Theoretical Approach to Dust Explosion

A number of experimental researches on dust explosion have been published with measured data of the explosive characteristics such as the ignition temperature, the flame propagation velocity, the upper and lower limit explosible concentrations and the rate of pressure rise. In this study the above-mentioned explosive characteristics are dealt with system atically based on a simple model of uniformly dispersed dust cloud consisting of particles of the same size, from the viewpoint of heat transfer process. Comparing the computed results with the empirical law and the experimental data, it is assured that the explosive characteristics can be predicted theoretically. In ad dition, the analysis of the area for the pressure relief venting is conducted to minimize the damage, leading to a new dimensionless vent ratio.


Prediction of ignition temperature
the heat emitting from the solid surface into the atmosphere by convection q 2 are taken on the ordinate. When q 1 is less than q 2 , the reaction is always kept at the level of oxidation. On the other hand, when q 1 is more than q 2 , the reaction of combustion makes continuous progress. A steady state can be kept only on the conditions ofq1 =q2 anddq1/dTs =dq2/dT 8 • The reading of the horizontal axis T (surrounding temperature) on this condition is experimentally defined as the ignition temperature.

1 Definition of ignition temperature')
The ignition temperature is defined as the gas temperature at which a mass of particles begin to burn, when they are heated in an oven with a certain shape and size. It has been measured with a great number of materials. In Fig. 1, the surface temperature of solid T 8 is displayed on the abscissa and the heat of reaction q 1 and  4 • 6 ) The reaction rate of oxidation of a single spherical particle with a diameter of Dp is given by Hottel eta!. s) as follows:

2 Ignition temperature of dust cloud
where m is mass of the particle, t time, ks rate constant of reaction, E activation energy and Cg concentration of oxygen. The number of the particles n in the dust cloud with a diameter land the dust concentration Cd is n = CdP !PsD/ (2) for the particle of a density Ps· The generated heat G in the lump of the particles and the emitted heat U from them are calculated in the following equations:  stant and film coefficient of heat transfer, respectively. The suffixes P, G and W represent particle, gas and wall respectively. The solution of T of the simultaneous equations dG dU (5) G = U and dTs dTs is compared with the empirical ignition tem-perature2·3) in Fig. 2. The value of A is plotted against the activation energy in

I Model of propagation
Let us consider a cloud of dust in the vast space where a series of particles can be regarded to line up in the direction of flame propagation excluding the case of a long pipe of small diameter. The average distance L between each particle with a diameter Dp is calculated as L = ( ; D;psfCd) 113 (6) with the dust concentration Cd and the solid density Ps· The time interval Mn between the ignition of the nth particle and that of the (n + 1 )th is the time required for the flame propagation. From these values, the so-called burning velocity is calculated. Meanwhile, if the left end of a vessel is closed and the right end is open, the particles in the second and the following sections will be pushed toward right owing to the pressure rise by the generated gas from the combustion of the 1st particle. The successive particles start burning one after another and cause the expansion of gas. In this way the expansion velocity is calculated as the sum of the transportation effect by the simultaneously burning particles.

2 Burning velocity
The equation of one-dimensional thermal diffusion and its solution with the boundary conditions of T = T 1 =constant flame temperature, and T = To at t = 0 are: where Tis the gas temperature with the suffix f for the flame and 0 for the initial stage and K represents the coefficient of thermal diffusion. The radius Rb of the flame is given as for a liquid droplet with a diameter Dp and equilibrium constant k* (k* = 1 ~ 1 0 em?~ Taking dynamic heat balance of a particle, The suffixes gL and dL represent the gas and the particle at the point of L respectively. The time required for the complete combustion of a single particle T is given as with the burning constant KD = 100 ~ 1000 (sec/cm 2 ) 8 >.

3 Expansion velocity, overall flame propagation velocity
The mass of a single particle m changes with time e as follows: When net nc moles of gas is generated by the combustion of one mole of the particles, the volume increases at the same pressure P and the temperature Tc from V 1 (=nNRT 0 /P)to V2=(nN+nc)RTc/P (14) Where nN is the number of mole of air per one mole of particle and nc the number of mole of gas evolved by the combustion of one mole of the particle. The velocity due to this expansion v (e) is

R dm v(e)= L2MP [(nN +nc)Tc -nNTo] de
M is the molecular weight of the particle. Dust cloud is regarded as a group of uniformly dispersed particles in the vast space with a constant mutual distance of L as shown in Fig. 7. The distance between each particle is given as Placing the dust cloud in the spherical coordination, first the particle in the center begins to burn and then the particles in the 2nd shell are heated by the emitted heat from the central particle. Supposing the particles in the 2nd shell reach the ignition temperature Tig and ignite at the moment when the central particle has finished its combustion, gives the limit condition for the flame propagation to be possible. The limit concentration Ca is calculated from the L satisfying the above-mentioned condition and the equation ( 1 7). This is the theoretical lower limit explosible concentration. and T= Ti at t = 0, the solution is From the heat balance of the particles, Assuming the relations 1) = t/T (21) and The predicted Cd with 1l = 1 (lower limit explosible concentration) is compared with the  Fig. 8. As for the eel-concentration is clearly defined as the one at lulose compounds both values agree well when the state where all the particles at least begin to k* is 10 em ll).
burn, not to say have finished burning. The 3. 2 Theoretical consideration on upper limit explosible concentration 12 ) Oxygen in a limited area must diffuse to catch up with the combustion of some particles in the dust cloud for the continuous burning. First the total amount of oxygen in the considered space is shared evenly among all the particles in it. Each particle must get enough heat for ignition from the adjoining burning particle, by the time when it. consumes its assigned oxygen. The upper limit explosible dust  13 • 14 ) is shown in Fig. 9. In this case explosion cannot take place with the oxygen concentration under 10%. Thus it is possible to conduct the fundamental calculation for "inerting ".

. 1 Cubical law
The pressure and the rate of pressure rise of dust explosion in a closed vessel are most immediate characteristics to show the degree of hazard. where V 0 is the volume of a spherical vessel and dP/dt is the rate of pressure rise.  Recently the above equation has become agreed empirically 15 ). Kcc has been found to be a constant for each dust, but to differ depending on the kind of material, particle size, dust concentration and so on, although the details of it still remain unknown. In the present report it will be discussed to understand the meaning of the constant Kcc and to enable to predict it for unknown materials and under any condition. Besides the effect on scale-up by the extention to the case of unspherical vessels and the possibility of comparing experimental data obtained with the testers of different size and shape will be also discussed in the followings.

2 Modeling of dust cloud in vessel
When the particles are dispersed as shown in   Therefore, the total mass of the material which started burning at t = 0 and has burned out by the time t is In case the oxygen is enough, the maximum amount M 0 of the burned out material in a vessel of volume V 0 and with the concentration If Cd is rather large and the oxygen is not enough, Where a is equal to Cd, when Cd is smaller than a definite value.

5 Pressure inside a vessel and maximum rate of pressure rise
P=RTc (nc/V) For the adiabatic system Combining the above equations, therefore This equation shows that the empirical cubical law has been deduced theoretically and that when the properties of the dust (chemical formula, concentration, particle size) are known, the value Kcc in the equation (24 ), namely the right hand side of the equation (39) is apparently constant. Though a spherical vessel has been dealt with here, the cubical law including a shape factor can be adaptable also for vessels of approximately cylindrical shape, which suggests the possibility of exchanging data with vessels of various shapes.

1 Vent ratio
The vent ratio is defined conventionally as the one of the areaS of the pressure relief vent to the volume V 0 of the vessel. Empirically it is used for designing pressure relief vent but not sound because it is not dimensionless.  Fig.   16 with different symbols for the various kinds of material. The theoretical values seem larger to some extent than the experimental ones, as the first term of rate of pressure rise in Eq. ( 40) is replaced by the maximum one to assure safety. But it is clearly understood from the figures that the tendency of the predicted values is in accordance with that of the experimental ones. Besides, the compared values range over the second power of I 0, which suggests the theoretical equation can be applied to the wide 12 n n nc mass of particle number of particles at nth spherical surface number of assumed spherical surface number of particles mole number of gas produced by combustion per one mole of particle mole number of air per one mole of particle pressure pressure ratio