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Integral_Momentum_Balance_web

# Integral_Momentum_Balance_web - out-X ³ ˙ m i ~ V i ´...

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ENU 4133 – Integral Conservation of Momentum January 25, 2010

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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 syst dt = X ~ F = d dt Z CV ~ V ρ d V + Z CS ~ V ρ ~ V r · ~ n dA (1) Comments: I Must use inertial reference frame . (Non-inertial frames, pp 164-167, 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 Z CV u ρ d V + Z CS u ρ ~ V r · ~ n dA (2)
Momentum Flux The momentum flux is defined as: ˙ M CS = Z CS ~ V ρ ~ V r · n 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 + Z CS ~ V ρ ~ V · ~ n dA (4) X F x = d dt Z CV u ρ d V + Z CS u ρ ~ 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 + X ˙ m i ~ V i out - X ˙ m

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Unformatted text preview: out-X ³ ˙ 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) Deﬁne 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 ρ ~ V · ~ ndA = Z ρ u 2 dA = ρ Z u 2 dA = β ˙ mV av (14) β > 1 (15) For laminar (usually slow speed or high viscosity or very small tube) ﬂow in a round tube/pipe, β = 4 / 3. (Not necessarily negligible.) For turbulent ﬂow, β ≈ 1 . 01-1 . 04. (Usually negligible.) In-class examples: 3-41, 3-49, 3-53, 3-92/3-93...
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