Simulations of turbulence, heat transfer and mixing across narrow gaps between rod-bundle subchannels

Turbulent flow and temperature fields were determined numerically in a rectangular duct containing a heated rod. As the spacing δ between the rod and the duct wall decreased from 0.10D (D is the rod diameter) to 0.03D, coherent turbulent kinetic energy and temperature fluctuations dramatically increased in the gap region, but, for δ = 0.01D, coherent fluctuations essentially disappeared. As δ/D → 0, the frequency of coherent fluctuations decreased and cross-gap mixing weakened, contrary to predictions based on extrapolated available empirical correlations.     

Analytical prediction of friction factors and Nusselt numbers of turbulent forced convection in rod bundles with smooth and rough surfaces

A simple analytical method was developed for the prediction of the friction factor, f, of fully developed turbulent flow and the Nusselt number, Nu, of fully developed turbulent forced convection in rod bundles arranged in square or hexagonal arrays. The friction factor equation for smooth rod bundles was presented in a form similar to the friction factor equation for turbulent flow in a circular pipe. An explicit equation for the Nusselt number of turbulent forced convection in rod bundles with smooth surface was developed. In addition, we extended the analysis to rod bundles with rough surface and provided a method for the prediction of the friction factor and the Nusselt number. The method was based on the law of the wall for velocity and the law of the wall for the temperature, which were integrated over the entire flow area to yield algebraic equations for the prediction of f and Nu. The present method is applicable to infinite rod bundles in square and hexagonal arrays with low pitch to rod diameter ratio, P/D<1.2.     

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.     

CFD investigation of vertical rod bundles of supercritical water-cooled nuclear reactor

The commercial CFD code STAR-CD v4.02 is used as the numerical simulation tool for the supercritical water-cooled nuclear reactor (SCWR). The numerical simulation is based on the real full 3D rod bundles’ geometry of the nuclear reactors. For satisfying the near-wall resolution of y⁺ ≤ 1, the structure mesh with the stretched fine mesh near wall is employed. The validation of the numerical simulation for mesh generation strategy and the turbulence model for the heat transfer of supercritical water is carried out to compare with 3D tube experiments. After the validation, the same mesh generation strategy and the turbulence model are employed to study three types of the geometry frame of the real rod bundles. Through the numerical investigations, it is found that the different arrangement of the rod bundles will induce the different temperature distribution at the rods’ walls. The wall temperature distributions are non-uniform along the wall and the values depend on the geometry frame. At the same flow conditions, downward flow gets higher wall temperature than upward flow. The hexagon geometry frame has the smallest wall temperature difference comparing with the others. The heat transfer is controlled by P/D ratio of the bundles.     

Evaluation of the conduction shape factor with a CFD code for a liquid–metal heat transfer in heated triangular rod bundles

A heat transfer due to conduction through a coolant itself is not negligible in a liquid–metal cooled reactor (LMR). This portion of a heat transfer is frequently described with a conduction shape factor during the thermal-hydraulic design of an LMR. The conduction shape factor, which is highly dependent on a pitch-to-diameter (P/D) ratio, is defined as the ratio of the local conduction heat flux at a gap between two subchannels to the reference heat flux calculated by the averaged subchannel temperatures. The shape factors in heated triangular rod arrays for three different pitch-to-diameter ratios are generated through CFX calculations in the present study. The flow paths of 1.0–2.0 m in length are meshed into 180,000–360,000 volumes depending on the flow velocities. The SSG Reynolds stress model is used as a turbulent model in the calculations. The evaluated data fell between the heated-rod data and the plane-source data obtained by theoretical investigations. The conduction shape factors were found to be independent of the heating pattern of the rod arrays. Based on the evaluated data, a correlation for a liquid sodium coolant is suggested, which will improve the accuracy of the subchannel analysis codes for the thermal-hydraulic design of an LMR. When it is compared with the existing correlations, the suggested correlation is expected to enhance the reliability of the conduction shape factor because the data is evaluated by a more realistic numerical experiment.     

Experimental study of the characteristics of the flow in the first rows of tube banks

This paper presents the experimental study of the flow instabilities in the first rows of tube banks. The study is performed using hot wire anemometry technique in an aerodynamic channel as well as flow visualizations in a water channel. In the wind channel three tube banks with square arrangement and pitch to diameter ratios P/D = 1.26, 1.4 and 1.6 were studied. The Reynolds number range for the velocities measurements, computed with the tube diameter and the flow velocity in the narrow gap between tubes was 7 × 10⁴–8 × 10⁴. Continuous and discrete wavelets were applied to decompose the velocity results, thus allowing the analysis of phenomena in time–frequency domain. Visualizations in a water channel complemented the analysis of the hot wire results. For this purpose, dye was injected in the flow in the water channel with a tube bank with P/D = 1.26. The range of the Reynolds number of the experiments was 3 × 10⁴–4 × 10⁴. The main results show the presence of instabilities, generated after the second row of the tube bank, which propagates to the interior of the bank. In the resulting flow, the three orthogonal components are equally significant. The three-dimensional behavior of the flow is responsible for a mass redistribution inside the bank that leads to velocity values not expected for the studied geometry, according to the known literature. The resulting flow process can be interpreted as a secondary flow which is characteristic of tube banks.     

Analysis of dynamical flow structure in a square arrayed rod bundle

Large eddy simulation (LES) of turbulent flow in a bare rod bundle was performed, and a new concept about the flow structure that enhances heat transport between subchannels was proposed. To investigate the geometrical effect, the LES was performed for three different values of rod diameter over pitch ratio (D/P = 0.7, 0.8, 0.9). The computational domain containing 4 subchannels was large enough to capture large-scale structures wide across subchannels. Lateral flow obtained was unconfined in a subchannel, and some flows indicated a pulsation through the rod gap between subchannels. The gap flow became strong as D/P increased, as existing experimental studies had reported. Turbulence intensity profile in the rod gap suggested that the pulsation was caused by the turbulence energy transferred from the main flow to the wall-tangential direction. This implied that the flow pulsation was an unsteady mode of the secondary flow and arose from the same geometrical effect of turbulence. This implication was supported by the analysis results: two-points correlation functions of fluctuating velocities indicated two length-scales, P-D and D, respectively of cross-sectional and longitudinal motions; turbulence stress in the cross-sectional mean flow contained a non-potential component, which represented energy injection through the unsteady longitudinal fluid motion.     

Numerical investigation of heat transfer in upward flows of supercritical water in circular tubes and tight fuel rod bundles

Heat transfer in upward flows of supercritical water in circular tubes and in tight fuel rod bundles is numerically investigated by using the commercial CFD code STAR-CD 3.24. The objective is to have more understandings about the phenomena happening in supercritical water and for designs of supercritical water cooled reactors. Some turbulence models are selected to carry out numerical simulations and the results are compared with experimental data and other correlations to find suitable models to predict heat transfer in supercritical water. The comparisons are not only in the low bulk temperature region, but also in the high bulk temperature region. The two-layer model (Hassid and Poreh) gives a better prediction to the heat transfer than other models, and the standard k– high Re model with the standard wall function also shows an acceptable predicting capability. Three-dimensional simulations are carried out in sub-channels of tight square lattice and triangular lattice fuel rod bundles at supercritical pressure. Results show that there is a strong non-uniformity of the circumferential distribution of the cladding surface temperature, in the square lattice bundle with a small pitch-to-diameter ratio (P/D). However, it does not occur in the triangular lattice bundle with a small P/D. It is found that this phenomenon is caused by the large non-uniformity of the flow area in the cross-section of sub-channels. Some improved designs are numerically studied and proved to be effective to avoid the large circumferential temperature gradient at the cladding surface.     

Simulation of flow pulsations in a twin rectangular sub-channel geometry using unsteady Reynolds Averaged Navier-Stokes modelling

An unsteady Reynolds Averaged Navier-Stokes (URANS) based turbulence model, the Spalart-Allmaras (SA) model, was used to investigate the flow pulsation phenomena in compound rectangular channels for isothermal flows. The studied geometry was composed of two rectangular sub-channels connected by a gap, on which experiments were conducted by Meyer and Rehme (1994) and were used for the validation of numerical results. Two case studies were selected to study the effect of the advection scheme. The results from the first order upwind advection scheme had clear symmetry and periodicity. The frequency of flow pulsations was under predicted by almost a factor of two. Due to inevitable numerical diffusion of the first order upwind scheme, a second order accurate in space advection scheme was also considered. The span-wise velocity contours, velocity vector plots, and time traces of the velocity components showed the expected cross-flow mixing between the sub-channels through the gap. The predicted kinetic energy in the unsteady velocity fluctuations showed two clear peaks at the edges of the gap. The dynamics of the flow pulsations were quantitatively described through temporal auto-correlations and power spectral functions. The numerical predictions were in agreement with the experiments. Studies on the effect of the Reynolds number and the computational length of the domain were also performed. The numerical results reproduced the relationship between the Reynolds number and the frequency of the flow pulsations. The impact of the channel length was tested by simulating a longer channel with respect to the base case. It was found that the channel length did not significantly affect the numerical predictions. Simulations were also performed using the standard k-? model. While the flow pulsations were predicted with this model, the frequency of the pulsation was in poor agreement with the experimentally measured value.     

Internal structure and interfacial velocity development for bubbly two-phase flow

This paper describes an experimental study of the internal structure of air-water flowing horizontally. The double-sensor resistivity probe technique was applied for measurements of local interfacial parameters, including void fraction, interfacial area concentration, bubble size distributions, bubble passing frequency and bubble interface velocity. Bubbly flow patterns at several flow conditions were examined at three axial locations, L/D = 25, 148 and 253, in which the first measurement represents the entrance region where the flow develops, and the second and third may represent near fully developed bubbly flow patterns. The experimental results are presented in three-dimensional perspective plots of the interfacial parameters over the cross-section. These multi-dimensional presentations showed that the local values of the void fraction, interfacial area concentration and bubble passing frequency were nearly constant over the cross-section at L/D = 25, with slight local peaking close to the channel wall. Although similar local peakings were observed at the second and third locations, the internal flow structure segregation due to buoyancy appeared to be very strong in the axial direction. A simple comparison of profiles of the interfacial parameters at the three locations indicated that the flow pattern development was a continuous process. Finally, it was shown that the so-called “fully developed” bubbly two-phase flow pattern cannot be established in a horizontal pipe and that there was no strong correspondence between void fraction and interface velocity profiles.     

Some problems for bundle CHF prediction based on CHF measurements in simple flow geometries

A comparison of critical heat flux (CHF) fuel bundles data with CHF data obtained in simple flow geometries was made. The base for the comparison was primary experimental data obtained in annular, circular, rectangular, triangular, and dumb-bell shaped channels cooled with water and R-134a. The investigated range of flow parameters (pressure, mass flux, and critical quality) in R-134a was chosen to be equivalent to modern nuclear reactor water flow conditions (p=7 and 10 MPa, G=350–5000 kg (m² s)⁻¹, x_cr=−0.1–1). The proper scaling laws were applied to convert the data from water to R-134a equivalent conditions and vise versa. The effects of flow parameters (p, G, x_cr) and the effects of geometric parameters (D, L) were evaluated during comparison. The comparison showed that no one simple flow geometry can be used for accurate and reliable bundle CHF prediction in wide range of flow parameters based on local (critical) conditions approach. The comparison also showed that the limiting critical quality phenomenon is unique characteristic for each flow geometry which depends on many factors: flow conditions (pressure and mass flux), geometrical parameters (diameter or surface curvature, gap size, etc.), flow obstructions (spacers, appendages, turbulizers, etc.) and others.     

From discovery to recognition of periodic large scale vortices in rod bundles as source of natural mixing between subchannels—A review

The mixing of cooling fluid in rod bundles from one subchannel to another through the gaps between the rods reduces the temperature differences in the coolant as well as along the perimeter of the rods. The phenomenon of natural mixing was first intensively investigated in the 1960s and remains a topic of research up to the present time. The paper describes the main stations on the way to understand the nature of the flow in rod bundles and generally in compound channels with the focus on work performed at Research Center Karlsruhe (FZK).¹Earlier, it was noticed that the mixing rates where higher than could be accounted for by turbulent diffusion alone. For more than 20 years attempts were made to prove experimentally and by code application that secondary flows could account for the measured mixing rates, although the measured secondary flow velocities were much too low. Measurements of the turbulence structure by hot wire anemometry confirmed the existence of cyclic flow pulsations, which had been postulated earlier on the basis of thermocouple measurements. More sophisticated hot wire measurements revealed the nature of these pulsations as periodic, coupled to gap width and Reynolds number. Finally, the extension of the investigation to other compound channel types and flow visualization revealed the true nature of the mixing process as a vortex train moving along the gap between rods or in the narrow part of a compound channel. These findings have been confirmed by LES calculations. Based on these results CFD codes with improved turbulence models calculated successfully the flow in rod bundles including the macroscopic oscillations.     

Axial-flow-induced vibration for a rod supported by translational springs at both ends

An axial-flow-induced vibration model was proposed for a rod supported by two translational springs at both ends in order to evaluate the sensitivity to spring stiffness on the FIV for a PWR fuel rod. For developing the model, a one-mode approximation was made based on the assumption that the first mode was dominant in vibration behavior of the single span rod. The first natural frequency and mode shape functions for the flow-induced vibration, called the FIV, model were derived by using Lagrange's method. The vibration displacements were calculated by both of the spring-supported rod and the simple-supported (SS) one. As a result, the vibration displacement for the spring-supported (50 kN m⁻¹) rod was 15–20% larger than that of the SS rod when the rods are in axial flow of 5–8 m s⁻¹ velocity. The discrepancy between both displacements became much larger as flow velocity increased, and that of the rod having the short span length was larger than that of the rod having the long span length although the displacement value itself of the long span rod was larger than that of the short one. The vibration displacement for the spring-supported rod appeared to decrease with the increase of the spring constant. Since single span beam supported by the two translational springs are focused on in this paper, further study will be needed to reflect more realistic supporting conditions of the PWR fuel rod such as two springs and four dimples and cross or swirling flow caused by the mixing vane of the spacer grid.     

Water flow rate pulsations with a flicker power spectrum in a commercial sodium steam generator

The spectral characteristics of the water flow rate pulsations in the evaporation module of a steam generator with sodium heating in the BN-600 power-generating unit at nominal power are investigated. The results show that the power spectrum of the flow rate pulsations vary inversely as the frequency (flicker pulsations). The appearance of the high-energy low-frequency pulsations of the water flow rate is attributed to a critical heat transfer regime (nonequilibrium phase transition) which is realized in the evaporation module.__________Translated from Atomnaya Énergiya, Vol. 98, No. 2, pp. 105–110, February, 2005.     

Flow structure and bubble characteristics of steam–water two-phase flow in a large-diameter pipe

The flow structure and bubble characteristics of steam–water two-phase upward flow were observed in a vertical pipe 155 mm in inner diameter. Experiments were conducted under volumetric flux conditions of J_G<0.25 m s⁻¹ and J_L<0.6 m s⁻¹, and three different inlet boundary conditions to investigate the developing state of the flow. The radial distributions of flow structure, such as void fraction, bubble chord length and gas velocity, were obtained by horizontally traversing optical dual void probes through the pipe. The spectra of bubble chord length and gas velocity were also obtained to study the characteristics of bubbles in detail. Overall, an empirical database of the multi-dimensional flow structure of two-phase flow in a large-diameter pipe was obtained. The void profiles converged to a so-called core-shaped distribution and the flow reached a quasi-developed state within a relatively short height-to-diameter aspect ratio of about H/D=4 compared to a small-diameter pipe flow. The PDF histogram profiles of bubble chord length and gas velocity could be approximated fairly well by a model function using a gamma distribution and log–normal distribution, respectively. Finally, the correlation of Sauter mean bubble diameter was derived as a function of local void fraction, pressure, surface tension and density. With this correlation, cross sectional averaged bubble diameter was predicted with high accuracy compared to the existing constitutive equation mainly being used in best-estimate codes.     

Fluid-to-fluid modeling of critical heat flux in 4 × 4 rod bundles

Fluid-to-fluid modeling of critical heat flux (CHF) is to simulate the CHF behaviors for water by employing low cost modeling fluid, and the flow scaling factor is the key to apply the technique to fuel bundles. The CHF experiments in 4×4 rod bundles have been carried out in Freon-12 loop in equivalent nuclear reactor water conditions (P=10.0–16.0 MPa, G=488.0–2100.0 kg/m² s, X_cr=−0.20–0.30). The models in fluid-to-fluid modeling of CHF is verified by the CHF data for Freon-12 obtained in the experiment and the CHF correlation for water obtained by Nuclear Power Institute of China (NPIC) in the same 4×4 rod bundles. It has been found that the S.Y. Ahmad Compensation Distortion model, the Lu Zhongqi model, the Groeneveld model and Stevens–Kirby model overpredict the bundles CHF values for water. Then an empirical correlation of flow scaling factor is proposed. Comparison of the CHF data in two kinds of test sections for Freon-12, in which the distance of the last grid away the end of heated length is different, shows that the spacer grid, which is located at 20 mm away from the end of the heated length, has evidently influenced on the CHF value in the 4×4 rod bundles for Freon-12. This is different from that for water, and the need for further work is required.     

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.     

Diffusion of uranium in molybdenum,niobium, zirconium,and titanium

The diffusion of uranium was studied in molybdenum, niobium, zirconium, and titanium. The diffusion coefficients were determined by measuring the over-all activity of the residue of the sample, using the a-radiation of uranium enriched with the U²³⁵ isotope to 90% at temperatures 1500 to 2000° C (molybdenum, niobium) and 915 to 1200° C (zirconium, titanium). The temperature dependence of the diffusion coefficients was given by the equationsD _Mo ^U=7.60.10³ exp (–76 400/RT) cm²/sec;D _Nb ^U=8.90.10^–2 exp (–76 800/RT) cm²/sec;D _Zr ^U=7.77,10^–5 exp (–25 800/RT) cm²/sec;D _Ti ^U=4.90. t0^–4 exp (–29 300/RT) cm²/sec.The considerable differences between the diffusion mobilities and activation energies of molybdenum and niobium on the one hand and zirconium and titanium on the other were probably due to the effects of lattice defects, for example, excess vacancies arising in zirconium and titanium during polymorphic transformations.Translated from Atomnaya Énergiya, Vol. 19, No. 6, pp. 521–523, December, 1965Report read by G. I. Budker at the International Conference on High-Energy Accelerators (Frascati, Italy).     

Local measurement of interfacial area, interfacial velocity and liquid turbulence in two-phase flow

Double sensor probe and hotfilm anemometry methods were developed for measuring local flow characteristics in bubbly flow. The formulation for the interfacial area concentration measurement was obtained by improving the formulation derived by Kataoka and Ishii. The assumptions used in the derivation of the equation were verified experimentally. The interfacial area concentration measured by the double sensor probe agreed well with one by the photographic method. The filter to validate the hotfilm anemometry for measuring the liquid velocity and turbulent intensity in bubbly flow was developed based on removing the signal due to the passing bubbles. The local void fraction, interfacial area concentration, interfacial velocity, Sauter mean diameter, liquid velocity, and turbulent intensity of vertical upward air–water flow in a round tube with an inner diameter of 50.8 mm were measured by using these methods. A total of 54 data sets were acquired consisting of three superficial gas flow rates, 0.015–0.076 m s⁻¹, and three superficial liquid flow rates, 0.600, 1.00, and 1.30 m s⁻¹. The measurements were performed at the three locations: L/D=2, 32, and 62. This data is expected to be used for the development of reliable constitutive relations which reflect the true transfer mechanisms in two-phase flow.     

Single and multiple discharge from a stratified two-phase region through small branches

Experimental data are presented for the mass flow rate and quality during single, dual and triple discharge from a stratified air–water region through small side branches (d=6.35 mm) installed on a semicircular wall. Dimensions of the semicircular wall and branches were chosen such that interaction among the branches is possible under certain flow conditions. All the branches were adjusted to have the same hydraulic resistance (R=1000 (kg m)^−1/2) and for the cases of dual and triple discharge, the same pressure drop ΔP was imposed across all active branches. Tests were conducted at two system pressures P₀=316 and 517 kPa and the pressure drop was varied within the range 40≤ΔP≤235 kPa. Data analysis is presented with emphasis on the effect of wall curvature and also the effect of additional discharges on the flow from a certain branch. The present data can serve as benchmark data for testing numerical safety codes and they should guide future research on the flow from two-phase headers.