# INSTRUCTOR: PM DR MAZLAN ABDUL WAHID

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1 SMJ 4463: HEAT TRANSFER INSTRUCTOR: PM ABDUL WAHID TEXT: Introduction to Heat Transfer by Incropera, DeWitt, Bergman, Lavine 5 th Edition, John Wiley and Sons Chapter 9 Natural Convection Assoc Prof. Dr Mazlan Abdul Wahid Faculty of Mechanical Engineering Universiti Teknologi Malaysia 1

2 Objectives When you finish studying this chapter, you should be able to: Understand the physical mechanism of natural convection, Derive the governing equations of natural convection, and obtain the dimensionless Grashof number by nondimensionalizing them, Evaluate the Nusselt number for natural convection associated with vertical, horizontal, and inclined plates as well as cylinders and spheres, Examine natural convection from finned surfaces, and determine the optimum fin spacing, Analyze natural convection inside enclosures such as double-pane windows, and Consider combined natural and forced convection, and assess the relative importance of each mode. 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 = = ( 1 K) V T ρ T P P (9-3) Viscous Force Buoyancy Force 2

3 volume expansion coefficient β The volume expansion coefficient can be expressed approximately by replacing differential quantities by differences as 1 ρ 1 ρ ρ β = ( at constant P) (9-4) ρ T ρ T T or ρ ρ = ρβ T T at constant P (9-5) ( ) ( ) For ideal gas 1 β ideal gas = T ( 1/K ) (9-6) Equation of Motion and the Grashof Number Consider a vertical hot flat plate immersed in a quiescent fluid body. Assumptions: steady, laminar, two-dimensional, Newtonian fluid, and constant properties, except the density difference ρ-ρ (Boussinesq approximation). g 3

4 a Consider a differential volume element. Newton s second law of motion δ m a = F (9-7) x x x ( dx dy 1) δ m = ρ The acceleration in the x-direction is obtained by taking the total differential of u(x, y) du u dx u dy = = + dt x dt y dt g a x u u = u + v x y (9-8) The net surface force acting in the x-direction Net viscous force Net pressure force Gravitational force τ P Fx = dy dx dx dy g dx dy y x ( 1) ( 1) ρ ( 1) 2 u P = µ ρg 2 y x ( dx dy 1) (9-9) Substituting Eqs. 9 8 and 9 9 into Eq. 9 7 and dividing by ρ dx dy 1 gives the conservation of momentum in the x-direction 2 u u u P ρ u + v = µ ρg 2 x y y x (9-10) 4

5 The x-momentum equation in the quiescent fluid outside the boundary layer (setting u=0) P = ρ g (9-11) 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, the force balance in the y-direction is P P P = 0 = = ρ g y x x Substituting into Eq u u u ρ u + v = µ + 2 ( ρ ρ ) g x y y (9-12) Substituting Eq. 9-5 it into Eq and dividing both sides by ρ gives 2 u u u u + v = ν + gβ 2 ( T T ) (9-13) 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. 5

6 The Grashof Number The governing equations of natural convection and the boundary conditions can be nondimensionalized u * 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 ( ) gβ T T L * * 3 * 2 * * u * u s c T 1 u + v = * * * x y ν ReL ReL y Gr L (9-14) The dimensionless parameter in the brackets represents the natural convection effects, and is called the Grashof number Gr L gβ T T L Gr ( ) 3 s c L = (9-15) 2 Gr L = ν Buoyancy force Viscous force Viscous force The flow regime in natural convection is governed by the Grashof number Gr L >10 9 flow is turbulent Buoyancy force 6

7 Natural Convection over Surfaces 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 ( Pr) n n Nu = = C GrL = C RaL (9-16) k Where Ra L is the Rayleigh number gβ ( T ) 3 s T Lc RaL = GrL Pr = Pr (9-17) 2 ν 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/2 is used for T s in the evaluation of the film temperature. Thus for uniform heat flux: hl q& sl Nu = = k k TL 2 T ( ) (9-27) 7

8 (9-26) Empirical correlations for Nu avg (9-26) (9-30) (9-31) (9-34) (9-35) Natural Convection from Finned Surfaces Natural convection flow through a channel formed by two parallel plates is commonly encountered in practice. Long Surface fully developed channel flow. Short surface or large spacing natural convection from two independent plates in a quiescent medium. 8

9 The recommended relation for the average Nusselt number for vertical isothermal parallel plates is 0.5 hs Nu = = k ( Ras S L) ( Ras S L) Closely packed fins greater surface area smaller heat transfer coefficient. Widely spaced fins higher heat transfer coefficient smaller surface area. Optimum fin spacing for a vertical heat sink (9-31) S opt S L L = = Ras Ra 0.25 L (9-32) Natural Convection Inside Enclosures 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. 9

10 Nusselt Number Correlations for Enclosures Simple power-law type relations in the form of n Nu = C Ra L 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. Combined Natural and Forced Convection 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): Nu Natural convection (vertical plate, laminar flow): Nu forced convection natural convection Re Gr Therefore, the parameter Gr/Re 2 represents the importance of natural convection relative to forced convection

11 Gr/Re 2 <0.1 natural convection is negligible. Gr/Re 2 >10 forced convection is negligible. 0.1<Gr/Re 2 <10 forced and natural convection are not negligible. Natural convection may help or hurt forced convection heat transfer depending on the relative directions of buoyancy-induced and the forced convection motions. hot isothermal vertical plate 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 (9-66) combined forced natural Nu forced and Nu natural are determined from the correlations for pure forced and pure natural convection, respectively. n 11

12 Chapter 9 NATURAL CONVECTION Dr. Mazlan Abdul Wahid Faculty of Mechanical Engineering Universiti Teknologi Malaysia 12

13 The free (natural) convection is originated a thermal instability, i.e., when a body force acts on a fluid in which there are density gradients. The net effect is a buoyancy force, which induces free convection currents. The density gradient is mainly due to a temperature gradient and the body force is due to the gravity. In free convection, the convection rate are smaller compared those in the forced convection. In many systems involving multimode heat transfer effects, free convection provides the largest resistance to heat transfer and thus plays an important role in the design or performance of the system. When it is desirable to minimize heat transfer rates or to minimize operating cost, free convection is often preferred to forced convection. 13

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