Transp Porous Med (2018) 124:509–531 https://doi.org/10.1007/s11242-018-1082-6
Comparison of Pore-Level and Volume-Averaged Computations in Highly Conductive Spherical-Void-Phase Porous Materials Jeremy Vu1 · Anthony Gerald Straatman1
Received: 29 November 2017 / Accepted: 6 May 2018 / Published online: 21 May 2018 © Springer Science+Business Media B.V., part of Springer Nature 2018
Abstract A study has been carried out to compare results obtained from pore-level simulations conducted on three-dimensional idealized spherical-void-phase geometric models to similar results obtained from a solver based on volume-averaging and local thermal nonequilibrium. The purpose of the comparison is to establish closure coefficients for the viscous and form drag terms in the volume-averaged momentum equations and the interstitial convective exchange coefficient required to couple the volume-averaged energy equations for the solid and fluid constituents. A method is also described for determining the solid-phase conduction shape factor, which is shown to be important for accurate volume-average simulation of highly conductive porous materials. The shape factor has been addressed in previous literature (using various terminology) and accounts in a bulk manner for resistance due to the elongated conduction path and for changes in the effective heat flow area along the conduction path. The conduction shape factor is a function of the geometry only and is found herein from a detailed comparison between pore-level and volume-averaged simulations of conjugate heat transfer. The conduction shape factor vastly improves volume-averaged predictions of the overall heat transfer and the temperature distributions in the porous material. Keywords Porous materials · Volume averaging · Conduction shape factor
1 Introduction Porous materials are prevalent in a wide variety of engineering and science applications. Some examples include catalytic converters and packed bed reactors; heat pipes and heat transfer elements; as well as flow through sands and soils. Highly conductive porous materials (like metal and graphitic foams) have seen increased use in heat transfer applications due to their high solid-phase conductivity and surface area-to-volume ratio. Such materials take on many
B 1
Anthony Gerald Straatman
[email protected] Department of Mechanical and Materials Engineering, Western University, London, ON N6A 5B9, Canada
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different shapes and structures. For example, many aluminum and titanium foams have a reticulated structure that takes the form of fused ligaments, while other foams have a nearly spherical void phase (SVP). Graphitic foam developed by Oak Ridge National Laboratory (Gallego and Klett 2003) is an SVP material that can be processed to have an extremely high solid-phase conductivity, much higher than that of aluminum or other common metallic foams. The combination of these attributes makes such materials an excellent choice for enhanced heat transfer applications such as heat sinks or heat exchangers. To make use of such foams, it is necessary that they be characterized in terms of properties such that their behavior can be modeled in design calculations for application. While mechanical and thermophysical properties of the solid-phase constituent are usually known, for permeable porous materials subjected to fluid infiltration, it is also critical to establish the thermofluid interactions between the interstitial fluid and the solid structure. At the porouscontinuum (volume-averaged) level, these interactions are typically characterized in terms of hydraulic permeability, interstitial convective exchange, and, for all but very weak flows, the inertial drag coefficient. These parameters are established by consideration of the porous material structure and are essential for making estimates of the pumping power required for a particular imposed flow and the resulting heat transfer between the porous structure and the interstitial fluid. For some conditions, and fluid–structure combinations, it is also important to characterize the thermal and momentum dispersion, the fluid- and solid-phase tortuosities, and coefficients to correct the heat flow in the conduction path. Several researchers have studied porous materials both experimentally and numerically to characterize the convective heat transfer and hydraulic permeability. For example, in terms of experiments, Antohe et al. (1997), Paek et al. (2000) and Boomsma and Poulikakos (2001) present results of hydraulic losses of normal and compressed aluminum foams to quantify the permeability and form drag coefficients for foams of different porosity and material properties. Calmidi and Mahajan (2000) studied forced convection in highly porous aluminum foams using experiments and computational fluid dynamics. Their paper reports on hydraulic losses, interstitial exchange, and thermal dispersion. In general, aluminum foams have a highly porous structure (92–96% void) that enables fluid to pass through relatively easily, resulting in little pressure penalty, but only modest heat transfer enhancement. Gallego and Klett (2003) performed experiments to evaluate heat transfer coefficients in sphericalvoid-phase graphitic foams. Straatman et al. (2006) determined hydraulic and heat transfer coefficients for four different graphitic foams and developed correlations for the interstitial heat transfer coefficient. Fu et al. (1998) established the interstitial heat transfer in cellular ceramics. An article by Kamiuto and Yee (2005) compiled past studies (Younis and Viskanta 1993; Ichimiya 1999) and found that a fairly accurate correlation exists when the strut diameter is used as a characteristic length in open cell (reticulated) foams. Ando et al. (2014) experimentally studied convective heat transfer for ceramic foams. More recently, with the availability of large-scale computing, it has become possible to conduct three-dimensional computational fluid dynamics (CFD) simulations at the pore-level of porous materials. Such computations provide insight into the pore-level exchanges and also enable the characterization parameters to be established by integration of the pore-level velocity and temperature fields in the fluid and solid constituents. One important prerequisite for such calculations—in addition to knowing the thermophysical properties of the solid and fluid constituents—is a geometric model of the pore-level structure and the assurance that the geometric model is a representative elemental volume (REV) of the porous material, meaning that the thermofluid activity averaged over that volume is representative of activity in any volume of that size (or larger) in the porous domain. A geometric model can either be established by directly scanning and reconstructing digital images of a porous media, or by
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developing an idealized mathematical representation of the porous structure. The method of direct scanning and reconstruction has been adopted [see, for example, Kumar et al. (2014), Ranut et al. (2014)], but is expensive and time-consuming and produces models that are not easily scalable, meaning that it is usually necessary to conduct the same process for all porous samples under consideration. In this respect, when possible, mathematical idealizations are a much more viable means for generating idealized geometries. The unit-cube model developed by Yu et al. (2006) is possibly the simplest threedimensional geometric idealization of a SVP porous material and is created by placing a hollow sphere inside a cube and increasing its diameter (beyond the cube side length) until the desired porosity is achieved. The geometry is extremely simple, which makes it easy to analytically calculate parameters such as porosity and interstitial surface area, but Yu et al. (2006) found that estimates of the permeability led to pressure drops that were unrealistically low, mainly due to the ordered orientation and size of the pore windows connecting the spherical voids. Karimian and Straatman (2009) conducted a CFD study using the unit-cube geometry to model flow and heat transfer to gain insight into axial dispersion. In that study, the unit cube was placed at 45° relative to all principle planes to give the essence of a more random structure, but they found that while convective heat transfer trends were in better agreement with experiments, the permeability remained unrealistically high unless blockage factors were applied. Another unit-cube-type geometric model was proposed by Leong and Li (2011), in which spheres are subtracted from the corners of a cube. This was used to evaluate geometric and thermal properties of carbon foam, and it was found that the effective thermal conductivity reduced considerably with increasing sphere diameter of the spheres and porosity. In addition, the fluid-phase conductivity was found to have almost no effect on the effective conductivity, due to it being many times lower than the solid phase. Research has also been done to improve on unit-cube-type geometric idealizations. Boomsma and Poulikakos (2001) proposed a tetrakaidecahedron shape (Kelvin cell), which was then used by Kumar et al. (2014) to calculate geometric parameters. Empirical correlations were proposed to calculate properties such as permeability and inertia coefficient. Kopanidis et al. (2010) used the Kelvin cell geometry to conduct conjugate heat transfer simulations involving fluid and solid phases, and found that this geometry produced results which matched reasonably well with experimental data—specifically heat transfer coefficient and pressure drop. Kirca et al. (2007) mathematically placed random spherical bubbles inside a cube, and modified the bubble radii, average bubble radius, and neighboring bubble interactions to obtain a desired porosity. Finite element analysis was then performed to obtain properties such as Young’s Modulus and Poisson’s Ratio, and the computed values matched closely with literature. Random-generation models were also proposed by Wang and Pan (2008) and Chueh et al. (2014), wherein cubic REVs could be generated to more closely mimic SVP porous materials. Dyck and Straatman (2015) used discrete element modeling (DEM) to create a random SVP REV which is spatially periodic over all three sets of faces, making it extremely versatile for CFD calculations. This model uses a contact law based on bubble physics, which handles the interaction and interference between spheres. The sphere diameters can also be specified as random within a specified interval, and the volume is compressed to obtain a desired porosity. Their calculations of convective heat transfer and permeability are in much closer correspondence to experiments on grahitic SVP foams. The value of pore-level CFD simulations on an accurate geometric REV is that the results can be used to evaluate closure parameters for volume-averaged solvers, which is the more typical engineering approach for studying porous materials. Comparisons at the pore-level make it possible to investigate processes separately—something that is virtually impossible using experimental results. For example, using the global result for convective heat transfer
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from an experiment does not enable you to separately understand the influences of conduction, interstitial convective exchange and thermal dispersion. Of particular importance in highly conductive porous materials is the ability to simulate both isothermal wall and conjugate heat transfer studies to separately investigate the heat flow in the fluid and solid constituents. In addition to being able to form interstitial exchange and dispersion expressions (from isothermal simulations), it is possible to also consider the influences of additional resistances along the conduction path in the solid (from conjugate simulations). Several recent works by Yang et al. point to the importance of including this influence in closed-cell porous metals. Yang et al. (2013a) coin the term “thermal stretching” to describe heat transfer reductions due to elongation and area changes along the conduction path in the solid phase of a closed-cell porous material. Their model is based on analysis of a simple circular cell trapped in a solid and shows that their thermal tortuosity (defined as the actual conduction path length divided by the straight-line distance between the heated surfaces) increases nearly monotonically with porosity. Yang et al. (2013b) present an analytical model for conduction tortuosity for simple closed-cell porous media with a non-conducting void phase. In another related article, Yang et al. (2013c) present analytical shape factors for closed-cell porous materials have cell shapes other than spheres using a circularity factor. Finally, Traxl et al. (2016) present a thin shell model to produce an analytical expression that corrects the effective thermal and electrical conductivities of closed-cell metal foams. This model extends the work of (Yang et al. 2013a, b, c) to porous media of much higher porosity, but is still only valid for simple, spherical, closed-cell structures. This article aims to study more closely the direct comparison between pore-level computations on a SVP porous material (of various solid-phase conductivity) and volume-averaged computations done on the same domain. Pore-level simulations are conducted on SVP geometries generated using the geometric modeling approach developed by Dyck and Straatman (2015) over a range of flow and heating conditions. Closure parameters are established from results of the pore-level simulations and then used directly in volume-averaged calculations of the same domain such that there is a precise analog between the two computations. In addition to the area-to-volume ratio, permeability and interstitial convection coefficient, it is shown that the additional resistances in the solid conduction path due to elongation and area changes must be accounted for to reproduce the heat transfer results obtained in the pore-level computations. An approach is described for establishing a solid-phase conduction shape factor for a complex geometric REV model.
2 Model Formulation Consider the schematic diagram in Fig. 1 illustrating how a small portion of a porous domain can be used to represent the activity in the larger domain. As mentioned in the introductory section, the small portion of the domain is much smaller than the full domain, but must be at a scale (1) that is large enough to be considered a representative elemental volume (REV). Since both pore-level and volume-averaged computations are presented in this article, the formulation for both is given below. In all cases, we are concerned with laminar, incompressible flow of a Newtonian fluid with energy exchanges between an interstitial fluid and a rigid solid. The equations for the transport of mass, momentum and energy in a conjugate fluid/solid domain with constant thermophysical properties are given as (Whitaker 1997): ∇ ·u0
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(1)
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Fig. 1 Image showing fluid and solid volume fractions within an REV
∂u + u · ∇u −∇ p + μf ∇ 2 u ρf ∂t ∂ Tf ρf C p,f + u · ∇Tf kf ∇ 2 Tf ∂t ∂ Ts ks ∇ 2 Ts ρs C p,s ∂t
(2) (3) (4)
where u is the velocity vector, T f and T s are fluid and solid-phase temperatures, respectively, and ρ, μ, C p and k are the density, dynamic viscosity, specific heat capacity and conductivity, respectively, of the fluid (f) and solid (s) phases. Equations (1)–(4) are solved directly to generate solution fields for flow and temperature in pore-level geometric models to understand the detailed interactions between the fluid and solid constituents of the porous media, and to generate parameters that characterize the porous material at the volume-averaged level. The domain under consideration and the boundary conditions are described in a subsequent section. Using the volume-averaging approach described in Whitaker et al. (1997), and considering local thermal non-equilibrium (i.e., different fluid and solid constituent temperatures in the REV), the mass, momentum and energy equations [Eqs. (1)–(4)] are transformed to (Betchen et al. 2006): ρf
∇ · u 0 (5) 1 ∂ u u ˜ nfs · (− I p˜ + μf ∇ u) ˜ dA − ρ f ∇ · u˜ u + · ∇ u −ε∇ pf + μf ∇ 2 u + ∂t ε V Afs
ρf C p,f
∂ Tf f + u · ∇ Tf f ∂t
⎛ ⎜1 kf ∇ 2 Tf f + ∇ · ⎝ V 1 + V
∂ Ts ∂t
⎟ nfs · kf Tf dA⎠
f nfs · kf ∇Tf · dA − ερf C p,f ∇ · u˜ T˜f
⎛
(1 − ε)ρs C p,s
(6)
⎞
Afs
Asf
s
⎜1 (1 − ε)ks ∇ 2 Ts s + ∇ · ⎝ V
Afs
⎞ ⎟ 1 nfs · ks Ts dA⎠ + V
(7)
nfs · ks ∇Ts dA (8) Asf
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where ε is the porosity, defined as the void volume per total REV volume; the terms in angled brackets represent the extrinsic average of a variable and f and s represent the intrinsic average over the fluid (f) or solid (s) constituent, respectively. The variables with a tilde (~) represent deviations of that variable inside the averaging volume (REV). The new unknowns that appear in Eqs. (5)–(8) are artifacts of the volume-averaging process and constitute a closure problem, which must be resolved by introducing models that describe the unknown processes in terms of known variables and closure coefficients. In the volumeaveraged momentum equation (Eq. 6), the last two terms represent interactions between the fluid and solid constituents and are replaced herein with the heuristic Darcy and Forchheimer models to account for viscous (Darcy) and inertial (Forchheimer) drag on the fluid [see, for example Vafai and Tien (1981) and Ward (1964)]. The closed, volume-averaged momentum equation is then given as: ερf C f ∂ u u εμf |u| u u − √ ρf (9) + · ∇ u −ε∇ pf + μf ∇ 2 u − ∂t ε K K where K is the permeability and C f is an inertial coefficient, both of which must be determined from experimental measurements or from the results of pore-level simulations on a particular porous structure. In the volume-averaged energy equations [Eqs. (7)–(8)], the integral terms and the terms involving temperature and velocity deviations must be resolved for closure. The second-last term on the right-hand side of Eq. (7) and the last term on the right-hand side of Eq. (8) represent convective heat transfer between the fluid and solid constituents and is modeled using a form of Newton’s Law of cooling (Vafai and Tien 1981): 1 nfs · kf ∇Tf dA −h sf Asf Tf f − Ts s (10) V Asf 1 (11) nfs · ks ∇Ts dA −h sf Asf Ts s − Tf f V Asf
where hsf is the interstitial heat transfer coefficient (in W/m2 K) and Asf is the interfacial surface area per unit of volume inside the averaging volume (in m2 /m3 ). The remaining terms in the fluid energy equation (Eq. 7) are the tortuosity and dispersion terms, respectively, which can be cast in the form:
f ∇ · (kf Λfs ) − ερf C p,f ∇ · u˜ T˜f (12) where Λfs G ∇ Tf f + σ ∇ Ts s
(13)
here G is a tortuosity closure parameter and σ k s /k f (Hsu 1999). Hsu (1999) showed that the value of G depends only on the local interfacial geometry and on the solid and fluid properties. He also showed that the value of G is always negative such that the tortuosity serves to reduce the (effective) conductive transport in the medium. For values of σ > 100, the value of G rapidly approaches zero, such that for highly conductive porous materials saturated interstitially with low-conductivity fluids (air or water), the tortuosity can be assumed to have no influence. The dispersion term can be modeled using, for example, relationships provided in Calmidi and Mahajan (2000) or DeGroot and Straatman (2012). Depending on the flow parameters, the dispersion can also be quite small compared to the conduction and interstitial
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convective transport. In this manner, the final form of the volume-averaged fluid-phase energy equation is: ∂ Tf f f + u · ∇ Tf kfe ∇ 2 Tf f − h sf Asf Tf f − Ts s ρf C p,f (14) ∂t where the fluid-phase conductivity tensor has been reduced to the vector k fe , which is comprised of molecular and dispersion effects as: kfe kf + kdisp
(15)
In the solid energy equation (Eq. 8) the only remaining term is the tortuosity term, which takes the form: ∇ · (ks fs )
(16)
where fs is given in Eq. (13). By the argument given above, the tortuosity coefficient G 0 for high solid to fluid conductivity ratios and can thus be ignored for highly conductive porous materials. Thus, the final form of the volume-averaged solid-phase energy equation is: ∂ Ts s (1 − ε)ρs C p,s kse ∇ 2 Ts s − h sf Asf Ts s − Tf f (17) ∂t where the effective conductivity of the solid phase, k se , is yet to be determined. Here it is important to note that the tortuosity described with respect to the volume-averaging process and resolved by Hsu (1999) does not hold the same meaning as the tortuosity described in Yang et al. (2013a). The tortuosity given in Eq. (13) is an interface effect that is due to temperature deviations inside the fluid and solid phases of the REV, and this effect disappears as the conduction ratio σ gets large, which is nearly always the case for porous metals. To avoid confusion with this tortuosity, the factor alluded to in Yang et al. (2013a) and Traxl et al. (2016) that accounts for elongation and area changes in the conduction path will herein be called a “conduction shape factor,” as this terminology already holds some physical relevance in accounting for such resistances in conduction heat transfer. Traxl et al. (2016) presents several model expressions—including their own—for closed-cell, spherical porous materials in which the fluid-phase conduction is negligible compared to the solid-phase conduction. However, all of the models can be cast into the form: ke F (18) ks where k e is the net effective conduction of a porous material and F is a shape factor expression that depends on the geometric model considered. To use Eq. (18) in the context of Eq. (17), the solid-phase effective conductivity is written as: ks F kse (19) (1 − ε) where we then define: F¯s F/(1 − ε)
(20)
as the solid-phase conduction shape factor. With reference to Yang et al. (2013a), their tortuosity factor τ¯ is (essentially) the inverse of F and is defined as the length of the conduction path divided by the straight-line path between the surfaces, which is why it is termed a “tortuosity” Even though both τ¯ and F were derived analytically for simple, closed-cell porous
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media with non-conducting voids, we argue herein that a factor of this form is also valid for permeable porous materials with complex conduction paths. In subsequent sections, we show the impact of the solid-phase shape factor and introduce a simple approach for establishing F¯s for domains that do not lend themselves easily to analysis. Finally, closure of Eqs. (14) and (17) also require specification of the interstitial exchange hsf , and the interstitial area, Asf , which must be derived from experimental data or pore-level simulations on a particular porous material.
3 Pore-Level Simulations Pore-level calculations are carried out in this study for two purposes; first, to determine the permeability, inertial coefficient and interstitial convective heat transfer coefficient as a function of flow and geometry to close the volume-averaged transport equations [Eqs. (9), (14), (17)]; and second for conjugate heat transfer cases that will be compared directly to similar results generated using the volume-averaged approach. In the first case, the entire solid portion of the porous media is considered isothermal so only the void portion of the REV requires discretization and simulation, while in the second case, the fluid and solid constituents must be discretized and solved simultaneously (conjugate simulations). The geometric modeling tool described in Dyck and Straatman (2015) (hereinafter referred to as YADE—Y et Another Development Engine) was used to generate representative elemental volumes of SVP foams for all pore-level simulations. The geometric tool requires specification of the number and size of pores (primitives) to be included in the volume. The tool initially places the primitives in a random pattern within a large cubic volume and then mathematically “squeezes” the cube until the target porosity is reached. During the “squeezing” operation, the pores move within the domain in response to a mathematical contact law and eventually intersect each other to a certain measure of interference based on a force balance. The result is a random SVP domain that has a specified pore diameter and porosity. A particularly unique feature of this digital domain generation tool is that it enforces spatial periodicity in all three principle directions and thus, the resulting REV can also be used as a building block in any direction to produce larger domains of the same geometric properties. Also, because the internal geometry is random, each REV can be run with flow in the x, y and z directions to produce three sets of results from which integral averages can be obtained, which is particularly useful for the isothermal cases where it is parameter evaluation that is sought. Figure 2 shows an image of a typical SVP foam generated using (Dyck and Straatman 2015), while Fig. 3 compares a digitally generated SVP foam with SEM images of graphitic foam. As illustrated, the digitally generated foams are a good representation of the graphitic foam structure in terms of the randomness of the pore positioning and the windows that connect the pores. In this manner, these geometric models are considered suitable for demonstrating the method for determining the closure parameters and the conduction shape factors; the approach presented can be adapted to any complex geometric model whether developed mathematically or by digital scanning. Geometric models were generated for pore diameters of 400 and 800 μm and porosities ranging from 0.70 to 0.85, as summarized in Table 1. The pore diameter and porosities were chosen as being the standard size ranges for graphitic foams used for in heat transfer applications [see, for example, Gallego and Klett (2003)]. In Table 1, d p is the pore diameter in μm, ε is the porosity, L is side length of the cubic REV in [m], V is the total volume of the
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Fig. 2 Digital representation of SVP foam generated using 100 spherical primitives
y
z x
Fig. 3 SEM images of a graphite foam specimen (a) and (b) (Gallego and Klett 2003) in comparison with a CAD model of the SVP geometry generated using the method of (Dyck and Straatman 2015) (c) and (d)
REV in [m3 ], A is the internal surface area of the REV in [m2 ], and Asf is the area per unit volume in [m2 /m3 ]. The geometric idealization produced from YADE is a cube with an edge length of approximately 1.5–3 mm, depending on the porosity specified during geometry generation, with 100 interconnected spherical cells, which is shown in Dyck and Straatman (2015) to be sufficient for the geometric model to be considered a REV. The geometric models were converted to CAD models in SolidWorks™ and then imported into ANSYS Workbench™ (http://www.ansys.com/products/fluids) where short inlet and outlet sections were added to avoid issues with recirculation and temperature periodicity in
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Table 1 Summary of geometric parameters for representative elemental volumes (REV) of SVP foams generated by YADE (Dyck and Straatman 2015) for the pore-level calculations of heat and fluid flow REV #
d p (μm)
ε
L (m)
V (m3 )
A (m2 )
Asf (m2 /m3 )
1
400
0.75
0.001632
4.35e−09
3.90e−05
8970
2
400
0.80
0.001589
4.01e−09
3.48e−05
8678
3
400
0.85
0.001544
3.68e−09
2.99e−05
8112
4
800
0.70
0.003342
3.73e−08
1.60e−04
4279
5
800
0.80
0.003159
3.15e−08
1.29e−04
4104
6
800
0.85
0.003059
2.86e−08
1.05e−04
3678
y
z x
Fig. 4 Image showing a wireframe of the computational domain, which includes short inlet and outlet sections
the flow direction; Fig. 4 shows the complete computational domain for one case. Preliminary calculations were made on double-length REVs to ensure that the addition of the inlet and outlet sections did not alter the integral results for permeability and interstitial exchange. Thus, the computational domain was a three-dimensional duct of cross section L × L, and length 2.5L (in the flow direction) with a porous plug of length L positioned 0.75L from the inlet plane. The domains were discretized using the ANSYS Meshing tool™ (http:// www.ansys.com/products/fluids), which produced tetrahedral grids that were fine near solid boundaries and gradually increased toward the pore centers. Figure 5 shows a center-plane (x–z) cross section of the domain for one case illustrating how the idealized REV is positioned between the inlet and outlet sections and grid distribution in the domain.
3.1 Isothermal Conditions Calculations were conducted to simulate flow and convective heat transfer for the case where the entire solid constituent was considered isothermal at a temperature T w . For these cases, only the fluid constituent of the REV required meshing, and grids of approximately 3,500,000 tetrahedral elements were used to produce grid-independent solutions to better than 5% based on overall heat transfer and pressure drop. The grid independency is estimated from a similar solution obtained on a grid of 7,000,000 tetrahedral elements. The commercial CFD software ANSYS CFX (http://www.ansys.com/products/fluids) was used to run simulations of air and water flow through the SVP models; the thermophysical properties used for air and water are
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Fig. 5 Planar (x–z) cross section of computational domain and mesh for case of d p = 400 μm and ε 0.85 (REV 3 in Table 1) Table 2 Fluid properties of air and water used in pore-level simulations
Property
Air
Water
Density (kg/m3 )
1.185
997.0
Dynamic viscosity (kg/m s)
1.831E−5
8.899E−4
C p (J/kg K)
1004.4
4181.7
Conductivity (W/m K)
0.0261
0.6069
given in Table 2. In all cases, the flow was considered laminar and advection in the momentum and energy equations was modeled using second-order up-winding. Steady-state simulations were converged to residual levels better than 1.0 × 10−6 . Even for these convergence levels, the solution converged stably and monotonically and convergence was achieved in less than 100 iterations, which corresponds to between 0.5 and 2 h of wall-clock time using a 4core Intel Core i7-3820 3.60 GHz CPU with 16 GB of RAM. The reader is now referred to Fig. 4 for the description of boundary conditions. The boundary conditions for the isothermal cases were that of periodicity on the pairs of lateral (x–y) faces and lateral (x–z) faces; the specification of a mass flow rate in the x-direction and zero-gradients at the domain outlet; and constant-temperature conditions for all interior surfaces. A temperature difference of 20 K between the solid and the incoming fluid was used for all calculations; i.e., T w 318 K, T i 298 K. As noted, similar cases were also run with flows in the y and z direction to generate a large body of results for obtaining parameters. For each geometric model listed in Table 1, cases were run for Redp {1, 5, 10, 15, 20, 40, 80, 120, 160, 200}. The permeability (K) and Forchheimer coefficient (C f ) for each geometric model were established by curve-fitting pressure-drop data with the Darcy–Forchheimer law: P μ ρCf (21) U + √ U 2, L K K where P is the pressure drop across the porous domain and L is the length of the porous block in the flow direction. The pressure difference was obtained by taking the difference between the area-averaged pressure at the inlet and outlet of the porous block (shown in Fig. 4). Figure 6 shows the pressure drop as a function of velocity for all the REVs given in Table 1. For REVs 1–3 having d p = 400 μm, results were obtained for Reynolds numbers in the range 1–200, while for REVs 4–6, a Reynolds number range of 1–20 was used on the basis that further cases did not impact the coefficients obtained to characterize the pressure
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Fig. 6 Pressure drop as a function of bulk inlet velocity for all pore-level isothermal simulations. a 400 μm geometric models. b 800 μm geometric models
12000
/ [kPa/m]
10000 8000
REV 1 REV 2 REV 3
6000 4000 2000 0 0.00
0.10
0.20
0.30
0.40
0.50
0.020
0.025
Inlet velocity U [m/s]
(a) 25
/ [kPa/m]
20 15
REV 4 REV 5 REV 6
10 5 0 0.000
0.005
0.010
0.015
Inlet velocity U [m/s]
(b)
Table 3 Summary of K and C f for all geometric models considered
REV #
Diameter
Porosity
Fluid
K
Cf
1
400
0.75
Air
1.09E−09
0.46
2
400
0.80
Air
1.44E−09
0.41
3
400
0.85
Air
2.62E−09
0.40
1
400
0.75
Water
7.00E−10
3.57
2
400
0.80
Water
1.09E−09
0.47
3
400
0.85
Water
1.69E−09
0.26
4
800
0.70
Water
2.67E−09
1.42
5
800
0.80
Water
5.00E−09
0.40
6
800
0.85
Water
7.96E−09
0.24
drop and heat transfer. In Fig. 6, the points represent data for the individual cases and the curve is a fit of Eq. (21) using the parameters derived from the points and summarized in Table 3. As can be observed, Eq. (21) accurately describes the trends observed for the 6 different REVs considered.
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The interstitial exchange coefficient, hsf , is established from the isothermal simulations by consideration of the bulk temperature variation for forced flow through an isothermal duct (Incropera and DeWitt 2002), which in this case is the foam block: Tw − Tmo h sf A exp − , (22) Tw − Tmi mC ˙ p This is similar to the approach used by DeGroot et al. (2009) to establish the heat transfer coefficient in foam filled finned heat sinks and is equally suitable here. In Eq. (22), T w is the solid-phase temperature, T mo is the bulk outlet temperature of the fluid, T mi is the bulk inlet temperature of the fluid, A is the interstitial area (in m2 ), m˙ is the mass flow rate of fluid (in kg/s) and C p is the specific heat of the fluid (in J/kg K). Solving Eq. 22 for the interstitial heat transfer coefficient gives: mC ˙ p Tw − Tmo h sf − , (23) ln A Tw − Tmi which is combined with the area per unit volume and the pore diameter to give the expression used for the Nusselt number: Nudp
Asf h sf d 2p kf
,
(24)
where k f is the fluid molecular conductivity. The bulk temperatures are obtained in a manner similar to that used for pressure, except that mass-averaging is used instead of area-averaging. Figure 7 summarizes the results for interstitial exchange for REVs 1–6 and shows that the heat transfer increases in all cases with increasing Redp . No single correlation describing the Nusselt number for the isothermal-solid cases is proposed herein, mainly because of the limited number of geometric cases and the Reynolds number range considered. This is not to say that such a correlation is not possible; the correlation is clearly a function of all of the geometric parameters, as is evident in Fig. 7. In the present study, the results for the permeability, Forchheimer coefficient and interstitial heat transfer coefficient are computed so that they can be used directly in volume-averaged simulations presented in Sect. 4 such that one-to-one comparisons can be made.
3.2 Conjugate Heat Transfer For the second part of the pore-level study, the REVs were assumed to be attached to a solid substrate and conjugate calculations were done to predict the flow and convective heat transfer in the fluid phase and the temperature field in the solid phase for the cases of a constanttemperature substrate and a constant heat flux at the substrate. The cases with the isothermal substrate are used in the detailed comparisons with volume-averaged computations, and the constant heat-flux calculations are used as additional validation of the final, calibrated system to demonstrate the viability of the model. The domain considered was identical to that used for the isothermal-solid simulations described above (see Figs. 4 and 5). As both the void and solid constituents of the REV now require meshing, grids comprised of approximately 6,500,000 tetrahedral elements are necessary to produce grid-independent results of overall heat transfer to within 5%. The grid independency is derived from similar calculations conducted on a grid of 13,000,000 elements. The boundary conditions were that of periodic conditions on the pair of lateral x–z planes, a symmetry condition on the upper x–y plane, and a wall condition on the lower x–y plane. Where the REV is in contact with the lower x–y plane, either a temperature T w is imposed
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Fig. 7 Nusselt number Nudp as a function of Redp for all pore-level isothermal simulations. a 400 μm models. b 800 μm models
80
Nudp
60 40 REV 1 REV 2
20
REV 3 0 0
50
100
150
200
250
Redp
(a) 40
Nudp
30 20 REV 4 REV 5
10
REV 6 0 0
5
10
15
20
25
Redp
(b) or a constant heat flux, while the lower plane of the inlet and outlet sections was specified as adiabatic. Under the isothermal condition, the temperature difference between the incoming fluid and the substrate was fixed at 20 K for all cases; under the constant heatflux condition, the heat flux was fixed at 925 [kW/m2 ]. This value was chosen as it would produce a similar temperature rise as in the constant-temperature cases. Computations of conjugate heat transfer were carried out for all REVs listed in Table 1 and each for solidphase conductivities of 50, 100, 237 and 400 [W/m K] and Reynolds numbers in the same range as described above for isothermal-solid simulations. A fringe plot of the fluid-phase temperature on the center (x–z) plane is given in Fig. 8 for REV 1 with k s = 237 [W/m K] and Redp = 80 for the isothermal-substrate condition. The appearance of seemingly trapped pores observed in the figure is due to the three-dimensional nature of the porous structure and the fact that this is a two-dimensional image. Referring to Figs. 2 and 3, it is clear that the connections between pores are from all sides making the foam highly permeable to the fluid. Figure 8 shows that the fluid temperature is the warmest near the substrate and rises gradually in both the stream wise (x) direction and the vertical (z) direction due to the temperature difference between the substrate (base) and the incoming fluid. Also, the fact that the fluid starts to warm very near the entrance at a substantial distance from the substrate is evidence of conduction into the solid matrix of the foam and subsequent convection to the fluid.
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Fig. 8 Temperature contours in the fluid phase obtained from a conjugate simulation at Redp 80, ε 0.75, 400 μm diameter (REV 1), k s 237 [W/m K] and under the isothermal-substrate condition
4 Volume-Averaged Simulations Computations were conducted again on the domain shown in Fig. 4, except with the porous region replaced by a simple box, which was subsequently discretized using hexahedral brick elements. The computational method described in Betchen et al. (2006) was used to solve Eqs. (5), (9), (14) and (17). This in-house code is based on finite-volume discretization on conjugate fluid/porous/solid domains and includes interface conditions to implicitly connect the regions of the conjugate domain. The reader is directed to (Betchen et al. 2006) for full details on the computational method. In this case, the total mesh consisted of 105 cells in the flow (x) direction, 55 cells in the z direction and 1 cell in the y direction due to the imposition of symmetry boundary conditions on the lateral (x–z) faces. The remaining boundary conditions are the same as those used for the pore-level simulations. In the volume-averaged region, the closure parameters obtained in Sect. 3 are imposed such that all possible aspects of the mathematically-generated porous region are accurately captured. In addition, since the volume-averaged model is formulated to account for local thermal non-equilibrium, the fluid and solid-phase conductivities are provided separately for Eqs. (14) and (17). Thermal dispersion in the fluid phase was ignored on the basis that it is negligible for all of the conditions considered. Steady-state solutions were obtained for the isothermal-substrate heating condition over the same Redp range as considered in the pore-level conjugate simulations for air and water. On average, both the energy and momentum equations were converged to mean residuals of less than 1.0 × 10−10 . A grid independence study showed that doubling the grid density led to changes that were less than 1% different in terms of overall pressure drop and total heat transfer. Computations of conjugate heat transfer using the volume-averaged approach required a small fraction of the time required for pore-level calculations of the same flow (approximately 30–120 s).
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Fig. 9 Plot showing comparisons of total heat transfer from results of pore-level computations and volume-averaged computations for conjugate case. Complete results are shown for REVs 1–3 using a solid-phase conductivity of k s 237 W/mK and Redp 1–200
J. Vu, A. G. Straatman
6
4
REV 1 REV 2 REV 3
2
0 0
2
4
6
8
Q [W] Pore-level Computations
Comparisons between pore-level and volume-averaged simulations are first made for the conjugate cases where a constant wall temperature was used at the substrate. For these cases, it is the energy transfer to the fluid that provides key information used to formulate the conduction shape factor introduced in Sect. 2. Comparisons of constant heat-flux cases are made following Sect. 4 to show that the shape factors also improve predictions of the temperature distribution under constant heat-flux conditions. Comparisons of the overall pressure drop from the pore-level and volume-averaged computations showed very little difference, owing to the fact that the permeability and Forchheimer coefficients derived from the pore-level cases were used directly. The heat transfer for the conjugate system was computed using the fluid and solid conductivities and the interstitial exchange coefficients obtained from the pore-level calculations, but despite the precise imposition of these parameters and the exact area per unit volume (Asf ), the results were as much as 20% in error, with the volume-averaged results overpredicting the heat transfer in all cases, as shown in Fig. 9. Figure 9 compares the results for overall heat transfer computed from volume-averaged simulations with the parameters obtained from the equivalent pore-level simulations. As such, an accurate correspondence would have all of the points on the diagonal line. It is felt that the over-prediction of heat transfer by the volumeaveraged model is mainly due to the lack of a correction for conduction path effects in the solid phase of the porous media. While the conduction path is automatically accounted for in the pore-level conjugate calculations by computing conduction in the actual solid structure, in the volume-averaged model the conduction path is inherently assumed to be straight and uniform, unless corrected by a solid-phase conduction shape factor. As noted in Sect. 2, the shape factor characterizes the additional resistance of the conduction path due to elongation and area changes of the solid structure, which are readily observed in the mathematical YADE models generated for this study (see Fig. 2).
4.1 Conduction Shape Factor In Sect. 2, a solid-phase conduction shape factor, F¯s was introduced as a means of providing a correction for the additional path resistances of conduction due to path elongation and
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Fig. 10 Image showing a wireframe of the computational domain given in Fig. 4, except with the SVP section replaced by 81 cylindrical pin fins of diameter 75 μm, yielding a porosity 0.85 and an area-to-volume ratio, Asf 7560 m2 /m3
changing area. The conduction shape factor has a negative influence on the heat transfer process and was presented in Eqs. (18)–(20) in terms of the shape factor F. To reinforce the fact that conduction path effects are (mainly) responsible for the difference between the porelevel and volume-averaged results shown in Fig. 9, and not some aggregate of other effects, a simple case of straight cylindrical pin fins was simulated with both the pore-level and volume-averaged solvers. The pin–fin case considered (see Fig. 10) replaces the SVP porous section shown in Fig. 4 with a 9 × 9 array of 75 μm diameter pins that extend from the lower to upper x–y planes (vertical pins of length 1.544 mm), yielding a porosity of 0.85 and, by definition, a conduction shape factor of 1. Pore-level computations were initially done using the isothermal (internal) solid conditions described in Sect. 3.1 to establish the parameters K, C f and hsf for the pin–fin geometry over the same range of Reynolds numbers considered for the SVP cases (based on the inlet mass flow). Pore-level conjugate simulations were then carried out using the approach described in Sect. 3.2 to yield heat transfer results over the same range of flow rates using the isothermal-substrate condition. Similar calculations were carried out using the volume-averaged solver with the parameters established in the pore-level isothermal cases described above. A comparison of the total heat transfer results from the conjugate pore-level and volume-averaged solvers is given in Fig. 11, which shows that the predicted heat transfer results are in excellent correspondence, confirming that for a simple geometric model with a straight, uniform conduction path, the volume-averaged solver faithfully reproduces the heat transfer when using the closure parameters derived using the pore-level approach described in Sect. 3.2. These results confirm that conduction path effects are largely responsible for the discrepancies observed in Fig. 9. The conduction shape factor can take many forms depending upon the structure of the porous media. For a porous material with trapped spherical voids of negligible fluid conductivity, Maxwell (1904) proposed: F
2 fs 3 − fs
(25)
where f s (1 − ε). This well-known expression was specifically developed for materials with low-porosity. Traxl et al. (2016) used an analytical approach to develop an expression
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Fig. 11 Plot comparing total heat transfer computed from results of pore-level computations and volume-averaged computations from the cylindrical pin–fin geometric model shown in Fig. 10
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6
4
2
0
0
2
4
6
8
Q [W] Pore-level computations
for much higher porosity materials, but still with a closed-cell geometry. Their expression takes the form; F
tanh−1
fs
π π + fs
(26)
while Eqs. (25), (26) are both strictly valid for closed-cell porous domains, we argue that a shape factor with the same physical meaning can be formed for a general complex porous structure. Figures 2 and 3 illustrate that both elongated paths and changing cross-sectional areas are prevalent in the SVP REVs considered, and for any similarly complex porous structure. Inspection of Figs. 2 and 3 also suggests that it would be nearly impossible to find the average solid-phase conduction shape factor by any analytical means by simply considering the physical geometric model. Here, it is worth reinforcing the critical point that the shape factor for any porous material is a geometric feature that is constant for a given configuration, regardless of the complexity of the structure. With this in mind, the solid-phase conduction shape factor can be established for a particular REV by comparing results from pore-level calculations with similar volumeaveraged calculations, and modifying the value of F¯s until the total heat transfer matches. This process is clearly only valid if the other closure coefficients (interstitial area and convective exchange) are accurately known, which is the case when pore-level results are available for an REV model of the domain. Once the shape factor is established for a particular geometric model, it can be verified by running a series of flow/heating conditions and solidphase conductivities on the same geometric model and then comparing to similar pore-level results. To this end, the conjugate cases computed using the isothermal-substrate temperature condition and simulated for Redp = 20 and k s 237 W/mK were compared for each of the six geometries indicated in Table 1 to produce the average solid-phase conduction shape factors summarized in Table 4. These factors were then used to re-run volume-averaged calculations for many of the other conditions of solid-phase conductivity and flow. The results are shown in Fig. 12, which when compared to Fig. 9 shows a significant enhancement in accuracy and, in this respect, confirms the viability of the approach for finding the solid-phase conduction shape factor and the importance of including this factor in the volume-averaged computations.
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Table 4 Summary of computed path-resistance factor for the geometric models listed in Table 1 REV #
Porosity
Pore diameter (μm)
Solid-phase conduction shape factor F¯s
1
0.75
400
0.481
2
0.80
400
0.450
3
0.85
400
0.400
4
0.70
800
0.474
5
0.80
800
0.448
6
0.85
800
0.385
8
Q [W] Volume-averaged Computaitons
Fig. 12 Plot comparing total heat transfer computed from results of pore-level computations and volume-averaged computations that now include the conduction shape factor in the solid-phase conductivity. Results are computed on at Redp 5–200 for REV 3
6
4
50 [W/m K] 100 [W/m K] 237 [W/m K]
2
400 [W/m K] 800 [W/m K] 0 0
2
4
6
8
Q [W] Pore-level computations
While differences as high as 20% are evident in Fig. 9, the pore-level and volume-averaged results are similar to better than 3% in Fig. 12. Note that by this approach, all of the geometric resistances (path length, expanding-contracting path) are captured by the average solid-phase conductivity factor F¯s . The shape factors established for the REVs considered in Table 1 are compared in Fig. 13 to those given in Eqs. (25), (26) (and converted to F¯s using Eq. 20) for low- and high-porosity closed-cell SVP porous materials. The figure shows that while the trend of decreasing shape factor with increasing porosity prevails for all models, the shape factors for the interconnected SVP geometries considered herein are considerably lower than those for the closed-cell SVP materials. Though the trend of the present data is similar, no fit line is proposed due to the small range of porosities over which the data are given. In addition, given the difference in pore size between REVs 1–3 and REVs 4–6, it is clear that the shape factor is mainly a function of porosity, similar to that proposed in Eqs. (25), (26) for closed-cells. In addition to the overall heat transfer, the proper modeling of solid-phase conduction in Eq. (17) leads to more accurate results of temperature distribution in the volume-averaged solutions. Figure 16 shows the results of solid-phase temperature as a function of z on the plane shown in Fig. 15 for the pore-level and volume-averaged calculations for REV 1 (with k s 237 W/mK) for the case of Redp = 20. For the pore-level cases, the temperature at a
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Fig. 13 Comparison of solid-phase conduction shape factors from present geometric models to those for closed-cell models in similar porosity range
1.0
Shape Factor
0.8
0.6
0.4
Present Data Maxwell [36]
0.2
Traxl et al. [26] 0.0 0.60
0.70
0.80
0.90
Porosity 320 Pore level Porous connuum
315
Temperature [K]
Fig. 14 Plot showing comparisons between average solid temperatures along z-axis computed by pore-level simulation and porous-continuum model for REV 1 (with k s 237 W/mK) for the case of Redp 20. The porous continuum (uncorrected) is the result is the volume-averaged result computed without the solid-phase conduction shape factor
Porous connuum (uncorrected)
310 305 300 295 0
0.0005
0.001
0.0015
0.002
z [m]
particular height is computed as the line-average at that height. This is compared to the temperature results at the same location obtained from volume-averaged computations. It is clear from the figure that the temperature variation predicted with the solid-phase conduction shape factor (porous continuum corrected) is much closer to the pore-level results. An additional comparison is made of conjugate heat transfer from the pore-level and volume-averaged codes using the constant heat-flux condition at the substrate. As mentioned in Sects. 3 and 4, a constant flux of 925 kW/m2 is imposed to produce approximately the same temperature rise(s) observed in the isothermal-substrate cases compared above. As the heat transfer into the fluid phase is fixed through the boundary condition, it is the temperature distribution that is compared to show the validity of the conduction shape factor for this case. Figure 16 gives comparisons of the solid- and fluid-phase temperatures from the pore-level and volume-averaged calculations on the plane shown in Fig. 15, where the temperatures from the pore-level results are obtained as described above with respect to Fig. 14. Once again, the volume-averaged results obtained from the corrected model shows improved results for both the fluid and solid phases, indicating that predictions of thermal non-equilibrium and the peak temperatures are more accurately predicted when the additional resistances along the conduction path are included.
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Fig. 15 Image showing locations of lines along which average temperature is calculated
314 Pore level
312
Temperature [K]
Porous connuum
310
Porous connuum (uncorrected)
308 306 304 302 300 298 0
0.0005
0.001
0.0015
0.002
z [m]
(a) 366 Pore level
356
Temperature [K]
Fig. 16 Plot comparing solid and fluid-phase temperature profiles for a case of a constant heat flux at the substrate on REV 1 with k s 237 W/mK and Redp 20. The temperature profiles are shown for the position indicated in Fig. 14. The porous continuum (uncorrected) is the volume-averaged result computed without the solid-phase conduction shape factor. a Solid-phase temperature distribution. b Fluid-phase temperature distribution
Porous connuum
346
Porous connuum (uncorrected)
336 326 316 306 296 0
0.0005
0.001
0.0015
0.002
z [m]
(b)
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5 Summary The present study compares the results of pore-level simulations conducted on idealized SVP geometric models with similar results obtained from volume-averaged simulations. For the purpose of this study, the results from the pore-level model were considered to be exact and were used to establish porosity, permeability, Forchheimer coefficients, interstitial exchange and interstitial surface area. These parameters were used as closure coefficients in a volume-averaged model that was then used to conduct similar computations. An approach was then described to establish the average solid-phase conduction shape factor, F¯s , which was shown to have a significant impact on the overall heat transfer and the temperature variation in the porous structure. The solid-phase conduction shape factor is dependent only on the geometric structure of the porous domain and reliably accounts for the additional resistances in the conduction path due to elongation and expansions and contractions along the path. Acknowledgements The authors gratefully acknowledge funding received from the Natural Sciences and Engineering Research Council of Canada.
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