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By: Quinton Rowson, Quest Integrity Group

As published in Energy Generation Jan-Mar 2012. Download the pdf.

The case study below of a compressor dovetail remnant life assessment demonstrates the implications of small aerodynamic loads. It also explains the use of computational fluid dynamics and finite element analysis stress modelling in obtaining the remnant life of compressor blades. Remnant life assessment determines the remaining life of a component due to a time- or cycledependent material deteriorating process, for example, creep or fatigue.

Components in the hot gas path are susceptible to deterioration. Manufacturers perform component life expectancy calculations, taking material deterioration factors into account based on design conditions.

Occasionally too much conservatism is included in those calculations, reducing the probability of failure, for example, from 1/1000 to 1/million. A plant could be running its equipment significantly below design conditions, extending the life of the component and reducing replacement and maintenance costs. The question then becomes: When operating below design conditions, how long can the component operate safely before failure? The purpose of remnant life analysis is to provide an answer.

Remnant life is often determined by the API/ASME 579:2007 Fitness-For-Service Standard. This employs fracture mechanics for the determination of critical crack sizes and fatigue crack growth analysis to determine how long it is going to take for the crack to appear and then grow to the critical crack size under cyclic loading.

The Standard also employs several creep analysis theories to determine how long a component will last before final creep void failure occurs at a certain temperature and time. This includes the widely known Larson-Miller parameter method and the Omega Law methodology.

Computational fluid dynamics (CFD) is used to determine the state of a fluid in terms of its pressure, velocity and density, at any point in time and at any location of the fluid in space when it is subjected to some disturbance. CFD can also be used to determine the reaction loads that fluid exerts in response to being pushed about.

A result of a CFD analysis (in plan view) of fluid passing a blade is shown in Figure 1. Note that the direction of the flow is given by the direction of the coloured markings, and the velocity is indicated by the colour of the markings. From this we can calculate the pressure the fluid exerts on the surface of the blade. The pressure gradients calculated over the blade surface can then be added over incrementally small finite areas to result in a system of forces. These forces can then be transposed to a datum point at the base of the blade. These loads can then be used in a finite element analysis to determine the critically stressed areas, which will be the life-limiting locations.

Depending on the component location and geometry, it may not make any difference whether these small secondary loads are included in the analysis of a remnant life. It is necessary to determine the appropriate amount of analysis required in order to ensure correct remnant life calculations are performed.

*Figure 1 Plan view of blade CFD velocity profiles*

Finite element analysis is used to determine the stress that exists in a component, due to external and internal loads, when the geometry is complicated. Where reasonable approximations cannot be made to simplify the geometry, classical hand calculation approaches cannot be used. Jeremy Astley, one of the great minds in the field of finite element analysis, was once quoted as saying "Finite element analysis is not a very elegant method, as it is a brute force method to solve a problem”. This is true. Finite element analysis can sometimes be likened to using a sledge hammer to crack a walnut, but it is invaluable when solving statically indeterminate problems.

This brute force idea comes from the fact that the geometry is broken up into small finite elements, of which a simplified "hooks law” can be written for the relationship between forces and displacements at nodes which exist within each element. This means the following relationship can be formed for each element:

[K]{u} = {f}.

[K] represents the stiffness matrix for each element. Each of these stiffness matrices can then be assembled to create a "global stiffness matrix” for the entire geometry of interest. For a given vector of forces {fn}, the global stiffness matrix can be inverted which allows the solution for the vector of displacements {un} to be determined. From this the strains of the model can be calculated. Using the Young’s modulus, the stresses within the model can then be determined.

Commercial finite element analysis codes are very user friendly these days, so that anyone can use them and get a result; however,

you need to know that it is the right result as there are many things that can be overlooked. An example of this is the number of elements required within specific volumes of the geometry to obtain convergence to the correct stress value. This mesh convergence is shown in Figure 2.

*Figure 2 Finite element mesh convergence*

The original analysis was undertaken without the presence of aerodynamic loading. Centrifugal loads which exist in the disc due to its spinning in operation (3300rpm) were applied. Connection loads, transmitted through the dovetail due to the centrifugal load (3300rpm) of the blades, were also applied. As a final step, boundary conditions to emulate the entire disc were applied.

The resulting stress profile in the dovetail region is shown in Figure 3.

*Figure 3 Dovetail region stress profile*

The analysis was repeated to examine the effect of tangential load: the blade loading tangential to the axis of the rotor was calculated using simple hand calculations, and was determined to be 0.76KN. This load was added to the model to determine if there was any

effect of this loading (given the non-symmetrical nature of the geometry). (Note: this is the aerodynamic load attributed to the torque generated by the compressor to make the turbine spin. As the torque from the compressor can be easily calculated, the torque associated with each disc and each blade on each disc can then be calculated.)

The results showed that the stress profiles in the disc were identical, although the peak stress with a single aerodynamic loading component (rotor torque) was decreased by 28.6MPa.

Remnant life calculations were performed in accordance with the API/ASME 579:2007 Fitness-For-Service Standard. A proprietary software tool called SignalTM Fitness-For-Service was used, as it automates the procedures in API579 and significantly reduces the required hand calculations.

The stress profile normal to the crack plane is extracted through the component along the crack growth path from the finite element

analysis results in order to determine the critical crack size as shown in Figure 4.

*Figure 4 Stress profile from finite element analysis*

The stress results through the thickness of the dovetail are plotted for the two cases of the analysis considered, with and without torque loading. See Figure 5 for an example stress plot.

These stress profiles were then used to determine the critical crack size and the number of machine starts to cause catastrophic failure; the results of which are shown in Figure 6.

*Figure 5 Comparison of dovetail stress ”" with and without torque loading*

*Figure 6 Crack depth and cycles to failure*

The depth of critical crack was determined to be 11.6mm deep, the full length of the dovetail radius. When aerodynamic torque loading was not considered, assuming an initial crack was already present which was 0.1mm deep, the dovetail fails at approximately 160,000 starts.

This remnant life analysis has shown that by determining just one (torque loading) of six of the reactions present at the base of the compressor blade, the critical flaw size has been increased from 11.6mm depth to 12.1mm depth. The operational result has been a corresponding increase in the number of starts to failure from approximately 160,000 starts to approximately 250,000 starts.

There is no guarantee that investments in additional analysis, as shown above, will result in greater accuracy in these calculations. It may be found that for the component of concern, improving the accuracy of the remnant life calculations reduces the life-to-failure prediction as opposed to increasing it. When dealing with critical equipment that has potentially catastrophic consequences related to failure, the most important factor is calculating the correct result, not just a result. The benefit of performing more detailed analysis provides more accurate predictions of component life times which thus provide increased confidence in extending part life.

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