Atomic Energy, Vol. 111, No. 4, February, 2012 (Russian Original Vol. 111, No. 4, October, 2011)

CFD CODE APPLICATION: CALCULATING HEAT TRANSFER IN A REACTOR WITH SUPERCRITICAL PARAMETERS

V. P. Smirnov, M. V. Papandin, A. Ya. Loninov, G. V. Vanyukova, and S. Yu. Afonin

UDC 621.039.546:536.24

Thermohydraulic calculations of isolated and communicating cells of a rod bundle were performed by the channel method for CANDU-X fuel assemblies and by a three-dimensional method. It was established that in solving the problem for the tightest cell in the case q = const the azimuthal nonuniformity of the temperature was found to decrease by 77°C but it too was inadmissibly large. The temperature distribution along the surface of a fuel element in the case q = const was found to be different from the solution of the adjoint problem. A region with elevated coolant temperature, impeding heat exchange between two neighboring cells, was found between two adjoining cells. It was found that to evaluate computational reliability an experimental study must be performed on rod assemblies with supercritical coolant parameters.

The heightened interest in reactors with supercritical parameters is due to the pursuit of higher efficiency. One variant of this direction is a technical work up of a CANDU-X type channel reactor [1]. The channel of this reactor is an assembly with 43 fuel elements – eight with outer diameter 13.5 mm and 35 with diameter 10.5 mm, which form a central group and two outer rows, respectively. The fuel assembly is encased in a channel tube with inner diameter 102 mm. The heated length of the channel is 5.77 m. Spacing lattices are used in the assembly. The conventional methods of hydraulic calculations are based on a representation of the reactor core as a system of parallel channels (channel method). In calculations, the thermohydraulic properties are averaged in the transverse section of the channels, which requires closure relations for the coefficients of hydraulic friction and heat emission and correction factors relating the parameters of the most highly heated cells and the fuel elements in a channel with the cross-sectional averages of the regime and geometric parameters. The development of channel methods for thermohydraulic calculations of rod assemblies makes it possible to obtain the spatial distribution of the properties. However, once again, the problem of the closure relations remains open but now at the level of the cells between fuel elements. Because no experiments have been done with rod assemblies, it is recommended that the relations obtained for tubes be used in the first approximation [1]. However, heat-transfer processes in assemblies more complicated than in tubes because of the strong change of the thermophysical properties of the coolant, the difference of the cell geometry, the nonuniformity of the energy release along and over the transverse cross section of the channel, and the presence of spacing lattices. In the present work, three-dimensional thermohydraulic calculations performed with the FLUENT code (USA) are used to study for the example of a CANDU-X channel the characteristics of heat transfer in a rod assembly cooled by water with supercritical parameters. Because we do not know the structural particulars of the spacing lattices in a CANDU-X channel the calculations were performed without modeling them.

Dollezhal Research and Development Institute of Power Engineering (NIKIET), Moscow. Translated from Atomnaya Énergiya, Vol. 111, No. 4, pp. 196–201, October, 2011. Original article submitted June 6, 2011. 252

1063-4258/12/11104-0252 ©2012 Springer Science+Business Media, Inc.

Fig. 1. Temperature of the inner surface of the wall, calculated in the k–ω model (A), lowReynolds k–ε model (B), k–ε model with a wall function (C), experimental (——), coolant temperature (– – –), pseudo-critical temperature (– · –) with ρu = 1002 kg/(m2·sec), Pin = 23.95 MPa, Tin = 350°C, q = 681 kW/m2 (a), 500 kg/(m2·sec), 24.156 MPa, 352°C, 340 kW/m2 (b).

First, the FLUENT code was tested on eight experiments with upward water flow in a vertical tube with inner diameter 10 mm, wall thickness 2 mm, heated length 4 m, and pressure 24 MPa in the mass velocity range ρu = 500–1500 kg/(m2·sec) [2]. The calculations were performed for three turbulence models: the standard k–ε model with a wall function, the low-Reynolds k–ε model, and the low-Reynolds Shear Stress Transport (SST) model – a variety of k–ω model. The calculations were performed for constant heat flux density q on the inner surface of the tube in an axisymmetric (two-dimensional) formulation. Constant velocity, temperature, pressure, and low turbulence – 3% of the squared average flow velocity – were prescribed at the entrance into the tube and “soft” conditions (derivatives of the desired functions equal to 0) at the exit. The convergence of the numerical solution was checked for the low-Reynolds models on 40 × 200, 80 × 400, and 160 × 800 grids and a model with 20 × 100, 40 × 200, and 80 × 400 grids and a wall function. A nonuniform difference grid with bunching near the wall according to a geometric progression was used in the direction of the normal to the wall. The grid was uniform along the tube. For grids with an average number of nodes, the numerical solution converged within 253

Fig. 2. Seven characteristic cells of a CANDU-X channel.

1–2°C with wall–liquid temperature difference 50–100°C. The accuracy of the numerical solution was checked by adherence to the condition ⏐(Φi – Φi–1)/Φi–1⏐ < ε, where Φi is the desired function at the ith iteration at an arbitrary node of the grid, and ε is an iteration convergence parameter. The calculations showed that the conventional convergence parameters ε = 10–3 for heat transfer near the pseudocritical temperature do not guarantee convergence of the numerical solution. It was shown that the iteration convergence parameter must be less than 10–4. As an illustration, the distribution of the wall temperature along the length is shown in Fig. 1 for two experiments with normal and degraded heat transfer, respectively [2]. For a normal heat-transfer regime (see Fig. 1a), the computation results obtained with low-Reynolds models of turbulence are in good agreement with experiment (the discrepancy is no greater than 5°C). The relative distance Δy+ of the center of a control volume near the wall does not exceed 0.05–0.1. The standard k–ε model with a wall function at the pseudo-critical temperature understates the wall temperature compared with experiment by 15–20°C. In the region of degraded heat transfer (see Fig. 1b), only the k–ω model describes the spike in the wall temperature caused by a decrease of the heat-emission coefficient. For this reason, this model was used in three-dimensional calculations of a CANDU-X channel. The calculations of heat transfer in a tube made it possible to estimate the required number N of control volumes: axisymmetric problem 80 × 400 = 3.2·104, regular cell 80 × 400 × 20 = 6.4·105, and CANDU-X channel 80 × 400 × 20 × 70 = 4.48·106. The estimates were made for a difference grid which is nonuniform in the direction of the normal to the wall. Such a grid makes it possible to describe the properties of flow in the laminar sublayer. This is of fundamental importance when the thermophysical properties of the coolant change strongly. Modeling of the spacing lattices will increase the total number of control volumes by at least a factor of 10. Therefore, a three-dimensional calculation of even a few-rod assembly, which a CANDU-X channel is, will require powerful supercomputers. In the present work, the analysis of heat transfer is limited to a three-dimensional calculation of seven properties of the cells of CANDU-X channel which are thermohydraulically isolated (Fig. 2). In addition, thermohydraulic calculations of the following were performed: 1) two neighboring cells to determine the character and the scale of the hydrodynamic thermal interaction; 2) the adjoint problem of heat conduction and convective heat transfer in the tightest cell No. 1, illustrating the role of heat conduction in a fuel element; and 3) triangular (relative step s/d = 1.15, 1.3) and square (s/d = 1.15, 1.3) regular cells in order to elucidate the effect of the relative step between fuel elements. The CANDU-X regime parameters are as follows [1]: pressure 25 MPa, mass velocity 860 kg/(m2·sec), heat flux density (constant along the length and over the transverse cross section) 670 kW/m2. The calculations were performed only for the entrance part of a horizontal channel (to z = 2.886 m), where a transition through the pseudo-critical temperature occurred. A hybrid multiblock difference grid, which was uniform along the 254

Fig. 3. Difference grid in a transverse section of cell 1.

length and bunched up near wet surfaces, was used for the cells studied. The near-wall step Δy+ along the normal did not exceed 0.05–0.1. The total number of control volumes for a cell was in the range 343,800–871,800 (Fig. 3). The results of the three-dimensional calculations were compared with calculations in each cell using the PUChOK BM code [3], in which the following closure relations were used: coefficient of friction [4] ξ = ξ0(ρw /ρb)0.4ψξ, where ξ0 = (0.7913·ln(Reb) – 1.64)–2 is the hydraulic resistance coefficient; ρw is the coolant density at the wall, kg/m3; ρb is the coolant density at the mass-average temperature, kg/m3; Reb is the Reynolds number at the mass-average coolant temperature; and ψξ is a form factor [4]; the Nusselt number [5] Nu = 0.00459Re0.923 Prw0.613(ρw /ρb)0.231ψNu, w where Rew and Prw are, respectively, the Reynolds and Nusselt numbers at the wall; ψNu = 1.13(s/d) – 0.26 (for triangular cells); s is the distance between the fuel elements in a bundle, m; d is the diameter of the fuel elements, m; and mixing ratio [6] –0.1 βm = βq = (dh,kn /Δkn)0.0062Rekn , where dh,kn is the average hydraulic diameter of two neighboring cells; Δkn is the gap between cells; and Rekn is the Reynolds number in two neighboring cells. Table 1 shows that substantial azimuthal nonuniformities of the surface temperature of the fuel elements ΔTϕ, max, equal to one or more wall–liquid temperature differences, are observed in all cells, except cell No. 4 with relative step between fuel elements s/d ≥ 1.29. According to the three-dimensional calculation, the surface of the fuel element in cell No. 1 has the maximum temperature, while according to the channel calculation this maximum temperature occurs in cell No. 3. It should also be noted that the maximum temperature calculated for cell No. 3 by the two different methods is the same. On average the maximum surface temperature of a fuel element calculated by the channel method is lower than the value obtained by the three-dimensional method. Figure 4 displays the temperature distribution as isolines along the surface of a fuel element in the first row, adjoining the tightest cell No. 1 (s/d = 1.12). The surface temperature was found to vary from 425 to 729°C. In the transverse sec255

TABLE 1. Maximum Surface Temperature Tmax of a Fuel Element, Its Average Nonuniformity ΔTϕ,max, and Maximum c Surface Temperature T max of a Fuel Element Calculated by the Channel Method, °C Cell No. Parameter 1

2

3

4

5

6

7

Tmax

729

688

572

494

572

664

523

c T max

510

505

572

469

544

500

456

ΔTϕ,max

306

161

84

28

137

142

100

Fig. 4. Distribution of the surface temperature of a fuel element in the first row in cell No. 1 (sweep in the axial and azimuthal directions).

Fig. 5. Coolant temperature distribution in the section z = 1.443 m in cell No. 1.

256

Fig. 6. Coolant temperature distribution in the sections z = 1.5 m (a) and 2.886 m (b) from the entrance (unheated central fuel element).

Fig. 7. Distribution of the temperature of the outer surface of a fuel element in the first row, adjoining the tightest cell No. 1 (sweep in the axial and azimuthal directions).

tion z = 1.443 m it reaches its maximum value (T = 729°C), and the azimuthal nonuniformity of the temperature of a fuel element is 306°C. In Fig. 5, the coolant temperature in the indicated transverse section is 385–580°C. The mass-average coolant temperature in cell No. 1 is close to the pseudo-critical temperature T = 384.9°C; the temperature in a narrow gap as well as the fuel-element temperature are considerably higher. If the degraded heat transfer in a tube is due to the suppression of turbulence near the wall and a decrease of the heat emission coefficient, then in a tight cell of the bundle the spike in the surface temperature of a fuel element is due to overheating of the coolant in the narrow gap. One can see in Fig. 6 that in the narrow space between neighboring cells there is a region where a spike is observed in the coolant temperature. Depending on where the peak temperature lies – to the left or right or at the center of the narrow 257

TABLE 2. Results of Three-Dimensional and Channel Calculations for Triangular and Square Cells Computational method Cell type

three-dimensional

channel for each cell

Tmax, °C

ΔTϕ,max, °C

s/d = 1.17

546

28

588

574

447

s/d = 1.3

471

11

489

484

390

s/d = 1.15

620

160

612

532

402

s/d = 1.3

490

46

475

452

386

Tmax, °C

Tin,out,°C

Triangular:

Square:

gap – the heat flux between the cells will be directed toward the right or left or it will equal zero. In any case, the heat flux between the cells will not be related directly with the difference of the average temperature in the cells. In this situation, in a channel calculation of a fuel assembly diffusive heat transfer between cells can be neglected completely or one can try to develop a method that takes account of the displacement of the region with elevated coolant temperature. Figure 7 shows as isolines the temperature distribution obtained by solving the adjoint problem of heat conduction in a fuel element and convective heat transfer in cell No. 1. Comparing with the similar distribution obtained for q = const shown in Fig. 4 showed that the maximum surface temperature decreased by 77°C, the position of the peak temperature of the fuel element shifted toward the exit from the computational region, and the maximum azimuthal nonuniformity of the cladding temperature of a fuel element decreased almost two-fold (by 143°C). Even though the decrease is appreciable, the azimuthal temperature nonuniformity once again remains inadmissibly large. To determine the effect of the relative step between fuel elements, calculations were performed for regular cells in addition to the three-dimensional calculations of seven irregular cells of a CANDU-X channel. The regime parameters and the cell length were taken to be the same as in cell No. 1 of a CANDU-X channel. The peak temperatures of the outer surface of a fuel element calculated by the three-dimensional and channel methods are compared in Table 2. The maximum temperature Tmax was obtained by increasing the average value of Tmax by half the azimuthal nonuniformity ΔTϕ, max determined in the three-dimensional calculations. In the channel calculations, the peak temperature Tmax of a triangular cell is approximately 20% higher than the value obtained in a three-dimensional calculation. For a square cell the situation is reversed. Hence it follows that the iteration approach to the thermohydraulic calculation of the core using three-dimension calculations to refine the closure relations in the channel method is best. A large azimuthal nonuniformity of the temperature of fuel-element cladding is possible in the closely packed rod assemblies (s/d ≤ 1.15) which are being considered for fast reactors with supercritical parameters. Three-dimensional thermohydraulic calculations in regular triangular and square cells with parameters close to those of CANDU-X were performed for light-water and fast reactor designs [7]. According to the calculations, the temperature of the outer surface of the fuel-element cladding was 750°C and the azimuthal nonuniformity was 330°C [7]. To decrease the latter it is recommended that a displacer be used between the fuel elements. In our view, spiral ribbing or wire coiling is more effective for decreasing the azimuthal nonuniformity and the temperature of the fuel-element cladding. As noted above, in tight cells a substantial increase of the surface temperature of fuel elements in narrow gaps leads coolant temperature growth because heat transfer is degraded. Calculations for regular cells (s/d = 1.3) have shown that heattransfer degradation is not observed for the CANDU-X parameters q = 670 kW/m2 and ρu = 860 kg/(m2·sec). The tendency for heat-transfer degradation to appear was noted only for q = 1200 kW/m2. The threshold value of q/ρu at which heat-transfer degradation starts in rod bundles can be much different from that in tubes. 258

Three-dimensional thermohydraulic calculations of isolated and interacting cells of a rod bundle have shown that for supercritical parameters near the pseudo-critical coolant temperature heat transfer processes in tight cells of an assembly differ greatly from those in tubes. Experiments on rod assemblies with supercritical parameters must be performed in order to confirm or reject the computational results. Specifically, the effectiveness of different flow-mixing devices must be checked: spiral ribbing and spacing lattices. From the computational standpoint, it is of interest to develop models of turbulence which incorporate the correlation of density and velocity pulsations, ordinarily neglected when the changes in the thermophysical properties of the coolant are moderate.

REFERENCES 1. 2. 3. 4. 5. 6. 7.

I. Pioro and R. Duffy, Literature Survey of Heat Transfer and Hydraulic Resistance of Water, Carbon Dioxide, Helium and Near-Critical Pressures, Chalk River Laboratories, Ontario KOJ170 (2003). P. L. Kirillov, V. V. Lozhkin, and V. V. Smirnov, Research on the Limits of Degraded Heat-Transfer Regimes at Supercritical Pressures, Preprint FEI-2988 (2003). G. S. Mingaleeva and Yu. V. Mirnov, “Thermohydraulic calculation of multirod fuel assemblies cooled by a singlephase coolant,” At. Énerg., 48, No. 5, 303–308 (1980). P. L. Kirillov, Yu. S. Yuriev, and V. P. Bobkov, Handbook of Thermohydraulic Calculations, Energoatomizdat, Moscow (1990). H. Swenson, J. Carver, and C. Karakava, “Heat transfer to supercritical water in smooth-bore tubes,” Trans. ASME J. C, No. 2, 477–484 (1965). D. Rowe, “A mechanical for transient mixing between rod bundle sub channels,” ibid., 12, No. 2, 805–807 (1969). Y. Ishiwatsri, I. Liu, Y. Oka, and I.Yoo, “Numerical investigation of heat transfer in upward flow of supercritical water in circular tubes and tight fuel rod bundles,” Nucl. Eng. Design, 234, 420–430 (2007).

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