Abstracts

EXACT EQUATIONS FOR SOIL PARTICLE-SIZE ANALYSIS BY GAMMA-RAY ATTENUATION

Elimoel Abraão Elias^1,3; Osny Oliveira Santos Bacchi^1,3; Klaus Reichardt^1,2,3,*

¹Laboratório de Física do Solo-CENA/USP, C.P. 96, CEP: 13400-970 - Piracicaba, SP.

²Depto. de Física e Meteorologia-ESALQ/USP, C.P. 9, CEP: 13418-970 - Piracicaba, SP.

³Bolsista do CNPq.

*e-mail: klaus@cena.usp.br

ABSTRACT: Soil particle-size analysis by gamma-ray attenuation was first suggested in 1992 and improved in 1997 by diminishing the measurement time and adapting it for automation. It is here demonstrated that when the mass attenuation coefficient m_w and the density D_w are replaced by m_s and D_s (where the subscripts w and s stand for water and solution) in the approximate equations used to estimate the concentration of suspended particles, they become exact. The demonstration is based by treating the dispersant and water solution as one single medium, instead of treating them as two media. In this way, six variables are reduced to only three. Physical considerations suggest that the precision in this analysis could be improved if other types of solutions and of photon energies would be used, so that the attenuation by the particles would differ more significantly from the attenuation by the solution.

Key words: granulometry, soil mechanical analysis, soil texture, gamma attenuation

EQUAÇÕES EXATAS PARA ANÁLISE DE TAMANHO DE PARTÍCULAS DE SOLO POR ATENUAÇÃO DA RADIAÇÃO GAMA

RESUMO: A análise de tamanho de partículas de solo por atenuação da radiação gama foi primeiramente sugerida em 1992 e melhorada em 1997 diminuindo o tempo de medida e a adaptando para automação. É demonstrado aqui que quando se substitui o coeficiente de atenuação de massa m_w e a densidade D_w por m_s e D_s. (onde os índices w e s representam água e solução) nas equações aproximadas utilizadas para o cálculo da concentração dos sólidos suspensos, estas tornam-se exatas. A demonstração baseia-se em tratar a água e o dispersante como um único meio, ao invés de tratar a solução como dois meios, ou seja, água e dispersante. Assim, seis variáveis são reduzidas a somente três. Considerações físicas sugerem que a precisão nesta análise poderia aumentar se outros tipos de solução e energias de fótons fossem utilizadas, tais que a atenuação pelas partículas se diferenciasse tanto quanto possível da atenuação pela solução.

Palavras-chave: granulometria, análise mecânica do solo, textura do solo, atenuação gama

INTRODUCTION

Soil-particle size analysis by gamma ray attenuation was first suggested by Vaz et al. (1992) and later improved by Oliveira et al. (1997). Oliveira et al. (1997) describe an experiment in which "...a gamma-ray beam passes through a soil suspension of water, dissolved dispersant and suspended soil particles within an acrylic container (...). At a time t after sedimentation has started, the attenuation of the gamma-ray beam is

I = I* exp( -m_w D_w W -m_h D_h H -m_p D_p P -m_c D_c X_c ) [1]

where I* and I are the count intensities of the gamma-ray beam in the absence of the absorbing medium (incident beam) and after passing through the absorbing medium (emergent beam), respectively. The mass attenuation coefficients, dependent on the chemical composition of the absorbing materials and the energy of the gamma rays, are m_w, m_h, m_p, m_c for water, hydroxide, soil particle and acrylic-container walls, respectively; D_w ,D_h ,D_p andD_c are the respective densities of the same four materials, and X_c is the thickness of the two container walls. The quantities of interspersed water, hydroxide, and soil particles within the gamma-ray beam are expressible as the rigorously equivalent pure-component lengths W, H and P, respectively. For a reference state of only water and hydroxide within the container and gamma-ray beam, then I_r = I and the attenuation equation becomes

I_r = I* exp( -m_w D_w W_r -m_h D_h H_r -m_c D_c X_c ) [2]

where I_r , W_rand H_rrepresent the reference state designated by subscript r." In sequence, the following equation is developed:

P = [ln(I_r /I)]/(m_p D_p - m_w D_w) [3]

Oliveira et al. (1997) admit that this equation is approximate and attempt to demonstrate that the approximation is adequate. This demonstration demanded a full article appendix. An exact equation is developed here, in a much simpler manner, without the need of approximations. Vaz et al. (1992) also use an equation which is basically the same as Equation [3], but no attempt is made to demonstrate that the approximation is adequate.

The equivalent length P of pure soil solid is converted to the interspersed particle concentration C according to the equation

P = CX_i/D_p[4]

where X_i , the beam length inside the container, is also the internal dimension of the container. Oliveira et al. (1997) use Equations [3] and [4] to eliminate P so that they finally obtain the equation for the concentration of suspended particles

C=[ln(I_r / I)]/[X_i(m _p -m_wD_w / D_p)] [5]

DEVELOPMENT OF THE EXACT EQUATIONS

The homogenous solution as one single medium: Oliveira et al. (1997) introduce an unnecessary complication by considering water and dispersant as two media, instead of regarding the solution as one single medium. First of all, the water and dissolved dispersant solution is a homogeneous mixture. In addition, the mixture's composition remains constant, which is correctly taken by Oliveira et al. (1997) when stating that "...the volume fraction composition of the hydroxide-water solution does hold constant". In fact, there is no reason for changes in composition of the solution. In gamma-ray attenuation, a homogenous medium of constant composition can always be regarded as one single medium of a certain attenuation coefficient m. For instance, the water can be regarded as one single medium, of attenuation coefficient m_w and it is not necessary to consider separately the attenuation coefficients of oxygen and hydrogen. A similar reasoning applies to plastic and specially soil, which is composed of several elements. We are thus allowed to consider the dispersant solution as one single medium, rather than considering it two media, namely, water and dispersant. In this way, six variables: m_w , D_w ,W, m_h ,D_h and H, used by Oliveira et al. (1997) are reduced to only three: m_s , D_s and S, where the subscript s stands for solution, and S=H+W is the length of interspersed solution within the gamma-ray beam.

A "picture-based" approach: Figure 1(a) represents a state at time t after sedimentation has started. and Figure 1(b) represents the reference state. A gamma-ray beam at point A and A' will have the same intensity I_A, whereas at points B and B' intensities will be:

I_B = I_A exp(-m_p D_pP) [1a]

I_B' = I_A exp(-m_s D_sP) [2a]

Equation [1a] and [2a] are similar to Equation [1] and [2], since I_B = I and I_B' = I_r .

Taking the ratio of Equations. [1a] and [1b], isolating P and noting that I_B = I and I_B' = I_r , yields

P = [ln(I_r /I)]/(m_p D_p - m_s D_s) [3a]

which is the exact form of Equation [3]. From Equation [3a] and [4], the exact equation for particle concentration becomes

C=[ln(I_r/I)]/[X_i(m _p -m_sD_s/D_p)] [5a]

which should replace Equation [5].

Figure 2 presents also the two states with the unnecessary complication of Oliveira et al. (1997). Here, a gamma-ray beam at point A and A' will not have the same intensity I_A and the simple deduction shown above cannot be performed.

An algebraic approach: The exact equation [3a] can also be obtained algebraically, without making resort to Figure 1. Regarding the solution as one single medium, the six variables are reduced to only three and Equations [1] and [2] can be replaced by [1b] and [2b] respectively:

I = I* exp(-m_s D_s S -m_p D_p P -m_c D_c X_c ) [1b]

I_r = I* exp(-m_s D_s S_r -m_c D_c X_c ) [2b]

where S_r is the length of interspersed solution within the gamma-ray beam at the reference state. S_r is exactly the internal dimension of the container X_i, so that for the reference state, we have

X_i = S_r [6]

and for the sedimenting state, X_i = S + P [7]

so that S_r = S + P [8]

And I_r = I* exp[-m_s D_s (S + P) - m_c D_c X_c ]

Hence I_r = I* exp(-m_s D_s S - m_c D_c X_c ) exp(-m_s D_s P) [9]

and Equation [1b] can be re-written as

I = I* exp(-m_s D_s S -m_c D_c X_c ) exp(-m_p D_p P) [10]

Taking the ratio of Equation [9] and Equation [10], and isolating P:

P = [ln(I_r /I)]/(m_p D_p - m_s D_s)

which is precisely Equation [3a].

It is worthwhile to observe that the product

I* exp(-m_s D_s S - m_c D_c X_c )

that appears in Equations [9] and [10] is exactly the same as I_A that appears in Equations [1a] and [2a].

DISCUSSION

It is relevant to find out the magnitude of the corrections in m_w and D_w when they are replaced by m_s and D_s. The corrections in m_w can be calculated as follows:

m_s = M_w m_w + M_h m_h = m_w(M_w + M_h m_h /m_w) = m_w(M_w + M_h - M_h + M_h m_h /m_w)

m_s = m_w[1 - M_h (1 - m_h /m_w)]= m_w[1 + M_h (m_h /m_w - 1)]

where M_w and M_h are the mass percentages of water and hydroxide, respectively, in the solution, so that M_w + M_h = 1. Noting that m_h /m_w = 1.049 and M_h @ 0.002, we have

m_s = m_w[1 + M_h (m_h /m_w - 1)]

m_s = m_w[1+ 0.002 ( 1.049 - 1)]

m_s= m_w ( 1 + 10^-4 ) @ m_w

Thus the correction in m_wis ~ 0.01% and would only be necessary at much higher dispersant concentrations and when the photon energy is such that m_h /m_w differs considerably from 1. With respect to the correction in D_w , for a concentration of 0.05N, the solution density D_s differs from D_w by ~ 0.2% .

Equation [3a] has an interesting physical interpretation. The difference (m_p D_p - m_s D_s) is equal to (m_pl - m_sl) where m_pl and m_sl are linear attenuation coefficients. This difference is a measure of how more effective the soil particles are as attenuators in relation to the solution, when a certain thickness P of solution is replaced by soil particles. If the particle attenuation would be exactly as effective as the water attenuation, i.e., m_pl = m_sl, then the difference m_pl - m_sl would obviously be zero, and we would have I_r = I and the analysis presented by Oliveira et al. (1997) would have no meaning. Hence an important condition to increase the precision in this analysis is that m_pl differs as much as possible from m_sl. This condition has not been previously explored in other studies, considering other kinds of solutions and other photon energies.

CONCLUSIONS

The correction introduced hereby in Equations [3] and [5] is quantitatively small, but very important from the theoretical point of view. One of the advantages of using the approach here presented is that it applies to any concentration of hydroxide and to any gamma-ray photon energy. Another advantage is that it allows a clearer and simpler understanding and "visualisation" of the experiment. Precision may increase using other kinds of solutions and photon energies.

ACKNOWLEDGEMENT

The authors are thankful to Dr. Fabio A. M. Cassaro and Dr. Raymond S. Pacovsky for commentaries on the manuscript.

Recebido para publicação em 10.05.98

Aceito para publicação em 16.11.98

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