Introduction

Our interest now is to achieve design or performance predictions of combustion devices. The word design refers to the determination of the geometry (length, diameter, shape) of the device and to ensuring safety for a desired product composition. Performance prediction implies the reverse – that is, determining product composition for a given geometry and safe operating conditions.

Combustion devices can be long and slender, as in a gas-turbine combustion chamber, a cement kiln, or a pulverized fuel (mainly coal)-fired boiler, in which air + fuel enter at one end and products leave at the other. Such devices can be idealized as plug-flow thermochemical reactors (PFTCRs). In a PFTCR operating under steady state, properties such as velocity, temperature, pressure, and mass fractions of species vary principally along the length of the reactor. Therefore, for such reactors, equations of mass, momentum, and energy may be simplified to their one-dimensional forms.

Devices known as furnaces (used in ceramic or steel making) are similar but rather stubby, with their spatial dimensions in the three directions being nearly equal. Such devices are formally called well-stirred thermochemical reactors (WSTCRs). Thus, the combustion space in a wood-burning cookstove may also be called a WSCTR. AWSTCR is a stubby PFTCR. Therefore, the governing equations can be reduced to a set of zero-dimensional algebraic equations.

Introduction

In air-conditioning and combustion applications, we encounter essentially gaseous mixtures. In air-conditioning, de-humidification, or humidification, the substance comprises a binary mixture of air + water vapor. The substance is called an inert mixture because no chemical reaction takes place between air and water vapor. In contrast, combustion mixtures are multicomponent mixtures because several chemical species (for example, CmHn, O2, CO, CO2, H2O, NOx, SOx) comprise them. They are not inert because under the right conditions of temperature and pressure, they can react chemically. Hence, they are called reacting mixtures. If the right conditions do not obtain, however, then they are inert mixtures.

Our interest in this chapter is to define mass, volume, and energy properties of gaseous mixtures. In the previous chapter, we noted that the state of a pure compressible substance requires specification of only two independent intensive properties. In contrast, the state of a mixture requires additional information that describes the composition of the mixture. Here, composition refers to the proportions of the components constituting the mixture.

In most applications of interest in this book, pressures are substantially low and temperatures substantially high, so the perfect gas range is obtained. As such, we shall assume that the mixture substance, as well as its components, behaves as a perfect gas. The intensive properties p, v, and T of a perfect gas are related by Equation 2.11.

Introduction

So far we have been concerned with combustion of gaseous fuels. However, solid (coal, wood, agricultural residues, or metals) and liquid (Diesel, kerosene, ethanol, etc.) fuels are routinely used in a variety of applications. Such fuels burn at a relatively slow flux rate, expressed in kg/m2-s. To achieve high heat release rates (J/s), it becomes necessary to increase the surface area of the fuel. This is achieved by atomizing liquid fuels and by pulverizing solid fuels. Liquid fuel atomization is achieved by employing high-pressure fuel injectors. Pulverized solid fuel is achieved by crushing/grinding operations to obtain tiny particles (typical diameter < 250 µm, in case of coal). This operation becomes increasingly expensive as the size of the particles is reduced.

Solid fuels are burned in a wide variety of ways. The most primitive systems burn fuels (typically wooden logs) by placing them on the ground and replacing (or feeding) fuel intermittently in a batch mode. More advanced versions include a fixed grate, as shown in Figure 10.1(a). The combustion air is drawn in by natural convection and the air flow rate is controlled by dampers. Such systems are used in small furnaces or cookstoves. These systems were improved considerably in power generation applications by having a fixed (sloping) or traveling endless grate on which solid fuel is fed continuously via a stoker and the combustion air is supplied by means of a fan/blower.

Importance of Chemical Kinetics

In the previous chapter, we analyzed chemical reactions in the manner of thermodynamics – that is, by considering the before and after of a chemical reaction. This analysis did not consider the rate of change of states during the process. Nonetheless, thermodynamic analysis coupled with the chemical equilibrium considerations enabled us to evaluate useful quantities, such as the (a) equilibrium composition of the postulated products, (b) heat of reaction or the calorific value of a fuel, and (c) adiabatic flame temperature.

In this chapter we introduce the empirical science of chemical kinetics, whose precise purpose is to determine the rate of a chemical reaction – in other words, the time rates of depletion of reactants and of formation of products. In this sense, the departure from chemical equilibrium thermodynamics to chemical kinetics may be viewed as analogous to the familiar departure made from thermodynamics to the empirical science of heat transfer. The latter, again, deals with rates of heat transfer across a finite temperature difference.

The science of heat transfer introduces rate quantities such as the heat transfer coefficient α, having units W/m2-K, and the transport property thermal conductivity K, having units W/m-K. Both these quantities are used in design of practical equipment. Thus, with the knowledge of α (which may be obtained from a correlation; see, for example, ref. [47]), we are able to size a heat exchanger by estimating the heat transfer surface area, determine the time required for a cooling process, or ensure that the temperature of solid elements in a furnace or a nuclear reactor is always well below safe limits.

Progress of a Chemical Reaction

In the previous chapter, we dealt with postulated products of a hydrocarbon chemical reaction for Φ ≤ 1 and for Φ > 1. The latter case was to be treated specially because it involves postulation of a two-step reaction mechanism (see Equations 3.46 and 3.47). The main point is that the postulated product composition was so far sensitized only to the value of Φ.

In reality, the product composition does not conform to the one postulated for any value of Φ. This is because, in addition to Φ, the product composition is governed by p, T, and the rate at which a reaction proceeds. As we have repeatedly mentioned, the central problem of combustion science is to predict product compositions accurately under conditions obtaining in practical devices. This inquiry into the most likely product composition is structured with increasing refinements in the following manner:

1. Product formation is determined by Φ only.

2. Product formation is determined by Φ, p, and T only.

3. Product formation is determined by Φ, p, T, and chemical kinetics only.

4. Product formation is determined by Φ, p, T, chemical kinetics, and fluid mixing.

The first type of product formation has already been discussed. In this chapter, our aim is to determine the effect of Φ, p, and T on the product formation process (second item in the list above).

Introduction

Our interest in this chapter is to look inside the thermochemical reactors discussed in the previous chapter. Inside the reactors, the formation of a visible flame is one obvious phenomenon. Flames are of two types, (a) premixed flames and (b) nonpremixed, or diffusion flames. A Bunsen burner, shown in Figure 8.1, is a very good example in which both types of flames are produced. In this burner, air and fuel are mixed in the mixing tube; this premixed mixture burns, forming a conical flame of finite thickness (typically, blue in color). This is called the premixed flame. It is so called because the oxygen required for combustion is obtained mainly from air, which is mixed with the fuel. This premixed combustion releases a variety of species, stable and unstable. Principally, the carbon monoxide resulting from the fuel-rich combustion burns in the outer diffusion flame. The oxygen required for combustion in this part of the flame is obtained from the surrounding air by diffusion (or by entrainment). The overall flame shape is determined by the magnitude of the mixture velocity and its profile as it escapes the burner tube, coupled with the extent of heat losses from the tube wall.

Unlike in premixed flames, in pure diffusion flames, the sources of fuel and oxidizer are physically separated. The candle flame shown in Figure 8.2(a) is an example of such a flame. When wax is melted, it flows up the wick and is vaporized as fuel.

Importance of Thermodynamics

Ever since humans learned to harness power from sources other than what their own muscles could provide, several important changes have occurred in the way in which societies are able to conduct civilized life. Modern agriculture and urban life are almost totally dependent on people's ability to control natural forces that are far greater than their muscles could exert. We can almost say that one of the main propellants of modern society is the increased availability of nonmuscle energy.

Pumping water to irrigate land and supply water to towns and cities; excavation of ores, oil, and coal, and their transport and processing; and transport of grains, foods, and building materials as well as passengers on land, water, and in the air are tasks that are exclusively amenable to nonmuscular energy. In fact, the extent of goods and services available to a society (measured in gross domestic product, GDP) are almost directly related to the per capita energy consumption. This is shown in Figure 1.1 for the years 1997 and 2004. The figure also shows that over these seven years, the GDP of all countries increased, but the developed countries achieved reduced per capita energy consumption through technological improvements, compared with the developing countries.

Introduction

The main purpose of this chapter is to briefly recapitulate what is learned in a first course in classical thermodynamics. As defined in Chapter 1, thermodynamics is the science of relationships among heat, work and the properties of the system. Our first task, therefore, will be to define the keywords in this definition: system, heat, work, and properties. The relationships among these quantities are embodied in the first and the second laws of thermodynamics. The laws enable one to evaluate change in the states of the system, as identified by the changes in its properties. In thermodynamics, this change is called a process, although, in common everyday language, the processes may be identified with terms such as cooling, heating, expansion, compression, phase-change (melting, solidification, evaporation, condensation), or chemical reaction (such as combustion or catalysis). As such, it is important to define two additional terms: state and process.

The last point is of particular importance, because the statements of thermodynamic problems are often couched in everyday language. However, with the help of the definitions given in the next section, it will become possible not only to to deal with all processes in a generalized way but also to recognize whether a given process can be dealt with by thermodynamics. Thus, adherence to definitions is very important in thermodynamics. Establishing a connection between the problem statement in everyday language and its mathematical representation is best helped by drawing a sketch by hand.

Introduction

Our interest now is to study chemical reactions in the presence of fluid motion so that fluid mixing effects can be incorporated in the determination of the product composition. In the study of transport phenomena in moving fluids, the fundamental laws of motion (conservation of mass and Newton's second law) and energy (first law of thermodynamics) are applied to an elemental fluid known as the control volume (CV).

The CV may be defined as an imaginary region in space, across the boundaries of which matter, energy, and momentum may flow. It is a region within which a source or a sink of the same quantities may prevail. Further, it is a region on which external forces may act.

In general, a CV may be large or infinitesimally small. However, consistent with the idea of a differential in a continuum, an infinitesimally small CV is considered. Thus, when the laws are to be expressed through differential equations, the CV is located within a moving fluid. In the Eulerian approach, the CV is assumed fixed in space and the fluid is assumed to flow through and around the CV. The advantage of this approach is that any measurements made using stationary instruments (velocity, temperature, mass fraction, etc.) can be compared directly with the solutions of differential equations.

Introduction

Diffusion flames were introduced in the previous chapter (see Figure 8.2) as flames in which fuel and air are physically separated. Diffusion flames occur in the presence of flowing or stagnant air such that the fuel moves outward and air moves inward to create a reaction zone at the flame periphery. Figure 9.1 shows a typical laminar diffusion flame configuration in stagnant surroundings.

Fuel issues from a round nozzle (diameter D) with axial velocity U0 into stagnant surroundings. As there are no effects present to impart three-dimensionality to the flow, the flow is two-dimensional and axisymmetric. The fuel burns by drawing air from the surroundings forming a flame that is visible to the eye. The flame develops to length Lf and assumes a shape characterized by flame radius rf(x), which is a function of axial distance x.

The energy release is thus principally governed by the mixing process between air and fuel. As such, the chemical kinetics are somewhat less important, being very fast. For this reason, diffusion flames are often termed physically controlled flames. Flames formed around burning volatile liquid droplets or solid particles (whose volatiles burn in the gaseous phase) are thus physically controlled diffusion flames.

The two main objectives of developing a theory of diffusion flames are to predict

1. The flame length Lf and

2. The shape of the flame, that is, the flame radius rf(x).

Introduction

As mentioned in Chapter 1, the main purpose of combustion science is to aid in performance prediction or design of practical equipment, as well as to aid in prevention of fires and pollution. This is done by analyzing chemical reactions with increasing refinements, namely, stoichiometry, chemical equilibrium, chemical kinetics, and, finally, fluid mechanics. Such, analyses of practical equipment are accomplished by modeling. The mathematical models can be designed to varying degrees of complexity, starting from simple zero-dimensional thermodynamic stirred-reactor models to one-dimensional plug-flow models and, finally, to complete three-dimensional models. At different levels of complexity, the models result in a system of a few algebraic equations, a system of ordinary differential equations, or a system of partial differential equations.

Practical combustion equipment typically has very complex geometry and flow paths. The governing equations of mass, momentum, and energy must be solved with considerable empirical inputs. The size of the mathematical problem is dictated by the reaction mechanism chosen and by the chosen dimensionality of the model. The choice of the model thus depends on the complexity of the combustion phenomenon to be analyzed, the objectives of the analysis, and the resources available to track the mathematical problem posed by the model.

In this chapter, a few relevant problems are formulated and solved as case studies to help in understanding of the art of modeling as applied to design and performance prediction of practical combustion equipment.