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Multi-group diffusion-age theory of neutron transport in homogeneous media of arbitrary composition Volchenko Ju.A. Contact

1. Introduction.

In earlier papers (1) it was shown that the problem of development of new neutron moisture meters for homogeneous inorganic materials reduces to definition of the optimal value of a source neutron energy by calculations for a sample with set parameters and to the choice of a neutron source with the most appropriate energy spectral distribution. The definition of the optimal source neutron energy has provided the multi-group diffusion-age approximation of the Boltzman kinetic equation solution, given in (1).
With the use of this approximation one can determine the optimal neutron energy, varying the energy of a mono-energy source. Besides it can also be used for direct comparison of poly-energetic neutron sources with different energy spectral neutron distribution. In this paper we try to extend the multi-group diffusion-age approximation for solution of the practical problems of neutron moisture monitoring in homogeneous materials under various geometrical testing conditions, namely: testing of given materials on tape transporters, in pipelines, batching hoppers, railway vans, etc.

2. Principal statements and results of the theory.

Set us assume that a homogeneous medium of arbitrary composition occupies some volume G with the boundary surface S. The mono-energetic power source Q, emitting neutron with the age , equal zero, is positioned in some point M_o of surface S. We again assume that a detector, used to record neutrons, is placed on the surface S and occupies some area S₁ with the center in a point M₁. In the multi-group approximation a neutron detector has the efficiency of neutron recording e , defined by the formula:

(1)

e _j - efficiency of j-group neutron recording;
j _in. - energy group of neutrons, emitted by a source;
j _k - energy group of thermal neutrons.
Then a reading of the detector, used for recording of all neutrons with energies from a source neutron energy to a thermal one can be described by

(2)

Q - neutron source power;
DY_j - number of j-group neutrons, fallen upon the detector area S₁ per time unit;
The function DY_j is defined by the formula:


(3)

q (M,) - the density of -age neutron moderation from a neutron source, positioned in the point M_o;

Integration is performed over the detector area S₁.
The moderation density q (M,) can be defined with solution of the age equation:


(4)

a ²( ) º S _a( ) /D( ) - the coefficient, considering absorption of -age neutron in a homogeneous medium.
S _a( ) - absorption cross-section of -age neutrons in a homogeneous medium.
D( ) - diffusion coefficient of the - age neutrons.
The initial condition for the age equation can be written in the form:


(5)

d (M - M₀) - Dirac delta function.
As a boundary condition we choose the condition of a neutron flux absence beyond region G through the surface S:


(6)

n - a normal to the surface S.
Taking into account that the neutron flux density F (M, ) per unit of -age can be written as [2]:


(7)

And solving the age equation (5) by a classical Fourier technique, we obtain the following results for different testing geometrys.

1. Rectangular sample.
   Placing the origin in a sample center and considering that | x| < a; | y| < b; | z| < c, where a, b, c - sample dimensions, we obtain:
   

   (8)

   where X_k(x), Y_l(y), Z_m(z) - eigenfunctions of the boundary value problem.
   
2. Conveyer belt
   

   (9)

   
   
3. Pipeline: 0 £ r£ r₀ ; - p < j £ p ;|Z| < ¥

   (10)

   
   boundary conditions of the third kind.

4. Half-space (batching hoppers, railway vans, etc.): 0 < x < ¥; |y|< ¥; |z|< ¥.

   (11)

   
   These formulae correspond to the case of the mono-energetic neutron source with power Q.
   A polyenergetic neutron source can be represented as superposition of mono-energetic sources:

   (12)

   and calculations are performed with formulae (8-11).
   

3. Some results of the theory application.

Using this theory one can determine an optimal energy of source neutrons for specific geometrical testing conditions and chemical composition of a homogeneous medium, varying the energy of a mono-energetic neutron source end using the criterion of the relative-absolute sensitivity, used for estimation of neutron moisture meter operation [3]:

(13)

P ₀ - neutron detector reading at minimum environmental moisture W_o.
Besides, one can directly compare neutron sources with different energy spectral characteristics under similar testing conditions and choose the best fitted neutron source.
Fig. 1a shows the choice of a neutron source for two cores of oil-bearing rocks with dimensions 100 and 150 mm both in height and diameter.
Fig. 1 b presents experimental testing of a proper source choice.


Fig 1: The choice of a neutron source for a sandy core according to the criterion h (a) and experimental testing of a proper source choice (b): - Æ 100mm; - - Æ 150mm.

It is evident, that Cf ²⁵² (E_av. = 2,5 MeV) source provides twice less errors compared with a Pu²³⁹- Be (E_av. = 5,5 MeV) source for the core 100 mm in diameter. But in the case of the core 150 mm in diameter the situation is quite opposite.
Fig. 2 shows the choice of a neutron source for a belt transporter 40cm wide, used for transportation of the concrete mix 10 cm thick.


Fig 2: The choice of a neutron source according to a neutron detector reading at a similar source power with different energy spectral neutron distribution: a-calculation; b-experiment.

It is evident that in this case the Cf ²⁵² source provides better results.
Fig. 3 illustrates the choice of the neutron recording method for determination of a mass fraction of the drilling mud liquid phase, pumped along a pipeline with inner diameter 120 mm.

Fig 3: The choice of the method for determination of a mass fraction of the drilling mud liquid phase in the pipeline with inner diameter 120 mm. The neutron source Pu-Be. The measurement geometry - "through" (a-calculation, b-experiment): 1-slow neutrons; 2-fast neutrons.

It is evident that the method of fast neutrons attenuation provides better results, compared with the method of moderated neutrons recording.
Some quantitative disagreement between readings of neutron detectors in calculations and experiments for a belt transporter and a pipeline can be explained by the fact that in calculations some volumetric characteristics of detectors (thickness, diameter) were not taken into account as well as thickness of a transporter belt or pipeline walls because they do not produce any significant effect upon the choice of both a neutron source type and a method of neutron recording, but complicate calculations.
This theory was used in development of neutron moisture meters for samples of free-flowing and plastic materials NI10VH, NI10VA, NI20VB, the core analyzer for oil-bearing rocks ANKR-2M and the sensor of the drilling mud liquid phase mass fraction for the complex of technical means for express analysis of the drilling mud parameters during drilling of oil-gas producing wells, named KIBR and show in Figs. 4-6.

Fig 4: General view of neutron moisture meters for samples of free-flowing and plastic inorganic materials NI10VH, NI10VA, NI20VB. 1-a measuring unit; 2-control panel; 3-container with a sample.

Fig 5: General view of core analyzer (moisture meter-densitometer-concentration meter) for oil-bearing rocks -ANKR-2M: 1-a measuring unit; 2-control panel; 3-container with a core; 4-a core.

Fig 6: General view of the complex of technical facilities for express testing of the drilling mud parameters KIBR: 1-sensor of a liquid phase mass fraction; 2-sensor of density; 3-thermometer; 4-sensor of salinity; 5-sensor of resistivity; 6-sensor of viscosity; 7-flowmeter; 8-principal control panel; 9-driller panel.

4. Conclusion.

The multi-group diffusion-age theory of neutron transport in homogeneous media of arbitrary chemical composition and geometrical dimensions, particular for practical neutron moisture measurement (samples, transporters, pipelines, batching hoppers, etc.) has been developed and can be used to choose the most appropriate neutron source for a neutron detector with the set recording efficiency.

References.

1. Ju. A. Volchenko. Proximate absorption Neutron Moisture Meters for samples of Inorganic Materials. 7^th ECNPT Conference and Exhibition, Copenhagen 26-29 May 1998.
2. B. Devison. The theory of neutron transport/ Translated from English by V.N. Morozov, O.A. Salnikov. Edited by prof, G.I. Marchuk - M: Atomizdat, 1960.
3. V.A. Artsybashev. Nuclear-geophysical prospecting / - M., Atomizdat, 1972.

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