Maximum power, ecological function and efficiency of an irreversible Carnot cycle. A cost and effectiveness optimization

0.28 MB·English

by  G Aragon-Gonzalez

Maximum power, ecological function and efficiency of an
7
0 irreversible Carnot cycle. A cost and effectiveness
0
2 optimization.

n
a
J G Arag´on-Gonz´alez, A. Canales-Palma,

9
A. Le´on-Galicia, J. R. Morales-G´omez
2
PDPA. UAM- Azcapotzalco. Av. San Pablo # 180. Col. Reynosa.

] Azcapotzalco, 02800, D.F. Tel´efono y FAX: [phone].
h
p e-mail: [email protected].
-
s
s February 2, 2008
a
l
c
.
s Abstract
c
i In this work we include, for the Carnot cycle, irreversibilities of linear finite rate of heat
s
transferences between the heat engine and its reservoirs, heat leak between the reservoirs and
y
internaldissipationsoftheworkingfluid. Afirstoptimizationofthepoweroutput,theefficiency
h
p and ecological function of an irreversible Carnot cycle, with respect to: internal temperature
[ ratio, time ratio for the heat exchange and the allocation ratio of the heat exchangers; is per-
formed. For the second and third optimizations, the optimum values for the time ratio and
1
internaltemperatureratio aresubstitutedinto theequation of powerand,then,theoptimiza-
v
tions with respect to the cost and effectiveness ratio of the heat exchangers are performed.
5
Finally, a criterion of partial optimization for theclass of irreversible Carnot engines is herein
2
presented.
3
1 Keywords: Internal and external irreversibilities,heat engines, finite time and size ther-
0 modynamics, cost and effectiveness optimization.
7
Nomenclature
0
/ I: internal irreversible factor.
s
Q: heat transfer.
c
i q: dimensionless heat transfer.
s
y W: work.
h P: power
p p: dimensionless power.
:
v S: entropy-generationrate.
i s: dimensionless entropy-generationrate.
X
K: thermal conductance for heat loss.
r
a t: time
x: internal temperatures ratio.
y: time ratio.
z: allocation ratio.
1
U: global heat transfer coefficient.
A: total heat transfer area.
L: thermal conductances ratio.
C: total cost.
Greek Symbols
α: thermal conductance of hot side.
β: thermal conductance of cold side.
η : efficiency.
σ: dimensionless dissipation.
ǫ : dimensionless ecologicalfunction.
µ : temperatures ratio of hot and cold sides.
Subscripts
C: Carnot.
CI: Carnot-like.
CA: endoreversible or Curzon-Ahlborn.
H: hot-side.
L: cold-side.
max: maximum.
mp: maximum power.
me: maximum efficiency.
mec: maximum ecological function.
Superscripts
: second optimization.
∗
: third optimization.
∗∗
1 Introduction
The thermal efficiency of a reversible Carnot cycle is an upper limit of efficiency for heat engines.
In according to classical thermodynamics, the Carnot efficiency is:
T
η =1 L (1)
C
− T
H
where T and T are the temperatures of the hot and cold reservoirs between which the heat
L H
engineoperates. Thethermalefficiencyη canonlybe achievedthroughthe infinitely slowprocess
C
requiredbythermodynamicequilibrium. Therefore,itisnotpossibleto obtainacertainamountof
poweroutputby usingheatexchangerswithfinite heattransferareas. Thus,the thermalefficiency
given in equation (1) does not have great significance and is a poor guide for the performances of
real heat engines.
A more realisticupper bound could be placedon the efficiency ofa heatengine operatingat its
maximum power point; the so-called CA efficiency (Curzon-Alhborn [1]):
T
L
η =1
CA
− T
r H
wherethe onlysourceofirreversibilityinthe engine is alinearfinite rateheattransferbetweenthe
working fluid and its two heat reservoirs.
2
Real heat engines are complex devices. Besides the irreversibility of finite-rate heat transfer
in finite time taken into account in the Curzon-Ahlborn engine (CA-engine), there are also other
sourcesofirreversibility,suchasheatleaks,dissipativeprocessesinsidetheworkingfluidandsoon.
Thus, it is necessary to investigate more comprehensively the influence of finite-rate heat transfer
together with other major irreversibilities on the performance of heat engines. For this aim, we
mustconsidergeneralirreversibleCarnotenginesincludingthreemajorirreversibilities,whichoften
exist in heat engines, and use it to optimize the performance of an irreversible Carnot engine for
several objective functions.
In the past decade some new models of irreversible Carnot engines which include other irre-
versibilities, besides thermal resistance, have been established: heat leak and internal dissipations
of the working fluid (see [2], [3], [4], [5] , [6], [7], [8], [9], [10] and included references there). Never-
theless,thereareanotherparametersinvolvedintheperformanceandoptimizationofanirreversible
Carnot cycle; for instance, the allocation ratio of the heat exchangers, cost and effectiveness ratio
of the heat exchangers and so on (see [8], [9] and [14]).
IntheoptimizationofCarnotcycles,includingthoseirreversibilities,hasappearedfourobjective
functions: power,efficiency,ecologicalandentropygeneration. Themaximumpowerandefficiency
have been obtained in [4], [5] and [10]. The maximum ecological function was obtained in [12]
for the CA-engine and in form more general in [7]. Bejan [13] has considered the minimization of
the entropy generation. In general, these optimizations were performed with respect to only one
characteristic parameter: internal temperature ratio.
In the first analysis of the CA-engine the time ratio of heat transfer from hot to cold side was
considered, but in further works this ratio was not taken into account (see [2] for details). In
this work this relation is considered as a parameter. On the other hand, [13] has performed the
optimization, also, with respect to other parameter: the allocation ratio of the heat exchangers;
and [14] has considered as parameters the cost and effectiveness ratio of the heat exchangers for
the CA-engine.
In this work we include, for the Carnot cycle, irreversibilities of linear finite rate of heat trans-
ferences between the heat engine and its reservoirs, heat leak between the reservoirs and internal
dissipations of the working fluid. A first optimization of the power output, the efficiency and eco-
logical function of an irreversible Carnot cycle, with respect to: internal temperature ratio, time
ratio for the heat exchange and the allocation ratio of the heat exchangers; is performed. For the
second and third optimizations, the optimum values for the time ratio and internal temperature
ratio are substituted into the equation of power and, then, the optimizations with respect to the
cost and effectiveness ratio of the heat exchangers are performed. Finally, a criterion of partial
optimization for the class of irreversible Carnot engines is herein presented.
This paper is organized as follows. In the section 2 the relations for the dimensionless power,
efficiency, entropy generation and ecological function of a class of irreversible Carnot engines are
presented. In the section 3, the optimal analytical expressions for the efficiencies corresponding to
powerandecologicalfunction;andmaximumefficiencyareshown. Insection4,theoptimumvalues
forthetimeratioandinternaltemperatureratioaresubstitutedintheexpressionfordimensionless
power. Then a second and third optimizations of dimensionless power, are performed with respect
to the cost and effectiveness ratio of the heat exchangers. In the section of Conclusions,a criterion
ofpartialoptimizationforpower,ecologicalfunction,efficiencyandentropygenerationispresented.
3
Figure 1: A Carnot cycle with heat leak, finite-rate heat transfer and internal dissipations of the
working fluid
2 Irreversible Carnot engine.
In considering the class of irreversibleCarnotengines (see [2]) shownin Figure 1,which satisfy the
following five conditions:
(i) The cycle ofthe engine consists of twoisothermalandtwo adiabatic processes. The temper-
atures of the working fluid in the hot and cold isothermal processes are, respectively, T and T ,
1 2
and the times of the two isothermal processes are, respectively, t and t . The temperatures of
H L
the hot and cold heat reservoirsare, respectively, T and T .
H L
(i) There is thermal resistance between the working fluid and the heat reservoirs.
(ii) There is a heat lost Q from the hot reservoir to the cold reservoir [13]. In real engines
leak
heat leaks are unavoidable,there are many features of an actual power plant which fall under that
kind ofirreversibility,suchas the heatlost throughthe walls ofa boiler, a combustionchamber,or
a heat exchanger, and heat flow through the cylinder walls of an internal combustion engine, and
so on.
(iii) All heat transfer is assumed to be linear in temperature differences, that is, Newtonian.
(iv) Besides thermal resistance and heat loss, there are other irreversibilities in the cycle, the
internal irreversibilities. For many devices such as gas turbines, automotive engines, and thermo-
electric generator,there are other loss mechanisms, like friction or generatorslosses, etc. that play
an important role, but are hard to model in detail. Some authors use the compressor (pump) and
turbineisentropicefficienciestomodeltheinternallossinthegasturbinesorsteamplants. Others,
in Carnot cycles, use simply one parameter to describe the internal losses. Such a parameter is
associatedwith the entropy produced inside the engine during a cycle. Specifically, this parameter
4
makes the Claussius inequality becomes an equality (for details see [2]):
Q Q
2 I 1 =0 (2)
T − T
2 1
where I = ∆S2 1 ([4]).
∆S1 ≥
Thus,the irreversibleCarnotengine operateswith fixedtime t allowedforeachcycle. The heat
leakage Q is ([13]):
leak
Q =K(T T )t
leak H L
−
The heats Q , Q transferred from the hot-cold reservoirs are given by:
H L
Q = Q +Q =α(T T )t +K(T T )t (3)
H 1 leak H 1 H H L
− −
Q = Q +Q =β(T T )t +K(T T )t (4)
L 2 leak 2 L L H L
− −
whereα,β andK arethe thermalconductancesandt ,t arethe time fortheheattransferinthe
H L
isothermalbranches,respectively. Theconnectingadiabaticbranchesareoftenassumedtoproceed
in negligible time ([3]), such that the cycle contact total time t is [11]:
t=t +t (5)
H L
By first law and combining equations (3) and (2) we obtain :
T (1 Ix) 1 µ
W =Q (1 Ix)= H − − x (6)
1 − 1 + I
αtH β(cid:0)tL (cid:1)
W
Q =Q +Q = +K(T T )t (7)
H 1 leak 1 Ix H − L
−
where µ= TL. And x= T2 is a characteristic parameter of the engine.
TH T1
Now, the equation (5) gives us the total time of the cycle, so it can be parametrized as:
t =yt; t =(1 y)t
H L
−
where y = tH = tH is other characteristic parameter of the engine.
t tH+tL
Another parameteris the allocationofthe exchangersheat[13]. The thermalconductancescan
be written as:
α=UA ; β =UA
H L
whereU isoverallheattransfercoefficientandA andA arethe availableareasforheattransfer.
H L
Then, anapproachmightbe to suppose that U is fixed, the same for the hotside andthe coldside
heat exchangers,and that the area A can be allocated between both. The optimization problem is
then selected, besides of the optimum temperature ratio and the time ratio, as the best allocation
ratio. To take UA as a fixed value can be justified in terms of the area purchased, and the fixed
running costs and capital costs that altogether determine the overall heat transfer coefficient (see
([14])). Thus, for the optimization we can take:
α β
+ =A (8)
U U
5
and parametrize it as:
α=zUA; β =(1 z)UA
−
α z
= (9)
β (1 z)
−
Therefore,thedimensionlesspoweroutput,p= W ,andthedimensionlessheattransferrate
AUtTH
q = QH are (by equations(6) and (7)):
H AUtTH
z(1 z)y(1 y)(1 Ix) 1 µ
p= − − − − x (10)
(1 z)(1 y)+zyI
− − (cid:0) (cid:1)
z(1 z)y(1 y) 1 µ
q = − − − x +L(1 µ) (11)
H
(1 z)(1 y)+zyI −
− − (cid:0) (cid:1)
where L = K . And z = α is the third characteristic parameter of the engine. The thermal
AU UA
efficiency is given by:
z(1 z)y(1 y)(1 Ix) 1 µ
η = − − − − x (12)
z(1 z)y(1 y) 1 µ +L(1 µ)((1 z)(1 y)+zyI)
− − − x − (cid:0)− (cid:1) −
Theentropy-generationrate,s =(cid:0) Sgen (cid:1),multipliedbythetemperatureofthecoldside,give
gen AUtTH
us a dimensionless function σ, which is (equations (10,11)):
q p q
σ =T s =T H − H =q (1 µ)2 p
L gen L H
T − T − −
(cid:18) L H(cid:19)
so,
z(1 z)y(1 y) 1 µ (Ix µ)
σ = − − − x − +L(1 µ)2 (13)
(1 z)(1 y)+zyI −
− −(cid:0) (cid:1)
Finally, the ecological function [12], when T is the environmental temperature, is:
L
2Ix 1 µ
ǫ=p σ =p − − +L(1 µ)
− Ix 1 −
−
then,
z(1 z)y(1 y) 1 µ (1 2Ix)
ǫ= − − − x − +L(1 µ) (14)
(1 z)(1 y)+zyI −
− −(cid:0) (cid:1)
when I =1 and L=0 the expressions for the CA-engine are obtained.
3 Maximum power, ecological function and efficiency.
In using the equation (10) and the extremes conditions:
∂p ∂p ∂p
=0; =0; =0
∂x|(xmp,ymp,zmp) ∂y|(xmp,ymp,zmp) ∂z|(xmp,ymp,zmp)
when the power reaches its maximum, x , y and z are given by:
mp mp mp
6
µ
x = (15)
mp I
r
1
y =z = (16)
mp mp
√3I+1
Clearly p reaches its maximum in (x ,y ,z ). Indeed, all the critical points are (necessary
mp mp mp
condition):
z =0,y =y,x=µ , x= 1,y =1,z =z , y =1,z =z,x=µ
{ } I { }
y =0,z =z,x=µ , x= 1,z =0,y =y , y =0,z =0,x=x
{ } (cid:8) I (cid:9) { }
z =1,y =1,x=x , z =1,y =y,x=µ , x= 1,z =1,y =y
{ } {(cid:8) }(cid:9) I
x= 1,y =0,z =z , x= µ,y = 1 ,z = 1
I ± I √(cid:8)3I+1 √3I+1 (cid:9)
(cid:8) (cid:9) n p o
In eliminating the solutions without physical meaning, we see that there is only one global
critical point given by the equations (15, 16). Moreover, at this critical point maximum power
developed. Indeed, a sufficient condition for maximum power is, the eingenvalues of the Hessian
( ∂2p ) must be negatives ([15]). It is clearly fulfilled that:
∂w∂u|(xmp,ymp,zmp) w,u=x,y,z
h i
3
2I2 0 0
−√µ(1+√3I)3
 2 1 √Iµ 2 
0 − 0
 0 − √3(cid:0)I(01+√3I(cid:1)) 2 1−√Iµ 2 
 − √3(cid:0)I(1+√3I(cid:1)) 
 
The efficiency that maximizes the power η is given by (see equation (12) and Figure 2),
mp
1 √Iµ
η = − (17)
mp 1+(cid:0)L(1−µ)(√3I(cid:1)+1)3
1 √Iµ
−
The generation of entropy is minimum when(cid:0)y and(cid:1)z are given by equation (16) and x =
µ . Nevertheless, for these values it is seen that the corresponding power does not have physical
√I
meaning. For x = µ (y = 0,1 or z = 0,1), makes the first term of the equation(13) zero. The
I
corresponding values of y,z are also without physical meaning. For x=µ (y =0,1 or z =0, 1) do
nothavephysicalmeaningeither. Therefore,forthis kindofCarnotengine,the entropygeneration
does not have a global minimum within the valid interval. In [16] an engine that corresponds with
thekindofirreversibleCarnotcycleshereinpresentedisanalyzedbutthecalculationsleadingtothe
minimizationofentropygenerationareatfault,sincetheydonothavephysicalmeaning. Itresults
that the obtained power is negative! Thus, it is only possible to minimize the entropy generation
partially for the variables y,z and those values are given by:
1
y =z = (18)
mσ mσ
√3I +1
7
In doing a analogousanalysis for the ecologicalfunction, we have by the equation(14) that the
unique critical point of ecological function solutions with physical meaning is:
µ(1+µ)
x = , (19)
mec
2I
r
1
y =z = (20)
mec mec
√3I+1
and newly can see that its Hessian has all its negative eingenvalues.
The efficiency that maximizes the ecologicalfunction η is given by (equation (12)):
mec
1 µ(1+µ)I
− 2
ηmec = (cid:18) L(1qµ)(√3I+1(cid:19))3 (21)
1+ −
1 2Iµ
− µ+1
(cid:16) p (cid:17)
Similarly, it’s easily seen that there is only one critical point, with physical meaning, for the
efficiency, and it is given by:
3 3
µ+ Lµ(1 µ) 1+√3I L √3I +1 (1 µ)+1 Iµ
s − − −
x = (cid:16) (cid:17) (cid:18) (cid:16) (cid:17) (cid:19) (22)
me
3
I2 L √3I+1 (1 µ)+1
−
(cid:18) (cid:16) (cid:17) (cid:19)
1
y =z = (23)
me me
√3I +1
To see, as above, that the efficiency reaches a maximum, becomes to cumbersome a task if the
solution of systems of equations are undertaken. Therefore,an alternativeway is presented in that
follows, to obtain equation (22). And when the efficiency reaches its maximum (x ,y ,z ) is
me me me
given by the equations(22,23).
Indeed,clearlythevaluesofy ,z givenbytheequation(23)fulfillthefollowingtwoextreme
me me
conditions:
∂η ∂η
=0; =0
∂y ∂z
Furthermore,as it was seen above,the optimal time ratio and the allocationratio are the same
for both maximum power and ecological function (equations (19), (20)). Therefore,
1
y =y =y =z =z =z =
mp mec me mp mec me
√3I +1
Thus, this values could be included in the equations of power and heat transfer (equations
(10,11))andproceedtooptimizestheefficiency(equation(12)bythefollowingcriterionvalidwhen
there is only one parameter([10]):
8
Criterion (Maximum efficiency) Let η = p . Suppose ∂2p = ∂2qH , for some x. Then the
qH ∂x2|x ∂x2 |x
maximum efficiency η is given by
max
∂p
η = ∂x|xme (24)
max ∂ qH
∂x |xme
where x is the point in which η achieves a maximum value.
me
Then, by the equations (10) and (3) we obtain the relationships of p and q with respect to x.
H
(1 Ix)(1 µ)
p= − − x
3
√3I+1
(cid:16) (cid:17)
(1 µ)
q = − x +L(1 µ)
H 3 −
√3I +1
The conditions of the criterion are cle(cid:16)arly sati(cid:17)sfied. Indeed,
∂2p ∂2q 2µ
H
= = <0
∂x2 ∂x2 − 3
x3 √3I +1
(cid:16) (cid:17)
since x>0.Therefore (equation(24)),
x2 I
η =1 me (25)
max − µ
where x must, by the second law, satisfies the inequality ([10]):
me
µ µ
x (26)
me
I ≤ ≤ I
r
if we apply the preceding statement and the equation (25), the following inequality is obtained
η =1 Iµ η 1 Iµ=η (27)
mp max CI
− ≤ ≤ −
where η = 1 √Iµ and η = 1 Iµpcorresponding to (Curzon-Ahlborn)-like and Carnot-like
mp CI
− −
efficiencies; which includes the internal irreversibilities in the I factor.
Nevertheless, we can calculate easily x from the following cubic equation:
me
1 x2meIS = p|xme = (1−xmeI)(1− xmµe)
− µ qH|xme (1 µ )+L(1 µ) √3I+1 3
− xme −
(cid:16) (cid:17)
In solving this equation and taking into account the inequality (26), we obtain the equation (22).
Finally, the maximum efficiency η is given by (equation (25) ):
max
2
3 3
√Iµ+ L(1 µ) 1+√3I L(1 µ) 1+√3I +1 Iµ
η =1  s − (cid:16) (cid:17) (cid:18) − (cid:16) (cid:17) − (cid:19) (28)
max − 3
 1+L(1 µ) √3I +1 
 − 
 
 (cid:16) (cid:17) 
 
9
Figure 2: Graphics of the efficiencies η , η and η versus µ when I =1.235 and L=0.01.
mp mec max
The behavior of the efficiencies η ,η and η is shown in the Figure 2.
mp mec max
In general it has been supposed that I 1; but sometimes can be considered that I = 1. In
≥
this case the internal irreversibilitiescan be physicallyinterpreted as part of the engine’s heat leak
that brings us to the engine modeled in [2] and [13]. So, substitution of I = 1 into equations (15;
19, 22) and (23) gives:
2
1 2µ 1 µ(1+µ)
− µ+1 − 2
xmp = õ; xmec = (cid:16) q (cid:17)(cid:18) q (cid:19) ;
8L(1 µ)+ 1 2µ
− − µ+1
(cid:16) q (cid:17)
µ+ 8Lµ(1 µ)(8L(1 µ)+1 Iµ)
x = − − −
me
8(8L(1 µ)+1)
p −
1
y =y =y =z =z =z =
mp mec me mp mec me
2
The equations:
1
xmp =õ and zmp =
2
are the same as the presented in [13] and
1
x =õ and y =
mp mp
2
correspondingto the CA-engine. Further, the following results are obtained(see equations (17; 21,
28)):
1 õ
η = −
mp 2+ 8L(1−µ)
1 õ
−
10