**INTRODUCTION**

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

**experienced by each internal phase particle of an emulsion or dispersion while it is within the gap between the rotor and stator.**

*shear stress*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

**. The real world seldom, if ever, conforms in every detail to our neat and orderly expectations. Our analysis is limited both by a less than perfect understanding of the physics involved and by the necessity to make certain assumptions in the mathematical treatment. Nevertheless, the colloid mill shear calculations performed for many years by the Bematek engineering team have proven to be very useful and reliable for designing colloid mills that perform up to the specified production requirements.**

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

**generated within the rotor/stator gap of a Bematek colloid mill can be calculated as a function of the following basic operating parameters:**

*shear stress*- 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.

** **

**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

*Figure 32-003-1).*

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

*Tek Talk*issue.

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

*for circular motion:*

**angular velocity**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 (

**P**) within the rotor/stator gap has coordinates

**(x, y)**with a shear velocity of

**u**. The situation is illustrated in

*Figure 32-003-2*.

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

**of its y coordinate:**

*linear function**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

**, leads to the following conclusions:**

*a fluid in contact with a solid object assumes the velocity of that object*- 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

**at our random point:**

*shear rate*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

**. Then, by substituting the shear rate from**

*Newtonian**Equation 32-003-4*above into this definition, the shear stress is given by:

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

*s***applied to any internal phase particle of an emulsion or dispersion passing through the rotor/stator gap, in terms of the basic operating parameters of the colloid mill.**

*hear stress***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.

** **

**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

**nature of most fluid viscosities and the**

*non-Newtonian***in Bematek’s colloid mills are considered.**

*conical rotor face*