VISUALIZATION AS A TOOL FOR PLUTONIUM PROLIFERATION PROTECTION ASSESSMENT
E.G. Kulikov
National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), Moscow, Russian Federation
Contents
2. Model of implosion-type HNED
3. Criteria defining implosion-type HNED non-functionality
4. Measures to prolong lifetime of implosion-type HNED
5. Why should transient warm-up of implosion-type HNED be considered?
6. Recommendations on plutonium proliferation protection
Abstract
The advancement of the nuclear power industry can come together with nuclear materials suitable both for peaceful and military applications circle. So, mathematical methodologies should be developed for the correct resolution of the following problem: what nuclear materials could be recognized as proliferation protected ones, i.e. what nuclear materials are unsuitable for manufacturing a nuclear explosive device (NED).
As known, there are two fundamentally different types of NED: gun-type and implosion-type [1]. The computational-theoretical model of gun-type device has been considered in author’s previous paper [2].
In the present paper computational-theoretical model has been developed and visualized, which aims at quantitative assessment of plutonium proliferation protection against creating implosion-type hypothetical nuclear explosive device (HNED). The main nuclear-physical and thermal processes that occur in implosion-type HNED are being modeled.
The model allows us to develop requirements that guarantee non-functionality of implosion-type HNED. This is achieved by introducing radioactive isotope (238Pu) into plutonium, the heat of alpha-decay of which could overheat HNED and render it non-functional.
The present paper is an essential development of previous works devoted to plutonium proliferation protection.
Firstly, it has been demonstrated that it is necessary to analyze the transient warming-up process of implosion-type HNED to determine how soon such a device loses its functionality taking into account various compositions of plutonium and other materials used, as well as different methods of heat dissipation.
Secondly, the possibility has been analyzed to slow down warming-up process of implosion-type HNED by means of:
· preliminary cooling of HNED components;
· encircling of HNED with a heat sink layer;
· introducing thermal isolating inter-layers into HNED structure for purposeful re-distribution of temperature field in such a way when device keeps its functionality the longest.
These aspects are not only fundamentally new, but extremely important, as they affect plutonium proliferation protection estimations drastically (several times difference).
As plutonium proliferation protection issues concern not only engineers, but also representatives of some of the humanities, visualization of these issues is essential. Moreover, the material presented in such a form is much easier for students to understand.
Key words: plutonium proliferation protection, implosion-type hypothetical nuclear explosive device.
1. Introduction
In 1977, it was announced that NED on the basis of reactor-grade plutonium has been successfully tested in the United States in 1962 [3]. So at the end of the 1970s, when strategies of closed nuclear fuel cycle were analyzed in the framework of IAEA [4], the question has arisen of reactor-grade plutonium proliferation protection. This problem has been investigated in detail by A. DeVolpi [1], who concluded that “it appears that the best strategy to prevent the use of plutonium for military purposes is the introduction of isotope 238Pu into it, as well as other available even isotopes of plutonium”.
Concluding that it seems perspective to ensure plutonium proliferation protection by introduction of isotope 238Pu into it, A. DeVolpi notes that quantitative evaluation of required 238Pu amount is needed.
Consideration of this issue (quantitative evaluation of 238Pu amount to ensure plutonium proliferation protection) was first given in paper [5] (1980). Since plutonium is a powerful source of spontaneous neutrons and it is impossible to create an effective gun-type HNED on its basis, the model of implosion-type HNED has been considered in paper [5]: the sphere of fissile material (plutonium), surrounded by layers of temper, chemical high explosive (HE) and outer casing. It has been concluded that introduction of at least 5% 238Pu into plutonium leads to such high temperatures that chemical HE is melted, and hence implosion-type HNED based on this plutonium composition would be non-functional [5].
Further development of the approach to assess 238Pu amount, required for plutonium proliferation protection, was represented in the paper of German scientist G. Kessler [6] (2007). Compared with earlier paper [5] the paper of G. Kessler has the following advantages:
· radius of plutonium sphere is chosen on the basis of criticality calculations (in contrast to [5], when radius of plutonium sphere was considered as a range of values without reference to the criticality);
· the analysis of different chemical HEs properties has been performed to provide NED non-functionality when using any of the available chemical HEs;
· the fact was taken into account that HNED can lose its functionality not only because of chemical HE melting, but also due to its self-ignition.
Paper [6] concludes that plutonium containing 6% 238Pu or slightly more can be considered as proliferation protected material.
It seems that the major disadvantage of papers [5, 6] is the fact that the conclusion on plutonium proliferation protection is based on the consideration of equilibrium (asymptotic) temperature profile in HNED layers, achieved after an infinite warm-up time. It is obvious that the loss of HNED functionality after an infinite warm-up time is not a sufficient condition of plutonium proliferation protection: the loss of HNED functionality should be achieved after a sufficiently short time of warm-up. This is the case when one can talk about unsuitability of such a device for practical use, and thus one can talk about plutonium proliferation protection.
In the present paper the analysis and visualization of transient process of HNED warm-up has been performed for the first time ever, to determine how soon such a device loses its functionality taking into account various compositions of plutonium and other materials used, as well as different methods of heat dissipation. With this in mind, it can be stated that the present work is fundamentally different from previous works on the subject, and it can be considered as the next step, expanding the capabilities of the analysis of this problem. Consideration of transient process of HNED warm-up also means the possibility of taking into account the material properties depending on temperature, which may vary within wide limits. In addition, the possibility of slowing down the process of HNED warm-up with additional measures has been analyzed (see section 4).
2. Model of implosion-type HNED
The principle model of implosion-type HNED is presented in Fig. 1 [6, 7].
Central spherical charge of plutonium is compressed by a blast of chemical HE. Converging detonation wave rapidly increases the density of plutonium and transfers it from subcritical state to supercritical one.
Strictly speaking, two chemical HEs with different detonation velocities are used in implosion-type HNED [6]. It is due to specially chosen structure of two chemical HEs with different detonation velocities that one can achieve a concave shape of detonation wave front, thereby ensuring spherically symmetric compression of plutonium.
The precise three-dimensional structure of two chemical HEs is not given in the open literature. Therefore, let us follow the approach used in paper [6], when homogeneous structure of HE is considered, which is justified by the fact that the most thermal conductive and heat-resistant chemical HE is chosen, which imposes the most stringent requirements on the heat source power (i.e. on the content of 238Pu) for plutonium proliferation protection.
Paper [6] suggests three technology levels of HNED (low, middle and high), which correspond to a different thickness of HNED layers (Fig. 1) and also to different thermal conductivity and thermal stability of HE. The high-technology option was chosen for a consideration, which corresponds to minimal sizes of HNED layers and to the use of HE with the highest thermal conductivity and thermal stability – TATB [8, 9].
Fig. 1. Principle model of implosion-type HNED
Implosion-type HNED can be brought into an ineffective state by introducing a heat source into the fissionable material for overheating. The isotope 238Pu is considered as the heat source (567 W/kg), because among other plutonium isotopes it is the most intense source of energy by the heat of α-decay [5, 6].
It is necessary to take into account all the measures that may extend the period of time during which HNED keeps functionality (let us call this period as lifetime of HNED and denote it as “Δτ”). So, the required level of plutonium proliferation protection by an admixture of isotope 238Pu as an intense heat source may be justified only after multiple numerical studies on dynamics of NED warm-up under various conditions of artificial heat removal. As a result, the most dangerous case must be revealed with the longest lifetime of HNED. Content of 238Pu in plutonium, which provides a short enough lifetime of HNED even in this most dangerous case, could be regarded as sufficient for protection from plutonium proliferation.
As it was already mentioned, the following measures could prolong lifetime of HNED (for details, see section 4):
· preliminary cooling of HNED components;
· encircling of HNED with a heat sink layer;
· introducing thermal isolating inter-layers into HNED structure for purposeful re-distribution of temperature field in such a way when device keeps its functionality the longest.
Let us assume that multi-layer implosion-type HNED has spherically symmetric geometry. Let us also assume that the source of α-decay heat is spherically symmetric, boundary conditions on the outer surface are also spherically symmetric, and the initial temperature distribution is also described by a spherically symmetric function.
In this case temperature distribution in HNED layers can be determined by solving the following non-stationary equation of heat transport in one-dimensional spherical geometry:
(1)
where: l (r, T), cV (r, T) – thermal conductivity and volumetric heat capacity, which depend on temperature field T (r, t);
T (r, t) – temperature in spatial point r at time t;
qV (r) – intensity of heat source.
Temperature distribution could be written as a finite series of continuous functions of separated variables:
, (2)
where spatial functions ψi(r) are eigenfunctions of operator equation (orthogonality should be taken into account):
, (3)
where αi – eigenvalues of operator equation (3).
Substituting temperature distribution as (2) into the
original equation (1), one can obtain the equation for time functions 
:
. (4)
The solution of equation (3) could be found using algorithm of “double-sweep”, which is a combination of “direct sweep” from the center to the periphery, within which the coefficients of the finite-difference quotient are being found, and “reverse sweep” from the periphery to the center to find the values of the unknown function ψi(r) at all spatial coordinates. For the calculations 10 coordinates per each layer of implosion-type HNED were considered. The above-described algorithm for the numerical solution of the equation (1) has been implemented as a computer code in programming language Fortran.
As for thermal conductivity and volumetric heat capacity dependences on temperature: for plutonium, uranium, aluminum and stainless steel one can use a linear interpolation of table data for a set of temperatures [7, 10]. Thermal conductivity of some chemical HEs at various temperatures is presented in paper [7], and there is the following formula for volumetric heat capacity of chemical HEs [11]:
(5)
where: 
= 4,31 J/(cm3∙K);
;
T0 = 298 K;
Tc = 560 K.
Boundary conditions are defined by conditions of heat removal:
1. “ideal” heat removal: T(RS, τ) = TS. Temperature of HNED outer surface (r = RS) is kept constant. In practice this is achieved that HNED is surrounded by a layer containing a substance undergoing a phase transition in the temperature range of warm-up. For example, HNED could be surrounded by a layer containing liquid nitrogen, which evaporates and goes into environment through special holes (Fig. 2a);
2. “ideal” thermal
isolation (Fig. 2b): 
3. heat removal by natural air convection and radiation (Fig. 2c).
|
a) “ideal” heat removal |
b) “ideal” thermal isolation |
c) heat removal by natural air convection and radiation |
Fig. 2. Boundary conditions
3. Criteria defining implosion-type HNED non-functionality
Chemical HE is the least thermostable component of implosion-type HNED. Chemical HE can lose its effectiveness not only at its melting or self-ignition point but also because of pyrolysis (high-temperature dissociation of chemical structure, not considered in papers [6 – 8]), as a result of which gaseous products are accumulated, that are able to destroy chemical HE.
According to [9], a criterion for chemical HE non-stability is 2% pyrolysis of its molecules. The rate of HE pyrolysis may be evaluated from the Arrhenius equation [12]:
, (6)
where: W(T) – rate of HE pyrolysis at temperature T;
 – pre-exponential factor;
Eact – activation energy for pyrolysis;
R0 – universal gas constant (8.31 J/(mol×K)).
The paper [9] has information about pre-exponential factor B and activation energy for pyrolysis Eact that are derived from experiments. Knowing rate of HE pyrolysis W(T), it is easy to determine the fraction of chemical HE, gone under pyrolysis at time τ:
(7)
For the majority of chemical HEs melting occurs at a lower temperature than self-ignition, i.e. melting is more stringent criterion. At the same time, for different chemical HEs melting can occur either prior or after 2% pyrolysis of its molecules. For instance, for TNT melting occurs prior to 2% pyrolysis of its molecules, while for TATB 2% pyrolysis of its molecules occurs prior to melting.
Thus, one can conclude that depending on the chemical HE the most stringent criterion defining implosion-type HNED non-functionality can be both chemical HE melting and 2% pyrolysis of its molecules.
4. Measures to prolong lifetime of implosion-type HNED
Preliminary cooling of NED components. It is an evident fact that, if fixed heat source is present in fissionable materials, then preliminarily cooled HNED is able to maintain its functionality (before overheating and failure) for a longer time interval as compared with HNED without a special preliminary cooling.
Apropos, heat capacity of nuclear materials under cryogenic temperature conditions is very small, as a rule. So, it makes no sense to tend to the utmost feasible pre-cooling of HNED components for reaching the utmost possible prolongation of HNED lifetime. Numerical evaluations showed that pre-cooling of HNED components below the temperature of liquid nitrogen (77 K) produced a relatively weak effect on the intensity of the heat source needed for plutonium proliferation protection: HNED pre-cooled to 4 K requires only a 3% more intense heat source than HNED pre-cooled to 77 K.
That is why the further studies considered the options with HNED components pre-cooling to 77 K, except for central plutonium charge. Initial temperature of plutonium charge was chosen to be 198 K because of the following reason. At temperatures below 198 K, plutonium stabilized in δ-phase can be converted into α'-phase with corresponding sharp change of volume, and the way back to δ-phase is not completely reversible for plutonium [13]. These phase transitions can result in partial or complete loss of HNED functionality.
After pre-cooling process, HNED components are isolated from the environmental thermal impacts for the utmost long conservation of the pre-cooled state.
Pre-cooling of HNED components to 77 K results in higher intensity of internal heat source needed for reliable proliferation protection of plutonium. Such a pre-cooling process increases the required intensity of heat source on about 50% as compared with no pre-cooling option.
Encircling of HNED by the heat sink layer. “Ideal” heat removal option (temperature of HNED outer surface is kept constant) may be provided by surrounding the HNED with a heat removing layer made of material with high thermal conductivity or phase transitions in the potential temperature range. Indeed, such a layer is able to absorb all the heat that comes without warming up because this heat is consumed by a melting or evaporation process.
Let’s assume the heat absorbing layer consists of two materials: for example, 25 vol. % - a metal with good heat conductivity (aluminum) and 75 vol. % - a material, which can undergo a phase transition at the temperature of preliminary cooling (liquid nitrogen).
Thickness of heat absorbing layer is chosen in a way that this layer is able to absorb all the heat that comes without any rise in temperature during the whole period of HNED warming up till the device loses its functionality.
Encircling of HNED by the heat sink layer to provide an “ideal” heat removal option results in higher (by about 15%) intensity of heat source needed for plutonium proliferation protection.
Introduction of thermal isolating inter-layers into HNED structure for purposeful re-distribution of temperature field. It has been found out that implosion-type HNED loses its effectiveness because of too high HE temperatures, while the temperatures of “internal” HNED layers (plutonium, uranium and aluminum) are far enough from their thermal limitations (from melting temperatures, for instance). So, it seems reasonable to apply preventive measures against heat transport from “internal” HNED layers to the HE layer for prolongation of HNED lifetime. For example, a thin inter-layer (5 mm) of material with low thermal conductivity may be introduced between the aluminum layer and the HE layer.
The thermal isolating inter-layer has to be made from a material with low thermal conductivity and high enough thermal stability. Silica aerogel could be a suitable material. It is used in construction area as an insulating and heat-retaining material [14]. Indeed, it has very small thermal conductivity of about 0.017 W/m∙K at normal pressure and it can be used at temperatures up to 1200° Ñ [14].
Introduction of a thermal isolating inter-layer into HNED structure for purposeful re-distribution of temperature field leads to an increase of the heat source, required for plutonium denaturing, by over 50%.
Altogether, it could be expected that applying all three considered measures at the same time, allowing the prolongation of HNED lifetime, would more than double the required heat source.
5. Why should transient warm-up of implosion-type HNED be considered?
Let’s demonstrate the need to consider transient process of implosion-type HNED warm-up (i.e., the need to consider the time) in the following example (Fig. 3). Plutonium of such isotopic composition is used as the fissionable material, which is the source of 710 W heat. Generated heat is removed from the outer surface by natural air convection and radiation. For clarity, in this example the following criterion of HNED non-functionality is considered: heating of HE (TATB) up to self-ignition temperature, which is 347 0Ñ.
Fig. 3. Visualization of transient warm-up of implosion-type HNED (heat removal by natural air convection and radiation)
It is assumed that at the initial moment of time all HNED layers have temperature of 27 0Ñ. Fig. 3 shows the temperature distribution in HNED layers after 2 and 5 hours of heating, and the asymptotic distribution. It can be seen that asymptotic equilibrium maximal temperature of HE (at its inner surface) is equal to 347 0Ñ, i.e. HE self-ignition and full loss of HNED effectiveness.
However, after a shorter time of heating (see curves for 2 and 5 hours) HE is not warmed up to such a high temperature, and hence HNED will keep its functionality. Therefore it would be wrong to state that plutonium, which is the source of 710 W heat, is proliferation protected material basing on the fact that HNED loses its functionality in the asymptotic state.
In order to achieve HE self-ignition temperature rather fast, more powerful heat source in plutonium is required, which can be determined only on the basis of transient warm-up of implosion-type HNED consideration.
Thus, asymptotic model, which is based on the temperature profile after an infinite time of heating, underestimates power of heat source, which is required for plutonium proliferation protection. Papers [6, 7] consider only asymptotic model, which due to above-mentioned reasons seems not quite justified: important is not just a fact of HNED non-functionality, but how fast it happens.
Let us note that in the framework of the current task, when spherically symmetric HNED is studied, for the means of scientific visualization two-dimensional graphics are the most convenient to use, as there is only one spatial coordinate – radius-vector r.
6. Recommendations on plutonium proliferation protection
The present section considers the possible simultaneous application of all the listed above measures for prolongation of HNED lifetime. This investigation can enable us to elaborate requirements for the magnitude of heat source intensity (238Pu content) for effective proliferation protection of plutonium.
As an example let us consider HNED lifetime being equal to 5 hours: it is unlikely that in such a short time it is possible to assemble and transport HNED.
Fig. 4 shows temperature profiles in HNED for two options:
a) plutonium melting results in full loss of HNED functionality: one heat isolating layer is introduced into the HNED structure, and, as a consequence, spatial temperature field is re-distributed in such a way that plutonium melting and 2%-dissociation of chemical HE take place simultaneously;
b) plutonium melting does not produce a negative effect on HNED functionality: three heat isolating layers are introduced into the HNED structure, and, as a consequence, the spatial temperature field is re-distributed in such a way that ultimate possible temperature of inner heat isolating layer, uranium melting, aluminum melting and 2%-dissociation of chemical HE take place simultaneously.
a) plutonium melting results in full loss of HNED functionality
b) plutonium melting does not produce a negative effect on HNED functionality
Fig. 4. Visualization of transient warm-up of implosion-type HNED (application of all the measures for prolongation of HNED lifetime)
Let us assume that plutonium melting does not produce a negative effect on HNED functionality, which is “upper estimation”. In this case on the basis of chosen criterion for plutonium proliferation protection (HNED on its basis should be functional less than 5 hours) one can conclude, that necessary heat power is 3100 W (Fig. 4b). Thus, it has been demonstrated that only plutonium containing, at least, 42% 238Pu may be regarded as a proliferation-protected plutonium.
7. Conclusion
On the basis of open literature data computational -theoretical model has been developed and visualized, which allows us to develop requirements that guarantee implosion-type HNED non-functionality and thus plutonium proliferation protection.
The novelty of the implosion-type HNED model is the analysis of transient warm-up of the device, as well as consideration of measures to slow down warm-up process: preliminary cooling, encircling with a heat sink layer, introducing thermal isolating inter-layers into HNED structure.
It is been demonstrated that plutonium can be considered as a proliferation-protected material if it contains at least 42% 238Pu (implosion-type HNED on its basis is functional less than 5 hours – it is unlikely that in such a short time it is possible to assemble and transport HNED).
Visualization of plutonium proliferation protection issues is essential, as this subject concerns not only engineers, but also representatives of some of the humanities. Also, students tend to understand the material easier when it is presented in this way.
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