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Design parameters of a radiative heat engine
0360-5442/93 $6.00 + 0.00 Copyright0 lW3 PergamonPressLtd
Ener~ Vol. 18, No. 6, pp. 651-6.55, 1993 Printed in Great Britnin. All rights reserved
DESIGN PARAMETERS OF A RADIATIVE ENGINE S. GBKTuN,?. t I.T.U.,
S. ~ZKAYNAK,$
HEAT
and H. YAVUZ$~
Maritime Faculty, 81704 Tuzla, Istanbul, $ Turkish Naval Academy, 81704 Tuzla, Istanbul, and 5 I.T.U., Institute for Nuclear Energy, 80626 Maslak, Istanbul, Turkey (Received
24 April
1992; received for publication
9 October
1992)
Abstract-By employing an endoreversible heat-engine model, the design parameters of a heat engine operating under radiative heat-transfer conditions were examined to find the maximum power output. It was found that the ratio of the cold to the hot reservoir temperature must be less than 0.2 for an optimal design. Increasing the heat-transfer area of the cold side rather than that of the hot side improves the thermal efficiency. When the temperature ratio is greater than 0.6, the efficiency of such a cycle approaches that of Curzon and Ahiborn.
INTRODUCTION
In classical thermodynamics,
the thermal efficiency of a heat engine is given by 11=
WlQ,
(1)
where # and 0, are the power generated by the heat engine and the rate of heat transfer from the heat source to the heat engine, respectively. The highest possible thermal efficiency that can be achieved by using a reversible Carnot cycle operating between the fixed reservoir temperatures T, and TH is qc = 1 - (T,/T,)
= 1 - t,
(3)
where 0 < t < 1. As explained by Curzon and Ahlborn,’ the power output of a Carnot cycle would be zero since it is necessary to run the heat engine infinitely slowly in order to ensure reversibility. If the heat engine runs infinitely fast, the power output is again zero. Between these two extremes, maximum power output is obtained. The thermal efficiency corresponding to the maximum power output is rjc_* = 1 - t”.“.
(4)
In the endoreversible engine model of Curzon and Ahlborn,’ as elaborated by Rubin,’ all losses are associated with heat transfer to and from a reversible heat engine. An endoreversible engine is operated not between the reservoir temperatures TH and TL but between the warm and cold sides of the heat engine Tw and Tc. Employing such a model, it has been shown’-” for several cases that the optimal performance of a real heat engine depends on the heat-transfer processes involved. Characteristics of solar-powered heat and thermionic engines have been determined by Wu,‘.s when they are operating at maximum power. Here, we consider an endoreversible engine operating under radiative heat-transfer conditions by following Vos.’ _..-__ ~To whom all correspondence should be addressed. 6.51
S. G~KTUN et al
652
LI
I
Fig. 1. General heat engine model.
THEORETICAL
MODEL
A general heat engine and its T-s diagram are shown in Figs. 1 and 2. The heat-flow rate from the high-temperature reservoir to the working substance is & = &+%-U~ Similarly, on the low-temperature
- CL),
n = 1, 4.
(5)
m = 1, 4.
(6)
side,
& = h&_(T?
- T),
Here, h and A are the heat-transfer coefficient and heat-exchanger surface area, respectively. The engine is endoreversible in the sense no entropy is being generated between the actual working temperatures Tw and Tc. It follows that
rid& = TwlTc.
(7)
The power output from Eq. (7) is
- (T,/T,)].
w = 0, - & = &[l If the temperature
(8)
ratio of the reversible cycle is defined as
w= WTw,
(9)
the power output can be written as W=&(l+)
(IO)
If the assumption n = m is made in Eqs. (5) and (6), the values of Tw and Tc can be obtained by using EqS. (5), (6), (9), and (10). Substituting Eqs. (5) and (6) in Eq. (7) and dividing by T;, Eq. (7) becomes @[I
T,
__
Gv/TH)“I= [G/&d“ - t”],
-
_ 2 ,
Warm
working
pid
(11)
temperature
t
TC
T,
c’ ---
___-__
1
Cool working fluid hnperature
Heat rink bxnperature
Entropy
Fig. 2. T - s diagram of an irreversible heat engine.
S
Design parameters of a radiative heat engine
653
where
Substituting (WW(WCI) for (To-Tu) in Eq. (ll),
= KGv/Gr)
(12)
we find
BV[l - (&v/&i)“] = [1CI”(TvITH)” - r’].
(13)
Solving Eq. (13) for (Z”/Tu)” yields (&v/T,)” = (BV + t”)l(Bly + V).
(14)
Using Eq. (14), the dimensionless heat-transfer rate to the cycle and the dimensionless output of the cycle can be obtained from Eqs. (5) and (lo), respectively, as $ = I;i&rAJ;I P = w/h,AHT”,
+ v”),
(15)
~/J)/(/?IJ + 1~“).
(16)
= (v” - 2”)/(83 = (9” - r”)(l-
power
The maximum amount of power can be extracted from the engine when dPld3
=0
(17)
or, from Eq. (16), ly” + n&l”+’ - (1 - n)(z” - /3)3” - nt”ly”-’
- t”j3 = 0.
(18)
The parameter rr describes the heat-transfer mode. For n = 1, Eq. (18) reduces to the form given in the literature and qyopt becomes independent of the ratio of heat-transfer coefficients /3, i.e.
hpt(C-A) = t0.5.
(19)
0.6
0
0.2
0.4
0.6
0.8
r
Fig. 3. The effect of the temperature ratio on the radiation efficiency.
1.0
654
S. GGKTUNet at
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
7 Fig. 4.
The effect of the temperature ratio on the maximum power ratio.
For n = 4, Eq. (18) becomes t/P + 4p35 + 3(t4 - fi)q” - @t/J4 -
424?&3 -
z"/? = 0.
(20)
For a given value of /3, Eq. (20) can be solved for ~,pl(RRj numerically in terms of r. &ax and ~~~~~~~~ in Eqs. (15) and (16), respectively. The thermal efficiency of the heat engine under radiative heat-transfer conditions is P,,,,, can then be found in terms of t by substituting
The variation of the design parameters and 4.
of an endoreversible
RESULTS
AND
heat engine are shown in Figs. 3
DISCUSSION
The Curzon and Ahlborn efficiency is independent of the plant size. By contrast, for radiative heat transfer, the efficiency depends on j3 (i.e. the heat-transfer area). An iteration is needed to determine the optimum design parameters. Examination of Figs. 3 and 4 yields guidelines for determining the design parameters. It may be seen from Fig. 3 that the nRR and I@_,~(~~) remain nearly constant when r < 0.2 for any given value of j3. Also Fig. 4 shows that the variation of power output in the same region is minor, in agreement with Fig. 3. This observation suggests that the condition r < 0.2 must be one of the design guides. It is also seen from Fig. 3 that in order to increase the thermal efficiency and the power output, /3 must be decreased. In other words, the heat-transfer area on the low-temperature side rather than that on the high-temperature side must be increased. For t >0.6, Fig. 3 shows that ly,pt~RR~and qRR approach the value of Curzon and Ahlborn. In this region, the influence of 6 is minor and the efliciency can be approximated by Eq. (4).
REFERENCES
1. F. L. Curzon and B. Ahlbom, Am. .I. Phys. 43, 22 (1975). 2. M. H. Rubin, Phys. Rev. A 19, 1272 (1979). 3. L. Chen and Z. Yand, J. Chem. Phys. 90,374.O (1989). 4. J. M. Gordon, Am. J. Phys. 58, 370 (1990).
Design parameters of a radiative heat engine
6.55
5. H. Yavuz, S. Gktun, and S. bzkaynak, “Sonlu Zaman Termodinamiginin Isi Makinalarina Uygulanmasi” (Finite Time Thermodynamics and Its Potential Heat Engine Application), Proc. of the Workshop on The Second Law of Thermodynamics 2, 30-1, Kayseri, Tiirkiye (1990). 6. A. Bejan, fnt. 1. Heat Mass Transfer 31, 1211 (1988). 7. C. Wu, Znt. J. Ambient Energy 9, 17 (1988). 8. C. Wu, Energy Convers. Mgmt 33,279 (1992). 9. A. D. Vos, Am. J. Phys. 53, 570 (1985).
NOMENCLATURE
A, = Heat-exchanger area at the hightemperature source side A, = Heat-exchanger area at the lowtemperature source side h, = Heat-transfer coefficient at the hightemperature source side h, = Heat-transfer coefficient at the lowtemperature source side m, n = Non-zero integers P = Dimensionless power Q, = Heat transferred to the heat engine Q, = Rate of heat transfer to the heat engine Q, = Rate of heat transfer from the heat engine s = Specific entropy
T = Temperature Tc = Cool working fluid temperature TH= Temperature of the heat source TL= Temperature of the heat sink T, = Warm working fluid temperature W = Heat engine work rk = Power output p = Ratio of hHAH to hLA, q = Thermal efficiency Al== Camot efficiency qC-A = Curzon & Ahlborn efficiency tl RR= Radiation efficiency t = Ratio of TLto TH I# = Ratio to Tc to Tw Cp= Dimensionless heat-transfer rate
Ener~ Vol. 18, No. 6, pp. 651-6.55, 1993 Printed in Great Britnin. All rights reserved
DESIGN PARAMETERS OF A RADIATIVE ENGINE S. GBKTuN,?. t I.T.U.,
S. ~ZKAYNAK,$
HEAT
and H. YAVUZ$~
Maritime Faculty, 81704 Tuzla, Istanbul, $ Turkish Naval Academy, 81704 Tuzla, Istanbul, and 5 I.T.U., Institute for Nuclear Energy, 80626 Maslak, Istanbul, Turkey (Received
24 April
1992; received for publication
9 October
1992)
Abstract-By employing an endoreversible heat-engine model, the design parameters of a heat engine operating under radiative heat-transfer conditions were examined to find the maximum power output. It was found that the ratio of the cold to the hot reservoir temperature must be less than 0.2 for an optimal design. Increasing the heat-transfer area of the cold side rather than that of the hot side improves the thermal efficiency. When the temperature ratio is greater than 0.6, the efficiency of such a cycle approaches that of Curzon and Ahiborn.
INTRODUCTION
In classical thermodynamics,
the thermal efficiency of a heat engine is given by 11=
WlQ,
(1)
where # and 0, are the power generated by the heat engine and the rate of heat transfer from the heat source to the heat engine, respectively. The highest possible thermal efficiency that can be achieved by using a reversible Carnot cycle operating between the fixed reservoir temperatures T, and TH is qc = 1 - (T,/T,)
= 1 - t,
(3)
where 0 < t < 1. As explained by Curzon and Ahlborn,’ the power output of a Carnot cycle would be zero since it is necessary to run the heat engine infinitely slowly in order to ensure reversibility. If the heat engine runs infinitely fast, the power output is again zero. Between these two extremes, maximum power output is obtained. The thermal efficiency corresponding to the maximum power output is rjc_* = 1 - t”.“.
(4)
In the endoreversible engine model of Curzon and Ahlborn,’ as elaborated by Rubin,’ all losses are associated with heat transfer to and from a reversible heat engine. An endoreversible engine is operated not between the reservoir temperatures TH and TL but between the warm and cold sides of the heat engine Tw and Tc. Employing such a model, it has been shown’-” for several cases that the optimal performance of a real heat engine depends on the heat-transfer processes involved. Characteristics of solar-powered heat and thermionic engines have been determined by Wu,‘.s when they are operating at maximum power. Here, we consider an endoreversible engine operating under radiative heat-transfer conditions by following Vos.’ _..-__ ~To whom all correspondence should be addressed. 6.51
S. G~KTUN et al
652
LI
I
Fig. 1. General heat engine model.
THEORETICAL
MODEL
A general heat engine and its T-s diagram are shown in Figs. 1 and 2. The heat-flow rate from the high-temperature reservoir to the working substance is & = &+%-U~ Similarly, on the low-temperature
- CL),
n = 1, 4.
(5)
m = 1, 4.
(6)
side,
& = h&_(T?
- T),
Here, h and A are the heat-transfer coefficient and heat-exchanger surface area, respectively. The engine is endoreversible in the sense no entropy is being generated between the actual working temperatures Tw and Tc. It follows that
rid& = TwlTc.
(7)
The power output from Eq. (7) is
- (T,/T,)].
w = 0, - & = &[l If the temperature
(8)
ratio of the reversible cycle is defined as
w= WTw,
(9)
the power output can be written as W=&(l+)
(IO)
If the assumption n = m is made in Eqs. (5) and (6), the values of Tw and Tc can be obtained by using EqS. (5), (6), (9), and (10). Substituting Eqs. (5) and (6) in Eq. (7) and dividing by T;, Eq. (7) becomes @[I
T,
__
Gv/TH)“I= [G/&d“ - t”],
-
_ 2 ,
Warm
working
pid
(11)
temperature
t
TC
T,
c’ ---
___-__
1
Cool working fluid hnperature
Heat rink bxnperature
Entropy
Fig. 2. T - s diagram of an irreversible heat engine.
S
Design parameters of a radiative heat engine
653
where
Substituting (WW(WCI) for (To-Tu) in Eq. (ll),
= KGv/Gr)
(12)
we find
BV[l - (&v/&i)“] = [1CI”(TvITH)” - r’].
(13)
Solving Eq. (13) for (Z”/Tu)” yields (&v/T,)” = (BV + t”)l(Bly + V).
(14)
Using Eq. (14), the dimensionless heat-transfer rate to the cycle and the dimensionless output of the cycle can be obtained from Eqs. (5) and (lo), respectively, as $ = I;i&rAJ;I P = w/h,AHT”,
+ v”),
(15)
~/J)/(/?IJ + 1~“).
(16)
= (v” - 2”)/(83 = (9” - r”)(l-
power
The maximum amount of power can be extracted from the engine when dPld3
=0
(17)
or, from Eq. (16), ly” + n&l”+’ - (1 - n)(z” - /3)3” - nt”ly”-’
- t”j3 = 0.
(18)
The parameter rr describes the heat-transfer mode. For n = 1, Eq. (18) reduces to the form given in the literature and qyopt becomes independent of the ratio of heat-transfer coefficients /3, i.e.
hpt(C-A) = t0.5.
(19)
0.6
0
0.2
0.4
0.6
0.8
r
Fig. 3. The effect of the temperature ratio on the radiation efficiency.
1.0
654
S. GGKTUNet at
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
7 Fig. 4.
The effect of the temperature ratio on the maximum power ratio.
For n = 4, Eq. (18) becomes t/P + 4p35 + 3(t4 - fi)q” - @t/J4 -
424?&3 -
z"/? = 0.
(20)
For a given value of /3, Eq. (20) can be solved for ~,pl(RRj numerically in terms of r. &ax and ~~~~~~~~ in Eqs. (15) and (16), respectively. The thermal efficiency of the heat engine under radiative heat-transfer conditions is P,,,,, can then be found in terms of t by substituting
The variation of the design parameters and 4.
of an endoreversible
RESULTS
AND
heat engine are shown in Figs. 3
DISCUSSION
The Curzon and Ahlborn efficiency is independent of the plant size. By contrast, for radiative heat transfer, the efficiency depends on j3 (i.e. the heat-transfer area). An iteration is needed to determine the optimum design parameters. Examination of Figs. 3 and 4 yields guidelines for determining the design parameters. It may be seen from Fig. 3 that the nRR and I@_,~(~~) remain nearly constant when r < 0.2 for any given value of j3. Also Fig. 4 shows that the variation of power output in the same region is minor, in agreement with Fig. 3. This observation suggests that the condition r < 0.2 must be one of the design guides. It is also seen from Fig. 3 that in order to increase the thermal efficiency and the power output, /3 must be decreased. In other words, the heat-transfer area on the low-temperature side rather than that on the high-temperature side must be increased. For t >0.6, Fig. 3 shows that ly,pt~RR~and qRR approach the value of Curzon and Ahlborn. In this region, the influence of 6 is minor and the efliciency can be approximated by Eq. (4).
REFERENCES
1. F. L. Curzon and B. Ahlbom, Am. .I. Phys. 43, 22 (1975). 2. M. H. Rubin, Phys. Rev. A 19, 1272 (1979). 3. L. Chen and Z. Yand, J. Chem. Phys. 90,374.O (1989). 4. J. M. Gordon, Am. J. Phys. 58, 370 (1990).
Design parameters of a radiative heat engine
6.55
5. H. Yavuz, S. Gktun, and S. bzkaynak, “Sonlu Zaman Termodinamiginin Isi Makinalarina Uygulanmasi” (Finite Time Thermodynamics and Its Potential Heat Engine Application), Proc. of the Workshop on The Second Law of Thermodynamics 2, 30-1, Kayseri, Tiirkiye (1990). 6. A. Bejan, fnt. 1. Heat Mass Transfer 31, 1211 (1988). 7. C. Wu, Znt. J. Ambient Energy 9, 17 (1988). 8. C. Wu, Energy Convers. Mgmt 33,279 (1992). 9. A. D. Vos, Am. J. Phys. 53, 570 (1985).
NOMENCLATURE
A, = Heat-exchanger area at the hightemperature source side A, = Heat-exchanger area at the lowtemperature source side h, = Heat-transfer coefficient at the hightemperature source side h, = Heat-transfer coefficient at the lowtemperature source side m, n = Non-zero integers P = Dimensionless power Q, = Heat transferred to the heat engine Q, = Rate of heat transfer to the heat engine Q, = Rate of heat transfer from the heat engine s = Specific entropy
T = Temperature Tc = Cool working fluid temperature TH= Temperature of the heat source TL= Temperature of the heat sink T, = Warm working fluid temperature W = Heat engine work rk = Power output p = Ratio of hHAH to hLA, q = Thermal efficiency Al== Camot efficiency qC-A = Curzon & Ahlborn efficiency tl RR= Radiation efficiency t = Ratio of TLto TH I# = Ratio to Tc to Tw Cp= Dimensionless heat-transfer rate