A technical study on the Spraytec performances: influence of multiple light scattering and multi-modal drop-size distribution measurements

Abstract

The present paper reports an experimental investigation on the performance of the Spraytec, the recent particle sizer commercialized by Malvern Instruments, Malvern, UK. As with the previous Malvern particle sizers, the Spraytec provides a volume-based drop-size distribution from the analysis of a diffraction pattern resulting from the interaction between a spray and a laser beam. This work focuses on the behavior of the Spraytec in two different situations. First, the influence of multiple scattering caused by the measurement of large and spatially inhomogeneous sprays is investigated. Second, a test of multi-modal drop-size distribution measurement is conducted. Contrary to the previous Malvern instrument, Spraytec's mathematical inversion procedure includes a patented multiple-scattering algorithm that allows measurements of very high concentrated sprays. This algorithm is tested here for large and inhomogeneous sprays for which multiple light scattering affects the measurement. The results show that the Spraytec algorithm cannot satisfactorily correct the measurements in this specific situation. It was also observed that the use of this algorithm is prohibited in the absence of multiple light scattering. In the second part of the study, a test of multi-modal drop-size distribution measurement is conducted. For the investigated situations it was found that the Malvern Spraytec is reliable in measuring such spray drop-size distributions.

Appendices

Appendix 1: Correction of Dodge (1984)

On the basis of a high number of measurements, Dodge (1984) developed a correction factor C _F for the Sauter mean diameter D ₃₂. This correction factor is defined by:

$$ {C_{{\rm{F}}} = {{D_{{32}} } \over {{D}'_{{32}} }}} $$

(2)

where D ₃₂ is the measured Sauter mean diameter and D′₃₂ represents the actual value of this diameter. The coefficient C _F is given by:

$$ {C_{{\rm{F}}} = 1 - a\exp {\left( { - bT} \right)}\exp ( - c{D}'_{{32}} )} $$

(3)

where T is the transmission (T=1−Obs) and the constants a, b and c are equal to:

$$ {\left\{ \matrix{ a = 0.9456 \hfill \cr b = 3.811 \hfill \cr c = 0.0204 \hfill \cr} \right.} $$

(4)

The drop-size distributions were calculated with the Rosin–Rammler model, and by varying the injection pressure, the Sauter mean diameter ranges from 19.2 µm to 53 µm.

Appendix 2: Correction of Gülder (1987, 1990)

The empirical correction factor C _F for the Sauter mean diameter deduced by Gülder (1987, 1990) is given by:

$$ {C_{{\rm{F}}} = {{D_{{32}} } \over {{D}'_{{32}} }} = 1.35\exp {\left( {F_{1} + F_{2} } \right)}} $$

(5)

where D ₃₂ is the measured Sauter mean diameter, D′₃₂ is the actual value of this diameter and F ₁ and F ₂ are given by:

$$ {\left\{ \matrix{ F_{1} = - 0.1184{\left( {{{D_{{32}} } \over {100}}} \right)}^{2} + 13.122{{{\rm{Obs}}} \over {D_{{32}} }} - 5.7474{{{\rm{Obs}}} \over {{\sqrt {D_{{32}} } }}} \hfill \cr F_{2} = 2.2389{\rm{Obs}}^{8} - 2.6077{\rm{Obs}}^{9} \hfill \cr} \right.} $$

(6)

This corrective equation is valid in the following range:

$$ {\left\{ \matrix{ 10{\rm{ - \mu m}} < D_{{32}} < 100{\rm{ - \mu m}} \hfill \cr 0.5 < {\rm{Obs}} < 0.98 \hfill \cr} \right.} $$

(7)

Appendix 3: Correction of Boyaval and Dumouchel (2001)

In this experimental investigation, correcting factors were empirically derived for the mean diameter D ₄₃ and the relative span factor Δ_v of the distribution. These characteristics are defined by:

$$ {\left\{ \matrix{ D_{{43}} = {\int\limits_0^\infty {f_{{\rm{v}}} {\left( D \right)}D{\rm{d}}D} } \hfill \cr \Delta _{{\rm{v}}} = {{D_{{{\rm{v}}0.9}} - D_{{{\rm{v}}0.1}} } \over {D_{{{\rm{v}}0.5}} }} \hfill \cr} \right.} $$

(8)

where f _v(D) represents the volume-based drop-size distribution, and D _vx is the drop diameter under which (100x)% of the total spray volume is contained. The correcting factors α ₄₃ and α _Δ are defined by:

$$ {\left\{ \matrix{ \alpha _{{43}} = {{D_{{43}} } \over {{D}'_{{43}} }} \hfill \cr \alpha _{\Delta } = {{\Delta _{v} } \over {{\Delta }'_{v} }} \hfill \cr} \right.} $$

(9)

where the symbols without a prime stand for the measured characteristics and the symbols with a prime represent the actual values. The two correcting factors are given by:

$$ {\left\{ \matrix{ \alpha _{{43}} = 1.0 - 4.5 \cdot 10^{{ - 4}} \exp {\left( {6.36\,{\rm{Obs}}} \right)} - 1.96 \cdot 10^{{ - 14}} \exp {\left( {29.85\,{\rm{Obs}}} \right)} \hfill \cr \alpha _{\Delta } = {1 \over {1.0 - 3.67 \cdot 10^{{ - 56}} \exp {\left( {126.53\,{\rm{Obs}}} \right)} - 2 \cdot 10^{{ - 6}} \exp {\left( {11.88\,{\rm{Obs}}} \right)}}} \hfill \cr} \right.} $$

(10)

In this work the mean diameter D ₄₃ ranged from 20 µm to 140 µm and the relative span factor Δ_v from 1 to 2.5.