Chapter 9

Chapter 9: Natural ConvectionYoav PelesDepartment of Mechanical, Aerospace and Nuclear Engineering Rensselaer Polytechnic Institute

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ObjectivesWhen 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 = (1 K ) (9-3) = V T P T PViscous Force Buoyancy Force

volume expansion coefficient () The volume expansion coefficient can be expressed approximately by replacing differential quantities by differences as

1 1 = ( ) T T T

( at constant P )

(9-4)

or

= (T T )1 = T

( at constant P )

(9-5)

For ideal gas (Pv = RT)ideal gas

(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 - (Boussinesq approximation).

g

Consider a differential volume element. Newtons second law of motion m ax = Fx (9-7) m = ( dx dy 1)

g

The acceleration in the x-direction is obtained by taking the total differential of u(x, y)

du u dx u dy ax = = + dt x dt y dt u u ax =u +v x y(9-8)

m in Zoo

P

The net surface force acting in the x-directionNet viscous force Net pressure force Gravitational force

P Fx = dy ( dx 1) dx ( dy 1) g ( dx dy 1) x y 2u P = 2 g ( dx dy 1) (9-9) x y Substituting Eqs. 98 and 99 into Eq. 97 and dividing by dxdy1 gives the conservation of momentum in the x-direction

u u 2u P u + v = 2 g y y x x

(9-10)

The x-momentum equation in the quiescent fluid outside the boundary layer (setting u = 0) P (9-11) = g () x Noting that v1708 Bnard Cells. Ra>3x105 turbulent flow.

Nusselt Number Correlations for Enclosures Simple power-law type relations in the form of

Nu = C Ra 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.n L

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):

Nuforced convection Re1 2

Natural convection (vertical plate, laminar flow):

Nunatural convection Gr1 4

Therefore, the parameter Gr/Re2 represents the importance of natural convection relative to forced convection.

Gr/Re2 < 0.1 : natural convection is negligible. Gr/Re2 > 10 : forced convection is negligible. 0.1 < Gr/Re2