transport phenomena in porous media iii || pore-scale transport phenomena in porous media

1 4 PORE-SCALE TRANSPORT PHENOMENA IN POROUS MEDIA

X.R PENG and H.L.WU

Laboratory of Phasechange and Interfacial Transport Phenomena, Department of Thermal Engineering, Tsinghua University, Beijing 100084, China

email: pxf~dtefimail.tsinghua.edu.cn and hailing.wufigmail.com

Abstract

The conjugated transport phenomena with pore and matrix strucnire widely exist in both the natural world and practical applications. It is of critical significance to understand these phenomena accounting for the dynamical processes and structure deformation taking place in the inner pores, as one of the mo.st important topics in the area of porous media. In this chapter, a series of different experimental observations and associated theoretical investigations have been conducted to understand the transport phenomena at the pore-scale level, including the transport phenomena with/without phase change and chemical reaction, and concerning a wide range of practical applications.

Keywords: pore scale, conjugated transport phenomena, boiling, bubble dynamics, two-phase flow, micro-CT, bead-packed structure, sludge, bio-tissues

14.1 INTRODUCTION

Transport phenomena in porous media commonly exist in the natural world and in various engineering applications. Classical porous media transport theory is based on a continuum approach, applying a statistical volume averaging method with the concept of 'representative elementary volume'. The associated formulations are primarily used to predict integral macroscopic characteristics of porous media, see Lin (1995), Ingham and Pop (1998) and Boer (2000). However, as is well known, the permeability and transport in porous media is in close relation with the detailed pore geometries, such as pore scale, shape, connectivity and tortuousness, see Dullien (1992) and Sahimi (1995), and can rarely be derived directly from macroscopic parameters such as porosity. The state-of-the-art in diverse scientific and technologic fields has endowed a more abundant meaning

366

X. F. PENG AND H. L. WU 367

and implied numerous motivations for research, with a variety of applications such as compact heat exchangers, composite materials, drying processes, and environmental and biochemical applications. Although porous media cover different scales, modeling should be initiated at the first scale of interest, typically the pore-scale, as a basis for successively upscaling to larger scales.

Recently there has been an explosion of interest in pore-scale modeling to meet the urgent demand of fundamental insight into transport mechanisms in porous media, see Blunt (2001). In many heat transfer applications, microstructure design of heat transfer enhancement is one of the key subjects in compact heat exchangers and micro fluid assemblies. The understanding of heat and mass transfer mechanism is sought to construct a tailored flow structure for optimal heat transfer. In composite casting processes, the macro physical and mechanical characteristics are controlled by the reinforcement-matrix microstructure and the morphology of the solidification interface evolution, see Michaud and Mortensen (2001). In the petroleum industry and environmental applications, the pore-scale displacement mechanisms for both imbibition and drainage are of great importance for predicting the percolation of oil and controlling underground pollutant diffusion. The knowledge of pore-scale transport phenomena is actually the foundation for all involved multi-physical and/or multi-scale coupling processes. Consequently, investigations on transport in porous media are no longer limited to the prediction of macroscopic parameters (permeability, porosity, etc.) of single phase or simple two-phase flow. Instead it is recognized as a comprehensive platform to adequately explore a far-ranging scope of phenomena. More and more attention is being paid to representative microstructures of the porous volume, and the pore-scale transport behavior in porous media, including capillarity, phase change, fluid-structure interaction, activated processes, etc.

As far as research work is concerned, understanding of pore-scale transport is divided into two aspects. Firstly, advanced microscopic techniques are necessary to measure and analyze the integrated three-dimensional information of pore-scale microstructures. Secondly, instead of macroscopic and simple pore geometries, predictive micro-modeling is used to develop the essence of the heat and mass transfer in complex pores in order to better understand the underlying transport mechanism in porous media, and to predict macroscopic behavior of various complicated porous media. Of course, this investigation would be highly dependent upon experimental technology, observations and measurements.

In this chapter a series of different experimental observations and associated theoretical investigations are presented on the pore-scale transport phenomena concerning a wide range of practical applications with/without phase change and chemical reaction.

14.2 CONJUGATED TRANSPORT PHENOMENA WITH PORE STRUCTURE

In this chapter we consider transport phenomena in porous media both with and without phase change. Chemical and biochemical reactions are usually conjugated with a pore structure variation, such as reduction of SO2 and NOx emissions. In these processes, the associated transport phenomena are highly dependent upon the pore structure. Inversely,

368 PORE-SCALE TRANSPORT PHENOMENA IN POROUS MEDU

the transport processes easily alter the pore structure and further influence the transport characteristics. In this section, the typical experimental evidence is presented with a detail discussion of the conjugated transport phenomena with pore structure.

14.2J Conjugated phenomena in sludge drying

As a typical example, the drying of sludge cakes has been investigated. The test facility, shown in Figure 14.1, consisted of an air duct, and data acquisition and image acquisition systems, with a steady air flow supplied by a fan, and a heating rate of the oncoming air controlled by an electrical heater. One thermocouple was arranged in the duct to measure the air temperature and another in the middle of the sludge cake tested to measure the interior temperature. Both the relative humidity and velocity of the air flow were measured by a humidity meter and an anemometer. The test samples were mechanically dewatered sludge cakes, with moisture content of 80-82%. Each was placed in an open container (80 mm in diameter and 5-13 mm in height), and was set on an electronic balance. Therefore the real-time weight of the sample was monitored. A CCD camera (WAT-505EX) connected to a personal computer recorded the appearance of sludge cakes for further processing by an image analyzer. The moisture distributions of the tested sludge cakes were continuously monitored during whole test by weighing and drying small pieces of samples collected from the dried cake. As a comparison, the sand cake samples were also performed under similar test conditions. The test conditions and results are illustrated in Table 14.1 and Figure 14.2, see Chen et al. (2002).

Figure 14.2 presents consecutive photographs of the cracks growing on the cake surface for cases a-2 and a-3. In the first 15 minutes, the sludge cake showed no noticeable change in its appearance, except for many initial defects or small seams randomly emerging on the surface. From 15 to 60 minutes, some of these small seams gradually grew up and

Temperature and i moisture sensor |

CCD

Anemometer

3 ^ t 213.5g P

Electronic balance

Figure 14.1 Experimental set-up.


Table 14.1 Experimental conditions and results of sludge.

Cake type

Sludge

Sand

*DS =

Case no.

a-1 a-2 a-3 a-4 a-5

S-1 S-2 S-3 S-4 S-5

dry sand

Wind Speed (m/s)

2.3 2.2 2.9 3-0 3.0

2.1 2.1 2.0 2.1 2.2

Wind temperature

CQ

30.0 40.0 41.0 41.0 40.5

40.2 39.9 39.6 42.6 40.9

Relative humidity (RH%)

26% 20% 21% 26% 14%

28% 23% 21% 28% 29%

Initial thickness

(nmi)

8 8 8 5 5

< 0 . 1 0.154^0.2 0.2--0.25

0.28<-0.35 0.355'N^0.45

Bound water content

(kg/kg-DS*)

2.4 2.1 1.8 1.8 1.8

0.025 0.025 0.025 0.03 0.025

Constant drying rate

(g/m^^ min)

26^0 32.0 35.0 33.0 34.0

23.5 24.0 26.0 24.0 22.5

tc (min)

80^ 85 95 80 80

145 145 140 170 160

Figure 14.2 The cracks growing during the first hour for (a) the case a-2, and (b) the case a-3, at (i) t = 15 min, (ii) t = 30 min, (iii) t = 45 min, and (iv) t = 60 min.

became cracks, and extended to meet others or the edge. Finally these cracks divided the cake into several smaller piles. In the following process, each part experienced individual drying, accompanied with further shrinkage. It is interesting to note that not all small seams become cracks. Some of them were initially preponderant in the drying and expanded faster than adjacent ones. Competition mechanism of the crack growth was highly depended upon the inner pore structure and moisture content. The cracks grew more rapid where there were favorable pore structures for water evaporation. The evaporation and the crack growth were mutually stimulated. Consequently, the preponderant seams or larger pores expanded as the moisture evaporated and escaped, resulting in much easier growth and the development of cracks. In contrast, the smaller pores in the adjacent zone were suppressed and remained almost unchanged, see the circled ones in Figure 14.2.

370 PORE-SCALE TRANSPORT PHENOMENA IN POROUS MEDIA

Zhang and Peng (2004) have investigated the inner pore structure behavior with moisture transport in a cake using a micro-CT, see Figure 14.3. They found that the pore structure of a dried cake was much altered after it had fully absorbed the water when compared with the original one. Many small inner pores were destroyed, while a few of the larger pores were extended. The pore structure of the original cake was more uniform than that of the dried cake. After drying, the solid matrix was denser and the pores became larger.

Figure 14.4 depicts the drying rates, averaged over a 4 minute period, of the sludge and sand cakes. For sand cakes, although there is a large data scatter, these curves resemble a conventional drying curve composed of a constant-rate period and a reducing-rate period, and the drying rates for the constant-rate period ranged from 26-36 g/m^ min. For sludge cakes, the humidity played a more crucial role, and the constant-rate period could be regarded as being evaporated at saturated condition (relative humidity, RH = 100%). Compared with sand, the drying curve fluctuated, and the transition from a constant-rate to a reducing-rate period cannot be clearly identified. Both Figure 14.4 and Table 14.1 indicate a much higher drying rate in the constant-rate period than that of sand cakes under similar conditions.

Vaxelaire et al. (2000) derived a parameter, namely the drying potential or DP* = 0.758w^'^DP, to evaluate the external drying conditions for a uniform media, and DP is the difference between the chemical potential of the saturated vapor at the wet bulb temperature (temperature at the surface of the wet material) and the vapor temperature at the dry bulb. The drying theory indicates that the constant drying rate is controlled by external drying conditions. However, the drying rate of sludge was larger than that of sand by 20'^40%, see Figure 14.5. This may be caused by the incidental cracking and associated changes of the inner structure in sludge cakes.

Accounting for the extra evaporation from cracks and other important effects, Chen et al (2002) explained the phenomena and modified the correlation for calculating the drying rate. The cracks on the surface increased the drying area and the change in the pore structure near the cracks significantly affected the escape of moisture. Figure 14.5 illustrates their results for the drying rates of sand cakes, sludge cakes and sludge cakes not

(b)

Figure 14.3 Images of the inner pore structure of a cake: (a) the original cake, and (b) a dried cake after fully absorbing water.


(a) 45

40

^ 30

"B) 25

I 20 £ 15

Q 10

5

0

t

(b) 30 H

^ 25

20

S 15

S. 10

—o— —A—

—V—

- 0 -

a-2 a-3 a-4 a-5

50 100 150

Time (min)

200 250

40 80 120

Time (min)

160 200

Sand 1 (<0.1mm) Sand 2 (0.154-0.2 mm) Sand 3 (0.2 ~ 0.25 mm) Sand 4 (0.28 ~ 0.35 mm) Sand 5 (0.355 ~ 0.45 mm)

Figure 14.4 The drying rate of (a) sludge cakes, and (b) sand cakes.

including cracks under the same drying potential. The drying rates of sludge cakes, not including cracks, are almost equal to those of the sand cakes. The drying of sludge cakes in the constant-rate region was still controlled by external conditions. The discrepancy in the drying rates, both with and without consideration of cracks was highly dependent upon the drying capability of the cracks on the cake surface.

14.2,2 Effect of inner evaporation on tlie pore structure

Conjugated transport phenomenon clearly exists in the loose porous media, and also in relatively dense or non-deformable matrix media. Hydration and calcination is a technology to reform appropriate pore structure inside the sorbent due to explosive evaporation and


C\J

36 1

34

§ 32 H

5 30

I 26H c o ^ 24

. • a - 3

a-2 a-4

a*-3

- • - Sand cakes - • - Sludge cakes

* Sludge cakes not including cracks

J S-1

280 300 320 340 360 380

DP* (kJ/Kg)

Figure 14.5 Constant drying rates as a function of the drying potential.

rapid escape of the water contained in or reactively released from hydrated CaO, and consequently improve the desulfurization performance, as one of its important applications, see Fu and Peng (2004). The origin material used in this study was a kind of pure Ca(0H)2 and was calcined in a furnace at a temperature of 700 °C for 15 minutes. CaO rapidly reacted with H2O to form Ca(0H)2 again in a sample container at a constant temperature. Five slurry samples (mol ratio of H2O to CaO) were selected, namely 1.40, 2.08, 2.77, 3.60 and 6.86, to produce hydrated samples. The slurry samples were directly calcined at 560 ""C, 760 °C and 950 ''C, and were intensely decomposed, and the contained water was evaporated and quickly escaped. This powerful process made a significant inner pore reconstruction in CaO particles.

Figures 14.6 and 14.7 illustrate the measured cumulative area and volume of the samples, respectively. The pore cumulative area performed a clear decreasing trend as the calcining

40 r

Mol ratio of H2O to CaO

Figure 14.6 Cumulative area of samples.


Mol ratio of HgO to CaO

Figure 14.7 Cumulative volume of samples.

^ "*" -^ -K-

560°C 760°C 950°C Ca(0H)2

100

Pore diameter (nm) 1000

100

Pore diameter (nm)

Origin 560°C/1.40 560°C/2.08 560°C/2.77 560°C/3.60 560°C/6.86

Origin 950°C/1.40 950°C/2.08 950°C/2.77 950°C/3.60 950°C/6.86

Figure 14.8 Pore size distribution calcined at (a) 560 " C, and (b) 950 ° C.


temperature was increased. CaO sample calcined at 950 °C obtained a much lower pore area than the origin one, while at a relatively lower calcinging temperature effectively enhanced the sample pore area. The mol ratio of H2O to CaO had a strong effect on the cumulative volume and the pore volume was significantly increased when the slurry samples were made with a weight mol ratio greater than about 2.77. Meanwhile, the CaO sample, calcined at 950 °C and hydrated at ratio of 2.08, exhibited a possible negative effect on the pore volume.

The pore distributions in the form of an incremental volume as a function of the pore diameter are presented in Figure 14.8. Those samples had three significant characteristics. First, the intensive heat and mass transfer made a considerable pore volume constriction. The dominant pore size region tended to be larger as the calcining temperature increased. Second, the small pores, diameter between 10 and 50 nm, decreased as the calcining temperature and mol ratio increased. In Figure 14.8(b) the pores of 10 to 40 nm for the original sample almost disappeared after the treatment. Third, large pores, 500 to 1000 nm, were considerably decreased. As the weight mol ratio reached 2.77, these phenomena became quite significant. Apparently, the intensive and rapid quick-evaporation and capillary effects are considered to be the important factors for the reconstruction of the micro pore configuration. When the water contained was evaporated quickly, the rapid decompounding of Ca(0H)2 also produced steam. The explosive release of steam induced an intense expansion of the micro pores. Meanwhile, the reduction of the liquid in the pores also produced a strong capillary effect, which inevitably caused the collapse of some of the larger pores.

14,3 TRANSPORT-REACTION PHENOMENA

14.3.1 Reaction in a porous solid

In solid-gas transport-reaction systems, gas reactant diffuses into the solid pore holes and simultaneously reacts with the solid reactant. Generally, the reaction performs in an interfacial zone where the solid reactant and solid product co-exist. These transport-reactions are expected to be associated with the pore structure. Figure 14.9 illustrates a series of SEM photos of the CaO particle reacting with SO2 at different times, see Yan (2003). The change of porous structure greatly influenced the gas diffusion and absorptive reaction, see Ishida and Wen (1968), and Wen and Ishida (1973). For a chemical reaction between a gas reactant (A) and a solid reactant (B), aA(g) + 6B(s) = cC(g) -f dD(s), Yan et al. (2003) presented a modified dual-zone model, assuming that the reaction at the second stage occurred in all the inner pores rather than reacted from the surface into the inner region, see Figure 14.10. As a result, the second-stage reaction is dependent upon the inner pore distribution.

Solving the mass transfer equation of gas reactant, the dimensionless reaction time of the surface product layer formation, and corresponding conversion rate, were obtained as

X. F.PENG AND H.L.WU

(b)

375

Figure 14.9 SEM photos of CaO absorbing SO2 at (a) i ^ 0 min (before reaction), (b) t = 10 min, (c) t = 30 min, and (d) t = 60 min.

Solid reactant

Reaction product

Figure 14.10 Modified dual zone reaction model: (a) first stage, formation of surface product, and (b) second stage, reaction in pores.

follows:

9c = 1 + Th[cosh(Th)/s inh(Th)]- l

Sh and X =

sinh(Th) K- < " • "

where the Thiele number Th = rQ{akyCBo/bDey^^ and the Sherwood number Sh = aoro/De. The Thiele number was explained to be the ratio of the chemical reaction to the diffusion rate. Crystallinely, the transport-reaction is controlled by the reaction resistance for Th < 1, while it is dependent on the diffusion when Th > 1. For the cases Th ~ 1,

376 PORE-SCALE TRANSPORT PHENOMENA IN POROUS MEDU

both the chemical reaction and diffusion are important. This demonstrates that the pore structure plays a significant role in the transfer-reaction processes.

During the second stage, the mass balance is given, see Yan (2003) and Yan et al. (2004), as follows:

dl {DyJ^)=2C^

dl J 1 \n{ry;/rpt)

A/7*-III Ue

-1

— f^sÂ ) (14.2)

where kg is the effective reaction constant, r^ and Vpt are the radii of the reacting surface and pore surface at time t, respectively, and Ds is the gas diffusive coefficient in the product layer. The rate of change in the pore radius is derived as follows:

dr. It

p _ ksCAd 2ar„ PD dpB J '

(14.3)

where PB and po are the density of the solid product and reactant, respectively. Accounting for the pore structure, the local reaction conversion rate, or the ratio of the reacted and original solid reactant for any pore rô in solid particle(s), is defined and derived as follows:

î^pO^t) = 7r{rl-rlQ)MipQ ^0

Accordingly, the total conversion rate is given by

a{r po) \rpoJ

where the pore distribution parameter is defined as follows:

; )

(14.4)

(14.5)

a(rpo) _ Virpo) _ r\ pO AZ

Vtotal Erpo^pO^^ (14.6)

Apparently, the final conversion rate and formation time of the product layer is mainly determined by the initial porosity. The pore structure, including the average pore radius and distribution, significantly affects the dynamic characteristics of the transfer and reaction processes.

14.3.2 Experimental investigation

In the experiments conducted, the reaction of SO2 with solid CaO particles was investigated using a thermogravity analyzer. The pore distributions of the samples were analyzed using a mercury porosimeter. The samples investigated have average diameters of about 4 jum and cumulative special pore volume is 0.40cm^/g (sample 1) and 0.39cm^/g (sample


2). The reacting temperature was 800 °C and the reaction period was 80 minutes. The concentration of SO2 in the flue gas was 2000 ppm in the facility shown in Figure 14.11.

Figures 14.12(a,b) illustrate the distribution of the pore characteristic parameters with the pore diameter, both having a very close final pore volume and quite different pore fraction

A/o

Flue gas

Balance \ Weight signal

Baffles

i Furnace

-.+ Temperature signal

Computer

"I

Figure 14.11 Diagram of the TGA system.

(a) 0.40

10 100 Pore diameter (nm)

Cumulative special pore volume

Pore fraction

(b) 0.40 r

10 100

Pore diameter (nm)

1000

Cumulative special pore volume

Pore fraction

Figure 14.12 Characteristic pore distribution for (a) sample 1, and (b) sample 2.


— Sample 1 - - Sample 2

20 30 40 Reaction time (min)

50 60

Figure 14.13 Desulfurization reaction conversion rate.

(a) 0.25

.9 0.20 h O <D

»^ c o o c w . 2 55

11 0.15

•g > 0.10 h

0.05

0.00

- t = . . . t = - - t = — t =

5 min 15min 30 min 60 min

100 Pore diameter (nm)

1000

— t = — t = — t = — t =

5 min 15 min 30 min 60 min

100 Pore diameter (nm)

1000

Figure 14.14 Distribution of the conversion rate for (a) sample 1, and (b) sample 2.

X. F.PENG AND H.L.WU 379

distribution. The final conversion rate was almost the same, while their transient dynamic conversion rates were quite different, see Figure 14.13. Clearly the final conversion was determined by the initial porosity and the dynamic reacting process was highly dependent on the pore distribution. Apparently, the coupled phenomena of the pore structure or distribution with the transport played very important roles in the reaction. Here a distributed conversion parameter is introduced to describe the pore influence on the reaction, i.e.

-fe)Vfe"<-'hfe)1)-(14.7)

This parameter means the ratio of the reacted solid reactant for a pore having an initial radius Vpo and the total reacted reactant at time t. Figures 14.14(a,b) present the distributed conversion parameters at different reaction times for the samples 1 and 2, respectively. Clearly, different pores have quite different reaction behaviors. The smaller pores exert dominant reacting areas at the early stages, while the larger pores make a greater contribution later. Yan (2003) found that this behavior greatly changed the gas transport in the pores due to the variation of the pore structure and distribution during the reaction process, and consequently the understanding of the reaction and transport phenomena at the pore scale would be very useful to explore the characteristics of the high-temperature desulfurization of the flue gases and improve the performance of the SO2 and NOx absorbents.

14.4 BOILING AND INTERFACIAL TRANSPORT

The contribution of interfacial effects on boiling heat transfer is extremely distinct in porous media, where unique phenomena may arise because the bubble interface is significantly affected by the porous structure, see Peng et al. (2002). Wang et al. (2002, 2004) and Wang and Peng (2004) have conducted a series of investigations for the boiling and interfacial phenomena in bead-packed structures.

14.4.1 Experimental observations

For water at ambient pressure and 18.4 °C, consecutive snapshots of a typical bubble growth-collapse cycle are presented in Figure 14.15 at 36.8 kW/m^. The bubble release frequency was very low due to high subcooling. Bubble sites in the narrow-gap comer (narrow region between a bead and surface) were more plentiful than in other regions. Small bubbles from the comers expanded and penetrated into the main cavities formed by neighboring beads, and merged to a primary bubble, see Figures 14.15(b,c). As the primary bubble grew, the pore size became the restriction for bubble growth, and the bubble distorted and elongated, see Figures 14.15(d,e). Subsequently, the bubble was truncated at the neck of the elongated bubble and quickly escaped, see Figure 14.15(f). Then, the pore space was re-occupied by the replenished liquid due to capillary effects,

380

(a)

PORESCALE TRANSPORT PHENOMENA IN POROUS MEDIA

(b) (c) (d)

Figure 14.15 Bubble behavior in a 7 mm bead-packed structure at g" = 36.8 kW/m^ (water) at (a) e = Os, (b) t = 0.15 s, (c) t = 0.36s, (d) t = 0.82 s, (e) t = 1.12s, (f) t = 3.41 s, (g) t - 3.48 s, and (h) t = 3.72 s.

see Figure 14.15(g), and another bubble cycle followed, see Figure 14.15(h). The growth of a fully-developed bubble was mainly controlled by the evaporative heat transfer on the liquid meniscus in the narrow comer. The replenishing liquid was pulled into the comer by the capillary pressure and it maintained an excellent wetting performance. The boiling heat transfer was enhanced with this process.

As the heat flux was increased, approaching to the CHF, the pore-scale phase-change behavior was highly disordered, see Figure 14.16 and Wang et aL (2002). The wall temperature was highly unsteady, and sometimes as high as 200 °C. However, dryout of the entire heated surface could not be observed, even at very high heat fluxes, and dry spots on the heated surface were observed to be rather unsteady because of the intensive *evaporation-replenishment' in the narrow-gap comers. Figure 14.16 illustrates that the bubble interface expanded horizontally. In particular, the pore space near the heated surface was vapor-filled except for the liquid menisci in the narrow-gap comers, and the vapor front on the heated surface tended to expand into the comer. The variation of the bubble interface would result in a capillary pressure gradient for the replenishing-liquid flow, and dry areas were then wetted. Clearly, the evaporation-replenishment became more intensive with increasing heat flux, and the narrow-gap comers were highly resistant

Figure 14.16 Bubble behavior in a 7 mm bead-packed structure at g" = 121.3 kW/m^ (water) at (a) < = 0 s, (b) t = 0.04 s, and (c) t = 0.12 s.


to dryout even at very high heat fluxes. The pore-scale bubbles on a wire in porous structures also exhibit some unique characteristics, see Figure 14.17 and Wang and Peng (2004).

14.4.2 Static description of primary bubble interface

The interface of a fully-developed static bubble is divided into three parts: top, concave and base sections, see Figure 14.18 and Wang et aL (2004). The top section interface is a

Figure 14.17 Bubble behavior on a wire in a bead-packed structure (water, bead diameter 4 mm).

Bubble interface A-A

Bubble

Figure 14.18 Bubble interface model.


spherical coronal, the base section interface is approximately a part of a hemisphere, and the concave section interface is assumed to touch the beads directly. The bubble profile can be described by several configuration parameters as follows:

i?i = (i?2 -f Rp) csc(7r - 62) -Rp, R2 = i?p(sec30° - 1), ]

J?3 = (sec^i - l)Rp , tan01 = R2^Rp (14.8)

Rp

These four relations indicate that the bubble shape is determined by only one parameter, 62, if the particle diameter is specified. For a given fluid, the departing bubble shape can be obtained through a force balance. Neglecting inertial effects, interfacial tension tends to hold the bubble, while the buoyancy acts to draw the bubble away, see Carey (1992). The departing bubble profile corresponding to the departure angle 62d is determined by the following equation, see Wang et al (2004):

27rR2a Ru ^ ' ^ ^^

where Vd and Khade are the volumes of the departing and shaded part of the bubble, respectively. The predictions were found to be in very good agreement with several experimental cases.

14.4.3 Replenishment and dynamic behavior of the interface

For boiling in the bead-packed structure, a downward replenishing-liquid flow is driven by the capillary pressure gradient. Peng et al, (20(X)) indicated that the liquid flow around the bubble is induced virtually by the evaporation and condensation at the bubble interface. The replenishment provides the necessary liquid supply required by the evaporation at the bubble interface in the narrow-gap comer area. With a view of the pores being a system of parallel, equilaterally-staggered capillary ducts, a preliminary analysis was conducted to determine the driving forces for the replenishing-liquid flow using Darcy*s approximation, see Figure 14.19. The following equation was obtained, see Wang et al. (2004):

where /3 is the vapor fraction in the pore, J ' the Leverett function, see Leverett (1941), and correlated by Udell (1983). Equation (14.10) can be integrated numerically to obtain the vapor fraction profile in the pore, or a dynamic description of the bubble interface.

14.4.4 Interfacial heat and mass transfer at pore level

At high heat fluxes, the bubble interface profile, especially the interface in the comer, varies with the applied heat flux. To balance the replenishment flux and the evaporation flux,


• Liquid phase

00 Solid particle

• Vapor phase

Heated wall

Narrow-gap corner area

Figure 14.19 Replenishment and liquid-vapor interface.

an appropriate capillary pressure gradient is established for the flow by the evaporative heat transfer. The vapor fraction increases continuously from the top of the bubble to the heated surface; and correspondingly, a negative capillary pressure gradient, driving a downward liquid flow. As Ca increases, the vapor fraction increases, i.e. the bubble interface expands horizontally, which is in agreement with the experimental observation in Figure 14.16. Consequently, a larger capillary pressure gradient is established for a higher value of Ca to maintain the mass balance between the replenishment and the evaporation, as described by Wang et al. (2004).

The replenishment is induced by the interfacial transport. The associated boiling heat transfer is improved by this 'evaporation-replenishment' mode. Following the analysis of Chien and Webb (1997), the pore-scale interfacial heat transfer can be idealized as the evaporation on the liquid meniscus, see Figure 14.19. The dimensionless evaporative heat transfer coefficient is defined as, see Wang et al (2004), /i* = Ca/AT^, where AT^ = (T^ - Ty)/Tv. As shown in Figure 14.20, as Ca (or heat flux) increasing, h* gradually increases, reaches a peak value, CapA:, and then falls rapidly. For Ca < Ca A;, the increase of h* may partly be attributed to the fact that the evaporating liquid film in the comer region becomes thinner. Additionally, the heat transfer is enhanced by the violent, repeated dryout-rewetting process, which results from the unceasing interactions between the replenishment and the interfacial evaporation. This dynamical process can accelerate the collapse of the dry regions. As the heat flux increases further (Ca > Ca^jt), most regions of the heated surface are covered by vapor, and /i* begins to reduce. The comer region may be surrounded by the vapor phase and the path for the replenishing-liquid flow may be disconnected. If this trend continues, dryout of the heated surface may occur. As


11.5

Dp = 3.5 mm

Figure 14.20 Dimensionless evaporative heat transfer coefficient as a function of the capillary number.

seen from Figure 14.21, the higher heat transfer coefficients are exhibited at the smaller pore sizes. This is due to the liquid replenishment being significantly intensified in smaller pores. However, it can be found that the Cdipk of a 3.5 mm pore structure is much lower than that of a 7 mm pore structure. Clearly, the spatial limitation becomes more serious with decreasing pore size.

Particle

- Interface

Water

Step motor

TL Light source TH

Figure 14.21 Test set-up of unidirectional freezing/thawing.


14.5 FREEZING AND THAWING

The development of modem medicine and bioengineering has increasingly demanded a high technology of cryo-conservation of biomaterials or cells. The demand is very urgent to understand the associated heat and mass transfer phenomena during the freezing/thawing processes, which has a determinative influence on the improvement on the survival rate or conservation quality of the biomaterials, see Rubinsky (1998). The freezing/thawing dewatering of the porous media with a high moisture is also widely employed in environmental and food processing applications. Motivated by these requirements, more attention is being paid to recognize and understand the transport phenomena and the dynamical behavior at the pore scale.

14.5.1 Experimental facility

Figure 14.21 illustrates the experimental facility employed, similar to the classical Bridg-man unidirectional freezing set-up, see Rubinsky (1998). A linear temperature gradient was generated between the high- and low-temperature zones. A semi-conductor served as a heat sink to adjust the temperature gradient and interface advancing rate. The image acquisition system included a microscope, a CCD camera, and a high-speed frame grabber. The image was captured by the image acquisition and transferred to a computer for online observation, see Tao et al (2003,2004a, 2004b) and Wu et al (2004a, 2004b).

14.5.2 Sludge agglomerates during freezing

During the sludge freezing, test slurry was placed in a shallow vessel with a 1 mm thick liquid film. Si02 powders, soil and crumb suspending in the solution were agglomerated as particles with diameter d of 0.02'>-0.5mm, see Tao et al (2004a, 2004b). As the freezing interface advanced, the loose agglomerates were repelled, engulfed or broken up by the advancing interface. There were typically two modes of the agglomerate merging. One is *low-rate merging', where agglomerates were repelled continuously, or repelled to a long distance and accumulated at the freezing interface before being trapped. The other mode is *high-rate merging', where agglomerates were entrapped immediately after being repelled to a short distance (less than the particle/agglomerate diameter), or even being engulfed without repulsion. Tao et al (2003) also observed a similar phenomena for rigid particles.

Structure alternation

Loose agglomerates are featured by their surface and inner structures, which are expected to result in some unique transport phenomena. Figure 14.22 illustrates the behavior of a large agglomerate (d = 300/um) at Vj = 13.8/xm/s. Some scraps were desquamated from the agglomerate surface and scattered along the interface. These scraps were engulfed immediately. From Figure 14.22, the agglomerate was reduced to about 3/4 of its original

PORE-SCALE TRANSPORT PHENOMENA IN POROUS MEDIA

p*

Figure 14.22 (b)t = 70s.

Desquamation from a loose surface of an agglomerate at (a) t = 0 s, and

size when being repelled for about 1 mm. This is expected to be the consequence of the inner structure alternation.

The loose structure of agglomerates also affected the interaction between the freezing interface and the agglomerates. Rigid particle always acted as a whole part at the interface, and therefore once one point of it was frozen, the whole particle would be halted and be engulfed by the interface, see Tao et al (2003,2004a). In contrast, for loose agglomerates, once any part was frozen and engulfed, the other part could still be repelled forward. This caused the agglomerates to be stretched and thinned, or to break up and be reconstructed if the agglomerates could not sustain the increasing inner stress. Additionally, the inside water would have an important action on the inner structure during freezing/thawing. Figure 14.23 presents a soil agglomerate being thinned and finally split into two parts at Vf = 7.8/im/s.

Internalflow

The internal pores in loose agglomerates have a great impact on both the flow field prior to the freezing interface and the interaction between the interface and the agglomerates. Figure 14.24 illustrates the flow pattern in front of the interface around the agglomerate

Figure 14.23 Change of loose stmcture during freezing at (a) t = 150 s, and (b) t = 225 s.


mi 100|im

Figure 14.24 Flow around a bread crumb.

(bread crumb) at Vj = 3.9/xm/s. The flow was traced with some tiny particles {d = 10/xm). Trajectories of these particles were identified from the continuous frames of experimental videos. Some particles went along the border of the agglomerate, and others went towards the agglomerate at a lower speed, indicating a weak flow through the agglomerate. Clearly, such flow penetration would reduce the repulsion force on the agglomerate, but it would exert a significant influence on the inner structure, and contribute to the disintegration of the agglomerate, such as desquamation on the surface, or breakup of the main body.

These observation and investigations demonstrate that the energy transfer through the solid matrix, mass, energy and momentum transport inside the inner pores and their conjugated transport phenomena would play critical roles during the freezing of the porous media, particularly loose porous media.

14.5*3 Botanical tissues during freezing

Wu et al. (2004a, 2004b) employed the facility shown in Figure 14.21, together with Micro-CT technology to observe the freezing-thawing characteristics of typical botanical tissues, and tried to understand the phenomena from the viewpoint of water morphology. The temperature gradient along the slide was generated between the two ends at a high temperature of 25 °C and a lower one of —10 °C, respectively.

Interfacial advancing behavior

Figure 14.25 illustrates snatched images of ice crystal growth and freezing interface in a thin pear slice. The interface front, denoted as white lines, did not advance at a uniform speed, or keep a straight and smooth line as commonly observed for a dilute solution. Instead, the interface advancing rate would alter spatially and instantaneously as it went on. Moreover, dendritic crystals formed in the vicinity of the interface, with coexistence of a few spicular or lumpish crystals. These observations just explore the heterogeneity of the transport and phase change in bio-tissues. It is likely that the freezing and advancing behavior exhibited a disparity in different regions, generally advancing faster in the region of looser matrix.

Sometimes dispersed freezing occurred and ice crystals abruptly formed in isolated front areas of the interface, see Figures 14.25(c-e). These scattered crystals would expand


(a) (b)

Figure 14.25 Ice crystal growth and interface advancing in a thin pear slice at (a) t = 1.833 s, (b) t = 1.867 s, (c) t = 3.833 s, (d) t = 4.467 s, and (e) t = 4.833 s.

rapidly and merge with the advancing interface. Therefore, the development of the palin-genetic interface was greatly influenced by these dispersed ice crystals, which usually appeared where the tissue exhibited appropriate pore configuration. Such appropriate geometrical configuration and tissue matrix could enhance the associated transport processes and promote the occurrence of phase change. From these experimental observations, it is reasonable to expect that the pore structure and water morphology in tissues would be important factors affecting the transport process, and therefore the survival or damage degree of biomaterials.

Inner structure

The growth of ice crystals caused mechanical stresses on the microstructure of the tissues, such as squeezing, dragging or even lacerating the pericellular membrane, contributing to the damage of the tissue structure integrity. Figure 14.26 shows the images of a thin

Figure 14.26 thawing.

Appearance images of a cabbage slice (a) before freezing, and (b) after


cabbage slice before freezing and after thawing. Apparently, the inner structure was more blurred in accordance with the tissue relaxation and softening.

Figure 14.27 illustrates the reconstructed sectional images of a pear slice obtained using a Micro-CT. Before freezing, the pear tissue has a reticular microstructure and probably contains various modes of bio-water. After thawing, many fine crescent cracks appear, see Figure 14.27(b). This might suggest that the freezing has effectively altered the water morphology and separated water from biomaterials. The cracks might be caused by damage to the tissue matrix because of the ice crystal growth and interface advancing.

In addition, the temperature distribution in the sample might be affected by the natural convection, which induces a slightly slower freezing rate near the top. Perhaps the gravity effect is also a significant reason for more cracks to appear in the upper part of the tissue. During the freezing-thawing process, micro bubbles were always observed, gathering on the surface of the tissue. Clearly, some non-condensable gas was dissolved in the bio-water solution, and as the tissue solution is frozen, the non-condensable gas is separated from the solution and forms bubbles. The gas is more apt to separate out and accumulate in the upper part due to the relatively slower freezing rate and the gravitation effect. This gas accumulation in some pores would probably cause further damage of the tissue matrix and result in micro cracks.

The effect of water morphology

Generally, the modes of water in biologic tissues could be simply classified into free, absorbed and combinative water. Typically, absorptive water means water attached to the surfaces of biological macromolecules and membranes, and combinative water in the interior of the cell compartments. For living systems, the confined water would affect the water transport and metabolism in biological activities, see Vogler (1998) and Jhon et al. (2003), while the free water plays a crucial role in the formation of the ice crystal. The probability for an ice crystal to form at any temperature is a function of the volume, see Tumbull (1969), and thereby the ice will most likely form in the free water during freezing.

As for high-hydrous tissues, there is an abundance of water existing in a free or absorbed mode. In the incipient freezing period, ice crystals preferentially form in regions rich of free water. Since the distributions of free water are uneven, an irregular zonal interface

Figure 14.27 CT images of a pear slice (a) before freezing, and (b) after thawing.


is always observed in a unidirectional freezing, see Figure 14.25. It also explains the abrupt appearance and expansion of dispersed crystals in front of the interface. On the other hand, the morphology of the water near the interface is conversely varied somewhat under the action of phase change and transfer. During freezing, the concentration of solutes increases in the solution around the ice crystal, which causes mass transfer. As a result, the absorbed, or even combinative water permeates out to the surrounding solution, and are said to in the free mode. Such hydration is visibly manifested by the change of the tissue macroscopic characteristics and the emergence of its inner micro cracks. For relatively denser tissues with less free water, such as animal hepatic tissues also investigated in the present experiments, the absorbed and combinative water is better conserved in the tissue structure. Hence the dehydration is comparatively less significant during the freezing/thawing.

Additionally, the whole freezing at an approximate cooling rate of about l ° C / s was conducted in a refrigerator for similar samples. Comparison of tissues undergoing unidirectional and whole freezing suggests that the latter has less appearance changes and lower dehydration rates. Ice forms firstly on the surface of the tissue, enveloping the tissue and restraining the loss of water. While in the case of unidirectional freezing, the advancing interface pushes the free water, as well as detrimentally separating the absorbed or combinative water in the tissue. Thus the dehydration performs more adequately than that in the whole freezing case. Further damages would be induced at a lower cooling rate because of the higher dehydration rate.

14.6 TWO-PHASE FLOW BEHAVIOR

The available literature related to two-phase flow always treats the vapor and liquid phases separately and regard the flow as a continuous medium, and normally both the interfacial interaction between two phases and the influence caused by the non-continuity of the vapor phase are not included in these approximate methods. In this section, we focus on the visually observed vapor phase transport in liquid-vapor flows through a bead-packed structure at the pore scale.

14.6.1 Experimental observation

The experimental rig employed is illustrated in Figure 14.28, consisting of a boiling section, visual section, heater, and water bath. The visual section was a quartz glass vessel, 200 x 100 x 20 mm rectangular cross-section, randomly filled with glass beads. The experiments were performed at different conditions (vapor/liquid mass ratios, saturation and bead-packed structure), see Fang et al. (2004a, 2004b).

There were two types of bubbly flow, small bubbles uniformly distributed and transported with the liquid, and flows with large bubbles having interface distortion and elongation. It was observed that the average bubble size was strongly influenced by the diameter of the


Visual section

Figure 14,28 The experimental system.

glass beads. In a large bead structure, the vapor phase was primarily transported in small bubbles, while in a small bead structure large bubbles apparently increased, see Figure 14.29, and bubbles were likely to be entrapped and turn into large bubbles. When the vapor fraction was not very high, small bubbles, with an average diameter of 0.1 mm, were the primary form of vapor transport, see Figure 14.30. These small bubbles periodically collided with the porous structure. After collision, the bubbles stopped immediately and then reaccelerated due to the liquid drag and buoyancy forces. In horizontal ducts the small bubbles finally reached the liquid velocity, while in non-horizontal ducts the final velocity of the small bubbles slightly exceeded the liquid velocity. When the vapor fraction

Figure 14.29 Bead size effect (dp mm).

Figure 14.30 Small bubble transport.

392 PORE^CALE TRANSPORT PHENOMENA IN POROUS MEDIA

increased, the average small bubble size and velocity remained nearly the same. However, the bubble number increased and the bubble distribution became denser. Compared with a non-horizontal duct, bubbles more easily tended to be entrapped by the porous structure and this led to bubbles coalescing and occasionally turning into huge bubbles covering a large area of the duct.

Large bubbles, see Figure 14.31, penetrated through the bead channels with distortion and elongation. The surface area change of the bubble would cause an increase in the surface energy. If the kinetic energy of a bubble exceeded the potential increase caused by the interface distortion, the bubble could go through the pore throat directly. Due to the geometrical configuration, large bubbles with sufficient energy would follow in a zigzag path in the porous structure. The other flow modes were also observed and discussed by FangetaL (2004a, 2004b).

14.6.2 Critical diameter

From the above experimental observations, the flow modes and the transport mechanisms of bubbles were greatly influenced by the diameter of the bubble and the pore size. A critical diameter was introduced to identify the small and large bubble flows having bubble size smaller and larger than the critical diameter, respectively. The critical diameter is defined as the maximum diameter at which a bubble can penetrate through the pore structure without interface deformation. Figure 14.32 depicts three typical bead arrangements and

Figure 14.31 Large bubble transport with distortion.

Figure 14.32 Critical diameter.

X. F.PENG AND H.L.WU 393

the corresponding critical diameters, where dp indicates the average diameter of the glass beads and d the critical diameter. Correspondingly, the critical diameter d is 0.154 dp to 0.414 dp. For an average bead diameter of 8 mm, the corresponding critical diameter is 1.23'^3.31mm.

14.6.3 lY-ansport of small bubbles

Normally, due to the drag force a bubble accelerates from rest to the liquid velocity, and then it moves at the same velocity as the liquid until it re-collides with the porous structure. The momentum equation and corresponding solutions are given as follows, see Fang et al. (2004a, 2004b):

k{uo — 5)^ = mt's, s = UQ -2mb = u.t^l'^\ S = UQI

K-^J T- C2 H-C4

{t " C2)K\ (14.11)

for the initial condition 5(0) = 0, s(0) = 0. In equation (14.11), m?, is the bubble mass, UQ the liquid phase velocity, k the resistance coefficient, and K = 5pw'7rdl/4y/dtpw/lJ'^ C2 - ~2mb/Ky/uE, C4 = {2mb/K)'^{l/c2).

Figure 14.33 depicts the transport of the bubbles having different radii. The time of acceleration is different for each size of bubble, and finally all bubbles approach the liquid velocity. The larger bubbles will take a longer time to speed up. Figure 14.34 illustrates the velocity-time relationship for bubbles at different liquid velocities. For the bubble in a high-velocity liquid, the drag is relatively high, and this corresponds to a quick acceleration process. For a bubble being transported in a non-horizontal duct, the buoyancy force F^ should be included. Accordingly, the momentum equation of a bubble is expressed as

U. 1^

0 1

-^ 0.08

.^ 0.06 8 > 0.04

0.02

n u 10

-

- 1 0

1 1 1 1 1

'hi' J J' 10"® 10"^ 10-"^ 10-2 1

Time (s)

d^ = 0.05 ^m

di^= 0.1 Jim

d^ = 0.5 im

Figure 1433 Change of bubble velocity with time.


10

E 6

o

>

2h

10-

1

L r /

1 v _ i — «

/ /

/

/ /

/

. — — •^i — 1 » -^ ^

--

"o =

"n =

^n =

"o =

^0 =

10m/s 5m/s 1 m/s 0.5 m/s 0.1 m/s

10- 10-6

Time (s) 10" 10-

foUows:

Figure 14.34 Velocity at various liquid velocities.

k(u^ — s)^ + Fb = TUb's or - fc(uo - s)^ -\- Ft = mt's. (14.12)

The latter expression is for the case of the liquid exerting a resistance on the bubble. Similar to a horizontal duct, analytical solutions of equation (14.12) can be obtained, see Fang et al. (2004b). In a non-horizontal duct, the final equilibrium velocity of the bubble was greater than the liquid velocity, and the excess velocity was proportional to the bubble radius.

For a non-continuous phase transport, the collision between the porous structure and the bubbles prevents bubbles from continuous movement and serves as the resistance on vapor transport. Meantime, the bubbles are propelled by the liquid phase, so the vapor phase also exerts an opposing force on the liquid. We can define the time interval, r , between two consecutive collisions as the average free time and the distance the bubble travels between two consecutive collisions as the average free distance. A, which is the order of the size of the pores. Given the average free time between two consecutive collisions, T, in which the average acceleration time is TQ, the average velocity of a bubble is u^. The average force of the liquid exerts on the bubble and the opposing force that the small bubbles exert on the liquid are derived as:

_ rritUb , rn rtsTaLAes rribUb ngLAcsmbUb . , A Fa = and Fu = = , with r = — .

Ta T Ta r Ub (14.13)

Finally, the additional pressure that small bubbles exert on the liquid is obtained as follows:

Pls = Fls QvLUbPg ^ . Ae XAe

(14.14)


where ^s denotes a parameter depending on the geometry. In the experiments on small bubbles in two phase flows, the pressure difference was measured for various vapor fractions 5 at a constant liquid flow rate of 1.67 x 10~^m^/s. Figure 14.35 illustrates the relationship between the pressure difference increase and the vapor fraction. When S is relatively low, the experimental data agrees well with the prediction of equation (14.14). For relatively large S, the effect of the large bubble distortion should be included in predicting the pressure difference.

14.6.4 TYansport of big bubbles

For bubbles having a size greater than the critical diameter, the shape of a bubble in its intermediate stage of cross-over is shown in Figure 14.36. Parts A and C are the portion of a sphere with radius r and part B an axis-symmetrical evolving body with a profile of the neighboring glass beads. The impetus of bubble transport comes solely from the liquid. The pressure of a cross-over bubble exerts on the liquid depends the surface energy of the distorted bubbles. This was also investigated in the work of Fang et al. (2004b).

CO

0)

o c 0

2 Q .

\il

10

8

6

4

2

-

•4>

T 1

-1-

+ ^

1 1

'

-H

+ y

1

' + '

y 1 /^

-

1 1 1

-I- Experimental data

Prediction

0.05 0.1 0.15 0.2 0.25 0.3 0.35

Vapor fraction

Figure 14.35 Pressure difference with vapor fraction.

Figure 14.36 Bubble during crossing-over.


14.7 CONCLUSION

The conjugated transport phenomena with a pore and matrix structure widely exist in both the natural world and practical applications. It is of critical significance to understand these phenomena accounting for the dynamical processes and structure deformation taking place in the inner pores. In these processes, the associated transport phenomena are highly dependent upon the pore structure. Inversely, the transport processes easily alter the pore structure and further influence the transport characteristics. In this chapter, a series of different experimental observations and associated theoretical investigations have been conducted to understand the transport phenomena at the pore-scale level, including the transport phenomena with/without phase change and chemical reaction, and concerning a wide range of practical applications. Several typical processes, namely drying, transport-reaction, boiling and two phase flow in porous media, freezing and thawing treatment of loose and bio materials, were visually observed and theoretically described at the pore-scale level. These investigations provide some new understanding and insights into the nature of transport phenomena in porous media from different points of views.

Without doubt, it is quite a new and challenging area to investigate the transport phenomena in porous media at the pore-scale level, and the present investigations are very preliminary and ongoing. In particular, more attempts should be addressed on the understanding and theoretical descriptions of the fundamentals. Of course, this kind of investigation would be highly dependent upon the experimental technology, observations, measurements and innovatory theoretical description. These will be the emphases in future investigations.

ACKNOWLEDGEMENT

This research was supported by the National Natural Science Foundation of China (Contract No. 50136020 and 50306009).

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transport phenomena in porous media iii || pore-scale transport phenomena in porous media

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