Regulatory Guide 4.4

Reporting Procedure for Mathematical Models Selected to Predict Heated Effluent Dispersion in Natural Water Bodies

ML003739535
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Issue date:	05/31/1974
From:	Office of Nuclear Regulatory Research
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RG-4.4
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May 1974 U.S ATOMIC ENERGY COMMISSION

REGULATORY (3UIDE

DIRECTORATE OF REGULATORY STANDARDS

REGULATORY GUIDt 4.4 REPORTING PROCEDURE FOR MATHEMATICAL MODELS

SELECTED TO PREDICT HEATED EFFLUENT DISPERSION

IN NATURAL WATER BODIES

A. INTRODUCTION

simulate the dispersion of cooling water effluent within the receiving water body. Because of the unique In accordance with Appendix D, "Interim properties of each plant site, no single available model is Statement of General Policy and Procedure: universally applicable to all site conditions. Therefore, a Implementation of the National Environmental Act of need exists to differentiate among existing models and

1969"' of 10 CFR Part 50, "Licensing of Production to identify those that will yield an optimum simulation and Utilization Facilities," and proposed Section 51.5 of of effluent/receiving water interactions at a particular

10 CFR Part 51, the Commission prepares an site. For background purposes, a qualitative account of environmental impact statement for consideration in the the basic physical principles and critical site factors upon licensing actions of certain nuclear facilities. In addition, which such interactions are founded is presented in each applicant for a permit to construct such a facility Appendix A. The fundamental differential equations, must submit an "Applicant's Environmental mathematical approximations, and solution techniques Report-Construction Permit Stage" (§51.20, 10 CFR for simulating turbulent transport processes are Part 51), which discusses the probable environmental discussed in Appendix B. The Regulatory staff regards impact of the proposed facility. Should the proposed Appendix B as representative of acceptable facility be a nuclear power plant, thermal effects from mathematical procedures used to model the dispersion the release of condenser cooling water or dosed cycle of heated effluent in aquatic systems.

blowdown to a natural water body can have a significant impact.

C. REGULATORY POSITION

Direct, quantitative measurements of environmental impact due to thermal discharge in natural water bodies an not possible during the preconstruction stages of To aid in the assessment of propoaed thermal proposed plants. Consequently, reasonable discharge mathematical models by the Regulatory staff, approximations to the interactions of thermal discharfs a uniform reporting format is desirable. Consequently, with the environment must be adopted .to provide a basis an itemized table of relevant modeling factors, such as for impact aseument. The applicant may select that shown in Exhibit 1, should accompany descriptive mathematical models as one such means of material for the one or more models submitted by an approximation. Section 5.1 of Regulatory Guide 4.2, applicant. The table is a logical extension of that

"Preparation of Environmental Reports for Nuclear presented in an existing model review,' essentially Power Plants," suggests that details of mathematical differihg only in the level of detail specified. Upop modeling methods should be given in an appendix to the completion of pertinent entries, the table functions as a Environmental Report. This guide describes a procedure comparative tool, enabling an analyst to assess the acceptable to the Regulatory staff for completing such an appendix.

B. DISCUSSION

I Polfastro, A. J., "Heated Effluent Dspersion in Large Lakes, State-of-the-Art of Analytical Modeling, Surface and Submerged Discharges," presented at Conference on Water As applied to nuclear power plants, thermal Quality Considerations: Siting and operating nuclear power discharge mathematical modeling attempts to accurately plants, Proceedinp Atonic Industrial Foruan, New York, 1972.

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proposed model (or models) against prototypical conditions. A tool of this type facilitates the model d. As presented in Exhibit 1, the MAT allows evaluation proos but completion of the table does not entries for four models is well w the prototype. This asure that the optimum simulation model of thou sructuring does not imply that four dffrent modes available will be selected, nor that the model (or models) should be sased; any number would be acceptable.

ultimately chosen by the applicant will be acceptable to the Regulatory staff. Alternatively, the applicant may e. Individual entries to the MAT usually consst devise other acceptable approximations to the of either a YES response or a numerical value. In casm interaction of a thermal discharge with the environment; where a numerical entry possesses units of measure, the this guide should not be construed as advocating the International System of Units (SI) is preferred. Under exclusive use of mathematical models. circumstanoes in which the entry is not applicable to the conditions of the prototype or model, a response of N/A

may be ued.

The Model Assessment Table (MAT) (Exhibit I) is organized relative to the four principal types of f. The values of a number of MAT entries characteristics pertinent to thermal discharge depend on which coordinate system is applied to a mathematical modeling: model. The terms longitudinal," "lateral," and

"vertical" have been propoded to define the ortholnel S Dischar9e Characteristics axes of a horizontal discharge that propagate Receiving Water Characteristics losibtudina~y in the discharge direction, laterally in the Discharge/Receiving Water Interactions horizontal transverse discharge direction, and vertically Model Characteristics In the tranverse discharge direction. The applicant shoud specify his choice of model coordinate system relative to local true North at the prototype.

Intructions for preparing ech section of the MAT

are presented below.

2. Dblcliw Cchactewktk

1. Genad Instruction Most descriptive properties of the disckarge are a. The basic intent of the table is to provide a supplied by a simple binary choice of entries (Exhlbit 1).

simple and direct means of comparison between Except where otherwise stated, a YES response in the prototypical conditions and thoe phenomena being appropriate MAT location is sufficient.

modeled. The MAT is not intended to suppwant the descriptive materia that normaily accompanies a. Type. Discha type should be identified as submisson of a modeL On the contrary, such descriptive either single port or multiport; a discharge canal is material should comprise the textual portions of the regarded as a single port. If the prototype dischare is a appendix to the Environmental Report and serve as a multitnmture and/or multiport variety, the total vehicle for justifying or elaborating entries made in the number and spacing of exit ports for each structure table. should be given in the descriptive text.

b. A secondary objective of the MAT is to b. SbWp The exit port shape should be ducidate unique properties and ranges of applicability of identified as either round or rectangular; a slot jet is each itemized model in order to establish a basis for considered rectangular. If a model uses a slot jet differentiating between models. As a result, certain approximation for a multiport diffuser, the entry should portions of the Discharge Characteristics and Receiving indicate a rectangular shape for the model. The linear dimension of each exit port should be specified, and Water Characteristics sections are irrelevant to the irregulady shaped discharge ports should be described in models being assessed, and the Model Characteristics the text.

section and portions of the DIshle/Receiving Water Interactions section are irrelevant to the prototype. In vueh parts of the table entries need not be made. c. Location. The discharge location is either shoreline or offshore. Shoreline is defined as the boundary between the land surface and the receiving c. As a rule, models are applie4to a set of water body. The decision for discharge ocation depends environmental conditions that may be characterized as upon the point of entry to the receiving water body.

wont probable cam. AU table a should be wad upon these conditlons and the consequances satenfs from them. In addition, the descriptive material should d. Pudtiam. The discharge postion is eaher indicate the environmental condition to which the table inarface or submerged. A surface discharge is defined as applies. If more than one set of conditions is required to one for which the water surface is a boundary at the cover the cases of interest, more than one MAT may be point of entry to the receiving water body. All other positions ar regarded as submerged. For a prototype necessary. with a submerged discharge, the text should state the

4.4-2

saline stratification at some time during the annual

ýertical distances from the discharge centerline to (1) the thermal cycle. If such is the case for the prototype, the bottom and (2) the surface of the receiving water body. fact should be noted on the MAT, and the extent to e. DbvctL In the case of the prototype, the which stratification affects the prototype should be direction should be specified as the described in the text along with a discussion of the horizontal discharge ability of each model to simulate this situation.

compass angle (degrees) of the initial horizontal component of discharge relative to local true North; the e. Natural Current. The predominant natural nonhorizontal discharge direction should be specified as current in the immediate vicinity of the discharge should the angle (degrees) produced by the initial discharge direction vector and the horizontal. If a model can be specified; model entries should include the validity range of current speed and direction. Direction is simulate both horizontal and nonhorizontal discharge directions, this should be indicated in the MAT. defined as the clockwise angle between the horizontal flow component and local true North. For systematic f. Volume Flow Rate. The anticipated operating natural current variations in more than one direction, range of volumetric flow rates for the prototype should multiple entries may be made. The detailed current be given. structure within that portion of the water body likely to be affected by the discharge should be discussed relative g. Dchiuge Vdocity. The normal operating to its impact on model applicability.

range of discharge velocities at the outfall should be specified. 4. Dichargf Receiving Water Interactions

1. Effluent Excess Temperature. Excess a. Jet Entrainment. The MAT should indicate whether jet mixing is the dorninant thermal dispersion temperature is defined as the difference between effluent temperature and ambient water temperature at mechanism in the near-field. The principal criterion for jet entrainment is the initial densimetric Froude number, the discharge point. The measured or expected value of the extreme operating range of which should be each temperature parameter (intake, discharge, ambient)

should be given in the text. specified in the table. Model entries should cite the densimetric Froude number limits of applicability for each model.

3. Receiving Water Chinactks A number of models utilize empirical a. Type. The type of receiving water body entminment coeftfcets to simulate jet mixing. In such should be identified in the MAT by the following cues, the numerical value(s) of coefficients applicable to number code: prototype conditions should be given for each

(1) lake orthogonal direction in which entrainment is modeled,

(2) rime, tidal and the text should contain the rationale for selection of

(3) river, nontidal each coefficient.

(4) estuary

(5) ocean b. Cross Flow. In the presence of an ambient

(6) cooling pond current, all discharges, regardless of initial flow

(7) reservoir direction, ultimately flow in the ambient current direction. If the current interacts in the near-field with a The entry for a model should include those noncoincidental discharge, cross flow conditions prevail, water bodies to which the model is applicable. and the MAT should reflect this fact. The principal criterion for cross flow is the velocity ratio defined as b. Depth at OutfaL Depth at outfall is defined as the ratio of initial discharge velocity to ambient current the naturally occumring extreme range of water depths, velocity. The MAT should contain the operating range of sich as might be caused by tides, likely to be velocity ratios for the prototype and the range of encountered at the discharge point. applicability for each model. If the initial velocity ratio is sufficiently small, presure drag effects on the c. Bottom Slope The near-field spatially discharge by the ambient current can be significant. In averaged angle of the receiving water bottom in the order to simulate the pressure force exerted on the direction of discharge should be specified for the discharge, many models employ an empirical relation prototype; the validity rane of bottom dope angles containing a drag coeficient. When such is the case, the should be given for each model. Also, a bathymetric map coefficient should be specified as a MAT entry, and the of the discharge aren should be idcluded with the text. functional form of the relation should be presented in Such a map would be a convenient medium on which to the text.

display the geometric properties of the discharge and the c. Natuba Turbulence. Natural water bodies orientations of model coordinate systems. possess a varying degree of natural turbulence.

Consequently, the effects of turbulence should be d. Natural Stratification. Some receiving water included in any thermal dispersion model which bodies, especially lakes, exhibit natural thermal and/or

4.4-3

purports to be valid outsde the near-field region. When Moda/Chauacteriltta N.

dealing with natural turbulence, many models employ an eddy diffualvity or dispersion coefficient to simulate This section of the MAT supplies no additional transport by turbulent diffusion. Under this information for understanding thermal dispersion by the circurmatafice, the assumed initial values of the prototype. However, in order to expand the basis for coefficient in the three orthogonal directions of the differentiation among models, the functional propertiet model's coordinate system should be specified, and the of each model should be included for completeness.

coefficient's functional form(s) and conditions of application should be discussed in the text. a. Fiad. The dispersion field to which the model applies should'be specified in the MAT by the following number code:

Although wind stress is typically responsible'

(1) Near-field for considerable induced turbulence occurring in water

(2) Intermediate-field bodies, this phenomenon can also directly affect the (3) Far-field surface velocity distribution. The average and extreme (4) Complete-field range of observed wind vectors for the worst probable conditions at the prototype should be included. A MAT b. Dimenion. This entry identifies the spatial entry should appear for those models that include wind properties of the model solution. The directions are stress. defined consistent with those previously utilized in the table. The Regulatory staff puts no constraint on the d. Buoyancy. Buoyant forces accelerate flow in number of dimensions to which a model applies.

the direction of discharge while enhancing lateral spreading. A detailed discussion of the importance of c. Mathematical Approach. The solution buoyancy relative to other thermal dispersion technique utilized by each model should be specified by mechanisms in the prototype should be presented; the the following number code:

means by which the effects of buoyancy are simulated in (I) Phenomenological each model should also be discussed. (2) Analytical

(3) Finite Difference e. Recrculation. When only a limited amount of (4) Integral entrainment water is available for mixing, such as in (5) Finite Element some shallow water discharges, partly diluted effluent (6) Stochastic may be re-entrained or recirculated into the discharge (7) Other (state)

plume. The inclusion of nonambient dilution water decreases thermal dispersion while increasing plume size. d. Approximation. Simplifying approximations The MAT should indicate if recirculation is fesaible for to the mathematical formulation of the model should be the prototype and whether the models can simulate this indicated by one or more of the following:

process. Substantiation for the entry should be (1) Steady State presented in the text. (2) Boussinesq

(3) Hydrostatic Pressure f. Surface Heat Tresfer. Every thermal (4) Buoyancy Decoupling discharge plume which is bounded by the water surface (5) Other (state)

experiences heat loss to the atmosphere. Since the quantity of heat dissipated in such a manner varies "Other" approximations applied to each directly with the surface area of the plume, surface heat model should be discumed in the text.

transfer is largely a far-field phenomenon. The simplest, e. . Model Verifiatibo. Efforts to verify the most direct approach for estimating surface heat transfer model by field and/or laboratory measurements should employs the excess temperature concept, which requires be indicated in the MAT and described in the text.

the adoption of an empirical surface heat transfer coefficient and often an equilibrum temperature. Under f. Computer Program. The availability of a such circumstances, the range of pertinent coefficients computerized version of the model should be specified and temperatures applicable to the prototype should be by one of the following:

specified in the MAT, and the coefficient's functional (1) Proprietary form, if any, should appear in the text. An alternative (2) Available on request method for modeling surface heat transfer may be (3) Available in open literature adopted, provided the method is described in detail. (4) No program exists.

4.4-4

MODEL >

ASSESSMENT o TABLE _ __ _

SINGLE PORT

TYPE MULTIPORT

ROUND

RECTANGULAR

LCTOSHORELINE O F H R

OFFSHORE_________ SLOCATION

SURFACE ____

O POSITION

(n SURMECE

SUBMERGED

oa HORIZONTAL

DIRECTION NON-HORIZONTAL

3 VOLUME FLOW RATE (M /SEC) ___L__

DISCHARGE VELOCITY (M/SEC)

EXCESS TEMPERATURE (deg C)

TYPE

z r DEPTH AT OUTFALL (Ml __,__,_

Lu BOTTOM SLOPE (dgeg)

W_~ NATURAL STRATIFICATION

S CRNSPEED (M/SEC)

R DIRECTION (degL)

JET ENTRAINMENT

FROUDE NUMBER

LONGITUDINAL ___-__________

ENTRAINMENT

LATERAL

COEFFICIENT LATERAL

__ _

Oz VERTICAL _

Z 0 CROSS FLOW

SVELOCITY RATIO

U <PRESSURE DRAG

W Lu DRAG COEFFICIENT _ _,_"_

Lj Z NATURAL TURBULENCE

EDDY LONGITUDINAL ,_______

1: cc DIFFUSIVITY LATERAL _,,_"___ ___

U) VERTICAL _______

S WIND STRESS

BUOYANCY "

RECIRCULATION

SURFACE HEAT TRANSFER

2 HEAT TRANSFER COEF (CALJM SEC-de C) ,_"_

EQUILIBRIUM TEMPERATURE (deg C) _,_____

FIELD , '

LONGITUDINAL *, "

DIMENSION LATERAL _ _ __

uj

0 1 VERTICAL ,' "

MATHEMATICAL APPROACH

f . PROXIMATIONS ,* .*'

MODEL VERIFICATION '

COMPUTER PROGRAM _ _ ___

EXHIBIT 1

4.4-5

APPENDIX A

BEHAVIOR OF THERMAL DISCHARGES

1. Nomenclature the discharge relative to the Intake, the volumetric flow rate of cooling water, and the initial discharge velocity.

From a phenomenological viewpoint, a thermal If not directly measurable, the discharge velocity can be discharge can be partitioned into three spatial fields, estimated from the volumetric flow rate and the exit ch characterized

pocesses. by a different set of dominant port cross-sectional area and orientation.

The "near-field" is marked by the interaction between the kinematic heated effluent and the receiving 3. Environmental Factors Governing Thermal water body. Since the effluent velocity usually exceeds Diprso the receiving water velocity, the discharge is referred to as a "jet." By definition, jet momentumn forces Thiree factors or physical processes govern the predominate in the near-field, however, buoyancy forces thermal dispersion of heated effluents in natural may also be important. In the "far-field," ambient flow environments:

determines the shape and position of the thermal "* entrainment discharge, which typically is called a "plume." The "* turbulent diffusion

11-defined region joining near- and far-fields has been "* surface heat exchange designated variously is the "interamelate-field" or transition zone. In this region the heated discarg Advection is another major process that directly changes from an active jet to a peshie plume as the influenrces tin size, shape, and distribution of heated effluent comes increasingly under the Influenc of the effluents. The interactive Woe played by advective receiving water body. In the tmanition region both the phenomena such as nilent currents Is discussed below, discharge flow and the ambient flow ame important. but only to the extent that advection affects dispersion This overview of discharp/lreceiving water proceses interactions serves to introduce the fundamental a.Esiralninnet terminology of thermal dischrg analysis, as well as to identify the grows behavior of heated effluents. Consider first the hypothetical case of a L. Engqneering Deailn Fact-ors Governing Therma nonbuoyant discharge into a stagnant homogeneous Disp rsion environment. (in the context of this discussion, a stagnant environment is one in which the magnitude of Outfall designs for aqueous thermal effluent from the Initia discharge velocity is much greater than any power plants have changed drastically in recent years. local ambient velocity.) The injection of a fluid as a jet Early structures usually consisted of simple shoreline into another fluid results in the generation of turbulent canals or pipes that discharged their contents at or just eddies due to shearing stresees; caused by the velocity below the water surface. Heated releases fromn structures difference between the two fields. Shearing stresses so of this type tend to produce thin surface plumes of large produced represen t the lateral flux of momentum, and areal extent, which are efficient in transferring heat to they are directly proportional to the velocity vector the stniosphre. Currently, many discharge structures difference.

are being designed to maximize dilution near the outfall while minimizing detectable surface plume area in the Eddy motion along the jet boundary yields a far-field. The result has been submerged structures with net mixing of jet fluid with ambient fluid, in effect

• Wge or multiple high-velocity exit ports. Outfalls of broadening and diluting the jet at increasing centerline this type are called diffusers, and they utilize a high distances from the outfall. This mixing and dilution is discharge velocity to produce intensive mechanical called entrainment, and the constant of proportionality mixing in the near-field. relating jet volume flux to velocity is referred to as the entrainment coefficient. Note that for a submerged jet Details of engineering design for outfall structures discharging into an infinite medium, entrainment largely ame generally unnecessary for thermal discharge modeling propagates transversely to the discharge direction. Ifthe problems. However, consideration must be given to jet exits at the suirface, entrainment is constrained by the outfall geometry, that is, the size and shape of the exit sir-water interface; if the vertical extent of the jet is port(s) or canal and its location and orientation relative equivalent: to the receiving water depth, vertical to some fixed coordinate system. The choice of enitrainment is nonexistent.

coordinate system is arbitrary, but a system is usually selected that simplifies the mathematics of the model. The entrainment-Induced transfer of jet momentum to the ambient medium progresses outward In addition to outfall geometry, several oth-r plant fronm the jet boundary, in effect altering the transverse design factors must be known for modleling purposes. velocity profile from a top-hat shape at the discharge These include the heat rejection rate, temperature rise of point to a normal or Gaussian distributio

n. Laboratory

4.4-6

By definition, for Froude numbers greater than e xpe rimental data have indicated that suitably unity, inertial forces (i.e., jet momentum) dominate; for time-averaged Gaussian velocity profiles are similar:' Froude numbers less than unity, buoyancy effects (i.e.,

that is, each transverse profile along the jet beyond the density stability) are predominant. This principle has point at which the centerline velocity begins to decay been corroborated by laboratory experiments, although has the general form field data 3 show that buoyancy effects at the discharge

2 (A-1) can be significant for initial densimetric Froude numbers

= Ue-n u as large as about 3. Obviously, in order to maximize thermal dispersion by entrainment, the preferred in which U. is the velocity at distance n normal to the approach would be to optimize plant engineering jet centerline, U is the jet centerline velocity, and a 2 is characteristics: increase discharge velocity and/or the jet velocity variance along n. This finding has decrease the vertical dimension of the outfall (i.e.,

permitted the simplification of many near-field thermal increase the densimetric Froude number).

discharge models, although the general applicability of the Gaussian approximation to real discharges remains Regardless of initial conditions at the discharge open to question.

port, jet velocity eventually decreases to a point where the local Froude number falls below one. When this For the case of a heated jet, the Gaussian happens, vertical mixing is suppressed, and the jet ceases approximation of the transverse temperature profile along the jet takes the form, to thicken. However, buoyancy also tends to induce lateral spreading due to the horizontal density gradient, in effect enhancing dilution while broadening the plume.

Tn = Te-n!/2X' aT 2 (A-2) With decreasing thermal plume temperature the normalized density difference, Ap/p, approaches zero, the Froude number increases above unity, and vertical where Tn and T are the temperature in :he transverse mixing can become significant again.

direction and the jet centerline temperature;

respectively, OT. is the temperature variance along n, and In the presence of an ambient current, the jet X is an adjustable constant. According to Taylor's discharge and natural flow interact to cause the jet to be theory,' heat diffuses more rapidly than momentum in deflected towards the direction of natural flow. This the transverse flow direction. Therefore, the value of X is cross-flow effect is important if the jet and current are always greater than unity.

initially perpendicular, becoming less so as the two flow Under most circumstances, heated effluent is directions approach coincidence. At some distance from the discharge point, the magnitude of which depends less dense than the receiving water in the immediate vicinity of the outfall. As a result of the density upon the ratio of discharge velocity to ambient velocity, the motion of the effluent completely follows that of disparity, there is a buoyant force on the jet acting both the ambient current. If the initial velocity ratio is less vertically and horizontally. A submerged buoyant jet

.ends to rise to the surface; a buoyant jet at the surface than about 10, cross-flow effects on the discharge jet are significant in the near-field."

fam= a stable density layer. For either case entrainment is reduced, especially in the vertical direction, and the dilution rate decreases. Ambient current motion around the discharge jet creates a pressure drag analogous to that induced on a The degree to which buoyancy influences a solid object inserted into a uniform flow field. A

fraction of initial discharge momentum is expended to heated jet is suggested by the densimetric Froude number, defined as maintain the integrity of the jet in response to the external pressure field. If the initial velocity ratio is below 2, pressure drag effects on the discharge jet by the

5 Fd

• 1/2 U- (A-3) ambient current can become important.

( P)I 3 R. A. Paddock, A. J. Policastro, A. A. Frigo, D. E. Frye, and J. V. Tokar, "Temperature and Velocity Measurements and where Uo is the initial jet velocity, g the gravitational Predictive Model Comparisons in the Near-Field Region of Surface Thermal Discharges," Center for Environmental Studies, acceleration, h the vertical thickness of the jet, Ap the Argonne National Laboratory, ANLUES-25, 1973.

density difference between effluent and ambient water,

4 and p the ambient water density. The nondimensional .LH. Carter, "A Preliminary Report on the Characteristics Froude number represents the ratio of inertial forces to of a Heated Jet Discharged Horizontally into a Transverse buoyant forces in the jet. Cu-rent; Part I-Constant Depth," Chesapeake Bay Institute Technical Report No. 61, November 1969.

5

'G. N. Abramovich, "The Theory of Turbulent Jets," B. A. Benedict, J. L. Anderson, and E. L. Yandell, Mamachusetts Institute of Technology Press, Cambridge, "Analytical Modeling of Thermal Discharges: A Review of the Massachumetts, 1963. State of the Art," Center for Environmental Studies, Argonne National Laboratory, ANL/ES-1 8, 1974.

4.4-7

In the case of a shallow-water, nearshore jet in over viscous iwr forces. The transformution from which the discharge extends from the water surface to laminar flow to turbulent flow in a fluid Is defined by the bottom, entrainment is effectively confined to the Reynold's criterion, upstream, offshore side of the jet. On the nearshore side in the downstream direction, the quantity of ambient dilution water can be severely limited, and recirculatlon Re = Pv_ JA (A-4)

()f partially diluted effluent becomes a definite possibility. The situation is less serious for a discharge in where v is fluid velocity, I represents a characteristic deep water. In this case, ambient water can move under dimension (typically the water depth), and p is dynamic as well as around the jet; pressure drag is reduced, and viscosity. The flow is turbulent if Re, the Reynolds fresh dilution water is availabl- nn the jet's downstream number, exceeds some critical value, which in most side. natural water bodies proves to be small.

The influence of ambient stratification on Turbulent flow is irregular in time, and any thermal discharges depends directly upon the density of dependent variable of that flow can be characterized by the heated effluent relative to the vertical density profile the sum of a mean component and an unsteady of the ambient water. As a result, the heated effluent component. If the dependent variable happens to be can assume any of a number of different configurations. temperature, If the heated effluent is less dense than the T = T + T' (A-5)

ambient water, the effluent forms a surface layer analogous to the unstratified cage. Moreover, natural where T is the instantaneous temperature, 'T corresponds stratification can enhance effects due to buoyancy and to the mean temperature, and T' is the unsteady can inhibit vertical mixing, effectively thinning the component.

surface layer while increasing its horizontal extent. If the effluent has an intermediate density relative to the Turbulence is manifested by eddy motions, the density range of the stratified ambient, the discharge jet size of which may vary up to the characteristic tends to rise or fall to the point of neutral density, dimension of the turbulent medium. Turbulent diffusion depending on whether the exit port is submerged or at by eddies can be expressed as a variation in the unsteady the surface. Usually, when the effluent sinks to the level component of the diffusing property. If the diffusing of density compensation, the phenomenon is referred to property is heat, as a "sinking plume." Note that effluent dilution by entrainment continually influences the density along the discharge trajectory. Dilution may be sufficiently s ds (A-6)

pronounced to affect the level at which the plume equilibrates. Once the neutral density level has been where UTT represents the time-averaged product of the achieved, the effluent is free to disperse horizontally, unsteady components of velocity and temperature, regulated by natural diffusion and any residual jet dTl/ds is the mean temperature gradient, and D, is the momentum. Under certain conditions the effluent may eddy diffusion coefficient. All quantities are defined be more dense than the receiving water and, as a result, relative to the s direction. The definition for eddy flow to the bottom. During the winter months this can diffusion given in Eq. (A-6) is analogous. to that for occur if the ambient water has equilibrated at a molecular diffusion in a Fickian substance.

temperature below the point of maximum water density.

Should the effluent form a bottom layer, dispersion can Because turbulent diffusion is scaled to eddies, be inhibited by bathymetry and bottom friction. the magnitude of the eddy diffusion coefficient depends b. Turbulent Diffusion directly upon eddy size, which in turn determines the size of a diffusing plume. As a result, an empirical 4/3 power law of plume width is often applied to estimate With the decay of jet momentum, heated horizontal eddy diffusion. The 4/3 law apparently holds effluent increasingly becomes subject to external only for a semi-infinite water body such as an ocean; for perturbations of the ambient water body. The influence finite systems such as lakes and rivers, a constant of advective motions such as currents has been discussed horizontal eddy diffusion coefficient may be preferable in the preceding section, but of greater importance to at large distances from the effluent source.'

heat dispersion is the effect of ambient turbulent Furthermore, since the eddy spectrum is limited by the diffusion. size of the system, boundary effects may appreciably diminish horizontal dispersion in near-shore or shallow areas. Similarly, the typical vertical turbulence structure, Essentially all water motions or transport processes in natural water bodies can be regarded as 6 G. T. Csanady, "Dispersal of Effluents in the Great turbulent; that is, inertial forces in the water dominate Lakes,"

Water Research, Vol. 4, No. 1, 1970.

4.4-8

with a maximum near the water surface due to wind with the surface area affected, the process becomes significant only over large plume areas. This condition is stress, usually results in decreasing horizontal diffusion with depth. satisfied at some distance from the discharge point, after the effluent has been cooled appreciably by jet Whereas horizontal eddy diffusion varies with entrainment and natural turbulent diffusion. Hence, the the scale of horizontal turbulence, eddy diffusion in the effects of surface heat exchange need be considered only vertical direction can be constrained by shallowness of for the far-field.

the water body and the interaction between turbulence.

and buoyancy. Buoyant forces result from ambient The processes that determine the amount of thermal stratification and/or heated discharge, and they surface cooling from a heated plume are identical to impose a density stability on the water column which those that prevail under typical ambient conditions. As a wind-induced turbulence must overcome. The result, heating by a discharge can be regarded as a interaction between turbulence and buoyancy is often perturbation on the normal thermal regime. Adoption of expressed by the Richardson number, this viewpoint leads to the concept of excess temperature, Te, the difference between the observed g dp plume temperature and the natural or ambient water p dz R -2 i p(A-7) temperature. On the basis of this definition, a heat ldU\ budget formulation may be applied to excess k\dz/) temperature to yield the rate of heat transfer across the air-water interface due to the plume:

where g represents the acceleration of gravity, dp/dz is H = KTe (A-8)

the vertical density gradient, and dU/dz is the vertical gradient of the horizontal mean velocity. The quantity (l/p)(dp/dz) represents buoyancy or density where the heat transfer coefficient, K, is primarily a stratification and is often referred to as the stability, function of wind speed, ambient temperature, and while dU/dz can be used as a measure of turbulence. As excess temperature. Tables are available from which K

Eq. (A-7) indicates, the Richardson number increases may be estimated, given the aovropriate meteorological with increasing stability, implying that vertical eddy parameters.7 diffusion varies inversely with Richardson number. A

number of formulations for vertical diffusion as a Note that Eq. (A-8) is based upon the premise function of Richardson number have been proposed, but that the ambient water temperature is equal to the no single relationship has received universal acceptance. equilibrium value (i.e., the temperature at which the net rate of heat transfer across the air-water interface is Natural turbulent diffusion occurs over all parts zero). This condition is rarely achieved in nature, but the of a heated discharge. However, in the near-field region, equation offers a simplified means of expressing surface mechanically induced turbulence due to jet momentum heat exchange independent of the full heat budget entrainment is the major source of heat dispersal; in the equation. Since the error introduced by assuming an far-field, surface heat exchange actively transfers heat equilibrium condition is believed to be inconsequential, from surface plumes. Therefore, natural turbulence is any disparities are ignored.

often masked by other dispersion mechanisms except in submerged plumes or perhaps the transition regions of Surface cooling, whereby the excess heat is surface discharges. passed to the atmosphere, is ultimately responsible for limiting the areal extent of heated plumes. If surface C. Surface Heat Exchange cooling were not utilized in far-field modeling, the iotherms of excess temperature would never achieve The third factor influencing the thermal closure.

dispersion of heated effluents is heat dissipation to the atmosphere from the surface of the water body. Surface 7D. W. Pritchard and H. H. Carter, "Design and Siting cooling is an inherent property of air-water coupling and Criteria for Once-Through Cooling Systems Based on a therefore acts over all portions of a surface effluent. First-Order Thermal Plume Model," Chesapeake Bay Institute However, since surface heat exchange varies directly Technical Report No. 75, 1972.

4.4-9

APPENDIX 8 MATHEMATICAL FORMULATION OF THERMAL DISPERSION

1. FWmxmlont SIul-iNo where AU thermal dispersin pbelso as govewed by Tm the basic laws, of moss, moamtum, and eerly conservation and an equation of taMe. Individu models specif heat at constant volume differ in formulation to the extent thdt approximations k coefficient of thermal conductivity

@ =

and usplyfYi aumptos are applied to the et of VIToUm energy dissipation function

"equation ezprMS thoseel The term p(Buj/8xj) represents the increase in internal enerwy due to compression of the fluid.

If one condden a fluid having velocity components uj 0 a 1, 2, 3) with density p a function of position xj

(-1,2, 3), the bade hydrodynamic and thermodynamic quadon of date equations governing thermal diuu-oW may be written in Cartesion tensor notation as folow6: 1 If one ipnores the effects on density of dssolved

9"d and restricts attention to temperatures above that Caemertion of me of maximum density, an approximate equation of state may be written as at+ ax (B-1) P = pO l1 - a(T - TO)N (114)

Cemuyatl of mo2entoak where a + PUj a8is - To = reference temperature at which p = Po

2cokpuJGk a - coefficient of thermal expansion.

p Lp + a r,.+2ss 2. Conuei agl Equaons with he Boeinaq where The derivation of the above equations was quite PM alin that the coefficients x, c, a, k, and the

A - coefficient of viscosity denalty p wer not assuned to be constant. In any Eljk- permutation (cyclic) tensor pMGacUW situation, however, these quantities are only the component of the earth's rotation v.c very sightly tempemture dependent and, with one tor in the k direction exception, may be treated a- constants. The one p- Pressure exception is the external force term, pXi, in the

- the i component ofamy external fore momentum equation. Heem, the density cannot be treted as a constant. Its variation, when multiplied by CAmWatiu of eat mery ( sby) Xi, can produce an acceleration comparable in mgneitude to that of the Inertial term. Hence, ;, cv, a, and k my be treated a consants wherever they appear, pi (cT) + p9i +/- (c,T) and p may be treated a a constant except when it axj WOWha In the tasdfiorce term. This treatment of the density variation is the Bouainesq approximation. Its physical significance becomes clear upon consideration (kE - pl + 0-) that, in practical situations, the external force term is the acceleration of gravity.

'S. CNdrs*klbr, "Hydrodymmic and Hydroosuagtic 2Effects of salinity have been ignored for the sake of brevity Stablity." Oxford Unlwrdty, Oxford at the Clbendon pr.m, of diacur*do. The smplest equation of state, including the

1961. rdksitywoulidbe ofthe form p a1p ll-(Tr-T )+p(S-S

o o )J,

whime S = salinity, 0 = coefficient o;?saline contraction.

4.4-10

Under the Bousuneq approximation, the equation 3. Conervtdon EqtL"one for Turbulent Flow for manuconservation becomes The above equations express the basic conservation principles in terms of the instantaneous fields of velocity, temperature, pressure, and density. In any natural system, the flow is turbulent. Because turbulence which Is the familiar expression of ,ncomPresability. is Inherently random, solution of the above equations is impractical for any real problem. The equations must be in the light of Eq. (B-S), the momentum equation subjected to an averaging operation that separates the becomes deterministic and stochastic components of the quantities in question.

'IV,+ Uau i=

2 Accordingly, the instantaneous quantities, ui, p, and at a9 ktJfk T, are each written as the sum of an average component (denoted by a bar over the symbol) and an instantaneous

1 ap fluctuation about the average (denoted by a primed

+ 6P X, + VV2u, PO ax, -k

'po) symbol):

where V2 is the Laplahcan operator p -* p'

axl 2

+ a2 a2 ax 3

2 T =T+T'

where the average is taken over a suitably chosen time v is the coefficient of kinematic viscosity - 'A period that is small compared with the time scales of P0

and interest. Pritchard 3 discusses certain advantages in performing ensemble rather than time averaging.

(B-7) However, the more conventional concept of time AP = -Po a (T- To) averaging is followed here. Comistent with the Bouulnesq approximation, the above equations do not The energy equation reduces to contain a term representin fluctuations in density.

8T + V2 T (B.8) If the above expresions are substituted into Eqs.

(5-4), (3-S), and (B.6) and the system is averaged, the following onsevation equations in mean quantities whee x - k is the thermometric conductivity. remst:

pocv Note that the viscous energy dissipation function,

4, has been dropped under the Bouinmeq Equa*oem of oiate approximation. For an Incompresible flud,, the by 7 = p. 11 - a(T- TO)] (W-9)

dissipation function is #v

,[aE,a uJ] 2 Commuvaliom i a9 a+ 8( = (B-10)

From Eqs. (B-6) and (B-7), the velocity scale is of the order (aeTJXld)l/ 2 where d isa characteristic length air-X X

scale and [Xi becomes the acceleration of gravity in any practical problem. Consequently, the scale of 0 is of the &U1+ ý U - 2% $in(*u order pnaTS/d. From Eq. (B-3), the scale of i relative to that of diffusion is pood/k, which for typical length

- I --a swales (d-l cm) is much smaller than unity. Hence, + a (B-11)

dropping the dissipation function has negligible effect - x-(jN 5 on the heat energ equation.

Equations (5-S) through (M-8) express the bai 3 D. W. Pritchard, "Threedimenional Models," in Estuarine conservation laws subject to the Bousunesq Modelb. An Asmigt, Eurkoumsatal Protection Aency, approximation. Water Pollution Control Resewah Series. 16070 DZV, 1971.

4.4-11

interpreted as the nine components of the turbulent a5+-G2 +213 sin(CO stres tensor Rj RiI = -pulul P10F ,v'U 1 a- (B-IS)

,2 (3.12)

The classical approximation is to relate the turbulent mofsentum flux in Eq. (3IS) to aldignts of the mean velocity field, giving rise to the concept of au 3 .1 .a eddy viscosity coefficients. Theme may be introduced at Jaxi through expressions of the form I aF Pg +-V-2g3 Po ax3

+ -1 (u-'a

,) ((&13) R-PUUj = A(i)5 + A(j) an' (B16)

P0 uj)

The momentum equations have been written in where A(i) and A(j) are interpreted as the lateral eddy component form for a Cartesian sytem with x 3 positive viscosity coefficient Ah for j

• 3, and the vertical eddy upward. The qutntities 0 and [13 are, respectlily, the viscosity coefficient A, for j = 3. The anisotropy of latitude and the locally vertical component of the earth's turbulent diffusion, discussed in Appendix A, leads to rotation vector. Rotational effects are negligible in the the use of separate coefficients for the horizontal and vertical momentum equation, and the corresponding vertical directions.

terms have been dropped for simplicity.

Similarly, in Eq. (B-14) it is customary to relate Coeevastion of enealy the turbulent diffusion of heat energy to gradients of the mean temperature, giving rise to eddy diffusion aT + jL V2T - a(u7) (B_14) coefficients Di for heat:

Equations (B-9) through (B-14) differ in form from ujT :DjF

aT (B-17)

their instantaneous counterparts only through inclusion of the terms representing time average of products of fluctuating quantities. These terms represent turbulent The turbulent momentum and heat transports diffusion of heat and momentum and arise from those shown above are much larger than their molecular stochastic components of the motion that have time counterparts, and the terms representing the latter may scales shorter than the averaging period. be dropped from Eqs. (W-I I) through (B- 14).

These equations, together -'ithappropriate boundary If the above approximations are uted, the conditions, are the fundamental relationships governing velocity and temperature appear explicitly in the thermal dispersion in a turbulent incompressible conservation equations only as mean valies., The effects medium and form the basis for any subsequent of turbulence are concealed in the eddy coefficients A,

deterministic mathematical models.

A4, D1 . This simplification is to a certain extent illusory since the eddy coefficients are functions of the

4. Approximutions to the Basic Eqemo stochastic part of the motion and cannot be determined pdoif. The usefulness of any predictive model using The conservation equations cannot be solved in eddy coefficients is severely limited by the reliability their complete forms as shown. The actual formulation with which the magnitudes and spatial variations of of any given model depends upon the simpllficatlom nd thee parametes can be determined beforehand.

assumptions invoked consistent with the time and space scales of interest in the dispersion proces, plant design b. Skady Stab parameters, and the flow field and gpometry of the receiving waters. For highly simplfied gpometo' and For analytical simplicity, thermal dispersion flow conditions, analytical solutions am polble in some modes often a= em that the velocity and temperature instances. In general, however, mom realistic model fields are in steady ate, i.e., that require solution by numerical methods.

The more common simplifying assmptions are

-

enumerated and dicussed briefly below. a 1-t =0

a. Eddy CoefI.ients The validity of this assumption depends upon a careful asement of the relative magnitudes of the The velocity correlation teom on the averaging period, the important time scales of variability right-hand side of Eqs. (M 1) through (3-13) any be in the velocity and temperature fields, and the time

4.4-12

period over which the model is intended to apply. For an initial velocity Vo moves in an inertia circle in the cxample, the assumption of a steady state clearly cannot absence of external forces and confining boundaries. The be made for a model intended to describe the behavior radius of the inertia circle at mean latitude 0o is of a thermal plume in a tidal river or estuary if the Vo/211 3sin 0o. The time required for the water to move averaging period is short compared with the tidal period. around the circle (i.e.. period) is 21r2/N3 sin~O. If. for a If, on the other hand, the average is taken over several particular problem, the distance and time scales ot tidal cycles, and if the time period of interest is less than interest are small compared with the circumference and the scale of seasonal fluctuations in fresh water flow period of the local inertia circle, ro.rational effects maý

(river flow), a steady state assumption isreasonable. It is be neglected.

important to realize, however, that in these two In the near-field, neglect of the Coriolis force is examples, the magnitudes of the eddy coefficients differ a valid approximation. In the far-field, Coriolii effects greatly. For a longer averaging period, larger scales of might become noticeable as a cum sol deflection of the motion are included in the turbulence, and eddy thermal plume. The practical importance of the latter coefficients are larger. depends upon the lateral dimensions of the receiving water and the time scale of temperature decay relative to c. Hydrostic Approximatlon the inertia period.

e. Heat Exchange Coefficient In a motionless fluid, the vertical equation of motion reduces to the hydrostatic equation expressing Equation (3-14) must satisfy the boundary an exact balance between the vertical pressure gradient condition that the heat flux is continuous across the and the acceletation of gravity, water surface. On the basis of Eq. (B-1 7), this boundary condition may be written I a g"

D3 a3 = surface heat flux.

=0

where p_ is a function of x 3 and is the density that would exist in the absence of motion. Whether or not As discussed in Appendix A, the surface heat this equation is valid as an approximation to a real flow flux is taken to be proportional to the product of a system depends on the velocity, space, and time scales surface heat exchange coefficient K and the difference of concern in the vertical momentum equation. A simple between the actual temperature of the water surface T.

scale analysis indicates that the hydrostatic assumption and its equilibrium temperature e. The above boundary is generally a valid approximation except possibly in the condition then becomes treatment of high-velocity discharges, in which cae the pressure field has a dynamic contribution that is not necessarily small compared with the hydrostatic D3 ;3=I = K@,s - e) (B-20)

aX33=0

component.

Equations (B-14), (3-17), and (&-20) describe In general, then, the vertical pressure gradient the temperature field in the presence of a thermal in Eq. (B3-13) may be written discharge. In the absence of a discharge, the equations would be identical in form, with the ambient p -g + lop * temperature T.' replacing the general temperature T.

P0 P 8x3 Note that the difference Ta Te is the excess P0 ax 3 temperature defined in Appendix A. If the assumptions where P is the difference between the actual pressure are made that (I) the spatial gradients of ambient and the hydrostatic pressure, and p. g is the gradient of temperature are small compared with those existing in hydrostatic pressure. Substitution of Eq. (8-18) into Eq. the priesence of the discharge, and (2) the heat exchange (B-13) results in a vertical equation of motion in which coefficient and the equilibrium temperature are the same buoyancy is contained explicitly in the gravity term: with or without the discharge, then the following equation may be written to describe the excess I ax temperature:

a%+ Uj.

-J a-xj

=

(Dj a) (B-2 I)

at i +axj

!T =

a.

2_P a3 (_

ax axJ

Pp- g --- + -(~g with the surface boundary condition d. Planetaiy Rotation D3 8TeIx-o = K(T. - T,) = KTe (B-22)

3X3 .0

A simple scale analysis can provide an estimate of the importance of rotational effects for a particular The form of Eq. (B-22), with the ambient tispersion problem. Theoretically, a parcel of water with temperature replacing the equilibrium temperiture, is

4.4-13

identical to that dimcussed in Section 3c of Appendix A.

This form is preferred by many workers because of the *au* vau [* o 1,,t.S (,*.ll practical difficulty of obtaining the equilibrium temperature. It should be noted, however, that this simpler form depends upon the validity of the two where u a time-averaged axial velocity component assumptions given above. v a time-veraged radial velocity component

5. techn.a. fr Solving On Hydrodem aWd [es0A] g = the axial component of Thermodynamic Equatiow s buoyancy acceleration.

The mathematical form of the hydrodynamic and Only the axial momentum equation is shown thermodynamic equations that govern thermal here, since this is sufficient to illustrate the approach dispersion having been discussed, it is desirable to review used in near-field analysis. This approach consists the methods most frequently applied to solve those essentially of three steps designed to reduce the equations For the most part, such methods tend to be complete set of partial differential equations to a set consequences of the scale size and simplifying of ordinary differential equations expressing the approximations used to construct the model. The centedine velocity components, temperature, and following discussion deals with common solution Cartesian position as functions of s. These three steps techniques as they are applied to the spatial dispersion consist of (1) removal of the r-dependence by fields of thermal discharL integration of the conservation equations over the jet cross section, (2) specification of the radial profiles of a. Nwe-Field Modeling temperature and velocity, and (3) introduction of an entrainment function.

The case of a buoyant axdsymmetric jet discharging into a cross-flowing ambient stream, while The radial integration of the first step places representing only one of a variety of possible discharge the equations in so-lled "integral" form. These are not configurations, contains the important dynamical integral equations in the strict mathematical sense, but processes and illustrates the bac assumptions and rather in the sense that radial distributions of properties techniques generally used in ner-field modeling. The have been integrated away. The simpler set of ordinary transformation of Eqs. (3.9) through (1&14) into simpler equations expresses only bulk properties of the flow as forms suitable for the axisymmetric jet is functions of jet trajectory. Retrieval of the lost details straightforward but tedious and is not shown here. A requires the second step, in which radial distributions of complete derivation of the governing equations has been velocity and temperature are secified. These are usually given by Hint,4 whose notation, with slight maumed to have a Gaussian form, as discussed in Section modification, is used, below. The important point to 3a of Appendix A. The third step is an attempt to note is that the near-field approximation is basically atpreu the radial flux of dilution water into the jet in equivalet to the Pradti boundary War ap*roimation tern of the axial flow.

applied to free turbulent shea flows, viz,, utmemwlse In accordance with the first step above, gradients are much smaller this raa gradients, integration of Eqs. (B-23) through (&-25) over the area streamnwise velocitf are much' Iager thea rdi 'Yea velocities, and the deviation from hydrostatic pressumre is approximately constant throughout the jet. df2hr f r dr] =d (N-26)

Subject to these approxisnations, the conservation equations may be written in a natural cylindrical coordinate system (sr) denoting the local axial and radial directions, rspectively. The angular dependence does not appear since the jet is d -[2x f7 V(T -T..)r dr]u dT_

axisymmetric. The equations of conservation of moss, heat energy, and axial momentum become -Q-dT- 2w lim (-r) (&27)

+f I a a + V)= 0 (B-23)

k--2w U2 r dr] d-s aT _a= ia.-.

uTs- +V - Thrv., (B-24)

4E. A. Hirt, "Analysi of Round, Turbulent, Buoyant Jets Discharged to Flowing Ambients,' ORNL-4685, Oak Ridge

=[2w f ag(1 - T_.) r d]

National Laboratory, 1971. 0S

4.4-14

+ E [u., - 2w tim (ruIv (B-28) equations in which the unknowns are un, Tm, and b (not considered here are the three additional equations relating the jet position to the fixed Cartesian where the subscript s indicates a component in the coordinates).

s-direction, and the upper integration limit, ", for In the third step, the entrainment function practical purlpses, is the physical edge of the jet.

must be expressed in terms of the centerline value of jet Equation (B-26) represents conservation of velocity, un. The simplest relationship that has been used is of the form, mass and states that the rate of change of volume flux Q along the jet trajectory a is equal to the rate of entrainment E = aum , (B-30)

- 21r lim (rV) = E which states that the entrainment rate is proportional to the centerline jet velocity. The constant of proportionality a is known as the entrainment of fluid at the jet edge.

coefficient. Such a relationship, at. best, satisfies the intuitive expectation that a fast jet will have a higher Equation (B-27) states that the rate of change entrainment rate than a slow jet. However, the correct of flux of excess temperature G along the jet trajectory choice of a for any given problem is open to is balanced by the rate of entrainment of fluid of considerable question. It is also unlikely that a simple ambient temperature T. and the turbulent flux proportional relationship such as Eq. (B-30) applies in

7 nature. lConsiderable theoretical and experimental work

- 27hr im ) over the put decade has resulted in increasingly more complex (and hopefully realistic) forms for the of heat energy at the jet edge. entrainment function. HirSt 4 presents the most general expression to date:

Equation (B-28) represents the conservation of E = (0.057 + 097 sin2 bulk momentum. It states that the rate of change of s-momentum flux M. along the jet path is balanced by

(1) the s-component of buoyancy (with temperature (B-31)

replacing density), (2) the entrainment E[u..]s of [blum - [u.I I + 9.0b Vu2ru-[U].I

ambient momentum by the mean radial flow at the jet edge, and (3) the turbulent flux where 02 is the elevation angle of the plume above the horizontal and FL is the local densimetric Froude

- 21 lim (rV number. This expression is similar to but considerably more sophisticated than that shown in Eq. (B-30). From the Froude number and angular dependence, it is seen of momentum at the jet edge. Note that in the buoyancy that the "entrainment coefficient" is a dynamic quantity term, the equation of state has been used to express the that changes with the evolution of the jet. Also, the density in terms of the temperature. secnd term in braces gives an enhanced entrainment rate for the case of the jet and ambient flow not being Clnmue of the set of conservation equations collinear.

requires that the radial distributions of jet velocity and temperature (step 2) and the entrainment function (step In keeping with the usual definition of the

3) be specified. near-field as being the region where dilution by jet entrainment dominates the dilution due to turbulent Beyond the zone of flow establishment (the diffusion, terms describing the latter are usually dropped reion in which the jet possess a potential core), all from Eqs. (B-27) and (B-28). Formally, this'amounts to radial profiles are assumed to have Gaussian forms, as the approximations discussed in Appendix A:

dT.

a = (u. - [U._,) e-(r/b) 2 + 1U.1J 2w lm(r << Q--

<)

2 T - T. = (Tm - T,.) e-(i/xb) (13.29)

2wrlim(ru'v) <<<E[u.],

where um and Tm represent centerline values, b is the jet half-width (b - o'4), X is the spreading ratio introduced in Appendix A, and [u. I s is the component of ambient However, since the turbulent transport terms velocity in the s-direction. Substitution of Eq. (B-29) involve undetermined eddy coefficients and since the into Eqs. (B-26) through (B-28) results in a set of form of the entrainment function E is not well

4.4-15

established, a proper scale analysis to substantiate the The most widely used numerical solution technique above inequalities is very difficult. On physical rounds, for far-field model equations is the method of finite it does not appear to be reasonable that the entrainment differences. The fundamental principle of this method is is always independent of the ambient turbulence level. the subdivision of the solution region into a number of discrete grid points at which the derivatives in the b. Far-Field Modeling governing equations are approximated by finite differences. There are several approximation schemes in The far-field is generally assumed to be that current use, the most popular of which are truncated region of ihe thermal plume sufficiently removed from Taylor's series and the treatment of individual grid the discharge that (0) ambient turbulence dominates the meshes as discrete control volumes. Sophisticated forms mixing process and (2) the excess temperature is of finite difference techniques permit variable time steps transported by the ambient flow as a passive and grid mesh sizes. This refinement provides contaminant. considerable computational efficiency and added flexibility in the solution of time-dependent problems The first of these assumptions results in the use with irregular shoreline geometry.

of eddy coefficients to represent the mixing. This introduces an indeterminancy into the problem in that In recent years, a somewhat different technique the values of these coefficients are not known known as the method of finite elements has emerged as beforehand. a powerful tool for the numerical solution of hydrodynamic transport equations. In this method, the The second assumption eliminates the effects of domain of interest is subdivided into a number of "finite discharge momentum and removes the nonlinearity due elements" interconnected at a discrete number of nodal to buoyancy coupling between the heat energy and momentum equations. This results in a considerable points. Within each element the dependent variables are mathematical simplification in that the convective approximated by known shape functions whose velocity field in Eq. (&-14) is known through either magnitudes are determined by assumed nodal values. In direct observation or prior solution of the appropriately early applications of the finite element method, simplified momentum equations. solutions were obtained from a discrete set of linear algebraic equations derived through minimization of a For discussion purposes, mathematical models functional for the governing differential equation. The of the fAr-field region may be divided roughly into application of this method to hydrothermal problems deterministic, stochastic, and phenomenological types. was limited because it was not always possible to find the proper functional.

(1) Delemnistic Models Recently, however, this restriction has been removed through use of Galerkin's method,S in Since the actual motion and temperature always which the set of algebraic equations is obtained directly have stochastic components, deterministic methods must from the governing differential equation. The relate these components to time-averaged quantities in approximate solution is obtained not by a variational order that closure of the governing equations can be principle, but rather by orthogonalization of" the achieved. This having been done, the solutions to the solution error with respect to the known shape basic hydrothermal equations may be determined either functions.

analytically or numerically.

The finite elements, usually triangular in shape, may Analytical solution refers to the closed form be arbitrary in size and arrangement. This flexibility integration of the governing equations. As disc'ussed provides a -twofold advantage in that computational earlier, this method is possible only for highly simplified resolution can be varied at will throughout the region of cases. It is seldom possible to obtain analytical solutions interest, and almost any boundary shape can be for time-dependent flow fields or complex receiving approximated by the proper choice of triangular water geometry. Consequently, the utility of any elements.

analytical solution should be very carefully assessed by the modeler to ascertain the conditions under which the (2) Stochatic Models model might be a valid predictive tool. From a practical point of view, the attractiveness and elegance of In stochastic dispersion models, the probabilistic analytical solutions are often vitiated by the fact that behavior of the flow field is handled directly. The most the models from which they stem have been simplified promising stochastic solution technique is the Monte to the point that they no longer adequately simulate the Cado 'method, in which the probabilistic behavior of the prototype. Hence, in predictive far-field modeling of complex systems, physically realistic solutions might 5

14,._.A. Lotiuk, J. C. Andenon, and T. Belytschko, require the rather early abandonment of analytical "H*]nna Analyis by the Finite Element Method," ASCE

solutions in favor of numerical methods. ,oemat of Hydmulks Dwinoa. VoL 98, No. YI11, November

1972.

4.4-16

fliw field is modeled directfly by the assignment of behavior of each particle to be simulated until it passes

,tochastic properties to dispersing "packets" of particles

% beyond the region of interest.

as they are tracked from the release point. The excess temperature within any given region is proportional to The Monte Carlo method has the advantage of the local particle density. direct dispersion simulation, and also the very appealing features of conceptual and programmatic simplicity. It The model associates a temperature contribution, a should be noted, however, that with this technique it is deterministic velocity, and a dispersion rate with each still necessary to specify eddy diffusion coefficients.

moving particle. As each particle is tracked over a given tiTe interval, it is assigned a total displacement (3) Phenomenological Models consisting of the sum of two vectors. The first vector displacement is caused by the known deterministic flow field and is given by [D ,At, 1 U2At]. The second In the sense used here, phenomenological far-field displacement, which represents dispersion, is a random models differ from either deterministic or stochastic Gaussian function with zero mean and is derived from models in that they do not derive directly from solution the relationship of the eddy coefficient Di to the of the basic transport equations, but rather from a dispersion rate: combination of theory and correlations of observed I d 2 plume behavior with known laboratory and field flow Di =)(B-32) conditions. These models are relatively lacking in mathematical elegance and sophistication, and it is where oi2 is the particle variance in the i-direction. difficult to extract basic information from them concerning the relative importance of individual From Eq. (B-32) the random displacement vector is transport processes within the framework of the given by 1(2D1At)'/ 2 ,(2D 2At)1/ 2]. governing equations. However, an important advantage is that they are relatively easy to use and are based on a By use of a digital computer, many particles are compilation of observed phenomena and a minimum released from the source, allowing the stochastic number of simplifying assumptions.

4.4-17