Heat Transfer. V2 4Jun15

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1 Heat Transfer V2 4Jun5

2 Heat Transfer Conduction Heat transfer through a solid object is done by conduction (Q) between two bodies is a function of the geometry (area and length) and thermal conductivity of the material Q Cond = k A l T Q heat flow in W k Material Conductivity, W m (function of the material) A Cross section Area (m2) (Function of the geometry) l Length (m) (Function of the geometry) ΔT Temperature difference between the two ends of the material, (Tfinal Tinitial) Note: Negative sign corrects for the ΔT sign (T final T initial ) where T initial > T final so that heat flow is positive from Initial to Final Bigger the area more heat can be transferred (thicker wire) The longer the object the less the heat transfer (longer wire) Parallel exits between thermal conduction and electrical conduction Heat flow is analogous to Current flow Temperature difference is analogous to voltage difference Thermal resistance decreases heat flow as electrical resistance reduces current flow Similar equations can be used for both T initial l A T final

3 Conductors in Series Q = k A l T m -Q l k A = T m = T m T initial K, A K 2, A 2 -Q R = T m T initial, where R = l k A -Q 2 R 2 = T final T m, where R 2 = l 2 k 2 A 2 T initial 2 T final T M l l 2 Q R + Q 2 R 2 = T m T initial + T final T m = T final T initial But Q = Q 2 = Q Q R + R 2 = T final T initial = T QR eff. = T; where R eff. = R i and R eff. = l eff. k eff. A eff. For conductors in series, effective thermal resistance (R eff. ) is the sum of the individual resistances (ΣR i )

4 Conductors in Parallel Q = k A l T m Q = T R, where R = l k A Q 2 = T R 2, where R 2 = l 2 k 2 A 2 T initial K, A K 2, A 2 2 T final Q total = Q + Q 2 = R + R 2 T l = l 2 Q total R + R2 = Q total R eff. = T, where R eff. = R i and R eff. = leff. k eff. A eff. For conductors in parallel, effective thermal resistance (R eff. ) is the reciprocal sum of the individual reciprocal resistances R i

5 Conductors in Parallel Q = k A l T m Q = T R, where R = l k A Q 2 = T R 2, where R 2 = l 2 k 2 A 2 T initial K, A K 2, A 2 2 T final Q total = Q + Q 2 = R + R 2 T l = l 2 Q total R + R2 = Q total R eff. = T, where R eff. = R i and R eff. = leff. k eff. A eff. For conductors in parallel, effective thermal resistance (R eff. ) is the reciprocal sum of the individual reciprocal resistances R i

6 Contact Resistance Surfaces are not perfectly smooth so contact area is much less than expected Higher resistance in the local area than in bulk of the material creates discontinuity in the temperature profile Increased pressure decreases thermal resistance Smoother surfaces decreases thermal resistance Conductive grease fills in the gaps and decreases thermal resistance Temperature gradient is found from: T Contact = R Contact Q Resistance can be treated as before

7 Heat Transfer Convection Convection Heat transfer Heat transfer (Q Conv ) between a surface and the surrounding environment at Q Cond = ha T Q heat flow in W h Convective heat transfer coefficient (Function of: Surface; Geometry, Orientation, fluid; speed, density, viscosity) A Cross section Area (m 2 ) (Function of the geometry) ΔT Temperature difference between the surface and the environment Convective Resistance: R conv = Such that: T = R ha ConvQ R conv can be treated as R cond Note: Negative sign corrects for the ΔT sign (T T Surf ) where T Surf > T so that heat flow is positive from high temperature to low temperature h calculated from non-dimensional parameters Reynolds number (Re) Ratio of velocity to viscosity and density Prandtl number (Pr) Ratio of heat capacity and viscosity to fluid thermal conductivity Nustle Number (Nu) Ratio of Convective heat transfer coefficient and length to the fluid thermal conductivity T surf Area (T <T surf ) T Q T T <T Surf

8 Heat Transfer Radiation Radiation Heat Transfer Heat transfer (Q Rad ) between two surfaces and does not require any contact between the surfaces i.e. surface can be in a vacuum Q Rad = F ε F G σa T 2 4 T 4, with T 2 > T Q Rad Radiation heat flow in W F ε Emissivity function (How the surface emits thermal radiation, Black Body: F ε =) F ε Geometric view factor (How the two surfaces see each other σ Boltzman Constant, 5.669x0-8 W/(m 2 K 4 ) A Area, m 2 T 2 Area For radiation incident on a surface, energy can take up to three paths depending on the nature of the material: = +ρ + τ T α Absorption factor also α=ε ρ Reflection factor τ Transition factor ε Emissivity, For blackbody ε=

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