Fractal reconstruction of microscopic rough surface for soot layer

Subscriber access provided by Kaohsiung Medical University

Article

Fractal reconstruction of microscopic rough surface for soot layer during ceramic filtration based on Weierstrass-Mandelbrot function Wei Zhang, Cheng Lu, Pengfei Dong, Yiwei Fang, Yanshan Yin, Zhangmao Hu, Huifang Xu, and Min Ruan Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.7b03845 • Publication Date (Web): 01 Mar 2018 Downloaded from http://pubs.acs.org on March 2, 2018

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

Industrial & Engineering Chemistry Research is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 29 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Industrial & Engineering Chemistry Research

Fractal reconstruction of microscopic rough surface for soot layer during ceramic filtration based on Weierstrass-Mandelbrot function Wei Zhanga, b, *, Cheng Lu a, b, Pengfei Dong a, b, Yiwei Fang a, b, Yanshan Yina, b, *, Zhangmao Hua, b, Huifang Xua, b, Min Ruana, b a

School of Energy and Power Engineering, Changsha University of Science &

Technology (CSUST), Changsha 410114, China b

Key Laboratory of Renewable Energy Electric-Technology of Hunan Province,

Changsha 410114, China Corresponding author phone: +86 13975803677; fax: +86 731 85258408; e-mail: [email protected] (W.Zhang), [email protected] (Y. S. Yin) ABSTRACT The microscopic surface of soot layer on the external surface of filtration elements is rather hard to be reconstructed. In this study, an incinerator-filter setup was designed to mimic the capture of soot particles in flue gas to achieve the samples to construct the rough soot-layer surface. The specific velocities around the ceramic cartridge were determined by particle image velocimetry (PIV) measurement. Resorting to box-counting method, the fractal dimensions (FDs) were determined by Richardson-Mandelbrot method with binary images of samples. Accordingly, the in situ thickness of soot layer was constructed with consideration of particle deposit and the microscopic rough surfaces were modeled by employing Weierstrass-Mandelbrot (W-M) function. Additionally, a comparison between the constructed surface and real surface achieved from the image taken by atomic force microscopy (AFM) was performed. The results suggest that all roughness deviations of constructed surface from real surface of soot layer not exceed 5%. KEYWORDS: Soot; Ceramic filtration; Fractal dimension; Weierstrass-Mandelbrot (W-M) function; Box-count 1. INTRODUCTION 1

ACS Paragon Plus Environment

Industrial & Engineering Chemistry Research 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Rigid ceramic filter (RCF) has become a potential technology to remove particulate matter from the hot flue gas of coal-fired power plants, biomass incinerator or biomass gasification 1 2 3. Advantages of this technology include high efficiency, low emissions, less waste, high temperature resistant, and resistance to corrosion, which is very important to protect the units in the exhaust treatment system 4. Most previous studies have focused on operational performance according to collection efficiency and pressure drop for the stability in practical applications 5. General, RCFs for hot gas cleaning can be classified into two filtration types: fibrous filtration and granular filtration 6. The two filtration types are mainly dependent on interception and deposition of flying particles. As a key intercept obstacle for fly ash, the characteristics of soot cake on the external surface of filtration element are very important. Thus, during the past decade, considerable efforts have been devoted into exploring mechanism and control of soot particles in the flue gas 7 8. It is well accepted that agglomeration of soot particles in flue gas on the surface of filtration element is the main cause of collision and adhesion between particles and filtration material 9, which is generally considered as a result of Van der Waals force, electrostatic force and liquid bridge between particles 10. Accordingly, the microscopic topology of cake layer is critical to the colliding and agglomerating process of soot particles 11. There exist great distinctions for microscopic surfaces of soot layer with different thicknesses. Kim et al confirmed that the pressure drop increased approximately linearly with the loading mass and soot agglomerates with different thicknesses are indistinguishable once deposited in the cake causing different roughness12. To study the microscopic surface at a certain point, it is vital to secure the thickness of soot layer. The pressure drop is pivotal in the formulation process of soot layer. Ergun 13 proposed one form of the pressure drop equation during one-dimensional flow through a packed bed of grain material, which is prevalent in chemical engineers 14-17 nowadays.

|| 150 1 − 1.75 1 − = + 1 2


Page 2 of 29

Page 3 of 29 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Here, ∆p is the pressure drop, µ dynamic fluid viscosity, ρ the fluid density,

porosity, uf face velocity, ac mean grain diameter of filtration material, and δ filtration depth. Previous researches indicate that incessant deposition of soot particles on the filter is constantly changing the porosity, specific surface areas and pore volumes. Namely, the in situ parameters should be employed to calculate real value of pressure drop. In order to investigate the microscopic morphology and microstructural properties of soot deposits, previous scholars had devoted great effort to deep into soot research in a microscopic scale. Konstandopoulos et al 18 proposed a measurement methodology to accurately determine the soot cake packing density and permeability with the application of first-principles measurement. Kim et al 12 investigated structural properties and filter loading characteristics of soot nanoparticle agglomerates generated from a diffusion burner by experiments. Saffaripour et al 19 investigated the effect of drive cycle and gasoline particulate filter on the morphology of soot particles emitted from a gasoline-direct-injection vehicle by transmission electron-microscope (TEM) image analysis. However, it is so difficult to quantify and model the surface morphology of soot layer by conventional methods. To reconstruct a microscopic rough surface for a specific material, the Weierstrass-Mandelbrot (W-M) function20 21 is often employed to model higher dimensional stochastic processes and the amplitudes is expressed as: =

ln

)

%

! "# ! $1

#'*

&'(%

− exp./01 & cos 5 − 6# 78exp./9#,& 7 01 & ;< ( 2

Where "# is amplitudes of anisotropy surface, & is geometrically spaced

frequencies ( > 1), ? is fractal dimension (FD) (2 < ? < 3), 9#,& is arbitrary

stochastic phases between 0 to 2π, 01 is a wavenumber that can be used to scale

horizontal variability. The normalizing factor, ln⁄ *⁄ is chosen to make the

limit of finite as → 1 and

→ ∞. / and 5 are the planar polar

coordinates of a surface point related to the Cartesian coordinates, E and F, by 3



/ = GE + F , 5 = arctan F/E . Additionally, for each index m there is an angle

6# ∈ M0, NO corresponding to the direction of a corrugation of the surface.

For the purpose of modeling rough surface, it is essential to secure the values of

crucial parameter, FDs. With the development of fractal theory over the years, there are many methodologies (such as box-counting algorithm 22, volume-surface algorithm 23, slit island method 24, frequency-power spectrum method 25, etc) to determine the values of FDs. To achieve the FDs from the extremely disordered morphology, it is necessary to utilize an appropriate fractal approach. Generally speaking, the previous works in literatures suggest that box-counting FD algorithm is an effective method to handle the image segmentation details 26. Accordingly, it is feasible to model rough surface with the FDs determined by box-counting approach. This study focuses on constructing the rough soot-layer surface on the external surface of ceramic cartridge in a RCF. Six longitudinal sampling points and six circumferential sampling points were firstly selected for FDs derivation and surface construction. Then the in situ soot-layer thickness was constructed by the pressure drop with consideration of particle deposit and filtration velocities achieved by PIV experiments. Next, the FDs were determined by box-counting approach and the microscopic rough surfaces were modeled according to W-M function with the secured FDs. To verify the validity of this methodology, a comparison was performed with the images taken by atomic force microscopy (AFM). This study will throw significant light on construction of soot surface topography. 2. MATERIALS AND METHODS 2.1 Experiments The schematic of the testing system for ceramic filter elements is shown as Figure 1(a). The system primarily consisted of combustion chamber and filtration chamber with testing system, which includes one filter element with the length of 1.2m (Kaibote Ceramic Inc., China). To mimic the real raw flue gas during power generation process, the premixed fuel of coal and air is conveyed into a combustion chamber and flue gas generated during combustion is introduced into filtration 4


Page 4 of 29

Page 5 of 29 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


chamber. A temperature controller with refrigerant in coiled heat exchanger is set in the flowing pipe between combustion chamber and filtration chamber to mimic the real temperature of flue gas (90°C-150°C) and ensure the safety of experimental setup because the shell of testing chamber is made of tempered glass (SBGX0013, Subei Optical Glass Inc., China), the normal working temperature of which is less than 288°C. Four differential pressure transducers (MODEL267, Setra Systems, Inc) are used to monitor the pressure drop across the filter elements. In order to compare the effect of ceramic material on the pressure drop, we employed 3 types of ceramic cartridges TCP-LG50, TCP-LG80 and TCP-LG100 and the properties are listed in Table 1. Six circumferential sampling points are set on the external surface of ceramic cartridge at b2 cutoff plane shown as Figure 1(b) and six longitudinal sampling points are set on the windward surface of ceramic cartridge shown as Figure 1(c). The FDs were estimated from the fitting logarithmic values of grid points of fractal micrographics taken by a JSM-6700F Field Emission Scanning Electron Microscope (Japan Electron Optics Laboratory Ltd. Corp) using box-counting dimension method. To study the surface morphologies of soot samples, an atomic force microscopy (AFM) (NT-MDT) was employed to characterize the microscopic rough surface. To ensure intactness of the thin raw sample, an alumina dish with the area of 10mm×10mm, coating Tempfix™ conductive adhesive (ZLK, Jolyc Technology, China) on the underlying surface, was stuck on the external surface of ceramic cartridge. In order to solidify the raw samples, they were set in a high temperature tube atmosphere furnace (HTL1400-80, Haoyue Inc., China) for calcination at 350°C after stripping from the filtration element. For the purpose of acquiring the in situ filtration velocity, a 45° stereo-PIV system consists of a CCD sensor camera (PCO.1200, PCO, Germany) equipped with a long-pass filter at 550 nm, with a maximum resolution of 1280*1024 pixels. A double-pulse Nd:YAG laser (EverGreen145@532nm, Quantel, USA) and compound lenses are employed to generate and shape a uniform laser light sheet (thickness 1.5-2 mm) in the orthogonal direction of CCD camera. 5



Page 6 of 29

Figure1. Schematic layout of the setup: (a) combustion chamber, filtration chamber with testing system; (b) distribution of circumferential sampling points; (c) distribution of longitudinal sampling points.

Table 1. Properties of filtration material used in the present model Filter

TCP-LG50

TCP-LG80

TCP-LG100

Wall thickness δ (mm)

10

15

20

Mean grain diameter ac(µm)

Initial porosity 1

257

214

154

0.53

0.47

0.42

Cartridge sizes (Dex|Din) (mm)

50|30

80|50

100|60

Total filtration area (m2)

14.47

23.32

29.29

Note: Dex|Din denotes (external diameter | internal diameter).

2.2 Derivation of soot thickness 2.2.1 Effect of soot deposit on pressure drop. For constant pressure condition, the pressure drop between export and inlet remains unchanged approximately in the filtration system. However, an ever-increasing dust preliminary layer can be formed on the surface of filter, which gradually becomes the main filtration layer. Generally speaking, porosity of filter material gradually will become smaller with the increase of specific deposit, i.e. the deposited-particles volume per unit volume of the filtration material 27. Consequently, pressure drop of ceramic filter keeps increasing with the decrease of porosity in actual process. In fact specific deposit can change many 6


Page 7 of 29 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


operating ambient conditions such as porosity, specific surface, face velocity and the depth. Hence, Ergun equation with constant pressure drop should be modified to adapt to the need of non-steady-state pressure drop due to specific deposit. To acquire the transient model from Eq.(1), we can introduce a parameter α to represent one-sixths of the weighted harmonic mean diameter of ceramic grain, and then synthetic porosity

= 1 − P⁄.1 − Q 7 and P is specific deposit (volume of particles deposited per

unit volume of ceramic filter), 1 ceramic porosity, Q soot porosity. On the basis

of Ergun’s equation13, the pressure drop with consideration of particles deposit can be expressed as:

T T U 1.75 S U 150 S |∆| 6 6 = + 3 T T 6 S1 − 6U 6 S1 − 6U

Here specific surface area T = 6 1 − , µ dynamic fluid viscosity, ρ fluid

density, uf filtration velocity.

According to Zamani et al’s empirical expression of specific surface 28, the specific surface of filter has linear relationships with filtration coefficient proposed by Ives 29. Filtration coefficient can be expressed as:

T V P YZ P Y[ P Y^ = = W1 + 6 X W1 − X W1 − X 4 T1 V1 1 1 P#\]

Here, S0 is initial specific surface, λ0 initial filtration coefficient and σmax the

maximum specific deposit, ka, kb and kc are dimensionless parameters with the value of 0 or 1. Consequently, the relationship between pressure drop and specific deposit can be expressed by the deduced model. To predict the variation of pressure drop in actual operating condition, it is necessary to set up a series of time-dependent expression for specific deposit. Based on experimental observations, Maroudas et al 30 proposed a clarification equation to depict the particle concentration profile through a filter for a slow filtration as following, which can be adopted to close equations and construct a time-independent-function model of pressure drop. 7



`a = −Va 5 ` Here, filtration coefficient V = V1 1 + 6P⁄1 1 − P⁄1 and δ is

thickness of ceramic cartridge. Combining and solving Eq.(4) and Eq.(5), the

changing rules of pressure drop with time may be derived with consideration of the influence of specific deposit. 2.2.2 In situ soot thickness construction. The macroscopic surface morphology is due to the formed thickness of soot layer on the external surface of ceramic cartridge during high-temperature filtration. The thickness of soot layer has an affinity with pressure drop. In terms of the conventional cake filtration theory, the relationship between total pressure drop and filtration velocity during ceramic filtration at high temperature can be characterized by the permeability resistance of filtration medium. Since part of flying soot is intercepted to form an incidental soot layer on the external surface of ceramic cartridge, the filtration medium is actually composed of two parts: ceramic material and soot-layer. Hence, permeability resistance comprises ceramic resistance and soot-layer resistance 31, which is expressed as:

∆b = 0* + 0 c 6

where 0* = e# , which is defined as the flow resistance of clean ceramic cartridge, and 0 = 6f, which is defined as the specific cake resistance. e# , the resistance of

porous ceramic, 6f, the average resistance of soot layer. Subsequently, the general expression of intercepted soot mass is: c=

∆b⁄ − 0* 7 0

Because the flow resistance value of clean ceramic cartridge remains constant, it can be decided by classical flowing resistance formula provided by Ergun et al 32 as follows:

0* = gh ∙ j (1.k l (*.k 8

where l is porosity of ceramic material, which can be derived by testing bulk density ρb and real density ρp, gh is regression constant, and K is permeability.

Comparably, the specific cake resistance is related with the filtration velocity. 8


Page 8 of 29

Page 9 of 29 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Dennis et al 33 reported that under certain operating conditions, k2 may be expressed by the empirical equation with filtration velocity u as independent variables:

0 = 1.81 1.k 9

Considering a local cake layer, the thickness can be described as: c o= 10 p "

where A is the local area of a certain sampling point. Thus, the in situ thickness of soot layer is expressed as: o=

∆b⁄ − gh ∙ j (1.k l (*.k 11 1.81 1.k p "

The thickness values of the longitudinal sampling points b1-b6 and the

circumferential sample points a1-a6 can be calculated with Eq.(10). The related parameters achieved from experiments list in Table 2. Table 2. Parameter values employed for calculating thickness of soot layer Parameters

Value

Unit

Inlet soot concentration, Cin

2.258

g/m3

Bulk density, ρb

1.325

g/cm3

Real density, ρp

1.014

g/cm3

Permeability, K

2.91×10-9

m2

Regression constant, Ct

1.16×10-7

Pa.s

2.3 Method of modelling surface morphology of soot layer 2.3.1 Construction of surface morphology. Although the randomly rough surface of soot layer is different with a rigorous mathematical model, the topological fractal characteristics of soot layer can be described by a two variable W-M function involved in the fractal geometry theory 21. Two properties of W-M function are that we want to preserve: homogeneity and scaling, which are statistical properties. When the frequency index is extended from 0 to ∞, the scaling property of W-M function

would only hold approximately. In spite of the choice of random phase, the constant two–variable scalar W-M function exhibits a degree of unsuitable regularity for modelling topography though it has both the scaling property and homogeneous. Then, 9



Ausloos et al 20 presents a useful generalization of univariate W-M function to satisfy weighted and random superposition of ridge-like surfaces. ⁄ ) &yZz

r(;< tu * ∆z E, F = (;< W X s − awx {|

! ! (.(; 1 is a

parameter that determines the density of frequencies in the profile; M denotes the number of superposed ridges used to construct the surface; n is a frequency index and the random phase 9#,& ∈ M0,2NO is used to generate microscopic anisotropy of

different frequencies at any point of the surface profile; ? .2 < ? < 37 is the FD

of rough microscopic surface.

For the purpose of application to the Hertz theory to determine the contact force at asperity microcontacts, only larger wavelength is permitted to calculate the force at an asperity microcontact 34. In this case, the smallest wavelength cannot be less than a cut-off length Ls, of the order of about 100 lattice distances. Thus, the highest frequency is set equal to 1/Ls, and the upper limit of n is given by u#\] = int

log r\ ⁄r 13 log

where intM⋯ O denotes the maximum integer value of the number in the brackets. The above method is then employed to model soot layer surface topography.

Table 3 lists the data used to model a surface morphology of soot layer on the external surface of ceramic cartridge with area of 1 µm2 by Eq. (12). To acquire natural surfaces, the typical statistical parameter values are applied as described in literature 20. Lin et al’s research indicates that the number of superposed ridges M should exceed 3 to approach realistic surface 35. Since the FD is a vital effect factor of microscopic surface topography, it is up to box counting method to decide it. 10


Page 10 of 29

Page 11 of 29 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


To maintain the microscopic anisotropy of different frequencies at any point of the surface of soot layer, the random phase 9#,& ∈ M0,2NO is generated by a random

number generator 36.

Table 3. Parameter values employed for modeling rough surface morphology of soot layer Parameter

Value

Unit

Height scaling parameter, s

1

µm

13.6

pm

1.5

-

Superposed ridges number,

10

-

0.01

µm

Sample length, r\

Spaced density frequency,

Cut-off length, r

2.3.2 Prediction of roughness. The mean deviation between the arithmetic averages of the absolute values of the ridge-like surface and the mean plane can represent the average value (AV) of microscopic surface height at the testing points, which can be expressed as: Δ

1 = !|∆ | 14 '*

The root mean square (RMS) of height deviations taken from the mean data

plane can denote the roughness of microscopic surface at the testing points, which is expressed as: Δ) =

∑ '* ∆ 15

For a rough soot layer surface, the surface topography is not homogeneous. In this case the ∆Zi should first be determined in the longitudinal and circumferential direction. The advantage of this approach is that the standard deviation of the uncorrelated distribution and of the microscopic surface height is inherited and accounts for the root mean square height. 3. RESULTS AND DISCUSSION 3.1 Evolution of FD. Figure 2a shows that microscopic morphologies of soot 11



particles approximately present spherical or oblong shape, with irregular edges, much obvious void fraction and a rough surface. In order to investigate the FD of soot layer, the gray-scale micrographs need to be converted to binary images with a reasonable threshold value (Figure 2b). To investigate the effect of threshold value and box scale on FD, diverse threshold values from 0.40 to 1.2 were employed and the logarithmic relationship of local dimension and box size was plotted (Figure 2c). The results suggest that the maximum deviation of FD attaches 12.5% when box size magnification < 1500×, and the deviation of FD is zero whatever is the threshold value. So, Figure 2a was converted to Figure 2b with the threshold value of 0.8 under high box size magnification, which is obtained by comparing the porosity of binary images with experimental data. FD can thereby be calculated with counting box method. Firstly, the micrographs were converted to grayscale images and a reasonable threshold value was set to binary images. The threshold value was determined by comparing statistical percentage of grid points in images (calculated by MATLAB software) with experimental porosities (estimated with the ratio of real density to stacking density of soot cake). Subsequently, we divided the images of Mcb×Mcb pixels into Lcb×Lcb sub-partitions and then the fractal scale was defined as r= Lcb /M. In each grid unit, there were a series of Lcb×Lcb×w boxes, where w is the number of black pixels in a box. If the number of black pixels in an image is Gcb, w= Gcb×Lcb /M. Assumed that the most concentrated and sparse black pixels were s and l in (i, j) grid unit respectively, the number of requirement boxes of covering this grid unit is

u /, = t − x + 1 16

Then the amount of requirement boxes of covering the whole image is:

= ! u /, 17 ,

log 18 log 1⁄

Therefore, the FDs were estimated by the following expression: ? = lim

Calculating Nr with assigning randomly a group of Lcb values, the FDs were linearly 12


Page 12 of 29

Page 13 of 29 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


fitted with logarithmic data pairs. As shown in the two-digit image (Figure 2b), it is obvious that different sizes and shapes of pores distribute over the real cross sections and similar patterns appear at different magnifications. This presents ‘‘self-similarity” on the particle size distribution. Figure 2d is the analytical result of box-counting method for at most 20 two-digit images. The dependence obeys a power law of

∼ (;< but two slight inflection points indicated by the arrows are found at ca.

17 µm and 65 µm, Df= 1.710.

Figure 2. Microscopic features of (a) soot particles and (b) two-digit image, (c) effect of threshold value and (d) numerical results of box-counting method (box count (log) vs. box size (log), R-square: 0.9972). The arrows show inflection points on the slope. Hereby, we can secure the FD distributions along the circumferential direction (Figure 3a) and the longitudinal direction (Figure 3b). Based on box-counting analysis, the FD values grow exponentially with the length of ceramic cartridge. In the region of 0-400mm, the change of FD value in the longitudinal direction was not obvious, and for L>400mm it grows rapidly with the length. It is indicated that most of the external surface of ceramic cartridge is cover by high-dimensional fine particles 13



because the fine particles, especial stokes particles, are more susceptible to being affected by van der Waals force, electrostatic force and liquid-bridging force to coagulate the external surface of ceramic cartridge, while the low-dimensional large particles are more influenced by gravity. Taking into consideration of longitudinal and circumferential sampling points on the external surface of ceramic cartridge, the FD surf was obtained by cubic interpolation with the data of samples shown as Figure 3c-3d. By virtue of the interpolation surf, the FD distribution of whole external surface is clearly presented in the Cartesian coordinates with longitudinal length and circumferential angle as independent variables. By analysis, the maximum FD value (Df=2.459) is situated at the utmost tip point in the leeward side and the minimum FD value (Df=2.026) is located the bottom point in the windward side. 2.6

2.5

θ θ θ θ θ θ

2.4 2.4

Df

Df

2.2 cutoff b6 cutoff b3 cutoff b5 cutoff b2 cutoff b4 cutoff b1

2

1.8 0

90

=30 ° =90 ° =150 ° =210 ° =370 ° =330 °

2.2

(b)

(a) 2

180

270

0

360

200

400

600

800

1000

1200

L /mm

θ /° 1000

(c)

2.5

2.4

800

2.35

2.4

L /mm

2.3

Df

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 14 of 29

2.2 2.1

2 1000

2.3

600

2.25

400

2.2 2.15

800

200 600

400

L /mm

200

0

0

90

180

270

360

(d)

0 0

θ /°

90

180

270

2.1 2.05

360

θ /°

Figure 3. FD distribution of soot layer on the external surface of ceramic cartridge. (a) circumferential direction; (b) longitudinal direction; (c) FD distribution surf; (d) FD distribution contour. 3.2 Derivation of soot layer thickness 3.2.1 Effect of soot deposit on pressure drop. Figure 4a shows ∆P-σ graphs of three types of filtration elements. By observation, the value of ∆P is growing gradually with increasing the value of σ. In the early stage, it exhibits a comparatively lower 14


Page 15 of 29 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


variation, and the growth rate of ∆P constantly rises in the later stage. Many previous investigators 31, 37 had provided the expression of total pressure drop of these two parts as:

∆P = ∆Pfilter +∆Pcake

(19)

where ∆Pfilter is the inherent pressure drop of filter and ∆Pcake is the external pressure

drop of dust cake. If using instead of ceramic porosity and cake porosity to calculate the total pressure drop instead of the sum of two pressure drop, the

computation will be simplified. Figure 4b shows the effect of caused by

particles deposit on pressure drop for the three types of filters. If controlling the

pressure drop on the allowable scope (∆P

Fractal reconstruction of microscopic rough surface for soot layer

Recommend Documents