VII-B-113

A MODEL FOR SHORE-ATTACHED THERMAL PLUMES IN RIVERS

P. P. PailyNALCO Environmental SciencesNorthbrook, Illinois U.S.A.

and

W. W. Sayre 'Iowa Institute of Hydraulic Research

The University of IowaIowa City, Iowa U.S.A.

ABSTRACT

Side discharge of power plant thermal effluents into natural riverscause the formation of thermal plumes, the shape and orientationof which depend upon both the discharge conditions and the ambientflow characteristics. When the turbulence level of the ambient flowis high, rapid vertical mixing is achieved between the effluentdischarge and the river flow, and the thermal plume becomes shore-attached with the maximum temperatures within the plume occurringalong the near shore. In order to determine the isotherm patternswithin shore-attached thermal plumes, a solution of the depth-inte-grated steady state convection-diffusion equation is obtained. Thesolution is cast in terms of space-cumulative discharge coordinates,thus allowing the inclusion of the effects of cross-channel flowinduced by channel curvature in the model. Comparison of the modelresults with the thermal plume data from power plant sites alongthe Missouri River indicates good agreement.

INTRODUCTION

In order to meet the increasing electrical energy requirements ofthe nation, steam-electric power plants of large capacities arebeing installed or being planned for installation by a number ofutilities. However, the low thermal efficiencies of these plantsnecessitate the rejection of large amounts of waste heat at thegenerating sites. In situations where it can be successfullydemonstrated that discharge of the waste heat into a natural bodyof water will not cause any harmful effects on the ecology of thereceiving aquatic environment, open-cycle cooling using river,lake, or sea water is the most desirable waste-heat-removal method.Closed-cycle cooling systems using cooling towers, cooling ponds,or spray canals involve enormous capital investments for construc-tion; these can be up to 15 times that for an open-cycle system.They may also require a considerable portion of the plant outputpower for operating the cooling systems, thereby imposing penaltieson plant efficiency. Closed-cycle systems also require extensive

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continuing maintenance. Finally, there is consumptive water lossdue to evaporation from these systems. Open-cycle systems/ on theother hand, are inexpensive to construct, place minimal demands onthe plant output, and evaporative water losses are much less. Thus,lower investment and maintenance costs, improved plant efficiencies,and less water loss all combine to make once-through cooling themost attractive method by far, from an economic point of view, fordissipating power plant waste heat.

The effects of the thermal discharge on a natural stream, and thephysical processes involved in heat transport within the flow systemare normally analyzed by separating the affected area into two re-gions: the far-field and the near-field. The determination of far-field temperature effects in a river is important in assessing thecumulative effects of several power plants distributed along thecourse of the river. On the other hand, the determination of near-field conditions is important in evaluating the possible effects ofthe thermal discharges on the aquatic biota and drifting organisms.In the vicinity of an outfall, the heated water forms a so-calledthermal plume until the combined actions of jet mixing, buoyant andconvective spreading, and turbulent mixing eventually lead to com-plete dilution of the heated water with the ambient flow. The ther-mal standards of the various regulatory agencies regarding thermaldischarges in natural waterways usually specify certain limitationson the size of and temperature rise in thermal plumes. It is there-fore important to have reliable methods for predicting configura-tions of and temperature patterns within thermal plumes.

Several computational methods are available for predicting thermalplume patterns resulting from power plant discharges. RecentlyJirka, Abraham and Harleman (4) have published a detailed reviewof these. Most of these techniques either assume or predict bell-shaped profiles for velocity and temperature along horizontal nor-mals to the plume axis. However, an examination of measured ther-mal plume patterns near power stations at Omaha, Fort Calhoun, andBrownville (Cooper) on the Missouri River indicates that the tem-perature profiles are not beel-shaped, but are attached to theshores, with the maximum temperatures in the plumes occurring nearor along the shore. None of the currently available thermal plumemodels predict plume patterns with shore attachment; this deficiencyis highlighted by Jirka, Abraham and Harleman (4).

The available thermal plume models fail to properly represent theshore-attached plume patterns because the effects of ambient flowcharacteristics such as bottom effects, variation of channel shape,non-uniform and varying flow conditions across the channel, andambient turbulence on the mixing phenomenon are not considered intheir formulation. If the turbulence level of the ambient riverflow is high, (e.g., the Missouri River), intense mixing betweenthe thermal jet and the ambient flow will be induced, leading torapid vertical mixing between the two flows. Thereafter, the jet

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is transported downstream as a coflowing stream, and the mixing ismainly due to turbulent diffusion and transverse convective spread-ing associated with bend-generated secondary circulation. Weil andFischer (13) have attempted to relate the effects of stream tur-bulence to plume behaviour. However/ their investigation was re-stricted to zero-momentum jets discharging at the centerline of theambient flow. Yeh (14) conducted a series of laboratory experimentsto investigate the transverse mixing process for a buoyant effluentin both near and far field regions of an open channel flow. Theresults showed that the mixing process in the near field region wasclosely related to the development and decay of buoyancy-generatedsecondary flow. However in the far field region the transverse mix-ing was found to be mainly due to turbulent diffusion.

The present study is concerned with the development of a predictivemodel, based on diffusion concepts, for determining transverse tem-perature profiles in shore-attached, vertically-mixed thermal plumesin rivers resulting from side discharges of thermal effluents. Thedetails of the analytical development of the model, and its applica-tion to field cases 'are presented in the following sections.

FORMULATION OF DIFFUSION PROBLEM

The formulation of the predictive model for determining the temper-ature distributions in shore-attached plumes is based on a model ofmixing of two rivers outlined by Sayre (9) and the transformationof the two-dimensional depth-integrated steady-state convection-diffusion equation into a simple diffusion equation as presentedby Yotsukura and Sayre (17). The solution of the transformed equa-tion is cast in terms of space-cumulative discharge coordinates,(x, p(x,z)), when x and z are the streamwise and cross-channel coor-dinates, and p(x,z) is the normalized cumulative discharge definedby

where QR = total river discharge; q = cumulative discharge passingthrough the river between z = 0 and z at section x; and q = totaldischarge per unit width. The use of the space-cumulative dischargecoordinate system which automatically follows the transverse shiftsand meander ings of the flow, was first introduced by Yotsukura andCobb (15). A procedure for synthesizing the normalized cumulativedischarge at cross sections of a river for which the shape is knownis outlined in the next section.

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The origin of coordinates, x = 0, p » 0, is located on the dischargebank of the river, just far enough downstream from the outfall sothat at the origin initial mixing has taken place, and the tempera-ture rise of the jet is 0.. The fraction of the total river dis-charge occupied by the jet at the origin is given by

Qo9o °0

where a = QO/QT = initial dilution factor; 0 = initial excess

temperature of the thermal discharge; and Q = volumetric rate ofeffluent discharge.

t

Let the variable £ represent the displacement from the origin in thep direction at x = 0, as shown in Fig. 1. Then the input transversetemperature distribution at x = 0 is given by

?a. o < e < »'0, P < 5 £ 1

(3)

Neglecting for the moment reflections from the banks, the normalizedtransverse temperature distribution function for the case of a con-tinuous vertical line source, extending over the depth of flow andconcentrated at (x = 0, p = 5) is fR(p~5; x) which is a Gaussian pro-bability distribution function, given by

735?

where s = p,/a is the standardized normal variable; and the standarddeviation in t8e p domain is

op = V2D x1 (5)

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In Eq.(5), D = overall transverse diffusion factor; and x1 = x - xi where x » longitudinal coordinate of the virtual source for any: subreacn determined from

where x. . = distance from the origin to the upstream end of j-thsubreacn/ and D. = value of D for the j-th subreach. For the firsubreach, x. '•'o.

The desired output temperature distribution for an input distribu-tion Q j ( ? , 0) is given by the convolution integral

rP=l/p ^\ _ r f / f, _Jp=0 R ' I

Using Eq. (3), Eq. (7) can be written

0-AT (p,x) = £• C fR(p-5;x) d? (8)

Taking into account now reflections from the banks, the final solu-tion can be obtained from Eq. (8) as

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where FR( ) is the standardized cumulative normal distributionfunction corresponding to the probability density function fR( ).The terms in the infinite series representing the bank reflectionsare negligible if Dx< 0.08.

The value of the initial dilution factor, a, introduced in Eq. (2)is mainly a function of the design and orientation of the dischargeoutfall structure, the initial jet or plume characteristics, andthe ambient flow properties. If the thermal effluent is dischargedalmost parallel to the river flow, and @Qis the temperature excessmeasured at the discharge canal outlet, the initial dilution factor,a, can have a value of one. For other situations, values of theinitial dilution factor between about 2 and 4 can be used (3,7),until better criteria are established.

Maximum Centerline Temperature

The maximum centerline temperature, AT , for a shore-attached plumeoccurs along the near shore (p = 0). The solution for AT , fromEq. (9) is c

AT = AT(0,X) = — (2Ff—) + 2 Z 1 F (-=--) (10)

The variations of QpAT /Q 9 with Dx for values of P equal to 0.05,0.10, and 0.20 are shown In Fig. 2. The limiting case, P = 0,corresponds to a source concentrated at the origin. The solutioncorresponding to Eq. (9) for this limiting case is

AT(p,X) = 20>(fR(P> + « { £*<2"-" + fR(2n+P'^ tl"

Up n=l

Substituting for the probability density function fR(?)

(12)

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the solution for maximum temperatures along the shore; (p » 0),corresponding to Eq. (10) for this case is

Q n i °° zATC - AT(0,x) = ®oo?^ * oT (1 + 2 Z exp(-25r-)) (13)WR P n=l p

Eq. (13) is represented by the curve labelled concentrated sourcein Fig. 2.

Diffusion Factor, D

The diffusion factor, D, in Eq. (5) can be_related to the dimension-less tranverse mixing coefficient, a= E_/du*, by the equation

Z

1d2 u dp) (14)

if it is assumed that the overall transverse mixing coefficient Edoes not vary across the channel. In Eq. (14) d = A/B = average, /2depth of flow section; A = area of river flow section; u*= (gdS) ' =shear velocity; g =» acceleration due to gravity; S = slope of energygradient; d = local flow depth; and u =» q/d = local depth-averagedvelocity of flow. Using the Manning formula for the river flow rate,

_ -B d (d) (S)

and approximating

1\ d2 u dp ~ (d) u = ^ *5'0 ~"

where q = d u = width-averaged value of q, the relation for thediffusion factor can also be written

D = « (d,1-49

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where n = Manning's coefficient. Values of D determined by Eg. (17)are less accurate compared to those by Eq. (14), mainly because ofthe approximation used in Eq. (16).

The transverse mixing coefficient, E or ct, for a natural riverdepends*upon the characteristics of ihe river channel and theambient flow. However, there are as yet no very reliable predic-tive relationships for evaluating its value. The most reliablemethod for determining the values of E for natural channels is bydirect experimentation in the field using tracers. Experiments inundistorted hydraulic models can also be used to determine E ;vertically distorted hydraulic models do not correctly simulatethe transverse mixing processes.

In laboratory flumes several investigators have found values ofa ranging from about 0.1 for B/d - 5 to about 0.2 for B/d - 60.In reasonably straight uniform reaches of natural channels some-what larger values of a have been found. This increase is pro-bably due to channel irregularities and weak secondary flows ratherthan to any scale effect. In sinuous channels, due to bend-inducedsecondary flow, avalues are typically much larger. As shown inFig. 3, average ctvalues ranging from 0.5 to 2.5 for bends influmes, and from 0.6 to 10 for bends in the Missouri River, havebeen found. The functional form of the relationships shown onFig. 3 was derived by Fischer (2) in a transverse convective dis-persion analysis which utilized Rozovskii's (9) radial velocitydistribution function for fully-developed secondary flow in thecentral portion of an idealized curved channel. The gap betweenthe laboratory flume data and the Missouri River data in Fig. 3indicates that there are still certain factors that are notaccounted for in evaluating the transverse mixing coefficient.The equation for the curve passing through the Missouri Riverdata is

0 - 4 < > c > 2 ( ) ' (13)

where u = QR/A = average river flow velocity; and R = radius ofcurvature or river bend. The value of the numerical coefficientin Eg. (18) may be different for other rivers.

SYNTHESIS OF TRANSVERSE FLOW DISTRIBUTION

The cumulative discharge, q , and the normalized cumulativedischarge, p = SrQp' in a ^iver section at a given flow dischargecan be synthesized from the transverse depth profile of the cross-section, as outlined by Sayre (10). The method is based on the

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hypothesis that the velocity distribution in natural streams is afunction of the transverse depth profile, and uses the relation

where d = local depth of flow; and b and b, are coefficients whosevalues according to the Manning formula (Eq. 19) are b =1.0, andb]^ = 5/3. Sium (12) investigated the applicability of Eq. (19) forsynthesizing the flow distribution in several natural rivers in-cluding the Missouri and Mississippi Rivers. That study indicatedthat the coefficients b and b, in Eq. (19) varied with the ratioof width to average depth, B/d . For cross-sections in straightriver reaches, approximate values of b and b, are

bQ = 1-0, b-, = 5/3; 50 < B/d < 700 ' L ~ _ (20)b = 0-92, bx = 7/4; B/d >, 70

For cross-sections in bends of sinuous and meandering reaches,approximate values of b vary_ between 0.95 and 0.80, and b,between 2.48 and 1.78, is B/d increases from 50 to 100. Usuallythe value of b must be adjusted to make p = 1 when z = B.

Fig. 4 shows the variation of measured q/q with d/d for threecross-sections in the Missouri RJ.VSE. together with the theoreticalcurve corresponding to q/q = (d/d) / . A comparison of the measuredand synthesized transverse distribution of unit discharge at oneof the cross-sections (Mile 410.20) is also shown in Fig. 4. Thedata shown in Fig. 4 correspond to a Missouri River discharge of61,300 cfs (1736 cu.ra.sec). Synthesized transverse flow distribu-tions at Mile 410.20 corresponding to a river flow rate of 35,000cfs (991. cu.m/sec) are shown in Fig. 5. To obtain the transversedistribution curve for depth d at the discharge of 35,000 cfs (991cu.m/sec), the water surface level was dropped by 3.8 ft (1.16 m)from that for 61,300 cfs (1736 cu.m/sec). This drop in watersurface elevation, corresponding to reduction in discharge from61,300 cfs (1736 cu.m/sec) to 35,000 cfs (991 cu.m/sec), wasdetermined from an estimated stage versus discharge curve for theMissouri River at Mile 411. The procedure thus assumes that thereis no significant change in the cross-sectional shape as the riverdischarge and depth are reduced, and also that there is a uniquestage-discharge relationship. For alluvial channels, theseassumptions are not strictly true. Changes in discharge can cause

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different amounts of aggradation and/or degradation of the bed indifferent parts of the cross-section. Also, seasonal shifts in thestage discharge relation may occur that evidently are associatedwith temperature-related effects on the bed configuration, amongother factors. Unfortunately, alluvial channel phenomena are notsufficiently well understood to quatitatively evaluate or signi-ficantly improve on these assumptions.

APPLICATIONS OF THE MODEL

The solutions of the thermal plume model, Eqs. (9) and (10), wereused to predict plume patterns in the Missouri River downstreamfrom the Fort Calhoun Power Station of the Omaha Public PowerStation District near Blair, Nebraska, and the Cooper NuclearStation of the Nebraska Public Power District near Brownville,Nebraska. Fig. 6 shows the measured (5) and the predicted (7) tem-perature-rise isotherms for the Fort Calhoun Station correspondingto two Missouri River discharges. A comparison of the measured(6) and predicted values of AT /8 for the Cooper Nuclear Stationthermal plumes on four different dates is shown in Fig. 7. Back-ground river flow and plant-discharge data corresponding to theseplumes are given in Table 1. The results shown in Figs. 6 and 7indicate that the model predicts thermal plumes in the MissouriRiver that are in quite good agreement with the measured plumes.

One major capability of the present model is that it permitsthe inclusion of the effect of the cross-channel flow induced bychannel curvature. Because of this, the model can predict expan-sions and contractions of the thermal plume as the river flowbecomes weighted towards one bank or the other. This feature isillustrated in Fig. 8 which shows the predicted thermal plumepattern near the Fort Calhoun Power Station for a Missouri Riverdischarge of 14,000 cfs (396 cu.m/sec). The computations for thiscase were performed for a 12-mile stretch of the river which wasdivided into ten subreaches, as shown in the insert map in Fig. 8.

CONCLUSIONS

A computational model, based on a diffusion equation, has beendeveloped for predicting the isotherm pattern -for shore-attachedthermal plumes in rivers. Comparison of the model results withfield measurements indicates reasonaoly good agreement. Furtherrefinements of the model, especially in defining the initial dilu-tion factor, need to be made based on laboratory investigations.The model can be used to assess the environmental impact of powerplant thermal discharges in natural rivers.

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Acknowledgement

Part of the work described in this study had been carried out duringa thermal plume modeling effort at the Iowa Institute of HydraulicResearch sponsored by the Omaha Public Power District. The computertime was provided by the Graduate College of the University of Iowaand NALCO Environmental Sciences. The writers are thankful of toProf. John F. Kennedy of the Iowa Institute of Hydraulic Researchand Dr. Richarch Johnson of NALCO Environmental Sciences for theirencouragement.

iLIST OF REFERENCES

1. Chang, Y.C., "Lateral Mixing in Meandering Channels," DoctoralDissertation, Department of Mechanics and Hydraulics,The University of Iowa, Iowa City, Iowa, 1971.

2. Fischer, H.B., "The Effect of Bends on Dispersion in Streams,"Water Resources Research, Vol. 5, No. 2, April 1969.

3. Giaquinta, A.R., and Sayre, W.W., "Effects of Outfall Design onthe Thermal Impact of Byron Station Slowdown Discharge,"IIHR Limited Distribution Report No. 39, Iowa Instituteof Hydraulic Research, The University of Iowa, Iowa City,Iowa, Feb. 1976.

4. Jirka, G.H., Abraham, G., and Harleman, D.R.F., "An Assessmentof Techniques for Hydrothermal Prediction," Report No.203, Ralph M. Parsons Laboratory for Water Resources andHydrodynamics, Massachusetts Institute of Technology,Cambridge, Massachusetts, July 1975.

5. OPPD, Nebraska City Power Station Unit No. 1 EnvironmentalAssessment Appendix for Section 10 Permit, Volume 2,June 1975, Omaha Public Power District, Omaha, Nebraska.

6. Paily, P.P., "Thermal Plume Surveys in the Missouri RiverNear the Cooper Nuclear Station," NALCO EnvironmentalSciences Report No. 550107642; Northbrook, Illinois,Jan. 1976.

7. Paily, P.P., "An investigation of the Thermal PlumeCharacteristics for the Fort Calhoun Power StationOutfall," IIHR Limited Distribution Report No. 30, IowaInstitute of Hydraulic Research, Iowa City, Iowa,June 1975.

8. Rozovskii, I.L. "Flow of Water in Bends of Open Channels,"Academy of Sciences of the Ukrainian Soviet SocialistRepublic, 1957, (Translation No. OTS60-51133, Office ofTechnical Services, U.S. Dept. of Commerce).

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9. Sayre, W.W., "Natural Mixing Processes in Rivers," Chapter 6in Environmental Impact on Rivers, Shen, H.w. (Ed.),Ft. Collins, Colorado, 1973.

10. Sayre, W.W., "Investigation of Surface-jet Thermal Outfall forlatan Steam Electric Generating Station," IIHR Report No.167, Iowa Institute of Hydraulic Research, The Universityof Iowa, Iowa City, Iowa, April 1975.

11. Sayre, W.W., and Yeh, T.P., "Transverse Mixing Characteristicsof the Missouri River Downstream from the Cooper NuclearStation," Iowa Institute of Hydraulic Research, TheUniversity of Iowa, IIHR Report No. 145, 1973.

12. Sium, 0., "Transverse Flow Distribution in Natural Streams asInfluenced by Cross-sectional Shape," M.S. Thesis, TheUniversity of Iowa, Iowa City, Iowa, July 1975.

13. Weil, J., and Fischer, H.B., "Effect of Stream Turbulence onHeated Water Plumes," Journal of Hydraulics Division,ASCE, Vol. 100, July 1974.

14. Yeh, T.P., "Transverse Mixing of Heated Effluents in Open-Channel Flow," Doctoral Dissertation, Department ofMechanics and Hydraulics, The University of Iowa, IowaCity, Iowa, May 1974.

15. Yotsukura, N., and Cobb, E.D., "Transverse Diffusion of Solutesin Natural Streams," U.S. Geological Survey ProfessionalPaper 582-C, 1972.

16. Yotsukura, N., Fischer, H.B., and Sayre, W.W., "Measurementof Mixing Characteristics of the Missouri River betweenSioux City, Iowa, and Plattsmouth, Nebraska," U.S.Geological Survey Water Supply Paper 1899-G, 1970.

17. Yotsukura, N, and Sayre, W.W., "Transverse Mixing in NaturalChannels," Water Resources Research, Vol. 12, No. 4,August 1976.

I

NOTATIONS

The following symbols are used in this paper:

A = area of river flow cross section;a = initial dilution factor;3 = width of river flow cross section;b = coefficient used in Eg. 19;b, = exponent used in Eq. 19;D = diffusion factor;

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d_ = local depth of flow;d = average depth of river flow cross section;E = overall transverse mixing coefficient;g = acceleration due to gravity;n = Manning coefficient;p = fraction of river flow occupied by plume;P = normalized cumulative discharge;Q = volumetric discharge rate of thermal effluent;QR = river discharge2 = discharge per unit width;q = width-averaged discharge per unit width;q = cumulative discharge from near bankR = radius of curvature of river bend;S = energy gradient;u = depth-averaged local velocity of flow;u = cross-sectional average velocity of flow;u* = shear velocity;x = longitudinal coordinate;x = position of virtual source;2 = transverse space coordinate;AT = excess temperature within plume;AT = centerline or bank excess temperature within plume;a = dimensionless transverse mixing coefficient;© = initial excess temperature;Gj = excess temperature after initial dilution;a = variance of the normal probability density functions,P Eq. 4.

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