Electronic Journal of Polish Agricultural Universities (EJPAU) founded by all Polish Agriculture Universities presents original papers and review articles relevant to all aspects of agricultural sciences. It is target for persons working both in science and industry,regulatory agencies or teaching in agricultural sector. Covered by IFIS Publishing (Food Science and Technology Abstracts), ELSEVIER Science - Food Science and Technology Program, CAS USA (Chemical Abstracts), CABI Publishing UK and ALPSP (Association of Learned and Professional Society Publisher - full membership). Presented in the Master List of Thomson ISI.
Volume 12
Issue 1
Environmental Development
Available Online: http://www.ejpau.media.pl/volume12/issue1/art-07.html


Beata Malczewska
Institute of Environmental Engineering, Wrocław University of Environmental and Life Sciences, Poland



The paper presents results of rheological measurements conducted on sewage sludge from settling tanks of several municipal sewage treatment plants. The basic purpose was to define rheological properties of the sludge. The experiments were conducted by using coaxial cylinder with a rotating torque. The sludges had solid concentrations in the range of 1.04% to 6.56%. The approximation was made after transforming pseudo-curve obtained from the measurements into the true flow curve, made according to the equation provided by Krieger, Elrod, Maron and Svec. In order to describe rheological characteristics, Herschel-Bulkleys' model was applied. The correlation between rheological parameters (tauo, k, n) and gravimetric concentration (Cs) was calculated. The research has allowed the dimensioning of the main transport installations for pumping sludge. Optimisation of the pump discharge pressure, when transporting viscous sludge in pipelines is of the main interest in wastewater industry. The determination of rheological parameters, especially yield stress (tauo), is important in sludge management, including designed parameters in transporting, storing, spreading.

Key words: sewage sludge, rheology, apparent flow curve, rheological model.


All techniques involved in the purification of wastewater are connected with sludge nascenting. Sludge disposal represents the major operating costs for wastewater treatment plants (WWTPs). WWTPs operations must also be considered according to the optimisation of sludge circumstances. The sludge characteristics may affect the hydraulic conditions of sludge as well as handling units, such as sludge thickening, dewatering and settling. A decrease in water content and therefore an increase in solid concentration which raises the overall unit cost in slurry pipelines technologies. Many studies have demonstrated that an increase in the solid slurries concentration leads to an increase in the frictional pressure loss. Some aspects of sludge hydraulic conditions are connected with reducing pipe friction and have been investigated by Heywood, Honey and Pretorius, Slatter, Klinksieg et. all, Chaari et. all [3,8,9,12,15]. The settling of solids inside pipelines and thus clogging is related with sewage concentration and flow velocity.

Among the characteristics of the sludge that can be used for design and operation, the particle size, the distribution of water and the flow properties have been extensively investigated recently [4,5,7,16,19]. Highly viscous sludge exhibits high shear. Therefore, sludge handling and management at wastewater treatment plants also depend on its rheological properties. The rheology describes the deformation of a body under the influence of stress. Rheological behaviour is often presented graphically by plotting the shear stress (tau) against shear rate (Gp) (often called rheogram or flow curve). Sewage sludge is a non-Newtonian fluid because its flow behaviour does not obey the Newtonian flow law. The law identifies the viscosity as a constant physical property, whereas the shear stress is non-linearly linked to the shear rate in non-Newtonians fluid. A non-Newtonian fluid like sewage sludge, cannot be based on Newtonians' model because of not a constant value in sludge viscosity under certain pressure and temperature conditions [10].

Several research groups studying the rheology of the sewage sludge have indicated the thixotropic nature of the sludge [8,20]. Many papers were devoted to exploring the correlation between sludge characteristics with solid concentrations, the temperature, the pH-value and chemical conditioning [1,6,17,18].

The main purpose of this investigation was to evaluate sensitivity of the rheological parameters according to the solid concentration of sewage sludge in connection with sludge quality variation (different chemical constitution).


The measurements were carried out using VT550 viscometer (manufactured by Haake) with inner rotating coaxial cylinder, under a constant temperature of 20oC ±1oC. The viscometer enabled measurement of the flow curves for a wide range of deformation velocity and shear stress. The device ensured the uniform stress distribution in the measuring gap and enabled continuous measurements of shear stress (tau) at defined shear rate Gp. The measurement system MV1 with measuring gap width 0.96 mm was used. The experiments were performed according to a method described in previous paper [10,11,13]. Measurements were carried out in steady flow so that laminar shear flow condition could be respected which means that the value Gp=300s-1 was not exceeded. Measuring conditions are summarised in the following procedure: stirring at a shear rate of 300s-1, decreasing shear rate to 0 and then increasing to 300s-1 which was repeated twice.

Several WWTPs were selected for this study. Five of them are situated in Poland, operated by different municipalities, and one is in Germany. All of the WWTPs used activated sludge processing. The samples of the sludge were collected from the secondary settling tank. The different solid contents were obtained by diluting the sludge with supernatant. The gravimetric concentration Cs was designated for each of the investigated samples and described as the mass ratio of dry component (ms) to the mass of a mixture (mm).

Pseudo-curves obtained from experimental data also depends on employed measuring systems therefore the transformation of a pseudo-curve into the true flow was calculated.

The transformation of a pseudo-curve into the true flow curve was calculated using the equation provided by Krieger, Maron, Elrod and Svec. Selection of the rheological model was performed on the basis of determined true flow curves, according to the method described by Czaban [2,11,13]. The pseudo-curves and correlation coefficient were used as a criterion in selecting the rheological model. The three most commonly adopted models were tested to select the one which fitted the experimental data best. The rheological parameters were extracted for each rheogram using least squares regression.

In order to describe rheological characteristics the 3-pararameter Herschel-Bulkley model was applied as follows:

tau = tau0 +kGn for tau>tau0         (1)
G = 0 for tau<tau0           

where: tau = shear stresses [Pa]
tau0 = yield stress [Pa]
n = flow behaviour index (dimensionless)
k = fluid consistency coefficient [Pa·m-2]
G = shear rate [s-1]


To investigate the rheological behaviour of the sludge, the following calculations were made:


The objective of this work was to characterise secondary sewage sludges treated aerobically. In order to investigate the rheological behaviour of sewage sludge at different concentrations Cs the shear stress vs. shear rate rheograms were plotted. Several authors have published works on the rheological properties of sludges and they have established that sewage sludge belongs to non-Newtonian fluids [4,5,6,15,18,19,20]. The rheograms indicate that the shear stress is non-linearly proportional to the shear rate and viscosity changes with the stress rate which is a typical for a non-Newtonian fluid. A correlation between rheological parameters and gravimetric concentrations Cs was made for each WWTPs investigated. Figs. 1, 2 shows an example for WWT5.

Fig. 1. Measured pseudo-curves of flow for various concentrations of sludge originating from WWT5

Fig. 2. Dependence of apparent viscosity on shear rate

A non-linear correlation between shear rate and apparent viscosity was found. There were variations of sludge viscosity with shear stress and concentration of sludge. These results were in a good agreement with Slatter, Dentel and Stain [4,16,18]. Various rheological models were suggested to interpret the observed rheological characteristics. [4,9,16,18]. The presence of yield stress in Bingham and Herschel-Bulkleys' models is due to resistance of sludge to deformation until sufficient stress is applied to exceed the yield stress of solid phase. This two rheological models were used to explain rheological characteristic of sludge. The rheological measurements showed that the best statistical approximations to the measured values are obtained with Herschel-Bulkley model. Correlation coefficient (0.97-0.99) indicated that the flow of sludge could be fitted best by the pseudoplastic model. Sludges exhibit a yield stress and the viscosity varired with the shear rate (Gp).

The viscometric research for various gravimetric concentrations of sludge allowed to describe dependence of gravimetric concentration on rheological parameters. Results are presented as diagrams of dependencies tau0 = f(Cs), k = f(Cs), and n =

The same calculation procedure was used to investigate each WWTPs. The graphical example of the relationship between rheological parameters and gravimetric concentration for sludge was taken from WWTPs is illustrated in Figs. 3, 4, 5.

Fig. 3. Dependence of flow yield stress on concentration for Herschel-Bulkley model

Fig. 4. Dependence of rigidity coefficient on concentration for Herschel-Bulkley model

Fig. 5. Dependence of structural number on concentration for Herschel-Bulkley model

Results indicated that the yield stress was present in a range of analysed gravimetric concentrations. The yield stress is described as a stress to be exceeded to form fluid flow. This means the knowledge of the yield point is crucial in designing the pumping pipeline installations. The research also indicated that the sludge exhibited pseudoplastic behaviour, what supports others’ discovering [5,16].

According to the viscosymetry method, rheological parameters for Herschel-Bulkley model (tau0, k, n), used in the gravimetric concentration function, were calculated for each treatment plant. The relationship between rheological parameters tauo, k, n and gravimetric concentration Cs were established and are as follows: yield stress and rigidity coefficient of Herschel-Bulkley model increase, while the structural number demonstrates the decreasing tendency with rising gravimetric concentration (Cs).

As it is shown in Figs. (3, 4, 5) presented above the rheological parameters are significantly dependent on the gravimetric concentration (Cs). Results confirmed that solid concentration is the main parameter affecting the sludge rheology, as stated in previous studies [5,13,16,19].

The critical concentration Cs,gr between Newtonian and non-Newtonian behaviour of the sludge was determined by depending on the function tauo(Cs), for tauo = 0 and was calculated using the least squares and power function [2,11,14]. The method is justified by the conversion of the generalised Herschel-Bulkley rheological model to the Newton model. The critical concentration Cs,gr for illustrated series of measurements is from 0.58% (WWTP 6) to 2.41% (WWTP 4).

Table 1. Dependence of gravimetric concentration (Cs) and critical concentration (Cs,gr.)
































Table 1. Dependence of gravimetric concentration (Cs) and critical concentration (Cs,gr.) The sludge with gravimetric concentration Cs < Cs,gr behaves like Newtonian fluid, whereas with the concentration of solid constituent Cs > Cs,gr appears as non-Newtonian fluid. After exceeding critical concentration the sludge behaves a the Herschel-Bulkley substance. The variability of sludge concentrations allowed the sludge to be described with the 3-parameter generalized Herschel-Bulkley model, which gets simplified to 2- or 1-parameter model, as the gravimetric concentrations Cs reduces.


In order to find an empirical equation illustrating the influence of concentration on the rheological parameters and evaluating the sensitivity of rheological parameters with sludge quality variation, the cumulative analysis was performed.

The cumulative diagram illustrated above presents the dependence of rheological parameters on gravimetric concentration for sludge. The dependency of the yield stress (tau0) increased with an increase in gravimetric concentration (Cs). The measured points clearly demonstrated dissipation tendency, especially in the range of higher gravimetric concentrations. Moreover the research allowed the assumption that the yield stress was unique, and time-dependent, therefore it should be measured individually for each treatment plant and sludge batch.

In case of the relationship of rigidity coefficient (k) along with structural number (n) for Herschel-Bulkley model on gravimetric concentration (Cs) there was an exponential correlation [13]. This correlation was calculated using least square method and it can be described by exponential function as follows:

for k = f(Cs):      k = 0.013332Cs3.273391         (2)

for n = f(Cs):      n = 0.592964Cs0.160462         (3)

Knowledge of these mathematical formulas can be used in order to evaluate rheological parameters, such as rigidity coefficient and structural number, for sewage sludge derived from similar WWTPs and sludge batches (assuming the similar range of gravimetric concentrations).


Sewage sludge behaves like non-Newtonian fluids. It is also found to behave as a pseudoplastic fluid in this study. The gravimetric concentration of sludge has the most significant effect on the rheological properties of sludge. With increasing gravimetric concentration, pseudoplastic viscosity increased sharply at high concentration. Sludge samples exhibited yield stress. Therefore, to describe rheological behaviour the 3-parameter Herschel-Bulkley model was applied. The relationship between rheological parameters tauo, k, n and gravimetric concentration Cs was established and is as follows: the yield stress and rigidity coefficient of Herschel-Bulkleys' model increases, while the structural number decreases with rising gravimetric concentration Cs.

The rigidity coefficient and structural number of Herschel-Bulkleys' model can be estimated with the proposed model although the yield stress should be measured individually for each sewage treatment plant and sludge batch. The mathematical models obtained by analysis of the experimental data predicted results reasonably well under experimental condition. Furthermore, it would be possible to evaluate mathematical model of rheological characteristic of sludge, which in turn could allow optimising pump discharge pressure.

The critical concentration Cs,gr between Newtonian and non-Newtonian behaviour of the sludge was determined by depending on the function tauo(Cs), for tauo = 0 and it is strongly depended on gravimetric concentration of sludge. For the concentration Cs < Cs,gr it behaves as Newtonian, whereas with the concentration of solid constituent Cs > Cs,gr it appears non-Newtonian fluid. The critical concentration Cs,gr which delimitates Newtonian and non-Newtonian fluid behaviour of sludge is Cs,gr.

Units and nomenclature



gravimetric concentration



critical concentration



shear rate



apparent shear rate, deformation rate



structural number



rigidity coefficient for Hershel-Bulkley model



shear stress



yield stress


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Accepted for print: 16.02.2009

Beata Malczewska
Institute of Environmental Engineering,
Wrocław University of Environmental and Life Sciences, Poland
Grunwaldzki Square 24, 50-365 Wroclaw, Poland
phone: +4871 3205519
email: beata.malczewska@up.wroc.pl

Responses to this article, comments are invited and should be submitted within three months of the publication of the article. If accepted for publication, they will be published in the chapter headed 'Discussions' and hyperlinked to the article.