LECTURER: DR. MAZLAN ABDUL WAHID HEAT TRANSFER

Transcription

1 SM 4463 LU: D. MZLN BDUL WID FULY OF MNIL NGINING UNIVSII KNOLOGI MLYSI SKUDI, JOO, MLYSI Mazlan 006 NSF D MZLN hapte Fundamental oncepts of onduction ssoc. of. D. Mazlan bdul Wahid UM Faculty of Mechanical ngineeing Univesiti enologi Malaysia FULY OF MNIL NGINING UNIVSII KNOLOGI MLYSI SKUDI, JOO, MLYSI Mazlan 006 NSF D MZLN 1

2 onduction eat ansfe In this chapte we will lean: he definition of impotant tanspot popeties and what govens themal conductivity in solids, liquids and gases he geneal fomulation of Fouie s law, applicable to any geomety and multiple dimensions ow to obtain tempeatue distibutions by using the heat diffusion equation. ow to apply bounday and initial conditions FULY OF MNIL NGINING UNIVSII KNOLOGI MLYSI SKUDI, JOO, MLYSI Mazlan 006 NSF D MZLN onduction is simply: ansfe of enegy fom moe enegetic to less enegetic paticles of a substance due to inteactions between paticles. onduction efes to the tanspot of enegy in a medium (solid, liquid o gas) due to a tempeatue gadient. he physical mechanism is andom atomic o molecula activity Govened by Fouie s law. Fom empiical obsevations (expeiments) FULY OF MNIL NGINING UNIVSII KNOLOGI MLYSI SKUDI, JOO, MLYSI Mazlan 006 NSF D MZLN

3 Fouie s Law: Fom empiical obsevations (expeiments) q cond q cond Mazlan 006 α L = L q: heat tansfe ate : coss-sectional aea L: length : themal conductivity : tempeatue diffeence acoss conducto FULY OF MNIL NGINING UNIVSII KNOLOGI MLYSI SKUDI, JOO, MLYSI NSF D MZLN hemal opeties of Matte ecall fom hapte 1, equation fo heat conduction: L " 1 q x = = L he popotionality constant is a tanspot popety, nown as themal conductivity (units W/m.K) Usually assumed to be isotopic (independent of the diection of tansfe): x = y = z = Is themal conductivity diffeent between gases, liquids and solids? hemal onductivity () povides an indication of the ate at which enegy is tansfeed by the diffusion pocess D MZLN FULY OF MNIL NGINING UNIVSII KNOLOGI MLYSI SKUDI, JOO, MLYSI Mazlan 006 NSF 3

4 hemal onductivity - he themal conductivity of a mateial is a measue of the ability of the mateial to conduct heat. igh value fo themal conductivity Low value FULY OF MNIL NGINING UNIVSII KNOLOGI MLYSI SKUDI, JOO, MLYSI Mazlan 006 good heat conducto poo heat conducto o insulato. NSF D MZLN hemal onductivities of Mateials he themal conductivities of gases such as ai vay by a facto of 10 4 fom those of pue metals such as coppe. ue cystals and metals have the highest themal conductivities, and gases and insulating mateials the lowest. FULY OF MNIL NGINING UNIVSII KNOLOGI MLYSI SKUDI, JOO, MLYSI Mazlan 006 NSF D MZLN 4

5 hemal onductivities and empeatue he themal conductivities of mateials vay with tempeatue. he tempeatue dependence of themal conductivity causes consideable complexity in conduction analysis. mateial is nomally assumed to be isotopic. FULY OF MNIL NGINING UNIVSII KNOLOGI MLYSI SKUDI, JOO, MLYSI Mazlan 006 NSF D MZLN hemal onductivity: Fluids hysical mechanisms contolling themal conductivity not well undestood in the liquid state Geneally deceases with inceasing tempeatue (exceptions glyceine and wate) deceases with inceasing molecula weight. Values tabulated as function of tempeatue. FULY OF MNIL NGINING UNIVSII KNOLOGI MLYSI SKUDI, JOO, MLYSI Mazlan 006 NSF D MZLN 5

6 hemal onductivity: Insulatos ow can we design a solid mateial with low themal conductivity? an dispese solid mateial thoughout an ai space fibe, powde and flae type insulations ellula insulation Foamed systems Seveal modes of heat tansfe involved (conduction, convection, adiation) ffective themal conductivity: depends on the themal conductivity and adiative popeties of solid mateial, volumetic faction of the ai space, stuctue/mophology (open vs. closed poes, poe volume, poe size etc.) Bul density (solid mass/total volume) depends stongly on the manne in which the solid mateial is inteconnected. FULY OF MNIL NGINING UNIVSII KNOLOGI MLYSI SKUDI, JOO, MLYSI Mazlan 006 NSF D MZLN FULY OF MNIL NGINING UNIVSII KNOLOGI MLYSI SKUDI, JOO, MLYSI Mazlan 006 NSF D MZLN 6

7 FULY OF MNIL NGINING UNIVSII KNOLOGI MLYSI SKUDI, JOO, MLYSI Mazlan 006 NSF D MZLN FULY OF MNIL NGINING UNIVSII KNOLOGI MLYSI SKUDI, JOO, MLYSI Mazlan 006 NSF D MZLN 7

8 FULY OF MNIL NGINING UNIVSII KNOLOGI MLYSI SKUDI, JOO, MLYSI Mazlan 006 NSF D MZLN hemal Diffusivity hemophysical popeties of matte: anspot popeties: (themal conductivity/heat tansfe), ν (inematic viscosity/momentum tansfe), D (diffusion coefficient/mass tansfe) hemodynamic popeties, elating to equilibium state of a system, such as density,ρ and specific heat c p. the volumetic heat capacity ρ c p (J/m 3.K) measues the ability of a mateial to stoe themal enegy. hemal diffusivity α is the atio of the themal conductivity to the heat capacity: FULY OF MNIL NGINING UNIVSII KNOLOGI MLYSI SKUDI, JOO, MLYSI Mazlan 006 α = ρc p NSF D MZLN 8

9 hemal Diffusivity eat conducted α = = eat stoed ρc (m s) he themal diffusivity epesents how fast heat diffuses though a mateial. ppeas in the tansient heat conduction analysis. mateial that has a high themal conductivity o a low heat capacity will have a lage themal diffusivity. he lage the themal diffusivity, the faste the popagation of heat into the medium. FULY OF MNIL NGINING UNIVSII KNOLOGI MLYSI SKUDI, JOO, MLYSI Mazlan 006 NSF p D MZLN he onduction ate quation ecall fom hapte 1: eat ate in the d Q x = dx q d q x = = dx x-diection q x eat flux in the x-diection 1 (high) q x Mazlan 006 x 1 x x FULY OF MNIL NGINING UNIVSII KNOLOGI MLYSI SKUDI, JOO, MLYSI (low) We assumed that vaies only in the x-diection, =(x) Diection of heat flux is nomal to coss sectional aea, whee is isothemal suface (plane nomal to x-diection) NSF D MZLN 9

10 he onduction ate quation In eality we must account fo heat tansfe in thee dimensions empeatue is a scala field (x,y,z) eat flux is a vecto quantity. In atesian coodinates: fo isotopic medium FULY OF MNIL NGINING UNIVSII KNOLOGI MLYSI SKUDI, JOO, MLYSI Mazlan 006 q = iq + jq + q x x y z q x =, q y =, qz = x y z q = i + j + = x y z Whee thee dimensional del opeato in catesian coodinates: = i + j + x y z NSF D MZLN Summay: Fouie s Law It is phenomenological, ie. based on expeimental evidence Is a vecto expession indicating that the heat flux is nomal to an isothem, in the diection of deceasing tempeatue pplies to all states of matte Defines the themal conductivity, ie. qx ( / x ) FULY OF MNIL NGINING UNIVSII KNOLOGI MLYSI SKUDI, JOO, MLYSI Mazlan 006 NSF D MZLN 10

11 he eat Diffusion quation Objective to detemine the tempeatue field, ie. tempeatue distibution within the medium. Based on nowledge of tempeatue distibution we can compute the conduction heat flux. eminde fom fluid mechanics: Diffeential contol volume. lement of volume: dx dy dz FULY OF MNIL NGINING UNIVSII KNOLOGI MLYSI SKUDI, JOO, MLYSI Mazlan 006 V (x,y,z) NSF We will apply the enegy consevation equation to the diffeential contol volume D MZLN negy onsevation quation q x q y Mazlan 006 z eat Diffusion quation y x FULY OF MNIL NGINING UNIVSII KNOLOGI MLYSI SKUDI, JOO, MLYSI q z + dz q z d dt & st in + & g & = out = q y + dy & out q x + dx & in = q x + q y + q z NSF & whee fom Fouie s law q x = x = ( dydz) x x q y = y = ( dxdz) y y q z = z = ( dxdy) zz zz st = q x+ dx + q y+ dy + q z+ dz D MZLN 11

12 eat Diffusion quation g = q V hemal enegy geneation due to an enegy souce: Manifestation of enegy convesion pocess (between themal enegy and chemical/electical/nuclea = q( dx dy dz) enegy) ositive (souce) if themal enegy is geneated Negative (sin) if themal enegy is consumed q is the ate at which enegy is geneated pe unit volume of the medium (W/m 3 ) negy stoage tem epesents the ate of change of themal enegy stoed in the matte in the absence of phase change. ρ c p FULY OF MNIL NGINING UNIVSII KNOLOGI MLYSI SKUDI, JOO, MLYSI Mazlan 006 / t & is the time ate of change of the sensible (themal) enegy of the medium pe unit volume (W/m 3 ) NSF st = ρc p ( dx dy dz) t t D MZLN eat Diffusion quation Substituting into q. (1.11c): x x y y y z Net conduction of heat into the V eat quation q = ρ c ate of enegy geneation pe unit volume p t time ate of change of themal enegy pe unit volume t any point in the medium the ate of enegy tansfe by conduction into a unit volume plus the volumetic ate of themal enegy geneation must equal the ate of change of themal enegy stoed within the volume D MZLN FULY OF MNIL NGINING UNIVSII KNOLOGI MLYSI SKUDI, JOO, MLYSI Mazlan 006 NSF 1

13 eat Diffusion quation- Othe foms If =constant q = x y z α t Fo steady state conditions x x Mazlan 006 FULY OF MNIL NGINING UNIVSII KNOLOGI MLYSI SKUDI, JOO, MLYSI y y α = ρ y z NSF is the themal diffusivity c p q = 0 Fo steady state conditions, one-dimensional tansfe in x-diection and no enegy geneation d d = 0 o x = 0 eat flux is constant in the diection of tansfe dx dx D MZLN dx dq q 1 + = α t 1 = α t q + eat quation- Othe foms = 0 Fouie Biot quation eat Diffusion quation oison quation FULY OF MNIL NGINING UNIVSII KNOLOGI MLYSI SKUDI, JOO, MLYSI Mazlan 006 = 0 NSF Laplace quation D MZLN 13

14 14 z y x + + = In ectangula coodinate: In diffeent coodinate z y x 1 1 z = φ In cylindical coodinate: Mazlan 006 FULY OF MNIL NGINING UNIVSII KNOLOGI MLYSI SKUDI, JOO, MLYSI NSF D MZLN + = θ θ θ θ φ θ sin sin 1 sin 1 1 In spheical coodinate: eat Diffusion quation In cylindical coodinates: c q = ρ 1 1 t c q z z p = ρ φ φ In spheical coodinates: t c q p = ρ θ θ θ θ φ φ θ sin sin 1 sin 1 1 Mazlan 006 FULY OF MNIL NGINING UNIVSII KNOLOGI MLYSI SKUDI, JOO, MLYSI NSF D MZLN t θ θ θ φ φ θ sin sin

15 ylindical oodinates with e gen = q 1 1 e & gen = ρc φ φ z z t (-43) FULY OF MNIL NGINING UNIVSII KNOLOGI MLYSI SKUDI, JOO, MLYSI Mazlan 006 NSF D MZLN g = q FULY OF MNIL NGINING UNIVSII KNOLOGI MLYSI SKUDI, JOO, MLYSI Mazlan 006 NSF D MZLN 15

16 Spheical oodinates with e gen = q sin θ e & gen = ρc sin θ φ φ sinθ θ θ t FULY OF MNIL NGINING UNIVSII KNOLOGI MLYSI SKUDI, JOO, MLYSI Mazlan 006 NSF D MZLN = q g FULY OF MNIL NGINING UNIVSII KNOLOGI MLYSI SKUDI, JOO, MLYSI Mazlan 006 NSF D MZLN 16

17 FULY OF MNIL NGINING UNIVSII KNOLOGI MLYSI SKUDI, JOO, MLYSI Mazlan 006 NSF D MZLN Bounday and Initial onditions eat equation is a diffeential equation: Second ode in spatial coodinates: Need bounday conditions Fist ode in time: Need 1 initial condition Bounday onditions 1) FIS KIND (DIIL ONDIION): escibed tempeatue xample: a suface is in contact with a melting solid o a boiling liquid s (x,t) FULY OF MNIL NGINING UNIVSII KNOLOGI MLYSI SKUDI, JOO, MLYSI Mazlan 006 NSF x D MZLN 17

18 Specified empeatue Bounday ondition Fo one-dimensional heat tansfe though a plane wall of thicness L, fo example, the specified tempeatue bounday conditions can be expessed as (0, t) = 1 (L, t) = he specified tempeatues can be constant, which is the case fo steady heat conduction, o may vay with time. FULY OF MNIL NGINING UNIVSII KNOLOGI MLYSI SKUDI, JOO, MLYSI Mazlan 006 NSF D MZLN Bounday and Initial onditions ) SOND KIND (NUMNN ONDIION): onstant heat flux at the suface xample: What happens when an electic heate is attached to a suface? What if the suface is pefectly insulated? q x x (x,t) FULY OF MNIL NGINING UNIVSII KNOLOGI MLYSI SKUDI, JOO, MLYSI Mazlan 006 NSF x (x,t) D MZLN 18

19 Specified eat Flux Bounday ondition he heat flux in the positive x- diection anywhee in the medium, including the boundaies, can be expessed by Fouie s law of heat conduction as d eat flux in q& = = dx the positive x-diection he sign of the specified heat flux is detemined by inspection: positive if the heat flux is in the positive diection of the coodinate axis, and negative if it is in the opposite diection. FULY OF MNIL NGINING UNIVSII KNOLOGI MLYSI SKUDI, JOO, MLYSI Mazlan 006 NSF D MZLN wo Special ases Insulated bounday hemal symmety (0, t) (0, t) = 0 o = 0 x x FULY OF MNIL NGINING UNIVSII KNOLOGI MLYSI SKUDI, JOO, MLYSI Mazlan 006 NSF ( L, t) x = 0 D MZLN 19

20 Bounday and Initial onditions 3) ID KIND (MIXD BOUNDY ONDIION) : When convective heat tansfe occus at the suface,h (0,t) x (x,t) FULY OF MNIL NGINING UNIVSII KNOLOGI MLYSI SKUDI, JOO, MLYSI Mazlan 006 NSF D MZLN Inteface Bounday onditions t the inteface the equiements ae: (1) two bodies in contact must have the same tempeatue at the aea of contact, () an inteface (which is a suface) cannot stoe any enegy, and thus the heat flux on the two sides of an inteface must be the same. and ( x 0, t ) B ( x 0, t ) = B x x FULY OF MNIL NGINING UNIVSII KNOLOGI MLYSI SKUDI, JOO, MLYSI Mazlan 006 (x 0, t) = B (x 0, t) NSF D MZLN 0

21 Genealized Bounday onditions In geneal a suface may involve convection, adiation, and specified heat flux simultaneously. he bounday condition in such cases is again obtained fom a suface enegy balance, expessed as FULY OF MNIL NGINING UNIVSII KNOLOGI MLYSI SKUDI, JOO, MLYSI Mazlan 006 eat tansfe to the suface in all modes = NSF eat tansfe fom the suface In all modes he quantities of majo inteest in a medium with heat geneation ae the suface tempeatue s and the maximum tempeatue max that occus in the medium in steady opeation. D MZLN Vaiable hemal onductivity, () he themal conductivity of a mateial, in geneal, vaies with tempeatue. n aveage value fo the themal conductivity is commonly used when the vaiation is mild. his is also common pactice fo othe tempeatuedependent popeties such as the density and specific heat. FULY OF MNIL NGINING UNIVSII KNOLOGI MLYSI SKUDI, JOO, MLYSI Mazlan 006 NSF D MZLN 1

22 Vaiable hemal onductivity fo One-Dimensional ases When the vaiation of themal conductivity with tempeatue () is nown, the aveage value of the themal conductivity in the tempeatue ange between 1 and can be detemined fom ( ) d 1 ave = 1 he vaiation in themal conductivity of a mateial with can often be appoximated as a linea function and expessed as ( ) = (1 + β ) 0 β the tempeatue coefficient of themal conductivity. FULY OF MNIL NGINING UNIVSII KNOLOGI MLYSI SKUDI, JOO, MLYSI Mazlan 006 NSF D MZLN Vaiable hemal onductivity Fo a plane wall the tempeatue vaies linealy duing steady one-dimensional heat conduction when the themal conductivity is constant. his is no longe the case when the themal conductivity changes with tempeatue (even linealy). FULY OF MNIL NGINING UNIVSII KNOLOGI MLYSI SKUDI, JOO, MLYSI Mazlan 006 NSF D MZLN

23 FULY OF MNIL NGINING UNIVSII KNOLOGI MLYSI SKUDI, JOO, MLYSI Mazlan 006 NSF D MZLN Steady vesus ansient eat ansfe Steady implies no change with time at any point within the medium ansient implies vaiation with time o time dependence FULY OF MNIL NGINING UNIVSII KNOLOGI MLYSI SKUDI, JOO, MLYSI Mazlan 006 NSF D MZLN 3

24 Multidimensional eat ansfe eat tansfe poblems ae also classified as being: one-dimensional, two dimensional, thee-dimensional. In the most geneal case, heat tansfe though a medium is thee-dimensional. oweve, some poblems can be classified as two- o onedimensional depending on the elative magnitudes of heat tansfe ates in diffeent diections and the level of accuacy desied. FULY OF MNIL NGINING UNIVSII KNOLOGI MLYSI SKUDI, JOO, MLYSI Mazlan 006 NSF D MZLN FULY OF MNIL NGINING UNIVSII KNOLOGI MLYSI SKUDI, JOO, MLYSI Mazlan 006 NSF D MZLN 4

25 FULY OF MNIL NGINING UNIVSII KNOLOGI MLYSI SKUDI, JOO, MLYSI Mazlan 006 NSF D MZLN FULY OF MNIL NGINING UNIVSII KNOLOGI MLYSI SKUDI, JOO, MLYSI Mazlan 006 NSF D MZLN 5

26 FULY OF MNIL NGINING UNIVSII KNOLOGI MLYSI SKUDI, JOO, MLYSI Mazlan 006 NSF D MZLN FULY OF MNIL NGINING UNIVSII KNOLOGI MLYSI SKUDI, JOO, MLYSI Mazlan 006 NSF D MZLN 6

27 FULY OF MNIL NGINING UNIVSII KNOLOGI MLYSI SKUDI, JOO, MLYSI Mazlan 006 NSF D MZLN FULY OF MNIL NGINING UNIVSII KNOLOGI MLYSI SKUDI, JOO, MLYSI Mazlan 006 NSF D MZLN 7

28 FULY OF MNIL NGINING UNIVSII KNOLOGI MLYSI SKUDI, JOO, MLYSI Mazlan 006 NSF D MZLN FULY OF MNIL NGINING UNIVSII KNOLOGI MLYSI SKUDI, JOO, MLYSI Mazlan 006 NSF D MZLN 8