Forced convection in micro-channels filled with porous media in local thermal non-equilibrium conditions

Highlights

• Analytic solution on fully developed forced convection in porous micro-channels in LTNE condition.

• Steady state regime for slip flows between two parallel plates with assigned heat flux.

• Heat transfer increases as Bi increases and reaches asymptotic values.

• Different trends are observed with respect to tangential momentum and thermal accommodation coefficients.

• Minimum entropy generation values are attained with respect to the Darcy number.

Abstract

An analytic solution on fully developed forced convection, in parallel plates porous micro-channels is accomplished in Local Thermal Non-Equilibrium (LTNE) condition. The analysis is realized in steady state regime for rarefied gaseous slip flows between two parallel plates with assigned heat flux. The Darcy–Brinkman model is considered in the momentum equation and two energy equations are used to evaluate solid and fluid temperatures. The entropy generation analysis is performed and the total entropy generation number is evaluated as a function of the different dimensionless parameters. The effect of tangential momentum and thermal accommodation coefficients are examined. Results are reported in terms of average Nusselt numbers, dimensionless temperature profiles and total entropy generation number as a function of Biot number (Bi), effective thermal conductive ratio (κ), Darcy number (Da), accommodation coefficients (tangential momentum and thermal) and Knudsen number. Results show that heat transfer increases as Bi increases and reaches asymptotic values. Different trends are observed with respect to tangential momentum accommodation coefficient for assigned thermal accommodation. Total entropy generation number as a function of Da^−0.5 presents minimum values with respect to Bi, accommodation coefficients and Brinkman number.

Introduction

In recent years, research activity in heat transfer at micro and nanoscale geometries has been strongly developing due to the incredible growth of micro-electro-mechanical systems (MEMS). Several engineering and biomedical applications have determined an increasing research interest in micro and nano flow, as recently reviewed in Refs. [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11]. A general introduction on the importance and developments of microfluidics is reported in Ref. [1]. A complete description of the fundamentals in microchannels is provided in Ref. [2]. A bio-MEMS application in terms of nanofluid flow in microchannels is presented in Ref. [3]. A review focused on molecular momentum transport at fluid–solid interfaces mainly related to microfluidics and nanofluidics in micro-/nano-electromechanical systems (MEMS/NEMS) is performed in Ref. [4]. A survey on existing methods for the characterization of mixing and flow in microchannels, micromixers and microreactors is accomplished in Ref. [5]. A review which emphasizes gas–liquid two-phase flow in minichannels or microchannels related to PEM fuel cell applications is reported in Ref. [6]. An analysis of published data on a gas–liquid two-phase flow in capillaries of various shapes to systematize the collected body of information is provided in Ref. [7]. A review on one and two phases flows and methods to improve heat transfer in microchannels is given in Ref. [8]. A comprehensive review on the work done regarding to heat transfer and fluid flow behaviors in microchannels heat exchanger using conventional fluids as well as nanofluids is described in Ref. [9]. The microscale transport processes that arise in the fabrication of advanced materials are reviewed in Ref. [10]. The basic considerations on transport phenomena at micro- or nanoscale levels are outlined. Different approaches found in literature for measuring microflow characteristics and the geometry of the microchannels are explored and categorized in Ref. [11]. It is pointed out the advantages and disadvantages inherent to each experimental technique.

At micro and nanoscales the phenomenon of rarefaction or Knudsen flow have been taken into account and the Navier–Stokes but energy equations in the continuum flow model could not be completely appropriated to describe the fluid flow and heat transfer [12], [13], [14], [15], [16], [17]. A critical review on the status of the understanding of fluid flow phenomena in microdevices is given in Ref. [12]. A new general boundary condition that accounts for the reduced momentum and heat exchange with wall surfaces in a wide range of Knudsen number at low Mach number is proposed and its validity is investigated in Ref. [13]. The importance of different approaches in terms of length scales for macro and micro scales is underlined in Refs. [14], [15]. A review of the available experimental works on the convective heat transfer through microchannels is presented in Ref. [16]. A survey on the main theoretical and experimental results about steady pressure-driven gas microflows is reported in Ref. [17].

The modeling in microflow depends on the Knudsen number, Kn, defined as the ratio of the fluid mean free path to the macroscopic length scale of the physical system, λ/L_ref. It allows to have a measure of the degree of rarefaction of gases which flow through very small channels and, consequently, of the degree of the validity of the continuum model. A classification of gas flow regimes depends on the Knudsen number values and the following expressions are usually accepted, as indicated in Refs. [2], [17], [18], [19]:

• Kn < 10⁻³ for the continuum no-slip flow,

• 10⁻³ < Kn < 10⁻¹ for the continuum slip flow,

• 10⁻¹ < Kn < 10¹ for the transition flow and

• Kn > 10 for the molecular regime.

A greater part of the investigations, both experimental and numerical, have been accomplished on forced convection gas flow in order to evaluate the pressure drop and the convective heat transfer in microchannels and microtubes [2], [5], [6], [9], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29].

Several studies and analysis on heat transfer in porous media are based on local thermal equilibrium (LTE) condition, assuming that fluid phase and solid phase local temperatures are equal and one energy equation model is considered. In many real cases temperature differences between the two phases, fluid and solid, are not negligible and the one equation model is not valid. In such cases, the solid phase and the fluid phase have different local temperatures and it is assumed a local thermal non-equilibrium (LTNE) condition in the porous medium which involves a two energy equations model, one for the fluid phase and one for the solid phase. The temperature difference determines a convective heat transfer between solid and fluid phase inside the porous medium, i.e. an inter-phase heat transfer or interstitial heat transfer coefficient.

Research interest has been recently developing in microchannels and microtubes filled with porous medium due to their applications in micro filtration, fractionation, catalysis and microbiology, as underlined in Ref. [30]. However, forced convection in porous channels and tubes, in rarefied condition or in porous microchannels and microtubes, have not been widely studied. Analytical and/or numerical solutions on different geometrical and thermal conditions should be examined. Moreover, some phenomenological aspects related to LTNE in porous media inside microchannels and microtubes have received only small attention.

One of the first papers on porous microchannel was accomplished by Haddad et al. [31]. Fluid dynamics and thermal behaviors of developing natural convection in a vertical open-ended parallel-plate microchannel, filled with porous media, were numerically investigated considering the slip flow regime. Local thermal non-equilibrium (LTNE) was assumed and the flow in porous medium was modeled employing the Darcy–Brinkman–Forchheimer model which accounts for viscous and inertia effects. An analytical solutions for forced convection with slip-flow in parallel plate channel and circular duct saturated by a rarefied gas was carried out by Kuznetsov and Nield [32]. They considered constant heat flux at the walls and studied the fully developed situation in local thermal equilibrium (LTE). Forced convection slip flow in parallel-plate channel, at uniform wall temperature, filled with porous media was studied numerically by Haddad et al. [33]. The analysis was accomplished in slip flow regime for Kn values from 10⁻³ to 10⁻¹, assuming LTE condition and the Darcy–Brinkman–Forchheimer model for momentum equation. Transient Couette flow, pulsating Poiseuille flow, Stokes second problem flow and transient natural convection flow in a porous media, in the slip flow regime, were investigated by Haddad et al. [34]. In the study the Darcy–Brinkman model, which accounts for viscous effects, and LTE condition were assumed. Results were carried out in the Knudsen number range 10⁻³–10⁻¹. The analysis of the problem investigated in Ref. [33] was extended to LTNE by Haddad et al. [35]. Forced convection slip flow in a circular micro-channel filled with porous media was accomplished by Haddad et al. [36]. Wall uniform temperature, Darcy–Brinkman–Forchheimer model and LTE condition were assumed. The investigation was performed for Kn values between 10⁻³ and 10⁻¹.

Stability analysis on a slip flow in a channel filled with a hyperporous medium, saturated by a rarefied gas, was studied by Avramenko et al. [37]. Numerical analysis for two and three-dimensional disturbances were carried out using the collocation method and the LTE was assumed. Fully developed forced convection in a rectangular microchannel, filled with or without a porous medium in LTE, was investigated analytically by Hooman [38], [39]. The Darcy–Brinkman flow model, slip flow and prescribed local heat flux, with H2 boundary condition [38] or H1 boundary condition [39], were assumed. The analysis was accomplished in the range: 10⁻³ ≤ Kn ≤ 10⁻¹. Forced convection in a parallel plate channel or a circular tube filled by a hyperporous medium saturated by a rarefied gas, with walls held at constant heat flux was investigated by Kuznetsov and Nield [40]. They found analytic solutions for thermally developing flow in LTE condition. Fully developed forced convection in a circular channel filled with a highly porous medium, saturated with a rarefied gas and in the slip-flow regime, was studied by Chauhan and Kumar [41]. Uniform heat flux at tube wall, Darcy–Brinkman–Forchheimer momentum equation and local thermal equilibrium were assumed. The entropy generation due to heat transfer and fluid friction was also evaluated. A numerical simulation of hydrodynamically developed and thermally developing forced convection in a circular micro/nano channel filled with porous media was performed by Shokouhmand et al. [30]. The Darcy–Brinkman–Forchheimer flow model, slip boundary condition and LTE condition were assumed. A unified flow model was considered for all flow regimes in a large Knudsen numbers range. Hydrodynamically and thermally fully developed flow of a dilute gas in forced convection in a porous annular microduct was investigated analytically by Hashemi et al. [42]. Darcy–Brinkman equation and local thermal equilibrium were used. Uniform heat flux at the outer cylinder and insulated inner cylinder and vice versa were considered as thermal boundary conditions.

The main contributions on the microchannels filled with porous media are summarized in Table 1. Except in Refs. [35], it seems that there are no studies on microchannels or channels in rarefied gas filled with the assumption of LTNE on porous media, mainly for assigned heat flux at the microchannel or channel walls.

As indicated by Hashemi et al. [42], several investigations on microchannel heat sinks analyze these devices by means of porous media models in LTNE [43], [44], [45], [46], [47], [48], [49], [50]. The no-slip condition is, however, assumed in all studied cases. Moreover, very few works in microchannels filled with porous media have been accomplished with the entropy generation analysis [41] which was carried out in LTE.

In the present study a hydrodynamically and thermally fully developed flow of a dilute gas in a parallel-plate microchannel, assuming the slip-flow and jump temperature on the microchannel walls, filled with a porous medium, is carried out adopting the LTNE condition on the porous medium. An analytical solution for uniform wall heat flux and Darcy–Brinkman model is evaluated and the entropy generation analysis is performed. Results, for Knudsen number, provided that 10⁻³ ≤ Kn_p ≤ 10⁻¹ (Kn_p = λ/K^1/2), in terms of dimensionless wall temperature profiles, Nusselt number, dimensionless fluid and solid phases and entropy generation number are given for various parameters values such as Darcy number, Biot number and thermal conductivity ratio. It should be underlined that the Biot number is referred to the convective heat transfer between the solid matrix and fluid at solid–fluid interface in the porous medium i.e. the interstitial heat transfer coefficient [35]. Moreover, the interstitial heat transfer coefficient is an assigned value.