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Neutron flux density profiling during iridium irradiation
~
Pergamon
Appl. Radiat. lsot. Vol.48, No. 10-12,pp. 1697-1701,1997 © 1997ElsevierScienceLtd. All rights reserved Printed in Great Britain PII: S0969-8043(97)00169-3 0969-8043/97 $17.00+ 0.00
Neutron Flux Density Profiling During Iridium Irradiation V. A. T A R A S O V a n d YU. G. T O P O R O V State ScientificCentre 'Research Institute of Atomic Reactors', Dimitrovgrad-10, 433510, Russia Using the perturbation method, the dependence of the specific activity of ~92Iron the variation of neutron flux density is investigated. It has been shown that sensitivity of the specific activity of ~92Irvalue at a given irradiation time depends on the time-point of neutron flux perturbation, being greater if perturbation is introduced at the end of the irradiation cycle. Practical realisation of a proposed scheme will allow an increase in ~92Irspecificactivity together with savings in reactor fuel, due to the lower levelof reactor power at the beginning of the cycle. © 1997 Elsevier Science Ltd. All rights reserved
Two research reactors (high-flux SM reactor and channel-type MIR reactor) are used at the Research Institute of Atomic Reactors (RIAR) for generation of '92Ir. The MIR reactor is characterised by ability of local power control in particular fuel assemblies (hereafter referred to as FA) and hence in irradiation channels. Such control is carried out by rods of the control and protection system (CPS) as well as a special arrangement of the reactor core. The maximum possible power in each particular irradiation channel may be varied from campaign-by-campaign and is determined by power restriction in both total reactor and particular FA. At first the base cycle of iridium irradiation in the MIR reactor was taken as follows: 25-day operation of irradiation channel at power of 1.8 MW, shut-down for fuel reloading and once more 25-day operation of irradiation channel at power of 1.8 MW (parameters of the base cycle are summarised in Table 1). Thus, the required value of tg'-Ir specific activity was obtained by the total power generation of 90 MW-days. As applied to the MIR reactor, the following problem has been formulated and solved: to provide the maximum possible economy of total power generation of the irradiation channel with respect to the base cycle (see above) while conserving the general duration of the irradiation cycle and the achieved value of t9qr specific activity. A precondition for resolving the problem was the possibility for power profiling in the irradiation channel during irradiation. It should be noted that such a problem is one of the undergenerate optimisation problems (with finite solution) in the field of kinetics of nuclide generation, some of which have been formulated by Kruglov and Rudik (1985).
As applied to the problem of 1921r irradiation, the radionuclide specific activity at a given irradiation time was determined as a function of power variation in the irradiation channel (or level of neutron flux). The investigations were performed by the methods of perturbation theory, for construction of which the formalism of adjoint functions was used. The method was based on the consistent solution of the system of differential equations describing the variation of nuclide concentration with time t: dn d-~ = [Aln(t)
(l)
and the systems of corresponding adjoint equations: dn dt - [A]Tn*(t)
(2)
where n(t) is the concentration of nuclides; [A] is the matrix of nuclear reaction rates; T is transposition and n*(t) is the importance function. Let us consider only the main idea of the perturbation theory without a full description of the methods for construction of the perturbation theory (the formalism is stated by Gandini et al. (1977) and Kallfelz et al. (1977)). The approximate solution of the perturbed problem can be performed at rather small perturbations by using only the solution results of the unperturbed problem. To change the concentration of ~92Irnuclide to the fixed irradiation time (Tit) due to the matrix perturbation of ([AA]j) nuclear reaction rates at the jth irradiation interval, we can write: 6nzj(Tir) = ['|'~'n*r(t)[AAbn(t)dt
(3)
i IJIi/
where t~j and t2j are the beginning and end of time intervals of perturbation action, respectively.
1697
1698
V.A. Tarasov and Yu. G. Toporov
Table 1. Parameters of iridium irradiation base cycle in the MIR reactor Parameters of cycle Value Irradiation time (days) 50 Density of thermal neutron flux (F) (cm 2 s ~) 1.05 × 10E4 Density of epithermal neutron flux (~p)(cm -' s-~) 1.1 x 10~ Temperature of Maxwell spectrum (K) 550 Optical thicknessof iridium specimens(cm) 0.06 Irradiation channel power (MW) 1.8 The values of neutron fluxes are brought to 1 MW power of FA. If there are N intervals, then: N
6n2(Tir) ,~ Z 6n2j
(4)
j=l
Let us refer to the mathematical formalism of the optimisation problem under consideration. When taking the base cycle of iridium irradiation as an unperturbed problem and denoting the coefficients of power profiling in the irradiation channel on j t h interval as K,, we can write: • to conserve the value of t92Ir specific activity for a fixed irradiation time N
cSn2(Tir)= ~ 6n2j = 0
(5)
j=l
• to limit the power of the irradiation channel: K~ m < K j _ < K m~x/for j =
1..... N
(6)
• for maximisation of total power generation economy: N
~" Kj = min
(7)
j=l
Taking equation (7) as an objective function and assuming the left part of equation (5) (for the problem under consideration) to be the sum of linear functions with respect to Kj, we obtain the optimisation problem to be resolved by the methods of procedures theory. It is known that for the linear chains of accumulation without reverse supply [(n,2n) reaction type] at the constant reaction rates the system of equation (1) (hence, adjoint system (2)) has a simple analytical solution, obtained by Bateman (1910). Using such solutions for n(t) and n*(t), equation (3) also has an analytical solution that allows us to write equation (4) in the form: N
Z sjn2j = s
(8)
j=l
where sj and S are numerical coefficients, the values of which are determined by the nuclear constants of chain and by the irradiation parameters. The considered ~gtlr(n,7)J92Ir chain is linear; however, the rates of nuclear reactions are varied in
time with variation of iridium composition under irradiation. To perform the constancy conditions of nuclear reaction rates it is necessary to determine their average values at the irradiation time. The average values of resonance self-shielding coefficients and the average value at irradiation time of the depression coefficient of thermal neutron flux in the volume of irradiated iridium specimens have been determined by 'accurate' calculation (considering the variation of every time-dependent parameter). The rate values of nuclear reactions were obtained by means of the nuclear constants (Mughabghab et al., 1981) and two-group representation for the base cycle of iridium irradiation (see Table 1): • depletion rate of ~gqr nuclei: a . = 9 . 5 x 10 -8 s ~; • accumulation rate of ~921r nuclei: a2~ = a H = 9.5 x 10 8 s - t ; • depletion rate of 192Ir nuclei: a n = 3 . 8 x 10 7s-a. The authors have developed a specific computer code using the averaged values of nuclear reaction rates as initial data and allowing us to determine concentrations, nuclide importance functions (or another factor of accumulation, e.g. specific activity) with perturbation of some irradiation parameters (e.g. power, group fluxes, nuclide cross-sections). The same computer code was used to calculate the coefficients in equation (8). Concerning the base cycle of iridium irradiation, the time-dependent ~92Irnuclei importance function is given in Fig. 1. Any perturbation introduced close to the end of the irradiation cycle will cause the most changes to the nuclei concentration of the nuclide. Indeed, concerning the sensitivity (the ratio of relative change of nuclide accumulation to relative change of varied parameter, %) of 192Ir specific activity as a function of introduced perturbation time (variation of irradiation channel power) it should be noted that the response to the perturbations performed at the beginning of the irradiation cycle is rather small and increases sharply for perturbations introduced at the end of cycle (Fig. 2). Based upon the data shown in Figs 1 and 2, the conclusion can be drawn that it is reasonable to decrease the power of the irradiation channel (relative to base one) at the beginning of irradiation and at the expense of its disproportional increase at the end of the cycle (because the response scale is greater, see Fig. 2) one can obtain economy of total power generation, achieving therewith the required value of ~92Ir specific activity. Let us consider a numerical solution. Using two-step profiling (easily realised in practice) and dropping calculations we obtain: 0.058K~ + 0.112K2 = 0.170 (equality (8));
K,,K2 > 0 (restrictions (6));
Neutron flux density during Ir irradiation
1699
1
/
0,9
/
J
0,8 0,7 ,..q 0,6 =. 0,5 h,.,
f
0
0,4 0,3 0,2 0,1 0
f 10
20
30
40
50
Irradiation time, days Fig. 1. Importance function vs irradiation time of 192[rnuclei. K~,K2 < K r~ax (restrictions
(6));
Kt + K2 = min (objective function (7)), where K~, K2 and K max are coefficients of irradiation channel power profiling during the first half of the iridium irradiation cycle, during the second half of the cycle, and maximum permissible cycle, respectively. The results of the numerical solution of the optimisation problem formalised by us for the different values of K max(in fact, K2 is always equal to
K max, which confirms the qualitative conclusion drawn above) are presented in Table 2. It should be noted that no first half of irradiation cycle is required in general at g maX= 1.5, while economy of total power generation reaches its maximum value at about 25% (later we'll return to discussion of data given in Table 2). To check the correctness of the applied perturbation method (actually method of small perturbations), a series of 'accurate' calculations of ~921r accumulation factors has been performed. In
10 9 8 7
3 2' 1 0 0
10
20
30
40
50
Irradiation time, days
Fig. 2. J92Irspecific activity sensitivity vs time of perturbation initiation (perturbation period is 5 days).
1700
V. A. Tarasov and Yu. G. Toporov Table 2. Economy of total power generation at two-step power profiling of irradiation channel (relative to base one) Profiling coefficient at Profiling coefficientat second irradiation stage, first irradiation Economy of total power K2 = Km'" (rel. unit) stage, K~ (rel. unit) generation (%) 1.05 0.90 2.5 1.10 0.81 5.5 1.20 0.6t 9.5 1.30 0.42 14.0 1.40 0.23 8.5 1.50 0,00 25.0
accordance with the functions s h o w n in Fig. 3, power profiling during irradiation allows the decrease in total power generation required for achievement of the given specific activity, whereas the conditions without profiling result in a significant decrease in the values o f 192Ir specific activity. It should be noted that in the 'accurate' calculations we used Kj values from Table 2. The high convergence of the results demonstrates the accuracy of the m e t h o d s applied to solve the optimisation problem. The above cited principles dealing with power profiling under irradiation are physically substantiated. Using B a t e m a n ' s formulae we can determine total 192Ir nuclei depletion by radioactive decay:
N~ =
2 × a21/(a22- aN)
× [exp( - allt) - exp( - a22t)]dt
E x p a n d i n g the exponential functions into a series (only the first three terms are used) one can o b t a i n the following: N~ ,~ 1/22 x a21 x Ti:r ~ F
N,.> ,,~ 1/2(a22 - 2) × a21 × T~ ~ F 2
Cycle ~ t h profiling )
/ 0
m
0,9
0,8
0,7 0
5
10
(10)
where F n e u t r o n flux density is p r o p o r t i o n a l to the power of the irradiation channel. In a similar way an expression m a y be obtained to determine ~9:Ir nuclei depletion by n e u t r o n capture:
1,1
f
(9)
15
20
25
Total ¢aotgy g~erate economy, %
Fig. 3. tg:Ir specific activity achieved at the cycle end as a function of total energy generates economy.
(1 1)
Neutron flux density during Ir irradiation
1701
Irradiation chennei p o ~ r profiling
600
500
400
'~ 300
d"3 200
100
0
5
10
15
20
25
30
35
40
45
50
55
Irradiation time, days Fig. 4. ~921raccumulation at base conditions and power profiling of irradiation channel. Taking into account that accumulation of ~92Ir nuclei is also proportional to the square of neutron flux, then with an increase in neutron flux (e.g. due to power growth), depletion of radioactive nuclei increases rather less relative to the accumulation. The absolute magnitude of the effect is determined by the nuclei quantity of the objective radionuclide in the specimen under irradiation (see above). This means that for any chain of accumulation we can formulate and solve the considered optimisation problem. The result obtained will be dependent on the technical abilities of power control of the irradiation channel (or the reactor as a whole), while the maximum gain is limited by a relative contribution of radioactive decay in the total depletion rate of the objective radionuclide. As an example, consider the conditions of power profiling realised in practice (Fig. 4). Economy of the total power generation for the cited two-step
conditions was about 7%. It has been possible to reach a maximum economy of up to 15% due to restriction of K .... (see Table 2).
References Bateman, H. (1910) The solution of a system of differential equations occurring in the theory of radioactive transformations. Proceedings of the Cambridge Philosophical Society 15, 423~427. Gandini, A., Salvatores, M. and Tondinelli, L. (1977) New developments in generalized perturbation theory. Nuclear Science Engineering 62, 339-345. Kallfelz, J. M., Bruna, G. B., Palmiotti, G. and Salvatores, M. (1977) Burnup calculations with time-dependent generalized perturbation theory. Nuclear Science Engineering 62, 304-309. Kruglov, A. K. and Rudik, A. P. (1985) Reactor Production of Radionuclides. Energoatomizdat, Moscow. Mughabghab, S. F., Divadeenam, M. and Holden N. E. (1981) Neutron Cross Section. Vol. 1, part A and B. Academic Press.
Pergamon
Appl. Radiat. lsot. Vol.48, No. 10-12,pp. 1697-1701,1997 © 1997ElsevierScienceLtd. All rights reserved Printed in Great Britain PII: S0969-8043(97)00169-3 0969-8043/97 $17.00+ 0.00
Neutron Flux Density Profiling During Iridium Irradiation V. A. T A R A S O V a n d YU. G. T O P O R O V State ScientificCentre 'Research Institute of Atomic Reactors', Dimitrovgrad-10, 433510, Russia Using the perturbation method, the dependence of the specific activity of ~92Iron the variation of neutron flux density is investigated. It has been shown that sensitivity of the specific activity of ~92Irvalue at a given irradiation time depends on the time-point of neutron flux perturbation, being greater if perturbation is introduced at the end of the irradiation cycle. Practical realisation of a proposed scheme will allow an increase in ~92Irspecificactivity together with savings in reactor fuel, due to the lower levelof reactor power at the beginning of the cycle. © 1997 Elsevier Science Ltd. All rights reserved
Two research reactors (high-flux SM reactor and channel-type MIR reactor) are used at the Research Institute of Atomic Reactors (RIAR) for generation of '92Ir. The MIR reactor is characterised by ability of local power control in particular fuel assemblies (hereafter referred to as FA) and hence in irradiation channels. Such control is carried out by rods of the control and protection system (CPS) as well as a special arrangement of the reactor core. The maximum possible power in each particular irradiation channel may be varied from campaign-by-campaign and is determined by power restriction in both total reactor and particular FA. At first the base cycle of iridium irradiation in the MIR reactor was taken as follows: 25-day operation of irradiation channel at power of 1.8 MW, shut-down for fuel reloading and once more 25-day operation of irradiation channel at power of 1.8 MW (parameters of the base cycle are summarised in Table 1). Thus, the required value of tg'-Ir specific activity was obtained by the total power generation of 90 MW-days. As applied to the MIR reactor, the following problem has been formulated and solved: to provide the maximum possible economy of total power generation of the irradiation channel with respect to the base cycle (see above) while conserving the general duration of the irradiation cycle and the achieved value of t9qr specific activity. A precondition for resolving the problem was the possibility for power profiling in the irradiation channel during irradiation. It should be noted that such a problem is one of the undergenerate optimisation problems (with finite solution) in the field of kinetics of nuclide generation, some of which have been formulated by Kruglov and Rudik (1985).
As applied to the problem of 1921r irradiation, the radionuclide specific activity at a given irradiation time was determined as a function of power variation in the irradiation channel (or level of neutron flux). The investigations were performed by the methods of perturbation theory, for construction of which the formalism of adjoint functions was used. The method was based on the consistent solution of the system of differential equations describing the variation of nuclide concentration with time t: dn d-~ = [Aln(t)
(l)
and the systems of corresponding adjoint equations: dn dt - [A]Tn*(t)
(2)
where n(t) is the concentration of nuclides; [A] is the matrix of nuclear reaction rates; T is transposition and n*(t) is the importance function. Let us consider only the main idea of the perturbation theory without a full description of the methods for construction of the perturbation theory (the formalism is stated by Gandini et al. (1977) and Kallfelz et al. (1977)). The approximate solution of the perturbed problem can be performed at rather small perturbations by using only the solution results of the unperturbed problem. To change the concentration of ~92Irnuclide to the fixed irradiation time (Tit) due to the matrix perturbation of ([AA]j) nuclear reaction rates at the jth irradiation interval, we can write: 6nzj(Tir) = ['|'~'n*r(t)[AAbn(t)dt
(3)
i IJIi/
where t~j and t2j are the beginning and end of time intervals of perturbation action, respectively.
1697
1698
V.A. Tarasov and Yu. G. Toporov
Table 1. Parameters of iridium irradiation base cycle in the MIR reactor Parameters of cycle Value Irradiation time (days) 50 Density of thermal neutron flux (F) (cm 2 s ~) 1.05 × 10E4 Density of epithermal neutron flux (~p)(cm -' s-~) 1.1 x 10~ Temperature of Maxwell spectrum (K) 550 Optical thicknessof iridium specimens(cm) 0.06 Irradiation channel power (MW) 1.8 The values of neutron fluxes are brought to 1 MW power of FA. If there are N intervals, then: N
6n2(Tir) ,~ Z 6n2j
(4)
j=l
Let us refer to the mathematical formalism of the optimisation problem under consideration. When taking the base cycle of iridium irradiation as an unperturbed problem and denoting the coefficients of power profiling in the irradiation channel on j t h interval as K,, we can write: • to conserve the value of t92Ir specific activity for a fixed irradiation time N
cSn2(Tir)= ~ 6n2j = 0
(5)
j=l
• to limit the power of the irradiation channel: K~ m < K j _ < K m~x/for j =
1..... N
(6)
• for maximisation of total power generation economy: N
~" Kj = min
(7)
j=l
Taking equation (7) as an objective function and assuming the left part of equation (5) (for the problem under consideration) to be the sum of linear functions with respect to Kj, we obtain the optimisation problem to be resolved by the methods of procedures theory. It is known that for the linear chains of accumulation without reverse supply [(n,2n) reaction type] at the constant reaction rates the system of equation (1) (hence, adjoint system (2)) has a simple analytical solution, obtained by Bateman (1910). Using such solutions for n(t) and n*(t), equation (3) also has an analytical solution that allows us to write equation (4) in the form: N
Z sjn2j = s
(8)
j=l
where sj and S are numerical coefficients, the values of which are determined by the nuclear constants of chain and by the irradiation parameters. The considered ~gtlr(n,7)J92Ir chain is linear; however, the rates of nuclear reactions are varied in
time with variation of iridium composition under irradiation. To perform the constancy conditions of nuclear reaction rates it is necessary to determine their average values at the irradiation time. The average values of resonance self-shielding coefficients and the average value at irradiation time of the depression coefficient of thermal neutron flux in the volume of irradiated iridium specimens have been determined by 'accurate' calculation (considering the variation of every time-dependent parameter). The rate values of nuclear reactions were obtained by means of the nuclear constants (Mughabghab et al., 1981) and two-group representation for the base cycle of iridium irradiation (see Table 1): • depletion rate of ~gqr nuclei: a . = 9 . 5 x 10 -8 s ~; • accumulation rate of ~921r nuclei: a2~ = a H = 9.5 x 10 8 s - t ; • depletion rate of 192Ir nuclei: a n = 3 . 8 x 10 7s-a. The authors have developed a specific computer code using the averaged values of nuclear reaction rates as initial data and allowing us to determine concentrations, nuclide importance functions (or another factor of accumulation, e.g. specific activity) with perturbation of some irradiation parameters (e.g. power, group fluxes, nuclide cross-sections). The same computer code was used to calculate the coefficients in equation (8). Concerning the base cycle of iridium irradiation, the time-dependent ~92Irnuclei importance function is given in Fig. 1. Any perturbation introduced close to the end of the irradiation cycle will cause the most changes to the nuclei concentration of the nuclide. Indeed, concerning the sensitivity (the ratio of relative change of nuclide accumulation to relative change of varied parameter, %) of 192Ir specific activity as a function of introduced perturbation time (variation of irradiation channel power) it should be noted that the response to the perturbations performed at the beginning of the irradiation cycle is rather small and increases sharply for perturbations introduced at the end of cycle (Fig. 2). Based upon the data shown in Figs 1 and 2, the conclusion can be drawn that it is reasonable to decrease the power of the irradiation channel (relative to base one) at the beginning of irradiation and at the expense of its disproportional increase at the end of the cycle (because the response scale is greater, see Fig. 2) one can obtain economy of total power generation, achieving therewith the required value of ~92Ir specific activity. Let us consider a numerical solution. Using two-step profiling (easily realised in practice) and dropping calculations we obtain: 0.058K~ + 0.112K2 = 0.170 (equality (8));
K,,K2 > 0 (restrictions (6));
Neutron flux density during Ir irradiation
1699
1
/
0,9
/
J
0,8 0,7 ,..q 0,6 =. 0,5 h,.,
f
0
0,4 0,3 0,2 0,1 0
f 10
20
30
40
50
Irradiation time, days Fig. 1. Importance function vs irradiation time of 192[rnuclei. K~,K2 < K r~ax (restrictions
(6));
Kt + K2 = min (objective function (7)), where K~, K2 and K max are coefficients of irradiation channel power profiling during the first half of the iridium irradiation cycle, during the second half of the cycle, and maximum permissible cycle, respectively. The results of the numerical solution of the optimisation problem formalised by us for the different values of K max(in fact, K2 is always equal to
K max, which confirms the qualitative conclusion drawn above) are presented in Table 2. It should be noted that no first half of irradiation cycle is required in general at g maX= 1.5, while economy of total power generation reaches its maximum value at about 25% (later we'll return to discussion of data given in Table 2). To check the correctness of the applied perturbation method (actually method of small perturbations), a series of 'accurate' calculations of ~921r accumulation factors has been performed. In
10 9 8 7
3 2' 1 0 0
10
20
30
40
50
Irradiation time, days
Fig. 2. J92Irspecific activity sensitivity vs time of perturbation initiation (perturbation period is 5 days).
1700
V. A. Tarasov and Yu. G. Toporov Table 2. Economy of total power generation at two-step power profiling of irradiation channel (relative to base one) Profiling coefficient at Profiling coefficientat second irradiation stage, first irradiation Economy of total power K2 = Km'" (rel. unit) stage, K~ (rel. unit) generation (%) 1.05 0.90 2.5 1.10 0.81 5.5 1.20 0.6t 9.5 1.30 0.42 14.0 1.40 0.23 8.5 1.50 0,00 25.0
accordance with the functions s h o w n in Fig. 3, power profiling during irradiation allows the decrease in total power generation required for achievement of the given specific activity, whereas the conditions without profiling result in a significant decrease in the values o f 192Ir specific activity. It should be noted that in the 'accurate' calculations we used Kj values from Table 2. The high convergence of the results demonstrates the accuracy of the m e t h o d s applied to solve the optimisation problem. The above cited principles dealing with power profiling under irradiation are physically substantiated. Using B a t e m a n ' s formulae we can determine total 192Ir nuclei depletion by radioactive decay:
N~ =
2 × a21/(a22- aN)
× [exp( - allt) - exp( - a22t)]dt
E x p a n d i n g the exponential functions into a series (only the first three terms are used) one can o b t a i n the following: N~ ,~ 1/22 x a21 x Ti:r ~ F
N,.> ,,~ 1/2(a22 - 2) × a21 × T~ ~ F 2
Cycle ~ t h profiling )
/ 0
m
0,9
0,8
0,7 0
5
10
(10)
where F n e u t r o n flux density is p r o p o r t i o n a l to the power of the irradiation channel. In a similar way an expression m a y be obtained to determine ~9:Ir nuclei depletion by n e u t r o n capture:
1,1
f
(9)
15
20
25
Total ¢aotgy g~erate economy, %
Fig. 3. tg:Ir specific activity achieved at the cycle end as a function of total energy generates economy.
(1 1)
Neutron flux density during Ir irradiation
1701
Irradiation chennei p o ~ r profiling
600
500
400
'~ 300
d"3 200
100
0
5
10
15
20
25
30
35
40
45
50
55
Irradiation time, days Fig. 4. ~921raccumulation at base conditions and power profiling of irradiation channel. Taking into account that accumulation of ~92Ir nuclei is also proportional to the square of neutron flux, then with an increase in neutron flux (e.g. due to power growth), depletion of radioactive nuclei increases rather less relative to the accumulation. The absolute magnitude of the effect is determined by the nuclei quantity of the objective radionuclide in the specimen under irradiation (see above). This means that for any chain of accumulation we can formulate and solve the considered optimisation problem. The result obtained will be dependent on the technical abilities of power control of the irradiation channel (or the reactor as a whole), while the maximum gain is limited by a relative contribution of radioactive decay in the total depletion rate of the objective radionuclide. As an example, consider the conditions of power profiling realised in practice (Fig. 4). Economy of the total power generation for the cited two-step
conditions was about 7%. It has been possible to reach a maximum economy of up to 15% due to restriction of K .... (see Table 2).
References Bateman, H. (1910) The solution of a system of differential equations occurring in the theory of radioactive transformations. Proceedings of the Cambridge Philosophical Society 15, 423~427. Gandini, A., Salvatores, M. and Tondinelli, L. (1977) New developments in generalized perturbation theory. Nuclear Science Engineering 62, 339-345. Kallfelz, J. M., Bruna, G. B., Palmiotti, G. and Salvatores, M. (1977) Burnup calculations with time-dependent generalized perturbation theory. Nuclear Science Engineering 62, 304-309. Kruglov, A. K. and Rudik, A. P. (1985) Reactor Production of Radionuclides. Energoatomizdat, Moscow. Mughabghab, S. F., Divadeenam, M. and Holden N. E. (1981) Neutron Cross Section. Vol. 1, part A and B. Academic Press.