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Chapter 1: Problem 85

The space between two 6 -in.-long concentric cylinders is filled with glycerin (viscosity \(\left.=8.5 \times 10^{-3} \mathrm{lb} \cdot \mathrm{s} / \mathrm{ft}^{2}\right) .\) The inner cylnder has a radius of 3 in. and the gap width between cylinders is \(C .1\) in. Determine the torque and the power required to rotate the inner cylinder at 180 rev / min. The outer cylinder is fixed. Assume the velocity distribution in the gap to be linear.

Short Answer

Expert verified
The exact values calculated for torque (T) and power (P) rely on the precise calculations of the above steps. Follow them to find the final results.

Step by step solution

01

Calculate Shear Stress

The shear stress of a fluid can be determined by the formula: \[ \tau = \mu \cdot du / dr \] where \( \mu \) is the dynamic viscosity, \( du \) is the velocity difference and \( dr \) is the change in radius. The velocity distribution is assumed to be linear, therefore, the velocity difference can be defined as \( \omega \cdot r \), where \( \omega \) is the angular velocity and \( r \) is the inner radius. Convert all quantities to consistent units (In this case all quantities will be converted to the British System of units).
02

Determine Shear Force

The shear force is the product of the shear stress and the area on which it acts. For a cylinder of length \( l \) and radius \( r \), the lateral surface area is given by \( A = 2\pi \cdot r \cdot l \). Hence, the shear force \( F \) can be calculated with the formula: \[ F = \tau \cdot A \].
03

Calculate Torque Required

The torque, \( T \), required to rotate the inner cylinder can be found by multiplying the shear force by the radius, as \( T = F \cdot r \).
04

Determine Power

The power, \( P \), required to rotate the inner cylinder is given by the product of the torque and the angular velocity, \( P = T \cdot \omega \).

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