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Jets and Plumes in Crossflow
273
C H A P T E R 11 JETS AND PLUMES IN CROSSFLOW
W h e n a j e t or forced p l u m e is injected transverse stream,
a
very
complicated three-dimensional
to a
flowfield
flowing results,
(Coelhoe a n d Hunt, 1989). T h e external stream is deflected a r o u n d
the
j e t creating a p r e s s u r e field w h i c h forces the j e t to b e n d
the
in
s t r e a m w i s e d i r e c t i o n . T h e m i x i n g o f t h e t w o fluids a n d t h e e x c h a n g e o f m o m e n t a a l s o r e s u l t i n flow d e f l e c t i o n , o u t w a r d for t h e fluid
a n d s t r e a m w i s e for t h e j e t
fluid.
external
In technical applications,
the
penetration depth is the m e a s u r e of the lateral distance from the j e t e x i t t o t h e m a x i m u m r e a c h o f t h e j e t fluid. S u c h a m a x i m u m ( i n a n a s y m p t o t i c s e n s e ) c a n b e d e f i n e d o n l y for a n o n b u o y a n t j e t . W i t h b u o y a n c y t h e j e t fluid w i l l c o n t i n u e t o r i s e , a l t h o u g h i n t h e f o r m o f a bent plume, while b e i n g carried d o w n s t r e a m with the velocity of the main
flow.
For this late bent-over stage, simple similarity solutions
exist w h i c h c a n b e u s e d to estimate p l u m e rise a n d rate of dilution. M a n y practical
problems
concerned with safety and
pollution
i n v o l v e j e t s or forced p l u m e s . T h e n e a r field, w i t h its c o m p l i c a t e d three-dimensional
flow
structure, will d e p e n d on the ratio of jet to
f r e e - s t r e a m m o m e n t u m . W h e n t h e fluids h a v e n e a r e q u a l d e n s i t y , t h e i m p o r t a n t flow p a r a m e t e r i s s i m p l y t h e v e l o c i t y r a t i o , j e t - e x i t t o f r e e s t r e a m v a l u e . A large b o d y o f e x p e r i m e n t a l i n f o r m a t i o n exists, for m o r e d e t a i l s s e e S c h e t z ( 1 9 8 0 ) a n d H e n d e r s o n - S e l l e r s ( 1 9 8 7 ) . It i s a c o m m o n experience that the initially round jet, develops a kidneyshaped
cross-section
counter rotating phenomenon
can
vortices
and
eventually
aligned
with
bifurcates
the
b e explained in terms
main
into
stream.
two This
of vortex dynamics,
see
Eskinazi (1975). For the b u o y a n t jet, the ratio o f inertia force to b u o y a n c y force (i.e. the densimetric F r o u d e n u m b e r ) , b e c o m e s a s e c o n d parameter and, at large distances from the exit, the m o r e important one. Negative well
as
positive buoyancy
involve most releases
are
is
of interest.
P r o b l e m s in
as
pollution
often positive b u o y a n c y w h e r e a s negatively b u o y a n t of frequent
interest
in safety w o r k . A s the
buoyant
r e l e a s e , far f r o m t h e e x i t , h a s a d e n s i t y n e a r l y e q u a l t o t h a t o f t h e ambient, d u e to entrainment,
the further
spread will b e passive, i.e.
driven b y the external turbulence. T h e methods discussed in Chapter 9 are then applicable. I n w h a t f o l l o w s w e w i l l first d i s c u s s t h e p a r a m e t e r s w h i c h c o n t r o l
274
Chapter 11
t h e flow i n t h e n e a r and the external
field,
flow
w h e r e the angle b e t w e e n the buoyant jet
i s l a r g e . W e w i l l t h e n c o n s i d e r t h e far
field,
w h e r e m o m e n t u m and b u o y a n c y both are important, but w h e r e the a n g l e b e t w e e n t h e t w o s t r e a m s i s s m a l l . H e r e it i s p o s s i b l e t o o b t a i n simple similarity solutions. T h e s e solutions will b e applicable also to negatively buoyant p l u m e s until such time as the p l u m e touches
the
g r o u n d . F r o m t h i s t i m e o n it w i l l s p r e a d a s a h e a v y c l o u d , a t o p i c t o b e discussed in Chapter 13. W e will deal primarily w i t h the case of a b u o y a n t release, i s s u i n g from a tall s t a c k in the f o r m o f a j e t
and
later developing into a p u r e p l u m e aligned w i t h the crossflow. T h e effects
of w i n d
shear
indirectly through entrainment.
and
turbulence
will
be
considered
only
t h e e m p i r i c a l r e s u l t s q u o t e d for p l u m e r i s e
and
T h e results are p r e s u m e d applicable also to a m o v i n g
source (such as a ship b u r n i n g refuse on the high seas). T h e actual source speed will b e important only in the near
11.1 The
field.
T h e Rise and Dilution of Forced P l u m e s f r o m Tall physical
problem,
the
coordinate
system
and
Stacks velocity
components used in the analysis, are illustrated in Fig. 11-1. This curvilinear coordinate system, w i t h s = length coordinate along the centerline
and
6
=
centerline
slope
relative
to
the
horizontal
d i r e c t i o n , w a s first p r o p o s e d b y Prof. F a y a n d c o w o r k e r s i n a s e r i e s o f p a p e r s ( 1 9 6 9 , 1 9 7 0 , 1 9 7 1 ) . It h a s b e e n a d o p t e d l a t e r i n H e n d e r s o n Sellers' computerized m o d e l (1987) as well as b y others. The system w a s proposed in the h o p e that interesting solutions could b e found in t e r m s o f t h e c u r v i l i n e a r d i s t a n c e s, b y e x t e n s i o n s o f t h e i d e a s
and
c o n c e p t s u s e d for p l u m e s i n still air. T h e e x t e n d e d a n a l y s i s is v a l i d o n l y for tall s t a c k s for w h i c h the
ambient
is relatively c a l m
and
unaffected b y m e c h a n i c a l turbulence d u e to vegetation, buildings etc. A rule of thumb states that the stack should b e two-and-a-half times taller t h a n the highest n e a r - b y building. (One' s faith in the validity o f t h i s "rule" i s s o m e w h a t s h a k e n b y S c o r e r ' s ( 1 9 7 8 , p . 3 8 2 ) r e v e l a t i o n t h a t it i s b a s e d o n o b s e r v a t i o n s o f b a l l o o n t r a j e c t o r i e s o v e r a h i l l . ) It a p p e a r s t h a t w i n d s h e a r a n d a t m o s p h e r i c t u r b u l e n c e h a v e little effect o n t h e r i s e a n d d i l u t i o n o f f o r c e d p l u m e s r e l e a s e d f r o m stacks,
and
this m a k e s
it e a s i e r
to conduct
relevant
tall
laboratory
experiments*. In these experiments the crossflow is uniform, laminar
A negatively buoyant plume approaching the ground, will be more influenced by wind shear and turbulence than the rising plume, as found by Schatzman et al. (1993).
Jets and Plumes in Cross/low
275
Pa
U
2b
t
,w , (
p
(
H Stack
Fig.
and
11-1:
most
Coordinate system in crossjlow.
often
of constant
and
velocity
density.
components,
In order
buoyant
to c o m p a r e
jet
such
e x p e r i m e n t s w i t h field o b s e r v a t i o n , it i s i m p o r t a n t t o u s e t h e c o r r e c t (nondimensional) scaling parameters
(see d i s c u s s i o n b y F a y et al.,
1 9 7 0 ) . T h e p r i m a r y " u n k n o w n s " a r e t h e height (z ) c
a n d t h e p l u m e radius
of the p l u m e centerline
(b) a s f u n c t i o n o f distance
The relevant scaling parameters
will be the lengths
d o w n w i n d (x). characterizing
e i t h e r t h e flow i n t h e m o m e n t u m - d o m i n a t e d r e g i o n c l o s e t o t h e s t a c k or
in
the
buoyancy-dominated
regions
further
downwind.
Stratification, w h e n present, introduces an additional length scale as discussed in Chapter 2.
11.1.1
Flow
Parameters
The geometric lengths,
stack height and
exit diameter,
would
a p p e a r a t first t o b e l e n g t h s c a l e s o f r e l e v a n c e . E x p e r i e n c e i n d i c a t e s t h a t t h e s t a c k h e i g h t i s i m p o r t a n t o n l y w h e n it i s s m a l l r e l a t i v e t o s c a l e s c h a r a c t e r i z i n g t h e flow a r o u n d it. A s l o n g a s t h e s t a c k r e a c h e s above a certain m i n i m u m height, the stack height as such has
no
Chapter 11
276
further
significance. T h e external stack
characteristics
diameter defines the w a k e
(vortex shedding etc.), but w h e n the velocity of the
upward-directed jet is sufficiently high, n o interaction
(downwash)
b e t w e e n t h e j e t flow a n d t h e w a k e o c c u r s . W e w i l l a s s u m e t h i s t o b e the case. T h e internal stack diameter 2 b defines the initial crosst
section of the j e t a n d also the exit R e y n o l d s number; R e = w b/ v . W e t
t
will a s s u m e that this R e y n o l d s n u m b e r is h i g h a n d that the
f
flow
in
the j e t a n d p l u m e is turbulent. T h e R e y n o l d s n u m b e r h a s otherwise n o direct importance. T h e exit diameter is important zone-of-flow
establishment
a l s o for
of the jet, i.e. a region a s s u m e d
the
to b e
small in comparison w i t h the heights of interest. The
ratio
of the jet m o m e n t u m
to that
of the
free
stream
is
o b v i o u s l y i m p o r t a n t for t h e j e t d e f l e c t i o n . T h i s r a t i o i s g i v e n b y
Pjbj
w
2
where £ When £
m
2 m
2
c h a r a c t e r i z e s a c h o s e n c r o s s - s e c t i o n o f the free
stream.
is c h o s e n so that the ratio o f m o m e n t a is unity a n d p « p {
a
we
have i £ ^ b ~ m U w
(11-1)
1 t
This is the m o m e n t u m
length, the relevant scaling length in
momentum-dominated
region.
As
the
vertical
m o m e n t a acting o n the j e t are equal at the height z = /
and m
the
horizontal
, w e expect the
j e t slope at this height to b e a b o u t 4 5 ° . T h e velocity ratio i R = -77 w
is
often
used
as
an
independent
(11-2)
parameter,
in
correlations
of
penetration depths, so that m
= bR
(11-3)
t
A t large distances from the source, b u o y a n c y , rather t h a n initial momentum,
is the m o r e i m p o r t a n t
parameter.
p a r a m e t e r F, at the source, is defined b y
The buoyancy
flux
Jets and Plumes in Crossflow
Pa - Pi
2
jtF = Jtb t
277
wg
t
(11-4)
t
Pa
and can b e considered constant in the absence of stratification thermochemical processes. The second important
parameter
is
and the
c r o s s f l o w v e l o c i t y U. T h e t w o parameters F a n d U can b e c o m b i n e d to give a quantity of f
dimension length. F r o m the dimensional equation
=
FIT
w e find q = 1 a n d p = - 3 . T h e scaling length
is d e n o t e d t h e b u o y a n c y
length and is defined b y
i
F
b
For an
i d e a l g a s , p /p t
i
b
2
U
w
i9
Pa
= T /T ,
a
[Pa-Pi]
U
a
and
t
by
using
the
Boussinesq
approximation, w e find
Pa-Pi Pa
JI- a T T
a
For plumes from thermal power plants,
the heat
flux
Q
t
rather
t h a n F is g i v e n t
9t = * b
2 t
p w c [T T ) t
t
p
r
(11-6)
a
It f o l l o w s t h a t
b
3
p
A physical interpretation
of
(11-7)
*PiC T U a
i s g i v e n b y F a y e t a l . ( 1 9 7 0 ) ; it i s a
m e a s u r e o f the radius o f curvature o f the p l u m e trajectory near
the
e x i t for a p u r e p l u m e o f n e g l i g i b l e i n i t i a l m o m e n t u m . A
characteristic
nondimensional
flow
parameter
of interest,
addition to the R e y n o l d s n u m b e r , is the densimetric F r o u d e
in
number
Fd defined b y the square root of the ratio of inertia force to b u o y a n c y force, per unit v o l u m e in the p l u m e , i.e.
Chapter 11
278
Pi
i ' 9{Pa-Pi) i
O
F=
,
2
d
d
W
(11-8)
b
Just as the Reynolds number
serves to characterize the
flow
state
(laminar, turbulent), the Froude n u m b e r characterizes the nature of t h e b u o y a n t j e t o r p l u m e . T h e v a l u e s for F
d
F = oo d
(pure jet)
range from
to
F =0 d
W e can also express £ in terms of R and F . D
(pure plume)
B y combining (11-2),
d
(11-5) and (11-8) w e obtain b
/
t
d
3
=
—
(
H
-
9
)
F
I n c o r r e l a t i n g d a t a a n d i n o u r t h e o r e t i c a l w o r k , w e w i l l s e a r c h for scaling laws of the type
where z
c
is the height of the p l u m e centerline, x
distance and / represents £
m
or
the
downwind
d e p e n d i n g o n flow r e g i m e .
T h e p r e s e n c e o f s t r a t i f i c a t i o n , dps/dz
* 0 m a k e s the b u o y a n c y
flux
F a f u n c t i o n o f h e i g h t , b u t it a l s o i n t r o d u c e s a n e w l e n g t h . W e c o u l d define this length as the height over w h i c h p
e
changes b y a specified
f r a c t i o n . It i s m o r e u s u a l t o m a k e u s e o f t h e B r u n t - V a i s a l a f r e q u e n c y (see Chapter 2 ) defined b y Pe
dz
T h e r a t i o o f c r o s s f l o w v e l o c i t y U a n d f r e q u e n c y co r e p r e s e n t s a l e n g t h , U/co.
W h e n this length is m a d e n o n d i m e n s i o n a l w i t h the
important
buoyancy parameter
the result is the
second
stratification
p a r a m e t e r S, S =- ^ co / o This parameter
defines
the
range
(11-11)
of validity of many
empirical
correlations o f the form Eq. ( 1 1 - 1 0 ) . S i n c e w " is a m e a s u r e o f the 1
Jets and Plumes in Crossflow
279
t i m e for the p l u m e o s c i l l a t i o n i n a stratified e n v i r o n m e n t , S c a n b e c o n s i d e r e d t h e r a t i o o f t h e f l o w t i m e r e q u i r e d for a p l u m e e l e m e n t t o r e a c h i t s f i n a l h e i g h t ( o r a m p l i t u d e ) t o t h e f l o w t i m e r e q u i r e d for t h e p l u m e to b e n d in the direction of the crossflow. For m a n y p l u m e s of practical interest, S will b e m u c h greater than unity.
11.1.2
Flow
Equations
The analysis of curved (bent-over) forced plumes represents extension
of the
integral
method
(with
entrainment)
an
of Morton,
T a y l o r a n d T u r n e r ( 1 9 5 6 ) , d i s c u s s e d i n C h a p t e r 10. T h e p r e s s u r e d r a g is not included, a n d as the p l u m e cross-section (circular) as well as t h e f l o w p r o f i l e s ( t o p h a t ) a r e s p e c i f i e d , it c a n n o t a c c o u n t complicated phenomena
(kidney-shaped cross-section,
for
the
bifurcation)
o b s e r v e d for r e a l p l u m e s . W e will follow here the analysis of Hoult a n d Weil (1972), but only the
unstratified
case will b e considered. T h e key element in
the
analysis, is the entrainment a s s u m p t i o n . M a s s is entrained d u e to the velocity difference b e t w e e n the p l u m e flow a n d the external flow b o t h for t h e a x i a l c o m p o n e n t ( a n a l o g o u s t o t h e c l a s s i c a l c a s e ) a n d for t h e component
normal
entrainment
to
the
parameters
plume
centerline.
On
denoting
the
a (classical) and p (new), w e can write the
e q u a t i o n for c o n s e r v a t i o n o f m a s s a l o n g the p l u m e c e n t e r l i n e ( s e e Fig. 11-1) ^
[jtb
u) = 2 Jtb a\u
2
- [ / c o s 0\ + 2 Tib p\Usin
0\
(11-12)
In accord with the Boussinesq approximation, the density does not appear in this equation. T w o m o m e n t u m equations are required, one a l o n g t h e p l u m e t r a j e c t o r y a n d o n e n o r m a l t o it. A l o n g t h e t r a j e c t o r y we
find
that
controlled
the
by
rate
the
of change
force
of momentum
associated
with
the
in
rate
plume
is
of change
the
of
m o m e n t u m entrained a n d b y the b u o y a n c y force:
4- [u ds
2
v
Jtb ) 2
'
= Ucose4-(u ds
nb ) 2
v
In the normal direction, the curvature
J
+J t b
2
g ^ — P
sin 6
(11-13)
a
dO/ds
implies a
centripetal
force w h i c h enters into the d y n a m i c balance b e t w e e n rate o f c h a n g e of m o m e n t u m a n d b u o y a n c y force
Chapter 11
280
u
2
Jtb
^ = - U s i n o4- ( " nb ) + ; r b g ds ds '
2
2
W i t h the a s s u m p t i o n
of constant
cos 0
2
K
P
(11-14)
a
a m b i e n t d e n s i t y p , there is n o a
change in b u o y a n c y along the p l u m e trajectory
* [
u
*
b
9
- p r r °
(
i
l
-
(The general integral equations, w i t h o u t profile assumptions,
l
5
)
and
w i t h a l l o w a n c e for s t r a t i f i c a t i o n , a r e g i v e n b y H e w e t t et al., 1 9 7 1 . ) T h e relations b e t w e e n the variables o f p r i m a r y interest, i.e. x a n d z , and the variables s and 0 are given b y c
=j
x
cos dds
z
c
=j
sin 0 d s
It a p p e a r s t h a t , e v e n w i t h t h e s i m p l i f i c a t i o n i n t r o d u c e d ,
(ll-16a,b)
Equations
(11-12) through (11-16) can b e solved only b y numerical integration. Following Hoult a n d Weil, w e will consider the region close to the stack w h e r e 6
jt/2
a n d the flow has the character of a pure jet, see
Fig. 1 1 - 1 . W e c a n linearize the equations w i t h reference to the initial conditions u = w
if
p =p
if
b = b , t o o b t a i n a n e q u a t i o n for t h e r a t e o f t
c h a n g e o f 0. F r o m ( 1 1 - 1 4 ) ,
w
2 1
Jtb
2 1
^ ds
= -u4-{u ds
nb ) 2
v
7
(11-17)
The right-hand side can b e approximated using (11-12)
[jtb
2
u) = 2 jcb
t
{aw
t
+ p U)
(11-18)
O n i n t e g r a t i n g a n d u s i n g 0 = jt/2, w e o b t a i n f
jt (aR + 6\ s 0/ = 7 7 - 2 — = - H - r 2 \ R J £ 1
(11-19)
Jets and Plumes in Cross/low
w h e r e R = Wi/U a n d £
m
281
= b R. t
F r o m ( l l - 1 6 a , b ) w e find and
from w h i c h w e w r i t e the e q u a t i o n for the trajectory in the f o r m ( 1 1 10), (11-20)
T h i s e q u a t i o n i s o b v i o u s l y v a l i d o n l y w h e n t h e flow is d o m i n a t e d b y t h e fluxes o f m a s s a n d m o m e n t u m a t t h e e x i t . T h e v a l u e o f a i s w e l l e s t a b l i s h e d for b o t h j e t s a n d p l u m e s ( s e e 10) b u t the n e w p a r a m e t e r p m u s t b e a s s i g n e d a n u m e r i c a l
Chapter value.
Hoult and
Weil suggest
p = 0.6
laboratory (nonstratified) whereas the
field
from experiments in
the
data e x a m i n e d b y F a y et
al. ( 1 9 7 0 ) i n d i c a t e p = 0 . 8 1 . For the r e g i o n s farther f r o m the source, w h e r e the p l u m e is n e a r l y aligned w i t h the w i n d , the flow m o d e l o f C h u a n d Goldberg ( 1 9 7 4 ) is a d v a n t a g e o u s a s it g i v e s a u n i f i e d s o l u t i o n for t h e m o m e n t u m -
and
buoyancy-dominated
and
coworkers,
it
does
flow regimes. Unlike the not
introduce
between a rising bent plume and
model of Fay
new parameters. a line thermal
The
analogy
(Fig. 11-3),
first
noticed b y Scorer (1978), can b e incorporated in this model, as s h o w n in the next section.
11.1.3 Plume Model It i s a s s u m e d
of Chu and
Goldberg
here that the horizontal velocity component of the
p l u m e i s n e a r l y e q u a l to t h e c r o s s f l o w v e l o c i t y a n d t h a t t h e p r e s s u r e drag* is s m a l l in c o m p a r i s o n w i t h that associated w i t h The
first
assumption
entrainment.
is m o r e restrictive t h a n the s e c o n d , b u t
both
require that the r e g i o n n e a r the exit b e e x c l u d e d . T h e effects o f the complex processes near the source could b e taken into account
by
r e p l a c i n g t h e r e a l efflux b y a v i r t u a l s o u r c e s u i t a b l y l o c a t e d n e a r t h e
* A pressure drag, as for a bluff body, has been introduced empirically by some investigators to give agreement with experiments. The idea is controversial; for a discussion see Coelhoe and Hunt (1989).
Chapter 11
282
exit, b u t
the a u t h o r s give n o information o n h o w to estimate
the
location
of this virtual
the
source
w h i c h is
also
the
origin of
c o o r d i n a t e s y s t e m ( s e e F i g . 11-2). W e consider a forced p l u m e of density p
t
and velocity w
which
t
enters vertically into a crossflow of velocity U and density p . T h e a
initial m o m e n t u m a n d b u o y a n c y atmosphere of constant
fluxes
M and F are given. For an t
t
density a n d in the absence of losses,
b u o y a n c y flux t h r o u g h e a c h v e r t i c a l c r o s s - s e c t i o n A r e m a i n s
the
constant
(11-21)
w h e r e p i s t h e p l u m e d e n s i t y a n d u t h e p l u m e v e l o c i t y , h e r e u = U. W h e n pressure
drag in the direction n o r m a l to the p l u m e axis is
n e g l e c t e d , t h e c h a n g e i n v e r t i c a l m o m e n t u m o v e r a d i s t a n c e dx
must
equal the b u o y a n c y force acting o n a p l u m e element of this length
O n c o m b i n i n g t h i s e x p r e s s i o n w i t h (11-21), w e o b t a i n
U
Pa
Fig.
11-2:
Flow geometry
for
Chu and Goldberg's
model
Jets and Plumes in Crossflow
dx
I puwdA
283
= -r, 1/
I
(11-22)
and upon integration wdA
JA
Ft
= JJX + M
(11-23)
t
A t t h i s p o i n t t h e p l u m e p r o f i l e s f o r p , u, I U , a s w e l l a s t h e p l u m e geometry, m u s t b e specified. W e a s s u m e a circular cross-section of radius b, and uniform velocity and s e c t i o n , i . e . u = U, w = W a n d p = p
a
density over the p l u m e
cross-
(Boussinesq approximation).
W e define the radius b so that
puwdA
= Jtp b UW
(11-24)
2
a
From geometric considerations w e have
^ dx
= 77 U
(11-25)
The entrainment assumption can best be introduced b y making use of the line-plume analogy. ( W e depart here from C h u a n d As
s h o w n in Fig. 11-3
three
cross-sections
Goldberg.)
( p o s i t i o n s x \ , *2» * 3 )
projected onto a vertical plane, can b e seen as sections of a line t h e r m a l a t s u c c e s s i v e t i m e s T\, t2, £3 w h e r e T\ = X\/U
etc. T h e line
t h e r m a l h a s s e l f s i m i l a r c h a r a c t e r a n d g r o w s a s db = f} dz
c
t h e e n t r a i n m e n t r a t e . A s dz
c
w h e r e j8 i s
- (W/ U) dx w e o b t a i n
(11-26) dx which
is
the
entrainment
'
U
relation
1
proposed
also
by
Chu
and
Goldberg. O n c o m b i n i n g ( 1 1 - 2 5 ) a n d ( 1 1 - 2 6 ) a n d a s s u m i n g /J t o b e a c o n s t a n t b = /3z
c
O n substituting (11-24), (11-25) a n d (11-27) into (11-23), w e get
(11-27)
Chapter 11
284
dz
F
(
(11-28)
W e integrate to obtain the p l u m e trajectory 1/3
1/ 2
4F
["Pa" } 3
f
\
(
1/3
4M,
(11-29)
[*P U
+
2
a
D e n o t i n g t h e o b v i o u s s c a l i n g l e n g t h s £B a n d fa w e h a v e 1/3
1/3
(11-30) 4 ^
where /
2
_ ^ _ 4 ^ V ^ _ ^
2
(11-31)
Jets and Plumes til Cross/low
4F
285
_ Ajtb?g[p -p^W
(
a
_ = 4
t
Jtp U"
(11-32)
a
and where the n e w scaling lengths are compared with those defined in Section 11.1.1. T h e p l u m e r a d i u s is g i v e n b y
b =
3 0
1/3
1 2
2
V
2 +
/
M
1
1/3
(11-33) *
B a s e d o n these results, C h u and Goldberg also define a
dilution
ratio Sd Pa~ Pi
(11-34)
Pa'P w h e r e the local "averaged" density difference is defined b y
f g(p -p)udA A
= gft~~p-)jib U
D e n o t i n g a s b e f o r e R = Wi/U, 4n/R = 4f/2R,
R
(36
= 4
=F
(11-35)
= (b/bj) /R
and, with b =
2
a
w e obtain
2\l/3
f)
R
2
MVM
£
M
t
2/3
t_X_
1 2
i
(11-36)
M
A distinct advantage o f the Chu-Goldberg results, is that they are valid over the w h o l e range of interest; the m o m e n t u m - d o m i n a t e d well
as
the
buoyancy-dominated
transition distance x
t r
regimes.
W e can
determine
w h e n the contributions from b u o y a n c y
as a and
m o m e n t u m are equal, from Eq. (11 -30), 2
x ^ - f for w h i c h
V B
(11-37)
Chapter 11
286
M
(11-38)
F o r x < Xfr w e c a n a p p r o x i m a t e 1/3 /
\l/3
v
(11-39) a n d for
x> x
tr
x I \HZf 3
Z„
5
For the entrainment parameter
(11-40)
v\2/3
C h u a n d G o l d b e r g s u g g e s t )S = 0.5
b a s e d o n their o w n e x p e r i m e n t s . E q u a t i o n s (11 -39) a n d (11 -40) are in a g r e e m e n t w i t h k n o w n c o r r e l a t i o n s f o r t h e n e a r f i e l d a n d far
field
respectively. There exist a large n u m b e r
of such correlations, based on
field
t e s t s a n d e x p e r i m e n t s . F a y e t al. ( 1 9 7 0 ) s u g g e s t \2/3
when
^ = 1 . 3 2 ' *
s
= 2.27
2/3,
l4
S
i< 1.55
(11-41)
i> 1.55
when
(11-42)
'ult
b
T h e s e c o r r e l a t i o n s i n c l u d e t h e e f f e c t s o f s t r a t i f i c a t i o n ; t h e i n d e x "ult" indicates
ultimate
plume
include wind shear and
rise.
Further
atmospheric
discussions,
turbulence,
which
C s a n a d y ( 1 9 7 3 ) w h o , w i t h S l a w s o n , first s h o w e d t h a t t h e x consistent with entrainment Prediction penetrating
methods
for
also
will be found 2 / 3
in
- l a w is
theory. ultimate
plume
rise
and
plumes
inversion layers, etc., are m a i n l y empirical, b a s e d
on
o b s e r v a t i o n s from tall s t a c k s a n d
high buoyancy fluxes. Attempts
have
in
been
advances
made in
our
to
incorporate
understanding
the
correlations
of atmospheric
the
recent
boundary-layer
physics, as discussed in Chapter 9. Weil (1985) suggests t w o models b y Briggs
(1984),
one
for
strong
convection
and
convection/strong winds. These models require
one
for
weak
knowledge of
the
Jets and Plumes in Crossflow
287
d e p t h o f t h e a t m o s p h e r i c b o u n d a r y l a y e r h, o f t h e v e l o c i t y s c a l e s for convection
a n d g r o u n d s h e a r u*. T h e y a r e t h e r e f o r e n o t s o e a s i l y
applied as the rise m o d e l s presented herein.
11.2
Negatively Buoyant Plumes from Tall
Stacks
N o s i m p l e a n a l y t i c m o d e l is k n o w n for the c a s e w h e n F < 0. It t
should b e possible to extend the m o d e l of C h u a n d Goldberg to include this case.
One would have
to consider
changes sign and that the entrainment modified
that the
vertical velocity
equation (11-26) should
be
to
The m a x i m u m height is o b t a i n e d from ( 1 1 - 2 8 ) , MU ='~iFf
Mz cMAX
V
From
this
height
the
(11-43)
F
,z
t
(negative)
plume
rise
can
be
calculated
c o n s i d e r i n g t h e effect o f t h e ( n e g a t i v e ) b u o y a n c y o n l y . B y c a l c u l a t i n g z
c
from (11-29) with x from (11-43), w e can estimate the highest point
of the plume, MAX
=
Z
H
+
cMAX
(11-44)
z
w h e r e H is the stack height. On substituting z
c
=z ^ x
i n t o (11 - 4 0 ) w e c a n e s t i m a t e t h e d i s t a n c e
Ax r e q u i r e d f o r t h e p l u m e t o fall, u n d e r i t s n e g a t i v e b u o y a n c y , h e i g h t ZMAX-
the
T h e point w h e r e the p l u m e touches the g r o u n d is located
at a d o w n w i n d d i s t a n c e f r o m t h e s t a c k e q u a l t o ( s e e F i g . 11 - 4 ) x
=(xL C
V
+Ax
(11-45)
cMAX
JZ
This ad hoc approach does not take into account the early history of the p l u m e . T h i s c o u l d b e rectified s o m e w h a t b y i n t r o d u c i n g
a
v i r t u a l s o u r c e w i t h h o r i z o n t a l m o m e n t u m (jet flow) u p s t r e a m o f t h e stack
at
height
z
M
A
X
-
The
location
and
initial
(streamwise)
m o m e n t u m at the virtual source, should b e c h o s e n to m a t c h the real p l u m e at x - x
Z c M A X
. Scorer h a s p r o p o s e d a different m o d e l (Fig. 11-5)
Chapter 11
288
Fig.
11-4:
Geometry negative
for plume buoyancy.
with
w i t h a virtual source at height z - 2 z
positive
c
M
A
X
-
T
initial
h
velocity
and
plume width
e
thereby b e matched, b u t the n e w p l u m e from the higher source
can has
m u c h too high d o w n w a r d velocity. The
virtual
specification
source
of an
with
additional
an
initial
entrainment
jet
flow
requires
parameter,
the
associated
with the axial velocity component. A l t h o u g h the entrainment
near
the high point of the trajectory will b e m a i n l y due to crossflow, an axial component will also be present,
for t a l l s t a c k s ,
as the
plume
falls t o w a r d s t h e g r o u n d . N u m e r i c a l m o d e l s for d e n s e p l u m e s h a v e b e e n d e v e l o p e d b y O o m s e t al. ( 1 9 7 4 ) , b y L i e t a l . ( 1 9 8 6 ) a n d b y H e n d e r s o n - S e l l e r s ( 1 9 8 7 ) . T h e m o s t comprehensive study of negatively b u o y a n t p l u m e s is that of S c h a t z m a n et al. ( 1 9 9 3 ) . T h i s s t u d y includes a c o m p l e t e d i m e n s i o n a l (similarity) analysis o f the p r o b l e m w h e r e all relevant are
included.
The
important
nondimensional
parameters parameters
(densimetric Froude number, velocity ratio, source height, discharge angle etc.) w e r e v a r i e d systematically in a series of w i n d experiments Substantial
including
turbulent
as
well
as
laminar
differences were observed between
the
laminar
turbulent cases w i t h regard to m a x i m u m p l u m e height, distance
and
concentration
level at
these
points
tunnel
crossflow. and
touchdown
of interest.
The
differences have practical implications as current regulatory m o d e l s
Jets and Plumes in Crossflow
Virtual origin of
Fig.
11-5:
289
plume
with negative
buoyancy
Model proposed plumes.
by Scorer
(1978) for
negatively
are b a s e d o n the laminar crossflow data of H o o t et al. ( 1 9 7 3 ) .
Special
Nomenclature
b
r a d i u s (jet o r p l u m e )
F
buoyancy
H
stack height
flux
buoyancy length, i s - ^ ^ b m o m e n t u m l e n g t h , £M - 2 l M
momentum
9
heat
m
flux
flux
R
velocity ratio, R =
s
curvilinear coordinate aligned with plume axis
w /U t
S
stratification parameter, Eq. ( 1 1 - 1 1 )
s
dilution ratio, Eq. (11-34)
U
d
velocity ( c o m p o n e n t in s-direction)
U
wind velocity
w
velocity ( z - c o m p o n e n t )
W
rise velocity (plume)
buoyant
Chapter 11
290
x
downwind distance
Xfr
point of transition, m o m e n t u m - to b u o y a n c y - d o m i n a t e d regime
z
vertical coordinate
z
height of plume centerline
a, /3
entrainment coefficients
6
slope of plume centerline
c
Indices a
ambient (atmospheric)
i
initial (stack exit)
REFERENCES B r i g g s , G . A . ( 1 9 8 4 ) P l u m e r i s e a n d b u o y a n c y effects. I n Science
and
Power
Prediction,
Chapter
Atmospheric
8, p p 3 1 7 - 3 6 6
(Ed D.
Randerson). U.S. Department of Energy, D O E / T I C - 2 7 6 0 1 . Chu, V . H . and Goldberg, M . B . (1974) B u o y a n t forced p l u m e s in cross flow. A.S.CE.
J. Hydraulics
Div. 100, N o H Y 9 .
Coelhoe, S.L.V. a n d Hunt, J.C.R. (1989) T h e dynamics of the near o f s t r o n g j e t s i n c r o s s f l o w s . J. Fluid C s a n a d y , G . T . ( 1 9 7 3 ) Turbulent
field
Mech. 2 0 0 , p p 9 5 - 1 2 0 .
Diffusion
in the Environment.
D . Reidel
Publ. Co. E s k i n a z i , S. ( 1 9 7 5 ) Fluid Environment
Mechanics
and
Fay, J.A., Escudier,
of
our
M . , H o u l t , D . P . ( 1 9 7 0 ) A correlation o f field
o b s e r v a t i o n s o f p l u m e r i s e . APCA
Journal
H e n d e r s o n - S e l l e r s , B . ( 1 9 8 7 ) Modeling The
Thermodynamics
Academic Press.
University
of
Salford
2 0 , N o 6, p p 3 9 1 - 3 9 7 .
of Plume
Model:
U.S.P.R.,
Rise
and
Dispersion
Lecture
Notes
in
Engineering, Springer Verlag. Hewett, T.A., Fay, J.A. and Hoult, D.P. (1971) Laboratory experiments o f s m o k e s t a c k p l u m e s i n a s t a b l e a t m o s p h e r e . Atm.
Environment
5,
pp 767-789. Hoot, T . G . , Meroney, R . N . and Peterka, J.A. (1973) W i n d tunnel tests of negatively
buoyant
Environmental 27711, USA.
plumes.
Report
Protection Agency,
No
EPA-65013-74-003.
Research Triangle Park,
NC
Jets and Plumes in Crossflow
291
Hoult, D.P., Fay, J.A., Farney, L.J. (1969) A theory of plume c o m p a r e d w i t h field o b s e r v a t i o n s . A P C A Journal
rise
19, N o 8, p p 5 8 5 -
590. Hoult, D.P. and Weil, J.C. (1972) Turbulent p l u m e in a laminar flow. Atm. Environment
cross
6, p p 5 1 3 - 5 9 0 .
Li, X . - Y . , L e j d e n s , H . a n d O o m s , G . ( 1 9 8 6 ) A n e x p e r i m e n t a l v e r i f i c a t i o n o f a t h e o r e t i c a l m o d e l for t h e d i s p e r s i o n o f a s t a c k p l u m e h e a v i e r t h a n air. Atm. Environment Morton,
B.R., Taylor,
gravitational
20, N o 6, p p
G.I. and
convection
from
1087-1094.
Turner,
J.S.
maintained
s o u r c e s . Proc. Roy. Soc. A 2 3 4 , p p
(1956) and
Turbulent
instantaneous
1-23.
O o m s , G., Mahieu, A . P . and Zelio, F. (1974) T h e p l u m e path of vent g a s e s h e a v i e r t h a n air. First Safety
Promotion
Schatzman,
Int. Symposium
in the Process
Industries.
on Loss
Prevention
and
The Hague.
M . , Snyder, W . H . and Lawson, R.E. (1993)
Experiments
with heavy gas jets in laminar and turbulent cross flows. A c c e p t e d for p u b l i c a t i o n , Atm.
Environment.
S c h e t z , J . A . ( 1 9 8 0 ) I n j e c t i o n a n d m i x i n g i n t u r b u l e n t f l o w . Progress Astronautics
and Aeronautics
S c o r e r , R . S . ( 1 9 7 8 ) Environmental
Aerodynamics.
Ellis H o r w o o d Ltd.
W e i l , J . C . ( 1 9 8 5 ) U p d a t i n g a p p l i e d d i f f u s i o n m o d e l s . J. Clim. Appl. 24, p p 1 1 1 1 - 1 1 3 0 .
in
68. Publ. ALAA, N e w Y o r k .
Met.
Chapter 11
292
PROBLEMS
P r o b l e m 1. A ship burning chemical waste products, travels at 10 knots (5 m/s) on the high seas under "no wind" conditions. It produces hot emission products (T = 250°C) at a rate of 5 m / s and an initial velocity of 10 m / s from a stack of 30 m height. Determine the rate of dilution of the resulting plume under the assumption of ideal gas and no chemical reactions. 3
g
P r o b l e m 2. Consider the same emission source as in Problem 1, but in this case the source is stationary in a wind of 5 m/s. (Neutral conditions m a y be assumed.) Consider next the moving source (10 knots) heading into a wind of equal strength. H o w would you solve this problem? Compare the answers for the three different conditions considered. P r o b l e m 3 . In the sketch an experiment is illustrated where a moving source in a stratified medium is used to simulate a rising plume in the atmosphere. Thefluidsare salt /water solutions and the concentration in the receiver varies as indicated. The heavy (source) flow is colored for visualization. Determine the relevant nondimensional parameters for this experiment. Find the analytic solution for the plume trajectory in terms of the flow parameters shown. W h y is this experiment not a realistic simulation (irrespective of scale) for a real (gaseous) plume rising in the atmosphere? u Pi
p(z)-
P r o b l e m 4. A tall stack (150 m ) emits hot exhaust (200°C) at a rate of 10 m / s and with initial velocity 15 m / s into an isothermal atmosphere of windspeed 7 m/s. Determine the plume trajectory 3
Jets and Plumes In Crossflow
293
b a s e d o n t h e m o d e l o f C h u a n d G o l d b e r g a s w e l l a s t h a t o f F a y et al., a n d c o m p a r e the results. T h e effluent c a n b e c o n s i d e r e d to h a v e the p r o p e r t i e s o f ( h o t ) air.
C H A P T E R 11 JETS AND PLUMES IN CROSSFLOW
W h e n a j e t or forced p l u m e is injected transverse stream,
a
very
complicated three-dimensional
to a
flowfield
flowing results,
(Coelhoe a n d Hunt, 1989). T h e external stream is deflected a r o u n d
the
j e t creating a p r e s s u r e field w h i c h forces the j e t to b e n d
the
in
s t r e a m w i s e d i r e c t i o n . T h e m i x i n g o f t h e t w o fluids a n d t h e e x c h a n g e o f m o m e n t a a l s o r e s u l t i n flow d e f l e c t i o n , o u t w a r d for t h e fluid
a n d s t r e a m w i s e for t h e j e t
fluid.
external
In technical applications,
the
penetration depth is the m e a s u r e of the lateral distance from the j e t e x i t t o t h e m a x i m u m r e a c h o f t h e j e t fluid. S u c h a m a x i m u m ( i n a n a s y m p t o t i c s e n s e ) c a n b e d e f i n e d o n l y for a n o n b u o y a n t j e t . W i t h b u o y a n c y t h e j e t fluid w i l l c o n t i n u e t o r i s e , a l t h o u g h i n t h e f o r m o f a bent plume, while b e i n g carried d o w n s t r e a m with the velocity of the main
flow.
For this late bent-over stage, simple similarity solutions
exist w h i c h c a n b e u s e d to estimate p l u m e rise a n d rate of dilution. M a n y practical
problems
concerned with safety and
pollution
i n v o l v e j e t s or forced p l u m e s . T h e n e a r field, w i t h its c o m p l i c a t e d three-dimensional
flow
structure, will d e p e n d on the ratio of jet to
f r e e - s t r e a m m o m e n t u m . W h e n t h e fluids h a v e n e a r e q u a l d e n s i t y , t h e i m p o r t a n t flow p a r a m e t e r i s s i m p l y t h e v e l o c i t y r a t i o , j e t - e x i t t o f r e e s t r e a m v a l u e . A large b o d y o f e x p e r i m e n t a l i n f o r m a t i o n exists, for m o r e d e t a i l s s e e S c h e t z ( 1 9 8 0 ) a n d H e n d e r s o n - S e l l e r s ( 1 9 8 7 ) . It i s a c o m m o n experience that the initially round jet, develops a kidneyshaped
cross-section
counter rotating phenomenon
can
vortices
and
eventually
aligned
with
bifurcates
the
b e explained in terms
main
into
stream.
two This
of vortex dynamics,
see
Eskinazi (1975). For the b u o y a n t jet, the ratio o f inertia force to b u o y a n c y force (i.e. the densimetric F r o u d e n u m b e r ) , b e c o m e s a s e c o n d parameter and, at large distances from the exit, the m o r e important one. Negative well
as
positive buoyancy
involve most releases
are
is
of interest.
P r o b l e m s in
as
pollution
often positive b u o y a n c y w h e r e a s negatively b u o y a n t of frequent
interest
in safety w o r k . A s the
buoyant
r e l e a s e , far f r o m t h e e x i t , h a s a d e n s i t y n e a r l y e q u a l t o t h a t o f t h e ambient, d u e to entrainment,
the further
spread will b e passive, i.e.
driven b y the external turbulence. T h e methods discussed in Chapter 9 are then applicable. I n w h a t f o l l o w s w e w i l l first d i s c u s s t h e p a r a m e t e r s w h i c h c o n t r o l
274
Chapter 11
t h e flow i n t h e n e a r and the external
field,
flow
w h e r e the angle b e t w e e n the buoyant jet
i s l a r g e . W e w i l l t h e n c o n s i d e r t h e far
field,
w h e r e m o m e n t u m and b u o y a n c y both are important, but w h e r e the a n g l e b e t w e e n t h e t w o s t r e a m s i s s m a l l . H e r e it i s p o s s i b l e t o o b t a i n simple similarity solutions. T h e s e solutions will b e applicable also to negatively buoyant p l u m e s until such time as the p l u m e touches
the
g r o u n d . F r o m t h i s t i m e o n it w i l l s p r e a d a s a h e a v y c l o u d , a t o p i c t o b e discussed in Chapter 13. W e will deal primarily w i t h the case of a b u o y a n t release, i s s u i n g from a tall s t a c k in the f o r m o f a j e t
and
later developing into a p u r e p l u m e aligned w i t h the crossflow. T h e effects
of w i n d
shear
indirectly through entrainment.
and
turbulence
will
be
considered
only
t h e e m p i r i c a l r e s u l t s q u o t e d for p l u m e r i s e
and
T h e results are p r e s u m e d applicable also to a m o v i n g
source (such as a ship b u r n i n g refuse on the high seas). T h e actual source speed will b e important only in the near
11.1 The
field.
T h e Rise and Dilution of Forced P l u m e s f r o m Tall physical
problem,
the
coordinate
system
and
Stacks velocity
components used in the analysis, are illustrated in Fig. 11-1. This curvilinear coordinate system, w i t h s = length coordinate along the centerline
and
6
=
centerline
slope
relative
to
the
horizontal
d i r e c t i o n , w a s first p r o p o s e d b y Prof. F a y a n d c o w o r k e r s i n a s e r i e s o f p a p e r s ( 1 9 6 9 , 1 9 7 0 , 1 9 7 1 ) . It h a s b e e n a d o p t e d l a t e r i n H e n d e r s o n Sellers' computerized m o d e l (1987) as well as b y others. The system w a s proposed in the h o p e that interesting solutions could b e found in t e r m s o f t h e c u r v i l i n e a r d i s t a n c e s, b y e x t e n s i o n s o f t h e i d e a s
and
c o n c e p t s u s e d for p l u m e s i n still air. T h e e x t e n d e d a n a l y s i s is v a l i d o n l y for tall s t a c k s for w h i c h the
ambient
is relatively c a l m
and
unaffected b y m e c h a n i c a l turbulence d u e to vegetation, buildings etc. A rule of thumb states that the stack should b e two-and-a-half times taller t h a n the highest n e a r - b y building. (One' s faith in the validity o f t h i s "rule" i s s o m e w h a t s h a k e n b y S c o r e r ' s ( 1 9 7 8 , p . 3 8 2 ) r e v e l a t i o n t h a t it i s b a s e d o n o b s e r v a t i o n s o f b a l l o o n t r a j e c t o r i e s o v e r a h i l l . ) It a p p e a r s t h a t w i n d s h e a r a n d a t m o s p h e r i c t u r b u l e n c e h a v e little effect o n t h e r i s e a n d d i l u t i o n o f f o r c e d p l u m e s r e l e a s e d f r o m stacks,
and
this m a k e s
it e a s i e r
to conduct
relevant
tall
laboratory
experiments*. In these experiments the crossflow is uniform, laminar
A negatively buoyant plume approaching the ground, will be more influenced by wind shear and turbulence than the rising plume, as found by Schatzman et al. (1993).
Jets and Plumes in Cross/low
275
Pa
U
2b
t
,w , (
p
(
H Stack
Fig.
and
11-1:
most
Coordinate system in crossjlow.
often
of constant
and
velocity
density.
components,
In order
buoyant
to c o m p a r e
jet
such
e x p e r i m e n t s w i t h field o b s e r v a t i o n , it i s i m p o r t a n t t o u s e t h e c o r r e c t (nondimensional) scaling parameters
(see d i s c u s s i o n b y F a y et al.,
1 9 7 0 ) . T h e p r i m a r y " u n k n o w n s " a r e t h e height (z ) c
a n d t h e p l u m e radius
of the p l u m e centerline
(b) a s f u n c t i o n o f distance
The relevant scaling parameters
will be the lengths
d o w n w i n d (x). characterizing
e i t h e r t h e flow i n t h e m o m e n t u m - d o m i n a t e d r e g i o n c l o s e t o t h e s t a c k or
in
the
buoyancy-dominated
regions
further
downwind.
Stratification, w h e n present, introduces an additional length scale as discussed in Chapter 2.
11.1.1
Flow
Parameters
The geometric lengths,
stack height and
exit diameter,
would
a p p e a r a t first t o b e l e n g t h s c a l e s o f r e l e v a n c e . E x p e r i e n c e i n d i c a t e s t h a t t h e s t a c k h e i g h t i s i m p o r t a n t o n l y w h e n it i s s m a l l r e l a t i v e t o s c a l e s c h a r a c t e r i z i n g t h e flow a r o u n d it. A s l o n g a s t h e s t a c k r e a c h e s above a certain m i n i m u m height, the stack height as such has
no
Chapter 11
276
further
significance. T h e external stack
characteristics
diameter defines the w a k e
(vortex shedding etc.), but w h e n the velocity of the
upward-directed jet is sufficiently high, n o interaction
(downwash)
b e t w e e n t h e j e t flow a n d t h e w a k e o c c u r s . W e w i l l a s s u m e t h i s t o b e the case. T h e internal stack diameter 2 b defines the initial crosst
section of the j e t a n d also the exit R e y n o l d s number; R e = w b/ v . W e t
t
will a s s u m e that this R e y n o l d s n u m b e r is h i g h a n d that the
f
flow
in
the j e t a n d p l u m e is turbulent. T h e R e y n o l d s n u m b e r h a s otherwise n o direct importance. T h e exit diameter is important zone-of-flow
establishment
a l s o for
of the jet, i.e. a region a s s u m e d
the
to b e
small in comparison w i t h the heights of interest. The
ratio
of the jet m o m e n t u m
to that
of the
free
stream
is
o b v i o u s l y i m p o r t a n t for t h e j e t d e f l e c t i o n . T h i s r a t i o i s g i v e n b y
Pjbj
w
2
where £ When £
m
2 m
2
c h a r a c t e r i z e s a c h o s e n c r o s s - s e c t i o n o f the free
stream.
is c h o s e n so that the ratio o f m o m e n t a is unity a n d p « p {
a
we
have i £ ^ b ~ m U w
(11-1)
1 t
This is the m o m e n t u m
length, the relevant scaling length in
momentum-dominated
region.
As
the
vertical
m o m e n t a acting o n the j e t are equal at the height z = /
and m
the
horizontal
, w e expect the
j e t slope at this height to b e a b o u t 4 5 ° . T h e velocity ratio i R = -77 w
is
often
used
as
an
independent
(11-2)
parameter,
in
correlations
of
penetration depths, so that m
= bR
(11-3)
t
A t large distances from the source, b u o y a n c y , rather t h a n initial momentum,
is the m o r e i m p o r t a n t
parameter.
p a r a m e t e r F, at the source, is defined b y
The buoyancy
flux
Jets and Plumes in Crossflow
Pa - Pi
2
jtF = Jtb t
277
wg
t
(11-4)
t
Pa
and can b e considered constant in the absence of stratification thermochemical processes. The second important
parameter
is
and the
c r o s s f l o w v e l o c i t y U. T h e t w o parameters F a n d U can b e c o m b i n e d to give a quantity of f
dimension length. F r o m the dimensional equation
=
FIT
w e find q = 1 a n d p = - 3 . T h e scaling length
is d e n o t e d t h e b u o y a n c y
length and is defined b y
i
F
b
For an
i d e a l g a s , p /p t
i
b
2
U
w
i9
Pa
= T /T ,
a
[Pa-Pi]
U
a
and
t
by
using
the
Boussinesq
approximation, w e find
Pa-Pi Pa
JI- a T T
a
For plumes from thermal power plants,
the heat
flux
Q
t
rather
t h a n F is g i v e n t
9t = * b
2 t
p w c [T T ) t
t
p
r
(11-6)
a
It f o l l o w s t h a t
b
3
p
A physical interpretation
of
(11-7)
*PiC T U a
i s g i v e n b y F a y e t a l . ( 1 9 7 0 ) ; it i s a
m e a s u r e o f the radius o f curvature o f the p l u m e trajectory near
the
e x i t for a p u r e p l u m e o f n e g l i g i b l e i n i t i a l m o m e n t u m . A
characteristic
nondimensional
flow
parameter
of interest,
addition to the R e y n o l d s n u m b e r , is the densimetric F r o u d e
in
number
Fd defined b y the square root of the ratio of inertia force to b u o y a n c y force, per unit v o l u m e in the p l u m e , i.e.
Chapter 11
278
Pi
i ' 9{Pa-Pi) i
O
F=
,
2
d
d
W
(11-8)
b
Just as the Reynolds number
serves to characterize the
flow
state
(laminar, turbulent), the Froude n u m b e r characterizes the nature of t h e b u o y a n t j e t o r p l u m e . T h e v a l u e s for F
d
F = oo d
(pure jet)
range from
to
F =0 d
W e can also express £ in terms of R and F . D
(pure plume)
B y combining (11-2),
d
(11-5) and (11-8) w e obtain b
/
t
d
3
=
—
(
H
-
9
)
F
I n c o r r e l a t i n g d a t a a n d i n o u r t h e o r e t i c a l w o r k , w e w i l l s e a r c h for scaling laws of the type
where z
c
is the height of the p l u m e centerline, x
distance and / represents £
m
or
the
downwind
d e p e n d i n g o n flow r e g i m e .
T h e p r e s e n c e o f s t r a t i f i c a t i o n , dps/dz
* 0 m a k e s the b u o y a n c y
flux
F a f u n c t i o n o f h e i g h t , b u t it a l s o i n t r o d u c e s a n e w l e n g t h . W e c o u l d define this length as the height over w h i c h p
e
changes b y a specified
f r a c t i o n . It i s m o r e u s u a l t o m a k e u s e o f t h e B r u n t - V a i s a l a f r e q u e n c y (see Chapter 2 ) defined b y Pe
dz
T h e r a t i o o f c r o s s f l o w v e l o c i t y U a n d f r e q u e n c y co r e p r e s e n t s a l e n g t h , U/co.
W h e n this length is m a d e n o n d i m e n s i o n a l w i t h the
important
buoyancy parameter
the result is the
second
stratification
p a r a m e t e r S, S =- ^ co / o This parameter
defines
the
range
(11-11)
of validity of many
empirical
correlations o f the form Eq. ( 1 1 - 1 0 ) . S i n c e w " is a m e a s u r e o f the 1
Jets and Plumes in Crossflow
279
t i m e for the p l u m e o s c i l l a t i o n i n a stratified e n v i r o n m e n t , S c a n b e c o n s i d e r e d t h e r a t i o o f t h e f l o w t i m e r e q u i r e d for a p l u m e e l e m e n t t o r e a c h i t s f i n a l h e i g h t ( o r a m p l i t u d e ) t o t h e f l o w t i m e r e q u i r e d for t h e p l u m e to b e n d in the direction of the crossflow. For m a n y p l u m e s of practical interest, S will b e m u c h greater than unity.
11.1.2
Flow
Equations
The analysis of curved (bent-over) forced plumes represents extension
of the
integral
method
(with
entrainment)
an
of Morton,
T a y l o r a n d T u r n e r ( 1 9 5 6 ) , d i s c u s s e d i n C h a p t e r 10. T h e p r e s s u r e d r a g is not included, a n d as the p l u m e cross-section (circular) as well as t h e f l o w p r o f i l e s ( t o p h a t ) a r e s p e c i f i e d , it c a n n o t a c c o u n t complicated phenomena
(kidney-shaped cross-section,
for
the
bifurcation)
o b s e r v e d for r e a l p l u m e s . W e will follow here the analysis of Hoult a n d Weil (1972), but only the
unstratified
case will b e considered. T h e key element in
the
analysis, is the entrainment a s s u m p t i o n . M a s s is entrained d u e to the velocity difference b e t w e e n the p l u m e flow a n d the external flow b o t h for t h e a x i a l c o m p o n e n t ( a n a l o g o u s t o t h e c l a s s i c a l c a s e ) a n d for t h e component
normal
entrainment
to
the
parameters
plume
centerline.
On
denoting
the
a (classical) and p (new), w e can write the
e q u a t i o n for c o n s e r v a t i o n o f m a s s a l o n g the p l u m e c e n t e r l i n e ( s e e Fig. 11-1) ^
[jtb
u) = 2 Jtb a\u
2
- [ / c o s 0\ + 2 Tib p\Usin
0\
(11-12)
In accord with the Boussinesq approximation, the density does not appear in this equation. T w o m o m e n t u m equations are required, one a l o n g t h e p l u m e t r a j e c t o r y a n d o n e n o r m a l t o it. A l o n g t h e t r a j e c t o r y we
find
that
controlled
the
by
rate
the
of change
force
of momentum
associated
with
the
in
rate
plume
is
of change
the
of
m o m e n t u m entrained a n d b y the b u o y a n c y force:
4- [u ds
2
v
Jtb ) 2
'
= Ucose4-(u ds
nb ) 2
v
In the normal direction, the curvature
J
+J t b
2
g ^ — P
sin 6
(11-13)
a
dO/ds
implies a
centripetal
force w h i c h enters into the d y n a m i c balance b e t w e e n rate o f c h a n g e of m o m e n t u m a n d b u o y a n c y force
Chapter 11
280
u
2
Jtb
^ = - U s i n o4- ( " nb ) + ; r b g ds ds '
2
2
W i t h the a s s u m p t i o n
of constant
cos 0
2
K
P
(11-14)
a
a m b i e n t d e n s i t y p , there is n o a
change in b u o y a n c y along the p l u m e trajectory
* [
u
*
b
9
- p r r °
(
i
l
-
(The general integral equations, w i t h o u t profile assumptions,
l
5
)
and
w i t h a l l o w a n c e for s t r a t i f i c a t i o n , a r e g i v e n b y H e w e t t et al., 1 9 7 1 . ) T h e relations b e t w e e n the variables o f p r i m a r y interest, i.e. x a n d z , and the variables s and 0 are given b y c
=j
x
cos dds
z
c
=j
sin 0 d s
It a p p e a r s t h a t , e v e n w i t h t h e s i m p l i f i c a t i o n i n t r o d u c e d ,
(ll-16a,b)
Equations
(11-12) through (11-16) can b e solved only b y numerical integration. Following Hoult a n d Weil, w e will consider the region close to the stack w h e r e 6
jt/2
a n d the flow has the character of a pure jet, see
Fig. 1 1 - 1 . W e c a n linearize the equations w i t h reference to the initial conditions u = w
if
p =p
if
b = b , t o o b t a i n a n e q u a t i o n for t h e r a t e o f t
c h a n g e o f 0. F r o m ( 1 1 - 1 4 ) ,
w
2 1
Jtb
2 1
^ ds
= -u4-{u ds
nb ) 2
v
7
(11-17)
The right-hand side can b e approximated using (11-12)
[jtb
2
u) = 2 jcb
t
{aw
t
+ p U)
(11-18)
O n i n t e g r a t i n g a n d u s i n g 0 = jt/2, w e o b t a i n f
jt (aR + 6\ s 0/ = 7 7 - 2 — = - H - r 2 \ R J £ 1
(11-19)
Jets and Plumes in Cross/low
w h e r e R = Wi/U a n d £
m
281
= b R. t
F r o m ( l l - 1 6 a , b ) w e find and
from w h i c h w e w r i t e the e q u a t i o n for the trajectory in the f o r m ( 1 1 10), (11-20)
T h i s e q u a t i o n i s o b v i o u s l y v a l i d o n l y w h e n t h e flow is d o m i n a t e d b y t h e fluxes o f m a s s a n d m o m e n t u m a t t h e e x i t . T h e v a l u e o f a i s w e l l e s t a b l i s h e d for b o t h j e t s a n d p l u m e s ( s e e 10) b u t the n e w p a r a m e t e r p m u s t b e a s s i g n e d a n u m e r i c a l
Chapter value.
Hoult and
Weil suggest
p = 0.6
laboratory (nonstratified) whereas the
field
from experiments in
the
data e x a m i n e d b y F a y et
al. ( 1 9 7 0 ) i n d i c a t e p = 0 . 8 1 . For the r e g i o n s farther f r o m the source, w h e r e the p l u m e is n e a r l y aligned w i t h the w i n d , the flow m o d e l o f C h u a n d Goldberg ( 1 9 7 4 ) is a d v a n t a g e o u s a s it g i v e s a u n i f i e d s o l u t i o n for t h e m o m e n t u m -
and
buoyancy-dominated
and
coworkers,
it
does
flow regimes. Unlike the not
introduce
between a rising bent plume and
model of Fay
new parameters. a line thermal
The
analogy
(Fig. 11-3),
first
noticed b y Scorer (1978), can b e incorporated in this model, as s h o w n in the next section.
11.1.3 Plume Model It i s a s s u m e d
of Chu and
Goldberg
here that the horizontal velocity component of the
p l u m e i s n e a r l y e q u a l to t h e c r o s s f l o w v e l o c i t y a n d t h a t t h e p r e s s u r e drag* is s m a l l in c o m p a r i s o n w i t h that associated w i t h The
first
assumption
entrainment.
is m o r e restrictive t h a n the s e c o n d , b u t
both
require that the r e g i o n n e a r the exit b e e x c l u d e d . T h e effects o f the complex processes near the source could b e taken into account
by
r e p l a c i n g t h e r e a l efflux b y a v i r t u a l s o u r c e s u i t a b l y l o c a t e d n e a r t h e
* A pressure drag, as for a bluff body, has been introduced empirically by some investigators to give agreement with experiments. The idea is controversial; for a discussion see Coelhoe and Hunt (1989).
Chapter 11
282
exit, b u t
the a u t h o r s give n o information o n h o w to estimate
the
location
of this virtual
the
source
w h i c h is
also
the
origin of
c o o r d i n a t e s y s t e m ( s e e F i g . 11-2). W e consider a forced p l u m e of density p
t
and velocity w
which
t
enters vertically into a crossflow of velocity U and density p . T h e a
initial m o m e n t u m a n d b u o y a n c y atmosphere of constant
fluxes
M and F are given. For an t
t
density a n d in the absence of losses,
b u o y a n c y flux t h r o u g h e a c h v e r t i c a l c r o s s - s e c t i o n A r e m a i n s
the
constant
(11-21)
w h e r e p i s t h e p l u m e d e n s i t y a n d u t h e p l u m e v e l o c i t y , h e r e u = U. W h e n pressure
drag in the direction n o r m a l to the p l u m e axis is
n e g l e c t e d , t h e c h a n g e i n v e r t i c a l m o m e n t u m o v e r a d i s t a n c e dx
must
equal the b u o y a n c y force acting o n a p l u m e element of this length
O n c o m b i n i n g t h i s e x p r e s s i o n w i t h (11-21), w e o b t a i n
U
Pa
Fig.
11-2:
Flow geometry
for
Chu and Goldberg's
model
Jets and Plumes in Crossflow
dx
I puwdA
283
= -r, 1/
I
(11-22)
and upon integration wdA
JA
Ft
= JJX + M
(11-23)
t
A t t h i s p o i n t t h e p l u m e p r o f i l e s f o r p , u, I U , a s w e l l a s t h e p l u m e geometry, m u s t b e specified. W e a s s u m e a circular cross-section of radius b, and uniform velocity and s e c t i o n , i . e . u = U, w = W a n d p = p
a
density over the p l u m e
cross-
(Boussinesq approximation).
W e define the radius b so that
puwdA
= Jtp b UW
(11-24)
2
a
From geometric considerations w e have
^ dx
= 77 U
(11-25)
The entrainment assumption can best be introduced b y making use of the line-plume analogy. ( W e depart here from C h u a n d As
s h o w n in Fig. 11-3
three
cross-sections
Goldberg.)
( p o s i t i o n s x \ , *2» * 3 )
projected onto a vertical plane, can b e seen as sections of a line t h e r m a l a t s u c c e s s i v e t i m e s T\, t2, £3 w h e r e T\ = X\/U
etc. T h e line
t h e r m a l h a s s e l f s i m i l a r c h a r a c t e r a n d g r o w s a s db = f} dz
c
t h e e n t r a i n m e n t r a t e . A s dz
c
w h e r e j8 i s
- (W/ U) dx w e o b t a i n
(11-26) dx which
is
the
entrainment
'
U
relation
1
proposed
also
by
Chu
and
Goldberg. O n c o m b i n i n g ( 1 1 - 2 5 ) a n d ( 1 1 - 2 6 ) a n d a s s u m i n g /J t o b e a c o n s t a n t b = /3z
c
O n substituting (11-24), (11-25) a n d (11-27) into (11-23), w e get
(11-27)
Chapter 11
284
dz
F
(
(11-28)
W e integrate to obtain the p l u m e trajectory 1/3
1/ 2
4F
["Pa" } 3
f
\
(
1/3
4M,
(11-29)
[*P U
+
2
a
D e n o t i n g t h e o b v i o u s s c a l i n g l e n g t h s £B a n d fa w e h a v e 1/3
1/3
(11-30) 4 ^
where /
2
_ ^ _ 4 ^ V ^ _ ^
2
(11-31)
Jets and Plumes til Cross/low
4F
285
_ Ajtb?g[p -p^W
(
a
_ = 4
t
Jtp U"
(11-32)
a
and where the n e w scaling lengths are compared with those defined in Section 11.1.1. T h e p l u m e r a d i u s is g i v e n b y
b =
3 0
1/3
1 2
2
V
2 +
/
M
1
1/3
(11-33) *
B a s e d o n these results, C h u and Goldberg also define a
dilution
ratio Sd Pa~ Pi
(11-34)
Pa'P w h e r e the local "averaged" density difference is defined b y
f g(p -p)udA A
= gft~~p-)jib U
D e n o t i n g a s b e f o r e R = Wi/U, 4n/R = 4f/2R,
R
(36
= 4
=F
(11-35)
= (b/bj) /R
and, with b =
2
a
w e obtain
2\l/3
f)
R
2
MVM
£
M
t
2/3
t_X_
1 2
i
(11-36)
M
A distinct advantage o f the Chu-Goldberg results, is that they are valid over the w h o l e range of interest; the m o m e n t u m - d o m i n a t e d well
as
the
buoyancy-dominated
transition distance x
t r
regimes.
W e can
determine
w h e n the contributions from b u o y a n c y
as a and
m o m e n t u m are equal, from Eq. (11 -30), 2
x ^ - f for w h i c h
V B
(11-37)
Chapter 11
286
M
(11-38)
F o r x < Xfr w e c a n a p p r o x i m a t e 1/3 /
\l/3
v
(11-39) a n d for
x> x
tr
x I \HZf 3
Z„
5
For the entrainment parameter
(11-40)
v\2/3
C h u a n d G o l d b e r g s u g g e s t )S = 0.5
b a s e d o n their o w n e x p e r i m e n t s . E q u a t i o n s (11 -39) a n d (11 -40) are in a g r e e m e n t w i t h k n o w n c o r r e l a t i o n s f o r t h e n e a r f i e l d a n d far
field
respectively. There exist a large n u m b e r
of such correlations, based on
field
t e s t s a n d e x p e r i m e n t s . F a y e t al. ( 1 9 7 0 ) s u g g e s t \2/3
when
^ = 1 . 3 2 ' *
s
= 2.27
2/3,
l4
S
i< 1.55
(11-41)
i> 1.55
when
(11-42)
'ult
b
T h e s e c o r r e l a t i o n s i n c l u d e t h e e f f e c t s o f s t r a t i f i c a t i o n ; t h e i n d e x "ult" indicates
ultimate
plume
include wind shear and
rise.
Further
atmospheric
discussions,
turbulence,
which
C s a n a d y ( 1 9 7 3 ) w h o , w i t h S l a w s o n , first s h o w e d t h a t t h e x consistent with entrainment Prediction penetrating
methods
for
also
will be found 2 / 3
in
- l a w is
theory. ultimate
plume
rise
and
plumes
inversion layers, etc., are m a i n l y empirical, b a s e d
on
o b s e r v a t i o n s from tall s t a c k s a n d
high buoyancy fluxes. Attempts
have
in
been
advances
made in
our
to
incorporate
understanding
the
correlations
of atmospheric
the
recent
boundary-layer
physics, as discussed in Chapter 9. Weil (1985) suggests t w o models b y Briggs
(1984),
one
for
strong
convection
and
convection/strong winds. These models require
one
for
weak
knowledge of
the
Jets and Plumes in Crossflow
287
d e p t h o f t h e a t m o s p h e r i c b o u n d a r y l a y e r h, o f t h e v e l o c i t y s c a l e s for convection
a n d g r o u n d s h e a r u*. T h e y a r e t h e r e f o r e n o t s o e a s i l y
applied as the rise m o d e l s presented herein.
11.2
Negatively Buoyant Plumes from Tall
Stacks
N o s i m p l e a n a l y t i c m o d e l is k n o w n for the c a s e w h e n F < 0. It t
should b e possible to extend the m o d e l of C h u a n d Goldberg to include this case.
One would have
to consider
changes sign and that the entrainment modified
that the
vertical velocity
equation (11-26) should
be
to
The m a x i m u m height is o b t a i n e d from ( 1 1 - 2 8 ) , MU ='~iFf
Mz cMAX
V
From
this
height
the
(11-43)
F
,z
t
(negative)
plume
rise
can
be
calculated
c o n s i d e r i n g t h e effect o f t h e ( n e g a t i v e ) b u o y a n c y o n l y . B y c a l c u l a t i n g z
c
from (11-29) with x from (11-43), w e can estimate the highest point
of the plume, MAX
=
Z
H
+
cMAX
(11-44)
z
w h e r e H is the stack height. On substituting z
c
=z ^ x
i n t o (11 - 4 0 ) w e c a n e s t i m a t e t h e d i s t a n c e
Ax r e q u i r e d f o r t h e p l u m e t o fall, u n d e r i t s n e g a t i v e b u o y a n c y , h e i g h t ZMAX-
the
T h e point w h e r e the p l u m e touches the g r o u n d is located
at a d o w n w i n d d i s t a n c e f r o m t h e s t a c k e q u a l t o ( s e e F i g . 11 - 4 ) x
=(xL C
V
+Ax
(11-45)
cMAX
JZ
This ad hoc approach does not take into account the early history of the p l u m e . T h i s c o u l d b e rectified s o m e w h a t b y i n t r o d u c i n g
a
v i r t u a l s o u r c e w i t h h o r i z o n t a l m o m e n t u m (jet flow) u p s t r e a m o f t h e stack
at
height
z
M
A
X
-
The
location
and
initial
(streamwise)
m o m e n t u m at the virtual source, should b e c h o s e n to m a t c h the real p l u m e at x - x
Z c M A X
. Scorer h a s p r o p o s e d a different m o d e l (Fig. 11-5)
Chapter 11
288
Fig.
11-4:
Geometry negative
for plume buoyancy.
with
w i t h a virtual source at height z - 2 z
positive
c
M
A
X
-
T
initial
h
velocity
and
plume width
e
thereby b e matched, b u t the n e w p l u m e from the higher source
can has
m u c h too high d o w n w a r d velocity. The
virtual
specification
source
of an
with
additional
an
initial
entrainment
jet
flow
requires
parameter,
the
associated
with the axial velocity component. A l t h o u g h the entrainment
near
the high point of the trajectory will b e m a i n l y due to crossflow, an axial component will also be present,
for t a l l s t a c k s ,
as the
plume
falls t o w a r d s t h e g r o u n d . N u m e r i c a l m o d e l s for d e n s e p l u m e s h a v e b e e n d e v e l o p e d b y O o m s e t al. ( 1 9 7 4 ) , b y L i e t a l . ( 1 9 8 6 ) a n d b y H e n d e r s o n - S e l l e r s ( 1 9 8 7 ) . T h e m o s t comprehensive study of negatively b u o y a n t p l u m e s is that of S c h a t z m a n et al. ( 1 9 9 3 ) . T h i s s t u d y includes a c o m p l e t e d i m e n s i o n a l (similarity) analysis o f the p r o b l e m w h e r e all relevant are
included.
The
important
nondimensional
parameters parameters
(densimetric Froude number, velocity ratio, source height, discharge angle etc.) w e r e v a r i e d systematically in a series of w i n d experiments Substantial
including
turbulent
as
well
as
laminar
differences were observed between
the
laminar
turbulent cases w i t h regard to m a x i m u m p l u m e height, distance
and
concentration
level at
these
points
tunnel
crossflow. and
touchdown
of interest.
The
differences have practical implications as current regulatory m o d e l s
Jets and Plumes in Crossflow
Virtual origin of
Fig.
11-5:
289
plume
with negative
buoyancy
Model proposed plumes.
by Scorer
(1978) for
negatively
are b a s e d o n the laminar crossflow data of H o o t et al. ( 1 9 7 3 ) .
Special
Nomenclature
b
r a d i u s (jet o r p l u m e )
F
buoyancy
H
stack height
flux
buoyancy length, i s - ^ ^ b m o m e n t u m l e n g t h , £M - 2 l M
momentum
9
heat
m
flux
flux
R
velocity ratio, R =
s
curvilinear coordinate aligned with plume axis
w /U t
S
stratification parameter, Eq. ( 1 1 - 1 1 )
s
dilution ratio, Eq. (11-34)
U
d
velocity ( c o m p o n e n t in s-direction)
U
wind velocity
w
velocity ( z - c o m p o n e n t )
W
rise velocity (plume)
buoyant
Chapter 11
290
x
downwind distance
Xfr
point of transition, m o m e n t u m - to b u o y a n c y - d o m i n a t e d regime
z
vertical coordinate
z
height of plume centerline
a, /3
entrainment coefficients
6
slope of plume centerline
c
Indices a
ambient (atmospheric)
i
initial (stack exit)
REFERENCES B r i g g s , G . A . ( 1 9 8 4 ) P l u m e r i s e a n d b u o y a n c y effects. I n Science
and
Power
Prediction,
Chapter
Atmospheric
8, p p 3 1 7 - 3 6 6
(Ed D.
Randerson). U.S. Department of Energy, D O E / T I C - 2 7 6 0 1 . Chu, V . H . and Goldberg, M . B . (1974) B u o y a n t forced p l u m e s in cross flow. A.S.CE.
J. Hydraulics
Div. 100, N o H Y 9 .
Coelhoe, S.L.V. a n d Hunt, J.C.R. (1989) T h e dynamics of the near o f s t r o n g j e t s i n c r o s s f l o w s . J. Fluid C s a n a d y , G . T . ( 1 9 7 3 ) Turbulent
field
Mech. 2 0 0 , p p 9 5 - 1 2 0 .
Diffusion
in the Environment.
D . Reidel
Publ. Co. E s k i n a z i , S. ( 1 9 7 5 ) Fluid Environment
Mechanics
and
Fay, J.A., Escudier,
of
our
M . , H o u l t , D . P . ( 1 9 7 0 ) A correlation o f field
o b s e r v a t i o n s o f p l u m e r i s e . APCA
Journal
H e n d e r s o n - S e l l e r s , B . ( 1 9 8 7 ) Modeling The
Thermodynamics
Academic Press.
University
of
Salford
2 0 , N o 6, p p 3 9 1 - 3 9 7 .
of Plume
Model:
U.S.P.R.,
Rise
and
Dispersion
Lecture
Notes
in
Engineering, Springer Verlag. Hewett, T.A., Fay, J.A. and Hoult, D.P. (1971) Laboratory experiments o f s m o k e s t a c k p l u m e s i n a s t a b l e a t m o s p h e r e . Atm.
Environment
5,
pp 767-789. Hoot, T . G . , Meroney, R . N . and Peterka, J.A. (1973) W i n d tunnel tests of negatively
buoyant
Environmental 27711, USA.
plumes.
Report
Protection Agency,
No
EPA-65013-74-003.
Research Triangle Park,
NC
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Hoult, D.P., Fay, J.A., Farney, L.J. (1969) A theory of plume c o m p a r e d w i t h field o b s e r v a t i o n s . A P C A Journal
rise
19, N o 8, p p 5 8 5 -
590. Hoult, D.P. and Weil, J.C. (1972) Turbulent p l u m e in a laminar flow. Atm. Environment
cross
6, p p 5 1 3 - 5 9 0 .
Li, X . - Y . , L e j d e n s , H . a n d O o m s , G . ( 1 9 8 6 ) A n e x p e r i m e n t a l v e r i f i c a t i o n o f a t h e o r e t i c a l m o d e l for t h e d i s p e r s i o n o f a s t a c k p l u m e h e a v i e r t h a n air. Atm. Environment Morton,
B.R., Taylor,
gravitational
20, N o 6, p p
G.I. and
convection
from
1087-1094.
Turner,
J.S.
maintained
s o u r c e s . Proc. Roy. Soc. A 2 3 4 , p p
(1956) and
Turbulent
instantaneous
1-23.
O o m s , G., Mahieu, A . P . and Zelio, F. (1974) T h e p l u m e path of vent g a s e s h e a v i e r t h a n air. First Safety
Promotion
Schatzman,
Int. Symposium
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Industries.
on Loss
Prevention
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The Hague.
M . , Snyder, W . H . and Lawson, R.E. (1993)
Experiments
with heavy gas jets in laminar and turbulent cross flows. A c c e p t e d for p u b l i c a t i o n , Atm.
Environment.
S c h e t z , J . A . ( 1 9 8 0 ) I n j e c t i o n a n d m i x i n g i n t u r b u l e n t f l o w . Progress Astronautics
and Aeronautics
S c o r e r , R . S . ( 1 9 7 8 ) Environmental
Aerodynamics.
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W e i l , J . C . ( 1 9 8 5 ) U p d a t i n g a p p l i e d d i f f u s i o n m o d e l s . J. Clim. Appl. 24, p p 1 1 1 1 - 1 1 3 0 .
in
68. Publ. ALAA, N e w Y o r k .
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Chapter 11
292
PROBLEMS
P r o b l e m 1. A ship burning chemical waste products, travels at 10 knots (5 m/s) on the high seas under "no wind" conditions. It produces hot emission products (T = 250°C) at a rate of 5 m / s and an initial velocity of 10 m / s from a stack of 30 m height. Determine the rate of dilution of the resulting plume under the assumption of ideal gas and no chemical reactions. 3
g
P r o b l e m 2. Consider the same emission source as in Problem 1, but in this case the source is stationary in a wind of 5 m/s. (Neutral conditions m a y be assumed.) Consider next the moving source (10 knots) heading into a wind of equal strength. H o w would you solve this problem? Compare the answers for the three different conditions considered. P r o b l e m 3 . In the sketch an experiment is illustrated where a moving source in a stratified medium is used to simulate a rising plume in the atmosphere. Thefluidsare salt /water solutions and the concentration in the receiver varies as indicated. The heavy (source) flow is colored for visualization. Determine the relevant nondimensional parameters for this experiment. Find the analytic solution for the plume trajectory in terms of the flow parameters shown. W h y is this experiment not a realistic simulation (irrespective of scale) for a real (gaseous) plume rising in the atmosphere? u Pi
p(z)-
P r o b l e m 4. A tall stack (150 m ) emits hot exhaust (200°C) at a rate of 10 m / s and with initial velocity 15 m / s into an isothermal atmosphere of windspeed 7 m/s. Determine the plume trajectory 3
Jets and Plumes In Crossflow
293
b a s e d o n t h e m o d e l o f C h u a n d G o l d b e r g a s w e l l a s t h a t o f F a y et al., a n d c o m p a r e the results. T h e effluent c a n b e c o n s i d e r e d to h a v e the p r o p e r t i e s o f ( h o t ) air.