temperature dynamics model of a pulsed fission material assembly

Temperature dynamics model of a pulsedfission material assembly

E.A. Bondarchenko, Yu.N. Pepyolyshev, A.K. Popov*

Joint institute for nuclear research, 141980 Dubna, Moscow region, Russia

Received 5 August 2003; accepted 3 October 2003

Abstract

Heat exchange process differential equations are considered for a subcritical fuel assemblywith an injector. The equations are obtained by means of the use of the Hermite polynomial.The model is created for modeling of temperature transitional processes. The parameters and

dynamics are estimated for hypothetical fuel assembly consisting of real mountings: thepowerful proton accelerator and the reactor IBR-2 core at its subcritical state.# 2003 Elsevier Ltd. All rights reserved.

1. Introduction

Interest in application of pulsed neutron sources (PNS) for various scientific andtechnological purposes has essentially increased in recent years. In parallel withunique instruments presently operating at pulsed nuclear reactors (IBR-2), powerfulproton accelerators with nonmultiplying targets (IPNS, LANCE, etc.), subcriticalsystems with electron accelerators (IBR-30), in Japan, Europe, Korea, China andRussia there exist a number of projects for the creation of PNS to obtain powerfulpulsed beams not only for research purposes but also for transmutation (‘‘burning’’)of long-living fission products. To do the latter, intense neutron fluxes are needed.They may be obtained, for example, using subcritical assemblies with a low multi-plication coefficient but a sufficiently powerful source of neutrons on an internal tar-get. The mean power generated by such an assembly may reach several megawatts. Ofspecial interest is therefore the determination of the temperature dynamics of ele-ments and, in the first place, of fission material (fuel) in the assembly. Naturally, the

Annals of Nuclear Energy 31 (2004) 601–617

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* Corresponding author. Fax: +7-9621-6-51-19.

E-mail address: [email protected] (A.K. Popov).

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solution of the problem depends on the PNS type. Our effort is aimed at modeling ofthe temperature dynamics of powerful pulsed subcritical systems on the exampleof some hypothetical subcritical assembly composed of the existing powerful non-multiplying neutron sources on the basis of proton accelerators and the IBR-2pulsed reactor active zone to create and complete a sufficiently universal model ofprospective PNS.1

2. Analytical representation of heat exchange processes

A subcritical assembly of fission material is designed for the multiplication ofneutrons generated periodically under the action of an accelerator of charged parti-cles. In the center of the assembly a target locates which can be made of tungsten forexample. The target is environed by fuel elements (FEs).Each FE is a rod (or a set of pellets) made of fission material. It is encased in a

hermetic metallic shell. Between the FE and the shell there is a dividing layer ofhelium for example. FEs in groups are packed into cassettes. The shell of each FE isbathed by a coolant, which may be liquid sodium, for example.

2.1. Equivalent FE (fuel element)

For the purpose of analytical estimation of heat exchange processes it is assumedthat

1. the assemblies consist of identical FEs with equal heat release,
2. the neutron density is uniformly distributed over the volume of the FE, 3. there is no heat transfer in the axial direction, 4. the flux of the coolant is uniformly distributed over all technological channels
of the assembly that are also assumed identical.

The assumptions allow one to believe that the processes in the assembly are iden-tical to those in the equivalent FE shown in Fig. 1.Since the assembly is in the state of deep subcriticality, it is possible not to discuss

separately the effect of shell heating on changes in the assembly reactivity. In thisconnection, for the sake of simplicity the dividing layer and the shell are consideredas part of the FE as their thickness is small. Fig. 2 depicts a qualitative distributionof steady-state temperatures over the radius of the equivalent FE.In a fission material assembly operating together with an accelerator of charged

particles heat exchange processes are analogous to similar processes in a pulsednuclear reactor (Popov and Rogov, 1976) and are described for radius-averagedtemperatures. To heat balance conditions there correspond the following partialdifferential equations

1 Work has been performed under ISTC project No. 1932.

602 E.A. Bondarchenko et al. / Annals of Nuclear Energy 31 (2004) 601–617

@�

@t¼ a1P� a2ð� � �Þ; ð1Þ

@�

@tþ v

@�

@x¼ a3ð� � �Þ; ð2Þ

Fig. 1. The equivalent FE. 1—FE; 10—dividing layer; 100—shell; 2—coolant channel.

Fig. 2. The distribution of the temperature over the radius. 1—FE; 10—dividing layer; 100—shell;

2—coolant; �; � —fuel temperature and coolant temperature averaged over the radius.

E.A. Bondarchenko et al. / Annals of Nuclear Energy 31 (2004) 601–617 603

ere
wh
P is the power�; � are the radius-averaged temperature of the fuel (FE) and of the coolant,respectivelyv is the speed of the coolantt is the timex is the axial (longitudinal) parameter of FE.

The parameters a1, a2, a3 are as follows


a1 ¼1

NL c1�1S1 þ c0� 0S0 þ c00� 00S00ð Þ; ð3Þ

a2 ¼ ka02 ð4Þ

a02 ¼�d0

c1�1S1 þ c0� 0S0 þ c00� 00S00; ð4aÞ

a3 ¼ ka03 ð5Þ

a03 ¼�d0

c2�2S2; ð5aÞ

where c1, c0, c00, c2 are the heat capacities, �1, �

0, � 00, �2 are the densities, and S1, S0,

S00, S2 are the cross section areas of the fuel, dividing layer, shell, coolant, respec-tively, N is the number of FEs, L is the length of the equivalent FE, d0 is the externaldiameter of the shell, k is the coefficient of heat transfer from fuel to coolant

k ¼1

r04l1

þr0r1

D0

l0þD00

l00

� �þ

1

�

; ð6Þ

where l1, l0, l00 are the thermal conductivity coefficients of the fuel, dividing layer,

shell, respectively, � is the coefficient of convective heat transfer from shell tocoolant, r1, r0 are the radius of the FE and the external radius of the shell,respectively, D0, D00 are the thickness of the dividing layer and the thickness of theshell, respectively. Eq. (6) is obtained by the traditional method (Arhan, 1970).In Eq. (2) the boundary conditions in height for the temperature � are the tem-

perature of the coolant at input (�1) and output (�2) of the channel (Fig. 1). Fig. 3shows the distribution of the radius-averaged temperature of the fuel � and of thecoolant � over the FEs height.

2.2. Transition to ordinary differential equations

Passing to height-averaged temperatures �� and �� (Fig. 3) in Eqs. (1) and (2) thereare obtained the following ordinary differential equations

d��

dt¼ a1P� a2 ��

� �; ð7Þ

d��

dt¼ a3 ��

� �þ a4v �1 � �2ð Þ; ð8Þ

where

a4 ¼ 1=L: ð9Þ

To have a complete system of equations, to Eqs. (7) and (8) there is usually addedthe equation

�� ¼ 0:5 �1 þ �2ð Þ; ð10Þ

under the assumption that the mean temperature of the coolant is an arithmeticmean of boundary temperatures.The system of Eqs. (7), (8), (10) is sufficiently simple and convenient for dynamics

analysis. It may appear, however, insufficient for analysis of some processes. So,from the system it follows that as the coolant temperature at input �1 changes jump-like its temperature at output �2 must also change jump-like but in the oppositedirection since the mean temperature cannot change jump-like according to Eq. (8).The indicated contradiction is removed if one uses the approximating Hermite

Fig. 3. The distribution of the radius-averaged temperature of the fuel � and of the coolant � over the height

of the equivalent FE. �1, �2, ��, �1, �2, �� are the boundary and height-averaged temperatures of the fuel and

coolant, respectively.


polynomial to perform the transition from the primary partial differential equations(Dorri, 1966). The method, proposed by Dorri (1966), involving the use of the Her-mite polynomial to perform the transition from partial differential to ordinary dif-ferential equations is possibly not widely known. Therefore, the Appendix containsa brief description of the method and the sequence of operations in application toour case.The Appendix presents seven ordinary differential equations that replace two

input partial differential equations as a result of application of the said method. Forthe convenience of modeling of transitional processes five equations, (A5), (A6),(A10), (A11), (A12), are combined into two differential, (11) and (12), and twoalgebraic equations, (11a) and (12a):

d

dt¼ �a2 �

d

dt�1 � �2ð Þ; ð11Þ

d

dt�1 � �2ð Þ ¼ a3 þ �: ð12Þ

In Eqs. (11) and (12) the variables c and g are as follows

¼ �1 � �1ð Þ � �2 � �2ð Þ; ð11aÞ

� ¼ 12a4v �1 � �� 1

2�1 � �2ð Þ

� �; ð12aÞ

where �1, �2, �1, �2 are the boundary temperatures of the fuel and of the coolant,respectively.So, two input partial differential equations, (1) and (2), are substituted by four

ordinary differential, (7), (8), (11), (12), and two algebraic equations, (11a) and(12a).

3. Simplified structural scheme for modeling of temperature dynamics

If the regime of the assembly’s operation is characterized by inessential changes ofthe fuel temperature, a simplified model may be used.Having combined Eqs. (11) and (12) it is convenient to present the system of Eqs.

(7), (8), (11) and (12) as follows

T�d��

dtþ �� ¼ kPPþ ��; ð13Þ

T�d��

dtþ �� ¼ �� þ k�v �1 � �2ð Þ; ð14Þ

T d2

dt2�1 � �2ð Þ þ

d

dt�1 � �2ð Þ ¼ k T�

d�

dtþ �

� �; ð15Þ


e the time constants
wher
T� ¼1

a2;T� ¼

1

a3;T ¼

T�T�T� þ T�

ð16Þ

and the transfer coefficients

kP ¼a1a2; k� ¼

a4a3; k ¼

T�T� þ T�

ð17Þ

are introduced. To Eqs. (13)–(15) there corresponds a block scheme in Fig. 4. Fig. 5shows the components of the block scheme in an expanded form. The transferfunctions of the components are written in rectangular boxes in Fig. 5, where s is theLaplace variable.

4. Dependence of parameters on mean fuel temeperature

From Eqs. (14), (15) and (4), (5) it follows that the time constants of the fuel tem-perature T� and of the coolant temperature T� as well as the transfer coefficients kPand k� are inversely proportional to the heat-transfer coefficient k which is a functionof the thermal conductivity of fuel l1 (6). In turn, l1 depends appreciably on the fueltemperature (Chirkin, 1968) or, in other words, on the power of the assembly.As an example we investigate an assembly analogous to the active zone of the

IBR-2 fast pulsed reactor where plutonium dioxide is used as the fuel. To the dif-ferent temperatures of the fuel there correspond the following values of the assemblyparameters (Table 1).

5. Structural scheme for modeling of temeprature dynamics over a wide range of

fuel temperature change

The dependence of the heat transfer coefficient from fuel to coolant is approxi-mated as the exponent

Fig. 4. The block scheme of the simplified model for the simulating of temperature transitional processes.


k ¼ k0exp b��

; ð18Þ

where k0 ¼ 1:38 � 103W=m2 K, b ¼ �4:29 � 10�41=K, ��—height-averaged tempera-ture of fuel in K.A linear approximation turns out to be not satisfactory because the model

becomes unstable.

Fig. 5. The components of the block scheme in Fig. 4: (a)—block A, (b)—block B̂, (c)—block C̃.

—multiplication element, —algebraic summation element.

Table 1

The parameters as a function of temperature

�� (K)
700 900 1100 1300 1500
l1 (W/m K)
2.9 2.4 2.0 1.7 1.4
k (W/m2 K)
1020 950 870 810 730
T� (s)
7.1 7.7 8.4 9.1 10
T� (s)
1.0 1.1 1.2 1.3 1.4

Fig. 6 shows the block scheme of a more complicated model and Fig. 7 presentsthat of its elements.

6. Estimation of the parameters of a hypothetical subcritical assembly

Let us assume the possibility of combining some existing powerful pulsed neutronsource based on a proton accelerator and nonmultiplying target and a subcriticalmultiplying assembly whose parameters are those of the active zone of the IBR-2reactor.As a pulsed source with a nonmultiplying target there is discussed ISIS (Chilton,

UK) and MLNSC (Los Alamos, USA) whose parameters are summarized in Table 2(Aksenov, 1998).A combination of such a source and a multiplying assembly analogous to the

IBR-2 active zone will have the parameters indicated in Table 3. It is accepted thatthe prompt neutron subcriticality " ¼ �0:05, i.e. 5% (with the multiplicationU ¼ 1= "j j ¼ 20), the portion of delay neutrons � ¼ 2:16 � 10�3 and the effective life-time of prompt neutrons ¼ 6 � 10�8s.Thus, the power of the subcritical assembly will increase 20 times in comparison

with that of a nonmultiplying target without changes in the pulse duration. At thesame time, 96% of the energy will be generated as pulse energy. Therefore, forassemblies with a power of 1 MW and higher the estimation of the temperaturedynamics is likely to become quite a topical issue.

Fig. 6. The block scheme of a more complicated model for the simulating of temperature transitional

processes.


Fig. 7. The components of the block scheme in Fig. 6. —multiplication element, —algebraic

summation element.


7. Examples of modeling of temperature transitional processes

The transitional processes in Figs. 8–11 correspond to the hypothetical subcriticalassembly ISIS+IBR-2. Fig. 8 shows simulated temperature transitional processescorresponding to the simplified model shown in Figs. 4 and 5 at a jump-like changein the coolant speed. The processes correspond to the most intensive regime (at amean temperature of 1500 K). The parameters of the model were the following:a1 ¼ 25:4K=MW s, a2 ¼ 0:0997 s�1, a3 ¼ 0:704 s�1, a4 ¼ 2:5 m�1, P ¼ 3:3 MW,�1 ¼ 573 K, v ¼ 3 m=s, kP=255 K/MW, k�=3.55 s/m, k =0.124, T�=10 s,T�=1.4 s, T =1.2 s.Figs. 9–11 show the simulated transitional processes corresponding to a more com-

plicated model (Figs. 6 and 7) at turning on and at short-lived turning off of theaccelerator and at jump-like change in the coolant speed. The parameters of the modelwere following: a1 ¼ 25:4 K=MW s, a02 ¼ 1:37 � 10�4 m2 K=W s,

a03 ¼ 9:69 � 10�4 m2 K=W s;

a4 ¼ 2:5 m�1;

k0 ¼ 1:38 � 103 W=m2 K;

b ¼ �4:29 � 10�4 K�1;

Table 2

The parameters of pulsed neutron sources

Source, construction

date

Target power (kW);

Beam energy (MeV)

Pulse duration (ms);Frequency (s�1)

ISIS, 1985
160; 800 20–30; 50
MLNSC, 1985
50; 800 20–30; 20
Table 3

The parameters of a hypothetical assembly

Assembly
ISIS+IBR-2
active zone

MLNSC+IBR-2

active zone

Target power (kW)
160 50
Energy generated in pulse (kJ)
64 50
Energy generated between pulses (background energy) (kJ)
3 2
Total energy over the pulse period (kJ)
67 52
Background energy proportion of total energy (%)
4 4
Assembly power (MW)
3.3 1.0
Assembly pulse duration (ms)
20–30 20–30
Pulse frequency (s�1)
50 20
Mean fuel temperature (K̂)
1500 900

Fig. 8. The change of the fuel temperature and coolant temperature (K) at jump-like change in the

coolant speed from 3 to 1.5 m/s (for the simplified model showed in Figs. 4 and 5).

Fig. 9. The transitional processes in the more complicated model shown in Figs. 6 and 7 at turning on of the

accelerator. P, �� , ��, �2, k are the power (MW), height-averaged temperatures of fuel and coolant, output

coolant temperature (K) and coefficient of heat transfer from fuel to coolant (W/m2 K), respectively.


Fig. 10. The transitional processes in the model showed in Figs. 6 and 7 at short-lived turning off of the

accelerator.

Fig. 11. The transitional processes in the model showed in Figs. 6 and 7 at the coolant speed jump (from 3

to 1.5 m/s).


3:3 MW, �1 ¼ 573 K, v ¼ 3 m=s. Comparison of Fig. 8 with Fig. 11 shows that
P ¼
the simulated transitional processes in both models at essential change of fuel tem-perature have the same qualitative character but essentially differ quantitatively.

8. Conclusion

The structural scheme for the modeling of temperature transitional processes inthe assembly is presented. This temperature dynamics model is the part of the pulsedfission material assembly including the kinetics block for simulating of power tran-sitional processes in the assembly.A many times increase in the power of a pulsed neutron source involving a pow-

erful proton accelerator and a nonmultiplying target if it is hypotetically combinedwith a multiplying assembly analogous to the active zone of the IBR-2 fast pulsedreactor is estimated.

Appendix. Brief description of method involving use of Hermite polynomial to

perform transition from partial differential to ordinary differential equations

A1. General idea of problem solution

There is investigated a certain linear partial differential equation of two indepen-dent variables, the spatial variable x and the time variable t. It is required to find thesolution y(x,t) in the interval x[0, L] at certain limiting conditions (Fig. A1).In addition, Fig. A1 shows the function y(x, t1) for a fixed value of t=t1. It is char-

acteristic of partial differential equations that at subsequent moments of time the solu-tion depends essentially on the entire curve y(x, t1). It is therefore extremely importantto have the function y=(x, t1) approximated appropriately in the interval [0, L].

Fig. A1. The graphical representation of a certain function y of two variables, x and t.


Instead of the spatial variable x in the interval [0, L] it is more convenient toconsider the dimensionless spatial variable

x~ ¼ x=L ðA1Þ

in the interval [0, 1]. In the interval [0, 1] the curve y ¼ x~; t1ð Þ is approximated usingthe curve ’ðx~Þ with the help of the Hermite polynomial. The result of such approx-imation is that the values of the approximating function and of its derivatives withrespect to x~ on the boundaries of the interval [0, 1] equal the corresponding values ofthe function y and its derivatives y0 with respect to x~ (Fig. A2).Also, an additional requirement is introduced: the [0, 1] interval-averaged values

of the functions ’ and y must be equal, i.e.
ð10
ydx~ ¼

ð10

’dx~ : ðA2Þ

The peculiarities of the approximating Hermite polynomials are discussed in detailin a work by Berezin and Zhidkov (1959). They describe how to find the polynomialcoefficients of different power, supply proof of uniform convergence and point toaccuracy of approximation of the Hermite polynomials ’ðx~Þ to the function yðx~Þ inthe interval [0, 1], if yðx~Þ is an integer function, i.e. if x~ is only subjected to operationof addition, subtraction and multiplication.If the function y has not more than one extremum in the interval [0, 1] (Fig. A2), it

is well approximated by the Hermite polynomial of the third power which has theform:

’3ðx~Þ ¼ a1 þ b1x~ þ �3a1 þ 3a2 � 2b1 � b2ð Þx~2 þ 2a1 � 2a2 þ b1 þ b2ð Þx~3; ðA3Þ

where a1 ¼ yð0Þ; a2 ¼ yð1Þ; b1 ¼ @y=@x~jx~¼0, b2 ¼ @y=@x~jx~¼1. The approximatingfunction ’ðx~Þ is integrated in accord with the condition (A2). In the equationobtained in result of the integration there appear the derivatives b1 and b2 with

Fig. A2. The graphical representation of the function yðx~Þ for a fixed t=t1, the approximating function

’3ðx~Þ and the tangents on the interval boundaries.


respect to x~ on the boundaries of the interval [0, 1]. The derivatives are founddirectly from the partial differential equations using the values of the function y andits derivatives with respect to t on the interval boundaries.

A2. Application of method for our case

The transition from partial differential Eqs. (1) and (2) to ordinary differentialequations is accomplished as follows.Upon integrating the Eq. (1) with respect to the spatial variable x~ between the limits

0 and 1 we obtain Eq. (A4) whose variables are the height-averaged temperatures:

d�ðtÞ

dt¼ a1PðtÞ � a2 �ðtÞ � �ðtÞ

� : ðA4Þ

Since it is accepted that the power is independent of x~ ,

PðtÞ ¼ PðtÞ:

From Eq. (1) there are obtained Eqs. (A5) and (A6) corresponding to the lower andupper boundary of the assembly:

d�1ðtÞ

dt¼ a1PðtÞ � a2 �1ðtÞ � �1ðtÞ½ �; ðA5Þ

d�2ðtÞ

dt¼ a1PðtÞ � a2 �2ðtÞ � �2ðtÞ½ �: ðA6Þ

The meaning of the variables is explained in Fig. 3. The indices 1 and 2 mark thevalues of the variables at the lower and upper end of the assembly, respectively.After transition from x to x~ (A1) we write Eq. (2) in the form:

@�

@t¼ a3ð� � �Þ � a4�

@�

@x~: ðA7Þ

Upon integrating Eq. (A7) with respect to x~ between the limits 0 and 1 we obtainthe ordinary differential Eq. (A8) for height-averaged variables:

d�ðtÞ

dt¼ a3 �ðtÞ � �ðtÞ

� þ a4� �1ðtÞ � �2ðtÞ½ �; ðA8Þ

where �1ðtÞ ¼ �ð0; tÞ; �2ðtÞ ¼ �ð1; tÞ are the boundary temperatures.The function �ðx~; tÞ is approximated using the Hermite polynomial by the formula

(A3):

�ðx~; tÞ ¼ �1 þ@�

@x~

� �1

x~ þ �3�1 þ 3�2 � 2@�

@x~

� �1

�@�

@x~

� �2

� �x~2

þ 2�1 � 2�2 þ@�

@x~

� �1

þd�

dx~

� �2

� �x~3: ðA9Þ


on integrating Eq. (A9) with respect to x~ between the limits 0 and 1 we obtain
Upthe equation for the height-averaged temperature of the coolant:

�ðtÞ ¼1

2�1ðtÞ þ �2ðtÞ½ � þ

1

12

@�

@x~

� �1

�@�

@x~

� �2

� �: ðA10Þ

It should be noted that the equation for the average temperature of the coolant(A10) differs from the widely used Eq. (10).Into Eq. (A10) there enter the derivatives with respect to x~ on the boundaries of

the interval [0, 1]. We obtain the expressions for those derivatives from the Eq. (A7)for x~=0 and x~=1:

@�

@x~

� �1

¼1

a4�a3 �1ðtÞ � �1ðtÞ½ � �

d�1ðtÞ

dt

�; ðA11Þ

@�

@x~

� �2

¼1

a4�a3 �2ðtÞ � �2ðtÞ½ � �

d�2ðtÞ

dt

�: ðA12Þ

Thus, in result of approximation using the Hermite polynomial of the third powerthe two input partial differential Eqs. (1) and (2), or, in other words, Eqs. (1) and(A7), have been replaced by seven ordinary differential Eqs., (A4), (A5), (A6), (A8),A(10), (A11), (A12).

References

Aksenov, V.L., 1998. Modern neutron sources. Surface Investigations 14, 151–159.

Arhan, R., 1970. SORA Dynamics and Control System Studies Using Mean-Value Neutron Kinetics

Equations. EUR-4408.

Berezin, I.S., Zhidkov, N.P., 1959. Methods of Calculations. Fizmatgiz, Moscow.

Chirkin, V.S., 1968. Thermophysical Properties of Nuclear Technology Materials. Atomizdat, Moscow.

Dorri, M.Kh., 1966. Approximate solution of some partial differential equations of heat exchange using

an analog computer. Avtomatika i Telemekhanika 8, 124–130.

Popov, A.K., Rogov A.D., 1976. A Program for the Computer Modeling of the Pulsed Reactor Dynamics.

JINR Deposit Publication B1-11-10120, Dubna.

temperature dynamics model of a pulsed fission material assembly

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