fundamentals of engineering supplied-reference handbook - Ventech!

Isentropic Flow Relationships
In an ideal gas for an isentropic process, the following
relationships exist between static properties at any two
points in the flow.
P ⎛ T ⎞ ( k −1)
2
⎛ ⎞
P ⎜ 2 ρ
=
T ⎟
⎜ 2
=
⎟
1 ⎝ 1 ⎠ ⎝ ρ1
⎠
k
The stagnation temperature, T0, at a point in the flow is
related to the static temperature as follows:
2
V
T0
= T +
2 ⋅ C
p
The relationship between the static and stagnation
properties (T0, P0, and ρ0) at any point in the flow can be
expressed as a function of the Mach number as follows:
T0 k −1
2
= 1+ ⋅ Ma
T 2
k
P0
⎛ T0
⎞ ( k −1)
⎛ k −1
2 ⎞ ( k −1)
= ⎜ ⎟ = ⎜1+
⋅ Ma ⎟
P ⎝ T ⎠ ⎝ 2 ⎠
1
1
ρ0 1
−
⎛ T0
⎞ ( k − ) ⎛ k −1
2 ⎞ ( k 1)
= ⎜ ⎟ = ⎜1+
⋅ Ma ⎟
ρ ⎝ T ⎠ ⎝ 2 ⎠
Compressible flows are often accelerated or decelerated
through a nozzle or diffuser. For subsonic flows, the
velocity decreases as the flow cross-sectional area increases
and vice versa. For supersonic flows, the velocity increases
as the flow cross-sectional area increases and decreases as
the flow cross-sectional area decreases. The point at which
the Mach number is sonic is called the throat and its area is
represented by the variable, A * . The following area ratio
holds for any Mach number.
where
A
A
*
=
1
Ma
⎡ 1
⎢1+
2
⎢
⎢
1
⎣ 2
( k −1)
k
( k + 1)
( k + 1)
2 ⎤ 2(
k −1)
Ma ⎥
⎥
⎥
⎦
A ≡ area [length 2 ]
A * ≡ area at the sonic point (Ma = 1.0)
Normal Shock Relationships
A normal shock wave is a physical mechanism that slows a
flow from supersonic to subsonic. It occurs over an
infinitesimal distance. The flow upstream of a normal
shock wave is always supersonic and the flow downstream
is always subsonic as depicted in the figure.
k
215
INLET
MECHANICAL ENGINEERING (continued)
1 2
Ma > 1 Ma < 1
EXIT
NORMAL SHOCK
The following equations relate downstream flow conditions
to upstream flow conditions for a normal shock wave.
Ma 2 =
2
1
T
T
2
1
P
P
ρ
ρ
=
2 ( k −1)
Ma1
+ 2
2k
Ma − ( k −1)
2
2 2k
Ma1
−
[ ( ) ] ( k −1)
2 + k −1
Ma1
2 2 ( k + 1)
Ma1
1 2 [ 2k
Ma − ( 1)
]
2 =
1 k + 1
1 k −
2
2 V1
( k + 1)
Ma1
= =
2
1 V2
( k −1)
Ma1
+ 2
T T =
01
02
Fluid Machines (Compressible)
Compressors
Compressors consume power in order to add energy to the
fluid being worked on. This energy addition shows up as an
increase in fluid pressure (head).
COMPRESSOR W in
For an adiabatic compressor with ∆PE = 0 and negligible
∆KE:
W� = − m�
h − h
comp
( )
For an ideal gas with constant specific heats:
W� = − m�
C T − T
comp
e
p
i
( )
Per unit mass:
w =
− C T − T
comp
p
e
( )
e
i
i