**INTRODUCTION**

Based on the known operating parameters of a Bematek colloid mill, it is possible to estimate the ** shear rate **and

Before presenting the calculations, it must be emphasized that any such attempt to develop a ** mathematical model **of a real world situation can only be viewed as an

Bearing in mind the cautions noted above, the ** shear rate **and

- D (ft) = rotor exit diameter
- N (rpm) = rotor rotational velocity
- h (ft) = rotor/stator gap
- μ (cP) = product viscosity

All of the other necessary variables can be derived from these fundamental quantities.

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**ROTOR TIP SPEED**

The first step is to calculate the tangential linear velocity of a point on the working surface of the rotor. This is called the ** peripheral rotor tip speed **(see

It is clear from *Figure 32-003-1 *that the diameter at the rotor surface increases from “d” to “D” as one travels through the rotor/stator gap. However, to arrive at a preliminary result for this basic analysis, we will consider the diameter of the rotor to be its ** exit diameter**. A more complete analysis is presented in another

Considering a random point (P) on the outer edge of the rotor, as shown in Figure 32-003-1, one complete revolution of the rotor travels a distance equal to the exit circumference of the rotor.

S= πD

Next, we apply an equation that relates ** linear velocity** to

v=S x N

By combining the two previous equations, we arrive at the desired result:

v =πDN/60 **(32-003-1)**

Where, v (ft/sec) = linear velocity at rotor surface

D (ft) = rotor exit diameter

N (rpm) = rotor rotational velocity.

**Note: ***A conversion factor of 60 has been applied in the above equation to permit the expression of “N” in the more common units of *rpm*.*

**SHEAR VELOCITY**

Having expressed the velocity of any point on the rotor surface in terms of known variables, we can now move on to an analysis of the phenomena within the rotor/stator gap. In order to accomplish this, we will approximate the rotor and stator surfaces in the immediate vicinity of any random point in the shear area by ** straight lines**. Since the rotor/stator gap is typically three orders of magnitude smaller than the rotor diameter, this is justifiable. Then, a coordinate system is chosen so that the x-axis lies along the stator surface, and a random point (

It is clear from the ** symmetry **of the situation that the velocity at the random point cannot be a function of its x coordinate, because any other point along a horizontal line through this point is subjected to exactly the same shear forces. Now, as an initial attempt, assume that the velocity at P can be expressed as a simple

*u = Ay + B (32-003-2)*

Where, u (ft/sec) = velocity at point P

y (ft) = y coordinate at point P

A, B = unknown constants.

In the equation above, the unknown constants must be determined from the ** boundary conditions **of the problem. A known empirical fact from the field of fluid mechanics, which states that

- the velocity of any point on the rotor surface is v

- the velocity of any point on the stator surface is 0

Thus, the following two equations can be used to express the boundary conditions:

*u**y=0 **= A **× **0 + B = 0*

*u**y=h **= A **× **h + B = v*

The solution of this pair of simultaneous equations requires that **B = 0 **and **A = v/h**. By combining *Equation*

*32-003-1 *and *Equation 32-003-2 *and inserting these values, a final expression for the velocity at our random point is obtained:

* * u =πDN/60 ∙ y/h **(32-003-3)**

Where, u (ft/sec) = velocity at point P

D (ft) = rotor exit diameter

N (rpm) = rotor rotational velocity

y (ft) = y coordinate at point P

h (ft) = rotor/stator gap.

Notice that at y = 0 and y = h, the values calculated for “**u**” do indeed satisfy the boundary conditions. The fact that values for the constants A and B were found which led to a final equation meeting the boundary requirements indicates that the assumption of linearity was valid.

**SHEAR RATE AND SHEAR STRESS**

With the ** shear velocity **equation completed, we are now prepared to complete the shear calculations. In the rectangular coordinate system, a velocity function that is independent of the x and z coordinates leads to the following expression for the

R =du/dy= πDN/60h **(32-003-4)**

Where, R (sec-1) = shear rate at point P

u (ft/sec) = velocity at point P

y (ft) = y coordinate at point P

D (ft) = rotor exit diameter

N (rpm) = rotor rotational velocity

h (ft) = rotor/stator gap.

Having calculated the shear rate, one final definition from the field of fluid mechanics completes the analysis:

Viscosity (μ) =(Sheer Stress (τ))/(Shear Rate (R))

Implicit in this definition is an assumption that viscosity is a constant that is independent of the shear rate. In other words the fluid ** viscosity **is assumed to be

τ =μπDN/60h

Finally, by combining all constants and converting viscosity to cP, the above equation becomes:

τ = (1.09 × 10^{–6} ) × μDN/h **(32-003-5)**

Where, τ (lb/ft2) = shear stress

μ (cP) = product absolute (dynamic) viscosity

D (ft) = rotor exit diameter

N (rpm) = rotor rotational velocity

h (ft) = rotor/stator gap.

*Equation 32-003-4 *and *Equation 32-003-5 *above complete our task of expressing the ** shear rate **and

**A TYPICAL EXAMPLE**

Now that all of the necessary equations have been developed, we can gain a feeling for the order of magnitude of some of these variables by applying the equations to a typical real world example. Let us assume that a hypothetical application has the following ** specifications**:

- N = 4800 rpm
- D = 4 i = 0.33 ft
- μ = 1 cP
- h = 0.010 in. = 0.00083 ft

Then, we can use the equations derived in the previous section to calculate the *s*** hear parameters **as follows:

v =(π x 0.33 x 4800)/60 = 83.8 ft/sec

R =(π x 0.33 x 4800)/(60 x 0.00083) = 100,531 sec^{-1}

τ =”(1.09 x 10^{-6})” x (1 x 0.33 x 4800)/( 0.00083) = 2.09 lb/ft2

From the above example, the ease with which these three important processing parameters can be calculated from the known basic specifications of the Bematek colloid mill should be obvious.

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**SUMMARY**

Although several assumptions and approximations were applied during the analysis presented here, the three important relationships derived have proven to be remarkably dependable. As a basic starting point, these equations may be used with complete confidence to accurately analyze a wide range of processing applications.

Nevertheless, Bematek’s efforts to further refine the ** hydraulic shear **analysis are ongoing. Additional improvements in the mathematical model may be achieved by eliminating some of the assumptions and approximations made here. Specifically, factors such as the