The temperature of a chemical reactor is to be regulated by adjusting the flow of heating fluid to a jacket.
The following data is available:
- Heating fluid maximum flowrate = 10kg/s
- Temperature measurement
- Minimum reading = 20oC
- Maximum reading = 220oC
The heating fluid valve is initially 20% open and the measured temperature is 140oC. The valve is then opened to 40% and the temperature finally settles down at 173oC.
- If the relationship between heating fluid flow and reactor temperature is assumed to be linear what is the relationship?
- What is the value of the dimensional process gain and what are its dimensions?
- What is the dimensionless process gain?
- If the reactor temperature is to be controlled at 180oC using a proportional-only controller, what percentage manual offset would you recommend?
- Experimental tuning of the process suggests the use of a dimensionless controller gain, standardised to a unity gain process, of 2.4. To what proportional band setting does this correspond?
- A theoretical investigation of the process suggests a dimensional controller gain of 0.2kg/s/oC. What proportional band does this represent?
Answers to Question
If the relationship between heating fluid flow and reactor temperature is assumed to be linear what is the relationship?
The equation for a straight line is
Temp = aFlow + b
So now we have to evaluate a and b. We know two points on the line
- Flow = 2 kg/s, Temp = 140 oC
- Flow = 4 kg/s, Temp = 173 oC
and so the equation can be evaluated to be:
Temp = 16.5 Flow + 107
What is the value of the dimensional process gain and what are its dimensions?
Dimensional process gain = 16.5 oC(kg/s)
The units are temperature/flow
What is the dimensionless process gain?
To get this multiply the dimensional gain by (kg/s per flow %) and divide by (oC per temperature %).
- Valve scaling is 0.1 (kg/s per flow %)
- Temperature range is 200 oC
- Temperature scaling is 2 (oC per temperature %)
Dimensionless gain = 16.5 * 0.1 / 2 = 0.825
Easier way…
The dimensionless gain really has units of temperature range % / flow range %.
- Change in temperature is 33 degrees which is 16.5% of range
- Change in flow is 20%
- Dimensionless gain = 16.5/20 = 0.825
If the reactor temperature is to be controlled at 180oC using a proportional-only controller, what percentage manual offset would you recommend?
For the answer to this question determine what flow will give a temperature of 180 oC and convert this to a valve position.
- 180 = 16.5 Flow + 107
- Flow = 4.42 kg/s
This represents 44.2% of the valve range and this must be the controller output with zero error to achieve the required temperature.
Experimental tuning of the process suggests the use of a dimensionless controller gain, standardised to a unity gain process, of 2.4. To what proportional band setting does this correspond?
The standard gain refers to a process with unity gain. Divide this by the process gain to get the actual required dimensionless gain:
- 2.4/0.825 = 2.91
This process has a gain of 0.825, which is less than one, so the required controller gain will need to be greater than the standard gain, so you need to divide by the process gain to increase it.
Proportional band is the reciprocal of dimensionless gain, expressed as a percentage:
PB = 1/2.91 * 100 = 34%
A theoretical investigation of the process suggests a dimensional controller gain of 0.2kg/s/oC. What proportional band does this represent?
This uses the same approach as the more involved procedure for the third part of this question above.
Dimensionless gain = dimensional gain {(kg/s)/oC } * {oC/temp%} / {(kg/s)/flow%
= 0.2 * 2 / 0.1
= 4
So PB = 100/4 = 25%