pool, pool surface temperature, and room concentration, all of which
change with time. NA can, however, be assumed to be nearly constant over a short
time interval. Equation (11) is integrated over a small time interval with constant NA
to obtain:
f
-NA(MWA)AOspii]
S pni
=
(12)
Equation (12) predicts spill area as a function of molar evaporative flux and
contaminant properties. Stated simply, it describes how the liquid pool 'shrinks up'
as it evaporates, ^i^ni is the area of the spill pool at the beginning of the time period
of integration (/]) and /lospiii is maximum pool area, which is assumed to occur
shortly after the spill.
The generation rate equation. Now that an equation for spill area has been
developed (Equation 12), it an be substituted for A^M in Equation (1), to obtain a
single expression for the evaporation rate, GA.
GA =
(]3)
Equation (13) can be used to predict generation rate provided that the molar
evaporative flux (A'A) is known or can be estimated.
Predicting molar evaporative flux. To calculate the generation rate (GA) from an
evaporating spill using Equation (13), molar evaporative flux (A^A) must be
predicted. Three approaches for predicting molar evaporative flux from pools of
liquid are compared for use in the model. These are the empirical flat plate mass
transfer method (Geankoplis, 1993), the theoretical Penetration Theory method
proposed by Higbie (1935), and the Mackay and Matsugu (1973) semi-empirical
method. These three methods are described in more detail in the Appendix. In each,
molar flux is a function of saturation mole fraction (yA,), mole fraction of
contaminant in the room air (yAroom), molar volume of the vapour phase
contaminant (V), and a mass transfer coefficient (k).
NA=f{yA.,
^Aroom, V, k).
(14)
Each of the three methods predicts a mass transfer coefficient from a different
function of air velocity (u), path length of air flow across the liquid pool (L),
Prediction of air contaminant concentrations
421
diffusion coefficient of the contaminant in air (D^r), and air properties (density,
pair, and viscosity, n^).
& = / ( « , L, £> Aal r, Main Pair)-
Predicting spill surface temperature. As indicated in Equation (14) and in Fig. 1,
the saturation mole fraction of the contaminant in air (y\t) must be known to
calculate molar flux (JVA.) and evaporation rate (GA)- According to Dalton's Law,
saturation mole fraction is a determined by the vapour pressure of the liquid,
y..=P-f
(16)
where P is ambient pressure. Vapour pressure is a strong function of temperature, so
that changes in liquid surface temperature will significantly influence evaporation
rate.
A pool of volatile liquid often cools to below room temperature because latent
heat is needed to vaporise the liquid which evaporates. Evaporative heat loss is
compensated by heat transferred to the liquid from the flowing air, from the floor,
and by radiation. Because cooling of the liquid pool influences the vapour pressure
and the evaporative flux, predictions of the liquid pool surface temperature must be
made before evaporation (generation) rate can be determined.
Two approaches to estimating spill surface temperature are used in this analysis.
The first assumes that spill pool surface temperature is unchanged during
evaporation. Pool surface temperature is set at ambient, and no additional spill
temperature calculations are made. This approach is referred to as the isothermal
spill method.
The second approach, the spill temperature method, uses a heat balance to
estimate changes in pool temperature with time. Heat transfer to and from a liquid
pool is illustrated in Fig. 2. A heat balance over the pool is written:
Rate of Accumulation = Rate In - Rate Out.
(17)
Rate of accumulation of heat in a liquid pool is written in terms of pool
temperature change to give:
1 rj-,
Rate of Accumulation = C p V^wP^wq — - ^ where
Cp
Fspii|
pAiiq
r spj |i
(18)
heat capacity of spilled liquid [J (kg-K)~']
volume of the spill pool at time t [m3]
density of the spilled liquid [kg m~ 3 ]
spill pool temperature [K]
Heat flow into the liquid includes convective flow from the air, conduction from
the floor, and radiative transfer from room surfaces. In this development, it is
assumed that heat transfer by radiation is negligible. This assumption is most
appropriate for rooms without windows, like the one used for the test spills
422
P. H. Reinke and L. M. Brosseau
AIR
Heat Transfer by Convection
= h a t r Aipiii (T.i r -
Evaporative Heat Transfer
Heat Transfer by R
N A A 8 pin H v a p
Heat Transfer by Conduction
=
^k ^ spill f a i r - T . p l l l )
AIR
(ROOM BELOW)
Fig. 2. Heat transfer to and from a liquid pool.
described later (see Fig. 4.) Radiation could influence spill temperature in a room
with windows, especially on a sunny day.
Other heat fluxes are estimated using the following convective heat transfer
relationships:
'spill)
(19)
R a t e In con duction = ^k^spill(?air ~ ^spill)
(20)
R a t e In conV ection — "air-^spillV-* air
where
(21)
and
O - Volumetric Mr Flow
B - Short Circuiting Factor
1- M o U Fraction
Fig. 3. Idealised well-mixed room with short-circuiting.
Prediction of air contaminant concentrations
423
overall heat transfer coefficient [W (m 2 -K)~']
transfer coefficient, other side of floor [W (m 2 -K)~']
thickness of floor [m]
thermal conductivity of floor [W (m-K)" 1 ]
air temperature [K]
Uk
hc
b(
kf
TaiT
Heat transfer coefficients in Equations (19) and (21) are obtained from an
empirical heat transfer correlation (Geankoplis, 1993):
Nu = ^
where
h
L
k air
Nu
Re L
Pr
= 0.0366(Re L )°- 8 (Pr) 1/3
(22)
bulk flow heat transfer coefficient [W (m 2 -K) - 1 ]
path length of flow [m]
thermal conductivity of air [W (m-K)~']
Nussult number for heat transfer [dimensionless]
Reynolds number of air flow [dimensionless]
Prandtl number of air [dimensionless] = (Cpair^ajr/kajr)
and
Cpa,,.
pmr
heat capacity of air stream [J (kg-mol-K)" 1 ]
viscosity of air stream [kg (m-s)" 1 ].
Heat leaving the spill due to evaporation is:
Rate Out = A^spUitfvap
where
H'vap
vap
(23)
latent heat of vaporization [J (kg-mol) ']
Substituting Equations (18), (19), (20) and (23) into Equation (17), the heat
balance becomes:
- C p KjPinpAiiq (
sp
j =
(24)
which may be written:
i
- r spi ||)
T,
at
=
NAHvap
I
F
O i i < p
{Uy + hml)
A 7o i i ( p
I y air - -<spill J
The depth of the unrestrained spill (ijpiu) is assumed constant. 7VA is function of
time, but is assumed constant over a small time interval. Equation (25) is integrated
to calculate a spill temperature after an increment of evaporation time, thus
T^spdi = r ^ , - - - ( r a i r - Tlspin - ^ ) exp[-a(r - tx)]
(26)
where 7"ispin is the temperature of the spill at the beginning of the time interval (/]),
P. H. Reinke and L. M. Brosseau
424
and
(27)
a =
(28)
In summary, the preceding equations allow estimation of the small spill
evaporation rate. Equations (26-28) predict the temperature of the liquid spill after
a short time interval of evaporation. Using rspin, the vapour pressure of the liquid
(PA*) can be determined from correlations or published data (see the Appendix).
Dalton's law is used to obtain the saturation mole fraction (Equation (16)), and if
the mole fraction of contaminant in the room 0>Aroom) is known or can be estimated,
the molar flux relationships can be used to predict JVA. With NA and Equation (13),
evaporation rate (GA) is calculated. Generation rate is now combined with a
dispersion approximation to predict room air concentrations after time increments
of spill evaporation.
Predicting dispersion
By way of illustration, consider the idealized room shown in Fig. 3. Dispersion in
this system may be approximated using a well-mixed box assumption modified to
account for short-circuiting of a portion of the incoming air.
A mass balance on the contaminant is used to obtain an equation which predicts
room air concentration after a period of spill evaporation. In the mass balance it is
EXHAUST
Ceiling Height:
2.74 m
AIR'
DOOR
AIR CURTAIN
0.307 m'/s EACH
7.62 m
Fig. 4. Test spill laboratory layout.
DIFFUSED
0.0646 m3/s
Prediction of air contaminant concentrations
425
assumed that the contaminant generation rate is low and incoming volumetric air
flow is approximately equal to outgoing volumetric air flow, Q. Flow into and out of
the contaminated space is Q{\ —S), where S is a short-circuiting factor (ASHRAE,
1989).
In this analysis, the short circuiting factor (5) represents that fraction of
incoming air which does not mix with the contaminated air in the well-mixed room.
In essence, the short-circuiting factor divides a room into two chambers or boxes.
All the contaminant is released into the first well-mixed box. Flow into and out of
this first chamber is Q{\ —S), and chamber volume is F room (l — S). Air in the second
chamber does not mix with air in the first, and does not contain any contaminant.
Flow into and out of the second box is QS, and its volume is KroomS.
For the low generation rate of a small spill, changes in room temperature and
pressure during evaporation are assumed to be negligible. When temperature and
pressure are constant, the ideal gas law states that the molar volume (V) of a gas is
also constant. In such conditions, the mass or mole balance on the vapour phase
contaminant can be written as a balance on the volume of vapour phase
contaminant, in which each term has units of m3 of contaminant (A) per second.
The balance is written:
Rate In + Rate of Generation = Rate Out + Rate of Accumulation.
(29)
Rates are expressed as follows:
(30)
Rate of Generation = GA V
(31)
Rate Out = (l-S)ej>Aroom
(33)
Rate of Accumulation = (1 - S) Vroom dyA'°°m
at
where
^Aroom
}>Ain
Q
S
Froom
GA
V
(32)
mole fraction A in mixed space [kg-mole A (kg-mole gas)" 1 ]
mole fraction A in incoming air [kg-mole A (kg-mole gas)~']
volumetric air flow rate [m 3 s ~ ' ]
short circuiting factor [dimensionless]
room volume [m3]
molar evaporation rate of component A [kg-mol s" 1 ]
gas phase molar volume of air and A [m3 (kg-mol)" 1 ].
Equations (29-33) are combined to give:
(1 - S)QyAin
+ GAV=(\-
S)QyAroom
+ (1 - 5) Vtoom ^
p
.
(34)
Assuming the incoming air contains no contaminant, Equation (34) becomes:
(35)
426
P. H. Reinlce and L. M. Brosseau
N o w , the expression for generation rate (GA) in Equation (13) is combined with
the differential contaminant mass balance above yielding the following first-order
linear differential mass balance.
Q
^room
ex
(l-S)VTO
P
7}
I I' ~ 'U
(36)
Equation (36) is integrated over a short time interval, assuming molar flux
constant over that interval, to obtain:
r
v
"i
- 01
(37)
(38)
(l-S)Vt
(39)
(40)
om is mole fraction of contaminant in room air at the start of the time interval
(t\). Equations (37-40) allow calculation of contaminant mole fraction in room air
using small time increments of spill evaporation.
In summary, all the model equations have now been presented. With G\ known
from the relationships in Section 3, changes in room mole fraction can be calculated
from Equations (36-40).
MODEL SUMMARY
Table 1 displays data entered into the model when the spill temperature method
is used. Parameters indicated with a superscript (a) are not required for the
isothermal spill method. Equations used in the model (spill temperature method) are
summarized in Table 2.
Equations are programmed into a Lotus 1,2,3 (Lotus Development Corporation,
Cambridge, MA, U.S.A.) spreadsheet. The user enters data and selects a short time
interval (t —1\) for the calculations. A time interval less than 20 s is recommended.
The program calculates and graphs predictions of room concentration against time.
The spill temperature program algorithm is as follows:
(1) Air properties V, paiT, fi^, Sc, DABir, and room properties hc and t/k are
calculated. These parameters remain constant for all calculations.
(2) Temperature dependent liquid properties pA,, Hvap are evaluated at rspin.
(^•piii is set to ambient temperature for the first time increment.) yA, is
calculated.
(3) Molar evaporative flux, JVA, is calculated.
(a) Flat plate mass transfer. k(p, and NA are calculated.
Prediction of air contaminant concentrations
427
Table 1. Spill temperature method input data
Parameter
Notation
Spill/room data
Volume spilled
Spill area
Room volume
Short-circuiting factor
Air velocity at floor
Volumetric air flow rate
Ambient air temperature
Ambient pressure
Floor thickness*
Floor composition*
^Ofpill
^OtpOl
V
' room
su
Q
T
P
be
Chemical properties
Molecular weight
Vapour pressure constants
Liquid density
Diffusion coefficient in air
Critical temperature
Critical pressure
Heat of vaporization*
Liquid heat capacity*
MW A
A, B, C, D
PAbq
£>A»lr
7*Ac
PAc
^vmp
cP
Air and floor properties
Viscosity at ambient T
Prandtl number at ambient 7*
Thermal conductivity at 7*
Thermal conductivity of floor*
Pr
Source
Spill information
Spill information
Building data
Data/estimate
Data/estimate
Building data
Building data
Building data
Building data
Building data
Chemical
Chemical
Chemical
Chemical
Chemical
Chemical
Chemical
Chemical
reference
reference
reference
reference
reference
reference
reference
reference
Chemical
Chemical
Chemical
Chemical
reference
reference
reference
reference
"Not required for isothermal spill method.
(b) Penetration Theory. kp and JVA are calculated.
(c) MacKay and Matsugu. k^ and 7VA are calculated.
(4) X, Y, Z, and ^Aroom at the end of the time interval are calculated.
(5) a, (J and Tspin at the end of the time interval are calculated.
(6) ^ipin at the end of the time interval is calculated.
(7) Results from steps 4-6 are recorded as room conditions at time t.
(8) Steps 2-7 are repeated for the subsequent time intervals. Predicted values of
.VAroom, T'.piii and AipM at the end of the preceding time increment are used as
initial conditions. Calculations continue until jAroom >s ' e s s than 1 ppm.
In the isothermal spill model, hc, Uk, and Hvap are not calculated, and step 5 is
eliminated.
TEST SPILLS
Solvent spills designed to test the model were made in a University of Minnesota
laboratory shown in Fig. 4. Air is exhausted from the laboratory through two
laboratory hoods. Makeup air is supplied by two ceiling diffusers, and an air curtain
flowing down the sash of each laboratory hood. Laboratory room volume is 40 m 3 ,
with an air flow of approximately 0.74 m 3 s~'.
P. H. Reinke and L. M. Brosseau
428
Table 2. Spill temperature method equation summary
A ir property calculations
Molar volume at ambient T
Density of air at ambient T
Heat transfer coefficient from air*
Schmidt number
Room property calculations
Heat transfer coefficient under floor*
Overall heat transfer coefficient under*
Chemical property calculations
Vapour pressure at T^a
Latent heat of vaporization at T^n
Diffusion coefficient at ambient T
Mass transfer calculations
Saturation mole fraction
Path length of flow
Flat Plate Mass Transfer Method
Reynolds' number of air flow
Mass transfer coefficient
Molar evaporative flux
Penetration Theory Method
Mass transfer coefficient
Molar Evaporative Flux
Mackay and Matsugu Method
Mass transfer coefficient
Molar evaporative flux
Spill temperature calculations
Equation parameters*
Spill temperature*
Room concentration calculations
Equation parameters
Room mole fraction
Spill area calculation
V
h*r
Sc
he
PA.
DAV,
yA.
L
Equation (12A)
Equation (13A)
Equation (22)
Equation (6A)
Equation (22)
Equation (21)
Equation (15A)
Equation (16A)
Equation (14A)
Equation (16)
Equation (7A)
NA
Equation (5A)
Equations (3A.4A)
Equation (IA)
*P
^A
Equation (9A)
Equation (8A)
NA
Equation (11 A)
Equation (10A)
Re L
«. P
7"«pill
X, Y, Z
Equations (27-28)
Equation (26)
^Aroom
Equations (38-40)
Equation (37)
^•pUl
Equation (12)
"Not required for isothermal spill method.
In the test spills, solvent was poured from a height of 60-76 cm onto the floor,
which was marked with a 12x12 in. grid (30.5x30.5 cm). The liquid stopped
spreading after approximately 30 s, and a sketch was made of the grid and the pool
at that time. Initial pool area was estimated from approximate number of grid
squares covered by liquid in the sketch.
Prior to each spill, detailed measures of air flows were made using a hot wire
anemometer (Model 8310/8315 VELOCICHECK Air Velocity Meter, TSI
Incorporated, St Paul, Minnesota, U.S.A.). Air velocity approximately 2 cm
above each square of the floor grid was measured. The median of the velocities
measured over the squares covered by the initial liquid pool was used in the model
equations.
Contaminant concentration was measured approximately 76 cm above the
center of the pool using a direct-reading dual flame ionisation detector (FID)/
photoionisation detector (PID) (Model TVA 1000A Toxic VaDor Analvzer. The
Prediction of air contaminant concentrations
429
Isopropyl alcohol Spill Modelled with Spill Temperature Method
2000
25
50
75
100
Time in minutes
Fig. 5. Isopropyl alcohol spill data compared to model predictions using spill temperature method and
each of the three mass transfer relationships.
Foxboro Company, Foxboro, Massachusetts, U.S.A.). PID and FID readings were
logged at 1 min intervals and corrected using manufacturer-furnished response
factors specific to the chemicals spilled.
Smoke tube measurements indicated that the air flow from one diffuser and that
from both the air curtains did not significantly mix with the air over the spill. The
short circuited flows were approximately 87% of the room air flow, so the shortcircuiting factor was set at 0.87.
RESULTS AND DISCUSSION
Figure 5 compares the time course of air concentration predicted by the
spreadsheet model using the spill temperature method to concentrations measured
following a 1000 ml spill of isopropyl alcohol. Figure 6 compares measured
concentrations to predictions made using the model and the assumption of an
isothermal spill. The spill in Fig. 5 and Fig. 6 had an initial area of 2.6 m2. The
average air velocity over the spill was 0.36 m s~'. The Schmidt number of isopropyl
alcohol in air at 21°C and 101.3 kPa is 1.48.
Figures 7 and 8 compare model and experimental results for a 1000 ml spill of nbutyl alcohol. The butyl alcohol spill covered approximately 2.14 m2. Air velocity
over the pool was approximately 0.41 m s" 1 . The Schmidt number for n-butyl
alcohol in air was 1.70.
Figures 5, 6, 7 and 8 show model predictions obtained using each of the three
evaporation rate methods introduced earlier. All three methods gave comparable
evaporation flux predictions, with all values within a factor of 1.5. In all of the
figures the Penetration Theory evaporative flux was only slightly higher than the
430
P. H. Reinke and L. M. Brosseau
Isopropyl alcohol Spill Modelled with Isothermal Method
2000
50
75
100
Time in minutes
Fig. 6. Isopropyl alcohol spill data compared to model predictions using isothermal spill method and each
of the three mass transfer relationships.
Mackay and Matsugu estimate, which resulted in similar predictions of room
contaminant concentration. The flat plate mass transfer method predicted a lower
evaporative flux than the other relationships, giving a somewhat lower peak and a
slower decay of room concentration.
n-Butyl alcohol Spill Modelled with Spill Temperature Method
500
25
50
100
Time in minutes
Fig. 7. n-Butyl alcohol spill data compared to model predictions using spill temperature method and each
of the three mass transfer relationships.
Prediction of air contaminant concentrations
431
n-Butyl alcohol Spill Modelled with Isothermal Method
500
50
75
100
Time in minutes
Fig. 8. n-Butyl alcohol spill data compared to model predictions using isothermal spill method and each
of the three mass transfer relationships.
In general, it appears that changing mass transfer method did not greatly
influence the final room concentration predicted by the spreadsheet model. The
method used to estimate spill temperature did, however, significantly impact the
predicted concentration. For both the isopropanol and the «-butanol spills, the
isothermal spill method (Fig. 6 and Fig. 8) predicts a higher peak concentration and
faster decay than the spill temperature method (Fig. 5 and Fig. 7). Because the pool
temperature assumed by the isothermal method is always greater than the pool
temperature predicted by the spill temperature method, the isothermal results in a
higher estimate of evaporation rate. This elevated evaporation rate creates the
higher peak and faster decay of room concentration predictions.
When the predicted curves are compared to the measured data for these
experiments, it appears that the isothermal model provides a more accurate
description of the peak and decay of room concentration. The higher evaporation
rate of the isothermal spill model more accurately describes this small sample of
experiments. Whether this observation is the consequence of the pool temperature
remaining nearly isothermal, or it is the result of higher evaporation rates than are
predicted by the mass transfer methods, cannot be determined from these data.
Measurements of spill temperature and evaporation rate during the experiment
would be required to ascertain why the isothermal method appears to give superior
predictions of evaporation rate.
These data indicate that although the spill temperature method is the product of
a more comprehensive analysis, its efficacy must be tested with experimental data.
The spill temperature method requires data on six parameters which are not needed
when the model uses the isothermal method. Four additional calculations per time
increment are also required. To justify its complexity, the spill temperature method
432
P. H. Reinke and L. M. Brosseau
must give superior predictions of room concentration, not the inferior predictions
observed in the two test spills.
Finally, it is noted that the well-mixed room with short-circuiting is only a fair
representation of true spatial dispersion in the room. Model predictions will depend
significantly upon the short-circuiting factor, and upon how closely the room
exhibits this two zone behaviour. It appears from the accuracy of the peak
concentration predicted by the model for both spills, that the well-mixed room with
a short-circuiting factor of 0.87 was a reasonable characterization of the area of the
laboratory where measurements were taken.
In conclusion, the model developed here and run on a computer spreadsheet
appears to provide a good first estimate of room concentration following an indoor
spill. The model can be used when the following are known or can be approximated:
spill identity, spilled volume, initial spill area, volumetric air flow, air velocity at the
spill surface, and short-circuiting factor. Data from two test spills indicate that the
model may perform best when the Penetration Theory or the Mackay and Matsugu
method are used to predict mass transfer, and when the spilled liquid pool is
assumed to remain at constant temperature throughout evaporation (isothermal
spill method).
REFERENCES
American Society of Heating, Refrigerating and Air-Conditioning Engineers Inc. (1989) Ventilation for
Acceptable Indoor Air Quality. ASHRAE Standard, 62-1989.
Braun, K. O. and Caplan, K.. J. (1989) Evaporation Rate of Volatile Liquids. Final Report, Second
Edition, NTIS Document PB92-232305.
Gcankoplis, C. J. (1972) Mass Transport Phenomena. Ohio State University Bookstores, Columbus, OH.
Geankophs, C. J. (1993) Transport Processes and Unit Operations, 3rd Edition. PTR Prentice Hall,
Englewood Cliffs, NJ.
Hemeon, W. C. L. (1963) Plant and Process Ventilation. The Industrial Press, New York.
Higbie, R. (1945) The rate of absorption of a pure gas into a still liquid during short periods of exposure.
Trans. A.I.Ch.E. 31, 365-389.
Hummel, A. A., Braun, K. O. and Fehrenbacher, M. C. (1996) Evaporation of a liquid in a flowing air
stream. American Industrial Hygiene Association Journal 57, 519—525.
Kawamura, I. and Mackay, D. (1987) The evaporation of volatile liquids. J. Hazard. Mater. 15, 343-364.
Mackay, D. and Matsugu, R.S. (1973) Evaporation rates of liquid hydrocarbon spills on land and water.
Can. J. chem. Engng 51, 434-^39.
Perry, R. H. and Chilton, C. H. (1987) Chemical Engineers' Handbook, 5th Edition. John Wiley and Sons,
Inc., New York.
Reid, R. C , Prausnitz, J. M. and Poling, B. E. (1987) The Properties of Gases and Liquids. McGraw-Hill
Inc., New York.
Roach, SA. (1981) On the role of turbulent diffusion in ventilation. Ann. occup. Hyg. 24, 105-132
Sherwood, T. K. and Pigford, R. L. (1952) Absorption and Extraction. McGraw-Hill Book Company,
Inc., New York.
Sutton, 0. G. (1953) Micromeleorology, A Study of Physical Processes in the Lowest Layers of the Earth's
Atmosphere. McGraw-Hill Book Company, Inc , New York.
APPENDIX
Mass transfer models
Smooth fiat plate correlation. When pure solvent is evaporating into an air stream
and air is insoluble in the solvent, the molar evaporative flux (NA) can be described
as diffusion of the contaminant through stagnant air (Geankoplis, 1972). The flux
Prediction of air contaminant concentrations
433
equation can be written:
where
k[P flat plate mass transfer coefficient [m s" 1 ]
^A«
mole fraction of A at liquid surface (saturation) [kg-mole A (kg-mol)" 1 gas]
J'Aroom mole fraction A in bulk air [kg-mol A (kg-mol) ~' gas]
V
molar volume of a gas [m3 (kg-mol)""1].
Equation (1A) describes mass transfer from the surface of the liquid pool into
the bulk air stream. y&t is mole fraction of contaminant at the liquid surface, which
is the saturation mole fraction of contaminant in air. j'Aroom is mole fraction of
contaminant in the well-mixed air above the spill.
Equation (1A) assumes that air behaves as an ideal gas, and remains at constant
temperature and pressure during evaporation. V, the molar volume, is then
constant. When contaminant concentrations are low and pressures are near
atmospheric, Dalton's law of partial pressures is used to determine _VA»:
PA*
J'A.
= -j-
/I A
\
(2A)
/?A«is the vapour pressure of contaminant A, and P is the ambient pressure.
To predict evaporative flux of a spilled liquid using Equation (1A), a mass
transfer coefficient must be determined. Experimental data have been used to
develop correlations which predict k$p from mass transfer geometry and flow
conditions. In the flat plate mass transfer relationship, mass transfer is modelled as
transfer from a solid flat plate dissolving into a flowing air stream. For this
geometry, Geankoplis (1993) recommends:
A:fp = 0.036(ReL)-°2M(Sc)"2/3
for 15 000 <ReL^ 300 000
(3A)
and
(4A)
where
(5A)
^air
Sc = Schmidt Number = — ^ —
and
(6A)
434
u
ReL
Sc
L
A\air
/i air
Pair
P H . Reinke and L. M. Brosseau
air velocity across plate [m s~']
Reynold's number for flow along flat surfaces [dimensionless]
Schmidt number [dimensionless]
plate length = pool length in flow direction [m]
Diffusivity of contaminant (A) in air [m2 s~']
air viscosity [kg (m-s)"1]
air stream density [kg m~ 3 ].
Length of pool in the air flow direction can be estimated using the square root of
the pool area.
L = \/ASPM
.
(7A)
Mass transfer coefficients predicted using Equations (3A) and (4A) have been
validated with experimental data on liquid evaporation rates. Data are summarized
in Sherwood and Pigford (1952).
Penetration Theory
The Penetration Theory (Higbie, 1945) mass transfer coefficient is used in the
same flux equation as that for flat plate mass transfer:
where
kp
Penetration Theory mass transfer coefficient [m s~']
Penetration Theory was developed from theoretical consideration of diffusion
from a gas into a thick flowing layer of liquid. When applied to the evaporating pool
of liquid, the Penetration Theory mass transfer coefficient has the form:
(9A)
A model based on the Penetration Theory was validated by Hummel et al. (1996)
using data from Braun and Caplan (1989).
Mackay and Matsugu
Mackay and Matsugu (1973) developed a correlation to predict evaporation
rates of oil spills under outdoor conditions. They propose a vapour phase resistance
model which can be written:
NA = 2.778 X If)"7 P ^ . - ^ A r o o m ) ]
where
km
^spiii
(10A)
Mackay and Matsugu mass transfer coefficient [m s~']
gas molar volume at pool temperature [m3 (kg-mol)"1]. km, the mass
transfer coefficient, is determined using:
Prediction of air contaminant concentrations
fcm=17.35U078L-°"Sc-067.
435
(HA)
The Mackay and Matsugu relationship was developed by combining a solution
set forth by Sutton (1953) with experimental data. The relationship was further
validated by Kawamura and Mackay (1987).
Predicting temperature dependent properties
It is assumed that air temperature and pressure do not change appreciably
during spill evaporation. Air density and molar volume are calculated at ambient
temperature using the ideal gas law.
03A)
(yy
Diffusion coefficients are obtained from Perry and Chilton (1987) and corrected
to ambient air temperature using (Geankoplis, 1972):
•
(14A)
Changes in temperature dependent properties of the liquid must be estimated in
the spill temperature model. Vapour pressure is calculated using the Wagner
equation (Reid et al., 1987) in the form:
\n(PM) _Ax
V Ac/
+ B 5
^
+ C^+Eh6
(T/T)
where
PAC
critical pressure of A [Pa]
7\c
critical temperature of A [K]
A, B, C, D
experientally determined constants.
(15A)
Latent heat of vaporization data is obtained from Perry and Chilton (1987) and
corrected for temperature using the Watson relation (Reid et al., 1987) in the form:
where (Hvap)n( is the reference latent heat of vaporization given at temperature T^.
Liquid density is assumed constant over the range of temperatures encountered.
Liquid heat capacity is not a strong function of temperature near ambient levels and
is also considered constant.