^{1,a)}, E. Andersson Sundén

^{1}, S. Jacobsson Svärd

^{1}and H. Sjöstrand

^{1}

### Abstract

Dynamic bias errors occur in transmission measurements, such as X-ray, gamma, or neutron radiography or tomography. This is observed when the properties of the object are not stationary in time and its average properties are assessed. The nonlinear measurement response to changes in transmission within the time scale of the measurement implies a bias, which can be difficult to correct for. A typical example is the tomographic or radiographic mapping of void content in dynamic two-phase flow systems. In this work, the dynamic bias error is described and a method to make a first-order correction is derived. A prerequisite for this method is variance estimates of the system dynamics, which can be obtained using high-speed, time-resolved data acquisition. However, in the absence of such acquisition, *a priori* knowledge might be used to substitute the time resolved data. Using synthetic data, a void fraction measurement case study has been simulated to demonstrate the performance of the suggested method. The transmission length of the radiation in the object under study and the type of fluctuation of the void fraction have been varied. Significant decreases in the dynamic bias error were achieved to the expense of marginal decreases in precision.

This work has been financed by the Swedish Center for Nuclear Technology, SKC.

I. INTRODUCTION

II. THE DYNAMIC BIAS ERROR

III. CORRECTING FOR THE DYNAMIC BIAS ERROR

A. Deriving an expression for the correction

B. Estimating the correction experimentally

IV. CASE STUDY: BASE CASE

A. Case study geometry and setup

B. Simulated data

C. Evaluation procedure

D. Correction of count rates

E. Correction of void fractions

V. STUDY OF THE INFLUENCE OF THE VOID FLUCTUATION CHARACTERISTICS

VI. DISCUSSION

VII. CONCLUSIONS AND OUTLOOK

### Key Topics

- Transmission measurement
- 11.0
- Multiphase flows
- 10.0
- Time measurement
- 7.0
- Neutrons
- 6.0
- Neutron tomography
- 4.0

## Figures

Flow modes of vertical two-phase flow. From left; bubbly flow, slug flow, churn flow, and annular flow.

Flow modes of vertical two-phase flow. From left; bubbly flow, slug flow, churn flow, and annular flow.

A cross section of the test case under study, illustrating the simulated geometry investigated with 13 beam lines marked in red in the figure. As the evaluated void fractions are presented below, these lines of sight are denoted 1–13 from the top downwards.

A cross section of the test case under study, illustrating the simulated geometry investigated with 13 beam lines marked in red in the figure. As the evaluated void fractions are presented below, these lines of sight are denoted 1–13 from the top downwards.

The modeled temporal void distribution is a square pulse, ranging from a void fraction of 0–1, i.e., with an average of 0.5 and a variance of 0.25. The period is 50 ms.

The modeled temporal void distribution is a square pulse, ranging from a void fraction of 0–1, i.e., with an average of 0.5 and a variance of 0.25. The period is 50 ms.

Schematic overview of the case study. Note that in a real measurement situation only steps (3) and (4) would be performed, the other steps are included here only for validation of the correction method.

Schematic overview of the case study. Note that in a real measurement situation only steps (3) and (4) would be performed, the other steps are included here only for validation of the correction method.

An example of the simulated distribution of N_{ τ } compared with the Poisson distribution (*λ* = 3.3). Bars represent 1*σ* uncertainty. It should be noted that although the distributions look very much alike, the Poisson distribution is slightly narrower, with a higher peak and lower tails. The sample variance of the simulated data is 2.6% larger than the variance of the Poisson distribution. This indicates an additional source of variance in the simulated data, i.e., the dynamic void fraction.

An example of the simulated distribution of N_{ τ } compared with the Poisson distribution (*λ* = 3.3). Bars represent 1*σ* uncertainty. It should be noted that although the distributions look very much alike, the Poisson distribution is slightly narrower, with a higher peak and lower tails. The sample variance of the simulated data is 2.6% larger than the variance of the Poisson distribution. This indicates an additional source of variance in the simulated data, i.e., the dynamic void fraction.

The distribution of the estimated variance of the void fraction in 1000 simulated measurements on line of sight # 7. The true (simulated) variance is 0.25.

The distribution of the estimated variance of the void fraction in 1000 simulated measurements on line of sight # 7. The true (simulated) variance is 0.25.

The results of the void fraction estimates in line of sight #7. The true value of the void fraction is 0.5, i.e., there still remains a bias error after the correction procedure. However, the suggested correction method leads to a significant improvement in the void estimate.

The results of the void fraction estimates in line of sight #7. The true value of the void fraction is 0.5, i.e., there still remains a bias error after the correction procedure. However, the suggested correction method leads to a significant improvement in the void estimate.

The estimated void fraction in the cylindrical two-phase flow system illustrated in Fig. 2 before and after applying the suggested dynamic bias correction on simulated data. The uncorrected void fraction estimates are presented with the solid curve, the corrected estimates using the suggested method are presented with the dashed curve and the true simulated void fraction is presented with the dotted curve. The bars in the figure represent the 1*σ* random errors of the void fraction estimates, based on the standard deviations of the void fraction estimates in the 1000 independent simulations.

The estimated void fraction in the cylindrical two-phase flow system illustrated in Fig. 2 before and after applying the suggested dynamic bias correction on simulated data. The uncorrected void fraction estimates are presented with the solid curve, the corrected estimates using the suggested method are presented with the dashed curve and the true simulated void fraction is presented with the dotted curve. The bars in the figure represent the 1*σ* random errors of the void fraction estimates, based on the standard deviations of the void fraction estimates in the 1000 independent simulations.

The bias errors of all the modeled temporal void distributions before and after correction. Note the different scales on the axes.

The bias errors of all the modeled temporal void distributions before and after correction. Note the different scales on the axes.

The bias errors with and without performing the proposed correction as a function of the variance of the void fraction. Bars represent 1*σ* uncertainty.

The bias errors with and without performing the proposed correction as a function of the variance of the void fraction. Bars represent 1*σ* uncertainty.

## Tables

Results of 1000 simulated void fraction measurements in 13 lines of sight through the simulated object. Errors are absolute errors expressed in void percent units.

Results of 1000 simulated void fraction measurements in 13 lines of sight through the simulated object. Errors are absolute errors expressed in void percent units.

The modeled temporal void functions and the result of the simulations of the bias error with and without the suggested correction procedure.

The modeled temporal void functions and the result of the simulations of the bias error with and without the suggested correction procedure.

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