Conceptual Design of a Fluidized Bed Nuclear Reactor

Transcription

1 NUCLEAR SCIENCE AND ENGINEERING: 139, ~2001! Conceptual Design of a Fluidized Bed Nuclear Reactor J. L. Kloosterman,* V. V. Golovko, H. van Dam, and T. H. J. J. van der Hagen Delft University of Technology, Interfaculty Reactor Institute Mekelweg 15, NL 2629 JB Delft, Netherlands Received December 7, 2000 Accepted March 26, 2001 Abstract A new type of nuclear reactor is presented that consists of a graphite-walled tube partly filled with TRISO-coated fuel particles. Helium is used as a coolant that flows from bottom to top through the tube, thereby fluidizing the particle bed. Only when the coolant flow is large enough does the reactor become critical because of the surrounding graphite that moderates and reflects the neutrons. The fuel particle designed for this reactor is strongly undermoderated and has a temperature coefficient of reactivity that is sufficiently negative. The outer diameter is 1 mm with a fuel kernel diameter of 0.26 mm. The fuel enrichment (16.7%) and the core inventory (120 kg of uranium) inherently limit the maximum power to 16 MW(thermal). A lumped-temperature point-kinetics model has been made that describes the fluidization of the particle bed, coupled to the thermal hydraulics and the neutronics of the core. The model has been linearized around the stationary solution, and the transfer function from coolant mass flow rate perturbations to reactor power fluctuations has been calculated. From a root-locus analysis, the reactor operation is shown to be stable with respect to small variations of the coolant mass flow rate around the stationary operation points. Transient analyses with the nonlinear reactor model show that for the three transients considered (a step in the coolant mass flow rate, a decrease of the coolant inlet temperature, and a loss of heat sink), the fuel temperature remains well below 16008C. Recommendations are made for further research. I. INTRODUCTION To meet the growing demand of electricity in the world, new nuclear reactors should be developed that are superior to the present ones with respect to safety, economics, waste management, nonproliferation, etc. For this reason, the U.S. Department of Energy and other organizations emphasized at the PHYSOR 2000 meeting the need for so-called fourth-generation reactors and have initiated the Nuclear Energy Research Initiative to stimulate the development of these reactors. 1 This paper describes a conceptual design of a fluidized bed nuclear reactor that can be used as a starting point for the development of a fourth-generation reactor. The integrity of the reactor is ensured by the fact that its safety philosophy is based on physics laws only and that its fuel consists of TRISO-coated particles that can withstand very high temperatures. Helium is used as a coolant, which offers the advantages of a high core outlet * J.L.Kloosterman@iri.tudelft.nl temperature and the use of a highly efficient direct-cycle gas turbine. The design is very simple, which allows for modularity during the manufacturing. Furthermore, no active devices are needed to control the reactor because it is a self-regulating thermostatic installation. In other words, the reactor s power is automatically adapted to the needs of the grid. The fluidized bed reactor concept is not new. Sefidvash published many papers 2 on the design of a fluidized bed nuclear reactor fueled with spherical fuel elements with a diameter of 8 mm that are lifted by an upwardly flowing light water coolant. Recently, the dynamic stability of such a reactor was investigated, 3 and it was shown that in this respect, a fluidized bed reactor is similar to conventional reactors. A similar concept with spherical fuel elements with a diameter of 10 mm has been published by Mizuno et al. 4 In this reactor concept, the water that is used for the fluidization of the fuel elements is allowed to boil. A concept more similar to ours is the pellet suspension reactor ~PSR! developed by Harms and his colleagues. 5 In this reactor, micro fuel 118

2 FLUIDIZED BED NUCLEAR REACTOR 119 pellets are suspended in an upwardly directed helium flow within cylindrical columns separated by an appropriate moderator material like graphite or low-pressure D 2 O into which control rods may be inserted for control and shutdown. All concepts have in common that the core collapses in case of a loss-of-coolant accident, which is considered an important safety aspect because it stops the fission process immediately. The conceptual design of the reactor presented here is very simple. The reactor consists of a vertically oriented graphite-walled tube with inner diameter of ;1 m and outer diameter ~o.d.! of 3 m. Also, the bottom and top reflectors of the tube are made of graphite with a thickness of 1 m. The exact dimensions are shown in Fig. 1. The tube is partly filled with TRISO-coated fuel particles made of a UO 2 fuel kernel with a 235 U enrichment of 16.76%. These TRISO-coated particles do not differ from the ones used in ordinary high-temperature gas-cooled reactors 6 ~HTGRs!, although the diameters of the layers are adapted to the needs of this reactor ~see Sec. II!. Helium is used as a coolant that flows from bottom to top through the tube. In fact, the helium flow not only cools the fuel particles but also lifts the particle bed. In other words, the helium flow fluidizes the particle bed, and it is only after the particle bed has been fluidized that the reactor produces power. In the remainder of this paper, we speak about the core to specify that part of the tube filled with particles, and the cavity, which is the remainder of the tube only filled with helium. The height of the core without fluidization, the collapsed core height, depends only on the fuel inventory of the reactor ~number of fuel particles!. The expanded core height, however, which is the height of the core when the particle bed is being fluidized, depends on the value of the collapsed core height and the mass flow rate of the helium. Its minimum value is the collapsed core height; its maximum value is the total inner height of the tube ~6 m!. For a given fuel inventory, the reactivity is determined only by the expanded core height. In a collapsed state, the reactor is subcritical. When the helium mass flow rate increases beyond the threshold value at which the lifting force of the coolant balances the buoyant weight of the particles, the particle bed starts to expand. When this occurs, more and more neutrons escape to the graphite walls where they are moderated and possibly reflected. As a result, the reactivity of the core increases. However, when the particle bed expands too much, too many neutrons leak away, and the reactivity decreases. Figure 2 shows a typical curve of the k eff as a function of the expanded core height for a reactor in the hot-zeropower ~HZP! ~T 543 K! condition. The k eff is larger than unity only for a limited range of the expanded core height. Fig. 1. Schematic presentation of the reactor. Both the axial and radial reflectors consist of 1-m-thick graphite. During operation, helium flows from bottom to top through the core, thereby fluidizing the particle bed and cooling the TRISOcoated fuel particles. The horizontal cross section of the core has a cylindrical shape. The fuel particles are not to scale. Fig. 2. The k eff of the reactor as a function of the expanded core height. For values of the expanded core height between 136 cm, which corresponds to a fully collapsed core, and ;270 cm, the reactivity increases with the expanded core height, while it decreases for expanded core heights.270 cm. More details about these calculations are given in Sec. III.

3 120 KLOOSTERMAN et al. The power that can be generated is inherently limited. When the k eff of the cold core exceeds unity, the reactor starts to produce power, and the fuel temperature will increase. With a proper design of the fuel particles such that the fuel temperature coefficient of reactivity is always negative, the reactivity introduced by the core expansion will completely be compensated for by the power reactivity defect. Because the reactivity is limited ~see Fig. 2!, the power is limited. Even more, left of the maximum in Fig. 2, the reactor power can increase only when the helium mass flow rate increases, which implies that the cooling capacity of the core increases. Of course, right of the maximum, the reverse is true, but in that operating regime, the ratio of the reactor power and the coolant mass flow rate is already much smaller. Having explained the basic physics principles of the reactor, in the following sections we discuss in detail the topics that are relevant for the safe operation of the reactor. Section II deals with the fuel particle design, which is important to ensure a negative temperature coefficient of reactivity. In Sec. III, we discuss the behavior of the k eff as a function of the expanded core height and explain the physics mechanisms behind this. In Sec. IV, we describe a reactor model that describes the fluidization of the particle bed coupled to the thermal hydraulics and the core neutronics. In Sec. V, the results of a linear stability study are shown, indicating that the reactor is stable in the linear sense for the whole operational range of the coolant mass flow rate, which is the parameter foreseen to be varied to control the reactor power. Results of transient analyses are shown in Sec. VI, showing that for all transients that can be envisaged at this stage, the fuel temperature remains below the limit valid for the TRISO-coated particles. We finish this paper with conclusions and with recommendations for further research. II. FUEL PARTICLE DESIGN It is very important to design the fuel particles such that the temperature coefficient of reactivity is always negative. In fact, only the power defect that exists by the grace of a negative temperature coefficient can compensate for the reactivity introduced by means of core expansion. The TRISO-coated fuel particles consist, as usual, of a UO 2 fuel kernel surrounded by three layers of pyrolytic carbon and silicium carbide 6,7 ~see Table I!. For the neutronics analyses, the three layers can be homogenized to one zone with volume averaged nuclide densities. This means that the fuel particles are modeled as a fuel kernel surrounded by one moderator shell. Figure 3 shows the k` as a function of the moderatorto-fuel ~MF! ratio with the diameter of the fuel kernel as a parameter. The MF ratio is the atomic ratio between the number of graphite atoms in the moderator shell and TABLE I Size and Composition of the Fuel Particles Finally Selected Material Density ~g0cm 3! Outer Diameter ~mm! UO 2 fuel kernel Porous carbon buffer layer PyC coating SiC coating PyC coating Fig. 3. The k` of the particles as a function of the MF ratio with the fuel kernel diameter as a parameter. The MF ratio has been varied by changing the o.d. of the particles and keeping constant the fuel kernel diameter. the number of uranium ~ 235 U 238 U! atoms in the fuel kernel. For each value of the fuel kernel diameter, the o.d. of the moderator shell was varied to change this ratio. The calculations have been performed with SCALE- 4.2 ~Ref. 8! using a 172-group data library based on JEF-2.2 ~Ref. 9!. A Dancoff factor of calculated with the DANCOFF-MC program 10 was used for all calculations, although in reality this parameter varies with the fuel kernel diameter and the MF ratio. We show later that the curves do not change qualitatively if a correct Dancoff factor is used for each point. At an MF ratio of ;600, the k` has a maximum. Right of this, the fuel is overmoderated because of the increasing neutron absorption by the moderator nuclides. Left of it, the fuel is

4 FLUIDIZED BED NUCLEAR REACTOR 121 TABLE II Values of the Factors of the Four-Factor Formula for a Fixed Fuel Kernel Diameter of 0.26 mm and a Varying Moderator Shell Thickness MF Ratio e P f h k` Fig. 4. The k` of the particles as a function of the fuel kernel diameter with the MF ratio as a parameter, which has been varied by changing the o.d. of the particle and keeping constant the fuel kernel diameter. undermoderated because of the increasing resonance absorption in the fuel. For fuel kernels big enough, the k` increases again if we reduce the MF ratio to a value below 40. Of course, this effect is seen only for large enough fuel enrichments. The same data are represented in Fig. 4, where the k` is shown as a function of the fuel kernel diameter with the MF ratio as a parameter. For modest values of the MF ratio, the k` increases with the fuel kernel diameter, indicating that resonance absorption is a significant effect. However, for very large values of the MF ratio, the k` does not depend on the fuel kernel diameter anymore, indicating that all neutrons are well thermalized by the moderator and that resonance absorption has become a minor effect. For several values of the MF ratio, Table II gives the factors of the four-factor formula 11 : the fast fission factor e, the resonance escape probability p, the thermal utilization f, and the reproduction factor h. These were calculated with the VAREX program. 12 In the overmoderated region with MF ratios.600, both e and p tend to unity because of the increasing fraction of neutrons that are thermalized in the moderator, while f decreases because the moderator absorbs more and more neutrons. The fuel particle finally selected has an o.d. of 1 mm. This size of fuel particle can easily be made with current fabrication technology and shows favorable fluidization properties. Having fixed the o.d. of the particles, the fuel kernel diameter should be selected somewhere between 0.22 and 0.30 mm to have an MF ratio between 300 and 100, respectively. In this range, expansion of the particle bed is expected to give an increase of the k eff because as a first very crude approximation, expansion can be considered as an increase of the MF ratio. The fuel kernel selected has an o.d. of 0.26 mm, which corresponds with an MF ratio of 160. The density and dimension of each layer are given in Table I. For the fuel particle selected, the temperature coefficient of reactivity is strongly negative. Figure 5 shows the uniform temperature coefficient as a function of the MF ratio. For the calculations, performed with SCALE- 4.2 using a 172 group data library based on JEF-2.2, the o.d. of the particle was kept constant at 1 mm, while the Fig. 5. The uniform particle temperature coefficient of reactivity as a function of the MF ratio, which has been varied by changing the fuel kernel diameter and keeping constant the particle o.d. at a value of 1 mm. Also shown are the contributions of the individual components that make up the fourfactor formula.

5 122 KLOOSTERMAN et al. fuel kernel diameter has been varied to change the MF ratio. The Dancoff factor was calculated for each point by means of an analytical expression. 13 Because the temperatures of the fuel kernel and the moderator shell are virtually the same and because the temperature coefficient of the fuel kernel is much larger than that of the moderator shell, only the results for the uniform particle temperature coefficient are shown. The separate contributions to the temperature coefficient of the fuel kernel and of the moderator shell have been reported elsewhere. 14 The temperature coefficient of reactivity shown in Fig. 5 has been calculated according to ~see Nomenclature on p. 135! a part 2 k hot k cold k hot k cold DT, ~1! which is the appropriate formula if the k` values are used. 15 The parameter DT is the temperature difference between 600 and 293 K. With VAREX, the contribution of each factor of the four-factor formula has been calculated according to a part 1 e ]e ]T 1 p ]p ]T 1 f ]f ]T 1 h ]h ]T. ~2! The effects due to e and p with opposite signs dominate the temperature coefficient. Because the effect due to resonance absorption ~ p! exceeds in magnitude that due to fast fission ~or better called nonthermal fission!, the temperature coefficient is negative for the whole range of the MF ratio. Only for very large values of the MF ratio ~MF. 1000! does the contribution due to f become significant because an increasing temperature shifts the Maxwell spectrum to higher energies. Because the capture and fission cross sections of 235 U fall off with increasing energy faster than the capture cross sections of 238 U and 12 C ~see Fig. 6!, a shift of the Maxwell spectrum to higher energy reduces the k`. Fig. 6. Cross sections as a function of energy of 235 U, 238 U, and 12 C normalized to the theoretical 10v cross section. In conclusion, a fuel particle with an o.d. of 1 mm and a fuel kernel diameter of 0.26 gives a particle temperature coefficient of reactivity that is strongly negative ~' K!. Although these studies have been conducted for the fresh fuel composition only, we expect that the temperature coefficient of reactivity remains strongly negative during the course of fuel depletion because of the strong contribution of the Doppler effect in 238 U, which is expected not to be very sensitive to the depletion of 235 U. However, this is to be investigated ~see Sec. VII for further work!. III. CORE DESIGN The k eff of the reactor depends strongly on the interaction between the core and reflector. Fast neutrons escape from the core and moderate in the reflector. Depending on the expanded core height, and thus on the average particle density in the core, these neutrons induce fission or will be absorbed by the moderator and reflector nuclides. The fraction of neutrons that leak away from the reactor is very small. It has been shown that an increase of the reflector thickness ~1 m!has a negligible effect on the k eff. Because the neutron spectrum varies strongly with the expansion of the particle bed and because the cavity above the core is virtually a void, the calculation of the k eff is not straightforward. For parametric design studies, we used the Bold-Venture two-dimensional diffusion code 16 with a 49-group data library based on JEF-2.2. The diffusion coefficient for the cavity was calculated by a formalism developed by Gerwin and Scherer, 17 and it varied between 35 cm for a collapsed bed to 29 cm for an expanded core height of 544 cm ~leaving 56 cm for the cavity!. Figure 7 shows the influence of the diffusion coefficient on the k eff with the expanded core height as a parameter. For an expanded core height of 136 cm, which corresponds in this case to a collapsed core ~see later in this section!, the k eff is very sensitive to the diffusion coefficient, while for an expanded core height of 544 cm, the result is almost independent of the diffusion coefficient. This means that a change of the diffusion coefficient changes not only the absolute value of the k eff as a function of the expanded core height but also the shape of these curves. For more detailed studies, we used the KENO Monte Carlo code 8 with a 172-group data library based on JEF-2.2. The MCNP-4B Monte Carlo code 18 was used to investigate the approximations introduced when using a multigroup data library. To this purpose, calculations were done with a 172-group data library and a pointwise library, both based on ENDF0B-VI. For core heights of 136 cm ~fully collapsed!, 272 cm, and 544 cm, the k eff showed no differences.0.3% ~Ref. 14!, which indicates that the 172-group results are very reliable. With Bold-Venture, it is possible to calculate the neutron flux

6 FLUIDIZED BED NUCLEAR REACTOR 123 Fig. 7. The k eff as a function of the diffusion coefficient with the expanded core height as a parameter. Clearly, the k eff of the collapsed core ~H 136 cm! is most sensitive to the diffusion coefficient, while the reverse is true for the k eff of the expanded core ~H 544 cm!. Fig. 8. The k eff as a function of the expanded core height with the uranium core inventory as a parameter. It is seen that with 16.76% enriched UO 2 fuel, a core loading of 120 kg of uranium is most appropriate because it gives k eff values,1 for a collapsed core and a fully expanded core and because it gives k eff values.1 for intermediate values of the expanded core height. These curves were calculated with the diffusion code Bold-Venture. density and the adjoint function in the core and reflector regions more easily than with a Monte Carlo code, although the results might be less accurate. Therefore, some calculations were repeated with the two-dimensional discrete ordinates code DORT with a 16-group data library based on JEF-2.2, although these results should not be rated automatically of higher quality because of the possible occurrence of ray effects in the cavity. Once the fuel particle design has been fixed, the uranium fuel inventory determined by the total number of fuel particles in the system is the main parameter to be selected. Of paramount importance is that the reactivity at room temperature should be sufficiently negative to ensure the shutdown condition, while at HZP, the maximum reactivity should be sufficiently positive to achieve desirable values for the reactor power and the coolant outlet temperature. Figure 8 shows the behavior of the k eff as a function of the expanded core height with the core fuel inventory as a parameter. After some iterations, a fuel inventory of 120 kg has been selected corresponding to a collapsed core height of 136 cm. The final choice is based on the assumption of a fresh fuel loading without too much consideration about the thermal efficiency of the reactor. At a later stage, the effects of fuel depletion should be included, and some kind of optimization should be performed to minimize the fuel cycle costs. In Sec. VII, some suggestions are given for further studies. To compare the code systems mentioned in the previous paragraph, the k` was calculated for a onedimensional, infinitely long reactor containing fuel particles with a density corresponding to a fully collapsed core ~136 cm! and an expanded core with height of 544 cm. As shown in Table III, the agreement between the codes is good; the largest difference is only 0.8%. As we have already shown in Fig. 2, the k eff is a function of the expanded core height for a uranium fuel inventory of 120 kg ~collapsed core height of 136 cm!. The core is at HZP with a temperature of 543 K, and the effective diffusion coefficient ranged from 34.7 cm for a collapsed core to 29.0 cm for an expanded core with a height of 544 cm, which is the maximum considered in this analysis. Note that only the axial component of the diffusion coefficient has been used. The actual value of the diffusion coefficient is slightly smaller because of the radial component that should be included. Experiments 19 have shown that the particle density distribution in the core is inhomogeneous and that the particle flow through the core is highly turbulent. The upward flow of particles occurs through the whole crosssectional area of the core, while the downward flow mainly takes place along the periphery of the core. Furthermore, clusters of particles and bubbles of helium move chaotically through the core in an alternating way. To investigate the influence of this complex space and

7 124 KLOOSTERMAN et al. TABLE III The k` for a One-Dimensional, Infinitely Long Reactor with a Particle Density Corresponding to a Fully Collapsed Core ~H 136 cm! and to an Expanded Core with 544-cm Height Code System Number of Groups H 136 cm H 544 cm XSDRNPM-S DORT Bold-Venture KENO V.a ~1 s! ~1 s! MCNP-4B ~1 s! ~1 s! time-dependent behavior of the particle distribution in the core, the k eff has been calculated as a function of the expanded core height for two other particle density distributions: a linearly decreasing distribution and a combination of a uniform and linearly decreasing one. In the latter distribution, which is closest to the real one, 15% of the particles is assumed to be present in jets that escape from the core-cavity boundary. Figure 9 shows the three distributions and the definition of the expanded core height in each case, while Fig. 10 shows the k eff as a function of the expanded core height calculated with the Monte Carlo code KENO Va. The curves for the uniform distribution and the uniform distribution with linear jets are very similar. The main difference stems from the fact that the expanded core height is larger for the distribution with linear jets, which shifts the curve in Fig. 10 to the right. The linearly decreasing particle distribution is more reactive than the other two because the particle density close to the bottom reflector, with a relatively large fuel importance, is larger. However, all distributions show a maximum, which means that the physics operational principles of the reactor are the same and that the reactivity and therefore the maximum power are limited in all cases. Although the uniform distribution with linear jets is most realistic, for ease of calculation, we used the uniform particle distribution in the remainder of this paper. Despite the fact that the Monte Carlo results are probably more trustworthy than the diffusion code results shown in Fig. 2, we used the latter ones in the remainder of this paper because Bold-Venture runs much faster and is better suited to calculate the temperature coefficients of reactivity and the adjoint function. Although the MF ratio of the particles has been chosen such that the particle temperature coefficient of reactivity is strongly negative, the influence of the graphite reflector on the neutron spectrum, and therefore on the temperature coefficient of reactivity, is very large. Figure 11 shows the temperature coefficient of reactivity for the reflector, for the fuel kernel of the particles, and for the moderator shell of the particles. Although the temperature coefficient of reactivity is positive for the reflector, the temperature increase of the reflector usually will be very small and very slow compared to the temperature increase of the fuel and moderator. Therefore, any power transient will be determined by the sum of the temperature coefficients of the fuel and moderator, which is sufficiently negative for all values of the expanded core height, as can be seen in Fig. 11. Note that the temperature coefficient of the fuel and moderator together is a factor of 2 smaller in magnitude than the particle temperature coefficient shown in Fig. 5. Apparently, the fraction of neutrons that gets thermalized in the reflector is so large that the temperature coefficient of the particles in the reactor corresponds to the particle temperature coefficient in Fig. 5 at a much higher MF ratio than 160. In any case, the temperature coefficient is still strongly negative for all values of the expanded core height. IV. REACTOR MODEL In this section, we describe a reactor model containing a fluidization model that describes the void fraction in the core as a function of the coolant mass flow rate, a thermal-hydraulics model that gives the coolant and fuel temperatures, and a neutronics model that describes the reactor power as a function of the void fraction and the temperatures. In Sec. IV.D, we show that the combined reactor model is able to operate, i.e., that there exists a unique stationary solution for each value of the coolant mass flow rate, which is the parameter foreseen to be varied to control the reactor power. IV.A. Fluidization Model Fluidization is a suspension of fuel particles in an upwardly flowing fluid, producing a fluidized bed. Fluidization can be classified by the number of phases involved. The present paper is concerned with two-phase fluidization: a solid ~fuel particle! and a gas ~helium!. On the vertically upward flow of the gas coolant, the particle bed undergoes little visible change with increasing superficial velocity of the continuous phase U until it

8 FLUIDIZED BED NUCLEAR REACTOR 125 Fig. 10. The k eff as a function of the expanded core height for the three particle density distributions defined in Fig. 9. All results are calculated with the KENO Monte Carlo code at a uniform particle temperature of 543 K ~the HZP condition!. Fig. 9. The three particle density distributions in the core: uniform ~top!, linearly decreasing ~middle!, and the mixture of the two ~bottom!. In all cases, H min corresponds to the collapsed core height, and H corresponds to the expanded core height. For the linearly decreasing density ~middle!, the particle distribution is partly uniform and partly linearly decreasing for values of H up to two times H min, while it is only linearly decreasing for H 2{H min. For the mixture of the two distributions ~bottom!, the particle density can be calculated by assuming that first the particle density decreases uniformly, after which the density of 15% of the particles in the upper part of the core is assumed to decrease linearly. reaches the minimum fluidization velocity for a twophase bed U m, at which the bed becomes fluidized. In reality, the fluidization process is not clear-cut but occurs over a range of velocities due to particle clusters and channeling. The Ergun relation 20 describes the pressure drop per unit length of the fluidized bed DP0H as the sum of that due to viscous and kinetic energy losses: Fig. 11. Temperature coefficient of reactivity for the reflector, the fuel kernel of the particles, the moderator zone of the particles, and the sum of the latter two.

9 126 KLOOSTERMAN et al. DP 150n ~1 e v! 2 U 1.75r c ~1 e v! U H d f e v d f e v, ~3! where e v void fraction of the particle bed d f fuel particle diameter n fluid dynamic viscosity r c coolant density U flow velocity. At minimum fluidization, the pressure drop exactly balances the buoyant weight of the particles: DP H ~1 e v!~r f r c!g, ~4! where r f ~2725 kg0m 3! is the fuel particle density and g is the gravitational acceleration. Equating formulas ~3! and ~4! gives the minimum fluidization velocity U m. In a column with flow velocities U m, an increase of the coolant velocity will cause the bed to expand with a constant pressure drop. Generally, columnar expansion follows the relation of Richardson and Zaki 21 : U U t e n v ] e v U U t 10n, ~5! where U t is the effective terminal velocity of the coolant at maximal porosity ~velocity above which the particles are blown out of the core! and n is the Richardson-Zaki constant, which has a value of ;2.4 for values of the Reynolds number.500 ~see Ref. 21!. For the whole operational regime of the reactor, the Reynolds number is of the order 10 4 to 10 5, which validates the use of Eq. ~5! with n 2.4. The terminal velocity U t can be obtained by equating the upwardly directed drag force and the downwardly directed gravitational force. Using the concept of the drag coefficient C d of a particle, and assuming that all the fuel particles have an ideal spherical form, the drag force reads as follows: p F d C d 4 d 1 f 2 2 r cu 2, ~6! which gives for the terminal velocity U t : U t 4gd f ~r f r c! 3C d r c 4gd f r f 3C d r c. ~7! Although in theory, the drag coefficient C d is a function of the Reynolds number of the gas flow, we used a constant value of 0.445, which is fairly accurate for Reynolds numbers between 750 and ~see Ref. 5!. Again, this range of Reynolds number covers well the operational regime of the reactor. The approximation in Eq. ~7! is valid because the density of the fuel particles is much higher than that of the gas coolant. Furthermore, we can safely assume that the gas coolant behaves like an ideal gas, which means that we can use for its equation of state the following: r c PM, ~8! RT c where P pressure of the helium ~ Pa! M molar mass of the helium R gas constant T c lumped gas coolant temperature. As mentioned earlier, the pressure drop during fluidization remains constant, which implies that the lumped gas pressure remains constant during the fluidization process. The temperature of the gas has to be derived from the thermal-hydraulics model presented in Sec. IV.B. Finally, knowing the density of the gas coolant, we can calculate the superficial velocity of the gas U from the formula for the coolant mass flow rate G: G r c UA, ~9! where A is the cross-sectional area of the core ~1 m 2!. From Eqs. ~3! and ~4!, we now know the minimum velocity U m above which fluidization occurs, the maximum velocity possible in the reactor U ~7!#, and the void fraction in the core as a function of the actual velocity ~5!#. The void fraction in the core varies between 0.48, the value corresponding to a collapsed cubic-packed particle bed, and 0.88, the value corresponding to a fully expanded core with a height of 600 cm. Although in reality, the particles in a collapsed bed will be stochastically distributed with a void fraction of ;0.43, a cubic-packed particle bed is easier to model in the calculations and gives a conservative estimate of the shutdown reactivity ~because the shutdown reactivity becomes more strongly negative for smaller values of the void fraction in the collapsed state!. In the remainder of this section, the bed is assumed to expand uniformly and continuously as the coolant velocity increases. IV.B. Thermal-Hydraulics Model For the thermal-hydraulics model, we consider two regions ~fuel particles and coolant!, each characterized by a region-averaged or lumped temperature. From the energy balance in the core, we can derive the equations for the temperature of the fuel particles T f and the temperature of the coolant T c : ~mc! f dt f dt N ha f ~T f T c!

10 FLUIDIZED BED NUCLEAR REACTOR 127 and dt c ~mc! c dt where ha f ~T f T c! Gc c ~T out T in!, ~10! N reactor power h heat transfer coefficient from the particles to the coolant A f heat transfer area of all the fuel particles in the core ~ m 2! ~mc! f,~mc! c heat capacities of all the fuel particles and all the helium in the active core region, respectively c c specific heat capacity of the coolant ~5193 J0kg{K! T out outlet temperature of the coolant T in inlet temperature chosen to be 543 K. Although the heat transfer coefficient depends strongly on the coolant mass flow rate and the thermal properties of the coolant, here we use a constant value of W0m 2 {K ~Ref. 22!. This assumption is allowed because the temperature difference between the fuel and the coolant is very small, which reduces the needed accuracy for this number. Assuming that the coolant temperature has a point-symmetric distribution relative to the active core midplane, we can rewrite Eqs. ~10! as dt f ~mc! f dt and N ha f ~T f T c! dt c ~mc! c ha f ~T f T c! 2Gc c ~T c T in!. ~11! dt Knowing the coolant mass flow rate G and the inlet temperature T in, we can now calculate the temperatures of the fuel and the coolant when the reactor power is known. IV.C. Neutron Kinetics Model The ordinary point-kinetics equations with one delayed neutron group describe the time-dependent behavior of the reactor power and precursor concentration 11 : and dn dt dc dt r b eff L b eff L N lc N lc, ~12! where C precursor density ~expressed in latent power; same units as N! r reactivity b eff effective delayed neutron fraction ~0.69%! L mean neutron generation time ~ s! l precursor decay constant for one group of delayed neutrons ~0.079 s 1! and b eff, L, and l have been calculated with the PERT-V code. 23 The neutronics model is coupled to the fluidization model and the thermal-hydraulics model via reactivity feedback: i 4 r r~e v! a D ~T f T in! ( a i e i v a D ~T f T in!, ~13! where a D is the temperature coefficient of reactivity of the fuel and moderator ~see Fig. 11! that is assumed not to depend on the void fraction and r~e v! is the reactivity as a function of the void fraction adequately approximated with a fourth-order polynomial. IV.D. Combined Reactor Model Equations ~5!, ~7!, ~8!, ~9!, ~11!, ~12!, and ~13! constitute a set of ~differential! equations, which must be combined to describe the dynamic behavior of the reactor. The equilibrium state of the reactor can be obtained by setting the time derivatives in Eqs. ~11! and ~12! to zero. In the remainder of this section, we use the subscript zero to indicate this equilibrium state of the reactor. In fact, we have to find the equilibrium values for T f, T c, r c, U t, U, e v, and N given the values of the coolant mass flow rate G and the helium inlet temperature T in. By straightforward algebra, the solution can be found from Eqs. ~14!: T f 0 T in r~e v0!, a D i 0 T c0 ha f T f 0 2G 0 c c T in ha f 2G 0 c c, r c0 P 0 M RT c0, 4gd U t0 f r f 3C d r co, U 0 U t0 ~e v0! n, G 0 r c0 U 0 A, C 0 b L N 0,

11 128 KLOOSTERMAN et al. and N 0 ha f ~T f 0 T c0!. ~14! There are two solutions possible: a trivial one corresponding to the shutdown condition of the reactor ~T c0 T f 0 T in, N 0 0, C 0 0!, and a solution corresponding to normal operation. The equilibrium power N 0 and the fuel temperature T f 0 during normal operation are shown in Fig. 12 as a function of the coolant mass flow rate G. Clearly, there is a unique value for the reactor power for each value of the coolant mass flow rate. It is seen that the reactor power reaches a maximum at a coolant mass flow rate of ;9 kg0s. The fuel temperature peaks at a flow rate of ;8 kg0s, which corresponds to the value at which the inserted reactivity reaches its maximum. This is due to the fact that the reactivity inserted by means of the core expansion is mainly compensated for by an increase of the fuel temperature. For a flow rate,9 kg0s, the reactor power increases with the flow rate, while the reverse is true for a larger flow rate. With the current fuel particle design and core fuel inventory, the fuel temperature does not exceed 730 K. To enhance the thermal efficiency of the reactor, a higher fuel temperature will be needed, which can be achieved by increasing the core fuel inventory and adding absorber nuclides to the bottom reflector ~see Sec. VII!. However, this option has not been considered in the current study. Note that because of the excellent heat transfer properties of the graphite, the coolant temperature equals that of the fuel particles within 1 K. V. LINEAR STABILITY ANALYSIS The fluidized bed nuclear reactor differs from ordinary solid fuel reactors because of the strong interaction between the coolant mass flow rate and the core neutronics. This is clear from Fig. 2, which shows the relation between the expanded core height, which is determined by the coolant mass flow rate, and the k eff of the core. Because of this interaction, it is not clear beforehand that the reactor is stable under all circumstances. In this section, we study the dynamic stability of the reactor by means of linear stability analysis, which is valid for small deviations from the equilibrium operational points calculated in Sec. IV.D. Large deviations ~transients! are investigated in Sec. VI by means of numerical analyses. Because the reactor power is foreseen to be controlled by varying the coolant mass flow rate and because fluctuations in this flow rate can be expected because of nonstationary operation of the coolant pumps, we decided to calculate the transfer function from coolant mass flow rate variations to reactor power fluctuations and to study the stability of the reactor with respect to coolant mass flow rate variations. To investigate the stability of the reactor in a linear sense, we must reduce Eqs. ~5!, ~7!, ~8!, ~9!, ~11!, ~12!, and ~13! to a set of linear equations correct to first order in the deviations from equilibrium. To this purpose, we substitute G G 0 dg, e v e v0 de v, r c r c0 dr c, U U 0 du, U t U t0 du t, T c T c0 dt c, T f T f 0 dt f, N N 0 dn, C C 0 dc, and r r 0 dr into Eqs. ~5!, ~7!, ~8!, ~9!, ~11!, ~12!, and ~13!; expand all terms; and retain only the linear ones. Proceeding this way, we find the following for the first-order variations of G, T f, T c, r c, U t, U, e v, r, N, and C: dg G 0 dr c r c0 du U 0, dr c r c0 dt c T c0, de v 1 e v0 n du du t U 0 U t0, Fig. 12. The equilibrium reactor power and fuel temperature as a function of the coolant mass flow rate. For the minimum and maximum values of the coolant mass flow rate shown in this figure ~;5 and 13 kg0s!, the expanded core height equals 157 and 416 cm, respectively. The power peaks at an expanded core height of 272 cm. ~mc! f ddt f dt du t U t0 1 2 dr c r c0, dn ha f ~dt f dt c!,

12 FLUIDIZED BED NUCLEAR REACTOR 129 and ddc b dn ldc, ~15! dt L where as mentioned before, the subscript zero denotes the equilibrium state of the reactor. The next step is to Laplace transform Eqs. ~15! to obtain the transfer function from one variable to another and, eventually, from coolant mass flow rate variations to reactor power fluctuations. For those formulas in Eqs. ~15! not containing a time derivative, this is straightforward. Here, only the results are given for the Laplace transforms of dt f, dt c, dn, and dc. For the fuel and coolant temperatures, these read as follows: Fig. 13. Block diagram of the reactor showing the dependence between one variable and the other. The formulas for the transfer functions H1 through H13 are given in Table IV. ddt c ~mc! c ha f ~dt f dt c! 2dGc c ~T c0 T in! dt 2G 0 c c dt c, dr ( a i ie i 1 v0 de v a D dt f, i E E E E E E dt f dn 1 h 1 dt c s h 1 s h 1 and b 2 h 2 dt c dg dt f s h 2 h 3 s h 2 h 3, ~16! where the tilde above a symbol indicates the Laplace transformed variable and where the constants have the following meaning: b 1 1, b 2 2c c~t c0 T in!, ~mc! f ~mc! c ddn dt r 0 b L dn N 0 L dr ldc, h 1 ha f ~mc! f, h 2 ha f ~mc! c, h 3 2G 0 c c ~mc! c. ~17! TABLE IV The Transfer Functions Between the Variables for the Block Diagram Presented in Fig. 13 H 1 N 0~s l! Ls~s b0l! Zero-power reactivity to power H 2 ( ie i 1 0 a i i 4 i 1 Void fraction to reactivity H 3 a D Fuel temperature to reactivity H 4 e 0 nu 0 H 5 e 0 nu t0 Gas terminal velocity to void fraction H 6 b 1 s h 1 H 7 U 0 r c0 Coolant density to gas velocity H 8 U 0 G 0 H 9 U t0 2r c0 Coolant density to gas terminal velocity H 10 r c0 T c0 H 11 b 2 s h 2 h 3 Coolant mass flow to gas temperature H 12 h 2 s h 2 h 3 Coolant velocity to void fraction Reactor power to fuel particle Gas velocity to coolant mass-flow Gas temperature to coolant density Coolant temperature to fuel temperature H 13 h 1 s h 1 Fuel temperature to coolant temperature

13 130 KLOOSTERMAN et al. When we Laplace transform the equations for dn and dc, we obtain the well-known zero power transfer function 24 from reactivity perturbations dr to reactor power fluctuations dn: dne dri N 0 ~s l! Ls~s l b0l! N 0~s l! Ls~s b0l!, ~18! where the approximation is valid because l b0l. Figure 13 presents the block diagram of the reactor showing the dependence of the variables, and Table IV presents the corresponding transfer functions H1 through H13, which have been derived from the Laplacetransformed Eq. ~15!. From the block diagram and the transfer functions, we can derive the overall transfer function Y~s! from the coolant mass flow variations to reactor power fluctuations, which, when expressed in the transfer functions H1 through H13, reads as follows: Y~s! dne dge H 1~H 2 ~H 4 ~H H 12 H 13# H 7 H 11 H 10! H 5 H 9 H 10 H 11! H 3 H 11 H 12!. ~19! 1 H 12 H 13 H 1 H 6 ~H 2 H 4 H 7 H 10 H 5 H 9 H 10 # H 3! The next step is to expand the numerator and denominator and to calculate the poles of the transfer function by equating the denominator, which is a sixth-order polynomial in the complex variable s, to zero. This is described in more detail in Ref. 25 and will not be repeated here. Instead, we focus on the results of this analysis. Figure 14 shows the root-locus plot of all the six poles of the transfer function Y~s! from coolant mass flow rate perturbations to reactor power fluctuations when the coolant mass flow rate is varied between 5 and 13 kg0s. The poles differ several orders in magnitude because of the parameters in the model that have values widely spread. Fortunately, all poles have negative real values, which is a prerequisite for stable reactor operation. Otherwise, if a pole would have a positive real part, the reactor response in the time domain would grow exponentially after a small reactivity insertion, which, of course, is not allowable. Because Fig. 14 does not show how the loci of the poles change with the coolant mass flow rate, the real and imaginary parts of the poles as a function of the coolant mass flow rate are shown in Fig. 15. It is clear from Fig. 15 that the real parts of poles 1, 2 and 3 are strongly negative for all values of the coolant mass flow rate. The imaginary parts of these poles, as well as that of pole 6, are zero, as could already be deduced from Fig. 14. With increasing coolant mass flow rate, the real parts of poles 4 and 5 attract each other until they meet at the s 2s 1 for G 10 kg0s. Then, these two poles become complex conjugates for 10, G, 11.2 kg0s, after which they become real valued again. All poles have a rather peculiar locus in the complex plane: With increasing coolant mass flow rate, they all seem to return to their starting point, while classical theory predicts that the loci start at the poles of the open-loop transfer function and approach the zeroes for an increasing value of the feedback gain. 24 The difference here is that the transfer function shown in Eq. ~19! is a rather complex function of the equilibrium value of the coolant mass flow rate G 0. Not only does the transfer function H 8 depend on G 0 ~see Table IV!, but also the h 3 defined in Eq. ~17! as well as most of the equilibrium values defined in Eq. ~14! depend on the value of G 0. Although this is interesting from a theoretical point of view, the complex trajectories of poles 4 and 5 have no consequences for the dynamic behavior of the reactor. When two poles are complex conjugates of each other, the impulse response of the linearized system shows an exponentially decreasing or increasing sinusoidal behavior. The ratio between two consecutive maxima of the impulse response is called the decay ratio and can be calculated from DR exp 2ps 6v6, ~20! Fig. 14. Loci of the poles of the transfer function from coolant mass flow rate variations to reactor power fluctuations when the coolant mass flow rate is varied from 5 to 13 kg0s. where s and v are the real and imaginary parts of the complex conjugate poles. For poles 4 and 5, the decay ratio is negligible because the ratio of s and 6v6 is strongly

14 FLUIDIZED BED NUCLEAR REACTOR 131 Fig. 15. Imaginary and real parts of the poles of the transfer function from coolant mass flow rate variations to reactor power fluctuations when the coolant mass flow rate is varied from 5 to 13 kg0s.

15 132 KLOOSTERMAN et al. negative. 25 This means that the oscillation superimposed on the exponentially decaying trend is strongly damped. The most critical locus, which has its real part closest to the origin, is that of pole 6. However, it is seen that with an increasing flow rate, the real part of this pole becomes more strongly negative until G ' 10 kg0s after which it becomes smaller again. Fortunately, for no realistic value of the coolant mass flow rate, the real part of pole 6 becomes positive, which implies that the reactor is stable in the linear sense for all realistic values of the coolant mass flow rate. It was mentioned before that the temperature difference between the fuel particles and the coolant, as well as the time constant corresponding to the heat transfer rate from the fuel to the coolant, is very small. This implies that the block diagram in Fig. 13 can be simplified by assuming one effective temperature averaged over the whole core region ~fuel, moderator, and coolant homogeneously mixed!. It has been shown 26 that also in this case, the loci of all the poles of the transfer function have negative real parts. The magnitude of the transfer function at an equilibrium reactor power of 16 MW is shown in Fig. 16 as a function of the radial frequency v. For low frequencies ~,0.1 rad0s!, the transfer function has a magnitude of ;2 MW0kg{s, which corresponds very well to the plot of the equilibrium power as a function of the coolant mass flow rate G shown in Fig. 12. For coolant mass flow rate variations at higher frequencies ~.1 rad0s!, the reactor power variations are much smaller. Figure 16 has been made for a conservative value of the temperature coefficient of reactivity of K 1. It has been shown, however, that with the actual value of the temperature coefficient of reactivity ~the sum of the fuel and moderator temperature coefficients shown in Fig. 11!, the magnitude and phase of the transfer function from coolant mass flow rate variations to reactor power fluctuations hardly changes 26. VI. TRANSIENT ANALYSIS The linear stability analysis presented in Sec. V is valid only for small perturbations from the equilibrium operational point. Only then can the second-order effects be neglected. In this section, we present the numerical results calculated with the nonlinear reactor model presented in Sec. IV. Because the reactor model presented in Sec. IV is rather simple and because there are only two parameters that can be varied, namely, the coolant mass flow rate G and the coolant inlet temperature T in, the number of transients is rather limited. Here, we consider three transients: a step in the coolant mass flow rate ~CMF!, a decrease of the coolant inlet temperature ~CIT!, and a loss of heat sink ~LOHS!. Table V gives the parameters that have been varied to simulate the transient. For all transients, we assumed a conservative value for the temperature coefficient of reactivity of K 1 ~see Fig. 11!. For all the transients, we used the reactivity curve as a function of the expanded core height ~Fig. 10! that corresponds to a uniform particle distribution in the core. This means that the reactivity variations due to the stochastic distribution of the particles in the core are not included in this analysis ~see Sec. VII!. VI.A. Coolant Mass Flow Rate Transient The results for the CMF transient are shown in Fig. 17. The left side of Fig. 17 shows the reactor power and the fuel temperature as a function of time, while the right side shows the expanded core height and the effective multiplication factor. It is seen that because of the sudden increase of the expanded core height, the reactivity of the reactor increases from a subcritical level ~collapsed bed! to slightly less than 1.009, which is the maximum reactivity that can be introduced this way ~see Fig. 2!. This gives rise to a strong increase of the reactor TABLE V Initial and Final States of the Transients Considered Fig. 16. Magnitude of the transfer function from coolant mass flow rate variations to reactor power fluctuations with a simplified thermal-hydraulics model containing one lumped temperature for both the fuel and moderator zones. The plot was made using a conservative constant value of the temperature coefficient of reactivity, and with an equilibrium reactor power of 16 MW~thermal!. Transient Scenario Initial State Final State CMF G 3kg0s G 9kg0s CIT T in 543 K T in 523 K LOHS T in 543 K T in ~t! T out ~t t!, t 5s