Modelling and Experimental Characterization of Unsaturated Flow in Absorbent and Swelling Porous Media: Material Characterization

Abstract

A comprehensive experimental protocol is presented for the characterization of physico-chemical and morphological properties of absorbent hygiene products (AHPs), composite materials, mainly made of super absorbent polymer (SAP) granules blended with cellulose fibres (fluff). The protocol is based on a combination of experimental methods for the characterization of the material properties varying SAP/fluff ratio (SFR): absorption rate, porosity, hydraulic conductivity, retention model and swelling. Major findings were that: (1) liquid absorption rate by the SAP particles was nonlinear; (2) the hydraulic conductivity could be expressed as a function of the porosity of the composite medium for any sample and liquid uptake; and (3) the retention model showed moderate variability with SFR, in the range investigated. Experimental results have been used to determine the constitutive equations for the multiphase flow model developed in the literature (Diersch et al. in Theory Transp Porous Media 83:437–464, 2010. https://doi.org/10.1007/s11242-009-9454-6) for the prediction of the performance of AHPs during cycles of imbibition and drainage (Chem Eng Sci 224:115765, 2020. https://doi.org/10.1016/j.ces.2020.115765).

Appendix

The model developed by Diersch et al. (2010) is derived from mass and momentum balances of the liquid, gas and solid phases. It is a continuum model, which implies the identification of a representative element volume (REV), in which balance and constitutive equations are meaningful. In this sense, each material particle of the continuum domain corresponds to a REV. Due to swelling of SAP particles, the model allows the domain to swell in the spatial frame. Thus, in the spatial frame, a displacement function is associated with the material particles, depending on the swelling extent.

The phases fractions are expressed with $\varepsilon^{\alpha }$, where $\alpha$ defines the correspondent phase ($s$, $l$ and $g$, for solid, liquid and gas, respectively). Moreover, $\varepsilon$ is the porosity of the material; thus,

$$\varepsilon^{\text{l}} = \varepsilon s^{\text{l}} \quad\varepsilon^{\text{g}} = \varepsilon \left( {1 - s^{\text{l}} } \right)\quad\varepsilon^{\text{s}} = 1 - \varepsilon,$$

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where $s^{{\rm l}}$ is the saturation, defined as the ratio of liquid and void volume.

Considering the phase $\alpha$, the spatial and the material frame are connected through the displacement function ($\varvec{u}^{\varvec{\alpha}}$):

$$\varvec{x} = \varvec{u}^{\varvec{\alpha}} \left( {\varvec{X}^{\varvec{\alpha}} ,t} \right),$$

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where $\varvec{x}$ and $\varvec{X}$ are the spatial and the material point coordinates, respectively. This is a bijective function and its Jacobian is always positive. Referring to the solid-phase displacement, the Jacobian ($J^{\text{s}}$) represents also the deformation (swelling or shrinkage) of the porous media as the ratio between the current control-space volume ($\left| {\varOmega ^{\text{s}} } \right|\left( {\varvec{x},t} \right)$) and the initial one ($\left| {\varOmega _{0}^{\text{s}} } \right|\left( {\varvec{X}^{{\varvec{s}}} ,0} \right)$) (Diersch et al. 2010; Masoodi and Pillai 2012):

$$J^{\text{s}} = \det \left( {\nabla \varvec{u}^{\varvec{{s}}} } \right) = \frac{{\left| {\varOmega ^{\text{s}} } \right|\left( {\varvec{x},t} \right)}}{{\left| {\varOmega _{0}^{\text{s}} } \right|\left( {\varvec{X}^{\text{s}} ,0} \right)}}.$$

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Stress-free deformation of the solid phase is considered because of the absorption process. Accordingly, nonlinear terms related to plastic deformation are avoided and the solid strain is only a function of the swelling (Coussy 1995; Masoodi and Pillai 2012). Furthermore, shear effects are neglected, and the strain has only normal components. Accordingly, the momentum balance on the solid phase reads:

$$\nabla \cdot \varvec{u}^{\varvec{s}} = \varvec{ }\frac{{\left| {\varOmega ^{\text{s}} } \right|\left( {\varvec{x},t} \right) - \left| {\varOmega _{0}^{\text{s}} } \right|\left( {\varvec{X}^{\varvec{s}} ,0} \right)}}{{\left| {\varOmega _{0}^{s} } \right|\left( {\varvec{X}^{\varvec{s}} ,0} \right)}} = J^{\text{s}} - 1.$$

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Using Cartesian coordinate system, if the strain is unidirectional, that is, the material is laterally confined and can swell only along one direction, the scalar volumetric strain coincides with the strain in that direction and no approximations are needed (Diersch et al. 2010).

For the gas phase, momentum balance becomes trivial because the gas is considered stagnant and its velocity assumes the same value of the solid-phase velocity.

As regards the mass balances, every phase can be treated with the same general multispecies conservation law. Mass exchanges between solid and liquid phases are defined by the absorption chemical reaction. The absorption process is an interfacial reaction, between SAP particles and liquid phase, and then, it is proportional to contact surface between phases, especially at the beginning of the reaction. It is assumed that all the SAP inside a material point can be reached by liquid, without considering the diffusional processes inside a single particle of SAP. The interfacial area is a function of the liquid saturation inside the porous matrix. In first approximation, the specific wet area could be set equal to the effective saturation, described below. The liquid density at the interface remains the same and approximately equal to the water density (no Donnan effects are considered) (Huyghe and Janssen 1997). The maximum amount of water that SAP can absorb is related to the chemical potential equilibrium on the interfacial surface between SAP particles and liquid phase, and it depends on the salinity of water. Regarding the gas phase, it is not necessary to consider a mass balance, because it can be obtained by difference from the mass balance of the liquid phase. Developing the general mass equation for the liquid and solid phases, neglecting the compressibility of liquid, a Richards-type flow equation is obtained:

$$\varepsilon \frac{{\partial s^{\text{l}} \left( {\psi^{\text{l}} } \right)}}{\partial t} - \nabla \cdot \left[ {\varvec{K}^{\varvec{l}} \cdot \left( {\nabla \psi^{\text{l}} - \varvec{e}} \right)} \right] = - \left[ {\frac{{\bar{C}_{{{\text{SAP}}_{0} }}^{\text{s}} }}{{\rho^{\text{l}} J^{\text{s}} }} + \varepsilon \frac{{\partial s^{\text{l}} }}{{\partial m_{2}^{\text{s}} }} + s^{\text{l}} \left( {\frac{\partial \varepsilon }{{\partial m_{2}^{\text{s}} }} + \frac{\varepsilon }{{J^{\text{s}} }}\frac{{\partial J^{\text{s}} }}{{\partial m_{2}^{\text{s}} }}} \right)} \right]\frac{{\partial m_{2}^{\text{s}} }}{\partial t}.$$

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For the solid mass balance, solid cannot migrate in the domain (velocity of the material particles in the material frame is zero, for each particle) or pass through the boundary conditions. The only changes in time of the mass solid phase are due to the absorption of water by SAP particles. Therefore, the mass balance for solid phase is reduced to the solely absorption reaction of SAP particles (Diersch et al. 2010, Santagata et al. 2020):

$$\frac{{\partial m_{2}^{\text{s}} }}{\partial t} = \frac{{m_{{2{ {\rm max} }}}^{\text{s}} \left( {1 - \frac{{m_{2}^{\text{s}} }}{{m_{{2{ {\rm max} }}}^{\text{s}} }}} \right)^{k} a^{\text{sl}} \left( { s^{\text{l}} } \right)}}{\tau },$$

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where $m_{{2{ {\rm max} }}}^{\text{s}}$ is the maximum absorbance capacity of SAP, $\tau$ is the characteristic time of the absorption reaction and $a^{\text{sl}}$ is the fraction of wet SAP granules interfacial area, which is dependent on saturation. To simplify the relation, in this study the fraction of wet interfacial area is equal to saturation, $a^{\text{sl}} = s^{\text{l}}$. The absorption kinetic is then dependent on the swelling and the liquid saturation of the material particle. Thus, the fluid dynamic of swelling porous system is then a coupled problem.