Page 326 - Bird R.B. Transport phenomena

310 Chapter 10 Shell Energy Balances and Temperature Distributions in Solids and Laminar Flow

SOLUTION
The thermocouple well wall of thickness В is in contact with the gas stream on one side only,
and the tube thickness is small compared with the diameter. Hence the temperature distribu-
tion along this wall will be about the same as that along a bar of thickness 2B, in contact with
the gas stream on both sides. According to Eq. 10.7-13, the temperature at the end of the well
(that registered by the thermocouple) satisfies

T\-T _ cosh 0 _ 1
a
2
T - T cosh N coshVhL /kB
w a
1
coshV(l02X0.2)7(60)(^ . 0.08)
1 1 (10.7-17)
cosh(2V3)
Hence the actual ambient gas temperature is obtained by solving this equation for T :
a
Л 7
350 - T 16.0 °'
a
and the result is
T = 510°F (10.7-19)
n
Therefore, the reading is 10 F° too low.
This example has focused on one kind of error that can occur in thermometry. Fre-
quently a simple analysis, such as the foregoing, can be used to estimate the measurement
errors. 4

§10.8 FORCED CONVECTION

In the preceding sections the emphasis has been placed on heat conduction in solids.
In this and the following section we study two limiting types of heat transport in flu-
ids: forced convection and free convection (also called natural convection). The main dif-
ferences between these two modes of convection are shown in Fig. 10.8-1. Most
industrial heat transfer problems are usually put into either one or the other of these
two limiting categories. In some problems, however, both effects must be taken into
account, and then we speak of mixed convection (see §14.6 for some empiricisms for
handling this situation).
In this section we consider forced convection in a circular tube, a limiting case of
which is simple enough to be solved analytically. ' 1 2 A viscous fluid with physical prop-
erties (/x, k, p, C ) assumed constant is in laminar flow in a circular tube of radius JR. For
p
z < 0 the fluid temperature is uniform at the inlet temperature Т\. For z > 0 there is a
constant radial heat flux q = -q at the wall. Such a situation exists, for example, when a
o
r
pipe is wrapped uniformly with an electrical heating coil, in which case q is positive. If
0
the pipe is being chilled, then q has to be taken as negative.
0
As indicated in Fig. 10.8-1, the first step in solving a forced convection heat transfer
problem is the calculation of the velocity profiles in the system. We have seen in §2.3

4
For further discussion, see M. Jakob, Heat Transfer, Vol. II, Wiley, New York (1949), Chapter 33,
pp.147-201.
1 A. Eagle and R. M. Ferguson, Proc. Roy. Soc. (London), A127, 540-566 (1930).
2 S. Goldstein, Modern Developments in Fluid Dynamics, Oxford University Press (1938), Dover
Edition (1965), Vol. II, p. 622.