In classical sense, the constitutive equation assumes point neutron flux, withv as the average speed of the neutrons passing through an area with n neutrons per unit volume as neutron density. The vector quantity representing the neutron flux is J . The following will elucidate the classical statements.
68 4 Concept of Fractional Divergence and Fractional Curl
−
→J =nv φ=nv
−
→J = −D∇φ
Consider a closed volume, the loss of neutrons from the closed surface is given as surface integral of neutron current, J.d S. The loss occurring in the volume by absorption is given by absorption cross-section and then taking volume integral ofΣaφd V . This total loss, when equated to the source term, gives the classical constitutive neutron diffusion equation, as depicted below.
S
J.d S+
V
Σaφd V =
V
Sd V
The above integral form when converted to volume integral is:
V
(∇.J+
aφ−S)d V =0 or ∇.J+Σaφ−S=0 for equilibrium Using the expression of J = −D∇φ, we obtain the following:
−D∇2φ+
aφ−S=0
In the steady state, the RHS of above constitutive equation is zero, and if there is time changing flux, then that is put in the RHS as
D∇2φ−
aφ+S= ∂n
∂t = 1 v
dφ dt
4.5.1 Discussion on Classical Constitutive Equations
The classical neutron diffusion constitutive equation as described is based on the classical divergence of the divergence of a vector field. The divergence is defined as the ratio of total flux through a closed surface to the volume enclosed by the surface, when volume shrinks toward zero.
divJ = lim
V→0
1 V
S
J.d S
Where J is the flux vector, V is an arbitrary volume enclosed by surface S. The dot product of vector J with the surface dS is obvious; this is valid only if the flux is indeed a “point” quantity relative to the scale of observation. Neutron diffusion is counterexample; this is primarily, due to velocity fluctuations (even at constant
energy/temperature) that arise only as the observation space grows larger, invali- dating the limit. Also the neutrons are no longer in homogeneous medium. The dispersive fluxes for a given volume are typically averaged in some fashion (vol- umetric, statistical) and are approximated by Fick’s first law as we have obtained in deriving the classical constitutive equation for neutron diffusion,−→
J = −D∇φ.
As the control volume shrinks to zero, the velocity fluctuations and the dispersive flux disappear. Therefore in a true sense, the classical divergence theorem discounts the real effects of macroscopic in-homogeneity and the fluctuations associated with neutron diffusion in a reactor.
Because of the limit in the divergence definition, the classical Gauss divergence theorem discounts the effect of large volume until the dispersive flux can be approx- imated by a point quantity.
4.5.2 Graphical Explanation
Refer Fig. 4.1. In Fig. 4.1a, it is shown that the surface flux with respect to volume of the observation space is of constant slope line. Figure 4.1b plots the ratio of the surface flux with respect to the control volume (first derivative of Fig. 4.1a).
Figure 4.1 shows, in simplistic manner, that if the surface flux of neutrons with average constant velocity grows in linear fashion with respect to the volume of the observation space, then in this case, the ratio of the surface flux to the control volume remains fixed. In this particular (ideal) case, making the control volume shrink to zero will yield ideal definition of the divergence of the vector flux (neutron current density). This simplistic picture neglects the effect of in-homogeneous medium and macroscopic dispersion, fluctuating velocity effects and effects due to neighbor- hood, neutron currents.
Figure 4.2 is an extension of Fig. 4.1 showing the macroscopic effects of surface flux manifestaion as the control volume is enlarged. The observation space when enlarged as shown in Fig. 4.2b captures dispersive effect of neutrons as magnified by the staircase type of ratio of surface flux to the volume figure. The effect can be seen as surface flux gets manifested as some power of observation space (vol- ume). Figure 4.2b is the first derivative of Fig. 4.2a and shows that at quasi large observation space (control volume), one gets seemingly constant ratio of surface flux to volume, therefore yielding a non-local divergence of the neutron flux vector.
This definition of non-local divergence is what contradicts the classical divergence, where the control volume is made to shrink to zero.
4.5.3 About Surface Flux Curvature
Refer Fig. 4.2a. The curvature is concave in nature as the observation space (con- trol volume) is made bigger. Contention could have been that why the curvature is taken as concave instead of convex. Here some practical reasoning will elucidate
70 4 Concept of Fractional Divergence and Fractional Curl the nature of the curve shown in Fig. 4.2. For a very small observation space area, the surface flux is the product of neutron current and that area. As the area is made larger, the neighbouring neutrons effect the neutron current in the wider area of measurement. This gives the larger value of the neutron current for the newer area considered. This increment in the neutron current is what gets integrated in the sur- face integral giving the concave shape (Fig. 4.2a).
This is elaborated in Fig. 4.3. Let the observation surface area for measurement of neutron surface flux be divided into squares as shown. Assume that each center of the square is having one neutron. If all the neutrons are at rest without any velocity fluctutaions, then there will not be any finite probabilty that it may jump across to the next adjacent box. However, the case is not so, as there always is a finite probabilty of having neutrons designated for a particular box finding into the adjacent box.
However, if the area of observation is very small as depicted by smaller circles inside each box, the fluctutaion effect of neutron velocity will not be observed. Therefore with the smaller circles in the observation space measures a smaller neutron current (solely due to the presence of its own neutron in the squrae box). However, the obser- vation area is made into larger circles as shown in Fig. 4.3. Here we see that with enlarged area the effect of neutrons in the adjacent square will enhance the neutron current compared to the first smaller area. Also this bigger circle will catch the effect of velocity fluctuations and therefore will show larger magnitude of neutron current.
This simplistic explanation is justified for the observation that the shape of the sur- face flux with respect to the observation space (Fig. 4.2) is concave and not convex.
4.5.4 Statistical and Geometrical Explanation for Non-local Divergence
Figure 4.3 divides the space into grids. The fluctuations in velocity cause the vio- lation of classical limit of volume shrink to zero, for classical divergence also
Fig. 4.3 The effect of growing observation space modifying the neutron current
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.