*2.2. Energy Analysis*

The energy analysis of the modified solar thermal power generation cycle is applied by considering the energy balance for all of the units of the investigated cycle. For the energy analysis, a number of usual assumptions considered in literature studies [4,5,12,29,30], are also considered here.


According to these common assumptions, by applying the first law of thermodynamics to all of the equipment of the investigated cycle, energy analysis is considered.

Equations (1)–(3) show the applied energy balances for the turbine, condenser, and evaporator, respectively.

$$\left| \dot{\mathcal{W}}\_s \right| = \dot{m}\_r (h\_1 - h\_2) = \dot{m}\_r (h\_{1'} - h\_2) \tag{1}$$

where *h*<sup>1</sup> and *h*<sup>2</sup> are the specific enthalpies of the inlet and outlet streams of the turbine, respectively. Additionally, *h*<sup>1</sup> and *h*1 are considered for the day and night, respectively. . *mr* and . *Ws* are the mass flow rate of the working fluid and the produced power of the Rankine cycle, respectively.

For the condenser,

$$\left| \dot{Q}\_{\mathcal{L}} \right| = \dot{m}\_r (h\_2 - h\_3) \tag{2}$$

where *h*<sup>2</sup> and *h3* are the specific enthalpies of the inlet and outlet streams of the condenser, respectively. . *Qc* is the desorbed heat from the working fluid of the Rankine cycle.

For the evaporator, which is used only during the day,

$$
\dot{m}\_w(h\_6 - h\_5) = \dot{m}\_r(h\_1 - h\_4) \tag{3}
$$

where . *mw* represents the mass flow rate of the heating fluid (water), and *h*<sup>6</sup> and *h*<sup>5</sup> are the specific enthalpies of the inlet and outlet heating working fluid streams (water) of the evaporator, respectively. *h*<sup>4</sup> and *h*<sup>1</sup> are the specific enthalpies of the inlet and outlet Rankine cycle working fluid streams (R134a) of the evaporator, respectively.

For the PCM tank, the energy balance is investigated separately for day and night.

A. *During the Day* The PCM tank is charged during the day by absorbing heat from the heating fluid (water). Accordingly, the energy balance of the PCM tank during the day follows Equation (4).

$$Q\_{\rm PCM,day} = m\_{\rm PCM} \Delta h\_{\rm fus,PCM} = t\_{\rm charging} \dot{m}\_w (h\_{10} - h\_7) \tag{4}$$

where *mPCM* is the mass of the PCM, Δ*hf us*,*PCM* is the PCM enthalpy of fusion, and *tcharging* is the charging time in the day, equal to 12 h. *QPCM*,*day* is the heat absorbed by the PCM from the heating fluid (water) during the day. Streams 4 and 1 are shut down during the day and the only inlet and outlet streams of the PCM tank are Streams 10 and 7, whose specific enthalpies are shown as *h*<sup>10</sup> and *h*7.

B. *During the Night* The PCM tank is discharged during the night by desorbing heat to the Rankine cycle working fluid (R134a). Therefore, the energy balance of the PCM tank during the night follows Equation (5).

$$\left|Q\_{\text{PCM},\text{night}}\right| = m\_{\text{PCM}} \Delta h\_{\text{fus},\text{PCM}} = t\_{\text{discharge}} \dot{m}\_r \left|h\_{1'} - h\_{4'}\right|\tag{5}$$

where *tdischarging* is the discharging time during the night, equal to 12 h. *QPCM*,*night* is the desorbed heat by the PCM to the Rankine cycle working fluid (R134a) during the night. Streams 10 and 7 are shut down at night, therefore, the only inlet and outlet streams of the PCM tank are Streams 4 and 1- , with specific enthalpies of *h*4 and *h*1-, respectively.

For the water tank, which is used only during the day, the energy balance is,

$$m\_{\overline{w}} \mathbb{C}\_{p\_{\overline{w}}} \frac{dT\_{\overline{w}}}{dt} = \dot{Q}\_{\overline{s}} + \dot{m}\_{\overline{w}\mathfrak{g}} (h\_{\overline{s}} - h\_{\overline{\mathfrak{g}}}) \tag{6}$$

where *h*<sup>8</sup> and *h*<sup>9</sup> are the specific enthalpies of the inlet and outlet streams of the water tank, respectively. . *Qs* is the collected solar energy. . *mw*<sup>9</sup> is the mass flow rate of Stream 9 and based on the proposed assumptions, it is twice the mass flow rate of Streams 10 (or 6). Therefore,

$$
\dot{m}\_{w\_9} = \mathcal{Z}\dot{m}\_w\tag{7}
$$

In Equations (6) and (7), . *mw* and *Tw* are the total mass and the temperature of water in the water tank, respectively, and *Cpw* is the heat capacity of water. In this study, it is assumed that the collected solar energy is controlled carefully using controlling collectors, therefore, the water tank during the day is at a thermal steady state. In this way, the unsteady state term of Equation (6) can be neglected. This assumption is, in fact, easily obtainable because during the day, the amount of collected solar energy which is transferred to the water tank is controlled in a way to keep the water at its boiling point, and since a pure component boils at a constant temperature, the temperature of water in the water tank remains constant. In this way, there is no temperature change in the water tank. So, during the day, Equation (6) can be simplified as follows.

$$
\dot{Q}\_s = \dot{m}\_{w\_\theta} (h\_\theta - h\_8) \tag{8}
$$
