Ecological Research Series
Analysis of Multiple Cell Mechanical
Draft Cooling Towers
National Environmental Research Center
Office of Research and Development
U.S. Environmental Protection Agency
Corvallis, Oregon 97330
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EPA-660/3-75-039
JUNE 1975
ANALYSIS OF MULTIPLE CELL MECHANICAL DRAFT COOLING TOWERS
by
Lorin R. Davis
Pacific Northwest Environmental Research Laboratory
National Environmental Research Center
Corvallis, Oregon 97330
Program Element 1BA032
ROAP/Task No. 21AJH/41
NATIONAL ENVIRONMENTAL RESEARCH CENTER
OFFICE OF RESEARCH AND DEVELOPMENT
U.S. ENVIRONMENTAL PROTECTION AGENCY
CORVALLIS, OREGON 97330
For Sale by the National ^Technical Information Service,
U.S. Department of Commerce, Springfield, VA 22151
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ABSTRACT
This report presents the development of a mathematical model
designed to calculate the rise and dilution of plumes from multiple
cell mechanical draft cooling towers. The model uses integral methods
and includes the initial development zone, the individual single plume
zone, and the zone of merging multiple plumes.
Although the governing equations for moist plumes are presented,
the final working equations are for dry plumes only. Techniques are
used that allow for a gradual merging of plumes without a discontinuity
in the calculation of plume properties. Entrainment techniques that
include the interference of unmerged plumes and the reduction of en-
trainment surfaces after merging are presented. The entrainment expres-
sion includes coefficients that need to be determined by tuning the
model with experimental data.
This report was submitted by the Pacific Northwest Environmental
Research Laboratory of the Environmental Protection Agency. Work was
completed as of May 1975.
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CONTENTS
Sections Page
ABSTRACT 11
CONTENTS 111
I CONCLUSIONS 1
II INTRODUCTION 2
III MULTIPLE PLUME ANALYSIS - GENERAL 7
IV INTEGRAL ANALYSIS OF MULTIPLE DRY PLUMES 12
V ENTRAPMENT 30
VI REFERENCES 32
VII NOMENCLATURE 34
111
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SECTION I
CONCLUSIONS
This report presents a mathematical model capable of calculating
plume rise and dilution from multiple cell mechanical draft cooling
towers with the wind normal to the tower line. The model includes
calculation techniques for each mode of plume development: 1) the
zone of flow establishment, 2) the zone of fully developed single plumes,
3) the zone of merging multiple plumes, and 4) the zone of completely
merged plumes. However, end effects have been left for future work.
The model is particularly significant because calculations proceed
smoothly from one zone to another without a discontinuity in plume prop-
erties. The entrainment functions-presented include the effects of ;'
plume interference and variable entrainment surfaces on merging. In
order to tune the model, however, the coefficients in the entrainment
function need to be determined from suitable field or laboratory data.
Although the present version is for dry plumes, the report includes the
equations and simple modifications required to convert the model for
moist plumes.
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SECTION II
INTRODUCTION
A mechanical draft cooling tower for a modern power plant consists
of clusters of tower cells arranged in rows. Each cell has its own
discharge port and fan. The plumes resulting from each row of cells
(i.e., tower) consist of a row of individual plumes that gradually
merge together forming a long, oblong plume cross-section.
Most multiple cell plumes interfere with each other before rising
very far into the environment. Interference affects individual plumes
by reducing entrainment and changing the shape of the plume and distribu-
tion of its properties. Entrainment...pumping environmental fluid into
and thereby diluting the plume...is caused by fluid shear at the bound-
aries of the plume where it contacts the environment and by interaction
with the wind. Even before the plumes merge, neighboring plumes compete
with one another by trying to entrain the same fluid that is between
them, thus reducing the amount of dilution that occurs. Merging
further reduces entrainment since only a portion of each plume subse-
quently contacts the environment.
Handling multiple port discharges mathematically is very difficult
due to the non-symmetrical nature of the plume. For single plumes, one
practical analytical method has been an integral analysis based on
assumed symmetrical velocity, liquid, vapor, and temperature profiles.
Entrainment is calculated from the plume size, relative velocity between
plume and wind, and an entrainment coefficient. The conservation equa-
tions are integrated and solved stepwise for the desired properties along
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the plumes's trajectory.
Koh and Fan* made one of the first attempts to solve the multiple
plume problem, assuming single round plumes up to a selected point of
transition and then a two-dimensional slot plume after that. This approach
was used to generate the nomograms for multiple port discharge in the
no
"Workbook on Thermal Plume Prediction, Volume I . The problem with this
approach is that continuity of plume center!ine properties and conservation
of mass, momentum, and energy cannot be obtained through the transition
region. As a result, a sudden drop in plume centerline temperature was
predicted using this method.
Another approach, suggested by Jirka and Harleman , ignores transi-
tion and assumes an equivalent slot discharge all the way from the source.
The size of the equivalent slot is one having the same mass flux and
momentum as the multi-port system. However, this approach overestimates
dilution except for plumes that initially are very close together.
4 5
Briggs ' modified his single plume equation by an "enhancement
factor" to account for multiple sources. His equations are good if you
want a quick, approximate answer.
Meyer, et al. also used a modified version of the Briggs formula to
predict plume rise and dispersion from a multiple cell tower. They found
their equations fairly accurate in predicting visible plume length, but
less accurate for trajectory predictions. Although they are working on a
more rigorous mathematical model that includes the effects of merging, the
details of that model presently are unknown.
Several existing single cell plume models provide insight into plume
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78 9
characteristics. A few are those by Slawson and Csanady ' , Csanady ,
Wigley and Slawson10, Lee11, Weil12, Hirst13, Hoult, et al.14, and Hanna15.
Data from multiple port discharges are scarce. One study by Carpenter,
et al. at TVA gives field information on plume characteristics for
discharge from as many as nine stacks and as few as one in operation. Meyer,
et al. have been collecting data from the tower at Potomac Electric
Power Company's Benning Road plant. Those data cover discharges from a
tower with up to eight cells in operation.
In an ongoing study, Kannberg and Davis at Oregon State University
in Corvallis are conducting laboratory experiments on multiple port
discharges. In this study, the characteristics of cooling tower plumes
as well as those of submerged multi-port diffusers are being investigated
by discharging hot water into cool water from selected discharge configur-
ations. Since the study is interested in discharges from diffusers as well
as from multiple cell cooling towers, discharge angles other than 90° have
also been investigated. The parameters varied are the discharge densimetric
Froude number, discharge port spacing, discharge angle, and ambient-to-
discharge velocity ratio. The ambient has been neutrally stratified in all
cases. Experiments .to date indicate that dispersion of individual plumes
from a row of multiple ports is significantly less than single port dis-
charges of the same diameter and Froude number when the ports are spaced
less then ten diameters apart. A complete report giving the results of
this study is forthcoming. Figure 1 is an example of the results.
The following sections present a mathematical model for multiple
cell cooling towers. The model, based on an integral analysis, includes
-------
U
LJ
0
R=0.10
10 15 20 25
HORIZONTAL X/D
30
35
Fig. 1 Temperature - trajectory plot for multiport discharge with different
current velocities.
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the development and merging zones. The report presents the basic equations
for moist plumes (including the effects of condensation) but only the final
working equations for dry plumes. The experimental data of Kannberg and
Davis and available field data will be used to tune the model in subsequent
work.
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SECTION III
MULTIPLE PLUME ANALYSIS - GENERAL
In order to analyze the plumes merging from a multiple cell cooling
tower, the plume is divided into four major regions: 1) The zone of flow
establishment, 2) the zone of fully developed single plumes, 3) the
merging zone, and 4) the zone of completely merged plumes.
In the zone of flow establishment, the velocity, temperature, density,
and moisture profiles change from those at the tower exit to fully developed
profiles at some point downstream. In the fully developed single plume
zone, the profiles within the plume retain their characteristic shapes,
changing only in magnitude. The length of this zone largely depends on
how close the discharge ports are. For cooling tower plumes with cells close
together, this zone will probably only be a few diameters in length.
•
Long before they touch, the plumes actually interfere with one another
by attempting to entrain the same fluid between them. The most dramatic
effects of merging, however, are after the plumes touch and begin to diffuse
into one another. Both the shape of the property profiles within the plumes
and the surface area available for entrainment changes during this process.
Merging continues until the properties at the mid-point between plumes
equal those at the plume centerlines. Beyond this point, the plumes behave
essentially like the discharge from a long slot.
The present analysis assumes that knowledge of the properties in the
central plumes is desired since they differ the greatest from ambient
conditions. It is further assumed that these central plumes are affected
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only slightly by the total number of cells in the towers. For this reason,
end effects are ignored and left for future work.
Figure 2 gives the coordinate system and defines the angles used. The
integral form of the governing conservation equations can be written as
Hirst13 and Weil12:
MERGING
PLUMES
WIND NORMAL
TO TOWER
Fig. 2 Sketch of cooling tower with definitions of coordinates
and plume angle.
8
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Continuity equation
. E <»
where E is the entrainment.
Energy equation
U r r R^
J U (a-a0)dA - - 3- J UdA + J ^ dA (4)
do
0 00
Combining equati ons (3 ) and (4) with the energy equation (2) yields
4 J (" U(T-T.) + ^ (q-qj 1 dA = - j^+ + ^ +r sin 6] JudA (5)
0
-------
Combining just (3) and (4) yields
oo oa
H r r i rdci daoo i r
Is I U (q~qJ + (0"O dA = " ds^ + ds~ / UdA (6)
L J L J -^
0 0
S - momentum equation
d
cfe
f U2 dA = E U cos 6 + / —^- g sin 6 dA (7)
J J P
0 0
Curvature equation (combined r and s momentum)
cose / — gdA - E U Sine
ie o " (8)
ds "
J U2dA - E2/4
0
If thermodynamic equiTibrium is assumed in the plume, the Clausius - Clapeyron
relation can be used to yield a relation for the plume centerline vapor content.
(q - q ) = (l-«>)q + 1 (9)
L oo ooc o
where
cj) is the local ambient relative humidity at height z.
q is the ambient saturation humidity at height z.
°°s
is the ambient saturation humidity at the top of the tower.
R is the vapor gas constant.
T is the ambient temperature at the top of the tower.
10
-------
q is given by
005
Vi
do)
.10
Moisture effects on plume trajectory are only secondary . For this
reason the above equations first will be solved for merging dry plumes
and later expanded to wet plumes.
11
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SECTION IV
INTEGRAL ANALYSIS OF MULTIPLE DRY PLUMES
ZONE OF FLOW ESTABLISHMENT
Because of the fan and its hub, the distribution of velocity at the
tower exit usually is shaped as shown in Figure 3.
To represent the actual profiles at discharge, this analysis uses
approximate top-hat profiles determined by mean discharge values. This
assumes that these profiles change to the fully developed, bell-shaped
profiles at the end of the zone of flow establishment as shown on Figure 3.
Profiles assumed within the zone of flow establishment are:
U=UQ , r ru (n)
, r> rt
Pco - p = 3(T - TJ
(12)
(13)
where r and r. are the radii to the edge of the velocity central core
u u
and to the temperature core, respectively, and U^ is the free ambient
velocity. This expression for velocity and temperature profiles was
selected over exponential or gaussian profiles because of the definite
12
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EDGE OF
PLUME
ASSUMED
TOP-HAT
DISCHARGE
PROFILE
'/ENDOF
^VELOCITY
DEVELOPMENT
ZONE
F
1
U
"'
lo '
1
1
1
; n~
' \ 1 i
1
1 i/
. ' IL
USUAL SHAPE OF
DISCHARGE VELOCITY
PROFILE
EXIT
Fig. 3 Sketch of assumed velocity profiles within the
development zone.
13
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plume edge definition inherent in this expression. This profile property
will become important later when the plumes begin to merge and will allow
a smooth transition. With these assumptions, equations (1), (2), (7)
and (8) can be integrated to yield:
Continuity
TUr2 b21
d | Vu + d.br + d b I
Hs[ — l u 27J
= E (14)
Energy
iKATo^T- + *Vt bd3 + ATo 7- d4
dT. -IP U r 2 d b2
Momentum
2
_;" L_ , 11: d6)
[u ~r
JLT
= E Uoo cos 6 + g I,- sin
Curvature equation
— 9 J5 cos 9 - E U- sin 6
ds " U 2r 2 2
14
-------
where
d, = .45U + .55U«, cos 6
1 o
d. = .2571U + .7429Uoo cos e
Z 0
d, = .31558U + .13442U«, cos 6
3 0
d4 = .13352UQ + .12362Ueo cos e
dc = .31558U 2 + .26885U U» cos 6 + .41558Uoo2 cos2
5 o o
de = .13352U 2 + .24724U Um cos e + .61924UO,2 cos2
DO 0
and
/r r 1
*" p dA = SAll -|- + .12857b2 + .45rtb
(18)
U I C. LI
0 p
g = - — (-5^)1 the coefficient of thermal expansion. For an ideal
°i
gas 3 = i .
Differentiating equations (14), (15) and (16) yields:
Conti nui ty
Uoru + dlb) ru' + (dlru + d2b) b' = E
Energy
rt Mrtd3 + bd4b =
dTo° T T U« 9 9 K2
e + - -? ft^u > + b(rtd3 ' rudl^ + T~ «fV (20)
15
-------
Momentum
(U r + bdJ r' + (r d,. + bdj b = E U«> cos 9 + g IK sin 9 (21)
vj LI O U LI 0 O 0
Equations (17), (19), (20), and (21) can be solved for the four unknowns,
r , r. , b , and 9 . These variables can be integrated step-wise to
yield r , r., b and 9 in the development zone.
U t ,\~r
Since the change in r depends on r and —g— , it may disappear before
or after r disappears. If r goes to zero first r. = 0 and r > 0,
U "C t U "•*
then AT will change as r continues to decrease. In this case, equations
(18) and (20) become:
I5 = 3ATc (0.12857 b^) (22)
2
b2 J - ' - - - - rdT°° + rsin 9] |~ YSL. +dbr t ^ } W
TdToo
"•- s-
Using these two equations, (17), (19), and (21) yield b , ru , 92 and ATc .
Should r go to zero first r = 0 and r > 0, AU = U -Uoo cos 9
U U t ~" C C
will change as rt continues to decrease. In this case the equations become:
Continuity
.12855b2AU ' + [ .2571AU b + A858Um cos 9 b 1 b1 = E (24)
C L C J
16
-------
Momentum
[ .13352b2AUc + .25714b2Uoo cos 6 ] All^
+ [ .13352bAUc2 + .51824bAUcUoo cos 6 + bU^2 cos2 e ] b1
= EUoo cos 6 + g I5 sin 9 (25)
Energy
, , rrt" h2i
[AUc + Uco cos 9 ) rt + d?b J rt + -|- + .31588rtb + .13552 J- AUC
r i ' 1 [ dT« 1 T h2
[ d?rt + dgb J b = ^- ^- + r sin 0 - J- (.2571AUC + .4858Uco cos 0
r .2 "I
(AU + Uoo cos 6 )-i- + r.bd? + S- dfl (26)
c i t 7 2 8J
where AU = U - Ua, cos 6
c c
d? = ,31558AUc + .45UCO cos 6
dQ = .13352AU + .25714 U«, cos 6
o C
de glgcose - EUooSine /27)
|- (.13352AUC2 + .51428AUcUcoCose + Uoo2cos26 ) - E2/4
17
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The integral IT is given by equation (18).
Equations (24), (25), (26), and (27) are sufficient to solve for r ,
AU , b', and 9 .
c
All of the above equations depend on the entrainment, E. This
variable can be approximated using an expression similar to that used
by Hirst , modified to account for the different profiles used in
this work.
18
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FULLY DEVELOPED SINGLE PLUME
After r. and r both go to zero and before neighboring plumes merge,
each individual plume will resemble a single fully developed plume.
In this region assumptions are:
U =AU + U cos 9 (28)
00
r v.3/2 1
where AU = AU 1 - (£)
C I D «J
f r 3/2 1
i-O ]
and AT = ATC j 1 - (£•) | (29)
Using this type of profile and assuming that velocity and temperature
profiles are the same allows the merging process to be calculated smoothly
without a discontinuity in either the conservation equation or the plume
temperature, velocity, or width. Figure 5 compares this profile to the
more popular gaussian profile given by AU = AUC e~^bj with b. = .529b
(^f. =0.5 for both).
With these assumptions the governing equations for the fully developed
dry single plume become:
Continuity
ds~ | ""c" xl T f~ U~ cos 9 I = E (30)
Momentum
<> h2
2AU b^ cos 9 I. + £-
= -3ATcb2g sin e Ij + EU^ cos 8 (31)
19
-------
=O.53b
ro
o
0.2 0.4 0.6 0.8 1.0
Fig. 5 Comparison of assumed to gaussian profile.
-------
Energy
ATcAUcb2I2 + ATcb\ cos
[
rdT» i r ? b2 1
= - + Fsin 6 AUcb Tx + Y~ um cos e| (32)
Curvature
de cos 9 Wjl - EU^sin 9
•37 = 2 ^ '
AU 2b2I, + 2AU b2I.U cos 6 + 5- U 2 cos26 - E2/.
c2 cl°° 2^°° 4
where
2
I, = / (l-n3/2) ndn = 0.12857 (34)
1 CM
and
I2 = T (l-n3/2) ndn = 0.06676 (35)
The solutions to equations (30-33) yield AU , AT , b, and 6, assuming
an appropriate entrainment function E. From 6 and ds, the trajectory
can be determined since:
dz = ds sin 9 (36)
dy = ds cos 6 (37)
21
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MERGING PLUMES
To this point the present analysis has been essentially the same as
Hirst's with a different profile and the addition of adiabatic expansion
effects of a compressible fluid. Once the plumes start to merge, however,
symmetrical profile no longer can be assumed.
Figure 6 shows a cross-section of merging plumes, and figure 7 shows
the approximate shape of the temperature distribution along the line
connecting the center of each individual plume.
In order to continue the integral analysis through this merging zone,
profiles in the merging plumes must be known. As seen on Figure 7, the
temperature at the mid-point between two plumes does not drop to the
ambient temperature. In fact, limited data indicate that the excess
temperature at the mid-point between plumes is approximately twice the
excess temperature at an equal distance from a plume center!ine in a direc-
tion normal to the plume connecting line. If this is assumed true, the
profiles must satisfy the following: Along the connecting line the profiles
must be smooth curves with zero derivative at the mid-point and plume
centerline with a value of AT twice the single plume value at the same
distance from the center. In the limit as the plumes just start to merge,
the profiles must be the same as the single plume.
Referring to Figure 6 for the definitions of the n and r direction and
the terms c, b and L, the profile assumed in this analysis that satisfies
the above condition is:
3/2 n2
L - b
22
-------
Entrainment
Surface
Fig. 6 Cross-section of merging plumes.
Temperature
Fig. 7 Temperature profile along connecting line of merging pi
times,
23
-------
=AT
3/2'
^3/2
b '
After AT - AT at r = -5-, it is assumed that AT = AT . In the n
I C L~ I C
direction it is assumed that:
AT = AT
n r
(39)
where c =Vb -r
Making the same assumption for velocity yields:
AUn = AUr
COS 8
(40)
where
AUr = AUc
r ,r3/2l
I !-(£) , 0 < r £ L-t
= AU <
c
and AU = AU after AU - AU. at r =
re re
_ L
, L-b < r <
-' -2
(41)
24
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MERGING PLUMES - INTEGRAL EQUATIONS
The continuity equation is still given by equation (1), but the
integral must include the distorted profile. Therefore, the integral in
equation (1) (called F.) is:
L/2 c
udA • dr * = F
i
o o
In the single plume calculation, the integral / urdr was evaluated
which is 1 I UdA. Substituting (40) and (41) into (42), then integrating
2? J
yields:
Fl = I ^cVlW + IT Uo° cos e f3(A) (43)
where A = L/b,
__ 9
fl (l-x3/2) dx + f
(l-x3/2) dx + f7Vl^2 [l-(A-x)3/2] dx (44)
A-l
A _
- .....«_
f3(A) = 2 f Vl^2 dx = fV1-^ + ^"^ ^ (45)
o
and
h3/1
I3 = /) l-T!07^ I dn = 0.45 (46)
After AU_ = AUr
i C
25
-------
/o
UdA = £- ("
AUCI3 + Uro cos 6
4 1-4)2 + sin-1(^)l (47)
The variable A, represents the degree of merging. For A = 2 the
plumes are just starting to merge. For A = 1 the plumes have nearly
merged together.
In a similar manner, the value of
(48)
can be found
where
A/2
f2(A)=
+
cos 6 +
A/2
22 2
Uoo COS 9
f3(A) (49)
and
[l-(A-x)3/2]4dx
1
I4 = / [ 1-X3/2 ] dx = 0.31558
(50)
(51)
After AU = AU
r c
26
-------
cos 92 + Uc= cos 92 f3(A) (52)
For the energy equation
L/2
r r
J- J UAT dA = i J dr AUdn = (53)
o
which when integrated becomes
F3 = | b2ATcAUcI4f2(A) + | b^I^ cos e fj(A) (54)
After AT = AT and AU = AT (which because of similar profile assumptions,
i C i C
occur at the same place):
h2
F3 = F- ATc [ AUcJ4 + ^"o. COS 8 ] f3(A) (55)
The density integral also must be evaluated.
(56)
«,
Since - = £AT this integral can be evaluated and becomes:
F4 = b3ATcVA) (57)
After ATr = AT this becomes
27
-------
.2
D T ^«-r f I n\ fRR^
The values of f, (A), f9(A) and f,(A) can be tabulated as a function of A.
L C. O
With these integrals evaluated, the conservation equation can be
written as:
Continuity
Momentum
^- F2 = Ella, cos 9 + F4g sin 6 (60)
Energy
_ F3=rs1n0+ FI (61)
and Curvature
de F4g cos 6 - EU^Sin 6 , g.
~ i i ._ .1 - _ * '
2,
F2 E /4
Equations (59)-(62) are sufficient to solve for AU , AT , b and 8 and
28
-------
thus the trajectory when an appropriate entrainment function is
introduced.
29
-------
SECTION V
ENTRAPMENT
In order to achieve closure, the analysis presented in the previous
sections requires an expression for the entrained ambient fluid. As far
as this analysis is concerned, any realistic expression for entrainment
could be used; however, the accuracy of the solutions depends to a large
extent on how accurately the entrainment is calculated. The following
13
expressions, developed according to the traditional line of Hirst ,
have been modified to include the effect of neighboring plumes and profile
definitions.
For the development zone, the suggested expression is:
.0204 + .0144b/r 1.0 - R cos 9 (1-
,Uo= [
If™ M
1.0 + Y
Ar
(63)
Where R is the velocity ratio IL/U , A-, A~, and A- are empirical
0 J. £- O
coefficients included to account for the effect of cell spacing, L/r ,
ambient cross current, R sin 9 and local densimetric Froude Number, Fr.
For the fully developed single plumes, the suggested expression is:
E = (C, + r£) b K-U» cos 6 (1- -r ) + M^b sin e (64)
c-o0 - T 4
in e
The values of Cj, C2 and C^ as suggested by Hirst are 0.0289,
0.492, and 4.56, respectively, when adjusted to account for the different
definition of plume size, b, in this study. However, these values may
have to be adjusted for multiple plumes to give best results. The value
30
-------
of C, needs to be determined from multiple cell discharge data.
O
Once the plumes start to merge, i.e., when b=p the entrainment surface
exposed to ambient fluid decreases with the degree of merging (see Figure
6). This arc length exposed to the ambient on each side of the plume is
approximated by A = bir - 2 cos " (t b) . Therefore, an entrainment
function of the following form is suggested after the plumes start to
merge:
J b-ucos
^ sin 9 1
(65)
The above expressions seem reasonable since they include the parameters
that possibly could affect entrainment. The values of the coefficients,
however, would have to be determined from selected data so the effects of
individual parameters could be isolated.
Wind directions other than normal to the line of towers and the drag
induced on the plume by the tower wake have not been included in the model.
How large a deviation in wind direction can be tolerated in the analysis
remains to be determined. The drag induced by the tower wake on the plume
could easily be included in the momentum equations when more information
becomes available as to the magnitude of an appropriate drag coefficient
for a particulate tower structure.
31
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SECTION VI
REFERENCES
1. Koh, R. C. Y., L. N. Fan. Mathematical Models for the Prediction of
Temperature Distributions Resulting from the Discharge of Heated Water
into Large Bodies of Water. Environmental Protection Agency, Water
Quality Control Office, Water Pollution Control Research Series
Report 16130 DWO 10/70. October 1970.
2. Shirazi, M. A., and L. R. Davis. Workbook of Thermal Plume Prediction
Vol. 1: Submerged Dishcharge. Environmental Protection Agency, Environ-
mental Protection Technology Series. EPA-R2-72-005a. August 1972.
3. Jirka, G., and D. R. F. Harleman. The Mechanics of Submerged Multiport
Diffusers for Buoyant Discharges in Shallow Water. MIT Ralph M. Parsons
Laboratory for Water Resources and Hydrodynamics. Report Number 169.
March 1974.
4. Briggs, G. A. Plume Rise from Multiple Sources. Cooling Tower Environ-
ment-74. CONF-74032, ERDA Symposium Series. National Technical Exchange
Service. U. S. Dept. of Commerce, Springhill, VA. pp. 161-179.
5. Briggs, G. A. Plume Rise. AEC Critical Review Series Report. Report
number TID-25075. November 1969.
6. Meyer, J. H., T. W. Eagles, L. C. Kohlenstein, J. A. Kagan, and
W. D. Stanbro. Mechanical Draft Cooling Tower Visible Plume Behavior:
Measurements, Models, Predictions. Cooling Tower Environment-74.
CONF-74032, ERDA Symposium Series. National Technical Exchange
Service. U. S. Dept. of Commerce, Springhill, VA. pp. 307-352.
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Over Chimney Plumes. Journal of Fluid Mechanics. 28^:311, 1967.
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l_0:36-42, 1971.
10. Wigley, T. M. L., and P. R. Slawson. On the Condensation of Buoyant
Moist, Bent-Over Plumes. Journal of Applied Meteorology. 10:253-259,
1971. ~~
11. Lee, J. The Lagrangian Vapor Plume Model - Version 3. NUS Corporation.
Report Number NUS-TM-S-184. July 1974.
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Laboratories Report. Baltimore, Maryland. January 1974.
32
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13. Hirst, E. A. Analysis of Buoyant Jets Within the Zone of Flow
Establishment. Oak Ridge National Laboratory. Report Number ORNL-
TM-3470. August 1971, and Analysis of Round, Turbulent, Buoyant
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14. Hoult, D. P., J. A. Fay, and L. J. Forney. A Theory of Plume Rise
Compared with Field Observations. Journal of Air Pollution Control
Association. 19.: 585-590, 1969.
15. Hanna, S. R. Rise and Condensation of Large Cooling Tower Plumes.
Journal of Applied Meteorology, llj 793-799, 1972.
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Study of Plume Rise at Large Electric Generating Stations. TVA,
Muscle Shoals, Alabama. 1968.
17. Kannberg, L. D., and L. R. Davis. An Experimental Investigation of
Multiple Buoyant Plumes. Work in progress under EPA Grant R-800818.
To be completed December, 1975.
33
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SECTION VII
NOMENCLATURE
A = Local aspect ratio, L/b, also used in integral equations as
area and appears as *dA.
b = characteristic size of plume.
c = specific heat
D = discharge port diameter
d., d2>.. = coefficients defined in text
E = entrainment function
Fr = Froude Number = U /(Ap/pgD)1/2
g = gravitational constant
Ij, I2... = integral constants defined in text
L = distance between discharge ports, also used as latent heat
of vaporization.
q = vapor concentration
r = plume coordinate normal to centerline
ru, rt = size of velocity and temperature central core in development
zone
RC = condensation rate
R = gas constant of vapor
s = distance along plume centerline
U = velocity
y, z = horizontal and vertical distance from discharge
3 = coefficient of thermal expansion
r = adiabatic lapse rate
n = plume coordinate normal to tower connecting line
34
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6 = plume angle relative to horizontal
p = density
a = moisture content
= relative humidity
Subscripts
c = at plume centerline
o = discharge conditions
r = in r direction
r\ = in n direction
s = saturation conditions
00 = free stream conditions
1 = at discharge level
35
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TECHNICAL REPORT DATA
(Please read Instructions on the reverse before completing)
1. REPORT NO.
EPA-660/3-75-039
2.
3. RECIPIENT'S ACCESSION NO.
4. TITLE AND SUBTITLE
Analysis of Multiple Cell Mechanical Draft Cooling
Towers
5. REPORT DATE
May 1975
6. PERFORMING ORGANIZATION CODE
7. AUTHOR(S)
Lorin R. Davis
8. PERFORMING ORGANIZATION REPORT NO.
9. PERFORMING ORGANIZATION NAME AND ADDRESS
10. PROGRAM ELEMENT NO.
Pacific Northwest Environmental Research Laboratory
Environmental Protection Agency
200 S.W. 35th
Corvallis, OR 97330
1BA032
11. CONTRACT/GRANT NO.
12. SPONSORING AGENCY NAME AND ADDRESS
13. TYPE OF REPORT AND PERIOD COVERED
Interim
14. SPONSORING AGENCY CODE
15. SUPPLEMENTARY NOTES
16. ABSTRACT
This report presents the development of a mathematical model designed to
calculate the rise and dilution of plumes from multiple cell mechanical draft
cooling towers. The model uses integral methods and includes the initial devel-
opment zone, the individual single plume zone, and the zone of merging multiple
pi umes.
Although the governing equations for moist plumes are presented, the final
working equations are for dry plumes only. Techniques are used that allow for
a gradual merging of plumes without a discontinuity in the calculation of plume
properties. Entrainment techniques that include the interference of unmerged
plumes and the reduction of entrainment surfaces after merging are presented.
The entrainment expression includes coefficients that need to be determined by
tuning the model with experimental data.
17.
KEY WORDS AND DOCUMENT ANALYSIS
DESCRIPTORS
b.lDENTIFIERS/OPEN ENDED TERMS C. COSATI Field/Group
*Cooling Towers
*P1umes
*Mechanical Draft
Merging
Thermal Pollution
Cooling Water
Mathematical Model
13B
18. DISTRIBUTION STATEMENT
19. SECURITY CLASS (ThisReport)'
21. NO. OF PAGES
39
20. SECURITY CLASS (Thispage)
22. PRICE
EPA Form 2220-1 (9-73)
A U. S. GOVERNMENT PRINTING OFFICE: 1975—699-140 (23 REGION 10
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