Concentration measurement in the wake of a free rising bubble using planar laser-induced fluorescence (PLIF) with a calibration taking into account fluorescence extinction variations

Abstract

A planar laser-induced fluorescence (PLIF) data treatment taking into account variations of the fluorescence extinction due to pH is presented. It is shown that the proposed method needs to be implemented when the product of extinction coefficient variations, path length travelled by the laser and fluorescence concentration become large as in large tank experiments. The influence of extinction coefficient variations has been evaluated on CO₂ concentration measurement for a test case in the wake of a free rising bubble.

Appendices

Appendix 1: Reaction time

As written in Gibbons and Edsall (1963) about CO₂ reaction kinetics with water : “although the acid–base equilibrium is essentially instantaneous, the dehydration rate of H₂CO₃ is relatively slow" implying that time to reach equilibrium for H₂CO₃ and pH may not be short enough to track CO₂ concentration variations [cited in Asher and Litchendorf (2009)]. The kinetic of H₂CO₃/CO₂ reaction has been modeled with the coupled differential equations system (7) where k _dehyd is the dehydration rate of H₂CO₃ and k _hyd is the hydration rate of CO₂.

In (Asher and Litchendorf 2009), the system has been solved analytically with Maple using the source term $F_{{{\text{CO}}_{2} }}$/δ (where $F_{{{\text{CO}}_{2} }}$ is the molecular gas flux and δ is a diffusive length scale of CO₂) considered as a constant and applied during a given time referred to here as exposure time.

$$\left\{ {\begin{array}{*{20}c} {\frac{{{\text{d}}[{\text{CO}}_{2} ]}}{{{\text{d}}t}} = \frac{{F_{{{\text{CO}}_{2} }} }}{\delta } + k_{{{\text{dehyd}}}} [{\text{H}}_{2} {\text{CO}}_{3} ] - k_{{{\text{hyd}}}} [{\text{CO}}_{2} ]} \\ {\frac{{{\text{d}}[{\text{H}}_{2} {\text{CO}}_{3} ]}}{{{\text{d}}t}} = k_{{{\text{hyd}}}} [{\text{CO}}_{2} ] - k_{{{\text{dehyd}}}} [{\text{H}}_{2} {\text{CO}}_{3} ]} \\ \end{array} } \right.$$

(7)

In order to find the time for the reaction to reach equilibrium, the system has been solved using the same reaction rates values as in (Asher and Litchendorf 2009). Nevertheless, the initial condition of the CO₂ spot has been changed in order to have a more realistic contact with the bubble : The source term $F_{{{\text{CO}}_{2} }}$ /δ has been modeled as a function of time. It was a shifted cosine of maximum $F_{{{\text{CO}}_{2} }}$/δ with a semi-period equal to the exposure time requisite to reach the maximum concentration of CO₂ in the wake of the bubble determined from Fig. 7.

It is shown in Fig. 8 that for the used continuous source term, [CO₂] _{t=t_e}  − [CO₂] _t=0 is very close to the integral of the source term $\int_0^{t_e} \frac{F_{\text{CO}_2}(t)}{\delta} dt$ meaning that for initial given concentrations, the corresponding concentration evolutions are mainly related to the evolution of the different source terms. Therefore, the time to reach equilibrium is very close to the exposure time (1 ms). The presented analysis suggests that the reaction has been fully completed between two frames (10 ms).

Fig. 8

Time evolution of CO₂ concentration from the reaction kinetic differential equations system (7) solved for the source term $F_{{{\text{CO}}_{2} }}$/δ and the influence of the source term on the CO₂ concentration

Appendix 2: Intensity attenuation due to the diverging lens

The decreasing intensity due to the diverging lens has to be corrected during the extinction coefficient variation calibration as a function of pH. Assuming no fluorescent dye and a constant thickness of the laser sheet, and considering light flux conservation, the ratio of intensities (per unit area) received by two pixel columns k and l can be determined geometrically :

$$ \frac{I_k}{I_l}=\frac{h_k}{h_l}=\frac{L_d+L_w}{L_d} $$

(8)

where I _k and I _l are the light intensities received by the pixels of two pixel columns k and l, belonging to the ROI, of respective height h _k and h _l . L _w is the width of the camera observation area, and L _d is the distance between this area and the diverging lens, as shown in Fig. (4).

This contribution has been taken into account in Eq. (6). As it is a constant for any pair of pixel columns, it has no influence on the extinction coefficient variations ($\varepsilon_{\text{ref}} - \varepsilon$) as a function of pH.

Appendix 3: Data acquisition and data treatment procedures

The data acquisition and data treatment procedures from the acquired fluorescence signal to CO₂ concentration maps are presented in the following Fig. 9.

Fig. 9

Data acquisition and data treatment procedures