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Chapter 2: Problem 72

A concentric-cylinder viscometer is shown. Viscous torque is produced by the annular gap around the inner cylinder. Additional viscous torque is produced by the flat bottom of the inner cylinder as it rotates above the flat bottom of the stationary outer cylinder. Obtain an algebraic expression for the viscous torque due to flow in the annular gap of width \(a\). Obtain an algebraic expression for the viscous torque due to flow in the bottom clearance gap of height \(b\). Prepare a plot showing the ratio, \(b / a,\) required to hold the bottom torque to 1 percent or less of the annulus torque, versus the other geometric variables. What are the design implications? What modifications to the design can you recommend?

Short Answer

Expert verified
The viscous torque in the annular gap is given by \(T1 = 2 \pi \mu L a^2 \Delta \Omega\) and in the bottom clearance gap is given by \(T2 = \pi \mu R^2 b \Delta \Omega\). The ratio \(b / a\) required to hold the bottom torque to 1% or less of the annulus torque increases with the increase in the geometric variable \(L / R^2\). To maintain the bottom torque at this limit, it would be beneficial to reduce the height of the bottom clearance gap or increase the width of the annular gap.

Step by step solution

01

Determine the Viscous Torque in the Annular Gap

The viscous torque produced in the annular gap, \(T1\), can be determined using the equation \(T1 = 2 \pi \mu L a^2 \Delta \Omega\), where \(\mu\) is the dynamic viscosity of the fluid, \(L\) is the length of the annular gap, \(a\) is the width of the annular gap, and \(\Delta \Omega\) is the difference in angular velocity between the inner and outer cylinders.
02

Determine the Viscous Torque in the Bottom Clearance Gap

The viscous torque in the bottom clearance gap, \(T2\), can be calculated using the equation \(T2 = \pi \mu R^2 b \Delta \Omega\), where \(R\) is the radius of the inner cylinder, \(b\) is the height of the bottom clearance gap.
03

Plot the Ratio \(b / a\) vs Geometric Variables

By setting \(T2 = 0.01 T1\) (since we want the bottom torque to be 1% or less of the annulus torque) and substituting \(T1\) and \(T2\) from previous steps, we get the equation \(b/a = 0.01 (2 L a /R^2)\). Plotting this equation will give the required \(b/a\) ratio versus the geometric variable \(L/R^2\).
04

Analyze the Implications and Suggestions

Analyzing the plot, it can be observed that as the geometric variable \(L / R^2\) increases, the value of \(b / a\) also increases. This has implications in terms of design and operation of a concentric-cylinder viscometer. It suggests that for values within the 1% limit, the bottom gap should be significantly smaller than the annular gap. This implies that the design could be modified by reducing the bottom gap height or increasing the annular gap width to achieve the least bottom torque.

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