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.

**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.

*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.

**Figure 1 - Simple Blender**

**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**

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.

As you can see in this example there are:

- No. of Streams = 3
- No. of Extra Phases (interfaces) = 1
- CDF = 3 - 1 = 2

This time there are:

- No. of Streams = 3
- No. of Extra Phases (interfaces) = 1
- CDF = 3 - 1 = 2

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

**Figure 3 - Countercurrent Cascade**

- 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**

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

- 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.
- 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

- Material balances will always be somehow balanced.
- An adjustment which affects only inventory is
**not**as such a control degree of freedom.

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.

- In the case of holdup dependent reaction conversion.
- In situations where inventory determines pressure.
- 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.