# More on Degrees of Freedom

## Introduction

When faced with the task of devising a control scheme for a process it is necessary to know how many of the process variables am I entitled to attempt to regulate. By process variables we mean temperatures, pressures, compositions, flowrates or component flowrates. The answer arrived at is known as the number of degrees of freedom.

What follows is a summary of a method for analysis of the number of degrees of freedom which is described in:

Ponton JW, 1994, Degrees of Freedom Analysis in Process Control, Chemical Engineering Science, Vol. 49, No. 13, pp 1089 - 1095.

In fact it is possible to understand the key concepts from a much simpler viewpoint. This is described here.

## Definitions

Before we start analysing the degrees of freedom it is perhaps wise to define a few terms.
Degrees of Freedom of a process
The number of process variables which can be set by the designer, operator or control system.
Control Degrees of Freedom
• The number of the above types of process variable which may be set once non-adjustable design variables have been fixed.
• The number of manipulated variables which may be used in control loops.
• The number of SISO control loops.
• The number of regulated variables in the control loops.
The procedure which will be described is a counting process which identifies potential manipulations, but does not directly identify variables which may be regulated.

## Derivation

The technique can be derived by applying the Kwauk method. This was first developed by Kwauk [1952] and later by Smith [1963]. The equation to be solved is

Degrees of Freedom = Unknowns - Equations

Unknowns

There are two types of streams - process streams and energy streams. If we take a process stream containing C components then the unknowns are C flowrates, temperature and pressure. An energy stream has 1 unknown associated with it.

Equations

The equations are best thought of as types of equations i.e. component balances, energy balances, equilibrium equations, equivalent T and P etc.

Degrees of Freedom

Once the final number of D.O.F. have been found then it is necessary to decide which of these are already fixed and which can be controlled.

## Example 1 - Simple Blender

Let us take, for our first example, a simple blending process. Assume that the mixing takes place in the vapour phase. There two inputs and one output as shown in the figure below.

Figure 1 - Simple Blender

It is possible to apply the above equation to evaluate the number of degrees of freedom

Unknowns

3 streams with C+2 unknowns = 3C + 6 Unknowns

Equations

• C Mass Balance Equations
• 1 Energy Balance Equation
• C+1 Equations

Degrees of Freedom

D.O.F = 2C + 5

However in this system we know the composition, temperature and pressure of the two input streams. This adds on a further

2C + 2 Constraints

Hence there are 3 D.O.F and so we can fix the output composition, pressure and throughput.

A typical control scheme is shown below.

Figure 2 - Simple Blender with Control

## Alternative Equation

The question now is Is there a relationship between the control degrees of freedom, the number of streams and the number of phases present. In the paper it is proved theoretically that there is such a relationship.

It is possible to

• Number all the streams associated with the unit
• Count all the phases
• Determine the number of extra phases ie the (number of phases - 1)
• This number is also the number of interfaces
• Subtract interface count from stream count giving answer

Hence for the example discussed earlier

Blender

• No. of Streams = 3
• No. of Extra Phases (interfaces) = 0
• CDF = 3 - 0 = 3 - as before

This equation is extremely simple to use and discards the need to think about unknowns and equations.

## Example 2 - Adiabatic Flash

The next example we will look at is the adiabatic flash. Note that we will be returning to this example in a later section. A diagram is shown below.

As you can see in this example there are:

• No. of Streams = 3
• No. of Extra Phases (interfaces) = 1
• CDF = 3 - 1 = 2
Therefore it is possible to control two strategic variables in an adiabatic flash eg feed flowrate and pressure, leaving the third stream free to determine the inventory.

## Example 3 - Total Condenser

The third example of this technique is a total condenser.

This time there are:

• No. of Streams = 3
• No. of Extra Phases (interfaces) = 1
• CDF = 3 - 1 = 2
Again it is possible to control two strategic variables.

## Example 4 - Countercurrent Cascade

A more complex unit that we can look at is a countercurrent cascade.

First consider a single countercurrent equilibrium stage:

• No. of Streams = 4
• No. of Extra Phases (interfaces) = 1
• CDF = 4 - 1 = 3

A stack of N such units built into a cascade will have 2N + 2 streams, N two phase elements and thus N + 2 apparant degrees of freedom. These would never all be used in practise however, but it is, in principle, possible to maintain each stage at a different pressure by some valve-like arrangement between the trays.

If there is a fixed rather than adjustable restriction between trays then a potential degree of freedom is lost for each of N-1 vapour interstreams, so the practical degrees of freedom for this device is [N + 2] - [N - 1] = 3.

This is the same number of degrees of freedom as would be calculated by simplistically counting the streams to the complete cascade, 4, and subtracting 1 for the presence of two phases in the device, again giving 3 Control Degrees of Freedom

## Application To Complete Processes

Determination of the total degrees of freedom for a complete process is now trivial. Connected units sharing a stream lose one degree of freedom from the sum of those for the individual units. Thus the stream related degrees of freedom is simply equal to the total number of material and energy streams in the process.

The degrees of freedom for the complete process may be determined by either of two equivalent procedures.

1. Using the approach above determine the degrees of freedom for each unit. Sum these and then subtract the number of shared streams to obtain the final count.
2. Count all the streams in the process. Separately count the total number of extra phases i.e. add up all occurances of phases greater than one in all units.

Method 2 is shown in the example below. All streams represent potential degrees of freedom and possible adjustable variables, but beside each unit is written the number lost as a result of the presence of multiple phases in the unit.

Figure 4 - Absorption Process with Solvent Recovery Flash Separator

• Total Streams = 12
• Extra Phases = 3
• Total D.O.F = 9

## Inventory

In the preceeding analysis it has been assumed that
1. Material balances will always be somehow balanced.
2. An adjustment which affects only inventory is not as such a control degree of freedom.
An alternative view of 2 is that inventory control should be regarded as a taking up a degree of freedom. Under this assumption the number of degrees of freedom will always equal the number the material and energy streams. Thus for the countercurrent cascade there would be 4 degrees of freedom rather than 3, the additional one being the inventory of the liquid phase. This alternative approach does have the advantage of identifying all the potential control loops, explicit or implicit, which will exist in a process.

It is quite straightforward to identify when and where inventory regulation should be provided. Two trivial but useful rules are used in this decision.

• If n streams join together in a process or part of a process over which mass must be conserved - normally any process - then the flow of only [n-1] these may be set by control loops other than one regulating inventory within the process or part process.
• In any unit containing P phases, P control loops - explicit or implicit - must be provided to maintain P inventories, which may include combinations of phases. This number allows for a loop to regulate total inventory.

Finally there appears to be three circumstances in which inventory affects strategic process variables and thus should be counted as a control degree of freedom.

1. In the case of holdup dependent reaction conversion.
2. In situations where inventory determines pressure.
3. The construction of certain types of equipment means that a change in holdup will affect performance e.g. by changing the surface area in a flooded condenser.

It should be noted that where inventory affects several strategic variables there will still only be one degree of freedom.