Conduction: Theory of Etended Surfaces
Why etended surface? h, T ha( T T ) s Increasing h Increasing A 2
Fins as etended surfaces A fin is a thin component or appendage attached to a larger body or structure In the contet of heat transfer also, these are components protruding out of a heated (or cold) surface Hot surface Fin 3
What happens in a fin? conv (+ rad )=ha s (T-T ) cond conv = ha s (T-T ) cond = -ka c dt/d An etended surface (combined conduction-convection system) is a solid within which heat transfer by conduction is assumed to be one dimensional, while heat is also transferred by convection (and/or radiation) from the surface in a direction transverse to that of conduction. 4
What happens in a fin? An etended surface (combined conduction-convection system) is a solid within which heat transfer by conduction is assumed to be one dimensional, while heat is also transferred by convection (and/or radiation) from the surface in a direction transverse to that of conduction. 5
Eamples of Fins 6
Question time Heat is transferred from hot water flowing inside a tube to cooling air flowing over the tube. To enhance heat transfer rate, which side should the fins be installed? Hot water @90C Air @ 25C Fins are most beneficial where h is low Fin dimensions and k are critical design parameters Liuid side Air side 7
Summary Etended surfaces help in enhancing heat dissipation Increases surface area for heat echange Fins: most common embodiment of etended surface Can be of varied shape and forms Fin peformance = f (h, material, size) 8
Fin Analysis T b P: the fin perimeter A c : the fin cross-sectional area ka dt d C d d d d d h( da )( T T ), where da is the surface area of the element conv Energy balance: S d S d conv d d hp( T T ) 9
Fin Analysis (cont.) d d dt ( kac ) hp( T T ) d 0 A c A c () d T T ( d hp A ) c 0 d d k For a constant cross-section A c 2 d 2 d m 2 0, m 2 hp ka c ( ) m C1e C2 e m Need two boundary conditions Base: b at 0 Tip: 4 scenarios at = L 10
Temperature profiles Case Tip Condition Temp. Distribution Fin heat transfer A Convection heat coshm( L ) ( h )sinh m( L ) sinh ml ( h )coshml transfer: mk M mk b h(l)=-k(d/d) h =L coshml ( )sinh ml coshml ( h )sinh ml mk mk B C D Adiabatic (d/d) =L =0 Given temperature: (L)= L Infinitely long fin (L)=0 coshm( L ) M b tanh ml coshml L ( )sinh m( L ) sinh m( L ) (coshml L ) b b Mb sinh ml sinh ml m e M b T T b (0), T b m 2 T, hp ka C M hpka C b 11
Eample Problem An Aluminum pot is used to boil water as shown below. The handle of the pot is 20-cm long, 3-cm wide, and 0.5-cm thick. The pot is eposed to room air at 25C, and the convection coefficient is 5 W/m 2 -K. Assume no heat transfer at the end of the handle. Question: can you touch the handle when the water is boiling? (k for Al = 237 W/m-K) 100 C T = 25 C h = 5 W/ m 2 C Steps Treat handle as fin Identify tip condition Adiabatic Calculate P, m, M Get temp. profile Calculate T at =L= 20 cm 12
Eample (contd ) Temperature distribution along the pot handle 100 100 95 75 T( ) T( ) 50 90 Al handle 25 Stainless Steel handle 85 0 0 0 0.05 0.1 0.15 0.2 0.05 0.1 0.15 0.2 Temperature at the tip = 87.3 C Not safe to touch Why? What can you do? 13
Triangular fin Question time? T = 75C T = 75C T = 25 C h = 5 W/ m 2 -K How can we get the temp profile? 1. What are the boundary conditions? 2. How will the temp. profile look like? 3. Any other way we can think of? 14
Fin parameters: Effectiveness (e f ) Fin effectiveness (e f ): how effective is the fin Ratio of heat transferred in presence of fin to in its absence e f ha f c, b b >1 fin is effective <1 should not include fin e with h, k and A / P f c Remember we wanted fins on the air side!! 15
Fin parameters: Efficiency (h f ) Fin efficiency (h f ): how close to ideal scenario is the fin Ratio of heat transferred to that if entire fin were at base temp h f f ha f,ma f b f T b Real situation Ideal situation b f For a fin array with N fins, A B : total base area A b, A t : base and tip area of fin A f : surface area of a single fin (ecluding tip) f h( A NA Nh ( A A )) s b f f t b 16
Fin array in heat sinks Parallel fins Radial fins Pin fins Circular heat sink Dovetail fins 17
Summary Solved ODE for 1-D heat conduction to get temp profile 4 different tip conditions Fin performance metrics Effectiveness (e f ) & Efficiency (h f ) Fin arrays used to design heat sinks Commonly used in computing products 18
Thank you!! 19