Heat Transfer. V4 3June16

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1 Heat Transfer V4 3June16

2 Heat Transfer Heat transfer occurs between two surfaces or bodies when there is a temperature difference Heat transfer depends on material properites of the object and the medium through which the heat is transferred Three modes of heat transfer Conduction Transfer trhough solid Linear function of thetemperature difference Q = k A l T where ΔT=T 2-T 1 with T 1 > T 2 Convection Transfer to or from a fluid (gas or liquid) Linear function of the temperature difference Q = ha T where ΔT=T 2 -T 1 with T 1 > T 2 Radiation Transfer without contact using electro-magnetig spectrum Function of difference of T 4 Q = F emissivity F Geometry σ T 2 4 T 1 4 with T 1 > T 2

3 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

4 Conductors in Series Q 1 = k 1 A 1 l 1 T m -Q 1 l 1 k 1 A 1 = T m = T m T initial K 1, A 1 K 2, A 2 -Q 1 R 1 = T m T initial, where R 1 = l 1 k 1 A 1 -Q 2 R 2 = T final T m, where R 2 = l 2 k 2 A 2 T initial 1 2 T final T M l 1 l 2 Q 1 R 1 + Q 2 R 2 = T m T initial + T final T m = T final T initial But Q 1 = Q 2 = Q Q R 1 + 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 )

5 Conductors in Parallel Q 1 = k 1 A 1 l 1 T m Q 1 = T R 1, where R 1 = l 1 k 1 A 1 Q 2 = T R 2, where R 2 = l 2 k 2 A 2 T initial K 1, A 1 K 2, A T final Q total = Q 1 + Q 2 = 1 R R 2 T l 1= l 2 Q total 1 R1 + 1 R2 = Q total R eff. = T, where R eff. = 1 1 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 1 1 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 Conv = 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 = 1 Such that: T = R ha ConvQ R conv can be treated as R cond (T <T surf ) 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 T T surf Area Q T T <T Surf

8 Convective Heat Transfer Coefficient h h is a factor of many variables and is determined experimentally Units: W/m 2 C Factors include: thermal conductivity of the fluid, thermal capacity of the fluid, air speed, viscosity, density, buoyancy, surface orientation, key dimension (length, diameter, etc.) Non-dimensional parameters generalize experimental results to a wide range of applications Reynolds number (Re) Ratio of inertia to viscous forces Natural convection Fluid flow induced by change in buoyancy caused by temperature differential 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

9 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 1 4, with T 2 > T 1 Q Rad Radiation heat flow in W F ε Emissivity function (How the surface emits thermal radiation, Black Body: F ε =1) F ε Geometric view factor (How the two surfaces see each other) σ Boltzman Constant, 5.669x10-8 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: 1 = +ρ + τ T 1 α Absorption factor also α=ε ρ Reflection factor τ Transmission factor

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