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Enhancement of Heat Transfer P M V Subbarao Associate Professor Mechanical Engineering Department IIT Delhi Invention of Compact Heat Transfer Devices Heat transfer enhancement • Enhancement • Increase the convection coefficient Introduce surface roughness to enhance turbulence. Induce swirl. • Increase the convection surface area Longitudinal fins, spiral fins or ribs. Heat Transfer Enhancement using Inserts Heat Transfer Enhancement using Inserts Heat transfer enhancement :Coiling • Helically coiled tube • Without inducing turbulence or additional heat transfer surface area. • Secondary flow FREE CONVECTION P M V Subbarao Associate Professor Mechanical Engineering Department IIT Delhi Its free, No operating cost!…….. Its Natural ….. Natural Convection Where we’ve been …… • Up to now, have considered forced convection, that is an external driving force causes the flow. Where we’re going: • Consider the case where fluid movement is by buoyancy effects caused by temperature differential Events due to natural convection • • • • Weather events such as a thunderstorm Glider planes Radiator heaters Hot air balloon • Heat flow through and on outside of a double pane window • Oceanic and atmospheric motions • Coffee cup example …. Small velocity Natural Convection • New terms – Volumetric thermal expansion coefficient – Grashof number – Rayleigh number • Buoyancy is the driving force – Stable versus unstable conditions • Nusselt number relationship for laminar free convection on hot or cold surface • Boundary layer impacts: laminar turbulent Buoyancy is the driving force • Buoyancy is due to combination of – Differences in fluid density – Body force proportional to density – Body forces namely, gravity, also Coriolis force in atmosphere and oceans • Convection flow is driven by buoyancy in unstable conditions • Fluid motion may be (no constraining surface) or along a surface Buoyancy is the driving force • Free boundary layer flow Heated wire or hot pipe A heated vertical plate • We focus on free convection flows bounded by a surface. • The classic example is Ts T Extensive, quiescent fluid u(x,y) Ts T g x y u v Governing Equations • The difference between the two flows (forced flow and free flow) is that, in free convection, a major role is played by buoyancy forces. X g Very important •Consider the x-momentum equation. u u 1 P u u v g 2 x y x y 2 •As we know, p / y 0 , hence the x-pressure gradient in the boundary layer must equal that in the quiescent region outside the boundary layer. Pascal Law : P - g x u u 1 u u v g g 2 x y y 2 u u u 2 u v g x y y 2 Buoyancy force Governing Equations • Define , the volumetric thermal expansion coefficient. 1 T P For an ideal gas : P RT 1 Thus : T P RT Not for liquids and non-ideal gases 1 1 T T T (T T ) Density gradient is due to the temperature gradient Governing Equations (cont’d) • Now, we can see buoyancy effects replace pressure gradient in the momentum equation. u u 2u u v g (T T ) v 2 x y y •The buoyancy effects are confined to the momentum equation, so the mass and energy equations are the same. u v 0 x y T T T u v 2 x y y c p y 2 u 2 Strongly coupled and must be solved simultaneously Dimensionless Similarity Parameter x x L u u u0 and and y y L v v u0 T T T Ts T * where L is a characteri stic length, and u 0 is an arbitrary reference velocity • The x-momentum and energy equations are * * 2 * g ( T T ) L u u 1 u * * * s u v T * * 2 x y u0 Re L y*2 * * 2 * T T 1 T u * * v* * x y Re L Pr y*2 Dimensionless Similarity Parameter (cont’d) • Define new dimensionless parameter, g (Ts T ) L u0 L g (Ts T ) L GrL 2 2 u0 2 3 •Grashof number in natural convection is analogous to the Reynolds number in forced convection. •Grashof number indicates the ratio of the buoyancy force to the viscous force. •Higher Gr number means increased natural convection flow GrL 1 forced 2 Re L GrL 1 2 Re L natural Laminar Free Convection on Vertical Surface • As y : u = 0, T = T • As y 0 : u = 0, T = Ts Ts T u(x,y) Ts • With little or no external driving flow, Re 0 and forced convection effects can be safely neglects T g x y u v GrL 1 2 Re L Nu L f (GrL , Pr) Analytical similarity solution for the local Nusselt number in laminar free convection 1/ 4 hx GrL Nu x k 4 f (Pr) Where f Pr 0.75 Pr 0.609 1.221 Average Nusselt # = Pr 1.238 Pr 1/ 4 1/ 4 h L 4 GrL NuL k 3 4 f (Pr) Effects of Turbulence • Just like in forced convection flow, hydrodynamic instabilities may result in the flow. • For example, illustrated for a heated vertical surface: • Define the Rayleigh number for relativemagnitude of buoyancy and viscous forces Ra x ,c Grx ,c Pr g (Ts T ) x 3 Ts T Effects of Turbulence • Transition to turbulent flow greatly effects heat transfer rate. Empirical Correlations Typical correlations for heat transfer coefficient developed from experimental data are expressed as: hL NuL CRa Ln k Ra L GrL Pr g Ts T L3 n 1 / 4 n 1 / 3 For Turbulent For Laminar Vertical Plate at constant Ts Log10 Nu L Log10 RaL •Alternative applicable to entire Rayleigh number range (for constant Ts) Nu L 0.825 1 (0.492 / Pr) 9 /16 0.387 Ra1L/ 6 8 / 27 Vertical Cylinders •Use same correlations for vertical flat plate if: D ~ 35 1/ 4 L GrL 2 Inclined Plate Horizontal Plate Cold Plate (Ts < T) Hot Plate (Ts > T) Empirical Correlations : Horizontal Plate •Define the characteristic length, L as As L P •Upper surface of heated plate, or Lower surface of cooled plate : Nu L 0.54 Ra1L/ 4 Nu L 0.15 Ra1L/ 3 104 RaL 107 107 RaL 1011 •Lower surface of heated plate, or Upper surface of cooled plate : 1/ 4 Nu L 0.27 RaL 10 5 RaL 10 Note: Use fluid properties at the film temperature 10 Ts T Tf 2 Empirical Correlations : Long Horizontal Cylinder •Very common geometry (pipes, wires) •For isothermal cylinder surface, use general form equation for computing Nusselt # hD NuD CRa Dn k Constants for general Nusselt number Equation RaD C n 1010 - 10 2 0.675 0.058 10 2 - 10 2 1.02 0.148 102 - 104 0.850 0.188 104 - 107 0.480 0.250 107 - 1012 0.125 0.333