End effects in rotational viscometry I. No-slip shear-thinning samples in the Z40 DIN sensor

Ondrej Wein; Marek Vecer; Jaromir Havlica

doi:10.1007/s00397-007-0180-0

Rheologica Acta

May 2007, Volume 46, Issue 5, pp 765–772 | Cite as

End effects in rotational viscometry I. No-slip shear-thinning samples in the Z40 DIN sensor

Authors
Authors and affiliations

Ondrej WeinEmail author
Marek Vecer
Jaromir Havlica

Original Contribution

First Online: 19 April 2007

Received: 31 May 2006

Accepted: 23 February 2007

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6 Citations

Abstract

Neglect of end effects in Couette rotational viscometry introduces a 10–30% error in the estimate of shear stress at the spindle surface. Actual deviations depend on the shear-thinning level of a given sample. We tackle the end effect for the standard sensor Z40 DIN according to the ISO 3219 by solving the related 2D boundary-value problem for a class of shear-thinning viscosity functions. The pseudosimilarity method of treating the primary data leaves an error of about 0.5% in shear stresses. Further reduction in the errors needs a full numerical simulation for each point of the primary data based on a suitable wide-range representation of the viscosity function. To support a high accuracy of torque calibrations, the effect of inertia on torque for Newtonian liquids in standard sensor Z40 DIN at Re < 500 is calculated using the FLUENT 6.2 commercial software.

Keywords

Rotational Couette viscometry Shear-thinning fluids End effects Inertia effects

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Notes

Acknowledgment

This work was supported by the Grant Agency of the Czech Republic under contract nos. 104/04/0826, 104/06/P287, and 104/05/P554.

List of symbols

C: total torque, Pa m³
c _L: correction on end effects, c _L = T / T _L, see also Eq. 2
D = d / R: normalized distance between a rim and an axial border, Fig. 1
E = η _∞ / η ₀: viscosity ratio in the normalized Cross model, Eq. 22
G = |∇ V| = [(∂_X V)² + (∂_Z V)²]^1/2: normalized module of velocity gradient
h: gap thickness, m
H = h/R: normalized gap thickness, Fig. 1
K: consistency coefficient for the power-law viscosity function, Pa sⁿ
k _H: estimate of the transient region length within the gap, Fig. 1
l: length of the gap between coaxial cylinders, m
L = l / R: normalized length of the gap, Fig. 1
n: global flow index for the power-law viscosity function
n′: local flow-behavior index, Eq. 4
N: intermediate flow index in the normalized Cross model, Eq. 22
\( q = {{\mathop \gamma \limits^. }} \mathord{\left/ {\vphantom {{{\mathop \gamma \limits^. }} {{\mathop \gamma \limits^. }_{I} }}} \right. \kern-\nulldelimiterspace} {{\mathop \gamma \limits^. }_{I} } \): normalized shear rate
Q = R _shaft/R: normalized shaft radius, Fig. 1
R: radius of the spindle (inner rotating cylinder), m
Re = ΩR ²/ν: Reynolds number, ν -kinematic viscosity, m² s⁻¹
s = σ / σ _I: normalized shear stress
s[q] = Y[q] q: normalized viscosity function
S = σ _R / σ _I: normalized shear stress at the spindle surface for ideal Couette flow
T = C / 2πσ _I R ³: normalized torque
T _L: T for ideal Couette flow, T _L = S L
T _k,p: contributions to T (k = C , G , A; p = srf, btm); Fig. 1, Eq. 10
V = ω / Ω: normalized velocity field
\( W = \Omega \mathord{\left/ {\vphantom {\Omega {{\mathop {\gamma _{I} }\limits^. }}}} \right. \kern-\nulldelimiterspace} {{\mathop {\gamma _{I} }\limits^. }} \): normalized speed
X = r/R, Z = z/R: normalized meridional coordinates
\( Y = {\eta {\mathop \gamma \limits^. }_{I} } \mathord{\left/ {\vphantom {{\eta {\mathop \gamma \limits^. }_{I} } {\sigma _{I} }}} \right. \kern-\nulldelimiterspace} {\sigma _{I} } \): normalized viscosity
Y[q] = s[q] / q.: normalized viscosity function
α: angle of the spindle cone, Fig. 1
\( {\mathop \gamma \limits^. } \): viscometric shear rate, s⁻¹
\( {\mathop {\gamma _{I} }\limits^. } \): characteristic shear rate (a model parameter), s⁻¹
Γ: spindle contour in normalized meridional plane (X, Z)
\( \eta = \sigma \mathord{\left/ {\vphantom {\sigma {{\mathop \gamma \limits^. }}}} \right. \kern-\nulldelimiterspace} {{\mathop \gamma \limits^. }} \): viscosity, Pa s
κ = R / (h + R): ratio of radii of the inner to outer cylinder
σ: viscometric shear stress, Pa
σ _I: characteristic shear stress (a model parameter), Pa
σ _R: shear stress at the spindle surface for ideal Couette flow, Pa, Eq. (2)
ω(r, z): meridional field of angular speed in liquid, rad s⁻¹
Ω: angular speed of the spindle, rad s⁻¹
[..]: any representation of the viscosity function
Subscripts, superscripts
L, C, G, A: contours of the sensor in the (X, Y) plane, Fig. 1: ideaL gap, Cylinder faces, Gap entrance, Axis or shAft;
srf, btm: dividing the overall domain at the surface and bottom parts, Fig. 1

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Authors and Affiliations

Ondrej Wein
- 1
Email author
Marek Vecer
- 1
Jaromir Havlica
- 1

1.Institute of Chemical Process Fundamentals ASCR, v.v.i.PragueCzech Republic

End effects in rotational viscometry I. No-slip shear-thinning samples in the Z40 DIN sensor

Abstract

Keywords

Notes

Acknowledgment

References

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