3a-r - One-Dimensional Steady-State Conduction without...

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One-Dimensional, Steady-State One-Dimensional, Steady-State Conduction without Conduction without Thermal Energy Generation Thermal Energy Generation Chapter Three Chapter Three Sections 3.1 through 3.4 Sections 3.1 through 3.4

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Methodology Specify appropriate form of the heat equation . Solve for the temperature distribution . Apply Fourier’s law to determine the heat flux . Simplest Case: One-Dimensional , Steady-State Conduction with No Thermal Energy Generation . Common Geometries: The Plane Wall : Described in rectangular ( x ) coordinate. Area perpendicular to direction of heat transfer is constant (independent of x ). The Tube Wall : Radial conduction through tube wall. The Spherical Shell : Radial conduction through shell wall. Methodology of a Conduction Analysis
Plane Wall Consider a plane wall between two fluids of different temperature: The Plane Wall Implications: 0 d dT k dx dx = (3.1) Heat Equation: ( 29 Heat flux is independent of . x q x ′′ ( 29 Heat rate is independent of . x q x Boundary Conditions: ( 29 ( 29 ,1 ,2 0 , s s T T T L T = = Temperature Distribution for Constant : ( 29 ( 29 ,1 ,2 ,1 s s s x T x T T T L = + - (3.3) k

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Plane Wall (cont.) Heat Flux and Heat Rate : ( 29 ,1 ,2 x s s dT k q k T T dx L ′′ = - = - (3.5) ( 29 ,1 ,2 x s s dT kA q kA T T dx L = - = - (3.4) Thermal Resistances and Thermal Circuits: t T R q = Conduction in a plane wall: , t cond L R kA = (3.6) Convection: , 1 t conv R hA = (3.9) Thermal circuit for plane wall with adjoining fluids: 1 2 1 1 tot L R h A kA h A = + + (3.12) ,1 ,2 x tot T T q R - = (3.11)
Plane Wall (cont.) Thermal Resistance for Unit Surface Area : , t cond L R k ′′ = , 1 t conv R h ′′ = Units: K/W t R 2 m K/W t R ′′ Radiation Resistance : , 1 t rad r R h A = , 1 t rad r R h ′′ = ( 29 ( 29 2 2 r s sur s sur h T T T

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