This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Monday January 13, 2008 Monday January 13, 2008 CHEM ENG 2A04 CHEM ENG 2A04 CONDUCTION CONDUCTION HEAT EQUATION HEAT EQUATION • 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 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. T T T q k i k j k k r r z φ → → → → ∂ ∂ ∂ ′′ =    ∂ ∂ ∂ r q ′′ q φ ′′ z q ′′ • Cylindrical Coordinates: ( 29 , , T r z φ sin T T T q k i k j k k r r r θ θ φ → → → → ∂ ∂ ∂ ′′ =    ∂ ∂ ∂ r q ′′ q θ ′′ q φ ′′ • 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 ′′ • 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 onedimensional, 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 π ′ ′ ′′ ′′ = = – Sphere 2 4 r r r r q A q r q π ′′ ′′ = = 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 (inflowoutflow) p T T T T k k k q c x x y y z z t ρ ∂ ∂ ∂ ∂ ∂ ∂ ∂ + + + = ∂ ∂ ∂ ∂ ∂ ∂ ∂ • Thermal energy generation Change in thermal energy storage 2...
View
Full Document
 Winter '10
 TODDHOARE
 Thermodynamics, Heat, Heat Transfer, Energy storage, TW

Click to edit the document details