421 5 Semiempirical Models Chapter Overview implications it had for flow of air and fuel in burners. This allowed us to make good physical models requiring one or no adjustable parameters. In this chapter, the physics become more complicated and involve thermodynamic, chemical, and kinetic quantities. For this reason, our semiempirical models will require more adjust- able parameters. Notwithstanding, such expressions will be capa- ble of correlating complicated behavior with engineering accuracy. With the techniques the reader has already learned, he will be able to regress such semiempirical models handily from facility data or planned experiments. We begin the chapter with a discussion of various NOx forma- tion mechanisms. Along the way, we introduce the reader to what- ever thermodynamics or kinetics the situation may require. NOx formation naturally progresses to NOx reduction, so here we discuss nearly a dozen such strategies. These lead to a generic semiempirical model for NOx and NOx reduction. We then develop a parallel approach for CO. Throughout, we focus on the practical. The heat flux profile from a combustion system has now gained importance. Modeling of such profiles is important for certain reactors such as ethylene cracking units (ECUs) and hydrogen and ammonia reformers. We develop the general three-parameter equation from a simplified analysis of fuel jets and the heat bal- ance. We can reduce this to a two-parameter model by normaliz- ing the heat flux. Such a model also allows us to consider the qualitative response of heat flux to various operational factors. Two important responses we consider are run length between decoking cycles and process efficiency. The heat flux model also allows us to develop a similarity criterion for test and field units. © 2006 by Taylor & Francis Group, LLC In Chapter 2, we discussed the mechanical energy balance and 422 Modeling of Combustion Systems: A Practical Approach Next, we consider how one may measure flame length. We discuss a flame model and show its semiempirical analog. Some fuel and combustion characteristics also cause other problems, such as plume formation and corrosion through acid dew point elevation. We treat these briefly in the final section. By the con- clusion of this chapter, the reader will have an arsenal of practical techniques for modeling important aspects of fired units. 5.1 NOx and Kinetics 5.1.1 NOx: Some General Comments NOx produced from combustion is a terribly inefficient process. The high- temperature reaction of oxygen and nitrogen produces most of it. The com- bustion air contains both. If we were trying to manufacture NOx in this way, we would go broke. For every million volumes of air, we would produce only a few hundred volumes of nitric oxide (NO) and only a few tens of volumes of nitrogen dioxide (NO 2 ). Collectively, we refer to these as NOx. Notwithstanding this paucity, 100 ppm is above regulated limits for most combustion sources; at the time of this writing, limits are 25 to 50 ppm for most processes and ever declining. 5.1.2 The Thermal NOx Mechanism At very high temperature, we may cause nitrogen to react with oxygen: N 2 + O 2 → 2NO (5.1) Many elemental reactions contribute to NO formation. In the simplest analysis, we may look at two: 1 O + N 2 = NO + N (5.2) N + O 2 = NO + O (5.3) N 2 + O 2 → 2NO Oxygen is the second most reactive gas in the periodic table. Only fluorine is more reactive. Oxygen dissociates relatively easily under high heat from a diatomic molecule to an atomic entity: 1/2 O 2 = O (5.4) © 2006 by Taylor & Francis Group, LLC Semiempirical Models 423 Atomic oxygen is very reactive and can rupture the very strong N≡N triple bond. This frees a nitrogen atom in the process (Reaction 5.2). Atomic nitro- gen goes on to attack the ambient oxygen (Reaction 5.3). The net reaction produces NO (Reaction 5.1). Since Reaction 5.2 involves the rupture of N≡N, it is the slowest reaction and the rate-determining step. The rate-determining step is the slowest reac- tion in a chain that paces the entire sequence. We always look for these where possible because they reduce the analysis to fewer equations. Now we may write the rate of the forward reaction in terms of the rate-limiting quantity for a differential amount of substance: where k f is the forward reaction rate constant and k r is the reverse reaction rate constant. The brackets [ ] denote the concentration of the enclosed spe- is a general kinetic expression. The forward reaction involves a forward rate constant and the mathematical products of the reactant concentrations. From this, we subtract the rate of the reverse reaction involving the reverse rate constant and the mathematical product of the product concentrations.) NO and N are present in very low concentrations. Therefore, we can safely presume that the forward reaction will dwarf the reverse because N 2 is present in a thousand times greater concentration. The equation simplifies to The wet concentration of N 2 is practically constant throughout the com- bustion reaction; subtracting a few ppm of NO via Reaction 5.2 from ~80,000 ppm (80%) nitrogen makes virtually no difference. However, the atomic oxygen concentration is a different story. To solve for this concentration, we presume an equilibrium relation between molecular and atomic oxygen (called partial equilibrium because we consider this part independent of the whole fabric of concurrent equilibrium reactions). Equation 5.4 leads to the following equilibrium relation according to: (Equilibrium constants always involve the mathematical product of the reac- tion products divided by the product of the reactants. Stoichiometric coeffi- cients become exponents. Also, the equilibrium reaction is the ratio of the forward to reverse reactions: .) Solving for [O] and combining con- stants in k, we derive d dt kk fr NO ON NON 2 ⎡ ⎣ ⎤ ⎦ = ⎡ ⎣ ⎤ ⎦ ⎡ ⎣ ⎤ ⎦ − ⎡ ⎣ ⎤ ⎦ ⎡ ⎣ ⎤ ⎦ d dt k f NO ON 2 ⎡ ⎣ ⎤ ⎦ = ⎡ ⎣ ⎤ ⎦ ⎡ ⎣ ⎤ ⎦ K O = ⎡ ⎣ ⎤ ⎦ ⎡ ⎣ ⎤ ⎦ O O 2 1 2 Kkk fr = © 2006 by Taylor & Francis Group, LLC cies. (See Appendix G for a useful summary of formulating rate laws.) (This 424 Modeling of Combustion Systems: A Practical Approach (5.5) Now we may integrate this relation. Presuming [N 2 ] is approximately constant, we have Finally, we note that k (like the equilibrium constant, K ) has an Arrhenius relation to temperature . With this substitution, we have the follow- ing form: (5.6) where A is a preexponential coefficient termed the frequency factor , and in this case it has the units of √ [ L 3 / N ], and b is a constant [ T ] related to the activation energy of the reaction. Equation 5.6 gives us a basic relation for NOx production. From it, we may deduce that NOx is a strong (exponential) function of temperature, and a weaker function of oxygen concentration and time. 5.1.3 The Fuel-Bound Nitrogen Mechanism Most refinery fuels and natural gas do not contain significant amounts of nitrogen bound in the fuel molecule. If that is the case, then the thermal NOx mechanism described above will account for most of the NOx. If the fuel contains significant nitrogen compounds, the fuel-bound mechanism predom- inates. It is important to reemphasize that the fuel-bound mechanism sub- sumes nitrogen bound as part of the fuel molecule. Diluting fuel gas with diatomic nitrogen will not increase NOx, nor result in fuel-bound NOx, because it is not chemically bound to the fuel. In fact, it will likely have the opposite Some compounds can elevate NOx via the fuel-bound mechanism. They include the following: • Ammonia (NH 3 ) and related compounds (e.g., ammonium hydroxide also known as aqua ammonia — NH 4 OH and urea (NH 2 ) 2 CO). • Amines — there are three kinds. – Primary amines have one organic group attached to the nitrogen (R–NH 2 ) where R stands for an organic group. In fuels, there are usually hydrocarbons (e.g., if R is CH 3 CH 2 then R–NH 2 is ethyl- amine CH 3 CH 2 NH 2 ). d dt k NO ON 2 ⎡ ⎣ ⎤ ⎦ = ⎡ ⎣ ⎤ ⎦ ⎡ ⎣ ⎤ ⎦ 2 NO N O 2 ⎡ ⎣ ⎤ ⎦ = ⎡ ⎣ ⎤ ⎦ ⎡ ⎣ ⎤ ⎦ ⌠ ⌡ ⎮ kdt 2 () (/ ) Ae bT− NO N O 2 ⎡ ⎣ ⎤ ⎦ = ⎡ ⎣ ⎤ ⎦ ⎡ ⎣ ⎤ ⎦ − ⌠ ⌡ ⎮ ⎮ Ae dt b T 2 © 2006 by Taylor & Francis Group, LLC effect for reasons we will discuss in Section 5.2.6 and elsewhere. Semiempirical Models 425 – Secondary amines have two organic groups attached to the ni- trogen (R 1 R 2 NH). An example would be methyl ethylamine (MEA), CH 3 NHCH 2 CH 3 . Since MEA is used in some refinery operations, it may be entrained as a mist into the fuel supply, termed amine carryover. This is expensive because the MEA needs to be replaced, can exacerbate corrosion, and, most impor- tantly for the topic of this chapter, can greatly elevate NOx from the fuel-bound mechanism. – Tertiary amines have no hydrogen atoms attached to the nitro- gen. They may be of two types: R 1 R 2 R 3 N, for example, dimethyl ethylamine (DMEA), (CH 3 ) 2 NCH 2 CH 3 , or aromatic types, such as pyridine C 6 H 5 N. Pyridine is often used as a surrogate to spike the fuel-bound nitrogen content of a base oil for the purpose of experimental investigations into fuel-bound NOx formation. Heavy fuel oils contain related compounds that elevate NOx. As far as the NOx chemistry is concerned, it makes little difference how the fuel binds the nitrogen because it will pyrolyze to form HCN and CN fuel fragments; that is, C n H m N → CN + HCN from the high heat radiating from the downstream flame. The pyrolized fragments oxidize to hydrogen and carbon monoxide. Finally, late in the process, the hydrogen and carbon monoxide oxidize to H 2 O and CO 2 . This provides the bulk of the heat to pyrolyze new fuel. If there is no nitrogen in the fuel, then there will be no CN or HCN and there can be no fuel-bound NOx. Returning to NOx formation, CN and HCN exist in partial equilibrium: H + CN ↔ HCN. One fate of these fuel fragments is to generate atomic nitrogen, which can oxidize to NO: HCN = CH + N (5.7) Then, NO formation occurs via N attack on diatomic oxygen; we gave this relatively facile reaction earlier in Equation 5.3: N + O 2 = NO + O Because Equation 5.7 avoids the rupture of the N≡N triple bond, it has a lower energy pathway for the generation of atomic N. Therefore, the pres- ence of fuel-bound nitrogen greatly accelerates NOx kinetics. To formulate a rate law, let us presume the following in Equation 5.3: • The fuel pyrolysis is fast. • The atomic nitrogen concentration is proportional to the concentra- tion of nitrogen in the starting fuel. • Reaction 5.3 is the rate-determining step. © 2006 by Taylor & Francis Group, LLC 426 Modeling of Combustion Systems: A Practical Approach Then a possible rate law is (5.8) 5.1.4 The Prompt NOx Mechanism If there is no fuel-bound nitrogen in the fuel, then there can be no fuel-bound NOx. In such a case, the thermal-bound NOx mechanism discussed earlier will generate most of the NOx. However, there is still another mechanism. 1 Suppose that a fuel fragment attacks diatomic nitrogen. C n H m → CH + C n–1 H m–1 CH + N 2 = HCN + N (5.9) Reaction 5.9 is clearly rate limiting because it involves the rupture of an N≡N triple bond by a fuel fragment. However, Reaction 5.2 is much faster. But on the fuel side of the flame front, where there is no oxygen, Reaction 5.9 is the only real possibility. As one might imagine, N from this source is meager and prompt NOx forms only 10 to 20 ppm NO at most. However, modern ultra-low-NOx burners can produce 20 ppm NOx under some prac- tical conditions. In such a case, prompt NOx must be a substantial contrib- utor to the total NOx budget. With the following presumptions, we may establish a rate law: • The fuel pyrolysis is fast. • CH is proportional to the number of carbons in the fuel molecule. • Reaction 5.9 is the rate-limiting step. This leads to the following rate equation: (5.10) Since [N 2 ] is essentially constant, prompt NOx is proportional to the amount of carbon in the fuel. Note that it is impossible for pure hydrogen to form any prompt NOx even though it may generate much thermal NOx (owing to hydrogen’s very high flame temperature). In fact, the thermal NOx mechanism causes hydrogen-combusting vehicles to generate more NOx than even gasoline engines. d dt k nm NO CH N O 2 ⎡ ⎣ ⎤ ⎦ = ⎡ ⎣ ⎤ ⎦ ⎡ ⎣ ⎤ ⎦ d dt k nm n NO CH N ⎡ ⎣ ⎤ ⎦ = ⎡ ⎣ ⎤ ⎦ ⎡ ⎣ ⎤ ⎦ 2 © 2006 by Taylor & Francis Group, LLC Semiempirical Models 427 5.1.5 Chemical Kinetic Effects for NOx in Diffusion Flames Diffusion flames require mixing of the fuel and air external to the burner. In such a case, additional air not only increases the oxygen concentration, but aids in fuel mixing. The additional facility of oxygen transport to the flame zone increases the availability of oxygen for NOx production. Hence, addi- tional air increases NOx. At some point, there is sufficient oxygen transport, and the further addition of air does not increase the NOx production rate enough to offset the cooling effect of more air. At this point, the NOx begins to fall. This phenomenon occurs between 5 and 8% oxygen for most indus- trial flames. At high bridgewall temperatures oxygen as high as 8% may still increase NOx because the kinetics are still fast, whereas for low bridgewall temperatures, oxygen above 5% tends to reduce NOx production, as it is easier to cool the flame. With premixed flames, additional oxygen does not significantly enhance the availability of oxygen for the NOx reaction — at least not enough to offset the cooling effect of the additional air. Therefore, additional air decreases NOx production in premixed flames. We begin by considering the NOx response for diffusion flames to oxygen, temperature, and fuel composition. Then we shift our attention to the behav- ior of premixed flames regarding these factors. 5.1.5.1 NOx Response to Air in Diffusion Flames The addition of oxygen has two effects on NOx: the first is the pure dilution measured NOx concentration. We can account for this exactly using dilution correction. A mass balance grounds the dilution correction on solid theoret- ical footing without the need for any adjustable parameters in the final equations. Additionally, oxygen participates in NOx chemistry. Unlike dilu- tion correction, estimation of this effect (chemical kinetic) necessarily involves some empiricism: the amount of excess air affects the availability of oxygen and the flame temperature. Small increases in excess air do not abstract significant heat from the flame and they aid in flame mixing, thereby increasing the availability of oxygen and increasing the NOx (see Equation 5.6). Theoretically, the local flame stoichiometry remains at unity regardless of the global oxygen concentration for a diffusion flame. This would certainly be so for normal variations in excess air between 0 and 5%. If this is so, then we may place the temperature effect outside the integral. Nitrogen will vary little for small differences in oxygen, and we place it outside the integral as well. This was the basis for Equation 5.6, developed earlier: NO N O 2 ⎡ ⎣ ⎤ ⎦ = ⎡ ⎣ ⎤ ⎦ ⎡ ⎣ ⎤ ⎦ − ⌠ ⌡ ⎮ ⎮ Ae dt b T 2 © 2006 by Taylor & Francis Group, LLC effect of adding air. As we have seen in Section 2.4.10, this acts to reduce the 428 Modeling of Combustion Systems: A Practical Approach Making use of the ideal gas law, we may recast Equation 5.6 in terms of mole fractions. This gives or (5.11) We do not know precise oxygen–time history in a diffusion flame. How- ever, we may define a dimensionless reaction coordinate, χ, such that (5.12) and , where y O2,b is the wet mole fraction of oxygen in the windbox or burner plenum (initial O 2 ), y O2,wet is the wet mole fraction of oxygen along the reaction coordinate, y O2,g is the wet mole fraction of oxygen in the effluent flue gas (final O 2 ), t is the time along the reaction coordinate, and θ is the total reaction time. Then and . Making these substitutions into Equation 5.11 gives or where (5.13) y P RT Ae y P RT P RT ydt b T NO = − ⌠ ⌡ ⎮ ⎮ ⎮ N2 O2 yAey P RT ydt b T wet wetNO = − ⌠ ⌡ ⎮ ⎮ N2, O2, χ θ = − − = yy yy t bwet bg OO OO 22 22 ,, ,, 01≤≤χ yy y wet b gOO O22 2 1 ,, , =− () +χχdt d=θ χ yAey P RT yyd b T bgNO N2 O2 O2 =− () + − ⌠ ⌡ ⎮ ⎮ θχχχ ,, 1 0 1 yAey Py RT b T b NO N2 O2 =+ () − ⌠ ⌡ ⎮ ⎮ θχζ , –11 0 1 ζ= = y y y y wet b g b O2 O2 O2 O2 , , , , © 2006 by Taylor & Francis Group, LLC Semiempirical Models 429 where y O2,g is the final oxygen concentration (wet) of the flue gas. The integral resolves to (5.14) Note that because ; likewise, because ; there- fore, . For practical combustion problems ζ is close to zero; e.g., for y O2,b = 21% and y O2,g = 3%, ζ = 1/7 and . For small ζ, a two-term Taylor series gives This substitution reduces the equation to If the total reaction time is an Arrhenius function of temperature, then we may combine constants and take the log to give (5.15) There are strong theoretical reasons to use log NOx rather than NOx. For example, data from planned experiments 2 show that the distribution of error is lognormal. Hence, one should use the log transform to correlate NOx data if fitting semiempirical correlations of NOx. Other statisticians have made more general statements about the log transform for environmental data 3 ; such data have zero as their lower limit but no theoretical upper bound. yAey Py RT d b T b NO N2 O2 =− + − () ⎡ ⎣ ⎤ ⎦ +− − θχζχζ , 1111 32 (() ⎡ ⎣ ⎤ ⎦ ⌠ ⌡ ⎮ ⎮ 1 0 yAey Py RT b T b NO N2 O2 = − − ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ − 2 3 1 1 32 θ ζ ζ , ζ<1 yy gbO2 O2,, < ζ>0 y gO2, > 0 01<<ζ ζ 32 118 0<≈ ζ ζ ζ 32 1 1 1 − − ≈+ yAy Py RT e b b T NO N2 O2 = ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ + () − , θζ 2 3 1 ln ln ln ln , yAy Py R T b T b NO N2 O2 = ′ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ −− ′ + 1 2 2 3 11+ () ζ © 2006 by Taylor & Francis Group, LLC 430 Modeling of Combustion Systems: A Practical Approach Therefore, the log-transformed data distribute more normally than untrans- formed data for emissions. From Equation 5.15, a semiempirical equation to correlate NOx with oxy- gen from diffusion flames would have the form where For small ζ, we may even linearize the log function and write (5.16) If there is no flue gas recirculation to the windbox, then y O2,a = y O2,g and . Letting , Equation 5.16 becomes (without FGR) (5.17) Then an estimate for NOx from one oxygen condition to a reference con- dition becomes With suitable adjustment of we may use either wet or dry concentrations for oxygen. Theoretically, which is too low. However, our goal was to find an approximate model and use the data to adjust the parameter. Data for low-NOx burners using dry oxygen values yield , and we shall use . (without FGR) (5.18) ln lnya NO =+ + () 0 2 3 1 ζ aAy Py R T b T b 0 1 2 = ′ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ −− ′ ln ln , N2 O2 ln ya NO =+ 0 2 3 ζ ζ=yy gaO2 O2,, ay a1 23= () ,O2 ln , yaay gNO O =+ 012 ln ,, y y ay y ref g ref NO NO, OO2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ =− () 12 a 1 a 1 2 3 100 21 3=≈()( ), 12 16 1 <<a a 1 14= ln ,, y y yy ref O g ref NO NO, O2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ≈− () 14 2 © 2006 by Taylor & Francis Group, LLC [...]... temperature and the major NOx influence for composition changes That is, for a given furnace temperature, the flame temperature should scale as a linear function of the adiabatic flame temperature Therefore, we may consider adiabatic flame temperature ratios as indicators of relative NOx production from the thermal mechanism Appendix A, Table A. 2 shows the difference among fuels If we presume that the... constant-composition fuels or for those fuels whose composition changes only slowly The strategy requires automated furnace stack dampers (or automated fan) and automated burner registers Usually, refinery process heaters lack automatic airflow control and use manual burner registers © 2006 by Taylor & Francis Group, LLC 444 Modeling of Combustion Systems: A Practical Approach O2 SP CO SP from CO analyzer... flue gas and airstreams Combining Equation 5.41 with Equation 5.42 we obtain RFG ,e = © 2006 by Taylor & Francis Group, LLC mr ma wO 2 , a − wO 2 ,b ma = ma mg wO 2 ,b − wO 2 , g mg 450 Modeling of Combustion Systems: A Practical Approach But ma na Wa = mg ng Wg where γ= ng na is given by Equation 2.22, Wa is the molecular weight of air, and Wg is the molecular weight of the flue gas Thus, we have RFG... 448 Modeling of Combustion Systems: A Practical Approach · · · mg = mf + ma System Boundary Flue Gas · · mr + mg mr Stack mr + mg · · · ma + mf + mr = · · mb + mf · mb = Air · ma · mf · · mr + mg = · · · ma + mf + mr = · · mb + mf Furnace · · ma + mr Windbox Fan Fuel FIGURE 5.4 Mass balance for a typical FGR system This figure shows a typical FGR system and its associated mass balance Recirculated... Solving for T as a Function of αw Now we know from Equation 2.93 that TAFT = © 2006 by Taylor & Francis Group, LLC ΔH c + Ta Cp 1 + α w ( ) (5.32) 442 Modeling of Combustion Systems: A Practical Approach where αw is a function only of fuel composition (ψ) and excess air rate (ε), and Ta is the air temperature Presuming once again that the flame temperature is given by T = (TAFT + TBWT)/2, we obtain T≈ 5.1.6.5... thermal and prompt mechanisms Outside the system boundary as drawn, FIR affects no concentrations or flows * COOLburn is a trademark of John Zink LLC, Tulsa, OK for their patented technology © 2006 by Taylor & Francis Group, LLC 454 Modeling of Combustion Systems: A Practical Approach fuel, the mixture could represent a flammable mixture upstream of the burner For normal flue gas concentrations near 3%... setpoint based on a CO analysis of the furnace flue gas Figure 5.2 shows a schematic of the general CO trim concept The air/fuel controller attempts to maintain a constant air/fuel ratio Oxygen control biases this air/fuel ratio based on a target oxygen concentration Thus, as the air density or fuel composition varies over the course of the day, oxygen control maintains oxygen level and furnace efficiency... In fact, steam is more effective in reducing NOx than flue gas because water is a three-body molecule (H—O—H) capable of absorbing infrared energy and thereby abstracting heat (Diatomic molecules such as N2 are not active in the infrared For this reason, CO2 (O=C=O) reduces NOx more strongly than N2.) Steam also has a high heat capacity (Cp = 0.53 Btu/lbm °F — about double that of air However, steam... represents an efficiency penalty because one requires external energy to generate steam and any heat abstracted from the flame comes at the expense of the process Generally, steam injection is limited to small percentages as a final NOx reduction strategy For example, suppose one must meet a legal requirement of 40 ppm NOx but the combustion process generates 42 ppm In such a case, a small amount of steam injection... flue gas must be recirculating through the venturi than guaranteed, so the manufacturer has met his claim Equation 5.61 gives the actual recirculation ratio: nr 2.1 = = 2.33 n f 3 − 2.1 © 2006 by Taylor & Francis Group, LLC 456 Modeling of Combustion Systems: A Practical Approach 5.2.9 Steam or Water Injection Flue gas is not the only diluent that is effective for NOx reduction One may inject steam as . LLC effect of adding air. As we have seen in Section 2.4.10, this acts to reduce the 428 Modeling of Combustion Systems: A Practical Approach Making use of the ideal gas law, we may recast Equation. Chapter 2, we discussed the mechanical energy balance and 422 Modeling of Combustion Systems: A Practical Approach Next, we consider how one may measure flame length. We discuss a flame model and. by Taylor & Francis Group, LLC cies. (See Appendix G for a useful summary of formulating rate laws.) (This 424 Modeling of Combustion Systems: A Practical Approach (5.5) Now we may integrate