3 - University of Ottawa CHG 2314 B Kruczek 1 CHG 2314 Heat...

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Unformatted text preview: University of Ottawa, CHG 2314, B. Kruczek 1 CHG 2314 Heat Transfer Part 2b Steady-state 1-D Heat Conduction- Conduction and thermal resistance- Conduction with heat generation- Conduction in extended surfaces University of Ottawa, CHG 2314, B. Kruczek 2 Conduction across the plane wall Note: The case of 1-D steady conduction across the plane wall was partly discussed in Part 1 | The temperature distribution requires solving the heat diffusion equation z For steady-state 1-D conduction with no heat generation the heat diffusion equation simplifies to: z If k is constant, the temperature distribution can be obtained by double integration of the above equation: where C 1 and C 2 are the integration constants, which depend on how we specify the required two boundary conditions = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ dx dT k dx d (3.1) ( ) 2 1 C x C x T + = (3.2) University of Ottawa, CHG 2314, B. Kruczek 3 Conduction across the plane wall | For conditions 1 and 2 the temperature distribution becomes: | Application of Fourier’s law to Eq. (3.3) leads to: ( ) ( ) 2 , 2 , 2 1 , 1 , 1 2 , 1 , ∞ = ∞ = = = − = − − = − = = T T h dx dT k T T h dx dT k T T T T s L x s x s L x s x 4. 3. 2. 1. : conditions boundary Common ( ) ( ) 1 , 1 , 2 , s s s T L x T T x T + − = (3.3) ( ) 2 , 1 , s s x T T L kA dx dT kA q − = − = (3.4) University of Ottawa, CHG 2314, B. Kruczek 4 Thermal resistance | Eq. (3.4) can be alternatively written in the following form: where R t,cond is the thermal resistance for conduction given by: | If heat transfer involves convection and radiation, there is also thermal resistance for convection ( R t,conv ) and thermal resistance for radiation ( R t,rad ) given by: ( ) cond t s s x R T T q , 2 , 1 , − = kA L R cond t = , (3.6) A h R hA R rad rad t conv t 1 1 , , = = (3.13) (3.9) University of Ottawa, CHG 2314, B. Kruczek 5 Thermal resistance | In general, the rate of heat transfer in any 1-D steady-state process without heat generation may be expressed by: where ∆ T is the temperature driving force and Σ R t = R tot is the total thermal resistance | Total resistance z Resistances in series: z Resistances in parallel: R A R B R B R A B A tot R R R + = B A tot R R R 1 1 1 + = tot t x R T R T q Δ = Σ Δ = University of Ottawa, CHG 2314, B. Kruczek 6 Thermal resistance | At steady state, the rate of heat transfer in the system can be expressed in several ways, for example: | It is easier to measure the temperature of fluids than of solids, consequently: where U is the overall heat transfer coefficient ( ) ( ) ( ) ( ) CF conv t cond t HF conv t cond t HF conv t s HF conv t s cond t s s x R R R T T R R T T R T T R T T q , , , , , 2 , 1 , , , , 2 , 1 , , , 1 , 1 , , 2 , 1 , + + − = + − = − = − = ∞ ∞ ∞ ∞ ( ) ( ) 2 , 1 , 2 , 1 , ∞ ∞ ∞ ∞ − = − = T T UA q R T T q x tot x or ∑ = Δ = = UA q T R R t tot 1 (3.19) University of Ottawa, CHG 2314, B. Kruczek 7 Composite wall | Consider a composite wall as shown: z The concept of total thermal resistance...
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This note was uploaded on 05/30/2011 for the course CHG 2314 taught by Professor Kruz. during the Spring '10 term at University of Ottawa.

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3 - University of Ottawa CHG 2314 B Kruczek 1 CHG 2314 Heat...

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