Dust Explosions in Spherical Vessels:
prediction of the pressure evolution and determination of the
burning velocity and flame thickness
A. E. Dahoe, J.F. Zevenbergen, P.J.T. Verheijen, S.M. Lemkowitz and B. Scarlett
Abstract
A well known limitation of the ’cube-root-law’ is that it becomes invalid when the flame thickness is significant with respect to the vessel radius. In the literature flame thicknesses in dust-air mixtures ranging from 15 to 80 centimeters have been reported [1], which exceed the radii of the 20-1itre sphere and the 1 m3 vessel. Therefore we have developed a model (the three zone model) for the pressure evolution of confined dust explosions in spherical vessels which takes the flame thickness into account. The pressure-time curves that are generated with this model show a good resemblance with those measured in practice. It is shown by numerical simulations that the maximum rate of pressure rise can be normalized with respect to the vessel volume as well as to the flame thickness and that the ’cube-root-law’ becomes inaccurate for relative flame thicknesses exceeding 1%. Furthermore, the actual burning velocity and the flame thickness during real dust explosions can be obtained by fitting the model to the experimental pressure-time curve.
1
Introduction
An important dust explosion characteristic for assessing venting requirements and for spe-cifying automatic dust explosion suppression equipment is the maximum rate of pressure rise,
(alP/dr)max, obtained from closed vessel experiments. Since (alP/dr)max depends on the size of
the vessel, it is normalized with respect to the volume of the vessel in question according to the ’cube-root-law’:
The validity of the cube-root-law as a scale-up relationship between differently sized vessels is
based on the following restrictions [2, p.347]:
* The vessels in question are geometrically similar.
* The flame thickness is negligible with respect to the vessel radius.
. The burning ve!ocity is the same in all volumes.
. Point ignition occurs at the centre of the vessels.
In practice, the cube-root-law is used as a scale-up relationship for applying standard test
results from laboratory-sized vessels to plant-sized equipment. Inaccuracy of the cube-root-law
will therefore irrevocably lead to overdimensioning or underdesign of equipment, both having
financial consequences. Of the restrictions mentioned above, only the role of the flame thickness
is treated in this work.
In the literature some theoretical background can be found for the ’cube-root-law’ [3, 4, 5].
Crucial in these derivations of the cube-root-law is the assumption that the maximum rate
of pressure rise, (dP/dt)m~::, occurs when the pressure attains its maximum value, Pm~z. Such
behaviour is indeed found in practice, as, for example, is shown in the hydrogen/oxygen
pressure-time curve of Figure 1. More usually however, the pressure-pressure-time curve of explosions in closed
spherical vessels exhibits an inflection point, the maximum rate of pressure rise occurring before
the end of the explosion. Such behaviour is typically found in dust explosions, as shown, for
°Delft University of Technology, Division of Particle Technology, Julianalaan 136, 2628 BL Delft, The
Nether-lands, email: dahoe@cpt6.stm.tudelft.nl,
Dust Explosions in Spherical Vessels: Prediction of the Pressure Evolution and Determination of the Burning Velocity and Flame Thickness
example, in the middle part of Figure 1. The right part of the figure shows that gas
explosi-ons may also exhibit an inflection point in the pressure-curve. Researchers have related this
behaviour to the flame thickness by reasoning that in the final stage of the explosion a certain
part of the flame front has already reached the vessel wall, while the remaining part is still
propagating through the vessel. No researchers, however, have as yet developed theory capable
of quantitatively describing this behaviour.
hydrogen/oxygen time 2 1 0 8 7 2 I 1 50 100 150 200 250 time (ms) I ’ I I ’ I ’ I 0 50 100 150 200 250 time (ms)
Figure !. Pressure-time recordings of an explosion of a hydrogen-oxygen mixture diluted
with helium (reproduced from [6, p. 620]), a cod! dust explosion and a methane exp!osion
(reproduced from [5]).
An important issue, dealt with in this work is how to handle the concept of the flame
thickness. Clearly, the flame zone is a region where combustion takes place, and the flame
thickness could be the width or a similar length scale of the region in question. But what are the
boundaries of the flame zone on which to base this width? The definition of the flame thickness
is to a certain extent arbitrary and various researchers have used a number of approaches in
setting the boundaries [7, 6, 8].
The burning zone is a region where a transition takes place from unburnt to burnt mixture.
Therefore we follow Van der Wel [9, Chap. 6] by assuming that a transition from completely
unburnt into completely burnt mixture takes place in the burning zone according to a linear
relationship which is a unique function of the flame thickness. The fresh particles that are to
be combusted enter the burning zone with a certain velocity S~ (the burning velocity), and
a certain time re (the chemical time scale or burning time) is required for their combustion.
Therefore a suitable expression for the flame thickness would be the product of this velocity and
the burning time for such a particle:
In this article the idea of treating the burning zone as a transition region where unburnt mixture
is converted into burnt mixture is incorporated into a model for predicting the pressure evolution
di~ring a confined dust explosion in order to assess the influence of the flame thickness on the
maximum rate of pressure rise.
2
Formulation of the problem
Since the objective of this work is to investigate only the role of the flame thickness in the
normalization of the ma~mum rate of pressure rise from tests with differently sized vessels, the
problem is formulated as follows:
Suppose that we have two differently sized spherical dust explosion vessels, both
containing the same dust-air mixture, having the same isotropic, invariant turbulent
flow field. Also, suppose that the burning velocity is not influenced by the
pres-sure, the temperature and the dust concentration. If the mixtures were ignited to
deflagration by a point ignition at the centre of the spheres, would we obtain the
same Kst value or something else? And, if we measure different Ks~-values, how
important is this difference with respect to the validity of the ’cube-root-lXw’ ?
Dust Explosions in Spherical Vessels: Prediction of the Pressure Evolution and Determination of the Burning Velocity and Flatne Thickness
This precise formulation of the problem will be further clarified by considering the flame propa-gation during the course of a closed vessel explosion.
After ignition, a flame front moves through the dust cloud with the flame speed, 5’] (see Figure 2). This velocity is the sum of three additive velocities, namely, a velocity due to the expansion of the burnt mixture, S~, a velocity due to the change in the number of gas molecules
by the conversion of unburnt into burnt mixture, S~, and the burning velocity, S~, (which is regarded to have a specific value for a certain dust cloud, provided that the thermostatic state and flow conditions remain invariant). The unburnt mixture ahead of the flame front moves with the sum of S~ and S,~. Hence, the flame front enters the unburnt ~nixture with the burning velocity Su, which determines the rate of heat production and the mass consumption rate of the unburnt mixture. The burning velocity may therefore be regarded as a key parameter in modelling deflagrations, and changes in its value will influence the pressure development and consequently the maximum rate of pressure rise. The flame thickness is also considered as a key parameter in this work since it causes the maximum rate of pressure rise to occur at an earlier time than the maximum explosion pressure. As equation (2) shows, the flame thickness is influenced by changes in both the burning velocity as well as the burning time of the particles.
(a
Figure 2. Two models for the moving flame front during a closed vessel deflagration with: (a) an infinitely thin burning zone and (b) a burning zone of finite thickness.
It is known that rate of pressure rise depends on a number of factors concerning the state of the dust cloud. Some of these factors are the thermostatic state (P0 and To), the dust concentration, properties of the dust (particle size, composition, caloric properties), properties of the gas (oxygen content, thermal properties) [10, 5, 11, 12] and the turbulent flow conditions of the dust cloud [3, 5, 13, 14, 15, 16, 1]. By using the burning velocity and the flame thickness as the key parameters in modelling the rate of pressure rise, these factors can be included in a simple way, once their effect on the flame thickness and the burning velocity is known from further experiments.
Three particularly influential factors require further explanation:
¯ The thermostatic state. Because of the rapidness of an explosion, the expansion of the
burnt mixture causes the unburnt mixture ahead of the flame zone to be compressed adiabatically. Therefore the unburnt mixture, encountered by the burning zone at each instant of time, will have a different pressure and temperature, resulting in a different
burning velocity and flame thickness. Furthermore, if the same amount of unburnt mixture is combusted in two differently sized vessels, the unburnt mixture in the larger vessel will undergo less adiabatic compression than in the smaller vessel.
¯ The dust concentration. The expansion of the burnt mixture causes the unburnt mixture
ahead of the flame front to flow towards the vessel wall, dragging the dispersed particles along. This causes the dust concentration in the unburnt mixture to increase duri~tg the course of the explosion. Due to the fact that the velocity in a smaller vessel is greater than in a larger vessel, this effect is more pronounced as the vessel size decreases.
¯ The turbulent flow conditions. The burning velocity and the flame thickness are known to
be altered by turbulence [17, 4]. In standard test procedures this turbulence is produced by the air blast used to disperse the dust prior to ignition [18],[!9, p.61]. After completion
Burning Velocity attd Flame Thickness
of the air blast the turbulent flow field starts to decay and after ignition, the explosion itself influences the turbulent flow field. Consequently. the unburnt mixture encountered by the burning zone has. a different state of turbulence at every instant of tinle during the
explosion.
Based on the burning velocity and the flame thickness as the key parameters, two models are presented in this work, namely, the thin flame model and the three zone model (Figure 2). The first model concerns an infinitely thin flame zone and serves mainly to guide the line of thought in the second model, which is based on a flame zone of finite thickness. Both models need the initial pressure of the dust cloud, P0, and the final explosion pressure, Pc, as input.
3
The thin flame model
During an explosion the contents of the vessel is assumed to consist of a spherical inner region of completely burnt mixture, encapsulated by an outer region of completely unburnt mixture. The regions are separated by an infinitely thin spherical flame front (of which the radial position is denoted by ryt~m~). The flame front is then a surface where a discontinuous transition takes place from unburnt to burnt mixture and propagates radially from the point of ignition towards the vessel wall. Further assumptions in this model are:
¯ The unburnt as well as the burnt mixture are treated as ideal gases.
¯ The specific heats of both the unburnt and the burnt mixture are the same and remain constant during the explosion.
¯ The transition of the unburnt into burnt mixture occurs through a single-step, irreversible chemical reaction which can be described by a global reaction rate expression.
¯ The temperature of the unburnt mixture, Tu, continually increases as’a consequence of the compression, which is assumed to be adiabatic.
¯ The burning velocity remains constant during the explosion (i.e., it does not depend on the pressure, temperature, dust concentration and state of turbulence during the explosion). ¯ Point ignition at the centre of the dust cloud occurs with a negligible energy input. Lewis and yon Elbe [6, p.388] give an approximate expression which relates the mass fraction of burnt mixture in the vessel to the fractional pressure rise. Based on this equation the fraction of unburnt mass can be expressed as
m ,o Pc - Po
and differentiation of equation (3) with ~espect to time yields(3)
dP Pc - Po dm~
dt muo dt
(4)
The combustion wave moves with a velocity that is the sum of the expansion velocity Se, the conversion velocity Sn, and the burning velocity Su. Since unburnt mixture immediately ahead of the flame front moves with Se + Sn, the velocity at which the unburnt mixture enters the
combustion wave is minus the burning velocity. Therefore the mass consumption rate of the unburnt mixture can be expressed as
and a relationship can be established between the rate of pressure rise and the burning velocity. By substitution of (5) into equation (4), the following relationship is obtained:
dP
P~-Po 2 ~ ~
~ = 47r--ryt~,~,~~
(6)
dt
rn~o
The next step is to express the density of the unburnt mixture Pu, and the location of the flame
front, rIlarae, in terms of known variables. For adiabatic compression of the unburnt mixture
Dust Eaplosions in Spherical Vessels." Prediction of the Pressure Evolution attd Deterntination of the Burning Velocity attd Flame Thickness
Pp-" = constant and hence
1
Furthermore Vb = Vv~ssd - Vu, which can be rewritten into
(T)
p (8)
Since p-1 = t~T/P where R~ denotes the specific gas constant in J/kg ¯ K, the volume of the unburnt mixture can be expressed as
& - Po
(9)
and equation (8) yields the following expression for the location of the flame front:
(io)
By inserting the equations (10) and (7) into equation (6) and by noting that rn~,0 = the following ordinary differential equation is obtained for the rate of pressure rise:
2
(ii)
A similar result is given by Bradley and Mitcheson [20]. The solution of equation (11) for vessels of several sizes is depicted in Figure 3.
~6 i i ... .: :. ’ I --20 litre ... 1 m3 ... lOm3 200 400 600 time (ms) -- 20 litre ... lm3 ... lOm3 ’ ’1’" ’ I ’ I’ 200 400 600 time (ms)
Figure 3. The predicted pressure evolution and rate of pressure rise with the thin flame model for three spherical explosion vessels: V = 20 litre, V = 1 m3 and V = 10 m3. Su = 0.6 m/s; P0 = 1 bar; Pc = 8 bar; 7 = 1.4.
From equation (11) it can be seen that (dP/dt) increases monotonically with P and hence
the ma~mum rate of pressure rise is attained when P = Pc. By substituting P = Pe into { 11)
and by multiplying both sides with the cube root of the vessel volume, the following expression
is found for the Ks,-value,
Burning Velocity atzd Flame Thickness
which is a normalization of the maximum rate of pressure rise with respect to the vessel volume. The effect of such a normalization is shown in Figure 4. A similar result is reported by Hertzberg and Cashdollar [5] namely,
Kst=4.84(P~-Po) ~oo S. Note that (367r)~/3 ~ 4.84. 8O ~_ 6o ~ 20 o o.o ,,’ lOm.~ 200.0 400.0 600.0 time (ms)
Figure 4. The effect of the normalization of the rate of pressure rise (of Figure 3) with respect to the vessel volume.
4
The three zone model
This model is called the three zone model for the following reason. When the flame fi’ont is fully developed, the contents of the explosion vessel consists of three zones: a spherical inner region of completely burnt mixture, a flame zone of a finite thickness 5, that consists of burnt as well as unburnt mixture and an outer region that consists of completely unburnt mixture. In this model the flame zone is a region where a continuous transition takes place from completely burnt to completely unburnt mixture. For the sake of simplicity the transition is assumed to be linear with position. The assumptions in this model are:
¯ A spherical surface r = constant within the sphere is thought to consist of only two species, namely, the unburnt mixture with a mass fraction f(r) and the burnt mixture with a mass fraction 1 - f(r). For a fully developed flame zone, f(r) = 0 in the competely burnt mixture (v < rr), f(r) is a linear function of r in the combustion wave (r~ <_ r <_ rI), and
f(r) = 1 in the completely unburnt mixture (r >
¯ The unburnt as well as the burnt mixture are treated as ideal gases.
¯ The specific heats of both the unburnt and the burnt mixture are the same and remain constant during the explosion.
¯ The transition of the unburnt into burnt mixture occurs through a single-step, irreversible chemical reaction which can be described by a global reaction rate expression.
¯ The temperature of the unburnt mixture, Tu, continually increases as a consequence of the compression which is assumed to be adiabatic.
¯ A constant burning velocity exists during the explosion.
¯ Point ignition at the centre of the dust cloud occurs with a negligible energy input. The derivation of the three zone model is identical to that of the thin flame approach up to equation (4) and analogous from thereon. The mass of the unburnt mixture at any instant of time can be expressed as
= fff
f(r)
dV
(13)V
and hence the mass consumption rate of the unburnt mixture is found to be
dt - dt f(r) dV + p~-~ f(r) dV
V V
Oust Explosions in Sphertcal Vessels." Prediction of the Pressure Evolution attd Determination of the Burning Velocity and Flame Thickness
The contribution to the mass consumption rate which is expressed by the time derivative of the second integral, consists of two effects, namely, consumption of unburnt mixture by the flame and the volume change of the unburnt mixture due to compression:
d fr
d[f[ d
fff (
)dV = Pu’~jjj f(r)dV + Pu-~ f(r)dV (15)v
v~
v
consumption
compression
The first effect occurs in the flame zone only and can be evaluated by means of the burning velocity as will be shown below. The second effect however, can only be taken into account by reasoning that it should be linearly proportional to dmu/dt. Hence, equation (14) can be rewritten into
dt -dt f(r) dV + v
where k is a constant that will be determined by demanding the three zone model to yield equation (12) when the flame thickness tends to zero.
Since the first integral on the right hand side of (16) is the volume of the unburnt mixture, the final result in equation (9) can be substituted for it. From equation (7) an expression can be derived for the rate of change of the density of the unburnt mixture:
dt dt
(17)
After inserting (17) into equation (14), substitution of the latter into (4) yields a differential equation for the pressure evolution with time,
(18)
where the integral on the right hand side needs some further treatment. Because f is formally a scalar function of location r and time t, Leibnitz formula t can be applied to the total time derivative of the integral:
d Of(r,t)
:- fff
f(r)dV = /// Ot dr+//f(r,t) (v_~. n)dS (19) Because f = f(r(t)), this equation may be rewritten intod fr
()dr
fff Of dr+ ff
n) dS
ddd
(2o)
:Leibnitz formula for differentiating a triple integral: If V is a closed moving region in space, surrounded by
the surface S, and v~ is the velocity of any surface element, then, if s(x,y, z,t) is a scalar function of position
and time,
~ sdV= -~dV+ s(v,. :~) dS
Relative to a fixed observer, the integration limits, r, and r f, are propagating with the flame speed SI, and their position can be calculated at any instant of time. If the unburnt mixture in the flame zone were also moving with the flame speed, dr/dr and v_~ ¯ n would be equal to
S]. However, the unburnt mixture in the flame zone is moving with a velocity equal to S~ +
Sn, which is smaller than S] (otherwise no unburnt mixture would be consumed by the flame). Therefore (20) is evaluated with respect to an observer standing on the combustion wave and consequently dr/dt and v_~ ¯ n must be set equal to
Formally, the compression of the unburnt mixture ought to be evaluated in the same fashion as the mass consumption by the flame. However, dr/dr and v_s ¯ n are equal to the flame speed,
S], in this treatment. Since S] is not known in advance this effect was taken into consideration
by rewriting equation (14) into equation (16) on the basis of arguments mentioned earlier.
1,0--0.8:
0.6’ S 0.2 ~ o.o 0 ’1’1’1!1’1’1’1’1 rf Ri
1.0 -~phase 2a/,,~V 0.8 0.6 0.4 0.2 0.0 ~,~,~,~, ’ I’I’I rf R ~ l.Oqphaselb ~ ~ 1.0 ~phase2b /0.4
~’~ ’ I i ’ I ’ I ’ i I ’ I 0.0 i ’ I ’ I ’ I ’ I ’ | ’ i ’ I 0 rr rf R 0 R 1.0 "] ~ 1,0 I phase lc ~o.8
0.8
0.6 = 0.6 0.4 o=0.4-0.2
~
0.2-O.O~,~,~,,,~,~,l~ ~0.0 0 rr R 0 9hase 2c ’I’I’I’Iii’I’I’I rr RFigure 5. The fraction of unburnt mixture along the radial coordinate.
Table 1. An overview of the different phases that the flame front goes through during an
explosion.
casel:
case2:
phase la: rr = 0.0 phase 2a: rr = 0.0
rf<~
phase lb: rr =ri-5
phase 2b: rr = 0.0rf=R
phase lc: 5 _< rr __< R phase 2c: O.O < rr <_R r/=R
For the sake of clearness, a classification has been made in the description of the evolution
of the flame front. First, a distinction is made between the case when the flame thickness is
smaller than the vessel radius and the case when the flame thickness exceeds the vessel radius.
In both cases the evolution of the flame front is subdivided into three phases of which the criteria
are presented schematically in Table 1. Figure 5 depicts the fraction of unburnt mixture during
these phases.
Dust Explosions in Spherical Vessels: Prediction of the Pressure Evolution attd Determination of the Burning Velocity and Flame Thickness
In case 1 the flame thickness is less than the radius of the vessel in which the explosion is occurring.
Phase la: At the start of a dust explosion, the flame zone starts to develop with its front end moving toward the vessel wall, while its rear end remains located at the centre of the vessel. The position of the front end of the flame zone can be determined from the fact that
Vb = V.~ss~t - V~,
(21)
The mass fraction of the unburnt mixture as a function of the radial position can be expressed
as
f(r)=~l-~r]+~r if 0<,<,I
(22)
if rf<r<R
which allows us to find an expression for the volume occupied by the burnt mixture:
ry
V (23)
By using the equations (21) and (23) together with (9), the followi~g expression is obtained for the location of the front end of the flame zone
1
=
-
_---2E°
(24)
In this phase, the time evolution of the pressure can be described by inserting the equations
(22) and (20)into (18):
At the rear end of the flame zone, which remains located at the centre of the sphere, the
mass fraction of the unburnt mixture is continually decreasing with time. When the fi’action of
unburnt mixture in the centre of the sphere has become zero, the flame propagation enters a
next phase.
Phase lb: In this phase there is a fully developed flame zone with its front as well as its
rear end propagating towards the vessel wall and the mass fraction of the unburnt mixture can
be expressed as
if 0_<r<rr
if
r, < r < r/
{26)
if
r/<r_<R
By integrating (26) over the volume of the vessel, an expression is found for the volume of the
unburnt mixture:
r! R
= ~r[4 ~l 47rR3
3
Burnin~ V~locit~ ~nd Fl~me Tlzid~n~zz
Then, from equation (9) it follows that
(28)
At any instant of time the location of the front end of the flame zone can be found by solving this equation for r]. The back end of the flame zone is, of course, located at r] - 5. Insertion of the equations (26) and (20) into (18) yields the following expression for the rate of pressure rise in this phase:When the back end of the flame zone is at the centre of the explosion sphere, this equation becomes exactly the same as equation (25). Equation (29) can be rewritten into
()1
(Po -
P
dP -~o 47rS~ L 35 35 5
_
=
(30)
and it will be shown that it yields the equation for the pressure evolution of the final phase when ry equals R.
Phase lc: When the front end of the flame zone reaches the vessel wall, the explosion enters its final phase. The issue here is to determine the location of the back end of the flame zone at any instant of time. Therefore the fraction of unburnt mixture is expressed as
[-~O}
~ if 0_<r<rr
(31)
f(r)=
rr+ r if r~<r<R
and the volume of the unburnt mixture is found by integrating (31) over the vessel volume:
R
4 rr (r~ - R3r~)
:
Together with equation (9), this result gives
(32)
which can be solved for the location of the back end of the flame zone. For the rate of pressure
rise, equations (31) and (20) are inserted into (18) to yield:
P ~ [4Ra r~
dP(Pe - Po) ~ 4~rS~ [ -~ 35
(33)
which is the same as equation (30) when r] = R.
In case 2 the flame thickness is larger than the radius of the vessel in which the explosion
is occurring. This case could exist for large particles of relatively slow burning materials which
are deflagrating in a small vessel (e.g. a 20-1itre sphere).
l.~ttst l~xplostoltS tit bl)nertcat Cessets." t~reatctton o.f the Pressure Evolution attd Determination of the Burning Velocity attd Flame Thickness
Phase 2a: This phase is identical to phase la. The rear boundary of the flame zone remains fixed at the centre of the vessel and the front end propagates towards the vessel wall. The location of r] can be determined by means of (24) and the rate of pressure rise is described by equation (25).
Phase 2b: During this phase r] remains at the vessel wall and r~ remains at the centre of the vessel. However, the mass fraction of the unburnt mixture at these locations continually decreases and can be expressed as
1
f(r)=a+~r if 0<r<_R
(34)
where a is a constant that needs to be determined at any instant of time. Since the volume integral of (34) equals the volume of the unburnt mixture, this constant can be determined by means of equation (9):
The rate of pressure rise can then be predicted by inserting (34) and (20) into (18):
47rS~ aR2 + dP
-- = (36)
dt
V~s~t [k + I P~ P,]
Phase 2c: This phase is identical to phase lc. The location of the back end of the flame zone can be determined from (32) and the rate of pressure rise is expressed by (33).
5
Determination of the constant
The constant k in equation 18 can be evaluated by demanding the three zone model to yield the same result as the thin flame model does for the Kst-value. Therefore the limit situation,
is considered with a proper equation of the three zone model. An infinitely thin flame zone always satisfies the conditions of phase lb and therefore the entire pressure evolution is described by equation (29) only:
If 5 ~ 0 then r~ ~ r]. Also, in the case of an infinitely thin flame zone, the maximum rate of pressure rise is achieved when the flame reaches the vessel wall and when the explosion pressure reaches its maximum. Consequently, rr ~ ry = R and P -~ P~. After inserting equation (29) into (37), 8 -~ 0 and P ~ P~ are treated separately by splitting the right hand side into two parts. Hence,
Burning Velocity and Flame Thickness and 1 k+ 7 P
(39)
iMultiplication of 3Vves~et with the product of the right hand sides of (38) and (39) then yields
the following expression for the It’st-value:
Since we demanded this equation to be the same as equation (12), k must be equal to 2.
6
Prediction of the pressure evolution
The three zone model has been used to assess the influence of the flame thickness on the pressure development and the Kst-value for spherical explosion vessels of three different sizes, namely, 20-litre, 1 m3 and 10 m3 (figures 6 to 8). Depending on the phase which the explosion goes through, one of the differential equations (25), (29), (33) or (g6) was solved by means of a fourth order Runga-Kutta procedure [21]. At each time step, the locations of the boundaries of the flame front were determined by solving equation (28) or (32) with the method of Newton-Raphson for root finding. All simulations were done with an assumed burning velocity of 0.6 m/s, and a value of 1.4 has been taken for the specific heat ratio 7. The behaviour of the Kst-value as a function of the flame thickness is shown in the figures 9 and 10.
8 6 4 2 0 ’ I ’ 1 ’ I ’ I 0.0 O.l 0.2 0.3 0.4 time (s)
Figure 6. Simulations with the three of the flame thickness, P0 = 1.0 bar,
350- 300-~50~ 200 -150" I00-o 0.0 o.1 --1 mm -- 2.5 cm -- lOcm ... 20 cm .... 50 cm I ’ I ’ I 0.2 0.3 0.4 time (s)
zone model for the 20-1itre sphere with different values Pe = 8.0 bar, 7 = 1.4 and Su = 0.6 m/s. 8 -- lOcm ~ " ---- 200cm/#Cm ~ 80 -. .... 5 ’~ 60 - 2o-o --Imm 10 cm -- 20 cm .... 50 cm 0 ’ I ’ I I ’ I ’ I ’ I I ’ I 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 time (s) time (s)
Figure 7. Simulations with the three zone model for the 1
m3vessel with different values of
the flame thickness, P0 = 1.0 bar, Pe = 8.0 bar, 7 = 1.4 and Su = 0.6 m/s.
Dust Explosions in Spherical Vessels: Prediction of the Pressure Evolution attd Determinatiot, o.f the Burnbtg Velocity and Flame Thickness
8 ~ 4 2 1 mm I mm ~ I0 crn ~ r~ 40" ~ 10 cm ~ 20 cm //I I az"-" , ~ 20 cm .... 50crn llJ// .~~. 30- ---50cm 10 o I I I ’ I 0.0 0.4 0.8 1.2 1.6 0.0 0.4 0.8 1.2 1.6 time (s) time (s)
Figure 8. Simulations with the three zone model for the 10 m3 vessel with different values of the flame thickness, P0 = 1.0 bar, Pe = 8.0 bar, 7 = 1.4 and Su = 0.6 m/s.
Figures 6 to 8 show a number of interesting results. The left sides of these figures illustrate the effect of the flame thickness on the pressure-time behaviour occurring during an explosion. The flame thickness exerts two effects. Firstly, as the flame thickness increases, the maximal slope of the pressure-time curve decreases. This effect becomes relatively greater with increasing flame thickness and decreasing vessel size. Secondly, the shape of the pressure-time curve also changes with increasing flame thickness. In the extreme case when the flame thickness is zero, the slope of the pressure-time curve reaches its maximum at the end of the explosion. With increasing flame thickness, an inflection point appears in the curve. Furthermore the position of the inflection point shifts to an earlier time as the flame thickness increases.
The right sides of Figures 6 to 8 explicitly illustrate the effect of the flame thickness on the maximum rate of pressure rise, with a varying vessel size. Clearly seen is how an increasing flame thickness decreases the rate of pressure rise. Again, the greater the flame thickness and the smaller the vessel size, the greater the relative reduction in the maximum rate of pressure rise. These results can be transformed into Kst-values. This is done in Table 2. The reduction of the Kst-values becomes progressively greater as the flame thickness increases and the vessel size decreases. The values of Table 2 are plotted in the figures 9a and 9b. These figures suggest that the practical effect of the flame thickness in influencing Kst-values can be quite substantial. This effect will be the largest in the smallest explosion vessel currently in use, the 20-litre sphere, when employed with large particles which burn slowly (i.e. large flame thicknesses).
Table 2. An overview of the of the Kst-values obtained by simulations of which a part is presented in Figures 6, 7 and 8. The explosion pressure at which the maximum rate of pressure rise occurs, p*, is also given.
20 litre sphere 1 m3 vessel i0 m3 vessel
Kst Kst
(m)
(bar) (bar.m/s) (bar) (bar.m/s) (bar) (bar.m/s)0.001
7.615
84.572 7.859 87.795 7.902 88.3660.005
6.822
73.470 7.656 84.811 7.775 86.7200.01
5.998
61.085 7.277 80.250 7.601 84.4480.025
5.075
40.448 6.495 68.566 7.190 79.1260.05
5.060
28.388 5.379 54.815 6.477 69.7240.I
5.068
19.868 5.070 38.789 5.483 56.6660.2
5.103
13.864 5.031 27.214 5.032 40.2910.3
5.106
11.221 5.070 22.091 5.059 32.7610.5
5.035
8.610 5.075 16.961 5.063 25.214The results of these figures and table also suggest that explosion tests carried out in diffe-rently sized spherical vessels will yield different Ks~-values, even if the mixtures and turbulence levels are identical. It has been shown [16] that the similar Kst-values measured in differently sized vessels are, at least in part, reached by adjusting the turbulent flow conditions in these vessels. In other words, to compensate for the smaller Kst-values measured in the smaller vessel
Burning Velocity attd Flame Thickness
(e.g. the 20-1itre sphere), the turbulent flow conditions of the dust cloud are manipulated, via the injection procedure, to yield the same Kst-value as the 1 m3 vessel. Or, from our viewpoint, the burning velocity and the flame thickness are influenced by adjusting the injection procedure.
From a practical point of view it is desirable to make use of the cube-root-law, on which a great deal of practical safety procedures are based, while taking the effect of the flame thickness into consideration. This can be achieved by plotting the Kst-values as a function of the relative flame thickness, ~/R, at constant burning velocities. This is done in the figures 10a and 10b and it can be seen that all values now form a single curve. This figure seems to be potentially of great practical importance, since it plots Kst-values which are independent of flame thickness and vessel size.
100-] + 20 litre sphere 80 .~ ~lL +1 m3 vessel 60t~7I0 msvessel
4o1
20 0 I ’ I ° I ’ I ’ I ’ I 0.0 0.1 0.2 0.3 0.4 0.5 100 2O ~ 10 m3 vessel....I ...I ...I
¯ 0.001 0.01 0. t
(a) flame thickness (m) (b) flame thickness (m)
Figure 9. Kst-values as a function of the absolute flame thickness on (a) a linear scale and (b) a a°log scale. Po = 1.0 bar, Pe = 8.0 bar, 7 = 1.4 and S~ = 0.6 m/s.
100],~’~ ¯ ’~m ~ ~ 60 ~ 40 ~,~ ~d i 201itre sphere 20t : lrn3vessel 10 m~vessel 0.001 0.01 0.1
(a) relative flame thickness (-) (b) relative flame thickness (-)
Figure 10. Kst-values as a function of the relative flame thickness on (a) a linear scale and (b) a ~°log scale. P0 = 1.0 bar, Pe = 8.0 bar, 7 = 1.4 and S~ = 0.6 m/s.
In fact Figure 10 can serve as a nomogram that can be used to predict the maximum rate of pressure rise. Since the Kst-values appear to be a unique function of the relative flame thickness, there is also the possibility of normalizing them with respect to the latter. To do so, this function must be known analytically. In fact, any function which goes through these points would satisfy for the domain of flame thicknesses in question. In order to take the limiting behaviour of the Kst-value when ~ --~ 0 and ~ -~ ~ into account, a matching procedure was carried out as is shown in Figure 11. The curves in this figure are tangent to each other when ~/R ~ 8%. This matching procedure leads to the following normaHz~tion of the ma~mum rate of pressure rise with respect to explosion volume and relative flame thickness:
dP
where
Of course, in order to ~pply it ia practice, the v~lue of the flame thickness ~nd the burning velocity need to be known. Ia the next section we discuss how the theory developed here c~n be
Dust Explosions in Sphertcal Vessels: l#’ediction of the Pressure Evolution attd Determination of the Burning Velocity attd Flame Thickness
applied to experimental data in order to determine the required values of the burning velocity and the flame thickness.
lOO
80
60
40-
20-o
0.001 " 0.01 0.1 1curve 1 :
curve 2 :
relative flame thickness [-]
Figure 11. Matching the Kst-value as a function of the relative flame thickness.
350
-250 ;
200 -".150
-.
100 5O + 20 litre sphere + I m3 vessel + 10 m3vessel I ’ I ’ I ’ I ’ I , I , I~0.0 0.5 1.0 1.5 2.0 2,5 3.0
350 -300 250200
150100
5O
-0 0.001 + 20 litre sphere ~ + 1 m~vessel ~"1 ’ ..."1 ... "1 ’ ..."1 ’ ’O.Ol O.l
1
relative flame thickness (-) relative flame thickness (-)
Figure 12. (dp/dt)m~ as a function of the relative flame thickness on a linear scale and a
1°log scale. P0 = 1.0 bar, Pe = 8.0 bar, 7 = 1.4 and S~ = 0.6 m/s.
One might wonder whether a single curve would also be obtained by plotting the maximum
rate of pressure rise, instead of the Kst-value, against the relative flame thickness. Because of
the compexity of the equations of the three zone model, this question can only be answered by
making such a plot. This is done in Figure 12, and it is shown that there is no single curve. This
implies that the ’cube-root-law’ is a step in the right direction to achieve normalization of dust
explosion severity. However, it requires more than volume-normalization to define an invariant
property for dust explosion severity.
7
Determination of flame thickness and burning velocity
We now demonstrate the application of the three zone model in the determination of the
burning velocity and the flame thickness of real confined deflagrations. For this purpose
ex-perimental dust explosion data of maize starch, measured in the strengthened 20-1itre sphere
[22, 23], are used.
In the previous section the burning velocity and the flame thickness were assumed to remain
constant during the course of a dust explosion. For the application of the three zone model to real
dust exp!osions, it is necessary to incorporate the dependence of these parameters on influencing
factors like the pressure, temperature etc. In order to apply this model to a cornstarch/air
explosion, the effect of the pressure and the temperature on the burning velocity was included
as follows. The dependence of S~ on the pressure and the temperature was taken as [2, p.301]:
Burning Velocity and Flame Thickness
where the index 0 refers to a reference state of 300 K and atmospheric pressure, T~ denotes the temperature of the unburnt mixture and/3 is an empericM constant. For adiabatic compression,
(43)
and equation (42) may be rewritten into
with a = 2 - 2/7 -/3. Equation (44) was incorporated into the three zone model and the differenti!l equations were fitted to the experimental explosion curve by means of the Levenberg-Marquardt method (see Figure 13). It should be noticed that, despite the incorporation of equation (44), Su must be regarded as a ’pressure and temperature dependent average burning velocity’ since the influences of the ’varying turbulence’ and altering dust concentration during the course of the explosion are not included.
8.0--6.0"
4,0"
2,0"
0.0
measured po!nts with
’ I ’ I ’ I ’ I ’ I ’ I
0.00 0.01 0.02 0.03 0.04 0.05 0.06 time (s)
Figure 13. Application of the three zone model to a 500 g/m3 cornstarch/air mixture, Po
= 1,0 bar, P, = 7.467 bar and 7 = 1.4.
In the early part of the curve in Figure (13), the fit of the model to the experimental data
is excellent. In the latter part of the curve, which is magnified, the fit is less good. This is due
to the fact that the influence of changes of the turbulent flow field and the effect of changes in
the dust concentration are not included in the key parameters. For this reason, it would even
be a bad sign if the model fitted perfectly to the entire pressure curve.
The fit results are shown in Table 3. As equation (44) suggests, the burning velocity, S~,,
varies during the course of the explosion. According to equation (2), the flame thickness should
also be variable during the explosion. Instead, we assumed that the variable flame thickness
could be represented by an average, but constant, flame thickness. Another possibility was to
maintain a variable flame thickness according to equation (2), but then we had to assume a
value for the burning time of a particle, ~-c.
Table 3. Fit results.
Parameter
Value + StdErr
(300.2 + 4.0)E-02 m/s
(-55.6 + 1.9)E-02
(43.6 + 1.6)E-03 m
Fit Method: Marquardt Non-Linear Fit. Sum of Res. squared: 6.0534 with 128 degr. of
freedom. This has a P-value of 1.0000. For a 95.0% Confidence interval, multiply StdErr
by 1.9787
The regression analysis yields a value of 3.00 m/s for the initial burning velocity, S~0, which
is in agreement with that found in the literature [2, p.302] namely, 3.15
m/s.
The value of a
however, implies that /3 = 1.13, which is not in agreement with that found in the literature
Dust Explosions in Spherical Vessels." Predictiot, of the Pressure Evolution attd Determination of the Burning Velocity and Flame Thickness
(/3 = 0.36 [2]). With /3 = 1.13, equation (42) reveals that the temperature and the pressure have opposite effects on the burning velocity. The right part of Figure 14 shows the temperature increase of the unburnt mixture due to compression and by comparing it with Figure 13 it can be seen that (Tu/To) << (P/Po). Therefore the overall effect is a decrease in the burning velocity (as shown in the left part of Figure 14).
10 20 30 40 50 60 70 time (ms)
Figure 14.
mixture.
250 0 ’ I ’ I ’ I ’ | ’ I ’ i ’ I l0 20 30 40 50 60 70 time (ms)The behaviour of the burning velocity and the temperature of the unburnt
For the flame thickness, ~, a value of about 4 cm has been found by the fit procedure. The interpretation of this result is difficult since we have no reference values to compare with. Therefore we validate this result-as follows. With equation (2) and the result for the burning velocity, the burning time of a cornstarch particle is calculated to be 14.5 miliseconds. From the literature [24] we have the following relationship for the combustion time, rc, of a particle
where dp denotes the diameter of the particle and K is an emperical constant that equals about 2000 s cm-~ for solid particles on the average. Since the cornstach particles have an average diameter of 15/zm, the burning time on the basis of (45) equals 4.5 miliseconds. This value for the burning time of a particle and the value on the basis of the flame thickness differ less than one order of magnitude.
8
Conclusions
1. A model is presented which is capable of predicting the pressure evolution during confined dust explosions in spherical vessels (the three zone model). The advantages of the model can be summarized as follows:
a) The model can be fitted to experimental dust explosion pressure-time recordings to determine the burning velocity and the flame thickness, which is demonstrated in the previous paragraph. In this way a simple and cheap simulation tool has been develo-ped to determine these key parameters. Another advantage is that old pressure-time recordings of dust explosion experiments can be utilized for the determination of the flame thickness and the burning velocity. This is interesting for expensive powders. b) The model can be used to predict the pressure development of dust explosions for
safety purposes. Simulations with this model (e.g. Figure 6) show a good resem-blance to experimentally observed pressure-time curves (Figure 1). Once the burning velocity and the flame thickness are known by fitting the model to the data of one ex-periment, the pressure evolution in another situation can be predicted by simulating the model. However, the influence of variations of the flow field and the dust con-centraion during the course of the explosion must be included in the flame thickness and the burning velocity.
c) The model provides the possibility of investigating the influence of a single parameter or influential factor on the maximum rate of pressure rise. In practice it is very difficult, if not impossible, to vary a single influential factor while keeping everything
Dust Explosiotts in hphertcat vessets: ~reatc[ton of ttte l~ressure Evolution attd l_)etertntttatton Burning Velocity and Flame Thickness
else the same. In this work the role of the flame thickness has been investigated and it is shown that the ma~mum rate of pressure rise can be normalized with respect to the vessel volume as well as to the flame thickness (Figure 10), ~vhich can be expressed as
where
From Figure 10b it can be seen that for relative flame thicknesses of up to 1%, the maximum rate of pressure rise can be predicted by the ’cube-root-law’. For greater flame thicknesses, a behaviour can be observed, which deviates significantly from the ’cube-root-law’. As the flame thickness increases, the Ks~-value decreases. This effect is of great practical importance, since it indicates that explosion testing of dusts with large, slowly burning particles in small vessels, such as the 20-1itre sphere, results in Kst-values which are too low.
The compression of the unburnt mixture and its effect on the dust concentration, as described in Section 2, raises an interesting point regarding the concept of ’overdriving’ an explosion. If an ignition source has sufficient energy to cause the concentration of dust particles ahead of the advancing ignition front to incre’ase to within the minimum explosible concentration, then propagation may occur. A lesser ignition source strength may not achieve the same effect.
9
List of symbols
7
specific heat ratio Cp/Cp
5
flame thickness
m
Kst
volume normalized maximum rate of pressure
bar m/s
rise
P0
initial pressure
N/m2
Pe
final pressure after explosion
N/m~
?’flame
location of the thin flame front
m
ry
location of the flame front boundary at the
m
side of the unburnt mixture
r~
location
of
the flame front boundary at the
m
side of the burnt mixture
R
vessel radius
m
R~
specific gas constant
J/kg K
p~
density of the unburnt mixture
kg rn-3Sy
flame speed
m s-~
S~,
burning velocity
m ~-~
S~
expansion velocity
m s-~
S~
velocity due to conversion
m s-1
Tu
temperature of the unburnt mixture
K
r~
chemical time scale
s
~
volume occupied by the unburnt mixture
m3
Vb
volume occupied by the burnt mixture
m3z~usg zz.~plostotts tn 3pt~ertcat vessets: l~reatcttott oJ ttte Pressure lz’voluttott atta I_)etertntttatton o] the Burning Velocity attd Flame Thickness
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