A new method to calculate sedimentation velocity of particle group with three-plane ECT based on the best correlation function

ABSTRACT

A novel method to calculate the flow velocity of solid in a solid–liquid two-phase system is induced in this paper by electrical capacitance tomography (ECT). Taking the slime settlement as an example, the concentration change of three planes in the settling tank is measured using ECT. Moreover, the settling velocity of continuous frames in different sensor planes employing the best correlation function is calculated. The results show that slime has a higher and more uniform settlement velocity in the center area of the settlement tank. A 0.02–0.04 cm/s deviation is found between the calculated velocity of clarifying interfacial layer when slime passing through the ECT sensor and the experimental data. Remarkably, the time point is in good agreement with the experimental result which is between 7 and 13 s, indicating that the method can be used to estimate the settling velocity of solid in a two-phase flow.

KEYWORDS:

• Slime
• settling velocity
• the best correlation function
• electrical capacitance tomography
• two-phase flow

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Nos. 51604273, 51304194, 51304195) and the Jiangsu Natural Science Foundation financially, BK20130181; it is also part of a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.

Nomenclature

ε the equivalent dielectric constant between fluids

permittivity of the continuous phase of the fluid

${\epsilon }_2$       permittivity of the discrete phase of the fluid

$\mathrm{V}$        the total volume of the fluid

${\mathrm{V}}_1$        the volume fraction of the continuous phase

${\mathrm{V}}_2$        the volume fraction of the discrete phase

$\gamma$         the concentration of discrete phase

$\tau$         the time delay between the two signals

T         the observation time

D       the distance between adjacent two sensors

P        the index of the pixel

N        the number of images for which the cross-correlation is calculated

k        the shift number in time sequence of frames

$C_{X_{\mathrm{A}}}\left[i\right]$        the concentration values associated with pixel p from image i obtained from sensor $X_{\mathrm{A}}$

$C_{X_{\mathrm{B}}}\left[i+k\right]$       the concentration values associated with pixel p from image i+k obtained from sensor $X_{\mathrm{B}}$

$C_{X_{\mathrm{A}}\left[m,n\right]}\left[\mathrm{i}\right]$        the concentration values with the pixel (m, n) from image i obtained from on planes $X_{\mathrm{A}}$

$C_{X_{\mathrm{B}}\left[m-\mathrm{\Delta }\mathrm{m},n-\mathrm{\Delta }\mathrm{n}\right]}\left[\mathrm{i}+\mathrm{k}\right]$   the concentration values with the pixel ($m-\mathrm{\Delta }\mathrm{m},n-\mathrm{\Delta }\mathrm{n}$) from image i+k obtained from on planes $X_{\mathrm{B}}$ and $\left(\mathrm{\Delta }\mathrm{m},\mathrm{\Delta }\mathrm{n}\right)\in \mathrm{Y}$. Y is the neighborhood of pixel (m, n) on plane $X_{\mathrm{B}}$

$C_{k\left[m,n\right]}$     the concentration values of the pixel (m, n) in kth frame

$C_{k+1\left[i,j\right]}$     the concentration values for coordinate (i, j) of (k+1)th frame

Additional information

Funding

This work was supported by the Jiangsu Natural Science Foundation financially [BK20130181]; the National Natural Science Foundation of China [51304194,51304195,51604273].