Final answer:
To determine the torque and power required to rotate the inner cylinder, we use the formulas for torque and power. The torque is calculated using the force, which is determined using the linear velocity distribution in the gap. The power is then calculated by multiplying the torque by the angular velocity.
Step-by-step explanation:
To determine the torque required to rotate the inner cylinder, we can use the formula:
Torque = force x distance
The force can be calculated using the linear velocity distribution in the gap. Since the velocity distribution is linear, the velocity of the fluid at the inner cylinder's surface is zero and at the outer cylinder's surface is twice the average velocity. Therefore, the average velocity is half of the velocity at the outer cylinder's surface. Using the formula for linear velocity distribution, we can calculate the average velocity as:
Average velocity = (2 x rev/min x pi x outer radius x gap width) / (60 x 12)
Next, we can calculate the force using the formula:
Force = viscosity x area x velocity gradient
The area can be calculated as the difference between the areas of the outer and inner cylinders:
Area = pi x (outer radius^2 - inner radius^2)
The velocity gradient can be calculated as the ratio of the average velocity to the gap width:
Velocity gradient = average velocity / gap width
Finally, we can calculate the torque as:
Torque = force x distance = (viscosity x area x velocity gradient) x 2pi x inner radius
To determine the power required to rotate the inner cylinder, we can use the formula:
Power = torque x angular velocity
First, we need to convert the angular velocity from rev/min to rad/s:
Angular velocity (rad/s) = (angular velocity (rev/min) x 2pi) / 60
Then, we can calculate the power:
Power = torque x angular velocity