Heat Transfer and Heat
Exchangers
Introduction
The engineer is frequently called upon to transfer
energy from one fluid to another. The
transfer is most often effected by a heat exchanger. A steam power plant, for
example, includes several major heat exchangers: boiler, superheater,
economizer, and condenser. It also
includes minor heat exchangers such as the gland exhaust condenser and the lube
oil cooler. The principal heat
exchangers in an air conditioning system are the evaporator and the condenser. Other applications abound throughout the
mechanical industry.
Theoretical Considerations and the Heat Transfer Problem
The typical heat transfer problem involves energy
transfer from a fluid stream at an assumed average temperature, through a series
of “resistances”, to a destination fluid stream also at an assumed average
temperature. For transfer through a pipe
wall, the resistances consist of, in order, a thermal boundary layer (TBL), a
scale or soot layer, the pipe wall, another scale layer, and the destination
TBL.
Each TBL is characterized by a
convection (or film) coefficient
designated h. Newton’s Law of Cooling models heat transfer
through the TBL:
where: 
= heat flux or rate of heat transfer per unit area 
= heat flux or rate of heat transfer per unit area
h = convection (film) coefficient 
= average temperature of the fluid stream
Ts = surface temperature of
the pipe
Fourier’s Law governs transfer by conduction through the
pipe wall:
where: k = thermal
conductivity (
)
)
The negative sign results from the decreasing temperature
gradient. Integration yields:
where: q = heat transfer rate 
L = length of pipe through which heat
transfer occurs 
The Overall Heat Transfer Coefficient
Heat transfer through the various
layers can be represented by an overall
heat transfer coefficient, U, such that:
where: A = heat transfer surface area based on
either the inside or outside pipe surface
U = overall heat transfer coefficient (Btu/h-ft2-R)
D = log mean temperature difference,
derived below
In analogy with electrical resistance, we can express the
Newton and Fourier laws as:
where:
We can then write:
Selecting either the inside or outside area as a basis,
factoring out 1/A1 (for
example), and letting A1 = 2pr1L,
we have:
Log Mean Temperature Difference
What temperature difference is to be used for
D? Because the temperature varies on
each side of the heat exchanger as heat transfer occurs, we must derive a sort
of average temperature difference. Consider a counterflow heat exchanger, which
we represent as shown in the figure below:
Let: T =
temperature of the hot fluid
t =
temperature of the cold fluid
h = hot
fluid
C= heat
capacity rate 
i = entrance condition
o = outlet
condition
D = log
mean temperature difference
For
multipass and cross-flow heat exchangers, the expression for D is modified by a multiple F, which depends upon heat exchange
geometry, number of tube passes and whether the fluid is mixed (flow transverse
to the fluid direction is not prevented) or unmixed (transverse flow ix
prevented by fins or separate tubes).
The F-factor can be found in
the technical literature.[2]
Extended Surfaces (Fins)
Extended surfaces, or fins, are
frequently used to increase the heat transfer rate. Fins can be justified economically when fin
effectiveness, defined by the below approximate equation, is greater than about
2.
where: k = thermal conductivity
P = perimeter of fin
h = convection coefficient
Ac = cross sectional area of
fin
Because the temperature varies from the base of the fin to
its tip, an adjustment is made to the total heat transfer surface area, which
consists of the uncovered base area plus the fin area. Multiplying the total area, At, by
an overall surface efficiency, ho,[3] which is less than unity, gives:
where: N = number of fins
Af = area of a
single fin
At = total area including
base
hf
= fin efficiency
qt = total heat transfer rate
from At
h = convection coefficient
q
= Tb - T¥
Tb = base temperature
T¥ =
bulk temperature
Fin efficiency is a function of fin
geometry, convection coefficient and thermal conductivity. Values of fin efficiency for various shapes
can be found in the technical literature.[4]
Having found both fin and overall surface
efficiency, the overall heat transfer coefficient is modified by multiplying
each area (fin plus uncovered base) by the overall surface efficiency. This step results in the following relation
for U:
From the definition of fin effectiveness,
it is deduced that fins with a high thermal conductivity, low ratio of
perimeter to cross-sectional area and small convection coefficient are
desirable. Because a small convection
coefficient improves fin performance, fins are frequently placed on the gas
side of a gas-to-liquid heat exchanger.
Automobile radiators and air conditioning evaporators and condensers are
common examples. Fins are often placed
on economizers in boiler designs.
THE CONVECTION HEAT TRANSFER COEFFICIENT
(FILM COEFFICIENT)
Convection
coefficients are determined from empirical correlations found in the
literature. Incropera, for example,
includes summaries of convection coefficients for external and internal flow in
the back of chapters 7 and 8. The classical correlation for internal flow is
the Colburn equation:
where: 
(the Nusselt
number) k = thermal conductivity of fluid
(the Nusselt
number) k = thermal conductivity of fluid 
(the
Reynolds number) n =
kinematic viscosity 
(the Prandtl
number) 
(thermal
diffusivity)
A cautionary note.
The engineer must exercise care in choosing and using a
correlation. Particular attention must
be given to limits of applicability. Is
the flow fully developed? Laminar or
turbulent? What temperature should be
used to find properties? What are the
applicable ranges for the Reynolds and Prandtl numbers and the length to
diameter ratio?
DESIGN PROCEDURE
·
Select an appropriate heat exchanger type.
·
Select appropriate convection coefficients.
·
Determine needed thermal properties.
·
Solve the fundamental equations. Iteration may be required.
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