Page 459 - Bird R.B. Transport phenomena

§14.4 Heat Transfer Coefficients for Forced Convection Around Submerged Objects 439

As shown in Table 12.4-1, a more accurate value of the numerical coefficient in Eq. 14.4-2
is that of Pohlhausen—namely, 0.332. If we use this value, then Eq. 14.4-2 gives

Since the numerical coefficient in Eq. 14.4-3 is the same as that in Eq. 14.4-1, we then
get

/H,IOC = 5/ioc = 0.332 Re; 1/2 (14.4-4)
for the Colburn analogy between heat transfer and fluid friction. This was to be ex-
pected, because there is no "form drag" in this flow geometry.
1
Equation 14.4-4 was derived for fluids with constant physical properties. When the
physical properties are evaluated at the film temperature T f = \(T 0 + TJ, Eq. 14.4-3 is
2
known to work well for gases. The analogy of Eq. 14.4-4 is accurate within 2% for Pr >
0.6, but becomes inaccurate at lower Prandtl numbers.
For highly turbulent flows, the Colburn analogy still holds with fair accuracy, with
/ loc given by the empirical curve in Fig. 14.4-1. The transition between laminar and turbu-
lent flow resembles that for pipes in Fig. 14.3-1, but the limits of the transition region are
harder to predict. For smooth, sharp-edged flat plates in an isothermal flow the transi-
tion usually begins at a Reynolds number Re x = xv^p/fi of 100,000 to 300,000 and is al-
most complete at a 50% higher Reynolds number.

Flow Around a Sphere

In Problem 10B.1 it is shown that the Nusselt number for a sphere in a stationary fluid is
2. For the sphere with constant surface temperature T o in a flowing fluid approaching
with a uniform velocity v^, the mean Nusselt number is given by the following empiri-
cism 3
1/2
= 2 + 0.60 Re Pr 1/3 (14.4-5)
Nu w
This result is useful for predicting the heat transfer to or from droplets or bubbles.
Another correlation that has proven successful 4 is
04
2/3
1/2
= 2 + (0.4 Re + 0.06Re )Pr ( ^ 1 (14.4-6)
Nu m
in which the physical properties appearing in Nu m, Re, and Pr are evaluated at the ap-
proaching stream temperature. This correlation is recommended for 3.5 < Re < 7.6 X
4
10 , 0.71 < Pr < 380, and 1.0 < ^«V^o < 3.2. In contrast to Eq. 14.4-5, it is not valid in the
limit that Pr -» oo.

1
The result in Eq. 14.4-1 was first obtained by H. Blasius, Z. Math. Phys., 56,1-37 (1908), and that in
Eq. 14.4-3 by E. Pohlhausen, Z. angew. Math. Mech., 1,115-121 (1921).
2
E. R. G. Eckert, Trans. ASME, 56,1273-1283 (1956). This article also includes high-velocity flows,
for which compressibility and viscous dissipation become important.
3
W. E. Ranz and W. R. Marshall, Jr., Chem. Eng. Prog., 48,141-146,173-180 (1952). N. Frossling,
Gerlands Beitr. Geophys., 52,170-216 (1938), first gave a correlation of this form, with a coefficient of 0.552
in lieu of 0.60 in the last term.
4
S. Whitaker, Fundamental Principles of Heat Transfer, Krieger Publishing Co., Malabar, Fla. (1977),
pp. 340-342; AIChE Journal, 18, 361-371 (1972).