燃烧学讲义(11液体燃烧)

错误！未找到图形项目表。

0.绪论
1.燃料
2.燃料的燃烧计算
3.输运现象[5][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
5.燃烧学基本方程
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.预混可燃气着火理论
7.层流预混可燃气火焰传播理论
8.非预混燃烧
9.火焰稳定
这一章与第6章的着火相对，考虑熄火问题。

10.湍流燃烧
11.液体燃料燃烧
19世纪末20世纪初，内燃机的发明使液体燃料开始得到大规模应用，人们才开始对液体燃料的燃烧机理、燃烧过程进行研究。

In many combustion applications the fuel is originally present as either liquid or solid. 液体燃料的燃烧很大程度上取决于其蒸发与扩散速度，而蒸发速度与热交换情况、物性以及表面积有关。

1mm液滴，燃尽时间约为1s；若雾化成10微米颗粒，表面积增大100倍，燃烬时间仅为10-4 s。

液体燃料需要先雾化后燃烧。

In order to facilitate mixing and the overall burning rate, the condensed fuel is frequently first atomized or pulverized, and then sprayed or dispersed in the combustion chamber. Consequently, in these devices combustion actually takes place in a two-phase medium, consisting of the dispersed fuel droplets or particles in a primarily oxidizing gas.
A description of two-phase combustion consists of three components, namely the gasification and dynamics of individual and groups of droplets (or particles); a statistical characterization of the spray; and the collective interaction of these droplets with the bulk gaseous medium through the description of the two-phase flow in terms of heat, mass, and momentum transfer.
11.1. 相对静止气流中液滴蒸发与燃烧
液雾颗粒从喷嘴喷射出来，与空气之间具有很大的相对速度，但是经过一段距离之后由于摩擦效应会逐渐滞慢下来，此时相对速度几乎消失。

具有相对速度的一段叫做“动力段”，在该段液滴完成受热升温，一般对于10～40微米的液滴这段时间仅为千分之几秒。

没有相对速度的称为“静力段”，蒸发汽化、燃烧过程在该段完成。

由于该段相对速度很小，理论分析时，一般简化为球形液滴在静止气流中的燃烧过程。

早期人们认为液体燃料的燃烧仅仅发生在液体的表面，是燃料分子与氧分子在燃料表面进行的化学反应。

但实际情况是，液体燃料的燃烧是在蒸汽状态下的燃烧：由于液体燃料的汽化温度比着火温度低得多，因此在着火之前已经蒸发汽化，所以液体燃料燃烧实质上是油气与空气的燃烧反应。

因此蒸发过程对于液体燃料的燃烧起着决定性作用，加强蒸发汽化是强化液体燃料燃烧的主要手段。

液体燃料的燃烧具有扩散燃烧的特点。

一般情况下，液体燃料燃烧时反应速度很快，相对来说，蒸发汽化与扩散混合却慢得多，因此属于扩散燃烧。

由于受到高温辐射，液滴表面首先蒸发形成油气向周围气体扩散并被点燃，形成距离液滴表面一段距离的球状火焰前锋，火焰前锋将空气与氧气完全隔离开来，内侧只有燃料气，外侧只有氧气和产物。

因此这是一个典型的非预混燃烧过程（Except at the very lowest of pressures）。

11.1.1.斯蒂芬（Stefan）流和相分界面的内移
Such a flow frequently arises as a consequence of the gasification of a condensed fuel. Here a fuel vapor source is present at the surface of the condensed fuel, and the ambience, or the flame, which is located away from the surface, is typically low in the fuel vapor concentration and thus represents a sink for the fuel vapor. The fuel vapor then continuously diffuses from the surface to
either the ambience or the flame. Since the consequence of this diffusion is a net transport of mass, a convective motion, that is, the Stefan flow, is induced. This motion can be quite significant and has to be accounted for, especially for rapid rates of vaporization when the condensed fuel is volatile, or when it is placed in a hot environment or undergoes combustion. This phenomenon is also particularly relevant for nonpremixed combustion because practical fuels are frequently present in the condensed phase, for example fuel droplets and coal particles.
例如，水面蒸发时产生斯蒂芬流。

环境温度为T ，环境压力为p 的静止空气，放置一个装有水温为T0 （T >>T0）的开口容器。

通过容器底部的细管不断
补充水，以保持液面的稳定。

坐标取法
如图所示。

这样在容器上方的空间中，
只有水蒸气及空气两种成分，沿 x 方向没有浓度梯度（均匀的），只在 y 方向有浓度梯度存在。

若用YH2O 、YA 分别
表示水蒸气及空气的相对浓度，则它们的分布如图所示。

水面上有Y H2O + Y A =1。

这时相分界面处水蒸气分子的扩散流是：
H2O H2O 000Y J D y ∂⎛⎫
=- ⎪∂⎝⎭
(11.1)
与此同时，相分界面处的空气浓度梯度也将导致空气分子的扩散：
A A 000
Y J D y ∂⎛⎫
=- ⎪∂⎝⎭
(11.2)
就是说有一个流向相分界面的空气扩散流。

空气是不会被水吸收，这个流向相分界面的空气扩散流哪儿去了呢？这说明，在相分界面处除有扩散流之外，还存在一个与空气扩散流方向相反的空气－水蒸气混合气的整体质量流，使得空气在相分界面上的总物质流等于零。

假设空气-水蒸气混合气的总体质量流以流速v0流动，则每一种组份的质量流都可分成两部分：一部分是该组份由于浓度梯度造成的扩散物质流，另一部分是由于混合气总体质量流所携带的该组份的物质流： ()
22222H O,0H O,00H O,0000H O 0H O,000
/m J Y v D Y y
Y v =+=-∂∂+
(11.3)
(),0,00,0000,0
,000
/0A A A AO A m J Y v D Y y Y v =+=-∂∂+= (11.4)
在相分界面处：20H O,0,0A m m m =+，但是mA0 = 0，所以m0 = mH2O,0，整理后得
()
2200H O 00H O,00
/(1)
D Y y
v Y -∂∂=
- (11.5)
由此可以看出，斯蒂芬流0m 包括扩散物质流部分，和随着混合气总体流动所携带的该组份的物质流部分。

又例如，碳平板在纯氧中的燃烧。

一个无限大的碳平板置于纯氧中燃烧，只有在碳平面的法线方向有浓度梯度变化，与碳平板平行的方向上是均匀的。

碳平板表面发生
x
0Y
Y H2O Y A y v 0
ρ1
1
T ∞
p ∞
T 0ρ0
Y CO2
Y O2
y
反应生成CO2。

扩散流只能输运一部分CO2离开碳表面，另一部分CO2将随着与CO2扩散方向相同的O2 & CO2的总体流动而离开碳表面。

这个混合气的总体质量就是斯蒂芬流。

Stefan 流就是碳表面烧掉的质量，也正是碳的燃烧速率。

通过上面的例子，清楚地说明斯蒂芬流产生的条件是，在相分界面处既有扩散现象存在，又有物理或化学过程存在。

相分界面的内移
相分界面的内移是一种普遍现象，液面蒸发、固体升华、液体或固体燃料表面燃烧，都要消耗液体或固体物质，引起相分界面的内移。

假设：0v 是相分界面处气相物质相对于静止坐标系的速度，0
v ' 是气相物质相对于相分界面的速度，0
v ''是凝聚相（液相或固相）边界面内移速度 那么就有下列关系式:
00v v v '''=+ (11.6)
假设相分界面处气相和凝聚相的密度分别为0和1，则有：
"
'
"
100000
0()v v v v ==
+
(11.7)
在一般情况下：1>>
，所以"'
00
v v <<。

所以
"
10
00v v ≈
(11.8)
11.1.2. Droplet Vaporization and combustion
Since droplet gasification is basically a
two-phase flow problem, a complete analysis will involve the four interacting processes consisting of liquid-phase transport, gasphase transport, phase change at liquid –gas interface, and chemical reactions in the gas phase. The last component is absent for pure vaporization.
In this situation the droplet is treated
as a constant source of single-component fuel vapor, implying that we do not need to be concerned with the heat and mass transfer processes within the droplet interior.
Furthermore, external forced and natural convection is absent such that spherical symmetry prevails.
Finally, because of the significant density disparity between liquid and gas, the liquid possesses great inertia such that its properties at the droplet surface, for example, the regression rate as well as the temperature and species concentrations in more complicated situations, change at rates much slower than those of the gas-phase transport processes. Since the ambience is also
assumed to be constant, the gas-phase processes can therefore be treated as steady, with the boundary variations occurring at longer time scales （τliquid , τsurface >>τgas ，因此在处理气相问题时可以把液面处的物理过程看成是与时间无关的。

）. This is called the quasisteady assumption , which is frequently invoked in problems involving gasification and combustion of condensed-phase materials. 为求解这个问题，假设:
● 单组分液滴，忽略液滴内部输运
● 液滴周围与环境无相对速度，只有斯蒂芬流引起的球对称径向一维流动 ● 忽略热辐射和离解
● 过程是准定常的，即不考虑表面的内移效应
● 由于一般燃料的DI 数较大，故可以将火焰面简化为几何面 ✓ 燃料气由液滴表面向火焰面扩散，但不能穿过火焰面 ✓ 氧气由环境向火焰面扩散，但不能穿过火焰面 ✓ 即在火焰面上有： YO =YF = 0 ✓ 燃烧产物则由火焰面分别向液滴表面和环境扩散
图 11-1液滴蒸发与燃烧的物理模型
这样就形成了如图 11-1所示的物理模型。

首先求解蒸发问题。

d2-Law of Droplet Vaporization
Here a droplet of radius rs vaporizes in an environment of temperature T ∞ and mass fraction Y1,∞ of the vaporizing species 1, which we shall designate as the fuel; the rest is air. Thus in the case of vaporization the relatively cold droplet receives heat from the hot ambience and gasifies. The gasified fuel is transported to the ambience, which has a lower concentration of the fuel vapor. The transport is through both diffusion and Stefan convection, causing the continuous “shrinking” of the droplet. The reverse holds for condensation. We aim to determine the vaporization or condensation rate.
Continuity, d(r 2ρu)/dr = 0, yields the constant mass flow rate
24v m r u =
(11.9)
where u is the radial velocity. The fuel conservation equation is
2112d d 1d d d d Y Y u
r D r r r r ⎛⎫
= ⎪⎝⎭
(11.10)
两边乘4πr 2，物理意义更清楚：
22
11d d d 44d d d Y Y r u
r D r r r ⎛⎫= ⎪⎝⎭
(11.11)
利用(11.9)，上式变为
22
11d d 440d d Y r uY r D r r ⎛⎫-= ⎪⎝⎭
(11.12)
integrating twice,
()21
11112d 4d 1
ln 4v v v
Y m Y r D
C r
D
m Y C C r m -=-=-+ (11.13)
subject to the boundary conditions Y1(rs) = Y1,s and Y1(∞) = Y1,∞ yields
()
()
()111,1,1,exp 11
exp 1
v s s v m r Y Y Y Y m -∞⎡⎤--⎣⎦=+--
(11.14)
where we have defined the nondimensional quantities []4v v s m m Dr =，s r r r =。

假定Le
数为1，则
(
)4
v v p s m m c r ⎡⎤=⎣⎦
(11.15)
由于净质量流率 = 燃料的蒸发速率 = 燃料的输运速率，所以有
21
1d 4d v v Y m m Y r D
r
=- (11.16)
上式除以4s Dr ，变为
2
1
1d d v v Y m m Y r r
=- (11.17)
将(11.14)代入(11.17)可得蒸发速率为 (),ln 1v m v m B =+
(11.18)
其中
1,1,,1,1s m v s
Y Y B Y ∞-=
- (11.19)
为Spalding 传质数。

This shows that the mass transfer number so identified is a universal parameter independent of the geometry of the system.
Working with energy conservation it can also be shown that
(),ln 1v h v m B =+
(11.20)
(),p s h v v
c T T B q ∞-=
(11.21)
,h v B 为Spalding 传热数。

Expressing Eq. (11.20)in dimensional form,
(),4
ln 1v s h v p
m r B c =+
(11.22)
so v
s m r .
Finally, we need to determine the instantaneous droplet size, rs. By definition mv is the rate of change of the droplet mass,
3d 4d 3v s l m r t ⎛⎫=-
⎪⎝⎭
(11.23)
Assuming ρl= constant,
2
d 2
d s v l s
r m r t
=- (11.24)
Equating Eqs.(11.24) and(11.22), we have
()2
,d 2ln 1d p
s h v l
c r B t
=-+
(11.25)
Noting that the quantity on the RHS of Eq. (11.25)is independent of the droplet size, a vaporization rate constant can be defined as
(),2
ln 1p
v h v l
c K B =+
(11.26)
which is also called the surface regression rate when it is multiplied by 4π. Equation (6.4.13) can be readily integrated, using the initial condition that rs = rs,o at t = 0, yielding the instantaneous droplet size
22,0s s v r r K t =-
(11.27)
Equation (11.27) shows that the square of the
droplet radius decreases linearly with time (Figure 6.4.2). This is the d2-law of droplet vaporization, where d stands for the droplet diameter. This result is physically reasonable because the phenomenon of interest is a spherically symmetric, diffusion-controlled process in which quantities vary with the surface
area of the propagating sphere of influence. The d2-variation has been
repeatedly shown through experimentation to be largely correct.
The time τv needed to completel y vaporize a droplet of initial size rs,o is obtained by setting rs = 0
2,0s v
v
r K =
(11.28)
which illustrates the importance of fine atomization in that the
time to achieve complete
vaporization decreases quadratically with the initial droplet size.
d2-Law of Droplet Combustion
图 11-2 Schematic showing spherically symmetric droplet combustion.
图 11-3 Concentration and temperature profiles for the reaction-sheet combustion of a droplet.
图 11-2shows the spherically symmetric droplet combustion process.The droplet gasification mechanism is basically the same as that of droplet vaporization, except now the heat source is the flame instead of the ambience. The flame also serves as the sink for the outwardly transported fuel vapor and inwardly transported oxidizer gas. 图 11-3shows representative temperature and concentration profiles.
Continuity yields the constant mass burning rate mc, given by
24c m r u =
(11.29)
组分质量守恒
221,,i i
i dY dY d u
r D w i O F dr dr dr
r ⎛⎫
=-= ⎪⎝⎭
(11.30)
能量方程
221p
F c dT d dT uc r w q dr dr dr r ⎛⎫
=+ ⎪⎝⎭
(11.31)
For the reaction
F O P
F O P ''''+→ (11.32)
We define
()
()
F F F i i i i
W W '''-='''- (11.33)
O F
F O F,B O F,B
,Y Y Y Y Y Y =
= (11.34)
F,B
p c c T T q Y =
(11.35)
F,B Y is the boundary value of F Y 。

利用Le=1的假定，则组分质量方程和能量方程变为
22F,B 1,,i i F
p dY dY w d u r i O F dr dr c dr Y r ⎛⎫=-= ⎪ ⎪⎝⎭
(11.36)
22F,B
1F
p w dT d dT u r dr dr c dr Y r ⎛⎫=+ ⎪ ⎪⎝⎭ (11.37)
定义一个耦合函数
i
i T Y =+
(11.38)
则组分质量方程和能量方程变为一个关于耦合函数的方程
221i i
p d d d u r dr dr c dr r ⎛⎫= ⎪ ⎪⎝⎭ (11.39)
利用质量守恒(11.29)，上式变为
22
440i
i
p d d r u r dr c dr
⎛
⎫-= ⎪ ⎪⎝
⎭
(11.40)
对上式积分两次，有
()2
1,1,2,d d exp i
i c i i c
i i c r c m r
c m r c m r
=-+⎛⎫
=+-
⎪⎝⎭
(11.41)
where (
)4
,c c p s s m m c r r r r ⎡⎤==⎣
⎦ and c1,i and c2,i are the integration constants to be
evaluated by applying the boundary conditions, ,,,1
11:
,0,d d 1:0,d d d ,d O O F O F c O s c F s c r r c v s r r Y Y Y T T Y Y r m Y m Y m r r T m q T T r ∞∞===→∞===⎛⎫⎛⎫→-=-= ⎪ ⎪⎝⎭⎝⎭⎛⎫== ⎪⎝⎭ (11.42) where v v c q q q =,v q 气化潜热，而c q 为燃料发热量。

令F,B 1Y =，则,F F O O O Y Y Y Y ==。

The first three relations in (11.42) respectively state that, at the droplet surface, the net oxidizer convective –diffusive transport vanishes because there is no oxidizer penetration into the liquid, the net fuel vapor transport is the fuel gasification rate, and the heat conduction from the flame is used to effect fuel gasification.
Applying the boundary conditions to Eqs. (11.41), for i = O, F, we obtain
()(){},c m r O O s v s v O T Y T q T T q Y e -∞∞⎡⎤=+=-+--+⎣⎦ (11.43) ()(){}11c m r
F F s v s v T Y T q T T q e -∞⎡⎤⎡⎤=+=+-+-+-⎣⎦⎣⎦ (11.44)
Equations (11.43) and (11.44) show that ( ˜T s − q˜v) appears as a group, providing an indication of the energy levels of the problem. In particular, it implies that the need for gasification is equivalent to lowering the droplet enthalpy ˜T s by the latent heat of gasification q˜v. This is physically reasonable.
To determine the mass burning rate m˜ c, the reaction sheet standoff ratio r˜f , and the flame temperature ˜Tf, we now apply the reaction sheet requirements,
()()0,0O f F f Y r Y r == (11.45) Furthermore, as a consequence of ()0O f Y r =， the relation
()10O Y = (11.46) obviously also holds because if there is no leakage of the oxidizer across the reaction sheet, the oxidizer concentration in the inner region to the reaction sheet, in particular that at the droplet surface, must also vanish. Applying Eqs. (11.45)and (11.46) to Eqs. (11.43)and(11.44), we obtain (),ln 1c h c m B =+
(11.47) ()()(),,ln 11ln 1ln 1f s v c f O O T T q m r Y Y ∞∞⎡⎤+-⎣⎦==+++ (11.48)
()(),,1O c v p f s p f O O O Y q q c T T c T T Y ∞∞∞⎛⎫-=+-+-+ ⎪ ⎪⎝⎭ (11.49)
In the above,
()(),,,s O p s c O O h c v v T T Y c T T q Y B q q ∞∞∞∞-+-+== (11.50)
is the heat transfer number for combustion. Compared to the heat transfer number for pure vaporization, B h,v given by Eq. (6.3.15), we see that the driving potential for gasification now consists of an additional source, (YO,∞/σO)qc, representing chemical heat release. For combustion of practical fuels, the chemical contribution is usually much larger than the thermal contribution, cp(T ∞ − Ts ), especially when th e environment is cold such that T ∞ is close to Ts . Numerical values of B h,c typically range between 1 and 10.
The flame temperature is deliberately expressed in the dimensional, implicit form of Eq. (11.49)in order to demonstrate the fact that it is again the stoichiometric adiabatic flame temperature of the system. Specifically, Eq. (11.49) shows that the heat release qc by unit mass of fuel is equal to the amount needed to first gasify it and then heat it from the droplet temperature Ts t o the flame temperature Tf , plus the amount needed to heat the stoichiometric, σO unit of oxidizer and the remaining [(1 − YO,∞)/YO,∞]σO unit of inert from the ambient temperature T ∞ to the flame temperature Tf 。

The above expressions are defined to within ˜Ts （Ts 还需待定）. Following the same procedure as discussed in Section 6.3 for pure vaporization, ˜Ts can b e determined by evaluating Eq. (11.44) at ˜ r = 1 to yield an expression for YF,s = YF,s( ˜Ts ),
(),11c m F s v s v Y q T T q e -∞⎡⎤=-+-+-⎣⎦ (11.51) From Eq.(11.51), a mass transfer number Bm,c can be defined through m˜ c = ln(1 + Bm,c) such that by equating Bh,c and Bm,c, and by using the Clausius –Clapeyron relation, ˜Ts can be iteratively solved.
An accurate knowledge of ˜Ts , however, is frequently not necessary in the evaluation of the bulk combustion parameters ˜ mc, ˜rf , and ˜T f because the enthalpy contribution from Ts is usually much smaller than the chemical source term, as just mentioned. Furthermore, realizing that the droplet is expected to be close to its boiling state under the situation of intense heating during steady burning, it is then frequently adequate to assume that
s b T T = (11.52) where Tb is the liquid’s boiling point under the prevailing pressure. Using Eq. (6.4.33), the bulk combustion parameters can be obtained through straightforward evaluation. It is, however, also important to recognize that the adoption of Eq. (6.4.33) falsifies the phase-change process and thereby necessitates the abandonment of the proper phase change description, for example the Clausius –Clapeyron relation, and all of its physical implications. Indeed, because of the presence of species other than fuel vapor at the droplet surface, for example nitrogen from the air and the combustion products generated at the reaction zone, the state of boiling can never be attained for the droplet, theoretically as well as in realistic situations.
It is also of interest to note that the present results specialize to those of pure vaporization, in an environment free of fuel vapor.
It is appropriate to recognize at this point the similarity between droplet vaporization and droplet combustion. Apart from the gas-phase reactions, the gasification process at the droplet surface is qualitatively the same in both cases.Thus during combustion, the droplet simply
perceives the flame as a hot “ambience” located at ˜ r f . Consequently, understanding gained from studying droplet combustion frequently can be applied to the modeling of droplet vaporization. Indeed, from an experimental design point of view, droplet vaporization in a high temperature environment can be usefully simulated by studying droplet burning in a cold environment. The flame now conveniently serves as a high-temperature, constant-pressure “chamber” within which vaporization takes place.
To relate m˜ c to the rate of decrease of the droplet size, by definition the droplet gasification rate m˜ v is given by Eq. (11.23).If we now assume that the instantaneous rate of fuel gasification at the droplet surface is equal to that of fuel consumption at the reaction sheet, then
()()3,d 44ln 1d 3v s l c p s h c c m r m c r B t ⎛⎫=-==+ ⎪⎝⎭ (11.53) Then we have
2d d s c r K t =- (11.54) where ()
(),2ln 1p c h c l c K B =+ (11.55)
is the droplet burning rate constant. Integrating Eq. (11.54)with the initial condition rs(t = 0) = rs,o yields
22,s s o c r r K t =- (11.56) which is the analogue of the d2-law for droplet vaporization. It is also of interest to note that since (λ/cp)g ∼ (ρD)g, and since Bc = O(1 ∼ 10) such that ln(1 + Bh,c) = O(1), Eq. (11.55)shows that
g g c
l K D (11.57)
where the subscript g designates gas-phase property. U nder atmospheric pressure (ρg/ρl) = O(10−3 ∼ 10−2), therefore the rate of surface regression is much slower than that of gas-phase diffusion.This is in agreement with the assumption of gas-phase quasi-steadiness, as it should be. （但在高压下(ρg/ρl) ∼1，准定常假定就不适用了）Furthermore, if we take Dg = O(100 cm2/sec), then Kc = O(10−3 ∼ 10−2 cm2/sec), which is the typical order of the burning rate constants for fuel droplets determined experimentally. For the pure vaporization case, the above estimates need to be modified by the factor ln(1 + Bh,v) ≈ Bh,v because it is usually less than unity.
Summarizing the above results, the d2-law states that, during quasi-steady droplet combustion, the droplet surface regression rate, the reaction sheet standoff ratio (r f /rs ), and the flame temperature (Tf ), remain as constants, and that Tf is also the stoichiometric adiabatic flame temperature of the fuel –oxidizer system.
Experimental Results of d2-Law
图11-4Experimental measurements of the droplet and flame sizes for an octane droplet burning in the air environment at 0.1–0.15 atm pressure (Law, Chung & Srinivasan 1980).
图11-4show the experimental data (Law, Chung & Srinivasan 1980) on the temporal variations of the square of the normalized droplet radius, R2 s = (rs/rs,o)2, the nondimensional flame radius, Rf = (r f /rs,o), and the flamefront standoff ratio, ˜ rf = (r f /rs ), for the spark-ignited, nearly spherically symmetric burning of an octane droplet in air and pure oxygen environments, respectively. The experiments were conducted under reduced pressure and thereby reduced buoyancy situations. The flame location was taken to be the midpoint of the luminous zone.
The experimental results show that after ignition a period exists during which the burning rate is very slow as indicated by the almost lack of variation in R2. This period is generally quite short, spanning about 5 to 10 percent ofthe droplet lifetime depending on the fuel boiling point and the ignition duration. After this initial period R2 s varies almost linearly with time. In the pure oxygen environment it is so short that it falls within the period during which the droplet has not resumed its spherical shape from the disturbance caused by the spark discharge.
The flamefront standoff ratio exhibits two distinct behaviors. At low oxygen concentrations, ˜ rf continuously increases, with the increase being actually faster towards the end of the droplet lifetime. For high oxygen concentrations, the increase instead levels off. The actual flame size, Rf , first increases and then decreases for both cases.
The experimental results show a number of inadequacies of the d2-law that, however, have all been satisfactorily explained. The presence of the short, initial period during which the droplet size hardly changes, indicating very slow rate of droplet gasification, signifies the need to heat up the initially cold droplet to close to the liquid boiling point for steady burning. Thus much of the heat arriving the droplet surface is used for droplet heating instead of liquid gasification. This feature is not captured by the d2-law which does not describe influences due to initial conditions except the initial droplet size.
The movement of the flamefront is also a consequence of suppressing the initial condition. Here the d2-law prescribes that upon ignition, the flame instantaneously assumes its quasi-steady value of ˜ rf . However, in order to support a flame of this size, considerable amount of fuel vapor needs to be present in the inner region to the flame in accordance to the d2-law fuel vapor concentration profile. This amount of fuel vapor is not present initially and therefore needs to be
gradually built up, leading to the corresponding gradual increase in the flame size. In a low-oxygen environment the amount of fuel vapor that needs to be accumulated is large because of the larger flame size; hence the continuous increase of ˜ rf as observed recognizing, of course, that rs also continuously decreases. By the same reasoning ˜ rf levels off in a high-oxygen environment because of the smaller ˜ rf
A corollary of this fuel vapor accumulation phenomenon is that because part of the fuel gasified is accumulated in the inner region to the flame instead of being instantaneously reacted, mv is actually not equal to mc. Thus the heat release rate from the droplet flame cannot be directly related to the droplet size in the manner of Eq. (6.4.42). This can have significant implications in the modeling of spray combustion.
The d2-law prediction of the flamefront standoff ratio has also been found to be much larger than experimentally observed value. Such a large discrepancy has been found to be caused by the unity Lewis number assumption in the d2-law. A much smaller flame size is predicted by allowing for unequal rates of heat and mass diffusion, especially for the slow diffusion rate of the large fuel molecules in the inner region. Physically, it is reasonable to expect that a slower fuel diffusion rate would lead to the flame located closer to the fuel region.
Droplet Heating
We have shown that during the steady vaporization and combustion of a purecomponent droplet the droplet temperature assumes a unique value for a given system. This temperature is usually much higher than the initial droplet temperature at the instant of injection or ignition. Therefore there must exist a transient droplet heating period during which the heat transferred to the droplet surface is used for both gasification as well as droplet heating, causing a reduction in the droplet gasification rate.
Since droplet heating involves the change of a liquid property, and therefore occurs over a longer period than that of gas-phase transport, gas-phase quasi-steadiness can still be assumed. Thus the only modification of the gas-phase solution such as the d2-law is to substitute qv by an effective latent heat of gasification, qv,eff, defined as
Evaluation of the last term requires knowledge of the droplet temperature distribution T(r, t). In the absence of internal recirculatory motion, the unsteady heat transfer process within the droplet is simply given by the spherically symmetric heat conduction equation
subject to
There are three sources of unsteadiness in the above equations, namely the unsteady conduction term, the continuously regressing droplet surface rs(t), and the continuously changing surface temperature Ts(t). To simplify the analysis, frequently the droplet temperature is assumed
to be spatially uniform but temporally varying (Law 1976). Then energy conservation at the droplet surface is simply
The above two models represent extreme rates of liquid-phase transport. Equation (13.2.2) only allows heat diffusion, which is always present, and is therefore the slowest limit. On the other hand Eq. (13.2.4) implies that the conductivity is infinitely large such that spatial variations are perpetually uniformized. Thus it represents the fastest possible limit. These two models are respectively referred to as the diffusion limit and the infinite conductivity limit; the latter is also conventionally called the batch distillation limit because it is analogous to the chemical process of distillation.
图11-5Temporal variation of the surface and center temperatures of an octane droplet after ignition, demonstrating the rapidity in the heating up of the droplet surface layer in both the diffusion and distillation limits.
In图11-5, predictions from both models are shown for an octane droplet burning in the standard atmosphere, with an initial droplet temperature of 300 K. 图11-5 shows the variations of the surface and center temperatures with a nondimensional time ˜ t = [(ρgDg)/(ρlr 2 s,o)]t. It is seen that in the diffusion limit the surface temperature initially increases rapidly while the core region slowly starts to heat up. This heating may or may not persist throughout the droplet lifetime, depending on the liquid-phase thermal diffusivity. In the distillation limit the increase in the uniform droplet temperature essentially follows that of the surface temperature in the diffusion limit. The increase is initially slower because of the additional heat needed for the core region, although heating of the complete droplet is finished earlier.
Summarizing, the following conclusions can be made regarding droplet heating for a single-component fuel in a constant environment, whose pressure is also sufficiently below the critical pressure. First, active droplet heating and gasification occur somewhat sequentially, with the former mostly over in the initial 5–10 percent of the droplet lifetime depending on the fuel volatility and the initial droplet temperature. The fact that these two processes occur somewhat sequentially is also physically reasonable because active droplet heating takes place when the droplet temperature is low.Alow droplet temperature implies a low fuel vapor concentration at the surface and consequently a slower gasification rate. As the droplet temperature is increased close。