Solution Found!
The viscosity of some fluids, called magnetorheologi- cal
Chapter 2, Problem 77P(choose chapter or problem)
Problem 77P
The viscosity of some fluids, called magnetorheologi- cal (MR) fluids, changes when a magnetic field is applied. Such fluids involve micron-sized magnetizable particles suspended in an appropriate carrier liquid, and are suitable for use in controllable hydraulic clutches. See Fig. P2-49. The MR fluids can have much higher viscosities than the ER fluids, and they often exhibit shear-thinning behavior in which the viscosity of the fluid decreases as the applied shear force increases. This behavior is also known as pseudoplastic behavior, and can be successfully represented by Herschel-Bulkley constitutive model expressed as τ =τy + K(du/dy)m. Here τ is the shear stress applied, τy is the yield stress,K is the consistency index, and m is the power index. For a Herschel-Bulkley fluid with τy, = 900 Pa,K = 58 Pa ∙ sm, and m = 0.82, (a) find a relationship for the torque transmitted by an MR clutch for N plates attached to the input shaft when the input shaft is rotating at an angular speed of ω while the output shaft is stationary and (b) calculate the torque transmitted by such a clutch with N = 11 plates for R1 = 50 mm,R2 = 200 mm,ṅ = 3000 rpm, and h = 1.5 mm.
Questions & Answers
QUESTION:
Problem 77P
The viscosity of some fluids, called magnetorheologi- cal (MR) fluids, changes when a magnetic field is applied. Such fluids involve micron-sized magnetizable particles suspended in an appropriate carrier liquid, and are suitable for use in controllable hydraulic clutches. See Fig. P2-49. The MR fluids can have much higher viscosities than the ER fluids, and they often exhibit shear-thinning behavior in which the viscosity of the fluid decreases as the applied shear force increases. This behavior is also known as pseudoplastic behavior, and can be successfully represented by Herschel-Bulkley constitutive model expressed as τ =τy + K(du/dy)m. Here τ is the shear stress applied, τy is the yield stress,K is the consistency index, and m is the power index. For a Herschel-Bulkley fluid with τy, = 900 Pa,K = 58 Pa ∙ sm, and m = 0.82, (a) find a relationship for the torque transmitted by an MR clutch for N plates attached to the input shaft when the input shaft is rotating at an angular speed of ω while the output shaft is stationary and (b) calculate the torque transmitted by such a clutch with N = 11 plates for R1 = 50 mm,R2 = 200 mm,ṅ = 3000 rpm, and h = 1.5 mm.
ANSWER:
Solution
Step 1 of 7
- When we apply magnetic field to some of the fluids, its viscosity changes, such a phenomenon is known as the magnetorheological effect and such fluids are referred as MR fluids.
The Herschel- Buckley model gives the shear stress for such materials as,
…………..1
Where is yield stress, K is consistency index, m is power index,
is velocity gradient of fluid.
-
The one of the best application such fluids is in MR-clutch which typically contains a multidisk with equally spaced steel disks of inner diameter
and outer diameter
with N number of disks attached to the input shaft. The gap between the parallel disks is h and is filled with viscous fluid as shown in the figure below,
