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- About
- Editorial board
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**Research article**
06 Apr 2018

**Research article** | 06 Apr 2018

Experimental study of forced convection heat transport in porous media

^{1}DICATECh, Department of Civil, Environmental, Building Engineering, and Chemistry, Politecnico di Bari, Bari, Italy^{2}Department of Physics & Earth Sciences, University of Ferrara, via Saragat 1, 44122 Ferrara, Italy^{3}School of Civil Engineering, University of Queensland, St Lucia, Brisbane 4072, Australia^{4}New Energies And environment Company (NEA), via Saragat, 1, 44122 Ferrara, Italy

^{1}DICATECh, Department of Civil, Environmental, Building Engineering, and Chemistry, Politecnico di Bari, Bari, Italy^{2}Department of Physics & Earth Sciences, University of Ferrara, via Saragat 1, 44122 Ferrara, Italy^{3}School of Civil Engineering, University of Queensland, St Lucia, Brisbane 4072, Australia^{4}New Energies And environment Company (NEA), via Saragat, 1, 44122 Ferrara, Italy

Abstract

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The present study is aimed at extending this thematic issue through heat transport experiments and their interpretation at laboratory scale. An experimental study to evaluate the dynamics of forced convection heat transfer in a thermally isolated column filled with porous medium has been carried out. The behavior of two porous media with different grain sizes and specific surfaces has been observed. The experimental data have been compared with an analytical solution for one-dimensional heat transport for local nonthermal equilibrium condition. The interpretation of the experimental data shows that the heterogeneity of the porous medium affects heat transport dynamics, causing a channeling effect which has consequences on thermal dispersion phenomena and heat transfer between fluid and solid phases, limiting the capacity to store or dissipate heat in the porous medium.

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How to cite.

Pastore, N., Cherubini, C., Rapti, D., and Giasi, C. I.: Experimental study of forced convection heat transport in porous media, Nonlin. Processes Geophys., 25, 279-290, https://doi.org/10.5194/npg-25-279-2018, 2018.

1 Introduction

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The European Climate and Energy Framework for 2050 aims to shift from the massive use of fossil-fuel sources to others characterized by very low emissions. Among the renewable sources, geothermal energy is the only one which is available basically everywhere and at any time.

For this reason, in recent years the use of groundwater as a low-enthalpy geothermal resource for heating and cooling of buildings and for agricultural and industrial processes has been growing.

One of the main limitations for the development of low-enthalpy geothermal systems concerns the high cost of investment. Installation of geothermal energy systems requires high upfront capital investments that often exceed the expectations of depreciation expense, so the investment is therefore inconvenient and the economic benefits can only occur after a long time. It is therefore of extreme importance to further the understanding of the behavior of hydrological systems, particularly concerning heat transport. Studying heat transfer phenomena takes advantage of the fact that the governing partial differential equations used to describe flow and transport processes in porous media are based on the same form of mass and/or energy conservation laws.

Several studies have been already carried out in this context with the aim of enhancing heat transfer phenomena in porous media for engineering processes. Theoretical and numerical research on convection heat transfer in porous media has used two different models for the energy equation: the local thermal equilibrium (LTE) model and the local thermal nonequilibrium model.

Most of the studies have been focused on investigating the validity of the local thermal equilibrium assumption between the solid and fluid phase, the influence of nonlinear flow patterns, and the existing relationship between thermal dispersion and flow velocity.

Koh and Colony (1974) carried out an analytical investigation of the performance of a heat exchanger containing a conductive porous medium using the Darcy flow model, while Koh and Stevens (1975) performed an experimental study of the same problem. They have shown that for a constant heat flux boundary condition the wall temperature is significantly decreased by using a porous material in the channel.

Vafai and Tien (1981) have formulated a general mathematical model that takes into consideration the boundary and inertial (non-Darcian) effects on flow and heat transfer in porous media. In analyzing these effects, they considered three flow resistances: the bulk damping resistance due to the porous structure, the viscous resistance due to the boundary, and the resistance due to the inertial forces.

Later, Vafai and Tien (1982) performed a numerical and experimental investigation of the effects of the presence of a solid boundary and inertial forces on mass transfer in porous media.

Kaviany (1985) studied laminar flow through a porous channel bounded by two parallel plates maintained at a constant and equal temperature by applying a modified Darcy model for transport of momentum.

Vafai and Kim (1989) considered fully developed forced convection in a porous channel bounded by parallel plates by applying a Brinkman–Forchheimer extended Darcy model to obtain a closed-form analytical solution.

Lauriat and Vafai (1991) presented a comprehensive review on flow and heat transfer through porous media for two basic geometries: flow over a flat plate embedded in a porous medium and flow through a channel filled with a porous medium.

Hadim (1994) carried out a numerical study to analyze steady laminar forced convection in a (1) fully porous and (2) partially porous channel filled with a fluid-saturated porous medium and containing discrete heat sources on the bottom wall.

He modeled the flow in the porous medium using the Brinkman–Forchheimer extended Darcy model.

Kamiuto and Saitoh (1994) examined theoretically the effects of several system parameters on the heat transfer characteristics of fully developed forced convection flow in a cylindrical packed bed with constant wall temperatures. They developed a two-dimensional model incorporating the effects of non-Darcian variable porosity and radial thermal dispersion.

Hwang et al. (1995) performed a study of non-Darcian forced convection in an asymmetric heating sintered porous channel to investigate the feasibility of using this channel as a heat sink. The study showed that the particle's Reynolds number significantly affected the solid-to-fluid heat transfer coefficients.

A review of the literature indicates that the local thermal equilibrium assumption between the solid and fluid phase is used in the majority of heat transfer applications involving porous media. Minkowycz et al. (1999) proposed a modified energy equation that can be solved for very early departures from LTE conditions. Their results confirmed that local thermal equilibrium in a fluidized bed depends on the size of the layer, mean pore size, interstitial heat transfer coefficient, and thermophysical properties. They concluded that for a porous medium subject to rapid transient heating, the existence of the local thermal equilibrium depends on the magnitude of a dimensionless quantity (which they called the Sparrow number) containing the contributions of the flow in porous media, interstitial heat transfer, and general thermal conduction.

An in-depth analysis of nonthermal equilibrium is provided by Amiri and Vafai (1994, 1998).

Amiri and Vafai (1994) carried out a steady-state analysis of incompressible flow through a bed of uniform solid sphere particles packed randomly. The investigation was aimed at exploring the influence of a variety of phenomena such as the inertial effects, boundary effects, and the effect of the porosity variation model together with the thermal dispersion effect on the momentum and energy transport in a confined porous bed. They also proved the validity of the LTE assumption and the two-dimensionality effects on transport processes in porous media.

In a subsequent study, Amiri and Vafai (1998) realized a rigorous and flexible model to explore the heat transfer aspects in a packed bed made of randomly oriented spherical particles. Along with the generalized momentum equation they used a two-energy equation model to describe the thermal response of a packed bed. They explored the temporal impact of the non- Darcian terms and the thermal dispersion effects on energy transport. In addition, they investigated on the LTE condition and the one-dimensional approach under transient conditions by formulating dimensionless variables that serve as instruments in depicting the pertinent characteristics of energy transport in a packed bed.

Khalil et al. (2000) performed a numerical investigation of forced convection heat transfer through a packed pipe heated at the surface under constant heat flux, showing the effects of a particle's Reynolds number, pipe-to-particle diameter ratios, and the Prandtl number. They showed that the average Nusselt number increases with both the particle's Reynolds number and the Prandtl number. They concluded that packing pipes with a porous medium can provide heat transfer enhancement for the same pumping power.

Wu and Hwang (1998) investigated, experimentally and theoretically, flow and heat transfer dynamics inside an artificial porous matrix by using a modified version of the local thermal nonequilibrium (LTNE) model, which neglected the effects of thermal dispersion in both fluid and solid. The results showed a highly non-Fourier behavior that combined rapid thermal breakthrough with extremely long-tailing, which was attributed to disequilibrium between the fluid and the porous matrix. However, the adopted model was unable to fully capture the thermal breakthrough observed in some experimental runs. They concluded that heat transfer coefficient increases with the decrease in porosity and the increase in the particle's Reynolds number.

Emmanuel and Berkowitz (2007) were able to successfully fit the thermal breakthrough curves obtained by Wu and Hwang (1998) by applying the continuous time random walk (CTRW), which provided an alternative description of heat transport in porous media. They argued that larger scale spatial heterogeneities in porous media present obstacles to both the equilibrium and the LTNE models and that CTRW would be particularly applicable to the quantification of heat transfer in naturally heterogeneous geological systems, such as soils and geothermal reservoirs.

Geological media are typically characterized by heterogeneities on many scales, resulting in a wide range of fluid velocities, porosities, and effective thermal conductivities.

Despite the uncertainty and contradiction in defining the thermal dispersion, several studies addressed the effects of thermal dispersion in porous media, and different approaches have been developed to describe it (Hsu and Cheng, 1990; Anderson, 2005; Molina-Giraldo et al., 2011).

Thermal dispersion is generally defined as a function of fluid velocity and grain size (Lu et al., 2009; Sauty et al., 1982; Nield and Bejan, 2006).

According to Sauty et al. (1982) and Molina-Giraldo et al. (2011), the thermal dispersion is a linear function of flow velocity and relates to the anisotropy of the velocity field whereas Rau et al. (2012) proposed a dispersion model as a function of the square of the thermal front velocity.

The literature also contains conflicting theories about the magnitude of thermal dispersivity. Smith and Chapman (1983) state that it has the same order of magnitude as solute dispersivities, while Ingebritsen and Sanford (1999) neglect it. According to Vandenbohede et al. (2009) thermal dispersivities are small in comparison to solute dispersivities and are less scale dependent.

Mori et al. (2005) showed experimentally that, for water fluxes ranging
between 0.6 × 10^{−6} and 0.3 × 10^{−3} (m s^{−1})
thermal dispersion was nearly independent of water flow and its effects were
insignificant.

According to Rau et al. (2012), the effect of thermal dispersion on heat transport is significant for high values of thermal Peclet number. Metzger et al. (2004) also introduced a dispersion model based on the thermal Peclet number.

Koch et al. (1989) obtained an analytical expression for the dispersion tensor for a regular arrangement of cylinders or spheres. They found that for high values of Peclet numbers, the ratio of longitudinal total thermal diffusivity to the fluid thermal diffusivity was proportional to the square of the Peclet number while maintaining the transverse dispersion constant. The analytic finding was in good concordance with the experimental measurements of Gunn and Pryce (1969).

Eidsath et al. (1983) quantified the longitudinal thermal dispersion and
stressed that the streamwise ratio of longitudinal total thermal diffusivity
to the fluid thermal diffusivity was proportional to *Pe*^{1.7}.

Ait Saada et al. (2006) investigated the behavior of microscopic inertia and thermal dispersion in a porous medium with a periodic structure by using a local approach at the pore scale to evaluate the velocity and temperature fields as well as their intrinsic velocity and temperature fluctuations in a typical unit cell of the porous medium under study. They concluded that nonlinear effects characterizing microscopic inertia might be the definitive cause of thermal dispersion depending on the nature of the porous medium and in certain situations can exceed 50 % toward the contribution of thermal dispersion. Particularly for a highly conducting fluid moving with high Peclet numbers, microscopic inertial effects were shown to take a great part in the heat transfer duty. They concluded that a considerable interaction between the velocity and thermal fields exists.

This work is aimed at studying the dynamics of forced convection heat transport in porous media, allowing the understanding of how the grain size and the specific surface affect heat transport in terms of macrodispersion phenomena, and heat transfer between solid and fluid phases, and heat storage properties. In particular, the present study involves the experimental investigation of heat transport through a thermally isolated column filled with a porous medium. Several heat tracer tests have been carried out using porous media with different grain sizes. The experimental observed breakthrough curves have been compared with the one-dimensional analytical solution for the forced convection heat transport under local thermal nonequilibrium conditions. The results highlight the effects of grain size and the specific surface on forced convection heat transport dynamics in porous media.

2 Theoretical background

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In several studies examining the flow dynamics through porous media it is
assumed that flow is described by Darcy's law, which expresses a linear
relationship between pressure gradient and flow rate. Darcy's law has been
demonstrated to be valid at low flow regimes (*Re* < 1),
whereas for *Re* ≫ 1 a nonlinear flow behavior is likely to
occur. As velocity increases, the inertial effects start dominating the flow
field. In order to take these inertial effects into account,
Forchheimer (1901) introduced an inertial term representing the kinetic
energy of the fluid to the Darcy equation. The Forchheimer equation for
one-dimensional flow in terms of hydraulic head *h* (L) is given as follows:

$$\begin{array}{}\text{(1)}& -{\displaystyle \frac{\mathrm{d}h}{\mathrm{d}x}}={\displaystyle \frac{\mathit{\mu}}{\mathit{\rho}gk}}q+{\displaystyle \frac{{\mathit{\beta}}_{F}}{g}}{q}^{\mathrm{2}},\end{array}$$

where *x* (m) is the coordinate parallel to the axis of the one-dimensional porous medium, *k* (L^{2}) is the
permeability, *μ* (M L^{−1} T^{−1}) is the viscosity, *ρ*
(M L^{−3}) is the density, *q* (L T^{−1}) is the Darcy velocity, and
*β*_{F} (L^{−1}) is called the non-Darcy coefficient.

Ergun (1952) derived a model for high-velocity pressure loss in a porous medium from the Forchheimer equation by correlating the permeability and inertial resistance dimensionally to the porosity and the equivalent sphere diameter of rough particles. The permeability and inertial coefficient are interpreted in terms of spatial parameters as follows:

$$\begin{array}{}\text{(2)}& {\displaystyle}& {\displaystyle}k={\displaystyle \frac{{d}_{\mathrm{p}}^{\mathrm{2}}{n}^{\mathrm{3}}}{A{\left(\mathrm{1}-n\right)}^{\mathrm{2}}}},\text{(3)}& {\displaystyle}& {\displaystyle}{\mathit{\beta}}_{F}={\displaystyle \frac{B\left(\mathrm{1}-n\right)}{{d}_{\mathrm{p}}{n}^{\mathrm{3}}}},\end{array}$$

where *d*_{p} (L) is the average particle diameter, *n* (–) is the porosity, and
the coefficients *A* = 180 and *B* = 1.8 are empirical values and were derived
by averaging the Navier–Stokes equations for a cubic representative unit
volume.

The behavior of convective heat transport in porous media is strongly dependent on the fluid velocity and the kinetics of the heat transfer process between fluid and solid phases.

Given a packed bed, within a thermally isolated column of length *L* (L) in
which a fluid flows with a specific flow rate *q* (L T^{−1}) and then with an
average fluid velocity *q*∕*n*, the initial temperature in the column is *T*_{0} (K)
and a continuous flow injection transports heat energy along the column. For
a small ratio of column diameter *D* (L) to the length *L* and large fluid
velocity, the radial heat transport dynamics can be neglected in comparison
with the axial dynamics. Then the heat transport dynamics in the porous
medium column can be represented by a one-dimensional model.

If the solid and fluid phases are in contact for a sufficient period of time, there is the possibility to establish a local thermal equilibrium condition. In such case, only one energy equation is sufficient for the description of the convective heat transport through the porous medium. Assuming that porosity, densities, and heat capacities are constant in time the energy equation for the fluid and solid phases are combined into a single equation as follows:

$$\begin{array}{}\text{(4)}& {\left(\mathit{\rho}c\right)}_{\mathrm{sf}}{\displaystyle \frac{\partial {T}_{\mathrm{f}}}{\partial t}}={\displaystyle \frac{\partial}{\partial x}}\cdot \left[-vn{\mathit{\rho}}_{\mathrm{f}}{c}_{\mathrm{f}}{T}_{\mathrm{f}}+{k}_{\mathrm{sf}}{\displaystyle \frac{\partial {T}_{\mathrm{f}}}{\partial x}}\right],\end{array}$$

with

$$\begin{array}{}\text{(5)}& {\displaystyle}& {\displaystyle}{\left(\mathit{\rho}c\right)}_{\mathrm{sf}}=\left(\mathrm{1}-n\right){\mathit{\rho}}_{\mathrm{s}}{c}_{\mathrm{s}}+n{\mathit{\rho}}_{\mathrm{f}}{c}_{\mathrm{f}}\text{(6)}& {\displaystyle}& {\displaystyle}{k}_{\mathrm{sf}}=\left(\mathrm{1}-n\right){k}_{\mathrm{s}}+n{k}_{\mathrm{f}},\end{array}$$

where *T*_{f} (K) is the temperature of the fluid, *ρ*_{f} (M L^{−3})
is the density of the fluid, *ρ*_{s} (M L^{−3}) is the density of the
solid, *c*_{f} (L T^{2} K^{−1}) is the thermal capacitance of the fluid,
*c*_{s} (L T^{2} K^{−1}) is the thermal capacitance of the solid, *k*_{f}
(M L T^{−3} K^{−1}) is the thermal conductivity of the fluid, *k*_{s}
(M L T^{−3} K^{−1}) is the thermal conductivity of the solid, and
(*ρ**c*)_{sf} and *k*_{sf} represent the equivalent heat
capacity and thermal conductivity of the porous domain, respectively,
including porosity and thermal properties of solid and fluid.

If the interaction between solid and fluid phase is rapid the solid and fluid phase cannot exchange sufficient amounts of energy to establish local thermal equilibrium. At a given location solid and fluid phases have different temperatures. In the LTNE conditions each phase needs an energy equation for the description of heat transport. Assuming that porosity, densities, and heat capacities are constant in time, the energy equations can be written for the fluid and solid phase:

$$\begin{array}{}\text{(7)}& {\displaystyle}& {\displaystyle}n{\mathit{\rho}}_{\mathrm{f}}{c}_{\mathrm{f}}{\displaystyle \frac{\partial {T}_{\mathrm{f}}}{\partial t}}={\displaystyle \frac{\partial}{\partial x}}\cdot \left[-vn{\mathit{\rho}}_{\mathrm{f}}{c}_{\mathrm{f}}{T}_{\mathrm{f}}+n{k}_{\mathrm{f}}{\displaystyle \frac{\partial {T}_{\mathrm{f}}}{\partial x}}\right]+{q}_{\mathrm{fs}},\text{(8)}& {\displaystyle}& {\displaystyle}\left(\mathrm{1}-n\right){\mathit{\rho}}_{\mathrm{s}}{c}_{\mathrm{s}}{\displaystyle \frac{\partial {T}_{\mathrm{s}}}{\partial t}}={\displaystyle \frac{\partial}{\partial x}}\cdot \left[\left(\mathrm{1}-n\right){k}_{\mathrm{s}}{\displaystyle \frac{\partial {T}_{\mathrm{s}}}{\partial x}}\right]-{q}_{\mathrm{fs}},\end{array}$$

The interaction between the two phases is represented by the sink–source
terms *q*_{fs} given by the following equation:

$$\begin{array}{}\text{(9)}& {q}_{\mathrm{fs}}=h{s}_{\mathrm{f}}\left({T}_{\mathrm{s}}-{T}_{\mathrm{f}}\right),\end{array}$$

where *h* (M T^{−3} K^{−1}) is the convective heat transfer coefficient and
*s*_{f} (L^{−1}) is the specific surface area given by

$$\begin{array}{}\text{(10)}& {s}_{\mathrm{fs}}={\displaystyle \frac{\mathrm{6}(\mathrm{1}-n)}{{d}_{\mathrm{p}}}}.\end{array}$$

The convective heat transfer coefficient is related to the Nusselt number
*Nu* that for the porous medium can be expressed as follows:

$$\begin{array}{}\text{(11)}& \mathit{\text{Nu}}={\displaystyle \frac{{q}_{\mathrm{fs}}{d}_{\mathrm{p}}}{{k}_{\mathrm{f}}\left({T}_{\mathrm{f}}-{T}_{\mathrm{s}}\right)}}={\displaystyle \frac{h{d}_{\mathrm{p}}}{{k}_{\mathrm{f}}}}.\end{array}$$

Heat transfer dynamics can also be represented by the volumetric Nusselt
number *Nu*_{v}:

$$\begin{array}{}\text{(12)}& {\mathit{\text{Nu}}}_{\mathrm{v}}={\displaystyle \frac{h{s}_{\mathrm{f}}{d}_{\mathrm{p}}^{\mathrm{2}}}{{k}_{\mathrm{f}}}}.\end{array}$$

Wakao et al. (1979) found the following correlation for the volumetric Nusselt number:

$$\begin{array}{}\text{(13)}& {\mathit{\text{Nu}}}_{\mathrm{v}}=\mathrm{2.0}+\mathrm{1.1}{\mathit{\text{Re}}}^{\mathrm{0.6}}{\mathit{\text{Pr}}}^{{\scriptscriptstyle \frac{\mathrm{1}}{\mathrm{3}}}}.\end{array}$$

Fu et al. (1998), Kamiuto and Yeem (2005), and Ando et al. (2013) found, based on experimental data, a correlation between the volumetric Nusselt number and Reynolds number:

$$\begin{array}{}\text{(14)}& {\mathit{\text{Nu}}}_{\mathrm{v}}=C{\mathit{\text{Re}}}^{m}.\end{array}$$

The hydrodynamic mixing of the interstitial fluid at the pore scale gives
rise to significant thermal dispersion phenomena. Generally, the
hydrodynamic mixing is due to the presence of obstruction, flow restriction,
and turbulent flow. Therefore, the equivalent thermal conductivity in
Eq. (4) and thermal conductivity in Eq. (7) is replaced with the
effective thermal conductivity *k*_{eff} which is the sum of thermal
conductivity and thermal dispersion conductivity. The effective thermal
conductivity depends on various parameters such as mass flow rate, porosity,
shape of pores, temperature gradient, and solid and fluid thermal properties
(Kaviany, 1995). The following equation can be used to estimate *k*_{eff}.

$$\begin{array}{}\text{(15)}& {\displaystyle \frac{{k}_{\mathrm{eff}}}{{k}_{\mathrm{f}}}}={\displaystyle \frac{k}{{k}_{\mathrm{f}}}}+K\cdot {\mathit{\text{Pe}}}^{a}\end{array}$$

*Pe* represents the Peclet number defined as the product between the Reynolds
number *Re* and Prandtl number *Pr*.

$$\begin{array}{}\text{(16)}& \mathit{\text{Pe}}=\mathit{\text{Re}}\times \mathit{\text{Pr}}={\displaystyle \frac{{\mathit{\rho}}_{\mathrm{f}}v{d}_{\mathrm{p}}}{\mathit{\mu}}}\times {\displaystyle \frac{{c}_{\mathrm{f}}\mathit{\mu}}{{k}_{\mathrm{f}}}}={\displaystyle \frac{v{d}_{\mathrm{p}}}{{D}_{\mathrm{f}}}}\end{array}$$

The energy equation representative of the local thermal nonequilibrium can be written as follows:

$$\begin{array}{}\text{(17)}& {\displaystyle}& {\displaystyle \frac{\partial {T}_{\mathrm{f}}}{\partial t}}=-v{\displaystyle \frac{\partial {T}_{\mathrm{f}}}{\partial x}}+{D}_{\mathrm{eff}}{\displaystyle \frac{{\partial}^{\mathrm{2}}{T}_{\mathrm{f}}}{\partial x}}+\mathit{\alpha}\left({T}_{\mathrm{s}}-{T}_{\mathrm{f}}\right),\text{(18)}& {\displaystyle}& {\displaystyle \frac{\mathrm{1}-n}{n}}{\displaystyle \frac{{\mathit{\rho}}_{\mathrm{s}}{c}_{\mathrm{s}}}{{\mathit{\rho}}_{\mathrm{f}}{c}_{\mathrm{f}}}}{\displaystyle \frac{\partial {T}_{\mathrm{s}}}{\partial t}}={\displaystyle \frac{\mathrm{1}-n}{n}}{\displaystyle \frac{{k}_{\mathrm{s}}}{{\mathit{\rho}}_{\mathrm{f}}{c}_{\mathrm{f}}}}{\displaystyle \frac{{\partial}^{\mathrm{2}}{T}_{\mathrm{s}}}{\partial x}}-\mathit{\alpha}\left({T}_{\mathrm{s}}-{T}_{\mathrm{f}}\right),\end{array}$$

with

$$\begin{array}{}\text{(19)}& {\displaystyle}& {\displaystyle}{D}_{\mathrm{eff}}={\displaystyle \frac{{k}_{\mathrm{eff}}}{{\mathit{\rho}}_{\mathrm{f}}{C}_{\mathrm{f}}}},\text{(20)}& {\displaystyle}& {\displaystyle}\mathit{\alpha}={\displaystyle \frac{h{s}_{\mathrm{f}}}{n{\mathit{\rho}}_{\mathrm{f}}{C}_{\mathrm{f}}}}.\end{array}$$

*D*_{eff} (L^{2} T^{−1}) is the thermal dispersion and *α* (T^{−1})
is the exchange coefficient.

The thermal dispersion happens due to hydrodynamic mixing of fluid at the pore scale caused by the nature of the porous medium. Greenkorn (1983) found nine mechanisms responsible of most of the mixing, among which are the following: (1) mixing caused by the tortuosity of the flow channels due to obstructions – fluid elements starting a given distance from each other and proceeding at the same velocity will not remain the same distance apart; (2) existence of autocorrelation in flow paths – in this case, all pores in a porous medium may not be accessible to a fluid element after it has entered a particular flow path; (3) recirculation due to local regions of reduced pressure due to the conversion of pressure energy into kinetic energy; (4) hydrodynamic dispersion in a capillary caused by the velocity profile produced by the adhering of the fluid to the wall; (5) molecular diffusion into dead-end pores as solute rich front passes the pore. After the front passes, the solute will diffuse back out, thus dispersing.

Using the analogy with the solute transport the Damköhler
number *Da* (Leij et al., 2012) can be introduced in order to evaluate the
influence of heat transfer between the fluid and solid phases on the
convection phenomena:

$$\begin{array}{}\text{(21)}& \mathit{\text{Da}}={\displaystyle \frac{\mathit{\alpha}L}{v}}.\end{array}$$

When *Da* reaches the unit the heat transfer timescale is comparable with the
convection timescale and the LTNE exists between solid and fluid phases. At
very high values of *Da* the heat transfer timescale is much lower than
the convective timescale and the LTE condition exists between solid and fluid
phases. Finally, at very low values of *Da* the heat transfer phenomena
between solid and fluid phase can be neglected.

Neglecting the first term on the right side of Eq. (18), the
analytical solution of the system equations describing one-dimensional heat transport in
a semi-infinite domain for instantaneous temperature injection is given by
Goltz and Roberts (1986). According to this analytical solution, the
probability of density function PDF_{LTNE} of the residence time for LTNE
condition can be written as follows:

$$\begin{array}{ll}{\displaystyle}& {\displaystyle}{\mathrm{PDF}}_{\mathrm{LTNE}}\left(x,t\right)=\\ \text{(22)}& {\displaystyle}& {\displaystyle}\phantom{\rule{1em}{0ex}}{e}^{\mathit{\alpha}t}{\mathrm{PDF}}_{\mathrm{0}}\left(x,t\right)+\mathit{\alpha}\underset{\mathrm{0}}{\overset{t}{\int}}H\left(t,\mathit{\tau}\right){\mathrm{PDF}}_{\mathrm{0}}\left(x,t\right)\mathrm{d}\mathit{\tau},\end{array}$$

with

$$\begin{array}{}\text{(23)}& {\displaystyle}& {\displaystyle}{\mathrm{PDF}}_{\mathrm{0}}\left(x,t\right)={\displaystyle \frac{\mathrm{1}}{\sqrt{\mathit{\pi}{D}_{\mathrm{eff}}t}}}\mathrm{exp}\left({\displaystyle \frac{x-vt}{\mathrm{4}{D}_{\mathrm{eff}}t}}\right),\text{(24)}& {\displaystyle}& {\displaystyle}H\left(t,\mathit{\tau}\right)={e}^{-\frac{\mathit{\alpha}}{\mathit{\beta}}\left(t-\mathit{\tau}\right)-\mathit{\alpha}\mathit{\tau}}{\displaystyle \frac{\mathit{\tau}{I}_{\mathrm{1}}\left(\frac{\mathrm{2}\mathit{\alpha}}{\mathit{\beta}}\sqrt{\mathit{\beta}\left(t-\mathit{\tau}\right)\mathit{\tau}}\right)}{\sqrt{\mathit{\beta}\left(t-\mathit{\tau}\right)\mathit{\tau}}}},\text{(25)}& {\displaystyle}& {\displaystyle}\mathit{\beta}={\displaystyle \frac{\mathrm{1}-n}{n}}{\displaystyle \frac{{\mathit{\rho}}_{\mathrm{s}}{c}_{\mathrm{s}}}{{\mathit{\rho}}_{\mathrm{f}}{c}_{\mathrm{f}}}},\end{array}$$

where PDF_{0}(*x*,*t*) represents the probability density function of the
residence time without heat transfer between the solid and fluid phase. The
parameter *β* (–) represents the ratio of the volume-specific heat
capacity of the solid phase to the fluid, and *I*_{1} is the modified
Bessel function of order 1.

The coefficient *α* can be viewed as the reciprocal of the exchange
time required to transfer energy from fluid to solid phase and vice versa.
The effect of local thermal nonequilibrium is stronger when the exchange
time is the same order of magnitude of the transport time. The local thermal
nonequilibrium is characterized by a thermal distribution profile with a
tailing effect.

The observed temperature function *T*_{obs}(*t*) at a generic distance *x* from the
injection temperature function *T*_{inj}(*t*) can be obtained using the
convolution theorem:

$$\begin{array}{}\text{(26)}& {T}_{\mathrm{obs}}(x,t)={T}_{\mathrm{inj}}(\mathrm{0},t)\cdot {\mathrm{PDF}}_{\mathrm{LTNE}}\left(x,t\right).\end{array}$$

3 Experimental setup

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The test on convective heat transport in the porous medium has been
conducted on a laboratory physical model. Figure 1 shows a sketch of the
experimental setup. A plastic circular pipe characterized by a diameter of
*D* = 0.11 m and height of *H* = 1.66 m has been thermally insulated using a
roll of elastomeric foam with a thickness of *s* = 0.04 m and a thermal
conductivity of *λ* = 0.037 W m^{−1} K^{−1}. The pipe can be
filled with different porous materials with different grain sizes and
hydrothermal properties. Seven thermocouples have been equally placed along
the axis of the pipe with a reciprocal distance of 0.185 m. The first
thermocouple is located at a distance of 0.435 m from the inlet of the
water. The TC08 Thermocouple Data Logger (Pico Technology) with sampling rate
equal to 1 s has been connected with the thermocouples. An adaptable
constant head reservoir and an outlet reservoir permit a
constant head to be maintained during the test, and water within the pipe flows from the
bottom to the top. An ultrasonic velocimeter (DOP3000 by Signal Processing)
is used to measure the instantaneous flow rate. An electric water boiler
characterized by a volume equal to 0.01 m^{3} has been used to heat the
water flowing through the pipe.

A medium gravel (*M*_{1}) (Soil Survey Staff, 1975) and a very coarse gravel
(*M*_{2}) (Soil Survey Staff, 1975) have been used. Figure 2 shows the tested
materials whereas in Table 1 the hydraulic and the thermal parameters of each
material are reported. The grain size and the specific surface of each porous
material is directly estimated on a sample of 100 grains, whereas the
porosity is estimated by the ratio of the volume of void space to the total
volume of the filled plastic circular pipe. The volume of the void space is
obtained by measuring the amount of water which enters the pipe until full
saturation. The thermal characteristics reported in the Table 1 are
literature values (Engineering ToolBox, 2009). The temperature tracer tests
involve the observation of the thermal breakthrough curves (BTCs) monitored
by the seven thermocouples. Initially cold water flows through the pipe
filled with the porous medium in order to have a constant temperature *T*_{0}
along the pipe. Subsequently hot water flows through the pipe, maintaining a
constant head condition during the test.

4 Discussion

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For each tested porous medium four thermal tracer tests have been carried out
varying *Re* in the range 5.7–22.5 for *M*_{1} and 23.5–105.5 for
*M*_{2}. The thermal BTCs observed at different distances have been fitted
together using Eq. (19). The root mean square error (RMSE) and the
determination coefficient (*r*^{2}) have been used as criteria to evaluate
the goodness of fit. The parameters *v*, *D*_{eff}, and *α* have
been individually fitted for each thermal tracer test whereas *β* has
been imposed constant for all tracer tests of each tested porous medium.
Table 2 shows the estimated values of the heat transport parameters, the
RMSE, and *r*^{2}, whereas Figs. 3 and 4 show the fitting results of the
observed temperature distribution along the porous column for *M*_{1} and
*M*_{2} respectively. Table 3 shows the dimensionless numbers *Pe*,
*k*_{eff}∕*k*_{f}, *Nu*, and *Da* evaluated for the
different values of *Re*.

As shown in Table 2, the fluid velocity *q*∕*n* is systematically higher than the
estimated thermal convective velocity *v* for the medium gravel *M*_{1};
on the contrary, for the very coarse gravel, *M*_{2}*q*∕*n* is systematically lower than
*v*.

This phenomenon for the coarser material might be attributable to the fact that the heat propagates through both the solid and fluid phase (Anderson, 2005; Rau et al., 2012) and to the existence of channeling phenomena that might also have an influence in increasing the convective heat.

Even for finer grained materials (2 mm), Rau et al. (2012) also found values of thermal velocity systematically lower than solute velocity, coherent with Bodvarssoon (1972), Oldenburg and Pruess (1998), and Geiger et al. (2006).

Another discrepancy has been observed when comparing the values of the porosity
presented in Table 1 and the value of porosity obtained from Eq. (18)
equal to 0.467 and 0.469 respectively for *M*_{1} and *M*_{2}. For *M*_{1}
the value of porosity presented in Table 1 reaches the value derived from
*β*, whereas for *M*_{2} the value presented in Table 1 is higher than
the value derived from *β*.

These results highlight that for *M*_{2} there is the existence of stagnant
zones which reduce the amount of porosity that contributes to fluid flow. In
other words, in *M*_{1} the total porosity reaches to the effective
porosity, whereas in *M*_{2} the effective porosity is less than total
porosity.

In other words, because the coarser material *M*_{2} is less well sorted
than the less coarse material *M*_{1}, not all pores of the former are actually
interconnected. In geologic materials, based on the connectivity of pores,
consequently, the void space can be divided into interconnected pore,
isolated pore, and blind pore (Hu et al., 2017). Only the pores that are
well interconnected provide continuous channels for heat and mass transfer
and fluid flow, while the pores that are not part of a continuous channel
network do not contribute. These pores are known as noneffective pores,
namely, they provide no space for fluid flow and heat transfer in
reservoirs.

Figure 5 shows the relationships between *Pe* and the ratio
of the effective thermal conductivity to the fluid thermal conductivity
*k*_{eff}∕*k*_{f}. The experimental results show a nonlinear
behavior well represented by Eq. (15). A change of slope is evident when changing
from *M*_{1} to *M*_{2}. The latter material shows a more pronounced thermal
dispersion caused by the hydrodynamic mixing of fluid at the pore scale. Some
mixing is caused by the tortuosity of the flow paths due to the presence of
obstructions: the fluid elements starting a given distance from each other
and proceeding at the same velocity will not remain at the same distance
apart. The high level of flow path heterogeneity gives rise to a higher
velocity variation at pore scale as well as the presence of the preferential
flow paths that enhance the effect of macrodispersion. Mixing can also be
caused by recirculation caused by local regions of reduced pressure arising
from flow restrictions.

Further mixing can arise from the fact that all pores in a porous medium may not be accessible to a fluid element after it has entered a particular flow path.

These results are coherent with those obtained by Rau et al. (2012) who found that the thermal dispersion was transitioning between not depending on the flow velocity and a nonlinear increase with velocity. They affirmed that the location of the transition zone is a function of the thermal properties of the solid and the sedimentological architecture.

In Fig. 6 the obtained experimental relationship between *Pe* and
*k*_{eff}∕*k*_{f} has been compared with the results obtained by
several authors (Levec and Carbonell, 1985a, b; Gunn and Pryce, 1969;
Pfannkuch, 1963; Ebach and White, 1958). For the range of *Pe*
investigated the experimental results presents the same order of magnitude of
*k*_{eff}∕*k*_{f}. For low Peclet numbers the experimental value
of *k*_{eff}∕*k*_{f} is systematically greater than the value of
the trend line. This phenomenon can be attributable to the density gradients
which altered the flow pattern. Given that the water flows from the bottom to
the top and the hot water is fed from the bottom, the buoyancy effect adds to
the diffusion effect. This effect seems relevant from low *Pe* values.

Figure 7 highlights the experimental correlation between *Nu*_{v} and *Re*.
Equation (13) fails to represent the experimental results, especially for
*M*_{1} where they are underestimated by an order of magnitude. For *M*_{2}
the theoretical model reaches the experimental results with a percentage
error of 5–35 %.

According to Ando et al. (2013) the volumetric Nusselt number is well
represented by Eq. (14). The exponent *m* approaches the unit whereas the
constant *C* assumes equal values for *M*_{1} and for *M*_{2}. According to
Ando et al. (2013) the coefficient *C* decreases as the pore diameter
(correlated with the particle diameter) increases.

Figure 8 shows the relationship between *Pe* and *Nu*. The
experimental results highlight that the Nusselt number can be represented by
an equation like *Nu* = *C* × *Pe*, where *C* (–)
is a coefficient that assumes a value equal to 0.41 for *M*_{1} and 0.03 for
*M*_{2}.

The physical meaning of the ratio of the surface of the grains in
contact with the active flow path that transports heat to the total surface
of the grain can be attributed to this coefficient. *M*_{2}, in comparison to
*M*_{1}, is characterized by the presence of preferential flow paths and then
an equal number of *Pe* corresponding to a lower *Nu* because the surface of the
grain available to exchange heat between the fluid and solid phase is lower.

As shown in Table 3 the Damköhler number *Da* calculated for *M*_{1} is
greater than the unit. Heat exchange is so rapid that it gives rise to an
instantaneous equilibrium between the solid and fluid phase. The heat has enough
time to diffuse in the solid phase. On the contrary *Da* calculated for *M*_{2} is
close to the unit, and there is the presence of the local thermal nonequilibrium condition.

A comparison of the heat *J* (M L^{2} T^{−2}) stored in the porous column per
unit temperature difference $\mathrm{\Delta}T={T}_{\mathrm{inj}}-{T}_{\mathrm{0}}$ (K) when varying the
specific discharge *q* for each tested porous medium can be evaluated
considering a continuous temperature injection function as follows:

$$\begin{array}{}\text{(27)}& {\displaystyle \frac{J}{\mathrm{\Delta}T}}={\mathit{\rho}}_{\mathrm{f}}{c}_{\mathrm{f}}Q\underset{\mathrm{0}}{\overset{\mathrm{\infty}}{\int}}\left(\mathrm{1}-\underset{\mathrm{0}}{\overset{t}{\int}}\mathrm{PDF}\left(L,\mathit{\tau}\right)\mathrm{d}\mathit{\tau}\right)\mathrm{d}t.\end{array}$$

The combined effects of the flow rate and the particle diameter on heat
transfer are illustrated in Fig. 9, which shows the variation of the heat
stored in the column per unit temperature difference when varying the specific
flow rate for *M*_{1} and *M*_{2}. As the flow rate increases, the stored
heat increases, and the porous medium with a smaller particle diameter
generates a higher increase in heat transfer enhancement than that with a
larger particle diameter. This is coherent with the results obtained by
Dehghan and Aliparast (2011). *M*_{1} permits the storage of more
heat than *M*_{2}. The former is characterized by a more homogeneous flow
path distribution that allows a greater interaction between fluid and solid
phase. On the contrary *M*_{2} has a more heterogeneous flow path
distribution that increases thermal macrodispersion phenomena at the pore
scale and at the same time reduces the interaction between the fluid and
solid phase.

In order to put into evidence the performance of the heat transfer
enhancement of the porous materials it can be useful to compare the Nusselt
number and the hydraulic head loss d*h*∕d*x* evaluated by Eq. (1). Figure 10
shows the ratio of the Nusselt number to head loss as function of the
Peclet number. Despite *M*_{1} presenting a higher heat transfer enhancement
with respect to *M*_{2}, the head losses are higher and then the ratio of *Nu*
to d*h*∕d*x* is lower. Furthermore, as *Pe* increases, the heat transfer enhancement
increases more rapidly than the head loss for the fine material *M*_{1},
whereas for the coarser material *M*_{2} the opposite happens: increasing
the Peclet number causes a weak increase in the Nusselt number due to the presence of
channeling phenomena that reduce the heat exchange area between fluid and
solid phases.

5 Conclusion

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In this study a laboratory physical model has been set up to analyze the behavior of forced convective heat transport in two porous media characterized by different grain sizes and specific surfaces. For each material four tracer tests have been carried out and they have been compared with the one-dimensional analytical solution of LTNE model. The flow path heterogeneity that characterizes the coarser material gives rise to a higher velocity variation at pore scale with a channeling effect which causes (1) the increase in the macrodispersion phenomena in the forced convection heat transport, (2) the decrease in the surface of the grain available to exchange heat between the fluid and solid phase, (3) the presence of the local thermal nonequilibrium condition (4) the decrease in the amount of heat that can be stored in the porous medium, and (5) a weak growth of heat transfer enhancement respect to the head loss as convective phenomena increases.

The finer material *M*_{1} has a more homogeneous flow path distribution
that allows a greater interaction between fluid and solid phase and
therefore allows the storage of more heat than the coarser one.

This can also be seen when analyzing the ratio of the Nusselt number to the
head loss as function of Peclet number for both materials. Even though the
coarser material *M*_{2} is more permeable than *M*_{1} as the advective
phenomena increase, the head loss increases more rapidly than the heat
transfer enhancement due to the channeling effect that increases the
macrodispersion phenomena and reduces the heat transfer between fluid and
solid phase.

The main contribution of this study is to investigate the optimal thermal energy storage of porous materials by analyzing how the grain size and the specific surface affect heat storage properties as well as heat transport in terms of macrodispersion phenomena, heat transfer between solid and fluid phases. This is relevant in order to optimize the efficiency of geothermal installations in aquifers.

The experimental results emphasize the differences between porous and fractured media. As observed by Cherubini at al. (2017) a fractured medium with a high density of fractures and then with a higher specific surface is not efficient to store thermal energy because the fractures are surrounded by a matrix with a more limited capacity to store heat. An opposite behavior has been observed in porous media in which a higher specific surface corresponds to a higher capacity to store heat. For porous media, as the specific surface decreases the macrodispersion phenomena increase due essentially to the channeling effect, and then the surface of the grain available to exchange heat between the fluid and solid phase decreases. However, for the fractured media this statement is not true because the macrodispersion phenomena are, in contrast, more related to the roughness and aperture variation of each single fracture as well as to the connectivity of the fracture network.

The study has increased the understanding of heat transfer processes in the subsurface, encouraging the investigation into how further parameters such as the shape and the roughness of the grain of porous media affect the amount of energy that can be stored. This is important to maximize the efficiency and minimize the environmental impact of the geothermal installations in groundwater.

Data availability

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Data availability.

The experiment data can be found in the Supplement.

Supplement

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Supplement.

The supplement related to this article is available online at: https://doi.org/10.5194/npg-25-279-2018-supplement.

Competing interests

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Competing interests.

The authors declare that they have no conflict of interest.

Acknowledgements

Back to toptop
Acknowledgements.

This research was funded by the regional program in support of the smart specialization
and social and environmental sustainability – FutureInResearch.

Edited by: Norbert Marwan

Reviewed by: two anonymous referees

References

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Short summary

The use of groundwater as a low-enthalpy geothermal resource for heating and cooling of buildings and for agricultural and industrial processes is growing. The study has increased the understanding of heat transfer processes in the subsurface, encouraging the investigation into how further parameters affect the amount of energy that can be stored and dissipated. This is important to maximize the efficiency and minimize the environmental impact of the geothermal installations in groundwater.

The use of groundwater as a low-enthalpy geothermal resource for heating and cooling of...

Nonlinear Processes in Geophysics

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