High Temperature Reactors (HTRs) employ so-called TRISO fuel particles embedded in graphite. The particles consist of a small fuel kernel made of uranium-dioxide with a typical diameter of 500 microns, surrounded with a buffer layer that can accommodate the gaseous fission products escaping from the fuel kernel, and three containment layers. The latter consist of an inner PyC layer, a SiC layer and an outer PyC layer. The SiC layer usually is under compression to assure the gas-tightness of the containment around the fuel kernel. The TRISO fuel particles can be embedded in a graphite sphere with a typical diameter of 5 cm surrounded with a 0.5 cm thick layer of graphite, or in small cylindrical pins with typical diameter of 1 cm, which are inserted in a graphite block. The first is called pebble-type fuel, while the second is called prismatic fuels.
HTRs differs from Light Water Reactors (LWRs) in various aspects. Because the TRISO particles are dispersed in graphite pebbles or compacts, the HTR fuel has a double heterogeneous geometry, which requires special treatment in reactor physics analyses. Because graphite can withstand very high temperatures and helium is used as a coolant, the HTR operate at a much higher temperature than the LWR. Because the TRISO fuel particles can withstand high pressure, the fuel burnup in a HTR can be higher than in LWRs. Of course this requires a higher initial enrichment. The actual core power density in an HTR is much lower than in an LWR, because of the dispersion of fuel in graphite. Due to the inherent safety requirement of modern HTRs, the power density in HTR is restricted even more.
Fission neutrons are produced with energy of 2 MeV on average and need therefore to be moderated to thermal energies where the movement of neutrons is in equilibrium with the vibration energy of the graphite moderator. In that energy range the fission cross section of the fissile nuclides is highest. In HTR, carbon is used as a moderator, which has some specific consequences for the reactor design, as will be shown later on.
If you add carbon to uranium-235, first the critical mass will increase. This is because the neutron reproduction factor of uranium-235, this is the number of fission neutrons produced upon absorption of a thermal neutron, decreases when the spectrum softens slightly. Only when more carbon is added, the critical mass decreases due to the beneficial effects of moderation. At some point, when all thermal neutrons are in thermal equilibrium with the graphite, adding more graphite will only increase the absorption of thermal neutrons, and the critical mass increases again. In that region the reactor is over-moderated.
Moderation of neutrons is mainly achieved by elastic scattering of neutrons with the carbon nuclei. This process is very similar to billiard ball collisions. The lighter the moderator nuclide is, the larger the energy transfer can be. For two particles the same in mass, all energy of the colliding particle can be transferred to the other. This is the case for a neutron colliding with a proton (hydrogen). Under certain conditions, like isotropic scattering in the centre-of-mass frame, no absorption and a constant scatter cross section, the probability for a neutron to scatter to certain energy is uniform for a certain energy interval determined by the mass of the scattering nuclide.
After elastic collision with a light nuclide, the energy of the neutron is proportional to the incoming energy. In other words, the neutron loses on average a fixed fraction of its energy. For example, a neutron with energy of 1 Mev that loses 90% of its energy will collide at average energies of 100 keV, 10 keV, 1 keV, 100 eV, etc. This suggests that the collision process can better be described using the logarithm of energy. To do so, the unit of lethargy is introduced, which is the logarithm of the initial energy of the neutron divided by the exit energy. A colliding neutron thus loses energy but gains lethargy. With the lethargy as a parameter, the neutron loses a fixed amount of lethargy at each collision, not a fixed fraction. It can easily be derived that the average amount of lethargy a neutron loses in a collision is fixed. Now the average number of collisions needed for a neutron to reach certain lethargy, is just the total lethargy interval divided by the average lethargy gain per collision.
For a neutron of 1 MeV to reach the thermal energy region at 1 eV, the number of collisions with hydrogen is 14, while this is 88 for carbon. A neutron thus needs many more collision with graphite as a moderator than with water or another hydrogen containing material, like polythene or paraffin. Together with the much larger mean free path travelled between interactions, about 1 cm in water and 2.5 cm in graphite, it can be concluded that the total distance travelled by a neutron in a graphite water moderated reactor is many times larger than in a water moderated reactor.
To derive a relation between the distance traveled and the energy or lethargy of e neutron, we can do the following simulation in one dimension. We have neutrons originating at a plane and starting either to the left or to the right. A neutron travels a fixed distance in space, one interval, and at each interval it can be absorbed or can scatter. In the latter case it gains a fixed amount of lethargy and can exit the collision either to the left or to the right. If we plot the resulting distribution of the neutrons, we get a bell-shaped curve that becomes broader and more flat when the lethargy of the neutrons increases. Apparently there is a relation between the average distance a neutron travels and its lethargy. Because the neutron gains a fixed amount of lethargy in each collision, the higher the lethargy the older the neutron will be. That is why the lethargy is sometimes also called the Fermi-age of the neutron, and the theory relating the distance traveled by diffusion of neutrons and the corresponding lethargy distribution the Fermi-age model for neutron slowing down. The stronger the diffusion process, the faster a neutron grows "old". Finally one can show that the Fermi-age of a neutron is proportional to the mean squared distance the neutron has traveled. The unit of Fermi-age is therefore that of an area.
Once the neutron has reached the upper boundary of the thermal energy range, the Fermi-age theory cannot be used anymore because the neutron can also gain energy (lose lethargy) in a collision. Therefore the transport process in the thermal energy range has to be described by diffusion theory properly. The thermal scattering kernel has a uniform distribution for high initial neutron energy, but gets a tail at the high energy part of the spectrum as well when the initial energy of the neutron becomes lower. This tail reflects up-scattering of neutrons upon collision in the thermal range.
When including the effects of up-scattering and assuming no absorption, the neutron density gets a so-called Maxwell distribution, with average energy proportional to the temperature of the moderating material. The most probable energy of the neutron flux density distribution is equal to the product of Boltzmann's constant and the temperature, and equals 0.025 eV for a moderator at room temperature. The corresponding velocity equals 2200 m/s. Because many thermal neutron interaction processes were measured in the past at this temperature, these "2200 m/s" values of the cross sections can often be found in tables in text books. The neutron distribution in the thermal energy range will never reach a real Maxwell distribution because of new thermal neutrons entering the thermal range at the upper boundary, and because of neutron absorption which takes preferentially place at the low energy region. The neutron spectrum will therefore shift to higher energy compared to the Maxwell spectrum at that temperature. In other words, the effective temperature of the neutron distribution is higher than the real temperature of the moderator.
From diffusion theory, it can be shown that the diffusion length squared is proportional to the mean squared distance a neutron travels as a thermal neutron until absorption. Furthermore the average distance between birth of a thermal neutron, this is the point where the neutron slows down beyond the upper boundary of the thermal energy region, and absorption is twice the value of the diffusion length.
By combining Fermi-age theory for slowing down of neutrons and diffusion theory for the thermal region, the total squared distance a neutron travels is just the sum of the Fermi-age and the squared diffusion length. This sum is called the migration area. The root of this area, the migration length, is a measure of the root-mean-squared distance a neutron travels between its birth as a fission neutron and absorption in the thermal range. For water, the migration length is just below 6 cm, while for carbon this is over 60 cm. Again this shows that the core of a graphite moderated reactor should be many times larger than a light-water cooled reactor.
There are three fundamental functions to ensure the safety of a nuclear reactor: 1) control of reactivity, 2) heat removal from the core and, 3) confinement of radioactive material. These three safety functions shall be performed both in normal and off-normal conditions. The "defense in depth" principle is implemented to ensure these safety functions work properly in off-normal conditions as well. This principle is applied to prevent, control and mitigate any off-normal event.
The control of reactivity is done by control rods in the inner and outer reflectors and, if necessary, by a reserve shut-down system that mostly consists of B4C pellets or balls. The cooling of the reactor is usually done by the main cooling system, and by an auxiliary cooling system that starts up when the reactor is scrammed. Although modern HTR designs can dissipate their residual heat to the environment via conduction and radiation, the auxiliary system provides the flexibility to cool down the reactor much quicker than would be possible by passive means. The latter option is used when all forced cooling systems are no longer available, like in a Depressurized Loss of Forced Cooling (DLOFC) incident, caused for instance by a rupture of a primary circuit pipe. The reactor vessel cooling function is always in operation to cool the reactor shielding wall.
The HTR differs from more conventional Light Water Reactors (LWRs) because the fission product containment is assured at the level of the TRISO fuel particles with a diameter of one millimeter contained in graphite spheres or cylinders (called compacts). In LWR's, the containment is enclosing the reactor pressure vessel, while in HTRs the containment is surrounding every single TRISO particle. Although the Japanese authorities requested a containment for the HTTR reactor encapsulating the reactor pressure vessel as well, the fuel has performed so well that it is now generally accepted that future HTRs don't need containment vessels comparable to those of LWRs.
Experiments have shown that TRISO fuel particles behave very well up to temperatures of 1800 degrees Celcius. Below this temperature, the fuel failure fraction is very small and the fission product leakage negligible. Because the temperature limit depends on the fuel composition and fuel burnup, a generally accepted limit for the maximum temperature of TRISO fuel particles is 1600 degrees Celcius. To ensure this limit is never exceeded the heat production during normal operation and the afterheat during shutdown (planned or unplanned) should always be assured. The decay heat is due to multiple components of which the most important one is fission product decay (beta particles and gamma rays releasing about 15 MeV/fission). Decay heat removal is especially important at depressurized conditions with a loss of forced coolant flow.
We can now start with the design of an HTR by inserting fuel pebbles into a cavity surrounded by a graphite reflector. The thickness of the reflector is determined by the reflector savings, which is the core size reduction obtained when a bare critical core is surrounded by a reflector and made critical again by reducing the core size. A reflector thickness of two times the neutron diffusion length is sufficient to maximize the gain. A thicker reflector would not reflect more neutrons into the core, but absorbs the neutrons that would otherwise be lost by leakage. For a graphite reflector, this implies that the thickness should be one meter at least, while this would only be 6 cm for water!
Having fixed the reflector thickness, the neutron leakage from a bare cylindrical core can be minimized by minimizing the buckling of the neutron flux density. For a fixed core volume, one can determine that the ratio between the height and radius of a bare cylindrical core should be 1.85. In other words, a cylinder with height a little bit less than the diameter minimizes the neutron leakage from the core. This would give the optimal dimensions for a cylindrical core from the economic point of view.
However, there is one "but" in this result. A thick core is not able to conduct quickly enough the fission product decay heat to the environment in case of a Depressurized Loss of Forced Cooling incident (DLOFC), and the temperature at the centre of the core could easily exceed the limit of 1600 degrees Celcius. The total volume of a reactor core with optimal shape from the economical point of view would therefore be very small! Only very small reactors like the Chinese HTR-10 can have a small reactor core without violating the passive decay heat removal requirement. That's why passive safe HTR cores with a large power output are usually long and thin. They need to be slender to remove the decay heat by conduction and radiation to the surroundings, while they are long to provide sufficiently large volume to produce enough power.
The maximum radius of the cylindrical core is usually fixed by practical considerations. The maximum radius of the Reactor Pressure Vessel (RPV) is limited to somewhere between 6 and 6.5 m, because of transport considerations from the factory to the reactor site. The PBMR design team, for example, has opted for a vessel of 6.2 m in diameter. As explained before, in cylindrical core geometry it is not possible to increase the core radius to increase core power. If the radius increases, the average power density needs to reduce in order not to exceed the temperature limit in a DLOFC incident. For example, an increase of the radius from 3 to 4 meters, which almost doubles the core volume, would allow only for a 20% increase in reactor power. On the other hand, by actually applying a central fuel-free column and keeping the core volume constant, the power density and the total power output can be raised without exceeding the temperature limit in a DLOFC. This is possible because the power production in the core is concentrated near the outer reflector, which can easily dissipate and transport the decay heat to the surroundings.
Besides practical considerations, the maximum height of the reactor core is limited by reactor physics limits as well caused by the sensitivity of large cores to Xenon oscillations. Xenon is a highly absorbing fission product mainly produced by the decay of Iodine, which decays to Xenon with a half life of about 6.6 hours. Iodine is directly produced from fission with an effective yield of 6.6%. Xenon disappears by neutron capture, for which it has a high absorption cross section of several millions of barns, and by decay with a half life of 9.2 hours to a virtually stable and non-absorbing isotope. If the neutron flux density increases in one part of the core, the Iodine production increases, but the Xenon production remains virtually constant for a couple of hours due to the 6 hours half life of Iodine. The neutron capture by the Xenon, however, is directly proportional to the neutron flux density. As a result, a flux increase leads to a reduction of the Xenon concentration in the core, which leads to less neutron capture by Xenon and to an increase of the local power density. This is a positive feedback effect, which is usually taken care of by the reactor control system, but which can lead to reactor power oscillations between one part of the core and the other. For this reason, the maximum core size is limited to about 30 migration lengths, which corresponds to about 10 meters for a pebble-bed type HTR and to 8 meters for a prismatic type reactor core (the migration lengths for a prismatic and pebble-bed type HTR are about 20 and 30 cm, respectively). The limit of 30 migration lengths can be understood from the analysis based on micro-kinetics. The reactor power can be thought of as being built up by prompt fission chains induced by the emission of delayed neutrons. In a critical reactor, the average number of neutrons in a prompt fission chain equals 1 divided by the delayed neutron fraction beta (0.007 for U-235). As the mean squared distance a neutron travels from birth to absorption equals six times the migration area (migration length squared), the average distance travelled by the neutrons in a prompt fission chain, as the crow flies, equals 30 migration lengths.
Another parameter limiting the height of reactors is the pumping power required to pump the gas coolant, which is proportional to the pressure drop across the core, and the mass flow. For pebble-bed reactors, the pressure drop is proportional to the Q2 H3 where Q is the power density and H is the height of the reactor core. To limit the pressure drop, one either has to reduce the core height or the core power density.
A typical layout for an HTR core would look like the following figure. Cold helium enters the outer reflector from the bottom and flows upward to cool the reflector and other internals. Then it is directed towards the core through which it flows from the top to the bottom to remove the heat from the fuel (either pebbles or prisms). The central column of graphite, if present, will have a typical diameter of about 2 meters and can be made of fuel-free graphite pebbles or a solid central column. The first option leads to a large bypass flow through the central column, which needs extra pumping power. The advantage, however, is that the graphite pebbles can be cycled slowly through the core and replaced by fresh ones if needed. To replace the solid central column, the reactor would have to be stopped and the vessel be opened.
The maximum temperature in a Depressurized Loss of Flow Condition (DLOFC) occurs at a height of about two-third from the bottom. The temperature in the pebble then reaches a value just below 1600 degrees Celcius, which is considered a safe limit to not damage the TRISO fuel particles. The maximum temperature occurs at about 40 hours after the initiating event. In case the reactor core remains pressurized, in a so-called Pressurized Loss of Flow Condition (PLOFC), the resulting temperatures are even lower because of the transfer of heat by the helium coolant from hot spots to the colder parts in the reactor core.
For all modern HTRs, the choice for the coolant is helium. The optimum coolant is able to remove as much heat as possible from the core. A certain amount of power per unit frontal core area can be removed most efficiently by hydrogen molecules, because of the low molar mass of this gas and the relatively high heat capacity due to the fact it is a diatomic gas. Carbon dioxide is second best, but with a value only one half of that of hydrogen. Other optimization parameters like the heat transfer area for a fixed flow area, or the pressure losses in the core due to friction, show that helium is the best choice as a coolant. In conclusion, there is not a clear choice for the coolant gas, but helium is among the best ones. Combined with the inertness of this gas, both chemically and neutronically, helium seems to be a very good choice indeed.
The slides can be downloaded here.
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