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Brij Bhooshan Asst. Professor B.S.A College of Engg. & Technology, Mathura (India)

Copyright by Brij Bhooshan @ 2013

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In This Chapter We Cover the Following Topics

Art. Content Page

7.1 Concept of Buoyancy Force 2

7.2 Dimensionless Parameters of Natural Convection 3

7.3 An Approximate Analysis of Laminar Natural Convection on a Vertical Plate 5

7.4 Free Convection from Vertical Planes and Cylinders 9

7.5 Free Convection from Horizontal Cylinders 12

7.6 Free Convection from Horizontal Plates 12

7.7 Free Convection from Spheres 13

7.8 Free Convection in Enclosed Spaces 13

7.9 Rotating Cylinders, Disks and Spheres 17

7.10 Combined Forced and Natural Convection 18

References:

1- J. P. Holman, Heat Transfer, 9th Edn, MaGraw-Hill, New York, 2002.

2- James R. Welty, Charles E. Wicks, Robert E. Wilson, Gregory L. Rorrer

Fundamentals of Momentum, Heat, and Mass Transfer, 5th Edn, John Wiley & Sons,

Inc., 2008.

3- F. Kreith and M. S. Bohn, Principal of Heat Transfer, 5th Edn, PWS Publishing Co.,

Boston, 1997.

4- P. K. Nag, Heat and Mass Transfer, 2nd Edn, MaGraw-Hill, New Delhi 2005.

Please welcome for any correction or misprint in the entire manuscript and your

valuable suggestions kindly mail us [email protected].




2 Chapter 7: Emparical Relation for Free Convection

The flow velocity in free convection is much smaller than that encountered in forced

convection. Therefore, heat transfer by free convection is much smaller than that by

forced convection. Diagram 7.1 (a) illustrates the development of velocity field infront of

a hot vertical plate owing to the buoyancy force. The heated fluid in front of the hot plate

rises, entraining fluid from the quiescent outer region.

Diagram 7.1 (b) shows a cold vertical plate in a hot fluid, where the direction of motion

is reversed, the fluid in front of the plate being heavier moves vertically down, again

entraining fluid from the quiescent outer region.

Diagram 7.1 Laminar and turbulent velocity boundary layer for natural convection on a vertical plate

In both cases a velocity boundary layer is develops with a certain peak in it. The velocity

is zero both the plate surface and at the edge of boundary layer is laminar, then at a

certain distance from the leading edge the transition to turbulent layer occurs, and

finally a fully developed turbulent layer is established.

7.1 CONCEPT OF BUOYANCY FORCE

We now consider a fluid contained in the space between two parallel horizontal plates

(Diagram 7.2(a)).

Diagram 7.2

Suppose the lower plate is maintained at a temperature higher than that of the upper

plate (Tl > T2). A temperature gradient will be established in the vertical direction. The

layer will be top-heavy, since the density of the cold fluid at the top is higher than that

of the hot fluid at the bottom. If the temperature difference is increased beyond a certain

critical value, the viscous forces within the fluid can no longer sustain the buoyancy

forces, and a convection motion is set up.

Suppose in Diagram 7.2(b), the lower plate is cold and the upper plate is hot (i.e. T l <

T2). Here, the density of the top layer is less than that of the bottom layer. The fluid is

then always stable, and no natural convection currents are set up.

(a) Lower plate hot

Unstable fluid

Hot wall

Cold wall

(b) Lower plate hot

Stable

fluid

Cold wall

Hot wall

(a) Hot wall

Turbulent

Laminar

(b) Cold wall

Turbulent

Laminar




3 Heat and Mass Transfer By Brij Bhooshan

Diagram 7.3 Buoyancy-driven flows on horizontal cold (Tw < T) and hot (Tw > T) plates,

(a) Top surface of cold plate, (b) bottom surface of cold plate, (c) top surface of hot plate and (d) bottom

surface of hot plate

Diagram 7.3 shows the directions of convection currents for horizontal plates, heated or

cooled, facing up or down.

7.2 DIMENSIONLESS PARAMETERS OF NATURAL CONVECTION

To develop the principal dimensionless parameters of natural convection, we consider

the natural convection on a vertical plate, as illustrated in Diagram 7.1. For simplicity

analysis, we assume that the boundary layer flow is steady and laminar. Since small

flow velocities are associated with natural convection, the viscous energy dissipation

term in the energy equation can be neglected. Then to governing continuity, momentum

and energy equations are obtained from the boundary layer equations, as derived in the

last chapter, and the appropriate buoyancy term is introduced in the momentum

equation:

Continuity:

Momentum:

Energy:

Here the term ρg on the right hand side of the momentum equation represents the

body force exerted on the fluid element in the negative x-direction.

For small temperature differences, the density ρ in the buoyancy term is considered to

vary with temperature, whereas the density appearing elsewhere in these equations is

considered constant. This is often referred to as Boussinesq approximation.

To determine the pressure gradient term dp/dx, the x-momentum equation, Eq. (7.2)

is evaluated at the edge of the velocity boundary layer, where u → 0 and ρ → ρ. We

obtain

where ρ is the fluid density outside the boundary layer. Then the term ρg dP/dx

appearing in the momentum equation, Eq. (7.2) becomes

If β denotes the volumetric coefficient in thermal expansion of the fluid,

(a)

(b)

(d)

(c)





Using equation (7.4) and (7.7)

To find the dimensionless parameters that governs heat transfer in natural convection.

Suppose

Using Eqns. (7.1) and (7.9)

Using Eqns. (7.2), (7.8) and (7.9)

Using Eqns. (7.3) and (7.9)

Here, the Reynolds and Prandtl numbers are defined as

The dimensionless group in the momentum equation can be rearranged as

where the Grashof number Gr is defined as

The Grashof number represents the ratio of the buoyancy force to the viscous force

acting on the fluid.

Equation (7.11) imply that when the effects of natural and forced convection are of

comparable magnitude, the Nusselt number depends on Re, Pr and Gr, or

The parameter Gr/Re2, defined by Eq. (5.13), is a measure of the relative importance of

natural convection in relation to forced convection.

When Gr/Re2 ≌ 1, natural and forced convection are of the same order of magnitude;

hence both must be considered.

If (Gr/Re2) << 1, flow is primarily by forced convection.

If (Gr/Re2) >> 1, natural convection becomes dominant and the Nusselt number depends

on Gr and Pr only:





In natural convection, flow velocities are produced by the buoyancy forces only; hence

there are no externally induced flow velocities. As a result, the Nusselt number does not

depend on the Reynolds number.

Sometimes another dimensionless parameter, called the Rayleigh number (Ra), which is

defined as

The Grashof number to correlate heat transfer in natural convection. Then the Nusselt

number relation (Eq. (7.16)), becomes

For three-dimensional shapes such as short cylinders and blocks the characteristic

length L may be determined from

where Lx is the height and Ly the average horizontal dimension of the body.

7.3 AN APPROXIMATE ANALYSIS OF LAMINAR NATURAL CONVECTION ON

A VERTICAL PLATE

Let us consider Tw and T be, respectively, the temperature of the wall surface and the

bulk temperature of the fluid (Diagram 7.4). The fluid moves upward along the plate for

Tw > T and flow downwards for Tw < T, as shown in Diagram 7.1. Within the boundary

layer temperature decrease from Tw to < T of the undisturbed or quiescent fluid outside

the heated region.

Diagram 7.4 Temperature and velocity profile for free convection on a hot vertical plate

Let θ = T T. When y = 0, θ = θw = Tw T, and when y = δ, θ = θ = 0. If y = 0, u = 0,

and if y = δ, u = 0.

The velocity and temperature profiles in the neighbourhood of the plate are shown. The

integral boundary layer equations for momentum and energy will be used to calculate

the heat transfer in natural convection.

Temperature profile

To solve the boundary layer equation, the temperature profile is approximated by a

parabolic equation of the form

At y = 0, T = Tw = C

At y = δ = δt, T = T, (∂T/∂y)y = δ = 0

Boundary layer





It is we assume δ = δt i.e. equal velocity and thermal boundary layer thicknesses.

Substituting in Eq. (7.20),

Therefore

The velocity profile may be assumed to be a cubical parabola given by

where u1 is a reference velocity and is a function of x.

At y = 0, u = 0.

At y = δ = δt, u = 0, ∂u/∂y = 0

Using the first boundary condition u = 0 = au1

Since u1 ≠ 0, d = 0

Now

At y = δ = δt, u = 0, ∂u/∂y = 0

From momentum equation,

Therefore

when y = 0, θ = θw, then we have

From Eqns. (7.23a), (7.23b) and (7.24a), we have





where

The velocity profile given by

There is a certain value of y where u is maximum

After solving we get y = δ or δ/3.

Since u = 0 at y = δ, therefore, u will be maximum when y = δ/3.Therefore

Analysis

Let us consider a control volume differential element dx at a distance from the bottom

edge within the boundary layer as shown in Diagram 7.5.

Diagram 7.5 Control volume in the boundary layer

Momentum flux across BC is zero.

Rate of increase of momentum

= Forces acting on the element

Integrating is limited to δ, as before

Energy equation for volume element gives

L

C.V.

C

B





Integrating is limited to δ = δt,

Now

Substituting these values in momentum equation (7.27)

and energy equation

But u0 and δ are function of x. Suppose

Now using equation (7.29), then

Again energy equation (7.30)

Equation (7.31) and (7.32) are valid for any value of x.

Equating the corresponding component of x in Eq. (7.31) and (7.32), then

Put the values of m and n in Eq. (7.31) and (7.32),





Now using Eqns. (7.33a) and (7.33b),

Heat flux

As x increases, δ also increases.

h 1/x1/4, as increases, h decreases.

For air Pr = 0.714, then using equation (7.35)

Exact solution gives the constant as 0.360, then

Fluid properties are evaluated at the film temperature T* = (Tw + T)/2.

Transition from laminar to turbulent flow occurs at Rax,c = 109.

EMPIRICAL RELATIONS FOR FREE CONVECTION

7.4 FREE CONVECTION FROM VERTICAL PLANES AND CYLINDERS





Isothermal Surfaces

The local value of heat transfer coefficient from equation (7.34) and (7.35)

The average value of the heat transfer coefficient for a height L is obtained

For air, Pr = 0.714

which is almost the same as Eq. (7.37).

McAdams recommends the relation for natural convection over a vertical flat plate or

vertical cylinder in the turbulent region (GrL > 109)

For laminar flow (valid for 104 < Gr.Pr < 109)

For laminar flow (valid for 109 < Gr.Pr < 1012)

The general criterion is that a vertical cylinder may be treated as a vertical flat plate,

when

for gases with Pr =0.7 indicates that the flat plate results for the average heat-transfer

coefficient should be multiplied by a factor F to account for the curvature, where

Churchill and Chu that are applicable over wider ranges of the Rayleigh number:

above equation (7.44) applies to laminar flow only and holds for all values of the Prandtl

number and for 10-1 < RaL < 109.

above equation (7.45) applies for both laminar and turbulent flow and for 10-1 < RaL <

109.

The physical properties are evaluated at the film temperature T* = (Tw + T)/2.





Diagram 7.6 (a) heating surface facing downwards, (b) Cold surface facing upwards

For a long vertical plate or a long cylinder tilted at an angle θ from the vertical with the

heated surface facing downward (Diagram 7.6(a)) or cooled surface facing upward

(Diagram 7.6 (b)), the following equation can be used:

for 105 < Gr.Pr cos θ < 1011; and 0 ≤ θ ≤ 89°.

Constant-Heat-Flux Surfaces

Extensive experiments have been reported for free convection from vertical and inclined

surfaces to water under constant-heat-flux conditions.

Suppose modified Grashof number, Gr* is defined as:

qw = being the constant wall heat flux.

Natural convection on a vertical plate subject to uniform heat flux at the wall surface

was investigated by Sparrow and Gregg, Vliet and Liu and Vliet. On the basis of their

experimental data, the following correlations were proposed:

Local heat transfer coefficient for laminar flow

Local heat transfer coefficient for turbulent region

Now local heat-transfer form gives

Inserting Grx = Gr*x/Nux gives

Thus, when the “characteristic” values of m for laminar and turbulent flow are

compared to the exponents on Gr*x, we obtain

Insulation

Plate

g

Insulation





Churchill and Chu show that the constant-heat-flux case if the average Nusselt number

is based on the wall heat flux and the temperature difference at the center of the plate (x

= L/2). The result is

where

7.5 FREE CONVECTION FROM HORIZONTAL CYLINDERS

Churchill and Chu proposed:

for 10-5 < Gr Pr < 1012

For laminar flow range of 10-6 < Gr Pr < 109.

The physical properties are evaluated at the film temperature T* = (Tw + T)/2.

Heat transfer from horizontal cylinders to liquid metals may be calculated from

Mc Adams proposed the relation

7.6 FREE CONVECTION FROM HORIZONTAL PLATES

Uniform wall temperature

The mean Nusselt number for natural convection on a horizontal plate as correlated by

McAdams is

For hot surface facing up or cold surface facing down:

(a) In the range (laminar) 105 < Ra < 2 107, C = 0.54, n = 1/4, and

(b) In the range (turbulent) 2 107 < Ra < 3 1010, C = 0.14, n = 1/3.

For hot surface facing down or cold surface facing up:

(a) In the range (laminar) 3 105 < Ra < 3 1010, C = 0.27, n = 1/4.

(b) In the range (turbulent) 7 106 < Ra < 11 109, C = 0.107, n = 1/3.

Uniform heat flux

For the horizontal plate with heated surface facing upward:

For the horizontal plate with the heated surface facing downward:





Properties are evaluated at T* = Tw 0.25 (Tw + T).

7.7 FREE CONVECTION FROM SPHERES

For natural convection on a single isothermal sphere for fluids having Pr ≌ 1

For higher ranges of the Rayleigh number the experiments of Amato and Tien with

water suggest the following correlation:

for 3 105 < Rad < 8 108 and 10 < Nud < 90

Yuge recommends the following empirical relation for free-convection heat transfer from

spheres to air:

Churchill suggests a more general formula for spheres, applicable over a wider range of

Rayleigh numbers:

for Rad < 1011 and Pr >0.5.

7.8 FREE CONVECTION IN ENCLOSED SPACES

The free-convection flow phenomena inside an enclosed space are interesting examples

of very complex fluid systems that may yield to analytical, empirical, and numerical

solutions.

Consider the system shown in Diagram 7.7(a), where a fluid is contained between two

vertical plates separated by the distance δ. As a temperature difference ΔTw = T1 − T2 is

impressed on the fluid, a heat transfer will be experienced with the approximate flow

regions shown in Diagram 7.7(b).

Diagram 7.7

According to MacGregor and Emery, in this Diagram 7.7(b), the Grashof number is

calculated as

(b)

(a)

Nu

Laminar

boundary layer

flow

Asymptotic flow

Turbulent boundary layer flow

Typical velocity

temperature profile

L

T T

L





As the Grashof number is increased, different flow regimes are encountered, as shown,

with a progressively increasing heat transfer as expressed through the Nusselt number

The heat transfer to a number of liquids under constant-heat-flux conditions, the

empirical correlations obtained were:

L/δ is aspect ratio

Valid for qw = const., T = 90°; 104 < Grδ Pr < 107; 1 < Pr < 20,000; 10 < L/δ < 40

Valid for qw = const., 106 < Grδ Pr < 109; 1 < Pr < 20; 1 < L/δ< 40

As Gr increases, the flow becomes more of a boundary layer type with fluid rising in a

layer near the heated surface, turning the corner at the top, and flowing downward in a

layer near the cooled surface. The boundary layer thickness decreases with Grd1/4, and

the core region is more or less inactive and thermally stratified. For the geometry in

Diagram 7.8,

Diagram 7.8 Natural convection in inclined enclosed spaces

For 2 < L/ < 10, Pr < 10 and Ra < 1010 and

in the range 1 < L/ < 2, 103 < Pr < 105, and

Tilted vertical enclose 0 < < 90° or 0 < τ < 90°

For H/L ≥ 12, 0 < < 70°

For H/L > 12, 70° < < 90°

The heat flux is calculated as

Rotating cell L

L

HOT





The results are sometimes expressed in the alternate form of an effective or apparent

thermal conductivity ke, defined by

By comparing Equations (7.65) and (7.66), we see that

In the building industry the heat transfer across an air gap is sometimes expressed in

terms of the R values, so that

Evans and Stefany have shown that transient natural-convection heating or cooling in

closed vertical or horizontal cylindrical enclosures may be calculated with

for the range 0.75 < L/d < 2.0. The Grashof number is formed with the length of the

cylinder L.

The effective thermal conductivity for fluids between concentric spheres with the

relation

where now the gap spacing is δ = r0 − ri. The effective thermal conductivity given by

Equation (7.70) is to be used with the conventional relation for steady-state conduction

in a spherical shell:

Equation (7.70) is valid for 0.25 ≤ δ/ri ≤ 1.5 and 1.2 × 102 < Gr Pr < 1.1 × 109, 0.7 < Pr <

4150

Properties are evaluated at a volume mean temperature Tm defined by

where rm =(ri + r0)/2.

Experimental results for free convection in enclosures are not always in agreement, but

we can express them in a general form as

For the annulus space the heat transfer is based on





where L is the length of the annulus and the gap spacing is δ = r0 − ri.

Diagram 7.9 Natural convection heat transfer in the annular space between two concentric cylinders or

concentric spheres

For natural convection heat transfer across the gap between two horizontal concentric

cylinders (Diagram 7.9) the following correlation is suggested for heat flow per unit

length (W/m)

where the effective thermal conductivity ke given by

Which is valid in range 0.7 < Pr < 6000, 10 ≤ Racyc ≤ 107.

For concentric spheres the following correlation is recommended

Valid for 102 ≤ Ra*cyc ≤ 104

Jacobs has suggested the correlation for vertical enclosed air space shown diagram

Valid for 2000 ≤ GrL ≤ 2 104

Valid for 2 104 ≤ GrL ≤ 11 106

where

Gr based on thickness of air space L.

Flow pattern

Outer cylinder

Inner cylinder

L





Jacobs has suggested the correlation for horizontal enclosed air space

Valid for 104 ≤ GrL ≤ 4 104

Valid for 4 105 ≤ GrL

where

Globe and Dropkin gives the relation for liquid contained in horizontal space

Valid for 3 x 105 ≤ Gr.Pr ≤ 7 x 109

Radiation R-Value for a Gap

The radiation transfer across a gap separating two large parallel planes may be

calculated with

Using the concept of the R-value

and thus could determine an R-value for the radiation heat transfer in conjunction with

Equation (7.82).

so that

The total R-value for the combined radiation and convection across the space would be

written as

7.9 ROTATING CYLINDERS, DISKS AND SPHERES

Heat transfer by convection between a rotating body and a surrounding fluid is of

importance in the thermal analysis of flywheels, turbine rotors and other rotating

components of various machines. With heat transfer, a critical velocity is reached when

the circumferential speed of the cylinder surface becomes approximately equal to the

upward natural convection velocity at the side of a heated stationary cylinder. Below the

critical velocity, simple natural convection, characterised by the conventional Grashof

number, [gβ(Tw T)D3]/ν2 controls the rate of heat transfer. At speeds greater than

critical (Rew > 8000 in air) the peripheral-speed Reynolds number D2w/ becomes the

controlling parameter. The combined effects of the Reynolds, Prandtl and Grashof

numbers on the average Nusselt number for a horizontal cylinder rotating in air above





the critical velocity as shown in diagram (Diagram 7.10) cab be expressed by empirical

equation.

Diagram 7.10 Horizontal cylinder rotating in air

The boundary layer on a rotating disk is laminar and of uniform thickness at rotational

Reynolds numbers wD2/ below about 106. At higher Reynolds numbers the flow

becomes turbulent near the outer edge, and as Rew is increased, the transition point

moves radially inward. The boundary layer thickens with increasing radius (Diagram

7.11).

Diagram 7.11 Velocity and boundary layer profiles for a disk rotating in an infinite environment

For the laminar regime in air

for wD2/ < 106

In the turbulent flow regime (wD2/ν > 106) of a disk in air, the local value at a radius r is

For a sphere of diameter D rotating in an infinite environment with Pr > 0.7 in laminar

regime (Rew = wD2/ν > 5 104), the average Nusselt number (hcD/k) can be obtained

from

while in the regime 5 104 < Rew < 7 105 the equation is

7.10 COMBINED FORCED AND NATURAL CONVECTION

The relative magnitude of the dimensionless parameter Gr/Re governs the relative

importance of natural convection in relation to forced convection where

which represents the ratio of the buoyancy forces to inertia forces. When this ratio is of

the order of unity, i.e. Gr ≌ Re2, the natural and forced convection are of comparable

magnitude, and hence they should be analysed simultaneously. If

Gr / Re2 >> 1: Natural convection dominates

Transition

g

D





Gr / Re2 ≌ 1: Natural and forced convection are of comparable magnitude

Gr / Re2 << 1: Forced convection dominates

For combined free and forced convection in the laminar flow regime inside a circular

tube. Brown and Gauvin recommend the following correlation for the Nusselt number

where Gz is the Graetz number, defined as

where Grd and Red are based on the tubes inside diameter with ΔT = Tw T difference

between tube wall and fluid bulk temperature.

External flow

Nux for mixed convection on vertical plate is given by

If (Grx/Rex-2) ≤ A

If (Grx/Rex-2) > A

A ≌ 0.6 for Pr < 10; A ≌ 1.0 for Pr = 100

For horizontal plate when (Grx/Rex2.5) ≤ 0.083 the equation for forced convection

Internal flow

For mixed convection in turbulent flow in horizontal tubes

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