*4.2. Exergy Analysis of the Integrated Systems*

Exergy is defined as the maximum amount of useful energy that can be obtained from a stream when it reaches an equilibrium condition with the reference environment while interacting only with this environment [67]. The exergy analysis is applied to measure the exergy destruction and exergy efficiency for each component of the system proposed in this study. The equation of exergy destruction for each component can be derived from the equation of exergy balance. Considering a control volume at steady state, the general form of exergy balance can be written as follows:

$$
\sum \dot{\mathbf{E}} \mathbf{x}\_{\mathrm{Q}} - \sum \dot{\mathbf{E}} \mathbf{x}\_{\mathrm{w}} + \sum \dot{\mathbf{E}} \mathbf{x}\_{\mathrm{flow, in}} - \sum \dot{\mathbf{E}} \mathbf{x}\_{\mathrm{flow, out}} - \sum \dot{\mathbf{E}} \mathbf{x}\_{\mathrm{dest}} = \mathbf{0} \tag{19}
$$

where ExQ represents the rate of exergy transfer due to heat exchange with the environment, Exw is the rate of exergy transfer related to work, Exdest represents exergy destruction, and Exflow corresponds to the exergy transfer rate associated with the flow of the stream. . Ex represents the stream exergy, which can be expressed as follows [68]:

$$\dot{\mathbf{E}} = \mathbf{F} \cdot (\mathbf{E} \mathbf{x}\_{\text{chem}} + \mathbf{E} \mathbf{x}\_{\text{phys}} + \boldsymbol{\Delta}\_{\text{mix}} \mathbf{E} \mathbf{x}) \tag{20}$$

F, Exchem, Exphys, and ΔmixEx denote the molar flow rate and the physical, chemical, and mixing exergy of the stream.

The chemical exergy of the stream, accounting for phases and their composition at the reference condition, is given by Equation (21), as follows [68]:

$$\text{Ex}\_{\text{chem}} = \text{L}\_{0} \cdot \sum \text{x}\_{0,i} \cdot \text{Ex}\_{\text{chem},i}^{\text{ol}} + \text{V}\_{0} \cdot \sum \text{y}\_{0,i} \cdot \text{Ex}\_{\text{chem},i}^{\text{olv}} \tag{21}$$

L0 and V0 are the liquid and vapor mole fraction of the stream at the reference condition, respectively. x*<sup>i</sup>* and y*<sup>i</sup>* are the mole fraction of species *i* in the liquid and vapor phases, respectively. Ex0*<sup>l</sup>* chem,*<sup>i</sup>* and Ex0*<sup>v</sup>* chem,*<sup>i</sup>* denote the standard chemical exergy of component *i* in the liquid and vapor phases, respectively. The superscripts l and v denote liquid and vapor phases, respectively, and subscript 0 refers to the reference conditions. The reference conditions, T0 and P0, are set to 25 ◦C and 101.325 kPa in this work.

The physical exergy of the stream can be calculated from the enthalpies and entropies of the pure components, the amount of each phase, and their respective compositions, as follows [68]:

$$\begin{array}{l} \text{Ext}\_{\text{phys}} = \left[ \text{L} \cdot \sum \left( \mathbf{x}\_{i} \cdot \mathbf{H}\_{i}^{I} - \mathbf{T}\_{0} \cdot \sum \mathbf{x}\_{i} \cdot \mathbf{S}\_{i}^{I} \right) + \text{V} \cdot \sum \left( y\_{i} \cdot \mathbf{H}\_{i}^{v} - \mathbf{T}\_{0} \cdot \sum y\_{i} \cdot \mathbf{S}\_{i}^{v} \right) \right] \text{actual T,P} - \left[ \text{L}\_{0} \\ \quad \cdot \sum \left( \mathbf{x}\_{i} \cdot \mathbf{H}\_{i}^{I} - \mathbf{T}\_{0} \cdot \sum \mathbf{x}\_{i} \cdot \mathbf{S}\_{i}^{I} \right) + \text{V}\_{0} \cdot \sum \left( y\_{i} \cdot \mathbf{H}\_{i}^{v} - \mathbf{T}\_{0} \cdot \sum y\_{i} \cdot \mathbf{S}\_{i}^{v} \right) \right] \text{T}\_{0,P\_{0}} \end{array} \tag{22}$$

Hi and Si are the molar enthalpy and molar entropy of pure component *i*, respectively. The mixing exergy, which has always a negative value, can be found from [68]:

$$
\Delta\_{\text{mix}} \text{Ex} = \Delta\_{\text{mix}} \text{H} - \text{T}\_0 \cdot \Delta\_{\text{mix}} \text{S} \tag{23}
$$

where

$$\Delta\_{\rm mix} \mathbf{M} = \mathbf{L} \cdot \left( \mathbf{M}^{l} - \sum \mathbf{x}\_{i} \cdot \mathbf{M}^{l} \right) + \mathbf{V} \cdot \left( \mathbf{M}^{\upsilon} - \sum \mathbf{y}\_{i} \cdot \mathbf{M}^{\upsilon} \right) \tag{24}$$

in which M is any thermodynamic property of the mixture and pure component *i*, respectively. The detailed methodology of exergy calculation of the stream can be referred from reference [68]. The equations for the exergy destruction of each component are summarized in Table 4.

To define exergy efficiency, the input and output of the system should be defined. In these integrated systems, output is the summation of the net electrical power generated and stream exergy change of the available hot water, whereas input is the summation of chemical exergy of the feed and fuel entering the system. The exergy efficiencies of the systems are expressed as follows [54]:

$$\eta\_{\text{ex,CH}\_4,\text{sys}} = \frac{\text{P}\_{\text{net, electrical}} + \dot{\text{Ex}}\_{\text{net, hotwater}}}{\dot{\text{Ex}}\_{\text{feed,CH}\_4} + \dot{\text{Ex}}\_{\text{fuel,CH}\_4}} \tag{25}$$

$$\dot{\mathbf{n}}\_{\text{ex,CH4,sys}} = \frac{\mathbf{P}\_{\text{net, electrical}} + \dot{\mathbf{E}} \mathbf{x}\_{\text{net, hotwater}}}{\dot{\mathbf{E}} \mathbf{x}\_{\text{feed,CH4,OH}}} \tag{26}$$

$$
\dot{\mathbf{E}}\mathbf{x}\_{\text{net, steam}} = \dot{\mathbf{E}}\mathbf{x}\_{\text{hot water generated}} - \dot{\mathbf{E}}\mathbf{x}\_{\text{water supplied}}\tag{27}
$$

.


**Table 4.** Equation of exergy destruction and efficiency of the components.
