egr 334 thermmodynamcis chapter 3: section 15 lecture 11: polytropic processes quiz today?
TRANSCRIPT
EGR 334 ThermmodynamcisChapter 3: Section 15
Lecture 11: Polytropic Processes Quiz Today?
Main concepts for today’s lecture:
• Polytropic Process is defined and explained• Let’s do some example problems.
Reading Assignment:
Homework Assignment:
• Read Chap 4: Sections 1-3
From Chap 3: 138, 142,144,147
3Sec 3.10.2 : Incompressible Substance Model
or
What is a polytropic process?
12 1 1 1 2 1 2 2 1 2ln( / ) ln( / ) ln( / )W pdV C V V C V V pV V V p V p p
The exponent, N, may take on any value from -∞ to + ∞, but some values of N are more interesting than others.
N= 1: Isothermal process
constantNpv constantNpV
pv C
N= 0: Isobaric process p C
2 1( )W pdV p V V
4Sec 3.10.2 : Incompressible Substance Model
or
What is a polytropic process?
0Q
The exponent, N, may take on any value from -∞ to + ∞, but some values of N are more interesting than others.
N= k: Adiabatic process
constantNpv constantNpV
kpv C
1 12 1( )
1
N NN
N
C V VCW pdV dV C V dV
V N
N≠ 1: most polytropic processes
2 1
1 11 12 2 1 12 1 2 2 1 1
( ) ( )
1 1 1
N N N NN N p V V pV VCV CV p V pV
N N N
5
Problem 3.148 T or F.a) T or F: The change in specific volume from saturated liquid to saturated vapor (vg - vf) at a specified saturation pressure increases as the pressure decreases. T
b) T or F: A two phase liquid-vapor mixture with equal volumes of saturated liquid and saturated vapor has a quality of 50%. F
c) T or F: The following assumptions apply for a liquid modeled as incompressible: the specific volume is constant and the specific internal energy is a function only of temperature. T d) T or F: Carbon dioxide (CO2) at 320 K and 55 bar can be modeled as an ideal gas. ( pr = 0.75 Tr = 1.05 Z = 0.74 ) F
e) T or F: When an ideal gas undergoes a polytropic process with n=1, the gas temperature remains constant. T
6
Example Problem: (3.139) One kilogram of air in a piston cylinder assembly undergoes two processes in series from an initial state where p1_gage = 0.5 MPa, and T1 = 227 deg C.
Process 1: Constant temperature expansion until the volume is twice the initial volume.Process 2: Constant volume heating until the pressure is again 0.5 MPa.
Sketch the two processes in series on a p-v diagram. Assuming ideal gas behavior, determine a) the pressure at state 2. b) the temperature at state 3c) the work and heat transfer for each process.
7
P-v diagram
v
p
2
13State 1:
p1 = 0.5 MPa+0.1013MPa = 0.6013MPaT1 = 227 deg C.= 500 K
State 2: T2 = T1 = 227 deg C. = 500 KV2 = 2V1 State 3: p3 = p1 = 0.6013MPaV3 = V2
m = 1 kg of air
Apply 1st Law of Thermo:
Process 1-2: constant T ΔU1-2=Q1-2 - W1-2
Process 2-3: constant V ΔU2-3=Q2-3 - W2-3
8
v
p
2
1 3
State 1: p1 = 0.6013 MPa T1 = 500K
m = 1 kg of air R= 0.2870 kJ/kg-KApply Ideal Gas Law: pV mRT
311 6 2
1
(1 )(0.2870 / )(500 ) 10000.2386
0.6013 10 /
mRT kg kJ kg K K MPa N mV m
p MPa N m kJ
State 2: T2 = 500 K V2 = 2V1 = 0.4773 m3
22 3 6 2
2
(1 )(0.2870 / )(500 ) 10000.3006
0.4773 10 /
mRT kg kJ kg K K MPa N mp MPa
V m N m kJ
State 3: p3 = p1 = 0.6013MPa V3 = V2 = 0.4773 m3
3 6 23 3
3
(0.6013 )(0.4773 ) 10 /1000
(1 )(0.2870 / ) 1000
p V MPa m N m kJT K
mR kg kJ kg K MPa N m
9
State 1: p1 = 0.6013 MPa T1 = 500K V1=0.2386 m3
m = 1 kg of air R= 0.2870 kJ/kg-K
2 1 1 2 1 2U U Q W
State 2: p2= 0.3006 MPa T2 = 500 K V2 = 0.4773 m3 State 3: p3 = 0.6013MPa T3 = 1000 V3 = 0.4773 m3
---------------------------------------------------------------------------------Process 1-2:
where: 2 1 2 1 2 1( ) ( ) 0vU U m u u mc T T
constantpV mRT
1 2 1 2 99.45Q W kJ
for constant T: and
2 21 2 1 1
1 1
ln lnV VC
W pdV dV C pVV V V
CpV
6 23 0.4773 10 /
(0.6013 )(0.2386 ) ln( ) 99.450.2386 1000
N m kJMPa m kJ
MPa N m
10
State 1: p1 = 0.6013 MPa T1 = 500K V1=0.2386 m3
m = 1 kg of air R= 0.2870 kJ/kg-K
State 2: p2= 0.3006 MPa T2 = 500 K V2 = 0.4773 m3 State 3: p3 = 0.6013MPa T3 = 1000 V3 = 0.4773 m3
---------------------------------------------------------------------------------
3 2 2 3 2 3U U Q W Process 2-3:
where:
so
3 2 3 2 3 2( ) ( )vU U m u u mc T T
onstantV c
2 3 3 2 358.75Q U U kJ
for constant T: 2 3 0W
0.28700.7175 /
1 1.4 1v
Rc kJ kg K
k
3 2 (1 )(0.7175 / )(1000 500) 358.75U U kg kJ kg K K kJ
11
End of slides for Lecture 11