6.3 BASIC REGULATION THEORY 

It follows from the centrifugal compressor theory, by the law of similarities, that the surge points in a single stage compressor at the various speeds form a parabola with the apex at the origin and axis coinciding with the axis of ordinatescn1the polytropic head - volumetric capacity plane. In fact the following laws apply for the surge point at any speed:

(1) Q₁ = K' N

(2) H_p = KN²

From equations (1) and (2), by substituting N, one gets the equation for the surge curve:

(3) H_p = K" Q²

It is known that the polytropic head is obtained from the following formula:

Should the suction capacity be measured with a calibrated orifice, we have:

where; h = orifice pressure loss in mm_H2O

Substituting in equation (5) the Y1 value obtained from equation (6), one has:

where h₁ is in Kg/cm² 

Again substituting the n. (4) and (7) formules in equation (3):

Simplifying:

or:

To avoid surging in any suction or speed conditions of the machine, it can be noted from this formula that it is sufficient to see that:

or:

 Formula (11) is important for the following reasons:

1. It is independent of the speed of the compressor
2. It requires only measurement of the calibrated orifice pressure loss (h), without any pressure, temperature, gas composition compensation or linearization.
3. It is influenced by the gas composition only through n; however variation of n has little influence on the quantity.

In short, formula (11) employs the compression ratio sensitive to the suction and speed conditions to compensate the capacity. Although considerably simplified formula (11) is still too complex to be realized with very simple instruments. If the compression ratio is small enough, as in the case of single-stage and two-stage compressors, it is possible

to substitute the quantity

with the quantity p - 1 without making any great error. However, any error is to the advantage of greater safet

being

Therefore formula (11) becomes

let us make

The percentage error which is made is obtained from the following expression:

 Expression n. (13) involves a percentage error in the suction capacity

In general one can say that the error in capacity si approximately half of that made substituting p - 1 with

Formula (12) can also be written as:

or:

This is the equation of a parabola (in the plane p,Q₁) with the origin at the point with coordinates p = 1; Q₁ = 0 (see figure. 6.2)

It demonstrates the extreme simplicity of this type of antisurge system which collects all the information necessary to protect the machine from the process with only two instruments (two differential transmitters) .

The two signals are sent to a divider which works out the ratio. This ratio is sent to a regulator, on which the K" ' value, i.e. the antisurge regulation set-point, is set.

The regulator, acting on the by-pass valve, keeps the ratio "from going below the established set-point, thus fulfilling the requirements.

Hereafter (see figure. 6.3) a typical diagram

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