Modelling Lifted Methane Jet Fires Using the Boundary Layer Equations

Transcription

1 Modelling Lited Methane Jet Fires Using the Boundary Layer Equations By.S. Cumber * and M. Spearpoint ** Heriot-Watt University * School o Engineering and hysical sciences Riccarton Campus Edinburgh United Kingdom Dept o Civil Engineering ** University o Canterbury rivate Bag 48 Christchurch 82 New Zealand Abstract The ocus o this paper is turbulent lited jet ires. The main objective is to present a lited jet ire methodology using the boundary layer equations as a basis. The advantages o this are inite volume mesh independent predictions o the mean low ields can be calculated on readily available computer resources which leads to rigorous model calibration. A number o lit-o models are evaluated. The model o choice is one based on the laminar lamelet quenching concept combined with a model or the large-scale strain rate. KEYWORDS: lited jet ires, parabolic low model, lamelet quenching, strain rate modelling, combustion Corresponding author: Address p.s.cumber@hw.ac.uk

2 Nomenclature C, C 2, C μ C p,i C s,, C s,2 d 2 k K a l t b burn c d q Rad r r st s T U U V Y i z z L arameters in the k-ε turbulence model Speciic heat o species i Calibration constants in strain rate models Nozzle diameter Mixture raction Variance o mixture raction Turbulence kinetic energy Absorption coeicient Turbulence length scale Turbulence production source term, robability density unction robability o burning Composite probability o burning ercolation threshold robability o burning relating to the location o the luctuating lame base Radiation heat lux Radial co-ordinate Radial location o the stoichiometric concentration Strain rate Temperature Stream-wise velocity Source velocity Radial velocity Mass raction o species i Axial co-ordinate Lit-o height Greek Symbols: Δh Enthalpy perturbation ε Dissipation rate o turbulence kinetic energy

3 ϕ ϕ b ϕ m κ μ e μ l μ t ν l ρ σ σ Δh σ ε σ k Generic low variable Burning lamelet Isothermal mixing lamelet Von Karman Constant Eective viscosity Laminar viscosity Turbulent viscosity Kinematic viscosity Density Stean Boltzmann constant Turbulent randtl number or speciic enthalpy perturbation Turbulent randtl number or the dissipation rate o turbulence kinetic energy Turbulent randtl number or the turbulence kinetic energy Subscripts: adia Adiabatic property amb Ambient value q Quench value st roperty at stoichiometric conditions Initial condition or source condition Over bars - Reynolds averaged quantity ~ Favre averaged quantity 2

4 Introduction Turbulent jet lames occur in many areas o industry either by accident such as ollowing the ignition o a leak rom a high pressure plant processing hydrocarbons or in a controlled environment such as a urnace or industrial burner. In a turbulent jet the low is characterised by the diameter o the nozzle or pipe, the uel composition and the speed o the uel at the nozzle exit. For a given uel and nozzle diameter, i the jet speed is suiciently low the lame is attached or rim-stabilised. I the jet speed is gradually increased then at some critical velocity the jet lits o and the lame structure undamentally changes. Immediately downstream o the nozzle the low is non-reacting, urther downstream combustion is initiated at a luctuating lame base ollowed by the main body o the jet lame. For most uels and nozzle diameters the critical velocity is relatively low, or example, [] give a critical velocity o 5.4 m/s or a propane jet ire with a nozzle diameter o 8.74 mm. Thereore the understanding o lited jet ires is o interest to industrial combustion engineers. A wide range o small-scale [2-5] and large-scale experimental and theoretical studies o lited ires [6-8] exist in the open literature. In some situations lited lames are encouraged as it has been shown that lited lames can reduce NO x levels in urnaces, [9]. The computational modelling o rim-stabilised ires is mature with good agreement between the predicted and measured mean temperature and major chemical species established, [-2]. This is not the case or lited jets where computational studies have ocussed on understanding the mechanism or the location o the combusting lame base and good agreement between predicted and measured low properties is not universal. The possible mechanisms or the location o the lit-o height will be considered urther below. Where turbulence-radiation eects are small, which has been shown to be the case or laboratory scale methane jets [], predicted received radiation heat luxes surrounding a rim-stabilised ire are in reasonable agreement with measured luxes [,3,4], whereas or lited ires this as yet has not been considered seriously. The mechanism or lame lit-o is a competition between chemical and mixing processes characterised by a chemical reaction time and a low residence time. The 3

5 ratio o these time-scales is deined as the Damköhler number. As the jet velocity increases the low residence time decreases. At the critical velocity or Damköhler number there is insuicient time or the chemical reaction to take place and lit-o occurs, [4]. This is the accepted physical explanation or lame lit-o but the mechanism involved in lame stabilisation at the lit-o height ater over three decades o research is still debated vigorously. The early accepted theory was the lito height was determined by the downstream location on the stoichiometric contour where the mean jet velocity equals the turbulent burning velocity. The location o the lame base is governed by the premixed nature o the uel air mixture. Computational models using this criterion were able to predict lit-o heights or the jet ires considered [5]. Experimentalists were also able to correlate measured lit-o heights or a range o uels using the premixed assumption and the turbulent burning velocity, [2]. eters and Williams [6] were the irst to question this theory suggesting that insuicient premixing could occur or the premixed theory to be valid in the lited jet ires they considered. eters and Williams based their analysis on itts [7] experiments, proposing another mechanism, based on laminar lamelet quenching that relies on an analysis o laminar lamelets distorted by the turbulence ield. At the luctuating lame base the lited lame can be considered as an ensemble o laminar lamelets in the instantaneous ield; the laminar lamelets varying rom air-uel mixtures at or near stoichiometric conditions to ully entrained air. Each o the lamelets can be considered to be burning or non-reacting depending on the local instantaneous composition and level o turbulent distortion or stretch the lamelet is subjected to. The location o the lame base is determined by the proportion o burning to non-reacting lamelets. A candidate property or characterising the degree o stretch o the lamelets is the scalar dissipation rate, [8], although Sanders and Lamers [4] present a convincing argument against the scalar dissipation rate and suggest the strain rate as the appropriate parameter. Muller et al. [5] use a model or the strain rate o the largest eddies to predict the lit-o height, whereas Sanders and Lamers avour the small-scale strain rate. Sanders and Lamers base their conclusions on a calibration o lit-o height using isothermal simulations, the argument being the lit-o height is primarily determined by the low upstream o the lame base. As well as the two theories discussed above itts [3] analysed a number o lited jet ires and 4

6 suggests that the location o the lit-o height is determined by large-scale rather than small-scale structures in the jet, and isothermal mixing upstream o the lit-o region is important in determining the location o the lame base. Even though itts presents a compelling case or his avoured mechanism he still states that models based on a premixed theory or small-scale turbulent structures can not be discounted. More recently Upatnicks et al. [9] using a high speed photography technique, Cinema IV identiied another mechanism or the location o the lit-o height based on an analysis o the edge o the lame. One reason or the lack o clarity as to the correct mechanism is all theories show some degree o agreement with observation, and in many experiments there is considerable overlap in prerequisite conditions or each theory to apply. In this paper a model originally proposed by Sanders and Lamers [4] based on the laminar lamelet quenching concept is implemented although the conclusion as to the appropriate scale o the strain rate model diers rom Sanders and Lamers. The main objective o this paper is to present a lited jet ire methodology based on a parabolic low model and as such the use o laminar lamelet quenching is a convenient lit-o model. The methodology presented with minor modiication could be implemented with other lit-o models. The overall computational ramework is a parabolic low solver, utilising the boundary layer equations, similar to GENMIX, [2], extended to include elliptic eatures ound in lited ires. In lited jet ires the important elliptic eatures are the eedback mechanisms downstream o the lit-o region that determines the lit-o height and local lame structure. In the context o the lit-o model implemented here, Sanders and Lamer s model, [4], the elliptic eedback mechanism is introduced through the probability o burning ield. The details o implementation are considered urther below. As jet ires have a dominant low direction it is common or the lame structure o a rim-stabilised jet ire to be calculated in this way, [-2]. This is the irst time lited lame predictions based on a parabolic low model have been reported. The computational advantages o a parabolic low model over a ully elliptic solver are clear, as no iteration cycle o a pressure correction algorithm is required in its solution as the dominant low direction has a time like quality allowing a marching procedure in the dominant low direction. This makes it possible to calculate complex lame structures using readily available computational resources. For example Wang and Chen, [2] have recently reported 5

7 the computation o a laboratory scale rim-stabilised lame, Flame D in re. [22]; the simulation is calculated using a DF transport model with detailed chemistry, including 53 species and 32 elemental reactions and a multi-time-scale k-ε turbulence model. Either o these modelling approaches in isolation would make the jet ire calculation prohibitively computationally intensive when combined with an elliptic solver on a standard C. Using a low solver based on the boundary layer equations Wang and Chen report a run-time o the order o week on a relatively high speciication C. The use o a parabolic solver is not only an issue o convenience as it makes it relatively simple to demonstrate mesh independence in any predicted low ields presented. In addition any model calibration is purely dependent on the quality o the experimental data used in the calibration. The issue o robust calibration is a particularly important one as without it, model development looses rigour. In this paper the inal issue addressed is the received radiation heat lux distributions sensitivity to the representation o the lit-o region. In the next section the mathematical model is presented. The details or the most part ollow Sanders and Lamers [4] ormulation but diers in a number o key areas. The model basis as described by Sanders and Lamers is repeated here or completeness and convenience. Mathematical Model The basis o the low equations is the parabolized Favre averaged Navier Stokes equations in an axisymmetric coordinate system. The system is closed using a variant o the k-ε turbulence model. ~ ~ ρ U k rρ V k + z r r μ e = r r r σ k k + ρ r ( ε ) ~ ~ ρ U ε rρ V ε μe + = r z r r r r σ ε ~ U = μ e r 2 ε ε + ρ r k ( C C ε ) 2 6

8 C μe = μl + C μ ρ k ε ~ k du = dz ε C μ =.9, C 2 =.9, σ k =, and σ ε = cl The version o the k-ε turbulence model given above is a modiication o the typically implemented variant o the k-ε turbulence model (C =.44, C 2 =.92) to take account o the round jet/ plane jet anomaly, [23] where the spreading rate o round jets tends to be over predicted by the standard version o the turbulence model. This is a well known limitation o most two equation turbulence models and 2 nd moment closure models unless some modiication is introduced to account or the reduced spreading rate. Indeed the 2 nd moment closure model o Jones and Mussonge [24] was also applied to the jet ires considered here with little or no improvement. The modiication to the k-ε model introduced above is due to Morse, [25] and has been used successully in previous rim-stabilised ire simulation, [25]. The axisymmetric correction is used here as it gives an appropriate balance between model complexity and predictive capability. In addition a urther modiication to the turbulence model to account or buoyancy induced turbulence was implemented, [26] but ultimately rejected as the improvement in the mean temperature ield was marginal at best. Turbulent Combustion Model The turbulent combustion model is a laminar lamelet combustion model, with two lamelet libraries, one or combustion and the other or isothermal mixing. Combustion is assumed to be ininitely ast with a prescribed probability density unction, a β unction, [27]. The shape o the β unction at any spatial location is determined by a conserved scalar, the mixture raction and its variance 2, which are calculated using modelled transport equations, []. The combusting lamelet is calculated using a laminar counter low non-premixed combustion simulation using a detailed kinetic scheme at a strain rate o 9 sec -, []. Any mean property can be calculated as a weighted average o the burning and isothermal mixing lamelet weighted by the pd and integrated over instantaneous mixture raction, 7

9 8 ( ) ( ) ( ) ( ) ( ) + = ~ d d m b d b b d ϕ ϕ ϕ where b is a probability o burning deined below and d is a probability that the axial location is above the luctuating lame base. ( ) > = L L base d dz z z Sanders and Lamers prescribe the pd or the location o the instantaneous lame base, z L to have a triangular shape, the apex located at the mean lit-o height and the base o the triangle is taken to be ive diameters. It should be noted that this is an assumption o convenience rather than one based on observation, however a sensitivity study has shown that the overall lame structure is insensitive to this aspect o the model. The mean density and mean adiabatic temperature are given by the relations, ( ) ( ) ( ) ( ) + = d d m b d b b d ρ ρ ρ ( ) ( ) ( ) ( ) ( ) ( ) ( ) + = d T d T T m m m b d b b b b d adia ρ ρ ρ ρ To account or radiation heat loss a transport equation or a speciic enthalpy perturbation is solved, ( ) ~ ~ amb a h e T T K r h r r r r h V r r z h U + Δ = Δ + Δ Δ σ σ μ ρ ρ where radiation heat loss is introduced using the optically thin approximation, [28]. The absorption coeicient, K a is adjusted to gain agreement between the model and laboratory scale jet ire temperature measurements. The optically thin approximation is valid or laboratory scale methane jets. This approach has been used successully in other computational studies [, 2]. The mean temperature is then calculated as, Δ = i i p i adia C Y h T T,

10 Y i is the mass raction o species i and the speciic heat at constant pressure, C p,i is evaluated rom curve its in temperature to JANNAF thermodynamic property tables or each species, [29]. To close the system the probability o burning and the mean lit-o height must be modelled. The probability o burning is given by, b s = q () s ds s is the strain rate, s q is the quenching strain rate and (s) is a quasi Gaussian pd or the strain rate. Sanders and Lamers [4] give a quench strain rate o 565 sec - derived rom an analysis o diluted methane-air counter low diusion lames. The strain rate can either be taken to be the strain rate o the small-scale turbulence, s = C s, ε 2 ν l / 2 Or the strain rate o the large-scale turbulence. s = C s,2 ε k Where C s, and C s,2 are calibration constants. The laminar dynamic viscosity is given by Sutherland s law, the constants in the ormula can be ound in Kalghati [2]. The mean lit-o height is prescribed using percolation theory, [6] and the probability o burning on the stoichiometric contour, L ( ) z = b c s r= r st = c s q / 2 π the above equation is an approximation to the error unction and implicitly gives the percolation threshold, c, see Sanders and Lamers[4] or more details o the lit-o model. As stated above the combustion model implemented is the one proposed by Sanders and Lamers, with some crucial dierences that ultimately eects the overall conclusions o this article. It is thereore o interest to consider the dierences 9

11 between the mathematical models described above and Sanders and Lamers approach. The main dierences are the model described above includes radiation heat loss via the speciic enthalpy perturbation transport equation and the turbulence model implemented accounts or the round jet/plane jet anomaly, [23]. These dierences in model basis modiy the predicted mean temperature ield, and the turbulence ields. To appreciate the importance o taking account o the reduced spreading rate in round jets compared to planar jets Figure shows the predicted and measured temperature ield at three downstream locations or a jet ire. This is a rim-stabilised methane jet ire measured by Jeng et al. [], the jet has a source diameter o 5 mm and a source velocity o 49.8 m/s giving a Reynolds number o,7. For each axial measuring station two predictions are shown, one using the round jet correction and the other using the standard turbulence model constants. The important dierence between the two model predictions is the dierent spreading rate, with the corrected model giving more accurate predictions o the mean temperature ield. There is some evidence or the over prediction o the spreading rate by Sanders and Lamers model, Figure 2 in [4] where the predicted radial position o the stoichiometric concentration is compared with Horch s data [3], good agreement is exhibited between Sanders and Lamers model and Horch s measurements or the irst 2 diameters ater which the agreement deteriorates.

12 a) 2 T/K 5 Measured Axisymmetric correction Standard Constants r/d b) 2 5 T/K r/d c) 2 5 T/K r/d Figure. Radial distributions o mean temperature at a) z/d=52.5, b) z/d=2 and c) z/d=5.

13 A third lit-o model is considered, based on a correlation or lit o height using a turbulence time-scale threshold. z L k 5. E 3 ε The turbulence time-scale is evaluated on the jet axis. This lit-o model is considered here as it is consistent with a parabolic low model without modiication and has been used in the past to predict the lit-o height o methane jet ires in a cross-low, [27, 3-33]. The turbulence time-scale correlation was derived by Chakravarty et al., [34]. In this model the transition rom no reaction to combustion is instantaneous at the lit-o height. Boundary Conditions and Numerical arameters In all o the simulations presented below the bulk inlet conditions are given by the nozzle diameter and the average source velocity. The mean stream-wise velocity distribution and radial velocity distribution are taken to be consistent with ully developed pipe low, that is a /7th power law is prescribed or the stream-wise velocity distribution and zero or the radial velocity component. Nozzle exit turbulence proiles are given by the ormulae, U k =.5 l t d d min κ r,. 2 2 = C.75 μ ε = k. 5 l t For all simulations in this article 4 control volumes in the radial co-ordinate direction spanning the jet radius are used to calculate the lame structure, with a maximum ractional step in the axial direction o less than 2% o the radial control volume spacing. A number o simulations using 8 control volumes were also completed to conirm that the predictions presented are independent o urther mesh reinement. It 2

14 is estimated that the predicted lit-o heights are within 2% o the ully mesh converged values. Thermal Radiation Modelling The external radiation heat lux distribution is modelled using a variant o the discrete transer method, [35], modiied using staggered ray meshes, [36, 37] to minimise the ray eect, [38]. Spectral emission rom the ire is modelled using a statistical narrow band model, RADCAL, [39]. In the predictions o the external radiation heat lux presented in the ollowing sections the spectrally integrated intensity is evaluated rom integrations over the spectral window, μm- μm with approximately 3-4 narrow bands. The predicted heat lux distributions are evaluated with 768 rays per receiver. 768 rays/ receiver was ound to be suicient or the predicted radiation heat lux distributions to be independent o urther ray reinement, as increasing the number o rays used to 372 rays/ receiver changed the lux by less than 5%. Description o the Numerical Algorithm When considering how a lited lame diers rom a rim-stabilised lame in the context o the mathematical model described above, the key dierence is the composite probability o burning ield, burn = b d In a rim-stabilised lame burn is one everywhere, whereas in a lited lame this takes a value between zero and one. Zero below the lit-o region, between zero and one in the lit-o region and one above it. Thereore an algorithm based on a guess and correct approach or the composite probability o burning ield suggests itsel as one way o extending a parabolic low model suitable or simulating rim-stabilised jet ires to simulate lited jet ires. An overview o the algorithm in the orm o a low chart is given in Figure 2. To initialise the process some estimate o the composite probability o burning ield must be prescribed. There are a number o possibilities, or example the ire could start o as being rim-stabilised, hence the initial composite probability o burning ield could be set to one everywhere. An alternative choice is to prescribe the jet to be isothermal, with a composite probability o burning o zero 3

15 everywhere. Both options have some appeal as in the initial rim-stabilised ire approach ( burn, =), this is how a lited jet ire would initiate, alternatively the analysis o itts [3] and Sanders and Lamers [4] work on isothermal jets indicates that the non-reacting isothermal region upstream o the lame base is important in the lame stabilisation process, avouring the second approach ( burn, =). Start Initialise burn, ield and mean lit-o height (z L, ) Calculate Flame Structure Store ields relevant to z L prediction and burn Calculate new z L and burn Change in lit-o height is large? Yes No Stop Figure 2. Flow chart o the lited let ire methodology. 4

16 Once the composite probability o burning ield is prescribed, an estimate o the lame structure can be calculated by running the parabolic low model. As the lame structure is calculated low ields required to estimate the lit-o height and the composite probability o burning ield, such as the mean mixture raction, mean temperature, turbulence kinetic energy, and the dissipation rate o the turbulence kinetic energy, are stored. Once the lame structure is complete, a new estimate o the lit-o height is calculated and the composite probability o burning ield recalculated. I the change in lit-o height is small the algorithm is terminated otherwise the lame structure is recalculated and the process continues. Large-Scale Strain Rate In the description given above there are a number o issues identiied or statements made that require urther clariication? Figure 3 shows the convergence history or our simulations. The relative change in mean lit-o height is plotted against the iteration number. The jet ire simulated is the same in each case, (nozzle diameter 8mm, initial jet velocity o 7 m/s), or two o the simulations the large-scale strain rate sub-model is used, with dierent initial composite probability o burning ields and or the other two simulations the small-scale strain rate sub-model is used, again with two dierent initial composite probability o burning ields. In the convergence histories labelled as hot start the initial guess or the composite probability o burning is one everywhere. The convergence histories labelled as cold start the initial guess or the composite probability o burning is zero everywhere. Considering the simulations using the large-scale strain rate sub-model little sensitivity to the initial guess or the composite probability o burning is shown. The cold start simulation is marginally superior. Note that convergence is monotonic or the irst twenty iterations ater which round-o error prevents urther convergence. However the dierences in predicted lit-o heights or successive iterations is less than.% and changes o less than % are suicient or the lit-o height to be converged to the visual resolution o Figure 4. In Figure 4 the lit-o height as a unction o iteration number is shown or the our simulations. Figure 3 and Figure 4 taken together suggest the algorithm takes around 5-6 iterations to converge when the large-scale strain rate sub-model is used in the lit o model. 5

17 Relative dierence.e+.e-.e-2.e-3.e-4.e-5.e-6 Large-scale strain rate Small-scale strain rate 'Hot' start 'Cold' start Iteration number Figure 3. Convergence histories or a lited methane jet ire 'Hot' start 'Cold' start zl/m Iteration number Figure 4. redicted lit-o height as a unction o iteration number. Small-Scale Strain Rate Considering the convergence histories or the small-scale strain rate simulations, the situation is less satisactory. Monotonic convergence occurs or the irst ive iterations, ater which no urther convergence is achieved, with the change in lit-o height tending to approximately cycle through three dierent values all within % o 6

18 each other. The poor convergence behaviour o the simulations using the small-scale strain rate sub-model is believed to be due to its dependence on the dissipation rate o turbulence and the kinematic viscosity. These two properties are weakly correlated and when the lit-o height changes these two properties respond dierently to the change in the mean temperature ield. This convergence problem is lessened when more control volumes are introduced in the radial co-ordinate direction, with o course an increase in computational cost. The large-scale strain rate sub-model does not suer rom this problem as its dependency on turbulence parameters alone means that it responds in a consistent way to changes in the mean temperature ield brought about by changes in the predicted mean lit-o height. The small-scale strain rate simulations exhibit sensitivity to the initial composite probability o burning although both the hot and cold start simulations ultimately converge to the same lit-o height. The reason or the sensitivity can be explained by considering the predicted lit-o height as a unction o iteration number, see Figure 4. For the small-scale strain rate simulation with a cold start, the irst estimate o the mean lit-o height is over twice the converged value and on successive iterations decreases to the mean lit-o height, taking approximately orty iterations to reach it. Figure 4 also shows that alse convergence is possible i just the change in mean lito height or successive iterations is used to monitor convergence. Comparing Figure 3 with Figure 4, or the small-scale strain rate simulation with a cold start, Figure 3 suggests that convergence is achieved in approximately twenty iterations i a convergence criterion o % change in lit-o height in successive iterations is prescribed; whereas in Figure 4 it is clear that convergence requires at least orty iterations. Lit-o Height Model Calibration Similar to Sanders and Lamers [4] the lited jet ire model is calibrated using Wittmer s measurements o lit-o height or a methane jet with an initial velocity o 7 m/s and a nozzle diameter o 8 mm. Fitting the models to this data gives values o C s, =.33 and C s,2 =6 7

19 or the calibration constants in the strain rate sub-models. Sanders and Lamers only calibrated the small-scale strain rate sub-model or combusting jet simulations, giving a value o C s, =.27. For the large-scale strain rate sub model Sanders and Lamers give a calibration constant o C s,2 =6.4 but this was using isothermal jet simulations to predict the lit-o height. Sanders and Lamers did not calibrate the large scale strain rate model using combusting jet simulations due to the large computational cost o using an elliptic solver. The calibration constants used here are o the same order as those given by Sanders and Lamers, the dierences are due primarily to the variant o the k-ε turbulence model used and the numerical resolution o the simulations. Model redictions Figure 5 shows a comparison o the measured and predicted mean stream-wise velocity along the axis or the lited jet ire used to calibrate the strain rate submodels. Three predictions are shown, one using the turbulent time-scale sub-model or the lit-o height, a prediction using the large-scale strain rate sub-model and a prediction using the small-scale strain rate sub-model. Below the lit-o region all three models are identical as expected; urther downstream the dierence between the three models is small. Overall all three models agree with the measurements, the turbulent time-scale model giving slightly better agreement. For convenience Sanders and Lamer s prediction o mean stream-wise velocity using the small-scale strain rate model is also shown in Figure 5. Sanders and Lamer s prediction has a noticeable spike in the axial velocity at the lit-o height. Sanders and Lamers suggest this is due to the acceleration o the low at the lame base because o the hot gas expansion due to initiation o combustion. The low would accelerate to some degree due to the hot gases expanding, however this is not consistent with Wittmer s measurements although the spacing between measuring stations may be too ar apart to capture the spike, i it were present. 8

20 U/U Measured Large scale strain rate Small scale strain rate Turbulence timescale Sanders and Lamer's prediction z/d Figure 5. Stream-wise velocity on the jet axis (d=8mm, U =7 m/s). The predicted and measured lit-o height as a unction o source velocity is shown in Figure 6. Again three predictions are shown, the lit-o height based on the turbulence time-scale, and the lit-o height predicted using the small-scale and largescale strain rate sub-models. The turbulence time-scale model gives a reasonable approximation o the lit-o height, however it should be noted that i this model were recalibrated to predict the high speed jet ire then its agreement or the other jet ires would worsen. O the other two predictions the large-scale strain rate model gives the closest agreement to the measured lit-o height. The error or the lited jet with a source velocity o 4 m/s using the large-scale strain rate sub-model is less than 5% compared to the small-scale strain rate model prediction, where the error is over 3%. erhaps a more useul measure o agreement is a comparison o the gradient o the lines, d z L du 9

21 4 3 zl/d 2 Measured Large scale strain rate Small scale strain rate Turbulence timescale U/m s - Figure 6. Lit-o height vs. source velocity. The measured gradient is 4.3x -3 sec compared to the predicted gradient o 3.45x -3 sec using the large-scale strain rate sub-model, 5.2x -3 sec using the smallscale strain rate and.84x -3 sec or the predicted lit-o heights based on the turbulence time-scale model. As mentioned above Sanders and Lamers compared the two strain rate sub-models ability to predict Wittmer s lit-o data and based on isothermal simulations concluded the small-scale strain rate was the superior o the two strain rate submodels. To investigate the dierence in conclusion between Sanders and Lamers, [4] and the present work, the turbulence model constants, C and C 2 were changed to the values used by Sanders and Lamers and the strain rate sub-models recalibrated. The good agreement o the predicted lit-o height using the small-scale strain rate submodel with Wittmer s data and the under prediction o the gradient, dz L /du or the large-scale strain rate sub-model was reproduced; see Figure 3 in [4]. However it must be emphasised the turbulence model constants used by Sanders and Lamers are not appropriate or axisymmetric ree jets as discussed above. 2

22 Figure 7 shows the predicted temperature ield at the lame base or all three lit-o models. The predicted temperature ield where the turbulence time-scale threshold is used to predict the lit-o height is unrealistic in that the combustion model takes no account o the luctuating nature o the lame base hence the step change in temperature at the lame base. The other two predicted temperature ields are more realistic in that no jump in temperature exists. Figure 8 shows the temperature ield or the whole lame or all three lit-o models. In each case the temperature contours plotted are the same showing that the temperature ield downstream o the lit-o region is insensitive to the way the lit-o region is modelled. a) b) c) z/d 4 z/d 4 z/d r/d 2 r/d 2 r/d Figure 7. Mean temperature ield in the lit-o region, a) Large-scale strain rate prediction, b) Small-scale strain rate prediction, and c) turbulence time-scale prediction. 2

23 a) b) c) z/d z/d z/d r/d 2 r/d 2 r/d Figure 8. Mean temperature ield. a) Large-scale strain rate prediction, b) Small-scale strain rate prediction, and c) turbulence time-scale prediction. Radiation Field o a Lited Jet Fire One inal consideration is the inluence o the lit-o model on the received radiation ield surrounding a jet ire. This is o interest as in many applications the lame structure is only o interest as a requirement or evaluating the received radiation ield surrounding the ire. As the predicted mean temperature ields downstream o the lito region are similar or all three simulations, see Figure 8, the thermal radiation distribution in the ar ield should be similar. Figure 9 conirms this is the case. The dierences in radiation heat lux distributions are consistent with the dierent mean temperature distributions in the lit-o region. In most saety applications it is the ar ield radiation heat lux distribution that is o major concern. The simulations in this paper suggest that the mean temperature ield in the lit-o region is only important i 22

24 the near ield thermal radiation distribution is o interest, such as i eedback to a lare tip were an issue. 6 qrad / kw m Large scale strain rate Small scale strain rate Turbulence timescale r/m Figure 9. Heat lux distribution in a horizontal plane through the nozzle surrounding a jet ire. Conclusion In this article an algorithm or calculating the lame structure o lited jet ires is presented. The algorithm is based on an extension o a ire model used to simulate rim-stabilised lames. An important characteristic o the lited jet ire methodology is the lame structure is calculated using the boundary layer approximation. The advantage o this is inite volume mesh independent predictions o the mean low ields can be calculated on readily available computer resources. The simulations presented here typically required o the order o 5 minutes run-time on a C with a 45 MHz entium III processor. This has allowed a rigorous calibration o the lit-o models considered. Three lit-o models were implemented and evaluated. The simplest uses a turbulence time-scale threshold to determine the lit-o height. This model is consistent with the boundary layer equations and can be implemented easily, but leads to unrealistic predictions o the mean temperature in the lit-o region. The other two lit-o models were ormulated by Sanders and Lamers, [4] and are based on laminar 23

25 lamelet quenching or the lame stabilisation mechanism at the lame base. The two lit-o models are dierentiated by the strain rate sub-model implemented. One is a small-scale strain rate sub-model; whereas the other is a large-scale strain rate submodel. The use o the boundary layer equations means the lit-o models calibration is rigorous. The lit-o models were evaluated by comparing predicted lit-o predictions and measurements taken by Wittmer. In this study the large-scale strain rate sub-model proved to be more accurate and the most numerically stable o the two strain rate sub-models. This is in contradiction to Sanders and Lamers who avour the small-scale strain rate sub-model. The major dierence between Sanders and Lamers analysis and the present article is the turbulence model implemented. The use o an axisymmetric correction to the turbulence model proved critical in evaluating the capabilities o the lit-o models. The next step is to validate the lit-o models using a wider range o measurements, both other lit-o data and detailed measurements o the mean temperature and chemical species in the lit-o region. This task is in hand and will be reported in a subsequent publication. Finally the inluence o the lit-o model on the external radiation heat lux distribution was considered. It was ound the radiation heat lux distributions in the ar ield were insensitive to the lit-o model implemented, consistent with the prediction o the mean temperature ield calculated using each lit-o model. Acknowledgements This research was undertaken while Dr eter Cumber was on a research visit to the Department o Civil Engineering, University o Canterbury, Christchurch, New Zealand. He would like to thank the Royal Academy o Engineering or the Global Research Award and the ESRC or the Overseas travel Grant, GR/T22735 that made this secondment possible. 24

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