Question 1.4: Determine the viscosity of a fluid in a basic cone-plate vis......
Determine the viscosity of a fluid in a basic cone-plate viscometer with given cone angle and radius, applied torque and resulting constant angular velocity. Plot the result as T = T(R,μ ).
| Sketch: | Assumptions: | Concepts: |
 |
• Steady 1-D flow | • Differential approach \rm dT=rdF=rτ_{wall}dA |
| • θ << 1, i.e., \rm v_θ(z) is linear and cos(θ)≈1 | • Integrate over “wetted surface” dA=2πrds; \rm ds=\frac{dr}{\cos \theta } \approx dr | |
| • Constant T, ω_o, μ | ||
| • No endeffects, i.e., ω_o<1 |
• From the graph, the linear circumferential velocity \rm v_θ can be deduced as:
• \rm{v _{\theta }}=({r\omega _{o}})\frac{{z}}{{h}}=\frac{{r\omega _{o}}}{{rtan\theta}}z
• \rm dT=rdF;\,dF =τ_w dA ;\,\tau_{{{w}}}=\mu \frac{{d}{ v}_{{\theta }}}{{dz}}\big|_{z={h}}
• \rm\therefore\;{dT}=2\pi\mu{\frac{\omega _{{o}}}{\tan\theta}}{r}^{2}{d}{r}
• or
• \rm T={K}\int_{0}^{R}{r}^{2}\,{d}{r}={\frac{{K}}{3}}\,{R}^{3}
• Finally, with \rm K ≡ 2πμω_0 / \tanθ , the test-fluid viscosity is:
• \rm\mu =\frac{\frac{3}{2} T\tan \theta }{\pi \omega _oR^3}
Graph:
Comments:
• Because of the stated assumptions, \rm v_θ=v_θ(z) only and \rm T=T(R,μ,ω_o,θ).
• As expected, the device size, in terms of R, has the strongest influence on T.
