Transactions on Ecology and the Environment vol 8, © 1996 WIT Press, www.witpress.com, ISSN 1743-3541
Development of an integral model for the
dispersion of hot and moist gases
M.A.C. Teixeira, P.M.A. Miranda
Centra de Geofisica da Universidade de Lisboa, Rua da Escola
Politecnica, 58, 1250 Lisboa, Portugal
Abstract
Buoyant plumes can be simulated with reasonable accuracy and simplicity using
integral models. This paper attempts to analyze some aspects of buoyant plume
dispersion using one of the most sophisticated integral models available,
developed by Schatzmann. The model is studied in some detail and compared to
a slightly different version, proposed by Davidson. It is found that differences
put in the establishment of the two models lead to significant changes in the
plume behaviour. Some modifications to Schatzmann's model are suggested, in
which sheared wind, radiation, moisture and phase transitions are taken into
account. Finally, an alternative, simpler, turbulence closure is proposed for the
parametrization of entrainment, which is the strongest constraint on the
performance of any plume model but also its less well founded aspect.
1. Introduction
Integral models are based on the fundamental laws of Fluid Mechanics:
conservation of mass, conservation of momentum, conservation of entropy and
equation of state. A steady average flow and some axisymmetry assumptions are
accepted, and the equations are integrated over the cross-section of the plume.
A set of ordinary differential equations is thus obtained, describing the evolution
of certain integral quantities.
These models are designed to simulate the initial phase of the dispersion of
buoyant plumes, when buoyancy is quite relevant and the turbulence generated
by shear in the jet flow dominates the mixing process. Although there exist
some attempts to include ambient turbulence (Slawson and Csanady^,
Netterville**), the range of applicability of integral models is always limited by
Transactions on Ecology and the Environment vol 8, © 1996 WIT Press, www.witpress.com, ISSN 1743-3541
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Air Pollution Monitoring, Simulation and Control
the effects of buoyancy, the mean lateral flow and the anisotropy of this kind of
turbulence, which tend to destroy axisymmetry.
2. Slawson and Csanady's model
Slawson and Csanady^ developed an integral model which extends the work of
Morton et al.* to the case of a moving environment. Their basic assumptions
are: i) Coriolis force neglected; ii) Viscous forces neglected; iii) Adiabatic
thermodynamic processes; iv) Horizontally homogeneous average ambient
properties; v) Pressure everywhere horizontally homogeneous, and in
hydrostatic equilibrium in the environment; vi) Steady state: mean fields not
depending on time; vii) Atmosphere with a constant Brunt-Vaisalla frequency;
viii) Ambient mean flow horizontal and constant; ix) Entrainment described by
an inward-pointing velocity at the plume's outer edge; x) Entrainment velocity
defined by v, = aw where w is the mean vertical velocity inside the plume and a
is a dimensionless constant; xi) Horizontal component of velocity equal
everywhere; xii) Top-hat profiles adopted for plume properties: temperature,
vertical velocity and, if that is the case, concentration of a tracer; xiii)
Boussinesq approximation: density treated as a constant except in the buoyancy
terms.
Using these simplifications, the authors obtain a set of 3 equations with
analytic solutions, which can be easily tested (Middleton*).
Several models have been developed, using the same integral approach but
with different assumptions for the plume geometry and its interaction with the
environment. Fox* developed an integral model for a gaussian plume with a new
parametrization of the entrainment process, based on a form of the kinetic
energy budget equation. Hirst^ generalized the work of Fox and developed an
integral model which allowed for three-dimensional trajectories. Schatzmann"
and Davidson^ tried to produce more rigorous models, taking out the
Boussinesq approximation and making a careful analysis of all the implications
of the assumed geometry.
3. Schatzmann's model
A sophisticated integral model has been presented by Schatzmann^'", where
assumptions i) to ix) of the previous section were maintained, but assumptions
x) to xiii) were rejected. Schatzmann started by adopting a curvilinear
coordinate system first introduced by Hirst* (figure 1). For the velocity excess
of the plume, he admitted a self-similar gaussian profile. Assumption x), the
entrainment definition, was replaced by a much more complex expression,
making use of the kinetic energy equation, following, again, Hirst'.
For a non-turbulent atmosphere, the model's equations are: conservation of
mass, ^-momentum budget, conservation of entropy and 0-equation.
Transactions on Ecology and the Environment vol 8, © 1996 WIT Press, www.witpress.com, ISSN 1743-3541
Air Pollution Monitoring, Simulation and Control
91
t :
where p^ and 0, are ambient density and potential temperature, p* and ®*
are their maximum excesses, C/, is the ambient flow velocity, u is the
maximum velocity excess, 0 is the steepness of the plume axis, b is a parameter
proportional to plume width and s is the along-plume coordinate. The radius of
the plume, R, is defined as R* =2b*. Outside this limit, it is assumed that
ambient values prevail.
/I = 1.16 is the ratio of temperature and momentum plume widths
(Morton™), g is the acceleration of gravity, c^=2.5 is a drag coefficient and v,
is the entrapment velocity. To close the set, the equation of state used by
Schatzmann and Policastro^ and the entrapment hypothesis (Schatzmann^)
are:
0*
(5)
=-/>,
0+0*
(6)
where A^ = 0.057, A^ = -0.67, ^ =10 e A < = 2 are dimensionless
constants, £ = ^!v^ / bu is a dimensionless entrainment coefficient and J is the
densimetric Froude number, defined as 3* = u** I (gbp*/p^).
An equation formally identical to eqn.(3) can be added to the set, to
describe the dispersion of a passive tracer. With q^ defined as the mass ratio of
pollutant in the environment and #* as its maximum excess, these two variables
have to replace 0^ and 0% respectively, in eqn.(3).
Transactions on Ecology and the Environment vol 8, © 1996 WIT Press, www.witpress.com, ISSN 1743-3541
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Air Pollution Monitoring, Simulation and Control
Figure 1: The plume coordinate system
4. Davidson's model
The equation set developed by Davidson^ differs from Schatzmann's only in the
formulation of the ^-momentum budget and the 0-equation. It seems that the
only significant difference between Schatzmann's and Davidson's assumptions
concerns the velocityfield.Schatzmann assumes that the mean flow inside the
plume contains a total ambient component whereas Davidson prescribes an
entirely axisymmetric flow pattern by just adding to the gaussian excess the
longitudinal component of the ambient wind.
200
400
600
800
Figure 2: Plume trajectories and radii for the conditions: U« - 1ms',
, = 283K, d®Jdz = 0.015Km', u] = 14.5ms', ®]',= 28.3K, R. = 0.75m
and 9j - 90°. Subscript./ identifies parameters at the source.
Davidson's model, like Slawson and Csanady's, formally violates the
continuity equation at the plume edge when it admits a jump in the transverse
component of velocity. Schatzmann's model also violates the fundamental
equations, but not at the plume boundary. As the flow includes a nonaxisymmetric component (the total ambient wind), it tends to destroy the
Transactions on Ecology and the Environment vol 8, © 1996 WIT Press, www.witpress.com, ISSN 1743-3541
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93
symmetry of the gaussian distribution of tracers, whose shape, in spite of that, is
imposed. The plume is forced to move in the direction of its velocity excess
instead of following the global velocity field. Such violations are not serious
because, in terms of integrated fluxes, all fundamental laws are respected. The
purpose of integral models is to simulate the gross beahaviour of plumes, but,
because of the symmetry assumption, it would be extremely difficult to respect
the physical laws locally.
Figure 2 shows that the differences between Schatzmann's and Davidson's
formulations become quite significant beyond the point of maximum plume
height. In that region, Davidson's model shows large oscillations in the velocity
excess, which lead to large oscillations in the radius. At the same time, the
wavelengths of the residual oscillation in plume height are different in the two
models by a factor of 2, and only Davidson's model displays a typical internal
gravity wave value of about InU^ IN where N is the Brunt-Vaisalla frequency.
The slight difference in the equilibrium heights could certainly be overcome by
an adjustment in empirical parameters.
5. Modifications to Schatzmann's model
5.1. Wind profile with shear
An ambient wind U^ which varies in strength with altitude can easily be
accounted for, as long as its scale of variation is significantly larger than the
radius of the plume. For that purpose, it is necessary to sum to the left-hand side
of eqn.(l), to the right-hand side of eqn.(2) and to the numerator of eqn.(4) the
following 3 terms, respectively:
(7)
as
<6
-COS0
(8)
(
ds
P
\
(
P
\
(9)
leaving unchanged the entropy and pollutant mass budgets. The modifications in
the behaviour of the plume induced by shear are relatively obvious, resulting
essentially from the necessity to maintain momentum balance in those somewhat
more general conditions.
5.2. The introduction of radiation
In the case of a very hot plume in a crossflow, the assumption of gaussian shape
Transactions on Ecology and the Environment vol 8, © 1996 WIT Press, www.witpress.com, ISSN 1743-3541
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simultaneously for the temperature and density profiles becomes a bad
approximation. Using the exact equation of state implies, however, that some
integrals become non-analytical, unless it is assumed that A = 1. This
approximation has been adopted here, since, in fact, A differs only slightly from
1. In these conditions, Schatzmann's equation set suffers some changes.
Concerning directly radiation, the approach of Shestopal and Grubits^ has
been adapted to Schatzmann's model. Those authors admit that the plume emits
radiation essentially in the radial direction and in an isotropic way. The
absorbing coefficient is assumed to have a gaussian distribution similar to those
used for tracers. The following term has to be added to the right-hand side of
the entropy equation:
' \^
v;
r
^
1
•^-J '' **Ajx[£ + exp(-x*)]exp -^J[^exp(-x^)]^'J x
'-**)]*-l)<fc
where %/= 0* /0,,
^ = K^/K\
(10)
K^ is the absorption coefficient in the
environment and K* the corresponding maximum excess, a is the StefanBolzmann constant, p is pressure, p^ = 1000mb, R^ is the ideal gas constant
for dry air and c^ is the specific heat of air at constant pressure.
Except in the case of very hot plumes, with carefully chosen absorption
coeficients, the impact of radiative cooling on plume behaviour was found to be
almost negligible. It can thus be concluded that, in this model, the effect of
mixing tends to be strongly predominant over that of radiation.
5.3. Water and phase transitions
Schatzmann and Policastro^ made further developments to Schatzmann's
model, turning it able to simulate cooling tower plumes. They accounted, in
particular, for the effect of downwash, induced by the cooling tower wake
(Carhart et al/) and slightly modified the entrainment definition.
Cooling tower plumes are generally not very hot (i.e. initial temperature
excesses of less than 3OK) so the Boussinesq equation set can be used. Total
water can be treated as a tracer, being described by an equation similar to that
of a pollutant and having as well a gaussian profile. Its partitioning between the
liquid and vapour phases obeys: #„=#,, q» = 0 if #, < q, and #„=#,,
q^-q^- q^ if q^ >#,, where <?, is the total water mass ratio, q^ is the
specific humidity, <?„ the liquid water mass ratio and #, the saturation specific
humidity.
q^ can be determined as a function of pressure and temperature making use
of the Clausius-Clapeyron equation. In the present paper, a linear dependence of
Transactions on Ecology and the Environment vol 8, © 1996 WIT Press, www.witpress.com, ISSN 1743-3541
Air Pollution Monitoring, Simulation and Control
95
9, on 0 is assumed, making the profile of q^ gaussian. To ensure that q^ is
evaluated correctly in the middle of the plume, the Taylor series suggested by
Schatzmann and Policastro is replaced by a linear fit between the ambient and
maximum temperatures; this approach avoids errors in the prediction of the
existence of a liquid water plume although it will certainly introduce them in the
prediction of its width.
The entropy equation is established having in mind that the quantity
Cp& + L^ is approximately conserved, as long as the specific heat and latent
heat of vaporization L^ of air are accepted to be constants and that the ratio
T 1 0 is very close to 1 .
With all these assumptions, the integrated entropy equation reads
cos* +
(ii)
where q^ = q,(&^,p) is the saturation specific humidity in the environment,
Q*s ~Qs(®a +®*»P)~?w is the corresponding maximum excess, q^ is the
mass ratio of total water in the environment and q* its maximum excess. The
ideal gas equation takes the form suggested by StulF, where both water vapour
and liquid water are taken into account.
The use of gaussian profiles for the plume properties turns out to be an
important aspect when moisture is introduced, being a less artificial scheme than
the use of a "peak factor" (Hanna*) or the assumption of a smaller width for the
water distribution, in order to limit condensation to the central portion of the
plume (Orville et al.^, Carhart and Policastro*).
In figure 3, trajectories of plumes with and without water content are
compared, and the contours of the liquid water plumes are presented. This
contours correspond to the points where q^ changes from zero to a positive
value. In case 1, condensation enhances plume rise, while in case 2, the moist
plume sinks, because of evaporative cooling, illustrating a phenomenon
described in Scorer^.
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Air Pollution Monitoring, Simulation and Control
100
400
200
500
X(m)
Figure 3: Behaviour of dry and moist plumes for the same conditions as figure 2
except d&Jdz = O.OOTKnT'. Case 1: q^ = 0.0075 and q\ = 0.03 . Case 2:
g,, = 0.007 and q* = 0.04.
5.4. A first-order closure scheme for turbulence
It is easy to derive an alternative entrainment definition, using a first order
turbulence closure scheme to substitute the correlation of radial and alongplume velocity perturbations in the kinetic energy equation. Manipulating that
equation and the remainder of Schatzmann's set, with the Boussinesq
approximation, all derivatives can be eliminated, yielding.
3U.
-2%
.
- + — r cos<9
2u
3^ri (7,
-- ^ - + -f
bu V2 u
.
. .
- + -- f cos# -f s
4 2u
) u
4
2u
(12)
in which K is a turbulent diffusivity that may depend on the s coordinate. This
dependance enables the simulation of the differing behaviour of dispersion for
small or large lagrangian times.
Transactions on Ecology and the Environment vol 8, © 1996 WIT Press, www.witpress.com, ISSN 1743-3541
Air Pollution Monitoring, Simulation and Control
200
400
200
400
600
800
97
1C
600
X(m)
Figure 4: Behaviour of dry plumes using Schatzmann's entrainment and the 1st
order closure, for the conditions of figure 2, except Rj = 2.5m. K has been
calibrated so that, for Rj = 0.75m, thefinalheight of the 3 plumes is the same.
Figure 4 shows that admitting a constant K produces results that are quite
different from those of Schatzmann. The most obvious hypothesis suggested by
dimensional analysis, which seems to be adequate for turbulence generated by
the plume, since it implies £ = 0 and db/ds = Q when u =0, is Kccbu*
(Slawson and Csanady^). With this definition, and a suitably chosen
proportionality constant, the behaviour of the model becomes relatively similar
to Schatzmann's. Since Schatzmann's model was calibrated against
experimental data, this agreement gives some confidence on the performance of
the new closure. Another advantage of this approach is that the calibration
becomes easier, since only one constant has to be adjusted instead of
Schatzmann's four constants in the entrainment definition.
Acknowledgments
The authors acknowledge the support given by Professor Jose Pinto Peixoto at
the Institute Geofisico do Infante D. Luis, and by Professor Mendes Victor at
the Centre de Geofisica da Universidade de Lisboa, where all the calculations
were performed. This work has been accomplished with thefinancialsupport of
JNICT under the Grant PBIC/C/CEN/1082/92.
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Transactions on Ecology and the Environment vol 8, © 1996 WIT Press, www.witpress.com, ISSN 1743-3541
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Air Pollution Monitoring, Simulation and Control
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