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HW_11_Soln

# HW_11_Soln - ME 309 Fall 2008 Section 2(Merkle Homework 11...

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ME 309 Fall 2008 Section 2 (Merkle) Homework # 11 Due Mon 22 September 2008 Problem 11-1 Find the mass and momentum flow across the right and left faces of a deforming hexahedral control volume and express conservation of mass and conservation of momentum for the control volume as viewed from the specified observation frames for the following cases: a) The left side of the control volume is fixed in space, but its right side moves to the right at a velocity of 10 m/s with respect to the ground. The air is motionless relative to the ground. The observer sits at a fixed position on the ground. b) The left side of the control volume is fixed in space, but its right side moves to the right at a velocity of 10 m/s with respect to the ground. The air is motionless relative to the ground. The observer sits in the frame of reference that moves with the right side of the control volume. c) The left side of the control volume moves to the right at a velocity of 10 m/s with respect to the ground while its right side toward the right at a velocity of 20 m/s with respect to the ground. The air is motionless relative to the ground. The observer sits in the frame reference that is fixed with respect to the ground. d) The left side of the control volume moves to the right at a velocity of 10 m/s with respect to the ground while its right side toward the right at a velocity of 20 m/s with respect to the ground. The air is moving to the right at 15 m/s . The observer sits in the frame reference that is fixed with respect to the right side of the control volume. The areas of the right and left faces are both 0.01 m 2 . The density is 1 kg/s. Be careful to specify the velocity and relative velocity on each face in each case. As there is no velocity in the other two directions, consider only variations in the x direction and only the x- momentum component. Solution: Assumptions: Incompressible, uniform flow Separate sketch for each part Analysis: Equations: Conservation of mass: (depends on relative velocity): ( ) = - CV CS rel dVol dt d dA ρ ρ n V Conservation of momentum: ( ) = + - CV CS rel dVol dt d F dA ρ ρ V n V V ( V is velocity in observer’s frame of reference)

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