Physical examples of first- order systems
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❑Liquid Level
➢ Consider the system shown in Fig. 5–1 , which consists of a tank of uniform
cross sectional area A to which is attached a flow resistance R such as a valve, a
pipe, or a weir.
➢ Assume that q o , 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 q of liquid of constant density r enters the tank.
➢ Determine the transfer function that relates head to flow. We can analyze this
system by writing a transient mass balance around the tank:
➢ In terms of the variables used in this analysis, the mass balance becomes
2
➢ Combining Eqs. (1) and (2) to eliminate q o ( t ) gives the following linear
differential equation:
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3
➢ the process is operating at steady state, which means that dh / dt = 0 and we can
write Eq. (3) as
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➢ Subtracting Eq. (4) from Eq. (3) gives
5
➢ If we define the deviation variables as
➢ then Eq. (5) can be written
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➢ Taking the transform of Eq. (6) gives
7
➢ Notice that H (0) is zero, and therefore the transform of dH/dt is simply sH ( s ).
➢ Equation (7) can be rearranged into the standard form of the first-order lag to
give
8
➢ where t = AR.
❖ In comparing the transfer function of the tank given by Eq. (8) with the transfer
function for the thermometer , we see that Eq. (8) contains the factor R.
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➢ The term R is simply the conversion factor that relates h (t ) to q ( t ) when the
system is at steady state.
➢ when the flow rate Q ( t ) changes according to a unit-step change; thus
➢ where u ( t ) is the symbol for the unit-step change. The transform of Q ( t ) is
➢ Combining this forcing function with Eq. (8) gives
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➢
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. (1)
9
➢ Subtracting Eq. (9) from Eq. (1) and using the deviation variable Qo =qo –qos
give
10
➢ Taking the transform of Eq. (10) gives
11
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❖
Combining Eqs. (11) and (8) to eliminate H ( s ) gives
➢ 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 q o ( 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, and the impulse function can be
approximated by suddenly increasing the flow to a large value for a very
short time; that is, we may pour very quickly a volume of liquid into the
tank.
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❑ Liquid-Level Process with Constant-Flow Outlet
❖ An example of a transfer function that often arises in control systems may be
developed by considering the liquid-level system shown in Fig. 5–3 .
❖ The resistance shown in Fig. 5–1 is replaced by a constant-flow pump.
❖ The same assumptions of constant cross sectional area and constant density
that were used before also apply here.
❖ For this system, Eq. (2) still applies, but q o ( t ) is now a constant; thus
15
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➢
At steady state, Eq. (15) becomes
16
➢ Subtracting Eq. (16) from Eq. (15) and introducing the deviation variables Q
= q - qs and H =h - hs give
17
➢ Taking the Laplace transform of each side of Eq. (17) and solving for H/Q
give
18
➢ Therefore, the solution of Eq. (18) is
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➢ Clearly, if we increase the inlet flow to the tank, the level will increase because
the outlet flow remains constant.
➢ The excess volumetric flow rate into the tank accumulates, and the level rises.
➢ For instance, if a step change Q ( t ) = u ( t ) were applied to the system shown
in Fig. 5–3 the result would be
➢ The transfer function for the liquid-level system with constant outlet flow given
by Eq. (18) can be considered as a special case of Eq. (8) as R ∞ .
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❑ Mixing Process
❖ Consider the mixing process shown in Fig. 5–4 in which a stream of solution
containing dissolved salt flows at a constant volumetric flow rate q into a tank
of constant holdup volume V.
❖ The concentration of the salt in the entering stream x (mass of salt/volume)
varies with time.
❖ It is desired to determine the transfer function relating the outlet concentration
y to the inlet concentration x.
❖ If we assume the density of the solution to be constant, the flow rate in must
equal the flow rate out, since the holdup volume is fixed.
❖ We may analyse this system by writing a transient mass balance for the salt;
thus
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➢
Expressing this mass balance in terms of symbols gives
21
➢ At steady state, Eq. (21) may be written
22
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➢
Subtracting Eq. (22) from Eq. (21) and introducing the deviation variables
➢
Taking the Laplace transform of this expression and rearranging the result
give
23
➢ This mixing process is, therefore, another first-order process for which the
dynamics are now well known.
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❑ Heating Process
➢ Consider the heating process shown in Fig. 5–5 . A stream at temperature Ti is
fed to the tank.
➢ Heat is added to the tank by means of an electric heater. The tank is well
mixed, and the temperature of the exiting stream is T.
➢ The flow rate to the tank is constant at w lb/h.
➢ A transient energy balance on the tank yields
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➢
Converting this energy balance to symbols results in
24
➢ where T ref is the reference temperature and C is the heat capacity of the fluid. At
steady state, dT / dt is zero, and Eq. (24) can be written
25
➢ where the subscript s has been used to indicate steady state. Subtracting Eq. (25)
from Eq. (24) gives
26
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➢
If we assume that T i is constant (and so Ti =T is ) and introduce the
deviation variables
➢ Eq. (26) becomes
27
➢ Taking Laplace transforms of Eq. (27) gives
28
➢ Rearranging Eq. (28) produces the following first-order transfer function
relating T ( s ) and Q ( s ):
29
➢ Thus, this process exhibits first-order dynamics as the tank temperature T
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responds to changes in the heat input to the tank.
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