600 Chapter 18 Mathematics in Engineering
18.4 Exponential and Logarithmic Models
In this section, we will discuss exponential and logarithmic models and their basic characteristics.
The Cooling of Steel Plates In an annealing process — a process wherein materials such as glass and
metal are heated to high temperatures and then cooled slowly to toughen them — thin steel plates
(kthermal conductivity 40 W/m#K,rdensity 7800 kg /m
3
, andcspecific heat
400 J/ kg#K) are heated to temperatures of 900 C and then cooled in an environment with a tem-
perature of 35 C and a heat transfer coefficient ofh25 W/m(^2) #
K. Each plate has a thickness of
L5 cm. We are interested in determining what the temperature of the plate is after one hour.
Those of you who will pursue aerospace, chemical, mechanical, or materials engineering
will learn about the underlying concepts that lead to the solution in your heat-transfer class. For
now, in order to determine the temperature of a plate after one hour, we use the following expo-
nential equation.
(18.10)
In Equation (18.10),Trepresents the temperature of the plate at timet. Using Equation (18.10),
we have calculated the temperature of the plate after each 12 minute (0.2 hr) interval. The
corresponding temperature distribution is shown in Figure 18.12.
From examining Figure 18.12, we note that the temperature of the plate after one hour is
308 C. Moreover, Figure 18.12 shows that, during the first hour, the temperature of the plate
TTenvironment
TinitialTenvironment
exp a
2 h
rcL
tb
14
12
10
8
6
4
2
0
4 3 2 10
f(x)
123 456
x
xf(x) x^2 4
313
28
15
04
15
28
313
■Figure 18.11 A function with imaginary roots.
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