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

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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

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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

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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

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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

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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

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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

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                                                                    =O.53b
ro
o
                                    0.2     0.4     0.6    0.8     1.0
                             Fig. 5     Comparison of assumed to  gaussian profile.

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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

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                                       Entrainment
                                       Surface
           Fig. 6     Cross-section of merging plumes.
                                             Temperature
Fig. 7    Temperature profile along connecting line of merging pi
times,
                           23

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           =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

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                    /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

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                                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

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          .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

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thus the trajectory when an appropriate entrainment function is
introduced.
                                   29

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                                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

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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.

7.   Slawson, P. R., and G. T. Csanady.  On the Mean Path  of Buoyant Bent-
     Over Chimney Plumes.  Journal of Fluid Mechanics.  28^:311, 1967.

8.   Slawson, P. R., and G. T. Csanady.  The Effect of Atmospheric
     Conditions on Plume Rise.  Journal of Fluid Mechanics.  47:33-49, 1971.

9.   Csanady, G. T.  Bent-Over Vapor Plumes.  Journal of Applied Meteorology.
     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.

12.  Weil, J. C.  The Rise of Moist, Buoyant Plumes.  Martin Marietta
     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
     Jets Discharged into Flowing Stratified Ambients.   Oak  Ridge National
     Laboratory.   Report NumberORNL-4685.  June  1971.

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.

16.   Carpenter, S. R., F. w. Thomas,  and R.  E. Gartrell.  Full-Scale
     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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