Basic Equations for Internal Compressible Flow:
Assume:
• Steady state, one-dimensional flow
(or uniform flow and properties at all cross-sections)
• Ideal gas with constant specific heats
• Negligible body forces
2
flow
Basic Equations for control volume
from Section (1) to Section (2)
of finite width and differential width, dx:
Conservation
of Mass
Momentum
Balance
Conservation
of Energy
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Equation
of State
2nd Law of
Thermodynamics
x
heat transfer
Finite Control
Volume
Differential Control
Volume
ρ1 V1 A1 = ρ2 V2 A2
dρ dV dA
+
+
=0
ρ
V
A
m˙ (V2 − V1 ) = p1 A1 − p2 A2 − pw ( A1 − A2 ) − Rx
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⎛ V22 V12 ⎞
c p (T2 − T1 ) + ⎜ − ⎟ = qnet in
2 ⎠
⎝ 2
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p1
p2
=
ρ1 T1 ρ 2 T2
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T ( s2 − s1 ) − qnet in ≥ 0
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Variables:
Rx
1
Geometry
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Net heat transfer per unit mass
Net pressure on walls
Reaction force due to friction
Upstream flow conditions €
Downstream flow conditions
dV
dp
dRx
=− 2 −
V
ρV
ρV 2 A
δqnet in
dT
dV 2
+
=
T 2c p T
cp T
dp dρ dT
=
+
p
ρ
T
T ds − δqnet in ≥ 0
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A1, A2
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qnet in = Q˙ net in m˙
pw
Rx
ρ1, p1, T1, V1
ρ 2 , p2 , T2 , V2
Problem Statement: given geometry, heat transfer rate, reaction force, and upstream flow
€ for downstream flow conditions.
conditions for specified control volume, solve
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