Fluid Mechanics: Fundamentals and Applications
Fluid Mechanics: Fundamentals and Applications
4th Edition
ISBN: 9781259696534
Author: Yunus A. Cengel Dr., John M. Cimbala
Publisher: McGraw-Hill Education
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Chapter 2, Problem 129P
To determine

(a)

The Relationship for the torque transmitted by an MR clutch for N plates attached to the input shaft rotating at an angular speed of ω.

Expert Solution
Check Mark

Answer to Problem 129P

The Relationship for the torque transmitted by an MR clutch for N plates attached to the input shaft rotating at an angular speed of ω is T=4πN[τy(13)(R23R13)+K( ω h)0.82(13.82)(R23.82R13.82)].

Explanation of Solution

Given:

The shear stress used for the Herschel-Bulkley constitute model is τ=τν+K( du dy)m, Yield stress is 900kPa, consistency index is 58Pasm, Power index is 0.82, the number of clutch plates attached to input shaft are N, angular shaft is stationary, and angular speed of the input shaft is ω.

Write the expression for the velocity gradient.

  Vh=dudy   ...... (I)

Here, velocity is V, fluid thickness is h and velocity gradient is du/dy.

Write the expression for shear stress.

  τ=τy+K( du dy)m   ...... (II)

Here, yield stress is is τy, shear stress is τ and consistency index is K.

Power index is m.

Substitute Equation in Equation (II).

  τ=τy+K(Vh)0.82   ...... (III)

Write the expression for the velocity of shaft.

  V=rω   ...... (IV)

Here, angular velocity is ω and radius of the shaft is r.

Substitute Equation (IV) in Equation (III).

  τ=τy+K( rωh)0.82   ...... (V)

Write the expression for the force.

  dF=τdA   ...... (VI)

Here, area is dA.

Substitute τy+K( rωh)0.82 for τ in Equation (VI).

  dF=[τy+K( rω h)0.82](2πrdr)   ...... (VII)

Write the expression for the torque generated.

  dT=rdF   ...... (VIII)

Substitute [τy+K( rω h)0.82](2πrdr) for dF in Equation (VIII).

  dT=r[τy+K( rω h)0.82](2πrdr)   ...... (IX)

Calculation:

Integrate both sides of the Equation (IX).

  dT= R 1 R 2r[ τ y+K ( rω h ) 0.82]( 2πrdr)=2π[τy R 1 R 2r2dr+K ( ω h ) 0.82 R 1 R 2 r 2.82 dr]T=2π[τy(13)(R23R13)+K( ω h)0.82(13.82)(R23.82R13.82)]

Write the expression for the torque generated by the both surfaces of the plate.

  T=2[2π{τy(13)(R23R13)+K( ω h )0.82(1 3.82)(R2 3.82R1 3.82)}]

Write the expression for the torque generated by the N number of clutch plate.

  T=N[4π{τy( 1 3 )( R 2 3 R 1 3 )+K ( ω h ) 0.82( 1 3.82 )( R 2 3.82 R 1 3.82 )}]=4πN[τy( 1 3)( R 2 3 R 1 3)+K( ω h )0.82( 1 3.82)( R 2 3.82 R 1 3.82)]

Conclusion:

The torque generated by the MR clutch is 4πN[τy(13)(R23R13)+K( ω h)0.82(13.82)(R23.82R13.82)].

To determine

(b)

Torque transmitted by the clutch.

Expert Solution
Check Mark

Answer to Problem 129P

Torque transmitted by the clutch is 81517.71Nm.

Explanation of Solution

Given information:

The number of plate clutch is 11, the inner radius of clutch is 50mm, the outer radius of clutch is 200mm, the speed of the input shaft is 3000rpm and fluid thickness is 1.5mm.

Write the expression for the angular speed of the shaft.

  ω=2πn·60   ...... (X)

Here, speed of the shaft is n·.

Write the expression for the torque generated.

  T=4πN[τy(13)(R23R13)+K( ω h)0.82(13.82)(R23.82R13.82)]   ...... (XI)

Calculation:

Substitute 2400rpm for n· in Equation (X).

  ω=2π( 2400rpm)60=15079.64460=251.31rad/s

Substitute 11 for N, 50mm for R1, 200mm for R2, 251.31rad/s FOR ω, 1.5mm for h, 58 for K, 900Pa for τy In Equation (XI)

  T=4π(11)[( 900Pa)( 1 3 )( ( 200mm ) 3 ( 50mm ) 3 )+58 ( 251.32 rad/s 1.5mm ) 0.82( 1 3.82 )( ( 200mm ) 3.82 ( 50mm ) 3.82 )]=4π(11)[( 900Pa)( 1 3 )( ( 200mm( 10 3 m 1mm ) ) 3 ( 50mm( 10 3 m 1mm ) ) 3 )+58 ( 251.32 rad/s 1.5mm( 10 3 m 1mm ) ) 0.82( 1 3.82 )( ( 200mm( 10 3 m 1mm ) ) 3.82 ( 50mm( 10 3 m 1mm ) ) 3.82 )]T=4π(11)[( 900Pa)( 1 3 )( 0.2 3 m 3 0.05 3 m 3 )+58 ( ( 251.31 rad/s ) 0.0015m ) 0.82( 1 3.82 )( 0.2 3.82 m 3 0.05 3.82 m 3 )]=44π[(300Pa×0.0078)m3Pa+(291837.97×0.02169)m3Pa]=44π[2.36m3Pa+587.36m3Pa]( 1N/ m 2 1Pa)=81517.71Nm

Conclusion:

The torque generated is 81517.71Nm.

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Chapter 2 Solutions

Fluid Mechanics: Fundamentals and Applications

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