Thermal-Fluids I. Chapter 18 Transient heat conduction. Dr. Primal Fernando Ph: (850)

Transcription

1 hermal-fluds I Chapter 18 ransent heat conducton Dr. Prmal Fernando prmal@eng.fsu.edu Ph: (850)

2 ransent heat conducton In general, he temperature of a body vares wth tme as well as poston. In rectangular co-ordnates ths varaton s expressed as (x,y,z,t) x,y,z varatons n x,y,z drectons t varaton wth tme he studes n ths chapter s focused on Lumped system analyss ransent heat conducton n large plane walls, long cylnders and spheres wth spatal effects ransent heat conducton n sem-nfnte solds ransent heat conducton n mult-dmensonal systems 2

3 BROAD OBJECIVE: INVESIGAE HE PROBLEM OF HOW DO SPHERES COMING OU OF A OVEN COOL? 3

4 Consder An engneer, a psychologst, and a physcst were asked to make recommendatons to mprove the productvty of an under-producng dary farm Engneer: more technology Psychologst: mprove envronment Physcst 4

5 Consder a sphercal cow (t) Great engneers and physcsts are able to approprately smplfy problems to extract the physcs! 5

6 Lumped system A lumped system s one n whch the dependent varables of nterest are a functon of tme alone. In general, ths wll mean solvng a set of ordnary dfferental equatons (ODEs) A dstrbuted system s one n whch all dependent varables are functons of tme and one or more spatal varables. In ths case, we wll be solvng partal dfferental equatons (PDEs) 6

7 Lumped system Consder a small hot copper ball comng out from an oven. emperature change wth tme. emperature does not change much wth poston at any gven tme. Lumped system analyss are applcable to ths system. 7

8 Lumped system Consder a large roast n an oven. emperature dstrbuton not even. emperature does change much wth poston at any gven tme. Lumped system analyss are not applcable to ths system. 8

9 Consder a body of arbtrary shape of mass m, volume V, surface area A s, densty ρ, and specfc heat C p ntally at a unform temperature of. At tme t0, the body s placed nto a medum at temperature Heat transfer take place between body and ts envronment emperature of the body change wth the tme and the temperature of the body at a gven tme (t) Heat transfer nto the body at any gven tme (t) Q ha s [ (t)] 9

10 Heat transfer nto the body at temperature Q has ( ) Heat transfer ntothe body durng a tme perod he ncrease n the energy dt of the body durng tme dt ha s ( ) dt mc p 10

11 ha ( ) dt s m ρv mc p d ( d ) has ρvc p dt ( t) d ( ) t 0 has ρvc p dt ln( ( t) ) has ρvc p t t 0 ln ( t) has ρvc p t ( t) e has ρvc p t 11

12 12 t Vc ha p s e t ρ ) ( bt e t ) ( where s 1 unts Vc ha b p s ρ me constant

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14 Crtera for lumped system analyss ( t) e bt b has ρ Vc p unts 1 s Characterstc length L c V A s Bot number B B hl k c B Lc / k 1/ h Conducton resstance wthn the body Convecton resstance at the surface of the body 14

15 B Lc / k 1/ h Conducton resstance wthn the body Convecton resstance at the surface of the body Small B number ndcate low conducton resstance, and therefore small thermal gradent wthn the body Lumped system s exact when B 0 Generally accepted lumped system analyss when, B 0. 1 If B < 0.1, there s a ± 5% error or less n estmatng temperature throughout body as a sngle-valued functon of tme (t) 15

16 Remember 1 st Major Assumpton emperature s unform throughout sphere. - emperature gradents are small nsde sphere. - Resstance to conducton wthn sold much less than resstance to convecton across flud boundary layer. B Lc / k 1/ h Conducton resstance wthn the body Convecton resstance at the surface of the body 16

17 Remember 2 nd Major Assumpton Heat transfer coeffcent s assumed not to be a functon of. Rate of heat energy passng through sphere Q - h A s ( s - ) (W) (W/[m 2 -K o ])(m 2 )(K o ) 17

18 Problem: Steel balls 12 mm n dameter are annealed by heatng to 1150 K and then slowly coolng to 400 K n an ar envronment for whch 325 K and h20 W/m 2 K. Assumng the propertes of the steel to be k40 W/mK, ρ7800 kg/m 3, and c p 600 J/kgK, estmate the tme requred for the coolng process. 18

19 Soluton Bot number B B hl k c Characterstc length L V 4 3 π r 3 r 3 3 c ( 2 ) As 4π r B hl k c 20 ( ) 40 W 2 m K W mk ( m) < 0. 1 herefore, temperature of the steel balls reman approxmately unform: lumped system analyss applcable 19

20 20 t h L c t p c ) ( ln ρ bt e t ) ( p c p s L c h Vc ha b ρ ρ t b 1 t ) ( ln ( ) mn. ln ) ( ln s K m W kgk J m m kg t h L c t p c ρ

21 21 B number provde-measure of temperature drop n sold relatve to temperature dfference between the surface and the flud the body Convecton resstance at the surface of Conducton resstance wthn the body h 1 k L B c / / Q Q B 2 s 2 s 1 s 2 s 2 s 1 s,,,,,, Steady state system

22 ransent heat conducton n large plane walls, long cylnders and spheres wth spatal effects In ths secton varaton of temperature wth tme and poston n one dmensonal problems such as those assocated wth large plane wall, long cylnder and sphere. A dstrbuted system s one n whch all dependent varables are functons of tme and one or more spatal varables. In ths case, we wll be solvng partal dfferental equatons (PDEs) 22

23 ransent heat conducton n large plane walls, long cylnders and spheres wth spatal effects < at t 0 23

24 large plane walls wth spatal effects ransent temperature dstrbuton (x,t) n a wall results n a partal dfferental equaton, whch can be solved usng advanced mathematcal technques. he soluton however, normally nvolves nfnte seres, whch are nconvenent and tmeconsumng to evaluate. herefore, there s a clear motvaton to present the soluton n tabular or graphcal form. Soluton nvolves so many parameters such as x, L, t, k, α, h, and. In order to reduce the number of parameters, t s defned dmensonless quanttes 24

25 Dmensonless parameters Dmensonless temperature θ ( x, t) Dmensonless dstance from the center ( x, t) X Dmensonless heat transfer coeffcent Dmensonless tme α t τ 2 L x L B r o for cylnder and sphere (not V/A) hl k (Fourer number) (Bot number) Nondmensonalzaton enables us to present temperature data n terms of X, B and τ he above defned dmensonless quanttes can be used for cylnder or sphere by replacng the varable x by r and L by r o. 25

26 One dmensonal transent heat conducton problem: For above geometres, solutons nvolve n fnte seres, whch are dffcult to deal wth. Solutons usng one dmensonal approxmaton Plane wall: Cylnder: Sphere: ( x, t) 2 λ τ θ ( x, t) wall A1e cos( λ1x / L), τ > ( r, t) 2 λ τ θ ( r, t) cyl A1e J0 ( λ1r / r0 ), τ > ( r, t) 2 λ τ sn( λ1r / r0 ) θ ( r, t) sph A1e, τ > λ r / r A 1 and λ 1 are functons of B and ther values are lsted n able 18-1 J 0 s the Zeroth order Bessel functon and values are lsted n able

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28 emperature of the center of the plane wall, cylnder and sphere Plane wall: Cylnder: Sphere: ( 0, t) θ ( 0, t) wall A1e 2 1 λ τ ( 0, t) θ ( 0, t) cyl A1e 2 1 λ τ ( 0, t) θ ( 0, t) sph A1e 2 1 λ τ Once B number s known, these relatons can be use to determne the temperature anywhere n the medum (nterpolatons may be requred to determne ntermedate values). 28

29 Hesler charts M.P. Hesler 1947 here are 3 charts assocated wth each geometry Chart 1: Determne the temperature of the of the center of the geometry ( 0 ) at a gven tme t. Chart 2: Determne the temperature of another locaton () n terms of ( 0 ) center temperature. Chart 3: Determne the total amount of heat transfer up to the tme. Note: these plots are vald to τ>0.2 29

30 Hesler charts - Large plane walls (18-13) (chart 1) Fourer number 30

31 Hesler charts - Large plane walls (18-13) (chart 2) 31

32 Hesler charts - Large plane walls (18-13) (chart 3) Note: Q max mc p ( - ) otal mount of heat transfer durng whole perod Q s amount of heat transfer at fnte tme perod t 32

33 Hesler charts Long cylnder (18-14) Hesler charts - Sphere (18-15) 33

34 ransent heat conducton n sem-nfnte solds Sem-nfnte sold s an dealzed body that has an sngle plane surface and extends to nfnty n all other drectons he earth A thck wall 34

35 ransent heat conducton n sem-nfnte solds- Graphcal representaton Nondmensonalze d temperature 35

36 ransent heat conducton n multdmensonal systems he presented charts can be used to determne the temperature dstrbuton and heat transfer n one dmensonal heat conducton problems assocated wth, large plane wall, a long cylnder, a sphere and a sem nfnte medum. Usng a superposton approach call product soluton, these charts can also be used to construct solutons for two dmensonal transent heat conducton problems encountered n geometres such as short cylnder, a long rectangular bar, or sem-nfnte cylnder or plate and even three dmensonal problems assocated wth geometres such as a rectangular prsm or sem-nfnte rectangular bar provded that: all surfaces of the sold are subjected to convecton to the same flud at a wth heat transfer coeffcent h and no heat generaton n the body. 36

37 37 Product soluton - short cylnder t r x t x t r cylnder nte wall plane cylnder short ), ( ), ( ),, ( nf Short cylnder

38 38 Product soluton rectangular profle t y x t x t r wall plane wall plane bar gular rec ), ( ), ( ),, ( tan

39 Look the examples n the book, Frday.. work on problems, wll gve the home work, probably a quz 39