Prediction of the Turbine Tip Convective Heat Flux Using Discrete Green's Functions

2018 ◽  
Vol 140 (7) ◽  
Author(s):  
Valeria Andreoli ◽  
David G. Cuadrado ◽  
Guillermo Paniagua
Keyword(s):  
Heat Flux ◽  
Convective Heat ◽  
Wall Temperature ◽  
Green's Functions ◽  
Green’S Functions ◽  
Transfer Problem ◽  
Turbine Rotor ◽  
Heat Fluxes ◽  
Viscous Effects ◽  
Adiabatic Wall

The heat fluxes across the turbine tip gap are characterized by large unsteady pressure gradients and shear from the viscous effects. The classical Newton heat convection equation, based on the turbine inlet total temperature, is inadequate. Previous research from our team relied on the use of the adiabatic wall temperature. In this paper, we propose an alternative approach to predict the convective heat transfer problem across the turbine rotor tip using discrete Green's functions (DGF). The linearity of the energy equation in the solid domain with constant thermal properties can be applied with a superposition technique to measure the data extracted from flow simulations to determine the Green's function distribution. The DGF is a matrix of coefficients that relate the temperature spatial (GF) distribution with the heat flux. This methodology is first applied to a backward facing step, validated using experimental data. The final aim of this paper is to demonstrate the method in the rotor turbine tip. A turbine stage at engine-like conditions was assessed using cfd software. The heat flux pulses were applied at different locations in the rotor tip geometry, and the increment of temperature in this zone was evaluated for different clearances, with a consequent variation of the DGF coefficients. Ultimately, a detailed uncertainty analysis of the methodology was included based on the magnitude of the heat flux pulses used in the DGF coefficients calculation and the uncertainty in the experimental measurements of the wall temperature.

Analysis of the Heat Transfer Driving Parameters in Tight Rotor Blade Tip Clearances

2015 ◽  
Vol 138 (1) ◽  
Author(s):  
Sergio Lavagnoli ◽  
Cis De Maesschalck ◽  
Guillermo Paniagua
Keyword(s):  
Heat Transfer ◽  
Heat Flux ◽  
Convective Heat Transfer ◽  
Convective Heat ◽  
Wall Temperature ◽  
Turbine Rotor ◽  
Tip Clearance ◽  
Transfer Coefficients ◽  
Adiabatic Wall ◽  
Flow Parameters

Turbine rotor tips and casings are vulnerable to mechanical failures due to the extreme thermal loads they undergo during engine service. In addition to the heat flux variations during the engine transient operation, periodic unsteadiness occurs at every rotor passage, with amplitude dependent on the tip gap. The development of appropriate predictive tools and cooling schemes requires the precise understanding of the heat transfer mechanisms. The present paper analyses the nature of the overtip flow in transonic turbine rotors running at tight clearances and explores a methodology to determine the relevant flow parameters that model the heat transfer. Steady-state three-dimensional Reynolds-averaged Navier–Stokes (RANS) calculations were performed to simulate engine-like conditions considering two rotor tip gaps, 0.1% and 1%, of the blade span. At tight tip clearance, the adiabatic wall temperature is no longer independent of the solid thermal boundary conditions. The adiabatic wall temperature predicted with the linear Newton's cooling law was observed to rise to unphysical levels in certain regions within the rotor tip gap, resulting in unreliable convective heat transfer coefficients (HTCs). This paper investigates different approaches to estimate the relevant flow parameters that drive the heat transfer. A novel four-coefficient nonlinear cooling law is proposed to model the effects of temperature-dependent gas properties and of the heat transfer history. The four-parameter correlation provided reliable estimates of the convective heat transfer descriptors for the 1% tip clearance case, but failed to model the tip heat transfer of the 0.1% tip gap rotor. The present study allows experimentalists to retrieve information on the gap flow temperature and convective HTC based on the use of wall heat flux measurements. The use of nonlinear cooling laws is sought to improve the evaluation of the rotor heat transfer data while enhancing the understanding of tight-clearance overtip flows.

Scalable Heat Transfer Characterization on Film Cooled Geometries Based on Discrete Green’s Functions

2020 ◽  
Author(s):  
Jorge Saavedra ◽  
Venkat Athmanathan ◽  
Guillermo Paniagua ◽  
Terrence Meyer ◽  
Doug Straub ◽  
...  
Keyword(s):  
Heat Transfer ◽  
Heat Flux ◽  
Flat Plate ◽  
Convective Heat Transfer ◽  
Convective Heat ◽  
Green's Functions ◽  
Green’S Functions ◽  
Numerical Procedure ◽  
Local Heat Transfer ◽  
Sensitivity Matrix

Abstract The aerothermal characterization of film cooled geometries is traditionally performed at reduced temperature conditions, which then requires a debatable procedure to scale the convective heat transfer performance to engine conditions. This paper describes an alternative engine-scalable approach, based on Discrete Green’s Functions (DGF) to evaluate the convective heat flux along film cooled geometries. The DGF method relies on the determination of a sensitivity matrix that accounts for the convective heat transfer propagation across the different elements in the domain. To characterize a given test article, the surface is discretized in multiple elements that are independently exposed to perturbations in heat flux to retrieve the sensitivity of adjacent elements, exploiting the linearized superposition. The local heat transfer augmentation on each segment of the domain is normalized by the exposed thermal conditions and the given heat input. The resulting DGF matrix becomes independent from the thermal boundary conditions, and the heat flux measurements can be scaled to any conditions given that Reynolds number, Mach number, and temperature ratios are maintained. The procedure is applied to two different geometries, a cantilever flat plate and a film cooled flat plate with a 30 degree 0.125” cylindrical injection orifice with length-to-diameter ratio of 6. First, a numerical procedure is applied based on conjugate 3D Unsteady Reynolds Averaged Navier Stokes simulations to assess the applicability and accuracy of this approach. Finally, experiments performed on a flat plate geometry are described to validate the method and its applicability. Wall-mounted thermocouples are used to monitor the surface temperature evolution, while a 10 kHz burst-mode laser is used to generate heat flux addition on each of the discretized elements of the DGF sensitivity matrix.

Scalable Heat Transfer Characterization on Film Cooled Geometries Based on Discrete Green’s Functions

2021 ◽  
Vol 143 (2) ◽  
Author(s):  
Jorge Saavedra ◽  
Venkat Athmanathan ◽  
Guillermo Paniagua ◽  
Terrence Meyer ◽  
Doug Straub ◽  
...  
Keyword(s):  
Heat Transfer ◽  
Heat Flux ◽  
Flat Plate ◽  
Convective Heat Transfer ◽  
Convective Heat ◽  
Green's Functions ◽  
Green’S Functions ◽  
Numerical Procedure ◽  
Local Heat Transfer ◽  
Sensitivity Matrix

Abstract The aerothermal characterization of film-cooled geometries is traditionally performed at reduced temperature conditions, which then requires a debatable procedure to scale the convective heat transfer performance to engine conditions. This paper describes an alternative engine-scalable approach, based on Discrete Green’s Functions (DGF) to evaluate the convective heat flux along film-cooled geometries. The DGF method relies on the determination of a sensitivity matrix that accounts for the convective heat transfer propagation across the different elements in the domain. To characterize a given test article, the surface is discretized in multiple elements that are independently exposed to perturbations in heat flux to retrieve the sensitivity of adjacent elements, exploiting the linearized superposition. The local heat transfer augmentation on each segment of the domain is normalized by the exposed thermal conditions and the given heat input. The resulting DGF matrix becomes independent from the thermal boundary conditions, and the heat flux measurements can be scaled to any conditions given that Reynolds number, Mach number, and temperature ratios are maintained. The procedure is applied to two different geometries, a cantilever flat plate and a film-cooled flat plate with a 30 degree 0.125 in. cylindrical injection orifice with length-to-diameter ratio of 6. First, a numerical procedure is applied based on conjugate 3D unsteady Reynolds-averaged Navier–Stokes (URANS) simulations to assess the applicability and accuracy of this approach. Finally, experiments performed on a flat plate geometry are described to validate the method and its applicability. Wall-mounted thermocouples are used to monitor the surface temperature evolution, while a 10 kHz burst-mode laser is used to generate heat flux addition on each of the discretized elements of the DGF sensitivity matrix.

Analysis of the Heat Transfer Driving Parameters in Tight Rotor Blade Tip Clearances

2014 ◽  
Author(s):  
S. Lavagnoli ◽  
C. De Maesschalck ◽  
G. Paniagua
Keyword(s):  
Heat Transfer ◽  
Heat Flux ◽  
Convective Heat Transfer ◽  
Convective Heat ◽  
Wall Temperature ◽  
Convective Heat Transfer Coefficient ◽  
Tip Clearance ◽  
Adiabatic Wall ◽  
Engine Operation ◽  
Flow Parameters

Turbine rotor tips and casings are vulnerable to mechanical failures due to the extreme thermal loads they undergo during engine operation. In addition to the heat flux variations during the transient phase, high-frequency unsteadiness occurs at every rotor passage, with amplitude dependent on the tip gap. The development of appropriate predictive tools and cooling schemes requires the precise understanding of the heat transfer mechanisms. The present paper analyzes the nature of the overtip flow in transonic turbine rotors running at tight clearances, and explores a methodology to determine the relevant flow parameters that model the heat transfer. Steady-state three-dimensional Reynolds-Averaged Navier-Stokes calculations were performed to simulate engine-like conditions considering two rotor tip gaps, 0.1% and 1% of the blade span. At tight tip clearance, the adiabatic wall temperature is not anymore independent of the solid thermal boundary conditions. The adiabatic wall temperature predicted with the linear Newton’s cooling law was observed to rise to non-physical levels in certain regions within the rotor tip gap, resulting in unreliable convective heat transfer coefficients. This paper investigates different approaches to estimate the relevant flow parameters that drive the heat transfer. The present study allows experimentalists to retrieve information on the gap flow temperature and convective heat transfer coefficient based on the use of wall heat flux measurements. Such approach is required to improve the accuracy in the evaluation of the heat transfer data while enhancing the understanding of tight-clearance overtip flows.

Engine-Scalable Rotor Casing Convective Heat Flux Evaluation Using Inverse Heat Transfer Methods

2018 ◽  
Vol 141 (1) ◽  
Author(s):  
David Gonzalez Cuadrado ◽  
Francisco Lozano ◽  
Valeria Andreoli ◽  
Guillermo Paniagua
Keyword(s):  
Heat Transfer ◽  
Heat Flux ◽  
Green’S Function ◽  
Convective Heat ◽  
Green's Function ◽  
Green's Functions ◽  
Green’S Functions ◽  
Filter Method ◽  
Convective Heat Flux ◽  
Inverse Heat Transfer

In this paper, we propose a two-step methodology to evaluate the convective heat flux along the rotor casing using an engine-scalable approach based on discrete Green's functions . The first step consists in the use of an inverse heat transfer technique to retrieve the heat flux distribution on the shroud inner wall by measuring the temperature of the outside wall; the second step is the calculation of the convective heat flux at engine conditions, using the experimental heat flux and the Green functions engine-scalable technique. Inverse methodologies allow the determination of boundary conditions; in this case, the inner casing surface heat flux, based on measurements from outside of the system, which prevents aerothermal distortion caused by routing the instrumentation into the test article. The heat flux, retrieved from the inverse heat transfer methodology, is related to the rotor tip gap. Therefore, for a given geometry and tip gap, the pressure and temperature can also be retrieved. In this work, the digital filter method is applied in order to take advantage of the response of the temperature to heat flux pulses. The discrete Green's function approach employs a matrix to relate an arbitrary temperature distribution to a series of pulses of heat flux. In this procedure, the terms of the Green's function matrix are evaluated with the output of the inverse heat transfer method. Given that key dimensionless numbers are conserved, the Green's functions matrix can be extrapolated to engine-like conditions. A validation of the methodology is performed by imposing different arbitrary heat flux distributions, to finally demonstrate the scalability of the Green's function method to engine conditions. A detailed uncertainty analysis of the two-step routine is included based on the value of the pulse of heat flux, the temperature measurement uncertainty, the thermal properties of the material, and the physical properties of the rotor casing.

Prediction of the Turbine Tip Convective Heat Flux Using Discrete Green Functions

2017 ◽  
Author(s):  
Valeria Andreoli ◽  
David G. Cuadrado ◽  
Guillermo Paniagua
Keyword(s):  
Heat Flux ◽  
Green Function ◽  
Convective Heat ◽  
Wall Temperature ◽  
Transfer Problem ◽  
Bottom Wall ◽  
Random Pattern ◽  
Inlet Temperature ◽  
Green Functions ◽  
Backward Facing Step

The complex heat and mass transfer across the turbine tip gap requires a detailed analysis which cannot be expressed using the classical Newton heat convection approach. In this portion of the turbine, characterized by tight moving clearances, pressure gradients are counterbalanced by viscous effects. Hence, non-dimensional analysis, based on the boundary layer, is inadequate and therefore the use of an adiabatic wall temperature is questionable. In this paper, we propose an alternative approach to predict the convective heat transfer problem across the turbine rotor tip using Discrete Green Functions. The linearity of the energy equation can be applied with a superposition technique to measure the data extracted from flow simulations to determine the Green’s function distribution. The Discrete Green Function is a matrix of coefficients that relate the increment of temperature observed in a surface with the heat flux integrated on the same surface. These coefficients are independent on the inlet temperature of the flow and are associated to the geometry. The controlled surface is discretized into cells and each cell is associated to a vector of coefficients. The Discrete Green Function coefficients are calculated using the temperature response of the cell to a heat flux pulse imposed at different locations. The methodology was previously applied to a backward facing step to prove its validity. Several simulations were performed applying a representative pulse of heat flux in different locations in the bottom wall of the backward facing step. From these simulations, the increment of temperature in each node of the geometry was retrieved and the Discrete Green Function coefficients associated to the bottom wall were calculated. A numerical validation was performed imposing a random pattern of heat flux and predicting the increment of temperature on the bottom wall under different inlet flow conditions. The final aim of this paper is to demonstrate the method in the rotor turbine tip. A turbine stage at engine-like conditions was assessed using a CFD software. The heat flux pulses were applied at different locations in the rotor tip geometry, and the increment of temperature in this zone was evaluated for different clearances, with a consequent variation of the Discrete Green Function coefficients. A validation of the rotor tip heat flux was accomplished by imposing different heat flux distributions in the studied region. Ultimately, a detailed uncertainty analysis of the methodology was included based in the magnitude of the heat flux pulses used in the Discrete Green Function coefficients calculation and the uncertainty in the experimental measurements of the wall temperature.

Engine-Scalable Rotor Casing Convective Heat Flux Evaluation Using Inverse Heat Transfer Methods

2018 ◽  
Author(s):  
David G. Cuadrado ◽  
Francisco Lozano ◽  
Valeria Andreoli ◽  
Guillermo Paniagua
Keyword(s):  
Heat Transfer ◽  
Heat Flux ◽  
Green’S Function ◽  
Convective Heat ◽  
Green's Function ◽  
Green's Functions ◽  
Green’S Functions ◽  
Filter Method ◽  
Convective Heat Flux ◽  
Inverse Heat Transfer

In this paper, we propose a two-step methodology to evaluate the convective heat flux along the rotor casing using an engine-scalable approach based on Discrete Green’s Functions. The first step consists in the use of an inverse heat transfer technique to retrieve the heat flux distribution on the shroud inner wall by measuring the temperature of the outside wall; the second step is the calculation of the convective heat flux at engine conditions, using the experimental heat flux and the Green Functions engine-scalable technique. Inverse methodologies allow the determination of boundary conditions, in this case the inner casing surface heat flux, based on measurements from outside of the system, which prevents aerothermal distortion caused by routing the instrumentation into the test article. The heat flux, retrieved from the inverse heat transfer methodology, is related to the rotor tip gap. Therefore, for a given geometry and tip gap, the pressure and temperature can also be retrieved. In this work, the Digital Filter Method is applied in order to take advantage of the response of the temperature to heat flux pulses. The Discrete Green’s Function approach employs a matrix to relate an arbitrary temperature distribution to a series of pulses of heat flux. In the present procedure, the terms of the Green’s Function matrix are evaluated with the output of the inverse heat transfer method. Given that key dimensionless numbers are conserved, the Green’s Functions matrix can be extrapolated to engine-like conditions. A validation of the methodology is performed by imposing different arbitrary heat flux distributions, to finally demonstrate the scalability of the Green’s Function Method to engine conditions. A detailed uncertainty analysis of the two-step routine is included based on the value of the pulse of heat flux, the temperature measurement uncertainty, the thermal properties of the material and the physical properties of the rotor casing.

Experimental Study of Heat Transfer to Supercritical Pressure Water Flowing in a 2×2 Rod Bundle

2014 ◽  
Author(s):  
Han Wang ◽  
Qincheng Bi ◽  
Linchuan Wang ◽  
Haicai Lv ◽  
Laurence K. H. Leung
Keyword(s):  
Heat Transfer ◽  
Experimental Data ◽  
Heat Flux ◽  
Pressure Drop ◽  
Wall Temperature ◽  
Mass Flux ◽  
Heat Fluxes ◽  
Outer Diameter ◽  
Square Channel ◽  
Rod Bundle

An experiment has recently been performed at Xi’an Jiaotong University to study the wall temperature and pressure drop at supercritical pressures with upward flow of water inside a 2×2 rod bundle. A fuel-assembly simulator with four heated rods was installed inside a square channel with rounded corner. The outer diameter of each heated rod is 8 mm with an effective heated length of 600 mm. Experimental parameters covered the pressure of 23–28 MPa, mass flux of 350–1000 kg/m2s and heat flux on the rod surface of 200–1000 kW/m2. According to the experimental data, it was found that the circumferential wall temperature distribution of a heated rod is not uniform. The temperature difference between the maximum and the minimum varies with heat flux and/or mass flux. Heat transfer characteristics of supercritical water in bundle were discussed with respect to various heat fluxes. The effect of heat flux on heat transfer in rod bundles is similar with that in tubes or annuli. In addition, flow resistance reflected in the form of pressure loss has also been studied. Experimental results showed that the total pressure drop increases with bulk enthalpy and mass flux. Four heat transfer correlations developed for supercritical pressures water were compared with the present test data. Predictions of Jackson correlation agrees closely with the experimental data.

scholarly journals Solving bio-heat transfer multi-layer equation using Green’s Functions method

2021 ◽  
Vol 2090 (1) ◽  
pp. 012150
Author(s):  
de Oliveira Eduardo Peixoto ◽  
Gilmar Guimaräes
Keyword(s):  
Heat Transfer ◽  
Green's Functions ◽  
Green’S Functions ◽  
Transfer Problem ◽  
Blood Perfusion ◽  
The Body ◽  
Mathematical Background ◽  
Number Of Layers ◽  
Pennes Equation ◽  
Transfer Problems

Abstract An analytical method using Green’s Functions for obtaining solutions in bio-heat transfer problems, modeled by Pennes’ Equation, is presented. Mathematical background on how treating Pennes’ equation and its μ2T term is shown, and two contributions to the classical numbering system in heat conduction are proposed: inclusion of terms to specify the presence of the fin term, μ2T, and identify the biological heat transfer problem. The presentation of the solution is made for a general multi-layer domain, deriving and showing general approaches and Green’s Functions for such n number of layers. Numerical examples are presented to simplify human skin as a two-layer domain: dermis and epidermis, accounting metabolism as a heat source, and blood perfusion only at the dermis. Time-independent summations in the series-solution are written in closed forms, leading to better convergence along the boundaries. Details on obtaining the two-layer solution and its eigenvalues are presented for boundary conditions of prescribed temperature inside the body and convection at the surface, such as its intrinsic verification.

scholarly journals Quantifying pyroconvective injection heights using observations of fire energy: sensitivity of spaceborne observations of carbon monoxide

2015 ◽  
Vol 15 (8) ◽  
pp. 4339-4355 ◽  
Author(s):  
S. Gonzi ◽  
P. I. Palmer ◽  
R. Paugam ◽  
M. Wooster ◽  
M. N. Deeter
Keyword(s):  
Heat Flux ◽  
Carbon Monoxide ◽  
Convective Heat ◽  
Biomass Burning ◽  
Environmental Conditions ◽  
Heat Fluxes ◽  
Plume Rise ◽  
Convective Heat Flux ◽  
Active Fire ◽  
Fire Area

Abstract. We use observations of active fire area and fire radiative power (FRP) from the NASA Moderate Resolution Imaging Spectroradiometers (MODIS), together with a parameterized plume rise model, to estimate biomass burning injection heights during 2006. We use these injection heights in the GEOS-Chem (Goddard Earth Observing System Chemistry) atmospheric chemistry transport model to vertically distribute biomass burning emissions of carbon monoxide (CO) and to study the resulting atmospheric distribution. For 2006, we use over half a million FRP and fire area observations as input to the plume rise model. We find that convective heat fluxes and active fire area typically lie in the range of 1–100 kW m−2 and 0.001–100 ha, respectively, although in rare circumstances the convective heat flux can exceed 500 kW m−2. The resulting injection heights have a skewed probability distribution with approximately 80% of the injections remaining within the local boundary layer (BL), with occasional injection height exceeding 8 km. We do not find a strong correlation between the FRP-inferred surface convective heat flux and the resulting injection height, with environmental conditions often acting as a barrier to rapid vertical mixing even where the convective heat flux and active fire area are large. We also do not find a robust relationship between the underlying burnt vegetation type and the injection height. We find that CO columns calculated using the MODIS-inferred injection height (MODIS-INJ) are typically −9 to +6% different to the control calculation in which emissions are emitted into the BL, with differences typically largest over the point of emission. After applying MOPITT (Measurement of Pollution in the Troposphere) v5 scene-dependent averaging kernels we find that we are much less sensitive to our choice of injection height profile. The differences between the MOPITT and the model CO columns (max bias ~ 50%), due largely to uncertainties in emission inventories, are much larger than those introduced by the injection heights. We show that including a realistic diurnal variation in FRP (peaking in the afternoon) or accounting for subgrid-scale emission errors does not alter our main conclusions. Finally, we use a Bayesian maximum a posteriori approach constrained by MOPITT CO profiles to estimate the CO emissions but because of the inherent bias between model and MOPITT we find little impact on the resulting emission estimates. Studying the role of pyroconvection in the distribution of gases and particles in the atmosphere using global MOPITT CO observations (or any current spaceborne measurement of the atmosphere) is still associated with large errors, with the exception of a small subset of large fires and favourable environmental conditions, which will consequently lead to a bias in any analysis on a global scale.

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