HEAT CONDUCTION IN TECHNOLOGICAL PROCESSES

Transcription

1 Journal of Engineering Physics and Thermophysics, Vol. 84, No. 6, November, HEAT CONDUCTION IN TECHNOLOGICAL PROCESSES PROPAGATION OF HEAT IN THE SPACE AROUND A CYLINDRICAL SURFACE AS A NON-MARKOVIAN RANDOM PROCESS A. N. Morozov and A. V. Skripkin UDC 59.;536. Consideraion has been given o he propagaion of hea in he space around a cylindrically shaped body in he presence of flucuaions of he hea flux hrough is surface. I has been shown ha he corresponding random changes in he emperaure and he hea flux are described by inegral sochasic equaions and belong o he class of non-markovian processes. Saisical characerisics of he considered flucuaions, including one-dimensional and mulidimensional characerisic funcions, specral densiies, and probabiliy densiies, have been found. Keywords: hea conducion, non-markovian process, inegral sochasic equaions. Inroducion. Hea-conducion processes occurring in physical media are accompanied by flucuaions of heir parameers (emperaure and hea flux) due o he random changes in he power of he hea source, he hermal-conduciviy coefficien of he medium and he inensiy of he hermodynamic flow in i, and for oher reasons. Physical processes giving rise o such flucuaions can be exemplified by curren noise in conducors and semiconducors [] and by he flicker noise of he maerial s elecrical conducion []. One generally describes he propagaion of hea in a medium using differenial hea-conducion equaions wih corresponding iniial and boundary condiions and akes accoun of flucuaions of is emperaure and he hea fluxes in i by adding random funcions o he obained differenial equaions. Such an approach enables one o use a well-developed heory of sochasic differenial sysems [3]. Random changes in he physical quaniies used in he formulaion of such a problem represen Markovian processes. Acual physical media, however, possess herediary properies which can be characerized wih differenial operaors only approximaely. Alhough such an approximaion may be reckoned as saisfacory, i ofen becomes necessary o more generally invesigae he hea-conducion phenomenon for aking accoun of he non-markovian characer of flucuaions of emperaures and hea fluxes. I is proposed ha he corresponding inegral equaions [4, 5] whose kernels can in principle allow for he herediary properies of a non-markovian hea-conducion process be used insead of differenial sochasic equaions. Inegral ransformaions have been used in [6] for finding analyical soluions of boundary-value problems of hea conducion in an infinie domain bounded by a cylindrical surface. The mehod of soluion of nonlinear nonsaionary hea-conducion problems using nonlinear Volerra inegral equaions of he second kind has been proposed in [7]. We noe ha he indicaed approach o descripion of Brownian moion, he evaporaion of a liquid drople in he amosphere, and he moion of a paricle in a medium wih a flucuaing fricion facor makes i possible o subsanially refine resuls obained wih classical mehods [8, 9]. In his work, consideraion is given o he process of propagaion of hea in he medium around an infinie cylindrical surface he hea flux hrough which can randomly be changed wih ime. I is shown ha his phenomenon is described by he linear Volerra inegral equaion of he second kind, and he corresponding funcions are non-markovian. Saisical characerisics describing he problem of physical quaniies, among which are characerisic funcions, specral densiies, and probabiliy densiies, are found using he mehod of descripion of non-markovian random processes prescribed by he linear inegral ransformaions. N. E. Bauman Moscow Sae Universiy, 5 ya Baumanskaya Sr., Moscow, 55, Russia; Translaed from Inzhenerno-Fizicheskii Zhurnal, Vol. 84, No. 6, pp. 7, November December,. Original aricle submied July, ; revision submied May 4,. 6-5//846- Springer Science+Business Media, Inc.

2 Fig.. Hea conducion in he space around he cylindrical surface. Formulaion of he Problem. We consider an infiniely long cylindrical body of radius R manufacured from a maerial wih a high conduciviy, hea capaciy per uni volume C V, and hermal diffusiviy χ c ; he body is in an unbounded hea-conducing medium wih a hermal conduciviy κ and a hermal diffusiviy χ (Fig. ). We will assume ha he surface emperaure of he cylinder, a a given insan of ime, is everywhere he same and is a cerain funcion of ime: T(R, ) = T R (). We noe ha in acual pracice his condiion corresponds o he case of consideraion of he hea-conducion phenomenon in he medium around a cylindrical body of small radius a a disance smaller or of he order of R from is surface. Also, we will assume ha a he iniial insan of ime, he emperaure hroughou he space, including he emperaure of he cylinder maerial, is he same and equal o zero. Wihin he framework of he assumpions made, he emperaure of he medium ouside he cylinder is dependen jus on he disance r o is axis of symmery and obeys a differenial hea-conducion equaion of he form T (r, ) = χ T (r, ) r + r T (r, ), r > R, r () wih he iniial and boundary condiions T (r, ) = =, () T (r, ) = T R (). (3) r=r, > Sochasic Inegral Equaion. I can be shown ha if he hea Q(τ) is insananeously released, a a cerain insan of ime τ, on each uni lengh of he cylindrical surface in quesion, he medium s emperaure for r > R a he insan of ime will be deermined by he expression [] T (r, ) = χ Q (τ) G (r, R, τ), (4) κ where G(r, R, τ) is he influence funcion of he insananeous cylindrical hea source (Green s funcion) defined as G (r, R, τ) = 4πχ ( τ) exp r + R 4χ ( τ) I Rr χ ( τ). (5) If Q(τ)δ( τ) = πrq T (), in he case of an arbirary hea flux hrough he surface of he cylinder q T () he emperaure a a disance r from is axis of symmery will be deermined from he expression

3 T (r, ) = R κ qt (τ) τ exp r + R 4χ ( τ) I Rr dτ. (6) χ ( τ) T(r, ) We find he derivaive of, using expression (6) and he fac ha he derivaive of he Bessel funcion of zero r order is equal in argumen o he Bessel funcion of firs order: di (x) = I dx (x). We obain T (r, ) R = q T (τ) r 4χκ ( τ) exp r + R RI Rr 4χ ( τ) ri Rr χ ( τ) dτ. (7) χ ( τ) The hea flux hrough he surface of he cylindrical body q T () is deermined by he general relaion q T () = κ T (r, ) r + ξ qt (), (8) r=r where he funcion ξ qt () represens a random hea flux whose saisical properies are deermined by he characer of he flucuaion source; he mean value is ξ qt () =. A he same ime, such a flux can also be deermined according o he equaion q T () = C V R dt R (). (9) d I should be emphasized ha expression (9) holds rue for he case of he assumed presence of he high hermal conduciviy of he cylinder s maerial compared o he conduciviy of he medium, i.e., on condiion ha χ << χ c. From he obained relaion (7), wih accoun of formulas (8) and (9), we find C V R K ( τ) Z (τ) dτ = ξ qt (), () where Z () = dt R () d () and K ( τ) = δ ( τ) + R 4χ ( τ) exp I I. () χ ( τ) χ ( τ) χ ( τ) R R R Thus, he process of propagaion of hea in he space around he cylindrical surface of radius R in he presence of he random hea flux hrough i is described by he sochasic Volerra inegral equaion of he second kind (). The random process described by he inegral equaions of he form () wih nonexponenial kernels is no reduced o a sysem of differenial sochasic equaions and belongs o he class of non-markovian processes [5]. Consequenly, flucuaions of he quaniy Z() and changes in he emperaure of he cylinder surface T R () and he flux q T () possess herediary properies: heir saisical characerisics beginning wih a cerain ime τ are dependen on he behavior of hese funcions a < τ. We noe ha for large values of he radius R, expression () can be wrien using he approximae formula [] K ( τ) = δ ( τ) + χ R π ( τ), (3) 3

4 whose subsiuion ino Eq. () insead of relaion () yields an inegral equaion of he form C V R Z () + κ πχ τ Z (τ) dτ = ξ q (), (4) T his equaion has been invesigaed in [] as applied o he phenomenon of hea conducion in he half-space above a fla surface. Indeed, le us consider a cylindrical shell of radius R and such a hickness h ha h << R. Then, insead of Eq. (5), we obain he relaion q T () = C V h dt R() + ξ d qt (). Taking ino accoun he fac ha he produc C V h represens he hea capaciy of uni surface of such a shell C S, we should ake he value of C S insead of C VR in (4); his precisely leads o he inegral equaion considered in [] and describing hea conducion in a half-space bounded by a fla surface. I is noeworhy ha if he main source of hea conducion is hea ransfer, he upper limi of inegraion in Eq. () should, generally speaking, be aken o be δ, where δ is he small parameer equal o he ime of free moion of he paricles of he hea-conducing medium in order of magniude. Le us elucidae he saisical properies of he inroduced random hea flux ξ qt () hrough he surface of he cylindrical body. The funcion ξ qt () in many cases can be represened as whie noise wih an inensiy ν, which is bounded above by a cerain limiing frequency ω max ; his frequency can be evaluaed accurae o a consan using he formula ω max χ c R. (5) To evaluae ω max we will assume ha he cylinder has been manufacured from copper (for which he hermal diffusiviy is χ c = 4 m s) and having radius R = μm. We obain ha ω max 6 s. Thus, whie noise characerisic of flucuaions of he hea flux hrough he cylinder surface urns ou o be wide-band, as far as he specrum is concerned. Evaluaion of he inensiy ν can be carried ou from he formula ν κ R 3k B T. (6) The obained esimae follows from relaions (8) and (9) which, upon he replacemen of he derivaive T r by he raio T R R, lead o an equaion of he form C VR dt R () + κ d R T R() = C V R ξ κ q T () or T R () + C V R T R() = 4 C VR ξ q T (), which corresponds o he Langevin equaion for Brownian moion. The las formula makes i possible o find [3] an esimae of he inensiy of he Langevin source ξ qt () and is represened by relaion (6). A he emperaure T = 3 K, he esimae given by expression (6) leads o an inensiy of ν 4 J (m 4 s) for he copper cylindrical body of radius R = 5 nm placed in waer. Case of he Whie Noise of Flucuaions of he Hea Flux hrough he Cylindrical Surface of Small Radius. We consider hea conducion in he medium around he cylindrical shell in he case where he parameer R <<, which (as is seen from expression (5)) corresponds o he case of a very high boundary frequency of he χ specrum of flucuaions of he hea flux hrough he cylindrical surface (whie noise). Also, seing τ > δ, we obain, upon he series expansion of modified Bessel funcions and exponens and reenion of he firs erms of expansion, ha Z () R 4χ ( τ) Z (τ) dτ = C V R ξ q (). (7) T 4

5 Fig.. Funcion g (λ; ) vs. parameer λ a differen insans of ime: ) = 7 s, ) 6 s, and 3). The soluion of he Volerra inegral equaion of he second kind (7) in he general case can be wrien as [4] where he resolven F( τ) is deermined by he recurrence relaion Z () = C V R [δ ( τ) + F ( τ)] ξ qt (τ) dτ, (8) Here we have F ( τ) = F n ( τ). (9) n= F ( τ) = R 4χ ( τ), () δ F n ( τ) = F ( s) F n (s τ) ds, n >. () τ+δ The above condiion R R << makes series (9) rapidly convergen. On condiion ha << δ (cylindrical surfaces of χ χ small radius, or hin filamens), we can disregard he second erm and erms ha follow in series (9) compared o he firs erm and finally obain F ( τ) = R 4χ ( τ). () Saisical Characerisics. The iniial inegral equaion (8) for he case of he resolven F( τ) of he form () enables us o find any saisical characerisics of he process Z(), if we use he mehod developed in [4] for descripion of non-markovian random processes prescribed by he linear inegral ransformaions. Thus, for he one-dimensional g (λ; ) and mulidimensional g L (λ,..., λ L ;,..., L ) characerisic funcions of he process Z() on condiion ha he random hea flux ξ qt () represens whie noise of inensiy ν, we obain g (λ; ) = exp R νλ 4 χ C V δ 3 3, (3) 5

6 Fig. 3. Funcion K( τ, ) vs. parameer τ a differen insans of ime: ) = 7 s, ) 6 s, and 3) s. g L (λ,..., λ L ;,..., L ) = exp R ν 4 χ C V L k,l=, k<l λ l λ k ( l k ) δ + k l + l k + l k ( l k ) l k ln l δ k ( l k ). (4) The funcion g (λ; ) defined by Eq. (3) is presened graphically in Fig.. Here and in wha follows we use as examples copper cylindrical bodies of small radius placed in waer (R = 8 m, ν = 4 J (m 4 s), χ = 7 m s, and C V =.3 3 J (K m 3 )). Expressions for he characerisic funcions (3) and (4) enable us o find any saisical characerisics of he process Z(). For he mahemaical expecaion Z() and he variance D Z () of he process Z(), we find, from (3), Z () = g (λ; ) =, (5) i λ λ= D Z () = Z () = g (λ; ) λ λ= = R ν 4χ C V δ 3 3. (6) I is seen from he found formula (6) ha he seady-sae hea-conducion process ( ) is accompanied by a consan value of he variance D Z () of he rae of change in he emperaure of he cylindrical surface Z(), which is dependen on he parameers of he body and he medium and on he characerisic quaniy δ considered earlier: R ν D Z () = 4χ 3. (7) C V δ For he correlaion funcion K(, ) = Z( )Z( ), wih he wo-dimensional characerisic funcion g (λ, λ ;, ) deermined from (4), we obain K (, ) = g (λ, λ ;, ) = R ν λ λ λ =, 48 χ C V λ = ( ) δ ( ) ln δ ( ). (8) Dependence (8) yields ha a and (seady-sae hea-conducion process), he correlaion funcion K(, ) = K( ) = K(τ) akes he form 6

7 Fig. 4. One-sided specral densiy G Z (ω, ) vs. frequency of whie noise a differen insans of ime: ). and ) s. Fig. 5. Probabiliy densiy p(z, ) vs. rae of change in he emperaure of he cylinder surface a differen insans of ime: ) = 7 and ) 6 s. K (τ) = 48 R ν χ C V τ δ + τ + τ δ ln. (9) τ The funcion K( τ, ) defined by relaion (8) is presened graphically in Fig. 3. The found correlaion funcion (8) enables us o find one-sided specral densiy of he process Z(): G Z (ω, ) = K ( τ, ) cos ωτdτ. (3) Figure 4 displays a resul of numerical compuaion of he one-sided specral densiy G Z (ω, ) using formula (3) for differen imes. In he paricular case of small ω values and shor, wih accoun of he condiion δ <<, (8) and (3) yield ha G Z (ω, )symbolω = K ( τ, ) ω τ dτ = R ν 48 χ C V Le us find he expression for he one-dimensional probabiliy densiy funcion p(z, ): p (Z, ) = π g (λ; ) exp ( iλz) dλ = 4 ω. (3) δ δ πd () exp Z D (). (3) Figure 5 gives dependence (3) for differen. I is clearly seen ha he plo of he probabiliy densiy of Z() flucuaions is "smeared" along he Z axis, ending o a saionary Gaussian curve a. Conclusions. Sudy of he process of propagaion of hea even in he case of a relaively simple model (infinie cylinder of consan radius flucuaions of he hea flux hrough whose surface are independen of a poin on is surface) requires inegral sochasic equaions for saisical descripion, whereas he flucuaions hemselves of he quaniies should be considered as non-markovian random processes. The obained resuls are of imporance for descripion of random flucuaions of he emperaure of cylindrical bodies of micromeer and submicromeer radii in media wih a low hermal conduciviy. 7

8 NOTATION C S, hea capaciy of uni surface of he cylinder; C V, hea capaciy per uni volume of he cylinder maerial; I (x), modified Bessel funcion of zero order; k B, Bolzmann consan; q T (), hea flux hrough he surface of he cylindrical body; R, cylinder radius; r, disance o he axis of symmery of he cylinder;, ime variable; T R (), emperaure of he cylinder surface; x, auxiliary variable; Z, rae of change in he emperaure of he cylinder surface; κ, hermal conduciviy of he medium around he cylinder; λ, parameer of he characerisic funcion; ν, inensiy of flucuaions of he hea flux hrough he surface of he cylindrical body which have he form of whie noise; τ, inegraion variable (ime); χ, hermal diffusiviy of he medium around he cylinder; χ c, hermal diffusiviy of he cylinder maerial; ω, whie-noise frequency. Subscrip: c, cylinder. REFERENCES. M. Buckingham, Noise in Elecronic Devices and Sysems [Russian ranslaion], Mir, Moscow (986).. G. N. Bochkov and Yu. E. Kuzovlev, Innovaions in invesigaions of he /f noise, Usp. Fiz. Nauk, 4, 5 76 (983). 3. V. S. Pugachev and I. N. Sinisyn, Sochasic Differenial Sysems [in Russian], Nauka, Moscow (99). 4. A. N. Morozov, A mehod for describing non-markovian processes assigned by a linear inegral ransformaion, Vesn. MGTU, Esesv. Nauki, No. 3, (4). 5. A. N. Morozov, Irreversible Processes and Brownian Moion [in Russian], Izd. MGTU im. N. E. Baumana, Moscow (997). 6. E. M. Karashov, A class of inegral ransformaions for he generalized equaion of nonsaionary hea conducion, Inzh.-Fiz. Zh., 8, No., 3 3 (8). 7. V. D. Belik, B. A. Uryukov, G. A. Frolov, and G. V. Tkachenko, Numerical analyical mehod of solving he nonlinear nonsaionary hea conducion equaion, Inzh.-Fiz. Zh., 8, No. 6, (8). 8. A. N. Morozov and A. V. Skripkin, Applicaion of inegral ransformaions o descripion of he Brownian moion as a non-markovian random process, Izv. Vyssh. Uchebn. Zaved., Fiz., No., (9). 9. A. N. Morozov and A. V. Skripkin, Saisical descripion of an oscillaor exposed o he acion of a flucuaing fricion facor, Vesn. MGTU, Esesv. Nauki, No., 3 5 (8).. A. N. Tikhonov and A. A. Samarskii, Mahemaical Physics Equaions [in Russian], Nauka, Moscow (977).. A. F. Nikiforov and V. B. Uvarov, Special Funcions of Mahemaical Physics [in Russian], Nauka, Moscow (984).. A. N. Morozov and A. V. Skripkin, Applicaion of he Volerra second-kind equaion o descripion of viscous fricion and hermal conduciviy, Vesn. MGTU, Esesv. Nauki, No. 3, 6 7 (6). 3. Yu. L. Klimonovich, Saisical Physics [in Russian], Nauka, Moscow (98). 4. V. Volerra, Theory of Funcionals and of Inegral and Inegro-Differenial Equaions [Russian ranslaion], Nauka, Moscow (98). 8