OPTIMIZATION OF THE IRREVERSIBLE DIESEL CYCLE USING FINITE SPEED THERMODINAMICS AND THE DIRECT METHOD

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1 Bulletn of the Translvana Unversty of Braşov Vol. (51) - 9 Seres I: Engneerng Scences OPTIMIZATION OF THE IRREVERSIBLE DIESEL CYCLE USING FINITE SPEED THERMODINAMICS AND THE DIRECT METHOD S. PETRESCU 1 N. BORIARU 1 M. COSTEA 1 C. PETRE 1 A. STEFAN 1 C. IRIMIA 1 Abstract: The paper presents a Drect Method Approach n estmatng the performances of the rreversble Desel cycle wth fnte speed. The rreversbltes generated by the fnte speed of the processes, throttlng and frcton were taken nto account. The effcency and the power output of the Desel cycle were compared for several workng gases, whle studyng the nfluence of the pston speed and volume ncrease rato at combuston. Optmum values, as well as lmt values of these varables resulted. Key words: Desel Cycle Performances, Fnte Speed Thermodynamcs, Drect Method, Irreversblty Causes. 1. Introducton Equlbrum processes can be studed usng the Classcal Reversble Thermodynamcs. However, the equlbrum condtons restran the study to the reversble approach, whch means to processes that occur ether n nfnte tme, or wth the speed tendng to zero. In realty, thermal processes are developng n fnte tme [1-3] or very fast, ther speed (w) determnng the gap between reversble and real rreversble processes [4-16]. The objectve of ths paper s to study the performances (Effcency and Power) of the Irreversble Desel Cycle, n the framework of the Irreversble Thermodynamcs wth Fnte Speed. The reversblty factors taken nto account were: - The Fnte Speed of the Pston (FSIT) durng compresson and expanson strokes; - The Mechancal Frcton between the pston and the cylnder, generatng pston frcton pressure losses (PFPL) all around the cycle; - The Flud Frcton due to Throttlng of the Gases durng ntake and exhaust strokes, generatng throttlng pressure losses (THPL).. Irreversble Processes Approach by the Drect Method n Thermodynamcs wth Fnte Speed Several models for determnng the mathematcal expresson of the Irreversble Mechancal ork n Fnte Speed Processes were developed, startng wth the knetc-molecular explanaton model ntroduced by. Macke [17] and A. Sommerfeld [18], and also usng the 1 Dept. of Engneerng Thermodynamcs, Poltehnca Unversty of Bucharest.

2 88 Bulletn of the Translvana Unversty of Braşov Vol. (51) - 9 Seres I Onsager-Prgogne Lnear Irreversble Phenomenologcal Thermodynamcs (LIPT) [19,]. a) S. Petrescu and L. Stocescu [9,11] have got the frst expresson of the pressure on a movng pston (p p ) wth fnte speed (w): aw bw p p = p +... (1) c c where: p - the nstantaneous mean pressure n the gas; a and b - coeffcents, a =, b = 5; c = 3 m/s] - the average speed of R T the molecules computed at nstantaneous mean temperature, T. The sgn (+) s for compresson and the sgn (-) s for expanson processes. R s the constant of the gas. By substtutng equaton (1) n the expresson of the elementary work (p p dv) the followng relatonshp for rreversble fnte speed work has been obtaned: aw bw δ rr = p +... dv = p p dv c c () Equaton () was the frst mathematcal expresson obtaned for Irreversble ork n Processes wth Fnte Speed, for smple closed systems (pston machnes - lke nternal combuston engnes and pston compressors). b) The Advanced Knetc-Molecular Model [9,11,1] based on the Maxwell- Boltzmann dstrbuton of the molecules speed and the fnte relaxaton tme n the syste led to the expresson: 3 w w w δ rr = p dv 3 c c c (3) Ths was the second mathematcal expresson for the Irreversble ork n Processes wth Fnte Speed c) The Phenomenologcal Model of the nteracton between pston and gas [9, 19,, ] based on the hypothess of pressure wave propagaton and ther relaxaton generated by the Fnte Speed of the pston and the speed of the sound n the gas [1,3], led to: aw δ rr = p dv (4) c where: a = 3 and s the specfc heat rato. Equaton (4) can also be wrtten as a functon of the speed of sound, c S = RT m,, nstead of the average speed of the molecules, c, leadng to: w δrr = pm dv c, (5) S d) The Onsager-Prgogne [19,] Lnear Irreversble Phenomenologcal Thermodynamcs (LIPT) model [15], appled to fnte speed nteractons between the gas and pston, led to the expresson [9,1]: ( K w) dv δrr = p 1 1 (6) Here K l s a constant (called the Phenomenologcal Coeffcent) dependng on the propertes of the gas and the nstantaneous mean temperature of the gas, T. Comparng equaton (6) wth equaton (), K l can be expresses as a functon of R and T : Kl = [s/m] (7) 3RT

3 S. PETRESCU et al: Optmzaton of Irreversble Desel Cycle by Fnte Speed and Drect 89 By consequence, when the expresson of the rreversble mechancal work eq. () s replaced n the Frst Law of Thermodynamcs ( du = δq δrr ), the obtaned mathematcal expresson becomes a combnaton of the Frst and Second Laws of Thermodynamcs for Fnte Speed Processes n closed systems [9-1]: a w bw du = δq p +... dv c c (8) In ths expresson (8) there are dfferent sgns for compresson (+) and expanson (-) as a consequence of the prmarly cause of rreversblty n the processes wth fnte speed. e) The other two causes of rrevesbltes (frcton and throttlng), as manfestatons of the Second law of Thermodynamcs n real rreversble processes, specfc to pston machnes, were smlarly expressed n a Generalzed Equaton for Irreversble ork [1,13]: δ rr = p a w p p th f + b c p p dv (9) wth p th - the pressure drop due to gas throttlng at ntake and exhaust p f - the effect of mechancal frcton between pston and cylnder walls. th equaton (9) ntroduced n the Frst Law we get a General Equaton for Irreversble Processes n closed systems generated by Fnte Speed, Frcton and Throttlng: du = δq p aw p p f th + b f c p p dv (1) where f s the fracton of heat generated by frcton that remans n the workng gas. The last mathematcal expresson combnes the Frst and Second Laws of Thermodynamcs. It contans the man 3 causes of nternal rreversblty (Fnte Speed, Frcton and Throttlng) for any pston-cylnder real machne [1,13]. So, we call equaton (1) The Fundamental Equaton of Thermodynamcs wth Fnte Speed. 3. Irreversble Processes Approach by Fnte Speed Thermodynamcs To study the fnte speed process, both knds of pressure are necessary (nstead of one for the classcal reversble process). Pressure (p) Fg. 1. Irreversble ork at Compresson Pressure (p) p m p p p m p p Volume (V) Volume (V) Fg.. Irreversble ork at Expanson

4 9 Bulletn of the Translvana Unversty of Braşov Vol. (51) - 9 Seres I Ther evoluton durng compresson and expanson are descrbed n p-v coordnates by the curves [19,] shown n Fgure 1 and. Durng compresson, the pressure on the pston wll be hgher than the pressure at any other pont n the system (reachng ts mnmum on the cylnder head). Durng expanson, the opposte happens, namely the pressure on the pston wll be less than n any other pont of the system (reachng ts maxmum on the cylnder head). In the rreversble thermodynamcs, the process work s equal wth the area under the pressure on the pston (p p ) curve, n p-v dagram. 4. Adabatc Processes wth Fnte Speed The equatons of the rreversble adabatc (δq=) fnte speed compresson (+) and expanson (-) processes were obtaned [5,9,14-16] by analytcal ntegraton of equaton (8): V1 = T 1 c1 T c V 1 (11) = p 1 1 V1 p 1 V (1) c1 c 1 1 T 1 c1 = 1 p1 T c p 1 (13) th these equatons the Drect Method s used n order to obtan the performances (Effcency and Power) of Desel rreversble cycle wth Fnte Speed. 5. ork Loss due to FSIT The work loss due to FSIT s gven by the dfference between the work for the reversble cycle ( cycle r ) and that for the rreversble one ( cycle,r ): cycle, loss, FSIT cycle, r cycl, rr = [J] (14) 6. ork Loss due to the Throttlng Throttlng occurs when a flud passes through small orfces lke valves. The correspondng loss of pressure durng ntake and exhaust reduces the actual cycle work. The total throttlng pressure loss per cycle s estmated from [13]: p w THPL, cycle =.45 [bar] (15) 1 St where St s the pston stroke. The rreversble work loss due to throttlng s determned multplyng equaton (15) by the dsplacement volume (V S ): THPL, cycle = pthpl, cycle VS [J] (16) 7. ork Loss due to the Frcton The frcton generated heat durng the relatve moton of the pston nsde the cylnder s another mportant source of rreversblty. It ncreases gas temperature and pressure, as compared to the reversble process. The total frcton pressure loss per cycle s estmated from [3]: ( w) ppfpl, cycle = 45 [bar] (17) whch multpled by the dsplacement volume (V S ), gves the rreversble frcton work loss: PFPL, cycle = ppfpl, cycle VS [J] (18)

5 S. PETRESCU et al: Optmzaton of Irreversble Desel Cycle by Fnte Speed and Drect Cycle Effcency The Desel cycle effcences could be determned takng nto account, step by step, dfferent knds of rreversbltes. The followng relatons have been used: - Reversble Cycle Effcency (no rreversblty) cycle, r η r = (19) Q n - FSIT Irreversble Cycle Effcency cycle, rr rr, FSIT = Qn η () - FSIT & THPL Irreversble Cycle Effcency cycle, rr cycle, loss, THPL η rr, FSIT& THPL = (1) Q - Actual Irreversble Cycle (FSIT & THPL & PFPL) Effcency: cycle, rr, act rr, FSIT = Qn η () where the Irreversble Cycle Effectve ork s: cycle, rr, act, = cycle, r cycle, loss FSIT [J] (3) cycle, loss, THPL cycle, loss, PFPL 9. Results and Conclusons In ths work the engne s consdered as adabatcally nsulated, n accordance wth the condtons offered by the ceramc engne, a hgh effcency energetc soluton under ntensve research at present. Usng the equatons for rreversble processes prevously deduced wthn the n framework of the Fnte Speed Thermodynamcs, the processes composng the Desel cycle are covered successvely, determnng the work () and the heat exchange (Q) for each one, fnally gettng to the performances: the Effcency (η) and the produced Power (P). Power [k] w = 1 m/s w = m/s w = 3 m/s w = 4 m/s Volume ncrease rato at combuston, λ Fg. 3. Power versus volume ncrease rato at combuston, for ar, at dfferent mean pston speeds Effcency [-] w = m/s w = 1 m/s w = m/s w = 3 m/s w = 4 m/s Volume ncrease rato at combuston, λ Fg.4. Effcency versus volume ncrease rato at combuston, at dfferent mean pston speeds (ar). Fgures 3 and 4 are showng the nfluence of the volume ncrease rato at combuston (λ) on the power output and, respectvely, on the effcency of the

6 9 Bulletn of the Translvana Unversty of Braşov Vol. (51) - 9 Seres I rreversble Desel cycle, takng nto account the work loss due to the fnte speed of the pston alone. hle the power grows wth the volume ncrease rato (λ), the nfluence of the mean pston speed s opposte (see Fgure 3). The effcency (η) - volume ncrease rato (λ) curve (Fgure 4) has a dfferent shape: for the studed pston speeds >, t shows a maxmum pont, wth a sudden decrease towards lower values of λ. Another mportant observaton s that there s a certan value of λ under whch the produced work cannot cover the rreversblty losses ( λ mn 1.3 for ar at w = 3 m/s, for nstance). Takng nto account step by step, dfferent knds of rreversblty (Fgure 5), the maxmum of the effcency mgrates towards hgher values of λ, as well as the extncton pont (λ mn ). In the rreversble approach, the powerpston speed curve (Fgure 6) s a parabola, too. By consequence, the same power can be reached at dfferent pston speeds (wth lower effcency at the hgher pston speed). Takng nto account progressvely dfferent knds of rreversblty (Fgure 6), the maxmum of the power curve, s E ffcen cy [-] Effcency-Speed nfluence Effcency-Speed&Frcton nfluence Effcency-All nfluences Power-Speed nfluence Power-Speed&Frcton nfluence Power-All nfluences decreasng, as well as the value of the correspondng optmum speed. Ths dagram (Fgure 6) reveals also that there s a hgher lmt correspondng to the maxmum affordable speed over whch the engne couldn t cover anymore the work loss due to rreversblty. A smlar nfluence of the pston speed on the power and the effcency of the Desel engne have been obtaned for He, CO and H (see Fgures 7 and 8). Effcency [-] Reversble process Speed nfluence Speed and frcton nfluence Speed, frcton and throttlng Volume ncrease rato at combuston, λ Fg. 5. Effcency versus volume ncrease rato at combuston for ar at w = m/s P o w er [k ] Speed [m/s] Fg. 6. Pston speeds nfluence on power & effcency of a Desel (ar, λ = 1.8)

7 S. PETRESCU et al: Optmzaton of Irreversble Desel Cycle by Fnte Speed and Drect 93 Power [k] Ar Power All H Power All CO Power All He Power All parameters (temperatures, speed, compresson rato, dmensons, etc.) on the engne performances and gvng the opportunty for optmzng the workng cycle wth respect to these parameters [1-16]. It has also the power to compute the Internal Source of Entropy and correlate t wth the Performance Parameters of the Cycle Speed [m/s] Fg. 7. Power output versus mean pston speed for dfferent workng gases (all three rreversblty causes) Effcency [-] Ar Effcency All C Effcency All H Effcency All He Effcency All Speed [m/s] Fg. 8. Effcency versus mean pston speed for dfferent workng gases (all three rreversblty causes) The same behavor has been observed at dfferent values of λ = For the same λ, each workng gas has ts own optmum speed, correspondng to the maxmum of the power output (Fgure 7), as well as a maxmum (unsurpassable) workng speed. Ths knd of analyss gves to the desgner the means to mprove the engne performances by nsstng on the reducton of ts most sgnfcant work losses. It opens, also, the way towards senstvty studes revealng the nfluence of dfferent References 1. Petrescu, S., Harman, C., Costea, M., Fedt, M.: Thermodynamcs wth Fnte Speed versus Thermodynamcs n Fnte Tme n the Optmzaton of Carnot Cycle. In: Proceedngs of the 6- th ASME-JSME Thermal Eng. Jont Conference, Hawa, 3.. Andersen, B.: Fnte-Tme Thermodynamcs, Physcs Laboratory II, Unversty of Copenhagen, Fedt, M.: Thermodynamque et optmsaton énergétque. In : Technque et Documentaton Lavoser, Pars, Petrescu, S., Costea, M., Harman, C., Florea, T.: Applcaton of the Drect Method to Irreversble Strlng Cycles wth Fnte Speed. In: Internatonal Journal of Energy Research, 6 (), p Petrescu, S., Harman, C., Costea, M., Florea, T., Petre, C.: Advanced Energy Converson, Bucknell Unversty, Lewsburg, Petrescu, S., Stănescu, G., Costea, M.: A Drect Method for the Optmzaton of Irreversble Cycles usng a New Expresson for the Frst Law of Thermodynamcs for Processes wth Fnte Speed. In: Proc. of the 1st Int. Conf. on Energy ITEC'93, Marrakech, 1993, p Petrescu, S., Fedt, M., Harman, C., Costea, M.: Optmzaton of the Irreversble Carnot Cycle Engne for

8 94 Bulletn of the Translvana Unversty of Braşov Vol. (51) - 9 Seres I Maxmum Effcency and Maxmum Power through Use of Fnte Speed Thermodynamc Analyss, In: Proceedngs of the ECOS Conference, Berln,, Vol. II, p Stocescu, L., Petrescu, S.: Thermodynamc Cycles wth Fnte Speed, In: Polytechnc Insttute of Bucharest Bulletn XXVII (1965) No., p Petrescu, S.: Contrbutons to the study of nteractons and processes of nonequlbrum thermal machnes. In: Ph.D. Thess, Polytechnc Insttute of Bucharest, Petrescu, S.: An Expresson for ork n Processes wth Fnte Speed Based on Lnear Irreversble Thermo-dynamcs, In: Stud ş Cercetăr de Energetcă ş Electrotehncă, 19 (1969) No., Romanan Academy, p Stocescu, L., Petrescu, S.: The Frst Law of Thermodynamcs for Processes wth Fnte Speed, n Closed. In: Polytechnc Insttute of Bucharest Bulletn, XXVI (1964), No. 5, p Petrescu, S., Harman, C.: The Connecton between the Frst and Second Law of Thermodynamcs for Processes wth Fnte Speed. A Drect Method for Approachng and Optmzaton of Irreversble Processes, In: Journal of the Heat Transfer of Socety of Japan, (1994), p Petrescu, S., Stănescu, G., Iordache, R., Dobrovcescu, A.: The Frst Law of Thermodynamcs for Closed Systems, Consderng the Irreversbltes Generated by the Frcton Pston- Cylnder, the Throttlng of the orkng Medum and Fnte Speed of the Mechancal Interacton. In: ECOS 9 Conference, Zaragoza, Span, 199, p Stocescu, L., Petrescu, S.: Expermental verfcaton of the new expresson of the Frst Law of Thermodynamcs for processes wth fnte speed. In: Polytechnc Insttute of Bucharest Bulletn, XXVII (1965) No., p Stocescu, L., Petrescu, S.: Thermodynamc Processes wth Constant Fnte Speed. In: Polytechnc Insttute of Bucharest Bulletn, XXVI (1964) No. 6, p Stocescu, L., Petrescu, S.: Thermodynamc Processes wth Varable Fnte Speed. In: Polytechnc Insttute of Bucharest Bulletn, XXVII (1965) No. 1, p Macke,.: Thermodynamk und Statstk. Lepzg, Sommerfeld, A., Thermodynamcs and Statstcal Mechancs. Vol. 5, New York. Academc Press, Onsager, L.: Phys. Rev. 37 (1931), p. 45; 38 (1938), p Prgogne, I.: Introducton to thermodynamcs of rreversble processes. Charles Thomas, Publsher, Sprngfeld, IL, Petrescu, S.: Knetcal consderatons regardng the pressure on a pston wth fnte speed. In: Stud ş Cercetăr de Energetcă ş Electrotehncă 1 (1971) No. 1, Romanan Academy, p Carnot, S.: Reflectons sur la pussance motrce du feu et sur les machnes propres à développer cette pussance. Pars Heywood, J.B.: Internal Combuston Engne Fundamentals. New York. McGraw-Hll Book Company, 1988.