Letter to the Editor

Comments on ``Computational problems in the calculation of temperature e�ects forheterogeneous nuclear reactor unit cells'' (Goltsev et al., 2000. Annals of NuclearEnergy 27 (2), 175±183)

In their interesting paper, Computational problems in the calculation of tem-perature e�ects for heterogeneous nuclear reactor unit cells, Goltsev et al. (2000)deal with the problem of how to determine the e�ective fuel temperature in case of anonuniform radial temperature pro®le in a fuel pin. The e�ective fuel temperature isan equivalent uniform temperature used in reactor neutronics calculations. Goltsevet al. propose a formula to calculate the e�ective fuel temperature Teff for a fuel pin,based on:

Teff ��R0 T�r�'�r�dr�R

0 '�r�dr�

�R0

���������T�r�p

dr�R0 1=

���������T�r�

p� �dr; �1�

with T�r� the radial temperature pro®le, '�r� � 1=���������T�r�

pa weighting function, and R

the radius of the fuel pin.There are two reasons why this weighting function is not physically plausible:

1. The absorption rate is approximately proportional to����Tp

instead of 1=����Tp

(Hellstrand et al., 1960).2. The weighting procedure does not include the position of the zones in the fuel

pin. The outer fuel zones contribute more to the capture of neutrons than theinner fuel zones. The neutron ¯ux in the resonance region decreases towardsthe centre of the fuel pin.

A more physical derivation of a theoretical expression for the e�ective fuel tem-perature is given by De Kruijf and Janssen (1996) based on earlier work (Roe, 1954;Keane, 1958; Dresner, 1961; Reichel and Keane, 1961; Rowlands, 1962). The e�ectivefuel temperature is de®ned by equating the absorption in the fuel pin with a giventemperature pro®le with the absorption in a pin with uniform e�ective temperature.Assuming that the neutrons moderated in the water region enter a purely absorbingfuel pin, De Kruijf and Janssen (1996) show that in case of high absorption (blackresonances) the e�ective fuel temperature can be approximated by the chord-averagedfuel temperature as proposed by Rowlands (1962). In the case of low absorption and

Annals of Nuclear Energy 28 (2001) 385±387

www.elsevier.com/locate/anucene

0306-4549/01/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved.

PI I : S0306-4549(00 )00084 -0

isotropic incidence, one arrives at the volume-averaged fuel temperature as e�ectivefuel temperature as proposed by Dresner (1961).Goltsev's approach to compare the in®nite multiplication factor for a fuel pin with

a given parabolic temperature pro®le with the in®nite multiplication factor for a fuelpin with uniform temperature is interesting. Note that for such a special temperaturepro®le an expression for the e�ective fuel temperature can be found which containsonly the surface temperature and the central fuel temperature. For a parabolictemperature pro®le and isotropic incidence, Rowlands' (1962) expression reduces to:

Teff � TS � 4

9�T0 ÿ TS�; �2�

with TS the temperature on the outer surface of the fuel pin and T0 the temperaturein the centre of the fuel pin. Goltsev's expression reduces to:

Teff � 1

3T0 � TS �

�����������T0TS

p� ��3�

``Exact'' results are obtained by Goltsev et al. (2000) by equating the in®nitemultiplication factor for a fuel pin with a given parabolic temperature pro®le withthe in®nite multiplication factor for a fuel pin with a uniform temperature. Let uscompare the ``exact'' results with the theoretically predicted values. Table 1 showsthe results for the range of interest.1

It is seen that both expressions perform rather well and are better than taking justthe average fuel temperature. In view of the discussion above this is at ®rst sightsurprising for Goltsev's expression. However, the 1=

����Tp

weighting ensures that thecentral part with higher temperature gets less weight which is needed to arrive at acorrect e�ective fuel temperature. So, the performance of Eq. (3) is misleading.Rowlands' expression has a more robust basis. However, De Kruijf and Janssen(1996) have also shown that for individual resonances theoretical expressions cannotbe used. The good agreement for the whole slowing-down energy range of 238U is acoincidence. A detailed discussion is given in their paper.

Table 1

``Exact'' e�ective fuel temperature (Goltsev et al., 2000) and evaluated theoretical expressions for a para-

bolic temperature pro®le with constant outer surface fuel temperature of 300 K

Taverage (K) ``Exact'' Te� (K) Goltsev Te� (K) Rowlands Te� (K)

400 375 396 389

600 568 573 567

800 752 741 744

1000 928 904 922

1200 1101 1065 1100

1400 1245 1222 1278

1 These numbers di�er from the numbers given by Goldsev et al. (2000) because they used the tem-

perature in the inner and outer ring in Eq. (3), whereas the actual outer surface and central fuel tem-

perature should be used.

386 Letter to the Editor / Annals of Nuclear Energy 28 (2001) 385±387

References

De Kruijf, W.J.M., Janssen, A.J., 1996. The e�ective fuel temperature to be used for calculating resonance

absorption in a 238UO2 lump with a nonuniform temperature pro®le. Nuclear Science and Engineering

123 (1), 121±135.

Dresner, L., 1961. Some remarks on the e�ect of a nonuniform temperature distribution on the tempera-

ture dependence of resonance absorption. Nuclear Science and Engineering 11, 39.

Goltsev, A.O., Davidenko, V.D., Tsibulsky, V.F., Lekomtsev, A.A., 2000. Computational problems in the

calculation of temperature e�ects for heterogeneous nuclear reactor unit cells. Annals of Nuclear

Energy 27 (2), 175±183.

Hellstrand, E., Blomberg, P., Horner, S., 1960. The temperature coe�cient of the resonance integral for

uranium metal and oxide. Nuclear Science and Engineering 8, 497±506.

Keane, A., 1958. Resonance absorption in a slab with a parabolic temperature distribution. Report

AERE R/M 198, Atomic Energy Research Establishment.

Reichel, A., Keane, A., 1961. Resonance absorption in a cylindrical fuel rod with radial temperature

variation. Proc. R. Soc. N.S.W 94, 215.

Roe, G.M., 1954. The Absorption of Neutrons in Doppler Broadened Resonances. Report KAPL-1241,

General Electric Company, Knolls Atomic Power Laboratory.

Rowlands, G., 1962. Resonance absorption and non-uniform temperature distributions. Journal of

Nuclear Energy, Parts A and B 16, 235.

W.J.M. de KruijfInterfaculty Reactor Institute,

Delft University of Technology,Mekelweg 15, 2629 JB Delft, The Netherlands

E-mail address: kruijf@iri.tudelft.nl

Received 23 June 2000; accepted 31 July 2000

Letter to the Editor / Annals of Nuclear Energy 28 (2001) 385±387 387