Download - Ch. 11: Introduction to Compressible Flow
Ch. 11: Introduction to Compressible Flow
When a fixed mass of air is heated from 20oC to 100oC, what
is change in….
p1, h1, s1, 1, u1, Vol1
20oC
p2, h2, s2, 2, u2, Vol2
100oC
…. Constant s? constant p? constant volume?…
STATE1
STATE2
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?
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)
IDEAL, CALORICALLY PERFECT GAS
h = u + pv IDEAL GAS
h = u + RT dh = du + RdT
IDEAL GAS
du = cvdT & dh = cpdT cpdT = cvdT + R dT
cp – cv = REq. (11.4)
cp - cv = R (11.4)
k cp/cv ([k=]) (11.5)
cp = kR/(k-1) (11.6a)
cv = R/(k-1) (11.6b)
IDEAL GAS
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)
always truedq + dw = du ds = q/T|rev
Tds = du - pdv = dh – vdp
s2 – s1 = cvln(T2/T1) - Rln(2/1)
s2 – s1 = cpln(T2/T1) - Rln(p2/p1)
Ideal / Calorically Perfect Gas
Handy if need to find change in entropy
Ideal / Calorically Perfect GasCv = du/dT; Cp = dh/dT; p = RT = (1/v)RT
Tds = du + pdv = dh –vdpds = 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 GasCv = du/dT; Cp = dh/dT; p = RT = (1/v)RT
Tds = du + pdv = dh –vdpds = 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
Tds = du + pdv = dh –vdpds = dh/T – vdp/T
ds = CpdT/T - (RT/[pT])dp
s2 – s1 = Cpln(T2/T1) - Rln(p2/p1)
Ideal / Calorically Perfect GasCv = du/dT; Cp = dh/dT; p = RT = (1/v)RT
Tds = du + pdv = dh –vdpds = dh/T – vdp/T
ds = CpdT/T - (RT/[pT])dp
Ideal / Calorically Perfect GasCv = du/dT; Cp = dh/dT; p = RT = (1/v)RT
Note: don’t be alarmed that cp and dp are in same equation! cp = dh/dT is ALWAYS TRUE for ideal gas
IsentropicIdeal / Calorically Perfect Gas
Handy if
isentropic
2/1 = (T2/T1)1/(k-1)
p2/p1 = (T2/T1)k/(k-1)
(2/1)k = p2/p1; p2/2k = 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 = 0ln(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)
assumptionsISENROPIC & IDEAL GAS
& constant cp, cv
2/1 = (T2/T1)1/(k-1)
p2/p1 = (T2/T1)k/(k-1)
assumptionsISENROPIC & IDEAL GAS
& constant cp, cv
(2/1)k = p2/p1
p2/2k = p1/1
k = constant
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 inthat 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
(p2/2) + u2 + ½ V22 + gz2 = (p1/1) + u1 + ½ V1
2 + gz1
Isentropic process
0
Definition: h = u + pv = u + p/;
assume z2 = z1
h2 + ½ V22 = h1 + ½ V1
2
= ho + 0
ho – h1 = ½ V12
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} M2
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
p/p0 = (1 + {(k-1)/2} M2)k/(k-1)
Steady, no body forces, one-dimensional, frictionless, ideal, calorically perfect,
adiabatic, isentropic
p = RT; cp = dh/dT; cv = du/dT
s2 – s1 = cvln(T2/T1) - Rln(2/1)
s2 – s1 = cpln(T2/T1) - Rln(p2/p1)
2/1 = (T2/T1)1/(k-1); p2/p1 = (T2/T1)k/(k-1); p2/2
k = const; c = kRT
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
Ideal & constant cp & cv
Ideal & constant cp & cv & isentropic
Ideal & constant cp & cv & isentropic + …
• 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
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