Reflected Reactor

In previous chapters (nuclear chain reaction) it was derived, that leakage of neutrons from the system significantly influences the criticality of a reactor. For thermal reactors two variables are defined in order to describe the leakage process:

The fast non-leakage probability (Pf) and the thermal non-leakage probability (Pt) may be combined into one term that gives the fraction of all neutrons that do not leak out of the reactor core. This term is called the total non-leakage probability and is given the symbol PNL.

The fast non-leakage probability (Pf) and the thermal non-leakage probability (Pt) may be combined into one term that gives the fraction of all neutrons that do not leak out of the reactor core. This term is called the total non-leakage probability and is given the symbol PNL, and may be expressed by following equation:

fast non-leakage probability_3

For large reactors, we can rewrite this equation without a substantial loss of accuracy simply by replacing the diffusion length Ld and the fermi age τ by the migration length M in the one group equation. The term B4 is very small for large reactors and therefore it can be neglected. We may then write.

fast non-leakage probability_4

where M is the migration area (m2). The migration length is defined as the square root of the migration area. As can be seen the total non-leakage probability of large reactors is primarily a function of migration area.

Neutron Moderators - Parameters

Except research reactors, practically all power reactor cores are designed to minimize the neutron leakage. In order to minimize the leakage, neutron reflectors surround reactor cores. The neutron reflector scatters back (or reflects) into the core many neutrons that would otherwise escape. By reducing neutron leakage, the reflector increases keff and reduces the amount of fuel necessary to maintain the reactor critical for a long period.
Neutron Reflector

Neutron reflector inside a reactor core of LWR. It is only an illustrative example.

In LWRs the neutron reflector is installed for following purposes:

  • The neutron flux distribution is “flattened“, i.e., the ratio of the average flux to the maximum flux is increased. Therefore reflectors reduce the non-uniformity of the power distribution.
  • Because of the higher flux at the edge of the core, there is much better utilization in the peripheral fuel assemblies. This fuel, in the outer regions of the core, now contributes much more to the total power production.
  • The neutron reflector scatters back (or reflects) into the core many neutrons that would otherwise escape. The neutrons reflected back into the core are available for chain reaction. This means that the minimum critical size of the reactor is reduced. Alternatively, if the core size is maintained, the reflector makes additional reactivity available for higher fuel burnup. The decrease in the critical size of core required is known as the reflector savings.
  • Neutron reflectors reduce neutron leakage i.e. to reduce the neutron fluence on a reactor pressure vessel.
  • Neutron reflectors reduce a coolant flow bypass of a core.
  • Neutron reflectors serve as a thermal and radiation shield of a reactor core.
See also: Albedo Boundary Condition

The efficiency of a reflector is measured by the ratio of the number of neutrons reflected back into the reactor to the number entering the reflector. This ratio is called the albedo, or the reflection coefficient. The albedo can be expressed it in terms of neutron currents as:

albedo - reflection coefficient - equation

The value of the albedo will depend on the composition and thickness of the reflector. An infinite reflector will have the maximum albedo, but in all practical cases the geometry of a reflector must conform to the geometry of a reactor core and a reactor pressure vessel. For sufficiently thick reflectors, it can be derived, that albedo becomes:

albedo - reflection coefficient - equation2

where Drefl is the diffusion coefficient in the reflector and the Lrefl is the diffusion length in the reflector.

axial-and-radial-reflectorIn general, neutron reflector may be axial and radial, but, for example, in pressurized water reactors the axial reflector does not form any special device. The neutrons are simply reflected by core inlet and outlet coolant. On the other hand common water volume cannot be used as a reflector in radial direction, because it is of the highest importance to maintain high flow rates  in the core and not to bypass fuel assemblies. Therefore the radial neutron reflectors are installed in PWR and BWR reactor cores. Design of such radial neutron reflectors may vary, but we can distinguish between two basic types:

Flux in a Reflected Thermal Reactor – One-group Method

The mathematical treatment of a reflected thermal reactor can be illustrated most simply by considering a slab reactor of thickness a extending from x= -a/2 to x= +a/2 reflected on both sides by a non-multiplying slab (neutron reflector) of thickness b. Neutrons will be considered as monoenergetic with thermal energy. Although the one-group method may provide reasonable results for the reactor, this method does not accurately predict the flux.

For the determination of the flux distribution in the reactor and in the reflector, the following diffusion equations in these zones need to be solved:

reflected-reactor-equations

where a is the real width of the reactor and b the outer dimension of the reflector (bex is the outer dimension including the extrapolation distance). With problems involving two different diffusion media, the following boundary conditions must be satisfied:

1,2. Interface Conditions

At interfaces between two different diffusion media (such as between the reactor core and the neutron reflector), on physical grounds the neutron flux and the normal component of the neutron current must be continuous. In other words, φ and J are not allowed to show a jump.

interface condition - equations

It must be added, as J must be continuous, the flux gradient will show a jump if the diffusion coefficients in both media differ from each other. Since the solution of these two diffusion equations requires four boundary conditions, we have to use two boundary conditions more.

3. Finite Flux Condition

The solution must be finite in those regions where the equation is valid, except perhaps at artificial singular points of a source distribution.  This boundary condition can be written mathematically as:

finite flux condition - equation

4. Vacuum Boundary Condition

The vacuum boundary condition requires the relative neutron flux near the boundary to have a slope of -1/d, i.e., the flux would extrapolate linearly to 0 at a distance d beyond the boundary (d is the extrapolated length). This zero flux boundary condition is more straightforward and is can be written mathematically as:

reflected-reactor-boundary

If d is not negligible, physical dimensions of the reactor are increased by d and this condition can be written as Φ(a/2 + bex) =  0.

The solution in the core satisfying these boundary conditions is:

reflected-reactor-solution

where

reflected-reactor-geometric-factor

The solution in the reflector satisfying these boundary conditions is:

reflected-reactor-solution-reflector

Neutron flux in a finite slab reactor with and without reflector.

Neutron flux in a finite slab reactor according to the one-group method (with and without reflector).

Flux in a Reflected Thermal Reactor – Two-group Method

thermal-and-fast-constants-moderatorsIt was pointed out, the one-group method may provide reasonable results for the reactor, but this method does not accurately predict the flux. To be more accurate the energy of neutrons must be grouped into at least two groups. The two-group energy model leads to very interesting and important results, that must be considered in the design of nuclear reactors.

In this method, the entire range of neutron energies is divided into 2 intervals (fast group and thermal group). All of neutrons within each interval are lumped into a group and in this group all parameters such as the diffusion coefficients or cross-sections are averaged.

To find solution in a reactor core and in a reflector, it is necessary to solve the following diffusion equations for the fast and thermal energy groups are:

reflected-reactor-two-group-method

The solution of this system of homogeneous algebraic equations leads to a determinant of the coefficients. Such solution is shown in:

Reference: Ragheb, M. TWO GROUP DIFFUSION THEORY FOR BARE AND REFLECTED REACTORS, University of Illinois, 2006.

One of the striking results of such solution is that the thermal flux reaches local maximum near the core-reflector interface. This behaviour cannot be derived using one-group diffusion method, because it is caused just by thermalisation of fast neutrons. The fast neutrons, which are produced in the core can enter the reflector at high energy, are not absorbed as quickly in the reflector as neutrons thermalizing in the core, because absorption cross-sections in the reflector are much smaller than in the core (due to the absence of fuel). The thermal neutrons accumulates then near the core-reflector interface, resulting in the local maximum, that is usually known as the reflector peak. This also reduces the non-uniformity of the power distribution in the peripheral fuel assemblies and also reduces neutron leakage, i.e. increases keff of the system (or reduces the critical size of the reactor). This effect can be seen in the following figure.

The reflector peak can be seen only in thermal flux within the reflector. It is found that the fast flux does not show recovery peaks in the reflector, but rather drops off sharply inside the reflector (due to thermalization and absorption).

two-group-method-reflected-reactor

This figure shows the general effect of reflection in the thermal reactor system. Note that a reflector can raise the power density of the core periphery and thus increase the core average power level without changing the peak power.

Reflector Savings

The neutron reflector scatters back (or reflects) into the core many neutrons that would otherwise escape. The neutrons reflected back into the core are available for chain reaction. This means that the minimum critical size of the reactor is reduced. Alternatively, if the core size is maintained, the reflector makes additional reactivity available for higher fuel burnup. The decrease in the critical size of core required is known as the reflector savings. The concept of reflector savings is very important in many reactor calculations.

It can be derived that there is a difference in the reflected and unreflected critical dimensions of a reactor. This difference is known as the reflector savings, it is denoted by δ and for infinite slab reactor it is determined by:

reflector-savings-infinite

according to this formula the reflector savings as a function of the reflector thickness T can be calculated. As can be seen reflector savings are a function of the thickness and the reflector savings has a finite value for infinitely thick reflector, where:

reflector-savings-finite

This can be interpreted as follows: when the neutron reflector has a thickness of a few diffusion lengths, the probability that neutrons that have come into the outer shell of the neutron reflector are scattered back to the core is very small, so that an even thicker reflector hardly yields extra savings.

These effects, however, become less significant as the size of the reactor, measured in migration lengths, increases. The reflector savings amounts to a smaller fraction of the core dimension as its size becomes larger.

Adding a reflector to a reactor allows either the volume of the reactor or the requirements on fuel to be reduced, or some combination of the two. If the reflector savings is known, the calculation of the critical dimensions of a reflected reactor needs only the solution of the bare reactor, which is simpler problem. For example, for the cylindrical reactor, it is only necessary to determine the bare critical radius R0 and the reflected radius is simply R = R0 – δ, where R0  is critical diameter of a bare reactor.

Reflector Thickness

Logically, the reflector coefficient (albedo) increases, as the reflector thickness increases. But how thick should a reflector be? From engineering point of view, reflectors are, of course, limited by certain reactor design (e.g. by width of reactor pressure vessel). But, in general, very little is to be gained by increasing the thickness of the reflector beyond a value of 2L (2 x diffusion length).

With H20 a thickness of 2L is only about 5 cm, whereas for D20 it is about 300 cm. It would require a considerable amount of D20 to extend the moderator by 300 cm, since this additional D20 is being placed on the outside of the core. From these values, it is obvious the neutron reflector plays crucial role in heavy water or graphite reactors, in which the neutron leakage is higher than in LWRs.

Neutron Moderators - Parameters

Reflector Effects on Reactor Kinetics

Reflector savings are not the only significant effect of the neutron reflector in reactor cores. The neutron reflector also influences the mean generation time and therefore influences the reactor kinetics.

In the effect of a reflector on reactor kinetics, we are concerned with the time the neutrons spend in the reflector before returning to the core. In a thermal reactor, a reflector is generally chosen to help moderate the neutrons. Consequently, the neutrons that leak out of the core as thermal, epithermal, and fast neutrons are reflected back into the core as thermalized neutrons.

Note that the prompt neutron lifetime, l, is the average time from a prompt neutron emission to either its absorption (fission or radiative capture) or to its escape from the system. In an infinite reactor (without escape) prompt neutron lifetime is the sum of the slowing down time and the diffusion time.

l=ts + td

In an infinite thermal reactor ts << td and therefore l ≈ td. The typical prompt neutron lifetime in thermal reactors is on the order of 10−4 second. The neutron lifetime with delayed neutrons, ld, is the weighted average of the prompt generation times and a delayed neutron generation time. The delayed neutron generation time, τ, is the weighted average of mean precursor lifetimes of the six groups (or more groups) of delayed neutron precursors.

In view of the time delay in the process of diffusing in and out of the reflector, the kinetics effect of a reflector can be thought of as an effective increase in the neutron lifetime in the core under steady state conditions, or as an additional delayed neutron precursor under transient operations.

Nuclear and Reactor Physics:

  1. J. R. Lamarsh, Introduction to Nuclear Reactor Theory, 2nd ed., Addison-Wesley, Reading, MA (1983).
  2. J. R. Lamarsh, A. J. Baratta, Introduction to Nuclear Engineering, 3d ed., Prentice-Hall, 2001, ISBN: 0-201-82498-1.
  3. W. M. Stacey, Nuclear Reactor Physics, John Wiley & Sons, 2001, ISBN: 0- 471-39127-1.
  4. Glasstone, Sesonske. Nuclear Reactor Engineering: Reactor Systems Engineering, Springer; 4th edition, 1994, ISBN: 978-0412985317
  5. W.S.C. Williams. Nuclear and Particle Physics. Clarendon Press; 1 edition, 1991, ISBN: 978-0198520467
  6. Kenneth S. Krane. Introductory Nuclear Physics, 3rd Edition, Wiley, 1987, ISBN 978-0471805533
  7. G.R.Keepin. Physics of Nuclear Kinetics. Addison-Wesley Pub. Co; 1st edition, 1965
  8. Robert Reed Burn, Introduction to Nuclear Reactor Operation, 1988.
  9. U.S. Department of Energy, Nuclear Physics and Reactor Theory. DOE Fundamentals Handbook, Volume 1 and 2. January 1993.

Advanced Reactor Physics:

  1. K. O. Ott, W. A. Bezella, Introductory Nuclear Reactor Statics, American Nuclear Society, Revised edition (1989), 1989, ISBN: 0-894-48033-2.
  2. K. O. Ott, R. J. Neuhold, Introductory Nuclear Reactor Dynamics, American Nuclear Society, 1985, ISBN: 0-894-48029-4.
  3. D. L. Hetrick, Dynamics of Nuclear Reactors, American Nuclear Society, 1993, ISBN: 0-894-48453-2. 
  4. E. E. Lewis, W. F. Miller, Computational Methods of Neutron Transport, American Nuclear Society, 1993, ISBN: 0-894-48452-4.

See above: