SAFE DIAGRAM - A DESIGN AND RELIABILITYTOOL FOR TURBINE BLADING

Murari P. SinghJohn J. Vargo

Donald M. SchifferJames D. Dello

Dresser-Rand, Wellsville, NY, USA

ABSTRACTThe complex field of turbine blade vibration has long been in need of improved  tools to help predict the reliability of blading.

The SAFE interference diagram is presented as such a tool. It presents much more information than the widely used Campbell diagram. In evaluating interferences, the SAFE diagram compares not only the frequencies of exciting harmonies with natural frequencies of blades, but also the shape of these harmonics with the normal mode shapes of a completely bladed disc including packeted blading. Examples are given of cases where the Campbell diagram predicts a dangerous resonance while the SAFE diagram shows that no resonances exist which are supported by experience. Examples are also provided to show when the SAFE diagram can pinpoint what interference is likely to cause the largest blade vibration. Finally, it is shown how a simple change in packeting can be used to change the blade interference and to avoid dangerous operation.

INTRODUCTIONTo meet the objective of designing reliable and trouble free blading in turbomachinery, the blade natural frequency analysis is of utmost importance. This is due to the fact that almost all blade failures can be attributed to metal fatigue, which is caused by variable aerodynamic loads acting on the blading. Resulting dynamic stresses depend on the natural frequency and the mode shape of the blade, the frequency and shape of the exciting force, and the energy dissipating mechanism present in the system included as damping.In the early days of blade design, the natural frequency analysis was based on the assumption of a single beam cantilevered at the blade root. Prohl and Weaver [1] showed that in the case of a packeted assembly where a group of blades are connected by shrouding, many more natural frequencies and modes exist which could not be predicted by a single blade analysis. This is due to the fact that the group of blades behave as a system and are coupled through the shrouding. The magnitude of frequencies and the number of modes depends on the number of blades in the group and the stiffness of the shrouding. This type of analysis has explained many blade failures and helped designers to build more reliable blades.

The next important step in blade analysis was due to the realization that blades are mounted on a disk which can influence the dynamic behavior of blades. Calculations including the disk showed that frequencies can be affected and a new large number of modes exist which cannot be predicted by using a single packet analysis. In reality all blades and the disk constitute one system which may or may not respond to an exciting force.

Singh and Schiffer [2] presented a finite element analysis for a packeted bladed disk assembly. They showed and discussed the features of dynamic behavior

which is different than when the shrouding is 360 degrees.

Traditionally, a blade design is evaluated by using the Campbell Diagram. This paper discusses a method called the SAFE diagram used by the authors' company for design evaluation of packeted bladed disk assemblies. it has been used successfully not only in explaining failures and in developing reliable designs, which otherwise would not have been explained, based on the Campbell diagram.

MODE SHAPE OF A BLADED DISKAND FLUCTUATING FORCEA brief description of the mode shapes of a bladed disk and the nature of fluctuating forces encountered in a turbine is provided to help understand the mechanism of resonance and also to depict graphically the necessary conditions of resonance.

MODE SHAPETo define the dynamics of any mechanical structure, the eigenvalue (natural frequency) and the accompanying eigenvector (mode shape) are calculated. In the case of a bladed disk, the mode shapes have been described as nodal diameters and nodal circles. Singh and Schiffer [2] presented some calculated mode shapes for a packeted bladed disk in tangential vibration. The displacement, for example, of the tip of each blade when plotted with angular position, shows a sinusoidal characteristic which is designated as a diametral pattern. This is valid for either pure tangential or axial vibration of the blades. A bladed disk of 90 blades is shown in Figure 1 vibrating in what is called the first tangential mode. This wheel has 15 packets of blades with 6 blades in each packet. The important feature of the mode shape of Figure 1 is that not only are all six blades within each packet vibrating in phase but all 15 packets are also vibrating in phase. This is called the zero nodal diameter mode shape because there are no phase changes in the vibration of all 90 blades. The 1 nodal diameter mode shape of the same bladed disk is illustrated in Figure 2. There are two phase changes in the bladed disk which are noted by the radial lines forming the 1 nodal diameter mode shape. Notice that the six blades within each of the individual packets are still vibrating in phase with one another in the first tangential mode shape. The second bladed disk shown in Figure 2 has the same mode shape except the radial lines are shifted in location by 90 degrees.

Figure 1. Zero Nodal Diameter Mode.

Figure 2. One Nodal Diameter Mode.

Each nodal diameter mode shape occurs in a pair. The two nodal diameter mode shape of the same bladed disk is shown in Figure 3. Again, all the blades within each packet are still vibrating in phase with each other in the first tangential

mode shape. This trend will continue until each packet is out of phase with its neighbors. In this case where there is an odd number of packets (15) there will be a maximum of 14 phase changes between packets or seven nodal diameters.

Figure 3. Two Nodal Diameter Mode.

The remainder of the first tangential family of mode shapes for a six blade packet is shown in Figure 4. These mode shapes are commonly called in phaseout of phase modes or fixed-supported modes. The significant characteristic of these mode shapes is that the number of phase changes within the packet varies from one to five. The packet with one phase change will form bladed disc nodal diameters eight through 15. The 15 nodal diameter mode shape is shown in Figure 5. The packet with two phase changes will form nodal diameter mode shapes 15 through 23. This trend will continue until the packet with five phase changes forms bladed disk mode shapes 38 through 45. For a bladed disk with 90 blades, the maximum number of nodal diameters is 45, which occurs when a phase change occurs between each blade.

Figure 4. Tangential Mode Shapes For Six Blade Packet.

Figure 5. 15 Nodal Diameter Mode.

When this pattern shows a one sine (sine ) pattern, it is designated as one nodal diameter. If it shows a two sine (sine 2) pattern, it is designated as two nodal diameter. The maximum number of nodal diameters in a bladed disk assembly is half of the number of blades for an even number of blades. For a disk having an odd number of blades, the maximum nodal diameter is half of the total number of blades minus one. Mathematically, the mth mode at one speed can be described as:

m is the natural frequency and in is the number of nodal diameters. The frequency of the blading is dependent on the speed of the turbine, due to the centrifugal stiffening effect. Hence, the natural frequency, m nodal diameter (m), and the turbine speed are required to describe any one mode. Due to the three variables required, it can only be plotted as a surface. A typical plot for a packeted bladed disk is given in Figure 6.

Figure 6. Three Dimensional Representation for Modesof the Packeted Bladed Disc.

 FLUCTUATING FORCESForces imposed on the blade in one complete revolution emanate from the circumferential distortion in the flow. The pressure variation in the fluid flow causes the forces to vary circumferentially. These forces depend on the relative position of blading with respect to nozzles and any other interruptions like struts, etc. In each revolution, however, blades will experience the same force pattern making it a periodic force. The force can be broken into its harmonic contents using Fourier decomposition.

The frequency of the harmonics depends on the speed of the turbine and the number of interruptions in the annulus, e.g., the number of nozzles, and is expressed as:

 Fluctuating forces are due to two primary sources in turbines. The first is known as nozzle passing frequency excitation and the second is running speed harmonic excitation. Nozzle passing frequency excitation is caused by nozzle vane trailing edge wakes as the flow leaves the vanes. A blade passes through nozzle vane wakes a number of times equal to the number of nozzle vanes in one revolution. Examination of equation (3) reveals that the harmonics of forcing can only be described as a surface with three variables, namely frequency, m, a shape M and speed of the turbine N. A typical plot is shown in Figure 7.

Figure 7. Three Dimensional Representation ofForcing in Steam Turbines.

Running speed excitation is caused by circumferential variations in the fluid flow pressure usually due to extraction or exhaust ducts. Other low harmonic sources of excitation which are multiples of running speed, are support struts or diaphragm nonuniformities. For example, six symmetrically located support struts

will give a six times running speed (6 x RS) or a sin 6 excitation to the blades.

CONDITION OF RESONANCE IN A BLADED DISKA mathematical discussion of the condition of resonance of the bladed disk is provided here and some important conclusions are drawn which will help give an understanding of the evolution of the SAFE diagram.

In each revolution of the bladed disk, blades pass through a field of pressure fluctuation due to nozzle or any other interruptions in the flow field. This fluctuation in pressure imposes a time varying force on the blades. In general such forces can be broken into harmonic contents using Fourier analysis. The frequency of any harmonic is an integer multiple of rotational speed and the number of interruptions The nth harmonic of the force can be expressed as:

 

The condition of resonance is defined as when the forces imposed on the blade do positive work. When a force is applied on a moving mass the work is defined as:

The work done by the nth harmonic of the force on the mth nodal diameter mode of the bladed disk in one period can be expressed by using equations (1), (4), and (5) as

The first of the above results defines the condition of resonance for the bladed disk. It can be stated as follows:

The natural frequency wm of the bladed disk must be equal to the frequency of the exciting force

The number of nodal diameters must coincide with the harmonic of the force n.

The second result defines that the work done will be zero when either of the above conditions is not satisfied.

Summarizing the results for a true resonance in the case of a bladed disk, each of the conditions is necessary, but not sufficient. Both of these must exist together for a true resonance to occur.

TRUE RESONANCE IN THE PACKETED BLADEDDISK CAMPBELL DIAGRAM

Determination of probable resonances has been traditionally made by using a Campbell diagram. A Campbell diagram depicts turbine speed on the horizontal axis and frequency on the vertical axis. The natural frequencies and the frequencies of exciting forces are plotted. The coincidence of natural frequencies with the exciting frequencies have been used as the definition of resonance. A typical Campbell diagram and the depicted points where natural frequencies are equal to the exciting frequencies are shown in Figure 8. Experience based on turbines in the field and calculations of response have demonstrated that only a few points among the numerous points in figure 8 show high response levels hence providing concern for the reliability of the blading.

 

Figure 8. Campbell Diagram.

There is a need to eliminate the points which are of no concern in the evaluation of reliability and also to explain why these points are of no concern to designers. This aspect is considered in the following sections.

GRAPHICAL REPRESENTATION OF TRUE RESONANCEThe true resonance in a bladed disk of a turbine is defined in Equation (7) by combining the definition of modes given in Equation (1) and force defined in Equation (4). The three dimensional plots of Equation (1) and Equation (3) have been given in Figure 6 and Figure 7. Graphically, the condition of resonance can be denoted by the intersection of two surfaces shown in Figure 6 and Figure 7. Such a plot is provided in Figure 9. The interference points shown as circles are the true resonance points. Even though a three dimensional plot such as that in Figure 9 helps define conceptually the resonances, two dimensional views are desired to make design decisions. Three planar views of resonance points are given in Figures 10, 11, and 12.

Figure 9. Three Dimensional Interference Representation Of

Resonance Conditions.

Figure 10. Plot of InterferencePoints on Campbell Plane.

Figure 11. Plot of Interference Points on SAFE Plane.

Figure 12. Plan View of Interference Points.

SAFE DIAGRAMA closer look at Figure 10 shows that this is a Campbell diagram and the circles denote true resonances. One can argue here that one must be able to show only the true resonance points as in Figure 10 to avoid a lot of unnecessary concern and anxiety over superfluous frequency coincidence. As it turns out, if one must use a two dimensional depiction of a three dimensional surface, the plot in Figure 11 provides a clearer alternative. Diagrams like this have been in use by the authors' company for many years under the name of the SAFE diagram and have helped in explaining failures, and also successes unexplained based on the traditional Campbell diagram.

The SAFE (Singh's Advanced Frequency Evaluation) diagram differs from the Campbell diagram in that it plots frequency vs nodal diameters (mode shape) rather than frequency vs speed. The concept of mode shapes in turbine blade vibration can best be explained by looking at the blade vibration of an entire bladed disk rather than a single blade or a packet of blades. This can be done for blade vibration in either the axial direction or the tangential direction as explained earlier The first family of tangential modes is plotted on the SAFE diagram of Figure 11. The Campbell diagram of Figure 10 and the SAFE diagram of Figure 11 can be directly compared, since they are plotting identical blade frequencies.

CASE STUDIES USING THE SAFE DIAGRAMThree cases are chosen to demonstrate the use of the SAFE diagram These cases help to show various advantages in making reliability decisions.

Case 1-No Design Change Based on the SAFE DiagramThis case illustrates that by using the bladed disk concept and the SAFE diagram a costly design change was averted.

In the head end stages of steam turbines, the blade heights are relatively short (less than two inches in height) and when their natural frequencies are plotted on a traditional Campbell diagram a situation similar to that depicted in Figure 13 frequently occurs. What is shown in the Campbell diagram is an interference between the first harmonic of nozzle passing frequency (1 x NPF) and the three lowest natural frequency modes of a packeted bladed wheel within the operating speed range. These modes are the first tangential, first axial and axial-rocking modes. Since nozzle passing frequency is one of the prime sources of excitation in turbines and the lowest natural blade frequencies are the most easily excited, a large response would be expected. Frequently design changes have been made in order to avoid these interferences. The design changes made are either in the number of nozzle vanes to change the exciting frequency or in the blade to change the natural frequency. Sometimes, both changes are required to accomplish the goal of avoiding the interferences. These design changes often result in compromising the stage efficiency.

Figure 13. Campbell Diagram ShowingInterference with 1 x NPF for Case 1

When the same frequencies in the Campbell diagram of Figure 13 are plotted in the SAFE diagram of Figure 14, a different result occurs. The SAFE diagram shows that there is no resonance, since only the frequencies match but not the mode shape (nodal diameters) and, therefore, the response will be small. The shape of the 1 x NPF exiting frequency with 78 nozzles occurs at the 72 nodal diameter location of Figure 14 for this case (150 rotating blades), since it is reflected off the 75 nodal diameter line. Detailed description of the construction of the SAFE diagram and the validity of reflecting speed lines and NPP excitation is shown in the literature [3]. Since the number of nozzle vanes is generally about half the number of rotating blades, the shape of the 1 x NPF exciting frequency and the mode shapes of the first three packeted blade frequencies will never match, and a true resonance will not occur. The only occasion when 1 x NPF excitation would be resonant with the lowest three blade frequencies is when the number of nozzle vanes and the number of rotating blades are almost equal. Then both the frequency and mode shape would match.

Figure 14. SAFE Diagram for Case 1.

Stages with Campbell diagrams similar to Figure 13 have occurred-on numerous occasions in the authors' experience. However, the failure rate of these short blade stages has been negligible. Most of the turbines are mechanical drive units that operate over a large speed range, and therefore, the interferences between 1 x NPF and the lowest blade frequencies are inevitable at some time in operation. The only explanation for the excellent reliability of these blades is the SAFE diagram.

CASE 2-THE SAFE DIAGRAM HELPS PINPOINTAN AREA OF CONCERN QUICKLYThis case study is presented to illustrate how use of the SAFE diagram is used to quickly determine if a bladed disk interference exists. Bladed disk natural frequencies and mode shapes are calculated. By plotting these on a SAFE diagram along with rotor speed and exciting frequency data, a determination of interference is made. Finally, blade response will be calculated at the

interference frequency.

The turbine disk consists of 18 packets of six blades each. This disk is a uniform 1.0 in thick with a bore diameter of 6.0 in. Additional information is given below:

Number of Blades - 108Blade Height - 4.0 in

Disk Diameter - 19.0 inNumber of Nozzles - 20Rotor Speed - 8000 rpm

To begin this case study' a traditional Campbell diagram shown in Figure 15 is constructed from data from blade packet natural frequency calculations. For simplicity, only axial modes are considered here. The Campbell diagram is drawn here for comparison with the SAFE diagram which is introduced next. There is an interference in the operating speed range with 1 x NPF and the axial "U" mode. The axial "U" mode is characterized by a "U" shaped bending of the shroud in the axial direction. Nozzle passing frequency is 20 times running speed. That is, there are 20 nozzles and for every revolution a blade "feels" 20 impulses. Maximum deflection is at the shroud midspan and there are two node points along the length of the shroud.

 

Figure 15. Campbell Diagram for Case 2.

Construction of the SAFE diagram begins with calculation of bladed disk natural frequencies and mode shapes. This is an important difference; the source of data for construction of the Campbell diagram is a packet calculation while the SAFE diagram is constructed from data from all entire bladed disk calculation. The computer program TBDYNE [4] has been developed expressly for the purpose of solving bladed disk problems. TBDYNE is a finite element program which solves for the natural frequency and mode shapes of bladed disks. It is also capable of performing a forced response analysis.

The SAFE diagram for this case is shown in Figure 16. Again, for clarity, only axial modes are included. The model used in this case consists of three elements for the disk from the bore to the outer diameter. Therefore, disk flexibility is included in this analysis, which is another advantage of this type of analysis. This feature is evident because the frequencies of lower nodal diameter modes are significantly reduced by inclusion of disk flexibility as shown in Figure 16. Also, axial natural frequencies at higher nodal diameters are lower on the SAFE diagram than on the Campbell diagram, because the flexibility of the disk is included in the analytical model.

Figure 16. SAFE Diagram for Case 2.

The excitation line at 8000 rpm passes through the 20 nodal diameter mode. This identifies the location as a real resonance and design changes should be made to avoid the situation. Either the structure must be changed or the forcing mechanism must be removed. Listed below are changes that could be made to avoid this interference.

Change the number of blades in a packet

Change the number of nozzles

Modify blade flexibility

Modify wheel flexibility

Change shroud flexibility

Move the operating speed range

The first two choices are the most practical and the ones most often used. The next case examined will continue Case 2, and observe the effect of making the necessary changes to avoid resonances.

CASE 3-THE SAFE DIAGRAM WILL HELP MAKE ASIMPLE CHANGE TO AVOID RESONANCENow that the cause of the resonance has been identified, this case will show how changes can be made to avoid resonance. To verify the SAFE diagram and to show improvements in the bladed disk, a forced response analysis is performed. In this analysis the basic equation of motion is solved for displacement. Blade stress can then be calculated from the displacement solution. For this analysis, a unit force of 1.0 lb was applied at the blade tip to the first blade in the packet. Forces were applied to the other blades in the packet using the appropriate phase shift based on the circumferential locations.

A calculation was performed at the 20 nodal diameter interference point of Figure 16 of Case 2. The frequency at this point is 2674.7 Hz. Results are shown in Table 1 for stresses at the blade base.

Table 1. Response of Six Blade Packet Due to 20 ND Excitation at 2674 Hz.

Since the mode in interference is the axial "U" described earlier, the above pattern of stress in not unexpected. Blade stress is relatively low for blades two and five, because these blades are near node locations, and there is relatively little motion in these blades.

Changing the number of blades in a packet will eliminate the interference of Case 2. The SAFE diagram for the same bladed disk of Case 2 is shown in Figure 17, but the number of blades in a packet has been reduced to three. This change has two results, it increases the (frequency of the) axial mode of concern and it "rearranges" the nodal diameter pattern of the axial modes. if a response calculation is performed at the same frequency, stress levels should be greatly reduced. The results of the calculation are shown in Table 2.

Figure 17. SAFE Diagram for Case 3.

Table 2. Response of Three Blade Packet Due to 20 ND Excitation at 2674 Hz

These numbers represent greatly reduced stress levels from those of the six blade packet. Changing the packeted disk natural frequency has eliminated the interference and thus increased the reliability of the wheel design.

CONCLUSIONThe tools to calculate natural frequencies of blading have become faster, more economical and more accurate in recent years. Still, the unplanned and untimely outage of turbomachinery due to blade failures are numerous. The study presented herein and the proposed use of the SAFE diagram have helped to identify possible problem areas and have also helped in less costly, faster design changes. The concept of the SAFE diagram is based on the matching of natural frequency and its associated mode shape with the exciting frequency and its shape. This can be used on any rotating, symmetrical system where the frequency of excitation is a function of rotational speed. Such cases are encountered in axial compressors, gas turbines, pump, and fans.

The SAFE diagram provides more information in the blade reliability decision process than the traditional Campbell diagram. The use of the SAFE diagram is proposed in blade reliability evaluation.

REFERENCES1. Prohl, M.A., "A Method of Calculating Vibration Frequency and Stress of a Banded Group of Turbine Buckets," Trans of ASME, 56-A-116 (1956).

2. Singh, M. and Schiffer, D., "Vibrational Characteristics of Packeted Bladed Discs," ASME Paper No. 82-DET-137 (1982).

3. Singh, M., "SAFE Diagram," Technology Report ST 16, Dresser-Rand Company (1984).

4. TBDYNE - A Finite Element Based Bladed Disc Program Developed by Dresser-Rand Company (1987).

Refer questions to:Contact D-R

 

TOPIC_2

 

RankineCycler™ 

Advanced Micro Steam Turbine Power plant

GENERAL DESCRIPTION

The Turbine Technologies'  RankineCyclerTm is a complete and functional micro steam turbine power plant.   It is delivered and supported directly by the factory and can be readied to operate within minutes of uncrating. 

This micro steam plant allows students to readily view all components and eliminates lengthy operational preparations.  Special site preparations are also eliminated. 

RankineCyclerTm is not a simple demonstration piece, but a fully interactive multiple experiment teaching device.  Turbine load can

be varied, power input can be varied and 8 data points offer many cycle analysis options. RankineCyclerTm can be set up and operated virtually anywhere.

All components are mounted on a 48" by 28" tabletop stand.  The entire "power plant" weighs less than 100 pounds (dry weight).  Major components include a multiple pass flame-through boiler, an impulse steam turbine and generator set, boiler feedwater pump, automatically controlled gas burner with combustion blower and a condenser tower. 

All sensors are routed to a dedicated PC data acquisition station for data collection and analysis.   

POWER PLANT COMPONENT DESCRIPTIONS:

Boiler

RankineCyclerTm multiple pass flame through boiler accurately models full scale units.  Its burner operates on natural or LP gas.  Gas ignition and safety valves comply with U.L. and CSA safety standards.    The forced air blower, ignitor, and gas valve are all automatically controlled for start, operation and shut down.  Operating pressure is maintained at a constant level via this automatic regulation capability.

Rear boiler door open during firing depicts flame tubes and flame

holder

Both the front and rear doors are insulated and open easily to reveal the burner, flame tubes and general boiler construction.  The boiler walls are also insulated to minimize heat loss.  A sight glass shows water levels and is equipped with safety valves that instantly close in case of glass breakage. 

Turbine/Generator set

The micro steam turbine consists of a flanged two piece housing and an axial flow impulse turbine wheel.  Steam condition (temperature and pressure ports) are located on the turbine inlet and outlet.  Load and measurement terminals are provided and routed to transducers for data acquisition purposes.  The DC generator's output varies with turbine steam inlet conditions and the generator winding is constructed for a safe low voltage output.

Axial steam turbine housing image depicts steam inlet and outlet flanges  

Axial flow impulse steam turbine wheel

Condenser Tower

Turbine outlet steam is piped to the condenser tower and is admitted tangentially.  The tower is constructed of stainless steel. Condensate is returned to the boiler from the condesor tower sump via a custom manufactured steam driven feedwater pump.

BOILER FEEDWATER PUMP  

The feedwater pump is a fully functioning, dual-piston reciprocating pump.  It is independent of electric or other external power requirements and is driven by boiler steam.  An intensifier principle (large area steam piston/ small area feedwater piston) comprises the basic design.

Water is drawn from the condenser tower sump and then routed through the suction port and inlet valve of the feedwater cylinder assembly.  The double acting steam cylinder and feedwater pistons are mounted on a single connecting rod.  The reciprocating piston motion is controlled by a four way steam valve.

INSTRUMENTATION/DATA ACQUISITION DESCRIPTION

Eight installed sensors allow for full Rankine Cycle analysis and study.  On screen data includes:

Boiler Temperature Boiler Pressure Turbine Inlet temperature Turbine Exit temperature Turbine Inlet pressure Turbine Exit pressure Generator amperage Generator voltage

The computer is capable of logging all data points and replaying them

at a later time.  Data can be viewed as collected in a strip chart type presentation on the computer screen.     

Data Acquisition Station: 

68 pin rail mountable terminal block 16 channel signal conditioning unit 2-meter, 68-pin shielded cable 4 shelf cart with lockable castors 17" color display  2-button Mouse Keyboard 56.6 K PCI modem 64 Meg Ram 100 Mhz SDRAM 733 Mhz Intel Pentium III processor 10 Gig hard drive 48XCD-ROM drive Power protection/surge suppresser

 

 

Software related items:

National Instruments data acquisition software Signal Conditioning Driver software Windows 98 Microsoft Office 2000 Norton Anti-Virus 2001                  

 

 

 RankineCyclerTm

SpecificationsWeight: 

Dimensions:

Power required: 

Impulse turbine wheel diameter: 

Turbine housing O.D. 

Generator output: 

Fuel:

Complete  RankineCyclerTm system cost: 

60 pounds dry weight

 48"L   X   28"W    46"H

120Vac (standard wall outlet)

2.00 inches

3.50 inches

100 W maximum

LP or natural gas

$19,750 U.S.D. (FOB Factory)

TOPIC_3

APPLYING STEAM TURBINES FOR PUMP DRIVES John Elmore

Randy Palmer Dresser-Rand, Wellsville, N.Y., USA

Steam turbines are well-suited for pump drive service for a number of reasons.  First is their ability to operate across a wide speed range.  This enables them to do two things: operate at the pump's most efficient speed and match the system head curve at varying pump flows, thus reducing throttling losses.  This can make the turbine a good choice for marginal NPSH applications as it can avoid drastic speed reductions.  When operating at low speed, turbines reduce radial reaction in centrifugal pumps operating at low flows.

Turbines also work well as pump drives because they can be used safely in an explosive atmosphere, do not fail when overloaded, have high starting torque (useful for positive displacement pumps), and are rugged and reliable.

This discussion mainly addresses single-stage turbines 3,500 hp and smaller, but most comments apply to larger single- and multi-stage turbines as well.  Below is a review of steam cycle and turbine performance terms.  You should be familiar with these before trying to make performance comparisons and analyze selection data.

Hp/Rpm

Wheel Pitch Diameter

14 inches 20 inches 28 inches

Steam Rate-lbs/hp-hr Steam Rate-lbs/hp-hr Steam Rate-lbs/hp-hr 300/3600 39.8 31.4 26.7300/1800 High Exh. Velocity 53.4 41.2300/1200 High Exh. Velocity 77.6 57.950/3600 43.1 35.3 36.550/1800 77.3 59.4 42.850/1200 111.2 87.1 69.6

Inlet: 600 psig @750°F; exhaust: 40 psig.  Please keep in mind that the table reflects actual steam rates based on turbine efficiency only. Turbine performance is only a component of the customer's steam cycle performance (for the entire plant steam loop).

Table 1. Performance comparisons: wheel pitch diameter vs. speed (single-stage).

STEAM CYCLESNon-Condensing CycleThe non-condensing cycle (back pressure operation) involves taking medium-to-high-pressure steam into the turbine and exhausting it to a process header where the pressure is higher than atmospheric  Cycle efficiency is high because the turbine and process absorb most of the heat before the condensate returns to the boiler. Turbine arrangement is usually straight non-condensing (no turbine extraction), and it is the most commonly used cycle for pump drive service.  For pump ratings of 3,500 hp or less, steam turbines are usually single-stage.

Condensing Cycle

This cycle takes the turbine's exhaust steam to a condenser at below atmospheric pressure.   Because the condenser's cooling water absorbs most of the heat, cycle efficiency is low. Condensing cycles are sometimes used when taking low-pressure process exhaust steam (as low as 5 psig) to the turbine inlet through a multi-stage turbine, or when waste fuel is burned in the boiler.  Condensing cycles are not often used for pump drive service.  The cycle is not very efficient, and the cost of a multi-stage turbine, condenser, condenser pump and cooling water can be prohibitive.  Condenser cooling water availability is also a concern for many processes.

STEAM RATESTheoretical Steam Rate (TSR)This is the quantity of steam per unit of power required by an ideal Rankine cycle heat engine, which assumes a no-loss (isentropic) expansion between turbine inlet and exhaus.   TSR is expressed as pounds of steam per horsepower hour (or kg/kw-hr) and is universally recognized and used as a benchmark for measuring steam turbine performance.   You can look up TSRs in the latest ASME Steam Tables (the easiest method) or derive them from a Mollier chart for steam (which requires time and good eyesight).

Actual Steam RateThe actual steam rate reflects turbine efficiency and is expressed in the same units as the TSR.  Manufacturers determine its value using pump and steam conditions.  The product of actual steam rate and pump horsepower is the steam flow required to develop a given pump power.FACTORS AFFECTING TURBINE PERFORMANCESteam Conditions

Greater pressure drop across the turbine makes more heat energy (BTUs) available.

Less than 50 Btu/lb enthalpy drop through the turbine is not generally workable due to low turbine efficiency resulting in high steam flows. (This is a rule of thumb figure, not an exact one.)

Imposing a wide range of steam conditions usually causes the turbine manufacturer to provide excess inlet nozzle area than required for normal operating point.

Handvalves

These are used to reduce throttling across the governor valve when the system is operated for long periods at above or below normal pump power.

Part-load performance can be improved by closing handvalve(s) to decrease nozzle area.

Maximum load/overload capability performance is improved by opening handvalve(s) to increase nozzle area.

Handvalves are set either full open or full closed.   (They are not used for throttling.)

Power and Speed

Generally, higher powers and speeds result in greater turbine efficiencies.

Speeds slower than 1,800 rpm usually require attention to whether there is sufficient steam flow availability, the high exhaust temperature and the type of lubrication.

Wheel Pitch Diameter

Larger pitch diameters perform well at higher power ratings and lower operating speeds than smaller wheels.

Small pitch diameters are best at low power ratings (under 100 hp) and high exhaust pressures.  They also produce less exhaust loss.

Inlet and Exhaust Nozzle Size Keeping steam velocities below limits specified by NEMA Standards can

minimize friction losses. NEMA Standard SM-23 velocity limits are 175 ft/sec at turbine inlet and

250 ft/sec non-condensing exhaust (450 condensing).

TURBINE BASIC SELECTION DATA REQUIRED

Minimum requirements: o inlet steam pressure and temperature o exhaust steam pressure o pump power (rated max.-min.) o pump speed (rated max.-min.) and maximum allowable over-

speed for trip o speed control (manual or type of process signal) o site conditions: indoor/outdoor and ambient conditions o cooling water data (pressure, temp and cleanliness)

Any off-normal steam or pump operating conditions: o affects turbine steam path areas and ability to make power o low power conditions create high exhaust temperature (affects

lubrication method) Type of pump and service:

o centrifugal or positive displacement o normal or quick start o continuous or standby duty o site electrical rating (if electrical accessories are involved)

Specifications: o customer required scope and turbine shop tests o API-611/612-3rd or 4th editions (API data sheets required) o steam cost evaluation o sound level requirements

Reprinted from Pumps and Systems Magazine.Refer questions to:

Contact D-R

    

 

TOPIC_4

FREQUENCY EVALUATION OF A STEAM TURBINE BLADED DISK

Jim DelloDresser-Rand, Wellsville, NY, USA

Unplanned and untimely shutdowns of critical turbomachinery is costly-for both the user and original equipment manufacturer. Therefore, it is very important that the turbomachinery be designed and operated to assure trouble-free operation. This is not always easily accomplished. Because turbomachinery components, specifically airfoils, are subjected to high variable loads that can cause failure, designing reliable components requires in-depth vibration and stress analysis.

Identification of when the frequency of variable force match component's natural frequencies is the traditional method to design around potential blade failures. The identification of matching frequencies-the force with the structure-is commonly shown on a Campbell diagram. However, the Campbell diagram lacks some information when used to analyze the possibility of blade failure. To further help bladed disk designers, Dresser-Rand has initiated the use of the SAFESM

(Singh's Advanced Frequency Evaluation) diagram.

Natural Frequency and Mode ShapeNatural frequency is the frequency at which an object vibrates when excited by a force, such as a sharp blow from a hammer. At this frequency, the structure offers the least resistance to a force and if left uncontrolled, failure can occur. Mode shape is the way in which the object deflects at this frequency. An example of natural frequency and mode shape is given in the case of a guitar string. When struck, the string vibrates at a certain frequency and attains a deflected shape.

The frequency can be noted by the pitch coming from the string. Different string geometries (length and diameter) lead to different natural frequencies or notes.

By nature of its structure, a turbine bladed disk, shown in figure 1, usually has many natural frequencies and associated mode shapes. These frequencies and mode shapes are somewhat further complicated by the use of a shroud to connect groups of blades together. Shrouding groups of blades together serves two purposes-one is for aerodynamical reasons to reduce losses and the other is for blade strength required to sustain steam forces. Blades are banded in brackets with varying numbers in one packet. At Dresser Rand, a 6-blade packet is most commonly employed. The knowledge of packeting arrangements of blades into groups is very important when analyzing packeted-bladed disk vibration.

Figure 1

 Vibrational behavior or typical deflection mode shapes of a packet containing six blades is shown in figure 2. Natural frequency values will change depending on blade geometry; a tall and narrow (i.e., less stiff or more flexible) blade will have relatively low natural frequencies.

Figure 2

In some mode shapes, some of the blades within a packet are moving in the same direction; while in others, blades are moving opposite each other. This change in direction of motion or phase change is evident in some of the mode shapes. For example, figure 3 shows several modes and the location of their phase changes. Because there is relatively no motion at this location, this point is called a node point. The axial rocking mode has just one phase change or node point located at the center of the packet; the axial "U" mode has two nodes. The tangential fix-supported (TFS) family of modes which typically have closely spaced natural frequencies, will number one less than the number of blades. Therefore, for a 6-blade packet, there will be 5 TFS modes. The example shown in figure 3 shows the TFS mode that has three nodes within the packet. Simple bending modes such as the first tangential and the first axial have no phase changes within the packet. This means that all the blades within the packet move in one direction at the same time. This concept of phase change within a packet is a key aspect of the SAFE diagram and its importance will be explained later.

Figure 3

Turbine wheels or disks which hold the blades also have natural frequencies and mode shapes of their own depending on the wheel geometry. Some characteristic mode shapes are illustrated in figure 4. These disk mode shapes can be obtained by sand pattern testing. This is performed by placing sand on a horizontal disk and exciting it a certain frequency until the sand forms into a pattern. The pattern is caused by the sand migrating from an area of large amplitude vibration and seeking a location of minimal amplitude vibration. Just as before a point of little or no vibration is called a node. In the case of a disk, these nodes could be thought of as diametral lines across the disk and/or circular curves.

Figure 4

Since a packeted bladed disk is a structure, it has unique mode shape characteristics of its own. It contains both blade packet and disk mode shape characteristics mentioned before. The interaction between the blade packets and the disk is best shown in figure 5.

Figure 5

Observe that disk nodal diameter (ND) lines are coincident with blade motion. Specifically, a nodal diameter occurs where blade movement is out of phase. There can be as few as 0 ND where all the blade packets are in phase with one another or a maximum of one-half the number of blades which occurs when every blade moves out of phase with its neighbor. For example, if a disk contains 90 blades, there is a maximum of 45 nodel diameters possible.

Vibratory ForcesNow that natural frequency and mode shapes have been established, some alternating forces must exist to excite a structure to vibrate. These forces have inherent frequencies and shapes just as bladed disks do. In a steam turbine, the most common sources of excitation are nozzle passing frequencies and running speed harmonics.

Running speed harmonics occur due to interruptions in the fluid flow path. Frequencies of running speed are multiples of rotor operating speed. For example, the tenth harmonic of running speed would be a force that occurs ten times for every revolution of the wheel. For example, a turbine rotor running at a speed of 3600 RPM (60 cycles/sec or Hz) would have running speed harmonics occurring at 120 Hz, 180 Hz, 240 Hz ....

Nozzle passing frequency excitation is caused by steam flowing through a nozzle or diaphragm which is used to turn and direct steam flow onto the blades. Because of their design, nozzles have flow interruptions at regular or cyclic intervals which cause the force imparted on the blades to be cyclic. An analogous situation would be running a stick along a picket fence. Figure 6 shows what this force would be for the first couple of evolutions. This cyclic or periodic force can excite bladed disks into their characteristic mode shapes.

Figure 6

 ResonanceCombining the previously established concepts of natural frequency and periodic forces together will help explain resonance. Resonance is a condition where response (amplitude of vibration) is a maximum and resistance to an oscillating force is minimal. At this condition, the shape and frequency of a force must match the mode shape and natural frequency of the structure that is being excited. Resonance cannot occur if one of these four factors is not in place.

The best example of the destructive power that a resonance condition can cause is the classic Tacoma-Narrows bridge failure. The bridge as any structure had inherent natural frequencies and associated mode shapes. When the wind blew (forcing phenomena) at a certain speed, air buffeting (vortex shedding) that apparently had the same frequency as the bridge natural frequency caused the bridge to oscillate. Vibration became large and uncontrolled until the bridge failed. As shown by this example, resonance is a condition best avoided or at least controlled.

Campbell DiagramTraditionally, the Campbell diagram shown in figure 7 is used to illustrate interferences between blade packet natural frequencies and the common exciting forces described earlier. Specifically, in the case of a turbine, blade packet frequency is plotted against rotor running speed. Diagonal lines represent sources of excitation (e.g.,1xNPF, 2xNPF, and 10x running speed).

Figure 7

When a diagonal crosses a packet natural frequency within the turbine speed range, a resonance is said to occur such as the axial "S" mode in figure 7. This may or may not be the case. What was shown by this interference is that the natural frequency matches the forcing frequency. Resonance may occur at this location but it is not assured since the definition of resonance is not complete.

The difficulty in defining the real resonance is that it considers only one packet in its analysis. This assumes that the packets around the wheel are vibrating in

phase to each other at any time. This is just one possibility. Packets may move independently of one another creating many combinations of out-of-phase movements. A slightly different Campbell diagram could be drawn showing all the possible packet motions, figure 8, but this is very cluttered and hard to interpret, particularly in the speed range where most designs operate.

Figure 8

As mentioned before, the concept of nodal diameters relating to nodes within packets and packets moving relative to one another is important. It should be noted at this point that a disk may be stiff where a disk nodal pattern like that shown in figure 4 will not form, however, the blades and packets still move opposite each other. Even when this situation occurs, the mode shapes will be designated by a nodal diameter. Therefore, nodal diameter is the name given to the convention of blades and/or packets moving out of phase. It is this feature of disk mode shapes and the out-of-phase movement of blade packets that the traditional Campbell diagram does not show.

The different type of Campbell diagram, shown in figure 8, could be drawn for a bladed disk showing all the nodal diameter possibilities. For simplicity this plot shows a disk that would be completely shrouded by just one shroud. The frequency axis is then the location for specific nodal diameter. As shown this plot is very hard to interpret. To illustrate this, observe points A and B in figure 8, Point A indicates a point of possible resonance where natural frequency equals the frequency of the exciting force. Closer observation reveals that the excitation is 4x running speed; but the mode shape is 2 nodal diameters. No resonance will exist in this situation. There are many points similar to point A. Point B shows a situation where resonance could exist because the running speed harmonic and nodal diameter are 3. A line connects the few locations where resonances may exist.

SAFE DiagramAnother projection of the Campbell plane will show the same information as before in a clear and concise manner, This projection is the SAFE plane and is shown in a 3-D model of the Campbell and SAFE planes given in figure 9.

Figure 9

Notice that the Campbell and SAFE planes share the vertical or frequency axis. But instead of running speed as on the Campbell, the SAFE plane plots nodal diameter on this axis. Nodal diameters describe the mode shapes of the

packeted-bladed disk and also the shape of the varying force. Nodal diameter describes the shape of the force will be 46 nodal diameters at 1xNPF. Therefore, nodal diameter also describes the shape of excitation such as 1xNPF. This is a key point since this completes the definition of resonance-shape of the force matching the disk mode shape. Therefore, the SAFE diagram will reveal true resonances, not just probable resonances as the Campbell diagram shows.

Figure 10 helps illustrate how a Campbell and SAFE diagram can be used to evaluate packeted-bladed disk resonances. Two cases are included for contrast. The first case shows a Campbell diagram interference and the SAFE diagram confirms it. The second case also shows a Campbell diagram interference; however, this time the SAFE diagram shows that a resonance condition does not exist.

Figure 10

For simplicity and clarity, only the first family of tangential modes are plotted. Construction of the Campbell diagram was discussed earlier. To construct the SAFE diagram from the Campbell diagram only two additional pieces of information must be known-the number of blades and the number of packets. This must be known so the maximum number of nodal diameters (horizontal axis) for a specific wheel can be determined. Recall that nodal diameter (ND) is the node line running across a disk. The maximum number of nodal diameters for a given disk is one-half the number of blades. That is the maximum which would occur if a line were drawn across the wheel between each and every blade.

Circles on the SAFE diagram represent the various nodal diameters. Notice that each frequency has a discrete set of nodal diameters. For instance, the first tangential mode occurs from 0 to 9 ND. What this means is that the first tangential has only these possible packet movements. All packets are moving in the same direction at the same time (0 ND), or the eighteen packets of this wheel are all moving opposite each other (9 ND). This means 18 change of phase movements of the packets.

Closer examination reveals that other modes could be thought of in the same manner. The first TFS mode has one phase change among the blades in the packet so it can only have a minimum of 9 ND which means the TFS1 mode picks up where the first tangential mode leaves off. The fifth TFS mode with five phase changes within a packet can have the maximum number of nodal diameters, 54. That is, all the blades within the wheel are moving out of phase with one another.

Solid circles represent single modes and the unfilled circles represent double modes. This means that, except for the maximum and minimum ND (0 and 54)

nodal diameters or disk modes occur in pairs. A nodal diameter will occur 90° from the first. Except for 0 and 54 ND, the other single modes are really separated double modes. For example, the 9th ND could describe either the first tangential or the 1st TFS mode.

Completing construction of the SAFE diagram requires plotting of the excitation phenomena. On the Campbell diagram, possible excitations are presented by diagonal lines. However, on the SAFE diagram, diagonals represent speed; excitation is represented by the horizontal nodal diameter axis. In this case, it is 5 times running speed (60 Hz) and occurs at 5 ND.

Therefore, since the 5ND excitation point crosses the first tangential frequency, case no.1 given in figure 10 shows a SAFE interference of the first tangential at 5 ND. It is interesting to note that direct interference would be unlikely to occur if:

1. turbine speed were much lower (i.e., a lower sloped speed line) 2. the first tangential mode was higher in frequency 3. there were fewer than 9 packets so that the maximum nodal diameter for

the first tangential frequency is 4ND. This judgement may be made from information on the SAFE diagram.

The second case shows the shortcoming of the Campell diagram. This Campell diagram shows an interference of the first tangential mode with 1xNPF. Examination of the SAFE diagram shows that the Campbell diagram interference is highly unlikely. That is, the shape of the force, 46 ND, does not match the shape of the disk, 0 to 8 ND. This example illustrates that the SAFE diagram contains the needed information to accurately determine if a bladed disk resonance will occur.

CONCLUSIONPacketed bladed disks are unique structures that require specialized vibrational analysis. Traditionally, the Campbell diagram is used to analyze resonance of a single blade packet. However, the Campbell diagram does not contain enough information to analyze a complete packeted bladed disk. To help understand and analyze packeted bladed disk interferences, the SAFE diagram was developed.

The SAFE diagram is created from a third axis projection of the real resonance points. This third axis plots nodal diameters which describe the shape of a disk mode and shape of the forcing function. A Campbell diagram may be constructed from information contained on a SAFE diagram if desired.

The main advantage of the SAFE diagram over the Campbell diagram is that it allows a bladed disk designer to see if a true resonance exists in the disk. As traditionally plotted, the Campbell diagram reveals only possible resonances.

Refer questions to:Contact D-R