3.2 The ‘state detrmined’ library
3.2.5 Model of a constant pressure combustion chamber
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(a) (b)
Fig. 3.34. Hot drum response: pressure (a) and liquid volume fraction (b).
(a) (b)
(c) (d)
Fig. 3.35. Evaporator response: (a) transfer fluid temperature, (b) pipe wall temperature, (c) organic fluid temperature and (d) organic fluid mass flow rate. Distribution with time and as function evaporator abscissa.
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The component can be at a first view modeled as a constant pressure heat exchanger where a stream of gases of defined composition enters with a certain temperature and enthalpy for leaving the chamber with higher temperature, but at the same pressure. The rise in sensible specific enthalpy occurs due to the oxidation of a fuel that is added and contributes, with its energy content at increasing the system overall energy. This particular heat exchanger does not require an hot exchanging media to rise the working fluid temperature, as in the examples presented in the previous sections, but rather requires a combustible. To be noted that for the representation of a closed Joule cycle the single phase heat exchanger presented in Par. 3.2.2 can be used, while the component presented here is introduced to simulate those plants where internal combustion processes are considered.
The combustion chamber is modeled here as a constant volume capacity. Air/fuel mixture is assumed to be perfectly homogeneous in each point of the CC and the combustion reactions to be instantaneous [36]. Therefore, differently to the heat exchangers presented in the previous sections, a simplified 0D model is proposed here and the property distribution within the chamber is homogeneous. The component is still state determined and can be considered as white box since only cardinal equations are applied, even though the combustion process is modelled through an empirical combustion coefficient ηb.
A block diagram of the model of the CC is shown in Fig. 3.36.
Inputs of the system are the air mass flow rate entering the CC (min), its temperature (Tin), the fuel mass flow rate (m ) and the mass flow rate of gases leaving the combustion chamber (f mout).
The State vector of the system is represented by the air/fuel mass stored within the volume of the CC (mCC), its internal energy (UCC) and the internal energy stored as heat within the combustion chamber walls (Uw).
Outputs of the model are the pressure in the combustion chamber (pCC), the temperature of the exhaust gases leaving the system (Tout), and, if required, the combustion chamber wall temperature (Tw).
Fig. 3.36. Block Diagram of the combustion chamber model.
It should be noted that, while pressure is assumed to be uniform within the component at a given time instant (hence the heat exchange process is assumed to be isobaric) pressure may change in time and therefore the overall heat exchange may take place as a sequence of constant pressure processes. The pressure changes in time is due to the accumulation of mass that may occur in transients, due to accumulation of energy that may lead to an increase in the gas temperature. These phenomena may occur in unsteady operation conditions.
Since the vector of state variables has three elements, three state equations in scalar form are needed.
These equations are the mass conservation equation, the energy equation applied to the mass of burnt gas within the combustion chamber, and the energy equation applied to the walls of the combustion chamber applied in zero dimensional form. The spatial distribution of the state variables is therefore neglected and
F
( )
m t
in
( )
m t
( )
mout t
in
( )
T t
( ) ( ) ( )
CC
CC
w
m t
U t
U t
out
( )
T t
CC
( )
p t
( )
Tw t
inputs state variables outputs
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The CC has been modelled as a constant volume with semi-permeable walls. Application of the mass conservation equation in differential form to the control volume of the combustion chamber gives:
( ) ( ) ( ) ( )
CC
in F out
dm m t m t m t m t
dt =
∑
= + − (3.61)Application of the energy equation to the mass of burnt gases within the combustion chamber, assuming that the work exchanged is zero (the CC has constant volume), gives:
, CC
chem w in in out
dU q q H H
dt = − + − (3.62)
where:
chem F vi b
q = m H η (3.63)
and represent the heat provided to the system through the combustion of the fuel added at any instant of simulation. The combustion is assumed to be instantaneous and homogeneous but not perfect, since a combustion efficiency ηb is introduced.
The total enthalpy flow entering and leaving the combustion chamber are calculated from the temperatures according to the following:
H =mc T p (3.64)
qw,in in Eq. (3.62) is the heat flux from the burning zone to the combustion chamber walls. This term appears in the energy equation applied to the walls, whose temperature Tw is assumed uniform and constant:
, , ,
w w
w in w out w p w
dU dT
q q m c
dT = − = dt (3.65)
Terms qw,in and qw,out can be calculated assuming the convective heat fluxes between the burned gas within the CC and the chamber walls (Eq(3.66)), and between the walls and the environment (Eq.(3.67)):
, , ( 3 )
w in in w in w
q =h A T −T (3.66)
, , ( 1)
w out out w out w
q =h A T −T (3.67)
Definition and solution of the Output Equations are achieved through integration of the State Equations, at each time step t:
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0
,0 t
CC CC CC
t t
m m m dt
=
= +
∫
(3.68)0
,0 t
CC CC CC
t t
U U U dt
=
= +
∫
(3.69)0
,0 t
w w w
t t
U U U dt
=
= +
∫
(3.70)This requires the definition of initial values for the vector of state variables (initialization):
0 ( )0 [ CC,0, CC,0, w,0]
X = X t = m U U (3.71)
The Output Equation (that is algebraic) can now be solved at each time instant in order to generate the vector of output variables:
CC CC
CC
CC
p m RT
= V (3.72)
3
, CC
CC p g
T U
=m c (3.73)
, w w
w p w
T U
=m c (3.74)
In this case, seen the limited number of equations and states involved in the model, a different modelling approach has been adopted and instead that compiling code function, icon programming has been preferred using predefined Simulink® operational blocks. The resulting model of the combustion chamber is presented in Fig. 3.37 (a) while Fig. 3.37 (b) shows the block dialog window.
(a) (b)
Fig. 3.37. Simulink® model of the combustion chamber (a) and block dialog mask (b).
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