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HW_9_Soln - ME 309 Fall 2008 Section 2(Merkle Homework 9...

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ME 309 Fall 2008 Section 2 (Merkle) Homework # 9 Due Wed 15 Sep 2008 Problem 9-1 Incompressible fluid flows through a duct with a bend that turns the fluid through an angle, φ .with respect to the horizontal The area at the duct inlet is A 1 , while the area at the exit is A 2 . The velocity can be approximated as one-dimensional and uniform at both outlet and inlet. The velocity at the inlet is given as u 1 i , while the unit normal to the area is n 1 = i . The unit normal at the exit is given as n 2 = j i φ φ sin cos + and the exit velocity is normal to the exit area. The flow is steady and does not vary with time. a) Using the continuity and momentum equations in the form: dt dM m m CV out in = - & & and dt dMV F u m u m CV out out in in = + - & & Find the u and v components of the exit velocity and the resulting force that is needed to hold the bend motionless. b) Using the continuity and momentum equations in the form: ( ) ∫∫∫ = - CV CS dxdydz dt d A ρ ρ d n V and ( ) ∫∫∫ = + - CV CS dxdydz dt d F A V d n V V ρ ρ Find the integrals on the left hand side for the inlet and outlet surfaces and show that this integral form gives the same results as for the algebraic form given above. Note: these two approaches will be very similar. The purpose is to demonstrate that the two are equivalent. For the second part, simply put the vector forms inside the integrals, note that everything is constant (and, hence, comes outside the integrals), and gives the first result from a more formal procedure (that becomes very useful for more complex problems).
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