1D PHENOMENOLOGICAL MODEL ESTIMATING THE OVERPRESSURE WHICH COULD BE GENERATED BY GAS EXPLOSION IN A CONGESTED SPACE

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1 1D PHENOMENOLOGICAL MODEL ESTIMATING THE OVERPRESSURE WHICH COULD BE GENERATED BY GAS EXPLOSION IN A CONGESTED SPACE Shabunya, S.I. 1, Martynenko, V.V. 1, Till, M. 2 and Perrin, J. 2 1 Heat & Mass Transer Institute, P.Broka Str., Minsk, , Belarus 2 CRCD, Air Liquide, B.P. 126, Les Loges-en-Josas, 78354, France ABSTRACT A phenomenological approach is deeloped to calculate the elocity o lame propagation and to estimate the alue o pressure peak when igniting gaseous combustible mixtures in a congested space. The basic idea o this model is aterburning o the remanent uel in pockets o congested space behind the lame ront. The estimation o probable oerpressure peak is based on solution o one-dimensional problem o the piston (haing corresponding symmetry) moing with gien elocity in polytropic gas. Submitted work is the irst representation o such phenomenological approach and is realized or the simplest situation close to one-dimensional. 1. Introduction The problem o an estimation o hazard and consequences o industrial accidents caused by explosion o combustible gases is crucial. It is now stimulated by the deelopment o new applications o hydrogen as a uel material. Direct modeling o industrial accidents inoles key diiculties, which makes it a real challenge. Though the computational power continuously grows, CFD modeling o such tasks still requires huge resources and cannot guarantee the desired accuracy. In such a situation phenomenological models hae their own range o applicability since they are bound with simpliied consideration o real processes. The simpliication o the physical pattern allows estimating process parameters promptly, and phenomenological models are ery useul at the irst stage o problem inestigation. Although it is diicult to imagine that essentially new phenomenological approaches could appear or problems inestigated or many years, some elements o aailable models could be deeloped. For example the phenomenological estimate o the excess pressure due to an explosion is based on a model o burning in an enclosure with a ent hole. It is assumed that the bulk pressure is spatially uniorm and the eolution o pressure in time is controlled by burning and by the outlet elocity o the gas in the ent hole. The most known implementations o such approach are the models SCOPE (Shell Code or Oerpressure Prediction in gas Explosions) and CLICHE (Conined LInked CHamber Explosion). These models are described, or example, in [1,2] and [3,4]. Prima acie the presence o a ent hole is a key element in these models, and the hole size should be small in a comparison with the total cross-section o the channel, to proide a alue o excess pressure about hundreds mbars. In principle explosions in open congested areas cannot be described within this ramework. Moreoer this approach o an incompressible medium assumes that typical elocities in the considered area are smaller than the speed o sound. This assumption eliminates the opportunity to simulate modes with intensie shock waes, but the presence or absence o a closure and ent holes does not aect the correctness o such approach. It means that as long as the relations or incompressible medium can be applied to determine the position o lame ront (and the alue ront elocity), the model should yield reasonable estimations o low parameters. It is a motiation to reormulate a phenomenological model. A basic way to adjust the models SCOPE and CLICHE consists in tuning the parameters controlling the elocity o turbulent burning. As a matter o act, in a one-dimensional phenomenological 1

2 approach, the relations calculating the elocity o turbulent burning are used or model tuning. In such relations the usage o dimensionless criteria o Reynolds, Karlowitz, Markstein, Lewis does not look not so ruitul, especially in the case o a congested space, where the word turbulence coers all eatures o scale and geometry o a congestion. 2. Goerning equations In the work presented here, it is proposed to modiy the phenomenological description o burning in a congested space. For this purpose the model o turbulent burning is replaced by a model o aterburning o residual uel behind lame ront. In such an approach the considered space is separated into two areas: the area behind the lame ront and the area ahead o the ront. The concepts behind and ahead are bound with the direction o ront propagation. Unburnt uel remains partially behind the ront in stagnant (ortex) areas o congested space as well as in boundary layers in icinity o solid suraces (ig. 1). Figure 1. Propagation o lame ront through congestion (combustible mixture is shown by grey colour, biscuit one denotes combustion products). Within such approach the concept o turbulent combustion regime is not used at all. The congestion characteristics o the residual uel hae a geometrical explanation, and make more room or phenomenological intuition, than the concept o elocity o turbulent burning. The burning elocity in the proposed model diers rom the elocity o a laminar lame only by a parameter accounting or the ront wrinkling. This coeicient usually has alue within limits rom 2 up to 3, i.e. burning elocity (in air) or dierent combustibles is within range rom ~1 m/s (methane) up to ~7 m/s (hydrogen). Such alues considerably dier rom the parameters used in conentional approaches, where the elocity o turbulent burning reaches alues o tens meters per second. The acceleration o the lame ront happens owing to aterburning o residual uel behind the ront instead o a transition to a turbulent combustion regime. The combustion o this uel can be rather prompt, as the surace o burning can be ery large. In turn, the accelerated ront transits bigger olume o congested space igniting residual uel in it. The process occurs as long as the lame ront encounters combustible domains. The combustible mixture leaking in ree space is mixed with ambient air and other laws control the burning o this mixture. The processes o «exterior burning» is not considered here. 2

3 The assumption o a constant pressure in the considered area means, that gas motion occurs at elocities smaller than the speed o sound. In other words the hot gases hae time to extend so to equalize pressure in the area ater combustion o the uel. This approach restricts the range o model applicability to rather weak explosions. As it is known rom the practice o gas-dynamic calculations, when disregarding compressibility eects it is still possible to obtain reasonable computational results up to a characteristic elocity o 70-80% o the sound speed (~ m/s). I in the computational domain the elocity o the combustion ront exceeds the sound speed, the model is not correct, and another procedure is required to estimate the alue o the pressure peak. In order to describe the kinematics o a motion o lame ront let us ormulate the continuity equations, energy conseration law, and the state o ideal gas at constant pressure: ρ + (, ρu) = 0 t T cpρ + ( u, T) = Q t A ρ0t0 ρ = = T T (1) Substituting the equation o state, the continuity equation can be written as: T ( u, T) + T(, u) = 0 t (2) The equation combining low ield with heat sources is obtained rom equation (2) and the energy conseration law: Q = (3) c A (, u) p In the one-dimensional case equation (3) allows calculating the dependence o elocity on coordinate using known distribution o heat source. The heat source Q in reduced equations is perceied both as due to burning in lame ront and olumetric aterburning o residual uel. Equation (3) is quasistationary, as Q is unction o coordinates and time. The alue o A also can be unction o time, i pressure increase in area behind lame ront has to be taken into account (or example the burning in closed olume with a ent hole). In order to determine the low ield in 2D and 3D ariants o such model the additional condition o low potentiality is required. Such requirement (absence o ortexes) does not contradict the preious reasoning about residual uel in ortex areas. It is known that the potential solutions with discontinuous low oer obstacles describe well actual streamlines and elocity distributions (excluding alone ortex structures). Earlier we use double term: stagnant (ortex) areas to emphasize that the ortex ormations in actual low is stagnant areas in the potential solution. For the subsequent estimation o pressure peak it is necessary to take into account that such method o modeling approximates well a kinematics o low pattern, but underestimates its dynamic parameters. Spatial coordinate in the one-dimensional approach is perceied as a line a perpendicular to a surace o lame ront. Such interpretation allows considering more complex situation than one-dimensional geometry. Howeer it is more conenient to present the model at consideration o simple geometry (spherical symmetry, cylindrical symmetry, one-dimensional channel). In case o complex conigurations with seeral ent holes the determination o position o lame ront and its elocity becomes an additional problem, which is not considered at the present moment. The experiments with open areas are selected to compare the data with calculations, i.e. at calculation o a kinematics it is 3

4 possible to consider that pressure in computational domain is approximately equal to atmospheric pressure and leae out o account the change o A with time. Taking into account pressure increase due to impeded lowing out has not key diiculties, but it is not required or presentation o model. In proposed model the key element is the amount o unburnt uel remaining behind lame ront. Let us introduce the dimensionless unction Ψ to describe the eolution o residual uel. This unction characterizes a portion o unburnt uel behind the lame ront. As in considered problems the initiation o explosion always happens in some point, it is reasonable to ormulate unction Ψ in orm o spatial dependence on distance rom this point. The elements creating congestion hae own characteristic sizes. Thereore the characteristics o residual uel should depend on these sizes. Initial alue o unction Ψ, i.e. its alue at the moment o transit o lame ront depends on properties o congestion and ront elocity. The direction o lame propagation is important also, especially when the coniguration o congestion is ar rom isotropic one. The construction o one-dimensional unction Ψ by gien geometry o congestion is a separate task; its successul solution determines the accuracy o estimations made by this model. In this direction the irst steps are made only. Let us accept the ollowing unctional dependence Ψ (, ru ( tr),; tαi) or this unction. Here: r is the distance rom a point o mixture ignition, t r is the time, when the lame ront is in the point r, u ( t r ) is the elocity o lame ront at an instant t r, and α i are the parameters eaturing a congestion. It is clear that the unction Ψ is determined only or t tr, and describes elementary layer (spherical, cylindrical or plane), i.e. yields aeraged characteristic o residual uel in such layer disposed at a distance r rom ignition point. Ater transit o the lame ront, the residual uel burns down some time, thus Ψ aries rom initial alue Ψ (, ru ( tr),; tαi) down to zero. Let us use two characteristics to describe burning o residual uel: S(, r u ( tr),; t α i) surace area o burning in a unit olume and V elocity o lame propagation (relatie to motionless gas) or process o an aterburning o residual uel. Using these characteristics the equation or Ψ can be written as: d dt Ψ = S V (4) The equation (4) allows calculating the heat source q behind the lame ront. The ollowing expression or q can be written using d combustion o the mixture q = cpρ0δ Tad then Δ Tad o temperature increase at adiabatic - isobaric Ψ q q ΔTad dψ dψ = = ( Kex 1), dt ca cρ T T dt dt p p where K ex is expansion coeicient o mixture ater combustion. Fuel burning down in lame ront generates a local heat source q = cpρ0 ΔTadVδ ( r r), where V burning elocity in lame ront (with taking into account wrinkling coeicient, r is coordinate o lame ront, and δ ( r r ) is delta unction. Thus, the equation (3) can be rewritten in orm: dψ, u = ( Kex 1) V δ ( r r ) dt ( ) (5) 4

5 The equations (4, 5) allow calculating the dependence urt (,), the position o lame ront r () t, and the elocity o this ront u () t. Zero elocity in ignition point is used as initial condition or equation (5). 2 The estimation o pressure peak by ρ u 2 alue is essentially underalued, as gas acceleration is not taken into account. It is proposed to use the alue o pressure calculated at solution o one-dimensional problem o the piston (haing corresponding symmetry) moing with gien elocity in polytropic gas. Within ramework o a considered problem, the piston is perceied as propagating lame ront, and its elocity is boundary conditions or the problem o the piston. The equations o such model ormulated using Riemann inariants R and S is written as (see or example [7]): R R uc + ( u+ c) = к, t x x S S uc + ( u c) = к, t x x R=Ρ+ u, S =Ρ u, p dp p p Ρ=, = const, c = γ, γ ρ( p) c( p) ρ ρ p0 (6) where c is local sound elocity, and parameter k determines the type o geometry: 0 - lat, 1 - cylindrical, 2 - spherical. It is hardly reasonable to sole the equation (6) together with the equations (4, 5). It is enough to calculate the table distribution o pressure in a subsonic interal o elocities o the piston or three geometries, and urther to use these data or pressure estimation. Basically, such three cures can be ound and in the textbooks, or example, [8]. The symmetry type selected or pressure estimation should correspond to the conditions o deelopment o actual low. 3. Description o congested space and construction o unctions Ψ (, ru ( tr), tr; αi) and S(, r u( tr), tr; α i) In a phenomenological approach the inormation about congestion should correspond to the type o model. For a one-dimensional model one can use only one-dimensional unctions, een i the detailed design o all elements o congestion is actually three-dimensional,. The simplest parameter describing congestion is the porosity por() r, i.e. the portion o ree space in section r (when reerring to experimental situation the blockage ratio br() r = 1 por() r is oten used.). Knowing the congestion geometry and the direction o the lame ront propagation, it is basically possible to calculate the unction por() r. By implication o the model it is necessary to add stagnant areas to a geometrical porosity. The moing gas stream lows oer these stagnant areas has the elocity o the lame ront. Let us determine such eectie porosity as Por(, r u ; α i ). Additional complexity in the deinition o unction Por() r is the inluence o congestion element on stream at some distance both downstream and upstream. Such inluence depends on low rate. In other words the unction Por(, r u ; α i ) is not completely determined by the congestion geometry, but it should be deined more exactly iteratiely during solution procedure. The model o higher dimensionality (at least 2D) is desirable to improe the unction Por(, r u ; α ). i I the unction o eectie porosity is constructed, the unction o residual uel Ψ basically is proportional to a dierence o geometrical and eectie porosity 5

6 Ψ(, r u ( tr), tr; αi)~ por() r Por(, r u ; αi). Except or stagnant areas taken into account in such way, there is residual uel in boundary layers o suraces. It is basic eect at lame propagation in channels without obstacles. The intuition suggests that increasing ront elocity leads to increase o olume o stagnant areas, i.e. eectie porosity Por(, r u ; α ) should hae descending behaior. Another characteristics o congestion elements is the aeraged size o these elements dr () (or een the distribution unction on the sizes ( dr, ) ). Basically this parameter determines the maturity o the surace o residual uel combustion, i.e. the unction S(, r u ( tr),; t α i). At identical porosity in case o a congestion with small characteristic sizes the surace o residual uel burning is more deeloped, i.e. the alues o unction S(, r u ( tr),; t α i) is large. Apparently the maturity o the surace o residual uel combustion increases with the parameter u. When burning the residual uel its speciic surace decreases. The ollowing dependence should be proposed as irst approach: ξ β Ψ(, ru ( tr),; tαi) u S(, r u ( tr),; t αi) = S0 + S1, (7) dr () c which has dimension (m -1 ) required or S. The sound elocity c is used to make dimensionless ariable o elocity. The coeicients S 0, S 1 as well as β, ξ are tuning parameters. All o them hae to be positie and the alue o ξ has not exceeded 1. At the present moment it is necessary to limit consideration by arguing about choice o unctions Por, Ψ and S. As the experience in data interpretation will be accumulated, the technique o construction o these unctions and the range o ariation o their alues could be determined more precisely. When 2D model based on the same principles o soling procedure will be deeloped, the amount o ree parameters decreases and the accuracy o taking into account congestion geometry increases. 4. Comparison o simulation results and experimental data Actual explosion experiments o enough large scales in unclosed congested spaces are not so numerous, and their geometry has no suicient symmetry to consider a situation as one-dimensional. In essence this experimental actiity has not the purpose to approach to such symmetry. Usage o 2D approach would be more preerable, but such ersion o model is not ready yet. Thereore only the results o 1D modeling are presented. Experimental data [5] and [6] are used or the comparison. These experiments considerably dier in olume o combustible mixture (4000 and 18 м 3 in [5] and [6] correspondingly) as well as in geometry o experimental rig and diameters o tubes creating congestion (0.315 m in [5] and or m in [6]). Besides the peaks o pressure the maximum elocities o lame ront, temporal ariation o lame ront position and time eolution o signals o pressure sensors are gien in [5]. The schematic diagram o experimental rig [5] and position o gates in experiments [6] are gien in ig. 2 and 3 correspondingly. The experimental rig [6] comprises a steel lattice ramework o 1 -mcube cells. For the unconined tests a ramework which had 3 m 3 m ootprint, and which was 2 m high is used. The tables 1 and 2 show the data o experiments with natural gas selected rom [5] and [6]. These data are supplemented by modeling results. i 6

7 Figure 2. Schematic diagram o experimental rig [5]. The data on position o lame ront and its elocity measured by photodiodes are rather useul to tune the parameters o model. They enable to estimate rate and duration o inluence o each series o obstacles that allows selecting the parameters o residual uel and speciic surace area o burning. Though such measuring by photodiodes is conducted in both experiments, the releant data are gien in [5] only. Such data set allows better tuning the model. As a matter o act all procedure o model tuning is based on the analysis o experimental data [5]. The same alues o tuned parameters are used when experiments [6] are simulated using geometry rom [6]. Figure 3. Position o gates in experimental rig [6]. 7

8 Table 1. Comparison o calculated oerpressure and maximum lame speed with experimental data [5]. Gas mixture Obstacle coniguration Results Test Fuel Conc. Stoich. No Intergrid Blockage No o Grid Max. lame Oerpressure % ol o spacing ratio pipes height speed grids m % per grid m exper. estim. exper. estim. 2 NG NG NG NG NG NG NG NG ? Test Fuel Gate BR Table 2. Comparison o calculated oerpressure with experimental data [6]. Gate (N/S) Gate BR Gate (E/W) 8 Gate BR Gate (UP) Stoich. Peak oerpressure exper. estim % % % mbar G Methane 10 1,2,3,4 10 1,2,3,4 10 1,2,3, J Methane 10 1,2,3,4,5,6 10 1,2,3,4,5,6 10 1,2,3,4,5, L Methane 20 1,2,3 20 1,2,3 20 1,2, N Methane 20 1,2,3,4 20 1,2,3,4 20 1,2,3, Q Methane 20 1,2,3,4,5,6 20 1,2,3,4,5,6 20 1,2,3,4,5, BB * Methane 20 1,2,3,4 20 1,2,3,4 20 1,2,3, DD * Methane 20 1,3,5,7 20 1,3,5,7 20 1,3,5, DDD Methane 20 1,3,5,7 20 1,3,5,7 20 1,3,5, FFF Methane 30 1,3,5,7 30 1,3,5,7 30 1,3,5, * all obstacle diameters = 26mm except tests BB and DD (d=49mm) Though exterior geometry o experiment [5] is similar to a spherically symmetric case, the geometry o set obstacles is closer to cylindrical symmetry. The attempts to simulate these experiments in spherically symmetric approach hae not yielded to reasonable results. It could be explained that the actual low pattern (and burning o residual uel) is set by blockage and its symmetry. Thereore all reduced simulation data are obtained using cylindrical symmetry. Only initial stage o lame propagation (up to the irst series o obstacles) is considered as spherically symmetric. Except or relations reduced aboe, at model operation the experimental data on elocity o laminar lame propagation depending on stoichiometric ratio o mixture ( St ) and similar dependence Δ T ( St) (or K ( St )) obtained rom thermodynamic calculation are used. ad ex Coincidence o calculated and experimental data or [5] (table 1) could be considered as good enough, while the modeling is not so successul in case o [6] (table 2). Principal explanation o such situation is the absence o suicient symmetry in congestion o experiments [6]. Among 9 inerior cubic cells our o them do not contain lattices at all, our cells hae an identical blockage ratio, and lost one has doubled blockage ratio. Such construction is ar rom cylindrical symmetry as well as rom spherical one. The most essential moment is the double density o obstacles in an angular cube. This cube will determine maximum peak o pressure, especially at low lam

9 blockage ratio. Certainly, within the ramework o the one-dimensional approach it is impossible to take into account such design eatures. Since the model tuned on experiment o a large scale has not gien senseless results at simulation o small scale experiments, it could be considered as the positie moment. 5. Conclusions The phenomenological approach presented here eliminates the concept o turbulent combustion regime rom physical model o lame propagation. In co-ordinates bound with lame ront position the burning occurs at rather low speeds, namely with elocity o a laminar lame typical or considered mixture multiplied by wrinkling coeicient o a surace o lame ront. An acceleration o the lame ront and generated peak o pressure arise because o an aterburning o residual uel behind the lame ront in stagnant areas and boundary layers. Testing o the model has shown reasonable coincidence with experimental data [5] that is not surprising, as the model was tuned by these experiments. There is a considerable discrepancy between simulation results and data [6] that could be explained by applying o one-dimensional modeling to the experiments, which hae not right symmetry. In order to eriy proposed approach, it is necessary to analyze much more experimental data. Though 1D approach o deeloped model is presented here, kinematic calculation based on potential model o gas low could be implemented or higher dimensionality. The procedure determining the position o lame ront should be the basic additional unit o models o higher dimensionality. Een a 2D ersion o this model could essentially clariy the mechanism o residual uel ormation. A number o experiments and actual situations should be considered with taking into account ent holes. Unconditionally such modiication o the model will extend the range o its applicability. Howeer it is necessary to keep in mind that merely delagration situations can be modeled at used assumption o medium incompressibility. Neertheless such restriction remains broad enough range o actual situations, where such estimations are alid. REFERENCES 1. Cates, A. T., and Samuels, B., A Simple Assessment Methodology or Vented Explosions, Journal o Loss Preention in the Process Industries, 4, 1991, pp Puttock, J.S., Yardley, M.R., Cresswell, T.M., Prediction o Vapour Cloud Explosions Using the SCOPE Model, Journal o Loss Preention in the Process Industries, 13, 2000, pp Fairweather, M. and Vasey, M.W., A Mathematical Model or Prediction o Oerpressure Generated in Totally Conined and Vented Explosions, Proceedings o the Nineteenth International Symposium on Combustion, 1982, pp Chippett, S., Modeling o Vented Delagrations, Combustion and Flame, 55, 1984, pp Harrison A.J. and Eyre J.A., The Eect o Obstacle Arrays on the Combustion o Large Premixed Gas/Air Clouds, Combustion Science and Technology, 52, 1987, pp Snowdon, P., Puttock, J.S. Proost, E.T., Creswell, T.M., Rowson, J.J., Johnson, R.A., Masters, A.P. and Bimson, S.J., Critical Design o Validation Experiments or Vapor Cloud Explosion Assessment Methods, Proceedings o International Conerence and Workshop on Modeling the Consequence o Accidental Releases o Hazardous Materials, San Francisco, September 28 - October 1, 1999, pp Lojcjanskij L.G. Mehanika zhidkosti i gaza, 1973, Nauka, Moska (in Russian). 8. Sedo L.I. Metody podobija i razmernosti mehanike, 1965, Nauka, Moska (in Russian). 9