Reprinted with permission from CEP (Chemical Engineering Progress), June 2011.
Copyright © 2011 American Institute of Chemical Engineers (AIChE).
Computational Methods
Integrate Compressor
Performance Maps
into Process Simulation
Grant Stephenson
Honeywell Process Solutions
V
Transforming the performance maps to a
reduced coordinate system that is independent
of suction conditions and rotational speeds
allows these curves to be accurately incorporated
into a process simulator.
ariable-speed centrifugal compressors are used in a
wide variety of industries and applications, including
natural gas pipelines and processing plants, oil refineries and chemical plants, air-separation units, refrigeration
and air conditioning equipment refrigerant cycles, gas turbines and auxiliary power generators, and many more. Some
natural gas plants, for example, employ a propane-precooled,
mixed-refrigerant natural gas liquefaction process consisting
of a classical propane liquefaction cycle that precools both
the natural gas feed and the mixed refrigerant followed by a
mixed-refrigerant liquefaction cycle that provides the lowtemperature refrigeration needed to liquefy the natural gas.
Compression is a key element of both refrigeration cycles.
Dynamic process simulation offers many tangible and
intangible benefits (1). Integrating the turbomachinery
controls (TMC) into the dynamic simulation of a compressor
system enables the engineer to select properly sized equipment that can cope with transient conditions (surge conditions
in particular), reduces commissioning time through pre­tuning
of the regulatory controls, improves operator training by
exposing operators to realistic simulated operating scenarios,
and provides verified startup and shutdown procedures during
site acceptance testing before startup takes place (2).
Dynamic simulators for modeling processes that involve
gas compression require a rigorous unit-operation model of
a centrifugal compressor. The performance maps provided
by compressor manufacturers are a critical element of this
model. This article explains how to use the performance
maps for variable-speed, fixed-inlet guide-vane centrifugal
compressors for the dynamic simulation of a compressor
system.
42 www.aiche.org/cep June 2011 CEP
Performance maps and their use in simulation
A performance map describes how a compressor’s polytropic head and power vary with volumetric suction flowrate
and rotational speed for a specific set of suction conditions (i.e., fluid molecular weight, pressure, temperature,
compressibility, and isentropic exponent at the inlet of the
compressor). Manufacturers typically provide performance
maps as two sets of performance curves — a series of plots
of polytropic head vs. volumetric suction flowrate, and a corresponding series of power vs. flow plots. Each pair of curves
corresponds to a different rotational speed, and all pertain to
the same suction conditions.
Performance maps may be constructed for more than one
set of suction conditions. Figure 1 presents two performance
maps for a high-pressure mixed-refrigerant compressor for
a natural gas liquefaction process. The top performance
map corresponds to design operation, and the bottom map
corresponds to derime operation, which removes rime (the
granular ice that forms when supercooled droplets freeze
rapidly on contact with a cold surface) from the compressor.
These maps were generated from data obtained by manually
digitizing the performance maps provided by the manufacturer; the irregularities in the curves are a result of the manual
digitization process.
Process simulators combine the performance maps with
material and energy balances and thermodynamic relationships to predict the performance of the compressor under
various operating scenarios. However, the performance
maps supplied by the manufacturer are not broadly useful for
process simulation because, strictly speaking, they apply only
at the suction conditions and rotational speeds for which they
30,000
Design Operation
20,000
15,000
10,000
3,760 rpm
3,850 rpm
3,950 rpm
4,000 rpm
5,000
0
Design Operation
25,000
0
5,000
10,000
15,000
20,000
25,000
Power, hp
Polytropic Head, ft
25,000
20,000
15,000
10,000
3,760 rpm
3,850 rpm
3,950 rpm
4,000 rpm
5,000
0
30,000
0
5,000
Volumetric flowrate, cfm
900
Derime Operation
20,000
20,000
25,000
30,000
700
15,000
10,000
3,760 rpm
3,850 rpm
3,960 rpm
4,000 rpm
5,000
0
15,000
Derime Operation
800
0
5,000
10,000
15,000
20,000
Power, hp
Polytropic Head, ft
25,000
10,000
Volumetric flowrate, cfm
600
500
400
300
3,760 rpm
3,850 rpm
3,960 rpm
4,000 rpm
200
100
0
25,000
0
5,000
10,000
15,000
20,000
25,000
Volumetric flowrate, cfm
Volumetric flowrate, cfm
p Figure 1. These performance maps represent design (top) and derime (bottom) operation of a high-pressure, mixed-refrigerant compressor.
were constructed. Process simulation requires prediction of
the compressor’s performance not just at these suction conditions and speeds, but over the compressor’s entire operating
range, from surge to stonewall, at varying suction conditions
and rotational speeds.
Some process simulators ignore the fact that the performance maps relate to specific suction conditions, or allow
users to enter multiple performance maps in tabular form and
synthetically transition between them as the suction conditions change. However, as shown in Figure 2, the performance maps for different suction conditions can be quite
different. In this instance, the scales for power differ by more
30,000
Design and Derime Operation
20,000
15,000
10,000
Design
Derime
5,000
0
Design and Derime Operation
25,000
Power, hp
Polytropic Head, ft
25,000
than an order of magnitude. Using the performance map for
design operation at all operating conditions will certainly
yield poor results for derime operation.
Varying rotational speed can be handled somewhat better than varying suction conditions, for example, by linear
interpolation and extrapolation on an irregularly spaced
nonrectangular grid. Although linear interpolation can yield
reasonable results, linear extrapolation breaks down as the
rotational speed moves further away from the lowest and
highest speeds for which performance curves were constructed. This limits the utility of the performance maps to
normal operating speeds.
0
5,000
10,000
15,000
20,000
Volumetric flowrate, cfm
25,000
30,000
20,000
15,000
10,000
Design
Derime
5,000
0
0
5,000
10,000
15,000
20,000
25,000
30,000
Volumetric flowrate, cfm
p Figure 2. Superimposing performance maps reveals that the power vs. suction flowrate relationships for design and derime operation are very different.
Article continues on next page
CEP June 2011 www.aiche.org/cep 43
0.7
0.7
0.6
0.6
0.5
0.5
Reduced Power
Reduced Polytropic Head
Computational Methods
0.4
0.3
0.2
Design
Derime
0.1
0
0.4
0.3
0.2
Design
Derime
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0
0.1
0.2
0.3
0.4
0.5
0.6
Reduced Volumetric Suction Flow
Reduced Volumetric Suction Flow
p Figure 3. A reduced performance map aggregates a series of performance curves into a single curve in the reduced coordinate system.
If the unit-operation model of the compressor does not
adequately account for variation in the suction conditions
and rotational speed, the predictions of the process model in
which it is used — for example, the speed and shaft power
needed to achieve a specified (controlled) discharge pressure — will certainly be flawed, and may even be misleading.
If the process model is used in a dynamic simulation study,
the trajectories for the operating variables will be incorrect,
possibly significantly, which could lead to false conclusions.
Simulation-based training using a model that misrepresents
the actual process behavior could cause an operator to make
incorrect operating decisions.
Transforming the performance maps
The challenge, therefore, is to find a way to make the
performance maps applicable over the entire range of suction conditions and speeds encountered in the operation of
the compressor. This can be achieved by transforming the
performance maps to a coordinate system that is independent
of both suction conditions and rotational speed. This transformation involves two steps.
First, dimensional analysis is used to construct a coordinate system that is independent of suction conditions (3).
A key aspect of this analysis is the identification of a set
of dimensionless parameters in which the pertinent forces
can be accurately represented.
Functional relationships are set up in which polytropic
head (Hp) and shaft power (P) are the dependent variables
and volumetric flowrate (Q) and rotational speed (w) are
the independent variables. The suction conditions and the
compressor size are incorporated into the functions by including viscosity (m), density (r), and acoustic velocity (a) at the
suction of the compressor and the diameter (d) of the compressor as additional independent variables. These functional
relationships are represented as:
Hp = f0(Q, w, m, r, a, d) P = f1(Q, w, m, r, a, d) 44 www.aiche.org/cep June 2011 CEP
(1)
(2)
Each of these relationships involves seven dimensional
parameters. Using the Buckingham Pi Theorem and assuming turbulent flow, they can be simplified to:
Hp,n = f2(Qn, wn) Pn = f3(Qn, wn) (3)
(4)
where the subscript n denotes a dimensionless variable.
Although this dimensionless coordinate system was
developed for compressor control, it is equally applicable to
process simulation.
Next, a modified form of the Fan Laws is applied to
further transform the dimensionless coordinate system to one
that is also independent of rotational speed. The Fan Laws
of centrifugal pumps or fans, also known as the Affinity
Laws, express the influence of changes in rotational speed,
diameter, and density on volumetric flowrate, head (H) (or
pressure), and shaft power. When the fan diameter and the
fluid density at the fan suction are constant, the Fan Laws can
be expressed as:
Q1/Q2 = w 1/w2 H1/H2 = (w 1/w2)2 P1/P2 = (w 1/w2)3 (5a)
(5b)
(5c)
It follows from these relationships that when H/w2 and
P/w3 are plotted against Q/w, the performance curves for a
given set of suction conditions reduce to a single curve for
head and a single curve for power. However, because the Fan
Laws do not strictly apply to compressors, a modified form
of the Fan Laws must be used.
The resulting coordinate system, referred to as the
reduced coordinate system, has as its dependent variables
the transformed (reduced) polytropic head (Hp,r) and reduced
power (Pr), and as its independent variable the reduced volumetric suction flow (Qr). In the reduced coordinate system,
the performance curves for polytropic head and for power at
different suction conditions are each represented by a single
0.7
Design
Derime
Fitted
0.6
0.6
Reduced Power
Reduced Polytropic Head
0.7
0.5
0.4
0.3
0.2
0.5
0.4
0.3
Design
Derime
Fitted
0.2
0.1
0.1
0
0
0
0.2
0.4
0.6
Reduced Volumetric Suction Flow
0.8
0
0.2
0.4
0.6
0.8
Reduced Volumetric Suction Flow
p Figure 4. The surge, normal, and stonewall operation segments of the aggregate performance curves can be represented by different functions.
aggregate curve, as shown in Figure 3. These aggregate
curves are expressed as:
Hp,r = f4(Qr) Pr = f5(Qr) (6)
(7)
Transforming the performance maps to the reduced
coordinate system is very powerful. Consider the following
modeling scenario. Given the suction conditions, rotational
speed, and volumetric suction flow for a particular operating condition (real or simulated), the reduced volumetric
suction flow is calculated by applying the transformations, calculating first Qn and wn, and then Qr. The reduced
polytropic head and reduced power corresponding to the
reduced volumetric suction flow are then determined from
the aggregate performance curves, i.e., from f4(Qr) and
f5(Qr), respectively. The polytropic head and power in
engineering units are then easily calculated by inverting the
transformations, first calculating Hp,n and Pn, and then Hp
and P. Transforming the performance maps provided by the
compressor manufacturer, which strictly apply only at their
corresponding suction conditions and rotational speeds, to
reduced coordinates in this way enables performance curves
for polytropic head and power that are particular to different
operating conditions (i.e., suction conditions and rotational
speed) to be constructed.
To make a reduced performance map (i.e., the aggregate curves of reduced polytropic head and reduced power)
more useful, the aggregate curves need to be represented
by appropriate functions whose parameters are fitted to the
reduced data.
Functional representation of
the reduced performance map
Based on a visual inspection of Figure 3, a quadratic
function appears to be a good choice for representing the
aggregate performance curves for both polytropic head,
f4(Qr), and power, f5(Qr). However, experience has proven
Table 1. The aggregate performance curves
can be divided into segments,
each represented by a different functional form.
Curve Segment
Polytropic Head
Power
Surge
Exponential
Quadratic
Normal
Power
Power
Stonewall
Quadratic
Quadratic
otherwise. A quadratic function is not a suitable representation of the aggregate power curve, which is not symmetric
about the perpendicular line through its apex. In addition, a
quadratic function fitted to the data typically will not extrapolate through the origin — a necessary physical constraint. A
quadratic function is also not a suitable representation of the
aggregate curve of polytropic head at low values of reduced
volumetric suction flow, where it becomes nearly linear.
The next likely choice of functional form would be a
higher-order polynomial. However, high-order polynomials
can wobble, even within the range of the data being fitted,
and they may not extrapolate well — both significant disadvantages for process simulation.
Experience has shown that a very good fit is obtained
when the aggregate performance curves are divided into
three segments — representing the surge, normal, and stonewall regions of the performance curves — and the segments
are represented by the functions listed in Table 1.
The functions’ parameters and the segments’ end points
are determined by fitting the curves to the reduced data.
This yields smooth, piecewise-fitted curves, as shown in
Figure 4. The functions fit the curve segments well, and they
extrapolate well for both low and high reduced volumetric
suction flows. Although flowrates below the reduced surge
flow are not physically meaningful, reasonable extrapolation to low flows is required to ensure robust solution of the
compressor unit-operation model. Reasonable extrapolation
in the stonewall region facilitates robust solution up to the
stonewall flow.
Article continues on next page
CEP June 2011 www.aiche.org/cep 45
Computational Methods
R
14
VLV-100
K-100
Surge
Controller
1
VLV-103
3
E-100
5
7
PIC-100
V-100
MIX-101
V-101
2
12
VLV-101
Q-101
MIX-100
K-100
15
TEE-100 11
10
4
17
13
RCY-1
6
K-100
Speed
16
Q-100
LIC-100
8
VLV-102
9
p Figure 5. This compressor loop flowsheet was created by a process simulator that rigorously models compressors over their entire operating range.
p Figure 6. The Parameters page of the Design tab for compressor
K-100 provides design details for this equipment.
Head Curves
3.5e+004
2,800 rpm
3,200 rpm
3,400 rpm
2,800 rpm
Operating Point
3.0e+004
Head, m
2.5e+004
2.0e+004
1.5e+004
1.0e+004
5.0e+004
0.00
0
5,000
10,000
15,000
20,000
25,000
Flowrate, actual m3/h
p Figure 7. Compressor K-100’s performance map for polytropic head
was constructed from the reduced aggregate polytropic head curve at the
existing suction conditions, at various speeds.
46 www.aiche.org/cep June 2011 CEP
Using the reduced performance map
in dynamic process simulation
As stated previously, accurately simulating the behavior
of processes that include centrifugal compressors requires
process simulators that rigorously model these devices over
their entire range of suction conditions and rotational speeds,
from surge through stonewall operation.
Figure 5 is a typical compressor loop flowsheet developed using such a simulator. Figure 6 presents details for
compressor unit operation K-100, which has been configured
as a centrifugal compressor using the reduced coordinate system representation of the performance maps. The polytropic
head performance map for this compressor, Figure 7, was
constructed from the reduced aggregate performance curve
for polytropic head at the existing suction conditions. The
speeds for which performance curves are plotted were specified by the modeling engineer using the speed curves selector
on the right-hand side of the window.
The current operating point, represented by the red dot, is
superimposed on the performance map. In dynamic simulation, the operating point moves as the suction conditions,
rotational speed, and/or discharge pressure vary. For the
purpose of illustration, a transfer function (the K-100 speed
block of the flowsheet) was applied to cause the rotational
speed to oscillate, as depicted by the green line on the strip
chart in Figure 8. The red and blue lines in Figure 8 represent
the dynamic responses of the volumetric suction flow and the
polytropic head.
Volumetric Flowrate, m3/h
1.800e+004
3,360 (rpm)
1.500e+004
1.200e+004
1.328e+004 (m3)
9,000
2.152e+004 (m)
6,000
2,995
3,000
3,005
Minutes
p Figure 8. This strip chart records the dynamic responses of polytropic
head (blue) and volumetric suction flowrate (red) to variations in rotational
speed (green).
In addition to integrating reduced performance maps
into its mathematical model of the centrifugal compressor
unit operation, it is crucial that the process simulator also
provide a tool to assist the engineer in transforming the digitized performance curves into the reduced coordinate system, that is, in creating the aggregate performance curves
and fitting the parameters of the functions that represent
these curves. Without such a tool, this task would simply
be too daunting, and the value associated with the reduced
coordinate system representation of the performance maps
CEP
would not be realized.
Literature Cited
1. Stephenson, G., et al., “Profit More from Process Simulation,”
Chem. Processing, 72 (8), pp. 23–26 (Aug. 2009).
2. Willetts, I., and A. Nair, “Using High-Fidelity Dynamic Simulation to Model Compressor Systems,” Chem. Eng. Progress,
106 (4), pp. 44–48 (Apr. 2010).
3. Batson, B. W., “Invariant Coordinate Systems for Compressor
Control,” presented at the International Gas Turbine and Aeroengine Congress and Exhibition, Birmingham, U.K. (June 1996).
Grant Stephenson is an Engineering Fellow with Honeywell’s Automation Control Solutions business, where he serves as the global
process simulation architect for Honeywell Process Solutions. He
has worked in the field of process simulation for more than 35 years,
with particular interest in dynamic simulation, equation-oriented
modeling and simultaneous solution of flowsheet models, and the
application of modeling and optimization to plant operations. He
is the originator of the dynamic simulation engine of the Shadow
Plant simulator and is a pioneer of the hybrid solution architecture
and its application to large-scale dynamic simulation. Before joining
Honeywell, he held positions with DuPont Canada, Atomic Energy
of Canada, the Univ. of Western Ontario (in the engineering faculty’s
Systems Analysis Control and Design Activity, SACDA, group), and
SACDA Inc. He holds an MSc degree in applied mathematics from the
Univ. of Western Ontario.
CEP June 2011 www.aiche.org/cep 47