Page 102 - Modelling in Transport Phenomena A Conceptual Approach

82 CHAPTER 4. EVALUATION OF TRANSFER COEFFICIENTS

4.3.2 Heat Transfer Correlations
When a sphere is immersed in an infinite stagnant fluid, the analytical solution for
the steady-state conduction is possible* and the result is expressed in the form
Nu=2 (4.3-28)
In the case of fluid motion, contribution of the convective mechanism must be
included in Q. (4.3-28). Correlations for including convective heat transfer are as
follows:

Ranz-Marshall correlation
Ranz and Marshall (1952) proposed the following correlation for constant surface
temperature:
I Nu = 2 + 0.6 Re? Pr1l3 1 (4.3-29)

All properties in Eq. (4.3-29) must be evaluated at the film temperature.

Whitaker correlation
Whitaker (1972) considered heat transfer from the sphere to be a result of two
parallel processes occurring simultaneously. He assumed that the laminar and
turbulent contributions are additive and proposed the following equation:
I
1 Nu = 2 + (0.4 by2 + 0.06 by) (pm/pW)'/4 (4.3-30)

All properties except pw should be evaluated at Tm. Equation (4.3-30) is valid for
5
3.5 5 bP 7.6 x 104
0.71 5 Pr 5 380

4.3.2.1 Calculation of the heat transfer rate
Once the average heat transfer coefficient is estimated by using correlations, the
rate of heat transferred is calculated as

(4.3-31)

Example 4.7 An instrument is enclosed in a protective spherical shell, 5cm in
diameter, and submerged in a river to measure the concentrations of pollutants. The
temperature and the velocity of the river are 10 "C and 1.2 m/ s, respectively. To
prevent any damage to the instrument as a result of the cold river temperature, the
surface temperature is kept constant at 32°C by installing electrical heaters in the
protective shell. Calculate the electrical power dissipated under steady conditions.
'See Example 8.9 in Chapter 8.