н^н University of Maribor Faculty of Energy Technology Journal of ENERGY TECHNOLOGY Journal of ENERGY TECHNOLOGY _____л IME) VOLUME 8 / Issue 4 Revija Journal of Energy Technology (JET) je indeksirana v bazi INSPEC©. The Journal of Energy Technology (JET) is indexed and abstracted in database INSPEC©. ✓_____"N ш JOURNAL OF ENERGY TECHNOLOGY Ustanovitelj / FOUNDER Fakulteta za energetiko, UNIVERZA V MARIBORU / FACULTY OF ENERGY TECHNOLOGY, UNIVERSITY OF MARIBOR Izdajatelj / PUBLISHER Fakulteta za energetiko, UNIVERZA V MARIBORU / FACULTY OF ENERGY TECHNOLOGY, UNIVERSITY OF MARIBOR Glavni in odgovorni urednik / EDITOR-IN-CHIEF Jurij AVSEC Souredniki / CO-EDITORS Bruno CVIKL Miralem HADŽISELIMOVIĆ Gorazd HREN Zdravko PRAUNSEIS Sebastijan SEME Bojan ŠTUMBERGER Janez USENIK Peter VIRTIČ Ivan ŽAGAR Uredniški odbor / EDITORIAL BOARD Zasl. prof. dr. Dali ĐONLAGIĆ, Univerza v Mariboru, Slovenija, predsednik / University of Maribor, Slovenia, President Prof. ddr. Denis ĐONLAGIĆ, Univerza v Mariboru, Slovenija / University of Maribor, Slovenia Doc. dr. Željko HEDERIĆ, Sveučilište Josipa Jurja Strossmayera u Osijeku, Hrvatska / Josip Juraj Strossmayer University Osijek, Croatia Prof. dr. Ivan Aleksander KODELI, Institut Jožef Stefan, Slovenija / Jožef Stefan Institute, Slovenia Prof. dr. Milan MARČIČ, Univerza v Mariboru, Slovenija / University of Maribor, Slovenia Prof. dr. Greg NATERER, University of Ontario, Kanada / University of Ontario, Canada Prof. dr. Enrico NOBILE, Università degli Studi di Trieste, Italia / University of Trieste, Italy Prof. dr. Brane ŠIROK, Univerza v Ljubljani, Slovenija / University of Ljubljana, Slovenia Znan. sod. dr. Luka SNOJ, Institut Jožef Stefan, Slovenija / Jožef Stefan Institute, Slovenia Prof. dr. Mykhailo ZAGIRNYAK, Kremenchuk Mykhailo Ostrohradskyi National University, Ukrajina / Kremenchuk Mykhailo Ostrohradskyi National University, Ukraine, Tehnični urednik / TECHNICAL EDITOR Sonja Novak Tehnična podpora / TECHNICAL SUPPORT Tamara BREČKO BOGOVČIČ Izhajanje revije / PUBLISHING Revija izhaja štirikrat letno v nakladi 150 izvodov. Članki so dostopni na spletni strani revije -www.fe.um.si/si/jet.html / The journal is published four times a year. Articles are available at the journal's home page - www.fe.um.si/en/jet.html. Cena posameznega izvoda revije (brez DDV) / Price per issue (VAT not included in price): 50,00 EUR Informacije o naročninah / Subscription information: http://www.fe.um.si/en/jet/ subscriptions.html Lektoriranje / LANGUAGE EDITING Terry T. JACKSON Oblikovanje in tisk / DESIGN AND PRINT Fotografila, Boštjan Colarič s.p. Naslovna fotografija / COVER PHOTOGRAPH Jurij AVSEC Oblikovanje znaka revije / JOURNAL AND LOGO DESIGN Andrej PREDIN Ustanovni urednik / FOUNDING EDITOR Andrej PREDIN Izdajanje revije JET finančno podpira Javna agencija za raziskovalno dejavnost Republike Slovenije iz sredstev državnega proračuna iz naslova razpisa za sofinanciranje domačih znanstvenih periodičnih publikacij / The Journal of Energy Technology is co-financed by the Slovenian Research Agency. Spoštovani bralci revije Journal of energy technology (JET) Dear Readers of the Journal of Energy Technology (JET) Vodikove tehnologije in metanolove tehnologije predstavljajo enega ključnih dejavnikov pri načrtovanju sodobnih energetskih naprav. Vodik kot energent se uporablja pri nekaterih tipih vojaških podmornic, avtomobilih, raketni tehniki. Načrtujejo se tudi sistemi soproizvodnje toplotne in električne energije za procese ogrevanja, hlajenja ... V ta namen nekatera velika energetska podjetja, kot so Fronius, Panasonic, Vailland in Viessmann, razvijajo sisteme ogrevanja z elektrolizo in gorivnimi celicami. Slednje bi lahko uporabili za pogon računalnikov, ur, transportnih sredstev in sistemov za soproizvodnjo električne ter toplotne energije. Skeptiki opozarjajo na težave, ki jih vodikove tehnologije še vedno imajo, na primer na transport in shranjevanje. Od leta 2015 je v serijski proizvodnji prvi avtomobil Toyota Mirai, ki je na tržišču Severne Amerike postal prava uspešnica; avtomobil je bil razprodan. Na nemško tržišče naj bi avtomobili prišli v tem letu. Najbolj pomemben dejavnik za razmah vodikovih tehnologij je proizvodnja poceni vodika. Trenutna proizvodnja bazira na pretvorbi ogljikovodikov. Elektrolizni postopki omogočajo pridobivanje vodika iz vode in so zelo dragi. V svetu so v sklepni fazi razvoja termokemični postopki pridobivanja vodika s pomočjo toplotne energije in majhnega deleža električne energije. Tudi v Sloveniji imamo polnilnico vodika, in sicer v Lescah, obeta pa se izgradnja v Velenju. Srčno upam na velik razmah vodikovih tehnologij, tudi zaradi globalnih ekoloških problemov. Jurij AVSEC odgovorni urednik revije JET Hydrogen technologies and methanol technologies represent one of the key factors in the design of modern energy installations. As an energy carrier, hydrogen is already used in some types of military submarines, cars, trains, and rockets. Systems of combined heat and power processes of heating and cooling are planned. For this purpose, some large energy companies, including Fronius, Panasonic, and Vailland and Viessmann, have developed heating systems with electrolysis and fuel cells. Fuel cells could be used to power computers, clocks, transport systems, and systems for the cogeneration of electricity and heat. Skeptics point to the problems that hydrogen technologies still have, such as transportation and storage. Starting last year, the first hydrogen car in regular production, the Toyota Mirai, has been on sale, primarily in North America. On the German market, such cars should also be released this year. The most important factor for the growth of hydrogen technologies is the production of cheap hydrogen. Current hydrogen production is based on the conversion of hydrocarbons. The electrolysis process enables the production of hydrogen from water; this is very expensive. Research institutes around the world are in the final stages of the development of thermochemical processes production of hydrogen using heat and a small amount of electricity. In Slovenia, the first hydrogen filling station has opened in Lesce. It is hoped that the next hydrogen filling station will be in Velenje. I sincerely hope the great expansion of hydrogen technologies will continue, especially due to global ecological problems. Jurij AVSEC Editor-in-chief of JET Table of Contents / Kazalo (JET Iv* J Journal Of JE1 Volume 8 (2015) p.p. 11-22 Issue 4, December 2015 Typology of article 1.01 Technology www.fe.um.si/en/jet.html Determination of local brittle zones at the crack front in the HAZ of welded joint / Določitev krhkih področij na fronti razpoke v TVP zvarnega spoja Zdravko Praunseis, Marko Mlakar, Sonja Novak...........................11 The application of nanomechanics in energy technologies / Uporaba nanomehanike v energetskih tehnologijah Jurij Avsec, Urška Novosel........................................23 On the isochronicity of periodic solutions at a centre manifold / O izohronosti periodičnih rešitev na centralni mnogoterosti Brigita Ferčec, Matej Mencinger....................................43 Design of the ventilation and air-conditioning system in an office building / Načrtovanje sistema prezračevanja in klimatizacije v poslovni stavbi Franc Rihl.................................................57 Institut of Energy Technology......................................70 Instructions for authors.........................................73 DETERMINATION OF LOCAL BRITTLE ZONES AT THE CRACK FRONT IN THE HAZ OF WELDED JOINT DOLOČITEV KRHKIH PODROČIJ NA FRONTI RAZPOKE V TVP ZVARNEGA SPOJA Zdravko PraunseisR, Marko Mlakar1, Sonja Novak2 Keywords: steel welds, heat-affected zone, fracture toughness, crack front Abstract Welded joints are the weakest link in the construction of steel structures. The brittlest part of a multi-pass welded joint is the heat-affected zone, where brittle fractures of the structure often appear. In this article, examination of the whole heat-affected zone of high strength multipass X welded joints was performed. All microstructures and the local brittle zones' sizes at the crack tip front were precisely determined. Povzetek Zvarni spoji so pri gradnji energetskih komponent najšibkejši člen jeklene konstrukcije. Najbolj krhko področje večvarkovnega zvarnega spoja je toplotno vplivano področje, kjer se najpogosteje pojavi krhki zlom konstrukcije. R Corresponding author: Zdravko Praunseis, PhD, Faculty of Energy Technology, University of Maribor, Tel.: +386 31 743 753, Fax: +386 7 620 2222, Mailing address: Hočevarjev trg 1, Krško, Slovenia, E-mail address: zdravko. praunseis@um.si 1 University of Maribor, Faculty of Energy Technology, Hočevarjev trg 1, Krško, Slovenia 2 University of Maribor, Faculty of Energy Technology, Department Velenje, Koroška 62a, Velenje, Slovenia V članku je obravnavano celotno toplotno vplivano področje visokotrdnostnega večvarkovnega X zvarnega spoja. Natančno so določene vse krhke mikrostrukture in velikosti lokalno krhki področij ob fronti utrujenostne razpoke. 1 INTRODUCTION The crack tip opening displacement (CTOD) test has become a common method of measuring the fracture toughness of steel welds. Nevertheless, the commonly used fracture mechanics testing standards, including the CTOD testing standard such as BS 5762, [2], assume the use of metals with a high degree of homogeneity, although this is not explicitly stressed. As already mentioned, in reality, welded joints have typical macroscopic heterogeneity and residual stresses as a result of welding. In order to clarify the applicability of the common testing methods, a basis of knowledge taking the above heterogeneity into account must be established in addition to the standards. Recently, some activities have been conducted for establishing the CTOD testing procedure of steel welds, and some recommended practices/guidelines for CTOD tests of welds have been published. It is widely understood that the fracture toughness is considerably affected by the shape of the crack front of the fracture toughness specimen, [6-8]. Therefore, in the common fracture toughness specimen, attention is carefully paid to realize a straight crack front perpendicular to the plate surface. However, in the welded joint, it is sometimes very difficult to obtain a straight crack front of the fatigue pre-crack due to the existence of weld residual stresses. In order to avoid the confusion due to the irregularity of the crack front and to realize the reproducibility, the current standards require that as a straight crack front as possible be achieved, [1 ]. 2 SECTIONING PROCEDURE TECHNIQUE In order to achieve a uniform fatigue crack shape which meets the standard requirements, some treatments, i.e. residual stress relieving treatment, have to be applied to notched specimens of welded joints. A different method for relieving residual stresses is to impose a local plastic strain to the region suffering from residual stresses; the following techniques, [1, 6-8], are currently in use, - Local compression - Reverse bending - The use of a high R-ratio in the cycle and step-wise high R-ratio method - Both side holes method. Table 1 gives the summary of the relative merits of the three methods. In the Recommended Procedure proposed by The Welding Institute [8], the mechanical relieving residual stresses by local compression, where a plastic strain of 1% of the specimen thickness, is recommended. Moreover, "the use of reverse bending prior to fatigue precracking as a means of redistributing welding stresses is not recommended", [8]. Moreover, "the effect of a high R- ratio on the fracture toughness is not well understood and so until more work has been completed on this technique its use is not generally recommended." However, for very thick section welds, "the use of high R-ratios during fatigue precracking has been found to be successful in obtaining acceptable crack front profiles", [8]. Table 1: CetcthCecisCihs nf mtCecitls used it sCcess celievieg Advantages Disadvantages LOCAL COMPRESSION - method well published - method in use since 1975 - uses normal fatigue precracking procedures - requires extra operation - requires high capacity compression rig and tools - toughness may be conservative for some materials - specimen must be flat REVERS BENDING - special equipment not needed - conservative toughness measurements expected - uses conventional fatigue pre-cracking procedures - requires extra operation - toughness may be significantly lower - little information published USE OF HIGH R - RATIO - no extra operation needed - no extra equipment needed - required loads and R-ratios in conflict with limits of current standards - little information published - non-conservative assessments of toughness are expected In the common fracture toughness test, the use of a notch sharpened by a precrack produced by fatigue loading of the test piece is generally required in order to simulate sharp macroscopic defects in the structure and to provide a conservative assessment of toughness. In order to avoid the confusion and to realize the reproducibility, the condition of the fatigue precracking loading must be kept within limits. After the CTOD test is conducted, both halves (or the half containing the weld metal) of the broken specimen are sectioned and metallurgically examined. The cut into the fracture face is taken just behind but within 2.5 mm, of the fatigue-crack front. The cross section may contain a portion of the fracture surface near one or both surfaces due to fatigue-crack front curvature. Each such portion is not wider than 10% of the specimen thickness. For CTOD specimens that are notched to sample the coarse grain (CG) regions, quantification is as shown in Fig. 1, where the linear fraction of the CGHAZ region sampled by the fatigue crack is calculated. A similar procedure is used for the intercritical coarse grain (IC) and subcritical coarse grain (SC) HAZ areas. Fatigue-crack sampling calculations are made by examining enlarged photographs (3 to 6 times magnification) of the CTOD cross sections. Figure 1: Sectioning both halves of an HAZ CTOD specimen to calculate CGHAZ percentage Figure 2: Example of sections taken from an HAZ, through-thickness notched CTOD specimen to identify microstructures sampled by fatigue crack and at the fracture initiation point In the case of through-thickness notched specimens, this is best achieved by sectioning at a small distance behind the fatigue crack tip, so as to include as much of the fatigue crack front as possible (Fig. 2). With surface notched specimens, a similar approach could be used. However, when the region being sampled is small and/or the fatigue crack front is bowed, misleading results may be obtained. For this situation, a better approach is to section as shown in Fig. 3(b), and if necessary, take a series of sections. By using both halves of the broken CTOD specimen and enlarged photographs, fatigue-crack sampling calculations can be made with reasonable accuracy without microscopic examination. Each HAZ specimen should be sectioned to determine the regions of microstructure sampled by the fatigue crack. Figure 3: Example of sectioning techniques for a) through-thickness notched, and -b) surface notched specimens It is recommended that similar sectioning procedures be applied to all tests (HAZs and weld metals) carried out to measure the fracture toughness associated with known cracks, [1]. By agreement, there is the additional requirement to establish the microstructure at the fracture initiation point; detailed fractography is necessary to determine the microstructure at that point and hence locate the position from which the section has to be taken. 3 EXPERIMENTAL PROCEDURE Three different types of welded joints were tested for a comparison of mutual mechanical properties. The mechanical properties of the welds made on HSLA steel grade HT50 and HT80 and mild steel were evaluated using standard tensile, [6], Charpy, [7], and CTOD (Crack Tip Opening Displacement) tests, [8]. The HAZ specimens were taken from the welded steel plates in the rolling, thickness, and width directions. The differences in microstructures among material regions influence the mechanical properties, [6-8]. Thus, systematic experimental determination of material mechanical properties including fractographical and metallographic investigation of fracture surfaces is necessary. The chemical composition of the HSLA steel grade HT50 and HT80 and mild steel, with plate thicknesses of 40 mm are given in Table 2. All testing HAZ specimens were taken from the steel plates in the rolling direction (A), the thickness direction (B), and the width direction (C). The yield strength and tensile strength were obtained using round bar tensile specimens, as shown in Fig. 4. Tensile testing was done at room temperature (20 °C). Table 2: COemiht l hnmpnsirina nf COe HSLA steel gttfe HT50 taf HT80 taf mild steel Compositi on (%) HT50 HT80 Mild steel C 0.12 0.16 0.28 Si 0.55 0.68 0.52 Mn 0.67 0.75 0.71 P 0.015 0.020 0.011 S 0.002 0.003 0.007 Cr 0.70 0.79 0.05 Ni 0.07 0.09 0.01 Mo 0.042 0.032 0.013 Cu 0.19 0.24 0.62 Al 0.001 0.002 0.001 Figure 4: Teesile tnuef btt specimen B 5 x 25 Charpy - V testing was used to determine the impact toughness of steel plates. The shape and dimensions of a standard Charpy - V specimen, mechanically notched, are shown in Fig. 5. Testing was performed at -10 °C. For every test temperature, three specimens were fractured. 4 DISCUSSION OF RESULTS Mechanical properties of welds made on HSLA steel HT80 and HT50 and mild steel plate in the rolling direction (A), thickness direction (B) and the width direction (C) are presented in Table 3. The basic values of yield strain and tensile strain of testing steels, given in Table 3, were obtained from engineering stress (R) - strain (e) diagrams. It is known that engineering material curves cannot be used for analysis of material deformation characteristics and finite element calculations in the range of high plastic deformations. Average Charpy-V testing values of three fractured specimens are represented in Table 3. Table 3: MehOteihtl pcnpecCies nf welfs mtfe ne HSLA sCeel HT80 taf HT50 taf milZ sCeel pltCe it Che cnllieg ZicehCine (П), COihkeess ZicehCine (B) taf HifCO ZicehCine (C) Figure 5: Shape and dimensions of Charpy - V notch specimen CTOD fracture toughness of the welds made on HSLA steel grade HT50 and HT80 and mild steel was evaluated using standard static CTOD test, [4-5]. Specimen loading was carried out with constant crosshead speed v = 0.5 mm/min. The test temperature was -10 °C according to the recommendation of the OMAE (Offshore Mechanics and Artic Engineering) association. For CTOD testing, the single specimen method was used, [8]. To evaluate the fracture toughness of steels, standard fracture mechanics tensile specimens with shallow notches were used, as shown in Fig. 6. t >,— Figure 6: Direct measurement of CTOD values at the crack tip of fracture mechanics specimen For all specimens, the fatigue precracking was carried out with the Step-Wise High R ratio (SHR) method procedure, [4]. During the CTOD tests the potential drop technique was used for monitoring stable crack growth, [5]. The CTOD values were directly measured with a special clip gauge, [5], on the specimen side surfaces at the fatigue crack tip over a gauge length of 5 mm (see Fig. 6). Weld-Steel grade Measured direction Yield strain (MPa) Tensile strain (MPa) CTOD (mm) Charpy V (J) at -10 °C HT50 A 542 591 0.390 47, 68, 71 Av=62 HT80 A 693 830 0.401 69, 78, 64 Av=70 Mild steel A 452 497 0.423 42, 55, 62 Av=53 HT50 B 501 562 0.240 39, 41, 55 Av=45 HT80 B 657 799 0.253 53, 68, 66 Av=62 Mild steel B 439 471 0.231 39, 44, 61 Av=48 HT50 C 531 587 0.416 42, 76, 69 Av=62 HT80 C 665 811 0.478 67, 71, 63 Av=67 Mild steel C 447 478 0.443 40, 51, 65 Av=52 The lowest Charpy toughness was measured in the HAZ specimens taken from the steel plates in the thickness direction (B). The cause for low toughness was the appearance of inconvenient ferritic microstructures with distributed brittle martensite-austenite (M-A) constituents (Fig. 7). Figure 7: Ferritic microstructure with distributed brittle M-A constituents along ferrite grain boundaries of HAZ specimen The Charpy toughness of HAZ specimens taken from the steel plates in the rolling direction (A) is approximately equal to the Charpy toughness of HAZ specimens taken from the steel plates in the width direction (C). Directly measured CTOD values of HAZ fracture toughness for each type of steels are summarized in Table 3. The maximal CTOD toughness was measured in the HAZ specimens with the crack tip located in the width direction (C). In the case of CTOD testing of HAZ specimens with the crack tip located in the thickness direction (B), the lowest CTOD was measured due to the appearance of the first brittle fracture in the mainly ferritic microstructure with carbides (Fe3C), precipitated at the grain boundary (Fig. 8) and the appearance of brittle fracture initiation point, i.e. Al-Si-Mn inclusions (Fig. 9). For the correct identification of a brittle fracture initiation point, it is of utmost importance to apply Energy Disperse X-ray (EDX) analysis to both fracture surfaces. In the opposite case, it could happen that the EDX analysis detects some fictitious brittle fracture initiation point. Figure 8: Mtialy fecriCic mihcnsCtuhCute hCO httbifes (Fe3C) ptehipiCtCef ft COe HAZ gttia bnuaZfty Figure 9: Appetctthe taf EDX tat lysis nf btiCCle fracture ieCifCine pnieC, i.e. Пl-Si-Me inclusine it COe HAZ specimen 5 CONCLUSION Exact evaluation of real material mechanical properties is essential for the safe servicing of energy components. The presence of different HAZ microstructures along pre-crack fatigue fronts has significant effects on the critical crack tip opening displacement (CTOD). This value is the relevant parameter for the safe servicing of modern energy components. The mechanical properties of welds made on of HSLA steels grade HT50 and HT80 and mild steel are the lowest in the thickness direction of the steel plate due to the appearance of carbides (Fe3C) and Al-Si-Mn inclusions in the ferritic microstructure. The mechanical properties of welds made on HSLA steels grade HT50 and HT80 and mild steel are approximately equal in the rolling and in the width directions of the steel plates. JET Journal of Energy Technology JET Volume 8 (2015) p.p. 23-41 Issue 4, December 2015 Typology of article 1.01 www.fe.um.si/en/jet.html THE APPLICATION OF NANOMECHANICS IN ENERGY TECHNOLOGIES References [1] Mlakar, M.: Determination of local brittle zones et the creck front ik the HAZ of welded joint. Master Thesis, Faculty of Energy Technology, University of Maribor, Slovenia, 2015 [2] BS 5762, Methods for creck opening displacement (COD) testing, The British Standards Institution, London 1979 [3] ASTM E 1152-87, Standard test method for determining J-R corves, Annual Book of ASTM Standards, Vol. 03.01, American Society for Testing and Materials, Philadelphia, 1990 [4] ASTM E 1290-91, Standard test method for crack-tip opening displacement (CTOD) fracture toughness measurement, American Society for Testing and Materials, Philadelphia, 1991 [5] GKSS Forschungszentrum Geesthacht GMBH, GKSS-Displacement Gauge Systems for Applications in Fracture Mechanic [6] Praunseis, Z., Toyoda, M., Sundararajan, T.: Fracture behaviours of fracture toughness testing specimens with metallurgical heterogeneity along crack front. Steel res., 71, Vol. 9, Sep. 2000 [7] Praunseis, Z., Sundararajan, T., Toyoda, M., Ohata, M.: The influence of soft root on fracture behaviors of high-strength, low-alloyed (HSLA) steel weldments. Mater. manuf. process., Vol. 16, 2001 [8] Toyoda, M.: Fracture toughness evaluation of steel welds, Book of Mechanics, Osaka University, 1989 Acknowledgements The authors wish to acknowledge the financial support of the Slovenian Foundation of Science and Technology and the Japanese Promotion of Science. UPORABA NANOMEHANIKE V ENERGESKIH TEHNOLOGIJAH Jurij Avsec?, Urška Novosel1 Keywords: nanomechanics, nanotechnology, nanomaterials, nanoenergetics Abstract Nanotechnologies may be very efficient for improving and optimizing energy technologies. The following will be shown in this paper: nanomaterials are very important in energy technology. Nano-fluids and nanostructured materials may be applied in refrigerators, heaters, and similar equipment. In the micro- and nano-regions, the classic laws of mechanics are insufficiently accurate for the calculation of characteristics and properties. In this article, the application of microfluidics and vibration theory in the micro and nano-regions will be shown. Povzetek Nanotehnologije so lahko zelo učinkovite za izboljšanje in optimizacijo energetskih procesov. Razvoj nanomaterialov je izjemnega pomena za energetiko. Nanotekočine in nano nanostrukturirani materiali se lahko uporabljajo v hladilnih strojih, grelnikih.... V mikro in nano-področju zakoni klasične termomehanike v mnogih primerih niso dovolj natančni. V ta namen je tudi prikazana problematika v mikrofluidni tehniki in mehanskih nihanjih. R Corresponding author: Prof. Jurij Avsec, University of Maribor, Faculty of Energy Technology, Laboratory for Ther-momechanics, Applied Thermal Energy Technologies and Nanotechnologies , Tel.: +386-7-6202217, Fax: +386-2620-2222, Mailing address: Hočevarjev trg 1, 8270 Krško, Slovenia E-mail address: jurij.avsec@um.si 1 University of Maribor, Faculty of Energy Technology, Laboratory for Thermomechanics, Applied Thermal Energy Technologies and Nanotechnologies, Hočevarjev trg 1, SI-8270 Krško, Slovenia 1 INTRODUCTION Along with the development of increasingly larger devices, many inventors and scientists have attempted to penetrate the secrets of the nanoworld. For centuries, watchmakers were miniaturizing mechanical devices. The discovery of the microscope in the 17th century enabled people to observe microbial, plant, and animal cells. Only in the late 20th century, however, were the technologies of microdevices developed. The size of transistors in integrated circuits is currently 0.18 micrometres, and laboratories are already developing transistors of 10 nanometres. One of the great scientific and technical advancements at the turn of the 21st century is the creation of nanomaterials and nanotechnology. The area that covers all significant problems from that field is called "mechanics" in the broadest sense, which is divided into: • macromechanics 10"4-10"5 m, • mesomechanics 10-5-10-7 m, • micromechanics 10-7-10-8 m, • nanomechanics: 10-8-10-9 m. Since the atomic level (interatomic distance in a crystal lattice) has an order of one to several A (10-10 m), the nanolevel is restricted to 10-9 m. Nanotechnology currently allows the successful manufacture of micro-electro-mechanical systems (MEMS), nano-electro-mechanical systems (NEMS) and biological micro-electromechanical systems (BIO MEMS). Significant among these are microchips, micro sensors, micro mirrors biochips, and nanomaterials. Micro-electro-mechanical systems are devices with a characteristic length ranging between 1 mm and 1 micrometre and consisting of electric and mechanical components. Microdevices, for example, are smaller than the diameter of a human hair. Nanodevices (NEMS) are devices of even smaller dimensions, containing nanocomponents or devices based on special materials called nanomaterials. Nanomechanics and micromechanics are becoming even more important in modern industry. Ideas about extremely small aircraft, pumps and other technical devices have become reality today, [1-9]. At the same time, unforeseen problems have emerged. If a fluid flows through a pipe with the diameter of a few nanometres or micrometres, thermodynamic and transport properties of the fluid are modelled completely differently due to the high impact of the surface effects. Similarly, the equations of conventional hydromechanics do not apply in the entire area. In addition to temperature and pressure, the significance of the Knudsen number (Kn) is also growing. This number is the ratio of the mean free path length of the molecules to a characteristic length L, [2-5]. The Euler equation for the calculation of flow gives poor results in almost the entire range, from the Navier-Stokes equation at a Knudsen number of 0.1 to the Burnett equation at a Knudsen number of 10. To analyse the molecular free path in micro- and nano-channels, non-equilibrium mechanics and the original Boltzmann equation should be used to be able to calculate hydro-mechanical problems in the entire range of Knudsen numbers. Nanofluids of exceptional properties are increasingly economically important. In addition to nanofluids, the application of nanopipes with even better properties is also increasing in engineering practice. Nanotechnology will expand into many areas in the future. Digital data storage with much higher recording densities will be possible as well as analyses of individual cells in cases of serious illnesses as well as the manufacture of ultra-light materials with exceptional properties. The size of nanoparticles and nanowires is nearly the same as the size of biomolecules, such as DNA molecules and proteins. In the future, it will be possible to use nanoparticles in various areas of medicines. Consideration is being given to applying nanoparticles as a suitable probe for small DNA samples or proteins. 2 NANOMATERIALS Nanomaterials are classified in the next divisions, [1-3]: 1) Carbon-based nanomaterials 2) Metallic nano-materials 2) Nanocomposites 3) Metals & alloys 4) Biological nanomaterials 5) Nano-polymers 6) Nano-glasses 7) Nano-ceramics 8) Natural nanoparticles The term 'nanofluid' is envisioned as describing a solid-liquid mixture that consists of nanoparticles and a base liquid; this is one of the new challenges of nano-technology for thermo-sciences. The possible application area of nanofluids is in advanced cooling systems, and in micro/nano electromechanical systems. The investigation of the effective thermal conductivity of liquid with nanoparticles attract much more interest experimentally and theoretically. The effective thermal conductivity of nanoparticle suspension can be much higher than for the fluid without nanoparticles. The calculation of properties for nanofluids for real substances is possible with the use of classical and statistical mechanics. Classical mechanics has no insight into the microstructure of the substance. Statistical mechanics, in contrast, calculates the properties of a state on the basis of the molecular motions in a space, and on the basis of the intermolecular interactions. The equations obtained by means of classical thermodynamics are empirical and apply only in the region under observation. The main drawback of classical thermodynamics is that it lacks the insight into the substance of microstructures. In contrast to classical mechanics, statistical mechanics calculates the thermomechanical properties of state on the basis of intermolecular and intramolecular interactions between particles in the same system of molecules. It deals with the systems composed of a vast number of particles. Figure 1 and Figure 2 shows the thermal conductivity and viscosity for nanofluids obtained by statistical thermomechanics. Volume fraaction of nano parti с les I Figure 1: Thrsmal conductivity rf mixtusr brtwrrn rthylrnr glycol andJl2O3 nanrpasticlrs Figure 2: Thrsmal conductivity rf mixtusr brtwrrn rthylrnr glycol and Jl2O3 nanrpasticlrs 3 MICROFLUIDICS Fluid flow in channels, minichannels, and microchannels is driven due to the presence of electric fields, magnetic fields or pressure-driven flows and some other effects, [1]. Electrohydrodynamics (EHD), known as electrokinetics, is the theory of the flow of electrically charged fluids. It is the study of the motions of ionised particles or molecules and their interactions with electric fields and the surrounding fluid. For EHD flow, low electrical conducting fluids, such as organic fluids or alcohols, are required, [2-4]. The electrokinetic flow can be classified into the following types: electrophoresis, electroosmosis, streaming potential and sedimental potential, [2-5]. The fundamental concept for magnetohydrodynamics (MHD) is that magnetic fields can induce currents in a moving conductive fluid, which in turn creates forces on the fluid and also changes the magnetic field itself, [2]. For MHD flows, highly conductive fluids, such as plasmas, liquid metals, electrolytes and salt water, are needed. f the flow is transported by pressure differential, the phenomenon is called pressure hydrodynamics (PHD). Ferrohydrodynamics (FHD) is the theory of magnetic fluid flow. The term 'nanofluid' is envisioned as describing a solid-liquid mixture that consists of nanoparticles and a base liquid; this is one of the new challenges for thermo-sciences provided by nano-technology. The possible application area of nanofluids is in advanced cooling systems, and in micro/nano-electro-mechanical systems. The investigation of the effective thermal conductivity of liquid with nanoparticles has attracted much more interest experimentally and theoretically. The effective thermal conductivity of nanoparticle suspensions can be much higher than for the fluid without nanoparticles. The calculation of properties for nanofluids for real substances is done using classical and statistical mechanics. Classical mechanics has no insight into the microstructure of the substance. Statistical mechanics, in contrast, calculates the properties of the state on the basis of molecular motions in a space, and on the basis of the intermolecular interactions. The equations obtained by means of classical thermodynamics are empirical and apply only in the region under observation. The main drawback of classical thermodynamics is that it lacks insight into the substance of the microstructure. In contrast to classical mechanics, statistical mechanics calculates the thermomechanical properties of state on the basis of intermolecular and intramolecular interactions between particles in the same system of molecules. It deals with systems composed of a very large number of particles. Q-heat a) Electrokinetic flow Q-heat Q-heat Q-heat b) Pressure-driven flow Magnetic field from core c) Ferrofluid flow Chanel Heat from windings Insulator d) MHD flow Electrode Consider the MHD flow in rectangular and circular microchannels (Figure 3). The charged surface of a microchannel wall may attract ions of the opposite charge in the surrounding fluid. The general form of the momentum equation for MHD flow is: cV p— + pvVv = -Vp + V(/Nv )+i X B, Ct (3.1) where the last term represents the electromagnetic force, and i and B refer to the current density and magnetic field strength, respectively. For steady-state flow in a microchannel at small Reynolds numbers, the transient and inertia terms can be neglected, so Eq. (3.1) is simplified in the next equation: 0 = -Vp + VjVv)+ i X B, (3.2) Figure 4: Rectangular and circular microchannels Assuming the fluid velocity, magnetic field and current density are orthogonal, the reduced momentum equation becomes: dp d 2u 0 = - dX + + ^ (3.3) The terms represent pressure, viscous and electromagnetic forces in the liquid. Using Ohm's Law to express the current density in terms of fluid velocity: d 2u „ „ 2 dp + CeBzU = —, dz dx (3.4) where Ge and Bz refer to the electrical conductivity and magnetic field strength. For fully developed flow in a microchannel, the pressure gradient becomes constant and independent of the magnetic field strength. In terms of the Hartman number, Mh ( MH = aBce / j ), d 2u 2 Mh Л dp dx (3.5) Figure 3: Types of fluid flow P P 1 2 u = 2 a Applying the no-slip boundary conditions at z=0 and z=-2a, the analytical solution of Eq. (3.5) becomes: + *2(i) zMH -e a + - *2d>2MH MH> ■ (l+2e2MH)MH2^ (l+2e2MH)MH2^ The mean velocity within the microchannel becomes: —Z'mH e a (3.6) 1 2a — I u(z)dz 2a i Uo = a2(^l)("M"+Tanh(M")) Non-dimensionalizing this result (z*=z/a, u*=u/ub), we obtain the next equation: e2MH MHI -i+ ez MH (l+2e 2mh) ( i+2e 2mh) ; — Z MH (3.7) (-Мн+ГппЛ(Мн)) Without electromagnetic effects, Equations (3.5-3.7) transform into the following expressions: 2 ( dp Л a a I — I 1 f r V dx ) * 3 2- J U(z)dz u(z) = dp I dx 2az + z2 ),« , u = 2 (2 - z* У 2^ v r " 2a J -a For the circular microchannel without electromagnetic forces, the governing equation is dp = (d2u + l đu) dx _ ^(đr2 r đr) Solving the differential equation subject to boundary conditions, (3.8) (dp\ — —)R2 u = W(r2 -Д2), u0 = -l2 fRu2n-r -dr = -i^— u = 2 - 2r 4u 0 n:R2J0 8u 4д 8M (3.9) (3.10) If we wish to calculate the velocity profile for MHD flow in a circular channel, we have to solve the following differential equation: dp /a2u l đu\ MH2 — = —2 +---) ^u dx ^ Vđr2 r dr/ ^ R2 (3.11) The analytical solution of equation (3.11) is slightly more complicated. We have obtained the next solution of the differential equation with the boundary conditions (u(R)=0, u'(0)=0): R2(^)(-BesselI[0,M]+BesselI[0^|-]) M2^BesselI[0,M] u[r] = ■ l ,R R2(fg)BesseH[2,M] ub = TR2Jo u2^r - dr = - M2^BesselI[0,M] (3.12) (3.13) Previous investigations of the pressure gradient for electro-osmotic liquid flow in microchannels have generally used no-slip conditions, while for gas flow in microchannels, the slip boundary condition is taken into account. Some previous experimental investigations have demonstrated the existence of liquid slip on a microchannel wall, [2-5]. In addition, some previous studies were numerically performed considering a slip velocity for a liquid flow in a microchannel made from hydrophobic surfaces, taking into account the electric field and pressure gradient, [3-5]. Furthermore, some previous studies numerically predicted a slip velocity for liquid flow in a microchannel made from hydrophobic surfaces, taking into account an imposed electric field and a pressure gradient without heat transfer. On the basis of a slip velocity profile and boundary conditions in liquid microchannel flow, the velocity profile can be determined by u(0) =usl,u(2a) = us2 пди\ ndu\ usl=ß~~\ .us2=ß~r\ az\z=0 -zz=2a (3.14) (3.15) Figures 5-9 show the analytical results for the velocity profile in fluids in dependance of slip conditions, type of microchannel and type of fluid flow. 1.2 ' ■ - S 1.0 _f — — 0.8 sl ° 0.6 // w 7 v \ 0.4 \ \ 0.2 1 1 V 0.0 I 0.0 0.5 1.0 1.5 2.0 z a Figure 5: Velocity profila (n) without clip (rad lica) nod with clip в*=0.5 (blua lica) nt MH=5 ic rantncnulnr micinOnccal и i = b 1.05 1.00 0.95 ^ 0.90 0.85 0.80 0.75 U 1.0 z Figure 6: Velocity psrfilr with slip (в*=0.5) at MH=5 (blur linr), MH=10 (srd linr) and MH=30 (gsrrn linr) in srctangulasminichannrl with slip в*=0.5 s I Ц 1.0 0.0 r R Figure 7: Velocity psrfilr in a cisculas micsrchannrl without rlrctsrmagnrtij rffrcts (blur linr), with a slip with slip в*=0.35 (srd linr) and with slip в*=0.7 (gsrrn linr) i i i i i i i i ^^^ . i i i i i i 1.4 - --- - / 4 1.2 / \ /f 1.0 /7/ ft t YvV 0.6 /// \\\ 1/ \v 0.4 II il 0.2 У \ r T 0.0 f I ..... □ 1.0 CD.5 0.0 0.5 1.0 r R Figure 8: Velocity profila with clip (в*=0.5) nt MH=5 (blua lica), MH=10 (rad lica) nod MH=30 (grano lica) io nirnulnr minrtnOnccal without clip 1.10 1.05 1.00 0.95 0.90 0.85 0.80 □1.0 0.0 r R Figure 9: Valtnity profila with clip в*=0.5 nt MH=5 (blua lica), MH=10 (rad lica) ncd MH=30 (nraac lica) ic nirnulnr microcOrncal a 4 THE VIBRATIONS OF MICROBEAMS AND NANOBEAMS The vibrations of beams, microbeams and nanobeams are of vital importance in mechanical engineering. Mechanical machines very frequently operate under diverse temperature conditions. In internal combustion engines, rocket systems, satellites, as well as MEMS and NEMS, conditions are particularly temperature-sensitive. Thermodynamic effects are frequently ignored in research, which may yield completely incorrect results. Literature, [3], shows that even the slightest temperature change leads to significant alteration of the clamped-beam vibration properties. In the present paper the impact of a change in thermodynamic properties, which have to be taken into consideration with major temperature changes, is not neglected. Carbon nanotubes in dependence of the chiral angle can be classified into three types: armchair, zigzag, and chiral. Numerous studies are available on the physical properties of armchair and zigzag carbon nanotubes [14-15]. However, only a limited portion of the literature studied nanotubes in dependence of temperature field. This article develops a model that analyses the frequency of the chiral single-walled carbon nanotubes (SWCNTs) subjected to thermal vibrations by using the Timoshenko beam model, including the effect of rotary inertia and shear deformation. The Timoshenko model has been compared with the Euler model. Carbon nanotubes can be classified into single wall nanotubes (SWNT) and multi-wall nanotubes (MWNT). On the basis of molecular simulation, many researchers have found that the modulus of elasticity is no longer constant, and is dependent on the diameter of nanotube and thickness of nanotube [6-16]. On the basis of molecular dynamics calculation, equations for surface Young modulus and Poisson number for armchair SWNT can be expressed: 4/K„ л/з(А + 3 ц) (4.1) where: 7 - cos(W n) Х + £/ц 34 + 2cos(W n) , ц = iL Kpr0 (4.2) The above equations are obtained on the basis of continuous mechanics and molecular simulation, [6-8], where Ys means surface Young modulus, and v the Poisson number. From Fig. 3.10, we can see that material properties are both temperature and size dependent. Figure 10: Young modulus of armchair nanotubes The Young modulus of carbon nanotubes and the linear expansion coefficient are also dependent on the temperature field. On the basis of Prakash, [16], a molecular dynamics simulation is obtained and the next relation for modulus of elasticity and linear expansion coefficient a for SWNT: Ys = Ys (1 - 0.000075T ) (4.3) = -(-)-I dT 10-18T3 10-18T 2 - 2*10-15 * T +10-13 -10-15T 2 + 10-13T + 3*10-8 (4.4) 4.1 Local Euler-Bernoulli beam model under thermal stresses Let us assume that the support is homogenous, having the same temperature over its entire length. As a result of thermal expansion, an additional axial force FT occurs: Y s a v FT = adEA (4.5) In equation (3.7), a is the linear thermal extension coefficient, and 6 is the temperature difference between the actual and initial or reference temperature. The equation, by means of which we can resolve the problem using the axial force, is as follows according to Wear, Timoshenko and Young, [4]: EI д4w{x, t) + F д 2w(x, t) + p д 2w(x, t) dx 4 dx 2 + pA- dt2 ■ = 0, (4.6) where E means Young modulus, I area moment of inertia, A area, p density of material, t time and w the displacement. Using the method of separation of variables w (x, t ) = x (x)q(( ) and introducing the new functions, Equation (4.6) can be written down in a slightly less complicated way: 2 X""(x) „ X"(x) Q(() 2 с2-tV + 2y—H---^ = ю2 , X (x) X (x) Q (4.7) where the partial derivatives have been replaced with total derivatives. Q (( ) + ®2Q(t ) = 0 (4.8) X " " ( x) + 2yX " ( x) - ß4 X (x) = 0 (4.9) In Equation (4.7), the new symbols represent the following functional relations: ß2 = Ю,с2 = EL, r = ^ с pA 2EI (4.10) Thus, a general solution to Equations (4.8) and (4.9) is (Ä^ß* +r2 ): X(x) = Ci cos^Ä + rx)+ C2 cosh(jÄ—yx)+ C3 sin(^Ä + rx)+ C4 sinh(jÄ—yx) (4.11) Q(t ) = A sin(®t ) + B cos(mt ) (4.12) With the known angular frequencies ron of individual modes of vibration, it is possible to calculate Xn and Qn of individual modes of vibration. To determine the solution for the displacement, we have to solve the equation, [6-15]: Ax t )=Y,(An sin (Pnt )+Bn cos(o)nt ))xn (x), (4.13) where the modal shapes can be shown to be orthogonal: J Xn (x)Xm (x)dx = 0 for n Ф m (4.14) The model presented in our paper is fully analytical, but when compared with the measured results it points to a large deviation from reality, [6-15]. The biggest problem with this model is that the clamped wall can fully withstand the beam for the beam to have a constant length all the time in the mathematical model in question. The above assumption is not realistic. As a result, a new model was designed to reduce to at least to some extent the huge differences between the analytical results and the measured values. 4.2 Local Timoshenko beam model under thermal stresses The Timoshenko beam model, [6-10], includes the effect of rotary inertia and shear deformation. The Timoshenko vibrational beam model gives the next expression: EI д 4Y Ft д 4Y д - + —--- + I P д Y pA dx4 ' pA dx4 ' dt2 A ^ ' KG Jdx2dt2 ' A KG dt4 -1 fl + ■ E д 4Y „2^2 (4.15) K = 2(1 + M) 4 + 3ц (4.16) In Eq. (4.15), I is the dynamic moment of inertia of the beam, nanotube, K is the shear coefficient of the nanotube, and ц is the Poisson's ratio. Ft presents additional thermal force: 1 In the equation (4.11), the value of Ä (where the influence of angular frequency ю is hidden) and three of four constants of integration C1, C2, C3, and C4 are determined from the boundary conditions. The fourth constant is possible to find in the combination with the constants A and B in equation (4.11). For a given beam at a defined temperature, the values by Ä depend upon the boundary conditions, [5-9]. Using boundary conditions, the following solutions can be analytically computed (Г = L2y,Л = L2Ä ): Ft = aTEA (4.17) The solution of Eq. (4.15) could be expressed as: Y (x,t) = y(x)e~iwt (4.18) In Eq. (4.18), ш is the angular frequency. In the above approximations, the following dimensionless forms can be expressed as: + ( + ß)r 2 + 5]^ _(l-r 2aaß)r\ ас dC (4.19) y „ x n =—;C_ —,a--L L AL2 , ß - El KGAL2 Г : pAm2 L4 §_ FTL2 El El (4.20) (4.21) Where 5 represents the effect of thermal vibration of the frequency of SWCNT. The general solution of Eq. (4.19) could be expressed as r/(^) _ Ci cos(fX—y£j+ C2 cosh(^/X + ye)+ C3 sin(jX — ye)+ C4 sinh(jX + yej (4.22) When the solution integrates with the boundary condition for the support-simply support nanotube model we obtain the solution: 2<«+ß ) r4 - 1 + (пЖГ" r2 + (- „2^2 + 5_ 0 aß aß ' (4.23) For the case of Euler-Bernouli beam (a = ß = 0), we obtain the following equation: Г _ пж^-0+ (пж)2 (4.24) The presented mathematical model was used to calculate the thermodynamic properties of the state of a pure aluminium microbeam. Table 1 contains the main important data of the beam. The aluminium beam is particularly interesting due to its relatively high expansion coefficients. In the presented section, we have calculated the vibrational characteristics for the supported-simply supported systems. For carbon nanotubes, we have used data for Young modulus and linear expansion coefficient shown in the Prakash Thesis, [16]. Figures 11 and 12 show angular frequency for nanotubes for the first, second and third orders. From both figures, we see that when we have relatively long nanotubes (L/D>10), the results for Timoshenko and Euler-Bernouli model give similar results; in contrast, when we have short nanotubes the Timoshenko model gives much better results for first and higher orders. Table 1: Fucdnmactnl ntcctnctc for aluminium bsnm Beam Length (m) Width (m) Thickness (m) Young modulus (N/m2) Volume expansion coefficient (1/K) Spring constant (N/m) Density (kg/m3) 6.35-10"2 2.0440"2 1.62-10"3 6.9-1010 24-10"6 K"1 1.55340s 2780 £ 20 300 320 340 360 380 400 420 440 460 480 500 Temperature [K] Figure 11: Fucdnmactnl fraquacny for cncttuba with L/D=40 with TimtcOackt ncd Eular- Barctuli modal l 40 35 30 25 15 10 5 0 [8] [9] [10] [11] [12] [13] Figure 12: Fundamental frequency frs nanrtubr with L/D=4 with TimrshrnCr and Eulrs- [14] Brsnruli mrdrl References [16] [1] N. Savage: Super carbon, Nature, vol. 483, 2012 [2] Singhal, V., Garrimela, S., Raman, A.: Micsrscalr pumping technologies frs micsrchannrl cooling systems, Appl. Mech. Rev., Vol. 57, No.3, p.p. 191-222, p.p. 30-31 [3] Avsec, J., Oblak, M.: The calculation rf thesmal conductivity, viscosity and thesmrdynamic psopesties fos nanofluids on the basis of statistical nanomechanics, Int. j. heat mass tsansfes. [Print ed.], Vol. 50, Iss. 21/22, p.p. 4331-4341, Oct. 2007 [4] Naterer, G.F., Adeyinka, O.B.: Micsofluidic Exesgy Loss in a Non-Polarized Thesmomagnetic Field, International Journal of Heat and Mass tsansfes, Vol. 48, p.p. 3945-3956, 2005 [5] Avsec, J.: MHD, FHD and PHD fluid flow in micsochannels in dependence of tempesatuse and rlrjtsrmagnrtij field, V: RUBINACCI, G. (ur.). 15th International Symposium on Applied Electromagnetics and Mechanics, 7-9 September 2011, Napoli, Italy. Applied electromagnetics and mechanics : proceedings of the 15th International Symposium on Applied Electromagnetics and Mechanics, Napoli, Italy, 7-9 September 2011 [6] Avsec, J.: The vibsations of micsobeams andnanotubes, J. Vibroeng (Vilnius), Vol. 13, Iss. 4, p.p. 638-645, Dec. 2011 [7] Pastoeiza-Gallego, M.J. Lugo, L., Legido, J.L., Pineiro, M.M.: Thesmal conductivity and viscosity measusements of ethylene-glycol based Jl2O3 nanofluids, Nanoparticles research Letters, Vol. 221, No.6, p.p.1-11, 2011 G 3500 3000 2500 2000 1500 $ 1000 500 0 300 350 400 450 Temperature [K] Timoshenko-1st Euler-1st Timoshenko-2nd Euler-2nd Timoshenko-3rd Euler-3rd 500 Avsec, J.: TOa combined ncnlycic tf pOtato nod alantrtc Oant transfer msnOncicm to tOnrmnl ntcduntivity for cncttluidc, Int. j. heat mass transfer. [Print ed.], Vol. 51, Iss. 19/20, p.p. 4589-4598, Sep. 2008 Gad-el-Hak, Mohamed: TOs MEMS Oncdbttk, CRC, 2001 Avsec, J., Praunseis Z., Tršelič I., Novosel, U.: TOs Dsvaltemsct tf Mtdarc Nnct ncd Micro Ecargy TanOctltniac, Enre 2013 conference, Velenje, 2013 Guz, A., Rushitskii, Ya.: Nnctmntarinlc tc tOa ManOncinc tf Nnctmntarinlc, International Applied Mechanics, Vol. 39, No. 11, p.p. 1271-1293, 2003 Marques, R.F., Inman, D.J. Rade, D.J.: Accacmact tf ndnptiva tanOciquac ftr tOa ntctrtl tf ctrunturac cubjant tt tamparntura vnrintitcc, 12th ICSV Conference, Lisbon, Portugal, 2005 Avsec, J.: TOa Ictluacna tf Tamparntura Fiald tc Vibrntitc COnrnntarictinc tf Banmc ncd Rttnticn SOnftc, Journal of Vibroengineering. Vol. 11, Iss. 4, p.p. 665-672, 2009 Hsu, J.-C., Chang, R.-P., Chang W.-J.: Ractcncna fraquacny tf nOirnl cicnla-wnllad nnrbtc ncttubac ucicg TimtcOackt banm tOatry, Phys. Let. A, Vol. 372, p.p. 2757.2759, 2008 Chang, T., Geng, J., Guo, X.: Pradintitc tf nOirnlity-, ncdciza dapacdact alnctinertpartiac tf cicnla-wnllad nnrbtc cncttubac vin mtlanulnr manOncinc mtdal, proc. Roy. Soc.A, Vol. 462, p.p. 2524-2540, 2006 Prakash N.: Datarmicntitc tf ntaffiniact tf tOarmnl axenccifc tf cicnla-wnllad nnrbtc cncttubac ucicg mtlanulnr dycnminc cimulntitc, PhD Thesis, Florida state University, 2005 (JET Iv* J JOUrnal Of JE1 Volume 8 (2015) p.p. 43-55 Issue 4, December 2015 Typology of article 1.01 Technology www.fe.um.si/en/jet.html ON THE ISOCHRONICITY OF PERIODIC SOLUTIONS AT A CENTRE MANIFOLD O IZOHRONOSTI PERIODIČNIH REŠITEV NA CENTRALNI MNOGOTEROSTI Brigita Ferčec12R, Matej Mencinger34 Keywords: centre manifold, centre problem, isochronicity Abstract The problem of isochronicity is discussed from the historical and dynamical systems point of view. The model of Huygen's cycloidal chops is mathematically explained. We consider two dynamical systems arising from a three-dimensional system with a centre manifold. Based on the period function approach we find necessary and sufficient criteria on the coefficients of the system to distinguish between the cases of isochronous and non-isochronous oscillations. Povzetek Problem izohronosti je obravnavan s stališča dinamičnih sistemov ter iz zgodovinskega stališča. Matematično je razložen Huygensov model ure s cikloidnim nihalom. Na osnovi analize funkcije periode obravnavamo dva dinamična podsistema tridimenzionalnega sistema s centralno mnogoterostjo in poiščemo potrebne in zadostne pogoje za izohronost centra na raznoterosti določeni s koeficienti sistema. R Corresponding author: Matej Mencinger, Tel.: +386 2 229 4321, Mailing address: University of Maribor, Faculty of Civil Engineering, Transportation Engineering and Architecture, Smetanova ulica 17, 2000 Maribor, Slovenia. E-mail address: matej.mencinger@um.si 1 University of Maribor, Faculty of Energy Technology, Hočevarjev trg 1, 8270 Krško, Slovenia 2 Center for applied mathematics and theoretical physics, University of Maribor, Mladinska 3, SI-2000 Maribor, Slovenia 3 University of Maribor, Faculty of Civil Engineering, Transportation Engineering and Architecture, Smetanova ulica 17, 2000 Maribor, Slovenia 4 Institute of Mathematics, Physics and mechanics, Jadranska 19, 1000, Ljubljana, Slovenia 1 INTRODUCTION A simple pendulum is a bob of mass m on a massless rigid wire of length l (see Fig. 1). Assuming no damping (i.e. the only force acting on the system is the weight of the bob), the differential equation governing a simple pendulum is d^ g är + rsin«P = ° (1.1) Figure 1: A simple pendulum of length I driven by the weight F = —mg of the bob. There are several approximations for a period of a simple pendulum (see Fig.1) of length I. One of them (see [1]) is Т(1;ф0) = 2и(-^ Д , Vsin фо У yjg (1.2) where ^>0 is the initial angle displacement (amplitude) of the bob. Note that substituting x = , y and т = •t into (1.1) one can easily derive the following system dx y3 dy _=— siny^— y + _— ..., _ = x (1.3) The qualitative behaviour of this system is determined by how х(т) and y(T) behave with the change of т. It is convenient to indicate how a solution behaves in the phase plane, that is in the x,y —plane. The qualitative behavior is represented by a family of curves, directed with increasing т. These phase curves are called trajectories or orbits of system (1.1.3). The geometrical representation of the qualitative picture of orbits of system (1.1.3) is called its phase portrait. System (1.1.3) admits a centre at (0,0), that is, all the orbits close to the origin are closed or, in other words, the origin is enclosed with simple closed curves (solutions are periodic). Obviously, because of (1.1.2) the centre of (1.1.3) is not isochronous, i.e. not all periodic solutions have the same period. Isochronicity, as will be stated in the next section (see Definition 2), was not known at the time of the great Dutch astronomer, physicist and mathematician Christian Huygens. However, he was aware that if the amplitude of the pendulum's swing changed, the time of swing would also change, which means that the pendulum was not isochronous, [2]. In the limit of small amplitudes, the period (1.1.2) is approximated by T(l) = 2n • J. The following system is obtained from the equation of a mathematical pendulum (1.1) (again using the change of time dx dy dx= _y' dT = X (1.4) However, for larger amplitudes, the period T is a function of the amplitude. The approximation formula (1.1.2) is just one of many existing [1]. Huygens' ingenious idea, which he put into practice, was to vary the effective length of the pendulum by allowing its cord to wrap partially around an obstruction as it swings. What should the shape of this obstruction be in order to ensure that the period is strictly independent of the amplitude? In the middle of the 17th century, the accuracy of clocks was not measured in minutes; even the best of them gained or lost several minutes per day because they were driven by falling weights or springs. Galileo Galilei was probably the first who wanted to design a pendulum driven clock in order to achieve better accuracy. He believed that he had mathematically proven that a (simple) pendulum is isochronous. As already stated, for the case of the pendulum this simply means that the time it takes to complete one full swing is independent of the size of the swing. Nowadays, it is well-known that this is not the case, and that Galileo was mistaken (see approximation formula (1.1.2)). The first scientist who produced a pendulum-driven clock to keep time errors to within one minute per day was Christian Huygens in 1656; within two years the clocks with accuracy of about 10 seconds per day were produced (by Huygens and others). (xM.y(O) /$(t) Figure 2: A baud es u wiss of shops (x(t),y(t)). In order to solve the problem of an "isochronous pendulum", a generalization of the simple pendulum in the sense depicted in Fig. 2 was considered. Thus, if we allow the shape connecting (x0,y0) with (0,0) to be different from a semicircle (i.e. a general shape (x(t),y(t))), then Huygens's question (of isochronous swing) becomes for what shape will the bead descend to the origin in the same time regardless of its starting point (x0,y0)? Denoting by s(t) the distance along the string that the bead has traveled at time t and by v(t) = it = \(т0 + (У) the velocity of the bead (at time t) which is (in contrast) equal to v(t) = —^2g(y0 — y(t)), where g is the acceleration of a freely falling body due to gravity. Furthermore, assume that the functorial dependence of s(t) in terms of y(t) is s = f(y). Then the time taken by a bead to descend from (x0,y0) to (x(t),y(t)) becomes: ч f°dtds Г T(y;y°) =1 ^^dy =1 Уо dy L — 1 V 2g(y° — y) f'(y)dy. (1.5) Substituting a dimensionless variable z = — into (1.1.5) and rearranging we obtain Уо 1 r1f'(zy°)Vy V2gJ° Vl — z T(Z)=-Lf1f'(zy°)Vyo dz, (1.6) which is independent of y° (i.e. a constant function of y°) for 00. y (1.8) Recall that s = f(y) yielding dy = f'(y). Dividing = (g2 + (^)2 by g)2 yields (g) (f'(y))2 = (dy) +1 and finally from (1.1.8) 1.8) we obtain fj ■'Уо с —y dy. (1.9) The solution to (1.1.9) containing (x,y) = (0,0) can be parametrized (using y = C sin2 j, for 0< ф < n, and the "half angle identities") by х(Ф) = ^fa + s^^), У(Ф) = 2(1 " созФ), (1.10) which passes through (x0,y0) for ф = ф0 < tan lj (satisfying — = 1 cosy° ) and C= 2y° . г о ч ~r -t-° x° ф°+sinф°' 1-cosф The shape of the curve (x(t),y(t)) from (1.10) is called an (inverted) cycloid (see Fig. 3). Definition 1. The cycloid (studied and named by Galileo in 1599) is the locus of a point on the rim of a circle of radius a rolling along a straight line. If the point considered on a circumference of the rolling circle is initially in the origin (0,0), the cycloid has a cusp at the origin and its humps are oriented upward. Its parametric equation is x(t) =a(t- sint), y(t) = a(1 - cost) (1.11) Figure 3: Cycloid: The path followed by a point on the circumference of a circle as that circle rolls along a straight line. An involute is a curve obtained from another given curve by attaching an imaginary taut string to the given curve and tracing its free end as it is (un)wound onto that given curve. The evolute of a curve is the locus of all its centres of curvature. That is to say that when the centre of curvature of each point on a curve is drawn, the resultant shape will be the evolute of that curve. For instance: the evolute of a circle is a single point at its centre, the evolute of a quadratic parabola y = x2 is a semicubical parabola y = j + ^ ^4x2 . The only curve for which the evolute (and involute) is "of the same size" (neglecting translation) as the original curve is the cycloid (see Fig. 3 and Fig. 4). The evolute of an involute is the original curve (fewer portions of zero or undefined curvature). Alternatively, another way to construct the involute of a curve is to replace the taut string by a line segment that is tangent to the curve on one end, while the other end traces out the involute (Fig. 3). Figure 4: Inverted cycloid (black) with its involute (red) which is a (inverted) cycloid with the same parameter a, just translated for a vector (na, —2a). The above results led Huygens to invent a pendulum with cycloidal chops (Fig. 5). Figure 5: The cycloidal chops pendulum invented by C. Huygens. However, it is not possible to determine the centre of oscillation for pendula suspended between cycloids, as seen in Fig. 4, since the motion of a pendulum with cycloidal chops is not planar (which was known by Huygens; such a pendulum is a multiple system, i.e. a system having two degrees of freedom). Therefore, the next step of improvements of the pendulum clocks was the so-called escapement mechanism (which was driven primarily by the gravity force and later by the force of a spring). 2 ISOCHRONICITY OF PLANAR SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS We discuss differential systems with analytical right sides TO TO dx v1 ■ ■ dy v-1 ■ ■ dt=—y + Z PijXiy], dt=x + Z qiiXiyJ, (2.1) i+j = 2 i+j=2 where pij and qij are real constants. There has been a longstanding problem, [3], called the Poincaré centre-focus problem; for the system (2.1) to find explicit conditions under which (2.1) has a centre at the origin. The problem is equivalent to an analogue for a corresponding periodic equation dr Zr=iAi(9)ri+1 Y _ , , i Нф=1+1Г=^(фУ= YR^)r (2-2) to have periodic solutions. To see this let us note that the phase curves of (2.1) near the origin in polar coordinates x = r cos ф, y = r sin ф are determined by (2.2), where А^ф) and В^ф) are polynomials in cos ф and sin ф and ^(ф) are 2n —periodic functions of ф and the series on the right-hand side of (2.2) is convergent for all ф and all sufficiently small r, [4]. Since the closed orbits of (2.1) correspond to periodic solutions of (2.2), the planar analytic vector field (2.1) has a centre at (0,0) if and only if (2.2) has a centre at r = 0 ; that is, all the solutions nearby are periodic. All methods for determining the nature of the critical point of (2.1) theoretically require determining infinitely many coefficients of some function or as we will see later the corresponding ideal of the infinitely many polynomials, [4,5,6,7]. The most widely used method to resolve the Poincare centre-focus problem are the so-called Lyapunov method (combined with the Bautin method) and Mironenko's method, [8]. Solving the Poincaré centre-focus problem for a given system (2.1) may be a tedious job. However, once we know the origin of (2.1) is a centre, the problem of isochronicity becomes sensible, and the definition of the period function makes sense. Definition 2 ([4]). Suppose r* >0 is so small that it lies in the so-called period annulus (the maximal neighbourhood of the origin that is enclosed with simple closed curves). Let us consider the line segment X = {(x,y): 0 1 such that ideal I with infinitely many polynomials Tj(pjj, qjj) is the same as Im. 3 CENTRE MANIFOLDS The theory of centre manifolds, which originates from the works of V. A. Pliss, [10,11], and was then further developed by many others (see, e.g. [12,13] and references therein) is an extremely effective tool for studying the behaviour of trajectories of high-dimensional systems of ordinary differential equations. Note that one of the important applications of centre manifolds (also called the Pliss reduction principle, [11]) is that it allows the reduction of the system and consequently the study of the stability in lower dimensional phase space (thus, instead of analysing the original high-dimensional phase space we can consider the lower-dimensional subsystem). Consider an m + n-dimensional system of ordinary differential equations of the form x = Ax + u(x,y) y = By + v(x,y), (3.1) where x e!m,y e Rn , Re(a(A)) = 0, Re(a(B))^0, a(A) and ct(A) are spectrums of A and B, respectively, and u, v are Ck- functions, k>1 which vanish together with their first derivatives at the origin. By definitio, a Ck- manifold Wc = WAc(0, U) in a neighbourhood U of 0 is said to be a centre manifold of (3.1) if Wc is invariant under the flow as long as the solution remains in U nd Wc is the graph of a Ck- function y = h(x) which is tangent at (0,0) e RmxRn to the x-space. The following fundamental result shows that for the system (3.1), there is always a centre manifold in a neighbourhood of the origin. Theorem 1 ([12]). There exists a neighbourhood U of (0,0) e Rmxln such that there exists a local centre manifold Wc of (3.1) which is the graph of u Ck- function y = h(x). The simplest examples of the centre manifolds can be seen in low dimensions. For instance, consider two-dimensional system x = —x3, y =—y. (3.2) For any constants cbc2 the curve h(x; сг,с2), where i—c1e~2^?, x <0 0, x = 0 c2e~2x2, x > 0, consists of orbits of (3.2) and is therefore invariant. Furthermore, this curve is tangent to the x-axis at the origin, which implies that it is a centre manifold. From this example, we can also see that the centre manifold is not necessaily unique. Another simple two-dimensional system with a centre manifold is x = —x3, y = —y + x2. Similar as in the previous case, we see that although the system is analytic, it is not difficult to verify that the centre manifold is not analytic. In three dimension, a family of systems with a centre manifold is ù = —v +P(u,v,w) v=u + Q(u,v,w) (3.3) w = —Xw + R(u,v, w), where X is a positive real number and P, Q, R are polynomials without constant and linear terms. By Theorem 1, this system has a centre manifold w = f(u,v). There are many systems arising from physics (for instance, the Rikitake system [14], for Earth's magnetic field or the Hide-Acheson Dynamo, [15]) that possess a fixed point at which the linear part has one negative and two purely imaginary eigenvalues and are therefore of the form (3.3). Since system (3.3) has a centre manifold Wc and X>0, the trajectories in a small neighbourhood of the origin tend to the trajectories to the centre manifold as time increases. In systems (3.3), the phase portrait in a neighbourhood of the origin on can be, depending on the added nonlinear terms P and Q, either a centre in which case every trajectory (other than the origin itself) is an oval surrounding the origin, or a focus, in which case every trajectory spirals towards the origin or every trajectory spirals away from the origin as the time increases. The problem of determining the dynamical behaviour on Wc, that is, distinguishing between a centre and a focus on the centre manifold for a quadratic polynomial system of the form (3.3) was studied in [16]. 4 THE STUDY OF TWO EXAMPLES In [16], the authors studied the dynamics of trajectories at the centre manifolds for the system U = —v + au2 + av2 + cuw + dvw v = u + bu2 + bv2 + euw + fvw w = —w + Su2 + Sv2 + Tuw + Uvw, (4.1) where the coefficients a, b, c, d, e, f, S, T, U are real. They found five conditions for the existence of a centre on the centre manifold: 1. S = 0; 2. a = b = c + f=8c + T2 — U2=4(e — d) —T2U2 = 2(e + d) + TU and S = 1; 3. a = b = c = f= d + e = 0 and S=1; 4. d + e = c = f= T — 2a = U — 2b = 0 and S = 1; 5. c = d = e = f =0 and S=1. The next question that naturally arises is whether the centre at the centre manifold is isochronous, that is, whether all oscillations have the same period. In this paper we study two subfamilies of system (4.1) satisfying conditions 1 and 2 above. For each system, we compute the period function as described in section 2 and use it to find conditions for the centre to be isochronous. System (4.1) satisfying condition 1 above is written as u = —v + au2 + av2 + cuw + dvw v =u + bu2 + bv2 + euw + fvw w = —w + Tuw + Uvw. (4.2) It is proven (see [16]) that w = 0 isthe centre manifold for system (4.2). The corresponding 2D system (4.3) v =u + bu2 +bv2 has a centre at the origin for all a,b£l (see [16]). We now study the isochronicity problem for the above centre. Using the computer algebra software MATHEMATIC, we first turn to the computation of the period function T of the form (2.6). After introducing polar coordinates system, (4.3) becomes dr — = r2(a cos ф + b sin ф) "dt =1 + r(b cos ф — a sin ф). (4.4) Following the procedure described in the second section, we find that the first non-zero coefficient of the period function is T2 = 2u(a2 + b2). Thus we see that the necessary condition for the isochronicity of system (4.3) is a = b = 0, which, obviously is also the sufficient condition. In the case of conditions 2, under the same renaming of parameters c,d,e,f,T,U and using a,ß as in [16], the system (4.1) takes the form 1 1 2 ù = —v — 2 aßuw — 2 ß2vw v = u + 2a2uw + 2«ßvw W = —w + u2 + v2 + (a +ß)uw + (ß — a)vw. (4.5) A search for invariant algebraic surfaces led to the explicit equation of centre manifold Wc given u2+v2 by w = t au pv Inserting the expression for w into system (4.5), we obtain the system u = —v — ■ v = u + ß(au + ^v)(u2 + v2) 2(1 — au — ^v) a(au + ^v)(u2 + v2) 2(1 — au — ^v) Using Taylor series expansion up to the order five, we obtain the system where u = —v + 2ß(au + ßv)(u2 + v2) + 2ß(au + ßv)2(u2 + v2) + m! v = u — 2a(au + ßv)(u2 + v2) — 2a(au + ßv)2(u2 + v2) + m2, m! = 12o(—60a3ßu5 — 180a2ß2u4v+10u3v2(—6a3ß — 18aß3) + 10u2v3(—18a2ß2 — 6ß4) — 180aß3uv4 — 60ß4vs) (4.6) and m2 = 120 (60a4 u5 + 180a3ßu4v+10u3v2(6a4 + 18a2ß2) 33^ _L_ ЮП^ГЗ2..-^4 _L_ I4u5 2 3 3 + 10u2v3(18a3ß + 6aß3) + 180a2ß2uv4 + 60aß3v5). Further computation following the computational pattern described in section 2 yields T2 = - (a2 + ß2), yielding the following result. System (4.6) has an isochronous centre if and only if a =ß = 0. 5 CONCLUSIONS After more than 300 years, the isochronicity problem remains one of the central problems in the theory of dynamical systems. In addition to the period function approach, there are some other be too complex for some systems. Therefore, in some cases, we need to find other ways to solve the isochronicity problem. It turns out that the isochronicity problem for system (2.1) is equivalent to the linearizability problem in which we look for an analytic change of coordinates that reduces (2.1) to the canonical linear centre X = —y,y = x (see e.g. [4] for more details). The theory of the linearizability of planar systems of ordinary differential equations was applied [17] to system (4.1) corresponding to condition 4 above. In fact, for each of the five conditions above we can find the conditions under which the corresponding system can be reduced to canonical linear centre, and we see that they are the same as conditions obtained using the period function approach. References [1] G.E. Hite: Approximations for the Period of a Simple Pendulum, The Physics Teacher, Vol. 43, p.p. 290, 2005 [2] C. Huygens The pendulum clock or geometrical demonstrations concerning the motion of pendula as applied to clocks. Translated from the Latin and with a preface and notes by Richard J. Blackwell. With an introduction by H. J. M. Bos. Iowa State University Press, Ames, IA, 1986 [3] H. Poincaré: Mémoire sur les courbes déflnies par une équation différentielle, J. Math. Pures et. Appl. (Sér. 3), Vol. 7, p.p. 375-422, 1881; (Sér. 3), Vol. 8, p.p. 251-296, 1882; (Sér. 4), Voil. 1, p.p. 167-244, 1885; (Sér. 4), Vol. 2, p.p. 151-217, 1886 [4] V.G. Romanovski, D.S. Shafer: The Center and cyclicity Problems: A computational Algebra Approach, Boston: Birkhäuser, 2009 [5] B. Ferčec, X. Chen, V. Romanovski: Integrability conditions for complex systems with homogeneous quintic nonlinearities, Journal of applied analysis and computation, Vol. 1, Iss. 1, p.p. 9 - 20, 2011 [6] B. Ferčec, J. Giné, Y. Liu, V. Romanovski: Integrability conditions for Lotka-Volterra planar complex quartic systems having homogeneous nonlinearities, Acta applicandae mathematicae, Vol. 124, Iss. 1, p.p. 107 - 122, 2013 [7] B. Ferčec, J. Giné, M. Mencinger, R. Oliveira: The center problem for a 1 : -4 resonant quadratic system, Journal of mathematical analysis and applications, Vol. 420, Iss. 2, p.p. 1568 - 1591, 2014 [8] Z. Zhou: A Nsw Method fes Rsrsuseh es ths Csstss-Feecr Problem of Diffsssstiul Sortsmr, Abstract and Applied Analysis, Art. ID 926538, 5 p.p., 2014 [9] D. Cox, J. Little, D. O'Shea: Idsulr, Vusistisr, usd Algosithmr: As Istsomcetios to Computational Algsbsuie Gsomstso usd Commctutivs Algsbsu, New York: Springer, 3rd edition, 2007 [10] S.Yu. Pilyugian, G.R. Sell: A Biogsuthieul Sksteh of Vietos A. Plirr, Journal of Dynamics and Differential Equations, Vol.15, 2003 [11] V.A. Pliss: A ssmcetios psiseitls is ths thsoso of rtubilito of motios, Izv. Akad. Nauk SSSR Ser. Mat., Vol. 28, p.p. 1297-1324, 1964 [12] S. Chow, J. K. Hale: Msthodr of BiCcshutios Thsoso, Springer-Verlag, New York, 1982 [13] J. Sijbrand: Psotsrtisr of esstssmusifolmr, Trans.~Amer.~Soc., Vol. 289, 431-469, 1985 [14] R. Rikitake: Oreillutiosr of u rortsm of dire dosumor, Proc. Cambridge Philos. Soc., Vol. 54, p.p. 89-105, 1958 [15] R. Hide, A. C. Skeldon, D.J. Acheson: A rtcdo of two sovsl rslC-sxhitisg risgls-dirk homotolus dosumor: thsoso, Proc. R. Soc. Lond. A , Vol. 452, p.p. 1369-1395, 1996 [16] V.F. Edneral, A. Mahdi, V.G. Romanovski, D.S. Shafer: Ths esstss psoblsm os u esstss musifolm is E3, Nonlinear Anal., Vol. 75, p.p. 2614-2622, 2012 [17] V. Romanovski, M. Mencinger, B. Ferčec: Isvsrtigutios of esstss musifolmr of thsss-mimssriosulrortsmr crisg eomtctssulgsbsu, Programming and computer software, Vol. 39, iss. 2, p.p. 67-73, 2013 (JET Iv* J Journal Of JE1 Volume 8 (2015) p.p. 57-68 Issue 4, December 2015 Typology of article 1.04 Technology www.fe.um.si/en/jet.html DESIGN OF THE VENTILATION AND AIR-CONDITIONING SYSTEM IN AN OFFICE BUILDING NAČRTOVANJE SISTEMA PREZRAČEVANJA IN KLIMATIZACIJE V POSLOVNI STAVBI Franc RihlR Keywords: ventilation and air-conditioning, cooling, heating, ventilation ducts, Brestanica Thermal Power Plant (TEB). Abstract This article presents an office building within the Brestanica Thermal Power Plant for which a design of the ventilation and air-conditioning system has been made. First, a brief description of the existing heating and cooling system from the aspect of the operation and the investment contribution is made. Then a ventilation and air-conditioning system is designed based on the geometric characteristics and thermal needs of the building. Calculation of heat losses, selection of an appropriate ventilation and air-conditioning unit, and design of a ventilation duct system are included in the design. The entire system is evaluated from the aspect of investment value. Povzetek V članku je predstavljena poslovna stavba v Termoelektrarni Brestanica, za katero se je naredila študija prezračevanja in klimatizacije. Najprej je na kratko opisan obstoječi sistem ogrevanja in hlajenja iz vidika samega delovanja in investicijskega vložka. V nadaljevanju je načrtovan sistem prezračevanja in klimatizacije glede na geometrijske karakteristike in toplotne potrebe. R Corresponding author: Franc Rihl, Faculty of Energy Technology, Tel.: +386 40 792 479, Mailing address: Hočevarjev trg 1, 8270 Krško, E-mail address: franc.rihl1@um.si V načrtovanje je vključen preračun toplotnih potreb stavbe, izbira ustrezne prezračevalne in klimatske naprave in načrtovanje sistema prezračevalnih kanalov za transport klimatiziranega zraka. Celotni sistem je ovrednoten tudi iz vidika investicijske vrednosti. 1 INTRODUCTION The aim of this article is to examine the system for heating, cooling, and ventilation in an office building within TEB. The entire study is based on data obtained from TEB; the current embedded system for heating, cooling and ventilation is described based on this data. Furthermore, the total investment costs of both embedded systems are researched and presented. Based on the size of the business buildings and the calculation of the heat demand, the selection of an appropriate ventilation and air-conditioning unit and the design of the ventilation duct system is presented. This article concludes with an assessment of the investment costs of installing such a system. Table 1: Thn cnemntricol choroctnriotico ef on efficn anildisc PART A PART B Area of the thermal envelope [m2] 2234.22 3980.50 Conditioned volume [m3] 1741.38 6449.47 Net heated volume [m3] 1286.02 5281.44 Usable area [m2] 494.35 1578.40 Length of the zone [m] 28.42 64.90 Width of the zone [m] 11.50 26.05 Height of the floor [m] 2.40 3.16 Number of floors 2 2 Number of rooms 28 60 Climate characteristics depending on the location of the office building are a temperature deficit of 3100 Kday, a projected temperature of -13 °C and an average humidity of 78%. Depending on the projected temperature, this is the 3rd climate area in Slovenia, [1]. These data are used in the calculation of the transmission heat losses. 2 DESCRIPTION OF THE OFFICE BUILDING The office building comprises two separate buildings connected by a corridor on the first floor. Therefore, the smaller part of the office building is named 'Part A', where the reception and offices are, and the bigger one 'Part B', where workshops and a warehouse are. The office building is shown in Figure 1. Figure 1: The office building The office building is physically separated from the production part of TEB. All relevant information about the office building for further calculation were provided by the head of the TEB, and they are shown in Table 1. 3 THE EXISTING SYSTEM FOR HEATING, COOLING AND VENTILATION The office building does not have a common system for heating and cooling, but it has two entirely separate systems that are independently operating. The old system for heating was upgraded before the heating season in 2006. The old oil boiler system was replaced with a gas-condensing boiler system, with a total heat capacity of 400 kW. The system consists of four Buderus Logamax Plus GB162 boilers connected into a cascade and regulated with two Buderus Logamatic 4000 controllers. Consumption of natural gas is recorded by a remote system, which is directly connected to the connected gas pipeline. Monthly natural gas consumption between 2011 and 2013 is shown in Figure 1. Annual average natural gas consumption for heating of the office building is 25,732.5 Sm3. Figure 2: Consumption ef notnrol goo antwnns 2011 ond 2013 In addition to gas boilers for DHW heating, an air-water heat pump is used. The maximum heat capacity of the heat pump is 5 kW and COP around 2.8. In the event of failure, the old oil boiler system takes over heating, which is connected through a parallel connection and heat exchanger. The total value of the investment in the gas-condensing boiler system was €121,014.86. The cooling system was mounted in summer 2004. Based on the number of required rooms and necessary cooling capacity, a multi-split unit system was selected. This system consists of 25 units and it is vacuumised and filled with ecologically acceptable R407C liquid. In the entire system is approximately 43 kilos of this cooling fluid. The total value of the investment in the cooling system was €8,863.26. The office building does not have its own system for ventilation; thus, employees have to provide sufficient air exchange by opening windows and doors. Consequently, this leads to uncontrolled ventilation and heat losses. 4 VENTILATION AND AIR-CONDITIONING SYSTEM The ventilation and air-conditioning system is concerned with achieving and maintaining thermal comfort parameters in confined areas, [2]. It is a complete process, which includes the production, transport and the injection of the transmission medium into the air-conditioned area. Constant thermal comfort is maintained by regulating the temperature, relative humidity, air velocity, air quality, volume level and differential pressure in the area, [2]. Depending on the scope of operation, two main types of ventilation and air-conditioning system are knowns: - Comfort, and - industrial systems, [2]. 4.1 Central system for ventilation and air-conditioning This is the most basic type of a ventilation and air conditioning system from the technical point of view. The transmission medium in this type of system is air. It is accordingly prepared in the central unit; then it is supplied through ventilation ducts in all air-conditioned areas. At the same time, the exhaust air is transported out of the area through the return line of ventilation ducts. Fresh air can be mixed with exhaust air in the mixing chamber in order to reduce energy consumption. However, this mixing process should not significantly affect the thermal comfort in the air-conditioned areas, [2]. An example of the central system for ventilation and the air-conditioning structure is shown in Figure 2. MK Figure 3: Strnctnrn ef thn cnntrol oyotnm fer onctiloties ond oir-cecditiecisc, [2] 5 DESIGN OF VENTILATION AND AIR-CONDITIONING SYSTEM The design of a ventilation and air-conditioning system includes a calculation of the heating and cooling load of the building according to the information in Table 1. Then, based on the calculated loads and climate data, the size of the central ventilation and air-conditioning unit is chosen. This is done with a numeric program that determines the size of the heater, cooler, heat recovery system and other parts of the system. After the central ventilation and air-conditioning unit is chosen, a ventilation duct system for transporting air into air-conditioned areas must be designed. This is made by a calculation of pressure drop in the system. 5.1 Thermal and cooling load The thermal load is calculated on the basis of the transmission heat losses of the building and data on the climate in which the building is placed. Cooling load is calculated as the sum of the transmission heat losses and heat gains specified in the German standard VDI 2076. The heat gains described in VDI 2076 are those due to the presence of people, solar radiation gains, the heat of equipment and lighting. Transmission heat losses were calculated in the program URSA GF 4.0, which is made for calculating the thermal protection of buildings and energy use in buildings in accordance with the regulation on energy efficiency PURES 2010. The data in Table 2 are calculated using Equations 5.1 to 5.4 and the program URSA GF 4.0. Qhl =QT -АГ Q =K + Qp, )'N Qm = 4 • Au Qs =[ A • 'max +( A - /difmax ]• Ь • (5.1) (5.2) (5.3) (5.4) Table 2: Calculated data for thermal and cooling load PART A PART B Thermal load Transmission heat losses [W] 36,145.23 63,931.23 Cooling load Transmission heat losses [W] 6,571.86 11,624.46 Heat gains due to people presence [W] 4,945.00 9,890.00 Heat of the devices and lighting [W] 1,977.40 6,313.60 Solar radiation gains [W] 26,800.66 26,907.45 Total cooling load [W] 40,294.92 54,735.51 5.2 Selection of the ventilation and air-conditioning unit The company Menerga Ltd. has helped with the selection of a ventilation and air-conditioning unit. This company is primarily engaged in ventilation and air-conditioning systems. According to the data calculated in Table 2, they have selected an Adconair 76 ventilation and air-conditioning unit with the counterflow plate heat exchanger and an air compressor cooling system. For Part A, an Adconair Ad 761601 IMH unit was selected with a heat capacity of 55.99 kW, a cooling capacity of 41.87 kW and a 92% heat recovery rate. For Part B, an Adconair Ad 761501 IMH unit was selected with a heat capacity of 87.53 kW, a cooling capacity of 55.45 kW and a 93% heat recovery rate. Despite all the characteristics, units do not have an integrated system for heating the water heaters. A solution to this manner had to be found, which was the design of the pipeline from the gas condensing boilers to both units. 5.2.1 Design of the pipeline The pipeline has to connect gas condensing boilers and both units and provide the required heat capacity. The transmission medium in the pipeline is water, and the operating temperature is 70/50 °C. The pipeline consists of several main elements: - pipes, - heat exchanger, - expansion vessel, and - water pump. Due to the required heat capacity of the water heaters, the pressure drop in the pipeline, and the velocity of medium and thermal insulation of pipes, a Danfoss XB 59M-1 70 heat exchanger, an OPTIMA 200 expansion vessel, Uponor Ecoflex Thermo Twin 2x63x5.8/200 DN50 pipes and a Grundfos Magna3 50-40 water pump have been selected. 5.3 The dimensions of ventilation ducts In the office building, a dual ventilation duct system was designed for each part of the building. This was done on the basis of the technical characteristics of the ventilation and air-conditioning units. The first step was the selection of air velocity through different elements of the ventilation duct system. Recommended values of air velocity are shown in Table 3. Table 3: Rncemmnndnd oolnno ef oir onlecity through diSSnrnct nlnmnnto ef thn onctiloties dnct oyotnm, [3] Element The approximate values of the air speed [m/s] Comfort system Industrial system Protective gratings 2-4 4-6 Main ducts 4-8 8-12 Side ducts 3-5 5-8 Air diffusers 1,5-2,0 3-4 The second step is the calculation of the cross-section of the ventilation duct according to Equation 5.5 [3]. An, 1/ 3600•w (5.5) When the calculation of cross-sections of all ducts is done, it is necessary to select a sufficiently large ventilation duct. Recommended sizes of the rectangular ventilation ducts are shown in Table 4. Table 4: Recommended sizes of the ventilation ducts, [3] Designation b h Surface [m2] 300/150 300 150 0.0450 400/200 400 200 0.0800 500/250 500 250 0.1250 600/300 600 300 0.1800 600/350 600 350 0.2100 700/400 700 400 0.2800 800/500 800 500 0.4000 1000/500 1000 500 0.5000 Table 5: Colcnlotnd prnoonrn drepo in thn onctiloties dncto Place Pressure drop Supply [Pa] Outlet [Pa] Part A Office 1.1 477.78 400.90 Reception 398.64 307.90 Part B Office 1325 661.13 583.24 Warehouse 633.54 567.78 Protective gratings are selected using Equation 5.6, for which the required effective surface must be larger than the calculated [4]. 4f2 = B1 (H1 - 21 -( 7 • n)) (5.6) The air diffusers are selected according to the required effective surface and injecting speed of air via diagrams, which can be found for special types of diffuser, [4]. 5.4 The calculation of the pressure drop in the ventilation ducts For the ventilation duct system, it is necessary to calculate the maximum possible pressure drop and verify whether it can be covered with the air fans built in the ventilation and air-conditioning units. The maximum possible pressure drop typically occurs in the longest duct. The pressure drop is caused by two types of pressure drops: — linear, and — local. Typically, these pressure drops are calculated using Equations 5.7 to 5.10. ap = ZApli„ +^APlok APlin = e-p^ w dH •2 • L dH = 2 • a • b a + b (5.7) (5.8) (5.9) Aflok =C p^ w 2 (5.10) The rooms in which the maximum pressure drops in the system were calculated are shown in Table 5. All calculated pressure drops are smaller than the maximum allowed pressure drops of the air fans inside the ventilation and air-conditioning units, which means that the system of ventilation ducts is appropriately designed. 6 FINAL VALUE OF THE INVESTMENT The investment in the ventilation and air-conditioning system consists of several sets. The main sets are the ventilation and air-conditioning units and elements related to their operation, followed by the ventilation duct system, protective gratings, diffusers and works related to the installation of equipment. The values shown in Table 6 are based on data obtained from different manufacturers, such as Menerga Ltd., NKM Ltd., Hidria Ltd., Danfoss Ltd., Uponor Corporation, Štern Ltd. and Grundfos Ltd. Table 6: Tetol oolnn ef thn iconotmnct Type of costs Total value without VAT [€] Equipment - ventilation and air-conditioning units 186,147.00 Equipment - pipeline 7,223.39 Installation and mounting costs, additional work 19,500.00 Total value without VAT [€] 212,870.39 VAT 22 % [€] 46,831.49 The final value of the investment [€] 259,701.88 Certain values are assumed, since it is difficult to estimate the exact value of the services. In that manner, the final value of an investment may be different than it would have been if the project is actually performed. 7 CONCLUSION This paper describes the current situation in terms of thermal comfort in an office building. A major shortcoming is uncontrolled ventilation, an oversized gas condensing system, and insufficient cooling of the entire building. The consequences of such situation are higher maintenance costs, and deficits or surpluses of fresh air, which leads to uncontrolled heat losses and higher operating costs. In the ventilation and air-conditioning system, such problems do not occur, because the automation maintains optimal thermal comfort. From the aspect of investment, it can easily be seen that the investment in the designed system is almost twice as high as in the old one. Another disadvantage is the usage of the gas condensing system for heating the water heaters of the designed system. This application has reduced the investment costs. It can be concluded that the designed system would not be worth the investment, because such systems are typically installed in new buildings. References [1] Podatki za pravilnik o učinkoviti rabi energije, Državna meteorološka služba - METEO [world wide web], available at: http://meteo.arso.gov.si/met/sl/climate/tables/pravilnik-ucinkoviti-rabi-energije/(3. 9. 2015) [2] B. Evdvrvvić, Klimatizacija, Savez mašinskih i elektrotehničkih inženjera i tehničara Srbije (SMEITS), Beograd, 2005 [3] Recknagl, Šprenger, Šramek, Češerkvvić, Grejanje i klimatizacija, sedmo, izmenjeno i dopunjeno izdanje, Interklima, Vrnjačka Nanja, 2011 [4] Distribution and control air, brochure, Hidria Ltd. [world wide web], available at: http://si.hidria.com/si/klima/programi/distribucija-regulacija-zraka/ (25. 10. 2014) [5] F. Rihl, Načrtovanje sistema prezračevanja in klimatizacije v poslovni stavbi, master's thesis, 2015 Nomenclature (Symbols) (Symbol meaning) TEB Thermoelectric power plant Brestanica DHW domestic hot water SV fresh air T thermometer KM mixing chamber H hydrostat F filter V2 fan for air extraction PG preheater OdV used air HL cooler OtV exhaust air MK chamber for air humidification Žrv grate of the recycled air DG heater Žsv grate of the fresh air V1 fan for air supply Žov grate of the used air PV conditioned air DP differential pressure switch UV supply air in the area RV recycled air Qhl transmission heat losses [W] QT heat flux [W] AT temperature difference [K] Q heat gains due to people presence [W] Qp,r sensitive heat [W] О, latent heat [W] N number of people QM heat of the devices and lighting [W] Au usable area [m2] Q s solar radiation gains [W] A glass surface of the window [m2] ^max global solar radiation [W/m2] A total surface of the window [m2] ^difmax diffuse solar radiation [W/m2] b transmittance Sa the coefficient of accumulation A "PK cross-section of the duct [m2] 1/ volume air flow [m2/h] w air velocity [m/s] Af effective surfice of protective grating [m2] ßl width of protective grating [m] Hl height of protective grating [m] n number of lamells Ap total pressure drop [Pa] Aplin linear pressure drop [Pa] e the coefficient of the material roughness p density of the medium [kg/m3] dH hydraulic diameter [m] L length of the duct [m] a, b dimensions of the duct[m] Ap, Vk local pressure drop [Pa] £ the coefficient of the local resistance Institut za Institute of energetiko Energy Technology On December 2015, the Faculty of Energy Technologies of the University of Maribor completed the project to provide additional research infrastructure at the Institute of Energy, which include €4 million of research equipment in addition to the expanded facilities. The Faculty of Energy Technologies of the University of Maribor is situated in Krško and has a separate department in Velenje. The faculty is active at Slovenia's two largest power stations: the Šoštanj Thermal Power Plant and the Krško Nuclear Power I Plant. Furthermore, in I addition to these two I environments, it is directly or | indirectly active in very large т^—^Я enterprise production and ' —--V--\ service activities throughout /// i j j j— — - -\ \ '. ' the energy sector. The ' ../../ f--—-L™,_\ W integration of both local _ > \ Л \ communities with energy technologies represents a great added value to studying at the faculty: students have the opportunity to encounter - - new photovoltaic field at the Institute от Energy Technology various energy systems in r J J practice, and their acquired theoretical knowledge is thus further consolidated. Additional research infrastructure within the institute is a great asset to the faculty. It is possible to carry out research in the field of aero and hydro energy technologies, thermo-mechanics and thermal-energy, monitoring the efficiency of energy conversion, highly efficient electrical machines and drives, electric vehicles and vessels, the ___materials and their applied . ^ise of energy self-sufficient ^^ and smart electricity P——■ "fc _ networks, detecting the potential of renewable energy sources, computer-supported ^H HVj prototypes design, and more. Additional research capacity enables the implementation of interdisciplinary research with an emphasis on excellence. In the context of ■ r ^ ^ basicand applied research and 3D visualisation system rr new photovoltaic field at the Institute of Energy Technology the implementation of new knowledge, new products, processes and services that increase competitiveness of the economy are created. The aforementioned equipment is a great asset for students of the faculty. A significant proportion of study at the Faculty of Energy Technologies is carried out in laboratories with up-to- date research equipment that provides excellent conditions for the acquisition of the applicable knowledge of students. _ Students with new equipment * ft are enabled to have contact jA • л. * with state-of-the-art л ^ ^jjjjf^jjp»^^ technology in the very broad J^t^BK A field of energy, which is an iy important added value that J^^^J^hh students gain when entering jr , Л ^ д Д the labourmarket. The results of a survey ' ^ ' confirmed that the students of > ^ the Faculty of Energy ' t J^ ^ / * нГ Д Technologies are well received ^ by the labour market in terms of the employability of graduates and competencies according to survey 90% of all graduates are employed acquired duringtheirstudies. A total of 90% of all students who participated in the survey and who did not continue their studies at a higher level programs have a regular employment; 87% are active directly in the field of energy technologies. Only 4.9% all graduates from the first generation of students who completed the study are reported asjobseekersatthe Employment Service Institute, which is very encouraging. In addition to companies engaged in electricity generation, transmission, managing and trading energy and maintaining of energy systems, the students of energy technologies are also valued by the companies that are aware of the importance of the rational use of energy and the associated impacts on operating costs. Information on employment opportunities after the end of the studies is one of the main criteria on which young people decide on the type of study. Study at the Faculty of Energy Technologies is therefore the right choice for all who want to good employment opportunities after graduation, as well as for those who wish to upgrade their knowledge and to obtain higher education. The faculty offers a friendly study environment and fu II study support from the freshman year to completion of one's doctoral thesis. republika slovenija ministrstvo za izobraževanje, znanost in Sport i.,) I Naložba v vašo prihodnost Fakulteta za energetiko »Operacijo delno financira Evropska unija, in sicer iz Evropskega sklada za regionalni razvoj ter Ministrstvo za izobraževanje, znanost in šport. 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