Instrumentation and Process Control
(CHE-323)
Dr. Iftikhar A. Salarzai
School of Chemical & Materials Engineering
National University of Sciences and Technology,
Islamabad
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Recall: Modeling in Process Dynamics:
Step Forcing Function and Response
Example: A thermometer having a time constant of 0.1 min is at a steady-state
temperature of 90 ° F. At time t = 0, the thermometer is placed in a temperature
bath maintained at 100°F. Find response of the thermometer to the step change.
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Recall: Effect of value of time constant
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Step Response
If a step change of magnitude A is introduced into a first-order system, the
transform of X ( t ) is
The transfer function, we just derived, is
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Step Response (cont.…)
• Combining the two equations
This can be expanded by partial fractions to give
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Step Response (cont.…)
• Solving for the constants C1 and C2 gives C 1= A and C 2 = A.
Inserting these constants into the above equation and taking the
inverse transform give the time response for Y:
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Recall: Modeling in Process Dynamics:
Step Forcing Function and Response
Example: A thermometer having a time constant of 0.1 min is at a steady-state
temperature of 90 ° F. At time t = 0, the thermometer is placed in a temperature
bath maintained at 100°F. Find response of the thermometer to the step change.
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Recall: Effect of value of time constant
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Recall: Modeling in Process Dynamics:
Sinusoidal Forcing Function and Response
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Example: Sinusoidal Forcing Function and
Response
• A mercury thermometer having a time constant of 0.1 min is
placed in a temperature bath at 100°F and allowed to come to
equilibrium with the bath. At time t =0, the temperature of the bath
begins to vary sinusoidally about its average temperature of 100°F
with an amplitude of 2°F. If the frequency of oscillation is 10/Pi
cycles/min, plot the ultimate response of the thermometer reading as
a function of time. What is the phase lag?
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Solution: Sinusoidal Forcing Function and
Response
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Solution: Sinusoidal Forcing Function and
Response (cont…)
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Solution: Sinusoidal Forcing Function and
Response (cont…)
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Solution: Sinusoidal Forcing Function and
Response (cont…)
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Recall:
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Example: Sinusoidal Forcing Function and
Response
• A mercury thermometer having a time constant of 0.1 min is
placed in a temperature bath at 100°F. At time t =0, the temperature
of the bath begins to vary sinusoidally about its average temperature
of 100°F with an amplitude of 2°F. If the frequency of oscillation is
10/Pi cycles/min, plot the response of the thermometer reading as a
function of time using the following model.
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