Page 391 - Modelling in Transport Phenomena A Conceptual Approach

9.3. HEAT TRANSFER WITH CONVECTION 371

Therefore, the first approximation to the Nusselt number is

/'a1 - t2I2fl(t) e

Nu(1) = (9.3-68)
1' ((1 - t2)f,2(t) ds

Substitution of f1(() from Eq. (9.3-67) into Q. (9.3-68) and evaluation of the
integrals gives
NU = 3.663 (9.3-69)
On the other hand, the value of the Nusselt number, 8s calculated by Graetz (1883,
1885) and later independently by Nusselt (1910), is 3.66. Therefore, for a thermally
developed laminar flow in a circular pipe with constant wall temperature Nu = 3.66
for all practical purposes.

Example 9.8 Water flows through a circular pipe of 5 cm internal diameter with
an average velocity of 0.01 m/s. Determine the length of the pipe to increase the
water temperature from 20 "C to 60 "C for the following conditions:
a) Steam condenses on the outer surface of the pipe so as to keep the surface
temperature at 100 "C.
b) Electrical wires are wrapped around the outer surface of the pipe to provide a
constant wall heat flux of 1500 W/ m2.

Solution
Physical properties
The mean bulk temperature is (20 + 60)/2 = 40 "C (313 K).
p = 992 kg/ m3
p = 654 x kg/ m. s
For water at 313 K :
= 632 10-3 wl
m.
Pr = 4.32
Assumptions
1. Steady-state conditions prevail.

2. Flow b hydrodynamically and thermally fully developed.
Analysis

The Reynolds number is

- (0.05) (0.01) (992)
- = 758 3 Laminar flow
654 x