SIAM Journal on Applied Mathematics, Volume 80, Issue 2, Page 1003-1033, January 2020.
Microsilica particles arise as a byproduct of silicon furnace operation, created inside high temperature flames due to the combustion reaction of silicon monoxide with oxygen. These nanoparticles, which grow as silicon dioxide vapor condenses on the surface of existing particles, are used in a variety of composite materials. The size and quality of the particles affect the performance of the material used for such applications, and hence control of these quantities is of importance to manufacturers. Motivated by this, we derive a mathematical model that connects local thermal and chemical concentrations conditions to the formation and growth of microsilica particles. We consider two distinct reductions of our general model: the case of initially well-mixed or spatially homogeneous chemical species (modeling the region within the flame or reaction zone), and the case of initially spatially separated chemical species, in which diffusion will play a dominant role in providing material to a combustion front (modeling a larger cross section, which contains a reaction zone with limiting quantities of fuel which must diffuse into the reaction zone). In both cases, we provide asymptotic solutions for the temperature, chemical concentrations, and number density function of microsilica particles in the oxygen rich limit, and compare them to numerical simulations. Motivated by realistic furnace control mechanisms, we treat the relative quantity of oxygen to other fuel components and the saturation concentration of silicon dioxide as control parameters, discussing how each may be used to modify the properties (such as size and abundance) of microsilica particles formed. One physically interesting finding is the theoretical description of a bimodal distribution for microsilica particle size which was previously observed in experiments.

中文翻译：

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一氧化硅与氧气燃烧反应引起的微二氧化硅颗粒形成和生长的模拟

SIAM应用数学杂志，第80卷，第2期，第1003-1033页，2020年1月。
由于一氧化硅与氧气的燃烧反应，在高温火焰内部产生的硅微粉颗粒是硅炉操作的副产品。这些纳米粒子随着二氧化硅蒸气在现有粒子表面上的凝结而生长，可用于多种复合材料中。颗粒的尺寸和质量会影响用于此类应用的材料的性能，因此，控制这些数量对制造商而言很重要。因此，我们得出了一个数学模型，该模型将局部热化学浓度条件与微硅颗粒的形成和生长联系起来。我们考虑了通用模型的两个明显简化：最初混合均匀或空间均匀的化学物种（对火焰或反应区域内的区域进行建模）的情况，以及初始空间分离的化学物种的情况，其中扩散将在向燃烧前沿提供材料方面起主要作用（对较大的横截面进行建模，其中包含一个反应区，该反应区的燃料数量有限，必须扩散到反应区中。在这两种情况下，我们都为富氧极限内的微二氧化硅颗粒的温度，化学浓度和数量密度函数提供渐近解，并将它们与数值模拟进行比较。根据现实的炉子控制机制，我们将氧气相对于其他燃料成分的相对量和二氧化硅的饱和浓度作为控制参数，讨论如何分别使用它们来修饰所形成的微二氧化硅颗粒的性质（例如大小和丰度）。一个物理上有趣的发现是先前在实验中观察到的微硅粒径双峰分布的理论描述。