Elsevier

Powder Technology

Volume 237, March 2013, Pages 367-375
Powder Technology

Is the fish-hook effect in hydrocyclones a real phenomenon?

https://doi.org/10.1016/j.powtec.2012.12.017Get rights and content

Abstract

Although the fish-hook effect has been reported by many for a very long time, scientists and practitioners alike share contradictory opinions about this phenomenon. While some believe that it is of physical origin, others opine that it is the result of measurement errors. This article investigates the possibility that the fish-hook effect could indeed be measurement error related. Since all the experimental errors are embedded in the raw size distribution measurements, the paper first lays down the steps that lead to estimation of the partition function and confidence bounds, which are seldom reported in hydrocyclone literature, from the errors associated with the experimental size distribution measurements. Using several data sets generated using a 100 mm diameter hydrocyclone operating under controlled dilute to dense regimes, careful analysis of the partition functions following the developed methodology yields unambiguous evidence that the fish-hook effect is a real physical phenomenon. An attempt is also made to reunite some of the major contradictory views behind the existence of the fish-hook based on sound statistical arguments.

Graphical abstract

While some believe that the fish-hook effect is of physical origin, others opine that it is the result of measurement errors. Using several data sets generated with a 100 mm diameter hydrocyclone operating under controlled dilute to dense regimes, careful analysis of the partition functions and associated confidence bounds yields unambiguous evidence that the fish-hook effect is a real physical phenomenon.

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Highlights

► Confidence bounds of partition function are needed to assess hydrocyclone performance. ► Measurement errors are ruled out as a possible cause of the fish-hook phenomenon. ► Statistical arguments explain the ‘apparent’ randomness of the fish-hook phenomenon.

Introduction

There are a number of issues associated with the operation of a hydrocyclone that remain without clear explanations, often leading to some level of discord about their cause and effect. This is not the least surprising given the complexity of the separation that takes place inside this unit operation.

One issue that has been the subject of much debate over the years is what is referred to as the fish-hook effect. As the particle size separation in a hydrocyclone is performed using a fluid medium in dynamic conditions and usually the feed particles have wide size distributions, the separation is never perfect at any particle size. Many criteria have, therefore, been proposed to evaluate the performance of hydrocyclones but the graphical method of representing the recoveries of each particle size in hydrocyclone underflow in relation to their availability in feed as a function of their respective sizes (usually in a log-normal scale) is the most popular one. The data thus plotted is popularly known as partition curve or Tromp curve. This is normally an ‘S’ shaped curve and a variety of quantitative expressions have been used to describe the shape of the curve. However, partition curves do not always follow the conventional ‘S’ shaped pattern, rather they have fish-hook patterns as described in literature [1], [2], [3]. This means that the recoveries of relatively finer particle sizes are initially higher than the coarser particle sizes in underflow up to a critical particle size and after that the recovery increases with increase in particle sizes.

Although the fish-hook effect has been reported by many for a very long time, researchers and practitioners alike share different opinions about this phenomenon. There is one school of thought that believes the fish-hook effect to be a real physical phenomenon [4], [5], [6], [7], whereas a second one argues that it is measurement error related [8], [3].

Flintoff et al. [3] opined that this fish-hook effect was due to poor experimental procedures and/or agglomeration of fine particles. Nageswararao [8] summarized that this is a random and sporadic occurrence caused by the imprecision of measurement and it does not affect the hydrocyclone performance. A few others [2], [3], [4] however, have also proposed some empirical correlations to predict the fishhook effect in a hydrocyclone classifier with reasonable accuracy. But these models failed to explain whether this irregular behavior with ultra fine particle sizes in a hydrocyclone is a characteristic phenomenon or not. Majumder et al. [5] have provided a mechanistic argument supporting the occurrence of fish-hook in all centrifugal separators while treating fine and ultra fine particles. The basis of their argument is that in a centrifugal force field, there is a sudden drop in relatively coarser particles settling velocities due to Reynolds number restrictions.

As scientists and practitioners alike share different opinions about this phenomenon, an attempt has been made here to assess whether measurement errors could possibly be responsible, on their own account, for the fish-hook effect. For the sake of clarity, it is emphasized that the fish-hook effect that is being investigated in this work is that caused by particle size only. This work is not concerned with other factors that can also affect classification, such as density variations within a particle size class [9].

Section snippets

Mass balance solution with particle size distribution error

Mass balancing is an important topic in extractive metallurgy. This section recaps on the set of equations that can be used to reconcile the measured particle size distributions around a hydrocyclone. The purpose is to give utter transparency to the approach that is used in this paper for propagating the experimental particle size distributions' measurement errors right through to the estimation of the error associated with the partition function of the hydrocyclone. One side value of this

Proof of the “existence” of the fish-hook effect

The purpose of the previous section was to present and validate the mass balance calculation that yields estimation of the partition function and associated confidence interval, taking into account the particle size distribution measurement errors. For the sake of transparency, it is recalled that the same distribution RSD(fi°) was applied to all 3 streams as per Eq. (8), excluding the possibility that sampling and preparation errors may be different between the 3 streams. This is an assumption

Origin of the fish-hook

The conditions that will yield the fish-hook phenomenon to be significant can be numerous, and it should not be inferred from the data that were presented in the previous section that the fish-hook is most significant in the 20–30 wt.% region only. Indeed, many configurations and operating conditions can lead to the fish-hook effect. Further discussion on this is outside the scope of this article.

Appearance of randomness and sporadicity of the fish-hook effect

By refuting the possibility that measurement errors could be the reason beyond the fish-hook effect,

Conclusions

Based on the aforementioned results and discussion the following conclusions may be drawn:

  • Estimation of confidence bounds of the partition function is of significant value for analysis of the performance of a hydrocyclone.

  • Partition functions estimated from carefully generated experimental data set in a 100 mm diameter hydrocyclone followed by rigorous statistical analyses of the data have completely ruled out the possibility of measurement errors as the cause of the observed fish-hook effect.

Notations

    Symbols

    du

    Solid mass flow split to the underflow

    f

    Vector of mass fractions in the feed stream

    u

    Vector of mass fractions in the underflow stream

    o

    Vector of mass fractions in the overflow stream

    n

    Number of particle size classes

    Subscripts

    i,j

    Indices of particle size class

    Superscripts

    ^

    Circumflex symbol denotes the estimate of a random variable

    t

    Transpose of a vector or matrix

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