Transport and Diffusion Equation, Physics tutorial

Introduction:

It is necessary to be acquainted with spatial and energy Distributions of neutrons in field of nuclear fission reactor, D-T or D-D fusion reactor, or another nuclear reactor engaged with greater numbers of neutrons. It is understandable why the spatial distribution should be recognized, and due to neutron reactions fluctuate extensively with energy, the energy allotment is also a critical parameter. The neutron energy allocation is often termed as neutron spectrum. The neutron sharing satisfies transport equation. It is generally complicated to resolve this equation, and often approximated equation so considered as diffusion equation is solved as an alternative. In this part a summary of transport equation and diffusion equation of neutrons are offered.        

Fick's Law:

The original Fick's Law states that the rate of change of solute concentration is proportional to the negative gradient of the solute concentration that was originally used to account for chemical diffusion.

Neutrons behave in much the same way as a solute in a solution. This means that if the density flux of neutron is higher in one part of a reactor than at another, there is a net flow of neutron into the region of lower density of neutron. Such as assume the flux it is usual in nuclear manufacturing to construct calculations with flux, which is proportional to density of neutron, rather than with the density itself varies along the x-direction as described in the figure given below.

Then Fick's law is written as:                                                        

Jx = - D dφ/(dx)

In this equation Jx is equal to net number of neutrons that pass per unit time by a unit area perpendicular to x-direction; it has same units as flux, namely, neutrons/cm2-sec. The parameter D in the above equation is termed as diffusion coefficient.

2161_Neutron Flux and Current.jpg

The above equation illustrated that, in the above figure, here is a negative flux gradient, and then there is a net flow of neutrons along the positive x-direction as denoted in figure. To appreciate the origin of this flow consider neutrons moving by the plane at x = 0. These neutrons move by plane from left to right as an outcome of collisions to left of the plane; and on the contrary, they flow from right to left as outcome of collisions to right of the plane.  While flux is larger for negative values of x, there are more collisions per cm3/sec on the left than on the right. More neutrons are thus spreaded from left to right than the other manner round, with the consequence that there is a net flow of neutrons in the positive x-direction via the plane, just as prescribed by Equation (Jx = - D dφ/(dx)). It is significant to consider the neutrons don't flow from regions of high flux to low flux because they are in any sense 'pushed' the way. There are simply more neutrons spreaded in one direction than in the other.

The flux is normally a function of three spatial variables, and in this case Fick's Law is: 

 J = - D grad φ = D ∇ φ

Here, J is known as the neutron current density vector and grad = ∇ is the gradient operator. The physical significance of the vector J with a unit vector in the x-direction ax this gives the x-component of J, namely Jx:

J.ax = Jx                                      

This, as already noted, is equal to the net flow of neutrons per second per unit area normal to the x-direction.  It follows, therefore, that if 'n' is a unit vector pointing in an arbitrary direction then:

J.n = Jn   

This Ii equal to the net flow of neutrons per second per unit area normal to the direction of n.

Returning to the equations (Jx = - D dφ/(dx)) and (J = - D grad φ = D ∇ φ ), it may be noted that, since Jx and J have the same unit as, Dhas units of length. It can be shown by arguments which are too lengthy to be reproduced here that D is given approximately by the following formula:

D = λtr /3,

Here, λtr is called the transport men free path, and is given in turn by:

λtr = 1/ Σtr = 1/ Σs( 1 - µ‾)

In this equation, Σtr is called the macroscopic transport cross section, Σs is macroscopic dispersion cross section of  medium, and   is  average value of cosine of  angle at which  neutrons are spreaded in the medium. The value of   at most of neutron energies of interest in reactor calculations can be computed from simple formula:

μ‾ = 2/3A,                                            

Here, 'A' is atomic mass number of medium. It has to be emphasized that Fick's Law is not an exact relation. In particular, it is not valid:

a) In a medium that strongly absorbs neutrons.

b) Within regarding three mean free paths of neutron source or surface of a medium.

c) When spreading of neutrons are strongly anisotropic.

To some extent these restrictions are available in every practical reactor trouble. Nevertheless, as mentioned previously, Fick's law and diffusion supposition are used to approximate reactor properties.

Equation of Continuity:

The equation of continuity is mathematical statement of apparent fact as neutron don't disappear curiously, rate of time change in the number of neutron in a volume, V in a medium must be accounted for. In particular, it follows that: 

Rate of change of neutron, V = (Rate of production of neutron in the volume) - (Rate of absorption in the volume) - (Rate of leakage in the volume)                                                                        

 If η the density of neutrons at any point and time in volume 'v'. The total number of neutrons within a volume, dv is then given as ηdv, but total number of neutron inside volume 'v' is given as ∫ηdv.

Therefore rate of change in number of neutron is given as:

v(dη/dt)dv

Also, let 'S' be the rate at which neutrons are emitted from a source per cubic metre for volume, 'v'. The rate at which neutrons are created by 'v' is given as:

Production rate = ∫vSdv

The rate at which neutrons are gone by absorption per cm 3/sec is given as:

dv = Σa φ

Here, ∑a is absorption cross-section by the volume. 

Therefore, the total number of neutron lost due to absorption is represented as:

Absorption = ∫v Σa φdv

Consider flow of neutron in and out of volume 'v'. If 'J' is neutron current density vector on surface of v and n is a unit vector pointing external from the surface. Then dot product of J and n (J. n) is net number of neutrons passing outward through the surface percm3 per second. It follows the total rate of leakage of neutrons by the surface A is given as: 

A J.n dA     

Therefore, the equation becomes:

V = ∫v ∂η/∂t dv - ∫v Σa Φ dv - ∫∇. J dv

∂η/∂t ≡ S - Σa Φ - ∇. J

Whenever the neutron density is time dependent, then the expression density now becomes zero.

0 ≡ S - Σa Φ - ∇. J

The above equation is termed as the steady state equation of continuity.

Diffusion Equation:

The neutron diffusion equation is attained by substituting expression J = -D∇Φ (where D is diffusion coefficient) in equation. If D is not function of space variable i.e. x, y, z. This signifies D ≠ D(x, y, z). Then, we will have:

D∇2Φ - ΣaΦ + S = ∂η/∂t and Φ = nvi where v is velocity of the neutron, when η is independent of time, expression on left hand side becomes zero

I.e ∂η/∂t = 0 and it is known as steady state diffusion equation. But if dependent on time, then it is known as diffusion equation.

Dividing equation through by D, then we will have:

2Φ - ΣaΦ/D + S/D = 0

Substituting

Σa/D = 1/L2 in equation

2Φ - 1/L2Φ + S/D = 0

Therefore L2 = D/Σa

Quantity L seems regularly in nuclear engineering problems and is known as diffusion length: L2 is known as diffusion area. AS D and Σa have units of cm and cm-1, respectively, it follows from Equation which has units of cm2 and L has units of cm.

Boundary Conditions:

To obtain neutron flux Φ from diffusion equation, it is essential to state certain boundary condition that should be satisfied by solution i.e.

i) Φ should be real and non-negative function

ii) Φ should be finite except maybe at singular points of source distribution.

In several problems, neutrons diffuse in medium that has outer surface, i.e., surface between the medium and atmosphere. In immediate vicinity of such surface, Fick's Law is not valid that means diffusion equations is not valid there either. Exact (non diffusion theory) calculations show, though, if flux as computed from diffusion equation is supposed to vanished at a small distance d beyond surface, then flux determined from diffusion equation is very nearly equivalent to exact flux in the interior of medium, although not, of course, near surface.

Parameter d is called as extrapolation distance, and for most cases of interest it is provided by simple formula:

d = 0.171λtr

Where λtr is transport mean free path of medium.

2358_Extrapolation Distance at a Surface.jpg

Therefore, from Equation it will be seen that d is generally small compared with most reactor dimensions. It is frequently possible, thus, when solving diffusion equation, to suppose that flux vanishes at actual surface of system.

Boundary conditions at the interface between two different media (like between reactor core and reflector) should also be specified. Conditions are that both flux and component of current normal to surface should be continuous across boundary. Therefore, at the interface between two regions A and B, we should have:

φA = φB

(JA)n = (JB)n

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