Integral_Momentum_Balance_web

Integral_Momentum_Balance_web - X m i ~ V i in (7) Pressure...

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ENU 4133 – Integral Conservation of Momentum January 19, 2011
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Applying Reynolds Transport to Momentum For momentum, B = m ~ V , β = d ( m ~ V ) / dm = ~ V . Most general case (in scope of class), moving CV: d ± m ~ V ² syst dt = X ~ F = d dt ³Z CV ~ V ρ d V ´ + Z CS ~ V ρ ± ~ V r · ~ n ² dA (1) Comments: I Must use inertial reference frame . (Non-inertial frames not in scope of class). I ~ F includes both surface forces and body forces (usually gravity) in the volume. Note it is a vector . I Entire equation is a vector equation for ~ V – need to consider (up to) 3 components. e.g., u = V x : X F x = d dt ³Z CV u ρ d V ´ + Z CS u ρ ± ~ V r · ~ n ² dA (2)
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Momentum Flux The momentum flux is defined as: ˙ M CS = Z CS ~ V ρ ± ~ V r · n ² dA (3) In the text, the integral is taken over “sec” rather than “CS” (for some reason).
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Simplifications Typically, we are concerned with a fixed control volume such that V r = V . The equations simplify to: X ~ F = d dt ±Z CV ~ V ρ d V ² + Z CS ~ V ρ ³ ~ V · ~ n ´ dA (4) X F x = d dt ±Z CV u ρ d V ² + Z CS u ρ ³ ~ V · ~ n ´ dA (5) (similar for v and w ). In the case of a 1-D inlet/outlet with constant ρ ~ V : ˙ M i = ~ V i ( ρ i V ni A i ) = ˙ m i ~ V i (6) So, for 1-D inlets and outlets: X ~ F = d dt ±Z CV ~ V ρ d V ² + X ³ ˙ m i ~ V i ´ out -
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Unformatted text preview: X m i ~ V i in (7) Pressure Forces on Surface General form: ~ F press = Z CS p (-~ n ) dA (8) Uniform pressure, p a : ~ F UP = Z CS p a (-~ n ) dA (9) ~ F UP =-p a Z CS ~ ndA = 0 (10) Dene a gauge pressure and simplify equation for ~ F press : p gage = p-p a (11) ~ F press = Z CS ( p-p a ) (-~ n ) dA (12) ~ F press = Z CS p gage (-~ n ) dA (13) Non-Uniform Momentum Flux Actual outlets/inlets are ducts of various shapes, such that ~ V is not constant across their cross-section. As a result Z u ~ V ~ ndA = Z u 2 dA = Z u 2 dA = mV av (14) > 1 (15) For laminar (usually slow speed or high viscosity or very small tube) ow in a round tube/pipe, = 4 / 3. (Not necessarily negligible.) For turbulent ow, 1 . 01-1 . 04. (Usually negligible.) In-class examples: 3-41, 3-49, 3-53, 3-92...
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Integral_Momentum_Balance_web - X m i ~ V i in (7) Pressure...

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