## Abstract

High-porosity metal foams have been extensively studied as an attractive candidate for efficient and compact heat exchanger design. With the advancements in additive manufacturing, such foams can be manufactured with controlled topology to yield highly tailorable mechanical and transport properties. In this study, a lattice Boltzmann method (LBM)-based pore-scale model is implemented to simulate the fluid flow in additively manufactured (AM) metal foams with unit cell topologies of Cube, Face Diagonal (FD)-Cube, Tetrakaidecahedron (TKD), and Octet lattices. The pressure gradient versus average velocity profiles predicted by the LBM model were validated against in-house measurements on the AM lattice samples with the same unit cell topologies. Based on the simulation results, a novel hybrid model is proposed to accurately predict the volume averaged flow properties (permeability and inertial coefficients) of the four structures. Specifically, the linear LBM (neglecting inertial forces) is first implemented to obtain the intrinsic permeability, and then the standard LBM is applied to obtain the inertial coefficient. Convenient correlations for those flow properties as a function of porosity and fiber diameter are constructed. The effects of the AM print qualities on the flow properties are also discussed. The advantages of the hybrid model compared to the polynomial fitting approach for determining flow properties are discussed and compared quantitatively. The hybrid model and presented results are valuable for flow and thermal transport evaluation when designing new metal foams for specific applications and with different materials and topologies. The presented correlations based on pore-scale simulations can also be conveniently used in volume-averaged models to predict the macroscale flow behavior in such complex structures.

## 1 Introduction

Open cell metal foams are a subclass of porous media that promote fluid mixing because of their crosslinked fiber structure. Due to the inherent fluid mixing and high fluid–solid contact area, metal foams are desirable for the manufacturing of heat exchangers that are prone to weight and space limitations. Some examples include heat/mass exchangers for electronic components, chemical reactors, fuel cells, and heat recovery systems for automobiles [1,2]. Due to their high thermal performance and high stress resistance, metal foams are also used in volumetric solar receivers, low thermal grade heat sinks, and gas turbines. Additively manufactured (AM) lattice-structure metal foams are the next class of porous media being explored for thermal management and flow regulation purposes because of the freedom in manufacturing custom fiber shapes and sizes [3–7]. The AM process allows one to selectively design metal foams that meet required fluid flow characteristics (permeability, inertial coefficient, friction factors, etc.) through modification of the lattice geometry. Efficient prediction of these flow properties will enhance the understanding and design of AM metal foams [8–10].

While AM metal foams are still in their infancy, extensive work has been dedicated to understanding the flow characteristics and regimes in stochastic metal foams produced through melting or powder methods [11,12], thus, we find it valuable to review those here. One of the earliest experimental investigations was conducted by Beavers and Sparrow [13], who reported pressure drop versus velocity characteristics for three nickel metal-foam samples with water as the working fluid. The transition from Darcy to non-Darcy regime was observed to occur at Reynolds number (Re* _{K}* based on the length scale of

*K*

^{1/2}with

*K*being the permeability) from 1 to 10, consistent with purely viscous flow trends over spheres and cylinders. Mancin et al. [14] experimentally analyzed six different aluminum open-cell foams having different porosities (

*ϕ*–0.92, 0.93, 0.95) and pore densities (pores per inch (PPI)–5, 10, 20, 40). Using pressure drop-velocity data, the authors evaluated the permeability and inertial coefficients, and finally synthesized a general correlation for predicting pressure drop in metal foams. Kim et al. [15] investigated the effect of aluminum alloy-foams on pressure drop in a plate fin heat exchanger, where the permeability and inertial coefficients were obtained from a quadratic-extended Darcy equation. A similar approach was followed by Mancin et al. [14], Hwang et al. [16], and Bhattacharya et al. [17], to determine the characteristic flow coefficients of their respective samples. Several other authors [18–20] have also performed extensive investigations of pressure drop characteristics in metal foams and henceforth characterized various flow “regimes” for varied Re

*. For AM metal foams, experimental investigation has been conducted to determine the local flow fields using laser Doppler velocimetry and X-ray particle tracking velocimetry. The pressure drop due to surface roughness and the impact of surface roughness on the efficiency of the AM turbines have been studied [21,22].*

_{K}Several numerical studies have also investigated the various flow regimes and effective flow properties in metal foams, either through idealized geometry or reconstructed images. Liu et al. [23] used a unit Kelvin cell to simulate flow through an “idealized” metal foam and developed a pressure drop model for sphere and skeleton Kelvin cells for prediction of flow properties. Bai and Chung [24] also used an idealized Tetrakaidecahedron (TKD) unit cell to run flow simulations, reporting that TKD unit cells exhibited pressure drops one to two orders of magnitude higher than an open microchannel. Iasiello et al. [25] and Diani et al. [26] performed a numerical simulation of 3D reconstructed images of metal foams to predict flow properties and compared the difference between the idealized and 3D reconstructed models of metal foams. They suggested that the real geometry obtained from the 3D reconstructed images had better agreement with pressure drop and Nusselt number experimental data in literature than the idealized Kelvin unit cell.

Effective prediction of the flow properties in metal foams can be useful for predicting effective heat transfer coefficients. Trilok and Gnanasekaran [27] analyzed the Nusselt number performance of 63 metal foam samples with *ϕ* ranging from 0.8 to 0.97 and pore density from 5 PPI to 45 PPI. Four criteria were defined based on the Nusselt number and the friction factor to determine the best performing foam. They found that the foam with pore density 20 PPI showed a higher performance in the four criteria as compared to the other foams. Ambrosio et al. [28] investigated the effects of strut shape on convective heat transfer. Their idealized model was also compared to X-ray computed microtomography-based models, noting that both predicted similar Nusselt numbers. The real and ideal structures showed a decrease in the Nusselt number with a decrease in porosity and the convex strut shape maximized the convective heat transfer coefficient. Zafari et al. [29] also conducted 3D numerical analysis on X-ray computed microtomography-based metal foam samples of *ϕ* ranging from 0.85 to 0.95. They observed that the thermal conductivity, heat transfer coefficient, permeability, and inertial coefficient had a direct relationship with porosity and tortuosity of the metal foam. Higher porosities showed a reduced Nusselt number, inertial coefficient, and thermal conductivity whereas, the permeability increased with porosity. Zhu et al. [30] presented the validation for a porous medium approximation for compact heat exchangers to effectively model the overall and local pressure drop in a channel heat exchanger as compared to modeling all individual channels. Such porous media assumption and volume-averaged simulations require the determination of flow properties including the permeability and inertial coefficient.

The lattice Boltzmann method (LBM) is a powerful tool to simulate fluid flow and heat transfer in porous media and has been used extensively to study effective flow and thermal transport properties. Advantages of the LBM include a simple and explicit algorithm, ease in implementation, convenience in boundary condition treatment, and capability in preserving complex geometry [31,32]. Pan et al. [33] studied various types of LBM models using different boundary conditions (standard “bounce-back,” linear interpolation “bounce-back,” and multireflection schemes) on the evaluation of permeability using Darcy's law. LBM has also been used to calculate the permeability in random packed porous media, microstructures, and periodic body centered cubic (BCC), face centered cubic, and simple cubic structures [34–36]. A 2D study on the effect of pore-scale heterogeneity on non-Darcy coefficients (permeability and inertial coefficients) in porous media has been performed by Takeuchi et al. [37] using the LBM, where they observed higher values of inertial coefficients for more heterogeneous media and increased permeability for higher Reynolds numbers. Channeling flow paths due to the heterogeneity in the porous media were also studied and possible relationships were suggested between the inertial coefficient and the channeling effects.

Compared to stochastic metal foams, detailed studies related to the flow and heat transfer properties of AM metal foams are scarce in the literature. Ekade and Krishnan [38] conducted direct numerical studies to determine the effective thermal conductivity, permeability, inertial coefficient, friction factor, and Nusselt number with unit cell Octet lattices. They observed that the Octet lattice structure showed a higher permeability at similar porosity values, lower friction factor, and lower inertial coefficients than traditional foams. Dixit et al. [39] observed that Octet and Kelvin unit cells performed better thermally than open channels and stochastic metal foams as they exhibited a higher Nusselt number. Kaur and Singh [40] also observed better heat transfer performance for the Octet lattice structure compared with other AM metal foams and proposed interstitial heat transfer coefficient correlations for the Octet structure. Experimental studies were conducted by Chaudhari et al. [41] on Octet lattices at different porosities to observe the trends in effective thermal conductivity, permeability inertial coefficient, friction factor, and Nusselt number. They concluded that Octet-based AM foams are suitable for applications which require high heat dissipation. Wang et al. [42] studied the effective thermal conductivity in AM structures and developed empirical correlations based on their computational and experimental data. Although the previous studies [38–42] mention the effective thermal properties of AM metal foams, flow properties are integral in holistically evaluating the heat transfer performance of AM metal foams in heat exchanger applications. Hence, the determination of permeability and inertial coefficient is vital in coupled flow and heat transfer analysis in those foam structures.

Although several studies focusing on the pressure drop characteristics of traditional foams exist [43], the literature still lacks comprehensive results for AM lattices, such as Octet, Face-Diagonal Cube (FD-Cube), Cube, and TKD structures over a wide range of porosities. A comparison of pressure drop features of different lattice structures under similar flow conditions is still needed to fully characterize their thermal-hydraulic behavior. The present study aims to fill this knowledge gap by performing a numerical investigation over a wide range of porosities for four periodic fiber-based AM lattice structures: Cube, FD-Cube, TKD, and Octet to determine the permeability and inertial coefficients of the structures. Numerical results for the pressure drop versus average velocity are obtained with the lattice Boltzmann method and validated for each of the four structures against in-house experiments. Numerical correlations for the normalized permeability and inertial coefficients as functions of the porosity for all four structures are also developed. The rest of this paper is organized as follows. Section 2 describes the configurations of the unit cell lattice structures. Section 3 provides the details of the experimental procedure and Sec. 4 presents the details of the numerical method. The model verification and validation are shown in Sec. 5. The results and discussion are presented in Sec. 6 and concluding statements are given in Sec. 7.

## 2 Configurations of the Unit Cell Porous Lattice Structures

This study focuses on four different lattice topologies including the Cube, FD-Cube, TKD, and Octet periodic unit cells as shown in Fig. 1. Those same lattice structures were also studied in Ref. [42] for determining the effective thermal conductivity. The porosity *ϕ* of each structure is varied through the unit cell length *L* and the fiber diameter *d*. The TKD structure is generally considered as an idealized geometric representation of the traditional metal foams. The Octet structure has recently amassed interest in heat sink and thermal management applications [38–40] and is thus included in this study. The cube and FD-Cube lattices are much simpler unit cell structures used for comparison with the more complex TKD and Octet structures [44,45]. This work focuses on the numerical prediction of flow properties including the permeability and inertial coefficient in those four structures over the wide of porosity 0.5 < *ϕ* < 0.95. Validation of numerical results is conducted through flow tests in additively manufactured structures at selected porosities. The various cases with different *L*/*d* ratios and corresponding porosities are presented in Table 1 for the four structures. The cases with supscript “^{a}” denote the CAD models that were designed for additive manufacturing, and the actual porosities for the printed samples are also measured and presented in Sec. 6. For the numerical simulations, the unit-cell length in LB units is fixed and the fiber diameter of the structures is varied according to the *L*/*d* ratios given in Table 1 to vary the porosity of the four lattice structures. For a unit-cell length/grid size of 150*δx*, the respective fiber diameters are 24.71*δx* (Cube), 15.00*δx* (FD-Cube), 13.50*δx* (TKD), and 9.49*δx* (Octet) at *ϕ* = 0.95, which will be sufficient to effectively capture the fluid flow at the fiber scale at high porosities (see the model verification and validation in Sec. 5).

Structure | L/d | $\varphi $ |
---|---|---|

Cube | 1.77 | 0.50 |

2.05 | 0.60 | |

2.43 | 0.70 | |

2.70 | 0.75 | |

3.06 | 0.80 | |

4.27^{a} | 0.89^{a} | |

6.07 | 0.94 | |

FD-Cube | 2.72 | 0.50 |

3.16 | 0.60 | |

3.75 | 0.70 | |

4.29 | 0.76 | |

5.00 | 0.81 | |

6.67^{a} | 0.89^{a} | |

10.00 | 0.95 | |

TKD | 2.80 | 0.50 |

3.26 | 0.60 | |

4.00 | 0.70 | |

4.41 | 0.75 | |

5.00 | 0.80 | |

7.00^{a} | 0.89^{a} | |

11.11 | 0.95 | |

Octet | 4.22 | 0.51 |

4.83 | 0.61 | |

5.89 | 0.71 | |

6.52 | 0.76 | |

7.50 | 0.81 | |

10.71^{a} | 0.89^{a} | |

15.79 | 0.95 |

Structure | L/d | $\varphi $ |
---|---|---|

Cube | 1.77 | 0.50 |

2.05 | 0.60 | |

2.43 | 0.70 | |

2.70 | 0.75 | |

3.06 | 0.80 | |

4.27^{a} | 0.89^{a} | |

6.07 | 0.94 | |

FD-Cube | 2.72 | 0.50 |

3.16 | 0.60 | |

3.75 | 0.70 | |

4.29 | 0.76 | |

5.00 | 0.81 | |

6.67^{a} | 0.89^{a} | |

10.00 | 0.95 | |

TKD | 2.80 | 0.50 |

3.26 | 0.60 | |

4.00 | 0.70 | |

4.41 | 0.75 | |

5.00 | 0.80 | |

7.00^{a} | 0.89^{a} | |

11.11 | 0.95 | |

Octet | 4.22 | 0.51 |

4.83 | 0.61 | |

5.89 | 0.71 | |

6.52 | 0.76 | |

7.50 | 0.81 | |

10.71^{a} | 0.89^{a} | |

15.79 | 0.95 |

The corresponding CAD models are designed for structure additive manufacturing.

## 3 Experimental Apparatus, Procedure, and Uncertainty

Figure 2 shows the four additively manufactured metal foam configurations used in this study, which were manufactured via Binder Jetting process in 420 stainless steel material with 40% bronze infiltration. All the samples were examined for defects and the local roughness on the endwalls rendered by the manufacturing process was measured by optical profilometer. It was determined that the mean roughness on the endwalls was about 26 *μ*m. The fiber diameters for the additively manufactured metal foam samples as shown in Fig. 2 are *d*_{Cube} = 2.37 mm, *d*_{FD-Cube} = 1.54 mm, *d*_{TKD} = 1.43 mm, and *d*_{Octet} = 1.00 mm, respectively, along with a unit-cell length of 10.00 mm for each sample. Figure 3(a) shows the experimental setup used in the present study, air is chosen as the working fluid and is drawn from a compressor through a series of 1 in. (25.4 mm) diameter pipes. A pressure regulator was placed immediately downstream of the compressor to adjust the mass flowrate to a desired value in the pipeline. The regulated air then passed through a moisture separator to ensure that the dry air reached the test section. The dry air was then directed to an ASME orifice plate for flow metering. Differential pressure across the orifice meter, air pressure, and air temperature were measured using Dwyer 477AV-2 (0–10 kPa), Dwyer DPG-002 (0–100 kPa), and fast response T-type thermocouple, respectively, which were fed into an in-house MATLAB code to determine the mass flowrate which has an uncertainty of ±4%. The flow then passed through another fine-tune pressure regulator which maintained uniform pressure downstream. The flow after passing through a three-way valve entered the test section where the smooth transition of flow from circular pipe to square duct was executed by a diffuser. The three-way functionality of the three-way valve was not used.

The schematic of the test section used in the present study is shown in Fig. 3(b). The test section made from a 0.5-inch-thick Plexiglas sheet had a square cross section of 50.8 mm × 50.8 mm. The dimensions in the figure are defined in terms of the hydraulic diameter, *d _{h}* (50.8 mm). The total length of the test section was 16

*d*The lattice structures shown in Fig. 2 were placed at ∼10

_{h.}*d*from the entrance of the test section to ensure that the lattices encounter hydrodynamically fully developed flow. After passing through the lattice sample, which was ∼100 mm (nearly 2

_{h}*d*) in streamwise direction, the air traveled through ∼4

_{h}*d*test section exit length before it was vented out to the laboratory ambient. The static pressure was measured using pressure tap flushed against the inner plexiglass surface and positioned upstream of the lattice sample. The pressure drop measurements were conducted for a wide range of mass flow rates. Therefore, the pressure values at smaller flow rates were measured through a sensitive manometer Dwyer 477AV-000 (0–0.2 kPa) and higher-pressure drops were recorded using Dwyer 477AV-00 (0–1 kPa), to achieve high accuracy throughout the investigated range of flow velocities.

_{h}The pressure drop results obtained through experiments are later presented as pressure drop per unit length (in streamwise direction) versus average flow velocity in Figs. 4(a) and 4(b). The accuracy of the Dwyer 477AV-000 (0–0.2 kPa) and Dwyer 477AV-00 (0–1 kPa) pressure transducers was $\xb10.5%\u2009$ of full scale. Since the uncertainty in the reported quantity *dp*/*dz* depends on the pressure drop measurements alone, they were simply determined by the accuracy of the measurement devices.

## 4 Numerical Method

For the present study, the D3Q19 multiple-relaxation-time LBM models [46] are used to simulate the 3D fluid flow within the periodic lattice structures. This section briefly describes the LBM models, including both the regular quadratic model and the reduced linear model; in addition, numerical evaluation of the flow properties is explained in detail, where we propose a novel hybrid model for consistent evaluation, i.e., the linear LBM model is first used to determine the permeability, and then the regular quadratic LBM model is implemented to evaluate the inertial coefficient; furthermore, the conversion from the LBM unit to physical unit systems for the average velocity and the pressure gradient is provided to facilitate direct comparison between simulations and experimental measurements.

### 4.1 Lattice Boltzmann Models.

*f*(

_{α}**x**,

*t*) can be written as [46]

**x**is the spatial vector,

**ξ**is the particle velocity vector discretized to a small set of discrete velocities given by

*δt*is the time-step, $F\alpha $ is the discrete forcing term in the LBM model to account for the physical external force

**F**with

*δx*the lattice spacing,

*ρ*

_{0}is the mean density of the fluid, and the respective weight coefficients

*ω*are

_{α}**M**is a 19 × 19 transformation matrix used to convert the distribution functions

*f*(

_{α}**x**,

*t*) to their moments

**m**

*(*

_{α}**x**,

*t*), by

**m = Mf**and

**S**is the diagonal relaxation time coefficient (

*τ*) matrix [46]. The LB viscosity is given by

_{ij}*τ*is a dimensionless relaxation time in the LBM model. The matrix

**m**is given by

**j**

*=*(

*j*), and the equilibrium moments

_{x}, j_{y}, j_{z}**m**

^{(eq)}are given by

**m**

^{(eq)}moments neglected to follow Darcy's law. These moments are given by

where the nonlinear terms as compared to Eqs. (7*a*)–(7*g*) are set to zero in Eqs. (8*a*)–(8*e*). Further details of the linear LBM implementation can be found in Ref. [33].

where $f\u0302\alpha $ is the postcollision state of the distribution functions.

An in-house LBM code, written in Fortran 90 has been implemented to simulate 3D fluid flow in AM metal foams. Periodic flow boundary conditions are used in the *x*-, *y*-, and *z*-directions of the unit cell boundaries. The no-slip condition is enforced on the solid fibers of the unit cell via the standard bounce back scheme. To simulate the pressure differential across the unit cell, a constant pressure gradient, −$dp/dz$, is applied as a body force in the *z*-direction. The initial flow-field is set to **u **=** 0** for the entire fluid domain. Steady-state is considered achieved when the *L*_{2} norm error of the flow-field for consecutive iterations is less than 10^{−6}.

### 4.2 Permeability and Inertial Coefficient Evaluation.

*K*, is defined by Darcy's law and is given by

*μ*is the dynamic viscosity of the fluid, $u\xafz$ is the average velocity across the fluid section along the flow direction, and

*dp*/

*dz*is the pressure gradient across the porous media. Darcy's law is typically considered valid only for low Re

*flows [47] where the average flow velocity is directly proportional to the pressure drop across the porous media. For most real-world porous media applications, however, Eq. (13) is not accurate as the fluid velocities are too large to neglect inertial forces. This high velocity (high Re*

_{K}*> 10*

_{k}^{2}Ref. [47]) regime is typically termed the non-Darcy region/Forchheimer region. An additional term including the inertial coefficient,

*C*, is typically used in pressure drop relations to capture the Forchheimer region. In this region, the flow behavior changes from a linear profile to a quadratic profile and hence, Eq. (14) can be used to represent the pressure driven non-Darcy flow as

_{e}*A*and

*B*are coefficients used to determine the permeability and the inertial coefficients as follows:

*C*can then be determined through the least-squares method as

_{e}where the *i* index denotes all the data points from the LBM simulations.

## 5 Model Verification and Validation

In this section, the linear LBM model for determining the permeability and the proposed hybrid LBM model for determining the inertial coefficient are verified and validated. Specifically, the linear LBM model is verified with the permeability obtained from the analytical solution for flow through a BCC structure [33,48] as shown in Fig. 5. The hybrid LBM model is verified with the permeability and inertial coefficient data obtained from Ref. [38] for the Octet structure. And the simulated pressure gradient versus average velocity profiles for all four structures are validated with in-house experimental measurement data obtained using the AM samples shown in Fig. 2.

### 5.1 Linear Lattice Boltzmann Method Model Verification Using Body Centered Cubic Structure.

*L*is the length of the BCC unit cell,

*d*is the diameter of the spheres (the 2D projection of the BCC structure is shown in Fig. 5),

*G*is a constant (12 for BCC structure), and $CD*$ is the dimensionless drag coefficient that is obtained from Ref. [48] as follows:

*α*are given in Ref. [48],

_{n}*χ*is related to the ratio of the solid volume fraction

*c*to the maximum solid volume fraction

*c*

_{max}for the BCC structure. A normalized permeability is used for convenient comparison of the LBM results to analytical solution as

*L*=

*45*

*δx*, 90

*δx*, 120

*δx*, 150

*δx*, 180

*δx*, 210

*δx*) in Table 2 at the porosity

*ϕ*= 0.91, where the relative error (R.E.) is defined as

Grid size (L) | Normalized permeability LBM $kLBM*$ | R.E. |
---|---|---|

45δx | 0.17806 | 7.65% |

90δx | 0.17894 | 7.19% |

120δx | 0.18006 | 6.61% |

150δx | 0.18013 | 6.58% |

180δx | 0.18041 | 6.43% |

210δx | 0.18045 | 6.41% |

Grid size (L) | Normalized permeability LBM $kLBM*$ | R.E. |
---|---|---|

45δx | 0.17806 | 7.65% |

90δx | 0.17894 | 7.19% |

120δx | 0.18006 | 6.61% |

150δx | 0.18013 | 6.58% |

180δx | 0.18041 | 6.43% |

210δx | 0.18045 | 6.41% |

Clearly, the R.E. decreases as the resolution increases and appears to asymptote near *L *=* *150*δx*, thus a grid size of *L *=* *150*δx* is later chosen for all simulations performed at other porosities. It is noted that the no-slip boundary condition is implemented through the standard bounce back scheme in the LBM, which assumes the sphere boundaries to be at the halfway of each lattice link. This assumption is not appropriate for low resolution, as the curvature of the spheres is not accurately accounted for.

Figure 6 shows the comparison of the simulated results with *L *=* *150*δx* from the linear LBM and the analytical solutions, with good agreement being observed for the wide range of porosities. The slight deviations at *ϕ* > 0.75 are due to the resolution of the unit cell (150*δx*). At higher porosities, the curvature of the spheres is less resolved, i.e., the stair-stepping approximation, leading to a loss of accuracy. A grid size of *L *=* *150*δx* sufficiently captures the expected permeability and is used in this work for the four structures in Sec. 6.

### 5.2 Hybrid Lattice Boltzmann Method Model Verification Using Octet Structure.

To verify the hybrid model and ensure that the flow properties obtained are independent of the grid resolution, four different grid sizes (*L *=* *90*δx*, 120*δx*, 150*δx*, 180*δx*) are used to evaluate the normalized permeability at different porosity values for the Octet structure using the linear LBM according to Eq. (13). A comparison to the numerical results from Ref. [38] is given in Fig. 7(a), with less than 2% deviation between different grid sizes.

Using the permeability values obtained from the linear model, the inertial coefficients are calculated according to Eq. (16) and the results are shown in Fig. 7(b). The predicted inertial coefficients are compared with those from Ref. [38] and a similarity in trend is observed. The discrepancy at *ϕ* = 0.5–0.8 is due to the method of least squares used in Eq. (16) in the present hybrid model compared to the quadratic fitting in Ref. [38] following Eq. (15), which can be very sensitive to both the number and distribution of the pressure drop versus velocity data points selected, i.e., minor changes in the selection of the points could affect the determination of the constants *A* and *B* in Eqs. (14) and (15); while the present hybrid model using the least squares method is less sensitive. This will be further discussed in Sec. 6.3 for the four structures studied in this work. Overall, this two-step verification of the permeability and inertial coefficient verifies our hybrid LBM.

### 5.3 Hybrid Model Validation With in-House Experiments.

For validation of the hybrid model, the pressure drop is scaled by the fluid density and the numerically and experimentally obtained scaled pressure drop–average velocity curves for the Cube, FD-Cube, TKD, and Octet structures are shown in Figs. 8(a)–8(d), respectively. It should be noted that the LBM results are converted to those in physical units following Eqs. (17) and (18). The structures in the LBM simulations presented in this section all have a porosity *ϕ* = 0.89 (see Table 1). The same CAD design with *ϕ* = 0.89 was used for additive manufacturing, while due to slight manufacturing inaccuracy, the 3D printed samples shown in Fig. 2 had a measured porosity *ϕ* ∼ 0.87.

For the Cube and FD-Cube structures (Figs. 8(a) and 8(b)), the LBM predicted profiles agree well with the measured results at low average flow velocity, with more noticeable deviations observed at $u\xafz$ > 2 m/s. In contrast, the predicted results for the TKD and Octet structures (Figs. 8(c) and 8(d)) show more reasonable agreement with their corresponding experimental profiles over the entire range of $u\xafz$. The maximum relative errors between the predicted and measured values are in the range of ±15% for FD-Cube, TKD, and Octet structures and ±25% for the Cube structure. These differences between the simulations and experiments are mainly attributed to the imperfect geometry and surface roughness in the AM samples. The effect of surface roughness on porous media flow was well documented (e.g., Refs. [49] and [50]) with increased surface roughness typically resulting in an accelerated transition to the Forchheimer region. Comparing the four lattices, the TKD and Octet structures can be considered more “torturous” than the Cube and FD-Cube structures, i.e., the streamwise direction flow will vary more significantly within the unit cell. Thus, for the TKD and Octet structures, the flow characteristics are primarily controlled by the geometry itself and are not significantly affected by surface roughness. In contrast, the Cube and FD-Cube structures exhibit flow primarily in the *z*-direction. Thus, surface roughness and AM imperfections are expected to have a greater bearing in controlling the flow attributes.

## 6 Results and Discussion

This section presents results of both the experimental and numerical investigations performed. First, the local flow behavior is discussed in Sec. 6.1. The pressure drop–average velocity data obtained from experiments and LBM simulations for the four different structures is then presented in Sec. 6.2. Section 6.3 focuses on the determination of the permeability and inertial coefficients for all structures and explains the advantages of the hybrid LBM model. Numerical correlations for the normalized permeability and inertial coefficients are then given in Sec. 6.4.

### 6.1 Flow Fields Within the Lattice Structures.

To compare the flow fields within the unit cell lattice structures, representative simulation results for the four structures at the same porosity are examined. For illustration, Figs. 9 and 10 show the respective contours of the velocity component *u _{z}* in the streamwise

*z*-direction on the unit-cell outer surfaces and the interior central planes at

*z*=

*L*/2 and

*y*=

*L*/2 with

*ϕ*= 0.89 and

*dp*/

*dz*= −1 × 10

^{−8}

*δx*. The Cube structure (Figs. 9(a) and 10(a)) exhibits the highest velocity magnitude of the four structures. This is due to the geometry of the Cube structure exhibiting the lowest flow resistance. For the FD-Cube lattice, substantial resistance is present at the entry (

*z*/

*L*=

*0, 1) due to the fiber strands choking the flow (Fig. 9(b)). Thus, a much higher pressure gradient is required to attain the same average velocity for the FD-Cube structure compared to the Cube structure. Both the TKD (Fig. 9(c)) and the Octet (Fig. 9(d)) unit cells show excellent “mixing” of the fluid as more uniform distributions of high flow velocities are observed across the unit cell faces. Increased fluid mixing is advantageous for heat transfer applications, since the increase in convection heat transfer along with the high surface contact area in those foam structures will enhance the overall heat transfer in the porous lattice structures [39,40]. The interior*

*u*contours in Fig. 10 further demonstrate the differences in flow distributions in the Octet and TKD as compared to Cube and FD-Cube structures.

_{z}### 6.2 Pressure Drop Analysis.

The experimental results for the pressure gradient characteristics of the investigated lattice structures with respect to the mean flow velocity are first summarized in Fig. 4. LBM simulations are performed for a very large number of cases with their *L*/*d* ratios given in Table 1, corresponding to porosities ranging from 0.5 to 0.95. For brevity, the detailed scaled pressure drop versus velocity plots from the LBM simulations are shown in the Appendix (Figs. 22–25). Clearly, quadratic pressure drop profiles are noticed for all cases in Fig. 4. When comparing the pressure drop magnitude in those four structures, it is observed in Fig. 4 that the FD-Cube structure yields the highest pressure drop, followed by the Octet, TKD, and Cube structures when evaluated at the same average flow velocity $u\xafz$, this is due to the much more significant flow blockage in the FD-Cube structure. The numerical results in Fig. 11 show a similar trend among the four structures at comparable porosities, where the average flow velocity versus porosity is plotted for a constant pressure gradient applied. In addition, the significant effects of porosity on the average flow velocity are noted in Fig. 11 for each structure. There seem to be two distinct regimes: for 0.5 < *ϕ* < 0.8, a moderate increase in $u\xafz$ with *ϕ* is noticed, while for 0.8 < *ϕ* < 0.95, the increase is much sharper; and the FD-Cube structure also exhibits much steeper profiles for the lower porosity cases with *ϕ* < 0.81 compared to the other structures—this is again explained by the large flow blockage in the FD-Cube structure, which becomes more noticeable at lower porosities and results in a choked-flow scenario leading to significant flow impedance compared to the other structures.

### 6.3 Permeability and Inertial Coefficient.

Normalized permeability (*K*/*d*^{2}) and inertial coefficients (*C _{e}*) are presented in this section. Figure 12 shows the

*K*/

*d*

^{2}values based on the polynomial fitting method (Eqs. (14) and (15)) and the hybrid model (Eqs. (13) and (16)). The

*K*/

*d*

^{2}results from the hybrid model are greater than those from polynomial fitting. In the hybrid model, the determination of the permeability is independent of the selected velocity and pressure drop magnitude, i.e.,

*K*/

*d*

^{2}is the same regardless of the velocity/pressure used in the linear LBM simulation. In comparison, the polynomial fitting method requires the use of multiple data points encompassing both the Darcy and Forchheimer regimes. For this reason, the computed permeability of the polynomial fitting method can vary as a function of the selected data range and can be discrepant in comparison to the hybrid model, as observed in Fig. 12.

The inertial coefficients *C _{e}* based on the polynomial fitting method (Eqs. (14) and (15)) and the hybrid model (Eqs. (13) and (16)) are shown in Fig. 13. Both strategies predict similar

*C*for the porosity range and structures considered, however, the polynomial fitting method gives rise to a moderately rugged contour of

_{e}*C*(

_{e}*ϕ*) while the hybrid method produces smoother trends. Similar to the permeability determination, the polynomial fitting is strongly influenced by the input data range, leading to the irregular profile in Fig. 13. In contrast, the hybrid model is less sensitive to the input data range, making it also preferable for constructing numerical correlations for macroscopic flow properties.

As such, the hybrid model is applied to obtain the results of *K*/*d*^{2} (*ϕ*) and *C _{e}* (

*ϕ*) for all four structures as shown in Figs. 14 and 15, respectively. The computed permeability for the Cube, TKD, and Octet structures have comparable magnitudes with slight variations observed at high

*ϕ*(>85%). In comparison, the FD-Cube lattice exhibits lower

*K*/

*d*

^{2}than the other three structures. Similarly, the Cube, TKD, and Octet structures have comparable inertial coefficients. However, the FD-Cube structure exhibits considerably larger

*C*compared to the other structures. As shown in Fig. 4, the FD-Cube structure exudes the largest resistance to flow of the four structures.

_{e}### 6.4 Development of Permeability and Inertial Coefficient Correlations for Porous Lattice Structures.

Many researchers have proposed semi-empirical correlations for the permeability *K* and inertial coefficient *C _{e}* for traditional metal foams, typically expressed as functions related to metal-foam morphology (porosity, pore diameter, tortuosity, strut diameter, etc.). These correlations are either based on an idealized unit cell representation of the open-cell metal foams (Dodecahedron, Cube, and Tetrakaidecahedron) or empirically fitted experimental data. Comprehensive reviews of the different models available in the literature can be found in Refs. [51] and [52]. A few representative correlations relating the flow properties with structure porosity are discussed below.

*K*/

*d*

^{2}as

*A*is a constant which varies from 100 to 865 and

_{c}*d*is the strut diameter. The value of

*A*is 150 for Ergun's model [54], which has largely been used as a benchmark for the permeability of creeping flow (1 < Re

_{c}*< 10). Carman [55] also proposed a general form of Ergun's equation with*

_{K}*A*= 180, typically referred to as the Kozeny–Carman equation. A universal equation for

_{c}*C*was also given in Ref. [53] as

_{e}where *B _{c}* is a constant between 0.65 and 2.6. AM metal foams with high heat transfer performance [38–40] like Octet and TKD structures require permeability and inertial coefficient values for volume averaged studies which are unavailable in literature. Comparisons of

*K*/

*d*

^{2}and

*C*from the present LBM simulations for TKD and Octet structures to those according to the correlations in Eqs. (23) and (24) are shown in Figs. 16 and 17. Overall,

_{e}*K*/

*d*

^{2}from the correlations fall within the range of 5.0 × 10

^{−3}to 3.48 for

*ϕ*= 0.5–0.95. From the selected LBM simulations,

*K*/

*d*

^{2}ranges from 7.30 × 10

^{−3}to 1.81 for

*ϕ*= 0.5–0.95, demonstrating that the presented correlations can be used to characterize

*K*/

*d*

^{2}quantitatively for AM structures at

*ϕ*> 0.8 but a significant deviation is observed between the predicted

*K*/

*d*from the correlations and the LBM simulations at

^{2}*ϕ*< 0.8 (Fig. 16(a)). However, the

*C*values obtained from present LBM simulations range from 5.42 × 10

_{e}^{−2}to 0.53 and are not captured well with the range of

*B*suggested by Tadrist et al. [53] (

*C*= 2.42 × 10

_{e}^{−2}to 0.10). This encourages the development of alternative correlations for those flow properties for AM metal foams.

where it is assumed that all the fibers in each AM structure are of uniform diameter *d* as shown in Fig. 1, and all the coefficients, *a _{K}*,

*b*,

_{K}*c*, and

_{K}*d*for determining

_{K}*K*/

*d*

^{2}, and $ace$,$bce$, and $cce$for determining

*C*, are structure dependent constants. The values of these constants are given in Tables 3 and 4 for the four lattice structures. The correlations are considered valid for

_{e}*ϕ*between 0.5 and 0.95. Double exponential functions are used in this work for

*K*fitting, compared to Dukhan [56] who used a single exponential function. This is because a single exponential function does not effectively capture

*K*for the wide range of

*ϕ*considered in this work, which includes both slow and fast exponential growth regions. A single exponential function was more suitable for

*K*prediction of the

*ϕ*range that [56] studied, which was from 0.7 to 0.95. Similarly, Dukhan [56] used a linear fitting for

*C*(

_{e}*ϕ*) for

*ϕ*between 0.6 and 0.95. In this work, however, a power law fitting was found to better fit

*C*over the entire range of

_{e}*ϕ*.

Structure | $aK$ | $bK$ | $cK$ | $dK$ |
---|---|---|---|---|

Cube | 8.20 $\xd7$ 10^{−5} | 9.23 | 3.73 $\xd7$ 10^{−14} | 32.67 |

FD-Cube | 1.99 $\xd7$ 10^{−6} | 12.89 | 4.74 $\xd7$ 10^{−17} | 39.11 |

TKD | 5.34 $\xd7$ 10^{−5} | 9.76 | 7.84 $\xd7$ 10^{−15} | 34.35 |

Octet | 1.14 $\xd7$ 10^{−4} | 8.66 | 2.26 $\xd7$ 10^{−11} | 26.13 |

Structure | $aK$ | $bK$ | $cK$ | $dK$ |
---|---|---|---|---|

Cube | 8.20 $\xd7$ 10^{−5} | 9.23 | 3.73 $\xd7$ 10^{−14} | 32.67 |

FD-Cube | 1.99 $\xd7$ 10^{−6} | 12.89 | 4.74 $\xd7$ 10^{−17} | 39.11 |

TKD | 5.34 $\xd7$ 10^{−5} | 9.76 | 7.84 $\xd7$ 10^{−15} | 34.35 |

Octet | 1.14 $\xd7$ 10^{−4} | 8.66 | 2.26 $\xd7$ 10^{−11} | 26.13 |

Structure | $ace$ | $bce$ | $cce$ |
---|---|---|---|

Cube | 0.1258 | −2.128 | −0.06701 |

FD-Cube | 1.17 | −1.523 | −1.191 |

TKD | 0.4405 | −1.097 | −0.4029 |

Octet | 0.5725 | −0.6768 | −0.5341 |

Structure | $ace$ | $bce$ | $cce$ |
---|---|---|---|

Cube | 0.1258 | −2.128 | −0.06701 |

FD-Cube | 1.17 | −1.523 | −1.191 |

TKD | 0.4405 | −1.097 | −0.4029 |

Octet | 0.5725 | −0.6768 | −0.5341 |

Figures 18 and 19 give a comparison of simulation results and proposed correlations (Eqs. (25) and (26) and Tables 3 and 4). An excellent agreement between correlations and LBM simulations for both the permeability (*R*^{2} ∼ 100%) and the inertial coefficient (*R*^{2} > 98%) is observed. The *C _{e}* correlation for the FD-Cube structure is considered valid for

*ϕ*from 0.7 to 0.95. This is due to the very large

*C*at low porosities with the FD-Cube lattice, which are not suitable for most practical applications.

_{e}To further verify the applicability and accuracy of the present correlations, the developed correlations for *K* and *C _{e}* for the TKD structure are also compared with the

*K*and

*C*data and correlations in Refs. [16] and [56] for traditional foams. The TKD structure is considered an ideal repeating unit cell for traditional foams and is hence used in the comparison. Figure 20 gives this comparison of

_{e}*K*from Eq. (25) and the results of Refs. [16] and [56] for traditional foams. The previous data of Refs. [16] and [56] lie between the

*K*predicted by Eq. (25) for the TKD structure with unit cell lengths of 2.5 mm and 5.0 mm. The correlations in Ref. [56] do not capture the complete variation in permeability for the extended porosity range studied here, but the numerical values are reasonable. The comparison of

*C*from the correlation in Eq. (26) with results in Ref. [56] is shown in Fig. 21. The present results using the correlation for the TKD structure are close to the results in Ref. [56] for foams with 20 PPI. The comparisons between

_{e}*K*and

*C*from Eqs. (25) and (26) and those in Refs. [16] and [56] show that the correlations developed in this study are necessary to accurately predict the flow properties in AM metal foams. The reasonable agreement observed for

_{e}*K*and

*C*predictions with Ref. [56] also lends confidence to the proposed correlations.

_{e}## 7 Conclusions

In this work, we performed numerical and experimental studies for flow in four additively manufactured lattice structures with Cube, FD-Cube, TKD, and Octet unit cell topologies. A wide range of porosity, average flow velocity, and pressure drop values were considered to characterize the flow behavior. Simulations were performed with a validated lattice Boltzmann model, including both the traditional model and the linear form with inertial effects neglected. The results from the present numerical model were validated against available literature, with good agreement being observed. On the topic of flow resistance, both experiments and numerical analysis demonstrated that the FD-Cube structure exhibited the greatest flow impedance, i.e., a greater pressure differential is required to reach the same flow velocity as with the other structures. The Cube structure was shown to have the lowest flow impedance. The Octet and TKD structure geometries demonstrated excellent fluid mixing, suggesting that they may be better suited for heat exchanger applications requiring enhanced heat transfer. Overall, the present simulation results agreed well with in-house experiments. Minor differences were accredited to surface roughness and printing imperfections from the manufacturing process.

Using the simulation results, the characteristic flow properties (normalized permeability *K*/*d*^{2} with *d* the uniform fiber diameter and inertial coefficient *C _{e}*) were evaluated through a novel “hybrid” method. This method was found to be less sensitive to the range of velocities used for fitting in comparison to the common polynomial fitting strategy. It is also noted that while the present work considers repeating unit cells in the lattice structures, the proposed “hybrid” method for evaluating

*K/d*

^{2}and

*C*can be applied to metal foams with nonuniform or nonperiodic topologies if the geometry of the metal foam is preserved (e.g., via 3D CAD design or

_{e}*μ*CT tomography reconstruction of fabricated samples). The length scale for determining the normalized permeability (the fiber diameter is used in the present work) of such nonuniform topologies will depend on a representative length scale such as the fiber diameter, average pore diameter, unit-cell length, or the hydraulic diameter of the metal foam. Lastly, numerical correlations were proposed for

*K*/

*d*

^{2}and

*C*as functions of structure porosities. A double exponential function was used for permeability predictions to capture the slow and fast exponential growth regimes. For

_{e}*C*, a power law fitting was proposed which better captures the nonlinear behavior of

_{e}*C*(

_{e}*ϕ*) for the four AM structures, compared to standard linear fitting commonly used in the literature. A comparison of the present correlations for the TKD structure to those for traditional metal foams was also carried out. Reasonable agreement was found, lending confidence to the proposed correlations. These correlations can be used in future studies for characterizing, designing, and testing heat exchangers, flow regulation devices, and other focus areas that use AM open-cell metal foams.

## Funding Data

U.S. Department of Energy's Office of Energy Efficiency and Renewable Energy (EERE) under the Solar Energy Technologies Office (SETO) (Award No. DE-EE0009377; Funder ID: 10.13039/100011883).

U.S. National Science Foundation (NSF) (Award No. OIA 2031701; Funder ID: 10.13039/100000106).

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

## Nomenclature

*C*=_{e}inertial coefficient

- $CD*$ =
dimensionless drag coefficient

*d*=unit cell fiber diameter (m)

*d*=_{h}hydraulic diameter (m)

*dp/dz*=pressure gradient across the porous media (Pa/m)

**e**=_{α}discretized particle velocity vector

*f*(_{α}**x**,*t*) =microscopic distribution functions

*F*=_{α}body force term

- $f\u0302\alpha $ =
post collision state of the distribution functions

*H*=height (m)

**j**=momentum flux of the fluid

*K*=permeability (m

^{2})*K*/*d*=^{2}normalized permeability

*L*=length of the unit cell (m)

**M**=19 × 19 transformation matrix

**m**(_{α}**x**,*t*) =moments

*P*=pressure (Pa)

**S**=diagonal matrix

*u*=velocity (m/s)

**x**=spatial vector

*δt*=time step

*δx*=lattice spacing

- Δ
*P*/*L*= pressure drop across the porous media (Pa/m)

- $\mu $ =
dynamic viscosity of the fluid (kg/m s)

- $\nu $ =
viscosity of the fluid (m

^{2}/s)**ξ**=particle velocity vector

- $\rho $ =
density (kg/m

^{3})- $\tau $ =
relaxation coefficient

*ϕ*=porosity

*ω*=_{α}weight coefficients