Centrifugal compressor efficiency types and rational application

Centrifugal compressor efficiency types and rational application Y B Galerkin, A Drozdov, K V Soldatova Compressor Dept., TU Saint-Petersburg, Russia

ABSTRACT Expected efficiency of a compressor must be used in engineering calculations as it is not possible to predict head losses in a flow path. Various aspects of efficiency application (flow parameters in control planes definition, power consumption calculation) are shortly discussed. Presented samples show errors of different compressors comparison if inappropriate adiabatic efficiency is applied. The “real” efficiency based on calculation of head losses in elements of a flow path is presented. The “real” efficiency calculated by the Universal modeling method (1,7)) is compared with usual total polytropic efficiency. NOTATION

c f H Hp

- absolute velocity, m/s; - area, m2; - head, j/kg; - polytropic head, j/kg;

η=

ηr - real efficiency;

H i - internal head, j/kg;

π=

H d - dynamic head, j/kg;

ρ

H r - head losses j/kg;

k - Isentropic coefficient; m& - mass flow rate, kg/s; М - Mach number; u2 Mu = ; kRTinl tot

n - polytropic coefficient; N i - power, W; p R S T v

- pressure, Pa; - gas constant, j/kg/K; - entropy, j/kg/K; - temperature, K; - volume, m3;

___________________________________________ © The author(s) and/or their employer(s), 2013

ψp - polytropic efficiency; ψi

ψ=

pex - pressure ratio; pinl - gas density, kg/m3;

hp u22

- polytropic head coefficient;

ψ T = cu 2 / u2 - Euler work coefficient; ς - loss coefficient; Subscripts ad – adiabatic; des – design; ex – stage exit; inl – stage inlet; ideal – ideal; mean – mean; meas – measured; t – total parameter.

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EFFICIENCY FOR CALCULATION OF GAS PARAMETERS IN FLOW PATH ELEMENTS Flow parameters in control planes in the flow path are necessary to define velocity triangles (direct task) or area of the control planes in the flow path (reverse task). The compression process equation controls (for) the change of the parameters:

pv n = const .

(1)

In engineering calculations the exponent n is assumed to be constant in all flow path elements (10). The graphic representation of the process is shown as a T − S diagram in Fig. 1.

Fig. 1. T − S - diagram of compression process in a stage or in a non cooled multistage compressor A polytropic exponent n depends on a polytropic efficiency (static parameters):

n k . =η n −1 k −1

(2)

A polytropic efficiency depends on flow parameters measured at the inlet and exit control planes of a stage or a compressor:

η=

lg ( p2 / p1 ) . k lg (T2 / T1 ) k −1

(3)

The temperature Tmeas is measured by the thermometer or thermocouple located in inlet/exit planes. Its value is intermediate between the static temperature T and total temperature Tt . Regulations for calculating temperatures T , Tt for thermometers and thermocouples based on Tmeas were put forward in (2).

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For calculating flow velocities the following values are needed:

cinl =

m&

ρinl finl

cex =

,

m&

ρex f ex

.

(4)

Pressure and density change equations together with the compression process equation allow solving the problem of flow parameters definition in the stage control planes in principle: n

1

p2 ⎛ T2 ⎞ n −1 =⎜ ⎟ , p1 ⎝ T1 ⎠

ρ 2 ⎛ T2 ⎞ n −1 =⎜ ⎟ . ρ1 ⎝ T1 ⎠

(5)

Location of control planes is shown in Fig. 2. Static temperatures at control planes depend on total temperature and a flow velocity:

T = Tt −

c2 . 2k R k −1

(6)

If heat transfer can be neglected, total temperatures in all control planes previous of an impeller are equal to Tt inl and after the impeller they are equal to Tt ex . Total temperature rise due to mechanical work input is:

T2 t = T1t +

Hi . k R k −1

(7)

Fig. 2. Flow path of the industrial type centrifugal stage and control planes

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Application of total adiabatic efficiency for calculations is also possible and is not less convenient: k

H t ad

⎡ k ⎢⎛ pt ex RTt inl ⎢⎜ = ⎜p k −1 ⎢⎣⎝ t inl

k ⎤ ⎞ k −1 ⎥ ⎟ − 1⎥ , ⎟ ⎠ ⎥⎦

Hi =

H t ad k R (Tt ex − Tt inl ) , ηt ad = k −1 Hi

⎛ pt ex ⎞ k −1 ⎜⎜ ⎟ −1 pt inl ⎟⎠ ⎝ , (8) = Tt ex −1 Tt inl

the total pressure ratio in accordance with Eq. (8) is:

pt ex pt inl

⎡ ⎛ Tt iex ⎞⎤ = ⎢1 + ηt ad ⎜ − 1⎟ ⎥ ⎜ Tt inl ⎟⎥ ⎢⎣ ⎝ ⎠⎦

k −1 k

,

(9)

the static pressure in a control plane is: k

⎛ T ⎞ k −1 p = pt ⎜ ⎟ , p = pt pt ⎝ Tt ⎠

(10)

As velocities in all control planes depend on density (Eq. (4)), density depends on temperatures (Eq. (5)) and temperature depends on velocity (Eq. (6)), it is necessary to apply iterative processes. Calculation of efficiency by equations (3) or (8) requires taking into account the external heat transfer. Usually a compressor case has a higher temperature than the surrounding atmosphere, so the exit temperature is less than it would be if there were no heat transfer. The performance test code establishes measures to take heat transfer into account (12). POWER CONSUMPTION CALCULATION Mechanical power of the drive transferred to the compressed gas by the impellers is equal to:

& i. N i = mH

(11)

& is measured in compressor tests. Some general considerations Mass flow rate m concerning an internal head are briefly presented below. Internal head H i transmitted by the impellers in a flow path produces polytropic head H p , and dynamic head H d . H i is partly lost in a flow path due to different kinds of drag forces:

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Hi = H p + Hd + H r ,

(12)

n ⎡ ⎤ ⎛ pex ⎞ n −1 ⎥ n ⎢ Hp = RTinl ⎜ − 1 , ⎢ p ⎟ ⎥ n −1 ⎢⎣⎝ inl ⎠ ⎥⎦

(13)

Hd =

2 cex2 − cinl . 2

(14)

It is possible to use one of the total efficiencies to estimate the head loss H r . Polytropic total efficiency:

ηt =

H p + Hd Hi

,

Hi =

H p + Hd ηt

.

(15)

.

(16)

Adiabatic total efficiency:

ηt ad

⎡ k ⎢⎛ p RTt inl ⎢⎜ t ex ⎜p k −1 ⎢⎣⎝ t inl = Hi

⎞ ⎟⎟ ⎠

k −1 k

⎤ ⎥ − 1⎥ ⎥⎦

As known the adiabatic efficiency depends not only on the head loss but also on similarity criteria M , k . It is not easy to estimate the value of ηt ad accurately if there is no close analog for the designed compressor. Therefore, the total polytropic efficiency is used in industrial compressor analysis (12):

⎛p ⎞ lg ⎜ t ex ⎟ ⎜ pt inl ⎟ ⎝ ⎠ ηt = ⎛ Tt ex k lg ⎜ k − 1 ⎜⎝ Tt inl

⎞ ⎟⎟ ⎠

Tt ex

,

Tt inl

⎡ ⎛ pt ex ⎞ ⎤ ⎢ lg ⎜⎜ ⎟⎥ ⎛ Tt ex ⎞ k ⎢ ⎝ pt inl ⎟⎠ ⎥ = a log ⎢ ⎥ , H i = k − 1 RTt inl ⎜⎜ T − 1⎟⎟ . k ⎝ t inl ⎠ ⎢ ηt pol ⎥ ⎢ k −1 ⎥ ⎣ ⎦

(17)

COMPARISON OF COMPRESSORS AERODYNAMIC PERFECTNESS The efficiency of a better designed compressor seems to be higher. However, this may not be true, if adiabatic efficiency is used in order to compare compressors with different pressure ratios and adiabatic exponents. As dynamic head

H d = 0,5 ( c22 − c12 ) is small in turbo compressors it will be neglected in further

considerations. In accordance with known properties of T − S diagram the internal head H i is proportional to the area 6-2-3-4 in Fig. 1. The head loss is proportional to the area under the line of a pv = const process, i.e. to area 5-1-2-6. It means that total n

polytropic efficiency depends exclusively on ratio H r / H i (and nothing else):

ηt =

Hp Hi

=1−

Hr . Hi

(18)

Adiabatic head in Fig. 1 corresponds to area 5-2'-3-4. Adiabatic head ratio to the internal head (area 6-2-3-4) defines adiabatic efficiency that is always less than polytropic efficiency for the same compressor:

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ηad =

H ad H <1− r =η . Hi Hi

(19)

Adiabatic efficiency is unduly treated as head loss area 1-2-2 '. So, loss of efficiency Δη ad = 1 − η ad > Δη p does not reflect a share of head losses in a compression process. The relationship between adiabatic and polytropic efficiencies follows from the formulae for adiabatic and polytropic heads:

ηad =

π π

k −1 k k −1 kη

−1

.

(20)

−1

Table 1 demonstrates a sample of efficiency difference Δη = η − η ad that depends on

π, k , η

in accordance with equation (20). Efficiencies are compared for

compressors operated at different M u that leads to different pressure ratio as it is shown for a single stage compressor: k

π = (1 + (k − 1)ηψ i M u2 ) k −1 . η

(21)

The calculation of adiabatic efficiency is made for a compressor with polytropic efficiency η = 0,86 and for a compressor with η = 0,82 . Polytropic efficiencies are assumed to be independent of pressure ratio, i.e. are independent of M u in accordance with eq. (21). Table 1. Adiabatic efficiency change with change of pressure ratio ( k = 1,4)

η

π η ad

1.5

2.0

4.0

8.0

16.0

0.86

0.8517

0.8457

0.8308

0.8151

0.7989

0.82

η ad

0.8084

0.8017

0.7827

0.7628

0.7423

The difference Δη = η − η ad increases from 0,5 – 1% for compressors with

π = 1,5

up to 6-8% for compressors with π = 16. It demonstrates an adiabatic efficiency inability to reflect properly level of head losses in a compression process. LOSS ESTIMATION If polytropic efficiency is equal 0,86 for instance it means that 14% of an internal head is lost – eq. (18). However, for an efficiency calculated by eq. (3) it would be true if a polytropic exponent n would be constant. In fact a flow path consists of compression elements with different level of compression efficiency where always n > k . There are also confuser elements of a flow path where n < k . It means that n are different in different elements. The next equation connects polytropic and isentropic exponents through a head loss and kinetic energy:

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⎛ ⎞ ⎟ n k ⎜ hw k ⎛ 0.5ζc22 ⎞ ⎜1 + ⎟. = 1 + ⎜ ⎟= k n −1 k −1⎜ k − 1 ⎜ 0.5 ( c12 − c22 ) ⎟ ⎟ R T T − ⎝ ⎠ ( ) ⎜ 2 1 ⎟ k −1 ⎝ ⎠

(22)

The loss model of the TU SPb Universal modeling method (3,4) is based on summarizing losses in each element of a centrifugal compressor stage. Therefore the programs provide information to present T − S - diagram with n = var in a stage elements. Several “real” T − S - diagrams by Universal modeling method are presented below. Calculation of entropy change by well-known formula

S 2 − S1 =

R ⎛ p2 p1 ⎞ ⎜ ln k − ln k ⎟ (9) is not applicable for flow path elements where ρ1 ⎠ k − 1 ⎝ ρ2

change of pressure and density is too small. The simplified formula based on

dS =

dhr was used instead: T

ΔS =

p2 t ideal − p2 t 2 R ( p2 t ideal − p2 t ) hw . = = ρ meanTmean Tmean p2 + p1

(23)

The T − S diagrams for the 1-st stage of two-stage pipeline compressor at three flow rates are presented in Fig. 3. 340 T 335

360

T

350

315 T

310

330

340

325

330

320

305

315

320

310

310

305

300 295

300

300

295

290

290

290

0

10

20

30

40

50ΔS60

0

5

10

15

20 ΔS25

0

5 10 15 20 25 30ΔS35

Fig. 3. T − S - diagrams of the 1st stage of two-stage pipeline compressor.

& / m& des = 0,485 (close to surge), in the center – design flow rate, Left – m & / m& des = 1,526. n = var - solid lines, n = const - stroke lines right - m The area under a line of a compression process that is proportional to H r is visibly different for n = const and for n = var . The difference is minimal at the most effective design flow rate. As appeared, it is true also for compressors with different efficiency at design regimes too. The compressor with T − S - diagrams presented at Fig. 3 has not too high efficiency. The newly designed one-stage pipeline compressor is more effective. Its diagrams at three regimes are shown at Fig. 4.

539

310

305

300

T 305

T

T 298

300

296 294

300

292

295

295

290 288

290

290

286

285

284

285

0 5 10 15 20 25 30 35 40 45 ΔS

0

2

4

6

0

8 10 12 14

ΔS

2 4

6

8 10 12 14 16 18 20 ΔS

Fig. 4. T − S - diagrams of one stage high effective pipeline compressor.

& / m& des = 0,360 (close to surge), in the center – design flow rate, Left – m & / m& des = 1.435. n = var - solid lines, n = const - stroke lines right - m The T − S - diagram at a design regime for less effective multistage compressor is presented at Fig. 5. 380 T 370 360 350 340 330 320 310 300 290 280 0

20

40

60

80

100

120

140 ΔS 160

Fig. 5. T − S - diagram of 6 - stage comparatively low effective pipeline compressor. Design regime. n = var - solid lines, n = const - stroke lines th

The computer programs of Universal modeling method calculate efficiency that we name “real” polytropic efficiency:

ηt r =

540

H i − ∑ hr Hi

=1−

∑h

r

Hi

.

(24)

A polytropic efficiency that is used in engineering practice and that is measured in plant tests of compressors in accordance with (12) is defined by the next equation:

⎛p ⎞ lg ⎜ t ext ⎟ ⎜ pt inl ⎟ ⎝ ⎠ ηt = ⎛ Tt ex k lg ⎜ k − 1 ⎜⎝ Tt inl

⎞ ⎟⎟ ⎠

.

(25)

The efficiencies calculated by eq. (25) and (24) for stages and compressors at three regimes are presented Tables 2, 3, 4. Table 2. Total, real, static polytropic efficiencies of the 1st stage medium – effective pipeline compressor

m& / m& des

1.5258

1

0.4845

ηt

0.5467

0.8554

0.7517

ηt r

0.5480

0.8600

0.7600

ηt r /ηt

1.0023

1.0053

1.0110

Table 3. Total, real, static polytropic efficiencies of the 2nd stage medium – effective pipeline compressor

m& / m& des

1.6939

1

0.4743

ηt

0.2518

0.8497

0.7455

ηt r

0.2580

0.8550

0.7560

ηt r /ηt

1.0246

1.0062

1.0057

Table 4. Total, real, static polytropic efficiencies of the one- stage high effective pipeline compressor

m& / m& des

1.4336

1

0.3590

ηt

0.759

0.875

0.717

ηt r

0.762

0.880

0.727

ηt r / ηt

1.004

1.006

1.014

The “real” efficiency is higher in all cases. It is not important for engineering practice but it must be meant when results of modeling are compared with test data. CONCLUSION Well-known efficiency types – polytropic, adiabatic, total and static - serve well for proper to each of them kinds of application. Head losses are calculated in elements of a flow path and summarized in a process of a centrifugal compressor performance modeling. The “real” polytropic efficiency is defined by these summarized losses. This “real” efficiency appeared to be higher than a usual total

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polytropic efficiency in several cases presented in the paper. The difference is visible especially for off-design regimes. This must be taken into account when calculated and measured performances are compared. REFERENCES 1.

Galerkin Y. Turbo compressors. // LTD information and publishing center. Moscow. – 2010 (In Russian). 2. Galerkin, Y, Rekstin F. Methods of research of centrifugal compressors. Leningrad. – 1969 (In Russian). 3. Galerkin Y. Danilov K., Popova E. Design philosophy for industrial centrifugal compressor. International Conference on Compressors and their systems. – London: City University, UK. – 1999. 4. Galerkin Y.. Mitrofanov V., Geller M., Toews F. Experimental and numerical investigation of flow in industrial centrifugal impeller. International Conference on Compressors and their systems. – London: City University, UK. – 2001. 5. Galerkin, Y. B., Soldatova, K.V. Operational process modeling of industrial centrifugal compressors. Scientific bases, development stages, current state. Monograph. // Sankt-Peterburg. - SPbTU. – 2011 (In Russian). 6. Galerkin, Y.B., Soldatova, K.V., Drozdov, A.A. Specification of algorithm of calculation of parameters of a stream in the centrifugal compressor stage. [text] // Scientific and technical transactions of the TU SPb. – 2010. – No. 4. – Page – 150-157 (In Russian). 7. Galerkin Y., Drozdov A., Soldatova K. Development of computer programs of the Method of universal modeling of 1th level. Works 14 International scientific and technical conferences on compressor equipment. Volume 1. – Kazan. – 2011. – Page 276-284. -420 (In Russian). 8. Galerkin Y., Drozdov A., Soldatova K. Turbocompressor efficiency application and calculation. // Compressors & Pneumatics. – Moscow. - 2011. – № 8. – Page 18-24 (In Russian). 9. Lojtsanskij L. Mechanics of liquid and gas. – Moscow. – 1978 (In Russian). 10. Ris V. Centrifugal compressors. - Leningrad. – 1981 (In Russian). 11. Transactions of the TU SPb compressor school. –TU SPb. – 2010 (In Russian). 12. Turbocompressors - Performance test code// International Standard DD ISO 5389.1991.

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