End Correction. It is important to account for the influence of the bottom of the cylinder on the torque response of the system. This surface is in contact with the fluid but not taken into account in the force balance given by Eq. [3.1].
To determine the end correction, torque (or instrument scale divi-sion) is measured at a fixed rate of rotation when the annulus is filled to various heights (Fig. 3.5). Resulting data are plotted (Fig. 3.6) as torque versus the height of fluid in contact with the immersed length of the bob. The curve should be linear with the slope equal to the torque required to maintain the fixed rate of rotation per unit length of cylinder.
Effective height ( ) is determined from the intercept by extrapolating to a value of zero torque (Fig. 3.6). This technique is illustrated in Example Problem 3.8.14 for the tapered bob of a Hercules high-shear viscometer.
Figure 3.5. Illustration of values used in determining end correction.
Figure 3.6. End correction for a concentric cylinder system using a graphical technique to determine .
Effective height values are used in the previous equations developed for concentric cylinder systems. The Margules equation (Eq. [3.21]), for example, would be expressed as
h h h
1
2
3
h
-10 0 10 20 30 40 50
TORQUE, SCALE UNITS
x
x
x
-h0
Height of Bob in Contact with Fluid, mm
ho
[3.73]
The value of given in Eq. [3.1] is replaced by which, together, may be thought of as the effective height of the bob. An end correction calculated for a particular bob with a standard Newtonian fluid provides a general approximation for To obtain maximum accuracy, the end correction should be evaluated for each fluid and rotational speed under consideration. This procedure, however, is very laborious and not considered standard practice.
The end correction can also be evaluated in terms of an equivalent torque ( ) generated by a fluid in contact with the bottom of the sensor.
This idea is illustrated for three different speeds in Fig. 3.7. can be plotted as a function of to determine the relationship between the two parameters. The torque correction is subtracted from the measured torque in calculating the shear stress at the bob:
[3.74]
Correcting for end effects with or should yield identical results.
Various bob designs have been developed to minimize end effects.
Bobs can be made with a reservoir at the top and a recessed bottom.
This bob design, shown in Fig. 3.1, is based on a German standard (DIN 53018) developed by the German Institute for Standardization (Deutsches Institut für Normung). End effect problems can also be reduced by designing the bottom with a slight angle (called a Mooney-Couette bob, Fig. 3.8) in an effort to make the shear rate at the bottom equivalent to the shear rate in the annulus. The proper angle ( ) can be calculated by equating the annular shear rate to the shear rate in the gap (see Example Problem 3.8.15). Problems with large particulates and sensor alignment limit the usefulness of Mooney-Couette systems.
Other methods, such as using a mercury interface at the bottom of the bob (Princen, 1986), have also been proposed.
Ω = M
4πµ(h+ho)
1 Rb2− 1
Rc2
h h+ho
ho
Me
Me Ω
σb=M−Me 2πhRb
2
Me ho
θ
Figure 3.7. End correction for a concentric cylinder system using a graphical technique to determine a torque correction at different speeds.
Figure 3.8. Mooney-Couette bob design.
Viscous Heating. Temperature increase in a fluid during rheological testing can be caused by the viscous generation of heat. It may be a serious problem in some experiments because rheological properties are strongly influenced by temperature. The purpose of this section is to provide a means of determining if a significant temperature increase
1 2
e
e Torque Due to Bottom of Bob at
3
M
M 1
h1 h2 h3
Height of Fluid in Contact with Bob
Torque
1
may occur during testing. If the problem is serious (it will always exist) appropriate action must be taken. Most viscometers are designed with effective temperature control systems that minimize viscous heating problems by rapidly removing the excess heat generated during testing.
To address viscous heating, the case of uniform shearing between parallel plates (Fig. 1.9) may be considered (Dealy, 1982). A concentric cylinder system can be approximated using this idea when the gap is narrow ( ). This is a one dimensional problem where it is assumed that plates are separated by a distance , with at the bottom plate and at the top plate. Also, assume the fluid is Newtonian with a viscosity that does not vary with temperature. In this case the differential equation relating temperature and location, under steady state conditions, is
[3.75]
where is the fluid thermal conductivity. is the viscous energy generated per unit time per unit volume expressed in units of J s-1m-3. Shear rate is considered to be uniform throughout the gap. The solution to Eq. [3.75] is
[3.76]
where and are constants which depend on the boundary conditions of the problem being considered. Solutions for two different cases follow.
The various scenarios should be visualized in terms of the propensity of heat to move through the cup and (or) the bob surfaces during shearing.
Both surfaces are maintained at the same temperature ( ). In this situation the boundary conditions are at , and at , which allow the constants in Eq. [3.76] to be determined: , . Substituting these values back into Eq. [3.76] allows the temperature to be expressed as a function of position between the plates:
[3.77]
Rc−Rb Rb
s x2=0 x2=s
kd2T
dx22= − µ(˙γ)2
µ(γ)˙2 k
T=f(x2) = −µ(γ)˙2
2k x22+C1x2+C2 C1 C2
To
T=To x2=0 T=To x2=s C2=To C1= µ(γ)˙ 2s/(2k)
T=f(x2) =To+µ(γ)˙2x2 2k (s−x2)
meaning temperature distribution is parabolic and the maximum temperature occurs at the midplane:
[3.78]
Hence, the temperature rise in the gap is equal to indicating that minimizing the size of the gap ( ), analogous to having a smaller value of , is beneficial in reducing viscous heating problems.
One surface is adiabatic and the other surface is maintained at . In this case the boundary conditions are at and at allowing determination of the constants: and . Substitution of these values into Eq. [3.76] yields the temperature distribution function:
[3.79]
The distribution is parabolic with the maximum temperature occurring at the adiabatic surface where :
[3.80]
Comparing this result to the case where both surfaces are maintained at indicates the temperature variation in a sample may be four times greater when one surface is considered adiabatic. Eq. [3.80] has been applied to tomato ketchup in Example Problem 3.8.16.
Effect of temperature variation on viscosity. Problems associated with viscous heating will depend on the extent to which rheological properties are sensitive to temperature. Dealy (1982) gave an example in which viscosity was expressed as an exponential function of temperature:
[3.81]
where is the viscosity at and is a constant, numerically dependent on the fluid in question. Assuming is the wall temperature, the ratio of the maximum to the minimum viscosity is
[3.83]
Considering the case when both walls are maintained at , Eq. [3.78]
can be substituted into Eq. [3.83] yielding:
[3.84]
Meaning, for example, if the variation in viscosity during testing is to be less than 10%, then
[3.85]
making it necessary to maintain the following inequality:
[3.86]
In the situation where one wall is maintained at and the other wall is adiabatic, the viscosity ratio may be evaluated by combining Eq. [3.83]
and [3.80]:
[3.87]
To maintain a viscosity variation of less than 10%
[3.88]
Although the above calculations are not quantitatively exact for non-Newtonian fluids, they do illustrate the relative importance of different experimental variables. Analytical solutions for power law fluids in couette flow -where the consistency coefficient is expressed as a power series of temperature and the flow behavior index is assumed to be independent of temperature- are cumbersome, but available (Middleman, 1968).
Wall Effects (Slip). Wall effects due to separation in multiphase materials may cause errors in concentric cylinder systems similar to those discussed for tube viscometers in Sec. 2.5. Oldroyd (1956) sug-gested that slip may be considered in terms of the general expression for angular velocity (Eq. [3.16]) by adding a slip velocity ( ) that is a function of wall shear stress at the bob and the cup:
[3.89]
In the absence of slip, the slip velocity is zero and this equation reduces to Eq. [3.16]. Using the method of Mooney (1931), it is possible to correct for slip in concentric cylinder viscometers. The method requires numerous bobs because measurements are required at different values
of .
A simple slip evaluation method, requiring two series of measure-ments in two different measuring sets, that have different gap widths, has also been suggested (Kiljanski, 1989). Cheng and Parker (1976) presented a method of determining wall-slip based on the use of a smooth and a rough bob. They also urged caution in investigating slip because fluids exhibiting that phenomenon may have accompanying particulate behavior which may mask slip and complicate data treatment. A related procedure was proposed by Yoshimura and Prud’homme (1988). If slip is a serious problem, mixer viscometry should be evaluated as an alternate experimental method.
The Mooney technique was used by Qiu and Rao (1989) to evaluate slip in apple sauce, a typical food dispersion of solid particles in a liquid.
This work represents one of the few thorough studies dealing with slip in a food product. The investigators found that the wall slip correction did not significantly influence the flow behavior index (the average value for applesauce was 0.253), but increased the consistency coefficient: An average consistency coefficient equal to 37.53 Pa snwas found for typical applesauce and the slip correction caused this value to increase by an average of 5% to a value of 39.40 Pa sn. Qui and Rao (1989) also made a very interesting observation when they said "Due to the fortuitous opposite effects of correction for non-Newtonian behavior (it increases the magnitudes of shear rates) and correction for slip effects (it decreases the magnitudes of shear rates), it appears that for food suspensions Newtonian shear rates uncorrected for slip may be closer to the shear
us
Ω = −1 2
⌠⌡σb σc
f(σ)dσ
σ +(us)bob Rb +(us)cup
Rc
Rc/Rb
rates corrected for both non-Newtonian behavior and for wall slip." They cautioned that this conjecture should be verified before applying it to any particular product.
Secondary Flow. Equations developed for the analysis of rheological data assume that the streamlines are circular, i.e., flow is laminar.
When an inner cylinder rotates in a concentric cylinder system, the fluid near the inner surface tries to move outward due to centrifugal forces.
This movement may create non-streamline flow due to the presence of
"Taylor vortices" (G.I. Taylor (1923. Phil. Trans. Roy. Soc. (London), Ser.
A 223: 289). Such vortices may occur for Newtonian fluids when (Whorlow, 1992)
[3.90]
In a Couette type system where the outer surface (the cup) is rotated, the inertial forces have a stabilizing effect and flow is laminar at much higher shear rates. Consult Larson (1992) for a detailed analysis of flow instabilities in concentric cylinder systems. Application of Eq. [3.90] is presented in Example Problem 3.8.6 and 3.8.17.
Cavitation. The formation and collapse of vapor cavities, known as cavitation, may occur in a high shear environment when the radial pressure drop is sufficient to cause partial vaporization of the sample.
By considering the Bernoulli equation in terms of the mechanical energy balance (Eq. [2.105]), one finds (Sakiadis, 1984) that cavitation will occur when where is the linear velocity of the bob or cup, whichever is greater. The left hand side of the equation ( ) is related to the pressure drop ( ) across the gap: . If present, cavi-tation may cause erroneous torque responses in a concentric cylinder viscometer. Cavitation is not a significant problem in food rheology because it is usually not present when laminar flow conditions are maintained. The cavitation problem is examined in Example Problem 3.8.17.