chapter 9 gas power cycles - vinhqtang.com
TRANSCRIPT
Chapter 9
GAS POWER CYCLES(based on notes from Profs. Kaya & Gauthier)
Vinh Q. Tang, Ph.D., P.Eng.
Adjunct Recearch Professor
MAAE 2400
Thermodynamics and Heat Transfer
1
Carleton University
Department of Mechanical and Aerospace Engineering
Office: ME2186
Fall 2020
OUTLINE
9.1 Introduction (Review of ideal gas model)
9.2 Otto Cycle (For your interest)
9.3 Brayton Cycle
9.4 Summary
Reading Assignment: Related Sections in Chapter 9 in Fundamentals of
Engineering Thermodynamics, 7th Edition ; M. Moran and H. N. Shapiro, et al
2Fall 2020
9.1 INTRODUCTION
3
Equations of state :
𝒑𝒗 = 𝑹𝑻 𝒐𝒓 𝒑𝑽 = 𝒎𝑹𝑻
Fall 2020
Review of Ideal Gas Model (presented in Chapters 3 and 6)
4
Changes in “u” and “h”
𝒖𝟐 − 𝒖𝟏 = 𝒄𝒗(𝑻𝟐 − 𝑻𝟏)
𝒉𝟐 − 𝒉𝟏 = 𝒄𝑷(𝑻𝟐 − 𝑻𝟏)
1) Constant specific heats
2) Variable specific heats
u(T) and h(T) are evaluated from tables:
▪ Table A-22 for air (mass basis)
▪ Table A-23 for other gases (molar basis)
Fall 2020
5
Changes in entropy
1) Constant specific heats
2) Variable specific heats :
𝒔𝟐 − 𝒔𝟏 = 𝒄𝒗𝒍𝒏𝑻𝟐𝑻𝟏
+ 𝑹𝒍𝒏(𝒗𝟐𝒗𝟏)
𝒔𝟐 − 𝒔𝟏 = 𝒄𝑷𝒍𝒏𝑻𝟐𝑻𝟏
− 𝑹𝒍𝒏(𝑷𝟐
𝑷𝟏)
𝒔𝟐 − 𝒔𝟏 = 𝒔𝟐𝟎 − 𝒔𝟏
𝟎 − 𝑹𝒍𝒏(𝑷𝟐
𝑷𝟏)
s0 can be obtained from tables:
▪ Table A-22 for air (mass basis)
▪ Table A-23 for other gases (molar basis)
(See Tables A-20 & 21 for cv and cP data)
Fall 2020
6
For Isentropic Processes
1) Constant specific heats
𝑻𝟐𝑻𝟏
=𝑷𝟐
𝑷𝟏
𝒌−𝟏𝒌 𝑻𝟐
𝑻𝟏=
𝒗𝟏𝒗𝟐
𝒌−𝟏𝑷𝟐
𝑷𝟏=
𝒗𝟏𝒗𝟐
𝒌
Where 𝒌 =𝒄𝒑
𝒄𝒗𝒈𝒊𝒗𝒆𝒏 𝒊𝒏 𝑻𝒂𝒃𝒍𝒆 𝑨 − 𝟐𝟎
2) Variable specific heats
𝑷𝟐
𝑷𝟏=𝑷𝒓𝟐
𝑷𝒓𝟏
𝒗𝟐𝒗𝟏
=𝒗𝒓𝟐𝒗𝒓𝟏
Where Pr and vr are provided for air 𝐢𝐧 𝐓𝐚𝐛𝐥𝐞 𝐀 − 𝟐𝟐
for air only
Fall 2020
End of Review of Ideal Gas Model
• Starting section 9.2
in the next slide
Fall 2020 7
9.2 OTTO CYCLE (1)
• Used to model spark-
ignition engines
• In a spark-ignition
engine, the air-fuel
mixture is ignited by a
spark plug
8Fall 2020
9.2 OTTO CYCLE (2)
• Pressure-
displacement curve
for a four-stroke
reciprocating
internal combustion
engine
(TDC) (BDC)9Fall 2020
9.2 OTTO CYCLE (3)
1. A fixed mass of air is the working fluid throughout the entire cycle. Thus, there is no inlet and no exhaust process.
2. The internal combustion process is replaced by a heat transfer from an external source.
3. The cycle is completed by heat transfer to the surroundings.
4. All processes are internally reversible.
5. Air is an ideal gas with constant specific heats (Cold Air Standard Analysis).
AIR STANDARD ANALYSIS
10Fall 2020
9.2 OTTO CYCLE (4)
1-2 reversible adiabatic (isentropic) compression
2-3 constant volume heat transfer (heat added)
3-4 reversible adiabatic (isentropic) expansion
4-1 constant volume heat transfer (heat rejected)
2
3
4
v=c
v=c
1
11Fall 2020
9.2 OTTO CYCLE (5)
• Air Standard Otto Cycle (ASOC) is an ideal cycle
approximating a spark-ignition internal
combustion engine.
• Due to the simplifications in the ASOC, the
analysis is only valid on a QUALITATIVE BASIS.
12Fall 2020
9.2 OTTO CYCLE (6)
CONSEQUENCES OF THE Air Standard Otto Cycle
ASSUMPTIONS
• The working fluid is a fixed amount of air.
• Heat rejection and addition (combustion) are
assumed to take place instantaneously.
• There are no pressure drops due to friction.
• Air is assumed to be an ideal gas.
• All processes are internally reversible.
13Fall 2020
9.2 OTTO CYCLE (7)
• USING 1st and 2nd Laws for a closed system with
DKE, DPE=0
121221For uuw −=−
433443For uuw −=−
232332For uuq −=−
144114For uuq −=−
WQU −=D
NOTE: w’s and q’s are all positive quantities. 14Fall 2020
( ) ( ) ( ) ( )
23
14
23
1423
23
1243
23
1234
1uu
uu
uu
uuuu
uu
uuuu
q
ww
q
w
cycle
cycle
cycle
added
net
cycle
−
−−=
−
−−−=
−
−−−=
−=
=
9.2 OTTO CYCLE (8)
1212 uuw −=
4334 uuw −=2323 uuq −=
1441 uuq −=
We haveHence
15Fall 2020
9.2 OTTO CYCLE (9)
1
4
3
3
4
1
2
1
1
2 and
−−
=
=
kk
V
V
T
T
V
V
T
T
For isentropic processes of 1-2 and 3-4 and air is an ideal gas:
2
1
3
4ButV
V
V
V=
4
3
1
2HenceT
T
T
T=
2
3
1
4OrT
T
T
T=
16Fall 2020
9.2 OTTO CYCLE (10)
1
2
1
1
2
2
3
1
4
23
141
−
=
=
=
−
−−=
k
cycle
V
V
T
T
T
T
T
T
dTcdu
uu
uu
( ) 1
21
2
1
23
14
2
1
23
14
/
11
1
1/
1/11
−−=
−=
−
−−=
−
−−=
kcycle
cycle
cycle
VV
T
T
TT
TT
T
T
TT
TT
We have
Hence
17Fall 2020
9.2 OTTO CYCLE (11)
min
max
1
rationcompressioc)(volumetri
where
Cycle Otto1
1
V
Vr
r
v
k
v
cycle
=
−=−
Thermal efficiency of CASOC:
Remember that this is an ideal cycle, and does not
account for variation of properties with temperature,
friction, heat losses, combustion losses, intake and
exhaust processes, etc. 18Fall 2020
9.2 OTTO CYCLE (12)
1
11
−−=
k
v
thr
• As the rv increases, the efficiency also
increases. However, in practice, this may
lead to DETONATION due to auto-ignition
of the fuel-air mixture.
• Lead (tetraethyl lead) in gasoline
improves detonation characteristics, but
is not environmentally friendly.
• Non-leaded gasolines with good
detonation characteristics have been
developed to reduce atmospheric
contamination.
19Fall 2020
9.2 OTTO CYCLE (13)
We can plot the Otto cycle efficiency
1
11
−−=
k
v
thr
5 10 15 200
0.2
0.4
0.6
0.8
k = 1.5k = 1.4k = 1.3
Compression Ratio
Th
erm
al E
ffic
ien
cy
Real Spark Ignition Cycle
20Fall 2020
9.3 BRAYTON CYCLE
OPEN CYCLE CLOSED CYCLE
21Fall 2020
22
“Air-standard analysis” involves an idealization that
simplifies the study of open gas turbine power plants.
Two basic assumptions:
1. The working fluid is air that behaves as an
ideal gas.
2. The temperature rise that would be
brought about by combustion is
accomplished by a heat transfer from an
external source.
Air-standard analysis (For gas turbine open cycles)
More on assumptions
Fall 2020
1. The working fluid is air, which behaves as an ideal gas.
2. The mass and properties of the fuel are neglected.
3. The combustion process is replaced by a heat transfer
from an external source
[It regards heat as being added through a perfect heat
exchanger and removed by exhaust to an infinite
atmosphere (or another perfect heat exchanger)].
4. All processes are internally reversible.
23
For COLD AIR STANDARD : constant specific heat is assumed
MAIN ASSUMPTIONS FOR IDEAL AIR
STANDARD BRAYTON CYCLE ANALYSIS
Fall 2020
24
Air-standard Ideal Brayton Cycle
Fall 2020
1-2 Compression: reversible and adiabatic (isentropic)
2-3 Heat added at constant pressure
3-4 Expansion: reversible and adiabatic (isentropic)
4-1 Heat rejected at constant pressure
‘
2
3
4p= cb
p=ca
1
25
P-v and T-s diagrams
Fall 2020
Applying the first law for an open system to
each CV shown in diagram
Recall the first law:
26Fall 2020
𝒅𝑬𝑪𝑽𝒅𝒕
= ሶ𝑸𝑪𝑽 − ሶ𝑾𝑪𝑽 + ሶ𝒎 𝒉𝒊 − 𝒉𝒆 +𝑽𝒊𝟐 − 𝑽𝒆
𝟐
𝟐+ 𝒈(𝒛𝒊 − 𝒛𝒆)
Steady-state
ΔKE = ΔPE = 0
𝟎 = ሶ𝑸𝑪𝑽 − ሶ𝑾𝑪𝑽 + ሶ𝒎 𝒉𝒊 − 𝒉𝒆
27
Recall :
Fall 2020
𝟎 = ሶ𝑸𝑪𝑽 − ሶ𝑾𝑪𝑽 + ሶ𝒎 𝒉𝒊 − 𝒉𝒆
𝟎 =ሶ𝑸𝑪𝑽
ሶ𝒎−
ሶ𝑾𝑪𝑽
ሶ𝒎+ 𝒉𝒊 − 𝒉𝒆
𝟎 = 𝒒 −𝒘 + 𝒉𝒊 − 𝒉𝒆To be applied to each control volume
Applying to specific control volume
shown on the diagram, the energy
equation becomes :
1-2 Compression: isentropic
2-3 Heat added at constant pressure
3-4 Expansion: isentropic
4-1 Heat rejected at constant pressure
28Fall 2020
𝟎 = 𝒒 −𝒘 + 𝒉𝒊 − 𝒉𝒆Recall :
𝑭𝒐𝒓 𝒄𝒐𝒎𝒑𝒓𝒆𝒔𝒔𝒐𝒓: 𝒘𝒄 = 𝒉𝟐 − 𝒉𝟏
𝑭𝒐𝒓 𝑯𝑿𝑸𝒊𝒏: 𝒒𝒊𝒏 = 𝒉𝟑 − 𝒉𝟐
𝑭𝒐𝒓 𝑯𝑿𝑸𝒐𝒖𝒕: 𝒒𝒐𝒖𝒕 = 𝒉𝟒 − 𝒉𝟏
𝑭𝒐𝒓 𝑻𝒖𝒓𝒃𝒊𝒏𝒆: 𝒘𝒕 = 𝒉𝟑 − 𝒉𝟒
29
𝜼𝒄𝒚𝒄𝒍𝒆 =𝒘𝒏𝒆𝒕
𝒒𝒊𝒏𝜼𝒄𝒚𝒄𝒍𝒆 =
𝒘𝒕 −𝒘𝒄
𝒒𝒊𝒏
𝜼𝒄𝒚𝒄𝒍𝒆 =(𝒉𝟑 − 𝒉𝟒) − (𝒉𝟐 − 𝒉𝟏)
(𝒉𝟑 − 𝒉𝟐)
𝜼𝒄𝒚𝒄𝒍𝒆 =(𝒉𝟑 − 𝒉𝟐) − (𝒉𝟒 − 𝒉𝟏)
(𝒉𝟑 − 𝒉𝟐)
𝜼𝒄𝒚𝒄𝒍𝒆 = 𝟏 −(𝒉𝟒 − 𝒉𝟏)
(𝒉𝟑 − 𝒉𝟐)
Cycle Thermal Efficiency
Fall 2020
Evaluating cycle thermal efficiency
30
1) Air standard analysis: Using air table data
(not assuming constant specific heat)
2) Cold air standard analysis: assuming
constant specific heat
Two different types of analyses
Fall 2020
‘
‘‘
2
3
4
p=const
p=const
1
When Air Table Data are used, recall the
following relationships (from Eq. 6.41)
Note also that P2 = P3 and P1 = P431
isentropic
processes
𝑷𝒓𝟐
𝑷𝒓𝟏=𝑷𝟐
𝑷𝟏
𝑷𝒓𝟒
𝑷𝒓𝟑=𝑷𝟒
𝑷𝟑
Where the values of “Pr” can be found in the tables
1) For Air standard analysis, i.e.,
Using air table data (not assuming constant specific heat)
Fall 2020
32
2) For Cold Air Standard Brayton Cycle analysis
We will develop an expression for the cycle
thermal efficiency, assuming constant
specific heat (or using cold air standard
analysis)
But first we will review some useful
pressure-temperature relationships
i.e., Assuming constant specific heat
Fall 2020
4
3
1
2NoteP
P
P
P=
For isentropic processes of 1-2
(compressor) and 3-4 (turbine) and
air is an ideal gas:
‘
‘‘
2
3
4
p=c
p=c
1
Review P-T relationship
Note: Cold air standard analysis assumes
constant specific heat 33
v
p
c
ckand =
𝒑𝟐𝑷𝟏
=𝑻𝟐𝑻𝟏
𝒌𝒌−𝟏
𝒑𝟑𝑷𝟒
=𝑻𝟑𝑻𝟒
𝒌𝒌−𝟏
Fall 2020
34
‘
‘‘
2
3
4
p=c
p=c
1
Also note :
𝒑𝟐𝑷𝟏
=𝑻𝟐𝑻𝟏
𝒌𝒌−𝟏 𝒑𝟑
𝑷𝟒=
𝑻𝟑𝑻𝟒
𝒌𝒌−𝟏Recall
𝑷𝟐
𝑷𝟏=𝑷𝟑
𝑷𝟒
𝑻𝟐𝑻𝟏
=𝑻𝟑𝑻𝟒
𝑻𝟒𝑻𝟏
=𝑻𝟑𝑻𝟐
𝑻𝟐𝑻𝟏
=𝑷𝟐
𝑷𝟏
𝒌−𝟏𝒌
Fall 2020
Substitute in
35
𝜼𝒄𝒚𝒄𝒍𝒆 = 𝟏 −(𝒉𝟒 − 𝒉𝟏)
(𝒉𝟑 − 𝒉𝟐)
𝜼𝒄𝒚𝒄𝒍𝒆 = 𝟏 −(𝑻𝟒 − 𝑻𝟏)
(𝑻𝟑 − 𝑻𝟐)
𝜼𝒄𝒚𝒄𝒍𝒆 = 𝟏 −𝑻𝟏(
𝑻𝟒𝑻𝟏
− 𝟏)
𝑻𝟐(𝑻𝟑𝑻𝟐
− 𝟏)
Δh = cpΔT for constant cP
Fall 2020
Recall :
36
For Cold Air
Standard Brayton
Cycle analysis
(Previously found)
𝜼𝒄𝒚𝒄𝒍𝒆 = 𝟏 −𝑻𝟏(
𝑻𝟒𝑻𝟏
− 𝟏)
𝑻𝟐(𝑻𝟑𝑻𝟐
− 𝟏)
𝑇4𝑇1
=𝑇3𝑇2
𝜼𝒄𝒚𝒄𝒍𝒆 = 𝟏 −𝑻𝟏𝑻𝟐
Fall 2020
37v
p
c
ckNote =:
Recall :
For Cold Air
Standard Brayton
Cycle analysis
(Previously found)
𝜼𝒄𝒚𝒄𝒍𝒆 = 𝟏 −𝑻𝟏𝑻𝟐
𝑻𝟐𝑻𝟏
=𝑷𝟐
𝑷𝟏
𝒌−𝟏𝒌
𝜼𝒄𝒚𝒄𝒍𝒆 = 𝟏 −𝟏
𝑷𝟐𝑷𝟏
𝒌−𝟏𝒌
Fall 2020
Thermal efficiency using Cold Air Standard Brayton
Cycle analysis
Note also that it is for an ideal cycle, which does not account for
❑ variation of properties with temperature
❑ components inefficiency
❑ friction
❑ heat losses
❑ combustion losses, etc. 38
𝜼𝒄𝒚𝒄𝒍𝒆 = 𝟏 −𝟏
𝒓𝒑
𝒌−𝟏𝒌
Where rp is the pressure ratio
𝒓𝒑=𝑷𝒎𝒂𝒙
𝑷𝒎𝒊𝒏
𝜼𝒄𝒚𝒄𝒍𝒆 = 𝟏 −𝟏
𝑷𝟐𝑷𝟏
𝒌−𝟏𝒌
Recall :
Fall 2020
We can plot the Brayton cycle efficiency
5 10 15 200
0.2
0.4
0.6
0.8
k = 1.5k = 1.4k = 1.3
Pressure Ratio
Th
erm
al E
ffic
ien
cy
.
Real Gas Turbine Cycle
−−=
k
k
p
cycle
r
1
11
39Fall 2020
• Ideal Cycle
ηth
Pressure Ratio, rp
100%
0%1
Ideal- The efficiency
continually
increases with
increasing
pressure ratio
ηth
Pressure Ratio, rp
Increasing Tmax
Real Cycle
• Real Cycle
- Efficiency and
work output peek as
the pressure ratio
increases.
- Pressure ratio for
maximum
efficiency and work
output increase
with Tmax.
40Fall 2020
• Due to the irreversibilities and losses, the actual T-s diagrams would be as illustrated
41
Effects of Irreversibilities
Fall 2020
42
𝜼𝒄 =𝒘𝟏−𝟐𝒔
𝒘𝟏−𝟐=𝒉𝟐𝒔 − 𝒉𝟏𝒉𝟐 − 𝒉𝟏
𝜼𝒕 =𝒘𝟑−𝟒
𝒘𝟑−𝟒𝒔=
𝒉𝟑 − 𝒉𝟒𝒉𝟑 − 𝒉𝟒𝒔
Isentropic efficiency
For compressor For turbine
Fall 2020
Compressor
Combustor
Turbine
Inlet Air Exhaust
Power
Fuel
Shaft
• Brayton cycle is used in several gas turbine engines.
43
Applications of Brayton Cycle
Fall 2020
OGT 2500 Turboshaft
(Electricity generation)
T56 Turboshaft (Turboprop)
(Thrust generation)
Turbojet 44Fall 2020
Turboprop Turbofan
Ramjet or Scramjet
Different engine types:
45Fall 2020
2
expansion)c(isentropiexit at
expansion)(real
2
exita
V
V t
nozzle
Brayton cycle for jet engines (turbojet)
46Fall 2020
F119-PW-100 Afterburning Turbojet
47Fall 2020
Examples from textbook
Example 9.4: Analyzing
the Ideal Brayton Cycle
Example 9.6 : Evaluating
Performance of a
Brayton Cycle with
Irreversibilities
Fall 2020 48
Example 9.4: Analyzing the Ideal Brayton
Cycle
49
Find: Determine the thermal
efficiency, the back work
ratio, and the net power
developed
Known: An ideal air-
standard Brayton cycle
operates with given
compressor inlet conditions,
given turbine inlet
temperature, and a known
compressor pressure ratio.
Fall 2020
Example 9.4 (continued)
50Find: Determine the thermal efficiency, the back work
ratio, and the net power developedFall 2020
Solution Approach
51Fall 2020
Fixing the states
State 1:
𝑇1 = 300 𝐾 → 𝑇𝑎𝑏𝑙𝑒 𝐴. 22 → 𝒉𝟏 = 𝟑𝟎𝟎. 𝟏𝟗𝒌𝑱
𝒌𝒈; 𝑷𝒓𝟏 = 𝟏. 𝟑𝟖𝟔
State 2:
𝑃𝑟2
𝑃𝑟1=
𝑃2
𝑃1(for isentropic process only)
𝑃𝑟2 = 𝑃𝑟1𝑃2
𝑃1= 1.386 10 = 13.86
Pr2 = 13.86 → 𝑇𝑎𝑏𝑙𝑒 𝐴. 22 → 𝒉𝟐 = 𝟓𝟕𝟗. 𝟗𝒌𝑱
𝒌𝒈
52
State 3:
𝑇3 = 1400 𝐾 → 𝑇𝑎𝑏𝑙𝑒 𝐴. 22 → 𝒉𝟑 = 𝟏𝟓𝟏𝟓. 𝟒𝒌𝑱
𝒌𝒈; 𝑷𝒓𝟑 = 𝟒𝟓𝟎. 𝟓
State 4:
𝑃𝑟4
𝑃𝑟3=
𝑃4
𝑃3(for isentropic process only)
𝑃𝑟4 = 𝑃𝑟3𝑃4𝑃3
= 450.51
10= 45.05
𝑃𝑟4 = 45.05 → 𝑇𝑎𝑏𝑙𝑒 𝐴. 22 → 𝒉𝟒 = 𝟖𝟎𝟖. 𝟓𝒌𝑱
𝒌𝒈
Fixing the states (continued)
Fall 2020
Substituting values in solving equations
53
A) Find thermal efficiency
𝑤𝑡 = ℎ3 − ℎ4 = 1515.4 − 808.5𝑘𝐽
𝑘𝑔= 706.9
𝑘𝐽
𝑘𝑔
𝑤𝑐 = ℎ2 − ℎ1 = 579.9 − 300.19𝑘𝐽
𝑘𝑔= 279.7
𝑘𝐽
𝑘𝑔
𝑞𝑖𝑛 = ℎ3 − ℎ2 = 1515.4 − 579.9𝑘𝐽
𝑘𝑔= 935.5
𝑘𝐽
𝑘𝑔
𝜂𝑐𝑦𝑐𝑙𝑒 =𝑤𝑡−𝑤𝑐
𝑞𝑖𝑛=
𝟕𝟎𝟔.𝟗−𝟐𝟕𝟗.𝟕𝑘𝐽
𝑘𝑔
𝟗𝟑𝟓.𝟓𝑘𝐽
𝑘𝑔
𝜼𝒄𝒚𝒄𝒍𝒆 = 𝟒𝟓. 𝟕% → 𝑨𝑵𝑺𝑾𝑬𝑹 (𝒂)
Fall 2020
𝜂𝑐𝑦𝑐𝑙𝑒 =𝑤𝑡 −𝑤𝑐𝑞𝑖𝑛
B) Find Back Work
Ratio (bwr)
𝑏𝑤𝑟 =𝑤𝑐𝑤𝑡
=𝟐𝟕𝟗. 𝟕𝟏
𝑘𝐽𝑘𝑔
𝟕𝟎𝟔. 𝟗𝑘𝐽𝑘𝑔
𝒃𝒘𝒓 = 𝟑𝟗. 𝟔%→ 𝑨𝑵𝑺𝑾𝑬𝑹 (𝒃)
Fall 2020 54
55
C) Find Net Power Developed (in kW)
Fall 2020
ሶ𝑾𝑵𝒆𝒕 = ሶ𝑾𝒄𝒚𝒄𝒍𝒆 = ሶ𝑾𝒕 − ሶ𝑾𝒄 = ሶ𝒎(𝒘𝒕 −𝒘𝒄)
𝑾𝒆 𝒉𝒂𝒗𝒆 𝒂𝒍𝒓𝒆𝒂𝒅𝒚 𝒇𝒐𝒖𝒏𝒅 "𝒘𝒕" 𝒂𝒏𝒅 "𝒘𝒄"𝑵𝒐𝒘𝒘𝒆 𝒏𝒆𝒆𝒅 𝒕𝒐 𝒇𝒊𝒏𝒅 ሶ𝒎
ሶ𝒎 = ሶ𝒎𝟏 = 𝝆𝟏 𝑽𝑨 𝟏 =𝑽𝑨 𝟏
𝒗𝟏
56Fall 2020
𝑤ℎ𝑒𝑟𝑒 𝑣𝑜𝑙𝑢𝑚𝑒𝑚𝑒𝑡𝑟𝑖𝑐 𝑓𝑙𝑜𝑤 𝑟𝑎𝑡𝑒 𝑉𝐴 1 𝑖𝑠 𝑔𝑖𝑣𝑒𝑛
𝑣1 =𝑅𝑎𝑖𝑟𝑇1
𝑃1(𝑓𝑟𝑜𝑚 𝑝𝑣 = 𝑅𝑇)
Where 𝑅𝑎𝑖𝑟 =ത𝑅
𝑀𝑎𝑖𝑟=
8.314𝑘𝐽
𝑘𝑚𝑜𝑙.𝐾
28.97𝑘𝑔
𝑘𝑚𝑜𝑙
= 0.287𝑘𝐽
𝑘𝑔.𝐾
Recall : ሶ𝒎 = ሶ𝒎𝟏 = 𝝆𝟏 𝑽𝑨 𝟏 =𝑽𝑨 𝟏
𝒗𝟏
𝑣1 =0.287
𝑘𝐽
𝑘𝑔.𝐾300 𝐾
100 𝑘𝑃𝑎= 0.861 (
𝑘𝐽
𝑘𝑔.𝑘𝑃𝑎)
𝑘𝑁.𝑚
𝑘𝐽
𝑘𝑃𝑎
𝑘𝑁/𝑚2
𝑣1 = 0.861𝑚3
𝑘𝑔
𝑾𝒆 𝒉𝒂𝒗𝒆 𝒕𝒐 𝒇𝒊𝒏𝒅 𝒕𝒉𝒆 𝒔𝒑𝒆𝒄𝒊𝒇𝒊𝒄 𝒗𝒐𝒍𝒖𝒎𝒆 "𝒗𝟏"
57
ሶ𝑚 =𝑽𝑨 𝟏
𝒗𝟏=
5𝑚3
𝑠
0.861𝑚3
𝑘𝑔
= 5.807𝑘𝑔
𝑠
Substituting all values in solving equation:
ሶ𝑊𝑁𝑒𝑡 = ሶ𝑚(𝑤𝑡 − 𝑤𝑐)
ሶ𝑊𝑁𝑒𝑡 = (5.807𝑘𝑔
𝑠)(706.9 − 279.7)
𝑘𝐽
𝑘𝑔
ሶ𝑾𝑵𝒆𝒕 = 𝟐𝟒𝟖𝟏 𝒌𝑾 → 𝑨𝑵𝑺𝑾𝑬𝑹 (𝒄)
Fall 2020
ሶ𝑚 =𝑽𝑨 𝟏
𝒗𝟏Recall :
Example 9.6 : Evaluating Performance of a
Brayton Cycle with Irreversibilities
58
pr
Known: An air-standard Brayton cycle
operates with given compressor inlet
conditions, given turbine inlet temperature,
and known compressor pressure ratio. The
compressor and turbine each have an
isentropic efficiency of 80%.
Find: Determine the thermal efficiency, the
back work ratio, and the net power
developed, in kW
Fall 2020
Example 9.6 (continued)
59
pr
Find: Determine the thermal
efficiency, the back work
ratio, and the net power
developed, in kWFall 2020
Fixing the states
60
State 1:
𝑇1 = 300 𝐾 ⇒ 𝑇𝑎𝑏𝑙𝑒 𝐴. 22 ⇒ ℎ1 = 300.19𝑘𝐽
𝑘𝑔𝑃𝑟1 = 1.386
State 2s:𝑃𝑟2𝑠
𝑃𝑟1=
𝑃2𝑠
𝑃1(for isentropic process only)
𝑃𝑟2𝑠 = 𝑃𝑟1𝑃2𝑠
𝑃1= 1.386 10 = 13.86
𝑃𝑟2𝑠 = 13.86 ⇒ 𝑇𝑎𝑏𝑙𝑒 𝐴. 22 ⇒ ℎ2𝑠 = 579.9𝑘𝐽
𝑘𝑔
Fall 2020
61
State 2:
𝜂𝑐 =𝑤1−2𝑠
𝑤1−2=
ℎ2𝑠−ℎ1
ℎ2−ℎ1
ℎ2 = ℎ1 +ℎ2𝑠−ℎ1
𝜂𝑐
ℎ2 = 300.19𝑘𝐽
𝑘𝑔+
579.9 − 300.19𝑘𝐽𝑘𝑔
0.8
ℎ2 = 649.8𝑘𝐽
𝑘𝑔
Fall 2020
ℎ2𝑠 = 579.9𝑘𝐽
𝑘𝑔Recall :
Fall 2020 62
State 3:
𝑇3 = 1400 𝐾 ⇒ 𝑇𝑎𝑏𝑙𝑒 𝐴. 22 ⇒ ℎ3 = 1515.4𝑘𝐽
𝑘𝑔𝑃𝑟3 = 450.5
State 4s:𝑃𝑟4𝑠
𝑃𝑟3=
𝑃4𝑠
𝑃3(for isentropic process only)
𝑃𝑟4𝑠 = 𝑃𝑟3𝑃4𝑠𝑃3
= 450.51
10= 45.05
𝑃𝑟4𝑠 = 45.05 ⇒ 𝑇𝑎𝑏𝑙𝑒 𝐴. 22 ⇒ ℎ4𝑠 = 808.5𝑘𝐽
𝑘𝑔
Fall 2020 63
State 4:
𝜂𝑡 =𝑤3−4
𝑤3−4𝑠=
ℎ3−ℎ4
ℎ3−ℎ4𝑠
ℎ4 = ℎ3 − 𝜂𝑡(ℎ3 − ℎ4𝑠)
ℎ4 = 1515.4𝑘𝐽
𝑘𝑔− 0.8 1515.4 − 808.5
𝑘𝐽
𝑘𝑔
ℎ4 = 949.9𝑘𝐽
𝑘𝑔
ℎ4𝑠 = 808.5𝑘𝐽
𝑘𝑔Recall :
Substituting values in solving equations
64
A) Find thermal efficiency
𝑤𝑡 = ℎ3 − ℎ4 = 1515.4 − 949.9𝑘𝐽
𝑘𝑔= 565.5
𝑘𝐽
𝑘𝑔
𝑤𝑐 = ℎ2 − ℎ1 = 649.8 − 300.19𝑘𝐽
𝑘𝑔= 349.6
𝑘𝐽
𝑘𝑔
𝑞𝑖𝑛 = ℎ3 − ℎ2 = 1515.4 − 649.8𝑘𝐽
𝑘𝑔= 865.6
𝑘𝐽
𝑘𝑔
𝜂𝑐𝑦𝑐𝑙𝑒 =𝑤𝑡−𝑤𝑐
𝑞𝑖𝑛=
565.5−349.6𝑘𝐽
𝑘𝑔
865.6𝑘𝐽
𝑘𝑔
𝜼𝒄𝒚𝒄𝒍𝒆 = 𝟐𝟒. 𝟗% ⇒ 𝑨𝑵𝑺𝑾𝑬𝑹 (𝒂)
Fall 2020
65
B) Find Back Work Ratio (bwr)
𝑏𝑤𝑟 =𝑤𝑐𝑤𝑡
=349.6
𝑘𝐽𝑘𝑔
565.5𝑘𝐽𝑘𝑔
= 61.8% ⇒ 𝑨𝑵𝑺𝑾𝑬𝑹 (𝒃)
Fall 2020
66
C) Find Net Power Developed (kW)
ሶ𝑊𝑁𝑒𝑡 = ሶ𝑊𝑐𝑦𝑐𝑙𝑒 = ሶ𝑊𝑡 − ሶ𝑊𝑐 = ሶ𝑚(𝑤𝑡 − 𝑤𝑐)
ሶ𝑚 = ሶ𝑚1 = 𝜌1 𝑉𝐴 1 =𝑉𝐴 1
𝑣1𝑉𝑜𝑙𝑢𝑚𝑒𝑚𝑒𝑡𝑟𝑖𝑐 𝑓𝑙𝑜𝑤 𝑟𝑎𝑡𝑒 𝑉𝐴 1 𝑖𝑠 𝑔𝑖𝑣𝑒𝑛.
𝑣1 =𝑅𝑎𝑖𝑟𝑇1
𝑃1(𝑓𝑟𝑜𝑚 𝑝𝑣 = 𝑅𝑇)
with 𝑅𝑎𝑖𝑟 =ത𝑅
𝑀𝑎𝑖𝑟=
8.314𝑘𝐽
𝑘𝑚𝑜𝑙.𝐾
28.97𝑘𝑔
𝑘𝑚𝑜𝑙
= 0.287𝑘𝐽
𝑘𝑔.𝐾
Fall 2020
Where:
67
Substituting values:
𝑣1 =0.287
𝑘𝐽
𝑘𝑔.𝐾300 𝐾
100 𝑘𝑃𝑎= 0.861 (
𝑘𝐽
𝑘𝑔.𝑘𝑃𝑎)
𝑘𝑁.𝑚
𝑘𝐽
𝑘𝑃𝑎
𝑘𝑁/𝑚2
𝑣1 = 0.861𝑚3
𝑘𝑔
Fall 2020
𝑣1 =𝑅𝑎𝑖𝑟𝑇1𝑃1
(𝑓𝑟𝑜𝑚 𝑝𝑣 = 𝑅𝑇)Recall:
Fall 2020 68
Mass flow rate
ሶ𝑚 =𝑽𝑨 𝟏
𝒗𝟏=
5𝑚3
𝑠
0.861𝑚3
𝑘𝑔
= 5.8𝑘𝑔
𝑠
Substituting values in solving equation:ሶ𝑊𝑁𝑒𝑡 = ሶ𝑚(𝑤𝑡 −𝑤𝑐)
ሶ𝑊𝑁𝑒𝑡 = (5.8𝑘𝑔
𝑠)(565.5 − 349.6)
𝑘𝐽
𝑘𝑔
ሶ𝑾𝑵𝒆𝒕 = 𝟏𝟐𝟓𝟐 𝒌𝑾 ⇒ 𝑨𝑵𝑺𝑾𝑬𝑹 (𝒄)
End of Examples 9.4 & 9.6
Fall 2020 69
Regenerative Gas Turbines
70
Regenerator Effectivenessሶ𝑸𝒊𝒏
ሶ𝒎= 𝒒𝒊𝒏 = 𝒉𝟑 − 𝒉𝒙 𝜼𝒓𝒆𝒈. =
𝒉𝒙 − 𝒉𝟐𝒉𝟒 − 𝒉𝟐
ሶ𝑸𝒊𝒏 𝒇𝒓𝒐𝒎 𝒆𝒙𝒕𝒆𝒓𝒏𝒂𝒍 𝒔𝒐𝒖𝒓𝒄𝒆𝒔 (𝒊. 𝒆. , 𝒉𝒐𝒕 𝒓𝒆𝒔𝒆𝒓𝒗𝒐𝒊𝒓)
Regenerator
(a heat exchanger)
Fall 2020
71
Ideal Gas Turbines with Reheat
Fall 2020
72
Ideal Gas Turbines with Reheat (continued)
(From referenced textbook)
For metallurgical reasons, the temperature of the
gaseous combustion products entering the turbine
must be limited. This temperature can be controlled
by providing air in excess of the amount required to
burn the fuel in the combustor (see Chap. 13).
As a consequence, the gases exiting the combustor
contain sufficient air to support the combustion of
additional fuel.
Some gas turbine power plants take advantage of
the excess air by means of a multistage turbine
with a reheat combustor between the stages.Fall 2020
Ideal Gas Turbines with Reheat (continued)
(From referenced textbook)
73
Despite the increase in net work with
reheat, the cycle thermal efficiency would
not necessarily increase because a greater
total heat addition would be required
Fall 2020
Two-stage compression with intercooling
74
Fig. 9.18 Two-stage compression with intercoolingFall 2020
75
Two-stage compression with
intercooling (continued)
The compressor work input can
be reduced by multistage
compression with intercooling
Fall 2020
EXAMPLE FROM TEXTBOOK
Example 9.7 : Evaluating Thermal Efficiency of a Brayton
Cycle with Regeneration
Fall 2020 76
Example 9.7 : Evaluating Thermal Efficiency
of a Brayton Cycle with Regeneration
77
pr
Known: A regenerative gas turbine
operates with air as the working
fluid. The compressor inlet state,
turbine inlet temperature, and
compressor pressure ratio are
known.
Find: For a regenerator
effectiveness of 80%, determine the
thermal efficiency. Also plot the
thermal efficiency versus the
regenerator effectiveness ranging
from 0 to 80%.
Fall 2020
Example 9.7 (continued)
78
pr
Find: For a regenerator effectiveness of 80%, determine
the thermal efficiency Fall 2020
Fall 2020 79
𝜼𝒄𝒚𝒄𝒍𝒆 =𝒘𝒕 −𝒘𝒄
𝒒𝒊𝒏
Find Thermal Efficiency
𝜼𝒄𝒚𝒄𝒍𝒆 =𝒉𝟑 − 𝒉𝟒 − (𝒉𝟐 − 𝒉𝟏)
(𝒉𝟑−𝒉𝒙)
Note: h1, h2, h3 and h4 are
the same as in Example 9.4
We need to find hx.
Note 1: On the Regenerator
(or Heat Exchanger)
80
𝜂𝑟𝑒𝑔𝑒𝑛𝑒𝑟𝑎𝑡𝒐𝒓 =ℎ𝑥 − ℎ2ℎ4 − ℎ2
ℎ𝑥 − ℎ2 :Heat energy actually received by a unitmass of coolant in the “cooler stream"
ℎ4 − ℎ2 : Hypothetical maximum of heat energy that could be
received by a unit mass of coolant in the “cooler stream"
2
4
x
Hotter
stream
Cooler
stream
(4 to y)
(2 to x)
Efficiency of regenerator:
Where:
Fall 2020
81
൯𝒉𝒙 = 𝒉𝟐 + 𝜼𝒓𝒆𝒈(𝒉𝟒 − 𝒉𝟐
𝜂𝑟𝑒𝑔𝑒𝑛𝑒𝑟𝑎𝑡𝑜𝑟 =ℎ𝑥 − ℎ2ℎ4 − ℎ2
Recall:
𝑭𝒊𝒏𝒅𝒊𝒏𝒈 𝒉𝒙
Fall 2020
𝒉𝒙 = 𝟓𝟕𝟗. 𝟗𝒌𝑱
𝒌𝒈+ 𝟎. 𝟖 𝟖𝟎𝟖. 𝟓 − 𝟓𝟕𝟗. 𝟗
𝒌𝑱
𝒌𝒈
𝒉𝒙 = 𝟕𝟔𝟐. 𝟖𝒌𝑱
𝒌𝒈
Note: h1, h2, h3 and h4 are the same as in Example 9.4
2
4
x
Hotter
stream
Cooler
stream
(4 to y)
(2 to x)
Note 2: "𝒒𝒊𝒏"
82
𝜂𝑐𝑦𝑐𝑙𝑒 =𝑤𝑡 − 𝑤𝑐𝑞𝑖𝑛
Where 𝒒𝒊𝒏 =ሶ𝑸𝒊𝒏
ሶ𝒎= 𝒉𝟑 − 𝒉𝒙
ሶ𝑸𝒊𝒏 𝒇𝒓𝒐𝒎 𝒆𝒙𝒕𝒆𝒓𝒏𝒂𝒍 𝒔𝒐𝒖𝒓𝒄𝒆𝒔 (𝒊. 𝒆. , 𝒇𝒓𝒐𝒎 𝒂 𝒉𝒐𝒕 𝒓𝒆𝒔𝒆𝒓𝒗𝒐𝒊𝒓)
For Brayton Cycle with Regeneration
The heat energy that the working fluid receives in the
regenerator (𝐟𝐫𝐨𝐦 𝐬𝐭𝐚𝐭𝐢𝐨𝐧 𝟐 𝒕𝒐 𝒙) is not counted in the
cycle efficiency expression
It’s a free energy obtained from the exhaust gas !!!
Fall 2020
Fall 2020 83
𝜂𝑐𝑦𝑐𝑙𝑒 =𝑤𝑡 −𝑤𝑐𝑞𝑖𝑛
Substituting values
𝜼𝒄𝒚𝒄𝒍𝒆 =𝒉𝟑 − 𝒉𝟒 − (𝒉𝟐 − 𝒉𝟏)
(𝒉𝟑−𝒉𝒙)
𝜼𝒄𝒚𝒄𝒍𝒆 =𝟏𝟓𝟏𝟓. 𝟒 − 𝟖𝟎𝟖. 𝟓
𝒌𝑱𝒌𝒈
− 𝟓𝟕𝟗. 𝟗 − 𝟑𝟎𝟎. 𝟏𝟗𝒌𝑱𝒌𝒈
𝟏𝟓𝟏𝟓. 𝟒 − 𝟕𝟔𝟐. 𝟖𝒌𝑱𝒌𝒈
𝜼𝒄𝒚𝒄𝒍𝒆 = 𝟎. 𝟓𝟔𝟖 = 𝟓𝟔. 𝟖% ⇒ 𝑨𝒏𝒔𝒘𝒆𝒓
Related Notes on Regenerator
84
ሶ𝒎𝑯𝒐𝒕(𝒉𝟒 − 𝒉𝒚) ሶ𝒎𝑪𝒐𝒐𝒍(𝒉𝒙 − 𝒉𝟐)
Assuming the same mass flow rate in both hot and cool streams, then
The rate of heat
transfer from the
hotter stream
The rate of heat
received by the
cooler stream
(𝒉𝟒 − 𝒉𝒚) (𝒉𝒙 − 𝒉𝟐)
Ideally
Ideally
Ideally
Reality% 𝒐𝒇 (𝒉𝟒 − 𝒉𝒚) (𝒉𝒙 − 𝒉𝟐)
Fall 2020
2
4
x
Hotter
stream
Cooler
stream
9.4 SUMMARY (1)
• Cold Air Standard Otto Cycle
1
11
−−=
k
v
Ottor
- Used to model spark-ignition internal combustion
engines
rv: Volumetric
Compression Ratio
3
2
4
v=c
v=c
1
85Fall 2020
9.4 SUMMARY (2a)
121221For hhw −=−
433443For hhw −=−
232332For hhq −=−
144114For hhq −=−
NOTE: w’s and q’s are written such that all have positive values
Air Standard Brayton Cycle
1-2 Compression: isentropic
2-3 Heat added at constant pressure
3-4 Expansion: isentropic
4-1 Heat rejected at constant pressure
‘
‘‘
2
3
4p=c
p=c
1
- Used to model gas turbine engines
86Fall 2020
9.4 SUMMARY (2b)
87
)(
)(1
23
14
hh
hhcycle
−
−−=
Brayton’s Cycle thermal efficiency
Fall 2020
9.4 SUMMARY (3)
• Cold Air Standard Brayton Cycle
- Used to model gas turbine engines
−−=
k
k
p
Brayton
r
1
11
Pressure Ratio rp = pmax / pmin
‘
‘
3
2
4
p=c
p=c
1
88Fall 2020
Regenerative Gas Turbines
89
Regenerator Effectiveness
𝜼𝒓𝒆𝒈. =𝒉𝒙 − 𝒉𝟐𝒉𝟒 − 𝒉𝟐
9.4 SUMMARY (4)
Regenerator
(a heat exchanger)
ሶ𝑸𝒊𝒏
ሶ𝒎= 𝒒𝒊𝒏 = 𝒉𝟑 − 𝒉𝒙
ሶ𝑸𝒊𝒏 𝒇𝒓𝒐𝒎 𝒆𝒙𝒕𝒆𝒓𝒏𝒂𝒍 𝒔𝒐𝒖𝒓𝒄𝒆𝒔 (𝒊. 𝒆. , 𝒉𝒐𝒕 𝒓𝒆𝒔𝒆𝒓𝒗𝒐𝒊𝒓)Fall 2020