Integral_Mass_Balance_web

# Integral_Mass_Balance_web - considerably: General: 0 = Z CS...

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ENU 4133 – Integral Conservation of Mass January 19, 2011

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Applying Reynolds Transport to Mass For mass, B = m , β = dm / dm = 1. Most general case (in scope of class), moving CV: d dt ( B syst ) = d dt ±Z CV βρ d V ² + Z CS βρ ~ V r · ~ ndA (1) d dt ( m ) = d dt ±Z CV ρ d V ² + Z CS ρ ~ V r · ~ ndA (2) 0 = d dt ±Z CV ρ d V ² + Z CS ρ ~ V r · ~ ndA (3) Fixed CV: 0 = Z CV ∂ρ t d V + Z CS ρ ~ V · ~ ndA (4)
Special (Simpliﬁed) Cases 0 = Z CV ∂ρ t d V + Z CS ρ ~ V · ~ ndA (5) Finite number of one-dimensional inlets and outlets: 0 = Z CV ∂ρ t d V + X i ( ρ i A i V i ) out - X i ( ρ i A i V i ) in (6) 0 = Z CV ∂ρ t d V + X i ( ˙ m i ) out - X i ( ˙ m i ) in (7) Steady ﬂow: 0 = Z CV ∂ρ t d V (8) 0 = Z CS ρ ~ V r · ~ ndA (9)

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Special (Simpliﬁed) Cases (2) 0 = X i ( ˙ m i ) out - X i ( ˙ m i ) in (10) X i ( ˙ m i ) out = X i ( ˙ m i ) in (11) X i ( ρ i A i V i ) out = X i ( ρ i A i V i ) in (12)
Incompressible Flow Simpliﬁcation In many engineering systems, ρ is well-approximated as a constant. In this case, the equations for ﬁxed CV mass conservation simplify

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Unformatted text preview: considerably: General: 0 = Z CS ~ V ~ ndA (13) 1-D inlets &amp; outlets: X i ( A i V i ) out = X i ( A i V i ) in (14) X i Q i out = X i Q i in (15) Average velocity: V av = Q A (16) Warnings Equation 3.29 contains a term (crossed-out) with errors. It reads: d dt Z CV t dv (17) Should read: Z CV t d V (18) No real ow is truly incompressible. You must judge whether the variations in need to be considered. Sometimes ( e.g., liquids with narrow temperature range), it is obviously a good approximation. In other cases, must be compared to and considered more carefully. In-class examples: 3-14, 3-27...
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## Integral_Mass_Balance_web - considerably: General: 0 = Z CS...

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