燃烧学讲义（12固体燃烧）

科技与论语 MACROBUTTON MTEditEquationSection2 Equation Chapter 1 Section 1 错误！未找到图形项目表。 0.​ 绪论 1.​ 燃料 2.​ 燃料的燃烧计算 MACROBUTTON MTEditEquationSection2 Equation Chapter (Next) Section 1 3.​ 输运现象 MACROBUTTON MTEditEquationSection2 Equation Chapter (Next) Section 1 REF _Ref224634539 \r \h [5] REF _Ref224654757 \r \h [7][10] 输运什么呢？质量、动量、能量。The subject of transport phenomena includes three closely 4.​ 化学动力学基础 MACROBUTTON MTEditEquationSection2 Equation Chapter (Next) Section 1 5.​ 燃烧学基本方程 MACROBUTTON MTEditEquationSection2 Equation Chapter (Next) Section 1 任何一门成熟的学科都有它自己的数学模型体系，它能够在数学上描述这门学科所涉及的所有特征。人们通过解方程就可以预测这门学科所涉及的所有现象，比如在天体力学中通过解经典力学方程，就能确定行星（如海王星）的轨道；在分子物理中通过解薛定谔方程，就可以确定分子光谱。 这章介绍燃烧学基本方程。它是在流体力学方程基础上得到的。The dynamics and thermodynamics of a chemically reacting flow are governed by the conservation laws of mass, momentum, energy, and the concentration of the individual species. In this chapter, we shall first present a derivation of these conservation equations based on control volume considerations. 6.​  7.​ 预混可燃气着火理论 MACROBUTTON MTEditEquationSection2 Equation Chapter (Next) Section 1 8.​ 层流预混可燃气火焰传播理论 MACROBUTTON MTEditEquationSection2 Equation Chapter (Next) Section 1 9.​ 非预混燃烧 MACROBUTTON MTEditEquationSection2 Equation Chapter (Next) Section 1 10.​ 火焰稳定 MACROBUTTON MTEditEquationSection2 Equation Chapter (Next) Section 1 这一章与第6章的着火相对，考虑熄火问题。 11.​ 湍流燃烧 MACROBUTTON MTEditEquationSection2 Equation Chapter (Next) Section 1 12.​ 液体燃料燃烧 MACROBUTTON MTEditEquationSection2 Equation Chapter (Next) Section 1 remains unknown. 13.​ 固体燃料燃烧 MACROBUTTON MTEditEquationSection2 Equation Chapter (Next) Section 1 人类燃烧燃料的历史： 木材→煤炭→石油→煤炭→生物质。 18世纪、19世纪上半叶之前，木材为主；之后，工业发达国家，煤炭开始取代木材, 成了能源的主要支柱, 20世纪50年代末，美国、中东、北非相继发现了巨大的油田和气田，石油成为了能源的主要来源。石油可用50~100年，总有一天会耗尽， 煤炭储量大些，还可以开采300年左右。以后的能源是什么? 世界上，每年由于光合作用产生的生物质量是目前年能源消耗量的8倍，但分散\能量密度低\利用效率不高，如何高效利用生物质，是目前较有前途的研究方向之一。目前和不久的将来，固体燃料仍然会占有重要的地位。 固体燃料燃烧主要有以下几种类型： 1、表面燃烧：燃烧反应在燃料表面进行，通常发生在几乎不含有挥发分的燃料中，如焦炭、木炭，氧气与二氧化碳通过扩散作用到达燃料表面进行燃烧反应。 2、分解燃烧：一般发生在热分解温度较低的燃料中，热分解产生的挥发分在离开燃料后与氧气接触进行同相燃烧反应，如纸张、木材、煤等。 3、蒸发燃烧：发生在熔点较低的燃料中，先熔化呈液态，然后进行蒸发和燃烧，如石蜡等链烷烃燃料。 4、冒烟燃烧：一些易分解的固体燃料当温度较低挥发分未着火时，将产生大量浓烟，从而使大量可燃成分散失在烟雾中。如木材纸张在低温下的燃烧。 13.1.​ 煤的燃烧过程 煤都是由多种有机可燃质、不可燃的矿物质灰分和水分组成。包括碳（C）、氢（H）、硫（S）、氧（O）、氮（N）等元素和不可燃的矿物质灰分（A）和水分（M）（煤的惰性质）。对于燃烧过程，我们关心的是根据煤的工业分析，得到的水分（M）、挥发分（V）、灰分（A）和固定碳（FC）。 挥发分含量的多少对煤的燃烧过程有很大影响。挥发分大的煤，释放的可燃气体多，所以易于着火。因此，挥发分的大小是衡量煤是否易于燃烧的重要指标。但是在空气不足或低温下着火时，挥发分易产生炭黑粒子（冒黑烟）。当煤中含有的挥发份较高时，则着火首先由挥发份开始，然后才是碳的着火。但在某些特定条件下，有可能碳首先着火，例如煤粒极微细及加热速率很大时。灰份和水分，这两种成分在一般情况下对着火与燃烧没有好处。The formation of ash deposits, particularly those that melt (referred to as “ slagging ” deposits) or otherwise are diffi cult to remove, restricts the effective heat transfer from furnace gases to boiler tubes and plays an important role in boiler design and operation. Excessive moisture in the fuel reduces the fuel heating value and fl ame stability. Conversely, insufficient fuel moisture can lead to spontaneous ignition problems when storing and handling some reactive fuels. 煤的燃烧大致经历如下几个过程：预热干燥、挥发分析出并燃烧、焦炭的燃烧和燃尽。 1、煤受热，表面水分蒸发，干燥。温度继续升高，热分解反应，析出碳氢化合物和少量CO2，称为挥发分；在很短时间内就可以析出全部挥发分的90％左右，但需要较长时间才可全部析出。 2、温度足够高，挥发分着火燃烧，一方面加热焦炭，一方面争夺氧气，抑制焦炭不能燃烧，中心温度800左右。这90％的挥发分的燃烧时间占全部燃烧时间10％左右。 3、焦炭在大部分挥发分燃烧后才着火，温度达到1200度，出现蓝火焰，是CO燃烧。仍有少量挥发分析出，二至几乎同时燃尽。由于焦炭发热量很高，故在煤燃烧中起决定性作用。 挥发分对煤燃烧的影响 挥发分对粒煤燃烧具有双重影响： 有利影响：挥发分与空气的混合物着火温度很低，因此先于焦炭着火燃烧，并形成包络火焰，提高了焦炭的温度，为其着火燃烧提供了条件；同时焦炭温度升高也促进了挥发份的析出。挥发分析出在焦炭内部形成众多空洞，增加了焦炭反应表面积，利于提高燃烧速度。 不利影响：挥发分的燃烧消耗了大量的氧气，造成扩散到焦炭表面的氧气显著减少，从而降低了煤颗粒的燃烧速度，特别是燃烧初期，对煤粒燃烧的抑制作用尤其明显。 灰分对煤燃烧的影响 有利影响：当灰分中含有Na、K等元素时，对煤的燃烧有催化作用。 不利影响：当煤颗粒由外层逐渐烧向内层的时候，灰分有可能形成包裹颗粒的灰壳，妨碍了氧气的扩散，不仅降低了燃烧速度，甚至导致不能燃烬。另外灰分升温消耗了一部分热量，降低了燃烧温度并导致着火延迟。 13.2.​ 热解[4] The volatile gases burn much more rapidly than the remaining char particles and therefore are important for fl ame ignition and stability and play an important role in the formation of oxides of nitrogen (NOx). Moreover, the devolatilization process determines how much char remains to be burned as well as the physical characteristics of the resulting char, with subsequent impacts on the char combustion properties. Different coal types vary signifi cantly in their volatile content, ranging from a maximum of approximately 50% (by mass) for low- and mid-rank coals to just a few percent for anthracitic coals (which are graphite-like in character). Biomass always has a large volatile content, generally around 80%, as determined using the standard ASTM test method. 图 12‑1shows a characteristic molecular structure of coal, featuring an aromatic carbon backbone and a wide range of bond strengths. General plant matter (as distinct from the fruiting bodies that are often used as food and are primarily composed of starch and sugar) is known as lignocellulosic biomass and is composed of a more or less even mixture of cellulose, hemicellulose, and lignin. As shown in Fig. 9.20 , cellulose and hemicellulose are both composed of polymers of oxygen-containing ring compounds linked by relatively weak carbon–oxygen bonds. Lignin, in contrast, is composed of small aromatic units connected in weakly linked branched structures. As coal and biomass particles are heated, the internal structure of the carbonaceous material undergoes internal molecular rearrangements. Many weakly bonded moieties break their connecting bonds to the main structure and form gas molecules that aggregate within the solid and, after building suffi cient pressure, burst forth from the particle with substantial force. At the same time, some weakly bonded structures and some structures with intermediate-strength bonds pivot about their connecting bond and are able to form stronger, cross-linking bonds with neighboring regions of the structure. Thus there is an inherent competition between solid decomposition and char-forming reactions, with different characteristic activation energies and reaction times. As a consequence, the quantity and chemical composition of the volatile matter that is emitted from these fuels is highly dependent on the nature of the original fuel structure, the rate at which the particles are heated, and the fi nal temperature attained by the particles. For large particles, this also means that the devolatilization process differs as a function of the internal radius of the particle, because the local heating rate varies with the distance from the surface of the particle, where heat is being applied. 图 12‑1 Hypothetical coal molecule as represented by Solomon [39] The effect of heating rate on evolution of volatiles is most clearly evidenced in the case of woody biomass, which has been shown to have a volatile yield of greater than 90% when small particles are rapidly heated to 1200°C and to have a volatile yield of only 65% when large particles are slowly heated to 500°C in the commercial charcoal-making process. 13.2.1.​ 煤的热解模型 With the importance of the devolatilization process to solid particle combustion and the complexity of the chemical and physical processes involved in devolatilization, a wide variety of models have been developed to describe this process. 一个完整的热解模型应该是能用来描述煤或炭在所有的加热和挥发份的析出阶段的物理状态、挥发份的产量、成分及挥发份的析出速率。但由于挥发份析出过程的复杂性，要写出一个完整的、反映物理化学过程的方程是困难的。人们在描述煤的热解过程时，不得不做出许多假设，且不考虑热解的细节，仅从宏观角度建立宏观数学模型。 单方程模型 热解过程的微分方程： MACROBUTTON MTPlaceRef \* MERGEFORMAT (12.1) V是已析出的挥发份质量百分数；V∞是当时间t→∞时，最终析出挥发份的百分数；k、E分别是表观频率因子和活化能；VM是工业分析值 ；Q、Vc 是实验常数。Fits of Eq. GOTOBUTTON ZEqnNum363641 \* MERGEFORMAT (12.1)to the experimental data typically yield an effective activation energy of about 230 kJ/mol, which is consistent with the activation energy for rupturing an ethylene bridge between aromatic rings 该模型没有通用性，该某些特定的煤种是合适的，但不能预测任何煤种、在任何工况下的热解过程。 两个平行的反应模型 柯巴约希等人提出了用两个平行的、互相竞争的一级反应来描述热解过程，即两个平行反应同时将煤的一部分（α1，α2）热解成挥发份V1和V2，另一部分则变成炭R1和R2，即： 其中k1、k2是反应速度常数，分别为： MACROBUTTON MTPlaceRef \* MERGEFORMAT (12.2) 这一模型的特点是认为存在两个不同的活化能E1、E2和两个不同的频率因子k01、k02的热解反应，且认为 E2 > E1 ， k01 > k02。这样，在较低温度时第一反应起主要作用，而在较高温度时，第二个反应起主要作用。With this model, the instantaneous devolatilization rate is the sum of the two independent rates, MACROBUTTON MTPlaceRef \* MERGEFORMAT (12.3) where R refers to the mass of solid coal or char remaining. 两个平行的反应模型弥补了单方程模型只适用于有限温度范围的局限性，可以适用于较大的温度范围。 无限个平行反应模型 安东尼等人提出的模型认为热解是通过无限多个的平行反应进行的，并假定活化能是一连续的正态分布，而频率因子是一个常数，结果得到了如下表达式： MACROBUTTON MTPlaceRef \* MERGEFORMAT (12.4) MACROBUTTON MTPlaceRef \* MERGEFORMAT (12.5) the distribution of kinetic rates is characterized by a mean and standard deviation of the activation energy。这个模型对不同的煤种有不同的动力学参数，因此也没有通用性。 多组分单方程模型 煤粒热解的通用模型（Fu-Zhang模型） 13.3.​ 碳粒燃烧 Compared with the combustion of liquid hydrocarbon fuels, carbon combustion has the following distinctively different features: (a) The boiling points of conventional hydrocarbon liquids seldom exceed 700–800 K, while the sublimation temperature of carbon is in excess of 4,000 K. Thus carbon is very nonvolatile. (b) The high sublimation temperature implies that carbon can be heated to very high temperatures.When it is also actively reacting, its temperature can easily exceed the combustor temperature as just mentioned. (c) The continuous increase in the particle temperature is eventually arrested through radiative heat loss. Thus the particle temperature seldom exceeds 2,500–3,000 K and therefore is substantially below its sublimation temperature. (d) Because of the low particle temperature relative to the sublimation temperature, the carbon burning rate is expected to be less than those of liquid hydrocarbons, which can be heated to close to their boiling points. (e) Low volatility and high surface temperature indicate that surface reactions are important. (f) The existence of surface and gas-phase reactions implies the need for reaction schemes that describe the coupling between them. 煤中的挥发份析出后剩下的固体物质称为炭或者残碳，它是呈多孔结构，反应不仅可以在表面上进行，也可以在炭内部的空隙中进行。碳和炭的主要区别是炭中含有灰份和其它杂质。 在燃烧过程中，碳的反应包括初次反应（碳与氧）和二次反应（碳与二氧化碳及一氧化碳与氧）。燃烧过程包括：1、氧扩散到碳表面；2、氧吸附于碳表面；3、氧与碳进行表面反应；4、反应产物的脱附；5、解析后的生成物从反应表面扩散到周围环境。 13.3.1.​ 碳燃烧过程中的化学反应 1、初次反应：碳与氧反应，生成中间状态的碳氧络合物，然后离解为二氧化碳与一氧化碳。 MACROBUTTON MTPlaceRef \* MERGEFORMAT (12.6) 2、二次反应：二氧化碳在炽热碳表面进行还原反应，吸热，也是一个吸附、络合、分解的复杂过程；一氧化碳与氧气进行容积反应，放热。 MACROBUTTON MTPlaceRef \* MERGEFORMAT (12.7) MACROBUTTON MTPlaceRef \* MERGEFORMAT (12.8) 另外，如果存在水蒸汽，在碳表面还有如下反应 MACROBUTTON MTPlaceRef \* MERGEFORMAT (12.9) 是一个高温下的吸热反应。氢气也在碳表面与碳发生反应。 主要的反应过程如下所示。 A、温度低于700度：由于温度不高，CO2不能进行还原反应，CO也不能与氧反应。 B、温度700～1200度：CO2仍然不能进行还原反应，CO与氧可以反应生成CO2并形成火焰前锋，所以其浓度高于CO。 C、温度大于1200度：CO2的还原反应生成CO，故增大其扩散量。在其扩散过程中与氧反应形成火焰前锋，CO2达到最大。氧已经不能扩散到碳表面。 13.3.2.​ 碳粒燃烧速率 扩散-动力控制的碳粒表面燃烧的半经验 假设只发生碳表面上的反应 MACROBUTTON MTPlaceRef \* MERGEFORMAT (12.10) 则氧的消耗速率为 MACROBUTTON MTPlaceRef \* MERGEFORMAT (12.11) 其中 MACROBUTTON MTPlaceRef \* MERGEFORMAT (12.12) 为表面反应速率常数。那么按反应方程式 GOTOBUTTON ZEqnNum260520 \* MERGEFORMAT (12.10)，碳的消耗速率为 MACROBUTTON MTPlaceRef \* MERGEFORMAT (12.13) 另外，稳态过程反应速度应该等于反应表面气体扩散速度 。霍特尔类比对流传热系数的概念，引入了“对流扩散系数hD”的概念 MACROBUTTON MTPlaceRef \* MERGEFORMAT (12.14) GOTOBUTTON ZEqnNum142211 \* MERGEFORMAT (12.11)、 GOTOBUTTON ZEqnNum470891 \* MERGEFORMAT (12.14)两式相等，可得 MACROBUTTON MTPlaceRef \* MERGEFORMAT (12.15) 将上式代入式 GOTOBUTTON ZEqnNum142211 \* MERGEFORMAT (12.11)有 MACROBUTTON MTPlaceRef \* MERGEFORMAT (12.16) 式中K为等值阻力系数 MACROBUTTON MTPlaceRef \* MERGEFORMAT (12.17) 根据传热传质理论，可有近似关系式 MACROBUTTON MTPlaceRef \* MERGEFORMAT (12.18) MACROBUTTON MTPlaceRef \* MERGEFORMAT (12.19) 其中Sh为舍伍德数，D为氧气质量扩散系数，d为碳粒直径。 对于两种极限情况的分析结果如下： （1）如果温度很高(或压力很高)那么kO2很大，且环境气流与碳粒之间相对速度不是太大，有 ，因此 MACROBUTTON MTPlaceRef \* MERGEFORMAT (12.20) 于是1/ hD >>1/ Ko2，K≈1/ hD，所以由 GOTOBUTTON ZEqnNum293050 \* MERGEFORMAT (12.16)得 MACROBUTTON MTPlaceRef \* MERGEFORMAT (12.21) 这时化学反应速率很高，氧的消耗速率很大，碳粒表面的氧几乎全部被消耗，所以YO2，S ≈0。碳粒的燃烧速率主要取决于氧的扩散速率，称之为扩散燃烧区，此时温度的影响较小。 图 12‑2碳粒燃烧随温度的变化 则单位时间内碳粒的质量变化为 MACROBUTTON MTPlaceRef \* MERGEFORMAT (12.22) 另外，碳粒的燃烧速率可表示为 MACROBUTTON MTPlaceRef \* MERGEFORMAT (12.23) 则由 GOTOBUTTON ZEqnNum135408 \* MERGEFORMAT (12.22)与 GOTOBUTTON ZEqnNum260767 \* MERGEFORMAT (12.23)可得 MACROBUTTON MTPlaceRef \* MERGEFORMAT (12.24) 即 MACROBUTTON MTPlaceRef \* MERGEFORMAT (12.25) 燃尽时间与碳粒直径的平方成正比，服从颗粒直径燃烧平方规律。这一结果与单个油滴的燃烧过程相似。这时燃烧过程为纯扩散控制的燃烧。 （2）如果温度不高 (或压力不大) 则化学反应速率较小，如果Sh较大，这时便有1/ hD <<1/ kO2。那么 ，由 GOTOBUTTON ZEqnNum293050 \* MERGEFORMAT (12.16)得 MACROBUTTON MTPlaceRef \* MERGEFORMAT (12.26) 这种情况下，化学反应速率低，氧的消耗速率很小，因此 。碳粒的燃烧速率主要取决于化学反应速率，称之为动力燃烧区（图 12‑2），此时温度的影响很大。 此时单位时间内碳粒的质量变化为 MACROBUTTON MTPlaceRef \* MERGEFORMAT (12.27) 另外，碳粒的燃烧速率可表示为 MACROBUTTON MTPlaceRef \* MERGEFORMAT (12.28) 则由 GOTOBUTTON ZEqnNum135408 \* MERGEFORMAT (12.22)与 GOTOBUTTON ZEqnNum260767 \* MERGEFORMAT (12.23)可得 MACROBUTTON MTPlaceRef \* MERGEFORMAT (12.29) 即 MACROBUTTON MTPlaceRef \* MERGEFORMAT (12.30) 燃尽时间与碳粒直径正比 碳燃烧的二次反应 实验研究发现：当碳粒燃烧进入扩散区以后，其反应速度并非如图 12‑2所示与温度无关，而是随着温度的提高显著提高，这便是二次反应的影响。温度越高，CO2越容易还原为CO，CO在离开表面途中与迎面的氧反应生成CO2，达到峰值并向碳表面扩散，此时氧已经不能扩散到碳表面，所以是CO2在与碳进行反应。 考虑到容积反应，CK Law[5]给出了如下的模型。 具有容积反应的碳粒燃烧模型[5] In the problem to be analyzed we shall assume that diffusion of the various species to and from the carbon surface is uninhibited, implying that there is either no ash layer or, if there were one, it is sufficiently porous and hence has very low diffusional resistance. It is further assumed that surface reactions take place only at the particle surface in that there is no species diffusion into the pores of the particle causing internal burning. It is noted that internal burning can be quite significant because of the large surface area associated with the porous structure, especially when the particle temperature is low such that the gaseous reactants can readily diffuse into the pores without suffering much depletion of its concentration through reaction in the particle surface region. The quasi-steady gas-phase processes will be formulated and solved allowing for finite rates of the three major reactions and for a given surface temperature Ts , which can be separately determined through overall particle energy balance including, say, particle heating and radiative heat loss. The major surface reactions for carbon oxidation are： 表面反应： 正反应 (a) C+O2→CO2 正反应 (b) C+1/2O2→CO 副反应 (c) C+CO2→2CO 容积反应： (d) CO+1/2O2→CO2 限定所讨论的碳粒温度大于1000ºC，这时碳粒的表面反应以 (b)、(c)为主，忽略反应(a) 对于静止空气中燃烧的球形碳粒（具有空间反应）在球对称的情况下，其组分方程和能量方程简化为： MACROBUTTON MTPlaceRef \* MERGEFORMAT (12.31) 为碳粒燃烧时的质量流率。边界条件： MACROBUTTON MTPlaceRef \* MERGEFORMAT (12.32) MACROBUTTON MTPlaceRef \* MERGEFORMAT (12.33) 式中 M表示分子量； 、 分别表示由反应式(b)和(c)所产生的碳粒燃烧速率: MACROBUTTON MTPlaceRef \* MERGEFORMAT (12.34) The particle burning rate is thus MACROBUTTON MTPlaceRef \* MERGEFORMAT (12.35) Using the coupling function formulation, the quasi-steady gas-phase heat and mass conservation equations are MACROBUTTON MTPlaceRef \* MERGEFORMAT (12.36) MACROBUTTON MTPlaceRef \* MERGEFORMAT (12.37) where MACROBUTTON MTPlaceRef \* MERGEFORMAT (12.38) MACROBUTTON MTPlaceRef \* MERGEFORMAT (12.39) MACROBUTTON MTPlaceRef \* MERGEFORMAT (12.40) qc is the heat release per unit mass of CO consumed, σi is the stoichiometric mass ratio of the ith species to CO according to the CO–O2 reaction, δ = MCO/MC, ˜ wg is its nondimensional gas-phase reaction rate whose specification does not need to concern us here, and ρD is taken as a constant. Solving Eq. GOTOBUTTON ZEqnNum303116 \* MERGEFORMAT (12.36)subject to the boundary conditions, it can be shown that the various coupling functions are given by MACROBUTTON MTPlaceRef \* MERGEFORMAT (12.41) MACROBUTTON MTPlaceRef \* MERGEFORMAT (12.42) MACROBUTTON MTPlaceRef \* MERGEFORMAT (12.43) where MACROBUTTON MTPlaceRef \* MERGEFORMAT (12.44) The problem is therefore reduced to solving the energy equation GOTOBUTTON ZEqnNum926945 \* MERGEFORMAT (12.37) subject to the boundary conditions ˜T(1) = ˜T s and ˜T(∞) = ˜T ∞, using the coupling functions given by Eqs. GOTOBUTTON ZEqnNum428925 \* MERGEFORMAT (12.41) and m˜ given by Eq. GOTOBUTTON ZEqnNum686254 \* MERGEFORMAT (12.35). The solution including states of ignition and extinction can be obtained either computationally or through asymptotic analysis of the gas-phase reaction. The system behavior, however, can be bracketed by the following limiting solutions. Limiting Solutions （1）Frozen Limit: Here we have （2）Detached Flame Limit: When the gas-phase reaction is infinitely fast, we obtain a detached flame, characterized by 。In this detached flame limit the burning ratem˜ still depends on the finite reaction rate of the surface C–CO2 reaction. The surface C–O2 reaction is however suppressed because there is no oxygen leakage through the flame. （3）Attached Flame Limit: With m˜ continuously decreasing in the detached flame limit, a value will be reached at which ˜ r f = 1, implying that the flame is now contiguous to the surface. With further reduction in m˜ , the combustion evolves into one with an attached reaction sheet. In this case the gas-phase reaction is still infinitely fast such that there is no leakage of CO into the gas phase. The rate of generation of CO through the surface C–CO2 reaction, however, is not as fast as the situation in the detached reaction-sheet limit as to totally consume the oxygen at the surface. The presence of oxygen at the surface, then, also activates the surfaceC–O2 reaction. （4）Diffusion Limit: The three limits identified above all pertain to the limiting behavior of the gas-phase reaction, with one or both of the surface reactions activated with finite rates. We now consider situations limited by the gas-phase transport. We first consider the situation when the surface C–O2 and C–CO2 reactions both occur infinitely fast. This implies that ˜YO2,s = ˜YCO2,s = 0, which when applied to the coupling function Eq. GOTOBUTTON ZEqnNum299123 \* MERGEFORMAT (12.42)readily yields MACROBUTTON MTPlaceRef \* MERGEFORMAT (12.45) and thereby the burning rate MACROBUTTON MTPlaceRef \* MERGEFORMAT (12.46) This is the maximum burning rate attainable for given ambient O2 and CO2 concentrations, with the carbon consumption rate limited by diffusion. The second situation occurs when the gas-phase CO–O2 and the surface C–CO2 reactions proceed infinitely fast. Then we still require ˜YO2,s = 0 and ˜YCO2,s = 0, which again yields βmax given by Eq. GOTOBUTTON ZEqnNum867648 \* MERGEFORMAT (12.46). Thus the diffusion-limited behavior can be attained by requiring only two of the three reactions to proceed infinitely fast. It should also be pointed out that m˜ max here pertains to only the mass gasification rate of carbon at the surface without being specific to the identity of the final product, which still depends on the nature of the gas-phase reaction. Thus the combustion product is mainly CO when the gas-phase reaction rate is slow, while it becomes CO2 when it is fast. We also note that since , the fact that m˜ max is a constant implies that the mass burning rate mvaries with the instantaneous particle radius rs , which is just the diffusion-limited d2-law result, as should be the case. （5）Surface Reaction Limit: In this limit k1 << 1 and k2 << 1. Then the mass burning rate for the frozen and attached flame limits degenerate to MACROBUTTON MTPlaceRef \* MERGEFORMAT (12.47) while that for the detached flame limit becomes MACROBUTTON MTPlaceRef \* MERGEFORMAT (12.48) The above expressions were obtained by assuming, realistically, that burning under the present situation is necessarily very slow, that is, ≈ β << 1. It is of interest to note that Eqs. GOTOBUTTON ZEqnNum287802 \* MERGEFORMAT (12.47)and GOTOBUTTON ZEqnNum448373 \* MERGEFORMAT (12.48) show that m˜ ∼ k in this limit. Since m ∼ rsm˜ and k ∼ rs, we have m ∼ r 2 s . Furthermore, recognizing that m ∼ dr3 s /dt ∼ r 2 s drs/dt, we obtain the result that MACROBUTTON MTPlaceRef \* MERGEFORMAT (12.49) which can be considered as a d-law for particle burning controlled by surface reactions. It is important to mention again that this result is only for a solid carbon particle, without any internal pores. Since particles of char are usually porous, and if burning is kinetically controlled, then diffusion of the gas into the pores is efficient, resulting in reactions over the surface of the internal pores. Because o