Review: heat and mass transfer in porous medium, mathematic/numerical models and research directions

doi:10.15406/ipcse.2018.03.00082

International Journal of

eISSN: 2475-5559

Petrochemical Science & Engineering

Review Article Volume 3 Issue 3

Review: heat and mass transfer in porous medium, mathematic/numerical models and research directions

Junfeng Lu,¹

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Wen qiang Lu²

¹Key Laboratory of Cryogenics, Technical Institute of Physics and Chemistry of CAS, China
²School of Engineering Sciences, University of Chinese Academy of Sciences, China

Correspondence: Junfeng Lu, Key Laboratory of Cryogenics, Technical Institute of Physics and Chemistry of CAS, China

Received: May 31, 2018 | Published: June 12, 2018

Citation: Junfeng LU, Wen-qiang LU. Review: heat and mass transfer in porous medium, mathematic/numerical models and research directions. Int J Petrochem Sci Eng. 2018;3(3):97-100. DOI: 10.15406/ipcse.2018.03.00082

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Abstract

This paper is a mini-review for research studies (range from year 2000 to 2018) on porous medium in China. It includes basic mathematic and numerical models for heat and mass transfer through porous medium. And then, several advanced contemporary research topics on porous medium in China are outlined.

Keywords: porous medium, heat/mass transfer in porous medium, adsorption, filtration, combustion.

Introduction

As key functional material, porous medium was used in a wide range of applications due to its good thermal performance and semi-permeable properties. These applications involve aerospace,¹ semi-conductors^2–4, bio-sensors,^5–8 environmental energy saving,^9–13 medicine,^14,15 infrastructure construction,^16–18 fuel cell,^19,20 petrochemicals^21,22 and so on. Thermal research studies in the above areas were generally concerned on separation/purification process, heat preservation, chemical catalysis, medical engineering, etc.

Basic mathematic models for porous medium

In the analysis of mass transfer process for porous medium, the mathematic models normally used could be traced back to Darcy’s law (1856).²³ The law was formulated by Henry Darcy based on the results of experiments on the flow of water through beds of sand. Its application to form numerical calculation in porous medium is even populous in today’s research. After decades of development, Darcy’s law was gradually improved, for example, Forchheimer’s amendment²⁴ on inertia force, Brinkman’s amendment²⁵ on viscous force, and Knudsen diffusion or Klinkenberg effect,²⁶ etc. Those were mathematic descriptions on flow inside porous medium. Another sort of models directly describes trans-porous medium (or usually called trans-membrane) flow accompanied with membrane solute filtration. More comprehensive mathematic presentation for the filtration process was accomplished by Kedem-Katchalsky (K-K)’s equations.^27,28 Table 1 listed all these mathematic models for porous medium

Contemporary numerical studies for porous medium always inside a complicated system. In most research works, the porous material was majorly considered as uniform. Under such assumption, Navier-Stokes equations were introduced to describe the flow and thermal properties inside a porous medium. In a relatively comprehensive system, by combining Darcy-Forchheimer law, Jiang and Lu’s work²⁹ displayed how to use turbulence model to describe the flow and heat transfer inside sintered porous plate channels. In China, their work is representative for the followers and became a mile-stone for the studies inside porous medium.

Our research work on porous plate channels started from hemodialysis process.³⁰ It tells a story of both ultra filtration on sideway channel wall and solute distribution inside channels. Multiple-scale calculations were applied in such a system: the flow inside the straight channel was calculated by Simple R algorithm, meanwhile, the ultra filtration flow profile was determined by K-K equations.

Since this paper is a required two page mini-review, in the following, I will tip out most recent advanced research topics of porous medium only in the following two areas:

Adsorption /filtration
Combustion with porous medium

Darcy’s law	$q = - \frac{k}{μ} \nabla p$ $q = - \frac{k}{μ} \nabla p$	Where q is the flux, k is the cross sectional area, u is viscosity, and $\nabla p$ $\nabla p$ is the pressure gradient.
Darcy-Forchheimer law	$\frac{\partial p}{\partial x} = - \frac{μ}{k} q - \frac{ρ}{k_{1}} q^{2}$ $\frac{\partial p}{\partial x} = - \frac{μ}{k} q - \frac{ρ}{k_{1}} q^{2}$ ;	Where the additional term is known as inertial permeability.
Brinkman form of Darcy’s law	$- β \nabla^{2} q + q = - \frac{k}{μ} \nabla p$ $- β \nabla^{2} q + q = - \frac{k}{μ} \nabla p$	Where is an effective viscosity term.
Darcy’s law for gases in fine medium	$q = - \frac{k^{e f f}}{μ} \nabla p$ $q = - \frac{k^{e f f}}{μ} \nabla p$ $k^{e f f} = k (1 + \frac{b}{p})$ $k^{e f f} = k (1 + \frac{b}{p})$	Where b is known as the Klinkenberg parameter.
K-K equations for trans-membrane flow	< $J_{v} = L_{p} Δ p - σ L_{p} R T Δ C$ $J_{v} = L_{p} Δ p - σ L_{p} R T Δ C$ $J_{s} = (1 - σ) C_{m} J_{v} + P_{s} Δ C$ $J_{s} = (1 - σ) C_{m} J_{v} + P_{s} Δ C$	Where $L_{p}$ $L_{p}$ is hydraulic permeability, $Δ p$ $Δ p$ is trans-membrane pressure, $Δ C$ $Δ C$ is solute concentration difference, $C_{m}$ $C_{m}$ is average solute concentration inside the porous membrane, $J_{v}$ $J_{v}$ is ultra-filtration velocity, $J_{s}$ $J_{s}$ is solute flux, σ is solute reflection coefficient, as well $P_{s}$ $P_{s}$

Advanced studies

Adsorption/filtration

Surface adsorption occurs in many gas/fluid separation processes (such as nitrogen/oxygen separation, pressure-swing absorption process). The research in this field covers from single gas adsorption isotherms (Langmuir, BET, DR isotherms, etc.), real gas equations of state (mBWR, R-K, Viral, Van der Waals models, etc.), to predicting the equilibria of mixed-gas adsorption (Myers’ IAST ~RAST MIAST models.^30–32

Filtration process typically uses semi-permeable membrane or porous column; and these processes are divided into four categories:

Micro-filtration (filtrated particle size ranges from 1 micron to several hundred microns).
Ultra-filtration (submicron filtration process);
Nano-filtration (particle size usually located in several nano-meters).
Reverse osmosis (only water molecules can be filtrated).

K-K equations becomes dominant mathematic model to analyze filtration process.^33,34 Also, MD,³⁵ SPH³⁶ methods were adopted to calculate the phantoms inside nano/micro pores.

Most recent studies in Chinese government projects also include multiple phases during separation process (such as liquid nitrogen separation, helium separation in cryogenic temperature; and oil or sea water distillation using porous column). In this studies,^37-45 CSF/VOF models were adopted to trace phase surface change and SPH or CFD methods were used to calculate pure single phase flows.

A research inclination arriving recently in our lab is magnetic adsorption. In this initiative research field, adsorbents are magnified and controlled by gradient magnetic field.^46,47 To study the pure physical phenomenon of this peculiar adsorption process, DEM/DPM model is assumed to be adopted. Since the motion of the particles are controllable and go against the carrying solution, turbulence LES model is also concerned because of accumulation of some magnetic particles. Even the numerical study has not yet been put into run-way, the study is valuable not only because it could be industrially applied, but because it is much more significant in that it is an initiative pure physics problem (controllable opposite direction two-phase flows inside a porous adsorption system) that ancestors never touched.

Combustion with porous medium

Typically, most combustion research studies were related to gas engines (such as aerospace engine and vehicle engine). These studies usually use LES turbulence model to analyze the power efficiency produced by combustion. Meanwhile, chemical reactions are always introduced to CFD simulations by equations of reaction rates. Also, porous film cooling was studied inside aerospace engines.^1,48–50 These studies were majorly focused on gaseous porous film cooling.^51–55

Combustion inside porous medium is an innovative topic in combustion research. This new technology is superior to conventional combustion in that:

Due to the huge amount of tiny reaction spaces, the combustion is evenly fractionized, such that its thermal efficiency is very high.
its combustion discharge is very little
It does not require large combustion facilities. The key issue inside porous medium combustion is premixed combustion.^56–58

As typical example of combustion inside porous medium, sintering process played an important role in most of Chinese steel industry. And today, accompanied with high-speed economic development in China, it becomes a big pollution contributor due to its high energy consumption and low residual heat recovery. Flue gas recirculation sintering (FGRS) technology can reduce flue gas emissions and reuse waste heat effectively in iron ore sintering. Five FGRS systems have been built or transformed in China since 2013.⁵⁹ Chinese combustion researchers also get their directions from this point. Iron ore sintering process is a combustion process inside porous medium; it covers chemical reactions, heat radiation, convection, as well as conduction thermal processes. Due to the high-energy gases consumed and produced inside the pores, the research in this field becomes very complicated.

Conclusion & future aspects

This paper reviewed recent (after year 2000) research studies as well as projects deployed in the area of porous medium in China. The majorly used mathematic models as well as numerical calculation methods for porous medium were also browsed in this paper. Accompanied with parallel computing hardware development (such as GPU computing and quantum computing hardware), in my personal opinion, I would suggest that the future research studies should be more focused on heat & mass transfer details to disclose more and more real story inside porous medium:

Dynamic porous medium combustion

Since the porous material could also participate in the process of combustion,⁵⁹ its size shrinks while the hole’ space increases. The research studies herein are really interesting because the reaction products sometimes are not gases but solids which will cause the reaction speed slow-down. Moreover, the gases that generated inside porous medium could also stall the chemical reaction in a way that suffocates the reaction. So, how to promote the combustion efficiency becomes the key point of these studies.
The heat transfer inside the porous medium during combustion is also complicated. There includes conduction, convection as well as radiation thermal processes. However, the boundary conditions could not be assumed as constant or insulated since the combustion is dynamic according to vigorous chemical reactions inside different porous areas; such that even the combustion flows are premixed, the combustion in entrance area could be far different from that in outlet area.

Dynamic porous medium adsorption/filtration

As an example for adsorption, our lab already started research on magnetic adsorption as mentioned in this paper. It is a far more interesting pure physics study that it works on a moving “magnetic porous medium (composed by dispersed particles)” inside a flow field. Since the particles are sometimes accumulated in different size according to magnetic force, the flow field is turbulent. Under this condition, the adsorption process is also dynamic. The adsorption potential on solid surface is obviously influenced by turbulent vicious force. Needless to say, this study will also exploit the studies of multiphase turbulence flow.
For filtration process, the studies of molecules passing through porous medium will be more realistic. For example, as fundamental construction molecules of protein, the exchange of amino-acids through cell membranes or polymer membrane will be highly concerned in the study of hemodialysis process.⁶⁰