Contribution to Load-Frequency Regulation of a Hydropower Plant with Long Tail-Race Tunnel

In this paper, a hydroelectric power plant with long tail-race tunnel has been modelled for assessing its contribution to secondary regulation reserve. Cavitation problems, caused by the discharge conduit length, are expected downstream the turbine where low pressure appears during regulation manoeuvres. Therefore, governor's gains should be selected taking into account these phenomena. On the other hand, regulation services bidden by the plant operator should fulfil TSO (Transmission System Operator) quality requirements. A methodology for tuning governor PI gains is proposed and applied to a Hydro power plant in pre-design phase in northwest area of Spain. The PI gains adjustment proposed provides a proper plant response, according to some established indexes, while avoiding cavitation phenomena.

Integrated plant monitoring to improve plant operation strategy and results

The more and more demanding conditions in the power generation sector requires to apply all the available technologies to optimize processes and reduce costs. An integrated asset management strategy, combining technical analysis and operation and maintenance management can help to improve plant performance, flexibility and reliability. In order to deploy such a model it is necessary to combine plant data and specific equipment condition information, with different systems devoted to analyze performance and equipment condition, and take advantage of the results to support operation and maintenance decisions. This model that has been dealt in certain detail for electricity transmission and distribution networks, is not yet broadly extended in the power generation sector, as proposed in this study for the case of a combined power plant. Its application would turn in direct benefit for the operation and maintenance and for the interaction to the energy market

Impact assessment of an integrated approach for thermal power generation plants asset management in the current energy market

El sector energético, en España en particular, y de forma similar en los principales países de Europa, cuenta con una significativa sobrecapacidad de generación, debido al rápido y significativo crecimiento de las energías renovables en los últimos diez años y la reducción de la demanda energética, como consecuencia de la crisis económica. Esta situación ha hecho que las centrales térmicas de generación de electricidad, y en concreto los ciclos combinados de gas, operen con un factor de utilización extremadamente bajo, del orden del 10%. Además de la reducción de ingresos, esto supone para las plantas trabajar continuamente fuera del punto de diseño, provocando una significativa pérdida de rendimiento y mayores costes de explotación. En este escenario, cualquier contribución que ayude a mejorar la eficiencia y la condición de los equipos, es positivamente considerada. La gestión de activos está ganando relevancia como un proceso multidisciplinar e integrado, tal y como refleja la reciente publicación de las normas ISO 55000:2014. Como proceso global e integrado, la gestión de activos requiere el manejo de diversos procesos y grandes volúmenes de información, incluso en tiempo real. Para ello es necesario utilizar tecnologías de la información y aplicaciones de software. Esta tesis desarrolla un concepto integrado de gestión de activos (Integrated Plant Management – IPM) aplicado a centrales de ciclo combinado y una metodología para estimar el beneficio aportado por el mismo. Debido a las incertidumbres asociadas a la estimación del beneficio, se ha optado por un análisis probabilístico coste-beneficio. Así mismo, el análisis cuantitativo se ha completado con una validación cualitativa del beneficio aportado por las tecnologías incorporadas al concepto de gestión integrada de activos, mediante una entrevista realizada a expertos del sector de generación de energía. Los resultados del análisis coste-beneficio son positivos, incluso en el desfavorable escenario con un factor de utilización de sólo el 10% y muy prometedores para factores de utilización por encima del 30%. ABSTRACT The energy sector particularly in Spain, and in a similar way in Europe, has a significant overcapacity due to the big growth of the renewable energies in the last ten years, and it is seriously affected by the demand decrease due to the economic crisis. That situation has forced the thermal plants and in particular, the combined cycles to operate with extremely low annual average capacity factors, very close to 10%. Apart from the incomes reduction, working in out-of-design conditions, means getting a worse performance and higher costs than expected. In this scenario, anything that can be done to improve the efficiency and the equipment condition is positively received. Asset Management, as a multidisciplinary and integrated process, is gaining prominence, reflected in the recent publication of the ISO 55000 series in 2014. Dealing Asset Management as a global, integrated process needs to manage several processes and significant volumes of information, also in real time, that requires information technologies and software applications to support it. This thesis proposes an integrated asset management concept (Integrated Plant Management-IPM) applied to combined cycle power plants and develops a methodology to assess the benefit that it can provide. Due to the difficulties in getting deterministic benefit estimation, a statistical approach has been adopted for the cot-benefit analysis. As well, the quantitative analysis has been completed with a qualitative validation of the technologies included in the IPM and their contribution to key power plant challenges by power generation sector experts. The cost- benefit analysis provides positive results even in the negative scenario of annual average capacity factor close to 10% and is promising for capacity factors over 30%.

Fresnel-based modular solar fields for performance/cost optimization in solar thermal power plants: A comparison with parabolic trough collectors.

Linear Fresnel collectors are identified as a technology that should play a main role in order to reduce cost of Concentrating Solar Power. An optical and thermal analysis of the different blocks of the solar power plant is carried out, where Fresnel arrays are compared with the most extended linear technology: parabolic trough collectors. It is demonstrated that the optical performance of Fresnel array is very close to that of PTC, with similar values of maximum flux intensities. In addition, if the heat carrier fluid flows in series by the tubes of the receiver, relatively high thermal efficiencies are achieved. Thus, an annual solar to electricity efficiency of 19% is expected, which is similar to the state of the art in PTCs; this is done with a reduction of costs, thanks to lighter structures, that drives to an estimation of LCOE of around 6.5 c€/kWh.

The use of geographic information systems for the optimal location of biomass power plants in the Madrid Community (Spain)

This article has been extracted from the results of a thesis entitled “Potential bioelectricity production of the Madrid Community Agricultural Regions based on rye and triticale biomass.” The aim was, first, to quantify the potential of rye (Secale Cereale L.) and triticale ( Triticosecale Aestivum L.) biomass in each of the Madrid Community agricultural regions, and second, to locate the most suitable areas for the installation of power plants using biomass. At least 17,339.9 t d.m. of rye and triticale would be required to satisfy the biomass needs of a 2.2 MW power plant, (considering an efficiency of 21.5%, 8,000 expected operating hours/year and a biomass LCP of 4,060 kcal/kg for both crops), and 2,577 ha would be used (which represent 2.79% of the Madrid Community fallow dry land surface). Biomass yields that could be achieved in Madrid Community using 50% of the fallow dry land surface (46,150 ha representing 5.75% of the Community area), based on rye and triticale crops, are estimated at 84,855, 74,906, 70,109, 50,791, 13,481, and 943 t annually for the Campiña, Vegas, Sur Occidental, Área Metropolitana, Lozoya-Somosierra, and Guadarrama regions. The latter represents a bioelectricity potential of 10.77, 9.5, 8.9, 6.44, 1.71, and 0.12 MW, respectively.

Emulation of an OWC Ocean Energy Plant With PMSG and Irregular Wave Model

Ocean energy is a promising resource for renewable electricity generation that presents many advantages, such as being more predictable than wind energy, but also some disadvantages such as large and slow amplitude variations in the generated power. This paper presents a hardware-in-the-loop prototype that allows the study of the electric power profile generated by a wave power plant based on the oscillating water column (OWC) principle. In particular, it facilitates the development of new solutions to improve the intermittent profile of the power fed into the grid or the test of the OWC behavior when facing a voltage dip. Also, to obtain a more realistic model behavior, statistical models of real waves have been implemented.

Accident Management Actions In An Upper-Head Small-Break Loss-Of-Coolant Accident With High-Pressure Safety Injection Failed

Since the Three Mile Island accident, an important focus of pressurized water reactor (PWR) transient analyses has been a small-break loss-of-coolant accident (SBLOCA). In 2002, the discovery of thinning of the vessel head wall at the Davis Besse nuclear power plant reactor indicated the possibility of an SBLOCA in the upper head of the reactor vessel as a result of circumferential cracking of a control rod drive mechanism penetration nozzle - which has cast even greater importance on the study of SBLOCAs. Several experimental tests have been performed at the Large Scale Test Facility to simulate the behavior of a PWR during an upper-head SBLOCA. The last of these tests, Organisation for Economic Co-operation and Development Nuclear Energy Agency Rig of Safety Assessment (OECD/NEA ROSA) Test 6.1, was performed in 2005. This test was simulated with the TRACE 5.0 code, and good agreement with the experimental results was obtained. Additionally, a broad analysis of an upper-head SBLOCA with high-pressure safety injection failed in a Westinghouse PWR was performed taking into account different accident management actions and conditions in order to check their suitability. This issue has been analyzed also in the framework of the OECD/NEA ROSA project and the Code Applications and Maintenance Program (CAMP). The main conclusion is that the current emergency operating procedures for Westinghouse reactor design are adequate for these kinds of sequences, and they do not need to be modified.

An overview on armor research for the laser fusion project HiPER

During the current preparatory phase of the European laser fusion project HiPER, an intensive effort has being placed to identify an armour material able to protect the internal walls of the chamber against the high thermal loads and high fluxes of x-rays and ions produced during the fusion explosions. This poster addresses the different threats and limitations of a poly-crystalline Tungsten armour. The analysis is carried out under the conditions of an experimental chamber hypothetically constructed to demonstrate laser fusion in a repetitive mode, subjected to a few thousand 48MJ shock ignition shots during its entire lifetime. If compared to the literature, an extrapolation of the thermomechanical and atomistic effects obtained from the simulations of the experimental chamber to the conditions of a Demo reactor (working 24/7 at hundreds of MW) or a future power plant (producing GW) suggests that “standard” tungsten will not be a suitable armour. Thus, new materials based on nano-structured W and C are being investigated as possible candidates. The research programme launched by the HiPER material team is introduced.

Interactive Graphic Simulation: An Advanced Methodology to Improve the Teaching-Learning Process in Nuclear Engineering Education and Training

Nowadays, computer simulators are becoming basic tools for education and training in many engineering fields. In the nuclear industry, the role of simulation for training of operators of nuclear power plants is also recognized of the utmost relevance. As an example, the International Atomic Energy Agency sponsors the development of nuclear reactor simulators for education, and arranges the supply of such simulation programs. Aware of this, in 2008 Gas Natural Fenosa, a Spanish gas and electric utility that owns and operate nuclear power plants and promotes university education in the nuclear technology field, provided the Department of Nuclear Engineering of Universidad Politécnica de Madrid with the Interactive Graphic Simulator (IGS) of “José Cabrera” (Zorita) nuclear power plant, an industrial facility whose commercial operation ceased definitively in April 2006. It is a state-of-the-art full-scope real-time simulator that was used for training and qualification of the operators of the plant control room, as well as to understand and analyses the plant dynamics, and to develop, qualify and validate its emergency operating procedures.

Contribution of re-regulation reservoirs considering pumping capability to environmentally friendly hydropower operation

Environmental constraints imposed on hydropoweroperation are usually given in the form of minimum environmental flows and maximum and minimum rates of change of flows, or ramp rates. One solution proposed to mitigate the environmental impact caused by the flows discharged by a hydropower plant while reducing the economic impact of the above-mentioned constraints consists in building a re-regulationreservoir, or afterbay, downstream of the power plant. Adding pumpingcapability between the re-regulationreservoir and the main one could contribute both to reducing the size of the re-regulationreservoir, with the consequent environmental improvement, and to improving the economic feasibility of the project, always fulfilling the environmental constraints imposed to hydropoweroperation. The objective of this paper is studying the contribution of a re-regulationreservoir to fulfilling the environmental constraints while reducing the economic impact of said constraints. For that purpose, a revenue-driven optimization model based on mixed integer linear programming is used. Additionally, the advantages of adding pumpingcapability are analysed. In order to illustrate the applicability of the methodology, a case study based on a real hydropower plant is presented

Problemas de interacción suelo estructura en cimentaciones y muros de contención. Influencia de los fenómenos de despegue y deslizamiento

Los terremotos constituyen una de las más importantes fuentes productoras de cargas dinámicas que actúan sobre las estructuras y sus cimentaciones. Cuando se produce un terremoto la energía liberada genera movimientos del terreno en forma de ondas sísmicas que pueden provocar asientos en las cimentaciones de los edificios, empujes sobre los muros de contención, vuelco de las estructuras y el suelo puede licuar perdiendo su capacidad de soporte. Los efectos de los terremotos en estructuras constituyen unos de los aspectos que involucran por su condición de interacción sueloestructura, disciplinas diversas como el Análisis Estructural, la Mecánica de Suelo y la Ingeniería Sísmica. Uno de los aspectos que han sido poco estudiados en el cálculo de estructuras sometidas a la acciones de los terremotos son los efectos del comportamiento no lineal del suelo y de los movimientos que pueden producirse bajo la acción de cargas sísmicas, tales como posibles despegues y deslizamientos. En esta Tesis se estudian primero los empujes sísmicos y posibles deslizamientos de muros de contención y se comparan las predicciones de distintos tipos de cálculos: métodos pseudo-estáticos como el de Mononobe-Okabe (1929) con la contribución de Whitman-Liao (1985), y formulaciones analíticas como la desarrollada por Veletsos y Younan (1994). En segundo lugar se estudia el efecto del comportamiento no lineal del terreno en las rigideces de una losa de cimentación superficial y circular, como la correspondiente a la chimenea de una Central Térmica o al edificio del reactor de una Central Nuclear, considerando su variación con frecuencia y con el nivel de cargas. Finalmente se estudian los posibles deslizamientos y separación de las losas de estas dos estructuras bajo la acción de terremotos, siguiendo la formulación propuesta por Wolf (1988). Para estos estudios se han desarrollado una serie de programas específicos (MUROSIS, VELETSOS, INTESES y SEPARSE) cuyos listados y detalles se incluyen en los Apéndices. En el capítulo 6 se incluyen las conclusiones resultantes de estos estudios y recomendaciones para futuras investigaciones. ABSTRACT Earthquakes constitute one of the most important sources of dynamic loads that acting on structures and foundations. When an earthquake occurs the liberated energy generates seismic waves that can give rise to structural vibrations, settlements of the foundations of buildings, pressures on retaining walls, and possible sliding, uplifting or even overturning of structures. The soil can also liquefy losing its capacity of support The study of the effects of earthquakes on structures involve the use of diverse disciplines such as Structural Analysis, Soil Mechanics and Earthquake Engineering. Some aspects that have been the subject of limited research in relation to the behavior of structures subjected to earthquakes are the effects of nonlinear soil behavior and geometric nonlinearities such as sliding and uplifting of foundations. This Thesis starts with the study of the seismic pressures and potential displacements of retaining walls comparing the predictions of two types of formulations and assessing their range of applicability and limitations: pseudo-static methods as proposed by Mononobe-Okabe (1929), with the contribution of Whitman-Liao (1985), and analytical formulations as the one developed by Veletsos and Younan (1994) for rigid walls. The Thesis deals next with the effects of nonlinear soil behavior on the dynamic stiffness of circular mat foundations like the chimney of a Thermal Power Station or the reactor building of a Nuclear Power Plant, as a function of frequency and level of forces. Finally the seismic response of these two structures accounting for the potential sliding and uplifting of the foundation under a given earthquake are studied, following an approach suggested by Wolf (1988). In order to carry out these studies a number of special purposes computer programs were developed (MUROSIS, VELETSOS, INTESES and SEPARSE). The listing and details of these programs are included in the appendices. The conclusions derived from these studies and recommendations for future work are presented in Chapter 6.

Desarrollo de un modelo para la caracterización termoeconómica de ciclos combinados de turbinas de gas y de vapor en condiciones de carga variable

En la actualidad, el interés por las plantas de potencia de ciclo combinado de gas y vapor ha experimentado un notable aumento debido a su alto rendimiento, bajo coste de generación y rápida construcción. El objetivo fundamental de la tesis es profundizar en el conocimiento de esta tecnología, insuficientemente conocida hasta el momento debido al gran número de grados de libertad que existen en el diseño de este tipo de instalaciones. El estudio se realizó en varias fases. La primera consistió en analizar y estudiar las distintas tecnologías que se pueden emplear en este tipo de centrales, algunas muy recientes o en fase de investigación, como las turbinas de gas de geometría variable, las turbinas de gas refrigeradas con agua o vapor del ciclo de vapor o las calderas de paso único que trabajan con agua en condiciones supercríticas. Posteriormente se elaboraron los modelos matemáticos que permiten la simulación termodinámica de cada uno de los componentes que integran las plantas, tanto en el punto de diseño como a cargas parciales. Al mismo tiempo, se desarrolló una metodología novedosa que permite resolver el sistema de ecuaciones que resulta de la simulación de cualquier configuración posible de ciclo combinado. De esa forma se puede conocer el comportamiento de cualquier planta en cualquier punto de funcionamiento. Por último se desarrolló un modelo de atribución de costes para este tipo de centrales. Con dicho modelo, los estudios se pueden realizar no sólo desde un punto de vista termodinámico sino también termoeconómico, con lo que se pueden encontrar soluciones de compromiso entre rendimiento y coste, asignar costes de producción, determinar curvas de oferta, beneficios económicos de la planta y delimitar el rango de potencias donde la planta es rentable. El programa informático, desarrollado en paralelo con los modelos de simulación, se ha empleado para obtener resultados de forma intensiva. El estudio de los resultados permite profundizar ampliamente en el conocimiento de la tecnología y, así, desarrollar una metodología de diseño de este tipo de plantas bajo un criterio termoeconómico. ABSTRACT The growing energy demand and the need of shrinking costs have led to the design of high efficiency and quick installation power plants. The success of combined cycle gas turbine power plants lies on their high efficiency, low cost and short construction lead time. The main objective of the work is to study in detail this technology, which is not thoroughly known owing to the great number of degrees of freedom that exist in the design of this kind of power plants. The study is divided into three parts. Firstly, the different technologies and components that could be used in any configuration of a combined cycle gas turbine power plant are studied. Some of them could be of recent technology, such as the variable inlet guide vane compressors, the H-technology for gas turbine cooling or the once-through heat recovery steam generators, used with water at supercritical conditions. Secondly, a mathematical model has been developed to simulate at full and part load the components of the power plant. At the same time, a new methodology is proposed in order to solve the equation system resulting for any possible power plant configuration. Therefore, any combined cycle gas turbine could be simulated at any part load condition. Finally a themoeconomic model is proposed. This model allows studying the power plant not only from a thermodynamic point of view but also from a thermoeconomic one. Likewise, it allows determining the generating costs or the cash flow, thus achieving a trade off between efficiency and cost. Likewise, the model calculates the part load range where the power plant is profitable. Once the thermodynamic and thermoeconomic models are developed, they are intensively used in order to gain knowledge in the combined cycle gas turbine technology and, in this way, to propose a methodology aimed at the design of this kind of power plants from a thermoeconomic point of view.

Grid-connected roof solar plants in educational institutions for testing and electricity generation: a case in Spain.

The use of photovoltaic experimental plants in engineering educational buildings contributes to an increase in acceptance of this technology by future engineers. There are some photovoltaic (PV) systems in educational buildings in Spain, but they are usually limited to buildings in relation to electrical technologies or research areas. They are not common in other educational or official buildings. This paper presents the project of a grid-connected solar plant with two main objectives. First, different PV module technologies will be compared. Second, an emphasis on agronomical areas in educational settings will be reviewed in an attempt to facilitate student engagement in the use of the power plant. The system is grid-connected in order to pay-back the investment in the plant. In fact the electricity generated by the plant will be used by the installations of the building, as it is the closest consumer. This work intends to approximate photovoltaic technology to university degrees not directly related with it and at the same time research in comparison of systems with different technologies. This is a good example of an solar plant for both optimum production and educational purposes.

Optimización de la metodología de contratación de carbones térmicos con destino a la generación eléctrica

En los últimos quince años se ha producido una liberalización de los mercados eléctricos en los distintos países de ámbito occidental que ha ido acompañado de un incremento por la preocupación por la incidencia de las distintas tecnologías de generación en el entorno medioambiental. Ello se ha traducido en la aparición de un marco regulatorio más restrictivo sobre las tecnologías de generación fósiles, con mayor incidencia en las derivadas de productos petrolíferos y carbón. A nivel mundial han ido apareciendo cambios normativos relativos a las emisiones de distintos elementos contaminantes (CO2, SO2, NOx…), que hacen que en particular las centrales térmicas de carbón vean muy afectadas su rentabilidad y funcionamiento. Esta situación ha supuesto que la tecnología de generación eléctrica con carbón haya avanzado considerablemente en los últimos años (calderas supercríticas, sistemas de desulfuración, gasificación del carbón…). No obstante, el desarrollo de la generación con energías renovables, la generación con gas mediante centrales de ciclo combinado y la opinión social relativa a la generación con carbón, principalmente en Europa, suponen un serio obstáculo a la generación con carbón. Por consiguiente, se hace necesario buscar vías para optimizar la competitividad de las centrales de carbón y el camino más razonable es mejorar el margen esperado de estas plantas y en particular el coste de adquisición del carbón. Ello se hace aún más importante por el hecho de existir numerosas centrales de carbón y un elevado número de nuevos proyectos constructivos de centrales de carbón en países asiáticos. Por consiguiente, el objeto de la presente tesis doctoral se centra en definir una metodología para optimizar la compra de carbón, desde el punto de vista económico y técnico, con destino a su consumo en una central térmica, con ello reducir el coste del carbón consumido y mejorar su competitividad. También se enfoca a determinar que herramientas pueden ser utilizadas para optimizar la gestión del carbón después de su compra y con ello abrir la posibilidad de obtener márgenes adicionales para dicho carbón. De acuerdo con este objetivo, el autor de la presente Tesis Doctoral realiza tres aportaciones novedosas en el ámbito de la contratación de carbón térmico y su optimización posterior: - Evaluación de carbones para su adquisición considerando el efecto de la calidad del carbón en el coste de generación asociado a cada carbón ofertado. - Creación, desarrollo, implantación y utilización de una potente herramienta de planificación de Combustibles. Esta herramienta, está diseñada con el objeto de determinar la solución económica óptima de aprovisionamientos, consumos y niveles de existencias para un parque de generación con centrales de carbón y fuelóleo. - La extensión de una metodología contractual habitual en el mercado spot de Gas Natural Licuado, a la contratación spot de Carbón de Importación. Esta se basa en el desarrollo de Acuerdos Marcos de Compra/Venta de carbón, que por su flexibilidad permitan obtener resultados económicos adicionales después de la compra de un carbón. Abstract In the last fifteen years, a liberalization of the electrical markets has occurred in the western countries. This process has been accompanied by an increasing concern of the impact of the different generation technologies towards the environment. This has motivated a regulated framework restricting the use of fossil fuels, impacting a great deal in coal and oil based products. Worldwide, new legal changes have been arising related to the emissions of the different pollutants (CO2, SO2, NOx…). These changes have had a deep impact in the feasibility, profit and running of coal fired power plants. This situation has motivated the coal electrical generation technologies to move forward in an important way in the last few years (supercritical furnaces, desulphuration plants, coal gasification…). Nevertheless, the development of the renewable generation, the gas combined cycle generation and the social opinion related to the coal electrical generation, mainly in Europe, have created a serious obstacle to the generation of electricity by coal. Therefore it is necessary to look for new paths in order to optimize the competitiveness of the coal fired power plants and the most reasonable way is to improve the expected margin of these plants and particularly the coal purchase cost. All of the above needs to be taken into context with the large number of existing coal fired power plants and an important number of new projects in Asian countries. Therefore, the goal of the current doctoral dissertation is focused to define a methodology to be considered in order to optimize the coal purchase, from an economical and a technical point of view. This coal, destined for power plant consumption, permits the reduction of consumption coal cost and improves the plant’s competitiveness. This document is also focused to define what tools we can use to optimize the coal management after deal closing and therefore open the possibility to get further margins. According to this goal, the author of this doctoral dissertation provides three important new ideas in the ambit of contracting steam coal and the posterior optimization: - Evaluation of coal purchases, considering the effect of coal quality on the cost of generation associated with each type of coal offered. - The creation, development, deployment and use of a strong planning tool of fuels. This tool is designed for the purpose of determining the optimal economic solution of fuel supply, consumption and stock levels for a power generation portfolio using coal and fuel oil fired power plants. - The application of a common contractual methodology in the spot market of Liquid Natural Gas, for the contracting spot imported coal. This is based on the development of Framework Agreements for the Purchasing / Sale of coal, which because of its flexibility allows for the gain of additional financial results after the purchase of coal.

Control rod drop surveillance using two friction coefficients

In the case of large burnup, a control rod (CR) guide tube in the pressurized water reactor of a commercial nuclear power plant might bend. As a consequence, a CR drop experiment may indicate an event of a CR partially inserted and whether the CR should be deemed inoperable. Early prevention of such an event can be achieved by measuring two friction coefficients: the hydraulic coefficient and the sliding coefficient. The hydraulic coefficient hardly changes, so that the curvature of the guide tube can only be detected thanks to a variation of the sliding coefficient. A simple model for the CR drop is established and validated with CR drop experiments. If tmx denotes the instant of CR maximum velocity, a linear relationship between (tmx)_2 and the sliding coefficient is found.