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S. Kawai, [I. Watanabe](https://orcid.org/0000-0002-7693-1675)

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[Image-based finite element modeling of air flow and thermal transport in Al-fiber-sintered porous materials](https://mdr.nims.go.jp/datasets/30bea341-0e6e-440e-a311-ec9dea21a1fa)

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1 Image-based finite element modeling of air flow and thermal transport in 1 Al-fiber-sintered porous materials 2  3 S. Kawaia, * and I. Watanabeb 4 a Innovation Center, Mitsubishi Materials Corporation, 5 1002-14 Mukohyama, Naka, Ibaraki 311-0102, Japan 6 b Materials Modeling Group, Data-driven Materials Research Field, Center for Basic 7 Research on Materials, National Institute for Materials Science, 1-2-1 Sengen, Tsukuba, 8 Ibaraki 305-0047, Japan 9 *Corresponding Author: Tel: 81(0)-29-295-5705, Fax: 81(0)-29-295-5740 10 E-mail: skawai@mmc.co.jp 11  12 Abstract 13 In this study, the pressure drop and heat transfer in a heat-transfer tube filled with a sintered 14 porous medium comprising Al fibers were investigated using computational fluid dynamics 15 (CFD) simulations. We reconstructed the sintered fibrous porous structure and generated a 16 computational mesh using X-ray computed tomography, and the simulated pressure drop and 17 heat transfer agreed well with the results reported by Enoki et al. The differences in both the 18 form coefficient and the permeability between two different samples can be reasonably 19 explained by the differences in both the specific solid surface area and the porosity. Further, 20 the CFD simulations indicated that thermal conduction of the Al solid phase enhanced the 21 heat exchange between the air and Al fibers. To examine the effect of solid-phase thermal 22 conduction, we generated a model of a heat-transfer tube having an ideal wire mesh structure, 23 in which we introduced the interfacial thermal conductivity between Al fibers and the inner 24 tube wall as additional factors. We also performed CFD simulations with four different shell-25 mailto:skawai@mmc.co.jp 2 region thicknesses and found that even a very narrow gap of 5 to 10 µm heavily impacted the 26 heat-exchange performance because of the low thermal conductivity of the shell region. 27 Keywords: Heat transfer; Pressure drop; Computational fluid dynamics; X-ray computed 28 tomography; Image-based finite element modeling. 29  30 1. Introduction 31 Efficient heat transfer and thermal recovery are required to ensure both the reliability 32 and stability of devices such as heat exchangers and heat sinks for the cooling of integrated 33 circuit chips and power electronics, as well as to improve their energy efficiencies. To 34 enhance the heat-transfer characteristics of such devices, porous media can be added, and this 35 approach has drawn significant attention. Crucially, porous materials exhibit favorable 36 thermal and hydrodynamic performance for use in industry, for example, as heat exchangers, 37 heat sinks, and thermal energy storage units. Generally, porous materials have large heat-38 transfer areas per unit volume, and maximizing this parameter is key to enhancing the heat-39 transfer rate. However, large heat-transfer areas often cause large pressure drops owing to 40 frictional losses of the flowing fluid. Therefore, a precise understanding of the pressure drop 41 and amount of heat transferred within devices, such as heat-transfer tubes, containing porous 42 materials is crucial for achieving optimal and efficient device function. 43 To date, different forms of porous materials, including metal foams, fibers, and 44 particle beds, have been used to enhance the heat-transfer performance. Of these, metal foams 45 have a disordered and interconnected network structure with a tortuous flow path. Kim et al. 46 [1] experimentally examined the effect of Al foam on the flow and convective heat transfer in 47 an asymmetrically heated channel. In addition, Noh et al. [2] experimentally investigated 48 both the fluid flow and heat transfer in an annulus containing high-porosity Al foam. Further, 49 Adabi et al. [3] experimentally measured the heat transfer and pressure drop in a small Cu 50 tube filled with Cu foam. 51  3 Woven wire gauzes have also been used in heat-transfer equipment and processes. 52 For example, Tian et al. [4] experimentally examined the effect of the orientation of the air 53 flow on the overall pressure drop and heat-transfer performance of brazed Cu textile meshes 54 with periodic cellular topologies; Wu et al. [5] experimentally measured the pressure drop 55 through woven metal screens and formulated an empirical equation to describe their friction 56 characteristics; and Kolodziej et al. [6] experimentally investigated the fluid flow and heat 57 transfer in a tubular apparatus composed of stacked wire gauzes. In addition, Iwaniszyn et al. 58 [7] performed computational fluid dynamics (CFD) simulations of the fluid flow and heat 59 transfer of a single wire gauze using the representative volume element (RVE) model. 60 Heat-transfer tubes filled with solid particles have also been frequently employed 61 because they can be manufactured relatively easily by filling a channel or tube with particles 62 and powder. Jiang et al. [8] experimentally investigated the fluid flow and forced convective 63 heat transfer in a plate channel filled with glass or metallic spherical particles. In a different 64 study, they also performed similar experimental investigations on microchannel heat 65 exchangers and microporous heat exchangers [9]. Later, Jeng et al. [10] experimentally 66 investigated heat transfer in an asymmetrically heated rectangular channel filled with brass 67 beads, and Zhang et al. [11] analyzed the heat transfer associated with forced convection in a 68 heat-transfer tube filled with granular porous media. 69 Fibrous porous materials have also been studied where the porous media are 70 composed of fibers or fiber-like particles. For example, Tadrist et al. [12] reported the 71 thermal and hydrodynamic performance of two kinds of fibrous materials: randomly stacked 72 fibers and commercial Al foam, whereas Enoki et al. [13,14] conducted experiments on the 73 heat transfer and pressure drop in the Al heat-transfer tube filled with sintered Al fibrous 74 high-porosity media. Panerai et al. [15] experimentally measured the pressure gradient across 75 a sample composed of a commercial fibrous carbon media to obtain a benchmark dataset of 76 the permeability properties. Raudenský et al. [16] reported a new type of heat exchanger, 77  4 chaotised flexible polymeric fiber heat exchangers (CFPFHEs), where the flexible polymeric 78 hollow fibers were used as heat transfer elements. 79 In the studies mentioned above, correlations between the friction factor or heat-80 transfer coefficient and the fluid velocity (or the Reynolds number) were proposed to 81 understand the heat-transfer performance associated with forced convection [1–3,5–82 10,13,14]. However, the proposed correlation formulas were somewhat different for several 83 coefficient values for each kind of porous media, and the reasons underlying this difference 84 have been discussed considering porosity, specific surface area, equivalent pore diameter, 85 strut diameter, and tortuosity differences. Particularly, the equivalent pore diameter is a 86 crucial parameter used to estimate the characteristic length scale for the interstitial fluid flow. 87 Dietrich et al. [17] proposed an estimation procedure for the equivalent pore diameter using 88 the specific surface area via magnetic resonance imaging for ceramic open-cell foams, and 89 they also proposed correlations between the equivalent pore diameter and the nominal pore 90 per inch number provided by the foam manufacturer.  91 Compared to woven wire gauze and open-cell foams, in the case of fibrous porous 92 structure, insufficient information concerning the correlation between the equivalent pore 93 diameter and geometrical parameters such as the fiber diameter, fiber length, porosity, and 94 specific surface area has been elucidated. The equivalent pore diameter is closely related to 95 the tangle level of large number fibers, which depends on the processing conditions such as 96 fiber filling and sintering. For these reasons, several earlier studies employed the square root 97 of the permeability as the characteristic length scale instead of the equivalent pore diameter 98 [2,13,14]. However, tedious and time-consuming experimental measurements are required to 99 estimate the permeability. CFD simulations may therefore be used as alternative methods for 100 experimentally estimating the permeability within fibrous porous materials. 101 Recently, computational power has increased significantly, and various numerical 102 studies of fluid flow through idealized porous structures have been conducted. For example, 103  5 Boomsma et al. [18] performed CFD simulations of the fluid flow through an idealized open-104 cell metal foam (OCMF) using the RVE model. Kopanidis et al. [19] performed a CFD 105 simulation for the heat transfer associated with fluid flow through an idealized high-porosity 106 OCMF, in which eight cells composed of six tetrakaidekahedra and two dodecahedra were 107 used as the representative open-cell volume. Bai et al. [20] performed a CFD simulation of 108 fluid flow through a simplified OCMF comprising a one-sphere-centered tetrakaidecahedron 109 structure. Tang et al. [21] also performed a pore-scale CFD simulation of the heat transfer 110 and fluid flow characteristics in tetrakaidekahedron OCMF. Nie et al. [22] numerically 111 investigated the pressure drop and heat transfer through an OCMF having a three-112 dimensional (3D) structure created by Laguerre–Voronoi tessellation. Yang et al. [23] 113 performed a pore-scale CFD simulation of the fluid flow and heat transfer through a high-114 porosity OCMF under rotating conditions; specifically, a radially rotating channel filled with 115 a Weaire–Phelan foam cell was numerically modeled. Konduru et al. [24] also presented an 116 experimental and numerical studies on porous heat exchangers composed of Kelvin-cell-117 based metal foam (KMF) at high temperature by considering the interplay among conduction, 118 convection, and radiation. Recently, triply periodic minimal surface (TPMS) structure has 119 drawn significant attention toward the novel convective heat-transfer enhancement [25-29], 120 in which both the method to precisely control TPMS lattice structure and the effects of TPMS 121 lattice parameters on the thermo-hydraulic performance were reported. 122 Furthermore, recent numerical studies have focused on the effects of the metal foam 123 strut shape on convective heat transfer and pressure drop performance. For example, Moon et 124 al. [30] performed CFD simulations of the fluid flow and heat transfer through a modified 125 KMF with circular or elliptical struts. In addition, Calati et al. [31] further examined the fluid 126 flow and heat transfer through a KMF with elliptical struts and discussed the effect of the 127 strut orientation with respect to the main fluid-flow direction on the pressure drop and heat 128 transfer coefficient. Finally, Ambrosio et al. [32] numerically examined the effects of strut 129  6 shape on the convective heat transfer and pressure drop in ideal and real foams using the 130 RVE model. 131 Recently, the fluid-flow and heat-transfer rates in foams having realistic pore 132 structures have been numerically calculated using CFD combined with X-ray computed 133 tomography (CT) measurements. For example, Torre et al. [33] performed laminar CFD 134 simulations of the fluid flow through an Al foam. Using the Darcy–Forchheimer law [34,35] 135 and micro-CT image-based CFD simulations, they estimated both the permeability and form 136 coefficient. Ranut et al. [36] numerically investigated the fluid flow and heat transfer in an Al 137 metal foam using CFD simulations combined with the X-ray CT technique. In their study, a 138 small representative cubic volume was extracted as the RVE model. Meinicke et al. [37] 139 performed CFD simulations of the fluid flow inside a porous SiO2 glass sponge and also 140 carried out experimental measurements of the fluid velocity using the particle image 141 velocimetry technique. In their CFD simulations, the RVE of the sponge structure was 142 numerically modeled using high-resolution X-ray CT, and the calculated velocity fields were 143 compared with their own experimental measurements. Dixit et al. [38] performed CFD 144 simulations of the fluid flow and heat transfer through an open-cell Al foam and reported 145 small deviations between the experimental and calculated results that arose from the small 146 computational elemental size extracted from the original foam sample. Wang et al. [39] 147 investigated the fluid flow and heat transfer inside reticulated porous ceramics numerically 148 using unsteady-state CFD simulations combined with X-ray CT data, and they discussed the 149 local thermal equilibrium during the transient conjugate heat-transfer process. Liu et al. [40] 150 numerically investigated a pore-scale convective heat transfer characteristics in a Bentheimer 151 sandstone core sample, where the numerical model was reconstructed with micro-CT image 152 data. Recently, Zhang et al. [41] performed a pore-scale simulation of forced convection heat 153 transfer in metal foams with uniform and gradient porosities, where CFD models were 154 reconstructed with an X-ray CT scan of copper foam samples and the CFD model with 155  7 gradient porosity was constructed merging two separate models with different porosities 156 numerically. They examined the effect of the gradient orientation on the heat transfer 157 performance. Kuhlmann et al. [42] developed an open-source based workflow for converting 158 CT data to volume meshes appliable for finite volume method based CFD simulation. In 159 recent numerical studies [43-45], the Lattice Boltzmann Method (LBM) was used to 160 calculated fluid flow through metal foams reconstructed with micro-CT data, since the LBM 161 is a quasi-molecular method and considered as powerful tool for microscopic or mesoscopic 162 fluid flow phenomena. 163 Most studies mentioned above used a RVE model with symmetrical or periodic 164 boundary conditions. Few studies have investigated the fluid flow associated with heat 165 transfer over the entire structure of a realistic porous structure. However, Sadeghi et al. 166 performed a full-field CFD simulation of the gas flow over the entire structure of open-cell 167 foams [46] and regular catalytic monolithic structures [47]. In their studies, the open-cell 168 foam and ceramic honeycomb structures were numerically generated based on X-ray CT 169 data. They also performed experimental measurements of the fluid-flow pattern using 170 magnetic resonance velocimetry. The calculated spatially resolved flow patterns 171 corresponded well with their experimental measurements. Although the RVE model can 172 reduce computational time, it does not consider the effects of a heterogeneous pore 173 distribution within the porous medium on both the pressure drop and heat-transfer 174 characteristics. In particular, when a porous medium is manufactured by filling a container 175 with fibers or particles, they are confined by the container inner wall, which affects the 176 porosity distribution within the medium; this is the so-called wall-confinement effect [48,49]. 177 A full-field CFD simulation is needed to examine the wall-confinement effect on the fluid 178 flow associated with heat transfer. 179 Furthermore, in real heat-transfer tubes containing fibrous porous media, the tube 180 walls often serve as a heat sink or heat source, and the interface between the inner tube wall 181  8 and the fibers is closely associated with the solid-phase thermal conduction path between the 182 porous structure and the heat sink or source region. Few studies have thus far investigated the 183 heat-transfer performance within the porous media associated with the solid-phase thermal 184 conduction path from the heat sink or source region. However, Zavattoni et al. [50] recently 185 numerically examined the effect of “contact or non-contact” between the porous structure and 186 parting plate, which serves as the heat sink or source region. In their CFD model, a relatively 187 wide gap of 0.25 mm was employed due to the difficulty of welding two different ceramic 188 materials. Their study focused on the heat-transfer tube in relatively high-temperature 189 environment (600 to 1000 K) where the porous ceramic structures and parting plate or tube 190 were manufactured separately. Relatively wide clearance inevitably was observed when the 191 porous structures and parting plate or tube were coupled together in the final device form. 192 Conversely, metal materials such as Cu and Al have been often employed as heat-transfer 193 tubes in a relatively low-temperature environment (around 300 to 500 K). These metal heat-194 transfer tubes can be manufactured in a combined state using a sintering process, where a 195 very narrow gap region (10-5 to 10-6 m) between the porous structure and tube inner wall is 196 expected to be present. To date, no studies have focused on the effects of such a narrow gap 197 region between the porous structure and the heat sink or source region on the heat-transfer 198 performance of such systems.  199 In this study, we applied an X-ray CT-based CFD model to heat transfer associated 200 with fluid flow through the entire structure of a heat-transfer tube filled with a sintered 201 fibrous high-porosity medium for the quantitative estimation of both the permeability and 202 form coefficient. The entire structure of the actual heat-transfer tube filled with sintered 203 fibrous high-porosity medium was reconstructed and modeled using the X-ray CT technique. 204 The calculated pressure drop and amount of heat transferred were compared with the results 205 of previous measurements by Enoki et al [13,14]. In their study, the heat-transfer tube was 206 manufactured by the same process as that in the present study. Following the precedent set by 207  9 earlier experimental studies, the outer wall of the heat-transfer tube in the present CFD model 208 used was defined as the heat sink.  209 Furthermore, the Al heat-transfer tube filled with sintered fibrous high-porosity 210 media has very large specific fluid surface area Sv, fluid of 2000 to 3000 m-1, which is 211 approximately two-times larger than those reported in earlier CFD simulations using the X-212 ray CT technique (Sv, fluid of 300 to 1500 m-1 [31,32,36,39]). This is the first study to apply a 213 CFD model with X-ray CT to an entire Al heat-transfer tube filled with high-porosity media 214 whose Sv, fluid value is over 2000 m-1, which is the first novelty of the present study. Higher Sv, 215 fluid values inevitably produce very narrow pores and highly irregular surface shapes for the 216 porous structure composed of sintered fibers. In the present study, the finite element method 217 was selected for analysis because of its high suitability for the 3D modelling of the irregular 218 shapes of sintered fibrous high-porosity structures. Both the permeability and form 219 coefficient were estimated using the calculated results and the Darcy–Forchheimer law. The 220 effects of the differences in the porosity and the specific solid surface area between the 221 different Al heat-transfer tube samples on both the permeability and form coefficient were 222 discussed according to the Ergun equation. The second novelty of the present research is to 223 numerically examine and discuss relationship between the Ergun parameters and X-ray CT 224 based porous structure.  225 Further, to discuss the heat-transfer performance of the heat-transfer tube containing 226 a fibrous high-porosity medium, a heat-transfer tube having an idealized wire mesh structure 227 was generated, which has a Sv, fluid value of 3500 m-1. The effect of a very narrow gap region 228 on the order of 10-5 to 10-6 m between the fibers and the tube inner wall on the heat-transfer 229 performance was numerically examined, and the shell conduction model was employed to 230 achieve simulations with limited computational costs. The third novelty of this study is to 231 numerically examine such narrow gap effects on the heat transfer performance. 232  233  10 2. CFD model 234 In the present study, a YXLON Cheetah μHD multifocus X-ray CT instrument was 235 used to scan the sample. Two Al heat-transfer tubes having sizes of ∅18 × 25 mm and 236 ∅18 × 50 mm filled with a sintered fibrous high-porosity medium having a porosity of 237 approximately 80 % were manufactured at Mitsubishi Materials Corporation (MMC 238 Innovation Center, SAITAMA-SHI, Japan). Aluminum fiber having a diameter of 239 approximately 300 µm was used as the fibrous material. The inner and outer diameters of the 240 heat-transfer tube were approximately 18 and 20 mm, respectively. The process used for the 241 manufacture of the filled Al heat-transfer tube was the same as that used in previous 242 experimental studies [13,14]. To reconstruct 3D images and generate a computational mesh 243 from cross-sectional image data, the Amira–Avizo program [51] was employed.  244 Figure 1 shows a cross-sectional image of the Al heat-transfer tube filled with 245 sintered fibrous high-porosity medium obtained using X-ray CT. In this figure, the white and 246 black colors correspond to the solid Al and fluid air phases, respectively. The 3D image of 247 the Al heat-transfer tube filled with sintered fibrous high-porosity media was then 248 reconstructed from the stack of cross-sectional two-dimensional images containing voxels 249 having sides measuring 55.4 µm. A total of 400 cross-sectional images were used to generate 250 the CFD model of the tube having a size of ∅18 × 25 mm, whereas 900 cross-sectional 251 images were employed for the sample having a size of ∅18 × 50 mm. The specific fluid 252 surface area Sv, fluid, specific solid surface area Sv, solid, and porosity ε are listed in Table 1, 253 which were estimated using the CFD model. 254 Figure 2 shows the generated CFD model setup and boundary conditions for the Al 255 heat-transfer tube filled with sintered fibrous high-porosity media. This model setup was 256 determined based on the previous experimental studies by Enoki et al. [13,14]. Figure 2 also 257 shows the cross-sectional plane used to visualize the temperature and fluid-flow velocity 258 contours. The CFD model was set up within the ANSYS FLUENT framework. In this study, 259  11 Eqs. (1), (2), (3) and (4) are the continuity, momentum, and energy equations and were 260 solved numerically using the single-phase steady-state laminar model [19,30]. 261 Continuity equation: 262  ∇ ∙ (𝜌𝜌𝐮𝐮) = 0. (1) 263 Momentum equation: 264  ∇ ∙ (𝜌𝜌𝐮𝐮𝐮𝐮) = −∇𝑝𝑝 + 𝜂𝜂∇2𝐮𝐮. (2) 265 Energy transport equation in the fluid region: 266  ∇ ∙ �𝜌𝜌𝑐𝑐𝑝𝑝𝐮𝐮𝑇𝑇� = ∇(𝑘𝑘∇𝑇𝑇). (3) 267 Energy transport equation in the solid region: 268  ∇(𝑘𝑘∇𝑇𝑇) = 0. (4) 269 Here, ρ expresses the fluid density, u = (u,v,w) is the fluid velocity vector, p is the pressure, η 270 is the viscosity, cp is the specific heat capacity, T is the temperature, and k is thermal 271 conductivity. Unlike in earlier numerical studies [7,22-24,31-33,36,37,39,46,47], the 272 dependencies of the density, viscosity, specific heat capacity, and thermal conductivity of the 273 fluid on temperature were determined using Eqs. (5)–(8), respectively [52]. 274 𝜌𝜌 = −2.331 × 10−8 × 𝑇𝑇3 + 3.409 × 10−5 × 𝑇𝑇2 − 1.819 × 10−2 × 𝑇𝑇 + 4.196 (5) 275  𝜂𝜂 = 4.317 × 10−8 × 𝑇𝑇 + 5.529 × 10−6 (6) 276 𝑐𝑐𝑝𝑝 = −2.112 × 10−8 × 𝑇𝑇4 + 3.353 × 10−5 × 𝑇𝑇3 − 1.909 × 10−2 × 𝑇𝑇2 + 4.692 × 𝑇𝑇 +277 582.9 (7) 278  𝑘𝑘 = 7.128 × 10−5 × 𝑇𝑇 + 4.196 × 10−3 (8) 279 The material properties of the solid Al phase are listed in Table 2 and were assumed to be 280 constant [31]. 281 A fixed temperature of 273.15 K was applied to the outer tube wall surface as a 282 boundary condition. On the thick inlet and outlet side walls of the heat-transfer tube, the heat 283 flux was assumed to be zero, and a no-slip boundary condition was applied to the wall 284 surfaces. The mass-flow inlet boundary condition was adopted at the inlet interface, and a 285  12 constant mass-flow rate of 0.125–1.5 g/s was set. A fixed air temperature of 473.15 K was 286 applied to the inlet surface. A mass-flow outlet boundary condition with a specified mass-287 flow rate was applied at the outlet interface. 288 Preliminarily, the grid independence was tested for the same geometrical model with 289 different computational grid sizes. Figure 3 compared the pressure drop along the heat-290 transfer tube length L and the averaged outlet air temperature among three CFD models with 291 different computational grid sizes for the same heat-transfer tube having size of ∅18 × 25 292 mm. In these calculations, the air mass-flow rate was varied between 0.25 and 1.5 g/s. We 293 examined three different computational meshes composed of 10 million, 25 million, and 40 294 million computational cells. As seen in Figure 3(a), the calculated pressure drops in 10 295 million cells were lower than those obtained in both 25 million and 40 million ones in each 296 air flow velocity. Similarly, the calculated averaged outlet air temperatures in 10 million cells 297 were lower than those obtained in both 25 million and 40 million ones in each air flow 298 velocity. Using the calculated results in 40 million cells as a benchmark, the relative error 299 increased with the air flow velocity, which became up to 9 % for pressure drop and over 1 % 300 for outlet air temperature. Contrarily, both the pressure drops and outlet air temperatures 301 obtained in 25 million cells, corresponded well with those obtained in 40 million cells: the 302 averaged relative error was estimated to be within 2.5% for pressure drop and 0.5 % for outlet 303 air temperature. Thus, we considered that approximately 25 million and 50 million 304 computational cells were appropriate for the present CFD simulations for the heat-transfer 305 tubes having sizes of ∅18 × 25 mm and ∅18 × 50 mm, respectively. Figure 4 shows the 306 generated surface mesh for the heat-transfer tubes having sizes of ∅18 × 25 mm, and the 307 typical visualizations of cross-sectional computational mesh in XY, XZ, and XZ mid-planes. 308 In Figures 4(b)-(d), the red and blue colors correspond to the solid Al and fluid air phases, 309 respectively.  310  311  13 3. RESULTS 312 3.1 Numerical evaluation of the Al heat-transfer tube filled with a sintered fibrous 313 high-porosity medium 314 Figure 5 shows the computed air-flow patterns and temperature contours in the Al 315 heat-transfer tube having a size of ∅18 × 25 mm. For these calculations, the air mass-flow 316 rates were 0.5 and 1.0 g/s respectively. Figure 5(a) shows the streamlines of the air flow 317 through the heat-transfer tube, and the air temperature is shown as colored contours. The 318 complicated path line indicated the intricate porous structure of the sintered Al fibers, which 319 remain unchanged with change in the air mass-flow rate. The average air temperature on the 320 outlet surface was estimated to be 280 K at a flow rate of 0.5 g/s, whereas it was estimated to 321 be 295 K at a flow rate of 1.0 g/s, suggesting that the temperature of the downstream air 322 increased with the increase in the air mass-flow rate because of the limited heat exchange. 323 Figures 5(b) and 5(c) show the contours of the Al fiber and air temperatures, respectively. 324 Notably, as the air-flow rate increased, the temperature of the Al fibers increased, particularly 325 in the upstream region. This indicates that the Al fibers receive a significant amount of heat, 326 and, thus, the solid-phase thermal conduction cannot cool the fibers sufficiently. 327 Figure 6 shows typical examples of the calculated cross-sectional contours for the Al 328 heat-transfer tube having a size of ∅18 × 25 mm for air mass-flow rates of 0.5 and 1.0 g/s 329 respectively. Figure 6(a) shows the cross-sectional temperature contours, where the 330 temperatures of the air and solid Al phases are shown. Figure 6(b) shows the cross-sectional 331 contours of the fluid velocity. The fluid velocity vector is shown as a black arrow, and the 332 solid Al phase is shown in white. Figure 6(a) shows the Al fiber portion as a relatively lower-333 temperature contour owing to the solid-phase thermal conduction toward the outer wall of the 334 heat-transfer tube, indicating that solid-phase thermal conduction is a key factor in achieving 335 a high degree of heat exchange. Figure 6(b) shows the intricate air-flow stream, as indicated 336 by the black arrow. As shown, the air passes through the narrow gap between the Al fibers 337  14 with a relatively higher velocity and then bifurcates after collision with the downstream Al 338 fiber. Therefore, the Al fiber should receive a large amount of heat via heat exchange during 339 collisions with the air molecules. 340 At higher air flow rates, both the air and Al fibers had higher temperatures within the 341 heat-transfer tube owing to the limited heat exchange. Figure 6(b) shows that both local high- 342 and low-velocity regions are scattered within the heat-transfer tube as a result of the 343 heterogeneous distribution of the gap regions. Air having a local low-velocity in the wide 344 gaps is expected to flow through the medium without sufficient heat exchange, whereas air 345 with a local high-velocity in the narrow gaps is expected to flow through with a high degree 346 of heat exchange. In addition, the local high-velocity air flow inside the narrow gaps 347 contributes to a larger pressure drop owing to frictional losses. Thus, fabricating a heat-348 transfer tube without extremely narrow or wide gaps is desirable for enhancing heat exchange 349 with lower pressure drop. However, in the manufacturing process used in this study, the gap 350 distribution in the heat-transfer tube cannot be controlled. Therefore, the reproducibility of 351 both the pressure drop and heat-transfer performance is reduced, and this is a key 352 disadvantage of the sintered fibrous high-porosity medium used in this study. 353  354 3.2 Comparison of CFD simulation and literature results 355 Figure 7 shows a comparison of the calculated pressure drops with those reported in 356 Ref. [13]. In these calculations, the air mass-flow rate was varied between 0.125 and 1.5 g/s 357 for heat-transfer tubes having sizes of ∅18 × 25 and ∅18 × 50 mm. The vertical axis 358 indicates the pressure drop from the inlet to the outlet divided by L. The black circles and 359 black squares indicate the experimental results reported in Ref. [13]. As shown in Figure 7, 360 both the measured and calculated pressure drops can be expressed by the following 361 Forchheimer-extended Darcy equation [34,35]. 362  15  ∆𝑃𝑃𝐿𝐿= 𝜂𝜂𝐾𝐾𝑢𝑢 + 𝐶𝐶𝐶𝐶𝑢𝑢2 (9) 363 Here, u expresses the air-flow velocity, C is the form coefficient, and K is the permeability. 364 The first term is the pressure drop arising from viscous effects, whereas the second term is 365 the pressure drop arising from inertial effects. The red and blue dotted lines are the 366 approximations to experimental results obtained using Eq. (9) for heat-transfer tubes having 367 sizes of ∅18×25 and ∅18×50 mm, respectively. For the heat-transfer tube having a size of 368 ∅18×50 mm, our calculated results (blue line with square symbols) corresponded well with 369 the experimental results (black squares with blue dotted line). However, a slight deviation 370 between the calculated and measured values was observed for the tube having a size of 371 ∅18×25 mm. We speculate that the reason for this discrepancy is that the heat-transfer tube 372 samples used in this study were not the same as those used in Ref. [13]. According to Eq. (9), 373 both C and K were estimated from the CFD simulations. The estimated values are listed in 374 Table 3, where experimental measurement data obtained by Enoki et al. [13] are also 375 included. The C values estimated by the present CFD simulation corresponded well with the 376 measured values, whereas the estimated K values via the present CFD simulation were 377 slightly larger than the measured values. The deviation of the pressure drop between the 378 calculated and measured values was therefore ascribed to this difference in permeability as a 379 higher permeability in the CFD simulation corresponds to a decrease in the pressure drop, as 380 seen in Eq. (9). Next, we examined the effects of the differences in the geometrical 381 parameters of the porous structure on the differences in both C and K estimated using the 382 calculated results for two different heat-transfer tube samples of ∅18×25 and ∅18×50 mm. 383 It is well-known that Eq. (9) can be rewritten as the following Ergun equation [53-55]. 384 ∆𝑃𝑃𝐿𝐿= 𝛼𝛼 𝑆𝑆𝑣𝑣,𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠2(1−𝜀𝜀)2𝜀𝜀3𝜂𝜂𝜂𝜂 + 𝛽𝛽 𝑆𝑆𝑣𝑣,𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠(1−𝜀𝜀)𝜀𝜀3𝜌𝜌𝑢𝑢2                    (10) 385 Here, α and β express the Ergun parameters for viscous and inertial terms, respectively. 386 According to Tables 1 and 3, (α, β) can be estimated as (2.86, 0.35) for ∅18 × 25 mm and 387  16 (3.43, 0.33) for ∅18 × 50 mm. Similar values of β were obtained between ∅18 × 25 mm and 388 ∅18 × 50 mm, whereas the values of α differed from one another. This result indicates that 389 the way the air flows through the Al heat-transfer tube of ∅18 × 25 mm is quite similar to 390 that in the ∅18 × 50 mm tube, and α is more sensitive to slight differences in Sv, solid and ε 391 than β. Furthermore, the values of (α, β) estimated by the present CFD simulations differ 392 from the empirical values of (4.17, 0.292) estimated by Ergun [55], whereas they agree well 393 with the results of an earlier study by Inayat et al. [53], who estimated (α, β) as (3.65, 0.31) 394 around a porosity of 0.84 in the case of cylindrical struts. In a different study, Inayat et al. 395 also proposed the semi-empirical correlation for reticulated ceramics foams [56] which 396 estimated (α, β) as (2.60, 0.30) around a porosity of 0.84. These comparisons suggests that 397 the tortuous flow trajectories associated with the sintered fibrous high-porosity media are 398 clearly different from those in packed-bed porous media, and they are quite similar to those in 399 open-cell foams with cylindrical struts around a porosity of 0.84. Considering the above 400 discussion, we speculate that the discrepancy between the calculated and measured pressure 401 drops occurs because the differences in both Sv, solid and ε between the heat-transfer tube 402 samples used in the present study and those used in Ref. [13] results in differences in α and 403 K. The present CFD simulations using X-ray CT can reasonably predict both C and K, even 404 though no detailed information concerning Sv, solid and ε was identified in an earlier study 405 [13]. Furthermore, we speculate that a deviation of (α, β) between the results of this study 406 and those by Inayat et al. [53,56] may be ascribed to the difference in the averaged tortuous 407 flow-path length, which is closely related to the geometric parameters (Sv, solid and ε) and the 408 heterogeneous gap distribution within the heat-transfer tube. To improve the prediction 409 accuracy of (α, β), the effects of the intricate fluid-flow trajectory associated with the 410 geometric tortuosity of the porous structure, Sv, solid, and ε on (α, β) must be examined. 411 Figure 8 shows a comparison of the calculated amount of transferred heat Q with the 412 experimental values reported in Ref. [14]. In the present simulation, Q was estimated from 413  17 the total heat-transfer rate over the entire wall surface area available for heat transfer, and Q 414 increased with the increase in L in the region having a high air mass-flow rate. This result 415 suggests that the heat exchange is not maximized at an air-flow rate of 1.5 g/s for the tube 416 having a size of ∅18 × 50 mm. Further, good quantitative agreements between the 417 calculations and experimental results were observed despite the differences in the samples 418 used in our study and that of Ref. [14]. 419  420 4. Discussion 421 As discussed in Section 3.1, the solid-phase thermal conduction between the Al 422 fibers and outer wall of the heat-transfer tube contributes to the high heat exchange between 423 the air and Al fibers. To examine the importance of solid-phase thermal conduction on the 424 heat-transfer performance, an Al heat-transfer tube having an idealized wire mesh structure 425 was modeled. Moreover, the interfacial thermal conductivity between the Al fibers and the 426 inner wall of the heat-transfer tube was introduced to the simulation model, and the effect of 427 interfacial thermal conduction on the heat-transfer performance was numerically examined. 428 Figure 9 shows the generated computational model of the Al heat-transfer tube 429 having an idealized wire mesh structure, procedure used for the development of the 25-mm-430 long heat-transfer tube model, and generated CFD model of the 25-mm-long heat-transfer 431 tube. The Al heat-transfer tube having an idealized wire mesh structure contained Al fibers 432 having widths and thicknesses of 0.3 and 0.3 mm, respectively, and was generated as shown 433 in Figure 9(a). The 0.5-mm-long heat-transfer tube had approximately 80 % porosity, and the 434 25-mm-long heat-transfer tube model was composed of 50 0.5-mm-long heat-transfer tubes. 435 The 25-mm-long heat-transfer tube model has a specific solid surface area of 13500 m-1 and a 436 specific fluid surface area of 3500 m-1. Each tube was stacked in the axial (z) direction and 437 rotated at 30° with respect to its neighbor, as shown in Figure 9(b). Approximately 50 million 438 computational cells were used in the CFD simulation for the heat-transfer tube having a size 439  18 of ∅18 × 25 mm. The CFD model setup and boundary conditions were the same as those 440 described in Section 2. The constant mass-flow rates in the simulation were 0.5 and 1.0 g/s. 441 In this CFD model, the interfacial thermal conductivity between the Al fibers and the 442 inner wall of the heat-transfer tube was introduced using the shell conduction model in the 443 Ansys Fluent framework. This interfacial region is called the shell region and is indicated by 444 the red line in Figure 9(a). In this shell region, a constant thermal conductivity of 0.3 W/mK 445 was used, which was comparable to the thermal conductivity of air. The constant thickness of 446 the shell region was introduced as the input parameter and ranged from 1 to 10 µm. The 447 effect of the shell-region thickness on the heat-transfer performance was also examined. 448 Figure 10 shows typical results of the calculation of the cross-sectional temperature 449 and air-flow velocity contours when the interfacial thermal conductivity between the Al 450 fibers and tube inner wall was assumed to be zero. Similar to Figure 6(a), Figure 10(a) shows 451 the Al fiber portion as a relatively lower-temperature contour owing to the contribution of the 452 solid-phase thermal conduction between the Al fibers and the outer wall of the heat-transfer 453 tube. Comparing Figures 6(a) with 10(a), both the air and Al fibers have a lower temperature 454 in the downstream region of the heat-transfer tube with the idealized wire mesh structure; 455 hence, the idealized wire mesh structure is expected to achieve higher heat exchange than that 456 of the sintered fibrous high-porosity medium. 457 As shown in Figure 10(b), the air passes through the narrow gap between the wire 458 mesh with a relatively high velocity and then bifurcates after collision with the Al fibers. 459 Compared to that shown in Figure 6(b), the local high-velocity regions are evenly distributed 460 throughout the heat-transfer tube as a result of the designed constant narrow gap of the wire 461 mesh. As discussed in Section 3.1, the air flow at higher velocities is expected to achieve 462 higher heat exchange with the Al fibers owing to convective heat transfer despite the larger 463 pressure drop owing to the friction loss within the narrow gaps. 464  19 The pressure drop and Q in the heat-transfer tube having an idealized Al wire mesh 465 structure at 0.5 g/s were estimated to be 0.00844 kPa/mm and 101.3 W, respectively, by CFD 466 simulation. However, these quantities were estimated to be 0.00404 kPa/mm and 97.88 W, 467 respectively, in the case of the sintered fibrous high-porosity medium. Therefore, the heat-468 transfer tube having an idealized wire mesh structure exhibited superior heat-exchange 469 performance than that filled with the sintered fibrous high-porosity medium. The same 470 superior heat-transfer performance in terms of pressure drop and Q was obtained at 1.0 g/s: 471 0.0147 kPa/mm and 179.8 W for the sintered fibrous high-porosity media but 0.03101 472 kPa/mm and 201.3 W for the idealized wire mesh structure. Therefore, the idealized Al wire 473 mesh structure, which could be fabricated through additive manufacturing (for example, 474 metal 3D printing), is expected to overcome the aforementioned disadvantages of sintered 475 fibrous high-porosity media because additive manufacturing can precisely control the gap 476 distribution within the heat-transfer tube. Thus, we believe that the idealized Al wire mesh 477 structure is a promising candidate, especially if prepared through additive manufacturing, for 478 achieving superior heat-exchange performance with better reproducibility. This superior heat-479 exchange performance of the idealized Al wire mesh structure may be also ascribed to the 480 Ergun parameters (α, β), which are closely related to the intricate fluid-flow trajectory 481 associated with the geometric parameters of a porous structure. The overall heat-transfer 482 performance has been frequently analyzed in terms of an efficiency factor, which is defined 483 by the Colburn number (or Nusselt number) and the frictional factor [30,35]. Because the 484 frictional factor is dependent on the Ergun parameters (α, β), the efficiency factor is also 485 dependent on geometric parameters such as Sv, solid and ε. To optimize the Ergun parameters 486 (α, β) to achieve superior heat-transfer performance, it is necessary to examine the effects of 487 the Ergun parameters (α, β) and geometric parameters of the porous structure on the 488 efficiency factor, which will be considered in a future study.  489  20 Further, the effect of the interfacial thermal conduction between the Al fibers and the 490 inner tube wall surface on the heat-transfer performance was examined. Figure 11 shows the 491 dependence of the calculated cross-sectional temperature contours on the shell-region 492 thickness at 0.5 g/s. As shown, with the increase in the shell-region thickness, the radial heat 493 transfer from the Al fibers toward the tube outer wall is attenuated as a result of the low 494 thermal conductivity of the shell region, which results in a higher temperature in both the 495 downstream air and Al fibers. 496 Figure 12 shows the dependence of both the pressure drop and Q on the thickness of 497 the shell region at 0.5 g/s. The left and right vertical axes indicate the pressure drop and Q, 498 respectively. As shown, the pressure drop increases slightly with the increase in shell 499 thickness. This is because when the shell is thicker, the air has a higher temperature, and the 500 viscosity of the air increases with temperature, which results in a large friction loss. In 501 contrast to the dependence of the pressure drop on the shell thickness, Q clearly decreased 502 with increasing shell thickness. This is because the low thermal conductivity of the shell 503 region dampens the solid-phase thermal conduction between the Al fibers and tube outer 504 wall, resulting in inferior heat-exchange performance. An earlier study [50] also reported that 505 a narrow constant gap of 0.25 mm heavily impacted the thermal performance when high 506 thermal conductive materials were employed for heat exchanger manufacturing. The present 507 CFD simulations clearly indicate that even a very narrow gap of 5 to 10 µm heavily impacts 508 the heat-exchange performance, and the connectivity between the Al fibers and inner wall of 509 the heat-transfer tube is therefore a key factor affecting heat-exchange performance. In 510 particular, the high heat-exchange performance of the Al heat-transfer tubes filled with 511 sintered fibrous high-porosity media manufactured at the MMC can be attributed to the good 512 connectivity between the Al fibers and the tube inner wall. 513  514 5. Conclusion 515  21 In the present study, we numerically calculated the fluid flow and temperature fields 516 over the whole structure of a heat-transfer tube filled with a sintered fibrous high-porosity 517 medium using a CFD model based on X-ray CT data. Calculated pressure drop, the amount 518 of heat transferred, the form coefficient C, and the permeability K were compared with earlier 519 studies [13,14], and the discrepancies was discussed according to the Ergun equation. To 520 examine the importance of solid-phase thermal conduction on the heat-transfer performance, 521 an Al heat-transfer tube having an idealized wire mesh structure was modeled, where the 522 interfacial thermal conductivity between the Al fibers and the inner wall of the heat-transfer 523 tube was introduced. Major conclusions of this study are listed as below. 524  525  Calculated pressure drop, the amount of heat transferred, and the form coefficient 526 C corresponded well with the results of previous experimental studies. The 527 estimated permeability K via present CFD simulation were slightly larger than the 528 measured ones. Based on the quantitative investigation according to Ergun 529 equation, we speculated that the difference in K occurs because the differences in 530 both Sv, solid and ε between the heat-transfer tube samples used in the present study 531 and those used in previous experimental study. 532  The differences in both C and K estimated via present CFD simulations between 533 two different samples can be reasonably explained by the slight differences in 534 both Sv, solid and ε. α value in the Ergun equation was more sensitive to slight 535 differences in Sv, solid and ε than β, which was a key parameter to estimate the 536 permeability. The higher α value was required to achieve lower pressure drop and 537 lower permeability. 538  The results of the CFD simulations revealed that even a very narrow gap of 5 to 539 10 µm heavily impacted the heat-exchange performance because of the low 540 thermal conductivity of the shell region, and the connectivity between the Al 541  22 fibers and inner wall of the tube was a key factor in obtaining high heat-exchange 542 performance. 543  The heat-transfer tube having an idealized wire mesh structure exhibited superior 544 heat-exchange performance compared to that filled with a sintered fibrous high-545 porosity medium. We speculated such superior performance of the idealized Al 546 wire mesh structure may be ascribed to the presence of Ergun parameters that are 547 more suitable for obtaining a high heat-exchange performance than those filled 548 with a sintered fibrous high-porosity medium. 549  550 The present CFD simulation can reasonably predict both C and K. However, further 551 investigations into the relationship between the Ergun parameters, the averaged flow path 552 associated with the geometric parameters of the porous structure, and the heterogeneous gap 553 distribution within the heat-transfer tube are required to improve the prediction accuracy for 554 the pressure drop associated with both C and K. Furthermore, the effect of the Ergun 555 parameters on the efficiency factor associated with the heat-transfer performance concerning 556 nondimensional analysis was not disclosed systematically in the present research. These 557 investigations are currently in progress. 558  559 Acknowledgements 560 This work was performed at the MMC-NIMS Center of Excellence for Materials 561 Informatics Research. We would like to thank Editage for English language editing. 562  563 Conflicts of interest 564 There are no conflicts of interest to declare. 565  566 Author contributions 567  23 S. 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Cross-sectional image of the Al heat-transfer tube filled with the sintered fibrous 768 high-porosity medium obtained using X-ray CT. 769   770  34  771 Figure 2. CFD model of the Al heat-transfer tube filled with the sintered fibrous high-772 porosity medium generated from the X-ray CT data. The boundary conditions for the model 773 are also specified. 774   775  35  776 Figure 3. Comparison of (a) pressure drop with respect to tube length and (b) averaged outlet 777 air temperature among three CFD models with different computational grid sizes for the heat-778 transfer tube having a size of ∅18 × 25 mm. 779   780  36  781 Figure 4. Details of the computational cell used for the Al heat-transfer tube filled with the 782 sintered fibrous high-porosity medium having a size of ∅18 × 25 mm. (a) Generated surface 783 mesh of Al heat-transfer tube, (b)-(d): typical visualizations of cross-sectional computational 784 mesh in XY, XZ, and XZ mid-planes. 785  786   787  37  788 Figure 5. Computed air-flow patterns and temperature contours in the Al heat-transfer tube 789 having a size of ∅18 × 25 mm at different air flow rates of 0.5 and 1.0 g/s. (a) Streamlines of 790 the air flow through the heat-transfer tube, (b) temperature contours of Al fibers, and (c) 791 temperature contours of air within the heat-transfer tube. 792  793   794  38  795 Figure 6. Calculated cross-sectional temperature and air-flow velocity contours for the Al 796 heat-transfer tube of size ∅18 × 25 mm at air-flow rates of 0.5 and 1.0 g/s. (a) Cross-797 sectional temperature contour and (b) cross-sectional contour of the fluid velocity. The fluid 798 velocity vector is represented by the black arrow. 799  800   801  39  802 Figure 7. Comparison of the pressure drops obtained numerically (present work) and 803 experimentally [13]. The abscissa represents the air-flow velocity estimated by the mass-flow 804 rate between 0.125 and 1.5 g/s. The sizes of the heat-transfer tubes were ∅18 × 25 mm and 805 ∅18 × 50 mm. The dotted lines are approximations to the experimental measurements. 806   807  40  808 Figure 8. Comparison of amount of heat transferred (Q) at different air-flow rates obtained 809 numerically (present work) and experimentally [14]. 810  811   812  41  813 Figure 9. Generated computational model of the Al heat-transfer unit tube containing an 814 idealized wire mesh structure and 25-mm-long heat-transfer tube model in which the unit 815 tubes are stacked and rotated 30° with respect to the next. (a) 0.5-mm-long heat-transfer unit 816 tube model, (b) stacking procedure for the model containing 50 heat-transfer tubes, and (c) 817 25-mm-long heat-transfer tube model. In Figure 7(a), the shell region is indicated by a red 818 line. In Figure 7(b), 50 tubes are stacked and rotated 30° with respect to the axial (z) 819 direction. 820  821  822  42  823 Figure 10. Calculated cross-sectional temperature and air-flow velocity contours at air-flow 824 rates of 0.5 and 1.0 g/s. The interfacial thermal conductivity between the Al fiber and tube 825 inner wall is assumed to be zero. (a) Cross-sectional temperature contours and (b) cross-826 sectional contours of the fluid velocity. The fluid velocity vector is represented by a black 827 arrow. 828  829   830  43  831 Figure 11. Dependence of the calculated cross-sectional temperature contours on the shell-832 region thickness at an air-flow rate of 0.5 g/s. 833  834   835  44  836 Figure 12. Changes in the pressure drop and the amount of heat transferred with respect to the 837 thickness of the shell region at an air flow-rate of 0.5 g/s. 838  839   840  45 Table 1. Geometrical properties of the Al heat transfer tube with a sintered fibrous high-841 porosity medium used in the present CFD simulations. 842   ∅18 × 25 mm ∅18 × 50 mm Specific fluid surface area Sv,fluid [ m-1] 2274 2527 Specific solid surface area Sv,solid [m-1] 11830 12040 Porosity ε [vol%] 0.839 0.827  843   844  46 Table 2. Physical properties of the solid Al phase used in the present CFD simulations. 845 Density [kg/m3] 2719 Specific heat capacity [J/(kgK)] 871 Thermal conductivity [W/(mK)] 237  846   847  47 Table 3. Comparison of the form coefficient and permeability between the present CFD 848 simulations and experimental measurements presented in Ref. [13]. 849   Present simulations Experiments in Ref. [13] ∅18 × 25 mm ∅18 × 50 mm ∅18 × 25 mm ∅18 × 50 mm Form coefficient C [m-1] 1128 1227 1051 1018 Permeability K [m2] 5.67×10-8 3.78×10-8 1.25×10-8 1.28×10-8  850  851