THE IGNITION PARAMETER - A quantification of the probability of ignition

Transcription

1 THE IGNITION PARAMETER - A quantfcaton of the probablty of ton INFUB Topc: Modellng of fundamental processes Man author Nels Bjarne K. Rasmussen Dansh Gas Technology Centre (DGC) NBR@dgc.dk Co-author Thomas W. Sødrng Babcock & Wlcox Vølund, Denmark TWS@volund.dk Abstract When a mxture of combustble gases s facng or flowng past hot surfaces there s a probablty of ton. But how can ths probablty be quantfed? The work on quantfyng such a probablty of ton s presented n ths paper ncludng verfcaton of the model. The basc dea of ths model of the Iton Parameter, dentfed by I, s that when I 1 ton wll occur, and when I < 1 no ton wll occur. The model s based on emprcal data for ton of combustble gases. For hgh temperatures t seems that for each of the gases the tme delay for ton can be descrbed by a lnear graph n a logarthmc dagram. Ths leads to a smple Arrhenus expresson. To calculate the Iton Parameter I by CFD modellng the followng equaton s assumed to be vald: I t x x I I u I S x [kg/m 3 s] Ths equaton s the transport equaton for I. It hghlghts the varous transport processes: the rate of change term and the convecton term on the left sde and dffusve term and the source term on the rght sde. The expresson for the combned source and snk term s S E Aexp RT C D fuel ar B I [kg/m 3 s] Ths equaton s related to the ton tme delay and the Arrhenus expresson s recozed. The code CHEMKIN was used to calculate the autoton temperature for natural gas at dfferent gas mxtures and resdence tmes. The results were used as emprcal data for the Iton Parameter to calculate the emprcal constants A, B, C, D and E by data fttng. The nterpretaton of the second equaton s as follows. The frst term on the rght sde s due to chan branchng and radcals producton whle the second term s due to chan termnaton and decay of radcals. The model was compared to expermental data. A steel pellet was slowly heated electrcally untl ton of a steady state flow of a premx of gas and ar passng the pellet. The temperature of the pellet at ton was found. The data from these experments were compared to the CFD calculatons of the same cases. Ideally, the results for I should end at 1.0 as maxmum n the volume n each case. We have found that the model calculates the ton temperature of actual test cases wthn devatons of only 0-50K and on the lower sde. These devatons nclude all the possble uncertantes from the CHEMKIN modellng, from the derved one-equaton model, from the expermental data and, fnally, from the CFD modellng of the expermental setup. The model of the Iton Parameter s a novel model, whch gves very good results and may be a great help n predctng probabltes of ton of combustble gases close to hot surfaces. Keywords: ton parameter, probablty of ton, ton tme, delay, CFD, CHEMKIN

2 Introducton When a mxture of combustble gases s facng or flowng past hot surfaces there s a probablty of ton. But how can ths probablty be quantfed? The work on quantfyng such a probablty of ton s presented n ths paper ncludng verfcaton of the model. The basc dea of ths model of the Iton Parameter, dentfed by I, s that when I 1 ton wll occur I < 1 no ton wll occur As I ncreases towards 1.0 at any pont throughout the volume of nterest, the rsk of ton rses to 100%. If I s above 1.0, ton has occurred. If I keeps below 1.0, no ton occurs n the volume. The next step s to mplement ths dea nto a useful model /3/. Emprcal data on ton The model s based on emprcal data for ton of combustble gases. Fgure 1 shows an example of such data. Fgure 1 Iton tmes for a number of gases measured by Bureau of Mnes (Babrauskas, 2003) For hgh temperatures t seems that for each of the gases the tme delay for ton can be descrbed by a lnear graph n a logarthmc dagram. Ths leads to a smple Arrhenus expresson where the followng relaton between ton tme delay and temperature can be used: ln E RT 1 a E A exp RT (1)

3 a and A are expermental constants E s the actvaton energy [J/mol] R s the deal gas law constant [J/molK] T s the temperature [K] s the ton tme delay [s] As found n Fgure 1, most gases show lnes wth two slopes: one at hgher temperature and one at lower temperature regme. Interestngly, gven the large number of reactons takng place n real gases, the curves are all composed of two straght lnes, rather than beng complex curves. Furthermore, f the tme scale were extended further t mght be the case that an asymptotc vertcal lne would occur for each graph. Ths means that below a certan temperature no ton wll occur, no matter the exposure tme. That would be a vertcal lne n the dagram. If we also take nto account the concentraton of a combustble gas [fuel] mxed wth ar [ar], equaton (1) can be rewrtten and expanded to the followng n whch t s assumed that the dependency of concentratons can be descrbed by the exponents C and D : E 1 C D A exp fuel ar [-] (2) RT At ths pont we ntroduce the Iton Parameter I. If nstead s vewed as the resdence tme and assumng constant condtons on the rght sde of the equaton, the left sde can be vewed as an Iton Parameter I : I E A exp RT fuel C ar D [-] (3) vald for hgh temperatures only. So when I s greater than 1, correspondng to >, ton wll occur and when I s less than 1, correspondng to <, there s no ton. CHEMKIN calculatons The code CHEMKIN was used to calculate the ton temperature for natural gas at dfferent gas mxtures and resdence tmes. The results from these calculatons are shown n Fgure 2. In ths model, hundreds of speces and thousands of chemcal reactons were ncluded to gve the best possble result. The AIT, AutoIton Temperature, s the lowest temperature for whch autoton wll occur wth the gven mxture of gases and for a very long resdence tme ( sec).

4 Temperature [K] sek sek sek AIT 650 0,0 0,2 0,4 0,6 0,8 1,0 Natural gas n Ar Fgure 2 CHEMKIN calculated correlaton between temperature and ton tme for natural gas n ar Whle Fgure 1 shows the ton tme for a fxed mxture of fuel and ar for dfferent fuels, Fgure 2 shows the relaton between temperature and ton tme for a wde range of mxtures of natural gas and ar. A vertcal lne n Fgure 2 could be mplemented as a graph for natural gas n Fgure 1, the AIT beng the mentoned vertcal lne below whch temperature no ton would occur. Equatons for the Iton Parameter To calculate the Iton Parameter I by CFD modellng the followng equaton s assumed to be vald,.e. t s assumed that I can be solved lke any other varable as temperature or concentratons. I t x x I I u I S x [kg/m 3 s] (4) s the densty t s the tme s the dffuson coeffcent S s the source term for the varable I u s the velocty x s the dstance n the -drecton The equaton (4) s the transport equaton for I. It hghlghts the varous transport processes: the rate of change term and the convecton term on the left sde and dffusve term and the source term on the rght sde. The next problem s to fnd the source term S.

5 The source term at hgh temperatures Frst, we consder a steady state condton and assume a case where dffuson can be neglected. Equaton (4) s then reduced to: I u S [kg/m 3 x s] (5) Usng the contnuty equaton: u 0 x [kg/m3 s] (6) we reduce equaton (5) to: I u S [kg/m 3 s] (7) x Equaton (3) for hgh temperatures can be reformulated to: I f [-] (8) where s the resdence tme and f s the Arrhenus functon of the temperature and the concentraton of fuel and ar. Usng equaton (8) n (7) we get: f u x f S x [kg/m 3 s] (9) If, furthermore, we assume that the temperature and concentraton of the speces are constant for our present system then f s constant and we get: u f S x [kg/m 3 s] (10) At ths pont the followng approxmaton s made (related to 1-D condtons): u x 1 [s/m] (11) and ncludng ths nto (10) the followng s obtaned: S f [kg/m 3 s] (12) Wth the above assumptons we now have a source term for the Iton Parameter, I, descrbed by equaton (12) only vald for hgh temperatures. The equaton (12) was deduced for 1-D constant property condtons. However, t s assumed to be vald n 3-D real condtons.

6 We have now formulated a mathematcal model for the Iton Parameter that takes nto account the chemcal reacton of chan branchng at hgh temperatures. We also need a term for descrbng the chan termnaton of reactons, reducng the amount of radcals and the possblty of ton, especally for colder regons followng hot regons. The snk term for the Iton Parameter Imagne a flammable gas mxture flowng n an nfntely long ppe, whch n turn s heated and cooled along the length. If the Iton Parameter s only calculated by the source term of equaton (12), the model wll predct that the gas mxture wll eventually te. In a real stuaton the coolng of the gas mxture wll termnate some of the chemcal reactons and prolong the tme to ton or prevent the ton. To take nto account the effect of coolng a negatve source term s ntroduced: S B [kg/m 3 ] (13) nk I B s a postve constant and ths snk term wll reduce the value of I n areas where the gas mxture s cooled. The logc behnd equaton (13) s that radcals combne and thereby reduce the rsk of ton. Ths combnaton of radcals s assumed to be ndependent of temperature and solely dependent on the concentraton of radcals. The Iton Parameter I s a measure of the radcals concentraton and hence equaton (13). Combned source and snk term for all temperatures The new expresson for the combned source and snk term s then S f B [kg/m 3 s] (14) I replacng equaton (12) for hgh as well as low temperatures. By unfoldng the expresson for the combned source and snk term (eq. 14) the followng emerges: S E Aexp RT C D fuel ar B I [kg/m 3 s] (15) By the use of the above equatons and assumng deal gases t can be shown that the equaton (15) s related to the ton tme delay of Fgure 1 n the followng way: 1 E Aexp RT C fuel ar D RT B M p [1/s] (16) Ths could be called an extended Arrhenus expresson for the ton tme delay of combustble gases. Please relate to equaton (1) and (2). R s the deal gas constant T s the temperature M s the molar mass of the gas mxture p s the pressure B s a postve constant The results obtaned by CHEMKIN (Fgure 2) were used as emprcal data for the Iton Parameter to calculate the emprcal constants A, B, C, D and E by data fttng.

7 The values for A, E, C and D are found at hgh temperatures and small resdence tme, thereby neglectng the second term on the rght sde of the equaton. The value of B s found at the tabulated autoton temperature for the gas mxture,.e. when the ton tme s nfnte. The nterpretaton of equatons (15) and (16) s as follows. The frst term on the rght sde s due to chan branchng and producton of radcals whle the second term s due to chan termnaton and decay of radcals. For very hgh temperatures, s very small and the second term (decay) may be neglected. For low ton temperatures wth very hgh resdence tme, s very large and decay and producton of radcals are equal (correspondng to left sde = 0). Ths relates to Fgure 1. For hgh temperatures all graphs of ths fgure have straght lnes (left sde of graphs). However, at certan low temperatures all graphs approach asymptotcally a separate vertcal lne wth nfnte ton tme, whch corresponds to solvng equaton (16) for T wth left sde = 0. Expermental setup wth hot steel pellet n gas flow An expermental setup was establshed to fnd expermental data of ton for mxtures of natural gas and ar and to test the model of Iton Parameter. A steel pellet was slowly heated electrcally untl ton of a steady state flow of a premx of gas and ar passng the pellet. The temperature of the pellet at ton was found. Fve dfferent experments wth the steel pellet and dfferent gas/ar mxtures were establshed. The data from these experments were compared to the CFD calculatons of the same cases. An overvew of the test rg wth the test object s shown n Fgure 3. The heght of the test rg s about 2 m and the test object has the heght of 16 cm. y Flow F g Premx x z Test object T 1, T 2 and T 3 T 4 T 9 Fgure 3 Overvew of the geometry of the test rg The flow of premxed gas/ar was from top and down. Fgure 4 and 5 show the test pellet nstalled n the test rg, cold and glowng, respectvely.

8 Fgure 4 Cold test object n test tube Fgure 5 Glowng test object n test tube Results of CFD modellng and experments The cases of the test setup were modelled by CFD solvng the balance equaton (4) wth equaton (15) gvng the source term for I at ton condtons. I s solved for n the volume of nterest. If at any pont I exceeds 1.0, ton wll occur. On the other hand, a hot body n the volume of nterest could be adjusted n temperature untl the calculatons show only one pont wth I above 1.0, whch then ndcates the maxmum temperature (or just below) of the hot body to avod ton n that volume of nterest. The resdence tme s solved for lke I but wth densty as the source term.

9 The boundary condtons for both and I are zero at nlet and adabatc at sold walls. The CFD modellng ncludes flow of the premxed gases, heat transfer and thermal radaton. The radaton model used s the CRG model /4/. Ideally, the results for I should end at 1.0 as maxmum n the calculaton volume n each case. Table 1 shows the actual results for Max I. To estmate how much the CFD model s on the conservatve sde, the temperatures for the test object were reduced n the modellng cases so that the maxmum Iton Parameter (max. I ) was approxmately 1.0. In Table 1 dt s the necessary reducton of the temperatures n the CFD modellng compared to the test case to reach a Max I of 1.0 along the heated test object. Fgure 6 shows examples of CFD results n case 2. Frst results Wth reduced temperatures Test Natural gas Resdence Test object Max. I n ar tme mean temp. Max. I dt [-] [-] [K] [% vol] [s] [K] Table 1 The CFD-calculated Iton Parameter for the test object at autoton condtons n the test rg It should be noted that the Resdence tme of Table 1 s the calculated dfference of from the pont where the flow hts the pellet untl the ton pont wth maxmum I. n Fgure 6 s the absolute resdence tme from nlet of gas flow above the pellet.

10 Fgure 6 Examples of CFD results

11 The present work has shown that by usng ths smple one-equaton model a very good representaton of the ton tme delay (Fgure 2) was reached. Furthermore, usng ths model of Iton Parameter n CFD modellng we have found that the model calculates the ton temperature of actual test cases wthn devatons of only 0-50K and on the lower sde. These devatons nclude all the possble uncertantes from the CHEMKIN modellng, from the derved one-equaton model, from the expermental data and, fnally, from the CFD modellng of the expermental setup. Ths model for ton could be mplemented and appled for dfferent purposes. In the use of CFD wth combuston modellng, ton of the mxture s a crtcal problem. Usng ths ton model could gve a measure of the tme and locaton for ton. When the Iton Parameter n the model exceeds 1.0, ton could be ntated at that pont n the model. Ths would gve a farly good model for ton n the combuston modellng wth CFD. Furthermore, the model could be used n CFD-modellng of safety ssues. When new standards are to be found for explosve and combustble gases ths model could be used for test cases to predct the rsk of ton n each case. Conclusons The model of the Iton Parameter s a novel model, whch gves very good results and may be a great help n predctng probabltes of ton of combustble gases close to hot surfaces. From the results shown n ths paper we conclude that t s possble to calculate and predct the probablty of autoton for natural gas near a heated surface n relaton to a gven natural gas/ar mxture wthn the exploson lmts. The model of the Iton Parameter could be used n all cases where t s desred to know the probablty of ton of a gas/ar mxture. In the present case the model s used for natural gas and ar mxtures. However, any other combustble gas could be modelled n the same way by establshng the emprcal constants related to the model. Acknowledgment Ths project /1/, /2/ has been supported by the Dansh gas companes (The Executve Techncal Commttee) who are greatly acknowledged. References /1/ Sødrng, Thomas W., Autoton Temperature Part 1, Measurements for Natural Gas near a Heated Surface, Project report, DGC, November /2/ Sødrng, Thomas W., Autoton Temperature Part 2, CFD modellng of Natural Gas near a Heated Surface, Project report, DGC, November /3/ Rasmussen, Nels Bjarne, Internal notes, deas and equatons, DGC, /4/ Rasmussen, Nels Bjarne, The Composte Radosty and Gap model (CRG) model of thermal radaton, INFUB 2002 Conference, Estorl Lsbon - Portugal.