Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Free Convection: General Considerations and Results for Vertical and Horizontal Plates Ts T 1 Ts T General Considerations • Free convection refers to fluid motion induced by buoyancy forces. • Buoyancy forces may arise in a fluid for which there are density gradients and a body force that is proportional to density. • In heat transfer, density gradients are due to temperature gradients and the body force is gravitational. • Stable and Unstable Temperature Gradients 2 • Free Boundary Flows Occur in an extensive (in principle, infinite), quiescent (motionless at locations far from the source of buoyancy) fluid. Plumes and Buoyant Jets: • Free Convection Boundary Layers Boundary layer flow on a heated or cooled surface Ts T induced by buoyancy forces. 3 4 • Pertinent Dimensionless Parameters Grashof Number: GrL g Ts T L3 2 Buoyancy Force Viscous Force L characteristic length of surface thermal expansion coefficient (a thermodynamic property of the fluid) 1 T p Liquids: Tables A.5, A.6 Perfect Gas: =1/T K Pr Rayleigh Number : RaL GrL Pr g Ts T L3 Molecular diffusion momentum Molecular diffusion heat • Mixed Convection A condition for which forced and free convection effects are comparable. Exists if Gr 2 0 1 / Re L L - Free convection GrL / Re2L 1 - Forced convection GrL / Re 2L 1 Heat Transfer Correlations for Mixed Convection: n n Nu n NuFC Nu NC assisting and transverse flows - opposing flows n3 5 Vertical Plates 6 • Free Convection Boundary Layer Development on a Heated Plate: STEADY STATE Ascending flow with the maximum velocity occurring in the boundary layer and zero velocity at both the surface and outer edge. How do conditions differ from those associated with forced convection? How do conditions differ for a cooled plate Ts T ? • Form of the Continuity Equation (=cte) u v 0 x y • Form of the x-Momentum Equation for Laminar Flow (constant properties) 2 u u u g T T u2 x y y Net Momentum Fluxes Buoyancy Force ( Inertia Forces) Viscous Force Temperature dependence requires that solution for u (x,y) be obtained concurrently with solution of the boundary layer energy equation for T (x,y). 2 T T u T2 x y y – The solutions are said to be coupled. 7 7C • Non-dimensional equations u v 0 x y u * v* 0 * * x y u u u g T T u2 x y y 2 GrL g TS T L3 2 2 T T u T2 x y y ; * GrL * u* 1 2u* * u u v T * * 2 Re L y * 2 x y Re L * Re L U0 L * T* 1 2T* * T u v * * x y Pr Re L y* 2 * 8 • Similarity Solution Based on existence of a similarity variable, through which the x-momentum equation may be transformed from a partial differential equation with twoindependent variables ( x and y) to an ordinary differential equation expressed exclusively in terms of . 1/ 4 y Grx x 4 f ( ) ( x , y) 4 Grx 4 4 2 1/ 2 u Grx f ' y x v 2 y 1/ 2 Grx f ' x x 2 Transformed momentum and energy equations: f 3 ff 2 f T 0 2 T 3Pr fT 0 f df x Grx1/ 2 u d 2 T T T Ts T Numerical integration of the equations yields the following results for f and T : 9 Velocity boundary layer thickness 5 for Pr 0.6 1/ 4 Gr Pr 0.6 : 5 x x 4 1/ 4 7.07 x x1/ 4 1/ 4 Grx y Gr x x 4 10 Nusselt Numbers Nux and Nu L : 1/ 4 Gr Nu x hx x k 4 g Pr 1/ 4 dT d 0 Gr x 4 0.75 Pr1/ 2 0.609 1.221 Pr 1/ 2 h 1 h dx Nu L 4 NuL L 3 1/ 4 1 L k g Ts _ T h h dx L 0 L 4 2 4 Gr L 3 4 • Transition to Turbulence Amplification of disturbances depends on relative magnitudes of buoyancy and viscous forces. Transition occurs at a critical Rayleigh Number. g Ts T x3 0 Pr 1.238 Pr L o Rax , c Grx , c Pr g Pr 109 1/ 4 g Pr 1/ 4 gPr L 0 dx 1/4 x 11 • Empirical Heat Transfer Correlations Laminar Flow RaL 109 : Nu L 0.68 0.670 Ra1/L 4 1 0.492 / Pr 9 /16 4/9 All Conditions: 1/ 6 0.387 RaL Nu L 0.825 9 /16 4 / 9 1 0.492 / Pr 2 CHURCHILL E CHU What if the vertical plate is subjected to a constant heat flux (temperature is varying)? In this case, the same correlations are applied, but the non-dimensional parameters (Nusselt and Rayleigh numbers) are defined in terms of the temperature difference at the midpoint of the plate: q s'' h T1 / 2 T1 / 2 Ts x L / 2 T Horizontal Plates • Buoyancy force is normal, instead of parallel, to the plate. • Flow and heat transfer depend on whether the plate is heated or cooled and whether it is facing upward or downward. • Heated Surface Facing Upward or Cooled Surface Facing Downward Ts T Ts T 4 Nu L 0.54 Ra1/ L 3 Nu L 0.15 Ra1/ L 10 10 4 RaL 107 7 RaL 1011 3 How does h depend on L when Nu L Ra1/ L ? IT IS INDEPENDENT OF L ! 12 • Heated Surface Facing Downward or Cooled Surface Facing Upward Ts T 4 Nu L 0.27 Ra1/ L 13 Ts T 10 5 RaL 1010 Why do these flow conditions yield smaller heat transfer rates than those for a heated upper surface or cooled lower surface? BECAUSE THE PLATE IMPEDES THE ASCENDING/DESCENDING NATURAL CONVECTION FLOW THAT HAS TO MOVE HORIZONTALLY. THIS MAKES CONVECTION HEAT TRANSFER INEFFECTIVE. 7A •Continuity Equation u v 0 ( = constant) x y u v 0 x y •x- and y-Momentum Equations for Laminar Flow (constant properties) u u , y x u v , x v y u v u u 1 p v x y x 1 u 2 u v 1 u v g 2 x x 3 x y y y x u v v 1 p v x y y 1 v 2 u v 1 u v 2 y y 3 x y x y x p 0 y •Energy Equation for Laminar Flow (constant properties) u u , y x v , x v y T T y x T T T T C p u v k q k y x x y y x u v 2 u 2 v 2 2 u v 2 2 y x x y 3 x y 2 T T 2 T u u v 2 x y y C p y T T 2T u v x y y2 7B 8A •BOUNDARY CONDITIONS y 0: uv0 y* 0 : u * v* 0 y: u0 y* : 0: : T Ts T* 1 T T u* 0 f f' 0 T* 1 f' 0 T* 0 T* 0 f df x Grx1/ 2 u d 2 T T T Ts T 9A •Nusselt number 1/ 4 Gr Nu x hx x k 4 1/ 4 Gr dT x d 0 4 10A g Pr hx q '' x k Nu x k Ts T q '' hTs T Nu x q '' k T y y0 T y y0 T* Ts T 0 1/ 4 * 1 Gr T Ts T x y x 4 0 k Gr q Ts T x x 4 '' 1/ 4 k Gr Nu x Ts T x x 4 T* 1/ 4 T* 0 1/ 4 0 x Gr x k Ts T 4 T* 0