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2. Literature Review on HLM cooled wire-wrap bundles
A literature review on the thermal-hydraulics of HLM cooled bundle is presented in order to introduce the topic of the thesis in a proper way. Theoretical studies, experimental studies and numerical studies are reviewed critically from the ‘60s to the 2000’s years. A special section is devoted to pressure drop and heat transfer correlations.
2.1. Definitions
For our purpose we won’t discuss about the typical bundle of bare fuel pins but we focus on the wire wrap fuel pin bundle studied in its various configurations (7, 19, 37 and 217 fuel pin bundle) and shown in Figure 2.1 (a, b, c).
The geometry is completely defined by the pin diameter D, the wire diameter Dw, the bundle
a
b
c
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apothem a, number of pins in the bundle n, the pitch between pins P and the wire pinch H. Basing on these simple geometrical parameters, it is possible to define the following quantities:
the flowing area ,
= 2 3 (2.1)
the wet perimeter,
= ( + ) + 4 3 (2.2) and the hydraulic or equivalent diameter
= (2.3) The driving-force that moves the coolant across the bundle can be obtained by mechanical systems, like pumps or by a gas-lift system. In loops characterized by a difference in height between the bundle (HS, heat source) and the heat exchanger (HX, cold sink) there is also an additional pressure-head due to the buoyancy effect, that is proportional to the height difference H between the HS and HX and to the temperature drop T across the bundle:
= (2.4) In Eq.(2.4) ’ [K-1] is the thermal expansion coefficient and [kg/m3] is the density of the working fluid. The temperature drop T across the bundle coincides with the temperature drop across the HX in an adiabatic system or loop.
To a purely hydrodynamic point of view, neglecting the effect of the wrap, it is possible to consider that the hydraulic resistance of the bundle is in first approximation represented by the resistance across its elementary sub-channel:
= (2.5) Where L is the active length of the bundle, is the sub-channel bulk velocity and f is Darcy-Weisbach friction factor described by a large number of correlation later in this chapter. It is possible to define a sub-channel Reynolds number as:
= = (2.6) Where G is the total mass flow rate in [kg/s], Resc expresses the ratio between inertial and
viscous forces in the sub-channel and thus it is relevant to estimate the role of turbulence. For liquid metals, like lead or LBE, the most important parameters for heat transfer is the Peclet number, defined as the product of the Reynolds and the Prandtl numbers, and representing the ratio between inertia and molecular diffusion of heat, as:
= = (2.7) Most of the known heat transfer correlations for liquid metals contains the Peclet number as governing parameter.
In many experimental studies, the cross-flow through a gap is represented with a dimensionless variable:
10 Figure 2-2 Thermal gradient “type a”.
( , ) = ( , ) (2.8) w
is the average axial e azimuthal coordinate of the rod. By integrating it over the gap width ,L, it follows:
( ) = ( ) / tan (2.9)
The relationship between ( ) -function’.
The above parameters characterize the hydrodynamics and heat transfer in liquid metals at a first approximation. At a deeper look, the thermal field induced by the heating rods can influence the fluid dynamics and heat transfer in the sub-channel.
In a rod bundle, the temperature gradients are basically in two different directions: a) in the axial direction, the large-scale gradient is represented by the thermal-drop T; b) in the transversal direction (generic section), from the pin wall to the bulk.
These gradients drive buoyancy forces that must be evaluate and compared with the dominant inertial forces, described before. This evaluation can be obtained in different ways; one of these is to consider the velocities of gravity waves in steady stratifications known as Brünt-Vaisala velocity:
2 g ’ T L (2.10) where is the velocity of a grave falling from a height L in a ‘modified’ gravity field of buoyancy acceleration g· · T. The force per unit area associated to this velocity is of the same order of the pressure head, and formally coincides with the buoyancy driving-force expressed in Eq. (2.4):
= = (2.11)
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As evidenced by the dotted arrow in Figure 2-2, the buoyancy generates an additional pressure head which opposes the main motion in the axial direction, and at the same time induces a secondary circulation of large scale superimposed to wall-bulk velocity present in the generic section.
From these issues, the effect of the gradient “type a” on the main flow is generally taken into account as an additional negative pressure head across the channel. If it is necessary to describe the turbulent fluid behavior, it will be required to consider a suppressive effect on turbulence due to the fact that the fluid rises across a stable stratification; this fact implies that the exact production term of turbulent kinetic energy k by buoyancy must be included in the k governing equation of the model.
The “type b” transversal gradient induces a natural circulation superimposed to the main flow, and therefore it causes a ‘distortion’ of the velocity and temperature profiles, reinforcing the ascending velocity close to the pins and dumping the bulk velocity in the center of the sub-channel. An order of magnitude of this effect can be computed by means of the ratio between transversal buoyancy forces and viscous forces in the section, i.e. by a transversal Grashof number:
= (2.12)
In Eq. (2.10), Tt represents the wall-bulk temperature difference in the generic section, that
is proportional to the wall heat flux. It is possible to estimate the importance of transversal buoyancy forces respect to inertial forces by a transversal Richardson number expressed by:
= = (2.13) Eq. (2.11) allows evaluating the importance of transversal buoyancy effect in different flow conditions. For example, for a fuel bundle in nominal conditions, typical values for the quantities are = 50 °C, = 5 mm, = 1 m/s, ’ = 1.2·10-4 K-1. The resulting transversal Richardson number is Rit = 3·10-4 <<1, i.e. buoyancy forces are negligible with
respect to the inertial forces. In accidental conditions, as LOFA or flow blockage, due to the decrease of the bulk velocity Rit can be O(1) and it is necessary to include the buoyancy in the
numerical studies.
2.2. Experimental studies
Many experimental studies regarding fluid-dynamics of rod bundles have been presented in literature. Some of these analyzed the hydraulic features of flow of transparent fluids by using optical methods; other investigated the thermal behavior of LM by measurements of bulk and wall temperatures.
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The first experiments on pressure field in wire wrap bundle were performed in the ‘70s by J.Lafay and B. Menant [5] . Thanks to a cold water closed convection loop with a hexagonal array of 19 unheated pins in equilateral triangular arrangement instrumented with 100 pressure taps and shown in fig. 2.4 , they were able to measure local axial pressure, transverse pressure gradient between peripheral sub-channels and total pressure in a triangular sub-channel. The axial static pressure profile is not linear along a wire wrapped rod bundle and the non-linearity increases as the mass velocity increases. Such behavior is shown clearly in Figure 2-5. A low pressure zone exists downstream and a high pressure zone exists upstream of a wire-wrap. This pressure variation which causes deviation from a linear profile is cyclic and has a period equal to the wire-wrap pitch. As expected the pressure variations are reflected in the velocity distribution.
Similar experiment were performed in 1977 by T.B. Barholet et al. [6] for a 19 rod assembly having P/D =1.5 , H/D = 20.9 a 178 mm wire-wrap pitch and a rod diameter equal to 8.6 mm, whose sub-channel pressure distribution is shown in Figure 2-6.
Figure 2-4 Flow test section with pressure taps.
Figure 2-5 Axial static pressure in a peripheral sub-channel.
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For the test data shown, the axial flow appeared to stabilize between the third and fourth wire-wrap pitch. This is consistent with other data which indicate that a length longer than one wire-wrap pitch is required for the field to stabilize. Qualitatively the peripheral pressure distribution around a rod can be separated into the two regions shown in Figure 2-6. Higher local pressure exists upstream of the wire-wrap and lower level pressure exists downstream of the wire-wrap. The angular separation between the two pressure regions is approximately 180°.
Lafay and others have also obtained extensive static pressure distribution data in a 19-rod assembly. These data indicate that a large transverse static pressure occurs around the assembly hexagonal duct at a given elevation as shown in Figure 2-7. When the wire moves from the interior sub-channel towards the wall gap, local pressure is considerably higher than the mean axial pressure at the same elevation. Maximum and minimum peripheral pressures occur on the faces adjacent to that face where the wire is in the peripheral gap. Although local static pressure increases and decreases in relation to the mean values, the distribution is non-symmetrical over a wire-wrap pitch. Additional data for the assembly of Figure 2-6 confirms the results of Lafay regarding the behavior of the pressure distribution and its proportionality to the mass flow rate.
Detailed sub-channel velocity distribution data have been reported from different test programs. Interior, side and corner sub-channels have been examined.
Figure 2-6 Pressure distribution around wire-wrap rods.
Figure 2-7 Transverse static pressure at the peripheral
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Figure 2-8 Interior sub-channel velocity distribution.
The WARD 11 : 1 model [6] study which used a sector of a fuel-type subassembly (P/D = 1.23, 304 mm wire-wrap pitch) obtained the following important results:
The most extensive low axial velocity region occurs behind the wire-wrap as the wire passes through the gap;
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Both the axial sub-channel velocity and cross flow are cyclic. However, when averaged over an axial pitch, mixing flow across interior gaps is small and near zero. In contrast, peripheral sweeping flow is unidirectional and always positive when averaged over an axial wire-wrap pitch. The behavior of interior sub-channel ‘mixing flow’ is shown in Figure 2-9.
Significant results from the Tokai study on axial and transverse velocity (using a 5:1 full geometry assembly air loop instrumented with pressure probe) are listed below.
A large transverse flow exists behind the wire-wrap as the wire moves through the gap.
A large axial velocity exists in regions where transverse (sweeping) velocity is high.
There is a significant circular flow behind a wire as the wire moves into a gap.
When wire-wraps move out of a sub-channel, the sub-channel velocity distribution is similar to that for bare rod bundles.
When a wire-wrap is in an interior channel the velocity decreases in the sub-channel. (This observation appears to be in conflict with the experimental conclusion that sub-channel velocity is maximum when the wire is in the sub-channels.
The WARD corner and side sub-channel data, Figure 2-10, show the following :
Axial length required for various types of flow development increases for interior side and corner sub-channel with a longer length required for corner sub-channels.
Side and corner sub-channel exhibit larger circumferential gradients then interior ones who exhibit larger radial gradients. This confirms the deduction drawn from the pressure gradient and cross flow measurements that in the interior rods, the fluid tends to follow the wire while in the blanket assembly, the fluid is broken up in the much smaller sub-channel with a pronounced cross flow
No gross gap backflow was observed in corner sub-channels. Rather, there was strong positive sweeping flow and only high localized back flow.
Figure 2-9 Interior sub-channel flow sweeping data.
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b a
17 Figure 2-11 Side sub-channel flow seeping
data over an axial wire-wrap pitch.
Figure 2-12 Eddy diffusivity in a wire wrap bundle.
For side sub-channels, the mixing tests indicate some areas of back flow or reverse cross flow against the direction of the wire-wrap in the gap; there is a strong cross flow behind a wire-wrap;
Side sub-channel flow effects appear to extend only to the immediately adjacent interior sub-channels.
The integrated swirl flow parameter -wrap pitch, is comparable for side and corner sub-channels and is independent of Reynolds number between the range of 4500 and 22000.
Peripheral sub-channel sweeping results for a 19-rod assembly are shown in Figure 2-11. Of considerable interest to model development is the observations that the axial average sweeping is essentially equal for all
Figure 2-13 Average ( ) function for interior, side and corner gaps.
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three gaps. However, there is a phase shift even on the same hexagonal face at a given elevation.
The MIT side sub-channel flow data [8] confirmed that areas of back flow exist as stated previously. These data also showed that the flow sweeping is slightly higher in turbulent than in laminar flow. The data also indicated that the flow sweeping is slightly higher for a 305 mm wire-wrap pitch than for a 152 mm wire-wrap pitch. There was also substantiation that side sub-channels and interior sub-channels are not coupled strongly, at least in so far as flow sweeping is involved.
Turbulent intensity = / and the dimensionless eddy diffusivity (inverse Peclet number) / were measured in the wire-wrap tests WARD [7]. On the average, the turbulent intensity and the dimensionless eddy diffusivity ranged from 0.02 to 0.06 and 0.001 to 0.006 respectively. A sharp increase in turbulence occurred just downstream of the wire-wrap as shown in Figure 2-12, but the turbulence decreased rapidly within one wire-wire-wrap diameter. It was suggested that the wire-wrap contributed both to the generation and decay of turbulence.
Recently (2000), an experimental investigation was performed by Fernandez and Carajilescov [9] in a turbulent flow in a seven wire-wrapped rod bundle, mounted in an open air facility. Static pressure distributions are measured on central and peripheral rods. By using a Preston tube, the wall shear stress profiles are experimentally obtained along the perimeter of the rods. The geometric parameters of the test section are P/D = 1.20 and H/D = 15. The measuring section is located at L/D = 40 from the air inlet.
It is observed that the dimensionless static pressure (Figure 2-14) and wall shear stress profiles (Figure 2-15) are nearly independent of the Reynolds number and strongly dependent of the wire-spacer position, with abrupt variations of the parameters in the neighborhood of the wires. The results agree with other similar works [10].
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As we can see from Figure 2-14, when the surface of the rod is in contact with the wire of a neighboring rod maximum and minimum static pressures are observed on both sides of that spacer. In such configurations, that wire represents an obstacle to the flow, reducing its velocity and increasing locally the static pressure. Behind the wire, a wake region is created, reducing the static pressure. The lowest peak occurs at the peripheral rod. When the described contact is absent, the static pressure profiles do not show abrupt variations.
The main results are:
The static pressure profiles on the hexagonal duct wall and on the surface of the rods reveal the existence of a sharp pressure variation across the wire. As expected, the minimum static pressure occurs in the wake region of the wires.
The static pressure profiles on the central and peripheral rods are quite similar in qualitative sense.
The lowest pressure peak was observed at the surface of the peripheral rod, behind the wire that belongs to the neighboring peripheral rod.
A change in Reynolds number has no appreciable effect on these non-dimensional static pressure profiles.
Local wall shear stress peaks seem to agree with Lafay's observation that some of the flow spills over the wire.
Large variations, between 243% and 180%, occur on the wall shear stress. So, similar variations are expected on the local friction factor.
The negligible effect of the Reynolds number on the wall shear stress reveals that the transverse cross-flow generated by the wire-wrap overshadows any effect of secondary flows eventually induced by turbulence.
Recently a Japanese team [11] measured velocity distribution in an inner sub-channel of a 7 pins wire wrap bundle water loop using PIVs (Particle Image Velocimetry) located on the front and lateral sides of the duct tube.
Figure 2-15
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The time averaged velocity field in the vertical cross section showed that the flow coming over the wire had horizontal velocity component toward the wire, i.e. wire wrapping direction. The horizontal velocity toward the wire wrapping direction had peak value at the point 0.16 D apart from the wire horizontally. On the other hand, upward straight flows were observed just upstream side of the wire. It means that the effect of the wire on the flow direction was obvious in downstream side of the wire. The vertical velocity in the gap between the pins had lower value especially near the wire, so as to make an asymmetric velocity distribution in the transverse direction.
In Figure 2-17 the velocity fluctuation intensity of vertical component was measured around the wire. The intensity increased near the wire and especially in wide region downstream from the wire.
Horizontal velocity field in a sub-channel including the wire was constructed from the matrix of vertical velocity fields. The clockwise swirl flow around the wire was obviously found. The recirculation flow became strong just before the wire went out from the sub-channel where relatively large flow area was formed behind the wrapping wire.
In 1972, at the ORNL, M.H. Fontana et al. [12] studied the temperature distribution in a 19-rod simulated LMFBR (sodium) fuel assembly in a hexagonal duct. Ten different tests were performed and a huge amount of data was collected.
Some of those tests were on : isothermal pressure drop measurement, effect of flow at constant power, effect of power at constant flow, temperature measurement at nominal power (5 kW/ft
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and with a 10% power skew) showing that temperature asymmetries occurred at uniform power due to the effects of the wire-wrap spacers, temperature measurements at low flow and low power (DHR simulating), investigation of temperature mixing from central to peripheral sub-channels and transport of energy from an heated rod through the bundle.
2.3. Numerical studies
In the Indian Prototype Fast Breeder Reactor (PFBR) that is currently under construction, the Fuel Subassembly (FSA) is a 217 pin bundle generating a heat flux of ~ 1.5 MW/m2
. In order to determine the sodium flow distribution in the pin bundle with helical wire-wrapped spacer and predict the temperature distribution including the maximum clad temperature to respect its permissible limit (for normal operating condition, the limit is 973 K), several detailed thermal hydraulic investigations have been carried out.
Gajalpathy et al. [13] compared numerical solutions related to the same pin bundle (sodium cooled) with wire wrap spacers and with bare pins. From the comparison, the helically wrapped wire produces a uniformly distributed temperature profile thanks to secondary swirl flow motions induced by the same helical wire. The simulated model is a 7 pins bundle with 6.6 mm rod diameter, ranked in hexagonal lattice with 1.26 pitch to diameter ratio (P/D). The wire has a diameter of 1.65 mm and a150 mm pitch. The model is one wire pitch long and inlet mass flow imposed. The total number of nodes is 4.42 106. Finer mesh size was adopted on pins, wires and clad surfaces for a better investigation of the secondary flow field and its development. A sketch of the mesh distributions for the bare bundle and wire spacer cases can be seen in Figure 2-18.
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Turbulence is modeled by the standard k- model. As boundary condition, the hexagonal can is taken to be adiabatic, no slip condition is imposed on all solid surfaces and the Reynolds number is varied from 103 to 105 using an appropriate wall function for higher Reynolds number. The wall heat flux is conservatively imposed at 1.85 MW/m2.
Results were investigated at various cross section (0, 30, 50, 90 and 150 mm from inlet section) as shown in Figure 2-19 .
The imposed inlet mass flow condition is used for studying the thermal boundary layer and the hydraulic one, even if this is not a proper condition (in a real bundle, the active region is 3-4 wire pitch far from inlet region that guarantees a velocity field already developed). The study evidences that the coolant flow entering the bundle is developed in two directions : an axial and a cross-stream one, this last induced by helix and not found in the bare pin bundle simulation. The secondary flow is maximum near the wire. The secondary flow seems to be higher in the peripheral sub-channels (near the hexagonal can) and lower in the central ones. The swirl flow phenomenon promotes turbulence giving rise to a good mixing degree and a uniform temperature distribution in the outlet section Figure 2-20.
Figure 2-18 Mesh distribution in helical wire-wrapped pin bundle (left) and bare pin bundle (right).
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Pressure losses in wire-wrapped bundle are higher than “bare” bundle ones, Figure 2-21. The coolant axial velocity is higher in sub-channels where the wire is close to the hexagonal can. The temperature difference between central and peripheral sub-channels is 21 K for the wire-wrapped case and 86 K for the bare bundle case.
Figure 2-19 Transverse velocity distribution (m/s): (a) Z = 0 mm, (b) Z = 45 mm, (c) Z = 90mm and (d) Z = 150 mm.
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Figure 2-20 Outlet sodium temperature distribution in the pin bundle.
Other CFD simulations by Gajalpathy et al. [14] were focused on bundles with an higher number of pins. A detailed CFD simulation is computationally difficult if not impossible for a 217 pin bundle and 12 wire pitch length, anyway the thermal hydraulic features could be assessed from the study of fuel pin bundles with less number of pin (7, 19, 37). The wire-wrap pin bundle model has a wire pitch of 200 mm and an equal length. Particular attention was also devoted to secondary flow features and peak cladding temperatures. Turbulence model adopted is the standard k- . At the inlet, the cross-stream velocity components are set to zero, while axial velocity of flow is specified with 2 % turbulence intensity for k and the corresponding value for . At the outlet, all the gradients are set to zero, as the pressure value. The hexagonal can is taken to be adiabatic while the fuel pin surfaces are specified with a uniform heat flux of 1.85 MW/m2. The inlet flow velocity is maintained as 7.4 m/s and a sodium inlet temperature of 397 °C.
Figure 2-21 Comparison of pin bundle friction factor with and without wire.
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A cross-stream velocity component can be seen in the three configurations modelled as in Figure 2-22, higher in the peripheral sub-channels than in the central ones; the secondary flow is higher in the peripheral sub-channels near the hexagonal can where the wire is not present, while it’s really less where the wire occupies the sub-channel flow area (because the wire opposes to the flow). The 19 and 37 pin bundle show similar velocity distributions, while the 7 pin bundle profiles are completely different because the interior sub-channels are more close to the hexagonal can and an higher interaction with it: high and low velocity areas are independent from the number of pins when it’s higher than 7. Sodium temperature seems to be higher in the interior sub-channels than in the peripheral ones for all bundles simulated. Pick cladding temperatures can be observed near wire spacers for the lower flow area available.
The helical wire forces the hot sodium from the central sub-channels to move out to peripheral sub-channels to mix with the relatively cold sodium and direct the relatively cold sodium at the periphery toward Figure 2-22 Transverse velocity (m/s) at the same height
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the center . These inward and outward flows create good mixing of the hot sodium at the central sub-channels with the relatively colder sodium at the periphery hence giving a more uniform temperature profile across the FSA. Gajalpathy also observed that in the peripheral zones of the wire-wrap bundle axial velocity and sodium temperature are higher (if compared with bare pin bundle case) where spacer wire is present; although the velocity is high, the flow area is less due to the presence of the spacer wires in these hot zones, leading to lower mass flow rate compared to the heat deposited in the peripheral zones. Finally, a very good agreement is found from the comparison of the CFD friction factor results for the 3 bundle type studied and the Novendstern’s correlation with a maximum deviation of 15 % .
Other CFD simulations on sodium cooled bundles were conducted by Pointer et al. [15]. Those simulations were focused on wire-wrap pin bundle ranked in hexagonal lattice with 7, 19, 37 and 217 pins using the commercial Star-CD code. Looking for general results scalable with a simple dimensionless analysis, the authors considered a P/D= 1.135, Dw =0.95 P , H =
17.7 D. A realistic configuration could have D = 10 mm and H = 177 mm. The wire-wrap bundle is enclosed in an hexagonal clad. The model’s length is not clear (equal to a wire pitch length or including a developing region). All the simulations have adopted a RANS approach and the k- as turbulence model; for the 217 pin bundle the total number of cells is 44 106. The whole study is focused on heat transfer phenomena, heat transfer coefficient predictions, pressure drop predictions into the bundle and predict the formation of flow separation zones. A radially uniform heat flux is specified for all pin and wire surfaces with a cosine distribution in the axial direction, while constant inlet velocity of 5.8 m/s is specified.
The pressure drop results for the 19 and 37 pin bundles show a good agreement with Todreas and Chen correlation and Rehme one. The velocity profile analysis shows that a secondary cross-stream flow was detected and it’s higher in the peripheral zone than in the central ones; and it can be found both in RANS and LES simulations of the 7 pin bundle.
A simplified model of the pin-wire contact zone was developed and a mesh independence study was conducted. For sake of simplicity, a pin with a finite contact surface with the wrapped wire and not a single point was developed. The 217 pin bundle mesh was developed using polyhedral elements with different aspect-ratio ranging from 4 to 16 (and total number of nodes varying from 15 to 75 106). Pointer shows that the results obtained are insensitive to geometry simplification and mesh aspect-ratio (Table 2-1) both. A decrease in the axial number of nodes has (at the contrary) a huge impact on main-stream velocity component, while it has a lower impact on temperature profiles and cross-stream velocity component.
Nominal cell aspect ratio Number of cells Predicted Pressure Loss (Pa)
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1:8 9769722 77.0
1:4 14955523 82.01
Table 2-1 Predicted pressure loss in simulations using meshes with three nominal aspect ratio.
Another interesting work is the Fisher et al. LES [16] study for single sub-channel of a wire-wrap bundle. The domain choice was smart considering an hexagonal cell around the central pin centered in the 6 neighbor pins center. The Reynolds number range from 40000 to 65000 (fully turbulent regime). The geometry investigated has P/D = 1.154 and H/D = 13.4 . The sub-channel length investigated is equal to the wire pitch, with source of momentum and inlet-outlet periodic conditions. Through a post-processing of the results, the authors investigated the cross-stream velocity between sub-channels at different points finding an axial periodicity equal to the wire-pitch and maxima of 30 % the mainstream value (Figure 2-23).
Hamman and Berry [16] developed a CFD simulation on wire-wrap bundle in the framework of the future development of a sodium cooled fast reactor. The study gives a complete overview to the bundle modeling, starting from the CAD model drawing, mesh generation technics, the CFD code simulations and code validation. The main objectives are the velocity profile study, the temperature distribution and the pressure drop profile in the bundle. The simulated wire-wrap bundle has 19 pins ranked in hexagonal lattice; the rods diameter is 8 mm, a pitch to diameter ratio of 1.127 and a wire diameter of 1.03 mm. The turbulence was modelled with the k- SST and at least one node in the viscous sub-layer; all simulations are stationary and solved with a segregated solver and a second order upwind scheme. The sodium inlet temperature was assigned and equal to 407°C and the inlet velocity is 2 m/s. On pins surfaces a constant heat flux of 1 MW/m2 was imposed. The mesh was inflated on pins and hexagonal can surfaces and in the zones between pin and wire. The total number of cells is 85 106 (hybrid shape). Reynolds number is about 50000 and the average velocity is 2 m/s.
Figure 2-23 Time- and spanwise-averaged velocity along channel interfaces for H/D=20.1
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The obtained results for parallel cross section at different height show higher velocity in the peripheral sub-channels where the wire is faraway. As we can see from Figure 2-24, a strong cross-stream velocity takes place into the bundle thanks to the helically wrapped wire reaching
the 30% of the mainstream velocity in some points, as already seen in other studies. Thanks to these flows, the temperature profile (Figure 2-25) is more uniform at the outlet region of the bundle even if the interior sub-channel are hotter than the peripheral ones and hotspots could be seen between pin and wire surfaces .
The temperature distribution is produced by the higher velocities in the peripheral sub-channels (Figure 2-26), justifying other studies and an optimized design for these ones (a wrong distance between pin and clad in facts give rise to uneven temperature distributions. From the same simulations, we can see that the local pressure profile (Figure 2-27) has a maximum near pin and wire interface. This study remains an important milestone for future RANS simulations, thanks to the high accuracy degree.
Figure 2-24 Cross-stream velocity
vector plot (midplane face). Figure 2-25 Temperature contours (outlet face).
Figure 2-26 Axial velocity
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Natesan [18] made CFD studies on a wire-wrapped bundle of a sodium cooled LMFBR for the future development of the Indian Prototype sodium cooled FBR. The simulation considered a 19 pins bundle with a pin diameter of 6.6 mm, hexagonally ranked, with a P/D = 1.25 and a wire diameter of 1.65 mm. All the simulations adopted a stationary RANS approach and various turbulence models were investigated (k- , k- , and the second order RSM) and first order upwind scheme was adopted for convective terms. A constant heat flux of 1.6 MW/m2 is imposed on pins surfaces. Uniform velocity and temperature profiles are imposed at the inlet region, while a constant pressure value is imposed at the outlet one. No-slip condition is imposed on all surfaces and the hexagonal can surfaces has an adiabatic condition imposed. The axial length simulated is 200 mm; equivalent to the inlet/developing region of the bundle. Thanks to the low resolution, standard wall functions with y+ ranging from 40 to 100 were adopted.
In all the cases where secondary flows and strong swirl flows investigated, the k- model has higher reliability than the k- one . The turbulence models adopted give similar bundle friction factor and 6 % margin compared with the Novendstern’s correlation (Figure 2-29). The same behavior is shown for the Nusselt number (Figure 2-30), even if a 20 % margin is found with the experimental correlations. The friction factor is strongly influenced by the wire-wrap pitch (with a 60 % increase when the wire-wrap pitch is reduced from 300 mm to 100 mm), while the Nusselt number seems unaffected from it, as we can see in Figure 2-31. Moreover the pitch reduction increases the cross-stream velocities and the swirl flow and, consequently, sodium mixing.
Natesan concluded that the wire spacer solution increases pressure loss in the bundle, increasing coolant mixing that generates a more uniform temperature profile at the Figure 2-28 Cross flow field produced
by the wire wrap in the rod bundle.
Figure 2-29 Friction factor predicted for liquid sodium flow by various turbulence models.
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outlet region. A more uniform temperature profile, in facts, reduces the differential thermal expansion of the pins.
Even if a low-resolution approach was adopted, we can easily state that secondary motions are predicted both by first order models and Reynold stress one, because they are induced by remarkable geometric constrains. It could also be seen in Figure 2-28 where the cross-stream velocity is calculated through the k- model and the RSM. Even in this study, the author noticed that the distance between the peripheral pins and the hexagonal clad is remarkable for having a more uniform temperature profile in the outlet region.
Other Indian researchers, Raza and Kim [19], developed a sensitivity study on wire-wrap cross section geometry. Circular, hexagonal and rhombus cross-sectional shapes of wire-spacer have been tested. Macroscopic flow structures in the three tested assemblies with different shapes of wire-spacer are found to be similar to each other. But, there are some differences in the thermal field and heat transfer. Circular shape of wire-spacer shows the obvious advantages in overall pressure drop, maximum temperature, and uniformity of temperature in the assembly in comparison with the other two shapes. However, in the aspect of heat transfer, rhombus and hexagonal shapes show superiority to circular shape. Rhombus and hexagonal shapes show almost same maximum temperature and overall temperature difference in the assembly. However, rhombus gives better heat-transfer performance but higher pressure drop than hexagon.
Rolfo et al. [20] developed various CFD models on wire-wrap bundles with a pitch to diameter ratio of 1.1, encapsuled into an hexagonal Figure 2-30 Circumferentially averaged Nu predicted
for liquid sodium flow by various turbulence models.
Figure 2-31 Variation in friction factor and circumferentially averaged Nu with wire-wrap
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casing with a wall to diameter ratio of 1.1, the wire diameter is 95% the gap between two adjacent rods, one wire pitch long and periodic boundary conditions changing: the number of pins (7, 19 and 61), the wire pitch to diameter ratio (H/D) from 21 to 16.8, adopting two different turbulence models (standard k- the Second Momentum Closure –Rij-).
CFD simulations of thermal-hydraulic features of wire-wrapped fuel bundles show that a variation of number of fuel pins (from 7 to 19, 61 and 271) does not have a large influence on the main flow features, as seen in other studies. The flow presents a very strong secondary motion and it can be divided into an inner area and an outer area, which is mainly composed by edge and corner channels. This is also visible from rotation of the maximum axial velocity. With the larger bundles (61 and 271 pins), the global swirl stays limited to the edge region, with an increasingly large homogeneous core. The eddy viscosity and second moment closure turbulence models are in good agreement with experimental global correlations (Figure 2-33), in particular that of Cheng and Todreas (1986).
The SMC global friction factor is higher than that given by the k- but the difference is less than 5% except at low Reynolds number (but down-to the wall models should then be used). Local friction
coefficient profiles also show that the Rij generally gives higher value with
respect to the k- .
Rolfo concluded that heat transfer effects would merit finer investigations, possibly by DNS or LES, since no precise experimental data is available (i.e. the experimental correlations found in literature are for smooth walls). In the case of constant wall temperature imposed on the pins (Dirichlet condition) the Nusselt number profiles are greater than those obtained with a prescribed constant heat flux (Neumann condition), and profiles are very similar as the geometry enlarges, see Figure 2-32. On the contrary, in the case of a constant heat flux, as the number of pins Figure 2-33 Friction factor for the 19 pin geometry
(H/D = 21).
Figure 2-32 Nusselt number for the 19 pin geometry (H/D = 21).
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increases, the Nusselt number drops and the temperature field in the cross section is less homogeneous.
The heat transfer is insensitive to the choice of turbulence model, as expected when the Peclet number is small, however the conjugate heat transfer cases are now being computed.
Since there is a serious lack of experimental data, refined LES and DNS calculations with only one pin and tri-periodicity are relevant, and ongoing to provide valuable local heat transfer data for down to the wall RANS model, or hybrid RANS-LES approach validation, which can then be used at higher Re numbers.
Recently, the Indian researchers Rasu et al. [21] investigated flow and temperature field development in wire-wrapped fuel pin bundles of sodium cooled fast reactor for the future Indian Prototype Fast Breeder Reactor (PFBR). The diameter of the fuel pin is 6.6 mm and that of spacer wire is 1.65 mm, the wire pitch length is 200 mm. Models with different number of pins were considered: a 7-pin geometry with a CFD model axial length of 7 wire pitches, a 19-pin one with an axial length of 10 pitches and a 37-pin geometry with an axial length of 5 pitches.
Three different turbulence models have been tested for a sensitivity study : the RNG k- model, the SST k- model and the Reynolds Stress Model (RSM). Based on the investigations for a 7-pin bundle, it’s found that the predictions of friction factor by RNG k- model are not significantly different from the predictions by other models as depicted in Figure 2-34a .
Figure 2-34 (a) Prediction of friction factor development in 7-pin bundle by various tur-bulence
models (Re = 0.842 105). (b) Prediction of Nusselt number development in 7-pin bundle by various
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The corresponding Nusselt number predictions presented in Figure 2-34b. Nusselt number predicted by RNG k– SST k– out the axial length of the pin bundle. Hence, the RNG k– adopted for all the simulations.
The Reynolds number, pitch of helical spacer wire and the number of pins in the bundle are varied as parameters. The dimensions and the Reynolds number are chosen based on the conditions prevailing in a typical medium size fast reactor. Major findings of the present work are as follows:
The magnitude of mean cross-stream velocity at the fully developed region is directly proportional to the Reynolds number, inversely proportional to the helical pitch length and it is nearly independent of the number of pins, as we can see from Figure 2-35. But, there is a strong correlation between the locations of spacer wire and the peak-cross-stream velocity.
Flow is fully developed at L 70 Deq in all the cases investigated. Further, the flow
development length is not significantly affected by the helical pitch.
In the wire-wrap rod bundle, the friction factor is not constant and it fluctuates periodically over a mean value. The variation/fluctuation over each helical pitch corresponds to a specific position of helical wire with respect to hexagonal pipe wall. The mean value of the friction factor in the entrance region reduces below the mean
value in the fully developed region, contrary to that seen in ducted flows. The mean fully developed friction factor is inversely proportional to the helical pitch. But, it’s independent of the number of pins in the bundle(Figure 2-36).
Figure 2-35 Cross-stream velocity for different wire-pitch (left) and number of pins (right).
34
The Nusselt number passes through three minima before attaining fully developed periodic fluctuations. The development of Nusselt number (Figure 2-37) is slower as compared to flow development. It attains full development within L=200 Deq in all the
pin bundles with 7-pins. But, for the 19-pin bundle, the development length exceeds 500 Deq , which is equal to 10 times the helical wire pitch. For larger number of pins
thermal development length is larger.
Traditionally, correlations corresponding to fully developed flow are considered for core design. But, the present study indicates that this approach is not conservative. Further, the entrance region effects and the oscillation in the fully developed region have to be properly accounted in the core design. Nusselt number variation exhibits a strong dependence on helical pitch, similar to that of friction factor.
Figure 2-36 Friction factor coefficient for different wire-pitch (left) and number of pins (right).
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Figure 2-39 Friction factor f as a function of Re for different leads of wire-wrap H . The test parameters:
P/D=1.417 , n=19 rods. 2.4. Overall Pressure drop
In order to maintain a proper spacing between the fuel pins and promote coolant mixing various types of spacers have been proposed. Among the various spacers the helical type wire-spacers have been most widely used in the liquid metal reactors (LMR), in particular.
To use a sub-channel analysis code, a pressure drop correlation that is applicable for a wire-wrapped LMR rod bundle is needed. Because of the complex geometry mainly caused by the presence of wire, a
simple equivalent diameter technique is not adequate to accurately predict the pressure drop in a wire-wrapped fuel assembly. Therefore, a number of experiments have been made by several investigators in the past to develop a better correlation for the friction factor applicable in a wire-wrapped fuel assembly. These experiments were conducted using different coolants (water, sodium, air), different sets of fuel bundle parameters, by different scientists in different countries and organizations.
One of the firsts pressure drop experiments on wire wrap bundle was conducted by Rehme using a water loop facility.
Systematic measurements of the pressure loss performance of the wire-wrapped rod bundles in hexagonal arrays were carried out by K. Rehme [22]. The important parameters varied in that study are the pitch-to-diameter ratio of the rods by using different wire diameters with the same rod diameter, the lead of the wire wraps and the number of rods in the rod bundles. The pitch-to-diameter ratio studied ranged between 1.125 and 1.417, the lead of the wire-wraps between 100 and 600mm, and the number of rods between 7 and 61 rods.
Figure 2-38 Friction factor f as a function of Re for different leads of wire-wrap H . The test parameters:
36
The first set of the Rehme experimental data in Figure 2-38 demonstrates the effect of different leads of wire wraps on the friction factor for 19 rods and the highest pitch-to-diameter ratio tested, 1.417. It is obvious that the friction factor increases with decreasing wire wraps pitch. One can see a strong increase in the friction factor with a high pitch-to-diameter ratio of the rods.
The second set of experimental data shown in Figure 2-39 demonstrates the effect of different leads of wire wraps on the friction factor for 19 rods and the lowest pitch-to-diameter ratio tested, 1.125. It is obvious that the friction factor increase with decreasing wire wraps pitch is smaller with a smaller pitch-to-diameter ratio of the rods. This fact can be explained by the decreasing blockage of the flow area caused by the wire wraps with decreasing pitch-to-diameter ratio of the rods.
The third set of experimental data (Figure 2-40) demonstrates more clearly the effect of different pitch-to-diameter ratio on the friction factor for seven rods and the smallest pitch of wire wraps (100mm) configuration as a function of Reynolds number. The dependence of pitch-to-diameter ratio on the friction factor is much stronger for smaller pitch of wire wraps value (100mm) than for higher pitch of wire wraps value.
The fourth set of experimental data demonstrates how the friction factor depends on the number of rods in a rod bundle. The friction factor increases with increasing number of rods.
Figure 2-40 Friction factor f as a function of Re for different pitch-to-diameter ratio P/D . The test parameters: H=110 mm , n=19 rods.
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This effect can be explained by the fact that the influence of the smooth channel wall results in a lower pressure loss. Since rod bundles with only a few rods include a relatively higher part of channel wall with respect to the total wetted perimeter, the total pressure drop is lower.
2.4.1. Definitions
The total pressure drop in a fuel assembly is usually calculated using the following formula: pFA = pinlet + poutlet + porf + pfric + pspacer (2.14)
Fuel assembly inlet, outlet and orificing pressure losses are determined by
pinlet + poutlet + porf = (Kinlet + Koutlet + Korf) 0.5 v2 (2.15)
with being the density and v the velocity of the coolant and K as the associated pressure loss coefficients. Pressure loss due to the flow friction along a smooth pipe is calculated as
= 0.5 (2.16) where L is the tube length, Deq the hydraulic diameter of the flow channel, and f
(Darcy-Weisbach friction factor) for the turbulent single phase flow can be estimated using the Blasius formula, namely
= . . (2.17) where Re represents the Reynolds number of the flow channel. In a similar manner, the pressure loss due to the spacer (in this case due to the wire-wrap) is calculated as
= 0.5 (2.18) where f (friction factor) correlations for the wire-wrap spacer configuration will be discussed next.
2.4.2. Friction factor correlations for wire-wrapped fuel assemblies The various friction factor correlations for the wire-wrapped fuel bundles that are available today are summarized in this section.
1. Novendstern model
Friction factor for the wire-wrapped fuel bundle in the Novendstern model [23] is calculated based on the following correlations:
= (2.19)
where = ; = . .
Deq1 is the hydraulic diameter of the interior sub-channels; Res is the average Reynolds number
38 = . ( ) . + . ( ) . . ( ) . . (2.20)
Re1 is the Reynolds number for the center sub-channel of the hot SA in the wire-wrap
configuration with
= , = = , (2.21)
= . . , (2.22)
= + + (2.23) where all the various symbols are defined in the nomenclature section of this paper. For our analysis we assumed only one averaged sub-channel, i.e. X1 = 1 and Deq1 =Deq.
2. Rehme model
Friction factor for the wire-wrapped fuel bundle in the Rehme model [22] is calculated based on the following correlations:
= . + . . . ( ) (2.24) where = . + 7.6 . (2.25) where all the various symbols are defined in the nomenclature section.
Engel, Markley and Bishop model
Friction factors for the wire-wrapped fuel bundle in the Engel, Markley and Bishop model [24] is calculated based on the following correlations:
Laminar flow : f = , for Re < 400 (2.26) Turbulent flow : f = . . for Re >5000 (2.27) Transition flow : = 1 + . . for 400 < Re < 5000 (2.28) where = , and where all the various symbols are defined in the nomenclature section of this paper.
After performing the analysis using the different experimental data sets, the authors changed the coefficient for the turbulent flow from 0.55 to 0.37 in order to obtain an improved agreement with the available experimental data sets.
3. Chen and Todreas models: simplified and detailed
Friction factor for the wire-wrapped fuel bundle in the simplified Cheng and Todreas model [25] is calculated based on the following correlations:
Laminar flow : = for Re < Re L (2.29)
39
Transition flow : = (1 ) + . for Re L < Re < Re T (2.31)
Where log = 1.7 1 (2.32) log = 0.7 1 (2.33) = ( ) ( . ( ) . ) . ( ) (2.34) = 974.6 + 1612 598.5 . . ( ) (2.35) = 0.8063 0.9022 log + 0.3526 log . . (2.36)
and where all the various symbols are defined in the nomenclature section.
Friction factors for the wire-wrapped fuel bundle in the detailed Cheng and Todreas model is calculated based on the center, side and corner sub-channels equations that are described in more detail in Cheng and Todreas (1986) and not reported here for sake of simplicity.
4. Baxi and Dalle-Donne model
Friction factor for the wire-wrapped fuel bundle in the Baxi and Dalle-Donne model [26] is calculated based on the following correlations:
Laminar flow: Re 400
= , = . ; with H in cm (2.37 a,b) Turbulent flow: Re 5000
ft = fs M, where fs = smooth friction factor in a tube (Blasius) = 0.316/Re0.25,
= . ( ) . + . ( ) . . ( ) . . (2.38) Transition flow: 400 < Re < 5000 = (1 ) + (2.39) fl = laminar friction factor, ft = turbulent friction factor, = (Re 400)/5000, and where all the
various symbols are defined in the nomenclature section.
After performing the analysis using the different experimental data sets, the authors changed the coefficient for the laminar flow from 80 to 300 in order to obtain an improved agreement with the available experimental data sets.
5. Sobolev model
Friction factor for the wire-wrapped fuel bundle in the Sobolev model [27] is calculated based on the following equation:
= 1 + 600 1 .. 1 + 1 . (2.40) where all the various symbols are defined in the nomenclature section.
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Model Year n P/D H/D Re range Uncertanty
Novendstern 1972 19-217 1.06-1.42 8.0-96.0 Transition & turbulent (2600-105) ±14 % Rehme 1973 7-217 1.1-1.42 8.0-50.0 Transition & turbulent (1000-3· 105) ±8 % Engel - Engel modified 1979 2008 19-61 1.067-1.082 7.7-8.3 All regimes (50 – 105) ±15 %
Chen and Todreas simplified and detailed 1986 19-217 D 1.0-1.42 S 1.025-1.42 D 4.0-52.0 S 8.0-50.0 All regimes (50 – 106) D ±14 % S ±15 %
Baxi and Dalle-Donne + modified 1981 2008 19-217 1.06-1.42 8.0-96.0 All regimes (50 – 105) NA Sobotev 2006 NA NA NA NA NA
Table 2-2 Application range and uncertainty for the friction factor correlations illustrated.
2.4.3. Comparative analysis of the available correlations
Many authors have studied the friction factor correlations agreement with the huge amount of experimental data available, looking for the best-fitting one for the entire parameters range. Chun & Seo [28], for example, have used some of those pressure drop correlations and the data of a water loop experimental apparatus installed with four test sections of the 19-pin assembly that have different P/D and H/D, judging on their agreement and best usage.
They concluded that :
1. Novendstern’s (1972) correlation agrees fairly well with experimental data in the turbulent region. This model, therefore, should be used only in the turbulent flow region.
2. Rehme’s (1972) correlation appears that it does not represent the effects of P/D and H/D well enough, and this correlation consistently under predicts the friction factor for overall flow regions.
41
3. Engel et al.’s (1979) correlation depends only the Reynolds number and the effects of P/D and H/D are not properly represented in their model. This correlation greatly over predicts the friction factor for overall flow regions.
4. Both the original and the simplified Cheng and Todreas (1986) correlations show good agreements with experimental data for all flow regions and difference between the original and the simplified correlations is less than 10%.
E.Bubelis & M.Schikorr [29] compared friction factor predictions with tens experimental facility or CFD data separately, giving a ranking number (from 0 to 3) every time for the agreement of every correlation with the experimental data.
A general conclusion was subsequently attempted which is the most appropriate, universally applicable friction factor correlation (model) fitting best the available experimental data. For the various coolants the following ordering was deduced :
For water tests: Rehme, Sobolev, Novendstern, Engel (modified); For air tests: Rehme, Engel (modified);
For sodium tests [13] : Rehme, Engel (modified), Baxi and Dalle-Donne (modified).
Generally, the friction factor correlations providing a good fit to most of the available experimental data analyzed in this report (for three types of coolants) are in order: Rehme, Sobolev, Novendstern, Engel (modified), Baxi and Dalle-Donne (modified).
Most recently, S.K.Chen et al. [30] have found that the Todreas & Chen formula used in Bubelis paper was incorrect; in fact the original correlation refers to H / Dr and not to H / (Dr
+ Dw) as in the Bubelis paper (the same error can be seen in other correlations of the same
article). With this important correction, the over-all ranking is modified showing the Chen&Todreas correlation as best fitting for every test section (air, water, sodium).
In addition, S.K. Chen et al. have used all experimental data available judging all pressure drop correlations on this basis :
a) Agreement index : The RMS of the prediction error is defined as the square root of ((1/n) ( ) ),where n is the number of data points for each bundle. Using the RMS of the prediction error of one bundle data set, the Agreement Index(AI) for each correlation can be assigned accordingly. The correlation that has the highest total Agreement Index considering all bundle data was concluded to provide the best performance.
b) Credit score : The Credit Score (CS) method assigns a performance index based on the RMS error between the data and the correlation prediction. Starting from 10% error, the greater the error, the less the Credit Score in a linearly assigned manner. c) Prediction error distribution : The STD of the prediction error is the square root of
the sample variance of data points, where the sample variance = (1/(N
( ) , and N is total number of data points. Assuming that the error distribution is a normal distribution, one can calculate the plus and minus prediction error % with a given confidence level. A 90% confidence interval of ±X%
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can be interpreted as there is a 90% chance that the prediction error is within ± X %. This interval is a superior index for the goodness of a correlation given that the normal distribution assumption is valid.
All pressure drop correlations were then judged for every flow regime and sub-channel position data concluding that the detailed Cheng and Todreas correlation is identified to be the best fit wire-wrapped rod bundle friction factor correlation, except for the supplemental 108 bundle set case covering the transition and/or turbulent regimes, in which half of the 108 bundles are Rehme bundles. In general for the transition and/or turbulent regimes, the descending order for the best fit correlation is Cheng&Todreas (detailed), Baxi&DalleDonne, Rehme, Cheng&Todreas (simplified). These conclusions are consistent with the results of previous evaluations, e.g., Chun and Seo [28], and Tenchine [31].
2.5. Overall Heat transfer
The heat transfer to liquid metals significantly differs from the heat transfer to water. The main reason for this difference is that liquid metals have a very low Prandtl number (Pr). In other words, the contribution to the total heat transfer from the thermal conductivity (compared to the contribution from the convection) is much higher for liquid metal compared to water. Another experimentally proved feature of the heat transfer to the liquid metal is that the Nusselt number (Nu) can be correlated as follows:
= + (2.41) where , and are the constants and Pe is the Peclet number. The first and second terms in (2.41) describes the contributions from the thermal conductivity and from the convection, respectively. Experimentally, the constant is found to be close to 0.8, while and depend on the geometry of the heat exchange section (round tube, annulus, tube bundle, etc.).
The majority of the experimental work on the measurement of the heat exchange parameters for liquid metal flowing in a regular lattice of circular rods was conducted in the 1960 s. Recently such experiments were revived for specific advanced nuclear reactor core configurations.
The increased requirements to advanced safety make it necessary to provide as accurate as possible numerical simulation of the processes of heat transfer, fluid flow, etc. in the core and in the whole reactor system. In particular, modelling of the heat transfer to the liquid-metal coolant in the core, which is a large bundle of circular fuel pins, is an important issue. For example under transient
conditions the heat exchange rate in a sodium-cooled reactor core determines temperature margins to the onset of nucleate boiling which could have impact on both the fuel rod temperatures and
core reactivity, while under steady-state conditions the heat transfer rate in a lead-cooled reactor defines the temperature margins for corrosion-resistant cladding operation.
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The available test data and correlations relating to the heat transfer to liquid metal in tube lattices are reviewed in this section. The data points are presented as pairs of the Nusselt (Nu) and Peclet (Pe)
numbers and the correlations provide the functional dependence of the Nusselt number on the Peclet number and pitch-to-diameter ratio (x). The correlations are usually a best fit to experimental
data, but some of them are derived theoretically. The current study does not provide a complete set of the existing data and modelling approaches, but provides a set which is representative from viewpoint of different conditions (liquid metal type, temperature, geometry, velocity, etc.) to make a selection of the heat transfer correlation appropriate to liquid-metal cooled reactor core.
6. Correlations by Dwyer and Tu (1960) and Friedland and Bonilla (1961)
Two equations for heat transfer to liquid metal flowing in a triangular bundle of circular rods were derived by Dwyer and Tu [32] and Friedland and Bonilla [33] :
= 0.93 + 10.81 + 0.0252 . ( ) .
(2.42) = 7 + 3.8 . + 2.01 + 0.0252 . ( ) . (2.43)
current study this ratio was assumed equal to 1).
The main difference in the two approaches was different assumptions about the velocity profile. The correlation (2.40) is recommended for Peclet numbers of 70 up to 104 and pitch-to-diameter ratios of 1.375 up to 2.2, while the range of the applicability of the second correlation (2.41)was specified to be Pe = 0–105 and x = 1.3–10.
Test data and correlation by Mareska and Dwyer (1964)
Experimental results were obtained for 13 rods of 13mm o.d. arranged in an equilateral triangular lattice with the pitch-todiameter ratio of 1.75. The working fluid was mercury (Pr 0.02).
The total of 146 data pairs of Nu vs. Pe are given in the paper by Mareska and Dwyer [34] and used in the current analysis for the range of the Peclet numbers of 150–4000.
The following semi-empirical correlation was used in (Mareska and Dwyer, 1964) to describe the test data:
= 6.66 + 3.126 + 1.184 + 0.0155( ) . (2.44)
The correlation (2.42) is recommended for triangular bundles in the range of Peclet numbers of 70–104 and pitch-to-diameter ratios of 1.3–3. In the current study the ratio of the eddy diffusivity of heat to the eddy was assumed equal to 1.
44 Correlation by Subbotin et al. (1965)
The following correlation is recommended by Subbotin et al. [35] for the flow of liquid metal in a triangular lattice of rods with the pitch-to-diameter ratio of 1.1–1.5 and Peclet numbers of 80–4000:
= 0.58( ) . . (2.45)
where Dh and d are the hydraulic diameter and the rod diameter, respectively. For the
triangular lattice the correlation (2.43) becomes
= 0.58( 1) . . (2.46) Correlation by Ushakov et al. (1977)
The following correlation is recommended by Ushakov et al. [36] for the flow of liquid metal in a triangular lattice of rods with the pitch-to-diameter ration of 1.3–2.0 and Peclet numbers up to 4000:
= 7.55 + . 1
. .
. . .
(2.47)
where 6 is the “approximate criterion of thermal similarity of the fuel rods in the triangular
assembly” (Ushakov et al., 1977), Dh and d are the hydraulic diameter and the rod diameter,
respectively. As the
second term in the square brackets (see correlation (2.45)) for x = 1.3 and 6 =0 equals 0.002
and for higher values of x and 6 will be even smaller, it can be neglected for the sake of
simplicity. For the same reason the Peclet number power can be reduced to 0.56 + 0.19x. Correlation (2.45) then becomes
= 7.55 + . . . (2.48) Test data and correlation by Borishanski et al. (1969)
The working sections in the tests reported in Borishanski et al. [37] consisted of seven tubes of 22mm o.d. arranged in equilateral triangular bundles with different pitch-to-diameter ratios (1.1,
1.3. 1.4 and 1.5), the heated length of the test section being 800mm.
The temperatures varied from 206 to 236°C. The results for three groups of the experiments are presented. The working fluids are not explicitly specified in the paper, however Prandtl numbers for three groups of the presented results are given: 0.007, 0.03 and 0.024. The total of 230 data pairs of Nu vs. Pe are given by Borishanski et al. (1969) and used in the current analysis.
The following correlation was derived (Borishanski et al., 1969) on the basis of this data: = 24.15 log( 8.12 + 12.76 3.65 ) + 0.0174 1 ( ) , (2.49)
where B = 0 , Pe < 200 (2.50a) = (Pe -200 )0.9 , Pe (2.50b)
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Correlation (2.47) is recommended for triangular bundles in the range of Peclet numbers of 60–2200 and pitch-to-diameter ratios of 1.1–1.5.
Test data and correlation by Gräber and Rieger (1972)
Three sets of the experimental data were measured by Gräber and Rieger [38] with the test sections consisting of 31 tubes of 12mm o.d. arranged in equilateral triangular bundles with pitch to diameter ratios of 1.25, 1.6 and 1.95. The working fluid was 44%Na–56%K at temperatures from100 to 425 C. The Prandtl number varied with temperatures from 0.011 to 0.024. A total of 246 data pairs of Nu vs. Pe are given by Gräber and Rieger (1972) and are used in the current analysis for the range of the Peclet numbers of 110–4300. The experimental results were correlated (Gräber and Rieger, 1972) by the following equation:
Nu = 0.25 + 6.2x + (0.032x 0.007)Pe0.8 0.024x (2.51) Test data and correlation by Zhukov et al. (2002)
In frame of the BREST lead-cooled reactor project an extensive experimental program was conducted to study the heat transfer to liquid metal in a square lattice of round tubes [39]. The working section consisted of 25 tubes of 12mm o.d. Four sets of experimental data were measured for pitch-to-diameter ratios of 1.25, 1.28, 1.34 and 1.46. The working fluid was 22%Na–78%K at a temperature of about 50 C. The heated length of the assembly was 980mm. A total of 36 data pairs of Nu vs. Pe are given by Zhukov and used in the current analysis for the range of the Peclet numbers of 60–2000. The experimental results were correlated by the following equation:
= 7.55 14 + 0.007 . . (2.52)
A separate correlation (of the same form, but with slightly different coefficients) was recommended for a square lattice with spacers, but since the influence of the spacers on the heat transfer strongly depends on the specific design this correlation is not considered in the current review.
Correlation by Mikityuk (2009)
Comparing the agreement of the available heat transfer correlation with the experimental results of different campaigns [41], Mikityuk numerically developed a new best fitting correlation :
= 0.047(1 . ( ))( . + 250) (2.53) For a complete understanding of the heat transfer correlations illustrated, their geometry and application range is shown in Table 2-3.
Reference Pe x=P/D Experimental fluid Lattice
Dwyer and Tu 70 – 10000 1.375 – 2.2 Hg Triangle
Friedland and
