ch. 11: introduction to compressible flow
DESCRIPTION
Ch. 11: Introduction to Compressible Flow. When a fixed mass of air is heated from 20 o C to 100 o C, what is change in…. p 2 , h 2 , s 2 , 2 , u 2 , Vol 2 100 o C. STATE 2. p 1 , h 1 , s 1 , 1 , u 1 , Vol 1 20 o C. STATE 1. …. Constant s? constant p? constant volume?…. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Ch. 11: Introduction to Compressible Flow](https://reader035.vdocuments.us/reader035/viewer/2022062309/56815b13550346895dc8be65/html5/thumbnails/1.jpg)
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
![Page 2: Ch. 11: Introduction to Compressible Flow](https://reader035.vdocuments.us/reader035/viewer/2022062309/56815b13550346895dc8be65/html5/thumbnails/2.jpg)
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?
![Page 3: Ch. 11: Introduction to Compressible Flow](https://reader035.vdocuments.us/reader035/viewer/2022062309/56815b13550346895dc8be65/html5/thumbnails/3.jpg)
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
![Page 4: Ch. 11: Introduction to Compressible Flow](https://reader035.vdocuments.us/reader035/viewer/2022062309/56815b13550346895dc8be65/html5/thumbnails/4.jpg)
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)
![Page 5: Ch. 11: Introduction to Compressible Flow](https://reader035.vdocuments.us/reader035/viewer/2022062309/56815b13550346895dc8be65/html5/thumbnails/5.jpg)
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
![Page 6: Ch. 11: Introduction to Compressible Flow](https://reader035.vdocuments.us/reader035/viewer/2022062309/56815b13550346895dc8be65/html5/thumbnails/6.jpg)
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
![Page 7: Ch. 11: Introduction to Compressible Flow](https://reader035.vdocuments.us/reader035/viewer/2022062309/56815b13550346895dc8be65/html5/thumbnails/7.jpg)
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
![Page 8: Ch. 11: Introduction to Compressible Flow](https://reader035.vdocuments.us/reader035/viewer/2022062309/56815b13550346895dc8be65/html5/thumbnails/8.jpg)
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)
![Page 9: Ch. 11: Introduction to Compressible Flow](https://reader035.vdocuments.us/reader035/viewer/2022062309/56815b13550346895dc8be65/html5/thumbnails/9.jpg)
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
![Page 10: Ch. 11: Introduction to Compressible Flow](https://reader035.vdocuments.us/reader035/viewer/2022062309/56815b13550346895dc8be65/html5/thumbnails/10.jpg)
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
![Page 11: Ch. 11: Introduction to Compressible Flow](https://reader035.vdocuments.us/reader035/viewer/2022062309/56815b13550346895dc8be65/html5/thumbnails/11.jpg)
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
![Page 12: Ch. 11: Introduction to Compressible Flow](https://reader035.vdocuments.us/reader035/viewer/2022062309/56815b13550346895dc8be65/html5/thumbnails/12.jpg)
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
![Page 13: Ch. 11: Introduction to Compressible Flow](https://reader035.vdocuments.us/reader035/viewer/2022062309/56815b13550346895dc8be65/html5/thumbnails/13.jpg)
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
![Page 14: Ch. 11: Introduction to Compressible Flow](https://reader035.vdocuments.us/reader035/viewer/2022062309/56815b13550346895dc8be65/html5/thumbnails/14.jpg)
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
![Page 15: Ch. 11: Introduction to Compressible Flow](https://reader035.vdocuments.us/reader035/viewer/2022062309/56815b13550346895dc8be65/html5/thumbnails/15.jpg)
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
![Page 16: Ch. 11: Introduction to Compressible Flow](https://reader035.vdocuments.us/reader035/viewer/2022062309/56815b13550346895dc8be65/html5/thumbnails/16.jpg)
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)
![Page 17: Ch. 11: Introduction to Compressible Flow](https://reader035.vdocuments.us/reader035/viewer/2022062309/56815b13550346895dc8be65/html5/thumbnails/17.jpg)
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)
![Page 18: Ch. 11: Introduction to Compressible Flow](https://reader035.vdocuments.us/reader035/viewer/2022062309/56815b13550346895dc8be65/html5/thumbnails/18.jpg)
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)
![Page 19: Ch. 11: Introduction to Compressible Flow](https://reader035.vdocuments.us/reader035/viewer/2022062309/56815b13550346895dc8be65/html5/thumbnails/19.jpg)
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)
![Page 20: Ch. 11: Introduction to Compressible Flow](https://reader035.vdocuments.us/reader035/viewer/2022062309/56815b13550346895dc8be65/html5/thumbnails/20.jpg)
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)
![Page 21: Ch. 11: Introduction to Compressible Flow](https://reader035.vdocuments.us/reader035/viewer/2022062309/56815b13550346895dc8be65/html5/thumbnails/21.jpg)
Stagnation Reference (V=0)
(refers to “total” pressure (po), temperature (To) or density (o) if flow brought isentropically to rest)
![Page 22: Ch. 11: Introduction to Compressible Flow](https://reader035.vdocuments.us/reader035/viewer/2022062309/56815b13550346895dc8be65/html5/thumbnails/22.jpg)
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 .
![Page 23: Ch. 11: Introduction to Compressible Flow](https://reader035.vdocuments.us/reader035/viewer/2022062309/56815b13550346895dc8be65/html5/thumbnails/23.jpg)
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
![Page 24: Ch. 11: Introduction to Compressible Flow](https://reader035.vdocuments.us/reader035/viewer/2022062309/56815b13550346895dc8be65/html5/thumbnails/24.jpg)
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
![Page 25: Ch. 11: Introduction to Compressible Flow](https://reader035.vdocuments.us/reader035/viewer/2022062309/56815b13550346895dc8be65/html5/thumbnails/25.jpg)
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)
![Page 26: Ch. 11: Introduction to Compressible Flow](https://reader035.vdocuments.us/reader035/viewer/2022062309/56815b13550346895dc8be65/html5/thumbnails/26.jpg)
To/T = 1 + {(k-1)/2} M2
Steady, no body forces, one-dimensional, frictionless, ideal, calorically perfect,
adiabatic, isentropic
![Page 27: Ch. 11: Introduction to Compressible Flow](https://reader035.vdocuments.us/reader035/viewer/2022062309/56815b13550346895dc8be65/html5/thumbnails/27.jpg)
/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
![Page 28: Ch. 11: Introduction to Compressible Flow](https://reader035.vdocuments.us/reader035/viewer/2022062309/56815b13550346895dc8be65/html5/thumbnails/28.jpg)
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
![Page 29: Ch. 11: Introduction to Compressible Flow](https://reader035.vdocuments.us/reader035/viewer/2022062309/56815b13550346895dc8be65/html5/thumbnails/29.jpg)
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 + …
![Page 30: Ch. 11: Introduction to Compressible Flow](https://reader035.vdocuments.us/reader035/viewer/2022062309/56815b13550346895dc8be65/html5/thumbnails/30.jpg)
• 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