p1012

p1012 - approximate as occurring when r R = e − t √ ν...

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1332 CHAPTER 10. VORTICITY, VISCOSITY, LIFT AND DRAG 10.12 Chapter 10, Problem 12 Problem: Missiles carrying liquid fuel spin for directional stability. Detailed analysis shows that during spin up , the radial position, r , of the front as it moves from the missile’s inner surface to its centerline is given by r = Re t ν /h . R and h are fuel-compartment radius and height, respectively, ν is kinematic viscosity and is rotation rate. Determine the approximate spin-up time, i.e., the time to achieve nearly rigid-body rotation ( r/R 0 . 05 ), for the following. (a) A missile with R = 110 cm, h =200cm , ν =1 . 5 · 10 6 m 2 /sec and =10sec 1 . (b) An experiment with R = 3.5 in, h =3 . 5i n , ν =1 . 08 · 10 5 ft 2 /sec and =120rpm . (c) Due to turbulence, the spin-up time for Part (b) is only t true =2 .5sec .Assum ing the equation above is valid with an effective kinematic viscosity, ν e f ,rep lac

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Unformatted text preview: approximate as occurring when r R = e − t √ ν Ω /h = 0 . 05 Taking the natural log of both sides of this equation, we find − t √ ν Ω h ≈ − 3 = ⇒ t ≈ 3 h √ ν Ω (a) For a missile with the given values, the spin-up time is t = 3(2 m) ± (1 . 5 · 10 − 6 m 2 / sec) (10 sec − 1 ) = 1549 sec . = 26 min (b) For an experiment with the given values, the spin-up time is t = 3(3 . 5 in)(1 / 12 ft / in) 5 p 1 . 08 · 10 − 5 ft 2 / sec Q (120 rpm) (2 π / 60 sec − 1 / rpm) = 75 sec (c) Solving for ν from the equation derived above, ν = 1 Ω X 3 h t ~ 2 10.12. CHAPTER 10, PROBLEM 12 1333 Denoting the observed spin-up time by t true , there follows ν e f ν = w t t true W 2 Hence, for the observed time of t true = 2 . 5 sec, we have ν e f ν = w 75 sec 2 . 5 sec W 2 = 900...
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p1012 - approximate as occurring when r R = e − t √ ν...

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