Exergetic Assessment and Computational Modeling of a Solar-Powered Directly-Coupled Air Conditioning System: An Application in Library Cooling

The escalating global energy demand necessitates a shift towards renewable energy resources, a paramount determinant of social, economic, and cultural sustainability within urban environments and broader societies [1]. This critical transition to sustainable energy resources is inextricably linked to the concept of sustainable development. This dependency underscores the importance of enhancing energy efficiency in processes that leverage sustainable energy resources [2].

Air conditioning systems, integral for maintaining comfortable and healthy living conditions in regions characterized by severe temperatures or high humidity, contribute significantly to the total energy consumption in buildings [3, 4]. An effective approach to alleviating this energy consumption challenge involves powering these air conditioning systems using photovoltaic solar energy. This strategy transforms solar energy into electricity through photovoltaic solar panels, powering the compressor within the air conditioning system.

Belém, a city situated within the Amazon region, is subject to a typically hot, humid, equatorial climate due to its proximity to the Equator. Consistent high temperatures throughout the year, along with substantial precipitation and humidity, result in minimal variation between daytime and nighttime temperatures. The immense levels of solar radiation incident on buildings, combined with external temperatures exceeding 30℃, often culminate in conditions of thermal discomfort [5].

In contrast to the majority of Brazilian households where electricity is ubiquitously accessible, in many Amazonian localities, electricity - a key facilitator of productive, educational, and leisure activities - is absent, impacting individuals across all age groups [6]. Approximately 425,000 families are bereft of electricity access, rendering the investigation and deployment of photovoltaic solar energy-powered air conditioning systems of considerable interest in various regional locations.

Solar refrigeration, a specific application of renewable energy conversion systems, uses technologies underpinned by photovoltaic solar energy. This application is particularly beneficial as cooling demand tends to align with periods of maximal solar irradiance [7, 8]. The desirability of cooling is significantly magnified in hotter climates compared to cooler regions.

The most prevalent metric for evaluating the efficiency of vapor compression refrigeration systems is the energy efficiency ratio, adjusted to a coefficient of performance [9]. However, exergy analysis, a thermodynamic evaluation technique aimed at assessing energy quality within a system, may be more suitable for ascertaining maximum system performance. This method, used to identify irreversible energy losses in a process, can pinpoint prospective improvement opportunities throughout the system, providing a more realistic perspective of the process [10, 11].

Exergy analysis of an air conditioning system can be conducted by scrutinizing individual system components, identifying the primary exergy destruction points and directing them towards potential improvement areas. Both theoretical and experimental studies have been conducted on air conditioning systems powered by photovoltaic generation.

In terms of coupled systems' behavior, Bilgili [12] theoretically investigated an electric-solar refrigeration system using indirect coupling and battery bank for energy storage. In contrast, Aguilar et al. [13] experimentally examined a photovoltaic energy-powered air conditioning unit, supplemented by conventional energy sources when photovoltaic energy was insufficient.

For numerical analysis and simulation methods, Salilih and Birhane [8] proposed a method for analyzing and simulating a solar-powered vapor compression refrigeration system with a variable-speed compressor directly coupled to the photovoltaic generator. They demonstrated that the hourly variation of exergy mirrors the input power profile of the generator.

Ahamed et al. [14] utilized a thermodynamic analysis better suited for investigating conventionally powered thermal systems' efficiency. They found that exergy depends on evaporation, condensation, subcooling, compressor pressure, and ambient temperature. Anand and Tyagi [15] presented a detailed experimental analysis of the vapor compression refrigeration cycle for different refrigerant charge percentages, while Bayrakçi and Özgür [9] conducted an analysis varying the refrigerant type used.

For combined systems with thermal storage, Mosaffa et al. [16] studied a system comprising latent heat thermal storage and vapor compression refrigeration. They performed an advanced exergy analysis, demonstrating that all unit's exergy destruction is exclusively due to its irreversibilities.

In terms of system performance, Sogut [17] evaluated the exergy and environmental performance of commercially sold air conditioners, while Aman et al. [18] conducted an energy and exergy analysis to evaluate the performance and potential of a solar-powered water and ammonia absorption refrigeration system for residential air conditioning applications.

Although numerous studies have been conducted on solar-powered refrigeration systems, few consider the direct coupling of the air conditioning system with the photovoltaic system. This study addresses this gap, focusing on systems that operate based solely on the availability of the solar resource, without energy storage and auxiliary sources for support.

Moreover, most studies on solar cooling did not employ an exergy analysis to indicate inefficiencies in various components of the air conditioning system and the conversion of solar energy into thermal energy for cooling. The identification of the components contributing most to exergy destruction can guide optimization efforts.

This research represents a significant contribution to the field of air conditioning systems coupled to photovoltaic generators, without energy storage, targeting to meet a small library's cooling demand in the Amazon region. The study employs an exergy analysis approach through computational modeling, examining system performance under different solar irradiance profiles. This analysis provides valuable information to improve efficiency, optimize system design and operation, and guide research and development efforts towards more efficient and sustainable systems.

3.1 Meteorological data

The meteorological data used in this work were obtained in the city of Belém - PA, in the Brazilian Amazon region, whose coordinates are: 1º 27' S, longitude 48º 48' W, and altitude of 16 m, average wind velocity of 1.5 m/s. Data were obtained through a monitoring system (Datalogger) that collects data measured in a meteorological station located in GEDAE, using the following instrumentation: external temperature and relative humidity sensor, model HC2S3, and Kipp & Zonen model CMP6 pyranometer. Irradiance and outside air temperature data for typical sunny and cloudy days are used in this study. Measurements were taken at 10-minute intervals, following the methodology used by Santos et al. [19].

3.2 Modeling the photovoltaic generator

To obtain the I-V characteristic curves of the photovoltaic generator for different irradiance profiles was used the equivalent circuit model shown in Figure 2 [20].

2.png

Figure 2. Equivalent circuit of a photovoltaic cell, including the series and parallel resistances

The cell output current is given by Eq. (1). Bana and Saini [21]

$I=I_{P V}-I_0\left[\exp \left(\frac{V+R_S I}{V_t a}\right)-1\right]-\frac{V+R_S I}{R_p}$                 (1)

where, R_s and R_p are the equivalent resistances, in series and parallel, respectively, a is the ideality constant of the diode, in the range 1≤a ≤1.5, Carrero et al. [22], being the approximate value of 1.3 for polycrystalline silicon and V_t is the thermal strain of the array, given by Eq. (2).

$V_t=\frac{N_s k T}{q}$             (2)

where, N_s is the number of cells connected in series, k is the Boltzmann constant (1.3806503×10⁻²³ J/K), T is the p-n junction temperature, and q is the electron charge (1.60217646 × 10⁻¹⁹ C).

I_PV is the current generated by the light, in amperes, in the standard test condition (T_c,ref= 25℃ e G_ref = 1000 W/m²) and is calculated by Eq. (3).

$I_{P V}=\left(I_{P V, n}+K_I \Delta T\right) \frac{G}{G_{r e f}}$                 (3)

I_PV,n can be calculated as I_PV,n=[(R_p+R_s/R_p]I_sc,n where I_sc,n is the short circuit current, given in Table 1 and $\Delta T=T_c-T_{c, r e f}$, where the solar cell temperature can be obtained from the ambient temperature and is presented in Odeh et al. [23] as shown in Eq. (4)

$T_c=T_a+\frac{G}{800 \mathrm{~W} / \mathrm{m}^2}\left(T_{\text {NOCT }}-20^{\circ} \mathrm{C}\right)$                  (4)

T_a is the measured ambient temperature, G is the in-plane irradiance of the generator, and T_NOCT is the nominal operating temperature of the solar cell, usually provided by PV module manufacturers.

The saturation current I₀ is calculated from Eq. (5). Villalva et al. [20].

$I_0=\frac{I_{s c, n}\,\,+\, K_I \Delta T}{\exp \left(\left(V_{o c, n}\,\,+\,K_V \Delta T\right) / a V_t\right)-1}$                    (5)

K_I and K_V are current and voltage coefficients. V_oc,n is the open circuit voltage, found in Table 1.

It is possible to calculate the resistances, R_s and R_p, from an iterative solution proposed by Villalva et al. [20], where the relationship between R_s and R_p, which are the only unknowns in Eq. (1), can be found by making the maximum power calculated by the I–V model of Eq. (1) (W_max,m) equal to the maximum experimental power, found in the module datasheet (W_max,e), at the maximum power point MPP, and solving the resulting equation for R_s as shown in Eqs. (6) and (7).

$\begin{gathered}\dot{W}_{\text {máx }, m}=V_m\left\{I_{P V}-I_0\left[\exp \left(\frac{q}{k T} \frac{V_m+R_s I_m}{a N_s}\right)-1\right]-\right. \left.\frac{V_m+R_s I_m}{R_p}\right\}=\dot{W}_{\text {max }, e}\end{gathered}$             (6)

where, I_m and V_m are the current and voltage at the maximum power point.

$R_p=\frac{V_m\left(V_m+I_m R_s\right)}{\left\{V_m I_{P V}-V_m I_0 \exp \left[\frac{\left(V_m \, +\, I_m \,\, R_s \,\,\right) q}{N_s a} k T\right]+V_m I_0-W_{\text {max },e \,\,\,}\right\}}$          (7)

According to Pinho and Galdino [24], it is necessary to associate the modules in series or parallel to obtain voltage and current levels suitable for the operation. Similar devices subject to the same irradiance, connected in series, are summed the output voltages while the current remains unchanged, according to Eqs. (8) and (9). In this work, were simulated eight photovoltaic modules connected in series.

$V=\sum_{i=1}^n V_i$            (8)

$I=I_1=I_2=\cdots=I_n$              (9)

3.3 Determination of the maximum power of the generator

A model to determine the maximum power capable of being supplied by a PVG under a given operating condition is suggested by Rawat et al. [25], as shown in Eq. (10).

$\dot{W}_{P V}=P_1 G\left[1+\gamma_m\left(T_c-T_{c, r e f}\right)\right]$               (10)

where, γ_m corresponds to the temperature coefficient at the point of maximum power; T_c and T_c,ref are the solar cell temperatures under actual and standard test conditions (STC), respectively; G is the irradiance on the generator plane; P₁ is the ratio between PVG rated power and the irradiance in the STC, that is, 1,000 W/m², as shown in Eq. (11).

$P_1=\frac{P_{P V}^0}{G_{r e f}}$              (11)

This model was chosen due to its employability in photovoltaic energy engineering, perfectly meeting the purpose of this work. Furthermore, it considers the two main parameters that affect the PVG output power, the incident irradiance in the generator plane and the temperature of the solar cell.

The power generated by the photovoltaic system ($\dot{W}_{P V}$) is delivered to the set (DC-DC + ESC converter), whose efficiency was equal to 95%. It then activates the ACS compressor electric motor, providing mass flow variation of the refrigerant fluid according to the variation of the power generated.

3.4 Mathematical formulation for exergetic analysis of the photovoltaic system – PV

According to Bayrak et al. [26], the electrical exergy of the photovoltaic system aims to use the present energy as useful energy, where the exergy analysis takes into account the energy quality or capacity and can be expressed by a general balance of photovoltaic exergy, as presented in Eq. (12).

$\sum \dot{E} x_{\text {in }}=\sum \dot{E} x_{\text {out }}+\sum \dot{E} x_{\text {loss }}+\sum \dot{E} x_d$               (12) 

where, the input exergy of a photovoltaic system includes only the solar irradiance intensity exergy, as shown in Eqs. (13) and (14) (Bayrak et al. [26] and Petela [27]).

$E x_{i n}=G A\left[1-\frac{4}{3}\left(\frac{T_a}{T_s}\right)+\frac{1}{3}\left(\frac{T_a}{T_s}\right)^4\right]$               (13)

Or

$\dot{E} x_{i n}=\left(1-\frac{T_a}{T_s}\right) G A$             (14)

in which T_s is the temperature of the sun, considered as 5,777 K.

An expression, see Eq. (15), for the output exergy was presented by Bayrak et al. [26] and Joshi et al. [28].

$\dot{E} x_{o u t}=\dot{E} x_{e l}-\dot{E} x_{\text {ther }}$                 (15)

Electrical exergy can be expressed according to Eq. (16), presented in Pandey et al. [29].

$\dot{E} x_{e l}=V_{o c} I_{s c}-\dot{E} x_{d, e l}$             (16)

The exergetic loss of electrical energy is presented by Eq. (17).

$\dot{E} x_{d, e l}=V_{o c} I_{s c}-V_m I_m$                (17)

The photovoltaic system thermal exergy, $\dot{E} x_{\text {ther }}$, is constituted by the environment heat loss on the surface of the photovoltaic generator, which can be expressed as shown in Eq. (18) [26, 28, 29].

$\dot{E} x_{t h e r}=\left(1-\frac{T_a}{T_c}\right) \dot{Q}$                  (18)

in which $\dot{Q}$ is given by Eq. (19).

$\dot{Q}=h_{c a} A\left(T_c-T_a\right)$             (19)

The heat transfer coefficient h_ca (convective and radiative) from the solar cell to the environment can be calculated considering the wind velocity (v), the air density, and the surrounding conditions as shown in Eq. (20) [30].

 $h_{c a}=5,7+3,8 v$                  (20)

Eqs. (15) and (16) include internal and external energy losses, where internal losses are the destruction of electrical exergy, that is, $\dot{E} x_{d, e l}$ and external losses are heat loss, $\dot{E} x_{d, \text { ther }}$ which is numerically equal to $\dot{E} x_{\text {ther }}$ for the PV system [29]. Thus, the loss of exergy in the photovoltaic system is given by Eq. (21), in which the decrease in energy quality is called exergy destroyed or irreversibility.

$\dot{E} x_{d, P V}=\dot{E} x_{d, e l}+\dot{E} x_{t h e r}$                 (21)

Using Eqs. (16)-(20) the output exergy of the PV system can be calculated by Eq. (22).

$\dot{E} x_{o u t}=V_m I_m-\left(1-\frac{T_a}{T_c}\right) h_{c a} A\left(T_c-T_a\right)$                (22)

The exergy efficiency of the PV module is described as the ratio of the total output exergy to the total input exergy, presented by Pandey et al. [29], Bayrak et al. [26] and Joshi et al. [28], see Eqs. (23) and (24).

$\eta_{E x, P V}=\frac{\dot{E} x_{o u t}}{\dot{E} x_{i n}}$            (23)

$\eta_{E x, P V}=\frac{V_m I_m-\left(1-\frac{T_a}{T_c}\right) h_{c a} A\left(T_c-T_a\right)}{\left(1-\frac{T_a}{T_s}\right) G A}$              (24)

3.5 Mathematical formulation for exergetic analysis of the vapor compression air conditioning system

Some mathematical formulations are necessary to carry out the analyses, both of energy and exergy for the refrigeration cycle by vapor compression.

The vapor compression system has four main components: evaporator, compressor, condenser, and expansion valve. External energy (power) is supplied to the compressor, and heat is added to the system in the evaporator. In the condenser, heat is rejected from the system.

Exergy losses in various system components are not equal, as shown by Ahamed et al. [14], and Anand and Tyagi [15]. Reference temperature and pressure are denoted by T₀ and P₀, respectively, and are shown in Table 3.

Table 3. Air conditioning system operating conditions

3.png

Figure 3. Diagram T-s (Temperature-entropy) for the actual steam compression refrigeration cycle

Exergy is consumed or destroyed due to the entropy created as a result of associated processes, according to Sahin et al. [31]. To specify the exergy losses or destructions in the system, in this study, the assumptions considered for the PV-ACS engineering model are:

1. Steady-state conditions are maintained on all components;

2. Pressure losses in the pipes are neglected;

3. Heat gains and losses from or to the system are not considered;

4. Kinetic and potential energy losses are not considered.

In air conditioning systems, heat exchange can be explained based on the T-s diagram shown in Figure 3, where 1-2 is the isentropic compression in the compressor, 2-3, 3-4, and 4-1 are condensation, throttling in the expansion valve and evaporation in the evaporator, respectively.

During a process, the change in the exergy of a system is equal to the difference between the total exergy transfer across the system boundaries and the exergy destroyed within the system boundaries due to irreversibility or entropy production [32]. The overall exergy balance for any type of system and any process is expressed by Eq. (25).

$\dot{E} x_{\text {in }}-\dot{E} x_{\text {out }}-\dot{E} x_d=\Delta E_{\text {system }}$               (25)

where, $\dot{E} x_{\text {in }}$ is the amount of exergy entering a system, $\dot{E} x_{\text {out }}$ is the amount of exergy leaving a system, and $\dot{E} x_d$ is the amount of exergy being destroyed within a system's boundaries. For systems with the steady-state flow, the exergy balance can be expresse Eq. (26).

$\begin{gathered}\dot{E} x_{\text {heat }}-\dot{E} x_{\text {work }}+\dot{E} x_{m a s s, i n}-\dot{E} x_{m a s s, o u t}- \dot{E} x_d=0\end{gathered}$                    (26)

Eq. (26) can be expressed in more detail, according to Eq. (27).

$\sum\left(1-\frac{T_0}{T_k}\right) \dot{Q}_k-\dot{W}+\dot{m}\left(\psi_1-\psi_2\right)-\dot{E} x_d=0$            (27)

The mathematical formulation for analysis of exergy in different components is performed according to Ahamed et al. [14], Anand and Tyagi [15], and Bayrakçi and Özgür [9].

The specific exergy, in any state, can be given by Eq. (28).

$\psi=\left(h-h_0\right)-T_0\left(s-s_0\right)$                    (28)

The exergy destroyed of the system components is obtained with the exergy balance (availability) described by Eq. (29).

$\dot{E} x_d=\dot{E} x_{i n}-\dot{E} x_{o u t}$                 (29)

Considering Eqs. (28) and (29) it is possible to write the first and second laws of thermodynamics for each component of the system.

For the evaporator: The cooling capacity is given by Eq. (30).

$\dot{Q}_L=\dot{m}\left(h_1-h_4\right)$                (30)

in which: h₁ (kJ/kg) and h₄ (kJ/kg) are the specific enthalpies of states 1 and 4.

The exergy destroyed is given in Eq. (31).

$\dot{E} x_{d, e v}=\dot{m}\left[\left(h_4-h_1\right)-T_0\left(s_4-s_1\right)\right]+\dot{Q}_L\left(1-\frac{T_0}{T_L}\right)$             (31)

where, T_L is the temperature of the cold (conditioned) environment.

For the compressor: Compressor power is obtained from Eq. (32).

$\dot{W}_{\text {comp }}=\dot{m}\left(h_2-h_1\right)$             (32)

For a real refrigeration cycle, the specific enthalpy at the compressor output, state two, in Figure 3, is given by Eq. (33).

$h_2=h_1+\frac{h_{2 s-} h_1}{\eta_{C I}}$             (33)

The isentropic compression efficiency η_CI is calculated in Eq. (34) [32].

$\eta_{C I}=0.874-0.0135\left(\frac{P_{c o n d}}{P_{e v}}\right)$               (34)

As presented by Özgoren et al. [33], P_ev and P_cond are the evaporating and condensing pressures. For the R410A used in the equipment under study, the P_ev assumes the value of 10.85 bar, for T_ev equal to 10℃, and the P_cond variable from 20.94 bar to 25.02 bar for T_cond ranging from 35.41℃ to 42.8℃.

The electrical power is given by Eq. (35).

$\dot{W}_{e l}=\frac{\dot{W}_{c o m p}}{\eta_{C M} \,\, \eta_{e l}}$               (35)

The exergy destroyed in the compressor is calculated by Eq. (36).

$\dot{E} x_{d, \text { comp }}=\dot{m}\left[\left(h_1-h_2\right)-T_0\left(s_1-s_2\right)\right]+\dot{W}_{e l}$            (36)

For the condenser: The condensing capacity is given by Eq. (37).

$\dot{Q}_H=\dot{m}\left(h_2-h_3\right)$                 (37)

The exergy destroyed in the condenser is given by Eq. (38).

$\dot{E} x_{d, \text { cond }}=\dot{m}\left[\left(h_2-h_3\right)-T_0\left(s_2-s_3\right)\right]-\dot{Q}_H\left(1-\frac{T_0}{T_H}\right)$                (38)

where, T_H=T_a

For the expansion valve: The exergy destroyed is given by Eq. (39).

where, h₄=h₃,

$\dot{E} x_{d, \exp }=\dot{m}\left(s_4-s_3\right)$          (39)

Coefficient of Performance for the refrigeration cycle

The refrigeration cycle performance coefficient (COP) is the relationship between the evaporator cooling capacity and the compressor's electrical energy consumption, given by Eq. (40).

$C O P=\frac{\dot{Q}_L}{\dot{W}_{e l}}$                  (40)

And the exergy efficiency is calculated by Eq. (41) [8].

$\eta_{E x, A C S}=\frac{\dot{X}_{Q_L}}{\dot{W}_{e l}}$                      (41)

where, $\dot{X}_{Q_L}$ is the rate of positive exergy associated with the removal of heat from the low-temperature medium, which can be obtained by multiplying the Carnot factor ((T₀-T_L))⁄T_L by the cooling capacity $\dot{Q}_L$, where T₀, is the reference temperature and T_L, is the temperature of the conditioned environment, as presented by Çengel and Boles [34] and used in the work of Salilih and Birhane [8], as presented in Eq. (42).

$\dot{X}_{Q_L}=\frac{T_0-T_L}{T_L} \times \dot{Q}_L$            (42)

3.6 Thermodynamic analysis of the combined system

Figure 4 presents a schematic diagram of the coupling between the PV and ACS systems, called the combined system (sys). According to the Figure 4, the electrical power input to the compressor can be given by Eq. (43).

4.png

Figure 4. Schematic diagram of the coupling between the PV and ACS systems

$\dot{W}_{e l}=\dot{W}_{P V} \cdot \eta_{D C-D C} \, \cdot \, \eta_{E S C}$                   (43)

Combining Eqs. (10), (32), (33), (35), and (43), we obtain the coupling equation of the combined system Eq. (44) as the variation of the mass flow of the refrigerant from the power generated in the PVG.

$\dot{m}=\frac{P_{F V}^0 \frac{G}{G_{r e f}}\left[1+\gamma_m\left(T_c-T_{c, r e f}\,\, \right)\right] \cdot E f}{\left(h_{2 s}-h_1\right)}$                 (44)

where, $E f=\eta_{D C-D C} \cdot \, \eta_{E S C} \cdot \eta_{C M} \cdot \eta_{e l} \cdot \eta_{C I}$.

The COP_sys of the combined system (PV-ACS) can be evaluated as in the study of Salilih and Birhane [8] and Kaushik et al. [7], by Eq. (45).

$C O P_{s y s}=\frac{\dot{Q}_L}{G \times A}$                 (45)

where, G is the irradiance intensity in (W/m²) and A is the area in (m²) of the photovoltaic module.

The exergy efficiency of the combined system (PV-ACS) can be evaluated by Salilih and Birhane [8] and Kaushik et al. [7], by Eq. (46). For greater precision of the results, it is also possible to evaluate from the input exergy in the PV system (Eq. (14)), as shown in Eq. (47).

$\eta_{E x, s y s}=\frac{E x_{e v}}{E \times A}$                 (46)

$\eta_{E x, s y s}=\frac{\dot{X}_{Q_L}}{\left(1-\frac{T_a}{T_S}\right) G A}$                    (47)

For the combined system (PV-ACS), the total system exergy destroyed can be estimated according to Eq. (48). The irreversibilities of each refrigeration system component are added to the photovoltaic system irreversibility. Adding the exergy destroyed in each component of the air conditioning system and the photovoltaic generator, an estimate of the total exergy destroyed in the integrated system can be obtained. This measure can be used to assess the efficiency of the system and identify opportunities for improvement in its design and operation.

$\begin{gathered}\dot{E} x_{d, t o t a l}=\dot{E} x_{d, \text { comp }}+\dot{E} x_{d, e v}+\dot{E} x_{d, \text { cond }}+  \dot{E} x_{d, \text { exp }}+\dot{E} x_{d, P V}\end{gathered}$               (48)

3.7 Estimation of the temperature of the environment to be conditioned T_L

To simulate the PV-ACS operation, an environment of a small library, with an area of 23 m² and a height of 3 m, located in the GEDAE laboratory, is considered a building with average thermal inertia. The thermal load was calculated for the two analyzed days using the HAP software (Hourly Analysis Program 4.9). The Thermal Loads Table, as well as the library plant, can be found by Santos et al. [19].

To estimate the temperature profile (T_L) inside the library from the moment the system starts operating was used the transfer function shown in Eq. (49) [19, 35].

$\sum_{i=0}^1 p_i\left(\dot{Q}_{L, t-\Delta}-T L_{t-i \Delta}\right)=\sum_{i=0}^2 g_i\left(T_r-T_{L, t-i \Delta}\right)$        (49)

where, p_i, g_i = coefficients of the transfer function, presented in Table 4.

$\dot{Q}_{L, t-\Delta}$= cooling capacity in time t – Δ, (W);

TL_t-iΔ= simulated environment’s thermal load in time t – iΔ, (W);

T_r = reference temperature used in calculus of the thermal load, (℃);

T_L,t-iΔ = simulated environment’s internal temperature.

Table 4. Normalized coefficients of the transfer function [19]

The normalized coefficients of the transfer function, pi, and gi, can be adjusted to not-null global conductivity coefficients so that they become representatives of the simulated environment, as Eq. (50).

$g_{i, t}=g_i^* \cdot A_{a m b}+p_i\left(U A_{\text {global }}+1.23 \dot{V}_{\text {total }}\right)$              (50)

where, A_amb is the floor’s area, UA_globalis given by Eq. (51), V̇_total is the ventilation and infiltration flow to the external environment (l/s) and g_i^* is presented in Table 4.

$U A_{\text {global }}=U A_{\text {frontages }}+U A_{\text {glasses }}+U A_{\text {roof }}$                 (51)

U (W m^-2 ℃^-1) and A (frontages, glasses, and roof, m²) can be found in Santos et al. [19].

3.8 Estimation of relative humidity inside the conditioned environment (HR_L)

To estimate the relative humidity of the air in the conditioned environment, the operation of the ACS is considered as represented in Figure 5.

The process is considered to occur at a steady state, and therefore the mass flow rate of dry air remains constant throughout the process and is given by Eq. (52).

$\dot{m}_{\text {air }}=\frac{A V}{v_{\text {air }}}$           (52)

where, AV is the volumetric flow rate of air circulation, in the amount of 18 m³/min, according to the manufacturer's information, and v_air is the specific volume of dry air at inlet 1, which can be obtained by $v_{\text {air }}=\left(\bar{R} / M_{\text {air }}\right)\left(T_1 / P_{\text {air } 1}\right)$, where $\bar{R}=8314 \mathrm{~m}^3 . P a / \mathrm{kmol} . K, M_{\text {air }}=28.97 \mathrm{~kg} / \mathrm{kmol}, T_1$ the inlet temperature of the coil, and $P_{\text {air } 1}$ the partial pressure of dry air, calculated by $P_{\text {air } 1}=P-P_{v 1}$, where $\mathrm{P}$ is the atmospheric air pressure, $P_{v 1}$ is the partial pressure of water atmospheric air pressure, $P_{v 1}$ is the partial pressure of water vapor, given by Eq. (53).

$P_{v 1}=R H_1 \cdot P_{g 1}$                (53)

$R H_1$ is the relative humidity of the air at the inlet of the coil and $P_{g 1}$ is the saturation pressure of the water at temperature $T_1$. The specific humidity at the coil inlet can be estimated according to Eq. (54).

$\omega_1=\frac{0.622 P_{v 1}}{P-P_{v 1}}$                (54)

When passing through the cooling coil, part of the moisture is condensed, and the air is considered saturated, where RH₂=1 and the specific humidity at the coil exit can be obtained from RH₂ and T_ev in the diagram psychrometric.

When leaving for the conditioned environment, the air will have a temperature T_L, which is a function of the cooling capacity and the thermal load of the environment, Q_L and TL, respectively, as shown in Eq. (49), with the specific humidity ω_L estimated as being equal to ω₂ of the cooling coil outlet. From the T_L and ω_L properties, the relative humidity of the conditioned environment (RH_L) can be defined in the psychrometric diagram.

Air from the conditioned environment returns to the SCA, with a temperature T_L and relative humidity RH_L, restarting the cycle. Figure 5 shows the operating times of the SCA, as i=6:10 am, 6:20 am, …, 6:00 pm, taking into account the library's operating period, from 6:00 am to 6:00 pm.

5.png

Figure 5. Schematic diagram of the SCA cooling coil

This work proposed an exergetic analysis of an air conditioning system directly coupled to a photovoltaic generator (PV-ACS), without batteries or grid support. The PV-ACS operates according to the natural regulation of solar energy for two days, with different irradiance profiles (sunny and cloudy), following the methodology used by Santos et al. [19] in Belém, the capital of the state of Pará - Brazil.

The air conditioning system requires a minimum input power of approximately 0.30 kW to achieve the minimum cooling capacity, with an irradiance level of 240 W/m² being required in the PVG to generate this power.

On a sunny day, the proposed system can operate above the minimum cooling capacity for approximately nine hours and forty minutes uninterrupted, however, in just 25% of the library operating time, the temperature and relative humidity parameters would be within the zone of comfort established in [36]. As for the cloudy day, the operation takes place above the minimum cooling capacity for five hours and fifty minutes, considering four interruptions due to the low availability of solar resources, which leaves the system inoperative for one hour and forty minutes throughout the day. Under these conditions, the system can keep the environment within the comfort zone for the occupants, for only 12.3% of the library's operating time.

Due to the buoyancy of the system's cooling capacity throughout the day, component design and sizing would need to be carefully calculated to handle daily variations in solar energy availability.

The analysis and simulation of the PV-ACS system showed that the exergy destruction increases throughout the day for the system components, both on sunny and cloudy days, with the photovoltaic system being the most affected, followed by the compressor, condenser, evaporator, and expansion valve, on a sunny day.

The maximum exergy destruction in the photovoltaic system was 3.73 kWh on a sunny day and 1.36 kWh on a cloudy day, corresponding to about 33.1% and 26.10% of the total exergy destroyed in the PV-ACS system on sunny and cloudy days, respectively.

Increased exergy destruction in system components may be related to greater thermal and mechanical stress, which may reduce the lifetime of these components, leading to a more frequent need for replacement and increasing maintenance costs.

For the ACS, the component with the highest exergy destruction was the compressor, with a value of 3.29 kWh on a sunny day and 2.08 kWh on a cloudy day, corresponding to 29.2% and 39.9% of the total exergy destroyed by the combined system, respectively. Part of the exergy destroyed in the compressor occurs when the system cannot reach the minimum cooling capacity but is still powered by the electricity generated by the photovoltaic system.

The component with the lowest exergy loss was the expansion valve, with a maximum destruction of 1.09 kWh on a sunny day and 0.38 kWh on a cloudy day, corresponding to 9.70% and 9.30% of the total exergy destroyed in the combined system on sunny and cloudy days, respectively. Overall, the total exergy destroyed in the combined system (PV-ACS) was 11.27 kWh and 5.21 kWh for both days analyzed.

It was possible to verify that the exergetic efficiency of the ACS and the system increases throughout the day for both analyzed irradiance profiles, due to the increase in cooling capacity and the difference between external and conditioned environment temperatures.

For the photovoltaic system, the exergetic efficiency decreases throughout the day as irradiance and ambient temperature increase, causing an increase in exergy destruction due to electrical losses associated with the photovoltaic module characteristics and PVG heating. It was found that the reduction of the COPsys throughout the day occurs with the increase of irradiance and ambient temperature and that the COPsys increase with the increase of the ACS evaporation temperature.

The study showed that the exergetic efficiency of the combined system increases with the increase in evaporation temperature and ambient temperature, when the ambient temperature is considered as the reference temperature of the dead state, reaching a maximum of 1.97% for a sunny day and 0.60% for a cloudy day.

Although the system simulation has shown that the operation of the ACS can be regulated from the direct coupling to the PV system, without energy storage, the large variations in temperature and relative humidity can lead occupants to discomfort at various times of the day, especially on cloudy days.

Some practical applications of the proposed system may include small or medium-sized houses, where the energy demand for air conditioning is not excessively high. However, the building's energy efficiency and thermal insulation would be crucial factors to optimize the use of available solar energy. In addition, buildings with a low energy load, such as small clinics or offices, can be considered for this system, especially when the air conditioning energy demand is not high. Proper sizing and energy efficiency remain key to ensuring the maximum use of solar energy and system functionality in meeting these diverse needs.

The experimental validation of the proposed system is necessary for further studies, as well as the implementation of control logic that optimizes the operation in different irradiation conditions, to balance the efficient performance of the system and the thermal comfort of the occupants.

Based on the results found in this study, it was possible to evaluate the places in the system where the most significant exergy destruction occurs when the input power for the air conditioning system is variable, according to the availability of solar resources. This information can serve as a subsidy for the improvement, optimization, and development of new components for systems with configurations similar to the one proposed in this study.