E1C03 09/14/2010
15:24:52 Page 89
Example 3.3
Suppose a bulb thermometer originally indicating 20
C is suddenly exposed to a fluid temperature
of 37
C. Develop a model that simulates the thermometer output response.
KNOWN T 0
ð Þ ¼ 20
C
T 1 ¼ 37
C
FðtÞ ¼ T 1 À Tð0Þ
½
Š UðtÞ
ASSUMPTIONS To keep things simple, assume the following: no installation effects (neglect
conduction and radiation effects); sensor mass is mass of liquid in bulb only; uniform temperature
within bulb (lumped mass); and thermometer scale is calibrated to indicate temperature
FIND
T t
ð Þ
SOLUTION Consider the energy balance developed in Figure 3.9. According to the first law of
thermodynamics, the rate at which energy is exchanged between the sensor and its environment
through convection, _
Q, must be balanced by the storage of energy within the thermometer, dE=dt.
This conservation of energy is written as
dE
dt
¼ _
Q
Energy storage in the bulb is manifested by a change in bulb temperature so that for a constant mass
bulb, dE(t)/dt ¼ mc v dT(t)/dt. Energy exchange by convection between the bulb at T(t) and an
environment at T 1 has the form _
Q ¼ hA s DT. The first law can be written as
mc v
dT t
ð Þ
dt
¼ hA s T 1 À T t
ð Þ
½
Š
This equation can be written in the form
mc v
dT t
ð Þ
dt
þ hA s TðtÞ À Tð0Þ
½
Š¼ hA s FðtÞ ¼ hA s T 1 À Tð0Þ
½
Š UðtÞ
with initial condition T(0) and
m, c v , T(t)
T ∞
Q in
.
dE
dt
Bulb sensor
Control volume
Figure 3.9 Lumped parameter model of thermometer and its
energy balance (Ex. 3.3).
3.3 Special Cases of the General System Model 89