Chapter 7: Natural Convection


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1 71 Introduction 7 The Grashof Number 73 Natural Convection over Surfaces 74 Natural Convection Inside Enclosures 75 Similarity Solution 76 Integral Method 77 Combined Natural and Forced Convection
2 71 Introduction (1) Buoyancy forces are responsible for the fluid motion in natural convection. Viscous forces appose the fluid motion. Buoyancy forces are expressed in terms of fluid temperature differences through the volume expansion coefficient 1 V 1 ρ β = = V T ρ T P P ( 1 K) Viscous Force Buoyancy Force 71
3 volume expansion coefficient β The volume expansion coefficient can be expressed approximately by replacing differential quantities by differences as 1 Δρ 1 ρ ρ β = ρ ΔT ρ T T or 71 Introduction () at constant P ( ) ρ ρ = ρβ T T at constant P ( ) ( ) For ideal gas β = ideal gas 1 T 1/K ( ) 7
4 71 Introduction (3) Equation of Motion and the Grashof Number Consider a vertical hot flat plate g immersed in a quiescent fluid body. Assumptions: steady, laminar, twodimensional, Newtonian fluid, and constant properties, except the density difference rr (Boussinesq approximation). 73
5 71 Introduction (4) a Newton s second law of motion δ ma = x F x δm= ρ( dx dy 1) The acceleration in the xdirection is obtained by taking the total differential of u(x, y) x du u dx u dy = = + dt x dt y dt Zoom in g a x u = u + x u v y 74
6 71 Introduction (5) The net surface force acting in the xdirection Net viscous force Net pressure force Gravitational force τ P Fx = dy dx dx dy g dx dy y x ( 1) ( 1) ρ ( 1) u P = μ ρg y x ( dx dy 1) u u u P ρ u + v = μ ρg x y y x 75
7 71 Introduction (6) The xmomentum equation in the quiescent fluid outside the boundary layer (setting u=0) P = ρ g x Noting that v<<u in the boundary layer and thus v/ x v/ y 0, and there are no body forces (including gravity) in the y direction, P = 0 P P = = ρ g y the force balance in the ydirection is u u u ρ u + v = μ + ( ρ ρ) g x y y x x 76
8 71 Introduction (7) ρ ρ = ρβ T T at constant P ( ) ( ) u u u u + v = ν + gβ T T x y y ( ) The momentum equation involves the temperature, and thus the momentum and energy equations must be solved simultaneously. The set of three partial differential equations (the continuity, momentum, and the energy equations) that govern natural convection flow over vertical isothermal plates can be reduced to a set of two ordinary nonlinear differential equations by the introduction of a similarity variable. 77
9 7 The Grashof Number (1) The governing equations of natural convection and the boundary conditions can be nondimensionalized * x * y * u * v * T T x = ; y = ; u = ; v = ; T = L L V V T T c c s Substituting into the momentum equation and simplifying give u gβ T T L ( ) * * 3 * * * u * u s c T 1 u + v = * * + * x y ν ReL ReL y Gr L
10 7 The Grashof Number () The dimensionless parameter in the brackets represents the natural convection effects, and is called the Grashof number Gr L Gr L Gr L = = gβ Ts T L ν ( ) 3 Buoyancy force Viscous force c Viscous force The flow regime in natural convection is governed by the Grashof number Gr L >10 9 flow is turbulent Buoyancy force 79
11 73 Natural Convection over Surfaces (1) Natural convection heat transfer on a surface depends on geometry, orientation, variation of temperature on the surface, and thermophysical properties of the fluid. The simple empirical correlations for the average Nusselt number in natural convection are of the form hlc Nu = = C ( Gr Pr) n L = C Ra k Where Ra L is the Rayleigh number n L Ra L = Gr Pr = L gβ T T L ( ) 3 s ν c Pr 710
12 73 Natural Convection over Surfaces () The values of the constants C and n depend on the geometry of the surface and the flow regime (which depend on the Rayleigh number). All fluid properties are to be evaluated at the film temperature T f =(T s +T ). The Nusselt number relations for the constant surface temperature and constant surface heat flux cases are nearly identical. The relations for uniform heat flux is valid when the plate midpoint temperature T L/ is used for T s in the evaluation of the film temperature. hl ql & s Thus for uniform heat flux: Nu = = ( L ) k k T T 711
13 73 Natural Convection over Surfaces (3) Empirical correlations for Nu avg 71
14 74 Natural Convection Inside Enclosures (1) In a vertical enclosure, the fluid adjacent to the hotter surface rises and the fluid adjacent to the cooler one falls, setting off a rotationary motion within the enclosure that enhances heat transfer through the enclosure. Heat transfer through a horizontal enclosure hotter plate is at the top no convection currents (Nu=1). hotter plate is at the bottom Ra<1708 no convection currents (Nu=1). 3x10 5 >Ra>1708 Bénard Cells. Ra>3x10 5 turbulent flow. 713
15 74 Natural Convection Inside Enclosures () Nusselt Number Correlations for Enclosures: Simple powerlaw type relations in the form of where C and n are constants, are sufficiently accurate, but they are usually applicable to a narrow range of Prandtl and Rayleigh numbers and aspect ratios. Numerous correlations are widely available for horizontal rectangular enclosures, inclined rectangular enclosures, vertical rectangular enclosures, concentric cylinders, concentric spheres. n Nu = C Ra L 714
16 75 Similarity Solution (1) Similarity Transformation Transform three PDE to two ODE Introduce similarity variable η ( x, y) y ( x, y) = C βg 1/4 x η (7.8) 1/4 ( T T ) s C = 4v Grx y η 4 = (7.10) 0 ( x, y) = 0 ( η) x u = ψ y (7.14) 1 Gr 4 x ψ 4v ξ( η) 4 == (7.15) 1/
17 75 Similarity Solution () 716
18 76 Integral Method u( x,0) ν y δ + β g ( T T ) dy = 0 d dx 0 δ u dy (7.30) T y ( x,0) d dx δ ( x) α = u( T T ) dy (7.31)
19 77 Combined Natural and Forced Convection (1) Heat transfer coefficients in forced convection are typically much higher than in natural convection. The error involved in ignoring natural convection may be considerable at low velocities. Nusselt Number: Forced convection (flat plate, laminar flow): Natural convection (vertical plate, laminar flow): Nu Therefore, the parameter Gr/Re represents the importance of natural convection relative to forced convection. Nu forced convection natural convection Re Gr
20 77 Combined Natural and Forced Convection () Gr/Re <0.1 natural convection is negligible. Gr/Re >10 forced convection is negligible. 0.1<Gr/Re <10 forced and natural convection are not negligible. hot isothermal vertical plate Natural convection may help or hurt forced convection heat transfer depending on the relative directions of buoyancyinduced and the forced convection motions. 719
21 77 Combined Natural and Forced Convection (3) Nusselt Number for Combined Natural and Forced Convection: A review of experimental data suggests a Nusselt number correlation of the form ( n n ) 1 Nu = Nu ± Nu combined forced natural Nu forced and Nu natural are determined from the correlations for pure forced and pure natural convection, respectively. n 70
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