**2. Energy analysis**

98 Efficiency, Performance and Robustness of Gas Turbines

Indirect cooling by mechanical refrigeration methods can reduce the air temperature to any desirable value, even below the 15oC, regardless of the ambient humidity. There are two common approaches for air chilling: a) use of refrigeration units via chilled water coils, b) use of exhaust heat-powered absorption machines. As reported by Elliot (2001), application of mechanical air-cooling increases the net power on the expense of the thermal efficiency, 6% power boosting for 10oC drop in the inlet air temperature. Using absorption machines was examined for inlet air-cooling of cogeneration plants, Ondrays *et al* (1991), while Kakaras *et al.* (2004) presented a simulation model for NH3 waste heat driven absorption machine air cooler. A drawback of the mechanical chilling is the risk of ice formation either as ice crystals in the air or as solidified layer on surfaces, such as the bell mouth or inlet

Several studies have compared the evaporative and mechanical cooling methods; with better performance for mechanical cooling. Mercer (2002) stated that evaporative cooling has increased the GT power by 10–15%, while the improvement for refrigeration chillers has reached 25%. Alhazmy and Najjar (2004) concluded that the power boosting varied between 1-7% for spray cooling but reached10-18% for indirect air cooling. In a recent study by Alhazmy *et al* (2006), they introduced two generic dimensionless terms (power gain ratio (*PGR*) and thermal efficiency change factor (*TEC*)) for assessment of intake air cooling systems. They presented the results in general dimensionless working charts covering a wide range of working conditions. Zadpoor and Golshan (2006) discussed the effect of using desiccant-based evaporative cooling on gas turbine power output. They have developed a computer program to simulate the GT cycle and the NOx emission and showed that the power output could be increased by 2.1%. In another development Erickson (2003 & 2005) suggested combination of the methods combining waste driven absorption cooling with water injection into the combustion air for power boosting; the concept was termed the "*power fogger cycle*". A novel approach has been presented by Zaki *et al* (2007), where a reverse Joule-Brayton air cycle was used to reduce the air temperature at the compressor inlet. Their coupled cycle showed a range of parameters, where both the power and thermal

As revealed in the above account abundant studies are concerned with the first law analysis of air intake cooling but those focuses on the second law analysis are limited. The basics of the second law analysis have been established and employed on variety of thermal systems by number of researchers. Bejan (1987, 1997), Bejan et al 1996, Rosen and Dincer (2003 a, and 2003 b) Ranasinghe, *et al* (1987), Zaragut *et.al* (1988), Kotas *et al* (1991) and Jassim (2003a, 2003b, 2004, 2005 and 2006), Khir *et al* (2007) have dealt extensively with various aspects of heat transfer processes. Chen *et al* (1997) analyzed the performance of a regenerative closed Brayton power cycle then extended the method to a Brayton refrigeration cycle, Chen *et al* (1999). The analysis considered all the irreversibilities associated with heat transfer processes. The exergy analysis of Brayton refrigeration cycle has been considered by Chen and Su (2005) to set a condition for the maximum exergetic efficiency while Tyagi *et al* (2006) presented parametric study where the internal and external irreversibilities were considered. The maximum ecological function, which was defined as the power output minus the power loss was determined for Brayton cycle by Huang *et al* (2000). Their exergy analysis was based on an ecological optimization criterion and was carried out for an

guide vanes, Stewart and Patrick, (2000).

efficiency can be simultaneously improved.

irreversible Brayton cycle with external heat source.

### **2.1 Brayton power cycle**

Details of the first law energy analysis have been presented in a previous study by Zaki *et al* (2007) but the basic equations are briefly given here. Figure 1 shows the components of a coupled gas turbine cycle with Brayton refrigeration cycle. The power cycle is represented by states 1-2-3-4 and the reverse Brayton refrigeration cycle is represented by states 1-6-7-8- 1. Portion of the compressed air α *m*<sup>1</sup> at pressure *P6* is extracted from the mainstream and cooled in a heat exchanger to *T7* then expands to the atmospheric pressure and *T8* as seen in Fig.1. The ambient intake air stream at *To* mixes with the cold stream at *T8* before entering the compressor.

Figure 2 shows the combined cycle on the T-s diagram; states *o*- 2 -3-4 represent the power cycle without cooling, while the power cycle with air inlet cooling is presented by states 1-2- 3-4-1. States 1-6-7-8 present the reverse Brayton refrigeration cycle.

In the mixing chamber ambient air at *mo , To* and ω*<sup>o</sup>* enters the chamber and mixes with the cold air stream having mass flow rate of α *m*<sup>1</sup> at *T8*. Air leaves the chamber at *T1*, which depends on the ambient air conditions and the extraction ratioα.

The mass and energy balance for the mixing chamber gives the compressor inlet temperature as;

$$T\_1 = \frac{\left(1 - a\right)c\_{po}\ T\_o + a\ c\_{p8}\ T\_s}{c\_{p1}}\tag{1}$$

Air leaves the compressor at *P xP* 6 1 = flows through the Brayton refrigerator and the rest at *P rP* 2 1 = as the working fluid for the power cycle, *x* is defined as the extraction pressure ratio and *r* is the pressure ratio.

The temperature of the air leaving the compressor at states 6 and 2 can be estimated assuming irreversible compression processes between states 1-2 and 1-6 and introducing isentropic compressor efficiency so that;

$$T\_{\delta} = T\_1 + \frac{T\_1}{\eta\_{\alpha}} \left( \mathbf{x}^{\frac{\mathbf{y} - 1}{\mathbf{y}}} - \mathbf{1} \right) \tag{2}$$

and

$$T\_2 = T\_1 + \frac{T\_1}{\eta\_{cr}} \left( r^{\frac{\eta - 1}{\eta}} - 1 \right) \tag{3}$$

Energy and Exergy Analysis

to calculate the wet air properties.

**2.2 Brayton refrigeration cycle analysis** 

The power output of the expander, *Wexp*

limit for *T7* is determined by the ambient temperature.

Fig. 1. A schematic of a gas turbine with an air cooler cycle

*ma* by ρ

η

*exp* yields,

The heat release *Qout*

of Reverse Brayton Refrigerator for Gas Turbine Power Boosting 101

. The Engineering Equation Solver (*EES*) software (Klein and Alvarado, 2004) has been used

It is noted that the extraction pressure ratio *x* is the main parameter that determines the cold air temperature *T8* achievable by the air refrigeration cycle. The hot compressed air at *P6* and *T6* rejects its heat through a heat exchanger to cooling water. Similar to standard compressor intercoolers, state 7 will have the same pressure as *P6* for ideal case while *T8* depends on the cooling process. Irreversible expansion process between 7 and 8 with expansion efficiency

> 8 7 7 exp <sup>1</sup> *TTT* <sup>1</sup>

> > is

between states 6 and 7 is computed by:

η

<sup>−</sup> =− −

17 6 ( ) *Q mh h out* = − α

Since many of the desalination plants in the Gulf area use dual purpose combined GT units

for brine heating or any industrial process that requires low grade heat. In general, the lower

for water and power production, it is suggested here to utilize the rejected heat ( *Qout*

1

*Wexp 1 7 8* = − *<sup>α</sup>m (h h )* (11)

(12)

(10)

, Fig. 1)

γ

γ

*x*

*a a V* . In the present analysis the moist air density *<sup>a</sup> ρ* is assumed function of *T1* and

ω1

where, *η<sup>c</sup>* is the compressor isentropic efficiency that can be estimated at any pressure *z*  from Korakianitis and Wilson, (1994)

$$
\eta\_{cz} = 1 - \left( 0.04 + \frac{z - 1}{150} \right), \text{ where } z \text{ is either x or } r \tag{4}
$$

The compression power between states 1 and 2 with extraction at state 6, separating the effects of the dry air and water vapor can be written as, Zaki *et al* (2007):

$$\dot{\mathcal{W}}\_c = \dot{m}\_a \left[ c\_{pa} \frac{T\_1}{\eta\_{cr}} \left( r^{\frac{r-1}{r}} - 1 \right) + \alpha\_1 \left( h\_{v2} - h\_{v1} \right) + \left( \frac{a}{1 - a} \right) \left( c\_{pa} \frac{T\_1}{\eta\_{cr}} \left( x^{\frac{r-1}{r}} - 1 \right) + \alpha\_1 \left( h\_{v6} - h\_{v1} \right) \right) \right] \tag{5}$$

where *vn h* is the enthalpy of the saturated water vapor at the indicated state *n*. Equation 5 is a general expression for the compressor work for wet air; for dry air conditions <sup>1</sup> ω = 0 and for stand alone GT α= 0 .

Heat balance about the combustion chamber gives the heat rate supplied by fuel combustion (net calorific value *NCV*) as:

$$\dot{Q}\_{cc} = \dot{m}\_f \text{ NCV} = \left(\dot{m}\_a + \dot{m}\_f\right)c\_{pg}T\_3 - \dot{m}\_a c\_{pa}T\_2 + \dot{m}\_v \left(h\_{v3} - h\_{v2}\right) \tag{6}$$

where *hv2* and *hv3* are the enthalpies of water vapor at the combustion chamber inlet and exit states respectively and *f* is the fuel to air ratio *f a f mm* = (related to the dry air flow rate). The total gases mass flow rate at the turbine inlet *mt* = ) *m (1 a 1* + + *ω f* .

Substituting for *T2* from equation 3 gives the cycle heat input as:

$$\dot{Q}\_{\rm cc} = \dot{m}\_a T\_1 \left[ (1+f) c\_{pg} \frac{T\_3}{T\_1} - c\_{pa} \left( \frac{r^{\frac{\gamma - 1}{\gamma}} - 1}{\eta\_{\rm cr}} + 1 \right) + \frac{\alpha\_1}{T\_1} (h\_{v3} - h\_{v2}) \right] \tag{7}$$

The power produced by the turbine due to expansion of gases between states 3 and 4, Fig.1, is

$$\dot{\mathcal{W}}\_t = \dot{m}\_a \left( 1 + \alpha\_1 + f \right) c\_{pg} \eta\_t \ T\_3 \left( 1 - \frac{1}{r^{\frac{\gamma - 1}{r}}} \right) \tag{8}$$

The turbine isentropic efficiency may be estimated using the practical relation recommended by Korakianitis and Wilson (1994) as:

$$
\eta\_t = 1 - \left(0.03 + \frac{r - 1}{180}\right) \tag{9}
$$

Since, the gas turbine is almost constant volume machine at a specific rotating speed, the inlet air volumetric flow rate,*Va* is fixed regardless of the intake air conditions. Equation 7 can be written in terms of the volumetric flow rate at the compressor inlet state by replacing

*ma* by ρ*a a V* . In the present analysis the moist air density *<sup>a</sup> ρ* is assumed function of *T1* andω1 . The Engineering Equation Solver (*EES*) software (Klein and Alvarado, 2004) has been used to calculate the wet air properties.

### **2.2 Brayton refrigeration cycle analysis**

100 Efficiency, Performance and Robustness of Gas Turbines

where, *η<sup>c</sup>* is the compressor isentropic efficiency that can be estimated at any pressure *z* 

The compression power between states 1 and 2 with extraction at state 6, separating the

− − <sup>=</sup> −+ − + −+ − (5)

where *vn h* is the enthalpy of the saturated water vapor at the indicated state *n*. Equation 5 is

Heat balance about the combustion chamber gives the heat rate supplied by fuel combustion

where *hv2* and *hv3* are the enthalpies of water vapor at the combustion chamber inlet and exit states respectively and *f* is the fuel to air ratio *f a f mm* = (related to the dry air flow rate).

> ( ) ( ) <sup>1</sup> *1 3 1*

γ

γ

*1 cr 1*

*r 1*

*180*

(7)

*γ*

is fixed regardless of the intake air conditions. Equation 7

(9)

*r* −

 (8)

<sup>−</sup> <sup>−</sup>

*cc a pg pa v3 v2*

The power produced by the turbine due to expansion of gases between states 3 and 4, Fig.1, is

( ) *t a 1 pg t 3 <sup>γ</sup> <sup>1</sup>*

The turbine isentropic efficiency may be estimated using the practical relation

<sup>−</sup> =− +

Since, the gas turbine is almost constant volume machine at a specific rotating speed, the

can be written in terms of the volumetric flow rate at the compressor inlet state by replacing

*t*

*η 1 0.03*

= ++ −

*<sup>1</sup> W m1 <sup>ω</sup> f c <sup>η</sup> T 1*

= + − ++ −

*<sup>T</sup> r 1 <sup>ω</sup> Q mT 1 f c c 1 h h <sup>T</sup> <sup>η</sup> <sup>T</sup>*

*γ 1 γ 1 1 1 γ γ c a pa 1 v2 v1 pa 1 v6 v1*

*T T <sup>α</sup> W mc r 1 <sup>ω</sup> hh c x 1 <sup>ω</sup> h h*

, where *z* is either *x* or *r* (4)

( ) ( )

( ) ( ) *Q m NCV m m c T m c T m h h cc f* = =+ − + − *a f pg a pa v v v* 3 2 32 (6)

ω

= 0 and

*z 1*

*150*

*cr cx*

a general expression for the compressor work for wet air; for dry air conditions <sup>1</sup>

*η 1- α η*

from Korakianitis and Wilson, (1994)

α= 0 .

for stand alone GT

(net calorific value *NCV*) as:

*cz*

*η 1 0.04*

<sup>−</sup> =− +

effects of the dry air and water vapor can be written as, Zaki *et al* (2007):

The total gases mass flow rate at the turbine inlet *mt* = ) *m (1 a 1* + + *ω f* .

Substituting for *T2* from equation 3 gives the cycle heat input as:

recommended by Korakianitis and Wilson (1994) as:

inlet air volumetric flow rate,*Va*

It is noted that the extraction pressure ratio *x* is the main parameter that determines the cold air temperature *T8* achievable by the air refrigeration cycle. The hot compressed air at *P6* and *T6* rejects its heat through a heat exchanger to cooling water. Similar to standard compressor intercoolers, state 7 will have the same pressure as *P6* for ideal case while *T8* depends on the cooling process. Irreversible expansion process between 7 and 8 with expansion efficiency η*exp* yields,

$$T\_8 = T\_7 - T\_7 \eta\_{\exp} \left[ 1 - \left( \frac{1}{\chi} \right)^{\frac{r-1}{\chi}} \right] \tag{10}$$

The power output of the expander, *Wexp* is

$$
\dot{W}\_{\rm exp} = a \dot{m}\_1 (h\_\gamma - h\_\text{s}) \tag{11}
$$

The heat release *Qout* between states 6 and 7 is computed by:

$$
\dot{Q}\_{out} = \alpha \dot{m}\_1 (h\_7 - h\_6) \tag{12}
$$

Since many of the desalination plants in the Gulf area use dual purpose combined GT units for water and power production, it is suggested here to utilize the rejected heat ( *Qout* , Fig. 1) for brine heating or any industrial process that requires low grade heat. In general, the lower limit for *T7* is determined by the ambient temperature.

Fig. 1. A schematic of a gas turbine with an air cooler cycle

Energy and Exergy Analysis

**3.1 Mixing chamber** 

2004 as,

of Reverse Brayton Refrigerator for Gas Turbine Power Boosting 103

The rate of exergy destruction in the mixing chamber includes different forms of exergy; physical due to changes in the properties during mixing and that due to chemical reactions if any. The *physical exergy* of a mixture of *N* components can be evaluated from Jassim, *et al*

> *<sup>N</sup> <sup>Δ</sup><sup>T</sup> ph j j o*

> > *<sup>p</sup> <sup>p</sup> c and c* ) as;

*j 1 o o <sup>j</sup>*

*o j oj o j j j j*

*<sup>o</sup> ε* is the molar chemical exergy (*kJ/kmol*) evaluated from (Kotas, 1995, Table D3).

The irreversibility involved in mixing hot and cold air streams as shown in Fig. 1 assuming

The values of *ph ε* and *<sup>o</sup> ε* can be calculated from Eqs. (15b) and (16), respectively. Hence,

*mc 1 o,8 ph,8* = + ++ −+ *αm* ( ) *ε ε m m* 8 1 *o o,o ph,o o 1 o,1 ph,1* ( ) *ε ε m m* ( ) *ε ε m* (17d)

equations 17b and 17c give the total rate of exergy losses in the mixing chamber as,

<sup>=</sup> *T P*

*ε y ε RT n*

where *P* is the total pressure of the mixture. The equation may be written in another form,

The mean physical exergies for the mixing streams are calculated using equation 15b. The

*j 1 o*

<sup>=</sup> *P*

*P*

*T P*

*oM* due to changes in number of moles is given by the

*mc 8 o 1 I EEE* =+− (17a)

*mc 1 8 o o 1 1 I* = +− *αm ε m ε m ε* (17b)

*o ph ε εε* = + (17c)

*<sup>ε</sup> <sup>M</sup>* = + *<sup>y</sup> <sup>ε</sup> RT <sup>y</sup> n x (kJ/kmol)* (16)

*p, j c* and entropy ( ) *<sup>s</sup>*

*p, j c* have

= −− + (kJ/kmol) (15b)

 = + (15a)

( ) *<sup>M</sup>*

*<sup>N</sup> h s ph j p o op o*

values of mean molar isobaric exergy capacities for enthalpy ( ) *<sup>h</sup>*

where *yj* is the mole fraction of component *j* in a mixture

= 0) is,

In general, the specific exergy component for each of the terms in Eq. 17b is,

ε

*ε y c T T T c n RT n*

using tabulated values of properties ( *h s*

been evaluated from (Kotas, 1995, Table D3).

The chemical exergy of a mixture

following expression (Kotas, 1995)

adiabatic mixing process (*Qo*

*I* 

or

( ) ( ) *<sup>M</sup>*

The cold stream flow rate *<sup>1</sup> αm* proceeds to the mixing chamber to cool down the air at the compressor intake. The mixture temperature *T1* depends on the mass flow rate and the temperature of each stream as seen in Eq. 6.

Fig. 2. *T-s* diagram for the proposed cycle

### **3. Exergy analysis**

The exergy of a system is the maximum work obtainable as the system comes to equilibrium with the surrounding. The first law analysis (section 2) did not give information on heat availability at different temperatures. It is the second law which asserts that from engineering perspective. Therefore, exergy method is a technique based on both the first and second law of thermodynamics providing the locations where the exergy destruction and irreversibility are most. Ways and means can then be explored to reduce these exergy destructions to the practical minimum values.

The exergy destruction is a measure of thermodynamic imperfection of a process and is expressed in terms of lost work potential. In general, the expression for exergy destruction as articulated by Kotas (1995) is;

$$
\dot{I} = T\_o \left[ \left( \dot{S}\_{out} - \dot{S}\_{in} \right) - \sum\_{i=1}^{n} \frac{\dot{Q}\_i}{T\_i} \right] = T\_o \dot{I} \dot{I} \ge 0 \tag{13}
$$

The exergy balance for an open system is,

$$
\dot{E}\_{in} + \dot{E}^{Q} = \dot{E}\_{out} + \dot{V}\dot{V} + \dot{I} \tag{14}
$$

where *<sup>Q</sup> E* is the exergy associated with a given heat transfer rate, is termed as thermal exergy flow (kW). Quantities of lost work or rate of exergy destruction due to irreversibility can be estimated for each component of the coupled cycles as follows.

### **3.1 Mixing chamber**

102 Efficiency, Performance and Robustness of Gas Turbines

The cold stream flow rate *<sup>1</sup> αm* proceeds to the mixing chamber to cool down the air at the compressor intake. The mixture temperature *T1* depends on the mass flow rate and the

The exergy of a system is the maximum work obtainable as the system comes to equilibrium with the surrounding. The first law analysis (section 2) did not give information on heat availability at different temperatures. It is the second law which asserts that from engineering perspective. Therefore, exergy method is a technique based on both the first and second law of thermodynamics providing the locations where the exergy destruction and irreversibility are most. Ways and means can then be explored to reduce these exergy

The exergy destruction is a measure of thermodynamic imperfection of a process and is expressed in terms of lost work potential. In general, the expression for exergy destruction

*i*

(13)

*<sup>Q</sup> E E E WI in* += ++ *out* (14)

*i 1 i*

<sup>=</sup> *T*

( ) *<sup>n</sup>*

can be estimated for each component of the coupled cycles as follows.

*o out in o*

 = −− = ≥

*<sup>Q</sup> IT S S T <sup>Π</sup> <sup>0</sup>*

where *<sup>Q</sup> E* is the exergy associated with a given heat transfer rate, is termed as thermal exergy flow (kW). Quantities of lost work or rate of exergy destruction due to irreversibility

temperature of each stream as seen in Eq. 6.

Fig. 2. *T-s* diagram for the proposed cycle

destructions to the practical minimum values.

The exergy balance for an open system is,

as articulated by Kotas (1995) is;

**3. Exergy analysis** 

The rate of exergy destruction in the mixing chamber includes different forms of exergy; physical due to changes in the properties during mixing and that due to chemical reactions if any. The *physical exergy* of a mixture of *N* components can be evaluated from Jassim, *et al* 2004 as,

$$\left(\tilde{\boldsymbol{\varepsilon}}\_{ph}\right)\_{\boldsymbol{M}} = \sum\_{j=1}^{N} \boldsymbol{y}\_{j} \,\,\tilde{\boldsymbol{\varepsilon}}\_{j}^{\boldsymbol{\alpha}\boldsymbol{T}} + \tilde{\boldsymbol{R}} \,\,\boldsymbol{T}\_{o} \,\,\ell \ln\left(\frac{\boldsymbol{P}}{\boldsymbol{P}\_{o}}\right) \tag{15a}$$

where *P* is the total pressure of the mixture. The equation may be written in another form, using tabulated values of properties ( *h s <sup>p</sup> <sup>p</sup> c and c* ) as;

$$\left(\tilde{\varepsilon}\_{ph}\right)\_{\text{in}} = \sum\_{j=1}^{N} y\_j \left[ \tilde{\varepsilon}\_p^h \left(T - T\_o\right) - T\_o \tilde{\varepsilon}\_p^s \,\ell n \left(\frac{T}{T\_o}\right) \right]\_j + \tilde{R} \, T\_o \, \ell n \left(\frac{P}{P\_o}\right) \tag{15b} \tag{15b}$$

The mean physical exergies for the mixing streams are calculated using equation 15b. The values of mean molar isobaric exergy capacities for enthalpy ( ) *<sup>h</sup> p, j c* and entropy ( ) *<sup>s</sup> p, j c* have been evaluated from (Kotas, 1995, Table D3).

The chemical exergy of a mixture ε *oM* due to changes in number of moles is given by the following expression (Kotas, 1995)

$$\tilde{\boldsymbol{\varepsilon}}\_{oM} = \sum\_{j} \boldsymbol{y}\_{j} \,\tilde{\boldsymbol{\varepsilon}}\_{o\boldsymbol{j}} + \tilde{\boldsymbol{\mathcal{R}}} \, T\_{o} \sum\_{j} \boldsymbol{y}\_{j} \,\ell \, \mathbf{x}\_{j} \quad \text{(kJ/kmol)}\tag{16}$$

where *yj* is the mole fraction of component *j* in a mixture

*<sup>o</sup> ε* is the molar chemical exergy (*kJ/kmol*) evaluated from (Kotas, 1995, Table D3).

The irreversibility involved in mixing hot and cold air streams as shown in Fig. 1 assuming adiabatic mixing process (*Qo* = 0) is,

$$
\dot{I}\_{mc} = \dot{E}\_8 + \dot{E}\_o - \dot{E}\_1 \tag{17a}
$$

or

$$\dot{I}\_{mc} = a\dot{m}\_1\varepsilon\_8 + \dot{m}\_o\ \varepsilon\_o - \dot{m}\_1\varepsilon\_1 \tag{17b}$$

In general, the specific exergy component for each of the terms in Eq. 17b is,

$$
\varepsilon = \varepsilon\_o + \varepsilon\_{ph} \tag{17c}
$$

The values of *ph ε* and *<sup>o</sup> ε* can be calculated from Eqs. (15b) and (16), respectively. Hence, equations 17b and 17c give the total rate of exergy losses in the mixing chamber as,

$$\dot{I}\_{nc} = a \dot{m}\_1 \left(\varepsilon\_{o,8} + \varepsilon\_{ph,8}\right) \Big/ \tilde{m}\_8 + \dot{m}\_o \left(\varepsilon\_{a\rho} + \varepsilon\_{ph,o}\right) \Big/ \tilde{m}\_o - \dot{m}\_1 \left(\varepsilon\_{o,1} + \varepsilon\_{ph,1}\right) \Big/ \tilde{m}\_1 \tag{17d}$$

Energy and Exergy Analysis

states 2 and 3 due to chemical combustion reaction is;

Introduce the entropy difference at the reference state Δ*So* as

Substitute for the entropy change for reactants and products gives;

( )

( )

The total energy input *Qeff cc* , to the combustion chamber is then

The rate of exergy destruction in the gas turbine is,

and *fuel*, respectively (Nag, 2002). Then

The exergy destruction rate becomes

where ( ) <sup>1</sup> *pg g c*

γ

<sup>−</sup> <sup>=</sup>

γ

*R*

**3.4 Gas turbine** 

rate as;

of Reverse Brayton Refrigerator for Gas Turbine Power Boosting 105

The exergy destruction rate during the combustion process is the entropy change between

where ( ) () ( )<sup>2</sup> *Raf 2 2 S SS* = + and the subscripts *P, R, a,* and *f* represent *products*, *reactants*, *air*

() () *cc o P R* 3 2 *I TS S* = − (24a)

<sup>3</sup> ( ) ( ) <sup>2</sup> <sup>2</sup> ( ) *cc o p a f I TS S S* = −+ (24b)

( ) ( ) ( ) *oP f a o o <sup>o</sup> <sup>Δ</sup>SS S S* =− + (24c)

*Q QI eff cc cc cc* , = + (25)

3 3

(26)

*T P*  (24e)

*ITS S S S S cc o P P* = − − − +Δ { ( ) ( ) () () 3 2 *o o a a <sup>o</sup>*} (24d)

3 3

*o o*

*T P*

2 2

*o o*

*T P*

*1 pg g*

+ + − −

*T P 1 f <sup>ω</sup> c n Rn*

*cc a o o o*

( )( ) *t a 1 o* 4 3 *I m1 f* = ++ − *ω Ts s*

= ++ <sup>−</sup>

The effective power output of the turbine can be expressed in terms of exergy destruction

( ) 4 4

*a 1 o pg g T P m1 f <sup>ω</sup> Tc n R n*

= + Δ + − 

*I mT T S*

*1 pa a*

*T P <sup>1</sup> <sup>ω</sup> c n Rn*

$$\dot{I}\_{mc} = \dot{m}\_a \left( 1 + \alpha\_1 \right) \left[ \left( \frac{a}{1 - a} \right) \frac{\left( \varepsilon\_{o,8} + \varepsilon\_{ph,8} \right)}{\tilde{m}\_s} + \frac{\left( \varepsilon\_{o,o} + \varepsilon\_{ph,o} \right)}{\tilde{m}\_o} - \left( \frac{1}{1 - a} \right) \frac{\left( \varepsilon\_{o,1} + \varepsilon\_{ph,1} \right)}{\tilde{m}\_1} \right]$$

#### **3.2 Compressor**

The rate of exergy destruction in the compressor (see Figs 1 and 2) for the compression process 1-2, is,

$$\dot{I}\_c = \dot{m}\_o \, T\_o \left( s\_2 - s\_1 \right) \, +a \left( \frac{\dot{m}\_o}{1 - a} \right) T\_o \left( s\_6 - s\_1 \right) \tag{18}$$

Substituting the entropy change for gases, Cengel and Bolos, 2005 gives

$$\dot{W}\_c = \dot{m}\_a \left( \mathbf{1} + \alpha\_1 \right) T\_o \left| \mathbf{c}\_{pa} \,\ell n \left( \frac{T\_2}{T\_1} \right) - \mathbf{R}\_a \,\ell n \left( \frac{P\_2}{P\_1} \right) + \left( \frac{a}{\mathbf{1} \cdot \mathbf{a}} \right) \left( \mathbf{c}\_{pa} \,\ell n \left( \frac{T\_6}{T\_1} \right) - \mathbf{R}\_a \,\ell n \left( \frac{P\_6}{P\_1} \right) \right) \right| \tag{19}$$

The mixing chamber irreversibility is added to the compressor irreversibility as it is caused by air mixing before entering the compressor. Therefore the compressor effective power is,

$$
\dot{\mathcal{W}}\_{\rm eff,c} = \dot{\mathcal{W}}\_c + \dot{I}\_c + \dot{I}\_{\rm mc} \tag{20}
$$

### **3.3 Combustion chamber**

The combustion process irreversibility is often accomplished by heat transfer, fluid friction, mixing and chemical reaction. In principle it is difficult to evaluate an absolute value for the contribution of each process to the total irreversibility. However, the process of combustion can be examined by assuming that it takes place under adiabatic conditions and neglect irreversibilities due to friction and mixing. The important term of fuel combustion is the chemical exergy *<sup>o</sup> ε* which has been extensively dealt with by Kotas, 1995. The exergy due to chemical reaction of a combustible component (*k*) is function of other parameters as;

$$\mathcal{E}\_o = f(\Delta h, \Delta \mathbf{s}\_\prime \ T\_\bullet, X\_k) \tag{21}$$

The subscript *k* refers to the component's mass (mole) fraction of products composition.

The chemical exergy *<sup>o</sup> ε* may be expressed as percentage of the net calorific value (*NCV*) as*,*

$$
\varepsilon\_o = \mathfrak{q} \,\,\mathrm{NCV} \tag{22}
$$

The value of ϕ is function of the fuel components mass to Carbon ratio, (*O/C*) for Oxygen, (*H/C*) for hydrogen and (*N/C)* for Nitrogen. For liquid fuels additional ratio for the sulfur content is included (*S/C*), andϕ is in the order of 1.04 to 1.08. Typical values of ϕ for some industrial fuels are given in Kotas 1995. Therefore, the rate of exergy loss *T S o o* Δ in a combustion reaction is;

$$T\_{\
u} \Delta S\_{\
u} = \dot{m}\_f \times \text{NCV} \left(\varphi - 1\right) = \dot{m}\_a \times f \times \text{NCV} \left(\varphi - 1\right) \tag{23}$$

The exergy destruction rate during the combustion process is the entropy change between states 2 and 3 due to chemical combustion reaction is;

$$\dot{I}\_{cc} = T\_o \left[ \left( \mathbf{S}\_{\mathbb{P}} \right)\_3 - \left( \mathbf{S}\_{\mathbb{R}} \right)\_2 \right] \tag{24a}$$

where ( ) () ( )<sup>2</sup> *Raf 2 2 S SS* = + and the subscripts *P, R, a,* and *f* represent *products*, *reactants*, *air* and *fuel*, respectively (Nag, 2002). Then

$$\dot{I}\_{cc} = T\_o \left[ \left( \mathbf{S}\_p \right)\_3 - \left[ \left( \mathbf{S}\_a \right)\_2 + \left( \mathbf{S}\_{f} \right)\_2 \right] \right] \tag{24b}$$

Introduce the entropy difference at the reference state Δ*So* as

$$
\Delta S\_o = \left( S\_P \right)\_o - \left[ \left( S\_f \right)\_o + \left( S\_a \right)\_o \right] \tag{24c}
$$

The exergy destruction rate becomes

104 Efficiency, Performance and Robustness of Gas Turbines

*εε εε εε <sup>α</sup> <sup>1</sup> I m1 <sup>ω</sup>*

The rate of exergy destruction in the compressor (see Figs 1 and 2) for the compression

( ) *c oo* 2 1 *I mT s s* = − ( ) 6 1

( ) 2 2 6 6

 = + − + <sup>−</sup> (19)

The mixing chamber irreversibility is added to the compressor irreversibility as it is caused by air mixing before entering the compressor. Therefore the compressor effective power is,

The combustion process irreversibility is often accomplished by heat transfer, fluid friction, mixing and chemical reaction. In principle it is difficult to evaluate an absolute value for the contribution of each process to the total irreversibility. However, the process of combustion can be examined by assuming that it takes place under adiabatic conditions and neglect irreversibilities due to friction and mixing. The important term of fuel combustion is the chemical exergy *<sup>o</sup> ε* which has been extensively dealt with by Kotas, 1995. The exergy due to

(,,,) *<sup>o</sup> o k*

The subscript *k* refers to the component's mass (mole) fraction of products composition.

*<sup>o</sup> ε* = ϕ

The chemical exergy *<sup>o</sup> ε* may be expressed as percentage of the net calorific value (*NCV*) as*,*

(*H/C*) for hydrogen and (*N/C)* for Nitrogen. For liquid fuels additional ratio for the sulfur

industrial fuels are given in Kotas 1995. Therefore, the rate of exergy loss *T S o o* Δ in a

is function of the fuel components mass to Carbon ratio, (*O/C*) for Oxygen,

is in the order of 1.04 to 1.08. Typical values of

*T S o o* Δ = *m NCV f a* × − = ×× − () () *φ 1 m f NCV φ 1* (23)

chemical reaction of a combustible component (*k*) is function of other parameters as;

ε

ϕ

*c a 1 o pa a pa a T P <sup>α</sup> T P I m1 <sup>ω</sup> Tc n Rn c n Rn*

Substituting the entropy change for gases, Cengel and Bolos, 2005 gives

*mc a 1*

**3.2 Compressor** 

process 1-2, is,

**3.3 Combustion chamber** 

The value of

ϕ

content is included (*S/C*), and

combustion reaction is;

( ) ( ) *o,8 ph,8 o,o ph,o* ( ) ( ) *o,1 ph,1*

*1- α m m 1- α m* ++ + = + + − 

> *o o*

1 1 1 1

*W WII eff c c c mc* , = ++ (20)

=ΔΔ *f h sT X* (21)

*NCV* (22)

ϕ

for some

*T P 1-α T P*

*m α Ts s 1 α* + − −

*8o 1*

(18)

$$\dot{I}\_{cc} = T\_o \left\{ \left[ \left( S\_p \right)\_3 - \left( S\_p \right)\_o \right] - \left[ \left( S\_a \right)\_2 - \left( S\_a \right)\_o \right] + \Delta S\_o \right\} \tag{24d}$$

Substitute for the entropy change for reactants and products gives;

$$\dot{I}\_{cc} = \dot{m}\_a T\_o \left\{ \begin{aligned} \left( \mathbf{1} + f + \alpha\_1 \right) \left[ c\_{pg} \,\ell \ln \left( \frac{T\_3}{T\_o} \right) - R\_g \,\ell \ln \left( \frac{P\_3}{P\_o} \right) \right] - \\ \left( \mathbf{1} + \alpha\_1 \right) \left[ c\_{pu} \,\ell \ln \left( \frac{T\_2}{T\_o} \right) - R\_a \,\ell \ln \left( \frac{P\_2}{P\_o} \right) \right] \end{aligned} \right\} + T\_o \Delta S\_o \tag{24e}$$

where ( ) <sup>1</sup> *pg g c R* γ γ <sup>−</sup> <sup>=</sup>

The total energy input *Qeff cc* , to the combustion chamber is then

$$
\dot{Q}\_{\text{eff,cc}} = \dot{Q}\_{\text{cc}} + \dot{I}\_{\text{cc}} \tag{25}
$$

### **3.4 Gas turbine**

The rate of exergy destruction in the gas turbine is,

$$\begin{aligned} \dot{I}\_t &= \dot{m}\_a \left( 1 + f + \alpha\_1 \right) T\_o \left( s\_4 - s\_3 \right) \\\\ &= \dot{m}\_a \left( 1 + f + \alpha\_1 \right) T\_o \left[ \mathcal{L}\_{p\_\mathcal{R}} \left\ell n \left( \frac{T\_4}{T\_3} \right) - R\_g \left. \ell n \right| \left( \frac{P\_4}{P\_3} \right) \right] \end{aligned} \tag{26}$$

The effective power output of the turbine can be expressed in terms of exergy destruction rate as;

Energy and Exergy Analysis

by subsisting

effectiveness.

total exergy output ( ) *Eout*

ω ω

of Reverse Brayton Refrigerator for Gas Turbine Power Boosting 107

*th,u*

leads to the conventional thermal cycle efficiency expression (without cooling *ηth,nocooling* ).

Equation 5 to get *Wc no cooling* , . Similarly the turbine power without cooling *Wt no cooling* , is obtained

*net no cooling t c* , *no cooling*

Adopting the terminology proposed by Alhazmy *et al* (2006) and employed in our previous work Zaki, *et al (*2007) to evaluate the thermal efficiency augmentation; a thermal efficiency

*th,no cooling*

The subscript *u* refers to useful indicating that the GT plant is serving adjacent industrial process so that the conventional thermal efficiency definition is not applicable for this condition. For the present parametric analysis let us focus on a simple GT plant just coupled

eliminated and the performance of the coupled cycles may be determined from a power

, , ,

<sup>−</sup> = × 

*W W*

*W*

*net no cooling*

*th,no cooling η η TEC <sup>100</sup> η*

The *PGR* is a generic term that takes into account all the parameters of the gas turbine and the coupled cooling system. For a stand-alone gas turbine under specific climatic conditions; *PGR =* 0. If Brayton refrigerator is used, the *PGR* increases with the reduction of the intake temperature. However, the *PGR* gives the percentage enhancement in power generation; the *TEC* of a coupled system is an important parameter to describe the fuel utilization

From the second law analysis the performance of a system may be articulated by the exergetic efficiency. For the proposed system define the exergetic efficiency *ηex* as a ratio of

rates as;

100% *net withcooling net no cooling*

% *th,withcooling th,no cooling*

*η*

*net*

*useful W <sup>η</sup> Q*<sup>=</sup>

<sup>1</sup> = *<sup>o</sup>* , *T T* <sup>1</sup> = *<sup>o</sup>* and *f* is calculated from Eq. 8 using 2 *T* instead of *T*2 Therefore,

100% *th,u th,no cooling*

An integral useful thermal efficiency ( ) *ηth,u* may be defined as:

If the GT is in operation without the cooling cycle then 0 *Q*

change term based on the useful input energy *TECu* is defined as:

gain ratio (*PGR*) and thermal efficiency change (*TEC*) terms as:

*PGR*

to input( ) *Ein*

*u*

*η η TEC*

to a Brayton refrigerator without use of the cooling energy; the term *Qout*

The net power for the GT unit without cooling is obtained by introducing

*Q QQ useful cc out* = − (34)

*W WW* = − (36)

<sup>−</sup> = × (37)

(38)

<sup>−</sup> = × (39)

(35)

*out* <sup>→</sup> and exp *<sup>W</sup>* <sup>→</sup> <sup>0</sup> , equation 35

ω ω<sup>1</sup> = *<sup>o</sup>* in

in Eqs. 34 is

α= 0 and

$$
\dot{\mathcal{W}}\_{\rm eff,t} = \dot{\mathcal{W}}\_t - \dot{I}\_t \tag{27}
$$

### **3.5 Heat exchanger**

During the air cooling process 6-7 (Fig. 1), the rate of exergy destruction in the heat exchanger is,

$$\dot{I}\_{H\bar{\chi}} = a \dot{m}\_1 \ T\_o \left(\mathbf{s}\_6 - \mathbf{s}\_7\right) + \frac{T\_o}{T\_6} \dot{Q}\_{out} \tag{28}$$

where *Qout* is the rate of heat rejection from the heat exchanger at *T6*, = ( ) α*mh h* 16 7 − .

The net energy output in the *HX* is then;

$$
\dot{Q}\_{\text{eff},\text{HX}} = \dot{Q}\_{\text{out}} + \dot{I}\_{\text{HX}} \tag{29}
$$

### **3.6 Expander**

The rate of exergy destruction in the expander (process 7-8) is similar to that in the turbine and can be expressed as,

$$
\dot{I}\_{\rm cap} = a \dot{m}\_1 T\_o \left( s\_s - s\_\gamma \right)
$$

$$
= a \left( \frac{\dot{m}\_a}{1 - a} \right) \left( 1 + \alpha\_1 \right) T\_o \left[ c\_{pa} \,\ell \ln \left( \frac{T\_s}{T\_\gamma} \right) - R\_a \,\ell \ln \left( \frac{P\_s}{P\_\gamma} \right) \right] \tag{30}
$$

The effective power output of the expander is then

$$
\dot{\mathcal{W}}\_{\text{eff,cap}} = \dot{\mathcal{W}}\_{\text{exp}} - \dot{I}\_{\text{exp}} \tag{31}
$$

To that point the exergy destruction have been evaluated for all the system components and the total rate of exergy losses in the plant, Fig. 1 is;

$$
\sum \dot{I} = +\dot{I}\_{mc} + \dot{I}\_c + \dot{I}\_{cc} + \dot{I}\_t + \dot{I}\_{\ell X} + \dot{I}\_{\text{exp}} \tag{32}
$$

### **4. Performance of the integrated system**

For the proposed cycle the net power output and heat input, can be easily calculated using equations 5, 8 and 11. If the expander power *W*exp is recovered, then the net output of the cycle may be expressed as:

$$
\dot{\mathcal{W}}\_{net, with cooling} = \dot{\mathcal{W}}\_t - \left(\dot{\mathcal{W}}\_{comp} - \dot{\mathcal{W}}\_{\rm exp}\right) \tag{33}
$$

An advantage of the present cycle is the availability of useful energy *Qout* that may be utilized for any process. Let us define a term for the useful heat input as the net heat utilized to produce the shaft power, therefore, equations 7 and 12 give,

During the air cooling process 6-7 (Fig. 1), the rate of exergy destruction in the heat

( ) 6 7

*HX 1 o out <sup>T</sup> <sup>I</sup> <sup>α</sup>mT s s Q <sup>T</sup>*

is the rate of heat rejection from the heat exchanger at *T6*, = ( )

The rate of exergy destruction in the expander (process 7-8) is similar to that in the turbine

( ) exp *1o 8 7 I* = − *αmT s s*

*1 o pa a*

*m TP α 1 ω Tc n Rn 1-α T P* =+ −

*7 7*

*mc c cc t HX* exp *I I II I I I* =+ + + + + + (32)

*W WW W net withcooling t comp* , =− − ( ) exp (33)

(30)

*W WI eff* ,exp exp exp = − (31)

is recovered, then the net output of the

that may be

( ) *<sup>a</sup> 8 8*

To that point the exergy destruction have been evaluated for all the system components and

For the proposed cycle the net power output and heat input, can be easily calculated using

utilized for any process. Let us define a term for the useful heat input as the net heat utilized

An advantage of the present cycle is the availability of useful energy *Qout*

to produce the shaft power, therefore, equations 7 and 12 give,

6 *o*

**3.5 Heat exchanger** 

The net energy output in the *HX* is then;

The effective power output of the expander is then

the total rate of exergy losses in the plant, Fig. 1 is;

**4. Performance of the integrated system** 

equations 5, 8 and 11. If the expander power *W*exp

cycle may be expressed as:

exchanger is,

where *Qout*

**3.6 Expander** 

and can be expressed as,

*W WI eff t t t* , = − (27)

= −+ (28)

*Q QI eff HX out HX* , = + (29)

α

*mh h* 16 7 − .

$$
\dot{Q}\_{useful} = \dot{Q}\_{cc} - \dot{Q}\_{out} \tag{34}
$$

An integral useful thermal efficiency ( ) *ηth,u* may be defined as:

$$
\eta\_{th,u} = \frac{\dot{W}\_{\text{net}}}{\dot{Q}\_{\text{useful}}} \tag{35}
$$

If the GT is in operation without the cooling cycle then 0 *Q out* <sup>→</sup> and exp *<sup>W</sup>* <sup>→</sup> <sup>0</sup> , equation 35 leads to the conventional thermal cycle efficiency expression (without cooling *ηth,nocooling* ).

The net power for the GT unit without cooling is obtained by introducing α = 0 and ω ω <sup>1</sup> = *<sup>o</sup>* in Equation 5 to get *Wc no cooling* , . Similarly the turbine power without cooling *Wt no cooling* , is obtained by subsistingω ω<sup>1</sup> = *<sup>o</sup>* , *T T* <sup>1</sup> = *<sup>o</sup>* and *f* is calculated from Eq. 8 using 2 *T* instead of *T*2 Therefore,

$$\left| \dot{\mathcal{W}}\_{\text{net\\_no\\_cooling}} \right| = \left| \dot{\mathcal{W}}\_t - \dot{\mathcal{W}}\_c \right|\_{\text{no\\_cooling}} \tag{36}$$

Adopting the terminology proposed by Alhazmy *et al* (2006) and employed in our previous work Zaki, *et al (*2007) to evaluate the thermal efficiency augmentation; a thermal efficiency change term based on the useful input energy *TECu* is defined as:

$$TECC\_{\mu} = \frac{\eta\_{th,u} - \eta\_{th,no\text{ cooling}}}{\eta\_{th,uo\text{ cooling}}} \times 100\,\%\tag{37}$$

The subscript *u* refers to useful indicating that the GT plant is serving adjacent industrial process so that the conventional thermal efficiency definition is not applicable for this condition. For the present parametric analysis let us focus on a simple GT plant just coupled to a Brayton refrigerator without use of the cooling energy; the term *Qout* in Eqs. 34 is eliminated and the performance of the coupled cycles may be determined from a power gain ratio (*PGR*) and thermal efficiency change (*TEC*) terms as:

$$PGR = \frac{\dot{\mathcal{W}}\_{\text{net,with cooling}} - \dot{\mathcal{W}}\_{\text{net,cooling}}}{\dot{\mathcal{W}}\_{\text{net,no cooling}}} \times 100\,\%\tag{38}$$

$$TEC = \frac{\eta\_{th, with\text{cooling}} - \eta\_{th, \text{no cooling}}}{\eta\_{th, \text{no cooling}}} \times 100 \,\% \tag{39}$$

The *PGR* is a generic term that takes into account all the parameters of the gas turbine and the coupled cooling system. For a stand-alone gas turbine under specific climatic conditions; *PGR =* 0. If Brayton refrigerator is used, the *PGR* increases with the reduction of the intake temperature. However, the *PGR* gives the percentage enhancement in power generation; the *TEC* of a coupled system is an important parameter to describe the fuel utilization effectiveness.

From the second law analysis the performance of a system may be articulated by the exergetic efficiency. For the proposed system define the exergetic efficiency *ηex* as a ratio of total exergy output ( ) *Eout* to input( ) *Ein* rates as;

Energy and Exergy Analysis

mouth and expander outlet.

Ambient air

Gas Turbine

ratio (*x*) and the ratio between the mass rates (

Max. ambient air temperature, *Ta* 

Reference ambient temperature, *To*

Relative humidity, *RHo* Volumetric air flow rate, *Va*

Net Calorific Value, *NCV*

Turbine inlet temperature *T3* Specific heat ratio of gas,

Standard environment temp., *To* Extraction pressure ratio *Px/P1* Extracted mass ratio, *α* Specific heat ratio of air,

Specific heat of gas, *cpg*

Air Compressor

Heat exchanger

φ

Table 1. Range of parameters for the present analysis

Δ*T*

Fuel Fuel oils,

established that for a fixed extraction pressure (P6), increasing (

of Reverse Brayton Refrigerator for Gas Turbine Power Boosting 109

process; the coolant flow rate can be controlled to obtain different outlet temperature suitable for the industrial process. For 10K terminal temperature difference the value of *T7*

In the previous study by the authors Zaki *et al* (2007), the effect of the extraction pressure

to operate the GT close to the ISO standard (288.15 K the dotted line in Figure 4). It is seen

intake temperature drops from 318 K to values between 310 and 288 K. Extending the range to reach temperatures below 273 K is constrained by frost formation at the compressor

α

) has been investigated. It has been

α

) increases the chilling

αand *x* 

for which the

α

318.15 *K* 43.4 % 1 *m3 s-1* 42500 *kJ kg-1* 298.15 K

1373.15 K 1.333

298.15 K 2 → 9 0.1 → 0.5 1.4

10 *K*

1.08

1.147 kJ kg*-1* K*-1*

can be fixed to 55oC. Table I shows the range of parameters for the present analysis.

effect and hence *T1* decreases as seen in Fig. 4. It is possible to adjust the values of

that a reasonable range of operation is bounded by 2 4 < < *x* and 0.2 0.3 < <

**Parameter Range** 

Pressure ratio *P2/P1* 10

γ

γ

$$
\eta\_{c\varepsilon} = \frac{\dot{E}\_{out}}{\dot{E}\_{in}} \tag{40}
$$

Exergy balance for the system shown in Figure (1) can be expressed in the following form,

$$
\dot{E}\_{\text{out}} = \dot{\mathsf{V}}\_{\text{eff},t} + \dot{\mathsf{V}}\_{\text{eff,cap}} - \dot{\mathsf{V}}\_{\text{eff,c}} \tag{41}
$$

The total rate of exergy input is

$$
\dot{E}\_{in} = \dot{Q}\_{\text{eff}\,,cc} - \dot{Q}\_{\text{eff}\,\,\text{HX}} \tag{42}
$$

Expressions for the effective power terms *Weff c*, , *W , eff,t* and *Weff,exp* are given by equations 20, 27 and 31 respectively, while *Qeff cc* , and *Qeff HX* , are determined from equations 25 and 29.

The irreversibilities of heat exchangers ( ) *HX I* , expanders ( ) exp *I* and mixing chamber ( ) *mc I* are all equal to zero when the GT operates on a stand alone mode.

Similar to the criteria adopted for the energy analysis let us define dimensionless terms as the e*xergetic power gain ratio* (*PGRex*) and *exergetic thermal efficiency change* (*TECex*) as:

$$PGR\_{ex} = \frac{\left(\dot{E}\_{out}\right)\_{cooling} - \left(\dot{E}\_{out}\right)\_{cooling}}{\left(\dot{E}\_{out}\right)\_{cooling}} \times 100\,\%\tag{43}$$

$$TECC\_{ex} = \frac{\eta\_{ex,cooling} - \eta\_{ex,cooling}}{\eta\_{ex,cooling}} \times 100\,\%\tag{44}$$

### **5. Results and discussion**

In order to evaluate the integration of Brayton refrigerator for intake air-cooling on the GT performance, a computer program has been developed using EES (Klein and Alvarado 2004) software. The computation procedure was first verified for the benchmark case of simple open cycle with dry air as the working fluid, for whichα = 0 , 0 ω*<sup>o</sup>* = *f* = 0 and assuming isentropic compression and expansion processes, Eqs. 5, 7, 8 and 37 leads to the expression for the thermal efficiency of the air standard Brayton cycle.

$$\eta\_{th} = 1 - \frac{1}{r^{(\gamma - 1)/\gamma}} \tag{45}$$

Similarly from the exergy point of view and for the same ideal conditions, all exergy destruction terms (Equations 17d, 24e, 26, 28, 30) and the exergetic efficiency (equation 44) leads to the same expression, Eq. 45.

For the present analysis, the ambient air at Jeddah, Saudi Arabia (Latitude 22.30o N and Longitude 39.15o E) a typical city with over 40 GT plants operating under sever weather conditions was selected (ambient temperature of *To* = 45oC and ϕ*<sup>o</sup>* = 43.3%). The air temperature at the expander inlet *T7* is determined by the requirements of the heating

*ex*

The total rate of exergy input is

**5. Results and discussion** 

leads to the same expression, Eq. 45.

Expressions for the effective power terms *Weff c*, , *W , eff,t*

are all equal to zero when the GT operates on a stand alone mode.

*ex*

*ex*

*PGR*

open cycle with dry air as the working fluid, for which

for the thermal efficiency of the air standard Brayton cycle.

The irreversibilities of heat exchangers ( ) *HX I*

Exergy balance for the system shown in Figure (1) can be expressed in the following form,

*EQ Q in eff cc eff HX* = − , , (42)

, expanders ( ) exp *I*

( ) 100% *out out cooling nocooling*

20, 27 and 31 respectively, while *Qeff cc* , and *Qeff HX* , are determined from equations 25 and 29.

Similar to the criteria adopted for the energy analysis let us define dimensionless terms as

− = ×

*out nocooling*

*ex,cooling ex,nocooling*

*ex,nocooling η η TEC 100% η*

In order to evaluate the integration of Brayton refrigerator for intake air-cooling on the GT performance, a computer program has been developed using EES (Klein and Alvarado 2004) software. The computation procedure was first verified for the benchmark case of simple

isentropic compression and expansion processes, Eqs. 5, 7, 8 and 37 leads to the expression

<sup>1</sup> <sup>1</sup> *th <sup>r</sup>* γ γ

Similarly from the exergy point of view and for the same ideal conditions, all exergy destruction terms (Equations 17d, 24e, 26, 28, 30) and the exergetic efficiency (equation 44)

For the present analysis, the ambient air at Jeddah, Saudi Arabia (Latitude 22.30o N and Longitude 39.15o E) a typical city with over 40 GT plants operating under sever weather

temperature at the expander inlet *T7* is determined by the requirements of the heating

η

conditions was selected (ambient temperature of *To* = 45oC and

( 1)/

the e*xergetic power gain ratio* (*PGRex*) and *exergetic thermal efficiency change* (*TECex*) as:

() ()

*E E*

*E*

*out*

(40)

are given by equations

*<sup>o</sup>* = *f* = 0 and assuming

*<sup>o</sup>* = 43.3%). The air

and mixing chamber ( ) *mc I*

(43)

<sup>−</sup> = × (44)

 = 0 , 0 ω

<sup>−</sup> = − (45)

ϕ

α

*EW W W out eff t eff* =+ − , ,exp , *eff c* (41)

and *Weff,exp*

*in E <sup>η</sup> <sup>E</sup>* <sup>=</sup>

process; the coolant flow rate can be controlled to obtain different outlet temperature suitable for the industrial process. For 10K terminal temperature difference the value of *T7* can be fixed to 55oC. Table I shows the range of parameters for the present analysis.

In the previous study by the authors Zaki *et al* (2007), the effect of the extraction pressure ratio (*x*) and the ratio between the mass rates (α ) has been investigated. It has been established that for a fixed extraction pressure (P6), increasing (α ) increases the chilling effect and hence *T1* decreases as seen in Fig. 4. It is possible to adjust the values of α and *x*  to operate the GT close to the ISO standard (288.15 K the dotted line in Figure 4). It is seen that a reasonable range of operation is bounded by 2 4 < < *x* and 0.2 0.3 < < α for which the intake temperature drops from 318 K to values between 310 and 288 K. Extending the range to reach temperatures below 273 K is constrained by frost formation at the compressor mouth and expander outlet.


Table 1. Range of parameters for the present analysis

Energy and Exergy Analysis

descending trend in Figure 6 a. is expected.

Fig. 4. Effect of extraction pressure *x* and mass ratio

*288.15 K (15 <sup>o</sup> C)*

260

270

280

290

*T1, K*

300

310

320

temperature.

of Reverse Brayton Refrigerator for Gas Turbine Power Boosting 111

reduces the nominator of Eq. 38 and decreases the *PGR*. Therefore, the *PGR* ascending

α

 */P1*

1 2 3 4 5 6 7 8 9 10

*x = P6*

Fig. 5. Power enhancement (Eq.38) and thermal efficiency change (Eq. 39) for GT plant, *r* =10

on the gas turbine intake air

α = 0.1

α = 0.2

α = 0.3

α = 0.4

Figure 5 shows the *PGR* and *TEC* variations within pressure extraction ratio up to 4 and 0 0.5 ≤ ≤ α . It is seen that the power gain increases linearly withα , which means enhanced chilling effect due to the large air mass rate ( *m <sup>a</sup>* α ) passing through the refrigeration cycle. As seen in the figure the power is boosted up while the thermal efficiency (Eq. 39) decreases, at *x* = 3 and 4, due to the increase in fuel consumption rate. The drop in the *TEC* is quite large for high α and *x* values. For moderate values as α = 0.2 and extraction pressure of 4 bar, the power increases by 9.11%, while the thermal efficiency drops by only 1.34%. This result indicates that passing 20% of the intake air at 4 bars pressure thorough a Reverse Brayton refrigerator increases the power by 9.11% with only drop in thermal efficiency of 1.34%. Therefore, selection of the operation parameters is a trade off between power and efficiency, boosting the power on penalty of the thermal efficiency. It is of interest to determine the parameters at which the power is boosted while the thermal efficiency of the GT stays the same as that for the condition of a stand alone operation. This means that the *TEC* (Eq. 39) approaches zero. Solving Eq. 39 shows that within a narrow range of extraction pressure (*x* = 2.6 to 2.8) as shown in Figure 3, the plant thermal efficiency is slightly altered from the stand alone efficiency and *TEC* is independent on the extracted mass ratioα . Though, the efficiency does not change appreciably the power gain strongly depends onα.

Fig. 3. Solution of Eq. 39, *TEC* = 0 gives the conditions for constant thermal efficiency

The change of the *PGR* within a range of pressure ratios is presented in Fig 6 a, indicating increase in *PGR* up to certain α followed by a descending trend. For a constant value of α the power gain increases with the extraction pressure reaching a peak value then decreases. As seen in Fig. 6 b. the increase in *x* or *P6* while *T7* is constant means that the outlet temperature from the expander moves from *T*8 to 8 *T* , which means lower intake temperature ( *T T* <sup>8</sup> < <sup>8</sup> ) with tendency to increase the *PGR*. On the other hand increase in *x* results high net compressor work ( *W W comp exp* <sup>−</sup> ) in Equation 33. High net compressor work

Figure 5 shows the *PGR* and *TEC* variations within pressure extraction ratio up to 4 and

As seen in the figure the power is boosted up while the thermal efficiency (Eq. 39) decreases, at *x* = 3 and 4, due to the increase in fuel consumption rate. The drop in the *TEC* is quite

bar, the power increases by 9.11%, while the thermal efficiency drops by only 1.34%. This result indicates that passing 20% of the intake air at 4 bars pressure thorough a Reverse Brayton refrigerator increases the power by 9.11% with only drop in thermal efficiency of 1.34%. Therefore, selection of the operation parameters is a trade off between power and efficiency, boosting the power on penalty of the thermal efficiency. It is of interest to determine the parameters at which the power is boosted while the thermal efficiency of the GT stays the same as that for the condition of a stand alone operation. This means that the *TEC* (Eq. 39) approaches zero. Solving Eq. 39 shows that within a narrow range of extraction pressure (*x* = 2.6 to 2.8) as shown in Figure 3, the plant thermal efficiency is slightly altered from the stand alone efficiency and *TEC* is independent on the extracted mass ratio

Though, the efficiency does not change appreciably the power gain strongly depends on

2.795

Fig. 3. Solution of Eq. 39, *TEC* = 0 gives the conditions for constant thermal efficiency

α

The change of the *PGR* within a range of pressure ratios is presented in Fig 6 a, indicating

0 0.1 0.2 0.3 0.4 0.5

α

the power gain increases with the extraction pressure reaching a peak value then decreases. As seen in Fig. 6 b. the increase in *x* or *P6* while *T7* is constant means that the outlet temperature from the expander moves from *T*8 to 8 *T* , which means lower intake temperature ( *T T* <sup>8</sup> < <sup>8</sup> ) with tendency to increase the *PGR*. On the other hand increase in *x* results high net compressor work ( *W W comp exp* <sup>−</sup> ) in Equation 33. High net compressor work

followed by a descending trend. For a constant value of

α

α

2.725

α

) passing through the refrigeration cycle.

2.605

, which means enhanced

α.

α

α.

= 0.2 and extraction pressure of 4

. It is seen that the power gain increases linearly with

and *x* values. For moderate values as

2.775

chilling effect due to the large air mass rate ( *m <sup>a</sup>*

0 0.5 ≤ ≤ α

large for high

α

increase in *PGR* up to certain

2

2.2

2.4

*x = P*

 */ P*

*6*

*1*

2.6

2.8

3

reduces the nominator of Eq. 38 and decreases the *PGR*. Therefore, the *PGR* ascending descending trend in Figure 6 a. is expected.

Fig. 4. Effect of extraction pressure *x* and mass ratio α on the gas turbine intake air temperature.

Fig. 5. Power enhancement (Eq.38) and thermal efficiency change (Eq. 39) for GT plant, *r* =10

Energy and Exergy Analysis

*x* = 3.5 and variable

0

4

8

12

*PGR %*

16

*TECu*

*PGR %*

20

24

*α <sup>c</sup> I*

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 α.

*cc I*

21.4 21.76 22.18 22.65 23.2 23.83 24.56 25.42 26.44 27.66 29.14

of Reverse Brayton Refrigerator for Gas Turbine Power Boosting 113

has been deducted from the heat released by fuel. Though the result was expected, the values here can not be compared to the stand alone plant or even with the case where the cooling water energy is wasted. The figure shows also the *PGR* variation, which is the same

> αand *x*

α = 0.3 α = 0.4

exp *I*

0 0.3103 0.6644 1.07 1.539 2.082 2.718 3.468 4.361 5.437 6.753

*mc <sup>I</sup>*

0 0.6484 1.377 2.133 2.917 3.733 4.581 5.464 6.386 7.349 8.356

*total I*




0

2

*TECu %*

4

6

8

10

0 433 464.5 497.6 532.4 568.9 607.5 648.3 691.6 737.8 787.2

The results of the energy analysis showed clearly the advantages of the proposed cooling system and the expected magnitude of performance enhancement. However, the practical range of operation parameters was determined the contribution of each of the system components to the irreversibility is only determined through equations 17d, 19, 24e, 26, 28 and 30. Table 2 shows numerical values for the exergy rate of the different components at

1 2 3 4 5 6 7 8 9 10

*x = P6 / P1*

*HX I*

0 17.39 35.68 54.95 75.28 96.77 119.5 143.8 169.6 197.3 227.1

22.7 22.97 23.3 23.64 24.00 24.36 24.74 25.12 25.52 25.94 26.36

whether the energy is utilized or wasted as indicated by Equations 35 and 38.

Fig. 7. The useful thermal efficiency change as function of

α = 0.1 α = 0.2

*<sup>t</sup> I*

Table 2. Cycle components irreversibility values (*kW*) for *x* =3.5

360.8 369.9 381.3 393.2 405.4 418.2 431.4 445.1 459.3 474.1 489.5

Fig. 6a. Power gain ratio and thermal efficiency change factors for a gas turbine cooled by air Brayton refrigerator

Fig. 6b. *T-s* diagram to illustrate the effects of increasing the extraction pressure. Increasing the extraction pressure *P*6 reduces the compressor intake temperature and increases the difference ( *W W comp exp* <sup>−</sup> ).

To illustrate the effect of the heat rate ( *Qout* ) during the cooling process (process 6-7, Fig. 1) on the overall system performance, the useful thermal efficiency change (*TECu,* Eq. 37) is plotted with the control variables α and *x* in Figure 7. It is seen that for all operation conditions *TECu* is positive, which means that the energy utilized for an industrial process

Fig. 6a. Power gain ratio and thermal efficiency change factors for a gas turbine cooled by air

1 2 3 4 5 6 7 8 9 10

α = 0.1 α = 0.2 α = 0.3

*x = P6 / P1*

Fig. 6b. *T-s* diagram to illustrate the effects of increasing the extraction pressure. Increasing the extraction pressure *P*6 reduces the compressor intake temperature and increases the

on the overall system performance, the useful thermal efficiency change (*TECu,* Eq. 37) is

conditions *TECu* is positive, which means that the energy utilized for an industrial process

α

) during the cooling process (process 6-7, Fig. 1)


α = 0.4




*TEC %*


0

4

and *x* in Figure 7. It is seen that for all operation

Brayton refrigerator

0

4

8

12

*PGR %*

16

20

*TEC*

*PGR*

24

difference ( *W W comp exp* <sup>−</sup> ).

To illustrate the effect of the heat rate ( *Qout*

plotted with the control variables

has been deducted from the heat released by fuel. Though the result was expected, the values here can not be compared to the stand alone plant or even with the case where the cooling water energy is wasted. The figure shows also the *PGR* variation, which is the same whether the energy is utilized or wasted as indicated by Equations 35 and 38.

Fig. 7. The useful thermal efficiency change as function of αand *x*

The results of the energy analysis showed clearly the advantages of the proposed cooling system and the expected magnitude of performance enhancement. However, the practical range of operation parameters was determined the contribution of each of the system components to the irreversibility is only determined through equations 17d, 19, 24e, 26, 28 and 30. Table 2 shows numerical values for the exergy rate of the different components at *x* = 3.5 and variableα.


Table 2. Cycle components irreversibility values (*kW*) for *x* =3.5

Energy and Exergy Analysis

refrigerator

*PGR %*

of parameters 2 4 < < *x* and 0.2 0.3 < <

0

0.05

0.1

0.15

η

*ex*

*,* η*th*

0.2

0.25

0.3

0.35

0.4

α

*PGR*

*PGRex*

of Reverse Brayton Refrigerator for Gas Turbine Power Boosting 115

is affected by both the amount of air that circulate in the refrigeration loop and the coupling pressure ratio *x*. Figure 9 indicates clearly that the difference increases at high extraction ratio (more air is bleed from the main compressor) and high extraction pressure (large portion of the energy is wasted in the heat exchanger). Within the previously selected range

Fig. 9. Energetic and exergetic power gain ratio for a gas turbine cooled by air Brayton

α = 0.1 α = 0.2 α = 0.3

*PGRex*

*PGR*

*PGR*

1 2 3 4 5 6 7 8 9 10

*PGR*

*PGRex*

*x = P6 / P1*

*x = 3.5* <sup>η</sup>*th,u*

Fig. 10. Thermal efficiency, useful thermal efficiency and exergetic efficiencies dependency on

0 0.1 0.2 0.3 0.4 0.5

α

α.

the maximum difference is in the order of 5%.

*PGRex*

α = 0.4

ηex,u

η*ex*

ηth

It is seen that the major contribution comes from the combustion chamber where the irreversibility is the highest and presents 62 to 85% of the total irreversibility. The heat exchanger comes next with nearly 5 % to 46% of the magnitude of the combustion chamber. Both the compressor and turbine are of nearly equal magnitude. Therefore, optimization of the system should focus on the irreversibilities in the combustion chamber and heat exchanger. The mixing chamber irreversibility is found to be very small compared to the other components as seen in Table 2 and can be ignored in future studies.

Fig. 8. Dependence of exergetic power gain ratio and exergetic thermal efficiency on the extracted pressure ratio without taking *Qeff HX* , into account

The variations of the exergetic *PGR* and exergetic *TEC* (equations 43 and 44) are shown in Figure 8 for the same range of parameters as in Fig. 6 a. Clearly this figure shows that the irreversibilities give lower values for the power gain and efficiency as compared to the energy analysis. The difference, at the maximum *PGR*, conditions is in the order of 20 to 25% as seen in Table 3. For example, the maximum *PGR* is reduced from 9.37 in case of first law of analysis to 7.436 when the second law was considered.


Table 3. Effect of irreversibility on power gain ratio

The energy destruction terms in Table II show dependency on α at constant pressure ratio *x*. The difference between the gain in power as predicted by the energy and exergy analysis

It is seen that the major contribution comes from the combustion chamber where the irreversibility is the highest and presents 62 to 85% of the total irreversibility. The heat exchanger comes next with nearly 5 % to 46% of the magnitude of the combustion chamber. Both the compressor and turbine are of nearly equal magnitude. Therefore, optimization of the system should focus on the irreversibilities in the combustion chamber and heat exchanger. The mixing chamber irreversibility is found to be very small compared to the

α = 0.4 α = 0.3 α = 0.2 α = 0.1

Fig. 8. Dependence of exergetic power gain ratio and exergetic thermal efficiency on the

*x = P6 / P1*

1 2 3 4 5 6 7 8 9 10

*Qeff,HX is not considered*

The variations of the exergetic *PGR* and exergetic *TEC* (equations 43 and 44) are shown in Figure 8 for the same range of parameters as in Fig. 6 a. Clearly this figure shows that the irreversibilities give lower values for the power gain and efficiency as compared to the energy analysis. The difference, at the maximum *PGR*, conditions is in the order of 20 to 25% as seen in Table 3. For example, the maximum *PGR* is reduced from 9.37 in case of first law

(*PGRex*)max%

3.448 7.436 11.25 14.66

*x*. The difference between the gain in power as predicted by the energy and exergy analysis

Fig. 8 Reduction %

α

20.478 20.683 22.414 25.128

at constant pressure ratio





*TECex*

*TEC*

*ex*



0

5

extracted pressure ratio without taking *Qeff HX* , into account

of analysis to 7.436 when the second law was considered.

Fig. 6 a

4.336 9.375 14.5 19.58

The energy destruction terms in Table II show dependency on

<sup>α</sup>(*PGR*)*max%* 

Table 3. Effect of irreversibility on power gain ratio

0.1 0.2 0.3 0.4

*PGR*

*ex*

*PGR*

*ex*

other components as seen in Table 2 and can be ignored in future studies.

is affected by both the amount of air that circulate in the refrigeration loop and the coupling pressure ratio *x*. Figure 9 indicates clearly that the difference increases at high extraction ratio (more air is bleed from the main compressor) and high extraction pressure (large portion of the energy is wasted in the heat exchanger). Within the previously selected range of parameters 2 4 < < *x* and 0.2 0.3 < < αthe maximum difference is in the order of 5%.

Fig. 9. Energetic and exergetic power gain ratio for a gas turbine cooled by air Brayton refrigerator

Fig. 10. Thermal efficiency, useful thermal efficiency and exergetic efficiencies dependency onα.

Energy and Exergy Analysis

*P* = pressure, *kPa PGR* = power gain ratio

*Q* = heat rate, *kW* 

*T* = temperature, *K*

**Greek symbols** 

ε

*o* ε

ω

ωο

η

γ

**Subscripts** 

*o* = ambiant *a* = dry air *c* = compressor

*cy* = cycle *ex* = exergy *exp* = expander g = gas

**Superscripts** 

*Qout*

*m* = mass flow rate*, kg s-1 NCV* = net calorific value, *kJ kg-1*

*PGRex* = exergetic power gain ratio

*TEC* = thermal efficiency change factor *TECu* = useful thermal efficiency change factor *TECex* = exergetic thermal efficiency change *W* = output power (according to subscript), *kW*

*x* = extraction pressure ratio, *P6/P1 y* = mole fraction of fuel components

*r* = pressure ratio = *P2/P1* s = specific entropy, *kJ kg-1K-1*

= heat release rate from the condenser, *kW*

α = fraction of air mass flowing through the cooler cycle

= ambiant specific humidity, kg water vapor/kg dry air

= specific exergy (according to subscript)

= efficiency (according to subscript)

= specific heat ratio

*cc* = combustion chamber

*~* = mean molar value of a property

*HX* = heat exchanger *p* = products *pg* = product gas *mc* = mixing chamber *R* = reactants *t* = turbine *u* = useful *v* = water vapor

= exergy due to chemical reaction, *kJ mol-1*

= specific humidity, kg water vapor/kg dry air

of Reverse Brayton Refrigerator for Gas Turbine Power Boosting 117

Figure 10 shows the magnitudes of the thermal efficiency, the useful thermal efficiency (equation 35) and the exergetic thermal efficiency (equation 40) for 0 0.5 ≤ ≤ α and extraction pressure of 3.5 bars. It is clear that utilization of the cooling energy for any process enhances the use of the input fuel for which η*th u th* , >η and η*ex u ex* , >η as seen in the figure. In general, the thermal efficiency weights all thermal energy equally, whilst the exergetic efficiency acknowledges the usefulness of irreversibilities on its quality and quantity. Thus, exergetic efficiency is more suitable for determining the precise power gain ratio. The figure shows also that the useful efficiency whether energetic or exergetic is higher than the corresponding values when wasting the heat exchanger heat rejection *Qout* .
