Chapter 2 ONE DIMENSIONAL STEADY STATE CONDUCTION. Chapter 2 Chee 318 1

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1 hapte ONE DIMENSIONAL SEADY SAE ONDUION hapte hee 38

2 HEA ONDUION HOUGH OMPOSIE EANGULA WALLS empeatue pofile A B X X 3 X 3 4 X 4 Χ A Χ B Χ hapte hee 38

3 hemal conductivity Fouie s law ( is constant) A A d d ( ) Fouie s law ( is function of ) d A d ( β 0 0 A ( ) ) β ( hapte hee 38 3

4 hapte hee 38 4 HEA ONDUION HOUGH OMPOSIE EANGULA WALLS Enegy balance B A Χ Χ Χ Β c B Solving these thee euations simultaneously B A Χ Χ Χ Β Β 4 4

5 HEA ONDUION HOUGH OMPOSIE EANGULA WALLS themal potentioal diffence esistance oveall hapte hee 38 5

6 HEA ONDUION HOUGH OMPOSIE EANGULA WALLS (a) Sufaces nomal to the - diection ae isothemal Fo esistances in seies: tot n Fo esistances in paallel: tot / / / n (b) Sufaces paallel to - diection ae adiabatic hapte hee 38 6

7 adial Systems-ylindical oodinates onside a hollow cylinde empeatue distibution hapte hee 38 7

8 adial Systems-ylindical oodinates A d A d πl d (πl ) d Bounday conditions Solution πl ( i o ) ln( l ) hemal esistance?? themal potentioal diffence esistance hapte hee 38 8 oveall

9 hapte hee 38 9 omposite Walls? Epess the following geomety in tems of a an euivalent themal cicuit. Χ A A L ) / / ln( ) / / ln( ) ( 4 π L i o π ) / ln( B A Χ Χ Χ Β Β 4 4

10 omposite Walls? Epess the following geomety in tems of a an euivalent themal cicuit. ln( o / i ) π L ln( / ) / A πl( 4 ) ln( / ) / 3 B ln( 4 / ) / 3 hapte hee 38 0

11 adial Systems-Spheical oodinates Home eecise hapte hee 38

12 Eample. A thic tube of steel ( 9 W/m.) with cm inne diamete and 4 cm oute diamete is coveed with 3 cm laye of insulation ( 0. W/m.) Given that the inside 600 and the outside 00 ) calculate the heat loss pe mete of length. Also calculate the inteface. hapte hee 38

13 .5 Oveall Heat ansfe oefficient, old fluid,,h s, s,, / h A s, s, L / A s, s, / h, A,,h, In tems of oveall tempeatue diffeence: Hot fluid 0 L tot, tot h A, L A h A hapte hee 38 3

14 .5 Oveall Heat ansfe oefficient, / h A s, s, L / A s, s, / h, A,, / h A L / A / h A U / h UA L / / h hapte hee 38 4

15 .5 Oveall Heat ansfe oefficient? Epess the following geomety in tems of a an euivalent themal cicuit. hapte hee 38 5

16 .5 Oveall Heat ansfe oefficient Altenatively tot UA t UA whee U is the oveall heat tansfe coefficient and the oveall tempeatue diffeence. U tot A [(/ h ) ( LA / A) ( LB / B ) ( L / ) (/ h 4 )] hapte hee 38 6

17 .5 Oveall Heat ansfe oefficient onside a hollow cylinde, whose inne and oute sufaces ae eposed to fluids at diffeent tempeatues empeatue distibution hapte hee 38 7

18 .5 Oveall Heat ansfe oefficient Based on the pevious solution, the conduction hea tansfe ate can be calculated: Fouie s law: A (πl) const ( ) ( ) ( ) πl s, s, s, s, s, ln( / ) ln( / ) /(πl) d d In tems of euivalent themal cicuit: d d t, cond s, tot, h tot (π, L) ln( / ) πl h (π L) hapte hee 38 8

19 .5 Oveall Heat ansfe oefficient? Epess the following geomety in tems of a an euivalent themal cicuit. hapte hee 38 9

20 hapte hee Oveall Heat ansfe oefficient whee U is the oveall heat tansfe coefficient. If AA π L: ln ln ln h h U B A altenatively we can use A π L, A 3 π 3 L etc. In all cases: t A U A U A U A U

21 Eample.4 Wate 50 inside a.5 cm inside diamete tube such that h3500 w/m^.. he thicness of the tube is 0.8 cm and 6 w/m^.. he outside of the tube loses heat by fee convection with h7.6 w/m^.. alculate the oveall heat tansfe coefficient and the heat loss pe unit length to suounding ai at 0. i t o U h A i ln( do / d πl A o o i h A UA 9W o / W 3500π (0.05)() i) ln(0.066 / 0.05) π (6)().575 / W 7.6π (0.066)() 7.577W / m / W hapte hee 38

22 .6 itical thicness of insulation If A, is inceased, will incease. When insulation is added to a pipe, the outside suface aea of the pipe will incease. his would indicate an inceased ate of heat tansfe he insulation mateial has a low themal conductivity, it educes the conductive heat tansfe. his contadiction indicates that thee must be a citical thicness of insulation. he thicness of insulation must be geate than the citical thicness, so that the ate of heat loss is educed as desied. hapte hee 38

23 .6 itical thicness of insulation d d 0 0 πl( i ln( o / i o maimize h 0 ) ) h As the outside adius, o, inceases, then in the denominato, the fist tem inceases but the second tem deceases. hapte hee

24 Eample.5 alculate the citical adius of insulation fo asbestos [0. 7 W/m. ] suounding a pipe and eposed to oom ai at 0o with h3.0 W/m.o. alculate the heat loss fom a 00, 5.0-cm-diamete pipe when coveed with the citical adius of insulation and without insulation. Fom Euation (-8) we calculate o as 0.7 o m 5. 67cm h 3.0 he inside adius of the insulation is 5.0/.5 cm, so the heat tansfe is calculated fom Euation (-7) as L π (00 0) In(5.67 /.5) 0.7 (0.0567)(3.0) 05.7W \ m hapte hee 38 4

25 Eample.5 L π (00 0) In(5.67 /.5) 0.7 (0.0567)(3.0) 05.7W \ m Without insulation the convection fom the oute suface of the pipe is L h( π )( i o) (3.0)(π )(0.05)(00 0) 84.8W / m So, the addition of 3.7 cm ( ) of insulation actually inceases the heat tansfe by 5 pecent. hapte hee 38 5

26 .7 Heat Souce system. (Heat onduction Euation) Enegy onsevation Euation z de dt & st in E& g E& out E E& E& in out y E& st d y dy z (.) z dz z y y y z whee fom Fouie s law A A A y z ( dydz) ( ddz) y y z ( ddy) z hapte hee 38 6

27 .7 Heat Souce system. (Heat onduction Euation) hemal enegy geneation due to an enegy souce: Manifestation of enegy convesion pocess (between themal enegy and chemical/electical/nuclea enegy) Positive (souce) if themal enegy is geneated Negative (sin) if themal enegy is consumed & is the ate at which enegy is geneated pe unit volume of the medium (W/m 3 ) E& g & dv & ( d dy dz) Enegy stoage tem epesents the ate of change of themal enegy stoed in the matte in the absence of phase change. E& st ρc p t ( d dy dz) ρ / t c p is the time ate of change of the sensible (themal) enegy of the medium pe unit volume (W/m 3 ) hapte hee 38 7

28 .7 Heat Souce system. (Heat onduction Euation) y y y z & ρc t p Heat Euation Net conduction of heat into the V ate of enegy geneation pe unit volume time ate of change of themal enegy pe unit volume At 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 eual the ate of change of themal enegy stoed within the volume hapte hee 38 8

29 .7 Heat onduction Euation- Othe foms If constant & y z α Fo steady state conditions t α ρc p is the themal diffusivity y y y z & 0 Fo steady state conditions, one-dimensional tansfe in -diection and no enegy geneation d " d d 0 o 0 Heat flu is constant in the diection of tansfe d d d hapte hee 38 9

30 hapte hee Heat onduction Euation t z y α & 0 & tae integal twice (assume t cons tan * ) * Χ Χ solve with WO Bounday conditions ( ) ( ),, Χ Χ

31 hapte hee Heat onduction Euation tae integal twice (assume t cons tan * ) * Χ Χ solve with WO Bounday conditions ( ) ( ),, Χ Χ

32 .8 Heat onduction Euation In cylindical coodinates: φ φ z z & ρc t p In spheical coodinates: sin θ φ φ sin θ θ sin θ θ & ρc t p hapte hee 38 3

33 hapte hee Heat onduction Euation In cylindical coodinates t c z z p ρ φ φ & 0 & w w d d 4 0

34 hapte hee Heat onduction Euation In cylindical coodinates t c z z p ρ φ φ & 0 & ) ln( 4 ) / ( ) ln( i i i i i

35 Eample.6 A cuent of 00 A is passed though a stainless-steel wie [9W/m.o]3 mm in diamete. he esistivity of the steel may be taen as 70.cm, and the length of the wie is m. he wie is submeged in a liuid at 0 and epeiences a convection heattansfe coefficient of 4 W/m^.. alculate the cente tempeatue of the wie. hapte hee 38 35

36 All the powe geneated in the wie must be dissipated by convection to the liuid: P ha( w ) he esistance of the wie is calculated fom 6 L (70 0 )(00) ρ 0.099Ω A π (0.5) Whee ρ is the esistivity of the wie. he suface aea of the wie is πdl, fom Euation (a), 3 ( 00) ( 0.099) 4000π ( 3 0 )()( w 0) 3960W and o o w 5 [49 F] so he heat geneated pe unit volume is calculated fom so that P V & &π L & 560.MW / m [5.4 0 Btu / h. ft ] π 3 (.5 0 ) () Finally, the cente tempeatue of the wie is calculated fom Euation (-6): & ( )(.5 0 ) o o [449 F] o o w (4)(9) hapte hee 38 36

37 hapte hee HEMAL ONA ESISANE Imagine two solid bas bought into contact with the sides of the bas insulated so that heat flows only in the aial diection Β 3 3 ΧΒ Χ ΧΒ Χ Β Β 3 3 A B

38 . HEMAL ONA ESISANE he tempeatue dop acoss the inteface between mateials may be appeciable, due to suface oughness effects, leading to ai pocets. No eal suface is pefectly smooth, and the actual suface oughness is believed to play a cental ole in detemining the contact esistance. his facto can be etemely impotant in a numbe of applications because of the many heattansfe situations which involve mechanical joining of two mateials. hapte hee 38 38

39 hapte hee HEMAL ONA ESISANE 3 A 3 B ΧΒ Χ Β Β A h B 3 ΧΒ Χ Β h 3

40 . HEMAL ONA ESISANE A B 3 3 Χ Χ h A Β 3 h ΧΒ : themal contact esistance h h : contact coefficient B ΧΒ Β Β w Btu, o o m. h. ft. F 3 Fo design puposes the contact conductance values given in able - may be used in the absence of moe specific infomation. hapte hee 38 40

41 Eample. ow 3 cm diamete 304 stainless steel bas, 0 cm long, have gound suface and ae eposed to ai with a suface oughness of about µ m. If the sufaces ae pessed togethe with a pessue of 50 atm and the two ba combination is eposed to an oveall tempeatue diffeence of 00, calculate the aial heat flow and the tempeatue dop acoss the contact suface. Χ 3 h ΧΒ Β th A (0.)(4) (6.3) π (0.03) / W c h A c 4 (5.8 0 )(4) π (0.03) /W (able. fo h c ) hapte hee 38 4

42 Eample. Χ 3 h ΧΒ Β th A (0.)(4) (6.3) π (0.03) / W c the total 4 (5.8 0 )(4) h A π (0.03) th c themal esistance is ()(8.679) th 5.5 W /W (able. fo h c ) he tempeatue dop acoss the contactcan be found by taing the atio of c c th (0.747)(00) the contact esistance to the total themal esistance hapte hee 38 4

43 .9 onduction-convection systems o A Steady state Opeation No Heat Geneated onstant hemal onductivity () onstant onvection oefficient (h) Neglect adiation hapte hee 38 43

44 .9 onduction-convection systems Enegy in the left face d Enegy in the left face A d Enegy lost by convection hpd( d A d d d ) d d d hp A ( ) 0 hapte hee 38 44

45 .9 onduction-convection systems ASE : he fin is vey long, and temp. at the end of the fin is essentially that of the suounding fluid. ASE : he fin is of finite length and loses heat by convection fom its end. d d hp A ( ) 0 ASE 3: he end of the fin is insulated hapte hee 38 45

46 .9 onduction-convection systems aing 'A' to be Pd Whee P ( w h) (peimete) Q conv hp( inf )d Q cond, Q cond ( inf ) hp in the limit as goes to infinity we get d d d Q cond d A c A c d d d d ( inf ) hp 0 ( ) hp inf 0 if '' and 'A' ae constant then ( ) hp inf 0 let θ inf hapte hee 38 46

47 .9 onduction-convection systems a hp A c geneal solution d θ d a θ 0 θ( ) e a e a bounday conditions θ( 0) θ b b inf θ( L) ( L) inf 0 assuming at L, the tempeatue of the fin euals the tempeatue of the suounding ou solution becomes θ( ) θ b e a ( ) inf b inf e hp A c hapte hee 38 47

48 .0 Fins hapte hee 38 48

49 Pupose of a Fin Fins ae etended sufaces that ae utilized in the emoval of heat fom a body One can incease heat tansfe by inceasing the heat tansfe coefficient o inceasing the suface aea Finned sufaces ae manufactued by etuding, welding, o wapping a thin metal sheet on a suface Fins enhance heat tansfe fom a suface by eposing a lage suface aea to convection and adiation hapte hee 38 49

50 hapte hee 38 50

51 he fin efficiency Fin efficiency, h f ( ) actual heat tansfeed heat which wouldb tansfeed if entie fin h wee at base temp. actual f ( o ) otal suface aea of the fin actual η f h f ( o ) hapte hee 38 5

52 he fin efficiency Fin efficiency is defined as the atio of actual heat tansfe ate to the maimum possible heat tansfe ate he suface tempeatue of the fin deceases fom the base to the tip diection. he degee of vaiation depends on the dimensions and the themal conductivity of the fin. If the themal conductivity is vey lage, the suface tempeatue may be constant and euals to b. he heat tansfe ate fom the elemental aea, pd he total heat tansfe ate fom the entie fin dq& hpd( ) he maimum possible heat tansfe ate of the entie fin b constant L he fin efficiency & Q hp( ) d fin 0 Q& hplθ ha ma b fin b η & Q fin & Q ma a hapte hee 38 5 θ

53 he fin efficiency (case 3) Fo ASE 3 above (he end of the fin of unifom coss-sectional aea is insulated) tanh ml η f (-38) ml Pofile aea ml h 3 L m Lt m (-39) hapte hee 38 53

54 he fin efficiency (case 3) It has been shown that the efficiency euation fo case 3 (insulated tip) can be used fo case (finite length) without insulation) if a coected length is used η f tanh ml ml hapte hee 38 54

55 - oected length ( ase ) onvection tip bounday condition he total fin suface aea A t including the tip aea is A Lp A t c Dividing the above euation by the peimete, p, the coected fin length is L c L Ac P If L is eplaced by L c, the tip suface aea of the fin subjected to convection bounday condition is consideed () Fo a ectangula fin () Fo a cicula fin L c wt wt t Lc L L L ( w t) w L D 4 L c t t/ hapte hee 38 55

56 empeatue Distibution Heat conduction euation in the -diection fo steady-state conditions, with no enegy geneation: d d d d 0 d d Fouie s law: A (πl) const Bounday onditions: ( ) s,, ( ) s, empeatue pofile, assuming constant : ( ) s, s, ( ) ln s, ln( / ) Logaithmic tempeatue distibution (see above) hapte hee d d

57 hemal esistance Based on the pevious solution, the conduction heat tansfe ate can be calculated: Fouie s law: A (πl) const ( ) ( ) ( ) πl s, s, s, s, s, ln( / ) ln( / ) /(πl) d d In tems of euivalent themal cicuit: tot, t, cond hapte hee tot, ln( / ) πl d d s,

LECTURER: PM DR MAZLAN ABDUL WAHID PM Dr Mazlan Abdul Wahid

LECTURER: PM DR MAZLAN ABDUL WAHID PM Dr Mazlan Abdul Wahid M 445 LU: M D MZL BDUL WID http://www.fkm.utm.my/~mazlan hapte teady-tate tate One Dimensional eat onduction M bdul Wahid UM aculty of Mechanical ngineeing Univesiti eknologi Malaysia www.fkm.utm.my/~mazlan