THERMAL PROPERTIES OF FRACTAL STRUCTURE MATERIALS

Transcription

1 HERMAL PROPERIES OF FRACAL SRUCURE MAERIALS Oldřich Zmeškal, Mioslav Buchníček, Matin Nežádal Pavla Štefková, Radek Capoušek Institute of Physical and Applied Chemisty, Faculty of Chemisty, Bno Univesity of echnology, Pukyňova 118, CZ-61 Bno, Czech Republic Abstact he aticle is dealing with study of themal popeties of fibe mateials. Fo specific heat, themal diffusivity and themal conductivity detemination the tansient pulse method and step wise wee used. he evaluation was caied with the help of mathematical appaatus used fo study of popeties of factal stuctues. he esults that wee obtained ae the same as esults obtained by the classical methods. Key wods: factal stuctue, specific heat, themal diffusivity, themal conductivity, tansient pulse method, and stepwise method 1. Intoduction he aticle is dealing with the desciption of the new data evaluation method. he method comes out of genealized elations that wee designed fo the study of physical popeties of factal stuctues [1, ]. In the wok it is shown that these elations ae in a good ageement with the equations used fo the desciption of time esponses of tempeatue fo the pulse input of supplied heat [3, 4, 5]. hemal paametes (specific heat, themal diffusivity, themal conductivity) calculated by the both methods ae the same.. heoy In papes [1, ] the density of factal physical quantity ρ (), (e.g. specific yield of the heat souce q ( ) in J.m 3.s 1 fo tempeatue field) in E - dimensional Euclidean space E n (E = n) was defined D E q ( ) = ek, (1) whee is the adius of elementay quantity, K is a factal measue and D a factal dimension. Fom specific yield of the heat souce q ( ) (1) we can detemine density of heat flow ate q () (in J.m.s 1 ) and tempeatue () ( q = div q = div( λ gad ) = λ ), whee λ is constant themal conductivity (in J.m 1.s 1.K -1 ). Fo adial tempeatue field we can wite D E+ 1 D E+ ek q = ek, =. () D λ D( D E + ) By integating of eq. (1) ove volume V * = E of E-dimensional space we can calculate the powe of heat (in J. s 1 ) D E * E E Q () = q( ) dv = ek = q(), (3) D D V * whee dv * = d( E ) is an elementay volume of E-dimensional space.

2 If we suppose that heat pemeates though the suounding by the constant speed then the can be obseved as the size of the invaiant space-time vecto = c t, whee = x + y + z is the magnitude of the position vecto and c is the maximum of the heat pemeation speed (e.g. heat adiance in the vacuum, speed of light in the vacuum espectively). ( D E+ )/ ( D E+ )/ ek( ) = 1 ( ), (4) λ D D E + whee = c t. If the heat diffuses by the significantly lesse speed then the citical speed ( << ct, small distances o long times) the tems in paenthesis can be obseved as significant in the x ansion of onential function ( 1 x e ) and we can wite then ( D E+ )/ ek( + ) D E =. (5) λ D( D E + ) If we substitute fo the themal conductivity λ = cp ρ a = cpρ /[c( D E + )], whee c p is specific heat capacity afte constant pessue (in J.kg 1. K -1 ), ρ mass density (in kg.m 3 ) and a the coefficient of themal diffusivity of the body (in m. s -1 ), and fo the total heat tansfeed E D to the body fom the heat souce Q = ek /( cd) [π /( D E + )] we obtain ( D E)/ Q π + D E =, (6) cpρ D E + esp. Q =. (7) ( E D)/ cpρ (4π a t) 4a t hat is the elation fo factal dimension D =, 1, and topological dimension E = 3 published in [3, 4, 5]. he Figue epesents time-tempeatue dependencies (accoding eq.7) calculated fo spheical (D = ), cylindical (D = 1), plana (D = ), and cubic (D = 3) geomety of the heat souce (see Figue 1). he maximum position can be detemined by the deivation of equation (7) with the Figue 1 Heat flow geomety fo a) plane-paallel, b) time cylindical and c) spheical coodinates Euclidean space log D E = log + 4 t =.(8) at It is evident fom the Figue and fom the equation (8) that fo D = E the function meets maximum fo time t. In the othe cases the diffusivity a can be detemined fom the time when the tempeatue is maximal a = [ ( E D) tm ], (9) Figue ime dependency of the tempeatue esponse fo the Diac themal pulse (fo heat flow geomety fom figue 1) calculated by eq. (6)

3 whee f a = E D it is a coefficient that chaacteizes the defomation of the themal field [5]. If we substitute the value of diffusivity to the equation (7) we obtain the themal capacity: ( E D ) / Q E D cp = E D m (1), (1) ρ π themal conductivity of the studied factal stuctue espectively ( E D ) / Q E D λ = cpρ a = E D ( E D) mt m (1), (11) π whee m is the maximum tempeatue of the esponse fo Diac themal pulse. 3. Expeimental Fo the esponses to the pulse heat the hemophysical ansient este 1. was used. It was developed at Institute of Physics, Slovak Academy of Science [6]. he block diagam of automated measuement wokstation is pesented in Figue. 3. he measued sample, which was placed in the isothemal chambe, consisted of thee pats of cylindical shape. Between the fist and the second pat the heat souce was placed ( µm thick nickel folium in kapton, and adius R = 1, o 3 cm, (see on Figue 4) and fist of diffeentially connected themocouple (NiC-Ni). Between the second and thid pat one connection of diffeentially connected themocouple (NiC- Ni) was placed too. he second connections of both themocouples wee placed on heat exchange whee the constant tempeatue was kept with the help of themostat. he efeence tempeatue was measued by Figue 3 he block scheme of measuing appaatus platinum esistance (Pt1Ω). Figue 3 Cuent flow geomety: a) plane-paallel, B) point (fo diffeent atio of length contact espectively)

4 Heating-up of sample was povided by ectangula long cuent pulse fom the softwae diected souce Mesit Z-YE-3/x. he powe of supplied heat was computed fom the paametes of pulse (fom the voltage U and cuent I) Q = U I. (1) he changes of tempeatue between heat exchange and sample was measued by nanovoltmete Agilent HP4119A. PC caied the eiment contol via GPIB and RS3 bus and softwae equipment ceated by authos. 4. Measued samples Fibe glass wool is a lightweight, flexible, themal and acoustical insulation mateial designed to povide the ultimate noise eduction. It is fomed fom esin-bounded boosilicate glass fibes. It is wate and fie esistant, it has low density of combustion gas and low toxicity. It educes tanspot of heat and sound. Its density in non-pessed state is 5 kg.m -3, themal conductivity is.3.4 W.m -.K -1 in 1 C [6]. Samples wee ound shaped with adius R 1 = 3 cm, thei thickness was changed by pessing in the ange of h = (3 5) mm. In this aticle thee ae discussed esults of measuement by pulse and by stepwise method [7]. 5. Results he Figue 5 epesents the typical esponses of tempeatue fo the unitay step of inputted powe. he aveage powe of the heat souce was 1.17W lasting ove the whole time of measuement. (it is appoximately 1s) he uppe cuve shows the tempeatue of the heat souce; tempeatue was calculated fom the themal change of its esistance. Its tempeatue in the steady state elative to the tempeatue of the heat exchange (measued by platinum esistance) is 14.7K. he second cuve shows couse of tempeatue at the suface of kapton cling film in which is sealed the heat souce. It was measued by hemocouple 1. Between the esistance and the suface of kapton cling film thee was.65k diffeence. he thid dependency esses the couse of tempeatue on the opposite side of the sample. It was measue by the hemocouple. he diffeence of the steady state tempeatues elative to heat souce is 75.49K in this case. By the deivation of the tempeatue esponse (Figue 6) we obtain Figue 5 hemal esponse of fibe sample fo cuent pulse of 1.17W and duation 1 s Figue 6 Deivation of themal esponse of fibe sample fo cuent pulse of 1.17W and duation 1 s

5 tempeatue esponse fo the Diac themal pulse. In the left pat of the figue thee ae showed couses of tempeatue on the both sides of sample afte the switching the heat souce on. In the ight pat of the figue thee ae esponses fo switching the heat souce off. Fom these elations the diffusivity can be calculated (accoding to eq.8]. Its value is pesented in the able 1 togethe with the value obtained by pulse method [7] able I he compaison of the esults of pulse and step-wise method method h (mm) t m (s) m (K) f a [5] (E D) f c [5] λ (W.m 1.K 1 ) c (J.kg 1.K 1 ) a (m.s 1 ) pulse step Conclusion In this aticle thee ae pesented esults of measuements of themal paametes (themal conductivity, specific heat and themal diffusivity) of glass wool fibes. In the theoetical pat thee ae pesented equations fo computing paametes of themal systems in factal stuctues. Obtained equations ae compaed with equations used fo evaluation with the help of pulse and step-wise method [5]. By these equations the eimental values of these paametes wee calculated. he conguence in the theoy and the eimental shows, that suggested mathematical appaatus is suitable fo study of themal popeties of stuctues with the factal stuctue. Refeences [1] Zmeškal O., Nežádal M., Buchníček M.: Factal Cantoian Geomety, Hausdoff Dimension and the Fundamental Laws of Physics. Chaos, Solitons & Factals 3; 17: [] Zmeškal O., Nežádal M., Buchníček M.: Field and Potential of factal Cantoian stuctues and El Naschie's infinite theoy. Chaos, Solitons & Factals 4; 19: [3] Caslaw H.S., Jaege J. C.: Conduction of Heat in Solids. Claendon Pess London 1959, 496 pp. [4] Kempaský J.: Measuement of hemophysical Quantities. VEDA Batislava 1969, 87 pp. [5] Kubičá L.: Pulse Method of Measuing Basic hemophysical Paametes, VEDA Batislava and Elsevie Nedeland 199, 344 pp. [6] hemophysical ansient este Model R 1., Institute of Physic, Slovak Academy of Sciences [7] Zmeškal O., Nežádal M., Lapčík L.: hemal Conductivity of Boosilicate Glass Wool, hemophysics, Kočovce.