Natural convection heat transfer from horizontal rod bundles in liquid sodium. Part 2: Correlations for horizontal rod bundles based on theoretical results

Natural convection heat transfer from horizontal rod bundles in N_xm × N_ym arrays (N_xm, N_ym = 5–9) in liquid sodium was numerically analyzed for three types of the bundle geometry (in-line rows, staggered rows I and II). The unsteady laminar two-dimensional basic equations for natural convection heat transfer caused by a step heat flux were numerically solved until the solution reaches a steady state. The PHOENICS code was used for the calculation considering the temperature dependence of thermophysical properties concerned. The surface heat fluxes for each cylinder were equally given for a modified Rayleigh number, R_f, ranging from 0.0637 to 63.1 (q = 1×10⁴ to 7×10⁶ W/m²). S_x/D and S_y/D for the rod bundle, which are the ratios of the distance between center axes on the abscissa and the ordinate to the rod diameter, respectively, were ranged from 1.6 to 2.5 on each bundle geometry. The spatial distribution of Nusselt numbers, Nu, on horizontal rods of a bundle was clarified. The average value of Nusselt number, Nu_av, for three types of bundle geometry with various values of S_x/D and S_y/D were calculated to examine the effect of the array size, S/D and R_f on heat transfer. The bundle geometry for the higher Nu_av value under the condition of S_x/D×S_y/D = 4 was examined by changing the ratio of S_x/S_y. A correlation for Nu_av for the three types of bundle geometry above mentioned including the effects of S_x/D and S_y/D was developed. The correlation can describe the theoretical values of Nu_av for the three types of bundle geometry in N_xm × N_ym arrays (N_xm, N_ym = 5–9) for S_x/D and S_y/D ranging from 1.6 to 2.5 within 10% difference.     

Single phase transport within bare rod arrays at laminar, transition and turbulent flow conditions

A theoretical analysis has been performed to study molecular and turbulent transport phenomena between subchannels of infinite bare rod arrays at laminar, transition and turbulent flow conditions. For this investigation, the theoretical approach of Ramm and Johannsen for predicting turbulent momentum and heat transfer in rod bundles has been extended to evaluate three-dimensional temperature fields. Results are presented enabling the prediction of the onset and growth of laminarization in typical subchannels of square and triangular rod arrays. These results are further applied to interpret the characteristic effects of variations in Reynolds number, Prandtl number or geometric spacing on integral exchange parameters as the thermal mixing flow rate and mixing length scale. These results are of particular significance relative to the explanation of recent data from tracer-type mixing experiments and also exhibit the importance of secondary flow effects on turbulent intersubchannel energy transport. In view of these findings, the physical relevance of current correlations derived from integral-type experiments to numerically predict exchange coefficients for use in lumped parameter subchannel analysis codes is discussed.     

Prediction of the single-phase turbulent mixing rate between two parallel subchannels using a subchannel geometry factor

This paper presents a simple method for predicting the single-phase turbulent mixing rate between adjacent subchannels in nuclear fuel bundles. In this method, the mixing rate is computed as the sum of the two components of turbulent diffusion and convective transfer. Of these, the turbulent diffusion component is calculated using a newly defined subchannel geometry factor F^* and the mean turbulent diffusivity for each subchannel which is computed from Elder's equation. The convective transfer component is evaluated from a mixing Stanton number correlation obtained empirically in this study. In order to confirm the validity of the proposed method, experimental data on turbulent mixing rate were obtained using a tracer technique under adiabatic conditions with three test channels, each consisting of two subchannels. The range of Reynolds number covered was 5000–66 000. From comparisons of the predicted turbulent mixing rates with the experimental data of other investigators as well as the authors, it has been confirmed that the proposed method can predict the data in a range of gap clearance to rod diameter ratio of 0.02–0.4 within about ±25% for square array bundles and about ±35% for triangular array bundles.     

Turbulent flow simulation in a wire-wrap rod bundle of an LMFBR

The pressure drop and heat transfer characteristics of wire-wrapped 19-pin rod bundles in a nuclear reactor subassembly of liquid metal cooled fast breeder reactor (LMFBR) have been investigated through three-dimensional turbulent flow simulations. The predicted results of eddy viscosity based turbulence models (k-?, k-ω) and the Reynolds stress model are compared with those of experimental correlations for friction factor and Nusselt number. The Re is varied between 50,000 and 150,000 and the ratio of helical pitch of wire wrap to the rod diameter is varied from 15 to 45. All the three turbulence models considered yield similar results. The friction factor increases with reduction in the wire-wrap pitch while the heat transfer coefficient remains almost unaltered. However, reduction in the wire-wrap pitch also enhances the transverse flow velocity in the cross-sectional plane as well as the local turbulence intensity, thereby improving the thermal mixing of coolant. Consequently, the presence of wire wrap reduces temperature variation within each section of the subassembly. The associated reduction in differential thermal expansion of rods is expected to improve the structural integrity of the fuel subassembly.     

A numerical investigation of natural convection heat transfer in horizontal spent-fuel storage cask

Natural convection heat transfer in a horizontally placed dry spent-fuel storage cask is numerical investigated. The commercial computational fluid dynamics (CFD) code, -3.2 is used and the laminar and turbulent model are employed. The numerical predictions obtained are compared with the experimental data reported by Nishimura et al. [J. Nucl. Sci. Technol. 33 (1996) 821]. The computational results corresponding to laminar model agree well with the experimental data, but the calculated results of turbulent model are higher. The velocity pattern and the isotherms are drawn. With the increasing of Rayleigh number, the heat transfer in the cask changes from conduction dominant mode to convection dominant mode. In the condition of Ra_m=1.3×10⁹, turbulent model prevails. The convective heat transfer is so strong that almost all temperature changes take place in the region near the wall of the cask. The Rayleigh number Ra_m and the Nusselt number Nu_m characterized by maximum temperature difference are defined to depict the heat transfer characteristics. It is found laminar and turbulent models predict the same trend but different value. The flow patterns in the cask can be divided to three regimes. In these three regimes, modified Nusselt numbers are proportional to the 0.7, 0.25 and 0 power of the modified Rayleigh number, respectively.     

Natural convection heat transfer from horizontal rod bundles in liquid sodium. Part 1: Correlations for two parallel horizontal cylinders based on experimental and theoretical results

Natural convection heat transfer coefficients on two parallel horizontal test cylinders in liquid sodium were obtained experimentally and theoretically for various setting angles, γ, between vertical direction and the plane including both of these cylinders’ axes, over the range of 0°–90°. Both test cylinders are 7.6 mm in diameter and 50 mm in heated length with the ratio of the distance between each cylinder axis to the cylinder diameter, S/D, of 2. Theoretical equations for laminar natural convection heat transfer from the two horizontal cylinders were numerically solved for the same conditions as the experimental ones. The average Nusselt numbers Nu on the cylinders obtained experimentally were compared with the corresponding theoretical values on the Nu versus modified Rayleigh number R_f [= Gr*Pr²/(4?+?9Pr^1/2?+ 10Pr)] graph. The experimental values of Nu for the upper cylinder are about 20% lower than those for the lower cylinder at γ = 0° for the range of R_f tested here. The value of Nu for the upper cylinder becomes higher and approaches that for the lower cylinder with the increase in γ over the range of 0°–90°: the values for each cylinder agree with each other at γ = 90°. The values of Nu for the lower cylinder at each γ are almost in agreement with those for a single cylinder. The theoretical values of Nu on two cylinders except those for R_f < 4 at γ = 0° are in agreement with the experimental data at each γ with the deviations less than 15%. Correlations for two cylinders were obtained as functions of S/D and γ based on the theoretical solutions. A combined correlation for multi-cylinders in a vertical array based on the correlations for two cylinders was developed. The values by the correlation agree with the theoretical solution for the multi-cylinders for R_f ranging from 4.7 to 63 within 10% difference.     

绕丝棒束组件低流速时摩擦阻力实验研究

钠冷快堆在事故停堆余热排放期间,堆芯组件内钠流为自然循环流动,流速很低,因此准确确定绕丝棒束组件低流速时的摩擦阻力系数对钠冷快堆非能动余热排出系统的设计具有重要意义。本文以水为流动介质,准确测量了37棒和19棒绕丝棒束组件在低流速（Re<1 000）时的摩擦阻力系数。实验结果表明,随着流速的降低,绕丝棒束组件的摩擦阻力系数迅速升高,流动从层流向过渡流转变时,摩擦阻力系数有明显跃升。将实验测量值与绕丝棒束摩擦阻力系数经验公式的计算结果进行比较,发现在低流速时,经验公式计算结果较实验测量值明显偏小,同时经验公式计算的绕丝棒束层流向过渡流转变的临界Re较实验值偏大。     

CFD analysis of thermal–hydraulic behavior of heavy liquid metals in sub-channels

CFD analysis was carried out for thermal–hydraulic behavior of heavy liquid metal flows, especially lead–bismuth eutectic, in sub-channels of both triangular and square lattices. Effect of various parameters, e.g. turbulence models and pitch-to-diameter ratio, on the thermal–hydraulic behavior was investigated. Among the turbulence models selected, only the second order closure turbulence models reproduce the secondary flow. For the entire parameter range studied in this paper, the amplitude of the secondary flow is less than 1% of the mean flow. A strong anisotropic behavior of turbulence is observed. The turbulence behavior is similar in both triangular and square lattices. The average amplitude of the turbulent velocity fluctuation across the gap is about half of the shear velocity. It is only weakly dependent on Reynolds number and pitch-to-diameter ratio. A strong circumferential non-uniformity of heat transfer is observed in tight rod bundles, especially in square lattices. Related to the overall average Nusselt number, CFD codes give similar results for both triangular and square rod bundles. Comparison of the CFD results with bundle test data in mercury indicates that the turbulent Prandtl number for HLM flows in rod bundles is close to 1.0 at high Peclet number conditions, and increases by decreasing Peclet number. Based on the present results, the SSG Reynolds stress model with semi-fine mesh structures is recommended for the application of HLM flows in rod bundle geometries.     

Analytical prediction of turbulent friction factor for a rod bundle

An analytical calculation has been performed to predict the turbulent friction factor in a rod bundle. For each subchannel constituting a rod bundle, the geometry parameters are analytically derived by integrating the law of the wall over each subchannel with the consideration of a local shear stress distribution. The correlation equations for a local shear stress distribution are supplied from a numerical simulation for each subchannel. The explicit effect of a subchannel shape on the geometry parameter and the friction factor is reported. The friction factor of a corner subchannel converges to a constant value, while the friction factor of a central subchannel steadily increases with a rod distance ratio. The analysis for a rod bundle shows that the friction factor of a rod bundle is largely affected by the characteristics of each subchannel constituting a rod bundle. The present analytic calculations well predict the experimental results from the literature with rod bundles in circular, hexagonal, and square channels.     

Turbulent heat transfer to longitudinal flow through a triangular array of circular rods

Temperature distribution and heat transfer to longitudinal turbulent, fully developed flow through triangular arrays of smooth circular rods are analysed for liquids with Prandtl number 1 and 1. Nusselt number is plotted versus pitch and turbulence for constant heat flow and for constant temperature on the rod surface, and the optimum pitch is determined. The influence of Prandtl number on Nusselt number is analysed.     

A critical heat flux approach for square rod bundles using the 1995 Groeneveld CHF table and bundle data of heat transfer research facility

The critical heat flux (CHF) approach using CHF look-up tables has become a widely accepted CHF prediction technique. In these approaches, the CHF tables are developed based mostly on the data bank for flow in circular tubes. A set of correction factors was proposed by Groeneveld et al. [Groeneveld, D.C., Cheng, S.C., Doan, T., 1986. 1986 AECL-UO Critical Heat Flux lookup table. Heat Transf. Eng. 7(1–2), 46] to extend the application of the CHF table to other flow situations including flow in rod bundles. The proposed correction factors are based on a limited amount of data not specified in the original paper. The CHF approach of Groeneveld and co-workers is extensively used in the thermal hydraulic analysis of nuclear reactors. In 1996, Groeneveld et al. proposed a new CHF table to predict CHF in circular tubes [Groeneveld, D.C., et al., 1996. The 1995 look-up table for Critical Heat Flux. Nucl. Eng. Des. 163(1), 23]. In the present study, a set of correction factors is developed to extend the applicability of the new CHF table to flow in rod bundles of square array. The correction factors are developed by minimizing the statistical parameters of the ratio of the measured and predicted bundle CHF data from the Heat Transfer Research Facility. The proposed correction factors include: the hydraulic diameter factor (K_hy), the bundle factor (K_bf), the heated length factor (K_hl), the grid spacer factor (K_sp), the axial flux distribution factors (K_nu), the cold wall factor (K_cw) and the radial power distribution factor (K_rp). The value of constants in these correction factors is different when the heat balance method (HBM) and direct substitution method (DSM) are adopted to predict the experimental results of HTRF. With the 1995 Groeneveld CHF Table and the proposed correction factors, the average relative error is 0.1 and 0.0% for HBM and DSM, respectively, and the root mean square (RMS) error is 31.7% in DSM and 17.7% in HBM for 9852 square array data points of HTRF.     

A scale analysis of the turbulent mixing rate for various Prandtl number flow fields in rod bundles

The turbulent mixing rate is a very important variable in the thermal–hydraulic design of nuclear reactors. In this study, the turbulent mixing rate for the flow through rod bundles is estimated with the scale analysis on the flow pulsation generated by periodic vortices that is pointed out as a main cause of the mixing in rod bundles. Based upon the assumption that turbulent mixing is composed of molecular motion, isotropic turbulent motion (turbulent motion without the flow pulsation), and flow pulsation, the scale relation is derived as a function of P/D, Re, and Pr. The derived scale relation is compared with the published experimental results and shows good agreement. Since the scale relation is applicable to various Prandtl number fluid flows, it is expected to be useful for the thermal–hydraulic analysis of liquid metal coolant reactors as well as moderate Prandtl number coolant reactors.     

Preface: IAEA specialists' meetings on radiation damage units in graphite and ferritic and austenitic steel

A porous body model, new in its application for predicting temperature distributions in wire-wrapped fuel rod assemblies, has been developed. The model developed for thermal transport in wire-wrapped rod bundles is similar in principle to the one which has long been successfully used for heat transfer in fixed beds of packed solids. Although the model is applicable to bundles in forced and mixed (combined forced and free) convection, attention in this paper is confined to bundles operating in forced (negligible natural) convection only. The results obtained from this analysis were found to predict available data with as good a precision as does the more complex analysis.     

The structure of turbulent flow in triangular array rod bundles

A wind tunnel study of fully developed uniform-density turbulent flow through triangular array rod bundles is described. Measurements were made for three tube spacings (

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) over a Reynolds number range of 12 000–84 000. The data include friction factors, local wall shear stresses, and the distributions of mean axial velocity, Reynolds stresses and eddy diffusivities. The secondary flow pattern is from the available evidence.     

Computations of post-trip reactor core thermal hydraulics using a strain parameter turbulence model

Coolant flows in the cores of nuclear reactors consist of ascending vertical flows in a large number of parallel passages. Under post-trip conditions such heated turbulent flows may be modified strongly from the forced convection condition by the action of buoyancy, in particular exhibiting impaired levels of heat transfer with respect to corresponding forced convection cases. The heat transfer performance of these ‘mixed convection’ flows is investigated here using two physically distinct eddy viscosity turbulence models: the recent ‘strain parameter’ (or k––S) model of Cotton and Ismael [A strain parameter turbulence model and its application to homogeneous and thin shear flows. Int. J. Heat Fluid Flow 19 (1998) 326] is examined against the benchmark low-Reynolds-number k– model of Launder and Sharma [Application of the energy-dissipation model of turbulence to the calculation of flow near a spinning disc. Lett. Heat Mass Transfer 1 (1974) 131]. Comparison is made with three sets of heat transfer data for ascending mixed convection flows, and it is demonstrated that both turbulence models are generally successful in resolving the Nusselt number distributions occurring along the lengths of mixed convection flow passages. The mechanisms by which the strain parameter model generates reduced turbulence levels, and hence impaired heat transfer rates, is explored in comparison with a fourth set of experimental data for mixed convection flow profiles.     

Thermal-hydraulic performance analysis of a subchannel with square and triangle fuel rod arrangements using the entropy generation approach

The present paper discusses entropy generation in fully developed turbulent flows through a subchannel,arranged in square and triangle arrays. Entropy generation is due to contribution of both heat transfer and pressure drop. Our main objective is to study the effect of key parameters such as spacer grid, fuel rod power distribution,Reynolds number Re, dimensionless heat power ω, lengthto-fuel-diameter ratio λ, and pitch-to-diameter ratio ξ on subchannel entropy generation. The analysis explicitly shows the contribution of heat transfer and pressure drop to the total entropy generation. An analytical formulation is introduced to total entropy generation for situations with uniform and sinusoidal rod power distribution. It is concluded that power distribution affects entropy generation.A smoother power profile leads to less entropy generation.The entropy generation of square rod array bundles is more efficient than that of triangular rod arrays, and spacer grids generate more entropy.     

Laminar convection in wall sub-channels and transport rates for finite rod-bundle assemblies by superposition

Finite rod-clusters in circular, square and hexagonal shells have been divided along the symmetry lines into a number of interior, wall and corner subchannels. Laminar fluid flow and heat transfer results have been generated for these subchannels with varying pitch-to-diameter and wall distance-to-diameter ratios. Wall shear stress and temperature variations for typical wall subchannels are presented. Friction factor values for finite bundles are then obtained by superposing the results of the subchannels that constitute the bundles. The values so generated are in excellent agreement with previous work in the literature obtained by the method of symmetry sectors. Considerable disagreements were, however, observed in the superposed values of Nusselt number, apparently due to conduction effects.     

Heat transfer, momentum losses and flow mixing in a 61-tube bundle with wire-wrap

Friction factors and heat transfer coefficients were obtained in the laminar and turbulent regions for a 61-tube wire-wrapped hexagonal bundle in a water flow loop. Circumferential static pressure and temperature profiles of tubes, and the flow patterns produced by injection of dye at the periphery of the bundle revealed a strong local effect of the wire-wrap. The increase in heat and momentum transfer resulting from the wire-wrap was more pronounced in the laminar region than in the turbulent region. Correlations for the friction factors and Nusselt numbers were developed from the data and compared with the literature.     

An analytical method of evaluating the effect of rod displacement on circumferential temperature and heat flux distributions in turbulent flow through unbaffled rod bundles

An analytical method of evaluating the circumferential variations of temperature and heat flux fields inside and around a displaced fuel rod in triangular rod bundles in turbulent flow is presented with illustrative examples. The analysis consists mainly of the derivation of the simultaneous solutions of a set of heat conduction equations for fuel, cladding and coolant under the assumption of fully developed flow and heat transfer conditions. The local coolant velocity distribution, which is necessary for deriving the temperature field in coolant, is determined by solving the Navier-Stokes equation and the turbulent mixing of coolant is taken into consideration. The results show how the circumferential variations in the temperature and heat flux fields on the outer surface of the cladding increase the lower the ratio and the larger the fuel rod displacement due to thermal conduction and peripheral coolant flow velocity distribution.     

Developed single phase turbulent flow through a square-pitch rod cluster

An experimental study of developed single phase turbulent flow through a square pitched array of rod bundles is described. Measurements were made at two spacings (p/d = 1.194, 1.107) of the mean velocity distribution and wall shear stress variation, together with the six terms of the symmetrical Reynolds stress tensor. The departure of the turbulent flow structure from axisymmetric pipe flow, particularly in the rod gap region, was found to depend strongly on the (p/d) ratio.