Dimensioning of a Solar Adsorption-Powered Cooling Bed for Generating Relief Cooling

Solar adsorption cooler technology employs natural resources for operation rather than depending on costly power expenses, which helps to dramatically cut energy costs over time in addition to decreasing emissions [10, 11]. The adsorption cooling system uses two materials known as the "adsorption pair" adsorbent and adsorbate [12, 13].

Additionally, they use environmentally benign and natural refrigerants such as ethanol [14-16], methanol [16-21], CO₂ [22-24], water [25], ammonia [26] so on. Consequently, the extensive development of implementing an adsorption cooling system will aid in resolving several environmental and energy-related issues. Thus, the widespread development of adsorption cooling systems will contribute to the resolution of several environmental and energy-related issues.

Particularly in warm regions, solar-powered refrigeration devices appear promising, but low performance is a big worry [27, 28]. Therefore, many recent research aims to improve performance [29]. For improving the COP of solar adsorption cooler working with ethanol ACF, employing experimentally and numerical simulation investigated a ACFs-ethanol pair in addition to enhancing transferring mass & heating in the adsorber bed [30].

Additionally, these problems make thermally driven adsorption cooling machines less commercially viable [31]. Improving performance may include, optimizing the reactor adsorbent bed design, improving the conductivity of the adsorbent, using more stages than one, using fins or additives, and optimizing the bed's geometrical thickness. (SCP) and (COP) are two metrics that are frequently used to evaluate an adsorption-cooling machine's efficacy. Khanam et al. [32] created a transient two-dimensional axisymmetric, with Computational Fluid Dynamics (CFD) for studying the adsorber with finned-tube design performance. Modeling the AC/ethanol as the operating functioning couples with different adsorbent thicknesses to optimize cycle time. The cooling system was working at (15, 20, and 80)℃ for heating, cooling, and evaporation, respectively. The temperature profiles that were generated for various adsorbent thicknesses were verified with experimental data. The outcomes of, SCP and COP, were established as, 488 W/kg and 0.61, correspondingly, and the optimal cycle time was 800 s.

Simulation studied hybrid solar adsorption systems, three cities have been chosen: Kuwait City in Kuwait, Riyadh in Saudi Arabia, and Sharjah in the United Arab Emirates [33]. The system consists of three components: (i) photovoltaic-thermal (PVT) solar collectors, (ii) evacuated tube solar collectors (ETCs), and (iii) a single-stage double-bed silica gel/water-based adsorption chiller for chilling applications. The performance of the suggested system is theoretically investigated through the development and implementation of a MATLAB code.

Hakemzadeh et al. [34] and Bozorgi et al. [35] established a multilayer computational network, a mix of analytical and conceptual frameworks, to determine the most effective solar thermal system, to enhance the techno-economic potential of solar adsorption cooling in Malaysia. The solar adsorption cooling system was modeled and simulated using TRNSYS, MATLAB, and REFPROP software. MATLAB-implemented Particle Swarm Optimization (PSO) connected to TRNSYS. To provide the necessary thermal energy, two types of solar collectors—the evacuated tube and Wavy Direct Absorption Solar Collectors (WDASC)—were assessed.

To analyze the adsorptive duration impact on the improvement of performance effectiveness of a solar adsorptive refrigerator machine, Wang et al. [36] offers a thermodynamic model that has been numerically verified. The adsorption rate model has been utilized to assess the based on observed concentrated solar heating. The mathematical model yielded an ideal time of approximately (71 min) for the adsorptive time, which corresponds to a COP of 0.476 and SCP of 28.6 Wkg^-1.

Grabowska et al. [37] presented a novel idea of adding fluidized beds to the adsorption systems that have the potential to yield the best improvements in energy efficiency. The objective of the empirical research was to expand the basic understanding of fluidization at low pressure, which is a feature of the cyclic occupied of adsorptive machines. Five samples from adsorption beds were examined. The samples included additions of carbon or aluminum nanotubes of five and ten percent, respectively, whereas the reference sample was composed solely of silica gel.

Focuses on creating innovative poroused (ACs) from widely accessible crop leftovers and using them sustainably to address these problems. Chauhan et al. [38] produced the ACs by activating optimized biochar with KOH showed apparent areas of (2.002 m²/kg) and (1.241 m²/kg). The honey-comb construction was an indication of micro-pores formed during activation processes, according to the surface morphology research. The structural constants of the (D_A) (Dubinin-Astakhov) representations may be most precisely determined by using the experimental data on ethanol uptake. The mass of adsorbent to combustion fuel ratio was presented as a new performance metric by using ethanol refrigerated in an adsorptive cooler machine driven by renewable energy.

Chauhan et al. [39] reported the performance estimation of a small-scale biomass-based heating unit combined with a one-step adsorptive cooler machine that uses several activated carbon/ethanol as adsorbent/adsorbate beds for different applications. To determine the best adsorbent bed designs for each application, a thorough parametric investigation was conducted to highlight the effects of operating temperatures, heat exchanger to adsorbent mass ratio, coefficient of performance (COP), uptake efficiency, and mass of adsorbent to combustion fuel. The numerical results show that the flat-finned tube adsorber works better with biomass carbon-ethanol pair than the traditional-finned tube and tube & shell-type adsorber, reaching 19% and 30% higher COP, respectively. For refrigeration and space cooling, the H₂-treated Maxsorb-III exhibits the highest ideal cooling coefficient (0.73), with the biomass-carbon following closely after with 0.72.

Gado et al. [40] and Wang et al. [41] ambitious to enhance the efficiency of adsorption desalination/cooling systems (ADCs) by the use of triply periodic minimum surface (TPMS) structures to contain adsorbent materials. To enhance the efficiency of ADCs, solid and sheet forms of metallic networks-based TPMS (such as Gyroid and Diamond) were studied. Based on specific daily water production (SDWP) and (SCP), the finned tube-packed adsorber and TPMS structures were compared using CFD simulations. Variant porosity levels were also used to analyze such structures. The sheet network of diamonds reached an SDWP of 10.4 m³/ton at a porosity of 0.8, which was 19.4% higher than that of the finned tube structure. The diamond sheet network achieved SDWP and SCP of 20.1 m³/ton and 627.4 W/kg, respectively, at a porosity of 0.2.

Agarwal et al. [42] carried out extensive research to solve this problem, beginning with lab-scale experiments to full-scale commercial refrigeration system trials. Three composite samples (B1, B2, and B3) containing different amounts of water (20-45% by weight) were made using calcium chloride–expanded graphite (CaCl₂: EG/4: 1), and utilized a standard CaCl₂-activated carbon-white cement composite (CaCl₂: AC: WC/16: 4: 1, sample A). For this reason, the composites were placed above fin tubes that had wire mesh wrapped around them. The structural integrity of these composites was examined in a designed constant-volume single fin-tube adsorption/desorption bed for 300 continuous cycles.

In the laboratory, by using CFD, concluded that Zeolite-water adsorbent bed desorption is inspired by several factors, particle size, thermal conductivity, bed thickness, working pressure, and various regeneration temperatures [43-45]. It has been demonstrated that the system's COP is highly dependent on the adsorbent bed layer's thickness.

One way of optimizing the COP was by improving heating and mass transferring internally the adsorptive bed accelerating the adsorption and desorption processes. Enhancing the adsorber heat exchanger's design, raising the adsorbent's thermal conductivity, and suggesting a brand-new, improved mass transfer mode that uses a tiny vacuum pump and two condensers to lower the desorption pressure significantly were well-liked strategies to ensure sufficient heat & mass transmission inside the bed [46], optimizing the design of adsorber beds and a time-consuming process that requires many resources to carry out experimentally.

Jribi et al. [47] simulated activated carbon/ethanol beds for adsorption chiller validated by using 2D-axisymmetric geometry CFD. They have performed a simulation of a cylindrical finned adsorber. With adsorbent thicknesses of 0, 1, 5, and 10 mm from the exterior diameter of the bed. It is chilled with water to almost 30℃ while the ethanol pressure suddenly rises from 0.95 kPa to 6 kPa. The adsorbent temperature increased by up to 20°C from the starting point temperature in the simulated temperatures. At larger flow times, they were consistent with experimental findings and were somewhat higher at the beginning of adsorption.

Mohammed et al. [48] used computational fluid dynamics (CFD) numerically for the simulation of the Maxsorb/ethanol solar adsorption process. The findings of the simulation indicate that the adsorption of the Maxsorb adsorbent layer reduces with increasing layer thickness. Regarding packing density, when it reaches its maximum at 750 kg/m³, the amount of ethanol adsorbed per plate rises.

Elsheniti et al. [49] aim to enhance the conductive heat transfer across the adsorbent field. The effects of using a composite adsorbent in this investigation include System-level numerical analysis conducted on 50% activated carbon type Maxsorb III, 40% graphene Nanoplatelets, and 10% binder. For three instances of composite layer thicknesses at various cycle periods, the conflicting impacts of heat and mass transfer mechanisms inside the composite adsorbent on the operation of an adsorption ice production system were investigated. The outcomes demonstrated that for composite thicknesses of 2 and 5 mm and cycle lengths of 430 and 1230 s, the highest specific daily ice production and coefficient of performance of 33.27 kgice·kgads⁻¹·day⁻¹ and 0.3046 were reached. Correspondingly, the mass transfer resistances increased due to the larger composite thickness of 10 mm, overriding the improvement in heat transmission and decreasing overall performance.

El Fadar et al. [50] assumed the temperature of the cooling water and heat source were to be 100 and 32℃, respectively, and Tev=0℃ was used to get the simulated results. Operating and design circumstances presented a numerical investigation conducted into how some critical factors, such as the thickness of the adsorbent bed and the temperature of the heat source, affect system performance. For values in the range of 10-90 mm, the radial thickness of the adsorbent bed is a crucial component for system optimization. Its impact on specific cooling power (SCP) and refrigeration cycle performance COP.

One of the recommendations for potential system performance enhancements, Palomba et al. [51] presented a design analysis based on the experimental data. The acquired results showed that it is possible to achieve performance by using an activated carbon/ethanol working pair similar to other dangerous refrigerants like ammonia and methanol by using a non-toxic refrigerant like ethylene. It is advised to replace the rectangular adsorber vessels with cylindrical ones since the thickness of metal sheets needed for a vacuum. As a result, it would be reasonable to anticipate a notable decrease in heat losses and a notable increase in COP.

Even though researchers have proved that heat and mass transmission inside the bed are the primary determinants of bed performance, this knowledge still requires more explanation. For performance improvement through the design of the generation adsorbent bed, a significant decrease in the internal masses might result in a more effective design. On the other hand, several researchers also carried out experimental and computational studies to examine the impact of particle size and adsorbent layer thickness on kinetics. Researchers have noted that heat and mass transmission inside the bed were the primary determinants of bed performance; however, further clarification of this information is still needed.

A dynamic investigation of methanol adsorption in compact adsorbent layers was discussed by Girnik et al. [52] Adsorbent and binder, respectively, were polyvinyl alcohol and marketable AC called ACM-35.4. At a given binder concentration of 12 weight percent, investigated the effects of layer thickness and carbon particle size on the Volumetric Large Temperature Jump (V-LTJ) dynamics. It is obvious that in an adsorptive refrigerator bed, the binder can improve mass and heat transmission. In comparison to packed beds, the process accelerated by a ratio of 1.5-3.5 for the compact layers.

The Biot number represents the body’s internal conduction resistance divided by its external convection resistance. Since weakly conducting materials have greater resistance to heat conduction, their Biot numbers are more likely to be bigger. Some bodies are seen in heat transfer analysis to behave as a "lump" whose total body temperature is constant during a heat transfer process. The lumped system analysis is a type of heat transfer analysis that makes use of this idealization. The Fourier number measures the ratio of heat stored to heat transferred through a body. Therefore, a high Fourier number denotes a quicker rate of heat transmission throughout the body. Given that the Fourier number is a function of time, two times a term will result in two times the Fourier number [53].

Alam et al. [54] investigated the effect of design parameters for silica gel/water solar adsorption chiller on performance. The findings indicate that the bed adsorbent reactor design characteristics have a significant impact on the ideal switching frequency. The number of transfer units, the number of adsorbent bed Biots (Bi), the heat exchanger thickness / the fluid channel radius, the fluid alpha number), and the alpha number of the inert material were the characteristics that define the design parameters. Scherle and Nieken [55] introduced an optimization method based on models to enhance the performance of adsorption heat pumps. Two steps are involved in the optimization process. For a particular adsorption material, first, optimize the cycle duration, the thickness of the adsorbent, and the operating parameters. In the second stage, we forecast mass and heat transport as well as adsorption capacity based on structural material properties using a material model. Kundu [56] inspected parametrically the thermal analysis of a variable solar adsorption bed thickness of a flat plate solar collector. A novel analytical formulation has been established using the Differential Transform Method (DTM). The improved Bessel's function has been utilized to compare the outcomes generated by DTM. The effectiveness of the adsorber has been examined as a function of Biot number.

For performance improvement through the design of the generation adsorbent bed, a significant decrease in the internal masses might result in a more effective design. On the other hand, several researchers also carried out experimental and computational studies to examine the impact of particle size and adsorbent layer thickness on kinetics. Researchers have noted that heat and mass transmission inside the bed were the primary determinants of bed performance; however, further clarification of this information is still needed.

Bahrehmand et al. [57] carried out parametric research to examine the impact of critical factors, such as generation bed geometry, heat transfer characteristics, and cycle duration, on solar cooling adsorption performance. The analytical proposed solution optimized (i) fin thickness and height of fin, (ii) adsorbent bed thickness, (iii) fluid channel height, (iv) the quantity of thermally-conductive additive in the composite adsorbent, and (vi) cycle duration.

The literature demonstrates that another sophisticated choice, solar adsorptive refrigerators, and air conditioning technologies are vital, workable solutions that outweigh the drawbacks of traditional adsorptive engines. Investigation research on the solar adsorptive refrigeration machine has been done, both theoretically and experimentally. Because it is expensive and takes a lot of time to do experimentally, optimizing the design of absorber beds is particularly challenging.Different working operational pairings, design sizes, and heating resources can all be used to solve the problems. Depending on the operational environment, various working pairs provide varied results. Furthermore, it seems that not much study has been done on adsorption refrigeration systems, particularly in Iraq when using ethanol as the adsorbate. In this paper, the adsorbent thickness will be optimized to improve the performance and SCP. Depending on previous work [14] same cooling machine.

4.1 Adsorption equilibrium and kinetics

As the vapor of ethanol strikes the surface of the activated carbon, which is a solid, the gas molecules were struck within the solar adsorption bed. While some of the impacting molecules bounce back, others adhere to the surface and become adsorbed. Because of the significant exposed area at the beginning of this process, the adsorption rate is high. The adsorption rate decreases as more ethanol molecules cover the activated carbon surface [59]. Within the current model of solar adsorptive machine, the Dubinin-Astakhov equation is used to estimate the ethanol/AC uptake pair.

$\mathrm{X}(\mathrm{P}, \mathrm{T})=\omega_0\ \rho_m(\mathrm{T})\ \exp \left\{-\left(\frac{R}{\epsilon E_0}\right)^{n}\left(\mathrm{T} \cdot \ln \left(\frac{\mathrm{P}_{\text {sat }}(T)}{\mathrm{P}}\right)^{\mathrm{n}}\right)\right\}$                    (1)

${{\text{X}}_{0\text{ }\!\!~\!\!\text{ }}}$ represents the maximum adsorbent uptake of the adsorbate, and it is correlated with both the adsorbent's density $\omega_0\ \rho_m(\mathrm{T})$ and its limiting micropore volume ${{\text{ }\!\!\omega\!\!\text{ }}_{0}}$. The density is taken to remain constant for simplicity's sake. The introduction of$~D$, a characteristic constant for the working pair, is the second portion of the statement that may be made simpler. It is made up of the gas constant $R$, affinity coefficient $\epsilon $, n is the characteristic parameter, and the characteristic adsorption energy ${{E}_{o}}$ of the working pair.

where, $D=\left(\frac{R}{\epsilon\ E_0}\right),\ \mathrm{X}_0=\omega_0\ \rho_m(\mathrm{T})$                        (2)

The saturation pressure based on the temperature of the adsorbent is ${{\text{P}}_{\text{sat}}}$ (${{\text{T}}_{\text{Sat}}}$) [60].

$\mathrm{X}(\mathrm{P}, \mathrm{T})=\mathrm{X}_0\left[\exp \left\{-\mathrm{D}\left(\mathrm{T} \ln \frac{\mathrm{P}_{\text {sat }}}{\mathrm{P}}\right)^{\mathrm{n}}\right\}\right]$                        (3)

$\mathrm{X}\left(\mathrm{T}, \mathrm{T}_{\text {sat }}\right)=\mathrm{X}_0\left[\exp \left\{-\mathrm{K}\left(\frac{\mathrm{T}}{\mathrm{T}_{\text {Sat }}}-1\right)^{\mathrm{n}}\right\}\right]$                        (4)

The quantity of ethanol (refrigerant) adsorbed kg/kg of AC (adsor-bent) is characterized via the symbolization [X (P, T) (kg. kg^-1)]. The beds of the AC bed unit affect the ethanol uptake at generation temperature (T) and (P) the generation bed pressure. The maximum adsorpative uptake of ethanol (${{\text{X}}_{0}}$,) is strong-minded by the (D) structure factor distinctive toward the arrangement of refrigerant-adsor-bent. While ${{\text{P}}_{\text{sat}}}$ demonstrates ethanol-saturated pressure at the evaporator and adsorber bed temperatures; (P) designates the stabilizing pressure of the AC/ethanol refrigerant grouping. The microstructure of the AC material besides the affinity factor, D or (X) denotes the equilibrium adsorption (ethanol uptake) on activated carbon. The processes stated in section (3.1) provide the energy balance equation that is used in the mathematical thermodynamically modeling of the machine. The starting baseline limitations equally listed in the study of Mohammed and Alhialy [14], were used in the simulated program. Using the EES program, the saturation property of ethanol may be found.

4.2 Methodology for thermodynamic investigation

The discontinuous adsorptive cooler unit's performance is not optimized. The intermittency problem with single-bed systems is recommended to be resolved by a solar adsorptive cooler machine with two beds. This research focuses mostly on optimizing concluded two groups of key parameters, operational and geometrical parameters. The geometrical design parameters concluded of, the geometrical shapes of the two beds, bed generator thickness, density of the adsorbent, the thermal conductivity of the adsorbent (k), Biot number (Bi), and Fourier number (Fo). Whereas the operational key parameters in addition to the working-pair qualities of (AC/ethanol), operational parameters that affected the COP and SCP of the machine by optimizing cooling time. The simulation modeling takes into account the features of the bed, the distribution maximum bed generating temperature at different points, evaporator temperature, condenser temperature, and surrounding ambient conditions. The optimization was performed by modeling simulation by the EES program, which uses a one-dimension-transient distribution temperature profile for (4) locations at the generator adsorbent bed of 2-bed adsorption refrigerator systems to provide cooling effects powered by solar energy.

Due to the complexity of the heat and mass transmission mechanism in a granular adsorbent bed, some assumptions must be made to determine the governing equations. The mathematical model proposed in this study is primarily based on the following assumptions and simplifications:

• the adsorbent granules' size was uniform, resulting in a constant porosity of the bed; the adsorbent granules' thermal resistance was negligible;
• the adsorbent granule's temperature was equal to the surrounding adsorptive; The rate of heat transfer at the adsorptive inlet surface, R = Ri, was negligible.
• It is assumed that the vapor phase of the adsorbate is an ideal gas and the size of the adsorbent particles and the bed porosity is spatially uniform;
• radiative heat transfer, viscous dissipation, and the work done by pressure changes are neglected;
• properties such as thermal conductivities, and viscosity are not a function of temperature;
• the specific heat of the desorbed or adsorbed ethanol is considered as that of the bulk liquid ethanol due to the vapor condensation on the surfaces of the adsorbent pores.

4.2.1 Isosteric heating phase

The initial action is taken in the daytime. The heat from the sun is absorbed by the solar collector and added to the system. The metal encloses (cylinder & covering), AC, and the sensibility heating from ethanol make up the total heat contributed until the pressure hits ${{\text{P}}_{\text{con}}}$, and the bed's valve is closed, $({{\text{Q}}_{1-2}}$). The generator component's perceptible heat can be expressed as follows:

${{\text{Q}}_{1-2}}=\left( {{\text{Q}}_{\text{tube}}}_{1-2}+{{\text{Q}}_{\text{shell}}}_{1-2}+{{\text{Q}}_{\text{AC}.}}_{1-2} \right)\text{*}\left( {{\text{T}}_{\text{sd}}}-{{\text{T}}_{\text{ad}}} \right)+\text{ }\!\!~\!\!\text{ Q}\_\text{et}{{\text{h}}_{1-2}}$                      (5)

where, the sensibility heating of metal (cylinder &covering), and AC are [14]:

$\begin{aligned} \Delta \mathrm{U}_{\text {tube }}+\Delta \mathrm{U}_{\text {shell }} & +\Delta \mathrm{U}_{\mathrm{AC}}=\left(\mathrm{m}_{\text {tube }} \cdot \mathrm{c}_{\text {tube }}+\mathrm{m}_{\text {shell }} \cdot \mathrm{c}_{\text {shell }}\right.\left.+\ \mathrm{m}_{\mathrm{AC} .} * \mathrm{c}_{\mathrm{AC} .}\right)\left(\mathrm{T}_{\text {sd }}-\mathrm{T}_{\mathrm{ad}}\right)\end{aligned}$                     (6)

Using the EES software, the specific internal energy of liquid ethanol at a specific internal energy status, ${{\text{U}}_{\text{eth}{{.}_{2}}}}\text{ }\!\!~\!\!\text{ }\And \text{ }\!\!~\!\!\text{ }{{\text{U}}_{\text{eth}{{.}_{1}}}}$ may be directly determined by utilizing two properties: the saturation temperature and the dryness fraction $\left( \text{x} \right)$. ${{\text{T}}_{\text{sd}}}$ may be obtained by: since T/Tsat is constant lengthwise an isosteric process.

${{\text{Q}}_{1-2}}=\left( {{\text{m}}_{\text{tube}}}.{{\text{c}}_{\text{tube}}}+{{\text{m}}_{\text{shell}}}.{{\text{c}}_{\text{shell}}}+{{\text{m}}_{\text{AC}.}}.{{\text{c}}_{\text{AC}.}} \right)\left( {{\text{T}}_{\text{sd}}}-{{\text{T}}_{\text{ad}}} \right)+\text{ }\!\!~\!\!\text{ }{{\text{m}}_{\text{AC}.}}.{{\text{x}}_{\text{max}}}.\left( {{\text{U}}_{\text{eth}{{.}_{2}}}}-{{\text{U}}_{\text{eth}{{.}_{1}}}} \right)$                         (7)

4.2.2 Isobaric desorption phase

At this phase, the bed reactor goes through a phase of desorption when reaches condensation pressure. The bed will continue to desorb until it attains its maximum temperature, as a result, the adsorbate's uptake (${{\text{x}}_{\text{max}}})~$adsorbent bed decreases to its minimum limit (${{\text{x}}_{\text{min}}}$)). In this process, the heating is exchanged with the metal (tube and shell), the adsorbate (${{\text{Q}}_{2-3}}$ + ${{\text{Q}}_{\text{des}}}$), and the adsorbent activated carbon is added to determine the overall heat exchange (${{\text{Q}}_{2-3}}$). The following is an expression of the heat produced in this process:

$\begin{aligned} & Q_{2-3}=m_{A . C .} \cdot \Delta x . h_g+m_{A . C .} . \Delta x . \mathrm{h}_{a d}+m_{\text {A.C. }} . \Delta x .\left(U_{e t h_3-} U_{e t h_2}\right)+\left(m_{\text {tube }} \cdot c_{\text {tube }}+m_{\text {shell }} \cdot c_{\text {shell }}\right.\left.+\ m_{\text {A.C. }} \cdot c_{\text {A.C. }}\right)\left(T_{g e n}-T_{s d}\right) \\ & \end{aligned}$                     (8)

The overall energy balance formula via a controlled volume without any workable performed is:

$\delta\text{Q}=\sum\text{m}_\text{e}\text{h}_\text{e}-\sum\text{m}_\text{i}\text{h}_\text{i}+\delta\dot{\text{Q}}+\text{dU}_\text{CV}$                       (9)

The saturation pressure was attained in the desorption and adsorption processes, ${{\text{P}}_{\text{con}}}\text{ }\!\!~\!\!\text{ and }\!\!~\!\!\text{ }{{\text{P}}_{\text{eva}}}$ respectively.

H is the heating of desorption kJ /kg of adsorbate. Given the slope of lines (1$\longrightarrow$2) and (3$\longrightarrow$4) in Figure 3 [60].

The increment in internal energy of the C.V. is:

$\begin{aligned} & \Delta U_{\text {tube }_{2-3}}+\Delta U_{\text {shell }_{2-3}}+\Delta U_{\text {A.C. } 2_{2-3}}=\int_2^3\left(m_{\text {tube }} \cdot c_{\text {tube }}+m_{\text {shell }} \cdot c_{\text {shell }}\right.\left.+m_{\text {A.C. }}\cdot c_{\text {A.C. }}\right) d T=\left(m_{-} \text {tube.c_tube }\right.+m_{-} \text {shell.c_shell } \left.+m_{\text {A.C. }} \cdot c_{\text {A.C. }}\right)(\text { Tgen }-T s d)\end{aligned}$                  (10)

$Q_{\text {des }}=m_{A \cdot C \cdot} \cdot \Delta x \cdot \mathrm{h}_{a d}$               (11)

The sum of the isosteric and desorption heat is known as the production or regeneration energy, and it is stated as follows:

$\begin{aligned} & Q_{1-3}=Q_{1-2}+Q_{2-3} \\ & Q_{1-3}=\left(m_{\text {tube }} \cdot c_{\text {tube }}+m_{\text {shell }} \cdot c_{\text {shell }}\right. \left.+m_{\text {A.C. }} \cdot c_{\text {A.C. }}\right)\left(T_{g e n}-T_{a d}\right) +m_{\text {A.C. }} \cdot x_{\max }\left(U_{\text {eth. } 2}-U_{\text {eth. } 1}\right) +m_{\text {A.C. }} . \Delta x . h_g+m_{\text {A.C. }} \Delta x . \mathrm{h}_{a d} +m_{\text {A.C. }} . \Delta x .\left(U_{e t h_3-} U_{e t h_2}\right) \end{aligned}$                           (12)

4.2.3 Isosteric cooling phase

Cooling and depressurization, which is the third stage from state 3 to state 4, is similar to ${{Q}_{3-4}}$ the heat rejected, or the initial process (1→2). Comparable to the process of heating, outlines (3→4) show the uptake of ethanol stays unvarying. Even by the minimum level (${{x}_{min}}$). As the bed temperature drops to ${{T}_{sa}}$, the pressure also drops.

${{Q}_{3-4}}=\left( {{m}_{tube}}.{{c}_{tube}}+{{m}_{shell}}.{{c}_{shell}}+{{m}_{A.C.}}.{{c}_{A.C.}} \right)\left( {{T}_{gen}}-{{T}_{sa}} \right)+{{m}_{A.C.}}.{{x}_{min}}\left( {{U}_{eth{{.}_{4}}}}-{{U}_{eth{{.}_{3}}}} \right)$                   (13)

4.2.4 Isobaric adsorption phase

The energy equilibrium of CV is:

$\delta Q=d{{U}_{CV}}+{{h}_{e}}.d{{m}_{e}}$                  (14)

The bed cycle's fourth and final stage involves further cooling and ethanol adsorption. Ethanol liquid from the receiver is throttle going from high pressure (${{P}_{con\text{ }\!\!~\!\!\text{ }}}$) to low pressure (${{P}_{eva\text{ }\!\!~\!\!\text{ }}}$). Saturated ethanol's mass and enthalpy are signified by the variables$~{{\text{m}}_{\text{e}}},\text{ }\!\!~\!\!\text{ }{{\text{h}}_{\text{e}}}$. As a result, the ethanol mass refrigerant starting evaporation is:

$\begin{aligned} & m_{\text {eth }}=m_{\text {A.C. }}\left(x_{\max }-x_{\min }\right) =m_{\text {A.C. }} . \Delta x, \text { Where }\left(x_{\max }\right.\left.-x_{\min }\right)=\Delta x\end{aligned}$                         (15)

$\begin{aligned} & Q_{4-1}=\left(m_{\text {tube }} . c_{\text {tube }}+m_{\text {shell }} \cdot c_{\text {shell }}\right. \left.+m_{A . C .} c_{A . C .}\right)\left(T_{s a}\right. \left.-T_{a d}\right)+m_{\text {A.C. }} x_{\min }\left(U_{e t h .4}\right.\left.-U_{e t h .1}\right)+m_{\text {A.C. }} \Delta H \cdot \Delta x \\ & \end{aligned}$                    (16)

4.2.5 The condenser thermal state

The desorption vapor of ethanol is received by the condenser during the desorptive period. By taking the energy out of the superheated vapor in this unit, the ethanol vapor is de-superheated first. The vapor of ethanol starts condensation once one touches the saturation evaporation phase, dropping that one latent-heat energy toward the circumscribing heating drop through a variance equal to $({{T}_{con}}$-${{T}_{amb}}$). By rejecting heat to the surroundings, the evaporation is cooled near the temperature of ambient, this heat is provided by:

$\begin{gathered}Q_{\text {con }}=m_{\text {A.C. }} \Delta x\left(L_{\text {eth }}\right)+m_{\text {A.C. }} c_{\text {eth }} \int_{T_{\text {con }}}^{T_{\text {gen }}}\left(T_{\text {con }}-T\right) . \\ {\left[\frac{\partial x}{\partial T}\right]_{P_{\text {con }}} d T}\end{gathered}$                 (17)

4.2.6 The evaporator thermal state

The liquid ethanol in the evaporator element evaporates, producing cold. The cooling production at the evaporator is computed as follows:

$Q_{\text {eva }}=m_{\text {A.C. }} \cdot \Delta x\left(h_{g_e}-u_{r_f}\right)$                    (18)

The daily total amount of ice generated by the refrigerator machine. For icing requests, can use the following formulae to calculate the freezing water temperature and ice mass ($~{{m}_{ice}})$.

$\begin{aligned}\mathrm{m}_{\text {ice }}& =\frac{Q_{\text {eva }}}{\left(\mathrm{L}_{\text {ice }}+\mathrm{c}_{\mathrm{w}}\left(\mathrm{T}_{\text {iw }}-273.15\right)+\mathrm{c}_{\text {ice }}\left(273.15-\mathrm{T}_{\text {ice }}\right)\right)+\mathrm{m}_{\text {eva }} \mathrm{c}_{\text {eth }}\left(\mathrm{T}_{\mathrm{iw}}-\mathrm{T}_{\text {eva }}\right)}&\\&\mathrm{T}_{\text {eva }}<273^{\circ}\end{aligned}$                       (19)

If the cooling system serves as a water chiller, air conditioning application, or cold chamber for storing fruits and vegetables, the amount of chilled water produced each day is determined by:

${{\text{m}}_{\text{cwc}}}=\frac{{{\text{Q}}_{\text{eva}}}}{{{\text{c}}_{\text{w}}}\left( {{\text{T}}_{\text{iw}}}-{{\text{T}}_{\text{eva}}} \right)}~~{{T}_{eva}}>273{}^\circ $                   (20)

4.3 The performance estimation for solar adsorption cooling system

(a) Coefficient of performance (COP)

The performance coefficient (COP) is defined as the ratio of refrigeration delivered (Qeva) / input heat (Qadded):

$COP=\frac{{{Q}_{eva}}}{{{Q}_{added}}}$                   (21)

$\text{C}\mathbf{OP}=\frac{{{\mathbf{m}}_{\mathbf{A}.\mathbf{C}.}}.\mathbf{∆x}\left( {{\mathbf{h}}_{{{\mathbf{g}}_{\mathbf{e}}}}}-{{\mathbf{u}}_{{{\mathbf{r}}_{0}}}} \right)}{\left( \begin{matrix}\left( {{\mathbf{m}}_{\mathbf{tube}}}.{{\mathbf{c}}_{\mathbf{tube}}}+{{\mathbf{m}}_{\mathbf{shell}}}.{{\mathbf{c}}_{\mathbf{shell}}}+{{\mathbf{m}}_{\mathbf{A}.\mathbf{C}.}}.{{\mathbf{c}}_{\mathbf{A}.\mathbf{C}.}} \right)\left( {{\mathbf{T}}_{\mathbf{gen}}}-{{\mathbf{T}}_{\mathbf{ad}}} \right)+{{\mathbf{m}}_{\mathbf{A}.\mathbf{C}.}}.{{\mathbf{x}}_{\mathbf{max}}}\left( {{\mathbf{U}}_{\mathbf{eth}{{.}_{2}}}}-{{\mathbf{U}}_{\mathbf{eth}{{.}_{1}}}} \right) \\+{{\mathbf{m}}_{\mathbf{A}.\mathbf{C}.}}.\mathbf{∆x}.\left( {{\mathbf{U}}_{\mathbf{et}{{\mathbf{h}}_{3}}-}}{{\mathbf{U}}_{\mathbf{et}{{\mathbf{h}}_{2}}}} \right)+{{\mathbf{m}}_{\mathbf{A}.\mathbf{C}.}}.\mathbf{∆x}.{{\mathbf{h}}_{\mathbf{ad}}}\\\end{matrix} \right)}$                   (22)

(b) Cooling capacity (SCC)

The cycle's coefficient of performance (COP) and cooling capacity power (SCC) dictate how effectively the adsorption refrigeration system performs. The efficiency with which solar energy or heat added (Qadded) is converted into a cooling capacity related to (Qeva). It is thus feasible to write it as follows using the cooling capacity (SCC) for two beds:

$\mathrm{SCC}=\frac{\mathrm{Q}_{\mathrm{eva}}}{2 * \text { Cycle Time } * 3600}$ in $\mathrm{W}$,                   (23)

(c) Specific cooling effect (SCE)

The specific cooling effect can be defined by the cooling energy (Qeva) divided by the activated carbon mass as:

$SCE=\frac{{{Q}_{eva}}}{{{m}_{A.C}}}$  kJ/kg                   (24)

(d) Specific cooling power (SCP)

Depending on the system's capacity, different systems have different specific cooling powers (SCPs), evaluated upon the production of cold water, which is utilized to determine an adsorption's usefulness. Cooling systems in different dimensions. (SCP) was the ratio (SCC) to the AC mass-filled reactor bed:

$SCP=\frac{SCC}{{{m}_{A.C}}}~W/kg$                   (25)

Since Specific Cooling Power (SCP) is related to both the mass of the adsorbent and the cooling capacity, it may be used to determine the size of a machine by gauging the refrigeration created / mass of AC / cycle time. This section displays the size of the system. System stability is indicated by a greater SCP value for a nominal cooling load.

4.4 Equilibrium of energy throughout the entire system

The unit's total input energy (Qadded) and output production (Qrejected) may be calculated using the following formulas:

${{Q}_{added}}={{Q}_{1-2}}+{{Q}_{2-3}}+{{Q}_{eva}}$                   (26)

${{Q}_{rejected}}={{Q}_{3-4}}+{{Q}_{4-1}}+{{Q}_{con}}$ [14]                   (27)

4.5 Proposed bed geometry adsorbent bed

Although there are many varieties and styles, this technique primarily allows for two designs: the rectangular and tubular adsorbent bed. The adsorbent is evenly distributed in a rectangle geometry in the rectangular design, which is split by fins that distribute the heat similarly. The adsorbent will be dispersed in cylinders of the tubular configuration. Usually, metals with strong heat conductivity qualities are used to make these tubes. The features of a tubular design are its ability to tolerate pressure differences and its straightforward construction. However, the rectangular shape may make it simpler to disperse the heat, and one single bed can accommodate more activated carbon. 

4.5.1 Bed temperature distribution

The heat transfer coefficient of the system will have different values at cooling than at heating. Nevertheless, for the period of the cooling process, the temperature will not be evenly spread across the bed, where ${{T}_{new}}$ is the new temperature, ${{T}_{amb}}$ is the ambient temperature, and ${{T}_{old}}$ is the previous temperature, provides the transitory heat transfer [53].

$\theta =\frac{{{T}_{new}}-~{{T}_{amb}}~}{{{T}_{old}}-{{T}_{amb}}}$                   (28)

Eq. (36) provides the dimensionless temperature, denoted by $\theta$, which is dependent on both the Fourier and Biot numbers:

${{B}_{i}}=\frac{alpha~L}{k},~~{{F}_{o}}=\frac{\alpha t}{{{L}^{2}}}$                   (29)

$\frac{1}{tan{{\mu }_{n}}}=~\frac{{{\mu }_{n}}}{{{B}_{i}}}$                   (30)

The Biot number is symbolized by the symbol $~~{{B}_{i}}$, where $\text{alpha}~$ is the external heat transfer coefficient (or outside film coefficient) is a convection coefficient, in, $~\frac{W}{{{m}^{2}}K}, L~$ denotes the conduction length in, $~~m$ and $k~$represents the body's thermal conductivity in $\frac{W}{K.m}$ .${{F}_{o}}$ Is the Fourier number, where, $\alpha ~$ is thermal diffusivity in, $~\frac{{{m}^{2}}}{sec}$ and $t$ is time in sec. The thermal diffusivity is recognized as, $\alpha =\frac{k}{{{C}_{p}}~\rho }$ where ${{C}_{p}}$ specific heat and $\rho $ is the density of AC the roots ${{\mu }_{n}}$ is the solution Eigen (roots), for the parametric Eq. (30).

4.5.2 Flat temperature distribution profile 

For plane adsorbent bed geometry, for the range of Biot values 0.1 < Bi < 100. The following equation defines the distribution of bed temperature. Where, $\cos \left( {{\mu }_{n}}\frac{x}{L} \right)$ characterizes the precise location of the computed temperature:

$\theta=\sum_{n=1}^{n \rightarrow \infty} \frac{2 \sin \mu_n}{\mu_n+\sin \mu_n \cos \mu_n} \cos \left(\mu_n \frac{x}{L}\right) \exp \left(-\mu_n^2 F_o\right)$                   (31)

For plane bed design, at the central bed, $x=0$. After determining the temperature profile for the center, the temperature profile of the plate's surface is obtained by setting $x=L$, and setting $x=0.5\ L$ obtains the temperature profile of the plate's middle. It is important to find the average distribution of the plane temperature to estimate the loading heat outline. To find the average temperature profile, use the equation:

$\bar{\theta}=\sum_{n=1}^{n \rightarrow \infty} \frac{2 \sin ^2 \mu_n}{\mu_n^2+\mu_n \sin \mu_n \cos \mu_n}\ \exp \left(-\mu_n^2 F_o\right)$                   (32)

4.5.3 Cylindrical design temperature distribution profile

For the tubular bed geometry, the listed below are usable, which is the Bessel function is:

$\theta=\sum_{n=1}^{n \rightarrow \infty} \frac{2 \mathrm{~J}_1\left(\mu_n\right)}{\mu_n\left[J_0{ }^2\left(\mu_n\right)+J_1{ }^2\left(\mu_n\right)\right]} J_0\left(\mu_n \frac{x}{r}\right) \exp \left(-\mu_n^2 F_o\right)$                   (33)

The roots$~{{\mu }_{n}}$ is the solution of the parametric equations below:

${{y}_{n}}=\frac{{{\mu }_{n}}}{{{B}_{i}}},~~{{y}_{n}}=\frac{1}{\tan {{\mu }_{n}}}$                   (34)

In this modeling only three of roots $\left(\mu_n\right),\ (\mathrm{n}=3)$ because additional terms have neglected effects on adsorbent bed generation temperature profile, $\boldsymbol{\mu}_{\boldsymbol{n}}$ is reiterated. The ranges of these roots, $\mu_1=(0-2), \mu_2=(3-5), \mu_3=(6-8)$.

The average dimensionless temperature is calculated by the Equation below:

$\theta =\frac{{{T}_{t}}-~{{T}_{amb}}~}{{{T}_{t=0}}-{{T}_{amb}}}$                   (35)

To calculate the average temperature profile through the tabular bed, use the listed equation:

$\bar{\theta }=\underset{n=1}{\overset{n\to \infty }{\mathop \sum }}\,\frac{4B_{i}^{2}({{\mu }_{n}})}{{{\mu }_{n}}^{2}\left[ {{\mu }_{n}}^{2}+B_{i}^{2} \right]}\text{exp}\left( -\mu _{n}^{2}~{{F}_{o}} \right)$ [53]                   (36)

4.6 Transient analysis - cooling transient

For most performance and cold production improvement way of solar adsorption machines, is rapid heating and cooling of the systems. The features of granular activated carbon determine the temperature profile in the bed, or how long it takes the entire bed to cool down. The four key factors may be adjusted to varying ranges: the thickness of the bed (Ro-Ri), the density of adsorbent (ρ), the external heat transfer coefficient (alpha), and the conductivity (k). It is essential to understand that the adsorption heat is not taken into consideration. Just the bed is a cooling process, stage (3→4).

4.6.1 Bed cylindrical design

Although there are many varieties and styles, using tubes filled with activated carbon with ethanol, as an adsorbent bed. Figure 6. Depicts the cylindrical geometrical design of the adsorbent bed with inner and outer tubes. The region between the two tubes (tabular) is filled with activated carbon. Since the adsorbent bed's upper and lower surfaces were insulated, it was thought that mass and heat transmission only happened radially. Ethanol vapor, or the adsorptive material, moved from the inlet's surface (R=Ri) to the bed's exterior (R=Ro). There are several benefits to this design.

• Initially, the cylindrical form cools more quickly than heat may be discharged from the flat plate bed from all sides of the cylinder, not just one.
• Second, the adsorbate flows into and out of the adsorbent more freely thanks to the tube's center being removed. There is less adsorbent per unit area and more sun radiation.
• The adsorbent will be dispersed in cylinders of the tubular configuration.
• Usually, metals with strong heat conductivity qualities are used to make these tubes.
• The advantages of a tubular design include its easy construction and ability to sustain pressure differential.

In the current adsorbent bed design, the equivalent radius (Ro-Ri) of the tubular generator bed was 0.048 m, whereas the inner and outer radius of the annular adsorbent bed were 0.06 and 0.012 m, respectively.  Contains an AC. mass of 16 kg with a density of 2200 kg/m³, 1m length of the cylindrical bed, and (1×1) m² solar water collector serving as the fluid exchanger.

6.png

Figure 6. Bed cylindrical generator geometry

Under the climate conditions of Iraq, an optimization of two-bed solar adsorption cooling unit working with an activated carbon /ethanol pair is conducted. The optimal cycle performances of an adsorption system are assessed and simulated by EES with the desorption temperature range of 60 to 95℃ and the evaporator temperatures of 5℃ for cooling applications. The current investigation is continued research after the research paper by the same author [14] and the outcomes were as follows:

It was found that the variant of the adsorbent bed thickness affects the specific cooling power more than the cycle COP. Instead, it was put in evidence that the increase in the heat source temperature would increase cycle COP, solar COP, and specific cooling power. Within the ranges of heat source temperatures between 80℃ and 95℃, with small radial bed thickness (4.8–5 cm) with an optimum cooling time about one hour. The adsorption and desorption processes are prolonged because of the bed's high thermal resistance, which slows down heat transport.

More adsorbent is utilized as the thickness of the adsorbent bed grows, consequently, more heat is anticipated to be necessary to raise the unit's overall temperature; the thermal there is a noticeable increase in resistance across the adsorbent bed. Shorter cycle times and reduced adsorption capabilities. A similar result has been confirmed by earlier studies on adsorption systems in general.

The temperature change of the AC bed over time is observed to rise quickly at the start of the adsorption process, and then progressively drop until it approaches the bath temperature. This happened because there is a very big amount of ethanol adsorbed at the start of adsorption, which leads to a correspondingly large heat of adsorption.

Reducing the thickness of the adsorbent bed improves heat transmission in the radial direction. A thinner adsorbent bed will result in a shorter cycle time and more specific cooling power for design purposes. Raising the number of beds Thus, thickness negatively affects the bed's ability to transmit heat. There is a trade-off to be aware of: while reducing bed thickness improves heat transport over the adsorbent bed, it also reduces the mass of the adsorbent and, consequently, the adsorption capacity.

The adsorption capacity is improved when the bed thickness is reduced because the heat of the bed thickness squared determines the diffusion time scale. Furthermore, lowering the bed Thickness affects uploading more strongly than raising thermal conductivity. The reason for this is that additionally, less heat is produced during the adsorption process.

Thus, with the same heat transfer and coolant temperature consequently, a thinner bed will cool down considerably quicker than a bigger bed. However, it ought to be said that decreasing the width of the bed is not usually advised because it lowers the mass ratio of lower ethanol adsorption results from an increase in temperature at the top layers of the adsorbent as its thickness rises. Lower than 5 cm adsorbent thicknesses provide superior cooling and increased absorption.

To improve the thermal conductivity of AC nevertheless, the adsorber's thermal mass will also rise with the addition of metal fins. The adsorber's thermal reaction time will lengthen, and the energy needed to increase. SCP and COP could be negatively impacted by this. Consolidating high-thermal conductivity material with AC boosts the bed's thermal conductivity and enables an increase in bed thickness, which is a method of problem-solving.

The COP of the system is positively affected by the increasing value of the adsorbent bed thickness but the same effect on the SCP is not observed. The distributions of solid phase temperature and the adsorbate concentration strongly depend upon adsorbent bed thickness but not pressure. For better specific cooling power, it is necessary to lower the conductive and convective resistances to quickly attain the equilibrium conditions. The effect of Fourier number on SCP produced by different bed thicknesses. It is found that SCP increases as the Fourier number of the adsorbent-adsorbate thickness increases.