Slides 3 - Fourier's Law and the Heat Equation Chapter Two...

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Fourier Fourier s Law s Law and the and the Heat Equation Heat Equation Chapter Two Chapter Two
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Fourier’s Law •A rate equation that allows determination of the conduction heat flux from knowledge of the temperature distribution in a medium Fourier’s Law • Its most general (vector) form for multidimensional conduction is: qk T →→ ′′ = −∇ Implications: – Heat transfer is in the direction of decreasing temperature (basis for minus sign). – Direction of heat transfer is perpendicular to lines of constant temperature ( isotherms ). – Heat flux vector may be resolved into orthogonal components. – Fourier’s Law serves to define the thermal conductivity of the medium / kq T ⎛⎞ ≡− ⎜⎟ ⎝⎠
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Heat Flux Components (2.22) TTT qk i k j k k rr z φ →→ ∂∂∂ ′′ =− ∂∂ r q q z q • Cylindrical Coordinates: ( ) ,, Tr z sin TT T i k j k k r θ θφ (2.25) r q q q • Spherical Coordinates: ( ) Tr φθ • Cartesian Coordinates: ( ) Txyz i kj k k xyz x q y q z q (2.3)
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Heat Flux Components (cont.) • In angular coordinates , the temperature gradient is still based on temperature change over a length scale and hence has units of ° C/m and not ° C/deg. ( ) or , φ φθ Heat rate for one-dimensional, radial conduction in a cylinder or sphere: Cylinder 2 rr r r qA q r L q π ′′ == or, 2 r r q r q Sphere 2 4 r r q r q
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Heat Equation The Heat Equation • A differential equation whose solution provides the temperature distribution in a stationary medium. • Based on applying conservation of energy to a differential control volume through which energy transfer is exclusively by conduction. • Cartesian Coordinates: Net transfer of thermal energy into the control volume (inflow-outflow) p TTT T kkk q c x xy yz z t ρ ⎛⎞ ∂∂ +++ = ⎜⎟ ⎝⎠ (2.17) Thermal energy generation Change in thermal energy storage
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Heat Equation (Radial Systems) 2 11 p TT T T kr k k q c rr r z z t r ρ φφ ⎛⎞ ∂∂ ++ + = ⎜⎟ ⎝⎠ (2.24) • Spherical Coordinates: • Cylindrical Coordinates:
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Slides 3 - Fourier's Law and the Heat Equation Chapter Two...

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