Some basic problems of heat and mass transfer in thin materials
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https://www.eduzhai.net International Journal of Composite Materials 2014, 4(5A): 35-40 DOI: 10.5923/j.cmaterials.201401.05 Some Fundamental Problems of Heat-Mass-Transfer through the Thin Materials Anna V. Shchutska Kyiv College of Light Industry, Kyiv, Ukraine Abstract This paper develops the novel approach to problems arising at the study of permeability for the textile fabrics, natural leathers, thin layers of polymers and other thin porous structures (PS). There is serious gap between the experimental investigation of such PS and the relevant different models proposed to imitate the heat-mass-transfer (HMT) of moist air through the small thickness of PS. The formers are based on the water-vapor-transmission (WVT) rate dynamical parameter, which is not, as a matter of fact, the characteristic of PS itself. The same is true for the uncontrolled in the majority of experiments quantity termed as the gradient of partial moisture pressure. The well-established methodology proposed in this paper provides the reliable estimate of the generalized diffusion coefficient. It characterizes just the properties of PS itself. Keywords Water-vapor-transmission rate’s, Porous structure, Generalized diffusion coefficient 1. Introduction The specific property of a thin porous structure (PS) is two strong inequalities: l , b between its linear sizes l,b determining the normal test area A l b ( 0.3 0.3 m2, for example) and its thickness δ ( 104 103 m, in the most cases). The water vapor transmission (WVT) rate of any PS and the textile fabrics, in particular, is of considerable interest for many branches of the practice and science. Such materials provide, for example, the steady moisture transfer from the skin of the weaver through the textile layers into the environment during heavy work or in hot climate conditions. There is a variety of methods to measure WVT (Δm is mass change): WVT m At kg m2s (1) which give, however, the quite different results for the same fabric, or, more generally, the same thin PS. The obvious reason of such situation is an absence of the general conception of treatment and interpretation of the experimental data obtained by the different test methodologies, which vary with respect to construction mechanisms, test conditions and measurement parameters. The aim of this paper is to demonstrate that the general * Corresponding author: email@example.com (Anna V. Shchutska) Published online at https://www.eduzhai.net Copyright © 2014 Scientific & Academic Publishing. All Rights Reserved approach to the WVT-problem becomes still possible if one rejects its physically incorrect but widespread interpretation as the property of PS itself. In contrast, it is considered here as the standard density of flow: jm (i.e. the dynamical parameter) determined for the quasicontinuous PS-media by the non-equilibrium thermodynamics: WVT jm u kg m2s , (2) where m A kg m3 is the local density of a moist vapor while u t m s is the effective local kinematic velocity. The main problem of this interpretation is the reliable estimate of both quantities ρ and u separately raking into account the netto-mass mM and the respective density M mM A of a PS- material (M) itself. During the water vapor transport process its brutto-mass M mM A is increased due to the sorption ability ( mM m ) of any PS. Of course, these corrections are very different for the hydrophobic or weakly hydrophilic fabrics, on the one hand, and hydrophilic fabrics, on the other hand. Another fundamental problem of the proposed below approach based on the generalized for PS Fick’s law of diffusion: jm D*grad P D* P2 P1 (3) is the reliable estimate of the generalized diffusion coefficient D* s which has the dimensionality of time. 36 Anna V. Shchutska: Some Fundamental Problems of Heat-Mass-Transfer through the Thin Materials Indeed the comparison (1) and (3) leads to the interesting possibility to calculate u from (2) u t D* P A m (4) is the difference of partial pressures ΔP across PS can be specified (or measured) by experiment. Unfortunately, this is not the case for a majority of WVT-measurements. Our goal below is to demonstrate that some progress can be achieved in the framework of proposed approach too if one measures the same fabric sample by the different test methods. where Ret is the total evaporative resistance provided by the porous metal plate (imitated the body skin at 35ºC) fabric and boundary air layer L1 in (5); A - test area, m2 ; Ps (T ) - the water vapor pressure of the above plate surface, Pa; Pa - the water vapor pressure of the air, i.e. of the undetermined layer of L2 in (5); H – the heating power, W; He - the correction evaporative term for heating power, i.e. latent heat of vaporization, W. The intrinsic evaporative resistance of the fabric Ref can be found as the difference: 2. Physical Concepts of Approach Ref Ret Reb (7) The gradient of partial pressure grad P for a moist vapor in the postulated linear transport equation (3) is, as a rule, uncontrolled by the experimentalist parameter. Its formal definition for the steady process is based on the following reasonable assumption. One supposes the linear decrease of partial pressure on the path from the source moisture 1 (usually it is the saturated water existing in an admitted equilibrium with its saturated vapor at Ps (T ) ) through the tested fabric’s layer 1-2 with the thickness δ up to the presumably zero pressure P2 0 of an experimental vessel 2. Of course, the atmospheric pressure Patm 101kPa can be added to the both limiting values of the finite difference P Ps (T ) to make the realistic estimate of its magnitude in terms of the measurable pressure. Nevertheless the fundamental question remains: Is the saturated vapor pressure of water divided by an arbitrary total width of layers the actual thermodynamic force to provide the respective flow of a moisture j through the fabric in accordance with (1)? grad P P2 P1 Ps (T ) (5) L1 L2 where L1 is the measurable width of an air layer between the source of moisture and the internal surface 1 of a tested fabric while L2 for an outside air layer is a bad-defined parameter for any experiment or model. To elucidate the problem of an effective local kinematic velocity u (of its definition, first of all) in total, let us remind the main specific features of the standard permeability test. The most relevant one is, of course, the sweating guarded hot plate test specified in ISO 11092 . Its detailed description and results can be found elsewhere  and only the main working equation should be discussed now: Ret A Ps Pa H He m2 Pa W s m (6) where Reb is the evaporative resistance of the porous plate and boundary air layer L1. Just the reciprocal Ref -value provides in this arrangement, which drives heat and moisture transfer upward only along the specimen thickness direction, the appropriate value of u : u 1 Ref m s (8) Huang and Qian  have proposed the rather informative methodology to test, simultaneously, Ref - and WVT-values. It is based on the change of relative humidity, i.e. measurable water vapor concentration gradient considered as thermodynamic force for the moist vapor transfer. This HQ-approach is closer to the main concept of Fick’s law for selfdiffusion formulated originally for the dilute gases: jm D grad D 2 1 (9) where the approximate equality demonstrates its possible extension on the thin PS and D m2 s is the molecular-based diffusion coefficient: D u 3. (10) It is expressed here in terms of the average free path of particles (between two neighboring collisions) and u - the average velocity of molecular motion in any transport process realized in the pure dilute gas : 1 2 2n kBT 2 2P (a) u 8kBT m0 (b) (11) where n m0 N V - concentration of particles with the molecular mass m0 M NA [kg] and σ – effective diameter of particles. It is worthwhile to note that the authors of HQ-method  themselves did not consider its generalized molecular interpretation for PS based on (9-11). However, the remarkable correlation between their measurements obtained International Journal of Composite Materials 2014, 4(5A): 35-40 37 by two different (HQ and ISO 11092) methods confirms completely the possibility of such analysis. It is found by us in terms of the reciprocal resistances (8), i.e. the effective diffusion velocities and is shown in Table 1. Table 1. Correlation of Intrinsic Diffusion Reciprocal (8) Resistances for Six Fabrics Measured in  Sample code u (HQ) (m/s) u (ISO) (m/s) 1 3.135·10-2 3.846·10-1 2 1.314·10-2 2.564·10-1 3 3.191·10-3 7.463·10-2 4 2.630·10-3 6.623·10-2 5 3.199·10-4 3.630·10-3 6 2.537·10-4 2.748·10-3 u mean of ( (HQ)/u(ISO)) = 0.066 for 6 fabrics u u 0.082 0.051 0.043 0.040 0.088 0.092 While the absolute magnitudes are different in two orders for the both columns, their ratio is the oscillating value near its mean level 0.066. The Pearson correlation test was performed also to demonstrate the quite high level of correlation (0.999) between two discussed methods. It is interesting to remind here the works of Chyjkova et al  dated by 1980 and published only in Russian journal KOP. These authors studied the so-called capillar model of porosity in terms of Fick’s law (3). The following molecular-based equations  have been used for the small ( d ) and large ( d ) mean diameters of capillars, i.e. model of pores: jm 4 3 M dg P2 P1 2 RT (12) jm D M RT g P2 P1 (13) where D is the diffusion coefficient (10) for a mixture, M [kg/mol] – the molar mass of air ( Ma 29 103kg / mol ) and water ( MW 18103kg / mol ), R 8.31J mol K - the universal gas constant and g – porosity determined by the common area of the surface pores AP , i.e. g AP A NP d 2 / 4A . Unfortunately, the respective molecular approximations of generalized diffusion coefficient D*[s] : D* d 1 3 M NPd 3 2RT A (14) D* d DM 4RT NP d 2 A (15) are rather uncertain ones because the enormous number NP and the very small d either should be a priori given or should be measured by the quite formidable experiment . Besides, Chyjkova et al  have not measured the grad P-value in (12,13) but have predicted it on the base of some unreliable, to our mind, estimates for D* in (14,15) and calculated jm -values. There are several rather reliable testing equipments (see, for example UTX-3360) for the so-called air permeability (AP) which is simply the velocity of a dry air u flowing through the thickness of PS: u V At m VM t V VM u (16) where V is the measurable volume change and t – time during which ΔV occurred, VM A - the measurable volume of a porous material (M). At first sight, one obtains from (16) the alternative to (4) variant for determination of a kinematic velocity if the condition m V const is fulfilled: u VM u V VM u m VM j m (17) However, just this condition for the volume density [kg / m3] of an air stream becomes questionable when the certain mass of a moist vapor remains in PS due to the sorption phenomenon. One should take into account the discontinuity of a relative humidity and the respective density “jump” between the ingoing 1 and outgoing 2 flows of a moving through fabric vapor. Another serious problem of the relevant experimental WVT-methodologies is itself their interpretation. One considers the WVT-quantity as the fabric’s characteristic parameter. Evidently, that this concept is an elusive one because the density of flow jm (1) is a typical dynamic parameter of the linear non-equilibrium thermodynamics. In the formalism of a continuous media, it is simply a reply of system to the action of a thermodynamic force, which is the appropriate gradient. In other words, only the effective coefficient D* can be the fabric’s parameter. The non-equilibrium thermodynamics states that the actual diffusion coefficient D[m2 / s] from (9) plays also the role of kinematic velocity and/or temperature-conductivity in the Newton’s and Fourier’s laws, respectively ( [kg / m s] Pa s; [J / (ms K) W / m K ) at least, for a diluted gas: D (a) Cv D (b) (18) where Cv [J / (kg K)] is the specific isochoric heat capacity. We assumed now that the discussed here well-established 38 Anna V. Shchutska: Some Fundamental Problems of Heat-Mass-Transfer through the Thin Materials formalism for the small densities of bulk gases can be extended and applied to the dense fluid (most vapor) states and, even, to the gas-liquid-transition states if the following conditions are fulfilled: 1) the density profile ρ(x) along the chosen direction of a mass-transfer process should be given to determine the stationary (or, about the equilibrium for a phase-transition state) grad ρ in (9); 2) the density of a mass-flow jm along the same direction should be measured by experiment to evaluate then the diffusion coefficient D; 3) the dynamical viscosity η in (18a) as well as the heat transfer coefficient in (18b) have to be calculated from the measurable value of the heat capacity Cv and the above estimates of D- and ρ(x)-values; 4) the effective modified quantity ρ from (18) determined for the special thin δ-layer along the x-direction becomes in the proposed here methodology the main parameter if the divergence of its profile ρ(x) is locally possible just in this layer; 5) accordingly to the statement 4) the thin layer of any fabric with the surface A l b located normally to the x-direction of a mass-heat transfer is a reason to change sharply the local value of density ρ taking into account the porous (fractal) structure of its distribution. 3. Generalized Diffusion Coefficient of Fabrics To illustrate the usefulness of the proposed in this work approach (Section 2) the following (see also Table 1) set of six fabrics (Table 2) studied experimentally  by six different test methods (Table 3) have been chosen below. Huang and Qian represented graphically the comparison of other five methods 1)-5) with that 6) developed in . There were: 1) the upright cup method used according to ASTM E96, Procedure B ; 2) the inverted cup method used according to ASTM E96, Procedure BW ; 3) the desiccant inverted cup method used according to ISO standard ; 4) the dynamic moisture permeation cell (DMPC) method used according to ASTM F2298 ; 5) the sweating guarded hot plate method used according to ISO 11092 . Table 2. Specification of the Fabrics Sample code and description PTFE laminated to a 1 nylon tricot Cotton/polyester 2 knitted fabric Polyester fabric 3 laminated with PU film Densely woven 4 polyester fabric 5 Polyester polar fleece laminated with TPU film 6 Nylon rip stop weave laminated with TPU film Surface density kg/m2 94.5·10-3 237.8·10-3 148.3·10-3 96.9·10-3 275.8·10-3 125.8·10-3 M , δ, mm kg/m3 410.9 0.23 264.2 0.90 449.4 0.33 969.0 0.10 195.6 1.41 405.8 0.31 The results of latter one 5) has been analyzed in Section 2 (see Table 1). We have used below in Table 4 the same technique to test the reciprocal by ratio of WVT-values 1)-4) derived by the WVT-standard obtained by HQ-method. The only result of such comparison is the rather rough (in accuracy) confirmation of the linear Fick’s law. Besides, the such approximate graphic analysis contains an unphysical feature because the average slopes from Table 4 as well as in Figure 1  do not provide the correct limiting passing through the point at the coordinate origin ( jm 0 when grad P = 0). In opposite to the above-discussed comparison we propose the next steps to estimate the realistic D* -value: 1) It follows from (1) and (3) that an WVT-ratio (see Table 4) represents also the ratio of the respective gradients: ji j j grad Pi grad Pj Pi Pj (19) if the same δ-specimen of a fabric has been measured by the different methods; 2) as a result, one must estimate only one method-dependent gradient of pressure Pj to predict the others (effective) Pi -values for the different methods from the WVT-ratio: Pi ji j j Pj (20) Table 3. Comparison of Experimental Setup for Six Test Methods Descriptors relative humidity air velocity test temperature air layers (al) pressure gradient across PS ASTM E96 B 50% 2.8 m/s 23ºC al on either side of PS not controlled ASTM E96 BW 50% 2.8 m/s 23ºC ISO 15496 23% not controlled 23ºC external al no al not controlled not controlled ASTM F2298 5% not controlled 20ºC small al on either side of PS 0 ISO 11092 40% 1 m/s 35ºC boundary al subtracted out not controlled HQ 0 0.4 m/s 20ºC small al subtracted out not controlled International Journal of Composite Materials 2014, 4(5A): 35-40 39 3) then, the generalized diffusion coefficient of fabric can be estimated accordingly to (3): Di* ji i Pi (21) 4) we have used for the simplicity the standard change of pressure for the basic HQ-method: P 1105 Pa to calculate the Di* -values represented in Table 5. Nevertheless, Fig.1 demonstrates that the experimental WVT-ratio changes its values mainly monotonically but nonlinearly from one method to another one. The supposed here generalized reason of such behavior is the respective monotonic but non-regular change of the respective thermodynamic force i.e. of the gradient of partial pressure. 4. Conclusions Table 4. Ratio of the WVT-value from HQ-method 6) to those following from 1)-4) methods Sample code 1 2 3 4 5 6 average slopes WVT-ratio HQ/E96-B 7.88 4.85 4.21 3.41 1.55 2.43 WVT-ratio HQ/E96-BW 1.49 1.47 1.64 2.11 0.65 0.67 WVT-ratio HQ/F2298 1.45 1.16 1.30 1.03 0.58 0.91 WVT-ratio HQ/ISO 15496 0.38 0.50 0.74 0.75 0.44 0.45 4.06 1.34 1.07 0.54 D * Table 5. Generalized Diffusion Coefficient for Six Fabrics 1-6 from Table 2 This work is an attempt to classify the set of standard methods directed usually to estimate WVT. The purpose is to modify approach, in general, and to study the thin PS-property D* itself instead of the dynamic WVT-characteristic which is mainly determined by the appropriate thermodynamic force in the variants (1,3) of the generalized linear Fick’s law, formulated for PS. Obviously that the remaining problem is the measurement of pressure drop across the sample. An absence of such experimental parameter cannot be compensated, from our viewpoint, by its theoretical estimates. This arbitrary estimate may cause a large shift in diffusion resistance even if the WVT-value is reliably measured. A combination of AP- and WVT-measurements of the same sample at the same conditions may be essential to solve independently this fundamental problem. Specimen 1 2 3 4 5 6 D* 1015 s 2.84 8.34 1.36 34.8 74.4 13.0 1 2 3 4 6 5 3,4 2 1 5,6 Figure 1. Non-linearity of WVT-ratio for the chain of regular changes in the sequence of test methods1) REFERENCES  J. Huang and X. Qian “Comparison of Test Methods for Measuring Water Vapor Permeability of Fabrics” Textile Research Journal, vol. 78(4), pp.342-352, 2008.  ISO 11092 1993 “Textiles-Physiological Effects – Measurement of Thermal and Water Vapor Resistance under Steady-state Conditions (Sweating Guarded Hot Plate Test)” International Organization for Standardization, Geneva, Switzerland (1993).  J. Huang and X. Qian “A new test method for measuring water vapor permeability of fabrics” Meas.Sci.Technol., vol. 18(9), pp.3043-3047, 2007.  A.K. Kikoin and I.K. Kikoin, Molecular Physics, M., Nature, 1976 (in Russian).  L.A. Grivtzova, I.V. Gnevysheva, L.F.Chyjkova and B.V.Saytin “About water vapor mechanism of permeability through hydrophobic porous materials”, Kogevennoobyvnaya promushlennost (KOP), vol. 4, pp.48-49, 1980; L.F. Chyjkova, I.V.Gnevysheva, L.A. Grivtzova, N.L. Tankova, T.K. Shaposhnikova, B.V. Saytin and V.G. Kalashnikov “Interconnection between sorption of moisture and permeability materials”, KOP, vol. 9, pp.47-50, 1980; L.A.Grivtzova, I.V.Gnevysheva, N.L. Tankova, L.F. Chyjkova and B.V. Saytin “Influence of porosity on permeability of polymer materials”, KOP, vol. 10, pp.44-46, 1980.  ASTM E96, 1995. Standard Test Methods for Water Vapor 40 Anna V. Shchutska: Some Fundamental Problems of Heat-Mass-Transfer through the Thin Materials Transmission of Materials, in “Annual Book of ASTM Standards 4.06” American Society for Testing and Materials. West Conshohochen, PA, 1995.  ISO 15496 2004 “Test Measurement of Water Vapor Permeability of Textiles for the purpose of Quality Control” International Organization for Standardization, Geneva, Switzerland, 2004.  ASTM F2298 2003. Standard Test Methods for Water Vapor Diffusion Resistance and Air Flow Resistance of Clothing Materials Using the Dynamic Moisture Permeation Cell in “Annual Book of ASTM Standards 11.03” American Society for Testing and Materials. West Conshohochen, PA, 2003.
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