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Journal of Engineering Mathematics 44: 5782, 2002. 2002 Kluwer Academic Publishers. Printed in the Netherlands.Modelling gas motion in a rapid-compression machineM.G. MEERE1, B. GLEESON1and J.M. SIMMIE2Department of Mathematical Physics, NUI, Galway, Ireland2Department of Chemistry, NUI, Galway, IrelandReceived 25 July 2001; accepted in revised form 8 May 2002Abstract. In this paper, a model which describes the behaviour of the pressure, density and temperature of a gasmixture in a rapid compression machine is developed and analyzed. The model consists of a coupled system ofnonlinear partial differential equations, and both formal asymptotic and numerical solutions are presented. Usingasymptotictechniques, asimplediscretealgorithmwhichtracksthetimeevolution ofthepressure, temperatureanddensity of the gas in the chamber core is derived. The results which this algorithm predict are in good agreementwith experimental data.Key words: gasdynamics, rapid-compression machines, shock-waves, singular perturbation theory1. Introduction1.1. RAPID-COMPRESSION MACHINESA rapid-compression machine is a device used to study the auto-ignition of gas mixtures athigh pressures and temperatures, with particular reference to auto-ignition in internal combus-tion engines; see 13. A typical combustion engine is a very dirty and complex environment,and this has prompted the development of rapid-compression machines which enable thescientific study of compression and ignition in engines in a cleaner and simpler setting. InFigure 1 we schematically illustrate a two-piston rapid-compression machine, such as theone in the department of Chemistry at NUI, Galway. However, single-piston machines, witha piston at one end and a stationary solid wall at the other, are more typical. The analysisdeveloped in this paper is appropriate to both single- and two-piston machines.The operation of a rapid-compression machine is very simple - the pistons are simul-taneously driven in pneumatically, compressing the enclosed gas mixture, thereby causingthe gas pressure, temperature and density to rise quickly. In Figures 1(a), 1(b) and 1(c) weschematically represent a rapid-compression machine prior to, during, and after compression,respectively. The ratio of the final volume to the initial volume of the compression chamberfor the machine at NUI, Galway is about 1:12, this value being typical of other machines. Atthe end of the compression, the gas mixture will typically have been pushed into a pressureand temperature regime where auto-ignition can occur.In Figure 2, we display an experimental pressure profile for a H2/O2/N2/Ar mixture whichhas been taken from Brett et al. 4, with the kind permission of the authors. In this graph,the time t = 0 corresponds to the end of the compression time. We note that, for the greaterpart of the compression, the pressure in the chamber is rising quite gently, but that towards theend of the compression (that is, just before t = 0), there is a steep rise in the pressure. Aftercompression, the pressure profile levels off as expected; the extremely steep rise at the end of58M.G. Meere et al.Figure 1. Schematic illustrating the operation of a rapid-compression machine; we have shown the configuration(a) prior to compression, (b) during compression and (c) after compression.Figure 2. An experimental pressure profile for a gas mixture H2/O2/N2/Ar = 2/1/2/3, as measured in therapid-compression machine at NUI, Galway. It is taken from 4, and has an initial pressure of 005 MPa andan initial temperature of 344 K.Modelling gas motion in a rapid-compression machine59the profile corresponds to the ignition of the mixture. We note that the compression time andthe time delay to ignition after compression are both O(10) ms.The pressure history is the only quantity which is measured in experiments. However,the temperature in the core after compression is the quantity which is of primary interest tochemists since reaction rates depend mainly on temperature for almost all systems, althoughthere may also be some weaker pressure dependence. Measuring temperature accurately inthe core can be problematic because of the presence of a thermal boundary layer; see thecomments below on roll-up vortices. However, with the experimental pressure data in hand,the corresponding temperatures can be estimated using the isentropic relationln(p/pi) =?TTi(s)s(s) 1)ds,(1)where (Ti,pi) are the initial values for the core temperature and pressure, (T,p) are thesequantities at some later time, and (s) is the specific heat ratio at temperature s. In exper-iments, the initial core temperature is typically O(300 K), while the core temperature aftercompression is usually O(1000 K).In this paper, we shall consider only the behaviour of the gas mixture during compression;the post-compression behaviour is not considered here, but this will form the subject forfuture work. Nevertheless, the model presented here does provide a reasonable descriptionof the post-compression behaviour of a single species pure gas, or an inert gas mixture; seeSection . THE MODELWe suppose that the compression chamber is located along 0 x 0. This assumption is actually quite a strong one in this context since higher-dimensionaleffects are frequently observed in experiments, roll-up vortices near the corner regions definedby the piston heads and the chamber wall being particularly noteworthy; see, for example,5. These vortices arise due to the scraping by the pistons of the thermal boundary layerat the chamber wall, and they can, and frequently do, disturb the gas motion in the core ofthe compression chamber. However, the justification for the one-dimensional model studiedhere is twofold: (i) the corner vortices can be successfully suppressed by introducing crevicesat the piston heads which swallow the thermal boundary layer as the pistons move in (seeLee 6), rendering the gas motion away from the chamber walls one-dimensional to a goodapproximation, and, (ii) the study of the one-dimensional model provides a useful preliminaryto the study of higher-dimensional models.We now give the governing equations for our one-dimensional model. A reasonably com-plete derivation of the governing equations for a multi-component reacting gas can be foundin the appendices of 7; these standard derivations are not reproduced here. The model whichwe shall study includes a number of simplifying assumptions and these will be clearly statedas they arise.The equation expressing conservation of mass is given byt+x(v) = 0,60M.G. Meere et al.where = (x,t) and v = v(x,t) are the density and the velocity of the gas, respectively, atlocation x and time t. It should be emphasized that these quantities refer to a gas mixture, sothat if there are N different species in the mixture then =N?i=1i,where i= i(x,t) is the density of species i. Also, the velocity v above refers to the mass-averaged velocity of the mixture, that is,v =N?i=1Yivi,where Yi= i/ and vi= vi(x,t) are the mass fraction and velocity, respectively, of speciesi; see 7.Neglecting body forces and viscous effects, the equation expressing conservation of mo-mentum is given byvt+ vvx= 1px,where p = p(x,t) is the pressure. Weassume that the gas mixture is ideal, so that the equationof state is given byp =RMT,(2)where T = T(x,t) is the temperature, R is the universal gas constant (8314 JK1mol1), andM is the average molecular mass of the mixture. This last quantity is given byM =N?i=1niWi(mA),where niand Wigive the number fraction and molecular weight, respectively, of species i,m is the atomic mass unit (1661 1027kg) and A is Avogadros number (6022 1023molecules mol1).The equation expressing conservation of energy is given by (see 7 or 8)?ut+ vux?= M?qx+ pvx?,(3)where u = u(x,t) is the internal energy of the gas mixture and q = q(x,t) is the heat flux.We also have the thermodynamic identityu =N?i=1hiYi Mp/,(4)where the enthalpies hi= hi(T) are given byhi(T) = hi(T0) +?TT0cp,i(s)ds,i = 1,2,. ,N,(5)Modelling gas motion in a rapid-compression machine61where T0is some reference temperature and the cp,i(T) are the specific heats at constantpressure for the N species. When diffusion velocities and the radiant heat (again, see 7 formore details) are neglected, the expression for the heat flux is given byq = (T)Tx,(6)where (T) is the thermal conductivity.The mass fractions Yi= i/ are not necessarily constant since chemical reactions canchange the composition of the mixture. However, for many systems such chemical effectscan be neglected in the analysis of the compression because the gas mixture is cold formost of the compression time. The core temperature will only rise to a level where chemicalreactions can have a significant effect near the end of the compression, and the duration of thisperiod is typically very short (a couple of milliseconds usually). Nevertheless, it is possiblefor some chemical reactions to proceed sufficiently rapidly for them to significantly influencethe compression behaviour. However, we do not attempt to model systems which exhibit thisbehaviour here and take the Yito be constant during compression.Substituting (4) and (6) in (3), and using (5), we have the final form of the equationexpressing conservation of energy:Tt+ vTx=M(cp(T) R)?x?(T)Tx? pvx?,wherecp=N?i=1Yicp,iis the mass-averaged specific heat.1.3. BOUNDARY AND INITIAL CONDITIONSWe suppose that the left and right pistons move with constant velocities V0and V0, re-spectively, so that their motions are given by x = V0t and x = 2L V0t. In reality, thepistons in a rapid-compression machine will spend some of the compression time acceleratingfrom rest and decelerating to rest, and this is not difficult to incorporate into the analysis givenbelow. However, rather than complicate the analysis unnecessarily at the outset by consideringvariable piston velocity, we shall simply quote the results for general piston motion once theconstant velocity case has been completed; see Section 3.4. Throughout the compression, weassume that the temperature of the walls of the chamber remain at their initial constant value,which we denote by T0. Hence, at the left piston, we imposev = V0, T = T0on x = V0t,while at the right piston we setv = V0, T = T0on x = 2L V0t.The gas in the chamber is initially at rest and we suppose thatv = 0, T = T0, p = p0, = 0at t = 0,62M.G. Meere et al.where p0and 0are constants. Clearly, in view of (2), we havep0=RM0T0.However, the above are not quite the boundary and initial conditions that are considered inthis paper. For the conditions described above above we have the symmetryv(x,t) = v(2L x,t), T(x,t) = T(2L x,t), p(x,t) = p(2L x,t),(x,t) = (2L x,t).We exploit this behaviour by halving the spatial domain, considering the gas motion in V0t x q+, we have v = 0,p = = T = 1. For x 0, but this amounts to nothing more than requiringthat the pistons travel at velocities which do not exceed the speed of sound in the gas. Recallthat the maximum speed of the pistons is O(10 ms1), while the speed of sound in gases undertypical conditions is frequently O(300 ms1).Substituting (17) in (16)3, and integrating subject to the conditions v0 0, p0 1 asz +, we haveModelling gas motion in a rapid-compression machine71p0= 1 + q+0v0/.(19)Letting z in (19) we obtainPs(t) = 1 + q+0/,(20)so that,Tc(t) = (1 + q+0/)(1 1/ q+0),(21)both of which are constant since, as we shall now show, q+0is constant. The prediction that(p,T) are constant to leading order in the outer region behind the wave-front is clearlyconsistent with the numerical solution displayed in Figure 3.Substituting (17)and (19) in(16)4and integrating subject tov0 0,T0 1,T0/z0 as z +, we get q+0(T0) = (T0)T0z (v0+ q+0v20/2),(22)where(T0) =?T01ds(s) 1.Letting z in (22), we obtain q+0?(1 + q+0/)(1 1/ q+0)?= 1 + q+0/2,(23)which determines q+0, completing the specification of the leading order outer problem. It isclear that the solution for q+0to (23) does not depend on t, so that q+0has the form t where is a constant.Using (9), we have(T0) =11ln?0+ 1T0 10+ 1 1?,so that (23) becomes q+01ln?0 1 + 1(1 + q+0/)(1 1/ q+0)0+ 1 1?= 1 + q+0/2,(24)which is an equation that is easily solved numerically for q+0for given values of 0, 1and .In the limit 1 0 (so that (T) 0) this expression reduces to a quadratic in q+0whichcan be solved to give q+0=14?0+ 1 ?(0+ 1)2+ 160?,with the positive solution being clearly the relevant one here. The numerical solution of (24)for 1?= 0 is usually unnecessary. Recalling that = O(103) typically, and considering thebehaviour of (24) for ? 1, we can easily show that q+0?(0+ 1) for ? 1.(25)In dimensional terms, this expression for q+0is72M.G. Meere et al.?(T0)p00,which is the familiar expression for the speed of sound in an ideal gas. We favour the simplerexpression (25) over (24) for the algorithm described in Section 3.4.It is worth noting here that the relations (18), (20) and (23) could also have been obtainedusing integral forms for theconservation laws, and itisnot necessary (although itispreferable)to consider the detail of the transition layer. For example, conservation of mass implies thatddt?1t(x,t)dx?= 0,which at leading order givesddt?q+0t0(x,t)dx +?1q+01dx?= 0,and this leads to (18).3.1.4. SummaryThe motion of the wave-front, x = q+(t;), is such that as 0, q+(t;) q+0(t), whereq+0(t) is determined by solving (24) subject to q+0(0) = 0. For x q+, p = = T = 1 and v = 0.3.2. THE FIRST REFLECTION OF THE WAVE FROM THE CENTRE-LINEWhen the wave-front reaches the centre-line, it reflects off the identical opposing wave, andthen moves from right to left towards the incoming piston. The leading-order behaviour aheadof the wave is now known from the calculations of the previous subsection. A numericalsolution illustrating this case is given in Figure 4. The leading-order behaviour in the boundarylayer near the piston is clearly unchanged from that considered in Section 3.1.1 and requiresno further discussion.3.2.1. Outer regionWe denote the motion of the reflected wave by x = q(t;). For x q, we have v = o(1) and we posep p+0(x,t), +0(x,t), T T+0(x,t)to obtainp+0= +0T+0,+0t= 0,p+0x= 0,T+0t= 0,so thatp+0= Psr, +0= g(x), T+0= Psr/g(x),Modelling gas motion in a rapid-compression machine73where Psr, which is constant, and g(x) are determined below by matching.3.2.2. Transition regionThis is located at z= O(1) where x = q(t;) + z. It gives the location of the narrowregion over which v drops from v 1 to v = o(1); the transition region is also clearlyidentifiable in the solutions for p, and T; see Figure 4. In z= O(1) we poseq q0(t), p p0(z,t), 0(z,t), v v0(z,t), T T0(z,t),to obtain leading-order equations which have precisely the same form as (16). Integrating andmatching in a manner similar to that described in Section 3.1.3, we obtain0= q+0( q0 1)( q+0 1)( q0 v0),v0= 1 +( q+0 1) q+0( q0 1)(p0 1 q+0/), q+0( q0 1) q+0 1(Tc) (T0) =(T0)T0z?(1 + q+0/)(v0 1) + q+0( q0 1)2( q+0 1)(v20 1)?.(27)Letting z + in (27) givesg(x) q+0( q0 1) q0( q+0 1), Psr= 1 + q+0( q+0 q0)( q+0 1),T+0= q0 q+0( q0 1)? q+0 1 + q+0( q+0 q0)?,(28)where the constant reflected wave speed q0is determined as the negative solution to q+0( q0 1)1( q+0 1)ln?0 1 + 1Tc0 1 + 1T+0?= 1 + q+0 q+0( q0 1)2( q+0 1),(29)where Tcis given by (21) and T+0is given in (28). Considering the behaviour of this lastexpression for ? 1, we find that q0 (0+ 1), the negative solution being therelevant one now. It is this simpler form which we shall use for the algorithm described inSection .3. SummaryThe location of the reflected wave-front, x = q(t;), is such that as 0, q(t;) q0(t), where q0(t) is determined as the solution to (29). For x q, we have v = o(1) andp 1 + q+0( q+0 q0)( q+0 1), q+0( q0 1) q0( q+0 1),T q0 q+0( q0 1)? q+0 1 + q+0( q+0 q0)?.74M.G. Meere et al.3.3. THE WAVE TRAVELS OVER AND BACK IN THE CHAMBER FOR THENthTIMEMost of the notation required here has previously been introduced in Section .1. The wave travels down the chamber for the NthtimeDenoting the location of the wave-front by q+N, we have for x q+N, we have v = o(1) andp p2N2, 2N2, T T2N2,where (p2N2,2N2,T2N2) are constants. If we now consider the transition layer of thick-ness O() about q+N, and perform calculations which are almost identical to those of Sec-tion 3.1.3, we find that2N1= q+N0 q+N0 12N2,p2N1= p2N2+ q+N02N2,T2N1=p2N12N1, q+N02N2(T2N1) (T2N2) = p2N2+ q+N022N2.(30)Using (9), and considering the behaviour of (30)4for ? 1, we can easily show that q+N0?(T2N2)T2N2 =?(T2N2)p2N22N2for ? 1.We note from this expression that as the pistons compress the gas in the chamber core, thespeed of the wave increases in proportion to the square root of the rising temperature (for(T) constant). If (T) 0, then we can solve exactly for q+N0to obtain q+N0=14?0+ 1 +?(0+ 1)2+ 160T2N2?.(31)3.3.2. The wave reflects off the centre-line for the NthtimeWe denote the location of the wave-front by x = qN. As 0, we have for x qN, we have v = o(1) andp p2N, 2N, T T2N,with (p2N,2N,T2N) constant. Considering the transition layer about qN, it is readily shownthatModelling gas motion in a rapid-compression machine752N= qN0 1 qN02N1,p2N= p2N1 qN02N,T2N=p2N2N, qN02N(T2N1) (T2N) = p2N+ qN022N.(32)Considering the behaviour of (34)4for ? 1, we have qN0 ?(T2N1)T2N1 = ?(T2N1)p2N12N1for ? 1.For (T) 0, we have the exact expression qN0=14?3 0?(3 0)2+ 160T2N1+ 8(0 1)?.(33)3.4. THE ALGORITHM3.4.1. Constant piston velocityThe calculations of the previous subsections yield the following simple iterative algorithm forthe evolution of the pressure, density and temperature in the chamber core:p0= 0= T0= 1, t0= 0,and for N = 1,2,3,., we have q+N=?(T2N2)T2N2,2N1= q+N q+N 12N2,p2N1= p2N2+ q+N2N2,T2N1=p2N12N1,t2N1= t2N2+1 t2N2 q+N, qN= ?(T2N1)T2N1,2N= qN 1 qN2N1,p2N= p2N1 qN2N,T2N=p2N2N,t2N= t2N1+1 t2N11 qN.(34)It is clear that this iterative scheme is trivial to implement on a computer. Once a calculationbased on this algorithm has been completed, one could, for example, plot the pressure inthe chamber during compression by simply passing a smooth curve through the data points(pi,ti), i = 0,1,2,.In Figure 6(a), we have plotted curves through data points for (pi,ti), (i,ti) and (Ti,ti),as calculated using the algorithm (34). In experiments, the only quantity which is measuredthroughout the compression is the pressure. It is immediately obvious that the algorithm is76M.G. Meere et al.Figure 6. Pressure, temperature and density profiles in the chamber core as functions of time calculated using (a)the iterative scheme (34) which has Xp(t) = t, and (b) the iterative scheme (37) with Xp(t) given by (38). In bothcases, the values used were (T) 14 and = 1500.correctly reproducing the shape of the experimental pressure profiles, and in particular, therapid rise in the measured pressure towards the end of the compression. In fact, we shallsee below that the algorithm accurately reproduces the experimental compression behaviourin detail. Within the context of the model presented here, the rapid pressure rise is easilyexplained; near the end of compression, the wave, which carries with it an increase in pressure,passes through a particular point in the chamber with much increased frequency (principally)because the distance it has to travel to return to that point has reduced by a factor of O(10).Modelling gas motion in a rapid-compression machine77Table 2. Comparison of results generated by the algorithm with thenumerical results displayed in Figures 35. In the table, the first el-ement of a pair (,) gives the algorithm value, while the secondelement gives the corresponding numerical value.TpFigure 3(0241,0252)(0220,0222)(0514,0522)Figure 4(0270,0271)(0241,0250)(0694,0732)Figure 5(0300,0317)(0263,0262)(0913,0941)3.4.2. Comparison of algorithm with full numericsWe now give a comparison of the numerical results displayed in Figures 35 with the corre-sponding results generated by the algorithm (34). We denote by , T and p, the jumps indensity, temperature and pressure, respectively, across the shock-front. In Table 2, wecomparethe values for ,T and p, generated by the algorithm, with those that were obtained bynumerically integrating the full equations. Because (T) 0(a constant) in the numericalresults, we can use the exact expressions (31) and (33) for q+N0and qN0, respectively. Thecorrespondence between the two sets of results is good; the difference that there is in theresults is due primarily to the fact that the algorithm is based on a leading order asymptoticanalysis. There will also be some error in the numerical results.3.4.3. The continuum limit as We now consider the continuum limit of the discrete algorithm (34) for ? 1. This corre-sponds to considering the behaviour for a large number of reflections since the speed of soundin the gas is proportional to for ? 1.As , it is readily shown from (34) that q+N?(T2N2)T2N2, qN q+N,t2Nt2N22 q+N(1 t2N2),p2Np2N22 q+N2N2,2N2N22 q+N2N2.(35)We write t = t2N2, p(t) = p2N2, (t) = 2N2, T(t) = T2N2, anddZ(t)dt= lim?Z2N Z2N2t2N t2N2?,where Z = p,. Equations (35) then imply that78M.G. Meere et al.d(t)dt=(t)1 t,dp(t)dt=(T(t)p(t)1 t,p(t) = (t)T(t), (T) = 0+ 1T,p(0) = (0) = T(0) = 1,(36)and integration of these equations givesp(t) =1 01(1 t) + (1 0 1)(1 t)0, (t) =11 t,T(t) =1 01+ (1 0 1)(1 t)01.For 0, these expressions reduce top(t) =1(1 t)0, (t) =11 t, T(t) =1(1 t)01,and these simple results are consistent with the curves displayed in Figure 6(a). The expressionfor (t) states that the gas density is inversely proportional to the volume of the chamber.It follows easily from (36) thatdpp=(T)(T) 1)dTT,which is the isentropic relation (1) referred to in the introduction.3.4.4.Variable piston velocityObviously, during any compression, there are periods when the pistons are accelerating anddecelerating, and it is not difficult to incorporate variable piston motion, x = Xp(t), into thealgorithm. The background analysis is almost identical to that of the constant piston velocitycase, and we shall confine ourselves to quoting the results here. We now havep0= 0= T0= 1, t0= 0,with for N = 1,2,3,., q+N=?(T2N2)T2N2,2N1= q+N q+NXp(t2N2)2N2,p2N1= p2N2+ q+NXp(t2N2)2N2,T2N1=p2N12N1,t2N1= t2N2+1 Xp(t2N2) q+N, qN= ?(T2N1)T2N1,2N= qNXp(t2N1) qN2N1,p2N= p2N1 qNXp(t2N1)2N,T2N=p2N2N,t2N= t2N1+1 Xp(t2N1)Xp(t2N1) qN.(37)Modelling gas motion in a rapid-compression machine79It is clear that this algorithm reduces to the correct forms for the particular casesXp(t) = 1andXp(t) = 0.In Figure 6(b), we have plotted a solution calculated using the scheme (37) withXp(t) =125t2for 0 t 04,t 02 for 04 t 1,08 + (t 1) 25(t 1)2for 1 t 12,09 for t 12,(38)which corresponds to uniform acceleration from rest and uniform deceleration to rest at thebeginning and end of the piston motion, respectively. When the pistons stop, the algorithmpredicts that the pressure, temperature and density in the chamber core remain at the valuesattained at the end of the compression. Within the context of the model presented in this paper,the calculation of changes in (p,T) in the core post-compression requires the considerationof higher-order terms in the asymptotic expansions. However, chemical effects are frequentlysignificant post-compression, so that the model analyzed here may not be valid after thepistons have drawn to a stop; we shall return to this issue in more detail in Section 4.3.5. COMPARISON WITH EXPERIMENTAL DATAIn Figure 7 we give some comparisons between experimental results and the predictions of themodel. The experimental data is taken from Musch 16, and each of the three experimentalcurves has an initial temperature of 295 K and an initial pressure of 006 MPa. The results aredisplayed in non-dimensional form.An approximation for Xp(t) was obtained using measurements for the piston motion. Forthe machine at NUI, Galway, the pistons spend about 30% of the compression time accel-erating from rest, less than 15% of the time decelerating to rest, and the remainder of thetime at a constant maximum velocity of about 14 ms1. It is found that the first phase ofthe motion can be reasonably well described by assuming that the pistons are acceleratinguniformly. However, the final phase of the motion does not conform to uniform decelerationsince measurements indicate that there is a slight rebound in the pistons at the end of theirstroke. This rebound increases the final volume of the compression chamber slightly, causinga slight drop in the pressure; this accounts for the kink in the peak of the model pressureprofiles in Figure 7, an effect which is also clearly visible in the experimental profiles.Piston crevices were used in the generation of the experimental data illustrated in Figure 7,and care must be taken to account for the volume which the crevices add to the chambervolume when modelling the behaviour. For the results displayed here, the crevices increasethe initial volume of the chamber by about 15%. The parameter values used to generate themodel curves of Figure 7 are given in Table 3; these parameters were calculated using the datafor the experimental conditions and known data for the gases (9).It is clear from Figure 7 that the correspondence between theoretical prediction and ex-perimental result is excellent in all three cases for almost all of the compression time. ForNitrogen and Oxygen, the model also captures the behaviour very well towards the end ofthe compression, and post-compression. However, for Argon, the model significantly over-predicts the final pressure inthe chamber core, and this highlights aweakness of the algorithm.The algorithm predicts that, when the pistons stop, the temperature and pressure in the coreremain at the values attained at the end of compression, since this is what the leading-order80M.G. Meere et al.Figure 7. Comparison of model predictions with experimental data taken from 16 for (a) Nitrogen, (b) Oxygenand (c) Argon. The parameter values used in the model are given in Table 3, Section 3.5.Table 3. Parameter values used to generate the modelcurves in Figure 7.FigureGas(T)7(a)Nitrogen (N2)495143 003T7(b)Oxygen (O2)395142 003T7(c)Argon (Ar)316167asymptotic analysis of the governing equations predicts. However, heat in the chamber coreis continually being lost to the cold walls, and while this is not a leading-order effect in thechamber core, it can nevertheless have a significant effect on the peak pressure for gases, suchas Argon, which have relatively large values for (T). The algorithm is limited by the fact thatit is based on a leading order asymptotic analysis, while higher-order effects can sometimesbe significant.Modelling gas motion in a rapid-compression machine814. DiscussionIt should be emphasized that we have only modelled the compression phase of the operationof a rapid-compression machine in this paper. The nature of the post-compression problemis typically very different. The principal purpose of a rapid-compression machine is to veryquickly push the enclosed gas mixture into a pressure and temperature regime where auto-ignition can occur. An obvious first attempt to model the post-compression behaviour is toassume that the gas very rapidly comes to rest after the pistons stop; the model studied in thispaper predicts that the gas motion settles to rest (to a good approximation) over a time periodwhich is O(1/) ? 1 after the pistons draw to a halt. After compression, one would thenuse a system of ordinary differential equations to model the concentrations of the reactingchemical species and the temperature in the chamber core. This model would not be validclose to the walls of the chamber as the thermal boundary layer persists and grows after thepistons stop.For systems where this approach does prove to be an adequate representation of the behav-iour, the problem will have sp
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