5. OPERATING FEATURES OF CENTRIFUGAL COMPRESSORS

5 .1 GAS COMPRESSION

5.1.1 Representation of compression on the h,s diagram

It is possible to analyse the whole compression process that is achieved in a compressor stage on a Mollier (h,s) diagram of the handled fluid

The entire compression stage, or rather the whole process of handling the gas in the stage, has the aim of increasing the gas pressure from the initial pressure pA to the final pressure pB.

The process takes place, however, in several stages each of which is achieved in a different part of the machine.

The initial stage is taken as an example. The fluid which is atia pressure pA, temperature T_A etc. outside the machine, that is, in the conditions of point A in figure. 5.1, will reach the compressor suction inlet in 0 conditions, i.e. at C₀ velocity, P₀ pressure, T₀ temperature, etc.

The velocity will then be further increased owing to suction so that the fluid reaches the impeller inlet with velocity C1, pressure p1 etc. This increase in velocity occurs with losses therefore the point on the Mollier diagram representing fluid conditions at the impeller inlet is point 1.

Next the fluid enters.,the impeller where all the energy exchangebetween the machine and fluid takes place and leaves it at pressure p2, temperature T2 and an absolute./¢elocity.vC2, namely with an energy content per unit of weight equal to

Inside the impeller, therefore, the fluid unit of weight has undergone an increase in energy in potential form (h2 - h1), as well as a further increase due to the difference

representing the increase in energy had in kinetic form

After leaving the impeller, the fluid flows through the diffuser-return channel (at the stage of a multistage compressor) where some kinetic energy is changed into potential energy; then it goes out of the stage in conditions B (p_B, T_B etc.) and at velocity C_B.

The fluid leaves therefore the stage in conditions B, at velocity CB.

On the T,s diagram the transformation is usually represented as shown on Figure. 5.2, referring to conditions 1 and 2 at the stage inlet and outlet. 

5.1.2 Definition of the major physical quantities in compression

a. Actual head Heff

The actual head Heff of a compressor stage (as for a whole machine) is the real work L₁, 2 which is exchanged between fluid and environment per unit of weigth

Following this assumption, the actual head Heff will be expressed by the relation

or by the relation

which, in case of machines where supposition is very close to actualities, assumes the well known form

Eor the adiabatic transformation (Qe = 0)

The considerations resulting from a swift check of the above relations, are the following:

• only a share of the work L1,2 given (or actual head) is found in the form of increase in thermo-dynamic potential energy of the fluid, expressed by vdp (this value being called polytropic head H_pol we shall speak about later on) while a part La, is used to overcome passive resistance phenomena (losses due to friction, impact etc.) connected with the fluid flow and which, transformed into heat, remain within the fluid itself;

• the knowledge of the variation in the quantity "enthalpy h", inferable from gas pressure and temperature measured at compressor suction and discharge, allows to estimate the actual head H_eff (or specific work) exchanged between environment and machine per unit of weight.

b. Polytropic head H_pol

As previously stated, the polytropic head Hpol of . one stage, as well as of a whole machine, is defined to be the energy accumulated in the fluid always in the form of thermodynamic potential energy increase expressed by

The above relation for polytropic head H_pol is generally expressed in the form obtained by making the function bond explicit between pressure p and specific volume V of the gas, thus being generally given by the relation

pvⁿ = cost

whereas n is the average exponent of the polytropic transformation between points 1 and 2 at the beginning and at the end of the real compression process (Figure. 5.2).

If we replace above relation in the integral and develop it, we have the following relation

where

Z₁ is the compressibility factor calculated at the initial conditions;

R is the gas characteristic constant;

T₁ is the gas suction temperature (OK)

p₁ is the gas Suction pressure;

p₂ is the gas discharge pressure;

n is the polytropic transformation exponent assumed to be constant during the transformation.

The units of measurement with which the polytropic head Hpol is expressed, are meters (kp m/kp) = (m), if the technical system for unit of measurement is adopted or, more properly, the units for the specific en

ergy (J/Kg) if the international system of units of measurement is applied.

c. Polytropic efficienfry ⁿpol

The polytropic efficiency npol of a compressor is defined to be the ratio between the polytropic head Hpol, just defined, and the actual head Heff expended to compress each kp of aeriform substance. 

Therefore according to what previously stated and considering the expressions already given for Hpol and Heff, we have

this relation can be in the form

 for a perfect gas, whereas

n is the average exponent of compression polytropic transformation between initial and final conditions

k is the exponent of the relative isoentropic adiabatic transformation, at the same compression ratio and the real transformation considered

Always referring to Figure. 5.2, remembering that perfect gas is considered,

hence

is following, from which we can infer that is possible to estimate the polytropic efficiency, reliably and experimentally, by measuring the thermodynamic parameters at compressor suction (p1 and T₁) anddischarge (p2 and T₂) , provided that the handled gas could be assimilated to a perfect gas.

Otherwise, Hpol and Heff values shall be calculated, after measuring pressure and temperature inlet and outlet values experimentally and being gas composition known, according to the equations of state which represent the gas behaviour as real as possible.

The commonest equation of state applied to natural gas is the B.W.R.S. equation (Benedict; Webb, Rubbin, Starling).

d. Adiabatic head Had

The adiabatic head in a compressor (Had) is the energy that would remain in the fluid because of a reversible, and hence isoenthropic, adiabatic compression_process (transformation 1  -  2_is in Figure. 5.2) occurred between the same initial p1 and the final pg pressures, between which the real compres- sion process is performed.

Head H_is is then obtained from the relation

considering k = cost. during the transformation, where

k isvthe esponent binding pressures and specific volumes during isoentropic compression

Z₁ is the compressibility factor of the fluid at the beginning of reversible adiabatic transformation

R is the gas characteristic constant

T₁ is the gas suction temperature

p1 is the gas suction pressure

p2 is the gas discharge pressure

Also the quantity H_is, as for the polytropic head H_pol, is expressed in (m) or (J/kg) depending on the fact whether the technical system or the international system is applied as to the units of mea: surement.

e. Adiabatic efficiency n_ad

The adiabatic efficiency n_ad of a compressor is the ratio between adiabatic head H_ad, hereby defined, and the actual head Heff.

In spite of the considerations on polytropic efficiency, the adiabatic efficiency nad depends on the compression ratio pg/p1, besides on the machine and the nature of the fluid, as in case of perfect gas it is connected with the relation

 This relation shows that the adiabatic efficiency nad is always lower than the polytropic efficiency npol (in an adiabatic compression process); the more the compression ratio "r" tends to 1 the more the adiabatic efficiency tends to the polytropic value.

f. Power absorbed by the compressor

The available diagrams, supplied by the manufacturer, which give head H (polytropic or adiabatic) and the efficiency (polytropic or adiabatic) for certain values of the speed of rotation n, allow to trace back to the real specific work L₁,  2, (or actual head H_eff) by means of the relations

 Knowing the specific work L¹,2, it is possible to go back to the total power P required by compression, by the relation

where P_f represents the power lost owing to leakage and Pm the power lost owing to mechanical losses.

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