Research Article | Open Access

# Fractal Derivative Model for Air Permeability in Hierarchic Porous Media

**Academic Editor:**Chuandong Li

#### Abstract

Air permeability in hierarchic porous media does not obey Fick's equation or its modification because fractal objects have well-defined geometric properties, which are discrete and discontinuous. We propose a theoretical model dealing with, for the first time, a seemingly complex air permeability process using fractal derivative method. The fractal derivative model has been successfully applied to explain the novel air permeability phenomenon of cocoon. The theoretical analysis was in agreement with experimental results.

#### 1. Introduction

Air permeation and moisture vapor diffusion in textile fabric have a close relation to the comfort ability of clothing [1–3]. Since the skin is “breathing” all the time, water vapor is continuously released from the skin. If the moisture cannot be effectively delivered to the outside atmosphere, the body will feel sultry in summer and clammy in winter.

Clothing exhibits a multiscale inner structure constructed by textile fiber. The micropores among the thin fibers provide a winding passage for air and moisture vapor exchange between the microclimate (between the skin and fabric) and atmosphere. For certain textile apparel, the arrangement style of the fiber in the fabric may result in different air permeation efficiency. Recently, Blossman-Myer and Burggren investigated the water loss and oxygen diffusion through the silk cocoon. And the experimental data indicated that the cocoon does not have any impact on the oxygen and water vapor transportation [4]. This interesting finding suggests that cocoon has some special functional performance in gas permeation besides its basic purpose for protecting pupa against predation, biodegradation, and dehydration [5]. The fantastic air permeation performance of cocoon might be related to its configuration. Investigation on the mechanism of gas permeation in such a natural porous media composed by fiber is an inspiration in developing new functional clothing.

As a transport property, mechanism of gas permeation in different porous media has received continuous attention due to its significance in both science and engineering [6, 7], such as gas separation or gas purification [8, 9], chemical catalyst industry [10], proton exchange membrane fuel cell [11, 12], and textile material [1–3]. Since the microstructures of the hierarchic porous media are usually disordered and extremely complicated, it is very difficult to appropriately describe the random structures analytically.

In the past two decades, fractal theory has been introduced to solve problems of diffusion in media with hierarchic configuration since Mandelbrot’s pioneering work [13]. The fractal dimensions have been very effective in characterization of objects in terms of their self-similar properties, which allows analysis of physical phenomena in disordered structures to become much easier.

A number of fractal geometry models have been developed by many researchers. However, most of these models include continuous parameters, which were against the basic property of fractal geometry that fractal objects are discontinuous. As alternative approaches, fractional and fractal derivatives have been found effective in modeling permeation problem in hierarchic porous media [14–16]. There are many definitions of fractional derivative, but most fractional derivatives are very complex for engineering applications [17]. In order to better model an engineering problem in a discontinuous media, a new derivative is much needed. Comparing it with fractional derivative, fractal derivative is a much easier and efficient new method in developing the fractal model to deal with these complex problems [18].

In this work, a fractal model for air permeation in hierarchic porous media was developed based on a new fractal derivative theory. And the excellent gas permeation of cocoon was investigated with assistant of the novel fractal derivative model.

#### 2. Definition of the Fractal Derivative

Chen [19] firstly developed fractal derivative from fractal concept defined as

As an alternative modeling formalism of fractional derivative, fractal derivative presents the fractal space-time transforms to display explicitly how the fractal metric space-time influences physical behaviors in statistical description of the anomalous diffusion in diverse engineering fields.

Recently, He [20] introduced a new fractal derivative defined as where is a constant, is the fractal dimension, is the smallest measure, and is the distance between two points in the porous media. The distance between two points in a discontinuous space is where is a constant.

The new fractal derivative can convert fractional differential equations to ordinary differential equations, so that nonmathematicians can easily deal with fractional calculus.

#### 3. Fractal Model for Gas Permeation in Hierarchic Porous Media

In continuous media, gas permeation obeys Fick’s equation: where is the concentration gradient of gas and is the diffusion coefficient of gas.

When the concentration of gas does not change with time, the gas diffusion can be regarded as in one-dimensional steady-state case. Equation (3.1) became The solution of (3.2) is where is the gas flux and is the initial gas concentration.

Equation (3.2) depicts the gas permeation in a continuous diffusion path. However, gas diffusion path through the hierarchic porous media was rather different.

Gas permeation in the discontinuous hierarchic porous media can be expressed as where is the gas concentration in porous media and is the gas flux in the porous media.

According to the definition of fractal derivative where tends to zero, but to the smallest measure size , is a constant.

Submit (3.5) into (3.4), we get the solution of (3.4), where is the diffusion coefficient of gas, is the initial gas concentration and is the fractal dimension of the gas path.

Suppose the gas concentration of the inner side of a porous medium equals to that of the outer side of the porous medium, that is, ,. Dividing (3.6) by (3.3), we obtain Equation (3.7) indicated that the gas flux strongly depends on the configuration of the gas diffusion path.

#### 4. Discussion

For (3.7), if the fractal dimension of the porous medium , then . The porous medium has no effect on preventing the gas to permeate through the medium. Air can permeate through the medium freely as if the medium does not exist.

For the case the smallest measuring size is small, , that is, .

If , then , which means that gas permeation became even faster with the existence of the porous medium.

If , then , which means that the hierarchic porous medium performs a blocking effect on gas permeation.

To investigate the novel gas diffusion phenomenon of cocoon, we need first to analyze the structure of cocoon. SEM observation shows that the cocoon was composed of mainly three layers. Silk fiber in the out layer of cocoon wall was relatively thick with a mean diameter of 26 *μ*m, while the silk fiber in the inner layer was relatively thin with a mean diameter of 16 *μ*m. Moreover, the thin fiber in the inner layer of cocoon wall was densely packed, resulting in the tiny pores for gas and vapor diffusion [4]. The size of pores in different layers along the thickness direction of cocoon wall gradually changed, which endows the cocoon with a hierarchic gas path, shown in Figure 1.

In Figure 1, the rings stand for the pores at different fiber layers of cocoon. The bars between the rings stand for the gas path between different layers in the cocoon, and the length of the bar is equal to the fiber diameter. Suppose the included angle between two bars is , then the fractal dimension of the gas diffusion path in cocoon can be calculated as The constant can be expressed as where is the bifurcation number, is the number of hierarchic level, is the length of gas path in the primary level, and is the length ratio of gas path between the daughter level and parent level.

Considering the construction of cocoon, the bifurcation number equals to 2. Since the thickness of the cocoon is about 500 *μ*m and the average diameter of fiber is about 21 *μ*m, the cocoon is composed of about 24 layers of fiber, which means that the number of hierarchic level . The length of gas path in the primary level is . Suppose the diameter of silk fiber gradually decreases in a linear relationship from the outermost layer of fiber to the innermost layer of fiber, the length ratio of gas path is . In this case, the smallest measuring scale of the gas diffusion path is , which is equal to the smallest diameter of silk fiber. Submit the above value into the following equation, we obtain
The value of (4.3) is in close proximity to the fractal dimension of the gas diffusion path in cocoon, indicating that the porous medium of cocoon has almost no effect on preventing air or moisture vapor from passing through the multilayer silk fiber mat. And the air flux ratio became

The value of (4.4) is close to 1, but slightly larger than 1, which suggests that the air flux of cocoon is slightly larger than that of the continuous medium. The cocoon performs excellent gas permeability. In addition, the cocoon can somewhat promote exchange of gas between both sides of the cocoon wall.

The result obtained in this presentation can explain the experimental phenomenon observed by Blossman-Myer and Burggren. The gas permeation property of cocoon is attributed to the inner hierarchic porous configuration constructed by hierarchic assemble of silk fiber. The structure feature of cocoon provides us with an optimized template, which could be duplicated in biomimic fabric design to improve the heat-moisture comfort ability of apparel.

#### 5. Conclusions

A fractal derivative model for gas permeation has been proposed for the first time in hierarchic space. This model is able to describe a complex dynamic process completely from the theory, which is of critical importance for biomimic design in any fields, such as industry and biomaterial, functional textiles. The validity of the model has been proved in explaining the fascinating gas diffusion phenomenon of cocoon discovered by scientific experiments.

#### Acknowledgments

The work is supported by National Natural Science Foundation of China under Grant no. 51203114 and 10972053, China Postdoctoral Science Foundation Grant no. 2012M521122, Natural Science Funds of Tianjin under Grant No. 12JCQNJC01500, Subject of Science and Technology Development Fund at Universities in Tianjin under Grant no. 20110317, Natural Science Funds of Zhejiang Province under Grant no. 2012C21039, and PAPD (A Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions).

#### References

- J. Fan, Z. Luo, and Y. Li, “Heat and moisture transfer with sorption and condensation in porous clothing assemblies and numerical simulation,”
*International Journal of Heat and Mass Transfer*, vol. 43, no. 16, pp. 2989–3000, 2000. View at: Publisher Site | Google Scholar | Zentralblatt MATH - J. Fan and X. Wen, “Modeling heat and moisture transfer through fibrous insulation with phase change and mobile condensates,”
*International Journal of Heat and Mass Transfer*, vol. 45, no. 19, pp. 4045–4055, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH - X. Qian and J. Fan, “A quasi-physical model for predicting the thermal insulation and moisture vapour resistance of clothing,”
*Applied Ergonomics*, vol. 40, no. 4, pp. 577–590, 2009. View at: Publisher Site | Google Scholar - B. Blossman-Myer and W. W. Burggren, “The silk cocoon of the silkworm, Bombyx mori: macro structure and its influence on transmural diffusion of oxygen and water vapor,”
*Comparative Biochemistry and Physiology*, vol. 155, no. 2, pp. 259–263, 2010. View at: Publisher Site | Google Scholar - S. Weisman, H. E. Trueman, S. T. Mudie, J. S. Church, T. D. Sutherland, and V. S. Haritos, “An unlikely silk: the composite material of green lacewing cocoons,”
*Biomacromolecules*, vol. 9, no. 11, pp. 3065–3069, 2008. View at: Publisher Site | Google Scholar - L. Z. Zhang, “A fractal model for gas permeation through porous membranes,”
*International Journal of Heat and Mass Transfer*, vol. 51, no. 21-22, pp. 5288–5295, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH - Q. Zheng, B. Yu, S. Wang et al., “A diffusivity model for gas diffusion through fractal porous media,”
*Chemical Engineering Science*, vol. 68, pp. 650–655, 2012. View at: Publisher Site | Google Scholar - T. E. Rufford, S. Smart, G. C. Y. Watson et al., “The removal of CO
_{2}and N_{2}from natural gas: a review of conventional and emerging process technologies,”*Journal of Petroleum Science and Engineering*, vol. 94-95, no. 9, pp. 123–154, 2012. View at: Publisher Site | Google Scholar - M. Gassner, R. Baciocchi, F. Maréchal, and M. Mazzotti, “Integrated design of a gas separation system for the upgrade of crude SNG with membranes,”
*Chemical Engineering and Processing*, vol. 48, no. 9, pp. 1391–1404, 2009. View at: Publisher Site | Google Scholar - M. Giona, A. Adrover, W. A. Schwalm, and M. K. Schwalm, “Exact solution of linear transport equations in fractal media- III. Adsorption and chemical reaction,”
*Chemical Engineering Science*, vol. 51, no. 22, pp. 5065–5076, 1996. View at: Publisher Site | Google Scholar - Y. Shi, H. Wu, S. Quan, J. Xiao, and M. Pan, “Fractal model for predicting the effective binary oxygen diffusivity of the gas diffusion layer in proton exchange membrane fuel cells,”
*Journal of Power Sources*, vol. 195, no. 15, pp. 4865–4870, 2010. View at: Publisher Site | Google Scholar - R. Wu, Q. Liao, X. Zhu, and H. Wang, “A fractal model for determining oxygen effective diffusivity of gas diffusion layer under the dry and wet conditions,”
*International Journal of Heat and Mass Transfer*, vol. 54, no. 19-20, pp. 4341–4348, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH - B. B. Mandelbrot,
*The Fractal Geometry of Nature*, W. H. Freeman and Company, New York, NY, USA, 1983. View at: Zentralblatt MATH - W. Chen, H. Sun, X. Zhang, and D. Korošak, “Anomalous diffusion modeling by fractal and fractional derivatives,”
*Computers and Mathematics with Applications*, vol. 59, no. 5, pp. 1754–1758, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH - D. Shou, J. Fan, and F. Ding, “A difference-fractal model for the permeability of fibrous porous media,”
*Physics Letters A*, vol. 374, no. 10, pp. 1201–1204, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH - M. D. Qassim, K. M. Furati, and N.-E. Tatar, “On a differential equation involving Hilfer-Hadamard fractional derivative,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 391062, 17 pages, 2012. View at: Google Scholar - A. Ashyralyev, “A note on fractional derivatives and fractional powers of operators,”
*Journal of Mathematical Analysis and Applications*, vol. 357, no. 1, pp. 232–236, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH - Q. Wang, J. He, and Z. Li, “Fractional model for heat conduction in polar bear hairs,”
*Thermal Science*, vol. 15, pp. 1–5, 2011. View at: Publisher Site | Google Scholar - W. Chen, “Time-space fabric underlying anomalous diffusion,”
*Chaos, Solitons and Fractals*, vol. 28, no. 4, pp. 923–929, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH - J. H. He, “A new fractal derivation,”
*Thermal Science*, vol. 15, supplement 1, pp. S145–S147, 2011. View at: Publisher Site | Google Scholar

#### Copyright

Copyright © 2012 Jie Fan and Jihuan He. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.