Mathematical modeling

Also, the Herchel-Bulkley model indicated that reactor fluid A performed as a pseudo­Newtonian fluid called Bingham plastic, since the yield stress-value was > 0 (0.24 Pa) and a flow behaviour index of 1.06 (Table 4). Results obtained by the Ostwald and Bingham models confirmed a Bingham plastic behaviour of reactor A. However, since the X0-value was almost 0 and the n-value 1 it was also closely performing as a Newtonian fluid which is consistent with the flow curve appearance (Fig. 2). However, when studying the viscosity curve (Fig. 4) the results showed an initial viscosity decrease and then a constant viscosity indicating a pseudo-Newtonian fluid behaviour.

The Herschel-Bulkley and Ostwald models both indicated a pseudoplastic behaviour of reactor D, since the X0-value was 0 and n < 1 (Table 4). The Bingham model gave a yield stress of 0.33 Pa which did not indicate Newtonian or Bingham plastic behaviour. Thus, the common results for reactor D strongest indicate a pseudoplastic fluid behaviour.

Reactor B was hard to define also when modelling the rheogram data values of figure 4. The regression values were low for all three mathematical models (Table 4). However, the Herschel-Bulkely model had a flow behaviour index n>1 indicating that the fluid acted as a shear thickening (dilatant) fluid, but the Ostwald and Bingham models indicated pseudoplastic and Bingham plastic behaviours, respectively. When the static yield stress appeared in the reactor B rheogram (Figures 2 and 4), the flow behaviour index showed shear thickening fluid behaviour (n=3.4) and a limit viscosity of 8 mPa*s. This also corresponded to a low consistency value (5*1010). At the static yield stress of 24 Pa (Fig. 5), the flow behaviour index showed shear thickening fluid behaviour (n=1.41) and a limit viscosity of 22 mPa*s. This also corresponded to a low consistency value (5*10-4). As soon as the fluid was measured again, n decreased (0.70) showing a pseduoplastic behaviour and K increased (0.11) indicating that the consistency of the reactor material was higher. The limit viscosity was 17 mPa*s. These results showing a time dependency and structure recovery strengthen the arguments for a thixotropic fluid behaviour of reactor B. Once the stirring has ended and the fluid was at rest, the fluid structure starts to rebuild. Therefore, the viscosity become time dependent. This information is important to consider for biogas reactor performance, e. g. when applying semi-continuous mixing.

Herschel-Bulkley

Ostwald

Bingham

t0

n

K

R2

n

K

R2

t0

R2

A

0.24

1.06

0.003

0.93

0.69

0.35

0.84

0.21

0.92

B

2.57

3.40

5*10-10

0.45

0.08

2.28

0.002

1.88

0.12

C

2.89

0.59

0.42

0.99

0.44

1.23

0.99

6.36

0.95

D

0

0.65

0.04

0.88

0.64

0.04

0.87

0.33

0.95

E

2.38

0.49

0.98

0.96

0.39

1.98

0.96

8.31

0.91

Table 4. The results obtained from mathematical modelling of rheogram data of fluids from reactors A-E. t0: yield stress (Pa); n: flow behaviour index; K: Consistency index; R2: regression coefficient.

Also, fluids from reactor C and E were hard to define from modelling of the rheogram data because they gave indications for fluids being between pseudoplastic and Bingham plastic behaviours, i. e. the t0-values were >0 (2.89 and 2.38) and n <1 (Table 4).

2. Conclusion

The biogas reactor fluids investigated were behaving viscoplasticly, since they had yield stress and one of them was also thixotropic, due to its partial structure recovering. However, the reactor treating slaughterhouse waste was very close to act as a Newtonian fluid. Also, there was a difference in dynamic — and limit viscosities depending on the substrates used. The results demonstrated that similar TS values did not necessarily correspond to similar flow and viscosity behaviours. Nor, did biosludge from two different Swedish paper mill industries with similar TS show similar viscosity values.

To encounter problems related to involvement of new substrates and/or co-digestions in existing facilities, investigations for possible viscosity changes are needed. Ongoing research will hopefully provide an important basis for predictions of changes in rheology linked to the composition of the organic materials, which are translated in the process. This is important in order to achieve proper designs in relation to possible variation in substrate mixes in conjunction with new constructions, but also to better control material flows in the existing facilities to avoid disturbances in the reactor performance.