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Performance relationships
V
6
©
®
Performance relationships
Introduction Satisfying the objective Gas characteristics Compression head Impeller types and specific speed Efficiency Horsepower The Fan Laws
Introduction In this section we will cover the relationships that the COMPRESSOR VENDOR uses to determine the head produced, efficiency, horsepower required and overall design for a particular compressor application. As previously mentioned, the END USER'S or the PURCHASER'S objective is to deliver a specified amount of a given gas to the process. Therefore, the data that the compressor vendor obtains is required mass flow, inlet pressure, temperature conditions and gas composition. With this data a compressor manufacturer will calculate actual flow, the ideal energy required and the horsepower required to achieve that objective. The calculation for horsepower will require a specific compressor efficiency as well as compressor mechanical losses, that is bearing friction losses, seal losses and disc friction losses. Gas characteristics are defined in this chapter and useful relationships are presented to enable the reader to calculate various compressor requirements. Once the VENDOR obtains the data, the gas head can be calculated. Once the head and required flow are known, the impeller can be selected.
63
I
Compressors • • • • • • • • • • • ^
The principle of impeller design is chiefly based on that of specific speed. Specific speed is defined as the ratio of speed times the square root of the actual flow divided by head raised to the three quarters power. It can be shown that increasing values of specific speed will result in increasing impeller efficiencies. Therefore, having been given the required flow and energy (head) the only source of obtaining higher specific speed for the vendor is to increase the compressor speed. This fact is very significant, because while compressors have increased in efficiency over the years, the mechanical requirements have also increased significandy, I.E., higher bore impeller stresses, etc. resulting in potential reliability problems. Therefore, the design of the impeller is a very fine balance between the performance requirements and the mechanical constraints of the components used in the compressor design. Efficiency is presented as a ratio of ideal energy to actual energy as depicted on a typical MoUier Diagram. In addition, the fan laws are presented showing how increased impeller energy can be obtained via speed change in a compressor application.
Satisfying the objective The objective of the end user is to deliver a specified amount of a given gas. Refer to Figure 6.1 and note that his objective can best be stated by the relationship: Gas Flow Produced = Gas Flow Delivered Gas Delivered LBS/MIN
Gas Produced
f\
©
/ ^
LBS/MIN
@
U® - ^
- ^
\J
KJ
P2
X LBS/MIN P2 T2 K2 Z2 EFFICIENCY
X LBS/MIN Pi Ti Ki Zi EFFICIENCY
Pi
El
E2
ENERGY REQUIRED
Figure 6.1 The objective: to deliver a specified amount of a given gas
64
Performance Relationships
This incidentally is the reason why most process control systems monitor pressure in the process system and install a controller to either modulate flow via a control valve (change the head required by the process) or vary the speed of the compressor (change the head produced by the compressor). The vendor then, determines the head required by the process based on the parameters given by the contractor and end user on the equipment data sheet. It is very important to note that all possible sources should be used to confirm that the conditions stated on the data sheet are correct and realistic. This fact is especially true for dynamic compressors, since erroneous process conditions will impact the throughput of the compressor.
Gas characteristics Figure 6.2 presents the relationships used to calculate the design parameters for the compressor. Note that the same relationships are used regardless of the type of compressor (Positive Displacement or Dynamic). To achieve the client's objective the compressor vendor must calculate the actual flow to the compressor inlet the actual energy and work required Actual flow Volume flow rate (F^/MIN) = mass flow rate ( — - ) + density ( — )
/gas density ( depends on P^,
Energy (ideal) Compression
_ FOOT-LB's
HEADpoLYTROPic ~
LB MASS
Energy (ideal) to compress
. Depends on P^, T^,
and deliver one LB of gas
( Zg.g, Kg.g, MW P2,
from Pi to P2
^ efficiency
Work ^_^g.^ ^g Power (horsepower) = ideal energy ( ) mass f l o w ( — ) LB Min 33,000 (
) X efficiency I min-HP
Figure 6.2
The gas characteristics used in the determination of design parameters are defined in Figure 6.3.
65
Compressors
Gas characteristics Compressibility (Z)
-
Accounts for the deviation from an ideal gas
Specific heat (C)
-
The amount of heat to raise one mass of gas one degree
CpandCv
-
Specific heat at constant pressure and volume respectively
Specific heat ratio (K)
-
Cp/Cv
MW
-
Molecular weight
Polytropic exponent (n)
Used in polytropic head calculations n-1 n
_
k-1 k
, 1 '^ T] polytropic
Figure 6.3 Gas characteristics
Figure 6.4 shows useful relationships used in compressor calculations as well as the definitions for constants used. Useful relationships Actual flow - FT3/minute ACFM
mass flow ^ Density
where:
(LBS/MIN) (LBS/F3)
C = 33,000 =
HEAD FT-LBS LB
Density (LBS/FT3) = ' ^ ^ '
HD:
ACFM = SCFMx ' ^ ^ *
Mass flow ••
^
Energy (Ideal) - F-LB/LB Mass Use head equation, Polytropic is usually used
FT-LBS Min-H.P.
LBS Minute
Effy = corresponding efficiency (polytropic, isentropic, etc) P = pressure - psia
Efficiency - %
T = temperature - R*
Derived from impeller test results - does not include mechanical losses
*R = °F + 460 Z = compressibility
Work - horsepower R = 1545/mol. wgt Brake horsepower = gas horsepower + mech. losses ^ , (HDKMassflow) Gas horsepower = (Q) (^ff-y) Figure 6.4 Useful relationships
66
SCFM = standard FT3/min referenced to 60°Fand 14.7 psia
Compressors
LOW SPECIFIC
LOW SPECIFIC
MEDIUM SPECIFIC
MEDIUM SPECIFIC
SPEED-OPEN
SPEED-OPEN
SPEED-OPEN
SPEED-OPEN
RADIAL W/INDUCER
RADIAL W/ INDUCER
3-DIMENSIONAL W/ INDUCER
SECTION
SECTION
SECTION
(LARGER FLOW)
(LARGEST FLOW)
RADIAL
Figure 6.6 Compressor impellers
side) leakage and increased number of blade natural frequencies resulting from the cantilevered attachment of the blades to the hub. Most end users restrict the use of open impellers to plant and instrument air applications since the high speeds and intercooling offset the efficiency penalties caused by shroud leakage. Older design multistage centrifijgal compressors frequently used open impellers in the first stages since the high flows caused unacceptable side plate stresses in closed impeller design. Modern calculation (finite element) methods and manufacturing methods (attachment techniques machine welding, brazing, etc.) today make possible the use of enclosed first stage impellers for all multistage compressor applications. Finally, radial bladed impellers (whether open or enclosed) produce an extremely flat (almost horizontal) head curve. This characteristic renders these impellers unstable in process systems that do not contain much system resistance. Therefore, radial impellers are to be avoided in process systems that do not contain much system resistance (plant and instrument air compressors, charge gas compressors and refrigeration applications with side loads. Enclosed impellers
Enclosed impellers are shown in Figure 6.7. Note that the first stage impeller in any multistage configuration is always the widest. That is, it has the largest flow passage. As a result, the first stage impeller will usually be the highest stressed impeller. The exception is a refrigeration compressors with side loads (economizers). Dynamic compressor vendors use specific speed to select impellers based on the data given by the contractors and end user. The vendor is given the total head required by the process and the inlet volume flow. As previously discussed, at the stated inlet flow (rated flow) the head
68
Performance Relationships
Low Specific Speed 2 - Dimensional Closed Impeller
Medium Specific Speed 2 - Dimensional Closed Impeller
Medium Specific Speed 3 - Dimensional Closed Impeller
Medium High Specific Speed 3 - Dimensional Mixed Flow Closed Impeller
NOTE: All Impeller Vanes are Backward Lean Figure 6.7 Enclosed impellers (Courtesy of IMP Industries, Inc.)
required by the process is in equilibrium with the head produced by compressor. Vendor calculation methods then determine how many compressor impellers are required based on mechanical limitations (stresses) and performance requirements (quoted overall efficiency). Once the head required per stage is determined, the compressor speed is optimized for highest possible overall efficiency using the concept of specific speed as shown in Figure 6.8. It is a proven fact that the larger the specific speed, the higher the attainable efficiency. As shown, specific speed is a direct function of shaft speed and volume flow and an inverse function of produced head. Since the vendor at this point in the design knows the volume flow and head produced for each impeller, increasing the shaft speed will increase the specific speed and the compressor efficiency. However, the reader is cautioned that all mechanical design aspects (impeller stress, critical speeds, rotor stability, bearing and seal design) must be confirmed prior to acceptance of impeller selection. Often, too great an emphasis on performance (efficiency) results in decreased compressor reliability. One mechanical design problem can quickly offset any power savings realized by designing a compressor for a higher efficiency. Referring back to Figure 6.8, calculation of specific speed for the first impeller by the contractor or end user will give an indication of the type of dynamic compressor blading to be used. One other comment.
69
Compressors RADIAL FLOW 2D 3D
MIXED FLOW
AXIAL FLOW
^ IMPELLER TYPE
90 80 70
>
O
lU
o
60 50 SPECIFIC SPEED-
40 30
WHERE:
20
N Q HD
10 H
(N)(/Q)
SPEED(RPM) FLOW RATE (FT^/MIN) HEAD(FT-LB) LB
0 200
400
600
800
1000
1200 1400
1600 1800
2000
SPECIFIC SPEED - DIMENSIONLESS
Figure 6.8 Impeller geometry vs. specific speed
Sundstrand Corporation successfully employs an integral high speed gear box design for low flow, high head applications or for low specific speed applications. The use of a speed increasing gear box (for speeds up to 34,000 RPM) enables the specific speed to be increased and therefore resulting in higher efficiency and less complexity than would be obtained with a multistage compressor design approach.
Efficiency Compressor efficiency, regardless of the type of compressor, can best be understood by referring to a typical Mollier Diagram as depicted in Figure 6.9. All produced heads shown on performance curves (isothermal, isentropic and polytropic) represent the ideal reversible head produced to compress a given gas from P^ to P2. This then is the theoretical compression path of the gas. That is, the energy required to compress a gas if the efficiency is 100%. However due to friction, sudden expansion etc., the efficiency is less than 100%. Therefore the actual compression path requires more head (energy) to compress the gas from P^ to P2. The efficiency then is equal to: ^rr ' AE Ideal (Eo Ideal - Ei) Erriciency = 1-^ — AE Actual (E2 Actual - Ei)
70
Performance Relationships
Efficiency = Liquid Zone
AE ideai (E2 Ideai - EQ AE Actual (E2 Actuai - Ei) I
Vapor Zone ^one
%
^ (Ideal)
%
E2 (Actual)
Figure 6.9 Efficiency
Note that ( ) is used to represent any ideal reversible path (isothermal, isentropic, polytropic).
Horsepower Horsepower is defined as the total actual energy (work) required to compress a given gas from P^ to P2 when compressing a given mass flow: ft-lbf P M p _ Head () Ibm (Mass Flow - Ib/min) •" (33,000) (Efficiency)()
Note: ( ) must be for the same ideal reversible compression path. The brake horsepower is the sum of the gas horsepower and the mechanical losses of the compressor. B.H.P. = G.H.P + Mechanical losses The mechanical losses are the total of bearing, seal and windage (disc friction) losses and are provided by the compressor vendor. For estimating purposes, a conservative value of mechanical losses for one centrifugal or axial compressor case would be 150 H.P.
71
Compressors i
QF
Qi
HDp HD,
= i^)'
BHPp BHP,
= i^)'
130 120 110 100 90 801
100% SPEED 95% SPEED
Where: Q
Speed (RPM)
N HD
Flow Rate (FT3/MIN)
=
=
Gas Head (FT-LB/LB)
BHP =
Horsepower
1
Initial
F
Final
I s i
.100% SPEED 95% SPEED I
I I I I I I I
60 70 80 90100110120130
%FLOW
Figure 6.10 The Fan Laws
The Fan Laws These familiar relationships, sometimes called the affinity laws for pumps were originally derived for a single stage fan which is a low pressure compressor. The Fan Laws are presented in Figure 6.10. As shown, if speed is changed, the flow, head and horsepower vary by the first, second and third power of speed ratio respectively. The reader must be cautioned however that the Fan Laws are only an approximation to be used as an estimating tool. Their accuracy significantly decreases with increasing gas molecular weight and increase in the number of compression stages.
72
6
©
®
Performance relationships
Introduction Satisfying the objective Gas characteristics Compression head Impeller types and specific speed Efficiency Horsepower The Fan Laws
Introduction In this section we will cover the relationships that the COMPRESSOR VENDOR uses to determine the head produced, efficiency, horsepower required and overall design for a particular compressor application. As previously mentioned, the END USER'S or the PURCHASER'S objective is to deliver a specified amount of a given gas to the process. Therefore, the data that the compressor vendor obtains is required mass flow, inlet pressure, temperature conditions and gas composition. With this data a compressor manufacturer will calculate actual flow, the ideal energy required and the horsepower required to achieve that objective. The calculation for horsepower will require a specific compressor efficiency as well as compressor mechanical losses, that is bearing friction losses, seal losses and disc friction losses. Gas characteristics are defined in this chapter and useful relationships are presented to enable the reader to calculate various compressor requirements. Once the VENDOR obtains the data, the gas head can be calculated. Once the head and required flow are known, the impeller can be selected.
63
I
Compressors • • • • • • • • • • • ^
The principle of impeller design is chiefly based on that of specific speed. Specific speed is defined as the ratio of speed times the square root of the actual flow divided by head raised to the three quarters power. It can be shown that increasing values of specific speed will result in increasing impeller efficiencies. Therefore, having been given the required flow and energy (head) the only source of obtaining higher specific speed for the vendor is to increase the compressor speed. This fact is very significant, because while compressors have increased in efficiency over the years, the mechanical requirements have also increased significandy, I.E., higher bore impeller stresses, etc. resulting in potential reliability problems. Therefore, the design of the impeller is a very fine balance between the performance requirements and the mechanical constraints of the components used in the compressor design. Efficiency is presented as a ratio of ideal energy to actual energy as depicted on a typical MoUier Diagram. In addition, the fan laws are presented showing how increased impeller energy can be obtained via speed change in a compressor application.
Satisfying the objective The objective of the end user is to deliver a specified amount of a given gas. Refer to Figure 6.1 and note that his objective can best be stated by the relationship: Gas Flow Produced = Gas Flow Delivered Gas Delivered LBS/MIN
Gas Produced
f\
©
/ ^
LBS/MIN
@
U® - ^
- ^
\J
KJ
P2
X LBS/MIN P2 T2 K2 Z2 EFFICIENCY
X LBS/MIN Pi Ti Ki Zi EFFICIENCY
Pi
El
E2
ENERGY REQUIRED
Figure 6.1 The objective: to deliver a specified amount of a given gas
64
Performance Relationships
This incidentally is the reason why most process control systems monitor pressure in the process system and install a controller to either modulate flow via a control valve (change the head required by the process) or vary the speed of the compressor (change the head produced by the compressor). The vendor then, determines the head required by the process based on the parameters given by the contractor and end user on the equipment data sheet. It is very important to note that all possible sources should be used to confirm that the conditions stated on the data sheet are correct and realistic. This fact is especially true for dynamic compressors, since erroneous process conditions will impact the throughput of the compressor.
Gas characteristics Figure 6.2 presents the relationships used to calculate the design parameters for the compressor. Note that the same relationships are used regardless of the type of compressor (Positive Displacement or Dynamic). To achieve the client's objective the compressor vendor must calculate the actual flow to the compressor inlet the actual energy and work required Actual flow Volume flow rate (F^/MIN) = mass flow rate ( — - ) + density ( — )
/gas density ( depends on P^,
Energy (ideal) Compression
_ FOOT-LB's
HEADpoLYTROPic ~
LB MASS
Energy (ideal) to compress
. Depends on P^, T^,
and deliver one LB of gas
( Zg.g, Kg.g, MW P2,
from Pi to P2
^ efficiency
Work ^_^g.^ ^g Power (horsepower) = ideal energy ( ) mass f l o w ( — ) LB Min 33,000 (
) X efficiency I min-HP
Figure 6.2
The gas characteristics used in the determination of design parameters are defined in Figure 6.3.
65
Compressors
Gas characteristics Compressibility (Z)
-
Accounts for the deviation from an ideal gas
Specific heat (C)
-
The amount of heat to raise one mass of gas one degree
CpandCv
-
Specific heat at constant pressure and volume respectively
Specific heat ratio (K)
-
Cp/Cv
MW
-
Molecular weight
Polytropic exponent (n)
Used in polytropic head calculations n-1 n
_
k-1 k
, 1 '^ T] polytropic
Figure 6.3 Gas characteristics
Figure 6.4 shows useful relationships used in compressor calculations as well as the definitions for constants used. Useful relationships Actual flow - FT3/minute ACFM
mass flow ^ Density
where:
(LBS/MIN) (LBS/F3)
C = 33,000 =
HEAD FT-LBS LB
Density (LBS/FT3) = ' ^ ^ '
HD:
ACFM = SCFMx ' ^ ^ *
Mass flow ••
^
Energy (Ideal) - F-LB/LB Mass Use head equation, Polytropic is usually used
FT-LBS Min-H.P.
LBS Minute
Effy = corresponding efficiency (polytropic, isentropic, etc) P = pressure - psia
Efficiency - %
T = temperature - R*
Derived from impeller test results - does not include mechanical losses
*R = °F + 460 Z = compressibility
Work - horsepower R = 1545/mol. wgt Brake horsepower = gas horsepower + mech. losses ^ , (HDKMassflow) Gas horsepower = (Q) (^ff-y) Figure 6.4 Useful relationships
66
SCFM = standard FT3/min referenced to 60°Fand 14.7 psia
Compressors
LOW SPECIFIC
LOW SPECIFIC
MEDIUM SPECIFIC
MEDIUM SPECIFIC
SPEED-OPEN
SPEED-OPEN
SPEED-OPEN
SPEED-OPEN
RADIAL W/INDUCER
RADIAL W/ INDUCER
3-DIMENSIONAL W/ INDUCER
SECTION
SECTION
SECTION
(LARGER FLOW)
(LARGEST FLOW)
RADIAL
Figure 6.6 Compressor impellers
side) leakage and increased number of blade natural frequencies resulting from the cantilevered attachment of the blades to the hub. Most end users restrict the use of open impellers to plant and instrument air applications since the high speeds and intercooling offset the efficiency penalties caused by shroud leakage. Older design multistage centrifijgal compressors frequently used open impellers in the first stages since the high flows caused unacceptable side plate stresses in closed impeller design. Modern calculation (finite element) methods and manufacturing methods (attachment techniques machine welding, brazing, etc.) today make possible the use of enclosed first stage impellers for all multistage compressor applications. Finally, radial bladed impellers (whether open or enclosed) produce an extremely flat (almost horizontal) head curve. This characteristic renders these impellers unstable in process systems that do not contain much system resistance. Therefore, radial impellers are to be avoided in process systems that do not contain much system resistance (plant and instrument air compressors, charge gas compressors and refrigeration applications with side loads. Enclosed impellers
Enclosed impellers are shown in Figure 6.7. Note that the first stage impeller in any multistage configuration is always the widest. That is, it has the largest flow passage. As a result, the first stage impeller will usually be the highest stressed impeller. The exception is a refrigeration compressors with side loads (economizers). Dynamic compressor vendors use specific speed to select impellers based on the data given by the contractors and end user. The vendor is given the total head required by the process and the inlet volume flow. As previously discussed, at the stated inlet flow (rated flow) the head
68
Performance Relationships
Low Specific Speed 2 - Dimensional Closed Impeller
Medium Specific Speed 2 - Dimensional Closed Impeller
Medium Specific Speed 3 - Dimensional Closed Impeller
Medium High Specific Speed 3 - Dimensional Mixed Flow Closed Impeller
NOTE: All Impeller Vanes are Backward Lean Figure 6.7 Enclosed impellers (Courtesy of IMP Industries, Inc.)
required by the process is in equilibrium with the head produced by compressor. Vendor calculation methods then determine how many compressor impellers are required based on mechanical limitations (stresses) and performance requirements (quoted overall efficiency). Once the head required per stage is determined, the compressor speed is optimized for highest possible overall efficiency using the concept of specific speed as shown in Figure 6.8. It is a proven fact that the larger the specific speed, the higher the attainable efficiency. As shown, specific speed is a direct function of shaft speed and volume flow and an inverse function of produced head. Since the vendor at this point in the design knows the volume flow and head produced for each impeller, increasing the shaft speed will increase the specific speed and the compressor efficiency. However, the reader is cautioned that all mechanical design aspects (impeller stress, critical speeds, rotor stability, bearing and seal design) must be confirmed prior to acceptance of impeller selection. Often, too great an emphasis on performance (efficiency) results in decreased compressor reliability. One mechanical design problem can quickly offset any power savings realized by designing a compressor for a higher efficiency. Referring back to Figure 6.8, calculation of specific speed for the first impeller by the contractor or end user will give an indication of the type of dynamic compressor blading to be used. One other comment.
69
Compressors RADIAL FLOW 2D 3D
MIXED FLOW
AXIAL FLOW
^ IMPELLER TYPE
90 80 70
>
O
lU
o
60 50 SPECIFIC SPEED-
40 30
WHERE:
20
N Q HD
10 H
(N)(/Q)
SPEED(RPM) FLOW RATE (FT^/MIN) HEAD(FT-LB) LB
0 200
400
600
800
1000
1200 1400
1600 1800
2000
SPECIFIC SPEED - DIMENSIONLESS
Figure 6.8 Impeller geometry vs. specific speed
Sundstrand Corporation successfully employs an integral high speed gear box design for low flow, high head applications or for low specific speed applications. The use of a speed increasing gear box (for speeds up to 34,000 RPM) enables the specific speed to be increased and therefore resulting in higher efficiency and less complexity than would be obtained with a multistage compressor design approach.
Efficiency Compressor efficiency, regardless of the type of compressor, can best be understood by referring to a typical Mollier Diagram as depicted in Figure 6.9. All produced heads shown on performance curves (isothermal, isentropic and polytropic) represent the ideal reversible head produced to compress a given gas from P^ to P2. This then is the theoretical compression path of the gas. That is, the energy required to compress a gas if the efficiency is 100%. However due to friction, sudden expansion etc., the efficiency is less than 100%. Therefore the actual compression path requires more head (energy) to compress the gas from P^ to P2. The efficiency then is equal to: ^rr ' AE Ideal (Eo Ideal - Ei) Erriciency = 1-^ — AE Actual (E2 Actual - Ei)
70
Performance Relationships
Efficiency = Liquid Zone
AE ideai (E2 Ideai - EQ AE Actual (E2 Actuai - Ei) I
Vapor Zone ^one
%
^ (Ideal)
%
E2 (Actual)
Figure 6.9 Efficiency
Note that ( ) is used to represent any ideal reversible path (isothermal, isentropic, polytropic).
Horsepower Horsepower is defined as the total actual energy (work) required to compress a given gas from P^ to P2 when compressing a given mass flow: ft-lbf P M p _ Head () Ibm (Mass Flow - Ib/min) •" (33,000) (Efficiency)()
Note: ( ) must be for the same ideal reversible compression path. The brake horsepower is the sum of the gas horsepower and the mechanical losses of the compressor. B.H.P. = G.H.P + Mechanical losses The mechanical losses are the total of bearing, seal and windage (disc friction) losses and are provided by the compressor vendor. For estimating purposes, a conservative value of mechanical losses for one centrifugal or axial compressor case would be 150 H.P.
71
Compressors i
QF
Qi
HDp HD,
= i^)'
BHPp BHP,
= i^)'
130 120 110 100 90 801
100% SPEED 95% SPEED
Where: Q
Speed (RPM)
N HD
Flow Rate (FT3/MIN)
=
=
Gas Head (FT-LB/LB)
BHP =
Horsepower
1
Initial
F
Final
I s i
.100% SPEED 95% SPEED I
I I I I I I I
60 70 80 90100110120130
%FLOW
Figure 6.10 The Fan Laws
The Fan Laws These familiar relationships, sometimes called the affinity laws for pumps were originally derived for a single stage fan which is a low pressure compressor. The Fan Laws are presented in Figure 6.10. As shown, if speed is changed, the flow, head and horsepower vary by the first, second and third power of speed ratio respectively. The reader must be cautioned however that the Fan Laws are only an approximation to be used as an estimating tool. Their accuracy significantly decreases with increasing gas molecular weight and increase in the number of compression stages.
72