Page 269 - Modelling in Transport Phenomena A Conceptual Approach

8.2. ENERGY TRANSPORT WTHOUT CONVECTION 249

Solution

k
With TL = 35 "C, qo = 100,000 W/ m2, = 398 W/ m. K and L = 0.04m, Eq. (H)
in Table 8.2 gives the temperature distribution as
(100,000)(0.04)
T-35=
398
or,
T = 45.1 - 251.32.

Example 8.3 In rivers ice begins to form when water is cooled to 0°C and con-
tinues to lose heat to the atmosphere. The presence of ice on rivers not only causes
transportation problems but also floods on its melting. Once the ice cover is formed,
its thickening depends on the rate of heat transfed from the water, through the
ice cover, to the cold atmosphere. As an engineer you are asked to estimate the
increase in the thickness of the ice block as a function of time.

Solution

Assumptions
1. Pseudo-steady-state behavior.

2. River temperature is close to 0 "C and the heat transfed from water to ice
is negligible. This assumption implies that the major cause of ice thickening
is the conduction of heat through the ice.
Analysis

System: Ice block
Since the density of ice is less than that of water, it floats on the river as shown in
Figure 8.7. The temperatures T, and Ts represent the melting temperature (0 "C)
and the top surface temperature, respectively.

Air, T,
=s
f
Y ICe
Tln J -
Water

Figure 8.7 Ice block on a river.