Microstructure based model for permeability predictions of open-cell metallic foams via homogenization

doi:10.1016/j.msea.2007.03.046

Materials Science and Engineering: A

Volume 472, Issues 1–2, 15 January 2008, Pages 214-226

https://doi.org/10.1016/j.msea.2007.03.046 Get rights and content

Abstract

To predict effective permeabilities of open-cell foams, produced by a SlipReactionFoamSintering (SRFS)-process, a multi-scale approach based on the homogenization method is applied. This formulation allows to calculate effective Darcy permeabilities by solving special Stokes flow problems on a unit cell. Starting from a detailed foam description by tomographic images, a new procedure based on a spectral analysis of the foam microstructure is presented in order to derive symmetric and non-symmetric unit cell models of the foam. Effective permeabilities are determined by using these unit cell models for a representative iron-based SlipReaction (SR)-foam sample. The analysis of these predictions leads to define one, rather simple, unit cell model suitable for further stochastic homogenization simulations.

Introduction

Intensive cooling of blades and combustion chambers in combination with new porous material solutions are required to guarantee the performance and acceptable lifetime of such components in contact with hot gas. In order to increase the efficiency of combined cycle power plants, open-cell metallic foams with fine pores, suitable for transpiration cooling, are manufactured and characterized. A promising method to produce such open porous metallic foams is the SlipReactionFoamSintering (SRFS)-process [1]. This process generates a porous cell structure by a chemical reaction in the powder metallurgy process route. It allows producing metallic foams of a wide range of metallic alloys like iron, steel and nickel-based superalloys. Compared to other process routes for metallic foams, the SRFS-method provides several advantages. It allows the foam formation at room temperature, needs lean machinery and results in foams with a great variety of densities, pore sizes and form distributions. After a brief summary of the SRFS-process, the morphology of such SR-foams like pore structure and porosity is characterized in Section 2.

Due to the complexity of the random foam microstructure with small secondary wall pores and large primary pores, Section 3 is focused on the development of realistic 3D micromodels of SR-foams. The elaboration of such a model starts with a 3D computer tomographic imaging of the foam sample. Then, in order to specify the dimensions of the unit cell, called also representative volume element (RVE), a spectral analysis of the microstructure of the SR-foam is applied [2]. This method provides the most characteristic periods and subsequently specifies the “periodicity” of the random foam microstructure. In order to specify a RVE micromodel suitable for further Monte-Carlo simulations, different finite element unit cell models, based on these periods, are built and compared: symmetric and non-symmetric ones, central and shifted ones, full and reduced integrated ones.

For metallic open-cell foams, dedicated to serve as basic material of future combustion chamber walls, the prediction of their effective permeability is essential. For this prediction, a multi-scale approach based on the homogenization method [3], [4] has been adopted here. Since the pioneer work of Sanchez-Palencia [3] in the early eighties, several papers about homogenization technique applied on the heat and mass transport through porous media can be found in the literature [4], [5]. But, to our knowledge, none of them is numerically predicting the permeability of open-cell metallic foams. Otherwise, only semi-analytical approaches with idealized unit cells [6], [7] have been proposed recently in the literature to predict the permeability of open-cell foams.

The homogenization method, outlined in Section 4, allows to calculate effective equivalent thermophysical properties such as thermal conductivities [4], [8] and permeabilities [9] of any heterogeneous material and needs the definition of a periodic unit cell at the microscale. At first, an asymptotic expansion of the velocity and pressure field variables with respect to the ratio between the macro- and microscopic length scale, is introduced in the stationary Navier–Stokes equations of a fluid flow through a porous material. These expansions lead to define special microscopic Stokes flow problems for an incompressible fluid on the unit cell. To solve these problems, the well-known analogy between Stokes flow and incompressible elasticity is used [11]. Effective Darcy permeabilities are finally expressed by averaging the resulting velocity fields over the unit cell.

The numerical analysis in Section 5 is realized for a representative iron-based SR-foam. At first, the influence of some micromodel parameters like the diameter of closed wall pores on the foam periodicity and the position of the unit cell on the foam porosity is investigated. Numerical permeability predictions and the corresponding Stokes flows, obtained by the different unit cell models, then are discussed in detail.

Section snippets

Foam processing

The SRFS-process procedure starts from a metal suspension. Fine metallic powders of up to 200 μm grain size are mixed with a dispersant, solvent and concentrated phosphoric acid as binder (see Fig. 1). The phosphoric acid and the solvent wet the metallic particles and two reactions are taking place: at first, hydrogen evolves by the reaction between the acid and the metallic particles, causing the slurry to foam. At the same time, a phosphate based binder forms and solidifies, freezing the

Tomographic imaging

In order to derive a realistic 3D micromodel for the homogenization procedure (see Section 4), X-ray microtomography capabilities have been used to characterize the 3D microstructure of SR-foam samples. These computer tomographical (CT) measurements have been made at Carl Zeiss using a RayScan 200 Wälischmiller computer tomograph. For the measurement, the sample is placed on a rotary disc between the X-ray tube and a flat panel detector. The absorption capacity of each volume element, called

Asymptotic expansion for flow through a porous material

The open-cell metallic foam is filled with a compressible gas whose density is small and assumed to be constant: ρ^h = ρ₀, where p₀ is the density at the reference temperature T₀ and h a superscript of a field which acts on the heterogeneous material.

Then, following the standard setup of the homogenization technique for laminar Stokes flow in porous media [3], formal asymptotic expansions of the form: $v^{h} (x) = v (x, y)) = ε^{2} v^{0} (x, y) + ε^{3} v^{1} (x, y) + 0 (ε^{4})$ $p^{h} (x) = p (x, y) = p^{0} (x, y) + ε p^{1} (x, y) + 0 (ε^{2})$ with

•
ɛ: ratio between

Numerical analysis of a SR-foam

Before discussing permeability predictions of the representative foam sample, the influence of some micromodel parameters like the critical diameter of closed secondary pores and the position of RVE in the foam sample are more detailed.

Conclusions

The complex microstructure of open-cell metallic foams produced by the SRFS-process has been characterized by defining two porosity classes. This process allows to adjust the density and the pore form of the foams and various alloys can be used as basic material. A new methodology for the definition of realistic 3D micromodels of SR-foams has been developed. This procedure combines tomographic imaging with a spectral analysis of the microstructure, in order to specify the dimensions of a RVE,

Acknowledgments

This work was supported by Deutsche Forschungsgemeinschaft (DFG) within the collaborative research center 561 “Thermally high loaded, porous and cooled multi-layer system for combined cycle power plants”. The authors thank also O. Reutter from DLR (Institute of Technical Thermodynamics, Köln, Germany) for the discussion and the realization of the permeability measurements of the foam samples.

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Microstructure based model for permeability predictions of open-cell metallic foams via homogenization

Abstract

Introduction

Section snippets

Foam processing

Tomographic imaging

Asymptotic expansion for flow through a porous material

Numerical analysis of a SR-foam

Conclusions

Acknowledgments

Int. J. Eng. Sci.

Int. J. Heat Fluid Flow

Acta Mater.

Comput. Meth. Appl. Mech. Eng.

J. Comput. Appl. Math.

Int. J. Solids Struct.

Steel Res. Int.

Adv. Eng. Mater.