The treatment of the neutron transport as a diffusion process has only been validated. For example, in a Light Water Reactor (LWR) the mean free path of thermal neutrons is typically around 1 cm.

The fractional model has been derived for the NPK equations with n groups of delayed neutrons, is given by:

Where:

t is the relaxation time, k is the anomalous diffusion order (for sub-diffusion process: 0 < k < 1; while that for super-diffusion process: 1 < k < 2), n is the neutron density, Ci is the concentration of delayed neutron precursor, l is the prompt-neutron lifetime for finite media, K is the neutron generation time, в is the fraction of delayed neutrons, and p is the reactivity. When Tk ^ 0, the classic NPK equation is recovered.

The fractional model includes three additional terms relating to the classic equations which are contained of fractional derivatives (Gilberto and Espinosa, 2011):

dk+1n dtk+1 ‘

dkn

and:

dkCj dtk

The physical meaning of above terms suggests that for sub-diffusion processes, the first term has an important contribution for rapid changes in the neutron density (for example in the turbine trip in a BWR nuclear power plant (NPP)), while the second term represents an important contribution when the changes in the neutron density is almost slow, for example during startup in a NPP that involves operational maneuvers due to movement of control rod mechanism. The importance of third term is when the reactor sets in shutdown state, it
could also be important to understand the processes in the accelerator driven system (ADS), which is a subcritical system characterized by a low fraction of delayed neutrons and by a small Doppler reactivity coefficient and totally there are many interesting problems to consider under the view point of fractional differential equations (FDEs) (Gilberto and Espinosa, 2011).

The neutron point kinetics (NPK) equations are one of the most important reduced models of nuclear engineering, and they have been the subject of countless studies and applications to understand the neutron dynamics and its effects. These equations are shown below:

	dntt) = n(t) = p-en(t) + £ XC (t) dt л 7=1	(37)
and:	=C i (t)=в n(t) — XC (t) dt Л	(38)
Where:
n(t) is: the variations of neutrons density, pe is: injected reactivity to system’s transfer function, в is: delayed neutrons fraction, Л is: neutron generation time, Xi is: decay coefficient per density of each group of delayed neutrons, Ci is: density of each group of fission fragments which are as delayed neutrons generators.
If the partial variations per each system and control variables C(t) and C(t) are considered the these can write as following:	including: n(t), pe, n(t),
	Sn(t) =Sp в (n0 + Sn(t)) + X(C0 + SC(t)) Л	(39)
and:	SC (t) = e(n0 + Sn(t)) — X(C0 + SC(t)) Л	(40)

If these both Sn(t) and SC(t) equations to matrix format are written, then the matrix format of them is shown as below:

In the linear state can write last equation as the following:

In case the point kinetic equations of reactor from both neutron density and fission fragments aspects are considered, it can be written:

s + в — Я

Л

-в,

— s + Я Л

(43)

 

Where:

(44)

 

Therefore the transfer function will be as shown:

(45)

Where:

Pe(s) — P0 — Pf (s)

Also the equation of zero power transfer function is as following:

_ n0(s + Я)

s(s + Я + в) s(s + в) ^Л + й

Л Л

and for variation of fission fragment density as per error reactivity variation, it can write:

(48)

In case the point kinetic equations of reactor from neutron density and fission fragments aspects and also temperature aspect (according to Newton’s low of cooling) are considered, it can be expressed:

s + P	-A	ШІ0
Л		~Л
-P	s + A	0
Л
-k	0	s + a

[1 0 0] adj

Where:

In this stage according to the main transfer function that has mentioned in the last stage, G(p) is defined:

Therefore through G(p), the parts of G(p) will be defined and the stability condition through V function is applied.

So can write:

G(s) — cT 0(s)h

In large twin reactor’s cores, it can be written:

n(t) = p-en(t) + f^Q (t) + q{ (t)

л u

It is also supposed:

and:

a 12 — a21

q2 , q2 and pi are respectively as following:

qi(t) = a n2(t — T12), Л1

q2(t)=a2L n1(t — t21) Л2

and:

m ZjL«j (t — T) qi(t) = І—1*i Лі__________

So qi (t) divided by pi equals:

According to Newton’s low of cooling can write:

T (t) — — n(t) — aT(t) (61)

mcp

If the variations of neutron density than initial neutron numbers, fission fragment density than initial fission fragment density and temperature than initial temperature as system variables are considered respectively as below:
and:

z; (t) = ax; (t) — az; (t) (67)

If these system variables differentials are to matrix format are written, then can write:

n; «,■; N П; ai; ■ £ if. /=1« ";0 ^ /=1*i f)

(68)