Supplementary Material for Paper entitled “The Critical Flux Method for Reduced
Filter Membrane Fouling when Monitoring High-Solids Digesters”
Stephan Tait, Christopher R. Carney, and Damien J. Batstone
Advanced Water Management Centre, The University of Queensland, QLD, 4072, Australia
S.1 Data from the rheology tests
Figure S1 presents data from the rheological experiments on both the test sludges. All the
sludge samples showed a predominantly elastic response. That is, the elastic modulus G’ was
significantly higher than the viscous modulus G”. The ratio G’:G’’ ranged from five for the
lowest solids concentration sample to 10 for the highest solids concentration sample. All the
samples were strongly shear-thinning, i.e. viscosity decreased strongly with increasing shear
rates. Most importantly, the measured viscosity of the THP-sludge was much higher than that
of the co-digested sludge and viscosity of the co-digested sludge substantially increased with
up-concentration from 2% total solids to 4% total solids (approximately 10 times increase in
viscosity).
S.2 Calculations to Determine Prevailing Flow Regimes
Calculations were performed to determine prevailing flow regime for the test sludges flowing
at an average pipe-flow velocity of 1m.s-1 through the four 0.025m diameter tubes of the filter
unit used in the present study. The sludge viscosities measured and reported in this paper
were comparable to that reported previously by Esthiaghi et al. for their tests on anaerobic
digester sludges.1 The data of Esthiaghi et al. covered a significantly broader range of shear
rates and were thus preferred and were used instead during the further analysis in the present
study.1 This literature data was fitted with a Herschel-Bulkley model as described by
Skelland: 2
𝑑𝑢
1/𝑚
𝜏𝑟𝑥 − 𝜏𝑦 = [𝜂 (− 𝑑𝑟 )]
(Equation S1)
where the yield stress (τy) was as provided by Esthiaghi et al., and the stress (τrx) and strain
rate (-du/dr) flow curve was plotted with the Esthiaghi et al. data.1 The parameters m and η
were obtained from the slope and intercept of a logarithmic plot of (-du/dr) versus (τrx–τy).
The Herschel-Bulkley model parameters were then used to calculate the flow characteristic
8V/D from the following relationship:2
8𝑉
𝐷
4
= 𝜂𝜏
𝑤
3
(𝜏𝑤 − 𝜏𝑦 )𝑚+1 [
(𝜏𝑤 −𝜏𝑦 )2
𝑚+3
+
2𝜏𝑦 (𝜏𝑤 −𝜏𝑦 )
𝑚+2
𝜏𝑦 2
+ 𝑚+1]
(Equation S2)
for a nominal wall shear stress (τw). Here, V is the mean pipe flow velocity and D is the pipe
diameter.
Figure S1. Example data for rheological measurements, including strain sweeps at a constant
frequency (10 rad.s-1) and frequency sweeps at a constant amplitude for (a) codigested sludge with 2% total solids by wet mass, (b) co-digested sludge upconcentrated to 4.0% total solids, and (c) digestate of thermally hydrolyzed waste
activated sludge (THP-sludge) at 3.3% total solids.
Frequency sweeps were performed at an amplitude of 1% for (a), (b) and (c) and 0.1%
for (d). Measurements for (a) and (b) were performed at 38°C, and for (c) at 37°C. At the
very lowest strain amplitudes for (a), the measured torque fell outside of the validity
range for the instrument, so data are not presented at these low strain amplitudes. The
linear visco-elastic region extended to around 1% strain amplitude for the co-digested
sludge samples, and 0.1% for the THP-sludge, and these upper limits were selected for
the frequency sweeps at constant amplitude, in order to maximise the measured
torque/accuracy of the results. Note the logarithmic axes.
Continuation of S.2.: The relationship between τw and 8V/D in-turn provided K’ and n’ as
fitting parameters for the relationship:2
8𝑉 𝑛′
𝜏𝑤 = 𝐾′ ( 𝐷 )
(Equation S3)
where n’ was estimated from:2
𝑛′ =
𝑑 ln(𝜏𝑤 )
8𝑉
)
𝐷
(Equation S4)
𝑑 ln(
and K’ was determined for each value of τw.2 The generalized Reynold’s number (NRe,gen):2
𝑁𝑅𝑒,𝑔𝑒𝑛 =
𝐷 𝑛′ 𝑉 2−𝑛′ 𝜌
′
𝐾′8𝑛 −1
(Equation S5)
was then calculated for each pair of V, K’ and n’ and the value of V corresponding to a NRe,gen
= 2100 was taken to be the transition velocity from laminar flow to transition-to-turbulent.2
The sludge matrix density, ρ, was assumed to be equal to that of liquid water (1000 kg/m3).
Table S1 illustrates the calculation sequence for the sludge tested by Esthiaghi et al. which
had the lowest solids concentration (3.17% wet basis, τy=1.04 Pa).1
Table S1. Illustrated calculations using rheology data of
Esthiaghi et al. measured for an anaerobic digester sludge of
solids content 3.17% wet basis and an estimated yield stress
of τy=1.04 Pa.1
Shear rate
-du/dr (s-1)
ln(-du/dr)
Shear stress
(Pa)
ln(τrx–τy)
10
2.30
1.20
-1.83
20
3.00
1.32
-1.26
30
3.40
1.44
-0.92
40
3.69
1.55
-0.68
50
3.91
1.65
-0.50
105
4.65
2.16
0.11
160
5.08
2.63
0.46
320
5.77
3.86
1.04
490
6.19
5.05
1.39
Figure S1 presents a logarithmic plot of (-du/dr) versus (τrx–τy) for the data of Table S1. The
parameters m and η were determined from the slope and intercept shown in the figure.
Figure S1. A logarithmic plot of (-du/dr) versus (τrx–τy) for data of a sludge tested by
Esthiaghi et al.,1 which had a solids concentration of 3.17% wet basis and
estimated yield stress, τy=1.04 Pa.
Table S2 presents the remainder of the calculations for the same sludge as Table S1 flowing
through the 25mm tubes of the filter unit used in the present work. The results suggested that
the transition velocity (Re=2100) for this sludge would be between 2.25 and 2.3m.s-1 (well
above the 1m.s-1 flow in the present study).
Table S2. Continued illustrated calculations for the rheology data of Esthiaghi et al.,1 for
an anaerobic digester sludge of solids content 3.17% wet basis and an estimated yield
stress of τy=1.04 Pa.
V
8V/D τw (Pa) ln(τw) ln(8V/D)
d ln(τw)
(8V/D) n'
K'
8V/Da
NRe,genb
(m/s)
(guess)
/d ln(8V/D)
(recalculated)
0.9
288
9.8
2.3
5.7
0.717
58
0.17
288
660
1
320
10.6
2.4
5.8
0.731
68
0.16
320
750
2.1
672
18.5
2.9
6.5
0.767
148
0.13
672
1910
2.2
704
19.1
3.0
6.6
0.769
154
0.12
704
2030
2.25
720
19.5
3.0
6.6
0.770
158
0.12
720
2080
2.3
736
19.8
3.0
6.6
0.771
162
0.12
736
2140
2.4
768
20.5
3.0
6.6
0.640
70
0.29
768
2250
a
Calculated with Equation S2
b
Calculated with Equation S5
However, co-digested sludge tested in the present work had a lower solids concentration (2%
total solids) than the sludge with the lowest solids content tested by Esthiaghi et al. (3.17%
total solids).1 To confirm the prevailing flow regime for the 2% solids co-digested sludge
flowing through the 25mm filter tubes, empirical correlations provided by Esthiaghi et al.
were used to estimate the relevant Herschel-Bulkley model parameters for a hypothetical
sludge with the same solids content as the co-digested sludge,1 and then the sequence of
calculations outlined above were repeated. Table S3 provides the Herschel-Bulkley model
parameters (K, n and τy) together with calculated generalized Reynold’s numbers for pipeflow
of all the sludges of Esthiaghi et al. and the 2% solids hypothetical sludge (argued to be
similar to the co-digested sludge) at 1m.s-1 flow velocity through the 25mmm filter tubes. All
the calculated Reynold’s numbers were sufficiently low (<<2100) to conclude that laminar
flow conditions prevailed during the filtration tests of the present study.
Table S3. Calculated general Reynold’s number values for the sludges tested
by Esthiaghi et al. 1, if they were to flow through a 25mm tube at 1m.s-1
Total
solids
(% weight
percent
wet basis)
τy
(Pa)
K
(Pa.sn)
n
NRe,gen
Lowest solids
concentrationa
3.17
1.04
0.0239
0.827
750
Medium solids
concentrationa
4.66
3.13
0.240
0.632
280
Highest solids
concentrationa
6.62
12.0
0.366
0.664
131
Hypothetical
Sludgeb
2
0.999
1086
Description
0.265 0.00548
a
Herschel-Bulkley parameters were as provided by Esthiaghi et al.1
b
Values for τy, K and n were calculated using the empirical correlations provided
by Esthiaghi et al.1 The flow curve was then estimated and the further analysis
was carried out for this hypothetical sludge.
References
1.
2.
Eshtiaghi, N.; Markis, F.; Slatter, P., The laminar/turbulent transition in a sludge pipeline.
Water Sci Technol 2012, 65, (4), 697-702.
Skelland, A. H. P., Non-Newtonian flow and heat transfer. Wiley: New York, 1967; p 469.