Elementary Trigonometric Sums related to Quadratic Residues

Let p be a prime = 3 (mod 4). A number of elegant number-theoretical properties of the sums T(p) = \sqrt{p}sum_{n=1}^{(p-1)/2} tan(n^2\pi/p) and C(p) = \sqrt{p}sum_{n=1}^{(p-1)/2} cot(n^2\pi/p) are proved. For example, T(p) equals p times the excess of the odd quadratic residues over the even ones in the set {1,2,...,p-1}; this number is positive if p = 3 (mod 8) and negative if p = 7 (mod 8). In this revised version the connection of these sums with the class-number h(-p) is also discussed. For example, a very simple formula expressing h(-p) by means of the aforementioned excess is proved. The bibliography has been considerably enriched. This article is of an expository nature.Comment: A number of misprints have been corrected and one or two improvements have been done to the previous version of the paper with same title. The paper will appear to Elem. der Mat

Future perspectives on sustainable tribology

AbstractThis paper highlights the future perspectives of sustainable tribology by examining the economic, environmental and social impact of three tribological case studies. One case study examines the sustainability and durability of micro-CHP systems looking the tribological phenomena generated within a scroll expander system. The scroll is the main part of a specific micro-CHP system and experiences wear and cavitation damage. The tribological optimization of the scroll expander improves the sustainability of the micro-CHP unit while it has a serious economic and environmental impact to the consumers and to the society in general. Another case study is focused on friction and wear performance of lifeboat launch slipways. The causes of high friction and wear during the RNLI's lifeboat launches along an inclined slipway are investigated with a view to reducing the environmental impact due to slipway panel wear and lubricant release into the marine environment. The project encompasses the sustainable design of slipway panels using design modifications based on tribological investigations to double their lifespan, while environmental and economic impact was significantly reduced by the use of biodegradable greases and water as lubricants. The final case study involves an investigation of recycled plastic materials to replace polyurethane used on skateboard wheels, scooters and similar applications. Polyurethane (PU) is difficult to recycle. With the dwindling resources and environmental problems facing the world today, recycling for both waste reduction and resource preservation has become an increasingly important aspect of sustainability. The tribological results showed that recycled polycarbonate plastic can effectively act as a substitute to polyurethane wheels. Moreover, sustainability considerations showing the environmental benefits of the use of recycled plastics over PU include reducing the CO2 footprint by 50% and the energy consumed by 60%, among other benefits. These case studies emphasise the importance of sustainable tribology in our epoch showing that increased sustainability performance can be achieved through tribology to a significant extent in many cases, providing stability to our world and more viable long term growth to our societies

On px2+q2n=yppx^2 + q^{2n}= y^p and related Diophantine equations

The title equation, where $p>3$ is a prime number $\not\equiv 7 \pmod 8$, $q$ is an odd prime number and $x,y,n$ are positive integers with $x,y$ relatively prime, is studied. When $p\equiv 3\pmod 8$, we prove (Theorem 2.3) that there are no solutions. For $p\not\equiv 3\pmod 8$ the treatment of the equation turns out to be a difficult task. We focus our attention to $p=5$, by reason of an article by F. Abu Muriefah, published in this journal, vol. 128 (2008), 1670-1675. Our main result concerning this special equation is Theorem 1.1, whose proof is based on results around the Diophantine equation $5x^2-4=y^n$ (integer solutions), interesting in themselves, which are exposed in Sections 3 and 4. These last results are obtained by using tools such as Linear Forms in Two Logarithms and Hypergeometric Series.Comment: 23 pages, second version with minor revision

Evaluation of Cavitation Erosion Behavior of Commercial Steel Grades Used in the Design of Fluid Machinery

The erosion response under cavitation of different steel grades was assessed by studying the erosion rate, the volume removal, the roughness evolution, and the accumulated strain energy. A 20 kHz ltrasonic transducer with a probe diameter of 5 mm and peak-to-peak amplitude of 50 lm was deployed in distilled water to induce damage on the surface of commercial chromium and carbon steel samples. After a relatively short incubation period, cavitation induced the formation of pits, cracks, and craters whose features strongly depended on the hardness and composition of the tested steel. AISI 52100 chromium steel showed the best performance and is, therefore, a promising design candidate for replacing the existing fluid machinery materials that operate within potential cavitating environments

Experimental and analytical thermal study of PTFE composite sliding against high carbon steel as a function of the surface roughness, sliding velocity and applied load

This study investigates the contact temperatures caused by frictional heating of sliding parallel pairs. In this case the materials studied are a PTFE composite in contact with a high carbon steel plate. These materials are commonly used for industrial applications, in particular as the main contacting components within a scroll expander system. The expected contact temperature values are important to be quantified in order to predict failure mechanisms associated with excessive thermal effects caused by sliding friction. A rational and coherent interpretation of the thermal effects on the actual tribological contact is presented.Contact temperatures are monitored continuously using a high-precision infrared thermal imaging technique with a systematic variation in surface roughness of the high carbon steel material. These surface temperatures are investigated as a function of the friction coefficient, the sliding velocity and the applied load while the most influential parameter for the temperature rise is determined. Analytical results using conventional mathematical methodology are also produced. The analytical and experimental findings are then compared indicating interesting correlations within the macro- and micro-surface temperature regimes and the experimental conditions. Microscopic observations show that thermal effects can seriously affect fibers durability while transfer films formed across the steel counterpart can be beneficial for the operation of scroll systems under specific roughness and test conditions

Thermal analysis and tribological investigation on TPU and NBR elastomers applied to sealing applications

This study investigates the contact temperatures reached due to frictional heating on TPU (Thermoplastic polyurethane) and NBR (Nitrile butadiene rubber) seal surfaces during operation. These elastomers present limited thermal resistance so an excessive temperature rise may affect their tribological performance. Sliding tests of the elastomers against steel cylinders were carried out and the surface temperature evolution was acquired during the tests using a high precision infrared camera. Frictional behaviour and temperature curves were analyzed. The influence of the experimental parameters, such as the sealing material, sliding velocity, applied load and steel surface conditions was examined. Experimental thermal results were compared with those calculated through well-established analytical models, in order to determine the advantages and limitations of the latter

Calibration and performance assessment of an innovative high-temperature cavitometer

This paper describes a series of systematic experimental studies to evaluate the performance of a high-temperature cavitometer under well-controlled conditions. The cavitometer was specifically designed for measurements in liquid metals: it operates through a long tungsten waveguide (probe), providing thermal protection to the piezo sensing elements placed outside the hot area, and with sufficient bandwidth to enable the monitoring of broadband acoustic emissions associated with cavitation activity. It was calibrated electrically, and acoustically, at kHz and MHz frequencies, and so can be used to estimate acoustic pressures (in Pa), providing physical, and consequently practical, meaning to cavitation measurements within liquid metals. Results obtained from ultrasonic sources in a cylindrical vessel using water showed that the cavitometer is a reliable and robust device for characterizing direct field acoustic pressures and broadband emissions from the resulting cavitation. Additionally, preliminary characterization of the real-time acoustic pressures during ultrasonic processing of liquid aluminium (Al) in a standard clay-graphite crucible were performed for the first time. The use of the calibrated cavitometer will establish a more generalized approach for measuring the actual acoustic pressures over a broad range of liquid temperatures within a sonicated medium, demonstrating its potential use as a tool for optimizing, controlling, and scaling-up processes

Cavitation in Thermoplastic Melts: New Insights into Ultrasound-Assisted Fibre-Impregnation

The impregnation of continuous carbon fibre roving with thermoplastic melt is a challenging task due to the high viscosity and surface tension of the melt. A new technique is under development utilizing ultrasonic oscillations within the thermoplastic melt which encloses a fibre roving, to achieve fibre impregnation. Despite ultrasonic processing being very efficient, the specific conditions created in the thermoplastic melt have never been studied before. This study investigated whether cavitation effects could be present during ultrasound-assisted fibre impregnation. The observed acoustic effects allowed us to suggest the possible underlying mechanisms. For the purpose of the study a melt-bath impregnation setup with polylactide was built. To detect the cavitation effects and acoustic parameters a calibrated high-temperature cavitometer was used. The results showed the formation of small cavitation zones in the direct vicinity to the sonotrode tip where the fibre roving would be positioned. Therefore, the occurrence of cavitation was established, and induced effects like shock waves, microjets and microstreaming should be further considered for detailed investigation of the ultrasound-assisted impregnation mechanism.This research study was supported for IT, MK and DE by the UK Engineering and Physical Sciences Research Council (EPSRC) through the UltraMelt2 (grant numbers EP/R011001/1, EP/R011095/1 and EP/R011044/1) project