Strategic Optimization of Operational Parameters in a Low-Temperature Waste Heat Recovery System: A Numerical Approach

Abstract

In the transition to sustainable energy consumption, waste heat recovery and storage systems become key to advancing Europe’s energy efficiency and reducing carbon emissions, especially by harnessing thermal energy from low-temperature sources like wastewater. This study focuses on optimizing a heat recovery system that uses heat pipes for effective heat extraction and coconut oil as a phase change material for efficient thermal storage. A total of 12 numerical simulations were conducted to analyze the outcomes of varying operational parameters, including the diameter of the heat pipe, condenser size, secondary agent flow rate, coil length, and primary agent inlet temperature. The numerical findings indicate that reduced flow rates, in combination with smaller condenser diameters and increased primary agent temperatures, greatly improve the efficiency of heat absorption and transfer. Following a 4 h test period, the most successful outcome resulted in a melting fraction of 98.8% and a temperature increase of 18.95 °C in the output temperature of the secondary agent. In contrast, suboptimal conditions resulted in only a 2.21 °C rise and a 30.80% melting fraction. The study highlights the importance of component sizing and optimization, noting that strategic modifications and appropriate phase change materials can lead to highly efficient and scalable systems.

1. Introduction

The necessity to shift towards more sustainable energy consumption is growing in light of the challenges presented by economic expansion and environmental constraints [1]. Both national and European levels acknowledge decarbonizing the energy system as an essential objective in the fight against climate change. Reducing emissions, increasing energy efficiency, and moving Europe toward at least 40% final energy consumption from renewable sources are among the energy sector’s main goals by 2030 [2]. In light of the increasing global emphasis on clean energy and improved energy management, it is essential to underscore the significance of these approaches, especially in the European Union. Within this particular context, buildings play a substantial role in both energy consumption and carbon dioxide (CO₂) emissions, accounting for 40% of energy consumption and contributing significantly to 36% of emissions across the entire area [3]. The idea of recovering waste heat has been a prominent focus in current energy-related approaches, indicating an increasing level of attention and acknowledgment of its potential to capture, utilize, and reuse excess thermal energy [4]. Furthermore, the ongoing investigation and development of waste heat recovery systems are currently at the forefront of research efforts, as underscored by the studies conducted by [5,6,7,8]. Current research underscores the significant potential of waste heat recovery systems; yet, there are specific knowledge gaps in optimizing these systems for low-temperature applications, enhancing energy storage efficiency, and integrating multiple components for both heat recovery and storage. While recovering energy from buildings and the industrial sector has received a lot of interest, the process of water heating contributes to an important amount of energy usage [8]. The discharge of wastewater from various sources, including residential, commercial, and industrial activities, offers an important amount of thermal energy. This energy is typically overlooked as a potential source of heat loss in terms of energy recovery, but it presents promising opportunities for improving energy efficiency while making a valuable contribution to global efforts against climate change [9].

As noted by [10], several technologies, ranging from economizers to waste heat boilers and heat pipe systems, have demonstrated the potential to capture, recover, and exchange significant amounts of thermal energy that would normally be lost in different processes. Heat pipes, recognized for their remarkable efficiency and versatility, hold a pivotal position in heat recovery technologies [11]. Functioning as thermal superconductors, they facilitate rapid heat transfer with additional benefits such as low operational costs and easy maintenance [12]. From isothermal processes to heat flux transformation, the diverse applications of heat pipes make them a preferred choice across various industries for efficient heat recovery and management [13,14]. Heat pipes have undergone significant advancements and are now considered essential elements for successfully integrating into heat exchangers, due to their numerous advantages and capabilities. Heat pipe heat exchangers usually have a small design that is both cost-effective and allows for the effective recuperation of thermal energy without requiring additional power sources. Moreover, the distinct benefit of heat pipe heat exchangers lies in their nearly uniform temperature distribution, emphasizing their substantial value in a wide range of applications [14]. Furthermore, their compact design facilitates the efficient adjustment of heat flux transfer, allowing for versatility in applications like preheating or heating air in conditioning systems and generating domestic hot water [10,11]. As researchers continue to focus on optimizing heat pipe heat exchangers, numerous studies provide important insights that could enhance these systems for a wide variety of applications.

In their study, ref. [15] conducted a CFD (Computational Fluid Dynamics) heat transfer analysis for a heat pipe system designed for waste heat recovery in buildings. The study focused on developing an innovative heat exchanger utilizing the return of the existing heating system, demonstrating its adaptability to varying secondary agent flow rates and different primary agent temperatures. The system exhibited successful performance, achieving favorable secondary agent temperatures even with increased flow rates and lower temperatures for the primary agent. Additionally, the introduction of heat transfer rings further enhanced heat transfer efficiency. Ref. [16] explored the optimization of a water-to-water heat pipe heat exchanger for effective waste heat recovery in the steel industry. Notably, they found that an increase in wastewater flow rate from 0.83 m³/h to 1.87 m³/h enhanced exergy efficiency from 34% to 41%, despite a decrease in system effectiveness. The findings not only revealed the impact of varying wastewater mass flow rates but also identified optimal flow rates at 1.20 m³/h for wastewater and 3.00 m³/h for fresh water. Their research not only underlines the intricate balance between flow rates and system efficiency but also points towards the importance of maintenance and structural optimization in sustaining the system’s performance. Building upon this, the subsequent study [17] further analyzed heat pipe heat exchangers (HPHE) in the steel industry, particularly focusing on slag cooling. Consistent with the previous findings, increasing the wastewater flow rate improves effectiveness but reduces exergy efficiency, pinpointing optimal flow rates of wastewater at 1.40 m³/h and cold-water mass flow rates and 2.90 m³/h. Both studies confirm that online cleaning devices have a significant role in enhancing heat transfer performance, reinforcing the importance of maintenance for optimal HPHE operation. Expanding the application of heat pipe heat exchangers, a distinct study successfully integrated an HPHE into the aluminum industry, recovering 97 kW with a notable 35-month return on investment. Although the theoretical model exhibited a 20% deviation, it suggests the potential for widespread adoption of HPHE, significantly reducing primary energy consumption (476 MWh/year) and estimated CO₂ emissions (86 tCO₂/year) [18]. Another study conducted by [19] highlighted the benefits of incorporating a heat-pipe heat exchanger in a window-type air-conditioning system. The HPHE significantly reduced compressor power consumption, resulting in substantial power savings of 2.01% to 1.33% for different working fluids. This emphasizes the significant role of HPHEs in enhancing air conditioning efficiency and promoting energy savings. To improve overall energy efficiency even more, incorporating advanced energy storage technologies becomes crucial.

As highlighted in the study by [20], efficient thermal energy storage systems, such as those utilizing latent heat, contribute significantly to storing and utilizing industrial waste heat effectively. Studies, such as those by [21,22], emphasize the efficacy of incorporating heat pipes and increasing the number of heat transfer tubes in a phase change material (PCM) for enhancing heat transfer rates in latent heat thermal energy storage systems. The integration of these techniques proves advantageous, with heat pipes demonstrating a two-fold increase in energy transfer during solidification and multitube arrangements, as demonstrated by [22], showcasing faster PCM melting. In their study, ref. [23] developed a low-cost, in-house method for manufacturing heat pipes, achieving an average melting enhancement coefficient of 135% with coconut oil as the PCM, further underscoring the potential for cost-effective heat recovery applications. Furthermore, ref. [24] investigated dynamic thermal management for industrial waste heat recovery using PCM thermal storage and achieved a reduction in daily fossil fuel consumption from 6.81 tons to 6.34 tons and optimized fuel consumption ratios, underscoring the effectiveness of advanced optimization algorithms. Expanding on the insights gained from earlier research, particularly highlighting the importance of effective thermal energy storage systems leveraging latent heat, a recent investigation led by [25] delves into the exploration of an innovative two-stage heat recovery–storage system aimed at mitigating thermal energy losses in industrial settings. By employing heat pipes for recovery and an eco-friendly phase change material for storage, the system demonstrated a peak efficiency of 78.1%, showcasing its substantial capacity for thermal energy recovery. Additionally, a novel triplex-tube heat exchanger proposed by [26] further underscores the potential of new configurations in enhancing storage and recovery processes. Their design, featuring a dual-PCM configuration, significantly improved the rate of melting and solidification. The optimized arrangement achieved 23.43% and 18.87% enhancements in energy storage and recovery, respectively, compared to conventional single-PCM systems. This study highlights the importance of innovative designs and configurations in maximizing the efficiency of these systems intended for heat recovery and storage.

Despite progress in refining these systems to boost heat transfer and energy storage capabilities, ongoing challenges persist in achieving optimal efficiency for low-temperature applications. The current literature often focuses on optimizing single components of heat recovery systems, neglecting the potential benefits of a comprehensive multicomponent approach. At the same time, studies in the literature on waste heat recovery systems predominantly employ heat pipes, neglecting the possibility of storage. Nonetheless, these studies demonstrate the significant potential of heat recovery in various industrial sectors [27,28,29]. Our study aims to fill this gap by optimizing a waste heat recovery and storage system that utilizes heat pipes and coconut oil as a PCM, focusing on improving heat transfer efficiency for low-temperature applications. Employing numerical simulations, we explored various configurations of the system to identify the most efficient setup for improved performance. Integrating techniques like heat pipes, as investigated by [20,21,30,31], and optimizing the number of heat transfer tubes in PCM and arrangement, as highlighted by [22,32], further accentuates the potential to enhance heat transfer rates in latent heat thermal energy storage systems. Additionally, an experimental investigation underscores the importance of optimizing temperature gradients to improve efficiency, as the increase of temperature in thermal energy storage systems can effectively enhance heat transfer [33].

The unique contribution of our study lies in the detailed optimization of these components and configurations, providing a comprehensive analysis of their effects on the system. This approach not only addresses challenges in existing systems but also offers a practical framework for improving waste heat recovery and storage. In alignment with these principles, our study aims to contribute to ongoing advancements in efficiently recovering and storing waste thermal energy for sustainable energy consumption. Within the context of the European energy system, the optimization of these systems is particularly relevant. Given the substantial role of buildings in energy consumption and emissions, our research addresses a critical need for efficient heat recovery solutions in residential, commercial, and industrial settings. Specifically, our system can be developed for use in residential settings, coupled with solar thermal panels. The heat pipe heat recovery system can utilize the hot water generated by the solar thermal panels for heat recovery, producing instant hot water for domestic use and storing excess heat within a phase change material for prolonged hot water availability. Additionally, it can be applied in industrial settings, such as in the steel and aluminum industries, to recover heat from wastewater resulting from various processes, providing instant hot water for other processes while also offering storage capabilities, improving overall efficiency in industrial processes, and significantly reducing primary energy consumption and CO₂ emissions. We anticipate that the insights gained from this study will offer valuable guidance for optimizing the geometrical design and operational parameters of heat pipe heat recovery and storage systems.

2. Materials and Methods

2.1. The Design of the Heat Pipe Heat Recovery System

This study focuses on the optimization of a heat pipe heat recovery and storage system to enhance heat transfer performance and address the imperative need for energy efficiency. We developed 3D behavioral models to optimize the system’s design and gain useful insights into its behavior. We used the educational version of ANSYS 2022 for numerical simulations and Autodesk Inventor 2023 for 3D modeling. Numerical simulations were employed for investigating alternate coil placement and secondary agent flow rates, as well as various design and operating scenarios. The SIMPLE algorithm was applied for efficient pressure-velocity coupling, and second-order upwind schemes were used for the discretization of momentum and energy equations. Our analysis involved several steps to systematically evaluate the system’s performance under various conditions. Initially, we developed 3D behavioral models of the heat pipe heat recovery system and then employed numerical simulations to explore different operational scenarios. These simulations were categorized based on varying single parameters to isolate their effects on the system’s thermal behavior. Finally, we analyzed the simulation results to identify the most promising design and operational settings.

During this phase of simulations, we conducted the tests on a system comprised of two key sections: the evaporation zone and the condensation zone, both traversed vertically by a single heat pipe. The evaporation zone has a height of 400 mm and a width of 100 mm, while the cylindrical condensation zone reaches a height of 600 mm with a diameter of 150 mm. Both zones are vertically penetrated by a heat pipe, with a diameter of 15 mm and a length of 1000 mm. At first, in our research, we examined two distinct designs. In the first one (Design 1), the coil measures 1916 mm in length, with a total height of 474 mm, featuring 10 turns and a diameter of 15 mm. In contrast, in the second design (Design 2), the coil is 1096 mm long, with a height of 244 mm, a reduced turn count of 5, and a diameter of 15 mm. These designs with different coil lengths help understand how coil dimensions affect the PCM melting behavior and the end temperature of the secondary agent. The heat pipe, which serves as a crucial component in the system, is crafted from copper. The phase change material used is coconut oil, distinguished by its melting point of 25 °C. The coil is made of copper, while the housings for the evaporator and condenser are made of stainless steel. The system operates by transferring thermal energy from the primary agent in the evaporator to the heat pipe, which subsequently transfers the heat to the condenser zone. Simultaneously, the secondary agent flows through the coil in the condenser, extracting heat from the melted PCM within the condenser. This dual-functionality ensures efficient heat recovery and storage while simultaneously producing warm water that can be utilized for a range of purposes, including domestic hot water supply. The constructive details are illustrated in

Table 1. Constructive details of the components.

	Component Specifications
Design No.	Evaporator		Condenser		Heat Pipe		Coil
	Height [mm]	Width [mm]	Height [mm]	Diameter [mm]	Length [mm]	Diameter [mm]	Height [mm]	Length [mm]	Diameter [mm]
Design 1	400	100	600	150	1000	15	1916	474	15
Design 2							1096	244
Design 3						25	1916	474
Design 4				100
Design 5				85

.

A visual representation of the first two designs and their components is presented in Figure 1.

After our initial simulations on Design 1 and Design 2, we refined our approach, making adjustments to the condenser and heat pipe dimensions, resulting in three additional constructive models represented in Figure 2. In Design 3, we kept the diameter of 150 mm for the condenser, and simultaneously, we increased the heat pipe diameter from 15 mm to 25 mm, for enhanced heat transfer. Next, we optimized the heat exchange efficiency by reducing the condenser diameter from 150 mm to 100 mm, resulting in what we denote as Design 4. This reduction was anticipated to increase the rate of heat exchange, resulting in faster and more uniform melting by concentrating the thermal energy transfer. In Design 5, we pushed the boundaries by further decreasing the condenser diameter from 100 mm to 85 mm, aiming to achieve higher thermal efficiency and heat transfer by concentrating the thermal energy within a smaller volume. These adjustments reflect our commitment to refining the system’s performance under varying conditions, with Design 1 serving as the baseline for our subsequent optimizations and explorations.

To systematically evaluate the impact of each design and operational parameter, simulations were grouped based on single-variable variation, allowing for a clear analysis of each parameter’s influence on system performance. Group A investigated how varying the secondary agent flow rate affected system melting behavior and heat transfer characteristics, while Groups B and C examined the impact of different coil lengths on the system. Group D assessed how changing the heat pipe diameter influenced thermal transfer within the system, while Group E evaluated the effect of different condenser diameters on the system’s heat exchange efficiency. Groups F and G examined the thermal response of the system to varying primary agent inlet temperatures. These groups and their corresponding details are presented in Section 2.5 along with the relevant tables and parameters.

The criteria for evaluating the performance of each design were primarily based on two key metrics: the melting percentage of the phase change material and the outlet temperature of the secondary agent. A higher melting percentage of the PCM signifies higher thermal storage capacity, while elevated outlet temperatures of the secondary agent demonstrate more efficient heat transfer and absorption.

2.2. Numerical Approach

In the simulation stage, we imported the three-dimensional geometries that were initially created using Autodesk Inventor into the Ansys-Fluent framework so that we could perform precise analyses. The meshing process, essential for numerical simulations, unfolded within the constraints of the academic version, limited to 500,000 elements. Our mesh networks, consisting of 130,000–200,000 nodes, were created using a progressive refinement approach. This was carried out by initially generating a mesh with a lower element count and subsequently increasing the density in critical areas as needed. The aim was to create a high-quality mesh that guarantees accurate representation of thermal behavior while also considering computational efficiency. In our computational analysis, the nodal distribution within the Ansys-Fluent environment was meticulously defined to capture the intricate thermal behaviors in both the evaporation and condensation zones. Specifically, the nodes were concentrated in critical regions within the heat recovery system where temperature gradients, phase change, or fluid flow were expected to exhibit significant variations or complexities. Particular attention was given to areas around the heat pipe and coil interfaces due to their critical roles in heat transfer processes. In areas where finer details were crucial, such as around bends or junctions where heat and fluid flow might experience abrupt changes, the mesh was refined to increase node density, enhancing the local resolution of the simulation results. Given the different configurations in our study—Design 1 with a longer coil and more turns, and Design 2 with a shorter coil and fewer turns—the nodal distribution was adjusted to reflect the geometric complexities of each design. The node distribution plays an essential part in our simulation method, allowing us to obtain detailed and accurate insights into the heat transfer performance of the heat pipe system under different conceptual and operational scenarios.

2.3. Initial and Boundary Conditions

Our primary objective was to assess the system’s thermal dynamics and identify optimal configurations under varying operational conditions. The next step in initiating our simulations involved selecting the appropriate materials and setting the boundary conditions. We chose steel for the evaporator and condenser casings, and copper for the coil and heat pipe due to its large thermal conductivity. Recognizing the computational constraints and the broad scope of our investigation, we opted not to simulate the internal phase change processes. Thus, the heat pipe was modeled as a cylindrical material with a very high thermal conductivity to simulate its heat transfer capabilities within the overall system. The focus was on ensuring that the heat pipe’s role in transferring heat between system components was accurately represented, thereby simplifying the model while still capturing the essential functionality needed for our analysis. The working fluids, designated as primary and secondary agents, were both modeled as water to accurately simulate the flow dynamics within the system. Additionally, coconut oil was introduced as a phase change material (PCM) within the condenser.

Given the absence of coconut oil’s properties in the Fluent database, we customized its material profile based on the recent literature [34], ensuring an accurate representation of its thermophysical properties. For our simulations, a transient, pressure-based solver was employed to capture the time-dependent responses of the system to various operational scenarios. A laminar flow model was chosen, aligning with the expected flow conditions and the system’s physical scale. Boundary conditions were applied, with velocity inlets specified for the primary agent entering the evaporator and the secondary agent flowing through the condenser coil. The inlet parameters—velocity magnitude and temperature—were set for each simulation design to reflect different operational states, as shown in

Table 2. Testing parameters.

Design No.	Simulation No.	Boundary Conditions
		Primary Agent—Water		Secondary Agent—Water		Phase Change Material—Coconut Oil
		Inlet Temp. [°C]	Flow Rate [L/min]	Inlet Temp. [°C]	Flow Rate [L/min]	Melting Temp. [°C]	Solidification Temp. [°C]
Design 1	S1	70	24	15	1	25	15
	S2				0.1
	S3				0.05
Design 2	S4	70			1
	S5				0.1
Design 3	S6				0.05
Design 4	S7	70
	S8	90
	S9	50
Design 5	S10	70
	S11	50
	S12	90

.

Pressure outlets were defined for both the evaporator and condenser, allowing the fluids to exit the system under controlled conditions. We conducted our analytical exploration across five distinct designs, varying the flow rate of the secondary agent and the inlet temperature of the primary agent. This approach enabled us to systematically evaluate the system’s performance across a spectrum of conditions, from higher flow rates and temperatures to more conservative settings. The consistency of the secondary agent’s temperature at 15 °C and the primary agent’s flow rate at 24 L/min provided a stable baseline for our investigations. The numerical methods chosen ensured a balance between computational efficiency and the accuracy needed to discern the subtle impacts of design and operational adjustments on system behavior.

2.4. Mathematical Modeling

Our study utilizes ANSYS Fluent to simulate dynamic heat transfer and phase change within a heat recovery and storage system. To model our system, we can break it down into three main components (see Figure 2):

• The evaporator: In this part, the hot fluid, known as the primary fluid, circulates. The hot fluid flows at a specified flow rate of 24 L/min, enters at a given temperature (the temperature of the heat source), and exits at a temperature T₀ °C;
• The condenser: This section is filled with PCM and is traversed by the upper part of the heat pipe. A coil, immersed in the PCM and surrounding the heat pipe, allows cold water to enter and absorb energy from both the heat pipe and the PCM, thus heating up;
• Heat Pipe: To simplify the calculations, we modeled the heat pipe as a solid cylinder with very high thermal conductivity of 5000 W/m·K to simulate a heat transfer as efficient as that of a heat pipe;

This assembly is insulated; the thermal losses from the walls to the outside are negligible.

The fundamental equations of fluid mechanics and heat transfer serve as the foundation for the Ansys-Fluent simulations that we conducted as part of our analytical framework. The governing equations for these processes are derived and adapted from existing methodologies, such as fixed domain methods and the enthalpy method for phase change.

In fixed domain simulations, the computational domain remains constant, even as the phase changes occur. The primary governing equation for energy within these systems, accounting for the phase change, is given by:

$$\rho [{\displaystyle \frac{\partial h}{\partial t}}+\nabla ·(\overrightarrow{v}h)]=\nabla ·q-S$$

(1)

where:

${\displaystyle \frac{\partial h}{\partial t}}$—unsteady term;

$\nabla ·\left(\overrightarrow{v}h\right)$—convective transport of energy;

q—diffusive heat flux;

S—source term;

The Voller [35] enthalpy method is commonly used for modeling phase change and integrated in ANSYS. This approach simplifies the problem by considering the enthalpy as the sum of sensible and latent heat, eliminating the need for interface conditions. This approach reduces complexities, transforming the governing equation into one that is equivalent to a single-phase equation and allowing precise tracking of fusion and solidification fronts.

• The mass conservation equation is fundamental in fluid dynamics simulations and is expressed as:

$${\displaystyle \frac{\partial \rho }{\partial t}}+\nabla ·\left(\rho \overrightarrow{v}\right)=0$$

(2)

where $\rho$ is the density of the fluid and $\overrightarrow{v}$ represents the velocity field.

• Conservation of Momentum:

$$\rho \left({\displaystyle \frac{\partial v_x}{\partial t}}+\nabla ·\left(\overrightarrow{v}{v}_x\right)\right)=\nabla \cdot \left(\mu \nabla v_x\right)-{\displaystyle \frac{\partial P}{\partial x}}+S_x$$

(3)

$$\rho \left({\displaystyle \frac{\partial v_y}{\partial t}}+\nabla ·\left(\overrightarrow{v}{v}_y\right)\right)=\nabla \cdot \left(\mu \nabla v_y\right)-{\displaystyle \frac{\partial P}{\partial y}}+S_y$$

(4)

$$\rho \left({\displaystyle \frac{\partial v_z}{\partial t}}+\nabla ·\left(\overrightarrow{v}{v}_z\right)\right)=\nabla \cdot \left(\mu \nabla v_z\right)-{\displaystyle \frac{\partial P}{\partial y}}+S_z$$

(5)

where $\overrightarrow{S}$ includes source terms due to the phase change. These source terms are particularly influenced by the mushy zone model, where the PCM transitions between solid and liquid states. The source term $\overrightarrow{S}$ can be specifically expressed as:

$$\overrightarrow{S}={\displaystyle \frac{M{\left(1-f\right)}^2}{f^3+c}}\overrightarrow{v}$$

(6)

Here, $f$ is the liquid fraction, $M$ is the mushy zone constant, and $c$ is a small number preventing division by zero, ensuring numerical stability.

• Energy Conservation in the PCM:

$$\rho {\displaystyle \frac{\partial h}{\partial t}}=\nabla ·\overrightarrow{q}-\rho \Lambda {\displaystyle \frac{\partial f}{\partial t}}$$

(7)

where $\rho$ represents the density of the phase change material, ${\displaystyle \frac{\partial h}{\partial t}}$ is the rate of change of enthalpy over time, and ${\displaystyle \frac{\partial f}{\partial t}}$ represents the rate of change of the phase fraction.

For systems undergoing phase transitions, the total enthalpy is the sum of the sensible heat ($h_{sensible}\left(T\right)$) and the latent heat ${(h}_{latent}\left(T\right))$. This relationship is defined by the following equations:

$$h\left(T\right)=h_{sensible}\left(T\right)+h_{latent}\left(T\right)$$

(8)

The sensible heat represents the energy accumulated due to the temperature increase of the material, whether in solid or liquid state. It is calculated as:

$$h_{sensible}\left(T\right)={\int }_{T_{initial}}^{T_{fusion}}{C}_p,S^{dT}+{\int }_{T_{fusion}}^{T_{final}}{C}_p,L^{dT}$$

(9)

The latent heat component represents the energy absorbed or released during the phase transition. It is expressed as:

$$h_{latent}\left(T\right)=f\left(T\right)∆h_{S-L}$$

(10)

Here, $∆h_{S-L}$ is the latent heat of fusion and $f\left(T\right)$ is the temperature-dependent liquid fraction, which varies between 0 and 1 as the material transitions from solid to liquid, being defined as follows:

• $f\left(T\right)=0,$ $\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{d}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{e},\mathrm{i}\mathrm{f}T<T_{solid};$
• $f\left(T\right)=1,$ $\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{d}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{e},\mathrm{i}\mathrm{f}T>T_{liquid};$
• $0<f\left(T\right)={\displaystyle \frac{T-T_{solid}}{T_{liquid}-T_{solid}}}$ < 1, when the material changes the phase, if $T_{solid}<T<T_{liquid};$

Combining the sensible and latent heat equations, the energy conservation equation for a system experiencing phase change is given by:

$$\rho \left[{\displaystyle \frac{\partial h_{sensible}\left(T\right)}{\partial t}}+\nabla ·\left(\overrightarrow{v}{h}_{sensible}\left(T\right)\right)+∆h_{S-L}{\displaystyle \frac{\partial f\left(T\right)}{\partial t}}+∆h_{S-L}\nabla ·\left(\overrightarrow{v}f\left(T\right)\right)\right]$$

(11)

2.5. Simulation Parameters and Grouping Overview

To facilitate a nuanced understanding of how individual parameters affect system behavior, we strategically organized the simulations into categories based on a singularly varying parameter. Figure 3 illustrates the organized simulations and their respective groups, enhancing clarity through systematic categorization. The rationale for this approach was to isolate the influence of each variable on the system, which allowed us to draw precise conclusions about the impact of each varied parameter. Our results are presented in the format of comparative groups, reflecting the one-variable-at-a-time approach.

The organization of these simulation data is summarized in the accompanying tables, which outline the groups based on the varying parameters.

Our simulation study commenced with the establishment of a baseline scenario, denoted as simulation S1, and this initial setup is presented in

Table 3. Characteristics of Secondary Agent Flow Rate Variations in Group A Simulations.

Comparison Group	ID	Parameter Variation and Values		Constant Parameters
				Coil Height [mm]	Condenser Diameter [mm]	HP Diameter [mm]	Tpin [°C]
Group A	S1	Secondary Agent Flow Rate (Qsa [L/min])	1	1916	150	15	70
	S2		0.1
	S3		0.05

.

Building upon the foundational parameters of S1, simulations S2 and S3 were systematically conducted with the sole variation of the secondary agent flow rate (Q_sa). These modifications allowed for a focused investigation into the effects of flow rate changes on system efficiency and heat transfer characteristics, while maintaining all other system parameters constant to isolate the impact of this single variable. Extending from Group A’s findings, we examined coil length variations in Groups B and C to discern their effects on the system’s performance. The specific parameters that remained fixed and those that were varied are systematically detailed in

Table 4. Characteristics of Coil Length Variations in Group B and Group C Simulations.

Comparison Group	ID	Parameter Variation and Values		Constant Parameters
				Qsa [L/min]	Condenser Diameter [mm]	HP Diameter [mm]	Tpin [°C]
Group B	S1	Coil length [mm]	1916	1	150	15	70
	S4		1096
Group C	S2		1916	0.1
	S5		1096

.

Following the assessment of coil lengths, our attention in Group D shifted towards the diameter of the heat pipe. Simulations S3 and S6 in Group D explore the system’s behavior with a smaller and larger heat pipe diameter and a temperature of 70 °C for the primary agent (T_pin). The constant parameters across these groups, including condenser diameter, coil height, and secondary agent flow rate, ensure a targeted analysis of the heat pipe’s influence.

Table 5. Characteristics of Heat Pipe Diameter Variations in Group D Simulations.

Comparison Group	ID	Parameter Variation and Values		Constant Parameters
				Tpin [°C]	Condenser Diameter [mm]	Coil Height	Qsa [L/min]
Group D	S3	HP diameter [mm]	15	70	150	1916	0.05
	S6		25

presents the specific constants and varied parameters for clarity.

Building on the insights from the heat pipe diameter analysis in Group D,

Table 6. Characteristics of Condenser Diameter Variations in Group E Simulations.

Comparison Group	ID	Parameter Variation and Values		Constant Parameters
				Coil Height [mm]	Qsa [L/min]	HP Diameter [mm]	Tpin [°C]
Group E	S6	Condenser diameter [mm]	150	1916	0.05	25	70
	S7		100
	S10		85

presents Group E, where we focus on the effects of varying condenser diameters. Retaining the optimized heat pipe diameter of 25 mm from the previous comparisons, we evaluated how the system performance adjusts to condenser diameters of 150 mm, 100 mm, and 85 mm in Simulations S6, S7, and S10, respectively, while all other variables were held constant.

Groups F and G, detailed in

Table 7. Characteristics of Primary Agent Inlet Temperature Variations in Group F and Group G Simulations.

Comparison Group	ID	Parameter Variation and Values		Constant Parameters
				Condenser Diameter [mm]	Qsa [L/min]	HP Diameter [mm]	Coil Height [mm]
Group F	S8	Primary agent inlet temperature (Tpin [°C])	90	100	0.05	25	1916
	S7		70
	S9		50
Group G	S12		90	85
	S10		70
	S11		50

, explore the impact of varying primary agent inlet temperatures. With condenser diameters of 100 mm for Group F and 85 mm for Group G carried forward from previous studies, these groups investigate the thermal response of the system to inlet temperatures ranging from 50 °C to 70 °C and 90 °C, respectively, keeping all other system parameters constant. Simulations S8, S7, and S9 constitute Group F, while S12, S10, and S11 form Group G. These groups presented below enable us to isolate and examine the impact of the primary agent inlet temperature variations on the system’s thermal efficiency.

3. Results and Discussion

The results of the 12 numerical simulations conducted to assess our heat recovery and storage system are analyzed based on specific boundary conditions and configurations from Table 1 and Table 2. The figures presented below offer a visual depiction of the melting progression (MF) and temperature variations (T) across the system, captured through cross-sectional views, after 4 h of operation. Although the simulations we ran did not reach a stable state within the 4 h real-time duration, we deliberately chose this time frame with consideration of both computational efficiency and the scope of our study, which encompassed numerous designs. The 4 h window provided sufficient time to discern the effects of each parameter on the system’s early thermal response, providing valuable insights while managing the extensive computational demand required for multiple long-duration simulations.

In the forthcoming discussions, we will analyze the results according to the groups outlined in Table 3, Table 4, Table 5, Table 6 and Table 7 from the previous section, examining the specific effects of individual parameter changes on system performance. Graphical representations will be provided for each group to accompany these discussions, illustrating the percentage of melted phase change material and the temperature of the secondary agent at the coil’s outlet. In order to properly understand the flow characteristics within our system, we determined the Reynolds numbers for each simulated scenario. The results are presented in

Table 8. Reynolds number.

Secondary Agent
Simulation No.	Temperature [°C]	Volume Flow Rate [L/min]	Reynolds	Regime
S1	15	1	1240.6	Laminar
S2	15	0.1	124.1	Laminar
S3	15	0.05	62.0	Laminar
S4	15	1	1240.6	Laminar
S5	15	0.1	124.1	Laminar
S6	15	0.05	62.0	Laminar
S7	15	0.05	62.0	Laminar
S8	15	0.05	62.0	Laminar
S9	15	0.05	62.0	Laminar
S10	15	0.05	62.0	Laminar
S11	15	0.05	62.0	Laminar
S12	15	0.05	62.0	Laminar

, showing that every scenario maintains a laminar flow profile, as the Reynolds number is smaller than 2300.

3.1. Comparative Analysis: Secondary Agent Flow Rate Variation in Group A

Figure 4 showcases cross-sectional views that capture the system’s thermal behavior and melting progression after 14,400 s of real flow time, highlighting the thermal behavior and melting patterns for this group’s simulations.

In Group A, the variations in secondary agent flow rate yield insightful trends regarding the melting fraction of the phase change material (PCM) and the outlet temperature of the system. As shown in the provided data in Figure 5a,b, a clear dependency is observed between the flow rate and the system’s thermal performance. The melting fraction of the PCM, represented as a percentage of total mass, demonstrates a progressive increase over the 4 h duration for all flow rates. Notably, the simulation with the lowest flow rate, S3-MF (0.05 L/min), exhibits a significantly higher melting fraction, reaching 30.71% at 14,400 s, compared to 20.49% for S1-MF (1 L/min).

Consistent with the melting fraction observations, the outlet temperatures of the secondary agent also display an increase as the flow rate diminishes, as can be seen in Figure 5b. The S1-T’s output temperature, which slightly exceeds the initial temperature, indicates limited heat transfer. Conversely, S3-T reaches an outlet temperature of 21.15 °C, demonstrating a substantially more effective heat transfer process. This elevated temperature is indicative of the secondary agent’s greater thermal absorption, aligning with the system’s goal to maximize heat extraction.

3.2. Comparative Analysis: Coil Length Variation in Group B and Group C

Figure 6 provides cross-sectional views that depict the system’s thermal behavior and melting trends over 4 h of real flow time, accentuating the impact of coil length on melting patterns and thermal dynamics in Group B’s simulations.

The data for Group B indicate that the half-length coil in Simulation S4 leads to a higher melting fraction over time compared to the full-length coil in Simulation S1. At the 4 h mark, S4-MF achieves a melting fraction of 30.64%, which is significantly greater than the 20.49% seen in S1-MF. In contrast to the melting fraction data in Figure 7a, the final temperature of the secondary agent indicates that S1-T operates slightly better than S4-T, with a recorded temperature of 15.43 °C compared to 15.24 °C after 4 h, as shown in Figure 7b, suggesting that a longer coil may be more effective in transferring heat to the secondary agent.

Figure 8 presents the cross-sectional melting fraction (MF) and temperature (T) contours for Group C, where the secondary agent flow rate is set at 0.1 L/min, contrasting a full-length coil in S2 with a half-length coil in S5 to visualize their respective impacts on the system’s thermal efficiency and PCM melting behavior.

The data in Figure 9a illustrate that reducing the flow rate to 0.1 L/min impacts the melting fraction of the PCM. In S5-MF, with the half-length coil, we see a higher melting fraction across the time spectrum, peaking at 30.80% after 4 h, compared to 26.17% in S2-MF with the full-length coil. Upon examining the outlet temperatures for S2-T and S5-T, it becomes evident that the half-length coil of S5-T does not achieve a significant temperature increase at the outlet as the full-length coil of S2-T does. After 4 h, S5-T reaches an outlet temperature of 17.21 °C, whereas S2-T attains a marginally higher temperature of 18.67 °C. Using a half-length coil at a lower flow rate can increase the amount of phase change material that melts. However, a full-length coil may be more effective in heating the secondary agent to a higher exit temperature, which is advantageous in systems that require higher exit temperatures.

3.3. Comparative Analysis: Heat Pipe Diameter Variation in Group D

Figure 10 delves into a comparative analysis of heat pipe diameter variation in Group D. We have retained the optimized secondary agent flow rate of 0.05 L/min and the full-length coil from prior simulations while investigating the impact of enlarging the heat pipe diameter from 15 mm in S3 to 25 mm in S6.

The melting fraction data for Group D in Figure 11a show a distinct increase when using a heat pipe with a larger diameter. Simulation S6-MF, with a 25 mm diameter heat pipe, exhibits a consistently lower melting fraction compared to S3-MF, which has a 15 mm diameter. At the end of the 4 h period, S3-MF achieves a higher melting fraction of 30.71%, suggesting that the smaller diameter pipe is more effective in transferring heat to the PCM. The outlet temperature trends for S3-T and S6-T reveal that the secondary agent’s temperature is consistently higher in S3-T throughout the 4 h period. Ending at 21.15 °C, S3-T’s temperature is higher than S6-T’s (20.47 °C). This indicates that the heat pipe with the smaller diameter is more effective at heating the secondary agent within the given flow rate and coil configuration.

3.4. Comparative Analysis: Condenser Diameter Variation in Group E

Figure 12 shifts our focus to a comparative analysis of condenser diameter variation in Group E. Maintaining the previously optimized flow rate of 0.05 L/min and the full-length coil configuration, alongside a 25 mm heat pipe, we now examine the implications of condenser diameter changes. This group compares the effects of varying the condenser diameter from an initial 150 mm in S6 to 100 mm in S7 and finally to 85 mm in S10.

The data in Figure 13a for Group E show a pronounced effect of condenser diameter on the melting fraction. As the diameter decreases, there is a notable increase in the melting fraction percentage. By the end of the 4 h period, S10-MF with the smallest diameter (85 mm) reaches the highest melting fraction of 95.24%, followed by S7-MF (100 mm) with 88.32%, and S6-MF (150 mm) with 28.19%. The outlet temperatures represented in Figure 13b correspondingly reflect the effects of condenser diameter reduction. S10-T demonstrates the greatest increase in outlet temperature, concluding at 28.85 °C, indicative of the most significant heat absorption by the secondary agent. The final temperatures of S7-T and S6-T are 27.95 °C and 20.47 °C, respectively, demonstrating that the secondary agent leaves the system at a higher temperature after taking greater amounts of heat when the condenser diameter decreases.

3.5. Comparative Analysis: Primary Agent Inlet Temperature Variation in Group F and Group G

In Group F represented in Figure 14, we extend our examination by keeping constant the optimized parameters from previous groups: a secondary agent flow rate of 0.05 L/min, a full-length coil, a heat pipe diameter of 25 mm, and a condenser diameter of 100 mm. We shifted our focus to explore the impact of the primary agent’s inlet temperature variation, observing the effects at 70 °C in S7, increasing to 90 °C in S8, and decreasing to 50 °C in S9.

The melting fraction data in Figure 15a illustrate a clear trend: higher inlet temperatures facilitate a greater degree of the phase change material’s melting over time. Simulation S8-MF, operating at a 90 °C inlet temperature, achieves a significantly higher melting fraction, peaking at 95.22% after 4 h. This is in contrast to S7-MF, which reaches 88.32% at a 70 °C inlet temperature, and Simulation S9-MF, which only attains 68.64% at the lower 50 °C temperature. These results indicate that the phase change process is highly sensitive to the temperature of the primary agent, with higher temperatures accelerating the melting substantially.

As shown in Figure 15b, the outlet temperature trends follow a similar pattern. Simulation S8-T exhibits the greatest secondary agent outlet temperature, peaking at 33.66 °C, due to its higher inlet temperature for the primary agent. This is significantly higher compared to S7-T’s 27.95 °C and S9-T’s 22.51 °C, concluding that the temperature of the primary agent directly influences the secondary agent’s heat absorption capabilities. Notably, S8-T’s elevated outlet temperature suggests that the system is highly effective in transferring heat from the primary agent to the secondary agent, maximizing the potential for heat recovery.

Group G extends our systematic exploration by maintaining a condenser diameter of 85 mm and assessing the system’s response to varied primary agent inlet temperatures. This analysis explores the thermal efficiency and phase change behavior at three different inlet temperatures: 70 °C for S10, 90 °C for S12, and 50 °C for S11. The corresponding thermal effects and phase change process are visually captured in Figure 16, which presents the cross-sectional melting fraction (MF) and temperature (T) contours for Group G.

The melting fraction data for Group G in Figure 17a reveal that the PCM melting is significantly influenced by the primary agent’s inlet temperature, particularly within the smaller 85 mm condenser diameter.

Simulation S12-MF, with the highest inlet temperature of 90 °C, consistently exhibits the greatest melting fraction, achieving near completion with 98.80% melted after 4 h. Simulation S10-MF, at a moderate 70 °C, shows a substantial melting fraction as well, but Simulation S11-MF with the lowest temperature of 50 °C, reached only 84.14%. This clearly demonstrates that higher inlet temperatures are more conducive to the phase change process, especially when combined with a smaller condenser diameter. Addressing the outlet temperatures of the secondary agent, Simulation S12-T outperforms the others, achieving an ending temperature of 33.95 °C, as shown in Figure 17b. This suggests that the secondary agent is capable of absorbing greater amounts of heat at higher primary temperatures, regardless of the assembly’s smaller dimensions. Simulations S10-T and S11-T conform to the aforementioned pattern, concluding at temperatures of 28.85 °C and 23.46 °C, respectively. The findings indicate that using an 85 mm condenser diameter is effective in transferring the thermal energy of the primary agent to the secondary agent. Higher inlet temperatures lead to greater heat gain, which is advantageous for systems that aim to achieve higher outlet temperatures for the secondary agent.

3.6. Summary of Findings

Upon analyzing Group A, it is evident that varying the flow rate of the secondary agent significantly impacts the thermal dynamics of the system. A lower flow rate results in a higher melting fraction and elevated outlet temperatures, indicating improved heat absorption and transport capacity. Although a steady state was not reached within the 4 h simulation, the observed patterns provide valuable insights for enhancing the early stage effectiveness of the heat recovery system. Results from Group B demonstrate that while a shorter coil enables more energy storage by melting more PCM, it does not necessarily improve the system’s ability to efficiently transfer this stored energy to the secondary agent as effectively as a longer coil. Therefore, when both high energy storage and efficient heat transfer are desired, the coil length becomes a critical consideration. Balancing the melting fraction and heat transfer rate is essential for optimizing system performance. Group C’s simulations show that combining a lower secondary agent flow rate with a half-length coil enhances PCM melting, indicating improved energy storage efficiency. However, the longer coil proves more effective at raising the outlet temperature of the secondary agent, highlighting its superior heat transfer capabilities. This suggests that while a half-length coil maximizes phase change and storage, a full-length coil excels at heat transfer, requiring a balance to optimize both thermal storage and heat delivery.

In Group D, findings emphasize the importance of selecting the appropriate heat pipe diameter to balance effective heat absorption from the primary agent and efficient heat transfer to both the PCM and the secondary agent. A reduced diameter increases heat transfer efficiency, underscoring the need for careful consideration of this parameter during the design phase to maximize system effectiveness. The findings from Group E highlight the significant impact of condenser diameter on heat recovery system performance. Smaller diameters improve both PCM melting and secondary agent outlet temperatures. However, balancing these benefits with the overall system design is crucial, particularly concerning PCM volume, which affects energy storage capacity. Optimizing system performance without compromising energy storage requires careful consideration of both condenser diameter and PCM volume. Group F reveals that higher primary agent inlet temperatures substantially increase PCM melting efficiency and secondary agent outlet temperatures. This underscores the importance of selecting an appropriate temperature range for the primary heat source, considering the specific needs and capabilities of the heat recovery equipment. Using a higher temperature source maximizes thermal output and enhances overall heat transfer performance, emphasizing the need for proper primary agent temperature selection. Results from Groups F and G demonstrate that the primary agent’s inlet temperature is crucial in optimizing PCM melting and enhancing heat extraction capabilities. The correlation between primary agent inlet temperature and an optimized condenser diameter significantly amplifies the system’s heat transfer capacity, highlighting the importance of these parameters in maximizing system performance.

In general, it can be concluded that lower secondary agent flow rates, higher primary agent inlet temperatures, and smaller condenser diameters consistently enhance the system’s thermal efficiency. These findings can be generalized to similar heat recovery systems aiming for improved heat absorption and transfer efficiency. However, specific design parameters such as coil length and heat pipe diameter need to be carefully selected based on the particular configuration and operational conditions of the system. These specific findings may not be universally applicable without considering the unique aspects of each system’s design, operation, and intended application.

3.7. Limitations of the Study and Future Research Directions

While our study provides valuable insights into the optimization of heat recovery systems, several limitations should be acknowledged. The scope of our numerical simulations was limited to specific parameters and configurations, which may not encompass all real-world scenarios. Additionally, the study currently lacks experimental validation, as the ongoing experimental work will be presented in subsequent research. Our numerical model includes necessary assumptions and simplifications, such as imposed conditions and not including internal phase change processes within the heat pipe, to streamline the analysis. Furthermore, the academic version of ANSYS used in this study posed limitations on the element count. Despite these considerations, our results offer a robust framework for further exploration and development in heat recovery system optimization.

There are several research questions that remain unanswered, necessitating further investigation. Future studies should focus on experimentally validating the numerical findings to confirm the model’s accuracy and reliability under real-world conditions. Including the internal phase change processes within heat pipes in the overall system analysis could provide a more comprehensive understanding of heat transfer dynamics and improve model precision. Additionally, exploring a broader range of parameters and configurations can help generalize the findings and enhance their applicability to various domestic and industrial settings. Assessing the long-term performance and durability of the optimized heat recovery systems under continuous operation is important for understanding their practical viability and maintenance needs. Exploring other phase change materials with superior thermal storage properties could lead to improved thermal performance and efficiency. Moreover, integrating heat recovery systems with renewable energy sources, such as solar thermal panels, can enhance sustainability and overall energy efficiency. Finally, conducting a detailed economic analysis, including cost-benefit and return-on-investment studies, can provide insights into the financial feasibility and attractiveness of implementing these systems.

Future studies addressing these questions will contribute to a more comprehensive understanding and development of efficient heat recovery systems, facilitating the development of practical applications in both the residential and industrial sectors.

4. Conclusions

A comprehensive heat recovery system was systematically analyzed through a series of 12 simulations, with each one optimizing various operating parameters. The thermal performance of the system was analyzed by methodically changing variables such as the flow rate of the secondary agent (1 L/min, 0.1 L/min, and 0.05 L/min), the length of the coil (474 mm and 244 mm), the diameter of the heat pipe (15 mm and 25 mm), and the size of the condenser (150 mm, 100 mm, and 85 mm). This analysis concluded with an investigation of the primary agent’s inlet temperature (90 °C, 70 °C, and 50 °C). This methodical approach provided insight into the effects of the aforementioned variables on the efficiency of the system, both individually and collectively. Empirical data from numerical simulations established a robust framework for evaluating the system’s performance, revealing the following key insights:

• By varying the secondary agent’s flow rate, it was possible to observe that slower rates greatly improve heat absorption and transfer, as shown by higher final temperatures and melting fractions;
• Similarly, the choice of coil length was shown to affect energy storage and heat transfer, as shorter coils enhance the melting of PCM while longer coils facilitate superior heat transfer;
• Furthermore, optimizing the heat pipe and condenser diameters was found to be important, as smaller diameters facilitated better heat transfer and increased the melting fraction, though careful attention must be paid to the overall volume of PCM to avoid limiting energy storage capacity;
• The temperature of the primary agent also played an important role, with higher temperatures leading to more efficient melting of the PCM and enhanced thermal output;
• Among all the simulations conducted, Simulation S12 demonstrated the most promising results after a 4 h flow time, achieving a remarkable melting fraction of 98.8% and an increase of 18.95 °C in the secondary agent’s outlet temperature. In contrast, Simulation S5 highlighted areas for potential enhancement, achieving only a 30.80% melting fraction and a relatively small temperature gain of 2.21 °C for the secondary agent;

In summary, the study highlights the need for optimizing components in heat recovery systems. Strategic adjustments can lead to significant improvements in thermal efficiency. Such systems are beneficial in industrial applications, where they can recover thermal energy from wastewater processes, store surplus thermal energy in phase change materials, and use this energy to prepare domestic hot water or for other purposes, thereby providing prolonged hot water availability. Further investigations could explore the integration of multiple high-performance heat pipes and superior phase change materials to advance the system’s design. Adjusting constructive dimensions and experimenting with innovative materials could lead to systems that are not only highly efficient but also scalable and suitable for industrial or domestic applications. Moreover, employing advanced experimental design methods such as the Taguchi method could further enhance our ability to efficiently optimize these systems. This comprehensive approach will ensure that future heat recovery systems are optimized for real-world settings, offering cost-effectiveness and environmental sustainability.

To maximize the benefits of these systems, it is recommended that policymakers introduce incentives for the adoption of optimized waste heat recovery systems in both residential and industrial sectors. Such policies could drive the implementation of these efficient technologies, resulting in increased energy savings, reduced operational costs, and significant environmental advantages.