Ch. 11: Introduction to
Compressible Flow
When a fixed mass of air is
heated from 20oC to 100oC, what
is change in….
STATE1
p1, h1, s1,
1, u1, Vol1
20oC
STATE2
p 2, h 2, s 2,
2, u2, Vol2
100oC
…. Constant s? constant p? constant volume?…
Ch. 11: Introduction to
Compressible Flow
When a fixed mass of air is
heated from 20oC to 100oC –
What is the change in enthalpy?
Change in entropy (constant volume)?
Change in entropy (constant pressure)?
If isentropic change in pressure?
If isentropic change in density?
IDEAL, CALORICALLY PERFECT GAS
p = RT [R=Runiv/mmole] (11.1)
du = cvdT
(11.2)
u2- u1 = cv(T2 – T1) (11.7a)
dh = cpdT
(11.3)
h2- h1 = cp(T2 – T1) (11.7b)
h = u + pv
IDEAL GAS
h = u + RT
dh = du + RdT
IDEAL GAS
du = cvdT & dh = cpdT
cpdT = cvdT + R dT
cp – c v = R
Eq. (11.4)
IDEAL GAS
cp - cv = R
(11.4)
k  cp/cv ([k=])
(11.5)
cp = kR/(k-1)
(11.6a)
cv = R/(k-1)
(11.6b)
always true
dq + dw = du
ds = q/T|rev
Tds = du - pdv = dh – vdp
Ideal calorically perfect gas
– constant cp, cv
p =  RT;
cp = dh/dT; cv = du/dT
s2 – s1 = cvln(T2/T1) - Rln(2/1)
s2 – s1 = cpln(T2/T1) - Rln(p2/p1)
Ideal / Calorically Perfect Gas
s2 – s1 = cvln(T2/T1) - Rln(2/1)
s2 – s1 = cpln(T2/T1) - Rln(p2/p1)
Handy if need to find change in entropy
Ideal / Calorically Perfect Gas
Cv = du/dT; Cp = dh/dT; p = RT = (1/v)RT
Tds = du + pdv = dh –vdp
ds = du/T + RTdv/T
ds = cvdT/T + (R/v)dv
s2 – s1 = cvln(T2/T1) + Rln(v2/v1)
s2 – s1 = cvln(T2/T1) - Rln(2/1)
Ideal / Calorically Perfect Gas
Cv = du/dT; Cp = dh/dT; p = RT = (1/v)RT
Tds = du + pdv = dh –vdp
ds = du/T + RTdv/T
ds = cvdT/T + (R/v)dv
Note: don’t be alarmed that cv and dv in same equation!
cv = du/dT is ALWAYS TRUE for ideal gas
Ideal / Calorically Perfect Gas
Cv = du/dT; Cp = dh/dT; p = RT = (1/v)RT
Tds = du + pdv = dh –vdp
ds = dh/T – vdp/T
ds = CpdT/T - (RT/[pT])dp
s2 – s1 = Cpln(T2/T1) - Rln(p2/p1)
Ideal / Calorically Perfect Gas
Cv = du/dT; Cp = dh/dT; p = RT = (1/v)RT
Tds = du + pdv = dh –vdp
ds = dh/T – vdp/T
ds = CpdT/T - (RT/[pT])dp
Note: don’t be alarmed that cp and dp are in same equation!
cp = dh/dT is ALWAYS TRUE for ideal gas
Isentropic
Ideal / Calorically Perfect Gas
Handy
if
isentropic
2/1 = (T2/T1)1/(k-1)
p2/p1 = (T2/T1)k/(k-1)
k
k
(2/1) = p2/p1; p2/2 = const
c = kRT
s2 – s1 = Cvln(T2/T1) - Rln(2/1)
If isentropic s2 – s1 = 0
ln(T2/T1)Cv = ln(2/1)R
cp – cv = R; R/cv = k – 1
2/1 = (T2/T1)cv/R = (T2/T1)1/(k-1)
assumptions
ISENROPIC & IDEAL GAS
& constant cp, cv
s2 – s1 = cpln(T2/T1) - Rln(p2/p1)
If isentropic s2 – s1 = 0
ln(T2/T1)cp = ln(p2/p1)R
cp – cv = R; R/cp = 1- 1/k
p2/p1 = (T2/T1)cp/R = (T2/T1)k/(k-1)
assumptions
ISENROPIC & IDEAL GAS
& constant cp, cv
2/1 = (T2/T1
1/(k-1)
)
p2/p1 = (T2/T1)k/(k-1)
k
)
(2/1 = p2/p1
k
k
p2/2 = p1/1 = constant
assumptions
ISENROPIC & IDEAL GAS
& constant cp, cv
Ch. 11: Introduction to
Compressible Flow
When a fixed mass of air is heated from
20oC to 100oC –
What is the change in enthalpy?
h2 – h1 = Cp(T2- T1)
Ch. 11: Introduction to
Compressible Flow
When a fixed mass of air is heated from
20oC to 100oC –
Change in entropy (constant volume)?
s2 – s1 = Cvln(T2/T1)
Ch. 11: Introduction to
Compressible Flow
When a fixed mass of air is heated from
20oC to 100oC –
Change in entropy (constant pressure)?
s2 – s1 = Cpln(T2/T1)
Ch. 11: Introduction to
Compressible Flow
When a fixed mass of air is heated
from 20oC to 100oC –
If isentropic change in density?
2/1 = (T2/T1
1/(k-1)
)
Ch. 11: Introduction to
Compressible Flow
When a fixed mass of air is heated
from 20oC to 100oC –
If isentropic change in pressure?
p2/p1 = (T2/T1
k/(k-1)
)
Stagnation Reference (V=0)
(refers to “total” pressure (po), temperature (To) or density (o) if
flow brought isentropically to rest)
11-3 REFERENCE STATE:
LOCAL ISENTROPIC STAGNATION PROPERTIES
Since p, T, , u, h, s, V are all changing along the flow,
the concept of stagnation conditions is extremely useful in
that it defines a convenient reference state for a flowing fluid.
To obtain a useful final state, restrictions must be put on the
deceleration process. For an isentropic (adiabatic and no
friction) deceleration there are unique stagnation To, po, o,
uo, so, ho (Vo=0) properties .
1-D, energy equation for adiabatic and no shaft or
viscous work Eq. (8.28); hlT = [u2-u1] - Q/m
0
(p2/2) + u2 + ½ V22 + gz2 = (p1/1) + u1 + ½ V12 + gz1
Definition: h = u + pv = u + p/;
assume z2 = z1
h2 + ½ V22 = h1 + ½ V12
= ho + 0
ho – h1 = ½ V1
2
Isentropic
process
1-D, energy equation for adiabatic and no shaft or
viscous work (8.28, hlT = [u2-u1] - Q/m)
ho - h1 = ½ V12
ho – h1 = cp (To – T1)
½ V12 = cp (To – T1)
½ V12 + cpT1 = cp To
To = {½ V12 + cpT1}/cp
T0 = ½ V12/cp + T1 = ½ V2/cp + T
T0 = ½ V12/cp + T = T (1 + V2/[2cpT])
cp = kR/(k-1)
T0 = T (1 + V2/[2kRT/{(k-1)})
T0 = T (1 + (k-1)V2/[2kRT])
c2 = kRT
T0 = T (1 + (k-1)V2/[2c2])
M = V2/ c2
T0 = T (1 + [(k-1)/2] M2)
To/T = 1 + {(k-1)/2}
2
M
Steady, no body forces, one-dimensional,
frictionless, ideal, calorically perfect,
adiabatic, isentropic
/o = (T/To)1/(k-1)
To/T = 1 + {(k-1)/2} M2
/o = (1 + {(k-1)/2} M2 )1/(k-1)
Steady, no body forces, one-dimensional,
frictionless, ideal, calorically perfect,
adiabatic, isentropic
p/p0 = (T/To)k/(k-1)
To/T = 1 + {(k-1)/2} M2
2
k/(k-1)
p/p0 = (1 + {(k-1)/2} M )
Steady, no body forces, one-dimensional,
frictionless, ideal, calorically perfect,
adiabatic, isentropic
p =  RT;
Ideal & constant cp & cv
cp = dh/dT;
cv = du/dT
s2 – s1 = cvln(T2/T1) - Rln(2/1)
s2 – s1 = cpln(T2/T1) - Rln(p2/p1)
Ideal & constant cp & cv & isentropic
2/1 = (T2/T1)1/(k-1); p2/p1 = (T2/T1)k/(k-1);
p2/2k = const; c = kRT
Ideal & constant cp & cv & isentropic + …
p0/p = (1 + {(k-1)/2} M2)k/(k-1);
o/ = (1 + {(k-1)/2} M2 )1/(k-1)
To/T = 1 + {(k-1)/2} M2
p0/p = (1 + {(k-1)/2} M2)k/(k-1);
o/ = (1 + {(k-1)/2} M2 )1/(k-1)
To/T = 1 + {(k-1)/2} M2
• Stagnation condition not useful for velocity
• Use critical condition – when M = 1, V* = c*
(critical speed is the speed obtained when flow is
isentropically accelerated or decelerated until M = 1)
• At critical conditions, the isentropic stagnation quantities become:
p0/p* = (1+{(k-1)/2} 12)k/(k-1) = {(k+1)/2}k/(k-1)
o/ = (1+{(k-1)/2} 12 )1/(k-1) = {(k+1)/2}1/(k-1)
To/T = 1 + {(k-1)/2} 12 = (k+1)/2