Steven Chambers

Horia Flitan

Paul Cizmas1

Department of Aerospace Engineering,Texas A&M University,

College Station, TX 77843

Dennis Bachovchin

Thomas Lippert

Siemens-Westinghouse Power Corporation,Pittsburgh, PA 15235

David LittleSiemens-Westinghouse Power Corporation,

Orlando, FL 32826

The Influence of In Situ Reheat onTurbine-Combustor PerformanceThis paper presents a numerical and experimental investigation of the in situ reheatnecessary for the development of a turbine-combustor. The flow and combustion weremodeled by the Reynolds-averaged Navier-Stokes equations coupled with the speciesconservation equations. The chemistry model used herein was a two-step, global, finiterate combustion model for methane and combustion gases. A numerical simulation wasused to investigate the validity of the combustion model by comparing the numericalresults against experimental data obtained for an isolated vane with fuel injection at itstrailing edge. The numerical investigation was then used to explore the unsteady trans-port phenomena in a four-stage turbine-combustor. In situ reheat simulations investigatedthe influence of various fuel injection parameters on power increase, airfoil temperaturevariation, and turbine blade loading. The in situ reheat decreased the power of the firststage, but increased more the power of the following stages, such that the power of theturbine increased between 2.8% and 5.1%, depending on the parameters of the fuelinjection. The largest blade excitation in the turbine-combustor corresponded to thefourth-stage rotor, with or without combustion. In all cases analyzed, the highest excita-tion corresponded to the first blade passing frequency. �DOI: 10.1115/1.2135812�

1 Introduction

In the attempt to increase the thrust-to-weight ratio and de-crease the thrust specific fuel consumption, turbomachinery de-signers are facing the fact that the combustor residence time canbecome shorter than the time required to complete combustion. Asa result, combustion could continue in the turbine, which is oftenconsidered to be undesirable. A thermodynamic cycle analysis,however, demonstrated a long time ago the benefits of using re-heat in the turbine in order to increase specific power and thermalefficiency. Even better performance gains for specific power andthermal efficiency were predicted for power generation gas tur-bine engines when the turbine is coupled with a heat regenerator�1�. Starting in the 1960s, several patents were awarded for dif-ferent inventions that addressed various aspects related to turbinereheat �2–4�.

In spite of these advances, the technological challenges and thedifficulty of predicting and understanding the details of the trans-port phenomena inside the reheat turbine precluded the develop-ment of turbine-combustors. Herein, a turbine-combustor is de-fined as a turbine in which fuel is injected and combustion takesplace. The process of combustion in the turbine is called in situreheat.

Several challenges are associated with the combustion in theturbine-burner: mixed subsonic and supersonic flows, flows withlarge unsteadiness due to the rotating blades, hydrodynamic insta-bilities, and large straining of the flow due to the very large three-dimensional acceleration and stratified mixtures �1�. The obviousdrawback associated with the strained flows in the turbine-burneris that widely varying velocities can result in widely varying resi-dence time for different flow paths and, as a result, there areflammability difficulties for regions with shorter residence times.In addition, transverse variation in velocity and kinetic energy can

1To whom correspondence should be addressed.Contributed by the International Gas Turbine Institute �IGTI� of ASME for pub-

lication in the JOURNAL OF ENGINEERING FOR GAS TURBINES AND POWER. Manuscriptreceived October 1, 2003; final manuscript received March 1, 2004. IGTI ReviewChair: A. J. Strazisar. Paper presented at the International Gas Turbine andAeroengine Congress and Exhibition, Vienna, Austria, June 13–17, 2004, Paper No.

GT2004-54071.

560 / Vol. 128, JULY 2006 Copyright © 20

cause variations in entropy and stagnation entropy that impactheat transfer. The heat transfer and mixing may be enhanced byincreasing interface area due to strained flows.

Experimental data for conventional �i.e., without in situ reheat�gas-turbines have shown the existence of large radial and circum-ferential temperature gradients downstream of the combustor�5,6�. These temperature nonuniformities, called hot streaks, havea significant impact on the secondary flow and wall temperature ofthe entire turbine. Since the combustor exit flow may containregions where the temperature exceeds the allowable metal tem-perature by 460–930 K �7�, understanding the effects of tempera-ture nonuniformities on the flow and heat transfer in the turbine isessential for increasing vane and blade durability. It is estimatedthat an error of 100 K in predicting the time-averaged temperatureon a turbine rotor can result in an order of magnitude change inthe blade life �8,9�.

Temperature nonuniformities generated by the upstream com-bustor can be amplified in a turbine-burner. Consequently, it isexpected that not only will the secondary flow and wall tempera-ture be affected but also the blade loading due to the modifiedpressure distribution. Temperature nonuniformities in a turbine-burner can also affect the location of hot spots on airfoils and, asa result, can affect the internal and film cooling schemes.

Numerous experimental �7,10–14� and numerical �15–19� in-vestigations explored the influence of temperature nonuniformitieson the flow and heat transfer in a conventional turbine. To the bestknowledge of the authors, there are no data, however, available inthe open literature that illustrate the effects of in situ reheat onturbine-burners. The objective of this paper is to evaluate a nu-merical model for in situ reheat against experimental data for asingle-vane burner and to use this numerical model to investigatethe effects of combustion on the performance of a four-stageturbine-combustor. This numerical simulation is crucial for thedevelopment of turbine-burners which, in spite of their challenges,can provide significant performance gains for turbojet engines andpower generation gas turbine engines.

Section 2 presents the physical model used for the simulation offlow and combustion in a turbine-combustor. The governing equa-

tions and the chemistry model are presented. Section 3 describes

06 by ASME Transactions of the ASME

the numerical model. This section includes information about thegrid generation, boundary conditions, and numerical method. Thecomparison against experimental data and the results for a four-stage turbine are presented in Sec. 4.

2 Physical ModelThe effects of in situ reheat on �i� a single vane-burner, and �ii�

a multirow turbine-burner are modeled by the Reynolds-averagedNavier-Stokes equations and the species conservation equations.The model is three-dimensional for the single vane, and quasi-three-dimensional for the four-stage turbine-combustor, in order toreduce the computational time. This section will present the de-tails of the governing equations and the chemistry model.

2.1 Governing Equations. The unsteady, compressible flowthrough the turbine-combustor was modeled by the Reynolds-averaged Navier-Stokes equations. The flow was assumed to befully turbulent and the kinematic viscosity is computed usingSutherland’s law. The Reynolds-averaged Navier-Stokes equationsand species conservation equations were simplified by using thethin-layer assumption �20�.

In the hypothesis of unity Lewis number, both the Reynolds-averaged Navier-Stokes and species equations were written as�21�

�Q

��+

�F

��+

�G

��=

���M�

Re�

�S

��+ Sch �1�

Note that Eq. �1� was written in the body-fitted curvilinear coor-dinate system �� ,� ,��.

The state and flux vectors of the Reynolds-averaged Navier-Stokes equations in the Cartesian coordinates were

qns = ��

�u

�v

e�, fns = �

�u

�u2 + p

�uv

�e + p�u�, gns = �

�v

�uv

�v2 + p

�e + p�v�

The state and flux vectors of the species conservation equations inthe Cartesian coordinates were

Fig. 1 Experimental setup

Journal of Engineering for Gas Turbines and Power

qsp = ��y1

�y2

]

�yN

�, fsp = ��uy1

�uy2

]

�uyN

�, gsp = ��vy1

�vy2

]

�vyN

�Further details on the description of the viscous terms and chemi-cal source terms are presented in �22�.

2.2 Chemistry Model. The purpose of this investigation wasto determine the influence of in situ reheat on the performance ofa turbine-combustor, as opposed to predicting the complete set ofcombustion products. Consequently, the chemistry model usedherein was a two-step, global, finite rate combustion model formethane and combustion gases �23,24�

CH4 + 1.5O2 → CO + 2H2O�2�

CO + 0.5O2 → CO2

This reduced kinetics model was tuned to match the flame speedand heat released, as opposed to species concentrations �23�. Therate of progress �or Arrhenius-like reaction rate� for methane oxi-dation was given by

q1 = A1 exp�− E1/R/T��CH4�−0.3�O2�1.3 �3�

where A1=2.8�109 s−1, E1 /R=24,360 K. The reaction rate forthe CO/CO2 equilibrium was

q2 = A2 exp�− E2/R/T��CO��O2�0.25�H2O�0.5 �4�

with A2=2.249�1012 �m3/kmol�0.75 s−1 and E2 /R=20,130 K.The symbols in square brackets represent local molar concentra-tions of various species. The net formation/destruction rate of

each species due to all reactions was wi=�k=1Nr Mi�ikqk, where �ik

were the generalized stoichiometric coefficients. Note that thegeneralized stoichiometric coefficient is �ik=�ik� −�ik� , where �ik�and �ik� are stoichiometric coefficients in reaction k for species iappearing as a reactant or as a product. Additional details on theimplementation of the chemistry model can be found in �20�.

3 Numerical ModelThe numerical model used herein to simulate the flow and com-

bustion in the four-stage turbine-combustor was implemented inthe CORSI code �20� and was based on an algorithm developed forunsteady flows in turbomachinery �25�. The Reynolds-averaged

for a single-vane burner

JULY 2006, Vol. 128 / 561

Navier-Stokes equations and the species equations were written instrong conservation form. The fully implicit, finite-difference ap-proximation was solved iteratively at each time level, using anapproximate factorization method. Three Newton-Raphson subit-erations were used to reduce the linearization and factorizationerrors at each time step. The convective terms were evaluatedusing a third-order accurate upwind-biased Roe scheme �26�. Theviscous terms were evaluated using second-order accurate centraldifferences. The scheme was second-order accurate in time.

The size of the computational domain used to simulate the flowinside the turbine-combustor was reduced by taking into accountflow periodicity. Two types of grids were used to discretize theflow field surrounding the rotating and stationary airfoils, asshown later in Fig. 10. An O-grid was used to resolve the govern-ing equations near the airfoil, where the viscous effects were im-portant. An H-grid was used to discretize the governing equationsaway from the airfoil. The O-grid was generated using an ellipti-cal method. The H-grid was algebraically generated. The O- andH-grids were overlaid. The flow variables were communicatedbetween the O- and H-grids through bilinear interpolation. TheH-grids corresponding to consecutive rotor and stator airfoils wereallowed to slip past each other to simulate the relative motion.

The transport of chemical species was modeled by the mass,momentum, energy, and species balance equations. The governingequations of gas dynamics and chemistry were solved using afully decoupled implicit algorithm �21,27–29�. A correction tech-nique was developed to enforce the balance of mass fractions�20�. The governing equations were discretized using an implicit,approximate-factorization, finite difference scheme in delta form�30�. The discretized operational form of both the Reynolds-averaged Navier-Stokes and species conservation equations, com-bined in a Newton-Raphson algorithm �31�, were described in�20,22� where additional details on the implementation of the in-tercell numerical fluxes and on the Roe’s approximate Riemannsolver were presented.

Two classes of boundary conditions were enforced on the gridboundaries: �i� natural boundary conditions, and �ii� zonal bound-ary conditions. The natural boundaries included inlet, outlet, peri-odic, and the airfoil surfaces. The zonal boundaries included thepatched and overlaid boundaries.

At the inlet boundary conditions, the flow angle, average totalpressure, and downstream propagating Riemann invariant werespecified. The upstream propagating Riemann invariant was ex-trapolated from the interior of the domain. At the outlet, the aver-age static pressure was specified, while the downstream propagat-ing Riemann invariant, circumferential velocity, and entropy wereextrapolated from the interior of the domain. Periodicity was en-forced by matching flow conditions between the lower surface ofthe lowest H-grid of a row and the upper surface of the topmost

Fig. 2 Detail of the computation

H-grid of the same row. At the airfoil surface, the following

562 / Vol. 128, JULY 2006

boundary conditions were enforced: the no-slip condition, theadiabatic wall condition, and the zero normal pressure gradientcondition.

Data were transferred from the H-grid to the O-grid along theO-grid’s outermost grid line to impose the zonal boundary condi-tions of the overlaid boundaries. Data were then transferred backto the H-grid along its inner boundary. At the end of each itera-tion, an explicit, corrective, interpolation procedure was per-formed. The patch boundaries were treated similarly, using linearinterpolation to update data between adjoining grids �32�.

4 ResultsThis section starts with the evaluation of the combustion model

against experimental data for a single-vane burner. Selected re-sults of the numerical simulation of unsteady transport phenomenainside a four-stage turbine-combustor are subsequently presented.The section describing the four-stage turbine-combustor beginswith a description of the geometry and flow conditions, followedby a brief discussion of the accuracy of numerical results. The lastpart of this section presents the effects of in situ reheat on theunsteady flow, blade loading, and power increase in the turbine-combustor.

4.1 Single-Vane Burner. To verify the validity of the meth-ane combustion model for in situ reheat applications, a single-vane burner was experimentally investigated and numericallysimulated. In situ reheat tests were run in the Siemens Westing-house small-scale, full-pressure, combustion test facility, shown inFig. 1. Preheated air �0.20 kg/s� and natural gas were delivered toa low-NOx burner section, which was run at full pressure �typi-cally 14 bar�. Air preheat temperature and fuel/air ratio were ad-justed to give an exhaust gas stagnation temperature and compo-sition corresponding to a selected location in a turbine cascade.The exhaust gas was then passed through a pressure-reducing ori-fice to increase the Mach number in the injection and samplingsections to typical turbine levels. A back-pressure control valvewas used to set the sampling section pressure.

Using a calibrated orifice plate, airflow to the system was mea-sured with an accuracy of 2%. Natural gas flow was regulatedwith a mass flow controller with an accuracy of 1%. Gases weresampled at various locations downstream of the injection point,and compositions determined using a gas chromatograph, witherror limits of ±5%. The temperature was measured with thermo-couples, with error limits of ±2 K. Upstream of the vane burner,the mass flow rate of gases was 0.134 kg/s, the total temperaturewas 1507 K and the total pressure was 6.26 bar. The vane burnerwas placed in a 17.8 mm�25.4 mm pipe. The 17.8 mm�25.4 mm pipe reduces to a 17.8 mm�17.8 mm pipe at

omain of the single-vane burner

al d

69.8 mm downstream from the vane burner, as shown in Fig. 2.

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The static pressure upstream of the 17.8 mm�17.8 mm pipe was5.44 bar and at the exit was 4.6 bar. The composition by volumeof the gas mixture upstream of the vane burner was N2 73.48%,H2O 10.59%, O2 10.21%, CO2 4.84%, and Ar 0.88%. The Ar wasnot modeled by the combustion model. The composition by vol-ume of the fuel injected through the vane burner was CH4 96.1%,C2H6 2.0%, C3H8 0.9%, CO2 0.5%, and N2 0.5%. The fuel wasinjected at the temperature of 289 K and static pressure of5.84 bar. The mass flow rate of fuel was 0.416 g/s.

The flow and combustion in the single-vane burner were three-dimensionally modeled. The computational domain extended0.115 m upstream from the vane injection location and 1.071 mdownstream. A detail of the computational domain is shown inFig. 2. The shape of the vane burner was defined by the intersec-tion of two radii. The injection hole had a diameter of 0.66 mm.The injection hole was located at the center of the pipe, however,the shoulders of the vane were not equally spaced with respect tothe injection hole. A detail of the computational grid of the single-vane burner is shown in Fig. 3.

Wall functions were utilized to reduce the number of grid pointsin the boundary layer regions. Consequently, the number of gridcells was limited to 2.2 million. The grid was unstructured andwas generated with Gambit �33�.

The chemistry model used to simulate the in situ reheat was atwo-step finite rate combustion model for methane and combus-tion gases described by Eqs. �2�–�4�. The flow and combustion inthe single-vane burner were modeled with FLUENT �34� as opposedto the four-stage turbine-burner, which was modeled with theCORSI code described in the previous sections. Both FLUENT andCORSI codes had an identical chemistry model.

At inlet, the input data specified total pressure, initial staticpressure, total temperature, turbulence intensity, hydraulic diam-eter, and the composition of the gas mixture, as shown in Table 1.The input data at the injector location specified the same list ofvariables as at inlet. The values of these variables are also shownin Table 1. Note that the small quantities of ethane and propanewere lumped into methane in order to be able to use the two-reaction model presented above. The mass fraction of N2 was notan input datum for the problem. The value of the N2 was calcu-

Fig. 3 Detail of the s

lated such that the sum of all mass fraction species equaled 1. At

Journal of Engineering for Gas Turbines and Power

the outlet, the static pressure value of 4.6 bar was specified.The numerical results shown herein illustrate the spatial varia-

tion of methane and carbon monoxide mass fractions, and totaltemperature. Figure 4 shows the variation of methane mass frac-tion along the z=0 plane of the combustor and at four planesperpendicular to the x-axis located at 12, 15, 20, and 35 mmdownstream of the injector. The methane completely burned at70 mm downstream of the injector. Figure 5 shows methanemass fraction variation in the four planes described above. Thelack of symmetry of the contour plots of methane mass fractionwas due to the off-center position of the vane. All other variablesshow a similar lack of symmetry.

Figure 6 shows the variation of CO along the z=0 plane of thecombustor and at five planes perpendicular to the x-axis located at12, 35, 45, 79, and 94 mm downstream of the injector. The flamewas off center and closer to the lower wall. Figure 7 shows COvariation in the five planes described above. Note that the lastplane, located at 94 mm downstream of the injector, was situatedin the smaller section part of the pipe �17.8 mm�17.8 mm�.

Figure 8 shows the variation of total temperature along the z=0 plane of the combustor and at five planes perpendicular to thex-axis located at 12, 35, 79, 94, and 120 mm downstream of theinjector. The maximum total temperature was 1970 K. Figure 9

gle-vane burner grid

Table 1 Input data for the vane burner

Parameter Inlet Injection

Total pressure �bar� 6.26 7.95Initial static pressure �bar� 5.93 5.84Total temperature �K� 1507 311Turbulence intensity �%� 10 10Hydraulic diameter �m� 0.0254 0.00066Mass fraction

CH40.0 0.9778

O20.115 0.0

CO20.0754 0.01355

CO 0.0 0.0H2O 0.06755 0.0N2

0.74205 0.00865

in

JULY 2006, Vol. 128 / 563

shows total temperature variation in the five planes describedabove. The total temperature predicted by the numerical simula-tion along the centerline at 836 mm downstream of the injectorwas 1602 K. The measured total temperature at the same locationwas 1544 K. The predicted temperature was 58 K higher than themeasured temperature. There are several possible reasons for thetemperature difference, such as: �i� simplified kinetics scheme, �ii�limitations of the k- turbulence model, �iii� approximations dueto using binary diffusion coefficients, and �iv� adiabatic boundaryconditions used in the simulation neglected the wall surface heat

Fig. 4 Contour plots o

Fig. 5 Contour plots of methane ma

564 / Vol. 128, JULY 2006

transfer that occurred in the experiment.To improve temperature prediction, the combustion model was

extended to include the backward reaction of the carbon monox-ide oxidation. The rate of the backward reaction of the CO/CO2equilibrium was

q2b = A2b exp�− E2/R/T��CO2� �5�

with A2b=5�108 �m3/kmol�0.75 s−1 and E2 /R=20,130 K. Thetotal temperature predicted with this improved kinetics model

ethane mass fraction

f m

ss fraction at x=constant planes

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rbo

along the centerline, at 836 mm downstream of the injector, was1562 K. Consequently, the temperature difference between the ex-perimental results and the numerical results was reduced to 18 K.Note that the modeling of the backward reaction did not requiresolving for additional species, and as a result, the increase of thecomputational time was insignificant.

The accuracy of numerical prediction could also be increasedby improving turbulence modeling. The standard k- model usedherein produced better results than the renormalization group orthe realizable k- models �35�. It has been reported that either thestandard k- or the shear stress transport k- model producedslightly better results than the k- model, without requiring addi-tional computational time �36�. The Reynolds-stress turbulencemodel would probably produce more accurate results than the

Fig. 6 Contour plots of ca

Fig. 7 Contour plots of carbon monoxid

Journal of Engineering for Gas Turbines and Power

k- model, but with a higher computational cost. However, theoverall agreement between the measurements and the predictionsobtained with both the k- and Reynolds-stress turbulence modelsare reasonably good �37�. Regardless of the turbulence modelused, the uncertainty caused by turbulence modeling grows as thedistance downstream of the flame increases.

Gas chromatograph measurements at 0.311 m downstream ofthe injector found that the volume fraction of CH4 was 0.35%, andthe volume fraction of CO was 0.16%. At the same location, thenumerical simulation predicted values close to zero �smaller than10−4%� for methane. The carbon monoxide volume fraction pre-dicted by the chemical model �2� was 3�10−4%, whereas themodel that included the backward reaction predicted 0.69%. Thisdiscrepancy between the numerical and experimental results indi-

n monoxide mass fraction

e mass fraction at x=constant planes

JULY 2006, Vol. 128 / 565

Fig. 8 Contour plots of total temperature

Fig. 9 Contour plots of total temperature at x=constant planes

Table 2 Parameters of fuel Injection

Parameter Case 1 Case 2 Case 3

Injection velocity �m/s� 270.6 270.6 77Pressure �bar� 14.88 14.88 14.88Temperature �K� 313 590 313Injection slot size �mm� 0.54 0.54 1.36Fuel mass flow rate ��10−4 kg/s /vane/mm� 13.5 7.2 9.6

566 / Vol. 128, JULY 2006 Transactions of the ASME

cates that the methane oxidation happened more rapidly in thesimulation than in the experiment. The species prediction can beimproved by using a reduced chemical kinetics model that in-cludes more reactions �38–40�. The computational cost of such asimulation, however, will increase several times, depending on theadditional number of species modeled. Since the purpose of thissimulation was the prediction of the influence of heat release onturbine-combustor performance, as opposed to predicting the de-tailed composition of the combustion products, the two-reactionmodel was adopted herein.

The numerical simulation was done on an IBM Regatta pSeries690 computer using four processors. The computation convergedin 3500 iterations. The wall clock time for this run was 195 h.

4.2 Four-Stage Turbine Burner. Once the combustionmodel was tested for the single-vane burner, the next step was toinvestigate a four-stage turbine-burner. The purpose of this nu-merical investigation was to determine the influence of severalfuel injection parameters on the unsteady flow and combustion inthe turbine-burner. Since the computational time of a three-dimensional model for the four-stage turbine-burner would exceedthe computational time of the single-vane burner by a factor offour, and since a parametric analysis of the turbine-burner wasnecessary, it was decided to replace the three-dimensional modelby a less computational expensive quasi-three-dimensional model.A quasi-three-dimensional, as opposed to a two-dimensionalmodel, was needed in order to take into account the large radialvariation of the four-stage turbine. Since FLUENT did not have aquasi-three-dimensional model, the CORSI code was used instead.

4.3 Geometry and Flow Conditions. The blade count of the

Fig. 10 Detail of the medium grid „eshown…

Fig. 11 Variation of averaged tota

Journal of Engineering for Gas Turbines and Power

four-stage turbine-combustor required a full-annulus simulationfor a dimensionally accurate computation. To reduce the compu-tational effort, it was assumed that there were an equal number ofairfoils in each turbine row. As a result, all airfoils except for theinlet guide vane airfoils were rescaled by factors equal to thenumber of airfoils per row divided by the number of airfoils ofrow one. An investigation of the influence of airfoil count on theturbine flow showed that the unsteady effects were amplifiedwhen a simplified airfoil count 1:1 was used �41�. Consequently,the results obtained using the simplified airfoil count represent anupper limit for the unsteady effects.

The inlet temperature in the turbine-combustor exceeded1800 K and the inlet Mach number was 0.155. The inlet flowangle was 0 deg and the inlet Reynolds number was 7,640,000/m,based on the axial chord of the first-stage stator. The values of thespecies mass fractions at inlet in the turbine-burner were yCO2=0.0775, yH2O=0.068, yCO=5.98�10−06, yH2

=2.53�10−07, yO2=0.1131, yN2

=0.7288, and yAr=0.0125. The rotational speed ofthe test turbine-burner was 3600 rpm.

The effects of in situ reheat were investigated by comparing theperformances of a turbine-combustor for several cases of fuel in-jection against the performance of the same turbine without com-bustion. Three of the most representative cases are presentedherein. Pure methane was injected at the trailing edge of the firstvane in all the cases of in situ reheat presented herein. The param-eters that varied in the turbine-combustor were the injection ve-locity, methane temperature, and injection slot dimension. Theseparameters and the fuel mass flow rate per vane and span lengthare presented in Table 2.

ry other grid point in each direction

ve

l enthalpy „absolute or relative…

JULY 2006, Vol. 128 / 567

4.4 Accuracy of Numerical Results. To validate the accu-racy of the numerical results corresponding to the governing equa-tions used, it was necessary to show that the results were indepen-dent of the grid that discretizes the computational domain. Theverification of grid independence results was presented in �20�,where a one-stage turbine-combustor was simulated. Note that thegrids were generated such that, for the given flow conditions, they+ number was �1. Approximately 20 grid points were used todiscretize the boundary layer regions.

Based on the conclusions of accuracy investigation presented in�20�, the medium grid was used herein since it provided the bestcompromise between accuracy and computational cost. This gridhad 53 grid points normal to the airfoil, 225 grid points along theairfoil in the O-grid, 75 grid points in the axial direction, and 75grid points in the circumferential direction in the H-grid. The sta-tor airfoils and rotor airfoils had the same number of grid points.The inlet and outlet H-grids each had 36 grid points in the axialdirection and 75 grid points in the circumferential direction. Thegrid is shown in Fig. 10, where for clarity every other grid point ineach direction is shown.

The results presented in this paper were computed using threeNewton subiterations per time step and 2700 time steps per cycle.Here, a cycle is defined as the time required for a rotor to travel a

Fig. 12 Variation of stagnation tempcase without combustion and case 1

Fig. 13 Contour plots of methane mas

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distance equal to the pitch length at midspan. To ensure timeperiodicity, each simulation was run in excess of 80 cycles. Thenumerical simulation was done on a 64-processor SGI Origin3800 computer. The computational time for a run was 160 h.

4.5 Unsteady Temperature Variation. The variation of totalenthalpy for the three in situ reheat cases and for the no combus-tion case is shown in Fig. 11. The abscissa indicates the axiallocation. S1 denotes stator 1, R1 denotes rotor 1, etc. The totalenthalpy was calculated at the inlet and outlet of each row. De-pending on the row type, that is, stator or rotor, the total enthalpywas calculated using either the absolute or the relative velocity.The switch between using absolute or relative velocities generateddiscontinuities between rows. As shown in Fig. 11, for all fuelinjection cases, the total enthalpy increased compared to the nocombustion case. The largest enthalpy increase was located on thefirst rotor, where most of the combustion takes place. The com-bustion and heat release continued throughout the second statorand rotor, as indicated by the total enthalpy variation shown inFig. 11 �42�.

The stagnation temperature variation along the first row of ro-tors was strongly influenced by the in situ reheat, as shown in Fig.12. Figure 12 shows the averaged, minimum, and maximum stag-

ture along first row of rotors for thein situ reheat

eraof

s fraction „case 1, first three stages…

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nation temperatures for the flow without combustion and for case1 of flow with combustion. On the pressure side, the averagedtemperature of case 1 was 180 K larger than the no combustioncase temperature. At the leading edge, however, the averaged tem-perature of case 1 was 70 K lower than in the no combustioncase. On the suction side, the averaged temperature of case 1 wasslightly higher than in the no combustion case. On most of thesuction side, the averaged temperature of case 1 was approxi-mately 15 to 20 K larger than the no combustion casetemperature.

The averaged temperature indicates that combustion took placeon the pressure side of the rotor airfoil. This conclusion is alsosupported by snapshots of contour plots of methane and oxygenmass fraction shown in Figs. 13 and 14. The existence of smallregions where the averaged temperature of the case with combus-tion was lower than the average temperature of the case withoutcombustion indicates that combustion was not completed. Conse-quently, the low enthalpy of the fuel injected reduced the airfoiltemperature locally. The maximum temperature of the case withcombustion was larger than the maximum temperature of the nocombustion case over the entire airfoil. On the pressure side, theminimum temperature of the case with combustion was largerthan the minimum temperature of the case without combustion.On most of the suction side, however, the minimum temperatureof the case with combustion was smaller than the minimum tem-perature of the case without combustion, indicating that the un-burned, cold fuel injected was affecting this region �42�.

4.6 Unsteady Force Variation. The fuel injection in theturbine-combustor modified the tangential forces in the turbine, asshown in Table 3. In situ reheat decreased tangential force Fy onthe first blade row but increased tangential force on the subse-quent rows. Since the tangential force decrease on the first stagewas smaller than the increase on the subsequent stages, the powerof the turbine-combustor increased for all cases with combustion.The largest power increase was 5.1% and corresponded to case 1.Power increased by 2.8% in case 2 and 4.6% in case 3. Althoughthe variation of the averaged blade force Ftot was rather small, asshown in Table 3, the power increase was significant.

The time variation of the rotor blade tangential forces, shown inFig. 15, indicates that the largest amplitudes occurred in the lastrotor row and the smallest amplitudes occurred in the first rotorrow. This conclusion is valid for every combustion or no combus-tion case.

A phase shift caused by fuel injection is visible for the first andsecond rotor blades. The larger unsteadiness within the secondrotor makes this phenomenon more clearly distinguishable in Fig.15�b�. The patches of burning mixture and the reduced degree of

Fig. 14 Contour plots of oxygen ma

mixedness were the probable causes for this tangential force phase

Journal of Engineering for Gas Turbines and Power

shift in the upstream region.Figure 16 shows the fast Fourier transform of the tangential

forces. They have been nondimensionalized by the average tan-gential force obtained from the case without fuel injection. Theblades of the fourth rotor were excited the most. This excitationoccurred at the first blade passing frequency �BPF�, which was1920 Hz. For the rest of the blades, the excitation due to thesecond BPF was comparable in amplitude to the excitation of thefirst BPF. Except for the first rotor in case 1 and the third rotor incase 3, the fuel injection increased the excitation of the first BPF.The largest amplitude increase was 216% and occurred on thethird-row blades in case 2. The unsteady force, however, was50% of the maximum amplitude value that occurred on thefourth rotor blade at BPF �42�.

5 ConclusionsA two-reaction, global, finite rate combustion model was evalu-

ated against experimental data for a single-vane burner. This com-bustion model has been utilized to explore the effects of in situreheat in a four-stage turbine-combustor. The complexity of thetransport phenomena in a multistage turbine-combustor generateda challenging numerical simulation. The large unsteadiness andstraining of the flow along with the wide range of velocity varia-tion lead to a wide range of local characteristic time scales forflow and combustion, which strongly impacted the ongoing reac-tions.

The numerical simulation was used to predict the airfoil tem-perature variation and the unsteady blade loading in a four-stageturbine-combustor. The largest excitation of the four-stageturbine-combustor corresponded to the fourth-stage rotor, with or

Table 3 Forces on blades

No Combustion Case 1 Case 2 Case 3

Ftot1 �kN� 18.28 18.21 18.71 18.67�1 �deg� 38.4 36.4 36.1 36.3Fy1 �kN� 11.36 10.81 11.03 11.05Ftot2 �kN� 11.87 12.27 12.17 12.31�2 �deg� 60.3 61.7 61.9 62.7Fy2 �kN� 10.31 10.81 10.74 10.94Ftot3 �kN� 12.62 13.19 12.75 13.08�3 �deg� 62.2 65.0 63.9 63.8Fy3 �kN� 11.17 11.95 11.45 11.73Ftot4 �kN� 11.41 13.03 12.31 12.58�4 �deg� 65.5 65.5 65.7 66.1Fy4 �kN� 10.38 11.85 11.21 11.51

fraction „case 1, first three stages…

ss

JULY 2006, Vol. 128 / 569

without combustion. The highest excitation corresponded to thefirst blade passing frequency, for all cases analyzed.

The in situ reheat decreased the power of the first stage, butincreased more the power of the following stages. The power ofthe turbine increased between 2.8% and 5.1%, depending on theparameters of the fuel injection.

AcknowledgmentThis paper was prepared with the support of the U.S. Depart-

ment of Energy �DOE�, under Award No. DE-FC26-00NT40913.However, any opinions, findings, conclusions, or recommenda-tions expressed herein are those of the authors and do not neces-sarily reflect the views of the DOE. The Government reserves foritself and others acting on its behalf a royalty-free, nonexclusive,irrevocable, worldwide license for Governmental purposes to pub-lish, distribute, translate, duplicate, exhibit, and perform thiscopyrighted paper. Additional funding was provided by SiemensWestinghouse Power Corporation. The authors gratefully ac-knowledge the support of Charles Alsup, the DOE project man-ager. The authors also appreciate the support of the Texas A&MSupercomputing Center and the Super-Computing Science Con-

2

Fig. 15 Variation of tang

sortium �SC� who generously provided access to the computing

570 / Vol. 128, JULY 2006

resources of the Pittsburgh Supercomputing Center.

NomenclatureA Arrhenius factorE activation energye total intrinsic internal energy per unit volume

�F ,G� inviscid flux vector in curvilinear coordinates�f ,g� inviscid flux vector in Cartesian coordinates

M Mach numberM molar massNr number of reactionsp pressureQ state vector in curvilinear coordinatesq state vector in Cartesian coordinates or rate of

progressR universal gas constantRe Reynolds number

S viscous flux vectoru fluid velocity in the x directionv fluid velocity in the y directiony

tial forces on the rotors

en

i mass fraction of species i

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sf

w species net production rate� adiabatic exponent �ratio of specific heats�� stoichiometric coefficient� density� nondimensional time

�� ,�� curvilinear coordinates

Subscriptsch chemical source term

i species index� upstream infinity

Superscriptsns Navier-Stokessp species� products� reactants

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