Performance Optimization of a Silica Gel–Water Adsorption Chiller Using Grey Wolf-Based Multi-Objective Algorithms and Regression Analysis

Abstract

The growing need for cooling, combined with the environmental concerns surrounding conventional mechanical vapour compression (MVC) systems, has accelerated research for sustainable cooling solutions driven by low-grade heat. Single-stage dual-bed adsorption chillers (ADCs) using silica gel and water provide a promising approach due to their continuous cooling output, lower complexity, and the use of environmentally safe working fluids. However, limitations in their performance, specifically in the coefficient of performance (COP), cooling capacity (${\mathrm{Q}}_{\mathrm{c}\mathrm{c}}$), and waste heat recovery efficiency (${\mathsf{\eta }}_{\mathrm{e}}$), necessitate improvement through optimization. This study employs statistically validated regression-based objective functions to optimize ten decision variables using the single Grey Wolf Optimizer (GWO) and its multi-objective variant, Muilti-Objective Grey Wolf Optimization (MOGWO), for a silica gel–water single-stage dual-bed ADC. The results from the single-objective optimization showed a maximum coefficient of performance (COP) of 0.697, cooling capacity (${\mathrm{Q}}_{\mathrm{c}\mathrm{c}}$) of 20.76 kW, and waste heat recovery efficiency (ηe) of 0.125. The values from the Pareto-optimal solutions for the MOGWO ranged from 0.5123 to 0.6859 for COP, 12.45 to 20.73 kW for ${\mathrm{Q}}_{\mathrm{c}\mathrm{c}}$ and 8.24% to 12.48% for ${\mathsf{\eta }}_{\mathrm{e}}$, demonstrating superior performance compared to existing benchmarks. A one-at-a-time sensitivity analysis revealed non-linear and non-monotonic impacts of variables, confirming the robustness and physical realism of the MOGWO model. The developed MOGWO framework effectively enhances the performance of the single-stage dual-bed ADC and improves low-grade heat utilization, offering a robust decision-support tool for system design and optimization.

1. Introduction

In the 21st century, air conditioning has become a necessity rather than a luxury. In hot, humid regions, people rely on air conditioning to alleviate the effects of high temperatures, maintain thermal comfort indoors, and reduce heat-related health risks (such as heat stroke and heat syncope), especially among vulnerable groups [1].

However, the widespread use of air-conditioners has a significant impact on global energy consumption. According to a report from the International Institute of Refrigeration (IIR), the refrigeration and air-conditioning (RAC) sector already consumes approximately 20% of the world’s total electricity, and space cooling alone accounts for over 8% of worldwide electricity usage [2,3]. Under the International Energy Agency (IEA) “Baseline Scenario”, space cooling alone has the potential to triple electricity demand, and the IIR further projects electricity consumption to more than double by 2050 [2,4]. This is largely due to rising AC ownership in emerging economies (India, China, and Indonesia) and hotter ambient temperatures. In addition, the escalating rate in the world economy predicts a 2% to 3% yearly increment in electricity demand by 2030 [2,3].

Globally, mechanical vapour compression (MVC) systems are the prevalent RAC systems employed for cooling. MVC systems require a high quality of energy to run the compressor and initiate the cooling process, and typically use hydrofluorocarbon/hydrochlorofluorocarbon refrigerants with high ozone-depletion and global-warming potential [5]. Current sector emissions from the RAC account for 4.14 tonnes of carbon dioxide equivalent (GtCO2eq), representing 7.8% of total global greenhouse gas (GHG) emissions [6]. Without intervention, RAC-related carbon dioxide emissions are estimated to double due to their significant environmental footprint, with 63% stemming from indirect power-sector emissions and 37% global-warming impact from direct refrigerant leakages [2].

While increasing Seasonal Energy Efficiency Ratio (SEER) standards provides some mitigations to enhance the efficiency of MVC systems, many of the currently available air conditioners operate at only one-third of the efficiency of the best available technology, leaving the remaining two-thirds underutilized [3]. These drawbacks of MVC systems and energy issues highlight the pressing requirement for a more radical, energy-efficient, and sustainable cooling alternative.

The adsorption cooling chiller is the most promising technology alternative to conventional high-energy-consuming MVC systems. Adsorption chillers, particularly multi-bed configurations, have emerged as a promising solution. They are entirely or partly powered by low-grade energy sources like waste heat from industries, solar, biomass, etc., for heating and cooling applications [7], offering advantages such as the utilization of environmentally benign refrigerant, durability, quiet operation, lower energy demands, and the use of simple controls [8]. In addition, adsorption cooling systems can harness low-grade heat sources such as waste heat (which is usually discharged into the environment) or solar power (which is renewable) as heat sources, making them environmentally friendly [9]. Despite growing interest, suboptimal performance indicators like low COP and specific cooling power (SCP) and sensitivity to operational parameters like variations in temperature, flow rates, and working fluids have held back wide-scale deployment of ADC.

Addressing these hurdles requires a robust approach to optimize the performance of single-stage dual-bed ADCs and potentially extend the findings to complex bed adsorption chillers in the future.

Although the GWO, a nature-inspired meta-heuristic, is widely applied in various optimization contexts, it remains unexplored in the design, configuration, and operational optimization of a single-stage dual-bed adsorption chiller. This study employs GWO to demonstrate the algorithm’s efficacy in improving the coefficient of performance (COP), cooling capacity (${\mathrm{Q}}_{\mathrm{c}\mathrm{c}}$), and waste heat recovery efficiency and offers a comparative analysis of hand-tuned parameter sets against single-objective GWO and its multi-objective approach, MOGWO.

To address this gap, the present study will conduct the following:

• Develop a novel MOGWO-based approach for optimizing a single-stage dual-bed adsorption chiller;
• Optimize the coefficient of performance (COP), cooling capacity (${\mathrm{Q}}_{\mathrm{c}\mathrm{c}}$), and waste heat recovery efficiency (${\mathsf{\eta }}_{\mathrm{e}}$);
• Conduct a one-at-a-time (OAT) sensitivity analysis to quantify how key decision variables influence each objective.

Multi-bed systems have been reported to reduce temperature fluctuations in the chilled water outlet and improve the usage of waste heat for waste heat ADCs. Adding heat and mass recovery processes to the ADC has also been shown to improve the COP and cooling capacity of the ADC [10,11,12,13].

Table 1. Summary of the relevant studies on two-bed and multi-bed adsorption chiller systems.

Study	System Type	Working Pair	Heat Source	Cycle Configuration	KPIs Evaluated	Reported Performance	Optimization Used
Sah et al. [10]	Two-bed ADC	Silica gel–Water	Hot Water (85 °C)	Fixed Cycle (1600 s)	COP, $Q_{cc}$	$Q_{cc}$ = 5.95 kW at $T_{hw,in }$= 85 °C, $T_{cw,in}$= 25 °C, $T_{chw,in}$= 14 °C	No
Krzywanski et al. [11]	Re-heat Two-Stage ADC	Silica gel–Water	Hot Water	AI/ANFIS-based	Cooling Capacity	No explicit value(s) reported; ANFIS-based parametric modelling	Yes
Elsheniti et al. [12]	Two-bed ADC	Silica gel–Water	Hot Water	Geometry Variation	COP, SCC, $Q_{cc}$	COP ↑ 68% and SCC ↑ 42% under turbulent regime	No
Chorowski et al. [13]	Three-bed, Two-evap ADC	—	Hot Water	Switching control (cooperation unit)	COP, Operational Noise	Improved control strategy led to COP increase	No
Gado et al. [14]	Two-bed ADC	Silica gel–Water	Hot Water	Optimized Cycle Time	Qcc, Enhancement Ratio	$Q_{cc}$ ↑ 15.6% at $T_{hw,in }$= 95 °C, $T_{cw,in}$= 40 °C, $T_{chw,in}$= 10 °C	No
Qadir et al. [15]	Two-bed, Solar ADC	Silica gel–Water	Solar	Adaptive vs. Fixed	COP, SCP	Adaptive cycle ↑ SCP by 19% and COP by 66%; min $T_{evap}$= 0.7 °C	No
Qadir et al. [16]	Two-bed Solar ADC	Silica gel–Water	Solar	Optimal Fixed vs. Adaptive	SCP, COP	Adaptive cycle ↑ SCP by 19% and COP by 66%; min $T_{evap}$= 0.7 °C	No
Present Study	Dual-bed, Single-stage ADC	Silica gel–Water	Waste Heat	MOGWO + Sensitivity	COP, $Q_{cc}$, ηₑ	COP = 0.69, Qcc = 20.76 kW, ${\mathsf{\eta }}_{\mathrm{e}}$= 0.125	Yes (MOO + SA)

presents selected studies on two-bed, three-bed, and recovery-enhanced adsorbers, highlighting key performance indicators (KPIs).

2. Materials and Methods

The single-stage dual-bed ADC described in this paper is according to the study by Papoutsis et al. [17]. Objective functions for COP, Qcc, and ηₑ are adopted directly from Papoutsis et al. without any recalibration. Validation metrics are those reported therein. The main components, as seen in Figure 1, are an evaporator, two adsorbent beds, and a condenser. The condenser and evaporator are connected to the adsorbent by valves (VA–VD), and the adsorbent beds consist of fin and tube heat exchangers with adsorbent materials packed between their fins to enhance heat exchange.

In Mode A, valves VA and VD are closed, and VB and VC remain open. Adsorbent Bed 1 is connected to the evaporator to initiate the adsorption–evaporation process. The chilled water supplies heat to the adsorbate (water) to boil in the evaporator at a low pressure while reducing the temperature. This causes adsorption of the refrigerant vapours from Bed 1. Heat rejected during the adsorption process is sent to the cooling water circuit. Simultaneously, the desorption–condensation process also happens in Bed 2 and the condenser. Heat is supplied to Bed 2 to desorb the refrigerant collected in the adsorbent material, sending the heat of condensation to the cooling water circuit. Modes A and B can alternate as an adsorber or desorber according to the opening and closing of the valves and pressure gains or losses.

Based on the system description in Figure 1, the three linear regression equations used as objective functions for the single-stage dual-bed ADC with their adjusted coefficient ($R^2)$ are as follows:

1.

Maximize COP: For adsorption cycles, COP is a key performance indicator calculated by estimating the cooling and heating within the evaporator and condenser, respectively. The formula for the chiller’s COP can be expressed as in Equation (1) [18].

$$\mathrm{C}\mathrm{O}\mathrm{P}={\displaystyle \frac{{\int }_0^{{\mathrm{t}}_{\mathrm{h}\mathrm{c}}}{\dot{\mathrm{m}}}_{\mathrm{c}\mathrm{h}\mathrm{w}}{\mathrm{c}}_{\mathrm{p}\mathrm{w}}\left({\mathrm{T}}_{\mathrm{c}\mathrm{h}\mathrm{w},\mathrm{i}\mathrm{n}}-{\mathrm{T}}_{\mathrm{c}\mathrm{h}\mathrm{w},\mathrm{o}\mathrm{u}\mathrm{t}}\right)\mathrm{d}\mathrm{t}}{{\int }_0^{{\mathrm{t}}_{\mathrm{h}\mathrm{c}}}{\dot{\mathrm{m}}}_{\mathrm{h}\mathrm{w}}{\mathrm{c}}_{\mathrm{p}\mathrm{w}}\left({\mathrm{T}}_{\mathrm{h}\mathrm{w},\mathrm{i}\mathrm{n}}-{\mathrm{T}}_{\mathrm{h}\mathrm{w},\mathrm{o}\mathrm{u}\mathrm{t}}\right)\mathrm{d}\mathrm{t}}}$$

(1)

where

• ${\mathrm{t}}_{\mathrm{h}\mathrm{c}}$ = half cycle time;
• ${\dot{\mathrm{m}}}_{\mathrm{c}\mathrm{h}\mathrm{w}}$ = chilled water mass flow rate;
• ${\mathrm{c}}_{\mathrm{p}\mathrm{w}}$ = specific heat capacity of the water;
• ${\mathrm{T}}_{\mathrm{c}\mathrm{h}\mathrm{w},\mathrm{i}\mathrm{n}}$ = chilled water inlet temperature;
• ${\mathrm{T}}_{\mathrm{c}\mathrm{h}\mathrm{w},\mathrm{o}\mathrm{u}\mathrm{t}}$ = chilled water outlet temperature;
• ${\mathrm{T}}_{\mathrm{h}\mathrm{w},\mathrm{i}\mathrm{n}}$ = hot water inlet temperature;
• ${\mathrm{T}}_{\mathrm{h}\mathrm{w},\mathrm{o}\mathrm{u}\mathrm{t}}$ = hot water outlet temperature.
• The objective functions were adopted verbatim from Papoutsis et al. [17] with no recalibrations. The linear regression equation of COP for the single-stage dual-bed ADC is shown in Equation (2), according to Papoutsis et al. [17] and with an adjusted $R^2=0.8041$.

$$\begin{gathered} \mathrm{COP}=-1.1469+0.0014{\mathrm{T}}_{\mathrm{h}\mathrm{w},\mathrm{i}\mathrm{n}}-0.0085{\mathrm{T}}_{\mathrm{c}\mathrm{w},\mathrm{i}\mathrm{n}}+0.0124{\mathrm{T}}_{\mathrm{c}\mathrm{h}\mathrm{w},\mathrm{i}\mathrm{n}}+0.0050{\dot{\mathrm{m}}}_{\mathrm{h}\mathrm{w}}+0.0099{\dot{\mathrm{m}}}_{\mathrm{c}\mathrm{w},\mathrm{b}\mathrm{e}\mathrm{d}} \\ +0.0793{\dot{\mathrm{m}}}_{\mathrm{c}\mathrm{h}\mathrm{w}}+0.0092{\dot{\mathrm{m}}}_{\mathrm{c}\mathrm{w},\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}}+5.0687\times {10}^{-6}\mathrm{U}{\mathrm{A}}_{\mathrm{b}\mathrm{e}\mathrm{d}}+5.2952\times {10}^{-6}\mathrm{U}{\mathrm{A}}_{\mathrm{e}\mathrm{v}\mathrm{a}\mathrm{p}} \\ +4.6260\times {10}^{-7}\mathrm{U}{\mathrm{A}}_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}} \end{gathered}$$

(2)

where,

• ${\mathrm{T}}_{\mathrm{c}\mathrm{w},\mathrm{i}\mathrm{n}}$ = cooling water inlet temperature;
• ${\dot{\mathrm{m}}}_{\mathrm{h}\mathrm{w}}$ = mass flow rate of the hot water;
• ${\dot{\mathrm{m}}}_{\mathrm{c}\mathrm{w},\mathrm{b}\mathrm{e}\mathrm{d}}$ = cooling water mass flow rate of the bed;
• ${\dot{\mathrm{m}}}_{\mathrm{c}\mathrm{w},\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}}$ = cooling water mass flow rate of the condenser;
• $\mathrm{U}{\mathrm{A}}_{\mathrm{b}\mathrm{e}\mathrm{d}}$ = adsorbent bed overall thermal conductance;
• $\mathrm{U}{\mathrm{A}}_{\mathrm{e}\mathrm{v}\mathrm{a}\mathrm{p}}$ = evaporator overall thermal conductance;
• $\mathrm{U}{\mathrm{A}}_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}}$= condenser overall thermal conductance.

2.

Maximize Cooling Capacity (${\mathrm{Q}}_{\mathrm{c}\mathrm{c}}$): Cooling capacity is another primary indicator of adsorption chiller performance. Qcc is defined in Equation (3) [19].

$${\mathrm{Q}}_{\mathrm{c}\mathrm{c}}={\displaystyle \frac{{\int }_0^{{\mathrm{t}}_{\mathrm{h}\mathrm{c}}}{\dot{\mathrm{m}}}_{\mathrm{c}\mathrm{h}\mathrm{w}}{\mathrm{c}}_{\mathrm{p}\mathrm{w}}\left({\mathrm{T}}_{\mathrm{c}\mathrm{h}\mathrm{w},\mathrm{i}\mathrm{n}}-{\mathrm{T}}_{\mathrm{c}\mathrm{h}\mathrm{w},\mathrm{o}\mathrm{u}\mathrm{t}}\right)\mathrm{d}\mathrm{t}}{{\mathrm{t}}_{\mathrm{h}\mathrm{c}}}}$$

(3)

The linear regression representation of ${\mathrm{Q}}_{\mathrm{c}\mathrm{c}}$ for the single-stage dual-bed ADC is defined as in Equation (4), with an adjusted $R^2=0.9250$ [17].

$$\begin{gathered} {\mathrm{Q}}_{\mathrm{c}\mathrm{c}}=-64.6199+0.3107{\mathrm{T}}_{\mathrm{h}\mathrm{w},\mathrm{i}\mathrm{n}}-0.8625{\mathrm{T}}_{\mathrm{c}\mathrm{w},\mathrm{i}\mathrm{n}}+0.7601{\mathrm{T}}_{\mathrm{c}\mathrm{h}\mathrm{w},\mathrm{i}\mathrm{n}}+0.6108{\dot{\mathrm{m}}}_{\mathrm{h}\mathrm{w}}+0.9944{\dot{\mathrm{m}}}_{\mathrm{c}\mathrm{w},\mathrm{b}\mathrm{e}\mathrm{d}} \\ +4.4533{\dot{\mathrm{m}}}_{\mathrm{c}\mathrm{h}\mathrm{w}}+0.5967{\dot{\mathrm{m}}}_{\mathrm{c}\mathrm{w},\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}}+0.0006\mathrm{U}{\mathrm{A}}_{\mathrm{b}\mathrm{e}\mathrm{d}}+0.0003\mathrm{U}{\mathrm{A}}_{\mathrm{e}\mathrm{v}\mathrm{a}\mathrm{p}}+2.6623\times {10}^{-5}\mathrm{U}{\mathrm{A}}_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}} \end{gathered}$$

(4)

3.

Maximize waste heat recovery efficiency (${\mathsf{\eta }}_{\mathrm{e}}$): Effective heat recovery strategies are pivotal in enhancing the efficiency of ADCs, and heat recovery is shown to influence the overall system performance [19]. Thus, ${\mathsf{\eta }}_{\mathrm{e}}$ is defined as a performance indicator for the single-stage dual-bed ADC, according to Equation (5) [19].

$${\mathsf{\eta }}_{\mathrm{e}}={\displaystyle \frac{{\int }_0^{{\mathrm{t}}_{\mathrm{h}\mathrm{c}}}{\dot{\mathrm{m}}}_{\mathrm{c}\mathrm{h}\mathrm{w}}{\mathrm{c}}_{\mathrm{p}\mathrm{w}}\left({\mathrm{T}}_{\mathrm{c}\mathrm{h}\mathrm{w},\mathrm{i}\mathrm{n}}-{\mathrm{T}}_{\mathrm{c}\mathrm{h}\mathrm{w},\mathrm{o}\mathrm{u}\mathrm{t}}\right)\mathrm{d}\mathrm{t}}{{\int }_0^{{\mathrm{t}}_{\mathrm{h}\mathrm{c}}}{\dot{\mathrm{m}}}_{\mathrm{h}\mathrm{w}}{\mathrm{c}}_{\mathrm{p}\mathrm{w}}\left({\mathrm{T}}_{\mathrm{h}\mathrm{w},\mathrm{i}\mathrm{n}}-{\mathrm{T}}_{\mathrm{h}\mathrm{w},\mathrm{o}\mathrm{u}\mathrm{t}}\right)\mathrm{d}\mathrm{t}}}$$

(5)

The regression equation for ${\eta }_e$ is represented by Equation (6), with an adjusted $R^2=0.8371$ [17].

$$\begin{array}{cc}\hfill {\mathsf{\eta }}_{\mathrm{e}}=-0.2347-& 0.0003{\mathrm{T}}_{\mathrm{h}\mathrm{w},\mathrm{i}\mathrm{n}}-0.0019{\mathrm{T}}_{\mathrm{c}\mathrm{w},\mathrm{i}\mathrm{n}}+0.0026{\mathrm{T}}_{\mathrm{c}\mathrm{h}\mathrm{w},\mathrm{i}\mathrm{n}}+0.0277{\dot{\mathrm{m}}}_{\mathrm{h}\mathrm{w}}+0.0034{\dot{\mathrm{m}}}_{\mathrm{c}\mathrm{w},\mathrm{b}\mathrm{e}\mathrm{d}}\hfill \\ & +0.0150{\dot{\mathrm{m}}}_{\mathrm{c}\mathrm{h}\mathrm{w}}+0.0019{\dot{\mathrm{m}}}_{\mathrm{c}\mathrm{w},\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}}+2.0286{\times {10}^{-6}\mathrm{U}\mathrm{A}}_{\mathrm{b}\mathrm{e}\mathrm{d}}+1.0279\times {10}^{-6}\mathrm{U}{\mathrm{A}}_{\mathrm{e}\mathrm{v}\mathrm{a}\mathrm{p}}\hfill \\ & +6.8084\times {10}^{-8}\mathrm{U}{\mathrm{A}}_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}}\hfill \end{array}$$

(6)

Based on the objective functions from Equations (2), (4) and (6), the decision variables and bounds are presented in

Table 2. Decision variables and bounds.

Variable Description	Symbol	Range	Units
Hot water inlet temperature	${\mathrm{T}}_{\mathrm{h}\mathrm{w},\mathrm{i}\mathrm{n}}$	65–95	°C
Cooling water inlet temperature	${\mathrm{T}}_{\mathrm{c}\mathrm{w},\mathrm{i}\mathrm{n}}$	22–36	°C
Chilled water inlet temperature	${\mathrm{T}}_{\mathrm{c}\mathrm{h}\mathrm{w},\mathrm{i}\mathrm{n}}$	10–20	°C
Hot water mass flow rate	${\dot{\mathrm{m}}}_{\mathrm{h}\mathrm{w}}$	0.8–2.2	kg/s
Bed cooling water mass flow rate	${\dot{\mathrm{m}}}_{\mathrm{c}\mathrm{w},\mathrm{b}\mathrm{e}\mathrm{d}}$	0.8–2.2	kg/s
Chilled water mass flow rate	${\dot{\mathrm{m}}}_{\mathrm{c}\mathrm{h}\mathrm{w}}$	0.2–1.4	kg/s
Condenser cooling water mass flow rate	${\dot{\mathrm{m}}}_{\mathrm{c}\mathrm{w},\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}}$	0.8–2.2	kg/s
Adsorbent bed overall thermal conductance	${\mathrm{U}\mathrm{A}}_{\mathrm{b}\mathrm{e}\mathrm{d}}$	2000–10,000	W/K
Evaporator overall thermal conductance	$\mathrm{U}{\mathrm{A}}_{\mathrm{e}\mathrm{v}\mathrm{a}\mathrm{p}}$	2000–10,000	W/K
Condenser overall thermal conductance	$\mathrm{U}{\mathrm{A}}_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}}$	10,000–24,000	W/K

.

$R^2$ values, as published by Papoutsis et al. [17], for their regression equations are $0.8041$ for COP, $0.9250$ for ${\mathrm{Q}}_{\mathrm{c}\mathrm{c}}$ and $0.8371$ for ${\mathsf{\eta }}_{\mathrm{e}}.$ Comparatively, the high $R^2$ values across all three objective functions (COP, ${\mathrm{Q}}_{\mathrm{c}\mathrm{c}}$, and ${\mathsf{\eta }}_{\mathrm{e}}$) confirm the suitability of the chosen equations for modelling the relationships between the variables [20]. Therefore, the $R^2$ values are adopted without refitting.

2.1. Overview of Algorithmic Techniques

Figure 2 illustrates the hierarchical overview of various algorithmic techniques and the five broad categorisations of meta-heuristic algorithm techniques [21]. Meta-heuristic algorithms are broadly classified under five categories: swarm intelligence-based algorithms, bio-stimulated algorithms, evolutionary algorithms, nature-inspired algorithms, and physics-based algorithms. The Genetic Algorithm (GA), proposed by Holland [22] in 1992, is the most recognised evolutionary algorithm, justified in Darwin’s evolution theory. The GA has been successfully used to optimize a real-world control system [23].

2.2. Greywolf Optimization (GWO)

This study employs the GWO, a novel swarm-intelligence algorithm inspired by the grey wolves’ social dominance hierarchy and hunting strategy to find optimal solutions for optimization problems [24]. In the GWO framework, the best-performing solution is designated alpha (α), signifying the top rank in the wolf’s social structure. The subsequent two best solutions are designated beta (β) and delta (δ), respectively, and the remaining solutions are labelled omega (ω). The α, β, and δ wolves lead the hunting process, with the ω wolves following their lead to find the global optimum. Grey wolves begin encircling their prey during hunting by employing the equations shown below [24].

Equation (7) shows how the position vector, $\overrightarrow{M}$ of grey wolves is calculated.

$$\overrightarrow{M}=\left|\overrightarrow{Q}·{\overrightarrow{N}}_p\left(t\right)-\overrightarrow{N}(t)\right|$$

(7)

where

• $t$ = current iteration;
• $\overrightarrow{Q}$ = coefficient vector;
• $\overrightarrow{N}$ = position vector;
• ${\overrightarrow{N}}_p$ = position vector of the prey.

The wolf’s position at the next step, $\overrightarrow{N}\left(t+1\right)$, is then updated according to Equation (8).

$$\overrightarrow{N}\left(t+1\right)={\overrightarrow{N}}_p-\overrightarrow{H}· \overrightarrow{M}$$

(8)

where $\overrightarrow{H}$ = coefficient vector.

This update adjusts the wolf’s position to the prey’s position and the calculated vector $\overrightarrow{M}$, influencing the encircling behaviour.

Equations (9) and (10) are employed to calculate the coefficient vectors $\overrightarrow{Q}$ and $\overrightarrow{H}$, respectively.

$$\overrightarrow{Q}=2·\overrightarrow{r_2}$$

(9)

$$\overrightarrow{H}=2·\overrightarrow{h·}\overrightarrow{r_1}-\overrightarrow{h}$$

(10)

Components $\overrightarrow{r_1}$ and $\overrightarrow{r_2}$ contain random values between 0 and 1 and $\overrightarrow{h}$ decrease linearly from 2 to 0 during the iterative optimization process. The GWO algorithm generates a random set of solutions when optimization starts. The algorithm then saves the top three solutions and updates the positions of the remaining search agents to their optimum solutions. When the termination criteria are met, the alpha solution’s location and value define the global optimum. Equations (11)–(13) calculate how far away the search agent is from the three leader wolves. ${\overrightarrow{M}}_{\alpha }$, ${\overrightarrow{M}}_{\beta }$, and ${\overrightarrow{M}}_{\delta }$ represent the distance and direction of the current search agent (wolf) to the alpha (α), beta (β), and delta (δ) wolves, respectively. $\overrightarrow{N}$ is the current position of the search agent and $\overrightarrow{Q}$ is a coefficient vector. ${\overrightarrow{N}}_{\alpha }, {\overrightarrow{N}}_{\beta }$, and ${\overrightarrow{N}}_{\delta }$ are the positions of the alpha, beta, and delta wolves.

$${\overrightarrow{M}}_{\alpha }=\left|{\overrightarrow{Q}}_1·{\overrightarrow{N}}_{\alpha }-\overrightarrow{N}\right|$$

(11)

$${\overrightarrow{M}}_{\beta }=\left|{\overrightarrow{Q}}_1·{\overrightarrow{N}}_{\beta }-\overrightarrow{N}\right|$$

(12)

$${\overrightarrow{M}}_{\delta }=\left|{\overrightarrow{Q}}_1·{\overrightarrow{N}}_{\delta }-\overrightarrow{N}\right|$$

(13)

Equations (14)–(16) use the distance from each of the three top wolves to calculate a probable next position for the search. Thus, these equations update the position based on the leaders. ${\overrightarrow{N}}_1$, ${\overrightarrow{N}}_2$, and ${\overrightarrow{N}}_3$ denote the next position of the search agent according to the influence of the alpha, beta, and delta wolves, respectively. ${\overrightarrow{H}}_1$, ${\overrightarrow{H}}_2$, and ${\overrightarrow{H}}_3$ are coefficient vectors.

$${\overrightarrow{N}}_1={\overrightarrow{N}}_{\alpha }-{\overrightarrow{H}}_1·\left({\overrightarrow{M}}_{\alpha }\right)$$

(14)

$${\overrightarrow{N}}_2={\overrightarrow{N}}_{\beta }-{\overrightarrow{H}}_2·\left({\overrightarrow{M}}_{\beta }\right)$$

(15)

$${\overrightarrow{N}}_3={\overrightarrow{N}}_{\delta }-{\overrightarrow{H}}_3·\left({\overrightarrow{M}}_{\delta }\right)$$

(16)

Equation (17) determines the final position of the search agent by finding the average of ${\overrightarrow{N}}_1$, ${\overrightarrow{N}}_2$, and ${\overrightarrow{N}}_3$ at the next iteration, $\left(t+1\right)$, to balance the influence of the alpha, beta, and delta wolves.

$$\overrightarrow{N}\left(t+1\right)={\displaystyle \frac{{\overrightarrow{N}}_1+{\overrightarrow{N}}_2+{\overrightarrow{N}}_3 }{3}}$$

(17)

The optimization methodology followed by this study and the general GWO optimization flow chart for a single-objective optimization are illustrated in Figure 3 and Figure 4.

In energy systems, a problem could have multiple conflicting objective functions. In such instances, a multi-objective optimization approach can be used to simultaneously generate a set of alternative feasible solutions. These solutions are referred to as Pareto Front Optimal or non-dominated solutions. Figure 5 illustrates the flowchart for the multi-objective GWO technique. To adapt the standard GWO algorithm for MOGWO, two extra elements are included: the archive, to store the Pareto-optimal solutions that are non-dominated, and a leader selection mechanism, which chooses the alpha and beta wolves from the archive to head and direct the search.

2.3. Mathematical Formulation

The GWO is mathematically formulated according to Equations (18)–(20) by Mirjalili et. al. [24] to maximize the thermodynamic performance of a single-stage dual-bed silica gel–water ADC. These equations serve as a basis for implementing the GWO algorithm to improve each performance indicator individually.

The single-objective optimization of the COP is expressed as follows:

$$\mathrm{Maximize} {\mathcal{F}}_1=COP\left(T_{hw,in},T_{cw,in}, T_{chw,in},{\dot{m}}_{hw},{\dot{m}}_{cw,bed},{\dot{m}}_{chw},{\dot{m}}_{cw,cond},U{A}_{bed }, {UA}_{evap},{UA}_{cond }\right)$$

(18)

Similarly, the optimization of cooling capacity (${\mathrm{Q}}_{\mathrm{c}\mathrm{c}}$) is defined as follows:

$$\mathrm{Maximize} {\mathcal{F}}_2=Q_{cc}\left(T_{hw,in},T_{cw,in}, T_{chw,in},{\dot{m}}_{hw},{\dot{m}}_{cw,bed},{\dot{m}}_{chw},{\dot{m}}_{cw,cond},U{A}_{bed }, U{A}_{evap},U{A}_{cond }\right)$$

(19)

The objective function for waste heat recovery efficiency (${\eta }_e$) is written as follows:

$$\mathrm{Maximize} {\mathcal{F}}_3={\eta }_e\left(T_{hw,in},T_{cw,in}, T_{chw,in},{\dot{m}}_{hw},{\dot{m}}_{cw,bed},{\dot{m}}_{chw},{\dot{m}}_{cw,cond},U{A}_{bed }, U{A}_{evap},U{A}_{cond }\right)$$

(20)

After the single-objective optimization of each objective function, the optimal trends were reviewed to identify ‘conflicts’. From the analysis of SOO results section, an upward arrow (↑) indicates that the variable increases at optimality, a downward arrow (↓) means it decreases, while (≠) denotes that the trend is in conflict across the objective functions. This analysis method is necessary to underscore the need for multi-objective optimization.

The multi-objective optimization problem combines the three objectives as Equation (21) [25]:

$$\mathrm{Maximize} \mathcal{F}=\left\{{\mathcal{F}}_1,{\mathcal{F}}_2,{\mathcal{F}}_3\right\}$$

(21)

Equations (18)–(21) are subject to the following variable constraints:

• $65\le T_{hw,in}\le 95$ [°C];
• $22\le T_{cw,in}\le 36$ [°C];
• $10\le T_{chw,in}\le 20$ [°C];
• $0.8\le {\dot{m}}_{hw}\le 2.2$ [kg/s];
• $0.8\le {\dot{m}}_{cw,bed}\le 2.2$ [kg/s];
• $0.2\le {\dot{m}}_{chw}\le 1.4$ [kg/s];
• $0.8\le {\dot{m}}_{cw,cond}\le 2.2$ [kg/s];
• $2000\le U{A}_{bed}\le \mathrm{10,000}$ [W/K];
• $2000\le U{A}_{evap}\le \mathrm{10,000}$ [W/K];
• $\mathrm{10,000}\le U{A}_{cond}\le \mathrm{24,000}$ [W/K].

3. Results

3.1. Single-Objective Optimization

The core single-objective grey wolf optimization algorithm implemented in this study was adapted from the original work by Mirjalili et al. [26], which is available online. The algorithm was tailored to the characteristics of the three objective functions and the selected decision variables and was implemented on the MATLAB R2021a platform using a 64-bit operating system, an x64-based processor, 8 GB RAM, and a 12th Gen Intel(R) Core (TM) i7-1255U CPU @ 1.70 GHz laptop.

3.1.1. Coefficient of Performance (COP) Maximization

To identify the conditions for maximum COP, the GWO algorithm was used to solve Equation (2), with the decision variables and values from Table 2. The resulting data in

Table 3. Optimal decision variables for single-objective maximization of COP, $Q_{cc}$, and ${\eta }_e$.

Decision Variable	Symbol	Optimal Value (Maximum COP)	Optimal Value (Maximum ${\mathbf{Q}}_{\mathbf{c}\mathbf{c}}$)	Optimal Value (Maximum ${\mathit{\eta }}_{\mathit{e}}$)	Unit
Hot water inlet temperature	$T_{hw,in}$	95.00	95.00	65.00	°C
Cooling water inlet temperature	$T_{cw,in}$	22.00	22.00	22.00	°C
Chilled water inlet temperature	$T_{chw,in}$	20.00	19.99	19.98	°C
Hot water mass flow rate	${\dot{m}}_{hw}$	1.051	1.198	2.198	kg/s
Bed cooling water mass flow rate	${\dot{m}}_{cw,bed}$	1.388	1.750	1.658	kg/s
Chilled water mass flow rate	${\dot{m}}_{chw}$	1.400	1.390	1.396	kg/s
Condenser cooling water mass flow rate	${\dot{m}}_{cw,cond}$	1.364	1.126	1.244	kg/s
Adsorbent bed overall thermal conductance	$U{A}_{bed}$	9830.45	9890.71	9882.41	W/K
Evaporator overall thermal conductance	${UA}_{evap}$	9931.11	7677.01	6501.39	W/K
Condenser overall thermal conductance	${UA}_{cond}$	14,157.87	11,495.20	12,386.87	W/K
Maximized objective value	—	0.69695	20.7589	0.12527	—/kW/—

illustrate the control strategies identified based on the settings of the decision variables for COP, $Q_{cc}$, and ${\eta }_e$. To achieve the optimal COP of 0.69695, the following control strategies were identified based on the direction of the following decision variables:

• Increasing $T_{hw,in}$, $T_{chw,in}$, ${\dot{m}}_{chw}$, ${\dot{m}}_{cw,cond}$, $U{A}_{bed},U{A}_{evap}$ and $U{A}_{cond}$;
• Maintaining a lower value for $T_{cw,in}$;
• Moderate values for ${\dot{m}}_{hw}$ and ${\dot{m}}_{cw,bed}$.

3.1.2. Cooling Capacity ($Q_{cc}$) Maximization

To identify the conditions for maximum $Q_{cc}$, the GWO algorithm was used to solve Equation (4), utilising the decision variables and values from Table 2. To achieve a maximum $Q_{cc}$ of 20.76 kW, the following variable adjustments depicted in Table 3 are required:

• Maximizing $T_{hw,in}$, $T_{chw,in}$, ${\dot{m}}_{hw}$, ${\dot{m}}_{cw,bed}$, ${\dot{m}}_{chw}$, $U{A}_{bed}$ and $U{A}_{cond}$;
• Minimizing $T_{cw,in}$ and $U{A}_{evap}$;
• Moderating $U{A}_{evap}$.

3.1.3. Waste Heat Recovery Efficiency (${\eta }_e$) Maximization

To identify the conditions for maximum ${\eta }_e$, the GWO algorithm was used to solve Equation (6), utilising the decision variables and values from Table 2. To achieve a maximum ${\eta }_e$ of 0.12527, the following variable adjustments depicted in Table 3 are required:

• Maximizing $T_{cw,in}$;
• Minimizing $T_{hw,in}$, ${\dot{m}}_{cw,cond}$ and $U{A}_{evap}$;
• Moderating ${\dot{m}}_{hw}$, ${\dot{m}}_{cw,bed}$, ${\dot{m}}_{chw}$, $U{A}_{bed}$ and $U{A}_{cond}$.

3.1.4. Analysis of the Single-Objective Optimization Results

Table 4. Trend of decision variables under GWO-based single-objective optimization.

Decision Variable	COP	${\mathbf{Q}}_{\mathbf{c}\mathbf{c}}$	${\mathit{\eta }}_{\mathit{e}}$	Conflict
${\mathrm{T}}_{\mathrm{h}\mathrm{w},\mathrm{i}\mathrm{n}}$	↑	↑	↓	≠
${\mathrm{T}}_{\mathrm{c}\mathrm{w},\mathrm{i}\mathrm{n}}$	↓	↓	↓	✓
${\mathrm{T}}_{\mathrm{c}\mathrm{h}\mathrm{w},\mathrm{i}\mathrm{n}}$	↑	↑	↑	✓
${\dot{\mathrm{m}}}_{\mathrm{h}\mathrm{w}}$	↓	↑	↑	≠
${\dot{\mathrm{m}}}_{\mathrm{c}\mathrm{w},\mathrm{b}\mathrm{e}\mathrm{d}}$	↑	↑	↑	✓
${\dot{\mathrm{m}}}_{\mathrm{c}\mathrm{h}\mathrm{w}}$	↑	↑	↑	✓
${\dot{\mathrm{m}}}_{\mathrm{c}\mathrm{w},\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}}$	↑	↓	↑	≠
$\mathrm{U}{\mathrm{A}}_{\mathrm{b}\mathrm{e}\mathrm{d}}$	↑	↑	↑	✓
$\mathrm{U}{\mathrm{A}}_{\mathrm{e}\mathrm{v}\mathrm{a}\mathrm{p}}$	↑	↓	↓	≠
$\mathrm{U}{\mathrm{A}}_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}}$	↑	↑	↑	✓

presents a concise overview of the findings from optimizing the objective functions individually. The conflicts are identified, as explained under Section 2.3.

These conflicting trends from Table 4 suggest that a single set of operating parameters cannot simultaneously achieve the optimal performance for all three objectives. This requires the determination of the Pareto-optimal solutions through MOO [27].

3.2. Multi-Objective Optimization

The multi-objective grey wolf optimization algorithm implemented in this study was adapted from the original work by Mirjalili, which is available online [28]. The algorithm was tailored to the characteristics of the three objective functions and the selected decision variables and was implemented on the MATLAB platform using a 64-bit operating system, an x64-based processor, 8GB RAM, and a 12th Gen Intel(R) Core (TM) i7-1255U CPU@ 1.70 GHz laptop. The set values of the hyperparameters are provided in

Table 5. Set values of the hyperparameters.

Hyperparameter	Value
Grid inflation parameter, alpha	0.1
Leader selection pressure parameter, beta	4
Gamma	2
Archive size	100
Number of agents	100
Maximum iterations	50
Number of grids per dimension (nGrid)	100
Final a	0
Random seed	42
Leader selection	Roulette wheel based on hypercube crowding
Crowding distance	Crowding handled through a hypercube grid and the “DeleteFromRep” function

.

The MOGWO algorithm generated 100 sets of non-dominated solutions. The Pareto front from the simultaneous optimization of all three objective functions representing the best trade-offs is shown in Figure 6. Each solution on the Pareto front offers a potential trade-off, with the final selection dependent on the decision maker’s priorities. The relationships between the objective functions COP, ${\mathrm{Q}}_{\mathrm{c}\mathrm{c}}$, and ηₑ were explored through three pairwise scenarios. The resulting overall Pareto front for these three objectives is shown in Figure 6. For clarity and to highlight key interactions, two two-dimensional Pareto fronts derived from these results are subsequently analysed.

Figure 6 illustrates the achievable performance limits and the inherent trade-offs between COP, ${\mathrm{Q}}_{\mathrm{c}\mathrm{c}}$, and ${\eta }_e$. The Pareto front guides the decision-maker in identifying and selecting the most suitable operating points that could lead to improvement across the three objectives. The colour gradient on the surface indicates that attaining higher ${\mathrm{Q}}_{\mathrm{c}\mathrm{c}}$ requires a compromise on COP and may lead to a lower ${\mathsf{\eta }}_{\mathrm{e}}$. Additionally, a compromise on the potential ${\mathrm{Q}}_{\mathrm{c}\mathrm{c}}$ and the amount of waste heat recovered is necessary to achieve a higher COP. The colour gradient also illustrates how eta varies across the Pareto front, confirming that achieving a greater eta necessitates operating the chiller within specific ranges of ${\mathrm{Q}}_{\mathrm{c}\mathrm{c}}$ and COP, which may require a sacrifice in one or both areas.

The two-dimensional (2D) Pareto front shown in Figure 7 shows an inherent negative correlation between maximizing the chiller’s energy efficiency (COP) and its ability to recover waste heat (${\mathsf{\eta }}_{\mathrm{e}}$). The close grouping of the grey wolves around the Pareto front suggests that the MOGWO algorithm effectively found the optimal trade-off solutions for these two objectives. Based on the relative significance of COP and ${\mathsf{\eta }}_{\mathrm{e}}$, decision-makers must select a point along this Pareto front. They will tilt towards the left side of the Pareto front if their specific application requires energy savings and reduced operational cost, but not higher waste heat recovery efficiency. On the other hand, if optimizing the use of waste heat (higher ${\mathsf{\eta }}_{\mathrm{e}}$) is more important, they would lean towards a solution on the right side of the Pareto front, understanding that the ADC’s COP will be reduced. Middle-range solutions represent a compromise between these two objectives.

The two-dimensional Pareto front depicted in Figure 8 reveals a strong positive relationship between the COP and the ${\mathrm{Q}}_{\mathrm{c}\mathrm{c}}$. From Figure 8, operating the ADC at a solution that yields a higher $Q_{cc}$ corresponds to a higher COP. This observed positive correlation is a desirable trait for system performance, reflecting the thermodynamic synergy of evaporator-side processes. Importantly, this 2D plot is not intended to show trade-offs, but rather to confirm the co-benefit nature of COP and $Q_{cc}$. Therefore, the points on the left are not dominated in the conventional Pareto sense but represent valid configurations where $Q_{cc}$ and COP co-improve.

The 2D Pareto front in Figure 9 shows a clear negative correlation between ${\mathrm{Q}}_{\mathrm{c}\mathrm{c}}$ and ${\mathsf{\eta }}_{\mathrm{e}}$. Although the mid-range of both objectives seems to have a slightly sparser distribution, the non-dominated solutions form a rather smooth and distinct curve that effectively represents the ideal trade-offs between ${\mathrm{Q}}_{\mathrm{c}\mathrm{c}}$ and ${\mathsf{\eta }}_{\mathrm{e}}$. The black dots (representing the individual grey wolf search agents) are densely concentrated around the Pareto front, indicating the algorithm converges toward the best trade-off solutions. A solution that leans left will be most preferable if the decision-maker’s top priority is to maximize waste heat recovery, even at the expense of ${\mathrm{Q}}_{\mathrm{c}\mathrm{c}}$. If a high cooling capacity is more important, a solution on the right side of the Pareto front is ideal, understanding that ${\mathsf{\eta }}_{\mathrm{e}}$ will be lower. A solution in the middle represents a compromise between these two objectives.

To validate the MOGWO results, benchmark data from Chua, Ng, and Saha [29,30] for a two-bed, single-stage silica-gel/water ADC (${\mathrm{T}}_{\mathrm{h}\mathrm{w},\mathrm{i}\mathrm{n}}$ = 90 °C, ${\mathrm{T}}_{\mathrm{c}\mathrm{h}\mathrm{w},\mathrm{i}\mathrm{n}}$ = 25 °C) is used as the baseline against which the Pareto-optimal solutions were compared.

Table 6. Performance metrics comparison for a two-bed, single-stage silica-gel/water adsorption chiller.

Parameter	Chua, Ng, and Saha [29,30]	MOGWO—This Work
COP Range	0.50–0.65 (at $T_{hw,in }$ = 90 °C and $T_{chw,in }$= 25 °C)	0.5123–0.6859 (at $T_{hw,in}$ = 86.77 °C and $T_{chw,in }$) = 22.01 °C)
Cooling Capacity (Qcc)	6–10 kW (depending on cycle time and $T_{hw,in}$)	12.45–20.73 Kw
Waste-Heat Recovery Efficiency (ηₑ)	≈ 0.10–0.12 (i.e., 10–12% of heat input converted to cooling at 90 °C/25 °C)	0.0824–0.1248 (i.e., 8.24–12.48% of heat input recovered)
Operating Conditions	$T_{hw,in }$= 70–95 °C (optimal near 90 °C), $T_{chw,in }$ = 20–30 °C (focus 25 °C); two beds, ~1 kg/bed; finned-tube UA (~103 W/K)	$T_{hw,in }$= 86.77 °C, $T_{chw,in }$ = 22.01 °C; two beds; $U{A}_{evap}$ = 6000 W/K, $U{A}_{cond}$= 17,000 W/K

presents COP ranges, ${\mathrm{Q}}_{\mathrm{c}\mathrm{c}}$, ${\mathsf{\eta }}_{\mathrm{e}}$, and boundary condition details for the MOGWO Pareto-optimal results in comparison to Chua, Ng, and Saha’s model. As indicated in Table 6, the COP values for the MOGWO Pareto-optimal solutions range from 0.5123 to 0.6859, exceeding Chua, Ng, and Saha’s range of 0.50 to 0.65 for a two-bed silica-gel/water ADC at ${\mathrm{T}}_{\mathrm{h}\mathrm{w},\mathrm{i}\mathrm{n}}$ = 90 °C and ${\mathrm{T}}_{\mathrm{c}\mathrm{h}\mathrm{w},\mathrm{i}\mathrm{n}}$ = 25 °C. In contrast to Chua et al. [30] cooling capacity values, which range from 6 to 10 kW under similar conditions, the $Q_{cc}$ for MOGWO spans from 12.45 to 20.73 kW. Although Chua et al. [30] did not specifically mention ${\mathsf{\eta }}_{\mathrm{e}}$, they qualitatively stated that approximately 10 to 12% of the supplied heat becomes useful cooling at a $T_{hw,in}$ of 90 °C and ${\mathrm{T}}_{\mathrm{c}\mathrm{h}\mathrm{w},\mathrm{i}\mathrm{n}}$ of 25 °C. Meanwhile, the MOGWO front yielded ${\eta }_e$ values of approximately 0.0824 to 0.1248 (8.24 to 12.48%), which closely align with that benchmark. This validates consistent trends among improvements in COP, ${\mathrm{Q}}_{\mathrm{c}\mathrm{c}}$, and ${\mathsf{\eta }}_{\mathrm{e}}$.

Representative Design Points and Decision Variables

Equation (21) is the formulated multi-objective optimization problem that will be used to simultaneously maximize COP, $Q_{cc}$ and ${\eta }_e$. Usually, MOO establishes inherent trade-offs between conflicting objectives [31] but can also find synergistic relationships between positively correlated objectives. Both relationships give comprehensive details on the design of the system [31,32]. For instance, Figure 7 and Figure 9 point out typical trade-offs, but Figure 8 highlights a necessary synergistic relationship.

To support practical design decisions,

Table 7. Representative Pareto-optimal design points and corresponding decision variables from the MOGWO.

Variable/Metric	Max COP Point	$\mathbf{Max} {\mathbf{Q}}_{\mathbf{c}\mathbf{c}}$ Point	Max ηₑ Point	Balanced Point
${\mathrm{T}}_{\mathrm{h}\mathrm{w},\mathrm{i}\mathrm{n}}$ (°C)	95.00	95.00	65.00	80.00
${\mathrm{T}}_{\mathrm{c}\mathrm{w},\mathrm{i}\mathrm{n}}$ (°C)	22.00	22.00	22.00	29.00
${\mathrm{T}}_{\mathrm{c}\mathrm{h}\mathrm{w},\mathrm{i}\mathrm{n}}$ (°C)	20.00	19.99	19.98	15.00
${\dot{\mathrm{m}}}_{\mathrm{h}\mathrm{w}}$ (kg/s)	1.051	1.198	2.198	1.500
${\dot{\mathrm{m}}}_{\mathrm{c}\mathrm{w},\mathrm{b}\mathrm{e}\mathrm{d}}$ (kg/s)	1.388	1.750	1.658	1.500
${\dot{\mathrm{m}}}_{\mathrm{c}\mathrm{h}\mathrm{w}}$ (kg/s)	1.400	1.390	1.396	0.800
${\dot{\mathrm{m}}}_{\mathrm{c}\mathrm{w},\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}}$ (kg/s)	1.364	1.126	1.244	1.500
$\mathrm{U}{\mathrm{A}}_{\mathrm{b}\mathrm{e}\mathrm{d}}$ (W/K)	9830.45	9890.71	9882.41	6000.00
${\mathrm{U}\mathrm{A}}_{\mathrm{e}\mathrm{v}\mathrm{a}\mathrm{p}}$ (W/K)	9931.11	7677.01	6501.39	6000.00
$\mathrm{U}{\mathrm{A}}_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}}$ (W/K)	14,157.87	11,495.20	12,386.87	17,000.00
Achieved COP (-)	0.69695	0.6000	0.5123	0.6000
$\mathrm{Achieved} {\mathrm{Q}}_{\mathrm{c}\mathrm{c}}$ (kW)	12.45	20.7589	15.00	16.00
$\mathrm{Achieved} {\mathrm{\eta }}_{\mathrm{e}}$ (-)	0.0824	0.0900	0.12527	0.1000

presents a set of representative variable sets from the 100 non-dominated solutions generated by the MOGWO algorithm that optimize the three objectives. These include points where the individual objectives, COP, $Q_{cc}$, and ${\eta }_e$, are each maximized and offer a balanced trade-off with moderate compromise among all objectives. The values were taken directly from the MOGWO results and represent how a unique combination of the ten decision variables from Table 2 influences the ADC’s system performance. These representative cases can serve as practical benchmarks when selecting operating parameters based on different performance priorities. These specific points will also help to understand the variations in performance outcomes of the decision variables from the sensitivity analysis.

3.3. Sensitivity

Sensitivity analysis (SA) was used to assess how each decision variable influenced the COP, ${\mathrm{Q}}_{\mathrm{c}\mathrm{c}}$, and ηₑ. The goal was to test the robustness of the MOGWO model and identify the range of parameters with major impact on the ADC system’s behaviour [17,33]. The impact of each decision variable was isolated using the one-at-a-time (OAT) approach to eliminate confounding interactions. All other variables were kept at their baseline values, while each variable was changed across three representative levels: the lower bound, midpoint, and upper bound of its design range. The OAT approach is simple, reliable, and complements MOGWO, as it allows post-optimization perturbation testing of optimal solutions [34]. The MOGWO algorithm framework was modified for each sensitivity analysis, maintaining the hyperparameters as listed in Table 4. Twenty independent MOGWO runs were performed to guarantee statistically stable Pareto fronts [35,36]. Cubic polynomial fits were used to extract the cloud of Pareto-optimal points for the objectives generated after each run. These trends were plotted as ${\mathrm{Q}}_{\mathrm{c}\mathrm{c}}$ against COP, ηₑ against COP, and ηₑ against ${\mathrm{Q}}_{\mathrm{c}\mathrm{c}}$. The results from each plot were thoroughly presented and discussed.

3.3.1. The Effects of Varying the Hot Water Inlet Temperature

Figure 10 describes the sensitivity of the hot-water inlet temperature ${\mathrm{T}}_{\mathrm{h}\mathrm{w},\mathrm{i}\mathrm{n}}$ on system performance. Plot (a) shows a stratified trend. The optimal thermal performance is observed at 65 °C (blue), while both COP and ${\mathrm{Q}}_{\mathrm{c}\mathrm{c}}$ moderately decline at 80 °C (orange) and are significantly reduced at 95 °C (yellow). Plot (b) reveals an improvement in ${\mathsf{\eta }}_{\mathrm{e}}$ at 95 °C despite COP peaking at 65 °C. Plot (c) shows a thermodynamic trade-off as ${\mathrm{Q}}_{\mathrm{c}\mathrm{c}}$ reduces with increasing ${\mathrm{T}}_{\mathrm{h}\mathrm{w},\mathrm{i}\mathrm{n}}$ but ${\mathsf{\eta }}_{\mathrm{e}}$ performs better at higher ${\mathrm{T}}_{\mathrm{h}\mathrm{w},\mathrm{i}\mathrm{n}}$. The trend observed in plot (a) indicates that the COP does not increase monotonically with hot water temperatures. The ideal hot water temperature for optimising COP is often around 80 °C, as supported by Chang et al. [37] and Marlinda et al. [38]. Interestingly, the trends in (b) and (c) suggest that a higher hot water inlet temperature improves ${\mathsf{\eta }}_{\mathrm{e}}$. This can be attributed to better thermal energy absorption at higher driving temperatures and an increase in hot water and adsorption cooling water temperatures (∆T), resulting in faster and more efficient heat transfer. This makes the chiller absorb more heat from the waste source [39,40].

3.3.2. The Effects of Varying the Cooling Water Inlet Temperature

Figure 11 displays three subplots illustrating the effects of varying the cooling water inlet temperature. In plot (a), the ADC’s performance deteriorates as ${\mathrm{T}}_{\mathrm{c}\mathrm{w},\mathrm{i}\mathrm{n}}$ increases from 29 °C (orange) to 36 °C (yellow), shifting the points downward and to the left, while lower inlet temperatures (22 °C, blue) correspond to higher COP and ${\mathrm{Q}}_{\mathrm{c}\mathrm{c}}$. Conversely, plot (b) reveals a slight decline in both performance metrics as ${\mathrm{T}}_{\mathrm{c}\mathrm{w},\mathrm{i}\mathrm{n}}$ increases, indicating a reduction in the effectiveness of heat rejection. Plot (c) demonstrates a mild trade-off as increasing ${\mathrm{T}}_{\mathrm{c}\mathrm{w},\mathrm{i}\mathrm{n}}$ slightly improves ${\mathsf{\eta }}_{\mathrm{e}}$ but deteriorates ${\mathrm{Q}}_{\mathrm{c}\mathrm{c}}$. These patterns are consistent with Alsarayreh et al. [40], who attributed elevated exergy losses and decreased heat rejection efficiency as the cause of decreasing performance metrics (COP, ${\mathrm{Q}}_{\mathrm{c}\mathrm{c}}$) at increasing recooling temperatures.

3.3.3. The Effects of Varying the Chilled Water Inlet Temperature

Figure 12 presents three subplots illustrating the effects of varying the chilled water inlet temperature. In plot (a), both COP and ${\mathrm{Q}}_{\mathrm{c}\mathrm{c}}$ decline as ${\mathrm{T}}_{\mathrm{c}\mathrm{h}\mathrm{w},\mathrm{i}\mathrm{n}}$ increases from 10 °C (blue) to 20 °C (yellow), with the lowest ${\mathrm{T}}_{\mathrm{c}\mathrm{h}\mathrm{w},\mathrm{i}\mathrm{n}}$ yielding the best thermal performance. Plot (b) shows a similar downward trend for ${\mathsf{\eta }}_{\mathrm{e}}$, with ${\mathsf{\eta }}_{\mathrm{e}}$ declining in tandem with increasing ${\mathrm{T}}_{\mathrm{c}\mathrm{h}\mathrm{w},\mathrm{i}\mathrm{n}}$. This suggests a consistent reduction in waste heat recovery efficiency. Plot (c) further supports these observations, as ${\mathrm{Q}}_{\mathrm{c}\mathrm{c}}$ and ${\mathsf{\eta }}_{\mathrm{e}}$ are both high at the lowest ${\mathrm{T}}_{\mathrm{c}\mathrm{h}\mathrm{w},\mathrm{i}\mathrm{n}}$, indicating a clear disadvantage of operating at higher ${\mathrm{T}}_{\mathrm{c}\mathrm{h}\mathrm{w},\mathrm{i}\mathrm{n}}$. These trends align with the findings from Elsheniti et al. and Sah et al. [41,42] who reported that reducing the chilled water inlet temperature enhances the evaporators’ driving potential. This boosts the vapour uptake capacity and improves the overall ADC performance.

3.3.4. The Effects of Varying the Bed Cooling Water Mass Flow Rate

Figure 13 illustrates the effects of varying the adsorbent bed cooling water mass flow rate. Plot (a) shows that increasing ${\dot{\mathrm{m}}}_{\mathrm{c}\mathrm{w},\mathrm{b}\mathrm{e}\mathrm{d}}$ from 0.8 kg/s (red) to 2.2 kg/s (blue) reduces both ${\mathrm{Q}}_{\mathrm{c}\mathrm{c}}$ and COP compared to better performance observed at lower ${\dot{\mathrm{m}}}_{\mathrm{c}\mathrm{w},\mathrm{b}\mathrm{e}\mathrm{d}}$. Plot (b) shows a relatively stabilised ${\mathsf{\eta }}_{\mathrm{e}}$ that slightly declines as ${\dot{\mathrm{m}}}_{\mathrm{c}\mathrm{w},\mathrm{b}\mathrm{e}\mathrm{d}}$ increases, with the 0.8 kg/s (red) line falling above the others. Plot (c) reinforces the effects of a higher ${\dot{\mathrm{m}}}_{\mathrm{c}\mathrm{w},\mathrm{b}\mathrm{e}\mathrm{d}}$ on ${\mathrm{Q}}_{\mathrm{c}\mathrm{c}}$ and ${\mathsf{\eta }}_{\mathrm{e}}$. The red curve still shows superior performance compared to the green and blue, just like plot (b). The concept of optimal operating conditions is prevalent for ${\dot{\mathrm{m}}}_{\mathrm{c}\mathrm{w},\mathrm{b}\mathrm{e}\mathrm{d}}$, as the trends from plots (a) to (c) indicate that increasing cooling water mass flow rates beyond a threshold is disadvantageous to the ADC’s system performance [43]. Baudouy confirms that increasing fluid velocity beyond a certain limit might increase pumping power and pressure drops, which could reduce net system efficiency [44,45].

3.3.5. The Effects of Varying the Hot Water Mass Flow Rate

Figure 14 shows the effects of varying the hot water mass flow rate. Plot (a) exhibits the highest COP and ${\mathrm{Q}}_{\mathrm{c}\mathrm{c}}$ at the lowest flow rate of 0.8 kg/s (red). Performance degrades steadily as ${\dot{\mathrm{m}}}_{\mathrm{h}\mathrm{w}}$ increases to 1.5 kg/s (green) and 2.2 kg/s (blue), although there is a consistent upward trend across all levels. For plot (b), the red dots (0.8 kg/s) are clustered around the upper band, while the blue dots (2.2 kg/s) are concentrated at the lower ${\mathsf{\eta }}_{\mathrm{e}}$ values. This shows a systematic decline in ${\mathsf{\eta }}_{\mathrm{e}}$ as ${\dot{\mathrm{m}}}_{\mathrm{h}\mathrm{w}}$ increases. Plot (c) captures a performance trade-off. Although thermal input increases at a higher ${\dot{\mathrm{m}}}_{\mathrm{h}\mathrm{w}}$, it does not proportionally improve ${\mathrm{Q}}_{\mathrm{c}\mathrm{c}}$ and ${\mathsf{\eta }}_{\mathrm{e}}$. These trends from plots (a), (b) and (c) confirm the existence of an optimal operating ${\dot{\mathrm{m}}}_{\mathrm{h}\mathrm{w}}$ for ADCs [45]. Overly high ${\dot{\mathrm{m}}}_{\mathrm{h}\mathrm{w}}$ values raise entropy generation, limit internal heat transfer, increase heat imbalance between the adsorbent layers, and tend to worsen the overall system efficiency [14,43]. This finding supports a moderate ${\dot{\mathrm{m}}}_{\mathrm{h}\mathrm{w}}$, which is represented as 1.5 kg/s from Figure 14.

3.3.6. The Effects of Varying the Chilled Water Mass Flow Rate

Figure 15 explores the influence of varying the chilled water mass flow rate on system performance. In plot (a), the progressive downward and leftward shift of the blue (1.4 kg/s) cluster reveals how both the COP and ${\mathrm{Q}}_{\mathrm{c}\mathrm{c}}$ are impaired at high ${\dot{\mathrm{m}}}_{\mathrm{c}\mathrm{h}\mathrm{w}}$ values. The best performance for COP and ${\mathrm{Q}}_{\mathrm{c}\mathrm{c}}$ occurs from 0.2 to 0.8 kg/s. In plot (b), the COP and ${\mathsf{\eta }}_{\mathrm{e}}$ moderately increase at higher ${\dot{\mathrm{m}}}_{\mathrm{c}\mathrm{h}\mathrm{w}}$ values and rapidly decline at 0.2 kg/s compared to 0.8 kg/s. A characteristic trade-off is observed in plot (c). A lower ${\dot{\mathrm{m}}}_{\mathrm{c}\mathrm{h}\mathrm{w}}$ (red) retains a substantial portion of input heat as useful work to enhance both ${\mathsf{\eta }}_{\mathrm{e}}$ and ${\mathrm{Q}}_{\mathrm{c}\mathrm{c}}$, but the midrange flow (green) balances the performance of ${\mathrm{Q}}_{\mathrm{c}\mathrm{c}}$ and ${\mathsf{\eta }}_{\mathrm{e}}$. Thermodynamically, although a higher ${\dot{\mathrm{m}}}_{\mathrm{c}\mathrm{h}\mathrm{w}}$ enhances convective heat transfer [7,46], it could reduce the temperature gradient and increase parasitic energy input [47,48]. This could lead to initial improvements in heat transfer but diminishing efficiency above a certain threshold. This is consistent with the observed MOGWO results, where both lower and higher flows diminish ηₑ, but a mid-range hot-water flow maximizes both COP and ${\mathrm{Q}}_{\mathrm{c}\mathrm{c}}$.

3.3.7. The Effects of Varying the Condenser Cooling Water Mass Flow Rate

Figure 16 illustrates the effects of varying the condenser cooling water mass flow rate. In plot (a), the red cluster and curve (0.8 kg/s) show a consistent positive slope and increased COP and ${\mathrm{Q}}_{\mathrm{c}\mathrm{c}}$. The green (1.5 kg/s) and blue (2.2 kg/s) curves slightly flatten, indicating reduced performance at elevated ${\dot{\mathrm{m}}}_{\mathrm{c}\mathrm{w},\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}}$ values. A non-linear trend is observed in plot (b), as a higher ${\dot{\mathrm{m}}}_{\mathrm{c}\mathrm{w},\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}}$ suppresses ${\mathsf{\eta }}_{\mathrm{e}}$, although initially improving COP to some extent. Plot (c) shows an overlapping green and blue lines, with the red (0.8 kg/s) displaying a higher ${\mathsf{\eta }}_{\mathrm{e}}$ for a given ${\mathrm{Q}}_{\mathrm{c}\mathrm{c}}$. The overlapping green and blue lines indicate ${\mathsf{\eta }}_{\mathrm{e}}$ may not improve tremendously above moderate ${\dot{\mathrm{m}}}_{\mathrm{c}\mathrm{w},\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}}$ values. The diminishing returns of increasing fluid flow are supported by the literature [49]. Increasing fluid flow could increase entropy generation, irreversibilities, and limit the thermal interface of heat transfer, thereby increasing system inefficiencies [50].

3.3.8. The Effects Varying the Adsorbent Bed Overall Thermal Conductance

Figure 17 illustrates the effects of varying the bed’s overall thermal conductance. Plot (a) shows a declining trend in both COP and ${\mathrm{Q}}_{\mathrm{c}\mathrm{c}}$ at mid-range (6000 W/K, purple) and high (10,000 W/K, sky blue) values of $\mathrm{U}{\mathrm{A}}_{\mathrm{b}\mathrm{e}\mathrm{d}}$. However, a lower $\mathrm{U}{\mathrm{A}}_{\mathrm{b}\mathrm{e}\mathrm{d}}$ (2000 W/K, orange) results in superior COP and ${\mathrm{Q}}_{\mathrm{c}\mathrm{c}}$. In Plot (b), the orange cluster dominates the ${\mathsf{\eta }}_{\mathrm{e}}$ range, while the highest $\mathrm{U}{\mathrm{A}}_{\mathrm{b}\mathrm{e}\mathrm{d}}$ (10,000 W/K) points remain clustered at the bottom. This indicates a reduction in ${\mathsf{\eta }}_{\mathrm{e}}$ as $\mathrm{U}{\mathrm{A}}_{\mathrm{b}\mathrm{e}\mathrm{d}}$ exceeds mid-range values. Plot (c) reinforces plot (b) and demonstrates that ${\mathrm{Q}}_{\mathrm{c}\mathrm{c}}$ and eta are better balanced at the lower and mid-range $\mathrm{U}{\mathrm{A}}_{\mathrm{b}\mathrm{e}\mathrm{d}}$ values, confirming that excessive thermal conductance may be detrimental to the ADC’s performance. Although increasing $\mathrm{U}{\mathrm{A}}_{\mathrm{b}\mathrm{e}\mathrm{d}}$ enhances convection, thereby increasing ${\mathrm{Q}}_{\mathrm{c}\mathrm{c}}$ and COP, there always exists an optimal $\mathrm{U}{\mathrm{A}}_{\mathrm{b}\mathrm{e}\mathrm{d}}$ above which the performance deteriorates [45]. Therefore, excessively high $\mathrm{U}{\mathrm{A}}_{\mathrm{b}\mathrm{e}\mathrm{d}}$ values could reduce the essential internal temperature swings required for effective adsorption cycles [51,52]. There is a need to balance $\mathrm{U}{\mathrm{A}}_{\mathrm{b}\mathrm{e}\mathrm{d}}$ with adsorbent mass, sorbent thermal degradation, cost effectiveness, and other design factors [51].

3.3.9. The Effects of Varying the Evaporator Overall Thermal Conductance

Figure 18 illustrates the effects of varying the evaporator overall thermal conductance on the optimal objective functions. Plot (a) shows that increasing $\mathrm{U}{\mathrm{A}}_{\mathrm{e}\mathrm{v}\mathrm{a}\mathrm{p}}$ from 2000 W/K (orange) to 6000 W/K (purple) increases both COP and ${\mathrm{Q}}_{\mathrm{c}\mathrm{c}}$. A marginal improvement in ${\mathrm{Q}}_{\mathrm{c}\mathrm{c}}$ is observed at 10,000 W/K (sky blue) [53,54]. Plot (b) reveals an initial peak increment in ${\mathsf{\eta }}_{\mathrm{e}}$ around 6000 W/K before flattening at higher $\mathrm{U}{\mathrm{A}}_{\mathrm{e}\mathrm{v}\mathrm{a}\mathrm{p}}$ values. Plot (c) confirms declining returns above the mid-range conductance for ${\mathsf{\eta }}_{\mathrm{e}}$, while ${\mathrm{Q}}_{\mathrm{c}\mathrm{c}}$ modestly rises. Even though research consistently confirms the positive correlation between increased $\mathrm{U}{\mathrm{A}}_{\mathrm{e}\mathrm{v}\mathrm{a}\mathrm{p}}$ and ADC performance [54], overly high $\mathrm{U}{\mathrm{A}}_{\mathrm{e}\mathrm{v}\mathrm{a}\mathrm{p}}$ values can equilibrate temperatures quickly, which could reduce the driving force needed for phase change [53]. Figure 18 shows that 6000 W/K offers the best compromise between COP, ${\mathrm{Q}}_{\mathrm{c}\mathrm{c}}$, and ${\mathsf{\eta }}_{\mathrm{e}}$. Exceeding this threshold may compromise the ADC’s stability and negate performance gains.

3.3.10. The Effects of Varying Condenser Overall Thermal Conductance

Figure 19 illustrates the effects of varying the condenser’s overall thermal conductance on the optimal objective functions using three subplots. Plot (a) shows that COP and ${\mathrm{Q}}_{\mathrm{c}\mathrm{c}}$ are inversely proportional to $\mathrm{U}{\mathrm{A}}_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}}$, as indicated by the orange trajectory (2000 W/K), which yields the highest performance. In Plot (b), compared to COP, ${\mathsf{\eta }}_{\mathrm{e}}$ increases modestly around mid and high $\mathrm{U}{\mathrm{A}}_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}}$ values. Plot (c) shows a performance-efficiency trade-off as the distribution of ${\mathsf{\eta }}_{\mathrm{e}}$ is rather flat, even while ${\mathrm{Q}}_{\mathrm{c}\mathrm{c}}$ improves at lower $\mathrm{U}{\mathrm{A}}_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}}$ values. The observed trends in plots (a) and (b) are plausible for ADC systems [55]. Despite the lack of research to explicitly confirm the threshold of “6000 W/K” as the point above which the adversely diminishing excessively high conductance caused by COP and ${\mathsf{\eta }}_{\mathrm{e}}$ could lead to smaller temperature gradients across heat exchangers and undermine second-law efficiency. This could reduce ${\mathsf{\eta }}_{\mathrm{e}}$ performance beyond a certain point [39]. Figure 19 emphasises that there is a conductance threshold, likely between 2000 and 6000 W/K, beyond which the performance of the single-stage dual-bed ADC diminishes. This highlights the need to effectively design the condenser by balancing overall thermal conductance, temperature differentials, and reducing entropy generation.

4. Discussion

A glance at the source-reported $R^2$ values of 0.804 (COP), 0.925 ${(\mathrm{Q}}_{\mathrm{c}\mathrm{c}})$ and 0.837 (${\eta }_e$) confirms in-sample fit and does not imply validity outside the reported ranges. All optimization bounds were kept within the domain of the models’ calibration and do not claim new statistical estimation [56].

Linking the one-at-a-time sensitivity insights (Section 3) with the multi-objective outcomes (Table 7) clarifies the MOGWO trade-off behaviour of the ten decision variables. The temperature levers behave as thermodynamically expected. Higher hot water inlet temperatures $\left(T_{hw,in}\right)$ and mass flow rates (${\dot{m}}_{hw})$ are associated with higher waste-heat recovery efficiency ηₑ (Figure 10 and Figure 14).

In contrast, lower cooling water inlet temperatures $\left(T_{cw,in}\right)$ and chilled water inlet temperatures $T_{chw,in }$ maximize COP and ${\mathrm{Q}}_{\mathrm{c}\mathrm{c}}$ by easing heat rejection at the condenser and enhancing evaporator performance (Figure 11 and Figure 12). These outcomes are consistent with the work of Papoutsis et al. [17] and Alsarayreh et al. [40], who observed that increasing thermal driving potential enhances waste heat recovery and attributed the decline to diminished heat-rejection effectiveness.

The effects of mass flow rate are non-monotonic. Increasing ${\dot{m}}_{hw}$ or ${\dot{m}}_{cw }$ initially improves heat transfer, but this weakens beyond an optimum value. That is, parasitic heat input tends to reduce waste heat recovery efficiency at very high mass flow rates, causing ηₑ to fall or plateau. This reinforces the necessity for multi-objective optimization and validates the robustness of the regression-based MOGWO model.

Within the bounds explored, a balanced ADC configuration preserves COP at the maximum ${\mathrm{Q}}_{\mathrm{c}\mathrm{c}}$ level while trading approximately 23% of peak capacity for about an 11% gain in ${\eta }_e$. In comparison to maximum COP, a balanced configuration boosts ${\mathrm{Q}}_{\mathrm{c}\mathrm{c}}$ around 29% and ηₑ about 21% at an approximately 14% COP penalty.

Thus, in the design and operations of a single-stage dual-bed ADC, $T_{cw,in }$ should be kept as low as possible and $T_{chw,in }$ should be modest. In addition, over-driving of mass flow rates beyond the local optimum should be avoided and one should select a $T_{hw,in }$ high enough for regeneration while balancing COP–${\mathrm{Q}}_{\mathrm{c}\mathrm{c}}$–${\eta }_e$.

5. Conclusions

This study demonstrates the effectiveness of GWO and MOGWO in maximizing the performance of a silica gel–water single-stage dual-bed adsorption chiller. Regression models that have been statistically validated with adjusted R² values of 0.8041 for COP, 0.9250 for ${\mathrm{Q}}_{\mathrm{c}\mathrm{c}}$, and 0.8371 for ${\eta }_e$ were used to formulate objective functions for three key performance indicators: COP, $Q_{cc}$, and ${\eta }_e$. A one-at-a-time (OAT) sensitivity analysis was carried out to guarantee the model’s physical consistency and robustness. The study confirms the feasibility of integrating meta-heuristic algorithms into the design and improvement of low-grade heat thermally driven cooling systems.

The main findings and outcomes are as follows:

• A multi-objective optimization approach based on MOGWO was used to identify Pareto-optimal sets of decisions (inlet temperatures, mass flow rates, and overall thermal conductance) for a single-stage dual-bed ADC. Instead of a single “best” solution, this approach generated a set of trade-off solutions.
• The MOGWO front exhibits COP values ranging from 0.5123 to 0.6859 and ${\mathrm{Q}}_{\mathrm{c}\mathrm{c}}$ values from 12.45 to 20.73 kW, both surpassing Chua, Ng, and Saha’s [30] reported ranges of 0.50–0.65 COP and 6–10 kW $Q_{cc}$ at $T_{hw,in}$ = 90 °C and $T_{chw,in}$ = 25 °C. The attained ${\eta }_e$ values between 0.0824 and 0.1248 (8.24–12.48%) align well with the qualitatively reported ${\eta }_e$ range of approximately 10–12% by Chua, Ng, and Saha [30]. All MOGWO-selected decision variables fall within experimentally validated ranges, confirming the predictive accuracy of the regression models.
• Rather than a typical trade-off, the Pareto front for COP and ${\mathrm{Q}}_{\mathrm{c}\mathrm{c}}$ showed a strong positive correlation, indicating the existence of a suitable thermodynamic synergy where both objectives co-improve.
• Table 7 shows a detailed description of how specific combinations of decision variables can maximize individual objectives (COP, ${\mathrm{Q}}_{\mathrm{c}\mathrm{c}}$, ${\eta }_e$) or offer a balanced compromise. This is to equip designers with actionable knowledge on the generated representative Pareto-optimal variable sets.
• Results from the one-at-a-time (OAT) sensitivity analysis showed that higher hot water inlet temperatures and mass flow rates are crucial for maximizing ${\eta }_e$, while lower cooling and chilled water inlet temperatures are beneficial for COP and ${\mathrm{Q}}_{\mathrm{c}\mathrm{c}}$. These confirm complex trade-offs and the non-monotonic influences of decision variables.

Ultimately, this study combines algorithmic optimization, regression modelling, experimental validation, and detailed SA to provide invaluable insights for developing a robust and environmentally benign decision-support framework for ADC system design powered by low-grade waste heat.