PII:
Ann. occup. Hyg., Vol. 45, No. 6, pp. 437–445, 2001
Published by Elsevier Science Ltd on behalf of British Occupational Hygiene Society
Printed in Great Britain.
S0003-4878(00)00082-X
0003-4878/01/$20.00
Predicting Evaporation Rates and Times for Spills
of Chemical Mixtures
RAYMOND L. SMITH*
US Environmental Protection Agency, National Risk Management Research Laboratory, 26 West
Martin Luther King Drive, Cincinnati, OH 45268, USA
Spreadsheet and short-cut methods have been developed for predicting evaporation rates
and evaporation times for spills and constrained baths of chemical mixtures. Steady-state
and time-varying predictions of evaporation rates can be made for six-component mixtures,
including liquid-phase non-idealities as expressed through the UNIFAC method for activity
coefficients. A group-contribution method is also used to estimate vapor-phase diffusion coefficients, which makes the method completely predictive. The predictions are estimates that
require professional judgement in their application.
One application that the evaporation time calculations suggest is a method for labeling
chemical containers that allows one to quickly assess the time for complete evaporation of
spills of both pure components and mixtures. The labeling would take the form of an evaporation time that depends on the local environment. For instance, evaporation time depends
on indoor or outdoor conditions and the amount of each chemical among other parameters.
This labeling would provide rapid information and an opportunity to premeditate a response
before a spill occurs. Published by Elsevier Science Ltd on behalf of British Occupational
Hygiene Society
Keywords: spills; evaporation; labeling
INTRODUCTION
Chemicals in the liquid state are common in our
industrial society. In many processes liquids are used
as reactants or products, as well as coatings, solvents,
fuels, additives, etc. In all of these capacities a liquid
can evaporate to form a vapor. Once in the vapor
phase chemicals are easily transferred, which
increases environmental concerns because of their
potential effects on sensitive human and ecological
life.
To mitigate the potential negative effects of chemical vapors one can take precautions in their use. For
example, a chemical bath used to clean greasy parts
should have the proper equipment to gather the
vapors (e.g., Wadden et al., 1989). To know whether
equipment or protective gear is needed, it is important
to know the amount of chemicals in the surrounding
air. Determining this amount requires that one either
measure the concentration or calculate the rate of
evaporation from liquid sources.
Received 14 March 2000; in final form 30 August 2000.
*Tel.: +1-513-569-7161; fax: +1-513-569-7111; e-mail:
[email protected]
Another case where evaporation rates are useful is
when liquids are spilled. In such a case one may want
to know in advance how quickly the liquid(s) will
evaporate. For instance, it is useful to know whether
to clean up a liquid before significant evaporation
occurs or to stay out of the spilled area because evaporation will be fast. In this second case of fast evaporation, attempting to clean up the liquid spill could
be futile and perhaps dangerous. A calculation of the
evaporation rate could help determine the correct
course of action in the case of such a spill.
The previous examples motivate the calculation of
evaporation rates. However, evaporation rates are not
very useful representations at the time of a chemical
spill. For this reason evaporation rates of chemical
spills are represented here in terms of the time it takes
to evaporate. An individual can rapidly scan a label
for the time for evaporation, or have it memorized for
commonly used chemicals, and quickly determine (or
premeditate) an appropriate course of action. This
coincides well with the EPA guidelines for emergency responses to spills (EPA, 1993), which says that
one needs, ‘fast, reliable information under stressful
conditions so that it can be understood and immediately acted upon.’
437
438
R. L. Smith
BACKGROUND
A number of investigators have studied evaporation
rates. Nielsen et al. (1995) have reviewed the various
models and developed their own. Most of the models
they reviewed employed air velocities, diffusion coefficients in air, and the vapor pressure of the substance
of interest. Nielsen et al. (1995) added corrections for
bulk flow, variable density, and starting length for air
flow in front of a liquid pool. Reinke and Brosseau
(1997) have studied spills in the laboratory, comparing various models, and developing their own. The
models they considered were the flat plate, Mackay
and Matsugu (1973) model, and penetration theory
models. In addition, Reinke and Brosseau (1997)
modeled spill temperature as either isothermal or with
a heat balance, and they employed a short circuiting
factor in their model of laboratory air concentrations.
A model for the prediction of evaporation rates of
mixtures is available from Nielsen and Olsen (1995).
They review the literature on mixtures, perform
experiments, and develop their model based on
Nielsen et al. (1995). Their model calculates liquidphase activity coefficients with the UNIFAC
(UNIversal Functional-group Activity Coefficients)
method (e.g., Reid et al., 1987). With the inclusion
of activity coefficients the model and experimental
results are in reasonable agreement.
MODELING THEORY
Modeling the transport of one or more chemicals
across a vapor–liquid interface (see Fig. 1) can be
accomplished on a spreadsheet with a few assumptions. First, it is assumed that the liquid phase is well
mixed (i.e., mixing is fast relative to mass transfer),
so that the concentrations in the liquid-phase are averaged over the remaining volume. Another perspective
is that there are no concentration gradients in the
liquid phase. This is normally an appropriate assumption because concentration gradients are dissipated by
macroscopic and microscopic currents and diffusion.
A second assumption is that the liquid phase is in
equilibrium with the vapor phase at the interface. This
permits one to use well-known vapor–liquid equilibrium calculations to determine the vapor-phase concentrations. While there are many methods for calcu-
Fig. 1. Diagram of liquid(s) evaporating through a vapor-phase
thin film.
lating vapor–liquid equilibrium, in this work the
UNIFAC method for determining activity coefficients
is used. The activity coefficient, as applied here to
vapor–liquid equilibrium, is a measure of the propensity for each liquid component to be volatile. As its
name implies, the UNIFAC method is able to determine the activity coefficient of different chemicals in
a mixture based on functional groups of atoms. This
is a powerful aspect of the method since it allows
one to do calculations on chemicals for which vapor–
liquid interactions are unavailable (e.g., calculations
could be done for a complex pharmaceutical and solvent which have unknown interactions).
A third assumption is that mass transfer through
the vapor phase is adequately described by a binary
diffusion coefficient that can be approximated from
the properties of each component. The properties
needed for the calculation of the diffusion coefficient
include temperature, pressure, molecular weight, and
atomic volumes. The atomic volumes have been tabulated for common molecules, certain functional
groups, and atoms. Because the method uses building
blocks of functional groups and atoms to characterize
a component, the diffusion coefficient can be determined even when experimental values are not available.
(See Appendix A for the method of estimating diffusion coefficients.)
Finally, in the vapor phase it is assumed that a ‘thin
film’ of air separates the vapor–liquid interface from
a bulk vapor region that has a constant and lower concentration of the evaporating chemical (see Fig. 1).
The implication of a thin film is that mass transfer
from the interface across the film develops quickly,
and therefore the concentration profile across the film
is linear. This linear concentration profile over a
defined film thickness (with the associated diffusion
coefficient) permits one to calculate the flux of
material. In this work the film thickness is determined
using boundary layer theory, although another
method of determining the film thickness could be
used. Once the film thickness is specified one can calculate the rates of evaporation for the components of
a liquid mixture.
MATHEMATICAL MODEL
The evaporation rates calculated with a spreadsheet
are a result of a mathematical model based on the
assumptions described above for diffusion across a
thin film. Even though the liquid (and therefore
vapor) concentrations change, they are assumed to
change slowly enough so that the thin-film model of
mass transfer applies. For a more in-depth description
of combining steady-state fluxes and an unsteadystate mass balance see Cussler (1984, p. 28). The
mathematical model conserves the number of moles
of each component as it is transported from the liquid
phase to the interface and on to a point of negligible
concentration on the other side of the vapor-phase
Evaporation rates and times for spills
thin film (i.e., the bulk vapor concentration, cbi , is
negligible). Mathematically, this transport is
described by a balance equation on the number of
moles of component i, Ni, as
dNi
= ⫺Aji
dt
(1)
where t is time, A is the interfacial surface area, and
ji is the molar flux of i in moles per area per time. An
assumption made here is that the surface area remains
constant, which is a good assumption for constrained
baths. For chemical spills, constant surface area is a
poorer approximation, especially when considering
complete evaporation (where decreasing area would
lengthen the evaporation time), and a method for calculating a varying spill area is presented by Reinke
and Brosseau (1997). However, this work emphasizes
methods for predicting estimates of evaporation rates
and times. These predictive methods require a constant surface area, which could be estimated as an
average over the evaporation time to take decreasing
surface area into account. Although the constant area
assumption introduces error into the results, it also
enables one to realize the power of the methods (as
described below).
Assuming that the surface area and flux at the interface (the rate of evaporation) are equal to the area
and flux for diffusion through the air, one can write
the flux as described in Cussler (1984)
ji = ⫺Di
dci Di ∗ b
= (c ⫺ci )
dz li i
(2)
where Di is the vapor-phase diffusion coefficient of
component i, ci is the vapor-phase concentration of i,
z is the direction of diffusion, li is the thin-film thickness (which will be shown to be a function of Di in
Eq. (6)), and the superscripts * and b refer to interfacial and bulk concentrations, respectively. (See Fig.
1 for a diagram showing the film thickness and
concentrations.) This equation shows how the flux
varies inversely with the film thickness. Also, the flux
is proportional to the vapor-phase concentration at the
interface because it is assumed that the bulk concentration is negligible. The resulting balance on the
number of moles (combining Eqs (1) and (2)) is
Di
dNi
= ⫺A c∗i
dt
li
(3)
Note that those who prefer to define a length li where
there is a non-zero value for cbi can do so, but it
necessitates defining a value for cbi .
The vapor-phase diffusion coefficient used in these
calculations is determined by the method of Fuller et
al. (1969), which is also described in a review by
Reid et al. (1987). (See Appendix A for the method.)
439
The diffusion coefficients are assumed to be binary
coefficients, with the second vapor-phase material
being air. This assumes that the individual binary
coefficients are independent of each other.
To calculate vapor-phase concentrations at the
interface, which are needed in the mole balance
above, first the vapor–liquid equilibrium calculations
must be performed for the liquid components. (Air is
assumed to be only in the vapor phase.) At low pressures vapor–liquid equilibrium is described by
Pyi = Poigixi
(4)
where P is the total pressure, Poi is the vapor pressure,
gi is the activity coefficient, and yi and xi are the
vapor-phase and liquid-phase mole fractions, respectively (e.g., Smith and Van Ness, 1987). For a given
temperature and set of liquid-phase mole fractions
one can calculate the vapor pressures and the activity
coefficients. The vapor pressures are solely a function
of temperature, and methods for calculating them are
available (e.g., Reid et al., 1987). The activity coefficients depend on both temperature and the liquidphase mole fractions, and are determined using
UNIFAC (tables from Professor B. E. Poling, University of Missouri at Rolla). Once the activity coefficients are determined it is simple to calculate the
partial pressures for the liquid components in the
vapor phase, Pi = Poigixi. These partial pressures are
then used to calculate the vapor-phase mole fractions,
yi = Pi/P, and the mole fraction of air is obtained by
subtracting all the other vapor-phase mole fractions
from one. This method of obtaining the mole fraction
of air is possible because the total pressure, P, is
specified in these calculations. Note that if the sum
of the partial pressures of the liquid components in
the vapor phase is greater than the total pressure, then
the liquid is boiling. This work does not consider the
case of boiling components. Once the vapor-phase
mole fractions are known, then the interfacial concentration is calculated with the ideal gas law as
c∗i =
y iP
RT
(5)
To estimate the vapor-phase thin-film thickness we
use boundary layer theory. Cussler (1984, pp. 288–
96) describes how the laminar boundary layer is
related to the film thickness of the film theory. The
laminar and turbulent mass transfer correlations for
averages over the spill length are found, for example,
in Green and Maloney (1997, 5–55). For laminar flow
over a flat plate (using a film theory average mass
transfer coefficient, k̃i = Di/li) the film thickness is
llam =
L
0.646(UL/n)1/2(n/Di)1/3
(6)
where U and n are the velocity and kinematic vis-
440
R. L. Smith
cosity (n = m/r) of air (far) above the chemical spill
and L is the length of the spill (L = A1/2). Note that
a difference in the diffusion coefficient for a chemical
leads to a slightly different film thickness. In this
work r = 0.001161 g/cm3 and m = 0.000186 g/cm/s,
(Lide, 1997, 6-1, 6-194). For a turbulent boundary
layer the film thickness is
ltur =
L
0.0365(UL/n)4/5
(7)
and the average film thickness is calculated
(Sherwood et al., 1975, 201 have suggested a
weighted average of the laminar and turbulent boundary layers) by our method as
l=
Lcrllam + (L⫺Lcr)ltur
L
(8)
(9)
From this point the derivation for the evaporation
time depends on whether the evaporating substance
of interest is dilute or in abundance. For dilute substances xi is substituted with Ni/NT (where NT is the
total number of moles at any time). A useful assumption is that NT = NT0, or in other words, that the total
number of moles is essentially unaffected by the
evaporation of the dilute substance. This approximation is more reasonable when the number of moles
and the evaporation time of the dilute substance are
small in comparison to the abundant substance. The
results will be checked afterwards to determine if this
is so. This assumption allows the use of NT0 as a constant in the calculations. The numerator and denominator of the right hand side of Eq. (9) are multiplied
by the initial total volume, VT0, and after rearranging
one obtains
dNi
Ak̃i PoigiVT0
N
=⫺
dt
VT0 RT NT0 i
where VT0/NT0 =
冘
(10)
xm,0MWm/rm). Equation (10) is
m
first order in Ni, which results in an exponential decay
equation of the form
(11)
where t̄i is the time constant for first order decay. To
approximate the total evaporation time, the time for
95% evaporation is used, which defines the evaporation time for dilute substances as, ti,dil = 3t̄i, or
ti,dil = 3
VT0 RT NT0
Ak̃i Poigi VT0
(12)
When our interest is the time for evaporation of a
substance that is in abundance the derivation starting
from Eq. (9) is different. We multiply the numerator
and denominator by both VT0 and NT0, and the mole
fraction of i is assumed to be unity. The resulting
equation, analogous to Eq. (10), is
dNi
Ak̃i PoigiVT0
N
=⫺
dt
VT0 RT NT0 T0
where Lcr is the spill length at which the Reynolds
number reaches a critical value of 300 000 (e.g.,
Cussler, 1984). When the spill length is small enough
that the Reynolds number is below the critical value,
one simply uses the laminar film thickness.
Having defined the film thickness, the method for
calculating the evaporation time can now be
described. Substituting for Di/li and c∗i in Eq. (3) gives
a result in terms of the mole fraction of i in the vapor
phase. Employing the vapor–liquid equilibrium
relationship of Eq. (4) produces
Poigixi
dNi
= ⫺Ak̃i
dt
RT
Ni = Ni0e⫺t/t̄i
(13)
which is zeroth order in Ni. As a result, the time for
evaporation of a substance in abundance is
ti,abu =
VT0 RT NT0
Ak̃i PoigiVT0
(14)
Note that while the forms of ti,dil and ti,abu differ by
a factor of three due to Eqs (10) and (13) being first
and zeroth order, respectively, that the magnitudes of
k̃iPoigi in each equation can vary substantially. When
the dilute substance has a relatively large value of
k̃iPoigi, t̄i is small, and ti,dil( = 3t̄i) will be less than
ti,abu.
RESULTS AND DISCUSSION
Model comparison
A comparison is now done with experimental
results found by Nielsen and Olsen (1995) for both
pure components and mixtures of chemicals. For pure
components the authors reported experimental evaporation rates of 8.4 and 0.68 mmol/m2/s for 2-butanone and n-butylacetate, respectively. Using Eq. (9)
in a spreadsheet gives 11 and 1.2 mmol/m2/s for 2butanone and n-butylacetate. These calculated evaporation rates are high by 31 and 76%, respectively.
Evaporation rates of mixtures for our calculation
method are compared with Nielsen and Olsen (1995)
in Table 1. The table follows the results of Nielsen
and Olsen, and includes information on the size of
their pools and the mole fractions in each example.
The mole fractions were chosen to avoid multiple
liquid-phase behavior, which could be an important
concern for evaporating mixtures. Nielsen and Olsen
reported experimental evaporation rates which we
compare to calculated values from a spreadsheet.
(The effects of time step size are analyzed in Appendix B.)
Evaporation rates and times for spills
441
Table 1. Comparisons of experimental versus calculated rates of evaporation and spreadsheet versus ti-estimated times
for evaporation. Calculations follow the experiments of Nielsen and Olsen (1995), with the exception that the results for
n-butylacetate/water were calculated for a 10-min instead of a 15-min period. Air velocities were 0.17 m/s, the temperature
was 300 K, and the pressure was 101.3 kPa
Chemical
mixture
Liquid pool
L×W×D (mm)
Trichloroethylene 75×20×5
n-Butylacetate
2-Butanone
75×20×5
Toluene
Ethanol
75×20×5
2-Butanone
2-Butanone
75×20×5
Ethanol
Trichloroethylene 75×20×5
Ethanol
Ethanol
75×20×5
Trichloroethylene
Ethanol
75×20×5
Water
2-Butanone
125×20×5
Water
2-Butanone
125×20×15
Water
n-Butylacetate
125×20×5
Water
n-Butylacetate
125×20×15
Water
Mole fraction
xi,0
0.1
0.9
0.1
0.9
0.1
0.9
0.1
0.9
0.1
0.9
0.1
0.9
0.1
0.9
5.1×10⫺3
苲1.0
5.1×10⫺3
苲1.0
5.1×10⫺3
苲1.0
5.1×10⫺3
苲1.0
Evaporation rate (mmol/m2/s)
Evaporation time estimate(s)
Exp.
Calc.
Sheet
ti
0.37
0.70
0.93
2.20
1.30
10.0
1.50
6.80
1.80
7.80
3.00
6.00
1.90
–
0.57
–
0.52
–
0.13
–
0.21
–
0.69
1.45
1.84
3.76
2.29
13.0
2.87
10.5
4.53
10.7
5.98
9.61
3.90
7.01
1.09
5.78
1.42
5.78
0.28
5.80
0.47
5.80
18 900
22 860
7980
10 620
3780
3900
5520
6600
5040
6480
2280
5160
14 340
27 540
3360
46 560
10 140
139 680
780
47 520
2280
142 500
16 860
24 780
7260
11 520
7320
4020
8100
6960
4980
6900
2520
5460
17 160
29 100
2580
46 800
7620
140 340
960
47 520
2880
142 620
Table 1 shows that our calculated evaporation rates
are consistently higher than the experimental values.
For substances in abundance (xi,0 = 0.9) the average
over-prediction is 60%. One could fit these results by
changing the coefficients in Eqs (6) and (7) for calculating the film thickness, however the relationship
between the model and experimental results may vary
for different conditions and (combinations of) chemicals. For dilute substances the experimental evaporation rates are over-predicted by an average of 110%,
and the same caveats apply to fitting these results.
Note that correction terms, described in Nielsen et al.
(1995), could be applied for variable density, bulk
flow and entrance length. The focus here is on quick
estimates of rates and times.
The last two columns of Table 1 show estimates
of total evaporation times. Values in the first of these
columns were found through a repetitive spreadsheet
application of Eq. (9), while the values in the last
column were determined from Eqs (12) and (14). In
analyzing the results we will first examine the anomalies for the two cases involving ethanol and 2-butanone. Only in these cases is the evaporation time, ti,
for the dilute substance estimated to be above that for
the abundant substance. In these cases the time for
evaporation of the dilute substance is too long, and
the time for the abundant substance as estimated by
Eq. (14), should be used for both dilute and abundant substances.
In general, the results calculated by the spreadsheet
and Eqs (12) and (14) are fairly precise, as the differences between the two columns of Table 1 average
10% (excluding the dilute ethanol and 2-butanone
cases discussed above). To a certain degree, the similar results can be expected since the same model is
used as the basis of both methods, however, the τi
estimates do not require integration and are very simple and quick compared to the spreadsheet calculations. Thus, without doing the iterative calculations
of the spreadsheet one can obtain an evaporation time
estimate of similar precision.
An interesting result of the analysis shown in Table
1 is that, except for very dilute mixtures, the evaporation time for the abundant substance is a reasonable
approximation for the evaporation time of a dilute
substance. For the examples of Table 1 where
xi,0 = 0.10 the spreadsheet times for abundant and
dilute substances differ by an average of 27% (with
differences ranging 3–56%). Therefore, instead of
needing to know all about a spill (e.g., exactly how
much of each chemical, etc.), the above result suggests it may be sufficient to use the abundant substance as a rough approximation of the evaporation
time for the mixture. This generalization is less valuable if one substance is very dilute (e.g.,
xi,0 = 5.1×10⫺3)
or
the
relative
volatility,
Podilgdil/Poabugabu, and/or the ratio of mass transfer coefficients, k̃dil/k̃abu, are large (as in the cases of ethanol
in trichloroethylene or water).
442
R. L. Smith
Spill environments
To obtain estimates of the evaporation times one
needs air velocities and spill lengths, which are used
to calculate the average mass transfer coefficients. In
the results of this work air velocities of 0.50, 5.0, and
12 m/s (approximately 1, 11, and 27 mph) are used
as representative of common outdoor and indoor
environments. A comparison of these selected velocities to outdoor winds and indoor ventilation is
presented in Table 2.
The volumes used in this analysis will be common
container sizes: a 1.0-l. bottle, a 55-gall drum (208.2
l.), and a 6000-gall tank truck (22 710 l.). Spill
lengths used in the analysis will be of 1.6, 16, and
160 m (approximately 5, 50, and 500 ft). A length of
1.6 m for a spill of a liter is obtained from assuming
a square area for the spill experiments done by Reinke
and Brosseau (1997), while the other lengths were
successively increased by an order of magnitude.
Note that a smaller than original spill length could
be used to approximate a decreasing surface area for
evaporation. A smaller spill length than that expected
for a particular volume could also be obtained if a
spill is constrained (or the ‘spill’ is really a chemical
bath). Larger spill lengths than expected could be
obtained if a spill is spread, for instance by being well
mixed in a pool.
Spill examples
The estimated evaporation times for different spill
volumes, spill lengths, and air velocities for trichloroethylene are shown in Table 3. For each volume the
evaporation time decreases with increased spill length
and air velocity. As the volume increases the evaporation time also increases.
The evaporation times in Table 3 could be used as
a label on a container of trichloroethylene. For
Table 3. Evaporation times in seconds estimated by Eq.
(14) for trichloroethylene at a temperature of 300 K and a
pressure of 101.3 kPa
Container and
spill length
Air velocity
0.50 m/s
Bottle 1.6 m
1100
Bottle 16 m
23
Bottle 160 m
0.25
Drum 1.6 m
240 000
Drum 16 m
4800
Drum 160 m
52
Tank truck 1.6 m 26 000 000
Tank truck 16 m
530 000
Tank truck 160 m
5700
5.0 m/s
12 m/s
230
2.5
0.039
48 000
520
8.2
5 300 000
57 000
890
88
1.2
0.019
1 8000
260
4.1
2 000 000
28 000
440
example, on a 1.0-l. bottle used in a laboratory, the
bottle could indicate that for a normal spill 1.6×1.6
m) the evaporation time is 1100 s (18 min) for general
ventilation and 230 s (3.8 min) if under a hood. Likewise, a 55-gall drum of trichloroethylene used outside
could be labeled for a normal spill (16×16 m) as having evaporation times of 4800, 520, and 260 s (80,
8.7, and 4.3 min) for light, gentle, and strong winds
respectively. A tank truck which spills its contents
(assuming a 160×160 m spill) would have evaporation times of 5700, 890, and 440 s (95, 15, and 7.3
min) in the same winds. If the spills were constrained
the evaporation times would be considerably larger.
Assuming that proper protective gear is available,
the evaporation times could be used to estimate
whether a clean up operation can be performed or
whether people should vacate the area. Note that the
intention of this paper is only to describe possible
uses of the method: it is not the intention to advise
any specific course of action for the examples dis-
Table 2. Air velocities for outdoor and indoor environments. Outdoor air velocities have been taken from Wood (1998),
and indoor velocities are based on work by the Committee on Industrial Ventilation (1982). Note that the velocities
describing indoor environments are both discontinuous and overlapping (to convert to SI units, 1.0 fpm = 0.0051 m/s and
1.0 mph = 0.45 m/s)
Outdoor winds
Air velocities
used in this
work (m/s)
Indoor ventilation
Class
Description
Air velocity (mph)
Air velocity (fpm)
Description
Light
Calm
⬍1
⬍88
General ventilation
(15–300 fpm)
1–3
0.50
88–264
Gentle
Weather vanes do not
move
Weather vanes move
Light flags extend
4–7
8–12
5.0
352–616
704–1056
Dust/papers rise
13–18
Small trees sway
Hard to control umbrellas
Hard to walk into wind
19–24
25–31
32–38
Moderate
Fresh
Strong
1144–1584
12
1672–2112
2200–2728
2816–3344
Hoods (1000–1200
fpm)
Fans (700–3400
fpm)
Evaporation rates and times for spills
cussed. If the evaporation time is long then there is
time for a clean up operation, but if the evaporation
time is short then there is no time for clean up. The
key to the information transferred as the evaporation
time is that it is simple. When confronted with a spill
one has to act quickly. Perhaps the greatest advantage
of the evaporation time is that it could be memorized
before using the chemical, so that a premeditated
response is possible.
In some uses a particular chemical may be
employed in the vicinity of others. In such a case,
Table 4 shows evaporation times for a 1.0-l. bottle of
trichloroethylene (TCE) well-mixed into ethanol
(EtOH). (Longer spill lengths could have been used
to account for the greater total volume, but consistency in the lengths was maintained in the tables
instead.) To create a table for a mixture one needs to
know the amounts of the chemicals as well as reasonable spill lengths and air velocities. Comparing the
evaporation times in Table 4 with those in Table 3
shows that TCE in the mixture takes much longer to
evaporate, nearly an order of magnitude longer than
the same amount of pure TCE. Depending on the
times, different actions may be possible. For instance,
for a spill of 1.0 l. of TCE, a 0.50 m/s air velocity
and a 1.6×1.6 m spill, the time has increased from
1100 to 7200 s (18 to 120 min) with the addition of
the ethanol. Another example would be if we wanted
to wait until a hood had removed the TCE from the
area. Assuming a 5.0 m/s air velocity and a 1.6×1.6
m spill, the evaporation time increases from 230 to
1400 s (3.8 to 23 min) when ethanol is present.
Short-cut method for evaporation time
A short-cut method for determining the evaporation
time can be developed from the model described
above. To obtain these short-cut results the evaporation time for a substance in abundance is calculated as
ti,abu =
NT0RT
Ak̃ Poi
(15)
Table 4. Evaporation times in seconds estimated by Eqs.
(12) and (14) for a spill of 1.0 l. of trichloroethylene (TCE)
into ethanol (EtOH), where the initial mole fraction of trichloroethylene is 0.10, the temperature is 300 K, and the
pressure is 101.3 kPa
TCE volume
and spill length
Air velocity
0.50 m/s 5.0 m/s
Bottle 1.6 m
Bottle 16 m
Bottle 160 m
TCE
EtOH
TCE
EtOH
TCE
EtOH
7200
10 000
140
190
1.6
1.9
1400
1900
16
19
0.24
0.30
12 m/s
550
700
7.7
9.4
0.12
0.15
443
which is similar to Eq. (14) with the exception of
removing the activity coefficient (and VT0 has been
divided out). One can expect that the activity coefficient for a dilute substance is often far from unity,
and so the short-cut method for determining the evaporation time may have a larger error for dilute substances.
For the short-cut method, approximations are made
of the diffusion coefficient, the vapor pressure, and
in the calculation of the film thickness. Of course, if
more accurate values are available for any of these
approximations, including the activity coefficient,
they should certainly be used. In the absence of more
accurate information the diffusion coefficient for substances through air can be approximated as 0.10 cm2/s
(from a table in Cussler, 1984). Vapor pressures for
chemicals in Table 1 ranged from 0.017 to 0.13 atm
and can be approximated as the average of 0.07 atm.
(Clearly, this approximation for vapor pressure will
have a dramatic effect on the results, and vapor pressures are often available, e.g., Reid et al., 1987.) The
film thickness is calculated with Eqs (6)–(8) with the
exception that (n/D)1/3 is assumed to be unity in Eq.
(6). A decision about whether to weight the film
thickness through Eq. (8) or to simply use the laminar
or turbulent value is a judgement left to the user,
although weighting will be employed in the results
below.
To calculate the evaporation time by the short-cut
method the following steps can be taken using a calculator: (1) estimate the volume and length, L, of a
spill, multiplying the volumes of each chemical by
ri/MWi to obtain the total original number of moles,
NT0; (2) determine the critical length of the spill, Lcr
using ULcr/n = 300 000), and the film thickness, l; (3)
calculate the average mass transfer coefficient,
k̃ = D/l, and the spill area, A = L2; and (4) use
k̃, A, NT0 and the temperature, vapor pressure, and gas
constant in Eq. (15) to calculate the evaporation time.
For dilute substances the evaporation time is normally
multiplied by three. However, as was pointed out for
Eqs (12) and (14), only when the dilute substance has
a relatively large value of k̃iPoigi will the evaporation
time of the dilute substance be below that of the abundant substance. In this short-cut method we have estimated all of these parameters generically. Therefore,
the evaporation time determined by this method for
the dilute substance will be larger than that for the
abundant substance, and the time for the abundant
substance should be used to describe the evaporation
of the mixture.
For pure trichloroethylene, 1.0 l. spilled in a
1.6×1.6 m area under a 0.5 m/s air velocity, the shortcut method gives an evaporation time of 1700 s compared to 1100 s (as in Table 3) determined with Eq.
(14). The difference in times is due to using approximations for the diffusion coefficient, vapor pressure,
and film thickness. This result has fair precision for
such a quick approximation.
444
R. L. Smith
For a mixture of trichloroethylene and ethanol,
results of calculations are shown in Table 5. As in
Table 4, 1.0 l. of trichloroethylene is considered to
be well mixed with ethanol. The resulting mixture has
a trichloroethylene mole fraction of 0.10. Because the
activity coefficient is not included in this short-cut
method, and k̃i and Poi are approximated as being the
same for each component of the mixture, one should
simply use Eq. (15) as a short-cut approximation for
the time of evaporation of the mixture. While the
results in Tables 4 and 5 differ considerably, one can
see whether the evaporation time will be a few hours
or a fraction of a minute. In its favor, the short-cut
method does give order of magnitude estimates that
follow the correct trends.
APPENDIX A
Estimating diffusion coefficients
The method for estimating the diffusion coefficients of evaporated chemicals was developed by
Fuller et al. (1969). The method uses parameters
known as ‘atomic diffusion volumes,’ which the
authors have regressed from experimental data. A list
of the atomic diffusion volumes is given in Table 6.
Table 6 contains parameters for both atoms and simple molecules. When a molecule of interest is not
found in the table the user must add up the atoms in
the molecule (and structural parameters if an aromatic
or heterocyclic ring is present) to obtain a total dif-
fusion volume (e.g.,
冘
ni is the total diffusion volume
A
for chemical A). The total diffusion volume is used
in the following equation for the diffusion coefficient
CONCLUSIONS
A method for predicting the rates of evaporation of
mixtures has been developed. The results over-predict
the rate of evaporation when compared with laboratory experiments, but the model has not been fit to
the experiments, which were performed under laminar
(laboratory) conditions only. Both steady-state and
time-varying calculations can be done, and the calculations are completely predictive. Results can be
obtained for species for which data on vapor–liquid
equilibrium or diffusion coefficients are unavailable.
Such results are obtained with group contribution
methods: UNIFAC for activity coefficients and Fuller’s method (Fuller et al., 1969) for diffusion coefficients. This use of group contribution methods
makes the method generally accessible.
Another result is the development of methods for
determining the evaporation time for spills and constrained baths of chemical mixtures. The evaporation
times depend on whether a spill is a single component
or a mixture. For mixtures, the times are functions of
their concentrations, as abundant and dilute substances evaporate at different rates. Finally, a shortcut method is developed which requires very little
information about the chemicals involved, and an
order of magnitude evaporation time can be determined quickly with only a calculator.
Table 5. Evaporation times in seconds for a mixture of
trichloroethylene and ethanol using the short-cut method,
where the initial mole fraction of trichloroethylene is 0.10,
the temperature is 300 K, and the pressure is 101.3 kPa
Container and
spill length
Bottle 1.6 m
Bottle 16 m
Bottle 160 m
Air velocity
0.50 m/s
5.0 m/s
12 m/s
17 000
320
3.0
3200
30
0.46
1100
15
0.23
DAB =
0.00143T1.75
1/2
PMAB
[(
冘
ni)1/3 + (
A
冘
(A1)
ni)1/3]2
B
where MAB = 2[1/MA) + 1/MB)]⫺1, Mj is the molecular weight of molecule j, P is the pressure in bars, T
is the temperature in degrees Kelvin, and the resulting
diffusion coefficient, DAB, is in cm2/s. The form of
the equation presented here is taken from Reid et al.
(1987) and is equivalent to that given in Fuller et
al. (1969).
APPENDIX B
Time step calculations
In using a spreadsheet to generate results for this
work, it was necessary to determine the time step size
for convergence. An example is shown in Fig. 2,
Table 6. Atomic diffusion volumes reproduced from Fuller
et al. (1969)
Atomic diffusion volumes
Atomic and structural diffusion volume increments
C
15.90
F
14.70
H
2.31
Cl
21.00
O
6.11
Br
21.90
N
4.54
I
29.80
Aromatic or
⫺18.30
S
22.90
heterocyclic
ring
Diffusion volumes of simple molecules
He
2.67
CO
18.00
Ne
5.98
CO2
26.70
35.90
Ar
16.20
N2O
Kr
24.50
NH3
20.70
Xe
32.70
H2O
13.10
H2
6.12
SF6
71.30
6.84
Cl2
38.40
D2
N2
18.50
Br2
69.00
O2
16.30
SO2
41.80
Air
19.70
Evaporation rates and times for spills
445
Fig. 2. Evaporation of ethanol using a spreadsheet with different time steps.
where a time step of 100 s gives nearly the same
results as a 10-s time step. This order of magnitude
difference in the number of steps is very important
when repeatedly entering values in a spreadsheet.
Also, depending on how the results are employed, a
1000 s time step might be useful in some circumstances. Note that for simplicity in the spreadsheet
calculations, Euler’s method was employed to generate the results, and larger time steps could have been
used with a midpoint or fourth-order Runge–Kutta
method (Press et al., 1989, 547–77).
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