A vertical cylinder of diameter 180 mm rotates concentrically inside anor cylinder of diameter 181.2 mm. Both cylinders are 300 mm high. The space between cylinders is filled with a liquid whose viscosity is unknown. Determine viscosity of fluid if a torque of 20 Nm is required to rotate inner cylinder at 120 r.p.m.

SOLUTION:

Given,

Diameter of the inner rotating cylinder, d=180 mm=0.18 m

Diameter of outer cylinder, D=181.2 mm =0.1812 m

Height of each cylinder, h=300 mm =0.3 m

Speed of rotation of inner cylinder, N=120 rpm

Torque, T=20 Nm.

Thickness of liquid layer, `dy=(0.1812-0.1800)/2=0.0006 m`

Now,

Tangential velocity of the inner cylinder, `u=omega*r=2pi*N*r=2pi*120/60*0.18`

or, `u=1.13`m/s

Also, the surface  area of the inner rotating cylinder is,

`A=2pi*rL=pi*d*L`

`=pi`*0.18*0.3

`=0.1696 m^2`

SOLUTION:

Given,

Diameter of the inner rotating cylinder, d=180 mm=0.18 m

Diameter of outer cylinder, D=181.2 mm =0.1812 m

Height of each cylinder, h=300 mm =0.3 m

Speed of rotation of inner cylinder, N=120 rpm

Torque, T=20 Nm.

Thickness of liquid layer, `dy=(0.1812-0.1800)/2=0.0006 m`

Now,

Tangential velocity of the inner cylinder, `u=omega*r=2pi*N*r=2pi*120/60*0.18`

or, `u=1.13`m/s

Also, the surface  area of the inner rotating cylinder is,

`A=2pi*rL=pi*d*L`

`=pi`*0.18*0.3

`=0.1696 m^2`

Now,

Shear stress, `tau=F/A=(tau/r)/A`

or, `tau=T/(r*A)=20/(0.9*A)=131.027 N/m^2`

Again,

`tau=mu*(du)/(dy)`

or, `mu=tau*(dy)/(du)`

where, `du=1.13`

or, `mu=0.696` Ns/`m^2`

Thus, `mu=6.96` poise