1-D Steady Conduction: Plane Wall

Transcription

1 1-D Steady Conduction: Plane Wall Governing Equation: d 2 T dx 2 0 Dirichlet Boundary Conditions: T(0) T s,1 ; T(L) T s,2 Solution: Heat Flux: Heat Flow: T(x) T s,1 + ( T s,2 T s,1 ) x L q ʹ x ʹ k dt dx k ( L T T s,1 s,2) q x ka dt dx ka ( L T s,1 T s,2 ) temperature is not a function of k heat flux/flow are a function of k Notes: A is the cross-sectional area of the wall perpendicular to the heat flow both heat flux and heat flow are uniform è independent of position (x) temperature distribution is governed by boundary conditions and length of domain è independent of thermal conductivity (k) D. B. Go 1

2 1-D Steady Conduction: Cylinder Wall Governing Equation: Solution: Heat Flux: Heat Flow: T(r) T s,1 T s,2 ln r 1 r 2 q ʹ r ʹ k dt q r ka dt dr + T s,2 ( ) ln r r 2 ( ) ( ) dr k T s,1 T s,2 rln r 2 r 1 2πrL q ʹ r ʹ 2πLk T s,1 T s,2 ln( r 2 r 1 ) ( ) Dirichlet Boundary Conditions: T(r 1 ) T s,1 ; T(r 2 ) T s,2 ( ) heat flux is non-uniform heat flow is uniform q ʹ r q r L 2πk T s,1 T s,2 heat flow per unit length ln( r Notes: 2 r 1 ) heat flux is not uniform è function of position (r) both heat flow and heat flow per unit length are uniform è independent of position (r) 1 r D. B. Go 2 d dr! # " kr dt dr $ & 0 d! % dr r dt $ # & 0 " dr %

3 1-D Steady Conduction: Spherical Shell Governing Equation: 1 r 2 d! dt # dr " kr2 dr $ & 0 d! dt $ # % dr " r2 & 0 dr % Dirichlet Boundary Conditions: T(r 1 ) T s,1 ; T(r 2 ) T s,2 Solution: Heat Flux: Heat Flow: T(r) T s,1 T s,1 T s,2 q ʹ r ʹ k dt ( ) 1 ( r 1 r) 1 ( r 1 r 2 ) ( ) r 2 [( 1 r 1 ) ( 1 r 2 )] dr k T s,1 T s,2 ( ) ( ) q r ka dt dr 4πr2 q ʹ r ʹ 4πk T s,1 T s,2 ( 1 r 1 ) 1 r 2 heat flux is non-uniform heat flow is uniform Notes: heat flux is not uniform è function of position (r) heat flow is uniform è independent of position (r) D. B. Go 3

4 Thermal Resistance D. B. Go 4

5 Thermal Circuits: Composite Plane Wall Circuits based on assumption of (a) isothermal surfaces normal to x direction or (b) adiabatic surfaces parallel to x direction R tot L E k E A + k F A + k GA 2L F 2L G 1 + L H k H A R tot 2L E k E A + 2L F k F A + 2L H k H A 1 + 2L E k E A + 2L G k G A + 2L H k H A 1 1 Actual solution for the heat rate q is bracketed by these two approximations D. B. Go 5

6 Thermal Circuits: Contact Resistance In the real world, two surfaces in contact do not transfer heat perfectly R t,c ʹ ʹ T A T B R q ʹ x ʹ t,c ʹ ʹ R t,c A c Contact Resistance: values depend on materials (A and B), surface roughness, interstitial conditions, and contact pressure è typically calculated or looked up Equivalent total thermal resistance: R tot L A k A A c + ʹ ʹ R t,c A c + L B k B A c D. B. Go 6

7 D. B. Go 7

8 D. B. Go 8

9 Fins: The Fin Equation Solutions D. B. Go 9

10 Fins: Fin Performance Parameters Fin Efficiency the ratio of actual amount of heat removed by a fin to the ideal amount of heat removed if the fin was an isothermal body at the base temperature that is, the ratio the actual heat transfer from the fin to ideal heat transfer from the fin if the fin had no conduction resistance η f q f q f q f,max ha f θ b Fin Effectiveness ratio of the fin heat transfer rate to the heat transfer rate that would exist without the fin ε f q f q f,max q f ha c,b θ b R t,b R t, f Fin Resistance defined using the temperature difference between the base and fluid as the driving potential R t, f θ b 1 q f ha f η f D. B. Go 10

11 Fins: Efficiency D. B. Go 11

12 Fins: Efficiency D. B. Go 12

13 Fins: Arrays Arrays total surface area N number of fins A t NA f + A b A b exposed base surface (prime surface) total heat rate q t Nη f ha f θ b + ha b θ b η o ha t θ b θ b R t,o overall surface efficiency η o 1 NA f A t overall surface resistance R t,o θ b q t ( 1 η f ) 1 ha t η o D. B. Go 13

14 Fins: Thermal Circuit Equivalent Thermal Circuit Effect of Surface Contact Resistance q t η o(c) ha f θ b θ b R t,o(c ) η o(c) 1 NA $ f 1 η ' f & ) A t % C 1 ( % R $ ( C 1 1 η f ha t,c f ' * & A c,b ) R t,o 1 ha t η o(c ) D. B. Go 14

15 AME Int. Heat Trans. Fins: Overview Fins extended surfaces that enhance fluid heat transfer to/from a surface in large part by increasing the effective surface area of the body combine conduction through the fin and convection to/from the fin the conduction is assumed to be one-dimensional Applications fins are often used to enhance convection when h is small (a gas as the working fluid) fins can also be used to increase the surface area for radiation radiators (cars), heat sinks (PCs), heat exchangers (power plants), nature (stegosaurus) Straight fins of (a) uniform and (b) non-uniform cross sections; (c) annular fin, and (d) pin fin of nonuniform cross section. D. B. Go 15

16 Fins: The Fin Equation Solutions D. B. Go 16