### Energy and exergy analysis

The law of conservation of mass and the first and second laws of thermodynamics are used to analyze the energy and exergy of thermodynamic systems. Considering the steady-state and regardless of kinetic energy and potential, the equations of mass balance, energy balance and exergy balance for the components of the cycle are as follows (Mohtaram et al. 2021a, b; Omidi et al. 2019; Chen et al. 2021):

$$\sum {\dot{m}}_{i}=\sum {\dot{m}}_{e}$$

(1)

$$\dot{Q}+\sum {\dot{\mathrm{m}}}_{\mathrm{i}}{h}_{i}=\sum {\dot{\mathrm{m}}}_{\mathrm{e}}{h}_{e}+\dot{W}$$

(2)

$${\dot{E}}_{Q}+\sum {\dot{\mathrm{m}}}_{\mathrm{i}}{e}_{i}=\sum {\dot{\mathrm{m}}}_{\mathrm{e}}{e}_{e}+\dot{W}+{\dot{E}}_{D}$$

(3)

In the above equations, \(\sum {\dot{\mathrm{m}}}_{\mathrm{i}}{h}_{i}\) is the input enthalpy rate to the control volume, \(\sum {\dot{\mathrm{m}}}_{\mathrm{e}}{h}_{e}\) is the output enthalpy rate to the control volume, \(\sum {\dot{\mathrm{m}}}_{\mathrm{i}}{e}_{i}\) is the input exergy rate to the control volume, and \(\sum {\dot{\mathrm{m}}}_{\mathrm{e}}{e}_{e}\) is the output exergy rate to the control volume. \({\dot{E}}_{D}\) is the rate of exergy destruction and \({\dot{E}}_{Q}\) is the rate of exergy associated with heat transfer, which is defined as follows (Haj Assad et al. 2021):

$${\dot{E}}_{Q}=\sum (1-\frac{{T}_{0}}{T})\dot{Q}$$

(4)

Due to the absence of chemical changes and regardless of the ability to use kinetic energy and potential, current exergy includes only physical exergy:

$${e}_{ph}=\left(h-{T}_{0}s\right)-\left({h}_{0}-{T}_{0}{s}_{0}\right)$$

(5)

The efficiency of the first law and the exergy efficiency are defined as follows:

$${\eta }_{\mathrm{th}}=\frac{{\dot{W}}_{\mathrm{net}}}{{Q}_{R}}$$

(6)

$${\eta }_{\mathrm{ex}}=\frac{{\dot{W}}_{\mathrm{net}}}{{\dot{E}}_{QR}}$$

(7)

The equation used to study the exergy for the k’s component is as follows:

$${\dot{E}}_{D,k}={\dot{E}}_{F,k}-{\dot{E}}_{P,k}$$

(8)

$${\varepsilon }_{k}=\frac{{\dot{E}}_{P,k}}{{\dot{E}}_{F,k}}=1-\frac{{\dot{E}}_{D,k}}{{\dot{E}}_{F,k}}$$

(9)

In these equations \({\dot{E}}_{D,k}\), \({\dot{E}}_{F,k}\) and \({\dot{E}}_{P,k}\) are the exergy destruction rate, fuel exergy rate and product exergy rate are k’s components, respectively.

### Advanced exergy analysis

Advanced exergy analysis as a new concept in the exergy analysis of thermodynamic cycles states that the exergy destruction in a component is not only due to the irreversibility of the component itself but also due to the irreversible effect of other components of the cycle on the component.

In advanced exergy analysis, the rate of exergy destruction of component k is divided into endogenous and exogenous (Echeeri and Maalmi 2022; Sohrabi and Behbahaninia 2022):

$${\dot{E}}_{D,k}={\dot{E}}_{D,k}^{\mathrm{EN}}+{\dot{E}}_{D,k}^{\mathrm{EX}}$$

(10)

\({\dot{E}}_{D,k}^{EN}\) is part of the exergy destruction of the k’s component due to the internal irreversibility of the component itself, and \({\dot{E}}_{D,k}^{\mathrm{EX}}\) is the part of the exergy destruction that results from the irreversible effect of the other components of the cycle on the performance of the *k*’s component. The exogenous exergy destruction of the *k* component due to the irreversible effect of the (*n-*1) component of the n-components cycle can be examined in more detail.

Also, by dividing the exergy destruction into two parts, avoidable and unavoidable, we can have a better understanding of the potential to improve the efficiency of cycle components (Sherwani 2022; Hashemian et al. 2022):

$${\dot{E}}_{D,k}={\dot{E}}_{D,k}^{\mathrm{AV}}+{\dot{E}}_{D,k}^{\mathrm{UN}}$$

(11)

Unavoidable exergy destruction (\({\dot{E}}_{D,k}^{\mathrm{UN}}\)) is a part of exergy destruction that cannot be reduced due to technical limitations, and avoidable exergy destruction (\({\dot{E}}_{D,k}^{\mathrm{AV}}\)) is a part that can be reduced by upgrading and improving cycle components. Experimental destruction is achieved when the components of the cycle operate at their real unavoidable exergetic efficiency (\({\varepsilon }_{k}^{\mathrm{UN}}\)). It should be noted that the unavoidable exergetic efficiency is the maximum efficiency that can be achieved by considering the industrial constraints (Xie et al. 2022; Boodaghia et al. 2014).

According to the above, the destruction of avoidable and unavoidable exergy can be divided into the following two parts:

$${\dot{E}}_{D,k}^{\mathrm{AV}}={\dot{E}}_{D,k}^{\mathrm{EX},\mathrm{AV}}+{\dot{E}}_{D,k}^{\mathrm{EN},\mathrm{AV}}$$

(12)

$${\dot{E}}_{D,k}^{\mathrm{UN}}={\dot{E}}_{D,k}^{\mathrm{EX},\mathrm{UN}}+{\dot{E}}_{D,k}^{\mathrm{EN},\mathrm{UN}}$$

(13)

We also have the division of endogenous and exogenous exergy into two parts, avoidable and unavoidable:

$${\dot{E}}_{D,k}^{EN}={\dot{E}}_{D,k}^{\mathrm{EN},\mathrm{AV}}+{\dot{E}}_{D,k}^{\mathrm{EN},\mathrm{UN}}$$

(14)

$${\dot{E}}_{D,k}^{\mathrm{EX}}={\dot{E}}_{D,k}^{\mathrm{EX},\mathrm{AV}}+{\dot{E}}_{D,k}^{\mathrm{EX},\mathrm{UN}}$$

(15)

In the above equations, \({\dot{E}}_{D,k}^{\mathrm{EN},\mathrm{UN}}\) refers to the destruction of the internal exergy of the k’s component under unavoidable conditions, which is irreversible, and \({\dot{E}}_{D,k}^{\mathrm{EN},\mathrm{AV}}\) refers to the destruction of the internal exergy of the *k*’s component, which decreases as it improves. In addition, the destruction of exogenous exergy is the part of the exergy that is reduced by improving the structure of other components of the cycle, and the exorcism is unavoidable, which is irreversible due to technical limitations.

Various methods have been proposed in advanced exergy analysis, including the thermodynamic cycle method, the engineering method, the exergy balance method, the equivalent component method, and the structural theory method. In the present work, the method of thermodynamic cycles has been used due to its high accuracy and compatibility. It should be noted that in the cycle analysis in ideal and unavoidable conditions, the net power of the whole system is equal to the net power of the whole system in real conditions (Kelly 2008).