chapter 7: natural convection -...

Chapter 7: Natural ConvectionAdvanced Heat TransferY.C. Shih Spring 2009

Chapter 7: Natural Convection

7-1 Introduction7-2 The Grashof Number7-3 Natural Convection over Surfaces7-4 Natural Convection Inside Enclosures7-5 Similarity Solution7-6 Integral Method7-7 Combined Natural and Forced Convection


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 1 1 KP P

VV T T

ρβρ

∂ ∂⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠

ViscousForce

BuoyancyForce

7-1 Introduction (1)

7-1


volume expansion coefficient βThe volume expansion coefficient can be expressed approximately by replacing differential quantities by differences as

or

For ideal gas

( )1 1 at constant PT T T

ρ ρρβρ ρ

∞

∞

−Δ≈ − = −

Δ −

( ) ( ) at constant T T Pρ ρ ρβ∞ ∞− = −

( )ideal gas1 1/KT

β =


7-2


Equation of Motion and the Grashof NumberConsider a vertical hot flat plate immersed in a quiescent fluid body.Assumptions:

steady,laminar,two-dimensional, Newtonian fluid, andconstant properties, except the density difference r-r∞ (Boussinesqapproximation).

g


7-3


g

Zoom in

Newton’s second law of motion

( )1x xm a F

m dx dyδ

δ ρ

⋅ =

= ⋅ ⋅

xdu u dx u dyadt x dt y dt

∂ ∂= = +

∂ ∂

xu uax y

∂ ∂= +

∂ ∂

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

u v


7-4


The net surface force acting in the x-direction

( ) ( ) ( )

( )

Net pressure forceNet viscous forceGravitational force

2

2

1 1 1

1

xPF dy dx dx dy g dx dy

y x

u P g dx dyy x

τ ρ

μ ρ

⎛ ⎞∂ ∂⎛ ⎞= ⋅ − ⋅ − ⋅ ⋅⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠⎝ ⎠⎛ ⎞∂ ∂

= − − ⋅ ⋅⎜ ⎟∂ ∂⎝ ⎠

6447448 64474486447448

2

2

u u u Pu v gx y y x

ρ μ ρ⎛ ⎞∂ ∂ ∂ ∂

+ = − −⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠


7-5


The x-momentum equation in the quiescent fluid outside the boundary layer (setting u=0)

Noting that v<<u in the boundary layer and thus ∂v/ ∂x≈ ∂v/∂y ≈0, andthere are no body forces (including gravity) in the y-

direction,

the force balance in the y-direction is

P gx

ρ∞∞

∂= −

∂

0Py

∂=

∂PP g

x xρ∞

∞∂∂

= = −∂ ∂

( )2

2

u u uu v gx y y

ρ μ ρ ρ∞

⎛ ⎞∂ ∂ ∂+ = + −⎜ ⎟∂ ∂ ∂⎝ ⎠


7-6


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 theintroduction of a similarity variable.

( )2

2

u u uu v g T Tx y y

ν β ∞∂ ∂ ∂

+ = + −∂ ∂ ∂

( ) ( ) at constant T T Pρ ρ ρβ∞ ∞− = −


7-7


7-2 The Grashof Number (1)

The governing equations of natural convection and the boundary conditions can be nondimensionalized

Substituting into the momentum equation and simplifying give

* * * * * ; ; ; ; c c s

T Tx y u vx y u v TL L V V T T

∞

∞

−= = = = =

−

( )2

3* * * 2 ** *

* * 2 2 *

1Re Re

L

s c

L L

Gr

g T T Lu u T uu vx y y

βν

∞⎡ ⎤−∂ ∂ ∂+ = +⎢ ⎥∂ ∂ ∂⎣ ⎦144424443

7-8


The dimensionless parameter in the brackets represents the natural convection effects, and is called the Grashof number GrL

The flow regime in natural convection is governed by the Grashof number

GrL>109 flow is turbulent

( ) 3

2s c

L

g T T LGr

βν

∞−=

GrL=Buoyancy forceViscous force

Buoyancy force

Viscous force

7-2 The Grashof Number (2)

7-9


7-3 Natural Convection over Surfaces (1)Natural convection heat transfer on a surface depends on

geometry,orientation, variation of temperature on the surface, andthermophysical properties of the fluid.

The simple empirical correlations for the average Nusselt number in natural convection are of the form

Where RaL is the Rayleigh number

( )Pr n ncL L

hLNu C Gr C Rak

= = ⋅ ⋅ = ⋅

( ) 3

2Pr Prs cL L

g T T LRa Gr

βν

∞−= ⋅ =

7-10


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 temperatureTf=(Ts+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 TL/2 is used for Ts in the evaluation of the film temperature.Thus for uniform heat flux: ( )2

s

L

q LhLNuk k T T∞

= =−

&

7-3 Natural Convection over Surfaces (2)

7-11


Empirical correlations for Nuavg

7-3 Natural Convection over Surfaces (3)

7-12


7-4 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 rotationarymotion within the enclosure that enhances heat transfer through the enclosure.Heat transfer through a horizontal enclosure

hotter plate is at the top ─ no convectioncurrents (Nu=1).hotter plate is at the bottom

Ra<1708 no convection currents (Nu=1).3x105>Ra>1708 Bénard Cells.Ra>3x105 turbulent flow.

7-13


Nusselt Number Correlations for Enclosures:

Simple power-law 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.

nLNu C Ra= ⋅

7-4 Natural Convection Inside Enclosures (2)

7-14


7-5 Similarity Solution (1)

Similarity Transformation• Transform three PDE to two ODE

• Introduce similarity variable ),( yxη

4/1),(x

yCyx =η (7.8)( ) 4/1

24 ⎥⎦⎤

⎢⎣⎡ −

= ∞

vTTβgC s

xyGr

η x4/1

4⎟⎠⎞

⎜⎝⎛= (7.10) )(),( η00 =yx

yu

∂∂

=ψ

(7.14)

( )ηξGrvψ x41

44 ⎟

⎠⎞

⎜⎝⎛== (7.15)

7-15


7-5 Similarity Solution (2)

7-16


7-6 Integral Method

dyudxddyTTg

yxu

∫∫ =−+∂

∂− ∞

δδβν

00

2)()0,( (7.30)

( )∫ ∞−=

∂∂

−)(

0)(0, x

dyTTudxd

yxT δ

α (7.31)

7-17


7-7 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):

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

1 2forced convection ReNu ∝

1 4natural convectionNu Gr∝

7-18


Gr/Re2<0.1 natural convection is negligible.

Gr/Re2>10forced convection is negligible.

0.1<Gr/Re2<10forced 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


7-19


Nusselt Number for Combined Natural and Forced Convection:

A review of experimental data suggests a Nusselt number correlation of the form

Nuforced and Nunatural are determined from the correlations for pure forced and pure natural convection, respectively.

( )1

combined forced natural

nn nNu Nu Nu= ±


7-20

chapter 7: natural convection -...

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