Dimensional analysis offers a rational approach to the estimation of the complex phenomena of convective heat transfer rates.

**APPLICATION OF DIMENSIONAL ANALYSIS TO CONVECTIVE HEAT TRANSFER**

Dimensional
analysis offers a rational approach to the estimation of the complex phenomena
of convective heat transfer rates.

Free
convection describes heat transfer by the bulk movement of fluids induced by
buoyancy forces. These arise from the variation of fluid density with
temperature. If the surface in contact with the fluid is hotter, the fluid will
absorb heat, a local decrease in density will occur, and some of the fluid will
rise. If the surface is colder, the reverse takes place. For these conditions,
the fol-lowing factors will influence the heat transferred per unit area per
unit time, q. The dimensional form of these factors is given, using the
additional fundamental dimensions of temperature, [θ], and heat, [H]. The latter is
justified if inter-change of heat energy and mechanical energy is precluded.
This is approxi-mately true in the subject under discussion, the heat produced
by frictional effects, for example, being negligible.

·
The viscosity of the fluid, η: [ML^{-1}T^{-1}]

·
The thermal conductivity of the fluid, *k*: [HT^{-1}L^{-1}θ^{-1}]

·
The temperature difference between the surface and the bulk
of the fluid, ΔT:[θ]

·
The density, ρ: [ML^{-3}]

·
The specific heat, Cp: [HM^{-1}θ^{-1}]

·
The buoyancy forces. These depend on the product of the
coefficient of thermal expansion, a, and the acceleration due to gravity, *g*: [θ^{-1}LT^{-2}]

Normally,
only one dimension, of the physical dimensions of the surface, is important.
For example, the height of a plane vertical surface has greater significance
than the width that only determines the total area involved. The important
characteristic dimension is designated *l*
[L].

The
equation of dimensions is then

[*q*] = [ *l ^{x
}*Δ

or

[HL^{-2}T^{-1}]
= [L^{x}θ^{y}H^{z}T^{-z}L^{-z}θ^{-z}M^{p}L^{-p}T^{-p}H^{q}M^{-q}θ^{-q}θ^{-r}L^{r}T^{-2r}M^{s}L^{-3s}]

Equating
indices,

H
1*
= q + z*

L
-2 = *x - p + r - 3s - z*

T -1 = -*p
- 2r- z*

θ 0 = *y
- q - r - z*, and

M 0 = *p
- q + s *

Solving
for *x, y, z, p*, and s in terms of *q* and *r*,

z
= 1 - q

y
= r + 1

p
= q - 2r

s
= 2r, and

x
= 3r - 1

Therefore,

[q] = [ *l*^{3r-1} ΔT^{r}^{+}^{1}C^{q}_{p}k^{1-q} (*ag*)^{r} ρ^{2r ηq} 2*r*]

Collecting
into three groups, the variables to the power of one, the power, *q*, and the power, *r*, we can write

Heat
transfer by free convection can thus be presented as a relation between three
dimensionless groups. C_{p}η/*k*
is known as the Prandtl number; the combination *l*^{3}ΔT*ag**ρ*^{2}/η^{2} is known as the
Grashof number; and *ql*/Δ*Tk* is the Nusselt
number. Since the film coefficient, h, is given by q/ΔT, the Nusselt number may also be
written as *hl/k*.

The
specific relation in which these groups stand is established for a particular
system by experiment. Then, for the same geometric arrangement, in which heat
is transferred by free convection, the correlation allows the Nusselt number,
Nu, to be determined with reasonable accuracy from known values of the
variables that constitute the Grashof number, Gr, and the Prandtl number, Pr.
From Nu, the heat transferred per unit area per unit time, *q*, and the film coefficient, *h*,
can be determined.

The
fluid properties Cp, *k*, η, and *ρ* are themselves
temperature dependent. In establishing a correlation, the temperature at which
these properties are to be measured must be chosen. This is usually the
temperature of the main body of the fluid or the mean of this temperature and
the temperature of the surface.

Experimental
correlations for many surface configurations are available. The exponents r and
q are usually found to be equal to a value of 0.25 in streamline flow and 0.33
in turbulent flow. The constant varies with the physical configuration. As an
example, the heat transfer to gases and liquids from a large horizontal pipe by
free convection is described by the relation

The
linear dimension in this correlation is the pipe diameter, *d*. The fluid properties are to be measured at the mean of the wall
and bulk fluid temper-atures.

In
forced convection, the fluid is moved over the surface by a pump or blower. The
effects of natural convection are usually neglected. The study of forced
convection is of great practical importance, and a vast amount of data has been
documented for streamline and turbulent flow in pipes, across and parallel to
tubes, across plane surfaces, and in other important configurations such as
jackets and coils. Such data is again correlated by means of dimensionless
groups.

In
forced convection, the heat transferred per unit area per unit time, *q*, is determined by a linear dimension,
which characterizes the surface, *l*,
the tem-perature difference between the surface and the fluid, ΔT, the viscosity, η, the density, ρ, and the velocity, *u*, of the fluid, its conductivity, *k*, and its specific heat, Cp.
Dimensional analysis will yield the following relation:

where
*ql*/*k*Δ*T* is the Nusselt number, Nu, and C_{p}η/*k*
is the Prandtl number, Pr, and ulρ/η is Reynolds number, Re, a parameter
discussed in chapter 2. The values of the indices, x and y, and of the constant
are established for a particular system by experiment. In the case of turbulent
flow in pipes, the correlation for fluids of low viscosity is

Nu
= 0 023Pr^{x}Re^{0}^{:}^{8 } (3:13)^{}

where
*x* has the values 0.4 for heating and
0.3 for cooling. The linear dimension used to calculate Re or Nu is the pipe
diameter, and the physical properties of the fluid are to be measured at the
bulk fluid temperature. This relation shows that in a given system, the film
coefficient varies as the fluid velocity^{0.8}. If the flow velocity is
doubled, the film coefficient increases by a factor of 1.7.

Although
the correlations given above may appear complex, their use in practice is often
simple. A large quantity of tabulated data is available, and numerical values
of the variables and their dimensionless combinations are readily accessible.
The graphical presentation of these variables or groups will, in many cases,
permit an easy solution. In other cases, the correlation can be greatly
simplified if it is restricted to a particular system. Free convection to air
is an important example.

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