Foam Pore Density Research Articles

Phase change heat storage and enhanced heat transfer based on metal foam under unsteady rotation conditions

Phase change heat storage technology is an essential method for balancing supply and demand in solar energy heat utilization. In this study, a numerical model of the phase change heat storage process is built to explore the impact of non-constant rotation, with and without metal foam. Subsequently, an investigation into the influence of metal foam pore density and porosity on heat storage performance is conducted, with a focus on Taguchi design for statistical analysis. The research delves into the effects of foam parameters on heat storage rate, temperature response, heat storage time, and heat storage properties. Findings reveal significant resolidification in the melting process of this thermal storage structure without metal foam, due to periodic rotation speed hindrance in transferring heat to the outside. Taguchi design results underscore the greater influence of porosity on the melting time and average heat storage rate compared to pore density. Specifically, a reduction in melting time by 62.21 % and 30.49 %, and an increase in average heat storage rate by 237.56 % and 51.19 % is observed when pore density and porosity are 40 PPI and 0.97, compared with the ones with porosities of 0.99 and 0.98, demonstrating minimal impact on total heat absorption.

Brazing of Copper Foam Using Cu-4.0Sn-9.9Ni-7.8P Filler Foil: Effect of Brazing Temperature and Copper Foam Pore Density

Copper (Cu) foam is a promising material that owns a high surface area that can be utilized in a thermal application. In this research, the brazing of Cu substrate to Cu foam in the sandwich configuration using Cu alloy filler foil was carried out. The foam at different pore per inch (PPI) of 15, 25 and 50 are brazed at different brazing temperatures. Mechanical and microstructure analysis were conducted to investigate a suitable brazing temperature and the best pore density of foam. The compressive strength of brazed 50 PPI foam has yielded the highest due to the highly dense interconnected branches. While the highest shear strength of brazed interface using 15 PPI foam has been recorded. The large branch size of 15 PPI foam has contributed to the sound joint between the brazed joint interface of Cu substrate and foam. Both mechanicals analysis above exhibits a highest strength at 660 °C as a brazing temperature The shear stress-strain curve of Cu substrate and foam brazed joint interface shows a brittle behaviour which accordance with the discoverable brittle phases of Cu3P and Ni3P using X-ray diffraction (XRD). Scanning electron microscopy (SEM) and Energy dispersive X-ray spectroscopy (EDX) have presented the formation of Cu3P and Ni3P at the brazed joint interface of Cu substrate and foam.

Optimizing of a metal foam-assisted solar air heater performance: a thermo-hydraulic analysis of porous insert placement.

A numerical assessment of the heat transfer efficacy of a solar air heater (SAH) was carried out. The SAH is supplied with a porous metal foam layer to improve thermal mixing. Both the local thermal non-equilibrium (LTNE) and Darcy-extended Forchheimer (DEF) models were employed to forecast fluid and thermal transport within the partly filled SAH channel. The analysis was performed for various values of dimensionless foam layer lengths ( ), pore densities ( ), and Reynolds numbers ( ) at a fixed value of layer thickness ( ). Based on the position of the porous layer, three distinct arrangements, marked as Case 1, Case 2, and Case 3, were explored. Regarding the parameters examined, the findings indicate a definite improvement in the average Nusselt number ( ), but unfortunately, the friction factor also increases unfavorably. By reducing the length of the porous layer, a reasonable reduction in heat transfer rate and a significant decrease in pressure drop were noticed. The results showed about 26.64%, 48.73%, and 70.74% reductions in pressure drop by reducing the dimensionless foam length from 1 to 0.25, 0.5, and 0.75 respectively for at . On the other side, there are only about 11.05%, 23.11%, and 40.78% reductions in . The exhaustive analysis of the thermal performance of SAH was conducted using the thermal performance factor (TPF), which considers the trade-off between the SAH channel's potential for improved heat transmission and its cost for pressure loss. The TPF may reach a maximum of 2.82 compared to the empty channel when the metal foam layer is inserted with , for , and .

Numerical simulation of holomorphic microscopic metal foam as cathode flow field in proton exchange membrane fuel cells

ABSTRACT An enhanced flow field significantly improves reactant transfer and proton exchange membrane fuel cells (PEMFCs) performance. Metal foams with high porosity and electrical conductivity are promising flow field structures. This study aims to investigate the effect of metal foam porosity and pore density on PEMFC performance. A three-dimensional multi-physics PEMFC model with holomorphic microscopic metal foam (HMMF) as a cathode flow field was established using computational fluid dynamics (CFD). The HMMF with high porosity as a cathode flow field can enhance reactant transfer and accelerate the electrochemical reaction process. It is also observed that increasing the porosity of HMMF can significantly improve the peak power and current density of PEMFC. The simulation results show that the current density gains an increase of 15.3% by raising the porosity from 0.8 to 0.96. Besides, the current density increases by 4.3% when the pore density is 95 PPI compared to 21 PPI. Moreover, the effects of cathode stoichiometry ratios on oxygen molar fraction and polarization curves were studied to obtain better operating conditions for the HMMF flow field. The results indicate that increasing the cathode stoichiometry ratio can effectively enhance oxygen concentration. And the increase of the cathode stoichiometry ratio from 1.5 to 4.0 results in a 45.5% increment of current density.

Effect of enhancement in metal foam pore density on heat transfer of phase-change materials

To explore the effect of metal foam(MF)pore density on the melting and heat transfer of phase-change materials(PCMs), a semi-cylindrical visualization experimental setup was established to experimentally study copper MFs with different pore densities coupled with paraffin wax. The results showed that as pore density increases, the melting time of the composite PCMs decreases compared to that of pure paraffin wax by 11.5%, 16.2% and 18.1% for pore densities of 5, 20, and 30 PPI, respectively. Composite PCMs with a pore density of 30 PPI exhibited fastest melting and lowest maximum temperature difference during melting. Considering the relative thermal energy storage(TES) capacity and relative TES rate, composite PCMs with a pore density of 30 PPI exhibited optimal overall thermal storage performance. Furthermore, heat conduction was the primary means of thermal transmission during the melting of the bottom paraffin. However, the melting of the upper paraffin was nevertheless influenced by natural convection, Hence, as pore size decreased, the influence on the weakening of natural convection became more significant. The results of the study provide guidance for the optimal design of latent heat storage devices for semi-cylinders.

Reverse latent heat power generation properties of PCM/nanoparticles/metal foam at night

Nanoparticles and metal foam are doped with the phase change materials (PCMs) to improve the utilization efficiency of solar energy during the day. In order to investigate the latent heat power generation properties of this hybrid PCM at night, A photothermal conversion experimental platform was built in this paper, the effects of metal foam pore density (PPI=0, 5, 10, 15) and nanoparticle concentrations (ω = 0%, 0.5%, 0.8%, 1.0%) on the reverse temperature difference properties were investigated. Results indicates that PPI= 15 metal foam and ω = 1.0% nanoparticles can reduce the internal temperature of hybrid PCM by 35.92% and 20.93% respectively compared with pure paraffin. PPI= 15, 10, and 5 metal foam decreases the temperature difference, voltage, and power generation at most by 74.57%, 71.02%, and 74.54% compared with PPI= 0. Compared with ω = 0% nanoparticles, the temperature drop rates of ω = 1.0%, 0.8%, and 0.5% nanoparticles increase at most by 103.0%, 79.69% and 39.84% respectively. To a certain extent nanoparticles and metal foam are not conducive to reverse latent heat power generation although they are beneficial to photothermal performance. This research has reference and guiding significance for reverse latent heat power generation properties of PCMs doped with nanoparticles and metal foam at night.

Heat transfer enhancement of a bio-based PCM/metal foam composite heat sink

• The impacts of various PCM filling height ratios on the cooling performance of a PCM composite-based heat sink are examined experimentally. • The effects of three different pore sizes of copper foam on the heat transfer of PCM composites at varying heat loads are studied. • PCM/copper foam composite samples outperform pure PCM. • A PCM filling ratio of 1.0 results in little to no improvement in cooling performance. • The optimum cooling performance is provided by PCM/copper Foam-95PPI with a PCM filling ratio of 1.3. Effects of phase change materials (PCMs) filling height and copper foam pore densities on the cooling performance of a bio-PCM composite-based heat sink are investigated. In this respect, three filling height ratios of 1.0, 1.3, and 1.6 as well as three copper foam samples with pore densities of 35, 80, and 95 pores per inch (PPIs) are examined. Temperature profiles of the flat plate, the time required to reach specified temperatures, and the enhancement ratios at various temperatures and filling ratios are used to evaluate the thermal performance of the PCM composite-based heat sink. The investigation enables the determination of the optimal PCM filling height for maximum cooling performance of the heat sink. The results show that the optimal PCM filling height is 1.3 times the copper foam thickness and that a PCM/copper foam composite with 95 PPI pore density produces the best cooling performance with enhancement ratios of 1.54 and 1.44 under 10 and 15 W heat loads, respectively.

Influence of surface wettability on heat transfer and pressure drop characteristics of R1234ze(E) flow boiling in metal foam filled tubes

The influence of surface wettability on the flow boiling heat transfer and pressure drop characteristics of R1234ze(E) in metal foam filled tubes was experimentally investigated. The working conditions include metal foam pore densities from 10 to 40 PPI and porosities from 90% to 95%, mass fluxes from 90 to 180 kg·m−2·s−1, and heat fluxes from 12.4 to 18.6 kW·m−2. The experimental results show that the hydrophilic modification deteriorates the flow boiling heat transfer coefficient in metal foam filled tubes by 7%-13%, and reduces the pressure drop by 9%-20%, while the hydrophobic modification improves the flow boiling heat transfer performance of metal foam filled tubes by 3%-8% and increases the two-phase pressure drop by 8%-15%. Heat transfer and pressure drop correlations were proposed for flow boiling in metal foam filled tubes with different surface wettability.

Melting performance of a composite bio-based phase change material: An experimental evaluation of copper foam pore size

• PCM-based heat sinks were optimized by combining them with copper foam. • The cooling performance of five different PCM-Copper based heat sinks was studied. • Higher copper foam pore densities were found to yield improved cooling performance. • Increased heat loads triggered a faster thermal response from the heat sink. This paper presents an experimental study on the thermal performance of a composite heat sink consisting of a bio-based phase change material and copper foam. The experiments are carried out at three different heat loads (10, 15, and 20 W) using five copper metal foam samples with the same dimensions (10 × 9 × 0.3 cm), porosity (98%), and pore densities of 20, 35, 60, 80, and 95 pores per inch (PPI). The thermal performances are evaluated using the temperature profiles, the time required to reach specific temperatures, and the enhancement ratios of the heat sinks. The results favor the PCM-Copper composite sample with 95 PPI because it took the longest time to achieve a constant temperature when compared to its other pore density counterparts. Also, for the same sample under 20 W power input, the enhancement ratios are 1.29, 1.45, and 1.23 at critical temperatures of 50, 55, and 60 °C, respectively.

Experimental study on the effect of metal foams pore size in a phase change material based thermal energy storage tube

It is well known that latent thermal energy storages can help in store large quantities of energy in small volumes, but that their charging and discharging periods are prohibitively long. In this work it was experimentally demonstrated that the addition of a foam can drastically shorten the phase change materials charging and discharging times up to 10 and 5 times, respectively. A tube-in-tube heat exchanger, where water flows in the tube with an aluminum foam-saturated Phase Change Material inserted in the annulus, was experimentally tested and new data was added to the open literature. Testing three different foam samples, the influence of the foam pore density (10, 20, and 40 Pores per Inch), sample orientation as well as that of water flow rate (from 2 to 8 l min−1) and inlet temperature (from 45 to 55 °C), on charging and discharging processes was investigated. Few researchers investigated the influence of sample orientation, while published results do not agree on the pore size influence. In this work it was found that the 20 Pores per Inch sample showed the best performance leading to fastest charging and discharging processes. Moreover, it was observed that with foam-saturated Phase Change Materials, the charging and discharging times are not affected by the inclination angle. On the contrary, they are affected by water flow rate and temperature. A high inlet water temperature and a high water flow rate can shorten both phase transfers.

Experimental study of multilayer gradient copper foam effect on pool boiling heat transfer performance and gas-liquid behavior characteristics

The rapid growth of the internet of things has induced the integration of many microelectronic devices in physical objects, yet the generated heat decreases the performance and stability of microelectronic devices. Therefore, new materials such as gradient metal foams (GMFs) have been recently designed to improve heat transfer. In this paper, an experimental visualization setup was built to investigate the effect of the GMFs gradient layers number and the arrangement order on the pool boiling heat transfer performance. Results show that increasing the number of gradient layers enhances the heat transfer when the copper foam pore density is low. By contrast, at high pore density of 50 PPI, increasing layers hardly changes heat transfer. The bubbles dynamic behavior on the metal foams surface of with different gradient structures is also different. When bubbles detach upward, the temperature of the metal foam is lower, and the temperature gradient is higher. When bubbles detach sideward, the bubble escape is much shorter, and the bubble detachment frequency and size increase. Combined with the theoretical research, the metal foam gas-liquid flow heat transfer model were constructed. The advantages and disadvantages of GMFs with different structures and the applicable scenarios are analyzed.

Effect of nanoparticles and metal foams on heat transfer properties of PCMs

Phase change material (PCM) are often used to realize thermal energy storage, which is regarded as an effective way to solve the problem of time difference between energy supply and demand. However, conventional phase change materials hinder the energy exchange rate and affect the storage efficiency due to their low thermal conductivity. To solve this problem, this study proposed three types of composite materials “nanoparticle/paraffin”, “metal-foam/paraffin” and “nanoparticle/metal-foam/paraffin”. The heat transfer properties of proposed composite materials were investigated based on the heat flux method and the heat storage and release test. The effects of metal foams and nanoparticles on the properties of phase change materials with different parameters were obtained, and the effect of the pore density of metal foams on the natural convection process of phase change materials was considered. The results show that the heat transfer performance of the composite phase change material is significantly improved compared with the traditional phase change material. With the increase of metal foam pore density, the heat transfer performance first increases and then decreases which caused by the enhanced obstruction to natural convection, 40 PPI pore density shows better comprehensive performance. In terms of thermal conductivity, nanoparticle/paraffin, metal-foam/paraffin (40 PPI) and nanoparticle-metal foam-paraffin (40 PPI) have been enhanced in turn, which are 123.27%, 740.39%, 847.15% higher than pure paraffin, respectively. The melting time and solidification time are shortened by 6.74%, 62.72%, 72.61% compared with pure paraffin and 26.46%, 49.47%, 38.62%.

Oxygen transport in proton exchange membrane fuel cells with metal foam flow fields

Metal foam flow fields have shown great potential in improving the performance of proton exchange membrane (PEM) fuel cells, while their effect on the oxygen transport process remains inadequately understood. In this study, oxygen transport in metal foam flow fields (under zero-humidity operating conditions) is simulated by using a three-dimensional multi-species lattice Boltzmann model. Comparison is done between the metal foam flow field and the conventional channel-rib flow field, and parametric studies are conducted on the metal foam porosity, pore density, and compression ratio. Results show that the metal foam flow field enhances mass transfer of oxygen to the catalyst layer and improves the oxygen distribution homogeneity. Within the range of parameters considered, decrease in the metal foam porosity yields nonmonotonic variation of the mass transfer rate of oxygen to the catalyst layer, which increases at high inlet velocities (higher than 2 m/s) but decreases at low inlet velocities (less than 2 m/s). The increase in metal foam pore density and compression ratio leads to enhanced mass transfer of oxygen, which becomes increasingly prominent at high inlet velocity. The results of this study could be insightful for the implementation of metal foam flow fields in PEM fuel cells. • Oxygen transport in PEM ofuel cells is studied by using lattice Boltzmann method. • Metal foam flow field has stronger oxygen transport than channel-rib flow field. • Increasing metal foam pore density/compression ratio facilitates oxygen transport. • Reducing porosity suppresses/enhances oxygen transport at low/high inlet velocity.

Thermodynamic analysis of entropy generation in a horizontal pipe filled with high porosity metal foams

In the field of thermal management of electronic equipment , examining entropy generation properties is extremely useful. The entropy production experiments have been expanded to porous media using high porosity metal foams. The entropy production/generation for forced convection heat transfer in a tube is quantified via a numerical research. In the field of air stream direction, the horizontal pipe is entirely filled with nickel metal foam of 0.6 m length. For the isotropic porous metal foam zone, the Darcy-extended Forchheimer (DEF) flow is used to capture the dynamics of flow and local thermal non-equilibrium (LTNE) model is used for analyzing the heat transport phenomenon, while the k-ɛ turbulent model is used for the non-foam porous region of the tube. The effect of fully filled nickel metallic foam with different pore densities of 10, 20, and 30 metal foam with a porosity of 0.85 is being investigated. The computational solutions presented here are supported by experimental results published in the literature. The outlet exergy of the system rises with higher flow rates and falls with higher metal foam pore densities. The results of entropy generation due to thermal and fluid friction and Bejan number conceptions are also shown and discussed.

Visualized-experimental investigation on the energy storage performance of PCM infiltrated in the metal foam with varying pore densities

To further enhance the melting rate of the metal foam composite phase change material (MFCPCM), we took partial and gradient optimizations on the pore densities of metal foams. The partially optimized models, including Partial-80-5-5 and Partial-5-5-80, were compared with the Uniform-5 model. Results show that the Partial-80-5-5 model has the most developed melting among the three models. It illustrates that enlarging the pore density in the top region is conducive to accelerating the whole melting process. Besides, through a further comparison of the Partial-80-5-5, Partial-40-5-5, and Partial-20-5-5 models, we concluded that the larger the pore-density in the top region is, the faster the melting is. Subsequently, the gradient optimizations, including Gradient-80-20-5 and Gradient-80-40-5 models, were experimented with and analyzed. It was obtained that the Gradient-80-40-5 model has the fastest melting rate among all models. The inhibition of the large pore density on natural convection at the top and middle regions causes a strong vortex at the bottom region, so the melting process is significantly reinforced. Through the optimizations on the metal foam's pore density, the energy storage rate can achieve a prominent enhancement.

Numerical Investigation of Metal Foam Pore Density Effect on Sensible and Latent Heats Storage through an Enthalpy-Based REV-Scale Lattice Boltzmann Method

In this work, an unsteady forced convection heat transfer in an open-ended channel incorporating a porous medium filled either with a phase change material (PCM; case 1) or with water (case 2) has been studied using a thermal lattice Boltzmann method (TLBM) at the representative elementary volume (REV) scale. The set of governing equations includes the dimensionless generalized Navier–Stokes equations and the two energy model transport equations based on local thermal non-equilibrium (LTNE). The enthalpy-based method is employed to cope with the phase change process. The pores per inch density (10≤PPI≤60) effects of the metal foam on the storage of sensible and latent heat were studied during charging/discharging processes at two Reynolds numbers (Re) of 200 and 400. The significant outcomes are discussed for the dynamic and thermal fields, the entropy generation rate (Ns), the LTNE intensity, and the energy and exergy efficiencies under the influence of Re. It can be stated that increasing the PPI improves the energy and exergy efficiencies of the latent heat model, reduces energy losses, and improves the stored energy quality. Likewise, at a moderate Re (=200), a low PPI (=10) would be suitable to reduce the system irreversibility during the charging period, while a high value (PPI = 60) might be advised for the discharging process. As becomes clear from the obtained findings, PPI and porosity are relevant factors. In conclusion, this paper further provides a first analysis of entropy generation during forced convection to improve the energy efficiency of various renewable energy systems.

Interfacial microstructure and strength of brazed copper/porous copper foam towards the effect of porous copper foam pore density and brazing holding time

AbstractThe porous copper foam was sandwiched between two coppers plate and then brazed using copper‐tin (9.7 %)‐nickel (5.7 %)‐phosphorus (7 %) filler foil. Brazing process was conducted to joint copper/porous copper foam by evaluating the effect of porous copper foam pore densities [pore per inch (PPI)] and brazing holding times. The brazed joint interface of copper and porous copper foam was characterised using Field emission scanning electron microscopy and Energy‐dispersive x‐ray spectroscopy for the microstructure and elemental composition analysis, respectively. X‐ray diffraction analysis was carried out on the shear fractured surfaces of brazed copper and porous copper foam for phase determination. The results exhibited distinct phases of copper (Cu), copper phosphide (Cu3P), nickel phosphide (Ni3P), and copper compound with tin (6 : 5) (Cu6Sn5). The filler layer was formed as an island‐shaped that consists of copper phosphide and nickel phosphide. Prolong brazing holding time causes a thinner filler layer in brazing seam. While the non‐uniform thickness of the filler layer was observed at different pore densities of porous copper foam. The shear strength of brazed copper/porous copper foam 15 PPI with a 10 min brazing holding time yield a maximum shear strength of 2.9 MPa.

Experimental study of condensation heat transfer characteristics on metal foam wrapped tubes under sloshing conditions

Metal foam wrapped tube has the advantages of large specific surface area, and thus it has great potential to enhance the condensation heat transfer for marine shell and tube condensers, which have the requirements of "high compactness" and "high reliability" under sloshing conditions. In the present study, the effects of the metal foam pore density and the sloshing conditions on the condensation heat transfer on metal foam wrapped tubes were experimentally investigated. The pore density of metal foam covers the widely used structure ranging from 5 PPI to 40 PPI; the sloshing conditions include six-freedom motions of surging, swaying, heaving, rolling, pitching and yawing; the sloshing displacement ranges from 0 to 120 mm; the sloshing angle ranges from 0 to 12°; and the sloshing frequency ranges from 0 to 0.33 Hz. The experiment results show that the condensation heat transfer coefficients of metal foam wrapped tubes increase firstly and then decrease with the increase of PPI, and the 10 PPI metal foam wrapped tube shows the optimal condensation performance. The rolling motion has significant effect on the condensation heat transfer of metal foam wrapped tube, while the other five sloshing motions have negligible effect on the condensation heat transfer of metal foam wrapped tube. With the increase of rolling angle from 0 to 12°, the condensation heat transfer performance of 10 PPI and 20 PPI metal foam wrapped tube gradually increases and the increase rate of the condensation heat transfer coefficient is up to 12%; while the condensation heat transfer performance of metal foam wrapped tube with 5 PPI increases as the rolling angle is larger than 10°. With the increase of rolling frequency, the condensation heat transfer performance of metal foam wrapped tube decreases, and the decrease rate of the condensation heat transfer coefficient is up to 8%.

Experimental and numerical investigations of latent thermal energy storage using combined flat micro-heat pipe array–metal foam configuration: Simultaneous charging and discharging

Simultaneous charging and discharging (SCD) of the latent thermal energy storage (LTES) can improve the flexibility of solar thermal systems and ensure the continuity of energy supply. Experiments and numerical simulation are conducted in this study to reveal the SCD thermal behavior of LTES device using flat micro-heat pipe array–metal foam composite structure. The effects of heat transfer fluid (HTF) temperature, volumetric flow, initial phase change material (PCM) state, and metal foam are analyzed. Results show that PCM absorbs heat from hot HTF or releases heat to cold HTF at the beginning depending on the PCM and HTF temperatures. The SCD eventually reaches an approximate steady state, cold HTF exchanges heat directly with hot HTF, power and PCM temperature remain unchanged. Low cold HTF temperature quickly reaches steady state and low final PCM temperature but has no influence on steady state power. PCM initial state and metal foam porosity do not affect the steady state power and final PCM temperature. However, they affect the time to reach steady state. Metal foam pore density has no obvious effect on SCD. SCD has more stable discharging performance than single discharging, especially when the cold HTF temperature is high.

Interfacial heat transfer in metal foam porous media (MFPM) under steady thermal conduction condition and extension of Lemlich foam conductivity theory

In this work, interfacial heat transfer characteristic in metal foam porous media (MFPM) under a steady thermal conduction condition (usually employed to determine the effective thermal conductivity of MFPM) is firstly investigated. To this end, thermal conduction in metal foam unit cells (represented by Weaire-Phelan foam geometry of different foam porosities and pore densities) saturated with filling mediums is directly simulated considering a wide range of thermal conductivity ratio between foam skeleton and filling medium; and the corresponding key heat flux information is obtained and analyzed. It is revealed that the local interfacial heat conduction in MFPM can be significant; however, the net total interfacial conductive heat transfer is negligibly small. More importantly, the negligible net total interfacial conduction is found unaffected by the discrepancy in thermal conductivity between foam skeleton and filling medium, which in fact is due to the intrinsically symmetric characteristic of foam structure enabling a result that the heat flowing out of foam skeleton offsets the heat flowing in it. Therefore, the respective heat conduction in foam skeleton region and filling medium region of MFPM can be considered in parallel. Then based on this conclusion, the Lemlich foam conductivity theory is reasonably extended to incorporate the influence of filling medium for predicting the effective thermal conductivity (ETC) of MFPM. Further comparisons with previously reported experimental data and ETC models show that the extended Lemlich theory not only well improves the ETC prediction accuracy of MFPMs, but also keeps a simple and elegant form of expression. The findings in this work can correct the previous misleading cognition on interfacial heat transfer in MFPM under steady thermal conduction condition, and can provide crucial clue to derive more efficient ETC models of MFPM.