Page 442 - Bird R.B. Transport phenomena

424 Chapter 14 Interphase Transport in Nonisothermal Systems

"1" Element of "2" Fig. 14.1-1. Heat trans-
surface area fer in a circular tube.
dA = irDdz
, Tube cross
2
/ r section TTD /4
Fluid enters at Fluid leaves at
bulk temperature bulk temperature
T n *
/2s|
Inner surface* Inner surface
atr 01 Heated section atT 02
with inner surface
temperature
T (z)
0

That is, hi is based on the temperature difference А7\ at the inlet, h is based on the arith-
n
metic mean AT a of the terminal temperature differences, and h ]n is based on the corre-
sponding logarithmic mean temperature difference AT . For most calculations h ln is
ln
preferable, because it is less dependent on L/D than the other two, although it is not al-
ways used. 1 In using heat transfer correlations from treatises and handbooks, one must
be careful to note the definitions of the heat transfer coefficients.
If the wall temperature distribution is initially unknown, or if the fluid properties
change appreciably along the pipe, it is difficult to predict the heat transfer coefficients
defined above. Under these conditions, it is customary to rewrite Eq. 14.1-2 in the differ-
ential form:
dQ = h (7rDdz)(T 0 - T ) = loc (14.1-5)
bc
b
Here dQ is the heat added to the fluid over a distance dz along the pipe, AT loc is the local
temperature difference (at position z), and h ]oc is the local heat transfer coefficient. This
equation is widely used in engineering design. Actually, the definition of /z, and AT U)C is
oc
not complete without specifying the shape of the element of area. In Eq. 14.1-5 we have
set dA = irDdz, which means that /z and AT loc are the mean values for the shaded area
loc
dA in Fig. 14.1-1.
As an example of flow around submerged objects, consider a fluid flowing around a
sphere of radius R, whose surface temperature is maintained at a uniform value T . Sup-
o
pose that the fluid approaches the sphere with a uniform temperature T . Then we
x
may define a mean heat transfer coefficient, h , for the entire surface of the sphere by the re-
m
lation
2
Q = /!, (47гК )(Т - TJ (14.1-6)
я 0
The characteristic area is here taken to be the heat transfer surface (as in Eqs. 14.1-2 to 5),
whereas in Eq. 6.1-5 we used the sphere cross section.
A local coefficient can also be defined for submerged objects by analogy with Eq.
14.1-5:
dQ = h (dA)(T - TJ (14.1-7)
loc Q
This coefficient is more informative than h because it predicts how the heat flux is dis-
m
tributed over the surface. However, most experimentalists report only h , which is easier
m
to measure.

is between 0.5 and 2.0, then AT may be substituted for ДГ and h for h , with a
]n
n
a
1п/
maximum error of 4%. This degree of accuracy is acceptable in most heat transfer calculations.