PHENOMENOLOGICAL MODELING OF DEFLAGRATION EXPLOSIONS
Shabunya, S.I.1, Martynenko, V.V.1, Till, M.2 and Perrin, J.2
1
Heat & Mass Transfer Institute, P.Brovka Str., Minsk, 220072, Belarus
2 CRCD, Air Liquide, B.P. 126, Les Loges-en-Josas, 78354, France
ABSTRACT
Submitted paper is the further development of phenomenological approach in simulation of
deflagration explosions. For the first time developed approach was presented on International
Conference on Hydrogen Safety held in Pisa, 2005 (reference number of the paper is 120132).
Momentum equation is added in the present version of phenomenological model. New version of the
model allows calculating the fields of pressure and velocity. Practically such improvement of the
model doesn’t affect the cost of computer time. Developed computer code is tested against
experimental data on unconfined deflagration of stoichiometric hydrogen-air mixture performed
previously the in Fraunhofer Institut Chemische Technologie, Germany, 1983. These experimental
data is proposed to HySafe partners as one of Standard Benchmark Exercise Problem (SBEP V2).
Comparison of computational and experimental results has shown good accuracy of predictions on
velocity of flame propagations and overpressure measured by pressure transducers.
1. Introduction
The developed phenomenological approach is the attempt to create mathematical model and computer
code for enough fast simulation of deflagration explosions. Development of the simplified modeling of
explosions in conditions of industrial sites is motivated by serious difficulties appeared at application
of CFD models to simulate this problem.
A big number of successful applications of CFD models are known when rather complex threedimensional flows are studied. The processes of combustion, supersonic flows and detonation can be
modeled numerically also. It would seem, that usage of powerful computers will resolve the problem
formulated above. The difficulties of direct CFD simulation of deflagration explosions in conditions of
industrial sites is connected with the principled issues bound up with the specificity of formulated
problem.
The description of all physical processes of flame propagation up to transition of burning into
detonation based on the first principles meets with a huge difference in scales of these phenomena.
This scale is fraction of millimeter in flame front, shock and detonation waves, while the characteristic
sizes of industrial equipment is the meters, and the site of a plant is tens and hundreds meters.
Computational cells of fraction of millimeter are required for adequate description of combustion
processes, formation of shock waves and detonation waves especially. The usage of such grid for
computational domain of complex geometry about ten meters is unreal task now. The expectancies to
apply “adaptive” grids in such case are rather doubtful, as spatial discretization should be adapted as a
minimum by 3-4 orders of magnitude.
The second difficulty facing complete CFD model is uncertainty of initial and boundary conditions for
a considered problem. Incorrect boundary condition related to concentration of combustible mixture in
modeled area affects computational results considerably. “Tuning” CFD code by data of experiments
having small scales (namely at small scales it is possible to control initial conditions with satisfactory
accuracy) does not result in good coincidence of model predictions with the results of big scale
experiments. (Big scales are perceived as spatial sizes ~ 10 meters and higher.) At such matching it is
difficult to assert unambiguously that could be the reason of such discrepancy: inaccuracy of used
model or incorrect initial conditions, which in principle cannot be controlled in experiment of big
scale.
The third issue related to application of CFD codes is a problem of “turbulence”. In case of technical
problems successfully resolved by CFD codes, the model of turbulence is selecting on the basis of
comparison of simulation results with experimental information. In case of problem formulated as
explosion on plant site such way is impossible. All CFD codes are sensitive to selection of turbulence
model (especially model of turbulent combustion) and formulation of initial conditions for such
models.
The fourth issue of CFD modeling is the simulation of chemical processes. The acknowledged (well
tested) kinetic equations describing oxidation of hydrocarbons contain several tens of components and
several hundreds of reactions. Even if to consider, that such kinetic schemes are correct, their usage in
three-dimensional calculations for considered problem of explosions is unreal owing to
incommensurability of spatial scales of chemical processes and computational area. Usage of
simplified schemes describing combustion actually means the loss of the main advantage of CFD
approach - its strictness.
Any modern versions of CFD codes intended for simulation of industrial explosions contain a plenty
of simplifications. The gravity of problems which has to be simplified does not allow evaluating the
effect of such simplifications on accuracy of model predictions in case of actual object. As the
matching of predictions given by existing CFD codes with the full-scale experimental tests [1]
demonstrate, the discrepancy in the values of pressure peak about two - three times is usual event. In
case of little bit stricter matching when both pressure peaks and their time instants are compared, the
accuracy of predictions is even worse.
Contrary to CFD modeling the phenomenological models are more capable for productive
development. This type of models is initially intended for restricted range of considered situations and
process parameters. Such models grounded on obvious physical ideas are clear to the broad audience
of the users, and the modeling using these models is very fast. The art of scientific computing using
phenomenological models consists in adequate description of a physical situation in concepts of model
and objective evaluating the limits of its applicability.
2. Basis of phenomenological approach
The physical problem initiated this work is an estimation of overpressure at hydrogen explosion in
conditions of refueling station. At that the consequences of accidental fire are evaluated excluding the
act of terrorism when the combustible cloud is blasted by solid explosive. Being limited by such
problem it is possible to simplify essentially the requirements to phenomenological model. As actual
situation implies open space (there are no walls, and considered site is under lightweight shed), from
the beginning it is possible to exclude from consideration all problems caused by detonation.
Moreover in such conditions deflagration process is the most probable only, when the typical values of
velocity and overpressure are rather low. In considered geometry the important factor controlling
pressure increase is the congestion of space by equipment. Summarizing these reasons our
requirements to phenomenological model are stated as follows.
 The model is developed for slow (relative to speed of sound) burning of combustible mixture
in open congested space.
 The model is oriented on a prediction of overpressure.
 Fuel concentration in space is given (it is calculated by other procedure).
 Ignition by small energy is always localized.
 Geometries of congestion and domain are taken into account by the model.
Two phenomenological models of explosions are mostly advanced. Therefore they are mostly known:
SCOPE (Shell Code for Over-pressure Prediction in gas Explosions) and CLICHE (Confined
LInked CHamber Explosion). Both models are oriented on simulation of explosions in vented indoors
and use the approach of incompressible (in gas-dynamic sense) medium. It is assumed that velocity of
gas medium inside considered domain is much less speed of sound. Therefore, inside domain gas
pressure has time to flatten, i.e. gas pressure depends on time only (in case of CLICHE several bound
areas having own pressure are considered). The overpressure in such model is explained by the fact
that the expanding gas has no time to outflow from vent holes. Such approach should give reasonable
results for slow combustion in both cases: completely closed indoor as well as at reasonable vent
areas. It is understandable that the case of open area can’t be described by such models.
The proposed phenomenological approach uses the following assumptions simplifying the
consideration of real physical processes:
 Flow is inviscid.
 Flow is incompressible in gas-dynamic sense.
 Flow is vortex-free.
 The combustion of fuel is described by two models, combustion in flame front and burning of
residual fuel behind front in stagnant regions of congested space.
The idea of two models of burning is proposed in [2], where the model of residual fuel is described. In
the present work the application of the model for simulation of the experiment [3] is described, when
hydrogen-air mixture is burned burning in a free space without congestion. Therefore two models of
burning is only mentioned here, while the advanced variant of the model describing the burning in
flame front is given in details.
At burning in free space the flame front separates combustion products and combustible mixture. In
congestion space there is some amount of combustible mixture behind “main” front of a flame. The
combustion of this residual combustible mixture results in acceleration of flame front. Resulting flame
velocity depends on amount of residual combustible mixture, its spatial distribution and distribution on
a surface of combustion. In such model there is no necessity to assume super-high burning velocity in
combustion front for originating essential overpressure as the required effect is reached due to
developed surface of burning of residual fuel.
The model of combustion in flame front is based on the following assumptions:
 There is adiabatic combustion of combustible mixture in flame front, the parameters of
combustion products are determined by conditions of isobaric thermodynamic equilibrium.
 Burning velocity V f (relative to stagnant gas) is set by empiric ratio based on approximation
of experimental data, for example:
V f  V f 0  C0  Turb  t  ,
 
where V f 0 C0
(1)
the dependence of laminar combustion velocity on concentration .
The function Turbt  models increasing burning velocity due to of rough surface of
combustion (shrinkage of flame front), and flow turbulization. The value of this function
changes monotonically from 1 up to maximum value evaluated by [4] as ( Kex 1) / 3 ,
where Kex is expansion coefficient of gases at combustion.
The modeling in flame front is connected with serious algorithmic difficulties because of small
thickness of this area. It would be logical to consider this type of burning as an internal boundary
condition formulated on a moving surface of flame front. Such consideration requires the body of
mathematics and the software capable to trace position of flame fronts and carry out the calculations
with singular conditions on the surfaces of these fronts.
The distribution of processes occurring on a surface of flame front on some narrow area close to this
front is the alternative approach to singular one. Considered thickness of this area should be narrow,
but it should exceed typical scale of a used grid which cannot be too detailed. In order to trace the
position of flame front (in terms of the distributed front it is an average surface of this area) the
proposed phenomenological model assumes two mechanisms of front motion. At the beginning
“diffusive” propagation of flame equal to   V f (t )  t occurs for a small time step t in direction
normal to flame front Then “instantly” extending (practically at constant pressure) combustion
products create flow field which takes into account both area of burning (source) and a configuration
of spatial borders. Convective motion of flame front for time equal to t is determined by this
velocity field.
In the geometry with high spatial symmetry where flow field is one-dimensional as a matter of fact,
convective step can be calculated analytically; however such way cannot be realized for twodimensional and three-dimensional situations. Though test experiment [3] has spherical symmetry,
which allows solving considered problem by the easiest way; our purpose is to develop and test
computational algorithm suitable for application in case of general formulation of such problem.
In the area with complex geometrical structure the flame front can be fallen to several pieces with
essentially different velocities of motion. The flame fronts moving together with flowing out
combustion products have the values of velocity conditioned by expansion of hot combustion
products, while the velocity of flame propagation is much less in those areas, where combustion
products are compelled to move towards to flame front is much less (approximately V f ). These
features are revealed in large-scale geometrical configurations of computational area resolved by a
grid used for simulation. The small-scale elements of space congestion (badly resolved by large scale
grid) should be taken into account by the model of residual fuel.
Above described features of combustion in two-dimensional area are graphically illustrated by pictures
shown in fig. 1. The position of flame front shown in first picture corresponds to a simple situation,
when only one fast front exists. In the second picture the front has fell to three pieces: two slow fronts
(a and b) and one fast one (c). Also in this picture a small scale obstacle (d) is shown, the combustion
in stagnant areas of this obstacle should be described by model of residual fuel. White arrows indicate
the direction of flame front propagation, while black one show the direction of motion of combustion
products, and grey arrows correspond to motion of combustible mixture. It is understandable, that the
actual objects can have more complex geometry and bigger amount of flame fronts.
c
d
a b
c
d
Figure 1. Positions of flame fronts in “complex” two-dimensional area.
Thus three types of combustion areas have to be modeled: the burning of fast flame front, slow flame
front and residual combustible mixture. At enough small value of t diffusive phase of flame front
motion is controlled by initial position of flame front (normally to initial front surface). Convective
phase of motion is controlled by initial position of flame front and geometry of area determining
velocity field. For example, front c is affected by several gases. This front is moved by gases
generated at combustion in this front, gases of fronts a and b and gases generated by combustion of
residual fuel in area d. The consideration of residual combustible mixture is out of scope of presented
work, and it is limited by modeling of flame front motion.
3. Flame propagation modeling
Diffusive phase of front motion is modeled by the algorithm developed on a basis of the equation of
thermal conductivity (2), which provides propagation of temperature profile along the gradient of
temperature.
C (T )
T
  ,  (T )T   Q(T )
t
(2)
At once let's point out that it is only the way to implement phenomenological model of combustion,
and the equation (2) has not any respect to actual thermal conductivity in gas medium. In order to
ensure the transfer of “standard” temperature profile on some distance by equation (1) and avoid
severe change of the shape of this profile the functions C (T ) ,  (T ) and Q(T ) should be selected by
a special way. Smooth stepwise profile shown in fig. 2 is used as standard one. The basic features of
used profiles are the plateau of maximum temperature corresponding to combustion products, and the
plateau with minimum temperature related to combustible mixture.
Tmax=Tad
Tmin=T0
Figure 2. Satndard temperature profile
Namely the values of temperatures have no meaning for the model, but for the sake of visualization it
is possible to select their values equal to natural one: isobaric - adiabatic temperature of burned down
combustible mixture Tad and initial temperature T0 of it. Initial temperature distribution is determined
by ignition area, which coincides with isotherm Tav  0.5  Tad  T0  . The principled moments of the
evolution of function T is maintaining of transition area localized in space, where transition occurs
from Tmax = Tad down Tmin = T0 and motion of profile for a distance  in direction normal to front
motion. At that time step (or several steps) ensuring the motion at solution equation (2) has not
physical meaning. The value of physical time t connected with such motion is determined by the
ratio   VF (t )  t .
Convective phase of flame front motion is described by the same thermal conductivity equation with
convective term:
C (T )
T
 C (T ) V , T   ,  (T )T   Q(T )
t


The field of velocity used in the equation (3) is calculated by continuity equation:
(3)

 ,  V  0 ,
t


(4)
where time derivative of density is rewritten using incompressibility approach. In narrow area having
width  between two positions of flame front (in presented algorithm between two isotherms Tavn and
Tavn 1 ) the density changes from density of combustible mixture  Fuel up to density of combustion
products  CP . In all other space the value of density does not vary essentially. Thus, the equation (4)
in zero approach can be transformed as:
CP   Fuel
t


 ,  V  0 ,
where source term
CP   Fuel
t
(5)
is absent everywhere except for area between isotherms Tavn and Tavn 1 .
In such approach the function  in (4) is equal  CP behind isotherm Tavn 1 ; the condition  =  Fuel is
ahead of this isotherm and up to the border separating combustible mixture and air; and further
 =  Air .
The hypothesis of vortex free flow allows rewriting (5), using potential of velocity ( V   ) written
as:
 ,    
CP   Fuel
t
 V f (t )
CP   Fuel
.

(6)
As function  is known, and the value of  is determined by procedure calculating function T , the
equation (6) allows determining potential  and velocity field without usage of momentum equation.
In vicinity of flame front the calculation of velocity by (6) should give reasonable values, i.e. to
describe motion of this front correctly enough. The equations of hyperbolic type are required to
simulate oscillation processes when all combustible mixture is burned out. Discrepancy in the values
of velocity calculated by (6) will arise far from flame front. In actual situation the motion of gas
medium exists behind of sound wave front only, but the equation (6) is elliptic, i.e. with infinite
velocity of disturbance propagation. The usage of elliptic equation instead of hyperbolic equations
system simplifies the computational algorithm, but it has consequences which should be taken into
account at the analysis and interpretation of computational results.
The algorithm calculating motion of flame front is based on the above mentioned equations and
consists of three steps:
1. Existing spatial distribution of combustion products described by temperature field T n is
transformed into new one T n 1 by solving the equation (2). The area having width  between two
isotherms Tavn and Tavn 1 is considered as area, where combustion occurs; the mass of burned fuel is
calculated.
2. The fields of potential and velocity are determined from the solution of the equation (6)
3. Calculated velocity field is used in the equation (3) which is solved until the mass balance of
combustion products will be reached.
The calculation of evolution of flame front for a physical time step t is crowned by step 3. The
following stage of calculations starts with step 1. Such algorithm allows calculating the evolution of
flame front (or fronts) up to complete combustion. The procedure calculating pressure field should be
based on an additional hypothesis.
4. Comparative calculations
Deflagration explosion can be separated by two stages: actual explosion, when there is combustion of
all combustible mixture, and inertial flow after explosion, when the stage of depression and the
subsequent fading pressure oscillations occur. Such pattern is typical for the experimental data [3].
During the first stage the increased values of pressure are realized in the area of combustion products,
which are kept behind front approximately at the same level up to combustion completion. Then there
are the oscillatory processes leveling pressure field.
These two stages of explosion essentially differ by key physical processes. At the first stage the fields
of pressure and flow are determined by burning. Thus the area of combustion products is practically
isobaric. Pressure in this area can slowly vary with time at changes of velocity of flame front and
cooling of combustion products. The area outside combustion front is adiabatic and can be modeled in
acoustic approach. At combustion stage pressure can be approximated by known solution of piston
(flame front) motion with given velocity.
In case of spherical piston moving with constant velocity U  csound using acoustic approximation of
gas dynamic equations it is easy to obtain self-similar solution. Within the framework of such solution
the normalized pressure can be written as follows:
P
 Rsound  r  R flame
P
2 U 2


,
P0 csound  csound  U  r  Rsound  R flame 
(7)
where  is adiabatic exponent, R flame is piston position at some time instant t and Rsound is the
position of sound front Rsound  csound t .
In comparative calculations with the data [3] at combustion stage the ratio (7) is used for calculation of
P ahead of flame front, i.e. R flame  r  Rsound , and the value of pressure equal to pressure in front is
accepted behind flame front. The velocity of flame front motion is used as U in (7).
The second stage of deflagration explosion is not described by above mentioned algorithm, as it has no
the prevailing phenomenon - burning. Another model with other forms of continuity and momentum
equations is required for the adequate taking into account the motion of combustion products and air
separated by a contact surface. The fields of pressure and velocity calculated at the moment of
combustion completion are used as initial condition for the second stage.
The necessity of modeling of the second stage within the framework of the phenomenological
approach is a disputable question. The model contains a number of simplifications, and the accuracy
of predictions of depression phase is especially difficult for estimating, as for this purpose all
background of flow is important. In case of simple geometry with high symmetry of computational
area the result can be good. However the question of calculation accuracy at depression phase remains
open, when real geometry of industrial sites is simulated. The situation with burning phase is more
certain, and it determines overpressure. Nevertheless for the completeness of comparison with
experimental data [3] both phases the combustion and depression are simulated. The last phase is
calculated by the model described below.
At adiabatic and acoustic approach the equations of gas dynamics can be rewritten by two linear
equations.
P
  , V  0
t
V P0
 P  0
t 0


(8)
As the hypothesis of vortex free flow has been accepted, it is possible to replace velocity by a gradient
of potential and integrate momentum equation. It results in:
P
   ,    0
t
 P0
 P  0
t
0

P
    0
t
  P0 


P  0

t

0



P
    0
t
 csound

P  f (t )
t

(9)
csound
As computational area consist of two media (combustion products and air), the factors  ,

and
f (t ) have different values in corresponding subareas. At solving (9) the smooth transition of these
factors is used on contact surface. Temperature T is used as argument of stepwise function, i.е. the
values of factors correspond to combustion products and air at T = Tad and T = T0 correspondingly.
Below in fig. 3 the results of pressure evolution calculated in installation sites of 6 pressure sensors are
compared with the data (labeled curves) measured in [3]. Without broad analysis of measurement
errors, it is possible to conclude, that the model well enough describes an experimental situation.
Comparison of results calculated by different codes made in [5], allows drawing the conclusion, that
the further development of proposed phenomenological model is meaningful. The accuracy of
predictions (for this test!) is not worse than using CFD codes and required computational time is
incommensurable.
10000
10000
10000
2 m.
Relative pressure, Pa
-5000
0
-5000
-10000
-10000
-15000
-15000
0
10000
0.1 0.2 0.3 0.4 0.5 0.6
Time, s
0
6000
Relative pressure, Pa
5000
-5000
-10000
0.1 0.2 0.3 0.4 0.5 0.6
Time, s
0.5 0.6
80 m.
4500
4500
3000
3000
1500
0
-1500
-3000
1500
0
-1500
-3000
-4500
-6000
0
0.3 0.4
Time, s
6000
-4500
-15000
0.1 0.2
35 m.
18 m.
0
-5000
-15000
0
0.1 0.2 0.3 0.4 0.5 0.6
Time, s
0
-10000
Relative pressure, Pa
Relative pressure, Pa
0
5000
Relative pressure, Pa
5000
5000
Relative pressure, Pa
8 m.
5 m.
-6000
0
0.1 0.2 0.3 0.4 0.5 0.6
Time, s
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
Time, s
Figure 3. Calculated and measured evolution of pressure sensors.
5. Conclusions
Successful modeling of this test by such phenomenological model is quite expected result, as actual
physical situation very favorable for proposed model in sense of geometry and flow parameters. It is
clear that others one-dimensional situations and transformed to them will be well described by
proposed model too.
Presented model is formulated in a vector form, and in this sense does not depend on dimension;
however as this model will consist of two parts (up combustion completion and after it) it will have
difficulties when the simulation of burning is required in one part of considered area, while already
there are relaxation processes in other part. In two-dimensional situations such problems do not seem
insuperable. This direction of researches is in progress now.
Complication of geometry up to essentially three-dimensional one creates serious problems for any
modeling. The idea of 3D phenomenological model is probably superfluous in principle. At
phenomenological approach the more natural way is separating initial area by several subareas which
can be modeled in approach of smaller dimension.
REFERENCES
1. C.J. Lea, H.S. Ledin, A Review of State of the Art in Gas Explosion Modeling.
Healh&Safety
Laboratory,
Fire
and
Explosion
Group.
www.hse.gov.uk/research/hsl_pdf/2002/hsl02-02.pdf
2. Shabunya S.I., Martynenko V.V., Till M., Perrin J., 1D phenomenological model estimating
the overpressure which could be generated by gas explosion in a congested space.,
International Conference on Hydrogen Safety, September 8-10, 2005, Paper ref. 120132
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in der gasfabrik und auswirkungen von Druckwellen auf das Containment. Dezember 1983.
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J. of Chemical Physics, Vol. 19, May 1951, pp. 541-547.
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deflagration of a large scale H2 – air mixture in open atmosphere, ref. 120003, proc. ICHS
conference, 2005,.