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%