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In this paper, data analysis and modeling of gamma ray tomography taken into account spatial resolution and source of errors and the attenuation coefficient measurement in row data from tomography process are presented. The results showed that this method is simple, effective and should be prior to any data treatment for opaque vessel reactor and by reconstruction algorithm in process imaging.

Comparing original and reconstruct image provides a fair evaluation of quality in whole process. This works quite well for medical X-Ray CT with hard and software in a full standardized process. Industrial Gamma Ray CT still has a long way for establishing a standardized methodology [

Experiments were carried out with a computerized scanner set up by translation-rotation motion for the gamma ray trajectories sampling positions. For transmission measurements with ^{137}Cs radioactive source (7.4 × 10^{8} Bq), a stainless steel tubes of 0.154 m internal diameter and NaI(Tl) scintillation detector of (51 × 51) × 10^{−3 m crys}- tal size coupled to a multichannel analyzer and Geniesoftware from Camberra. Source and detector collimators of cylindrical aperture of 5 × 5 × 10^{−3 m and }10 × 10^{−3 m were used. T}he irradiation geometry source-tubedetector in a fixed alignment, keeps a good quality gamma spectrum by means of adequate beam collimation. The gamma ray transmission measurements were carried out by the 0.662 MeV photopeak evaluations. Compton scattering contribution was minimized by collimator length of 60 × 10^{−3 m for source and }75 × 10^{−3 m for de}- tector [

A general view of the gamma ray scanner is given in

The scan interval, with r taken according to internal radius R, and gamma trajectories sampling positions

defined according to gamma beam diameter Δs. In the experiments, a pure aluminum half-moon, well defined shape with 12.0 cm length and 6.0 cm radius was used as test object.

By scanning an object the number of the gamma rays trajectories and the beam diameter choice: involve temporal, spatial and density resolutions as they are closely correlated parameters. For a third generation tomography process spatial resolution is strongly linked to the collimation of detectors, the number of detectors per projection and the number of projections. A question of measurement resolutions to time (speed of response), matter (e.g. density which defines the contrast in each pixel) and space (spatial resolution which defines how detailed the image is), as given in reference [

The main sources of errors in tomography process are: 1) Contribution of measurement system;

2) Geometric magnification factor;

3) Tomography reconstruction algorithms.

They are well accepted in literature, and for the FCCFluidized Catalytic Cracking process the irradiation geometry of the riser a stainless cylinder, tube wall effect is include as source of errors. Contribution of measurement system is given in [

The mathematical fundaments in gamma ray tomography process are described by means of Beer-Lambert based equations, as

In Equation (1) the gamma intensities I, I_{0} with and without absorber are related to the linear attenuation coefficient μ, and x is the gamma ray path length. The Equations (2) gives the integration of the attenuation coefficient μ(t) at position t, and logarithm and discrete form of integral are the expressed in Equation (3). This Equation (3) is a usual form to generate matrix data for computational algorithm reconstruction. In Equation (4), the gamma intensities were adapted to riser irradiation geometry as I_{V}, I_{F} for empty tube and at flow conditions, related to the mass attenuation coefficient α. The riser internal diameter is D and on the left side ρ_{m} is mean density along gamma ray path.

The gamma beam width and sampling procedure for the gamma ray trajectories are following requirements to data analysis in scanning process [

Taken tomography row data several information are available prior to calculation of parameters given in Equations (1) to (4). In Figures 2 and 3 it can be observed the whole data points from the scanner process. They are two different stain tubes; the one from

Attenuation coefficient is a prime parameter in all calculations given by Equations (1) to (4). Experimental and reconstructed data needs a metric to compare the approximation and RMSE-Root Mean Square Error is widely used for that evaluation. Appling for the row data, calculated with Equation (3) and comparing with attenuation coefficient from literature value [

where the two matrixes are, in this work, attenuation coefficient reference A and measured values X. For which difference the Frobenius norm is equal to the expression on the right side of Equation (5), according to known linear algebra theorem. Although the metric was applied correctly but RMSE value, estimation is depending of the ROI-region of interest. In this experiment a good approximation for mass attenuation was found by excluding extremes data. RMSE is recommended by [

A further evaluation of attenuation coefficient measurement was carried out, taken the whole matrix acquisition data with i = 12 and t trajectories with j = 61. Experimental matrix was of M(12, 97) size, but the attenuation coefficient calculation takes just the object region M(12, 61). In this region, by collecting a vector of values from one angle φ from the data acquisition matrix M(1, 61) the μ evaluation was carried out. At first calculating μ given in Equation (3), for known object length and then the mean value of the vector data

was obtained. Taken as reference mass attenuation coefficient α = = 0.07802 cm^{2}/g, and = (0.2098 ± 0.0004) cm^{−2}, with density ρ = 2.7 g/cm^{3}. The standard deviation of μ considers a statistical dispersion of 0.4% among the values given in literature [^{−1} was measured corresponding to an error of 8% error. The following step, takes the whole object region matrix M(12, 61), and again μ was calculated with the mean value of this matrix. This measure mean value μ = 0.2001 cm^{−1} is associated to a 5% error.

In addition, it might be considered that α literature values of aluminum are given for a long energy interval [^{2}/g value for mass attenuation coefficient of the 0.662 Mev photon, was found. Now, recalculating with α = 0.0720 cm^{2}/g value, the errors in measurig linear attenuation coefficient were of 0.8% in one vector and a 3% error for the whole matrix data. Attenuation coefficient evaluation in row data, as it is given in

Attenuation interval should be explicit as it gives a measure of object suface by means of sufficient scanned

data. By re-arranging Equation (4), in order to place length dimension on the left side as attenuation interval on the right side will remain object density ρ and α is mass attenuation coefficient as given in Equation (4).

Experimental aluminum half-moon data were used for evaluation of tomography process and to show spatial distribution. The model developed for simulate attenuation as a function of internal and external tube radii [

where B is a vector of modeled data, are rotation errors and the answer is the A vector of modeled data at the measured angle φ. The result, experimental data of the test object at spatial distribution of test object, can be visualized in Figures 7 and 8.

Whole scanning interval can be seen, in _{0} gamma intensity appears before and after steel tube. Two data sets are superposed empty tube and tube with object, I_{V}, I_{F} intensities for calculations with Equation (4). Superposition makes precision of the scanner operation easy to visualize.

In

In ^{−1}. Attenuation coefficient was determined also with measurement data in whole matrix as given in ^{−1}.

Object test, aluminum half moon mesured and calculated data, at zero angle φ, is seen in

Simulation of half-moon under scan rotation, fits object shape moving around origin of coordinete system, as

To visualize experimental data tourning around origin of coodinate system scanned object data was modeled and then under matrix rotation as it is shown in Figures 8 and 9.

The same data procedure is given also in

show object spatial distribution.

Spatial distribution in cross-section area is visualized inside internal tube diameter. Errors, due to modeling and matrix rotation calculation can be observed for departure from object shape in line, are of 11%. Comparing with

As industrial tomography reconstruction aims reduced number of data [

Attenuation coefficient is precisely measured in row data from tomography process. Space distribution is shown in attenuation length to compare with object shape, but surely, any other required parameter can be calculated and modeled. Analysis with row data is shown and it could be used for any matrix size at same conditions. The procedure is simple, effective and should be prior to any data treatment for opaque vessel reactor and by reconstruction algorithm in process imaging.

The authors are grateful to CNPq for the Scholarships and financial support and to Dr. Waldir Martignoni of Petrobrás for technical assistance.