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Analytical model-based energy and exergy analysis of a gas microturbine at part-load operation
Applied Thermal Engineering 57 (2013) 125e132
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Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng
Analytical model-based energy and exergy analysis of a gas microturbine at part-load operation Leszek Malinowski a, *, Monika Lewandowska b a b
Faculty of Maritime Technology and Transport, West Pomeranian University of Technology, 71-065 Szczecin, Al. Piastów 41, Poland Institute of Physics, West Pomeranian University of Technology, 70-311 Szczecin, Al. Piastów 48, Poland
h i g h l i g h t s Analytical model for part-load operation of a gas microturbine is elaborated. The model is based on heuristic part-load performance formulas. The model is validated by comparison with experimental and manufacturer’s data. Exergy destruction or loss for each microturbine component is calculated.
a r t i c l e i n f o
a b s t r a c t
Article history: Received 30 September 2012 Accepted 23 March 2013 Available online 10 April 2013
In this paper a universal analytical model for part-load operation of gas microturbines has been elaborated which is subsequently used in the energy and exergy analysis of a sample device. The model, based on the Brayton cycle and heuristic part-load performance formulas, takes into account: the temperature variation of working fluid specific heat at constant pressure in calculations of adiabatic processes, enthalpy, and exergy, the non-linear dependence of pressure drop on flow rate, and the cooling of generator by intake air. The model is validated using the manufacturer data for a commercially available microturbine of 30 kWe and results of measurements. The agreement is very good as for such a general simple analytical model. Exergy calculations based on the elaborated model show that the greatest potential for improving the efficiency of the microturbine lies in the combustion chamber and recuperator, as these components are characterized by the largest exergy destruction and loss. Ó 2013 Elsevier Ltd. All rights reserved.
Keywords: Gas microturbine Part load operation Analytical model Energy and exergy analysis
1. Introduction Small gas turbines whose mechanical power do not exceed several hundred kilowatts are classified as microturbines. In general gas microturbines operate according to the open Brayton cycle with or without heat regeneration. A typical microturbine is a single shaft design with the turbine, compressor and generator mounted on the same shaft. The pressure ratio is of the order 3.5 to 4.0 and the microturbine is controlled by fuel delivery alone. Usually, during load changes the rotational speed of the microturbine also changes while the temperature of the exhaust gas leaving the turbine is kept constant. Gas microturbines are increasingly used as a primary source of electricity and heat for individual objects, such as: hotels, sports
* Corresponding author. Tel.: þ48 91 449 4827. E-mail addresses: [email protected], [email protected] (L. Malinowski). 1359-4311/$ e see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.applthermaleng.2013.03.057
facilities, greenhouses, offices, small businesses, small houses and others. Such micro plants operate under varying demand for electricity and heat. However, running the microturbine at a reduced load results in loss of efficiency. In the literature, some attention has been paid to the problem of modeling of gas turbines operated at part load and to the analysis of their performance based on the first and second laws of thermodynamics. Zhang et al. [1] formulated an analytical model of a constant rotating speed single shaft gas turbine operated at part load and determined basic part-load characteristics of the turbine. Wang et al. [2] extended the model presented in Reference 1 for microturbines operated with variable rotational speed. They derived the optimal rotational speed for part-load operation and analyzed the effect of pressure and temperature ratios on the offdesign performance. Song et al. [3] presented an exergy-based performance analysis of a 150 MWe gas turbine based power plant for part-load operation. They investigated numerically the influence of the variable inlet guide vane and the blade cooling on
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Nomenclature B cp G h HHV k LHV n p Q R s T w W
exergy rate, W constant-pressure specific heat, J/(kg K) mass flow rate through the compressor and turbine, kg/s specific enthalpy, J/kg higher heating value, J/kg relative pressure loss lower heating value, J/kg rotating speed, 1/s pressure, Pa heat rate, W gas constant, J/(kg K) specific entropy, J/(kg K) temperature, K specific work, J/kg power, W
Greek symbols 3 effectiveness of the recuperator h efficiency P pressure ratio f total relative pressure loss
exergy destruction. Kim et al. [4] investigated the performance characteristics of recuperated gas turbines operating at partial load. They examined the influence of various design and operational factors on the part load efficiency. They considered simple (fuel only control), variable speed and variable inlet guide vane operation, constant and variable area nozzles (for two-shaft turbine). Their analysis was based on the manufacturer’s compressor and turbine performance maps. Aklilu et al. [5] developed a mathematical model to simulate a part-load operation of a single shaft gas turbine with variable geometry compressor. They created performance maps for the compressor and turbine based on manufacturer data and thermodynamic laws. In their model average values of the specific heat at constant pressure and isentropic exponent of the working gas were used. They tested their model with the data from a 4.2 MWe turbine. Badami et al. [6] performed an exergy analysis of a small 150 kWe combined cycle cogeneration plant with an internal combustion engine as a primary source of energy. They examined exergy and energy efficiency of the plant for part load operation. Ghaebi et al. [7] applied the first and second laws of thermodynamics as well as economic analysis to investigate a combined cooling, heating and power system with gas turbine of 19 MWe. They investigated the effect of selected parameters on the efficiency of heat, cold and power production for rated load. In their analysis, they used a constant heat capacity at constant pressure while calculating the enthalpy and exergy of the working gas. Khaliq et al. [8] studied, by the energy and exergy method, the influence of compressor intake air cooling on energy and exergy efficiency of a gas turbine cycle. They also examined exergy destruction rate in the power plant components as well as the effect of the pressure ratio and turbine inlet temperature on the energy and exergy efficiency of the cycle. They used an average value of the specific heat at constant pressure of the working gas and an average isentropic exponent in their model. Bakalis et al. [9] performed a part load exergetic analysis of a hybrid gas microturbine fuel cell system using the commercial AspenPlus software. They created the microturbine performance maps through the scaling of representative maps. Meybodi et al. [10] studied the optimum arrangement of prime movers in small scale
Subscripts atm atmospheric c compressor cc combustion chamber e electronics el electric f fuel g generator in inlet out outlet r recuperator rh recuperator, high pressure side rl recuperator, low pressure side s isentropic t turbine 0 design value Accents w
~ c ¼ Gc =G divided by design value, e.g. G c0 pffiffiffiffiffi reduced parameter, e.g. Gc ¼ G T1 =p1
Abbreviation TET turbine exit temperature
microturbine-based CHP systems. In their analysis, they used approximate equations for the relative efficiency, fuel consumption, exhaust mass flow rate, and exhaust temperature for partial-load conditions. These equations were derived based on the manufacturers’ data for five microturbines. Wei et al. [11] investigated the off-design performance of a small-sized humid air turbine. Their numerical calculation model was based on digitized performance maps. They validated the results from the model with own experimental data. Full and partial load performance tests of a small trigeneration pilot plant based on a microturbine were carried out by Rocha et al. [12]. Experimental results presented by the authors include the fuel consumption, electrical and thermal power, primary energy saving index, and energy utilization factor. This paper deals with the development of a universal analytical model of microturbines operated with partial load, at variable rotational speed, for use in energy and exergy analysis. The model is based on the heuristic part-load performance formulas proposed in References 1 and 2. Due to commercial reasons, the manufacturers usually do not publish performance maps of turbines and compressors in the open literature, hence the large usefulness of universal performance formulas. Thanks to the availability of such formulas, the microturbine’s suitability for the intended use can be thoroughly verified. To reduce calculation errors, we do not use an average isentropic coefficient in our model, but the temperature dependence of the specific heat at constant pressure of the working fluid is directly implemented into the equation of isentropic process. The condition Ds ¼ 0 brings us to the following isentropic equation c0 ln T þ c1 T Rln p ¼ const, where c0 and c1 are coefficients in the equation cp ðTÞ ¼ c0 þ c1 T describing the dependence of specific heat at constant pressure on temperature. The temperature dependence of specific heat at constant pressure is also taken into account during the calculation of enthalpy and exergy. When calculating the thermodynamic cycle parameters and exergy destruction and losses in the microturbine components, a non-linear dependence of pressure drop in the components on the microturbine load is accounted for. The model also allows to include the heat absorbed by intake air from the generator in the energy and exergy analysis.
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127
where
2. Basic assumption and expressions A schematic flow diagram for the modeled microturbine is presented in Fig. 1. When formulating the mathematical model of microturbine the following assumptions are adopted:
P ¼ pj =pi
(4)
Specific entropy of air in the state i is determined by the formula microturbine is operated as stand alone, turbine is controlled by fuel flow rate at constant turbine exit temperature (TET), compressor and turbine are of radial type, turbine and compressor are mounted on a common shaft, compression in the compressor and expansion in the turbine are adiabatic irreversible processes (i.e. non-isentropic), pressure losses in the recuperator, in the combustion chamber, at the inlet to the compressor, and at the outlet from the turbine are taken into account as well as their non-linear dependence on the turbine load, the air sucked in by the compressor is used for cooling the generator, the working medium is considered to be an ideal gas, air; the mass of fuel is disregarded because of a very high value of the excess air ratio, the specific heat at constant pressure of the working medium depends linearly on temperature, exergy and entropy of the air have a value of zero for pressure patm ¼ 0.11013 MPa and temperature Tatm ¼ 288 K. The following correlation for the specific heat of air at constant pressure is applied which is based on the data taken from Ref. [13]
cp ðTÞ ¼ 0:931046 þ 2:066697$104 T kJ=ðkg KÞ
(2)
Ti
Ds Ti ; Tj ; P ¼
BðT; pÞ ¼ G½DhðTatm ; TÞ Tatm DsðTatm ; T; p=patm Þ
(6)
Pressure, pj, at the outlet of the element in which there is a loss of pressure, is calculated from the equation
pj ¼ ð1 kÞpi
(7)
where pi is the pressure at the inlet to that element. Relative pressure losses, k, for part loads are calculated from the expression [14]
k ¼ k0 ðG=G0 Þ2 ðpi0 =pi Þ2 ðTi =Ti0 Þ
(8)
where index 0 indicates the value for the rated load. The measure of the overall system pressure loss is the coefficient
f ¼ ð1 kin Þð1 kout Þð1 kcc Þð1 krl Þð1 krh Þ
(9)
Pt ¼ p3 =p4
(10)
is related to the compressor pressure ratio
Pc ¼ p2 =p1
(11)
by the following equation
Pt ¼ fPc
Increase in specific entropy between states i and j is determined from the formula
Exergy rate is computed from
The turbine pressure ratio
ZTj cp ðTÞdT
(5)
(1)
It is assumed that the fuel is natural gas with LHV ¼ 49653 kJ=kg and HHV ¼ 54990 kJ=kg. Increase in specific enthalpy of the working gas between states i and j is calculated from
Dh Ti ; Tj ¼
si ¼ DsðTatm ; Ti ; pi =patm Þ
ZTj
cp ðTÞ dT RlnðPÞ T
(3)
Ti
(12)
A case study is performed for the data presented in Table 1. The following values of k, for rated load, are used in the calculations: kin0 ¼ 0.01, kout0 ¼ 0.01, kcc0 ¼ 0.02, krl0 ¼ 0.02, krh0 ¼ 0.02. For the analysis, this model was implemented in the software package Mathcad 15 [15].
Table 1 Data used in calculations.
Fig. 1. Schematic diagram of the modeled microturbine. C e compressor, R e heat exchanger, CC e combustion chamber, T e turbine, G e electric generator, E e electronics.
Description
Quantity
Value
Unit
Turbine load
Microturbine electric power Microturbine inlet temperature Microturbine exit temperature (TET) Ambient temperature Ambient pressure Compressor pressure ratio Recuperator effectiveness Combustion chamber efficiency Compressor isentropic efficiency Turbine isentropic efficiency Compressor mechanical efficiency Turbine mechanical efficiency Generator efficiency Electronics efficiency
Wel0 T30 T4 Tatm patm
30 1114 872 288 0.11013 3.6 0.79 0.98 0.78 0.83 0.995 0.995 0.96 0.96
kW K K K MPa e e e e e e e e e
Rated Rated Any Any Any Rated Rated Any Rated Rated Any Any Any Any
Pc0 30
hcc hc0 ht0 hcm htm hg he
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3. Energy calculations for part load operation
Similarly for the turbine
3.1. Analytical model for part load operation
Gt0 ¼ G0
In our model we make use of the performance formulas for compressor and turbine at part load operation, and a modified Flügel formula proposed in Refs. [1,2]. Performance formulas for compressor are
Gt ¼ G
pffiffiffiffiffiffiffi T30 =p30
(25)
pffiffiffiffiffi T3 =p3
~ t ¼ Gt =G G t0
(26)
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ðG=G0 Þðp30 =p3 Þ T3 =T30
(27)
The reduced compressor rotational speed at rated load
Pc =Pc0
~2 þ c G ~ ¼ c1 G 2 c þ c3 c
i h i h ~c 2 n ~c ~ c =G ~ c =G ~ c Þ2 n ¼ 1 c4 ð1 n
hc =hc0
(13)
pffiffiffiffiffiffiffi nc0 ¼ n0 = T10
(14)
and at part load
pffiffiffiffiffi nc ¼ n= T1
where
~ c =c c1 ¼ n c2 ¼ c3 ¼
p
(15) ~ 2c 2mn
. c
(16)
~ 3c ~ c þ m2 n pmn
. c
~ c ðn ~ c mÞ2 ~c Þ þ n c ¼ pð1 m=n i h i h ~t 2 n ~t ~ t =G ~ t =G ~ t Þ2 n ¼ 1 t4 ð1 n
pffiffiffiffiffiffiffi T10 =p10
(30)
Similarly for the turbine
pffiffiffiffiffi nt ¼ n= T3
(32)
~ t ¼ nt =nt0 ¼ nra n (19)
(20)
and at part load
(22)
(23)
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T30 =T3
(33)
where the ratio of microturbine shaft rotational speeds for partial and full load is
nra ¼ n=n0
(34)
The real rotational speeds of the turbine and compressor are the same because they are mounted on a common shaft, but the relative speeds differ. The generator produces a high-frequency AC power which is rectified to constant-voltage DC power. The DC power is then inverted to 50 or 60 Hz constant frequency AC power. In the present analysis the only independent variable is the ratio of microturbine shaft rotational speeds, nra. Each value of nra corresponds to a single value of microturbine load. For assumed values of nra, the following system of seven equations is solved: - Eq. (13), - Eq. (20) in which G/G0 is substituted by the expression determined from Eq. (27) and Pt is substituted by the RHS of Eq. (12), - equation of isentropic expansion in the turbine
Ds½T3 ; T4s ; 1=ðfPc Þ ¼ 0
(35)
- equation for the turbine isentropic efficiency
ht ¼ DhðT3 ; T4 Þ=DhðT3 ; T4s Þ
(36)
where ht for part load is calculated from Eq. (19), - Eq. (33), - equation in which expressions for the real mass flow rate, G, determined from Eqs. (24) and (27) are equated to each other
~ c ðp =p Þ G 1 10
The relative compressor air mass flow rate
~ c ¼ Gc =G G c0
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T10 =T1 ¼ nra T10 =T1
(18)
(21)
pffiffiffiffiffi Gc ¼ G T1 =p1
~ c ¼ nc =nc0 ¼ ðn=n0 Þ n
(31)
According to Refs. [1,2], Eqs. (13)e(20) are valid for the following equilibrium conditions: nt ¼ nc, Gt ¼ Gc, pt ¼ fpc . The relative variables labeled by tildes, occurring in Eqs. (13)e(19), are equal to the quotient of the reduced variables for partial and nominal load respectively. The reduced and relative variables are defined as follows [1,2]. The reduced compressor air mass flow rate at rated load
Gc0 ¼ G0
The relative compressor rotational speed
pffiffiffiffiffiffiffi nt0 ¼ n0 = T30
The coefficients: m, p, c4, and t4 appearing in Eqs. (14)e(19) are dimensionless and their numerical values are taken arbitrarily. The values of these coefficients are adjusted by the heuristic trial and error method in order to obtain the appropriate shape of compressor and turbine performance maps. For the same reason pffiffiffi the coefficients p and m should satisfy the condition 3 p 2m=3. The correctness of choice of these coefficients can be verified by comparing the results achieved from the model with experimental data and data published by a microturbine manufacturer. We have used the following values of these coefficients: m ¼ 1.8, p ¼ 1.8, c4 ¼ 0.1, and t4 ¼ 0.1 for our case study. A modified Flügel formula is
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Pt 1 P2t0 1 G=G0 ¼ a T30 =T3
(29)
(17)
The efficiency characteristic of the turbine is
ht =ht0
(28)
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~ t ðp =p Þ T =T T10 =T1 ¼ G 3 30 30 3
(37)
- equation for pressure in point 3 (see Fig. 1)
After taking into account Eqs. (21) and (22) we get
~ c ¼ ðG=G Þðp =p Þ G 0 10 1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T1 =T10
(24)
p3 ¼ ð1 kcc Þð1 krh ÞPc p1
(38)
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From above seven equations the values of the following vari~ c, G ~ t , Pc, p3, T3, T4s are determined. Then the remaining ~t , G ables: n temperatures and pressures in the characteristic points of cycle (see Fig. 1) are computed and the calculation of other quantities used in the analysis is carried out. 3.2. Further calculations
p1 ¼ patm ð1 kin Þ
(39)
The temperature of air, T1, after absorbing the heat from cooling the generator, Qatm, is initially assumed. After the gas flow rate, G, and Qatm have been determined, T1 is calculated from the equation
Qatm ¼ GDhðTatm ; T1 Þ
(40)
where Qatm is the rate of heat transferred from the generator to the compressor intake air. Qatm is computed from
Qatm ¼ Gwm 1 hg
It is assumed that the turbine is controlled in such a way that TET remains constant for all loads
T4 ¼ T40 ¼ const
(41)
(50)
The temperature of gas leaving the recuperator, T4r, is determined from the equation of energy balance in the recuperator
DhðT2 ; T2r Þ ¼ DhðT4r ; T4 Þ
The pressure of air at the inlet to the compressor is calculated by
129
(51)
Turbine expansion ratio is calculated from Eq. (10). The turbine isentropic efficiency for part load is computed from Eq. (19). The air mass flow rate, G, is determined from Eq. (24). The turbine internal power
Wt ¼ GDhðT4 ; T3 Þ
(52)
The compressor internal power
Wc ¼ GDhðT1 ; T2 Þ
(53)
The turbine-compressor set mechanical power
Wm ¼ G½Wt htm Wc =hcm
(54)
The microturbine electric power
where hg is the efficiency of generator. The value of temperature T1 and the values of coefficients kin, kout, kcc, krl, and krh which depend on p, T, and G (see Eq. (8)) are determined by iterative calculation. In each iteration the set of the above described seven equations is solved. The pressure of air leaving the compressor is calculated from Eq. (11). The temperature T2s is evaluated from the equation of isentropic expansion 1 2s
Wel ¼ Wm hg he
DsðT1 ; T2s ; Pc Þ ¼ 0
The rate of heat obtained after cooling the gas exiting the recuperator to temperature Tatm
(42)
(55)
where he is the efficiency of electronics. The rate of heat supplied to the combustion chamber is computed from
QH ¼ GDhðT2r ; T3 Þ=hcc
(56)
The temperature of air leaving the compressor, T2, is determined from the expression
QL ¼ GDhðTatm ; T4r Þ þ Gf qv
hc ¼ DhðT1 ; T2s Þ=DhðT1 ; T2 Þ
where qv is the condensation heat of water vapor produced by combustion of 1 kg of fuel and Gf is the fuel consumption. In the case study, the value of qv is 5337 kJ/kg. The overall electrical efficiency of the power plant
(43)
where the compressor isentropic efficiency, hc, at part load is calculated from Eq. (14). The pressure of air leaving the recuperator
p2r ¼ ð1 krh Þp2
(44)
The temperature of air at the recuperator outlet, T2r, is determined from the defining equation for effectiveness 3
¼ DhðT2 ; T2r Þ=DhðT2 ; T4 Þ
h ¼ Wel =QH
(57)
(58)
The fuel consumption
Gf ¼ QH =LHW
(59)
(45) 4. Model validation
where the effectiveness of the recuperator for part load is given by Ref. [2] 3
¼ 3 0 = 3 0 þ ð1 3 0 ÞðG=G0 Þz
(46)
The value of the exponent z is taken as 0.7. The pressure of exhaust gas leaving the recuperator is computed from
p4r ¼ patm =ð1 kout Þ
(47)
The gas pressure at the outlet from the turbine
p4 ¼ p4r =ð1 krl Þ
(48)
The temperature of turbine exhaust gas at rated power, T40, is determined from the equation
ht0 ¼ DhðT30 ; T40 Þ=DhðT30 ; T4s0 Þ
(49)
Using the model elaborated in the paper a case study is carried out. The data for the Capstone C30 microturbine available in the open literature and some typical data for microturbines were used. Both these data are collected in Table 1. The computations of selected parameters were performed for the microturbine part load ranging from 2 to 30 kWe. Although the model allows performing calculations for various ambient conditions, the calculations were carried out for ISO ambient conditions (15 C, 101.325 kPa) because of the availability of manufacturer’s data for part load operation at ISO conditions [16]. The results of calculation and their comparison with the Capstone C30 manufacturer data and the results of measurements [17] are presented in Figs. 2e5. The electrical efficiencies presented in Ref. [17] are based on HHV while the points in Fig. 2 correspond to efficiencies based on LHV. These efficiencies were calculated using the value of HHV/LHV which for the gas fuel used in the calculation equals to 1.107. The differences between the
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Fig. 2. Electrical efficiency of the plant at part-load. Fig. 4. Exhaust gas energy rate at part-load (heat of condensation of water vapor included).
measured and calculated values results in part from the different outside temperature conditions (measurements were taken at outside temperatures between 1.1 C and 3.9 C) and the errors of measurements. According to Pierce [17] the standard deviation does not exceed 0.08 for the electrical efficiency and 0.6 C for the gas exhaust temperature, in case of 68% or more measurements. As seen in Figs. 2 and 3, the electric efficiencies and fuel energy rates obtained from the model are in very good agreement with the manufacturer’s data. The calculated exhaust energy rates (Fig. 4) are slightly larger than those reported by the manufacturer. Fig. 5 shows that the gas exhaust temperature curve, T4r, obtained from the model lies below the manufacturer’s data, especially for the range of low loads. We were unable to compensate for the differences in Fig. 5 by an appropriate choice of the adjustable coefficients: m, p, c4, t4, z. Because the model provides larger values of the gas flow rate for smaller loads, as a result are the proper values of the exhaust energy rate. The results obtained can be considered as surprisingly accurate for such a simple analytical model based on heuristic performance maps. 5. Exergy analysis of the power plant Exergy rates in the characteristic points of the plant thermodynamic cycle are determined from Eq. (6). Particular rates of exergy destruction or loss are calculated as follows. The rate of internal exergy destruction in the compressor - from the GouyeStodola law
Fig. 3. Fuel energy rate at part-load.
dBci ¼ GTatm ½DsðT1 ; T2 ; p2 =p1 Þ
(60)
- from the exergy balance equation
dBci ¼ B1 þ Wc B2
(61)
The rate of exergy destruction in the compressor due to mechanical losses
dBcm ¼ Wc ð1=hcm 1Þ
(62)
The total rate of exergy destruction in the compressor
dBc ¼ dBci þ dBcm
(63)
The rate of internal exergy destruction in the turbine - from the GouyeStodola law
dBti ¼ GTatm ½DsðT3 ; T4 ; p4 =p3 Þ
(64)
- from the exergy balance equation
dBti ¼ B3 Wt B4
(65)
The rate of exergy destruction in the turbine due to mechanical losses
dBtm ¼ Wt ð1 htm Þ
(66)
Fig. 5. Temperature of exhaust gas at the outlet from the recuperator at part-load.
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131
The total rate of exergy destruction in the turbine
dBt ¼ dBti þ dBtm
(67)
The rate of exergy destruction in the generator
dBg ¼ Wm 1 hg
(68)
The rate of exergy destruction due to electronic losses
dBe ¼ Wel ð1=he 1Þ
(69)
The rate of internal exergy destruction in the recuperator determined from the exergy balance equation
dBr ¼ B2 þ B4 B2r B4r
(70)
The rate of physical and chemical exergy of fuel
h i Bf ¼ Gf Tatm Rf ln pf =patm þ gLHV
(71)
The value of the coefficient g is taken as 1.04 according to Kotas [18, p. 269]. The fuel gas pressure, pf, is assumed equal to 0.489 MPa. The rate of internal exergy destruction and external exergy losses in the combustion chamber
dBcc ¼ Bf þ B2r B3
(72)
Fig. 6. Exergy destruction and loss rates in the power plant components at part-load. Exhaust gas exergy rate curve with number 2 is plotted for comparison purposes.
Exergetic efficiencies (conventional) of - compressor including mechanical losses
hBc ¼ B2 =ðB1 þ Wc =hc Þ
(73)
- turbine including mechanical losses
hBt ¼ ðWt hm þ B4 Þ=B3
(74)
depending on the places of their origin. The most significant sources of exergy destruction and loss are the combustion chamber and the recuperator. The least exergy is destructed at the inlet to the turbine and in the generator with electronics. In Figs. 6 and 7 the curves of exhaust gas exergy rate are depicted for the sake of comparison. The exergy of the exhaust gas leaving the turbine is not treated as a loss as it can be partly utilized in a low-temperature engine. The turbine, heat exchanger and generator with electronics have the largest exergetic efficiencies while the compressor and combustion chamber feature the lowest exergetic efficiency.
- recuperator
hBr ¼ ðB2r þ B4r Þ=ðB2 þ B4 Þ
(75)
- combustion chamber
hBcc ¼ B3 = Bf þ B2r
(76)
- generator and electronics
hBge ¼ Wel =Wm
(77)
- microturbine power plant
hB ¼ ðWel þ B4r Þ=Bf
(78)
The results of exergy calculations are presented in Figs. 6e8. In Fig. 6 the rates of exergy destruction and loss in the components of the power system are presented for part load operation. Relative exergy destruction and loss rates, referenced to the fuel exergy, are shown in Fig. 7. In Fig. 8 the conventional exergetic efficiencies of the components are displayed. As seen in Fig. 6 the exergy destruction and loss rates decrease with decreasing load. Only in the case of compressor a slight increase in exergy destruction is noticeable for the lowest loads. However, the relative exergy destruction and loss rates increase or decrease with decreasing load
Fig. 7. Relative exergy destruction and loss rates in the power plant components at part-load referenced to the fuel exergy. Exhaust gas relative exergy rate curve with number 2 is plotted for comparison purposes.
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Exergy destructions and losses in these two components are mutually associated with each other. An increase in the efficiency of the recuperator results in an increase in the heated air temperature. A higher air temperature at the inlet to the combustion chamber means less exergy destruction. On these elements should be given particular attention during the design of a microturbine with high efficiency in the entire load range. According to the GouyeStodola law, these exergy losses directly translate into a decrease in power efficiency of the microturbine. Each microturbine component is characterized by an intrinsic exergy destruction due to construction constrains. For example, the heat transfer area of the recuperator is limited which restricts the increase of its effectiveness. The exergy destruction due to gas flow with friction increases with velocity, but high gas velocities are related to the principle of operation of a turbine. These intrinsic exergy destructions cannot be easily eliminated.
Acknowledgements
Fig. 8. Exergetic efficiency of the power plant components at part-load.
Exergy destruction rate in the combustion chamber is mainly due to the chemical reaction. Less exergy is destructed there when the air is preheated in the recuperator before entering the combustion chamber or when the oxidant-fuel ratio is reduced. The increase in the recuperator effectiveness not only reduces the destruction of exergy in the recuperator, but also reduces the exergy destruction in the combustion chamber by increasing the air temperature at the inlet to the chamber. Exergy destruction in the turbine and compressor is mainly due to internal losses during expansion or compression respectively - the lower the internal efficiency, the greater the exergy destruction. The exergy destruction in the turbine and compressor is also increased by friction in the bearings. The main cause of exergy destruction in the recuperator is heat transfer at a finite temperature difference. 6. Conclusions Essential elements of a model for the microturbine part-load operation are the compressor and turbine performance maps. These maps are usually not disclosed by the microturbine manufacturers. This difficulty can be overcome by using universal formulas with adjustable coefficients for the part-load operation of a microturbine. In the analytical model of a microturbine presented in this paper such formulas have been successfully used for a 30 kWe commercial microturbine. A surprisingly good agreement between results from the model and manufacturer’s data has been obtained. The model presented in the paper allows the calculation of thermodynamic properties at the characteristic cycle points, heat and work during the processes of the cycle, and exergy destruction and loss in the plant components for various loads. The model also makes it possible to examine the effect of the microturbine cycle parameters (e.g.: the pressure ratio, maximum temperature, ambient parameters) on the energy and exergy efficiency of the microturbine. The exergy analysis carried out in this paper reveals components of the microturbine in which the largest exergy destructions and losses occur. These are the combustion chamber and recuperator.
This study was partly supported by the following project: the strategic program of scientific research and experimental development of the National (Polish) Centre for Research and Development: “Advances Technologies for Energy Generation”; Task 4. “Elaboration of Integrated Technologies for the Production of Fuels and Energy from Biomass as well as from Agricultural and other Waste Materials”.
References [1] N. Zhang, R. Cai, Analytical solutions and typical characteristics of part-load performances of single shaft gas turbine and its cogeneration, Energy Convers. Manage. 43 (2002) 1323e1337. [2] W. Wang, R. Cai, N. Zhang, General characteristics of single shaft microturbine set at variable speed operation and its optimization, Appl. Therm. Eng. 24 (2004) 1851e1863. [3] T.W. Song, J.L. Sohn, J.H. Kim, T.S. Kim, S.T. Ro, Exergy-based performance analysis of the heavy-duty gas turbine in part-load operating conditions, Exergy Int. J. 2 (2002) 105e112. [4] T.S. Kim, S.H. Hwang, Part load performance analysis of recuperated gas turbines considering engine configuration and operation strategy, Energy 31 (2006) 260e277. [5] B.T. Aklilu, S.I. Gilani, Mathematical modeling and simulation of a cogeneration plant, Appl. Therm. Eng. 30 (2010) 2545e2554. [6] M. Badami, M. Mura, Exergetic analysis of an innovative small scale combined cycle cogeneration system, Energy 35 (2010) 2535e2543. [7] H. Ghaebi, M. Amidpour, S. Karimkashi, O. Rezayan, Energy, exergy and thermoeconomic analysis of a combined cooling, heating and power (CCHP) system with gas turbine prime mover, Int. J. Energy Res. 35 (2011) 697e709. [8] A. Khaliq, I. Dincer, Energetic and exergetic performance analyses of a combined heat and power plant with absorption inlet cooling and evaporative aftercooling, Energy 36 (2011) 2662e2670. [9] D.P. Bakalis, A.G. Stamatis, Full and part load exergetic analysis of a hybrid micro gas turbine fuel cell system based on existing components, Energy Convers. Manage. 64 (2012) 213e221. [10] M.A. Meybodi, M. Behnia, A study on the optimum arrangement of prime movers in small scale microturbine-based CHP systems, Appl. Therm. Eng. 48 (2012) 122e135. [11] C. Wei, S. Zang, Experimental investigation on the off-design performance of a small-sized humid air turbine, Appl. Therm. Eng. 51 (2013) 166e176. [12] M.S. Rocha, R. Andreos, J.R. Simões-Moreira, Performance tests of two small trigeneration pilot plants, Appl. Therm. Eng. 41 (2012) 84e91. [13] W.Z. Black, J.G. Hartley, Thermodynamics, Harper Collins Collage Publishers, New York, 1996. [14] J. Kaikko, J. Backman, Technical and economic performance analysis for a microturbine in combined heat and power generation, Energy 32 (2007) 378e387. [15] Parametric Technology Corporation, Mathcad 15.0 M010, User’s Guide (2011). [16] Capstone Turbine Corporation, Capstone Model C30 Performance (410004 Rev. D) (2006). http://docs.capstoneturbine.com. [17] F.E. Pierce, Summary of Results from Testing a 30-kW-microturbine and Combined Heat and Power (CHP) System, U.S. Department of Energy, Federal Technology Alert, 2007. DOE/EE-0316. [18] T.J. Kotas, The Exergy Method of Thermal Plant Analysis, Krieger Publishing Company, Malabar, Florida, 1995.
Contents lists available at SciVerse ScienceDirect
Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng
Analytical model-based energy and exergy analysis of a gas microturbine at part-load operation Leszek Malinowski a, *, Monika Lewandowska b a b
Faculty of Maritime Technology and Transport, West Pomeranian University of Technology, 71-065 Szczecin, Al. Piastów 41, Poland Institute of Physics, West Pomeranian University of Technology, 70-311 Szczecin, Al. Piastów 48, Poland
h i g h l i g h t s Analytical model for part-load operation of a gas microturbine is elaborated. The model is based on heuristic part-load performance formulas. The model is validated by comparison with experimental and manufacturer’s data. Exergy destruction or loss for each microturbine component is calculated.
a r t i c l e i n f o
a b s t r a c t
Article history: Received 30 September 2012 Accepted 23 March 2013 Available online 10 April 2013
In this paper a universal analytical model for part-load operation of gas microturbines has been elaborated which is subsequently used in the energy and exergy analysis of a sample device. The model, based on the Brayton cycle and heuristic part-load performance formulas, takes into account: the temperature variation of working fluid specific heat at constant pressure in calculations of adiabatic processes, enthalpy, and exergy, the non-linear dependence of pressure drop on flow rate, and the cooling of generator by intake air. The model is validated using the manufacturer data for a commercially available microturbine of 30 kWe and results of measurements. The agreement is very good as for such a general simple analytical model. Exergy calculations based on the elaborated model show that the greatest potential for improving the efficiency of the microturbine lies in the combustion chamber and recuperator, as these components are characterized by the largest exergy destruction and loss. Ó 2013 Elsevier Ltd. All rights reserved.
Keywords: Gas microturbine Part load operation Analytical model Energy and exergy analysis
1. Introduction Small gas turbines whose mechanical power do not exceed several hundred kilowatts are classified as microturbines. In general gas microturbines operate according to the open Brayton cycle with or without heat regeneration. A typical microturbine is a single shaft design with the turbine, compressor and generator mounted on the same shaft. The pressure ratio is of the order 3.5 to 4.0 and the microturbine is controlled by fuel delivery alone. Usually, during load changes the rotational speed of the microturbine also changes while the temperature of the exhaust gas leaving the turbine is kept constant. Gas microturbines are increasingly used as a primary source of electricity and heat for individual objects, such as: hotels, sports
* Corresponding author. Tel.: þ48 91 449 4827. E-mail addresses: [email protected], [email protected] (L. Malinowski). 1359-4311/$ e see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.applthermaleng.2013.03.057
facilities, greenhouses, offices, small businesses, small houses and others. Such micro plants operate under varying demand for electricity and heat. However, running the microturbine at a reduced load results in loss of efficiency. In the literature, some attention has been paid to the problem of modeling of gas turbines operated at part load and to the analysis of their performance based on the first and second laws of thermodynamics. Zhang et al. [1] formulated an analytical model of a constant rotating speed single shaft gas turbine operated at part load and determined basic part-load characteristics of the turbine. Wang et al. [2] extended the model presented in Reference 1 for microturbines operated with variable rotational speed. They derived the optimal rotational speed for part-load operation and analyzed the effect of pressure and temperature ratios on the offdesign performance. Song et al. [3] presented an exergy-based performance analysis of a 150 MWe gas turbine based power plant for part-load operation. They investigated numerically the influence of the variable inlet guide vane and the blade cooling on
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Nomenclature B cp G h HHV k LHV n p Q R s T w W
exergy rate, W constant-pressure specific heat, J/(kg K) mass flow rate through the compressor and turbine, kg/s specific enthalpy, J/kg higher heating value, J/kg relative pressure loss lower heating value, J/kg rotating speed, 1/s pressure, Pa heat rate, W gas constant, J/(kg K) specific entropy, J/(kg K) temperature, K specific work, J/kg power, W
Greek symbols 3 effectiveness of the recuperator h efficiency P pressure ratio f total relative pressure loss
exergy destruction. Kim et al. [4] investigated the performance characteristics of recuperated gas turbines operating at partial load. They examined the influence of various design and operational factors on the part load efficiency. They considered simple (fuel only control), variable speed and variable inlet guide vane operation, constant and variable area nozzles (for two-shaft turbine). Their analysis was based on the manufacturer’s compressor and turbine performance maps. Aklilu et al. [5] developed a mathematical model to simulate a part-load operation of a single shaft gas turbine with variable geometry compressor. They created performance maps for the compressor and turbine based on manufacturer data and thermodynamic laws. In their model average values of the specific heat at constant pressure and isentropic exponent of the working gas were used. They tested their model with the data from a 4.2 MWe turbine. Badami et al. [6] performed an exergy analysis of a small 150 kWe combined cycle cogeneration plant with an internal combustion engine as a primary source of energy. They examined exergy and energy efficiency of the plant for part load operation. Ghaebi et al. [7] applied the first and second laws of thermodynamics as well as economic analysis to investigate a combined cooling, heating and power system with gas turbine of 19 MWe. They investigated the effect of selected parameters on the efficiency of heat, cold and power production for rated load. In their analysis, they used a constant heat capacity at constant pressure while calculating the enthalpy and exergy of the working gas. Khaliq et al. [8] studied, by the energy and exergy method, the influence of compressor intake air cooling on energy and exergy efficiency of a gas turbine cycle. They also examined exergy destruction rate in the power plant components as well as the effect of the pressure ratio and turbine inlet temperature on the energy and exergy efficiency of the cycle. They used an average value of the specific heat at constant pressure of the working gas and an average isentropic exponent in their model. Bakalis et al. [9] performed a part load exergetic analysis of a hybrid gas microturbine fuel cell system using the commercial AspenPlus software. They created the microturbine performance maps through the scaling of representative maps. Meybodi et al. [10] studied the optimum arrangement of prime movers in small scale
Subscripts atm atmospheric c compressor cc combustion chamber e electronics el electric f fuel g generator in inlet out outlet r recuperator rh recuperator, high pressure side rl recuperator, low pressure side s isentropic t turbine 0 design value Accents w
~ c ¼ Gc =G divided by design value, e.g. G c0 pffiffiffiffiffi reduced parameter, e.g. Gc ¼ G T1 =p1
Abbreviation TET turbine exit temperature
microturbine-based CHP systems. In their analysis, they used approximate equations for the relative efficiency, fuel consumption, exhaust mass flow rate, and exhaust temperature for partial-load conditions. These equations were derived based on the manufacturers’ data for five microturbines. Wei et al. [11] investigated the off-design performance of a small-sized humid air turbine. Their numerical calculation model was based on digitized performance maps. They validated the results from the model with own experimental data. Full and partial load performance tests of a small trigeneration pilot plant based on a microturbine were carried out by Rocha et al. [12]. Experimental results presented by the authors include the fuel consumption, electrical and thermal power, primary energy saving index, and energy utilization factor. This paper deals with the development of a universal analytical model of microturbines operated with partial load, at variable rotational speed, for use in energy and exergy analysis. The model is based on the heuristic part-load performance formulas proposed in References 1 and 2. Due to commercial reasons, the manufacturers usually do not publish performance maps of turbines and compressors in the open literature, hence the large usefulness of universal performance formulas. Thanks to the availability of such formulas, the microturbine’s suitability for the intended use can be thoroughly verified. To reduce calculation errors, we do not use an average isentropic coefficient in our model, but the temperature dependence of the specific heat at constant pressure of the working fluid is directly implemented into the equation of isentropic process. The condition Ds ¼ 0 brings us to the following isentropic equation c0 ln T þ c1 T Rln p ¼ const, where c0 and c1 are coefficients in the equation cp ðTÞ ¼ c0 þ c1 T describing the dependence of specific heat at constant pressure on temperature. The temperature dependence of specific heat at constant pressure is also taken into account during the calculation of enthalpy and exergy. When calculating the thermodynamic cycle parameters and exergy destruction and losses in the microturbine components, a non-linear dependence of pressure drop in the components on the microturbine load is accounted for. The model also allows to include the heat absorbed by intake air from the generator in the energy and exergy analysis.
L. Malinowski, M. Lewandowska / Applied Thermal Engineering 57 (2013) 125e132
127
where
2. Basic assumption and expressions A schematic flow diagram for the modeled microturbine is presented in Fig. 1. When formulating the mathematical model of microturbine the following assumptions are adopted:
P ¼ pj =pi
(4)
Specific entropy of air in the state i is determined by the formula microturbine is operated as stand alone, turbine is controlled by fuel flow rate at constant turbine exit temperature (TET), compressor and turbine are of radial type, turbine and compressor are mounted on a common shaft, compression in the compressor and expansion in the turbine are adiabatic irreversible processes (i.e. non-isentropic), pressure losses in the recuperator, in the combustion chamber, at the inlet to the compressor, and at the outlet from the turbine are taken into account as well as their non-linear dependence on the turbine load, the air sucked in by the compressor is used for cooling the generator, the working medium is considered to be an ideal gas, air; the mass of fuel is disregarded because of a very high value of the excess air ratio, the specific heat at constant pressure of the working medium depends linearly on temperature, exergy and entropy of the air have a value of zero for pressure patm ¼ 0.11013 MPa and temperature Tatm ¼ 288 K. The following correlation for the specific heat of air at constant pressure is applied which is based on the data taken from Ref. [13]
cp ðTÞ ¼ 0:931046 þ 2:066697$104 T kJ=ðkg KÞ
(2)
Ti
Ds Ti ; Tj ; P ¼
BðT; pÞ ¼ G½DhðTatm ; TÞ Tatm DsðTatm ; T; p=patm Þ
(6)
Pressure, pj, at the outlet of the element in which there is a loss of pressure, is calculated from the equation
pj ¼ ð1 kÞpi
(7)
where pi is the pressure at the inlet to that element. Relative pressure losses, k, for part loads are calculated from the expression [14]
k ¼ k0 ðG=G0 Þ2 ðpi0 =pi Þ2 ðTi =Ti0 Þ
(8)
where index 0 indicates the value for the rated load. The measure of the overall system pressure loss is the coefficient
f ¼ ð1 kin Þð1 kout Þð1 kcc Þð1 krl Þð1 krh Þ
(9)
Pt ¼ p3 =p4
(10)
is related to the compressor pressure ratio
Pc ¼ p2 =p1
(11)
by the following equation
Pt ¼ fPc
Increase in specific entropy between states i and j is determined from the formula
Exergy rate is computed from
The turbine pressure ratio
ZTj cp ðTÞdT
(5)
(1)
It is assumed that the fuel is natural gas with LHV ¼ 49653 kJ=kg and HHV ¼ 54990 kJ=kg. Increase in specific enthalpy of the working gas between states i and j is calculated from
Dh Ti ; Tj ¼
si ¼ DsðTatm ; Ti ; pi =patm Þ
ZTj
cp ðTÞ dT RlnðPÞ T
(3)
Ti
(12)
A case study is performed for the data presented in Table 1. The following values of k, for rated load, are used in the calculations: kin0 ¼ 0.01, kout0 ¼ 0.01, kcc0 ¼ 0.02, krl0 ¼ 0.02, krh0 ¼ 0.02. For the analysis, this model was implemented in the software package Mathcad 15 [15].
Table 1 Data used in calculations.
Fig. 1. Schematic diagram of the modeled microturbine. C e compressor, R e heat exchanger, CC e combustion chamber, T e turbine, G e electric generator, E e electronics.
Description
Quantity
Value
Unit
Turbine load
Microturbine electric power Microturbine inlet temperature Microturbine exit temperature (TET) Ambient temperature Ambient pressure Compressor pressure ratio Recuperator effectiveness Combustion chamber efficiency Compressor isentropic efficiency Turbine isentropic efficiency Compressor mechanical efficiency Turbine mechanical efficiency Generator efficiency Electronics efficiency
Wel0 T30 T4 Tatm patm
30 1114 872 288 0.11013 3.6 0.79 0.98 0.78 0.83 0.995 0.995 0.96 0.96
kW K K K MPa e e e e e e e e e
Rated Rated Any Any Any Rated Rated Any Rated Rated Any Any Any Any
Pc0 30
hcc hc0 ht0 hcm htm hg he
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3. Energy calculations for part load operation
Similarly for the turbine
3.1. Analytical model for part load operation
Gt0 ¼ G0
In our model we make use of the performance formulas for compressor and turbine at part load operation, and a modified Flügel formula proposed in Refs. [1,2]. Performance formulas for compressor are
Gt ¼ G
pffiffiffiffiffiffiffi T30 =p30
(25)
pffiffiffiffiffi T3 =p3
~ t ¼ Gt =G G t0
(26)
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ðG=G0 Þðp30 =p3 Þ T3 =T30
(27)
The reduced compressor rotational speed at rated load
Pc =Pc0
~2 þ c G ~ ¼ c1 G 2 c þ c3 c
i h i h ~c 2 n ~c ~ c =G ~ c =G ~ c Þ2 n ¼ 1 c4 ð1 n
hc =hc0
(13)
pffiffiffiffiffiffiffi nc0 ¼ n0 = T10
(14)
and at part load
pffiffiffiffiffi nc ¼ n= T1
where
~ c =c c1 ¼ n c2 ¼ c3 ¼
p
(15) ~ 2c 2mn
. c
(16)
~ 3c ~ c þ m2 n pmn
. c
~ c ðn ~ c mÞ2 ~c Þ þ n c ¼ pð1 m=n i h i h ~t 2 n ~t ~ t =G ~ t =G ~ t Þ2 n ¼ 1 t4 ð1 n
pffiffiffiffiffiffiffi T10 =p10
(30)
Similarly for the turbine
pffiffiffiffiffi nt ¼ n= T3
(32)
~ t ¼ nt =nt0 ¼ nra n (19)
(20)
and at part load
(22)
(23)
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T30 =T3
(33)
where the ratio of microturbine shaft rotational speeds for partial and full load is
nra ¼ n=n0
(34)
The real rotational speeds of the turbine and compressor are the same because they are mounted on a common shaft, but the relative speeds differ. The generator produces a high-frequency AC power which is rectified to constant-voltage DC power. The DC power is then inverted to 50 or 60 Hz constant frequency AC power. In the present analysis the only independent variable is the ratio of microturbine shaft rotational speeds, nra. Each value of nra corresponds to a single value of microturbine load. For assumed values of nra, the following system of seven equations is solved: - Eq. (13), - Eq. (20) in which G/G0 is substituted by the expression determined from Eq. (27) and Pt is substituted by the RHS of Eq. (12), - equation of isentropic expansion in the turbine
Ds½T3 ; T4s ; 1=ðfPc Þ ¼ 0
(35)
- equation for the turbine isentropic efficiency
ht ¼ DhðT3 ; T4 Þ=DhðT3 ; T4s Þ
(36)
where ht for part load is calculated from Eq. (19), - Eq. (33), - equation in which expressions for the real mass flow rate, G, determined from Eqs. (24) and (27) are equated to each other
~ c ðp =p Þ G 1 10
The relative compressor air mass flow rate
~ c ¼ Gc =G G c0
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T10 =T1 ¼ nra T10 =T1
(18)
(21)
pffiffiffiffiffi Gc ¼ G T1 =p1
~ c ¼ nc =nc0 ¼ ðn=n0 Þ n
(31)
According to Refs. [1,2], Eqs. (13)e(20) are valid for the following equilibrium conditions: nt ¼ nc, Gt ¼ Gc, pt ¼ fpc . The relative variables labeled by tildes, occurring in Eqs. (13)e(19), are equal to the quotient of the reduced variables for partial and nominal load respectively. The reduced and relative variables are defined as follows [1,2]. The reduced compressor air mass flow rate at rated load
Gc0 ¼ G0
The relative compressor rotational speed
pffiffiffiffiffiffiffi nt0 ¼ n0 = T30
The coefficients: m, p, c4, and t4 appearing in Eqs. (14)e(19) are dimensionless and their numerical values are taken arbitrarily. The values of these coefficients are adjusted by the heuristic trial and error method in order to obtain the appropriate shape of compressor and turbine performance maps. For the same reason pffiffiffi the coefficients p and m should satisfy the condition 3 p 2m=3. The correctness of choice of these coefficients can be verified by comparing the results achieved from the model with experimental data and data published by a microturbine manufacturer. We have used the following values of these coefficients: m ¼ 1.8, p ¼ 1.8, c4 ¼ 0.1, and t4 ¼ 0.1 for our case study. A modified Flügel formula is
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Pt 1 P2t0 1 G=G0 ¼ a T30 =T3
(29)
(17)
The efficiency characteristic of the turbine is
ht =ht0
(28)
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~ t ðp =p Þ T =T T10 =T1 ¼ G 3 30 30 3
(37)
- equation for pressure in point 3 (see Fig. 1)
After taking into account Eqs. (21) and (22) we get
~ c ¼ ðG=G Þðp =p Þ G 0 10 1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T1 =T10
(24)
p3 ¼ ð1 kcc Þð1 krh ÞPc p1
(38)
L. Malinowski, M. Lewandowska / Applied Thermal Engineering 57 (2013) 125e132
From above seven equations the values of the following vari~ c, G ~ t , Pc, p3, T3, T4s are determined. Then the remaining ~t , G ables: n temperatures and pressures in the characteristic points of cycle (see Fig. 1) are computed and the calculation of other quantities used in the analysis is carried out. 3.2. Further calculations
p1 ¼ patm ð1 kin Þ
(39)
The temperature of air, T1, after absorbing the heat from cooling the generator, Qatm, is initially assumed. After the gas flow rate, G, and Qatm have been determined, T1 is calculated from the equation
Qatm ¼ GDhðTatm ; T1 Þ
(40)
where Qatm is the rate of heat transferred from the generator to the compressor intake air. Qatm is computed from
Qatm ¼ Gwm 1 hg
It is assumed that the turbine is controlled in such a way that TET remains constant for all loads
T4 ¼ T40 ¼ const
(41)
(50)
The temperature of gas leaving the recuperator, T4r, is determined from the equation of energy balance in the recuperator
DhðT2 ; T2r Þ ¼ DhðT4r ; T4 Þ
The pressure of air at the inlet to the compressor is calculated by
129
(51)
Turbine expansion ratio is calculated from Eq. (10). The turbine isentropic efficiency for part load is computed from Eq. (19). The air mass flow rate, G, is determined from Eq. (24). The turbine internal power
Wt ¼ GDhðT4 ; T3 Þ
(52)
The compressor internal power
Wc ¼ GDhðT1 ; T2 Þ
(53)
The turbine-compressor set mechanical power
Wm ¼ G½Wt htm Wc =hcm
(54)
The microturbine electric power
where hg is the efficiency of generator. The value of temperature T1 and the values of coefficients kin, kout, kcc, krl, and krh which depend on p, T, and G (see Eq. (8)) are determined by iterative calculation. In each iteration the set of the above described seven equations is solved. The pressure of air leaving the compressor is calculated from Eq. (11). The temperature T2s is evaluated from the equation of isentropic expansion 1 2s
Wel ¼ Wm hg he
DsðT1 ; T2s ; Pc Þ ¼ 0
The rate of heat obtained after cooling the gas exiting the recuperator to temperature Tatm
(42)
(55)
where he is the efficiency of electronics. The rate of heat supplied to the combustion chamber is computed from
QH ¼ GDhðT2r ; T3 Þ=hcc
(56)
The temperature of air leaving the compressor, T2, is determined from the expression
QL ¼ GDhðTatm ; T4r Þ þ Gf qv
hc ¼ DhðT1 ; T2s Þ=DhðT1 ; T2 Þ
where qv is the condensation heat of water vapor produced by combustion of 1 kg of fuel and Gf is the fuel consumption. In the case study, the value of qv is 5337 kJ/kg. The overall electrical efficiency of the power plant
(43)
where the compressor isentropic efficiency, hc, at part load is calculated from Eq. (14). The pressure of air leaving the recuperator
p2r ¼ ð1 krh Þp2
(44)
The temperature of air at the recuperator outlet, T2r, is determined from the defining equation for effectiveness 3
¼ DhðT2 ; T2r Þ=DhðT2 ; T4 Þ
h ¼ Wel =QH
(57)
(58)
The fuel consumption
Gf ¼ QH =LHW
(59)
(45) 4. Model validation
where the effectiveness of the recuperator for part load is given by Ref. [2] 3
¼ 3 0 = 3 0 þ ð1 3 0 ÞðG=G0 Þz
(46)
The value of the exponent z is taken as 0.7. The pressure of exhaust gas leaving the recuperator is computed from
p4r ¼ patm =ð1 kout Þ
(47)
The gas pressure at the outlet from the turbine
p4 ¼ p4r =ð1 krl Þ
(48)
The temperature of turbine exhaust gas at rated power, T40, is determined from the equation
ht0 ¼ DhðT30 ; T40 Þ=DhðT30 ; T4s0 Þ
(49)
Using the model elaborated in the paper a case study is carried out. The data for the Capstone C30 microturbine available in the open literature and some typical data for microturbines were used. Both these data are collected in Table 1. The computations of selected parameters were performed for the microturbine part load ranging from 2 to 30 kWe. Although the model allows performing calculations for various ambient conditions, the calculations were carried out for ISO ambient conditions (15 C, 101.325 kPa) because of the availability of manufacturer’s data for part load operation at ISO conditions [16]. The results of calculation and their comparison with the Capstone C30 manufacturer data and the results of measurements [17] are presented in Figs. 2e5. The electrical efficiencies presented in Ref. [17] are based on HHV while the points in Fig. 2 correspond to efficiencies based on LHV. These efficiencies were calculated using the value of HHV/LHV which for the gas fuel used in the calculation equals to 1.107. The differences between the
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L. Malinowski, M. Lewandowska / Applied Thermal Engineering 57 (2013) 125e132
Fig. 2. Electrical efficiency of the plant at part-load. Fig. 4. Exhaust gas energy rate at part-load (heat of condensation of water vapor included).
measured and calculated values results in part from the different outside temperature conditions (measurements were taken at outside temperatures between 1.1 C and 3.9 C) and the errors of measurements. According to Pierce [17] the standard deviation does not exceed 0.08 for the electrical efficiency and 0.6 C for the gas exhaust temperature, in case of 68% or more measurements. As seen in Figs. 2 and 3, the electric efficiencies and fuel energy rates obtained from the model are in very good agreement with the manufacturer’s data. The calculated exhaust energy rates (Fig. 4) are slightly larger than those reported by the manufacturer. Fig. 5 shows that the gas exhaust temperature curve, T4r, obtained from the model lies below the manufacturer’s data, especially for the range of low loads. We were unable to compensate for the differences in Fig. 5 by an appropriate choice of the adjustable coefficients: m, p, c4, t4, z. Because the model provides larger values of the gas flow rate for smaller loads, as a result are the proper values of the exhaust energy rate. The results obtained can be considered as surprisingly accurate for such a simple analytical model based on heuristic performance maps. 5. Exergy analysis of the power plant Exergy rates in the characteristic points of the plant thermodynamic cycle are determined from Eq. (6). Particular rates of exergy destruction or loss are calculated as follows. The rate of internal exergy destruction in the compressor - from the GouyeStodola law
Fig. 3. Fuel energy rate at part-load.
dBci ¼ GTatm ½DsðT1 ; T2 ; p2 =p1 Þ
(60)
- from the exergy balance equation
dBci ¼ B1 þ Wc B2
(61)
The rate of exergy destruction in the compressor due to mechanical losses
dBcm ¼ Wc ð1=hcm 1Þ
(62)
The total rate of exergy destruction in the compressor
dBc ¼ dBci þ dBcm
(63)
The rate of internal exergy destruction in the turbine - from the GouyeStodola law
dBti ¼ GTatm ½DsðT3 ; T4 ; p4 =p3 Þ
(64)
- from the exergy balance equation
dBti ¼ B3 Wt B4
(65)
The rate of exergy destruction in the turbine due to mechanical losses
dBtm ¼ Wt ð1 htm Þ
(66)
Fig. 5. Temperature of exhaust gas at the outlet from the recuperator at part-load.
L. Malinowski, M. Lewandowska / Applied Thermal Engineering 57 (2013) 125e132
131
The total rate of exergy destruction in the turbine
dBt ¼ dBti þ dBtm
(67)
The rate of exergy destruction in the generator
dBg ¼ Wm 1 hg
(68)
The rate of exergy destruction due to electronic losses
dBe ¼ Wel ð1=he 1Þ
(69)
The rate of internal exergy destruction in the recuperator determined from the exergy balance equation
dBr ¼ B2 þ B4 B2r B4r
(70)
The rate of physical and chemical exergy of fuel
h i Bf ¼ Gf Tatm Rf ln pf =patm þ gLHV
(71)
The value of the coefficient g is taken as 1.04 according to Kotas [18, p. 269]. The fuel gas pressure, pf, is assumed equal to 0.489 MPa. The rate of internal exergy destruction and external exergy losses in the combustion chamber
dBcc ¼ Bf þ B2r B3
(72)
Fig. 6. Exergy destruction and loss rates in the power plant components at part-load. Exhaust gas exergy rate curve with number 2 is plotted for comparison purposes.
Exergetic efficiencies (conventional) of - compressor including mechanical losses
hBc ¼ B2 =ðB1 þ Wc =hc Þ
(73)
- turbine including mechanical losses
hBt ¼ ðWt hm þ B4 Þ=B3
(74)
depending on the places of their origin. The most significant sources of exergy destruction and loss are the combustion chamber and the recuperator. The least exergy is destructed at the inlet to the turbine and in the generator with electronics. In Figs. 6 and 7 the curves of exhaust gas exergy rate are depicted for the sake of comparison. The exergy of the exhaust gas leaving the turbine is not treated as a loss as it can be partly utilized in a low-temperature engine. The turbine, heat exchanger and generator with electronics have the largest exergetic efficiencies while the compressor and combustion chamber feature the lowest exergetic efficiency.
- recuperator
hBr ¼ ðB2r þ B4r Þ=ðB2 þ B4 Þ
(75)
- combustion chamber
hBcc ¼ B3 = Bf þ B2r
(76)
- generator and electronics
hBge ¼ Wel =Wm
(77)
- microturbine power plant
hB ¼ ðWel þ B4r Þ=Bf
(78)
The results of exergy calculations are presented in Figs. 6e8. In Fig. 6 the rates of exergy destruction and loss in the components of the power system are presented for part load operation. Relative exergy destruction and loss rates, referenced to the fuel exergy, are shown in Fig. 7. In Fig. 8 the conventional exergetic efficiencies of the components are displayed. As seen in Fig. 6 the exergy destruction and loss rates decrease with decreasing load. Only in the case of compressor a slight increase in exergy destruction is noticeable for the lowest loads. However, the relative exergy destruction and loss rates increase or decrease with decreasing load
Fig. 7. Relative exergy destruction and loss rates in the power plant components at part-load referenced to the fuel exergy. Exhaust gas relative exergy rate curve with number 2 is plotted for comparison purposes.
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Exergy destructions and losses in these two components are mutually associated with each other. An increase in the efficiency of the recuperator results in an increase in the heated air temperature. A higher air temperature at the inlet to the combustion chamber means less exergy destruction. On these elements should be given particular attention during the design of a microturbine with high efficiency in the entire load range. According to the GouyeStodola law, these exergy losses directly translate into a decrease in power efficiency of the microturbine. Each microturbine component is characterized by an intrinsic exergy destruction due to construction constrains. For example, the heat transfer area of the recuperator is limited which restricts the increase of its effectiveness. The exergy destruction due to gas flow with friction increases with velocity, but high gas velocities are related to the principle of operation of a turbine. These intrinsic exergy destructions cannot be easily eliminated.
Acknowledgements
Fig. 8. Exergetic efficiency of the power plant components at part-load.
Exergy destruction rate in the combustion chamber is mainly due to the chemical reaction. Less exergy is destructed there when the air is preheated in the recuperator before entering the combustion chamber or when the oxidant-fuel ratio is reduced. The increase in the recuperator effectiveness not only reduces the destruction of exergy in the recuperator, but also reduces the exergy destruction in the combustion chamber by increasing the air temperature at the inlet to the chamber. Exergy destruction in the turbine and compressor is mainly due to internal losses during expansion or compression respectively - the lower the internal efficiency, the greater the exergy destruction. The exergy destruction in the turbine and compressor is also increased by friction in the bearings. The main cause of exergy destruction in the recuperator is heat transfer at a finite temperature difference. 6. Conclusions Essential elements of a model for the microturbine part-load operation are the compressor and turbine performance maps. These maps are usually not disclosed by the microturbine manufacturers. This difficulty can be overcome by using universal formulas with adjustable coefficients for the part-load operation of a microturbine. In the analytical model of a microturbine presented in this paper such formulas have been successfully used for a 30 kWe commercial microturbine. A surprisingly good agreement between results from the model and manufacturer’s data has been obtained. The model presented in the paper allows the calculation of thermodynamic properties at the characteristic cycle points, heat and work during the processes of the cycle, and exergy destruction and loss in the plant components for various loads. The model also makes it possible to examine the effect of the microturbine cycle parameters (e.g.: the pressure ratio, maximum temperature, ambient parameters) on the energy and exergy efficiency of the microturbine. The exergy analysis carried out in this paper reveals components of the microturbine in which the largest exergy destructions and losses occur. These are the combustion chamber and recuperator.
This study was partly supported by the following project: the strategic program of scientific research and experimental development of the National (Polish) Centre for Research and Development: “Advances Technologies for Energy Generation”; Task 4. “Elaboration of Integrated Technologies for the Production of Fuels and Energy from Biomass as well as from Agricultural and other Waste Materials”.
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