To ensure good thermal performance of a nuclear reactor a detailed knowledge of the heat transfer and fluid flow phenomena taking place within the core is required. Coolant flow rates in different parts of the complex rod bundles and the manner in which the single phase and the two-phase flows are distributed in the subchannels are very important for evaluating enthalpy distribution and performance parameters, such as the onset of boiling and critical heat flux.

This entry describes some of the fundamental features turbulent flow in rod bundle subchannels in a water cooled reactor, namely:

the momentum balance in subchannel geometry

intersubchannel flow

*mixing in rod bundles*the

*subchannel analysis*method for predicting the thermal hydraulic performance of rod bundles.

Accurate prediction of pressure drop is of essential importance in the design of a nuclear fuel rod bundle geometry for two reasons:

Pumping costs are a significant fraction of operating expenses.

The subchannel pressure drop is intimately related to coolant flow and enthalpy distribution.

In other words, for a good assessment of the thermal hydraulic behaviour of rod bundles by subchannel analysis it is necessary to formulate all the basic equations, including axial and momentum balance, as accurately as possible. The general form of axial momentum balance for each subchannel is:

where the subscript f is the frictional pressure gradient, e is the hydrostatic head loss, ace is the total acceleration loss (sum of the direct contribution of changes in subchannel mass flow rate and velocity and contribution of diversion cross flow), s is the pressure loss due to area restriction, and m is the effect of turbulent mixing on pressure change in axial direction.

The transverse momentum balance determines the magnitude of diversion crossflow which is caused by radial pressure gradients between adjacent subchannels; it indicates a strong dependence on the rate of change of the lateral velocity in an axial direction. To calculate the diversion crossflow rates for a given radial pressure difference, transverse resistance coefficients are required. Unfortunately, there is limited experimental data for these coefficients and a computer code must take an oversimplified form.

It is usual to describe turbulent flow phenomena and pressure drop in channels having a non-circular cross section, by introducing an equivalent hydraulic diameter and applying the laws established experimentally for circular tubes. This is why we start by reviewing some of the results obtained for this geometry. However, it must be borne in mind, that this is a rough estimate only and the true value varies with the geometry of the channel and in particular with the rod bundle configuration.

In a cylindrical duct, with hydraulic diameter DH defined as four times the cross sectional area divided by the periphery, the general form of the fraction factor f is defined by the relation:

where is the mean velocity.

**Friction factor correlation for laminar flow:**the pressure drop relationship is: f = K/Re, where the constant K depends on the geometry: for circular tubes K = 16 and f = 16/Re (Poiseuille Flow).**Friction factor correlation for turbulent flow:**For fully turbulent flow in a smooth circular tube, the friction factor is given with sufficient accuracy by the Blasius correlation which is a good approximation of Nikuradse's experiments and integration of the universal distribution law:in the range of Reynolds numbers 10

^{4}< Re < 10^{5}.**Friction factor correlation for rough pipe:**f is a function of the Reynolds number and the roughness parameter ε/D_{H}, where ε is a characteristic roughness height. In fully turbulent flow and of a high Reynolds numbers f is a function of ε/D_{H}only and is given by the relation:(3)

In the case of turbulent flows developed under uniform wall heat flux boundary conditions, the friction coefficient decreases markedly with heating. Accordingly, it is necessary to take into account the effect of the variable physical property of the water as a function of the temperature in the laminar boundary layer, especially changes in viscosity.

Most researchers have suggested the following correlation:

where m has been considered for many years as a constant value in the range 0.14 to 0.60.

An important improvement was achieved by the author [Lafay (1974)] who found that the dependence of the friction ratio on the viscosity ratio is more complex than a simple power law and proposed a more accurate parameter:

with the following correlation:

for

The dispersion is characterized by a standard deviation:

(See also Friction Factors For Single Phase Flow; Pressure Drop, Single Phase.

Theoretical and experimental studies have shown that the equivalent hydraulic diameter is not adequate to describe accurately the considerable influence of the geometry on the friction factor.

Rheme (1973) has proposed a method for calculating friction factors for turbulent flow in non-circular channels. All that is required is a knowledge of the geometry factor K of laminar flow (f = K/Re). For a number of noncircular channels, these factors have been accurately determined by numerical calculation procedures; this is the case more particularly for rod bundles in square and hexagonal subchannels for which the friction factor in turbulent flow is given by the relation:

where the geometry factors A and G* are a function of K.

Fuel designs usually include spacers for providing fuel rod support particularly to promote mixing and improvement of the thermal hydraulic performance of rod bundles.

A spacer causes significant local velocity perturbations in its wake, increasing axial turbulence intensity and introducing considerable perturbation of mass flow rate distribution in the subchannels. Consequently, the presence of a spacer in a rod bundle results in significant local pressure perturbations and increases the global pressure drop which depends to a large extent on spacer design. Two basically different types of spacers are used in the construction of reactor fuel elements:

grid spacers defining the distance between the fuel elements and (the gap) relative to the wall of the enclosing box; they are generally used for square and hexagonal arrays of fuel elements with a large distance between pins.

wire spacers wrapped around the fuel elements in a triangular arrangement used for a smaller distance between pins.

**Grid spacer**. The pressure drop at the spacer grid is related to the mean velocity V_{B} in the rod bundle by:

where C_{B} is the drag coefficient of the spacer which is presented as a function of the Reynolds number of the rod bundle:

As the drag coefficient of individual grids is very dissimilar in magnitude, Rehme (1973) proposes correlation of the quantity:

where S is the flow cross section of the grid and S_{G} the undisturbed flow section of the rod bundle.

The agreement of the modified loss coefficient is comparatively good, taking account the geometrical diversity of grid designs.

When the grid spacer has small wings at the exit edge in all subchannels and with different orientations, this method is probably not good and we must use experimental measurements for each case.

**Wire spacer**. With wire wrapped rod bundles the situation is quite different. Experimental studies [Lafay et al. (1975)] have showed that the helical wire spacer around each rod induces an important swirl flow and a significant local pressure perturbation; accurate exploration of the pressure field in peripheral subchannels shows that the pressure distribution in a cross section is not hydrostatic and the axial pressure profile is a periodic function of the axial position, the period corresponding to the helical wire pitch.

Therefore the axial pressure gradient on one pitch is the same as the mean pressure gradient for several pitches and it is possible to define a friction pressure drop coefficient, though the simple equivalent diameter process is not sufficient for accurate prediction.

Novendstern has developed a theoretical model which determines the flow distribution in subchannels and has obtained an empirical correction factor depending on fuel rod bundle dimensions and flow rate.

The friction factor is given by the relation:

where fiso is the friction factor for flow in a smooth circular tube, P/D is the pitch to diameter ratio, H/D is the helical wire pitch to diameter ratio, and Re is calculated with the mean velocity and equivalent diameter of the central subchannel.

It is obvious that the friction factor increases with decreasing wire wrapping pitch, while there is a significant increase with a high pitch-to-diameter ratio of the rods.

The two-phase pressure drop during a steadly flow of a two-phase mixture is the sum of three components: hydrostatic head, accelerational pressure drop and frictional pressure loss.

Evaluation of the two-phase friction pressure drop requires a void fraction correlation to calculate the hydrostatic and acceleration components, a single phase friction factor and a two-phase friction multiplier correlation.

Two principal types of flow models are used in the analysis of two-phase flow frictional pressure drops: the *homogeneous flow* model and the *separated flow* model.

In 1948, Martinelli and Nelson were the first to suggest an approach and an empirical correlation for the calculation of the two-phase frictional pressure drop in a tube based on the separated flow model. They correlated the ratio of local two-phase pressure gradient to pressure gradient 100% liquid flow as a function of quality and pressure

the two-phase friction multiplier and (dP/dl) is the single phase friction gradient calculated with the single phase friction factor of the Blasius correlation:

More recently, many verifications or improvements of the Martinelli–Nelson correlation based on experimental data, have been published. Some of the more prominent correlations among these are due to Thom, Baroczy, Chisholrn and Columbia University. They found that the two-phase friction multiplier is also a function of the mass flux. A comparative study of their performance shows that the Columbia correlation gives the more accurate friction factor. (See Pressure Drop, Two Phase Flows.)

To evaluate two-phase flow friction pressure drop in rod bundles we assume that the rod bundle behaves like a circular channel of equivalent hydraulic diameter; consequently we recommend using:

the friction factor to calculate the single-phase flow pres sure drop,

the

*Columbia*correlation which gives the more accurate two-phase flow friction multiplier

In order to improve the thermal hydraulic characteristics of the nuclear reactor core, a considerable amount of research has been carried out in order to obtain improved understanding of coolant flow and enthalpy distribution in rod bundle geometries.

One form of fundamental research has been the study of the mixing process between subchannels in this complex geometry.

In analyzing the effect of mixing on rod bundle temperature and pressure gradient it has generally been assumed that the mixing process is the result of several components.

#### Natural mixing (see also Mixing)

To describe turbulence in bundles of smooth bare rods (without protuberances) and which include both turbulent and diversion crossflow mixing:

Turbulent mixing which results from the oscillation component of flow in a transverse direction between two subchannels and can be characterized by the eddy diffusivity of momentum ε

_{M}.Diversion crossflow mixing is the rate of mass flow in a transverse direction though the gap between two subchannels caused by radial pressure gradients. The diversion crossflow contributes to subchannel flow rate in an axial direction.

The subchannel interchange is induced by the presence of spacers in the rod bundle; we distinguish:

Flow scattering refers to the nondirectional mixing effect associated with grid spacers which break up streamlines; the turbulence intensity increases immediately downstream from the device.

Flow sweeping refers to the directed crossflow effect associated with wire wrap spacers or grid spacers with vanes which give a net crossflow in a preferred direction.

Much research has been devoted to the study of turbulent flow processes and improved understanding of mixing. Most experimental work has focused on quantifying the mechanisms individually and their dependency. First, the relative importance of these processes in rod bundle performance vary significantly with bundle geometry characteristics and particularly with gap spacing and flow parameters. Secondly, though turbulence interchange is present in all situations, the major emphasis has been in determining turbulent crossflow (diversion and sweeping) because it is the most important means of momentum and energy transfer: it improves the thermal hydraulic performance of rod bundles as it provides an important mechanism for equalizing temperatures thoughout the bundle. An excellent review of the related aspects of turbulent mixing and diversion crossflow is given by Rogers and Todreas (1968); the reviewers describe the circumstances in which crossflow mixing are particularly developed:

in the entrance region of bundles,

in the region where boiling crisis begins and develops in the various subchannels,

in the region of physical distortion of the bundle elements.

Various experimental techniques have been used to study mixing in rod bundles. Chemical tracers, hot water injection and direct subchannel enthalpy measurement are generally used. Most experimental data give only the mean mixing rates over the test section, while the flow sweeping effect which is the predominant process in intersubchannel mixing is strongly dependent on the local conditions due to the grid spacer vanes or the periodic wire wrapped spacer position in the bundle. Therefore, a local description of the flow field is extremely important to assist the interpretation of coolant cross mixing.

The recent development of two component laser Doppler anemometry permits local fluid velocity measurements and consequently provides an important tool for improving fluid flow research. This technique is performed with light beams and has the advantage of not disturbing the flow field; furthermore, the control volume over which the measurement is performed is very small. This is a significant improvement over previous methods such as hot-wire anemometers and Pitot tubes. (See Anemometers, Laser Doppler.)

The experimental results obtained by Rowe in rod bundles without and with spacer has shown, in the first case, that rod gap spacing has a significant influence on the turbulent flow structure in rod bundles in a way that cannot be deduced from round pipes, whereas the velocity profiles are in reasonable agreement with universal profile in pipes of the same hydraulic diameter.

The grid spacer results in a significant change in axial mean velocity, turbulence intensity and turbulence scale in the wake of the grid spacer; these effects are weakly dependant upon the Reynolds number.

Concerning the concentration measurement technique, the use of a liquid tracer introduced in the rod bundle flow upstream from the grid spacer is also an important technical means for fluid flow research in spite of a small disturbance of the probe inside the subchannel. Sampling of the solution in all subchannels at different cross sections downstream from the grid and its analysis in a spectrofluorimeter indicates first the local concentration in all subchannels and its axial and transversal evolution and secondly the average mixing coefficient qualifying the grid spacer. (See also Tracer Methods.)

This two-measurement technique, velocimetry laser and concentration, has been used with success for a large range of grids with vanes in hexagonal and square geometry at the Heat Transfer Laboratory of the CEA at Grenoble with two fundamental and analytical experiments: AGATE and HYDROMEL. In two cases we observe that the flow behavior depends strongly on the position, dimensions, shapes and inclinations of the mixing vanes at the exit of the grid.

Experimental data available in the literature about the mixing rate under boiling flow conditions are limited.

Some investigations have been made with air-water mixtures to explain mechanisms of mixing in two-phase flow and to indicate the effects of various parameters such as the turbulent interchange rates. However, the principal information has been developed by Rowe though the use of thermal hydraulic code COBRA which constitutes a significant contribution to the knowledge of boiling flow behavior in rod bundles. It has been observed that boiling turbulent interchange appears to be a function of the channel geometry, quality and flow regime, with a maximum at low qualities in the transition region from bubbly to annular flow.

The subchannel analysis method is an important tool for predicting the thermal hydraulic performance of rod bundle nuclear fuel element. It considers a rod bundle to be a continuously interconnected set of parallel flow subchannels which are assumed to contain one dimensional flow and are coupled to each other by crossflow mixing; the axial length is divided into a number of increments such that the whole flow space of a rod bundle is divided into a number of nodes.

The principle of subchannel analysis is the application of continuity and conservation equations to the flow between these nodes. The conservation equations relate the local variations of velocity and enthalpy of each node to those of its neighboring ones. The relation between subchannel flow rate which is the mass flow rate in an axial direction through subchannel area and diversion cross-flow which is the mass flow in a transverse direction resulting from local pressure differences between two subchannels, is strongly governed by momentum balance in a transverse direction.

A correct formulation of momentum equations and good knowledge of the mixing process between subchannels are an absolute necessity, for obtaining reliable predictions using subchannel calculations.

#### REFERENCES

Courtaud, M., Ricque R., Martinet, B. (1966) Etude des pertes de charge dans ies conduites circulates contenant un faisceau de barreaux, *Chem. Eng. Sci.*, 21, 881–893. DOI: 10.1016/0009-2509(66)85082-0

Giot, M. (1981) Friction factors in single channels, *Thermohydraulics of Two-Phase Systems for Industrial Design and Nuclear Engineering*, Delhaye, J. M. et al., Eds., Hemisphere.

Grand, D. (1981) Pressure drops in rod bundles, *Thermohydraulics of Two-Phase Systems for Industrial Design and Nuclear Engineering*, Delhaye, J. M et al., Eds., Hemisphere.

Lafay, J. (1974) Influence de la variation de la viscosite avec la temperature sur le frottement avec transfert de chaleur en regime turbulent etabli, *Int. J. Heat and Mass Trans.*, 17, 815–834. DOI: 10.1016/0017-9310(74)90150-1

Lafay, J., Menant, B. and Barroil, J. (1975) Local pressure measurements and peripheral flow visualization in a water 19 rod bundle compared with FLICA IIB calculations: influence of helical wire wrap spacer system, *ASME Heat Transfer Conf., San Francisco* 75-HT-22.

Reddy, D. G. and Fighetti, C. R. (1983) Evaluation of two-phase pressure drop correlations for high pressure steam-water systems, *ASME Thermal Engineering Conf. Proc, Honolulu*, Vol. 1.

Rheme, K. (1973) Pressure drop correlations for fuel element spacers, *Nuclear Technology*, 17, 15–23, January.

Rogers, J. T. and Todreas, N. E. (1968) Coolant interchannel mixing in reactor fuel rod bundles single phase coolants,* Winter Annual Meeting of the ASME, New York*, December.

Rouhani, Z. (1973) A review of momentum balance in subchannel geometry, *European Two-Phase Flow Group Meeting Brussels*, June 4–7.

Rowe, D. S. (1973) Measurement of turbulent velocity, intensity and scale in rod bundle flow channels, *BNWL-1736*, May.

Tong, L. S., (1968) Pressure drop performance of a rod bundle, *Winter Annual Meeting of the ASME, New-York*, December.