This document discusses thermodynamic modeling of engine cycles from the 20th century. It presents: 1) Carnot's analysis that prime mover models should be cyclic and the highest temperature process generates the most work. Accurate temperature predictions improve model reliability. 2) Phenomenological models for engine cycles that consider the actual working fluid constituents like fuel-air rather than just air. This leads to more accurate property predictions. 3) Detailed analysis of processes in the ideal Otto cycle like variable properties in isentropic compression and constant volume combustion to maximize temperature increase and entropy reduction. The document emphasizes developing accurate thermodynamic models of engine cycles considering real fluid properties and temperature-dependent phenomena like collision theory

1. 20th Century Thermodynamic Modeling of Automotive
Prime Mover Cycles
P M V Subbarao
Professor
Mechanical Engineering Department
Respect True Nature of Substance…..

2. Theoretical Learnings from Carnot’s Analysis
• Any model developed for a prime mover be a cyclic model.
• The most important part of the model is the process that
generates the highest temperature.
• Very important to develop a model, which predicts the
temperatures more accurately.
• Higher the accuracy of temperature predictions, higher will be
the reliability of the predictions…
• Enhances the closeness between theory & Practice.
 










 
1
)
1
(
1
1
1 1
c
k
c
k
c
const
Dual
r
k
r
r
v 




3. Important Feature of An Artificial Horse
Air/fuel Ratio
Stoichiometric
Mixture

th
,
%
Lean Rich
Predictions by Air-standard Cycle
Actual Prime Mover

4. The Thermodynamics Importance of
Temperature
• From the Gibbsian equations, the change of specific entropy of any
substance during any reversible process.
vdp
dh
pdv
du
Tds 



• Consider a control mass executing a Isothermal heat addition
process as suggested by Carnot: pdv
du
Tds 

Heat addition at a highest absolute
temperature leads a lowest increase
in entropy for a given increase in
specific volume of a control mass.
ds
dv
p
ds
du
T 

• For an Ideal gas executing above process:
ds
dv
p
ds
dT
c
T v 

ds
dp
R
or
ds
dv
R
T 

Temperature is created by
mere Compression ??!!!!???

5. The Thermodynamics of Temperature
Creation : Otto’s Model
• From the Gibbsian equations, the change of specific entropy of any
substance during any reversible process.
vdp
dh
pdv
du
Tds 



• Consider a control mass executing a constant volume heat addition
process:
pdv
du
Tds 

constant




v
s
u
T
The relative change in internal energy of a control mass w.r.t.
change in entropy at constant volume is called as absolute
temperature.

6. The Thermodynamics of Temperature Creation :
Diesel’s Model
vdp
dh
Tds 

• Consider a control mass executing a reversible constant pressure
heat addition process:
constant




p
s
h
T
The relative change in enthalpy of a control volume w.r.t. change
in entropy at constant pressure is called as absolute temperature.

7. Phenomenological Models for Engine Cycles
• Fuel-air analysis is more accurate analysis when compared to
Air-standard cycle analysis.
• An accurate representation of constituents of working fluid is
considered.
• More accurate models are used for properties of each
constituents.
Process Otto’s Model Diesel’s Model
Intake Air+Fuel +Residual gas Air+ Residual gas
Compression Air+Fuel vapour +Residual gas Air + Residual gas
Expansion Combustion products Combustion Products
Exhaust Combustion products Combustion Products

8. Fuel-Air Model for Otto Cycle
Otto
Cycle
Air+Fuel
vapour
+Residual
gas
TC
BC
Compression
Process
Const volume
combustion
Process
Expansion
Process
Const volume
Blow down
Process
Products of
Combustin
Products of
Combustin

9. 20th Century Analysis of Ideal Otto Cycle
• This is known as Fuel-air Cycle.
• 1—2 Isentropic compression of a mixture of air, fuel vapour and
residual gas without change in chemical composition.
• 2—3 Complete combustion at constant volume, without heat loss,
with burned gases in chemical equilibrium.
• 3—4 Isentropic expansion of the burned gases which remain in
chemical equilibrium.
• 4—5 Ideal adiabatic blow down.

10. Isentropic Compression Process: 1 - 2
For a infinitesimal compression process: pdv
du
Tds 

pdv
du 

0
dv
v
T
dT
R
cv


Mass averaged properties for an ideal gas mixture:


 





n
i
i
i
n
i
i
v
i
v
n
i
i
p
i
p R
x
R
c
x
c
c
x
c
1
1
,
1
, &
&
0

 pdv
dT
cv
0

 dv
v
RT
dT
cv
v
dv
T
dT
R
cv 














0

 dv
v
RT
dT
cv
Assume ideal gas nature with variable properties:

11. Variation of Specific Heat of Ideal Gases
kgK
kJ
T
C
T
C
T
C
C
cp /
1000
1000
1000
3
3
2
2
1
0 















Gas C0 C1 C2 C3
Air 1.05 -0.365 0.85 -0.39
Methane 1.2 3.25 0.75 -0.71
CO2 0.45 1.67 -1.27 0.39
Steam 1.79 0.107 0.586 -0.20
O2 0.88 -0.0001 0.54 -0.33
N2 1.11 -0.48 0.96 -0.42

12. Variable Properties of Air
0.5
0.7
0.9
1.1
1.3
1.5
0 200 400 600 800 1000 1200 1400
Temperature,K
g
cp
cv

13. Properties of Fuels
kgK
kJ
T
C
T
C
T
C
T
C
C
C f
p /
1000
1000
1000 2
4
3
3
2
2
1
0
, 






















Fuel C0 C1 C2 C3 C4
Methane -0.29149 26.327 -10.610 1.5656 0.16573
Propane -1.4867 74.339 -39.065 8.0543 0.01219
Isooctane -0.55313 181.62 -97.787 20.402 -0.03095
Gasoline -24.078 256.63 -201.68 64.750 0.5808
Diesel -9.1063 246.97 -143.74 32.329 0.0518

14. Isentropic Compression model with variable
properties : 1 - 2
v
dv
T
dT
R
cv 













   
 


 2
2
sin
cos
1
2
1
1 





 R
R
r
V
V
c
   
 











 

 2
2
sin
cos
1
2
1
1 R
R
r
m
V
v c
   
 



v
T
R
p 

15. True Phenomenological Model for Isentropic
Compression
dv
v
RT
dT
cv 

v
dv
R
T
dT
cv 

kgK
kJ
T
c
T
c
T
c
c
c v
v
v
v
v /
3
3
,
2
2
,
1
,
0
, 



Let the mixture is modeled as:
      





























1
2
3
1
3
2
3
,
2
1
2
2
2
,
1
2
1
,
1
2
0
, ln
3
2
ln
v
v
R
T
T
c
T
T
c
T
T
c
T
T
c v
v
v
v
  
 




2
1
2
1
3
3
,
2
2
,
1
,
0
,
v
dv
R
T
dT
T
c
T
c
T
c
c v
v
v
v
      



























r
R
T
T
c
T
T
c
T
T
c
T
T
c v
v
v
v
1
ln
3
2
ln 3
1
3
2
3
,
2
1
2
2
2
,
1
2
1
,
1
2
0
,

16. Generalized First Order Models for Variable
Specific Heats
T
k
a
c p
p 

 1
T
k
b
c v
v 

 1
ap = 28.182 – 32.182 kJ/kmol.K
bv = 19.868 – 23.868 kJ/kmol.K
k1 = 0.003844–0.009844 kJ/kmol.K2
For design analysis of Engine Models:

17. Isentropic Compression model with variable
properties
• For compression from 1 to 2:
  
















1
2
1
2
1
2
1 ln
ln
v
v
R
T
T
b
T
T
k v
  
 


2
1
2
1
1
v
dv
R
T
dT
T
k
bv
  
















r
R
T
T
b
T
T
k v
1
ln
ln
1
2
1
2
1

18. The Role of Isentropic Compression
       
r
R
T
T
c
T
T
c
T
T
c
T
T
c v
v
v
v ln
3
2
ln 3
1
3
2
3
,
2
1
2
2
2
,
1
2
1
,
1
2
0
, 




















   
r
R
T
T
b
T
T
k v ln
ln
1
2
1
2
1 









Second order Property Model:
First order Property Model:
• Ready for combustion:
• In a combustion reaction, bonds are being broken and formed
between different atoms in molecules.
• The parts of the molecules that undergo bond breakage and
formation need to line up with each other.
• There needs to be the appropriate overlap in the orbitals that are
"donating" and "accepting" electrons.
• The probability of occurrence of appropriate overlap is
proportional to temperature of reacting molecules.

19. Collision Theory
• Collision theory says that ”in order for a chemical reaction to
happen, three separate things need to happen” :
• 1. The molecules have to hit each other
• 2. The molecules have to hit each other in the right way (both
have to be facing the right way)
• 3. The molecules have to hit each other with enough speed
(energy of motion, or "kinetic energy") to activate the reaction.
Number of successful collisions  Frequency of collisions  Time
available for collision.

20. Phenomenological Modeling of Combustion
• Engineering Objective of Combustion:
• To Create Maximum Possible Temperature through
conversion of free energy into microscopic kinetic energy.
Thermodynamic Strategy for conversion:
Constant temperature combustion
Constant volume combustion
Constant pressure combustion

21. Engineering Strategy to Utilize A Resource
• Engineering constraint: Both combustion and expansion have
to be finished in a single stroke.
• Rapid combustion : Constant Volume combustion
– Less time to combustion process.
– More time to adiabatic expansion
• Slow combustion : Constant pressure combustion
– More time to combustion process.
– Less time to adiabatic expansion