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Reflections on Thermal Energy, Reversible and Caloric Processes, Exergy and Entransy – (Lecture II) Institute of Engineering Thermophysics Tsinghua University Beijing, China, June 18, 2013 Prof. M. Kostic Mechanical Engineering NORTHERN ILLINOIS UNIVERSITY www.kostic.niu.edu Slide 1 Some Challenges in Thermoscience Research and Application Potentials Energy Ecology Economy Tsinghua University, XJTU, and HUST China 2013: Beijing, Xi’an, Wuhan, June 14-28, 2013 Prof. M. Kostic Mechanical Engineering NORTHERN ILLINOIS UNIVERSITY www.kostic.niu.edu Slide 2 Hello: Thank you for the opportunity to present a holistic, phenomenological reasoning of some challenging issues in Thermo-science. Discussions are informal and not finalized yet. Thus, respectful, open-minded arguments, and brainstorming are desired for better comprehension of tacit and often elusive thermal phenomena. •www.kostic.niu.edu 3 Slide 3 •Among distinguished invites were five keynote speakers from China and seven international keynote speakers: three from the USA and one each from Japan, United Kingdom, Singapore, and Spain; including four Academicians and •It has been my great pleasure and honor to meet Prof. ZY Guo and other distinguished colleagues, •and even more so to visit again and meet friends six university Presidents/vice-presidents. now! Slide 4 •www.kostic.niu.edu •www.kostic.niu.edu Slide 5 •www.kostic.niu.edu Slide 6 What is Energy ? •More important than what it appears to be •If one could expel all energy out of a physical system … … then empty, nothing will be left … •… so ENERGY is EVERYTHING … E=mc2 Mass (m) and energy (E) are manifestation of each other and are equivalent; they have a holistic meaning of “mass-energy” •www.kostic.niu.edu Slide 7 What is Energy ? •“From the Sovereign Sun to the deluge of photons out of the astounding compaction and increase of power-density in computer chips … Mass-Energy represents motion of a system structure, i.e., its representative particles at different space and time scales, and ultimately motion of photons. Where the Thermal Energy fits in? •www.kostic.niu.edu Slide 8 Energy Carriers & “Underlying” Energy Carriers Fundamental or “Underlying energy carriers” are the FOUR fundamental forces/interactions (and related particles) in physics: 1. 2. 3. 4. Strong nuclear Weak nuclear Electro-magnetic (EM), and Gravitational Underlying carriers for electro-chemical and thermo-mechanical energy are photons (EM), And “massive (‘convective’)” carriers may be electrons (or electron shells) and bulk matter, including crystal shell (phonons) •www.kostic.niu.edu Slide 9 Heat Transfer Is Unique and Universal: Heat transfer is a spontaneous irreversible process where all organized (structural) energies are disorganized or dissipated as thermal energy with irreversible loss of energy potential (from high to low temperature) and overall entropy increase. Thus, heat transfer and thermal energy are unique and universal manifestation of all natural and artificial (man-made) processes, … and thus … are vital for more efficient cooling and heating in new and critical applications, including energy production and utilization, environmental control and cleanup, and biomedical applications. •© M. Kostic •2009 January 10-12 Slide 10 2 …thus thermal & mechanical energies are coupled •www.kostic.niu.edu Slide 11 Heat Is Transfer of Thermal Energy …thus Q=Uin_transfer • Philosophically, you cannot transfer something that does not exist. For example, you cannot transfer water unless you have water. You cannot transfer energy (type) without having it somewhere (stored) to transfer and store it somewhere else. In the process (while transferring) you may convert/reprocess (modify the "original structure") while conserving the underlying substructure (true elementary particles): existential conservationism. • Some deny existence of ‘thermal energy.’ It is the same as denying existence of its (heat) transfer! © M. Kostic •041115 Slide 12 Interchangeability of heat and work? • I have reservation about accuracy of "Heat and Work Example" and “Thermal Energy” in: http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/heat.html • "This example of the interchangeability of heat and work as agents for adding energy to a system can help to dispel some misconceptions about heat.“ ??? •www.kostic.niu.edu Slide 13 Useful Energy: Work potential, Exergy (and Entransy) concept(s) Two systems in non-equilibrium have potential of extracting work (useful energy). The maximum work potential is if they are reversibly brought to mutual equilibrium while the work is extracted (thus re-arranging the nonequilibrium: entropy is conserved, thus over-all isentropic), otherwise part or in-whole that work potential (i.e., non-equilibrium) will dissipate via heat to thermal energy and generate entropy. If one system is fixed (an infinite thermal reservoir) and taken as a reference (like environment at To & Po), then that maximum work potential depends on the other system state, i.e., it is independent of the process path, thus, could be considerred the system property, called Exergy. Note that there will be a need to reversibly exchange heat (and entropy) at the reference temperature or reversibly regenerate heat internally, except for isentropic processes. •www.kostic.niu.edu Slide 14 “Work Potential” NOT path dependent (very important!) Or any path Also for any 1-2 states (non-isentropic) •www.kostic.niu.edu Slide 15 All reversible processes are “over-all isentropic” (entropy conserved)! Exergy analysis to minimize and optimize irreversibility Entransy analysis to maximize and optimize heat transfer •www.kostic.niu.edu Slide 16 Engineering Thermodynamics 7th Ed. By Moran et al, Wiley But not ALWAYS true: Irreversible work will increase entropy thus resulting in different state with the same internal energy as reversible work. It is, though, true if ALL work is irreversibly converted to heat and stored as thermal energy, as in isohoric processes (V=constant) with solids and liquids (as in the Jule’s experiments). NOT true! BUT so is E2-E1=Qin, if W=0 •www.kostic.niu.edu Slide 17 Howard DeVoe, Thermodynamics and Chemistry (electronic) textbook: www2.chem.umd.edu/thermobook/ on 11 April 2013 •www.kostic.niu.edu Slide 18 Mechanical and Thermal Energies Are Distinguishable Within Internal Energy! DU12s= DU12v U2s=U2v U=Uth+Umech(elastic) T2s=T2v (for Ideal Gas) BUT! 2s≠2v Ps>Pv; Ss<Sv Vs<Vv, Uth,s<Uth,v & Umech,s>Umech,v Exs>Exv etc. FORCE applied or HEAT applied •© M. Kostic •2009 January 10-12 Slide 19 Thermal and Mechanical energies © M. Kostic 1 kJ heating is NOT the SAME as 1 kJ compressing! Thermal and Mechanical energies are distinguishable, NOT the same Internal energy (as argued by some)! •041115 Slide 20 •www.kostic.niu.edu Slide 22 Note : Q Irr Q Gen W Loss T Ref S Gen ( any ) •www.kostic.niu.edu Slide 23 IRREVERSIBILITY AND REVERSIBLE HEAT TRANSFER: The Quest and Nature of Energy and Entropy Prof. M. Kostic Mechanical Engineering NORTHERN ILLINOIS UNIVERSITY © M. Kostic <www.kostic.niu.edu> 041115 Focus and Goal: Focuses on philosophical and practical aspects of energy and entropy, with emphasis on reversibility and irreversibility, and goal to establish the concept of “reversible heat transfer,” regardless that heat transfer is a typical irreversible process. © M. Kostic <www.kostic.niu.edu> 041115 Objective: … to emphasize known, but not so well-recognized issues about entropy, irreversibility and reversibility, as well as to put certain physical and philosophical concepts in perspective, and initiate discussion and arguments about the paper theme. © M. Kostic <www.kostic.niu.edu> 041115 Heat Transfer: Heat transfer like any other energy transfer, may be achieved from any-to-any temperature level, and in limit be reversible, if temperature of an intermediary cyclic substance is adjusted as needed, using isentropic compression and expansion © M. Kostic <www.kostic.niu.edu> 041115 This is practically demonstrated… This is practically demonstrated in refrigeration and heat pump devices, and enables further increase in energy efficiency. A dual power-and-heat-pump cycle is introduced and analyzed here, to provide for reversible heat transfer. It may be considered as a reversible heat-transfer transformer, from-any-to-any temperature levels. © M. Kostic <www.kostic.niu.edu> 041115 Limits and Practical Potentials: The reversible heat transfer limits are the most efficient and demonstrate limiting potentials for practical heat transfer processes. © M. Kostic <www.kostic.niu.edu> 041115 REVERSIBILITY AND IRREVERSIBILITY: ENERGY TRANSFER AND DISORGANIZATION, RATE AND TIME, AND ENTROPY GENERATION Net-energy transfer is in one direction only, from higher to lower potential (energy-forcing-potential), and the process cannot be reversed. Thus all real processes are irreversible in the direction of decreasing energy-forcing-potential, like pressure and temperature (forced displacement of mass-energy) © M. Kostic <www.kostic.niu.edu> 2009 January 10-12 Local-Instant & Quasi-Equilibrium: At instant (frozen) time, a locality around a point in space may be considered as ‘instant-equilibrium’ (including inertial forces) with instantaneous local-properties welldefined, regardless of non- uniformity. Quasi-equilibrium is due to very small energy fluxes due to very small gradients and/or very high impedances, so that changes are infinitely slow, for all practical purposes appearing as equilibrium with virtually net-zero energy exchange. © M. Kostic <www.kostic.niu.edu> 2009 January 10-12 REVERSIBILITY –Relativity of Time: Therefore, the changes are ‘fully reversible,’ and along with their rate of change and time, totally irrelevant (no irreversible-permanent change; it could be put/reversed back), as if nothing is effectively changing (no permanent-effect to the surroundings or universe) The time is irrelevant as if it does not exist, since it could be reversed or forwarded at will and at no ‘cost’ (no permanent change) and, thus, relativity of time. Real time cannot be reversed, it is a measure of permanent changes, like irreversibility, which is in turn measured by entropy generation. In this regard the time and entropy generation of the universe have to be related. © M. Kostic <www.kostic.niu.edu> 2009 January 10-12 Quasi-equilibrium Process : in limit, energy transfer process with infinitesimal potential difference (still from higher to infinitesimally lower potential, P). Then, if infinitesimal change of potential difference direction is reversed P+dP → P-dP with infinitesimally small external energy, since dP→0, the process will be reversed too, which is characterized with infinitesimal entropy generation, and in limit, without energy degradation (no further energy disorganization) and no entropy generation thus achieving a limiting reversible process. © M. Kostic <www.kostic.niu.edu> 2009 January 10-12 Entropy … … entropy of a system for a given state is the same, regardless whether it is reached by reversible heat transfer or irreversible heat or irreversible work transfer. However, the source entropy will decrease to a smaller extent over higher potential, thus resulting in overall entropy generation for the two interacting systems. © M. Kostic <www.kostic.niu.edu> 041115 (a) T 2S W12= Q12 SG=DSS -DSR=0 Could be reversed 1R 2R 2S 1S 1S SS S DSS R (b) T 2S W12= 0 Unrestricted expansion SG=DSS>0 Could NOT be reversed 1S 2S 1S S S Q12=0 DSS © M. Kostic <www.kostic.niu.edu> For eample: reversible expansion at DSR Q12>0 It is possible to obtain work from the equal amount of disorganized thermal energy or heat, if such process is reversible. constant internal energy, e.g. isothermal ideal-gas expansion, (dW=dQ), see Fig. 1a, and reversible adiabatic expansion (dW=-dU). Work potential is lost during unrestricted expansion (Fig. 1b) 041115 Heat Transfer and Irreversibility: T 2R (a) 1R DT > 0 (b) Multiple Heat Reservoirs with DT 0 2S ENTROPY RANSFER AND GENERATION 1R 2S 1S 2R SG =DSS - DSR > 0 Irreversible SG =DSS - DSR = 0 Could be reversed 1S S S SG DSR DSR DSS DSS (c) Variable T Reservoir with DT 0 1R 2S 2R SG =DSS - DSR = 0 Could be reversed 1S S DSR DSS © M. Kostic <www.kostic.niu.edu> 041115 Entropy … We could consider a system internal thermal energy and entropy, as being accumulated from absolute zero level, by disorganization of organized or higher level energy potential with the corresponding entropy generation. Thus entropy as system property is associated with its thermal energy. © M. Kostic <www.kostic.niu.edu> 041115 Entropy Primer: entropy could be transferred in reversible processes along with heat transfer, and additionally generated if any work potential (including thermal energy’s) are disorganized at the lower thermal potential during irreversible processes. Once a process completes, any generated entropy due to irreversibility becomes (permanent) system property and cannot be reversed/destroyed by any process in nature (thus, a permanent change). © M. Kostic <www.kostic.niu.edu> 041115 Entropy Primer (2): Thus, entropy transfer is due to reversible heat transfer and could be ether positive or negative, while entropy generation is always positive and always due to irreversibility. © M. Kostic <www.kostic.niu.edu> 041115 T 1H 2H TH PC TL CPC 2L 1L HPC QL=TLDSL=THDSH= QH T0 Reversible Heat Transfer and Practical Potentials: Dual Power-Heat Pump cycle SG=DSL-DSH>0 Irreversible SG DSH (a) S DSL T 1H 2H TH TL Eq . (1) Q H TH D S H Eq . ( 2 ) Q L TL D S L TL (D S H D S 0 ) Eq . ( 3 ) SG=0 Power Cycle Reversible Heat Transfer T C (T H T L ) D S H (T L T 0 ) D S 0 2L 2L′ 1L T Heat Pump Cycle C COP PHP T0 WPC =WHPC DSH Saved Energy QH TL TH 5 500 % S DS0 (b)Kostic D<www.kostic.niu.edu> DS′′L S′L © M. QL 041115 T H T0 T L T0 350 1050 1050 300 350 300 Eq .( 5 ) Coefficients of Performance for Three Typical Cases of Reversible Heat Transfer TABLE I: COEFFICIENTS OF PERFORMANCE FOR THREE TYPICAL CASES OF REVERSIBLE HEAT TRANSFER REVERSIBLE HEAT TRANSFER TYPE Heating from higher temperature source: Dual Power-Heat Pump Cycle (introduced here) COEFFICIENT OF PERFORMANCE for TH> TL> T0> TR QL COP PHP Cooling: Refrigeration or AirConditioning COP R Heating from lower temperature source: Heat Pump COP HP © M. Kostic <www.kostic.niu.edu> QH QR W QH W TL T H T0 TH T L T0 Eq. (4) TR Eq. (6) T0 T R TH T H T0 Eq. (7) 041115 the most efficient reversible heat transfer from system H at higher temperature TH to system L at lower temperature TL as presented on Fig. 3b may be obtained (as limiting case) by using a dual power-and-heat-pump cycle (PHP), which is governed by the following conditions (WPC = WHPC) Conclusion Energy is a fundamental concept indivisible from matter and space, and energy exchanges or transfers are associated with all processes (or changes), thus indivisible from time. Energy is “the building block” and fundamental property of matter and space, thus fundamental property of existence. For a given matter (system) and space (volume) energy defines the system equilibrium state, and vice versa. For a given system state (structure and phase) addition of energy will tend (spontaneously) to randomly distribute (disorganize) over the system microstructure and space it occupies, called internal thermal energy, increasing energy-potential (temperature) and/or energy-displacement (entropy), and vice versa. © M. Kostic <www.kostic.niu.edu> 041115 Conclusion (2): Energy and mass are conserved within interacting systems (all of which may be considered as a combined isolated system not interacting with others surrounding systems), and energy transfer (in time) is irreversible (in one direction) from higher to lower potential only, which then results in continuous generation (increase) of energy-displacement, called entropy generation, which is a fundamental measure of irreversibility, or permanent changes, the latter also measured with the passing time. Reversible energy transfer is only possible as limiting case of irreversible energy transfer at infinitesimally small energypotential differences, thus in quasiequilibrium processes, with conservation of entropy. Since such changes are reversible, they are not permanent (could be reversed without leaving any relevant effect on the surroundings) and, along with time, irrelevant (NOT permanent). © M. Kostic <www.kostic.niu.edu> 041115 Conclusion (3): Entropy may be transferred from system to system by reversible heat transfer and also generated due to irreversibility of heat and work transfer. Heat transfer, like any other energy transfer, may be achieved from any-to-any temperature level (performed in real power and refrigeration cycles), and in limit be reversible, if temperature of an intermediary cyclic substance is adjusted as needed, using isentropic compression and expansion. The reversible heat transfer limits are the most efficient and demonstrate limiting potentials for practical heat transfer processes. © M. Kostic <www.kostic.niu.edu> 041115 Conclusion (4): The “Dual Power-Heat Pump Cycle,” introduced here, may be considered as a reversible heat-transfer transformer, from-any-to-any temperature levels. The simple analysis of this dual, combined cycle (Eq. 4. and Fig. 3b), to achieve reversible heat transfer only (from higher to lower temperature system) and without any network produced or utilized, Presented emphasis (with analysis) of underlying physical phenomena, including several hypothesis, is intended contribution of this paper. © M. Kostic <www.kostic.niu.edu> 041115 © M. Kostic <www.kostic.niu.edu> 041115 Compressed Liquid water enthalpy corrections: It is custom to approximate solid and liquid properties as being function of temperature only, since they are virtually incompressible: Pdv compression work may be neglected. For isothermal compression processes: correction for liquid enthalpy approximation “pumping” work, vdP Analysis of water real properties shows that such a correction is unnecessary for intermediate pressures and temperatures, and it is even erroneous for higher temperatures and pressures, and thus counterproductive and misleading. © M. Kostic <www.kostic.niu.edu> 041114 Compressed Liquid water enthalpy corrections (2): Solids and liquids are virtually incompressible, work, Pdv, could be neglected, and properties will not be function of pressure but temperature only: thus compression u u (T ) and c p c v c (T ) •Even enthalpy … h w / O ( P , T ) h f (T ) h sat (T ) However, enthalpy is unique, since it is explicitly defined as a function of pressure: h u P v thus, © M. Kostic <www.kostic.niu.edu> h (T , P ) u (T ) P v 041114 REAL fluid enthalpy CORRECTIONS In all engineering references, and Thermodynamics textbooks [1, 2]: dh d ( u P v ) du P dv v dP corr . A corr . B du vdP cdT vdP dv 0 corr . C h corr ( P , T ) h f (T ) v f (T ) ( P Psat ) or h corr ( P , T ) h f (T ) v f (T ) ( P Psat ) correction C correction C However, REAL fluid enthalpy CORRECTIONS are not only due to change of pressure (cor.C), but also to change of internal energy (cor.A) , and volume (cor.B): corr . A D u u ( P , T ) u ( Psat , T ) corr . B P D v P dv corr .C v D P v dP Dh Du v dP v ( P , T ) v ( Psat , T ) © M. Kostic <www.kostic.niu.edu> 2 h ( P , T ) h ( Psat , T ) ( corr . A ) ( corr .C ) ( P Psat ) 041114 Isothermal vs. Isentropic compression of sat. liquid water For isentropic compression (q=0): Du=q+wcomp= Pdv 0, increase of internal energy and temperature (for 12 oC, from 260 oC to 272 oC, see last row in Table I). To maintain constant temperature in isothermal compression, there must be some cooling (q out), thus internal energy decrease (corr.A). © M. Kostic <www.kostic.niu.edu> 041114 Recommended enthalpy correction in the literature for isothermal compression Recommended enthalpy correction in the literature is more appropriate for the isentropic than for isothermal processes, due to erroneous assumption that internal energy is not, and enthalpy is, dependent on pressure. It is exactly opposite in Table I, see how the corresponding values (u & h) change with pressure at constant temperature of 260 oC. © M. Kostic <www.kostic.niu.edu> 041114 Compressed Liquid water properties and relevant enthalpy corrections Corr.B small –may be neglected. BUT Corr.A & B are comparable and opposite sign. So, take both or none, since it is more erroneous to take one (corr.C) only ! © M. Kostic <www.kostic.niu.edu> 041114 Compressed liquid enthalpies at different temperatures and pressures more in error for higher temperature and pressure than the corresponding saturated values about the same for the intermediate temperatures © M. Kostic <www.kostic.niu.edu> 041114 Conclusion Recommendations in the literature for improvement of enthalpy calculation of compressed liquids, by accounting for pressure dependence, are not generally valid. Those recommendations may be erroneous and thus counterproductive and misleading, as is the case for liquid water at higher temperatures and pressures. For intermediate pressures and temperatures, the recommended enthalpy corrections are unnecessary, since the errors are about the same in magnitude (but opposite in sign) as the corresponding saturated enthalpy values without any corrections. The recommended enthalpy corrections are only useful for smaller temperatures and pressures. © M. Kostic <www.kostic.niu.edu> 041114 The Concept of "Entransy" May Be More Important Than What It Appears at First … but it has to be "properly" related to existing concepts of Thermal energy (not precisely defined yet, see elsewhere), Exergy and Entropy, as well as irreversibility and reversibility. Entransy concept and analysis have some unique advantages over other approaches. There is a need to define Entransy as a property (how it relates to other thermodynamic properties) and as process energy flux (how it relates to heat & work transfer and entropy transfer & generation). We also could advance and synergize your "Thermomass" concept with my work in that area. www.kostic.niu.edu Stretching the mind further … Mass may be a special tensor-like quantity due to "over-allisotropic in all-directions" motion of elementary particles (that make up its structure) and thus give rise to inertia if accelerated in any direction, i.e., resisting change of motion in any and all directions with equal components (the isotropic mass inertia). There may be anisotropic masses, with bulk linear or rotational motion, being the extreme cases. Note that fundamental particles (without inertial mass, like photons and similar, but with relativistic masses E/c^2) has to always move with ultimate speed of light in vacuum, and such particles (some yet to be discovered) might be moving (orbiting with twisting, stringlike vibration and rotation) within virtually infinitesimal spaces and thus making-up other "massive" so-called elementary particles www.kostic.niu.edu Deterministic vs. Probabilistic All interactions in nature are physical and based on simple cause-and-effect conservation laws, thus deterministic and should be without any exceptional phenomenon. Due to diversity and complexity of large systems, we would never be able to observe deterministic phenomena with full details but have to use holistic and probabilistic approach for observation; therefore, our observation methodology is holistic and probabilistic, but phenomena have to be deterministic, not miraculous nor probabilistic www.kostic.niu.edu Elementary Particles: Electron? There is no proof that an electron, or any other elementary particle, has or does not have a structure. The concept of elementary particle is intrinsically problematic (just because we cannot observe or reason a structure which exhibits certain phenomena, does not mean it does not exist). Past and recent history proved us to be wrong every time. Particularly problematic is the current theory which requires elementary particle annihilation/creation (“miraculous creationism”) while using conservation laws. At the very least (in phenomenological view) the elementary particles should be conserved and be the building structure for other particles and systems. Note that many concepts (in modern physics) are "virtual" entities that are part of the mathematical theory, but are not directly observable. www.kostic.niu.edu Boundary Forces … There is no such thing as a unidirectional force or a force that acts on only one body (no imaginary boundary vectorforces). Put it very simply: a forcing (force-flux cause-and-effect phenomena) acts between an interface of pair of objects (forced interaction: action-reaction, including process-inertial forces), and not on a single object. The Newton Laws and the Laws of Thermodynamics imply that all forces are massenergy interactions (forced displacements with momentum and energy transfer and conservation) between different particulate bodies due to non-equilibrium (available energy or work potential, cause of forcing) towards the equilibrium. www.kostic.niu.edu No Perfect Rigidity … All matter must be somewhat elastic (can be compressed or stretched). If bodies could be perfectly rigid we'd have infinite forces acting with infinite speeds for infinitesimal times (if you pushed on one end of a perfectly rigid stick, the other end would move instantaneously). System components (bodies) that exert forces have to be massive (2nd Newton Law) and with accompanying reaction forces (3rd Newton Law). www.kostic.niu.edu Energy is bound by forced motion … Energy is possessed (thus equilibrium property) by material systems and redistributed (transferred) between and within system(s), due to systems' non-equilibrium, via forceddisplacement interactions (process) towards the equilibrium (equi-partition of energy over mass and space); thus energy is conserved (the 1st Law) but degraded (the 2nd Law). Effects are consequences of Causes except at Equilibrium they are equal (reversible). The existence in space and transformations in time are manifestations of perpetual mass-energy forced displacement processes: with net-zero mass-energy transfer in equilibrium (equilibrium process) and non-zero mass-energy transfer in nonequilibrium (active process) towards equilibrium. System components (particles and bodies) that exert forces have to be massive (2nd Newton Law) and with accompanying reaction forces (3rd Newton Law). www.kostic.niu.edu Force and Forcing … Force or Forcing is a process of exchanging useful-energy (forced displacement) with netzero exchange at forced equilibrium. The Second Law provides conditions and limits for process forcing (energy exchange direction www.kostic.niu.edu Processes … Miracles "Nothing occurs locally nor in the universe without mass-energy exchange/conversion and irreversible entropy production. It is crystal-clear (to me) that all confusions related to the far-reaching fundamental Laws of Thermodynamics, and especially the Second Law (Abstract), are due to the lack of their genuine and subtle comprehension." The miracles are until they are comprehended and understood. www.kostic.niu.edu For further Info you may contact Prof. Kostic at: [email protected] or on the Web: www.kostic.niu.edu Prof. M. Kostic Mechanical Engineering NORTHERN ILLINOIS UNIVERSITY © M. Kostic <www.kostic.niu.edu> 041115