Theory of a Cone and Plate Viscometer for Non-Newtonian Liquids
1966 Volume 15 Issue 152 Pages 294-296
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1966 Volume 15 Issue 152 Pages 294-296
In a cone and plate viscometer a cone with a wide vertical angle is placed on a horizontal flat plate. The generator of the cone makes a very small angle α with the plate which is of the order of one degree to two degrees. The wedge-like space between the cone and the plate is filled with liquid to be investigated. Let the cone be fixed; the plate rotates with a constant angular velocity Ω around the axis of the cone. It is the purpose of this paper to find a general relationship between the torque M on the plate and the angular velocity Ω for a non-Newtonian liquid specified by an arbitrary flow curve.
The relationship between M and Ω for a Newtonian steady flow has already been presented by several authors. Thus the formula M=(2πa3/3)·(ηΩ/α) has been derived, where a is the radius of the plate and η is the coefficient of viscosity. One of the most important characteristics of a cone and plate viscometer lies in the fact that the rate of shear is practically constant throughout one sample. Thus this type of viscometer is quite suitable for the measurement of non-Newtonian liquids.
With regard to the motion of the liquid, the following assumptions are made: (1) the liquid is incompressible: (2) the motion of the liquid is laminar: (3) the motion is steady: (4) no body force acts on the liquid; (5) the motion has an axial symmetry; (6) each liquid particle moves on a circle on the horizontal plane perpendicular to the axis of rotation; (7) there is no relative motion between the walls and the liquid in immediate contact with the walls; (8) the end-effect is neglected. The assumption (9) means utter disregard of centrifugal forces, and as well as the assumption (2) is allowable for small values of Ω.
We take a spherical coordinate system r, θ and φ whose origin is at the vertex of the cone. If we assume that the angular velocity ω of a liquid particle around the axis of the cone is a function of θ alone, then the shear stress τθφ is given by τθφ=c/sin2θ, where c is a constant. For a non-Newtonian liquid specified by an arbitrary flow curve f(τ), we get
ω=1/2∫c/cos2αc/sin2θf(τ)/√τ(τ-c)dτ, Ω=1/2∫c/cos2αcf(τ)/√τ(τ-c)dτ
The constant c is related with the torque M on the plate by the relationship M=c·2πa3/3. For a special case of a non-Newtonian liquid obeying power law flow curve f(τ)=kτn the following formula has been obtained:
M=2πa3/3(Ω/kα)1/n
We have also studied another special case of a Bingham body specified by a plastic viscosity and a yield value. For a non-Newtonian liquid specified by a flow curve f(τ)=Σn=1Anτn, An can be determined from the experimental relationship between Ω and M.
