This document describes mathematical models for simulating the temperature fields of gas turbine blades during convective cooling. It presents boundary integral equation methods (BIEM) and finite difference methods (FDM) for calculating the stationary and quasi-stationary temperature distribution on a blade profile with radial cooling channels. The BIEM approach formulates the problem as a system of boundary integral equations involving temperature values and heat transfer coefficients on the blade surface and cooling channel boundaries. Numerical methods are developed to solve these equations, including discrete logarithmic potential operators and non-uniform surface discretizations. The reliability of the proposed methods is confirmed through computational and experimental analysis of heat transfer for a gas turbine nozzle blade.
Simulation of gas turbine blade for enhancement of efficiency of gas turbine...IJMER
As day by day population of the world is increasing and our resources are frequently reducing
hence to meet this demand of the world of energy we have to move to a device which have a maximum
efficiency for the condition turbo-machinery are better suited machines having a good efficiency, in
which a Gas turbine is best example of turbo- machinery Turbine is the part of gas turbine which provide
the power to compressor to run or provide power to external source from where energy can be extracted
by attaching alternator in the shaft of Gas turbine. As in earlier a lot of work have been done by the
researcher to increase the efficiency and standard of Gas turbine by the method of film cooling, coating,
and curvature of blade to protect the blade from high temperature of 1200 C° inside the Gas turbine to
increase the life of blade without considering about the efficiency of the engine As in this work is to
enhancement of efficiency of Gas turbine. Gas turbine blade is very important component of engine as
they are attached to both turbine or compressor and turbine provide energy to compressor hence the
turbine blade are more important component to enhance the efficiency which will be analyzed on the
basis of blade height area of fluid flow , area of blade thickness and angles . This simulation is based on
the define value of temperature pressure density of fluid and solid used in blade construction will be
meshed in ANSYS and calculation on the basis of FEM and the result from this calculation over the
temperature and fluid flow inside the gas turbine of different number of blade is studied will be compare
to reach high efficiency point. By determent these value output is formulated on graph chart and will be
studied and result obtain
Rib roughened cooling passages for turbine coolingDevi Archana Das
In modern gas turbine engines, the continuous increase of power for an expected lifetime has resulted in a continuous increase of cycle pressure ratio and turbine inlet temperature. The later implies that advanced materials and cooling techniques must be adapted for a safe operation of the high pressure gas turbine blades and vanes. This need for high power and high efficiency gas turbine engines forces the designer to continuously increase the turbine inlet temperature. In recent military applications, the turbine inlet temperature could be as high as 2000K, far above the melting temperature of the most advanced vane and blade materials. Thus apart from the progress made in the metallurgical domain, a continuous cooling of blade of turbine 1st stages allows operating at temperatures which are far above material’s melting point without affecting component integrity and geometry. The efficiency of the blade cooling system is therefore strictly related to the safe operation of the engine and complete understanding of the convection mechanisms resulting from the cooling techniques is mandatory.
The most common internal heat transfer enhancement methods of heat transfer augmentation in gas turbine airfoils are ribs, pins, jet impingement, and flow disturbing inserts.To maintain the flow inside the cooling passage of turbine blade turbulent all these methods can be used. These devices act to increase turbulent mixing through the enhancement of turbulence.
Enhancement of Heat Transfer Analysis and Optimization of Engine Fins of Vary...ijtsrd
The Engine cylinder is one of the major automobile components, which is subjected to high temperature variations and thermal stresses. In order to cool the cylinder, fins are provided on the cylinder to increase the rate of heat transfer. By doing thermal analysis on the engine cylinder fins, it is helpful to know the heat dissipation inside the cylinder. The principle implemented in this project is to increase the heat dissipation rate by using the invisible working fluid, nothing but air. As know, by increasing the surface area we can increase the heat dissipation rate, so designing such a large complex engine is very difficult. The main purpose of using these cooling fins is to cool the engine cylinder by air. The main aim of the project is to analyse the thermal properties by varying geometry, material, distance between the fins and thickness of cylinder fins. Parametric models of cylinder with fins have been developed to predict the transient thermal behaviour. The models are created by varying the geometry circular and also by varying thickness of the fins for both geometries. The 3D modelling software used is Pro/Engineer. Thermal analysis is done on the cylinder fins to determine variation temperature distribution over time. The analysis is done using ANSYS. Thermal analysis determines temperatures and other thermal quantities. In this thesis, using materials cast iron, Copper and Aluminium alloy 6082 are also for cylinder fin body. Thermal analysis is done using all the three materials by changing geometries, distance between the fins and thickness of the fins for the actual model of the cylinder fin body. K. Karthikeyan | C. Saravanan | Dr. T. Senthil Kumar"Enhancement of Heat Transfer Analysis and Optimization of Engine Fins of Varying Geometry" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-2 | Issue-4 , June 2018, URL: http://www.ijtsrd.com/papers/ijtsrd14327.pdf http://www.ijtsrd.com/engineering/mechanical-engineering/14327/enhancement-of-heat-transfer-analysis-and-optimization-of-engine-fins-of-varying-geometry/k-karthikeyan
Natural convection heat transfer flow visualization of perforated fin arrays ...eSAT Journals
Abstract
The present paper reports, the validation of results of modeling and simulation in CFD by experiment on the fluid flow and heat
transfer characteristics of a fin arrays with lateral circular perforation and its external dimensionally equivalent solid fin arrays
equipped on horizontal flat surface a problem of natural convection. The simulation is carried out using the fluid flow (CFX)
workbench of ANSYS 12.0. In this study, results shows that formation of the stagnant layer around the solid fin array which slowdowns
the heat dissipation rate. Increase in the fluid flow movement around the fin results increase in the heat dissipation rate. It can
be achieved by adding perforation to the fins. Natural convection is a buoyancy driven phenomenon; the state of the art of CFX was
used to carry the study of fluid flow separation and velocity field over a fin array. New designed perforated fins have an improvement
in average Nusselt number, over its external dimensionally equivalent solid fin arrays.
Keywords: CFD simulation, perforated fins, Natural convection, Heat sink, Nusselt number, Flow Visualization
Review of heat transfer augmentation for cooling of turbine blade tip by geom...eSAT Publishing House
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
Simulation of gas turbine blade for enhancement of efficiency of gas turbine...IJMER
As day by day population of the world is increasing and our resources are frequently reducing
hence to meet this demand of the world of energy we have to move to a device which have a maximum
efficiency for the condition turbo-machinery are better suited machines having a good efficiency, in
which a Gas turbine is best example of turbo- machinery Turbine is the part of gas turbine which provide
the power to compressor to run or provide power to external source from where energy can be extracted
by attaching alternator in the shaft of Gas turbine. As in earlier a lot of work have been done by the
researcher to increase the efficiency and standard of Gas turbine by the method of film cooling, coating,
and curvature of blade to protect the blade from high temperature of 1200 C° inside the Gas turbine to
increase the life of blade without considering about the efficiency of the engine As in this work is to
enhancement of efficiency of Gas turbine. Gas turbine blade is very important component of engine as
they are attached to both turbine or compressor and turbine provide energy to compressor hence the
turbine blade are more important component to enhance the efficiency which will be analyzed on the
basis of blade height area of fluid flow , area of blade thickness and angles . This simulation is based on
the define value of temperature pressure density of fluid and solid used in blade construction will be
meshed in ANSYS and calculation on the basis of FEM and the result from this calculation over the
temperature and fluid flow inside the gas turbine of different number of blade is studied will be compare
to reach high efficiency point. By determent these value output is formulated on graph chart and will be
studied and result obtain
Rib roughened cooling passages for turbine coolingDevi Archana Das
In modern gas turbine engines, the continuous increase of power for an expected lifetime has resulted in a continuous increase of cycle pressure ratio and turbine inlet temperature. The later implies that advanced materials and cooling techniques must be adapted for a safe operation of the high pressure gas turbine blades and vanes. This need for high power and high efficiency gas turbine engines forces the designer to continuously increase the turbine inlet temperature. In recent military applications, the turbine inlet temperature could be as high as 2000K, far above the melting temperature of the most advanced vane and blade materials. Thus apart from the progress made in the metallurgical domain, a continuous cooling of blade of turbine 1st stages allows operating at temperatures which are far above material’s melting point without affecting component integrity and geometry. The efficiency of the blade cooling system is therefore strictly related to the safe operation of the engine and complete understanding of the convection mechanisms resulting from the cooling techniques is mandatory.
The most common internal heat transfer enhancement methods of heat transfer augmentation in gas turbine airfoils are ribs, pins, jet impingement, and flow disturbing inserts.To maintain the flow inside the cooling passage of turbine blade turbulent all these methods can be used. These devices act to increase turbulent mixing through the enhancement of turbulence.
Enhancement of Heat Transfer Analysis and Optimization of Engine Fins of Vary...ijtsrd
The Engine cylinder is one of the major automobile components, which is subjected to high temperature variations and thermal stresses. In order to cool the cylinder, fins are provided on the cylinder to increase the rate of heat transfer. By doing thermal analysis on the engine cylinder fins, it is helpful to know the heat dissipation inside the cylinder. The principle implemented in this project is to increase the heat dissipation rate by using the invisible working fluid, nothing but air. As know, by increasing the surface area we can increase the heat dissipation rate, so designing such a large complex engine is very difficult. The main purpose of using these cooling fins is to cool the engine cylinder by air. The main aim of the project is to analyse the thermal properties by varying geometry, material, distance between the fins and thickness of cylinder fins. Parametric models of cylinder with fins have been developed to predict the transient thermal behaviour. The models are created by varying the geometry circular and also by varying thickness of the fins for both geometries. The 3D modelling software used is Pro/Engineer. Thermal analysis is done on the cylinder fins to determine variation temperature distribution over time. The analysis is done using ANSYS. Thermal analysis determines temperatures and other thermal quantities. In this thesis, using materials cast iron, Copper and Aluminium alloy 6082 are also for cylinder fin body. Thermal analysis is done using all the three materials by changing geometries, distance between the fins and thickness of the fins for the actual model of the cylinder fin body. K. Karthikeyan | C. Saravanan | Dr. T. Senthil Kumar"Enhancement of Heat Transfer Analysis and Optimization of Engine Fins of Varying Geometry" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-2 | Issue-4 , June 2018, URL: http://www.ijtsrd.com/papers/ijtsrd14327.pdf http://www.ijtsrd.com/engineering/mechanical-engineering/14327/enhancement-of-heat-transfer-analysis-and-optimization-of-engine-fins-of-varying-geometry/k-karthikeyan
Natural convection heat transfer flow visualization of perforated fin arrays ...eSAT Journals
Abstract
The present paper reports, the validation of results of modeling and simulation in CFD by experiment on the fluid flow and heat
transfer characteristics of a fin arrays with lateral circular perforation and its external dimensionally equivalent solid fin arrays
equipped on horizontal flat surface a problem of natural convection. The simulation is carried out using the fluid flow (CFX)
workbench of ANSYS 12.0. In this study, results shows that formation of the stagnant layer around the solid fin array which slowdowns
the heat dissipation rate. Increase in the fluid flow movement around the fin results increase in the heat dissipation rate. It can
be achieved by adding perforation to the fins. Natural convection is a buoyancy driven phenomenon; the state of the art of CFX was
used to carry the study of fluid flow separation and velocity field over a fin array. New designed perforated fins have an improvement
in average Nusselt number, over its external dimensionally equivalent solid fin arrays.
Keywords: CFD simulation, perforated fins, Natural convection, Heat sink, Nusselt number, Flow Visualization
Review of heat transfer augmentation for cooling of turbine blade tip by geom...eSAT Publishing House
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
Computational analysis of heat transfer through four-stroke S. I. engine finsijsrd.com
it is important for an air-cooled engine to utilize fins for effective engine cooling to maintain uniform temperature in the cylinder periphery. Many experimental works has been done to improve the heat release of the cylinder and fin efficiency. In this study, heat release of an IC engine cylinder cooling with straight fins and with wavy fins is calculated numerically using commercially available CFD tool ANSYS. The IC engine is initially at 500⁰C and the heat release from the cylinder is analysed at a wind velocity of 60 km/hr to 100 km/hr. The heat release from both the cylinders is compared. With the help of the available numerically results, the design of the I. C. engine cooling fins can be modified for improving the heat release and efficiency.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
Heat Transfer Analysis and Optimization of Engine Cylinder Fins of Varying Ge...IOSR Journals
: The main aim of the project is to analyze the thermal properties by varying geometry, material and
thickness of cylinder fins. Parametric models of cylinder with fins have been developed to predict the transient
thermal behavior. The models are created by varying the geometry, rectangular, circular and curved shaped
fins and also by varying thickness of the fins. The 3D modeling software used is Pro/Engineer.The analysis is
done using ANSYS. Presently Material used for manufacturing cylinder fin body is Aluminum Alloy 204 which
has thermal conductivity of 110-150W/mk. We are analyzing the cylinder fins using this material and also using
Aluminum alloy 6061 and Magnesium alloy which have higher thermal conductivities.
Experimentation and analysis of heat transfer through perforated fins of diff...SharathKumar528
Engineering Project by Abhijath HB, Dashartha H S, Akshay Mohanraj and Sharath Kumar M S involving analysis of Fins( Heat exchanging extensions) with various geometrical perforations.
Analysis of Coiled-Tube Heat Exchangers to Improve Heat Transfer Rate With Sp...IJMER
Steady heat transfer enhancement has been studied in helically coiled-tube heat exchangers. The outer side of the wall of the heat exchanger contains a helical corrugation which makes a helical rib on the inner side of the tube wall to induce additional swirling motion of fluid particles. Numerical calculations have been carried out to examine different geometrical parameters and the impact of flow and thermal boundary conditions for the heat transfer rate in laminar and transitional flow regimes. Calculated results have been compared to existing empirical formula and experimental tests to investigate the validity of the numerical results in case of common helical tube heat exchanger and additionally results of the numerical computation of corrugated straight tubes for laminar and transition flow have been validated with experimental tests available in the literature. Comparison of the flow and temperature fields in case of common helical tube and the coil with spirally corrugated wall configuration are discussed. Heat exchanger coils with helically corrugated wall configuration show 80–100% increase for the inner side heat transfer rate due to the additionally developed swirling motion while the relative pressure drop is 10–600% larger compared to the common helically coiled heat exchangers. New empirical Co-relation has been proposed for the fully developed inner side heat transfer prediction in case of helically corrugated wall configuration.
Engineering Project involving design,manufacturing, testing of Fins( Heat exchanger for 2 wheeler). An analysis of heat transfer on fins with various geometrical perforations.
Numerical Analysis of Inverted Notched Fin Array Using Natural ConvectionIOSR Journals
Abstract:Geometry and orientation plays an important role in natural convection heat transfer. For
horizontal rectangular fin array a chimney flow pattern is developed due to density difference. This flow
pattern creates a stagnant zone near central bottom region. That portion does not contribute much towards
heat dissipation. This area is removed from fins and they became inverted notched fins. This modified geometry
reduces material cost, material weight without hampering heat transfer rate. Numerical models are prepaid to
investigate heat transfer characteristics in plane fins and inverted notched fins. This investigation is also
extended over different types of notches and their effectiveness comparison. Fin spacing, fin height, fin length,
heater input, percentage of area removed in the form of inverted notch are the parameters under
consideration. This analysis is done numerically using CFD package (Fluent). It is found that the heat transfer
coefficient of inverted notch fin array is 25% to 35% higher as compared with normal fin array. Also we found
that the triangular shape notch gives better result than trapezoidal and rectangular shape notch.
Key words: Inverted Notched Fin, Chimney Flow, Natural Convection, Heat transfer coefficient enhancement
Thermal analysis of water cooled charge air cooler in turbo charged diesel en...eSAT Journals
Abstract
Air-cooled and water-cooled charge air coolers are used as applications in automobiles. The cooling media include engine’s
coolant, ambient air or any other external coolant source. Charge air cooler with offset strip fin geometry is chosen on both hot
and cold sides with fin thickness 0.15 mm, length 3.2 mm, height 5.2 mm and frequency of 18 fins per inch. The flow rate on hot
side is taken as 0.3 kg/s and cold side as 2 kg/s for air-cooled charge air cooler and for water-cooled charge air cooler on hot
side it is 0.3 kg/s and cold side is 0.265 kg/s with inlet temperature on hot side 160 oC and cold side 40 oC. With the above
conditions, performance evaluation is done on both charge air coolers. Having found water-cooled charge air cooler is having
higher effectiveness than air-cooled type, performance evaluation is carried out on water-cooled charge air cooler with varying
mass flow rates on hot side from 0.1 to 0.6 kg/s and keeping cold fluid rate constant at 0.265 kg/s and the outputs obtained are
Colburn-j factor, Fanning friction factor, heat transfer coefficient, overall heat transfer coefficient and effectiveness of heat
exchanger. From the obtained results, graphs are drawn to assess the performance of the water charge air cooler.
Keywords: Charge Air Cooler, Plate Fin Heat Exchanger, Colburn-J Factor, Heat Transfer Coefficient, Fanning
Friction Factor, Effectiveness
Helically Coiled Tube with Different Geometry and Curvature Ratio on Convecti...AM Publications
A helically coil-tube heat exchanger is generally applied in industry applications due to its compact structure, larger heat transfer area and higher heat transfer capability. Several studies from literature have also indicated that heat transfer rate in helically coiled tube are superior to straight tube due to complex flow pattern exist inside helical pipe. The concept behind compact heat exchanger is to decrease size and increase heat load which is the typical feature of modern helical tube heat exchanger. While the heat transfer characteristics of helical coil heat exchangers are available in the literature, This paper elaborates a brief review on different curvature ratio and geometry of tubes in heat transfer through heat exchangers.
Heat transfer augmentation in different geometries of dimpled surface under n...eSAT Journals
Abstract The prime objective of present work is to study experimentally the heat transfer augmentation through various geometries of dimpled surfaces in longitudinal and lateral directions. In this paper horizontal rectangular plates of copper and aluminum with different dimpled geometries (like square, circular and triangular) for in-line arrangements were studied in natural convection with steady laminar external flow condition. The various parameters considered for study are Nusselt number, heat transfer coefficient and heat transfer rate for a constant Prandtl number (0.7) and Grashof number (104-107).It has been found that the heat transfer coefficient and heat transfer rate increases for various dimpled surfaces as compared to plane surface. It has been also found that the heat transfer coefficient and heat transfer rate increases along longitudinal direction as compared to lateral direction. And it is seen that heat transfer rate is maximum for triangular shape dimple when the apex of triangle is faced towards inlet of air flow Finally it is concluded that heat transfer enhancement takes place along the dimpled surface
Heat transfer augmentation in different geometries of dimpled surface under n...eSAT Publishing House
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology.
Improving boiler efficiency by using air preheaterNetha Jashuva
Abstract: Air pre-heater is a heat transfer surface in which air temperature is raised by transferring heat from other media such as flue gas .Hot air is necessary for rapid combustion in the furnace and also for drying coal in milling plants. So an essential boiler accessory which serves this purpose is air pre-heater. The air pre-heater are not essential for operation of steam generator, but they are used where a study of cost indicates that money can be saved or efficient combustion can be obtained by their use. The decision for its adoption can be made when the financial advantages is weighed against the capital cost of heater in the present paper we have taken up the operation and performance analysis of LJUNGSTROM AIRPREHEATER27VITM 1900 of 2x210 MW capacity Rayalaseema Thermal Power Plant, Kalamala and compare with Rothemuhle air pre-heater. In analysis of performance preventive measures for corrosion of heating elements has been studied, and also air heater leakage, corrected gas outlet temperature and finally gas efficiency has been calculated.
Computational analysis of heat transfer through four-stroke S. I. engine finsijsrd.com
it is important for an air-cooled engine to utilize fins for effective engine cooling to maintain uniform temperature in the cylinder periphery. Many experimental works has been done to improve the heat release of the cylinder and fin efficiency. In this study, heat release of an IC engine cylinder cooling with straight fins and with wavy fins is calculated numerically using commercially available CFD tool ANSYS. The IC engine is initially at 500⁰C and the heat release from the cylinder is analysed at a wind velocity of 60 km/hr to 100 km/hr. The heat release from both the cylinders is compared. With the help of the available numerically results, the design of the I. C. engine cooling fins can be modified for improving the heat release and efficiency.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
Heat Transfer Analysis and Optimization of Engine Cylinder Fins of Varying Ge...IOSR Journals
: The main aim of the project is to analyze the thermal properties by varying geometry, material and
thickness of cylinder fins. Parametric models of cylinder with fins have been developed to predict the transient
thermal behavior. The models are created by varying the geometry, rectangular, circular and curved shaped
fins and also by varying thickness of the fins. The 3D modeling software used is Pro/Engineer.The analysis is
done using ANSYS. Presently Material used for manufacturing cylinder fin body is Aluminum Alloy 204 which
has thermal conductivity of 110-150W/mk. We are analyzing the cylinder fins using this material and also using
Aluminum alloy 6061 and Magnesium alloy which have higher thermal conductivities.
Experimentation and analysis of heat transfer through perforated fins of diff...SharathKumar528
Engineering Project by Abhijath HB, Dashartha H S, Akshay Mohanraj and Sharath Kumar M S involving analysis of Fins( Heat exchanging extensions) with various geometrical perforations.
Analysis of Coiled-Tube Heat Exchangers to Improve Heat Transfer Rate With Sp...IJMER
Steady heat transfer enhancement has been studied in helically coiled-tube heat exchangers. The outer side of the wall of the heat exchanger contains a helical corrugation which makes a helical rib on the inner side of the tube wall to induce additional swirling motion of fluid particles. Numerical calculations have been carried out to examine different geometrical parameters and the impact of flow and thermal boundary conditions for the heat transfer rate in laminar and transitional flow regimes. Calculated results have been compared to existing empirical formula and experimental tests to investigate the validity of the numerical results in case of common helical tube heat exchanger and additionally results of the numerical computation of corrugated straight tubes for laminar and transition flow have been validated with experimental tests available in the literature. Comparison of the flow and temperature fields in case of common helical tube and the coil with spirally corrugated wall configuration are discussed. Heat exchanger coils with helically corrugated wall configuration show 80–100% increase for the inner side heat transfer rate due to the additionally developed swirling motion while the relative pressure drop is 10–600% larger compared to the common helically coiled heat exchangers. New empirical Co-relation has been proposed for the fully developed inner side heat transfer prediction in case of helically corrugated wall configuration.
Engineering Project involving design,manufacturing, testing of Fins( Heat exchanger for 2 wheeler). An analysis of heat transfer on fins with various geometrical perforations.
Numerical Analysis of Inverted Notched Fin Array Using Natural ConvectionIOSR Journals
Abstract:Geometry and orientation plays an important role in natural convection heat transfer. For
horizontal rectangular fin array a chimney flow pattern is developed due to density difference. This flow
pattern creates a stagnant zone near central bottom region. That portion does not contribute much towards
heat dissipation. This area is removed from fins and they became inverted notched fins. This modified geometry
reduces material cost, material weight without hampering heat transfer rate. Numerical models are prepaid to
investigate heat transfer characteristics in plane fins and inverted notched fins. This investigation is also
extended over different types of notches and their effectiveness comparison. Fin spacing, fin height, fin length,
heater input, percentage of area removed in the form of inverted notch are the parameters under
consideration. This analysis is done numerically using CFD package (Fluent). It is found that the heat transfer
coefficient of inverted notch fin array is 25% to 35% higher as compared with normal fin array. Also we found
that the triangular shape notch gives better result than trapezoidal and rectangular shape notch.
Key words: Inverted Notched Fin, Chimney Flow, Natural Convection, Heat transfer coefficient enhancement
Thermal analysis of water cooled charge air cooler in turbo charged diesel en...eSAT Journals
Abstract
Air-cooled and water-cooled charge air coolers are used as applications in automobiles. The cooling media include engine’s
coolant, ambient air or any other external coolant source. Charge air cooler with offset strip fin geometry is chosen on both hot
and cold sides with fin thickness 0.15 mm, length 3.2 mm, height 5.2 mm and frequency of 18 fins per inch. The flow rate on hot
side is taken as 0.3 kg/s and cold side as 2 kg/s for air-cooled charge air cooler and for water-cooled charge air cooler on hot
side it is 0.3 kg/s and cold side is 0.265 kg/s with inlet temperature on hot side 160 oC and cold side 40 oC. With the above
conditions, performance evaluation is done on both charge air coolers. Having found water-cooled charge air cooler is having
higher effectiveness than air-cooled type, performance evaluation is carried out on water-cooled charge air cooler with varying
mass flow rates on hot side from 0.1 to 0.6 kg/s and keeping cold fluid rate constant at 0.265 kg/s and the outputs obtained are
Colburn-j factor, Fanning friction factor, heat transfer coefficient, overall heat transfer coefficient and effectiveness of heat
exchanger. From the obtained results, graphs are drawn to assess the performance of the water charge air cooler.
Keywords: Charge Air Cooler, Plate Fin Heat Exchanger, Colburn-J Factor, Heat Transfer Coefficient, Fanning
Friction Factor, Effectiveness
Helically Coiled Tube with Different Geometry and Curvature Ratio on Convecti...AM Publications
A helically coil-tube heat exchanger is generally applied in industry applications due to its compact structure, larger heat transfer area and higher heat transfer capability. Several studies from literature have also indicated that heat transfer rate in helically coiled tube are superior to straight tube due to complex flow pattern exist inside helical pipe. The concept behind compact heat exchanger is to decrease size and increase heat load which is the typical feature of modern helical tube heat exchanger. While the heat transfer characteristics of helical coil heat exchangers are available in the literature, This paper elaborates a brief review on different curvature ratio and geometry of tubes in heat transfer through heat exchangers.
Heat transfer augmentation in different geometries of dimpled surface under n...eSAT Journals
Abstract The prime objective of present work is to study experimentally the heat transfer augmentation through various geometries of dimpled surfaces in longitudinal and lateral directions. In this paper horizontal rectangular plates of copper and aluminum with different dimpled geometries (like square, circular and triangular) for in-line arrangements were studied in natural convection with steady laminar external flow condition. The various parameters considered for study are Nusselt number, heat transfer coefficient and heat transfer rate for a constant Prandtl number (0.7) and Grashof number (104-107).It has been found that the heat transfer coefficient and heat transfer rate increases for various dimpled surfaces as compared to plane surface. It has been also found that the heat transfer coefficient and heat transfer rate increases along longitudinal direction as compared to lateral direction. And it is seen that heat transfer rate is maximum for triangular shape dimple when the apex of triangle is faced towards inlet of air flow Finally it is concluded that heat transfer enhancement takes place along the dimpled surface
Heat transfer augmentation in different geometries of dimpled surface under n...eSAT Publishing House
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology.
Improving boiler efficiency by using air preheaterNetha Jashuva
Abstract: Air pre-heater is a heat transfer surface in which air temperature is raised by transferring heat from other media such as flue gas .Hot air is necessary for rapid combustion in the furnace and also for drying coal in milling plants. So an essential boiler accessory which serves this purpose is air pre-heater. The air pre-heater are not essential for operation of steam generator, but they are used where a study of cost indicates that money can be saved or efficient combustion can be obtained by their use. The decision for its adoption can be made when the financial advantages is weighed against the capital cost of heater in the present paper we have taken up the operation and performance analysis of LJUNGSTROM AIRPREHEATER27VITM 1900 of 2x210 MW capacity Rayalaseema Thermal Power Plant, Kalamala and compare with Rothemuhle air pre-heater. In analysis of performance preventive measures for corrosion of heating elements has been studied, and also air heater leakage, corrected gas outlet temperature and finally gas efficiency has been calculated.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
Integral transform methods for inverse problem of heat conduction with known ...IJLT EMAS
Three dimensional inverse transient thermoelastic problem of a semi-infinite hollow cylinder is considered within the context of the theory of generalized thermoelasticity. The lower surface, upper surface and inner surface of the semi-infinite hollow cylinder occupying the space D={(x,y,z)E R<sup>3</sup>: a≤(x<sup>2</sup>+y<sup>2</sup>)<sup>1/2</sup> ≤b, 0≤z≤∞} are known boundary conditions. Finite Marchi-Zgrablich transform and Fourier sine transform techniques are used to determine the unknown temperature gradient, temperature distribution, displacement and thermal stresses on outer curved surface of a cylinder. The distribution of the considered physical variables are obtained and represented graphically.
Thermal Stress Analysis of a Thin Rectangular Plate With Internal Heat SourceIJLT EMAS
This paper deals with the determination of temperature
distribution, displacement function and thermal stresses of a thin
rectangular plate with internal heat source. A thin rectangular
plate is considered having zero initial temperature and the edges
of the plate are maintained at zero temperature whereas the thin
rectangular plate is subjected to arbitrary heat supply on the
edges y. The governing heat conduction equation has been solved
by the method of integral transform technique. The results are
obtained in a series form in terms of Bessel’s functions.
Thermal Stresses of a Semi-Infinite Rectangular Slab with Internal Heat Gener...IJLT EMAS
This paper is concerned with transient thermoelastic
problem in which we need to determine the temperature
distribution, displacement function and thermal stresses of a
semi- infinite rectangular slab when the boundary conditions are
known. Integral transform techniques are used to obtain the
solution of the problem.
In developing electric motors in general and induction motors in particular
temperature limit is a key factor affecting the efficiency of the overall design. Since
conventional loading of induction motors is often expensive, the estimation of temperature
rise by tools of mathematical modeling becomes increasingly important. Excepting for
providing a more accurate representation of the problem, the proposed model can also
reduce computing costs. The paper develops a three-dimensional transient thermal model in
polar co-ordinates using finite element formulation and arch shaped elements. A
temperature-time method is employed to evaluate the distribution of loss in various parts of
the machine. Using these loss distributions as an input for finite element analysis, more
accurate temperature distributions can be obtained. The model is applied to predict the
temperature rise in the stator of a squirrel cage 7.5 kW totally enclosed fan-cooled induction
motor. The temperature distribution has been determined considering convection from the
back of core surface, outer air gap surface and annular end surface of a totally enclosed
structure.
Measuring the Thermal Conductivities of Low Heat Conducting Disk Samples by M...IJERA Editor
This article aims to establish an experimental procedure to measure heat transmission coefficients in low heat conductive materials. The newly developed model takes as starting point the application of Fourier’s law to a disk sample when a temperature gradient is established between its faces. The power of a heating element is determined as the heat transfer coefficient of the problem disk. Initially, a glass vessel containing water is placed in direct contact with the heating element; then, a problem plastic disk is placed between this element and the glass vessel, treating the set as a composite wall. Prior to the above the water equivalent of a calorimetric set (vessel + water + accessories) and the thermal conductivity of the vessel must be determined. The thermal conductivity of the problem plastic disk sample is obtained for temperatures ranging from 30 to 70° C. The results reveal the existence of some type of structural transition for the problem material.
Experiment on single-mode feedback control of oscillatory thermocapillary con...IJERA Editor
Feedback control was carried out on nonlinear thermocapillary convections in a half-zone liquid bridge of a high
Prandtl number fluid under normal gravity. In the liquid bridge, the convection changed from a two-dimensional
steady flow to a three-dimensional oscillatory flow at a critical temperature difference. Feedback control was
realized by locally modifying the free surface temperature using local temperature measured at different
positions. The present study aims to confirm whether the control method can effectively suppress oscillatory
flows with every modal structure. Consequently, the control was theoretically verified to be effective for
oscillatory flows with every modal structure in a high Marangoni number range.
Radial Heat Transport in Packed Beds-III: Correlations of Effective Transport...inventionjournals
The reliability and accuracy of experimental with predictions data of two models ("MC model" Marshall and Coberly model, [1] and modified model by Ibrahim et al. [2] are investigated for the effective radial thermal conductivity (Ker), and the wall heat transfer coefficient (hw) in packed beds in the absence of chemical reactions. The results were evaluated by the modified mathematical model as to the boundary bed inlet temperature; (To) number of terms of the solution series and number of experimental points used in the estimate. Very satisfactory was attained between the predicted and measured temperature profiles for a range of experiments. These cover a range of tube to (equivalent) particle diameter ratios from dt /dp = 4 to 10; Reynolds numbers ranged between 3.8-218 for particle, and elevated pressure from 11 to 20 bar for particle catalyst pellets. In all cases the fluid flowing throughout the bed has been air. The results indicate to the choice of the inlet boundary condition can have a large impact on the values of obtained parameters. And model parameters have been shown to be dependent on the pressure inside the reactor. The following correlations for both (hw) and (Ker) respectively under a given conditions obtained by using multiple regressions of our results that based on the modified mathematical model: Nuw = 67.9Re0.883(dt /dp) -0.635(P/Po) -1.354 Ker = 0.2396 + 0.0041Re The results accuracy of these correlations obtained from the modified mathematical model are more than the results accuracy of correlations obtained from MC model with respect to experimental data; these accuracy of both correlations reach up to 91% and 65% for (hw) and (Ker) respectively; which these results indicate to the reliability
EXPERIMENTAL STUDY OF HEAT TRANSFER FROM PLATE FIN ARRAY IN MIXED CONVECTION ...ijiert bestjournal
The work summarized in this paper presents an exper imental study of heat transfer from plate fin in mixed convection mode enhancement by the us e of plate fins is presented. After a brief review of the basic methods used to enhance the hea t transfer by simultaneous increase of heat transfer surface area as well as the heat tran sfer coefficient,a simple experimental method to assess the heat transfer enhancement is p resented. The method is demonstrated on plate fins as elements for the heat transfer enhanc ement,but it can in principle be applied also to other fin forms. That is varying various paramet ers (height,spacing). The order of the magnitude of heat transfer enhancement obtained exp erimentally,it was found that by a direct comparison of Nu and Re no conclusion regarding the relative performances could be made. This is because the dimensionless variables are int roduced for the scaling of heat transfer and pressure drop results from laboratory to large scal e but not for the performance comparison. Therefore a literature survey of the performance co mparison methods used in the past was also performed. Experiments will carried out on mix ed convection heat transfer from plate fin heat sinks subject to the influence of its geometry and heat flux. A total of 9 plate fins were pasted into the upper surface of the base plate. Th e area of the base plate is 150mm by 150mm. The base plate and the fins were made of alu minum. For all tested plate fin heat sinks,however,the heat transfer performance for h eat sinks with plate fins was better than that of solid pins.
Double Diffusive Convection and the Improvement of Flow in Square Porous AnnulusIJERA Editor
There has been increased interest shown in recent years to investigate the behavior of heat and mass transfer in a square annulus with a porous medium fixed between the inner and outer walls. This paper aims to evaluate the Soret effect arising in the case of heat and mass transfer in a porous medium bounded by a square annulus and subjected to isothermal heating of the inner surfaces as well as the outer horizontal surfaces. The phenomenon is governed by 3 partial differential equations, the momentum, energy and concentration equations, that are coupled together and result in a situation where change in one variable affects the other equations and vice versa. The partial differential equations are converted into finite element equations with the help of the Galerkin method and then solved to predict solution variables such as temperature, stream function and concentration in the porous medium. It is found that the heat transfer rate at the hot wall decreases with increasing viscous dissipation effect in the porous medium.
Transcript: Selling digital books in 2024: Insights from industry leaders - T...BookNet Canada
The publishing industry has been selling digital audiobooks and ebooks for over a decade and has found its groove. What’s changed? What has stayed the same? Where do we go from here? Join a group of leading sales peers from across the industry for a conversation about the lessons learned since the popularization of digital books, best practices, digital book supply chain management, and more.
Link to video recording: https://bnctechforum.ca/sessions/selling-digital-books-in-2024-insights-from-industry-leaders/
Presented by BookNet Canada on May 28, 2024, with support from the Department of Canadian Heritage.
Smart TV Buyer Insights Survey 2024 by 91mobiles.pdf91mobiles
91mobiles recently conducted a Smart TV Buyer Insights Survey in which we asked over 3,000 respondents about the TV they own, aspects they look at on a new TV, and their TV buying preferences.
Key Trends Shaping the Future of Infrastructure.pdfCheryl Hung
Keynote at DIGIT West Expo, Glasgow on 29 May 2024.
Cheryl Hung, ochery.com
Sr Director, Infrastructure Ecosystem, Arm.
The key trends across hardware, cloud and open-source; exploring how these areas are likely to mature and develop over the short and long-term, and then considering how organisations can position themselves to adapt and thrive.
Essentials of Automations: Optimizing FME Workflows with ParametersSafe Software
Are you looking to streamline your workflows and boost your projects’ efficiency? Do you find yourself searching for ways to add flexibility and control over your FME workflows? If so, you’re in the right place.
Join us for an insightful dive into the world of FME parameters, a critical element in optimizing workflow efficiency. This webinar marks the beginning of our three-part “Essentials of Automation” series. This first webinar is designed to equip you with the knowledge and skills to utilize parameters effectively: enhancing the flexibility, maintainability, and user control of your FME projects.
Here’s what you’ll gain:
- Essentials of FME Parameters: Understand the pivotal role of parameters, including Reader/Writer, Transformer, User, and FME Flow categories. Discover how they are the key to unlocking automation and optimization within your workflows.
- Practical Applications in FME Form: Delve into key user parameter types including choice, connections, and file URLs. Allow users to control how a workflow runs, making your workflows more reusable. Learn to import values and deliver the best user experience for your workflows while enhancing accuracy.
- Optimization Strategies in FME Flow: Explore the creation and strategic deployment of parameters in FME Flow, including the use of deployment and geometry parameters, to maximize workflow efficiency.
- Pro Tips for Success: Gain insights on parameterizing connections and leveraging new features like Conditional Visibility for clarity and simplicity.
We’ll wrap up with a glimpse into future webinars, followed by a Q&A session to address your specific questions surrounding this topic.
Don’t miss this opportunity to elevate your FME expertise and drive your projects to new heights of efficiency.
Kubernetes & AI - Beauty and the Beast !?! @KCD Istanbul 2024Tobias Schneck
As AI technology is pushing into IT I was wondering myself, as an “infrastructure container kubernetes guy”, how get this fancy AI technology get managed from an infrastructure operational view? Is it possible to apply our lovely cloud native principals as well? What benefit’s both technologies could bring to each other?
Let me take this questions and provide you a short journey through existing deployment models and use cases for AI software. On practical examples, we discuss what cloud/on-premise strategy we may need for applying it to our own infrastructure to get it to work from an enterprise perspective. I want to give an overview about infrastructure requirements and technologies, what could be beneficial or limiting your AI use cases in an enterprise environment. An interactive Demo will give you some insides, what approaches I got already working for real.
Builder.ai Founder Sachin Dev Duggal's Strategic Approach to Create an Innova...Ramesh Iyer
In today's fast-changing business world, Companies that adapt and embrace new ideas often need help to keep up with the competition. However, fostering a culture of innovation takes much work. It takes vision, leadership and willingness to take risks in the right proportion. Sachin Dev Duggal, co-founder of Builder.ai, has perfected the art of this balance, creating a company culture where creativity and growth are nurtured at each stage.
Search and Society: Reimagining Information Access for Radical FuturesBhaskar Mitra
The field of Information retrieval (IR) is currently undergoing a transformative shift, at least partly due to the emerging applications of generative AI to information access. In this talk, we will deliberate on the sociotechnical implications of generative AI for information access. We will argue that there is both a critical necessity and an exciting opportunity for the IR community to re-center our research agendas on societal needs while dismantling the artificial separation between the work on fairness, accountability, transparency, and ethics in IR and the rest of IR research. Instead of adopting a reactionary strategy of trying to mitigate potential social harms from emerging technologies, the community should aim to proactively set the research agenda for the kinds of systems we should build inspired by diverse explicitly stated sociotechnical imaginaries. The sociotechnical imaginaries that underpin the design and development of information access technologies needs to be explicitly articulated, and we need to develop theories of change in context of these diverse perspectives. Our guiding future imaginaries must be informed by other academic fields, such as democratic theory and critical theory, and should be co-developed with social science scholars, legal scholars, civil rights and social justice activists, and artists, among others.
Accelerate your Kubernetes clusters with Varnish CachingThijs Feryn
A presentation about the usage and availability of Varnish on Kubernetes. This talk explores the capabilities of Varnish caching and shows how to use the Varnish Helm chart to deploy it to Kubernetes.
This presentation was delivered at K8SUG Singapore. See https://feryn.eu/presentations/accelerate-your-kubernetes-clusters-with-varnish-caching-k8sug-singapore-28-2024 for more details.
"Impact of front-end architecture on development cost", Viktor TurskyiFwdays
I have heard many times that architecture is not important for the front-end. Also, many times I have seen how developers implement features on the front-end just following the standard rules for a framework and think that this is enough to successfully launch the project, and then the project fails. How to prevent this and what approach to choose? I have launched dozens of complex projects and during the talk we will analyze which approaches have worked for me and which have not.
Dev Dives: Train smarter, not harder – active learning and UiPath LLMs for do...UiPathCommunity
💥 Speed, accuracy, and scaling – discover the superpowers of GenAI in action with UiPath Document Understanding and Communications Mining™:
See how to accelerate model training and optimize model performance with active learning
Learn about the latest enhancements to out-of-the-box document processing – with little to no training required
Get an exclusive demo of the new family of UiPath LLMs – GenAI models specialized for processing different types of documents and messages
This is a hands-on session specifically designed for automation developers and AI enthusiasts seeking to enhance their knowledge in leveraging the latest intelligent document processing capabilities offered by UiPath.
Speakers:
👨🏫 Andras Palfi, Senior Product Manager, UiPath
👩🏫 Lenka Dulovicova, Product Program Manager, UiPath
DevOps and Testing slides at DASA ConnectKari Kakkonen
My and Rik Marselis slides at 30.5.2024 DASA Connect conference. We discuss about what is testing, then what is agile testing and finally what is Testing in DevOps. Finally we had lovely workshop with the participants trying to find out different ways to think about quality and testing in different parts of the DevOps infinity loop.
GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using Deplo...James Anderson
Effective Application Security in Software Delivery lifecycle using Deployment Firewall and DBOM
The modern software delivery process (or the CI/CD process) includes many tools, distributed teams, open-source code, and cloud platforms. Constant focus on speed to release software to market, along with the traditional slow and manual security checks has caused gaps in continuous security as an important piece in the software supply chain. Today organizations feel more susceptible to external and internal cyber threats due to the vast attack surface in their applications supply chain and the lack of end-to-end governance and risk management.
The software team must secure its software delivery process to avoid vulnerability and security breaches. This needs to be achieved with existing tool chains and without extensive rework of the delivery processes. This talk will present strategies and techniques for providing visibility into the true risk of the existing vulnerabilities, preventing the introduction of security issues in the software, resolving vulnerabilities in production environments quickly, and capturing the deployment bill of materials (DBOM).
Speakers:
Bob Boule
Robert Boule is a technology enthusiast with PASSION for technology and making things work along with a knack for helping others understand how things work. He comes with around 20 years of solution engineering experience in application security, software continuous delivery, and SaaS platforms. He is known for his dynamic presentations in CI/CD and application security integrated in software delivery lifecycle.
Gopinath Rebala
Gopinath Rebala is the CTO of OpsMx, where he has overall responsibility for the machine learning and data processing architectures for Secure Software Delivery. Gopi also has a strong connection with our customers, leading design and architecture for strategic implementations. Gopi is a frequent speaker and well-known leader in continuous delivery and integrating security into software delivery.
1. World Academy of Science, Engineering and Technology
International Journal of Mathematical, Computational Science and Engineering Vol:2 No:1, 2008
Mathematical Modeling of Gas Turbine
Blade Cooling
А. Pashayev, C. Ardil, D. Askerov, R. Sadiqov, A. Samedov
International Science Index 13, 2008 waset.org/publications/14623
Azerbaijan National Academy of Aviation
AZ-1045, Bina, 25th km, Baku, Azerbaijan
Phone: (99450) 385-35-11; Fax:(99412) 497-28-29
e-mail: sadixov@mail.ru
αг
Abstract—In contrast to existing methods which do not take into
account multiconnectivity in a broad sense of this term, we develop
mathematical models and highly effective combination (BIEM and
FDM) numerical methods of calculation of stationary and quasistationary temperature field of a profile part of a blade with
convective cooling (from the point of view of realization on PC). The
theoretical substantiation of these methods is proved by appropriate
theorems. For it, converging quadrature processes have been
developed and the estimations of errors in the terms of A.Ziqmound
continuity modules have been received.
For visualization of profiles are used: the method of the least
squares with automatic conjecture, device spline, smooth
replenishment and neural nets. Boundary conditions of heat exchange
are determined from the solution of the corresponding integral
equations and empirical relationships. The reliability of designed
methods is proved by calculation and experimental investigations
heat and hydraulic characteristics of the gas turbine first stage nozzle
blade.
αВ
air convective heat exchange local
coefficient
quantity of outlines
M
x, y
coordinates of segments
n
external normal to outline or
quantity of outline segments
m
quantity of inline segments of all
cooling channels
variable at an integration of the
R
distance between fixed and
“running” points
ΔS
mean on sections of the partition
(Lτ ,ε f )( z ) two-parameter quadrature formula
for logarithmic double layer
potential
~
double layer logarithmic potential
f (z)
operator
(I τ ,ε f )( z ) two-parameter quadrature formula
for logarithmic potential simple
layer
simple layer logarithmic potential
f (z )
operator
ω f ( x)
module of a continuity
curve outline; the circulation of
Г
speed
ϕ
potential of speed
ψ
current function of speed
gas speed vectors mean on the
V∞
flowing
angle between the speed vector and
α∞
the profile cascade axis
angle that corresponds to the outlet
θВ
edge of the profile
ratio of turbulences
Tu
turbulences coefficient
εT
Keywords—Mathematical Modeling, Gas Turbine Blade
Cooling, Neural Networks, BIEM and FDM.
NOMENCLATURE
TГ
T
T0
Tγ0
Ti
Tγi
Tk
ρ
cv
λ
qv
α0
gas temperature
required temperature
temperature of environment at
i =0;
temperature on the outline γi at
i = 0 (outside outline of blade);
temperature of the environment at
i = 1, M (temperature of the cooler)
temperature on the outline γi at
i = 1, M (outline of cooling
channels)
temperature in the k point
material density,
density of a logarithmic potential
thermal capacity
heat conduction coefficient of
material or thermal conductivity of
material of the blade
internal source or drain of heat
heat transfer coefficient from gas to
a surface of a blade (at i = 0 )
gas local heat exchange coefficient
K
K
K
K
K
K
K
3
kq/m
J/kg·K
W/m·K
u
GВ
ψГ , ψВ
κф
gas and air temperature coefficients
coefficient of the form
μ В , λВ
W/m2
W/m2·K
air flow
cooler dynamic viscosity,
heat conductivity coefficient
Bio criterion
Вi
7
W/m2·K
W/m2·K
mm
m2/s
m2/s
m/s
deg
deg
kg/s
poise,
W/m·K
-
2. World Academy of Science, Engineering and Technology
International Journal of Mathematical, Computational Science and Engineering Vol:2 No:1, 2008
kg/m3
density of coolant flow
ξ ij
coefficient of hydraulic resistance
-
Nu
Re
International Science Index 13, 2008 waset.org/publications/14623
pij
Nusselt criterion
Reynolds criterion
-
blade and assigned by heat conduction in a skew field of a
blade.
−λ
temperature on an γi at i = 1, M (outline of cooling channels);
α 0 - heat transfer factor from gas to a surface of a blade (at
i = 0 ); α i - heat transfer factor from a blade to the cooling air
at i = 1, M ; λ - thermal conductivity of the material of a
blade; n - external normal on an outline of researched area.
III. APPLICATION OF BOUNDARY INTEGRATED
EQUATIONS METHOD FOR DEFINITION OF AGTE
ELEMENTS’ TEMPERATURE FIELDS
At present for the solution of this boundary problem (2)-(4)
four numerical methods are used: Methods of Finite
Differences (MFD), Finite Element Method (FEM),
probabilistic method (Monte-Carlo method), and Boundary
Integral Equations Method (BIEM) (or its discrete analog ─
Boundary Element Method (BEM)).
Let us consider BIEM application for the solution of
problem (2)-(4).
The function T = T (x, y ) , continuous with the derivatives up
to the second order, satisfying the Laplace equation in
M
considered area, including and its outline Г = ∪ γi , is harmonic.
∂t
∂x 2
∂ 2T
∂y 2
i =0
Consequence of the Grin integral formula for the researched
harmonic function T = T (x, y ) is the ratio:
Т(x, y) =
∂Tγ 0
∂n
1
∂( nR)
∂Т
− nR Г ]ds
∫ [Т Г
2π Г
∂n
∂n
(5)
where R - variable at an integration of the distance between
point K (x, y ) and “running” on the outline k - point; T Г temperature on the outline Г . The temperature value in some
point k lying on the boundary is determined (as limiting at
approach of point K (x, y ) to the boundary)
=0
Тk =
(2)
When determining particular temperature fields in gas
turbine elements are used boundary conditions of the third
kind, describing heat exchange between the skew field and the
environment (on the basis of a hypothesis of a NewtonRiemann). In that case, these boundary conditions will be
recorded as follows:
α 0 (T0 − Tγ 0 ) = λ
(4)
on an outline γi at i = 0 (outside outline of blade); Tγi -
where ρ , c v and λ - accordingly material density, thermal
capacity, and heat conduction; qv - internal source or drain of
heat, and T - is required temperature.
Research has established that the temperature condition of
the blade profile part with radial cooling channels can be
determined as two-dimensional [2]. Besides, if to suppose
constancy of physical properties and absence of internal
sources (drains) of heat, then the temperature field under fixed
conditions will depend only on the skew shape and on the
temperature distribution on the skew boundaries. In this case,
equation (1) will look like:
+
= α i (Tγ i − Ti )
- temperature of the environment at i = 1, M (temperature of
the cooler), where M - quantity of outlines; Tγ0 - temperature
II. PROBLEM FORMULATION
In classical statement a heat conduction differential equation
in common case for non-stationary process with distribution of
heat in multi–dimensional area (Fourier-Kirchhoff equation)
has a kind [1]:
∂( ρCvT )
(1)
= div(λ grad T) + qv ,
∂ 2T
∂n
Equation (4) characterizes the heat quantity assigned by
convection of the cooler, which is transmitted by heat
conduction of the blade material to the surface of cooling
channels: where T0 - temperature of environment at i = 0 ; Ti
I. INTRODUCTION
The development of aviation gas turbine engines (AGTE) at
the present stage is mainly reached by assimilation of high
values of gas temperature in front of the turbine ( T Г ). The
activities on gas temperature increase are conducted in several
directions. Assimilation of high ( T Г ) in AGTE is however
reached by refinement of cooling systems of turbine blades. It
is especially necessary to note, that with T Г increase the
requirement to accuracy of results will increase. In other
words,
at
allowed
values
of
AGTE
metal
temperature Tlim = (1100...1300K ) , the absolute error of
temperature calculation should be in limits ( 20 − 30 K ), that is
no more than 2-3% [2,3,5,6,12].
This is difficult to achieve (multiconnected fields with
various cooling channels, variables in time and coordinates
boundary conditions). Such problem solving requires
application of modern and perfect mathematical device.
ΔT =
∂Tγ i
∂( nRk )
∂Т
⎤
1 ⎡
ds − ∫ Г nRk ds ⎥
∫Т Г
2π ⎢ Г
∂n
Г ∂n
⎣
⎦
(6)
With allowance of the boundary conditions (2)-(3), after
collecting terms of terms and input of new factors, the ratio (6)
can be presented as a linear algebraic equation, computed for
the point R :
ϕ k 1Tγ01 + ϕ k 2 Tγ02 + ... + ϕ kn Tγ0 m −
(7)
− ϕ kγ0 T0 − ϕ kγi Ti − 2πTk = 0
(3)
where n is the quantity of sites of a partition of an outside
outline of a blade
γ 0 ( γ i on i = 0 ) on small sections
This following equation characterizes the quantity of heat
transmitted by convection from gas to unit of a surface of a
ΔS 0 ( ΔS i at i = 0 ) , m is the quantity of sites of a partition of
8
3. World Academy of Science, Engineering and Technology
International Journal of Mathematical, Computational Science and Engineering Vol:2 No:1, 2008
outside outlines of all cooling channels
γi
(i = 1, M ) on small
R(s,ξ ) = ((x(s) − x(ξ ))2 + ( y(s) − y(ξ ))2 )1/ 2 .
For the singular integral operators evaluation, which are
included in (12) the discrete operators of the logarithmic
potential with simple and double layer are investigated. Their
connection and the evaluations in modules term of the
continuity (evaluation such as assessments by A. Zigmound
are obtained) is shown Theorem (main)
Let
ωξ ( x)
sections ΔS i .
Let us note, that unknowns in the equation (7) except the
unknown of true value Tk in the k point are also mean on
sections of the outlines partition ΔS 0 and ΔS i temperatures
Tγ 01 , Tγ 02 ,..., Tγ 0 m and Tγ i 1 , Tγ i 2 ,..., Tγ im (total number n + m ).
From a ratio (7), we shall receive the required temperature for
any point, using the formula (5):
1
T(x, y) = [ϕk1Tγ01 +ϕk2Tγ02 + ...+ϕknTγ0n +
2π
(8)
+ ...+ϕkmTγim −ϕkγ0Tcp0 −ϕkγiTcpi ]
∫
and let the equation (12) have the solution f*∈CГ (the set of
continuous functions on Г). Then ∃Ν0∈Ν= {1, 2…} such that
the discrete system ∀N>N0, obtained from (12) by using the
discrete double layer potential operator (its properties has been
where
ϕ k1 = ∫
ΔS01
.
ϕ kn
International Science Index 13, 2008 waset.org/publications/14623
α
∂( nRk )
ds − 01 ∫ nRk ds
λ1 ΔS01
∂n
studied), has unique solution {f j(N) }, k = 1,m j ; j = 1,n;
k
.
.
.
.
α0m
∂( nRk )
ds −
= ∫
∫ nRk ds
λm ΔS0m
∂n
ΔS0m
ϕ kγ0
*
(N)
| f jk − f jk |≤ С(Г)(
+ε
L/2
∫
In activities [2] the discretization of aniline Г = ∪ γi by a
∫
i =0
∫
ΔS γi
∂n
,
(
{ε N }
∞
N =1 --
τN
∞
N =1
--the
the sequence of
τ N , , ε N ), satisfies the
2
)
(I τ δ f )( z ) − f ( z ) ≤ С ( Г )
(11)
,
⎛
⎜ f
⎝
M
i =0
C
δ ln
2d
δ
+ ω f ( τ ) + τ ln
2d
δ
+ f
C
⎞
⎠
ω Z ( τ )⎟;
ω f ( x)ω l ( x)
⎛
~
f )( z ) − f ( z ) ≤ ⎜ С ( Г ) ∫
dx +
⎜
x2
0
⎝
,
d
d
ω f ( x) ⎞
ω ( x)
dx ⎟
+ ω f ( τ )∫ l dx + τ ∫
⎟
x
x2
Δ
Δ
⎠
Г
(L
M
cooled channels; ρ = ∪ ρ i - density of a logarithmic potential
Ω, Г
i =0
.
i =0
M
Г = ∪ γi are positively oriented and are given in a
i =0
parametric kind: x = x(s ) ; y = y (s ) ; s ∈ [0 , L ]; L = ∫ ds .
Г
where
Using BIEM and expression (11) we shall put problem (2)-(4)
to the following system of boundary integral equations:
,
≥2
ψ ∈ C Г ( C Г - space of all functions continuous
on Г ) and z ∈ Г , ( z = x + iy )
where Г = ∪ γi -smooth closed Jordan curve; M -quantity of
∂
1
nR( s, ξ )dξ =
∫ ( ρ ( s ) − ρ (ξ ))
∂n
2π Г
τ
then for all
Г
ρ (s) −
dx +
dx ),
p′ ≥ δ
T(x, y) = ∫ ρ nR −1 ds ,
Thus curve
x
that, which is satisfied the condition
VI. NEW INTERPRETATION OF THE BIEM
In contrast to [4], we offer to decide the given boundary
value problem (2)-(4) as follows. We locate the distribution of
temperature T = T (x, y ) as follows:
M
x
0
f
condition 2 ≤ ε τ −−1 ≤ p .
Let δ ∈ 0, d , where d is diameter Г, and the splitting τ is
(10)
γi
S = ∪ si
f
ω * (x)
f
positive numbers such that the pair (
(where ΔS γi ∈ L = ∪ li ; li = ∫ ds )
uniformly distributed on γi
ω * (x)
L/2
dx + ω * ( τ N ) ∫
sequence of partitions of Г ;
M
i =0
∫
ωξ (x)ω * (x)
f
dx +
x
where C ( Г ) is constant, depending only on
γi
nRk ds ≈ nRk ΔS γ i
x
εN
many discrete point and integrals that are included in the
equations as logarithmic potentials, was calculated
approximately with the following ratios:
∂( nR k )
∂( nR k )
,
(9)
ds ≈
ΔS
∂n
f
L/2
+ τN
M
∫
ωξ (x)ω * (x)
εN
α01
αim
∫ nRk ds + ... +
∫ nRk ds
λ1 ΔSi1
λm ΔSim
ΔS γi
εN
0
α
α
= 01 ∫ nRk ds + ... + 0n ∫ nRk ds
λ1 ΔS01
λn ΔSn
ϕ kγii =
< +∞
x
0
⎛ f ( z k ,e +1 ) + f ( z k ,e )
⎞
⎜
− f ( z) ⎟ ⋅
⎜
⎟
2
z m , e∈τ ( z ) ⎝
⎠
− y k ,e )( x k ,e − x ) − ( x k ,e +1 − x k ,e )( y k ,e − y )
(Lτ ε f )( z ) = ∑
,
( y k ,e +1
(12)
z − z k ,e
α
= i (T − ∫ ρ ( s ) nR −1 ds )
2πλ
Г
(Lτ ε f )( z )
where
,
9
2
+ πf ( z )
4. World Academy of Science, Engineering and Technology
International Journal of Mathematical, Computational Science and Engineering Vol:2 No:1, 2008
- two-parameter quadrature formula (depending on τ and δ
parameters) for logarithmic double layer potential; ~ ( z ) f
double layer logarithmic potential operator; C ( Г ) – constant,
dependent only from a curve Г ; ω f ( x ) is a module of a
continuity of functions f;
(I τ ε f )( z ) = ∑
f ( z k , j +1 ) + f ( z k , j )
,
2
z m , e∈τ ( z )
⋅ ln
1
zk, j − z
⋅
The developed technique for the numerical solution of
stationary task of the heat conduction in cooled blades can be
distributed also to quasistationary case.
Let us consider a third boundary-value problem for the heat
conduction quasilines equation:
∂ ⎛
∂T ⎞ ∂ ⎛
∂T ⎞
(16)
⎟=0
⎜ λ (T )
⎟ + ⎜ λ (T )
∂x ⎝
α i (Tci − Tγi ) − λ (T )
z k , j +1 − z k , j
(I τ ε f )( z ) - two-parameter quadrature formula (depending on
T
z k ,e ∈ τ , z k ,e = xk ,e + iy k ,e
∂2 A
τ ( z ) = {z k ,e z k ,e − z > ε }
International Science Index 13, 2008 waset.org/publications/14623
k
τ = max z k , j +1 − z k , j
j =1, mk
Thus are developed effective from the point of view of
realization on computers the numerical methods basing on
constructed two-parametric quadratute processes for the
discrete operators logarithmic potential of the double and
simple layer. Their systematic errors are estimated, the
methods quadratures mathematically are proved for the
approximate solution Fredholm I and II boundary integral
equations using Tikhonov regularization and are proved
appropriate theorems [1].
V. SOLUTION TECHNIQUE OF DIRECT AND INVERSE
PROBLEMS OF HEAT CONDUCTIVITY
The given calculating technique of the blade temperature
field can be applied also to blades with the plug–in deflector.
On consideration blades with deflectors in addition to
boundary condition of the III kind adjoin also interfaces
conditions between segments of the outline partition as
equalities of temperatures and heat flows
(13)
Tv ( x, y ) = Tv +1 ( x, y ) ,
(14)
where ν - number of segments of the outline partition of the
blade cross-section; x, y- coordinates of segments. At finding
of cooler T best values, is necessary to solve the inverse
problem of heat conduction. For it is necessary at first to find
solution of the heat conduction direct problem with boundary
condition of the III kind from a gas leg and boundary
conditions I kinds from a cooling air leg
(15)
Tv (x, y) = Ti0
γ0
where Ti0 -the unknown optimum temperature of a wall of a
blade from a leg of a cooling air.
VI. APPLICATION OF BIEM TO QUASYSTATIONARY
PROBLEMS OF HEAT CONDUCTIVITY
+
∂2 A
(19)
=0
∂x
∂y 2
For preserving convection additives in boundary-value
condition (17), we shall accept in initial approximation
λ (T ) = λc . Then from (18) we have
(20)
T = A / λc
and the regional condition (17) will be transformed as follows:
∂Aγi
(21)
α i (Tci − Aγi / λc ) −
=0
∂n
So, the stationary problem (19) with (21) is solved by
boundary integrated equations method. If the solution L( x, y )
2
,
(18)
0
Then equation (16) is transformed into the following Laplace
equation:
f ( z ) -simple layer logarithmic potential operator;
k
∂Tγi
A = ∫ λ (ξ )dξ
τ and δ parameters) for logarithmic potential simple layer;
τ k = {z k ,1 ,..., z k ,m }, z k ,1 ≤ z k , 2 ≤ ... ≤ z k ,m
∂y ⎟
⎠
(17)
=0
∂n
For linearization of tasks (16) - (17) we shall use the Kirchhoff
permutation:
,
∂Tv ( x, y ) ∂Tv +1 ( x, y )
=
∂n
∂n
∂y ⎜
⎝
∂x ⎠
in the ( x, y ) point of the linear third boundary-value problem
(19), (21) for the Laplace equation substitute in (18) and after
integration to solve the appropriate algebraic equation, which
degree is higher than the degree of function λ (T ) with digit,
we shall receive meaning of temperature T ( x, y ) in the same
point. Thus in radicals is solved the algebraic equation with
non-above fourth degree
a 0 T 4 + a 1T 3 + a 2 T 2 + a 3 T + a 4 = A .
(22)
This corresponds to the λ (T ) which is the multinomial with
degree non-above third. In the result, the temperature field
will be determined on the first approximation, as the boundary
condition (17) took into account constant meaning heat
conduction λ c in convective thermal flows. According to it
we shall designate this solution T (1) (accordingly A (1) ). For
determining consequents approximations A (2 ) (accordingly
T (2 ) ), the function A(T ) is decomposing in Taylor series in
the neighborhood of T (1) and the linear members are left in it
only. In result is received a third boundary-value problem for
the Laplace equation relatively function A (2 ) . The temperature
T (2 ) is determined by the solution of the equation (20).
The multiples computing experiments with the using BIEM
for calculation the temperature fields of nozzle and working
blades with various amount and disposition of cooling
channels, having a complex configuration, is showed, that for
practical calculations in this approach, offered by us, the
discretization of the integrations areas can be conducted with
smaller quantity of discrete points. Thus the reactivity of the
10
5. World Academy of Science, Engineering and Technology
International Journal of Mathematical, Computational Science and Engineering Vol:2 No:1, 2008
algorithms developed and accuracy of evaluations is increased.
The accuracy of temperatures calculation, required
consumption of the cooling air, heat flows, losses from cooling
margins essentially depends on reliability of boundary
conditions, included in calculation of heat exchange.
VII. PROFILING OF COOLED BLADES
Piece-polynomial smoothing of cooled gas-turbine blade
structures with automatic conjecture is considered: the method
of the least squares, device spline, smooth replenishment, and
neural nets are used.
Let the equation of the cooled blade outline segments is the
third degree polynomial:
International Science Index 13, 2008 waset.org/publications/14623
y (x ) = a 0 + a 1 x + a 2 x 2 + a 3 x 3
(23)
The equation of measurements of the output coordinate has a
kind:
(24)
Z y = a0 + a1 x + a 2 x 2 a 3 x 3 + δ y
where Zy=║z1y, z2y, …, zny║T - vector of measurements of
output coordinate, n-amount of the points in the consideration
interval. For coefficients of polynomial (23) estimate the
method of the least squares of the following kind is used
θ
=(XTX)-1(XTZy),
(25)
Dθ =(XTX)-1σ2 ,
(26)
where
1 x1 x12 x13
X=
2
3
1 x2 x2 x2
2
3
1 x3 x 3 x3
- structural matrix;
............
2
3
1 xn xn xn
Dθ - dispersion matrix of errors; θ =║a0, a1, a2, a3║T - vector
of estimated coefficients.
Estimations of coefficients for the first segment is received
with using formula (25). Beginning with second segment, the
θ vectors components is calculated on experimental data from
this segment, but with the account of parameters found on the
previous segments. Thus, each subsequent segment of the
blade cross-section outline we shall choose with overlapping.
Thus, it is expedient to use the following linear connections
between the estimated parameters of the previous segment
2
3
⎡a0 N −1 + a1 N −1 xe + a 2 N −1 xe + a3 N −1 xe ⎤
⎢
⎥
2
⎥,
V= ⎢a1 N −1 + a 2 N −1 xe + 3a3 N −1 xe
(29)
⎢
⎥
⎢2 a 2 N −1 + 6 a 3 N −1 xe
⎥
⎣
⎦
e=(N-1)(n-L); L- number points of overlapping.
The expressions (27)-(29) describe communications, which
provide joining of segments of interpolation on function with
first and second degrees.
Taking into account the accuracy of measurements, the
problem of defining unknown coefficients of the model in this
case can be formulated as a problem conditional extremum:
minimization of the quadratic form (Zy-Xθ)Tσ2I(Zy-θ) under
the limiting condition (27). Here I is a individual matrix.
For the solution of such problems, usually are using the
method of Lagrange uncertain multipliers. In result, we shall
write down the following expressions for estimation vector of
coefficients at linear connections presence (27):
~T
θ
= θ T+(VT- θ TAT)[A(XTX)-1AT]-1A(XTX)-1
(30)
T
-1 T
T
-1 T -1
T
-1 2
Dθ~ = Dθ − (X X) A [A(X X) A ] A(X X) σ (31)
Substituting matrixes A and Х and vectors Zy and V in
expressions (25), (26), (30), and (31), we receive estimations
of the vector of coefficients for segment of the cooled blade
section with number N and also the dispersing matrix of
errors.
As a result of consecutive application of the described
procedure and with using of experimental data, we shall
receive peace-polynomial interpolation of the researched
segments with automatic conjecture.
Research showed that optimum overlapping in most cases is
the 50%-overlapping.
Besides peace-polynomial regression exist interpolation
splines which represent polynomial (low odd degrees - third,
fifth), subordinated to the condition of function and derivatives
(first and second in case of cubic spline) continuity in common
points of the next segments.
If the equation of the cooled gas-turbine blades profile is
described cubic spline submitted in obvious polynomial kind
(23), the coefficients а0, а1, а2, а3 determining j-th spline, i.e.
line connecting the points Zj=(xj, yj) and Zj+1=(xj+1, yj+1), are
calculating as follows:
a0 = z j ;
a 1 = z 'j ;
(32)
2
1
1
a 2 = z 'j' / 2 = 3( z j +1 − z j )h −+1 − 2 z 'j h −+1 − z 'j +1 h −+1 ;
j
j
j
θ N 1 and required θ N for N-th segment:
3
2
2
a 3 = z 'j'' / 6 = 2( z j − z j +1 )h −+1 + z 'j h −+1 + z 'j +1 h −+1 ;
j
j
j
AθN=V,
(27)
2
3
⎡1 xe xe xe ⎤
⎢
⎥
2
A= ⎢0 1 2 xe 3 xe ⎥ ,
⎢
⎥
⎢0 0 2 6 xe ⎥
⎣
⎦
(28)
where hj+1=|zj+1-zj|, j= 1, N − 1 .
Let us consider other way smooth replenishment of the cooled
gas-turbine blade profile on the precisely measured meaning of
coordinates in final system of discrete points, distinguishing
from spline-function method and also from the point of view
of effective realization on computers.
Let equation cooled blades profile segments are described by
the multinomial of the third degree of the type (23). By taking
advantage the smooth replenishment method (conditions of
function smooth and first derivative are carried out) we shall
define its coefficients:
11
6. World Academy of Science, Engineering and Technology
International Journal of Mathematical, Computational Science and Engineering Vol:2 No:1, 2008
a0 = z j ;
a 1 = ( z j +1 −
1
z j )h −+1 ;
j
a 2 = −(( z j + 2 −
1
z j +1 )h −+ 2
j
(33)
+ ( z j +1 −
1
1
z j )h −+1 )h −+1 ;
j
j
1
1
2
a 3 = (( z j + 2 − z j +1 )h −+ 2 − ( z j +1 − z j )h −+1 )h −+1 ;
j
j
j
j= 1, N −1 − S , S=1.
If it’s required carry out conditions of function smooth first
and second derivatives, i.e. corresponding to cubic splines
smooth, we shall deal with the multinomial of the fifth degree
(degree of the multinomial is equal 2S + 1, i.e. S = 2).
The advantage of such approach (smooth replenishment) is
that it’s not necessary to solve system of the linear algebraic
equations, as in case of the spline application, though the
degree of the multinomial is higher 2.
Let us consider new approach of profiles mathematical
models’ parameters identification. This approach is based on
Neural Networks (Soft Computing) [7-9]. Let us consider the
regression equations:
n
International Science Index 13, 2008 waset.org/publications/14623
∂E
,
∂ars
c
н
where ars , ars are the old and new values of NN parameters
and γ is training speed.
The structure of NN for identifying the parameters of the
equation (34) is given on Fig 2.
VIII. DEFINITION OF HEAT EXCHANGE BOUNDARY
CONDITIONS
For determining of the temperature fields of AGTE
elements, the problem of gas flow distribution on blades’
profile of the turbine cascade is considered. The solution is
based on the numerical realization of the Fredholm boundary
integrated equation II kind.
On the basis of the theory of the potential flow of cascades,
distribution of speed along the profile contour can be found by
solving of the following integrated equation [10]:
ϕ (x k , y k ) = V∞ (x k cos α ∞ + y k sin α ∞ ) ± 1 Гθ B ∓ 1 ∫ ϕ (S )dθ , (36)
2π
Yi = ∑aij x j ;i =1,m
(34)
r s
Yi = ∑arsx1 x2 ;r = 0,l;s = 0,l;r + s ≤ l
(35)
j=1
r ,s
where
ars are the required parameters (regression
coefficients).
The problem is put definition of values aij and a rs
parameters of equations (34) and (35) based on the statistical
experimental data, i.e. input x j and x1 , x2 , output coordinates
Y of the model.
Neural Network (NN) consists from connected between their
neurons sets. At using NN for the solving (34) and (35) input
signals of the network are accordingly values of
variables X = ( x1 , x2 ,..., xn ) , X = ( x1 , x2 ) and output Y .
As parameters of the network are aij and a rs parameters’
values.
At the solving of the identification problem of parameters aij
and ars for the equations (34) and (35) with using NN, the
basic problem is training the last.
We allow, there are statistical data from experiments. On
the basis of these input and output data we making training
pairs ( X , T ) for network training. For construction of the
model process on input of NN input signals X move and
outputs are compared with reference output signals Т .
After comparison, the deviation value is calculating by
formula
E=
н
c
ars = ars + γ
1 k
∑ (Y j − T j ) 2
2 j =1
If for all training pairs, deviation value Е less given then
training (correction) parameters of a network comes to end
(Fig 1). In opposite case it continues until value Е will not
reach minimum.
Correction of network parameters for left and right part is
carried out as follows:
2π
S+
where ϕ ( x k , y k ) -the value of speeds potential; V∞ - the gas
speed vectors mean on the flowing; α ∞ - the angle between
the vector V∞ and the profile cascade axis; Г - the circulation
of speed; θ В - the angle that corresponds to the outlet edge of
the profile.
For the numerical solution of the integrated equation (36) the
following approximating expression is received:
n
ϕ j ± ∑ ϕi (θ j ,i +1 − θ j ,i −1 ) = V∞ (xk cos α ∞ + yk sin α ∞ ) ± 1 Гθ j , B ,
i =1
j
j
2π
where i = 2n − 1 , j = 2n , n is the numbers of parts.
Distribution of speeds potential ϕ along the profile contour
received from the solution of linear algebraic equations
system.
The value of the gas flow speed is determined by the
derivation of speeds potential along the contour s , i.e.
V (s ) = dϕ ds .
Distribution of speed along the profile contour can be
determined by solving the integral equation for the current
function ψ [10, 11]:
ψ = V∞ ( y cos α ∞ − x sin α ∞ ) ∓
1
2 π
2 π
∫ V ln sh t (x − xk ) + sin t ( y − yk )ds
2π S +
,
taking it to simple algebraic type:
⎧
1 n
⎪
⎪
⎡ 2π
⎤⎫
⎡ 2π
⎤
ψ =ψ ∞ ∓ ∑Vi ln⎨ sh2 ⎢ (x − xk )⎥ − sin2 ⎢ ( y − yk )⎥ ⎬Δsi
2π i=1
t
t
⎣
⎦⎪
⎣
⎦
⎪
⎭
⎩
The data of speed distribution along the profile contour are
incoming for determining outer boundary heat exchange
conditions.
The method for finding the local heat transfer coefficient α г
in this case is given in [4].
At the thickened entrance edges characteristic of cooled gas
vanes, the outer local heat exchange is described by empirical
dependences offered by E.G.Roost [4]:
Nu x = 0,5 ⋅ Re 0,5 ⋅ ε Tu ,
x
12
7. World Academy of Science, Engineering and Technology
International Journal of Mathematical, Computational Science and Engineering Vol:2 No:1, 2008
where
at 1,0% < Tu < 3,5% : ε Tu = 1 + Tu
at 3,5% < Tu
1,4
: ε Tu = 1 + 4Tu
/ 10;
0 ,28
/ 10
The problem of determining inner boundary heat exchange
conditions is necessary. For example, to calculate heat transfer
in the cooling channel track of the vanes of deflector
construction usually is applied criterial relationships. The
mean coefficients on the inner surface of the carrier envelope
at the entrance edge zone under the condition of its spray
injection by the number of sprays from round holes in the nose
deflector were obtained by the equation [12]:
Nu = C Re 0.98 Pr 0.43 /( L / bequ ) ,
International Science Index 13, 2008 waset.org/publications/14623
2
where bequ = π d 0 2t 0 - the width of hole which is equivalent
by the trans area; d 0 , t 0 - diameter and pitch of the holes in
nose deflector. The Re criterion in this formula is determined
by the speed of the flow from the holes at the exit in the nose
deflector and the length L of the carrier wall in the entrance
edge zone.
The empirical criterion equation earlier received [12]:
(
)
2
Nu = 0.018 0.36δ − 0.34δ + 0.56 − 0.1h S x ⋅
,
(37)
k
⋅ (Gc f k Gk f c ) ⋅ Re 0.8
was used for the calculation of the mean coefficient of heat
transfer at the inner surface of the vane wall in the area of the
perforated deflector.
In this equation: δ = δ d - the relative width of the deflector;
h = h d - the relative height of the slot channel between
deflector and vane wall; S = S d - the relative longitudinal
step of perforations holes system; d -the diameter of
perforation; L = 0.75 − 0.45δ ; k = 0.25 + 0.5h . The Reynolds
criterion in the formula (37) is defined by hydraulic diameter
of cross-section channel and speed of cooler flow in the
channel after the zone of deflector perforation.
IX. SOLUTION OF PROBLEM OF THE COOLING
SYSTEMS’ INTERNAL HYDRODYNAMICS
At known geometry of the cooling scheme, for definition of
the convective heat exchange local coefficients α В of the
cooler by the standard empirical formulas, is necessary to have
income values of air flow distribution in cooling channels.
For example, for blades with deflector and with cross current,
the value of the airflow GВ for blade cooling is possible to
define with the following dependence:
⎡
GВ =
1
⎢
⎞n
μ В FВ ⎛ d В
⎜
⎟ ⎢
d В ⎜ λВ ⋅ С ⎟ ⎢
⎝
⎠
⎢1 −
⎢
⎣
α Г (ψ Г − 1) кф
2 Вi (ψ Г − 1) кф
1 + кф
1
⎤n
⎥
⎥ ,
⎥
−ψ В ⎥
⎥
⎦
where ψ Г , ψ В -the gas and air temperature coefficients; κ ф coefficient of the form; d В - the characteristic size in the
formula Re B ; μ В , λВ -cooler dynamic viscosity and heat
conductivity coefficients; Вi - the Bio criterion for the blade
wall; FВ - the total area of passage for air; C and n coefficient and exponent ratio in criteria formulas for
convective heat exchange Nu В = C Re n for considered
В
cooling parts.
To determine the distribution of flow in the blade cooling
system, an equivalent hydraulic scheme is built.
The construction of the equivalent hydraulic tract circuit of the
vane cooling is connected with the description of the cooled
vane design. The whole passage of coolant flow is divided in
some definite interconnected sections, the so-called typical
elements, and every one has the possibility of identical
definition of hydraulic resistance. The points of connection of
typical elements are changed by node points, in which the
streams, mergion or division of cooler flows is taking places
proposal without pressure change. All the typical elements and
node points are connected in the same sequence and order as
the tract sites of the cooled vane.
To describe the coolant flow at every inner node the 1st low by
Kirchhoff is used:
m
m
j =1
j =1
(
)
f 1 = ∑ Gij = ∑ sign Δp ij k ij Δp ij ; i = 1,2,3...n
(38)
where Gij is the discharge of coolant on the element, i − j ,
m are the e number of typical elements connected to i node of
the circuit, n is the number of inner nodes of hydraulic circuit,
Δp ij - losses of total pressure of the coolant on element i − j .
In this formula the coefficient of hydraulic conductivity of the
circuit element ( i − j ) is defined as:
k ij = 2 f ij2 ⋅ pij ξ ij ,
(39)
where f ij , p ij , ξ ij are the mean area of the cross-section
passage of elements ( i − j ), density of coolant flow in the
element, and coefficient of hydraulic resistance of this
element. The system of nonlinear algebraic equations (38) is
solved by the Zeidel method with acceleration, taken from:
k
pik +1 = pik − f i k (∂f ∂p ) ,
where k is the iteration number, p ik is the coolant pressure in
i node of the hydraulic circuit. The coefficients of hydraulic
resistance ξ ij used in (39) are defined by analytical
dependencies, which are in the literature available at present
[12].
For example, to calculate a part of the cooling tract that
includes the area of deflector perforation coefficients of
hydraulic resistance in spray [13]:
ξ cΣ =
(Gr Gk )m ,
2( f c f k ) − 0.1
m=
1
2( f c f k ) − 0.23
and in general channel:
ξ cΣ = 1.1(Gr Gk )n , n = 0.381 ln( f c f k ) − 0.3
In these formulas, G r , G k are cooling air consumption in the
spray stream through the perforation deflector holes and slot
channel between the deflector and vanes wall, and f r , f k - the
flow areas.
13
8. World Academy of Science, Engineering and Technology
International Journal of Mathematical, Computational Science and Engineering Vol:2 No:1, 2008
X. RESULTS
The developed techniques of profiling, calculation of
temperature fields and parameters of the cooler in cooling
systems are approved at research of the gas turbine Ist stage
nozzle blades of GTN-16 (Turbomachinery Plant Enterprise,
Yekaterinburg, Russia) thermal condition. Thus the following
geometrical and state parameters of the stage are used: step of
the cascade - t = 41.5 мм , inlet gas speed to cascade V1 = 156 м / s , outlet gas speed from cascade -
manufacturability, reliability of engine elements design, and
on acceleration characteristics of the engine has been studied.
REFERENCES
[1]
[2]
V2 = 512 м / s , inlet gas speed vector angle - α1 = 0.7 , gas
0
flow temperature and pressure: on the entrance to the stage -
International Science Index 13, 2008 waset.org/publications/14623
Tг* = 1333 K , p * = 1.2095 ⋅ 10 6 Pа , on the exit from stage г
Tг1 = 1005 K , p г 1 = 0.75 ⋅ 10 6 Pа ; relative gas speed on the
exit from the cascade - λ1аd = 0.891 .
The geometrical model of the nozzle blades (fig.3), diagrams
of speed distributions V and convective heat exchange local
coefficients of gas α г along profile contour (fig.4) are
received.
The geometrical model and the cooling tract equivalent
hydraulic scheme (fig.5) are developed. Cooler basics
parameters in the cooling system and temperature field of
blade cross section (fig.6) are determined [2].
XI. CONCLUSIONS
The reliability of the methods was proved by experimental
investigations heat and hydraulic characteristics of blades in
"Turbine Construction" (Laboratory in St. Petersburg, Russia).
Geometric model, equivalent hydraulic schemes of cooling
tracks have been obtained, cooler parameters and temperature
field of "Turbo machinery Plant" enterprise (Yekaterinburg,
Russia) gas turbine nozzle blade of the 1st stage have been
determined. Methods have demonstrated high efficiency at
repeated and polivariant calculations, on the basis of which has
been offered the way of blade cooling system modernization.
The application of perfect methods of calculation of
temperature fields of elements of gas turbines is one of the
actual problems of gas turbine engines design. The efficiency
of these methods in the total influences to operational
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
Pashaev A.M., Sadykhov R.A., Iskenderov M.G., Samedov A.S.,
Sadykhov A.H. The effeciency of potential theory method for solving of
the tasks of aircraft and rocket design. 10-th National
MechanicConference. Istanbul Technic University, Aeronautics and
astronautics faculty, Istanbul, Turkey, July, 1997, p.61-62.
Pashayev A.M., Askerov D.D., Sadiqov R.A., Samedov A.S. Numerical
modeling of gas turbine cooled blades. International Research Journal of
Vilnius Gediminas, Technical University. «Aviation», vol. IX, No3,
2005, p.9-18.
Pashaev A.M., Sadykhov R.A., Samedov A.S. Highly effective methods
of calculation temperature fields of blades of gas turbines. V
International Symposium an Aeronautical Sciences “New Aviation
Technologies of the XXI century”, A collection of technical papers,
section N3, Zhukovsky, Russia, august,1999.
L.N. Zisina-Molojen and etc., Heat exchange in turbomachines. M.,
1974.
G. Galicheiskiy, A thermal guard of blades, М.: МАI, 1996.
A.M.Pashaev, R.A.Sadiqov, C.M.Hajiev, The BEM Application in
development of Effective Cooling Schemes of Gas Turbine Blades. 6th
Bienial Conference on Engineering Systems Design and Analysis,
Istanbul, Turkey, July, 8-11,2002.
Abasov M.T., Sadiqov A.H., Aliyarov R.Y. Fuzzy neural networks in the
system of oil and gas geology and geophysics // Third International
Conference on Application of Fuzzy Systems and Soft computing/
Wiesbaden, Germany, 1998,- p.108-117.
Yager R.R., Zadeh L.A. (Eds). Fuzzy sets, neural networks and soft
computing. VAN Nostrand Reinhold. N.-Y. - № 4,1994.
Mohamad H. Hassoun. Fundamentals of artificial neutral networks / A
Bradford Book. The MIT press Cambridge, Massachusetts, London,
England, 1995.
Pashaev A.M., Sadykhov R.A., Samedov A.S., Mamedov R.N. The
solution of fluid dynamics direct problem of turbomachines cascades
with integral equations method. Proceed. NAA. Vol. 3, Baku, 2003.
V.S. Beknev, V.M. Epifanov, A.I. Leontyev, M.I. Osipov and ets. Fluid
dynamics. A mechanics of a fluid and gas. Moscow, MGTU nam. N.E.
Bauman, 1997, 671 p.
S.Z. Kopelev, A.F.Slitenko Construction and calculation of GTE cooling
systems. Kharkov, “Osnova”,1994, 240 p.
L.V. Arseniev, I.B. Mitryayev, N.P. Sokolov The flat channels hydraulic
resistances with a system of jets in a main stream . “Energetika”, 1985,
№ 5, p.85-89.
APPENDIX
Correction algorithm
X
Input
signals
NN
Parameters
Deviations
Random-number
Target
signals
Y
Training
quality
generator
Fig. 1 System for network-parameter (weights, threshold) training (with feedback)
14
9. World Academy of Science, Engineering and Technology
International Journal of Mathematical, Computational Science and Engineering Vol:2 No:1, 2008
αg,
x1
x
2
xj
a1
Vt
i
1
2
m K
λ
2
Yi
a2
i
aij
Fig. 2 Neural network structure for linear regression
Fig. 2 Neural network structure for multiple multiple linearequation
International Science Index 13, 2008 waset.org/publications/14623
у,
mm
Fig. 4 Distribution of the relative speeds λ (1)
and of gas convective heat exchange coefficients
α Г (2) along the periphery of the profile contour
100
95
90
85
80
75
70
65
60
55
50
45
40
35
Fig. 5 The equivalent hydraulic scheme of experimental
nozzle blade cooling system
30
25
20
ТB , К
15
1180
10
1160
5
1140
0
0
5 10 15 20 25 30 35 40
45 x, mm
Fig. 3 The cascade of profiles of the
nozzle cooled blade
1120
1100
1080
1060
1040
1020
1000
0
10
20
30
40
Fig. 6 Distribution of temperature along outside ( ) and internal (
the cooled nozzle blade
15
№ sections
) contours of