Page 256 - Fluid mechanics, heat transfer, and mass transfer

DIMENSIONLESS NUMBERS, TEMPERATURE MEASUREMENT, AND CONDUCTION HEAT TRANSFER
236

FIGURE 8.12 Heat transfer shape factors.
system, preserving the angle of intersection between
pairs of curves.
& A mapping or transformation of a set E of points in
Multilayer hollow cylinder.
the xy plane onto a set F in the uv plane is a
FIGURE 8.10
correspondence that is defined for each point (x, y)
& For example, for the case of a long (L w) circular
in E and sends it to a point (u, v)in F, so that each
cylinder centered in square solid of equal length,
point in F is the image of some point in E.
& A mapping is one to one if distinct points in E are
S ¼ 2pL=ln ð1:08 w=DÞ: ð8:18Þ
transformed to distinct points in F.
. What are Heisler charts? & A mapping is conformal if it is one to one and
& Heisler charts are a series of charts that depict preserves the magnitudes and orientations of the
unsteady-state temperature distributions in different angles between curves. Conformal mappings pre-
shaped bodies such as slabs, cylinders, and spheres. serve the shape but not the size of small figures.
. What is conformal mapping? & If the points (x, y) and (u, v) are viewed as the
& A special operation in mathematics in which a set of complex numbers z ¼ x þ iy and w ¼ u þ iv, the
points in one coordinate system is mapped or trans- mapping becomes a function of a complex variable:
formed into a corresponding set in another coordinate w ¼ f(z). It is an important fact that a one-to-one
mapping is conformal if and only if the function f is
0
analytic and its derivative f (z) is never equal to zero.
& Conformal mappings are important in two-dimen-
sional problems of fluid flow, heat conduction, and
potential theory. They provide suitable changes of
coordinates for the analysis of difficult problems. For
example, the problem of finding the steady-state
distribution of temperature in a conducting plate
requires the calculation of a harmonic function with
prescribed boundary values.
& The term conformal applies in a more general context
to the mapping of any surface onto another.

8.3.1 Thermal Insulation
. What are the reasons for insulating plant and
equipment?
& To save energy and consequent reduction in CO 2
emissions from fossil fuel burning.
FIGURE 8.11 Multilayer hollow sphere. & To help maintain process temperatures.