Exergy Analysis of a Shell and Tube Energy Storage Unit with Different Inclination Angles

Abstract

To optimize the utilization of solar energy in the latent heat thermal energy storage (LHTES) system, this study conducts exergy analysis on a paraffin-solar water shell and tube unit established in the literature to evaluate the effects of different inclination angles, inlet temperatures, original temperatures, and fluid flow rates on the exergy and exergy efficiency. Firstly, the thermodynamic characteristics of the water and the natural convection effects of the paraffin change with different inclination angles. When the inclination angle of the heat storage tank is less than 30°, the maximum exergy inlet rate rises from 0 to 144.6 W in a very short time, but it decreases to 65.7 W for an inclination angle of 60°. When the inclination angle is increased from 0° to 30°, the exergy efficiency rises from 86% to 89.7%, but it decreases from 94% to 89.9% with the inclination angle from 60° to 90°. Secondly, under the condition that the inclination angle of the energy storage unit is 60°, although increasing the inlet temperature of the solar water enhances the exergy inlet and storage and reduces the charging time, it increases the heat transfer temperature difference and the irreversible loss of the system, thus reducing the exergy efficiency. As the inlet water temperature is increased from 83 to 98 °C, the exergy efficiency decreases from 94.7% to 93.6%. Moreover, increasing the original temperature of the LHTES unit not only reduces the exergy inlet and storage rates but also decreases the available work capacity and exergy efficiency. Finally, increasing the inlet water flow rate increases the exergy inlet and storage rates slightly. The exergy efficiency decreases from 95.6% to 93.3% as the unit original temperature is increased from 15 to 30 °C, and it is enhanced from 94% to 94.6% as the inlet flow rate is increased from 0.085 to 0.34 kg/s with the unit inclination angle of 60°. It is found that arranging the shell and tube unit at an inclination angle is useful for improving the LHTES system’s thermal performance, and the exergy analysis conducted aims to reduce available energy dissipation and exergy loss in the thermal storage system. This study provides instructions for solar energy utilization and energy storage.

1. Introduction

Exergy may be used as a measure of meaningful labor in the design, modeling, and performance evaluation of various energy storage systems. Energy analysis only offers data on the amount of energy, but energy quality [1,2,3] is evaluated through the exergy analysis [4,5,6]. Exergy is defined as the maximum available capacity of a unit from a specific beginning state throughout a reversible process when the state is in equilibrium with the environment [7]. Exergy is the most beneficial work that a material or energy flow may produce [8]. Thus, for a given heat source, the exergy efficiency is determined by the available energy and waste heat utilization efficiencies [9]. The primary sources of engineering energy loss may be discovered by examining the exergy and studying the ability of each energy flow to create work [10]. The economic and environmental analyses suggested here can be used with several exergy-based analytical methodologies [11]. The investigation of exergy efficiency as well as economic [12,13,14] and environmental [15] concerns can offer a thorough understanding of how to pick a suitable solution.

The exergy study revealed which system component has the most potential for energy retrofitting [16]. The primary study of a thermal device design must incorporate exergy destruction or entropy generation estimates due to viscous friction and heat transmission [17]. Because of the rise in exergy destruction in the solar energy collector, the system’s exergy efficiency shows a decreasing fluctuation trend [18]. The influence of a turbulent flow condition on energy efficiency and exergy efficiency was studied [19]. When compared to water, the exergy efficiency of the solar energy storage system applying the hybrid nanofluid was improved [20]. Appropriate systematic dimensioning is critical for obtaining maximum exergy efficiency and reducing energy usage [21]. Thermal energy storage (TES) designs for phase change material (PCM) have been developed [22], and systems can recover the needed useful energy during the discharge [23]. The exergy efficiency of the multi-objective working condition was lower than that of the single-objective one, while the energy efficiency was higher [24]. The entropy increases with increasing irreversibility, reducing the system’s ability to be used [25]. Experiments are conducted to investigate the thermophysical characteristics, convective coefficient, entropy amount, and exergy efficiency under turbulent flow [26]. Entropy production refers to the exergy efficiency of a thermal system. Minimum entropy generation is the most important criterion for increasing the system’s exergy efficiency [27]. The effects of Reynolds number [28,29,30], inlet temperature [31], and design parameters [32] on exergy efficiency were investigated. Using the mathematical model [33], the variation laws of exergy loss and transfer coefficient at different testing places of the TES tank throughout the charging process were computed and studied. The model of Ding et al. [34] was used to observe the variation in exergy efficiency as well as the loss of exergy efficiency with different parameters and operating circumstances. According to Si et al. [35], the modeling results demonstrated that the overall thermal efficiency in both energy and exergy generated greater precision. A multi-objective utilization model for thermal systems was developed, using the system’s energy cost minimization and energy efficiency maximization as the optimization functions [36]. In conclusion, sophisticated exergy analysis gives the designer a thorough roadmap for the system’s optimization potential and economic savings [37].

It is crucial to identify the cause of the systematic exergy destruction because of component arrangement and relationships. Exergy analysis, which bases its conclusions on exergy efficiency and thermal performance, is a potent instrument for assessing energy quality. The tube structure played a more significant role in improving the thermal storage performance of the shell and tube LHTES unit [38] compared to the PCM thermal conductivity [39]. The exergy analysis is now studied by adopting the numerical data of the established shell and tube LHTES system [40]. On the one hand, the available work capacity is evaluated in terms of the quality of energy, and, on the other hand, the exergy destruction analysis indicates the entropy generation and the irreversible loss. Figure 1 shows the tube and shell thermal storage unit with four different inclination angles.

In this study, the thermal data of a paraffin-solar water shell and tube unit during the energy storage process with different inclination angles is used to perform exergy and exergy efficiency analyses for the solar energy storage. The sensitivity analysis is conducted by changing operating conditions. A novel way to improve the thermal performance of the LHTES is studied with different inclination angles. It is found that setting the inclination angle of the phase change unit properly is effective in improving the heat convection and thermal performance of the energy storage unit.

2. Mathematical Analysis and Operating Conditions

2.1. Mathematical Model

In this study, the mathematical model used to calculate the exergy rate of the TES unit is shown as follows:

$${\stackrel{·}{Ex}}_{in}={\stackrel{·}{Ex}}_{sto}+{\stackrel{·}{Ex}}_{loss}$$

(1)

The heat transfer fluid (HTF), whose inlet temperature is higher than the original temperature of the PCM, not only transfers heat to the TES unit but also supplies exergy due to the change in its own thermal state. The exergy inlet can be obtained from the calculation:

$${\stackrel{·}{Ex}}_{in}=c_{p,f}{\stackrel{·}{m}}_{{}_f}\left(T_{f,inlet}-T_{f,outlet}-T_o\ln \frac{T_{f,inlet}}{T_{f,outlet}}\right)$$

(2)

where, T_o is the original environment temperature.

The PCM stores exergy as the thermal state deviates from its original value during the temperature rise and phase change process. The exergy storage of the PCM is as follows:

$$\stackrel{·}{Q}=c_{p,f}{\stackrel{·}{m}}_{{}_f}\left(T_{f,inlet}-T_{f,outlet}\right)$$

(3)

$${\stackrel{·}{Ex}}_{sto}=\stackrel{·}{Q}\left(1-\frac{T_o}{T_p}\right)=c_{p,f}{\stackrel{·}{m}}_{{}_f}\left(T_{f,inlet}-T_{f,outlet}\right)\left(1-\frac{T_o}{T_p}\right)$$

(4)

$$E{x}_{sto}={\displaystyle \underset{0}{\overset{t}{\int }}\left(c_{p,f}{\stackrel{·}{m}}_{{}_f}(T_{f,inlet}-T_{f,outlet})(1-\frac{T_o}{T_p})\right)}dt$$

(5)

As the thermal transfer resistance and flowing loss exist, the exergy loss, which indicates the entropy generation of the TES unit, is calculated as follows:

$${\stackrel{·}{Ex}}_{loss}={\stackrel{·}{Ex}}_{in}-{\stackrel{·}{Ex}}_{sto}$$

(6)

As the potential work capacity is measured by the non-equilibrium degree with the original environment, the exergy efficiency is calculated as follows:

$$\psi =1-\frac{{\stackrel{·}{Ex}}_{loss}}{{\stackrel{·}{Ex}}_{in}}=\frac{{\stackrel{·}{Ex}}_{sto}}{{\stackrel{·}{Ex}}_{in}}$$

(7)

The ratio of the exergy loss to HTF exergy inlet is calculated as the entropy generation number [41]:

$$N_s=\frac{E{x}_{loss}}{E{x}_{in}}$$

(8)

The relationship between the exergy efficiency and the entropy generation number is:

$$\psi =1-N_s$$

(9)

The Reynolds number (Re) is defined as the dynamic intensity of the inlet flow. It is calculated as [4]:

$$Re=\frac{u_{f}d}{{\nu }_f}$$

(10)

The Stefan number (Ste) represents the heat transfer driving force between the HTF and the PCM [4]:

$$Ste=\frac{c_p\mathrm{\Delta }T}{H_m}$$

(11)

The boundary of the shell is adiabatic, the tube surface, which conducts convective heat transfer between HTF and PCM, is the coupled boundary, and the original conditions for HTF and PCM are as follows:

$$T_f=T_{inlet},att=0$$

(12)

$$T_p=T_o,att=0$$

(13)

2.2. Operating Conditions

In order to conduct the exergy analysis of a solar thermal storage system, a shell and tube TES unit with multiple operating conditions established by Peng et al. [40] was applied to evaluate the exergy and exergy efficiency for different inclination angles, inlet conditions, and original conditions. The configuration of different inclination angles studied in the reference and this paper is shown in Figure 1, and the variable and constant conditions are listed in

Table 1. Operating conditions in this study.

	Variable Conditions [40]	Constant Conditions [40]	Re	Ste
Effect of inclination angle	Inclination angles of 0°, 30°, 60°, and 90°.	Inlet water temperature of 83 °C, original temperature of 20 °C, and inlet flow rate of 0.085 kg/s.	25,278.7	0.478
Effect of inlet water temperature	Inlet water temperatures of 83, 88, 93, and 98 °C.	Inlet water temperature of 83 °C, original temperature of 20 °C, and inlet flow rate of 0.085 kg/s.	25,278.7	0.478, 0.55, 0.623, 0.695
Effect of original temperature	Original temperatures of 15, 20, 25, and 30 °C.	Inlet water temperature of 83 °C, original temperature of 20 °C, and inlet flow rate of 0.085 kg/s.	25,278.7	0.478
Effect of inlet flow rate	Inlet flow rates of 0.085, 0.17, 0.255, and 0.34 kg/s.	Inlet water temperature of 83 °C, original temperature of 20 °C, and inlet flow rate of 0.085 kg/s.	25,278.7, 50,557.5, 75,836.2, 101,115	0.478

. The variation pattern of the thermal performance and the mechanism of systematic thermodynamic loss are also discussed. Table 1 also displays the different effect cases, and the inlet intensity shown as Re and the heat transfer driving force shown as Ste are calculated to obtain the different working conditions.

3. Results and Discussion

3.1. Effect of Inclination Angle

Figure 2 demonstrates the variation of the exergy inlet rate of the solar water with different inclination angles. When the inclination angle of the shell and tube unit is between 0° and 30°, the exergy inlet trend of the solar water varies similarly, which is different from the inclination angle of 60° and 90°. It shows that the inclination angle varies the heat transfer process of the solar water and the natural convection of the PCM, which influences the transient exergy inlet and thus affects the exergy performance. When the inclination angle of the unit is 0°, the exergy inlet rate rises from 0 to 144.6 W in a very short time, and then it gradually decreases. The maximum exergy inlet rate is 150 W when the inclination angle of the unit is 30°. The exergy inlet rate is only 65.7 W for an inclination angle of 60°, but it decreases slowly. When the inclination angle increases to 90°, the exergy inlet rate is further reduced, and the trend is flatter. The increase in the inclination angle reduces the exergy inlet rate of the solar water and slows down the decreasing trend, which increases the useful energy of the solar water.

Figure 3 shows the exergy storage rate for different inclination angles. The exergy storage, which is influenced by the HTF, shows a similar variation trend. When the inclination angle is between 0° and 30°, the exergy storage rate of the PCM reaches its maximum of 55 W at about 120 min, then the exergy storage rate decreases rapidly, which is because the phase change process ends quickly. The exergy storage is finally completed at 480 min with an inclination angle of 0°. Due to the effect of the inlet exergy, the maximum exergy storage rate is only 23.6 W when the inclination angle is 60°, and the maximum exergy storage rate decreases to 21 W when the inclination angle increases to 90°. The time required for the latent heat TES system to complete the radiation increases to 540 min. It can be found that increasing the inclination angle decreases the exergy storage rate of the PCM and extends the exergy storage process.

Figure 4 displays the exergy of the system for four inclination angles. The exergy is only 354.3 kJ for the inclination angle of 0°, and 395.1 kJ for the inclination angle of 30°. As the inclination angle increases to 90°, the work capacity of the latent heat storage unit is 369.6 kJ, which is more than the inclination angle of 0° and less compared to the inclination angle of 30°. With an inclination angle of 60°, the exergy is 408.2 kJ, which is the maximum value that can be achieved for four inclination angles. The inclination angle affects the exergy that can be stored by the TES system.

Figure 5 shows the exergy efficiency for different inclination angles. For an inclination angle of 0°, the exergy efficiency is 86%, and the time required to complete the variation is 480 min. For an inclination angle of 30°, the exergy efficiency is 89.7%, and it takes 340 min to complete the process. When the inclination angle is 60°, the system obtains the exergy efficiency of 94%, but the time is extended to 540 min. As the inclination angle is 90°, the exergy efficiency of the system decreases to 89.9%, but the change time is also shortened to 520 min. When the inclination angle is less than 60°, increasing the inclination angle of the shell and tube unit increases the exergy efficiency. However, when the inclination angle is further increased, the exergy storage and inlet rate of the system decreases significantly, the required time increases, and the second law efficiency decreases, which affects the performance of the TES system. The exergy analysis of the shell and tube unit with different inclination angles indicates a novel approach for the reasonable arrangement of the solar thermal storage system.

It can be seen from Figure 2 and Figure 3 that HTF obtains solar energy to increase its own temperature and deviate from the original state, thereby being able to produce usable energy. The exergy inlet rate of HTF is much higher than the exergy storage rate of the PCM at the beginning, because HTF has the ability to change the surrounding environment and provide the exergy accumulated by the PCM. As HTF continuously delivers available energy to the PCM, the exergy storage rate of the PCM shows delay and peak attenuation compared to the exergy inlet. However, the thermal state of the PCM continues to approach, and the exergy of the system is continuously produced. In Figure 4, the exergy of the system eventually reaches a certain value. After the charging process is completed, as shown in Figure 5, the exergy efficiency is calculated to analyze the effectiveness of available energy transfer and storage. The changes in exergy and exergy efficiency vary greatly under different inclination angles, but changing the inlet and initial conditions, such as Re and Ste, can also have a positive or negative impact on the thermal performance of the system.

3.2. Effect of Inlet Water Temperature

Figure 6 shows the effects of fluid inlet temperatures on the exergy inlet in the charging process when the inclination angle of the phase change unit is 60°. It is found that increasing the temperature of the solar water significantly enhances the exergy inlet rate and reduces the time required for the HTF to supply the inlet exergy. As is shown in the diagram, the maximum exergy inlet rate is 58.5 W for an inlet temperature of 83 °C, and the time for maintaining the peak value is about 140 min. When the water temperature is 88 °C, the maximum exergy inlet rate of heat transfer fluid increases to 64.7 W, and the time for maintaining the peak value is shortened by 20 min. Increasing the inlet temperature to 93 and 98 °C will increase the maximum exergy inlet rate to 82.5 and 106.6 W, respectively. Increasing the inlet fluid temperature in the system will significantly improve the exergy inlet rate, which accelerates the heat exchange process.

Figure 7 depicts the effects of different water temperatures on the exergy storage rate of the PCM for an inclination angle of 60°. In general, as the fluid temperature increases, the exergy storage rate increases, and the maximum exergy storage rate is advanced. Increasing the water temperature from 83 to 98 °C will increase the maximum exergy storage rate from 21.1 to 29.7 W. The charging time is reduced from 600 to 420 min, which greatly facilitates the progress. The increase in the inlet temperature significantly enhances the heat transfer drive force, which promotes the convection intensity between the fluid and the paraffin through the tube wall.

The variation of the exergy for different inlet temperatures is shown in Figure 8. It can be seen that increasing the fluid temperature from 83 to 88 °C increases the exergy from 385.0 to 408.3 kJ. Increasing the fluid temperature from 88 to 93 °C expands the maximum exergy by 20.3 kJ. Raising the fluid temperature by 5 °C further increases the exergy by 23.2 kJ. It can be found that as the fluid temperature increases, the available energy of the system increases, and the variation trend grows. Increasing the fluid temperature causes a significant effect on the energy storage density because it promotes the effective thermal exchange and the PCM final temperature, expanding the deviation from the original state and thus increasing the thermal potential.

Figure 9 demonstrates the exergy efficiency for different inlet temperatures. As shown in the diagram, increasing the fluid temperature reduces the time to complete the thermal storage, thus reducing the variation process of exergy efficiency. For the inlet temperature of 83 °C, the time taken to complete the charging process is 600 min, but the system has the maximum exergy efficiency of 94.7%. Increasing the fluid temperature to 88 °C reduces the time to 540 min, but also reduces the exergy efficiency to 94%. Raising the fluid temperature further reduces the required time. When the fluid temperature is 98 °C, the time required for the system to complete the process is 460 min, and the exergy efficiency is 93.6%. Although increasing the fluid temperature improves the heat transfer process, it will also increase the irreversible energy loss and reduce the exergy efficiency.

3.3. Effect of Original Temperature

Figure 10 shows the effects of unit original temperatures on the exergy inlet rate when the inclination angle is 60°. The figure shows that increasing the original temperature of the unit reduces the time necessary for the solar water to supply the inlet exergy but also decreases the exergy inlet rate. The maximum exergy inlet rate is 75.7 W when the system’s original temperature is 15 °C. When the system’s original temperature is raised to 20 °C, the exergy inflow rate declines under the same inlet conditions, with a maximum value of 70.1 W. When the temperature is raised to 25 °C, the exergy inflow is reduced even further, and the maximum rate is 65.7 W. When the original temperature is 30 °C, the system has the lowest exergy inlet rate with a maximum value of 55.5 W. Increasing the fluid temperature reduces the heat transfer temperature difference between the solar water and the paraffin, which weakens the thermal transfer intensity and decreases the exergy inlet rate.

Figure 11 depicts the variation of the exergy storage rate for different original temperatures. As the original temperature of the unit increases, the exergy storage rate decreases due to the influence of the charging fluid. Although increasing the original temperature reduces the charging time, it decreases both the exergy storage rate and the exergy. As a result, raising the initial temperature by 15 °C decreases the maximum exergy storage rate of the phase change material from 29.9 to 18.0 W.

Figure 12 demonstrates the change of the exergy with different original temperatures. From the figure, increasing the original temperature decreases the total available energy gradually. When the PCM temperature is 15 °C, there is a temperature transfer difference between the water and the paraffin, and the system deviates from the original condition after completing the heat storage mostly, so the system maintains the maximum exergy, which is 610.5 kJ. As the PCM temperature is 20 °C, the thermal potential of the unit is weakened, and the exergy is 500 kJ. The original temperature is increased to 30 °C, the exergy is only 275.5 kJ. With the increase in the PCM original temperature, the potential capacity of the system is weakened, the available energy is reduced, and the reduction trend extends.

Figure 13 reflects the variation of the exergy efficiency with different original conditions. Increasing the original temperature decreases the charging time and decreases the exergy efficiency, which has a negative influence on the thermal performance. The exergy efficiency is 95.6% when the PCM temperature is 15 °C. The system’s exergy efficiency is 94.5% as the original temperature is raised to 20 °C. The exergy efficiency drops to 94% when the original temperature is 25 °C. As the original temperature is 30 °C, the system’s exergy efficiency is 93.3%. Increasing the original temperature of the latent heat unit reduces the useful energy and decreases the heat density of the PCM.

3.4. Effect of Inlet Flow Rate

To investigate the effects of inlet conditions on the thermal performance, the fluid flow rate of the solar water is changed when the inclination angle is 60°. Figure 14 shows the variation of the exergy inlet rate for different inlet flow rates. Although increasing the fluid rate enhances the Reynolds number and thus increases the convective intensity in the tube, increasing the flow rate improves the exergy inlet rate slightly for an inclination angle of 60°. When the fluid flow rate is 0.085 kg/s, the maximum exergy inlet rate is 65.7 W. As the flow rate is 0.170 kg/s, the maximum exergy inlet rate increases to 68.9 W. As the fluid rate rises to 0.255 and 0.340 kg/s, the values are 73.3 and 77 W, respectively.

Figure 15 depicts the variation of the exergy storage for different fluid flow rates. Due to the influence of the exergy inlet, the variation of the exergy storage rate of the paraffin is similar. When the fluid flow rate is 0.085 kg/s, the exergy storage rate is 23.6 W. When the fluid flow rate is 0.170 kg/s, the value increases to 27.7 W. As the fluid rate is 0.255 and 0.340 kg/s, the maximum exergy storage rate of the heat exchanger material stabilizes at about 27 W.

Figure 16 shows the effects of fluid flow rates on the exergy of the system. It can be found from the diagram that increasing the fluid rate from 0.085 to 0.340 kg/s has little influence on the work capacity, and the exergy is about 410 kJ.

Figure 17 demonstrates the variation of the exergy efficiency with different fluid flow rates. As can be seen, enhancing the fluid rate of the solar water improves the exergy efficiency, and the exergy efficiency is enhanced from 94% to 94.6%. When the inclination angle is 60°, the effect of gravity on the heat exchange process is prominent, while changing the fluid flow rate affects the exergy slightly.

3.5. Future Work

Through the exergy analysis of the low-temperature paraffin-solar water unit, it was found that the inclination angle plays a prominent role in optimizing the thermal performance. Therefore, in future works, a complete experimental platform and computational model will be established to conduct in depth research on medium and low temperature thermal storage systems with different inclination angles. The optimal inclination angle of the phase change unit will be found and the thermal performance during the charging and discharging cycles will be evaluated.

4. Conclusions

In this study, the mathematic analysis of the shell and tube unit with different inclination angles in a TES system established in the literature is conducted to evaluate the exergy performance.

The effects of variable inclination angles, inlet temperatures, original temperatures, and inlet fluid rates on the exergy rate and exergy efficiency are evaluated. As a result, there is an effective way to optimize the thermal efficiency of the solar energy system with different inclination angles, and the main conclusions are shown below.

The inclination angle affects the inlet exergy of the fluid and the exergy storage of the PCM due to the effects of energy transfer and natural convection. When the inclination angle is greater than 0° and less than 60°, increasing the inclination angle improves the exergy performance. However, when the inclination angle is greater than 60° but less than 90°, increasing the inclination angle reduces the exergy efficiency.

As the inclination angle of the heat storage tank is 60°, increasing the inlet temperature of the solar water improves the exergy rate and reduces the time required for the exergy storage. The heat transfer driving force, which is shown as the Stefan number, is promoted accordingly. However, it increases the heat transfer temperature difference and the irreversible energy loss, so that the exergy efficiency is reduced. Moreover, increasing the original temperature of the tank not only reduces the tube temperature difference between the fluid and the paraffin but also decreases the exergy inlet and storage rates. As a result, it decreases the work capacity and exergy efficiency. Raising the water rate enhances the exergy inlet and storage rates of the system slightly, and the improvement in the exergy performance is minimal.

Utilizing the inclination angle of the phase change unit is a novel way to improve the heat convection and thermal performance of the energy storage system. The exergy analysis of the paraffin-solar water shell and tube unit in the latent heat TES system indicates that the systematic exergy efficiency is related to the working conditions, inlet and original conditions, geometric features, and material parameters. Different operating processes have either positive or negative effects on the thermal efficiency. Setting the working conditions properly can not only achieve energy savings and cost reduction but also optimize the thermal performance and the operating efficiency.