Lumped System Analysis

Transcription

1 Lumped System Analysis T ( t) T T T i b e h A ρv C P bt Spring 00 ECE09: Chapters 9,10,11 1 Criteria for Lumped System Analysis Convection at thesurface of the body h T Biot Number Bi conduction within thebody T / L L Characteri stic Length V / A Bi hv A 0.1 Spring 00 ECE09: Chapters 9,10,11 1

2 Convection Heat Transfer Convection heat transfer strongly depends on the: fluid properties, dynamic viscosity µ, thermal conductivity, density ρ, specific heat Cp, fluid velocity V, geometry of the solid surface, type of fluid flow q & Conv q& Cond fluid T y y0 h ( T / y) fluid y 0 T T s Spring 00 ECE09: Chapters 9,10,11 Nusselt Number, Nu q& q& & q conv & q cond conv cond h T T δ hδ Nu The larger the Nusselt number, the more effective is the convection. A Nusselt number Nu 1 for a fluid layer represents heat transfer by pure conduction. Spring 00 ECE09: Chapters 9,10,11 4

3 Velocity Boundary Layer The region of the flow above the plate bounded by δ V in which the effects of the viscous shearing forces caused by fluid viscosity are felt is called the velocity boundary layer. The thicness of the boundary layer, δ v, is arbitrarily defined as the distance from the surface at which V 0.99 V. Spring 00 ECE09: Chapters 9,10,11 Laminar & Turbulent Flows Laminar flow is characterized by smooth streamlines and highly ordered motion, Turbulent Flow is characterized by velocity fluctuations and highly disordered motion. Re inertia forces viscouse fources V δ V Free - stream velocity [m/s] δ Characteristic length, [m] Kinematic viscosity ( µ / ρ),[m / s] Critical Reynolds number is the Reynolds number at which the flow becomes turbulent. Re critical, flat plate 10 Spring 00 ECE09: Chapters 9,10,11 6

4 Thermal Boundary Layer A thermal boundary layer develops when a fluid at a specified temperature flows over a surface which is at a different temperature. The thicness of the thermal boundary layer δt at any location is defined as the distance from the surface at which the temperature difference (T-Ts) equals 0.99(T-Ts). The thicness of the thermal boundary layer increases in the flow direction. The relative thicness of the velocity and the thermal boundary layers is described by andtl Number. α 1 << 1 >> 1 µc P MostGases, Liquid Metals, Oils. Spring 00 ECE09: Chapters 9,10,11 7 Flow Over Flat Plates Laminar: h 1/ Nu 0.Re hl Nu 1/ 0.664ReL 1/ 1/ ( 0.6) Turbulent: Combined: h 4 / Nu 0.096Re hl Nu 4 / 0.07ReL cr L 1 + h, la min ard h L 0 cr 1/ 1/ h, turbulent d Re ReL 10 Ts + T All properties are evaluated at film temperature as: T f Spring 00 ECE09: Chapters 9,10,11 8 4

5 Natural Convection, Physical Mechanism The magnitude of the natural convection heat transfer between a surface and a fluid is directly related to the mass flow rate of the fluid. The higher the mass flow rate, the higher is the heat transfer rate. The mass flow is established by the dynamic balance of buoyancy and friction. Spring 00 ECE09: Chapters 9,10,11 9 Grashof Number, Gr Grashof Number, Gr, is a measure of the relative magnitudes of the buoyancy force and the opposing friction force acting on the fluid. buoyancy forces g ρv Gr viscous forces ρ g β T V ρ gβ ( Ts T ) δ g Gravitatio nal accelerati on [m/s ] β Coeff. of volume epansion, 1/T,[1/K] T Ts T, [ C] δ Characteristic length, [m] inematic viscosity, µ / ρ,[m /s] The Grashof number provides the main criteria in determining whether the fluid flow is laminar or turbulent in natural convection. Gr 10 9 Gr > 10 9 Laminar Turbulent Spring 00 ECE09: Chapters 9,10,11 10

6 Natural Convection Over Surfaces h δ n n Nu C(Gr ) C Ra Where Rayleigh number, Ra, is gβ ( Ts T Ra Gr ) δ The values of the constants C and n depend on the geometry of the surface and the flow regime, The value of n is usually ¼ for laminar flow and 1/ for turbulent flow. The value of the constant C is normally less than 1. (Table 11-1, P 87) Spring 00 ECE09: Chapters 9,10,11 11 Natural Convection Inside Enclosures gβ ( T1 T ) δ Ra δ T 1, T Distance between the hot & cold plates [m] Temperatures of hot & cold surfaces [ C] Spring 00 ECE09: Chapters 9,10,11 1 6