Hamilton–Jacobi–Bellman equations and dynamic programming for power-optimization of radiative law multistage heat engine system

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International Journal of Energy and Environment (IJEE), Volume 3, Issue 3, 2012, pp.359-382
ISSN 2076-2895 (Print), ISSN 2076-2909 (Online) ©2012 International Energy & Environment Foundation. All rights reserved.

363
Let
1
α
and
2
α
be the heat transfer coefficients corresponding to the high- and low-temperature sides,
respectively,
1V
a
is the heat transfer area between the driving fluid per unit volume and the working fluid
of the heat engine at the high-temperature side, and
1
F
is the driving fluid cross-sectional area,
perpendicular to
x
. The above parameters are all known for the real systems. For the radiative heat
transfer law, one has
11B
α σε
=
, where
1
ε
is the emissivity of the photon flux. The first law of
thermodynamics gives

3
11
111
1111 11 11
64
3
hh
VBV V
GC dT GC dT
qTdT
k a Fvdt a Vdt c a dt
ασεε
−−
== =−

(16)

For the given integration section
[, ]
if
τ τ
, the boundary temperatures of the driving fluid are denoted as
11
()
ii
TT
τ
=
and
11
()
ff
TT
τ
=
, then the power output
W

and the entropy generation rate
s
σ
are,
respectively, given by

11
11
33
12 12
111
''
64 64
[(1)][(1)]
33
ff f
ii i
TT t
I
BB
h
TT t
VT T VT T
WGCdT dT Tdt
cT cT
σσ
η
=− =− − =− −
∫∫ ∫


(17)

11
11
3
1
11 1
' '
12 1
64
11 11
() () [ ()]
3
fff
iii
TTt
I
h
B
sh C
TTt
GC
VT
GC dT dT T dt
TT T c TT
σ
σηη
=− − =− − =− −
∫∫∫


(18)

where
11
/TdTd
τ
=

. The dot notation signifies the time derivative. The pressure
p
of the photon flux is a
function of the temperature
1
T
, which is given by
4
1
4/(3)
B
pTc
σ
=
according to the thermodynamics of
radiation. When effects of change of pressure
p
on the power output of the multistage heat engine
system are considered, another calculation expression of the power output
W

is given by [35-40]

1
1
1
1
2
1
'
1
33 33
12 1 112
11
''
[(1 ) ]
16 16 64 16
[(1) ] ( )
33
f
i
f f
i i
T
II
V
T
Tt
BB
Tt
T
dp
WGC dT
TdT
TT T TTT
VdTVTdt
cTc ccT
σσ
=− − +
=− − + =− −
∫
∫∫


(19)

1
1
3
1
11
''
11
16
11 11
() [ ()]
ff
ii
Tt
II
B
sV
Tt
VT
GC dT T dt
TT c TT
σ
σ
=− − =− −
∫∫


(20)

Refs. [35-40] calculated the maximum power output for the case with pseudo-Newtonian heat transfer
law based on Eq. (19). This paper will further considered two different cases with and without effects of
the pressure, and calculate the optimization results for radiative and pseudo-Newtonian heat transfer
laws. If the multistage endoreversible Carnot heat engine turns to reversible, Eqs. (17) and (19) further
give

44 33
11 211
16 ( ) 64 ( )
39
B if Bif
I
rev
VTT VTTT
W
cc
σσ
−−
=−


(21)

44 33
11 211
16 ( ) 16 ( )
33
B if Bif
II
rev
VTT VTTT
W
cc
σσ
−−
=−


(22)

In Eqs. (21) and (22),
rev
W

is the reversible power output performance limit. If
12f
TT=
further, Eqs. (21)
and (22), respectively, become

International Journal of Energy and Environment (IJEE), Volume 3, Issue 3, 2012, pp.359-382
ISSN 2076-2895 (Print), ISSN 2076-2909 (Online) ©2012 International Energy & Environment Foundation. All rights reserved.

364
44
4
11
22
P
11
16 16
44
[1 ( ) ]
333 3
I
Bi Bi
rev
ii
VT VT
TT
W
cTT c
σσ
η
=−+=


(23)

44
11
2
j
1
16 16
(1 )
33
II
Bi Bi
rev class
i
VT VT
T
WA
cT c
σσ
η
== −=


(24)

P
η
and
C
η
in Eqs. (23) and (24) are the named Petela’s efficiency and Jeter’s efficiency [47-51]. What
should be paid attention is that the form of the efficiency
j
η
derived by Jeter is the same as that of Carnot
efficiency.
class
A
in Eq. (24) is called classical thermodynamic exergy of radiation photon flux. For the
endoreversible Carnot heat engine system considered herein, there exists loss of irreversibility due to the
finite rate heat transfer, and the high-temperature driving fluid temperature can not decrease to the low-
temperature environment temperature
2
T
in a finite time, so the maximum value of Eq. (19) is smaller
than
class
A
of Eq. (24) consequentially. Combining Eq. (10) with Eq. (16) yields

4'
111
33' 1 3
111 1 22
(5 ) ( )
[( )( / ) / ( ) 1]
nn n
n
dT n T T T
dt T k T T T k T
β
−
−
−−
=−
+
(25)

where
11
3/64
V
ca
β ε
=
. Substituting Eq. (25) into Eqs. (17) and (19) yields

4'
11 2
3' 1 3 '
11 1 22
64 (5 ) ( )
{(1)}
3[( )( / ) /( ) 1]
f
i
nn n
t
I
B
n
t
VnTTTT
Wdt
ckT T T kT T
σβ
−
−
−−
=−
+
∫


(26)

4'
211
'3'13
11 1 22
(5 ) ( )
64 16
{( ) }
3[()(/)/()1]
f
i
nn n
t
II
B
n
t
TV nT TT
Wdt
ccT kTTT kT
σβ
−
−
−−
=−
+
∫


(27)

3. Optimization
The problem now is to determine the maximum values of Eqs. (26) and (27) subject to the constraint of
Eq. (25). The control variable is
'
21' 2'
/TTTT≡
, and the inequality
11'2'2
TTT T
>>>
always holds for the
heat engine, so one obtains
'
21
TTT
≤≤
. This optimal control problem belongs to a variational problem
whose control variable has the constraint of closed set, and the Pontryagin’s minimum value principle or
Bellman’s dynamic programming theory may be applied. When the state vector dimension of the optimal
control problem is small, the numerical optimization conducted by the dynamic programming theory is
very efficient. Let the optimal performance objective of the problem be
max 1 1
(,, ,)
ii f f
WT T
τ τ

, and the
admissible control set of the control variable
'
()Tt
is denoted as
Ω
. The performance objective of the
control problem can be expressed as follows

'
'
'
max 1 1 1 1 0 1
()
()
( , , , ) max[ ( , , , )] max[ ( , , ) ]
f
i
t
ii f f ii f f
t
Tt
Tt
WTtTt WTtTt fTTtdt
∈Ω
∈Ω
≡=
∫

(28)

The Hamilton-Jacobi-Bellman (HJB) control equation of the optimization problem is

'
''
max max
01 1
()
1
max{ ( , , ) ( , , )} 0
Tt
WW
fTTt fTTt
tT
∈Ω
∂∂
++ =
∂∂

(29)

where
'
01
(, ,)
fTTt
corresponds to integrands in Eqs. (26) and (27), and
'
1
(, ,)
fTTt
corresponds to the right
term of Eq. (25). Then HJB control equations corresponding to objectives of Eqs. (26) and (27) are,
respectively, given by

International Journal of Energy and Environment (IJEE), Volume 3, Issue 3, 2012, pp.359-382
ISSN 2076-2895 (Print), ISSN 2076-2909 (Online) ©2012 International Energy & Environment Foundation. All rights reserved.

365
'
4'
max max
211
33' 1 3
()
1111 1 22
(5 ) ( )
max{[ (1 ) ] } 0
'[()(/)/()1]
II
nn n
h
n
Tt
WW
TnTTT
GC
tTTTkTTTkT
β
−
−
∈Ω
∂∂
−−
+−− =
∂∂ +

(30)

'
4'
max max
211
'33'13
()
1111 1 22
(5 ) ( )
max{[( ) ] } 0
[( )( / ) / ( ) 1]
II II
nn n
hV
n
Tt
WW
TnTTT
GC GC
tTTTkTTTkT
β
−
−
∈Ω
∂∂
−−
+−− =
∂∂ +

(31)

There are only analytical solutions of Eqs. (30) and (31) for the special cases, while for the radiative heat
transfer law, one has to refer to numerical methods. Consider that the continuous differential equation
should be discretized for the numerical calculation performed on the computer, and then the discrete
equations are given based on Eqs. (25)-(27), as follows

4'
11 2
3' 1 3 '
1
11 1 22
64 (5 ) [( ) ( ) ]
() { (1 )}
3[ ( )( / ) /( ) 1]
i n in in
N
IN
B
iiin i
i
VnTTTT
W
ck T T T kT T
σβθ
−
−
=
−−
=−
+
∑


(32)

4'
211
'3'13
1
11 1 22
(5 )( ) [( ) ( ) ]
64 16
() {( ) }
3[()(/)/()1]
iininin
N
II N
B
iiiin
i
TV nT T T
W
ccT kT TT kT
σβθ
−
−
=
−−
=−
+
∑


(33)

4'
1
11
11
33' 13
111 1 22
(5 ) [( ) ( ) ]
()[()( / ) /( )1]
nin in
ii i
iiiin
nT T T
TT
TkTTT kT
β
θ
−
−
−
−−
−=−
+
(34)

1ii i
tt
θ
−
−=
(35)

The optimal control problem is to determine the maximum values of Eqs. (32) and (33) subject to the
constraints of discrete Eqs. (34) and (35). From Eqs. (32)-(35), the Bellman’s backward recurrence
equations corresponding to Eqs. (32) and (33) are, respectively, given by

'
4'
()
11 2
max 1
3' 1 3 '
,
11 1 22
4'
() 1
11
max 1
33' 13
111 1 22
64 (5 ) [( ) ( ) ]
(,)max{ (1 )
3[ ( )( / ) /( ) 1]
(5 ) [( ) ( ) ]
(,)}
()[()( / ) /( )1]
ii
i n in in
Ii i i
B
iiin i
T
nin in
Ii i i i i
iiiin
VnTTTT
WTt
ck T T T kT T
nT T T
WT t
TkTTT kT
θ
σβθ
β
θ θ
−
−
−
−
−
−−
=−
+
−−
++ −
+



(36)

'
4'
()
211
max 1
'3'13
,
11 1 22
4'
()1
11
max 1
33' 13
111 1 22
(5 )( ) [( ) ( ) ]64 16
(,)max{( )
3[()(/)/()1]
(5 ) [( ) ( ) ]
(,
()[()( / ) /( )1]
ii
iininin
II i i i
B
iiiin
T
nin in
II i i i i
iiiin
TV nT T T
WTt
ccT kT TT kT
nT T T
WT t
TkTTT kT
θ
σβθ
β
θ θ
−
−
−
−
−
−−
=−
+
−−
++ −
+



)}
i
(37)

4. Analysis for special cases
4.1 For pseudo-Newtonian heat transfer law
When
n1
=
, i.e. the heat transfer between the working fluid and the heat reservoir obeys pseudo-
Newtonian heat transfer law. From Appendix A, Refs. [11, 35-40] derived analytical solutions of
extremum power output and the optimal fluid temperature configuration based on pseudo-Newtonian
heat transfer law, i.e. Eqs. (A12) and (A14). However, Eqs. (A12) and (A14) were obtained based on the
condition that the total equivalent thermal conductance is a constant. This condition is very strictly,
which is due to that the total equivalent thermal conductance is a function of the reservoir temperature
1
T
. The temperature
1
T
changes along the fluid flow direction, so the condition that the total thermal
conductance is a constant is difficult to hold. Thus there are also no analytical solutions for the case with
the pseudo-Newtonian heat transfer law, but some algebra equations related to the optimal solutions can
be obtained. Eqs. (25), (30) and (31), respectively, become

'
11
33
11 2 2
4( )
[( ) / ( ) 1]
dT T T
dt k T k T
β
−
=−
+
(38)

International Journal of Energy and Environment (IJEE), Volume 3, Issue 3, 2012, pp.359-382
ISSN 2076-2895 (Print), ISSN 2076-2909 (Online) ©2012 International Energy & Environment Foundation. All rights reserved.

366
'
'
max max
21
33
()
11122
4( )
max{[ (1 ) ] } 0
'[()/()1]
II
h
Tt
WW
TTT
GC
tTTkTkT
β
∈Ω
∂∂
−
+−− =
∂∂+

(39)

'
'
max max
21
'33
()
11122
4( )
max{[( ) ] } 0
[( ) /( ) 1]
II II
hV
Tt
WW
TTT
GC GC
tTTkTkT
β
∈Ω
∂∂
−
+−− =
∂∂+

(40)

When
max
I
W
is chosen to be the optimization objective, maximizing the second term of Eq. (39) with
respect to
'T
yields

'1
12 max 1
/[1 ( ) ( / )/]
I
h
TTT GC W T
−
=−∂∂

(41)

Substituting Eq. (41) into Eq. (39) yields

12
max 1
max 1 2 1
33
11 2 2
4
{[1 ( )( / )] / } 0
[( ) / ( ) 1]
I
I
h
h
WGCT
GC W T T T
tkTkT
β
−
∂
+−∂∂−=
∂+


(42)

The second term of Eq. (42) is the extremum Hamilton function
1max1
(, / )
I
H TW T∂ ∂



12
1
1 max 1 max 1 2 1
33
11 2 2
4
(, / ) {[1( )( / )] / }
[( ) / ( ) 1]
I I
h
h
GC T
H TW T GC W T TT
kT kT
β
−
∂∂= − ∂∂−
+

(43)

From Eq. (43), one can see that
H
contains the variable
τ
inexplicitly, and the equation
//dH d H
τ τ
=∂ ∂
holds for the Hamilton function, so the Hamilton function is autonomous and
1max1
(, / )
I
H TW T∂∂

keeps constant along the optimal path. Let the constant be
h
, and one further obtains

12
1
max 1 2 1
33
11 2 2
4
{[1 ( )( / )] / }
[( ) /( ) 1]
I
h
h
GC T
GC W T T T h
kT kT
β
−
−∂∂− =
+

(44)

From Eq. (44), one obtains
max 1
/
I
WT∂∂

, as follows

33 2
max 1 1 1 2 2 1 2 1
/{1{[()/()1]/(4)/}}
I
hh
WTGC hkTkT GCTTT
β
∂∂= − + +

(45)

Substituting Eq. (45) into Eq. (41) yields

'33
11122 2
/{ [( )/( ) 1]/(4 ) 1}
h
TT hkT kT GCT
β
=++
(46)

Substituting Eq. (46) into Eq. (38) yields

33
11122 2
1
33 33
11 22 11 2 2 2
4[()/()1]/(4 )
[( ) / ( ) 1][ [( ) /( ) 1]/ (4 ) 1]
h
h
ThkT kT GCT
dT
dt
kT kT h kT kT GC T
ββ
β
+
=−
+++
(47)

For the given boundary conditions
11
()
ii
Tt T=
and
11
()
ff
Tt T=
, an equation related to the Hamiltonian
constant
h
is obtained by substituting
3
1
64 / (3 )
hB
GC V T c
σ
=

into Eq. (47)

1
1
33
12 11 22
33
1
11 11 1
3
22
4[()/()1]
1
()ln(/){ }
12 4
3
f
i
T
B
fi fi if
T
VTTkT kT
k
TT TT dTtt
kT
ch
σ
ββ
β
+
−+ + =−
∫

(48)

International Journal of Energy and Environment (IJEE), Volume 3, Issue 3, 2012, pp.359-382
ISSN 2076-2895 (Print), ISSN 2076-2909 (Online) ©2012 International Energy & Environment Foundation. All rights reserved.

367
The Hamiltonian constant
h
corresponding to the objective
max
I
W

is obtained from Eq. (48), and then
substituting
h
into Eq. (47). Eq. (47) becomes the problem of initial value of differential equation, and
the optimal temperature
1
T
versus the time
t
is obtained.
When
max
II
W
is chosen to be optimization objective and though some mathematical derivations, the similar
equations to Eqs. (47) and (48) are also obtained, which are, respectively, given by

33
11122 2
1
33 33
11 2 2 11 22 2
4[()/()1]/(4 )
[( ) /( ) 1]{ [( ) /( ) 1]/ (4 ) 1}
V
V
ThkT kT GCT
dT
dt
kT kT h kT kT GC T
ββ
β
+
=−
+++
(49)

1
1
33
33
11 1
12 11 22
11 1
3
22
()
2[()/()1]
1
ln( / ) { }
12 4
f
i
T
fi
B
fi if
T
kT T
VTTkT kT
TT dTtt
kT
ch
σ
ββ
β
−
+
++ =−
∫

(50)

For the given boundary conditions
11
()
ii
Tt T=
and
11
()
ff
Tt T=
, the Hamiltonian constant
h
corresponding
to the objective
max
I
W

is obtained from Eq. (50). And then substituting
h
into Eq. (49), and Eq. (49)
becomes the problem of initial value of differential equation, so the optimal temperature
1
T
versus the
time
t
is also obtained.
What should be paid attention is that the above methods are only suitable for the case with the fixed final
driving fluid temperature
1 f
T
. While for the case with the free
1 f
T
, one has to refer to dynamic
programming algorithm (Figure 2).



Figure 2. The dynamic programming schematic plan of the multistage discrete endoreversible Carnot
heat engines [36]

4.2 For Stefan-Boltzmann heat transfer law
When
4n =
, i.e. the heat transfer between the working fluid and the heat reservoir obeys Stefan-
Boltzmann heat transfer law. Eqs. (25), (30) and (31), respectively, become

4'4
11
3'3
112 2
()
[( / )( / ) 1]
dT T T
dt T k k T T
β
−
=−
+
(51)

'
34'4
max max
12 1
3'3
()
1112 1
64 ( )
max{[ (1 ) ] } 0
3' [(/)(/)1]
II
B
Tt
WW
VT T T T
tcTTTkkTT
σβ
∈Ω
∂∂
−
+−− =
∂∂+


(52)

'
33 4'4
max max
112 1
'3'3
()
1112 1
64 16 ( )
max{[( ) ] } 0
3[(/)(/)1]
II II
BB
Tt
WW
VT VTT T T
tccTTTkkTT
σσ β
∈Ω
∂∂
−
+−− =
∂∂+


(53)

There are no analytical solutions of Eqs. (51)-(53) for the radiative heat transfer law, and one has refer to
numerical methods. For numerical calculations, Eqs. (32)-(34), respectively, become

4'4
12
'3 '
1
12 2
64 [( ) ( ) ]
() { (1 )}
3[( / )( / ) 1]
ii i
N
IN
B
ii
i
VTT T
W
ck k T T T
σβθ
=
−
=−
+
∑


(54)

International Journal of Energy and Environment (IJEE), Volume 3, Issue 3, 2012, pp.359-382
ISSN 2076-2895 (Print), ISSN 2076-2909 (Online) ©2012 International Energy & Environment Foundation. All rights reserved.

368
4'4
21
''3
1
12 2
[( ) ( ) ]64 16
() {( ) }
3[(/)(/)1]
ii i
N
II N
B
ii
i
TV T T
W
ccT kkTT
σβθ
=
−
=−
+
∑


(55)

4'4
1
1
11
3'3
112 1
[( ) ( ) ]
()[(/)( /) 1]
ii
ii i
iii
TT
TT
TkkTT
β
θ
−
−
−=−
+
(56)

The Bellman’s backward recurrence equations corresponding to the objective functions
I
W

and
II
W

are,
respectively, given by

'
4'4
()
12
max 1
'3 '
,
12 2
4'4
() 1
1
max 1
3'3
112 2
64 [( ) ( ) ]
(,)max{ (1 )
3[( / )( / ) 1]
[( ) ( ) ]
(,)}
()[(/)( /) 1]
ii
ii i
Ii i i
B
ii
T
ii
Ii i i i i
ii
VTT T
WT
ck k T T T
TT
WT t
TkkTT
θ
σβθ
τ
β
θ θ
−
−
=−
+
−
++ −
+



(57)

'
4'4
()
21
max 1
''3
,
12 2
4'4
()1
1
max 1
3'3
112 2
[( ) ( ) ]64 16
(,)max{( )
3(/)(/)1
[( ) ( ) ]
(,)}
()[(/)( /) 1]
ii
ii i
II i i i
B
ii
T
ii
II i i i i i
ii
TV T T
WT
ccT kkTT
TT
WT t
TkkTT
θ
σβθ
τ
β
θ θ
−
−
=−
+
−
++ −
+



(58)

5. Numerical examples and discussions
Refs. [43, 44] show that the maximum power output of the multistage heat engine system is
max 2
rev s
WWT
σ
=−

. When the total process period is fixed (i.e. the total heat conductance of the driving
fluid at the high-temperature side is fixed), the final driving fluid temperature at the high-temperature
side can not decrease to the environment temperature, and there is a low limit value
1 f
T
. With the
decrease of the final temperature
1
f
T
, both the reversible power output
rev
W

and the total entropy
generation rate
s
σ
increase, so the relationship between
max
W

and
1
f
T
is unknown. Since
max
W

is the
continuous function of
1
f
T
, there is an optimal
*
1 f
T
during the closed section
11
[,]
fi
TT
for
max
W

to achieve
its maximum value. This was ignored in Refs. [5, 7, 11, 13-22, 34-42], which chose the low-temperature
environment temperature
2
T
as the final temperature. The same analysis methods as Refs. [43, 44] are
adopted herein, and numerical solutions for the radiative heat transfer law [
4
()qT∝∆
] are solved by
dynamic programming algorithm [52, 53] by taking the power output
I
W

of the system for example.
Two different boundary conditions including fixed and free final temperatures are considered herein, and
optimization results for the radiative heat transfer law are compared with those for the pseudo-Newtonian
heat transfer law.
According to Refs. [35, 36], the following calculation parameters are set: the volume flow rate of the
high-temperature radiation photo flux is
43
10 /
Vms
=

, the initial temperature is
10
5800TK=
, the
environment temperature at the low-temperature side is
2
300TK=
, the velocity of the light is
8
2.998 10 /
cms
=×
, Stefan-Boltzmann constant is
824
5.66667 10 / ( )
B
WmK
σ
−
=× ⋅
, Avogadro’s number is
23
6.0221367 10 (1/ )
v
A mol=×
, Boltzmann constant is
23
1.380658 10 /
B
kJK
−
=×
, the universal gas constant
is
8.314510 / ( )
Bv
RkA JmolK
== ⋅
, the emissivity are
12
1
ε ε
= =
. The grid division of the time coordinate
is linear. Since
11
3/64
V
ca
β ε
=
and its unit is
1/
s
,
i
βθ
is a dimensionless quantity and
0.15
i
βθ
=
is set
herein. Let
21
kk
=
for the radiative heat transfer law, and
21
100
kk
=
for pseudo-Newtonian heat transfer
law.

5.1 Performance analysis for a single steady heat engine
Figure 3 shows the heat flux rate
1
q
absorbed by the heat engine versus Carnot temperature
'
T
for two
different heat transfer laws. From Figure 3, one can see that with the increase of Carnot temperature
'
T
,
the heat flux rate
1
q
for the pseudo-Newtonian heat transfer law decreases linearly, while that for the
radiative heat transfer law decreases non-linearly; for the same Carnot temperature
'
T
, the heat flux rate
International Journal of Energy and Environment (IJEE), Volume 3, Issue 3, 2012, pp.359-382
ISSN 2076-2895 (Print), ISSN 2076-2909 (Online) ©2012 International Energy & Environment Foundation. All rights reserved.

369
1
q
for the pseudo-Newtonian heat transfer law increases with the increase of the heat conductance at the
low-temperature side. Figure 4 shows the efficiency
η
of the heat engine versus Carnot temperature
'
T
.
Since
'
2
1/TT
η
=−
,
η
increases with the increase of
'
T
, but its relative increase amount decreases, which
is independent of heat transfer laws. Figure 5 shows the power
P
of the heat engine versus Carnot
temperature
'
T
. From Figure 5, one can see that there is an extremum for
P
with respect to Carnot
temperature
'
T
, and the optimal Carnot temperatures
'
T
corresponding to the maximum power output
for different heat transfer laws are different from each other; for the same Carnot temperature
'
T
, the
power
P
of the heat engine increases with the increase of the heat conductance at the low-temperature
side. Figure 6 shows the entropy generation rate
σ
versus Carnot temperature
'
T
. From Figure 6, one
can see that the entropy generation rate
σ
for different heat transfer laws decreases with the increase of
Carnot temperature
'
T
. Especially when Carnot temperature
'
T
is small, the entropy generation rate
decreases fast, and its change rate tends to be smoothly with the increase of Carnot temperature
'
T
. From
'
21' 2'
/
TTTT
≡
and when
'
2
300
TT K
==
, the heat-absorbed temperature
1'
T
of the working fluid in the
endoreversible Carnot heat engine is equal to its heat-released temperature
2'
T
, i.e. the limit Carnot cycle,
the heat flux rate
1
q
absorbed by the working fluid is equal to that released, the heat engine efficiency
η

is equal to zero as shown in Figure 4, the power output
P
of the heat engine is also equal to zero as
shown in Figure 5, and the entropy generation rate achieves its maximum value as shown in Figure 6.
While
'
1
5800
TT K
==
, the heat-absorbed temperature
1'
T
of the working fluid in the endoreversible
Carnot heat engine is equal to the high-temperature reservoir temperature
1
T
, and the heat-released
temperature of the working fluid is equal to the low-temperature reservoir temperature
2
T
, i.e. the
reversible Carnot cycle. The rate of heat absorbed
1
q
is equal to zero as shown in Figure 3, the heat
engine efficiency achieve its maximum value and equals to the Carnot efficiency
21
1/
C
TT
η
=−
as shown
in Figure 4, its power
P
is equal to zero as shown in Figure 5, and the entropy generation rate
σ
is also
equal to zero as shown in Figure 6.




Figure 3. The absorbed heat flux rate
1
q
of the single-stage heat engine versus Carnot temperature
'
T



International Journal of Energy and Environment (IJEE), Volume 3, Issue 3, 2012, pp.359-382
ISSN 2076-2895 (Print), ISSN 2076-2909 (Online) ©2012 International Energy & Environment Foundation. All rights reserved.

370


Figure 4. The efficiency
η
of the single-stage heat engine versus Carnot temperature
'
T




Figure 5. The power output
P
of the single-stage heat engine versus Carnot temperature
'
T




Figure 6. The entropy generation rate
σ
of the single-stage heat engine versus Carnot temperature
'
T

International Journal of Energy and Environment (IJEE), Volume 3, Issue 3, 2012, pp.359-382
ISSN 2076-2895 (Print), ISSN 2076-2909 (Online) ©2012 International Energy & Environment Foundation. All rights reserved.

371
5.2 Numerical examples for the multistage heat engine system with the radiative heat transfer law
5.2.1 For the fixed final temperature
When the final temperature
1 f
T
is fixed, the reversible power output
rev
W
is also fixed, and then
optimization for maximizing power output is equivalent to that for minimizing entropy generation due to
2rev s
WW T
σ
=−

. In order to analyze effects of the final temperature
1 f
T
on the optimization results, the
final temperature is set to be
1
500
f
TK
=
,
1
1000
f
TK
=
, and
1
1500
f
TK
=
. Figures 7 and 8 show the optimal
fluid temperature
1
T
and Carnot temperature
'
T
versus the time
t
β
. Figure 9 shows the optimal power
output
i
W

of the heat engine versus the stage
i
. In Figures 7-9, the continuous lines denote the analytical
optimization results, while the discrete points denote the numerical optimization results. The total stage
100
N
=
of heat engines are shown with the step of 2 in Figures 7-9. Table 1 lists optimization results of
the key parameters of the multistage endoreversible heat engine system with the radiative heat transfer
law. From Figure 7, one can see that the driving fluid temperature
1
T
decreases non-linearly with the
increase of the time
t
β
. From Figures 8 and 9, one can see that when
1
500
f
TK=
and
1
1000
f
TK=
, the
optimal Carnot temperature profiles consist of two segments: the heat engines in the former segment
have power output, while those in the latter segment have no power output due to
'300
TK
=
. What
should be paid attention is that the heat engines in the latter segment seem to be shortened so that the
fluid temperature at the high-temperature side decreases to the desired final temperature at the fast speed.
When
1
1500
f
TK=
, there is power output for each stage heat engine. From Table 1, one can see that
when
1
500
f
TK
=
, one obtains
'(0) 981.1
TK
=
and
3
max
6.88 10
WW
=×

; when
1
1000
f
TK
=
, one obtains
'(0) 1020.7
TK
=
and
3
max
7.05 10WW=×

; when
1
1500
f
TK
=
, one obtains
'(0) 1040.0
TK
=
and
3
max
7.13 10
WW
=×

, i.e. both the initial Carnot temperature
'(0)
T
and the maximum power output
max
W


increase with the increase of the final temperature
1 f
T
. Both the maximum power output of the
multistage heat engine system with the radiative heat transfer law and the corresponding optimal control
are different for the cases with different final fluid temperatures. From the above analysis, the boundary
temperature change has significant effects on the power output optimization results of the multistage heat
engine system.




Figure 7. The optimal driving fluid temperature
1
T
versus the dimensionless time
t
β
for Newtonian heat
transfer law (fixed
1 f
T
)

International Journal of Energy and Environment (IJEE), Volume 3, Issue 3, 2012, pp.359-382
ISSN 2076-2895 (Print), ISSN 2076-2909 (Online) ©2012 International Energy & Environment Foundation. All rights reserved.

372


Figure 8. The optimal Carnot temperature
'
T
versus the dimensionless time
t
β
for Newtonian heat
transfer law (fixed
1 f
T
)



Figure 9. The optimal power output
i
W

of each stage heat engine versus the stage
i
for Newtonian heat
transfer law (fixed
1 f
T
)

Table 1. Optimization results of the key parameters of the multistage endoreversible heat engine system
with the radiative heat transfer law

Key parameters
'(0)T

max
W


1
500
f
TK=

981.1K

3
6.88 10 W×

1
1000
f
TK=

1020.7K

3
7.05 10 W×

Fixed
1 f
T

(
1
150ts=
)
1
1500
f
TK=

1040.0K

3
7.13 10 W×

Key parameters
*
1
f
T

'(0)T

*
max
W


0.10
i
βθ
=

2626.9K

937.4K

3
6.57 10 W×

0.15
i
βθ
=

2286.0K

1070.2K

3
7.19 10 W×

Free
1 f
T

0.30
i
βθ
=

1770.6K

1346.9K

3
8.09 10 W×