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A TRANSFER FUNCTION APPROACH TO STRUCTURAL VIBRATIONS INDUCED BY THERMOACOUSTIC SOURCES
Rob Huls,Andr´e de Boer,Jim Kok
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University of Twente
P.O.Box217,7500AE Enschede,The Netherlands
江畔独步寻花的诗意是什么r.a.huls@utwente.nl
Abstract
To decrea NOx emissions from a combustion system,lean premixed combustion in combination with an annular combustor is ud.One of the disadvantages is an increa in sound pressure levels in the combustion system,resulting in an incread excitation of the surrounding structure,the liner.This caus fatigue,which limits the life time of the combustor.To model the interaction betweenflame,acoustics and structure,a transfer function approach is ud.In this approach,the components are reprented by the frequency dependent linear transfer between their inputs and outputs.For the flame a low passfilter with convective time delay is ud as transfer function between velocity perturbations at the burner outlet and theflame as acoustic volume source. The acoustic transfer from volume source to velocity perturbation at the burner outlet is obtained from a harmonic
finite element analysis,in which a temperaturefield from CFD calculations is ud.The calculated respon is subquently curve-fitted using a pole-zero model to allow for fast calculations.Thefinite element model includes the two way coupling between structural vibrations and acoustics,which allows extraction of the vibration levels.The different transfers arefinally coupled in one model.
Results show frequencies of high acoustic respon which are susceptible to ther-moacoustic instability.Damping mechanisms and the pha relation between the differ-ent components determine stable or unstable behavior and the amplitude of the resulting perturbations.Furthermore there are also frequencies of high structural respon.Es-pecially when the two coincide,the risk of structural damage is high,whereas when they move away from each other,the risk decreas.
INTRODUCTION
To decrea NOx emissions from an industrial gas turbine lean premixed combustion (using a surplus of air to burn the fuel)in combination with an annular combustor is
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parameter description value parameter description value E Young’s modulus130GPa D in inside width150mm νPoisson’s ratio0.30µdamping at outlet30%
nttρs density solid7800kg m−3ζstructural damping2%
jeans
make sb do stht liner thickness 1.5mm L chamber length 1.8m
Table1:Material properties and dimensions
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Flame model
The heat relea from theflame responds to the acoustic particle velocity perpendicular to the burner outlet,becau the perturbations cau modulations in the supply of combustible mixture to thefla
me.The transfer to the heat relea rate by theflame is a low passfilter combined with a time delay.The time delay is caud by the time it takes for a disturbance to convect from the burner outlet to theflame front.The low pass behavior is related to the size of theflame.A disturbance burns during its passage of theflame front.Disturbances which convect less than this distance over one period are averaged out,causing low pass behavior.Theflame model can be written as1
ˆq
= 11+iωτ1 ˆu B u0,B e iω(t−τ)(1)
q0
in which a subscript zero denotes a mean value and a hat the amplitude of the pertur-bation.Furthermore u B is the velocity at the burner outlet,q denotes heat relea rate,τ1determines the cutoff frequency of the low passfilter,t is time,τis the time delay (τ=x f/u0,B with x f theflame position relative to the burner outlet,or the time delay is determined directly from CFD results)andωthe angular frequency.To translate the unsteady heat relea rate to a volume source as input for thefinite element model the following relation is ud2residual
洛丽塔什么意思ˆQ=γ−1
ˆq(2)
ρc2
in whichˆQ denotes the equivalent acoustic volume source.Theflame transfer function (ˆQ/ˆu B)is depicted infigure4.The properties ud are listed in table2.
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parameter description value parameter description value c0cold speed of sound343m s−1u0,B burner outlet velocity27ms−1ρ0cold density of air 1.21kg m−3x f position offlame0.05m p0mean pressure 1.5bar q0mean thermal power150kW γratio spec.heats 1.4
Table2:Numerical values for theflame model
half life
The coupled vibroacoustic model is made of a combination of linear shell(shell63) and linear acoustic(fluid30)elements using thefinite element package Ansys R .An unstructured mesh made of triangular and tetrahedral elements made by Femap R is ud(figure5).
The acoustic and shell elements are coupled using special elements that include fluid structure interaction(figure5).The elements ensure that the acoustic particle velocity normal to the surface is the same as the structural normal velocity and that the dynamic equilibrium equations are
satisfied.
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a steady CFD RaNS calculation of a quarter ction is ud.It typically consists of cold zones,aflame front with very steep temperature gradients and hot zones.The temperaturefield is therefore divided in a hot and a cold ction and the properties of the gass are ud according to this spatial distribution,which is interpolated on the finite element mesh.The structural damping is taken as2%(constant stiffness matrix multiplier),which is an estimation of the damping introduced by the highly turbulent coolingflow between liner and pressure vesl.The properties of the structure are taken at elevated temperature as listed in table1.A harmonic analysis is performed on the model.The resulting frequency respon functions are curvefitted to obtain a pole zero description of the differe
nt parts.The can then be ud in the coupled description. The acoustic transfer function and thefit are depicted infigure6.
RESULTS
Unstable frequencies
There are two well-known types of methods from control theory to evaluate system stability,being root-locus methods and frequency domain methods.Whereas the root-locus methods are easier to understand and evaluate,the frequency domain method has some other advantages,two important ones being the ability to include measured data (for instance a measuredflame transfer function)and easier inclusion of the time delay in theflame transfer function.
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