1D STEADY STATE HEAT
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1 D SEADY SAE HEA CONDUCION () Pabal alukda Aociate Pofeo Depatment of Mecanical Engineeing II Deli Palukda/Mec-IID
2 emal Contact eitance empeatue ditibution and eat flow line along two olid plate peed againt eac ote fo te cae of pefect and impefect contact Palukda/Mec-IID
3 Conide eat tanfe toug two metal od of co-ectional aea A tat ae peed againt eac ote. Heat tanfe toug te inteface of tee two od i te um of te eat tanfe toug te olid contact pot and te gap in te noncontact aea and can be expeed a Q Q Q contact Q gap A Δ c int eface Mot expeimentally detemined value of te temal contact eitance fall between and m C/W (te coeponding ange of temal contact conductance i 000 to 00,000 W/m C). wee A i te appaent inteface aea (wic i te ame a te co- ectional aea of te od) and Δ inteface i te effective tempeatue diffeence at te inteface. e quantity c, wic coepond to te convection eat tanfe coefficient, i called te temal contact conductance and i expeed a eface It i elated to temal contact eitance by Palukda/Mec-IID c c Q Δ A int (W/m o C) Δ int eface (m o C/W) c Q A
4 Impotance of conideation e temal contact t eitance ange: between and m C/W Palukda/Mec-IID
5 wo paallel laye wo paallel laye ) ( Q Q Q total Q ' A k ' A k Palukda/Mec-IID total total wee
6 Combined eie-paallel Q total total 3 conv 3 conv ' ' k A k A 3 3 ' k A 3 3 conv A 3 Palukda/Mec-IID
7 Seie and paallel compoite wall and it temal cicuit D A C E B F C A B D E F 3 4 Q UA Δ (W) Palukda/Mec-IID wee U i te oveall eat tanfe coefficient UA total
8 Complex multi-dimenional poblem a -D poblem. Any plane wall nomal to te x-axi i iotemal. Any plane paallel to x-axi i adiabatic Palukda/Mec-IID
9 Heat conduction in cylinde Q cond, cyl ka d d A π Qcond,cyl d A kd Subtituting A π and pefoming te integation give Q cond,cyl πk ln( ) Q cond,cyl contant at teady tate Q cond,cyl cyl (W) Palukda/Mec-IID cyl ln( πk ) ln(oute adiu/inne adiu) π(lengt)(temal conductivity)
10 Heat conduction in pee Fo pee Q& cond,pee p p 4π k oute adiu - inne adiu 4π(oute adiu)(inne adiu)(temal conductivity) Palukda/Mec-IID
11 eitance Netwok cylindical total conv, cond conv, ln ( ) ( π ) π k ( π ) peical total conv, p conv, ( 4 π ) 4 π k ( 4 π ) e temal eitance netwok fo a cylindical (o peical) ell ubjected to convection fom bot te inne and te oute ide. Palukda/Mec-IID
12 Multilayeed cylinde total conv, cyl, cyl, cyl,3 conv, Palukda/Mec-IID ln ( ) ln ( ) ln ( ) A π k π k π k 3 A 4
13 adial eat conduction toug cylindical ytem k. φ k. φ k. z z g& ρc t d d d d 0 Integating te above equation twice, C ln C Subject to te bounday condition, at and at () ln ln ln ln ln Palukda/Mec-IID
14 C d C. k. d d ka Q π π ln )..( k. Q π ln ) k( Palukda/Mec-IID
15 Citical adiu of Inulation. Steady tate condition. One-dimenional eat flow only in te adial diection 3. Negligible temal eitance due to cylinde wall 4. Negligible adiation excange between oute uface of inulation and uounding Inulation, in wall Palukda/Mec-IID
16 Citical adiu of Inulation Pactically, it tun out tat adding inulation in cylindical and peical expoed wall can initially caue te temal eitance to deceae, teeby inceaing te eat tanfe ate becaue te outide aea fo convection eat tanfe i getting lage. At ome citical tickne, c, te temal eitance inceae again and conequently te eat tanfe i educed. o find an expeion fo c, conide te temal cicuit below fo an inulated cylindical wall wit temal conductivity k: Palukda/Mec-IID Inulation, in wall Q & ( ) ln πk t π
17 An inulated cylindical pipe expoed to convection fom te oute uface and te temal eitance netwok aociated wit it. Palukda/Mec-IID
18 o find c, et te oveall temal eitance d t /d 0 and olve fo : ln( ) i t πk π i inne adiu d d πk t c k π 0 Similaly fo a pee k c Fo inulation tickne le tat c te eat lo inceae wit inceaing and fo inulation tickne geate tat c te eat lo deceae wit inceaing If k 0.03 W/(m K) and 0 W/(m K): k 0.03W/(m K) cylinde c 0.003m 3mm 0W/(m K) pee Palukda/Mec-IID k c 6mm
19 Value of, and k ae contant o ee te condition maximize o minimize te total eitance d d total πk π 3 otal temal eitance pe unit lengt total Heat tanfe pe unit lengt At k/ d π( k ) ln πk π i total Q d total k k 3 πk Alway poitive, total eitance at k/ i minimum k c, cylinde (m) > 0 Optimum tickne i aociated wit, πk π 0 d total d k 0
20 o ( W m C) o ( W m C) k max, inulation 0.05 c,max 0.0m cm 5 min We can inulate ot wate pipe and team line witout woying te citical adiu of inulation Inulation of electic wie: -adiu of electic wie may be malle tan te citical adiu -Addition of inulation mateial inceae eat tanfe Citical adiu of inulation fo peical ell: c, pee k
21 Summay able Palukda/Mec-IID
22 D Conduction wit Heat Geneation Palukda/Mec-IID
23 0 q d 0 k dx ( ) C x C x q x Bounday condition: ( ) C x C x k x y ( ), C,, ( ),,, k q C Palukda/Mec-IID ) (,,,, x x k q x
24 ,, q x ( x ) k q (0) o k dx Put x 0 If te uface tempeatue of te eat geneating body i unknown and te uounding fluid tempeatue i d Find tempeatue gadient Uing enegy balance k x ( ) fom te above Eq. at x We can obtain te uface tempeatue Palukda/Mec-IID q
25 d d d d q k 0 d q d kk C q ( ) C ln C 4kk Bounday condition: d d C Palukda/Mec-IID ( o ) C q o 4k q o ( ) 4k o ( Π o ) (Π o )( ) q q o
1D STEADY STATE HEAT
D SEADY SAE HEA CONDUCION () Prabal alukdar Aociate Profeor Department of Mechanical Engineering II Delhi E-mail: [email protected] Convection Boundary Condition Heat conduction at the urface in a
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