thermally affected flows

THERMALLY AFFECTED FLOWS IN POWER PLANTS

J.H. KIMEnergy Conversion DivisionEPRI3412 Hillview AvenuePalo Alto, California 94304 USA

K.-W. YOUCenter for Advanced Studies in Energy and EnvironmentKorea Electric Power Research Institute/03-16, Munji-dong, Yusung-guTaejon, Korea 305-380

1. Introduction

Thermo-fluid dynamics plays an important role in power plant operations. A typicalfossil-powered or nuclear power plant uses a working fluid as a medium to transport heataround the plant system. As heat is added to or rejected from the fluid, thermal gradientsdevelop in the system and, under certain conditions, natural circulation can be induced.In another situation, hot and cold fluids may come in contact with each other, creatingthermal stratification, which can cause thermal fatigue on the piping materials. As afurther example, fluid is also discharged from power plants into the surroundingreservoir or estuaries, the thermal effluent dispersing over a wide area and possiblyimpacting on the ecological system around the power plant.When a thermal gradient is aligned with gravity, the buoyancy will naturally force

the warmer fluid to rise and the colder one to fall. This is the principle that is used forpromoting thermal (or natural) circulation. Natural circulation has importantapplications in fossil plant boilers and nuclear reactors.

If, on the other hand, the thermal gradient is against the gravity with the warm fluidlying over the cold one, thermal stratification can set in. If the stratification is severe,as in nuclear reactors in which the temperature difference between the two stratifiedlayers can often reach a few hundred degrees Fahrenheit (a couple of hundred degreesCelsius), it can cause thermal fatigue on materials and may pose a safety concern.Thermal effluents discharged from power plants are dispersed into the surrounding

reservoir such as a river, a lake or an ocean. This can create a horizontal as wen as avertical thermal gradient as the temperature of the effluent is different from that of thesurroundings.In this chapter, we win review flows with thermal gradient that are relevant to power

185A. Sejan et al. (eds.), Energy and the Environment, 185-198.© 1999 Kluwer Academic Publishers.

186 J.H. KIM AND K.-W. YOU

plant applications, in particular, natural (or thermal) circulation flow, thermallystratified flow, and thermal discharged flow.

2. Natural Circulation

Natural circulation refers to a bulk flow driven by buoyancy. Natural circulation playsan important role in many industrial applications [1], such as in nuclear reactors,thermosyphon reboilers, evaporators, solar water heaters, turbine blade cooling,electronic equipment cooling, and heat pipes.In a certain type of boiler of a fossil-fuel-frred power plant, the circulation of the

working fluid is produced by the difference in density between the steam-water mixturein the heated generating or "riser" tubes (heat source) and the water in the unheateddowncomers (heat sink); the two-phase mixture tends to rise due to its lower densitywhile the cooler and heavier water in the downcomers flows downwards, making acirculation in the flow loop. This is illustrated in Fig. la [2], and a typical naturalcirculation boiler circuit is schematically illustrated in Fig. 1b [3]. The thermal drivinghead is created by the difference in mean densities between the subcooled water in thedowncomer-supply tubes and the heated furnace evaporator tubes, and the driving head issufficient to overcome the flow resistances in the steam-water circuit. A drum is used toseparate the steam-water mixture before the separated steam is sent to the superheaterwhile the water is returned to the evaporator tubes via the downcomer. Since there is norecirculating pump, friction losses must be minimized around the circuit so that thethermal driving head alone can do the job. For this reason, relatively large diametertubes must be used and flow rate is adjusted during the design stage by appropriatelysizing the number and diameter of all tubes to provide adequate flow rates for all loadconditions. Flow rates vary directly with heat input (fuel firing rates) and tend tocompensate automatically for local heat input upsets.

,Downcomer '----Waterwall

Riser Tubes

Superheater(SH)

FurnaceWalls(Fum)

(a) conceptual drawing [2] (b) schematic circuit [3]

Figure 1. Natural circulation in boilers.

THERMALLY AFFECTED FLOWS IN POWER PLANTS 187

Natural circulation also plays an important role in long-term cooling of nuclearreactors under accident conditions. During a typical small-break loss-of-coolant accident(LOCA), the coolant can undergo both single-phase and two-phase natural circulation.With the recirculating pump of the nuclear reactor shut off, the cooling of the reactorcore must depend on natural circulation. In this situation, the natural circulation flowcan be sustained by the density gradients created by heating of the coolant in the core(heat source) and cooling of the same fluid in the steam generator (heat sink), so that thebuoyancy forces created by the density gradients drive the coolant to circulate around therecirculation loop. This is the same principle that is used in boilers as described earlier.Much study was conducted by the U.S. nuclear industry in the past [4] to resolvetechnical issues related to natural circulation in operating light water reactors. Thesestudies have demonstrated that cooling by natural circulation provides a viable heatremoval option under a variety of steady state and transient thermal-hydraulic conditions.

Air outlet

Air inlet---J~r--

oncrete --t--b mat

Wat r storage tank(-350,000 gall n )

~~~~~==]=*- Air int t

'oncreleshieldbuilding

Figure 2. An advanced PWR design utilizing natural circulation containment cooling.


In addition to such an application for operating nuclear reactors, the recent interest inthe design of next generation light water reactors further rekindled interest in naturalcirculation. In these new designs, natural circulation provides all or a large fraction ofthe required core flow rate even during normal operations. In the U.S., one of the designfeatures in the advanced light water reactor (ALWR) development program is its heavyreliance on natural circulation for core cooling during both normal operation andaccidents [5]. For instance, an advanced boiling water reactor (BWR) design reliesentirely on natural circulation for core cooling.Fig. 2 shows an advanced pressurized water reactor (PWR) design that relies on

natural circulation for containment cooling during an accident. During an accident, thehot steel containment vessel heats up the air to rise upward along the wall and exits tothe air outlet. This chimney effect draws the fresh air from outside the concrete shieldbuilding. The cooling of the containment is thus maintained by natural circulation.

3. Thermally Stratified Flow in Nuclear Power Plants

Thermal stratification phenomena may occur in the piping system of power plants.When two fluid streams of different temperatures come in contact, the density differencebetween them causes the cold and heavier water to settle at the bottom of the pipingwith the warmer and lighter water lying on top, thus creating a thermal stratification. Innuclear plants, these phenomena can occur in safety-related lines where there is potentialfor hot and cold fluids to come in contact with each other, potentially causing excessivethermal stress in the piping materials and posing safety concerns. Often, the phenomenaare caused by a leaky valve through which hot (or cold) water leaks into cold (or hot)water on the other side. Thermal stratification phenomena drew much attention becauseof the incidents at several nuclear power plants related to some fatigue failures that hadresulted in small amounts of radioactive leakage from unisolable branch pipelines [6].These branch pipes are connected to the main coolant pipe and the other ends of thebranch pipes are normally connected to check valves so that they effectively act as adeadend for the pipes. If the valve is leaky, however, cold water can leak into the branchpipe which is maintained at a much higher temperature. The cold fluid leaked into thebranch pipe naturally settles at the bottom of the pipe, creating a thermally stratifiedfluid layer. In a nuclear reactor, the temperature differential between the two fluid layerscan reach up to a few hundred degrees Fahrenheit (a couple of hundred degrees Celsius).Thermal stratification problems in nuclear power plants are very complicated.

However, they may be reduced to a couple of more tractable key components of thephenomenon. First, it is important to know the boundary between the thermally wellmixed region and the thermally stratified region in the branch pipe when cold fluid leaksinto the pipe. This requires the information on how far the main pipe flow turbulencecan penetrate into the branch pipe as this region of penetration will be thermally wellmixed while the remainder of the branch pipe will be thermally stratified. Once this isdetermined, one can calculate the temperature distribution within the thermally stratifiedportion of the piping. This second problem can be treated as natural convection in along horizontal enclosure and can be analyzed computationally.


3.1. TURBULENCE PENETRAnON INTO DEADEND BRANCH PIPE

189

Turbulence penetration into unisolable pipelines, or the transport of turbulence into adeadend piping branched from the reactor coolant system (RCS) line, represents amechanism for carrying hot RCS water into regions filled with cold stagnant water, thuspromoting mixing. How far turbulence can penetrate into the branch pipe defines theboundary between the regions of thermal mixing and thermal stratification.The experimental results of Kim et al. [7] typically showed that the turbulent

fluctuating velocities decayed exponentially with distance from the tee. Thisexponential decay can be deduced from the foIlowing simple argument:Following G.I. Taylor's postulation, the turbulent rate of dissipation can be given

(1)

where u is turbulent velocity, I is a characteristic eddy dimension, and ~ is a constant.This holds for all turbulent flows with a single scale length that do not requiregeneration of turbulent energy. Since in our deadend branch pipe we cannot generateturbulent energy as there is no pressure gradient, the above relation should be valid.Equating convected turbulent energy into the pipe to the dissipation rate, we have

(2)

since the bulk mean velocity in the branch pipe is zero. Now, the characteristic eddydimension I is of the order of the pipe diameter D so that Eq. (2) can be easily integratedto give

u = Uo exp (-~xID) (3)l2y2

In this experiment, the hot film anemometer output is presented as ~ Y (ftlsec),i.e., the RMS values of the turbulent eddy velocities were measured since there were nomeasurable steady velocities. The results are best summarized in Figure 3, which showsthe maximum tangential turbulent velocity measured at each cross-section as a functionof distance into the branch line measured from the main pipe junction. They appear tofall off exponentially with distance.The decay of the maximum turbulent velocities is exponential with increasing

distance from the tee. Following the argument presented above, the empirical decay lawthat fits the test data has been determined to be:

W = 4.26 exp (-0.315 xID) (4)

where x is distance into the branch pipe and D is its diameter.The results from this test are somewhat different from observations from some

limited in-plant testing. The in-plant data indicate that turbulence penetrates up to 22diameters into the branch lines. This apparent discrepancy is resolved in the following:

190 J.H.KIM ANDK.-W. YOU

In normal pipe flow, the walls of the pipe extract momentum from the flow whichmust be maintained through application of a pressure gradient doing work on the flow.Turbulent exchange of momentum can only be maintained in such a flow if theReynolds number is above a "critical" value of about 2000. If there is no net pressuregradient, such as is the case in the branch line, the amount of turbulent momentumentering the branch will only be sustained until the wall shear stresses reduce theReynolds number to below the critical value where the flow will become laminar. Inour problem, there is no mean velocity in the branch pipe but only the turbulentfluctuations. Nevertheless, in analogy with pipe flows, it is postulated that this wouldcall for the turbulence penetration velocity criterion to be written as:

Recrit = WD/v = 2000 (5)

where V is the turbulence velocity, D is the branch line diameter and v is the kinematicviscosity. At Reynolds numbers less than this value, momentum transport is bylaminar process and the tllrbulence bursts are no longer capable of being sustained. Ifthe empirical expression for the branch line turbulence, Eq. (4), is substituted into theabove expression, the following relationship results:

Recrit = 2000 = 4.26D/v exp (-0.315 x/D)

10. 0 r-r-rr-r-rr-r-rr-r--rr-r-rr-r--rr-\

1.0

o

..o

(6)

0.1

o.N =4.26 expo (-.315 X/D)

o I-A·I-Bo 2-Ao 3-Ao 4-A

OL..l.-L..L..1-L.-'-'-'--'-'-'-.L..1-'--'-'-'--:':-'o 10 12 14 16 18

X/D

Figure 3. Maximum tangential velocities for turbulence penetration tests [7].


Using y =0.8 X 10-5 felsec and D =0.25 ft, the critical turbulence penetration distancewhere the flow becomes "laminar" is found to be LID = 13.3. This compares well withthe observed experimental results.This empiricism may be extended further to explain the in-plant data which indicated

the penetration distance of 22 diameters. There, the main line mean velocity was 50ftIsec and the affected branch piping was an 8-inch line with a kinematic viscosity of0.145 x 10-5 felsec corresponding to 550°F. In our experiment, the mean velocity in

the main pipe, ii is estimated to be 0.89 x 35 ftIsec = 31.15 ft/sec where the factor 0.89represents the mean velocity to the centerline velocity ratio [8]. If the turbulent velocityis normalized with respect to the mean velocity, Eq. (4) yields the following results:

17Iii =0.137 exp (-0.315 xID) (7)

Let us assume the above correlation holds in general. Then, the critical conditionbecomes:

2000 =17Diy =0.137 ii Diy exp (-0.315 xlD) (8)

for any combination of ii , D and y where ii Diy is in fact a Reynolds number basedupon main channel velocity and branch line diameter. Now, for the in-plant condition,

setting ii = 50 ftIsec and retaining the other variable values, the projected turbulencepenetration is found to be LID = 23.4. This is in excellent agreement with the in-plantdata showing LID = 22. Based on this preliminary evaluation, these results look verypromising and may well fill the gap between laboratory model experiments and thoseperformed at plant operating conditions.

3.2. Natural Convection in Long Horizontal Enclosures

Now that the distance of turbulence penetration into the deadend branch pipe isestimated, the next step is to investigate the temperature distribution within theremainder of the pipe where thermal stratification persists.These branch lines are normally insulated and have length to diameter ratios much

greater than one. In addition, high temperatures result in Rayleigh numbers that arenormally greater than 108. Experimental and analytical evidences show that naturalconvection at high Rayleigh numbers is characterized by thin fluid layers at the top andbottom of the channel connecting the end boundary layers adjacent to the hot and coldend walls. The combination of high aspect ratio and high Rayleigh number poses agreat challenge to numerical simulations.In this section, numerical solutions are compared to the experimental data of Bejan et

at. [12] for flow between parallel plates, for Rayleigh numbers greater than 108.FLUENT V4.25 was employed in this study.


Consider a channel, such as a two dimensional enclosure, of length L and height H.The high temperature end is at Th and the colder end is at Te. The top and bottomsurfaces are insulated and serve as adiabatic walls. Fig. 4 illustrates this configuration.The solution for variation of velocity, temperature and pressure, within the

enclosure, can be expressed as a function of the following dimensionless parameters:

3 2Gr= Grashof Number = g~ (Th - Te)H Iv

Pr = Prandtl Number = vIa

A = Aspect Ratio = HIL

where T is fluid temperature, H, the height between plates, L, the plate length, g,acceleration due to gravity, v, kinematic viscosity, ~, coefficient of thermal expansion,and a, the thermal diffusivity. The Rayleigh number defined as Ra = g~ (Th - Te)H

3/(va)will be used to express the strength of the natural convection process. This is alsoconsistent with the majority of the published literature.Numerical studies of .Paolucci and Chenoweth [9] for natural convection in long

enclosures show that natural convection for Rayleigh numbers greater than about 108

falls in what may be termed as the "intrusion" layer regime [10]. Experimental resultsof Kimura and Bejan [11] and Bejan et at. [12] show this flow is characterized by heattransfer through the boundary layers at the hot and cold ends connected by a fullydeveloped flow in the core region, consisting of thin top and bottom intrusion layersconnected by an almost stationary core. The flow typically rises along the hot end,traverses the top plate toward the cold end, falls along the cold end; and traverses thelower plate toward the hot end. The validity of this flow pattern was also verified in anumber of approximate analytical solutions, e.g., those of Shiralkar et at. [13], Tichyand Gadgil [14], and Lubin and Kim [15].The open literature contains a number of examples of numerical solutions for natural

convection in long enclosures. In most numerical simulations, the effect of gravity isintroduced as a body force in the momentum equations. It is difficult to obtain stableand converged numerical solutions for natural convection problems with large body force

o

TeOlD +9

lr---

I ~

I"

~C- I I .. )bI ~~ I ~~ I B I ~

I ~~I I

/: - ~

TTHOT D

1INTRUSION lAYER REGION

END I....t---------------IENDBOUNDARY .. .. BOUNDARYlAYER lAYERREGION REGION

Figure 4. Natural convection in long horizontal enclosures at high Rayleigh numbers.


terms (which are characteristic of high Rayleigh number flows) using conventionaliterative schemes. Typically, as the Rayleigh number increases, the solution becomesunstable, thus making it very difficult to obtain a meaningful, converged solution.Some of these difficulties were circumvented by taking a number of appropriatemeasures in the numerical modeling used in this study [16].In the following, numerical simulations are obtained for the conditions of the

experiments of Bejan et al. [12]. Their experimental results were obtained for aninsulated parallel plate channel with an aspect ratio of 0.0625 and end temperatures of305.4°K and 283.5°K. Rayleigh numbers for their experiments ranged from 2x108 to2x109.

w•••1'MI

la)

Cold oftd

I , .0,; ,;

",

" ....." ", ..", .' ··u

"....

o

/

"

",/,/

;' ..../ ,; ....

.... ", ..••• '0

s

(b)

o PREDICI10

Temperature d,ff r nee IOC)

Figure 5. Comparison of predictions and experimental data of Bejan et al. [12].(a) streamlines (b) temperature difference vs. vertical height


Fig. 5a shows a comparison of data of Bejan et at. [12] with numerical predictionsfor stream lines at Ra=109. The patterns are in agreement with each other, showing thelarge area of low velocity flow on either side of the centerline.A comparison of the predicted (Ra=I.8x109) and measured (Ra=I.50x109) temperature

differences, between the top (ylH=l) and the bottom (ylH=O) plates, at the axial locationhalf-way between the ends, are shown in Fig. 5b. The agreement is shown to be quitegood at this value of Rayleigh number, demonstrating the feasibility of using a CFDcode for this challenging problem.

4. Dispersion of Thermal Discharge

This section addresses an important environmental aspect of flow with thermal gradient.Typically, these flows experience "horizontal" thermal gradients as well as verticalstratification. Thermal effluents belong to this category. Thermal effluents from powerplants can cause marine environmental problems. For instance, they may influencecoastal marine ecosystem. such as fishery production at power plant sites. Thus,prediction of thermal dispersed areas around power plants is important for better marineenvironmental management.Many techniques such as extensive field survey, infrared tracing, and numerical

modeling have been applied to assess and predict thermal discharged dispersion areas.The rapid advancement of numerical modeling and field monitoring techniques enable animproved predictive capability that integrates field measurements, infrared ray imagedata, and numerical model.As a pilot study to demonstrate such capability, an operating'power plant was

chosen for the present study. The power plant chosen for this study, Young KwangNuclear Power Plant (YKNPP), is located on the southwestern coast of South Koreafacing the Yellow Sea that lies between China and the Korean peninsula.The mean water depth in the seashore of the YKNPP is less than 2 m on the ebb tide

and about 6 m on the flood tide. The maximum water depth is about 10m at 12 kmoffshore from the channel of the thermal discharges. Along the shore of the YKNPPthere is about 1 km wide tidal flat composed of mud and sand. The general circulationpattern of the water adjacent to the YKNPP is to the NE on flood and SW on ebb tides,respectively, with winds occasionally producing other patterns. The range of meanvelocity is approximately 0.2 m/sec to 0.5 m/sec on 3 m depth of water. Because ofshallow waters around the coast of the YKNPP, the dominant forcing term is windstogether with tides. The tidal level is 6 to 6.5 m. The surface floating drogue test tosurvey the Lagrangian water movement shows that the flow is to the NE or NNE on theflood tide and SW or SSW on the ebb tide, which is parallel to the YKNPP shoreline.The present basic three-dimensional numerical model has been applied successfully

to study diverse regions [17-20] The model solves the finite-difference analogue of thefollowing set of equations assuming that the ocean water is incompressible andhydrostatic, and using the Boussinesq approximation:

V.n + dW/dZ = 0 (9)


d U /dt + u.Vu + fkxu = - Vp/Po + d(KMdU/dZ)/dZ = FM

pg = - dp/dZ

de/dt + uOVe + wde/dz = d(KHde/dz)/dZ + FH

p = p(T,S,p)

195

(10)

(11)

(12)

(13)

Here, u = (u,v) is the horizontal velocity vector and w is the vertical component ofvelocity; e denotes either the temperature T or the salinity S, which is related to pthrough the equation of state; f = 2Qsin<l> where Q = 7.292x1O,s/sec and <I> is thelatitude. FM and FH are Laplacian-type horizontal viscosity and diffusion termsrespectively, with coefficients AM and AH taken to be equal, modeled by theSmagorinsky diffusion formulation:

(14)

where C is a constant taken to be 0.05. This gives values of AM and AH near the coastof the order of 10 m2/s. Vertical turbulent mixing is modeled through the use of theeddy viscosity and diffusivity, KM and KH respectively, according to the level-2.5turbulence kinetic energy closure scheme. The model has a free surface, thus can includetides and atmospheric-induced sea level variations.At the free surface, Z = ~(x,y,t), wind stresses and heat fluxes can be specified but for

the present simulation they are not included. At open boundaries, we specify sea levelelevations using measured tidal records. Ideally, observed temperature and salinityboundary conditions with vertical structure would have been preferred. However, suchdata are not available at the present time. We use a mean temperature and salinity forthe spring and fall seasons from the observed recorded data available. The thermaldischarge rate from the YKNPP is approximately 225 m3/sec. At the ocean floor, Z = H(x,y), salt and heat fluxes are zero while momentum flux is balanced by matching thecomputed velocity, ub' nearest the bottom Z = zb with the logarithmic law of the wall,so that the drag coefficient CD = max [2.5x1O,3, {kIln(H=~)lzr}2], where k = 0.40 is thevon Karman's constant and zr = 1 cm is the roughness height.Since water is the only object whose emissivity is essentially constant under

different conditions, its temperature can be reliably estimated from its radiance. LandsatTheffi!.atic Mapper (TM) sensors collect data from various bands of visible and infraredlight spectra. Among these various bands, Band 6 data (wavelengths of 10.4 - 12.5!-lm) are the most useful in monitoring the water temperature of the cooling ponds ofnuclear power plants. A preliminary evaluation of Landsat IV infrared ray image dataaround the YKNPP typically showed that thermal plume discharged from the YKNPPgenerally would stretch northward along the coastline near the time of a maximum floodtide and southward near the time of a maximum ebb tide.The model simulation result for the flood tide is shown in Fig. 6 and a similar result

was also obtained for the ebb tide case. These simulated responses showed that the


dynamic changes of thermal plume according to the tidal changes are in qualitativeagreement with the multi-channel satellite images. This pilot study suggests that anintegration of methods such as field survey, infrared tracing, and improved numericalmodeling might provide a powerful tool for predicting thermal discharge dispersion.

zCft

inft

cC'f Chilsando

,t?

15'

o 4km

120· 30' E

Figure 6. Model surface temperature difference distribution relative to the ambient reference seatemperature near the time of a maximum flood.


5. Conclusion

197

We reviewed some thennally affected flows with emphasis on their applications topower plants. These flows include natural circulation, thennally stratified flow, andthennal discharge flow.Natural circulation is widely utilized in both fossil-fueled and nuclear power plants,

as the flow is driven by natural force and therefore can be circulated around the systemwithout resorting to recirculation pumps. Natural circulation has gained a renewedinterest in the design of some advanced nuclear reactors as it has proven to be aneffective mechanism for heat removal from reactors under both nonnal operation andaccident or emergency conditions.Thennal stratification has drawn much attention in nuclear industry as thennally

stratified flows can cause thennal fatigue and stress on the piping materials. Wepresented two technical issues that are relevant to resolving the problem: turbulencepenetration, and temperature distribution in long horizontal enclosures undergoingnatural convection. A working correlation for the turbulence penetration was developedand numerical capability for hanclling the natural convection was demonstrated.Thennal effluent from power plants is an important environmental issue. In this

chapter, we presented some preliminary computational results of a pilot study as a firststep towards integrating field survey, infrared image tracing, and advanced numericaltechniques.

6. References

I. Kim, J.H. and Hassan, Y.A.: Natural Circulation, ASME FED-Vol. 61, American Society ofMechanical Engineers, 1987.

2. Singer, J.G.: Combustion: Fossil Power Systems. Combustion Engineering, Inc., 19813. Kakac, S.: Boilers. Evaporators, and Condensers. John Wiley & Sons, Inc., 19914. Kim, J.H.: Heat removal by natural circulation in light water reactors, in U. Miiller and K. Rust (eds.),

Proc. Fourth International Topical Meeting on Nuclear Reactor Thermal-Hydraulics, G. BraunKarlsruhe (1989), 1, pp. 430-447.

5. EPRI Journal (1986), July/August 1986, Electric Power Research Institute.6. U.S. Nuclear Regulatory Commission: TherTTUlI stresses in piping connected to reactor coolant systems,

Bulletin 88-08 (with Supplements I, 2, and 3) (June 22, 1988 to April II, 1989).7. Kim, J.H., Roidt, R.M., and Deardorff, A.F.: Thermal stratification and reactor piping integrity,

Nuclear Engineering and Design 139 (1993),83-95.8. White, F.M.: Fluid Mechanics McGraw Hill, 1979.9. Paolucci, S. and Chenoweth, D.R.: Natural convection in shallow enclosures with differentially heated

end walls, Journal ofHeat Transfer 110 (1988), 625-634.10. Cormack, D.E., Leal, L.G., and Imberger, J.: Natural convection in a shallow cavity with differentially

heated end walls; Part I, Asymptotic theory; Part 2, Numerical results; Part 3, Experimental results,journal of Fluid Mechanics 65 (1974), 209-260.

II. Kimura, S. and Bejan, A.: Experimental study of natural convection in a horizontal cylinder withdifferent end temperatures, Int. J. Heat and Mass Transfer 23 (1980), 1117-1126.

12. Bejan, A., AI-Homoud, A.A., and Imberger, J.: Experimental study of high-Rayleigh-numberconvection in a horizontal cavity with different end temperatures, Journal of Fluid Mechanics 109(1981),283-299.

13. Shiralkar, G., Gadgh, A., and Tien, C.L.: High Rayleigh number convection in shallow enclosures withdifferent end temperatures, Int. J. ofHeat and Mass Transfer 169 (1981), 1621-1629.

14. Tichy, J., and Gadgil, A.: High Rayleigh number laminar convection in low aspect ratio enclosureswith adiabatic horizontal walls and differentially heated vertical walls, Journal of Heat Transfer 104(1982), 103-110.


15. Lubin, B.T. and Kim, J.H.: Prediction of wall temperature distributions in long lines undergoing naturalconvection at high Rayleigh numbers, in G.F. Hewitt (ed.), Heat Transfer 1994 (Proc. 10th Int. HeatTransfer Conf) (1994),5, pp. 519-524.

16. Kim, J.H., Chakrabarti, M., and Lubin, B.T.: Natural convection between long horizontal plates at highRayleigh numbers, Proc. Symp. on Thermal Science and Engineering ill Honor of Chancellor ChangLin Tien, University of California, Berkeley, California (1995), pp. 115-126.

17. Blumberg, A.F.: Numerical tidal model of Cheasapeake Bay, J. Hydraulic Engineering, ASCE 103(1977), 1-10.

18. Blumberg, A.F. and Mellor, G.L.: A simulation of the circulation in the Gulf of Mexico, Israel J. EarthSciences 34 (1985), 122-144.

19. Oey, L.Y., Mellor, G.L., and Hires, R.l.: A three-dimensional simulation of the Hudson-RaritanEstuary. Part 1: Description of the model and model simulations, J. Physical Oceanography 15 (1985),1676-1692.

20. Galperin, B. and Mellor, G.L.: A time-dependent, three-dimensional model of the Delaware Bay andRiver System. Part I: Description of the model and tidal analysis, Estuarine, Coastal, and Shelf Science31 (1990), 231-253.

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