Conductivity Of Phase Change Materials Research Articles

Optimization of a finned multi-tube latent heat storage system using new structure evaluation indexes

The latent heat thermal energy storage (LHTES) is one of the most promising ways of storing solar thermal energy. Since the thermal conductivity of phase change materials are low, traditional shell and tube heat exchangers tend to develop dead zones. Therefore, structural optimization is essential, and a finned multi-tube design is recommended. In this work, nineteen structures for a heat storage tank are designed to explore the influence of different specifications of heat exchange tubes and fins on heat storage/release performance. After establishing 3D physical models, ANSYS/Fluent simulation and a two-step optimization for charging and discharging processes were conducted. Among these structures, N3-M3-D10.2 emerged as the most efficient during charging process in terms of reducing the complete melting time by 25 % and increasing the 3-h heat storage volume by 2.95 times; however, it shows low performance during discharging process, especially with a 2.24 times non-uniformity factor. In the four preferred structures, the benefit-to-cost ratio of N3-M3-D10.2 is 48 %–86.6 % higher than that of the others. The results also show that the number of tubes and fins are negatively related to the melting rate, and positively related to the solidification rate. Moreover, the influence of fin diameter is greater than that of fin number and tube diameter on heat transfer rate. The novelty of this work is not only lies in both considering the charging and discharging processes, but also in using dimensionless normalized indexes and different weights based on customers’ actual requirements for optimization.

Numerical study on the combined application of multiple phase change materials and gradient metal foam in thermal energy storage device

Latent heat thermal energy storage (LHTES) offers significant energy-saving benefits, but its application is limited due to the low thermal conductivity of phase change material (PCM). To address this issue, this article studied the combined application of metal foam structures with different porosity gradients and multiple PCMs with different melting points through numerical simulation to accelerate the melting of PCM and enhance system efficiency. The results showed that the application of multiple PCMs improved the heat transfer performance of the system, reducing the complete melting time by 9.18% compared to using a single PCM with uniform metal foam. Based on the multi-PCM storage system with an average porosity of 0.90, this article designed metal foam structures with one-dimensional and two-dimensional porosity gradients and explored their impacts. The results indicated that in the structure with a one-dimensional porosity gradient along the heat flow direction, the positive gradient decreased thermal resistance, further reducing the complete melting time by 6.18%, while the negative gradient increased it by 19.78%. However, the temperature non-uniformity was lowest with the negative gradient and highest with the positive gradient. The optimal two-dimensional porosity gradient multi-PCM storage model not only reduced thermal resistance but also effectively solved the issue of uneven melting, reducing the complete melting time by 17.96% and increasing the energy storage efficiency by 20.16% compared to the single PCM system with uniform porosity. Furthermore, the article conducted a dimensionless analysis of the optimal structure and different gradient structures, establishing formulas for the liquid fraction concerning modified Fourier number, modified Stefan number, and modified Rayleigh number.

NH4Al(SO4)2·12H2O-Na2SO4 eutectic-based phase change materials with excellent light absorbance, enhanced thermal conductivity, and high latent heat performance

A novel eutectic phase change material with light absorption and high thermal conductivity was successfully prepared. The based phase change material was ammonium aluminum sulfate dodecahydrate‑sodium sulfate eutectic phase change materials. The black Ti2O3 after ball milling was selected as light thermal conversion material which can absorb sunlight effectively and simultaneously convert the light to thermal and the expanded graphite was added to further improve the thermal conductivity of phase change materials. The phase change temperature of the eutectic phase change material was effectively regulated to 80 °C compared with 93 °C of pure aluminum sulfate dodecahydrate. The thermal conductivity of eutectic phase change materials with expanded graphite was enhanced to 1.16 W·m−1·K−1 compared with 0.53 W·m−1·K−1 of the pure eutectic phase change material. By adding Ti2O3, the combination material shows the best solar absorbance capacity of 87.96 % compared with the pure eutectic phase change materials of 73.43 %. The latent heat of the novel material was 190.97 kJ/kg. Considering their huge superiority of highly efficient use of solar energy in natural resources, the new novel ammonium aluminum sulfate dodecahydrate‑sodium sulfate eutectic-based phase change material with excellent light absorbance, enhanced thermal conductivity, and high latent heat performance is a very promising material for solar-thermal conversion and energy storage.

Bidirectionally High Thermal Conductive Phase Change Materials with Arrayed Radially Oriented Boron Nitride Nanosheets for Battery Pack Thermal Management

Bidirectionally High Thermal Conductive Phase Change Materials with Arrayed Radially Oriented Boron Nitride Nanosheets for Battery Pack Thermal Management

Investigation on dynamic heat transfer characteristics and fin geometric parameters in latent heat storage system with vertical tubes and longitudinal fins

Latent heat storage plays an important role in the utilization of solar energy. However, the low thermal conductivity of phase change materials (PCM) significantly reduces the heat transfer efficiency of latent heat storage systems. To enhance its storage/release efficiency, optimizing the fin geometry is essential. This paper establishes a validated three-dimensional numerical model that considers PCM natural convection to study the effects of fin height and number on the heat transfer process. The fin volume of all models is kept constant, and the fin height is determined by the annular space. The impact of fin heights (0.3ΔR, 0.5ΔR, 0.7ΔR, 0.9ΔR) and numbers (4, 8, 10, 16) on heat transfer efficiency was investigated by analyzing the PCM temperature distribution on the shell section, the liquid fraction within the shell over time, and the average heat transfer rate and heat flux. The results show that increasing the fin height from 0.3ΔR to 0.9ΔR reduces the heat storage and release completion times by 61.16% and 45.43%, respectively. Similarly, increasing the number of fins from 4 to 16 reduces the heat storage and release completion times by 33.35% and 31.13%, respectively. The study concludes that increasing both the fin number and height dilutes the heat flux between the fin and PCM during both the heat storage and release processes, with fin number having a more significant effect on reducing heat flux than fin height. Therefore, when the fin volume remain constant, increasing fin height is more conducive to improving the heat transfer performance of the PCM. These findings will provide a foundation for the application of finned tube energy storage systems in building energy conservation and other fields.

Enhancing Thermal Conductivity of Phase Change Materials Using Nanomaterials and its Effectiveness in Cooling Solar Cells

The latent heat storage system is considered the most promising technology in thermal energy storage (TES) because of its many advantages. This technique uses phase change material (PCM) as a TES medium. However, the low thermal conductivity (TC) of PCM is the major drawback of the system. Previous studies have shown that the addition of nanomaterials to PCM generally increases its TC. On the other hand, researchers tend to study the possibility of using enhanced PCM to cool solar cells. In fact, raising the solar cell’s operating temperature negatively impacts its efficiency. This problem is considered one of the biggest problems in solar energy systems, and researchers are still working to solve it. This paper focused on the possibility of enhancing the TC of paraffin by adding different shapes of silver nanomaterials to it (silver nanoparticles, silver nanowires and nanohybrid with silver nanoparticles and silver nanowires). Nanomaterials were added at volume fractions of 0.5% and 1%. Then, the study examined the ability to use this improved material to reduce the temperature of solar cells. We used the “Solidworks” program to create a 3D system, and then we used the “Ansys Fluent” software to simulate five cases; PV without PCM, PV with PCM, PV with PCM enhanced by nanoparticles, PV with PCM enhanced by nanowires, PV with PCM enhanced by nanohybrid (a mixture of nanoparticles and nanowires). The results show that using PCM enhanced by nanowire at a volume fraction of 0.5% gave the best results in decreasing the temperature of PV. The average temperature of PV declined by 14.9[Formula: see text]. In addition, the efficiency of PV rose by about 7.6%. Followed by using PCM enhanced by nanoparticles at a volume fraction of 1%. The average temperature of PV declined by 12.9[Formula: see text]. Moreover, the efficiency of PV rose by about 6.6%.

Analysis on the effect of novel topological optimization fin structures considering eccentricity on the heat storage and release characteristics of shell and tube phase change heat accumulator

Shell and tube phase change accumulator (STPCA), as a common type of heat accumulator, can effectively solve the mismatching of time and space in solar thermal power generation by using phase change storage technology. However, the low thermal conductivity of phase change materials (PCMs) reduces its heat storage and release rate. To achieve a rapid latent heat energy storage and release, it needs to carefully modify the structure of STPCA. In the existing literature, insufficient attention has been given to simultaneously addressing the heat storage and release processes, while there is a scarcity of topological optimization methods employed in fin design. On one hand, it is difficult to find the optimal structure. On the other hand, the optimized structure is suitable for melting process but may not be suitable for solidification process. In this contribution, we initially integrated topological optimization and eccentricity to derive fin structures suitable for respective melting and solidification processes. Then, we compared and analyzed the effects of eccentricity and objective function on melting and solidification processes. Finally, we summarized the topology optimization process, objective function and eccentricity suitable for fast heat storage and release. The findings indicated that as the eccentricity increases, the rate of heat storage and release initially rises before subsequently declining. In this work, it is found that an eccentricity of 5 mm is the best. From the whole heat storage and release cycle, when the eccentricity is 5 mm, the objective function is the minimum temperature difference and the topological optimization fin structure calculated during solidification process shows the best heat storage and release performance. Compared with the topology optimization structure without eccentricity, the heat storage speed is increased by 36.43 %, and the heat release speed is increased by 15.53 %. The design law of the STPCA based on combined with the eccentricity influence and topology optimization method is summarized to help improve its heat storage and release performance.

Enhancing the phase change material based shell-tube thermal energy storage units with unique hybrid fins

The poor thermal conductivity of phase change material (PCM) has limited its application to thermal energy storage system. The present work aims to improve the performance of PCM in a vertical shell-tube energy storage unit through unique hybrid fins. The enthalpy-porosity approach is used to numerically investigate the phase change phenomenon. Based on the straight and spiral fin results, the novelty designs of double side spiral fin and hybrid fins, i.e. spiral/straight hybrid fin and straight/spiral hybrid fin, are proposed to further optimize the PCM charging process. The effects of different fin structure, hybrid fin proportion and spiral fin angle on the liquid fraction, temperature and average thermal energy storage rate are discussed. The fin structures can reduce the melting time of PCM up to 100% compared to the no-fin case. Although double side spiral fin outperforms the straight fin for the PCM melting behavior, the hybrid fin configurations shows the best enhancement, especially for the straight/spiral hybrid fin. The optimal fin design for the straight/spiral hybrid fin case is that with 0.7 fin proportion and 180° spiral angle, with up to 11.8% reduction of the melting time compared to the designs of other fin proportions and spiral angles. This work demonstrates the potential of this unique hybrid fin to be integrated with PCM for efficient thermal energy storage.

Progress and challenges of latent thermal energy storage through external field-dependent heat transfer enhancement methods

The development of Energy Storage technologies is critical to achieving a cleaner energy future. As one of the most widely used energy storage technologies, Latent Thermal Energy Storage (LTES) still suffers from poor charging and discharging performance subjected to the low thermal conductivity of Phase Change Materials (PCMs) and inefficient heat transfer process. Heat transfer enhancement techniques, such as using fins and adding highly conductive materials, have been widely developed and optimized to overcome these challenges. Recently, additional novel methods integrating adjustable external fields such as gravity, magnetic field, and electric field have been proposed to enhance the heat transfer performance of LTES, due to their advantages including easy adjustment of the field parameters according to the evolution of the heat storage process. Firstly, this work briefly summarizes the progress of conventional heat transfer enhancement methods. Secondly, the advancement of heat transfer enhancement by integrating adjustable external field effects has been analyzed and discussed. Finally, the potential of external fields to improve the heat storage performance of LTES under fluctuating thermal sources is discussed considering the wide existence of fluctuating energy supply and load in real applications. The main contributions from this review are: (a) The emerging heat transfer enhancement methods for LTES are comprehensively summarized and discussed for the first time; (b) The potential of different heat transfer enhancement techniques is demonstrated to overcome the poor heat storage performance under fluctuating thermal sources when considering real applications of LTES.

Investigation of Thermal Performance of Optimized Tree-Shaped Fins in Latent Heat Storage Units

An innovative tree-shaped fin is proposed to address the problem of low thermal conductivity of phase change materials (PCMs) in latent heat energy storage units. Numerical methods are applied to investigate the thermal properties of the melting process of PCMs. This paper investigates the effects of fin arrangement, fin geometry parameters, and temperature of the heat source on melting characteristics of PCMs in the latent heat energy storage unit. Results show that the melting time of PCMs with the inwardly-tilted fins at the top increases by 21.3% compared to the outwardly-tilted fins at the bottom. The response surface methodology is adopted to obtain the fin parameter combination with lower quality and better performance. The optimal fin dimensions are a thickness of 1.06 mm, a length of 3.38 mm, and a spacing of 12.61 mm. The reduction of the ambient temperature from 35 °C to 30 °C increases the melting time of the latent heat energy storage unit by 85%, and from 35 °C to 40 °C decreases the melting time by 29%.

The effect of fin geometry on the thermal performance of a conical shell double tube latent storage unit

Thermal energy storage systems are widely used in a variety of applications. Latent heat thermal energy storage units are generally recognized as one of the most efficient ways to utilize and store renewable energy because of their high energy density and constant charging/discharging temperature. Conversely, the low thermal conductivity of phase change materials in these units limits their applications. The present paper investigates the performance of a novel double-tube phase-change unit of a conical shape. To aid the melting rate, fins are added to enhance heat transfer. A two-dimensional ANSYS computational fluid dynamics model has been developed. The results have been validated against published data from the literature with very good agreement. The study considered different geometrical modifications in terms of fin angle and fin length. Their effect on melting rate and energy stored was carefully discussed. Results have shown that longer fins accelerate the PCM melting rate. The melting rate was enhanced by up to 42.6% for the 20 mm fin. However, the best orientation angle varies with the considered fin length. Energy and power have revealed the best performance for a short fin of length 5 mm and an angle of 60° from the vertical.

Heat transfer study of metal foam with partial filling method to strengthen phase change material

The advancement of ice-ball thermal energy storage systems is limited by the poor thermal conductivity of phase change materials(PCM). This paper presents a numerical investigation into enhancing heat transfer in ice balls by partially filling them with metal foam. Dynamic temperature changes, solid phase fraction, and cold storage capacity are analyzed for various filling radius ratios (2/13, 4/13, 6/13, 8/13, and 13/13). We quantitatively assess the specific impact of metal foam filling on heat transfer by calculating changes in the comprehensive thermal conductivity coefficient. Our findings reveal that the comprehensive thermal conductivity coefficient increases nonlinearly with the growing metal foam filling radius ratio, indicating that full filling may not the most optimal configuration. Furthermore, the energy storage capacity per unit time, per unit weight and per unit cost of the ice ball filled with metal foam under different radius ratios was evaluated by comprehensive evaluation criteria, and the optimal filling radius ratio was determined to be 6/13. Contrary to prior findings, this research highlights the efficacy of partial filling strategies, offering valuable insights for optimizing ice ball performance in thermal energy storage applications.

Improving thermal conductivity of styrene ethylene butylene styrene/paraffin/boron nitride phase change composite via the sacrificial template method for battery thermal management

To address the bottleneck of easy leakage and low thermal conductivity of phase change materials (PCMs) for battery thermal management. Phase change composite (PCCs) with paraffin (PA) as the PCM, styrene ethylene butylene styrene (SEBS) as the supporting skeleton, and boron nitride (BN) as the thermally conductive filler were fabricated by the sacrificial template method. Compared with the composite prepared by melt-blending method, the thermal conductivity of PCC was improved by 27.6 % and the leakage rate was reduced by 82.4 % because the BN was rubbed with Na2SO4 under the shear provided by internal mixer. As a result, the maximum temperature and maximum temperature difference of the battery pack with PCC cooling were 35.8 % and 62.5 % lower than those without PCC cooling at 2C discharge rate. In total, this work offers a strategy for improving the thermal conductivity and the leakage-proof ability of PCCs as well as for large-scale fabrication.

Analysis and comparative assessment of charging dynamics in vertical multi-channel latent heat storage system with corrugated wavy channels

Latent heat thermal energy storage (LHTES) is a promising technology, but thermal response limitations caused by the relatively poor thermal conductivity of phase change materials (PCMs) remains an issue. In recent years, computational fluid dynamics (CFD) has proven to be a powerful and reliable tool in design optimization of different systems and problems in the energy sector. Building on that, the present work will use CFD to investigate the thermal performance enhancements achieved using vertical double-tube LHTES systems with multi-channel sinusoidal wavy geometries. The main parameters investigated here are the wavy channel height (H), pitch (P) and the number of channels (N) and the results are assessed in terms of melting rates and heat transfer coefficients. The analysis found that increasing the height from 1 mm to 3 mm, accelerated the phase change by 21 %, while reducing the pitch from 12.5 mm to 5 mm, enhanced the heat storage rates by 28 %. In addition, increasing the number of channels from 3 to 7 improved the charging rate by 64 %. Compared to a conventional straight channel, the enhanced wavy channel (N = 7, H = 3 mm, P = 5 mm) achieved 80 % higher thermal storage rate and 42 % faster charging time. The above findings provide new insight into the potential of utilizing compact vertical LHTES systems via customized wavy channels, with significant impact towards enhancing the existing thermal energy storage systems.

Performance evaluation and optimization of compact tube-fin latent heat storage system in phase change nanoemulsion discharge mode

The low thermal conductivity of phase change materials (PCMs) and the reduced heat transfer temperature difference along a tube both limit the thermal performance of latent heat thermal energy storage (LHTES). Herein, a compact tube-fin LHTES system based on phase change nanoemulsions was developed and the heat transfer process of nanoemulsion and PCM was numerically investigated. The effects of the heat transfer structure of the heat exchanger and the thermophysical parameters of the heat transfer fluid (HTF) on the cold energy discharge performance of the system were quantified. The optimized model reduced the discharge time by 24.4 % in nanoemulsion mode relative to water. Heat transfer improvement on the PCM side could be achieved by decreasing the fin spacing and tube spacing and by increasing the fin thickness. Such improvement reinforced the heat transfer enhancement of the nanoemulsion. The heat transfer mechanism was explained by the temperature distributions of the HTF and PCM along the tube. The phase-change thermostatic properties of the nanoemulsion and PCM contributed to the formation of dual-phase-change coupled heat transfer (i.e., the heat transfer process when the fluid inside the tube and the PCM filled outside the tube undergo a simultaneous phase change) within the LHTES, increasing the heat transfer temperature difference and the magnitude of the driving force. Increasing the phase change temperature difference accelerated the onset of dual-phase-change heat transfer, while increasing the PCM content prolonged the process. Briefly, this study provided operational and design assistance for LHTES systems in nanoemulsion mode with the intention to accelerate heat transfer rates.

A graphene-CuNWs@PAMPS mixed aerogel for composite phase change materials with high thermal conductivity and excellent solar-thermal to electrical conversion

Phase change materials could store and release heat energy through phase change. However, the shortcomings of low thermal conductivity in the photothermal conversion process and easy leakage in the solid-liquid conversion seriously hinder the application of heat energy utilization. Graphene had high thermal conductivity, and it was an ideal material for improving the solar absorption, solar energy conversion and thermal conductivity of phase change materials. In this paper, a graphene aerogel composed of copper nanowires (CuNWs)/2-acrylamide-2-methylpropanesulfonic acid (AMPS) was designed. The thermal conductivity and shape stability of phase change materials for solar-thermal-electrical conversion applications were enhanced by composite CuNWs. Furthermore, PAMPS was coated on the surface of copper nanowires to avoid the oxidation of copper nanowires, and the sulfonic acid group of PAMPS was used to induce CuNWs to disperse uniformLy on the graphene sheets, which further constructed a good conductive network. After vacuum impregnation of PW, the best thermal conductivity phase change composite (PW-CuNWs@PAMPS-GA(1:1)) was obtained. The PCC loaded with CuNWs with the same graphene content achieved a thermal conductivity of 1.46 W/(m·K) and a high latent heat of 206.8 J/g. The latent heat retention rate was as high as 99.57 %. The photothermal conversion efficiency of PW-CuNWs@PAMPS-GA(1:1) was remarkable at 96.4 %. Due to its excellent solar-thermal-electrical conversion efficiency, the output voltage and output current were 800.8 mV and 85.8 mA, respectively, at a solar analog intensity of 5 kW/m2. Even after the sunlight was stopped, the voltage output could be maintained for a while by releasing the heat energy stored in the phase change composite.

Enhanced latent thermal energy battery with additive manufacturing

The low thermal conductivity of Phase Change Materials (PCMs), such as paraffin waxes, hinders efficient latent heat storage, especially for rapid charging and discharging cycles. To address this issue, this study explores experimentally and numerically the use of metal additive manufacturing to create a latent heat storage system operating at medium temperatures (around 90°C). A 3D Cartesian metal lattice is manufactured through laser powder bed fusion to optimize heat conduction within the PCM. Experimental tests show impressive specific power densities (approximately 714 ± 17 W kg−1 during charging and 1310 ± 48 W kg−1 during discharging). Moreover, the device exhibits stability over multiple cycles. Finally, the validated finite-element model has the potential to provides a basis for general design guidelines to boost the system’s performance further. Potential applications of this technology are highlighted in the automotive industry, where such systems could efficiently manage thermal energy, for instance, by capturing excess heat from an engine’s cooling radiator to expedite the warm-up process during a cold start, which is a critical phase for reducing pollutant emissions.

Enhanced study of phase change thermal storage using active rotating tank wall coupled fins combined with eccentric design

To address the low thermal conductivity of phase change materials and e improve heat storage, a novel rotating T-LHTESU structure is proposed. Numerical simulations systematically analyze the effects of rotation speed, various eccentricities and wall temperature on the performance of phase change heat storage. The results show that the introduction of rotation in the intermediate sleeve shortens the melting time with increasing rotation speed, reaching a maximum reduction of 61.66 % at 2 rps. Furthermore, the melting rate at 1 rps and 2rps first decreases and then increases with increasing eccentricity. At an eccentricity of 10 mm and 2 rps, the melting time is reduced by 7.23 % compared to the non-eccentric model. In addition, a higher wall temperature accelerates the change in the liquid phase fraction and increases the energy input. At a wall temperature of 371.15 K, the melting time is reduced by 73.56 % compared to the stationary model, with an increase in total heat by 2.52 %. This study provides theoretical guidelines for subsequent research on phase change heat transfer.

Radial assembly of graphene nanoplates via ice-template method to construct 3D thermally conductive phase change materials

Phase change material (PCM) with the capacity to store and release substantial thermal energy, has drawn significant attention. However, its large-scale application is still hindered by inadequate thermal management capacity and unsatisfactory thermal conductivity. In this study, graphene nanoplates were assembled into a 3D thermally conductive scaffold using a specialized ice template method. By virtue of the unique growth of ice crystals in radial pattern, a series of multidirectional thermal conduction pathways were successfully constructed within PCM, which has provided microscopically interconnected pathways for thermal transport. Consequently, a remarkable 5.2-fold increment in thermal conductivity (1.365 W m−1 K−1) in comparison to pure paraffin was achieved by incorporating a mere 6 vol% of GNP, which therefore enables excellent thermal cycling stability, remarkable latent heat storage capacity exceeding 175 J/g, together with leak-proof properties even under extended high-temperature conditions. Moreover, the solar-to-thermal energy storage efficiency is up to 85.8 % due to its efficient photothermal effect. This work presents an innovative approach to design PCMs with enhanced thermal conductivity and photothermal efficiency, offering promising applications in advanced thermal storage.

Effect of local mechanical oscillation on the performance of PCM thermal energy storage with various unit configurations

The low thermal conductivity of phase change materials is an essential restriction to their usage in thermal energy storage units. In this research, applying local oscillation by an oscillator on the portion of a side wall of RT35 enclosure is presented as a novel method to enhance the charging rate. The finite volume method is conducted for the numerical study on the effect of different configuration parameters including enclosure aspect ratios in the range of 1: 6, 1:2, 1:1, and 2:1, various oscillator lengths of 5, 10, 15, and 20 mm, as well as different oscillation frequencies of 10, 100 and 1000 Hz on the melting performance. Results indicate that applying local oscillation enhances the melting rate by more than 50 % as compared to without using oscillator because of enhancing convection heat transfer originating from vorticity intensification. The improvement of required time for melting the PCM is 91.55 %, 54.66 %, 67.7 % and 71.4 % for aspect ratios of 1:6, 1:2, 1:1 and 2:1, respectively.Results show that the melting performance is enhanced as the length of the oscillator increases. The required time for complete melting is obtained equal to 24, 26, 28, and 33 min at the oscillator length of 20, 15, 10, and 5 mm, respectively. The finding on the effect of oscillation frequency reveals that the liquid fraction, mean temperature, and heat flux at a specified time of melting process becomes higher as the oscillation frequency increases with significant enhancement of vorticity around the oscillator. At the frequency of 1000 Hz, the required time for melting is reduced by 9.5 % and 20.8 % as compared to frequencies of 100 Hz and 10 Hz, respectively. Other results also indicate that the higher Nusselt number is obtained for a higher enclosure aspect ratio.