Introduction to Combustion

2 Introduction to Combustion

Chapter Overview This chapter overviews some practical and theoretical aspects of  combustion. We first discuss burners generically, including fuel and air metering, flame stabilization and shaping, and some fundamental techniques for control of emissions such as NOx. So prepared, we move on to consider archetypical burners — those representative representative of the traditional classes of burners one might find in a refinery or petrochemicals plant, or that provide heat for steam. We build upon this foundation by next considering archetypical process units such as boilers, process heaters of various types, and reactors such as hydrogen reformers and cracking units. In order to lay the groundwork for more detailed combustion modeling, the chapter considers important combustion-related responses such as NOx emissions, flame length, noise, etc., and the factors that influence them. Historically, practitioners have defined a traditional test protocol for quantifying these effects, which we present. We We also consider some aspects aspec ts of thermoacousthermoacous tic instability; this has become a more important topic with the advent of ultralow NOx burners employing very fuel lean flames. In the latter third of the chapter, chapter, we develop stoichiometric and mass balance relations in considerable mathematical detail. We also consider energy-related quantities such as heat and work, adiabatic flame temperature, and heat capacity. As well, we relate the practical consequences of a mechanical energy balance as applied to combustion equipment. Such things include draft pressure, incompressible incompressibl e airflow, airflow, compressible fuel fu el flow, flow, and practical practi cal representations thereof — capacity curves for air and fuel.

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 Modeling of Combustion Combustion Systems: A Practical Approach Approach

General Overview

Combustion is the self-sustaining reaction between a fuel and oxidizer characterized by a flame and the liberation of heat. Usually, but not always, the flame is visible. A flame is the reaction zone between fuel and oxidizer; it typically comprises steep thermal and chemical gradients — the flame is often only a millimeter or so thick. On one side of the flame, there is fuel and oxidizer at low temperature; on the other side are combustion products at high temperature. Hydrogen and hydrocarbons in some combination are the typical fuels in the petrochemical and refining industries. Occasionally, due to some special refining operations, we find carbon monoxide in the fuel stream. Oxygen (in air) is the usual oxidizer. In practice, combustion reactions proceed to completion with the fuel as the limiting reagent — that is, with air in excess. A burner is a device for safely controlling the combustion reaction. It is typically part of a larger enclosure known as a furnace. A  process  process heater is any device that makes use of a flame and hot combustion products to produce some product or prepare a feed stream for later reaction. Examples of  such processes are the heating of crude oil in a crude unit, unit, the production of  hydrogen in a steam–methane catalytic reformer, reformer, and the production of ethylene in an ethylene cracking unit. unit . A boiler is a device that makes use of a flame or hot combustion products to produce steam. The  furnace is the portion of the process unit or boiler encompassing the flame. The radiant section comprises the furnace and process tubes with a view of the flame. In contrast, the convection section is the portion of the furnace that extracts heat to the process without a line of sight to the flame. Every industrial combustion process has some thermal source or sink.

2.1.1

The Burner

A burner is a device for safely controlling the combustion reaction. A diffusion burner is one where fuel and air do not mix before entering the furnace. If  fuel and air do mix before entering the furnace, then the device is a premix a  premix burner. Premix burners may mix all or some of the combustion air with the fuel. If one desires to distinguish between them, a  partial premix burner is one that mixes only part of the combustion air, with the remainder provided later. later. Most pilots are of the partial premix type to ensure that they will light under high excess air conditions typical of furnace start-up. Diffusion burners supply most of the heating duty in refinery and boiler applications; therefore, we discuss them first. Figure 2.1 shows the main features for accomplishing this. The particular version of burner shown in Figure 2.1 is a natural draft  burner.  burner. That is, a slight vacuum in the furnace (termed draft — 0.5 in. water column below atmospheric pressure pressure is a typical figure) and a relatively large opening (burner throat) allow enough air to enter the combustion zone to

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Introduction to Combustion

Bur ner Throat Refractory Tile

Furnace Floor

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Primar y Fuel Tip

Flame

Secondary Fuel Tips Burner Front Plate Cone (Throat Restriction)

Fuel Risers Inlet Damper

Noise Muffler Air Plenum

FIGURE 2.1 A typical industrial burner. The typical industrial burner has many features, which can be classified in the following groups: air metering, fuel metering, flame stabilization, and emissions control. Refer to the text for a discussion of each.

support the full firing capacity. The diffusion burner comprises a fuel a  fuel mani fold,  fold, risers, risers, tips, tips, orifices, orifices, tile, plenum, plenum, throat restriction, restriction, and damper. damper. Each diffusion burner type may differ in detailed construction, but all will possess these main functional parts. We discuss each in turn.

2.1.1.1 The Fu Fuel Sy System A fuel manifold is a device for distributing fuel. In the figure, one fuel inlet admits fuel to the manifold while several risers allow the fuel to exit. A riser is a fuel conduit. (In the boiler industry, risers are sometimes termed pokers termed  pokers,,  but the function is the same.) Each riser terminates in a tip; for some boiler  burners the tip is called a  poker shoe or just a shoe. shoe. A tip is a device designed to direct the fuel in a particular orientation and direction. It has one or more orifices, holes, or slots drilled at precise angles and size. An orifice is a small hole or slot that meters fuel — for a particular design fuel pressure, composition, and temperature, the orifice restricts the flow to the specified rate. Together, these parts comprise the fuel the  fuel system of a gas burner.

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2.1.1.2 About Fuels There are two main gaseous fuels for combustion processes: processes: natural gas and refinery gas. Commercially available natural gas has a stable composition comprising mostly methane (see Appendix A, Table A.9). A.9). Refinery gas, on the other hand, is capable of considerably more variation. In a sense, refinery gas is the “garbage dump” for the gas products in the refinery. Generally, whatever the refinery cannot use for some higher value-added process it consumes as fuel. It is quite typical for refineries to specify several different refinery fuels for combustion equipment — one representing normal conditions, another representing representing a normal auxiliary case, and perhaps two or three upset scenarios. The scenarios will vary in hydrogen hydrogen concentration, typically from 10 to 60%, giving the fuels quite different combustion properties. It is very important that the refinery weigh the likelihood of scenarios that represent represent widely varying heating values on a volumetric basis. For example, suppose a particular process unit can receive fuel according to four different different scenarios: • • • •

Fuel Fuel A: A: 620 Btu/ Btu/scf scf,, norma normall fuel fuel repr represe esenti nting ng 10% 10% of run run time time Fuel Fuel B: 760 760 Btu/sc Btu/scf, f, stand stand-by -by fuel fuel repr represe esenti nting ng 84% 84% of run run time time Fuel Fuel C: 840 840 Btu/scf Btu/scf,, start-u start-up p fuel repr represe esenti nting ng 5.98% 5.98% of of run time time Fuel D: 309 309 Btu/sc Btu/scf, f, high high hydrogen hydrogen upset upset case case repr represen esenting ting 0.02% of run time

Since fuel C is a start-up case, it does not matter how infrequently it occurs, the burners must operate on fuel C. However, the difference in volumetric flow rate among the fuels A, B, and C is small. On the other hand, fuel D represents represents a significant difference in hydrogen concentration. This will markedly affect major fuel properties such as flame speed, specific gravity, and flow rate through an orifice. Later, we shall develop the flow equations that show that fuel D represents the maximum flow (and maximum max imum fuel pressure) condition. One can obtain a burner to meet all these fuel conditions. However, for fuels A, B, and C, the pressure will necessarily be lower. Some possible consequences are lower fuel momentum and “lazier” longer flames when the facility is not running fuel D. In severe cases, the flue gas momentum will control the flame path. Thus, flames may waft into process tubes and will be generally poorer in shape — all for the sake of preserving good operation for an operating case representing only 0.02% of the run time. A far better scenario would be to design the burner to handle fuels A, B, and C. Then fuel B is the maximum pressure case, but the facility will not have enough pressure pressure to make maximum capacity with fuel D. Therefore, Therefore, 99.98% of the time the flames will be fine and the unit will operate properly; 0.02% of the time the unit will not be able to fire the full firing rate. In the opposing scenario, the operators will struggle with the unit 99.98% of the time, and the burner will run optimally only 0.02% of the time.

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2.1.1.3 Fuel Metering  A fuel control valve upstream of the burner generally does the fuel metering. A riser or  poker delivers fuel to the burner tip. The facility specifies the pressure the burner will receive at maximum (design) firing rate. The burner manufacturer then sizes the fuel orifices in the tips to ensure that burner will meet the maximum fuel capacity at the specified conditions. The burner manufacturer provides a series of capacity curves (one for each fuel) that show the firing rate vs. the pressure. Figure 2.2 gives an example. 3.5  B  l C  e l A  u e l  e  u  u  F  F  F

3.0

2.5

   W    M  ,   e2.0   s   a   e    l   e    R1.5    t   a   e    H

sonic flow transition to sonic flow subsonic flow

1.0

0.5

0.0

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

Fuel Pressure, bar(g) FIGURE 2.2 A typical capacity curve. Fuel capacity curves give heat release as a function of fuel pressure. They are quite accurate for a given fuel composition. However, if there are a range of fuels, each needs its own capacity curve. This example shows three. The typical range is two to five fuel scenarios, all represented on the same graph. As the flow transitions to sonic, the capacity curve becomes linear.

In natural draft burners, air is generally the limiting factor, and its controlling resistance is the burner throat. Thus, the only way to increase the overall firing capacity of the burner is to increase the throat (i.e., burner) size. This may or may not be possible depending on the available space in the heater and if the heater can handle the extra flue gas and heat that result. If not, the entire unit may require modifications, not just the burner.

2.1.1.4 Turndown Burners operate best at their maximum capacity. One measure of the flame stability of a burner design is the turndown ratio. The turndown ratio is the

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ratio of the full firing capacity to the actual firing capacity. The maximum turndown ratio is the max/min firing ratio. The turndown ratio is higher if  one modulates the air in proportion to the fuel. If the air dampers are manually controlled, then one is interested in the maximum unmodulated turndown ratio, because this is the more conservative case. A typical maximum turndown ratio for premix burners is 3:1. Diffusion burners often have turndown ratios of 5:1 without damper modulation, i.e., leaving the damper fully open despite lower fuel flow. One typically achieves 10:1 turndown ratios with automatic damper and fuel modulation. Multiple burner furnaces usually require a turndown ratio of somewhere between 3:1 and 5:1 with the air dampers fully open. To achieve greater turndown for the unit as a whole, one isolates some of the burners (fuel off, dampers closed). Turndown is easiest for a single fuel composition. Multiple fuel compositions always reduce the available pressure for some fuels and reduce the maximum turndown ratio.

2.1.1.5 The Air System For natural draft burners, air is the limiting reactant. That is, sufficient fuel pressure is available to allow the burner to run at virtually any capacity, but only so much air will flow through the burner throat at the maximum draft. The burner throat refers to the minimum airflow area; it represents the controlling resistance to airflow. Therefore, the airside capacity of the burner determines the burner’s overall size. Air may enter from the side (as shown in Figure 2.1) or in line with the  burner. Analogous to the fuel orifice, there are two metering devices for the airside. The first is the damper, upstream of the plenum. A damper assembly is a variable-area device used to meter the air to the burner. This is necessary  because the maximum airside pressure drop is limited and the firing rate modulates. Therefore, one must modulate the air to maintain the air/fuel ratio. The damper may require manual adjustment, or one may automate it  by means of an actuator. The inlet damper unavoidably creates turbulence and pressure fluctuations  behind it. The plenum is the chamber that redistributes the combustion airflow before allowing it to enter the burner throat. This redistribution does not need to be perfect, and there is a trade-off between uniform flow (larger plenum) and burner cost. In some cases, one may shorten the plenum by means of a turning vane — a curved device designed to redirect the airflow (not present in the figure). Burners are available in discrete standard sizes. To accommodate the infinite variety of potential capacities, manufacturers adjust the limiting airflow  by means of a restriction in the burner throat. A choke ring is an annular  blockage from the outside diameter inward. A baffle plate is a flat restriction originating from the center outward. A cone is an angled restriction originating from the center of the burner throat. Figure 2.1 shows a cone, but baffle plates or choke rings are also very common. The purpose of the throat

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restriction is to provide a point of known minimum area and pressure drop characteristic. Some codes actually specify what portion of the draft (airside pressure drop across the burner) that the damper and the burner throat must take. For example, one guideline says that the burner must use 90% of the available pressure and take 75% of the pressure drop across the throat.* The 90/75 rule, as it has come to be known, aims to create turbulence in the burner throat rather than at the damper to aid fuel–air mixing. In the author’s opinion, this is a misguided approach and the industry should abandon it for several reasons. First, if the burner performs well, the degree of turbulent mixing in the throat is immaterial. Second, if the burner performs well, it does not matter where the pressure drop occurs so long as the total pressure drop is correct. Third, this rule increases the time and cost of burner testing. More importantly, changes to the burner for the sake of meeting the 90/75 rule may actually make the burner perform worse. Therefore, one is consuming resources to meet an essentially useless rule. Very often, the end user will have to clean or inspect tips while the furnace is running. In multiple-burner furnaces comprising many burners, there is usually little danger in shutting off a single burner. In petroleum refineries, the usual practice is to specify a burner having replaceable tips that do not require burner removal from the heater (of course, one must still shut off  fuel flow to the individual burner before removing the tips). One must also take care to close the air damper during this procedure; otherwise, the furnace will admit tramp air. Tramp air is air admitted out of place. Air entering the furnace should participate fully in the combustion process, and tramp air enters the com bustion process too late to oxidize the fuel properly. Tramp air may come not only through unfired burners, but also through leaks in the furnace. One possible sign of tramp air is a high CO reading even with supposedly sufficient excess oxygen. CO is a product of incomplete combustion. Depending on the furnace temperature, 1 to 3% oxygen should represent enough excess air, and CO should be quite low under such conditions. But with tramp air, significant CO (>200 ppm) may still occur — even with 3% oxygen (or more) in the flue gas. The oxygen has entered the furnace somewhere, but it is not participating fully in the combustion reaction. As long as there is tramp air, the furnace will require higher oxygen levels — enough to provide both effectual air through the burner and ineffectual tramp air. In severe cases of tramp air leakage, even a wide-open damper cannot provide enough air to the burner and CO persists, though stack oxygen levels are 5% or more. The exit of the furnace radiant section is the relevant place to measure emissions for combustion purposes. The exit of the stack is the relevant place to measure emissions for compliance purposes. The stack exit is not generally useful for understanding what is happening in the combustion zone. * This requirement appears as a footnote to API Standard 560, Fired Heaters for General Refinery Service, 3rd ed., Washington, DC, American Petroleum Institute, May 2001, p. 65.

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2.1.1.6 The Flame Holder  Anchoring the flame to the burner is essential for the sake of performance and safety. A flame holder is a device designed to keep the leading edge (root) of the flame stationary in space. There are many different devices for accomplishing this. The most common device is the burner tile. A burner tile is a refractory flame holder designed to withstand the temperature of direct flame impingement. The tile ledge is the portion of the tile that anchors the flame (Figure 2.3).

FIGURE 2.3 The tile ledge as flame holder. (From Baukal, C.E., Jr., Ed., The John Zink Combustion Handbook , CRC Press, Boca Raton, FL, 2001.)

One type of flame holder is the bluff body — a nonstreamlined shape in the flow path — to present an obstruction to some of the flowing fuel–air mixture; the tile ledge qualifies. This obstruction generates a low-pressure, low-velocity zone at its trailing edge. The flame holder affects only a small portion of the flow, reducing its velocity to well below the flame speed. The velocity upstream of the flame holder is very low. Thus, the hot combustion products recirculate there, continually mixing fresh combustion products with an ignition source — the hot product gases. In this way, the flame holder anchors and stabilizes the flame over a wide turndown ratio. Figure 2.4 shows a 2-MW round-flame  burner employing a tile-stabilized flame. The shape of the burner and position of the fuel ports mold the flame into the desired shape.

2.1.1.7 Stabilizing and Shaping the Flame A flame has a very fast but finite reaction rate. One measure of the reaction rate is the flame speed. The laminar flame speed is the flame propagation rate [L/θ] in a combustible mixture of quiescent fuel and air. If the air and fuel mixture exceed the flame speed, then the flame will travel in the direction of the stream — a phenomenon known as liftoff. If the liftoff continues, the

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FIGURE 2.4 A gas burner in operation.

flame will be transported to a region of high flue gas concentration that cannot support combustion and the flame will extinguish. Air and fuel velocities in a typical industrial burner far exceed laminar flame speeds. But in the vicinity of the bluff body, the fluid speed is low; therefore, the flame anchors over wide turndown range and the burner is quite stable throughout its entire operation.

2.1.1.8 Controlling Emissions In the past, the only emission of concern was CO because it indicated incomplete combustion, combustion inefficiency, or a safety hazard. Nowadays, life is more complex and other emissions such as nitric oxides are important due to their role in the formation of ground-level ozone and photochemical smog. Technically, noise is also a regulated emission (e.g., 10-MW heat release per burner). 2.1.2.2 Round-Flame Gas Premix Burners Figure 2.5 shows an example of a round-flame floor-fired premixed burner. Some furnaces use premix burners in larger upfired applications requiring round flames. However, this has fallen into disfavor because the burners are usually loud (due to fuel jet noise) and sensitive to hydrogen concentration variations. In premix burners, air and fuel mixing occur prior to entering the furnace. Premixing has several advantages. First, premixed burner flames tend to be short, crisp, and well defined. The fuel jet provides momentum for fuel mixing and air entrainment prior to the burner exit. An advantage of this

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Secondary air

Primary air

Pilot Gas

FIGURE 2.5 A gas premix floor burner. (Courtesy of the American Petroleum Institute, Washington, DC.)

arrangement is that increased fuel flow results in increased airflow. However, premixed burners also have major disadvantages in the wrong application. First, the momentum of the fuel stream depends on the molecular weight of  the gas and the fuel pressure. As the H/C ratio of the fuel changes, so does the molecular weight, the fuel pressure for a given heat release, and the amount of educted air. Higher pressures tend to educt less than the proportional air, while lower fuel pressures educt higher than ideal airflows. Thus, the air/fuel ratio is not as constant as one might hope. Another serious drawback of premixed combustion is the potential for flashback. Flashback  is the upstream propagation of flame into the premix chamber of the burner. In a diffusion burner, it is impossible for a flame to propagate back into the fuel riser because there is no oxygen there to support combustion. However, in a premix burner, there is a combustible mixture inside the burner tip. Premix burners also have a much larger tip and orifices  because these have to accommodate a high-volume, low-pressure fuel–air mixture. If the flame speed significantly exceeds the fuel–air exit velocity, then the flame may flash back into the burner tip.

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Burner internals cannot withstand the high temperatures of combustion for long. Flashback is very sensitive to hydrogen concentration because the laminar flame speed of hydrogen is about three times that of hydrocarbons. Various techniques moderate flashback. One important consideration is the quench distance — a characteristic length for a given orifice geometry through which a flame cannot propagate. For small-diameter orifices, the edges will abstract sufficient heat from a propagating flame to extinguish it. The quench distance varies with orifice diameter — smaller orifices are more effective than larger ones. Appendix A, Table A.4 gives critical diameters and minimum slot widths for various fuels. Generally, for a given area, a circular orifice is more efficient for quenching flames than a rectangular one. However, smaller rectangular slots may have manufacturing advantages. In practice, both geometries are commercially available. The removal of heat from the flame via thermal conduction through the tip is a mechanism for quenching the flame and eliminating flashback. Since the tip temperature depends in some measure on the temperatures of the furnace, fuel, and combustion air, the tip’s ability to conduct heat away from a propagating flame also changes with temperature. Moreover, at higher temperatures, the orifices in the tip expand and one must take account of  this effect in burner design as well.

2.1.2.3 Flat-Flame Gas Diffusion Burners Figure 2.6 gives a typical example. These burners are usually floor fired against a flat wall that radiates heat to the process tubes (Figure 2.7). In other ways, these burners are similar to round gas diffusion burners. One uses these burners to maximize radiant heat transfer from the wall to the process. For example, high-temperature processes — such as production of hydrogen or ethylene — use the hot wall to radiate to process tubes containing feedstock. Since the reaction occurs along the length of the tube, the so-called heat flux profile can be important. Figure 2.8 gives an example. The burner manufacturer adjusts the heat flux profile according to furnace vendor specifications by changing the angle, size, and distribution of the fuel jets. NOx reduction with these burners is a challenge because the process operates at very high furnace temperatures (1000 to 1250°C). Another type of flat-flame diffusion burner is side-fired. The architecture of the burner resembles that of the side-fired premix burner, but the burner meters the air and fuel separately. Figure 2.9 gives one example, and Figure 2.10 shows some in operation. Ethylene reactors and wall-fired hydrogen reformers (described later) use these burners. The overall chemistry for hydrogen production is CHψ  + 1/2 O2

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→ CO + ψ /2 H2

(2.1)

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FIGURE 2.6 A flat-flame gas diffusion burner. The burner creates a flat flame used to heat a wall that radiates heat to the process.

Hydrogen plants often use pressure-swing adsorption (PSA) to purify hydrogen and separate it from the CO 2 by-product. The off-gas from this stream is mostly CO2, with some hydrogen, and to a lesser extent hydrocar bons. The high concentration of CO 2 dramatically lowers the flame temperature. Flat-flame side-fired diffusion burners have excellent stability, cannot flash back, and NOx emissions below 20 ppm are possible even with 400°C air preheat and 1200°C furnace temperatures. As Figure 2.9 and Figure 2.10 show, the fuels travel down a central riser; the slotted tip projects the fuel in a radial plane parallel to the wall. Preheated air comes through the large annular gap and enters the furnace in the same orientation. At the high furnace temperatures, the separate fuel and air streams react to generate a flame with very low NOx and uniform radiation.

2.1.2.4 Flat-Flame Premix Burners Flat-flame premix burners comprise side-fired ethylene or steam–methane reforming service (hydrogen production) almost exclusively. Figure 2.11 shows a common design. The burner tip is roughly 4 in. in diameter and 8 in. long. Slots or holes cover the end and admit premixed fuel and air to the furnace. Premix burners are prone to flashback, though proper design will ameliorate this.

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FIGURE 2.7 Floor-fired flat-flame burners. The burners, John Zink Model PXMR, are shown firing against a wall in an ethylene cracking furnace. The wall radiates heat to the process tubes (not shown).

2.1.2.5 Flashback  Flashback can only occur in premixed burners because they are the only type that has a combustible mixture inside the tip. Once the flame flashes back into the burner tip it can destroy it in minutes. Combustion inside the tip increases the mixture temperature downstream of the eductor. This backpressures the eductor and further reduces the air/fuel ratio. The richening of the fuel mixture and higher temperatures increase the flame speed. Therefore, once flashback occurs, there is no mechanism for moving the flame  back to the furnace side. The burner immediately experiences lower mass flow due to the reduced density of the gas during flashback; one may hear a gurgling sound from the combustion rumble in the tip. 2.1.2.6 Use of Secondary Fuel and Air  The tip receives its premixed fuel and air from a venturi ( Figure 2.12). A fuel jet ahead of the venturi induces surrounding air via the fuel’s forward momentum. Under some conditions the fuel educts only a portion of the combustion air. In that case, secondary air slots allow additional air to bypass the venturi (shown in Figure 2.11). In some cases, the premix  burner may have some nonpremixed fuel in addition to the fuel–air mixture, thus staging the fuel (Figure 2.11). Fuel not added to the immediate com bustion zone — the primary zone — is termed secondary fuel or staged fuel.

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% of max heat flux 90 80 70 60

50

  s   e    b   u    t   s   s   e   c   o   r   p

40 30 20 10

< floor burners >

FIGURE 2.8 A typical heat flux profile. Shown is a side view of one cell of a dual-cell floor-fired ethylene cracking unit (ECU) with its associated heat flux profile. The proper heat flux profile is a tradeoff between reduced fouling (flat heat flux profile) and maximized efficiency (heat release skewed toward furnace bottom). Burner vendors design floor-fired burners to provide the required flame shape for a given flux profile.

Secondary fuel injection is one technique for reducing NOx. Side-fired premixed burners are typically much smaller than floor-fired burners and weigh a mere 50 kg or so, including the tile. Firing rates for these burners are about 1 MMBtuh or 1/3 MW.

2.1.2.7 Round Combination Burners In some facilities, fuel oil can be a significant fuel stream. Normally, a refinery will want to burn as heavy a fuel as possible because other liquid fuels have greater value (e.g., transportation fuels for automobiles, trucks, and aviation). When sold commercially, fuel oils are widely available and graded as either number 2 or 6, with intermediates formed by blending. Fuel oil 2 is similar to automotive diesel. Fuel oil 6 is much heavier (also called residual fuel oil or, archaically, Bunker C oil). Marine and stationary boilers and some process heaters burn this fuel. Sometimes, the liquid fuel comprises rejected oil from other processes (waste oil) in whole or part. One can also burn pitch — a nondescript fuel from a variety of sources that is solid at room temperature. One must heat these fuels to reduce their viscosity in order for them to burn efficiently. Heavy liquid fuels do not atomize well even under pressure

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FIGURE 2.9 A flat-flame diffusion burner. Radial orifices admit fuel through the center pipe, while com bustion air flows through the outer pipe. Unlike premix designs, this radiant wall burner cannot flash back. The design accommodates high forced draft air preheat applications.

FIGURE 2.10 Wall-fired diffusion burners in operation. The photo shows an ethylene cracking furnace equipped with John Zink Model FPMR burners. The process tubes (right) are receiving heat radiated from the burner firing along the wall. (Photo courtesy of John Zink LLC, Tulsa, OK.)

(so-called mechanical atomization), so fuel guns make use of pressurized steam to produce the requisite atomization. Mechanical atomization is sufficient for light oils such as fuel oil 2. Sometimes, light liquid fuels use compressed air for atomization. This is the case if steam atomization could be detrimental or there is insufficient fuel oil pressure for mechanical atomization. For

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FIGURE 2.11 A flat-flame premix burner. The flame heats the refractory, which in turn radiates heat to the process tubes inside the furnace. Secondary air allows for a higher capacity, as the eductor need not inspirate all of the combustion air. Also shown is a secondary fuel nozzle at the burner tip. Some burners do not have all these features.

INLET BELL

THROAT

EXPANSION SECTION

EDUCTED AIR FUEL NOZZLE

FUEL/AIR MIXTURE TO TIP

EDUCTED AIR

VENTURI

FIGURE 2.12 Venturi section of a premix burner. The Venturi (more generically, an eductor) comprises an inlet bell, throat, and expansion section. The fuel jet induces a low-pressure zone along the jet surface. The surrounding atmospheric pressure pushes air into the low-pressure zone. The fuel and air mix and exit the venturi toward the tip outlet.

example, light naphtha fractions can prevaporize in the fuel oil gun. Prevaporization is unwanted because it leads to slug flow in the fuel gun, that is, alternate slugs of liquid and gaseous fuel going to the burner. This causes erratic flow and performance.

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Practitioners use the term oil gun to refer to the liquid fuel delivery and atomization assembly. Figure 2.13 shows a typical design.

FIGURE 2.13 An oil gun. Steam vaporizes oil droplets allowing for uniform combustion.

Steam shears heated oil into fine droplets — the fuel oil vapor is the phase that actually burns. The burning vapor provides heat, vaporizing even more droplets and recharging the combustion zone. Very often, one burns fuel oil and gas together (in so-called combination burners). This is sometimes to add fuel flexibility — perhaps the gas and the liquid fuel are available in different seasons. However, the more typical practice is to use the less expensive heavy oil with the gas fuel serving to make up the required process heat. Thus, both fuels fire simultaneously. A combination burner is a gas-fired burner augmented with a fuel oil gun. Figure 2.14 shows one common arrangement. The typical combustion scenario is a single fuel oil gun in the center of  the burner with gas firing at the periphery. When both fuels fire at once, flame lengths tend to be longer than when either fires alone. This is due to the peripheral combusting gas reducing the available oxygen for the fuel oil stream. To minimize (but not eliminate) this effect, separate air registers provide individualized airflow to each zone.

2.1.2.8 Burner Orientations One may fire burners in four basic orientations: up, down, sideways, and  balcony fired (horizontally mounted to the wall but having a vertical flame at right angles to the burner). 2.1.2.9 Upfired  This is the most typical firing arrangement (see Figures 2.1, 2.4, and 2.7). The  burner mounts to the floor. Pillars support the heater and allow sufficient clearance for personnel to walk under the burners and inspect and maintain them. A 1.8-m gap from the lowest part of the burner to the ground is adequate for personnel to walk under the heater without stooping. However, some units do not have this kind of clearance. Clearance is important for initial installation and because one requires sufficient room to extract, clean, and reinsert fuel oil guns, risers, and tips.

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REGEN TILE GAS RISERS (FOR COMBINATION FIRING)

OIL GUN TERTIARY AIR CONTROL

PRIMARY TILE

GAS PILOT

AIR INLET PLENUM PRIMARY AIR CONTROL

SECONDARY AIR CONTROL

GAS RISER MANIFOLD (FOR COMBINATION FIRING)

FIGURE 2.14 A combination burner. John Zink Model PLNC combination gas–oil burner. One may fire the  burner on gas only, oil only, or both. (Rendering courtesy of John Zink LLC, Tulsa, OK.)

2.1.2.10 Downfired  Most burner models can be adapted to fire downward with some kind of  tile case or support to hold the tile in place at the roof. Figure 2.15 shows a generic schematic. Hydrogen and ammonia reformers of this type are large furnaces often with more than 200 burners, each firing at ~2 MW. With so many burners in a furnace, first cost is important. Downfired operation affects the flame shape because the firing direction is opposite the buoyant force. Hence, forced draft is the preferred option for these burners. Forced draft operation helps to minimize the flame bending toward the tubes. The greater momentum of the forced air helps to overcome the buoyant effects and furnace currents. Furnace currents can be quite complicated. Downfiring increases NOx emissions by 15% or so as the burner receives hotter convective air at the roof than at the floor. Space is limited and roof burners are more difficult to access than floor burners. There are also limitations on the total burner weight because the furnace roof can support only so much. Therefore, a simple, reliable design is the order of  the day.

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FIGURE 2.15 A downfired burner for hydrogen reforming. This burner is equipped with a center gun for waste gas from a pressure-swing adsorption (PSA) unit.

2.1.2.11 Side-Fired  Some steam–methane reformers and ethylene cracking units (ECUs) use side-firing. The general idea is to present a uniform heat flux to the reactor tubes by using many small burners ( Vol % wet species in all cases 

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

60% 100%

ε

FIGURE 2.20 Flue gas relations. The figure gives relations for air in excess with combustion of a typical refinery gas comprising 25% H2, 50% CH4, and 25% C3H8.

2.4.3

Accounting for Moisture

One may also derive an equation for the moisture fraction of  Table 2.2:  y H 2O =

ψ  42ψ  = 100 ( ψ + 4) (1 + ε ) + 21ψ  2 ( ψ + 4) ( K wet + ε )

(2.34)

The following formula relates the wet and dry measurements:  ywet  ydry

100(ψ + 4)(1 + ε) − 21ψ  K dry + ε = = 100(ψ + 4)(1 + ε) + 21ψ  K wet + ε

= 1 − y H 2O

(2.35)

Here,  y is the mole fraction of any species of interest and the subscripts distinguish the wet or dry bases. From Equation 2.35, one can convert from dry to wet concentrations or vice versa if one knows the hydrocarbon ratio and the excess air. If the H/C ratio in the fuel varies (e.g., municipal solid waste (MSW), waste gases, refinery fuel gases, landfill and digester gases, etc.), one may use ratios of  the above equations to solve for ψ . For example, if we know moisture and CO2 in the flue gas, we can find ψ  directly, or if we know any two of the three, the following relations will determine the third:

ψ = 2  yH2O  yCO2 ,wet

© 2006 by Taylor & Francis Group, LLC

⎛  2 ⎞ ⎛   yH2O ⎞  =⎜ ⎝  yCO2 ,dry ⎟ ⎠ ⎜⎝ 1 − yH2O ⎟ ⎠ 

(2.36)

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ψ  yCO 2 ,dry  yH2O = 2ψ yCO2 ,wet = 2 + ψ   yCO2,dry

(2.37)

⎛  2 ⎞   yCO2 ,wet = ⎜ ⎟ yH2O ⎝ ψ ⎠ 

(2.38a)

⎛  2 ⎞ ⎛   yH2O ⎞   yCO2,dry = ⎜ ⎟ ⎜ ⎝ ψ ⎠ ⎝ 1 − yH2O ⎟ ⎠ 

(2.38b)

Figure 2.21 shows these graphically.

20%

18%

Ψ  =1

Ψ  =3 

Ψ  =2 

Ψ  =4 

16%   s    i   s 14%   a    b   y   r    d12%  ,    2

Ψ  =5 

Ψ  =6 

   O    C 10%

Ψ  =7  Ψ  =8 

8%

Ψ  =9  Ψ  =10 

6% 6%

8%

10%

12%

14%

16%

18%

20%

22%

24%

26%

% moisture

FIGURE 2.21 Wet–dry flue gas relations. Knowing ψ H2O and ψ CO2 dry, one may determine ψ , the H/C ratio of the fuel, using this chart. CO2 is given on a dry basis because CO2 analyzers usually report results on this basis. The chart requires a moisture determination. One may do this with a physical moisture analysis or knowing oxygen on both a wet and dry basis. See text for cautions with this approach.

To determine the moisture in flue gas, one draws a known volume of gas and condenses the water. However, in principle from Equation 2.35, one can find the moisture concentration if one measures both wet and dry oxygen:  yH2O = 1 −

© 2006 by Taylor & Francis Group, LLC

 yO 2 ,wet  yO 2 ,dry

(2.39)

Introduction to Combustion

Example 2.4

143

Determination of Moisture Content from Wet and Dry O 2 Measurements

Problem statement: The dry and wet oxygen concentrations of a flue gas are 2.95% and 2.56%, respectively. Determine the moisture content by Equation 2.39 above and compare it with the moisture content given in the table. Solution: From Equation 2.39, we know that

 yH2O = 1 −

2.56 − 1 = 13.1% 2.95

This agrees with the listed value in Example 2.3. However, despite this agreement, measurement errors can affect the calculation severely, as described presently. One should exercise caution in the application of Equation 2.39. The equation is perfectly accurate. However, oxygen measurements are subject to some error. The largest error comes when the analyzers sample wet and dry oxygen from different locations (typical). Very often, one measures wet oxygen in the furnace and dry oxygen in the stack (Figure 2.22). For negative and balanced draft units, air in-leakage can bias the results wildly. For example, even though we listed the dry oxygen as 2.95%, air inleakage could cause the measurement in the stack to be 5.0%. If we were to repeat Example 2.4 with this dry O2, the result would be 1 – 2.56/5.00 – 1 = 48.8% moisture, an absurd result. Even if we measure the wet and dry oxygen at the same location, small errors can still make a significant difference. For example, suppose the dry O 2 were biased upward by a 0.1% reading, and the wet O2 were biased downward by the same amount; then 1 – 2.46/3.05 – 1 = 19.3% moisture. In short, the procedure is subject to considerable error. Notwithstanding, it gives a good indication of moisture in the furnace gas if we draw the wet and dry samples from the same location and calibrate the analyzers before measurement.

2.4.4

Addition of Molecular Hydrogen to the Fuel

Refinery fuels may comprise more than just hydrocarbons; molecular hydrogen (H2) is a common addition. We distinguish molecular hydrogen from fuel-bound hydrogen, i.e., hydrogen that is part of a hydrocarbon fuel component (CHψ ). A typical simulation of refinery gas comprises 25% H2, 50% natural gas, and 25% C3H8. These fuels are readily available and easily blended. In the case of molecular hydrogen, one could adjust ψ to account for both contributions (i.e., r H2 + CHψ  = CHψ *,  where ψ * = ψ + 2r). Then ψ * represents

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stack

convection section

stack probe

convection tubes

furnace probe

shock tubes

“arch” or “bridgewall” radiant section radiant tubes

measure emissions here for regulatory purposes

measure emissions here for control purposes

burners

FIGURE 2.22 A process heater. The typical sections of a process heater comprise a (a) radiant section — the portion of the furnace that can “see” the flame; (b) arch or bridgewall — the transition section  between the radiant and convection sections; (c) convection section — where the primary heat transfer mechanism is via convection; and (d) the stack — the outlet conduit for the furnace. For control purposes, one must measure O2 and CO at the bridgewall. For regulatory purposes, one must measure emissions in the stack.

the total molar H/C ratio where H is counted from both CH ψ  and H2. This is a straightforward concept with intuitive physical meaning; however, if we combust pure hydrogen, then ψ → . If we desire a single equation to account for a blend of hydrogen and hydrocarbons — including possibly pure hydrogen — then we should transform x from its unbounded range, 0 < ψ < ∞, to another variable, say χ, that has finite bounds (e.g., 0 < χ < 1). We accomplish this by defining χ as the total mole fraction of hydrogen in the fuel, having the following relation to ψ *:  ∞ 

χ = ψ * = H 1 + ψ * H + C

(2.40a)

ψ * = χ = H 1− χ C

(2.40b)

Then we may recast our equations in terms of  χ for mixtures of hydrocar bons and hydrogen via the substitution given in Equation 2.40b. This leads to the following:

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K dry

145

⎛  ⎞  = ⎜ 4 − 3.21χ ⎟ , ⎝  4 − 3χ ⎠ 

K wet

⎛  ⎞  = ⎜ 4 − 2.79χ ⎟  ⎝  4 − 3χ ⎠ 

(2.41)

The K  factor form of the equations remains valid with either Equation 2.40a or Equation 2.40b defining the K factor. Recasting the remaining equations in terms of  χ gives the following:

χ=

 yH2O

2 yCO 2 ,wet + yH2O

 yH2O =

=

2 χ yCO 2 ,wet 1 − z

 yH2O 2 (1 − yH2O ) yCO 2 ,dry + yH2O

=

2 χ yCO 2 ,dry 1 + χ(2 yCO 2 ,dry − 1)

⎛ 1 − χ ⎞ ⎛   yH2O ⎞  (1 − χ) yCO 2 ,wet =  yCO 2 ,dry = ⎠ ⎜⎝ 1 − yH2O ⎟ ⎠  1 − χ (1 + 2 yCO 2 ,wet ) ⎜⎝  2χ ⎟  2.4.5

(2.42)

(2.43)

(2.44)

Addition of Flue Gas Components to Fuel

Besides hydrogen and hydrocarbons, other significant (nontrace) quantities in refinery fuel gas sometimes include carbon monoxide and diluents such as nitrogen and carbon dioxide. Sometimes oxygen and water vapor are also present, especially for certain waste streams or for digester or landfill gas — a 50/50 mixture of CO2 and H2O from anaerobic decomposition diluted with some air. In the case of landfill gas, a series of subterranean collectors exert a vacuum and extract gas from the landfill. If the vacuum is too high, air will migrate through the soil and dilute the landfill gas. If the vacuum is too low, landfill gas will escape, releasing methane to the atmosphere. The customary practice results in some small addition of air to the landfill gas, and this is the safer alternative to fugitive methane emissions. H2 and CO significantly increase certain fuel properties, such as flame speed and temperature (see Appendix A). Hydrogen reduces the heating value on a volumetric basis owing to its very low molecular weight. The heating value on a volumetric basis (rather than a mass basis) is the usual value of interest because burners meter fuel on a volumetric basis using a fuel pressure differential across orifices. For this purpose, the Wobbe index (ω) is a useful ratio:  W a ω = Δ H 

W  f 

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(2.45)

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 is the volumetric higher heating value (HHV; e.g., Btu/scf), W  is where Δ H  a the molecular weight of air (29 g/mol), and W  f  is the molecular weight of  the fuel (see Appendix A for tabulations of these quantities for some common

fuels). The ratio W f W a is known as the specific gravity of the fuel (abbreviated sp. g.). Thus, an alternate form for Equation 2.45 is

ω = Δ H  . The-

sp. g. oretically, the heat release from a fuel orifice scales as the Wobbe index. If H 2 and CO concentrations in the fuel vary during the course of normal operation, it may be better to make a separate accounting for them in the mass balance. As we have already seen, the addition of molecular hydrogen adds no real complexity to the mass balance. However, the addition of O 2, N2, CO2, CO, and H2O to the fuel complicates matters because the species now appear on both sides of the combustion equation. In that case, the stoichiometric equation becomes more involved: c CO2 + d N2 + f O2 + g H2O + r H2 + s CO + CHψ  + (1+ r/2 + s/2 + x/4 – f )(1 + ε) O2 + (79/21)(1+ r/2 + s/2 + x/4 – f )(1 + ε) N2 → (1 + c + s) CO2 + ( g + r + x/2) H2O + ε(1 + r/2 + s/2 + x/4 – f )O2 + [(79/21)(1 + r/2 + s/2 + x/4 – f )(1 + ε) + d] N2 (2.46) Here, c is the CO2/CHψ  mole ratio, d is the N2/CHψ  mole ratio, f is the O2/CHψ  mole ratio, g is the H2O/CHψ mole ratio, r is the H2/CHψ  mole ratio, s is the CO/CHψ  mole ratio, and ψ is the H/C ratio in the hydrocarbon stream only. We emphasize that these refer to the component ratios in the fuel, not the combustion products. Equation 2.46 is the most general form of the stoichiometric equation we will consider. Other combustibles such as H2S and NH3 may be present in trace amounts in the fuel stream, and we could modify Equation 2.46 to account for them. However, as regards the stoichiometry, trace quantities are insignificant. H2S and NH3 do affect SO2 and NOx emissions, respectively, so we will consider these species in this context later in this chapter. Since we are accounting for molecular hydrogen separately from the hydrocarbon portion of the fuel, there is no need to use a variable transform such as χ. Equations cast in terms of ψ or K have the following general form:  y =

a + bε K + ε

(2.47)

The reader may verify that if c, d, f , g, r, and s = 0, the following expressions will reduce to the original (simpler) relations:

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147

⎡⎣ 4 ( c + d + f + g ) + 2(s − r ) − ψ ⎤⎦ K dry = 1 + 0.21 ⎡⎣ 4 (1 −  f ) + 2 ( r + s ) + ψ ⎤⎦

(2.48a)

⎡⎣ 4 ( c + d + f + g ) + 2(r + s) + ψ ⎤⎦ K wet = 1 + 0.21 ⎡⎣ 4 (1 −  f ) + 2 ( r + s) + ψ ⎤⎦

(2.48b)

aO2

ε=

=

(2.49a)

 yO 2 ,dry =

0.21ε + aO 2 ε + K dry

(2.49b)

 yO 2 ,wet =

0.21ε + aO 2 ε + K wet

(2.49c)

 yO 2 ,dry Kdry − aO 2 0.21 − yO 2 ,dry aN2

=

=

 yO 2 ,wet Kwet − aO 2 0.21 − yO 2 ,wet

84d + 0.79 4 (1 −  f ) + 2 ( r + s ) + ψ 

(2.50a, b)

(2.51a)

 yN 2 ,dry =

0.79ε + aN2 ε + K dry

(2.51b)

 yN 2 ,wet =

0.79ε + aN 2 ε + K wet

(2.51c)

aCO2 =

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 f  4 (1 −  f ) + 2 ( r + s ) + ψ 

84 (1 + c + s ) 4 (1 −  f ) + 2 ( r + s ) + ψ 

(2.52a)

 yCO 2, dry =

aCO2 ε + K dry

(2.52b)

 yCO 2, wet =

aCO2 ε + K wet

(2.52c)

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2.4.6

 Modeling of Combustion Systems: A Practical Approach

Substoichiometric Combustion

So far, we have considered air in excess and fuel as the limiting reagent. This is the usual (and safe) combustion practice. Substoichiometric combustion is the fuel–oxygen reaction having oxygen as the limiting reagent. Substoichiometric combustion is extremely dangerous in conventional furnaces  because large volumes of hot combustibles may accumulate, find some air, and explode — a heater explosion can have remarkable destructive potential. When the combustion first becomes substoichiometric, the flame does not go out. Burners will stay lit even without the full complement of air because what air is available will combust enough fuel to keep the combustion reaction going and establish a flame. However, under these circumstances  burners can become unstable: the burners may go out, relight, or the flame may detach and reattach haphazardly. If one discovers substoichiometric combustion, the conventional wisdom is not to add more air, and especially not to add it quickly. Adding air to a hot furnace full of combustibles increases the likelihood of explosion. Rather, one should slowly reduce the fuel until oxygen reappears and then reaches its target level in the flue gas; only then should one add additional air and increase the firing rate. This is a good general rule of thumb, but it is not foolproof. So be warned, this situation is extremely dangerous and potentially deadly. The only safe option is to avoid substoichiometric combustion in the first place. Related to avoiding substoichiometric combustion is a lead-lag control strategy.

2.4.6.1 Lead-Lag Control  Whenever the unit is to increase in firing rate, the air increase should lead the fuel increase; that is, the air increase should occur first in time. Whenever the unit must decrease in firing rate, the air decrease should lag the fuel decrease; that is, the air decrease should occur last in time. This is the leadlag control scheme for combustion. This practice keeps the unit from encountering substoichiometric combustion. 2.4.6.2 Substoichiometric Equations It is possible to develop equations for substoichiometric combustion, recognizing the following limitations: 1. Practical combustion systems are not supposed to be operated substoichiometricly. 2. Equilibrium calculations neglect the formation of soot. 3. Some soot formation is likely. Even so, such equations can at least be instructive, and for these reasons we develop the relations that follow. (Note: Some reactors — partial oxidation reactors — deliberately employ substoichiometric combustion. However,

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partial oxidation reactors have specific design features to carry out the substoichiometric combustion safely. We do not consider them in this text per se. But if the partial oxidation reaction is with air, the following equations will be applicable nonetheless.) Earlier in the chapter, we defined the quantity 1 + ε as being the ratio of  theoretical air required for combustion. When ε > 0, some excess air exists in the furnace and the flue gas contains some O 2. When ε < 0, there is no relation between excess air and O2 because yO2 = 0 for all ε < 0. Accordingly, we define Φ as the fuel/air ratio (also known as the stoichiometric ratio), having the following relationship to ε:

Φ= 1 1+ ε

(2.53a)

ε = 1− Φ Φ

(2.53b)

When Φ < 1 (ε > 0), then the system has more than the required air for combustion and we term the combustion superstoichiometric. In superstoichiometric combustion, we know that all H will oxidize to H2O and all C to CO2. However, in substoichiometric combustion, H will go to H 2O — until it consumes much of the oxygen — and to H 2 thereafter. Likewise, C will first go to CO and thereafter to CO2. So, let us define the following molar ratios:

α = H2

(2.54a)

β = CO

(2.54b)

H2O

CO 2

Then we may recast Equation 2.15 in terms of  α and β: ⎡β + 2 ⎞  ψ ⎤ ⎛ 79 1 β CH ψ + ⎢ + CO 2 + CO ⎥ ⎜ O2 + N 2 ⎟ → 2 ⎢β+1 21 β+1 β +1 ⎠ 2 (1 + α ) ⎥ ⎝ ⎣ ⎦ 1

(2.55a)

⎞ ⎛ 1 ⎞ ⎛ 79 ⎞ ⎡ β + 2 1 ⎛ αψ ⎞  ψ ⎤ + H 2O + H2 + + ⎢ ⎥ N 2 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ 2 ⎝  ψ + α ⎠ 2 ⎝ 1+ α⎠ ⎝ 2 ⎠ ⎝ 21 ⎠ ⎢⎣ β + 1 2 (1 + α ) ⎥⎦ 1⎛

1

Turns7 notes that α and β are related by the equilibrium constant K wg for the water–gas shift reaction: CO + H 2O = CO2 + H2, and K wg = 0.19 provides

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good agreement with more rigorous equilibrium calculations for propane. That is, K wg =

α ≈ 0.19 β

(2.56)

Then we may recast Equation 2.55a as a function of  β: ⎡β + 2 ⎤⎛ ⎞  ψ β 79 1 ⎢ ⎥ ⎜ O2 + N 2 ⎟ → CH ψ + CO 2 + CO + β+1 β+1 2 ⎢β+1 21 ⎠ 2 (1 + K wgβ ) ⎥ ⎝ ⎣ ⎦ 1

⎤ ⎞ ⎞ ⎛ 1 ⎞ ⎛ 79 ⎞ ⎡ β + 2  ψ 1 ⎛ K wg  gβψ ⎢ ⎥ N 2 H O H + + + ⎜ ⎟ 2 ⎜ ⎟ 2 ⎜⎝ 2 ⎟⎠ ⎜⎝ 21 ⎟⎠ β + 1 2 ⎝  ψ + K wgβ ⎠ 2 ⎝ 1 + K wgβ ⎠ 2 (1 + K wgβ ) ⎥ ⎢⎣ ⎦ 1⎛

+

(2.55b)

1

Now we know that the total amount of oxygen consumed must by definition be

⎛ ψ + 4 ⎞  ⎛ ψ + 4 ⎞  + ε = 1 Φ ⎜⎝  4 ⎟ ⎠  ( ) ⎜⎝  4 ⎟ ⎠  1

Equating this with the coefficient for O 2 in Equation 2.55 yields

⎤ ⎛ ψ + 4 ⎞  1 ⎡ β + 2 ψ  ⎥ = ⎢ + ⎜ ⎟  Φ ⎝  4 ⎠  2 ⎢⎣ β + 1 2 ( K wgβ + 1) ⎥⎦

(2.57)

K wgβ + 1) (β + 1) ( ψ + 4) ( Φ (β, ψ ) = 2 ( K wgβ + 1) (β + 2 ) + (β + 1) ψ 

(2.58)

1

or

Then Φ or ε is a function of only one parameter, β. From Equation 2.55 we may calculate all the required species as a function of  β. Therefore, we may cast the total dry and wet products (TDP and TWP, respectively) as functions of  β and ψ  alone:

⎤ ψ  1 ⎛  K wgβψ  ⎞  79 1 ⎡ β + 2 ⎢ ⎥ + + TDP (β, ψ ) = 1 + ⎜ β + 2 ⎝ 1 + K wgβψ ⎟  21 2 1 2 ( K wgβ + 1) ⎥ ⎠  ⎢

(2.59)

⎤ ψ ψ  79 1 ⎡ β + 2 ⎢ ⎥ + TWP (β, ψ ) = 1 + + 2 21 2 ⎢ β + 1 2 ( K wgβ + 1) ⎥ ⎣ ⎦

(2.60a)

⎣

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⎦

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151

Making use of Equation 2.57, we recast TWP as a function of  Φ and ψ .

⎛  ⎝ 

TWP ( Φ, ψ ) = ⎜ 1 +

79 1 ⎞ ⎛  ψ ⎞  ψ  1 + ⎟  + ⎟  ⎜ 21 Φ ⎠ ⎝  4 ⎠  4

(2.60b)

The species expressions become  yO2 ,dry = 0

(2.61a)

 yO2 ,wet = 0

(2.61b)

 yN2 ,dry =

⎞  ψ  79 ⎛  1 ⎞ ⎛ β + 2 + ⎜ ⎟  ⎜ ⎟  21 ⎝ 2 TDP ⎠ ⎝ β + 1 2 ( K wgβ + 1) ⎠ 

(2.62a)

 yN2 ,wet =

⎞  ψ  79 ⎛  1 ⎞ ⎛ β + 2 + ⎜ ⎟  21 ⎜⎝ 2 TWP ⎟  ⎠ ⎝ β + 1 2 ( K wgβ + 1) ⎠ 

(2.62b)

 yCO2 ,dry =

1 ⎛  1 ⎞  TDP ⎜⎝ β + 1 ⎟  ⎠ 

(2.63a)

 yCO2 ,wet =

1 ⎛  1 ⎞  TWP ⎜⎝ β + 1 ⎟  ⎠ 

(2.63b)

1 ⎛  ψ  ⎞  2 TWP ⎜⎝ Kwg   β + 1 ⎟  ⎠ 

(2.64)

 yH2, dry =

1 ⎛  K wgβψ  ⎞  2 TDP ⎜⎝ K wgβψ + 1 ⎟  ⎠ 

(2.65a)

 yH2, wet =

1 ⎛  K wgβψ  ⎞  2 TWP ⎜⎝ K wgβψ + 1 ⎟  ⎠ 

(2.65b)

1 ⎛  β ⎞  TDP ⎜⎝ β + 1 ⎟  ⎠ 

(2.66a)

 yH2O =

 yCO, dry =

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 yCO,wet =

1 ⎛  β ⎞  TWP ⎜⎝ β + 1 ⎟  ⎠ 

(2.66b)

Combining Equations 2.57 and 2.66b gives  yCO, wet (β, ψ ) =

⎛  β ⎞  ⎛  79 1 ⎞ ⎛  ψ ⎞  ψ  ⎜⎝ β + 1 ⎟ ⎠  ⎜ 1 + 21 ⎟ ⎜⎝ 1 + 4 ⎟ ⎠ + 4 Φ (β, ψ ) ⎠  ⎝  1

(2.66c)

Equation 2.66c gives  yCO,wet (β, ψ ) and Equation 2.58 gives Φ(β, ψ ) . In principle, we could combine these to give Φ( yCO,wet , ψ ) and  yCO,wet (Φ, ψ ), but this results in a messy analytical expression. The easier solution is to use β as a parameter and plot Φ against ψ  as β varies, or solve for β numerically for a given Φ or yCO. As noted, we may use β as a parameter in Equation 2.58 and the species concentration equations above to yield the equilibrium relations. Combining the substoichiometric and superstoichiometric relations, we may estimate wet or dry species for any hydrogen–hydrocarbon blend. If desired, we may use the substitution of Equation 2.40b for very high hydrogen cases. Figure 2.23 shows an example for refinery gas. From the above equations or the figure, we make the following and immediate observations:

20% 18%

   )    2    N16%    t   p   e 14%   c   x   e    ( 12%   s   e    i   c 10%   e   p   s 8%    %    l   o 6%    V 4%

DryN2

H2O

CO2

2% Vol % dry > Vol % wet  species in all cases  0% 0.50 0.60 0.70 0.80

CO Wet N2 O2 H2

0.90

1.00

1.10

1.20

1.30

1.40

90% 88% 86% 84% 82% 80% 78% 76% 74% 72% 70% 68% 66% 64% 62% 60% 1.50

   )    2    N    %    l   o    V

Φ

FIGURE 2.23 Substoichiometric combustion relations. The figure gives relations for a typical refinery gas comprising 25% H2, 50% CH4, and 25% C3H8.

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• No matter how substoichiometric the combustion, oxygen is always zero. Therefore, the oxygen concentration cannot estimate ε or Φ, except merely to say that ε < 0 and Φ > 1. • For hydrocarbon fuels, CO is present whenever the combustion is substoichiometric. In principle, a CO analyzer could estimate Φ in the furnace. However, CO analyzers for combustion monitoring read no more than 5000 ppm. This represents only a very small amount of substoichiometric combustion. Therefore, this kind of CO measurement is not a useful quantitative measure of the degree of air starvation. However, CO measurement does provide a valuable early warning signal — CO skyrockets before entering and while in a dangerous substoichiometric operating region.

Example 2.5

Calculation of Combustion

Φ and ε for Substoichiometric

Problem statement: For a refinery gas having 25% H 2, 50% CH4, and 25% C3H8, calculate Φ and ε if the furnace gas shows 5000 ppm CO. Estimate the CO level if  Φ = 1.1. Comment on the suitability of a CO or combustibles analyzer having an upper limit of 5000 ppm for this kind of service. Solution: Since this reading comes from the furnace, we may presume it is an in situ reading and that the CO is on a wet basis. Solving for ψ  we obtain

ψ = 0.25(2) + 0.50( 4) + 0.25(8) = 3.6 0.25(0) + 0.50(1) + 0.225(3) Using Equation 2.66c to give  yCO, wet (β, ψ ) and Equation 2.58 to give Φ(β, ψ ) , we solve for β such that  yCO, wet (β, ψ ) = 5000 ppm and ψ = 3.6 using an iterative routine. Spreadsheets usually contain one or two such routines (e.g., Goal Seek™ in Excel™). This gives β = 0.0318 and Φ = 1.011. Thus, 5000 ppm CO represents only a small level of substoichiometric combustion (about Φ – 1 = 1.1% substoichiometric). If we use Equation 2.53b to calculate ε, we obtain the same result

ε = 1 − Φ = 1 − 1.011 = −0.0109 = −1.1% Φ 1.011 Using the same process, if Φ = 1.1 (ε = –0.10 = –10%), then β = 0.3197 and yCO,wet ≈ 41,000 ppm. Thus, 10% substoichiometric operation

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 Modeling of Combustion Systems: A Practical Approach results in more than 40,000 ppm CO. Therefore, a typical combustion analyzer having an upper limit of 5000 ppm would not be useful to indicate this level of substoichiometric combustion; however, it would allow us to know when we exceed ~1% substoichiometric combustion, giving us an early warning that something has gone awry.

Although we desire to have excess oxygen present, too much excess air robs efficiency from the unit. CO evidences how little O 2 is too little and quantifies a healthy balance between good efficiency and good combustion. For these reasons, we recommend that furnaces be equipped with wellmaintained and calibrated CO and O2 analyzers.

2.4.7

Conservation of Mass for Flow in a Furnace

As shown previously, combustion conserves total mass and individual atomic species but not molecular entities. However, we may write a general conservation equation for all molecular species, including those that the reaction consumes or produces. In a strict sense, we do not conserve such species, but rather account for them. However, the standard nomenclature is conservation of mass, not accounting of mass. Heuristically, the general conservation equation is in + gen = out + acc

(2.67)

where in is the flow rate into an arbitrary volume [M/θ], gen is the amount of substance produced per unit time within the volume [ M/θ], out is the flow rate out of the volume [M/θ], and acc is the amount of substance accumulating per unit time [M/θ]. In order to know the difference between in and out we need a system  boundary. We divide the universe into two pieces: system and surroundings. The system is any portion of the universe we are interested in; the surroundings comprise the rest. We shall always pick a convenient reference — one where we know the boundary conditions. Our sole purpose is to define a system so that we may solve the conservation equation and produce results that are meaningful to us. Mathematically, Equation 2.67 becomes

d ( ρv xv )   ρi xiVi + rV = ρe xeVe + V   dt In Equation 2.68 we have the following entities:

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(2.68)

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ρ is the density [M/L3]. x is the mass fraction [ ].  is the volumetric flow rate [L3/θ]. V  i is a subscript indicating influent flow. e is a subscript indicating effluent flow. v is a subscript indicating properties belonging to the control volume. r is the reaction rate [M/L3θ]. V is the system volume [L3]. t is the time [θ]. Therefore, the units of Equation 2.68 are [ M/θ]. We use the dot superscript  is the volumetric flow rate, to refer to flow rates in general. For example, V   is the molar flow rate, etc. Time is the unit in the  is the mass flow rate, N  m denominator for all rate equations.

2.4.8

Simplifying Assumptions (SAs)

Here we list some conditionals that can simplify Equation 2.68: 1. If there is no reaction, gen = 0 (rV = 0). 2. If the process is time invariant with position (steady state), then

⎛  d ( ρv xv ) ⎞  = 0⎟ . acc = 0 ⎜ V  dt ⎝  ⎠  3. If the process is well mixed, then

ρv = ρe and xv = xe .

4. If the density is time invariant, then V 

Example 2.6

d ( ρv xv ) dt

= ρvV dxv . dt

An Example of a Non-Steady-State Mass Balance

Problem statement: Consider a furnace with a control volume as shown in Figure 2.24.

Take the system to be that as shown by the dotted boundary in the figure. Beneath the system boundary is a combustion process producing some emission of concern. Above the system boundary is a sensor that measures the emission. Presume that the system volume is well mixed. What will be the time-varying behavior of  the analyzer to a sudden step change in the emission rate from a process upset? In other words, develop an equation for xθ, where xθ is the analyzer concentration at time t = θ.

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effluent flue gas sample probe ρe x eQe

V 

d (ρv x v) dt 

influent flue gas

control volume boundary

ρi x i Qi

FIGURE 2.24 A furnace control volume. The furnace diagram shows a control volume and elements of the associated mass balance. For ease of computation, the control volume is drawn such that there is no reaction within its boundary.

Solution: The process is well mixed, so simplifying assumption (SA) 3 applies. We have drawn the system volume above the reaction zone in the flame, so SA 1 applies. We shall presume that automated controls will operate to keep the temperature and composition constant. Therefore, SA 4 applies. Furnaces do not accumulate mass; otherwise, the furnace would become a pres. sure vessel. Since the acc and gen terms are zero, in = out, so Vi = V  e Dropping the subscripts for the variables that do not need to be distinguished, Equation 2.68 becomes

ρxiV = ρxV + ρV dx  dt

Dividing by ρV and multiplying by dt gives   V  V  xi dt = x dt + dx V  V 

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We can easily separate the variables and integrate the equation at once: θ

xθ

0

x0

 V  − dt = V 

∫ 

⌠ dx ⎮ ⎮ x−x ⌡ i

This yields the equation  ⎛ xθ − xi ⎞  V  − θ = ln ⎜ V  ⎝ x0 − xi ⎟ ⎠ 

(2.69)

or equivalently, xθ − xi x0 − xi



=e

− V θ V 

(2.70)

Equations 2.69 and 2.70 are dimensionless forms. We may rearrange the latter to give xθ in terms of the other variables.

(

xθ = xi 1 − e



− V θ V 

)+ x

0

(2.71)

In reactor engineering, V V  is known as the space velocity, having units of   is known as the space-time [θ]. In reciprocal time [θ–1]. Conversely, V V  process control engineering, V V is known as the first-order gain, while V V is known as the system time constant. (V V )θ gives the number of  volume changes for the furnace in a given time, where V is a volume of interest [L3], usually the furnace volume. The residence-time is the time that a quantity of  fluid of interest stays within a specified volume. For well-mixed volumes of  the type considered in the above example, the space-time and the residencetime are synonymous. Figure 2.25 shows the relation graphically. Equation 2.71 and Figure 2.25 point out some interesting facts. First, furnaces behave as integrators. The effluent is a time average of input condi , is the time it takes for the outlet value to achieve tions. The time constant, V V  1 − 1 e ≈ 63.2% of the final value. Second, after making a change, one must allow sufficient time to observe it. The time constants for furnace volumes are typically only a few minutes. However, they tie to processes that can have much longer time constants and that communicate to the furnace volume via a control system. Suppose the time constant for a furnace and  is 15 minutes. After 30 minutes, the system process is 15 minutes; i.e., V V  has reached about 85% of its final value. After 45 minutes, the system reaches

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1-e-3= 95.0%

0.9    0      x        i

1-e-2= 86.5%

0.8

 –      x

        θ

     x

0.7   n   o    i    t 0.6   a   r    t   n   e 0.5   c   n   o    C 0.4    d   e 0.3   z    i    l   a 0.2   m   r   o    N 0.1

1-e-1= 63.2%

0.0

0

1 Time Constant,

Q

2

3

V 

FIGURE 2.25 Concentration vs. time for well-mixed behavior. Concentration shows a law of diminishing returns over time. The effluent concentration, xθ , approaches but never quite reaches the new inlet concentration, xi.

95% of its final value. So it would be futile to make sense of an observation 20 minutes after effecting a process change. Many furnaces require 45 minutes to return to steady state after a significant step change, as a rule of  thumb. However, depending on the magnitude and nature of the distur bance, it may take even longer.

2.4.9

Ideal Gas Law

Except for very special applications, industrial combustion is a high-temperature, low-pressure affair. Therefore, the ideal gas law is applicable and quite accurate: (2.72) PV = nRT  where P is the pressure [ ML/θ2], V  is the volume [L3], n is the number of  moles [N], R is the gas constant [ L2/θ2T], and T  is the temperature [T]. In some cases, one may prefer to use the molar volume Vˆ = V/n [L3/N], in which case Equation 2.72 takes the form of Equation 2.73: PVˆ = RT  

(2.73)

Some cases will call for mass rather than molar quantities: PW = ρRT 

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(2.74)

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where W is the molecular weight [M/N] and ρ is the density [M/L3].

2.4.10 Dilution Correction Suppose a local agency regulates NOx to 40 ppm. What does this mean? Technically, it means that for every million volumes of flue gas, 40 will be NOx. If the NOx limit were not otherwise qualified, there would be two ways to meet it. One way would be to reduce the amount of NOx the combustion system generates; another way would be to increase the flue gas volume by adding dilution air. But this latter option does not reduce the amount of NOx that enters the atmosphere. Indeed, this dilution may occur without our even knowing it. For example, if the stack pressure is below atmospheric pressure, then any leakage will allow air to infiltrate the stack gas and reduce the NOx concentration. For this reason, air quality districts reference the NOx concentration to some oxygen percentage. For boilers and process heaters the typical reference is 3% oxygen. For gas turbines, the typical reference is 15%. Some districts use 0% O2 as the reference. A mass  balance allows us to develop a simple equation to correct any actual concentration to a reference concentration. Figure 2.26 shows the general logic. effluent

. V e  yNO ,e  yo2,e

total mass balance: . . . = + V e V i V a

. V a  yo2,a

mass balance on O 2: . . .  yo2,eV e = yo2,iV i + yo2,aV a

air in-leakage

mass balance on NOx: . . =  yNO ,eV e  yNO ,iV i . V i  yNO ,i  yo2,i

influent

FIGURE 2.26 Dilution correction. A mass balance provides a method for calculating the amount of air inleakage and the influent NOx concentration.

Now we can do a total mass balance as follows: in + gen = out + acc If the air in-leakage does not react with the emissions of concern (that is, it functions only as a diluent), then gen = 0. For a steady-state process, acc = 0. Therefore, in = out and we obtain i +m a m

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= m e

(2.75)

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 i is the influent mass flow rate [M/θ], m  a is the mass flow of inwhere m  e is the effluent mass flow rate [M/θ]. We leaking (diluting) air [M/θ], and m can use volumetric flow rates if we correct them to some standard temperature and pressure (STP), say 60°F and 1 atm. In that case, we have  Vi + Va = V  e

(2.76)

 is the influent volumetric standard flow rate [L3/θ], V   is the voluwhere V  i a  3 metric standard flow rate of in-leaking air [L /θ], and V e is the effluent volumetric standard flow rate [ L3/θ], all at STP. Now a mass balance for NOx  , but  y will be  yNO ,iVi + yNO , aVa = yNO, eV  e NO , a = 0 ; therefore, the equation  becomes   yNO ,iVi = yNO, eV  e

(2.77)

 directly, leading to We may solve Equation 2.77 for Vi V  e   y , V  NO e i =   y V  NO,i e

(2.78)

With respect to oxygen, the mass balance yields   yO 2 ,iVi + yO 2 , aVa = yO2 ,eV  e

Rearranging Equation 2.79, we have   V  V  i  yO 2 ,i  + yO 2 , a a = yO 2 ,e V e V e

But we know from Equation 2.76 that  Vi + V  a  V  e

=1

Therefore,  V  a  V  e

 V  = 1 − i V e

Substituting this into the foregoing equation gives

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(2.79)

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  ⎞  ⎛  V  V  i  yO 2 ,i  + yO 2 , a ⎜ 1 −  i ⎟  = yO 2 ,e V e ⎝  V e ⎠   yields Collecting Vi V  e  V  i  yO 2 ,i − yO 2 , a ) + yO 2 , a = yO 2 ,e (  V e  , we obtain Multiplying both sides by –1 and solving for Vi V  e   y V  O 2 , a − yO 2 , e i =   y V  O 2 , a − yO 2 ,i e

(2.80)

 and inverting the Equating Equations 2.78 and 2.80 to eliminate Vi V  e fractions gives

 yNO,i  yNO, e

=

 yO 2 , a − yO 2 ,i  yO 2 , a − yO 2 , e

(2.81)

The above equation gives us the inlet concentration ( yNO,i) if we know the effluent concentration ( yNO,e ). In fact, we can correct even for hypothetical influent concentrations, and this is the real power of Equation 2.81. Let us replace  yO 2, i and  yNO,i by  yO 2, ref   and  yNO , ref , respectively. In this way, we expressly declare that we are correcting our results to a hypothetical reference concentration. Then Equation 2.81 becomes

⎛  yO 2 , a − yO2 ,ref ⎞    yNO, e ⎟  ⎝  yO 2 , a − yO2 ,e ⎠ 

 yNO ,ref  = ⎜

(2.82)

This is the traditional form for correction to a hypothetical reference condition. Indeed, this dilution correction philosophy is perfectly generic for any emission of interest.

⎛  yO 2 , a − yO2 ,ref ⎞    yx ,ref  = ⎜  yx ,e ⎟  ⎝  yO 2 , a − yO2 ,e ⎠ 

(2.83)

where the subscript x refers to any emission of interest that is not found in the dilution air.

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Example 2.7

Emissions Corrections to Specified Reference Conditions

Problem statement: Emissions measurement at the stack reveals NOx and CO fractions of 32 and 18 ppm, respectively, at an outlet concentration of 5% oxygen. The legal limit is 40 ppm NOx and 200 ppm CO, corrected to 3% oxygen. Calculate the corrected values. Is the unit in compliance? Solution: The form of Equation 2.82 is perfectly generic for correcting the concentration of any species not found in the dilution air. Therefore, we have

⎛  yO 2 , a − yO2 ,ref ⎞   ⎛ 21 − 3 ⎞  = 32 ⎡⎣ ppm ⎤⎦ = 36 ⎡⎣ ppm ⎤⎦  yNO,3% = ⎜  y NO, e ⎟  ⎜ ⎟  ⎝ 21 − 5 ⎠  ⎝  yO 2 , a − yO2 ,e ⎠  ⎛  yO2 , a − yO2 ,ref ⎞   ⎛ 21 − 3 ⎞  = 18 ⎡⎣ ppm ⎤⎦ = 20.3 ⎡⎣ ppm ⎤⎦  yCO ,3% = ⎜  y CO , e ⎟  ⎜ ⎟  ⎝ 21 − 5 ⎠  ⎝  yO 2 , a − yO2 ,e ⎠  So the emissions are well within the legal limits. Dilution correction normalizes all emissions to the same reference basis and eliminates the dilution effects of air. Note: Many regulatory districts have their own opinion about how much oxygen air comprises. Some districts use 20.9% for the oxygen concentration, some use 20.8%. Be sure to replace 21 by whatever the district specifies. The following example highlights how different references affect the reported values.

Example 2.8

Corrections to Various Bases

Problem statement: One district corrects its NOx readings to 3% oxygen, while another corrects to 15% oxygen. Yet another corrects its readings to 0% oxygen. Show the equivalence of 40 ppm corrected to 3% with these other bases. Solution: We begin with NOx at 40 ppm corrected to 3%. If we correct this to 15% oxygen we have

⎛ 21 − 15 ⎞  40 ⎡⎣ ppm ⎤⎦ = 13.3 ⎡⎣ ppm ⎦⎤ ⎟  ⎝  21 − 3 ⎠ 

 yNO,15% = ⎜

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If we correct to 0% we have

⎛ 21 − 0 ⎞  40 ⎡ ppm ⎤⎦ = 46.7 ⎡⎣ ppm ⎤⎦  yNO,15% = ⎜ ⎝ 21 − 3 ⎟ ⎠  ⎣ Therefore, to convert emissions referenced at 3 to 15%, we divide  by 3 (or multiply by 1/3), that is,

⎛ 21 − 15 ⎞  6 1 ⎜⎝  21 − 3 ⎟ ⎠  = 18 = 3 To convert emissions referenced at 3% to a base of 15%, we multiply by 7/3, that is,

⎛ 21 − 0 ⎞  7 ⎜⎝ 21 − 3 ⎟ ⎠  = 3 The point of dilution correction is that dilution-corrected emissions are invariant to dilution by air. An example illustrates the point.

Example 2.9

Invariance of Dilution-Corrected Emissions

Problem statement: Dilution-corrected emissions are invariant to dilution by air. Prove this by considering 100 volumes of flue gas containing 100 ppm NOx at 0% O2. Then add 100 volumes of air to this and determine the oxygen and NOx at this condition. Correct both cases to 3%. What do you note? Solution: In the first case, we have 100 ppm of NOx. Doubling the volume by adding air will reduce the NOx emissions to 100/2 = 50 ppm at 21/2 = 10.5% oxygen. Correcting the first case to 3% oxygen, we obtain

⎛ 21 − 3 ⎞  100 ⎡⎣ ppm ⎤⎦ = 85.7 ⎡⎣ ppm ⎤⎦ ⎟  − 21 0 ⎝  ⎠ 

 yNO,3% = ⎜

corrected to 3% oxygen. Correcting the second case to 3% oxygen, we have

⎛  21 − 3 ⎞  50 ⎡⎣ ppm ⎤⎦ = 85.7 ⎡⎣ ppm ⎤⎦  yNO,3% = ⎜ ⎟  ⎝ 21 − 10.5 ⎠  corrected to 3% oxygen.

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 Modeling of Combustion Systems: A Practical Approach Therefore, in any case, the corrected emissions are identical, as the cases differ only by air dilution. This is the point of dilution correction.

2.5

Conservation of Energy

Mechanical energy (E) is force times distance (d), E = Fd. In turn, force is mass times acceleration, F = ma. Thus, E [=] [ML2/θ2], where [=] is read has units of. However, all energy terms have the same units. The fundamental SI unit for energy is the Joule (J) = kg m 2/sec2. The U.S. customary unit is the British thermal unit (Btu). The general conservation equation applies to energy as well as mass. There are two basic divisions for energy: work and heat. First, we consider heat and related quantities.

2.5.1

Heat and Related Quantities

 Heat is thermal energy in transit. Internal energy is thermal energy stored in a body. If a body acquires thermal energy, we no longer call it heat; it is internal energy. We may not know the absolute internal energy of a body. However, one may calculate the difference in internal energy from one state to another if we know the initial and final states of the body. Temperature is a measure of the average random kinetic energy of atoms and molecules in a body. At absolute zero, the average random kinetic energy is zero; this does not mean that all atomic and subatomic motion stops, only random kinetic motion. It is not true that absolute zero is unattainable or that calamity awaits us when we get there. However, we cannot get there  by purely thermal means because temperatures above absolute zero can never average to zero. One must not confuse the concepts of temperature, internal energy, and heat. They are all different but related things. To relate them, it is useful to first distinguish between intensive and extensive variables. An intensive variable describes some property of a substance or system that does not depend on mass. For example, pressure and temperature are inherently intensive variables. The temperature of a substance will tell us nothing of its extent, e.g., its mass, number of moles, or volume. Volume, energy, and power are examples of extensive variables. An extensive variable describes some property of a substance or system that depends on mass (extent). Twice the amount of hot water contains double the volume and double the internal energy. Therefore, both internal energy and volume are extensive variables. We can convert any extensive variable to an intensive one by dividing by some measure of extent. We will use the following nomenclature: U , internal

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energy (Btu or kJ, [ ML2/θ2]), an extensive property; U = U m, specific internal energy (e.g., Btu/lbm or kJ/kg [L2/θ2]), an intensive property; Uˆ = U n, molar internal energy (e.g., Btu/lbmol or J/mol [ML2/Nθ2]), an intensive ~ property; and, U = U V , volumetric internal energy (e.g., Btu/ft3 or kJ/m3, [M/Lθ2]) an intensive property. We shall use the same superscript symbols whenever we want to distinguish the molar or volumetric forms of any property, variable, or function; however, when unit dimensions are clear from context we may omit the superscripts. Equation 2.84 relates the internal energy to the temperature through a constant of proportionality known as the isometric (constant volume) heat capacity:

Δ 21Uˆ = Cˆ v Δ 21T 

(2.84)

where the prefix Δ21 is a difference operator signifying the difference between ˆ is the state 2 and state 1 of the succeeding variable (e.g., Δ21U = U 2 – U 1), U  molar internal energy, Cˆ v is the isometric heat capacity (e.g., Btu/lbmol °F or J/mol K [ML2/NTθ2]), and Δ21T is the temperature difference from state 2 to state 1 (T 2 – T 1).

2.5.2

Work

If the substance expands or contracts upon heating, then it performs work against the surrounding pressure. Work  is nonrandom energy in transit. Energy in transit that is not heat is work, and vice versa. The work may or may not do something useful. Regardless, it is a form of energy, and we must account for it in the energy conservation equation. A flame certainly produces heat. However, less obvious in a constant-pressure system is that the flame expands against the atmosphere and therefore performs pressure–volume work. Often, we harness some of this work to pump air in natural draft systems, i.e., natural draft. To account for both heat and pressure–volume work we define enthalpy. Enthalpy is internal energy plus pressure–volume work:  H = U + PV 

(2.85)

where  H  is the enthalpy [ ML2/θ2], P is the pressure [M/Lθ2], and V  is the system volume [L3]. For a constant-pressure system (e.g., a furnace), we may write

Δ 21 Hˆ = Cˆ p Δ 21T 

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(2.86)

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ˆ is the molar enthalpy difference from state 2 to state 1 (e.g., Btu/ where Δ 21 H  lbmol or J/mol [ML2/Nθ2]) and Cˆ p is the isobaric (constant-pressure) heat capacity (e.g., Btu/lbmol °F or J/mol K [ ML2/Nθ2T]), which accounts for differences in both the internal energy and volume. For an ideal gas Cˆ p = Cˆ v + R The reader should note that the ideal gas constant has the same units as the molar heat capacity, e.g., (lit·atm/mol·K) or (psia·ft 3/lbmol·°R), i.e., [ML2/ Nθ2T].

2.5.3

Heating Value

Combustion enthalpy (more commonly called heating value) is the state function used to the account for energy in constant-pressure combustion systems such as furnaces and boilers. State functions are path-independent functions; they depend only on the initial and final states. State functions offer great convenience because we are free to construct the path in any theoretical way we choose, so long as we start and end at the specified initial and final states. The heating value is the total enthalpy difference of the fuel and flue gas products referenced to a standard state. An initial state comprises a quantity of fuel and stoichiometric oxygen at standard conditions (usually 25°C and atmospheric pressure) and ends with products at the same temperature and pressure. Thus, heating values comprise interval data and ratios of heating values have no intrinsic meaning, depending as they do on an arbitrary reference state. The lower heating value presumes that water vapor generated in the combustion reaction does not condense, even at the reference condition. The higher heating value presumes that the water of  combustion does condense, thereby adding the enthalpy of vaporization to the total heating value. Hence, the higher and lower heating values differ  by the heat of vaporization of water. Fuels with no hydrogen (e.g., CO) have identical higher and lower heating values, as their combustion produces no water. Because thermal energy can pass through solid walls (something mass cannot do), we need to modify the generic conservation equation to account for this feature. Then the generic equation for conservation of energy  becomes in + gen + trans = out + acc

(2.87)

where trans denotes the energy transferred without mass flow across the system boundary into the system [ ML2/θ3]. The following two processes deserve special attention:

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1. An isothermal processes is one whose temperature does not change. In such a case, the inlet and outlet temperatures must be equal (T i = T e). 2. An adiabatic process is one that does not gain or lose heat. In such a case, trans = 0. It is possible for a system to be both adiabatic and isothermal, for example, nonreacting flow in a perfectly insulated pipe. It is also possible for a system to be isothermal but not adiabatic, for example, water and steam in a boiler tube. Additionally, it is possible for a system to be adiabatic but not isothermal, for example, reacting flow in a perfectly insulated pipe. Finally, a system may be neither adiabatic nor isothermal, for example, reacting flow in an imperfectly insulated pipe. Simplifying assumptions: 1. If there is no reaction, gen = 0. 2. If the process is time invariant with position (steady state), then acc = 0. 3. If the process is adiabatic, then trans = 0. 4. If the process is isothermal, then the inlet and outlet temperatures are equal (T i = T e).

2.5.4

Adiabatic Flame Temperature

Suppose we wish to estimate the adiabatic flame temperature. Then we presume an adiabatic steady-state process, and the energy balance reduces to in + gen = out or   f Cp , f (Tf m

− Tref ) + m aCp ,a (Ta − Tref ) + m f ΔHc = ( m  f  f + m a ) Cp, g ( TAFT − Tref  )   (2.88)

  f  is the flow of fuel [M/θ]; C p , f  is the specific heat capacity of the where m fuel, Btu/lbm °F or kJ/kg K [ L2/θ2T]; T  f  is the temperature of the fuel; T ref is  a is the flow of air; C p , a is the heat an arbitrary reference temperature; m capacity of the air; T a is the temperature of the air; Δ H c is the specific heat of combustion of the fuel; C p , g is the heat capacity of the flue gas; and T  AFT  is the adiabatic flame temperature for which we wish to solve. We may put this equation into dimensionless form also:

C p , f  T f − T ref  C T − T  Δ H c + α w  p,a a ref   + = 1 + αw C p , g T AFT − T ref  C p , g T  A C p , g (TAFT − T ref  )  AFT − T  ref 

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(2.89)

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where  α w = m a

m f 

At the high temperatures found in combustion, species such as CO 2 and H2O dissociate in the flame — enough to reduce the adiabatic flame temperature. However, for our purposes, we only need an approximate calculation, so we will neglect this effect. We can introduce an additional simplification by presuming that the heat capacity on a mass basis [ L2/θ2T] is approximately the same for hydrocarbon fuels, air, and flue gas, C p = C p , f  = C p , a = C p , g (0.25 Btu/lbm °F is one typical   f Cp , f (1 + α w ) and solving for T AFT − T ref  we have value). Dividing m T f − T ref  ⎛  α w ⎞  Δ H c = T − T  +⎜ − + T T  ( ) ( ) 1 + α w ⎝ 1 + α w ⎟  ⎠  a ref   C p (1 + α w )  AFT ref 

(2.90)

Or in dimensionless form,

⎛  T f − T ref  ⎞  ⎛  Ta − T ref   ⎞  Δ H c + α + ⎜⎝ T − T  ⎟ ⎠  w ⎜⎝ T − T  ⎟ ⎠  C T − T  = 1 + α w  AFT ref   AFT ref   p , g ( AFT ref  )

(2.91)

Now if the air and fuel are at ambient temperature, and we use the ambient temperature for the reference temperature, then T f = T ref  and Ta = T ref ,  and the first two terms in Equation 2.90 vanish and we have T AFT − T ref  =

Δ H c C p (1 + α w )

(2.92)

We may rearrange this to solve for T  AFT : T  AFT  =

Δ H c + T ref  C p (1 + α w )

(2.93)

Now αw is purely a function of the fuel composition and excess air. We may derive it as a function of both by noting the following: 1. The mass ratio of wet flue gas to fuel must equal the mass of fuel plus the mass of air divided by the mass of fuel, i.e.,   g m   f  m

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=

  f + m a m   f  m

= 1 + αw

(2.94)

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2. We may derive this quantity directly from Reaction 2.15: 1 + αw

⎛  ⎞  = W CO2 + ψ W CO + 79 W N2 ⎜ 1 + ψ ⎟ (1 + ε ) + W O2 ε W CHψ 2 W CHψ 21 W CHψ  ⎝  4 ⎠  W CHψ 

(2.95)

So, substituting Equation 2.93 into Equation 2.95 gives the adiabatic flame temperature as a function of fuel composition ( ψ ) and excess air ( ε). If we want a more precise value for the adiabatic flame temperature, we can use individual heat capacities.

Δ H c

T  AFT  = C p ,CO2

W CO2 W CHψ

+ ψ Cp,CO 2

⎛  ⎞  W CO 79 + C p,N2 W N2 ⎜ 1 + ψ ⎟ (1 + ε ) + Cp,O2 W O2 ε W CHψ  21 W CHψ ⎝  4 ⎠  W CHψ 

+ T ref  (2.96)

We can also obtain αw directly from the volumetric air/fuel ratio, α, and the molecular weight of the fuel, W  f  [M/N], and air, W a [M/N].

α W a = α w W  f 

2.5.5

(2.97)

Heat Capacity as a Function of Temperature

Heat capacity is a weak function of temperature. However, for combustion calculations, the temperature difference between the influent air (and fuel) and the flue gas emerging from the flame is quite large. Therefore, we consider heat capacity as a function of temperature. In general, we may estimate the heat capacity of an ideal gas at any temperature by the formula C p = a0 + a1T + a2T 2 + a3T 3  

(2.98)

Appendix A, Table A.5 gives the constants for calculation of some important flue gas species. Then the heat capacity of the flue gas species will be n

C p , g

=

∑

wk ( a0 ,k + a1 ,kT + a2 ,kT 2 + a3 ,k T 3 )  

(2.99a)

yk ( b0 ,k + b1,kT + b2 ,k T 2 + b3 ,k T 3 )  

(2.99b)

k =1

n

Cˆ p , g

=

∑ k =1

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where wk is the mass fraction and yk is the mole or volume fraction of the k th flue gas component. The overall heat capacity of the flue gas will be C p , g = α 0 + α1T + α 2T 2 + α 3T 3 or Cˆ p , g = β0 + β1T + β 2T 2 + β 3T 3 , where   n

α j =

∑w a

k j ,k 

k =1

and n

β j =

∑ y b

k j , k 

k =1

Since all the mass and mole fractions are functions of  ε and ψ  only, α and β are also functions of  ε and ψ  only. Therefore, for a given temperature C p , g = φ(ε, ψ ), and we may calculate C p, g for any stoichiometry. We may do the same for molecular weight calculations. This will greatly simplify thermodynamic calculations with flue gas.

Example 2.10 Calculation of Adiabatic Flame Temperature Problem statement: Find the approximate adiabatic flame temperature for a fuel having W = 16 lbm/lbmol, ψ  = 4, and Δ H c = 22,000 Btu/lbm in 15% and 10% excess air for C p = 0.25 Btu/lbm °F. Solution: From Equation 2.32,

α=

100 ⎛  ψ ⎞  100 ⎛  4 ⎞  1+ (1 + ε ) = 21 ⎜⎝ 1 + 4 ⎟ ⎠ (1 + 0.15) = 10.95 21 ⎜⎝  4 ⎟  ⎠ 

We convert this to αw by multiplying by the molecular weight ratio of fuel and air according to Equation 2.97. That is, α w = (10.95)( 29 16) = 19. 85. From Equation 2.93 T  AFT  =

Δ H c + T ref  C p (1 + α w )

Substituting all this into Equation 2.93 gives

⎡ Btu ⎤ 22, 000 ⎢ lbm ⎥⎦ Δ  H c ⎣ + T ref  = + 60 °F = 4, 280 °F T  AFT  = ⎡ Btu ⎤ C p (1 + α w ) 0.25 5⎢ ⎥ (1 + 19.85) ° lbm F ⎣ ⎦

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At 10% excess air

⎛  ⎞  ⎛  ⎞  α = 100 ⎜ 1 + ψ ⎟ (1 + ε ) = 100 ⎜ 1 + 4 ⎟ (1 + 0.10 ) = 10.48 21 ⎝  4 ⎠  21 ⎝  4 ⎠  and

⎛ 29 ⎞  α w = (10.48) ⎜ ⎟  = 18.99 ⎝ 16 ⎠  Then the calculation becomes

⎡ Btu ⎤ lbm ⎥⎦ Δ  H c ⎣ + T ref  = + 60 °F = 4460 °F T  AFT  = ⎡ Btu ⎤ C p (1 + α w ) 0.25 5⎢ 1 + 18.99) ( ⎥ ⎣ lbm °F ⎦ 22, 000 ⎢

Thus, the adiabatic flame temperature has risen about 180°F due to the lower excess air.

2.5.6

Adiabatic Flame Temperature with Preheated Air

If the air is preheated, then we may no longer consider the combustion air temperature equal to the reference temperature, though T f = T ref  still applies. Then Equation 2.90 becomes

⎛  α w ⎞  Δ H c − + T AFT − Tref  = ⎜ T T   ( ) ⎝ 1 + α w ⎟ ⎠  a ref   C p (1 + α w )

(2.100)

or in dimensionless form, T AFT − T ref  Ta − T ref  

⎛  α w ⎞  Δ H c =⎜ + ⎝ 1 + α w ⎟ ⎠  C p (1 + α w ) (T a − T ref  )

(2.101)

⎛  α w ⎞  Δ H c + − + T T  T ref  ( a ref )  ⎟  + α 1 ⎝  C p (1 + α w ) w ⎠ 

(2.102)

and T AFT  = ⎜

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Note that if  T a = T ref , then the first term on the right vanishes and the equation reduces to Equation 2.93.

Example 2.11 Adiabatic Flame Temperature with Preheated Air Problem statement: Repeat the calculation of Example 2.10 with 400°C air preheat. Solution: This is very high air preheat — about as high as it gets in industry. The calculation of interest likely references a hightemperature reactor such as a hydrogen reformer. In the interest of efficiency, one must recover this energy by heat exchange with some process stream. One such convenient stream is the inlet combustion air.

To begin, we will convert the air preheat temperature from the problem statement to degrees Fahrenheit so that we have directly comparable results with the previous example: 400[°C](1.8)[°F/°C] + 32[°F] = 752°F. Then from Equation 2.102 we have

⎛  α w ⎞  Δ H c + T  − + T AFT  = ⎜ T T  ( ) ⎝ 1 + α w ⎟ ⎠  a ref   C p (1 + α w ) ref  The latter term is the adiabatic flame temperature without air preheat. So the results will differ only by the first term:

⎛  α w ⎞  ⎛  19.85 ⎞  − = T T  ⎜⎝ 1 + α w ⎟ ⎠ ( a ref )  ⎜⎝ 1 + 19.85 ⎟ ⎠ (752 − 60) °F = 659°F Adding this product to the previous example’s results gives 4280°F + 660°F = 4940°F at 15% excess air and 4460°F + 660°F = 5120°F at 10% excess air These results ignore dissociation, which would lower the temperatures. Notwithstanding, as the problem shows, air preheat can significantly increase the flame temperature.

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2.6

173

Mechanical Energy Balance

Besides thermal energy, combustion equipment makes use of mechanical energy. Mechanical energy for burners primarily includes air and fuel flow under pressure. Conservation of energy allows us to relate pressure and flow.

2.6.1

Work Terms

Since area times distance is volume ( Ad = V ), pressure times volume (PV) has units of [ML2/θ2]. Therefore, PV work is an energy term. As we have stated, any energy or work term will have dimensions of [ ML2/θ2]. For example, the kinetic energy of a moving body or fluid is 1/2 mv2 [ML2/θ2], and the potential energy of a fluid under gravitation is mgh [ML2/θ2], where g is the gravitational constant [L/θ2] and h is the height of the fluid column [ L]. We shall consider three kinds of mechanical energy: pressure–volume work ( PV ), gravitational work (mgh), and kinetic work (1 2 mv2 ), each having units of [ML2/θ2]. If there are no other sources or sinks of energy, then the mechanical energy balance comprises all the energy the system has to perform work. Combustion reactions conserve energy. Therefore, in going from state 1 to state 2, we have P1V +

1 2 1 mv1 + mgh1 = P2V + mv22 + mgh2 2 2

(2.103)

Since mass [M] appears in all energy terms, each is an extensive variable. To cast them as intensive variables, we normalize by the mass. Specific energy is the term for energy normalized by mass. Dividing by m, the specific energy terms become P1

ρ

v12 + + gh1 = P2 ρ 2

v22 + + gh2 2

or more succinctly,

ΔP + Δv 2 + gΔh = 0 ρ 2

(2.104)

This famous equation, the Bernoulli equation,8 applies to steady flow and neglects nonidealities such as compressible flow or friction. Nonetheless, it is a very useful starting point and it applies to real situations with some modifications, which we will discuss shortly.

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The reader should note carefully the distinction between Δv 2 = v22 − v12 and 2 v = (v2 − v1 )2. The above equation uses the former designation and meaning. The units for Equation 2.104 are not energy units; they are specific energy units [L2/θ2], often referred to as head. For example, 30 ft of head actually denotes 30 ft-lbf/lbm. It is the equivalent potential energy of a 1-lb weight at a height (head) of 30 ft above a reference plane. Notwithstanding, industry convention calls Equation 2.104 the mechanical energy balance, not a head  balance, or specific energy balance.

2.6.2

Theoretical Mechanical Models

In this section, we derive theoretical models relevant to combustion equipment. We begin from first principles of momentum, mass, and energy conservation.

2.6.2.1 Units of Pressure From Newton’s second law, gravitational force (F) is equal to mass (m) times gravitational acceleration ( g): F = mg

(2.105)

The SI unit for force is the Newton (N), N = 1 kg m/sec 2. The customary unit is the pound-force (lbf). All force units have dimensions of [ ML/θ2]. By definition, pressure (P) is force divided by area ( A): P = F/A

(2.106)

The SI unit for pressure is the kilopascal (kPa), kPa = 1000 kg/m sec 2; the customary unit is the pound-force per square inch (psi). All pressure units have dimensions of [M/Lθ2]. There are three different references for pressure, e.g., psi absolute (psia), psi gauge (psig), and psi differential (psid). • The absolute pressure is the only pressure with a true zero — it is a ratio variable. The gauge pressure has a relative zero at 1 atm at sea level (e.g., 101 kPa or 14.7 psig). • The gauge pressure is equal to absolute pressure minus atmospheric pressure. Gauge pressures are interval variables; therefore, ratios of  psig or kPa(g) do not necessarily have meaning. • The final pressure category is the differential pressure. It is also an interval variable. In the case of differential pressure, the zero is the downstream pressure. Thus, if a fuel jet at 40 psia passes across an orifice such that its downstream pressure is 22 psia, then one may measure 18 psid (40 – 22 = 18).

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Note that so long as the upstream and downstream pressures have the same relative zero, it makes no difference which unit one uses to derive the differential measure. For example, to repeat the previous calculation in gauge units, 25.3 psig – 7.3 psig = 18 psid.

2.6.2.2 Natural Draft Model  Equating PV work to kinetic energy determines the draft pressure and maximum draft velocity that a flame or furnace can create. Natural draft is the pressure gradient created by a temperature (i.e., density) difference. In our case, the temperature–density difference is between the system (furnace) and its surroundings (ambient air). Ultimately, draft units are pressure units, no matter how well disguised. The two most common are inches water column (“w.c., in. w.c., i.w.c., or "H 2O) and millimeters water column (mm w.c. or mm H2O), referring to the differential pressure required to elevate a column of water so many inches or millimeters. Since we wish to know what velocity or draft pressure is possible due to natural draft, we equate the terms PV = 1 2 mv 2 = mgh . In order to cast the equation in terms of intensive variables, we divide by m: P

ρ

=

v2 =  gh 2

(2.107)

Equation 2.107 has units of [L2/θ2].

2.6.2.3 Draft Pressure in a Furnace We shall solve first for the draft pressure, by multiplying Equation 2.107 by ρ:

ρ v2 = ρ gh P= 2

(2.108)

To specify the gradient, we apply the difference operator,

Δ 21P = Δ 21ρgh

(2.109)

and obtain P2 − P1 = (ρ2 − ρ1 ) gh . Note that g is invariant over the height of the furnace, and h is implicitly Δh because one always measures height relative to a floor or zero datum. (We shall use state 2 to signify the furnace and state 1 to signify the surroundings. However, when the difference in states is obvious from context, we shall omit the subscripts.) Thus, a negative draft pressure corresponds to a furnace pressure that is lower than the surroundings. By the ideal gas law

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⎛  ⎞  Δρ = ⎛ ⎜ PM ⎞ ⎟ ⎜ 1 − 1 ⎟  ⎝  R ⎠ ⎝ T2 T 1 ⎠  the final equation becomes

⎛  ⎞  Δ P = ⎛ ⎜ PM ⎞ ⎟ ⎜ 1 − 1 ⎟  gh ⎝  R ⎠ ⎝ T2 T 1 ⎠ 

(2.110)

Therefore, if we know the furnace temperature (T 1), the ambient temperature (T 2), and the height of the furnace (h), we may determine the draft at the floor or any other furnace elevation. Often, heater vendors express draft units in terms of an equivalent column of water at standard temperature and pressure under normal gravitation. There are 406.8 in. w.c. in a standard atmosphere. This unit looks more like a distance rather than a unit of pressure, but Equation 2.109 shows what is unstated, with ρ being the density of water and g the gravitational acceleration. A typical draft pressure is –0.5 in. w.c. It is amazing what this seemingly feeble force per unit area can do.

Example 2.12 Calculation of Draft Pressure Problem statement: What is the maximum draft pressure at the floor of a 10-m furnace operating at 800°C? Presume that the control system adjusts the pressure at the top of the furnace to a draft of 3 mm w.c. via a stack damper. Solution: From Equation 2.110, at 1 atm of pressure (101 kPa) we have

⎛  ⎡ g ⎤ ⎞  ⎜ 101 ⎡⎣ kPa⎤⎦ 29 ⎢ mol ⎥ ⎟ ⎛  ⎞  ⎡ m ⎤ 1 1 ⎣ ⎦ ⎜ ⎟  ΔP = ⎜ − 9.8 ⎢ 2 ⎥ 10 ⎡⎣ m ⎤⎦ ⎜ ⎟  3 ⎟  ⎡ m ⋅ Pa ⎤ ⎝ (800 + 273) ⎡⎣ K ⎤⎦ ( 25 + 273) ⎡⎣ K ⎤⎦ ⎠  ⎣ s ⎦ ⎜⎜ 8.314 ⎢ ⎟ ⎟  ⎥ ⎝  ⎣ mol ⋅ K ⎦ ⎠  = −54.6 ⎡⎣ Pa⎤⎦ = −5.6 ⎡⎣ mm w.c. ⎤⎦ The negative sign reminds us that the pressure is below atmospheric. Adding the –3 mm w.c. pressure (draft) we have at the  bridgewall gives a total of –8.6 mm w.c. pressure at the floor (or 8.6 mm w.c. draft).

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2.6.2.4 Air Velocity Due to Natural Draft  We may solve for the maximum air velocity in terms of the available draft pressure. We are usually interested in the velocity of the air into the furnace; there, the density of the air is approximately that of the surroundings ( ρ). With these assumptions, we obtain ΔP = −ρv 2 2 yielding v = 2 ΔP ρ. The maximum velocity occurs when there is no resistance to flow (dampers wide open, frictionless flow). In practice, one can realize the theoretical draft pressure because it occurs at the no-flow condition, i.e., without frictional losses. However, one cannot attain the maximum theoretical velocity because flowing air is subject to friction. In order to approximate inlet losses, one usually uses a flow coefficient, Co [ ], with values typically ranging from 0.6 to 0.9 for simple orifices. v = Co

2 ΔP

ρ

(2.111)

2.6.2.5 Airflow through a Diffusion Burner  Consider Figure 2.27.

FIGURE 2.27 Simplified airflow analysis of burner. A simplified flow analysis shows the many twists and turns that the air must take. Flow around the center riser has been neglected, and only the restriction of the cone is taken into account. Also, we presume that normal operation of the  burner is with the damper wide open, so we neglect the flow disturbance across the damper. The velocity at Pt 9 is presumed to be expanded by the furnace temperature to a higher velocity. The momentum of the fuel from the center riser (which will help inspire airflow) is also neglected for this simplified analysis. Also, as the duct is short, we presume that the skin friction (N F ) is negligible.

The continuity equation gives the mass flow in terms of flow area and density:

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Modeling of Combustion Systems: A Practical Approach  = ρAv m

(2.112)

 is the mass flow [M/θ], A is the area [L2] of the controlling resistance where m (e.g., burner throat or damper), and v is the air velocity [L/θ]. Combining Equations 2.108 and 2.112, we obtain the mass flow relation for a given draft  = A 2ρ1Δ 21P. However, this is the maximum possible mass flow. pressure: m The actual mass flow will be less because when gases flow across an orifice, the flow is not perfectly streamlined. Some of the energy goes to sending wakes of gas around in circles (known as eddies). So we augment the equation with a loss coefficient, Cb:  = Cb A 2ρ1Δ 21P m

(2.113)

Theoretically, we can determine Cb from the burner geometry alone using loss factors for various geometries, but as the figure shows, the flow path in  burners can be quite complicated. Appendix B, Table B.3 lists some examples of flow obstructions. The K factor is a dimensionless parameter that modifies the kinetic energy term to account for friction. We begin with Equation 2.109 and modify it to include two important kinds of resistance: skin friction and form drag. Skin friction is the resistance due to surfaces parallel to the flow path, while form drag is the resistance due to bodies and fittings normal to the flow path. Equation 2.109 becomes

ΔP = v 2 ⎛ 1 + N  L + F ρ 2 ⎜⎝  D

⎞  K k ⎟  ⎠  k =1 n

∑

(2.114)

where N F is the frictional factor [ ] and K k  [ ] represents the loss for the k th fitting or obstruction to flow. Rewriting the equation in terms of velocity gives v=

2 ΔP

1 L 1 + N F + D

ρ

n

∑ K 

(2.115)

k 

k =1

In Equation 2.115, the first quantity to the right of the equal sign is the  burner coefficient for natural draft, Cb. That is, 1

Cb = 1 + N F

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L + D

n

∑ K 

k 

k =1

(2.116)

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Thus, we may also write Equation 2.115 as

ΔP =

1 ρv 2 Cb2 2

(2.117)

Example 2.13 Calculation of Pressure Loss in a Burner Problem statement: Consider the burner shown in Figure 2.27. Calculate the pressure loss resulting from an airflow through the  burner of 3 m/sec. Presume that ambient temperature is 25°C and that the furnace temperature is 800°C. What is the burner airflow coefficient? Solution: Figure 2.27 shows a simplified analysis of the effects. Equation 2.116 becomes

Cb

=

1

⎛ 800 + 273 ⎞  1 + 0 + 3.5 + 1.0 ⎜ ⎝  25 + 273 ⎟ ⎠ 

= 0.35

Note that we have adjusted the last term for the greater velocity of the expanded gas because the Cb references the ambient temperature and pressure. Note also that Cb is much lower than Co for simple orifices because of the tortured flow path and sudden expansion caused by heating of the air from the furnace. We have neglected skin friction losses through the duct because losses from expansions and flow direction changes dwarf them. Then, from Equation 2.117

ΔP =

1 ρv 2 Cb2 2

The density of air is approximately 1.18 kg/m3 at 25°C. Therefore, 2

ΔP =

⎡ kg ⎤ ⎛  ⎡ m ⎤⎞  1.18 ⎢ 3 ⎥ ⎜ 3 ⎢ ⎥⎟  ⎣ m ⎦ ⎝  ⎣ s ⎦⎠ 

1

( 0.35)

2

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2

= 43.3 ⎡⎣ Pa ⎤⎦ = 4.4 ⎡⎣ mm w.c.⎤⎦

180

Modeling of Combustion Systems: A Practical Approach The foregoing calculations provide a good a priori estimate of   burner capacity, but accurate airside capacity relations require experimental determination due to a number of factors. First, the maximum resistance to flow occurs in the burner throat, so one should evaluate the density and pressure drop there. A better procedure would be to integrate along the entire flow path. However, the factors vary along the flow path and depend in a complex way on the burner design. Reradiation from the burner tile and the burner internals and momentum effects from the fuel jets affect the mass flow of air into the burner. Therefore, the practical procedure is to evaluate the factors at convenient locations and adjust Cb to match the actual measured flow. Generally, we take  A to be the burner throat area (without restriction or baffle), ρ is the air density at the inlet, and ΔP is the natural draft pressure at the furnace floor. A semiempirical correlation is an input–output relation whose model form is theoretically derived, but which has one or more experimentally determined coefficients (adjustable parameters). For example, if we determine Cb experimentally, then Equation 2.117 is a semiempirical equation. Figure 2.28 gives an example of capacity curves based on a semiempirical model. Often, the manufacturer presents the curves on a log–log scale to linearize the relation. This permits easier extrapolation and interpolation. To correct burners to actual conditions, one may make use of the following dimensionless relation:

⎛  Δ ref P ⎞ ⎛  Pref ⎞ ⎛  ρref ⎞   ⎛  T  ⎞ ⎛ Cb ,ref  ⎞ ⎛  Aref  ⎞ ⎛  1 + ε ⎞ ⎛  Q ⎞  ⎜⎜ ΔP ⎟ ⎟ ⎜⎜ P ⎟ ⎟ ⎜⎜ ρ ⎟ ⎟ ⎜ T  ⎟ ⎜ C ⎟ ⎜  A ⎟ ⎜ 1 + ε ⎟ ⎜  ⎟  = 1 ⎝  ⎠ ⎝  ⎠ ⎝  ⎠ ⎝  ref  ⎠ ⎝  b ⎠ ⎝  ⎠ ⎝  re f  ⎠ ⎝ Qref  ⎠ 

(2.118)

where subscript ref refers to the reference conditions required by the capacity curve. For example, suppose the burner capacity is 8 MMBtuh (millions of British thermal units per hour) and one wants to know what will happen to the burner capacity if the ambient air temperature decreases from 100 to 10°F, ceteris paribus. This may reference daytime summer temperatures to nighttime winter temperatures for a particular locality. Then all ratios in Equation 2.118 become unity except for the Q and T  ratios. Therefore, Equation 2.118 reduces to

⎛  T  ⎞ ⎛  Q ⎞  ⎜ T  ⎟ ⎜ Q ⎟  = 1 ⎝  ref  ⎠ ⎝  ref  ⎠ 

© 2006 by Taylor & Francis Group, LLC

Introduction to Combustion For model XYZ burner requiring 15% excess air at sea level and 60˚F

10 9 8 7    h   u    t    B    M    M  ,   e   s   a   e    l   e   r    t   a   e    H

181

20 19   s   e    h   c   n    i  ,   e   z    i    S    t   a   o   r    h    T   r   e   n   r   u    B

6 5 4 3

2

18 17 16 15 14 13 12

1

.1

.2

.3

.4

.5 .6 .7 .8 .9 1

2

3

4

5

6 7 8 9 10

Air side pressure drop

FIGURE 2.28 A typical airside capacity (cap) curve. The log-log scale linearizes the square root relation  between burner capacity (heat release) and pressure drop. Thus, the capacity curve allows one to easily select the appropriate burner size as a function of the available pressure drop. One must correct the cap curve for altitude, air temperature, or excess air conditions that are different than the stated reference conditions. Due to differences among burner coefficients for various models, the cap curve is specific for sizes within a particular burner family only.

Solving for Q one obtains T ref    Q = Qref  T  Therefore, the burner capacity rises to 460 + 100 Q = 8 ⎡⎣ MMBtuh ⎤⎦ 460 + 10

= 8.7 ⎡⎣MMBtuh ⎤⎦

To correct for barometric pressure at elevation, one may use the relation of Equation 2.107 in differential form and integrate it  between some reference height ( zref ) and some elevation z: P

 z

Pref

 zref

∫

∫ 

− dP =  g dh ρ

© 2006 by Taylor & Francis Group, LLC

 

182

 Modeling of Combustion Systems: A Practical Approach The negative sign is necessary because barometric pressure decreases with elevation. Substituting Equation 2.74 into this leads to

− RT   M

P

∫

Pref

dP =  g P

 z

⎛  P ⎞  Wa g =− dh → ln ⎜ ( z − zref  ) → ⎟  P RT  ⎝  ref  ⎠ 

∫ 

 zref

 

Wg ⎛ Pref  ⎞  Wa g Pref  ( z− z RT  = =e ln ⎜  z − zref  ) → ( ⎟  P ⎝  P ⎠  RT  a

Pref  P

=e

ref 

)

→

Wa g ⎛  z − zref  ⎞  ⎜ ⎟  RT  ⎝  2 ⎠ 

(2.119)

One can then substitute Equation 2.119 into Equation 2.118 to correct for barometric pressure at another elevation. To use Equation 2.119, one must presume some average temperature for the air column, say 60°F.

2.6.2.6 Airflow through Adjustable Dampers We may model a damper as a variable orifice. Consider the damper shown in Figure 2.29. Here the free damper area ( A = Ao –  AN ) is related to the damper angle normal to the flow direction by the following formula: = m

Cd A 2ρΔP 1 − cos θ

(2.120)

. m = ρ Av1 = ρ A N v2 = ρ A(1–cos θ)v2 . m=

C o A

1–cos θ

√ 2ρΔP

θ

AN =A(cos θ)

Ad 

Ao

FIGURE 2.29 Single-blade damper. The free damper area ( A) is the total duct area ( Ao) minus the normal area projection of the damper ( AN ). © 2006 by Taylor & Francis Group, LLC

Introduction to Combustion

183

where Cd is the damper coefficient, which we presume constant, and A is the free area of the duct with the damper fully open. We reference the damper angle normal to the flow so that 0° corresponds to a closed damper and 90° (π/2) to a fully open one. We may express Equation 2.120 in the following dimensionless form:

⎧⎪ m ⎛  1 ⎞  1 ρv 2 1 ⎫⎪ = = = C ⎨ ⎬ d ⎜⎝ 1 − cos θ ⎟ ⎠  Δ 2 P ρ Δ 2 2  A P N  ⎪⎩ Eu ⎪ ⎭

(2.121)

Therefore, a plot of 1 ( 2 N Eu ) vs. 1 (1 − cos θ) will yield a line of slope Cd . We refer to 1 (1 − cos θ) as the damper function. It generally does a good job of linearizing airflow through a damper (Figure 2.30). 3500

80° 70° 60° ▲

3000

   h    /    m    N  ,   w    o    l    f    r    i    A

40°

30°

▲ ▲

=20 mm ΔP=20 ▲

■

   3

50°

θ

airflow

water column

■ ■

2500

▲

■

◆

◆

ΔP=15

mm w.c.

◆

2000

■

◆

ΔP=10

mm w.c.

◆

▲

1500

■

Damper function 1/(1-cos θ), θ=deg. from vertical

◆

1000

1

2

3

4

5

6

7

8

FIGURE 2.30 Airflow vs. damper function. The damper function does a good job of linearizing airflow across a damper.

2.6.2.7 Unknown Damper Characteristics For a complex damper characteristic having a sigmoid shape, one may use an empirical function to linearize it. The squash or  probit function is an empirical equation that does a good job of linearizing the flow characteristic: ex  y = S(x) = 1 + ex

−∞ < x < ∞

0 < S(x) < 1

The inverse of the squash function is the logit function:

© 2006 by Taylor & Francis Group, LLC

(2.122a)

184

 Modeling of Combustion Systems: A Practical Approach

⎛   y ⎞  ⎝ 1 − y ⎟ ⎠ 

x = L( y) = ln ⎜

0 1.85 to 1.89), we will have compressible flow and sonic velocity at the fuel orifice.

2.6.2.10 The Fuel Capacity Curve Revisited  Mass is a conserved quantity in combustion reactions as presented earlier:  = ρAv m

(2.112)

If one repeatedly integrates with respect to velocity, one obtains a heuristic for deriving the conserved quantities. For example, mass integrated with  → mv  . respect to velocity gives momentum, also a conserved quantity: ∫ mdv

© 2006 by Taylor & Francis Group, LLC

Introduction to Combustion

187

In a similar manner, if momentum is integrated with respect to velocity, then  → 1 2 mv  2 . Repeated application generates one obtains kinetic energy: ∫ mv  3 6 , mv  4 24, … mv  n n! . However, practitioners have not found the series mv any physical relevance for these higher homologs. Notwithstanding, the heuristic is convenient for remembering the conserved quantities m, mv, and 1/2 mv2, and therefore for deriving the respective dimensionality for mass, momentum, and energy: [M], [ML/θ], and [ML2/θ2]. When the pressure ratio exceeds the critical pressure ratio, then the exit velocity becomes sonic. No matter what the upstream pressure in excess of  the critical pressure, the velocity from a simple orifice cannot exceed sonic velocity. (It is possible to obtain supersonic flow from a converging–diverging nozzle; however, manufacturers do not include them in burner designs  because they are expensive to machine and because they create shock waves and loud noise. However, some steam-assisted flares use supersonic nozzles to increase the air entrainment rate.) Despite the limitation of sonic velocity, the mass flow always increases with increasing upstream pressure, even for simple orifices in choked (sonic) flow. From Equation 2.112 with v = c and constant area, the resulting increase in mass flow can only result from compression of the fluid to higher density. The Mach number relates the actual velocity to the sonic velocity: N  M

=v c

(2.127)

If  N  M