one primary direction in which heat transfers (generally the smallest dimension) simple model good representation for solving engineering problems

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1 CHAPTER 3: One-Dimenional Steady-State Conduction one pimay diection in which heat tanfe (geneally the mallet dimenion) imple model good epeentation fo olving engineeing poblem 3. Plane Wall 3.. hot fluid T, h cold fluid T, h 0 Fo one-dimenional teady tate conduction with no enegy geneation, the heat euation educe to: E in E out + E g E t T k + ρc Aume an iotopic medium (k contant) and integate to detemine the tempeatue ditibution T(): p T t d k 0 C T ( ) C + C 3.

2 Apply bounday condition to olve fo contant: T(0)T ; T()T T (0) T T ( ) T C C + T C T T The eulting tempeatue ditibution i: T ( ) ( T T ) + T and vaie linealy with. Applying Fouie law: heat tanfe ate: heat flu: T T A ka ka T T k k Theefoe, both the heat tanfe ate and heat flu ae independent of. Note: Altenative bounday condition might involve convection if uface tempeatue ae unknown. e.g., k T (0) T h T C [ ( ) T ], and T() C + C ; olve fo C 3.

3 3.. Themal Reitance A eitance can be defined a the atio of a diving potential to a coeponding tanfe ate. e.g., Analogy: electical eitance i to conduction of electicity a themal eitance i to conduction of heat themal eitance fo conduction: cond T ka T themal eitance fo convection: ha( T T ) conv 4 4 themal eitance fo adiation echange: εσ A( T T ) ad h whee εσa T εσa T εσa T 4 4 ( Tu ) ( Tu )( T + Tu ) ( T )( T + T )( T + T ) ( T ) A T h u u εσ ( T + T )( T + T ) u u u u ad u 3.3

4 The euivalent themal cicuit fo a plane wall The total ate of heat tanfe i: 3..3 The Compoite Wall Euivalent themal cicuit analyi can alo be ued fo comple ytem uch a compoite wall Often, it i convenient to define an oveall heat tanfe coefficient, U: whee T i the oveall tempeatue diffeence, uch that: 3.4

5 3..4 Contact Reitance Realitically, the heat flu at the inteface of two olid uface i continuou but the tempeatue i not. Tempeatue vaie at the inteface of two uface becaue of impefect contact between the olid. The tempeatue diffeence i attibuted to a themal contact eitance: TA TB Rt, c The euivalent themal cicuit fo impefect contact: Themal contact eitance can be educed by (e.g.) (Omit 3., An Altenative Conduction Analyi) 3.5

6 3.3 Radial Sytem 3.3. cold fluid T,, h, hot fluid T,, h, Fo one-dimenional, teady-tate condition with no enegy geneation, the heat euation educe to: d k 0 d d Aume an iotopic medium and integate to detemine the tempeatue ditibution, T(): k d 0 d d C d T ( ) C ln + C Apply bounday condition to olve fo contant C and C : T( ) T C ln + C T( ) T C ln + C ubtacting T T : ubtitute C into T( ): C C T ln, T ( ) T, T,, T, ln ln ( ) Note: The value of C and C can have othe fomulation, depending on the bounday condition ued to olve fo T(). 3.6

7 The eulting tempeatue ditibution i: T T T + and vaie logaithmically with. Applying Fouie law: heat tanfe ate: ka k d heat flu: A,, ( ) ln T, ln( ) πk ( π) T ln T πk ln,, ( T T ),, T ( ) ( ) T,, ( ) k T, T ( π) ln( ) ( ) ln, Thu, the heat tanfe ate i contant and independent of. The heat flu i a function of the adiu. The euivalent themal cicuit fo a cylinde themal eitance fo conduction: cond π T k ln, T, ( ) themal eitance fo convection: πh( T T ) conv 3.7

8 3.3. Sphee (aume hollow cente) Fo one-dimenional, teady-tate condition with no enegy geneation, the heat euation educe to: d k d d Aume an iotopic medium and integate to detemine the tempeatue diffeence T(): k d d d C T ( ) C C 0 0 d Apply bounday condition: T( )T and T( )T The eulting tempeatue ditibution i: T ( ) T, ( T T ), The coeponding heat tanfe ate i: ( T T ) 4πk,, themal eitance fo conduction: R t, cond 4πk, ( ) ( ) 3.8

9 3.5 Conduction with Themal Enegy Geneation Conide the effect of a poce occuing within a medium uch a themal enegy geneation, E, e.g, conveion of electical to themal enegy g 3.5. Plane Wall T, T, T,, h T,, h - Fo one-dimenional, teady-tate conduction in an iotopic medium: d k + 0 T ( ) + C + C k Note: The tempeatue ditibution i paabolic in. Applying Fouie law: heat tanfe ate: heat flu: A ka ka + C k k k + C k Thu, / i a function of, and theefoe both the heat tanfe ate and heat flu ae dependent on fo a medium with enegy geneation. 3.9

10 Cae #: T, > T, T, > T, h h bounday condition: T(-) T, T() T, T,, h T,, h - Fom and the above bounday condition: T ( ) k T +, T, T +, + T, To detemine whee the tempeatue i a maimum: Cae #: T, T, T T, T, T h h h bounday condition: T(-) T T() T,, h T,, h Fom and the above bounday condition: T ( ) + k T - 3.0

11 Cae #3: at 0, adiabatic uface at, T T bounday condition: T,, h 0 0 T ( ) T (i.e., 0) The inulated wall i analogou to a ymmetic wall, and the ame tempeatue ditibution i found a Cae #: T ( ) + k T 3.

12 3.5. Radial Sytem T, h T o Fo one-dimenional, teady-tate conduction in an iotopic medium: k d d d T ( ) 4k C ln + C Apply bounday condition to olve fo the contant C and C : d 0 o 0 C 0 T ( o ) T C 4k The eulting tempeatue ditibution i: o + T and vaie paabolically in. T + T o ( ) k 4 o Caution. Note that it i incoect to ue themal eitance concept within a medium when the heat tanfe ate i NOT contant (enegy geneation). 3.

13 3.6 Heat Tanfe fom Etended Suface involve conduction though a olid medium a well a convection and/o adiation enegy tanfe goal i to enhance heat tanfe between a olid and a fluid conv ( ) ha T T Poibilitie: - inceae heat tanfe coefficient - inceae uface tempeatue - deceae fluid tempeatue - inceae uface aea The mot common way to enhance heat tanfe i by inceaing the uface aea fo convection via an etenion fom a olid medium: fin Conide a long, lende pin fin attached to a bae: T, h D T b 3.3

14 3.6. Geneal Conduction Analyi Aume one-dimenional condition in diection, i.e., the fin i thin, thu Aume teady-tate condition, iotopic medium, neglect adiation, and thee i no enegy geneation E t 0 E in + E g [ + d ] + E out conv ka ( ) + d conv c hda + ka ( ) [ T ( ) T ] c k d Ac ( ) then, finally, d A c h k da ( T ) 0 T whee, A c, A, and T can be function of. 3.4

15 3.6. Fin of Unifom Co-Sectional Aea Conide a taight fin: at the bae ( 0), T(0)T b A c i contant A P, P i the peimete canceling and eaanging appopiate tem fom the geneal euation: define d T hp A k c ( T T ) 0 then, Bounday condition: at 0; T(0)T b at (efe to each cae, A.-D.; ame a Table 3.4): A. Convection at tip; cond conv ka final olution i: [ T ( T ] c hac ) θ T T θ T T b b coh [ m( ) ] + ( h mk) inh[ m( ) ] coh( m) + ( h mk) inh( m) 3.5

16 B. Adiabatic fin tip; 0 final olution i: θ θ b coh coh [ m( ) ] ( m) C. Fied tempeatue at tip; T() T final olution i: θ θ b ( θ θ b ) inh( m) + inh[ m( ) ] inh( m) D. Infinitely long fin; final olution i: T( ) T θ e θ b m 3.6

17 Analyze Cae B (adiabatic tip) to detemine othe infomation about the fin: total heat tanfe fom the fin: f ka c dθ ka c 0 0 Note: the ame olution can be obtained by: [ ( T ] h T ) f da f whee A f i the uface aea of the fin. fin effectivene: A f ε f f b fin efficiency, η f : atio of the actual fin heat tanfe ate to the ate pedicted fo an iothemal fin η f f b io 3.7

18 Regadle of the tip bounday condition, the olution will povide the tempeatue ditibution though the fin, θ/θ b. Due to the deceaing tempeatue though the fin, the conduction heat tanfe (locally) deceae a a eult of continuou convection loe The total heat tanfe by the fin: T, h tip T b bae 3.8

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