Estimation of settling velocities
Introduction
This paper deals with the estimation of the settling velocity density of solid particles based on settling column measurements. We have taken a quite general approach which was originally motivated by the study of particles transported by rainfall over urban areas. Settling process is one kind of efficient treatment means for removing such pollution. In order to evaluate the efficiency of settling tanks there is a need to estimate the settling characteristics of the solids involved. We have used a very simple settling process model, in initially neglecting important aspects of the particle dynamics such as flocculation or diffusion process.
In this article we formalize the mathematical identification procedures which are naturally present in the various settling apparatus that have been developed for measuring the settling velocity distribution of representative samples of storm sewage (Michelbach and Wohrle, 1993; Tyack et al., 1996; Lucas-Aiguier et al., 1996).
In addition, assuming that there are statistical errors in the measurements, we are able to provide statistical error characteristics of the estimated settling velocities. For simplicity sake, we will assume that measurement errors only appear in solid mass measurement (errors in the weight of samples).
Section snippets
Solids settling characterization methods
Firstly, we will briefly present the different methods which are used to estimate settling velocities. A more precise description of the physical apparatus can be found in (Lucas-Aiguier et al., 1996). A settling velocity distribution curve is a cumulative graph which represents the proportion of solids by weight with settling velocities less or equal to a given settling velocity. The characterization methods aim at estimating this curve.
Two classes of methods are generally used for measuring
Modelisation: Generality
All the methods presented in the previous section can be summarized as follows. A physical apparatus (type x) is used to obtain a set of discrete measurements Mx(ti), i=1, N and these data must be used to identify an unknown quantity for example ρ(dv) (the mass of particles with settling velocities between v and v+dv). In order to exhibit a mathematical relation between measures and unknown function, we must use a model from our settling apparatus. We will show in the sequel that with a simple
The φ function (Eq. (3)) for the type a column
In the first method that we will study, we measure Ma(t), the mass of settling particles accumulated at time t at the bottom of the settler. Using the model of Section 3.2this is exactly the mass contained in the domain {z<0} at time t.
Proposition 4.1. Ma(t) can also be written as follows:Proof. If we use Eq. (2)and the definition of Ma(t), we get:An explicit solution of Eq. (2)is g(t, z, v)=g(0, z+vt, v). Thus:
The φ function (Eq. (3)) for the type b column
In this method, the concentration is measured at different levels at sampling times. We will denote by p1<p2<…<pK<h the different depths at which measurements are made and of course it's possible to assume that parameter K=1 if only one level is used. Using Eq. (2)the concentration measurement at depth pk gives way to an evaluation of ∫0∞g(t, h−pk, v)dv which in turn leads to:We are once again faced with an estimation
Type c column
We will, only briefly, consider the type c method. Even if it is quite different from type b method it leads to a relation between measured quantities Mc(t) and unknown function ρ(dv) which is almost the same.
Discussion on the models and measurements
Here, we reformulate the estimation problem to show that it belongs to a general class of identification problems. This indicates that taking into account new phenomenons (for example diffusion) is not out of the context for the exposed methodology and can give some guidance for future developments.
We prove that all the quantities which are measured on the settler can be expressed as follows:where Ω stands for a part of the boundary of the
Conclusion
We have tried, in this paper to state and solve the estimation of settling velocity density functions for three different types of columns. The settling model which we have used (Eq. (2)) is very simple. However, this model reflects authors implicit assumptions when estimating settling velocities.
The main advantage of this model comes from the fact that we can compute explicit solution of the particle density inside the settler and thus we can obtain explicit solutions for the estimators. Our
Acknowledgements
The authors are indebted to M. Jean-Marie Mouchel for his help and fruitful discussions.
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