Phase Change Material Volume Fraction Research Articles

Multiscale Finite Element Modeling of the Effect of Macro-Encapsulated Phase-Change Materials on the Thermal Performance of Hydronic Floor Heating Systems

Phase-change materials (PCMs) are commonly employed in building service equipment to regulate indoor temperatures and reduce energy consumption. This study conducted multi-scale finite element modeling to analyze the steady-state and dynamic thermal behavior of a hydronic radiant floor heating system integrated with macro-encapsulated PCMs. It predicted performance values for hydronic floor heating with and without macro-encapsulated PCMs. The study assessed the impact of the PCM volume fraction, heating water temperature, capsule thermal conductivity, and shape on the thermal performance of hydronic floor heating through various finite element models. The predictive capability of the finite element model was validated using experimental data, showing good agreement. Although the inclusion of PCMs lowered the floor temperature, it improved temperature distribution and retained heat when the system was inactive. The PCM volume fraction significantly influenced the performance of the hydronic floor. However, the shape of the macro-encapsulated PCM and thermal conductivity of the shell had minimal effects in the studied case. For instance, increasing the thermal conductivity of the shell of the PCM capsule fifty times from 0.3 to 15 W m−1 K−1 resulted in an increase in surface temperature by 1.2 °C.

Enhancing solar panel cooling efficiency: a study on the influence of nanofluid inclusion and pin fin shape during melting and freezing of phase change materials

The present article presents a 3D simulation of a solar thermal panel containing phase change materials (PCMs). Two pipes are devised in the panel, and several pin fins (PFs) are applied to each pipe. Organic PCMs are encapsulated in a compartment around the PFs and pipes. The variable is PF shape, which includes four types, i.e., square, rectangular, triangular, and circular. Nanofluid (NFD) is used within the pipes. The study is carried out transiently and continued until the stabilization of outlets. Utilizing an FEM method based on a weak form, namely, Galerkin, to find a numerical solution for mathematical modeling. The artificial intelligent results indicate that using triangular, square, rectangular, and circular PFs provides the highest NFD temperature in the outlet, respectively. Circular PFs lead to a lower heat transfer coefficient (HC) compared to other PFs. The comparison between various PF shapes shows that the use of circular and triangular PFs results in the lowest and highest panel temperature, respectively. Moreover, the highest and lowest volume fraction of melting PCMs around the pipe is obtained through the use of triangular and circular PFs, respectively.

Manipulation of the thermo-rheological properties of stable Fe3O4 nanoparticles-embedded PCM nanoemulsions

Phase Change Material (PCM) nanoemulsions offer numerous benefits as thermal energy storage systems, including high energy storage capacity, rapid thermal response, enhanced stability, and easy integration into existing thermal management systems. This study focuses on producing nanoemulsions with fine droplets of commercial paraffin RT42, stabilized with the anionic surfactant sodium dodecyl sulfate (SDS) using the sonication method. The impact of formulation of PCM nanoemulsions on its thermophysical and rheological characterization has been investigated. The results indicated that the degree of supercooling declined from 12.8 °C to 8.6 °C as the PCM volume fraction varied from 10 to 45 vol%. This decrease was attributed to the enlargement of PCM droplet size from 110 to 164 nm while maintaining a constant PCM-to-surfactant ratio of 10:1. By incorporating Fe3O4 nanoparticles as a nucleating agent at two different concentrations of 1 and 4 %, the supercooling degree decreased from 10.2 °C to 8.2 °C when 4 % Fe3O4 nanoparticles were added. The samples exhibited remarkable stability throughout a period of one year of storage and under dynamic conditions, including 50 freeze-thaw cycles. Rheological investigations and the gelation mechanism through estimation of the attractive and repulsive interaction forces among droplets revealed that nanoemulsions with volume fractions ≥30 % transformed from fluid-like states to viscoelastic gels due to the depletion attractions caused by SDS micelles in the continuous phase. These findings contribute to a better understanding of the factors influencing the formation of viscoelastic PCM nanoemulsion gels and their potential applications across various fields.

Artificial intelligent modeling and optimization of pin-fin configuration on the thermal performance of two tubes inside phase change material for cooling of a solar panel using Ag-water nanofluid

This paper performs a two phase numerical analysis on a solar thermal panel of dimensions 80 × 100 cm in the presence of Ag-water nanofluid flow. The nanofluid flows in two individual tubes beneath the panel, and there are some circular pin fins on the tubes. Tubes and pin fins under the panel are surrounded with a phase change material (PCM), paraffin wax. By increasing the length of pin fins from 1 to 7 cm, the volume fraction (VOF) of molten PCM, outlet nanofluid temperature (T-NF), and heat transfer coefficient (HTC) are examined from the transient state up to reaching the steady state. Utilizing a FEM based on a weak form, namely Galerkin, to find a numerical solution for mathematical modeling. Results of this study indicated that increasing the length of pin fins from 3 to 7 cm maximally increases the outlet nanofluid flow up to 2.11 °C. In the transient conditions, increasing the length of pin fins decreases the VOF of molten PCM. When the problem is evaluated in the transient conditions, increasing the length of pin fins decreases the mean temperature of the panel (T-PL), but in the steady-state conditions, the T-PL is the same for different lengths of pin fins. The lowest value of the HTC occurs for pin fins of length 3 cm, and in most cases, the average HTC is maximum for pin fins of length 1 cm.

Two phase simulation of solar still in the presence of phase change materials in its bottom and aluminum nanoparticles in the water

This article performs a numerical study on a transient solar still (SOST) by utilizing a variable heat flux using COMSOL software. A phase change material (PCM) layer with a thickness of 10–50 mm is placed at the bottom of the desalination. Some aluminum nanoparticles are added to the water in the water desalination, and the two-phase method is used to simulate this part of the desalination water. Other variables include the angle of the glass that changes from 10 to 45⁰ and the heat transfer coefficient (HTFC) on the glass that varies from 5 to 300 W/m2. The effect of these variables on the average PCM temperature (T-PCM), PCM volume fraction (VOF-PCM), average moisture (AV-MO) temperature, and AV-MO concentration is examined in 12 h. The simulations are done using the finite element method (FEM). The results demonstrate that the average temperature and VOF-PCM, AV-MO concentration, and AV-MO temperature are enhanced from morning to noon and decreased from noon to evening. An increment in the thickness of PCM causes the VOF-PCM in desalination water to reduce by 35%. An enhancement in the glass angle reduces the average temperature of PCM, especially in the morning and afternoon. The change in PCM thickness, especially in the evening, significantly enhances the AV-MO temperature.

A two-phase analysis of the use of water-aluminum nanofluid in a solar still with a layer of phase change materials

This study employed the finite element method to simulate a solar still three-dimensionally. Aluminum nanoparticles were used in the still's tank, at the bottom of which was a layer of the N-auxin Phase Change Material (PCM). The variables of the problem included the degree of solar radiation for 12 different hours a day, the 10–45˚ angle of the front glass plate of the still, and the ambient temperature ranging from 293˚K to 330˚K. The effects of these variables on the air temperature, glass, PCM, and PCM volume fraction were examined. A two-phase method was used to analyze the nanofluid in the tank. The results showed that solar radiation generally had the highest effect on the maximum PCM temperature, maximum air temperature, and PCM volume fraction. However, the drop in the ambient temperature had the maximum impact on the maximum glass temperature. The PCM inside the solar still made the maximum air temperature slightly higher in the evenings than in the mornings. As a result, the maximum glass temperature could reach 330˚K at the ambient temperature of 330˚K and maximum solar radiation. In the last hours of the study, a decline in the angle of the solar still's front plate reduced the volume fraction of the molten PCM. Besides, at the 10˚ angle of the still's front plate, a drop in the outside temperature led the PCM to freeze faster.

Parameter optimization and sensitivity analysis of a Lithium-ion battery thermal management system integrated with composite phase change material

In order to ensure the performance and to extend the life cycle of power battery module within electric vehicle, a battery thermal management system (BTMS) integrated with composite phase change materials (PCM) was proposed. The heat dissipation performance of BTMS using paraffin/expanded graphite (EG) composite PCM with different mass fraction was investigated. The results showed the optimal heat dissipation performance was observed using composite PCM with 20 wt% EG. The combined effect of liquid volume fraction and thermophysical parameters of composite PCM on heat dissipation performance of battery module was revealed. The phase transition temperature of composite PCM was found to have the largest impact on the maximum temperature and temperature uniformity of battery module, followed by latent heat and thermal conductivity of composite PCM. A 10 % decrease in the phase transition temperature led to a 7.9 % in the maximum temperature of battery module. Both the thermal conductivity and latent heat growth contributed to the reduction in the maximum temperature of battery module. Composite PCM with the thermal conductivity of 3 W m−1 K−1, phase change temperature of 38 °C, and latent heat of 220 J g−1, was found to achieve the optimal heat dissipation performance of BTMS and a high utilization rate of 97.3 % for composite PCM. The findings of this work could be beneficial for future development of highly-efficient and economical composite PCM for BTMS.

Impact of the number of tubes containing nanofluid flow on the melting and freezing of phase change materials in the thermal management of plate lithium-ion batteries

Phase change material (PCM) and nanofluids (NFs) flow are used in three-dimensional simulations for the thermal management (TMGT) of a plate battery. On each side of the battery, there are two enclosures in which the PCM is housed. Within the PCM, there is a tube that allows alumina/water NFs to flow. The two-phase model is used to simulate the NFs flow, the number of tubes inside the PCM is varied from 1 to 4, and the NFs velocity (VNF) is changed from 10 to 30 mm/s. Also, the distance of the tubes from the battery center changes to determine the volume fraction of molten PCM (VFMPC) and the maximum temperature of the battery (T-BT). For the simulations, the finite element technique (FEM) is used. The findings show that the lowest maximum T-BT and VFMPC are caused by a 5-mm distance between the tubes, whereas the greatest maximum temperature (T-MX) and VFMPC are caused by an 8-mm gap. The use of more tubes in PCM causes the VFMPC to be decreased. Besides, the maximum T-BT is reduced with time. Enhancing the VNF causes the maximum T-BT to decrease. The maximum T-BT is decreased by 45.37% in 1200 s by adjusting the VNF from 10 to 30 mm/s.

Performance optimization of metal hydride hydrogen storage reactors based on PCM thermal management

Metal hydrides (MH) are considered promising hydrogen storage materials. In applications, the heat transfer characteristics of MH hydrogen storage reactors are closely related to their storage performance. Phase change materials (PCM) were proven to integrate with MH reactors for effective thermal management. Nevertheless, the PCM based MH reactors (MH-PCM reactors) still need to overcome the heat transfer resistance and the loss of volumetric hydrogen storage capacity caused by the PCM volume proportion. The present work established a mathematical model to describe the transfer and reaction process in the MH-PCM reactor. Various parameters were discussed in the current work. It was found that the increased PCM's amount and latent heat have a restricted effect on the enhancement of heat transfer and absorption reaction rate. The higher PCM's thermal conductivity and hydrogen supply pressure effectively enhanced the heat transfer and reaction rate. Furthermore, the increased absorption pressure can improve PCM's sensible heat storage capacity, which provides a new approach for optimizing PCM consumption. Finally, the quantitative relationship among the volumetric hydrogen storage capacity, the volume fraction and thermophysical properties of PCM, and the absorption pressure were established by systematic analysis, which provides an essential theoretical basis for the optimization of MH-PCM reactors.

Effect of tube location on the temperature of plate lithium-ion battery applicable in the aerospace industry in the presence of two-phase nanofluid flow inside a channel placed in phase change material

This article presents three-dimensional simulations of the cooling system of a plate lithium-ion battery (BTY) using the finite element method (FEM). Since weight is of particular importance in the aerospace industry, this type of batteries can be used. The cooling system consists of two enclosures containing phase change material (PCM) on both sides of the BTY cell and alumina/water nanofluid flows in a tube in the middle of the PCM. This simulation is performed using the two-phase model. By changing the nanofluid velocity (VNF) from 10 to 30 mm/s and changing the location of the tube in the PCM enclosure, the temperature contours of the BTY, PCM, and tube, as well as the contours of PCM melting and volume fraction (VFN) of molten PCM are provided. According to the findings, increasing the VNF from 10 to 30 mm/s decreases the average temperature of the BTY (TBT) by 14.8% and the VFN of molten PCM by 18.9% in 1200 s. Also, when the tube distance is 8 mm, the TBT is minimal and the amount of solid PCM is maximal. When the tube distance is 11 mm, the TBT is maximal and the VFN of solid PCM is minimal.

Effect of Phase Change Materials on Lithium-Ion Plate Batteries

This paper presents the simulations of the cooling system of a battery pack (BTPC) consisting of lithium-ion (LIN) plate batteries. The BTPC includes six battery cells (BTCL) in two rows with three BTCLs, which are placed in a channel with one inlet and two outlets. The laminar and steady airflow flows in the channel. Phase-change material (PCM)-filled rectangular cubic enclosures enclose every BTCL. Transiently adjusting the cavity aspect ratio (AR) every 6000 s is how this investigation is conducted. For four values of AR, the values of the PCM volume percentage surrounding each BTCL in the BTPC, and the temperature of each BTCL are calculated. The simulations are performed using the FEM and COMSOL software. The results demonstrate that the maximum changes in temperature of the battery (TOB) pack by changing the AR occur when the TOB pack is reduced. The maximum temperature reduction at this time is 1.88 °C which occurs between AR2 and AR4 at 720 s. The maximum temperature corresponds to AR3 and AR4 and the minimum one is related to AR1 and AR2. From 1260 to 3500 s, the effect of AR on PCM volume fraction is maximal. The value of solid PCM for AR1 and AR2 is higher than that for AR3 and AR4 at different times. Additionally, an increment in the value of the AR enhances the amount of channel pressure drop by 14%.

ANALYSIS OF PLATE-FIN HEAT SINK INFUSED WITH PHASE CHANGE MATERIALS FOR INTERMITTENT SPACE MISSIONS

In this work, we numerically investigated the heat transfer effectiveness of different phase change materials (PCMs) when infused in a plate-fin heat sink with a fixed volume fraction of thermal conductivity enhancer. The PCM's ability to absorb and release large amounts of thermal energy at constant temperature is a desired feature in transient electronics cooling applications. In this study, we focused on examining the effect of the number of fins, type of PCM, heat flux, PCM volume fraction, and heat sink bottom wall thickness. The results showed that increasing the number of fins improved the performance of the PCM-infused heat sink. When a heat flux of 4000 W/m<sup>2</sup> was applied for 30 minutes on a plate-fin heat sink infused with paraffin wax, the maximum temperature did not exceed 70°C in the four-fin design, while it exceeded 80°C in the two-fin design. A salt hydrate PCM outperformed paraffin wax and RT35. The bottom wall of the heat sink acted as a thermal spreader and a nonlinear relationship existed between the bottom wall thickness and the maximum electronics temperature. Compared to the two- and four-fin heat sink models, the zero-fin model required the longest time to fully melt the entire PCM due to the additional amount of PCM present in the heat sink gaps.

Investigation on Thermal Performance of a Novel Passive Phase Change Material-Based Fin Heat Exchanger

Abstract This work presents a numerical investigation of the thermal control performance of a phase change material (PCM) composite-filling fin heat exchanger under natural convection and transient heat flux shocks. First, the temperature control performance of the PCM-based fin heat exchanger and the fin heat exchanger without PCM is compared. Then, the influence of the PCM filling patterns, liquid phase change temperature, and the PCM filling volume fraction on the temperature control performance are studied. The results show that filling the PCM in fins could significantly improve the temperature control performance by absorbing heat during the solid–liquid phase change process. Compared with the fin heat exchangers without PCM, the temperature control performance of the designed PCM-based fin heat exchangers could be improved by up to 13%. PCM filling patterns influence the temperature control performance of fin heat exchangers to a certain extent, and the asymmetric filling pattern is better than the symmetric filling pattern. The results of different phase change temperatures of PCM show that, within a certain temperature range, the lower the liquid phase change temperature results in a better temperature control performance since more PCM would melt when suffering the heat flux shock. Besides, when the volume fraction of PCM is small, it also greatly influences the temperature control efficiency of the heat exchanger; however, when the volume fraction of PCM is larger than 75%, its influence could be neglected.

Machine Learning-Based Approach for Modeling the Nanofluid Flow in a Solar Thermal Panel in the Presence of Phase Change Materials

Considering the importance of environmental protection and renewable energy resources, particularly solar energy, the present study investigates the temperature control of a solar panel using a nanofluid (NFD) flow with eco-friendly nanoparticles (NPs) and a phase change material (PCM). The PCM was used under the solar panel, and the NFD flowed through pipes within the PCM. A number of straight fins (three fins) were exploited on the pipes, and the output flow temperature, heat transfer (HTR) coefficient, and melted PCM volume fraction were measured for different pipe diameters (D_Pipe) from 4 mm to 8 mm at various time points (from 0 to 100 min). Additionally, with the use of artificial intelligence and machine learning, the best conditions for obtaining the lowest panel temperature and the highest output NFD temperature at the lowest pressure drop have been determined. While the porosity approach was used to model the PCM melt front, a two-phase mixture was used to simulate NFD flow. It was discovered that the solar panel temperature and output temperature both increased considerably between t = 0 and t = 10 min before beginning to rise at varying rates, depending on the D_Pipe. The HTR coefficient increased over time, showing similar behavior to the panel temperature. The entire PCM melted within a short time for D_Pipes of 4 and 6 mm, while a large fraction of the PCM remained un-melted for a long time for a D_Pipe of 8 mm. An increase in D_Pipe, particularly from 4 to 6 mm, reduced the maximum and average panel temperatures, leading to a lower output flow temperature. Furthermore, the increased D_Pipe reduced the HTR coefficient, with the PCM remaining un-melted for a longer time under the panel.

Thermal performance prediction of a phase change material based heat-sink cooling system for a printed circuit board, using response surface method

The main objective of this experimental study is investigation of using phase change material (PCM) on the performance of a printed circuit board (PCB), as an electronic chipset. To this end, by a comparative study, the effects of using two types of PCMs (RT42 and RT35/CH), which have different melting points and latent heat of fusions, are examined on the cooling performance of the module under both free and forced convection. Experiments are conducted under two heat fluxes of 45 and 60 W. After selecting the appropriate PCM in terms of time to reach the critical temperature of 80 °C, the response surface method (RSM) is utilized to extract predictive models for considered responses. The predictive equations estimate the cooling performance of the PCM-based heat sink by considering PCM volume fraction and fan speed as two important independent design parameters in this study. By using RSM, the interaction effects of different factors on responses, and their grade of importance are presented. Under free convection condition, using PCM could significantly improve the usage time of the module and the enhancement ratio, especially in low heat flux of 45 W. Increasing the PCM volume fraction enhanced the usage time for both considered PCMs, although the RT42 showed better performance in all studied volume fractions compared to the RT35 HC. Comparison of two PCMs shows the superiority of using RT42 under both applied heat fluxes and operating condition. For the heat flux of 45 W, the highest thermal effectiveness is found for RT45 PCM/30 % (9.73) and for the heat flux of 60 it is found for RT45 PCM/90 % (3.67). According to the RSM, the PCM volume fraction significantly affects the cooling performance in comparison with the fan voltage.

RETRACTED:The investigation of battery thermal management via effects of using phase change materials in the oval packages around the lithium-ion battery cells with an airflow

Due to the ability of phase change materials (PCM) in heat storage and thermal management system (TMS) of various devices, in this paper, the numerical study of the effect of using these materials on TMS of lithium-ion batteries was studied. There is also airflow in the battery pack (BAP) to control the battery temperature (T-B). Around the batteries, CaCl2.6H2O phase change agent was used in oval packages to control the T-B. A transient study performed for the time after melting, 100 %. PCM, which cools the airflow and CaCl2.6H2O at the same time as the battery is running, so that the amount of molten PCM is reduced so that the PCM can control the T-Bs again. The T-Bs was also monitored by establishing a space between the rows of batteries. The numerical analysis was carried out using the finite element technique and the COMSOL program. The findings revealed that the volume fraction (VLF) of PCM melt surrounding the batteries reduced with time. The decrease in melt PCM was significant in the first column but had little effect in the third column. Changing the vertical distance of the batteries from each other in the third column of batteries is very effective, while in the first column has little effect on the temperature and VLF of PCM. At a distance of 7 mm from the battery rows, there is the highest amount of T-B and molten PCM. The lowest T-B pack and molten PCM occurs at a distance of 9 mm from the battery rows.

RETRACTED: Study of circular, horizontal and vertical elliptical enclosures filled with phase change material in thermal management of lithium-ion batteries in an air-cooled system

In this paper, the battery cooling is examined by placing a battery cooling system in the airflow path. The cooling system consists of 9 cylindrical battery cells so that a phase change material (PCM) surrounds them. PCM is housed in horizontal and vertical elliptical (ELL) and circular enclosures. The amount of PCM used in enclosures is equal and the effect of enclosure shapes on battery temperature (TBat) and the rate of phase change of PCM is evaluated. COMSOL software is employed to simulate the battery cooling system. The results demonstrate that the use of PCM enclosures with vertical ELL shape causes the highest maximum temperature (Tmax-b) and the average temperature (Tav-b) of battery cells, while horizontal ELL ones produce the lowest Tmax-b and the Tav-b. The maximum amount of liquid PCM at different times corresponds to vertical ELL enclosures and the minimum amount of average volume fraction of PCM is related to horizontal ELL enclosures. Utilizing the enclosures with horizontal ELL shapes creates the minimum outlet air temperature. The circular ones have the maximum outlet air temperature.

Numerical simulation of dimensions and arrangement of triangular fins mounted on cylindrical lithium-ion batteries in passive thermal management

This paper examines a battery pack consisting of three cylindrical lithium-ion batteries in an enclosure saturated with Phase Change Material (PCM) using COMSOL software. A two-dimensional longitudinal section of the enclosure is evaluated transiently. There are 6 circular fins on the batteries, which are triangular in two dimensions. The temperature of the battery cells, the liquid volume fraction of PCM, and the heat transfer coefficient (HTCO) are studied using two fin arrangements on the battery. The FEM is used to solve the equations. The results show that the use of a fin with a width of 1 cm to 4 cm reduces the maximum temperature (Tma) of the battery pack by 2.04 and 2.15, respectively, for models F1 and F2. Also, using a 1-cm wide fin instead of a 4-cm one enhances the HTCO for models F1 and F2 by 20% and 19.4%, respectively. However, the use of a fin with a width of 4 cm compared to 1 cm intensifies the amount of liquid PCM at different times. The values of battery temperature (TBa), the amount of liquid PCM, and the amount of HTCO are intensified with time.

Experimental investigation on the performance of RT-44HC-nickel foam-based heat sinks for thermal management of electronic gadgets

A large amount of heat is generated during the operation of electronic gadgets. Efficient thermal management is crucial for their safe operation and reliability. Phase change materials serve the role to maintain almost constant temperature during the phase change process by absorbing the generated heat. Therefore, the present experimental investigation is performed to single out the thermal performance of nickel foam-based heat sink embedded with RT-44HC paraffin as phase change material (PCM). Nickel foam is used for a high heat transfer area with a minimum reduction in the latent heat of a composite PCM. Results revealed that at the end of the charging process, the heat sink base temperature is reduced by 18.6% when a PCM volume fraction of 0.6 was added to the nickel foam. Furthermore, as the PCM fraction increased, the base temperature decreased further. An 11.6% additional decrease in temperature of the PCM was observed for volume fraction increment from 0.6 to 0.8. The effective thermal conductivity of the composite PCM was noticed to be enhanced by six times as that of a pristine PCM. The discharging process of the composite PCM was delayed compared with that of nickel foam without PCM. However, the sink temperature lies within the safe limits for the composite PCM. The latent heat of the composite PCM was diminished by 23% when effective thermal conductivity was enhanced. Thus, a nickel foam-PCM based heat sink is an efficient source to maintain the electronics temperature within the safe limits.

Thermal management evaluation of Li-ion battery employing multiple phase change materials integrated thin heat sinks for hybrid electric vehicles

The optimal performance of a Li-ion battery is directly impacted by temperature. In order to control the temperature rise and provide even temperature distributions in the battery pack, a thermal management scheme comprises thin heat sinks with multiple phase change materials (PCMs) and air channels is investigated in this paper. The cooling performance and temperature homogenization of the battery thermal management (BTM) system are carefully studied under various configurations of PCMs. The results show that increasing the air inlet velocity has less effect in suppressing the temperature rise at early discharge stage, but ameliorates as the discharge prolonged to 3600 s. The standard deviation of the temperature (STDV) and maximum temperature of the batteries can be decreased by arranging PCMs with a lower melting temperature at the midsection and a higher melting temperature at the air outlet region of the heat sinks. In addition, for volume fraction of PCMs, Case IV, having a PCM with a higher melting point adjacent to the air outlet region and occupying one-half the height of the heat sink, illustrates a lower temperature rise and decreases the maximum temperature in the battery module by 1.024 K, 2.186 K, and 2.553 K, compared to Case I, II, and III, respectively.