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PHYSICAL EXAMPLES OF FIRST-ORDER SYSTEMS
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In the first part of this chapter, we shall consider several physical systems that can be represented by a first-order transfer function. In the second part, a method for approximating the dynamic response of a nonlinear system by a linear response will be presented. This approximation is called linearization.
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EXAMPLES OF FIRST-ORDER SYSTEMS Liquid Level
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Consider the system shown in Fig. 6.1, which consists of a tank of uniform crosssectional area A to which is attached a flow resistance R such as a valve, a pipe, or a weir. Assume that qO, the volumetric flow rate (volume/time) through the resistance, is related to the head h by the linear relationship
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A resistance that has this linear relationship between flow and head is referred to as a linear resistance. A time-varying volumetric flow 4 of liquid of constant density p enters the tank. Determine the transfer function that relates head to flow.
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*A pipe is a linear resistance if the flow is in the laminar range. A specially contoured heir, called a Sutro weir, produces a linear head-flow relationship. Formulas used to prepare the shape of
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tank:
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FIGURE61
Liquid-level system.
We can analyze this system by writing a transient mass balance around the Mass flow in - mass flow out = rate of accumulation of mass in the tank In terms of the variables used in this analysis, the mass balance becomes
Pm - pqm = y q(t) - q&) = A$
(6.2)
Combining Eqs. (6.1) and (6.2) to eliminate qO(t) gives the following linear differential equation: q-;=A$ (6.3)
We shall introduce deviation variables into the analysis before proceeding to the transfer function. Initially, the process is operating at steady state, which means that dhldt = 0 and we can write Eq. (6.3) as
where the subscript s has been used to indicate the steady-state value of the variable. Subtracting Eq. (6.4) from Eq. (6.3) gives (4 - qs) = ;(h - h,) + Adchd- hs)
(6.5)
such a weir have been mported in the literature; & Soucek, Howe, and Mavis (1936). tibulent flow through pipes and valves is generally proportional to &. Flow through weirs having simple geometric shapes can be expressed as K/a , where K and n are positive constants. For example, the Bow through a rectangular-shaped weir is proportional to h .
LINEAR OPEN-LOOP SYSTEMS
If we define the deviation variables as
e=4-4s
H = h - h ,
Eq. (6.5) can be written
Q=;H+Az (6.6)
Taking the transform of Eq. (6.6) gives
Q(s) = ;H(s) + ASH(S)
(6.7)
Notice that H(0) is zero and therefore the transform of dH/dt is simply sH(s). Equation (6.7) can be rearranged into the standard form of the first-order lag to give R H(s) (6.8) 7s + 1 Q(s) where r = AR. In comparing the transfer function of the tank given by Eq. (6.8) with the transfer function for the thermometer given by Eq. (5.7), we see that Eq. (6.8) contains the factor R. The term R is simply the conversion factor that relates h(t) to q(t) when the system is at steady state. For this reason, a factor K in the transfer function Kl(rs + 1) is often called the steady-state gain. We can readily show this name to be appropriate by applying the final-value theorem of Chap. 4 to the determination of the steady-state value of H when the flow rate Q(t) changes according to a unit-step change; thus
Q(t) = u(t)
where u(t) is the symbol for the unit-step change. The transform of Q(t) is
Q(s) = f
Combining this forcing function with Eq. (6.8) gives
H(s) = f--&
Applying the final-value theorem, proved in Chap. 4, to H(s) gives
f-f(t) t--*m = liio[sH(s)] = liio --& = R
This shows that the ultimate change in H(t) for a unit change in Q(t) is simply R. If the transfer function relating the inlet flow q(t) to the outlet flow is desired, note that we have from Eq. (6.1)
PHYSICAL EXAMPLES OF FIRST-ORDER SYSTEMS
67 (6.9)
Subtracting Eq. (6.9) from Eq. (6.1) and using the deviation variable Q, = q. - qo, gives
Qo = #
Taking the transform of Eq. (6.10) gives
(6.10)
Combining Eqs. (6.11) and (6.8) to eliminate H(s) gives
1Qo(s> _ 7s + 1 Q(s)
(6.12)
Notice that the steady-state gain for this transfer function is dimensionless, which is to be expected because the input variable q(t) and the output variable qO(t) have the same units (volume/time). The possibility of approximating an impulse forcing function in the flow rate to the liquid-level system is quite real. Recall that the unit-impulse function is defined as a pulse of unit area as the duration of the pulse approaches zero, the impulse function can be approximated by suddenly increasing the flow to a large value for a very short time; i.e. we may pour very quickly a volume of liquid into the tank. The nature of the impulse response for a liquid-level system will be described by the following example.
Example 6.1. A tank having a time constant of 1 min and a resistance of i ft/cfm is operating at steady state with an inlet flow of 10 ft3/min. At time t = 0, the flow is suddenly increased to 100 ft3/min for 0.1 min by adding an additional 9 ft3 of water to the tank uniformly over a period of 0.1 min. (See Fig. 6.2 for this input disturbance.) Plot the response in tank level and compare with the impulse response. Before proceeding with the details of the computation, we should observe that, as the time interval over which the 9 ft3 of water is added to the tank is shortened, the input approaches an impulse function having a magnitude of 9. From the data given in this example, the transfer function of the process is H(s) = 1 - 1 9s+l Q(s) The input may be expressed as the difference in step functions, as was done in Example 4.5. Q(t) = 90[u(t) - u(t - O.l)] The transform of this is
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