1 Conduction Heat Transfer

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1 Eng690 - Formula Sheet 2 Conduction Heat Transfer. Cartesian Co-ordinates q x xa x A x dt dx R th A 2 T x T y T z 2 + q T T x) plane wall of thicness 2, x 0 at centerline, T s, at x, T s,2 at x, steady state, D, uniform source and properties: ) T x) q2 x 2 ) Ts,2 T s, x + 2 ) 2 + T s, + T s,2 2.2 Polar-Cylindrical Co-ordinates q r 2πr dt dr R th lnr 2/r ) 2π 2 T r 2 + T r r + 2 T r 2 θ T z 2 + q T T r) in pipe wall r 2 > r ), T s, at r, T s,2 at r 2, D radial, steady state, no source, uniform properties: T r) T ) s, T s,2 r lnr /r 2 ) ln + T s,2 r 2 T r) solid rod of radius r o, T s at r o, D radial, steady state, uniform source and properties:.3 Spherical Co-ordinates r 2 r 2 T ) + r r T r) qr2 o 4 q r 4πr 2 dt dr r 2 sin θ 2 Convection Heat Transfer ) ) r 2 + T s ro sin θ T ) + θ θ R th ) 4π r r 2 2 T r 2 sin 2 θ φ 2 + q T q ha surf T s T ) R th ha surf 3 Radiation Heat Transfer q rad F G ɛσa surf T 4 T 4 2 ) h r ɛσt s + T sur )T 2 s + T 2 sur) R th h r A surf σ W/m 2 K 4

2 Eng690 - Formula Sheet Fins hp θ x T x T m η f q f q f A c q max ha f θ b q f η f ha f θ b 4. Case A: Convection Heat oss from the Tip of the Fin θ coshm x)) + h/m) sinhm x)) θ b coshm) + h/m) sinhm) η f m 4.2 Case B: Insulated Tip [ ] sinhm) + h/m) coshm) coshm) + h/m) sinhm) q f hp A c θ b sinhm) + h/m) coshm) coshm) + h/m) sinhm) θ coshm x)) θ b coshm) q f hp A c θ b tanhm) η f tanhm) m 4.3 Case C: Specified Tip Temperature θ θ /θ b ) sinhmx) + sinhm x)) θ b sinhm) q f hp A c θ b coshm) θ /θ b sinhm) 4.4 Case D: Very ong or Infinite) Fin θ e mx q f hp A c θ b η f θ b m 4.5 Case B Approximation of Case A η f tanhm c) c + A tip /P A f P c tanhx) ex e x m c e x + e x 4.6 Fin Resistance 5 Contact Resistance q f η f ha f θ b θ b R t,f R t,f η f ha f R t,c h c A contact R t,c A contact

3 Eng690 - Formula Sheet Forced Convection - Flat Plate 6. Parameters Nu x h xx Nu h Re x ρu x Re ρu Re D ρu D P r c p 6.2 aminar Boundary ayer T s constant: C f,x 0.664Re /2 x C f.328re /2 Nu x 0.332Re /2 x P r /3 Nu 0.664Re /2 P r/3 P r 0.6 Heating starts ξ from the leading edge: s constant: Nu x 0.332Re /2 x P r /3 [ ] ξ 3/4 /3 x) Nu x 0.453Re /2 x P r /3 Nu 0.680Re /2 P r/3 P r Turbulent Boundary ayer T s constant: s 3 2 h T s T ) C f,x Re /5 x Nu x Re 4/5 x P r /3 0.6 < P r < 60, Re x < 0 8 Heating starts ξ from the leading edge: Nu x Re 4/5 x P r /3 [ ξ/x) 9/0 ] /9 0.6 < P r < 60, Re x < 0 8 s constant: Nu x Re 4/5 x P r /3 0.6 P r Mixed Boundary ayer Conditions When Re x,c : Nu 0.037Re 4/5 A)P r/3 A 0.037Rex,c 4/ Rex,c /2 Nu 0.037Re 4/5 87)P r/3 C f Completely turbulent boundary layer: Re /5 Nu 0.037Re 4/5 P r/3 C f 0.074Re /5 742 Re 0.6 < P r < 60, < Re 0 8

4 Eng690 - Formula Sheet Forced Convection - Flow over Cylinders and Spheres 7. Parameters Nu D hd Re D ρu D 7.2 Flow over Cylinders 7.3 Flow over Spheres Nu D CRe m DP r /3 P r 0.6 Re D C m , , , Nu D Re /2 D ) / Re2/3 D )P r0.4 s 8 Forced Convection - Internal Flows 8. Parameters 8.2 Energy Balance T s constant: Nu D hd Nu D hd Re D ρu md q conv ṁc p T m,o T m,i ) UA T lm T lm T s T m,o ) T s T m,i ) ln [T s T m,o )/T s T m,i )] s const: T m,o T s T s T m,i ) exp T m x) T m,i + q s P ṁc p x UA ) ṁc p UA ΣR th

5 Eng690 - Formula Sheet aminar Flow Fully Developed: Nu D 4.36 s const Nu D 3.66 T s const Entry Region /D < 0.05Re D P r, constant T s ) 8.4 Turbulent Flow Nu D.86 Fully Developed /D 0, constant T s or s ): ) ReD P r /3 ) 0.4 /D s Nu D 0.023Re 4/5 D P rn n 0.4, T s > T m n 0.3, T s < T m Entry Region /D < 0, constant T s or s ): ) Nu D 0.036Re 4/5 D D P r/3 8.5 Flows in Noncircular Tubes Replace D in all parameters with the hydraulic diameter, D H : D H 4A c P

1 Conduction Heat Transfer

1 Conduction Heat Transfer Eng6901 - Formula Sheet 3 (December 1, 2015) 1 1 Conduction Heat Transfer 1.1 Cartesian Co-ordinates q x = q xa x = ka x dt dx R th = L ka 2 T x 2 + 2 T y 2 + 2 T z 2 + q k = 1 T α t T (x) plane wall of

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