Theoretical modeling of water vapor transport in cellulose-based materials

Abstract

The theory of mass transport in porous media is of fundamental importance for different applications such as food, paper packaging, textiles, and wood for building materials. In this study, a theoretical water vapor transport model has been developed for cellulose-based materials, such as paper and regenerated cellulose film. Pore diffusivities were determined from the dynamic moisture breakthrough experiments comprising a stack of paper sheets and regenerated cellulose films in a configuration similar to a packed adsorption column. Other mass transfer parameters were determined from transient moisture uptake rate measurements. The model incorporates pore and surface diffusion as a lump parameter into a variable effective diffusion coefficient. The mass transport, involving both pore and surface diffusions, is evaluated independently. The theoretical water vapor transmission rates (WVTRs) obtained from the model were compared with experimentally determined WVTRs measured under steady-state conditions. The theoretical model, based on intrinsic diffusion, stipulates higher WVTR values compared to the experimental results. However, the theoretical water vapor transfer rates agree well with the experimental results when external mass transfer resistance is incorporated in the model.

Appendix: Calculated result for the Knudsen diffusion coefficient

Knudsen diffusion is a means of molecular transport through pores that are small in comparison to the mean free path of the gas. For straight and round pores, the Knudsen diffusion coefficient can be predicted from the diameter of the pore by the expression (Hines and Maddox 1985):

$$\frac{1}{{D_{\text{A,k}} }} = 97.0 r\left( {\frac{T}{{M_{\text{A}} }}} \right)^{1/2}$$

(24)

where r is the pore radius, m, T is temperature, K, D _A,K is the Knudsen diffusion coefficient, m²/s, and M _A is the molecular weight of component A.

In order to account for the tortuosity path of the molecule (τ) and the porosity of the material (ε), an effective Knudsen diffusivity (D _A,K,e) can be expressed as:

$$D_{\text{A,K,e}} = D_{\text{A,k}} \frac{\varepsilon }{\tau }$$

(25)

Regenerated cellulose film has a small mean pore size and ranges between 0.5 to 2.0 nm (Tolle 1971; Ichwan and Son 2011). It is possible that the regenerated cellulose film exhibits a dominant Knudsen diffusion mechanism. Molecules collide more often with the pore walls than with other molecules in the Knudsen diffusion mechanism. Assuming straight and round pores in the cellulose film, the effective diffusivity can be calculated using Eq. 26.

$$D_{\text{A,K,e}} = 97.0\left( {\frac{\varepsilon }{\tau }} \right) r\left( {\frac{T}{{M_{\text{A}} }}} \right)^{1/2}$$

(26)

The calculated D _A,K,e is found to be around 5.3 × 10⁻⁹ m²/s using the porosity value, ε = 0.20, tortuosity, τ = 15, molecular weight of the water vapor molecule, M _A = 18 g/mol, T = 298 K, and pore radius, r ≈ 1.25 nm.