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Derivation of the Basic Equation. Numerous assumptions are required in developing the Rabinowitch-Mooney equation: flow is lam-inar and steady, end effects are negligible, fluid is incompressible, properties are not a function of pressure or time, temperature is constant, there is no slip at the wall of the tube meaning that the velocity of the fluid is zero at the wall-fluid interface, and radial and tangential velocity components are zero.

The starting point in the derivation of a tube viscometer equation is a force balance. Consider a fluid flowing through a horizontal tube of length ( ) and inside radius ( ). A pressure drop ( ) over a fixed length ( ) is causing flow. The force balance, equating the shear stress causing flow to the shear stress resisting flow (i.e., the fluid), over a core of fluid (Fig. 2.4) with radius and length yields

[2.1]

which can be solved for the shear stress:

[2.2]

Eq. [2.2] can also be obtained by starting from the general conservation of momentum equations (Bird et al., 1960; Brodkey and Hershey, 1988;

Darby, 1976; Denn, 1980) as discussed in Example Problem 2.12.1. Eq.

[2.2] depicts the shear stress varying over the pipe from zero at the center ( ) to a maximum at the wall ( ) where the equation may be written as

L R δP

L

r L

(δP)πr²= σ2πr L

σ =f(r) =(δP)πr² 2πr L =(δP)r

2L

r=0 r=R

Figure 2.4. Core of fluid in tube flow geometry.

[2.3]

To develop the shear rate equations, a differential flow element ( ) must be evaluated. This can be expressed by considering the steady, laminar flow of fluid moving through an annulus located between the core, with radius , and the position :

[2.4]

where is the linear velocity at . The total volumetric flow rate is found by integrating Eq. [2.4] over the radius:

[2.5]

Recognizing that , so that becomes the variable of integra-tion, allows Eq. [2.5] to be written as

[2.6]

The right hand side of this equation can be integrated by parts:

[2.7]

and simplified by applying the no slip boundary condition which stipulates that the fluid velocity is zero at the wall of the pipe or,

mathematically, :

R

r

L Flow

Flow fluid core

σ_w=(δP)R 2L

dQ

r r+dr

dQ=u2πr dr

u r

Q= ⌠⌡0 Q

dQ= π ⌠⌡r=0 r=R

u2r dr 2r dr=dr² r²

Q= π ⌠⌡_r²₌₀

r²=R²

(u)dr²

Q= πur²|

r²=0

r²=R²− π ⌠⌡_r²₌₀

r²=R²

r²du

u=0 at r=R

[2.8]

To further evaluate Eq. [2.8], a number of items must be noted. First, by assuming steady laminar flow, we know the shear rate is some function of the shear stress:

[2.9]

or

[2.10]

The negative sign is required in Eq. [2.9] because we assume, as indicated in Eq. [2.1], the positive direction of to be opposite the direction of flow. Second, Eq. [2.2] and [2.3] can be combined to give

[2.11]

which, when differentiated, yields

[2.12]

Taking the expression for given by Eq. [2.10], and inserting Eq. [2.12]

for , gives

[2.13]

Using Eq. [2.13] and noting, from Eq. [2.11], that allows Eq. [2.8] to be rewritten as

[2.14]

Observe the change in the limits of integration: goes from 0 to as goes from 0 to . Simplifying this expression gives the final general equation relating shear stress and shear rate:

[2.15]

Eq. [2.15] may be evaluated by differentiation using Leibnitz’ rule, which allows an integral of the form

Q= π[(0)R²−u(0)²] − π ⌠⌡_r²₌₀

[2.16]

to be written as

[2.17]

to differentiate the integral. Writing, Eq. [2.15] as

[2.18]

then, applying Leibnitz’ rule on the right hand side, and differentiating both sides with respect to (which is ) gives

[2.19]

Solving Eq. [2.19] for the shear rate at the wall ( ) yields the well known Rabinowitsch-Mooney equation:

[2.20]

where the derivative is evaluated at a particular value of . Application of this equation is demonstrated for soy dough in Example Problem 2.12.2.

Eq. [2.20] can also be expressed in terms of the apparent wall shear rate, :

[2.21]

where . Further manipulation gives

[2.22]

or

[2.23]

that can be rewritten in the following simplified format:

[2.24]

[2.25]

showing that is the point slope of the versus . If the fluid behaves as a power law material, the slope of the derivative is a straight line and . Eq. [2.24] is a convenient form of the Rabinowitsch-Mooney equation because power law behavior is frequently observed with fluid foods. Also, slight curvature in the logarithmic plot can often be ignored. Application of Eq. [2.24] and [2.25] is illustrated for a 1.5%

solution of sodium carboxymethylcellulose in Example Problem 2.12.3.

Newtonian Fluids. In developing the Rabinowitsch-Mooney equation a general expression relating shear stress to shear rate, Eq. [2.15], was developed:

This can be solved for a Newtonian fluid by inserting the Newtonian definition for shear rate, :

[2.26]

Integration of Eq. [2.26] gives

[2.27]

Substituting the shear stress at the wall ( ) into Eq. [2.27]

results in the Poiseuille-Hagen equation:

[2.28]

Eq. [2.28] indicates that the radius has a very strong influence on the behavior of the system since it is raised to the power four. Also, if this equation is written in terms of the definition of a Newtonian fluid ( ), then the formula for the shear rate at the wall may be determined:

This expression is identical to the one given in Eq. [1.90] (and Fig. 1.27) for estimating the maximum shear rate of a Newtonian fluid in tube flow.

Power Law Fluids. Eq. [2.15] can be solved for a power law fluid by inserting into the equation:

[2.30]

Integration and substitution of gives

[2.31]

If this equation is written in terms of the definition of a power law fluid ( ), then the formula for the shear rate at the wall may be determined:

[2.32]

Eq. [2.32] is an exact solution for a power law fluid and also useful as an estimate of the maximum shear rate in tube flow (as indicated in Eq.

[1.91]) for a wide range of fluid foods.

Bingham Plastic Fluids. In a Bingham plastic fluid, the shear rate is defined in terms of the plastic viscosity and the yield stress:

. This function is discontinuous because there is no shearing flow at points in the tube near the center where the shear stress is below the yield stress. Mathematically, for in

the central plug region and for in the outer

portion of the tube. Given the above, it is clear that Eq. [2.15] must be integrated for each region of the tube to determine the total volumetric flow rate:

[2.33]

Since shear rate is zero when the shear stress is below the yield stress, Eq. [2.33] can be simplified to

[2.34]

which, when integrated, yields the Buckingham-Reiner equation for flow of a Bingham plastic material in a pipe:

[2.35]

The same technique discussed for Bingham plastics is used to derive the flow rate equation for Casson fluids in Example Problem 2.12.4.

Bulkley Fluids. The volumetric flow rate for a Herschel-Bulkley fluid ( ), is found using the same method discussed for the Bingham plastic material:

[2.36]