Question

Q2. A concentric cylinder viscometer shown below measures the viscosity of the liquid. The radius of the inner cylinder is R

1
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=37.5 mm, the clearance of the gap is d=0.02 mm and the height of the viscometer is h=150 mm. The inner cylinder rotates at ω=100rpm under the torque of T=0.021Nm. Please find the viscosity of the liquid in the clearance gap.

Answer 1

To find the viscosity of the liquid in the clearance gap of a concentric cylinder viscometer, we can use Stokes' law. The torque acting on the inner cylinder is equal to the drag force, which can be calculated using the formula T = Fd. By finding the drag force and using Stokes' law, we can determine the viscosity of the liquid.

Step-by-step explanation:

To find the viscosity of the liquid in the clearance gap, we can use Stokes' law. Stokes' law relates the drag force experienced by a small sphere moving through a viscous fluid to its terminal velocity. The formula is given by: F = 6πηrv, where F is the drag force, η is the viscosity of the fluid, r is the radius of the sphere, and v is the terminal velocity.

In this case, the small sphere can be represented by the gap between the inner and outer cylinders. The torque acting on the inner cylinder is equal to the drag force. The torque can be calculated using the formula T = Fd, where T is the torque, F is the drag force, and d is the clearance of the gap.

Given that the inner cylinder rotates at ω = 100rpm (convert to rad/s), and the torque is T = 0.021Nm, we can find the drag force and then calculate the viscosity of the liquid using Stokes' law.

Answer 2

The viscosity of the liquid in a concentric cylinder viscometer can be calculated using the formula for viscous torque with the provided measurements of torque, angular velocity, inner cylinder radius, viscometer height, and clearance.

Step-by-step explanation:

To find the viscosity of the liquid using a viscometer, we can apply the formula for viscous torque in a cylindrical geometry:

Given:
T = 0.021 Nm
R1 = 37.5 mm = 0.0375 m
h = 150 mm = 0.15 m
ω = 100 rpm = 100 * 2π / 60 rad/s
d = 0.02 mm = 0.00002 m

First, convert the angular velocity to rad/s:

ω = 100 * 2π / 60 rad/s = 10.47 rad/s

Now we can rearrange the formula to solve for viscosity:

η = T / (2πωhR2d)

Substitute the given values into the equation to calculate η:

η = 0.021 / (2π * 10.47 * 0.15 * (0.0375)2 * 0.00002)

After calculating the above expression, we find the viscosity of the liquid in the clearance gap.