0.1 0.2 0.3 0.4 0.5 0.6 0.7 A/V 2/3 FIGURE 15. - Pressure rise versus vent ratio (A/V2/3) for venting stoichiometric CH4-air ignitions with 0, 10, 20, and 30 pct added N2. (Calculations by equation 10 for mixtures initially at 300° K and 1 atm.) such inerted explosions is not fully valid since the flame propagations will not necessarily be spherical, particularly for the highly inerted ignitions which tend to display large buoyancy effects. However, this uncertainty is largely offset by the conservative assumptions made in the derivation of this venting equation. For example, the assumption of adiabatic combustion and that the maximum S of the mixture is attained at the instant of ignition may not be realized in most instances but is prudent from the standpoint of safety. The predicted freeventing requirements for "spherical" explosions of highly inerted mixtures were much lower than those for uninerted mixtures. Figure 15 shows the predicted pressure rise as a function of the vent ratio (A/v2/3) for stoichiometric CH -air ignitions with 0 to 30 pct added N. As noted, the pressure rise at a given A/v/3 ratio is reduced by about fiftyfold when the CH-air ignitions are inerted with 30 pct N2; this reduction is attributable to the much lower S, and E values associated with the inerted mixture. The above figure was obtained using the S, data in figure 7 and the corresponding E values that were calculated for each mixture assuming adiabatic combustion. The venting requirements in figure 15 are primarily applicable to the early stages of combustion when pressure increases are still relatively small. At the same time, venting during this stage is of the greatest importance as a protection against explosions in large industrial facilities, including those of the automobile shredder industry. Furthermore, when the explosions involve inerted mixtures such as those in figure 1, both the spatial and temporal requirements for venting become less stringent. Unfortunately, venting data were not available in the literature for inerted systems and, therefore, a comparison could not be made between experiment and theory and present data should be applied only to normal flame propagation since venting requirements are generally greater for turbulent conditions. CONCLUSIONS The pressure development of near-stoichiometric CH1-air ignitions in a large spherical enclosure roughly approximates the cubic law (AP = kt3) during the early period of combustion but tends to deviate from this law with the addition of excess air or inert because of buoyancy effects. The downward limit-of-flammability was about 5.6 pct CH4 in air and the corresponding inerting limit with downward propagation was about 24 pct N2 at an optimum fuel-air ratio; flame front distortion is extensive at such limits. Generally, Na dilution had a greater inhibiting effect on flame speed (S,) and burning velocity (S1) than did air dilution. The S values increase with burning duration and yield an S of approximately 44 cm/sec during the prepressure period for a 9.4 pct CH, -air mixture. A correlation of flame speed and buoyant velocity data indicates that a stoichiometric CH -air mixture should display accelerated burning when the fireball diameter is larger than 440 cm. In addition, theoretical expressions are derived for predicting the unrestricted venting requirements for spherical explosions of hydrocarbon vapor-air-diluent mixtures in differently sized enclosures. Predicted venting requirements for ignitions of CH -air and other mixtures are at least as conservative as those found experimentally by other investigators for unrestricted venting conditions. APPENDIX A. --DERIVATION OF VENTING EQUATIONS The venting expressions in this report were derived by modifying Perlee's mathematical treatment of constant volume combustion in a spherical enclosure to accommodate venting. Since the present treatment entails additional assump tions and requires different mass balance equations, essentially complete derivations of the basic descriptive expressions are given here. Initially, the case of venting unburned gas is considered and, subsequently, that of burned gas; venting criteria are then derived for these two cases. following assumptions are made in deriving the basic expressions: The 2. Spherically symmetric geometry, that is, neglect distortion due to buoyancy and venting. 3. Adiabatic compression of burned and unburned gases. 4. 5. 6. 7. 8. 9. Flame thickness is negligible in attainment of local thermodynamic equilibrium. Flame speed (S) is less than 0.1 speed of sound in the gas mixture so that no significant pressure change occurs across the flame front. Pressure (P) is uniform in the sphere and a function only of time for the given gas mixture. Vent opens instantaneously at a given pressure or time. Ideal flow through vent as determined by the discharge coefficient. Temperature (Tu, T), density (Pu, Pb), and specific heat ratio Case of Venting Unburned Gas The ignition of a flammable gas mixture in a spherical enclosure (V.) is assumed to propagate as a spherically expanding flame. Following ignition, the gas being burned during each successive time interval is expanded to a larger volume but, simultaneously, the entire volume of unburned (V.) and burned (V) gas mixture is compressed, resulting in an increase of pressure (P). The ideal equations of state for the unburned and burned gases at any time can be written as follows: P Vn R Tu, น (1) 1Perlee, H. E., F. N. Fuller, and C. H. Saul. BuMines RI 7839, 1973, 24 pp. Constant Volume Propagation. Differentiating equations 1 and 2 with respect to time (t) and eliminating dt, one obtains If unburned gas is being vented, the mass balance at any time is given by where m is the initial mass and muy is the total mass of unburned gas. differentiation, dm/at or dm = o and (2) (3) (4) (5) (6) Upon (7) -dmuc, that is, the change in mass of burned gas is equal to the decrease of unburned gas due to combustion, and converting to moles (n1), By letting dm2 = -dmuc Since the mole change of unburned gas due to combustion (dnuc) is defined as a negative quantity and that due to venting (dnv) as a positive quantity, dnu is actually the sum of two negative terms in equation 8. For the burned gas, the mole change is given by Muc dnuc. The temperature of the burned gas at any time is assumed to be equal to that of the burned gas at the center of the sphere and is described here by the differential form of Poisson's equation; this equation is also applicable to the unburned gas, Substituting in equations 4 and 5 for Vu, dVu, dTu, dTь, and dn,, the respective equations for unburned and burned gas become. Furthermore, if Sdt is the distance the spherical flame front advances into the unburned gas relative to the flame front prior to expansion, the mole change of unburned gas consumed can be defined during the time interval dt as น (15) where R is flame radius and S1 is burning velocity; Su, Pu, and M, are functions of P and Tu. Also, if the unburned gas is being vented through a circular vent of radius R, and at a velocity Vu, it can be written as follows: where ca is a discharge coefficient, Pu is density of unburned gas, is the pressure drop across the vent. (16) (17) and P-P. (18) |