3.3. Chapter III. Human colon organoids as an assay platform for antitumor drugs
3.3.5. Drug - calcitriol interactions
giving
2.12.8. Turbulent Flow - Power Law Fluid
Determine the maximum velocity of a power law fluid in a pipe given
the following information: Pa sn; m; m/s;
kg/m3; .
The Reynolds number (Eq. [2.52]) is 1
µ =(.0348/2) (.1107)1250
.012 =200.64
allowing to be calculated (equation presented in Table 2.3) from:
yielding
Velocity is maximum at the center-line where . The friction velocity (Eq. [2.127]) is
is calculated using Eq. [2.132] as:
Then, is determined from Eq. [2.131]:
The maximum velocity is found from the definition (Eq. [2.125]) of the turbulent velocity ( ) as
f 1
√f =
4 n0.75
log10[(NRe,PL)f(1− (n/2))] −
0.4 n1.2
=
4 (.45)0.75
log10[(6736.6)f(1− (.45/2))] −
0.4 (.45)1.2
f=0.0052
y=R
u*=u
√
2f =1.66√
.00522 =.0846 m/sy+
y+=yn(u*)2−nρ/K= (.0348/2).45(.0846)2−.45(1250)/.25=17.6 u+
u+= 5.66
(.45).75log10(17.6) −0.566 (.45)1.2+3.475
(.45).75
1.960+0.815(.45) −1.628(.45)log10
3+ 1 (.45)
=22.75 u+=u/u*
umax=u+u*= (22.64).0846=1.92 m/s
3.1. Introduction
Traditional rotational viscometers include cone and plate, parallel plate, and concentric cylinder units operated under steady shear con-ditions Fig. 1.1). They may also be capable of operating in an oscillatory mode which will be considered in the discussion of viscoelasticity, Chapter 5. Cone and plate systems are sometimes capable of deter-mining normal stress differences. Concentric cylinder systems have been used in research to evaluate these differences (Padden and DeWitt, 1954); however, commercial instruments of this type are not available.
Mixer viscometry, a "less traditional" method in rotational viscometry, is also presented because it has excellent utility in solving many rheo-logical problems found in the food industry.
3.2. Concentric Cylinder Viscometry 3.2.1. Derivation of the Basic Equation
The concentric cylinder viscometer is a very common instrument that will operate in a moderate shear rate range making it a good choice for collecting data used in many engineering calculations. A number of assumptions are made in developing the mathematical relationships describing instrument performance: flow is laminar and steady, end effects are negligible, test fluid is incompressible, properties are not a function of pressure, temperature is constant, there is no slip at the walls of the instrument, and radial and axial velocity components are zero. The derivation presented here is based on a physical setup known as the Searle system where the bob rotates and the cup is stationary:
It is also applicable to a Couette-type system in which the cup rotates and the bob is stationary. Most concentric cylinder viscometers are Searle-type systems. Unfortunately, it is not uncommon for the word
"Couette" to be used in referring to any concentric cylinder system.
When the bob rotates at a constant speed and the cup is stationary (Fig. 3.1), the instrument measures the torque ( ) required to maintain a constant angular velocity of the bob ( ). The opposing torque comes from the shear stress exerted on the bob by the fluid. A force balance yields
[3.1]
M Ω
M=2πr hrσ =2πhr2σ
where is any location in the fluid, . Solving Eq. [3.1] for the shear stress shows that decreases in moving from the bob to the cup:
[3.2]
Utilizing Eq. [3.2], the shear stress at the bob ( ) can be defined as [3.3]
Figure 3.1. Typical concentric cylinder testing apparatus (based in DIN 53018) showing a bob with recessed top and bottom to minimize end effects.
To determine shear rate, consider the linear velocity at in terms of the angular velocity ( ) at :
[3.4]
The derivative of the velocity with respect to the radius is
[3.5]
r Rb≤r≤Rc
σ
σ =f(r) = M 2πhr2
r=Rb σb= M
2πhRb2
h
Rb Rc
r
ω r
u=rω
du dr =r dω
dr + ω
Since is related to the rotation of the entire body, it does not relate to internal shearing; therefore, Eq. [3.5] can be written as
[3.6]
Using the definition of shear rate developed in Eq. [2.9], may be defined in terms of :
[3.7]
To relate angular velocity to shear stress, note that torque is constant with steady flow so an expression for may be determined from Eq.
[3.2]:
[3.8]
Differentiating Eq. [3.8] with respect to the shear stress yields [3.9]
Substituting the value of torque defined by Eq. [3.1] into Eq. [3.9] gives [3.10]
or, with simplification,
[3.11]
The shear rate is some function of the shear stress, hence,
[3.12]
Solving Eq. [3.12] for the differential of the angular velocity yields [3.13]
which can be expressed in terms of by substituting Eq. [3.11] into Eq. [3.13]:
Integrating Eq. [3.14] over the fluid present in the annulus results in a general expression for the angular velocity of the bob ( ) as a function of the shear stress in the gap:
[3.15]
Note that the limits of integration are an expression of the no slip boundary condition assumed in the derivation: Angular velocity is zero at the cup (the stationary surface), and equal to at the bob (the moving surface). The left hand side of Eq. [3.15] is easily integrated resulting in the following equation relating angular velocity to shear stress:
[3.16]
The solution of Eq. [3.16] depends on which is dictated by the behavior of the fluid in question. It can be solved directly if the functional relationship between shear stress and shear rate is known. Eq. [3.15]
is used as the starting point in Example Problem 3.8.5 to find the velocity profile of a power law fluid in a concentric cylinder system.
Eq. [3.15] reflects a general solution for concentric cylinder vis-cometers because the limits of the integral could be easily changed to the case where the bob is stationary and the cup rotates (torque is equal in magnitude, but opposite in sign if measured on the cup) or even a situation where the bob and cup are both rotating. It is important to recognize the fact that Eq. [3.16] is analogous to the general solution (Eq. [2.15]) developed for tube viscometers. Both provide an overall starting point in developing mathematical relationships for specific types of fluids.
Application to Newtonian Fluids. The relationship between shear stress and shear rate for a Newtonian fluid is, by definition,
[3.17]
Substituting this into the general expression for given by Eq. [3.16]
yields
then,
[3.19]
Using Eq. [3.2] for shear stress allows Eq. [3.19] to be written in terms of the system geometry and the torque response of the instrument:
[3.20]
Rearrangement gives a simplified expression, called the Margules equation, describing the behavior of a Newtonian fluid in a concentric cylinder system:
[3.21]
This equation clearly indicates that experimental data for Newtonian fluids will show torque to be directly proportional to bob speed.
Application to Power Law Fluids. With a power law fluid, the relationship between shear stress and shear rate is
[3.22]
which can be substituted into Eq. [3.16] yielding
[3.23]
or, after integration,
[3.24]
Using Eq. [3.2], an alternative expression for the power law fluid is obtained:
[3.25]
Eq. [3.25] reveals that torque is not directly proportional to bob speed because it is strongly influenced by the flow behavior index.
Application to Bingham Plastic Fluids. A Bingham plastic fluid has the following relationship between shear stress and shear rate:
Ω = 1
[3.26]
Substituting Eq. [3.26] into Eq. [3.16] yields
[3.27]
Integration and substitution of Eq. [3.2] provides the general relation-ship (known as the Reiner-Riwlin equation) between the torque, angular velocity, and system geometry:
[3.28]
This equation is valid only when the yield stress is exceeded at all points in the fluid meaning that the minimum shear stress must greater than the yield stress:
[3.29]
is the minimum torque required to overcome the yield stress. If evaluating fluid behavior near the limits described by Eq. [3.29], the yield stress should be determined before conducting standard tests in Searle-type concentric cylinder viscometers. This can be accomplished with various techniques, such as the vane method discussed in Sec.
3.7.3, With that data, one can calculate the minimum rotational speed of the bob required to insure shearing throughout the cylindrical gap (see Example Problem 3.8.1). In Couette systems, applying sufficient torque to rotate the cup assures shear flow in the entire annulus because the minimum shear stress occurs at .