414 advanced computational methods in heat transfer · simulation method developed is based on the...

Numerical simulations of two-phase fluid flow

and heat transfer in a condenser

A. Bokil, C. Zhang

Department of Mechanical and Materials Engineering, University

of Windsor, Windsor, Ontario, Canada

Abstract

A quasi-three dimensional algorithm is developed to simulate two-phase fluidflow and heat transfer in the shell-side of power plant condensers. Thesimulation method developed is based on the fundamental governingconservation equations of mass and momentum for both vapor-phase andliquid-phase, and air mass fraction conservation equation. In the proposednumerical method, the condenser shell-side is subdivided into number ofsectors normal to the cooling water flow direction. The three-dimensionaleffects due to the cooling water temperature difference are taken into accountby a series of two dimensional calculations, each being for one sector. Porousmedia concept is employed to model the tube bundle. The pressure dropbalance concept is used to determine the inlet mass flow rate for each domain.A finite volume discretization procedure is used to solve the governingequations.

The numerical predictions of an experimental steam condenser arecompared with the experimental results. The predicted results are in goodagreement with the experimental data. The predicted results also show animprovement over the results obtained using single-phase model.

1 Introduction

A condenser is an important component that affects the efficiency andperformance of power plants. Accurate and detailed predictions of the shell-side flow distribution and heat transfer are of primary importance inperforming the hydraulic design analysis. An efficient and viable approach to

Transactions on Engineering Sciences vol 12, © 1996 WIT Press, www.witpress.com, ISSN 1743-3533

414 Advanced Computational Methods in Heat Transfer

improve the design tools is through the development of advanced numericalmethods using fundamental conservation equations and appropriate constitutiverelations. The advantage of numerical simulations is that they can provide amore detailed picture of the flow and include the effects of baffles, drip trays,steam lanes in the tube bundle and other mechanical details. Until now, thesesimulations have been carried by using either single-phase two-dimensionalmodels [1,2,3], or single-phase quasi-three-dimensional models [4], or two-phase two-dimensional models [5,6].

The shell-side flow within large power plant condensers, usually, is three-dimensional. The steam flow in different sections along the cooling water flowdirection will differ due to the rise of the cooling water temperature. Thequasi-three-dimensional approach takes into consideration the three-dimensionalaspects of the flow due to the cooling water temperature difference.

The liquid phase affects the fluid flow and heat transfer in the condenserby virtue of interphase friction force and forming films on the tubes. There isa need to understand the physical behaviour of the condensate, involving itsform and its effect, in condensers. In the design of condensers, the assumptionis usually made that heat transfer is reduced in the lower tubes of a tube bundleby the increase in film thickness because of condensate raining from the tubesabove. This condensate would inundate the steam space producing thickerwater coatings on the cooling tubes causing a decrease in the heat transfer.The numerical simulations by using the single-phase model show the region ofminimum heat transfer to be present in the lower half the tube bundle. Recentexperimental results [5,6], however, have shown that the region of minimumheat transfer is in the upper half the tube bundle.

The purpose of the present work is to develop a numerical simulationmodel that includes the effects of both three-dimensional and two-phase flowto predict the fluid flow and heat transfer in a condenser more accurately.

The flow situation and experimental data of a steam surface condenser aretaken from Al-Sanea et al. [5] and Bush et al. [6]. The numerical resultsobtained by the proposed numerical model are compared against theexperimental data and the predicted results by using the single-phase numericalmodel.

2 Condenser Configuration

The configuration of the experimental steam surface condenser at NEI Parsonsis depicted in Figure 1. The tube bundle is composed of 20x20 tubes ofequilateral triangular arrangement. The cooling water is arranged to flow ina single pass through the condenser. This experimental condenser was designedto demonstrate the effects of air pockets. It is not a good example of condenser


Advanced Computational Methods in Heat Transfer 415

t

Steaminlet -»"

0.0

r \ITube bundle j

Vent0

i IL<

2m

4;<j>0

Tube bundleboundary

Calculation planes

Cooling wateroutlet

Figure 1. Experimental Condenser

Table 1. Geometrical and Operating Parameters

Geometrical parameters

Condenser length (m)Condenser depth (m)Condenser height (m)Tube outer diameter (mm)Tube wall thickness (mm)Tube pitch (mm)

Operating parameters

Inlet temp, of cooling water (° C)Inlet velocity of cooling water (m/s)Inlet pressure of steam (Pa)Inlet steam flow rate (kg/s)Inlet air flow rate (kg/s) 2

1.2191.020.7825.41.2534.9

17.81.19276702.032

.48x10-4

design, however, thiscondenser can be used to testnumerical models. Thesteam enters the condenserfrom the left-hand side. Airand uncondensed steam areextracted to the internal ventas shown in the figure. Thecondensate is removed fromthe bottom of the condenser.The geometrical andoperating parameters arelisted in Table 1. Measure-ments of mean heat fluxwere obtained along the 3"*,8*, 13*, and 18* tube rowsfrom the bottom of the tubebundle [5,6].

3 Numerical Model

In steam condensers, steamand air mixture forms thevapor-phase while thecondensed water forms theliquid-phase. The liquidcondensate exists as film onthe tubes and as droplets or

columns between thetubes. Severalmechanisms of heatand mass transfer takeplace between thephases. There is anentrapment of liquidfrom the films to formdroplets. The dropletsimpinge onto the tubes.Heat transfer fromvapor to condensatefilm takes place due tocondensation. Inter-phase friction existsbetween the vapor andthe droplets. Similarly



friction exists between the vapor and the film and between the film and thetubes. Also, three-dimensional effects occur in condensers primarily due to thecooling water temperature gradients, which lead to a space-variable sinkpotential. The fluid flow and heat transfer will differ in the cooling water flowdirection.

3.1 Physical Representation

Usually, the flow behaviour within the tube bundle of a steam condenser isvery complicated. Making a number of assumptions is therefore necessarybased on the likely characteristics of the flow. In this study, the followingmajor elements are proposed for the complete physical model.• The vapor-phase comprises a mixture of steam and air, the proportions

being defined by the air mass fraction.• Mixture of air and steam is considered as a perfect gas.• Both steam and liquid condensate are assumed to be saturated.• Liquid condensate exists as droplets of single size so there will be no

diffusion terms for the condensate.• There is no heat transfer between the steam and the droplets.• Pressure is assumed common to both phases.• Mass sink term for vapor associated with condensation is equal to the

mass source term for the liquid.• The turbulent diffusivity is equal to the turbulent viscosity, i.e., Schmidt

number is equal to one.• Pressure drop from inlet to vent for all sectors must be the same.

3.2 Mathematical Formulation

The simulation method developed in this study is based on the fundamentalgoverning equations of mass, momentum and air mass fraction conservationwith flow, heat and mass transfer resistances. Coupled momentum equationsfor each phase are solved along with the continuity equations to obtain thevolume fractions, velocity fields, shared pressure, and air mass fraction. Theeffect of liquid-phase on the vapor-phase is through the interphase friction.The condenser shell-side is divided into several sectors along the cooling waterflow direction and the flow is assumed to be two-dimensional in each sector.The three-dimensional effects due to the cooling water temperature gradientsare taken into account by a series of step by step two-dimensional calculations,each being for one sector.

The porous medium concept is used in the simulations. A porosity factoris incorporated into the governing equations to account for the flow volumereduction due to the tube bundles, baffles and other internal obstacles. In theliquid-phase momentum equations, diffusive terms are neglected. A pressurecorrection equation is obtained from the vapor-phase continuity equation and



vapor-phase momentum equations. Liquid volume fractions are obtained fromthe liquid-phase continuity equation. The vapor volume fractions are obtainedby using an auxiliary equation.

3.2.1 Governing equationsThe governing equations are the equations of conservation of mass, momentumfor both vapor-phase and liquid-phase, and air mass fraction.

Vapor-phase mass conservation equation

Vaor-hase momentum conservation equations

2_d_

3 dx

'—(•dx^

'I3z 9yjJ 9y dx

—IR o u v \ + —IR p v v \ = — \R n -^ + —* *

2 d

' dy) 3 dyR

dx[ * % & a*9% 3v

(2)

(3)

Liquid-phase mass conservation equation

(4)

Liuid-hase momentum conservation equations

£/>,-«,)-*/*•«< (5)

v, - v,) - % + R,p,g (6)

Conservation of air mass fraction

The dependent variables are velocity components, Ug, Vg, u, and v,, pressure,p, liquid volume fraction, R,, and air mass fraction, $.



3.2.2 Auxiliary relations3.2.2.1 Volume fractions The isotropic porosity, 0, which is employed todescribe the flow volume reduction due to the tube bundles, is defined as theratio of the volume occupied by the fluid to the total volume. Since the tubebundle is laid out in an equilateral triangular pattern, the porosity in the tubebundle region, ft, is given as

(8)

In the tube bundle region, 0=ft; in the untubed region, 0=1. The vaporvolume fraction and liquid volume fraction are defined as the ratio of thevolume occupied by the vapor to the total volume, and the ratio of the volumeoccupied by the liquid to the total volume, respectively. Thus,

V*, = P (9)

3.2.2.2 Momentum source terms The interphase friction forces inmomentum equations are related to the interphase friction coefficient, Q,which is given as:

9. 4 P/K, !(",-",) I(10)

where:

1.5 R.A. = '

Friction factor, fy, for spherical objects is obtained from an empiricalcorrelation given by Clift et al. [7]. The local hydraulic flow resistance forces,F^ and F^, in the vapor-phase and liquid-phase momentum equations due tothe tube bundle are given as:

where and £^ are the pressure loss coefficients. The expressions proposedby Rhodes and Carlucci [8] for the loss coefficients are used.

3.2.2.3 Mass source term The mass source term, m, is the steam masscondensation rate per unit volume. The condensation takes place only on thecooling tubes since no condensation or re-evaporation from the condensatedroplets is assumed. The steam mass condensation rate per unit volume, m,can be calculated by equating the phase change enthalpy with the heat transferrate from the steam to the cooling water flowing in the tubes, as follows:

T-T*A (12)

R



The cooling water temperature, T^, is obtained by the heat balance betweenthe steam and the cooling water in each control volume. The overall thermalresistance for each control volume, R, is the sum of all individual resistancescalculated from various semi-empirical heat transfer correlations,

a = a^+a, + a, + a,+a. (13)

For the water side thermal resistance, R^, the relationship given by Ditus andBoelter [9] can be employed,

(14)

The tube wall resistance, % is calculated from the following expression:

«,,W°<) (15)2fc,

The thermal resistance for condensation heat transfer, %, is determined byusing modified Nusselt's correlation [10,11,12].

J_ =tf«±[i + 0.0095 (Re (16)C O

where:

Nu = 0.725 f—) ; Gr =

I*.The resistance from the presence of air film is evaluated via a mass transfercoefficient as reported by Herman and Fuks [13].

(17)(P-P.) \"r

where:a = 0.52, b = 0.7 for Re,<350; a = 0.82, b = 0.6 for Re,>350;

3.2.2.4 Other properties The density of the mixture varies locally accordingto the perfect gas equation of state. Turbulent viscosity, %, is taken as aconstant (forty times the value of the dynamic viscosity) throughout. Thedroplet diameter is set to equal to 1 mm as suggested by Al-Sanea et al. [5].

3.3 Numerical Solution

The discretization of the differential equations, Eqns.(l)-(7), is carried out byintegrating them over small control volumes in a staggered grid. The resulting



180-

160-

140-

120-

1= 100-

- 80-

^ 60-

40-

20-

0-I I I I I I I

0.31 0.41 0.51 0.61 0.71 0.81 0.91 1.01

x (m)

QOO Single-Phase A A A Two-Phase### Experiment

Figure 2 Heat Flux Along the 3"* Rowfrom the Bottom of the Tube Bundle

180-

160-

140-

120-

100-

80-

60-

40-

20-

0-

# #

I I I I I I I T0.31 0.41 0.51 0.61 0.71 0.81 0.91 1.01

X (m)

OOO Single-Phase A A A Two-Phase# # # Experiment

Figure 3 Heat Flux Along the 8*^ Rowfrom the Bottom of the Tube Bundle

discretized equations aresolved in primitive variables.These equations are coupledand are highly nonlinear.Thus, an iterative approachis used for their solution.

The condenser is dividedinto five sectors as shown inFigure 1. The sectorsinteract with each otherthrough the "thermalmemory" of the coolingwater on the tube side.Calculations are made foreach plane, which is locatedat the middle of the sectorand normal to the coolingwater flow direction.

4 Results

Comparisons between thepredictions obtained by theproposed quasi-three-dimensional, two-phasenumerical model and themeasurements are made forheat flux distributions.Figures 2 to 5 show thecomparison of the heat fluxalong four of the tube rowswith the experimental data.Since the experimental heatflux at each plane is notavailable and the experi-mental values are meanvalues along the coolingwater flow direction, thepredicted data used here areaverage values of these fromthe five sectors. The valuesof heat fluxes along all thefour tube rows show that thevariation is similar to that in



180-

160-

140-

120-

100-

80-

60-

40-

20-

0-

0.31 0.41 0.51 0.61 0.71 0.81 0.91 1.01

X (m)

OOO Single-Phase A A A Two-Phase• • • Experiment

Figure 4 Heat Flux Along the 13* Rowfrom the Bottom of the Tube Bundle

1 1 1 1 r .0.31 0.41 0.51 0.61 0.71 0.81 0.91 1.01

x (m)

OOO Single- Phase A A A Two-Phase• • • Experiment

Figure 5 Heat Flux Along the 18* Rowfrom the Bottom of the Tube Bundle

the experimental case. Theagreement with experimentresults is good for the upperthree tube rows. There issome over prediction of heatflux for the bottommost tuberow. It is noted from theexperimental data and thepredicted results that theregions of minimum heattransfer are in the upper halfthe tube bundle, i.e., alongthe 13* and 18* tube rowsfrom the bottom of the tubebundle, as shown in thefigures. This contradicts theusual assumption made in thecondenser design that heattransfer is reduced in thelower tube rows of thecondenser due to theinundation effect. Onepossible explanation is thatthe turbulence of thecondensate film on the lowertubes in the tube bundle isstrongly enhanced due tosplashing of condensatedroplets from above.

The predicted resultsusing the proposed numericalmodel are also comparedwith the results obtained byusing the quasi-three-dimensional single-phasenumerical model [4] asshown in Figures 2 to 5. Asit can be seen from thesefigures, there is a goodagreement between theexperiments and thepredictions using both single-phase and two-phase modelsin the upper half the



condenser (13* and 18* tube rows from the bottom). The reason for this isthat the effect of the liquid is not significant in the upper tube bundle.However, for the lower half the tube bundle (3"* and 8* tube rows from thebottom), the heat fluxes obtained by single-phase are under predicted in thecentral part of the tubular region. The liquid has strong influence in the lowerpart of the tubular region because most of the condensate accumulates there.In the single-phase model, the effect of liquid droplets is not taken intoconsideration. This is probably the reason for discrepancies in the single-phasemodel results.

5 Conclusion

A quasi-three-dimensional two-phase numerical model to predict the fluidflow and heat transfer in condensers has been developed. The simulations havebeen carried out for an experimental condenser. The numerical modelproposed in the present study has shown the capability of predicting theperformance of condensers. The agreement with the experimental results isgood in most regions of the tube bundle. On comparing the results from thequasi-three-dimensional two-phase model with those from the quasi-three-dimensional single-phase model, it has been seen that the quasi-three-dimensional two-phase model gives better results in the lower part of thetubular region where the liquid plays a major role. In the upper part of thetubular region, results from the two-phase model are similar to those from thesingle-phase model.

Nomenclature

A heat transfer area of a given F^ flow resistance force for in thecontrol volume y-momentum equation

A<, total projected area of droplets f friction factorin a given control volume Gr Grashof number

Cf inteiphase friction coefficient g gravitational accelerationCp specific heat at constant pressure H non-dimensional number ofc condensation rate in the control condensation

volume under consideration k thermal conductivityD diffusivity of air in steam L latent heat of condensationDj droplet diameter m steam condensation rate per unitDg effective diffusivity of air in volume

steam (=D+DJ Nu Nusselt numberDj inner diameter of tube Pr Prandtl numberDO outer diameter of tube P< tube pitchD( turbulent diffusivity of air in p pressure

steam q" heat fluxFy flow resistance force for in the Re Reynolds number

x-momentum equation R* air resistance



RC thermal resistance forcondensation heat transfer

Rf fouling resistanceRg volume fraction of vapor-phaseR| volume fraction of liquid-phaseRt tube wall resistanceRw water-side thermal resistanceT temperatureUp velocity magnitude (=u*+v*)u velocity component in the x-

directionV volumev velocity component in the y-

directionw total amount of condensate

leaving the particular controlvolume

x steam maincoordinate

y steam crosscoordinate

Greek Letters0 local porosity

References

p<t>

laminar dynamic viscosityeffective viscosity (=turbulent viscositypressure loss coefficientdensityair mass fraction

Subscriptsa airc condensatecs steam-condensate interfaced dropletg vapor-phasei phase in question1 liquid-phases steamt tube

flow direction u

flow direction v

wx

y

parameter in x-momentumequationparameter in y-momentumequationcooling watervariable in x-directionvariable in y-direction

[1] Davidson, B.J. and Rowe, M., Simulation of Power CondenserPerformance By Computational Method: An Overview, PowerCondenser Heat Transfer Technology, eds., P.Marto and R.Nunn, pp. 17-49, Hemisphere, Washington, 1981.

[2] Caremoli, C., Numerical Computation of Steam Flows in Power PlantCondensers, Numerical Methods In Thermal Problems, eds., R.W.Lewisand K. Morgan, pp.315-325, Pineridge Press, Swansea, U.K., 1985.

[3] Al-Sanea, S., Rhodes, N., Tatchell, D.G. and Wilkinson, T.S., AComputer Model for Detailed Calculation of the Flow in Power StationCondensers, Condenser: Theory and Practice, Int. Chem. E. SymposiumSeries, No.75, pp.70-88, Pergamon Press, Oxford, 1983.

[4] Zhang, C. and Zhang, Y., A Quasi-Three Dimensional Approach toPredict the Performance of Steam Surface Condensers, Journal of EnergyResources Technology, 1993, Vol.115, No.3, pp.213-220.



[5] Al-Sanea, S., Rhodes, N. and Wilkinson, T.S., Mathematical Modellingof Two-Phase Condenser Flows, pp. 19-21, Proceedings of the Int.Conference on Multiphase F/ow, London, England, 1985.

[6] Bush, A. W., Marshall, G.S. and Wilkinson, T.S., A Prediction of SteamCondensation Using A Three Component Solution Algorithm, pp.223-234, Proceedings of The Second International Symposium on Condensersand Condensation, U. of Bath, U.K., 1990.

[7] Clift, R., Grace, J.R. and Weber, M.E., Bubbles, Drops and Panicles,Academic Press, 1978.

[8] Rhodes, D.B. and Carlucci, L.N., Predicted and Measured VelocityDistributions in a Model Heat Exchanger, International Conference onNumerical Methods in Nuclear Engineering, Canadian NuclearSociety/American Nuclear Society, Montreal, 1983.

[9] Ditus, F.W. and Boelter, L.M.K., Heat Transfer in AutomobileRadiators of the Tubular Type, U. of California at Berkeley, Publ. inEngineering, Vol.2, 1930, pp.443-461.

[10] Grant, I.D.R. and Osment D.J., The Effect of Condensate Drainageon Condenser Performance, NEL Report No. 350, April, 1968.

[11] Fuks, S.N., Heat Transfer with Condensation of Steam Flowing in aHorizontal Tube Bundle, Teploenergetika, Vol. 4, 1957, pp.35-39.

[12] Wilson, J.L., The Design of Condensers by Digital Computers, 7.Chem. E. Symposium Series, No.35, 1972, pp. 132-151.

[13] Berman, L.D. and Fuks, S.N., Mass Transfer in Condensers withHorizontal Tube When the Steam Contains Air, Teploenergetika, Vol.5,No.8, 1958, pp.66-74.


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