HeatTransfer-I-Section-5

# 2 we can use just one term in the summaons j0 and j1

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Unformatted text preview: ∞ T - T∞ 2 Temperature = ∑ Cn cos ζn x L exp( −ζn FoL ) Ti - T∞ n =1 2 sin ζn Fourier Coefficient Cn = ζn + cos ζn sinζn Eigenvalues ζn tan ζn = BiL ∞ 2 k (Ti - T∞ ) ζn sin2 ζn Surface Heat Flux (One Side) q' ' = ∑ ζ + cosζ sinζ exp(−ζn2FoL ) L n n n =1 n ∞ 2 Q 1 sin ζn 2 Total Heat Flow = 1 − 2∑ exp( −ζn FoL ) Qi n =1 ζn ζn + cos ζn sin ζn ( ) Initial Energy Content Qi = 2 ALρc p (Ti − T∞ ) •  Single Term Approxima/ons € –  When Fo > 0.2 we can use just one term in the summa/ons. 16 Transient Conduc/on 17 •  Inﬁnite Cylinder (L > 10R) ∞ Temperature Fourier Coefficient Eigenvalues T - T∞ 2 = ∑ Cn J 0 ζn r R exp( −ζn FoR ) Ti - T∞ n =1 J1 (ζn ) 2 Cn = 2 ζn J 0 (ζn ) + J12 (ζn ) ( ζn ∞ ) J1 (ζn ) = BiR J 0 (ζn ) J12 (ζn ) Surface Heat Flux 2 k (Ti − T∞ ) q' ' = ∑ J 2 (ζ ) + J 2 (ζ ) exp(−ζn2FoR ) R n =1 0 n 1 n Total Heat Flow ∞ J12 (ζn ) Q 2 = 1 − 4∑ 2 2 exp( −ζn FoR ) 2 Qi n =1 ζn J 0 (ζn ) + J1 (ζn ) [ Ini...
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