730 Journal of Chemical Engineering of Japan Copyright © 2015 The Society of Chemical Engineers, Japan

Journal of Chemical Engineering of Japan, Vol. 48, No. 9, pp. 730–741, 2015

E�ect of Aperture Structure of Dutch Weave Mesh on Flow Resistivity

Yuichi Yoshida1,2, Yohei Inoue3, Atsuko Shimosaka2, Yoshiyuki Shirakawa2 and Jusuke Hidaka2

1 Technical Section, Kansai Wire Netting Co., Ltd., 2-7-8 Inari, Naniwa-ku, Osaka 556-0023, Japan2 Department of Chemical Engineering and Materials Science, Faculty of Science and Engineering, Doshisha University, 1-3 Tatara Miyakodani, Kyotanabe-shi, Kyoto 610-0321, Japan

3 Department of Mechanical Engineering and Intelligent Systems, Faculty of Informatics and Engineering, �e University of Electro-Communications, 1-5-1 Chofugaoka, Chofu-shi, Tokyo 182-8585, Japan

Keywords: Dutch Weave Mesh, Pressure Drop, Aperture Size, IB–LBM Simulation, Kozeny–Carman Equation

Plain Dutch weave and twilled Dutch weave meshes are superior �lter media in terms of their high mechanical strength and tiny apertures. However, because they have high �ow resistivity due to their complex �ow paths, it is crucial to predict the pressure drop with high accuracy for a �ltration process. We, therefore, investigated the e�ect of the aper-ture structure of a Dutch weave mesh on the �ow resistivity. First, we proposed a calculation model for estimating the aperture size of a twilled Dutch weave mesh to thoroughly understand the aperture structure; whereas, the aperture structure of a plain Dutch weave mesh has already been clari�ed. Next, numerical simulations were performed using a combination of the lattice Boltzmann and immersed boundary methods. It was found that the drag force of the Dutch weave mesh increased at the inside aperture where the volume fraction increased, and in the twilled Dutch weave mesh, the drag force at the center also varied with the local torsion of the �ow path. Based on these �ndings, we derived an equation for estimating the pressure drop across the Dutch weave mesh, and experimentally veri�ed its validity. This enables a rational and highly accurate prediction of the pressure drop.

Introduction

Because of their very accurate aperture size, high pressure and heat resistance, and reusability, metal woven meshes are often used in particle filtration processes in which the target particle size is over a few micrometers. There are four typical types of woven meshes: plain weave, twilled weave, plain Dutch weave, and twilled Dutch weave meshes. In particular, the plain Dutch weave and twilled Dutch weave meshes are superior filter media because of their high me-chanical strength and tiny apertures. However, the Dutch weave mesh has a high flow resistivity due to its complex flow path. The pressure drop across a woven mesh is a key factor in estimating the filtration life and required operating power, and it is desirable for this pressure drop to be as low as possible. It is, therefore, necessary to select an appropriate filter medium (Dutch weave mesh) or develop one using an accurate prediction of the pressure drop.

Many equations for evaluating the flow resistivity of a woven mesh have been proposed (Wieghardt, 1953; Taka-hashi, 1980; Osaka et al., 1986; Amaki et al., 2008; Kolodziej and Lojewska, 2009). Armour and Cannon (1968) proposed an equation for estimating the pressure drop across a Dutch

weave mesh by considering the woven mesh as a particle aggregate during laminar flow and as a pipeline during tur-bulent flow. Wu et al. (2005) applied Ergun’s equation, which represents the flow resistivity of a particle bed to the pressure drop across several types of woven meshes, includ-ing a Dutch weave mesh. However, because it is difficult to investigate the relationship between the aperture structure and the flow resistivity in detail using an experiment, there has been limited application of the existing equations. Some experimental verification is, therefore, often required for highly accurate prediction of the pressure drop to design a filtration process.

It is expected that computational fluid dynamics (CFD) would make it possible to clarify the relationship between the aperture structure of a Dutch weave mesh and the flow resistivity. A numerical simulation has the advantages of revealing the flow on a microscopic scale and enabling the creation of a fictive woven mesh using an arbitrary wire diameter and number of meshes. Teitel (2010) applied it to the flow through an insect-proof mesh to verify the existing equations for estimating the pressure drop, and Wiegmann et al. (2010) noted its applicability to the design of a multi-filament fiber mesh and its filter element.

It is also crucial to understand the detailed aperture struc-ture of a Dutch weave mesh. Yamamoto et al. (1986) clari-fied the aperture structure of a plain Dutch weave mesh and derived an equation for estimating the aperture size. There have been few studies on the aperture structure of a twilled Dutch weave mesh. Kobayashi et al. (2012) developed a simulation that makes it possible to evaluate the aperture

Received on May 15, 2014; accepted on November 4, 2014DOI: 10.1252/jcej.14we168Presented in part at the SCEJ 79th Annual Meeting (Contribution No. R215) at Gifu, March 18–20, 2014Correspondence concerning this article should be addressed to Y. Yoshida (E-mail address: Y-Yoshida@kwn.co.jp).

Research Paper

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size of a twilled Dutch weave mesh using particles of vari-ous sizes passing through a three-dimensional model of the mesh. However, the aperture structure was not discussed in sufficient detail.

In the present study, we first proposed a calculation model for estimating the aperture size of a twilled Dutch weave mesh to understand its structure in detail. Next, the effect of the aperture structure of the Dutch weave mesh on the flow resistivity was clarified using a numerical simula-tion. The lattice Boltzmann method was used for the fluid calculations, and the immersed boundary method was ap-plied to the solid boundary conditions of the mesh. Based on the simulation results, we derived an equation for es-timating the pressure drop across plain Dutch weave and twilled Dutch weave meshes.

1.　Calculation Model for Estimating Aperture Size of Twilled Dutch Weave Mesh

A plain Dutch weave mesh was made from warps and wefts that crossed each other, where the wefts were in con-tact with each other, and the warps were located at intervals that were set to the warp pitch. A twilled Dutch weave mesh was made from warps and wefts that crossed at least two wires, where the wefts were in contact with each other, and the warps had a set interval. In this study, warps and wefts with two crossing wires were used because this is the config-uration used in the majority of twilled Dutch weave meshes.

Figure 1 shows examples of the magnified photos of the plain Dutch weave and twilled Dutch weave meshes. Yama-moto et al. (1986) found that there were two areas that provided apertures in a plain Dutch weave mesh (the space between two wefts and the space between a warp and two wefts) and derived an equation for estimating these aperture

sizes using a geometrical model. Each aperture size (the first aperture size, δA1 and second aperture size, δA2) was given by the maximum diameter of a spherical particle passing through the aperture.

In the twilled Dutch weave mesh (Figure 1), it can be seen that there are three areas that provide apertures: area 1, the space made between wefts L1 and L1′; area 2, the space made between wefts L1, L2, and L3; and area 3, the space between wefts L2 and L3, and warp L4. We proposed the calculation model for estimating each aperture size by considering the maximum diameter of a spherical particle passing through each aperture.

1.1　Aperture in area 1Wefts L1 and L1′ are parallel with a certain pitch. The

weft pitch, ps, is calculated using the number of meshes in the weft direction, ns (number of wefts per inch), according to Eq. (1).

=ss

25.4[mm]/ 2p n (1)

If the diameters of the wefts are maintained when they are set, the weft pitch will be equal to the weft diameter. How-ever, in most cases, the weft pitch is smaller than the weft diameter because the wefts are slightly compressed by their contact. Because the wefts are considered to maintain their diameters at locations where they are not in contact, the maximum diameter of a spherical particle passing between wefts L1 and L1′, δA1, is determined by Eq. (2).

−=A1 s s2δ p d (2)

The particle diameter, δA1, was given as the aperture size in area 1.

1.2　Aperture in area 2The diameter of a spherical particle passing between wefts

L1 and L2 increases from warp L4 to warp L5; whereas, the diameter of a particle passing between wefts L1 and L3 de-creases. Therefore, the size at which these particle diameters were balanced, δA2, was given as the aperture size in area 2.

Figure 2 shows the geometrical model for area 2, where l1, l2, l3, l4, and l5 are the center lines of wefts L1, L2, and L3, and warps L4 and L5, respectively. The start and end points of line segments l1, l2, and l3 are located just above or under line segments l4 and l5, and their coordinates are described by Eq. (3).

−

s T s w

s T s w w T s T

s w s T

w T s T

s T

s T s w w T

A ( cos , , 0)

B (2 cos , , sec sin )

C (0, , sin )

D (0, 0, sec sin )

E ( cos , 0, 0)

F ( cos , , sec )

p θ d d

p θ d d p θ p θ

d d d θ

p θ p θ

p θ

p θ d d p θ

= +

= +

= +

= +

=

= +

(3)

Fig. 1 Apertures of plain Dutch weave and twilled Dutch weave meshes

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Where, dw is the warp diameter, pw is the warp pitch, θT is the angle between the perpendicular of l4 and the z axis in the xz plane, and θS is the angle between l2 and the z axis in the yz plane.

−

−

=

+=

s1T

w

s w1S

w T

/ 2tan

tan sec

pθ p

d dθ p θ

(4)

When a spherical particle (diameter=δj,z) is in contact with weft L1 at a certain z, the relationship between the par-ticle center, J(x, y, z), and l1 is given by the following ellipse equation.

2 2G G2 2 2

s j, T s j,

( ) ( ) 1( / 2 / 2) sec 2 ( /2 /2)z z

x x y yd δ θ d δ

− −+ =

+ + (5)

The center of the ellipse, G(xG, yG, z), is determined from the following relationship: GG A AB.k ⋅= +

G s G T(1 ) cosx p k θ+= (6)

G s wy d d= + (7)

Gw T s Tsec sin

zk p θ p θ−= (8)

When the same particle is also in contact with weft L2, the relationship between the particle center and l2 is given by the following ellipse equation.

2 2H H

2 2 2s j, s j, S

( ) ( ) 1( / 2 / 2) ( /2 / 2) secz z

x x y yd δ d δ θ

− −+ +

+ = (9)

The center of the ellipse, H(xH, yH, z), is determined from the following relationship: ⋅

HH C CD.k= +

H 0x = (10)

H s w H( )(1 )y d d k−= + (11)

s TH

w T

sinsec

z p θk p θ−

= (12)

Using Eqs. (5) and (9), the relationships between the x and y coordinates of the particle and the particle diameter, δj,z are given as follows.

2H2 2 2 2 2

G G T G T G 2S

2T

( )(1 sec 2 ) sec 2 ( )

sec

1 sec 2

y yx x θ x θ y y

θx

θ

−± − − − −

−

+

=

(13)

2G 2

j, s G2T

( )2 ( )sec 2zx xδ d y y

θ−

− −= + + (14)

The maximum diameter of a particle passing between wefts L1 and L2 at a certain z is determined by calculating the minimum δj,z using Eqs. (13) and (14). Because the equa-tions are complex, the minimum δj,z was calculated by as-signing a value to y in a particular y region. Namely, because the center of a particle that gives the minimum δj,z was posi-tioned at the region: 0≤y≤ds+dw, the y region was divided into 10,000 points, and the minimum δj,z was determined from each value of δj,z calculated for each point. The mini-mum δj,z was the maximum diameter of the particle passing between wefts L1 and L2 at a certain z, δ12,max.

In the same way, the maximum diameter of a particle passing between wefts L1 and L3 at a certain z, δ13,max was determined, and the variations of δ12,max and δ13,max in the z direction were calculated (Figure 3). The size given at the intersection of the curves of δ12,max and δ13,max was the maxi-mum diameter of a particle passing through wefts L1, L2, and L3, δA2, and was the aperture size for area 2.

1.3　Aperture in area 3The aperture in area 3 is similar to the aperture of the

plain Dutch weave mesh. Thus, we applied the calculation method proposed by Yamamoto et al. (1986).

Fig. 2 Geometrical model for calculating aperture size in area 2 of twilled Dutch weave mesh

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First, to simplify the calculation, the origin O in Figure 2 was moved. The point at the intersection of l2 and l3 pro-jected to the yz plane is described by Eq. (15).

s w s w T s TT

w

sec sin( , ) 1 sin(2 ) ,2 2d d p p θ p θy z θp

+ ⋅

+= +

(15)

Because (ps/pw) sin(2θT) is structurally quite small, Eq. (15) can be approximated by Eq. (16).

s w w T s Tsec sin( , ) ,2 2d d p θ p θy z

+ += (16)

The y and z coordinates of the origin O were moved to the position described in Eq. (16), and its x coordinate was moved +(ps/2) cosθT. The new origin is shown in Figure 4.

In Figure 4, when the diameter of a particle that is in con-tact with both wefts L2 and L3 is at a minimum, the particle center is positioned at x=0, y=0. When the particle is also in contact with warp L4, the particle center position is deter-mined using Eq. (17).

w w A3T0, 0, sec2

p d δ θ − − −

(17)

Where, δA3 is the particle diameter. The relationship be-tween the particle center and l2 is described using the fol-lowing ellipse equation, and the maximum diameter of the particle passing through wefts L2 and L3, and warp L4 is given in Eq. (19).

22w w A3 T Ss T

2 2 2s A3 s A3 S

(0 {( ) / 2} sec tan )(0 ( / 2)cos )1

( / 2 / 2) ( / 2 / 2) secp d δ θ θp θ

d δ d δ θ− − −+

+ =+ +

(18)

2

A3

2 2 2T S S

2 2 2w w T S s S

2 2 2 2 2 2 2w w T S s T s S

sec tan sec

( ) sec tan sec

( ) sec tan ( cos ) sec

B B ACδ AA θ θ θ

B d p θ θ d θ

C p d θ θ p θ d θ

− − −

−

− −

− −

=

=

=

= +

(19)

The particle diameter, δA3, is given as the aperture size in area 3.

1.4　Substantive aperture sizeAll three of the apertures determined in areas 1–3 are

connected in the twilled Dutch weave mesh, and particles that are larger than at least one of the apertures cannot pass through the twilled Dutch weave. We, therefore, determined that the substantive aperture size, δsub, was the smallest of the three aperture sizes.

2.　Simulation

The lattice Boltzmann method (LBM) was used to simu-late the incompressible fluid field (Tsutahara et al., 1999, 2007; Inamuro, 2001). The computational lattice was cubic, and a three-dimensional 19-velocity (D3Q19) model was used to depict the advection directions of the ideal particles that represented the fluid motion. The grid spacing, Δ, and time step, Δt were both 1. The lattice Boltzmann equation with an external force term and BGK approximation is given as Eq. (20).

(0)

ext3( 1, ) ( , ) ( , )2

i ii i i i i

g gg t g t E tτ−

= − ⋅+ + +x c x c F x

(20)

Where, ci is the advection velocity of the particles in the i-direction (i=0–18), t is the time, x is the position vec-tor of the lattice point, gi is the distribution function of the particles in the i-advection direction, gi

(0) is the equilibrium distribution function, τ is the single time relaxation param-eter, Ei is the weight coefficient in the i-advection direction

Fig. 3　Example of variation in δ12,max and δ13,max in z direction

Fig. 4 Geometrical model for calculating aperture size in area 3 of twilled Dutch weave mesh

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for a D3Q19 model, and Fext is the external force. The body-force-type immersed boundary method (Yuki et al., 2007) was applied to the solid boundary of the Dutch weave mesh. The external force was obtained from the volume fraction, α of the Dutch weave mesh and the relative velocity between the fluid and the solid.

ext s( )αρ= −F u u (21)

Where, us is the solid velocity vector, u is the fluid veloc-ity vector, and ρ is the density of the particles. In this study, the physical quantities were normalized by the grid spacing (Δ=1), the flow velocity at the outlet (U0), and the charac-teristic velocity of the particle (c=1).

Figure 5 shows the simulation system (note that the x, y, and z axes are different from those of the calcula-tion model for estimating the aperture size of the twilled Dutch weave mesh in Section 1). The flow field was di-vided into 128×128×512 grids, which corresponded to the numbers of lattices in the x, y, and z directions, respec-tively (nL,x=128, nL,y=128, nL,z=512). The Dutch weave mesh of the simulation had four warps in the x direction (warp pitch, pw=nL,x/4) and was located at z=3nL,z/4. The boundary conditions were as follows: at the inlet, density ρ0=constant and velocity gradient=0; at the outlet, velocity U0=constant and density gradient=0, and there were pe-riodic boundaries at the four sides. The flow state was con-trolled by the outlet velocity, U0, and the tentative Reynolds number, Ret, for which the warp pitch, pw, was used as the characteristic length.

0 wt

U pRe ν⋅

= (22)

Where, ν is the kinematic viscosity. The simulation condi-tions are given in Table 1.

The design of the model of the Dutch weave mesh com-prised a combination of a horizontal cylinder and a declined cylinder (Figure 6). The horizontal cylinder was used at the intersection of the wires (length=wire diameter). The declined cylinder was located where the wire bent obliquely from one intersection to the next, with the wire making contact with the cylinder at the top or bottom surface of the mesh. However, where the wire ran straight between the intersections, the horizontal cylinder at the intersection

was elongated to the next intersection. Figure 7 shows the models of the plain Dutch weave and twilled Dutch weave meshes used for the simulation.

3.　Experiment

3.1　Challenge test for evaluating aperture sizeA challenge test proposed by Rideal and Storey (2011)

was performed at Whitehouse Scientific Ltd. to evaluate the aperture size of the twilled Dutch weave mesh. A cut point of a mesh was measured by the dry method of the chal-lenge test using a sonic filter tester. In this method, intense oscillating air currents fluidized well-calibrated glass mi-crospheres through the twilled Dutch weave mesh, and the particles has sufficient contact with the apertures for screen-ing. 50% and 97% separation sizes (D50 and D97, respec-tively) were measured by the wet method of the challenge test using a suspension challenge test apparatus. A slurry of well-dispersed glass microspheres was filtrated using this method.

3.2　Air permeability testAn air permeability test was performed to measure the

pressure drop across the Dutch weave mesh (Figure 8). A disk-shaped mesh made from stainless steel was placed in the cylindrical pipe of internal diameter 35.7 mm, and a fully developed circular pipe airflow was passed through

Fig. 5　Simulation system

Table 1　Simulation conditions

nx×ny×nz [—] 128×128×512ν [—] 0.064τ [—] 0.692ρ0 at inlet [—] 1U0 at outlet —] 0.004, 0.01, 0.02, 0.04Ret [—] 2, 5, 10, 20

Fig. 6　Wire location of Dutch weave mesh in simulation

Fig. 7　Dutch weave mesh models in simulation

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the mesh. The pressure drop across the mesh was measured using a differential pressure gauge (PT105A-A or PT103B-A, Cosmo Instruments Co., Ltd.) for a certain flow rate. The flow rate was controlled by a valve to within 5–50 l/min, and the exit value was measured by a laminar flow meter (DF-2810 A, Cosmo Instruments Co., Ltd.) placed at the outlet of the pipe. The experimental conditions are given in Table 2. The averages of the warp diameter, dw, weft di-ameter, ds, number of meshes in the warp direction, nw, and number of meshes in the weft direction, ns, were calculated from 10 measurements using a profile projector (V-12B, Nikon Corp.).

4.　Results and Discussion

4.1　Validity of calculation model for estimating aperture size of twilled Dutch weave mesh

The validity of the calculation model for estimating the aperture size of the twilled Dutch weave mesh was veri-fied by comparing the calculated aperture sizes with the results of the experiments to evaluate the aperture sizes. The twilled Dutch weave mesh specifications are listed in Table 3, which also gives the average values of the experimentally determined warp diameter, dw, weft diameter, ds, number of meshes in the warp direction, nw, and number of meshes in the weft direction, ns. The aperture sizes were calculated using these average values.

Table 4 lists the results of a comparison of the calculated aperture sizes and the challenge test results. The calculated substantive aperture size is in good agreement with the 50% separation size, D50, which verifies the validity of the calculation model. The cut point corresponds with the 97% separation size, D97, and the calculated substantive aperture size is smaller than the cut point or 97% separation size, D97. This is because the twilled Dutch weave mesh has an aperture size distribution as a result of the variabilities in the wire diameter and number of meshes. The substantive aper-ture size corresponds with D50 because of using the average values of the wire diameter and number of meshes.

4.2　Reliability of simulationThe reliability of the simulation was verified by compar-

ing its results with those of experiments. The Dutch weave mesh specifications used for the verification are listed in Table 5, which also gives the average values of the experi-mentally determined warp diameter, dw, weft diameter, ds, number of meshes in the warp direction, nw, and number of meshes in the weft direction, ns.

The drag coefficient was calculated from the drag force determined from the simulation results using Eq. (23).

DD 2

0

2 ( / )f AC

ρU= (23)

Where, CD is the drag coefficient, fD is the drag force of the Dutch weave mesh, and A is the permeating area of flow. The drag force, fD was calculated as the sum of the external forces in the z (mainstream) direction.

Fig. 8　Schematic air permeability test device

Table 2　Experimental conditions

Fluid AirTemperature [°C] 20±5Density, ρf [kg/m3] 1.21Viscosity, μf [Pa·s] 18.1×10−6

Velocity, Uf [m/s] 0.0833–0.833Permeation area [m2] 1.00×10−3

Table 3　Twilled Dutch weave mesh specifications for experiments

Sample No. Material Weave type

Wire diameter Number of meshes Measurement

Warp [mm]

Weft [mm]

Warp direction [in−1]

Weft direction [in−1]

dw [mm]

ds [mm]

nw [in−1]

ns [in−1]

TD-1 SUS316 Twilled Dutch 0.15 0.12 32 450 0.143 0.119 32.0 421TD-2 SUS316 Twilled Dutch 0.08 0.055 120 1000 0.078 0.052 120 902TD-3 SUS316 Twilled Dutch 0.07 0.04 165 1400 0.068 0.038 167 1331TD-4 AISI316L Twilled Dutch 0.038 0.025 325 2300 0.035 0.024 326 2040TD-5 SUS304 Twilled Dutch 0.025 0.015 508 3600 0.025 0.014 504 3613

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L

D ext,zn

f F= (24)

The experimental drag coefficient was also calculated using Eq. (25).

D 2f f

2ΔPCρ U

= (25)

Where, ΔP is the experimentally determined pressure drop across the Dutch weave mesh, ρf is the fluid density, and Uf is the fluid velocity at the outlet.

Figure 9 shows the relationship between the calculated drag coefficient, CD, and the tentative Reynolds number, Ret, calculated using Eq. (22). The simulation results are in good agreement with those of the experiments, which verifies the reliability of the simulation. For the plain Dutch weave mesh, there is a slight difference between the simulation and experimental results. The reason for this difference is that, in the experiment, the number of meshes in the weft direc-tion, ns, was larger than that in the simulation. The number of meshes in the weft direction in the simulation model and the number in the actual Dutch weave mesh were different because the diameter of the wefts was held constant in the simulation; whereas, the wefts of the actual Dutch weave mesh became thinner and partially compressed during the weaving process. A large value of ns indicates a high weft density, with a consequent increase in the flow resistiv-ity. It is difficult to consider deformation of the wefts in a simulation, but the difference between the simulation and the experimental results can be explained logically. Thus, the simulation model was believed to adequately reflect the structure of Dutch weave meshes.

4.3　E�ect of aperture structure of Dutch weave mesh on �ow resistivity

The effect of the aperture structure of the Dutch weave

mesh on the flow resistivity was investigated. Table 6 lists the Dutch weave mesh specifications and calculated aper-ture sizes used for the simulations. The Dutch weave meshes had the same aperture size but different wire diameters and numbers of meshes. The aperture size of the plain Dutch weave mesh was calculated using the equation proposed by Yamamoto et al. (1986), and the aperture size of the twilled Dutch weave mesh was calculated using the proposed calcu-lation model. Simulations were performed at Ret=5 because filtrations using a Dutch weave mesh are mostly performed at low Reynolds number.

Figures 10 and 11 show examples of the flow states around a plain Dutch weave mesh and a twilled Dutch weave mesh, respectively. The fluid flowed through the three-dimensional spatial aperture produced by a separation in the wires. In the plain Dutch weave mesh, the drag force was large at the center in the thickness direction; whereas, in the twilled Dutch weave mesh, the drag force was large, not only at the center, but also at the foreside and backside in the thickness direction. The plain Dutch weave mesh had

Fig. 9 Comparison of simulation and experimental results for drag coefficient CD

Table 4　Comparison of calculated aperture sizes with challenge test results for twilled Dutch weave meshes

Sample No.Calculated aperture size

Challenge test

Dry method Wet method

δA1 [µm] δA2 [µm] δA3 [µm] δsub [µm] Cut point [µm] D50 [µm] D97 [µm]

TD-1 123 58.4 93.3 58.4 75 — —TD-2 60.6 29.9 26.9 26.9 30 23.0 29.7TD-3 38.3 23.3 16.2 16.2 19 16.6 20.6TD-4 25.8 11.8 8.34 8.34 — 8.5 9.9TD-5 14.1 8.21 4.81 4.81 — 4.7 6.0

No value: outside the applicable range of the test.

Table 5　Dutch weave mesh specifications for simulation and experiment

Material (Experiment) Weave type

Wire diameter Number of meshes Measurement (Experiment)

Warp [mm]

Weft [mm]

Warp direction [in−1]

Weft direction [in−1]

dw [mm]

ds [mm]

nw [in−1]

ns [in−1]

PD97.7 SUS304 Plain Dutch 0.38 0.26 24 97.7 0.359 0.239 23.9 109TD923.6 SUS316 Twilled Dutch 0.08 0.055 120 923.6 0.078 0.052 120 902

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two types of apertures (the weft–weft surface aperture and the weft–warp–weft inside aperture), and the fluid flowed through the surface aperture→inside aperture→surface ap-erture. The twilled Dutch weave mesh had three types of apertures (the surface aperture in area 1 and the inside apertures in areas 2 and 3, as discussed in section 1), and the fluid flowed through areas 1→2→3→2→1. Considering

these flow paths, it was found that the drag force was large at these inside apertures in both the plain Dutch weave and twilled Dutch weave meshes.

The variation in the average drag force fD in the xy plane in the thickness direction of the Dutch weave mesh is shown in Figure 12. As is the case with the flow state observation, the variation in the drag force for the plain Dutch weave mesh has a single peak at the center of the mesh; whereas, that for the twilled Dutch weave mesh is trimodally distrib-uted, with the peaks of the variation occurring at the center, foreside, and backside of the mesh. Figure 13 shows the variation in the average volume fraction in the xy plane in the thickness direction of the Dutch weave mesh. The peaks of the variation in the volume fraction for the Dutch weave mesh correspond with the peaks of the variation in the drag force. Thus, it can be seen that the volume fraction signifi-cantly affects the drag force. Moreover, because the inside aperture is a narrow space in the flow path, the wire density

Fig. 10 Flow state around a plain Dutch weave mesh (PD97.7, Ret=5)

Fig. 11 Flow state around a twilled Dutch weave mesh (TD923.6, Ret=5)

Fig. 12　Variation in average drag force in xy plane in z direction

Table 6　Dutch weave mesh specifications for simulations

Weave typeWire diameter Number of meshes Aperture size

Warp [mm]

Weft [mm]

Warp direction [in−1]

Weft direction [in−1]

δA1 [µm]

δA2 [µm]

δA3 [µm]

PD70.6 Plain Dutch 0.38 0.36 24.4 70.6 360 151PD97.7 Plain Dutch 0.38 0.26 24 97.7 260 151PD158.8 Plain Dutch 0.38 0.16 24.2 158.8 160 151

TD725.7 Twilled Dutch 0.08 0.07 116.7 725.7 70.0 26.1 24.5TD923.6 Twilled Dutch 0.08 0.055 120 923.6 55.0 27.7 24.5TD1270 Twilled Dutch 0.08 0.04 124 1270 40.0 29.2 24.5

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is high, with a consequent increase in the volume fraction. Therefore, the drag force is large at the inside aperture be-cause of its high volume fraction.

In the plain Dutch weave meshes, the difference in the drag forces was larger than the difference in the volume fractions. The tail of the volume fraction distribution is dif-ferent for each specification; whereas, the tails of the drag force distributions are almost the same for the different specifications. Therefore, it is believed that the drag force is increased by the wide tail of the volume fraction distribu-tion.

In the twilled Dutch weave meshes, the tail of the volume fraction distribution in each specification was almost the same as that for the drag fraction. However, the difference in the drag force at the center was larger than the difference in the volume fraction there. As observed in the flow state for each specification shown in Figure 14, a higher drag force results in a longer flow path at the center.

To evaluate this local flow path quantitatively, the length of the local flow path, lq, and the local torsion of the path, q, were derived using the proposed calculation model for esti-mating the aperture size of the twilled Dutch weave mesh. The length of the local flow path was determined from the position of the particle at the aperture in area 2, (xp,A2, yp,A2, zp,A2), and the position of the particle at the aperture in area 3, (xp,A3, yp,A3, zp,A3), using Eq. (26).

2 2 2p,A2 p,A3 p,A2 p,A3 p,A2 p,A3( ) ( ) ( )ql x x y y z z− − −= + +

(26)

There are two positions for particles at the aperture in area 2: between wefts L1 and L2, and between wefts L1 and L3, and the position that was the closest to the position of the particle in area 3 was adopted for the position, (xp,A2, yp,A2, zp,A2). The local torsion, q, was determined from the ratio of the length of the local flow path to its perpendicular length using Eq. (27).

p,A2 p,A3| |qlq y y−

= (27)

Figure 14 also shows the calculated local torsion. The local torsion correlates with the drag force at the center of the twilled Dutch weave mesh, which indicates the useful-ness of the local torsion for estimating the flow resistivity of the twilled Dutch weave mesh.

4.4　Equation for estimating pressure drop across Dutch weave mesh

From the simulation results, it was found that the Dutch weave mesh had a complex flow path as a result of its three–dimensional spatial aperture. Then, we applied the Kozeny–Carman equation (Miwa, 1981), which represents the flow resistivity of a particle bed (also having a complex flow path) in a laminar flow, to the equation for estimating the pressure drop across the Dutch weave mesh.

The following equation is the Hagen–Poiseuille equation, which represents the relationship between the flow velocity and the pressure drop in a pipe flow.

2pipe

ff pipe

Δ32d PU μ l⋅= (28)

Where, dpipe is the internal diameter of the pipe, and lpipe is the length of the pipe. The effective flow velocity inside the Dutch weave mesh given by Eq. (29), the hydraulic radius of the Dutch weave mesh, 4m, given by Eq. (30), and the thick-ness of the mesh, tm, as the flow path were considered for the Hagen–Poiseuille equation.

ff,e

UU ε= (29)

4 4 εm a⋅= (30)

We then derived the equation for estimating the pressure drop across the Dutch weave mesh as Eq. (31).

Fig. 13 Variation in average volume fraction in xy plane in z direc-tion

Fig. 14　Flow state inside twilled weave mesh (Ret=5)

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2f m f2Δ ka μ t UP εε

⋅= (31)

Where, k is the Kozeny constant, which correlates with the shape of the flow path, ε is the porosity of the Dutch weave mesh, and a is the surface area of the mesh per unit volume. ε and a can be calculated from the wire diameter and number of meshes using the simple geometrical models of a Dutch weave mesh proposed by Armour and Cannon (1968).

Furthermore, using the Reynolds number of the Dutch weave mesh, Ref, given by Eq. (32), Eq. (31) was nondimen-sionalized as Eq. (33).

f f f ff

f f

( / )4 4ρ U ε m ρ URe μ μ a= = (32)

31

f2f f m

Δ4

Pεf kReρ U t a

−= = (33)

Where, f is the dimensionless flow resistivity of the Dutch weave mesh.

To verify Eq. (33), the relationship between the dimen-sionless flow resistivity, f, and the Reynolds number of the Dutch weave mesh, Ref, was investigated using the experi-mental results. The specifications of the plain Dutch weave mesh used for the experiments are listed in Table 7, and the specifications of the twilled Dutch weave mesh used for the experiments are listed in Table 3. Table 8 gives the calculated thickness of the mesh, tm, porosity of the mesh, ε, surface area of the mesh per unit volume, a, and hydraulic radius, 4m.

Figure 15 shows the relationship between f and Ref. The relationship plotted in the logarithmic chart shows the linearity at Ref≤15 in the plain Dutch weave mesh and at Ref≤7 in the twilled Dutch weave mesh. The slope is ap-proximately −1, which verifies the validity of the proposed equation for estimating the pressure drop of the Dutch weave mesh in these Ref ranges. When Ref is greater than these ranges, the flow state through the mesh is considered to transition to a turbulent flow, which will require a modi-fication of the proposed equation. We believe that further study should be conducted to derive the equation for esti-mating the pressure drop considering this turbulent flow.

As previously mentioned, filtrations using a Dutch weave mesh are mostly performed at low Reynolds numbers, namely at laminar flows. Therefore, the proposed equation

for estimating the pressure drop will be useful for design-ing a filtration process and developing a filter medium. The proposed equation suggests that the pressure drop across the Dutch weave mesh can be effectively decreased by increas-ing Ref (which implies a decrease in the surface area of the Dutch weave mesh per unit volume, a while maintaining the same fluid condition) or by decreasing the porosity of the mesh, ε or its thickness, tm.

In the plain Dutch weave mesh, because the relationship between f and Ref was plotted as a single curve, the Kozeny constant, k, hardly varied with the mesh specification. In contrast, for the twilled Dutch weave mesh, the relationship was plotted as various curves for each mesh specification, which means the Kozeny constant, k, in the twilled Dutch weave mesh varied with the mesh specification. From the simulation results, it was found that the local torsion inside the twilled Dutch weave mesh affected the flow resistiv-ity. Thus, the relationship between the Kozeny constant, k,

Table 8 Dutch weave mesh properties calculated from experimental results

Sample No. tm [m] ε [—] a [m2/m3] 4m [m]

PD-1 1.38×10−3 0.639 3.30×103 7.75×10−4

PD-2 8.37×10−4 0.623 5.68×103 4.38×10−4

PD-3 5.49×10−4 0.654 7.83×103 3.34×10−4

PD-4 4.29×10−4 0.645 1.02×104 2.52×10−4

PD-5 3.28×10−4 0.670 1.26×104 2.13×10−4

TD-1 3.81×10−4 0.451 1.82×104 9.93×10−5

TD-2 1.81×10−4 0.430 4.10×104 4.19×10−5

TD-3 1.44×10−4 0.376 5.80×104 2.59×10−5

TD-4 8.29×10−5 0.361 9.90×104 1.46×10−5

TD-5 5.26×10−5 0.350 1.64×105 8.57×10−6

Fig. 15　Relationship between Ref and f

Table 7　Plain Dutch weave mesh specifications for experiments

Sample No. Material Weave type

Wire diameter Number of meshes Measurement

Warp [mm]

Weft [mm]

Warp direction [in−1]

Weft direction [in−1]

dw [mm]

ds [mm]

nw [in−1]

ns [in−1]

PD-1 SUS316L Plain Dutch 0.6 0.42 12 64 0.559 0.410 12.0 66.7PD-2 SUS304 Plain Dutch 0.38 0.26 24 110 0.359 0.239 23.9 109PD-3 SUS316 Plain Dutch 0.23 0.18 30 150 0.211 0.169 29.9 154PD-4 SUS304 Plain Dutch 0.18 0.14 40 200 0.163 0.133 40.3 197PD-5 SUS304 Plain Dutch 0.14 0.11 50 250 0.130 0.099 50.1 247

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determined from the experimental results and the local tor-sion, q, calculated using Eqs. (26) and (27) was investigated. Figure 16 shows the relationship between the Kozeny con-stant, k, and the local torsion, q, in the twilled Dutch weave mesh. The Kozeny constant increases with an increase in the local torsion, with a consequent increase in the flow resistiv-ity. This is the same tendency as the simulation results.

The approximate value of the Kozeny constant, k, in the plain Dutch weave and the curve for the relationship be-tween the Kozeny constant, k, and the local torsion, q, are represented by Eqs. (34) and (35).Plain Dutch weave mesh:

12.6k= (34)

Twilled Dutch weave mesh:

2.34k q= (35)

Equations (31), (34) and (35) are practical equations for es-timating the pressure drop across a Dutch weave mesh. The pressure drop can also be effectively reduced by reducing the local torsion, q, for a twilled Dutch weave mesh.

The Kozeny constant, k, of the twilled Dutch weave mesh was relatively small compared to that of the plain Dutch weave mesh (although the value depends on the local tor-sion). This suggests that the flow resistivity of the twilled Dutch weave mesh is lower than that of the plain Dutch weave mesh at the same Reynolds number, Ref. From Eq. (32), the Reynolds number and flow conditions being the same means that the surface area per unit volume, a, is the same for the plain Dutch weave and twilled Dutch weave meshes. However, the dead space for the flow is thought to be larger in the twilled Dutch weave mesh (i.e. the effective surface area of the flow path is smaller) because its wefts are structurally denser than those of the plain Dutch weave mesh. Thus, the flow resistivity of the twilled Dutch weave mesh was relatively small compared to that of the plain Dutch weave mesh, and the value of the Kozeny constant was smaller. Further study should be conducted to evaluate the dead space of the Dutch weave mesh.

Conclusion

To determine the effect of the aperture structure of a Dutch weave mesh on the flow resistivity, we first proposed a calculation model for estimating the aperture size of a twilled Dutch weave mesh to thoroughly understand the ap-erture structure. Next, we performed numerical simulations using a combination of the lattice Boltzmann and immersed boundary methods, and an equation for estimating the pres-sure drop across a Dutch weave mesh was derived.

According to the simulation results, the drag force in-creased at the inside aperture, where the volume fraction increased. Therefore, the variation in the drag force of the plain Dutch weave mesh in the thickness direction had a single peak; whereas, the drag force of the twilled Dutch weave mesh was trimodally distributed, and the drag force at the center varied with the local torsion of the flow path inside the mesh.

We applied the Kozeny–Carman equation to represent the flow resistivity and derived an equation for estimating the pressure drop. The validity of this equation for a lami-nar flow was verified experimentally. The Kozeny constant of the plain Dutch weave mesh remained almost constant; whereas, that of the twilled Dutch weave mesh varied with the local torsion. The Kozeny constant of the twilled Dutch weave mesh was relatively small compared to that of the plain Dutch weave mesh.

A rational and sophisticated design for a filtration process using a Dutch weave mesh could be developed using the proposed equation for estimating the pressure drop and the model for calculation of the aperture size.

Nomenclature

A = permeating area in simulation [—]a = surface area of Dutch weave mesh per unit

volume [m2/m3]CD = drag coefficient [—]c = characteristic velocity of ideal particles [—]ci = advection velocity of ideal particles [—]D50 = 50% separation size [µm]D97 = 97% separation size [µm]d = wire diameter [mm]dpipe = internal diameter of pipe [m]Ei = weight coefficients [—]Fext = external force vector [—]fD = drag force of Dutch weave mesh [—]gi = distribution function of ideal particles [—]gi

(0) = equilibrium distribution function [—]k = Kozeny constant [—]kG = factor to determine center of ellipse given in Eq. (5) [—]kH = factor to determine center of ellipse given in Eq. (9) [—]lpipe = length of pipe [m]lq = length of local flow path in twilled Dutch weave

mesh [mm]m = hydraulic radius of Dutch weave mesh [m]n = number of wires per inch [in−1]nL = number of lattices [—]p = wire pitch in simulation or experiment [—] or [mm]

Fig. 16　Relationship between torsion q and Kozeny constant k

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q = local torsion in twilled Dutch weave mesh [—]Ret = tentative Reynolds number [—]Ref = Reynolds number of Dutch weave mesh [—]t = time [—]tm = thickness of Dutch weave mesh [m]u = fluid velocity vector [—]us = solid velocity vector [—]U0 = flow velocity at outlet in simulation [—]Uf = flow velocity at outlet in experiment [m/s]x = position vector of lattice point [—]x = x position in calculation model for estimating

aperture size [mm]y = y position in calculation model for estimating

aperture size [mm]z = z position in calculation model for estimating

aperture size or simulation [mm] or [—]

α = volume fraction of woven mesh [—]ΔP = pressure drop across Dutch weave mesh [Pa]Δt = time step [—]δ = aperture size [mm]δj,z = diameter of particle which is in contact with wefts

L1 and L2 or L3 [mm]δ12 = diameter of particle passing between wefts L1 and

L2 [mm]δ13 = diameter of particle passing between wefts L1 and

L3 [mm]Δ = grid spacing [—]ε = porosity of woven mesh [—]θT = angle between perpendicular of center line l4 and z

axis in xz plane in Figure 2 [rad]θS = angle between center line l2 and z axis in yz plane in

Figure 2 [rad]μf = fluid viscosity [Pa·s]ν = fluid kinematic viscosity in simulation [—]ρ = density of ideal particles [—]ρ0 = density of ideal particles at inlet [—]ρf = fluid density [kg/m3]τ = single time relaxation parameter [—]

‹Subscript›A1 = first apertureA2 = second apertureA3 = third aperturee = effectiveG = center of ellipse given in Eq. (5)H = center of ellipse given in Eq. (9)i = advection direction of ideal particlesmax = maximump = particles = weftsub = substantivew = warp

x = x directiony = y directionz = z direction

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