M98-P2 (formerly C98-P1) Non-Newtonian Fluid Flow through Fabrics Matthew W. Dunn Philadelphia University
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1 1 Non-Newtonian Fluid Flow through Fabrics Matthew W. Dunn Philadelphia University Goal Statement The overall objectives of this program are to Model fabric porosity based on material properties Develop a turbulence-free fabric permeability tester Model the flow of Newtonian fluid through fabrics Model the flow of non-newtonian fluid through fabrics Compare experimental and theoretical predictions Abstract The flow of a fluid through textiles has been studied for a number of years. Understanding the flow phenomena along with the corresponding environmental changes (pressure drop, concentration gradient, etc.) becomes important in many fields, such as liquid transport through geomembranes, effluent movement through filtration devices, and chemical movement through protective apparel. This project is focussed on appreciating the difference between Newtonian and non-newtonian fluids and how they differ in their resistance to forced movement through a fabric. In order to accurately measure the predicted fluid flow parameters, a novel permeability tester is under development. The apparatus was inspired by the classical Joule-Thomson thermodynamic experiment. The tester has been designed to use a constant fluid volume and velocity while the pressure drop across a fabric is measured. The fluid velocity can then be varied over a range of values. A double piston design has been utilized to minimize turbulence in fluid movement. Fluid Movement Fluids differ from solids in that they undergo continuous deformation as long as shear is applied. The amount of resistance offered to deformation under applied shear stress (τ) is the viscosity (µ), defined by τ yx = µ u y = µ γ yx where u y is the velocity gradient. Normally this gradient is replaced by the term γ yx, or the shear strain rate. This relationship is known as Newton s Law of Viscosity, and fluids that follow this linear behavior are termed Newtonian fluids. Not all fluids follow this relationship. As viscosity is measured, normally a force is applied and the resulting shear strain rate is measured (the opposite is of course entirely feasible, but less common). The relationship between the shear stress and shear strain rate is then plotted and differences appear among fluids (Figure 1). 1
2 2 Some fluids become easier to deform under increasing shear strain rate. Molten polymer solutions are one example of this behavior, since it requires less force to stir the solution as the speed of stirring increases. Fluids such as this are termed shear thinning (or pseudoplastic). Other fluids, like paint, become harder to stir as the speed of stirring increases. These fluids are termed shear thickening (or dilatent). Sh re a Th cki ing e n N tonia e w n τ (MPa) Sh re a Th ing n γ (s -1 ) Figure 1: Different types of fluid flow under shear For each of these non-newtonian cases, curve fitting yields a power law relationship between shear stress and shear strain rate: τ = K γ n where K is the consistency index and n is the power law exponent. Shear thinning fluids would have a power law exponent below one, while the exponent would be greater than one for shear thickening fluids. In the case of a fluid described by the power law, the viscosity relationship is found by substitution to be µ app = τ n-1 = K γ γ where µ app is the apparent viscosity. This definition of viscosity removes the limitation of describing only fluids with a linear shear stress-shear strain rate response. Permeability Modeling Filtration can be described mathematically in many ways. One approach is to describe the flow of air or fluid through a filtration media based on the properties of the air or fluid penetrant. Fluid flow will be used to describe both air and fluid flow henceforth, since air can be assumed to be a very low viscosity fluid. Fluid flow through a porous membrane can be described by the relationship 2
3 3 B = µtv P where µ is the viscosity of the fluid, t the thickness of the membrane, V the velocity of the fluid, and P the pressure drop across the membrane. This is one form of Darcy s Law [1], named for the French engineer who published an equivalent relation based on experiments with the water supplies for the fountains of the city of Dijon [2]. Here B is considered as a constant for the relationship between V and P, and is called Darcy s constant or (more commonly) the permeability of the membrane. The value for B will depend on the type of porous media and the pore geometry. Darcy s Law assumes the viscosity to be a single value, i.e. Newtonian. Non-Newtonian fluids can be accounted for by substituting the power law equation, so that B = Kγ n-1 tv P Following the reasoning that the void content in a porous media is a primary factor in the media permeability, the Kozeny-Carman equation was developed to provide a description of fluid flow based on the filtration media properties. One form of this equation [3] is B = 1 K o S o 2 Φ 3 (1-Φ) 2 where K 0 is the Kozeny Constant, S 0 a shape factor, and Φ the media porosity. The shape factor is found from surface area of solid phase S o = volume of solid phase where the solid phase characteristics are based on the construction and content of the filtration media. It has been found that image analysis methods for porosity evaluation may yield reasonable results when predicting the Kozeny-Carman parameters [4]. Pierce [5] attempted to describe the resistance a textile material will exhibit based on material properties: R = S ρ o S2 2 (1-Φ) 2 Φ 3 where R is the resistance to flow, S the surface of void channels per unit mass (or the total specific surface of the media mass), and ρ the overall mass density of constituents within the media. Pierce noted that this is in fact a method of extreme simplification, and that there is no pretense that the form assumed is geometrically similar to the form to be studied. He suggested that the shape factor be determined empirically, with the other variables being calculated from derived relationships. This equation was the first published attempt at describing fluid flow specifically through textile materials. 3
4 4 Ergun [6], working with experimental results gathered from filtration media based on different shape parameters, developed the following relationship for filtration media with cylindrical solid constituents: Φ B = (φd)2 3 g c 150(1-Φ) 2 where φ is a shape factor, d the diameter of the cylinders, and g c the gravitational constant. MacGregor [7] extended the Kozeny-Carman equation for a textile assembly in order to model the flow of dyes through textile yarn packages. If the solid phase is composed of circular fibers with diameter d and length l, it can be easily found that for textile beds: S o = 4πdl πd 2 l = 4 d By substitution: d 2 Φ 3 B = 16K o (1-Φ) 2 The MacGregor equation provides a method to predict permeability based on constituent properties of a subjected fabric. The MacGregor equation yields the importance of fiber size on permeability. Consider two unit cells, each with a porosity of 50% (Figure 2). Since cell B has constituents of half the size of those in cell A, the MacGregor permeability prediction varies with the diameter squared (Figure 3 and Figure 4). L L L L A B Figure 2: Cells A and B have equal porosity but differ in permeability 4
5 5 80% 70% 60% 50% Porosity 40% 30% PERMEABILITY SCALES WITH DIAMETER SQUARED 20% 10% 10 µm diameter 20 µm diameter 0% 0.00E E E E E E E-07 Permeability (m 2 ) Figure 3: Variation of MacGregor permeability prediction for different diameter fiber 100% 40 90% 35 80% 70% 60% constant diameter (10 mm) constant porosity (50%) Porosity 50% 20 Diameter (µm) 40% 15 30% 20% 10 10% 5 0% E E E E E-12 Permeability (m 2 ) Figure 4: MacGregor permeability prediction at constant porosity vs. constant diameter 5
6 6 Porosity Modeling Prediction of porosity can be undertaken by accounting for the physical characteristics of a fabric. The areal weight (Γ ) can be found from Γ = (1-φ) ρ t where φ is the porosity of the fabric, ρ the fiber density, and t the fabric thickness. Porosity can now be found based on measured values: φ = 1- Γ tρ Since porosity may be targeted for a certain application, these physical parameters can serve as the basis for fabric design. To target a thickness, the contribution of each yarn system must be addressed. For yarn area (in cm 2 ), N A i = i [9x10 5 ]ρ i ψ where N i is the linear density (in denier) and ψ is the yarn packing factor, given by the ratio of total fiber area to total yarn area. To find the expected fabric thickness for a woven fabric, it is convenient to assume elliptical yarn cross-sections such that A i = πa i b i where A i is the yarn cross-sectional area, a i = minor axis dimension for i th yarn system, and b i the major axis dimension for i th yarn system. Since the yarn flattening will occur normal to the fabric formation direction, the minor axis becomes the important factor for fabric thickness. a i b i Figure 5: Major (b i ) and minor (a i ) axis for an elliptical cross-section Factoring for this direction yields a i = A i πb i. Each yarn now has a thickness contribution of 2a i. For a single layer weave with warp (subscript 1) and filling (subscript 2), t = 2a 1 + 2a 2. If the warp and filling direction are equivalent, then t is simply 4a. It is convenient to introduce the aspect ratio taken by the yarn (R i ), or the ratio of the major axis to the minor axis. The minor axis of each yarn can now be seen as a i = A y Rπ. Assuming that each yarn direction will undergo the same amount of spreading so that R 1 and R 2 are equivalent, the subscript will be dropped. If the fabric thickness and yarn sizes are known, then calculating the value of R is straight forward. 6
7 7 As an example, consider a fabric to be constructed of 500 denier multifilament textured polyester (with a density of 1.38 g/cm 3 ) for both the warp and filling directions. For a packing factor of 0.5, each yarn area would be 500 A = [9x10 5 ]1.38(0.5) = 8.05 x 10-4 cm 2 If the aspect ratio is expected to be 4, then the predicted fabric thickness becomes A t = 4a = 4 = 0.32 mm. 4π Fluid Permeability Testing A novel tester is currently under development in order to evaluate the fluid permeability of a fabric (Figure 6, Figure 7). The tester has been inspired by the Joule-Thomson experiment, which was concerned with the changing temperature of a gas as the volume was varied. The present tester has been designed to use a constant volume of fluid in an enclosed chamber while the pressure drop across a fabric is measured at controlled flow. The constant volume approach should lead to permeability values untainted by turbulence. Figure 6: Schematic of novel fluid permeability testing device 7
8 8 Figure 7: Fluid permeability tester The tester has been designed to have multiple apertures to provide for a range of velocities. Pressure transducers will monitor the pressure on either side of the fabric. The tester is not limited in terms of fluid type or viscosity. Refinement of the testing equipment will remain a focus of this project. Air Permeability Testing In lieu of completion of the fluid permeability tester, air permeability was used as a preliminary investigation in fluid flow. Air can be considered a low viscosity (1.75 x 10-7 Pa!sec) fluid that is Newtonian in the range of testing. Air permeability is measured as the resistance to air flow at a controlled velocity and pressure. Data was obtained using the Kawabata KES-F8-API Air Permeability Tester. The fabric is subjected to a constant velocity of air by single piston movement, and the pressure drop across the face of the fabric is measured. The output is the air resistance R, measured in kilopascals times second per meter (kpa!sec/m), found from: R = (P 1 -P 2 ) V = P V where P is the pressure drop and V the air velocity. The air permeability measurement assumed laminar (non-turbulent) flow through the fabric. This is true where the pressure drop is due to frictional loss, defined as K = P V where K is a constant for the specimen. In this case, K is equivalent to R and is linear with respect for the specimen. A material with this response can be considered a linear resistor. If the specimen exhibits turbulent flow, then Bernoulli s Law holds true, where K = P V. 2 8
9 9 In this case R is no longer equivalent to K, but instead R = K V. Here R is not constant since it is now a function of changing velocity. Such a material is considered as a nonlinear resistor. Previous studies have shown that many fabrics are in fact nonlinear resistors [8]. Fabric Examples Three example fabrics have been made for initial investigation. Each fabric used 340 denier polyester for the warp at 90 ends per inch, and 170 denier polyester for the filling at 42 picks per inch. The three fabrics are summarized in Table 1. Porosity values were calculated from the measured values of areal density and thickness. The fabrics can be viewed in Figure 8. Table 1: Investigated fabric physical parameters fabric type plain weave 2/1 twill 8/1 steep twill fabric weight (g/m2) warp crimp 8.3% 5.3% 4.6% fill crimp 2.2% 3.0% 3.9% thickness (mm) porosity 69.5% 74.9% 77.0% Figure 8: Photomicrographs of investigated fabrics (30x magnification) (a) plain weave, (b) 2/1 twill, (c) 8/1 steep twill These fabrics were tested for air permeability at an air velocity of 4 cm/sec. Using the Darcy equation, the permeability values were calculated. The predicted air permeability values were also calculated from the MacGregor equation (at K=8). These values are compared together in Figure 9. What can be viewed from the curves is that within the range of porosities tested, the measured air permeability values do not appear to be linear. This implies that some account for fabric geometry should be made. It would be expected that fluid permeability testing using a nonlinear fluid penetrant would yield further nonlinearity. 9
10 10 7E-12 6E-12 Darcy Permeability MacGregor Prediction 5E-12 Permeability (m 2 ) 4E-12 3E-12 2E-12 1E % 66% 68% 70% 72% 74% Porosity Figure 9: Measured permeability from Darcy compared to predicted MacGregor values Future Work Future work will now concentrate on completion of the fluid permeability tester. Using Newtonian and non-newtonian fluids, engineered fabrics will be evaluated. A predictive model will be refined that accounts for nonlinear fluids and different fabric geometries. Acknowledgements Moishe Garfinkle has served as Research Associate on this project, and has done the majority of work in refining the fluid permeability tester. Mir Quddus is the Graduate Research Assistant. References 1. H. Darcy, Les Fontaines Publiques de la Ville de Dijon, Paris, J. Daily and D. Harleman, Fluid Dynamics, Addison-Wesley, Reading, Mass., 1966, pp A. Scheidegger, The Physics of Flow Through Porous Media, revised ed., University of Toronto Press, Toronto, J. Berryman and S. Blair, Journal of Applied Physics, February 1987, pp F. T. Pierce, Textile Research Journal, Vol. 17, No. 3, pp S. Ergun, Chemical Engineering Progress, Vol. 48, No. 2, Feb. 1952, pp R. McGregor, Journal of the Society of Dyers and Colourists, Vol. 81, October 1965, pp M. Dunn, Masters Thesis, Renitent Textile Composites: Formation and Permeability Analysis of Porous Media. Philadelphia College of Textiles & Science,
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