Mathematical Models of Continuous Combustion

Abstract

The Problem Considered — Economical design and operation of combustion systems can be greatly facilitated by prior predictions of performance by way of a mathematical model, incorporated into a digital-computer program. Morever, since the emission of pollutants is more sensitive to detailed design changes than is the overall heat transfer or power output of the system, the refined insight provided by predictions of concentration distributions is especially valuable nowadays. Suitable mathematical models exist for simple combustion systems and are being developed for others.

Purpose of the Paper — The difficulty of developing such models has two origins: mathematical and physical. The former is a result of the facts that: most combustion-chamber flows are three-dimensional; the differential equations describing them are numerous, simultaneous and non-linear; and neither digital computers of sufficient power nor knowledge of the optimal solution procedures are widely available. The physical source of difficulty is the complexity of the laws governing turbulent transport, the chemical kinetics of laminar and turbulent gases, formation and disappearance of condensed-phase particles, and thermal radiation through absorbing and scattering media.

The purpose of the paper is to illuminate these difficulties, to demonstrate that many useful predictions can already be made, and to indicate in what areas further research may be useful.

Topics Referred to — The paper will touch on the following detailed topics, and will illustrate them by examples of predictions:

• 1. Computational procedures for solving two-dimensional and three-dimensional steady-flow problems, both with and without recirculation.

• 2. Turbulence models for the prediction of the distribution through the combustion space of such statistical properties of turbulence as the kinetic energy of the fluctuating motion, the average length scale, the Reynolds stresses, and the root-mean-square fluctuations of concentration.

• 3. Hypotheses for the rates of the main combustion reaction, and of the NOX -formation reaction, when turbulent fluctuations of temperature and concentration are present.

• 4. The calculation of the variation through a combustion space of the size distribution of a condensed-phase material.

• 5. The calculation of radiative transfer.