Three-Dimensional Simulation of a Simplified Advanced Gas-cooled Reactor Fuel Element

Transcription

1 Three-Dimensional Simulation of a Simplified Advanced Gas-cooled Reactor Fuel Element Amir Keshmiri 1,2 1 School of Mechanical, Aerospace and Civil Engineering (MACE), The University of Manchester, Manchester, M13 9PL, U.K. 2 Department of Engineering and Technology, Manchester Metropolitan University, Manchester, M1 5GD, U.K. Abstract Tel: +44 (0) ; fax: +44 (0) address: a.keshmiri@mmu.ac.uk The present work involves simulations of a simplified three-dimensional representation of the UK fleet of Advanced Gas-cooled Reactors (AGR) fuel element using a 30 sector configuration. The computations were carried out using the v 2 -f formulation which was shown to be one of the most accurate turbulence models in earlier simulations of two-dimensional rib-roughened channels. In the present work main features of the mean flow and heat transfer in the fuel element were identified and discussed. The pressure loss and friction factor were also calculated where good agreement with the experimental correlations was found. Further comparisons were made against simulations of a 2D rib-roughened channel in order to assess the validity and relevance of a 2D approximation approach. It was shown that although a two-dimensional approach is very useful and economical for parametric studies, it does not provide an accurate representation of a 3D fuel element configuration, especially for the velocity and pressure coefficient distributions, where large discrepancies were found between the results of the 2D channel and azimuthal planes of the 3D configuration. Keywords: AGR, CFD, Eddy Viscosity Model, Fuel element, RANS, Rib roughness, v 2 -f. 1. Introduction Advanced Gas-cooled Reactors (AGRs) are the second generation of British gas-cooled reactors which were developed from the Magnox reactor, but which are capable of operating at higher temperatures, thus have higher thermal efficiencies. AGRs have a graphite moderator and use pressurized CO 2 as coolant. The reactor core consists of over 300 fuel assemblies. Eight fuel elements are held together vertically by a tie bar which passes through the centres of the elements to form a fuel assembly. Each fuel element (as shown in Fig. 1) comprises 36 stainless steel fuel pins, which are housed in graphite sleeves. The 36 fuel pins and a central guide tube are supported by a stainless steel grid and two stainless steel braces 1

2 which maintain the spatial arrangement of the fuel pins. The main structural component of the fuel element is the graphite sleeve which supports the pin cluster and the whole of the stringer above it. A typical fuel element assembly weighs approximately 85 kg of which 28 kg is made up by the graphite sleeve. An important feature of AGR fuel pins is the rib-roughening applied to the heat transfer surface which can be seen in Fig. 2. Applying roughness to the fuel pins of a nuclear reactor could potentially result in either an improvement or reduction in performance, depending upon the relative increases in heat transfer and flow resistance. The rib-roughened fuel pins of the UK fleet of AGRs can be divided into the following two categories: Multi-Start Configuration as shown in Fig. 3. In this design 12 helices configured at an angle of approximately 50 o to the horizontal cover the outer surface of the fuel pin. As indicated in Fig. 3 (c), the fuel elements of this design consist of a combination of 36 right- and left-hand fuel pins. The minimum section to be simulated for this configuration would be a 120 o sector (with rotational periodicity), as indicated in the figure. Transverse Configuration as shown in Fig. 4 (and also in Fig. 2). In this design a single-start continuous helix at a small angle of pitch to the horizontal covers the outer surface of the fuel pin. As indicated in Fig. 4 (c), the fuel elements of this design consist of 36 identical fuel pins. Therefore, the minimum section to be simulated for this configuration would be a 60 o sector (with rotational periodicity). The transverse configuration was the first design used in AGRs. However, this configuration was later replaced by the multi-start design following the findings of a number of researchers in this field including White and Wilkie (1967) and Wilkie (1983a, b), who revealed the advantages of using the multi-start configuration in place of its predecessor. In comparing transverse and multi-start designs, Wilkie found that the principal advantage of the multi-start configuration is that the much enhanced coolant mixing of heat due to swirl induced by the ribs reduces across-channel temperature gradients and hence reduces peak temperature. In addition, hotspots in the anti-stacking grooves 1 are eliminated. The good mixing also flattens out the coolant temperature profiles in all channels making the central guide tube cooler. Since the ribs are higher, they are less sensitive to any rib wear, rib oxidation and deposition while their strength is increased. Also, the fuel pin temperatures are less sensitive to pin displacement (due to pin bow). The only certain disadvantage of the 1 Anti-stacking grooves, as shown in Fig. 1, are indentations which are spaced along the length of each fuel pin. For transverse design, the spacing between these grooves is usually about 80 mm. 2

3 multi-start design is the increased neutron absorption, although this could be offset by a small reduction in fuel pin thickness or by reducing the rib width. Regardless of the fuel pin design, generally in a 36-fuel pin passage (representing a complete AGR fuel element), three different sub-channels (defined as the area surrounded by fuel pins and outer casing) exist including square, triangular and wall sub-channels (indicated in Fig. 5). This makes the simulations conducted using simple 2D channels or even triangular or square sub-channels dubious (Chang and Tavoularis, 2007). Therefore, three-dimensional simulations of the fuel element are required to capture possible interactions among different fuel pins and sub-channels. A 3D representation of a fuel element is necessary in order to compute the pressure, temperature and velocity distributions across the whole fuel element, which are in fact crucial for design and safety purposes (Morrison, 2003). Computation of the 3D case can also provide a fuller picture and valuable detailed information concerning the thermal-hydraulics of AGR core coolant flows. Note that in order to simulate a multi-start fuel pin configuration, one would require a long computation domain and an unsteady solution in order to capture the 3D swirling effects and unsteady evolution of possible coherent structures (Hussain, 1986) 2 which can be important in heat transfer calculations. The coherent structures emerge as a result of strong azimuthal flow pulsations associated with large-scale vortices which form quasi-periodically in pairs on either side of the gap between the fuel pins (Chang and Tavoularis, 2007). In the present work a simplified design of an AGR fuel element is chosen and is simulated using a three-dimensional approach and comparison is made against the results of a twodimensional rib-roughened channel as well as correlations. To the author s knowledge, to date there is no numerical simulation of AGR fuel elements and, therefore, the present work represents the first attempt to compute this type of fuel elements. However, in terms of ribroughness design, the simplified configuration studied here is merely an idealized example of an AGR fuel element and does not exist in reality. Moreover, since it was not possible to extract experimental data for these fuel elements, the main conclusions of this work are based on the present CFD results. 2. Background In the past 6 decades, a number of experimental works have been carried out on various designs of AGR fuel elements, mostly by the UK Atomic Energy Authority (UKAEA) and the 2 A coherent structure is a connected turbulent fluid mass with instantaneously phase-correlated vorticity over its spatial extent. 3

4 British Energy. Most of these experiments were based on measurements of heat transfer and pressure drop from annuli with heated roughened inner and unheated smooth outer surface. As there are currently no numerical publications on AGR fuel elements, only the available experimental works are briefly reviewed here. One of the earliest experiments in this field was carried out by Wilkie (1966). In a singlestart transverse rib-roughened fuel pin, Wilkie tested the effects of varying the rib height (k/d e ), rib pitch-to-height ratio (P/k), rib profile, rib width, rib helix angle and the Reynolds number. In a similar experiment, White and Wilkie (1967) measured heat transfer and friction factor over a range of Reynolds number (Re = 60, ,000) for eight tubes roughened externally by square-section wires in the form of single-start and multi-start rib design. White and Wilkie studied the effects of varying the P/k ratio and rib helix angle. In fact, the findings of Wilkie (1966) and White and Wilkie (1967) played a major role in making the decision of replacing the single-start (transverse) fuel pins by the multi-start ones. A few years later, Wilkie (1983a, b) carried out new experiments to compare the singleand multi-start designs to a design with longitudinal fins with approximately triangular crosssection (Fig. 6) and listed some of the advantages and disadvantages of each design. Some of the works done by the British Energy on the AGR fuel elements were reported in Pirie (1987), Morrison (2003) and Gotts and Xu (2006). Pirie (1987) carried out tests to measure the pressure drop of a fuel element containing multi-start fuel pins. The measurements were used to calculate the resistance of a complete stringer. Comparison was also made with the recommended value of resistance of the same fuel element but with transverse fuel pins. One of the main findings of this work was that the flow resistance of the stringers with multi-start and transverse fuel pins was found to be nearly the same for most of the flow range with the maximum difference of 2.2%. In an attempt to optimize the fuel resistance of the multi-start fuel elements, Morrison (2003) calculated the effects of: 1) Slightly widening the bore of the graphite sleeve, 2) Replacing the fabricated streamlined brace with the machined streamlined brace and 3) Using thinner and optimized fuel pin walls. Calculations of Morrison showed that, by incorporating the above changes to the fuel element, the fuel resistance can be reduced by 19%. However, in practice, only the last two of these changes were made to the existing AGRs (Gotts, 2009). Later, Gotts and Xu (2006) investigated the problem of carbon particle deposition on fuel pins with the aim of understanding the nature of the problem and recommending how it can be allowed for in safety assessments. Gotts and Xu carried out their analysis based on an 4

5 empirical equation developed by Mantle and his colleagues at British Energy. This empirical equation calculates the Heat Transfer Impairment (HTI) through the following equation: 1.4 m& ( δ 0.015δ ) HTI = (1) λ where λ is the deposit conductivity [W/mK], δ the deposit thickness between ribs [µm] and m& the mass flow rate [kg/s]. The above equation is valid for the following range: λ = [W/mK] δ = [µm] m& = [kg/s] While all the works reviewed so far are concerned with AGRs, there are several experimental and numerical contributions in the literature which have been carried out on other nuclear reactor designs including Pressurized Water Reactors (PWRs), Pressurized Heavy Water Reactors (PHWRs) and various types of Fast Breeder Reactors (FBRs). A brief overview of some of the numerical works carried out on these types of reactors using 3D modelling is presented here. Chang and Tavoularis (2007) carried out CFD simulation on the fuel bundle of a typical PWR. This was the first attempt to use unsteady RANS approach (URANS) combined with a Reynolds Stress Model (RSM) in these types of simulations. Chang and Tavoularis solved the isothermal flow in a 60 sector of a 37-rod bundle and reported time average velocity and fluctuations as well as high correlations between turbulent structures in the entire geometry. Abbasian et al. (2009) studied the turbulent flow inside a 43-rod bundle of a PHWR (CANada Deuterium-Uranium - CANDU) both experimentally and numerically. In their CFD simulations, a 60 sector was chosen and three different CFD methods were employed: 1) Large Eddy Simulation (LES), 2) Detached Eddy Simulation (DES) and 3) RSM. All three CFD methods lead to nearly identical outcome when compared to their experimental results. However, for instantaneous flow analysis, LES was shown to be more accurate. Recently, Péniguel et al. (2010) investigated the thermal-hydraulic of a wire-wrapped fuel bundle of a typical FBR with liquid metal coolant. In this design, the fuel rods are wrapped with a wire spacer which follows a helical pattern around the rod s axis to avoid collision between adjacent pins and to reduce vibration. In Péniguel et al. (2010) two different configurations with 7 and 19 fuel rods were simulated using the k-ε model and RSM. The near-wall modelling was based on the Scalable Wall Function (Grotjans and Menter, 1998). Friction factor and Nusselt number were compared against the experimental data, where both models had relatively good agreement. Due to lack of experimental data, Péniguel et al. 5

6 suggested to use low-reynolds-number RANS models or LES to simulate this type of problems more accurately. 3. Physical and Numerical Formulations 3.1. Mean Flow Equations Adopting Cartesian tensor notation, the mean flow equations read as follows: Continuity: U x j j = 0 (2) Momentum: U i 1 p U i U j = + ( ν + ν t ) x j ρ xi x j x j (3) Energy: U j T x j = x j ν Pr ν t T + σ t x j where, following standard modelling practice, e.g. Launder and Sharma (1974), the turbulent Prandtl number is set to a constant value, σ t = The v 2 -f turbulence model The turbulence model to be used in the present work is the (4) v 2 f model (or simply v 2 -f ) which was first proposed by Durbin (1991) (v is the wall-normal fluctuating velocity). The v 2 - f model is of Eddy Viscosity type. In Eddy Viscosity Models (EVMs) the Reynolds stress tensor is modelled by the Boussinesq hypothesis which assumes the turbulent stress to be proportional to the mean strain rate, via an eddy viscosity, ν t, that is 2 uiu j = 2ν t Sij kδ ij (5) 3 where δ ij is the Kronecker delta (which is equal to 1 when i and j are equal, and zero otherwise) and k is the turbulent kinetic energy. The mean strain rate tensor, S ij, is defined as S ij 1 U U i = + 2 x j xi The turbulent (eddy) viscosity is defined as j (6) v k 2 ν t = C µ k T s (7) 6

7 Damping in the v 2 -f scheme is related to the diminution of the v 2 /k that occurs as a wall is approached (see Launder, 1986, for example). The large-scale turbulence timescale is truncated on the Kolmogorov scale, 0.5 k v Ts = max, CkT ε ε (8) The v 2 -f formulation does not include any direct dependence on wall distance. In fact, in an earlier version of the general form of Iaccarino (2001), Durbin (1995) included a y-dependent function in the production term of the ε-equation, but that was subsequently modified by Iaccarino to eliminate wall distance dependence. In addition, an elliptic equation for the redistribution term in the v 2 equation, f 22, is included to account for near-wall and non-local effects. STAR-CD uses a variant of Durbin (1995) version of the v 2 -f model. The governing equations of this particular model are given in Iaccarino (2001) as follows: Dk Dt = P k + x j ν t ν + σ k k x j ε (9) D z ε Cε1 ν t ε Cε = Pk + ν 2 ε Dt Ts x + j σ (10) ε x j Ts D v Dt 2 2 ν t v 2 ε = ν + k f 22 6v x + j σ (11) k x j k where (1 C1 ) 2 v Pk v L f22 f22 = C 2 5 (12) T s 3 k k kts is the Laplacian operator and [ k ] 3/ / ε, C ( ν / ε ) 1/ L = C max (13) L η 2 ε 1 = k / v (14) C z + The coefficients of the v 2 -f model in STAR-CD are given in Table 1. In earlier works by the author and their colleagues, the v 2 -f model as implemented in STAR-CD has been checked against the same model implementation in an industrial code (Code_Saturne) and very similar results were obtained (Keshmiri et al., 2008a; Keshmiri et al., 2008b). 4. Case Description 7

8 In the present work, a simplified fuel element consisting of fuel pins with parallel and uniformly spaced ribs is simulated (Fig. 7). In the present configuration, a simple square rib profile is chosen since it was shown by Keshmiri et al. (2009) and Keshmiri and Gotts (in press) that a rib profile has an insignificant effect on the mean flow and heat transfer. Table 2 compares the dimensions of the present domain to the real dimensions of the transverse and multi-start configurations provided by the British Energy (Gotts, 2008). In Table 2 it is seen generally both the transverse and multi-start configurations have the same dimensions, except for those corresponding to their rib profiles, pitch and height. Taking advantage of azimuthal symmetry of a complete fuel element in order to reduce the computational requirements, the present simulation employed a symmetry boundary condition on the two azimuthal faces and computed the flow in only a 30 sector of the circular cross section (Fig. 8). (Note that for the present configuration if a rotational periodicity was to be used at the azimuthal faces, a 60 sector would have been required, even though the results of both cases would still be the same.) Streamwise periodic boundary condition maintaining a constant mass flow rate and a constant bulk temperature was applied at the top and bottom faces of the domain. The computational domain was chosen to be of length 2P (i.e. 4.2 mm) in the streamwise direction to include 2 complete ribs. The latter length combined with a streamwise periodic boundary condition was found to be sufficiently long for the attainment of fully developed flow. It is worth noting that in applying periodic boundary conditions, special treatment of the transport equations is required; by adding extra source terms to the momentum and energy equations, the streamwise pressure drop and temperature increase due to heating are accounted for. The reader is referred to the work done by Liou et al. (1993) for further details on the implementation of the periodic boundary condition in the present work. Temperature in the present computations is solved as a passive scalar. The thermal boundary conditions at all ribbed walls (i.e. fuel pin numbers 1-5 in Fig. 9) consist of the same uniform wall heat flux, while both the guide tube and outer graphite sleeve faces were set as adiabatic walls. Dimensions of the present configuration are given in Fig. 10. The results presented in this work were generated using the v 2 -f model (see Section 3.2, above) which was found to be generally the most accurate turbulence model in the precursor 2D simulations (Keshmiri et al., 2009; Keshmiri and Gotts, in press). Since this is a low- Reynolds-number turbulence model, the grid was generated so as to be very fine near the wall. After carrying out a series of mesh sensitivity tests, an unstructured hybrid mesh was chosen (shown in Fig. 11) with approximately 821,000 cells. The mesh is composed of 8

9 prismatic elements for the near-wall (the wall-adjacent cell typically extends only to y + 0.5) and polyhedral elements for the core regions. The present mesh was generated using STAR- CCM+ Version (CD-Adapco, 2008). Further information on how the governing equations are discretised and treated in unstructured grids in STAR-CD can be found in its user manual (CD-Adapco, 2006) as well as Patankar (1980) and Warsi (1981). All computations were carried out using the commercial code, STAR-CD Version 4.02 (CD-Adapco, 2006). The Reynolds number based on hydraulic diameter (defined below) was fixed at Re = 30,000 and Prandtl number was set to Pr = All fluid properties were assumed to be constant. The momentum and turbulence transport equations were discretized using first-order upwind differencing scheme. The passive scalar transport equation (i.e. energy equation) was discretized using the Monotone Advection and Reconstruction Scheme (MARS) (CD-Adapco, 2006). The SIMPLE algorithm was adopted for pressurevelocity correction. All the present simulations were carried out using a steady-state algorithm and the convergence criterion was set to The hydraulic diameter of the present geometry is defined as D e 4A = (15) P T where A is the free flow area and P T the total wetted perimeter (Table 2). The free flow area for a 30 sector is calculated as [( D ) (36 D ) ( D )] A = α π s p g (16) 4 and the total wetted perimeter is given by P T = α 36π Dp + π ( Ds + Dg ) (17) rough perimeter smooth perimeter where α =1/12 for the present 30 sector. The rough perimeter is that of the fuel pins, while the smooth perimeter corresponds to the guide tube and graphite sleeve. Morrison (2003) has suggested an improved definition of the wetted perimeter in which the perimeters are weighted according to their friction factors, thus giving more weight to the pin roughened surfaces: P T = α 36π D + + p 0.2π ( Ds Dg ) (18) rough perimeter smooth perimeter 9

10 In the present work, however, the wetted perimeter was calculated according to Eqn. (17) in order to be consistent with earlier 2D analyses reported in Keshmiri et al. (2009) and Keshmiri and Gotts (in press). 5. Results 5.1. Three-Dimensional Results In this section, results are presented in the form of contours of the normalized streamwise velocity, temperature and pressure obtained at streamwise mid-plane (Fig. 12) and four other planes along one of the azimuthal faces (hereafter referred to as the azimuthal planes ; Fig. 13). Note that in all the contours presented in this section, the flow direction is from bottom to top of page. Fig. 14 shows the magnitude of the normalized streamwise velocity at the mid-plane (presented with two different scales for more clarity). As one would expect, local maxima (U/U b 1.2) form in the centres of all sub-channels, while the flow is slowed down near fuel pins, guide tube and graphite sleeve. As was mentioned earlier, three different sub-channels including triangular, square and wall sub-channels can be identified in Fig. 14. The highest local velocity maxima is observed to occur in the square sub-channel between fuel pins in the middle and outer rings, followed by that in the wall sub-channel between the two fuel pins in the outer ring. The local maxima surrounding the central guide tube, however, has an average velocity magnitude of approximately 0.8 which is about 30% lower than that found in the other sub-channels. It can also be seen that excluding boundary layers, the variation of the streamwise velocity is relatively mild in the sub-channels. The inset to Fig. 14 provides an overview of the streamwise velocity distribution over the whole computational domain. Contours of the normalized streamwise velocity at the azimuthal planes are shown in Fig. 15. The first picture to emerge from comparing the contours in Fig. 15 is that the maximum velocity magnitude in Plane2 is the largest compared to the other 3 planes since it has the largest channel height, H and consequently the lowest k/h ratio (this will be further discussed in connection with Fig. 23 and Fig. 24, below). It should be noted that the magnitude of the average streamwise velocity in each plane is not only a function of plane height ( y), but is also significantly affected by the value of local maxima in the near sub-channel. This is why the velocity contours in Fig. 15 (b) and (c) are different in spite of having similar plane height (see Table 3 for the dimensions of the azimuthal planes). Within the inter-rib cavity, however, all 4 planes show similar trends, indicating a recirculation bubble downstream of the first rib. 10

11 Attention is turned next to the thermal-field results. A contour of the relative temperature (T-T ref ) at the mid-plane is shown in Fig. 16. Iso-thermal lines are also plotted in the figure. It is seen that since in the inner rings the gaps between different fuel pins are smaller, much higher temperatures and lower streamwise velocities occur in these regions, resulting in fuel pin numbers 3, 2 and 4 having the highest temperature levels. Clearly, the lowest temperatures can be found around the guide tube and graphite sleeve, at which adiabatic wall boundary condition is imposed. The inset to Fig. 16 shows a temperature contour of the whole computational domain and it is seen how the temperature levels vary between the fuel pins in the inner and outer rings. Temperature contours at the azimuthal planes are shown in Fig. 17. Each plane shows a considerably different range of temperatures which depends on three main factors: 1) The type of thermal boundary condition imposed on the plane, 2) The position of the surrounding fuel pin within the fuel element (e.g. in the inner, middle or outer rings) and 3) The magnitude of the average streamwise velocity in the plane. As was mentioned above, the third factor itself depends on the local velocity maxima in the near sub-channel and the k/h ratio. In Fig. 17 it is clear that Plane1 has the lowest temperature levels due to an adiabatic wall boundary condition being imposed at the graphite sleeve, in addition to a relatively large distance between the two fuel pins in the outer ring. The importance of the latter can be better understood by comparing Fig. 17 (a) and (d), where it is seen that much higher temperature levels are obtained in Plane4 despite both planes having similar thermal boundary condition and streamwise velocity magnitudes. Results of the relative pressure are next considered. Fig. 18 shows a contour of the relative pressure at the mid-plane. From the values of the relative pressure, it can be seen that pressure at regions further away from the ribbed walls is barely affected by the ribs i.e. P-P ref 0. The inset to Fig. 18 shows the pressure contour from a different angle, in which high- and lowpressure regions near the ribs are more visible. A better representation of the pressure variations near the ribs is given in Fig. 19 where the relative pressure at the azimuthal planes is shown. It can be seen that all four planes show very similar patterns which include high-pressure zones upstream and low-pressure zones downstream of the ribs. From design point of view, it is of great interest to compute the friction factor (or pressure loss) in the present configuration and compare it against the experimental data or correlations. The friction factor (sometimes referred to as Fanning friction factor ) here is linked to the pressure drop ( p) through the following equation: 11

12 p. De f = 2 (19) 2ρU L b where D e is the hydraulic diameter defined in Eqn. (15). The friction factor for an equivalent smooth geometry, f 0 may be obtained through the Blasius equation: 0.25 f = 0. Re (20) The friction factor values obtained for the present geometry can also be compared against the correlations of Ravigururajan and Bergles (1996). These correlations are based on a number of experimental data obtained for a wide range of roughness and flow parameters. Several factors are taken into account in defining these correlations including Reynolds number and various geometrical variables. For the present case, the Ravigururajan and Bergles ( RB ) correlation for calculating normalized friction factor is simplified to (21) The values of the parameters given in the above equation can be found in Table 2. Table 4 shows the friction factor magnitude obtained for the present 3D configuration and is compared against friction factors found using the Blasius equation and Ravigururajan and Bergles correlation. It can be seen that the friction factor (and similarly the pressure drop) in the simulated 30 sector of an AGR fuel element is approximately 4.3 times higher than its equivalent geometry with smooth fuel pins. From Table 4, it can also be seen that the normalized friction factor (f/f 0 ) obtained for the current AGR fuel element is in good agreement with the normalized friction factor calculated by the Ravigururajan and Bergles correlation (f RB /f 0 ) Comparison against a Two-Dimensional Simulation In the previous section, the results of the 3D test case were reported qualitatively by showing contours of normalized streamwise velocity, temperature and pressure. In this section, the aim is to compare the results obtained using the present 3D configuration with a 2D channel simulation. This comparison would be an assessment on the validity and relevance of a 2D channel approximation used as a simplified representation of 3D flow problems (see Keshmiri et al., 2009; Keshmiri and Gotts, in press, for example). 12

13 The first step for carrying out a 2D simulation here is to choose one of the azimuthal planes of the present 3D configuration and then run a separate 2D simulation for that particular plane. As shown in Fig. 20 the azimuthal face between fuel pin numbers 1 and 2 (which contains Plane2 ) is selected. In order to generate a 2D mesh, the selected face is extruded in the spanwise direction (z-direction) to have 1 cell thickness. Symmetry boundary condition is then imposed on the spanwise faces. Other boundary conditions used are also indicated in Fig. 20. The present 2D grid would now have the same geometrical properties (i.e. the same mesh resolution and P/k and k/h ratios) as Plane2 defined earlier in Fig. 13. This grid consists of 1,800 cells and the wall-adjacent cell extends only to y The computations are then carried out on this grid using STAR-CD Version 2.04 with the same numerical inputs (including the same turbulence model i.e. the v 2 -f model) as in the 3D case. Some of the features of the cases compared in this section are listed in Table 3. Note that, as can be seen in Table 3, the Reynolds number based on hydraulic diameter is different for the 2D simulation, that is due to the fact that the hydraulic diameter of the 2D channel is smaller (D e = 2H = ) compared to that of the present 3D case (D e = ), while the value of the bulk velocity, U b for both configurations are kept the same. Fig. 21 shows a contour of the streamwise velocity obtained using the present 2D channel. Comparing the results of Fig. 21 and Fig. 15, it can be seen that while similar patterns are present between 2D and 3D results, the maximum velocity magnitude predicted by the 2D mesh is generally higher than that found at the azimuthal planes of the 3D case (this will be discussed further below in connection with Fig. 23 and Fig. 24). Fig. 22 shows the normalized streamwise velocity at one-tenth of the rib height (y/k = 0.1). Two recirculation regions, represented by two negative velocity peaks are evident in all cases. All four azimuthal planes have similar recirculation bubbles, while the magnitude of the primary recirculation bubble obtained by the 2D case is nearly twice of that found by Plane2. In addition, the velocity distributions found by azimuthal planes 1-3 indicate that the flow remains reversed at this elevation. The velocity distributions for the 2D channel and Plane4, however, suggest that the flow may reattach within the cavity; this is indicated by a small region between x/k where U/U b 0. Results for the streamwise velocity profiles over the rib-top (x = 0) and at the middle of the inter-rib cavity (x/k = 3.5) are shown in Fig. 23 and Fig. 24, respectively. It can be seen that the velocity magnitude of the 2D channel is generally higher than that of the azimuthal planes (the maximum of 20% difference compared to the distributions of Plane2). As was discussed earlier, the velocity magnitude of the azimuthal planes depends on both the magnitude of the 13

14 velocity maxima in the near sub-channel and the k/h ratio. This has resulted in Plane2 and Plane4 having respectively, the highest and lowest velocity magnitudes compared to the other azimuthal planes. Furthermore, the inset to Fig. 24 shows more clearly the recirculation regions (represented by U/U b < 0). It can be seen that except for Plane1, the recirculation length at the middle of the inter-rib cavity is relatively similar for all cases. The recirculation length of Plane1, however, is the largest followed by that of Plane4. It is noted that the single-side ribbed planes (i.e. Planes 1 and 4) show larger recirculation regions compared to the double-side ribbed planes (i.e. Planes 2 and 3). The size of the recirculation bubble is also affected by the k/h ratio. In addition, the results shown in the inset of Fig. 24 are also consistent with the magnitude of the streamwise velocity in Fig. 22 at x/k = 3.5. Fig. 25 shows the wall-normal velocity distributions at rib height (y/k = 1). It can be seen that while the magnitude of the wall-normal velocity is similar for all four azimuthal planes, the 2D channel generally returns higher velocity magnitudes especially within the primary recirculation region (x/k = ). The discrepancies between the results of the 2D channel and azimuthal planes in Fig. 25 are in part related to the differences in the streamwise velocity magnitude within the inter-rib cavity (seen in Fig. 22 to Fig. 24). The discrepancies found above (in Fig. 22 to Fig. 25) between the mean flow results of the 2D channel and Plane2 of the 3D case are mainly related to the 3D effects and spanwise velocity component which is non-zero in the azimuthal planes. In the case of a 2D channel, however, there is no velocity component in the spanwise direction due to a symmetry boundary condition being imposed on both spanwise faces. Attention is turned next to the results of the relative temperature. Fig. 26 shows a contour of the relative temperature obtained using the 2D grid. Comparing this figure with the contours obtained for the azimuthal planes (Fig. 17) reveals a considerably different heat transfer levels between 2D and 3D results (note that the scales are different in Fig. 17 and Fig. 26). It is seen that the 2D configuration has approximately 60% lower temperature levels compared to Plane2, due to higher streamwise velocity magnitudes in the 2D configuration (see Fig. 23 and Fig. 24) which results in higher turbulence mixing and consequently higher heat transfer levels. Fig. 27 compares the distributions of the local Nusselt number, Nu ( = q& D /[ λ( T T )] ). e w b between the ribs for both 2D and 3D configurations. It is seen that Plane1 has the highest levels of heat transfer (since T is small) which is again mainly due to the effects of imposing 14

15 adiabatic wall boundary condition at the graphite sleeve. As could be expected from the contours presented in Fig. 17, Plane3 has the lowest levels of heat transfer due to the plane being heated from both sides (from fuel pin numbers 2 and 3) and to a lesser degree due to slightly lower streamwise velocity magnitude in this plane. An interesting picture to emerge from Fig. 27 is the excellent agreement between the results of the 2D channel and Plane2. It should be noted, however, that such good agreement is obtained in spite of having different temperature levels (compare Fig. 26 to Fig. 17b). This is due to the fact in the 2D simulation, the smaller temperature difference (T w T b ) in the denominator of the Nusselt number equation is compensated by the smaller hydraulic diameter in its numerator i.e. such good agreement between the results of the 2D channel and Plane2 should be viewed as being largely fortuitous. Results of the relative pressure are next discussed. Fig. 28 shows a contour of the relative pressure obtained using the 2D channel. It can be seen that while the patterns of the pressure distributions are broadly similar in Fig. 28 and Fig. 19 (i.e. high-pressure zones upstream and low-pressure zones downstream of the ribs), the pressure levels especially downstream of the ribs are much lower in the 2D channel compared to the azimuthal planes. This can be better quantified in the next figure. Inter-rib pressure distributions represented by the pressure coefficient, C p 2 ( = ( p p ref )/(0.5ρU b ) ) are shown in Fig. 29 (note that C p in all cases is offset to zero at the middle of the cavity). It can be seen that all four azimuthal planes of the 3D configuration have similar distributions. However, the pressure at the downstream face of the first rib is much lower in the 2D channel compared to the azimuthal planes, resulting in much higher form drag in the 2D case. This is due to the 2D channel having a higher streamwise velocity magnitude (Fig. 24) within the cavity in comparison with the azimuthal planes. In the inset to Fig. 29 the pressure coefficient distributions of the azimuthal planes are re-plotted but now C p is offset to the same value at x/k = 0.5 for all four cases. It is seen that the pressure difference within the cavity has the same order as the streamwise velocity magnitude (Fig. 24) i.e. higher the velocity, higher the pressure at the upstream face of the rib. Therefore, Plane2 and Plane4 have respectively, the maximum and minimum pressure differences within the inter-rib cavity. 6. Concluding Remarks 15

16 In the present work, a three-dimensional simulation of a simplified design of a fuel element employed in the UK fleet of Advanced Gas-cooled Reactors was presented using a 30 sector geometry. The computations were undertaken using the commercial CFD package, STAR-CD Version All the simulations were carried out using the v 2 -f formulation. The following conclusions could be drawn from the present work: The results of the streamwise velocity in the 3D case showed that local maxima form in the centres of all sub-channels, while the flow is slowed down near fuel pins, guide tube and graphite sleeve. The results of the thermal field showed that the maximum temperature generally occurs around fuel pins, especially near fuel pins in the inner and middle rings, where the gap between the fuel pins is relatively small. The lowest temperatures were found to be near the guide tube and graphite sleeve, at which adiabatic wall boundary condition was imposed. The normalized friction factor calculated for the present configuration was in good agreement with that found using the correlations of Ravigururajan and Bergles. By comparing azimuthal planes of the 3D case against the results of a 2D configuration (which had the same dimensions as Plane2), it was found that magnitudes of the streamwise and wall-normal velocities were generally higher in the 2D simulation. Pressure difference within the inter-rib cavity was also significantly higher in the 2D channel compared to the azimuthal planes of the 3D configuration. However, heat transfer levels of the 2D plane were found to be in quite close agreement with the results of Plane2, even though this agreement was shown to be largely fortuitous. Although a 2D approach can be extremely useful and economical for parametric studies, it does not provide an accurate representation of a 3D fuel element configuration, especially for the velocity and pressure coefficient distributions, where large discrepancies were found between the results of the 2D channel and azimuthal planes of the 3D case. The main source of discrepancy between the results of the 2D channel and Plane2 of the 3D case was related to the 3D effects which can only be captured by a 3D configuration. (While the 3D configuration was shown to produce better results when compared to the 2D computations, it has to be noted that the accuracy of the 3D results can only be evaluated by comparison against experimental measurements done on the AGR fuel element, which are currently not available.) The results presented in this work have important implications that deserve consideration in the analyses of nuclear reactor operation and safety. It appears that 16

17 conventional CFD methods including RANS are attractive alternatives to experiments, at least in the stages of proof-of-concept and optimization of new fuel element designs. Although the configuration studied in this work was somewhat idealized, the present approach could be followed and applied in simulating more realistic and complex designs such as the multi-start configuration. Acknowledgements The author is grateful to UK Engineering and Physical Sciences Research Council for funding under grant EP/C549465/1 and for the PhD+ award. Mr. J. Gotts and Dr. S.A. Fairbairn of British Energy kindly supplied information concerning AGR design and operating conditions. The author is also pleased to acknowledge the value of discussions with his colleagues Dr. M.A. Cotton, Dr. Y. Addad and Professor D. Laurence. Nomenclature A Cross-sectional area of the channel c f Local friction coefficient c p Specific heat coefficient at constant pressure C p Pressure coefficient, (p s P ref )/(0.5ρU 2 b ) D Diameter D e Hydraulic diameter, 4A/P T f Friction factor H Channel height k Height of the rib or turbulent kinetic energy L Length scale q& D λ( T T ) Nu Nusselt number, [ ] p P Pressure Pitch e / w b P k Rate of shear production of k, u u ( U / x ) p s P T Static pressure at the wall Total wetted perimeter Pr Prandtl number, c pµ / λ q& Wall heat flux Re Reynolds number, U / ν b D e 2 Re t Turbulent Reynolds number, k /( νε) S ij T T s U i,u i Mean strain rate tensor Temperature Turbulent timescale Mean, fluctuating velocity components in Cartesian tensors i j i j 17

18 U τ Friction velocity, ( / ρ) τ w 1/ 2 x,y Streamwise and wall-normal coordinates y + Dimensionless distance from the wall, yu τ /ν Greek Symbols δ Carbon deposition thickness δ ij Kronecker delta ε Dissipation rate of k λ Thermal conductivity µ Dynamic viscosity ν Kinematic viscosity, µ/ρ ν t Turbulent viscosity ρ Density σ t Turbulent Prandtl number Wall shear stress τ w Subscripts b Bulk ref Reference t Turbulent w Wall Additional symbols are defined in the text. References Abbasian, F., Yu, S.D., Cao, J., Experimental and numerical investigations of threedimensional turbulent flow of water surrounding a CANDU simulation fuel bundle structure inside a channel. Nuclear Eng. Design 239, CD-Adapco, STAR-CD Methodology, Version CD-Adapco, STAR-CCM+ User Guide, Version Chang, D., Tavoularis, S., Numerical simulation of turbulent flow in a 37-rod bundle. Nuclear Eng. Design 237, Durbin, P.A., Near-wall turbulence closure modeling without damping functions. Theoret. Comput. Fluid Dynamics 3, Durbin, P.A., Separated flow computations with the k-ε-v 2 model. AIAA J. 33, Gotts, J., Personal communication. Gotts, J., Personal communication. Gotts, J., Xu, B., An investigation of heat transfer impairment modelling arising from fuel pin carbon deposition, British Energy Generation Limited, Engineering Division, Report E/REP/BBDB/0057/AGR/05, RAWG/P(05)08. 18

19 Grotjans, H., Menter, F.R., Wall function for general application CFD codes, in Computational Fluid Dynamics, Proc. Fourth European CFD Conf. ECCOMAS, Wiley, Chichester, England. Hussain, A.K.M.F., Coherent structures and turbulence. J. Fluid Mechanics 173, Iaccarino, G., Predictions of a turbulent separated flow using commercial CFD codes. J. Fluids Eng. 123, Keshmiri, A., Addad, Y., Cotton, M.A., Laurence, D.R., Billard, F., 2008a. Refined eddy viscosity schemes and large eddy simulations for ascending mixed convection flows, Proc. 4th Int. Symp. on Advances in Computational Heat Transfer, 'CHT08', Paper CHT , Marrakech, Morocco. Keshmiri, A., Cotton, M.A., Addad, Y., Laurence, D.R., Thermal-hydraulic analysis of rib-roughened fuel pin performance in gas-cooled nuclear reactors, Proc. 6th International Symposium on Turbulence, Heat and Mass Transfer, Rome, Italy, pp Keshmiri, A., Cotton, M.A., Addad, Y., Rolfo, S., Billard, F., 2008b. RANS and LES investigations of vertical flows in the fuel passages of gas-cooled nuclear reactors, Proc. 16th ASME Int. Conf. on Nuclear Engineering, ICONE16, Paper ICONE , Orlando, Florida, USA. Keshmiri, A., Gotts, J., in press. Thermal-hydraulic analysis of four geometrical design parameters in rib-roughened channels. Numerical Heat Transfer; Part A: Applications. Launder, B.E., Low-Reynolds-number turbulence near walls, Dept. of Mechanical Engineering, UMIST (now School of MACE, The University of Manchester), Report TFD/86/4, Manchester, UK. Launder, B.E., Sharma, B.I., Application of the energy dissipation model of turbulence to the calculation of flow near a spinning disc. Lett. Heat Mass Transfer 1, Liou, T.M., Hwang, J.J., Chen, S.H., Simulation and measurement of enhanced turbulent heat transfer in a channel with periodic ribs on one principal wall. Int. J. Heat Mass Transfer 36, Morrison, A.M., Task team F phase 3: Fuel resistance for increased sleeve bore and machined brace, British Energy Generation Limited, Engineering Division, Report E/REP/BBDB/0002/AGR/03, RAWG/P(03)02. Patankar, S.V., Numerical Heat Transfer and Fluid Flow, Hemisphere, Washington, D.C. Péniguel, C., Rupp, I., Juhel, J., Guillaud, M., Gervais, N., Rolfo, S., Three dimensional conjugated heat transfer analysis in Sodium Fast Reactor wire-wrapped fuel assembly, International Congress on Advances in Nuclear Power Plants (ICAPP), San Diego, CA, USA. Pirie, M.A.M., Pressure drop measurements on nominal 7.5 inch diameter stage 2 CAGR fuel elements containing CE05 multi-start fuel cans, Report TPRD/B/PS/589/M88, HTSG/P(87)27, FADC/P

20 Ravigururajan, T.S., Bergles, A.E., Development and verification of general correlations for pressure drop and heat transfer in single-phase turbulent flow in enhanced tubes. Experimental Thermal and Fluid Science 13, Warsi, Z.U.A., Conservation form of the Navier-Stokes equations in general non-steady coordinates. AIAA Journal 19, White, L., Wilkie, D., The heat transfer and pressure loss characteristics of some multistart ribbed surfaces, United Kingdom Atomic Energy Authority, TRG Report 1504 (W). Wilkie, D., Forced convection heat transfer from surfaces roughened by transverse ribs, Third International Heat Transfer Conference, 1-19 August American Institute of Chemical Engineers, Paper 1, Chicago, USA. Wilkie, D., 1983a. Alternative heat transfer surfaces for AGR fuel pins, Proc. Gas-cooled reactors today, BNES, September 1982, Bristol, UK, Vol. 4, pp Wilkie, D., 1983b. Heat transfer development of AGR fuel, Proc. Gas-cooled reactors today, BNES, September 1982, Bristol, UK, Vol. 4, pp

21 Tables C µ σ k σ ε C ε1 C ε2 C 1 C 2 C L C η C kt Parameter Table 1 Constants appearing in the v 2 -f model. Notation Multi-Start (360 sector) Transverse (360 sector) Present (30 sector) Rib width at base b 0.30 Rib width at half rib height w h/ Rib width at tip w Rib height k Radius at base R Radius at tip R Pitch-to-height ratio P/k * * 7.0 Graphite sleeve inner diameter D s Fuel pin diameter D p Guide tube diameter D g Free flow area A Rough perimeter P R Smooth perimeter P S Total perimeter Eqn. (17) P T Total perimeter Eqn. (18) P T Hydraulic diameter using P T in Eqn. (17) D e Hydraulic diameter using P T in Eqn. (18) D e Length of ribbed surface/computational domain L * The exact value depending upon whether pitch is measured normal to the ribs or in the axial direction. Table 2 Geometrical data for the multi-start, transverse and the present configurations (all dimensions are given in mm and mm 2 ). Case Number of rough walls H [m] k/h P/k Re Plane ,000 Plane ,000 Plane ,000 Plane ,000 2D simulation ,000 Table 3 Comparison of the 4 planes on the symmetry line of the 3D simulation with the 2D case. 21

22 Variable Definition Value D e Hydraulic diameter calculated based on P T defined in Eqn. (17) [m] p Pressure loss along the present 3D configuration [Pa] f Friction factor calculated for the present 3D configuration Eqn. (19) f 0 Friction factor of an equivalent smooth configuration calculated using the Blasius equation Eqn. (20) f / f 0 Friction factor of the present 3D configuration normalized using the Blasius equation f RB / f 0 Ravigururajan & Bergles correlation calculated for the present configuration Eqn. (21) Table 4 Comparison of the friction factor in the present configuration with correlations. 22

23 Figures Fig. 1. Schematic of an AGR fuel element. Fig. 2 An early design of AGR fuel pins with transverse ribs. (Pictures have been captured by the author from a fuel element sample at the Dalton Nuclear Institute of the University of Manchester.) (a) (b) (c) Fig. 3 The Multi-Start configuration (a) Schematic of a fuel pin (b) Rib profile (c) Schematic of a fuel element and the minimum sector to be simulated. (Fig. 3 (a) and (b) have been provided by J. Gotts from British Energy.) 23

24 (a) (b) (c) Fig. 4 The Transverse configuration (a) Schematic of a fuel pin (b) Rib profile (c) Schematic of a fuel element and the minimum sector to be simulated. (Fig. 4 (a) and (b) have been provided by J. Gotts from British Energy.) Fig. 5 Schematic of three typical sub-channels which may exist in an AGR fuel element. Fig. 6 Schematics of transverse-ribbed, multi-start ribbed and longitudinally finned fuel pins (from Wilkie, 1983a) 24

25 (a) (b) (c) Fig. 7 The present configuration (a) Schematic of a fuel pin (b) Rib profile (c) Schematic of a fuel element and the minimum sector to be simulated. Fig. 8 Boundary conditions used in the present configuration. Fig. 9 Notations of different elements in the present configuration. 25

26 Fig. 10 Dimensions of the present configuration. Fig. 11 Schematic of the mesh used for the present 3D computations. 26

27 Fig. 12 Definition of domain s streamwise mid-plane. Fig. 13 Definition of azimuthal planes (Planes 1-4). 27

28 Fig. 14 Contour of the normalized streamwise velocity at the mid-plane (iso-contours vary from -0.2 to 1.2 with an increment of 0.1). (a) Plane1 (b) Plane2 (c) Plane3 (d) Plane4 Fig. 15 Contours of the normalized streamwise velocity at the azimuthal planes (isocontours vary with an increment of 0.1). 28

29 Fig. 16 Contour of the relative temperature at the mid-plane (iso-contours vary from 0 to 8, with an increment of 1). (a) Plane1 (b) Plane2 (c) Plane3 (d) Plane4 Fig. 17 Contours of the relative temperature at the azimuthal planes (iso-contours vary with an increment of 1). 29

30 Fig. 18 Contour of the relative pressure at the mid-plane. (a) Plane1 (b) Plane2 (c) Plane3 (d) Plane4 Fig. 19 Contours of the relative pressure at the azimuthal planes. 30

31 Fig. 20 Schematic of the mesh used for the 2D simulation and the monitoring plane. Fig. 21 Contour of the normalized streamwise velocity obtained using the 2D simulation (iso-contours vary from -0.2 to 1.2 with an increment of 0.1).. Fig. 22 Streamwise velocity distributions at y/k = 0.1 for the azimuthal planes compared against the 2D simulation. 31

32 Fig. 23 Streamwise velocity profiles on the rib-top for the azimuthal planes compared against the 2D simulation. Fig. 24 Streamwise velocity profiles at the middle of the cavity for the azimuthal planes compared against the 2D simulation. Fig. 25 Wall-normal velocity distributions at y/k = 1 for the azimuthal planes compared against the 2D simulation. 32

33 Fig. 26 Contour of the relative temperature obtained using the 2D simulation (iso-contours vary from 0 to 3.5 with an increment of 0.5). Fig. 27 Nusselt number distributions for the azimuthal planes compared against the 2D simulation. 33

34 Fig. 28 Contour of the relative pressure obtained using the 2D simulation. Fig. 29 Pressure coefficient distributions for the azimuthal planes compared against the 2D simulation. 34