A centrifugation method for the assessment of low pressure compressibility of particulate suspensions

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Abstract

This work describes an easy methodology for assessing the compressibility of particulate suspensions at relatively low pressures. The material used in this study was an anaerobically digested waste water sludge. Using a centrifuge, the equilibrium sediment bed height was recorded as a function of rotational speed during the centrifugation. This approach avoids errors due to expansion of the sediment bed after the centrifugation is stopped, a phenomenon typically seen with waste water sludges. A straightforward numerical method was used to solve a one-dimensional centrifugation model describing the equilibrium sediment bed height as a function of rotational speed. It has been shown that this approach yields realistic results. The one-dimensional model is able to describe the compressibility of the anaerobically digested sludge. Besides yielding a close fit for the measured sediment bed heights, extrapolating these results to a different initial solids volume fraction resulted in a good prediction of the bed height as well. Different constitutive equations for the relation between solids pressure and solids volume fraction were assessed. The typical power law functions [F. Tiller, W. Leu, Basic data fitting in filtration, Journal of the Chinese Institute of Chemical Engineers 11 (1980) 61–70], often used in filtration modelling, were the most adequate for describing the compressional behaviour of the sludge. A functional relation based on the osmotic pressure within the solids network, suggested for describing the phenomenology of filtration dewatering [K. Keiding, M. Rasmussen, Osmotic effects in sludge dewatering, Advances in Environmental Research 7 (2003) 641–645], could not be used to adequately describe the behaviour of the sludge at the low pressures generated during centrifugation.

Introduction

Waste water treatment produces enormous amounts of excess activated sludge and/or anaerobically digested sludges as a sidestream. Before the sludge is disposed of, the solids concentration is increased through a series of different operations in order to reduce the transportation and disposal costs [3]. With regard to dewatering, mechanical processes are economically more favourable than thermal ones [4]. The main processes used for increasing the solids concentration are gravitational thickening, centrifugation and pressure filtration. Because of the high associated costs, understanding and predicting the behaviour of these processes is of great importance to the waste water treatment industry.

The first major attempt to derive a mathematical background for describing filtration kinetics was made by Ruth in the early 1930’s [5], [6]. Since then, the modelling of dewatering processes has been a major topic of interest. Even though Ruth acknowledged the impact of compressibility on the solids distribution within the cake, solids were assumed to be incompressible in the earliest work. Incompressible solids lead to a filter cake with a homogeneous solids concentration and permeability that is insensitive to the applied pressure. Although the basic concepts of these theories have proven to be valid, most materials show compressible behaviour to some extent. Activated sludge, and more generally biological materials, are known to possess a very high compressibility [7], [8] and a low permeability or, accordingly, a high specific resistance at moderate to high pressures. Hence, most of the mathematical models for dewatering comprise two material properties, being a compressibility term and a permeability term.

Various approaches to describing filtration behaviour start from an adapted form of the Darcy equation:q=1μαdpsdx,with q the superficial liquid velocity [m/s], μ the liquid viscosity [Pa s], α the specific resistance [1/m2], ps the solids pressure [Pa] and x the height within the cake [m]. In pressure dewatering, the local solids pressure is the difference between the total applied pressure and the local liquid pressure. In the case of an incompressible cake, the porosity does not depend on the solids pressure, and hence the porosity and specific resistance are constant throughout the cake. In the case of a compressible system, however, an increase in solids pressure causes a decrease in porosity and an increase in specific resistance. To relate the porosity and the specific resistance to the solids pressure, different forms of constitutive equations have been suggested. The so-called power law functions proposed by Tiller and Leu [1] are probably the most widely used:ϕ=ϕg1+pspaβ,andα=α01+pspan.ϕ is the solids volume fraction, equal to (1ɛ) with ɛ the porosity []. ϕg and α0 are the volume fraction and specific resistance when the solids are in contact without being compressed (ps=0). ϕg is also called the gel point, as it is the volume fraction at which a continuous solids network is formed. β, n and pa are fitting parameters that allow for fitting the equations to measured data.

Some other functional forms that have been suggested are [9], [10]:ϕ=ϕgpspaβ,ϕ=ϕg1+pspaβ,ande=AB×ln(ps),where e=(1ϕ)/ϕ.

A more extensive listing of compressibility equations used in literature can be found in Olivier et al. [10].

Buscall and White [11] developed a unified theory of compressional rheology for the dewatering of flocculated suspensions. Key factors in this theory are the compressive yield stress Py(ϕ) and the hindered settling factor r(ϕ). The latter relates the liquid velocity through the cake to the free settling velocity of the individual particles, while Py(ϕ) is the pressure needed to collapse the cake irreversibly to a solids volume fraction ϕ. It can be shown that r(ϕ) is related to the specific resistance α [12], [13]. Under the assumptions used in the further development of the theory [14], [15], [16], Py(ϕ) is equivalent to ps(ϕ). Buscall and White [11] regard these properties as material properties, rather than filtration properties and use their theoretical framework for settling and centrifugation as well as filtration.

Whereas most constitutive relationships were established empirically, Keiding and Rasmussen [2] suggested an interesting approach by interpreting the relationship between the solids volume fraction and the solids pressure in a more fundamental, physical way. They suggested that for slurries such as activated sludge, the osmotic pressure in the sludge matrix – due to the presence of charged polymeric substances – could be responsible for the high compressibility. They suggested that the solids pressure is equal to the osmotic pressure, and is given byps=σρϕ1ϕRT,with σ the charge density [eq/kg], ρ the solids density [kg/m3], R the universal gas constant [J K−1 mole−1] and T the temperature [K]. The authors found that this approach could reproduce the typical phenomenology of filtration dewatering of sludge.

Buscall and White [11] showed that the compressive yield stress function Py(ϕ) (i.e. the constitutive equation) can be obtained from centrifuge experiments. They determined the final sediment bed height for different rotational speeds. Within their compressional rheology framework, they derived an approximate solution for Py and ϕ at the bottom of the tube as a function of the centrifugal acceleration at the bottom of the tube. Green et al. [17] noted that it does not appear possible to show theoretically that this solution is an accurate approximation for all functions of Py(ϕ). Green et al. [17] suggested an iterative solution method, which includes numerically solving a set of differential equations. Again, this solution method yields values for Py and ϕ at the bottom of the tube as a function of the centrifugal acceleration at the bottom of the tube.

In this work, we will develop a novel methodology for assessing the compressibility behaviour of particulate suspensions using the equilibrium sediment bed height during centrifugation. The relationship between solids pressure and solids volume fraction is obtained by parameter optimisation of a simple numerical model, based on straightforward physical principles. We will apply this methodology to an anaerobically digested sludge and we will assess various functional forms for the relationship between solids pressure and solids volume fraction at relatively low solids pressures. These pressures are relevant for gravity settling and centrifugation, and to a lesser extent for the filtration stage in pressure dewatering.

Section snippets

Sludge

The sludge used for this work was a mesophilic anaerobically digested sludge from a waste water treatment plant at Carrum, Australia. During storage, biotic sludges tend to show a change in filtration behaviour due to the occurrence of biological processes. For activated sludge, it was seen that changes in extracellular polymeric substances slow down significantly towards the end of a 14-day storage period [18]. It was decided to use anaerobic sludge, and to keep it in a fridge (at 4 °C) for 14

One-dimensional equilibrium state centrifugation model

The centrifugal force is a reactive force, acting in the radial direction. It is convenient to depict centrifugation using a rotating frame of reference, with its origin in the centre of the centrifuge and with the same angular speed as the centrifugation itself, as shown in Fig. 3. The bottom figure shows a scaled representation of the real dimensions of the sediment bed in the centrifuge. During centrifugation at an angular speed of ω, the centrifugal force Fc acting on a particle along the X

Fitting

To assess the validity of the model, samples with two different solids mass fractions (i.e. 1.0 and 2.0%, w/w) were centrifuged. Eq. (2) was used as the constitutive equation in the model; the other equations will be discussed later. The density of water at the operational temperature (i.e. 1000  kg/m3) was selected for the liquid density (ρliq), whereas a value of 1590 kg/m3 was assumed for the solids density (ρsol). Assuming that the water which is removed from the sludge during the

Conclusions

It was shown that a relatively simple numerical model can predict the equilibrium sediment bed height as a function of rotational speed during centrifugation. The advantages over previous methods are the direct measurement of the sediment bed height during filtration, and the relative ease of implementation due to the use of straightforward mathematical and physical principles. Different functional forms for the relationship between solids pressure and solids volume fraction – typically used

Acknowledgements

One of us, D. Curvers, acknowledges the financial support he receives from FWO Vlaanderen (Scientific Research Fund of Flanders) as an aspirant of FWO. He is also greatly indebted to D. Huygens and P. Saveyn for invaluable discussions. Further funding assistance for the work was from the Particulate Fluids Processing Centre, a Special Research centre of the Australian Research Council.

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