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Dispersion modelling of an elevated high momentum release forming aerosols
Dispersion modelling of an elevated high momentum release forming aerosols John L. Woodward Ecology and Environment, Inc. 368 Pleasantview Drive, Lancaster, NY 14086, USA Pressurized releases of liquids generally form an aerosol phase. The presence of liquid droplets decreases the overall density of the resulting vapour cloud, both by a mass effect and by depressing the cloud temperature until the droplets evaporate. The droplet temperature generally falls considerably below the boiling point temperature of the liquid because of evaporation. The elevated, high momentum model of Ooms and coworkers has been extended to treat single component aerosols. The model predicts hazard zones for toxic and flammable releases which are more accurate than all-vapour model predictions. (Keywords: dispersion; modelling; release)
Accidental releases of liquids under pressure quite readily entertain liquid droplets as an aerosol. In addition, liquids may escape through a pressure relief valve as a result of: 0 overfilling the vessel; 0 thermal expansion of liquid, particularly during a runaway reaction; 0 bubble formation in the liquid; 0 entrainment of liquid during rapid depressurization. Not all pressure relief valves are connected to a flare, so there is a potential hazard if releases of heavy aerosol reach the ground before diluting below flammable or toxic limits. The presence of liquid droplets imparts heavy vapour characteristics to the dispersion of the release, often greatly extending the hazardous range of the release. For example, experiments with ammonia’*2, and with hydrogen fluoride3, showed that essentially all of the total mass in the release is entrained as an aerosol. The resulting aerosol clouds were found to persist longer and extend farther than a corresponding dry vapour cloud. The model by Ooms, et al. 4 was originally developed for elevated releases of heavy vapour and was made available in the public domain by the US Coast Guard as the ONDEK model 5*6. By a straightforward extension, this model can be made to treat aerosols without droplet rainout, using homogeneous equilibrium assumptions. The extended model is called the EJET model. The further extension to treat rainout is planned.
Received 12 October 1988 0950~4230/89/010022-11$3.00 0 1989 Butterworth & Co. (Publishers) Ltd 22 J. Loss Prev. Prbcess Ind., 1989, Vol2, January
Liquid breakup and rainout Liquid droplets larger than 1 mm are known to be unstable in a gas stream with appreciable velocity ( > 10 m s-l). The onset of breakup is usually tabulated at a ‘critical’ Weber number, defining Weber number as: We =
pvu2d,
(1) . ,
UL
Breakup occurs by the milder ‘bag’ mechanism above We = 10, and by successively more chaotic mechanisms at higher Weber number e.g. by bag-jet mechanism around We = 20, by shear or stripping off mechanism around We > 65 (Ref. 8). The maximum stable drop size according to Brodkey’ is, for low speed jets: dp =
1.89& (1 + 3We 1’2Rej”2) 1’2
(2) where Dj is the diameter of the jet Reynolds number defined by
For higher-speed jets, Brodkey recommends calculating the maximum drop size from We = 20. The fallout of droplets in a quiessant or at least laminar flow environment is well characterized by a force balance involving the drag coefficient lo. Terminal droplet velocities are readily found from an implicit Reynold’s number expression”: qCo(Reo)Rei=
4d;cPL - Pabag
(4) 3 P.2 where Co is the drag coefficient for a rigid spherical drop and n is a correction factor for the motion of fluid
Dispersion modelling
of an elevated high momentum release forming aerosols: John i. Woodward
within the drop given by Batchelor “:
1 + APL
(5)
’ = 1 + 2/3 P~/PL
Wheatley13 plots the terminal gravitational settling velocity of the largest stable drop as predicted by Equations (3), (4) and (5) against the jet velocity. He then applies a simple geometric argument with a top hat jet model to suggest that the smallest drop size likely to fall outside the jet cone is = 600 pm. These size drops occur with jet velocities up to = 30 ms-‘. For higher jet velocities, smaller drops will be formed, and no rainout is likely. Similar arguments can be made based on observations of turbulent flow. Successful horizontal pneumatic conveying of solids is reported to be possible with fairly low transport velocities. Perry”, citing Dalla ValleL4, suggests a correlation for the minimum carrying velocity needed to transport solid particles < 8100 pm of density less than 2,640 kilogramsm-3 as:
1
I-
Figure 1
d$&a, Thus, solid material with the density of chlorine, for example, would remain in suspension with particle sizes up to 1,000 pm with only 5 ms-’ horizontal velocity. Observations of aerosol jets showed substantial liquid fallout primarily when the jet impinges on the ground or a surfacet3. Thus, for nonimpinging jets, except for releases at quite low pressure and velocity, rainout is unlikely.
Elevated release model Releases of vapour or aerosol are treated for an arbitrary release angle. Within a short zone of initial development just outside the release, liquid breakup produces a droplet size distribution which would be stable except for subsequent evaporation (see Figure I). Droplets are also accelerated to the vapour velocity. Thus, thereafter the assumptions of the homogenous equilibrium model (HEM) apply. That is, vapour liquid equilibrium and equal vapour liquid velocities occur at each radial position across an axisymmetric cross section. The composition of the plume can be described quite generally using four components: dry air, water, a condensable component C, and a noncondensable component I, which is present in the source. For multicomC would be considered a ponent mixtures, pseudocomponent and the active pollutant may be one of several actual components making up C. The results would have to be scaled appropriately to account for such a case. Concentration is expressed hereafter in terms of the actual or pesudocomponent C. Following Ooms, et aL4, after the initial flow establishment zone, profiles for velocity, density, concentration, and enthalpy are assumed to have a Gaussian, cylindrically symmetric shape in sections normal to the plume axis. The similarity profiles are given respectively
ll
Notation
for elevated jet momentum release EJETmodel, after Ooms et al., 1972
by: U(S, r, 8) = ua cos 0 + U*(s)e-“‘b””
(7a)
pm(s, r, 8) = pa + pm*(s)e-r”x’b’(‘)
U’b)
c(s, r, 0) = c*(s)e-‘z’X’b’(s)
(7c)
AH(s, r, 8 ) = AH*(s)e-“‘XL b’(s)
(7d)
with the plume coordinates given by: dx - cos 9 ds-
(8a)
dz - sin 0 iiiThe plume coordinates, s and 9 represent the distance along the plume centreline and the angle of the centreline with respect to horizontal. For a falling plume 0 is negative. Equations (7) and (8) are the same for a vapour or an aerosol. In the flow establishment zone where vapour/ liquid slip takes place, the aerosol density is given by:
-I=..,=us~+uL~2
Pm
where urn, u,, UL are the respective reciprocal densities for the mixture, vapour, and liquid, qv, is the vapour quality or mass fraction of vapour in the aerosol and erg is the volume fraction of vapour in the aerosol: ar
=
+(’ - 4"NuL/u")"2 -’
1
!3 [
4v
1
(10)
In the homogeneous equilibrium zone, Equation (9) reduces to: -'=um=tr,q,+uL(l-q") Pm
(11)
Species concentration is given by: c(s, r, @ ) =
WcPm (s, r. @ 1
(12)
J. Loss Prev. Process lnd., 1989, Vol2, J a n u a r y 2 3
Dispersion modelling of an elevated high momentum release forming aerosols: John L. Woodward
where w, is the mass ratio of component c in the plume, consisting of a vapour phase component, W,V + W~L. The equations for conservation of species mass, momentum, and enthalpy are integrated across a cross section, thereby simplifying the problem to that of solving a set of nonlinear ordinary differental equations. These equations are: Conservation of species: d .,2b -.I o ds
c*(s,
r, 9)u(s, r, 8)2*r dr = 0
(13a)
Conservation of energy: d
.rb
ds s 0
AH*(s, r, 8)u(s, r, 8)2~r dr = 0
(13b)
Conservation of mass: d .zb P,,,(s, r, 0)u(s, r, 8)2nr dr ds s 0 = 2~rb(s)p,[crr 1 urn ) + (~2 ua 1 sin 9 1 cos 8 + ol3U’l ( 1 3 c ) Conservation of horizontal (x-directed) momentum: d ,zb p,,,(s, r, 9 )u2(s, r, 8)cos 82sr dr dso s = 2rb(s)u, [a, ) urn 1 + (~~24~ 1 sin 8 ( cos 8 + a3u'l + Cdrrb(s)p,u~ 1 sin3 8 1 (13d) Conservation of vertical (z-directed) momentum: d .zb P,,,(s, r, e)u*(s, r, 8)sin 82rr dr -s ds o = + Cn?rb(s)p,uZ sin’ 8 cos +
s
8
g(p, - p,)2nr dr
wa =
gmwaz
wW = gm ww2 = g, war Hum wc
=
w1=
(1 - &n)Wl
in which Hum is the absolute humidity of ambient air. Thus, composition is expressed in terms of a single variable, the mixing ratio, g,,,. It is similarly shown in Appendix A that ideal enthalpy can also be expressed in terms of g,,, and physical properties with known temperature dependence. Nonideal mixture enthalpies vary with liquid composition, but can be treated within the logic framework based on ideal enthalpies. As a consequence of the dependence between density, concentrations, and enthalpy on a single variable, the mixing ratio, there are only four independent equations (taken as 13a, c, d and e) to be solved. Mixture density is eliminated in favour of centreline concentration by dividing Equation (7~) by Equation (7b) using Equation (12) and evaluating at the centreline to give:
c*(s,r,e)=$$p,(s,r,e) where
nc* = Apt::
yc =
(14)
so the composition at any point will be given by: wn = (1 - &ll)W”l +
gmwnz
Y
(17b)
For an isothermal vapour, Equation (17b) reduces to a constant value5
(13e)
Equations (13) are not entirely independent. Equations (13a) and (13b) have essentially the same form, and, indeed concentration and enthalpy are linearly related for an adiabatic mixing process such as occurs with plume dilution. The process, for a homogeneous equilibrium model, can be considered as the mixing of two components, mass ml of the released aerosol, consisting of inerts plus active component C (which can be in a liquid and vapour phase), and the other component, mass mz of humid and possibly misty or snow-laden air. At any point in the plume, the mixing ratio of air will be:
in which n a = dry air, I = inerts, ambient air
WP~~S, r, e ) = DCR ~1
P&, r, 8) - ps
yc,~Mc M./I - I’&,
(18)
where, in general the mole fraction yc. is related to the mass fraction, w, by:
et a14.
ml + m2
(17a)
m
wc+ c
The derivation of these equations is given by Ooms,
gm = m2
(16)
(1 - gnl)wcl
DC R isothermal =
,2b 0
WC2 = 0:
(15)
refers to the four lumped components, w = water, C = active component, and 1 refers to vented conditions, and 2 to conditions. Specifically, since w,r = 0 and
24 J. Loss Rev. Process lnd., 1989, Vol2, J a n u a r y
making use of the average molecular weight of the aerosol mixture, Mmix as: I _=wL+w, Mmi.x
ML
Mv
For a vapour as in Equation (18). w, = 1 .O, yc = y,,, Mmi, = M,. For an aerosol, Equation (17b) is not constant, and y must be treated as a slowly-varying parameter. Since the relationship between pm and w, is implicit, through Equations (7), (1 l), and (17). and the phase relationships that are found by enthalpy considerations, we chose a direct approach to solve this system of equations. In the EJET model an adiabatic mixing calculation is made, apriori, over an arbitrary range of values of the mixing ratio, g,. This produces a consistent set of compositions, phase fractions, enthalpy, y and mixture density, pm, or pX. Then Equations (13a,c,d,e) are solved at the next integration step forward. The resulting value of c* is used as the independent variable
Dispersion modeling of an elevated high momentum release forming aerosols: John 1. Woodward
to interpolate the remaining variables from the table developed from adiabatic mixing. The expansion of Equations (13a,c,d,e) are given in Appendix B. Nonideal heat of mixing is treated by using correlations for excess enthalpy and by solving flash equations for the liquid and corresponding vapour compositions. For ammonia and hydrogen fluoride, the approach given by Wheatley ‘3,‘5 is recommended. This uses thermodynamics relationships for the saturated vapour pressure of water and ammonia or hydrogen fluoride over an aqueous solution and will, when incorporated, enable comparison with the data of Goldwire, et al. ‘,2 and Blewitt, ef al. 3 Figure 2 EJET model comparison with wind tunnel experiment
Model validation
by Eodurtha
Unfortunately, data are very limited for high momentum, elevated releases of aerosols. The wind tunnel data by Hoot, et al. 16, and the wind tunnel data by Bodurtha”, were used by Ooms, et a1.4 to verify their choice of model parameters. These data are for vertical (release angle 90”), elevated releases of isothermal gases (freon-air mixtures) over a fairly mild range of vent velocities. Small-scale tests with ammonia and propane aerosols were reported by Pfenning, et al. “. These involve high release velocities for a nearly horizontal release near the ground. The Desert Tortoise Series ammonia release experiments are large-scale tests of a similar type’s’. The EJET model was compared with each of the above-mentioned experimental data. Figure 2 shows a
Table 1
Case
typical wind tunnel experiment using Freon-l 14/sir mixtures by Bodurtha”. Oil drops were used to make the emission visible. Bodurtha did not report concentrations, rather only visual observations of plume ‘touchdown’. The uncertainty in making such observations is apparent. The EJET model predictions shown superimposed on Figure 2 are drawn for the 100 ppm contour. The point of plume centreline touchdown is reported for EJET model predictions. Hoot, et al. I6 made similar wind tunnel measurements using Freon 12-air mixtures, and also reported a plume touchdown distance. Their data are compared in Tub/e I with EJET model predictions. Figure3 is a parity plot for the data in Table I, showing generally
Verification of EJET predictions with wind tunnel experiments for pure vapours
Specific gravity
5 6
1.572 1.572
; 9 10 11 12 13 14 15 16 17 18 21 22 23 24 25 26 27 28
1.572 1.572 2.158 2.158 2.158 2.126 2.174 3.316 3.278 3.290 4.160 4.160 1.572 1.572 2.158 2.158 3.316 3.278 4.160 4.160
Vent velocity (inches s ‘J 45.06 90.13 135.2 180.3 44.5 89.1 133.6 180.3 180.3 44.5 89.1 133.6 44.5 90.1 45.1 90.1 44.5 89.1 44.5 89.1 44.5 so. 1
Rise height (inches)
X to touch down (inches)
Wind velocity (inches s-‘1
Stack height (inches)
8.70 8.70
3.0 3.0
3.05 5.21
1.86 4.32
19.5 28.2
16.5 29.2
0.70 8.70 8.70 8.73 8.73 8.70 8.70 8.73 8.73 8.77 8.73 8.73 8.75 8.75 8.55 8.55 8.98 8.98 8.98 8.96
3.0 3.0 3.0 :::
10.8 8.15 2.71 5.22 7.76 10.82 10.30 2.37 5.24 8.18 2.36 5.12 2.30 5.42 2.81 5.0 2.07 5.00 2.24 5.14
6.83 9.40 1.58 3.82 6.17 8.71 8.66 1.30 3.52 5.87 1.17 3.50 1.83 4.32 1.57 3.82 1.26 3.50 1.18 3.47
44.2 56.0 12.64 23.75 26.2 38.8 32.6 8.64 13.35 19.60 4.85 9.68 23.4 37.4 19.6 28.7 9.68 18.0 8.84 13.8
42.3 55.2 10.9 17.1 25.6 34.6 34.4 7.0 12.9 18.8 5.7 11.1 36.2 44.6 18.6 26.5 11.3 17.1 9.6 14.4
3.0 3.0 3.0 3.0 3.0 3.0 3.0 6.0 6.0 6.0 ::: 6.0 6.0 6.0
Experimental
EJET
Experimental
EJET
Using data from Ref. 16 Vent diameter = 0.25 inches Model parameters: CQ = 0.045, (Y* = 0.20, 01~ = 0.25. Very similar results are obtained with a, = 0.035, al = 0.20, a3 = 0.25 with slightly more bias toward underprediction
J. Loss Prev. Process Ind., 1989, Vol2, January 25
Dispersion modeling of an elevated high momentum release forming aerosols: John L. Woodward
Figure 3 Parity plot comparing EJET model predictions with wind tunnel isothermal vapour data of Hoot et a/., 1973, modal parameters; a, = 0.045, 012 = 0.20, a3 = 0.25
good agreement. The EJET model tends to underpredict the touchdown distances for the low vent height cases, but overall no bias is apparent. A similar comparison for the Bodurtha data is shown in Table 2 and Figure 4. The EJET model is able to describe both sets of wind tunnel data quite well, using the following adjustable parameter values: (~1 = (jet mixing parameter) = 0.035-0.045 (~2 = (plume drag parameter) = 0.20 a3 = (ambient wind turbulence parameter) = 0.25 These differ from the values used in the ONDEK model
Table 2 Selected runs by Eodurtha17 0.61 m and stack height = 30.5 m
Case
10 11 12 13 27 28 29 47 48 49 65 66 67
using vent diameter =
x to touchdown meters
Specific gravity
Vent velocity (ms ‘)
Wind velocity (m-x’)
Experimental
EJT
1.17 1.17 1.17 1.17 1.52 1.52 1.52 2.96 2.96 2.96 5.17 5.17 5.17
6.10 6.10 6.10 15.24 6.10 6.10 6.10 6.10 6.10 6.10 6.10 6.10 6.10
3.05 6.10 4.47 3.05 3.05 6.10 4.47 3.05 6.10 4.47 3.05 6.10 4.47
57.9 252.9 244 58 58 197 380 30.5 64.0 140 31.3 70 122
88 250 290 32 46 130 358 24 59 147 18 38 102
Model parameters aI = 0.035, az = 0.020. CQ = 0.25
26 J. Loss Prev. Process Ind., 1989, Vol2, J a n u a r y
Figure4 Parity plot comparing EJET model predictions with wind tunnel isothermal vapour data of Sodurtha, 1961, model parameters; IX, = 0.035, cx2 = 0.20, CQ = 0.25 using 200 ppm contours
which are CYI = 0.057, (~2 = 0.50, CY~ = 1.0. These coefficients were reportedly obtained from experimental data of Albertson, et al. 19, but no data comparisons were shown by Ooms, er aI.4. However, in my comparison, the ONDEK parameters give too much entrainment and a poor fit to the wind tunnel data. Havens and Spicer2o also reported agreement between the Ooms model and that of Hoot, et al. Since Havens and Spicer report different formulae for coefficients azl, a3,. a32, a34. &I, a42 and a44 than are given in Appendix A, the EJET model predictions differ from their reported results. As shown in Table 3, the EJET model agrees reasonably well with Hoot, et al., i n p r e d i c t i n g maximum rise height and the ‘distance to maximum ground level concentration’ compared with the ‘touchdown distances’ predicted by Hoot’s correlation. The maximum ground level concentration predicted by the EJET model is below the Hoot model predictions for the low vent diameter, low vent velocity case, but above for the high vent diameter, high vent velocity case. The EJET model predicts lower peak ground level concentrations with increasing wind speed, in contrast with Havens and Spicer’s realization of Ooms model. Hoot, ef aI.% model predicts decreasing peak ground level concentrations with increasing wind speed. Model stability is enhanced by using the correlations proposed by Morrow ” . These equations modify the Kamotani and Greber 22 correlation originally used with the ONDEK model to describe the region of developing flow. They apply for the first short distance past the end
Dispersion modeling of an elevated high momentum release forming aerosols: John L. Woodward Table 3 Comparison of elevated jet model predictions Maximum plume centreline height (ml above vent
Downwind distance’ to maximum ground level concentrations (m)
Low vent diameter (0.06 m), low vent velocity (30.6 ms-‘) Wind speed ims-‘I 3 6 EJETb 2.3 1.3 Havens”’ 2.4 1.6 Hoot 3.0 2.2 ONDEK’ 1.1 0.6 Medium vent diameter (0.2 m), medium vent velocity 197.1 ms-‘) Wind speed Ims-‘) 3 6 EJETb 23.5 15.2 Have&O 18.0 13.8 Hoot 23.8 19.0 ONDEK’ 12.6 7.3 High vent diameter (0.5 m), high vent velocity (213.9 ms-‘) Wind speed (ms-‘1 3 6 EJETb 99.2 70.7 Havens” 77.0 61.0 Hoot 97.0 76.6 ONDEKE 56.7 37.2
Maximum ground level concentration (kg mm3 x 103)
3 120 375 155 23
6 125 325 382 29
3 0.21 0.8 1.3 0.79
6 0.096 1.2 0.74 0.44
3 192 350 175 57
6 248 305 368 43
3 4.44 3.7 5.05 4.28
6 1.26 6.1 3.63 8.0
3 406 650 325 171
6 644 705 660 114
3 10.3 4.3 5.6 15.1
6 2.83 6.2 4.35 14.0
Density of vented material = 4.0 kg m-’ aHavens and Hoot values are downwind distance to plume touchdown “EJET model parameters u, = 0.035, 01~ = 0.20, Q~ =b.25 ‘Using correction by Morrowzo
of the vent and provide the initial conditions for the ordinary differential equations that are solved in the ONDEK and EJET models. By setting the initial plume angle, momentum, and location, the Morrow correlation extends the range of model stability to lower Froude numbers (specifically Froude number < 1 .O). The initial conditions can be set at the vent exit conditions for Froude numbers > 1.0 and only minor differences in model predictions result. Morrow’s correlation is a function of the momentum ratio of vent gases to wind at the stack exit height, or:
(21)
zone, or the ratio of this diameter, D, to t h e v e n t diameter, D+ Otherwise, for the Pfenning, et al. experiments where Dvo = 0.0254 m, the vent velocities are very high, in fact exceeding the two-phase choked (sonic) flow values predicted by the correlation of H e n r y23 . Such very high initial vent velocities produce high jet momentum mixing, and the model predictions are too low as Figure 5 illustrates. The figure plots the EJET model predictions for the ammonia release experiment of Pfenning, et al. as a function of the assumed ratio Dv/Dvo of developed-zone diameter to vent (nozzle) diameter. Good agreement with experimental observations is obtained for D,/D,IJ = 5 or initial
and depends on the densimetric Froude Number (Fr): Fr =
U,2P, g(P, - p=)Dv
(22)
After Kamotani and Greber”: x
z
-
D”
( D, >
-_=a”
0.4
(23)
with: fdr 0.0 < Fr < 1.7 a = %
18.5195 for J< exp(-3.5219+0.3726lnFr) Iexp(0.2476 + 0.3016 1nFr + 0.2438 1nJ) for J > exp( - 3.5219 + 0.3726 1nFr)
For Fr < 0.0 or Fr > 1.7 18.519/ av = t exp(0.405465 + 0.24386 In J)
for J < 0.036 for .I > 0.036
The concept of a ‘developing flow zone’ used in Equations (21)-(23) must be extended to treat the high momentum aerosol releases of Pfenning, et af. It is also necessary to define the diameter of the developed flow
000 * s . 6 OCYELOTrDnucolurrr~~r2rr DI_Tw!
a
7
$$I
Figure 5 Effect of assumptions of initial condition for developed plume diameter on EJET model predictions for ammonia experiment of Pfenning et al., 1387
J. Loss Prev. Process lnd., 7989, Vol2, January 27
Disoersion modellino of an elevated high momentum release formina aerosols: John L. Woodward
centreline velocities in the developed flow zone below 110 m s- ‘. The same concept was needed also to obtain good agreement with Pfenning’s propane and ammonia release data, as Table 4 shows.
Table 4 Experimental conditions and model predictions for high pressure releases of ammonia and propane Test
1.2
Release Release rate (kg s ‘) Mass fraction liquid
NH3 4.05 0.825
Vent diameter (m) Vent length (ml Vent height Iml Tank pressure (kPa)
0.0254 3.048 1.22 825.27
Ambient temperature ICI
18.9
Relative humidity (%)
49
Wind speed (ms ‘) Atmosphere stability
7.62 Slightly unstable
Visible concentration, contour (~01%) Maximum distance of visible cloud cm)
3-5 Propane 1.93 0.675, 0.662 0.0209 1.37 0.8 1032.1, 1066.6 15.6. 17.8 51. 42 5.08 Slightly unstable 2.003. 2.655 24.4, 21.3, 18.1 3.0 2.1, 1 .a
1.627 45.2, 45.7
Maximum visible cloud height (m)
3.4, 3.7
EJET model predictions maximum distance of visible cloud (m) Maximum visible cloud height (m)
Example cases Figures 6,7, and 8 illustrate the importance of modelling aerosol effects for jet momentum releases. In Figure 6, a 90” vent releasing ammonia is shown for assumed aerosol mass fractions, _fLIQ, of 0.6, 0.4, and 0.2 and the vent and ambient conditions listed. The 1,000 vppm vertical profile contours touch down for aerosol mass fractions of 0.60 and 0.40 but not for 0.20. All three aerosol fractions develop a hazardous zone at ground level where concentrations exceed the 500 vppm IDLH (immediately dangerous to life and health) level. With the EJET model formulation, the model is able to tolerate positively buoyant conditions over a fraction of the modelling range. Figure9 illustrates this by plotting the aerosol mixture density for ammonia for ftro = 0.20 over the entire range of ammonia-air mixture derived for the assumption that the aerosol temperature remains constant at the normal boiling point (NBP) until all liquid evaporates. This is not correct, of
18.2, 18.1 2.6, 3.1
43 4.4
Figure 6 EJET model predictions for elevated release of ammonia aerosol
EJET model parameters OL, = 0.035, 012 = 0.020, ulg = 0.25 I
I
I 0
I
d-W
/’ /
I
I \
\
\
\
I
I
I
Vent Height = 10 m, Vent Diameter = 0.20 m, Vent Velocity = 50 m/s Wind = 5 m/s at 10 m, Atmosphere Stability = D, Releese Angle = 96 \
\
‘\
1!
\
HAZARD ZONE
w
1000 “P&ml Contours
40
80
120
160 2aa DOWNWIND DISTANCE Im)
J. Loss Prev. Process tnd., 7 989. Vol2, January
240
280
320
Dispersion modeling of an elevated high momentum release forming aerosols: John 1. Woodward
course, since a dew point calculation shows aerosol temperatures below the NBP. A horizontal, 0” vent release of n-pentane is modelled in Figure 7 for assumed aerosol mass fractions, fLro, of 0.0, 0.30 and 0.60 and the listed vent and ambient conditions. With a release height of 10 m, the plume is diluted below the lower flammable limit (LFL) of 1.4% in all three cases. However, for fLIQ = 0.60 the plume touches down with concentrations above l/2 of the LFL. Chlorine releases require considerable dilution to dilute below toxic limits. Figure 8 plots the side view and plan view profiles at the STEL of 25 vppm and at 100 and 500 vppm for chlorine vented at 10 m from a 0.1 m diameter vent at 50 ms-’ into a 5 ms-’ wind. Aerosol mass fractions of 0 and 0.40 are shown, for a chlorine release temperature of 7°C with an ambient temperature of 25°C. The aerosol is predicted to evaporate completely before the plume touches down. Again, the aerosol case extends significantly further
90
Figure 7 EJET model predictions, horizontal jet of 80% pentanc 3, 20% nitrogen, vent velocity = 50 ms I, vent diameter = 0.2 m, vent height = 10 m, wind = 5 ms ‘, temperature = 25OC
Figure 8 Comparison of EJECT model predictions for aerosol and vapour release of chlorine
CONTOURS AT 1.68 m E L E V A T I O N -
I
180
I 200
I 300
-
-
I 400 DOWNWIND
-
-
I 600 DISTANCE
I 600
I 700
I 800
(METERS)
J. Loss Prev. Process Ind., 1989, Vol2, January 29
Dispersion modelling of an elevated high momentum release forming aerosols: John L. Woodwerd
- .24
2.2 -
2.0 -
MASS
Figure 9
FRACTION NH3 IN MIXTURE
Density of ammonia-air mixture assuming evaporation at the boiling point
than the nonaerosol case. Unfortunately, experimental data are lacking for high momentum releases of pentane and chlorine.
Conclusions The EJET model is shown to be stable for all cases attempted, using the initial conditions suggested by Morrow*‘. The extension to treat aerosol releases is shown to tolerate positive buoyancy over a fraction of the solution range. The concept of a developed flow zone is necessary for treating small diameter releases at very high velocities. Until a correlation for the diameter of the flow establishment zone, initial condition for the EJET model, is developed, it is necessary to run the model over a range of diameters to find the furthestextending plume predictions and use these to be conservative. The validated range of release velocities at this point is c 110 ms-‘.
References I
Goldwire, H. C., Jr. et al. Desert Tortoise Series Data Report
1983, Pressurized Ammonia Spills, Report UCRL-70562, Dec. 1985 2 Goldwire, H. C. Jr. et 01. Desert Tortoise Series Data Report 1983, Pressurized Ammonia Spills UCRL-20562, Lawrence Livermore National Laboratory, Livermore, CA, USA, April, 1986 3 Blewitt, D. N., Yohn, J. F., Koopman. R. P. and Brown, T. C. in
30 J. Loss Prev. Process lnd., 1989, Vol2, J a n u a r y
‘International Conference on Vapor Cloud Modeling,’ (Ed. J. L. Woodward), AIChE, New York, NY, 1987, pp. 1-38 4 Ooms, G., Mahieu, A. P. and Zelis, F. The Plume Path of Vent Gases Heavier Than Air, First International Symposium on Loss Prevention and Safety Promotion in the Process Industries (Ed. C. H. Buschman), Elsevier, Amsterdam, 1974, May 29-30 5 ‘VAX ONDEK 84 System Documentation’. prepared for US Coast Guard, Washington, D.C. under contract W6903-07G by Computer Data Systems, Inc., Rockville, MD, March, 1985 6 Astleford, W. J., Morrow, T. B. and Buckingham, J. C. ‘Hazardous Chemical Vapor Handbook for Marine Tank Vessels, Final Report No. CC-D-12-83, April, 1983 7 Colenbrander. G. W., 3rd International Symposium on Loss Prevention and Safety Promotion in the Process Industries, Basel, Switzerland, 1980 8 Krzeckowski, S. A. ht. J. Multiphase Flow 1980, 6, 227 9 Brodkey, R. S. ‘The Phenomena of Fluid Motions’, AddisonWesley, London, 1%7 1 0 Lapple, C. E. and Shepherd, C. B. Ind. and Eng. Chem 1940,32, 605 11 Perry, R. H. (Ed.) ‘Perry’s Chemical Engineer’s Handbook, McGraw-Hill, New York, 1984, p. 5-66 1 2 Batchelor. G. K. ‘An lntioduction to Fluid Dynamics,’ Cambridge University Press, 1967 13 Wheatley, C. J. Discharge of Liquid Ammonia to Moist Atmospheres-Survey of Experimental Data and Model for Estimating Initial Conditions for Dispersion Calculations, SRD/HSE Report R 410, UK Atomic Energy Authority. 1987 1 4 Dalla Valle Heat Piping and Air Cond. 1932, 4, 639 IS Wheatley. C. J., SRD Report R357, April, 1986 16 Hoot, T. G., Meroney. R. N. and Peterka, J. A. Report CER7374TGH-RNM-JAPI3. Fluid Dynamics and Diffusion Laboratory, Colorado State University, October, 1973 17 Bodurtha, F. T. LAPCA 1961, 11, 431 18 Pfenning, D. B., Millsap, S. B. and Johnson, D. W. in ‘International Conference on Vapor Cloud Modeling’ (Ed. J. L. Woodward) AIChE, New York, NY, USA, 1987, pp. 81-l 15
Dispersion modelling of an elevated high momentum release forming aerosols: John L. Woodward 19 Alberston, M. L., Dai, Y. B., Jenson, R. A. and Rouse. H. AXE Transactions 1950, 115, 639 20 Havens, J. A. and Spicer, T. 0. International Conference on Vapor Cloud Modeling (Woodwared Ed. J. L.), AIChE, New York, NY, USA 1987, p. 568 21 Morrow, T. B., SwRI Project 06.5686. investigation of the Hazards Posed by Chemical Vapours Released in Marine Operations-Phase II, July 26, 1985 22 Kamotani, Y. and Greber, 1. AIAA J. 1972, 10 23 Henry, R. E. Proceedings of the Japan-U.S. Seminar, 1979, on Two-Phase Flow Dynamics, McGraw-Hill, Toronto, 1981
Nomenclature A b CD c
coefficients in matrix, see Appendix B nominal width of jet plume (m) drag coefficient on droplet (dimensionless, 0.30) concentration of component C (kg rnm3) heat capacity at constant pressure (J/(kg “K)) droplet diameter (m) 2; Djct diameter of jet plume (m) diameter of vent (m) D” DC&! parameter relating c* to p,?,, Equation (17b) gravitational acceleration (ms-‘) B mixing ratio of air in plume (kg kg-‘) grn AHE enthalpy of emission (J kg-‘) AH, enthalpy of water entrained or taken up by mass transfer (J kg?‘) AH, enthalpy of dry air = 0 since air temperature is taken as the reference temperature AH enthalpy of plume relative fo reference temperature (J kg-‘) vaporization AH, _. (J kg- ‘) __ heat of._. AH, heat of fusion (J kg-‘) Hum absolute humidity (kg water/kg dry air) mass of air in any parcel of’piume (kg) ml mass of emitted material in any parcel of plume (kg) m2 M” molecular weight of component n (kg/kgmole) w e i g h t o f wet air = JQM, + (1 - Y~z)M, Mw molecular (kg/k’gmole) average molecular weight of vented material (kg/kgmole) M, vapour quality (kg vapour/kg emitted) 4q radius of plume(m) droplet Reynold’s number (dimensionless) ;(eo Rej jet Reynold’s number (dimensionless) distance along plume centreline (m) s t time (5) T temperature (K) u velocity of plume (ms- ’ ) wind speed at r(ms-‘) Ua ~t,,,,i” mmimum horizontal gas velocity to suspend particles in pneumatic conveying (ms-‘) atmospheric turbulence (ms-‘) vent velocity (ms-‘) specific volume of vapour (m’ kg-‘) specific volume of liquid (m” kg- ‘) specific volume of two phase mixture (m’ kg-‘) mass fraction of component n in mixture Weber number, Equatiqn (I), dimensionless downwind distance (m) crosswind distance (m) mole fraction component n in aerosol mole fraction component n in vapour phase mole fraction component n in liquid phase vertical height (m) vent height (m) volume fraction of vapour in aerosol coefficient in entrainment correlation, Equation (13d) turbulent Schmidt number relatlonshlp between C* and p:. Equation (17)L correction factor to drag coefficient, Equation (5) viscosity of air (Nm s-‘) viscosity of liquid (Nm s-‘) density of ambient air (kg m-‘) density of liquid (kg m-s) density of aerosol mixture (kg m-‘) density of vapour (kg rn-“) angle of emission relative to horizontal angle of plume centreline relative fo horizontal
Subscripts air component C (active pollutant) inert constituent liquid aerosol mixture droplet or particle vapour water source (vent) conditions ambient air conditions plume conditions
: I L m. P ” w 1 2 3
*Superscripts centreline properties of plume
Appendix A Aerosol enthalpy treatment
The plume mixing process mixes vented material at conditions (1) with moist air at conditions (2). Both materials are brought to a reference temperature, T,, and then to the plume mixture temperature, T3 (s, r, 9). Thus, all heats of vaporization and heats of fusion are evaluated at the reference temperature, T,, which is taken as the ambient air temperature T&z. For enthalpies, we need to know the phase relationships (vapour, liquid, solid respectively): WC = WC” + WCl WW = WW” +
WWL + wws
to obtain the enthalpy of material vented at T,: AHi= (wcytCpcv+
WLI&L
+ WIC,I,)(TI - Tr)
- W~LI AHcv(Tr)
(Al)
The initial air mixture can be treated generally as a two or even three-phase aerosol. To do so, specify the relative humidity at some temperature, T,,, and then let the air temperature be Ta2 which is below the dew-point and/or freezing point of water. Making use of the saturation humidity relationship in the Degadis model’: sat =
0.622 Pvw(Ta~) P amb
(A21
- Pvw(Ta~)
define the absolute humidity at the initial unsaturated temperature as: Hum = RH(T,,) Sat(T,I)
(A3)
so wW = w, Hum The saturated mass fraction of water is given by:
At any temperature, T, the mass fraction of water condensing, and freezing is, respectively: 1 - wW..,,/wW if wW > w,,,, if wW C w,,,,
fsw =
z.1 1.0
(Tmp_ T) ii
,‘; Fp (freezing point) .
mp
J. Loss Prev. Process lnd., 1989, Vol2, January 31
Dispersion modelling
of an elevated high momentum release forming aerosols: John L. Woodward
The enthalpy of moist air, relative to a reference temperature is:
A32 = 2 COS 8[2(#, COS e)‘+ kGu*ua COS 8 + k,u*2] +g [k.&acos8)‘+ ksu*uacosC3+ ksuf2]
AHz = ( wazCpa + wwv2Cpwv + W.,LZ c PWl + wws2Cpw*)(Ta2-
Tr)
- WILL AH,v(T,) - wwsz A&, (Tr)
(AS)
A33 = b cos 0 kau, cos 8 + k,ou
*
YC* +-(ksuacos8+k,,u*)
1
which is nonzero when liquid and solid are present even when T, is taken as Tar. The enthalpy of any parcel in the plume is given for an ideal mixture by: AH! = (1 - g,) AH, + g, AH,
(A61
Pa
A34 = -b sin 8 16 u,’ cos’ 8 + k12 u*ua cos 8
or for a nonideal mixture
+ k,u*2] + $ [kqu*2 + k,z,u*ua cos
AH3 = AH! + AHmix(T,, Xc)
+
k,,(u, cos 0)‘]
>
Aq = b sin 8 I-[k4(Ua cos e)2 Pa
Appendix B
+ ksu*ua cos 6 + ksu*‘]y
Expansion of Ooms model Using the similarity profiles, Equations (7) in the integral form equations of motion, Equation (13), the balance equations are integrated over the plume radius to give a set of ordinary differential equations:
A42 = 2 sin 8 [2(u, cos e)’ + k6u*u, cos 8 + k,u*21 (
+ $ [k4(& cos e)2 + ksu*u, cos 8 + kgu*‘]
All
>
A21
A31
L&I
A42 A43 A4441
dejds
A43 = b sin 8 k&la cos 8 + klou [
*
-Yc* (kSu, cos 0 + k,,u*) + Pa
where: AII = b(klu, cos 8 + kzu*) A12 =
8
(A7)
The enthalpy of mixing is a function of the liquid compostion (X,) as well a5 temperature.
I
.
2 c*(k,ua cos e + kru*)
A44 =
b [Zu: cos f3(1 - 3 sin’ 0) ( + k&*U,(COS2 8 - sin2 0) + k,u*2 cos
A,3 = k2bc* A14 = - klbc*u, sin 8 -rb
A21 = - (ha cos 8 +
Pa
1
+ 5 [kdui
e]
cos e(l - 3 sin2 9)
ksu*) + kgu*ua(cos2 8 - sin2 8) + ksu*2 cos 81)
>
BI =0
+
$ (k4ua cos 8 + ksu*) >
A23=b(k3+kr$)
B2=2[~I~u*(+~2ua/sin8~cose+~3u’l B3 = ua [Bz] + CDU~ / sin3 8 1 B4 = - k4bgyc*/p, I? CD(U, sin 8 )2 cos 9
and A24= - bu, sin63(2+k4 5) A31 = yb cos 8[k4(ue cos 0j2+ ksu*u, cos 8 + kgu*‘]/pa
32 J. Loss Prev. Process Ind., 7989, Vol2, January
k, = 0.772699; k2 = 0.412442; k, = 0.864665; ks = X2kI; k5 = X2kz; kg = 2k3; k, = 0 . 4 9 0 8 4 2 ; ks = 2X2k2; kg = 0 . 3 6 3 3 4 6 ; k,o = 2k7; k,, = 2ks; k,2 = 4ks; k,s = 2X2k,; k14 = 4X2k2; with X2 = 1.35
Accidental releases of liquids under pressure quite readily entertain liquid droplets as an aerosol. In addition, liquids may escape through a pressure relief valve as a result of: 0 overfilling the vessel; 0 thermal expansion of liquid, particularly during a runaway reaction; 0 bubble formation in the liquid; 0 entrainment of liquid during rapid depressurization. Not all pressure relief valves are connected to a flare, so there is a potential hazard if releases of heavy aerosol reach the ground before diluting below flammable or toxic limits. The presence of liquid droplets imparts heavy vapour characteristics to the dispersion of the release, often greatly extending the hazardous range of the release. For example, experiments with ammonia’*2, and with hydrogen fluoride3, showed that essentially all of the total mass in the release is entrained as an aerosol. The resulting aerosol clouds were found to persist longer and extend farther than a corresponding dry vapour cloud. The model by Ooms, et al. 4 was originally developed for elevated releases of heavy vapour and was made available in the public domain by the US Coast Guard as the ONDEK model 5*6. By a straightforward extension, this model can be made to treat aerosols without droplet rainout, using homogeneous equilibrium assumptions. The extended model is called the EJET model. The further extension to treat rainout is planned.
Received 12 October 1988 0950~4230/89/010022-11$3.00 0 1989 Butterworth & Co. (Publishers) Ltd 22 J. Loss Prev. Prbcess Ind., 1989, Vol2, January
Liquid breakup and rainout Liquid droplets larger than 1 mm are known to be unstable in a gas stream with appreciable velocity ( > 10 m s-l). The onset of breakup is usually tabulated at a ‘critical’ Weber number, defining Weber number as: We =
pvu2d,
(1) . ,
UL
Breakup occurs by the milder ‘bag’ mechanism above We = 10, and by successively more chaotic mechanisms at higher Weber number e.g. by bag-jet mechanism around We = 20, by shear or stripping off mechanism around We > 65 (Ref. 8). The maximum stable drop size according to Brodkey’ is, for low speed jets: dp =
1.89& (1 + 3We 1’2Rej”2) 1’2
(2) where Dj is the diameter of the jet Reynolds number defined by
For higher-speed jets, Brodkey recommends calculating the maximum drop size from We = 20. The fallout of droplets in a quiessant or at least laminar flow environment is well characterized by a force balance involving the drag coefficient lo. Terminal droplet velocities are readily found from an implicit Reynold’s number expression”: qCo(Reo)Rei=
4d;cPL - Pabag
(4) 3 P.2 where Co is the drag coefficient for a rigid spherical drop and n is a correction factor for the motion of fluid
Dispersion modelling
of an elevated high momentum release forming aerosols: John i. Woodward
within the drop given by Batchelor “:
1 + APL
(5)
’ = 1 + 2/3 P~/PL
Wheatley13 plots the terminal gravitational settling velocity of the largest stable drop as predicted by Equations (3), (4) and (5) against the jet velocity. He then applies a simple geometric argument with a top hat jet model to suggest that the smallest drop size likely to fall outside the jet cone is = 600 pm. These size drops occur with jet velocities up to = 30 ms-‘. For higher jet velocities, smaller drops will be formed, and no rainout is likely. Similar arguments can be made based on observations of turbulent flow. Successful horizontal pneumatic conveying of solids is reported to be possible with fairly low transport velocities. Perry”, citing Dalla ValleL4, suggests a correlation for the minimum carrying velocity needed to transport solid particles < 8100 pm of density less than 2,640 kilogramsm-3 as:
1
I-
Figure 1
d$&a, Thus, solid material with the density of chlorine, for example, would remain in suspension with particle sizes up to 1,000 pm with only 5 ms-’ horizontal velocity. Observations of aerosol jets showed substantial liquid fallout primarily when the jet impinges on the ground or a surfacet3. Thus, for nonimpinging jets, except for releases at quite low pressure and velocity, rainout is unlikely.
Elevated release model Releases of vapour or aerosol are treated for an arbitrary release angle. Within a short zone of initial development just outside the release, liquid breakup produces a droplet size distribution which would be stable except for subsequent evaporation (see Figure I). Droplets are also accelerated to the vapour velocity. Thus, thereafter the assumptions of the homogenous equilibrium model (HEM) apply. That is, vapour liquid equilibrium and equal vapour liquid velocities occur at each radial position across an axisymmetric cross section. The composition of the plume can be described quite generally using four components: dry air, water, a condensable component C, and a noncondensable component I, which is present in the source. For multicomC would be considered a ponent mixtures, pseudocomponent and the active pollutant may be one of several actual components making up C. The results would have to be scaled appropriately to account for such a case. Concentration is expressed hereafter in terms of the actual or pesudocomponent C. Following Ooms, et aL4, after the initial flow establishment zone, profiles for velocity, density, concentration, and enthalpy are assumed to have a Gaussian, cylindrically symmetric shape in sections normal to the plume axis. The similarity profiles are given respectively
ll
Notation
for elevated jet momentum release EJETmodel, after Ooms et al., 1972
by: U(S, r, 8) = ua cos 0 + U*(s)e-“‘b””
(7a)
pm(s, r, 8) = pa + pm*(s)e-r”x’b’(‘)
U’b)
c(s, r, 0) = c*(s)e-‘z’X’b’(s)
(7c)
AH(s, r, 8 ) = AH*(s)e-“‘XL b’(s)
(7d)
with the plume coordinates given by: dx - cos 9 ds-
(8a)
dz - sin 0 iiiThe plume coordinates, s and 9 represent the distance along the plume centreline and the angle of the centreline with respect to horizontal. For a falling plume 0 is negative. Equations (7) and (8) are the same for a vapour or an aerosol. In the flow establishment zone where vapour/ liquid slip takes place, the aerosol density is given by:
-I=..,=us~+uL~2
Pm
where urn, u,, UL are the respective reciprocal densities for the mixture, vapour, and liquid, qv, is the vapour quality or mass fraction of vapour in the aerosol and erg is the volume fraction of vapour in the aerosol: ar
=
+(’ - 4"NuL/u")"2 -’
1
!3 [
4v
1
(10)
In the homogeneous equilibrium zone, Equation (9) reduces to: -'=um=tr,q,+uL(l-q") Pm
(11)
Species concentration is given by: c(s, r, @ ) =
WcPm (s, r. @ 1
(12)
J. Loss Prev. Process lnd., 1989, Vol2, J a n u a r y 2 3
Dispersion modelling of an elevated high momentum release forming aerosols: John L. Woodward
where w, is the mass ratio of component c in the plume, consisting of a vapour phase component, W,V + W~L. The equations for conservation of species mass, momentum, and enthalpy are integrated across a cross section, thereby simplifying the problem to that of solving a set of nonlinear ordinary differental equations. These equations are: Conservation of species: d .,2b -.I o ds
c*(s,
r, 9)u(s, r, 8)2*r dr = 0
(13a)
Conservation of energy: d
.rb
ds s 0
AH*(s, r, 8)u(s, r, 8)2~r dr = 0
(13b)
Conservation of mass: d .zb P,,,(s, r, 0)u(s, r, 8)2nr dr ds s 0 = 2~rb(s)p,[crr 1 urn ) + (~2 ua 1 sin 9 1 cos 8 + ol3U’l ( 1 3 c ) Conservation of horizontal (x-directed) momentum: d ,zb p,,,(s, r, 9 )u2(s, r, 8)cos 82sr dr dso s = 2rb(s)u, [a, ) urn 1 + (~~24~ 1 sin 8 ( cos 8 + a3u'l + Cdrrb(s)p,u~ 1 sin3 8 1 (13d) Conservation of vertical (z-directed) momentum: d .zb P,,,(s, r, e)u*(s, r, 8)sin 82rr dr -s ds o = + Cn?rb(s)p,uZ sin’ 8 cos +
s
8
g(p, - p,)2nr dr
wa =
gmwaz
wW = gm ww2 = g, war Hum wc
=
w1=
(1 - &n)Wl
in which Hum is the absolute humidity of ambient air. Thus, composition is expressed in terms of a single variable, the mixing ratio, g,,,. It is similarly shown in Appendix A that ideal enthalpy can also be expressed in terms of g,,, and physical properties with known temperature dependence. Nonideal mixture enthalpies vary with liquid composition, but can be treated within the logic framework based on ideal enthalpies. As a consequence of the dependence between density, concentrations, and enthalpy on a single variable, the mixing ratio, there are only four independent equations (taken as 13a, c, d and e) to be solved. Mixture density is eliminated in favour of centreline concentration by dividing Equation (7~) by Equation (7b) using Equation (12) and evaluating at the centreline to give:
c*(s,r,e)=$$p,(s,r,e) where
nc* = Apt::
yc =
(14)
so the composition at any point will be given by: wn = (1 - &ll)W”l +
gmwnz
Y
(17b)
For an isothermal vapour, Equation (17b) reduces to a constant value5
(13e)
Equations (13) are not entirely independent. Equations (13a) and (13b) have essentially the same form, and, indeed concentration and enthalpy are linearly related for an adiabatic mixing process such as occurs with plume dilution. The process, for a homogeneous equilibrium model, can be considered as the mixing of two components, mass ml of the released aerosol, consisting of inerts plus active component C (which can be in a liquid and vapour phase), and the other component, mass mz of humid and possibly misty or snow-laden air. At any point in the plume, the mixing ratio of air will be:
in which n a = dry air, I = inerts, ambient air
WP~~S, r, e ) = DCR ~1
P&, r, 8) - ps
yc,~Mc M./I - I’&,
(18)
where, in general the mole fraction yc. is related to the mass fraction, w, by:
et a14.
ml + m2
(17a)
m
wc+ c
The derivation of these equations is given by Ooms,
gm = m2
(16)
(1 - gnl)wcl
DC R isothermal =
,2b 0
WC2 = 0:
(15)
refers to the four lumped components, w = water, C = active component, and 1 refers to vented conditions, and 2 to conditions. Specifically, since w,r = 0 and
24 J. Loss Rev. Process lnd., 1989, Vol2, J a n u a r y
making use of the average molecular weight of the aerosol mixture, Mmix as: I _=wL+w, Mmi.x
ML
Mv
For a vapour as in Equation (18). w, = 1 .O, yc = y,,, Mmi, = M,. For an aerosol, Equation (17b) is not constant, and y must be treated as a slowly-varying parameter. Since the relationship between pm and w, is implicit, through Equations (7), (1 l), and (17). and the phase relationships that are found by enthalpy considerations, we chose a direct approach to solve this system of equations. In the EJET model an adiabatic mixing calculation is made, apriori, over an arbitrary range of values of the mixing ratio, g,. This produces a consistent set of compositions, phase fractions, enthalpy, y and mixture density, pm, or pX. Then Equations (13a,c,d,e) are solved at the next integration step forward. The resulting value of c* is used as the independent variable
Dispersion modeling of an elevated high momentum release forming aerosols: John 1. Woodward
to interpolate the remaining variables from the table developed from adiabatic mixing. The expansion of Equations (13a,c,d,e) are given in Appendix B. Nonideal heat of mixing is treated by using correlations for excess enthalpy and by solving flash equations for the liquid and corresponding vapour compositions. For ammonia and hydrogen fluoride, the approach given by Wheatley ‘3,‘5 is recommended. This uses thermodynamics relationships for the saturated vapour pressure of water and ammonia or hydrogen fluoride over an aqueous solution and will, when incorporated, enable comparison with the data of Goldwire, et al. ‘,2 and Blewitt, ef al. 3 Figure 2 EJET model comparison with wind tunnel experiment
Model validation
by Eodurtha
Unfortunately, data are very limited for high momentum, elevated releases of aerosols. The wind tunnel data by Hoot, et al. 16, and the wind tunnel data by Bodurtha”, were used by Ooms, et a1.4 to verify their choice of model parameters. These data are for vertical (release angle 90”), elevated releases of isothermal gases (freon-air mixtures) over a fairly mild range of vent velocities. Small-scale tests with ammonia and propane aerosols were reported by Pfenning, et al. “. These involve high release velocities for a nearly horizontal release near the ground. The Desert Tortoise Series ammonia release experiments are large-scale tests of a similar type’s’. The EJET model was compared with each of the above-mentioned experimental data. Figure 2 shows a
Table 1
Case
typical wind tunnel experiment using Freon-l 14/sir mixtures by Bodurtha”. Oil drops were used to make the emission visible. Bodurtha did not report concentrations, rather only visual observations of plume ‘touchdown’. The uncertainty in making such observations is apparent. The EJET model predictions shown superimposed on Figure 2 are drawn for the 100 ppm contour. The point of plume centreline touchdown is reported for EJET model predictions. Hoot, et al. I6 made similar wind tunnel measurements using Freon 12-air mixtures, and also reported a plume touchdown distance. Their data are compared in Tub/e I with EJET model predictions. Figure3 is a parity plot for the data in Table I, showing generally
Verification of EJET predictions with wind tunnel experiments for pure vapours
Specific gravity
5 6
1.572 1.572
; 9 10 11 12 13 14 15 16 17 18 21 22 23 24 25 26 27 28
1.572 1.572 2.158 2.158 2.158 2.126 2.174 3.316 3.278 3.290 4.160 4.160 1.572 1.572 2.158 2.158 3.316 3.278 4.160 4.160
Vent velocity (inches s ‘J 45.06 90.13 135.2 180.3 44.5 89.1 133.6 180.3 180.3 44.5 89.1 133.6 44.5 90.1 45.1 90.1 44.5 89.1 44.5 89.1 44.5 so. 1
Rise height (inches)
X to touch down (inches)
Wind velocity (inches s-‘1
Stack height (inches)
8.70 8.70
3.0 3.0
3.05 5.21
1.86 4.32
19.5 28.2
16.5 29.2
0.70 8.70 8.70 8.73 8.73 8.70 8.70 8.73 8.73 8.77 8.73 8.73 8.75 8.75 8.55 8.55 8.98 8.98 8.98 8.96
3.0 3.0 3.0 :::
10.8 8.15 2.71 5.22 7.76 10.82 10.30 2.37 5.24 8.18 2.36 5.12 2.30 5.42 2.81 5.0 2.07 5.00 2.24 5.14
6.83 9.40 1.58 3.82 6.17 8.71 8.66 1.30 3.52 5.87 1.17 3.50 1.83 4.32 1.57 3.82 1.26 3.50 1.18 3.47
44.2 56.0 12.64 23.75 26.2 38.8 32.6 8.64 13.35 19.60 4.85 9.68 23.4 37.4 19.6 28.7 9.68 18.0 8.84 13.8
42.3 55.2 10.9 17.1 25.6 34.6 34.4 7.0 12.9 18.8 5.7 11.1 36.2 44.6 18.6 26.5 11.3 17.1 9.6 14.4
3.0 3.0 3.0 3.0 3.0 3.0 3.0 6.0 6.0 6.0 ::: 6.0 6.0 6.0
Experimental
EJET
Experimental
EJET
Using data from Ref. 16 Vent diameter = 0.25 inches Model parameters: CQ = 0.045, (Y* = 0.20, 01~ = 0.25. Very similar results are obtained with a, = 0.035, al = 0.20, a3 = 0.25 with slightly more bias toward underprediction
J. Loss Prev. Process Ind., 1989, Vol2, January 25
Dispersion modeling of an elevated high momentum release forming aerosols: John L. Woodward
Figure 3 Parity plot comparing EJET model predictions with wind tunnel isothermal vapour data of Hoot et a/., 1973, modal parameters; a, = 0.045, 012 = 0.20, a3 = 0.25
good agreement. The EJET model tends to underpredict the touchdown distances for the low vent height cases, but overall no bias is apparent. A similar comparison for the Bodurtha data is shown in Table 2 and Figure 4. The EJET model is able to describe both sets of wind tunnel data quite well, using the following adjustable parameter values: (~1 = (jet mixing parameter) = 0.035-0.045 (~2 = (plume drag parameter) = 0.20 a3 = (ambient wind turbulence parameter) = 0.25 These differ from the values used in the ONDEK model
Table 2 Selected runs by Eodurtha17 0.61 m and stack height = 30.5 m
Case
10 11 12 13 27 28 29 47 48 49 65 66 67
using vent diameter =
x to touchdown meters
Specific gravity
Vent velocity (ms ‘)
Wind velocity (m-x’)
Experimental
EJT
1.17 1.17 1.17 1.17 1.52 1.52 1.52 2.96 2.96 2.96 5.17 5.17 5.17
6.10 6.10 6.10 15.24 6.10 6.10 6.10 6.10 6.10 6.10 6.10 6.10 6.10
3.05 6.10 4.47 3.05 3.05 6.10 4.47 3.05 6.10 4.47 3.05 6.10 4.47
57.9 252.9 244 58 58 197 380 30.5 64.0 140 31.3 70 122
88 250 290 32 46 130 358 24 59 147 18 38 102
Model parameters aI = 0.035, az = 0.020. CQ = 0.25
26 J. Loss Prev. Process Ind., 1989, Vol2, J a n u a r y
Figure4 Parity plot comparing EJET model predictions with wind tunnel isothermal vapour data of Sodurtha, 1961, model parameters; IX, = 0.035, cx2 = 0.20, CQ = 0.25 using 200 ppm contours
which are CYI = 0.057, (~2 = 0.50, CY~ = 1.0. These coefficients were reportedly obtained from experimental data of Albertson, et al. 19, but no data comparisons were shown by Ooms, er aI.4. However, in my comparison, the ONDEK parameters give too much entrainment and a poor fit to the wind tunnel data. Havens and Spicer2o also reported agreement between the Ooms model and that of Hoot, et al. Since Havens and Spicer report different formulae for coefficients azl, a3,. a32, a34. &I, a42 and a44 than are given in Appendix A, the EJET model predictions differ from their reported results. As shown in Table 3, the EJET model agrees reasonably well with Hoot, et al., i n p r e d i c t i n g maximum rise height and the ‘distance to maximum ground level concentration’ compared with the ‘touchdown distances’ predicted by Hoot’s correlation. The maximum ground level concentration predicted by the EJET model is below the Hoot model predictions for the low vent diameter, low vent velocity case, but above for the high vent diameter, high vent velocity case. The EJET model predicts lower peak ground level concentrations with increasing wind speed, in contrast with Havens and Spicer’s realization of Ooms model. Hoot, ef aI.% model predicts decreasing peak ground level concentrations with increasing wind speed. Model stability is enhanced by using the correlations proposed by Morrow ” . These equations modify the Kamotani and Greber 22 correlation originally used with the ONDEK model to describe the region of developing flow. They apply for the first short distance past the end
Dispersion modeling of an elevated high momentum release forming aerosols: John L. Woodward Table 3 Comparison of elevated jet model predictions Maximum plume centreline height (ml above vent
Downwind distance’ to maximum ground level concentrations (m)
Low vent diameter (0.06 m), low vent velocity (30.6 ms-‘) Wind speed ims-‘I 3 6 EJETb 2.3 1.3 Havens”’ 2.4 1.6 Hoot 3.0 2.2 ONDEK’ 1.1 0.6 Medium vent diameter (0.2 m), medium vent velocity 197.1 ms-‘) Wind speed Ims-‘) 3 6 EJETb 23.5 15.2 Have&O 18.0 13.8 Hoot 23.8 19.0 ONDEK’ 12.6 7.3 High vent diameter (0.5 m), high vent velocity (213.9 ms-‘) Wind speed (ms-‘1 3 6 EJETb 99.2 70.7 Havens” 77.0 61.0 Hoot 97.0 76.6 ONDEKE 56.7 37.2
Maximum ground level concentration (kg mm3 x 103)
3 120 375 155 23
6 125 325 382 29
3 0.21 0.8 1.3 0.79
6 0.096 1.2 0.74 0.44
3 192 350 175 57
6 248 305 368 43
3 4.44 3.7 5.05 4.28
6 1.26 6.1 3.63 8.0
3 406 650 325 171
6 644 705 660 114
3 10.3 4.3 5.6 15.1
6 2.83 6.2 4.35 14.0
Density of vented material = 4.0 kg m-’ aHavens and Hoot values are downwind distance to plume touchdown “EJET model parameters u, = 0.035, 01~ = 0.20, Q~ =b.25 ‘Using correction by Morrowzo
of the vent and provide the initial conditions for the ordinary differential equations that are solved in the ONDEK and EJET models. By setting the initial plume angle, momentum, and location, the Morrow correlation extends the range of model stability to lower Froude numbers (specifically Froude number < 1 .O). The initial conditions can be set at the vent exit conditions for Froude numbers > 1.0 and only minor differences in model predictions result. Morrow’s correlation is a function of the momentum ratio of vent gases to wind at the stack exit height, or:
(21)
zone, or the ratio of this diameter, D, to t h e v e n t diameter, D+ Otherwise, for the Pfenning, et al. experiments where Dvo = 0.0254 m, the vent velocities are very high, in fact exceeding the two-phase choked (sonic) flow values predicted by the correlation of H e n r y23 . Such very high initial vent velocities produce high jet momentum mixing, and the model predictions are too low as Figure 5 illustrates. The figure plots the EJET model predictions for the ammonia release experiment of Pfenning, et al. as a function of the assumed ratio Dv/Dvo of developed-zone diameter to vent (nozzle) diameter. Good agreement with experimental observations is obtained for D,/D,IJ = 5 or initial
and depends on the densimetric Froude Number (Fr): Fr =
U,2P, g(P, - p=)Dv
(22)
After Kamotani and Greber”: x
z
-
D”
( D, >
-_=a”
0.4
(23)
with: fdr 0.0 < Fr < 1.7 a = %
18.5195 for J< exp(-3.5219+0.3726lnFr) Iexp(0.2476 + 0.3016 1nFr + 0.2438 1nJ) for J > exp( - 3.5219 + 0.3726 1nFr)
For Fr < 0.0 or Fr > 1.7 18.519/ av = t exp(0.405465 + 0.24386 In J)
for J < 0.036 for .I > 0.036
The concept of a ‘developing flow zone’ used in Equations (21)-(23) must be extended to treat the high momentum aerosol releases of Pfenning, et af. It is also necessary to define the diameter of the developed flow
000 * s . 6 OCYELOTrDnucolurrr~~r2rr DI_Tw!
a
7
$$I
Figure 5 Effect of assumptions of initial condition for developed plume diameter on EJET model predictions for ammonia experiment of Pfenning et al., 1387
J. Loss Prev. Process lnd., 7989, Vol2, January 27
Disoersion modellino of an elevated high momentum release formina aerosols: John L. Woodward
centreline velocities in the developed flow zone below 110 m s- ‘. The same concept was needed also to obtain good agreement with Pfenning’s propane and ammonia release data, as Table 4 shows.
Table 4 Experimental conditions and model predictions for high pressure releases of ammonia and propane Test
1.2
Release Release rate (kg s ‘) Mass fraction liquid
NH3 4.05 0.825
Vent diameter (m) Vent length (ml Vent height Iml Tank pressure (kPa)
0.0254 3.048 1.22 825.27
Ambient temperature ICI
18.9
Relative humidity (%)
49
Wind speed (ms ‘) Atmosphere stability
7.62 Slightly unstable
Visible concentration, contour (~01%) Maximum distance of visible cloud cm)
3-5 Propane 1.93 0.675, 0.662 0.0209 1.37 0.8 1032.1, 1066.6 15.6. 17.8 51. 42 5.08 Slightly unstable 2.003. 2.655 24.4, 21.3, 18.1 3.0 2.1, 1 .a
1.627 45.2, 45.7
Maximum visible cloud height (m)
3.4, 3.7
EJET model predictions maximum distance of visible cloud (m) Maximum visible cloud height (m)
Example cases Figures 6,7, and 8 illustrate the importance of modelling aerosol effects for jet momentum releases. In Figure 6, a 90” vent releasing ammonia is shown for assumed aerosol mass fractions, _fLIQ, of 0.6, 0.4, and 0.2 and the vent and ambient conditions listed. The 1,000 vppm vertical profile contours touch down for aerosol mass fractions of 0.60 and 0.40 but not for 0.20. All three aerosol fractions develop a hazardous zone at ground level where concentrations exceed the 500 vppm IDLH (immediately dangerous to life and health) level. With the EJET model formulation, the model is able to tolerate positively buoyant conditions over a fraction of the modelling range. Figure9 illustrates this by plotting the aerosol mixture density for ammonia for ftro = 0.20 over the entire range of ammonia-air mixture derived for the assumption that the aerosol temperature remains constant at the normal boiling point (NBP) until all liquid evaporates. This is not correct, of
18.2, 18.1 2.6, 3.1
43 4.4
Figure 6 EJET model predictions for elevated release of ammonia aerosol
EJET model parameters OL, = 0.035, 012 = 0.020, ulg = 0.25 I
I
I 0
I
d-W
/’ /
I
I \
\
\
\
I
I
I
Vent Height = 10 m, Vent Diameter = 0.20 m, Vent Velocity = 50 m/s Wind = 5 m/s at 10 m, Atmosphere Stability = D, Releese Angle = 96 \
\
‘\
1!
\
HAZARD ZONE
w
1000 “P&ml Contours
40
80
120
160 2aa DOWNWIND DISTANCE Im)
J. Loss Prev. Process tnd., 7 989. Vol2, January
240
280
320
Dispersion modeling of an elevated high momentum release forming aerosols: John 1. Woodward
course, since a dew point calculation shows aerosol temperatures below the NBP. A horizontal, 0” vent release of n-pentane is modelled in Figure 7 for assumed aerosol mass fractions, fLro, of 0.0, 0.30 and 0.60 and the listed vent and ambient conditions. With a release height of 10 m, the plume is diluted below the lower flammable limit (LFL) of 1.4% in all three cases. However, for fLIQ = 0.60 the plume touches down with concentrations above l/2 of the LFL. Chlorine releases require considerable dilution to dilute below toxic limits. Figure 8 plots the side view and plan view profiles at the STEL of 25 vppm and at 100 and 500 vppm for chlorine vented at 10 m from a 0.1 m diameter vent at 50 ms-’ into a 5 ms-’ wind. Aerosol mass fractions of 0 and 0.40 are shown, for a chlorine release temperature of 7°C with an ambient temperature of 25°C. The aerosol is predicted to evaporate completely before the plume touches down. Again, the aerosol case extends significantly further
90
Figure 7 EJET model predictions, horizontal jet of 80% pentanc 3, 20% nitrogen, vent velocity = 50 ms I, vent diameter = 0.2 m, vent height = 10 m, wind = 5 ms ‘, temperature = 25OC
Figure 8 Comparison of EJECT model predictions for aerosol and vapour release of chlorine
CONTOURS AT 1.68 m E L E V A T I O N -
I
180
I 200
I 300
-
-
I 400 DOWNWIND
-
-
I 600 DISTANCE
I 600
I 700
I 800
(METERS)
J. Loss Prev. Process Ind., 1989, Vol2, January 29
Dispersion modelling of an elevated high momentum release forming aerosols: John L. Woodwerd
- .24
2.2 -
2.0 -
MASS
Figure 9
FRACTION NH3 IN MIXTURE
Density of ammonia-air mixture assuming evaporation at the boiling point
than the nonaerosol case. Unfortunately, experimental data are lacking for high momentum releases of pentane and chlorine.
Conclusions The EJET model is shown to be stable for all cases attempted, using the initial conditions suggested by Morrow*‘. The extension to treat aerosol releases is shown to tolerate positive buoyancy over a fraction of the solution range. The concept of a developed flow zone is necessary for treating small diameter releases at very high velocities. Until a correlation for the diameter of the flow establishment zone, initial condition for the EJET model, is developed, it is necessary to run the model over a range of diameters to find the furthestextending plume predictions and use these to be conservative. The validated range of release velocities at this point is c 110 ms-‘.
References I
Goldwire, H. C., Jr. et al. Desert Tortoise Series Data Report
1983, Pressurized Ammonia Spills, Report UCRL-70562, Dec. 1985 2 Goldwire, H. C. Jr. et 01. Desert Tortoise Series Data Report 1983, Pressurized Ammonia Spills UCRL-20562, Lawrence Livermore National Laboratory, Livermore, CA, USA, April, 1986 3 Blewitt, D. N., Yohn, J. F., Koopman. R. P. and Brown, T. C. in
30 J. Loss Prev. Process lnd., 1989, Vol2, J a n u a r y
‘International Conference on Vapor Cloud Modeling,’ (Ed. J. L. Woodward), AIChE, New York, NY, 1987, pp. 1-38 4 Ooms, G., Mahieu, A. P. and Zelis, F. The Plume Path of Vent Gases Heavier Than Air, First International Symposium on Loss Prevention and Safety Promotion in the Process Industries (Ed. C. H. Buschman), Elsevier, Amsterdam, 1974, May 29-30 5 ‘VAX ONDEK 84 System Documentation’. prepared for US Coast Guard, Washington, D.C. under contract W6903-07G by Computer Data Systems, Inc., Rockville, MD, March, 1985 6 Astleford, W. J., Morrow, T. B. and Buckingham, J. C. ‘Hazardous Chemical Vapor Handbook for Marine Tank Vessels, Final Report No. CC-D-12-83, April, 1983 7 Colenbrander. G. W., 3rd International Symposium on Loss Prevention and Safety Promotion in the Process Industries, Basel, Switzerland, 1980 8 Krzeckowski, S. A. ht. J. Multiphase Flow 1980, 6, 227 9 Brodkey, R. S. ‘The Phenomena of Fluid Motions’, AddisonWesley, London, 1%7 1 0 Lapple, C. E. and Shepherd, C. B. Ind. and Eng. Chem 1940,32, 605 11 Perry, R. H. (Ed.) ‘Perry’s Chemical Engineer’s Handbook, McGraw-Hill, New York, 1984, p. 5-66 1 2 Batchelor. G. K. ‘An lntioduction to Fluid Dynamics,’ Cambridge University Press, 1967 13 Wheatley, C. J. Discharge of Liquid Ammonia to Moist Atmospheres-Survey of Experimental Data and Model for Estimating Initial Conditions for Dispersion Calculations, SRD/HSE Report R 410, UK Atomic Energy Authority. 1987 1 4 Dalla Valle Heat Piping and Air Cond. 1932, 4, 639 IS Wheatley. C. J., SRD Report R357, April, 1986 16 Hoot, T. G., Meroney. R. N. and Peterka, J. A. Report CER7374TGH-RNM-JAPI3. Fluid Dynamics and Diffusion Laboratory, Colorado State University, October, 1973 17 Bodurtha, F. T. LAPCA 1961, 11, 431 18 Pfenning, D. B., Millsap, S. B. and Johnson, D. W. in ‘International Conference on Vapor Cloud Modeling’ (Ed. J. L. Woodward) AIChE, New York, NY, USA, 1987, pp. 81-l 15
Dispersion modelling of an elevated high momentum release forming aerosols: John L. Woodward 19 Alberston, M. L., Dai, Y. B., Jenson, R. A. and Rouse. H. AXE Transactions 1950, 115, 639 20 Havens, J. A. and Spicer, T. 0. International Conference on Vapor Cloud Modeling (Woodwared Ed. J. L.), AIChE, New York, NY, USA 1987, p. 568 21 Morrow, T. B., SwRI Project 06.5686. investigation of the Hazards Posed by Chemical Vapours Released in Marine Operations-Phase II, July 26, 1985 22 Kamotani, Y. and Greber, 1. AIAA J. 1972, 10 23 Henry, R. E. Proceedings of the Japan-U.S. Seminar, 1979, on Two-Phase Flow Dynamics, McGraw-Hill, Toronto, 1981
Nomenclature A b CD c
coefficients in matrix, see Appendix B nominal width of jet plume (m) drag coefficient on droplet (dimensionless, 0.30) concentration of component C (kg rnm3) heat capacity at constant pressure (J/(kg “K)) droplet diameter (m) 2; Djct diameter of jet plume (m) diameter of vent (m) D” DC&! parameter relating c* to p,?,, Equation (17b) gravitational acceleration (ms-‘) B mixing ratio of air in plume (kg kg-‘) grn AHE enthalpy of emission (J kg-‘) AH, enthalpy of water entrained or taken up by mass transfer (J kg?‘) AH, enthalpy of dry air = 0 since air temperature is taken as the reference temperature AH enthalpy of plume relative fo reference temperature (J kg-‘) vaporization AH, _. (J kg- ‘) __ heat of._. AH, heat of fusion (J kg-‘) Hum absolute humidity (kg water/kg dry air) mass of air in any parcel of’piume (kg) ml mass of emitted material in any parcel of plume (kg) m2 M” molecular weight of component n (kg/kgmole) w e i g h t o f wet air = JQM, + (1 - Y~z)M, Mw molecular (kg/k’gmole) average molecular weight of vented material (kg/kgmole) M, vapour quality (kg vapour/kg emitted) 4q radius of plume(m) droplet Reynold’s number (dimensionless) ;(eo Rej jet Reynold’s number (dimensionless) distance along plume centreline (m) s t time (5) T temperature (K) u velocity of plume (ms- ’ ) wind speed at r(ms-‘) Ua ~t,,,,i” mmimum horizontal gas velocity to suspend particles in pneumatic conveying (ms-‘) atmospheric turbulence (ms-‘) vent velocity (ms-‘) specific volume of vapour (m’ kg-‘) specific volume of liquid (m” kg- ‘) specific volume of two phase mixture (m’ kg-‘) mass fraction of component n in mixture Weber number, Equatiqn (I), dimensionless downwind distance (m) crosswind distance (m) mole fraction component n in aerosol mole fraction component n in vapour phase mole fraction component n in liquid phase vertical height (m) vent height (m) volume fraction of vapour in aerosol coefficient in entrainment correlation, Equation (13d) turbulent Schmidt number relatlonshlp between C* and p:. Equation (17)L correction factor to drag coefficient, Equation (5) viscosity of air (Nm s-‘) viscosity of liquid (Nm s-‘) density of ambient air (kg m-‘) density of liquid (kg m-s) density of aerosol mixture (kg m-‘) density of vapour (kg rn-“) angle of emission relative to horizontal angle of plume centreline relative fo horizontal
Subscripts air component C (active pollutant) inert constituent liquid aerosol mixture droplet or particle vapour water source (vent) conditions ambient air conditions plume conditions
: I L m. P ” w 1 2 3
*Superscripts centreline properties of plume
Appendix A Aerosol enthalpy treatment
The plume mixing process mixes vented material at conditions (1) with moist air at conditions (2). Both materials are brought to a reference temperature, T,, and then to the plume mixture temperature, T3 (s, r, 9). Thus, all heats of vaporization and heats of fusion are evaluated at the reference temperature, T,, which is taken as the ambient air temperature T&z. For enthalpies, we need to know the phase relationships (vapour, liquid, solid respectively): WC = WC” + WCl WW = WW” +
WWL + wws
to obtain the enthalpy of material vented at T,: AHi= (wcytCpcv+
WLI&L
+ WIC,I,)(TI - Tr)
- W~LI AHcv(Tr)
(Al)
The initial air mixture can be treated generally as a two or even three-phase aerosol. To do so, specify the relative humidity at some temperature, T,,, and then let the air temperature be Ta2 which is below the dew-point and/or freezing point of water. Making use of the saturation humidity relationship in the Degadis model’: sat =
0.622 Pvw(Ta~) P amb
(A21
- Pvw(Ta~)
define the absolute humidity at the initial unsaturated temperature as: Hum = RH(T,,) Sat(T,I)
(A3)
so wW = w, Hum The saturated mass fraction of water is given by:
At any temperature, T, the mass fraction of water condensing, and freezing is, respectively: 1 - wW..,,/wW if wW > w,,,, if wW C w,,,,
fsw =
z.1 1.0
(Tmp_ T) ii
,‘; Fp (freezing point) .
mp
J. Loss Prev. Process lnd., 1989, Vol2, January 31
Dispersion modelling
of an elevated high momentum release forming aerosols: John L. Woodward
The enthalpy of moist air, relative to a reference temperature is:
A32 = 2 COS 8[2(#, COS e)‘+ kGu*ua COS 8 + k,u*2] +g [k.&acos8)‘+ ksu*uacosC3+ ksuf2]
AHz = ( wazCpa + wwv2Cpwv + W.,LZ c PWl + wws2Cpw*)(Ta2-
Tr)
- WILL AH,v(T,) - wwsz A&, (Tr)
(AS)
A33 = b cos 0 kau, cos 8 + k,ou
*
YC* +-(ksuacos8+k,,u*)
1
which is nonzero when liquid and solid are present even when T, is taken as Tar. The enthalpy of any parcel in the plume is given for an ideal mixture by: AH! = (1 - g,) AH, + g, AH,
(A61
Pa
A34 = -b sin 8 16 u,’ cos’ 8 + k12 u*ua cos 8
or for a nonideal mixture
+ k,u*2] + $ [kqu*2 + k,z,u*ua cos
AH3 = AH! + AHmix(T,, Xc)
+
k,,(u, cos 0)‘]
>
Aq = b sin 8 I-[k4(Ua cos e)2 Pa
Appendix B
+ ksu*ua cos 6 + ksu*‘]y
Expansion of Ooms model Using the similarity profiles, Equations (7) in the integral form equations of motion, Equation (13), the balance equations are integrated over the plume radius to give a set of ordinary differential equations:
A42 = 2 sin 8 [2(u, cos e)’ + k6u*u, cos 8 + k,u*21 (
+ $ [k4(& cos e)2 + ksu*u, cos 8 + kgu*‘]
All
>
A21
A31
L&I
A42 A43 A4441
dejds
A43 = b sin 8 k&la cos 8 + klou [
*
-Yc* (kSu, cos 0 + k,,u*) + Pa
where: AII = b(klu, cos 8 + kzu*) A12 =
8
(A7)
The enthalpy of mixing is a function of the liquid compostion (X,) as well a5 temperature.
I
.
2 c*(k,ua cos e + kru*)
A44 =
b [Zu: cos f3(1 - 3 sin’ 0) ( + k&*U,(COS2 8 - sin2 0) + k,u*2 cos
A,3 = k2bc* A14 = - klbc*u, sin 8 -rb
A21 = - (ha cos 8 +
Pa
1
+ 5 [kdui
e]
cos e(l - 3 sin2 9)
ksu*) + kgu*ua(cos2 8 - sin2 8) + ksu*2 cos 81)
>
BI =0
+
$ (k4ua cos 8 + ksu*) >
A23=b(k3+kr$)
B2=2[~I~u*(+~2ua/sin8~cose+~3u’l B3 = ua [Bz] + CDU~ / sin3 8 1 B4 = - k4bgyc*/p, I? CD(U, sin 8 )2 cos 9
and A24= - bu, sin63(2+k4 5) A31 = yb cos 8[k4(ue cos 0j2+ ksu*u, cos 8 + kgu*‘]/pa
32 J. Loss Prev. Process Ind., 7989, Vol2, January
k, = 0.772699; k2 = 0.412442; k, = 0.864665; ks = X2kI; k5 = X2kz; kg = 2k3; k, = 0 . 4 9 0 8 4 2 ; ks = 2X2k2; kg = 0 . 3 6 3 3 4 6 ; k,o = 2k7; k,, = 2ks; k,2 = 4ks; k,s = 2X2k,; k14 = 4X2k2; with X2 = 1.35