ML20041B933
| ML20041B933 | |
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|---|---|
| Issue date: | 02/10/1982 |
| From: | Marotta C NRC OFFICE OF NUCLEAR MATERIAL SAFETY & SAFEGUARDS (NMSS) |
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| NUDOCS 8202250418 | |
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i ENTROPYLIKE FUNCTION FOR A NEUTRON CHAIN-REACTING SYSTEM tn
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% A hY' w a6 19 by Charles R. Marotta Office of Nuclear Material Safety and Safeguards U.S. Nuclear Regulatory Commission l
8202250418 820210 PDR MISC.
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2 ABSTRACT A thermodynamic analog of the neutron chain-reacting process is presented without regard to the energy-heat production aspects of that process. Sinple heuristic arguments are used to show that a relative entropy S of the chain reacting system is equivalent to the time a neutron resides without contributing to 7
the chain-reaction in isolated fissionable systems. The inverse 1
of S, derived from a Monte Carlo forward solution of the Boltzmann equation shows an apparent correspondence with the Adjoint-Importance spectral function. Some inconsistencies in reactor theory surfaced by this development are noted.
)
3 ENTROPYLIKE FUNCTION FOR A NEUTRON CHAIN-REACTING SYSTEM by Charles R. Marotta U.S. Nuclear Regulatory Commission I.
INTRODUCTION Reactor transport theory, from its inception, was founded and developed on the physical principles and mathematical methods of the kinetic theory of dilute gases and the formalism of statistical mechanics.
Ironically, the closely related thermodynamic bases and aspects of this subject appear to have been neglected; a review of the literature shows only one(I} formal attempt at developing thermodynamics for a neutron gas mixing within the atomic structure of a fixed host medium effecting a neutron chain reaction.
It is the purpose of this paper to show through plausibility and heuristic reasoning that there is a basic connection which can be fruitfully exploited between elementary neutron transport theory and thermo-dynamics for a neutron chain-reacting system.
The aforementioned connection relies on previously published work of the author (2,3,4) in which the algebraic difference between two differently (but appropriately) defined average neutron lifetimes was shown to be the average time a neutron spends unproductively not contributing to the chain-reaction of the fissionable system in which it resides. This average time was previously labelled the " excess time"(2) E(secs) of the neutron; the development in this paper l
l l
4 purports to show that this E qualifies as a relative-entropy function S to be associated with a neutron chain-reacting process without regard to the energy-heat production aspects of that process.
It would also appear that the neglect of a thermodynamica1 approach and development stems from the successful use of the concept and methods of Importance-Adjoint methods in reactor theory.
This elegant and practical calculational device possibly aborted the interest in a thermodynamic description of the neutron chain-reaction. This is especially true when one considers that the principle of neutron Importance (adjoint function) could be thought of occupying the position as " manager" of the chain-reaction, while the principle of reactivity (keff) plays the role of " book-keeper" performing the bookkeeping and balancing of neutrons.
Replacement of the principle of Importance by the principle of Entropy and the principle of reactivity by the principle of Energy shows the fundamental thermodynamical analogy for the neutron transport phenomena.
II. THE MODEL AND BASIC OBSERVATIONS The model used for this thermodynamical analysis is nuclear j
fuel stored in a standard water storage pool. This particular model of the chain reaction, when analyzed by the Monte Carlo solution of the Boltzmann conservation equation, is shown to be much more fruitful than that of fuel closely packed as in a LWR nuclear reactor, flore basic chain-reacting information can be j
5 derived in a model of separated fuel assemblies in which the water density between the assemblies as well as among the fuel assembly rods is varied from 1.0 gram per cc to a void, zero density.
In addition to bringing the process of reactivity-coupling (i.e., a form of cooperative phenomena) of assemblies to the fore, the stochastic Monte Carlo numerical analogue experimentation also appears to give more insight and an inde-pendent examination of the fundamental assumptions and parameters used in reactor theory, both statics and kinetics. A single fuel assembly, namely, the basic sub-unit of the above cooperative-phenomena type array will also be analyzed by varying the moderator density. These models were developed and analyzed in detail in references 2 and 4 and will be repeated as necessary in condensed form in what follows.
In reference (2), it was noted that the average non-productive time of a neutron in a fissionable system, the excess time E, was calculated as the algebraic difference between the neutron lifetime t (average time before one neutron is eliminated from the reactor by leakage or absorption) and the ncutron reproduction time A (average time for one neutron to produce another neutron) 1.e.,
l E=t-A.
This form chosen for E was based on the physical argument that it gives through this difference the most meaningful infor-l l
mation on neutron utilization in a chain reacting process.
l Intuitively, as the unproductive time E becomes larger the more a neutron mixes (without productive interaction) among the host l
1
6 atoms giving a more chaotic (less ordered) character to the system of neutrons towards effecting a chain reaction.
It is this latter characteristic of E with thermodynamic entropy S which affords the direct analogy between these quantities.
However, since the multiplication factor k f a nuclear eff reactor is independent of the power level, there is no information on the amount of energy produced by the fissionable system at a given state of k Because of this, the above S must be eff.
considered as an entropy-like quantity (rather than the classical entropy in thermodynamics directly related to the energy of the system) frore naturally related to the chain-reacting process l
itsel f.
Thus, the heating of the fuel-clad-moderator is of no relevance in the model and all temperature feedbacks are neglected making the model linear in this respect. The only non-linearity comes from the coupling-interaction between fuel assemblies in the array effecting a chain reaction.
The actual model chosen is an array of 200 (20 x 10 x 1) light water reactor (LWR) fuel assemblies [each a 17 x 17 rod lattice, Zircaloy clad U(3)0 ]
stored in normal-density water 2
on a 53-cm (21-in.) center-to-center assembly separation giving overall array dimensions of 10.7 x 5.3 x 3.7 m (35 x 17.5 x 12 ft.), A water region ~31.8 cm (12.5 in.) thick exists across the fuel assembly flats; a 30.5-cm (1-ft)-thick water reflector surrounds the entire array. Since each assembly is separated by more than 30.5 cm (1 ft) of normal-density water, neutronic isolation is ensured and the k,ffof the array is essentially
7 that of a single LWR fuel assembly immersed and reflected by water, i.e., the array is subcritical.
Some preliminary neutron physics observations can be made concerning the array. We note that neutrons generated by fissioning in the assembly rods diffuse some distance in the water regions outside the assembly. Thus, E is numerically
[
larger than A under these conditions.
If the moderator density is now decreased, i.e., between rods, assemblies, and in reflector regions, absorption of neutrons is decreased and neutrons diffuse further from the assemblies (increasing E);
fewer neutrons are produced because fewer neutrons are moderated, thus increasing the time between successive fissions (increasing A). Hence, both I and A are increasing together for the moderator perturbation considered. Accordingly, k,ff becomes progressively lower in value with the exception of the possibility that neutrons from one fuel assembly have a potential eventually for interacting with their neighboring fuel assemblies as the moderator density is further reduced. This latter interaction, when realized, makes the crucial difference in reactivity (versus the environment in a compact nonmoderator-separated fuel assembly reactor) since the neutrons enter neighboring assemblies at sufficiently lower (important) energies resulting in a sharp increase in k,ff. This then is the multispectra characteristic of the array absent in a LWR reactor, i.e., fuel assemblies clumped together without any appreciable assembly-assembly separating moderator.
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8 III. MONTE CARLO ANALYSIS OF MODEL Since the Monte Carlo procedure is based on tracking and recording the neutron's life history in a realistic fashion in a fissionable system with a minimum of nuclear and geometric approximations, it was chosen as the method to analyze the above model of a complex multispectra system.
In particular, the 6
KENO IV Monte Carlo program was used together with the 123-neutron group GAM-THERMOS neutron cross-section set.
This program routinely calculates for a well-defined geometric-material fissionable configuration the system k,ff as well as the associ-ated t and A of the system; these latter two quantities are defined in the Appendix of this paper exactly as calculated by KENO. 15,000 neutron histories were used in estimating all k
's to within 0.004 in k f r one standard deviation; I's eff eff
-6 and A's are calculated to 1.5 x 10 seconds for one standard f
deviation. k
's and E's and 1's and A's thus calculated are eff plotted in Figure 1 and Figure 2 respectively as solid lines for the described array when the moderator density is varied 3
from 1.0 gms/cm to a void of zero density.
In using the Monte Carlo method, a renormalization is performed to the system reactivity (keff) after each generation ( A) calculation.
Thus, criticality is forced artificially by assuming that the number of neutrons emitted in a fission is v/k rather than v.
eff l
l
l 9
9 In this sense, all solutions (even though sub or super critical) represent equilibrium steady-state neutron configurations in the array for the moderator density considered.
In a similar fashion of moderator density variation, one of the fuel assemblies of the array was analyzed by Monte Carlo.
k
's and E's and t's and A's thus calculated for the single eff fuel assembly are plotted in Figure 1 and Figure 2 respectively i
as dashed lines as a function of moderator density.
The results for the array as plotted in Figures 1 and 2 indicate that as the spectrum becomes progressively harder from the reduction in moderator density, the simultaneous increase in r
1 and A does not give any sensible information as to what is happening to k f the system. However, ( t-A ) would be eff a measure of the coupling between fuel assemblies that would
[
l increase k At the beginning of the density reduction, eff.
( t-A) would increase (k decreases) and eventually ( t-A )
eff would reach a positive maximum at some reduced moderator density (k
minimum). After this point (maximum E), the coupling eff starts (through effective transmission of somewhat thermalized neutrons) between the once-isolated fuel assemblies, increasing keff [( - A) decreasing] until ( t-A ) reaches a minimum, namely zero, i.e., ( t= A)(maximum keff). The reduction of moderator density also increases the probability of fast fissioning in j
the U-238 component of the fuel. The above point of maximum J
positive E is a transition point tagging the inception of the l
10 interaction process taking control of the system for the lower moderator densities. After reaching maximum keff
- 0) the effective'thermalizing-coupling of neutrons between (E
=
assemblies is reduced due to the hydrogenous deficiency at this density and further moderator reduction effectively makes the system a tremendous voided fast fuel assembly that cannot 235 (due to low 0 content) sustain a k greater than unity.
eff I"
The fast fission in U-238 is the major contributor to keff this extremely low moderator density range. Thus, keff plummets value. In to some low suberitical value past this maximum keff this regime, A>t (since absorption has been significantly reduced and leakage has been greatly enhanced) and E becomes negative. This latter feature of the parameter would be lost if the absolute value were used to define E.
For E = 0, it will be observed that the array system is utilizing neutrons in the most efficient manner (from assembly-assembly interactions as well as fast-fission in U-238) to effect a chain reaction over the range of moderator perturbations performed.
The single assembly (sub-unit of array) Monte Carlo results also given in Figures 1 and 2 illustrate the same qualitative behavior of the array chain-reactor without the coupling effect.
As the moderator density is decreased from unity, the single assembly k,ff goes down monotonically as expected.
Correspondingly, the excess time E increases as k decreases with the decrease eff of moderator density until the fast fission in U-238 begins to 1
I
11 contribute to k,ff.
Since the single assembly represents 1/200 of the array volume, the effects of higher leakage and absence of hydrogenous material as an effective neutron moderator are j
more detrimental to the system reactivity at higher moderator l.
densities. Similarly, the effectiveness of the fast fission in U-238 appears at a higher moderator density than for the array reactivity. This is discernible from the sharp slope change in k at the very low moderator density. This, eff however, is the regime in which for the neutrons present in the system, utilization of these neutrons is most efficient in effecting (however futile) a chain reaction through the fast fission process. Correspondingly, E decreases to zero at this slope change giving the same qualitative efficient neutron utilization as in the array model which because of its size was effected at a lower moderator density.
IV THERMODYNAMIC ANALYSIS OF THE MODEL A nuclear reactor is a neutron machine producing neutrons via the fission chain-reaction process. The thermodynamic develop-ment for this chain-reaction will be based on: a) the phenomenon of random neutron diffusion (mixing) among the fixed host atoms-l the more time a neutron exists in a fissionable system-the greater i
the absence of a chain-reaction pattern will result; b) the l
recognition that not k,77, but the inverse of the, excess time E l
l 1s the measure of the degree of order which the system has attained the chain reacting process; c) that E can be equated to
12 a dimensionless relative-entropy S; d) that interpretation is l
possible of the transport theory Monte Carlo results of scetion III (i.e., kgff, E as a function of moderator density) as l
thermodynamic equilibrium steady state solutions for the systems f
I specified; e) that the information content of the degree to which the system has attained the chain-reacting process is contained in E (or S).
We note that k f unity plays no special role in the eff above picture.
Excess time E will be shown to have the necessary properties of entropy 5, and hence, S will be a measure of the neutron disorder in attaining a chain-reaction system. Since 10 15 the host atomic density is of the order of from 10 to 10 larger (depending on the neutron energy) than the neutron density of any system, neutrons will not interact with one another in any way while mixing with the host atoms and thus the neutrons can be considered a dilute gas whose population calculation in the fissionable system becomes a linear problem.
In addition, since k is independent of the fission power eff level, the heat production and physical temperature of the chain-reacting system are not considered germane for the view-I point taken here for the thermodynamics of the neutron chain reaction.
l The physical delimited systems are the previously discussed i
l fissionable array or single assembly with the assocjated water j
moderator-separator and reflectors and are isolated systems from l
l l
13 other neutron chain-reacting systems and in addition have no external non-fissionable sources within them. The only process under consideration is the neutron chain reaction without delayed neutrons and the process by its nature is considered irreversible. Since the models will be perturbed by varying the moderator density, S, the ability to mix unproductively will not, a priori, be predictable. This is not at variance with a formulation of the second law of thermodynamics which states that a natural reversible process taking place in an isolated system never has decreasing entropy.
The Monte Carlo results for the single fuel assembly model given in Figure 1 show that as the water moderator-reflector density decreases from 1.0 gms/cc, the k g es down monotoni-eff
-cally as expected since fewer neutrons are moderated hence, fewer fissions occur and neutron leakage of all energies is enhanced. The spectrum, that is the average energy of the neutron causing most of the system fissions as well as the number of these neutrons in the system is getting harder as the density progresses to zero. Hence, there is more unproductive mixing of neutrons among the host fissile atoms and the excess time as indicated by the E or S curve starts to increase indicating a larger disorder (hence larger entropy S) relative for the chain reaction up to the moderator density of about 0,4 gms/cc. At this density, some fast fission in U-238 can take place, hence, E (or S) starts decreasing up to about 0.2 gms/cc at which point l
l
4 14 l
l E (or S) becomes zero - indicating zero disorder (state of perfect order) towards effecting a chain reaction.
In this density range of 0.4 - 0.2 gms/cc there are more and more fissions fractionally 1
being produced by the U-238 fast fission (even though k keeps eff on dropping) there is more order in effecting a chain reaction i
than at higher k $s Further reduction of the moderator density eff.
past 0.2 gms/cc enhances the very fast neutron population such I
that now all the neutrons present are able to contribute to fast fission in U-238 excepting that leakage is so great at these energies that the fissioning seldom results.
E(ors) is negative in this region due to the definition of E or S and avoids the discontinuity that would be presented if the absolute l
value of the difference between 2. and A were used. This approach also makes E and S a continuous function going through zero in a natural manner.
l The Monte Carlo results in Figure 1 for the array model indicate as the water moderator-separator-reflector density i
decreases from 1.0 gm/cc, the keff goes down, E or S (unproduc-tivity to the chain reaction) increases up to a density of about 0.3 gms/cc. Other than a higher keff (since this is 200 times the volume size of the single assembly) the qualitative behavior of the S curve is the same as the single assembly model demon-strating the additivity property of entropy:
sum of subunit entropies equal the total entropy of system. The maximum of S for the array occurs at a lower moderator-density than the maximum S for the single assembly. This also is due to the
I 15 j
larger system size of the array which permits more mixing and less leakage. As the density is further reduced k starts eff climbing due to the somewhat thermalized neutrons interacting with neighboring assemblies at important lower energies creating more fissions. Also, now the fast fissions in U-238 are possible and start contributing to k S me neutroa order is eff.
being established for the system chain reaction as indicated by the monotonic decrease in S or E.
E eventually becomes zero at a density of about 0.07 gm/cc moderator density at maximum keff indicating a state of perfect order for the array at this point.
+
Further density decrease reduces the number of thermalized neutrons significantly and enhances the population of the very high energy riutrons which could lead to fast fissions in U-238.
E or S is negative in this region for the same reasons as cited above in the analysis for the single fuel assembly.
The above two models have shown by plausibility and heuristic arguments and analysis that the quantity E, excess time defined as the algebraic difference between the average neutron lifetime (t) and the average neutron reproduction time (A) fits the quali-l tative behavior of an entropy-like parameter for the neutron chain-reacting process.
Inverse S actually is a spectral importance for each state of the system. The S curve also expresses the degree to which the energy of the system has ceased to be available energy.
In essence, thermodynamically speaking, reactivity or k
--criticality relates to the first law which gf7 tells us which neutron process can take place while the second i
{
i l
i 16 law-excess time or relative entropy-tells us in which direction these processes will go.
I V SUMNMRY AND CONCLUSIONS The above non-rigorous relative-entropy development of a neutron chain reaction can be viewed as an independent formulation of an importance function developed entirely from the forward (rather than the adjoint) solution of the Boltzmann equation.
This development is simple without the necessary complex mathematical contrivances and interpretations attendant in the Adjoint-Importance concepts of reactor theory. This latter aspect is a virtue by itself.
i Although much more theoretical and computational analyses is wanting in the present development, it has surfaced some interesting inconsistencies which warrant resolution.
For example, a basic simplifying assumption in the standard neutron kinetics solution is to assume a one group reactor at critical and take A equal to 1 For real reactors, which are multigroup A does not equal 1 at critical. To what extent this inconsis-tency effects Reactor Noise theory applied to kinetics would be interesting and could possibly elucidate this fundamental dis-cipline. The above inconsistency is usually resolved in safety kinetic studies by using the most conservative A and also noting that definitions of basic parameters can be conveniently altered since the behavior of the solution of the kinetic equations depends on f/A and pi/A rather than on f, A or pi separately.
i 17 In addition, the fundamental approach to reactor control has classically relied on designs with maximum prompt neutron reproduction time, A.
The present analysis indicates that E(=2.-A) should be maximized to generate the smallest change in k,77 relative to the changing parameter (which was the moderator density in the models).
P f
l l
18 l
APPENDIX For details of the Monte Carlo iterative technique used to generate kgff [e.g., solving the Boltzmann equation in neutron generation format, which defines k as the ratio of the number eff of neutrons in the (n + 1) generation to the number of neutrons 6
in the n generation], the KENO IV manual should be consulted.
For lifetime (t) and reproduction ( A) in KENO IV (reproduction time A is called generation g in KENO IV program) the following quantities are necessary and are defined for the calculation of these quantities.
Fundamental is that of the neutron's velocity v, which is defined as fyy)1/2 v-T kl where v = velocity, cm/s Y = average energy in eV at the midpoint of the lethargy interval
-24 M = mass of the neutron,1.67 X 10
-12 l
(There are 1.602 X 10 erg /ey), therefore v = 1.3859 X 10 Y, where l
Y = [(E,3)(E )]l/2 j
j l
The quantities E,) and E, in units of eV, are the energies at j
9 the bounds of the group interval.
1 l
19 l
Lifetime and reproduction time calculations utilize an elapsed time, t, defined es the distance traveled by the e
neutron multiplied by 1/v. This time is collected at boundary crossings and is utilized whenever a collision occurs.
The reproduction time, A is defined as fe NNG NOG f
.NNG 3
g N0G where factor)
W = fission weight of the neutron (includes vrf f
t = elapsed time as defined above e
NOG = number of generations that were calculated NNG = number of neutrons per generation.
Thus, reproduction time is the average time it takes a neutron to produce another neutron via fission in the system.
The neutren lifetime, t, is defined as (W,t, + W t,)
L NNG i
N0G N0G -
L where N0G W, = absorption weight of the neutron W = weight. associated with a neutron that leaks from the g
system.
t, = elapsed Mme as above
20 and where NOG and NNG are defined as above.
Thus, the lifetime is the average life span of a neutron in the system i.e.,
until it escapes from the system or is absorbed.
One caution is cited in the KENO-IV manual.
If differential slbedos are used, the lifetime and reproduction time are incorrect, Calculation of the two models in this paper did not use differential al bedos.
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l.
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21 REFERENCES 1.
R.K. Osborn and S, Yip, "The Foundations of Neutron Transport Theory", Gordon & Breach, New York,1966 2.
C.R. Marotta, Nucl. Sci. Eng., 77, 107 (1981) 3.
J.D. Lewins, Nucl. Sci. Eng., 78, 105 (1981) e.
4.
C.R. Marotta, Nucl. Sci. Eng., 78, 106 (1981) 5.
Here, U(3) designates uranium enriched in the U
isotope to 3 wt%.
6.
L.M. Petrie and N.F. Cross, " KEN 0 IV, An Improved Monte Carlo Criticality Program," ORNL-4938 Oak Ridge National Laboratory (Nov.1975).
7.
"123 Group GAM-THERMOS Neutron Cross Section Set,"
available from Oak Ridge National Laboratory Reactor Shielding Information Center.
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i ACKNOWLEDGEMENT i
,i It is with great pleasure that I acknowledge fruitful technical discussions in the area of thermodynamics as well as constant encouragement 1
from my colleagues Jerry E. Jackson and William H. Lake at NRC.
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