08-performance optimization of a new combined power

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Performance optimization of a new combined power

cycle based on power density analysis of the dual cycle

Bahri Sahin a,*, Ugur Kesgin a, Ali Kodal b, Nurten Vardar a

a Department of Naval Architecture, Yildiz Technical University, Besiktas, 80750 Istanbul, Turkeyb Department of Aeronautical Engineering, Istanbul Technical University, Maslak, 80626 Istanbul, Turkey

Received 30 March 2001; accepted 14 August 2001

Abstract

In this paper, maximum power density (MPD) analysis of an air standard internal combustion Dual

cycle has been performed. Based on the obtained results for MPD analysis of the Dual cycle, a new

combined power cycle model (Dual Joule-Brayton) has been introduced and optimized. Optimal per-formance and design parameters are obtained analytically under the MPD conditions of the Dual cycle.

The obtained results are discussed in terms of thermal efficiency, power and engine sizes. It is shown that for

the combined cycle, a design based on the MPD conditions is more advantageous from the point of view of

engine sizes and thermal efficiency. 2002 Elsevier Science Ltd. All rights reserved.

Keywords: Maximum power density; Performance optimization; Dual cycle; Joule-Brayton cycle; Combined cycle.

1. Introduction

During the last decade, many optimization studies for heat engines based on endoreversible and

irreversible models have been conducted by considering finite time and finite size constrains undervarious heat transfer modes, mainly linear and non-linear [1,2]. In these studies, the objective

functions chosen for optimization are usually power output, thermal efficiency, entropy genera-tion and ecological benefit. The proper optimization criterion to be chosen for the optimal designof heat engines may differ depending on their purpose and working conditions. For power plants,

in which fuel consumption is the main concern, the maximum thermal efficiency criterion isvery important, whereas for aerospace vehicles, for which propulsion is of great importance,the maximum power (MP) output criterion has great significance. On the other hand, for ship

Energy Conversion and Management 43 (2002) 20192031

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* Corresponding author. Fax: +90-212-258-2157.

E-mail address: [email protected] (B. Sahin).

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2002 Elsevier Science Ltd. All rights reserved.P I I : S0196- 8904( 01) 00149- 2


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propulsion systems, both fuel consumption and thrust gain may be equally important, so in such

a case, both the maximum power output and maximum thermal efficiency criteria have to beconsidered together in the design. However, performance analyses based on the above optimi-

zation criteria do not take the effect of the engine size as related to the investment cost into ac-count. The maximum power density (MPD) analysis, considering the size of the heat engine, hasbeen introduced and applied to the reversible and irreversible Joule-Brayton (JB) heat enginesby Sahin et al. [35]. In their study, it is shown that a design based on MPD conditions is more

advantageous from the points of view of engine size and thermal efficiency. Erbay and Yavuz[6,7], Medina et al. [8] and Chen et al. [9] applied the MPD technique to the Ericsson, Stirling,regenerative JB and Atkinson heat engines, respectively. Sahin et al. [10] applied the MPD

technique to the endoreversible Carnot heat engine, which can be considered as a theoreticalcomparison standard for all real heat engines in finite time thermodynamics, and thus, generalized

the endoreversible MPD analysis results. Kodal et al. [11] extended the MPD analysis to an ir-reversible Carnot heat engine model by considering internal irreversibility and heat leakage effects.

Kodal [12] applied the MPD technique to irreversible combined Carnot heat engines.Much interest has been recently paid to the optimization of air standard Otto, Diesel and Dual

cycles [1320]. In these optimization studies, optimal design and operation parameters under MP

conditions were investigated. No performance analysis study of internal combustion engine cycles(Otto, Diesel and Dual cycles) using the MPD technique appears to be published in the literatureyet. Therefore, we have performed a performance analysis on the Dual cycles using the MPD

criterion. Also, a new combined power cycle model (Dual JB) has been introduced and opti-mized based on the obtained results of the MPD optimization of the Dual cycle.

2. The theoretical model

ThepVandTSdiagrams of the considered combined cycle model are shown in Fig. 1. It isa combination of a modified air standard Dual cycle (MDC) and JB cycle in a cascade. In the

model, the classical air standard Dual cycle (CDC) (12345) is also given for comparisonreasons. In modern times, the Atkinson ideal cycle has been used as a model for the combinationof a free piston engine/compressor with a gas turbine [9]. The combined cycle model given in Fig. 1

may be considered as a new theoretical model for the combination of a reciprocating heat enginewith a gas turbine system. The proposed combined cycle model considering finite rate heat transfer

irreversibility between both heat engines is more realistic than the Atkinson cycle.The modified Dual cycle in Fig. 1 involves two constant volume (23, xy), one constant

pressure (34) and two isentropic (y2, 4x) processes. It should be noted that the final condition

of the expansion process is the statex for the modified Dual cycle and the state 5 for the classicalDual cycle. Statexis located between state 4 and state 5, and it has an optimal location in terms of

MPD.The heat rate inputs to the Dual cycle are _QQ23 and _QQ34, and the heat rate input to the JB cycle

is the waste heat rate, _QQxyof the Dual cycle. The JB cycle that is combined with the Dual cycleinvolves two isentropic and two constant pressure processes. The process 1y is an isentropic

compression by an ideal compressor, process y6 is an isobaric heat rate addition (waste heat ratefrom the Dual cycle), process 67 is an isentropic expansion by an ideal turbine and process 71 is

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an isobaric heat rate extraction. The state 1 is the initial state of the compression process for JBcycle and classical Dual cycle, which is known for a fixed atmosphere.

The working fluid in each cycle system flows continuously so that the combined cycle operatesin a steady state. Assuming the working fluid is an ideal gas with constant specific heat, the powerof the modified Dual cycle can be written approximately in the form

_WWMDC _mmcVT3T2 kT4T3 TxTy; 1where _mmis the mass flow rate, cV is the constant volume specific heat capacity, kis the ratio of the

specific heat capacitiescP=cVand T2,T3,T4,Tx andTyare temperatures at states 2, 3, 4, x and y,respectively.Let us define the pressure ratio b, the cut-off ratio q and the expansion ratio Xas:

bp3p2

T3T2

; 2

qV4V3

T4T3

; 3

X TxT4

pxp4

k1=k: 4

Let us also define the cycle temperature ratio as:

aT4T1

; 5

that is, the ratio at the maximum temperature to the minimum temperature in the combined cycle.

In terms of these parameters for the modified Dual cycle, it is easy to find the dimensionlesspower as:

_WWMDC_WWMDC

_mmcVT1a kb q1 b1

bq

bq

k1bqk

X

: 6

Fig. 1.pV and T

Sdiagram for the combined (DualJB) cycle.

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Following Sahin et al. [3], we define the power density, _WWMDCd, as the power per maximumvolume in the MDC, i.e.

_WWMDCd_WWMDC

Vx: 7

By using the definitions in Eqs. (2)(5), the maximum volume of the modified Dual cycle, Vx, canbe written as:

VxqV1 X abq

1=1k: 8

The dimensionless power density of the modified Dual cycle is found by dividing Eq. (6) by Eq. (8)as follows:

_WWMDCd _WWMDC

d_mmcVT1=V1 a

b k=k1 kbq 1 b1q2k1=k1 X1=k1

bqk

1

qk2=k1Xk=k1

: 9

One can maximize the power density given in Eq. (9) with respect to the expansion ratio pa-

rameter, Xand, at MPD, can find it as:

Xqk1kbq1 b1

kbqk1 : 10

The dimensionless MPD now can be found by substituting Eq. (10) into Eq. (9), i.e.

_WWMDCd max k 1bqk1 akbq1 b1

kbq

bqk

1

k=k1: 11

By substituting Eq. (10) into Eq. (6), the dimensionless power at the MPD conditions can also beobtained as:

_WWMDC

_WWMDC_mmcVT1

a k1k

k

q1 q

b1

bq

: 12

The thermal efficiency of the modified Dual cycle can be written in terms of the definitions in Eqs.(2)(5) as,

gMDC1 bqk1

qk1

b

1

kb

q

1

X: 13

By substituting Eq. (10) into Eq. (13), the thermal efficiency of the modified Dual cycle at MPD

conditions becomes

gMDCk1

k ; 14

which is only a function ofk, the ratio of the specific heat capacities. It is very interesting to findthat gMDC is independent of the cycle parameters.

The variation of the MPD,_WWMDCd max given in Eq. (11) with respect to q and b is shown inFig. 2. As it can be seen from the figure, there are optimal values for q and b which further



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maximize _WWMDCd max. The optimal values forq and b can be obtained analytically asb1 andfrom the solution of the equation: qkq20, e.g. for k1:4, q1:33. This result indi-cates that the highest power density can be obtained by the Diesel cycle ( b1).

The maximum volume in the cycle can characterize the size of a reciprocating heat engine (i.e.the sum of the clearance and displacement volumes). Using Eq. (10) in Eq. (8), the ratio of the

maximum volume in the modified Dual cycle at MPD (VxVx) to that of the classical Dual cycle(VxV1) can be written as

VxV1

akbq1 b1kbqbqk1

1=1k: 15

For a Dual cycle, the above analytically obtained performance results show that, although adesign based on the MPD criterion has the advantage of smaller size, it has the disadvantage oflower power output and thermal efficiency than a classical design (see Fig. 3ac). However, by

using the waste heat from the Dual topping cycle working at MPD conditions as the heat sourceof a JB bottoming cycle as shown in Fig. 1, the power output and the thermal efficiency can beimproved considerably.

Assuming an ideal gas with constant specific heats, the power produced by the JB bottomingcycle according to the first law of thermodynamics can be written in the form

_WWJB _QQy6 _QQ71 _mmcPT6Ty T7T1: 16The heat gained by the working fluid of the JB cycle during process (y6) is equal to the heatrejected by the working fluid of the Dual cycle during process (xy), i.e. _QQxy _QQy6. Using this

Fig. 2. Variation of the MPD of the modified Dual cycle _WWMDCd max with respect toqfor variousbvalues (a) and withrespect to b for various q values (b) (k1:4, a7).



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equality and assuming the two cycles have the same mass flow rate, the maximum temperature inthe JB cycle, T6 can be written as,

T6TxTyk

Ty: 17

Fig. 3. Variations ofVx=V1 (a), _WWMDC=_WWCDC (b) and g

MDC=gCDC (c) with respect to the cut-off ratio, q for various b

values (k1:4, a7).



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Since the processes 1y and 67 are isentropic and there are no pressure losses in the heatinteractions, the temperature ratios can be given as T6=T7Ty=T1. By using the Dual cycle pa-rameters defined in Eqs. (2)(5), we have

Tx

T1aX; 18

Ty

T1 aX

bqk: 19

By using Eqs. (17)(19), the dimensionless power output for the JB bottoming cycle can be ob-tained as

_WWJB_WWJB

_mmcVT1 bqk1 a

bqkX

1

: 20

By summing Eqs. (6) and (20), we can obtain the power of the combined Dual and JB cycles as

_WWC_WWMDC _WWJB

_mmcVT1akq1

q a

q

b 1b

bqk 1: 21

The thermal efficiency of the combined system is

gC_WWC

_QQ23 _QQ341 bq

k 1a kq1

q b1

qb

: 22

When we examine Eqs. (21) and (22), we see that the power and the thermal efficiency are thesame as those of the classical Dual cycle [21]. If we optimize Eq. (21) with respect to q and b, weget

o _WWC

ob 0!b2qk1 a0; 23

o _WWC

oq 0!aka b1

b kbqk1 0; 24

which yieldb1 andqa1=k1. If we use these results in Eqs. (21) and (22), we get the MP ofthe combined cycle as

_WWCmax 1 ka 1kak=k1 25and the thermal efficiency at this MP as

gCmp1 1 ak=k1ka1=k11 : 26

It should be noted that the optimum condition of the combined cycle is the same as for the Dieselcycle [16].

The performance disadvantage of a reciprocating engine designed as a modified Dual cycle thathas an important size advantage with respect to the classical design is removed by a combined



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system (modified Dual cycleJB). The added gas turbine cycle to the system will increase theoverall size. However, this will not totally remove the size advantage of the modified Dual cyclebecause for a given power, the sizes of the turbo-machinery engines are relatively quite smaller

than those of the reciprocating engines [22,23]. Therefore, the proposed combined cycle systemmodel can have a size advantage and have the same performance as the classical Dual cycle for thesame q and b values. In addition, in the application of the proposed combined cycle model, thepiston speed of the reciprocating engine will be reduced due to the smaller size for a given engine

speed. When it is required to work at the same piston speed as the classical reciprocating enginedesign, the proposed model gives the availability to increase the revolutions or power.

3. Results and discussion

The variations of the dimensionless power density,_WWMDCd with respect to thermal efficiencyfor various b and q are shown in Fig. 4a and b, respectively. We observe from Fig. 4a that the

power density increases as the pressure ratio (b) decreases, and it reaches higher values for b1,i.e. the Diesel cycle. There is also an optimum value of the cut-off ratio (q) at which the power

density has higher values (Fig. 4b). This case can be seen more clearly from Fig. 2. The optimalvalues at which the MPD has its highest value are obtained asq1:33 andb1 fork1:4. Atthese obtained optimal design parameters (X; q and b), the size of the modified Dual cycle (thetopping cycle of the combined system) is minimum for a given power. The dimensionless powerthat can be obtained from the modified Dual cycle at the highest value of the MPD is found from

Eq. (12) as _WWMDCjqq; bb0:09924812a. By considering Eq. (9) for a special case ofq1 (Otto

Fig. 4. Variation of the power density of the modified Dual cycle_WWMDCd with respect to the thermal efficiency forvarious b values (q1:2) (a) and for various q values (b1:2) (b) (k1:4, a7).



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cycle), the MPD analysis yields the optimum pressure ratio asbkand the optimum expansionratio as X1=k. At these optimal values the MPD is

_

WWOd max k12

k

2k=

1

k

ak=

k

1

27and the corresponding power is

_WWOa

k1k

2ag2: 28

The thermal efficiency at MPD conditions is independent of the cycle parameters (X; q; b), so itis the same for the Diesel, Otto and Dual cycles (Eq. (14)). An internal combustion engine cycle

(Dual, Diesel, Otto) designed by using the MPD criterion can provide an important size ad-vantage with respect to the classical cycles. This case can be observed from Fig. 3a. For example,for q1:8 and b1:6, the engine size can be reduced as much as 70%. Although the design atMPD conditions has a major size advantage with respect to the classical design, it has power andthermal efficiency drawbacks. Forq1:8 andb1:6, the decrease in the power and the thermalefficiency is about 50% (Fig. 3b and c). Asq and b increase, the performance at MPD gets better,

however in this case, the size advantage reduces. By using the waste heat from the Dual toppingcycle working at MPD conditions as the heat source of a JB cycle (Fig. 1), the power and thermal

efficiency can be equalized to those of a classical Dual cycle as is shown in Eqs. (21) and (22), andthe size advantage of the MPD conditions will be conserved considerably.

The effects of q and b on the performance of the combined cycle (Dual JB) are shown inFigs. 5 and 6. As b decreases the MP and the corresponding thermal efficiency of the combinedcycle will increase (Fig. 5). There is an optimum value of q that maximizes the power of the

combined cycle (Fig. 6). The optimal values can be found analytically as bmp

1 andqmpa1=k1 following Eqs. (23) and (24). In Fig. 7, the variations of the normalized power of the

Fig. 5. The effect ofb on the performance characteristics (varying q) of the combined cycle (k1:4, a7).



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combined cycle and the normalized MPD of the topping Dual cycle with respect to the thermalefficiency of the combined cycle by varyingqare shown. The point c corresponds to the conditions

of the highest value of the MPD of the topping Dual cycle (b1, q1:33), and the point bcorresponds to the performance of the combined cycle at these conditions. The power and the

thermal efficiency at point b can be obtained by substituting b1 andq1:33 into Eqs. (21) and(22). The point a indicates the conditions of the MP of the combined cycle. As can be observed

Fig. 6. The effect ofq on the performance characteristics (varying b) of the combined cycle (k1:4, a7).

Fig. 7. Comparison of the normalized power of the combined cycle and the normalized MPD of the MDC with respectto the combined cycles thermal efficiency (varying q) (k1:4, a7, b1).



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from the figure, a combined cycle consisting of the Dual cycle designed at MPD conditions has

higher thermal efficiency but lower power with respect to the combined cycle designed at MPconditions. By considering the power, the thermal efficiency and the size of the combined system

together, the optimal design region will be the arc ad, i.e.1P _WWC= _WWC maxP 0; gmp6 gC6 gmax: 29

The choice of the optimal design parameters depends on the working place and purpose of thecombined system. That is, if the power is the major concern, the design parameters should bechosen close to the point a, or close to the point b for less engine size or close to the point d forbetter thermal efficiency. For q1, the topping cycle becomes the Otto cycle and the MP con-ditions of the combined cycle yieldsbmp

ffiffiffia

p. By substitutingq1 andbmp

ffiffiffia

p into Eqs. (21)

and (22), the MP and the corresponding thermal efficiency are obtained, respectively, as

_WWC

max

a 1

1

ffiffiffiap

2

;

30

gCmp1

1ffiffiffia

p : 31When the topping Otto cycle works at MPD conditions, we have already obtained bk. Usingq1 andbkin Eqs. (21) and (22), we can obtain the power and the thermal efficiency of thecombined cycle corresponding to MPD conditions of the topping Otto cycle, i.e.

_WWC

k1k

ak; 32

gC

1

k

a

:

33

Fig. 8. Variation of the ratio of the Dual cycles power to that of the combined cycle at MPD conditions of the Dual

cycle with respect to q for various b values (k1:4, a7).



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The effects of the design parameters, q and b on the ratio of the power of the topping Dual cyclewhenX X (Eq. (12)) to the power of the combined cycle (Eq. (21)) are shown in Fig. 8. As canbe seen from the figure, the power percentage of the topping Dual cycle increases as q and b

increase. For the conditions of the highest value of the MPD of the topping Dual cycle (b1,q1:33), the power percentage is about 36% for a7.

4. Conclusions

A new kind of combined cycle (Dual JB) has been proposed based on air standard internalcombustion Dual cycles MPD conditions. In this perspective, the design parameters (q; b, X) thatmaximize the power density of the Dual topping cycle have been investigated. The performance ofthe Dual cycle under MPD conditions are discussed and compared with respect to the classical

Dual cycle in terms of power, thermal efficiency and engine size. It is shown that for a Dual cycle,although a design based on the MPD criterion has the advantage of smaller size, it has the dis-

advantage of lower power output and thermal efficiency than a classical design. The power andthermal efficiency disadvantages of the Dual cycle working at MPD conditions are removed by theproposed combined system (DualJB). In the proposed combined cycle, the JB bottoming cycleuses the waste heat from the Dual topping cycle working at MPD conditions as the heat source. Itis shown in Eqs. (21) and (22) that the power and the thermal efficiency can be equalized to those

of a classical Dual cycle for the same q and b values while conserving the size advantage of theMPD conditions. The proposed combined cycle consisting the Dual topping cycle designed at thehighest value of the MPD conditions (b1, q1:33) has higher thermal efficiency but lowerpower output with respect to the combined cycle designed at MP conditions (bmp

1 and qmp

a1=k1).The results obtained in this analysis may provide a new theoretical basis for the optimal design

of the proposed combined cycle that is the combination of a reciprocating heat engine with a gasturbine system.

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08-performance optimization of a new combined power

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