vvEPA
United States
Environmental Protection
Agency
Office of Air Quality
Planning and Standards
Research Triangle Park NC 27711
EPA-450/4-88-006a
April 1988
Air
A Dispersion Model For
Elevated Dense Gas
Jet Chemical Releases
Volume I.
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EPA-450/4-88-006a
April 1988
A Dispersion Model For Elevated Dense
Gas Jet Chemical Releases
Volume I.
U.S. ENVIRONMENTAL PROTECTION AGENCY
Office of Air and Radiation
Office of Air Quality Planning and Standards
Research Triangle Park, North Carolina 27711
U.S. Environmental Protection
Segion 5, Library (5PL-16)
230 S. Dearborn Street, Room j
Chicago, -1L 60604
-------
DISCLAIMER
This report has been reviewed by the Office of Air Quality Planning and Standards, U.S. Environmental
Protection Agency, and approved for publication as received from Dr. Jerry Havens. Approval does not
signify that the contents necessarily reflect the views and policies of the U. S. Environmental Protection
Agency, nor does mention of trade names or commercial products constitute endorsement or
recommendation for use. Copies of this report are available from the National Technical Information Service
(NTIS).
ACKNOWLEDGEMENTS
The elevated dense gas jet model incorporates methodology published by Ooms and his colleagues at
the Technological University Delft, The Netherlands, along with the DEGADIS dense gas dispersion
model developed at the University of Arkansas. Tom Spicer, my coauthor of DEGADIS, contributed to
the development and was responsible for the modifications to DEGADIS required for interfacing with
Ooms' model.
Jerry Havens
-------
PREFACE
This version of the elevated dense gas dispersion model, Ooms/DEGADIS,
has been developed by Dr. Jerry Havens and Dr. Tom Spicer of the
University of Arkansas with the support of funding from the United States
Environmental Protection Agency (EPA). It represents intermediate
development of a dense gas modeling package which is undergoing further
refinement through additional EPA support. While this model has not been
extensively tested against field data, and is subject to specific
limitations and uncertainties, the EPA is making it publicly available
through the National Technical Information Service (NTIS) as an interim
research tool pending further model evaluation and development.
The Ooms/DEGADIS model has been written in FORTRAN with specific
intent for compilation and execution on a Digital Equipment Corporation
VAX computer. Implementation of this model on any other computer system
may be attempted at the risk of the user. Considerations for such
implementation, however, are discussed in Appendix B of Volume II.
To facilitate dissemination of the model, it is being provided on two
PC-compatible diskettes. The model should be uploaded via modem from a PC
terminal to a host VAX computer, and several files must then be renamed
prior to compilation and execution. Specific information on this process
is contained in the file AAREADME.TXT. Print this file and the
compilation batch file, BUILD.COM, prior to attempting compilation.
It is the concern of the EPA that this model be applied only within
the framework of its intended use. To this end the user is referred to
the specific recommendations in Volume I, Section VII for model
application. These recommendations take advantage of the fact that, in
this version of the Ooms/DEGADIS model, the portion of the model adapted
from Ooms and his colleagues can be executed as a standalone model, as can
the DEGADIS portion. To begin any particular simulation, it is
recommended that the Ooms portion of the model be executed by itself.
-------
This can be accomplished by setting the input variable equal to 1.
If the output from this simulation predicts that the plume will touch down
less than 1 kilometer from the source, the complete Ooms/DEGADIS model may
be appropriately applied (set equal to zero or greater). If the
plume is not predicted to touch down within 1 kilometer, this model should
not be used.
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VOLUME I
TABLE OF CONTENTS
Chapter Page
List of Figures ^^
v
List of Tables
vii
List of Symbols
Summary
X1X
I. Introduction
II. Characterizing Gas Density Effects in Dispersion 3
III. Description of the Ooms Model 7
IV. Evaluation of the Ooms Model 15
V. Description of the DEGADIS Model 23
VI. Interfacing the Ooms and DEGADIS Models 55
VII. Conclusions and Recommendations 57
References 61
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LIST OF FIGURES
Figure 'Page
III.l Schematic Diagram of Corns' Model 7
V.I Schematic Diagram of DEGADIS Dense Gas Dispersion Model 24
V.2 Schematic Diagram of a Radially Spreading Cloud 26
V.3 The Unsteady Gravity Current 28
V.4 The Head of a Steady Gravity Current 31
111
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LIST OF TABLES
Table Page
II.1 Criteria for Determining Whether Jet Effects
Dominate a Ground-Level Release 6
IV.1 Specification of Gas Release Rates for Modeling 15
IV.2 Comparison of Ooms Model Prediction with
Hoot et al.'s Wind-Tunnel Data Correlation 17
IV.3 Sensitivity of Ooms Model Prediction to Variation
of Entrainment Coefficients a]_ and 02 with no
Atmospheric Turbulence Entrainment 19
IV.4 Sensitivity of Ooms Model Prediction to Variation
of Entrainment Coefficients 20
V.I Typical Atmospheric Boundary Layer Stability and
Wind Profile Concentrations 45
V.2 Coefficient 5 in Gaussian Dispersion Model for Use in
ay = fix/3 with ft = 0.894 and ay and x in Meters 51
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LIST OF SYMBOLS
NOTE: This list of symbols is divided into numbered sections
corresponding to the sections of this report.
II. Characterizing Gas Density Effects on Dispersion
D diameter of release (m)
g gravitational acceleration (m/s)
H characteristic height (depth) of release (m)
HL total layer depth in DEGADIS model (m)
k von Karman's constant, 0.35
Lg buoyancy length scale (m)
Ma mass of air (kg)
Q volumetric release rate (nr/s)
R radius of jet (m)
Ric release Richardson number
Ri* Richardson number in Equation (V.78)
u ambient (wind) velocity (m/s)
UL average transport velocity associated with HL (m/s)
u^ friction velocity (m/s)
V jet velocity (m/s)
x distance along jet axis (m)
-------
II. Characterizing Gas Density Effects on Dispersion (symbols
concluded)
pa density of air (kg/m )
o
Pe density of released gas (kg/m )
PL vertically averaged layer density (kg/m )
SL empirical constant (2.15) in Equation (V.53)
Q constant in power law wind profile
density stratification effect function in Equation (V.76)
Vlll
-------
III. Description of the Ooms Model
bi characteristic plume width (radius = b^/2), m
J J
c local concentration, kg/m3
c^0 concentration of jet at exit, kg/m
c% concentration on plume axis, kg/m
c^ drag coefficient, 0.3
C local heat capacity of plume, joule/kg K
C heat capacity of air, joule/kg K
"Cry" heat capacity of jet at exit, joule/kg K
O
g gravitational acceleration, m/s
P atmospheric pressure, newtons
r radial distance to jet/plume axis, m
R ideal gas law constant, 8314 joules/kg
s distance along plume axis, m
u local velocity in direction of plume axis, m/s
ua wind velocity, m/s
u* plume excess velocity at plume axis, u(r=0) - uacos0, m/s
u' entrainment velocity due to atmospheric turbulence, m/s
T local temperature of plume, K
Ta temperature of air, K
TJO temperature of jet at exit, K
T' reference temperature (Ta at jet exit elevation)
x horizontal coordinate, m
y vertical coordinate, m
IX
-------
III. Description of the Poms Model (symbols concluded)
ai jet entrainment coefficient
«2 line thermal entrainment coefficient
0:3 atmospheric turbulence entrainment coefficient
8 angle between plume axis and horizontal, radians
A turbulence Schmidt number, 1.16
p local density, kg/m
pa air density, kg/m
p'a density of air at elevation of jet exit
"3? *^
p plume excess density at plume axis, p(r=0) - pa, kg/m
p. local molecular weight of plume, kg/kg mol
p.: molecular weight of jet at exit, kg/kg mol
A*_ molecular weight of air, 28.9 kg/kg mol
-------
V. Description of the DEGADIS Model
a empirical constant (1.3) in Equations (V.6) and (V.27)
B,,-- effective width of gas plume (m)
Err
B'. local half width of source seen by observer i (m)
b half width of horizontally homogeneous central section
of gas plume (m)
b empirical constant (1.2) in Equations (V.8) and (V.28)
C constant (1.15) in density intrusion (spreading) relation
£j
C heat capacity (J/kg K)
C heat capacity of air (J/kg K)
^a
C heat capacity of contaminant (J/kg K)
PC
C heat capacity of water (liquid phase) (J/kg K)
"w
3
c concentration (kg/m )
c centerline, ground-level concentration (kg/m )
c T vertically averaged layer concentration (kg/m )
c, L
cf friction coefficient
c' centerline, ground-level concentration corrected for
x-direction dispersion (kg/m )
D source diameter (m)
D added enthalpy (J/kg)
2
25 diffusivity (m /s)
d empirical constant (0.64) in Equations (V.4) and (V.12)
E plume strength (kg/s)
E(t) contaminant primary source rate (kg contaminant/s)
e empirical constant (20.) in Equations (V.5) and (V.15)
2
F overall mass transfer coefficient (kg/m s)
XI
-------
V. Description of the DEGADIS Model (symbols continued)
F,. mass transfer coefficient due to forced convection
(kg/m s)
F mass transfer coefficient due to natural convection
(kg/m s)
Gr Grashoff number
2
g acceleration of gravity (m/s )
H height or depth of density intrusion or cloud (m)
H ambient absolute humidity (kg water/kg dry air)
3.
H___ effective cloud depth (m)
trr
H, height of head in density-driven flow (m)
HT total layer depth (m)
J_i
H height of tail in density-driven flow (m)
H.. average depth of gravity current head (m)
H, depth of inward internal flow in a gravity current head (m)
h enthalpy of source blanket (J/kg)
h enthalpy of ambient humid air (J/kg)
3,
hp enthalpy associated with primary source mass rate (J/kg)
c*
h,. heat transfer coefficient due to forced convection
(J/m s K)
hT enthalpy of vertically averaged layer (J/kg)
LI
h heat transfer coefficient due to natural convection
n (J/m s K)
2
h., overall heat transfer coefficient (J/m s K)
h enthalpy of primary source (J/kg)
h enthalpy associated with mass flux of water from surface
w
(JAg)
KQ constant in Equation (V.96) (m )
Xll
-------
V. Description of the DEGADIS Model (symbols continued)
2
K horizontal turbulent diffusivity (m /s)
2
K vertical turbulent diffusivity (m /s)
k von Karman's constant, 0.35
k constant in Equation (V.13)
k0 constant in Equation (V.14)
L source length (m)
M total cloud mass (kg)
M total mass of air in the cloud (kg)
3.
M total mass of contaminant in the cloud (kg)
M. initial cloud mass (kg)
MM molecular weight
•
M mass rate of air entrainment into the cloud (kg/s)
3.
M mass rate of water transfer to the cloud from the water
w s
' surface under the source (kg/s)
N number of observers
Nu Nusselt number
P cloud momentum (kg m/s)
P, momentum of head in density-driven flow (kg m/s)
P momentum of tail in density-driven flow (kg m/s)
P^ virtual momentum due to acceleration reaction (kg m/s)
Pr Prandtl number
p atmospheric pressure (atm)
p partial pressure of water in the cloud (atm)
w, c
*
p vapor pressure of water at the surface temperature (atm)
Q volumetric release rate (m /s)
XI Ll
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V. Description of the DEGADIS Model (symbols continued)
2
Q source mass flux (kg/m s)
0 volumetric entrainment flux (m/s)
e
Q1 flux of ambient fluid into front of gravity current
head (m/s)
•
Q rate of heat transfer from the surface (J/s)
2
Q^ atmospheric takeup flux (kg/m s)
9
Q^ r maximum atmospheric takeup flux of contaminant (kg/m" s)
2
q surface heat flux (J/m s)
R gas source cloud radius (m)
R, inner radius of head in density-drive flow (m)
2
R value of R when (?rR 0.) is a maximum (m)
m *
R maximum radius of the cloud (m)
max v '
R primary source radius (m)
Ri,. Richardson number associated with the front velocity,
Equation (V.ll)
Ri_ Richardson number associated with temperature differences,
Equation (V.88)
Ri^. Richardson number associated with density differences
corrected for convective scale velocity
Ri^ Richardson number associated with density differences,
Equation (V.78)
Sc Schmidt number
Sh Sherwood number
Stu Stanton number for heat transfer
n
St.. Stanton number for mass transfer
M
S horizontal concentration scaling parameter (m)
S vertical concentration scaling parameter (m)
S n S at the downwind edge of the source (x - L/2) (m)
xiv
-------
V. Description of the DEGADIS Model (symbols continued)
2
S n value of S Q when (*-R Q^) is a maximum (m)
m
T temperature associated with source blanket enthalpy (K)
T temperature associated with layer-averaged enthalpy (K)
c, L
T surface temperature (K)
t time (s)
t specified time (s)
S
t, time when observer i encounters downwind edge (s)
dn.
i
t time when observer i encounters upwind edge (s)
upi
u ambient average velocity (m/s)
3.
u horizontal or frontal entrainment velocity (m/s)
u effective cloud advection velocity (m/s)
u_ cloud front velocity (m/s)
u. velocity of observer i (m/s)
u average transport velocity associated with H (m/s)
Li L
u wind velocity, along x-direction (m/s)
X
u ' wind velocity measured at z = Z- (m/s)
u internal flow out of gravity current head (m/s)
u internal flow into gravity current head (m/s)
u^ friction velocity (m/s)
u characteristic average velocity (m/s)
V heat transfer velocity (0.0125 m/s) in Equation (V.38) (m/s)
w mass fraction of air
a
w mass fraction of contaminant
c
w mass fraction of contaminant in primary source
c ,p
w
e
vertical entrainment velocity associated with H (m/s)
Jj
XV
-------
V. Description of the DEGADIS Model (symbols continued)
w convective scale velocity (m/s)
w' entrainment velocity associated with £!_„„ (m/s)
e br r
x.(t) x position of observer i at time t (m)
x . position of puff center due to observer i (m)
x downwind distance where gravity spreading terminates (m)
x virtual point source distance (m)
x, x position of downwind edge of source for observer i
i
x x position of upwind edge of source for observer i
Ul?i
x,y,z Cartesian coordinates (m)
x,. downwind edge of the gas source (m)
z_ surface roughness (m)
K.
zn reference height in wind velocity profile specification (m)
a constant in power law wind profile
0 constant in a correlation in Equation (V.97)
F gamma function
7 ratio of (p - pj/c
ci C
7.. constant in Equation (V.96)
A ratio of (p - p )/p
3.
AT temperature driving force (K) (T - T ) or (T - T)
s c, Li s
A' ratio of (p - p )/p
3. 3.
5 constant in a correlation in Equation (V.97)
5T empirical constant (2.15) in Equation (V.53)
Li
5 constant (0.20) in Equation (V.25)
€ frontal entrainment coefficient (0.59) in Equation (V.33)
xv i
-------
V. Description of the DEGADIS Model (symbols concluded)
f collection of terms defined by Equation (V.62) (m )
A Monin-Obukhov length (m)
p, viscosity (kg/m s)
p density of gas-air mixture (kg/m )
p ambient density (kg/m )
3.
p cloud density (kg/m )
p vertically averaged layer density (kg/m )
J_*
pn density of contaminant's saturated vapor at Tn (kg/m )
cr Pasquill-Gifford x-direction dispersion coefficient (m)
X,
a Pasquill-Gifford y-direction dispersion coefficient (m)
a Pasquill-Gifford z-direction dispersion coefficient (m)
function describing influence of stable density
stratification on vertical diffusion, Equation (V.76)
A
integrated source entrainment function
^ logarithmic velocity profile correction function
xv 11
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xviii
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Summary
A mathematical model was developed for estimating ambient air
concentrations downwind of elevated, denser-than-air gas jet-type
releases. Ooms' model is used to predict the trajectory and dilution,
to ground contact, of a denser-than-air jet/plume. The output of Corns'
model interfaces with the DEGADIS dense gas dispersion model Co predict
the ensuing ground level dispersion. The model incorporates momentum
and heat transfer important to turbulent diffusion in the surface layer
of the atmospheric boundary layer and provides for:
• inputting data directly from external files
• treatment of ground-level or elevated sources
• estimation of maximum concentrations at fixed sites
• iteration over discrete meteorological conditions
• estimation of concentration-time history at fixed sites.
This report and the accompanying User's Guide (Volume II):
• document the theoretical basis of the model
• discuss its applicability and limitations
• discuss criteria for estimating the importance of gas density
effects on a jet release from an elevated source
• define and describe all input variables and provide appropriate
guidance for their specification
• identify and describe all output files and provide appropriate
guidance for their interpretation
• provide user instructions for executing the code
• illustrate the model usage with example applications.
xix
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I. INTRODUCTION
Episodic releases of hazardous chemical gases in chemical process
pressure relief operations may pose significant hazards to public
health, and methods are required for assessing their consequences.
Conventional air pollutant dispersion models may not be applicable to
such releases, particularly when the gases released are denser than air.
Although there has been considerable recent development of dense gas
dispersion models, such models have only been demonstrated for
predicting dispersion of gases released at ground level on flat,
obstacle-free terrain. The DEGADIS model (Havens and Spicer, 1985) was
developed for simulating dispersion of zero momentum, ground-level,
heavy gas releases. DEGADIS describes the dispersion processes which
accompany the ensuing gravity-driven flow and entrainment of the gas
into the atmospheric boundary layer. DEGADIS has been verified by
comparison with a wide range of laboratory and field-scale heavy gas
release/dispersion data. However, DEGADIS makes no provision for
processes which occur in high velocity releases, as from pressure relief
valves.
Corns, Mahieu, and Zelis (1974) reported a mathematical model for
estimating trajectory and dilution of dense gas vent jets. The model
comprises simplified balance equations for mass, momentum, and energy,
with Gaussian similarity profiles for velocity, density, and concen-
tration in the jet.
The purpose of this work was to evaluate Ooms' model for prediction
of the trajectory (to ground contact) and dilution of elevated dense gas
jet releases, and to provide for interfacing Corns' model with DEGADIS.
The Ooms model can be used to predict the downwind distance where the
dense jet/plume falls to ground level and the plume concentration at
ground contact, and the Ooms model output can be used as initial (input)
conditions to DEGADIS for prediction of the ensuing ground-level plume
dispersion.
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II. CHARACTERIZING GAS DENSITY EFFECTS ON ATMOSPHERIC DISPERSION
Atmospheric dispersion of gases released with low momentum may be
characterized by three more-or-less distinct fluid flow regimes:
buoyancy-dominated • stably stratified • passive dispersion
The three regimes, which may be present to different degrees depending
on the rate and (characteristic) dimensions of the release, the gas
density, and the characteristics of the atmospheric flow, must be
accounted for if a model is to be generally applicable. Estimation of
atmospheric dispersion when the gas is released at high momentum
(velocity) may require consideration of (additional) air entrainment due
to the accompanying jetting effect.
In rapid releases of large quantities of dense gas (with little
initial momentum) a cloud having similar vertical and horizontal
dimensions may form. In this "buoyancy-dominated regime", (gravity-
induced) slumping and lateral spreading motion ensues until the kinetic
energy of the buoyancy-driven flow is dissipated. The gravity-induced
flow may effect mixing (primarily at the advancing vapor cloud front)
which can be an important determinant of the shape and extent of the gas
cloud. After the buoyancy-induced kinetic energy is dissipated, the
dispersion process which follows can be described as a "stably
stratified" plume (or cloud) embedded in the mean wind flow. The
density stratification present in this regime, which can be much
stronger than that occurring naturally in the atmospheric boundary
layer, tends to damp turbulence and reduce vertical mixing. As the
dispersion proceeds, the stable stratification due to the dense gas
decreases until the dispersion process can be represented as a neutrally
buoyant plume (or cloud) in the mean wind flow. This final regime of
"passive" dispersion (by pre-existing turbulence) in the atmosphere can
be predicted using trace contaminant dispersion theory. For low-
momentum dense gas releases at ground level on uniform, level terrain
with unobstructed atmospheric flow, the buoyancy-dominated, stably-
stratified, and passive dispersion regimes can be modeled with DEGADIS
(Havens and Spicer, 1985).
-------
Based on water tunnel experiments reported by Britter (1980), Havens
and Spicer (1985) suggested criteria for determining the importance of
each of the three flow regimes described above for zero-initial-
momentum, ground-level releases. In Britter's experiment brine was
released at floor level into a water tunnel flow, and the lateral and
upwind extent of the brine/water plume was recorded as a function of the
buoyancy length scale
Lg ** Qg(p£ - Pa)/(p£u )
where Q was the volumetric (brine) emission rate and u was the water-
tunnel flow velocity. Britter's data indicated releases were passive
from the source when Lg/D 10 Ri (II-2)
u
This criterion has the characteristic that as the density of the
released fluid increases, the release is less readily dominated by jet
effects (since the rate of air entrainment would increase in the absence
-------
of the jet effects due to the entrainment associated with the gravity
front).
The air entrainment into a turbulent free jet can be estimated as
suggested by Wheatley (1986).
dM
~ - I ^^ I I ~ I V p_ (II-3)
The rate of (vertical) air entrainment (per unit width) for the stably
stratified shear flow and passive dispersion regimes predicted by
DEGADIS is
Pa«Lku5(r(l + a)
where <£(Ri*) = 0.88 + 0.099 Ri*. Using an effective width of 27rR, a
typical value of a = 0.2, SL = 2.1, k = 0.35, and (u/u*) = 30 (typical
of atmospheric boundary layers), the above equations can be combined to
show that jet effects dominate the stably-stratified flow regime when
J > 16/(19 + Ric) (II-5)
This criterion has the characteristic that as the density of the
released fluid increases, the release is more readily dominated by jet
effects (since the rate of air entrainment would decrease in the absence
of the jet effects). When RiQ ^ 1 the criterion indicates that jet
effects dominate the passive dispersion regime when (V/u) 5 0.8, which
is consistent with the criterion suggested by Cude (1974) and Wheatley
(1986).
Summarizing, the following procedure is suggested for determining
which dispersion regime is dominant from the start of a release:
(1) Calculate Ric - g(p£ - pa)H/(pau£).
(2) Determine the dominant nonjet dispersion regime in the
absence of a jet using Equation (II-l).
(3) Determine if ground-level jet effects dominate the dominant
nonjet regime determined from (2) using the relationships
summarized in Table II.1.
-------
TABLE II. 1
CRITERIA FOR DETERMINING WHETHER JET EFFECTS
DOMINATE A GROUND-LEVEL RELEASE
Ground-level jet effects dominate:
Negative buoyancy-dominated regime when
V_
u.
> 10 Ri
Stably stratified shear flow regime when V/u > 16/(19 + Ri )
Passive dispersion phase when
V/u > 0.8
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III. DESCRIPTION OF THE OOMS MODEL
Ooms, Mahieu, and Zelis' (1974) model comprises simplified balance
equations for mass and momentum to describe the jet illustrated in
Figure III.l.
Figure III.l. Schematic diagram of Corns' model.
Gaussian similarity profiles for velocity, density, and concentra-
tion are assumed to apply in the developed jet:
u(s,r,0) - u cos0 + u*(s) e
3,
"r
2 „ 2
p(s,r,9)
2 22
/A bj(s)
(III.l)
(III.2)
,r,*) = c*(s)
(III.3)
-------
Balance equations for mass and horizontal and vertical momentum are:
Component Mass
d_
ds
fb.72
J 2?rrcudr = 0
(III.4)
Overall Mass
ds
4 P. "
a, |u (s) I + a u |sin#|cos0 + a,u'
X ^ ci J
Horizontal (x-direction) Momentum
(III.5)
d_
ds
dr - 2?rb.p u -^ 0,
j a a I 1
_
2 a1
+ a-u'
o o
.jrb.p u I sin I
d jra a1
(III.6)
Vertical (v-direction) Momentum
d_
ds
2 .
sin# dr
- p)dr
2 2
- sign(0) c ;rb.p u sin 0 cos#
O 3.
(III.7)
Energy
_
ds
(T - T')dr - 2?rb.p C (T - T')
P J a pav a x
(s)
(III.8)
-------
The temperatures of the plume and air are calculated from the ideal
gas law, with the plume (gas-air mixture) molecular weight and heat
capacity estimated using ideal gas mixing relations:
T - and T
Rp a
(III.9)
"
cT
cT
j c. T.
J Jo J>
cT
Vpj ^~T~
J FJ jo jo
4- /j. C
a pa
^
1 -
cT
c. T.
jo jo
(III.10)
(III.11)
Substitution of Equations (III.9-III.il) in the energy balance
equation (III.8) gives
(vy2
ds
'"
2?rru
27Tb.
1 - £-
pa
C.
- 1
QI|U
dr =
+ a u'
(III.12)
Approximating the product of plume molecular weight and heat
capacity using the relation (following Ooms et al., 1974)
_P_ = i
j. C 2
a pa
M-
a pa
(III.13)
and utilizing the similarity profiles and the ideal gas equation of
state, the balance equations were integrated over the radius of the
plume to give a set of ordinary differential equations:
-------
10
where
-
All A12
A A
A21 A22
A31 A32
A41 A42
A51 A52
\., - b(k,u
11 1
*
A... - 2c (k
Iz
*
A. - k^c b
13 2
~
A A A
13 A14 A15
A A A
A23 24 A25
A33 A34 A35
A43 A44 A45
A53 A54 A55
*
dc /ds
db/ds
*
du /ds
dd/ds
*
dp /ds
=
-
V
BJ2
BJ3
BJ4
BJ5
*
costf + k^u )
a 2.
*
,u cos0 -f k«u )
la 2
A14 =
-k..u c b sintf
1 a
(III.14)
A^, - 0
* *
2(2u cos? + k,u + k p u cosd/p + k p u /p )
a. J ^f 3 £L J 3.
b(k + k
J a.
- b(-2u sin^ - k.p u sinfl/p )
a 4^ a 'a
A25 - b(k4u£
+k,.u
A31"°
•i
A-- - 2u
- + k-u + p /p (kQu
D 3. / 3. Q 3.
kQu ))
7
* *
- b cos^(kfiu cos^ + k1nu +p /p (kQu cos0 + k^u ))
O 3 i.U 3 O 3 J_-L
34
35
bu
sin0(-6u
- k u - k.. _u cosd p /p
12 lj a a
u P /Pa) - bu
22 *
b(k u cos 6 cosd + k u cos0 u
*2
+ kgU cos<9)/pa
-------
11
2 * 2 *
A . - 2cos0 sin0(2u cosff + k,u u + k u cosS p /p
{4 2. 3 03. M- 3. 3.
^t ^t ^"9 7^*2 ^
+ kguau p /pa) + 2sin0(kyu + kgu p /pa)
•>t "5t •&
A.- - b sin0(k,u cos0 + k, nu + p /p (kQu cos0 + k.nu ))
43 o a IU a o a Li.
2 2 * *2
A. = bu cos0(l-3sin 5)(2 + k.p /p ) .+ bk u costf
H-^f 3 4- 3 /
2 2 * * *2 *
+ b(cos ^-sin 5)u u (k,+ k0p /p ) + bk u cos5 p /p
a o o a y
22 * *2
A. - b(k u cos tf sin# + kflu sinS cosff u + kQu siuS~)/p
^•j^-S o 3.7 3.
2 2 *
-k1,.groupl group4 u b.cosd - k17groupl group4 b.u
J J
2b.(u group3 cos& - k.,u cosd p /p'
J 3 J_I) 3
- kncgroupl group4 u c cos0 + k1fgroup3 u
l-> a lo
* * * *
- k17(u p /p' + groupl group4 c u )
2 * *
A,-0 = b.(k»,group3 - k^-Cp /p' + c
j j J ID 1 /
2 2 *
54 ° " a j 15 jUa^ x
2 *
- k..,.groupl group4 b. u c)
J
A55 " "k!5V
B2 =
0
B, = 2u (a1 |u | + a u |sin5|cos^ + a,u') -(- c u Isin ^1
•jsj. £.3. j da
* 22
B, = -k b gp /p + c u sin 5 cos0
"-?• J " do.
B_ - rtside/(7r(group2 + 1.))
and
A* /i.C.
/ia - mol weight, air groupl - — - 1 group2 - -J° El
a pa
p. - mol weight, jet group3 = 1 - — group4 = —1—
p' c p'
-------
12
k
k
k
^
k
k
1
4 -
-
M
10
13=
1 /-
0
1
0
0
3
0
.772699
.043144
.490842
.981684
.129432
.432332
k2 =
k5 =
k8 =
kn
11
k!4
fci-,
0
0
1
=
=
=.
.412442
0
0
0
556796
113593
.726692
.227186
.278398
k
k
k
k
k
3
6
9
12
15
- 0
- 1
- 0
- 3
-
.864665
.729329
.363346
.458658
0.521572
Initial conditions for Corns' model were specified at the beginning of
the "developed flow" region of the jet (Figure III.l). The trajectory
of the jet to the developed flow region was calculated using wind-tunnel
data correlations by Kamotani and Greber (1972). The differential
equations were solved with the subroutine SIMUL and integrated using the
subroutines RUNGE and HAMING, described by Carnahan, Luther, and Wilkes
(1969).
The balance equations for mass and momentum incorporate empirical
coefficients for estimating air entrainment. The coefficients a^, a.^,
and 0:3 provide for entrainment as follows:
Oj_ is the entrainment coefficient for a turbulent free jet. The
value 0.057 was incorporated by Ooms, after Albertson, et al.
(1950) .
c*2 accounts for entrainment into the plume at a sufficiently long
distance downwind of the vent where the velocity of the plume
approaches the wind velocity. The value 0.5 was incorporated by
Ooms, after Richards (1963).
a-j accounts for entrainment due to atmospheric turbulence. Ooms
suggested estimation of the entrainment velocity as u' - (eb^) / ,
with specification of e (the eddy energy dissipation) as a function
of height, wind velocity, and atmospheric stability. The eddy
energy dissipation for a neutral atmosphere was recommended by
Briggs (1969):
-------
13
£ - 0.0677 ua/z (m2/s3) for z < 300 m
and for "unstable" and "stable" atmospheres by Kaimal et al. (1976)
0 "}
€ - 0.004 (m /s ) for unstable atmospheres
e *• 0 for stable atmospheres
The value 1.0 for 03 was incorporated by Ooms, after Briggs (1969).
-------
IV. EVALUATION OF THE OOMS MODEL
A series of simulations were made with the Ooms model to:
• compare the model predictions with wind tunnel dense gas jet
trajectory and dilution data
• characterize the sensitivity of the model to the specification
of entrainment coefficients Q-J_, a.^, and 0:3
Table IV.1 shows the "typical" jet releases simulated.
TABLE IV.1
SPECIFICATION OF GAS RELEASE RATES FOR MODELING
Jet Diameter Jet Velocity Gas Release Rate
m m/s kg/s
0.05 (-2 in) 30.6 (-100 ft/s) 0.24
0.2 (-8 in) 91.7 (-300 ft/s) 11.52
0.5 (-20 in) 213.9 (-700 ft/s) 168.0
Comparison with Wind Tunnel Test Data
Hoot, Meroney, and Peterka (1973) reported plume rise, downwind
distance to plume touchdown, and dilution at touchdown for wind tunnel
jet releases of Freon-12/air mixtures. The ranges of experimental
variables studied were:
gas specific gravity (air - 1) 1.1-4.6
gas exit diameter, cm 0.32 and 0.64
gas exit height, cm 7.6 and 15.2
gas exit velocity/wind velocity ratio 2.5 - 25
wind (tunnel) velocity, m/s 0.23 and 0.46
-------
16
Correlations (of the wind tunnel data) were presented for plume
rise, distance to plume touchdown, and plume centerline concentration ac
ground contact, in a laminar crosswind:
Plume Rise
H - 1.32 D [(u/u )(p/p )]1/3Fr2/3
3 3
Downwind Distance to Maximum Rise
X - (D u /u) Fr 2
3
Downwind Distance to Centerline Ground Contact
X - 0.56 D {(H/D)3[(2 + H /H)3- l]u /u}1/2Fr, + X
S 3 ft
Centerline Concentration at Ground Contact
X - 3.1 (Q/u D2) [(2H + H )/D]"1>95
3 S
where D — jet diameter (initial), m
Fr - Froude number, u/[g D [ (p-pa)/p]1//2
Frh - "horizontal" Froude number, ufl/[g D [(p-pa)/pa]1/2
H - maximum height of plume rise, m
Hs - exit height, m
Q - jet rate (initial), kg/s
ua - wind velocity, m/s
u - jet velocity (initial), m/s
X — downwind distance to ground contact (centerline), m
X — downwind distance to maximum rise, m
pa — ambient air density, kg/m
o
p - jet density (initial), kg/m
X — jet centerline concentration, kg/m
Table IV.2 compares the Ooms model predictions for plume rise,
downwind distance to ground contact, and concentration at ground contact
with the results of Hoot et al.'s wind-tunnel data correlations for the
"low diameter/low velocity", "typical diameter/typical velocity," and
"high diameter/high velocity" cases (Table IV.1) in 3 and 6 m/s winds.
-------
17
TABLE IV.2
COMPARISON OF OOMS MODEL PREDICTION WITH
HOOT ET AL.'S WIND-TUNNEL DATA CORRELATION
Gas Jet Density: 4.0 kg/m3 Gas Jet Elevation: 10 m
(Hoot et al. correlation / Ooms model prediction)
Centerline
Distance to Concentration
Maximum Ground at Ground Contact
Height, m Contact, m kg/nr x 103
(3, 6 mps) (3, 6 mps) (3, 6 mps)
Low Diameter (0.05 m) / Low Velocity (30.6 m/s)
3.0/2.2 2.4/1.6 150/140 375/325 1.3/1.9 0.8/1.2
Typical Diameter (0.2 m) / Typical Velocity (97.1 m/s)
23.0/19.0 18.0/13.8 165/170 350/305 5.2/7.5 3.7/6.1
High Diameter (0.5 m) / High Velocity (213.9 m/s)
97.0/79.0 77.0/61.0 320/415 650/705 5.6/7.6 4.3/6.2
The Ooms model predictions assumed a power law vertical velocity profile
with the 3 and 6 m/s velocities at 10 m elevation and a power law wind
profile constant 0.142. Since the wind tunnel correlations were for
jets directed into a laminar crosswind, the jet model predictions shown
in Table IV.2 were made with the entrainment coefficient accounting for
atmospheric turbulence, Q-J , set to zero.
The model predictions of maximum rise, distance to ground contact,
and concentration at ground contact are in good agreement with
predictions from the wind tunnel data correlations, considering the
uncertainty in representing the (tunnel) wind profile.
Sensitivity __to Entrainment Coefficient Specification
The model sensitivity to the entrainment coefficients a-^ and a.^
(with or-j - 0) was determined first. Specification of ctj equal to zero
-------
IS
is equivalent to specifying stable stratification of the atmospheric
flow (with zero turbulent entrainment). The model sensitivity to the
coefficients Q]_ and a.^ was determined by systematic variation around the
values recommended by Ooms. All simulations were of heavy gas jets
(exit density 4.0 kg/m ) exiting vertically upward at 10 meters
.elevation. The predictions assumed a power law vertical velocity
profile with the 3 and 6 m/s velocities at 10 m elevation and a power
law wind profile constant 0.142. Table IV.3 shows the effect of
individually varying the entrainment coefficients a-\ and a.^ by factors
of two below and above the values recommended by Ooms, for the "low
diameter/low velocity", "typical diameter/typical velocity," and "high
diameter/high velocity" cases (Table IV.1) in 3 and 6 m/s winds.
The predictions of maximum rise and distance to ground contact are
relatively insensitive to factor-of-four variation in the entrainment
coefficient a^ over the range of release conditions in Table IV.1. The
predictions of concentration at ground contact are very insensitive to
the same variation in a^. The predictions of maximum rise and distance
to ground contact are more sensitive to factor-of-four variation in the
coefficient OL^ > anc* c^e concentration at ground contact changes by
factors of about 4, 6, and 10 for the high diameter/high velocity,
typical diameter/typical velocity, and low diameter/ low velocity cases,
respectively.
The model sensitivity to all three entrainment coefficients (Q-J_ , a^,
and 03) was also determined by systematic variation around the values
recommended by Ooms. Again, all simulations were of heavy gas jets
o
(exit density 4.0 kg/m ) exiting vertically upward at 10 meters
elevation, assuming a power law vertical velocity profile with the 3 and
6 m/s velocities at 10 m elevation and a power law wind profile constant
0.142. Table IV.4 shows the effect of individually varying the
entrainment coefficients a^, c^, and a-j by factors of two below and
above the values recommended by Ooms, for the "low diameter/low
velocity", "typical diameter/typical velocity," and "high diameter/high
velocity" cases in 3 and 6 m/s winds.
Again, the predictions of maximum rise are relatively insensitive to
factor-of-four variation in the entrainment coefficient a over the
-------
19
TABLE IV.3
SENSITIVITY OF OOMS MODEL PREDICTION TO
VARIATION OF ENTRAINMENT COEFFICIENTS Q]_ and «2
WITH NO ATMOSPHERIC TURBULENCE ENTRAINMENT
Gas Jet Density: 4.0 kg/m3
Gas Jet Elevation: 10 m
Entrainment
Coefficients
Ql a2 a3
0.03
0.057
0.12
0.057
0.057
0.057
Low
0.5
0.5
0.5
0.25
0.5
1.0
Typical
0.03
0.057
0.12
0.057
0.057
0.057
0.03
0.057
0.12
0.057
0.057
0.057
0.5
0.5
0.5
0.25
0.5
1.0
Hi eh
0.5
0.5
0.5
0.25
0.5
1.0
Maximum
Height, m
(3, 6 mps)
Diameter
0.0
0.0
0.0
0.0
0.0
0.0
12
12
11
12
12
11
Diameter
0.0
0.0
0.0
0.0
0.0
0.0
32
28
24
31
28
25
Diameter
0.0
0.0
0.0
0.0
0.0
0.0
107
89
' 71
97
89
77
(0.05
.6 11
.2 11
.9 11
.6 11
.2 11
.8 11
(0.2
.8 25
.9 21
.9 21
.5 26
.9 21
.6 20
m)
.7
.5
.4
.9
.5
.2
m)
.9
.9
.4
.2
.9
.9
(0.5 m)
.0 82
.0 70
.4 59
.7 80
.0 70
.1 59
.0
.9
.3
.1
.9
.4
Distance to
Ground
Contact, m
(3, 6 mps)
/ Low Velocity
145
140
140
100
140
215
/ Typical
185
170
155
150
170
200
325
325
320
220
325
520
at
(30
0.
0.
0.
0.
0.
0.
Velocitv
325
305
295
265
305
380
/ High Velocity
470
415
370
395
415
460
780
705
645
650
705
790
0.
0.
0.
0.
0.
0.
Centerline
Concentration
Ground Contact
kg/m3
(3 , 6 mps)
.6 m/s:
00190
00192
00185
00522
00192
00062
(97.1
00716
00746
00752
01460
00746
00339
I
0
0
0
0
0
0
.00124
.00122
.00124
.00363
.00122
.00037
m/s)
0
0
0
0
0
0
.0059
.0061
.0060
.0121
.0061
.0026
(213.9 m/s)
0.
0.
0.
0.
0.
0.
00706
00753
00733
01274
00753
00365
0
0
0
0
0
0
.0057
.0062
.0063
.0111
.0062
.0031
-------
20
TABLE IV.4
SENSITIVITY OF OOMS MODEL PREDICTION TO
VARIATION OF ENTRAINMENT COEFFICIENTS
Gas Jet Density: 4.0 kg/m3
Gas Jet Elevation: 10 ra
Entrainment
Coefficients
ai ay 0:3
Maximum
Height, m
(3, 6 mps)
Low Diameter (0.05 m)
0.03 0.5 1.0
0.057 0.5 1.0
0.12 0.5 1.0
0.057 0.25 1.0
0.057 0.5 1.0
0.057 1.0 1.0
0.057 0.5 0.5
0.057 0.5 1.0
0.057 1.0 2.0
12.3 11.5
12.1 11.4
11.7 11.2
12.4 11.7
12.1 11.4
11.7 11.1
12.2 11.5
12.1 11.4
11.9 11.3
Tvoical Diameter (0.2 m)
0.03 0.5 1.0
0.057 0.5 1.0
0.12 0.5 1.0
0.057 0.25 1.0
0.057 0.5 1.0
0.057 1.0 1.0
0.057 0.5 0.5
0.057 0'. 5 1.0
0.057 0.5 2.0
30.7 24.4
27.3 22.5
23.7 20.4
29.4 24.6
27.3 22.5
24.5 20.1
28.1 23.1
27.3 22.5
25.9 21.5
High Diameter (0.5 m)
0.03 0.5 1.0
0.057 0.5 1.0
0.12 0.5 1.0
0.057 0.25 1.0
0.057 0.5 1.0
0.057 1.0 1.0
0.057 0.5 0.5
0.057 0.5 1.0
0.057 0.5 2.0
99.5 72.8
83.4 63.6
71.4 55.6
90.7 74.2
83.4 63.6
73.1 56.3
86.1 68.5
83.4 63.6
78.4 62.2
Distance to
Ground
Contact, m
(3, 6 mps)
/ Low Velocity
>1000 >1000
>1000 >1000
>1000 >1000
>1000 >1000
>1000 >1000
>1000 >1000
>1000 >1000
>1000 >1000
>1000 >1000
Centerline
Concentration
at Ground Contact
kg/m3
(3, 6 mps)
(30.6 m/s)
0.000001*
0.000001*
0.000001*
0.000001*
0.000001*
o.booooi*
0.000002*
0.000001*
0.000001*
0.000002*
0.000002*
0.000002*
0.000002*
0.000002*
0.000002*
0.000008*
0.000002*
0 . 000000*
/ Typical Velocity (91.7 m/s)
500 >1000
425 >1000
370 >1000
330 >1000
425 >1000
615 >1000
225 550
425 >1000
>1000 >1000
/ Hig;h Velocity
645 >1000
560 >1000
370 >1000
510 >1000
560 >1000
650 >1000
470 895
560 >1000
>1000 >1000
0.000083
0.000130
0.000180
0.000300
0.000130
0.000030
0.002100
0.000130
0.000007*
(213.9 m/s)
0.001220
0.001490
0.007330
0.002530
0.001490
0.000690
0.003740
0.001490
0.000120*
0.000060*
0.000060*
0.000050*
0.000060*
0.000060*
0.000050*
0.000540
0.000060*
0.000017*
0.001250*
0.001100*
0.000840*
0.001300*
0.001100*
0.000780*
0.002000
0.001100*
0.000440*
-------
21
range of release conditions (Table IV.1), with greater sensitivity at
lower wind speeds. The predictions of concentration at ground contact
are more sensitive to the same variation in Q^_ in the presence of
atmospheric turbulence entrainment, because the additional entrainment
results in the plume remaining aloft to a greater downwind distance.
The inclusion of atmospheric turbulence entrainment (assumed Pasquill
Stability Class D - Neutral) results in all of the low diameter/low
velocity plumes remaining aloft for downwind distances greater than 10CO
meters. A similar result is seen for the typical diameter/typical
velocity and high diameter/high velocity plume in 6 mps wind. At 3 mps
wind, the typical diameter/typical velocity and high diameter/high
velocity plumes (centerline) contacts ground at distances less than 1000
m in all except the two cases with the high value for a^ (2.0). The
predictions of maximum rise and distance to ground contact are more
sensitive to factor-of-four variation in the coefficient a^, and the
concentration at ground contact changes by factors of about 10 and 4 for
the typical diameter/typical velocity, and high diameter/high velocity
cases, respectively (when the plume contacts the ground within 1000
meters downwind). Finally, the predictions of maximum rise, distance to
ground contact, and concentration at ground contact are sensitive to the
prescribed factor-of-four variation in a-j, with the greatest sensitivity
to those cases where the plume dilution is dominated by atmospheric
turbulence entrainment.
Observations
Corns' model predictions of maximum rise, distance to ground contact,
and concentration at ground contact for a range of elevated dense gas
jets considered typical of industrial pressure-relief venting
operations, assumed to be released in an atmospheric flow with little or
no atmospheric turbulence, are in good agreement with predictions based
on the wind tunnel data correlations of Hoot et al.
Corns' model predictions of the trajectory and dilution of elevated
dense gas jets considered typical of industrial pressure-relief venting
operations are relatively insensitive to the specification of the jet
entrainment coefficient a-^. Model predictions are more sensitive to the
-------
22
specification of the intermediate field entrainment coefficient 0.2', the
concentration at ground contact changes by factors of about 4, 6, and 10
for the high diameter/high velocity, typical diameter/typical velocity,
and low diameter/ low velocity cases investigated, respectively, when
atmospheric turbulence entrainment is discounted (03 - 0).
Predictions of maximum rise, distance to ground contact, and
concentration at ground contact are sensitive to uncertainty in 03, with
the greatest sensitivity to those cases where the plume dilution is
dominated by atmospheric turbulence entrainment. For cases where the
plume remains aloft, the dilution of the plume is controlled by
atmospheric turbulence entrainment after the velocity of the jet
decreases to approach the wind velocity. Application of the version of
Ooms model presented here to such cases is not recommended, for two
reasons:
• the jet/plume cross section is assumed to be circular, rather than
elliptical as would be expected (because of lesser entrainment in
the vertical than horizontal direction)
• no provision is made for decreased air entrainment which occurs if
the bottom of the plume impinges on the ground.
-------
23
V. DESCRIPTION OF THE
DEGADIS DENSE GAS DISPERSION MODEL
The DEGADIS (DEnse GAs DISpersion) model was developed from research
sponsored by the U.S. Coast Guard and the Gas Research Institute (Havens
and Spicer, 1985). DEGADIS is an adaptation of the Shell HEGADAS model
described by Colenbrander (1980) and Colenbrander and Puttock (1983).
DEGADIS also incorporates some techniques used by van Ulden (1983).
If the primary source (gas) release rate exceeds the maximum
atmospheric takeup rate, a denser-than-air gas blanket is formed over the
primary source. This near-field, buoyancy-dominated regime is modeled
using a lumped parameter model of a denser-than-air gas "secondary
source" cloud which incorporates air entrainment at the gravity-spreading
front using a frontal entrainment velocity. If the primary source
release rate does not exceed the maximum atmospheric takeup rate, the
released gas is taken up directly by the atmosphere and dispersed
downwind. For either source condition, the downwind dispersion phase of
the calculation assumes a power law concentration distribution in the
vertical direction and a modified Gaussian profile in the horizontal
direction with a power law specification for the wind profile (Figure
V.I). The source model represents a spatially averaged concentration of
gas present over the primary source, while the downwind dispersion phase
of the calculation models an ensemble average of the concentration
downwind of the source.
Denser-than-Air Gas Source Cloud Formation
A lumped parameter model of the formation of the denser-than-air gas
source cloud or blanket, which may be formed from a primary source such
as an evaporating liquid pool or otherwise specified ground-level
emission source, or by an initially specified gas volume of prescribed
dimensions for an instantaneous release, is illustrated in Figure V.I.
The gas blanket is represented as a cylindrical gas volume which spreads
laterally as a density-driven flow with entrainment from the top of the
source blanket by wind shear and air entrainment into the advancing
-------
Ambient
wind
(can be
zero)
24
Input to
dovnvind dispersion.
model
(t)
V^
If'
T(t), C(t), p(t)
Secondary Source Formation
Frontal
entrainment
velocity u
C(x,y,z)«Cc(x) exp
|y|-b(x)\2 / z
~ ~
/|y|-b(x)f /
\ Sv(x) / \
C(x,y,z)» CQ(X) exp
"Uz(x)l
A1*8
ISO CONCENTRATION \
CONTOURS \
FOR C = CU
Downwind Dispersion
Figure V.I. Schematic diagram of DEGADIS dense gas dispersion model.
-------
25
front edge. The source blanket will continue to grow over the primary
source until the atmospheric takeup rate from the top is matched by the
air entrainment rate from the side and, if applicable, by the rate of gas
addition from under the blanket. Of course, the blanket is not formed if
the atmospheric takeup rate is greater than the evolution rate of the
primary source. For application of the downwind calculation procedure,
the blanket is modeled as being stationary over the center of the source
(x - 0).
Secondary Source Blanket Extent for Ground Level Releases
If a denser-than-air gas blanket is present, the (downwind) emission
rate from the blanket is equal to the maximum atmospheric takeup rate.
•o
That is, for E(t)/rrRr(t) > Q*max, a source blanket is formed over the
primary source. The blanket frontal (spreading) velocity is modeled as
u
f
P -
P
a.
H (V.I)
where p is the average density of the source blanket. This gravity
intrusion relationship is applicable only for p > pa; the value of Cg
used is 1.15 based on laboratory measurements of cloud spreading velocity
(Havens and Spicer, 1985).
The blanket radius R as a function of time is determined by
integrating dR/dt — u^. When the total mass of the cloud is decreasing
with time, the radius is assumed to decrease according to
(dR/dt)/R = (dH/dt)/H for ground-level sources. The radius of the
blanket is constrained to be greater than or equal to the radius R^ of
any primary (liquid) source present.
Secondary Source Blanket Extent for Instantaneous Releases
The gravity intrusion relationship (Equation (V.I)) will overpredict
initial velocities for instantaneous, aboveground releases of a denser-
than-air gas since no initial acceleration phase is included. In this
-------
26
case, the following procedure adapted from van Ulden (1983) is
recommended.
For instantaneous gas releases, the radially symmetric cloud is
considered to be composed of a tail section with height Ht and radius R^
and a head section with height Hjj (Figure V.2). A momentum balance is
used to account for the acceleration of the cloud from rest; the effect
of ambient (wind) momentum is ignored. Although the following equations
are derived assuming the primary source emission rate is zero, the
resulting equations are assumed to model the secondary source cloud
development when the primary source rate is nonzero. When the frontal
velocity from the momentum balance is the same as Equation (V.I), the
momentum_b_alance is no longer applied and the frontal velocity is given
by Equation (V.I).
T
I
M
E
i - 0
•| s Q
Ht
I
Ht
LI i
; H < H
/ h
f r^~~
Rh
-T-1
Ht
Hh >Ht
_L Gravity
H Slumping
Figure V.2. Schematic diagram of a radially spreading cloud.
-------
27
There are three main forces acting on the cloud: a static pressure
force (F0), a dynamic drag force (F^), and a force which accounts for the
acceleration reaction of the ambient fluid, represented as a rate of
virtual momentum change with respect to time (-dPv/dt). Denoting the
momentum of the head and tail as P^ and Pt, respectively, the momentum
balance is
dP
2£ _ 2_ (p + P ) - F + F, - ~ (V.2)
dt dt v h t' p d dt 7
or
— (p + p + P ) = F + F
dt ^ h t v' p d (V.3)
The terms in the momentum balance are evaluated differently for early
times before a gravity current head has developed (H^ < Ht) and for times
after the head has developed but the cloud is still accelerating (Figure
V.2). Because the gravity current head develops so rapidly, the model
equations describing the times after the gravity current head forms
(H^ > Ht) are derived first. The model equations describing earlier
times (H^ < Ht) use simplification of the equations for H^ > Ht.
Unsteady Gravity Current
When the cloud accelerates to the point that H^ > Ht (Figures V.2,
V.3), the frontal velocity is determined from the momentum balance
(Equation (V.2)) as follows.
-------
28
Figure V.3. The unsteady gravity current (van Ulden, 1983).
The static pressure force, obtained by integrating the static
pressure over the boundary of the current, is
(V.4)
Neglecting the shear stress at the bottom, the dynamic force on the
current is the sum of the drag force on the head of the current and the
lift force that arises due to asymmetry in the ambient flow around the
head. The drag force is represented by
F_ = - ~ p u2. I 27rR,a H, | = -a d TrRH p UJ (V.5)
D 2 a f I hvtij vv h^a f
where dy. is an effective drag coefficient and the constant av is an
empirical ratio of the average head depth H^ to H^ (a^ = H-^/H^) .
The horizontal acceleration reaction (-dPv/dt) is approximated by the
reaction to an accelerating elliptical cylinder with an aspect ratio H/R
(Batchelor, 1967):
-------
dP
dT
R
- - 5L f k,
dt (_ r
(V.6)
and the vertical acceleration reaction is represented as
d
d_
dt
(V.7)
where k-j and
are coefficients of order one. Using a single constant
Equations (V.6) and (V.7) give
d
- e irp
v *
d(RH uf)
dt
(V.8)
Using Equations (V.4), (V.5), and (V.8), the momentum balance (Equation
(V.2)) becomes
dP
d(RH
(V.9)
Following van Ulden (1979, 1983), it is assumed that the potential
energy decrease due to slumping of the cloud is offset by the production
of kinetic energy, which through the action of shear, is partly
transformed to turbulent kinetic energy. Part of the turbulent kinetic
energy is transformed back into potential energy due to entrainment of
air by the cloud. This "buoyant destruction" of kinetic energy is
assumed to be proportional to the rate of production of turbulent kinetic
energy, and following Simpson and Britter (1979) it is assumed that the
turbulent kinetic energy production rate scales as 7rpaHRuf. Then,
dV ,.„ 3
which can be written
(V.10)
dV
dt
e(2jrRH)u- f(27rRH)u.
(V.ll)
Ri
-------
30
where e is an empirically determined coefficient. Noting dV/dt
represents the air entrainment rate,
M
_a
a
gApH
2
(V.12)
where M represents the air entrainment mass rate.
3.
The volume integral
R
V - 2-K \ h(r,t)rdr
J0
(V.13)
where h(r,t) is to be expressed in terms of H, and H , and the
momentum integral
f
J (
P - 2?r pu(r,t)h(r,t)rdr - P + P,
!0 t h
(V.14)
are then approximated with separate analyses of the head and tail of the
current.
In the tail of the current, the shallow water equations are assumed
applicable. It is assumed that the shape of the current is quasi-
stationary in time, and the layer -averaged density difference is assumed
horizontally uniform. It follows that the volume and momentum of the
tail are given by
t -f > (f Ht
A momentum balance for the head region, Figure V.4, assuming quasi -
steady state, indicates that the static and dynamic pressure forces on
the head should be balanced by the net flux of momentum due to flow into
and out of the head. The static pressure and drag are, respectively
Fp -
-------
31
T? , I 1 ,/
FD " ' dv | 2 PaUi
- - a d p u^rR-H, (V.18)
v v a f n n
Near the surface, the inward flow (u^ in Figure V.4) carries momentum
into the head, while the return flow (u^ in Figure V.4) carries momentum
out of the head. Assuming u^ - u^, H^ = 1/2 H^, and u^ = 5VU£, the
momentum flux into the head is approximately
Figure V.4. The head of a steady gravity current
(Simpson and Britter, 1979; van Ulden,
1983).
Upon rearranging, the momentum balance on the head gives
2
2 1 2
Vv * 25v - CE ^V-2°)
-------
32
when 5V = 0.2 and d^. - 0.64; Equation (V.20) then specifies the head
velocity boundary condition. The volume of the head is determined by
assuming that the head length scales with Hi. It follows that
where bv is an empirical constant, and the volume of the head becomes
^ (V.22)
If the layer-averaged velocity is assumed to increase linearly with
r, it follows that
(V.23)
IL J- r>.
and
(V.24)
Along with the definition of U£,
g-uf, (V.25)
Equations (V.9)', (V.ll), (V.20), (V.21), (V.23), and (V.25) are solved to
determine p, Ht, H^, V, P^, and Pt when H^ > Hfc.
The constants a^, bv, d^, ev, and e are assigned values 1.3, 1.2,
0.64, 20., and 0.59, respectively, based on analysis of the still-air
denser-than-air gas release experiments of Havens and Spicer (1985).
Initial Gravity Current Development
In order to model the initial cloud shape, the tail and head height
are considered constant with respect to radius. The momentum balance on
the cloud is then given by
-------
33
R. H + a b
n t v
<}
v h J
dP
v
(V.26)
where the first term on the right-hand side represents the static
pressure force on the head and the second term represents the drag force
on the bottom surface of the cloud. The third force is the acceleration
reaction by the ambient fluid, represented by Equation (V.8).
The dimensions of the head are again given by
R, - R - a b H,
n v v n
(V.27)
and
u.
[gAp/p ]
c±
(V.28)
When the height of the tail Ht is assumed uniform with respect to radius.
it follows that
Ht°
M 2 2
- - wa b (R + R, ) HT
p v v TV n
(V.29)
where M is the total mass of the cloud. The momentum of the head Pi. and
tail Pt are then
P, - -r ?ra
h 3 v
(V.30)
R
and
2
t ' 3 *
U
(V.31)
Equations (V.26) through (V.31) determine the momentum of the blanket as
a function of time, and thus the frontal velocity Uf. The cloud
accelerates from rest because H^ - 0 initially.
-------
34
Material and Energy Balances
The balance on the total mass of gas in the source blanket
(M - 7rR2H/0 is
dM d r -.2,, ] E(t) , • • w*max f _,2 1
— - -r- ?rR Hp - —- . • + M + M - TrR
dt dt I I w (t) a w,s w I J
c,p ^ c J
(V.32)
where E(t) is the contaminant evolution rate from the primary (liquid)
source and w (t) is the contaminant mass fraction in the primary
source. For spills over water, the water entrainment term (M )
w, s
is included in the source blanket description and is calculated
from Equation (V.46), and the (humid) air entrainment rate (Equation
(V.12)) is
/ f o 1
(V.33)
The balance on the mass of contaminant in the source blanket
(M = w 7rR2H/j) is
c c
dM
r -i
ar - It [ v1*2"* I
E(t) - Q*max UR) (v-34)
and the mass balance on the air in the source blanket
(M - w *R2H/3) is
3. 3.
M
a
' 1 + H
a
where the ambient humidity is Ha and the mass fraction of contaminant and
air are wc - MC/M and w& - Ma/M, respectively. Note that any dilution
with air of the primary source is assumed to be with the ambient
humidity.
-------
35
o
The energy balance on the source blanket (hn-R Hp) gives
d r 2 i hP E(t)
— hwR Hp •• —- -,—r + h M + h M
dc w (t) aa ww.s
L J c,p
(V.36)
where h is the enthalpy of the primary source gas, ha is the enthalpy of
the ambient humid air, and hw is the enthalpy of any water vapor entrained
by the blanket if over water. There are three alternate submodels
included for the heat transfer (Q ) from the surface to the cloud.
The simplest method for calculating the heat transfer between the
substrate and the gas cloud is to specify a constant heat transfer
coefficient for the heat transfer relation
where Q is the rate of heat transfer to the cloud, q is the heat flux,
and AT is the temperature difference between the cloud and the surface.
For the calculation of heat transfer over the source, the temperature
difference is based on the average temperature of the blanket.
In the evaluation of the Burro and Coyote series of experiments,
Koopman et al . (1981) proposed the following empirical heat transfer
coefficient relationship for heat transfer between a cold LNG cloud and
the ground
ho-VGP (v-38)
where the value of Vjj was estimated to be 0.0125 m/s . This constant can
be varied in the model.
From the heat transfer coefficient descriptions for heat transfer
from a flat plate, the following relationships can be applied. For
natural convection, the heat transfer coefficient is estimated using the
Nusselt (Nu) , Grashoff (Gr) , and Schmidt (Sc) numbers (McAdams , 1954)
from
Or
Nu - 0.14 (Gr Sc)1/3 (V.39)
-------
36
n
0.14
&P CM
T Pr
1/3
(V.40)
where hn is the heat transfer coefficient due to natural convection and
Pr is the Prandtl number. In order to simplify the calculations, the
parameter group
1/3
Pr"2 I V*7 I t MW T
13 f M 1
J [ MW T J
(V.41)
is estimated to be 60 in mks units. The actual value of the group is
47.25, 58.5, and 73.4 for air, methane, and propane, respectively.
Equation (V.40) becomes
h - 18
n
MW I AT
1/3
(V.42)
where the density p, molecular weight MW, and temperature difference AT
are based on the average composition of the gas blanket.
For forced convection, the Colburn analogy (Treybal, 1980) is applied
to a flat plate using the Stanton number for heat transfer Stjj and the
Prandtl number as
Or
StHPr
hf =
2/3
u
Pr
-2/3
u.
u
(V.43)
(V.44)
where h^ is the heat transfer coefficient due to forced convection. If
the velocity is evaluated at z - H/2 and Pr is estimated to be 0.741,
(V.45)
2 - 1
u*
1.22 —
uo
22o
H
a
^ PC
If H/2 < z^, then the velocity is evaluated at z
-------
37
The overall heat transfer coefficient is then the maximum of the
forced and natural coefficients, i.e. hg = max(hf,hn). The heat flux and
transfer rate are then estimated by Equation (V.37).
If the gas blanket is formed over water, water will be transferred
from the surface to the cloud by a partial pressure driving force
associated with the temperature difference between the surface and the
gas blanket. The rate of mass transfer of water is
^ f p* . p 1 U [ R2 - R2 1 ]
p I Fw,s Fw,c J L I P J J
Mw,s^p fw.s *w,c J ["I" "P J J (v-46)
where FQ is the overall mass transfer coefficient. The driving force is
the difference of the -vapor pressure of water at the surface temperature
pw s and the partial pressure of water in the cloud, pw c. (The water
partial pressure in the cloud is the minimum of: (a) the water mole
fraction times the ambient pressure; or (b) the water vapor pressure at
the cloud temperature (pw c).) The natural convection coefficient is
based on the heat transfer coefficient and the analogy between the
Sherwood number (Sh) and the Nusselt number (Nu) suggested by Bird et al
(1960)
Sh - Nu - 0.14 (Gr Sc)1/3 — f — 1 (V.47)
If the Schmidt number is taken as 0.6, and =r-— is estimated to
.9 (_ T MW
be 2.2 x 10" in mks units,
F = 9.9 x 10
-3
n
\ P-
L MW
2
AT
1/3
(V.48)
For forced convection, Treybal (1980) suggests that the Stanton number
for mass transfer St^ and the Stanton number for heat transfer St^ are
related by
f Pr 1 2/3
M = StH 1 S7 J -1'15StH (
St
-------
38
' 20.7 hn
F (V 50)
f MW C V ;
P
The overall mass transfer coefficient FQ is calculated as the larger of
the natural and forced convection coefficients.
For the case when the primary (liquid) source emission rate E(t) is
larger than the atmospheric takeup rate Q*max7rR^, Equations (V.32),
(V.34), (V.35), and (V.36) are integrated for the mass, concentration,
and enthalpy of the gas blanket along with an appropriate equation of
state (i.e. relationship between enthalpy and temperature and between
temperature and density).
For the case when the emission rate is not sufficient to form a gas
blanket, the flux of contaminant is not determined by the maximum
atmospheric takeup rate. Consider the boundary layer formed by the
emission of gas into the atmosphere above the primary source. If the
source is modeled to have a uniform width 2b and entrain no air along the
sides of the layer, the balance on the total material (PL^LHL) in a
differential slice of the layer is
d f 1
-T- />, u_HT = p w +
dx (_ ^L L L J a e
w
C
'p
(V.51)
where we is the vertical rate of of air entrainment into the layer given
by Equation (V.83), p-^ is the average density of the slice, and (Q*/wc)
is the total flux of gas from the primary (liquid) source. The balance
on the mass flow rate of contaminant (^PLU^H^) at any (x - x^,) is
With an equation of state to relate GC ^ and p-^, Equation (V.51) is
integrated from the upwind edge of the source (x - xup) to the downwind
edge (x - L + xup) .
In order to generate the initial conditions for the downwind
dispersion calculations, the maximum concentration cc and the vertical
dispersion parameter Sz are needed. Since Equations (V.51) and (V.52)
are written for a vertically averaged layer, consider the vertical
-------
39
average of the power law distribution. The height of the layer H^ is the
height to some concentration level, say 10% of the maximum. Although
strictly a function of a, this value is modeled by
(V'53)
H
*LHEFF
where Hppp is the effective height defined by Equation (V.79) and 6-^ is
2.15. The vertically averaged concentration cc ^ can ^e defined by
LHL=r
,L L JQ
cdz
(V.54)
And similarly, the effective transport velocity u is defined by
TUTHT " I
.LLL JQ
cu dz
X
(V.55)
With Equation (V.53) and defining relations for H£pp and
(Equations (V.79) and (V.93), respectively), it follows that
c - 5Tc
c L c,L
Vo_
1 + a
l+a
(V.56)
(V.57)
and
5Tw'
L e
w
(V.58)
where w' is given by Equation (V.83).
Maximum Atmospheric Takeup Rate
The maximum atmospheric takeup rate will be the largest takeup rate
which satisfies Equations (V.51) and (V.52). As well, the maximum
concentration of contaminant in the power law profile at the downwind
edge of the source will be the source contaminant concentration (CG)S.
If Equations (V.51) and (V.52) are combined along with the assumption of
adiabatic mixing of ideal gases with the same constant molal heat
-------
40
capacity (i.e.
modeled by
*max c s
P - P.
7 = constant), the maximum takeup flux is
a)
(V.59)
where
dx
(V.60)
where ^(Ri ) is given in Equation (V.76) for p > p .
* a
An upper bound of the atmospheric takeup flux can be characterized by
the condition where the source begins to spread as a gravity intrusion
against the approach flow. In water flume experiments, Britter (1980)
measured the upstream and lateral extent of a steady-state plume from a
circular source as a function of Ri*. A significant upstream spread was
obtained for Ri* > 32, and lateral spreading at the center of the source
was insignificant for Ri* < 8. The presence of any significant lateral
spreading represents a lower bound on the conditions of the maximum
takeup flux.
The integral of Equation (V.60) is calculated using a local
Richardson number of
Ri^(x)
- V
1
1+a
(V.61)
where
p - p
and ^ is 3.1 (corresponding to
Ri (x), Equation (V.60) is
20(8 <
1+a
(V.62)
< 32)). Using this
-------
dx
1.04
0.88 + 0.099 f1'04 x1+a
In order to simplify the numerical problem, the integral is
approximated as
0.099 L
.1.04
In
0.88 + 0.099 f
1.04
0.88
(V.63)
which then specifies the maximum atmospheric takeup flux.
Transient Denser-than-Air Gas Release Simulation
If a steady-state spill is being simulated, the transient source
calculation is carried out until the source characteristics are no longer
varying significantly with time. The maximum centerline concentration
cc> the horizontal and vertical dispersion parameters S and S2, the half
width b, and if necessary, the enthalpy h are used as initial conditions
for the downwind calculation specified in a transient spill.
If a transient spill is being simulated, the spill is modeled as a
series of pseudo-steady-state releases. Consider a series of observers
traveling with the wind over the transient gas source described above;
each observer originates from the point which corresponds with the
maximum upwind extent of the gas blanket (x - -R^x) • The desired
EFF
observer velocity is the average transport velocity of the gas U£FF from
Equation (V.93); however, the value of UEFF will differ from observer to
observer with the consequence that some observers may be overtaken by
others. For a neutrally buoyant cloud, UgFF becomes a function of
downwind distance alone which circumvents this problem. With this
functionality, Colenbrander (1980) models the observer velocity as
-------
u (-) U°
i f 1 }
S
2 Om
zo
Q
x + R
max
ir R + R
2 m max
(V.64)
where S is the value of S when the averaged source rate
2 Z0m Z0
(?rR Q ) is a maximum and the subscript i denotes observer i. Noting
that u.(x) - dx./dt, observer position and velocity as functions of
time are determined.
A pseudo-steady-state approximation of the transient source is
obtained as each observer passes over the source. If t and t
Upi
dn.
denote the times when observer i encounters the upwind and downwind
edges of the source respectively, then the source fetch seen by
observer i is:
L. - x - x,
i up. dn.
*
The width of the source 2Z.(t) is defined by
222
Bf *• / *_ \ V r i-\ v f +-\
\t^ ~ K ^.t^ - X.^tj
Then the gas source area seen by observer i is
t
/• dn.
2L.b. - 2 1 B'.u.dt
11 J._ 11
up.
where 2b. is the average width.
The takeup rate of contaminant 2(Q Lb). is calculated as
^ i
p dn.
. - 2 L Q^BIu.dt
i Jt v* i i
The total mass flux rate from the source is
t
- 2
•dn.
UP^
p w' +
a e
w
Blu.dt
l i
(V.65)
(V.66)
(V.67)
(V.68)
(V.69)
-------
With these equations, the average composition of the layer can
Letermined at each
of the layer given by
be determined at each x - x over the source. With the enthalpy
2(hlAULHLb)i
r dn.
2It lh
w
B'.u.dt
i i
(V.70)
(due to the choice of the reference temperature as the ambient
temperature) and with a suitable equation of state relating enthalpy,
temperature, and density, the source can be averaged for each observer.
After the average composition of the layer is determined at the downwind
edge, an adiabatic mixing calculation is performed between this gas and
the ambient air. This calculation represents the function between
density and concentration for the remainder of the calculation if the
calculation is adiabatic; it represents the adiabatic mixing condition if
heat transfer is included in the downwind calculation.
For each of several observers released successively from
x - -Rjnax, the observed dimensions L and b, the downwind edge of the
source x^n, the average vertical dispersion coefficient S2, the average
takeup flux Q*, the centerline concentration cc, and if applicable, the
average enthalpy hL can be determined for each observer. With these
input values, a steady-state calculation is made for each observer. The
distribution parameters for any specified time ts are determined by
locating the position of the series of observers at time t_ , i.e. xn-(t_).
o X o
The corresponding concentration distribution is then computed from the
assumed profiles.
-------
44
Steady-State Downwind Dispersion
The model treats dispersion of gas entrained into the wind field frorc
an idealized, rectangularly shaped source of width 2b and length L. The
circular source cloud is represented as an equivalent area rectangle
(TTR^ = 2bL) with equivalent fetch (L - 2R) . Similarity forms for the
concentration profiles are assumed which represent the plume as being
composed of a horizontally homogeneous section with Gaussian
concentration profile edges as follows:
c(x,y,z) =
exp
- b(x)
Sy(x)
c (x) exp
l+a
A power law wind velocity profile is assumed
- rt I —
x 0 I z.
l+a
for |y| > b
for |y| < b
(V.71)
(V.72)
where the value of a is determined by a weighted least-squares fit
of the logarithmic profile
u — -—
x k
z + z
In
R
(V.73)
Functional forms for V" a^d typical values of a are given in Table V.I for
different Pasquill stability categories. With these profiles, the
parameters of Equation (V.71) are constrained by ordinary differential
equations.
Vertical Dispersion
The vertical dispersion parameter Sz is determined by requiring that
it satisfy the diffusion equation
-------
CO
z
o
H
Crf
H
Z
O
Z
o
u
-J
Eu
O
C£
CU
a
z
»
Q
Z
"*
H
M
a
H
co
Crf
u
?"J
^
>J
t£\
^
BOUND/
u
02
J™
CU
CO
o
S
H
<
*"T
^
^•J
CU
H
.
^
U
I-J
«
^J
H
'-v
f^
• 1
3 «5
w o> a-
O iH •_,
tn o g
O -H ^ S
oi S >,
u -c .a o>
M J-J u
Sugg.
60^ S
•J ° t)
5
U W)
0) 4J ^^
3 e ts
O 0) r*-
CM C C 1
O -H ffl
rH O. «-»
, cfl
H J
en
U)
0)
c
> W £
O (0 C 60
•5 ° D
3 ^< u oi ^
_a ~-^ o S
o c
tq u n)
Cu co U
(0
v-'
i
i
m
.
I fJ-n) | ^^1 f^
1 0 f-
C tH
i-i <-^ n o-
i
+ n -3-
n
1 j ^ _^
4- cvi "5
L± — 1 "5
c
CM +
n
^.
ao ts O CM fi rn
O •" * CM
-------
46
3c _ 3_ 3c
x 3x 5z z 5z
(V.74)
with the vertical turbulent diffusivity given by
K
ku*z
(V.75)
The function <£(Ri*) is a curve fit of laboratory-scale data for
vertical mixing in stably density-stratified fluid flows reported by
Kantha et al. (1977), Lofquist (1960), and McQuaid (1976) for Ri* > 0.
For Ri* < 0, the function <£(Ri*) is taken from Colenbrander and Puttock
(1983) and has been modified so the passive limit of the two functions
agree as follows:
0.88 + 0.099
0.88/(1 + 0.65 |Ri,
i0.6.
i> 0
Ri*<°
(V.76)
The friction velocity is calculated using Equation (V.73) from a
known velocity UQ at a specific height ZQ. Combining the assumed
similarity forms for concentration and velocity, Equations (V.71),
(V.72), (V.74), and (V.75) give
(V.77)
d
dx
'Vo
1 + a
» j
' S
z
I zo .
1+a "
ku^(l + a)
*(Ri*)
where the Richardson number Ri is computed as
- /»
EFF
u.
(V.78)
and the effective cloud depth is defined as
"
EFF
r ( r^ ] I^rs
(V.79)
-------
Equation (V.77) can be viewed as a volumetric balance on a
differential slice of material downwind of the source. For a mass
balance over the same slice,
§- f />TuTHT I - p v
dx L KL L L j 'a e
(V.80)
which is the same result as Equation (V.51) without the source term.
With Equations (V.57) and (V.58), this becomes
£- f PLUEFFHEFF ] - p V
dx L J a e
(V.81)
Using the assumption of adiabatic mixing of ideal gases with the
same constant molal heat capacity (i.e.
P - P.
constant) along
with the contaminant material balance, the mass balance becomes
d
dx [ UEFFHEFF ) We
which leads to
w ku.(l + a)
(V.82)
(V.83)
Equations (V.81) and (V.83) are combined to give
) "
d_ f u „ 1 "aku*(1 + Q)
(V.84)
Furthermore, Equation (V.84) is assumed to apply when (p - p )/c
3. C
is not constant.
When heat transfer from the surface is present, vertical mixing will
be enhanced by the convection turbulence due to heat transfer. Zeman and
Tennekes (1977) model the resulting vertical turbulent velocity as
1/2
w
»* 2
(V.85)
where w^ is the convective scale velocity described as
-------
48
u,
[l . I 1
Is c,Lj
u*u
c,L
If u is is evaluated at H
EFF'
where
Ts - Tc,L
Tc,L
U*U0
2/3
(V.86)
(V.87)
H
EFF
(V.88)
and Tc T is the temperature obtained from the energy balance of Equations
(V.103) and (V.104). Equation (V.84) is modified to account for this
enhanced mixing by
d_ f H
dx [ ^EFI^EFF
(V.89)
where Ri' - Ri.
* *
u.
w
Although derived for two-dimensional dispersion, this is extended for
application to a denser-than-air gas plume which spreads laterally as a
density intrusion:
d_
dx
( "LUEFFJ!EFFBEFF )
p kw(l + Q)
B
EFF
(V.90)
where the plume effective half width is defined by
BEFF " b + ~2 Sy
and determined using the gravity intrusion relation
1/2
(V.91)
IBEFF r
dt E
g
' P - pa '
I '* \
HEFF
(V.92)
-------
The average transport velocity in the plume is defined by
u.
r
Jo
cu dz
x
EFF p®
cdz
J0
u
0 7.,
(V.93)
and the lateral spread of the cloud is modeled by
dB
EFF
dB
EFF
dx u£FF dt
-
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50
S
dS 4/3 ,2
y dx TT EFF
(V.98)
where Equation (V.98) is also assumed applicable for determining Sy when
b is not zero.
At the downwind distance xt where b - 0, the crosswind concentration
profile is assumed Gaussian with Sy given by
S - /2 S(x + x )^ (V.99)
where x is a virtual source distance determined as
S (xt) - Jl £(*£_+ xv/ . (V.100)
The gravity spreading calculation is terminated for x > xt.
Although the effect of averaging time on the maximum downwind
concentration for steady releases is still open to some question, the
most important effect is assumed to be a result of plume meander for the
purposes of the model. This behavior is reflected in the values of the
dispersion parameters used in the Gaussian plume mode; Turner (1970) and
Seals (1971) report different dispersion parameters for 10 minute average
plume behavior and instantaneous or puff behavior. Because of plume
meander, ay depends on the averaging time, while the value of az is
essentially unaffected by the averaging time (Beals, 1971). Therefore,
the variation of cry is modeled as
where ay(x;t]_) and uy(x;t2) are the values of a associated with the
averaging times t^ and t£ , respectively. The value of p is a function of
averaging time; Hino (1968) found p - 0.2 for averaging times less than
10 minutes and p = 0 . 5 for times greater than 10 minutes. Obviously,
Equation (V.101) will not be appropriate as the averaging time goes to
zero (i.e. for a puff). Using the values reported by Turner (1970), if
the puff and 10 minute plume values of ay are compared for D stability,
the equivalent averaging time for the puff coefficient is about 20 s.
This result indicates that the width of a steady-state plume would not
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Dl
vary significantly due to meander over any given 20 s period. (Note that
the same statement would not be appropriate for the maximum
concentration.) If the value of cr is parameterized as ay - 5x^ where a
and x are in meters, then the value of 0 can be approximated as being the
same for the plume and puff values. (Seinfeld (1983) used £ = 0.894 for
the plume a values.) Using the same power /3, the parameterizations for
plume and puff <7y values are shown in Table V.2 where the averaging time
for the puff coefficient is taken to be 20 s for all stabilities. For
further discussion, see Spicer (1987) or Spicer and Havens (1987) .
TABLE V.2
COEFFICIENT S IN GAUSSIAN DISPERSION MODEL FOR USE IN
o - Sx-9 WITH /3 - 0.894 AND a AND x IN METERS
Averaging
Time
Stability Class
A B C D E
F
10 min 0.443 0.324 0.216 0.141 0.105 0.071
20 s or less 0.224 0.164 0.109 0.071 0.053 0.036
For a steady plume, the centerline concentration c is determined
from the material balance
nr
cuxdydz - 2cc T-r-
S
z
z
0
1+a
B£FF (V.102)
where E is the plume source strength.
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52
Energy Balance
For some simulations of cryogenic gas releases, heat transfer to the
plume in the downwind dispersion calculation may be important,
particularly in low wind conditions. The source calculation determines a
gas/air mixture initial condition for the downwind dispersion problem.
Air entrained into the plume is assumed to mix adiabatically. Heat
transfer to the plume downwind of the source adds additional heat. This
added heat per unit mass D^ is determined by an energy balance on a
uniform cross-section as
S [ VLUEFFHEFF] - V5L (V-103)
where q is determined by Equation (V.37) along with the desired
method of calculating h_. Equation (V.103) is applied when b = 0 and
is extended to
S [ VLUEFE»EFFBEFF J - %>W5L (v-104)
when b > 0. Since the average density of the layer p-, cannot be
determined until the temperature (i.e. D^) is known, a trial and error
procedure is required.
Equations (V.77), (V.78), (V.79), (V.87)-(V.91), (V.94), and (V.98)-
(V.104) are combined with an equation of state relating cloud density to
gas concentration and temperature and are solved simultaneously to
predict Sz, Sy, cc, and b as functions of downwind distance beginning at
the downwind edge of the gas source.
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53
Correction for Along-Wind Dispersion
Following Colenbrander (1980), an adjustment to GC is applied to
account for dispersion parallel to the wind direction. The calcu-
lated centerline concentration cc(x) is considered to have resulted from
the release of successive planar puffs of gas (cc(x)Ax) without any
dispersion in the x-direction. If it is assumed that each puff diffuses
in the x-direction as the puff moves downwind independently of any other
puff and that the dispersion is one-dimensional and Gaussian, the
x-direction concentration dependence is given by
c (x )Ax.
Pi
exp
1
2
x - x
Pi
a
x
2 "
(V.105)
where x denotes the position of the puff center due to observer i.
After Reals (1971), the x-direction dispersion coefficient crx is
assumed to be a function of distance from the downwind edge of the gas
source (X - x - XQ) and atmospheric stability given by
a. 22
a (X) - 0.02 X
x
- 0.04 X
- 0.17
1.14
,0.97
unstable, x > 130 m
neutral, x > 100 m
stable, x > 50 m
(V.106)
where (X - x - XQ) and a are in meters. The concentration at x is
then determined by superposition, i.e., the contribution to c at a
given x from neighboring puffs is added to give an x-direction
corrected value of c'. For N observers,
cc(x)
N
2
exp
"
1
2
x - x
pi
a
x
2 "
Ax.
(V.107)
and for large N,
:<
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54
The corrected centerline concentration cl is used in the assumed
\f
profiles in place of cc, along with the distribution parameters Sy, S
and b.
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55
VI. INTERFACING THE OOMS AND DEGADIS MODELS
The Ooms model presented here is intended to be used to predict the
trajectory and dilution, to the point of (downwind) ground contact, of a
dense gas jet released vertically upward into the atmospheric surface
layer. The Ooms model prediction is terminated when the lower edge of
the plume impinges the ground. The resulting downwind distance, plume
centerline concentration and temperature, and plume radius (/2b^) are
used as input to DEGADIS. The ground level gas source input to DEGADIS
is a circular area source with radius J2bi and concentration and
temperature equal to the centerline values output from the Ooms model.
Application of the Ooms model to prediction of the trajectory and
dilution of a dense gas jet, with subsequent input to DEGADIS for
prediction of the ensuing ground level dispersion, is straightforward
when the plume falls to the ground within a short distance downwind of
the release point. If the plume remains aloft (because atmospheric
entrainment results in dilution of the plume to an essentially passive
state), it will continue to entrain air and grow in (circular) cross-
section until the lower edge of the plume impinges the ground. For
releases at elevations of a few meters (typical of chemical storage
vessel vents), particularly under conditions of high atmospheric
turbulence entrainment, the lower edge of the plume may impinge the
ground at a short distance downwind. Since the plume centerline is
still aloft it is questionable to input the resulting Ooms model output
to DEGADIS (which assumes a ground level', area source). For this reason
it is recommended that the jet release be simulated initially with the
Ooms model to determine if the plume falls to the ground or remains
aloft. If the plume centerline would fall to the ground within a short
distance (say 1 kilometer) of the release point, the Ooms model output
should be input to the DEGADIS model for prediction of the ensuing
ground level dispersion. If the plume is predicted to remain aloft, the
use of DEGADIS is not presently recommended, since the Ooms model does
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56
not account for ground reflection of the plume and the "matching" of the
Ooms output to a ground level gas source required by DEGADIS is not
straightforward.
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57
VII. CONCLUSIONS AND RECOMMENDATIONS
A mathematical model was developed for estimating ambient air
concentrations downwind of elevated, denser-than-air gas jet-type
releases. Ooms' model is used to predict the trajectory and dilution,
to ground contact, of a denser-than-air jet/plume. The output of Ooms'
model interfaces with the DEGADIS dense gas dispersion model to predict
the ensuing ground level dispersion. The model incorporates momentum
and heat transfer important to turbulent diffusion in the surface layer
of the atmospheric boundary layer, and provides for:
• inputting data directly from external files
• treatment of ground-level or elevated sources
• estimation of maximum concentrations at fixed sites
• iteration over discrete meteorological conditions
• estimation of concentration-time history at fixed sites.
This report and the accompanying User's Guide (Volume II):
• document the theoretical basis of the model
• discuss its applicability and limitations
• discuss criteria for estimating the importance of gas density
effects on a jet release from an elevated source
• define and describe all input variables and provide appropriate
guidance for their specification
• identify and describe all output files and provide appropriate
guidance for their interpretation
• provide user instructions for executing the code
• illustrate the model usage with example applications.
Application of the Ooms model to prediction of the trajectory and
dilution of a dense gas jet, with subsequent input to DEGADIS for
prediction of the ensuing ground level dispersion, is straightforward
when the plume falls to the ground within a short distance downwind of
the release point. If the plume remains aloft, it will continue to
entrain air and grow in (circular) cross-section until the lower edge of
the plume impinges the ground. For releases at elevations of a few
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58
meters, the lower edge of the plume may impinge the ground at a short
distance downwind. For jet/plumes which remain aloft, it is
questionable to input the resulting Ooms model output to DEGADIS (which
presently assumes a ground level, area source). Consequently, it is
recommended that a jet release be simulated initially with the Ooms
model to determine if the plume falls to the ground. If the plume
(centerline) returns to the ground within a short distance (say 1
kilometer) of the release point, the Ooms model output can be input to
the DEGADIS model for prediction of the ensuing ground level dispersion.
If the plume is predicted to remain aloft, the use of DEGADIS is not
presently recommended since the Ooms model does not account for ground
reflection of the plume and the "matching" of the Ooms output to a
ground level gas source required by DEGADIS is not straightforward.
The Ooms model is strictly applicable to steady state. However, in
the intended application to denser-than-air jets, it is used to simulate
the development of a jet-plume which would be time-limited (for example,
a jet release of duration 5 minutes). The output of the jet model then
provides a time-limited gas source for DEGADIS, which predicts the
resulting (transient) downwind gas concentration history. Interfacing
the Ooms model with DEGADIS for such applications can be problematical,
particularly when the plume (gas-air) output of Ooms is very near
ambient density. Although resolution of such difficulties appears to be
relatively straightforward, time constraints in this effort were
prohibitive. Additional work should be undertaken to resolve these
difficulties, with the specific tasks:
• provide for "matching" of the Ooms model output mass flux rate
with the DEGADIS atmospheric takeup at the Ooms-DEGADIS
interface
• modify the Ooms model atmospheric turbulence entrainment
specification to provide an elliptical cross-section and insure
that the entrainment so specified is consistent with the
Gaussian (Pasquill-Gifford) dispersion coefficient
representation of atmospheric turbulence entrainment.
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59
Address of these two (related) tasks would provide extension of the
applicability of the model to description of plumes which become
neutrally (or positively) buoyant.
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61
REFERENCES
Albertson M. L., Y. B. Dai, R. A. Jenson, and H. Rouse, "Diffusion of
Submerged Jets," Transactions of American Society of Civil
Engineers, 115. (1950).
Batchelor, G. K., An Introduction to Fluid Dynamics. Cambridge
University Press, Cambridge, UK, 1967.
Seals, G. A., "A Guide to Local Dispersion of Air Pollutants," Air
Weather Service Technical Report 214, April, 1971.
Bird, R. B., W. E. Stewart, and E. N. Lightfoot, Transport Phenomena.
John Wiley and Sons, New York~TT960.
Briggs, G. A., "Plume Rise," AEC Critical Review Series, USAEC Division
of Technical Information Extension, Oak Ridge, Tennessee, 1969.
Britter, R. E., "The Ground Level Extent of a Negatively Buoyant Plume
in a Turbulent Boundary Layer," Atmospheric Environment. 14, 1980.
Britter, R. E., unpublished monograph, 1980.
Businger, J. A., J. C. Wyngaard, Y. Izumi, and E. F. Bradley, "Flux-
Profile Relationships in the Atmospheric Surface Layer," Journal of
the Atmospheric Sciences. 28. March 1971.
Carnahan, B., H. A. Luther, and J. 0. Wilkes, Applied Numerical Methods,
John Wiley and Sons, 1969.
Colenbrander, G. W., "A Mathematical Model for the Transient Behavior of
Dense Vapor Clouds," 3rd International Symposium on Loss Prevention
and Safety Promotion in the Process Industries, Basel, Switzerland,
1980.
Colenbrander, G. W. and J. S. Puttock, "Dense Gas Dispersion Behavior:
Experimental Observations and Model Developments," International
Symposium on Loss Prevention and Safety Promotion in the Process
Industries, Harrogate, England, September 1983.
Cude, II. L., "Dispersion of Gases Vented to Atmosphere from Relief
Valves," The Chemical Engineer. October, 1974.
Havens, J. A. and T. 0. Spicer, "Development of an Atmospheric
Dispersion Model for Heavier-than-Air Gas Mixtures," Final Report to
U.S. Coast Guard, CG-D-23-85, USCG HQ, Washington, DC, May 1985.
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62
Hoot, T. G., R. N. Meroney, and J. A. Peterka, "Wind Tunnel Tests of
Negatively Buoyant Plumes," Report CER73-74TGH-RNM-JAP-13, Fluid
Dynamics and Diffusion Laboratory, Colorado State University,
October 1973.
Hino, Mikio, "Maximum Ground-Level Concentration and Sampling Time,"
Atmospheric Environment. 2, pp. 149-165, 1968.
Kaimal, J. C., J. C. Wyngaard, D. A. Haugen, 0. R. Cote, and Y. Izumi,
"Turbulence Structure in the Convective Boundary Layer," J. Atmos.
Sci. , 33, 1976.
Kamotani, Yasuhiro and Isaac Greber, "Experiments on a Turbulent Jet in
a Cross Flow," AIAA Journal, Vol. 10, No. 11, November 1972.
Kantha, H. L., 0. M. Phillips, and R. S. Azad, "On Turbulent Entrainment
at a Stable Density Interface," Journal of Fluid Mechanics. 79.
1977, pp. 753-768.
Koopman, R. P. et al., "Description and Analysis of Burro Series
40-nr LNG Spill Experiments," Lawrence Livermore National
Laboratories Report UCRL-53186, August 14, 1981.
Lofquist, Karl, "Flow and Stress Near an Interface Between Stratified
Liquids," Physics of Fluids. 3, No. 2, March-April 1960.
McAdams, W. H., Heat Transmission. McGraw-Hill, New York, 1954.
McQuaid, James, "Some Experiments on the Structure of Stably Stratified
Shear Flows," Technical Paper P21, Safety in Mines Research
Establishment, Sheffield, UK, 1976.
Morrow, T., "Analytical and Experimental Study to Improve Computer
Models for Mixing and Dilution of Soluble Hazardous Chemicals,"
Final Report, Contract DOT-CG-920622-A with Southwest Research
Institute, U.S. Coast Guard, August 1982.
Ooms, G., A. P. Mahieu, and F. Zelis, "The Plume Path of Vent Gases
Heavier than Air," First International Symposium on Loss Prevention
and Safety Promotion in the Process Industries, (C. H. Buschman,
Editor), Elsevier Press, 1974.
Pasquill, F., Atmospheric Diffusion. 2nd edition, Halstead Press, New
York, 1974.
Richards, J. M., "Experiments on the Motion of an Isolated Cylindrical
Thermal Through Unstratified Surroundings," International Journal of
Air and Water Pollution, 7., 1963.
Seinfeld, J. H., "Atmospheric Diffusion Theory," Advances in Chemical
Engineering. 12, pp. 209-299, 1983.
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63
Simpson, J. E. and R. E. Britter, "The Dynamics of the Head of a Gravity
Current Advancing over a Horizontal Surface," Journal of Fluid
Mechanics. 94, Part 3, 1979.
Spicer, T. 0., "Using Different Time-Averaging Periods in DEGADIS,"
Report to the Exxon Education Foundation, June 1987.
Spicer, T. 0. and J. A. Havens, "Development of Vapor Dispersion Models
for Non-Neutrally Buoyant Gas Mixtures--Analysis of USAF/^O^ Test
Data," USAF Engineering and Services Laboratory, Draft Final Report,
February, 1986.
Spicer, T. 0. and J. A. Havens, "Development of Vapor Dispersion Models
for Nonneutrally Buoyant Gas Mixtures--Analysis of TFI/NH-j Test
Data," UASF Engineering and Services Laboratory, Draft Final Report,
July 1987.
Treybal, R. E., Mass Transfer Operations. 3rd edition, McGraw-Hill, New
York, 1980.
Turner, D. B., "Workbook of Atmospheric Dispersion Estimates," USEPA
999-AP-26, U.S. Environmental Protection Agency, Washington DC,
1970.
van Ulden, A. P., "The Unsteady Gravity Spread of a Dense Cloud in a
Calm Environment," 10th International Technical Meeting on Air
Pollution Modeling and its Applications, NATO-CCMS, Rome, Italy,
October 1979.
van Ulden, A. P., "A New Bulk Model for Dense Gas Dispersion: Two-
Dimensional Spread in Still Air," I.U.T.A.M. Symposium on
Atmospheric Dispersion of Heavy Gases and Small Particles, Delft
University of Technology, The Netherlands, August 29-September 2,
1983.
Wheatley, C. J., "Discharge of Ammonia to Moist Atmospheres--Survey of
Experimental Data and Model for Estimating Initial Conditions for
Dispersion Calculations," UK Atomic Energy Authority Report SRD
R410, 1986.
Zeman, 0. and H. Tennekes, "Parameterization of the Turbulent Energy
Budget at the Top of the Daytime Atmospheric Boundary Layer,"
Journal of the Atmospheric Sciences. January 1977.
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TECHNICAL REPORT DATA
(Please read Instructions on the reverse before completing)
1. REPORT NO.
EPA 450/4-88-006a
2.
3. RECIPIENT'S ACCESSION NO.
4. TITLE AND SUBTITLE
A Dispersion Model for Elevated Dense Gas Jet
Chemical Releases—Volume I
5. REPORT DATE
April 1988
6. PERFORMING ORGANIZATION CODE
7. AUTHOR(S)
8. PERFORMING ORGANIZATION REPORT NO.
Dr. Jerry Havens
9. PERFORMING ORGANIZATION NAME AND ADDRESS
10. PROGRAM ELEMENT NO.
11. CONTRACT/GRANT NO.
P.O. #6D2746NASA
12. SPONSORING AGENCY NAME AND ADDRESS
13. TYPE OF REPORT AND PERIOD COVERED
U.S. Environmental Protection Agency
Office of Air Quality Planning and Standards
Source Receptor Analysis Branch
Research Triangle Park, N.C. 27711
14. SPONSORING AGENCY CODE
15. SUPPLEMENTARY NOTES
EPA Project Officer: Dave Guinnup
16. ABSTRACT
This document is the first of two volumes describing the development and use of
a computer program designed to model the dispersion of heavier-than-air gases which
are emitted into the atmosphere with significant velocity through elevated ports.
The program incorporates the sequential execution of two models. The first one (Ooms)
calculates the trajectory and dispersion of the gas plume as it falls to the ground.
The second (DEGADIS) calculates the downwind dispersion after the plume touches
ground.
This first volume discusses the development of both models and establishes the
mathematical framework for the calculations. In addition, the trajectory portion of
the model is evaluated in reference to wind tunnel data.
17.
KEY WORDS AND DOCUMENT ANALYSIS
DESCRIPTORS
b.lDENTIFIERS/OPEN ENDED TERMS
c. COSATI Field/Group
Air pollution
Dense gas
Mathematical model
Computer model
Dispersion
Elevated sources
18. DISTRIBUTION STATEMENT-
Release unlimited
19. SECURITY CLASS (TIlis Report)
21. NO. OF PAGES
88
20. SECURITY CLASS (Tliis page)
22 PRICE
EPA Form 2220-1 (Rev. 4-77) PREVIOUS EDIT i -• is OBSOLETE
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