Permeability of mono-and bi-dispersed porous media

In this study, the permeability of monoand bi-dispersed porous media is considered. The effects of the particle size distribution and the packing structure of particles on the permeability are investigated experimentally and analytically. Both experimental and analytic results suggest that the particle size distribution is close to the log-normal distribution, and the permeability of the mono-dispersed porous media quasi-linearly decreases as the range of the particle size distribution increases. On the other hand, the effect of packing structure of particles on the permeability is shown to be negligible. The permeability of the bidispersed porous media quasi-linearly decreases as the range of cluster size increases, and nearly independent of the particle size distribution. The present model is valid over the range of parameters typically found in heat transfer applications.


Introduction
Porous media with sintered microsized particles, have received considerable interests for heat transfer applications due to its ability to provide high capillary pumping pressure and high effective thermal conductivity [1,2].The heat pipe is a well-known heat transfer device which is based on the capillary pumping of the porous media [3].A key parameter that determines the capillary pumping ability of the porous media is the permeability [4], which relates the velocity of the flow and the applied pressure drop [5].In order to predict the permeability of the porous media, the Blake-Kozeny's equation (equation 1) has been widely used [6].
where K is permeability, ε is porosity, and d is the particle size.As shown in equation 1, the Blake-Kozeny's equation assumes that the particle diameter is constant.However, the microsized particles that constitute the porous media have a significant size distribution since they are generally produced by high pressure atomization and subsequent sieving process.Therefore, the Blake-Kozeny's equation may result in a significant error when the range of the particle size distribution is not negligible compared with its mean value [8][9][10].
In addition, the packing structure of particles in sintered porous media is different from that of the packed bed and it is highly dependent on the sintering process.For instance, in the sintered porous media, the particles are not connected at a point contact but have bonded areas.To the authors' knowledge, no investigation has been performed to address the aforementioned concerns.In order to accurately predict the permeability of the sintered porous media, a model which takes into account the effects of particle size distribution and the packing structure should be developed.
This study aims to investigate the permeability of the sintered porous media with taking into account the particle size distribution and the packing structure.In this study, two distinct types of porous media are considered: The mono-dispersed porous media and the bi-dispersed porous media.The mono-dispersed porous media are conventional types of porous media, which consist of particles.The prefix 'mono' indicates that the microstructure has one characteristic length represented by the particle size.On the other hand, the bi-dispersed porous media have two distinct characteristic lengths and generally consist of agglomerated clusters of small metal particles.The two characteristic lengths indicate the particle size and the cluster size, accordingly.The bidispersed porous media are known to provide attractive properties such as high permeability and efficient vapor escape.Typical configurations of mono-and bi-dispersed porous media are illustrated in figure 1.In this study, the effects of the particle size distributions and the packing structure of mono-and bi-dispersed porous media are investigated experimentally and analytically.The porous media with various particle size distributions are prepared by sieving the raw particles into various fractions.The size distribution of the particles that constitute each porous medium is quantitatively measured using a particle size analyzer and the distribution parameters are extracted from the result.Based on the estimated particle size distribution, an analytic model for predicting the permeability of porous media is developed.In addition, the effect of the packing structure of the particles on the permeability of the porous media is investigated.

Experiment
To fabricate the mono-and bi-dispersed porous media, spherical copper particles (ACupowder, Inc.) are sintered into a porous medium in a high temperature furnace with a reducing atmosphere of nitrogen and hydrogen.The mono-dispersed porous media is fabricated by directly sintering the as-received particles.To fabricate the bidispersed porous media, the mono-dispersed porous medium is grinded into clusters, sieved into specific sizes, and sintered again.In this study, cylindrical samples are fabricated with 11 mm diameter and 30 mm length.Figure 1 shows SEM images of the fabricated monodispersed and bi-dispersed porous media.The particle size and a measure of the size distribution range normalized by its mean value are summarized in Table 1, where D p is mean particle diameter, and ΔD p * the standard deviation of particle size distribution normalized by the mean particle diameter.Accordingly, D c and ΔD c * (Table 2) are the mean cluster size and the standard deviation normalized by mean cluster size, respectively.In this study, 9 mono-dispersed porous media and 10 bidispersed porous media (D p = 117 μm and D c = 675 μm) are tested.The size distributions of the particles and clusters are determined using the particle size analyzer (Beckman Coulter, LS230) which is based on the laser diffraction principle.
The experimental apparatus is illustrated in figure 2. It consists of a constant temperature water bath, a micro manometer, a mass flow meter, a PC, and stainless steel tubing.Water flow is maintained at a constant temperature using the water circulation bath (JeioTech, HTBC-2320AT).A mass flow meter (MFC, WinTEC) is used to measure the flow rate.Its accuracy and repeatability are ±1% and ±0.15%, respectively.The stainless steel tube and the glass tube which is the shell of the test sample are carefully attached each other and sealed.The outer diameters of stainless steel tube and glass tube are both set to be 12.7 mm to eliminate the head pressure loss.To measure the pressure at the inlet and the outlet of the test sample, a micro-manometer (KIMO, MP112) is used.The measurement range and the accuracy of the micro-manometer is 200,000 Pa and ±0.5%, respectively.The velocity at the test sample can be readily evaluated from the mass flow rate and the cross sectional area of the test sample.From the measured velocity and the pressure difference between inlet and outlet, the permeability of the test sample can be obtained using the Darcy's equation: where K is permeability, u is the area-averaged velocity of water, μ is dynamic viscosity, ΔP is pressure loss along the sample, and Δx is the length of the sample.

Analysis
To predict the permeability of packed beds consisting of multi-sized particles, MacDonald et al. proposed an analytic expression based on the probability density theory [12]: where ε is the porosity, and M i is ith moment of the particle size distribution defined as follows: where D indicates D c or D p , and f(D) represents the particle size distribution (fraction of particles whose diameters range between D and D + dD).As the particle size distribution becomes narrower, the ratio M 2 /M 1 in equation 3 becomes close to a mean particle diameter and thus, this equation approaches the Blake-Kozeny's equation.
The log-normal distribution is often used to approximate the particle size distribution of atomized particles, aerosols, aquatic particles and pulverized material [13].
where D m is median value, σ is deviation constant and A is curve area.M 1 and M 2 can be obtained by using equations 4-5.Mean, mode and standard deviation of the log-normal distribution can be obtained in analytic forms as shown in the following equations, respectively [14].
In Table 3, analytically (equation 6-8) and experimentally obtained distribution parameters are compared.As shown in this table, they match each other well.Figure 3 shows the experimentally obtained particle size distributions with best fits provided by equation 5.The mean absolute deviation (MAD) of the best fit is determined to be less than 4%.Therefore, the size distribution of sieved particles can be well-characterized by the log-normal distribution.In the numeri fix the Reyno in micro-flow Figure 7 curve repres shown in thi shown to pre media well re bonded area.which has be can be also u have irregula zero particle particle size d an accurate pr

Fig. 1
Fig. 1 SEM images of fabricated porous media

Fig. 2
Fig. 2 Schematic diagram of the experimental apparatus

Fig
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Table 1 .
Size ranges of fabricated mono-dispersed porous media

Table 3 .
Comparison between analytically and experimentally obtained distribution parameters Analysis Experiment D mode D mean σ stan D mode D mean