More on Degrees of Freedom
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.
This particular problem has exercised the minds of quite a few clever people
over the years. It is non-trivial if we start (as everyone did) by considering
all possible measurements. This is because not everything that
can be measured can be regulated independently. For example, if a
piece of equipment
has a single input input and a single output, only one of these can
be flow controlled, since a mass balance about the unit must be
maintained.
However, it is much easier to understand if we consider the number of
quantities we can adjust. Since in practice:
- all adjustments on a chemical plant are made by changing the
flow of a stream by means of a valve, and
- only one valve can be placed on any one pipe,
then the maximum number of adjustable quantities is the
same as the number of streams on the process flowsheet.
Hence the following are all the same:
- Number of streams
- Maximum number of adjustments
- Maximum number of control loops
- Maximum number of meaurements
This is the maximum number since some material balances will
balance automatically, e.g. a simple mixing tee. In this case
we do not need a control loop and valve are not required
on the stream and must not be put there for any other purpose.
However we must check that there is either an explicit (control loop)
or implicit (self regulating) mechanism to ensure that all mas balances
balance about each processing unit.
Where there are two or more phases in a unit, e.g. a liquid and a vapour,
we must maintain an interface between them, which is equivalent to
maintaining a mass balance on each phase. this will normally
require an interface level control system, although some devices,
such as distillation trays, are designed so that a hydaulic balance can
maintain the interface over a limited range of operation.
When counting streams, it is convenient to count utilities
such as cooling water and steam, as a single stream,
although they both enter and leave a unit. An incompressible
utility stream will have a self-regulating mass balance, and
steam heating coils and jackets are usually equipped with a
steam trap, a self contained device which maintains a steam-condensate
interface independent of other control systems.
Counting Degrees of Freedom
The fuller treatment of this concept describes how to count the degrees
of freedom for both individual units and complete processes. This
procedure is a means to an end, to enable the designer to determine
how many control systems must be provided.
In practice, it is better to work as follows. It is useful to determine
both the total number of control loops
(total count T) and the number of loops associated with
regulating `strategic' quantities, i.e. other than inventories,
interfaces and levels (strategic count, S).
- Count the total number of streams in the process; this
is the upper limit on the number of control systems required.
This is the initial value of both T and S.
Now look at each unit in turn.
- Each will require at
least one material balance:
- deduct 1 from S
- Note those which will have self-regulating
mass balances:
- deduct one from T
- Note where an interface level has to be maintained:
- deduct one from S
Finally:
- S is the number of required non-inventory loops
- T - S is the number of explicit inventoru loops
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to Start of Part 2: Further Developments