Abstract
The recently proposed short back extrusion (SBE) method is an improved immersed-type BE method. A translational concentric cylinder rheometer is used. As the measurement position is inside the sample, the upflow in the annular space is smooth even over a short distance (5 to 15 mm). Measurement over a short distance decreases the amount of sample adhering to the plunger; this facilitates repeated measurements, as the removal of the adhering sample after each measurement is not needed. The rheometer analysis program can perform automated Newtonian, power-law, and Herschel–Bulkley flow analyses by introducing novel mathematical solutions and obtaining constitutive equations for the various flow types. Herschel–Bulkley fluids exhibit semisolid properties and thixotropic flow characteristics, specifically, stress growth or time-dependent behavior. Currently, SBE viscometry is the only available method to evaluate viscosity and yield stress simply and simultaneously and this study presented the results compared with rotational cone-plate viscometry.
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Data availability
The datasets analyzed during the current study are available from the author on reasonable request.
Abbreviations
- A :
-
Upflow position of the liquid, m
- AB:
-
Distance traveled by the plunger (AB equals L1), m
- C :
-
Initial position of the plunger bottom, m
- D :
-
Travelled position of the plunger bottom, m
- dγ/dt :
-
Shear rate, s−1
- F b :
-
Buoyancy force, N
- F cb :
-
Force corrected for buoyancy, N
- F T :
-
Maximum force just before the plunger is stopped, N
- F Te :
-
Convergence force after the plunger is stopped, N
- F 0 :
-
Yield force applied to the plunger, N
- g :
-
Acceleration due to gravity, m s−2
- K :
-
Consistency coefficient, Pa sn
- L :
-
Distance of the dipped plunger, m
- L o :
-
Initial dipped distance of the plunger, m
- L 1 :
-
Distance traveled by the plunger (AB), m
- L b :
-
Distance between the plunger bottom and the cup bottom, m
- n :
-
Flow behavior index, dimensionless
- n a :
-
Assumed flow behavior index, dimensionless
- O :
-
Initial fluid level before the plunger is forced down into the sample, m
- OB:
-
Distance traveled by the plunger, m
- P :
-
Pressure drop per unit of length, Pa m−1
- r :
-
Radial coordinate, m
- R i :
-
Radius of the plunger (equals κ × Ro), m
- R o :
-
Radius of the cup, m
- s :
-
Reciprocal of n, dimensionless
- T :
-
Dimensionless shear stress, dimensionless
- T 0 :
-
Dimensionless yield stress, dimensionless
- T w :
-
Dimensionless shear stress at the plunger wall, dimensionless
- v p :
-
Velocity of the plunger, m s−1
- α 1 :
-
Morgan’s geometric constant, dimensionless
- α 2 :
-
Suzuki’s geometric constant, dimensionless
- ΔL :
-
Distance traveled by the plunger (equals OB), m
- κ :
-
Ratio of Ri to Ro (i.e., Ri/Ro), dimensionless
- λ :
-
Value of the dimensionless radial coordinate ξ, for which the shear stress is zero, dimensionless
- λ−, λ+ :
-
Limits of the plug flow region in Herschel–Bulkley flow, dimensionless
- μ :
-
Viscosity, Pa s
- μ a :
-
Apparent viscosity, Pa s
- ρ :
-
Sample density, kg m−3
- σ :
-
Shear stress, Pa
- σ 0 :
-
Yield stress at the plunger wall, Pa
- σ rz :
-
Shear stress of the rz–component, Pa
- σ m :
-
Maximum shear stress in rotational cone-plate viscometry, Pa
- σ S :
-
Steady-state shear stress in rotational cone-plate viscometry, Pa
- σ w :
-
Shear stress at the plunger wall in SBE viscometry, Pa
- φ :
-
Dimensionless velocity, dimensionless
- φ p :
-
Dimensionless velocity at the plunger wall, dimensionless
- Φ :
-
Dimensionless flow rate, dimensionless
- ξ :
-
Dimensionless radial coordinate, dimensionless
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The author would like to thank Editage (www.editage.jp) for the English language editing services provided.
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All rights of the “SBE pro” (Fujitsu Ltd. (2015) “SBE Pro” for Aohata Corporation) program code and software were preserved by Fujitsu Ltd.
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Appendix. Result comparison using the root mean square error
Appendix. Result comparison using the root mean square error
The dispersion or difference between the experimental values σi and reference values \( \overline{\upsigma} \) was investigated using the RMSE, which is defined below.
The differences between the sums of the experimental values σi and the reference \( \overline{\upsigma} \) were also assessed. Positive and negative RMSEs indicated values above and below the ideal line, respectively.
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Hoshino, T. Analysis of the flow properties of a Herschel–Bulkley fluid using short back extrusion viscometry and considering time-dependent and stress growth behaviors. Rheol Acta 59, 809–819 (2020). https://doi.org/10.1007/s00397-020-01243-3
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DOI: https://doi.org/10.1007/s00397-020-01243-3