The Foundation of Critical and Non-critical Neutrons Equilibrium Equations in Uniformity Dispersion

International Journal of Nuclear Energy Science and Engineering Volume 4 Issue 4, December 2014 doi: 10.14355/ijnese.2014.0404.01

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The Foundation of Critical and Non-critical Neutrons Equilibrium Equations in Uniformity Dispersion Reactor with Delayed Release Neutrons and the Calculation of Relevant Parameters Deqiang Pei Equipment Branch of Nuclear Island System Department of Engineering Design Company of China Guang Dong Nuclear Power of Shen Zhen (A-212 01 Building on Site of Tai Shan Nuclear Power), Tai Shan city Guang Dong Province People’s Republic of China peideqiang@cgnpc.com.cn Abstract I have written the paper: “Definition and expression of thermal neutron effective proliferation coefficient in the uniformity dispersion nuclear reactor with uniformity dispersion source neutrons(see reference II)” and “The introduction to the solution method of thermal neutron effective proliferation coefficient of any point in the non-uniformity dispersion reactor with non-uniformity dispersion source neutrons”(see reference III), but no consideration of the existence of delayed release neutrons.

However, actually there are delayed release neutrons in the reactor, so I write the paper in order to take the consideration of the effect of delayed release neutrons.

Key Word: Delayed Release Neutrons; Equilibrium Equations; Non-critical Uniformity Dispersion Reactor

process of fission, these neutrons’ pioneering element nucleus’ generating rate is βi ( k / p ) ∑ 2c ⋅Φ 2c . If

Ci (r , t ) is the space time distribution for the density of pioneering elements of the group i of delayed release neutrons, then the normal radioactive rate of spallation is λi Ci (r , t ) for the number of nucleus in every cubic centimeters at every second. The value of λi at every second is the corresponding spallation constant.

Supposing the space time distribution for thermal neutrons is Φ 2 (r , t ) , then the dynamic equilibrium equation for pioneering elements in group i of delayed release neutrons is the following formula.

∂Ci (r , t ) k = −λi Ci (r , t ) + βi ∑ 2c ⋅Φ 2c (r , t ) ∂t p

(1)

So the total generating rate for whole delayed release

The Foundation for Two Group Neutrons’ Equilibrium Equations in Non-Critical Uniformity Dispersion Reactor with Delayed Release Neutrons without Source Neutrons

delayed release neutrons, but the decelerating rate for

In the first we build the generating and disappearing equilibrium equations of delayed release neutrons.

The number of neutrons, which instant generating neutrons are decelerating into the thermal group in unit time, is (1 − β ) ⋅ k ⋅ Ps ⋅ Φ 2c (r , t )∑ 2c . At the same

The total generating rate for neutrons with nuclear fission is ∑ 2c ⋅Φ 2c ⋅ f ⋅η ⋅ ε , that is ( k / p ) ∑ 2c ⋅Φ 2c . If we suppose the value of βi is the share group i for delayed release neutrons in whole fission neutrons. Then in the

m

neutrons is ∑ λi Ci (r , t ) , m is the number of group of i =1

m

these neutrons to thermal districts is P ⋅ Ps ∑ λi Ci (r , t ) . i =1

time, the generating rate of fast neutrons decelerating into the thermal group can also be expressed as ∑1c ⋅Φ1c (r , t ) . Then the dynamic equilibrium equation of thermal neutrons in the reactor is the following

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

Supposing

formula.

D2c ∇ 2 Φ 2c (r , t ) − ∑ 2c ⋅Φ 2c (r , t ) +

m

w = (1 − β ) ⋅ k ⋅ Ps ∑ 2c + P ⋅ Ps ⋅ ∑ λiα i − ∑ 2c / Pd i =1

m

(1 − β ) ⋅ k ⋅ Ps ∑ 2c ⋅Φ 2c (r , t ) + P ⋅ Ps ⋅ ∑ λi Ci (r , t )

So formula (10) can change into

i =1

=

m

(1 − β ) ⋅ k ⋅ Ps ∑ 2c ⋅Φ 2c (r , t ) + P ⋅ Ps ⋅ ∑ λi Ci (r , t )

Φ 2c (t ) = e w⋅v ⋅t

1 d Φ 2c (r , t ) dt V2c

(2)

This can also be expressed as the following formula.

m

k (1 − β ) ∑ 2c ⋅Φ 2c (r ) + P ∑ λi ⋅ α i ⋅ Φ 2c (r ))Φ1c (t )

m

(1 − β ) ⋅ k ⋅ Ps ∑ 2c ⋅Φ 2c (r , t ) + P ⋅ Ps ⋅ ∑ λi Ci (r , t )

i =1

i =1

−Φ 2c (r , t ) ⋅ ∑ 2c / Pd =

1 d Φ 2c (r , t ) dt V2c

(3)=

The value of r is the space distribution. As for pillar shape reactor, the value of r is (r,z). From the former part of formula (2) and (3), we can get the following formula.

) ∑1c ⋅Φ1c (r , t =

w ⋅ V2 Φ1c (r )Φ1c (t ) Φ1c (r ) d Φ1c (t ) = dt V1 V1

+ P ⋅ Ps ⋅ ∑ λi Ci (r , t )

(4)

i =1

If the above formula holds, the space distribution of Φ1c (r , t ), Φ 2c (r , t ) and Ci (r , t ) must depart from the time varying section. Then

(13)

The first part of the above formula can changes into

 w ⋅ V2 D1c ∇ 2 Φ1c (r ) −  ∑1 + V1 

  Φ1c (r ) + 

m    k (1 − β ) ∑ 2c + P ∑ λi ⋅ α i  ⋅ Φ 2c (r ) = 0 i =1  

(1 − β ) ⋅ k ⋅ Ps ∑ 2c ⋅Φ 2c (r , t ) m

(14)

We substitute formula (6), (7), (8), (9) and (11) into the first part formula of formula (3), and eliminate Φ1c (t ) , Φ 2c (t ) and Ci (t ) , then

D2c ∇ 2 Φ 2c (r ) − ∑ 2c ⋅Φ 2c (r ) + ∑1c ⋅Φ1c (r ) =

W ⋅ Φ 2c (r )

Φ1c (r , t ) = Φ1c (r ) ⋅ Φ1c (t )

(5)

Φ 2c (r , t ) = Φ 2c (r ) ⋅ Φ 2c (t )

(6)

Ci (= r , t ) Ci (r ) ⋅ Ci (t )

(7)

D2c ∇ 2 Φ 2c (r ) − ( ∑ 2c +W ) Φ 2c (r ) + ∑1c ⋅Φ1c (r ) = 0 (16)

(8)

In addition, we substitute formula (6), (7), (8), (9) and (12) into formula (1), then after arrangement

And

Φ1c (t ) = Φ 2c (t ) = Ci (t )

So Φ1c (r ), Φ 2c (r ) and Ci (r ) have the same distribution, and it means that they are the identical distribution.

Supposing

(15)

The above formula can changes into

αi =

k ⋅ βi ⋅ ∑ 2c

(

P w ⋅ V2 + λi

(17)

)

Then formula (14) can changes into

Ci (r= ) α i ⋅ Φ 2c (r )

(9)

We substitute formula (6), (7), (8) and (9) into the last part formula of formula (3), and we eliminate Φ 2c (r ) ,then We can gain : m  1− β ⋅ k ⋅ P  ) ( s ∑ 2 c + P ⋅ Ps ⋅ ∑ λiα i − ∑ 2 c / Pd  i =1   1 d Φ 2c (t ) ⋅Φ 2c (t ) = dt V2c

82

From formula (5), (6), (7),(8),(9)and (12), we get the result that the dynamic equilibrium formula of fast neutrons is the following.

( D1c ∇ 2 Φ1c (r ) − ∑1c ⋅Φ1c (r ) +

D2c ∇ 2 Φ 2c (r , t ) − ∑ 2c ⋅Φ 2c (r , t ) + ∑1c ⋅Φ1c (r , t ) =

(12)

2

i =1

−Φ 2c (r , t ) ⋅ ∑ 2c / Pd =

(11)

(10)

 w ⋅ V2 D1c ∇ 2 Φ1c (r ) −  ∑1 + V1   m  k (1 − β ) ∑ 2c + ∑ k βi λi ⋅ ∑ 2c  i =1 w ⋅ V + λ i 2 

(

  Φ1c (r ) + 

)

  ⋅ Φ (r ) = 0  2c 

(18)

So formula (16) and (18) are the corresponding equilibrium equations of non-critical reactor with delayed release neutrons.

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

The Solution And Corresponding Parameters’ Confirmation Of Two Group Neutrons’ Equilibrium Equation In The Uniformity Dispersion Non-Critical Nuclear Reactor With Delayed Release Neutrons

After neutrons leaving the singular point region of formula (24) (all the WV2 + λ1 → 0 region), and between the region for greater or less than singular point region of W we can calculate the solution of W. Supposing

Similar to the formula7.10.3 in reference Ⅰ, formula (16) and (18) can change into

∇ 2 Φ 2c (r ) + B12 Φ 2c (r ) =0

(19)

∇ 2 Φ1c (r ) + B12 Φ1c (r ) =0

(20)

= b

In consideration of formula (19) and (20), we put

∇ Φ 2c (r ), ∇ Φ1c (r ) into formula (16) and (18), then 2

2

 w ⋅ V2 −  D1c B12 + ∑1 + V1   m  k (1 − β ) ∑ 2c + ∑ k βi λi ⋅ ∑ 2c  i =1 w ⋅ V + λ 2 i 

(

  ⋅ Φ (r ) = 0  2c 

(

)

2 0 ∑1c Φ1c (r ) − ∑ 2c +W + D2c B1 Φ 2c (r ) =

(21)

(22)

following formula.

w ⋅ V2 V1

∑1c

m

k (1 − β ) ∑ 2c + k ∑

i =1

=

∑ 2c +W

βi λi ⋅ ∑ 2c w ⋅ V2 + λi

+D2c B12

Supposing

τ th =

D1c

L22c =

∑1c

(1 + τ

(1 + τ

2 th B1

)

2 th B1 + ∑ 2 c

)(1 + L

2 2 2 c B1

 1 + L22c B12  W + 

  m βi λi ∑ ∑ V  k (1 − β ) 1c 2c 1 + k  ∑  V2  i =1 WV2 + λi 

)

1c ∑ 2 c V1

V2

) + ∑ (1 + L 2c

2 2 2 c B1

)

)(

(

)

c= 1 + τ th B12 1 + L22c B12 ⋅

∑1c ∑ 2c V1 − V2

  m βi λi ∑ ∑ V  k (1 − β ) 1c 2c 1 + k  ∑  V2  i =1 WV2 + λi  So formula (24) formula.

 ∑1c ∑ 2c V1     V2  

can changes into the following

W 2 + bW + c = 0 b 1 2 So W =− ± b − 4c 2 2

(23)

(25)

If the reactor is super critical reactor, then the value of W is over zero, and, formula (24) is equivalent to a quadratic equation when after calculated m

∑

βi λi

i =1 WV2

+ λi

And then we give a positive value of W

.into formula (25). According to formula (25), we calculate the quadratic equation and get a new positive value of W ' . If W ' > W , we increase the value of W to calculate again until the value of W converges to some value. If W ' < W , we decrease the value of W to calculate again until the value of W converges to some value. This is the result of W when W > 0

When W is less zero ,the result of W should add a formula as below:

∑ 2c

(

)⋅ ∑

2 th B1

At this time ,the C must less zero.

D2c

So formula (23) can change into

V ∑ W 2 +  1 1c  V2

(1 + τ

b 1 2 W2 =− + b − 4c 2 2

The condition for the meaningful solution to Φ1c (r ) and Φ 2c (r ) in the above two formulas is the

D1c B12 + ∑1 +

V2

When W is positive value the result is

  Φ1c (r ) + 

)

V1 ∑1c

So b > 0

The value of B12 in the above formula is the geometric curvature of the uniformity dispersion non-critical nuclear reactor. Therefore, at this moment the corresponding distribution of two group neutrons’ flux doesn’t change in the reactor with delayed release neutrons, but the part of W in the varying part for time has changed.

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−

b 1 2 W2 =− − b − 4c 2 2

(26)

We will discuss the calculating result of W when W is less zero as below:

At this time , WV2 + λ1 is over zero and less than the  ∑1c ∑ 2c V1  WV2 + λ1 ,when W is over zero. So at this time C is less 0 (24) =   V 2  zero And when we introduce C into the formula (25), 

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

the result of W for calculating will be over zero. It’s conflict with the hypothesis which W is less zero. So formula (25)is not applicable for the calculation of W When W is less zero. Only formula (26) is applicable.

The result of calculation using formula (26) is the right result of the right side of singular point region of W when W is less zero. Otherwise, we suppose at some point inside some singular point region (from λi to

λi +1 ), the C is equal to zero, then the right region of

The Determination Of K eff and Corresponding Parameters in the Uniformity Dispersion Non-Critical Nuclear Reactor with Delayed Release Neutrons From formula (11) m

P ⋅ Ps ∑ ⋅λi ⋅ α i = W − (1 − β )k ⋅ Ps ⋅ ∑ 2c + ∑ 2c / Pd i =1

(27)

From formula (17) this point ,the C is over zero, and at the left region of this point ,the C is less zero. Then when C is over zero, k (28) W ⋅ V2 ⋅ α i =−λi ⋅ α i + ⋅ βi ∑ 2c because the region which C is over zero, the W is P meeting the formula (25),(26) at this time., there are In consideration of formula (27) and (28)，then: two calculation result which satisfy formula (25),(26) m m k m for W. And when the region which C is less zero, the − ∑ λiα i + ∑ ⋅βi ⋅ ∑ 2c W ⋅ V2 ∑ α i = W is only meeting the formula (26) at this time. That is, Pi 1 =i 1 =i 1 = there is only one calculation result which satisfy (1 − β ) ⋅ k ⋅ Ps ⋅ ∑ 2c −W − ∑ 2c / Pd k formula (26) for W.. And at whole singular point (29) = + ⋅ β ⋅ ∑ 2c P ⋅ Ps P region ,there are 3(m-1) entries result for W. At the left side of singular point region, because C is over zero, We substitute formula (17) into the above formula, the W is meeting the formula (25),(26), at this time. after the arrangement There are two calculation result which satisfy formula m K eff − 1 W ⋅ Pd βi (25),(26) for W. Namely for formula (24), there are (30) = + W ⋅ V2 ∑ K eff K eff ∑ 2c i =1 WV2 + λi 1+3(m-1)+2=3m entries results. From the 10.52 of reference I we know, at the right side of singular point From the above formula region when W is less zero, the result which is calculating is the right result. The method of ∑ 2c +W ⋅ Pd (31) K eff = m βi calculation is below: − W ⋅ V ∑ 2c 2 ∑ 2c ∑ i =1 WV2 + λi .If the reactor is the secondary critical reactor, the solution of W will be less than zero. So at this time formula(26) is the negative root equation. At this time there are one calculation methods in order to make sense for the result to the value of W . In the right side of singular point region, if the minimum value of

λi is λ1 , then Wmin = −

λ1 V2

W after calculation. If W < W , we can decrease the value of W and calculate again, until the vale of

absorbing rate of thermal neutrons + leaking rate of thermal neutrons

1 ∑ 2c Φ 2c (r ) …(32) = 2 2 2 ∑ 2c Φ 2c (r ) + D2c B1 Φ 2c (r ) 1 + L2c B1

Meanwhile, supposing

'

W converges to some value. However, if W ' > W , we can increase the value of W and calculate again, until the vale of W converges to some value. The obtained value of W is the right result to the equation (24) at this moment.

84

Pd = the absorbing rate of thermal neutrons/ the

. Then if we suppose W > Wmin , Pd =

we substitute it to formula (26), which is the result to formula (24). We will get a new negative value of '

In the formula (30) and (31) Pd is the following formula

ρ=

k eff − 1 k eff

(33)

Reference to 10.22, 28, 29 of reference Ⅰ and formula (10), (11) and (12) of this paper. So

1 W ⋅ V2 = − T

(34)

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

So formula (30)can change

the following formula

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(

)

m β i ⋅ WV2 + λi m βi ⋅ λi βi ⋅ λi =1 − ∑ +∑ m i 1 = i 1 WV2 + λi =i 1 = Pd βi WV2 + λi WV2 + λi (35) = +∑ ρ m m m T λ 1 + 1 i = k eff T ∑ 2c V2 βi βi ⋅ λi βi ⋅ λi i = −∑ +∑ 1 − WV2 ∑ =i 1 = = i 1 WV2 + λi WV2 + λi i 1 WV 2 + λi 1 in 10.18.1 of reference Reference to formula l0 = m βi = 1 − WV2 ∑ ∑ 2c V2 i =1 WV2 + λi m

1− β + ∑

Ⅰ, so the above formula becomes

ρ =

m l0 Pd βi +∑ k eff ⋅ T i =1 1 + λiT

(36)

(

formula 10.23.2 in referenceⅠ

l0

(37)

1 + L22c B12

)

WV2 ) ∑ 2c +W + D2c B12 V1 from the k= m   βi ∑1C ⋅∑ 2c ⋅ 1 − WV2 ∑  i =1 WV2 + λi   definition of k eff and formula 7.30.2 and 7.62.1 in ( D1c B12 + ∑1C +

However, Pd is defined in formula (32), so from

l=

We substitute it to the former formula, then

referenceⅠ,then

Then

k eff 1 =k ⋅ Pd ⋅ Ps

ρ =

l k eff ⋅ T

βi i =1 1 + λi T m

(38)

+∑

This formula is the same as the formula 10.29.1 in reference I.

k eff =

2 ∑ 2c +W + D2c B1 m   βi 2 2 ∑ 2c 1 − WV2 ∑  1 + L2c B1 i =1 WV2 + λi  

(

)

k=

(

WV2 ) ∑ 2c +W + D2c B12 V1 m  βi ⋅ λi  ∑1C ⋅∑ 2c 1 − β + ∑  i =1 WV2 + λi   

)

k eff =

=

(1 + τ

2 th B1

)( ∑

2 c +W

 βi ∑ 2c 1 − WV2 ∑ i =1 WV2 + λi  m

(

) )(1 + τ

+ D2c B12

 2 2  1 + L2c B1 

(

) )(1 + τ

2 th B1

(∑1C + D1c B12 ) ∑ 2c +W + D2c B12 m  βi ∑1c ⋅∑ 2c 1 − WV2 ∑ i =1 WV2 + λi 

 2 2  1 + L2c B1 

(

2 th B1

)

)

(41)

In according to the definition, then

k eff 1 = k eff

(42)

V2

is the first item in the

However, in fact the factor W

V1

numerator of formula (40) more than the first item of formula (41). Therefore, the two items are different. So what is the reason? We will analyze for the question.

Supposing every generation of thermal fission instant neutrons will produce at t0 , and decelerate to thermal

From formula (23) in this paper

( D1c B12 + ∑1C +

(39)

)(

But formula (39) can change the following formula.

above formulas into formula (31) and (35), so we get the values of k eff and ρ in the uniformity dispersion

Although we get the above result, we still have an embarrassing phenomenon. We substitute formula (32) into formula (31), then

)

(

We substitute the values of W , Pd and T from the

non-critical nuclear reactor with delayed release neutrons at this moment. But in the formula (38) of this paper is the same as formula 10.29.1 in referenceⅠ.

The proof of this paper on quantity relationship and result of generation and disappearance of thermal neutrons and fast neutrons is the same as the result widely acknowledged. It is feasible, and can be applied.

(

WV2 ) ∑ 2c +W + D2c B12 V1 (40) = m   βi 2 2 2 ∑1C ⋅∑ 2c ⋅ 1 − WV2 ∑  1 + L2c B1 1 + τ1h B1 i =1 WV2 + λi   ( D1c B12 + ∑1C +

)

region after t2 . They disappear after t3 . From the right side of the former equation of formula (2) of thermal fission neutrons generating and disappearing equilibrium equation, we can see that delayed release m

so

neutrons ∑ λi ⋅ Ci ( r , t0 ) being generated at t0 ,after t2 i =1

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

being changed fission:

into thermal neutrons of thermal m

P ⋅ Ps ⋅ ∑ λi ⋅ Ci ( r , t0 + t2 ) i =1

and

disappear

at t0 + t 2 + t3 . So delayed release neutrons will generate at

the

same

time

with

instant

neutrons

m

(as ∑ λi ⋅ Ci ( r , t0 ) ), and at the same time will arrive at i =1

thermal

region

m

(as P ⋅ Ps ⋅ ∑ λi ⋅ Ci ( r , t0 + t2 ) ) i =1

and

at t0 + t 2 + t3 will disappear at one time. λi ⋅ Ci ( r , t ) will generate and disappears at the same time with instant neutrons of thermal fission. That is to say, generally speaking, the delayed time of delayed release neutrons don’t influence the time of produce and disappearance of delayed release neutrons for λi ⋅ Ci ( r , t ) . This time is same as the generating and disappearing time of every generation neutrons of instant neutrons.

Then we look at the formula (3) in this paper,.

Supposing Φ 2c ( r , t1 ) is the flux of thermal neutrons at t1 , and t2 is the decelerating time for fission neutrons reducing into the thermal region. So the generating rate (1 − β ) k ⋅ Ps ∑ 2c Φ 2c (r , t1 ) should generate

pass the process of Ps , that is to say,

these neutrons will arrive in the thermal region after t2 that is they will generate after t2 ,but is not now.

The generating rate of thermal neutrons in the reactor at t1 equals the generating rate of neutrons, which have disappeared before t2 . Its flux is Φ 2c (r , t1 − t 2 ) . Delayed release neutrons have the same situation. The density Ci ( r , ti ) of delayed release neutrons at t1 will arrives to thermal region after t2 . However, the delayed release neutrons, which arrive at thermal region at t1 , are issued by delayed release neutrons before t2 . Its

density is Ci ( r , t1 − t2 ) . Because of the difference, formula (40) is different from formula (41). But for light-water reactor the generation neutrons time is 2.1× 10−4 second, the time t2 for neutrons decelerating is 10−5 second(see table 6.147 in referenceⅠ. t2 have a proportion for less than 5% of the generation neutrons time. Therefore, it’s negligible for the difference between Φ 2c (r , t1 ) and Φ 2c (r , t1 + t 2 ) So t2 is negligible. So when there are delayed release neutrons, it’s feasible, correct and acceptable for the result, that is the quantity relationship equilibrium and the expression of keff of the generation and disappearance for thermal neutrons and fast neutrons, 86

gotten from the usual method at present. But the actual the non-uniformity dispersion reactor has delayed release neutrons.

Therefore, using the above method for it to get the solution is credible and acceptable.

The Determination of Critical Condition and Foundation for Two Group Neutrons Equilibrium Equation in the Uniformity Dispersion Nuclear Reactor with Delayed Release Neutrons As the critical reactor, formula (1),(3),the part of time are constant, and the left side of formula (1) is zero, and:

λ= i Ci ( r )

k βi ∑ 2c ⋅Φ 2c ( r ) p

(43)

After accumulate it, we substitute it into formula (3), then

D2c ∇ 2 Φ 2c (r ) − ∑ 2c ⋅Φ 2c (r ) + ∑1c ⋅Φ1c (r ) =

(1 − β ) k ⋅ Ps ∑ 2c Φ 2c (r ) + k ⋅ Ps ∑ 2c βΦ 2c (r ) −Φ 2c (r )∑ 2c / Pd = 0

(44)

The above formula can change into

D2c ∇ 2 Φ 2c (r ) − ∑ 2c ⋅Φ 2c (r ) + ∑1c ⋅Φ1c (r ) = 0

(45)

This is the equilibrium equation of thermal neutrons at core part in the critical reactor. For fast neutrons we have the same corresponding result. So in the critical condition the uniformity dispersion nuclear reactor has the same equilibrium equation, critical condition and flux space distribution of neutrons with or without delayed release neutrons.

The Expression of Effective Proliferation Coefficient for Thermal Neutrons of Thermal Fission in the Uniformity Dispersion Non-Critical Reactor with Delayed Release Neutrons Because it is non-critical reactor, only when introducing neutron source, the reactor can be stable operation.

Because at the moment the reactor is in stable condition, the left side of formula (1) should be zero.

We accumulate formula (1), and because of the stabilization and no relationship with time. Then m

∑ λ= i Ci ( r )

i =1

k β ∑ 2c ⋅Φ 2c ( r ) p

(46)

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

After introducing neutron source, according to the definition of effective proliferation coefficient of thermal neutrons of thermal fission (see literatureⅡ), we have the following formula.

(1 − β ) ⋅ k ⋅ Ps ⋅ ∑ 2c ⋅Φ 2c ( r ) + m

' keff =

P ⋅ Ps ∑ λi ⋅ Ci ( r ) + Φ 0 ( r ) ⋅ ∑ 2c ⋅ f ⋅η ⋅ ε ⋅ P ⋅ Ps i =1

Φ 2c ( r ) ⋅ ∑ 2c / Pd

(47)

Because of formula (46), formula (47) can changes into the following formula. ' = keff keff (1 + Φ 0 ( r ) / Φ 2c ( r ) )

(48)

' keff =1

' . This paper also verifies the relation expression of keff

formula between T and ρ .

Meanwhile, this paper verifies the calculation is acceptable for the quantity relationship equilibrium equations and the expression of keff of the generation and disappearance for thermal neutrons and fast neutrons in condition of delayed release neutrons.

Until now, all the deductions have finished.

REFERENCE

Deqiang. Pei “ Definition and Expression of Thermal Neutron

As in the stable condition, then (49)

The above result is the same as literature Ⅱ.

The Determination Of keff

www.ijnese.org

Effective

Proliferation

Coefficient

in

the

Uniformity Dispersion Nuclear Reactor with Uniformity Dispersion Source Neutrons “Vol.4.ISS.1 of International Journal of Nuclear Energy Science and Engineering.

Deqiang. Pei” The Introduction to the Solution Method of Thermal Neutron Effective Proliferation Coefficient of

The formula (40) of keff deduced from formula (23) of

any Point in the Non-uniformity Dispersion Reactor with

this paper, which is inferred using the usual method.

Therefore, we suggest that keff take the value as

Non-uniformity Dispersion Source Neutrons” Vol.4.ISS.3

formula (40).

Engineering.

Conclusion This paper gives the calculation formula for W, T, keff and ρ in the condition of delayed release neutrons,

of International Journal of Nuclear Energy Science and Glassden/Adelen. “Atom Nuclear Reactor Theory Outline” Lamash J.R. “Nuclear Reactor Theory Introduction” Zhongsheng Xie “Nuclear Reactor Theory Introduction”

and gives the deduction and determination of the

87