Performance analysis of brayton heat engine at maximum efficient power using temperature dependent s

Abstract

Keywords: Brayton cycle; Efficient power; Finite time thermodynamics; Temperature dependent specific heat

1. Introduction

Brayton cycles have been widely used in gas power plants and aircrafts. As shown in Fig. 1, the cycle involves reversible adiabatic compression followed by isobaric heat addition. The expansion process takes place isentropically, and finally it has isobaric heat rejection. Leff [2] analysed an endoreversible Brayton heat engine following Curzon & Ahlborn [1] and found that impositions of maximum work put in limits on the Brayton cycle temperatures. Wu [3] optimized the power of an

endoreversible Brayton gas heat engine while Wu & Kiang [4] incorporated non-isentropic nature of compressor and turbine to Brayton heat engine and found that engine power and engine efficiency are sturdy functions of the compressor and turbine efficiencies. Sahin et al. [5] analyzed Joule-Brayton cycle at maximum power density. Using the MPD criterion, they found that the efficiency at MPD conditions is greater than that at maximum power output. Sahin et al. [6] also applied the MPD methods to the endoreversible Carnot heat engine while Sahin and Yilmaz [7] extended the work to irreversible Joule-Brayton engine from the view of finite time thermodynamics or entropy generation minimization [8]. Finite time thermodynamics has explored many new fronts since this technique had been used to analyze and optimize the performance of real thermodynamic processes, devices and cycles. In past, many optimization studies for Brayton heat engines based on endoreversible and irreversible mode have been carried out by number of researchers [9-18]. Yilmaz [19] examined the performance of Brayton cycle based on the efficient power criterion and found a compromise between engine power and thermal efficiency on maximizing the efficient power function. Usually constant specific heats of the working fluid were used in their studies. Recently, many researchers [20-22] were involved in performing thermodynamic analysis using variable specific heats of the working fluid. It was proved and verified that variable specific heats give better approximation to actual cycles than using constant specific heat of the working fluid. In this paper, we have presented a performance analysis of Brayton heat engine with variable specific heat of the working fluid under maximum efficient power conditions as the literature available has not covered the variable specific heat consideration for Brayton cycle performance analysis. So, actual performance of proposed model has not been obtained. The

authors obtained expressions for maximum power output, maximum power density, maximum efficient power and the corresponding thermal efficiency of the cycle. The effect of various efficiencies, isentropic temperature ratio, maximum pressure ratios, maximum cycle temperature ratio, maximum volume ratios have been studied in detail and the results are presented on the graphs. The results obtained with temperature dependent specific heat are also compared with those obtained by using constant specific heat of the working fluid.

2. Thermodynamic Analysis

An air standard Joule-Brayton cycle involving two isobaric and two reversible adiabatic processes is shown in Fig. 1. The compression and expansion processes are reversible adiabatic as shown by process (1-2) and process (3-4) respectively on pressure-volume (p-V) and Temperature-Entropy (T-S) diagrams. The heat addition and heat rejection processes are isobaric as shown by process (2-3) and process (4-1) respectively in Fig. 1.

Maximum temperature ratio (τ) and isentropic temperature ratio (θ) for Brayton heat engine cycle is defined as: τ =

3. Analysis Of Results

The above derived formulae are used and plotted in order to analyze performance of Brayton heat engine with variable specific heats of the working fluid. According to references [18-20], the following constants and ranges of parameters are used in the calculation:

c) Fig. 2 Variations of the (a) normalized power, (b) normalized power density and (c) normalized efficient power with respect to thermal efficiency

A1=0.9521 kJ/kg-K, A2=0.0002 kJ/kg-K, T1 = 298 K, γ = 1.4, τ = 1-25 Variations of the normalized power (W / Wmp), normalized power density (Wpd / Wmpd) and normalized efficient power (Wep / Wmep) with respect to thermal efficiency are shown in Figs 2(a)

As one can see from Fig. 2 that, thermal efficiency at maximum normalized power density (ηmpd) and thermal efficiency at maximum normalized efficient power (ηmep) are always greater than the thermal efficiency at maximum normalized power conditions (ηmp).It can also be observed that the value of thermal efficiency at above maximum normalized conditions increase with an increasing value of maximum cycle temperature ratio (τ). Normalized power, normalized power density and normalized efficient power are plotted together for τ = 2.5 and τ = 10 in Fig. 3 (a) and Fig. 3 (b), respectively.

where (V4)mpd is volume at maximum power density and (V4)mp is volume at maximum power criterion. Similarly, the ratio of maximum volume at maximum efficient power to that at maximum power can be written as:

The size of a Brayton heat engine can be characterized by the maximum volume in the cycle i.e. V4. Using equation (16), one can write

Fig. 4 Variations of the various efficiencies (ηmp, ηmpd, ηmep) with respect to τ

Fig. 3 Normalized power, normalized power density and normalized efficient power are plotted together for τ = 2.5 and (b) τ = 10 It can be concluded from these figures that, ηmep > ηmpd for τ = 2.5 but ηmpd becomes greater than ηmep for τ = 10. These findings can be seen more clearly from Fig. 4, which shows the comparison of three maximum efficiencies with respect to various values of τ. It can be seen from Fig. 4 that ηmep > ηmpd for 1< τ<2.7 and after this value ηmpd becomes greater than ηmep. For example, at τ = 2.5, ηmp = 0.3939, ηmpd = 0.4578, ηmep = 0.4625 while at τ = 10, ηmp = 0.7469, ηmpd = 0.8706, ηmep = 0.8021

Fig. 5 Maximum volume ratios variations with respect to cycle temperature ratio (τ) Maximum volume ratio variations with respect to cycle temperature ratio are shown in Fig. 5. It can be concluded that Brayton heat engine size designed at maximum power density and maximum efficient power criterion is smaller compared with engine designed at maximum power criterion. It can also be observed that increasing value of maximum cycle temperature ratio, results in further reduction in size of Brayton heat engine designed at maximum power density conditions

while in maximum efficient power conditions, size of Brayton engine remains more or less the same. Maximum pressure ratio at maximum power density to pressure ratio at maximum power is defined as: (h$ )BC

where (P3)mpd is pressure at maximum power density and (P3)mp is the pressure at maximum power criterion.

Fig. 7 Isentropic temperature ratios variations at maximum power, maximum power density, max. efficient power with respect to maximum cycle temperature ratio 1 τ = 2.5 τ = 5.0 τ = 7.5 τ = 10.0

where (P3)mep is the maximum pressure at maximum efficient power conditions. Maximum pressure ratios variations with respect to maximum cycle temperature ratio are shown in Fig. 6. It has been observed that increased value of τ results in increase of maximum pressure ratios and Fig. 6 also reflects that engine designed at maximum power density conditions operates at higher pressure ratios than those designed at maximum efficient power conditions. Therefore, tougher material is required for Brayton heat engine designed at maximum power density conditions to with stand higher pressure ratio than those designed at maximum efficient power conditions.

Fig. 8 Normalized efficient power (Wep/Wmep) variations with respect to pressure ratio (r)

Fig. 6 Maximum pressure ratios variations with respect to cycle temperature ratio (τ) Fig. 7 shows the variations of the isentropic temperature ratios at maximum power (θmp), maximum power density (θmpd), and maximum efficient power (θmep) for variable specific heat of the working fluid with cycle temperature ratio (τ). As can be seen from Fig. 7, that the values of various isentropic temperature ratios go on increasing with an increasing value of the cycle temperature ratio (τ).It can also be observed that θmep = θmpd at τ = 2.7.

Fig. 9 Normalized power density (Wpd/Wmpd) variations with respect to pressure ratio (r)

Variations of normalized efficient power (Wep/Wmep) and normalized power density (Wpd/Wmpd) with respect to pressure ratio (r) are shown in Fig. 8 and Fig. 9 respectively. The following observations are made from these figures: i) For a given value of pressure ratio, normalized power density and normalized efficient power increases with decreasing value of maximum cycle temperature ratio. ii) For any value of τ and r (less than 40), the normalized efficient power is more as compared with normalized power density. iii) For a value of τ = 2.5, the normalized power density and normalized efficient power touches its maximum value at a pressure ratio of 8. But normalized efficient power always remains more as compared with normalized power density.

Fig. 11 Normalized power output variations with respect to maximum cycle temperature ratio

Fig. 10 Thermal efficiency variations with respect to pressure ratio (r)

Variations of thermal efficiency with pressure ratio for various maximum cycle temperature ratios are shown in Fig. 10. It has been observed that thermal efficiency increases with an increasing value of maximum cycle temperature ratio and pressure ratio. The value increases sharply at pressure ratio of (r ≤ 5) compared with at pressure ratio of (r > 5).

Fig. 12 Variations of normalised power density with respect to maximum cycle temperature ratio 0.7 Constant Temperature Dependent

Using Constant Specific HEAT Of The

WORKING FLUID: The formulae derived above also apply to an isentropic Brayton heat engine with constant specific heat of working fluid provided one sets A1 equals to Cp and A2 equals to zero. Thus, for Brayton heat engine cycle with constant specific heat of the working fluid, maximum power output, maximum power density, maximum efficient power and the corresponding efficiencies can be found as: *= =

(29) Fig. 13 Variations of normalised efficient power with respect to maximum cycle temperature ratio

The above equations (29)-(34) replicate the results obtained by Yilmaz [17] in which Joule-Brayton engine with constant specific heat was examined. These equations are plotted in the view of comparison of Brayton heat engine with temperature dependent specific heat with those results

assuming a constant specific heat as shown in Figures 11 through 17. 1

The following constants and range of τ are selected: γ = 1.4, T1 = 298, τ = 1-25. Fig. 11 – Fig. 13 show the variations of normalised power output, normalised power density, normalised efficient power for variable and constant specific heats with maximum cycle temperature ratio (τ). The dotted lines represent the various normalised powers with constant specific heat while dark lines represents with temperature dependent specific heat of the working fluid. The deviation between W/ Wmp, Wpd/ Wmpd, Wep/Wmep using the temperature dependent and constant specific heat increases with increasing τ. The values of W/ Wmp, Wpd/ Wmpd, Wep/Wmep using constant specific heat are superior than using temperature dependent specific heat of working fluid.

c) Fig. 14 Variations of the (a) normalized power (W / Wmp), (b) normalized power density (Wpd / Wmpd) and normalized efficient power (Wep / Wmep) with respect to thermal efficiency for various τ values

Fig. 15 Normalized power, Normalized power density and normalized efficient power are plotted together for (a) τ =

5. Conclusion

A comparative performance analysis for Brayton heat engine with temperature dependent specific heat based on maximum power, maximum power density and maximum efficient power conditions has been done by introducing new criterion. The results show that more efficient engines are designed at maximum efficient power conditions than those at maximum power and maximum power density conditions. Although the Brayton heat engine designed at maximum power density conditions requires smaller size but higher pressure ratio in comparison with those designed at maximum efficient power conditions. Brayton heat engine with temperature dependent specific heat of the working fluid gives realistic prediction of engine efficiency and power output than does the reversible Brayton heat engine with constant specific heat. The results obtained from this work can be useful in the optimal design and operation of real Brayton heat engines. This work can further be extended by taking both internal and external irreversibilities and optimizing the system using evolutionary techniques.

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