Definition and Expression of Thermal Neutron Effective Proliferation Coefficient in the Uniformity D

International Journal of Nuclear Energy Science and Engineering Volume 4 Issue 1, March 2014 doi: 10.14355/ijnese.2014.0401.02

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Definition and Expression of Thermal Neutron Effective Proliferation Coefficient in the Uniformity Dispersion Nuclear Reactor with Uniformity Dispersion Source Neutrons Deqiang Pei Nuclear Island Branch of Engineering Department of Tai Shan Nuclear Power Joint-Limited Company (A-212 01 Building on Site), Chi Xi Town Tai Shan City Guang Dong Province People’s Republic of China peideqiang@cgnpc.com.cn Abstract This paper will introduce the definition and expression of thermal neutron effective proliferation coefficient keff after modification in the uniformity dispersion nuclear reactor with uniformity dispersion source neutrons and when studying the nuclear reactor physical process, we can exactly explain the phenomenon in the actual physical process and prove it for use.

Keywords Thermal Neutron Effective Proliferation Coefficient; Definition and Expression; Critical; Subcritical; Supercritical

Introduction The introduced thermal neutron effective proliferation coefficient keff in the literature is the accurate definition of the change of neutrons in the reactor without sources neutrons and corresponding accurate expression. It can precisely explain and describe six changing situation of neutron’s physical process in the reactor. However, when it is used as a united definition and expression in reactor with source neutrons for describing, explaining and calculating the actual physical processes , in some situation it can’t exactly, clearly and correctly explain corresponding physical processes, and its expression can’t be used as a united expression to get the corresponding calculation results according to its definition either.

Thus this paper was aimed to give an actual, correct and exact corresponding definition and expression for the thermal neutron effective proliferation coefficient.

Notes: The discussion object of the paper is: the uniformity dispersion nuclear reactor with uniformity dispersion source neutrons; and “the uniformity dispersion nuclear reactor” is called as “the reactor”,

“the uniformity dispersion source neutrons” as “the source neutrons”, and the representative means of source thermal neutrons is same as source neutrons, in the following contents.

The Raised Problem Now in the literature thermal neutron effective proliferation coefficient in the reactor is commonly regarded as: keff  the number of thermal neutrons in the new generation of the reactor/ the number of thermal neutrons in the ahead generation of the reactor (1) In fact its definition is: keff = the generating rate of thermal neutrons in the reactor/ the disappearing rate of thermal neutrons in the reactor (2) Its specific expression is: keff      P  f  Ps  Pd

(3)

But the thermal neutron effective proliferation coefficient’s definition in the reactor should be: k 'eff =thermal neutron effective proliferation coefficient of thermal fission

(4)

Its concrete definition is: k 'eff = the generating rate of thermal neutrons of thermal fission in the reactor/ the disappearing rate of thermal neutrons of thermal fission in the reactor (5) Notes: The definition of neutrons of thermal fission is to differentiate the neutrons generated from source neutrons in the  process from the neutrons of thermal fission, but they still belong to the class of source neutrons.

Its concrete expression is:

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k 'eff      P  f  Ps  Pd 1  0 /  

(6)

Notes: 0 is the thermal neutron flux produced by source neutrons and it isn’t absorbed by thermal fission, but it contains the flux in the  process of source neutron and the flux comes from the part of thermal neutrons produced by the fission of fast neutron. It is called the flux of source thermal neutron in abbreviation. The neutron in 0 is called source thermal neutron in short.

 is the thermal neutrons flux produced from the fission of thermal neutrons in the reactor. It is called the flux of the thermal neutron of thermal fission. The neutron in  is called the thermal neutron of thermal fission.

From the definitions above it can be concluded that the thermal neutrons in the reactor is either the thermal neutrons of thermal fission, or source thermal neutrons.

There are no other thermal neutrons. Thus the definitions above contain all the neutrons in the reactor.

In the following we will discuss the characteristic of producing rate and disappearing rate of both types of neutrons in the thermal district.

The Discussion of Concrete Specification of Generating Rate and Disappearing Rate of Thermal Neutrons of Thermal Fission and Source Thermal Neutrons in the Reactor with Sources Neutrons The Discussion of the Definition of Original keff In perspective of keff , the thermal neutrons of thermal fission and source thermal neutrons in the reactor are the same, both perceived as the thermal neutrons in the reactor. However, it can be noticed that in the thermal neutrons of interior reactor, the generating rate of the thermal neutrons of thermal fission,(that is, generating rate of the thermal neutrons after thermal fission,) is:    f     P  Ps  0   f     P  Ps , and the disappearing rate is    c / Pd .But the generating rate of source thermal neutrons is S    P  Ps , the disappearing rate of source thermal neutrons is 0   c / Pd . We can see the compositions of generating rate and disappearing rates of thermal neutrons of thermal fission and source thermal neutrons are different. In the paper, I will prove from constitutes the generating rate and disappearing rates of thermal neutrons of thermal fission and source thermal neutrons: the actual expression of keff in the reactor

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with sources neutrons deduced from the definition of keff is different from the original expression. That is to say, compared with the original definition, the expression of keff in the reactor with sources neutrons is incorrect. Additionally in according with the definition of keff , source thermal neutrons are also interior reactor thermal neutrons. So if the value of keff excesses one, as the definition of keff , the flux of source thermal neutrons is continually increasing. But in fact because: S    P  Ps  0   c / Pd

(7)

Therefore, the flux of source thermal neutrons will not continually increase because the value of keff excesses one, just being up to a certain value according to the value of  c . The same situation will happen if the value of keff is below one. It can be concluded that qualitatively speaking, it is incorrect, if we don’t distinguish the meaning not in line with the definition of source thermal neutrons from unclearly definition of thermal neutrons in the interior reactor.

The Discussion of Definition in the Paper The definition in the paper will be independent from whether thermal fission continues or thermal neutrons of thermal fission related to the power change of thermal fission in interior reactor neutrons.

Furthermore, in perspective of the definition of k 'eff , when the value of k 'eff excesses one, the flux of thermal neutrons of thermal fission will increase continually because of the value of k 'eff excesses one, unless the value of k 'eff

excessesing one changes

because the value of  increase or the change of f.

The flux of source thermal neutrons will be changing with equation (7). When the value of k 'eff equates one, from the definition, it is known that the flux of thermal neutrons of thermal fission will be unchanged. Then the flux of source thermal neutrons will be also unchanged because of equation (7) and k 'eff =1. When the value of k 'eff is below one, from the definition of k 'eff it can be learnt that the flux of thermal neutrons of

thermal fission will decrease continually. Unless the condition of the value of k 'eff below one changes because of the decrease of  or the change of f, the trend will continue. However, the flux of source thermal neutrons will change with equation (7). These

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descriptions are consistent with the reality. So the definition in the paper could correctly explain and describe the changing situation of two parts(that is to say, the flux changes of thermal neutrons of thermal fission and source thermal neutrons. in the reactor), in stable conditions, in the process of the increasing and decreasing power. Therefore qualitatively speaking, the definition in the paper is right.

of source neutrons in unit time.

(3)    f in equation (9) is absorbing rate of fuel to

The Deduction of Expression in k eff

of thermal neutrons.

'

f: the absorbing effective rate of thermal neutrons in the reactor, that is to say, the absorption of fuel to thermal neutrons make the proportion of the absorptions of all the materials to thermal neutrons.

Pd : The non let out probability in the diffusion process

   f / f  / Pd Is

Introductions

the total the thermal neutrons of

thermal fission disappearing rate in the reactor.

The generating rate of thermal neutrons of thermal fission is I, and the disappearing rate of thermal neutrons of thermal fission is Ⅱ.

Derivations Consider expression (5), in accordance with the definition of I II. We have:   the generating rate of thermal neutrons of thermal fission (8)     f     P  Ps  0   f     P  Ps Ⅱ= the disappearing rate of thermal neutrons of thermal fission



thermal fission thermal neutrons in the reactor.



    f / f / Pd

(9)

Then: k 'eff   / Ⅱ

(10)

That is k 'eff  (   f     P  Ps  0   f     P  Ps ) /    f / f / Pd      P  f  Ps  Pd 1  0 /  





Ps : The non let out probability in the slowed process of fast neutrons.

 : The average producing amount of neutrons after the absorption and fission of thermal neutrons.

 : considering the modified factor after being captured from fission of U-238 fast neutrons.

P: considering the modified factor after the resonance absorption of the slowed process of fast neutrons.

In equation (8), it is supposed that the macro absorbed section of fuel atomic to source thermal neutrons is the same as the macro absorbed section of fuel atomic to thermal fission thermal neutrons.

Note of Φ≠0

t1

t2

t1

t2 (11)

t0

t3 t

So the expression (6) above is right.

FIGURE 1

The equation (11) actually is thermal neutron effective proliferation coefficient in the reactor. If it is used to calculate the physical parameters in various conditions in the reactor, we can get the same result conforming to the reality. Then it has already been proved that the definitions in equation (5) and equation (11) are right.

And in stable condition. Thus k 'eff =1

(12)

Explanations (1)    f     P  Ps in equation (8) is the number of neutrons entering the hot district after thermal fission for thermal fission neutrons in unit time.

(2) 0   f     P  Ps in equation (8) is the number of neutrons entering the hot district after thermal fissions

The explanations of the above figure1 are that: t0 represents at the start moment of neutrons generating of each generation, t1 represents at the start moment of thermal neutrons generating of each generation and t2 represents at the start moment of the thermal neutrons disappearing of each generation; t represents the life time of the each generation neutrons.

Therefore, when we put in source neutrons in the reactor, the first generations of source neutrons generates at the time t0 , and disappears at the time t2 .

At the time t2 , the first generation of source neutrons disappears just the moment, and the time of first generation of neutrons of thermal fission generates then, and after a time of t1 , the first generation of

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thermal fission neutrons will becomes thermal neutrons of thermal fission quickly. Then if after a time of t3 , and 0  t3  t2 ; now thermal neutrons of thermal fission haven’t disappeared. However, at the time thermal neutrons of thermal fission have already generated, and their flux aren’t zero. Until the time of t3  t2 , the value of  : the flux of thermal neutrons of thermal fission isn’t zero. Therefore, the worry isn’t necessary that the flux of thermal neutrons of thermal fission is zero in the scope of them.

The Applied Verification for the Result Equation (11) in the Paper The Applied Verification If the reactor is in some subcritical condition, then (13) keff  k  1 Notes: keff      P  f  Ps  Pd

(14)

In the paper we still use the expression in order to conveniently discuss and compare with the result in original teaching books. If the value of keff is stable, the reactor will be in steady condition after a while and in the first time we can put in the source neutrons, which release the number S of fast neutrons every second in unit volume in the reactor, into the reactor.

Then in the reactor the flux of thermal neutrons of thermal fission is  , and the flux of source thermal neutrons is 0 .

It is well learnt from equations (9-12) in references that the number of neutrons in hot district produced by thermal neutrons of thermal fission and sources thermal neutrons in unit time after thermal fission absorbed is: N '热 =S    P  Ps  k  S    P  Ps  k 2   S    P  Ps S    P  Ps  k n    S    P  Ps (21) 1 k S    P  Ps  k  1 k We temporarily suppose the definition for keff is right.

So because of the definition of keff , the above equation can be written as: N '热 = 0   c / Pd   k  0   c / Pd   k 2   0   c / Pd   k n     c / Pd  0  S    P  Ps 1 k As the equations (7), (21) and (22), we can get   c k N '热 = 0 (1  k ) Pd

(22)

(23)

So N热 =N '热 The expression of k 'eff in the paper can be proved right by the case, because the equations (21) and (22) are the correct result, and the two equations are the sum of the generating rate of thermal neutrons of thermal fission in hot district after source thermal neutrons being absorbed and the generating rate of thermal neutrons of thermal fission being absorbed from generation to generation every moment in steady condition. Meanwhile it has been proved that equations (7) are right too.

Because of equation (14), equation (11) actually is: k 'eff  keff 1  0 /  

(15)

and keff by Calculating the Above Result in the

Due to equation (13):

k 'eff  k 1  0 /  

(16)

As in steady condition this time, equation (12) is effective. Then k 'eff  1 , so k 1  0 /    1

(17)

We can get the following result for the above equation: k (18)  0 1 k We know from the definition of

k 'eff that the

generating rate of neutrons of thermal fission in hot district is: (19) N热 =k 'eff     c / Pd We know from equations (16), (18) and k eff  1 that: '

N热 =0   c  k / (1  k ) Pd

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The Compared Explanation for the Definition of k 'eff

(20)

Reactor By the above-mentioned calculation, it is noticed that equation (20) can be acquired by the definition and expression of k 'eff . Equation (20) is the same as the result from equation (23). From the definition of keff , we sum every item of generating rate in hot district for neutrons of thermal fission produced by the fission of absorbed thermal neutrons (source thermal neutrons and every generation of thermal neutrons of thermal fission). So the calculations are both right when k 'eff and keff are used in the nuclear reactor, which is impossible.

First of all, equation (20) is the result after once calculation for the united definition and perspective

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expression of k 'eff . It correctly and precisely explains

keff constantly. The six factors, which constitute keff ,

the relationship between the generating rate and disappearing rate for thermal neutrons of thermal fission, from the definition and expression of k 'eff .

correctly reflect quantitative change relationship of neutrons in the reactor, no matter neutrons produced by thermal fission neutrons or the source neutrons reflected in the six factors. So the result and conclusion of the paper won’t deny the correction of six concrete physical processes reflected in six factors. Therefore, the expression of keff and the six factors, which

Equation (23) calculated by summing every item of generating rate of thermal neutrons of thermal fission is not the result calculated by one summation through the united definition and perspective expression. We can’t distinguish source thermal neutrons and thermal neutrons of thermal fission by the definition of keff . We don’t distinguish and make alone thermal neutrons of thermal fission from thermal neutrons in interior reactor either. Therefore, the result from the united definition of keff and corresponding expression is the generating rate or disappearing rate of thermal neutrons in the reactor. However, equation (23) is actually the generating rate of thermal neutrons of thermal fission. So the result of equation (23) is failed to obtain by the definition and expression of keff through just one time calculation. So the correction of equations (20) and (23) can’t support the conclusion that keff actually is the neutron effective proliferation coefficient in the reactor with sources neutrons and keff could correctly describe and express the relationship between the generating and disappearing rates in thermal neutrons in the reactor. It is just illustrated that equation (23) is calculated in a correct way and used by a correct method of calculation and it is a right result.

Secondly, it can be noticed in equation (22), that except the first item 0   c / Pd   k is the generating rate of thermal neutrons of thermal fission in hot district after the absorption of source thermal neutrons in thermal fission, the other items are the generating rate of new generation thermal neutrons of thermal fission in hot district after the thermal fission absorption of every generation thermal neutrons of thermal fission.

Equation (22) reflects the fact that interior reactor materials (including fuel) absorb some amount of neutrons, 0   c or    c , then it will produce new

constitute keff , are still very important. keff is a very useful expression, factor and concept. However, it should be expressed by: keff = the corresponding generating rate of thermal neutrons of thermal fission/the disappearing rate of one-item thermal neutrons.

(24) Namely: the disappearing rate of one-item thermal neutrons (sources thermal neutrons or thermal neutrons of thermal fission)* keff = the generating rate of thermal neutrons of thermal fission from absorbed thermal fission.

(25) When keff is used to describe the relationship between the generating and disappearing rate of neutrons in the reactor, it is right for source neutrons or thermal neutrons of thermal fission. But when applying equation (24), it must be noticed that there is a very important concept: absorbed thermal neutrons of thermal fission still produce thermal neutrons of thermal fission. However, absorbed source thermal neutrons produce thermal neutrons of thermal fission (But for both thermal neutrons, the change law are different).That is to say, the expression of thermal neutrons of thermal fission produced by source neutrons after being absorbed by the materials of the reactor: (26) 0   f     P  Ps  k  0   c / Pd The above equation should be right. Thus when it is believed that the definition of keff could be applied to source neutrons, and when it can be found that the definition of keff is applied to source thermal neutrons, the numerator of keff is 0   f     P  Ps and the

thermal neutrons of thermal fission 0   c / Pd   k or

denominator is 0   c / Pd . The numerator and

   c / Pd   k , that is to say, using

denominator are not the same class of neutrons.

Although the expression of keff can be linked by the

keff to describe the

thermal neutrons of thermal fission after the absorption of thermal neutrons, no matter the absorption is source thermal neutrons or thermal neutrons of thermal fission, and the value between the generating and disappearing rates can stand for

same expression, by which the expression of keff can link the numerator and denominator as the linkage of the generating and disappearing rates of thermal neutrons of thermal fission. However in theory, it is

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incorrect that the definition of keff is applied to describe the relationship between the generating and disappearing rates of source neutrons. But the results of calculation seem to be the same. Equation (7) is the only right expression, which describe and calculates the relationship between the generating and disappearing rates of source thermal neutrons. It is also the explanation in theory of the result and phenomenon that the definition of keff discussed in the front of the paper is applied to describe the generating and disappearing rates of source thermal neutrons and the changing flux of source thermal neutrons and both conclusions are incorrect. So it is not right that the definition of keff is utilized to describe the relationship between the generating and disappearing rates of source thermal neutrons and the changing flux of source thermal neutrons.

Now it is

supposed that the definition of keff is

applied to describe the relationships of the generating and disappearing rates of thermal neutrons of thermal fission and source thermal neutrons and it is right.

That is to say, when we describe the relationship of the generating and disappearing rates of thermal neutrons in the reactor, it is right too. However, it’s incorrect if we actually use the definition of keff to describe the relationship of the generating and disappearing rates of source thermal neutrons. Therefore, it’s also incorrect that we use the definition of keff to describe the relationship between the generating and disappearing rates of thermal neutrons in the reactor.

The Discussion of the Consistent Problems Which Happened in the Reactor with the Source Neutrons Using the Definition and Expression of keff The

definition

and

expression

of

keff are

the

accordance in the reactor with no source neutrons. But in the reactor with source neutrons, because of source neutrons, what is the expression deduced from the definition of keff ? It can be noticed that the flux  of thermal neutrons of thermal fission defined in the paper and the flux 0 of source thermal neutrons exist simultaneously. Thermal neutrons in the reactor equal the sum of thermal neutrons of thermal fission and source thermal neutrons. We first suppose the generating and disappearing rates of thermal neutrons in the reactor, namely the generating and disappearing rates of thermal neutrons of thermal fission and source

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thermal neutrons are:

   f     P  Ps  0   f     P  Ps and    c / Pd  0   c / Pd . Then the expression of keff is equation (3) derived from the definition of keff (equation (2)). Equation (3), the expression of keff ,

is still effective in the reactor with source neutrons. But it must be noted that as neutrons in the reactor, source thermal neutrons have a generating rate S    P  Ps in hot district for the reason they directly are thermal to hot districts, besides the disappearing rate 0   c / Pd .

Obviously, derived from equation (2), which is also the definition of keff , its expression in the reactor is: keff 

  0   f     P  Ps  S    P  Ps   0   c / Pd

(27)

Obviously equation (3) isn’t equal to equation (27).

So the definition and expression of keff are not the same in the reactor. Compared with the definition, the expression of keff is not right in the reactor. In the front of the paper it has already been proved that the definition of keff in the reactor with source neutrons is not correct and precise. Compared with the definition, the expression of keff is also incorrect in the reactor with source neutrons. Therefore, the definition and expression of thermal neutron effective proliferation coefficient keff in the nuclear reactor with source neutrons in previous texts are not correct and can’t be applied.

Verification of the Definition and Expression of the Paper and keff in Differential Equation Perspective Verification of the Definition and Expression of the Paper

keff will keep constant if the value of f in the reactor of the discussed chapter is constant. If source neutrons are induced in the reactor at the moment t=0 at the first time and the source neutrons release the number S of fast neutrons in unit volume every second, after a while, at the moment ti , the flux of thermal neutrons of thermal fission in anywhere in the reactor is Φ, and the corresponding density of thermal neutrons of thermal fission is n. The flux of source thermal neutrons in anywhere in the reactor is 0 . Then according to the definition of the paper, we have in unit volume in anywhere in the reactor: The generating rate of thermal neutrons of thermal fission – the disappearing rate of thermal neutrons of

International Journal of Nuclear Energy Science and Engineering Volume 4 Issue 1, March 2014

thermal fission = the change rate of the density for the thermal neutrons of thermal fission (28) We apply equation (28) at anywhere in the reactor, so

   f     P  Ps  0   f     P  Ps     c / Pd dn dt By equation (8-23) of references [1]: 

  2  nVT / 

(29)

(30)

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When keff =1 , equation (33) could be:

d (38) dt obviously   0 , so the solution to

0   f     P  Ps  V  When t= t  t1

equation (38) in the initial conditions is:   0   f     P  Ps V  t

Equation (7), (14), (35) and the hypothesis of keff  1 can make equation (39) become:

  S    P  Ps  Pd  t / td   c 

VT : The most probably speed of thermal neutrons

(39)

(40)

If V  2VT / 

(31)

If the flux of thermal neutrons in the reactor is T , then

So   nV

(32)

we have the followings according to the definition of the paper: (41) T    0

According to equations (32) and (14), (29):it can be expressed by:

0   f     P  Ps   keff  1     c / Pd 1 d 

(33)

Because equations (40) and (41), we have T  S    P  Ps  Pd  t /  td  c   0

(42)

V dt According to the assumption of the chapter, obviously, as it is well known from figure 1 that , in the reactor when we put in source neutrons at the time t0 , so after

Equations (9-8) in references [1] is the solution for Equation (9-6) in references [1] in the hypothesis of this chapter in the initial condition of t=0 and T  0 .

a time t of one generation for source neutrons will disappear and neutrons of thermal fission will generate. The generated neutrons of thermal fission after a time t1 , will generate thermal neutrons of

equation (41). We will prove the consistence of T in

thermal fission. At the time t  t1 after we put in source neutrons, the flux of thermal neutrons of thermal fission is zero. But after a while the flux will generate and change in the way of equation (33). So the initial condition for equation (33) is that the time t  t1 after we put in source neutrons means the time t=0. Therefore,   0 at the time for initial conditions. The solution to the above equation in the initial conditions is: 0  keff exp(V  c (keff  1)t / Pd )  1 (34)  (k  1)  eff

If td    Pd / 2 VT   c  Pd / V   c 

(35)

Equation (34) could be: 0  keff   t (36)  exp((keff  1) )  1 (keff  1)  td  Discussion: when keff  1 and t ∞ , equation (36) could be:



0  keff (1  keff )

(37)

The results of this equation and equation (18) are the same, so it is proved that when keff is smaller than one, equation (28) and its result equation (36) are right in the definition of the paper.

when keff  1 , and the meaning of T is consistent with reference [1] with equation (40) in item 3 in this chapter of the paper, but the result of the paper is equation (42). Whichever is right? We will see the followings that if we keep the same assumption as the front paper, then when t  0,   0, 0  0, T    0  0 ; but if the aforementioned source neutrons are supposed to be induced in the reactor without fuel first, after the reactor is in stable condition, if the flux of source neutrons in the reactor is 0 , then we again induce the fuel, which make the value of f be constant, in the reactor. If slow stuff and structural materials have already been installed before the fuel induced, when we put in the fuel, the absorbed section of other materials in the reactor is lessened by  f to keep

 c constant, other conditions are the same with afore mentioned. If we make the moment we put in the fuel materials being t=0, then   0 , and the flux of source neutrons is 0 , then T    0  0 . No matter what we put in the fuel first then put in source neutrons, or put in source neutrons first then the fuel, as for the reaction of nuclear fission in the reactor, the physical process of nuclear fission in the reactor is the same, that is to say, the changing law of  is the same. Thus the result should be the same, and only just one. And it is noticed that when t=0 in the front and rear situations we have constantly: T (0)  0 (43) The equation is effective. Equation (43) should be the

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International Journal of Nuclear Energy Science and Engineering Volume 4 Issue 1, March 2014

initial condition for equations (9-6) of references.

Under this condition the solution to equations (9-6) is same as equation (42). Therefore, the result of the paper is correct. Simultaneously because of equation (41), when t=0, which is the initial condition in references, T  0 is correct. So: Namely

T    0  0

(44)

  0

(45)

Obviously if we first put in source neutrons in the reactor then put in the fuel, it can be seen that when t=0, it’s incorrect that  is minus value in perspective of the result from initial condition, namely equation (44), of equations (9-6) in references. Because the initial condition of equations (9-8) in references is incorrect, equations (9-8) is not right neither.

Now we again directly build differential balance equation for neutrons in hot districts in perspective of the definition of k 'eff , then dn (46) k 'eff    c / Pd   c / Pd  dt Namely,

0   f     P  Ps   keff  1     c / Pd (47) 1 d 

V dt Equations (47) and (33) are the same, and it is illustrated that the definition and expression of k 'eff are

the same in perspective of differential balance equation of thermal neutrons of thermal fission. It is also right.

But it can be seen from equations (40) and (42) that when keff  1 , the flux of thermal neutrons of thermal fission increases steadily. Because of the expression of k 'eff in the paper k 'eff  keff (1  0 /  )  1  0 /   1

(48)

So when keff  1 , the value of k 'eff is over one in the reactor with source neutrons. Therefore, the flux of thermal neutrons of thermal fission increases steadily.

So the definition of the paper not only correctly explain the change laws for thermal neutrons of thermal fission and source thermal neutrons in actual physical processes in the reactor, but also the corresponding expression with the definition of the paper precisely calculates the correct result. So when keff  1 and keff  1 , the definition and expression of the paper are both right, no matter what the value or the changing law of the calculated result discussed in differential balanced equation of thermal neutrons of thermal fission. But when keff  1 , it is not right for the result of references.

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Simultaneously we should notice the third item of the conclusion in P206 of references, that is to say, when keff  1 and there are source neutrons, the flux of thermal neutrons in the reactor increases with time, which is the result of equation (42). Although the result is proper, it is illustrated that when we use the original definition of keff in the reactor with source neutrons, the original definition can’t explain correctly the changing law of the flux of thermal neutrons of the reactor. Thus its corresponding expression can’t be the united and unanimous expression with the definition to precisely describe and calculate the changing law of the flux of thermal neutrons in the reactor.

When keff  1 , the flux of thermal neutrons of thermal fission will change with equation (36). Whether the results of the paper and references are correct or not will be discussed in the third item of this chapter.

Verification of the Result of the Paper, the Definition and Expression of keff Are Right or not in Perspective of Differential Balanced Equation of Thermal Neutrons in the Reactor When discussing the item, it is supposed that the contents of various items in the reactor and the first item of this chapter are the same. In addition, suppose that the density of source neutrons in the hot districts is n 0 .

According to the definition, the differential balanced equation of thermal neutrons in unit volume anywhere in the reactor is: The generating rate of thermal neutrons in the reactor —the disappearing rate of thermal neutrons in the reactor = the changing rate of the density of thermal neutrons in the reactor.

(49) Because thermal neutrons of thermal fission and source thermal neutrons constitute thermal neutrons in the reactor, the above equation could be written as:

(  0 )   f     P  Ps  S    P  Ps  (  0 )   c / Pd 

d  n  n0 

(50)

dt If the spectrum of source neutrons and neutrons of thermal fission is the same, then by equations (30), (31) and (32): (51) 0  n0V And because of equations (41) and (50):

1 dT (52) V dt So we can similarly use the first item of this chapter, and the figure 1, that the time t  t1 after we put in S    P  Ps  (keff  1)T   c / Pd 

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source neutrons means the time t=0. And with initial conditions of differential balance equation (52) is equation (43), with equation(7), So the solution of equation (52) is  0  t (53) T   keff exp((keff  1) )  1 keff  1  td  Discussions: Because equation (49) is right, and the following deduction is correct, equation (53) is also right.

When keff  1 and t ∞ ,

T 

0 1  keff

(54)

We add the above equation into the result of equation (41), then 0  keff (55)  1  keff Namely the equations (55) are same with the correct results of equations (18) and (37). Then the result discussed in the differential balanced equation of thermal neutrons in the reactor and the correct result are the same, no matter with the object of thermal neutrons of thermal fission or thermal neutrons in the reactor.

When keff  1 , with equation (43) as the initial condition, the solution to equation (52) is equation (42).

When keff  1 , because of equation (41), equation (53) can be changed to: 0  keff  1  keff

  t (56) exp((keff  1) )  1 t d   So equations (56) and (36) are perfectly identical.

Therefore, the result can be obtained that the result discussed in the differential balanced equation of thermal neutrons in the reactor and the correct result are the same, no matter with the object of thermal neutrons of thermal fission or thermal neutrons in the reactor. The result is the same as the correct result in both conditions of keff  1 and keff  1 .

Now we turn to equation (50), because n 0 will keep constant when

 c is constant. So dn0  0 , and

because of equation (7), the equation (50) actually is:

0   f     P  Ps  (keff  1)   c / Pd 1 d 

(57)

V dt That is identical with equation (33). So we aren’t curious about the results we take thermal neutrons of thermal fission and the thermal neutrons in the reactor as the descriptions are the same in the perspective of

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the discussion in differential balanced equation of thermal neutrons. It is also proved that in the front of the paper equations (43) and (7) are both correct, that is, supposing that t=0 initial conditions of differential balanced equation (52) is correct. With the discussion in the front of the paper, the conclusion can be made that the result we take the neutrons in the reactor as the description in perspective of differential balanced equation of thermal neutrons is the same as the result we get in the differential balanced equation of the thermal neutrons of thermal fission in the reactor in perspective of the definition and expression of k 'eff .

Thus it can be said that we prove the definition, expression and corresponding conclusion we get from differential balanced equation of thermal neutrons are right, at least When keff  1 and keff  1 are right.

The Comparison and Discussion of the Result of the Paper and the Corresponding Result in References when keff  1 It is indicated that equations (9-7) in reference [1] in perspective of differential balanced equation of thermal neutrons is when keff  1 :

T 

0 keff

  t exp((keff  1) )  1 1  td 

(58)

It is gotten from equations (9-6) in reference, which is in accordance with outward of equation (52), in initial condition of t  0andT (0)  0 . Equations (9-6) and (52) seem similar, because all parameters in the expressions of them are the same, and both represent the flux of thermal neutrons by T . But T in equation (52) is the flux of thermal neutrons in the reactor constituted by the flux  of thermal neutrons of thermal fission and the flux 0 of source thermal neutrons. But T in equations (9-6) vaguely represents thermal neutrons in the reactor, which is not analyzed and indicated in detail that it is constituted by two neutrons with different characters, that is to say, thermal neutrons of thermal fission and source thermal neutrons. The characters are different for the two types of neutrons, because the changing laws of the two types of neutrons are different in the processes of increasing and decreasing powers. However, the actual contents are the same for T in equation (52) and T in equations (9-6). So the results should be identical for the two equations.

Now if equation (58) is right, that is, equations (9-7) in figure [1] is right, then the following discussions are

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made.

condition, we have keff  1 . According to the original

First we suppose that T in equations (9-7) in figure [1]

definition of keff in texts, the flux of thermal neutrons in

doesn’t include source neutrons.

the reactor will decrease steadily because of keff  1 then. At last the flux of thermal neutrons in

When keff  1 and t ∞ , equation (58) is: 0 T  1  keff

(59)

Although it accords with the result of equations (9-11) in references [1], S    P  Ps in equations (9-12) is the number of source neutrons in hot districts in unit time.

Obviously the meaning for T of equations (9-11) (according to (9-12)) isn’t identical with the hypothesis for T in the front equation (59). It is wrong that T of equation (59) excludes source neutrons. Therefore, T in equation (59) satisfies equation (41).

We pull equation (41) in equation (59), then 0  keff  1  keff

can continue in steady condition.

If the power is lowered, meaning that the value of f decreases, when the value of f decreases until some stable value, the value of k 'eff will be smaller than one.

So because of the definition of k 'eff in the paper, the

(60)

So equation (60) accords with the correct result of equations (18), (37) and (55). A problem arises. The result from equation (53) and equation (58) in different initial conditions is the same as the result when keff  1 in perspective of the same differential balanced equation of thermal neutrons. It is impossible that both are correct. However, which one is the truly and precisely correct one? It is implied that the actual difference for them is their initial conditions, and that the result of equation (59) can prove that equation (41) is right. But the initial condition for equation (59) is that when t=0 and T  0 , T satisfied equation (41). So (when t=0)，the equation(44), equation(45) are correct.

So, according to equation (44), equation (45) and later discussion a conclusion can be reached, that is: the initial conditions of equations (9-7) in reference are wrong. So the result of equations (53) of the paper is correct, while the result of equations (9-7) is wrong, when keff  1 .

In sum, the verified result from the definition and expression of the paper in perspective of differential balanced equation of thermal neutrons proves that the definition, expression and the corresponding result are correct, precise and consistent.

The Application for the Phase of Power Operation and Ascending and Decreasing Powers in the Paper The Decreasing Power in Secondary Critical Phase If the reactor is in secondary critical phase and steady

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the reactor will be zero. It contradicts with the reality.

But, in fact because of the existence of source neutrons, when the reactor is in steady condition, the generating rate equals the disappearing rate of thermal neutrons of thermal fission in the reactor in any time. So according to the definition of the paper and equation (12), we have k 'eff  1 , the thermal fission in the reactor

flux  of thermal neutrons of thermal fission will decrease continually. In addition, although the decrease of the value of f and that is the increase of  c leads to the decrease of 0 either, the value of 0 is decreases until some stable value because of the corresponding equation (7). However, the value of  will decrease continually because of the value of k 'eff smaller than

one. Therefore, the value

of 0 /  will increases steadily, leading to the fact that the value of k 'eff will increase continually from the value below one. When it increases until k 'eff  1 , the reactor will be in steady condition again, and the parameters in the reactor will be stable and unchanged.

The Increasing Power of the Reactor in Secondary Critical Condition When the reactor is in some secondary critical condition and steady, then k 'eff  1 .

Because the reactor is in secondary critical condition, the value of keff is some stable value below one. When the power increases, the value of f will increase, as well making the value of k 'eff over one. If the value of f increases until some value f i and is unchanged. The value of  will increase continually because k 'eff  1 .

Although the value of 0 will increase with the increment of the value of f and that is the decreasing of  c up to some stable value because of the corresponding equation (7). So the value of 0 /  will decrease steadily. The increase of  will lead to the fact

International Journal of Nuclear Energy Science and Engineering Volume 4 Issue 1, March 2014

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that the value of k 'eff will eventually return to the

phase, the value of keff must be below one in

condition of k 'eff  1 from the value over one. Then it

anywhere in the reactor. That is to say, when in power operation phase, only in secondary critical condition, the reactor can run steadily.

returns to the stabilization.

steady

condition

and

maintain

The Application of the Result of the Paper in the Power Operation 1) The Description of the Reactor in Steady Condition As known, for the reactor without source neutrons, in theory, when it starts to take the reaction of fission in some way, and keff  1 , the reactor can run smoothly. But after a while, the value of f will decrease in the six factors of keff , and keff  1 .

Furthermore, the values of f in the reactor are different for the reason of distribution of the flux.

So the values of keff in the reactor are different.

Therefore, if we want to make it take the reaction of fission continually, source neutrons must exist in the reactor. The fission could be balanced then.

Therefore, after the introduction of source neutrons, the reactor can run eventually in steady condition.

2) The Description of Value of keff

3) The Determination of the Flux Distribution for Thermal Neutrons in the Reactor Because of equation (15), we know the values of keff and the fluxes of source thermal neutrons in anywhere in the reactor, so we can get the flux distribution of thermal neutrons of thermal fission and the power distribution in the reactor.

In addition, we can raise the flux of thermal neutrons of thermal fission, that is, raising the rating operation power of reactor, by raising the flux of source thermal neutrons in the situation of doing not change the fuel concentration, that is, do not change the keff . In the situation of same concentration and load of fuel, we can raise the power capacity of reactor by raising the flux of source thermal neutrons.

Conclusions

in Power

Then we can get the conclusion that when the original definition and expression of keff used in the reactor

In power operation phase in steady condition, if keff  1 at somewhere in the reactor, the flux of

with source neutrons are wrong and vague, and the definition conflicts. But when the definition and expression of the paper are used in the reactor with source neutrons, the definition is correct and the expression is precise. And the definition and expression are consistent.

Operation Phase

thermal neutrons will increase steadily and is unstable there. So this phenomenon can’t happen in the reactor.

If keff  1 , the phenomenon will appear in the 1) of this section. But if keff  1 at somewhere in the reactor and keff  1 at other where, the reaction of fission where keff  1 , can be balanced after the introduction of source neutrons. But after introduction of source neutrons, the reaction of fission, where keff  1 , will make its flux increase

At this point, the definition, explanation and expression deduction for k 'eff , the verification for the correction of the expression and the application of the result have already introduced until now.

REFERENCES

Lamash J.R. “Nuclear Reactor Theory Introduction”

continually, but unstable. The phenomenon won’t appear in the reactor either.

Therefore, in summary even if in power operation

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