1. The Explosion Limits of Confined Gaseous Reactive Mixtures in
Porous Media
Miguel Becerra, BS in progress, Mechanical Engineering, California Polytechnic State University San Luis Obispo, San Luis Obispo, CA
ABSTRACT
Reactive flow in porous media is important for its technological applications, such
as catalytic pellets for propulsion systems, and its implications regarding safety
hazards, an example being explosions in grain silos. The focus of the present
investigation is on the effects of the buoyancy-driven flow on the resulting system
response. As a model problem, we consider steady reactive solutions in an
enclosed spherical vessel with constant-temperature walls. Besides the Damköhler
number 𝐷𝑎, defined as the ratio of the conduction time across the vessel to the
characteristic chemical time, the solution is seen to depend on the Rayleigh
number 𝑅𝑎, which characterizes the effect of the buoyant motion on the heat
transfer to the walls. Analytic solutions are sought using an asymptotic expansion
with the Rayleigh number 𝑅𝑎 as the small parameter. The analysis provides
predictions for critical explosion conditions accounting for buoyancy effects.
BACKGROUND
Grain Dust Acting as a Porous Media and its Potential to Explode
• Grain dust suspended in the air within an enclosed space can explode when it reacts
with oxygen and if there is a form of heat conduction that ignites the explosion
Kansas Grain Silo Explosion and Significance
• Six workers were killed after a deadly grain elevator explosion in Kansas back in
October of 2011
• In order to avoid future grain silo explosions, there is a need for factories to develop
strong safety factors that will maintain a secure work environment and community
OBJECTIVES
• Investigate through numerical analysis the conditions that trigger an explosion
of a gaseous reactive mixture through a porous medium
• Investigate the heat distribution that occurs when buoyancy-induced motion is
applied and when buoyancy-induced motion is not applied
• Gain a better understanding of the conditions that trigger grain dust explosions
• Conditions:
• Analysis performed on a small, enclosed spherical vessel
• Walls are kept at a constant temperature difference
METHODS
Mother Equations, Derived in Spherical Coordinates
• Continuity Equation, 𝛻 ⋅ 𝑉 = 0
• Momentum Equation, 𝑉 = −𝛻𝑝 + 𝜙𝑒 𝑧
• Energy Equation, 𝑅𝑎 𝑉 ⋅ 𝛻𝜙 = 𝛻2 𝜙 + 𝐷𝑎 𝑒 𝜙
Important Variables
• Rayleigh Number (𝑅𝑎) represents a dimensionless number associated with
buoyancy-driven flow, where 𝑅𝑎 ≪ 1 if the effect of buoyancy-induced
motion is small, and 𝑅𝑎 ≫ 1 if the effect of buoyancy-induced motion is
high
• Damköhler Number (𝐷𝑎) represents the ratio of the heat conduction time
across the vessel to the chemical reaction time, and the ratio will also
determine the point of ignition at which an explosion begins to occur
• 𝜙 represents the temperature difference within the vessel as a function of
𝑟 and 𝜃, where 𝑟 represents the radius from the center of the sphere and
𝜃 represents the angle with respect to the vertical (𝑧) axis
• 𝜓 represents the streamline function
Defining the Regular Perturbation Series
• 𝜙 = 𝜙 𝑜 + 𝑅𝑎 𝜙1 + 𝑅𝑎2
𝜙2 , 𝜓 = 𝜓 𝑜 + 𝑅𝑎 𝜓1 + 𝑅𝑎2
𝜓2
• Leading order starts at 𝜙 = 𝜙 𝑜 and 𝜓 = 𝜓0 , where 𝑅𝑎 ≪ 1
• Leading order represents zero buoyancy-induced motion
At Zero Buoyancy-Induced Motion, 𝑹𝒂 = 𝟎
• Energy equation converts to a 2nd order ordinary differential equation
• 𝜙′′
+
2
𝑟
𝜙′
+ 𝐷𝑎 𝑒 𝜙
= 0 , Equation is a function of 𝑟, with domain of (0,1)
• 𝑟 = 0 represents a singular point, 𝜙 0 = ∞ , ∅ 1 = 0 , ∅′
0 = 0
RESULTS
RESULTS
CONCLUSIONS
• At 𝑅𝑎 = 0, a fire and ignition begin to occur at 𝐷𝑎 = 3.322, and heat
distribution begins from the center of the sphere outward in the form of heat
conduction
• Based on the 𝐷𝑎 number, the chemical reaction time is very small compared to
the heat conduction time
• Relating the research to grain silos, it is important for factories to enforce strict
regulations as to the temperature that is maintained within the enclosed
system and making sure that no metallic debris is present in order to avoid any
type of fire ignition from heat conduction
• For future research, one would examine what occurs at the next orders of the
regular perturbation series and at larger Rayleigh numbers
• Important to understand how the Damköhler number is effected when buoyancy-
induced motion is introduced
• Analyze if heat transfer will be in the form of conduction or convection
• In addition, one would analyze an enclosed system using cylindrical
coordinates in order to have a better understanding of explosion limits within
grain silos
• These graphs represent different Damköhler numbers and the temperature difference
at each value
• Through these graphs, it is concluded that 𝐷𝑎 = 3.322 represents the critical
Damköhler number at which a fire begins to occur at ∅ = 1.609
• These graphs represent the heat flow velocity, the streamlines at different velocities, and
the vortex that depicts heat transfer from the center of the sphere to the walls
• Through further numerical analysis, it is seen that at 𝐷𝑎 = 2.5 the temperature difference
can equal 0.6502 and 3.504
• These graphs depict the different explosion points that occur at different Damköhler numbers
ACKNOWLEDGEMENTS
The work reported here was carried out at the Department of Mechanical and
Aerospace Engineering at UC San Diego during a summer research stay funded
through the STARS program. My mentors Daniel Moreno-Boza and Antonio L.
Sánchez are gratefully acknowledged for suggesting the problem and for
guiding my research efforts.