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Extended Modelica Model for Heat Transfer of Two-Phase Flowsin Pipes Considering Various Flow Patterns

Timm Hoppe1 Friedrich Gottelt1 Stefan Wischhusen1

1XRG Simulation GmbH, Harburger Schlossstr. 6-12, 21079 Hamburg Germany,{hoppe,gottelt,wischhusen}@xrg-simulation.de

AbstractBoiling in vertical and horizontal pipes is a complex pro-cess defining transient and static performance of varioustechnical applications. This work presents an extendedheat transfer model which takes the complete boiling pro-cess into account. Models from the literature for the differ-ent boiling regimes are evaluated with respect to accuracyand suitability for system simulation application. A setof sub-models for each of the existing boiling phenomenais implemented and applied to the global boiling model.Special attention is paid to smooth transition between thesub-models and to numerical efficient solutions with re-spect to the consideration of the boiling crisis. The simu-lation results show good accordance with literature data.Keywords: boiling model, heat transfer, two phase flow,pipe flow, evaporation, critical heat flux, boiling crisis,subcooled boiling, saturated boiling, flow pattern, Fluid-Dissipation, ClaRa

1 IntroductionThe heat transfer from an evaporator wall to a flowingtwo-phase fluid is called flow boiling. During flow boil-ing three different regimes can be identified. The first oneis the subcooled boiling regime. The bulk of the fluid isstill subcooled but bubbles can already form in the walllayer, are cooled by the surrounding liquid and thus en-hance the heat transfer. It is followed by the saturatedboiling regime in which the wall is predominantly cov-ered by the liquid phase. Bubbles are formed there, leavethe wall and exchange heat and mass with the surround-ing liquid at saturation temperature. The critical heat fluxmarks the beginning of the post critical heat flux regime,where the gas phase becomes the dominating phase in thewall layer. In all of these regimes the slope of the pipeplays an important role. For example, a vertical pipe has adistinct point at which the regime changes from saturatedboiling to the post critical heat flux regime. In a horizontalpipe there is a gradually transition, as the top of the pipemay be already covered by steam, while the bottom of thepipe is still cooled by liquid.

During flow boiling of fluid flows high heat fluxes canbe transferred at low temperature differences. Flow boil-ing occurs in many industrial applications, such as thermalpower plants, air conditioning or heat exchangers in the

process industry. Exact knowledge about the two-phaseheat transfer and pressure loss is important to determinebehaviour of these applications. However, it is quite com-mon to reduce the boiling process to the saturated boil-ing regime in system simulation, although several boilingregimes can be identified which require specific models.This assumption neglects important effects like the criti-cal heat flux at which the heat transfer coefficient dropsby magnitudes.

An industrial example where it is important to includethis effect are different kinds of thermal power plants likeconventional coal fired steam generators, solar steam gen-erators, natural circulation heat recovery steam generatorsor nuclear steam generators.

In coal fired once-through steam generators the criticalboiling state occurs during normal operation at the end ofthe evaporator. In normal operation the mass flow rate ishigh enough to sustain a sufficient cooling of the pipes, see(Brinkmeier, 2015). However, in abnormal working con-ditions, e.g. maldistribution of mass flow between parallelpipes or failure of feed water supply systems the mass flowrate may be significantly lower than during nominal oper-ation. A detailed calculation of the heat transfer is neededto determine the wall temperatures in this abnormal situa-tions.

2 State of the Art of Two-phase HeatTransfer Modelling

2.1 Literature ReviewThe actual steam quality of the flow is described by thelocal vapour mass fraction defined by the ratio of vapourmass flow mvap to total mass flow m:

xact =mvap

m(1)

If thermodynamic equilibrium is assumed the steam qual-ity can as well be defined by the local specific enthalpyh, the specific enthalpy at bubble point h′ and the specificevaporation enthalpy ∆hV :

xeq ≡ x =h−h′

∆hV(2)

The thermodynamic equilibrium steam quality is used inthe further course of the paper and will be referred to as

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x. In the subcooled area it can also be negative. Theoverall boiling process with the different boiling regimesis displayed schematically for a horizontal pipe in Figure1. Two different heating situations are shown, one witha high heat flux resulting in a nucleate boiling dominatedflow and one with a low heat flux resulting in a convec-tive boiling dominated flow. The circumferential averagedheat transfer coefficient is plotted against the actual steamquality and the steam quality assuming thermodynamicequilibrium.

1-PhaseFlow

1-PhaseFlow

SubcooledBoiling

Saturated Boiling

Post-chfBoiling

Figure 1. Schematic circumferential averaged heat transfer co-efficient for an evaporating flow in a horizontal pipe with a highheat flux (nucleate boiling dominated) and a low heat flux (con-vective boiling dominated), according to (Steiner, 2002), flowregime descriptions refer to nucleate boiling dominated flow

Subcooled BoilingBubble formation starts in the wall layer of the currentalready at bulk enthalpies below the bubble enthalpy, i.e.x < 0. These bubbles deteriorate as they move to the sub-cooled core of the flow, thus enhancing the heat transfercoefficient. This is known as subcooled boiling, see Fig-ure 1 for a schematic behaviour of the heat transfer coeffi-cient in the subcooled boiling regime of a nucleate boilingdominated flow.

In general the literature on subcooled boiling is sparsecompared to other boiling regimes. A comparison ofseveral subcooled boiling models was done by Spindler(K. Spindler, 1990), the model proposed in the VDI heatatlas (Schröder, 2002) shows the least error in predictionof the measurements. At the end of the subcooled boilingregime at a steam quality x = 0 the VDI model is by con-struction equivalent to the saturated boiling heat transfermodel for a vertical pipe from the VDI heat atlas. There-fore, a smooth transition to the following boiling regimeis ensured.

Saturated BoilingIn the saturated boiling regime two different boilingmodes can be observed. This is convective boiling on theone hand and nucleate boiling on the other hand. Con-vective boiling describes the convective process betweenthe wall and the liquid phase, whereas nucleate boiling de-scribes the heat transfer induced by formation, growth and

departure of the bubbles. In horizontal pipes a stratifica-tion of the fluid can occur, depending on the present flowpattern. As the upper pipe wall is partly dry, i.e. only cov-ered by the gaseous phase, the circumferential heat trans-fer coefficient is lower compared to flow patterns whichcover the wall completely with the liquid phase.

All relevant saturated boiling models are a function ofthe convective boiling heat transfer coefficient, basicallycalculated with convective one phase heat transfer corre-lations and the nucleate boiling heat transfer coefficient,basically calculated with pool boiling heat transfer corre-lations. A simple addition of the coefficients is done bythe Chen model (Chen, 1966). He introduced a boilingsurpression factor and a two phase multiplier to fit poolboiling heat transfer correlation and the one phase con-vective heat transfer correlation to his flow boiling data.Shah (Shah, 1982) proposed a model which is not su-perimposing the convective and the nucleate heat trans-fer coefficients but chooses the larger of the two. Gungerand Winterton (Winterton, 1986) developed a new formof the Chen model, basing on a larger database. Theylater also proposed a simpler version of the model (Gun-gor and Winterton, 1987). The VDI heat atlas (Steiner,2002) proposes an asymptotic model which incorporatesnatural limitations of the flow boiling coefficients, insteadof surpression and two phase factors as used by the othermodels. It is also refered to as the Steiner-Taborek modelas it bases on a publication of the two authors (Steiner andTaborek, 1992). The model is recommended by ASHRAE(Owen, 2005) and Thome (Thome, 2006a).

Critical Heat FluxIn the further course of the evaporation process the criti-cal boiling state xcrit , also known as critical heat flux, isreached and the gas phase becomes the dominating phaseat the wall. In a vertical pipe it is a distinct point atwhich the heat transfer coefficient drops suddenly by mag-nitudes. In a horizontal pipe the critical boiling state at thetop of the pipe can already be reached while the bottom ofthe pipe is still covered with liquid. The circumferentialaveraged heat transfer coefficient in such a situation canbe seen in Figure 1.

Several approaches for prediction of the critical heatflux can be found in literature. This are on the one hand thelocal hypothesis correlations which take the local condi-tions at the point of boiling crisis into account. Then thereare the global hypothesis correlations which consider theconditions at the pipe inlet. A third method are the look-uptables.

The VDI heat atlas (Auracher et al., 2002) gives theGroeneveld tables (Groeneveld, 2007) as reference forlook-up tables and the Katto-Ohno model (Katto Y.,1984) as reference for the global hypothesis correlations.The Katto-Ohno correlation is also proposed by Thome(Thome, 2006a). For medium to high pressures and massflow rates the VDI heat atlas proposes the local hypothesiscorrelations of Doroshchuk and Kon’kov. They are con-

Extended Modelica Model for Heat Transfer of Two-Phase Flows in Pipes Considering Various Flow Patterns

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sidered as superior to the other two methods. Furthermore,they differentiate between dry out and departure from nu-cleate boiling.

Post Critical Heat Flux BoilingAt least two boiling regimes can be recognized after thecritical boiling state. On the one hand there is the dry outof the surface. The wall is not necessarily dry, as the re-maining liquid is entrained as droplets in the vapour flow,also referred to as mist flow. Droplets may then hit thewall and wet it temporarily. The second regime is calledfilm boiling or departure from nucleate boiling. It is char-acterized by the formation of a gas film, the correspondingflow pattern is called inverted annular flow. The heat trans-fer coefficient is significantly lower than in mist flow. Foreach regime different heat transfer models are required.If the wall temperature is very high compared to the gastemperature radiation will also play an important role inthe heat transfer.

For dry out there are two different kinds of correlations.The simpler correlations assume thermodynamic equilib-rium between gas and liquid phase. The model of Groen-eveld (Groeneveld, 1973) is proposed by the VDI heat at-las (Katsaounis, 2002) and by Thome (Thome, 2006a).

The more complicated models account for thermody-namic non-equilibrium effects by an apparent superheatedgas temperature. The Köhler model (Köhler, 1983) pro-posed by the VDI heat atlas is to mention here. A modelwhich also includes radiation was published by Ganic andRohsenow (Ganic and Rohsenow, 1977).

SummaryThe VDI heat atlas proposes correlations for all differentboiling mechanisms, thus the complete boiling process iscovered by the VDI proposal. In some extent the mod-els refer to each other. For example one model mergesby construction into the model for the following boilingregime. This smooth transition between the different mod-els is important in system simulation. It should be con-tinuous, continuous differentiable, with low gradients andwithout hysteresis. The proposed models are also con-sidered among the most reliable ones by other authors.Therefore, the models of the VDI heat atlas are imple-mented. Where it is possible the references between themodels are used to smooth or simplify the transitions.

2.2 Library ReviewIt is not known to the authors that an existing Modelicalibrary incorporates a heat transfer model for the completeboiling process. The AirConditioning library and the TILlibrary, which are commercially used for simulation ofautomotive refrigeration cycles, cover only the saturatedboiling regime. The AirConditioning includes implemen-tations of the simple Gungor-Winterton and the Chen cor-relation. The TIL library includes the Chen correlation andthe convective boiling correlation of the Steiner-Taborekmodel. In the freely available ThermoCycle library theShah and the simple Gungor-Winterton correlation are im-

plemented.In power plant modelling the critial heat flux plays

an important role. In the freely available ClaRa li-brary only the saturated boiling regime is covered. Alsoother libraries, the freely available ThermoSysPro, thefreely available ThermoPower and the commercial Ther-malPower, provide no correlations for determination ofthe boiling crisis.

3 ImplementationThe different boiling regime correlations are implementedin the FluidDissipation which is an open source li-brary and freely available, see (XRG Simulation). Forimplementation the functional approach described in(Vahlenkamp and Wischhusen, 2009) is used. The mainaspects of the approach are:

• Independence of thermo-hydraulic framework

• Use of function calls

• Inputs are delivered by records

In contrast to the models already implemented in the Flu-idDissipation each boiling regime is implemented in aseparate model. Thus, the overall heat transfer models arebuilt outside of the FluidDissipation. The advantage ofthis procedure is that models can be developed which aretailored to the corresponding problem. In the followingthe implementation of the separate models is shortly de-scribed focussing on main declarative equations and dif-ferences of the implementation to the VDI heat atlas mod-els. For detailed description of the models see the Fluid-Dissipation documentation.

3.1 Subcooled Boiling Heat TransferFrom the VDI heat atlas (Schröder, 2002) the models fordetermination of the position, in terms of a steam quality,of initial bubble formation xi and of net vapour forming xnare implemented. Both values are by definition of the sub-cooled boiling regime always negative. For xi < x < xn theproposed model for calculation of the heat transfer coeffi-cient from VDI heat atlas is used. For xn < x < 0 the VDIheat atlas proposes a superposition of the nucleate boil-ing heat transfer coefficient and the subcooled convectiveheat transfer coefficient. However, this model incorpo-rates the wall temperature in the heat transfer coefficientcalculation, which would introduce additional iterationsfor the solver. Furthermore, the transition to the saturatedboiling heat transfer coefficient is only smooth if the in-fluence of the flow pattern on the saturated boiling heattransfer coefficient is neglected. To simplify this model,thus making it numerical more stable, a simple smooth-ing from the subcooled convective heat transfer coeffi-cient αsc(x = xn) to the saturated heat transfer coefficientαsat(x = 0) is done. The Stepsmoother from the FluidDis-sipation is used which makes use of the smooth transition

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of the following equation

yreg = (1+ tanh(tan(x)))/2 (3)

The output yreg is between 0 and 1 for x=[-π/2,π/2].

3.2 Saturated Boiling Heat TransferThe Steiner-Taborek model from the VDI heat atlas(Steiner, 2002) is implemented. The following equationresults in an asymptotic combination of the convectiveboiling αconv and the nucleate boiling heat transfer coef-ficient αnclt

αsat =3√

α3conv +α3

nclt . (4)

Convective BoilingThe VDI heat atlas distinguishes between horizontal andvertical pipes, in consequence two different correlationsare provided. Both depend on the one phase heat transfercoefficients for liquid and steam αliq and αvap, the densityratio ρ ′/ρ ′′ and the steam quality x. For horizontal pipesthe VDI correlation includes a correction for stratifiedflow patterns. The information of the present flow patternand the angle of the unwetted circumference of thepipe is needed. The correction weights the one phaseliquid and steam heat transfer coefficients accordingto the unwetted angle. In the implementation of thesefunctions a smoothing with the stepsmoother from theFluidDissipation is applied for the transition betweendifferent flow patterns.

Nucleate BoilingThe used correlation from the VDI heat atlas is a functionof heat flux q, pressure p, pipe diameter d and surfaceroughness W . For horizontal pipes the correlation showsan additional dependency on the mass flow rate m and onthe steam quality x. Furthermore, a correction includesthe dependency of the present flow pattern. The functionexpects the present flow pattern of the flow as an inputand aligns a correction factor to each flow pattern.These correction factors depend on the thickness andconductivity of the wall. Also in this implementation thetransition between flow patterns is smoothed.

3.3 Boiling CrisisThe position of the critical boiling state in terms of a crit-ical steam quality xcrit is determined by the correlationswhich are explained in this subchapter. The critical steamquality marks the end of the saturated boiling regime andthe begin of the post critical heat flux regime and thereforedetermines in an overall boiling model at which point theheat transfer coefficient calculation has to change from thesaturated boiling correlation to the post critical heat fluxcorrelation.

The local thesis correlations of the VDI heat atlas (Au-racher et al., 2002) of Konkov and Doroshchuk for waterare implemented. They depend on the local mass flow

rate, the local pressure, diameter of the pipe and the localheat flow rate.

The Konkov correlation predicts the critical steam qual-ity for dry out, the Doroshchuk correlation the criticalsteam quality for departure from nucleate boiling. A logicchooses the smaller of the two. The used correlation deter-mines whether dry out or departure from nucleate boilingis present. Thus, it can be decided which post critical heatflux correlation has to be used.

For horizontal pipes gravitational effects have to beconsidered. With the modified Froude number Fr theseeffects can be described

Fr =xcritm√

ρ ′′√

9.81d(ρ ′−ρ ′′)cos(θ). (5)

The angle θ is the inclination of the pipe. For Froudenumbers < 10 the stratification of the flow induces an oc-currence of the boiling crisis at the upper side of the pipexcrit,up at much lower steam qualities than at the bottomside of the pipe xcrit,low. This difference ∆xcrit can be de-termined with the modified Froude number

∆xcrit = xcrit,low− xcrit,up =16

(2+Fr)2 . (6)

With the difference ∆xcrit the transition zone in horizontalpipes from the saturated boiling regime to the post criticalheat flux regime can be determined.

3.4 Post Dry-Out Heat TransferTwo different kinds of models exist for the post dry outheat transfer. One type assumes thermodynamic equilib-rium between the liquid and the vapour phase, i.e. thetemperatures are equal. The other type calculates a super-heated gas temperature, which determines the temperaturedifference for the heat transfer.

Thermodynamic EquilibriumThe proposed model from the VDI heat atlas (Katsaou-nis, 2002) is implemented. According to the heat at-las the scope of the model is only for large mass fluxesm > 2000 kg/(m2s) and high pressures with ρ ′/ρ ′′ ≤ 6.However, the model bases on data of a much wider range,according to (Thome, 2006b).

Thermodynamic Non-EquilibriumThe proposed model for thermodynamic non-equilibriumfrom the VDI heat atlas (Katsaounis, 2002) is imple-mented. The assumption of the vapour liquid equilibriumis not valid in this boiling regime. The heat transferredfrom the wall to the fluid is not used completely for vapor-ising the liquid but also for superheating the gas phase.Thus, the temperature difference for heat flow rate cal-culation is calculated with the temperature of the super-heated gas and the wall temperature within the scope ofthe model, i.e.:

Q = αch f A(Tw−Tg). (7)



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For calculation of the gas temperature the model providesan empirical correlation. It is no result of energy balanc-ing.

The steam quality xlim gives the end of validity of themodel in terms of steam quality and is an output of themodel. For x > xlim the original source of the model (Köh-ler, 1983) suggests to keep the wall temperatures constant.As the object-oriented modelling approach does not allowa direct manipulation of the wall temperature calculation,this behaviour is approximated implicitly by keeping thesuperheated gas temperature constant. This temperature isused for calculation of the heat flux until the equilibriumtemperature of the fluid is greater.

3.5 Flow Pattern MapThe calculation bases on the flow pattern map model de-scribed in the VDI heat atlas (Steiner, 2002). To determinethe flow pattern the angle of the unwetted circumferenceof the pipe ϕ is needed. The VDI heat atlas model usesan iteration process to calculate the unwetted angle. Toreduce iterations and make the model suitable for systemsimulations some modifications have been made. The un-wetted angle is calculated directly with the approximationsuggested by Biberg (Biberg, 1999).

The output of the function is the unwetted angle ϕ anda real variable flowPattern. To each number a flow patternis aligned to, see Figure 2. The variable flowPattern andthe unwetted angle are used by the saturated boiling heattransfer model, see section 3.2. They identify with thevalue of the variable the present flow pattern and correctthe heat transfer coefficient, accordingly. The transitionbetween two flow patterns is smoothed using the Fluid-Dissipation Stepsmoother see, equation 3.

• Bubble flow→ 6

• Stratified flow→ 1

• Wavy flow→ 2

• Slug flow→ 3

• Annular flow→ 4

Figure 2. Flow patterns for two phase flow in horizontal straightpipes.

4 ApplicationIn this section an examplary application of the functionsis described. The overall boiling process including flowpattern effects is implemented in the model. The combi-nation of the subfunctions is done in a replaceable modelapplying the FluidDissipation functions. This makes theintroduction of additional states possible. These states arenecessary due to numerical reasons. The thermo-hydraulic

framework of the ClaRa library, (The ClaRa developmentteam) and (Brunnemann et al., 2012), is used. The ap-plication is done for the three conservation equation pipemodel. The main features of the pipe model are:

• homogeneous single phase, i.e. thermodynamicequilibrium between the phases and same velocitiesfor gas and vapour phase

• one dimensional flow direction

• dynamic energy and mass balances, static momen-tum balance

• balance equation spatially discretised in flow direc-tion

4.1 Transition of subfunctionsMain task of the application model is to ensure a smoothtransition between the different heat transfer modes. Thesmoothing function, defined by equation 3, is used to en-sure a continuous and numerical stable transition. In Fig-ure 3 an example for the transition from saturated boilingto post critical heat flux boiling is given. In the examplethe transition zone is defined by

tz = ∆xcrit (8)

Figure 3. Schematic picure for transition from saturated to postcritical heat flux boiling

In the following the used submodels and the switchingbetween the submodels is described in detail. Figure 4gives an overview of the used models and the selectionsequence which is run by the model for determining theheat transfer mode. The corresponding transition zone isdisplayed in the figure as well. The chosen values of thetransition zone are a trade-off between accuracy and nu-merical performance.

1. Superheated one-phase flow: The one-phase heattransfer model (Dittus-Bölter) from the FluidDissi-pation is used. The transition zone depends on thechoice of the post critical heat flux heat transfermodel. For the equilibrium model the transition zoneis defined from x = 0.9 to x = 1.


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supe

rhea

ted flo

w

subc

ooled flo

w

subc

ooled

boiling

satu

rate

d bo

iling

nuclea

te b

oilin

g

conv

ectiv

e

boilin

g

post

CHF

boilin

g

1

2

3

4

4

5

Figure 4. Transition between different Boiling Regimes

2. Subcooled one-phase flow: The one-phase heattransfer model (Dittus-Bölter) from the FluidDissi-pation is used. The point of initial bubble forma-tion xi is calculated by the subcooled boiling modelfrom section 3.1. The transition zone is defined fromxi−0.02 to xi +0.02.

3. Subcooled boiling: The model for subcooled boilingdescribed in the previous section is used. At x = 0the model is merged by construction of the equationsinto the saturated boiling model described in the pre-vious section.

4. Saturated boiling: The model for saturated boil-ing described in the previous section is used. If theminimum heat flux for onset of nucleate boiling isexceeded the saturated boiling heat transfer coeffi-cient is calculated according to equation 4 otherwiseonly convective boiling is present. The critical steamquality xcrit is calculated by the boiling crisis model.The transition area in terms of a steam quality dif-ference is provided for a horizontally oriented tubeby the boiling crisis model. For vertically orientedtubes a fixed value of ∆xcrit = 0.05 is set. Thus, thetransition area is defined by tz=±∆xcrit .

5. Post-CHF boiling: The models described in the pre-vious section are used. It can be decided whether thethermodynamic equilibrium or the non-equilibriummodel is used.

4.2 Calculation of critical heat fluxThe position of the critical heat flux is calculated usingthe functions of Konkov and Doroshchuk (Auracher et al.,

2002) which assume a local hypothesis for the critical heatflux. However, the position is calculated for the total pipeand not for each cell. Thus, situations are avoided in whichthe model calculates several boiling crises at a time in dif-ferent cells. Otherwise inconsistent results could be ob-tained with the previously described selection sequenceof the model. As inputs to the critical heat flux modelthe pressure and mass flow rate of the last cell are used.The heat flow rate is averaged over the total pipe. Thus,a strong numerical coupling between the wall tempera-tures, heat flows and the heat transfer coefficients of allcells is introduced. To uncouple these variables and helpthe simulator to break up the resulting non-linear systemsof equations a stabilizer state for the heat flow rate Q_ isintroduced with the following equation:

dQ_dt

=Q− Q_

τ(9)

During stationary conditions both heat fluxes are equal,thus no new information is included, the stationary resultsdo not change. During transient conditions the additionalstate lags behind the real heat flux Q with the time constantτ = 0.1s.

In consonance with the calculation of the critical heatflux position also the point of initial bubble formation xiand the point of net vapour generation xn are calculated forthe total pipe and not for each cell. The fluid propertiesof the first cell are handed over to the two models. Theaverage stabilizer state heat flow rate of equation 9 is usedas a heat flow rate in the models.



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5 Verification and Discussion of firstResults

5.1 Verification of overall heat transfer modelFor verification of the total model which is described insection 4 two simulations of evaporation of water in a hori-zontal pipe are conducted. A screenshot of the used modelbuilt with the ClaRa library is shown in Figure 5. The re-

Figure 5. Screenshot of test model

sults should show the principle behaviour of the heat trans-fer coefficient as can be seen in Figure 1. The boundaryconditions and geometric parameters are shown in Table1, column "Sim. 1". The first values in column one re-fer to the extreme case of a convective boiling dominatedflow, the second values to the extreme case of a nucleateboiling dominated flow. Furthermore, an independency ofthe flow pattern is assumed, which occurs in pipe wallswith a good conductivity (Steiner, 2002). A circumferen-tial uniform heating is assumed. The pipe is discretisedwith 60 control volumes. The pipe length is chosen suchthat the fluid is evaporated completely in both cases.

The simulation results are shown in Figure 6. The cir-

Table 1. Boundary conditions and geometric parameters of ver-ification simulation

Parameter Sim. 1 Sim. 2 Unit

Mass flow 500/400 500 kg/(m2s)Heat flow 20/750 600 kW/(m2)Outlet pressure 50 50 barInlet spec. enthalpy 550 800 kJ/kgHyd. diameter 0.032 0.032 mPipe length 600/15 15 mWall conductivity high low -

cumferential averaged heat transfer coefficient is plottedagainst the steam quality. The convective boiling domi-nated case shows a behaviour which is to be expected fromFigure 1. The heat transfer coefficient rises with a steamquality of zero, thus no subcooled boiling is present. Itrises further with rising steam quality until it peaks at a

-0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1steam quality x [-]

0

10000

20000

30000

40000

50000

heat

transf

er

coeff

icie

nt α

[W

/(m

2K

)]

convective boiling dominated

nucleate boiling dominated

Figure 6. Simulation results of evaporation of water in a hori-zontal pipe

steam quality of x= 0.8. From that point on the heat trans-fer coefficient drops until the one phase heat transfer at asteam quality of 1 is reached, thus no post critical heatflux boiling is present. The nucleate boiling dominatedcase shows a distinct subcooled boiling regime. A plateauof the heat transfer coefficient follows at which a limit ofthe effect of mass flow is reached, according to the nucle-ate boiling model from the VDI (Steiner, 2002). After thatplateau the nucleate boiling heat transfer coefficient dropsas one could expect from the schematic Figure 1. At asteam quality of x = 0.65 the critical heat flux at the upperside of the pipe is reached. At x = 0.9 the critical heat fluxis reached also at the bottom side, a post critical heat fluxregime follows.

In summary it can be said that the results for both ex-treme conditions are in very good consonance with theschematic figure. The model predicts the occurrence ofthe different heat transfer regimes correctly. Also the tran-sition between the different subfunctions is handled in aplausible way.

5.2 Comparison with a simple saturated boil-ing heat transfer model

A simulation with the boundary conditions from Table1, column "Sim. 2", is conducted with the overall heattransfer model described in section 4 and the Gungor-Winterton 1986 (Winterton, 1986) heat transfer modelwhich is widely used in system simulation. The boundaryconditions are chosen such that they lie in between the ex-treme conditions of a nucleate and a convective dominatedflow. Furthermore, a pipe with a low thermal conductivityof the wall is used.

The results are shown in Figure 7. As an interpreta-tion of the results a schematic pipe with the present flowpattern is drawn below the plot. The flow pattern is cal-culated from the functions described in section 3.5. Theoverall heat transfer model predicts a distinct subcooledboiling regime. This boiling regime is neclected in theGungor-Winterton model and the one phase heat transfercoefficient is used instead. Thus, the heat transfer coef-ficient of the overall model is over a wide range multipletimes larger than the one of the simple Gungor-Winterton


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BubbleFlow

SlugFlow

AnnularFlow

Dry-OutStratifiedFlow

-0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2steam quality x [-]

0

5000

10000

15000

20000

25000

30000

35000

40000

45000

heat

transf

er

coeff

icie

ntα

[W/(

m2K

)]

Overall boiling

Gungor-Winterton

Figure 7. Simulation results of evaporation of water in a hori-zontal pipe

model.At steam qualities slightly below zero the heat trans-

fer coefficient of the overall model peaks before it dropsagain at steam qualities larger than zero. This is due to thefact that the subcooled boiling model does not depend onthe flow pattern. Whereas with begin of saturated boilingthe heat transfer coefficient shows a high dependency onthe flow pattern. At low steam qualities a stratified flowis present. As the upper surface of the pipe is not wet-ted due to stratification, the circumferential averaged heattransfer coefficient is significantly lower than the Gungor-Winterton model. It predicts a up to 60% larger heat trans-fer coefficient, as it has no correction due to stratificationeffects of the flow.

In the further course of the evaporation process theoverall boiling model predicts two more different flow pat-terns. Both induce only a partial wetting of the upper pipewall, thus the heat transfer coefficient is reduced comparedto the simple Gunger-Winterton model.

In the overall heat transfer model the saturated boilingregime ends at a steam quality of x= 0.65. The dry out be-gins at the upper side of the pipe. At the end of the dry outprocess at a steam quality of x = 0.9 the heat transfer co-efficient is approximately 10 times smaller than predictedby the simple Gungor-Winterton model.

The results show that the neglection of effects can leadto over- or underestimation of the heat transfer coefficientby magnitudes. For applications, in which crucial vari-ables depend strongly on the heat transfer coefficient, sim-ple models fail to produce reliable results. For example,this may be the case for wall temperatures in the post dryout regime. In that case it is important to include morecomplex heat transfer models which cover the overall boil-ing process.

5.3 Validation with measurementsAs a validation case the Becker experiments (Abel-Larsenet al., 1985) are presented. They are a series of steady statecritical heat flux experiments, all of them with a dry-out

heat crisis. The test section is a vertical pipe, electricallyheated with a length of 7 m and a diameter of 0.0149 m.

In Figure 8 the results of a simulation with the over-all model using the thermodynamic equilibrium model forthe post critical heat flux heat transfer are shown. The

Figure 8. Becker Experiment Case 3 Equilibrium Model

wall temperatures before the heat crisis and the locationof the heat crisis are predicted correctly. The predictedwall temperatures after the heat crisis until a length of ap-proximately 5.5 m match also the measurement. Beyondthat point the post critical heat flux heat transfer model isnot valid, the overall boiling model changes to the one-phase heat transfer correlation. This change leads to anunderestimation of the wall temperature, as the gas phaseis superheated in the post critical heat flux area. This un-derestimation has two main reasons. On the one hand thedefinite temperature is underestimated as the temperaturedifference for calculation of the heat flux is formed withthe equilibrium temperature. On the other hand, the heattransfer coefficient is calculated with fluid properties at theequilibrium temperature, the fluid properties at the super-heated gas temperature should be used instead.

In Figure 9 the test case is simulated with the overallmodel using the thermodynamic non-equilibrium model.The effect of the superheating of the wall near gas phaseis included in the non-equilibrium model. Thus, it matchesthe measurement much better than the simple equilibriummodel. At a length of 6 m the model keeps the wall tem-peratures constant. This assumption fits also better to themeasurement than the use of the one phase heat trans-fer coefficient as it is done by the simple thermodynamicequilibrium model.

The thermodynamic non-equilibrium model is able topredict the wall temperatures more accurate than the sim-ple equilibrium model. Especially, the transition to thepurely convective one phase heat transfer regime fits muchbetter to the experiment data. However, this advantageis dearly bought by a loss of numerical stability and ef-ficiency. This is due to the introduction of a fictive gas



DOI10.3384/ecp17132467

Figure 9. Becker Experiment Case 3 Non-Equilibrium Model

temperature and the indirect manipulation of the wall tem-peratures. These means are introduced to overcome thedrawback of an equilibrium temperature for gas and liquidphase which is used for calculation of the heat balance inthe three equation pipe model. For a more accurate mod-elling of the effects in the post critical heat flux regime aseparate energy balancing of the liquid and the gas phase isnecessary. This would require a thermo-hydraulic frame-work which includes volume models for separate balanc-ing of the phases, the so called five or six conservationequation models, see (Hänninen and Ylijoki, 1992).

6 Summary

In this paper an implementation of an extended heat trans-fer model for two phase flow in pipes is presented.

The various correlations of the different boiling regimesare implemented in separate subfunctions. The functionalbased approach of the FluidDissipation (XRG Simulation)is used. It enables users to create models which are tai-lored to their problem.

An exemplary application within the thermo-hydraulicframework of the ClaRa library is given. This model isused to verify the implementation. A comparison with awidely used saturated boiling model is conducted. Thepredicted heat transfer coefficients of the two models dif-fer by magnitudes. These results show the importance ofcomplex heat transfer models in system simulation. Thesemodels are relevant in applications, in which crucial vari-ables depend strongly on the heat transfer coefficient, e.g.a safety analysis concerning the pipe wall temperatures.

For validation of the extended heat transfer experimentdata from the literature are used. The simulations matchthe measurement data well. Especially, the position of thecritical boiling state is predicted correctly. For more accu-rate results concerning the post critical heat flux regime aseparate balancing of the phases is necessary, which is afeature of the six conservation equations models.

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