Determination of Permeability and Inertial Coefficients of Sintered

applied sciences

Article Determination of Permeability and Inertial Coefﬁcients of Sintered Metal Porous Media Using an Isothermal Chamber

Wei Zhong 1,2,* , Xiang Ji 1,2, Chong Li 1,2 , Jiwen Fang 1,2 and Fanghua Liu 1,2 1 School of Mechanical Engineering, Jiangsu University of Science and Technology, Zhenjiang 212003, China; [email protected] (X.J.); [email protected] (C.L.); [email protected] (J.F.); [email protected] (F.L.) 2 Jiangsu Provincial Key Laboratory of Advanced Manufacture and Process for Marine Mechanical Equipment, Jiangsu University of Science and Technology, Zhenjiang 212003, China * Correspondence: [email protected]; Tel.: +86-511-8444-5385

Received: 7 August 2018; Accepted: 10 September 2018; Published: 15 September 2018

Abstract: Sintered metal porous media are widely used in a broad range of industrial equipment. Generally, the flow properties in porous media are represented by an incompressible Darcy-Forchheimer regime. This study uses a modified Forchheimer equation to represent the flow rate characteristics, which are then experimentally and theoretically investigated using a few samples of sintered metal porous media. The traditional steady-state method has a long testing time and considerable air consumption. With this in mind, a discharge method based on an isothermal chamber filled with copper wires is proposed to simultaneously determine the permeability and inertial coefficient. The flow rate discharged from the isothermal chamber is calculated by differentiating the measured pressure, and a paired dataset of pressure difference and flow rate is available. The theoretical representations of pressure difference versus flow rate show good agreement with the steady-state results. Finally, the volume limit of the isothermal chamber is addressed to ensure sufficient accuracy.

Keywords: porous media; permeability; inertial coefﬁcient; isothermal chamber; discharge method

1. Introduction Air flow through porous media is universal in a broad range of engineering activity, which includes such diverse fields as pneumatics, filtration, soil mechanics, and petroleum engineering. Pressure drops occur when such porous media are connected in an air circuit because they involve extremely complicated flow channels. Thus, accurate prediction of the correlation of pressure drop with flow rate is crucial for performance evaluation, since it determines the energy requirements of the supplying compressors. Some researchers have demonstrated the physics of flow through porous media. They modeled the flow behavior by a linear relation of pressure gradient versus flow velocity (Darcy regime) in cases where the flow is dominated by viscous effects. When the flow velocity becomes adequately large, they used a nonlinear relation to include the inertial effects [1–6]. Some researchers attempt to use theoretical methods to correlate with experimental data. The two best-known theoretical relations are the Forchheimer equation [7,8] and Ergun or Ergun-type equations [9,10]. Both are widely used [8], and there is no specific reason for choosing one rather than the other. Early works by Beavers et al. [11,12] reported that the flow pressure characteristics of porous media can be represented by a dimensionless expression related to the square root of the permeability. Montillet et al. [13] extended the applicability of an existing correlating equation to predict the pressure drop through packed beds of spheres. Antohe et al. [14] proposed a more precise method based on curve fitting to calculate the permeability and inertia coefficient of porous matrices

Appl. Sci. 2018, 8, 1670; doi:10.3390/app8091670 www.mdpi.com/journal/applsci Appl. Sci. 2018, 8, 1670 2 of 17 in terms of a Forchheimer flow model. Dukhan and Minjeur [15] found that the permeability for the Darcy regime differs from that for the Forchheimer regime. Liu [16] developed an Ergun-type empirical equation to correlate the dimensionless pressure drop with flow velocity for several types of foam matrix porous media. Dukhan and Patel [17] correlated the permeability and the form drag coefficient using an Ergun model with the reciprocal of the surface area density, which has a unit of m3/m2, indicating that it is used as an equivalent length scale. Dietrich et al. [18] presented experimental data of pressure drop measurements of different ceramic sponges and determined the two constants of the Ergun equation, which are verified to be independent of the material and porosity. The Forchheimer and Ergun equations look very similar in that they both have a viscous term and an inertia term. The difference is that the Forchheimer equation uses an analogy with pipe flow while the Ergun equation models the space between packed beds of spheres as parallel capillaries. An overview of the literatures is provided in Table1. Air compressibility is negligible and flow velocity can be treated as a constant through porous media with a large porosity (over 0.8) [19–21]. Existing studies [22–27] proved that the Darcy-Forchheimer-based theory can be applied to both compressible and incompressible fluids. Belforte et al. [22,23] measured considerable pressure differences during the operation of compressed air bearings equipped with sintered bronze porous pads. Nicoletti et al. [24] predicted the pressure-flow relationship of the aerostatic porous bearing based on the quadratic Forchheimer assumption. Amano et al. [25], Oiwa et al. [26], and Miyatake et al. [27] have developed air conveyor systems with porous air supply pads. In these applications, the pressure drop is significant (400 kPa or more) and the air compressibility needs to be considered. The authors have developed theoretical equations and represented the pressure drop characteristic of sintered metal porous media in terms of experimentally determined property coefficients [28–31]. However, the measurements are conducted under steady flow condition and this inevitably lengthens the test time and consumes much energy. Kawashima et al. [32] proposed a method using an isothermal chamber to obtain the flow rate characteristics of solenoid valves by the ISO 6358 equation. The equation depicts the flow rate characteristics with two independent parameters: sonic conductance C and critical pressure ratio b, which is determined in the choked flow state and subsonic flow state, respectively (the equation is detailed in the Appendix). The isothermal chamber is stuffed with steel or copper wool to enlarge the heat transfer area so that the air state change can be considered isothermal. Therefore, instantaneous flow rate charged into or discharged from the chamber can be indirectly calculated by differentiating the pressure in the chamber. If this approach can be applied to the test of porous media, the property coefficients can be ascertained more conveniently and effectively. To date, the discharge method has been used by other researchers for determining the flow rate characteristics of some pneumatic components (valves, etc.) in terms of the ISO 6358 equation. However, the porous media have very different properties to the common pneumatic components, and therefore the operating procedure needs to be reconstructed to extend its application to porous media. In this study, air is employed as the working fluid in order to investigate the flow rate characteristics of the porous media for some pneumatic applications (e.g., air bearing). The objective of the study is to use a simple and convenient method to measure and evaluate the permeability and inertial coefficient to represent the flow rate characteristics. Accordingly, a discharge method based on an isothermal chamber is proposed to conveniently ascertain the coefficients. This method accomplishes the test within the discharge process (only a few seconds), and of course reduces the air consumption. Three groups of sintered metal porous media made by different class of powder were tested. The validity of the discharge method was verified because the flow rate characteristics can be well represented with the determined coefficients. Appl. Sci. 2018, 8, 1670 3 of 17

2. Materials and Methods For flowing through porous media with high enough velocity, the pressure gradient dP/dx versus flow velocity is described using the Forchheimer equation: Appl. Sci. 2018, 8, x FOR PEER REVIEW 3 of 17 dP µυ ρυ2 − = + β 2 dP  1/2 , (1) dx K  K , (1) dxKK 1/2 where µ is the air viscosity, v is the velocity, ρ is the density, K is the permeability and β is the inertial where μ is the air viscosity, v is the velocity, ρ is the density, K is the permeability and β is the inertial coefficient. The Forchheimer equation includes two terms denoting the viscous effect and the inertial coefficient. The Forchheimer equation includes two terms denoting the viscous effect and the inertial effect. Removing the quadratic term yields the Darcy equation. The Forchheimer equation employs effect. Removing the quadratic term yields the Darcy equation. The Forchheimer equation employs the squarethe square root root of permeability of permeability as as equivalent equivalent lengthlength characteristics in inorder order to agree to agree with with the linear the linear DarcyDarcy law at law low at velocities, low velocities, andthe and dimensionless the dimensionless coefficient coefficient in the in quadratic the quadratic term term is used is used to represent to the inertialrepresent effect. the inertial effect. FigureFigure1 schematically 1 schematically shows shows the physicalthe physical model model for for air air flow flow through through a porousa porous medium. medium. The The flow is assumedflow is assumed to be steady to be steady and fullyand fully developed, developed, and and only only inin the length length direction direction (x coordinate). (x coordinate). Actually,Actually, the distributionthe distribution of poresof pores with with respect respect toto shape and and size size is isirregular irregular in the in theporous porous medium. medium. On theOn microscopicthe microscopic scale, scale, the the flowflow quantities quantities (velocity, (velocity, pressure, pressure, etc.) etc.)are also are clearly also irregular. clearly irregular.The Forchheimer regime models the flow through a porous medium as a pipe flow, and therefore the The Forchheimer regime models the flow through a porous medium as a pipe flow, and therefore quantities of interest are described using the space-averaged quantities for representative areas that the quantities of interest are described using the space-averaged quantities for representative areas cross many pores. That is, as shown in Figure 1, the pressure and velocity are considered to be that crossuniformly many distributed pores. That along is, as a shown tiny length in Figure (dx).1 , Moreover, the pressure the and porous velocity medium are isconsidered considered to be uniformlyisotropic distributed and homogeneous, along a tiny and length therefore (dx the). Moreover, ratio of the the circulation porous area medium and the is consideredtotal area equals isotropic and homogeneous,the porosity. and therefore the ratio of the circulation area and the total area equals the porosity.

Representative volume dx A G P1 P2

x φ

L Fluid phase Solid phase Porous medium FigureFigure 1. Schematic1. Schematic diagram diagram ofof airﬂowairflow through through a aporous porous medium medium..

Accordingly,Accordingly, the meanthe mean ﬂow flow velocity velocity in in each each thin thin section section (d (dx)x) varies varies withwith thethe airair density, and and can be calculatedcan be calculated by the following by the following expression: expression:

G υ= ,, (2) (2) ρAAϕ where G is the mass flow rate, A is the cross-sectional area, and φ is the porosity. where G is the mass ﬂow rate, A is the cross-sectional area, and ϕ is the porosity. According to the ideal gas state equation, the air density ρ can be expressed as below: According to the ideal gas state equation, the air density ρ can be expressed as below: P   , (3) RTP ρ = , (3) RT where R is the gas constant and T is the temperature. where R isSubstituting the gas constant Equations and (2T) andis the (3) temperature. into Equation (1) and integrating the pressure with respect to Substitutinglength (x-coordinate) Equations, with (2) the and boundary (3) into conditions Equation (1)P = andP1 (x integrating = 0 [m]), P = the P2 pressure(x = L [m]), with weobtain respect to the following modified Forchheimer equation: length (x-coordinate), with the boundary conditions P = P1 (x = 0 [m]), P = P2 (x = L [m]), weobtain the 22 following modiﬁed Forchheimer equation:RT2 RT P21 P GG   0 . (4) KA 22 KAL 2 βRT µRT P2 − P2 √ G2 + G + 2 1 = 0. (4) The square root of the permeabilityKϕ2 A2 is employedKϕA as a characteristic2L dimension and a Reynolds number is defined as

Appl. Sci. 2018, 8, 1670 4 of 17

The square root of the permeability is employed as a characteristic dimension and a Reynolds numberAppl. isSci. defined 2018, 8, x asFOR PEER REVIEW 4 of 17 ρυK1/2 Re = . (5) µK1/2 Re  . (5) A dimensionless friction factor f is defined as  A dimensionless friction factor f is defined as √ (−dP/dx) K f = . (6) d/dPxK2 f  ρυ . (6)  2 Comparing Equation (6) with the Darcy–Weisbach equation for the case of flowing in a cylindrical Comparing Equation (6) with the Darcy–Weisbach equation for the case of flowing in a pipe, it is observed that the square root of permeability is actually equivalent to two times the hydraulic cylindrical pipe, it is observed that the square root of permeability is actually equivalent to two diameter of the pipe. Substituting Equations (5) and (6) into Equation (1) obtains the following times the hydraulic diameter of the pipe. Substituting Equations (5) and (6) into Equation (1) correlationobtains for thethe following friction correlation factor as for a function the friction of factor the Reynolds as a function number: of the Reynolds number: 1 f f=+β. (7) (7) ReRe

3. Results 3. Results 3.1. Test Medium 3.1. Test Medium The porous media used were fabricated by sintering SUS316L powder at a temperature above ◦ The porous media used were fabricated by sintering SUS316L powder at a temperature above 1000 1000C. Three°C . Three pieces pieces of of porous porous pad pad (Figure (Figure2 2aa),), withwith a diameter of of 40 40 mm mm and and a length a length of 3 ofmm, 3 mm, werewere firstly firstly made made with with existing existing models models using using differentdifferent class classeses of of powder powder to form to form different different pore poresizes. sizes. However,However, a large a large size size not onlynot only compounds compounds the the difficulty difficulty in in sealing sealing but but also also requires requires aa flowflow metermeter with a largewith measurement a large measurement range. Hence, range. Hence, each original each original medium medium is cut isinto cut into five five small small pieces pieces to to form form a a group (a totalgroup of three (a total groups) of three to groups) perform to theperform experiments the experiments separately. separately Some. Some studies studies [33,34 [33,] revealed34] revealed that the size effectthat the would size effect influence would the influence property the coefficients. property coefficient In this regard,s. In this as regard, shown as in shown Figure in2 b,Figure small 2b pieces, of samplessmall pieces are selected of samples with are a selected diameter with of a 7, diameter 8, 9, 10, of and 7, 8, 11 9, mm,10, and which 11 mm, can which be regarded can be regarded as infinitely as infinitely large compared with the pore size so that the wall effect in the porous media is large compared with the pore size so that the wall effect in the porous media is negligible. negligible.

40 mm

#1 #2 #3

(a) (b)

FigureFigure 2. Photographs 2. Photograph ofs of the the porous porous medium: medium: (a) Original Original porous porous media media;; (b) (tbest) test porous porous media media..

DuringDuring the manufacturingthe manufacturing process, process, particle particle size size changes changesas as aa resultresult ofof the shrinkage caused caused by by the process’sthe process’s linking effect. linking Therefore, effect. Therefore, we use we SEM use imaging SEM imaging (JEOL, (JEOL, Japan, Japan, JSM-6480) JSM-6480) to roughly to roughly estimate estimate the particle diameter for indication of the pore size. Three specimens for SEM testing are the particle diameter for indication of the pore size. Three specimens for SEM testing are extracted at extracted at random from each group and the SEM photographs are shown in Figure 3. The particle random from each group and the SEM photographs are shown in Figure3. The particle diameters for diameters for the three groups of porous media are visually discernible in the figure, approximately the threein the groups range of 40 porous–50 μm, media 55–65 are μm visually, and 80– discernible90 μm, respectively. in the ﬁgure, Regarding approximately the porosity in φ the in rangethe of 40–50theoreticalµm, 55–65 equations,µm, and it 80–90 can beµ definedm, respectively. as the ratio Regarding of void space the to porosity the totalϕ volumein the theoreticaland calculated equations, as it canbelow be deﬁned: as the ratio of void space to the total volume and calculated as below: m ϕ = 1 − , (8) γV

Appl. Sci. 2018, 8, x FOR PEER REVIEW 5 of 17

m  1 , (8) Appl. Sci. 2018, 8, 1670 V 5 of 17 where γ is the metal density (8.03 × 103 kg/m3), m is the mass of the medium, and V is the total where γ is the metal density (8.03 × 103 kg/m3), m is the mass of the medium, and V is the total volume. volume. Table 2 lists the estimated particle diameter, dimensions, and porosity of the test porous Table2 lists the estimated particle diameter, dimensions, and porosity of the test porous media. media.

Group 1 Group 2 Group 3

m 0 μ 5 6 5 μm m 88 μ

82 μm

μm m 45 μ 61

58 μm

m 4 μ 7 μ 4 m 8

Figure 3. SEM images of magniﬁedmagnified particles for some porous media.

Table 1.1. ClassiﬁcationClassification ofof thethe references.references.

ModelModel AuthorsAuthors Ma Materialterial Working Working Fluid Fluid Liao etLiao al., (2016) et al., [1 ,(22]016) [1,2] SinteredSintered metal metal porous porous Air Air Lage etLage al., (1997) et al., [3 ](1997) [3] AluminumAluminum porous porous media media Air Air Newtonian, Andrade et al., (1999) [4] Assumed disorder porous media Andrade et al., Assumed disorder Newtonian,Incompressible fluid Boomsma(19 et99 al.,) [4 (2002)] [5] Open-cellporous media aluminum foamsIncompressible Water fluid MedrajBoomsma et al., (2007) et [6 al.], metallic foams Air Beavers & Sparrow, (1969) [11] MetallicOpen-cell fibers aluminum foams Water Liquid Forchheimer or (2002) [5] Beavers et al., (1973) [12] Packed beds of spheres Liquid Forchheimer-type equation Medraj et al., (2007) [6] metallic foams Air Poly-alpha-olefin fluid AntoheetBeavers al, (1997) & [Sparrow14] , Compressed aluminum porous matrices Metallic fibers LiquidAir Dukhan(1 &96 Minjeur,9) [11] (2011) [15] open-cell aluminum foam Air Jin & Kai,Beavers (2008) et [20 a]l., Open-cell metal foams Air Mancin et al., (2010) [21] aluminumPacked beds open-cell of spheres foam Liquid Air Rodrigo(1 et97 al.,3) (2008)[12] [24] Bronze sintered porous bearing Air Forchheimer or ZhongAntohe et al., (2014)et al [,28 (1]997) SinteredCompressed metal porousaluminum media Poly Air-alpha-olefin fluid Forchheimer-type equation Dukhan[1 &4] Ali, (2012) [33] open-cellporous matrices aluminum foam Air Air Dukhan & Minjeur, Packed spheres porous media Nihad et al., (2014) [8] open-cell aluminum foam Air Water (2011) [15] aluminum foam Liuet al, (2006) [16] foam matrixes Air Ergun or Ergun-type equation Jin & Kai, (2008) [20] Open-cell metal foams Air Dukhan & Patel, (2008) [17] Metal foam Air Mancin et al., Dietrich et al., (2009) [18] Ceramicaluminum sponges open-cell foam Air Air Zhong(2 et0 al.,10 (2016)) [21] [31] Sintered metal porous media Air Rodrigo et al., Bronze sintered porous A correlating equation Montillet et al., (2007) [13] packed beds of spheres Air Liquid (2008) [24] Porousbearing Medium-like Cylinder A force balance flow model Kim & Lu, (2008) [19] Air Zhong et al., BundlesSintered metal porous Air Amano(2 et0 al.,14)(2011) [28] [25] Porousmedia bearing pad Air Darcy Law Oiwa etDukhan al., (2012) & [ 26Ali,] Porousopen-cell bearing aluminum pad foam Air Air (2012) [33]

Appl. Sci. 2018, 8, x FOR PEER REVIEW 6 of 17

Packed spheres Nihad et al., (2014) [8] porous media Water aluminum foam Liuet al, (2006) [16] foam matrixes Air Dukhan & Patel, Metal foam Ergun or Ergun-type equation Air (2008) [17] Dietrich et al., Ceramic sponges Air (2009) [18] Zhong et al., Sintered metal Air (2016) [31] porous media Montillet et al., packed beds of spheres A correlating equation Liquid (2007) [13] Porous Medium-like A force balance flow model Kim & Lu, (2008) [19] Cylinder Air Bundles Amano et al., Porous bearing pad Air Appl.Darcy Sci. 2018 Law, 8, 1670 (2011) [25] 6 of 17 Oiwa et al., (2012) [26] Porous bearing pad Air

3.23.2.. Steady Steady-State-State M Methodethod Figure4 4 shows shows the the pneumatic pneumatic circuit circuit for for the the measurement. measurement. Akbarnejad Akbarnejad et al.et [al35. [35]] stated stated that that if the if thetest test medium medium is not is wellnot well sealed sealed the measuredthe measured parameters parameters deviates deviates greatly greatly from from its real its value.real value. In this In work,this work, the porous the porous medium medium is placed is placed in a cylindrical in a cylindrical rubber and rubber two and end two covers end are covers used to are compress used to compressthe rubber. the The rubber rubber. The has rubber a trend has of a reductiontrend of reduction in the inner in the diameter inner diameter when subjected when subjected to the axial to thecompressing axial compressing force. If the force. rubber If the is compressedrubber is compressed tightly enough, tightly the enough, test medium the test can medium be well can sealed. be wellPressurized sealed. airPressurized is adjusted air by is a adjusted regulator by and a regulator a buffer tank and with a buffer a volume tank with of 10 a L volume is in use. of A 10 pressure L is in ussensore. A pressure (GE Druck, sensor UK, (GE PTX5012, Druck, range: UK, PTX5012, 0–1 MPa, range: accuracy: 0–1 MPa, 0.1%), accuracy: which is0. carefully1%), which calibrated is carefully by calibrateda high-precision by a high pressure-precision controller pressure (GE controller Druck, UK, (GE PACE6000), Druck, UK, is PACE6000), used to measure is used the to upstream measure pressure,the upstream and the pressure downstream, and pressurethe downstream is treated as pressure atmospheric is treated pressure as throughout. atmospheric The pressure throughoutsensor is installed. The pressure at a sufficient sensor isdistance installed fromat a sufficient the test elementdistance tofrom eliminate the test element the entrance to eliminate effects. Anthe adjustableentrance effects. valve isAn installed adjustable after valve the buffer is installed tank to after vary the buffer flow rate. tank The to flowvary ratethe inflow the rate. circuit The is 3 flowmeasured rate in by the a dry circuit test gasis measured meter (DS-6A, by a Shinagawa dry test gas Co., meter Ltd., (DS Japan,-6A, range: Shinagawa 0.04~6 Co., m /h, Ltd. accuracy:, Japan, range:0.3%). 0.04 During~6 m the3/h, experiments,accuracy: 0.3% the). During paired datathe experiments, of pressure differencethe paired anddata flowof pressure rate are difference recorded andas the flow opening rate are of recorded the valve as is adjusted.the opening For of minute the valve flow is rate,adjusted. a wet-type For minute gas meter flow (W-NKDa-0.5A,rate, a wet-type gasShinagawa meter (W Co.,-NKDa Ltd.,-0.5A, Japan, Shinagawa range: 0.016-5 Co., L/min,Ltd., Japan, accuracy: range: 0.1%) 0.016 is‒5 used, L/min and,, accuracy: correspondingly, 0.1%) is used, the pressureand, correspondingly, difference is measured the pressure by a differential difference pressureis measured transducer by a differential (PA-100-500D-S, pressure Copal transducer Co., Ltd, Japan,(PA-100 range:-500D 0–5-S, Copal kPa, accuracy: Co., Ltd, 0.5%).Japan, range: 0–5 kPa, accuracy: 0.5%).

Porous media Rubber

Computer IN OUT Data Acquisition Device

Test element Pressure transducer Buffer tank Air supply

P1 Pa Adjustable Test element Gas meter valve Regulator Figure 4.4. Schematic of the apparatus for steady steady-state-state method.

3.3. Discharge Method with an Isothermal Chamber Figure 5 shows the experimental setup for the discharge method. An isothermal chamber that can provide almost isothermal condition is achieved by filling it with copper wire, which has a diameter of approximately 50 μm. The volume of the filled wire is about 5% compared with that of the chamber. The used isothermal chamber has a volume of 0.5 L, 0.65 L, and 0.8 L for the test samples of groups 1, 2 and 3, respectively. The procedure is simple in comparison with the steady-state method because it is not required to record data point by point. First, the solenoid valve is closed and the hand valve is opened to charge the chamber until its pressure reaches the regulated level. Then, close the hand valve and send a control signal to the solenoid valve to start the discharge process until the pressure in the chamber reaches the atmospheric pressure. A calibrated pressure sensor (GE Druck, UK, PTX5012, range: 0–1 MPa, accuracy: 0.1%) is used to measureAppl. Sci. 2018 the, 8 , cham1670 ber pressure. The pressure data were transferred into a personal computer7 of 17 through a 14-bit A/D board (Advantech 4704), and the sampling time was 5 ms. The flow rate discharged from the isothermal chamber can be calculated according to the differentiated pressure, astime below was: 5 ms. The ﬂow rate discharged from the isothermal chamber can be calculated according to the differentiated pressure, as below: VPV dP GG= ccc c ,, (9 (9)) RtRθ dt where Pc is the pressure in the chamber, t is the time, Vc is the volume of the chamber, R is gas constant, where Pc is the pressure in the chamber, t is the time, Vc is the volume of the chamber, R is gas constantand θ is the, and room θ is temperature.the room temperature. It should should be be noted noted that, that, in intheory, theory, both both the charge the charge method method and the and discharge the discharge method method are capable are capableof obtaining of obtaining the ﬂow ratethe flowcharacteristics rate characteristics of the test element.of the test The element. discharge The method discharge is to method discharge is air to dischargethrough the air test through resistance the test into resistance the atmosphere into the from atmosphere a pressurized from chamber, a pressurized while chamber the charge, while method the chargeuses a constantmethod airuses supply a constant to charge air supply air into to the charge chamber. air into However, the chamber. in this work,However, the discharge in this work, method the dischargeis preferred method since the is chargepreferred method since exhibits the charge a sudden method drop exhibits in the a upstream sudden pressuredrop in the at the upstream start of pressthe chargingure at the process. start of the charging process.

D/A

A/D PC Pressure Hand sensor Porous valve resistance

Air supply Pressure Isothermal Solenoid regulator chamber valve Figure 5 5.. Schematic of the apparatus for discharge method method.. 4. Discussion 4. Discussion 4.1. Results of the Steady-State Method 4.1. Results of the Steady-State Method Figure6 plots the pressure difference as a function of the mass flow rate for all the test media. Figure 6 plots the pressure difference as a function of the mass flow rate for all the test media. The pressure difference increases with the flow rate and appears slightly nonlinear as the flow rate The pressure difference increases with the flow rate and appears slightly nonlinear as the flow rate becomes adequately large. The viscous effect and the inertial effect can be viewed by permeability K becomes adequately large. The viscous effect and the inertial effect can be viewed by permeability K and inertial coefficient β, which can be determined according to the modified Forchheimer equation. and inertial coefficient β, which can be determined according to the modified Forchheimer equation. Observing these coefficients provides more insight than observing the pressure drop. ObservingAppl. Sci. 2018 these, 8, x FOR coefficients PEER REVIEW provides more insight than observing the pressure drop. 8 of 17

Group 1 Group 2 Group 3

600 600 600 )

a 7 mm

P 8 mm

k 500 500 500

(

e 9 mm c

n 400 400 400 10 mm

e r

e 11 mm

f f

i 300 300 300

e r

u 200 200 200

s e

r 100 100 100 P

0 0 0 0 2 4 6 8 10 12 14 0 3 6 9 12 15 0 3 6 9 12 15 18 Flow rate (10-4 kg/s) Flow rate (10-4 kg/s) Flow rate (10-4 kg/s) FigureFigure 6. VariationVariation ofof measuredmeasured pressurepressure differencedifference withwith massmass ﬂowflow rate.rate.

TheThe viscousviscous effecteffect dominates dominates the the flow flow pattern pattern at aat sufficiently a sufficiently small small flow flow velocity, velocity, and inand this in casethis thecase Darcy the Darcy equation equation can be can used be to used characterize to characterize the flow the behavior. flow behavior. However, However, how small how flow small velocity flow canvelocity be used can as be the used boundary as the for boundary the transition for the from transition viscous flowfrom to viscous inertial flowflow isto unknown. inertial flow To the is bestunknown. knowledge To the of the best authors, knowledge it is almost of the impossible authors, to it define is almost a universal impossible boundary to define for the a transition universal ofboundary regimes forapplicable the transition to all kinds of regimes of porous applicable media. Although to all kinds some of researchers porous media. presented Although a definite some boundaryresearchers of thepresented Reynolds a definite number forboundary the regime of the transition, Reynolds it depends number on for the the properties regime of transitio the porousn, it depends on the properties of the porous material and the working fluid they used. For example, Dybbs and Edwards [36] reported that the boundary for viscous flow regime and inertial flow regime is Re = 1; however, it cannot be used in this work. For this problem, firstly, experiments are conducted under a slight pressure drop less than 3 kPa, where the air compressibility can be neglected and the pressure difference should exhibit a linear relationship with the flow rate. Figure 7 shows the measuring results in the form of pressure difference versus the mass flow rate. The data points are fitted using a least squares straight line, and the permeability can be calculated by the slope. The selected flow region proved appropriate since the R2 value of the fitting line exceeds 0.98.

Group 1 Group 2 Group 3 3 3 3

7 mm ) a 8 mm

P 2.5 2.5 2.5 k

( 9 mm

c 2 2 2 10 mm n

e 11 mm

r e

f 1.5 1.5 1.5 Fitting

e 1 1 1

s s

e 0.5 0.5 0.5

r P 0 0 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 1 2 3 4 5 6 7 8 0 2 4 6 8 10 Flow rate (10-6 kg/s) [10-6] Flow rate (10-6 kg/s) [10-6] Flow rate (10-6 kg/s) Figure 7. Variation of measured pressure difference with mass flow rate under small pressure drops (<3 kPa).

Then, experiments are conducted with large pressure drops. With the resulting permeability K, the inertial coefficient β is determined by the data points close to the maximum flow rate, where the inertial effect is regarded as completely dominating the flow pattern. Table 3 lists the determined coefficients K and β. In Figure 8, the experimental data for the samples of group 2 are presented in the form of friction factor versus Reynolds number, and the axes are plotted on a log-log scale. The Darcy equation in the form of f-Re is represented by a linear dashed line. Clearly, the data points significantly depart from the Darcy regime when Re > 0.1. Therefore, Re = 0.1 can be considered as the boundary separating the viscous flow from the inertial flow. Actually, there exists a transitional region between the viscous and inertial regimes; however, the region is uncertain and thus neglected in order to facilitate analysis. The maximum flow rate used for determination of the permeability for the media 7–11 mm (group 2) corresponds to the Reynolds number of 0.019, 0.024, 0.033, 0.039, and 0.052, respectively, and thus the reliability of the obtained permeability can be confirmed. Moreover, the friction factor converges to the inertial coefficient as the Reynolds number is adequately large. Taking group 2 as an example, the data points for the sample of 11 mm, 10 mm,

Appl. Sci. 2018, 8, x FOR PEER REVIEW 8 of 17

Group 1 Group 2 Group 3

600 600 600 )

a 7 mm

P 8 mm

k 500 500 500

(

e 9 mm c

n 400 400 400 10 mm

e r

e 11 mm

f f

i 300 300 300

e r

u 200 200 200

s e

r 100 100 100 P

0 0 0 0 2 4 6 8 10 12 14 0 3 6 9 12 15 0 3 6 9 12 15 18 Flow rate (10-4 kg/s) Flow rate (10-4 kg/s) Flow rate (10-4 kg/s) Figure 6. Variation of measured pressure difference with mass flow rate.

The viscous effect dominates the flow pattern at a sufficiently small flow velocity, and in this case the Darcy equation can be used to characterize the flow behavior. However, how small flow velocity can be used as the boundary for the transition from viscous flow to inertial flow is Appl.unknown. Sci. 2018 , To8, 1670 the best knowledge of the authors, it is almost impossible to define a universal8 of 17 boundary for the transition of regimes applicable to all kinds of porous media. Although some researchers presented a definite boundary of the Reynolds number for the regime transition, it materialdepends and on the the properties working fluid of the they porous used. material For example, and the Dybbs working and Edwards fluid they [36 used.] reported For example, that the boundaryDybbs and for Edwards viscous flow[36] reported regime and that inertial the boundary flow regime for is viscousRe = 1; flow however, regime it cannot and inertial be used flow in thisregime work. is Re For = this1; however, problem, it firstly, cannot experiments be used in arethis conductedwork. For underthis problem, a slight pressurefirstly, experiments drop less than are 3conducted kPa, where under the air a compressibilityslight pressure candrop be less neglected than 3 andkPa, the where pressure the difference air compressibility should exhibit can be a linearneglected relationship and the pressure with the difference flow rate. should Figure 7exhibit shows a the linear measuring relationship results with in thethe formflow ofrate. pressure Figure difference7 shows the versus measuring the mass results flow in rate. the Theform data of pressure points are difference fitted using versus a least the squaresmass flow straight rate. The line, data and thepoints permeability are fitted can using be calculateda least squares by the straight slope. The line, selected and the flow permeability region proved can appropriatebe calculated since by the 2 Rslope.value The of selected the fitting flow line region exceeds proved 0.98. appropriate since the R2 value of the fitting line exceeds 0.98.

Group 1 Group 2 Group 3 3 3 3

7 mm ) a 8 mm

P 2.5 2.5 2.5 k

( 9 mm

c 2 2 2 10 mm n

e 11 mm

r e

f 1.5 1.5 1.5 Fitting

e 1 1 1

s s

e 0.5 0.5 0.5

r P 0 0 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 1 2 3 4 5 6 7 8 0 2 4 6 8 10 Flow rate (10-6 kg/s) [10-6] Flow rate (10-6 kg/s) [10-6] Flow rate (10-6 kg/s)

FigureFigure 7. Variation of measured pressure difference with mass ﬂowflow rate under small pressure drops (<3(<3 kPa).

Then, experiments are conducted with large pressure drops.drops. With thethe resulting permeability K,, the inertial coefficientcoefficient β is determined by the data points close to the maximum flow flow rate, where the inertial effect is regardedregarded asas completelycompletely dominating the flowflow pattern.pattern. Table 33 lists lists thethe determined determined coefficientscoefficients K and ββ.. In In Figure Figure 8,8, the the experimental experimental data data for for the the samples samples of of gro groupup 2 2are are presented presented in inthe the form form of friction of friction factor factor versus versus Reynolds Reynolds number, number, and andthe axes the axes are plotted are plotted on a onlog a-log log-log scale. scale. The TheDarcy Darcy equation equation in the in theform form of f of-Ref -Re is represented is represented by by a alinear linear dashed dashed line. line. Clearly, Clearly, the the data data points significantlysignificantly depart from the DarcyDarcy regime whenwhen ReRe >> 0.1. Therefore, Re = 0.1 can be considered as the boundary separat separatinging the the viscous viscous flow flow from from the the inertial flow. flow. Actually, there existsexists aa transitionaltransitional region betweenbetween the the viscous viscous and and inertial inertial regimes; regimes however,; however, the region the is region uncertain is and uncertain thus neglected and thus in orderneglected to facilitate in order analysis. to facilitate The maximum analysis. flowThe rate maximum used for flow determination rate used of for the determination permeability for of the mediapermeability 7–11 mm for (groupthe media 2) corresponds 7–11 mm (group to the Reynolds2) corresponds number to the of 0.019, Reynolds 0.024, number 0.033, 0.039, of 0.019, and 0.024, 0.052, respectively,0.033, 0.039, and thus 0.052, the respectively, reliability of and the thus obtained the reliabil permeabilityity of the can obtained be confirmed. permeability Moreover, can the be frictionconfirmed factor. Moreover, converges the to friction the inertial factor coefficient converges as to the the Reynolds inertial coefficient number is adequatelyas the Reyno large.lds number Taking groupis adequately 2 as an large. example, Taking the datagroup points 2 as an for example, the sample the of data 11 mm, points 10 mm,for the 9 mm,sample 8 mm, of 11 and mm, 7 mm10 mm, can be well represented by analytical expressions f = 1/Re + 0.623, f = 1/Re + 0.578, f = 1/Re + 0.613, f = 1/Re + 0.585, and f = 1/Re + 0.522, respectively.

Table 2. Properties of the test porous media in this study.

Estimated Particle Diameter Porous Media Length (mm) Porosity Diameter (µm) (mm) Group 1 40–50 µm 3 7, 8, 9, 10, 11 0.395, 0.392, 0.388, 0.389, 0.397 Group 2 55–65 µm 3 7, 8, 9, 10, 11 0.418, 0.419, 0.418, 0.418, 0.416 Group 3 80–90 µm 3 7, 8, 9, 10, 11 0.420, 0.412, 0.413, 0.420, 0.425 Appl. Sci. 2018, 8, x FOR PEER REVIEW 9 of 17

9 mm, 8 mm, and 7 mm can be well represented by analytical expressions f = 1/Re + 0.623, f = 1/Re + 0.578, f = 1/Re + 0.613, f = 1/Re + 0.585, and f = 1/Re + 0.522, respectively.

Table 2. Properties of the test porous media in this study.

Porous Estimated Particle Length Diameter Porosity Media Diameter (μm) (mm) (mm) Group 1 40–50 μm 3 7, 8, 9, 10, 11 0.395, 0.392, 0.388, 0.389, 0.397 Group 2 55–65 μm 3 7, 8, 9, 10, 11 0.418, 0.419, 0.418, 0.418, 0.416 Group 3 80–90 μm 3 7, 8, 9, 10, 11 0.420, 0.412, 0.413, 0.420, 0.425

Appl. Sci. 2018, 8, 1670 9 of 17

100 50 11 mm 9 mm 10 mm 8 mm

7 mm

r 10

o t

c 5

c i

r 1 F 0.5 f = 1/Re

0.1 0.01 0.05 0.1 0.5 1 5 10

Reynolds number Re (a)

100 50 11 mm 9 mm 10 mm 8 mm

7 mm

r 10

t c

a 5

c i

r 1 F 0.5 f = 1/Re

0.1 0.01 0.05 0.1 0.5 1 5 10 Appl. Sci. 2018, 8, x FOR PEER REVIEW 10 of 17 Reynolds number Re (b) 100 50 11 mm 9 mm 10 mm 8 mm

7 mm

r 10

t c

a 5

c i

r 1 F 0.5 f = 1/Re 0.1 0.01 0.05 0.1 0.5 1 5 10

Reynolds number Re (c)

FigureFigure 8. 8Friction. Friction factor factor f asf as a a function function of of the the Reynolds Reynolds number number Re Re for for the the porous porous media. media (. a()a Group) Group 1; 1; (b()b Group) Group 2; 2; (c )(c Group) Group 3. 3.

4.2. Results of the Discharge Method In the steady-state case, the permeability is first determined in the Darcy regime, and then the inertial coefficient is determined in the Forchheimer regime with the resulting permeability. This treatment is laborious since at least two types of flow sensors and pressure sensors with different ranges and accuracies are needed to satisfy the requirements of measurements across different flow ranges. During the discharge process, the instantaneous flow rate can be indirectly obtained thanks to the isothermal chamber, and the permeability and inertial coefficient can be simultaneously determined by Equation (4). If this method works, it will undoubtedly shortens the measuring time and greatly reduce the air consumption. Figure 9 shows the pressure variation in the chamber during the discharge process for the samples of group 2. The initial data immediately after the opening of the valve are left because those data might include pressure disturbance. The acquired pressure data were first smoothed by the moving average method using 10 points, and then differentiated numerically with eight points to calculate the flow rate. Figure 10 shows the relation of chamber pressure versus the calculated mass flow rate. A subscript i, as the sequence number, is used to refer to any data point (e.g., chamber pressure Pi, mass flow rate Gi) in the dataset, and the number of the total data points is denoted as n. To facilitate iterative computation, Equation (4) is written in the following form:

2 AAKAPP   2  2() 2 2 G   1 a  f(,,) P K  , (10) 22KK 2RL

where Pa is the atmospheric pressure. Then, the permeability and inertial coefficient can be determined by the Gauss‒Newton fitting method using the following procedure:

(1) Initial value K0 = 1 × 10−12 and β0 = 1 are given for the Forchheimer flow rate equation, Equation (10): GfP K(,,)  . 2 (2) The square residual error Spre is calculated by: (Gii f ( P , K , )) . Here, The Gi values are obtained by differentiating the measured pressure in the chamber. f f (3) The elements of the first derivative , ,and the residualsG f(,,) P K  are calculated K  ii for i =1~n.

Appl. Sci. 2018, 8, 1670 10 of 17

4.2. Results of the Discharge Method In the steady-state case, the permeability is first determined in the Darcy regime, and then the inertial coefficient is determined in the Forchheimer regime with the resulting permeability. This treatment is laborious since at least two types of flow sensors and pressure sensors with different ranges and accuracies are needed to satisfy the requirements of measurements across different flow ranges. During the discharge process, the instantaneous flow rate can be indirectly obtained thanks to the isothermal chamber, and the permeability and inertial coefficient can be simultaneously determined by Equation (4). If this method works, it will undoubtedly shortens the measuring time and greatly reduce the air consumption. Figure9 shows the pressure variation in the chamber during the discharge process for the samples of group 2. The initial data immediately after the opening of the valve are left because those data might include pressure disturbance. The acquired pressure data were first smoothed by the moving average method using 10 points, and then differentiated numerically with eight points to calculate the flow rate. Figure 10 shows the relation of chamber pressure versus the calculated mass flow rate. A subscript i, as the sequence number, is used to refer to any data point (e.g., chamber pressure Pi, mass flow rate Gi) in the dataset, and the number of the total data points is denoted as n. To facilitate iterative computation, Equation (4) is written in the following form: v u !2 √ µAϕ u µAϕ KA2 ϕ2(P2 − P2) G = − √ + t √ + 1 a = f (P, K, β), (10) 2 Kβ 2 Kβ 2βRθL where Pa is the atmospheric pressure. Then, the permeability and inertial coefficient can be determined by the Gauss-Newton fitting method using the following procedure:

−12 (1) Initial value K0 = 1 × 10 and β0 = 1 are given for the Forchheimer ﬂow rate equation, Equation (10): G = f (P, K, β). 2 (2) The square residual error Spre is calculated by: ∑ (Gi − f (Pi, K, β)) . Here, The Gi values are obtained by differentiating the measured pressure in the chamber. ∂ f ∂ f (3) The elements of the ﬁrst derivative ∂K , ∂β , and the residuals Gi − f (Pi, K, β) are calculated for i =1~n. (4) Solve the following two linear equations to obtain ∆K and ∆β.

∂ f 2 ∂ f ∂ f ∂ f i ∆K + i i ∆β = i (G − f (P , K, β)), ∑ ∂K ∑ ∂K ∂β ∑ ∂K i i

∂ f ∂ f ∂ f 2 ∂ f i i ∆K + i ∆β = i (G − f (P , K, β)), ∑ ∂K ∂β ∑ ∂β ∑ ∂β i i

(5) Update K and β by K = K0 + ∆K and β = β0 + ∆β. 2 (6) Calculate S = ∑ (Gi − f (Pi, K, β)) , if this value satisﬁes the convergence condition, i.e., |S − Spre| < ε, the determination process is ended. If it does not satisfy the condition, the aforementioned process is repeated with K0 and β0 replaced by K and β. Appl. Sci. 2018, 8, x FOR PEER REVIEW 11 of 17

(4) Solve the following two linear equations to obtain ΔK and Δβ.

2 ffffiiii KGfPK  ii(,,)  , KKK 

2 ffffiiii   KGfPK ii(,,)  , K  

(5) Update K and β by K = K0 + ΔK and β = β0 + Δβ.

2 (6) Calculate S = ((,,))GfPKii  , if this value satisfies the convergence condition, i.e., |S − Spre| < ε, the determination process is ended. If it does not satisfy the condition, the Appl. Sci. 2018aforementioned, 8, 1670 process is repeated with K0 and β0 replaced by K and β. 11 of 17

600 7 mm

) 8 mm

a P

k 500 9 mm

(

e 10 mm r

u 11 mm

s 400

e 300

m a

h 200 C

100 0 10 20 30 40 50 Time (s)

Figure 9. PressureFigure 9. inPressure the chamber in the chamber versus timeversus during time during discharge discharge process process for thefor the ﬁve five porous porous media media of of group 2. group 2. In Figure 10, five pressure subsections over the whole range, denoted as I, II, III, IV, and V, are uniformlyIn divided Figure from10, five 100 pressure kPa to subsections 600 kPa toover calculate the whole the range, permeability. denoted as The I, II, measuringIII, IV, and V, accuracy are changesuniformly at different divided stages from of 1 the00 kPa discharge to 600 processkPa to calculate since the the rate permeability. of pressure The drop measuring is always accuracy changing. changes at different stages of the discharge process since the rate of pressure drop is always The pressure bands are uniformly determined in order to find a satisfactory section to measure the changing. The pressure bands are uniformly determined in order to find a satisfactory section to K permeabilitymeasure accurately. the permeability Figure 11accurately. shows the Figure permeability 11 shows thedetermined permeability by theK determined discharge method by the in differentdischarge pressure method subsections in different for pressure group 2. subsec Objectively,tions for the group permeability 2. Objectively, should the permeability not change should too much for differentnot change samples too of much the same for different group; however, samples for of high-pressurethe same group; subsections however, (IV for and high V),-pressure it exhibits significantsubsections alterations. (IV and That V), isit probablyexhibits significant because thealterations. collected That data is probably do not truly because reflect the thecollected flow ratedata due to the rapiddo not change truly reflect of pressure the flow in ratethe chamber. due to the For rapid the change steady-state of pressure case, the in the measured chamber. permeability For the rangessteady from-state 3.55 tocase, 3.73. the measured A gray belt permeability that indicates ranges the from varying 3.55 to range 3.73. A is gray plotted belt that in Figure indicates 11 thefor an easy comparison.varying range Clearly, is plotted the in permeability Figure 11 for an determined easy comparison. using aClea medium-pressurerly, the permeability dataset determined (II and III) accordsusing with a thatmedium of the-pressure steady-state dataset results. (II and InIII) theory, accords the with permeability that of the steady should-state be determinedresults. In theory, in small the permeability should be determined in small flow regions. However, when the pressure in the flow regions. However, when the pressure in the chamber changes too slowly, it is probable that the chamber changes too slowly, it is probable that the measurement noise might be magnified and the measurementlow signal noise-to-noise might ratio be wouldmagnified result and in significant the low signal-to-noise uncertainties. Considering ratio would these, result the in datasets significant uncertainties.with the Considering pressure ranging these, from the datasets 250 kPa with to 350 the kPa pressure are extracted ranging and from used 250 to kPa determi to 350ne kPa the are extractedpermeability and used K to and determine inertial coefficient the permeability β, whichK areand listed inertial in Table coefficient 4. Actuallyβ, which, the are dataset listed with in Table the 4. Actually,pressureAppl. the Sci. dataset ranging 2018, 8, x with FORfrom PEER the 250 REVIEW pressure kPa to 350 ranging kPa is only from a 250small kPa subset to 350 of kPathe whole is only range. a small Taking subset12 ofthe 17 of11 the mm sample (group 2) as an example, 250 kPa–350 kPa corresponds to the Reynolds number of whole range.1.13–1.87. Taking It isthe thus 11 concluded mm sample that (group the data 2) in as the an form example, of f versus 250 kPa–350 Re taken kPawith corresponds the discharge to the Reynoldsmethod number cannot of 1.13–1.87. completely It reflect is thus the concluded characteristics that compared the data toin the the steady form-state of fmethod. versus Re taken with the discharge method cannot completely reflect the characteristics compared to the steady-state method.

700

) 600

a P

k V

(

e 500

r u

s IV

s e

r 400 p

7 mm r III e 8 mm b 300

m 9 mm a

h II 10 mm C 200 11 mm I 100 0 3 6 9 12 15 18 -4 Flow rate (10 kg/s) Figure 10.FigPressureure 10. Pressure in the in chamber the chamber versus versus mass mass ﬂow flow rate rate obtainedobtained by by the the discharge discharge method method for the for the ﬁve porousfive media porous ofmedia group of group 2. 2.

4.5

4 3.73 K

y t i

l 3.5 3.55 i b a e m

r 3 7 mm e

P 8 mm 9 mm 2.5 10 mm 11 mm 2 0 1I I2I I3II I4V V5 6 Pressure subsection Figure 11. Permeability obtained by the discharge method in different pressure subsections for the five porous media of group 2.

Table 3. Permeability and inertia coefficients of the porous media by the steady-state method.

Porous Group 1 Group 2 Group 3 Medium K (10−12 m2) β K (10−12 m2) β K (10−12 m2) β 7 mm 2.15 0.512 3.73 0.522 5.02 0.608 8 mm 2.02 0.515 3.64 0.585 4.99 0.617 9 mm 2.02 0.486 3.65 0.613 4.86 0.588 10 mm 2.048 0.467 3.68 0.578 5.05 0.598 11 mm 2.095 0.491 3.55 0.623 4.93 0.615

Table 4. Results of fitting coefficients K and β by the discharge method.

Porous Group 1 Group 2 Group 3

Appl. Sci. 2018, 8, x FOR PEER REVIEW 12 of 17

1.13–1.87. It is thus concluded that the data in the form of f versus Re taken with the discharge method cannot completely reflect the characteristics compared to the steady-state method.

700

) 600

a P

k V

(

e 500

r u

s IV

s e

r 400 p

7 mm r III e 8 mm b 300

m 9 mm a

h II 10 mm C 200 11 mm I 100 0 3 6 9 12 15 18 -4 Flow rate (10 kg/s) Figure 10. Pressure in the chamber versus mass flow rate obtained by the discharge method for the

Appl. Sci.five2018 porous, 8, 1670 media of group 2. 12 of 17

4.5

3.73

t i

l 3.5 3.55

e m

r 3 7 mm e

P 8 mm 9 mm 2.5 10 mm 11 mm 2 0 1I I2I I3II I4V V5 6 Pressure subsection FigureFigure 11. 11Permeability. Permeability obtained obtained by by the the discharge discharge method method in different in different pressure pressure subsections subsections for the for ﬁve the porousfive porous media media of group of group 2. 2.

Table 3. Permeability and inertia coefficients of the porous media by the steady-state method. Table 3. Permeability and inertia coefficients of the porous media by the steady-state method. Group 1 Group 2 Group 3 PorousPorous Medium Group 1 Group 2 Group 3 −12 2 −12 2 −12 2 Medium K (10K (10−12 m2) m ) β β K K(10(10−12 m2)m ) β β KK (10(10−12 mm2) ) ββ 77 mm 2.15 2.15 0.512 0.512 3.73 3.73 0.522 0.522 5.02 5.02 0.608 0.608 88 mm 2.02 2.02 0.515 0.515 3.64 3.64 0.585 0.585 4.99 4.99 0.617 0.617 9 mm 2.02 0.486 3.65 0.613 4.86 0.588 109 mm mm 2.02 2.048 0.486 0.467 3.6 3.685 0.613 0.578 4.86 5.05 0.588 0.598 1011 mm 2.048 2.095 0.467 0.491 3.68 3.55 0.578 0.623 5.05 4.93 0.598 0.615 11 mm 2.095 0.491 3.55 0.623 4.93 0.615 Table 4. Results of fitting coefficients K and β by the discharge method. Table 4. Results of fitting coefficients K and β by the discharge method. Group 1 Group 2 Group 3 Porous MediumPorous Group 1 Group 2 Group 3 K (10−12 m2) β K (10−12 m2) β K (10−12 m2) β

7 mm 1.96 0.411 3.71 0.562 5.03 0.468 8 mm 1.97 0.467 3.68 0.641 4.91 0.537 9 mm 2.12 0.509 3.63 0.658 4.98 0.582 10 mm 1.98 0.467 3.66 0.621 4.95 0.562 11 mm 2.08 0.574 3.54 0.739 5.01 0.621

4.3. Comparison of the Steady-State Method and the Discharge Method Figure 12 compares the steady-state results and the calculated results with the determined coefficients by the discharge method. The calculated curves agree with the steady-state data points on the whole, indicating that the discharge method works well for obtaining the property coefficients. To evaluate the discharge method, the accuracies of the method are calculated and compared with that of the steady-state method, which was proven viable in a previous work [28]. The error E is expressed as v u 2 u ∑ G(exp)i − G(cal)i = u × E t 2 100%, (11) ∑ G(exp)i where G(exp)i is for the steady-state flow rate, and G(cal)i is for the calculated flow rate. Appl. Sci. 2018, 8, x FOR PEER REVIEW 13 of 17

Medium K (10−12 m2) β K (10−12 m2) β K (10−12 m2) β 7 mm 1.96 0.411 3.71 0.562 5.03 0.468 8 mm 1.97 0.467 3.68 0.641 4.91 0.537 9 mm 2.12 0.509 3.63 0.658 4.98 0.582 10 mm 1.98 0.467 3.66 0.621 4.95 0.562 11 mm 2.08 0.574 3.54 0.739 5.01 0.621

4.3. Comparison of the Steady-State Method and the Discharge Method Figure 12 compares the steady-state results and the calculated results with the determined coefficients by the discharge method. The calculated curves agree with the steady-state data points on the whole, indicating that the discharge method works well for obtaining the property coefficients. To evaluate the discharge method, the accuracies of the method are calculated and compared with that of the steady-state method, which was proven viable in a previous work [28]. The error E is expressed as

2 (GG(exp)ii− (cal) )  (11) E =2 100% , G(exp)i

Appl.where Sci. 2018G(exp), 8i ,is 1670 for the steady-state flow rate, and G(cal)i is for the calculated flow rate. 13 of 17

600 )

a 500

(

c 400

e f

f 300

e 7 mm: Exp. Cal. r

u 200 8 mm: Exp. Cal.

s s

e 9 mm: Exp. Cal. r

P 100 10 mm: Exp. Cal. 11 mm: Exp. Cal. 0 0 3 6 9 12 15 18 Flow rate (10-4 kg/s)

Figure 12. Comparison of the theoretical and experimental flow rate characteristics for the five porous mediaFigure of 12 group. Comparison 2. of the theoretical and experimental flow rate characteristics for the five porous media of group 2. The results in Table5 show that the calculated results can achieve an accuracy within 3%. However,Table referring 5. Deviations to one from of the the experimental previous works data of by the the authors, steady-state the method calculated and theresults discharge with the propertymethod coefficients. determined by the steady-state method can achieve an accuracy within 1% in the Forchheimer flowE (Group region. This1) is probably becauseE (Group the flow 2) rate in the currentE work(Group is obtained3) byPorous differentiating the measured pressure with respect to time, and this treatment would magnify the Steady-State Discharge Steady-State Discharge Steady-State Discharge noiseMedium and deteriorate the measuring accuracy. Taking one of the previous works by Kawashima et al. Method Method Method Method Method Method as another example, the sonic conductance C determined by the discharge method following the ISO 7 mm 0.97% 2.16% 0.88% 2.39% 1.03% 2.55% 6358 equation can achieve an accuracy within 2% while the permeability in the current work is 5–10%. 8 mm 0.59% 0.80% 1.09% 1.64% 1.21% 1.78% Despite a larger deviation in the coefficients, the accuracy of the theoretical flow rate over the entire 9 mm 1.94% 1.18% 0.51% 2.34% 0.77% 1.28% range is sufficient for most applications. 10 mm 1.60% 2.26% 0.93% 1.91% 0.62% 1.25% 11 mm 2.08% 2.74% 1.02% 2.31% 0.58% 0.76% Table 5. Deviations from the experimental data by the steady-state method and the discharge method.

E (Group 1) E (Group 2) E (Group 3) Porous Medium Steady-State Discharge Steady-State Discharge Steady-State Discharge Method Method Method Method Method Method 7 mm 0.97% 2.16% 0.88% 2.39% 1.03% 2.55% 8 mm 0.59% 0.80% 1.09% 1.64% 1.21% 1.78% 9 mm 1.94% 1.18% 0.51% 2.34% 0.77% 1.28% 10 mm 1.60% 2.26% 0.93% 1.91% 0.62% 1.25% 11 mm 2.08% 2.74% 1.02% 2.31% 0.58% 0.76%

The key point of the discharge method is that the isothermal chamber must remain isothermal during the measuring process. If the selected chamber has a very small volume, the isothermal condition cannot be maintained due to the rapidly changing pressure. Previous research [32] implies that the temperature variation can be limited within 3K when the pressure change is less than 100 kPa/s. That is, for group 2, the lower limit volumes for 7–11 mm samples are 0.28 L, 0.34 L, 0.43 L, 0.48 L and 0.62 L, respectively. Inversely, if the chamber volume is too large, the measured pressure change is almost equal to the sensor resolution, and this also deteriorates the measuring accuracy. In the experiments, it is considered that at least 20 kPa/s is needed. Therefore, the upper limit volumes for 7–11 mm samples (group 2) are 0.85 L, 1.04 L, 1.27 L, 1.46 L, and 1.88 L, respectively. In Figure 12, careful observation shows that the 11-mm sample exhibits more apparent deviation from the steady-state data points than others. This is because the selected volume of the isothermal chamber is very close Appl. Sci. 2018, 8, 1670 14 of 17 to the lower limit, and probably the isothermal condition cannot be strictly maintained during the discharge process. The discharge method greatly reduces the test time and air consumption. In the steady-state case, flow measurement is maintained for a while to reduce the measuring error. Taking the sample of 11 mm (group 2) as an example, the maximum flow is maintained for at least 30 s to ascertain the inertial coefficient, and thus the air consumption is about 39 L (ANR). Noted that ANR (Atmosphere Normale de Reference) after the unit depicts air volume at conditions 101.325 kPa absolute, 20 ◦C, and 65% relative humidity. For the discharge method, the testing procedure is comparatively simple because it can be finished in a few seconds, and the consumed air is the amount of compressible air initially stored in the chamber, which is about 4 L (ANR).

5. Conclusions This study proposes a discharge method to determine the flow characteristics of sintered metal porous media using an isothermal chamber. The pressure in the chamber is measured and then differentiated to calculate the discharged flow rate. Fifteen samples, categorized into three groups, are employed as the test element to perform the experiments. A modified Forchheimer equation is constructed to represent the relationship between flow rate and pressure difference in terms of the permeability and inertial coefficients. First, the coefficients are determined per the steady-state method by determining first the permeability in the Darcy region and then the inertial coefficient in the Forchheimer region. The Darcy and Forchheimer regions were defined with a Reynolds number of 0.1 as the boundary. Then, with the discharge method, the coefficients are simultaneously determined in five uniformly distributed pressure subsections. The results of permeability in comparison with the steady-state results indicate that using a medium-pressure dataset (250 kPa–350 kPa) improves the accuracy. Although the permeability and inertial coefficients determined by the discharge method somewhat deviate from those of the steady-state method, the theoretical calculation can represent the flow rate characteristics within 3% accuracy compared with the experimental results indicating that the traditional method can be replaced by the discharge method, which greatly shortens the measuring time, with only 10% air consumption as in the steady-state method. However, an appropriate volume for the isothermal chamber is necessary to ensure the measuring accuracy because the discharge method requires a reasonable discharging speed. The presented results indicate that the discharge method has potential for use in industry. As an example, in air-bearing systems, a large number of porous pads are employed and their flow properties are required to be as consistent as possible. A lot of testing and analysis are needed to help evaluate the manufacturing process. In this case, the discharge method is very useful since it facilitates the procedures and greatly reduces the air consumption under the premise of ensuring the accuracy. The limitations of the present study lie in the following aspects: (1) The test samples are mainly tight porous media, which have a permeability on the order of 10−12 m2 and are generally applied in air-bearing systems. (2) An isothermal condition for the chamber is necessary during the discharge process. Otherwise, the calculated flow rate would exhibit large error. This means that a reasonable discharge speed is required. (3) The theoretical model is somewhat realistic since the inner structure of the actual porous media is not uniform. Therefore, the determined coefficients might not reflect the true properties of the media. (4) Only the permeability and inertial coefficients are discussed. The influences of other parameters such as porosity ϕ, cross-sectional area A, and length of the medium L are not included.

Author Contributions: Conceptualization, W.Z.; Data curation, X.J.; Investigation, J.F.; Methodology, X.J. and C.L.; Resources, C.L.; Validation, J.F.; Writing—original draft, W.Z.; Writing—review & editing, F.L. Funding: This study was supported by the National Natural Science Foundation of China (Grant No. 51675247). Appl. Sci. 2018, 8, 1670 15 of 17

Conﬂicts of Interest: The authors declare no conﬂict of interest.

Nomenclature

A cross-sectional area (m2) G mass ﬂow rate (kg/s)

G(cal)i calculated flow rate (kg/s) G(exp)i steady-state flow rate (kg/s) K permeability (m2) L length of the test medium (m) m mass of the medium (kg) P pressure (Pa) P1 upstream pressure (Pa) P2 downstream pressure (Pa) Pa atmospheric pressure (Pa) Pc pressure in the chamber (Pa) R gas constant t time (s) T temperature (K) V total volume (m3) 3 Vc volume of the chamber (m ) β inertial coefficient γ metal density (8.03 × 103 kg/m3) Θ room temperature (K) µ air viscosity (Pa·s) v velocity (m/s) ρ density (kg/m3) ϕ porosity

Appendix A ISO 6358 is an international standard that specifies a method for testing pneumatic components to represent their flow rate characteristics under steady-state conditions. The ISO 6358 equation, which is deduced from isentropic flow through a perfect nozzle, is provided in the ISO 6358 standard. The ISO 6358 equation depicts flow rate characteristics with two independent parameters: sonic conductance C and critical pressure ratio b. Two flow states, choked flow and subsonic flow, are defined according to the relationship between the pressure ratio P2/P1 and the critical pressure ratio b. When P2/P1 ≤ b, choked flow occurs; when P2/P1>b, subsonic flow occurs. The ISO 6358 equation is written in the following form:  q 293 P2  Cρ0P1 T P ≤ b  v 1 u P !2 G = q 2 −b , 293 u P1 P2  Cρ0P1 · t1 − > b  T 1−b P1 where C is the sonic conductance, b is the critical pressure ratio, ρ0 is the air density under normal condition, P1 is the upstream pressure, P2 is the downstream pressure, and T is the temperature.

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