Effective Thermal Conductivity of a Wet Porous Medium—Presence of Hysteresis When Modeling the Spatial Water Distribution for the Pendular Regime

[+] Author and Article Information
Édouard Canot

Campus de Beaulieu,
Rennes 35000, France
e-mail: edouard.canot@irisa.fr

Renaud Delannay

Campus de Beaulieu,
Rennes 35000, France
e-mail: renaud.delannay@univ-rennes1.fr

Salwa Mansour

Campus de Beaulieu,
Rennes 35000, France
e-mail: salwa.mansour@inria.fr

Mohamad Muhieddine

Nabatieh Campus Section V,
Nabatieh, Lebanon
e-mail: mohamad.muhieddine@liu.edu.lb

Ramiro March

Campus de Beaulieu,
Rennes 35000, France
e-mail: ramiro.march@univ-rennes1.fr

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received September 9, 2014; final manuscript received September 9, 2015; published online June 1, 2016. Assoc. Editor: Dennis A. Siginer.

J. Heat Transfer 138(9), 091011 (Jun 01, 2016) (7 pages) Paper No: HT-14-1599; doi: 10.1115/1.4032950 History: Received September 09, 2014; Revised September 09, 2015

This paper deals with the heat transfer between two spherical grains separated by a small gap; dry air is located around the grains and a liquid water meniscus is supposed to be present between them. This problem can be seen as a microscale cell of an assembly of solid grains, for which we are looking for the effective thermal conductivity. For a fixed contact angle and according to the volume of the liquid meniscus, two different shapes are possible for the meniscus, giving a “contacting” state (when the liquid makes a true bridge between the two spheres) and a “noncontacting” one (when the liquid is split in two different drops, separated by a thin air layer); the transition between these two states occurs at different times when increasing or decreasing the liquid volume, thus leading to a hysteresis behavior when computing the thermal flux across the domain.

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Fig. 2

Simplification of the grains' assembly. The dashed rectangle is the computational domain used in this paper.

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Fig. 1

Real granular medium in the “pendular regime”: dry air is located around the grains (in white), whereas liquid water menisci (in light gray) is present as liquid bridges between them

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Fig. 6

This represents the family of the liquid meniscus when the volume varies, in the contacting state. Each curve is the boundary of the half of a bridge linking the two grains.

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Fig. 7

This represents the family of the liquid meniscus when the volume varies, in the noncontacting state. Each curve is the boundary of a sessile drop, and the same symmetric drop is on the top of the other grain (not represented).

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Fig. 3

Sketch of the cylindrical computational domain (taking into account all the symmetries). Boundary conditions are of homogeneous Dirichlet type on top and bottom sides and of zero Neumann type on the external vertical sides.

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Fig. 4

A scheme of the liquid meniscus (solid grain in dark gray and liquid water in light gray). A is an arbitrary point defined as the beginning of the integration process.

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Fig. 5

A zoom-in of the liquid–solid interface (solid grain in dark gray and liquid water in light gray)

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Fig. 8

Example of contour curves of the temperature obtained after a numerical computation on a 500 × 500 mesh, in the caseϵ/R=0.1, θ = 30 deg. Due to Dirichlet boundary condition (bottom: T = 0 and top: T = 1), each contour line from bottom totop corresponds to the temperature: T=0.1,0.2,…,0.9, respectively.

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Fig. 9

A representation of a top (noncorner) cell with its neighbor cells

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Fig. 10

Thermal heat flux with respect to the liquid fraction, for the contacting state (o) and the noncontacting state ( + ). ϵ/R=0.1, θ = 30 deg. Note the great increase in the heat flux due to the presence of water: about a factor of 5 when humidity is only 0.12.

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Fig. 11

This zoom-in of Fig. 10 shows with more evidence the hysteresis behavior when liquid fraction is increasing or decreasing. The jump in the heat flux occurs at the dashed line; this jump is more pronounced in the increasing case, i.e., when liquid water increases. The arrows show the direction in the hysteresis cycle.

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Fig. 12

Thermal heat flux with respect to the liquid fraction, for the contacting state only, when θ = 30 deg. The curves show the influence of the distance ϵ.

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Fig. 13

Thermal heat flux with respect to the liquid fraction, for the contacting state only, when ϵ/R=0.1. The curves show the influence of the contact angle θ.



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