Integral_Momentum_Balance_web

# Integral_Momentum_Balance_web - X ³ ˙ m i ~ V i ´ in(7...

This preview shows pages 1–6. Sign up to view the full content.

ENU 4133 – Integral Conservation of Momentum January 19, 2011

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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)
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).

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 -

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 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...
View Full Document

## This note was uploaded on 10/22/2011 for the course ENU 4134 taught by Professor Schubring during the Fall '11 term at University of Florida.

### Page1 / 6

Integral_Momentum_Balance_web - X ³ ˙ m i ~ V i ´ in(7...

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online