A MODEL FOR PREDICTING THE RADIAL POWER PROFILE IN A FUEL PIN LD. PALMER, K.W. HESKETH, P.A. JACKSON British Nuclear Fuels Ltd, Springfields Works, Salwick, Preston, United Kingdom
A simple, fast naming computer m u t r m for calculating radial power profiles, throughout U f a , is both standard aad duplex fuel pellets for all types of thermal reactor has been developed. The code sub-divides the pellet into a number of ? ? ^ for each of which it solves for the concentrations of uraniua and plutoniua and hence calculates a mean inverse diffusion length. The diffusion equation is solved in terms of Bessel functions and the resulting flux profile multiplied by the concentration profiles to give a radial rating profile which is normalised to unity. The model shows good agreement with the results of detailed physics calculations for different thermal reactors over a wide burn-up range. Its incorporation into the H0TR0L-4C and SIHJT3-SEER-77 fuel performance codes has led to a negligible increase in running times. UJTBODCTCTIQK The variation with burnup of the radial power profile in a thermal reactor fuel pin is a complicated function both of pin design parameters, such as geometry and initial enrichment, and of the reactor operating conditions. In the RADAR model described here (Rating Depression Analysis Routine) an approximation to this complex variation has been formulated using a simple and economical computer routine which is sufficiently physically based that it can be applied reliably to fuel of any design, operated in any type of thermal reactor. General purpose fuel performance codes such as BOSRODv1/ and SIEUTH-SEERN^) require a model of the radial power profile in order that the temperature distribution within the fuel may be determined accurately. At present, these codes calculate a satisfactory start of the life rating profile, but make little or no attempt to modify its shape as burnup proceeds. This means that they are neglecting the build up of plutonium in a thin layer near the pellet surface and the consequential effects on centre temperature and the rate of crack closure. In the case of duplex pellets (for a description of the performance of duplex fuel see ref (3)), they can be overlooking a change of up to an order of magnitude in the rating ratio at the duplex boundary. The incorporation of RADAR into the codes is aimed at overcoming these limitations. THEORY The RASAS model divides the fuel pellet into a nucber of concentric annul i and breaks up the irradiation history into small burnup steps. The size of the steps and the number of asnuli axe specified by the user and effectively determine the accuracy of the final results. At the start of life, no plutoniua is considered present so that the fuel pellet consists of sioishiomevric UO2 with a U 2 35 enrichment and dimensions supplied by the user. At subsequent busnup steps equations are formulated in each annulus independently and the solutions related through a volume weighting normalisation process.
In each annulus the burn-out of TJ235 during a bumrup step, AB, i s governed by the equation
AU=-Uh t AB
where T i s concentration of U 35 atoms and 7
where * i s thermal flux and o* 235 i s the U235 absorption cross section appropriately weighted to reflect a typical reactor neutron spectrum (and taken here as 347 barns). R i s the relative specific rating vhich i s given by
R= ? ? ^ m o 1 ( * t ? ? U . 8 ' f M , p j 1.06E
where P is the Pu239 concentration, the o*f's are the weighted fission cross sections (taken as 455 barns for U 2 35 and 534 barns for Pu 2 39), S is the energy liberated per fission (200 Me7), nmol is the molecular nuaber density (2.445 x 10 2 2 cm-3), o. is the UO2 theoretical density (1O.96 gca-3) and the factor 1.08 is an allowance for the contribution of U 2 38 fast fissions. By using small burnup steps the previous step values for the uraniun and plutoaium concentrations, Ho and Po, can be employed in equations (2) and (3). thus giving the TJ235 burn-out as
? 4-10* Uo* 9.5 -10* P. .
The equations governing the plutonium concentration are nore complicated than for uranium because i n addition to bum-out, two production mechanisms are also considered. Firstly, thermal neutrons are captured by U22 nuclei! which sub33 sequently decay via two short half-life beta decays to Pu 39, whose production rate i s proportional to the thermal flux, * , and the U 2 3 s thermal capture cross section, o* 238 (taken as 2.14 bams). Allowing fox bum-out, the Plutonium concentration i s then given by:P=P
°* ( & f i r
where d* 239 *?? the plutonium absorption cross section (822 barns) and h 2 = 9 r 2 3 9 <I>/R Secondly, the TT3B absorption spectrum contains large resonant peaks in the ?pi-thermal energy region, giving rise to enhanced plutonium production near the surface of the fuel pellet. For a given buznu? step the amount produced across the -whole pellet i s given by
- f -
8 6 4 0 0 - 238
( 1 - p j F i A B -v g. 6.6J-10"
where ?%> is the average somber of fast neutrons liberated per fission (2.44), F is the fast leakage factor & p is the resonance escape probability. F L and p are user-input parameters which are readily available for any individual reactor (4). She program distributes this plutonium amongst the various fuel aanuli according to a very steep inverse exponential function viz
APRES = i * 3 exp (-9.? s/7r=r)
subject to the normalisation condition
A P * e s - V = APTOT
where 7 i s the volume of an anuulus whose mean radius i s r, and r2 i s the pellet outer radius. Ibis distribution was found to give good agreement with observation!5).
Sinple diffusion theory gives that the thermal neutron inverse diffusion length, bt , i s deteadned by the transport and absorption seen free paths \xr. ?nd Aobs.
K = f3/Xtr
For large atomic weights Xtr is wall approximated by the scattering m a n free path, Xs ? and banc*
nmol ( 9 $ ) HEAVY ? 2 flmol (^s) OXYGEN
where 8s ia the scattering cross section (iC barns for uranium and plutooius) and 4. barns for oxygen). Likewise, the absorption mean free path is just the reciprocal of the macroscopic absorption cross section
= n m o I far 2 3 5 u * <r239 P*<r 23e
Acos \ combining equations (10), (11) and (12) gives
?* = ( 1*6 U * 26.6P* 0.069 V' 7
k value of oc is calculated for each amnios and these values are averaged, on a volucs weighting basis, to give either a single mean value, oT , for a standard pellet or a mean value for each of the two regions of a duplex pellet, c*1 and o<2 . The flux profile is the solution of the resulting diffusion equations which are solved in terms of modified 3essel Functions, I and K, the solutions taking on different forms according to whether the pellet is solid, hollow or duplex:-
? = Io (M
^ ; 1
?? r * 2 o ^ r 3r r2 o S r Sro roir *r2
4> = a Io (5cir) $= b Io(K2r)*CKo(^2r)
o ? r <*n r, $ r S r 2
whart r? and fj art the inner and outer fuel radii, ti i s the duplex radius and a, b and c are constants calculated to give continuity of flux and flux gradient at the duplex boundary. 2he program has thus calculated concentration profiles for the two fissionable atoms, -which i t combines together with weightings given by the fission cross sections. She rating profile i s then
| U**f 2 3 o P )
which is normalised to unity for the whole pellet. The actual level of specific rating is dictated by the user, from which the buxnup profile is incremented. VALUATION QF TBS R H U S MODEL The model's predictions of radial power profile have been investigated for several types of thermal reactor over the whole applicable bumup sange, by comparing then with the results of the detailed physios code WIMS-E(°J. (VDB-? is a scheme for neutronics calculations first implemented in the UK in 1969 and. is a development from the earlier Vinfrith Improved Multi-Group
Schesef?}; its validation now run* to s o w hundreds of comparisons ranging from simple lattices to operating reactors). Presented hex* are comparisons between KA2U5 and VI35-E for two types of therntl reactor, the results being typical of all those considered. Fig 1 shows the predicted profiles for standard, solid PWR futl at start of lift and aftsr 32 3Wd/Te buxanp. Tb* cods employed 15 equally spaced radial — ' ^ in the fuel and divided ths irradiation into 20 staps of 1.5 OTd/Te bunmp. Very food agreement was found at all points in life, tht only discrepancy sting near the surface of the pellet where the very steep rise in the profile is difficult to predict accurately without employing arbitrarily spaced radial nodes (a feature which would sake the model less compatible with fuel perfoxmasct codes). Fig 2 shows that the seat good agree—at exists ia the <rM*?'Pii>c of duplex fuel. The particular design considered consisted of a core of natural uranius surrounded by an equal volume aanulus of 5>5fc enriched uranium. The change ia the proflit as bursup proceeds is clearly substantial and would have a very significant tffect on the performasca of the fuel. The under-aoderation ia a PWR leads to a low resonance escape probability and hence to a large plutoniun build up at tht fuel surface. In the ACS this is not the case and the radial power depression is consequently much smaller in magsitude. This is shown in Pigs 5 *nd 4 for the slightly enriched initial fuel and for tht sore highly enriched feed fuel; thest figures denonstrate that the program perfanas equally well when modelling a very different reactor esvironaent to that of the PVR. USE 0? T52 RlLAE KD22L V H S J B A FUEL FEHFOBMABCE CODE The aodtl has been incorporated as a subroutine into the thermal reactor fuel performance codes 2CZBQS-4C and SX3JTE-SEEB-77. In both casts the only extra. inputs required for the sodel were tht pellet enrichment and the resonanct escape probability. The increase is running costs for tht codes was always lesa than five percent and was often ouch less than this figure. The first cast considered was an artificial example chosen to demonstrate the magnitude of the tffect of an evolving radial power profile on the performance of a futl pin. Standard and duplex PWR fuel (the rating profiles for which are shown in Figs 1 and 2) were modelled for a constant power irradiation history using EDT30D-4C/EA2IAR and with an infinite gap conductance (ie a constant fuel surface temperature} imposed. The resulting temperature profiles at 10 SWd/Te burnup intervals are shown is Pig 5. Without the new model, HCTRCD-4C employs an unchanging power profile, which in the absence of other information is taken to be the profile applicable at tht start of life. Fig 6 illustrates that the effect of this approximation on the prediction of temperatures in standard futl is significant; in the case of duplex fuel (Fig 7) the effect is dramatic. C0NCUJSI0KS The neglect of changes in the radial power profile has been shown to lead to serious errors (of 100°C or more) in the calculation of futl temperatures is duplex (dual enrichment) fuel; in standard single enrichment fuel appreciable errors of several tens of degrees Centigrade have bets demonstrated. Gtntral purpose thermal reactor futl performance computer codes thus require some method of calculating the evolution of the futl pis radial power profile. The RADAR model described is this paptr has been shown to provide an effective, reliable and accurate method for determing radial power profiles is thermal reactor fuel pins without incurring a significant increase is computing costs.
gwarrgg 1. I P X BA7BS A eoapaxison of B T O cote predictions with PIE data. OB S Paper presented at tte tMS Heating on Water Baaetor Foal Performance, Chicago, May 1977. J R OTTOS and 0 A 30ML ?xo(taaa la using parforannce aodels. Paper pxaaaatad at the ABS Topical Heeting an Water Baactor Foal Performance, St Cbarlaa, I l l i n o i s , 1977. J ^ AJBSCOOGS, D K COUCILL, D 1 B W , A JBBB and I KESFEUff. O L Duplex foal - British and Danish experience and eraluatien. Papar to bo prasentad at the ASS Topical Heating : 1 B Srttnded Bnxaop - ?oal M Perforaance and Utilisation, Williamsburg, April 1982. gfTFBmTTOBAL ATOKIC SKSGT AG9CT Directory of Buclear Beactors.
H CAHLSFB and D H SAB Badial concentration and effect on temperature of plutcniun fo=aed in TO2 (hiring irradiation. Beclear Technology 7ol 55 P587 December 1981. I J a?T.giT.T. A&SW-B 1289 Pis-call calculations for P B duplex fuel designs. W October 1979 WIMS-E A Sehaae for neutronics calculations
J B ASKSV and K J B T OH AEEW-B1315.
WIMS-E RAOA* 32 GWd/TeBURNUP
F1G.1. COMPARISON OF RADAR'S PREDICTIONS AGAINST THOSE OF WIMS-E FOR STANDARD SOLID 3-1%ENRICHED PWR FUEL
32 6Wd/T? BURNUP
FIG.2. COMPARISON OF RADAR'S PREDICTIONS AGAINST THOSE OF WIMS-E FOR DUAL ENRICHED (NATURAL CORE 5-5 ?/. ANNULUS? PWR FUEL
WIMS-E R A OAR
IS GWd/Te BURNUP
PELLET OUTER RADIUS
FIG. 3, COMPARISON OF RADAR'S PREDICTIONS AGAINST THOSE OF WIMS-E FOR 1 16% ENRICHED HOLLOW AGR INITIAL FUEL
s o r
10 ZERO BURNUP
25 OWd/Te BURNUP
FIG-4. COMPARISON OF RADARS PREDICTIONS AGAINST THOSE OF W I M S - E FOR 2-6'/* ENRICHED HOLLOW AGR FEED FUEL 355
ZERO BURNUP 10 GWd/ Tt 20 GW?/ Tt 30GWd/T? 40 GWd/ Tt
EVOLUTION OF RADIAL TEMPERATURE PROFILE FOR SOLID AND DUPLEX PWR FUEL
* 5 o o
O UJ X
3 Z 0C 3 O
< Z IT ui
(T 0- O UJ
u o o a:
3 t'iWniVMWH31 3U1N33
900 ? U*INU RADAR WITHOUT RAOAR
20 BURNUP, GWd/T?
FIG. 7. HOTROD CENTRE TEMPERATURE PREDICTIONS WITH AND WITHOUT NEW MODEL FOR DUAL ENRICHEDt NATURAL CORE, 5-5*4 ANNULUS)PWR FUEL.