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International Heat Transfer Conference Digital Library International Centre for Heat and Mass Transfer Digital Library Begell House Journals Annual Review of Heat Transfer


DOI: 10.1615/AtoZ.r.rod_bundles_heat_transfer_in


This article describes heat transfer in nuclear fuel rod bundles in light water nuclear reactors under normal, incidental and accidental conditions.

Relevant basic information on Heat Transfer, Forced Convective Boiling, Nucleate Boiling, Burnout and Rewetting of Hot Surfaces is given in other entries.

Laminar Flow

Laminar flow occurs during reactor shutdown and removes the residual power (less than 0.5% of the nominal power).

Although it is mainly of academic interest, laminar flow equations have been analytically resolved (without spacer grids) and lead to some constant Nusselt Numbers in steady-state and fully developed flow [Lach (1986)], depending on the pitch to diameter ratio P/D (see Table 1).

Table 1. Nusselt Numbers (constant heat flux) for fully developed laminar flow in rod bundles.

Single-Phase Turbulent Flow

This is the most usual case encountered in normal Pressurized Water Reactor (PWR) operation. It is important to know the temperature field in the reactor core not only in order to be able to calculate the neutron flux field, but also because a difference of some degrees may have noticeable consequences in the long term on the chemical and metallurgical behavior of the cladding of the fuel rods.

Usually, the single phase turbulent heat transfer coefficients are estimated using the same correlations as for tubes, using the equivalent hydraulic diameter instead of the tube diameter. According to many authors, this approach underestimates the actual heat transfer coefficient. Tong et al. (1979) proposed modifying the constant C = 0.023 in the well-known Dittus-Boelter correlation and recommended:

  • for a triangular-pitch lattice

  • for a square-pitch lattice

where P/D is the pitch over diameter ratio.

This increase in the heat transfer coefficient seems to be due mainly to the mixing grids that enhance the turbulence downstream, and, consequently, the heat transfer; some local measurements have demonstrated that the heat transfer coefficient decreases as the distance to the mixing grid increases. A precise heat transfer correlation in this geometry would depend closely on the grid geometry and so would probably be proprietary. However, the overestimation of the wall temperature obtained by a tube correlation is probably less than the inaccuracy due to the subchannel analysis!

Nucleate Boiling Flow

Nucleate boiling occurs in the hottest locations of a PWR core in normal operation and in the greater part of a BWR core. It may also occur in the greater part of a PWR core during transients associated with accidents.

As long as nucleate boiling occurs, the wall temperature is Tsat + ΔTsat, and ΔTsat is only a few degrees. The correlations used for calculating ΔTsat are usually the same as for the tubes. The Onset of Nucleate Boiling (ONB) may be calculated as the point at which the wall temperature, as estimated by a single-phase turbulent heat transfer correlation, reaches Tsat + ΔTsat; a subchannel analysis code is needed to estimate this position. For more details on nucleate and saturated boiling flow [see Groeneveld (1986)].

Boiling Crisis

The boiling crisis, or Departure from Nucleate Boiling (DNB) is one of the phenomena limiting the power of a water nuclear reactor.

Although some boiling crises, like dryout, lead to only moderate temperature rises, others, like Burnout at low quality, may result in a dramatic and very rapid increase of the rod wall temperature. Consequently, in many countries, the rules are that all boiling crises must be avoided in normal and abnormal situations, up to incidental transients of class 2.

The basic boiling crisis phenomena have been modeled in tubes (ex. Whalley for dryout, Weissman et al. or Katto for burnout), but the phenomenon is far more complicated in rod bundles due to the presence of crossflows, mixing grids, complex geometry, nonuniform heat-flux, and the tube models give only a very rough approximation of the boiling crisis conditions in a reactor.

There are several ways to obtain a local Critical Heat Flux, CHF, prediction:

  • a quick-and-approximate method is to use a tube CHF prediction and adjust the result to the appropriate rod bundle geometry using correcting factors. Among the several hundreds of existing CHF correlations for tubes, annular space and other simple geometry the Groeneveld (1993) CHF tables are recommended.

  • a more accurate method is to use a specific CHF correlation (or table, smoothing splines, etc.) especially fitted for the specific geometry and mixing grids used. This involves car rying out several CHF tests with the specific geometry and the appropriate parameter range, analyzing these tests with the same subchannel computer code and building a CHF predictor. Since these experiments are difficult, expensive and long, and apply only to specific mixing grids and geometries, most of the CHF predictors are proprietary and not openly published.

  • an intermediate method is to use a reference general purpose rod bundle CHF correlation, perform some CHF tests with the appropriate geometry and mixing grid, calculate the average and standard deviations of the experimental results with respect to the reference CHF correlation, and then use these deviations to modify the safety margin. The use of this methodology seems surprising as it is less efficient than the former one, and the missing step—the construction of a specific CHF predictor based on expensive and hard-won experimental data—is relatively easy, quick, cheap and more precise. However, this methodology is very commonly used, especially for new fuel to be loaded into existing plant, because the on-line computer software and, above all, the existing regulations (including the reference CHF correlation) seem very difficult to modify.

The geometry and the grid spacer have a considerable influence on the CHF values. For the same thermohydraulic local conditions (pressure, mass velocity, quality), the CHF may vary by a factor of 2 or 3! The most obvious effects on the CHF are due to:

  • the grid spacing: the CHF increases significantly as the grid spacing decreases;

  • the geometry of the mixing grid, especially that of the mixing vanes: it may have a considerable influence on CHF especially at low quality and high heat flux but most of the results in this field are proprietary;

  • the guide thimble: it influences CHF and acts both as a cold wall effect and as a hydraulic diameter effect;

  • the equivalent hydraulic diameter: as in tubes, CHF generally increases when the hydraulic diameter decreases for constant local thermal-hydraulic conditions. This sensitivity is usually greater for low quality and tends to zero for higher quality.

Quenching in Rod Bundles

Quenching of the core may occur after a Loss Of Coolant Accident. Here the specificity of the rod bundle case is directly related to radial steam and water crossflows, to the complex geometry and heterogeneities of the core. (See Blowdown and Reflood.)

Steam and water crossflows

Analytical experiments in large 2D test sections have demonstrated that strong steam and water crossflows occur between neighboring assemblies with different residual powers (radial peaking factor as high as 1.8). Considering, for instance, a "hot" assembly between two "cold" ones [Deruaz et al. (1984) and Housiadas et al. (1989)] perfect steam radial mixing is observed in the rewetted region indicating existence of steam crossflows from the hot to the cold assemblies. At the same time, water cross-flows occur from the cold to the hot assembly with an intensity which can reach values of up to 60% of the feeding rate. Liquid crossflows are negligible in the dry region whereas steam escapes to the cold assemblies due to the flow resistance inherent in a larger amount of liquid in the dry zone of the hot assembly. This complex behavior has a noticeable impact on quench front progression and heat transfer in the assemblies. In the hot assembly, quench velocity is lower and precursory cooling is enhanced due to the larger amount of liquid in the dry zone. (See Rewetting of Hot Surfaces.)

In steady or quasi-steady situations (boil-up or boil-off cases), level swell is uniform and depends only upon the averaged residual power (and not upon its distribution between the assemblies). This result is consistent with the existence of perfect steam radial mixing.

If exit steam velocities are not too large an additional possible 2D phenomenon could be preferential fall back of water from the upper plenum to the cold assemblies. Due to mixing of steam in the core this type of phenomenon does not occur.

Ballooning and grid effects

Extensive experiments have shown that both spacer grids [Clement et al. (1982) and Hochreiter et al. (1992)] and flow blockages (Hochreiter et al., 1992) (due to ballooning of some heater rods) can significantly alter post-CHF heat transfer. Similarly, they exhibit local precursory rewetting and induce downstream effects due to convection enhancement of the continuous phase and breakup of entrained droplets.

Cold rod effects

The presence of unheated rods in a bundle (control rod guide thimbles in the standard PWR for example) has a significant effect on the cooling efficiency viewed in terms of overall quench time (the time necessary to quench the full bundle) or of maximum wall temperature [Veteau et al. (1994)]. The cold rods exhibit very early quenching, modifying hydrodynamics in the subchannels and enhancing radiative cooling of the adjacent heating rods.


Clement P., Deruaz, R., and Veteau, J. M. (1982) Reflooding of a PWR bundle: Effect of inlet flowrate oscillations and spacer grids, NUREG/CP-0027, Vol. 3, Proceedings of the International Meeting on Thermal Nuclear Reactor Safety, Chicago, August 29–Sept. 2, 1982.

Deruaz, R., Clement, P., and Veteau, J. M. (1985) 2D effects in the core during the reflooding case of a LOCA, Safety of Thermal Water Reactors, Proceedings of a Seminar on the results of the European Communities' Indirect Action Research Programme, on Safety of Thermal Water Reactors, held in Brussels, 1–3 October 1984, E. Skupinsky, B. Tolley and J. Vilain Eds., Graham and Trotman Limited Publishers.

Groeneveld, D. C. and Snoek, C. W. (1986) A comprehensive examination of heat transfer correlations suitable for reactor safety analysis, Multiphase Science Technology, Ch. 3, Vol. 2, G. F. Hewitt, J. M. Delhaye, and N. Zuber. Eds. Hemisphere Publishing Corporation.

Groeneveld, D. C, Erbacher, F. J., Kirillov, P. L., Zeggel, W. et al. (1993) An improved table look-up method for predicting critical heat flux, 6th International Topical Meeting on Nuclear Reactor Thermal Hydraulics, Grenoble, France, October.

Housiadas, C, Veteau, J. M., and Deruaz, R. (1989) Two-dimensional quench front progression in a multi-assembly rod bundle, Nuclear Engineering and Design, 113, 87–98. DOI: 10.1016/0029-5493(89)90299-9

Hochreiter, L. E., Lotos, M. J., Erbacher, F. J., Hile, P., and Rust K. (1992) Post CHF effects of spacer grids and blockages in rod bundles, Post dry-out heat transfer, Multiphase Science and Technology, G. F. Hewitt, J. M. Delhaye and N, Zuber, Eds., CRC Press.

Lach, J., Kielkiewicz, M., and Kosinski M. (1986) Heal Transport in Nuclear Reactor Channels of Heat Transfer Operations, Vol. 1. Ch. 36, N. P. Cheremisinoff Ed., Gulf Publishing Company.

Tong, L. S. and Weisman, J. (1979) Thermal Analysis of Pressurized Water Reactors, 2nd edn., American Nuclear Society, La Grange Park, Illinois.

Veteau, J. M., Digonnet, A., and Deruaz, R, (1994) Reflooding of tight lattice bundles, Nuclear Technology, 107, July 1994.

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