Acoustic probe for solid-gas-liquid suspensions. 1997 annual progress report

Abstract

'Acoustic probes have shown promise to be quite effective in determining the solid content in solid-liquid suspensions. However, the presence of small amounts of gas in the waste slurries stored in tanks across the DOE complex prevents straightforward application for characterization of these slurries. The proposed research will develop an acoustic probe for monitoring particle size and volume fraction in slurries in the absence and the presence of gas bubbles. Theoretical Analysis Accomplished: Attenuation of sound waves depends on the size distribution of the solids and the volume fraction of solids. These can in principle be calculated from attenuation measured over a range of frequencies. However, small amounts of bubbles distort the measured attenuation. A typical result from theoretical analysis for the attenuation of solid- gas-liquid systems is given in Figure 1. The total attenuation of a sound wave v(o) equals the sum of contributions by a large number of ''bins'' of particle sizes. This notion yields the following equation for the (hitherto) unknown number density of solid particles as a function of particle radius N(a): j k(o,a)N(a)da = v(o), where the kernel k(o,a) is obtained from analysis. If N(a) is given, the above equation is used to calculate the attenuation v(o). This is referred to as solving the ''forward problem''. Solving for N(a) with v(o) given is the ''inverse problem''. A complication that one faces when trying to solve the inverse problem is that the stated problem is mathematically ill-posed, i.e., small fluctuations in v(o) cause large fluctuations in the result for the number density. Therefore the problem needs to be ''regularized'', i.e., the stated problem needs to be changed slightly such as to make it well-posed. This has been done by others for gas-liquid systems in the past. This approach is currently being applied in the present project to solid-liquid systems. As is shown in Figure 2, it successfully recovers the number density that has been used in the forward problem to generate attenuation data. Having this solution technique giving reliable results for the inverse problems of both gas-liquid and solid-liquid systems, the authors shall apply this method in the near future to solid-gas-liquid systems.'