Numerical investigation on the thermal non-equilibrium in low-velocity reacting flow within porous media

https://doi.org/10.1016/j.ijheatmasstransfer.2014.05.043Get rights and content

Abstract

This paper addresses the thermal non-equilibrium problem for a low-velocity reacting flow within isotropic porous media. The method of volume-averaging is employed to derive the macroscopic thermal transport equation for the fluid phase inside the porous medium including a heat source due to a homogenous chemical reaction, which is then closed by representing the temperature deviation through a constitutive equation. Theoretical treatment indicates that the contribution of the non-equilibrium due to reaction heat to the macroscopic heat transfer comes up in terms of an additional energy source rather than affecting thermal transport properties. Through dimensional analysis, this term can be interpreted as a convective transport of the reaction heat. Numerical computations are conducted by solving the closure problems at the pore scale for the fluid phase in a spatially periodic representative elementary volume (REV). Simulation results show that the convective coefficient related with the reaction heat is dependent on the Thiele modulus number, and the arrangement of cylinders. For the inline case, when the Péclet number is less than 10, a part of reaction heat will feed back to the upstream due to local conductivity, for Péclet number greater than 10 this part of energy will be transmitted to the downstream. For the staggered case, the same conclusion holds for smaller Péclet number, while the non-equilibrium will gradually decay to zero as the Péclet number increases. In addition, the other effective coefficients, such as effective conductivity, surface convective heat transfer coefficient are calculated with and without inertia.

Introduction

According to the difference of the superficial averaging temperatures between fluid and solid phases, models for heat transfer occurring in porous media can be categorized into local thermal equilibrium and non-equilibrium models. Strictly speaking, when the difference among the local temperature of the phases is comparable in magnitude to the temperature difference across the length scale the local thermal non-equilibrium occurs [1]. For a heat exchanger, since there is no heat generation, so generally, it is treated as the local thermal equilibrium in the case that far from the system boundaries. However, as stated in Kaviany’s monograph [2], when a significant heat generation is of existence, the local thermal equilibrium will break down, and double separated phase averaging equations for solid and fluid will be necessary for the description of the heat transfer process. Moreover, Quintard and Whitaker [3] provided a number of physical situations that the local thermal equilibrium could be failed in their researches.

The thermal non-equilibrium can be characterized by the heat transfer between the solid and fluid phases, which is closely related to the Darcy velocity and geometrical characteristic of the matrix as well as thermal physical properties, such as specific heat capacity, thermal conductivity, and so on. For the creeping flow or low Reynolds number laminar flow, the thermal conduction is the most important factor affecting heat transfer, so the total thermal conductivity comprising of thermal dispersion conductivity and stagnant effective conductivity is sufficient to describe the process. With the Darcy velocity increasing, the convection will gradually replace thermal conduction, and become a dominant element. Simultaneously, the thermal dispersion due to hydraulic mixing overcomes molecular diffusion and plays a significant role on the enhancement of heat transfer.

Dispersion phenomena including two aspects of heat and mass have been extensively studied both theoretically and experimentally. Hsu and Cheng [4] proposed that axial thermal dispersion conductivity is linearly proportional to the Péclet number in the range of high Reynolds numbers, while it has a quadratic dependence on the velocity for low Reynolds numbers. Kuwahara and Nakayama [5] considered the effect of porosity and proposed a set of empirical correlations, which also have the same dependence on the Péclet number as Hsu and Cheng, though some distinction existing in the expressions for the longitudinal dispersion within different range of the Péclet number. However, for the transversal dispersion, if the Péclet number based on the diameter is less than 10, Nakayama’s result also shows a quadratic dependence on the Péclet number. Pedras and de Lemos [6] investigated thermal dispersion and showed that thermal dispersion is a function of solid-to-fluid conductivity ratio. For the longitudinal dispersion, it decreases with the ratio increasing in an exponential relationship, Pem, where Pe is based on the length of unit cell, m equals 1.65; while for the transversal dispersion, m takes values of 0.94 and 0.88 for ks/kf = 10 and 2, respectively. Moyne et al. [7] employed a new concept, namely average temperature of the medium by replacing the condition that the mean temperature deviations is equal to zero, getting a one-equation model for describing the thermal dispersion process inside porous media. In addition, Nakayama et al. [8] developed an equation for thermal dispersion flux with the volume-averaging method which provides rigid constraints that the gradient diffusion hypothesis can be satisfied. A series of empirical correlations were established by Delgado [9] based on more than five hundred experimental data and through an exhaustive compilation and a critical analysis of the mass dispersion data, which are measured in packed beds of mono-sized particles and constant voidage, with both air and water as the fluid phase. For the case of liquid flows, the author used a division into five and four dispersion regimes to obtain expressions for longitudinal and transversal dispersion, respectively. In the case of gas, due to its Schmidt number is close to unity, the author quoted the results of previous studies. It is worth to note that the tortuosity was treated as an important parameter affecting mass dispersion in his study. Furthermore, Wood [10] paid a particular attention on the influence of inertial contribution on the mass dispersion at moderated Reynolds numbers. Their results show that the inertial effect on the longitudinal dispersion coefficient is relatively small, but for the transversal dispersion coefficient, the inertial effect can increase it up to 40 times as big as that predicted by Stokes flow. More elaborate reviews on dispersion can be found in Refs. [9], [11].

On convective heat transfer in porous media, many studies have been conducted in the last five decades. Most of the obtained empirical correlations are in the form of RemPrn, for instance the result by Wakao and Kaguei [12]. Kuwahara et al. [13] resorted to the numerical method to determine the convective heat transfer coefficient of porous media with periodic structure. In their correlation, the porosity was introduced as an independent parameter. Later on, in Saito and de Lemos’s [14] research, the influence of porosity on the convective heat transfer was confirmed again. Izadpanah et al. [15] experimentally investigated the convective heat transfer in a cylindrical porous medium with the conclusion that the natural convection could be neglected with respect to the forced convection and it is related to the flow velocity rather than to the thermal flux. Alshare et al. [16] considered the influence of flow angles on convective heat transfer for the square rod arrangement. Contrast to the in-line configuration, it is more closely dependent on the Reynolds number for the case of staggered one. Recently, Aguilar Madera et al. [17] investigated the heat transfer at the fluid-porous medium boundary and a one-domain approach was utilized in their work. More reviews on this issue can be found in Refs. [18], [19] and will not be presented here due to the space limitation.

Previous studies have discussed the heat transfer either in terms of thermal equilibrium or not, yet in which no effect of chemical reaction has been taken into account, so the question how the reaction would affect the heat transfer mechanism is not very clear. It is noted that so far there has been few researches on the effect of heat generation within the fluid phase on the thermal non-equilibrium including the dispersion, convective heat transfer, and other related phenomena. Recently, Valdés-Parada et al. [20], [21] studied the effects of homogeneous and heterogeneous chemical reactions on the mass dispersion in porous media and showed that the effective diffusivity is an increasing function of the Thiele modulus. Based on the relation between mass and heat transfer, it is almost assured that chemical reaction would denote somewhat to the heat transfer.

To this end, our attention is mainly focused on the effect of chemical reaction on the heat transfer. The method of volume-averaging is employed to derive the macroscopic thermal transport equation for the fluid phase inside the porous medium including a heat source due to a homogenous chemical reaction, which is then closed by representing the temperature deviation through a constitutive equation. Many macroscopic transport properties such as effective thermal conductivity, surface convective heat transfer coefficient and convective velocity are obtained conveniently through the solution of pore-scale closure problems. A representative, periodic unit cells is employed to avoid time consuming of the calculation for a whole domain. Based on the simulation results, effects of conductivity ratio, inertia and reaction rate on these coefficients are discussed respectively. Therefore, the current study may be seen as an extension of the work by Quintard et al. [22], while no chemical reaction was involved in their study.

Section snippets

The method of local volume-averaging

Up-scaling technique is a typical approach for treating the transport processes in porous media, which includes the methods of moment, the local volume averaging, the homogenization, the ensemble averaging, the thermodynamically constrained averaging theory (TCAT) and so on. In this work, the local volume averaging method is selected for the development of the closure equation. To elaborate the process, some necessary relations are given in the following.

Definition of the local volume averaging

Results and discussion

As known, the value of 0.38 for the porosity of a packing bed, is widely employed in the literature, which is adopted here for the aim of comparison. Two cases of the porous medium geometry, i.e., the in-line and staggered cylinders arrangements, are considered here, respectively. The flows enter the computational zones along the horizontal direction in both cases (see Fig. 2).

The two-dimensional computational domains are discretized into triangular and quadrilateral elements by the COMSOL

Conclusions

In this work, the thermal non-equilibrium problem for the low-velocity reacting flow within isotropic porous media is numerically studied. The volume-averaging method is employed to derive the macroscopic thermal transport equation for the fluid phase inside the porous medium including a heat source due to a homogenous chemical reaction. The equation is then closed by representing the temperature deviation through a constitutive equation, and some expressions for effective coefficients are

Conflict of interest

None declared.

Acknowledgments

The authors are thankful to Francisco Jose Valdés Parada for the utilization of the COMSOL Multiphysics software, with whose help this work could be completed smoothly. The support by the National Natural Science Foundation of China (NSFC Grant No. 51176021) and Petro China Innovation Foundation (2013D-5006-0208) is acknowledged.

References (27)

  • C.-L. Tien et al.

    Convective and radiative heat transfer in porous media

    Adv. Appl. Mech.

    (1989)
  • F. Valdés-Parada et al.

    On diffusion, dispersion and reaction in porous media

    Chem. Eng. Sci.

    (2011)
  • F.J. Valdes-Parada et al.

    On the effective diffusivity under chemical reaction in porous media

    Chem. Eng. Sci.

    (2010)
  • Cited by (5)

    • Pore-level numerical simulation of methane-air combustion in a simplified two-layer porous burner

      2021, Chinese Journal of Chemical Engineering
      Citation Excerpt :

      The obvious thermal non-equilibrium for the intra-phase and heterogeneous phase were observed except for the inlet and outlet regions of the burner. A two-scale method was proposed by Chen et al. [20] to study the dispersion effect and turbulent premixed flame characteristics in a porous burner at the pore level and system level. They analyzed the thermal non-equilibrium of low-velocity reaction flows in isotropic porous media.

    • Heat and mass transfer in Fischer–Tropsch catalytic granule with localized cobalt microparticles

      2018, International Journal of Heat and Mass Transfer
      Citation Excerpt :

      The pore size is given by the normalized Rayleigh distribution [8]. To simulate heterogeneous chemical reactions between the liquid and solid phases inside the granule, numerical methods are developed [9]. In all the models listed above, a homogeneous approximation is realized, that is, a chemical reaction occurs at any point of the active surface.

    • Pore-scale simulation of vortex characteristics in randomly packed beds using LES/RANS models

      2018, Chemical Engineering Science
      Citation Excerpt :

      Due to the stochastic structure, high pressure gradient appears in some regions, so the pressure was calculated by the PRESTO! algorithm (Chen and Xie, 2014), and for the momentum, turbulent kinetic energy and dissipation rate the second order upwind scheme was used, and for the time-marching second order implicit scheme was used (Inigo and Simone, 2017), it was verified by a numerical test that for the judgment criterion of 10−6 (Pakrouh and Hosseini, 2017), the error of the calculation results could be negligible. In order to obtain accurate results, before a local mesh refinement of the packed bed is carried out, a grid independence was checked first.

    • Analytical prediction of coal spontaneous combustion tendency: Velocity range with high possibility of self-ignition

      2017, Fuel Processing Technology
      Citation Excerpt :

      The rate of reaction obeying an Arrhenius temperature dependence [34]. There is no volume change due to reactions or heating [16] and the porosity is constant [41]. Air is forced into coal bulk from one end, and flows through it with a constant rate [7,8,42].

    • Convection in porous media

      2017, Convection in Porous Media
    View full text