elucidated by the fact that at a particular space the neutrons will have spatial long- tailed distributions. The effect of this long-tailed statistical probability distribution will thus get enhanced by the use of non-local divergence, and this reality effect will thus be shown with avoidance of volume shrinkage to zero. Also in reality, the coupling between various zones in the reactor takes place. The non-local diver- gence with the principle of non-zero volume therefore is the apt tool for constitutive equation for neutron diffusion equation for reactor description.
Refer the classical neutron diffusion equation and Fig. 4.2. The surface flux is
S
J.d S; then the ratio of the surface flux to the volume is taken to consider the leakage through surface term as ∇.J , at volume shrinking towards zero.
Then with Fick’s law, we get the leakage of neutrons through a closed surface as
∇.(−D∇φ)= −D∇2φ. This term in one dimension case is
−Dd2φ d x2
Examining the above and relating this to Fig. 4.2, one may say that divergence is the slope of the surface flux (Fig. 4.2a). If this curvature has square law variation in the shape, then the double derivative will be constant, and we have the entire curvature captured in that constant. If the curvature of the surface flux (Fig. 4.2a) is not having
≈x2variation, but say has≈x1.5variation, then double derivative will not capture the information about the curvature in a constant.
72 4 Concept of Fractional Divergence and Fractional Curl
Dd1+αφ
d x1+α −Σaφ+S =0, . . .0< α <1 Ddβφ
d xβ −Σaφ+S =0, . . .1< β <1
One may interpret the simplified form of∇α.J is that a fractional divergence oper- ator is applied to Fickian dispersion term. For illustration of how fractional deriva- tives relate to the definition of divergence in neutron transport, consider two simple functions f (x )=x2and f (x )=x1.5. Recalling the fundamental of the derivatives, we recall that the derivatives of function give the idea of the curvature contained in the curve. Successive derivatives will strip of the independent variable and sub- sequently will show up the curvature contents. Take the example f (x ) = x2, the first and second derivatives are f(1)(x ) = 2x and f(2)(x ) = 2, respectively. In this example, the second derivative contains all the information about the function and that is constant (i.e., 2). This argument favors well for well-behaved integer order functions. Now let us change the function and take any real order func- tion, say f (x ) = x1.5; the first and second derivative is f(1)(x ) = 1.5x0.5 and f(2)(x ) = 0.75x−0.5. In both the derivatives, for this real ordered function, the derivatives vary with the independent variable. Refer Fig. 4.2b; the effect of growing control volume is approximated as steps. Each step signifies growing value of D.
The shape emerged as dotted curve in Fig. 4.2b, approximates the first derivative of Fig. 4.2a. Here the first derivative or any other higher order integer derivative fails to contain the curvature information.
In the functions with real order (other than integer order), if the concepts of fractional calculus are applied then we can contain the curvature information of f (x )=x1.5by taking 1.5 derivative of the f (x ), symbolically d1.5x1.5/d x1.5=1.33.
dα
d xαxu = Γ(u+1) Γ(u−α+1)xu−α d1.5
d x1.5x1.5= Γ(1.5+1)
Γ(1.5−1.5+1)x1.5−1.5=Γ(2.5)=1.33
Therefore if the fractional differential operator is chosen in which the fractional order of differentiation matches the power law scaling of the function, then the curvature is reduced to a constant and all the scaling information is contained in the order of the derivative and that constant. If the neutron plume is traveling through material with evolving heterogeneity, then a fractional divergence might account for the increased dispersive flux over larger range of measurement scale.
This argument sets the stage for writing neutron diffusion equation with frac- tional calculus. The above derivation and discussion completes the description of neutron diffusion equations in fractional calculus.
In deriving the fractional differential equations for neutron diffusion in an enclosed volume, we argued the basis of taking a larger observation space for defining divergence. The non-local formation of the divergence thus gives the effect
of macroscopic effects caused by velocity fluctuations, the coupling effect of nearby zonal neighborhood neutrons (refer Fig. 4.3). This therefore is making the consti- tutive descriptive equations closer to reality. The classical integer order constitutive differential equations are approximations to everything as “point” quantity in time or in space. The classical integer order methods do not thus take into account the space history or time history and therefore cannot represent the natural laws close to reality. Fractional calculus does take of all these reality and therefore is more appropriate for representation of natural phenomena. Refer Fig. 4.3, as an outside observer, let us try to visualize space squares as depicted in the figure. The squares without any neutrons in them look different to an outside observer as compared to the squares with the neutrons, while observer sitting in the same squares will not notice the difference in the squares with or without neutrons. So the observer in the same space will apply point quantity and will try to describe the neutron balance by classical integer order calculus. Whereas to the outside observer the squares or the space will appear transformed with or without the presence of neutrons. The outside observer thus will apply this space transformation correction factor and obtain some different results, and that result will be close to reality.
4.6.1 Solution of Classical Constitutive Neutron Diffusion Equation (Integer Order)
This section will serve as a revision to simple classical solution of the diffusion equation. Then in the next section, we will solve the fractional differential equa- tion obtained. This we will demonstrate the space variables in one dimension for simplicity.
D∇2φ−Σaφ+S =dn dt = 1
v dφ
dt S=k∞Σaφ
D∇2φ+(k∞−1)Σaφ=1 v
dφ dt
The flux term is variable of space and time. The source term multiplication law governs S. The separation of variables will give the following for the flux term which can be substituted in the basic constitutive equation, and following expressions will emerge.
φ=φ(r )e−Λt
D∇2φ(r )+(k∞−1)Σaφ(r )= −Λ v φ(r )
Λis positive for sub-critical, negative for super critical, and zero for critical equilib- rium reactor. We replace the space coordinate r , by x and with substitution of B as
74 4 Concept of Fractional Divergence and Fractional Curl geometric buckling, we get following simple form. The temporal solution is avoided for simplicity.
B2=(k∞−1)Σa+Λv D d2φ(x )
d x2 +B2φ(x )=0
Here we can apply standard Laplace method with initial conditions at x =0 at the center point of the reactor geometry having constant flux and at the walls at x =a zero flux. General Laplace formula for derivative of function is indicated below and is applied to have polynomial form.
s2Φ(s)− 1
k=0
skd2−k−1φ(x )
d x2−k−1 ]at x=0+B2Φ(s)=0 s2Φ(s)−dφ(x )
d x ]at x=0−sφ(x )]at x=0+B2Φ(s)=0 dφ(x )
d x ]at x=0=0 andφ(x )]at x=0 =C
The above initial condition gives simple equation as s2Φ(s)−sC+B2Φ(s)=0 Φ(s)= sC
s2+B2 taking the inverseφ(x )=C cos B x
4.6.2 Solution of Fractional Divergence Based Neutron Diffusion Equation (Fractional Order)
With the extension of the above method, we try to solve the fractional differential equation:
dβφ(x )
d xβ +B2φ(x )=0 1< β <2
L dαf (x ) d xα
=sαF(s)−
n−1
k=0
skdα−k−1f (x ) dα−k−1 ]at x=0
sβΦ(s)−dβ−1φ(x )
d xβ−1 ]at x=0−sdβ−2φ(x )
d xβ−2 ]at x=0+B2Φ(s)=0
The above is Laplace transformation for LHD definition of the fractional derivative.
In this expression, the second and the third term of the left hand side has frac- tional derivative of the flux at initial point, which is physically difficult to define and to realize the same by experimental measurements is difficult at this stage. Let us try to make use of Laplace transformation of RHD Caputo definition, as given below:
L dα d xα f (x )
=sαF(s)−
n−1
k=0
sα−k−1 dk
d xk f (x )at x=0
sβΦ(s)−sβ−1φ(x )at x=0−sβ−2 d
d xφ(x )at x=0+B2Φ(s)=0
We relate the above expression physically to the earlier initial condition taking sec- ond term as C and third term as zero as done in Sect. 4.6.1. Here the integer order derivative comes as initial condition, therefore physically realizable from measure- ments and observations.
Φ(s)= sβ−1C sβ+B2 φ(x )=C.L−1
sβ−1 sβ+B2
The solution of the fractional differential equation for the constitutive neutron bal- ance equation therefore is with Laplace identity
L(Eα(−λtα)= sα−1 sα+λ, we obtain
φ(x )=C.Eβ(−B2xβ)=C ∞ k=0
(−B2xβ)k Γ(βk+1) φ(x )=C+C(−B2xβ)
Γ(β+1)+C (B4x2β)
Γ(2β+1)+C(−B6x3β) Γ(3β+1)+. . .
The above is flux mapping obtained by the solution of fractional order neutron con- stitutive equations that is obtained by the concept of fractional divergence.
Let us see what classical flux pattern and fractional order flux pattern are same when we take the fractional order equal to 2, the integer order.
Solution in the classical form is cosine function and series representation of the same is
76 4 Concept of Fractional Divergence and Fractional Curl
cos(x )=1−x2 2!+x4
4! −x6 6! +. . . β =2, φβ=2(x )=C Eβ(−B2x2)=C
∞ k=0
(−B2x2)k Γ(2k+1)
=C
1−(B2x2)
Γ(3) +(B2x2)2
Γ(5) −(B2x2)3 Γ(7) +. . .
Γ(n+1)=n!
φβ=2(x )=C
1−(B x )2
2! +(B x )4
4! −(B x )6 6! +. . .
≈C.cos(B x )
Therefore when the fractional order equals the integer order, we get classical flux profile. This is proof of our assumption that indeed neutron flux being not a point quantity be represented as fractional divergence of order less than unity.
4.6.3 Fractional Geometrical Buckling and Non-point Reactor Kinetics
The above concept of fractional divergence gave a deviation from ideal flux map (cosine). The term geometrical buckling is indicative of the flux profile of neutron flux inside the reactor. Measuring the actual flux distribution and then controlling the power of reactor is one mode of reactor control. Now if the control computer is kept with a map of cosine table and the neutron spatial detectors are mapping in each control cycle, then a deviation, the unwarranted correction cycles, will keep the control devises moving. Actually the correction may not be called for if the control computer is programed with actual fractional geometrical buckling data. The frac- tional divergence has given the new thought of “fractional geometrical buckling,”
which in turn when used with basic multiplying factor k∞gives rise to a concept of fractional criticality. The describing reactor kinetics with fractional divergence will give the concept of non-point kinetic description.