TRANSIENT HEAT CONDUCTION IN A TWO-STROKE DIESEL ENGINE PISTON

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1 TRANSIENT HEAT CONDUCTION IN A TWO-STROKE DIESEL ENGINE PISTON Leonardo F. Saker Marcelo J. Colaço Helco R.B. Orlande Programa de Engenhara Mecânca, EE / COPPE / UFRJ - Caxa Postal 6853, Ro de Janero - RJ - Brasl Abstract: In ths paper we study the transent heat conducton n a pston of a desel engne, subjected to a perodc boundary condton on the surface n contact wth the combuston gases. An ellptc scheme of numercal grd generaton was used, so that the rregular shaped pston n the physcal doman was transformed nto a cylnder n a computatonal doman. The tmewse varatons of the temperature of several ponts n the pston are examned for dfferent pston materals, as well as for motored and fred engnes. Keywords: Desel engne, fnte-dfferences, numercal grd generaton INTRODUCTION The soluton of heat transfer problems n nternal combuston engnes s very complcated for several reasons, ncludng, among others: the cyclc temperature varaton of gases nsde the engne; the parts nvolved, such as pstons, do not have a regular shape; such parts are subjected to dfferent heat transfer coeffcents from the top, bottom and lateral sdes, whch may vary durng the cycle; and the estmaton of heat transfer coeffcents consttute, n tself, a problem. A revew of avalable theoretcal and expermental works on the subject was presented by Borman and Nshwak (1987). In the present paper we perform a two-dmensonal axally symmetrc fnte-dfference analyss of the transent heat conducton n a pston of a two-stroke desel engne. For such an analyss, we transformed the rregular shaped pston from the physcal doman nto a cylnder n a computatonal doman. The transent heat conducton equaton was transformed nto the computatonal doman, where t was solved wth fnte-dfferences by usng the ADI (Alternatng Drecton Implct) method (Peaceman and Rachford, 1955). The computer code used n ths work s an mprovement over another code developed by our group n the past for the soluton of a smlar problem (Colaço and Orlande, 1996). The problem addressed by Colaço and Orlande (1996) also nvolved the transent analyss of a desel engne pston; but consdered for the computatons tme-averaged values for the heat transfer coeffcent between the gas and the pston surface, as well as for the temperature of the gas nsde the cylnder. In the present work we consder the perodc varatons of the heat transfer coeffcent between the gas and the pston surface and of the temperature of the gas nsde the cylnder, for the computaton of the transent temperature feld nsde the pston. In ths work we used the correlaton of Echelberg (Borman and Nshwak, 1987; Prasad and Samra, 199) due to ts smplcty and because t does

2 not nvolve many emprcal constants. The temperature varaton of the gas nsde the cylnder was computed by usng a double -Webe functon for the heat release durng combuston (Ramos, 1989). PHYSICAL PROBLEM The physcal problem consdered here s the transent heat conducton n a desel engne pston. The pston s assumed to be ax-symmetrc, so that asymmetres due to the pston pn and ol coolng channels are neglected. The pston consdered n the present work s the same studed by Prasad and Samra (199). The pston geometry wth coordnates (n mllmeters) relevant for ths study are presented n fgure 1: Fgure 1. Geometry and coordnates The pston s heated through ts top surface by the gas nsde the combuston chamber. The gas temperature (T gas ) and the heat transfer coeffcent between gases and pston (h gas ) are assumed to vary wthn each engne cycle. The pston s cooled by ol on ts botom surfaces and by a coolant flud flowng through passages n the cylnder wall. The ol temperature (T ol ), as well as the heat transfer coeffcent between ol and pston (h ol ) are supposed to be constant. The heat transfer to the coolant flud s taken care by usng a constant overall heat transfer coeffcent (h ), whch takes nto account the heat transfer from the pston to the cylnder wall, conducton through the wall, and convecton from the wall to the coolant flud. The flud temperature (T ) s assumed to be constant. The mathematcal formulaton of such physcal problem s gven by: 1 α* T( r, t ) t T K = n h 2 = T( r, t ) [ T t ] ( ) ( ) t T n the regon, for t > on the boundary surface T r = on the symmetry axs (r = ), for t > T T ( r ) (1.a) Γ, for t > (1.b) (1.c) = for t = n the regon (1.d) where h, T and T n are, respectvely, the heat transfer coeffcent, the flud temperature and the normal dervatve of temperature at each of the boundary surfaces Γ. α* and k are the thermal dffusvty and thermal conductvty, respectvely. ANALYSIS

3 The dscretzaton of the pston presented n the fgure 1 s dffcult due to ts rregular shape. In order to overcome such dffculty, we neglected the effects of the pston rngs and transformed the rregular pston n the physcal doman (z,r) nto a cylnder n the computatonal doman (,η), as shown n fgure 2. (2) PHYSICAL DOMAIN (3) (4) (3) COMPUTATIONAL DOMAIN (4) r (5) (1) (6) (1).5.1 Z (7) Fgure 2. Physcal and computatonal doman (6) In fgure 2, M and N are the number of lnes of and η varables, respectvely. The transformaton above s defned by the soluton of two ellptc partal dfferental equatons, used to generate the fnte-dfference grd for the pston (Thompson et al, 1985; Malska, 1995; Özsk, 1994). The problem gven by Eqs. (1) s transformed nto the computatonal doman (,η), where t s solved for the temperatures T(,η,t). In the computatonal doman, problem (1) takes the form: 1 T α* (, η,t ) 1 [ ] [ ] = αt 2βT +γt + PT + QT t 2 J η ηη k ( αt βtη ) = hgas (t)[t Tgas (t)] J α k ( αt βt η ) = h ol (T T ol ) J α η at = 1;1 < η < N; t at = M;1 < η > < N; t > ( γ Tη βt ) = at η = 1; 1 < < M; t > (2.a) (2.b) (2.c) (2.d) k ( γtη βt ) = h (T T ) at η = N;1 < < M; t > (2.e) J γ T = T (, η) where 2 2 α = η + rη for t = ;1 < < M;1 < η < N 2 2 z, β = z zη + r rη, γ = z + r, J = z r η + r z η (2.f) (3.a,b,c,d)

4 BOUNDARY CONDITION AT THE PISTON-GAS INTERFACE Several correlatons for the heat transfer coeffcent at the gas-pston surface are avalable n the lterature (Heywood, 1988; Ramos, 1989; Borman and Nshwak, 1987). In ths paper, we preferred to use the correlaton of Echelberg, due to ts smplcty and because t does not nvolve many emprcal constants. Such correlaton was developed for naturally-asprated large two-stroke and four-stroke desel engnes, such as the one under pcture n ths work. Echelberg s correlaton s gven by h gas ( t) = [P (t) T (t)] 1 (c ) kw / m K (4) where c m s the mean-pston-speed n m/s, whle T and P are the nstantaneous temperature n Kelvn and pressure n kpa, respectvely, of the gas nsde the cylnder. For the calculaton of the nstantaneous temperature and pressure nsde the cylnder, we assume that the compresson and expanson processes are polytropc, wth polytropc ndex of 1.3 (Ferguson, 1986). The mass of gas nsde the cylnder s supposed constant and t s assumed to be at atmospherc condtons (P = 1 5 Pa and T = 298 K) when the pston s at the bottom-deadcenter. Snce we are dealng wth a two-stroke engne, we assume here that exhauston and admsson take place smultaneously at the bottom-dead-center, so that, at the end of the expanson stroke, the gas nsde the cylnder returns nstantaneously to the ntal atmospherc condtons. Durng combuston, the relaton between the cylnder volume, gas pressure and the rate of heat release can be expressed as (Ferguson, 1986): dp d? m P dv (? 1) dq =? + (5) V d? V d? where θ s the crankshaft angle n degrees. The rate of heat release durng combuston can be obtaned from a double-webe functon n the form (Ramos,1989) dq Q = 6.9 dθ θ p p θ θ θp p g g ( ) Q d M + 1 exp ( M + 1) p θ θ θp M + 1 M 1 θ d d θ θ θd g θ θ exp 6.9 θd where the subscrpts p and d refer to premxed and dffusve combuston, respectvely; Mp and Md are shape factors correspondng to premxed and dffusve combuston, respectvely; θ p and θ d are the duratons of the energy release n premxed and dffusve combuston, respectvely; and Q p and Q d characterze the heat release n premxed and dffusve combuston, respectvely; θ g s the gnton angle. Such parameters are functons of the njecton angle and can be obtaned n Ramos (1989). After the fuel s njected nto the cylnder, several physcal and chemcal phenomena take place before combuston can start. Such phenomena result on a delay tme that can be represented n the form of an Arrhenus-type expresson such as ( T T( K) ) n t ( ms) = A p( atm) exp (7) d a g d + (6)

5 For the case under pcture n ths work, we used the followng values for the constants appearng n equaton (7), obtaned from Ramos (1989): A=53.5, n=1.23, T a = K. RESULTS AND DISCUSSION For the results presented below, the values of varous parameters were chosen as follows (Prasad and Samra, 199):() Intal temperature: T o =2 o C; () Ol temperature: T ol =85 o C; () Heat transfer coeffcent to the ol: h ol =175W/m 2o C; (v) Coolng water temperature: 85 o C; (v) Overall heat transfer coeffcent to the coolng water: h = 1 W/m 2o C; (v) Engne Speed: 85 RPM; (v) Compresson rato: 17; (v) Pston dameter:.23 m; (x) Stroke:.3 m; (x) Injecton angle: -2 o. Four test-cases were examned n the present work, dependng on the pston materal and f the engne s motored or fred, as summarzed n table 1. For the case of a motored engne, the compresson and expanson processes were also assumed polytropc, wth polytropc ndex of 1.3. Table 1. Test-cases Test-case Pston materal Engne condton 1 Alumnum Fred 2 Alumnum Motored 3 Cast ron Fred 4 Cast ron Motored The thermal conductvty and thermal dffusvty of alumnum and cast ron were taken, respectvely as, 24 W/m o C, x 1-5 m 2 /s, 54 W/m o C and.97 x 1-5 m 2 /s (Ozsk, 1993). Before obtanng results for the pston transent temperature feld by usng the present numercal approach, a grd convergence analyss s requred n order to assess the numercal error nvolved n the soluton. Nne dfferent grds were generated. The number M of lnes and N of η lnes of each grd are presented n table 2, whle fgure 2 shows grd G5, wth M=51 and N=66. Table 2. Fnte dfference grds Grd M N G G G G G G G G G The temperatures of the frst 6 ponts shown n fgure 2 were compared for the grds presented n table 2. Such temperatures were obtaned for tme t=5 s and for a motored engne wth an alumnum pston. The tme step used was t=1x1-3 s. The relatve dfferences n percent for the temperatures computed wth the dfferent grds used n ths study are shown n table 3. Ths table shows that generally the grds are not converged wth N=56, because dfferences of the order of 2% can be observed for pont 2, when N s ncreased to 66, rrespectve of the number of lnes (M) utlzed. On the other, dfferences of less than.5% can be notced when N s ncreased to 84, as compared to N=66. By takng nto analyss now the grds wth N=66 (G4, G5 and G6), we note that the grd s bascally converged n the drecton wth M=51. A maxmum dfference of 1.1% s observed for pont 6, when M s ncreased from 51 to 61

6 (grds G5 and G6, respectvely). From the examnaton of table 3, we decded to use grd G5 for the foregong analyss, wth M=51 and N=66. Such grd s bascally converged n the and η drectons. Also, ts CPU tme was 4 mn and 23 s as compared to 5 mn and 7 s for grd G6, thus enablng substantal savngs on computer tme, wthout loss of accuracy. The CPU tmes correspond to a Pentum 2 MMX, wth 64 Mb of RAM memory, runnng under the Mcrosoft Fortran PowerStaton 4. platform. Table 3. Relatve temperature dfference n percent between grds Pont G1-G2 G2-G3 G4-G1 G5-G2 G6-G3 G4-G5 G5-G6 G7-G4 G8-G5 G9-G6 G7-G8 G8-G After choosng the grd, we performed an analyss of the tmewse varaton of the temperature n the pston for the dfferent test-cases summarzed n table 1. Fgure 3.a-d show the varaton for the temperature wth tme, untl the quas-steady-state s reached, of selected ponts n the pston for cases 1-4, respectvely. These fgures show that, ntally, the temperature of the ponts near the cylnder wall s larger than for the other ponts. Such s the case because ntally the pston s assumed to be at a temperature smaller than that of the coolng flud and of the ol. By comparng fgures 3.a-b wth fgures 3.c-d, we note that the quas-steady-state s reached faster for the alumnum pston than for the cast-ron pston, as a result of the larger thermal conductvty and thermal dffusvty for alumnum. Temperature( C) Pont 1 Pont 3 Pont 4 Pont 6 CASE 1 (a) Temperature ( C) Pont 1 Pont 3 Pont 4 Pont 6 CASE 2 (b) tme (s) tme (s)

7 Temperature ( C) CASE 3 (c) Pont 1 Pont 3 Pont 4 Pont tme (s) Temperature ( C) CASE 4 (d) Pont 1 Pont 3 Pont 4 Pont 6 Fgure 3. Temperature varaton of selected ponts tme (s) The quas-steady-state temperature dstrbutons n the pston are shown n fgures 4.a-d, for test-cases 1 to 4, respectvely. Note n these fgures the hgher temperatures n the cast-ron pston than n alumnum pston. Also, note the hgher temperature gradents n the radal drecton n the top regon of the cast-ron pston, resultng n larger thermal stresses than for the alumnum pston. CASE 1 (a) CASE 2 (b) CASE 3 (c) CASE 4 (d) Fgure 4. Quas-steady state temperature dstrbuton n the pston

8 Fgure 5 presents the temperature varaton of ponts 1 and 7 n the pston for test-case 1, nvolvng an alumnum pston n a fred engne, when the quas-steady-state s establshed. Fgure 5 shows that the ampltude of varaton for the temperature of pont 1 s 1.94 o C. For pont 7, located 1.39 mm below the gas-pston nterface on the pston center-lne, such ampltude of varaton s less than 1 o C. Much smaller ampltudes of varaton were observed for the temperature of the cast-ron pston, due to ts smaller thermal dffusvty as compared to alumnum. Also, the ampltudes for motored engnes are much smaller than for fred engnes. These results are extremely nterestng because they reveal the hgh-level of dffculty for the soluton of the nverse problem of estmatng the perodc heat flux at the gas-pston nterface, by usng temperature measurements taken below the nterface. Such s the case because the ampltudes of temperature varaton are of the same order of magntude of the expected levels for the measurement errors. Hence, t would be mpossble to dstngush f the measured temperature varatons result from the measurement errors or from the perodc boundary condtons, and no useful nformaton would be recovered from the temperature measurements. CONCLUSIONS Fgure 5. Temperature varatons of ponts 1 and 7 for the test-case 1. The analyss performed reveals that, for the cases studed, the steady-state was reached earler for alumnum pstons than for cast-ron pstons. Also, cast-ron pstons are subjected to hgher temperatures and larger temperature gradents, thus resultng n larger thermal stresses. Generally, the temperature varatons, resultant from the perodc boundary condton at the gaspston nterface, are largely damped wthn a qute small dstance below the nterface. Therefore, the soluton of the nverse heat conducton problem of estmatng the perodc boundary heat flux, by usng temperature measurements below the surface, s qute dffcult. REFERENCES Borman, G., Nshwak, K., 1987, Internal Combuston Engne Heat Transfer, Prog. Energy Combust. Sc., Vol. 13, pp Colaço, M.J., Orlande, H.R.B., 1996, Transent Heat Transfer Analyss od Desel Engne Pston, Procedngs of VI ENCIT / VI LATCYM, UFSC, Vol.1. p Ferguson, C. R., 1986, 1986, Internal Combuston Engnes, Wley, New York.

9 Heywood, J.B., 1988, Internal Combuston Engne - Fundamentals, MC Graw-Hll, New York. Malska, C.R., 1995, Transferênca de Calor e Mecânca dos Fludos Computaconal, LTC, Ro de Janero. Özsk, M.N., 1994, Fnte Dfference Methods n Heat Transfer, CRC Press, Boca Raton. Özsk, M.N., 1993, Heat Conducton, 2 nd edton, McGraw-Hll Internatonal Edtons, New York. Peaceman, D.W. and Rachford, H.H., 1955, The Numercal Soluton for Parabolc and Ellptc Dfferental Equatons, J. Soc. Ind. Appl. Math, Vol.3, pp Prasad, R. and Samra, N.K., 199, Transent Heat Transfer n an Internal Combuston Engne Pston, Computers & Structures, Vol. 34, no. 5, pp Ramos, J. I., 1989, Internal Combuston Engne Modelng, H P C. Thompson, J.F., Wars, Z.U.A. and Mastn, C.W., 1987, Numercal Grd Generaton, North- Holland, New York.