A cylinder of 0.122-m radius rotates concentrically inside a fixed cylinder of 0.128-m radius. Both cylinders are 0.305 m long. Determine the viscosity of the liquid that fills the space between the cylinders if a torque of 0.881 N • m is required to maintain an angular velocity of 60 revolutions p

Question

A cylinder of 0.122-m radius rotates concentrically inside a fixed cylinder of 0.128-m radius. Both cylinders are 0.305 m long. Determine the viscosity of the liquid that fills the space between the cylinders if a torque of 0.881 N • m is required to maintain an angular velocity of 60 revolutions per minute.

Accepted Answer

(a) The torque is transmitted through the fluid layers to the outer cylinder. Since the gap between the cylinders is small, the calculation can be made without integration.

tangential velocity of the inner cylinder = rω = (0.122 m)(2π rad/s) = 0.767 m/s

For the small space between cylinders, the velocity gradient can be assumed to be a straight line, and the mean radius can be used. Then dV/dy = (0.767 m)/(0.128 - 0.122) = 127.8 s[latex]^{-1}[/latex].

torque applied = torque resisting

0.881 = τ(area)(arm) = τ(2π × 0.125 x 0.305)(0.125) and τ = 29.4 Pa

Then [latex]\mu=\frac{ au}{dV/dy}=\frac{29.4}{127.8}=0.230[/latex] Pa•s.

(b) The more exact mathematical approach uses calculus, as follows.

As before, 0.881 = τ(2πr × 0.305)r, from which τ = 0.4597/r[latex]^{2}[/latex].

Now [latex]\frac{dV}{dy}=\frac{ au}{\mu}=\frac{0.4597}{\mu r^{2}}[/latex], where the variables are velocity V and radius r. The velocity is 0.767 m/s at the inner radius and zero at the outer radius.

Rearranging the above expression and substituting — dr for dy (the minus sign indicates that r decreases as V increases), we obtain

[latex]\int_{V_{outer}}^{V_{inner}}dV=\frac{0.4597}{\mu} \int_{0.128}^{0.122}{\frac{-dr}{r^{2}}}[/latex] and [latex]V_{inner} - V_{outer}=\frac{0.4597}{\mu}\left[\frac{1}{r} ight]_{0.128}^{0.122}[/latex]

Then [latex](0.767 - 0)=\left(\frac{0.4597}{\mu} ight)\left(\frac{1}{0.122}-\frac{1}{0.128} ight)[/latex], from which μ = 0.230 Pa • s.

Question 1.12: A cylinder of 0.122-m radius rotates concentrically inside a...

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