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CHE 185 – PROCESS CONTROL AND DYNAMICS SECOND AND HIGHER ORDER PROCESSES

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2 SECOND ORDER PROCESSES CHARACTERIZATION CAN RESULT FROM TWO FIRST ORDER OR ONE SECOND ORDER ODE GENERAL FORM OF THE SECOND ORDER EQUATION AND THE ASSOCIATED TRANSFER FUNCTION

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CHARACTERISTIC EQUATION POLYNOMIAL FORMED FROM THE COEFFICIENTS OF THE EQUATION IN TERMS OF y: THREE POSSIBLE SOLUTIONS FOR THE STEP RESPONSE OF PROCESSES DESCRIBED BY THIS EQUATION. USING THE NORMAL QUADRATIC SOLUTION FORMULA:

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7 SECOND ORDER PROCESSES CHARACTERIZATION NOTE THAT THE GAIN, TIME CONSTANT, AND THE DAMPING FACTOR DEFINE THE DYNAMIC BEHAVIOR OF 2ND ORDER PROCESS.

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DAMPING FACTORS, ζ DAMPING FACTORS, ζ, ARE REPRESENTED BY FIGURES THROUGH IN THE TEXT, FOR A STEP CHANGE TYPES OF DAMPING FACTORS –UNDERDAMPED –CRITICALLY DAMPED –OVERDAMPED 8

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UNDERDAMPED CHARACTERISTICS 9

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EFFECT OF ζ (0.1 TO 1.0) ON UNDERDAMPED RESPONSE: 10

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UNDERDAMPED CHARACTERISTICS EFFECT OF ζ (0.0 TO -0.1) ON UNDERDAMPED RESPONSE: 11

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OVERDAMPED CHARACTERISTICS 12

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CRITICALLY DAMPED CHARACTERISTICS 13

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CHARACTERISTICS OF AN UNDERDAMPED RESPONSE RISE TIME OVERSHOOT (B) DECAY RATIO (C/B) SETTLING OR RESPONSE TIME PERIOD (T) FIGURE 6.4.4

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EXAMPLES OF 2 ND ORDER SYSTEMS THE GRAVITY DRAINED TANKS AND THE HEAT EXCHANGER IN THE SIMULATION PROGRAM ARE EXAMPLES OF SECOND ORDER SYSTEMS PROCESSES WITH INTEGRATING FUNCTIONS ARE ALSO SECOND ORDER.

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2ND ORDER PROCESS EXAMPLE THE CLOSED LOOP PERFORMANCE OF A PROCESS WITH A PI CONTROLLER CAN BEHAVE AS A SECOND ORDER PROCESS. WHEN THE AGGRESSIVENESS OF THE CONTROLLER IS VERY LOW, THE RESPONSE WILL BE OVERDAMPED. AS THE AGGRESSIVENESS OF THE CONTROLLER IS INCREASED, THE RESPONSE WILL BECOME UNDERDAMPED.

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DETERMINING THE PARAMETERS OF A 2ND ORDER SYSTEM SEE EXAMPLE 6.6 TO SEE METHOD FOR OBTAINING VALUES FROM TRANSFER FUNCTION SEE EXAMPLE 6.7 TO SEE METHOD FOR OBTAINING VALUES FROM MEASURED DATA

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2 ND ORDER PROCESS RISE TIME TIME REQUIRED FOR CONTROLLED VARIABLE TO REACH NEW STEADY STATE VALUE AFTER A STEP CHANGE NOTE THE EFFECT FOR VALUES OF ζ FOR UNDER, OVER AND CRITICALLY DAMPED SYSTEMS. SHORT RISE TIMES ARE PREFERRED

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2 ND ORDER PROCESS OVERSHOOT MAXIMUM AMOUNT THE CONTROLLED VARIABLE EXCEEDS THE NEW STEADY STATE VALUE THIS VALUE BECOMES IMPORTANT IF THE OVERSHOOT RESULTS IN EITHER DEGRADATION OF EQUIPMENT OR UNDUE STRESS ON THE SYSTEM

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2 ND ORDER PROCESS DECAY RATIO RATIO OF THE MAGNITUDE OF SUCCESSIVE PEAKS IN THE RESPONSE A SMALL DECAY RATIO IS PREFERRED

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2 ND ORDER PROCESS OSCILLATORY PERIOD THE OSCILLATORY PERIOD OF A CYCLE IMPORTANT CHARACTERISTIC OF A CLOSED LOOP SYSTEM

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2 ND ORDER PROCESS RESPONSE OR SETTLING TIME TIME REQUIRED TO ACHIEVE 95% OR MORE OF THE FINAL STEP VALUE RELATED TO RISE TIME AND DECAY RATIO SHORT TIME IS NORMALLY THE TARGET

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HIGHER ORDER PROCESSES MAY BE CONSIDERED AS FIRST ORDER FUNCTIONS GENERAL FORM

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HIGHER ORDER PROCESSES THE LARGER n, THE MORE SLUGGISH THE PROCESS RESPONSE (I.E., THE LARGER THE EFFECTIVE DEADTIME TRANSFER FUNCTION

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