kinetics 3

7/31/2019 Kinetics 3

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Chapter 3 Non-stationary transport and diffusion equations

3.1 Delayed neutrons balance

As discussed earlier, delayed neutrons are the result of the decay of fission products

through beta emission, followed by a decay by neutron of their daughters. Furthermore,fission products, and, thus, delayed neutrons, are grouped in accordance with their ti halflife and characterized by the effective delayed neutrons fraction they produce in eachgroup, i. Half life (average of a set of precursors) and effective fraction depend both onthe isotope that produced it (mainly) and on the energy of the neutron that produced it(less drastically). The temporal variation of the concentration of precursor group i will

be given by the difference between the decay of the precursor and its productionthrough fissions. If this precursor concentration is designated as Ci, the balance will begiven by the following equation:

(3.1.1)

As the groups of precursors are arbitrarily defined in accordance with their averagedecay constants, the previous one and the most general case shall include all fissileisotopes in the system. The contribution of the different j fissile isotopes aremodulated with the effective fraction that each contributes to the group of precursors:

Equation 3.1.1 is the balance of the concentration of precursors of the i group, andthus represents a series of equations, one for each group of precursors. As the delayedneutrons of each group have to do with the decay of the precursors, the source ofdelayed neutrons corresponding to the i group will be given by the neutron losscomponent (as source, positive term) of equation 3.1.1:

(3.1.2)

This is one of the source terms of the transport equation, which contributes to thebalance of neutrons with i fraction, as this fraction is implicit by being explicit in theproduction of the precursors, equation 3.1.1.


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3.2 Transport and diffusion equations

The non-stationary transport equation represents a temporal balance betweenproductions and leaks. When the description of the delayed population is not included,an equation will be sufficient. When the contribution of delayed neutrons is included,

the transport equation shall include an explicit source term due to these neutrons, andshall be related to the precursors balance equations, which are in turn delayed neutronsources. If the delayed neutrons groups are Nd, we will have a system of Nd + 1equations, namely:

General balance, one equation:

=

+dN

i

iid trcE1

),()(r

(3.2.1)

Balance of each group of precursors, source of delayed neutrons for each group, Ndequations:

(3.2.2)

Equation 3.2.1 is the description of the balance between productions and leaks to definethe temporal variation of the neutron population, considering all system sources. Theseries of equations characterized by equation 3.2.2 is identical to equation 3.1.1 shownin the previous item, numbered again for completeness purposes, and corresponds to thedelayed neutron precursors source, which indicates the precursors balance required todefine the source of delayed neutrons of equation 3.2.1.

We may see that if we rule out the contribution of the precursors, we will have = 0 in

equation 3.2.1. Moreover, d(E) = 0, p(E) = (E) corresponds to this situation, and thusequation 3.2.1 represents the temporal balance without delayed neutrons used in otherreactor physics areas.

We should also note that in equation 3.2.1, the source of delayed neutrons is represented

by addend =

dN

i

iid trcE1

),()(r

. The fraction of delayed neutrons of each group is implicit

in this addend, for the decay of precursors is part of equation 3.2.2. We should also

emphasize the fact that, for simplicity reasons, it was assumed that all groups of delayedneutrons are modulated by the same delayed spectrum d(E). This is an approximation


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to the actual situation in which each group of delayed neutrons has a typical emissionspectrum, as seen with relation to Figure 1.2.2.

When the diffusion approximations are valid, equation 3.2.1 may be simplified to reachthe diffusion equation, using:

In this situation, equation 3.2.1 may be written as:

=++

dN

i iidt

trcEtErtErDtErtr1

),()(),,(),,(),,(),(rrrrrr

(3.2.3)

Equation 3.2.2 is not altered. The equations system given by the diffusion equation(transport in the most general case) and the evolution equations of the precursorsrepresents the reactors temporal evolution due to variations in the reactor parameters.The equations shown correspond to the continuum in energy, due to which its resolutionin real systems calls for an adequate multi-group representation. We must add to this thecontribution of the delayed neutrons, for which a description in six temporal groups isgenerally sufficient in most cases (it may be necessary to describe them in forty groups,for example). Furthermore, the balance is confirmed in each system volume element.This implies that a detailed analysis of the problem requires managing a large numberof balance equations. However, for the purposes of the physical description of the

problem, it suffices in practice to focus all efforts on the temporal description,considering the reactor as a spatial assembly (specific reactor) and one energy group.The approximations required, the consideration of the problem, and its limitations will

be discussed in further sections.

3.3 Point kinetics equations

For the temporal description of the evolution of the neutron population, we must solvethe equations system given by the series represented in equation 3.2.2, which defines for

each precursor group its concentration and, thus, the delayed neutrons source for thebalance equation, given by equation 3.2.1 or equation 3.2.3 should the diffusionapproximation be valid.

We will further work on the diffusion equation, as the approximation relating flux andcurrent through Ficks law is valid in most cases when considering the reactor as anassembly. The following are thus the equations on which other simplifications will bemade (3.2.3 and 3.2.3 are repeated):


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=

++dN

i

iidt trcEtErtErDtErtr

1

),()(),,(),,(),,(),(rrrrrr

(3.3.1)

(3.3.2)

These equations imply the simultaneous analysis of the continuum in energy, thetemporal variation of a discrete series of precursors as neutron sources, and the spatialvariable, also expressed in this case as a continuum. We further discuss theapproximations on each one of these variables.

Analysis of the energy variable: Multigroup equations

The energy variable is dealt with in the usual way within the field of reactor physics,dividing the range into energy groups whose amount and sections will depend on the

problem to consider and the required degree of detail.

Assuming few energy groups, equation 3.3.1 becomes a finite set of equations. Whenthe number of energy groups is G, we will have G equations, with the following shapefor each group:

=

++dN

i

iigdggga trctrrDtrr1

, ),(),()(),()(rrrrrr

(3.3.3)

Equation 3.3.2 is not altered, with the exception of the integral, which is represented asa sum on the G energy groups. It will also represent a set of Nd equations, one for eachgroup of delayed neutrons:

(3.3.4)

Spatial variable treatment: Point reactor

For the spatial description, it is necessary to analyze the balances of each volume

element chosen to represent the system. In many cases, and for the purposes ofdescribing the reactor response as a whole due to variations in its properties, it usually


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suffices to make do without the exact geometric description and use an approximationof the point reactor, which leads to the point kinetics equations. In this approximation,we assume that the geometric representation of all functions is the same, correspondingto the fundamental mode , which is modulated by a temporal function, i.e.

(3.3.5)

The spatial function is given by the fundamental mode, due to which:

(3.3.6)

In turn, considering a constant diffusion coefficient and the above:

(3.3.7)

Replacing sets 3.3.5 and 3.3.7 in equations 3.3.3 and 3.3.4, the spatial modulation iscancelled and the multi-group equations are as follows, also replacing the flux with itsdefinition :

(3.3.8)

(3.3.9)

As it is of interest to obtain the temporal description of the population, the set ofequations given by equation 3.3.8 (which represents G equations, one for each energygroup), will be simplified and a group will be used. This affects the source of the setgiven by equation 3.3.9 (which represents Nd equations, one for each group of

precursors). The following are thus the monoenergetic equations:


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=

++dN

i

iif tctnv1

)()()1( (3.3.10)

(3.3.11)

In these, the expressions given in Section 2.2 for finite systems are replaced:

Replacing and regrouping:

=

++=

dN

i

ii tctntSdt

tdn

1

)()()1(

)()(

l (3.3.12)

(3.3.13)

3.4 Validity of point kinetics equations

Regardless of the number of delayed neutron precursors - which in the series of specifickinetics equations may be chosen at our convenience there are at least two strongapproximations in the treatment described in the previous section. Equation 3.3.12 is toone energy group, and it was assumed Equations 3.3.5 to 3.3.7 - that spacedependence is the same for all intervening functions, and uncoupled from the temporalvariable. These approximations are not strictly necessary, since if transport equationswere strictly applied instead of applying the diffusion theory, the same series ofequations would be obtained, with some variables in the definitions of intervening

parameters: , , .

Since they are equations to one group, the energy effects are included in themodifications of the intervening parameters, e.g. Section 1.4 discusses the modification

on the fraction of delayed neutrons to consider that these have better chances to producethermal fissions than prompt neutrons, as there is a higher fast non-leakage probability .

The same criterion yields effective value pdiieffi Fp ,, = .

The most significant approximation is that of assuming that the space form of thefunctions is the same. This function corresponds to the systems fundamental mode, onwhich a reactivity change has been applied. The flux shape corresponding to the case ofthe critical reactor is generally used when the reactivity change has not taken it muchfarther from this state. When temporal space distribution variations cannot be ruled out,a more detailed description must replace the specific kinetics model.


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3.5 Solution to one group of delayed neutrons

Equation 3.3.12, general balance of the neutron population, and Equation 3.3.13,representing a set of Nd equations, will allow defining the neutron density evolution dueto changes in system properties, either source or reactivity variations. The first

approximation will consider only a group of delayed neutrons, i.e. all delayed neutronsin fraction emitted with the same half life. This will allow identifying the typical

behavior of the system and certain key parameters in its evolution, and will begeneralizable to the most detailed case of several delayed neutron groups.

When working with one group of delayed neutrons, the Nd groups shall be adequatelyaveraged to obtain the constants to one group. The total fraction of delayed neutrons andthe decay constant to one group shall be:

==dN

i

i

1

=

=

=

===d

d

i

i

d N

iN

i

i

i

N

i

iitt1

1

1

Equations 3.3.12 and 3.3.13 are in this case:

(3.5.1)

(3.5.2)

This system has thus two unknown functions, neutron density and delayed neutronprecursor concentration, coupled by productions. To solve it, we may use the Laplacetransform, which allows passing from a differential equation system to an algebraic

equation system. Its definition is as follows:


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The following are some general expressions of the Laplace transform:

Derivative in t

Integral

Derivative in s

Exponential transform t

Exponential transform s

Initial value theorem

Final value theorem

Convolution,overlapping principle

Some usual functions:

Using this tool, we will determine the evolution of the population as from the ratiosgiven in Equations 3.5.1 and 3.5.2. We assume that the cross sections do not vary withtime (constant , and variable reactivity only through external action, i.e. insertion orremoval or absorbent elements). Without external sources, we have:

(3.5.3)


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(3.5.4)

In initial stationary state, with zero derivative for neutron density and delayed neutron

precursor concentration, we obtain the initial concentration ratio as from 3.5.4:

(3.5.5)

Replacing in 3.5.4, the equation allows evaluating the following:

(3.5.6)

Replacing 3.5.6 in 3.5.3, we obtain an equation in the neutron density, which whenregrouping becomes:

(3.5.7)

We must note that equation 3.5.7 represents a polynomial quotient in s, linear in thenumerator, and quadratic in the denominator. This implies that it may be factored as:

(3.5.8)

This implies that, in temporal terms, density is:

(3.5.9)

The values of the coefficients in the exponents are the roots of the denominator of

equation 3.5.7, as equation 3.5.8 is its factoring. Hence, given a value of reactivity ,the values of1 and 2 are the s values that render:

(3.5.10)

Clearing in terms of reactivity, we obtain the following equation, known as inhourequation:

(3.5.11)

3-6 Solution analysis

When graphing equation 3.5.11, inhour equation, we obtain the discontinuous curveschematized in Figure 3.6.1, in which three well-defined branches appear. We mark on

the dotted line the value of = 1 as real physical limit, for by its definitions keff 0, ityields:


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In other words, the branch above unit reactivity is not physically reachable.

From the analysis of equation 3.5.11, we may confirm that the equation has asymptotic

values in s = and in s = 1 / l . In all physically reachable ranges, i.e. for < 1, theequation has always two roots.

Figure 3.6.2 repeats the previous figure, although stressing two cases of reactivityinsertion, a positive one and a negative one, with the resulting roots. These roots are thevalues of1 and 2 that appear in equations 3.5.8 and 3.5.9, and are designated withsupra indexes + and for their direct identification in the figure.

Figure 3.6.1: Inhour equation to one group of delayed neutron precursors.

s

-10 s-11/ -104 s-1

Real

s

> 0

< 0

Figure 3.6.2: Roots of the inhour equation to one group of delayed neutron precursors


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In equation 3.5.9, repeated below, both exponentials decay (with a different constant) inthe case if negative reactivity insertion, and one decay and the other persists in the caseof positive reactivity insertion.

(3.6.1)

In the case of both positive and negative insertions, the largest root prevails (designatedas 1

+ and 2-, respectively). Figure 3.6.3 schematizes the relation between population

n(t) and the initial value after the insertion of positive and negative reactivities. After acertain period, we may consider that one of the roots prevails, whose inverse is knownas stable reactor period, i.e. both for the case of positive and negative insertions, thestable period will be T = 1 / 1.

We must also note that both the time required to reach the stable period and the stableperiod itself depend on the reactivity inserted to the system, which in turn defines thevalues of the roots. We have marked its approximate location in the figure, the latter

showing that the transition area is not the same for both insertions. Given theasymmetry of inhour equation branches, we observe that even when the absolute valuesof positive and negative insertions were equal, the root values are not. The fact that the

period (and, thus, the response) of the reactor depends on its properties and on thereactivity introduced had been anticipated in section 2.1 through the use of a model ofless sophistication.

3.7 Reactor response due to reactivity changes

Certain extreme cases in reactivity insertions will be analyzed on the basis of equation3.5.11, inhour equation, repeated below:

Figure 3.6.3: Neutron population behavior following reactivity insertions.

t [s]

n(t)/n(0)


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(3.7.1)

3.7.1 Small reactivity insertions


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Figure 3.7.1 shows the reactor period variation for reactivity insertions in dollars.Although the change in $ = 1 is not a step, there is a strong reduction of the reactor

period when that value is exceeded. The graphs are parametrics in the mean life of an l generation for U233, U235, and Pu239 for positive reactivities. We provide schematic

indication of the asymptotic values for the negative insertion period, which will bediscussed in 3.7.4.

Figure 3.7.1: Reactor period versus reactivity for fissile isotope.

3.7.3 Approximations to the evolution in the initial moment

Immediately after a perturbation either positive or negative - is introduced, there is aninstant response from the system due to the prompt neutrons. As seen in Figure 3.6.3,there is an initial response before the reactor reaches its stable period. In many cases, adetailed description of the initial evolution is not required, as the neutron density valuesas from which the reactor will evolve with its stable period are of significance.Assuming that the delayed neutron precursor density will not vary at the initial moment,we apply the prompt jump approximation to determine the approximate population

value after the initial transient, and prior to varying with the stable period.

We will carry out the approximations on equation 3.5.1 (repeated below without source)to obtain a relation at the initial moment:

At the moment prior to the insertion of reactivity, the derivative and the reactivity itselfare zero, due to which at t = 0 the following is confirmed:

< 0


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Thus, assuming that the concentration of delayed neutron precursors will not varysignificantly in short periods, the neutron population variation may be uncoupled fromthat of the delayed neutron precursors to obtain the following:

In the previous one, = ( - ) / (1 - ), it may be a function of time when we have (t).Assuming a prompt reactivity insertion, the solution for n(t) is:

(3.7.2)

Note that there are no constraints on whether the reactivity insertion is positive ornegative. However, as theprompt criticalcondition was considered in the previous item

as limiting, the following analysis will be restricted to - < < . In this case, we willalways have the < 0 condition. If we can consider that:

then we have that in equation 3.7.2 the exponent is controlled solely by the promptneutrons, i.e.

This implies that in a short interval known as tpj, this approximation indicates that thepopulation will reach the initial value as from which it will decay with the stable reactorperiod, i.e.

(3.7.3)

The level and corresponding periods for positive and negative reactivity insertions areschematically indicated in Figure 3.7.2, similar to the already-presented 3.6.3. Once the

prompt level is reached, approximate in this scheme, the population evolves with thestable period. Many applications do not require knowing the exact behavior of the

population at the initial moment, due to which it suffices to determine this prompt jumpas if it were a step and further follow up its evolution using the stable reactor period.


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Theprompt jump approximation should not be taken for the prompt criticalcondition.In prompt critical, the reactor only responds with prompt neutrons due to a positiveinsertion above the delayed neutrons fraction, and this is a physical situation whichshould be avoided, as system control capacity is lost. Prompt jump omits the descriptionof the delayed population at the initial moment to produce a simple model of the

evolution of the total population as from the reaching of the stable period, but this isonly a simplified model when the description of the initial detail is not required.

3.7.4 Reactor shutdown

A significant consequence of the behavior after reactivity insertions is the situation in

which the reactor is shut down. In such situation, 0)tpj ( < 0)


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extinguish the population by ten orders of magnitude. In reactors with heavy water or

beryllium, where reactions such us (, n) type are significant, the extinction of theneutron population calls for longer periods.

3.8 Inhour equation

The inhour equation (inverse hours, from the units in which the period in a reactor wasinitially expressed) is expressed and used currently in terms of the delayed neutron

fraction. For the presentation, it was chosen to express it in terms of reactivity , leaving explicit, and other amounts, so that there would be direct relation with previously-

presented parameters. Its expression in terms of reactivity in dollars makes use ofpreviously-established definitions and some additional ones:

Using these expressions, the following relations are thus equivalent (the first of thembeing equation 3.5.11, which is repeated):

(3.8.1)

(3.8.2)

(3.8.3)

3.9 Treatment to several groups of delayed neutron precursors

In our previous discussions, we have considered delayed neutron precursors as membersof a single group of delayed neutron sources. In the development of previous sections,the existence of Nd groups of delayed neutrons or precursors has always beenconsidered. In standard practice, six groups are used (as discussed with the presentationof tables in section 1.2, although tens of them may be used depending on the detailrequired.) The line of reasoning is similar, the uncoupling in the sources is carried outwith a larger number of equations, and, for example, the expressions of the previoussection will be, for the inhour equation, the following:

= +++

+=

dN

i i

i

s

s

ss

s

1 )()1(

1

1

ll

l

(3.9.1)


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++= =

dN

ii

i

s

bs

1 )(*$

;

iib = (3.9.2)

The final result will be equivalent, for since in one group of delayed neutrons thereappeared two roots that gave place to two exponentials, with Nd groups there willappear Nd + 1 roots, with Nd + 1 exponentials. Figure 3.9.1 schematizes the branches ofthe inhour equation to six groups of delayed neutrons (Nd = 6). We may note that due to

positive insertions there prevails the positive period of the possible roots (seven in all,six being negative), while in the case of negative insertions there are seven negative

roots, of which the largest will be within asymptote 1, corresponding to the first groupof delayed neutron precursors (the figure shows only the roots for positive reactivity toavoid too many references within the same figure). When considering higher temporaldetail, the value of1 corresponds to a longer decay period, and since s represents afrequency, 1 (six groups)

kinetics 3

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