E1C03 09/14/2010
15:24:52 Page 90
where
m ¼ mass of liquid within thermometer
c v ¼ specific heat of liquid within thermometer
h ¼ convection heat transfer coefficient between bulb and environment
A s ¼ thermometer surface area
The term hA s controls the rate at which energy can be transferred between a fluid and a body; it is
analogous to electrical conductance. By comparison with Equation 3.3, a 0 ¼ hA s , a 1 ¼ mc v , and
b 0 ¼ hA s . Rewriting for t ! 0
þ and simplifying yields
mc v
hA s
dT t
ð Þ
dt
þ T t
ð Þ ¼ T 1
From Equation 3.4, this implies that the time constant and static sensitivity are
t ¼
mc v
hA s
K ¼
hA s
hA s
¼ 1
Direct comparison with Equation 3.5 yields this thermometer response:
T t
ð Þ ¼ T 1 þ Tð0Þ À T 1
½
e
Àt=t
¼ 37 À 17e
Àt=t
½
C
COMMENT Two interactive examples of the thermometer problem (from Exs 3.3–3.5) are
available. In the program FirstOrd, the user can choose input functions and study the system
response. In the LabView program Temperature_response, the user can apply interactively a step
change in temperature and study the first-order system response of a thermal sensor.
Clearly the time constant, t, of the thermometer can be reduced by decreasing its mass-to-area ratio
or by increasing h (for example, increasing the fluid velocity aroundthe sensor). Without modeling, such
information could be ascertained only by trial and error, a time-consuming and costly method with no
assurance of success. Also, it is significant that we found that the response of the temperature
measurement system in this case depends on the environmental conditions of the measurement that
control h, because the magnitude of h affects the magnitude of t. If h is not controlled during response
tests (i.e., if it is an extraneous variable), ambiguous results are possible. For example, the curve of
Figure 3.8 will become nonlinear, or replications will not yield the same values for t.
Review of this example should make it apparent that the results of a well-executed step
calibration may not be indicative of an instrument’s performance during a measurement if the
measurement conditions differ from those existing during the step calibration.
Example 3.4
For the thermometer in Example 3.3 subjected to a step change in input, calculate the 90% rise time
in terms of t=t.
KNOWN Same as Example 3.3
ASSUMPTIONS Same as Example 3.3
FIND 90% response time in terms of t=t
90 Chapter 3 Measurement System Behavior
15:24:52 Page 90
where
m ¼ mass of liquid within thermometer
c v ¼ specific heat of liquid within thermometer
h ¼ convection heat transfer coefficient between bulb and environment
A s ¼ thermometer surface area
The term hA s controls the rate at which energy can be transferred between a fluid and a body; it is
analogous to electrical conductance. By comparison with Equation 3.3, a 0 ¼ hA s , a 1 ¼ mc v , and
b 0 ¼ hA s . Rewriting for t ! 0
þ and simplifying yields
mc v
hA s
dT t
ð Þ
dt
þ T t
ð Þ ¼ T 1
From Equation 3.4, this implies that the time constant and static sensitivity are
t ¼
mc v
hA s
K ¼
hA s
hA s
¼ 1
Direct comparison with Equation 3.5 yields this thermometer response:
T t
ð Þ ¼ T 1 þ Tð0Þ À T 1
½
e
Àt=t
¼ 37 À 17e
Àt=t
½
C
COMMENT Two interactive examples of the thermometer problem (from Exs 3.3–3.5) are
available. In the program FirstOrd, the user can choose input functions and study the system
response. In the LabView program Temperature_response, the user can apply interactively a step
change in temperature and study the first-order system response of a thermal sensor.
Clearly the time constant, t, of the thermometer can be reduced by decreasing its mass-to-area ratio
or by increasing h (for example, increasing the fluid velocity aroundthe sensor). Without modeling, such
information could be ascertained only by trial and error, a time-consuming and costly method with no
assurance of success. Also, it is significant that we found that the response of the temperature
measurement system in this case depends on the environmental conditions of the measurement that
control h, because the magnitude of h affects the magnitude of t. If h is not controlled during response
tests (i.e., if it is an extraneous variable), ambiguous results are possible. For example, the curve of
Figure 3.8 will become nonlinear, or replications will not yield the same values for t.
Review of this example should make it apparent that the results of a well-executed step
calibration may not be indicative of an instrument’s performance during a measurement if the
measurement conditions differ from those existing during the step calibration.
Example 3.4
For the thermometer in Example 3.3 subjected to a step change in input, calculate the 90% rise time
in terms of t=t.
KNOWN Same as Example 3.3
ASSUMPTIONS Same as Example 3.3
FIND 90% response time in terms of t=t
90 Chapter 3 Measurement System Behavior