Tutorial 5: Dynamic Behavior of First-order and Second-order Processes
In addition to the standard inputs discussed in Section 5.1, other input functions occasionally are useful for special purposes. One, the so-called doublet pulse, is shown in Fig. E5.1.
a. Find the Laplace transform of this function by first expressing it as a composite of functions whose transforms you already know.
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b. What would be the response of a process having a first-order transfer function K/((s+1) to this input? Of the integrating process K/s?
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c. From these results, can you determine what special property this input offers?
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Two liquid storage systems are shown in Fig.E5.9.Each tank is 4 feet in diameter. For system I, the valve acts as a linear resistance with the flow-head relation q = 8.33 h, where q is in gal/min and h is in feet. For System II, variations in liquid level h do not affect exit flow rate q. Suppose that each system is initially at steady-state with
and
and that at time t = 0 the inlet flow rate suddenly changes from 50 to 70 gal/min. For each of the system, determine the following information:
The transfer function
where the primes denote deviation variables.
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The transient response h(t).
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The new steady-state levels?
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If each tank is 8 ft tall, which tank overflows first? when?
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The dynamic behavior of the liquid level in each leg of a manometer tube, responding to a change in pressure, is given by



where h`(t) is the level of liquid measured with respect to the initial steady-state value, p`(t) is the pressure change, and R, L, q, (, and ( are constants.
Rearrange this equation into standard gain-time constant form and find expression for K, (, ( in terms of the physical constants.
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For what values of the physical constants does the manometer response oscillate?
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Would changing the manometer fluid so that ρ (density) is larger make its response more or less oscillatory? Repeat the analysis for an increase in ( (viscosity).
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A step change from 15 to 31 psi in actual pressure results in the measured response from a pressure indicating element shown in Fig. E5.14.
Assuming second-order dynamics, calculate all important parameters and write and approximate transfer function in the form


where R` is the instrument output deviation (mm), P` is the actual pressure deviation (psi).
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Write an equivalent differential equation model in terms of actual (not deviation) variables.
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