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Lecture-Week-3-on

# Lecture-Week-3-on - II Conduction A Resistance concept 1...

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Unformatted text preview: II. Conduction A. Resistance concept 1. Plane wall (1-D) no heat source Consider T 1 T i = ? specified " q A q = T specified x L 1 L 2 K 1 K 2 If you take any C.V 1 st law of thermo " 1 q " 2 q in steady state " " 2 " 1 q q q = = ⇒ q ” = constant Linear temperature profile in each material Fourier law = = dx dT k q 1 " const ⇒ T: linear 2 2 1 1 1 " ) ( ) ( L T T k L T T k q i i- =- = if 2 1 k k ≠ slopes are different but flux is the same so ∆ T 1 ∆ T 2 1 ) ( ) ( 2 2 1 1 1 T T L Ak T T L Ak q i i- =- = ⇒ resistence Thermal Difference Potential Thermal R T R T q 2 2 1 1 = ∆ = ∆ = where 1 1 1 Ak L R = 2 2 2 Ak L R = Thermal resistences d b c a d c b a q + + = = = Algebra . T 1 R 1 ∆ T 1 q . ∆ T = T 1 – T 0 ∑ ∆ = k k R T q T i Resistances in series R 2 ∆ T 2 . T ⇒ 2 2 1 1 1 Ak L Ak L T T q +- = 2 ↓ ↑ ↓ ↑ ↑ ↑ ⇒ = R k R A R L Ak L R , , Consider now the situation T fluid @ T ∞ q h watt heat transfer coefficient T q = hA (T – T ∞ ) h is calculated from fluid side (convection) ⇒ Ah R R T q 1 = ⇒ ∆ = ↓ ↑ ↓ ↑ R h R A , 3 So now we can look at: T 1 ∙&#2; ∧∧∧ ∙&#2; ∧∧∧ ∙&#2; ∧∧∧ ∙&#2; ∧∧∧ ∙ T ∞ Find q = ? fluid R 1 R 2 R 3 R 4 T ∞ L 1 , k 1 L 2 , k 2 L 3 , k 3 h T 1 x , Ak L R k k = , K=1,…...3 Ah R 1 4 = k k R T T q 4 1 1 = ∞ Σ- = = Ah Ak L Ak L Ak L T T 1 3 3 2 2 1 1 1 + + +- ∞ 4 It is often convenient to compare convective with conductive resistances, entional Non Biot B k hL Ah Ak L R R i conv cond dim # 1- = = = = resistence convection resistence conduction k hL B i = = If B i << 1 No need to consider the “walls” only h T 1 T ∞ If B i >> 1 Neglect h and only consider the walls T 1 T ∞ For comparison: Orders of magnitude of h (W/m² ° C) Natural convection 5 Hot air Air 0(10) rises naturally Water 0(100) Forced convection Air 100 U ∞ δ Water 100 – 1000 δ T Boiling 1000 –10000 6 Think of convection as conductive resistance T ∞ δ T fluid T w fluid T kA q T k q T T ∆ = ⇒ ∆ = δ δ " Note δ T replaces L = = T T con k h kA R δ δ Motion decreases ( δ T ) or reduce its size ) ( ↑ ↓ ↓⇒ h R T δ 7 Convention: Define an overall heat transfer coefficient by T UA q ∆ = T 1 T ∞ ) ( 1 ∞- = ∆ T T T but R T q Σ ∆ = ∆ T = (T 1- T ∞ ) R A U Σ = ⇒ 1 (always define which A U is based on) Resistance concept for 1-D steady situations can be used to estimate behavior of move complex systems of composite 2-D walls 8 Conduction of heat 1-D...
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Lecture-Week-3-on - II Conduction A Resistance concept 1...

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