A three-dimension approach to the porous surface of screens
Abstract
Currently, the porous surface of the screens is measured on digital images taken by microscope representing the orthogonal projection of the textiles. It is known that this way of measuring the porous surface underestimate largely the real hole surface. To improve this aspect, in this work the hole surface is identified as a specific region of the hyperbolic paraboloid and a methodology is developed to address its calculation. Indeed, the results show that the porous surface measured on orthogonal projection is significantly less than the real hole surface. However, the application of this methodology is very complex and for this reason an approximate alternative method that considerably simplifies the difficulty of the problem is proposed. The results obtained by one and another method have small discrepancies so that the approximate method is also a good option for the calculation of the porous surface of these textiles.
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Full Terms & Conditions of access and use can be found at http://www.tandfonline.com/action/journalInformation?journalCode=tjti20 The Journal of The Textile Institute ISSN: 0040-5000 (Print) 1754-2340 (Online) Journal homepage: http://www.tandfonline.com/loi/tjti20 A three-dimensional approach to the porous surface of screens A. J. Álvarez, R. M. Oliva, A. Jiménez-Vargas & M. Villegas-Vallecillos To cite this article: A. J. Álvarez, R. M. Oliva, A. Jiménez-Vargas & M. Villegas-Vallecillos (2018): A three-dimensional approach to the porous surface of screens, The Journal of The Textile Institute, DOI: 10.1080/00405000.2018.1500740 To link to this article: https://doi.org/10.1080/00405000.2018.1500740 Published online: 04 Dec 2018. Submit your article to this journal View Crossmark data
A three-dimensional approach to the porous surface of screens A. J. Alvarez a , R. M. Oliva a , A. Jim enez-Vargas b and M. Villegas-Vallecillos c a Department of Engineering, University of Almer ıa, Almer ıa, Spain; b Department of Mathematics, University of Almer ıa, Almer ıa, Spain; c Department of Mathematics, University of C adiz, Puerto Real, C adiz, Spain ABSTRACT Currently, the porous surface of the screens is measured on digital images taken by microscope representing the orthogonal projection of the textiles. It is known that this way of measuring the porous surface underestimate largely the real hole surface. To improve this aspect, in this work the hole surface is identified as a specific region of the hyperbolic paraboloid and a methodology is developed to address its calculation. Indeed, the results show that the porous surface measured on orthogonal projection is significantly less than the real hole surface. However, the application of this methodology is very complex and for this reason an approximate alternative method that considerably simplifies the difficulty of the problem is proposed. The results obtained by one and another method have small discrepancies so that the approximate method is also a good option for the calculation of the porous surface of these textiles. ARTICLE HISTORY Received 12 September 2017 Revised 9 July 2018 Accepted 10 July 2018 Published online 12 September 2018 KEYWORDS Agrotextiles; crop protection; screens; threedimensional porous surface 1. Introduction Screens have very different applications such as crop protection or its use in ducts for flow experiments. In the first case, for example, insect-proof screens are a physical method of crop protection whose use has become widespread in many parts of the world over recent decades. They are installed at the side and roof vents of greenhouses with a view to impeding or reducing the access of insects to the crop. The benefits of protection screens are sufficiently proven. However, there are still many knowledge gaps to get an optimal design. Their design is a very complex matter and besides its optimal can be addressed from different viewpoints that very often are opposing solutions ( Alvarez, 2010). If tiny holes are required to avoid the entry of insects inside the greenhouse, the ventilation rate is reduced as the porous surface decreases (Bailey et al., 2003; Bartzanas, Boulard, & Kittas, 2002;Dierickx,1998; Linker, Tarnopolsky, & Seginer, 2002;Mu ~ noz, Montero, Ant on, & Giuffrida, 1999;Soni, Salokhe, & Tantau, 2005) and produces imbalances in the greenhouse microclimate with negative consequences for crop development (Kittas, Boulard, Bartzanas, Katsoulas, & Mermier, 2002; Teitel, 2010). For this reason, the aerodynamic study of these textiles is also essential. The structure of the weave of screens is determined by two sets of threads (weft and warp) which interweave perpendicularly. The separation of the threads in each direction means that the geometry of each hole is generally rectangular, since the threads making up the warp are usually closer together than those of the weft. The number of threads per unit length establishes the density of threads (number of threads per unit length) of the screen in each direction. The diameter (thickness) of the threads is another variable that defines the geometry of the screen. Image analysis can be defined as the extraction of meaningful information from images by means of digital processing techniques (Solomon & Breckon, 2011). Image processing has been proved to be an efficient method of analyzing fabric structures (Jeong & Jang, 2005). These techniques are very important to many applications within the textile industry and their use is widespread and has been used extensively to obtain textile data ( Alvarez, Oliva, & Valera, 2012; Cardamone, Damert, Phillips, & Marmer, 2002; Gan, Bickerton, & Battley, 2012; Kang, Choi, Kim, & Oh, 2001; Shin, Cho, Seo, & Kim, 2008). Various techniques have been used including optical scanning, optical microscopy, confocal microscopy, optical coherence tomography, and X-ray microtomography (Sherburn, 2007). Accurate measurements of the geometry of woven textiles are essential for quality control of the weaving process (Lim & Kim, 2011), in textile modeling techniques for the prediction of material properties (Lomov et al., 2001; Zeng, Brown, Endruweit, Matveev, & Long, 2014) or for characterization of geometric parameters. Information regarding to the thickness of the threads, dimensions of the holes, number of threads per unit length, shape parameters as well as the quantitative and objective measurement of complex properties can be obtained (Kang, Kim, & Oh, 1999; Lim & Kim, 2011; Sherburn, 2007). Analysis of the geometry of protection screens is important to characterize their effectiveness to prevent insect entry inside the greenhouse ( Alvarez, Valera, & Molina-Aiz, 2006). The capacity of screens to keep insects out is CONTACT A. J. Alvarez [email protected] Department of Engineering, University of Almer ıa, Almer ıa, Spain ß2018 The Textile Institute THE JOURNAL OF THE TEXTILE INSTITUTE https://doi.org/10.1080/00405000.2018.1500740
determined comparing the dimensions of the holes with the usual size of the most damaging pest species. Therefore, the design of protection screens is carried out according to the body size of the smallest insect pest whose presence inside the greenhouse is intended to avoid (Bailey, 2003). The efficacy of a screen should not be predicted by comparing only the mesh size and the insect body size (Bethke & Paine, 1991) because other variables such as insect ability, temperature or air velocity are involved (Oliva & Alvarez, 2017). However, the relationship between insect body size and real hole surface is critical. The geometry of screens is also important to determine the resistance offered by the textile to the airflow. Screens are porous media since they have a solid structure combined with a void space. An important property of these materials is their porosity that can be defined as the ratio between the surface area of holes A h and the total surface area A t . But hitherto the surface area of holes A h is always underestimated. Due to the small size of the holes, geometric characteristics of screens are obtained on digital images taken by microscope. These images are orthogonal projections of the fabrics and therefore the measurements taken do not reflect the reality since the spatial arrangement of the threads does not conform to a plane. The opening left between the threads is larger than that obtained in the measurements on orthogonal projections. This has a direct impact both in the determination of the hole size as in the calculation of the open surface area. Considering the three-dimensional (3D) reality, the insects have more space to pass through the holes than the one initially supposed. Likewise, an air stream has more section to flow. Figure 1 shows both the top and perspective view of a screen. Almost all the protection screens present a rectangular hole geometry that is based on what we denominate like prison bars effect ( Alvarez & Oliva, 2017). Considering the previous, the warp represents a cage whose bars are not in the same plane as the Figure 1 shows. This determines that the limiting dimension for insect exclusion is not the separation between warp threads measured on orthogonal projection images but a greater distance. The objective pursued by the manufacturers with the design based on the prison bars effect is to restrict the entry of insect by limiting the distance between warp threads and, in turn, to avoid that the porosity of the textiles is too low increasing the distance between weft threads. An insect-proof screen will not fulfil its purpose if the only criterion of design considered is to establish the distance between warp threads (in orthogonal projection) lower than the insect thorax size. An exhaustive analysis of the real distance that the threads leave between them will allow to predict with more certainty the possibility for an insect crosses through a hole. In the next phase of study, it will be necessary to take into account that insects are living structures and therefore their abilities will also have to be considered and not only their size. In the other hand, the calculation of the hole surface area considering the 3D structure of the textile will allow to improve the models explaining the aerodynamic resistance that screens offer to airflow. There are hardly any references in the literature dealing with the issue of the 3D surface area of the screen holes. Pinker and Herbert (1967), in their study of the pressure drop that square hole screens cause on the airflow, proposed two alternatives to the porosity obtained as a result of the relationship between the hole surface area and the total surface area measured on orthogonal projections. Their method to measure the 3D surface of the holes with square geometry assumes that both weft and warp threads undergo the same deformation and for this reason the Pinker and Herbert’s starting assumptions are wrong. The distance between threads for the screens with square hole geometry is the same both in warp and weft directions but the curvature of the warp and weft threads is not similar and therefore both sets of threads can be distinguished since the warp threads “embrace”the weft ones. The differences between the measured dimensions on orthogonal projections and the spatial measurements for screens with square and rectangular holes are of the same nature because for both types of screens the deformations of the threads are similar. This work presents a theoretical study of the geometry of screens from a 3D point of view consisting of a method to calculate the 3D surface that the spatial crossing between two consecutive warp threads defines in the context of a hole (Figure 2). Many applications may have this new development: in the field of physical barriers (textiles used to combat insects harmful for crops) this approach can improve the prediction of efficacy of the textiles against insects and the calculation of this spatial surface can also improve the methods that describe the airflow through these porous media. 2. Theory The 3D representation of the porous surface left between adjacent threads is shown in Figure 2. The fabric structure Figure 1. Representation of a screen in orthogonal projection (left) and in perspective (right). 2 A. J. ALVAREZ ET AL.
determines that the warp threads “embrace”the weft ones and therefore the first ones have greater deformations. In the representation of Figure 2 has been considered that the cross-section of the threads is circular and that the axis of the cylindrical body of the warp threads remains approximately straight along the distance between the crossings with the weft threads (in general, the observation demonstrates that both assumptions are very close to the reality). The spatial surface of a hole is defined by the two closest generatrices of two adjacent warp threads (Figure 2). These two segments define a doubly ruled surface since for each of its points pass two straight lines completely contained in the surface. The surface can be included in a parallelepiped whose length L py , width L px and height Dz–D hy are known and described in Figure 2 (where Dzis the thickness of the screen and D hy the thickness of the warp threads). This surface is a region of the hyperbolic paraboloid and to calculate its surface area is proposed the following procedure based on the fundamentals of the analytic geometry and an approximate method to simplify the calculations. 2.1. Surface area of the defined region of the hyperbolic paraboloid Let a,c,d2R þ be three parameters with ad c. Let rbe the straight line passing for the points (Figure 3): p¼c;cþda;d2þ2cd a (1) q¼cþda;c;d22cd a (2) and sthe straight line containing to the points (Figure 3): p0¼cda;c;d22cd a (3) q0¼c;cda;d2þ2cd a (4) The relationship between a,c, and dwith the length L py , the width L px and the height DzD hy of the parallelepiped containing the surface area under study is obtained solving a system with three equations and three unknowns. The result is the following (Figures 2 and 3): a¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi LpxLpy 䉭zDhy s(5) c¼Lpx þLpy 2ffiffiffi 2 p(6) d¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Lpx 䉭zDhy 2Lpy s(7) Let T 1 and T 2 be two triangles (Figure 4): Figure 2. 3D representation of a hole (left) and surface area that the threads leave between them (right); the rectangle in bold represents the orthogonal projection of the hole. Figure 3. Definition of the segments r and s. THE JOURNAL OF THE TEXTILE INSTITUTE 3
T1¼nx;y ðÞ 2R2:x2c;cþda ½ ;xda yx þ2cdao T2¼x;y ðÞ 2R2:x2cda;c ½ ;x2cþda yxþda no and R 0 the rhomboid (Figure 4): R0¼x;y ðÞ 2R2:x2cþda;cda ½ ;xda yxþda no Joining T 1 ,T 2 , and R 0 , a rectangle Ris obtained (Figure 4). In that rectangle can be defined the function f:R!R given by: fx;y ðÞ ¼1 a2x2y2 ;8x;y ðÞ 2R(8) The graph of the function fis a region Rof the hyperbolic paraboloid containing the segment endpoints pand qand the segment endpoints p 0 and q 0 (Figure 3) and can be defined by the surfaces S 1 ,S 2 ,S 3, and S 4 (Figure 5). Using an appropriate change of variable, this description of Rallows to calculate the surface area of the graph of the function fas follows. The surface area A 3D of the graph fis given by the integral: A3D ¼ðð Rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi of oxx;y ðÞ 2 þof oyx;y ðÞ 2 þ1 sdx;y ðÞ ¼ ¼ðð S1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 a4x2þy2 þ1 rdx;y ðÞ þðð S2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 a4x2þy2 þ1 rdx;y ðÞ þðð S3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 a4x2þy2 þ1 rdx;y ðÞ þðð S4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 a4x2þy2 þ1 rdx;y ðÞ (9) It can be shown that by solving the previous four integrals the following expression is obtained: A3D ¼a4 3F1 ðÞ Fcþad c þGad 2cad G0 ðÞ p 2 (10) with F1 ðÞ¼p 2d3a2þ2d2 ðÞ a3ffiffiffi 2 pln dffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2a2þ4d2 p (11) Ft ðÞ¼d2t1 ðÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2tþ1 ðÞ 2þ4d2t2þ1 ðÞ qa3tþ1 ðÞ 2 tan1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2tþ1 ðÞ 2þ4d2t2þ1 ðÞ qat1 ðÞ 0 @1 A d3a2þ2d2 ðÞ a3ffiffiffi 2 p ln 2d2t1 ðÞ þdffiffiffi 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2tþ1 ðÞ 2þ4d2 qt2þ1 ðÞ tþ1 ! (12) Gt ðÞ¼2cad ðÞ 2 a6tffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a4þ22cad ðÞ 2t2þ1 ðÞ q þtan 1a2t ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a4þ22cad ðÞ 2t2þ1 ðÞ q 0 @1 A þ2cad ðÞ 3a4þ22cad ðÞ 2 a6ffiffiffi 2 p ln 2 2cad ðÞ tþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2a4þ42cad ðÞ 2t2þ1 ðÞ q (13) 2.2. An alternative and approximate method The hyperbolic paraboloid is a warped surface, that is, a non-developable doubly ruled surface since two consecutive positions of the generatrix are not coplanar. In other words, this kind of surfaces cannot be extended on a plane as in the case of the surface of a cylinder or a cone. The approximate method proposed is based on “flattening”the defined Figure 4. The projection on the orthogonal plane. Figure 5. Definition of the region R. 4 A. J. ALVAREZ ET AL.
surface of the hyperbolic paraboloid to measure its surface area more easily than with the previous procedure. The results obtained will always underestimate the real solution because this is a non-developable surface. This approximate method is simpler although involves a small loss of accuracy that we will assess a little later. To “flatten”the studied region of hyperbolic paraboloid the generatrices parallel to x-axis are rotated around an axis parallel to y-axis as shown in Figure 6. In this way, it is obtained a flat surface defined by a hyperbola whose surface area A 3D can be calculated by the following expression: A3D ¼A1þ4A2(14) where A 1 is the surface area of the hole orthogonal projection, that is, obtained by multiplying L px and L py ; the other summand 4A 2 is the surface area enclosed between the curves of the hyperbola and the rectangle representing the hole orthogonal projection (Figure 6). One of the branches of the hyperbola can be isolated to calculate the surface area A 2 . For that, we can turn the coordinate system so that the x-axis coincides with the direction of the rectangle length representing the orthogonal projection of the hole (Figures 6 and 7). The isolated branch can be described as a second-degree polynomial y¼kx 2 þmxþn. The quadratic polynomial can be fitted measuring some generatrices of the hyperbolic paraboloid in space. The chosen generatrices (Figure 6) are defined by the points p 1 ,p 2 ,q 1 ,q 2 ,r 1, and r 2 as shown below: p1¼0;0;DzDhy 2 ;p2¼Lpx;0; DzDhy 2 (15) q1¼0; Lpy 4;DzDhy 4 ;q2¼Lpx; Lpy 4; DzDhy 2 (16) r1¼0; Lpy 2;0 ;r2¼Lpx; Lpy 2;0 (17) From the previous points, the generatrices lengths d 1 ,d 2, and d 3 can be calculated as the distances between p 1 and p 2 , q 1 and q 2 , and r 1 and r 2 , respectively (Figure 6): d1¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L2 px þDzDhy 2 q(18) d2¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L2 px þDzDhy 2 2 s(19) d3¼Lpx (20) After obtaining the previous lengths, we need to calculate the ordinates b y1 ,b y2, and b y3 of the required points to do the polynomic adjustment (Figures 6 and 7): by1¼d1Lpx 2(21) by2¼d2Lpx 2(22) by3¼d3Lpx 2¼LpxLpx 2¼0 (23) Finally, the coordinates of the points are the following (Figure 7): 0;by1 ¼0;ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L2 px þDzDhy 2 qLpx 2 ! (24) Lpy 4;by2 ¼Lpy 4;ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L2 px þDzDhy 2 2 rLpx 2 0 @1 A(25) Lpy 2;by3 ¼Lpy 2;0 (26) The surface area A 2 is the result of solving the definite integral of the polynomic adjustment between the limits 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, the surface area A 3D of the studied region of the hyperbolic paraboloid measured according to the approximate method is: A3D ¼A1þ4A2¼LpxLpy þkL3 py 6þmL2 py 2þ2nLpy (28) Figure 6. Scheme to “flatten”the hyperbolic paraboloid. THE JOURNAL OF THE TEXTILE INSTITUTE 5
3. Results and discussion 3.1. Characterization of screens To apply the methodology exposed in the previous section, five experimental insect-proof screens have been used. The geometric characterization of these agrotextiles has been performed following the methodology proposed by Alvarez et al. (2012). The images have been taken by an optical microscope (B3, Motic) with a built-in digital camera (Moticam 2, Motic). The thickness of the screens has been measured by a micrometer (Micromaster, Tesa). The data obtained are shown in Table 1. The five screens used have about the same number of warp threads per unit length and their densities of weft vary between 14 and 18 threads cm 1 approximately. To weave these screens two different diameters of threads have been used. The screens 1 and 2 were woven with the threads of approximately 110 lm, the agrotextiles 4 and 5 were woven with the threads of approximately 163 lm and the screen 3 combines the thinnest threads in weft and the thickest ones in warp. The textiles of this set are similar in many aspects and, in this way, it is possible compare between them to check the influence of the variable changing. 3.2. Calculation of the 3D surface The length L py and width L px of the holes and the thickness of the warp threads D hy are obtained from the images of the screens taken with a microscope (orthogonal projections). These values along with the thickness Dzof the screens allow to obtain the parameters a,cand dusing equations (5) (6), and (7). Then, we have the value F(1) in equation (11), and it is possible to calculate the values F((cþad)/c) by using equation (12), and G(0) and G(ad/(2cad)) by applying equation (13). Once all these values have been computed, the surface area A 3D of the studied region of the hyperbolic paraboloid can be determined by equation (10). The results obtained are shown in Table 2. Using the same data, that is, mean width L px and length L py of holes, thickness of the warp threads D hy and thickness of the screens Dzcan be obtained by equation (14) the surface area A 3D of the studied region of the hyperbolic paraboloid by the above exposed approximate method. The results are shown in Tables 3 and 4. As we had predicted, the mean surface area of the holes calculated by the approximate method underestimates the values obtained calculating the referred area of the hyperbolic paraboloid. However, the ratios A 3D /A 3D are very close to one and for this reason the approximate method is a reliable alternative and it is necessary to bear in mind that its application is much simpler. The hole surface area measured in orthogonal projection is obtained by multiplying the width L px by the length L py (A 1 in Table 4). These values significantly underestimate the porous surface of the Table 1. Measured values of the screens. Screen q x q y (threads cm 2 )L px ±r(lm) L py ±r(lm) D hx ±r(lm) D hy ±r(lm) Dz±r(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 surface area A 3D of holes (region of the hyperbolic paraboloid). Screen cadF(1) F(t) G(t) 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. Calculation of the parameters by applying the approximate method. Screen d 1 (lm) d 2 (lm) b y1 (lm) b y2 (lm) k10 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. Surface area A 3D obtained by the approximate method and comparison between surface areas. Screen 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 Figure 7. Adjustment of one of the branches of the hyperbola to a second-degree polynomial. 6 A. J. ALVAREZ ET AL.
hyperbolic paraboloid A 3D as the ratios A 1 /A 3D show (Table 4). This justifies quantitatively the importance of considering the real surface (3D) of the holes instead of the surface related to its orthogonal projection. The differences between the real surfaces and the measures in orthogonal projection are influenced by the thread thickness and therefore by the thickness of the screen. As the thickness of the screens increases, these differences are more pronounced (A 1 /A 3D ). The thickness of the screens is a particularly complex issue because it does not depend only on the thicknesses of the threads. If the longitudinal axis of the weft threads does not undergo any deformation, the thickness of the screens would be approximately the sum of twice the thickness of the warp threads plus the thickness of the weft threads, but this is not so since the weft threads are deformed and therein lies the complication. Considering the above, the difference between the thickness of the screens 2 and 3 (Table 1) are in line with the logic since, if the thickness of the screen 2 is 270 lm, the thickness of the screen 3 is 109 lm greater and this approximately coincides with twice the increase of thickness of the warp threads (50 lm). However, the thicknesses of the screen 3 and 4 are practically the same but it was expected a difference of approximately 50 lm (they have the same warp threads, but the screen 4 is woven with weft threads 50 lm thicker). Figure 8 shows the longitudinal profile of the weft and warp threads of the screens 2, 3, and 4. The images show how the deformations of the threads are very different and this explains the differences in the thickness of the screens above mentioned. The conditions of the threads in the loom during the manufacture of the screen are possibly the main reason for these variations in the deformation of the weft and warp threads. In any case, this aspect requires an in-depth study. On the other hand, regarding to Pinker and Herbert’s proposals (1967), these are limited to the case of screens with holes of square geometry and consider identical deformations for warp and weft threads which is completely unrealistic (Figure 8) even in the case o square screens. In addition, the approach proposed does not take into account the thickness of the screen that is a variable factor that cannot be ignored in the calculation of the hole dimensions considering the 3D structure of the textile. 3.3. New criterion for choosing protective screens against insects As we learn more about the interaction between insects and screens, each insect species will have a differentiated treatment but nowadays the treatment is completely general. Currently, the common criterion for choosing a screen is based on hole width, relative to insect thorax size. The screen is chosen such that specific insects, with a given thorax size, would not be able to cross holes of a given width. However, although the hole width is lower than the thorax size, the insect will be able to cross the screen through the space left between the segment d 3 (L px in Table 1) and the segment d 1 (Figure 8,Table 3) if the hole length is too large (prison bars effect). Theoretically, it can be say that the hole length is too large if half of the hole length is greater than the cross section of the thorax size. In this case, the insect could cross the hole if the distance d 2 (Figure 6,Table 3) is greater than the thorax size. With the currently criterion the theoretical results are more promising than the real results because the orthogonal hole width L px is always lower than the generatrix d 2 (Table 1 and 3). For this reason, the theoretical efficacy of a screen will be more accurate if it is considered the distance d 2 instead of the hole width L px . 4. Conclusions The purpose of this work was to develop a method to calculate the real surface of the holes of woven textiles because this is a very important matter for many specific applications. So far, the dimensions of the holes have been measured on digital images taken by electronic devices such as microscopes or scanners. However, these orthogonal images underestimate the real open surface because the threads form a spatial structure. The surface between two consecutive warp threads in the context of a hole is a warped surface and has been identified as a region of the hyperbolic paraboloid. The calculation of this surface is a complex mathematical problem that has been solved by applying the fundamentals of the analytic geometry resulting a complicated function that depends on the thickness of the screen, the width and length of the holes and the thickness of the warp threads. All these parameters can be easily measured Figure 8. Weft threads (left) and warp threads (right) of the screens 3, 4, and 5 (sorted from top to bottom). THE JOURNAL OF THE TEXTILE INSTITUTE 7
by traditional methods (orthogonal images and a micrometer). An alternative and approximate method has been proposed to obtain the same result by means of a simpler mathematical procedure. With this second alternative a simpler function has been obtained and the variables that define it are the same as in the previous case. This second method slightly underestimates the surface area calculated by the first method but the results obtained are very similar, so both methods are valid for the calculation of the porous surface. The results show how the surface area obtained measuring on orthogonal images significantly underestimate the real porous surface obtained by the proposed methods. The consideration of the 3D surface of the holes and its generatrices is crucial in fields such as crop protection and can help improve the models that predict the aerodynamic behavior of the woven textiles. A new design criterion is given for the theoretical prediction of the efficacy of the screens against insects harmful to crops. This criterion consists in the consideration of the generatrix d 2 instead of the orthogonal width L px and it explains why in some cases insects cross the screen holes when the hole width is lower than the size of their bodies. Acknowledgments Authors are thankful to Criado y L opez S.L. for manufacturing and providing the samples for the present study. ORCID A. J. Alvarez http://orcid.org/0000-0001-6281-9394 R. M. Oliva http://orcid.org/0000-0002-3924-5983 A. Jim enez-Vargas http://orcid.org/0000-0002-0572-1697 M. Villegas-Vallecillos http://orcid.org/0000-0002-6004-4836 References Alvarez, A. J. (2010). Estudio de las caracter ısticas geom etricas y del comportamiento aerodin amico de las mallas antiinsectos utilizadas en los invernaderos como medida de protecci on vegetal (Doctoral dissertation). University of Almer ıa (in Spanish). 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