-. international agreement: report

NUREG/JA-0073-. ABEEW-M2590

InternationalAgreement: Report

Time Step and Mesh SizeDependencies in the HeatConduction Solution of aSemi-Implicit, Finite DifferenceScheme for Transient Two-Phase Flow

Prepared byR. O'Mahoney

Winfrith Technology CentreUnited Kingdom Atomic Energy AuthorityDorchester, Dorset, DT2 8DHUnited Kingdom

Office of Nuclear Regulatory Research" U.S. Nuclear Regulatory CommissionWashington, DC 20555

April 1992

Prepared as part ofThe Agreement on Research Participation and Technical Exchangeunder the International Thermal-Hydraulic Code Assessmentand Application Program ([CAP)

Published byU.S. Nuclear Regulatory Commission

NOTICE

This report was prepared under an international cooperativeagreement for the exchange of technical information.ý Neitherthe United States Government nor any agency thereof, or any oftheir employees, makes any warranty, expressed or implied, orassumes any legal liability or responsibility for any third party'suse, or the results of such use, of any information, apparatus pro-duct or process disclosed in this report, or represents that its useby such third party would not infringe privately owned rights.

Available from

Superintendent of DocumentsU.S. Government Printing Office

P.O. Box 37082Washington, D.C. 20013-7082

and

National Technical Information ServiceSpringfield, VA 22161

NUREG/IA-0073AEEW-M2590

InternationalAgreement Report

Time Step and Mesh SizeDependencies in the HeatConduction Solution of aSemi-Implicit, Finite DifferenceScheme for Transient Two-Phase Flow

Prepared byR. O'Mahoney


Office of Nuclear Regulatory ResearchU.S. Nuclear Regulatory CommissionWashington, DC 20555

April 1992

Prepared as part ofTe Ageemtent on Resac Partciaton and Te ,chca Exchange

une th ntrational Thra-yrulicCoeAss enand Application Program (ICAP)

Published byU.S. Nuclear Regulatory Commission

NOTICE

This report is based on work performed under the sponsorship of the

United Kingdom Atomic Energy Authority. The information in this

report has been provided to the USNRC under the terms of the

International Code Assessment and Application Program (ICAP)

between the United States and the United Kingdom (Administrative

Agreement - WH 36047 between the United States Nuclear Regulatory

Commission and the United Kingdom Atomic Energy Authority Relating

to Collaboration in the Field of Modelling of Loss of Coolant

Accidents, February 1985). The United Kingdom has consented to the

publication of this report as a USNRC document in order to allow

the widest possible circulation among the reactor safety community.

Neither the United States Government nor the United Kingdom or any

agency thereof, or any of their employees, makes any warranty,

expressed or implied, or assumes any legal liability of

responsibility for any third party's use, or the results of such

use, or any information, apparatus, product or process disclosed

in this report, or represents that its use by such third party

would not infringe privately owned rights.

AEEW - M 2590

TIME STEP AND MESH SIZE DEPENDENCIES IN THE HEAT CONDUCTIONSOLUTION OF A SEMI-IMPLICIT, FINITE DIFFERENCE SCHEME FOR

TRANSIENT TWO-PHASE FLOW

R O'Mahoney

Summary

This report examines, and establishes the causes of, previouslyidentified time step and mesh size dependencies. Thesedependencies were observed in the solution of a coupled system ofheat conduction and fluid flow equations as used in theTRAC-PFl/MODl computer code.

The TRAC-PF1/MODl computer code employs a semi-implicit, finitedifference solution scheme to solve the differential equationsdescribing heat transfer and two-phase fluid flow; it is commonlyused to analyse loss-of-coolant accidents in Pressurised WaterReactors.

The report shows that a significant time step size dependency canarise in calculations of the quenching of a previously unwettedsurface. The cause of this dependency is shown to be theexplicit evaluation, and subsequent smoothing, of the term whichcouples the heat transfer and fluid flow equations. An axialmesh size dependency is also identified, but this is very muchsmaller than the time step size dependency.

The report concludes that the time step size dependencyrepresents a potential limitation on the use of large time stepsizes for the types of calculation discussed. This limitationaffects the present TRAC-PFl/MODl computer code and may similarlyaffect other semi-implicit finite difference codes that employsimilar techniques. It is likely to be of greatest significancein codes where multi-step techniques are used to allow the use oflarge time steps.

Safety and Engineering Science DivisionWinfrith Technology Centre

July 1989

AEEW - M. 2590 i

CONTENTS

PAGE

SECTION 1 INTRODUCTION 1

2 DESCRIPTION OF TRAC-PFI/MODl 1

3 TRAC-PFI/MOD1 QUENCHI NG RESULTS 2

4 DETAILED EXAMINATION OF CONDUCTION TERMS 4

4.1 Finite Difference Equation 4

4.2 Surface-to Fluid Effects 4

4.2.1 Explicit Evaluation and 5

Smoothing Effects

4.2.2 Time Step Size Effects 6

4.3 Axial Conduction Effects 6

4.4 Quench Front Profiles 7

4.4.1 Calculation with 0.25 mm 7Minimum Axial Mesh

4.4.2 Calculation with 0.25 mm 8Mesh and 0.3 ms Time Step

4.4.3 AXIAL Term Time Step8Dependency

4.4.4 Calculations with no Axial 9Conduction

4.4.5 Axial Mesh Size Effects 10

5 SUMMARY AND CONCLUSIONS 10

6 REFERENCES 12

AEEW - M 2590V

FIGURES

PAGE

1 Rod surface temperatures at 5 elevations, for 4 different 13mesh sizes. Min axial meshs are: CONT=2.5 mm,SHORT=0.25 mm, LONG=0.1 mm, THICK=0.05 mm.

2 Axial profile of rod surface temperature at 4 different 14times. Profiles at 0, 10, 20 and 30 seconds, for 0.1 mmaxial mesh.

3 Rod surface temperatures at 5 elevations, for 3 15calculations. Calculations are: CONT=0.25 mm,SHORT=0.05 mm, LONG=O.25 mm + 0.3 ms step.

4 Surface-to-fluid heat transfer coefficient vs temperature, 16at 20 seconds. TRAC calculation with 0.25 mm min mesh +theoretical heat transfer.

5 Surface-to-fluid heat flux vs temperature, at 20 seconds. 17TRAC calculation with 0.25 mm min mesh + theoretical heatflux.

6 Surface-to-fluid heat transfer coefficent vs temperature, 18at 20 seconds. TRAC calculation with 0.25 mm min mesh, nosmoothing + theoretical heat transfer.

7 Surface-to-fluid heat tranfer coefficent vs temperature, at 1920 seconds. TRAC calculation with 0.25 mm min mesh, 0.3 mstime step + theoretical heat transfer.

8 Surface-to-fluid heat flux vs temperature, at 20 seconds. 20TRAC calculation with 0.25 mm min mesh, 0.3 ms time step +theoretical heat transfer.

9 Rod surface temperatures at 7 elevations, for 3 no-axial 21calculations. Calculations are: CONTO0.25 mm,SHORTO0.25 mm. + 0.3 ins, LONG=0.05 mm + 0.3 ins.

10 Surface-to-fluid heat flux vs temperature, at 20 seconds. 22TRAC calculation, no axial conduction, 0.25 mm minioum mesh+ theoretical heat flux.

11 Surface-to-fluid heat flux vs temperature, at 20 seconds. 23TRAC calculation, no axial conduction, 0.25 mm min mesh,0.3 ms time step + theoretical heat flux.

12 Heat conduction equation: quench front profile at 2420 seconds. TRAC calculation with 0.25 mm, min mesh.

13 Heat conduction equation: quench front profile at 2520 seconds. TRAC calculation with 0.25 mm min mesh(exploded view).

AEEW - M 2590 iVii

FIGURES (Continued)PAGE

14 Heat conduction equation: quench front profile at 2620 seconds. TRAC calculation with 0.25 mm min mesh, 0.3 mstime step.

15 Heat conduction equation: quench front profile at 2720 seconds. TRAC calculation with 0.25 mm min mesh, 0.3 mstime step (exploded view).

16 Surface-to-fluid heat flux vs temperature, at 20 seconds. 28TRAC calculation with 0.25 mm min mesh, 0.3 ms step,reduced CHF + theoretical heat flux.

17 Heat conduction equation: quench front profile at 2920 seconds. TRAC calculation with 0.25 mm min mesh, 0.3 msstep, reduced heat flux.

18 Heat conduction equation: quench front profile at 3020 seconds. TRAC calculation with no axial conduction,0.25 mm min mesh.

19 Heat conduction equation: quench front profile at 3120 seconds. TRAC calculation with no axial conduction,0.25 mm min mesh (exploded view).

20 Heat conduction equation: quench front profile at 3220 seconds. TRAC calculation with no axial conduction,0.25 mm min mesh, 0.3 ins, time step.

21 Heat conduction equation: quench front profile at 3320 seconds. TRAC with no axial conduction, 0.25 mm minmesh, 0.3 ms step (exploded view).

22 Surface-to-fluid heat flux vs, temperature, at 20 seconds. 34TRAC calculation with 0.1 mm min mesh, 0.3 mns time step +theoretical heat transfer.

23 Heat conduction equation: quench front profile at 3520 seconds. TRAC calculation with 0.1 mm min mesh, 0.3 mstime step.

24 Heat conduction equation: quench front profile at 3620 seconds. TRAC calculation with 0.1 mm min mesh, 0.3 mstime step (exploded view).

25 Rod surface temperatures at 5 elevations, for 3 37calculations. Calculations are: CONTO0.25 mm,SHORT=0.25 mm + 0.3 mns, LONG=0.l mm + 0.3 ins.

AEEW - M 2590 viViii

NOMENCLATURE

A Surface area

Cp Specific heat at constant pressure

h Heat transfer coefficient

K Thermal conductivity

qK Volumetric heat generation rate

Q Heat transfer (energy)

r Radial cylindrical coordinate

p Density

T Temperature

t Time

z Axial cylindrical coordinate

(SI units)

AEEW - M 2590 ii X

1 INTRODUCTION

A previous study, [1], examined certain axial effects in the heatconduction solution of the transient, two-phase flow computercode TRAC-PFl/MODl [2]. Calculations which simulated thequenching of the surface of a nuclear fuel rod were seen to havetime step size and, to a lesser extent, axial mesh sizedependencies. The purpose of the present paper is to examine andexplain these dependencies. Similar dependencies may well arisein other computer codes which employ semi-implicit, finitedifference solution schemes.

Section 2 of this paper gives a brief description of theTRAC-PFl/MODl computer code. This section concentrates on theparticular aspects of the code that are relevant to this study.

Section 3 presents some results from the TRAC-PFl/MODlcalculations which demonstrate the time step size and axial meshsize dependencies.

In Section 4 a more detailed examination is made of theindividual terms that contribute to the heat conduction equation.Various graphical surfaces are generated by over-plotting theresults from several successive time steps.

Finally, Section 5 presents the overall conclusions of thisstudy.

2 DESCRIPTION OF TRAC-PFl/140D1

TRAC-PFl/MODl is used to perform analyses of Loss-of-Coolantaccidents and other transient's in Pressurised Water Reactors(PWR's). It is also used to analyse a wide range of relatedthermal-hydraulic experiments.

The basic operation of the code is to solve the time-dependentpartial differential equations describing two-phase flow (waterand steam) and heat transfer, by finite difference methods. Theheat transfer equations are treated by using a semi-implicitdifferencing technique. The fluid dynamics equations are solvedfor one-dimensional components, such as pipes, using a mnultistepprocedure that allows the material Courant condition to beviolated. For a three-dimensional component, such as the reactorvessel, a semi-implicit differencing scheme is used. Thecombined finite-difference equations form a system of coupled,non-linear equations. They are solved by a Newton iterationprocedure for each time step.

One aspect of the numerical scheme that is relevant to thesubsequent discussion in this paper relates to the couplingbetwen the beat transfer equations and the hydrodynamicequations. The heat transfer equations might, for example, beused to model the two-dimensional heat conduction within a heatedcylindrical rod. The coupling with the hydrodynamics equations

AEEW - M 25901 1

takes place via the surface heat transfer between the rod and thesurrounding fluid. This surface heat transfer will be dependenton the rod surface temperature and several of the fluid'sproperties; it provides a surface boundary condition for the heatconduction equation and contributes to the energy and massconservation equations for the fluid. The surface boundarycondition for the heat conduction equation, at time step (n4-l),is of the form:-

hT n +1 - Tn+l) (1)K br h sTurface - fluid)

The surface to fluid heat transfer contribution to the energyequation, for time step (n+l), is of the form:-

0 sufae hn (surface - Tftliid) At (2)to fluid

The point of particular significance in this heat transfercoupling is that the surface heat transfer coefficient isevaluated explicitly; it is calculated using rod and fluidconditions from the previous time step. In later sections ofthis paper it is shown that this explicit evaluation, takentogether with the smoothing that is applied to the heat transfercoefficient, can significantly affect the calculated surface heattransfer.

3 TRAC-PFl/MODI QUENCHING RESULTS

The calculations originally reported in [1] were hypotheticalsimulations of a 1 m, vertical, length of nuclear fuel rod insidea cylindrical pipe. The calculations were initialised with the.rod temperatures sufficiently high that the surface, forelevations above the very bottom, could not be wetted. Aconstant flow of water was introduced at th *e bottom of the pipe;the resulting cooling and ultimate quenching of the rod surfaceby the fluid, was then calculated.

Some typical results from the TRAC-PFl/MODl calculation arepresented in Figure 1. This Figure shows rod surfacetemperatures, at five elevations, plotted against time, for fourseparate calculations. The differences between the fourcalculations lie in the size of the smallest axial mesh used inthe finite difference representation of the fuel rod. This meshis separate from the mesh used to solve the fluid flow equations,which was unchanged. It can been seen from Figure 1 that thereis a wide variation in the times at which the rod surfacetemperature, for any particular elevation, quenches (ie fallsrapidly to the fluid saturation temperature). It is notimmediately apparent why changing the axial mesh size should havethis effect.

AEEW - M 25902 2

The reason for wanting to change the axial mesh can best beexplained by reference to Figure 2. This Figure shows axialprofiles of the rod surface temperature at successive times, forone of the calculations represented in Figure 1. It can be seenfrom Figure 2 that shortly after the start of the calculatedtransient a sharp, or steep, temperature gradient develops; thisgradient, or quench front, effectively separates the hotunquenched region from the cooler quenched region. As thetransient continues this quench front progress along the rod.The reason for changing the axial mesh size in the originalTRAC-PFl/MODl calculation was to identify and examine the effectsit might have on the quench front progression.

The quench front region itself is typically only a fewmillimetres wide. The TRAC-PFl/MODl solution scheme attempts toresolve this very steep temperature gradient by inserting anextra row of heat conduction mesh points, wherever thetemperature difference between adjacent surface nodes exceeds auser-input value. This value is typically 3*K for mesh points inthe vicinity of the quench front. In order to prevent anexcessively large number of mesh points being used the user alsospecifies a lower bound on the axial mesh spacing that can 'havean extra row of mesh points inserted. The four calculationsrepresented in Figure I used differing values of this lowerbound; the effective minimum mesh sizes were 2.5 mm, 0.25 mm,0.1 mm and 0.05 mm. Figure 1 shows that reducing the lower boundcauses the quench front to progress more quickly; it also causesthe quenching to occur at slighly higher surface temperatures.

The semi-implicit nature of the heat conduction solution inTRAC-PFl/MODl leads to additional complications in trying tounderstand the apparent mesh size dependency.

TRAC-PFI/MODl uses a two-dimensional (r,z) cylindrical heatconduction equation. Azimuthal symmetry is assumed. Thedifferential equation can be written as

6T .4SS 1 a T a 6

The finite difference form of equation (3), implemented inTRAC-PFlIMOD1, has implicit differencing in the radial (r)direction and explicit differencing in the axial (z) direction.The explicit differencing used for the axial term in Equation (3)leads to a stability restriction on the maximum time step size(Atmax) for a particular minimum axial mesh size (Azmin). Thisrestriction is of the form -

Atmax '" AZ2min

AEEW - M 25903 3

Thus, for the four calculations represented in Figure 1, changingthe lower bound on the mesh size has also changed the time stepsize in the calculations. The effect of reducing the time stepalone can be judged from Figure 3. This Figure shows resultsfrom three calculations; a large mesh size case, a small meshsize case, and a case with a large mesh size but a time steprestricted to 0.3 millisecs. (0.3 millisecs was the average timestep size of the small mesh size calculation). Figure 3 showsthat most of the effect seen in reducing the mesh size is in factdue to the resultant reduction in time step. This time stepsize, and to a lesser extent mesh size, dependency is furtherexamined and explained in Section 4.

4 DETAILED EXAMINATION OF CONDUCTION TERMS

The previous''section highl~ighte'd the fact that reducing' the timestep size us'ed in the quenching calculatio~ns had changed theresults. In particular, it had caused the rod surface to quenchat a faster rate and from a higher temperature. To a lesserextent, reducing the axial mesh size had a similar effect. Thisbehaviour is now examined in more detail by considering theindividual terms of the heat conduction solution.

4.1 Finite Difference Equation

TRAC-PFl/MODl solves a finite difference form of equation (3);this is obtained by applying an integral method to an appropriatedifferential volume. If the resulting finite difference equation

for each node is divided by pCp and by the node volume, then an

equation of the form:

TOTAL = GENERATION-+ RADIAL + AXIAL (3a)

(where each term is in *K/sec)

can be written for each node in turn. The heat generation occurs

internally within the rod so that for the surface nodes theGENERATION term in Equation (3a) will be zero. For the nodes ofintere-st iEn this section, ie close to the quench front, theautomatic mesh refinement will cause all the node sizes to beclose to the minimum allowed.

4.2 Surface-to-Fluid. Effects

Plots presented later in this section show the individual termsof Equation (3a), for the surface nodes, drawn as a function ofthe wall temperature. First, however, it is useful to examineone component of the RADIAL term, namely the surface heattransfer between the rod and the coolant.

Figure 4 shows a plot of surface heat transfer coefficient versus

surface temperature. Results from the TRAC-PFl/MODl calculationwith a 0.25 mm. minimum axial mesh are displayed as a sequence ofpoints, drawn as numbers. The results are taken from eachsurface node for 11 consecutive time steps at approximately20 seconds into the calculation. The fluid conditions will

AEEW - M 25904 4

normally only change slightly during 11 time steps so it isreasonable to expect that the points representing heat transfercoefficient versus wall temperature will lie on a curve. InFigure 4 the points labelled "1" are from time step 1 of thesequence and so on. Points labelled ""~ and "A" are for timestops 10 and 11 respectively. The curve traced out by the pointslabelled "I" to "A" is the effective heat transfer curve for thisparticular calculation,' at 20 seconds. Figure 4 also shows thetheoretical heat transfer curve derived for the particular fluidconditions present in the TRAC-PFl/MODl calculation. This curvewas calculated by evaluating the TRAC-PFI/MODl heat transfercorrelation separately, in a stand-alone manner, for the range ofsurface temperatures of interest.

Figure 4 shows that once a surface node is cooled belowapproximately 600*K its surface heat transfer coefficientincreases sharply. A theoretical maximum is shown to be reachedat appoximately 470*K; this corresponds to the point of criticalheat flux. However, the most striking feature of Figure 4 is thefact that the achieved, or effective, heat transfer curve issignificantly below the theoretical curve. Many values are40-50% below the theoretical values and the critical heat fluxtemperature appears to be 20*K lower. These differences arefurther highlighted in Figure 5 which shows the surface heat fluxvalues corresponding to the coefficients given in Figure 4.

4.2.1 Explicit Evaluation and Smoothing Effects

The differences observed between the effective and theoreticalheat transfer curves arise from two separate aspects of theTRAC-PFl/MOD1 solution scheme. Firstly the explicit evaluationof the surface heat transfer coefficients; this means, forexample, that the surface temperature from the previous time stepis used to evaluate the new coefficient. Secondly, the smoothingand limiting techniques applied to the calculated heat transfercoefficient; 55% under-relaxation is used (55% old-time value +-45% new-time value), followed by the restriction that,essentially, the resulting new value is no more than twice theold-time value. These techniques are applied on a per-time stepbasis and not on a per-unit time basis; thus, for example, duringthe rapid increase in coefficient shown in Figure 4 some timestep size dependency will occur.

The TRAC-PFl/MODl results shown in Figure 6 will allow these twoaspects of the solution scheme to be considered separately. Theresults shown in Figure 6 are from a calculation in which thesurface heat transfer smoothing and limiting have been removed.The theoretical heat transfer curve has been derived for thefluid conditions present at the end of the time step sequence.The effect of the explicit evaluation of the surface heattransfer coefficient can be clearly seen in Figure 6 for timesteps 3 onwards (ie points numbered 3-9, * and A). For example,the point marked 'W4', at approximately 485*K, has a heat transfervalue that corresponds to the theoretical curve evaluated at thetemperature of the point marked "3", close to 500*K. Similarlythe point "5' value corresponds to the point "4" temperatures and

AEEW - M 25905 5

so on. (This correspondence does not work in Figure 6 for thepoints marked '3", "2" and "1" because the fluid conditions atthose time steps were slightly different to those used to derivethe theoretical curve). In other words the surface temperatureT, is at time step (n+l), but the heat transfer coefficient h, isat time step (n). This point is confirmed in the formulation ofequation (1).

Figure 6 demonstrates that, in a region where the heat transfercoefficient is changing rapidly, the explicit evaluation of thecoefficient can lead to a significant deviation of the effectiveheat transfer curve from the theoretical one. In Figure 4 thedeviation also includes the under-relaxation and limitingeffects; the difference between the effective and theoreticalheat transfer is greater, particularly with regard to the peakvalue.

4.2.2 Time Step Size Effects

The calculation for which results were* presented in Figure 4 usedtime steps that were in the range of 5-10 milliseconds. Figure 7shows results from an equivalent calculation in which the timestep was constrained to be no greater than 0.3 milliseconds. TheTRAC-PFl/MODl calculated values have been drawn every 24 timesteps, ie every 7.2 milliseconds, as this corresponds to theaverage time step size of the earlier calculation. Figure 7shows that reducing the time step size has caused the effectiveheat transfer curve to follow closely the theoretical curve.Figure 8 shows the surface heat flux values corresponding to thecoefficients given in Figure 7. A comparison with Figure 5emphasises the effect of reducing the time step size.

Clearly, reducing the time step size has led to an increase inthe effective surface heat flux for surface temperatures betweenapproximately 450*K and 620*K. This is likely to be asignificant factor in explaining the time step size effect seenin Figure 2, for example. However, as the next subsection shows,the presence of axial effects must also be taken into account.

4.3 Axial Conduction Effects

In an attempt to isolate the separate contributions of the RADIALand AXIAL terms of equation (3a) several TRAC-PFl/MODlcalculations were carried out with the AXIAL term artificiallyset to zero. This prevents any axial conduction of heat withinthe rod. The results, shown in Figure 9, are somewhatsurprising, With the AXIAL term removed the calculations showvirtually no sensitivity to either time step or axial mesh size.

Figure 10 shows the surface heat flux, plotted as a function ofsurface temperature, for the first NO-AXIAL conductioncalculation. The TRAC-PFl/MODl results are similar to thoseshown in Figure 5 for the standard calculation; the effectiveheat flux curve is again significantly below the theoreticalcurve. Figure 11 shows the equivalent results from the NO-AXIALconduction calculation with the time step restricted to.

AEEW - M 25906 6

'0.3 milliseconds. The TRAC-PFl/MODl results now closely followthe theoretical curve in a similar manner to the standardcalculation results presented in Figure 8.

This shows that reducing the time step size in a calculationwithout axial conduction causes the effective surface heat fluxcurve to follow closely the theoretical curve. However, thisdoes not affect the overall quenching behaviour to anysignificant effect. The time step size effect seen in thestandard calculations must, therefore, depend on more than justthe change in the effective surface heat transfer.

4.4 Quench Front Profiles

In previous sections effective surface heat transfer curves havebeen generated by over-plotting heat transfer values from asequence of consecutive time steps. A similar technique can beused to generate a quench front profile of the individual termsof equation (3a), for nodes at the rod surface.

4.4.1 Calculation With 0.25 mm Minimum Axial Mesh

Figure 12 shows the quench front profile at 20 seconds for thestandard TRAC-PFlIMOD1 calculation with a 0.25 mm minimum axialmesh. Points labelled "A" represent the magnitude of the AXIALterm of equation (3a), points labelled "R" represent the RADIALterm and points labelled "T" represent the TOTAL term, ie-tF-esum of the AILand RADIAL term. For the sequence of 11 timesteps plott-e-d in Figu-re 12 the points representing the separateterms trace out an effective quench front profile.

The role of the AXIAL term can be readily seen from Figure 12.At the high temperature end of the region the AXIAL term isnegative ie tending to cool the rod surface. ThIn-ct fortemperatures above 5500K the AXIAL term makes up almost all ofthe TOTAL term. (The RADIAL It-e-r-m as positive values aboveapproximately 585*K because the heat being transferred frominside the rod to the surface exceeds that being transferred fromthe surface to the fluid). At the low temperature end of theregion the AXIAL term is positive ie it is opposing the coolingrate genera-ted by the larger negative RADIAL term. Thus, theoverall effect of the AXIAL term is to transfer heat from thehigh temperature end to -the low temperature end where the RADIALterm, largely governed by the surface-to-fluid heat flux, islarge and negative.

In Figure 12 the magnitude of the TOTAL (WT/bt) term becomessmall, for temperatures above approx=imately 655*K. Thiscorresponds to the temperature of the "knee" in the temperatureversus time plot for the 2.5 millimetre minimum mesh calculation,shown in Figure 1. For temperatures above this value the rodsurface is cooled comparatively slowly. However, fortemperatures below this value the rate of temperature fallincreases very rapidly, until the surface is quenched. It can beseen from Figure 12 that, at least for this calculation, the

AEEW - M 25907 7

-temperature at which this knee occurs is governed by the onset ofthe large negative AXIAL values.

The movement of the quench front region along the rod canconveniently be characterised by the movement along the rod ofthe knee in the temperature profile. The actual temperature atwhich the knee is maintained will be dependent on the details ofthe heat conduction solution within the quench front regionitself. Figure 13 presents an exploded view of the AXIAL andRADIAL terms taken from Figure 12 in the region of the knee. Fortemperatures above 660*K it can be seen that the AXIAL term isessentially zero and the RADIAL term is negative (jie cooling thesurface) and increasing i7n -magnitude with increasing surfacetemperature. For temperatures below 660*K the RADIAL termbecomes negligible and the AXIAL term very rapidl-ybecomes largeand negative. This large.neg7t"Ive AXIAL term rapidly cools thecladding surface and allows the que-nc-h front or temperature kneeto move forward.

4.4.2 Calculation With 0.25 mm Minimum Mesh and 0.3 ms Time Step

Figure 14 presents the quench front profile at 20 seconds f or thestandard calculation with the reduced time step size. Comparisonwith Figure 12 shows that the magnitude of the peak negativeRADIAL term has increased significantly; this is in line with theincreaed surface heat flux seen by comparing Figure 8 withFigure S. The magnitudes of the peak AXIAL terms (positive atlow temperatures, negative at high temperatures) have alsoincreased significantly, leading to increased magnitude TOTALterm values. In particular, the increased magnitude nega~tiv-eAXIAL terms at high temperatures have moved the temperature ofthe knee up by approximatley 20*K. This is borne out bycomparing the temperature versus time profiles shown in Figure 3.An exploded view of the AXIAL and RADIAL terms close to the kneeis given in Figure 15. C-omparison -wit-hFigure 13 shows that theRADIAL terms ahead of the knee, from the two calculations, lieapproximately on the same curve.

Figure 14 shows that with the reduced time step the magnitude ofthe AXIAL (and hence TOTAL) term increases more rapidly, as thesurf-ace temperature f-alls below the knee temperature, than forthe standard calculation shown in Figure 12. This is consistentwith the observed faster progression of the knee in the smalltime step calculation.

Thus the observed time step size dependency appears to be relatedto the increased magnitude AXIAL terms at high temperatures. Twoquestions remain unresolved however: why are the AXIAL termsincreased in magnitude, and why does the calculatio-nwith noaxial conduction show no time step size dependency. These twoquestions are now addressed in turn.

4.4.3 AXIAL Term Time Step Dependency

The significant increase in the AXIAL term magnitude shown inFigure 14 could be due to two possirble effects. Firstly, the

AEEW - M 25908 8

large increase in the peak RADIAL term magnitude will havechanged the axial temperature profile in the quench front region.This is likely to change the AXIAL term values as they are,essentially, derived from the axial temperature profile.Secondly, reducing the time step size may in itself have changedthe AXIAL term values as they are evaluated explicitly. Toresolve this issue a calculation has been performed using thereduced time step size but with the surface-to-fluid heat fluxmodified so that it remains at the level shown in Figure 5 ratherthan the increased level shown in Figure S. This was achieved byreducing the critical heat flux value (CHF) used by TRAC-PFl/MODlin evaluating the heat transfer coefficients.

Figure 16 shows the effective surface heat flux curve from thisnew reduced-time step calculation. It is in fact quite close tothe effective curve presented in Figure 5 for the originalcalculation. Figure 17 shows the quench front profile for thenew calculation at 20 seconds. Both the RADIAL and AXIAL termcurves are very similar to the correspond'in-gcurves sh-own inFigure 12 for the original calculation. Thus the AXIAL termvalues have no time step size dependency of their -own (Twithinthe time step range considered) but rather they reflect the timestep size dependency of the RADIAL term. This in turn reflectsthe time step size dependency of-the surface-to-fluid heat flux;as previously shown this is due to the explicit heat transferevaluation and smoothing techniques inherent in the solutionscheme.

4.4.4 Calculations With No Axial Conduction

Figures 10 and 11 showed the effective surface-to-fluid heat fluxcurves for two calculations with no axial conduction. Reducingthe time step size to 0.3 milliseconds caused the effective curveto follow the theoretical curve (Figure 11) but did not, however,change the overall quenching behaviour (Figure 9).

Figure 18 shows the quench front profile for the large time stepcalculation. As the AXIAL term is zero the TOTAL term is simplyequal to the RADIAL term. The effective RADIAL term curve inFigure 18 is _sir`=har to the RADIAL term curve shown in Figure 12for the standard calculation. However, the lack of an AXIAL termmeans that the TOTAL term becomes small at a lower temperaturethan in Figure flT=,e the temperature knee is maintained at alower temperature. This is confirmed by the temperature versustime profiles shown in Figure 9. Figure 18 also shows that themagnitude of the TOTAL term increases slightly less rapidly, asthe surface temperature falls below the knee temperature, than.for the standard calculation. This is consistent with the slowerprogression of the quench front in the calculation with no axialconduction. Figure 19 shows an exploded view of the TOTAL termin the region of the temperature knee. A comparison wi~ththestandard calculation results, given in Figure 13, shows that theRADIAL term values lie essentially on the same effective curve.Howevr, in the no axial conduction calculation the knee ismaintained at a lower temperature.

AEEW - M 25909 9

'The quench front profile for the no axial conduction calculationwith the reduced time step size is presented in Figure 20. Thepeak magnitude of the RADIAL term has increased, compared toFigure 18, in line with -the increase in the surface-to-fluid heatflux shown in Figure 11. However, at the high temperature end ofthe region the values are unchanged. This is further borne outin the exploded view shown in Figure 21.

Thus in a calculation with no axial conduction, although the peakTOTAL term magnitude is increased, the TOTAL term values at thehigh temperature end of the quench front region are unchangedwhen a small time step size is used. This is consistent with theobservation that the overall quench front movement is unchangedwhen a small time step size is used.

4.4.5 Axial Mesh Size Effects

Having established and examined the time step size dependency itis now worthwhile examining any mesh size effects. TRAC-PFl/MoDlwill normally automatically reduce the time step size when smallaxial mesh sizes are used, because of the explicit evaluation ofthe axial terms. Therefore, to establish any genuine mesh sizeeffects a comparison has to be made with a calculation thatalready uses a sufficiently small time step size.

A calculation has been performed using a 0.1 mm. minimum axialmesh and a 0.3 millisecond time step size. Figure 22 shows theeffective surface-to-fluid heat flux curve for this calculation;this can be compared to the curve in Figure 8, which used a0.25 mm minimum mesh. The two effective curves are very similar;

the smaller mesh curve lies slightly closer to the theoreticalcurve at the peak value. The smaller mesh size gives more nodes,and hence a better resolution, in the region of peak surface-to-fluid heat transfer values.

Figure 23 shows the quench front profile for the new calculation;this can be compared to Figure 14 for the larger mesh sizeresults. The results are again very similar apart from the peakAXIAL and RADIAL values at the low temperature end of the region.An exploded vie-w of the AXIAL and RADIAL terms at the hightemperature end of the region is shown in Figure 24.Threutare very similar to those shown in Figure 15 for the larger meshsize calculation. This suggests that the overall quench frontprogress should be very similar for the two calculations.

.The surface temperatures versus time for the new calculation areshown in Figure 25. A comparison is made with the larger meshsize calculation and also the original larger time step sizecalculation. The new calculation shows that there is a smallaxial mesh size dependency, but that it is very small compared tothe time step size dependency.

5 SUMMARY AND CONCLUSIONS

The purpose of this paper is to examine and explain the time stepand mesh size dependencies observed in calculations of the

ABEW - M 2590 110

quenching of a nuclear fuel rod. Both effects have been shown toarise from an underlying dependency in the surface-to-fluid heattransfer. The time step dependency occurs because the heattransfer coefficient is evaluated explicitly, ie using valuesfrom the previous time step, and because under-relaxation isapplied to the newly calculated coefficient. This dependencywill be particularly noticeable whenever the heat transfercoefficient is changing significantly from one time step to thenext, such as occurs during quenching. The smaller mesh sizedependency appears to arise from changes in the spatialresolution at the calculated heat transfer coefficient close toits peak value.

The paper has shown that changes in the surface-to-fluid heattransfer affect the overall quenching behaviour by virtue ofchanging the axial temperature profile; this changes the axialconduction terms in the overall rod conduction equation. It ischanges in the axial conduction terms, at the high temperatureend of the quench front region, that alter the overall quenchfront progression. In calculations where the axial conductionterm was artificially removed, changes to the surface-to-fluidheat transfer did not affect the overall quenching behaviour.

The findings can be summarised in the following conclusions:-

5.1 The studies described in this report have identified asignificant time step size dependency in the solutionobtained from a coupled system of heat transfer and two-phase flow partial differential equations.

5.2 The time step size dependency of the solution arises fromthe time step size dependency of the surface-to-fluid heatflux; this flux is the coupling between the heat transferequations and the fluid flow equations. The dependencyoccurs as a result of the explicit evaluation of thesurface-to-fluid heat transfer coefficient, and as a resultof the time 6tep-to-time step smoothing techniques appliedto the coefficient.

5.3 For the TRAC-PFl/MODl quenching calculations described inthe report the time step size dependency of the solutiondissappears if the axial conduction term of the heatconduction equation is removed. This is because thesurface-to-fluid heat flux time step dependency affects theoverall solution only by changing the size of the axialconduction terms.

5.4 The studies described in this report have also identified asmall axial mesh size dependency; this is, however, muchsmaller than the time step size dependency. Thisdependency again appears to arise from a small mesh-sizedependency of the surface-to-fluid heat transfer, 'mainly inthe region of high and rapidly changing heat transfervalues.

AEEW - M 2590 111

5.*5 The time step size dependency represents a potentialproblem in the use of the TRAC-PFl/MODl code, with regardto running times. The numerical solution scheme for one-dimensional components employs a multistep procedure thatallows the material Courant condition to be violated. Thisability to use large time step sizes will be restricted ifsmall time steps are needed for the heat transferevaluation part of the scheme. Further work is needed toimprove or replace the explicit heat transfer evaluationand to remove the time step size dependency from the heattransfer smoothing techniques.

6 REFERENCES

(1) O'Mahoney, R. A Study of Axial Effects in theTRAC-PFl/MOD1 Heat Conduction Solution During Quenching.AEEW - M 2552, PWR/HTWG/P(89)686, June 1989.

(2) TRAC-PFl/MODl. An Advanced Best-Estimate Computer Programfor Pressurised Water Reactor Analysis. Los AlamosNational Laboratory Report. LA-10157-MS, NUREG/CR-3858.

AEEW - M 2590 112

TRAC-PF1/MOD1 013.0 (bO3e)

0

900-

8501

800.

750-

700-

650

S600

550-

500-

450-

4001.

FIGURE 1

HIGH TEMPS

LOW FLOWS

5 10 15 20 25REACTOR TIME ,SECONDS

30

lWinfr it h

XI

ROD SURFACE TEMPERATURES AT 5 E~LEVATIONS, FOR 4 DIFFR NT ESH SIZESMIN AXIAL MESMS ARE: CONT=2. 5MM, SHORT=O. 25MM, LONG-O. 1MM, THICKO0.05MM

TRAC-PFI/MODI 013.0 (bO3e)

I,

rn

1100

1000

900

800

700

600

500

Annf

. ......... --

-- - - - -- -- - - - - - - - - - -- -- - --- - - - - - - - - - - -- - - - -

-- --- ----- ----- ----- ----- -- - - - - -t . . . .. . - - - - - - -

-- -- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

T- ,01 T-0T3

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...........

HIGH TEMPS

LOW FLOWS

-u

0 20 40DISTANCE ALONG IROD

60 80 100

FIGURE 2WinEri

-11

AXIAL PROFILE OF ROD SURFACE TEMPERATURE AT 4 DIFFERENT TIMES

PROFILES AT 0, 10, 20 & 30 SECS, FOR 0.1M MAIN AXIAL MESH

TRAC-PF1/MODI 013.0 (bO3e)

0

lN,a

-4

N(A

900 -

850-

800-

750-

700

s1 650

S600-

550-

500-

450-

400 -

0

FIGURE 3

HIGH TEMPS

LOW FLOWS

W in Fr It h

5 10 15 20 25 30REACTOR TIME ,SECONDS

-q,

mL4

ROD SURFACE TEMPERATURES AT 5 IELEVATIONS, FOR 3 CALCULATIONSCALCULATIONS ARE: CONT=O.25MM, SHORT=O0.O5NM, LONG=O0.25MM + O.3MS DT

TRAC-PFl/MOD1 013.0 (bQ3e)120000 I

to

1000001

800001

C14 60000

-- -- -- - - \- --- -- - - - - ---- - - - - -- --------U--------------A

< THEORETICAL HEAT TRANSFER

-- --- - - - - - .4. ----- -- - - - - - - - - - ---. . . . .

:1 SYMBOLS:1 - 91 *p A ARE TRAC CALCUJýATION VALUES

------------------------ - -- - I - - - - - - -- - - - - - - - - - - - -

----- -----I N,---- -------- -------------- - -----------

40000~

20000

OL400 450 500 550 600

DEG. K650 700

SURFACE TEMPERATURE

FIGURE 4Winfrithl

-r,

SURFACE-TO-FLUID HEAT TRANSFER COEFF. VS TEMPERATURE, AT 20 SECONDS

TRAC CALCULATION WITH 0.25MM MIN MESH + THEORETICAL HEAT TRANSFER


mqI.'

('Ito0

7000000

6000000

5000000

4000000

4cI-I-.4

-o

3~

-4

0

~00,to

-.4I~3U'

3000000

2000000 -

1000000 --

0 -400 450 500 550 600 650 700

SURFACE TEMPERATURE IDEG. KFIGURE 5 Win fri thl

0n

Lc

SURFACE-TO-FLUID HEAT FLUX VS TEMPERATURE , AT 20 SECONDSTRAC CALCULATION WITH 0.25MM MIN MESH + THEORETICAL HEAT FLUX

TRAC-PF1/MOD1 013.0 (bO3e), NO HEAT TRANSFER SMOOTHING120000

mn

rn

IQ

00

-o

C)

coto

100000k --------

:6-

, I-I.

I'

'I

"8'* - . - I. - -

'I47

I,

- - -I --------- - - - -

I THERETICAL HEAT TRASFER

-- - - - - - - - - - - - - - - - - - - ----------80000k --------

04'A

IIliI~bz I - ýl I F iia TKRJ4L i ~JAX4ALUN VAWtkS

- -- - - - - - - - - - - -.- - ---.- - - - - - --.- - - - - - - - - -60000k ---------7/40000

9

20000k1---- -- ---- - - - - - - - - - - - - - - - - - - -

*A-4

* ~~~7 Q It D7 A~f~ -0140

I

i

0 450 500 550 600DEG. K

650 700SURFACE TEM4PERATURE

FIGURE 6 lWinfrithl

c-)

SURFACE-TO-FLUID HEAT TRANSFER COEFF. VS TEMPERATURE, AT 20 SECONDS

TRAC CALC. WITH 0.251MM MIN MESH, N MOHN HOEIA .TNO SMOOTHING + THEORETICAL H. T.


rnf-n

to0

120000

100000 -----------

i

-- -- - - -- - - ------------------------- - - - - - -

SYMBOLS;1 - 9p *1 A ARE TRAC CMOWUI~fON VALUES

--------

80000

clqI 60000k ---------I,

40000 ..--

-o

-4

L"

5\

20000 k-.7 --------------- -------------------------

0140

I Ion NX M&LOANtahrmMiLAL-AL-m-m

0 450 500 550 600,DEG. K

650 700SURFACE TEMP2ERATURE

FIGURE 7 Wtinfrit

-n SURFACE-TO-FLUID HEAT TRANSFER COEFF. VS TEMPERATURE, AT 20 SECONDSTRAC CALC. WITH 0.25NM MIN MESH,03M TISEP+ HORICLH.0.3MS TIMESTEP + THEORETICAL H.T.


0

8'

7000000

6000000

5000000

4000000

3000000

00

-4

2000000-

1000000 -

0-400 450 500 550 600 650 700

SURFACE TEM4PERATUJRE I DEG. K

FIGURE -8 lWinfritdi

-Ti

C)CP1

02

SURFACE-TO-FLUID HEAT FLUX VS TEMPERATURE , AT 20 SECONDS

TRAC CALC. WITH 0.25MM. MIN MESH, 0.3MS TIMESTEP + THEORETICAL H.T.

TRAC-PFl/MOD1 013.0 (bO3e), NO AXIAL CONDUCTION

mn

coIto

900-

850-

800-

750

700

650

600-

550-

500-

450-

400L0

HIGH TEMPS

LOW FLOWS

5 10 15 20 25 30REACTOR TIME ,SECONDS

FIGURE" 9 Iintril

ROD SURFACE TEMPERATURES AT 7 ELEVATIONS, FOR 3 NO-AXI.AL CAMCSCALCULATIONS ARE: CONT=O .25MM, SHORT=O .25NM+ . 3M5, LONG-. O5NM+ . 3MS

Nd

-In

mto

TRAC-PF1/MOD1 013.0 (bO3e), NO AXIAL CONDUCTION7000000

'Ii~1~

01to0

cq4'

-u

0-%

CA01 OEII-I -wm400 450 500 550 600 650 700

SURFACE TEMPERATURE r DEG. K

FIGURE-10 1WInfitjh

C-nC)

SURFACE-TO-FLUID HEAT FLUX VS TEMPERATURE, AT 20 SECS

TRAC CALC, NO AXIAL CONDN, TRA CAC, O AIALCONN,0.25MM MIN MESH + THEORETICAL HEAT FLUX

TRAC-PF1/MOD1 013.0 (bO3e), NO AXIAL CONDUCTION

rn

to

0

7000000 -

6000000 -

5000000 -

4000000 -

3000000 -

2000000 -

1000000 -

0-400 700

SURFACE TEMPERATURE I DEG. KFIGURE 11 Winfr'it hý

-I,

C

I,,

SURFACE-TO--FLUID HEAT FLUJX VS TEMPERATURE,TRAC, NO AXIAL CONDN, 0.25MM MIN MESH, 0.3

AT 20 SECONDS,MS STEP + THEORETICAL H.T.


rn

#0

to

-4

00

to

14)

U)

CDrz~Q

6000

4000

2000

0

-2000

-4000

-6000

-8000

-1000044

* SYMBOLS As R, T ARE TRAC CALCULAT~ION VALUES

A < AXIAL TERM (A).

*R e2RADIAL TERM (R)

-- - -- - - - - - - - - I -- - - - - - - - - - - - - - - - - - - - - - - - -

00 450 500 550 600DEG. K

650 700SURFACE TEPERATURE

FIGURE 12

HEAT CONDUCTION EQUATION: QUENCH FRO]

TRAC CALCULATION WITH O.25NvM MIN MESH

Winfrithi

-n,

0

NTPROFILE AT 20 SECONDS


5 It

20 -- - - - - - - - - - I -- - - -A1- - - - -1 - -- - - - - - - -. . . . .

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HEA CODCTO EQAIN QUNC FRN PRFL AT2EOD

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- r9

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P1

I.3Utto0

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2000

-----------

A

---------- :4--

SYMBOLSA, R, T AR~E TRAC CALMULTI101 VALUJES

A 1 AXIAL TERM (A)A

-A...A.---------------------------------------.... ...- .... ........

An ca R:R RR RRW R; IU ~ ~ J I Ii - I 1=~~~~~ -

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r-4

0

U)

rzl-2000

-4000

-6000

-8000

-1000040

-... .. .. .4 ----

0 P

A R

ARA

R~ TT

T I IT T I TOTAL DTDT (T)T T T1 PT

-- - - - -- - - ---- - - - - - ------ -- -

R < RADIAL 'TERM (R)

-- - - - - - - - - - - ----9 ---- - - - - -- - - - - - - - - - - - - -- - - - - - - - - -----------

0 450 500 550SURFACE TEM~'PERATURPE

600DEG. K

650 700

FIGURE 14 lwinfith

~1~

C

I,,

HEAT CONDUCTION EQUATION: QUENCH FRONT PROFILE AT 20 SECONDSTRAC CALCULATION WITH 0.25I1M4 MIN MESH, 0.3MS TIMESTEP

TRAC-PF1/MOD1 013.0 (bO3e)[I.

IT'

mn

R

RRR

AXIA EWM (A)A A0 I I II IA

A'

A - a

-51----- ------------- * --- ------ --- --- --- --- --- --if Urn* if lyw a

* *-* "if

I)J U .LhLL

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-15 k------ 1--------- I--------

-o

-4

-20 -S- - - --.. J .. ... --- --- ---

--------- -------- ---------

A .

-2560'0 620 640 660 680 700

DEG. K720 740

SURFACE TEM4PERATUREFIGURE 15 1WInfrit

~'15cm-A

U,

HEAT CONDUCTION EQUATION: QUENCH FRONT PROFILE AT 20 SECONDSTRAC, CALCULATION WITH 0.25MM MIN MESH, 0. 3MS TIMESTEP [EXPLODED VIEW]

TRAC-PF1/MOD1 013.0 (bO3e), REDUCED HEAT FLUX7000000

:SYMBLS 1 - %~ *,, A ARE PEDUCBD-CH MW: CALCATICU MMIE

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5000000

4000000 L-(~~l

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* I THE~OPMEICAL WWEVA!FD (UMODIFIEP)

--- - -- -

-N - - - - - - - -- - -- -- ---.- -- --- --.-.- - -- --

---- ---- -... ...----- ---- --- - -- ----. . .. . . . . . . . ..--- ---. ---- --

9

* ~7 *--- I ..--------- ------.......------5

2000000W-

00

10000001--

0L40

* 4 A 3 ~*-.

0 450 500 550SURFACE TEMIPERATURE

600DEG. K

650 700

FIGURE 16 1Winfrithl

-I,

SURFACE-TO-FLUID HEAT. FLUX VS TEMvPERATURE, AT 20 SECONDS

TRAC CALC WITH 0.25MM MIN MESH, 0. 3MS STEP, REDUCED CHF + THEORETICAL

TRAC-PF1/MOD1 013.0 (bO3e), REDUCED HEAT FLUXI

rv,

to0

6000

4000SYMOLS: At R, T AME TRAC C LATIO VALUE

~AAA AA <AXIAL TERM :(A)

A A'20001

A A:

0 n n9I n n

m m I nh1 r----------.1 -- WEB.........

C,,

-2000 k----

R~

A AA

ARR

It R K K ' IAm~

Av~~ Av~ At'T T~ T ~ T~i ' LJLLU IIUL 9.L_

R *

RR RADIAL TERM (R)

-4000ý- --

-u

Nt-4

C-,N-u

toS.-.

NU,

-6000

-8000 -----------

I I.. . . . . . . . . . . . . . . . . .

-100001U400 450 500 550

SURFACE TEMPRATURE600

DEG. K650 700

1WInfr'ithiFIGURE 17

C)

-4

HEAT CONDUCTION EQUATION: QUENCH FRONT PROFILE AT 20 SECONDS

TPAC CALCULATION WITH 0.2 5M! MIN MESH, . MS ST-EP, EDCE HFAT FLUX


0

6000

4000

--------------------------------------------------------- -------

2000 F ---------------- --------------- - - - - - - - - - - . . . . . . . .

0 isissal i -. ITI [a T ST ITT

U)

T:

: T T-2000 k- - < TOTM 6Mln (T) = RAatAL TERM

.T T. TIý- - - - -- - -- - - - - - -- - - - -

-4000 k---

0

0-%

N(if

-6000k --------

-8000 -- - -- -- - -- --* I I* I I

* I I

* I

* I I

* I I

-10000140 0 450 500 550

SURFACE TEMPERATURE600

DEG. K650 700

FIGURE 18 1WInfrithl

cxo

HEAT CONDUCTION EQUATION: QUENCH FRCThAC CALCULAION WITH NO AXIAL CONDN,

)TPROFILE AT 20 SECONDS0.25MM MINMESH

TRAC-PF1/MOD1 013.0 (bO3e), NO AXIAL CONDUCTION2~).

mP1

I~JU'~00

(.A~I-.

0

-5

TI T I I I

T I I1

fCo

C,rz~

-101-----------...TOTAL DT~ir (T) = RADEAL TEPM

T

f V

-151 - -- ------ -T--- - ---- ------ - --- -- .... .. :-- -- -

-U

=-g

C)

-U

£0

U'

-20 -- - - - - - - - -

ITI- - - - - - -

-2516(0 620 640 660 680 700

DEG. K720 740

SUJRFACE TEMAPERATUREFIGURE, 19 lWinfrdith

C

HEAT CONDUCTION EQUATION:QUTRAC CALC WITH NO AXIAL CONDN,

,NCH FRONT PROFILE AT 20 SECONDS0.25MM MIN MESH 0.21~'4 MN MSH t EXPLODED VIEW ]

TRAC-PF1/MOD1 Y13.0 (bO3e), NO AXIAL CONDUCTIONi

6000fyi

r'-U~00

(J~

-U

Nz-g

C,N-uw(0

-.1

01

SYMBOLS: T AR~E TRAC CAICULAMICN VALUES

4000 k- - - - - ---- - - - - - .

2000

0

.*.. ...... . ...... - - - - - - - - - - - - - - - - - - - - - - - -

49MM" i i iI1 TIT N muTTT=T9X :: :

U)aT

-------- ------2000

*~ 1T T T

T

TTOTAL DTDT (T) = RWDIAL TERM

-40001ý - - -- --- a - - - - - -I- - - - - - -- - - - - - - - - - - -:- - - - - -

T s

-6000 I- - - - - - .. . . .. . .- - - - --- - - - -- - - - -

-8000

-i Aflflf

-- -- - - - . . . . . . . . . : - - - - - - - - - - -

- - - - - - - - - - - -

4--~00 450 500 550 600,DEG. K

650 700SUR~FACE TEMPERATURE

FIGURE 20 1WInfri th1

-yi

C)CP1

0

HEAT CONDUCTION EQUATION:QU

TRAC CALC WITH NO AXIAL CONDN,

NCH FRONT PROFILE AT 20 SECONDS

0.25MM MIN MESH, 0. 3MS TIMESTEP


rn 0

-5 -- - - - - - - - -

----------

'T Ti .......... ------'I

U)

-10 T .. . .TOTAL DTdT (T) =RADALTIM 4j T y

- - -- - - - - - - - - - - - - - - - - - - - - - - - - -

-15

--4

14N

CA

-201 --------

T

--- -- --- --- - - ----- - - ---- --- - - - - - - --..- -.-..

-251600 620 640

SURFACE660 680 700 720 740

TEMPERATURE I DEG. K 1WInfrihFIGURE 21

In

t'5

HEAT CONDUCTION EQUATION:TRAC WITH NO AXIAL COND,

QUENCH FRONT PROFILE AT 20 SECONDS0.25MM MIN MESHFO.3MS STEP [EXPLODED VIEW]

TRAC-PF1/MOD1 013.0 (bO3e)7000000

6000000

I'301

0

50000001

40000001C4J

3000000

- - - - ---------------- ~---- ------------- ------------ ------------ ......

6 SIMB0LS 1 - 9, *1 A jRE TRA CMOJAIION OE

*~ V-

-- ----------- I---6x--------L-------------

------------.-------------------- A...........

- - -1 -

2000000t-u

N-I

CI)N-uw10

U'

1000000

0t400 450 500 550 600

DEG. K650 700

SURFACE TEMPERATUJRE

FIGURE 22 1WInfrithl

-I, SURFACE-TO-FLUID HEAT FLUX VSThAC CALC WITH 0.1MM MIN MESH,

ITEM1PERATURE,

0. 3MS TINESTEAPAT 20 SECONDS

+ THEORETICAL H.T.


*A

6000m

~0

A*A SYMBOLS:At R, T ARE TRAC CALCULATION VALUES

4000 k----------- IA i

2000

0

-- - -- - - - - - - -

( i---- AXIA TEM (A)

A~~~~ ~ ~ : I

AAA

A m S al

cl)

--- --- --- 7 T -T. rT -

A A

-2000 A - - - - - - - - -

* P t~lTyT T TTT rTOTAL DTDT (T)

-40001k------

* RR- - - - - - - -- - - - - - -- - - - -

* R A

c"to

*-4

0,

-60001. ------

-8000 -- - - - -- -- - -- --~< : ADIAL:TEFNM(R)

R

-100001L40

I - - a I a

0 450 500 550 600,DEG. K

650 700SURFACE TEPERATURE

FIGURE 23 1WInfrith1

c);o

HEAT CONDUCTION EQUATION: QUENCH FRONT PROFILE AT 20 SECONDS

TRAC CALCULATION WITH 0.1WM MIN MESH,0.3STESE0.3MS TIMESTEP


ON

(~)

-u

-4

-u M4 660 680 700SURFACE TEMPERATURJE ,DEG.IK

FIGURE 24

HEAT CONDUCTION EQUATION:QUTRAC CALC WITH 0. 114 MIN MESH,

1Winfrithl

-I,

'NCH FRONT PROFILE AT 20 SECONDS

0. 3MS TRIESTEP [ EXPLODED VIEW

TRAC-PF1/MOD1 013.0 (bO3e)800

r~im

NU'to0

Li.)

-U

NI-4

C,N

(0

NC,'

750 HIGH TEMPS

700 LOW FLOWS

650

600

550

500

450

4001L0 2 4 6 8

REACTOR TIME10 12 14

rSECONDS16 18 20

FIGURE 25

-I,

;omn

ROD SU1WACE TBP4ERATURES AT 5 IELEVATIONS, FOR 3 CALUT.ATIONSCALCS ARBE: CONT=O.25NMISHORT=O.25MM + O.3M3 DT,LONG-O.1MM + O.3MS DT

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NRC FORM 335 U.S. NUCLEAR REGULATORY COMMISSION 1. REPORT NUMBER(2-89) lAsiged by NRC. Add Vol., Sig. Rv..NRCM 1102.,p ddnu NVbr . n.3201.3202 BIBLIOGRAPHIC DATA SHEET NUREGICA-007umes.If3nr

(See inlstrutitons on the feverse) AEREW-M259002. TITLE AND SUBTITLE

I AE-29

Time Step and Mesh Size Dependencies in the Heat Conduction 3. DATE REPORT PUBLISHED

Solution of a Semi-Implicit, Finite Difference Scheme for MONTH YEAR

Transient Two-Phase Flow April 19924. FIN OR GRANT NUMBER

____ ___ ___ ___ ____ ___ ___ ___ ____ ___ ___ ___ ____ ___ ___ ___ A4682

5. AUTHORIS) 6. TYPE OF REPORT

R. O'Mahoney Technical7. PERIOD COVERED (minciese Dares)

0 PER~FORMING ORGANAIZATIONJ - O.5AUC Atdf ArlflDC0C .... , .,. - ..

nsme end mailing address.) I - r, U - ,ufeo ein . ulwRgltr omsin n ~~n ~e;i ojw rvd


9. SPONSORING ORGANIZATION - NAME AND ADDRESS (if NRC. type me as. e it contrac vo,~ provide NRC Division~ Offsce or Region. U.~ Nuclear Regulatory Commission.

and mailing address.)

Office of Nuclear Regulatory ResearchU.S. Nuclear Regulatory CommissionWashington, DC 20555

10. SUPPLEMENTAIRPY NOTES

11. ABSTRACT 1200 words or less)

This report examines, and establishes the causes of, previously identified time stepand mesh size dependencies. These dependencies were observed in the solution of acoupled system of heat conductionand fluid flow equations as used in the TRAC-PF1/MQD1computer code. The report shows that a significant time step size dependency canarise in calculations of the quenching of a previously unwetted surface. The cause ofthis dependency is shown to be the explicit evaluation, and subsequent smoothing, ofthe term which couples the heat transfer and fluid flow equations. An axial mesh sizedependency is also identified, but this is very much smaller than the time step sizedependency. The report concludes that the time step size dependency represents apotential limitation on the use of large time step sizes for the types of calculationdiscussed. This limitation affects the present TRAC-PF1/MOD1 computer code and maysimilarly affect other semi-implicit finite difference codes that employ similartechniques. It is likely to be of greatest significance in codes where multi-steptechniques are used to allow the use of large time steps.

12. KEY WORDS/DESCR:PTORS (List wordsoriphrases that will wasist esearchers In locating the report.) 13. AVAILABILITY STATEMENT

Unl imi tedTime Step, Mesh Size, Dependencies, Heat Conduction Solution, 14. SECURITY CLASSIFICATION

Transient Two-Phase Flow (This Pag)

Uncl assi fied(This Report)

Uncl assi fied15. NUMBER OF PAGES

16. PRICE

NRC FORM 3351249)

THlIS DOCUM2ENT WAS PRINTED USING RECYCLED PAPER

NUREG/IA-0073 TIME STEP AND MESH SIZE DEPENDENCIES IN THE HEAT CONDUCTION SOLUTION OFA SEMI-IMPLICIT, FINlTE DIFFERENCE SCHEME FOR TRANSIENT TWO-PHASE FLOW

APRIL 199

UNITED STATESNUCLEAR REGULATORY COMMISSION

WASHINGTON, D.C. 20555

FIRST CLASS MAILPOSTAGE AND FEES PAID

USNRCPERMIT NO. G-67

OFFICIAL BUSINESSPENALTY FOR PRIVATE USE, $300