Effects of Zeta Potential and Fiber Diameter on Coupled Heat and Liquid Moisture Transfer in Porous Polymer Materials

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1 Journa of Fiber Bioengineering and Informatics Reguar Artice Effects of Zeta Potentia and Fiber Diameter on Couped Heat and Liquid Moisture Transfer in Porous Poymer Qing-Yong Zhu *, Jing Yang, Ming-Hui Xie Schoo of Engineering, Sun Yat-sen University, Guangzhou, Guangdong, 51075, China Abstract: This study is a theoretica investigation of the iquid water transport couped with moisture and heat transfer in porous poymer materias. The soid surface zeta potentia and the fiber diameter are investigated to revea the mechanisms of the couping effects. Based on the Poisson- Botzmann equation for eectric doube ayers and Navier-Stokes equation of iquid fows, a mathematica mode to simuate the compex couped heat and moisture transfer in porous poymer materias is reported in this paper. Keywords: Eectric doube ayer, couped heat, iquid moisture transfer, porous poymer materias. 1. Introduction Porous poymer materias are widey used in many fieds, such as artificia skins in tissue engineering. So far, the couped heat and iquid moisture transfer in porous poymer materias have aroused wide concern. Pores with various sizes, which can be considered as interconnecting micro-channes with different diameters, exist from the porous poymer materias. The fows in porous materias are generay a compex process which invoves simutaneous, couped heat and mass transfers. The heat transfer mechanisms in porous poymer materias incude conduction by the soid materia of the fibers, conduction by intervening air, radiation and convection. Meanwhie, the heat transfer process is couped with phase changes, such as moisture sorption, moisture desorption, evaporation and condensation. Liquid and moisture transfer mechanisms incude water vapor diffusion in the void spaces and iquid diffusion by capiary action. The rearrangement of the charges on the soid surface and the baancing charges in the iquid is caed the eectrica doube ayer (EDL). It has been proved that the fows in micro-scae are quite different from those in macro-scae. Generay for macro-channe fow the EDL can be negected due to the very sma thickness of the EDL compared with the characteristic size of macro-channes. However, for the micro-channe fow the thickness of the EDL is compared with the characteristic size of fow channes. It means that the EDL effects must be taken into account in micro-channe fow. A coupe of schoars, such as Li and Zhu, have made numerous research in this fied [1-11]. A satisfactory mode to simuate the coupe heat and mass transfer in porous materias has been estabished by Li and Zhu and a great number of meaningfu concusions have been made. But the resistance effect of EDL on the couped heat and iquid moisture transfer in porous poymer materias is sedom taken into account. The eectroosmosis phenomenon of the eectroytic soution which fows through micro-channes comprising a bunch of cyinders was researched [1] and it was shown that the EDL has a significant effect on the fow fied and the smaer the ratio of the channe diameter to the thickness of EDL is, the bigger the effect of EDL is. The heat effect of the eectro osmotic fow in capiaries was studied [13] and it was pointed out that the EDL has a crucia effect on the heat transfer in capiaries as we. In this paper, taking the resistance effect of the EDL into account, an improved mode is deduced from the basic equations, incuding Poisson-Botzmann equation for eectric doube ayers and Navier-Stokes equation for iquid fows. The diffusion coefficient of the eectroytic soution and a dimensioness number that aims to describe the resistance effect of the EDL are obtained through theoretica derivation. With specification of initia and boundary conditions, the distribution of the moisture content and the temperature in the porous poymer materias with different soid surface zeta potentias and fiber diameters are numericay computed. Effects of the soid surface zeta potentia and fiber diameter on the couped heat and iquid moisture transfer in porous poymer materias are discussed. The comparison with a series of experimenta measurements shows the superiority of this mode in resoving the couped heat and iquid moisture transfer in porous poymer materias. The resuts iustrate that the heat and moisture transfer process is infuenced by the soid surface zeta potentia and the fiber diameter of the porous poymer materias. * Corresponding author s emai: mcszqy@mai.sysu.edu.cn JFBI Vo. 3 No doi: /jfbi

Nomencature C a [Kg/m 3 ] water vapor concentration in the air fiing the inter-fiber void space C f [Kg/m 3 ] water vapor concentration in the fiber c v [KJ/m 3 K] voumetric heat capacity of the

2 Nomencature C a [Kg/m 3 ] water vapor concentration in the air fiing the inter-fiber void space C f [Kg/m 3 ] water vapor concentration in the fiber c v [KJ/m 3 K] voumetric heat capacity of the fabric C* [Kg/m 3 ] saturated water vapor concentration D a [m /s] diffusion coefficient of water vapor in the air of the fabric D f [m /s] diffusion coefficient of water vapor in the fibers of the fabric D [m /s] diffusion coefficient of iquid water in the fabric d [m] fiber diameter d cmax [m] maxima radius of the pores e [C] fundamenta charge h -g [m/s] mass transfer coefficient k b [-] Botzmann constant K mix [W/mK] therma conductivity of the fabric n 0 [-] buk number concentration of ions in the eectroyte soution p [N/m ] pressure in the capiary S v [m -1 ] specific voume of the fibers T [K] absoute temperature t [s] rea time from change in conditions x [-] X-coordinate Greek symbos ρ [Kg/m3] density of the fibers ρ e [C/m3] the net charge density per unit voume ρ [Kg/m3] density of the iquid water τ a [KJ/Kg] atent heat of evaporation of water τ a [KJ/Kg] heat of sorption or desorption of iquid water by fibers τ av [KJ/Kg] heat of sorption or desorption of vapor by fibers ω 1 [-] proportion of the sorption of water vapor by fibers ω [-] proportion of the sorption of iquid water by fibers σ [N/m] surface tension μ [Kg/ms] Dynamic viscosity ε [C/Vm] permittivity of the eectroyte ε [-] voume fraction of iquid phase γ [-] the average ange of the capiaries in Fabrics φ [-] contact ange χ [-] vaence of the ions of the eectroyte soution Ф [-] porosity of the fabric Ψ [C] oca eectrica potentia. Numerica mode This paper ays emphasis on the porous poymer materias that is considered as containing a countess number of yarns woven out of fibers. The yarn is made up of a custer of fibers twisted up. The capiaries are formed by these pores in yarns woven into porous poymer materias. The eectroyte soution fows from the regions of higher eectroyte soution concentration to regions of ower eectroyte soution concentration due to the fiber-iquid moecuar attraction at its surface, which is determined by the surface tension and the effective capiary pore distribution. The iquid fow is couped with heat transfer, moisture transfer and phase changes such as evaporation and condensation. The foowing assumptions are made in this mode: 1. Loca therma is assumed to be equa among a phases due to the reativey ow veocities considered and the sma dimension of the constituting fibers.. The inertia of each phase in porous poymer materias is ignored because of the reativey ow veocities. 3. Sweing of fibers whie making porous poymer materias is negected. Figure 1 Structure of porous poymer materias. When the eectroyte soution diffuses in porous poymer materias resuting from capiary action, the mass baance of the water vapor is estabished by equation (1), the mass baance of the iquid moisture is estabished by equation () and the energy baance is estabished by equation (3). The governing equations can be expressed as 17

3 ( C ε ) ( C fε f ) a a + ϖ 1 ( Caε a) = Da τ x x a * ( ( ) ) 1 h gsv' C T Ca (1) ( ρε ) ( C fε f ) + ϖ + 1 ( ρε ) = D( ε) + a( ε) τ x x x * ( ( ) ) h gsv' C T Ca ( ρε ) ( Cfε f ) ( Cfε f ) () T cv = ϖι 1 av + ϖι a * T ι ah gsv '( C ( T) Ca ) + Kmix x x (3) ε ε + ε = 1 (4) + a f Sorption and desorption of moisture by the fibers obey the Fickian aw: Cf (,,) xrt 1 Cf (,,) xrt = rdf (,) x t r r r (5) The momentum equation subject to an eectrica body force is as foows: 1 u p µ r = ρee r r r x Based on the Poisson-Botzmann Equation, the eectrostatic potentia Ψ and charge density e ρ obey the foowing equations: ψ χen χψ e sinh ε kt b 0 = χψ e ρe = ε ψ = χen 0 sinh kt b (6) (7) (8) The diffusion coefficient of the iquid water is expressed as [10]: D ( ε ) = 3 σ cosϕ sin γε λdcmax 1 ( + eθ ) 3µλφ 4ε ρ φ (9) where ρ = ρ /( εζ / r ) is dimensioness expression of e e c the charge density ρ. e Θ= ε ζ / µλd is a cmax dimensioness number to describe the resistance effect of the EDL, which is proportiona to the square of the iquid dieectric constant and the soid surface zeta potentia, and inversey proportiona to the iquid dynamic viscosity, eectric conductivity and the square of the effective pore size. According to the expression of Θ, we know that the magnitude of Θ is affected by the soid surface zeta potentia and the maximum the radius of the pores in porous poymer materias and the bigger the magnitude of Θ is, the more significant the resistance effect of EDL is. 3. Computationa and experimenta Resuts According to the improved mathematica mode estabished above, the soid surface zeta potentia and the fiber diameter are investigated to revea the mechanisms of the couping effects on the couped heat and mass transfer in porous poymer materias. The porous poymer materias are suspended in containers fied with water at T 1 =1 C. The initia conditions are: T 0 = C and RH=70%. At the time t=0 a sudden boundary condition changes at the surface x=0 to RH=100% and T 1 =1 C. The reative humidity and temperature of the air surrounding the materias at x=5cm are 70% and C respectivey that is the same as the initia conditions. In order to research the effects of the soid surface zeta potentia and fiber diameter on the simutaneous and couped heat and mass transfer in porous poymer materias, the distribution of the moisture content and the temperature in porous poymer materias with different soid surface zeta potentias and fiber diameters are obtained. Figure shows the distribution of the moisture content in fibers of the watch-point which is ocated at x=cm as shown in Figure 3. With the fows of the eectroyte soution due to capiary action, the eectroyte soution reaches at the watch-point quicky and the moisture sorption by the fibers amost starts at the same time. Comparativey, the eectroyte soution reaches the watch-point of the porous poymer materias with ζ= 3mV at 0.6min which is shorter than 0.75min cost by the materia with ζ= 33mV. According to the definition of the dimensiona number Θ, the arger the absoute vaue of zeta potentia is, the arger the magnitude of Θ is, and consequenty the smaer the coefficient of diffusion is. In other words, for the porous poymer materias of same fiber diameter, the arger soid surface zeta potentia means more obvious resistance effect of EDL and vice versa. 18

4 The iquid fow is obviousy affected by EDL. However, the effect of fiber diameters is not cear. The process of the moisture diffusion into fibers is a reativey sow process. It is found that the distribution of the moisture content in fibers is cosey reated with the fiber diameters and the EDL does not have significant infuences on the sorption of the moisture by the fibers. The moisture content of the porous poymer materias with fiber diameter d=1μm due to the sorption of moisture by fibers takes shorter time to reach the equiibrium state than that with fiber diameter d=0μm. The moisture content in fibers with fiber diameter d=1μm reaches equiibrium in amost 0.1min, whie the moisture content in fibers with fiber diameter d=0μm reaches equiibrium in more than 1.5min. The distribution of the water content in fibers is obviousy affected by fiber diameters and the resistance effect of EDL is not quite cear. materias with different soid surface zeta potentias and fiber diameters. The porous poymer materia sampe is suspended in containers fied with the eectroyte soution at T 1 =1 C. Then as a resut of the capiary action, the eectroyte soution penetrates into the porous poymer materias. The experimenta instaation is shown in Figure 3. Figure 3 Experimenta instaation. (a)ζ= 3mV (b)ζ= 33mV Figure Distribution of moisture content in fibers of the watch-point. In order to verify our mathematica mode, a series of experiments are carried out with porous poymer The watch-point is ocated at x=5cm and its temperature is measured by an infrared camera. The comparisons between computationa and experimenta resuts of the distributions of temperature at the watch-point are shown in Figure 4. At the beginning, the temperature is the same as the initia vaue. Due to capiary action, iquid reaches the watch-point and then fibers amost absorb moisture at the same time. The initia rise in temperature is due to the heat reeased during moisture sorption by the fibers. The heat absorbed during the evaporation of iquid water aso affects the temperature. The temperature graduay decines to a vaue which is same as that of the temperature of the iquid due to the couping effects of moisture sorption by fibers and evaporation of iquid water. The maximum temperature of the porous poymer materias with a arger soid surface zeta potentia comes reativey in onger time than that with a smaer soid surface zeta potentia. It is proved that the arger the soid surface zeta potentia is, the more obvious the resistance effect of EDL is. Because EDL resists the iquid fows and the eectroyte soution reaches the watch-point in a onger time and the maximum temperature is reached ater comparativey. By comparing Figure 4(a) and Figure 4(b), the smaer the fiber diameter is, the arger the maximum temperature is, because the heat of adsorption is inversey proportiona to the fiber diameters and the smaer the fiber diameter is, more heat is reeased 19

5 during the moisture sorption of fibers. The predictions of temperature changes at the watch-point are compared with experimenta measurements, a good agreement is observed between the two. (a) d=1μm on the Poisson-Botzmann equation for eectric doube ayer and Navier-Stokes equation for iquid fows, a mathematica mode to simuate the compex couped heat and moisture transfer in porous poymer materias is presented and a dimensioness number Θ is given to describe the resistance effect of the EDL, which is proportiona to the square of the iquid dieectric constant and the soid surface zeta potentia, and inversey proportiona to the iquid dynamic viscosity, eectric conductivity and the square of the effective pore size. With specification of initia and boundary conditions, the distributions of the moisture content and the temperature in the porous poymer materias with different soid surface zeta potentias and fiber diameters are numericay computed. Comparison of these resuts with a series of experimenta measurements shows the superiority of this mode in investigating the couped heat and iquid moisture transfer in porous poymer materias. The resuts iustrate that the heat and moisture transfer process is infuenced by the soid surface zeta potentia and the fiber diameter of the porous poymer materias. References: (b) d=0μm Figure 4 Comparison between computationa and experimenta resuts of temperature. 4. Concusion This study is a theoretica investigation of the iquid water transport couped with moisture and heat transfer in porous poymer materias. The soid surface zeta potentia and the fiber diameter are investigated to revea the mechanisms of the couping effects. Based [1] Li Y, Li FZ, Zhu QY. Numerica simuation of virus diffusion in facemask during breathing cyces. Int. J. Heat Mass Transfer 005; 48: [] Zhu QY, Li Y. Effects of pore size distribution and fiber diameter on the couped heat and iquid moisture transfer in porous fibrous media. Int. J. Heat Mass Transfer 003; 46: [3] Li Y, Zhu QY, Yeung KW. Infuence of thickness and porosity on the couped heat and iquid moisture transfer in porous fibrous media. Textie Res. J. 00; 46: [4] Li Y, Zhu QY. Simutaneous heat and moisture transfer with moisture sorption, condensation and capiary iquid diffusion in porous fibrous media. Textie Res. J. 003; 37: [5] Li Y, Zhu QY. A mode of couped iquid moisture and heat transfer in porous texties with consideration of gravity. Numer. Heat Transfer, A 003; 43: [6] Zhu QY, Li Y. Mathematica mode of the heat and moisture transfer in porous organic texties. Chin. J.Comput. Mech. 003; 0: [7] Li Y, Luo Z. An improved mathematica simuation of the couped diffusion of moisture and heat in woo fabric. Textie Res. J. 1999; 69:

6 [8] Li Y, Hocombe BV. A two-stage sorption mode of the couped diffusion of moisture and heat in woo fabrics. Textie Res. J. 199; 6: [9] Li Y, Pante BV, Hocombe BV. Fiber hygroscopicity and perceptions of dampness partⅡ: physica mechanisms, Textie Res. J. 1995; 65: [10] Zhu QY, Li Y. Numerica simuation of the transient heat and iquid moisture transfer through porous texties with consideration of eectric doube ayer, Int. J. Heat Mass Transfer 010; 53(7-8): [11] Zhu QY, Xie MH, Yang J, Li Y. Investigation of the 3D mode of couped heat and mass moisture transfer in hygroscopic porous fibrous media, Int. J. Heat Mass Transfer 010; 53: [1] Hsu JP, Ting CC, Lee DJ, Tseng S, Chen CJ, Su A. Residence time distribution for eectrokinetic fow through a microchanne comprising a bunde of cyinders. J. Cooid Interface Sci. 007; 307: [13] Xuan XC, Sinton D, Li DQ. Therma end effects on eectroosmotic fow in a capiary. Int. J. Heat Mass Transfer 004; 47:

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