Brode

L. Siggel and S. Brode, technical data, BASF AG, Ludwigshafen, Germany, 1992. 

Finite-difference schemes used to solve Lagrangean gas dynamics have been described many times (Richtmyer and Morton 1967 Brode 1955, 1959 Oppenheim 1973 Luckritz 1977 MacKenzie and Martin 1982 Van Wingerden 1984 and Van den Berg 1984). 

In the earliest applications of numerical methods for the computation of blast waves, the burst of a pressurized sphere was computed. As the sphere s diameter is reduced and its initial pressure increased, the problem more closely approaches a point-source explosion problem. Brode (1955,1959) used the Lagrangean artificial-viscosity approach, which was the state of the art of that time. He analyzed blasts produced by both aforementioned sources. The decaying blast wave was simulated, and blast wave properties were registered as a function of distance. The code reproduced experimentally observed phenomena, such as overexpansion, subsequent recompression, and the formation of a secondary wave. It was found that the shape of the blast wave at some distance was independent of source properties. 

Brode, H. L. 1955. Numerical solutions of a spherical blast wave. J. Appl. Phys. 26 766-775. 

Brode, H. L. 1959. Blast wave from a spherical charge. Physics of Fluids. 2(2) 217-229. 

Many numerical methods have been proposed for this problem, most of them finite-difference methods. Using a finite-difference technique, Brode (1955) analyzed the expansion of hot and cold air spheres with pressures of 2000 bar and 1210 bar. The detailed results allowed Brode to describe precisely the shock formation process and to explain the occurrence of a second shock. 

This energy measure is equal to Brode s definition of the energy, multiplied by a factor 2. The reason for the multiplication is that the Brode definition applies to free-air burst, while Eq. (6.3.15) is for a surface burst. In a fiee-air burst, explosion energy is spread over twice the volume of air. 

In Figure 6.35, lines have been added for a sphere bursting into 2 or 100 pieces for pi/po = 50 and 10, in accordance with Figure 6.33. Obviously, the simple relations proposed by Brode (1959) and Baum (1984) predict the highest velocity. Differences between models become significant for small values of scaled energy E, in the following equation ... 

In the relationships proposed by Brode (1959) and in Figure 6.33, velocity has no upper limit, although Figure 6.33 is approximately bounded by scaled pressures of 0.05 and 0.2 (scaled energies of approximately 0.1 and 0.7). Baum (1984) states, however, that there is an upper limit to velocity, as follows The maximum velocity... 

It is not clear which measure of explosion energy is most suitable. Note that, in the method presented in Section 6.3, the energy of gas-filled pressure vessel bursts is calculated by use of Brode s formula, and for vessels filled with vapor, by use of the formula for work done in expansion. 

Boyer, D. W., H. L. Brode, I. I. Glass, and J. G. Hall. 1958. Blast from a pressurized sphere. UTIA Report No. 48. Toronto Institute of Aerophysics, University of Toronto. 

The total energy of a vessel s contents is a measure of the strength of the explosion following rupture. For both the statistical and the theoretical models, a value for this energy must be calculated. The first equation for a vessel filled with an ideal gas was derived by Brode (1959) ... 

First, energy must be calculated. Di erences among results from the various equations are illustrated here by the application of each to the problem. Brode [Eq. (9.3.1)] gives... 

EJIC913 D. Enders, H. Gielen, J. Runsink, K. Breuer, S. Brode, and K. Boehn,... 

Enders D, Gielen H, Runsink J, Breuer K, Brode S, Boehn K (1998) Eur J Inorg Chem... 

Enders D, Breuer K, Raade G, Runsink J, Teles JH, Melder JP, Ebel K, Brode S (1995) Angew Chem Int Ed Engl 34 1021-1023... 

Four methods are used to estimate the energy of explosion for a pressurized gas Brode s equation, isentropic expansion, isothermal expansion, and thermodynamic availability. Brode s method21 is perhaps the simplest approach. It determines the energy required to raise the pressure of the gas at constant volume from atmospheric pressure to the final gas pressure in the vessel. The resulting expression is... 

H. L. Brode, Blast Waves from a Spherical Charge, Physics of Fluids (1959), 2 17. 

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