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03_piston - 1 On Choosing and Using Control Volumes Six...

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Unformatted text preview: 1 On Choosing and Using Control Volumes: Six Ways of Applying the Integral Mass Conservation Theorem to a Simple Problem Ain A. Sonin, MIT 2001, 6 pages Reference : Ain A. Sonin, Fundamental Laws of Motion for Particles, Material Volumes, and Control Volumes , 2001 We shall use a very simple example to illustrate the variety of ways in which a control volume theorem can be applied to a particular application, depending on the choice of control volume and which of the two alternative forms of the control volume theorem is used. The exercise provides a few basic insights into the thought processes that are used in control volume analysis. Figure 1 depicts something like a cylindrical syringe, or a grease gun, in which a solid piston of radius R 1 is pushed at a speed U(t) into a fluid-filled cylinder with the same internal radius, forcing the fluid out through a tube with internal radius R 2 and length L . The piston, cylinder, and tube are inflexible and made of material with density ρ s ; the fluid has density ρ and can be considered incompressible. Fig. 1: The system and the control volume (broken red line) for Methods 1 and 2. This control volume is fixed in the inertial reference frame of the cylinder. 2 Given the aforementioned quantities, what is the average flow speed V(t) of the fluid at the exit plane? The governing principle is clearly mass conservation, which can be written for a control volume in two alternative forms, Form A d dt ρ dV CV ( t ) ∫ + ρ v rn dA = CS ( t ) ∫ (1) Form B ∂ρ ∂ t CV ( t ) ∫ dV + ρ v n dA = CS ( t ) ∫ (2) where v n = r v ⋅ r n = v cos θ is the outward normal velocity component of the fluid at the control surface, and v rn = v v − v v c ( ) ⋅ v n is the outward normal component of the fluid...
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03_piston - 1 On Choosing and Using Control Volumes Six...

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