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The use of on-off control offers significant economic advantages over proportional control or other more sophisticated modes of control. The control mechanism is simply a relay that turns on or off depending on the value of the measured variable. The disadvantage is usually that the quality of control is inferior to that realized with proportional control. Consider the stirred-tank heater of Fig. 32.4. Water is being heated to a controlled temperature by mixing with steam. It is assumed for the analysis that the cold-water input rate is constant. Heated water overflows into an outlet pipe at the top of the tank, so that no accumulation of mass occurs in the tank. Most of the steam is added, at a fixed flow rate, from the main steam supply. However, this amount of steam is set at a value somewhat less than the amount required to heat the cold water to the desired temperature. An additional amount of steam may be added whenever the solenoid valve is opened. When this additional steam is admitted, the sum of the two steam inputs is enough to heat the water to a temperature somewhat in excess of the desired temperature. A temperaturemeasuring device such as a thermocouple or vapor-pressure bulb transmits the tank temperature to the relay. When this temperature is below the set point, the relay closes, which opens the solenoid valve, thus admitting more steam. Eventually, the additional steam will result in the temperature exceeding the set point, the relay will open, the valve will close cutting off the additional steam, and the temperature will fall again.
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FIGURE 32-4
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It is apparent that an oscillating control will be achieved. In fact, from the discussion in the previous section, we recognize that a limit cycle will occur. We consider now a numerical example of this type of control system. Water at 40 F, at a rate of 100 lb/min, is to be heated to 150 E The main steam supply is to be set so that it will heat this much water to 125oF, while additional steam, through the controlled solenoid valve, is available to heat the water another 50 F. This means that the steady-state temperatures with the solenoid closed and open, 125 to 17S F, are equally spaced about the set point. Heat losses to the surroundings are negligible. The volume of the tank is 1.6 ft3. The relay control system has a vapor-pressure bulb for measurement of temperature. This measuring system has a time constant of 30 sec. The solenoid valve is very rapid in response. We first analyze this system considering the relay to behave ideally. This means that it opens precisely at the instant the temperature exceeds the set point and closes similarly. Later, we shall correct this to conform more closely to the behavior of actual relays. If the tank is perfectly stirred, it is a first-order system with a time constant of
7 _ PV _ W(1.6)
= 1.0 min
and its transfer function relating changes in the steam input rate to temperature is
where 10 (OF) (min)/(lb) is the change in steady-state temperature per unit change in steady-state steam flow. The necessary fixed and controlled steam rates are (using 1,000 BtuAb for latent heat)
METHODS
OF PHASE-PLANE ANALYSIS
Qfixed
Q controlled
(125 - 4OWO) = 8 5 lb,min 1,ooO (175 - 125)UOO) = 5 o lb,min
1,ooO
The amount of steam that would be necessary to maintain the water at a steadystate temperature of 150 F is = (150 - 4OWO) = 11.0 lb/min 1,ooO Hence, in terms of deviation variables, the controller output may be taken as *2.5 lb/min of steam. A block diagram may now be constructed for this system, as shown in Fig. 32.5. This diagram uses deviations from 150 F as temperature variables, so the set point is taken as zero. The action of the relay is symbolized by the input-output relations, indicating that +2.5 lb/min of steam are admitted when the error is positive and -2.5 lb/mm when the error is negative, again in deviation variables. The transduction from the vapor-pressure bulb to a temperature reading is included implicitly in Fig. 32.5 in the comparator. The comparator is physically a device that balances the pressure generated by the bulb against a mechanical tension caused by positioning the set point. It need not be explicitly shown because its dynamics are very fast. It is convenient to use a dimensionless version of Fig. 32.5. This is provided in Fig. 32.6, where the changes s Q
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