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Fig. 10 — Parameters k and n of the Ostwald de Vaele model, estimated as function of MLSS (a) and parameters k, n and ‘o of the Herschel—Bulkley model as function of MLSS (b) (Rosenberger et al., 2002). This danger of over parameterization is clearly illustrated by Rosenberger et al. (2002), who compared the Herschel—Bulkley approach (3-parameter model) to the Ostwald equation (2-parameter model) for measurements on AS samples from nine MBR installations (municipal and industrial). Since rheograms were measured for different MLSS concentrations, the parame- ters of both models were calculated in function of this MLSS concentration. For the Ostwald model, satisfactory regressions were found for both k and n (Fig. 10a). However, including a third parameter to, as required in the Herschel—Bulkley model,

Figure 10 — Parameters k and n of the Ostwald de Vaele model, estimated as function of MLSS (a) and parameters k, n and ‘o of the Herschel—Bulkley model as function of MLSS (b) (Rosenberger et al., 2002). This danger of over parameterization is clearly illustrated by Rosenberger et al. (2002), who compared the Herschel—Bulkley approach (3-parameter model) to the Ostwald equation (2-parameter model) for measurements on AS samples from nine MBR installations (municipal and industrial). Since rheograms were measured for different MLSS concentrations, the parame- ters of both models were calculated in function of this MLSS concentration. For the Ostwald model, satisfactory regressions were found for both k and n (Fig. 10a). However, including a third parameter to, as required in the Herschel—Bulkley model,

Related Figures (18)

Fig. 1 — Friction factor for AS analysed as a Bingham plastic (see Eq. (4)) in function of the velocity (u) and the tube diameter (2). The dimensionless numbers are defined by He=pd71,/k and Re=pud/k (Tchobanoglous et al., 2003).
Fig. 1 — Friction factor for AS analysed as a Bingham plastic (see Eq. (4)) in function of the velocity (u) and the tube diameter (2). The dimensionless numbers are defined by He=pd71,/k and Re=pud/k (Tchobanoglous et al., 2003).
Fig. 2 — (a) Bubble characteristics in a low viscosity system (water). The width of the bubble plume is approximately equal to the diameter of the sparger (Ds), and (b) Bubble characteristics in a high viscosity system (Carboxymethy] cellulose, CMC). The width of the bubble plume is significantly smaller than the diameter of the sparger and the mean bubble size is much larger (Fabiyi and Novak, 2008).
Fig. 2 — (a) Bubble characteristics in a low viscosity system (water). The width of the bubble plume is approximately equal to the diameter of the sparger (Ds), and (b) Bubble characteristics in a high viscosity system (Carboxymethy] cellulose, CMC). The width of the bubble plume is significantly smaller than the diameter of the sparger and the mean bubble size is much larger (Fabiyi and Novak, 2008).
Table 1 — Different widely used non-Newtonian shear stress—rate relationships or explicit apparent viscosity—shear rate relationships. 79 is the yield stress (Pa), n the flow behaviour index (—), k the flow consistency index (Pa s"), ... the infinite rate apparent viscosity (Pa s), Mg the zero shear apparent viscosity (Pa s), A the (Cross) time constant (s) and m the Cross rate constant (—).
Table 1 — Different widely used non-Newtonian shear stress—rate relationships or explicit apparent viscosity—shear rate relationships. 79 is the yield stress (Pa), n the flow behaviour index (—), k the flow consistency index (Pa s"), ... the infinite rate apparent viscosity (Pa s), Mg the zero shear apparent viscosity (Pa s), A the (Cross) time constant (s) and m the Cross rate constant (—).
Fig. 3 — Shear stress—shear rate diagram: 1: Shear- thickening (dilatant), 2: Newtonian, 3: Shear-thinning (pseudoplastic), 4: Bingham plastic and 5: Casson plastic fluids.
Fig. 3 — Shear stress—shear rate diagram: 1: Shear- thickening (dilatant), 2: Newtonian, 3: Shear-thinning (pseudoplastic), 4: Bingham plastic and 5: Casson plastic fluids.
system to system (Spinosa and Lotito, 2003; Yang et al. 2009). The viscous characteristic is usually modelled using either the pseudoplastic power-law or Bingham plastic rheological models (see Table 3). Hence, the general rheological behaviour of AS is not really an issue, but the exact apparent viscosity value, which is typically the most critical, under different circumstances and in different systems shows a broad vari- ability in the literature. Indeed, this is problematic when the exact value is required for design and operation purposes. The influence of aeration on the rheological behaviour (elonga- tional rheology) is not a thoroughly studied domain. As an exception on this, Seyssiecq et al. (2008) studied the influence of aerational shear on top of the traditional rheometer shearing and found that aeration provided a significant decrease in apparent viscosity measurements in the low shear rate range. However, this decrease was rather independent of aeration intensity. For high shear rates, no impact of aeration was concluded for the entire range of aeration intensities studied.
system to system (Spinosa and Lotito, 2003; Yang et al. 2009). The viscous characteristic is usually modelled using either the pseudoplastic power-law or Bingham plastic rheological models (see Table 3). Hence, the general rheological behaviour of AS is not really an issue, but the exact apparent viscosity value, which is typically the most critical, under different circumstances and in different systems shows a broad vari- ability in the literature. Indeed, this is problematic when the exact value is required for design and operation purposes. The influence of aeration on the rheological behaviour (elonga- tional rheology) is not a thoroughly studied domain. As an exception on this, Seyssiecq et al. (2008) studied the influence of aerational shear on top of the traditional rheometer shearing and found that aeration provided a significant decrease in apparent viscosity measurements in the low shear rate range. However, this decrease was rather independent of aeration intensity. For high shear rates, no impact of aeration was concluded for the entire range of aeration intensities studied.
Table 3 — Overview on results of 42 publications regarding AS rheology.
Table 3 — Overview on results of 42 publications regarding AS rheology.
Fig. 4 — Microscopic pictures of AS a) from a conventional AS plant witha MLSS of 3.9 g/L and b) from a pilot scale membrane bioreactor with a MLSS of 12.2 g/L; magnification: 40x (Picture: Osnabriick University of Applied Sciences).
Fig. 4 — Microscopic pictures of AS a) from a conventional AS plant witha MLSS of 3.9 g/L and b) from a pilot scale membrane bioreactor with a MLSS of 12.2 g/L; magnification: 40x (Picture: Osnabriick University of Applied Sciences).
Fig. 5 — Taylor vortices development for shear rates beyond the critical shear rate TA rheometer type AR2000 (gap 1 mm) (Data: Ghent University). These three examples demonstrate the sensitivity of experimental results of AS apparent viscosity to equipment and measurement procedure. When measuring identical sludge samples either in different rheometers (see case i) or in the same rheometer but by varying measurement protocols (see case ii and iii), different results regarding values of apparent viscosity for certain shear rates, and rheological measurement, the dynamic yield stress is not observed. (iii) Fig. 8 shows apparent viscosity curves measured with different shear rate ramps. Chosen time intervals for each reading were logarithmic and identical in both measurements. In Fig. 8a the apparent viscosity was increased from 1—1000 s * and then lowered back to 1 s-*, while the hysteresis in Fig. 8b was 1-100 s-' and back to 1 s~*. It becomes apparent that the history of experienced shear (time and maximum shear rate) has an impact on current AS apparent viscosity.
Fig. 5 — Taylor vortices development for shear rates beyond the critical shear rate TA rheometer type AR2000 (gap 1 mm) (Data: Ghent University). These three examples demonstrate the sensitivity of experimental results of AS apparent viscosity to equipment and measurement procedure. When measuring identical sludge samples either in different rheometers (see case i) or in the same rheometer but by varying measurement protocols (see case ii and iii), different results regarding values of apparent viscosity for certain shear rates, and rheological measurement, the dynamic yield stress is not observed. (iii) Fig. 8 shows apparent viscosity curves measured with different shear rate ramps. Chosen time intervals for each reading were logarithmic and identical in both measurements. In Fig. 8a the apparent viscosity was increased from 1—1000 s * and then lowered back to 1 s-*, while the hysteresis in Fig. 8b was 1-100 s-' and back to 1 s~*. It becomes apparent that the history of experienced shear (time and maximum shear rate) has an impact on current AS apparent viscosity.
Fig. 7 — Comparison of shear-rate and shear-stress controlled measurement protocol with an MBR sludge with 5.8 g/L, a) apparent viscosity curves, b) flow curves; measured in an Anton Paar rheometer type MCR101 with double gap cylinder (0.5 mm gap size) and air bearing (Data: Osnabriick University of Applied Sciences). Fig. 6 — Shear rate vs. shear stress of an MBR AS for two MLSS-concentrations measured with the Bohlin rheometer type CVO50 (gap 1.25 mm) and with the TA rheometer type AR2000 (gap 1 mm). (Data: Ghent University).
Fig. 7 — Comparison of shear-rate and shear-stress controlled measurement protocol with an MBR sludge with 5.8 g/L, a) apparent viscosity curves, b) flow curves; measured in an Anton Paar rheometer type MCR101 with double gap cylinder (0.5 mm gap size) and air bearing (Data: Osnabriick University of Applied Sciences). Fig. 6 — Shear rate vs. shear stress of an MBR AS for two MLSS-concentrations measured with the Bohlin rheometer type CVO50 (gap 1.25 mm) and with the TA rheometer type AR2000 (gap 1 mm). (Data: Ghent University).
Fig. 8 — Apparent viscosity curves of an MBR AS with 9.7 g/L. The same sample was measured a) with a shear rate ramp of 1—1000—1 and b) 1—100—1 s~* in an Anton Paar rheometer type MCR101 with double gap cylinder 0.5 mm gap size) and air bearing (Data: Osnabriick University of Applied Sciences).
Fig. 8 — Apparent viscosity curves of an MBR AS with 9.7 g/L. The same sample was measured a) with a shear rate ramp of 1—1000—1 and b) 1—100—1 s~* in an Anton Paar rheometer type MCR101 with double gap cylinder 0.5 mm gap size) and air bearing (Data: Osnabriick University of Applied Sciences).
Fig. 9 — Different steps in a general modelling scheme. models. Hence, there is not such a thing as “the best model” to describe a certain system. Next, the framework has to be set up (type of equations, system boundaries, level of detail, ...) to build the model structure and a correct numerical solution needs to be pursued, also a potential source of errors (Biirger et al., 2011). If several candidate models are available, one has to select the most appropriate model, using the model selection techniques at hand (Dochain and Vanrolleghem, 2001). Depending on the selection technique used, the model parameters need to be estimated a priori or a posteriori. Note that all steps, except for the goal definition, require knowledge and/or experimental data, which need to be collected with care since ‘garbage in leads to garbage out’. Finally, because the entire modelling process exhibits several dangers and pitfalls (overparameterization, unrepresentative calibration data, parameter identifiability issues, low data quality,...), itis important that the model is properly validated after its
Fig. 9 — Different steps in a general modelling scheme. models. Hence, there is not such a thing as “the best model” to describe a certain system. Next, the framework has to be set up (type of equations, system boundaries, level of detail, ...) to build the model structure and a correct numerical solution needs to be pursued, also a potential source of errors (Biirger et al., 2011). If several candidate models are available, one has to select the most appropriate model, using the model selection techniques at hand (Dochain and Vanrolleghem, 2001). Depending on the selection technique used, the model parameters need to be estimated a priori or a posteriori. Note that all steps, except for the goal definition, require knowledge and/or experimental data, which need to be collected with care since ‘garbage in leads to garbage out’. Finally, because the entire modelling process exhibits several dangers and pitfalls (overparameterization, unrepresentative calibration data, parameter identifiability issues, low data quality,...), itis important that the model is properly validated after its
Fig. 11 — Illustration of the inappropriateness of only using R? as a model selection criterion: both the Bingham and Ostwald solution provide an equal R? value, but the data clearly show a cavity: residuals of the linear model are not equally distributed (Laera et al., 2007).
Fig. 11 — Illustration of the inappropriateness of only using R? as a model selection criterion: both the Bingham and Ostwald solution provide an equal R? value, but the data clearly show a cavity: residuals of the linear model are not equally distributed (Laera et al., 2007).
Fig. 12 — Mean and standard deviation of parameter estimates for to and k of the Herschel—Bulkley model with variable n (a and b) and fixed n (c and d) (Petersen et al., 2008). All of the problems above can be detected when the model is exposed to a validation data set, and its performance is eval- uated based on a predefined criterion. This step should not be forgotten in the modelling process, since it can test the predictive power of the model and point out flaws for which one needs to iterate back to previous steps. However, in From the same parameter estimation variance—covariance matrix, the correlation between parameters can be derived. Highly correlated parameters should never be estimated simultaneously. This is clearly shown in Petersen et al. (2008), where the Herschel—Bulkley model was fitted to the data but showed highly correlated parameters. Since the flow behaviour
Fig. 12 — Mean and standard deviation of parameter estimates for to and k of the Herschel—Bulkley model with variable n (a and b) and fixed n (c and d) (Petersen et al., 2008). All of the problems above can be detected when the model is exposed to a validation data set, and its performance is eval- uated based on a predefined criterion. This step should not be forgotten in the modelling process, since it can test the predictive power of the model and point out flaws for which one needs to iterate back to previous steps. However, in From the same parameter estimation variance—covariance matrix, the correlation between parameters can be derived. Highly correlated parameters should never be estimated simultaneously. This is clearly shown in Petersen et al. (2008), where the Herschel—Bulkley model was fitted to the data but showed highly correlated parameters. Since the flow behaviour
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EngineeringMedicineMembrane BioreactorsReproducibility of ResultsApparent Viscosity
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