# 2a-r - Fouriers Law and the Heat Equation Chapter Two...

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Fourier’s Law Fourier’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: q k 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 / k q T → → ′′ ≡ -
Heat Flux Components (2.22) T T T q k i k j k k r r z φ ′′ = - - - r q ′′ q ′′ z q ′′ Cylindrical Coordinates: ( 29 , , T r z sin T T T q k i k j k k r r r θ θ φ ′′ = - - - (2.25) r q ′′ q ′′ q ′′ Spherical Coordinates: ( 29 , , T r φ θ Cartesian Coordinates: ( 29 , , T x y z T T T q k i k j k k x y z ′′ = - - - 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. ( 29 or , φ φ θ Heat rate for one-dimensional, radial conduction in a cylinder or sphere: Cylinder 2 r r r r q A q rLq π ′′ ′′ = = or, 2 r r r r q A q rq ′ ′′ ′′ = = Sphere 2 4 r r r r q A q r q ′′ ′′ = =
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:

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## This note was uploaded on 02/21/2009 for the course MAE 310 taught by Professor Kuznetsov during the Spring '08 term at N.C. State.

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2a-r - Fouriers Law and the Heat Equation Chapter Two...

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