Campbell Tip of the Month

Author: Joe Honeywell

  • Why NPSHR Changes With Impeller Diameter?

    Introduction

    Confusion sometimes results when reviewing published NPSHR curves.  This is especially true when faced with trimming the impeller diameter to match changing operating conditions.  A well known fact is that the head-flow relationship varies with the diameter.  This can be accurately approximated by the affinity laws.  However, what happens to the NPSHR-flow relationship when the diameter changes?  This relationship is frequently over looked and can lead to pump cavitation.  This Tip of the Month examines the relationship of NPSHR to the impeller diameter and clarifies other misconceptions regarding pump NPSHR curves.

    Background

    Pump performance may be shown for a single impeller or a range of impeller diameters.  In the latter case the pump performance may be shown as multiple curves from the maximum to the minimum diameter, and may show several intermediate impeller sizes.  In addition, the pump performance characteristics may show curves for NPSHR, efficiency and required power.   The representation of the pump performance varies widely depending on many factors and can lead to design errors and possible confusion.

    An example of a typical pump performance curve frequently seen in publications is shown in Figure 1.  The pump flow rate is plotted on the horizontal axis, and the head and NPSHR curves, which are a function of flow rate, plotted on the vertical axles.  Note that a single-line NPSHR curve starts at the no-flow condition and continually rises to the maximum flow rate.  For several reasons that will be discussed later, this type of NPSHR curve is incorrect and can lead to design errors and possible cavitation problems.

    Pump cavitation is a complex subject and the topic of many technical papers and books.  However, it is widely accepted that this phenomenon begins at the pump inlet.  It basically results from the increased velocity and reduced pressure as the fluid enters the impeller.  If the fluid static pressure drops below the vapor pressure, gas bubbles form and later collapse as the fluid flows along the impeller vanes.  These vapor bubbles can have a significant effect on the head produced by the pump.

    It is important to note that fluid temperature also plays an important part in pump cavitation.  Obviously, the fluid vapor pressure will vary with temperature.  The fluid temperature will also vary with pump efficiency.  Temperature rise due to pump efficiency is not significant in the high to mid-range flow rates, however, can be very significant at low flow rates.  This is why pump NPSHR values are not given at low flow conditions.

    Figure 1

    Figure 1 – Pump Performance Curve for a Range of Impeller Diameters

    Another important factor in pump cavitation is the fluid velocity.  Fluid entering a pump will continually increase in velocity as it passes to the impeller eye.  This increase in velocity causes a drop in the fluid static pressure and is analogous to lift on an airfoil.   At high to mid-range flow rates the incoming fluid velocity and the impeller rotational velocity are compatible and contributes to stable flow through the pump.  However at low flow rates the entering velocity is well below the rotational velocity and may cause the fluid to “recirculation” at the impeller inlet.  Fluid recirculation is another form of pump cavitation.   This is another reason why NPSHR is not given at low flow rates.

    NPSHR Testing

    Understanding how NPSHR tests are conducted and how the impeller diameter influences the produced head will help eliminate confusion and possible errors.  Pump manufacturers determine the characteristic shape of the NPSHR curve for each impeller through carefully controlled shop testing, hydraulic modeling and computer simulation.  Hydraulic Institute Standard 1.6 gives strict guidelines for conducting shop testing and is used by most pump manufactures.  Pumps are normally connected to closed-loop piping circuit where water flows from a suction tank (or sump) through the pump and then back to the tank. The discharge flow rate, temperature and pressure are carefully measured and controlled throughout the test.    Basically the test is conducted at a fixed flow rate and speed while the suction pressure is reduced.   By reducing the suction pressure a point is reached when the water begins to vaporize thus causing the pump to cavitate.  The characteristic “cavitation” point is the flow rate that is exhibited by a small drop in head.  The test is conducted again at another fixed flow rate and again the resulting suction pressure and flow rate value are recorded at the “cavitation” point.  Once the series of tests are completed, a smooth line is drawn through the recorded data and plotted.   Figure 2 illustrates a typical series of test results and the resulting NPSHR curve.

    Figure 2

    Figure 2 – NPSHR Test Curve

    A pump cavitation point can be difficult to define.  The formation of vapor bubbles is a gradual process, starting slowly and increasing with flow rate.  The API-610 defines the cavitation point as a three percent drop in head.  This is not to say that pump cavitation does not occur at smaller values, it is just difficult to accurately measure at smaller values.     To obtain a single point it is necessary to run a pump for a period of time and allow the testing circuit to stabilize to the reducing suction pressure.  Remember, vapor bubbles are forming and instruments need time to react to the fluid dynamics.

    Impeller Diameter and Head Relationship

    Larger pump impellers produce greater values of head for a given speed.   This is because the head is proportional to the tip speed.  The relationship of head to tip speed can be approximated by Equation 1.

    Equation 1

    (Eq. 1)

    Tip velocity can also be related to impeller diameter and rotating speed by Equation 2.

    Equation 2

    (Eq. 2)

    From Equations 1 and 2 it can be seen that changes in impeller diameter will have a direct effect on the pump head.  For example, reducing the impeller diameter will lower the pump head by a factor of four.   Since the cavitation point is identified by a three percent drop in pump head, it is logical that any change in impeller diameter will have a direct effect on the NPSHR value.  For this reason, most pump manufacturers provide a single NPSHR curve for a given impeller diameter.  Figures 3 and 4 are typical pump performance curves for a range of impeller diameters.  Note that a separate NPSHR curve is given for each diameter.

    Figure 3

    Figure 3 – Typical Pump Performance Curve for a Range of Diameters

    Figure 4

    Figure 4 – Optional Pump Performance Curve for a Range of Diameters

    Conclusions

    The following conclusions can be reached from the previous discussion.

    1. Each impeller will have a characteristic NPSHR curve.  It will depend on many design factors including the diameter.
    2. At a given flow rate, the NPSHR increases as the impeller diameter is reduced.
    3. The NPSHR is never tested at the shut-off point.  The fluid temperature continually rises as the flow rates decreases.  This prevents the system from stabilizing sufficiently to obtain accurate measurements.
    4. Pumps may cavitate at low flow rates due to recirculation of fluid at the impeller eye.
    5. The shape of the NPSHR curve is a U-shape.  There is a slight rise in values as the flow is reduced and again at higher values.  The NPSHR is lowest in the mid-range values.

    By: Joe Honeywell

    Legend

    A          Conversion constant = 720 ft/sec (600 m/s)

    D          Impeller diameter, inches (cm)

    H          Total pump head, ft (m)

    g          Gravitational constant, 32.17 ft/sec2 (9.81 m/s2)

    n          Rotational speed, rev/min

    V          Impeller tip velocity, ft/sec (m/s)

    References

    1. American Petroleum Institute Standard 610, Centrifugal Pumps for Petroleum, Petrochemical and Natural Gas Industries, 10th Ed.
    2. Hydraulic Institute Standard 1.6, Centrifugal Pump Tests, 2000
    3. Terry Henshaw, Pumps and Systems, May 2009

  • Important Aspects of Centrifugal Compressor Testing – Part 2

    This is the final part of a two part Tip of the Month (TOTM) series on important aspects related to centrifugal compressor performance testing.  The first part dealt with the review of the testing procedure presented in ASME PTC-10 (also referred to as the Code), selection criteria for test gases and factors to consider in a performance testing.  This TOTM will review the basic assumptions and performance relationships required for an accurate test.  Also discussed are three important principles: volume ratio, Machine Mach Number and Machine Reynolds Number, which also influence the accuracy of the test results.

    Introduction
    The Code recognizes that the actual testing conditions and the specified design conditions may not be identical.  Basic assumptions are made so that test results can be compared to the original design or some other baseline datum.  For example, a compressor can have a different efficiency depending on where it is operating on a head-flow curve.  However, if the gas composition and operating condition are not the same as the original design, then how accurate are the results?  This question will be discussed below.

    There are other important parameters utilized by the Code to analyze compressor performance.  The first two are called flow coefficient and work coefficient.  These are dimensionless parameters that are useful in the interpretation of test results, especially when comparing the test results to the original design or some other datum.  Three more important parameters are called volume ratio, Machine Mach Number, and Machine Reynolds Number. These parameters assure that the aerodynamic properties of a compressor are maintained whenever test gases or alternate operating conditions are used.  In addition, they establish limits on the operating range and help correct head and efficiency for friction losses.   Each parameter will be briefly discussed.

    Dimensionless Parameters
    Most likely the actual testing conditions and specified design conditions are not identical.  To compensate for the differences, the Code utilizes dimensionless parameters called flow coefficient, work coefficient and total work coefficient.   The Code also makes assumptions regarding each coefficient and their equivalency at test and specified conditions.  Table 1 lists the Code’s principle parameters and the assumptions used to convert test data into values at specified design conditions.

    Changes in compressor performance can be determined whenever the speed fluctuates by simply utilizing the affinity laws.  If the compressor flow, head and efficiency characteristics are known at a given speed, then merely applying the affinity laws at an alternate speed will produce a new curve representing the compressor performance at that speed.   This is the same concept behind head and flow coefficients.  In essence, the flow coefficient represents the “normalized flow rate” of the compressor at any speed.  Similarly, the work coefficient and total work coefficient represents the “normalized head” of the compressor at any speed.  The affinity laws also imply that the efficiency represented at the two equivalent conditions will remain the same.  These properties play a major role in shop and field testing of centrifugal compressors.

    Table 1
    Dimensionless Parameter Assumptions

    Dimensionless
    Parameter1    		 Description Mathematical Description1
    Flow coefficient Flow coefficient of the test gas and specified gas are equal using ideal and real gas methods. Equation
    Work input coefficient – enthalpy method Work input coefficients of the test gas and specified gas are equal.  Ideal or real gas laws apply. Equation
    Work input coefficient – isentropic or polytropic methods Work input coefficient of the test gas is corrected for the Machine Reynolds Number to obtain the specified work input coefficient.  Ideal or real gas laws apply. Equation
    Efficiency –isentropic or polytropic methods The efficiency at the test operating condition is corrected by the Machine Reynolds Number to obtain the specified operating condition. Equation
    Total work input coefficient – heat balance or shaft balance methods The total work input coefficient is equal for test and specified gases. Equation

    NOTE:
    1.     See ASME PTC-10 for complete mathematical description of the coefficients.

    Basic Performance Relationships

    Equations
    Equations

    The Code recognizes three methods of determining compressor work (also called head).  The first is the enthalpy method and is defined by Equation 2.  It represents the difference in the inlet and discharge enthalpy, and results in theactual work supplied to the gas.  The next method of determining work is by the isentropic method.  This method only determines the ideal compressor work and may be calculated utilizing Equation 3 and 4.  The last relationship for determining compressor work is the polytropic method.  Only the ideal work is found by this method and may be calculated using Equations 5 and 6.  All three methods are commonly used by compressor users and manufacturers.

    Volume Ratio
    The volume ratio is an important aerodynamic parameter.  It maintains similar flow conditions as gas properties and operating conditions change.  The best way to describe volume ratio is to consider a multi-stage compressor.  The mass of gas entering the first impeller must equal the mass entering other impellers.  However, the actual gas volume entering the first stage is not the same for other impellers.  The gas is compressed and heated, which results in a reduction of volume.  If the gas properties and operating conditions of the test gas are different from the specified gas, then the volume entering and leaving each stage will also be different.  Therefore, to duplicate the aerodynamic performance of a compressor at the specified design condition it is important to simulate the equivalent flow of gas through the impellers by carefully matching the volume ratio.

    A centrifugal compressor performance test is frequently performed with a gas other than the specified gas.  In addition, the compressor may operate at conditions other than the original design.  To assure an accurate performance test that simulates the original design, the volume ratio of the specified gas must match the volume ratio of the test gas at the respective operating conditions.  Equations 1-6 can be used to determine the conditions that match the test and specified volume ratio.  The Code sets limits on deviations of the test gas properties and operating conditions, which is found in Table 2 of Part 1.

    Seven variables define the volume ratio relationship between a test gas and the specified gas.  The variables and the influence each has to increase or decrease the volume ratio is shown in Table 2.  For example, if the k-value of the test gas is greater than the specified gas, the volume ratio will decrease.  Similarly, if the test gas suction temperature is less then the volume ratio will increase.  Also note another important fact, and that is changes in the suction pressure of the test gas have no effect on volume ratio.

    Table 2 – Variable Influence on Volume Ratio

    Variable Change Volume Ratio Change Volume Ratio
    Head Increase Increase Decrease Decrease
    Molecular Weight Increase Increase Decrease Decrease
    Suction Temperature Increase Decrease Decrease Increase
    Compressibility Increase Decrease Decrease Increase
    k-value Increase Decrease Decrease Increase
    Speed Increase Increase Decrease Decrease
    Suction pressure Increase No change Decrease No change

    As previously mentioned, the volume ratio of the specified gas must match the volume ratio of the test gas.  So if each of the physical properties of the test gas can change the volume ratio, what can be done so that the two volume ratios match?  A common practice is to change the test speed to compensate for the mismatch of volume ratios.  This practice is illustrated in Figure 1.  Note how the compressor speed is decreased so that the volume ratio changes imposed by other variables add up to zero.

    Figure 1

    In summary, the operating conditions and physical properties of a performance test should be carefully examined.  It is critical that the test gas volume ratio closely match the volume ratio of the specified gas.  The closer the test gas volume ratio is to the specified gas, the more accurate are the performance test results.

    Mach Number
    The Mach number influences the maximum amount of gas that can be compressed for a given impeller speed.  The limiting flow is known as stonewall (also called choke flow) and is typically found on the compressor characteristic head-flow curve at maximum flow condition for a given speed.  As the gas flow rate increases so does the velocity within the compressor’s internal flow path until it approaches the fluid acoustic velocity, thus limiting the flow.  Therefore, gas velocities that approach a Mach number of one indicate choke flow inside the compressor.

    The Code defines a term called the Machine Mach Number which is the ratio of the outlet blade tip velocity of the first stage impeller to the acoustic velocity at inlet conditions.  The Code also sets allowable limits on the deviation between the specified and test gas Machine Mach Numbers.  This helps assure the accuracy of the performance test.  When shop testing a compressor, the Machine Mach Number at the operating condition is calculated and compared to the difference of the specified gas and test gas.  See Figure 2 for allowable deviation limits.  If the value exceeds the permitted deviation the test gas operating conditions may need adjusting to comply with to these limits.

    Figure 2

    Figure 2 – Allowable Deviations for Machine Mach Number

    Reynolds Number
    The effect that the Reynolds Number has on a compressor is similar to the effect it has on pipes.  The gas flowing through the internal passages of a compressor produce friction and energy loss which influences the machine efficiency. For centrifugal compressors, the Code defines a term called the Machine Reynolds Number and places limits on the allowable values during a performance test and is defined by Equation 8.  If the Machine Reynolds Number for the test condition and specified condition differs then a correction factor is applied to the test efficiency and head values. See Equation 9 for the correction factor.
    Equation
    The allowable Machine Reynolds Number departure limits between the test gas and specified gas are given in Figure 3.

    By Joe Honeywell

    Figure 3

    Figure 3 – Allowable Machine Reynolds Number Departures
    References

    1. ASME PTC-10, “Performance test Code on Compressors and Exhausters”, 1997
    2. Short Course “Centrifugal Compressors 201”, Colby, G.M., et al. 38th Turbomachinery Symposium, 2009.

     

    Nomenclature
    Nomenclature

  • Important Aspects of Centrifugal Compressor Testing-Part 1

    Every centrifugal compressor, whether it is new or has been in service for many years will most likely be tested to verify its thermodynamic performance.  For a new machine the testing may be conducted in the manufacturer’s facility under strict controlled conditions or in the field at actual operating conditions.  Older compressors that have been placed in service after maintenance or have been operating for an extended period of time may require testing to verify the efficiency and normal operation.  This TOTM will review ASME PTC-10 (also referred to as the Code) testing procedure and other topics that contribute to an accurate centrifugal compressor test results.

    This two-part series will review the salient aspects of a performance test.  Part 1 will review the thermodynamic performance test objectives established in the Code as well as other factors to consider in a testing procedure.  While this code is primarily applicable to shop testing it can also apply to field testing.  Part 2 will review the Code assumptions and basic performance relationships.  It will also examine the three important principles that influence the operating conditions and ultimately influence the accuracy of the performance test.  They are volume ratio, Machine Mach Number and Machine Reynolds Number.

    Introduction

    The purpose of a performance test is to verify that a centrifugal compressor will perform in accordance with the manufacturer’s design at the operating conditions given in the specifications.  It also provides a method of confirming the shape of the compressor head-flow curve, efficiency, and the maximum and minimum flow limits at various speeds.  Frequently a performance test is conducted under field conditions with the specified gas and operating conditions.  However, if the performance test is conducted in the shop it may not be possible to test the compressor with the specified gas because of safety concerns or testing facility limitations.  Whether the test is conducted in the field or in the shop, proof of the compressor design is recommended and often necessary to demonstrate contractual obligations and mechanical integrity.

    Frequently the gas composition used to confirm a compressor performance differs from the specified gas.  This is often the case regardless if the test is conducted in the field or in the shop.  For field tests, where the gas composition and operating conditions are set by the process, adjustments must be made in the calculations to confirm the compressor design specifications.  Typically, a shop test is conducted with a carefully selected mixture of gases blended together to form a gas that has physical properties that closely resemble the specified gas.  Even with a substitute gas, differences remain which influence the test results.

    The original compressor design places limits on the thermodynamic performance.  The most important of these limits include flow rate, power, temperature, pressure and speed.  There are other design restraints which are not as commonly known but will also influence the compressor performance.  Such factors are volume ratio, Mach number and Reynolds number.  These limits were incorporated in the compressor design and are influenced by gas properties, operating conditions and the mechanical design.  To verify the design and operating limits for a compressor, it is necessary to test the machine.  For new machines, these tests are commonly performed in the manufacturer’s facility; however, the testing is sometimes performed in the field.  It may also be helpful to periodically test a compressor to trend the machine performance.  Testing conducted during commissioning will establish a baseline of performance.  Periodic field tests are often conducted to verify the overall performance and signal changes that may predict mechanical damage, internal fouling, or other deteriorating conditions.

    Summary of ASME PTC-10 – Performance Test Code
    The procedure presented in the Code provides a method of verifying the thermodynamic performance of centrifugal and axial compressors.  This code offers two types of tests which are based on the deviation between test and specified conditions.  A detail procedure is given for calculating and correcting results for differences in gas properties and test conditions.  The following briefly describes the guiding principles of the Code.

    • Type 1 test is conducted with the specified gas at or very near to the specified operating conditions.  While the actual and test operating conditions may differ, the permissible deviations are limited.  See Table 1, 2 and 3 for deviation limits of testing variables of a Type 1 test.
    • Type 2 test is conducted with either the specified gas or a substitute gas.  The test operating conditions will often differ significantly from the specified conditions.  The operating conditions are subject to limitations based on the compressor aerodynamic design.  See Table 2 and 3 for permissible deviations of operating conditions and test gas properties.
    • The calculation method of a Type 1 and Type 2 test may conform to either Ideal or Real Gas laws.  Physical property limitations are given in Table 3 if Ideal Gas Law methodology is used.


    Tables 1 and 2
    Table 3

    The Code also gives procedures for calculating and correcting test results for difference between the test conditions and specified conditions.  It also gives recommendations for accurate testing including compressor testing schemes, instrumentation, piping configuration and test value uncertainties.  The following summarizes each topic.

    • Thermodynamic calculations may utilize either enthalpy, isentropic or polytropic methods.  The Code provides equations and examples for determining compressor work (also referred to as head), gas and overall efficiencies, gas and shaft power, and parasitic losses.
    • The Code gives a correction procedure for test gases and test operating conditions that deviated from the specified operating conditions.
    • Compressor testing may be open-loop or closed-loop; however, the test results are subject to limits that may give preference to the test arrangement.
    • Instrumentation methods and measurement uncertainties (refer to PTC-19 series of standards) used to test compressors are given.
    • Recommendations for piping layout are also included.

    Test Gas Selection
    There are many gases commonly used to test compressors.  They are selected based on physical properties, toxicity, flammability and environmental concerns.  See Table 4 for a list of the most frequently used gases.  The manufacturers will sometimes blend the various gases to match the equivalency criteria and the test facilities limitations.  Following are recommendations to consider when selecting a test gas.

    • The compressor mechanical design may impose constraints on the test.  Consider the machine rotor dynamics, overspeed, maximum temperature and power limitations when selecting a test gas.
    • Avoid flow rate mismatch of impellers.  The volume ratio equivalency is the most important parameter in selecting a test gas.  This may also place limitations on the operating conditions.  More on this subject in Part 2. of this series.
    • The test gas molecular weight should closely match the molecular weight of the specified gas.
    • The test gas k-value should closely match the specified gas to duplicate the Machine Mach Number.  If this is not practical then the test k-value should be slightly greater to avoid possible stonewall limitations.
    • Select a test gas with minimum Reynolds Number deviation from the specified gas.  This will minimize the efficiency and head correction factors.  This is especially important for machines with a low Machine Reynolds Number.

    Table 4
    Typical Test Gas Mediums (1)

    Test Gas Molecular Weight k-Value (2) Absolute Viscosity-cP (2)
    Helium 4.003 1.667 0.0194
    Nitrogen 28.014 1.401 0.0174
    Air (dry) 28.959 1.401 0.0175
    Carbon Dioxide 44.010 1.299 0.0145
    R134a 102.0 1.124 0.0114
    Natural Gas (4) 17.1  (3) 1.26  (3) 0.010  (3)
    Propane 44.096 1.141 0.00789

    Note:

    1. From “Compressors 201” course at Turbomachinery Conference, 2009
    2. Values from National Institute of Standards and Technology and Gas Processors Suppliers Association
    3. Values at 60 0F (15.6 C) and 14.696 psia (101.3 kPa)
    4. Gas composition and physical properties varies with local utility

    Test Objectives
    The following are some factors to consider as part of the performance test procedure.

    • API 617 requires a minimum of five test points to be taken at the operating speed to demonstrate the surge point, stonewall, required operating point and two alternate points.  The user may optionally request additional test points to verify compressor performance at alternate speeds.  For example, extra data points may be needed to verify the surge line or critical process operating conditions for variable speed machines.
    • The test may be performed as a Type 1 or Type 2 test.  Type 1 is normally more accurate and is typically reserved when test conditions can be made to closely match the specified operating conditions.  A Type 2 test is typically a shop test utilizing a substitute gas.
    • If a Type 2 test is recommended, the test gas may be a pure gas such as those listed in Table 4, or a mixture of gases.  The composition of the test gas should be agreed upon before testing.  In addition, the composition of the test gas should be sampled before, during and after the test.  Some gas mixtures tend to stratify and give erroneous results.
    • The physical properties of the test gas are critical to the outcome especially if it is a mixture of selected gases.  An agreement on the physical properties is recommended.
    • Normally an agreement is made as to the “equation of state” used to calculate the results of the test.  Not all EOS programs give the same results, nor is there industry agreement as to which method is best.
    • Discuss the specific driver used in the test.  Will a shop driver or the specified driver be used?  Will the driver be fixed or variable speed?  If it is variable speed, will it be motor, gas turbine or steam turbine?
    • If a gear is part of the test, will it be manufacturer or user supplied?  Is the efficiency of the gear known?  Tests can be performed to verify gear efficiency.
    • Will the gas be cooled with a water-cooled or air-cooled exchanger?  Is there temperature limitations on the coolant used in the test?
    • Is the allowable working pressure of equipment and piping systems adequate for the test?  Will a pressure safety valve be needed to protect the system and is it properly sized?
    • An agreement on how the input power will be measured is important.  Options include, heat balance, calibrated driver, dynamometer, and torque meter.  Review the specific method of measuring input power with the manufacturer.
    • A piping and instrument schematic is recommended.  The drawing should show details of the test loop including the placement of major equipment, number and location of instruments, and piping size.  This is especially important for compressors with multiple sections, inlet sidestream, or back-to-back configuration.
    • Before proceeding with a performance test a written procedure is recommended that outlines how the test will be conducted.  The procedure should clearly convey the scope of the test, the responsibilities of each party, test piping and instrument arrangement, measurement methods, uncertainty limits, calibration, taking of test data and how to interpret results, and acceptance criteria.

    By Joe Honeywell

    References

    1. ASME PTC-10, “Performance test Code on Compressors and Exhausters”, 1997.
    2. API Standard 617, “Centrifugal compressors for Petroleum, Chemical, and Gas Services Industries”, 1995.
    3. Kurz, R., Brun, K, & Legrand, D.D., “Field Performance Testing of Gas Turbine Driven Compressor Sets”, Proceeding of the 28th Turbomachinery Symposium, 1999.
    4. Short Course “Centrifugal Compressors 201”, Colby, G.M., et al. 38th Turbomachinery Symposium, 2009.
    5. National Institute of Standards and Technology, Web Site for Properties of Fluids.

  • The Sensitivity of k-Values on Compressor Performance

    One of the most important physical properties of a gas is the ratio of specific heats.  It is used in the design and evaluation of many processes.  For compressors, it is used in the design of components and determination of the overall performance of the machine.  Engineers are frequently asked to evaluate a compressor performance utilizing traditional equations of head, power and discharge temperature.  While these simplified equations may not give exact results, they give useful information needed to troubleshoot a machine, predict operating conditions, or a long-term trend analysis.  The accuracy of the performance information will depend on the proper selection of the ratio of specific heats.  This Tip of the Month (TOTM) will investigate the application of the ratio of specific heats to compressors, its sensitivity to the determination of machine performance and give recommendations for improved accuracy.

    Background of k-value

    The ratio of specific heats is a physical property of pure gases and gas mixtures and is known by many other names including: adiabatic exponent, isentropic exponent, and k-value.   It is used to define basic gas processes including adiabatic and polytropic compression.  It also appears in many of the traditional equations commonly used to determine a compressor head, gas discharge temperature, gas power, and polytropic exponent.  The k-value also influences the operating speed of a compressor, but we will simplify the present analysis by deleting speed from our evaluation.  The following commonly used compressor performance equations show how the k-value is utilized in the design and evaluation of compressors.

    Equations

    Note:    The actual Z-value will vary from the suction to discharge conditions.  ZS is sometimes replaced with ZAVE to approximate the variations in compressibility value [1, 5]. See the nomenclature at the end of this TOTM.

    The above equations are written in terms of the adiabatic process with the exception of Equation 5, which refers to the polytropic process.  Both compression processes are similar and will give the same actual results.  The adiabatic and polytropic methods are extensively used by manufacturers to design compressors, and make use of k-values to calculate their performance.  However, as will be seen, the effect of the k-value and the calculated results will influence both compression processes alike.  For simplicity, this Tip of the Month will use the adiabatic process.
    It can be seen from Equations 1-5 that the k-value has an effect on a compressor head, temperature, power, and polytropic exponent.  In order to determine how small changes in the k-value can influence a compressor performance, let us first define the k-value of a pure gas.  The thermodynamic definition of a gas k-value is given by Equation 6.  It shows the relationship to the specific heat at constant volume, CV and specific heat at constant pressure, CP.  Both values vary with temperature and pressure.

    Equation

    For a pure gas there are many references that give CP and CV values at various conditions.  One useful source is National Institute of Standards and Technology.  Their website is http://webbook.nist.gov/chemistry/fluid/

    The method of determining the k-value for gas mixtures is more complex.  The major difference is that a gas mixture does not behave as any one of its components but as an “equivalent” gas.  Therefore, to determine the k-value of the mixture, we must know the mole fraction of each component, Yi and the molar specific heat at constant pressure for each component, M CPi.   Equation 7 can be used to determine the k-value of an ideal gas mixture [1, 5].  Real gases may deviate from the calculated value.

    Equation

    While Equations 1-7 are applicable for manual calculations methods, it is important to note that process simulation packages determine the compressor head and discharge temperature utilizing equations of state.  The results are the same but the methods are very different.

    K-value Sensitivity Analysis

    In the compression process the temperature and pressure of the process gas both increase.  Not knowing what k-value to select for evaluating the compression process can lead to errors.  For example, a typical propane compressor may have a k-value at suction conditions of 1.195.  At the compressor discharge conditions the k-value is 1.254.  The difference in the two values varies by 4.94 percent and can have a significant influence in the performance evaluation.  The following example illustrates how minor changes in the k-value can influence the calculated compressor head, temperature, power and the polytropic coefficient.

    Example 1: A natural gas compressor is operating at the conditions given below.  Only the k-value is varied from 1.20 to 1.28, all other given parameters remain constant.   Figure 1 illustrates how the “apparent” performance of a compressor can change by varying the k-value.

    Figure 1

    It can be seen from Figure 1 that the discharge temperature deviated over 18.8 percent by only changing the k-value by 6.7 percent.  In this case the k-value varied from a value of 1.20 to 1.28; which is the typical range for natural gas.  Similarly, the power changed by 2.5 percent, polytropic exponent by 9.5 percent, and adiabatic head by 2.5 percent for the same variation of the k-value.  The changes in compressor performance described in Figure 1 can be much larger depending on the gas composition and the operating temperature and pressure.

    Corrected k-Value Recommendations

    The k-value sensitivity for a single-stage machine is not nearly the problem as a multi-stage compressor.  For a single-stage machine, the pressure ratio is typically lower and the temperature and pressure changes are less.  As a result the changes in k-value are not as great and accurate results can be obtained by approximating the k-value at the suction conditions.  However, for multi-stage machines, where the pressure and temperature ratios are higher, the k-value sensitivity is more of a factor in evaluating compressor performance. Most compressor manufacturers calculate the k-value for each stage of compression and avoid errors introduced by utilizing an overall k-value. Without their software, we are left with a corrected k-value by empirical methods.

    There are many useful approximations that will correct for changes in the k-value as the process gas passes through the compressor.  Normally the k-value will decrease during compression but not always.  Utilizing the suction conditions to estimate the k-value will generally give higher values of temperature, heat, and power.  The polytropic exponent generally decreases as the adiabatic exponent decreases.  To avoid potential discrepancies, a k-value correct may be warranted.  The following are six methods of determining the corrected k-value commonly used in industry.

    1. At TS and PS:  This method determines the k-value at suction conditions and is useful for single stage compressors or applications where there is little change in the k-value.  The k-value is easy to determine and tends to overestimate results, especially if the temperature and pressure do not change significantly.  For greater values of RP the results may become so conservative they become useless.k = ks at suction conditions
    2. At TD and PD:  This method determines the k-value at discharge conditions.  The k-value is less conservative and tends to underestimate results.  The k-value may be difficult to determine, especially if the discharge temperature is unknown.    For gases with highly variable k-values, an iterative solution may be required to estimate the discharge temperature and corrected k-value.k = kD at discharge conditions
    3. At TAVE and PSTD [5]:  This method utilizes the average operating temperature at standard pressure and determines the k-value.  Numerous reference books propose this method.  Errors are introduced because the k-value at standard pressure may not accurately represent values at the operating pressure.k = at average operating temperature and standard pressure
    4. At TAVE and PAVE:  This method utilizes the k-value at the average operating temperature and pressure.k = at average operating temperature and pressure
    5. Average value [1, 3]:  This empirical method takes the average k-value at compressor inlet conditions and outlet conditions.  Utilizing the average k-value will result in performance values that are closer to the actual performance of the compressor.Equation
    6. Weighted average value [4]: This empirical method takes the weighted average of the suction, mid-point and discharge conditions.  Note that the mid-pressure is determined by equivalent pressure ratios, Equation.  The mid-temperature is estimated from the mid-pressure.  This method considers the staged k-value to change with diverging isentropic and pressure lines shown on a Mollier chart.
    Equation

    Example 2 illustrates the various methods used to determine corrected k-values given above.  It also compares the range of the resulting values.

    Example 2: A propane compressor is operating at the given conditions shown below.  Table 1 lists the k-values attributed to various operating and reference conditions [6].

    Table 1

    Summary

    This Tip of the Month has defined the physical property of process gases called the k-value or ratio of specific heats.  It has shown that small changes in the k-value can have a significant effect on the calculated values of head, power, gas discharge temperature, and polytropic exponent.  Recommendations were also given to improve the accuracy by utilizing different k-value methods.

    To learn more about similar cases, we suggest attending our ME44 (Overview of Pumps and Compressors in Oil and Gas Facilities), ME46 (Compressor Systems – Mechanical Design and Specification), PL4 (Fundamental Pipeline Engineering), G40 (Process/Facility Fundamentals), G4 (Gas Conditioning and Processing), and PF4 (Oil Production and Processing Facilities) courses.

    By: Joe Honeywell

    Nomenclature

    References

    1. Ronald P Lapina, Estimating Centrifugal Compressor Performance, Vol. 1, Gulf Publishing, 1982.
    2. John M. Campbell, Gas Conditioning and Processing, Vol. 2, John M. Campbell & Co., 8th Edition.
    3. Elliott Compressor Refresher Course,
    4. John M. Schultz, “The Polytropic Analysis of Centrifugal Compressors”, Journal of Engineering for Power, January 1962.
    5. Gas Processor Suppliers Association, Engineering Data Book, Section 13, 2004
    6. National Institute of Standards and Technology, Web Site for Properties of Propane, Fluid Data.
    7. ASME PTC10-1997, Performance Test Codes, “Compressors and Exhausters”, R2003

  • Friction Pressure Drop Calculation

    Introduction

    Engineers are frequently asked to calculate the fluid pressure drop in a piping system. Many software programs are available for solving complicated hydraulic problems; however’ they can be complex and difficult to use. In addition, there are many tables or shortcut methods that give adequate answers but they usually apply to predefined conditions which are sometimes misleading or less accurate. This “Tip of the Month” discusses a method of calculating friction pressure losses for liquid lines. A spreadsheet is presented that gives friction losses based on this method.

    Background Information

    Equation 1 is known as the Darcy-Weisbach (sometimes called the Darcy) equation and has been used by engineers for over 100 years to calculate fluid flow pressure loss in pipe. This equation is derived by dimensional analysis and relates the various parameters that contribute to the friction loss. A correction factor, called the Moody friction factor, is included which compensate theoretical results with the experimental results.

    Equation 1

    Where:

    hL = Head loss due to friction, m [ft]
    f = Moody friction factor
    L = Pipe length, m [ft]
    V = Velocity, m/s [ft/sec]
    g = Gravitational acceleration, 9.81 m/sec2 [32.2 ft/sec2]
    D = Inside diameter, m [ft]

    The task of determining the friction factor can be difficult due to the many variables that influence flow behavior. For example, the friction factor is significantly different if the fluid flow exhibits Newtonian or non-Newtonian behavior, or if the flow is laminar or turbulent. Other variables that influence the friction factor are properties of the pipe represented by absolute roughness and inside diameter, and fluid parameters such as flow rate, viscosity and density.

    The Moody diagram given in Figure 1 is a classical representation of the fluid behavior of Newtonian fluids and is used throughout industry to predict fluid flow losses. It graphically represents the various factors used to determine the friction factor. For example, fluids with a Reynolds number of 2000 and less, the flow behavior is considered a stable laminar fluid, and the friction factor is only dependent on the Reynolds number. The friction factor for the Laminar Zone is represented by Equation 2. Fluids with a Reynolds number between 2000 and 4000 are considered unstable and can exhibit either laminar or turbulent behavior. This region commonly referred to as the Critical Zone, and the friction factor can be difficult to accurately predict. Judgment should be used if accurate predictions of fluid loss are required in this region. Either Equation 2 or 3 are commonly used in the Critical Zone. Beyond 4000, the fluid is considered turbulent and the friction factor is dependent on the Reynolds number and relative roughness. For Reynolds numbers beyond 4000, the Moody diagram identifies two regions, Transition Zone and Completely Turbulent Zone. The friction factor represented in this region is given by Equation 3.

    Graph 1

    Figure 1. Moody Friction Factor Diagram

    Equations

    Where:

    Re = Reynolds Number
    V = Fluid velocity, m/s [ft/sec]
    D = Inside diameter, m [ft]
    e = absolute pipe roughness, m [ft]
    ? = Fluid density, kg/m3 [lbm/ft3]
    µ = Fluid viscosity, kg/(m-s) [lbm/(ft-sec)]

    The Method

    The Colebrook formula, Equation 3, is used throughout industry and accurately represents the Transition and Turbulent flow regions of the Moody Diagram. However, this implicit equation is difficult to solve by manual methods. Typically an iterative method is used to solve the Colebrook equation. One method of solving this equation is with numerical analysis technique called Newton-Raphson’s1 Method. This successive approximation approach is represented by Equation 5, and involves 1) the Colebrook formula, 2) the first derivative of the Colebrook formula and 3) an initial guess. Since the Colebrook formula is a convergent equation, the solution is usually determined with less than four iterations.

    Equation 5

    Where:

    fn = nth iteration friction factor
    fn+1 = (n+1)th iteration friction factor
    g(fn) = Colebrook equation
    g'(fn) = First derivative of Colebrook equation

    A macro that solves the Colebrook formula is given in this spreadsheet. It is easily adapted to programmable calculators. The iterative method assumes that the following input variables are available:

    Pipe inside diameter – mm [in]
    Pipe length – m [ft]
    Absolute roughness – m [ft]
    Absolute viscosity – cP
    Fluid relative density
    Fluid flowrate – m3/h [gpm]

    Example Problem

    The macro begins with inputting the variables needed to solve for the Moody friction factor. Next, the macro determines the Reynolds Number. If the Reynolds value is below 2000 the flow is considered laminar and a simplified friction formula shown in Equation 2 is used. Above 2000 the flow is considered turbulent and the Colebrook formula is used. Finally, the Moody friction factor is determined and combined with the Darcy formula, Equation 1, to determine the fluid friction losses.

    Results

    Numerous results were checked against values given in “Cameron Hydraulic Data Book”2 and found to vary by less than one percent. A term called “Delta-F” is also given in the spreadsheet which gives an indication of the variance in the Colebrook equation and the calculated value. Values of Delta-F less then 0.05 indicates an accuracy of three or more decimal places.

    Alternate Method:

    An alternate method of determining the friction factor is given by Chen3. His method of calculating the friction factor is explicit and does not require iterations to solve. This method has been by studied by Gregory and Fogarasi4, and found to give satisfactory values compared to the Colebrook equation. For those interested in this alternate approach, see Equation 6.

    Equations 6 and 7

    Where:

    f = Fanning friction factor (1/4 of Moody friction factor)
    D = Inside diameter, m [ft]
    e = absolute pipe roughness, m [ft]
    Re = Reynolds Number

    To learn more about friction factor and its impact on piping and pipeline calculation, design and surveillance, refer to JMC books and enroll in our ME41, PL4, PL61, and G4 courses.

    By: Joe Honeywell
    Instructor & Consultant

    References:

    1. “Elementary Numerical Analysis”, by S. D. Conte, McGraw-Hill Book Company, 1965, pp 30
    2. “Cameron Hydraulic Data Book”, by Ingersoll-Rand Company, Woodcliff, N. J., 15 ed., pp 3-49 to 3-85
    3. Chen, N.H., An Explicit Equation for Friction Factor in Pipe, Ind. Eng. Chem. Fund., 18, 296,1979
    4. Gregory, G.A. and Fogarasi, F., Alternate to Standard Friction Factor Equation, Oil & Gas Jour. Apr. 1 1985, pp 127.

    Excel Program Input and Output

    ResultsResults

  • Effect of Viscosity on Pump Performance

    Many years ago a generalized procedure was developed by the Hydraulic Institute (HI) for correcting centrifugal pump performance when handling viscous fluids. Their procedure was universally accepted by pump manufactures and industry users. HI has recently issued a revised procedure for predicting pump performance based on new research and field test data [1]. The revised procedure also corrects the shortcomings in the earlier method and has added mathematical formulas for use with computers. This month we will discuss how viscous fluids affect centrifugal pump performance and the changes recently made by the HI.

    It is widely accepted that viscous fluids affect the performance of centrifugal pumps. Since the performance of most centrifugal pumps is determined from water, a procedure is needed to correct the performance curves when pumping viscous fluids. Figure 1 illustrates a typical pump performance curve based on water and how it is affected when pumping viscous fluids. In many applications the difference in water and viscous performance are significant.

    Figure 1

    The previous HI procedure was based on a graphical representation that provided correction factors at four operating points: 60, 80, 100 and 120 percent of the best efficiency point (BEP). Using these correction factors at the four operating points, the user was able to prepare a corrected pump performance chart of flow versus total head (Q–H) and flow versus efficiency (Q-?), when handling viscous fluids. The procedure had limitations but proved to be accurate under a wide range of conditions.

    The recently published HI procedure has many of the same features as the older method. However, many improvements were made to improve the accuracy and correct problems in the earlier procedure. Some of the noteworthy changes made in the latest version are presented below:

    1. HI previously had two separate graphs for determining pump viscous effects, one for pump flow rates less then 100 gpm and another for flow rates over 100 gpm. The two graphs gave conflicting results if the pump operated at 100 gpm. The new procedure has only one procedure regardless of the flowrate.
    2. HI now uses a new basis for determining the correction factors CH, CQ, and C??. The new method is based on a performance factor, called Parameter B, and includes terms for viscosity, speed, flow rate and total head.
      3. Equation 1
      Key 1
    3. The latest HI procedure allows the user to determine viscous performance from graphs or equations. With the previous method only graphical methods were used to calculate correction factors. Figures 2 and 3 illustrate the revised graphical method and gives the flow rate correction factor (CQ), head correction factor (CH) and efficiency correction factor (C?), based on the performance Parameter B given above.
      4. Figure 2
      Figure 2b
    4. The following equations are used with the correction factors to determine the corrected pump performance for viscous fluids.
      5. Key 2
    5. The previous method required the user to find the pump BEP before calculating corrections factors at 60, 80, 100 and 120 percent of BEP. The revised procedure uses a similar approach of finding the pump BEP but corrections factors can be determined, with caution, at any flow rate, not necessarily at 60, 80, 100 and 120 percent of BEP.
    6. The revised HI procedure provides a method of estimating the Net Positive Suction Head Required (NPSHR) when pumping viscous fluids. This procedure is based on pump performance measured with water and corrected for viscous fluids.

    One final comment on HI’s procedure for correcting centrifugal pump performance: The question frequently asked is, “When should a pump performance curve be corrected for fluid viscosity.” The pump manufacturer is the best source to answer this question. They can provide performance curves for any pumping condition including viscous fluids. However, with available computer software, the user should always check how fluid viscosity will affect the pump flow rate, total head and most importantly power.

    The following are some factors derived from the latest addition of the HI procedure for correcting pump performance. They may be helpful when considering when to correct pump water-performance curves for viscous fluids.

    • The performance factor, Parameter B, is a good indicator of how viscosity influences pump performance. Parameter B takes into consideration fluid viscosity, speed, total head and flow rate at BEP.
    • Whenever the performance Parameter B is one or less the fluid has no affect on the head curve, Q-H. Figure 2 shows that the head and flow correction factors are less then three percent when Parameter B equals three and about ten percent when Parameter B equals seven.
    • Viscous fluids have a more dramatic effect on pump power. Figure 3 shows that the efficiency decreases by almost 12 percent when Parameter B equals three and over 30 percent when Parameter B equals seven.

    For more information about pumps refer to Chapter 14, Volume 2,”Gas Conditioning and Processing”. We also suggest attending the JMC courses such as G4, ME62.

    Joe Honeywell
    Instructor/Consultant

    References:

    1. ANSI HI 9.6.7-2004, “Effects of Liquid Viscosity on Rotodynamic (Centrifugal and Vertical) Pump Performance”