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Simulation Model of a Variable Speed Centrifugal Compressor. (by A.E. for mychemengmusings.wordpress.com) (newly updated and improved editing Jan 2020) In practice there are opportunities to optimize or modify conditions or re-design operational systems where a variable speed centrifugal compressor is involved. An example is the case where we have an existing refrigeration system that needs to be improved and or optimized or cooling conditions or energy consumption needs to be minimized. To study the existing system and identify possible ways to improve the operation a simulation model can be of great help. This post gives an example of how a simulation model of an actual existing variable speed centrifugal compressor (vsccr) can be made and implemented in excel. The developed model includes accounting for the compression of a non-ideal gas, as well as the

dealing with the fact that in a real, actual rotating machine the compression process proceeds in a non-ideal fashion requiring more ‘shaft-power’ than theoretically calculated. The degree of non-ideality of the gas handled by the compressor, reflected in the “gas compressibility factor ‘Z’ “, is described with the “Zv-correlation” (presented in earlier posts). The energetic losses incurred (due to non-ideal compression) during the operation of the existing compressor machine are accounted for by a newly developed correlation for the “adiabatic efficiency factor” , allowing the actual, real shaft-power required to operate the machine to be determined. Let us begin with reviewing what we know of the existing variable speed centrifugal compressor “vscCR-01”. But first as this post will be long I have divided it the following sections: ‘1. The variable speed centrifugal CompressoR “vscCR-01”. ‘1.1 The Compressor’s Performance Chart ‘1.2 The Gas inlet conditions and the Gas Quality ‘1.3 The Power Consumption Chart of “vscCR-01”. ‘2. A Compressor’s Operating Window (in general). ‘3. A simple Model to describe the surge-line in a compressor map ‘3.1 The surge point. ‘3.2 Modeling of the conditions at the point of incipient surge. ‘3.3 Describing the surge curve on a head versus volumetric flow rate

map. ‘4. Compressor Modeling considerations. ‘5. The Adiabatic Head, Adiabatic Efficiency and the Adiabatic Temperature rise ‘6. A Compressor’s operation at off-design conditions ‘6.1 The influence of speed on the volumetric flow rate ‘6.2 The influence of speed on the Adiabatic Head ‘6.3 Summary and the “Fan Laws” ‘7. Modeling the existing “vscCR-01”. ‘7.1 Fingerprinting the Head versus Flow characteristic of the machine ‘7.2 Fingerprinting the machine’s Adiabatic Efficiency Factor

‘8. The Simulation of the existing compressor “vscCR-01”. ‘8.1 Formulating the Simulation Model and its steps in excel. ‘8.2 Excel Spreadsheet print out of the Simulation Model calculations

‘9. The Simulation Model Results ‘9.1 Comparing Simulation results with actual Manufacturers performance

Charts for Discharge pressure and Shaft Power versus Volumetric suction flow

‘9.2 Inter – and extrapolated Performance Charts ’10. Conclusion. Note: a new pdf version with added updates and improved editing of this post is here

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1. The variable speed centrifugal CompressoR “vscCR-01” 1.1 The existing Compressor’s Performance Chart.

The Manufacturer had provided us with a chart with a set of performance curves showing how the machine behaves. The curves in Diagram-1 show the discharge pressure versus the volumetric gas inlet flow for the machine. In general the manufacturer may also provide a chart showing the outlet- over inlet pressure ratio (P2 / P1) versus the gas volume inlet flow. The compressor has been designed for a given set of operating conditions specified by the flow rate, the temperature, gas quality and inlet and outlet pressure. At gas flow rates higher than the design point the discharge pressure diminishes. At flow rates lower than that point the pressure increases. Below is shown the manufacturer’s chart as Diagram-1, showing the outlet pressure versus the inlet suction volume flow rate performance for three speeds of the existing machine “vscCR-01”.

Diagram -1 Manufacturer’s performance chart of “vscCR-01”.

1.2 The Gas inlet conditions and the Gas quality

The inlet conditions and the quality of the gas for which this compressor is designed and to which the above performance curves apply is summarized below:

Suction conditions Gas Quality

Inlet Pressure Inlet Temper. Mol Weight Compressibility

Factor

P1 in Barabs T1 in Deg.C MW Z1 K

10.5 27 41.74 0.838 1.1644

Performance Curves of Variable Speed Centrifugal Compressor vscCR-01

02468

1012141618202224262830323436384042

1500 2000 2500 3000 3500 4000 4500 5000

Suction Volumetic Gas Flow Q1 ------> m3 / hour

Dis

ch

arg

e P

res

su

re P

2 -

----

--->

B

ar

ab

so

lute

5870 rpm

7000 rpm

8330 rpm

3

1.3 The Power consumption Chart of “vscCR-01”. The shaft power required to run the compressor for each of these same three speeds as given by the manufacturer in the previous chart is shown in the next graph. Sometimes manufacturers combine the two charts into one. Note: these power numbers do not include the mechanical losses in gearbox and the inefficiencies of the driver of the compressor (either e.g. electric motor or steam turbine).

Diagram -2 Manufacturer’s Shaft Power for “vscCR-01”.

2 A Compressor’s Operating Window (in general)

A compressor is designed for a specific operating point, that is for a given gas flow rate, gas quality, suction and discharge conditions at its best energetic efficiency point. The variable flow rate – at design rotation speed – is at the low end bounded by the ‘surge’ point, while on the high end it is limited by conditions at which the flow reaches ‘choke’ conditions at which point the discharge pressure starts to steeply drop at increasing flow rates. Choke conditions arise at a particular point along the flow path inside the machine when the local velocity of the gas approaches the speed of sound. These limiting flow rates are portrayed in the following diagram of pressure ratio versus suction volumetric flow rate:

Power Consumption of Variable Speed Centrifugal Compressor vscCR-01

0

500

1000

1500

2000

2500

1500 2000 2500 3000 3500 4000 4500 5000

Suction Volumetric Gas Flow ---------> m3 / hour

Sh

aft

Po

we

r --

----

-- >

kW

att

4

Diagram -3 Outlet over inlet pressure versus inlet volumetric flow rate

For other speeds of rotation similarly the operating points form a curved line bounded on two sides by the ‘surge’ point, the minimum flow rate below which no stable compressor operation is possible and at the high end the maximum flow rate above which the compressor starts to “choke”, manifested by a sharp drop in the outlet pressure at increased the gas flows caused by the gas velocities inside the machine approaching the speed of sound. Such set of curves is often called the Performance Map or Chart of the vsccr ( see Diagram 1 )

In general the area in which the compressor is allowed to operate, the ‘Operating Window’ is bounded on four sides. On the left hand side the operating area is bounded by the surge line, while on the right hand side the choke points giving the maximum gas flow rates the compressor can handle. The operating area is furthermore constrained by the minimum and maximum allowable speeds. When considering lowering operating speed a limitation maybe imposed by the ‘self resonance’ speed of the rotor, the “rotor’s critical speed”. For operation of speeds at the high end, a limit may be posed by the occurrence of high frequency vibrations, by shaft torque limitations or the driver’s maximum power limitations.

3 A Simple model to describe the surge–line in a compressor map 3.1 The surge point

Consider a centrifugal compressor with a performance curve like the one shown in the previous Diagram continuously operating at a speed “S”. If the flow is being reduced below its ‘normal’ operating point, the compressor’s discharge pressure increases. As the flow is reduced further, a point is reached where the machine approaches an unstable condition at which an incipient surging of the compressor is imminent. This condition is often accompanied by noise, a surging sound, vibrations and in the extreme case, if the starving of feed gas is prolonged and intensified, by shaking of the entire machine on its foundation! This condition, obviously, has to be avoided.

P2/P1

Suction Volumetric Flow Rate

surge line

point of incipient

surge

choke

point

constant

speed S

operating

point

5

The model developed below is based on the notion that at the surge point a gross back flow inside the machine occurs. In other words, surge occurs when the pressure difference between the compressor’s discharge and suction cannot be sustained at the low flow rate and the developed higher outlet pressure P2 overcomes the internal back flow resistance along the flow path inside the machine and the gas starts to flow in reverse! Whereas normally the gas flows from the inlet, through the first impeller onto the next impellers mounted on the shaft, at the surge point the gas starts to flow from the discharge side pressure P2 back towards the suction side at P1.

In other words, at the surge point, the pressure difference of outlet - minus inlet pressure overcomes the dynamic internal flow resistance of the internals in the centrifugal machine. It is assumed here that the flow path over a series of impellers can be lumped together. When the pressure ratio over the machine is large and inter-stage cooling is used then the present the model has to be applied over each of the non- cooled sections.

3.2 Modeling the conditions at the point of incipient surge

In this simple model the operating point at which surge occurs is equated with the condition at which the pressure difference P2 minus P1 overcomes the flow resistance of the machine’s internal flow-path. This resistance is chosen to be described with a “Fanning friction factor” type equation. The condition (flow “F” at speed “S”) at which the compressor goes into surge than is

described as : 2

122

1vKPPP R

in which equation the gas density, ‘Rho’, and the velocity, ‘v’, are based on an average between inlet and outlet. The average velocity can be related to the average volumetric gas flow (for the mass flow of “F” kg/s) divided by the cross sectional area A characterizing the internal flow path

2

122

1

avg

avgA

QKPP

The steady state continuity of mass-flow along the flow path from suction to outlet is:

FQQQ avgavg2211

and hence 2

122

1

A

FKPP

avg

avg

in which ‘rho–avg’ symbolizes the average gas density over the inlet and outlet of the compressor, and re-arranging:

P 1 ,

P2

F kg/s

T1

6

2

2122

1

avg

wF

A

KPP

The average gas density is taken as the algebraic average of inlet and outlet gas density

2

21

212

2

2

1wF

A

KPP

This simple model says that in a P2 - P1 versus mass-flow diagram, the surge points vary with the square of the mass flow rate, forming a single line with a parabolic shape. The factor ½ * K/A^2 characterizes the internal back flow resistance of the gas through the compressor from suction to outlet and hence is assumed to be constant for the particular machine and gas under consideration.

When determining the (incipient) surge points of a compressor in the field, a plot of the pressure difference or the discharge pressure versus the mass flow rate through the machine, will show the surge points as a forming one line with an upward parabolic shape.

Note (1): If the compression ratio is high and the absolute discharge pressure is significant with respect to the gas’ critical pressure the average gas density increases more than in strict proportion with the absolute discharge pressure; this additional increase is accounted for by the ‘gas compressibility factor “Z” ‘. As a result the parabolic shaped surge curve tends to be “flattened”. In this case the Fw * ( 1/ rhoavg))^0.5 needs to be plotted. Example: propylene at from 10 -> 30 bara ; critical pressure 46 bara

Note (2): determining the actual surge point on an installed machine is not a routine field test, but it may need to be done when the anti-surge protection control system needs to be upgraded to allow operation closer to the surge line, in order to save energy operating costs. (energy cost saving incentive has to be carefully weighed against the risk of damaging the machine!)

Note (3): in multistage compressors with high pressure ratio’s the surge line can deviate from the parabolic profile when at different speeds the impeller in which the surge occurs may shift to another impeller. In a well designed compressor the different impellers are appropriately designed for the different speeds

The compressor performance curves and surge points can be presented in a variety of ways. Often the manufacturer will provide compressor curves including the surge line in a P2/P1 versus the suction volumetric gas flow diagram or shown in a head versus volumetric suction gas flow rate diagram. The particular expressions for the surge line in different type of diagrams can be derived from the above equations.

3.3 Describing the surge curve in a head versus volumetric flow map

Taking again equation (1) as a starting point, the head developed over the machine relates to the pressure difference as:

2

1

2

1

21

212

2

2

1Q

A

KPP

converts into :

7

1

21

2

12

1

12 2

2

1Q

A

KPP

Note that the dimension of the left hand side expression is Nm/kg. Combining with the real gas density relation (that introduces compressibility factor “Z”) gives:

221

112

2

12

1

12

1

2

2

1

TZP

TZPQ

A

KPP

re-writing it as :

22

12

1

12

2

1hQ

A

KPP

with “eps-h” being defined as:

221

1121

2

TZP

TZPSQRTh

Thus the surge line is of parabolic shape if we plot Q1* eps-h versus the head !

4 Compressor Modeling Considerations In general terms we know what the purpose of the rotating machine is: the raising of the pressure of a stream of gas through a compression process. We know that it is a centrifugal compressor machine which is able to handle large volumetric flows of gas. The internal free volume for the gas to pass through the machine is small. Hence the time span the gas resides inside the machine is short and the time to exchange heat between the flowing gas and the rotor and housing is short. Our first assumption is therefore the process of compression occurs adiabatic, i.e. no heat is transferred from or into the gas.

In a well designed machine the flow path and the sizing of the impeller(s) channels and the diffuser(s) are streamlined and dimensioned such that the energy flow-losses are kept to a minimum for operation at the ‘design point’. In other words the amount of losses of energy associated with pressure- or kinetic energy of the moving gas being converted into an (additional) gas temperature rise are minimized. The gas involved in the compression process is, for our case of the machine as given in section 1, clearly a real gas as the compressibility factor at inlet is stated by the manufacturer to be smaller than 1 and given as: 0.838. A detailed, stage to stage description and modeling of the compressor is not possible because of lack of detail of the shaping and sizing of the internal rotary and stationary parts of the compressor, details manufacturers are (very) reticent to share. For example we have no information on how many stages of compression are implemented (how many impellers mounted on the same rotor) inside the machine and other details on the construction and sizing of the internals.

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We therefore will take the following modeling approach. We are defining an ‘equivalent compressor’ consisting of a one stage ‘ideal’ isentropic compression step combined with a coupled ‘energy loss process’ step to mirror the actual loss processes going on inside the actual machine. This loss process can be characterized by an overall efficiency factor that links the ideal, isentropic compression power with the actual shaft power consumption of the actual machine operating under the conditions of flow, pressure ratio and speed. The ‘ideal’ compression step is thought of to be an isentropic compression process. The implicit assumptions underlying such process are that it proceeds in a reversible and adiabatic manner. In other words the gas is compressed in a quasi-static fashion without addition or removal of heat and hence the gas is brought from its inlet state at the suction end of the machine to its final state at the discharge end of the machine following a (thermodynamical) path (‘line’) of constant entropy. The energy losses that occur during the compression in the actual machine are accounted for by the ‘Adiabatic Efficiency’ factor. This factor thus varies, as said, with feed rate, suction – and discharge pressure and speed!

The following Diagram -4 summarizes the modeling approach with the help of the following block flow diagram:

Note: In this simulation model we have chosen to describe the gas compression as an isentropic compression process inside the machine. An alternate choice is to describe the compression as a polytropic gas compression process. Such model may be given in a separate presentation.

5 Isentropic Compression: the Adiabatic Head, the Adiabatic Efficiency and the Adiabatic temperature rise In a continuously operating centrifugal compressor the increase in pressure is created by the rotating impellers mounted on the same shaft. We are looking at the behavior of the machine from an overall point of view and describe the compression across the entire machine, from inlet to outlet on a one stage process basis. The compressor brings the gas from the inlet state (1), along an isentropic path to outlet state (2). In larger industrial compressors the mass-flow rate is substantial and the speed at which the compression process takes place is sufficiently fast (short residence time), that (almost) no heat

Adiabatic (isentropic)

compression

Energy Loss

Process

Simulated

Machine Discharge Flow

at P2 Suction Flow at

P1

Shaft

Power

Speed

Actual

Machine

Suction Flow at

P1

Shaft Power

Discharge Flow

at P2 Speed

Variable Speed Centrifugal

Compressor

9

exchange with the environment can take place. In other words the process is adiabatic. Work required for the isentropic compression process is given by the following equation :

11

1

1

211

k

k

adP

P

k

kT

MW

RZH

in which Had = the energy required to compress the gas from pressure P1 to pressure P2. Expressed in Joule/kg. Also called the adiabatic head of the machine; in earlier post called the Ideal Compression Energy (ICE).

Z1 = the compressibility factor of the gas at inlet conditions MW = the molecular weight of the gas fed to the machine in kg/kmol R = the universal gas constant equal to 8.3145 kJ/kmol k = the specific heat capacity ratio of the gas Cp/Cv

The power to be added to the flow of gas during compression then is:

adwad HFG

Gad = adiabatic gas compression power (gas horse power) is expressed in kiloJoule/kg x kg/s or kWatt.

Fw = mass flow of gas to the compressor, kg/s

The Gas Power ‘Gad’ is the energy flow required to sustain the compressor’s pressure ratio for the given gas feed flow rate according to an isentropic process. In reality, in an actual compressor the rotating impeller(s) on the shaft impart energy to the flowing gas, while the gas is following its ‘tortuous’ path through the machine. The entropy of the gas in reality will not stay constant due to friction losses, convection losses and shock losses (‘velocity vector mismatches along the flow path) in particular during off-design operating conditions. Hence in actuality a larger power is required to drive the compressor shaft than reflected by the formulas above. Therefore , the energetic in-efficiencies we are gong to account for it by way of the “Adiabatic Efficiency Factor”. The actual power required at the shaft of the compressor is:

ad

adw

ad

ads

HFGP

in which Ps = the actual shaft power, Nm/s eta-ad = the adiabatic efficiency of the compressor

Note that the adiabatic efficiency is linked to the particular compressor machine under consideration and the particular gas being compressed at the measured operating point (P2/P1, Q1 and rotating speed S). The usefulness of this efficiency factor is that the gas horsepower Gad can be directly computed from first principles (isentropic compression formula) and the efficiency factor encapsulates all deviations from the isentropic process in the real compressor. Hence, when the adiabatic efficiency is known for a given compressor and a given gas, the shaft power required to drive the machine can be straightforwardly calculated.

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The temperature increase due to the isentropic compression can be calculated from

k

k

P

PTT

1

1

212

in which: T2 = the absolute discharge temperature , oK T1 = the absolute suction temperature, oK

k = the specific heat capacity ratio of the gas Cp/Cv

6 A Compressor’s Operation at off-design conditions. The map that was provided by the manufacturer of how the compressor behaves for the volume feed rate range and the three speeds already gives an impression of how the compressor performance looks at operating points other than the original design point for the specified design feed gas quality. Now, if we want to describe the behavior of the compressor for other speeds than we could do a simple inter- or even extrapolation of the curves given as a very rough first approach. If however we also want to predict the behavior of the machine not only for other speeds but also for other feed gas conditions or even other molecular weights than such approach is not good enough if not very risky. We need therefore to take a closer look at what happens inside a centrifugal compressor. 6.1 The influence of speed on the volumetric flow Consider a radial impeller with diameter ‘D’ and the ‘eye’ of the impeller (gas inlet annular opening) with diameter ‘d’. The impeller has been constructed with a certain number of vanes creating ‘channels’ through which the gas flows through these channels towards the outer diameter of the impeller. The area of the impeller over which the gas flows is equal to Pi/4 * D^2 (ignoring for the moment the area the vanes and the ‘eye’ ). If the impeller is constructed with a channel depth ‘w’, then the total (static) volume of the channels is Pi/4* D^2 * w. If upon every full rotation the impeller can displace and refresh this channel volume once then at a rotational speed ‘S’ the impeller moves a volume at a rate that is proportional to:

SwDQimp

24/

The width of the channel is a design parameter of the impeller which we could characterize as fraction of the impeller diameter

SDwDQimp /4/ 3

Whether upon every revolution each channel is ‘emptied and refreshed’ once that will in reality depend on the actual flow pattern and the path the gas follows over the impeller. The movement of gas over the impeller is determined by the forces – generated by the rotation – acting on it. To describe the path a pocket of gas takes over the impeller we have to shift our frame of reference by stepping –in mind - onto the impeller as it were and look at it as if from a rotating observer’s point of view. In this reference frame we ‘see’ two apparent forces working on a pocket of gas. The first force acts in the radial direction: the centrifugal force. The second force, called the Coriolis force, acts in a direction that is opposite to the direction of rotation. As a result of these

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two combined forces we see the gas moving over the impeller along a path that is curved in a way opposite to the direction of rotation. The total, actual suction flow ‘Q1’ to the inlet of the impeller then is determined by the relative velocities and flow pattern over the impeller. We can thus write :

impQfQ1 or SDwDfQ /4/ 3

1

In which ‘Q1’ is the actual total suction volume flow rate and ‘f ‘ is a factor reflecting the actual

flow pattern. Alternatively we can express this relation as:

~ constant

Thus this expression is approximately constant as long as the ‘flow pattern’ remains ‘similar’. The factor ‘f ‘ then characterizes this flow pattern. In other words we could say that ‘f ’ is like a ‘fingerprint’ of the flow pattern created by the dimensioning and design of the impeller operating at speed ‘S’. At this point we can draw one clear conclusion from these considerations:

===> In essence a centrifugal Compressor is a “ volume moving machine” Note: in API 67 the flow factor ‘phi’ is defined as: Q1/ (Pi/4* D^2 * U ) in which ‘U’ is equal to the tip speed of the impeller. And as ‘U’ = omega* D/2 , hence ‘phi’ is proportional to D^3.

6.2 The influence of speed on the adiabatic head. The centrifugal force acting on a gas pocket with mass ‘m’ performs ‘work’ on it while the pocket moves from inlet of the impeller to its outer edge. The centrifugal force varies with the distance ‘r ‘ from the centre of rotation and is thus is a function of the position ‘r ‘ for a given angular velocity ‘omega’ in radian per second :

drrmFce

2

The differential work done is: drrmdrFdE ce

2

upon integration: 22 2/1 DmE

the Head (units: Nm/kg or J/kg) for the pocket of gas increases to:

222/1/ DmE

The total Head increase of the gas produced by the impeller then is:

22 DH imp

The angular velocity ‘omega’ (in radian/sec) is related to the rotational speed by the relation: omega = 2 * Pi * S and thus we can write:

fS

DwD

Q3

1

4

12

2224 DSH imp

This head increase of the gas is reflected by both the increase in pressure (gas density) and the increase in gas velocity, or in other words by the increase in both potential energy as well as kinetic energy. The ‘velocity head‘ associated with the kinetic energy of the gas exiting the impeller is next, along the flow path, converted by the diffuser, located in the stationary part of the machine, where it gets, for the greater part, converted back into additional ‘pressure head’ thus adding to the overall ‘pressure head’ produced by the compressor machine. For the total pressure head developed by the compressor machine (impeller+diffuser) we can write:

impad HhfH or 2224 DShfH ad

which expression we alternatively we can write as :

~ constant

This expression is approximately constant as long as the ‘overall flow pattern’ in the machine remains ‘similar’. The factor ‘hf ’ then can be said to characterize this overall flow pattern.

From the above considerations we can conclude that: =====> in essence a centrifugal compressor is a ‘head’ producing machine Note: API 67 defines the “head coefficient mu “ as : Had/ (S^2 * D^2)

6.3 Summary and the “Fan Laws”

From the expression given in section 6.1 giving the relationship between the rotational speed and the volumetric gas flow rate we have defined the flow factor ‘f’ that characterizes the flow pattern over the impeller(s). In section 6.2 giving the relation between rotational speed and the ‘head’ generated we have defined the ‘head factor ‘hf’ that characterizes the overall flow pattern in the machine. These two factors are coupled together via the prevailing flow pattern. We can take a “fingerprint“ of the machine under consideration and its ‘head producing capability’ by plotting how the head factor ‘hf ‘ varies with the flow factor ‘f ‘. Such plot is unique to the machine and independent of speed of rotation, gas MW, and inlet conditions (gas density)! The numerical relationship between ‘ hf ‘ and ‘ f ‘ can be captured

through regression fitting a quadratic or a polynomial mathematical expression through the data points of ‘hf’ versus ‘f’. Fan Laws

The relationships described in the previous sections about the influence of speed on volumetric flow and head developed, is sometimes presented in a form that is called “The Fan Laws”. We have seen that the suction volumetric flow ‘Q1’ divided by the speed ‘S’ is only dependent on the geometric configuration of the compressor internals and the inside ‘flow-pattern’. Therefore for

hfDS

Had

2224

13

conditions where the flow-pattern is the same or similar, we can say that for small variations in speed and volume flow rate that:

Q/S = constant and hence ===> 2

2

1

1

S

Q

S

Q

and similarly for the adiabatic head ‘Had’ developed we can write:

Had / S^2 = constant and hence ===> 2

2

2

2

1

1

S

H

S

H adad

in which the subscripts 1 and 2 here refer to two different conditions and not to inlet and outlet per se. The ‘Fan Laws’ are often applied to predict changes in operating performance for running the machine under off-design conditions. These ‘laws’ are valid , as discussed in sections 6.1 and 6.2, as long as such deviations from design conditions are not too drastic, and hence the ‘sameness’ or ‘similarity’ of flow- and head patterns assumptions are not violated.

7 Modeling of the existing “vscCR-01” compressor 7.1 Fingerprinting the Head versus Flow characteristic of the machine

We want to create a simulation model of our “vscCr-01” compressor whose performance charts, provided by the manufacturer of the machine, has been shown in section 1. To do that we first need to take the “Fingerprint” of this compressor. By making use of what has been given in sections 4, 5, 6. We begin by defining two variables as follows:

S

QQ 1* and

2*

S

HH ad

in which ‘Q1’ is the suction volumetric flow rate ‘S’ the speed of rotation and ‘Had’ the adiabatic head. With the relations from section 6.1 and 6.2 we can see that:

and

The right hand side of both equations contains particular data about the detailed construction and sizing of the specific internals of our compressor, which Information however we do not have available. Nevertheless, by using the two new variables Qstar “Q* “ and Hstar “H* “ we are still able to determine and plot the relationship between ‘hf ‘ and ‘f ‘under the proviso that this is valid only for this machine; and that is no problem at all , it is in fact exactly what we want in making a simulation model for our particular machine “vscCR-01” ! The first step therefore is to calculate the values of H* and Q* from each of the points along the Manufacturer’s performance curves. The next step is to plot these values of H* and Q* in a single graph. In doing so we should find all original three (speed) curves to merge into one single curve. With this plot of a single curve of H* versus Q* , we will have found the unique fingerprint of the Head versus Volume flow rate of this machine vscCR-01.

DwDfS

QQ /4/* 31

2

24* Dhf

S

HH ad

14

The shape of this H* versus Q* curve is influenced , in general, by the number of stages, the detailed geometric design of impellers, the diffuser design and the shaping of the internal flow channels in the machine. To obtain values for H* between zero and one and work with simple numbers we have introduced a constant factor of 1000 in the calculation of the square of the rotation speed as follows H* = Had / (S/1000)^2. For the calculation of Q* no such factor was (necessary) to be included. Thus Q* was calculated as Q1/ S , with Q1 in m3/hr and speed S expressed in rev/min ( rpm). These calculated values were plotted and the following Graph was obtained: Diagram -5 Plotting H* versus Q* (according to initial definitions of these variables) Only the two curves with the highest speeds are shown in this Graph. We can observe from it that H* values look in a similar range for both speeds, however these two curves have not merged into a single line! Why is this so? The speed change does not appear too drastic. Upon further thinking about this we can see that in our preliminary definition of Q* we have based it on the inlet flow Q1, but the ‘fingerprint’ is characteristic for what flow-pattern happens inside the machine not what enters the machine. In other words H* should be related to the average volume flows inside the machine. This reminds me of how we have developed the relation for the surge line through the use of the average volume flow through the machine. Another consideration is that the gas is strongly non- ideal and the effects of the gas compressibility factor Z should be taken into account. Therefore we are re-defining Q* as:

hS

QQ

1

1*

in which eps-h is equal to (see section 3.)

Had / (S/1000)^2 versus Q1 /S for vscCR-01

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6

Q1/ S ---------> m3/h / rpm

Ha

d / (

S/1

00

0)^

2 --

----

-->

(k

J/k

g)

/ (r

pm

/10

00

^2

)

7000 rpm

8330 rpm

15

221

1121

2

TZP

TZPSqrth

A minor modification was already made to H* too, as explained earlier, but this is just a constant scaling factor to obtain H* values between 0 and 1 as follows:

21000/

*S

HH ad with ‘S’ being the speed in rpm.

These last two adjustments to H* and Q* are the final definitions we are working with to determine the “fingerprint” of the machine .

After re-calculating the values for the newly defined H* and Q* variables for the points in the manufacturer’s performance charts, we get the following new Graph: Diagram -6 Plotting the newly defined H* versus Q* The two curves have now coalesced into one, single relationship. This relation forms the “fingerprint” of the machine. This “fingerprint” relation we can capture through regression on the data for H* versus Q* for 8330 rpm (has most data points). This fingerprint relation will be referred to by the symbol “H*fp”.

The (quadratic) regression yields the following equation:

Had / (S/1000)^2 versus Q1 /S * eps-h for vscCR-01

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.2 0.25 0.3 0.35 0.4 0.45

Q1/ S * eps-h -----------> m3/h / rpm

Had

/ (

S/1

000)^

2 --

----

-->

(kJ/k

g)

/ (r

pm

/1000^

2)

7000 rpm

8330 rpmsurge point

4985.06042.3][5658.7 *2* QQH fp

16

It should be pointed out that in this graph of the fingerprint ‘H*fp’ versus Q* , the surge line has been reduced to one point. From its H*fp - Q* coordinates the whole surge curve can be re-developed by applying the reduction transformation in reverse. 7.2 Fingerprinting the machine’s Adiabatic Efficiency Factor To be able to predict the shaft power we need the adiabatic efficiencies of the operation of this machine. The power absorbed by the adiabatic isentropic compression of the gas, Gad, is equal to the product of the calculated adiabatic head and the mass flow rate. ( kW = kg/s * kJ/kg )

adwad HFG

From the Manufacturers data for the required shaft power (section 1) together with the calculated Gas-power absorbed by the adiabatic isentropic compression of the gas the Adiabatic Efficiency Factor was back calculated for all the manufacturer’s data points , a total of 25 points ( for the three speeds).

PsGadad /

With these efficiency data in hand the question now became: can these efficiencies be related to the operating conditions in particular to Q* ? If so how? As discussed before no theoretical relation for the adiabatic efficiency for the off-design operating conditions and/or machine design parameters exists to our knowledge. Therefore, by trial and error it was found that the adiabatic efficiency, Eta-ad, could be correlated with Q* (defined as Q1/S * eps-h) as follows:

QSQ

Sad 4779.80793.6/

1000/

1

39.0

A graphical representation of this equation is shown below: Diagram -7 Correlation of Adiabatic Efficiency factor with Q*

Correlating Adiabatic Efficiency with Q1/S*eps-h for vscCR-01

y = -8.4779x + 6.0793

R2 = 0.9966

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

Q1/S * eps-h ----------->

Eta

/(Q

1/S

)*(S

/10

00

)^0

.39

--

----

--->

Eta/(Q1/S) * (S/1000)^0.39 =

-8.4779*Q1/S * eps-h + 6.0793

for three speeds 5870 , 7000 and 8330 rpm

25 datapoints

17

8 Simulation of the existing Compressor “vscCR-01” Now that we have the two key ‘fingerprint’ correlations characterizing the existing compressor (model identification) we can formulate and set up the calculation sequence to simulate the compressor’s discharge pressure and gas density as well as the shaft power required to drive the compressor shaft for a given volumetric feed rate and speed. We can set up the calculation steps of the simulation model and implement these in excel as follows.

8.1 Formulating the Simulation Model and its calculation steps in excel.

The sequence of calculation steps and the equations involved in this simulation model are shown in the next Diagram. There are two parts to the Model. The first part ‘A’ deals with the calculation of the adiabatic head for the speed ‘S’ and volumetric feed flow ‘Q1’ from which the ratio of discharge- over inlet pressure can be calculated followed by the absolute discharge pressure P2 and T2 and the compressibility factor Z2 and so on . The simulation model is ‘solved’ by iterative calculations using simple substitution as convergence technique. Note that excel needs to be put into ‘Manual Calculation’ mode. In part B ,once the outlet conditions P2, T2 , Z2 have been arrived at, the Adiabatic efficiency Factor can be calculated together with the Adiabatic Gas power. Next the shaft power predicted by the simulation model can be calculated.

Diagram 8 Simulation Model Calculation steps showing iteration

CALCULATION STEPS OF THE SIMULATION MODEL

PART -A- THE ADIABATIC ISENTROPIC HEAD Had and Pressure Ratio P2/P1

Steps Description

Input : Q1 suction vol feed rate in m3/hour

Input rotational speed S in rpm

Input inlet pressure P1 (10.5 Barabs)

Input : inlet temperature T1 ( 300.16 oK)

Input : inlet compressibility factor Z1 ( 0.838)

Calculate Feed mass flow rate in kg/s

Assume values for the discharge conditions P2 ( bara) and T2 (degr. K)

Calculate the Reduced outlet Pressure and Reduced outlet Temperature

Calculate outlet compressibility Factor Z2 using the Zv correlation

Calculate eps-h factor from eps-h=Sqrt[ 2/(1+P1*Z2*T2/(P2*Z1*T1)) ]

Calculate Q* = Q1/S * eps-h

Calculate H* = -7.5658 * (Q*)^2 +3.6043 * (Q*) + 0.4985

Calculate pressure ratio P2/P1 from H*

P2 / P1 = [ (H*)*(S/1000)^2 *MW / (Z1*R*T1*) *(k-1)/k + 1 ]^(k/(k-1))

Calculate outlet temperature from T2 = T1 *( P2/P1)^((k-1)/k)

(re)-Calculate P2

PART -B- THE ADIABATIC EFFICIENCY FACTOR AND SHAFT POWER

Calculate the Adiabatic Head from Had = (H*) * (S/1000)^2

Calculate the Adiabatic Gas Power Gad from from Gad = Had * Fw

Calculate the Adiabatic Efficiency Factor from

Eta-ad = [ -8.4779 * (Q*) + 6.0793 ] * Q1/S * ( S/1000)^(-0.39)

Calculate Simulated Shaft Power Ps = Gad / Eta-ad in kW

(P2/Pcrit ; T2/Tcrit)

+ P2*Z1*T1/(P1*Z2*T2)) ]

18

In the next section a copy of the excel worksheet with the implementation of these steps and the resulting calculations for the model is given.

8.2 Excel Spreadsheet printout of the Simulation Model (steps)

The constants in the two ‘fingerprint’ equations for H*fp and the adiabatic efficiency Eta-ad are

shown in green above the respective columns. A pseudo Zcrit parameter of 0.26 is used for the feed gas mixture used in the Zv equation.

Diagram -9 Copy of excel worksheet for compressor simulation model

The calculated P2 values can be compared with the manufacturer’ P2 data shown on the right hand side in bold numbers. The absolute percentage error in the calculated discharge pressure P2 is on average 0.6%. See the penultimate column. The model’s calculation results for the Adiabatic Efficiency factor and the calculated Shaft Power, required to run the compressor, are shown on the very right hand side columns.

9 Simulation Model Results

The results obtained with the simulation model of “vscCR-01” are presented in the form of a series of Graphical representations where simulation results are co-plotted with manufacturer’s data for both discharge pressure and shaft power consumption.

0.39

-7.5658 Zcrit = -8.4779

3.6042 0.26 6.0793

0.4985

Mass inlet speed Vol Eff Q/S * H~ P2/P1 P2 T2 calc P2red T2red Z2 P2 P2/P1 Shaft Eta-ad P2 calc Calc

Flow capacity rpm eps-h eps-h calc calc calc Manufact Manufact Power Calc Error% Shaft Power

kg/sec m3/hr Barabs kW kW

Fw Q S P2 Ps calc Ps

13.97 2400 7000 0.77 0.262 0.923 2.340 24.57 65.3 0.53 0.93 0.720 24.32 2.316 1000 0.619 1.0 1022

14.56 2500 7000 0.77 0.274 0.918 2.329 24.46 65.1 0.53 0.93 0.720 24.17 2.302 1024 0.628 1.2 1042

15.72 2700 7000 0.77 0.297 0.901 2.295 24.10 64.4 0.52 0.92 0.723 23.87 2.273 1067 0.643 1.0 1080

16.89 2900 7000 0.78 0.322 0.874 2.242 23.54 63.2 0.51 0.92 0.728 23.39 2.228 1100 0.649 0.6 1114

17.47 3000 7000 0.78 0.336 0.856 2.206 23.17 62.5 0.50 0.92 0.731 23.09 2.199 1120 0.649 0.3 1129

18.63 3200 7000 0.80 0.364 0.809 2.117 22.23 60.5 0.48 0.91 0.738 22.36 2.130 1143 0.641 0.6 1151

19.22 3300 7000 0.80 0.379 0.778 2.061 21.64 59.3 0.47 0.91 0.743 21.82 2.078 1152 0.633 0.8 1157

19.80 3400 7000 0.81 0.395 0.742 1.996 20.96 57.8 0.46 0.91 0.748 21.32 2.030 1162 0.621 1.7 1159

20.38 3500 7000 0.82 0.412 0.699 1.921 20.17 56.0 0.44 0.90 0.754 20.29 1.933 1172 0.606 0.6 1153

18.92 3250 8330 0.66 0.257 0.925 3.246 34.09 81.3 0.74 0.97 0.640 34.18 3.255 1829 0.665 0.3 1826

19.22 3300 8330 0.66 0.261 0.924 3.241 34.03 81.2 0.74 0.97 0.641 34.03 3.241 1843 0.669 0.0 1840

20.38 3500 8330 0.66 0.279 0.915 3.210 33.70 80.7 0.73 0.97 0.644 33.66 3.206 1906 0.683 0.1 1895

20.96 3600 8330 0.67 0.288 0.909 3.187 33.46 80.4 0.73 0.97 0.646 33.39 3.180 1937 0.688 0.2 1922

22.13 3800 8330 0.67 0.307 0.892 3.123 32.79 79.4 0.71 0.97 0.652 32.71 3.115 1980 0.694 0.3 1973

23.29 4000 8330 0.68 0.328 0.867 3.032 31.84 77.9 0.69 0.96 0.660 31.86 3.034 2027 0.693 0.1 2021

23.87 4100 8330 0.69 0.339 0.851 2.974 31.23 76.9 0.68 0.96 0.665 31.26 2.977 2044 0.690 0.1 2042

24.46 4200 8330 0.70 0.351 0.831 2.905 30.50 75.8 0.66 0.96 0.671 30.50 2.905 2060 0.684 0.0 2061

25.62 4400 8330 0.72 0.379 0.778 2.727 28.63 72.7 0.62 0.95 0.687 28.73 2.736 2074 0.663 0.3 2087

26.20 4500 8330 0.73 0.395 0.742 2.610 27.41 70.5 0.60 0.94 0.697 27.46 2.615 2078 0.645 0.2 2090

26.78 4600 8330 0.75 0.414 0.695 2.467 25.90 67.8 0.56 0.93 0.709 25.81 2.458 2073 0.622 0.3 2078

27.37 4700 8330 0.77 0.436 0.632 2.282 23.96 64.1 0.52 0.92 0.724 23.93 2.279 2067 0.588 0.1 2040

11.65 2000 5870 0.84 0.286 0.911 1.822 19.13 53.5 0.42 0.89 0.762 19.20 1.829 595 0.625 0.3 585

12.81 2200 5870 0.84 0.316 0.881 1.789 18.78 52.7 0.41 0.89 0.765 18.90 1.800 618 0.638 0.6 609

13.39 2300 5870 0.85 0.332 0.861 1.766 18.54 52.1 0.40 0.89 0.767 18.70 1.781 628 0.641 0.9 620

14.56 2500 5870 0.86 0.365 0.805 1.704 17.90 50.5 0.39 0.89 0.772 18.10 1.724 650 0.637 1.1 634

15.14 2600 5870 0.86 0.383 0.769 1.666 17.49 49.4 0.38 0.88 0.775 17.70 1.686 662 0.629 1.2 638

this facor is calculated

iteratively

H*fp

19

9.1 The discharge pressure and shaft power simulation results are shown in the next Diagrams (click on picture to enlarge):

Diagram -10 Discharge pressure versus volumetric flow for 3 speeds

Diagram -11 Shaft Power versus volumetric gas flow for the same three speeds.

Simulated and Actual Power Consumption of Compressor vscCR-01

0

500

1000

1500

2000

2500

1500 2000 2500 3000 3500 4000 4500 5000 5500

Suction Volumetric Gas Flow Q1 ---------> m3 / hour

Sh

aft

Po

we

r --

----

-- >

kW

att

Simulated and Actual Performance Curves of compressor vscCR-01

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

32

34

36

38

40

42

1500 2000 2500 3000 3500 4000 4500 5000

Suction Volumetic Gas Flow Q1 ------> m3 / hour

Dis

ch

arg

e P

ressu

re

----

----

>

Bar

ab

so

lute

Manufacturer's Data

Simulation Model Data

Surge Curve

20

9.2 Simulation results for inter- and extrapolated compressor speeds The simulation model can be used to predict the discharge pressure performance curves for other speeds:

Diagram -12 Discharge pressure versus volumetric gas flow for 5 speeds

Diagram -13 Shaft Power versus Volumetric gas flow for 5 speeds

Simulated and Actual Power Consumption of Compressor vscCR-01

0

500

1000

1500

2000

2500

3000

1500 2000 2500 3000 3500 4000 4500 5000 5500

Suction Volumetric Gas Flow Q1 -------> m3/hour

Sh

aft

Po

we

r

----

----

->

kW

Simulated and Actual Perormance Curves Centrifugal Compressor vscCR-01

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

32

34

36

38

40

42

1500 2000 2500 3000 3500 4000 4500 5000 5500

Suction Volumetic Gas Flow Q1 ------> m3 / hour

Dis

ch

arg

e P

ressu

re

----

----

>

Bar

ab

so

lute

Manufacturers data

Simulation Model Data

Surge Curve

5780 rpm

7000 rpm

7800 rpm 8330 rpm

8800 rpm

21

10. Conclusion In this post a model is developed and described that allows an existing variable speed compressor to be simulated with good accuracy. In addition a simple description of the surge line is discussed. The actual compressor’s performance, in terms of volume flow processed, pressure rise achieved and shaft power consumption required, is simulated by modeling the behavior of the machine as the result of two concurrently proceeding parallel processes. In one process a ‘real’ gas is compressed in an adiabatic and isentropic fashion. In the other process the energetic ‘loss’ behavior is described in relation to the operating performance and characterized by the “Adiabatic Efficiency Factor”. This factor relates the calculated adiabatic compression power with the actual shaft power of the actual compressor machine A ‘fingerprint’ of the machine’s characteristic adiabatic head production capability (hf) is taken as a function of the machine’s “volume- moving” capability (or flow factor “f” as defined by Q1/S * eps-h ) (model identification). The energetic loss processes occurring in the actual compressor are captured with the Adiabatic Efficiency Factor. Similarly a ‘fingerprint’ of the Adiabatic Efficiency Factor was taken with the help of a newly developed relation that correlates this factor the flow factor “f” . It should be noted that this correlation is strictly valid only for our particular machine under consideration “vscCR-01”. The simulation model developed here can be made to apply to other compressors as well as long as the particular ‘fingerprints” for the machine under study are determined. To be precise as long as the expression for ‘H*fp’ versus Q* is determined and the ‘adiabatic efficiency factor’ are correlated to Q* for that machine under study anew. The correlation we found to apply to our machine may roughly apply to other machines and may be a good start to develop your own : for example it may require just some “tuning” or at worst require you to develop of a new correlation for your machine under study! It should be noted that the model includes the accounting for the fact that the gas being handled is a real gas with large deviations from ideal (from a value of 1.0) These Z values are calculated with the ‘Zv correlation’ presented in earlier posts.