Application of the maximum entropy technique in tomographic reconstruction from laser diffraction data to determine local spray drop size distribution

Abstract

This work proposes a new deconvolution technique to obtain local drop size distributions from line-of-sight intensity data measured by laser diffraction technique. The tomographic reconstruction, based on the maximum entropy (ME) technique, is applied to forward scattered light signal from a laser beam scanning horizontally through the spray on each plane from the center to the edge of spray, resulting in the reconstructed scattered light intensities at particular points in the spray. These reconstructed intensities are in turn converted to local drop size distributions. Unlike the classical method of the onion peeling technique or other mathematical transformation techniques that yield unrealistic negative scattered light intensity solutions, the maximum entropy constraints ensure positive light intensity. Experimental validations to the reconstructed results are achieved by using phase Doppler particle analyzer (PDPA). The results from the PDPA measurements agree very well with the proposed ME tomographic reconstruction.

Appendix: Bevensee’s iterative algorithm

The algorithm for an iterative solution starts with an initial guess β ⁽⁰⁾ _ij and a reasonable estimate of I ⁽⁰⁾ _tj . β ⁽⁰⁾ _ij are constant which provide the minimum error. In this study, all of β ⁽⁰⁾ _ij are guessed with the same value between −10 and 10. Because I _tj is implicit function of β _ij , I _tj will converge to a unique value and its initial guess is 1 for this study. The following procedures are then performed:

• (1) Calculate I ⁽¹⁾ _j (r _k ) from β ⁽⁰⁾ _ij by Eq. (11).

• (2) Compute ${\bar{I}_{j}^{{(1)}} (y_{i})}$ from Eq. (2).

• (3) I ⁽⁰⁾ _ij is scaled to I ⁽¹⁾ _ij by multiplying it by

  $$ c_{j} = \frac{{{\left({{\sum {\beta^{{(0)}}_{{ij}} \bar{I}_{j} (y_{i})}}} \right)}}}{{{\left({{\sum {\beta^{{(0)}}_{{ij}} \bar{I}^{{(1)}}_{j} (y_{i})}}} \right)}}} $$

  (A1)

• (4) I ⁽¹⁾ _j (r _k ) and ${\bar{I}^{{(1)}}_{j} (y_{i})}$ are also scaled by the factor c _j (this scale do not change the values of entropy S _j ).

  $$ I^{{(1)}}_{j} (r_{k}) \to c_{j} I^{{(1)}}_{j} (r_{k}), \bar{I}^{{(1)}}_{j} (y_{i}) \to c_{j} \bar{I}^{{(1)}}_{j} (y_{i}) $$

  (A2)

• (5) To obtain the computed ${\bar{I}_{j} (y_{i})}$ closer to the measured ${\bar{I}_{j} (y_{i}), \beta_{ij}^{(0)}}$ is updated by Δβ ⁽¹⁾ _ij

  $$ \beta_{ij}^{(1)} = \beta_{ij}^{(0)} + \Delta\beta_{ij}^{(1)} $$

  (A3)

  where Δβ ⁽¹⁾ _ij is computed by substituting Eq. (11) into Eq. (2) and differentiating ${\bar{I}_{j} (y_{i})}$ in Eq. (2), which is the set of I equations

  $$ \Delta \bar{I}^{{(1)}}_{j} (y_{i}) = \bar{I}_{j} (y_{i}) - \bar{I}^{{(1)}}_{j} (y_{i}) = {\sum\limits_{m = 1}^I {\frac{{\partial \bar{I}^{{(1)}}_{j} (y_{i})}}{{\partial \beta_{{mj}}}}\Delta \beta^{{(1)}}_{{mj}}}} $$

  (A4)

  or (in the matrix form)

  $$ {\left[ {\Delta \bar{I}^{{(1)}}_{j} (y)} \right]} = {\left[ {L{\left(\begin{aligned}&\, X^{{(1)}} \\ & - \frac{1}{{{\left({I_{t}} \right)}_{j}}}{\left[ {\bar{I}^{{(1)}}_{j} (y)} \right]}{\left[ {\bar{I}^{{(1)}}_{j} (y)} \right]}^{t} \end{aligned} \right)}L^{t}} \right] }\cdot {\left[ {\Delta \beta_{j}^{{(1)}}} \right]} $$

  (A5)

  where L is the I × I matrix of 2L _ik , X is the diagonal matrix that X _kk is I _j (r _k ), L ^t is the transpose matrix of L, and [ ] is the matrix. Δβ ⁽¹⁾ _ij are computed based on singular-value decomposition (Press et al. 2002). Now all of I ⁽¹⁾ _j (r _k ), ${\bar{I}^{{(1)}}_{j} (y_{i}),}$ I ⁽¹⁾ _ij and β ⁽¹⁾ _ij are known and the same procedure from step (1) to (5) is conducted iteratively with the new β _ij until the convergence of ${\bar{I}_{j} (y_{i})}$ is acceptable.