Environmental Engineering Reference
In-Depth Information
The first model input is the inducted gas composition, which is formulated as
the ratio of moles of reinducted product (i.e., previously exhausted combustion
gas) to moles of inducted reactant charge (i.e., fuel and air), denoted
. The
second model input is the final valve closure, which dictates the volume, V
1
=
V(
1
), at the start of compression and therefore the effective compression ratio.
Model outputs are the peak pressure, P, and the volume at the onset of
combustion, V
23
= V (
23
), which acts as a proxy for combustion timing. By
linking the thermodynamic states of the system together, a dynamic model of
peak pressure, P, and phasing,
23
, for residual-affected LTC is formulated.
Here the modeling techniques are applied to propane-fueled LTC. The model
will also apply to other fuels by making appropriate changes to the fuel-specific
model constants (in particular the lower heating value). The LTC control model
dynamics for single-cylinder, constant engine speed operation have the follow-
ing mathematical form:
ð
1
þ
1
Þ
P
1
V
1
;
1
V
23
:
1
V
1
:
V
1
1
;
V
1
23
;
V
23
:
þ
ð
1
"Þ
LHV
C
3
H
8
V
1
:
P
1
=
1
ð
1
e
Þ
LHV
C
3
H
8
þ
c
T
in
V
23
:
þ
ð
1
"Þ
LHV
C
3
H
8
P
1
=
1
P
¼
c
T
in
ð
1
þ
1
Þ
xP
1
V
1
;
1
V
23
;
1
(4
:
1)
¼
f
1
ð
states; inputes
Þ
¼
f
1
ð
P
1
;
23
;
;
23
;
1
;
;
1
;
1
;
;
1
;
1
Þ
Here the subscript k and k-1 denote the variable value at the current and
previous engine cycles, respectively. Other parameters include
c
v
- average constant volume specific heat for the reactant and residual gases
T
in
- temperature of the incoming reactant gas, assumed constant
LHV
C3H8
- lower heating value of propane for a given number of moles of
reactant (a measure of the amount of energy released during combustion),
constant assuming a fixed equivalence ratio
"
,
- constants related to in-cylinder and exhaust manifold heat transfer
- specific heat ratio
The presence of cycle-to-cycle dynamics is evident by inspection of Eq. 4.1, as
the current peak pressure P
k
depends on the previous cycle peak pressure P
k-1
and combustion timing
23,k-1
. This is a very powerful expression as it relates a
desired model output, the peak pressure, to the model inputs, the molar ratio of
the reinducted products and reactants,
, and the final valve closure timing
1,k
(via V
k,1
). Additionally, note the dependence on the combustion timing (repre-
sented by the combustion volume, V
23
). What is now required is a physics-based
expression for the combustion timing.
By using a simplified version of the integrated Arrhenius rate used in the
simulation model, a nonlinear dynamic model of the following form can be
derived for the combustion timing:
