Chapter 9

NATURAL CONVECTION

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PHYSICAL MECHANISM OF NATURAL CONVECTION
Many familiar heat transfer applications involve natural convection as the primary
mechanism of heat transfer. Examples?
Natural convection in gases is usually accompanied by radiation of comparable
magnitude except for low-emissivity surfaces.
The motion that results from the continual replacement of the heated air in the
vicinity of the egg by the cooler air nearby is called a natural convection current,
and the heat transfer that is enhanced as a result of this current is called natural
convection heat transfer.

The warming up
of a cold drink in a
warmer
environment by
The cooling of a boiled egg in a cooler natural
environment by natural convection. convection. 2
Buoyancy force: The upward force exerted by a fluid on a body completely or
partially immersed in it in a gravitational field. The magnitude of the buoyancy
force is equal to the weight of the fluid displaced by the body.

The net vertical force acting on a body

Archimedes’ principle: A body

immersed in a fluid will experience
a “weight loss” in an amount equal
to the weight of the fluid it
displaces.

The “chimney effect” that induces

the upward flow of hot combustion
gases through a chimney is due
to the buoyancy effect.
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 Volume expansion coefficient: Variation of
the density of a fluid with temperature at
constant pressure.

ideal gas

The larger the temperature

difference between the fluid adjacent
to a hot (or cold) surface and the
fluid away from it, the larger the
buoyancy force and the stronger the
The coefficient of volume expansion natural convection currents, and thus
is a measure of the change in the higher the heat transfer rate.
volume of a substance with
temperature at constant pressure. 4
In natural convection, no blowers are used, and
therefore the flow rate cannot be controlled externally.
The flow rate in this case is established by the dynamic
balance of buoyancy and friction.

An interferometer produces a
map of interference fringes,
which can be interpreted as
lines of constant temperature.
The smooth and parallel lines
in (a) indicate that the flow is
laminar, whereas the eddies
and irregularities in (b) indicate
that the flow is turbulent.
The lines are closest near the
surface, indicating a higher
temperature gradient.
Isotherms in natural convection
over a hot plate in air.
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EQUATION OF MOTION AND THE GRASHOF NUMBER
The thickness of the boundary layer
increases in the flow direction.
Unlike forced convection, the fluid
velocity is zero at the outer edge of the
velocity boundary layer as well as at the
surface of the plate.
At the surface, the fluid temperature is
equal to the plate temperature, and
gradually decreases to the temperature
of the surrounding fluid at a distance
sufficiently far from the surface.
In the case of cold surfaces, the shape
of the velocity and temperature profiles
remains the same but their direction is
reversed.

Typical velocity and temperature

profiles for natural convection
flow over a hot vertical plate at
temperature Ts inserted in a fluid
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at temperature T.
Derivation of the
equation of motion
that governs the
natural convection
flow in laminar
boundary layer

This is the equation that governs the fluid motion in

the boundary layer due to the effect of buoyancy. The
Forces acting on a differential momentum equation involves the temperature, and
volume element in the natural thus the momentum and energy equations must be
solved simultaneously.
convection boundary layer
over a vertical flat plate. 7
The complete set of conservation equations, continuity, momentum, and
energy that govern natural convection flow over vertical isothermal plates are:

Solving as shown before using integral eqn, boundary conditions, the

boundary layer thickness

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 Buoyancy forces/viscous forces

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• The Grashof number provides the main criterion in determining whether the
fluid flow is laminar or turbulent in natural convection.
• For vertical plates, the critical Grashof number is observed to be about 109.

When a surface is subjected to external

flow, the problem involves both natural
and forced convection.
The relative importance of each mode of
heat transfer is determined by the
value of the coefficient Gr/Re2:
• Natural convection effects are
negligible if Gr/Re2 << 1.
• Free convection dominates and the
forced convection effects are negligible
if Gr/Re2 >> 1.
The Grashof number Gr is a • Both effects are significant and must
measure of the relative be considered if Gr/Re2  1 (mixed
magnitudes of the buoyancy convection).
force and the opposing viscous
force acting on the fluid.
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11
NATURAL CONVECTION OVER SURFACES
Natural convection heat transfer on a surface depends on the geometry of the
surface as well as its orientation, the variation of temperature on the surface and
the thermophysical properties of the fluid involved.
With the exception of some simple cases, heat transfer relations in natural
convection are based on experimental studies.

Rayleigh
number
The constants C and n depend on the
geometry of the surface and the flow
regime, which is characterized by the
range of the Rayleigh number.
The value of n is usually 1/4 for Natural convection heat transfer
laminar flow and 1/3 for turbulent flow. correlations are usually expressed
in terms of the Rayleigh number
All fluid properties are to be evaluated raised to a constant n multiplied
at the film temperature Tf = (Ts + T)/2. by another constant C, both of
which are determined
experimentally. 12
For vertical surfaces, the Nusselt and Grashof numbers are formed with L, the
height of the surface as the characteristic dimension.

If the boundary-layer thickness is not large compared with the diameter of the
cylinder, the heat transfer may be calculated with the same relations used for
vertical plates.

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

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14
Free convection is studied from vertical and inclined surfaces to water under
constant-heat-flux conditions. In such experiments, the results are presented in
terms of a modified Grashof number, Gr∗:

Boundary-layer transition was observed to begin between Gr∗x Pr =3×1012 and

4×1013 and to end between 2×1013 and 1014. Fully developed turbulent flow was
present by Gr∗x Pr =1014

Although expts were conducted with water they hold fairly well for air.

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For laminar range

For turbulent range

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 Churchill and Chu
A simpler equation is available but is
restricted to the laminar range of 10−6
<Gr Pr <109:

Heat transfer from horizontal cylinders to liquid metals may be calculated

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Horizontal Plates For a hot surface in a cooler environment,
the net force acts upward, forcing the
heated fluid to rise.
If the hot surface is facing upward, the
heated fluid rises freely, inducing strong
natural convection currents and thus
effective heat transfer.
But if the hot surface is facing downward,
the plate blocks the heated fluid that tends
to rise, impeding heat transfer.
The opposite is true for a cold plate in a
warmer environment since the net force
(weight minus buoyancy force) in this case
Natural convection flows on acts downward, and the cooled fluid near
the upper and lower surfaces the plate tends to descend.
of a horizontal hot plate.
Characteristic
length

Lc = a/4 for a horizontal square surface of length a

Lc = D/4 for a horizontal circular surface of diameter D 18
For heated surface
facing upwards

For heated surface

facing downwards

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 For the inclined plate facing downward with approx
constant heat flux, the following correlation was
obtained for the average Nusselt number:

All properties except β are evaluated at a reference temp Te

defined by

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21
22
Horizontal Cylinders and Spheres
The boundary layer over a hot horizontal
cylinder starts to develop at the bottom,
increasing in thickness along the
circumference, and forming a rising plume at
the top.
Therefore, the local Nusselt number is
highest at the bottom, and lowest at the top of
the cylinder when the boundary layer flow
remains laminar.
The opposite is true in the case of a cold
Natural convection horizontal cylinder in a warmer medium, and
flow over a horizontal the boundary layer in this case starts to
hot cylinder. develop at the top of the cylinder and ending
with a descending plume at the bottom.

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NATURAL CONVECTION FROM FINNED SURFACES AND PCBs
The plates could be the fins of a finned heat
sink, or the PCBs of an electronic device.
The plates can be approximated as being
isothermal (Ts = constant) in the first case, and
isoflux (qs = constant) in the second case.
Boundary layers start to develop at the lower
ends of opposing surfaces, and eventually
merge at the midplane if the plates are vertical
and sufficiently long. In this case, we will have
fully developed channel flow after the merger of
the boundary layers, and the natural convection
flow is analyzed as channel flow.
But when the plates are short or the spacing is
large, the boundary layers of opposing surfaces
never reach each other, and the natural
Natural convection flow convection flow on a surface is not affected by
through a channel the presence of the opposing surface. In that
between two isothermal case, the problem should be analyzed as natural
vertical plates. convection from two independent plates in a
quiescent medium.
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NATURAL CONVECTION INSIDE ENCLOSURES
Enclosures are frequently encountered in practice, and heat transfer through them is of
practical interest. 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.
Lc charecteristic length: the distance between the hot
and cold surfaces
T1 and T2: the temperatures of the hot and cold surfaces
Nu = 1
Ra > 1708, natural
convection currents
Ra > 3106, turbulent
fluid motion
Fluid properties at

Convective currents
in a horizontal
enclosure with (a)
Convective currents in a hot plate at the top
vertical rectangular and (b) hot plate at
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enclosure. the bottom.
Effective Thermal Conductivity

effective thermal
conductivity
The fluid in an enclosure behaves like
a fluid whose thermal conductivity is
kNu as a result of convection currents.
Nu = 1, the effective thermal
conductivity of the enclosure is equal
to the conductivity of the fluid. This
case corresponds to pure conduction.
Numerous correlations for the Nusselt number
exist. Simple power-law type relations in the
form of Nu = CRan, where C and n are
constants, are sufficiently accurate, but they are
usually applicable to a narrow range of Prandtl
A Nusselt number of 3 for an enclosure and Rayleigh numbers and aspect ratios.
indicates that heat transfer through the
enclosure by natural convection is three
times that by pure conduction. 26
For confined spaces in buildings an effect of radiation is also important as free
convections heat transfer rates are typically small. In such cases a total
resistance is used

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Horizontal Rectangular Enclosures
For horizontal enclosures that
contain air, These relations can
also be used for other gases with
0.5 < Pr < 2.
For water, silicone
oil, and mercury

[ ]+ only positive
values to be used
Based on experiments with air. It may be used for liquids with
moderate Prandtl numbers for RaL < 105.

When the hotter plate

is at the top, Nu = 1.

A horizontal rectangular
enclosure with
isothermal surfaces. 28
For values of Grδ below about 1700, pure conduction is still observed and
Nuδ =1.0. As convection begins, a pattern of hexagonal cells is formed as
shown in Figure 7-10. These patterns are called Benard cells. Turbulence
begins at about Grδ =50,000 and destroys the cellular pattern.

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Inclined Rectangular Enclosures

An inclined rectangular enclosure

with isothermal surfaces. 30
Vertical
Rectangular
Enclosures

Again, all fluid properties are

to be evaluated at the average
temperature (T1+T2)/2.
A vertical rectangular
enclosure with
isothermal surfaces. 31
For free convection in enclosures we have seen that when the product Grδ Pr is
sufficiently small, usually less than about 2000, the fluid layer behaves as
if pure conduction were involved and ke/k→1.0.

Asmall value of Grδ can result from either lowering the fluid pressure (density) or
by reducing the spacing δ. If the pressure of a gas is reduced sufficiently, we
refer to the situation as a low-density problem, which is influenced by the mean
free path of the molecules and by individual molecular impacts.

The mean free path of the gas molecules is no longer small in comparison with a
characteristic dimension of the heat-transfer surface.

where r is the effective molecular radius for

collisions and n is the molecular density
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For very low densities (high vacuum) the mean free path may become very large
compared to the plate separation distance and the conduction-convection heat
transfer will approach zero. However, radiation will occur.

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Combined Natural Convection and Radiation
Gases are nearly transparent to radiation, and thus heat transfer through a
gas layer is by simultaneous convection (or conduction) and radiation.
Radiation is usually disregarded in forced convection problems, but it must be
considered in natural convection problems that involve a gas. This is especially
the case for surfaces with high emissivities.

Radiation heat transfer from a surface at temperature Ts surrounded by

surfaces at a temperature Tsurr is
 = 5.67  108 W/m2K4
Stefan–Boltzmann constant
Radiation heat transfer between two large parallel plates is

When T < Ts and Tsurr > Ts, convection and

radiation heat transfers are in opposite
directions and subtracted from each other.
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 COMBINED FREE AND FORCED CONVECTION

Aiding flow means that the forced- and free-convection currents are in the
same direction, while opposing flow means that they are in the opposite
direction.

mixed-convection, laminar flow region

Forced convection is negligible when Gr/Re2 > 10

Free convection negligible when Gr/Re2 , 0.1
Neither is negligible when 0.1 , Gr/Re2 < 10

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