A three-dimension approach to the porous surface of screens

Full Te ms & Condi ions o access and use can be ound a h p://www. and online.com/ac ion/jou nalIn o ma ion?jou nalCode= j i20 The Jou nal o The Tex ile Ins i u e ISSN: 0040-5000 (P in ) 1754-2340 (Online) Jou nal homepage: h p://www. and online.com/loi/ j i20 A h ee-dimensional app oach o he po ous su ace o sc eens A. J. Ál a ez, R. M. Oli a, A. Jiménez-Va gas & M. Villegas-Vallecillos To ci e his a icle: A. J. Ál a ez, R. M. Oli a, A. Jiménez-Va gas & M. Villegas-Vallecillos (2018): A h ee-dimensional app oach o he po ous su ace o sc eens, The Jou nal o The Tex ile Ins i u e, DOI: 10.1080/00405000.2018.1500740 To link o his a icle: h ps://doi.o g/10.1080/00405000.2018.1500740 Published online: 04 Dec 2018. Submi you a icle o his jou nal View C ossma k da a A h ee-dimensional app oach o he po ous su ace o sc eens A. J.  Al a ez a , R. M. Oli a a , A. Jim enez-Va gas b and M. Villegas-Vallecillos c a Depa men o Enginee ing, Uni e si y o Alme  ıa, Alme  ıa, Spain; b Depa men o Ma hema ics, Uni e si y o Alme  ıa, Alme  ıa, Spain; c Depa men o Ma hema ics, Uni e si y o C adiz, Pue o Real, C adiz, Spain ABSTRACT Cu en ly, he po ous su ace o he sc eens is measu ed on digi al images aken by mic oscope ep e- sen ing he o hogonal p ojec ion o he ex iles. I is known ha his way o measu ing he po ous su ace unde es ima e la gely he eal hole su ace. To imp o e his aspec , in his wo k he hole su - ace is iden i ied as a speci ic egion o he hype bolic pa aboloid and a me hodology is de eloped o add ess i s calcula ion. Indeed, he esul s show ha he po ous su ace measu ed on o hogonal p o- jec ion is signi ican ly less han he eal hole su ace. Howe e , he applica ion o his me hodology is e y complex and o his eason an app oxima e al e na i e me hod ha conside ably simpli ies he di icul y o he p oblem is p oposed. The esul s ob ained by one and ano he me hod ha e small dis- c epancies so ha he app oxima e me hod is also a good op ion o he calcula ion o he po ous su ace o hese ex iles. ARTICLE HISTORY Recei ed 12 Sep embe 2017 Re ised 9 July 2018 Accep ed 10 July 2018 Published online 12 Sep em- be 2018 KEYWORDS Ag o ex iles; c op p o ec ion; sc eens; h ee- dimensional po ous su ace 1. In oduc ion Sc eens ha e e y di e en applica ions such as c op p o ec- ion o i s use in duc s o low expe imen s. In he i s case, o example, insec -p oo sc eens a e a physical me hod o c op p o ec ion whose use has become wide- sp ead in many pa s o he wo ld o e ecen decades. They a e ins alled a he side and oo en s o g eenhouses wi h a iew o impeding o educing he access o insec s o he c op. The bene i s o p o ec ion sc eens a e su icien ly p o en. Howe e , he e a e s ill many knowledge gaps o ge an op imal design. Thei design is a e y complex ma e and besides i s op imal can be add essed om di e en iew- poin s ha e y o en a e opposing solu ions ( Al a ez, 2010). I iny holes a e equi ed o a oid he en y o insec s inside he g eenhouse, he en ila ion a e is educed as he po ous su ace dec eases (Bailey e al., 2003; Ba zanas, Boula d, & Ki as, 2002;Die ickx,1998; Linke , Ta nopolsky, & Segine , 2002;Mu ~ noz, Mon e o, An  on, & Giu ida, 1999;Soni, Salokhe, & Tan au, 2005) and p oduces imbalances in he g eenhouse mic oclima e wi h nega i e consequences o c op de elopmen (Ki as, Boula d, Ba zanas, Ka soulas, & Me mie , 2002; Tei el, 2010). Fo his eason, he ae o- dynamic s udy o hese ex iles is also essen ial. The s uc u e o he wea e o sc eens is de e mined by wo se s o h eads (we and wa p) which in e wea e pe - pendicula ly. The sepa a ion o he h eads in each di ec ion means ha he geome y o each hole is gene ally ec angu- la , since he h eads making up he wa p a e usually close oge he han hose o he we . The numbe o h eads pe uni leng h es ablishes he densi y o h eads (numbe o h eads pe uni leng h) o he sc een in each di ec ion. The diame e ( hickness) o he h eads is ano he a iable ha de ines he geome y o he sc een. Image analysis can be de ined as he ex ac ion o mean- ing ul in o ma ion om images by means o digi al p ocess- ing echniques (Solomon & B eckon, 2011). Image p ocessing has been p o ed o be an e icien me hod o analyzing ab ic s uc u es (Jeong & Jang, 2005). These ech- niques a e e y impo an o many applica ions wi hin he ex ile indus y and hei use is widesp ead and has been used ex ensi ely o ob ain ex ile da a ( Al a ez, Oli a, & Vale a, 2012; Ca damone, Dame , Phillips, & Ma me , 2002; Gan, Bicke on, & Ba ley, 2012; Kang, Choi, Kim, & Oh, 2001; Shin, Cho, Seo, & Kim, 2008). Va ious echniques ha e been used including op ical scanning, op ical mic os- copy, con ocal mic oscopy, op ical cohe ence omog aphy, and X- ay mic o omog aphy (She bu n, 2007). Accu a e measu emen s o he geome y o wo en ex iles a e essen ial o quali y con ol o he wea ing p ocess (Lim & Kim, 2011), in ex ile modeling echniques o he p edic ion o ma e ial p ope ies (Lomo e al., 2001; Zeng, B own, End uwei , Ma ee , & Long, 2014) o o cha ac e - iza ion o geome ic pa ame e s. In o ma ion ega ding o he hickness o he h eads, dimensions o he holes, num- be o h eads pe uni leng h, shape pa ame e s as well as he quan i a i e and objec i e measu emen o complex p ope ies can be ob ained (Kang, Kim, & Oh, 1999; Lim & Kim, 2011; She bu n, 2007). Analysis o he geome y o p o ec ion sc eens is impo - an o cha ac e ize hei e ec i eness o p e en insec en y inside he g eenhouse ( Al a ez, Vale a, & Molina-Aiz, 2006). The capaci y o sc eens o keep insec s ou is CONTACT A. J.  Al a ez [email p o ec ed] Depa men o Enginee ing, Uni e si y o Alme  ıa, Alme  ıa, Spain ß2018 The Tex ile Ins i u e THE JOURNAL OF THE TEXTILE INSTITUTE h ps://doi.o g/10.1080/00405000.2018.1500740 de e mined compa ing he dimensions o he holes wi h he usual size o he mos damaging pes species. The e o e, he design o p o ec ion sc eens is ca ied ou acco ding o he body size o he smalles insec pes whose p esence inside he g eenhouse is in ended o a oid (Bailey, 2003). The e icacy o a sc een should no be p edic ed by compa - ing only he mesh size and he insec body size (Be hke & Paine, 1991) because o he a iables such as insec abili y, empe a u e o ai eloci y a e in ol ed (Oli a &  Al a ez, 2017). Howe e , he ela ionship be ween insec body size and eal hole su ace is c i ical. The geome y o sc eens is also impo an o de e mine he esis ance o e ed by he ex ile o he ai low. Sc eens a e po ous media since hey ha e a solid s uc u e combined wi h a oid space. An impo an p ope y o hese ma e ials is hei po osi y ha can be de ined as he a io be ween he su ace a ea o holes A h and he o al su ace a ea A . Bu hi he o he su ace a ea o holes A h is always unde es ima ed. Due o he small size o he holes, geome ic cha ac e is- ics o sc eens a e ob ained on digi al images aken by mic oscope. These images a e o hogonal p ojec ions o he ab ics and he e o e he measu emen s aken do no e lec he eali y since he spa ial a angemen o he h eads does no con o m o a plane. The opening le be ween he h eads is la ge han ha ob ained in he measu emen s on o hogonal p ojec ions. This has a di ec impac bo h in he de e mina ion o he hole size as in he calcula ion o he open su ace a ea. Conside ing he h ee-dimensional (3D) eali y, he insec s ha e mo e space o pass h ough he holes han he one ini ially supposed. Likewise, an ai s eam has mo e sec ion o low. Figu e 1 shows bo h he op and pe spec i e iew o a sc een. Almos all he p o ec ion sc eens p esen a ec angu- la hole geome y ha is based on wha we denomina e like p ison ba s e ec ( Al a ez & Oli a, 2017). Conside ing he p e ious, he wa p ep esen s a cage whose ba s a e no in he same plane as he Figu e 1 shows. This de e mines ha he limi ing dimension o insec exclusion is no he sepa - a ion be ween wa p h eads measu ed on o hogonal p ojec- ion images bu a g ea e dis ance. The objec i e pu sued by he manu ac u e s wi h he design based on he p ison ba s e ec is o es ic he en y o insec by limi ing he dis- ance be ween wa p h eads and, in u n, o a oid ha he po osi y o he ex iles is oo low inc easing he dis ance be ween we h eads. An insec -p oo sc een will no ul il i s pu pose i he only c i e ion o design conside ed is o es ablish he dis ance be ween wa p h eads (in o hogonal p ojec ion) lowe han he insec ho ax size. An exhaus i e analysis o he eal dis ance ha he h eads lea e be ween hem will allow o p edic wi h mo e ce ain y he possibili y o an insec c osses h ough a hole. In he nex phase o s udy, i will be necessa y o ake in o accoun ha insec s a e li ing s uc u es and he e o e hei abili ies will also ha e o be conside ed and no only hei size. In he o he hand, he calcula ion o he hole su ace a ea conside ing he 3D s uc u e o he ex ile will allow o imp o e he models explaining he ae odynamic esis ance ha sc eens o e o ai low. The e a e ha dly any e e ences in he li e a u e dealing wi h he issue o he 3D su ace a ea o he sc een holes. Pinke and He be (1967), in hei s udy o he p essu e d op ha squa e hole sc eens cause on he ai low, p oposed wo al e na i es o he po osi y ob ained as a esul o he ela ionship be ween he hole su ace a ea and he o al su - ace a ea measu ed on o hogonal p ojec ions. Thei me hod o measu e he 3D su ace o he holes wi h squa e geom- e y assumes ha bo h we and wa p h eads unde go he same de o ma ion and o his eason he Pinke and He be ’s s a ing assump ions a e w ong. The dis ance be ween h eads o he sc eens wi h squa e hole geome y is he same bo h in wa p and we di ec ions bu he cu a- u e o he wa p and we h eads is no simila and he e- o e bo h se s o h eads can be dis inguished since he wa p h eads “emb ace” he we ones. The di e ences be ween he measu ed dimensions on o hogonal p ojec ions and he spa ial measu emen s o sc eens wi h squa e and ec angu- la holes a e o he same na u e because o bo h ypes o sc eens he de o ma ions o he h eads a e simila . This wo k p esen s a heo e ical s udy o he geome y o sc eens om a 3D poin o iew consis ing o a me hod o calcula e he 3D su ace ha he spa ial c ossing be ween wo consecu i e wa p h eads de ines in he con ex o a hole (Figu e 2). Many applica ions may ha e his new de el- opmen : in he ield o physical ba ie s ( ex iles used o comba insec s ha m ul o c ops) his app oach can imp o e he p edic ion o e icacy o he ex iles agains insec s and he calcula ion o his spa ial su ace can also imp o e he me hods ha desc ibe he ai low h ough hese po ous media. 2. Theo y The 3D ep esen a ion o he po ous su ace le be ween adjacen h eads is shown in Figu e 2. The ab ic s uc u e Figu e 1. Rep esen a ion o a sc een in o hogonal p ojec ion (le ) and in pe spec i e ( igh ). 2 A. J.  ALVAREZ ET AL. de e mines ha he wa p h eads “emb ace” he we ones and he e o e he i s ones ha e g ea e de o ma ions. In he ep esen a ion o Figu e 2 has been conside ed ha he c oss-sec ion o he h eads is ci cula and ha he axis o he cylind ical body o he wa p h eads emains app oxi- ma ely s aigh along he dis ance be ween he c ossings wi h he we h eads (in gene al, he obse a ion demon- s a es ha bo h assump ions a e e y close o he eali y). The spa ial su ace o a hole is de ined by he wo closes gene a ices o wo adjacen wa p h eads (Figu e 2). These wo segmen s de ine a doubly uled su ace since o each o i s poin s pass wo s aigh lines comple ely con ained in he su ace. The su ace can be included in a pa allelepiped whose leng h L py , wid h L px and heigh Dz–D hy a e known and desc ibed in Figu e 2 (whe e Dzis he hickness o he sc een and D hy he hickness o he wa p h eads). This su - ace is a egion o he hype bolic pa aboloid and o calcula e i s su ace a ea is p oposed he ollowing p ocedu e based on he undamen als o he analy ic geome y and an app oxima e me hod o simpli y he calcula ions. 2.1. Su ace a ea o he de ined egion o he hype bolic pa aboloid Le a,c,d2R þ be h ee pa ame e s wi h ad c. Le be he s aigh line passing o he poin s (Figu e 3): p¼c;cþda;d2þ2cd a  (1) q¼cþda;c;d22cd a  (2) and s he s aigh line con aining o he poin s (Figu e 3): p0¼cda;c;d22cd a  (3) q0¼c;cda;d2þ2cd a  (4) The ela ionship be ween a,c, and dwi h he leng h L py , he wid h L px and he heigh DzD hy o he pa allelepiped con aining he su ace a ea unde s udy is ob ained sol ing a sys em wi h h ee equa ions and h ee unknowns. The esul is he ollowing (Figu es 2 and 3): a¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi LpxLpy 䉭zDhy s(5) c¼Lpx þLpy 2ffiffiffi 2 p(6) d¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Lpx 䉭zDhy  2Lpy s(7) Le T 1 and T 2 be wo iangles (Figu e 4): Figu e 2. 3D ep esen a ion o a hole (le ) and su ace a ea ha he h eads lea e be ween hem ( igh ); he ec angle in bold ep esen s he o hogonal p ojec- ion o he hole. Figu e 3. De ini ion o he segmen s and s. THE JOURNAL OF THE TEXTILE INSTITUTE 3 T1¼nx;y ðÞ 2R2:x2c;cþda ½ ;xda yx þ2cdao T2¼x;y ðÞ 2R2:x2cda;c ½ ;x2cþda yxþda no and R 0 he homboid (Figu e 4): R0¼x;y ðÞ 2R2:x2cþda;cda ½ ;xda yxþda no Joining T 1 ,T 2 , and R 0 , a ec angle Ris ob ained (Figu e 4). In ha ec angle can be de ined he unc ion :R!R gi en by: x;y ðÞ ¼1 a2x2y2  ;8x;y ðÞ 2R(8) The g aph o he unc ion is a egion Ro he hype bolic pa aboloid con aining he segmen endpoin s pand qand he segmen endpoin s p 0 and q 0 (Figu e 3) and can be de ined by he su aces S 1 ,S 2 ,S 3, and S 4 (Figu e 5). Using an app op ia e change o a iable, his desc ip ion o Rallows o calcula e he su ace a ea o he g aph o he unc ion as ollows. The su ace a ea A 3D o he g aph is gi en by he in eg al: A3D ¼ðð Rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o oxx;y ðÞ  2 þo oyx;y ðÞ  2 þ1 sdx;y ðÞ ¼ ¼ðð S1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 a4x2þy2  þ1 dx;y ðÞ þðð S2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 a4x2þy2  þ1 dx;y ðÞ þðð S3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 a4x2þy2  þ1 dx;y ðÞ þðð S4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 a4x2þy2  þ1 dx;y ðÞ (9) I can be shown ha by sol ing he p e ious ou in eg als he ollowing exp ession is ob ained: A3D ¼a4 3F1 ðÞ Fcþad c  þGad 2cad  G0 ðÞ p 2  (10) wi h F1 ðÞ¼p 2d3a2þ2d2 ðÞ a3ffiffiffi 2 pln dffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2a2þ4d2 p  (11) F ðÞ¼d2 1 ðÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 þ1 ðÞ 2þ4d2 2þ1 ðÞ qa3 þ1 ðÞ 2  an1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 þ1 ðÞ 2þ4d2 2þ1 ðÞ qa 1 ðÞ 0 @1 A d3a2þ2d2 ðÞ a3ffiffiffi 2 p ln 2d2 1 ðÞ þdffiffiffi 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 þ1 ðÞ 2þ4d2 q 2þ1 ðÞ þ1 ! (12) G ðÞ¼2cad ðÞ 2 a6 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a4þ22cad ðÞ 2 2þ1 ðÞ q þ an 1a2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a4þ22cad ðÞ 2 2þ1 ðÞ q 0 @1 A þ2cad ðÞ 3a4þ22cad ðÞ 2  a6ffiffiffi 2 p ln 2 2cad ðÞ þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2a4þ42cad ðÞ 2 2þ1 ðÞ q  (13) 2.2. An al e na i e and app oxima e me hod The hype bolic pa aboloid is a wa ped su ace, ha is, a non-de elopable doubly uled su ace since wo consecu i e posi ions o he gene a ix a e no coplana . In o he wo ds, his kind o su aces canno be ex ended on a plane as in he case o he su ace o a cylinde o a cone. The app oxi- ma e me hod p oposed is based on “ la ening” he de ined Figu e 4. The p ojec ion on he o hogonal plane. Figu e 5. De ini ion o he egion R. 4 A. J.  ALVAREZ ET AL. su ace o he hype bolic pa aboloid o measu e i s su ace a ea mo e easily han wi h he p e ious p ocedu e. The esul s ob ained will always unde es ima e he eal solu ion because his is a non-de elopable su ace. This app oxima e me hod is simple al hough in ol es a small loss o accu acy ha we will assess a li le la e . To “ la en” he s udied egion o hype bolic pa aboloid he gene a ices pa allel o x-axis a e o a ed a ound an axis pa allel o y-axis as shown in Figu e 6. In his way, i is ob ained a la su ace de ined by a hype bola whose su ace a ea A 3D can be calcula ed by he ollowing exp ession: A3D ¼A1þ4A2(14) whe e A 1 is he su ace a ea o he hole o hogonal p ojec- ion, ha is, ob ained by mul iplying L px and L py ; he o he summand 4A 2 is he su ace a ea enclosed be ween he cu es o he hype bola and he ec angle ep esen ing he hole o hogonal p ojec ion (Figu e 6). One o he b anches o he hype bola can be isola ed o calcula e he su ace a ea A 2 . Fo ha , we can u n he coo dina e sys em so ha he x-axis coincides wi h he di - ec ion o he ec angle leng h ep esen ing he o hogonal p ojec ion o he hole (Figu es 6 and 7). The isola ed b anch can be desc ibed as a second-deg ee polynomial y¼kx 2 þmxþn. The quad a ic polynomial can be i ed measu ing some gene a ices o he hype bolic pa aboloid in space. The chosen gene a ices (Figu e 6) a e de ined by he poin s p 1 ,p 2 ,q 1 ,q 2 , 1, and 2 as shown below: p1¼0;0;DzDhy 2  ;p2¼Lpx;0; DzDhy 2  (15) q1¼0; Lpy 4;DzDhy 4  ;q2¼Lpx; Lpy 4; DzDhy 2  (16) 1¼0; Lpy 2;0  ; 2¼Lpx; Lpy 2;0  (17) F om he p e ious poin s, he gene a ices leng hs d 1 ,d 2, and d 3 can be calcula ed as he dis ances be ween p 1 and p 2 , q 1 and q 2 , and 1 and 2 , espec i ely (Figu e 6): d1¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L2 px þDzDhy  2 q(18) d2¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L2 px þDzDhy 2  2 s(19) d3¼Lpx (20) A e ob aining he p e ious leng hs, we need o calcula e he o dina es b y1 ,b y2, and b y3 o he equi ed poin s o do he polynomic adjus men (Figu es 6 and 7): by1¼d1Lpx 2(21) by2¼d2Lpx 2(22) by3¼d3Lpx 2¼LpxLpx 2¼0 (23) Finally, he coo dina es o he poin s a e he ollowing (Figu e 7): 0;by1  ¼0;ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L2 px þDzDhy  2 qLpx 2 ! (24) Lpy 4;by2  ¼Lpy 4;ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L2 px þDzDhy 2  2 Lpx 2 0 @1 A(25) Lpy 2;by3  ¼Lpy 2;0  (26) The su ace a ea A 2 is he esul o sol ing he de ini e in e- g al o he polynomic adjus men be ween he limi s 0; Lpy 2 hi : A2¼ð Lpy 2 0 kx2þmx þn ðÞ dx¼kx3 3þmx2 2þnx  Lpy 2 0 ¼kL3 py 24 þmL2 py 8þnLpy 2 (27) Finally, he su ace a ea A 3D o he s udied egion o he hype bolic pa aboloid measu ed acco ding o he app oxi- ma e me hod is: A3D ¼A1þ4A2¼LpxLpy þkL3 py 6þmL2 py 2þ2nLpy (28) Figu e 6. Scheme o “ la en” he hype bolic pa aboloid. THE JOURNAL OF THE TEXTILE INSTITUTE 5 3. Resul s and discussion 3.1. Cha ac e iza ion o sc eens To apply he me hodology exposed in he p e ious sec ion, i e expe imen al insec -p oo sc eens ha e been used. The geome ic cha ac e iza ion o hese ag o ex iles has been pe o med ollowing he me hodology p oposed by  Al a ez e al. (2012). The images ha e been aken by an op ical mic oscope (B3, Mo ic) wi h a buil -in digi al came a (Mo icam 2, Mo ic). The hickness o he sc eens has been measu ed by a mic ome e (Mic omas e , Tesa). The da a ob ained a e shown in Table 1. The i e sc eens used ha e abou he same numbe o wa p h eads pe uni leng h and hei densi ies o we a y be ween 14 and 18 h eads cm 1 app oxima ely. To wea e hese sc eens wo di e en diame e s o h eads ha e been used. The sc eens 1 and 2 we e wo en wi h he h eads o app oxima ely 110 lm, he ag o ex iles 4 and 5 we e wo en wi h he h eads o app oxima ely 163 lm and he sc een 3 combines he hinnes h eads in we and he hickes ones in wa p. The ex iles o his se a e simila in many aspec s and, in his way, i is possible compa e be ween hem o check he in luence o he a iable changing. 3.2. Calcula ion o he 3D su ace The leng h L py and wid h L px o he holes and he hickness o he wa p h eads D hy a e ob ained om he images o he sc eens aken wi h a mic oscope (o hogonal p ojec ions). These alues along wi h he hickness Dzo he sc eens allow o ob ain he pa ame e s a,cand dusing equa ions (5) (6), and (7). Then, we ha e he alue F(1) in equa ion (11), and i is possible o calcula e he alues F((cþad)/c) by using equa ion (12), and G(0) and G(ad/(2cad)) by applying equa ion (13). Once all hese alues ha e been compu ed, he su ace a ea A 3D o he s udied egion o he hype bolic pa aboloid can be de e mined by equa ion (10). The esul s ob ained a e shown in Table 2. Using he same da a, ha is, mean wid h L px and leng h L py o holes, hickness o he wa p h eads D hy and hickness o he sc eens Dzcan be ob ained by equa ion (14) he su - ace a ea A 3D o he s udied egion o he hype bolic pa - aboloid by he abo e exposed app oxima e me hod. The esul s a e shown in Tables 3 and 4. As we had p edic ed, he mean su ace a ea o he holes calcula ed by he app oxima e me hod unde es ima es he alues ob ained calcula ing he e e ed a ea o he hype - bolic pa aboloid. Howe e , he a ios A 3D /A 3D a e e y close o one and o his eason he app oxima e me hod is a eliable al e na i e and i is necessa y o bea in mind ha i s applica ion is much simple . The hole su ace a ea meas- u ed in o hogonal p ojec ion is ob ained by mul iplying he wid h L px by he leng h L py (A 1 in Table 4). These alues signi ican ly unde es ima e he po ous su ace o he Table 1. Measu ed alues o he sc eens. Sc een q x q y ( h eads cm 2 )L px ± (lm) L py ± (lm) D hx ± (lm) D hy ± (lm) Dz± (lm) 1 15.2 30.2 222 ± 20 549 ± 9 111 ± 4 110 ± 5 271 ± 2 2 18.6 31.3 209 ± 12 428 ± 8 111 ± 4 110 ± 4 269 ± 3 3 16.2 30.3 168 ± 11 510 ± 34 109 ± 4 163 ± 5 378 ± 2 4 16.1 30.8 162 ± 11 458 ± 18 163 ± 5 163 ± 6 379 ± 3 5 14.2 30.7 163 ± 11 541 ± 18 160 ± 6 164 ± 7 385 ± 3 Table 2. Mean su ace a ea A 3D o holes ( egion o he hype - bolic pa aboloid). Sc een cadF(1) F( ) G( ) G(0) A 3D (mm 2 ) 1 272.59 27.51 5.71 0.89 2.42 9.93 9.19 133 450 2 225.21 23.72 6.23 1.58 3.15 10.34 9.40 99 319 3 239.71 19.96 5.95 1.92 4.11 21.65 20.24 107 338 4 219.20 18.53 6.18 2.37 4.71 22.97 21.33 94 612 5 248.90 19.98 5.77 1.77 4.02 23.93 22.49 112 422 Table 3. Calcula ion o he pa ame e s by applying he app oxima e me hod. Sc een d 1 (lm) d 2 (lm) b y1 (lm) b y2 (lm) k10 4 mn 1 274.2 236.1 26.1 7.1 3.18 0.182 26.12 2 262.6 223.6 26.8 7.3 5.33 0.239 26.80 3 272.9 199.4 52.4 15.7 6.45 0.370 52.43 4 270.0 194.7 54.0 16.3 8.12 0.422 54.00 5 274.6 196.9 55.8 17.0 5.98 0.368 55.80 Table 4. Su ace a ea A 3D ob ained by he app oxima e me hod and com- pa ison be ween su ace a eas. Sc een A 1 (lm 2 )A 2 (lm 2 )A 3D (mm 2 )A 1 /A 3D A 3D /A 3D 1 121 878 2489 131 834 0.92 0.99 2 89 452 1998 97 445 0.92 0.98 3 85 680 4901 105 285 0.81 0.98 4 74 196 4557 92 424 0.80 0.98 5 88 183 5575 110 482 0.80 0.98 Figu e 7. Adjus men o one o he b anches o he hype bola o a second-deg ee polynomial. 6 A. J.  ALVAREZ ET AL. hype bolic pa aboloid A 3D as he a ios A 1 /A 3D show (Table 4). This jus i ies quan i a i ely he impo ance o conside - ing he eal su ace (3D) o he holes ins ead o he su ace ela ed o i s o hogonal p ojec ion. The di e ences be ween he eal su aces and he measu es in o hogonal p ojec ion a e in luenced by he h ead hickness and he e o e by he hickness o he sc een. As he hickness o he sc eens inc eases, hese di e ences a e mo e p onounced (A 1 /A 3D ). The hickness o he sc eens is a pa icula ly complex issue because i does no depend only on he hicknesses o he h eads. I he longi udinal axis o he we h eads does no unde go any de o ma ion, he hickness o he sc eens would be app oxima ely he sum o wice he hickness o he wa p h eads plus he hickness o he we h eads, bu his is no so since he we h eads a e de o med and he ein lies he complica ion. Conside ing he abo e, he di e ence be ween he hick- ness o he sc eens 2 and 3 (Table 1) a e in line wi h he logic since, i he hickness o he sc een 2 is 270 lm, he hickness o he sc een 3 is 109 lm g ea e and his app oxi- ma ely coincides wi h wice he inc ease o hickness o he wa p h eads (50 lm). Howe e , he hicknesses o he sc een 3 and 4 a e p ac ically he same bu i was expec ed a di e ence o app oxima ely 50 lm ( hey ha e he same wa p h eads, bu he sc een 4 is wo en wi h we h eads 50 lm hicke ). Figu e 8 shows he longi udinal p o ile o he we and wa p h eads o he sc eens 2, 3, and 4. The images show how he de o ma ions o he h eads a e e y di e en and his explains he di e ences in he hickness o he sc eens abo e men ioned. The condi ions o he h eads in he loom du ing he manu ac u e o he sc een a e pos- sibly he main eason o hese a ia ions in he de o ma ion o he we and wa p h eads. In any case, his aspec equi es an in-dep h s udy. On he o he hand, ega ding o Pinke and He be ’s p oposals (1967), hese a e limi ed o he case o sc eens wi h holes o squa e geome y and conside iden ical de o - ma ions o wa p and we h eads which is comple ely un ealis ic (Figu e 8) e en in he case o squa e sc eens. In addi ion, he app oach p oposed does no ake in o accoun he hickness o he sc een ha is a a iable ac o ha can- no be igno ed in he calcula ion o he hole dimensions conside ing he 3D s uc u e o he ex ile. 3.3. New c i e ion o choosing p o ec i e sc eens agains insec s As we lea n mo e abou he in e ac ion be ween insec s and sc eens, each insec species will ha e a di e en ia ed ea - men bu nowadays he ea men is comple ely gene al. Cu en ly, he common c i e ion o choosing a sc een is based on hole wid h, ela i e o insec ho ax size. The sc een is chosen such ha speci ic insec s, wi h a gi en ho ax size, would no be able o c oss holes o a gi en wid h. Howe e , al hough he hole wid h is lowe han he ho ax size, he insec will be able o c oss he sc een h ough he space le be ween he segmen d 3 (L px in Table 1) and he segmen d 1 (Figu e 8,Table 3) i he hole leng h is oo la ge (p ison ba s e ec ). Theo e ically, i can be say ha he hole leng h is oo la ge i hal o he hole leng h is g ea e han he c oss sec ion o he ho ax size. In his case, he insec could c oss he hole i he dis ance d 2 (Figu e 6,Table 3) is g ea e han he ho ax size. Wi h he cu en ly c i e ion he heo e ical esul s a e mo e p omising han he eal esul s because he o hogonal hole wid h L px is always lowe han he gene a ix d 2 (Table 1 and 3). Fo his eason, he heo e ical e icacy o a sc een will be mo e accu a e i i is conside ed he dis ance d 2 ins ead o he hole wid h L px . 4. Conclusions The pu pose o his wo k was o de elop a me hod o calcu- la e he eal su ace o he holes o wo en ex iles because his is a e y impo an ma e o many speci ic applica- ions. So a , he dimensions o he holes ha e been meas- u ed on digi al images aken by elec onic de ices such as mic oscopes o scanne s. Howe e , hese o hogonal images unde es ima e he eal open su ace because he h eads o m a spa ial s uc u e. The su ace be ween wo consecu- i e wa p h eads in he con ex o a hole is a wa ped su - ace and has been iden i ied as a egion o he hype bolic pa aboloid. The calcula ion o his su ace is a complex ma hema ical p oblem ha has been sol ed by applying he undamen als o he analy ic geome y esul ing a compli- ca ed unc ion ha depends on he hickness o he sc een, he wid h and leng h o he holes and he hickness o he wa p h eads. All hese pa ame e s can be easily measu ed Figu e 8. We h eads (le ) and wa p h eads ( igh ) o he sc eens 3, 4, and 5 (so ed om op o bo om). THE JOURNAL OF THE TEXTILE INSTITUTE 7 by adi ional me hods (o hogonal images and a mic om- e e ). An al e na i e and app oxima e me hod has been p o- posed o ob ain he same esul by means o a simple ma hema ical p ocedu e. Wi h his second al e na i e a sim- ple unc ion has been ob ained and he a iables ha de ine i a e he same as in he p e ious case. This second me hod sligh ly unde es ima es he su ace a ea calcula ed by he i s me hod bu he esul s ob ained a e e y simila , so bo h me hods a e alid o he calcula ion o he po ous su ace. The esul s show how he su ace a ea ob ained measu ing on o hogonal images signi ican ly unde es ima e he eal po ous su ace ob ained by he p oposed me hods. The conside a ion o he 3D su ace o he holes and i s gene a ices is c ucial in ields such as c op p o ec ion and can help imp o e he models ha p edic he ae odynamic beha io o he wo en ex iles. A new design c i e ion is gi en o he heo e ical p edic ion o he e icacy o he sc eens agains insec s ha m ul o c ops. This c i e ion con- sis s in he conside a ion o he gene a ix d 2 ins ead o he o hogonal wid h L px and i explains why in some cases insec s c oss he sc een holes when he hole wid h is lowe han he size o hei bodies. Acknowledgmen s Au ho s a e hank ul o C iado y L opez S.L. o manu ac u ing and p o iding he samples o he p esen s udy. ORCID A. J.  Al a ez h p://o cid.o g/0000-0001-6281-9394 R. M. Oli a h p://o cid.o g/0000-0002-3924-5983 A. Jim enez-Va gas h p://o cid.o g/0000-0002-0572-1697 M. 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