# Scaling Analysis on Pulsating Flame Spread over Liquids

Authors: Kozue Takahashi; Akihiko Ito; Yuji Kudo; Tadashi Konishi; Kozo Saito
Journal: International Journal of Chemical Engineering
 (2008)

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## Abstract

Scaling analyses based on subsurface layer instability were performed to explore the role of three independent (surface tension, gravity, and viscosity) influences on the mechanism of pulsating flame spread under normal and microgravity conditions. These three influences form two independent pi-numbers: the Marangoni (Ma) number and Grashof (Gr) number, which include the characteristic length scale ratio (depth of subsurface circulation)/(horizontal length of preheated liquid surface). The Prandtl (Pr) number was introduced to compensate for the different thermal diffusivity and kinematic viscosity of different liquids. Also a nondimensional flame spread rate,V/VD (= Vδ/D, where δ is the quenching distance and D is the diffusivity of fuel vapor) was introduced. Using these nondimensional parameters, the flame spread mechanism was divided into two separate regimes: for the shallow liquid pool the nondimensional flame spread rate was correlated with {Gr0.2/(Ma·Pr)}1.0, while for the deep liquid pool it was correlated with {Gr0.2/(Ma·Pr)}1.5.


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## Body

## 1. Introduction

Flame
spread over liquids is one of our current interests, because of both its relevance to fire safety and our basic
curiosity about its complex mechanism. A series of studies was performed to
understand the flame spread mechanism on alcohols [1–12] and hydrocarbon fuels
[4, 10]. Based on these results, Glassman and Dryer [13] wrote a review that
includes explanations of the mechanism of pulsating spread over subflash point
liquids. According to Glassman and Dryer [13], pulsating flame spread could be a
result of coupled gas-liquid phase convection effects, since it occurs only
when subsurface liquid convection has been generated. Recently, Ross and Miller
conducted flame spread experiments in microgravity on shallow liquid pools [14, 15] and on deep liquid pools [16]. Including these results for microgravity
flame spread, Ross wrote a review in a book he edited [17] stating that the
rate and frequency in pulsating flame spread depend on gravity, liquid pool
depth, initial fuel temperature, gas flow rate, the kind of liquid, and so on, indicating
that the mechanism of pulsating flame spread is not yet completely understood.Over the past ten years, a series of experimental studies
of flame spread over liquids has been conducted [14–29], creating a rich
experimental database. With six different experimental techniques (laser sheet
particle tracking, smoke tracing, single and dual wavelength holographic
interferometry, infrared thermography, and high-speed photography), we measured
the detailed thermal, fluid dynamic and chemical structures of both gas, and
liquid phases near the flame's leading edge to understand the mechanism of pulsating
flame spread [26–29]. We found a dual pulsation structure for flame pulsation,
consisting of a main pulsation of about 0.5 to 1 Hz, and a subpulsation of about 5 to 10 Hz [29]. The main pulsation generated and eliminated a
cold temperature valley in the liquid surface ahead of the flame [28]. Based on
these experimental data, we proposed a pulsating flame spread model [28]
involving subsurface layer circulation and a surface wave. Figure 1 shows a
series of photographs of a pulsating flame and schematic side views of liquid
and gas flow structures for the four steps of the cycle (a)–(d). Step (a) is the
onset of pulsation. There is little flow in the liquid phase, while there is
air flow that is induced by buoyancy ahead of the flame leading edge. The flame
spreads slowly in this crawl phase of the pulsation cycle. Step (b) is the
formation of a cold temperature valley. Rapid interfacial flow develops due to thermocapillarity
toward the upstream direction and produces convective motion inside the bulk
liquid and also a small-scale wave on the liquid surface. A cold temperature
valley where the minimum temperature is below the flash point exists ahead of
the flame leading edge. Step (c) is fuel vapor buildup. Some of the warm liquid
flows on the surface ahead of the flame, the upstream liquid fuel surface
temperature rises above the flash point and then the cold temperature valley
disappears. The liquid fuel evaporates and diffuses into the recirculation
cell. The fuel vapor in the cell accumulates with time and reaches a nearly
uniform flammable concentration. Step (d) is flame jumping. When the height of
the flammable lean limit exceeds that of the quenching layer, the flame jumps
forward to the flammable layer. In the process of jumping, the combusting gases
expand thermally and destroy the recirculation cell. The flame front nearly
stops at the end of the flammable layer and then begins to spread slowly [returns to step (a)] and the cycle repeats itself.Step in a single
cycle of pulsating flame spread and surface wave in step (b) [29].To
benefit from all these experimental and numerical studies both under normal
gravity and microgravity and to update the theory of flame spread over liquids
formulated by Williams [30], we conducted stability based on the basic
reference [31] and scaling analyses on these results in the hope of finding a
unified theoretical framework within
which to relate them [32–34]. Our instability analysis was focused on the
liquid subsurface layer ahead of a flame’s leading edge, where a surface wave
is initiated and a cold temperature valley is created. The four pi-numbers,
Marangoni (Ma) number, Grashof (Gr) number, Weber (We) number, and Froude (Fr) number, and a characteristic length
scale ratio (hT/L) were introduced for scaling analysis.
The characteristic length scale ratio, hT/L consists of a characteristic thermal
fluid length in the liquid surface, and a subsurface liquid circulation (or
thermal structure) depth both created by the spreading flame. Figure 2 shows
thermal and flow characteristic depths hT and hF, and thermal characteristic length L, determined from a shadowgraph image
and a particle-track laser-sheet image [34].Figure 2
A shadowgraph
and a particle laser-sheet image defining thermal characteristic length,L, and thermal and flow depth, hT and hF, of subsurface layer circulation (fuel is n-butanol, H=50mm and the initial fuel
temperature =23.2C°).William
used the liquid pool depth as an approximation for the characteristic length in
his flame spread theory and pointed out that it is both possible and desirable
to pursue more careful analysis to obtain improved estimates of spread rate
using an appropriate characteristic length [30]. In this study, we
experimentally measured both thermal and fluid characteristic lengths and
depths and improved our previous instability and scaling analyses using these
results. We propose a new formula for the nondimensional flame spread rate as a
function of Gr0.2/(Ma⋅Pr) on both deep and shallow pools, and also we discuss the effect
of gravity on pulsating flame spread.


## 2. Theoretical Analysis

### 2.1. Basic Equation

Our
objective is to study the instability of laminar subsurface layer flow ahead of
a spreading flame. The temperature coefficient of surface tension and other
physical properties are treated as constants. Our instability analysis is based
on the previous study [31–34]. A schematic of subsurface layer flow is
shown in Figure 3. The depth of subsurface layer [18] that we conduct our
analysis is an order of 2 mm (very thin); we use the fully developed flow
assumption. All symbols used are defined in the nomenclature section. The
liquid temperature just ahead of the flame leading edge is higher than the bulk
liquid temperature because the high-temperature liquid generated by the flame
moves in the upstream direction due to surface tension. The continuity and
momentum equations for the subsurface layer can be, respectively, expressed as (1)ht+(hu)x=0,(2)(hu)t+(∫0hU2dy+12gh2)x+(σk2δhρ)x=1ρ(τi−τb). The
third term on the LHS in (2) is due to surface tension, where δ=Aexpik(x−ct) is the
expression for the surface wave, k(≡2π/λ) is the wave number, λ is the wavelength, c is the
wave velocity, and all other symbols are defined in the nomenclature section.
In (2), τi is
the surface shear stress due to the Marangoni effect and is expressed as (3)τi=∂σ∂x=σT∂T∂x,σT≡dσdT. If the
momentum displacement thickness: β(≡∫0hU2dy/u2h) and the continuity equation, (1), are substituted into (2), we can
obtain the following equation: (4)hut+(2β−1)huux+(β−1)u2hx+βxu2h+ghhx+σk2ρ(δxh+δhx)=1ρ(τi−τb). The
perturbation at the gas-liquid interface is very small and the independent
variables (u,h,τ) are expressed as (5)u=u¯+u′,τb=τ¯b+τb′,h=h¯+δ,τi=τ¯i+τi′, where
the superscripts bar and prime indicate time-averaged and perturbation
quantities, respectively. If the velocity remains similar on perturbation, then β=β¯. With the substitution of (5) and (1) into (4),
and if the velocity is fully developed, u¯x=h¯x=0,τ¯i=0,τ¯b=0,     and the higher-order perturbation terms are
negligible; the continuity and momentum equations can then, respectively, be
written as (6)δt+u¯δx+h¯ux′=0,(7)ut′+(2β¯−1)u¯ux′+(β¯−1)u¯2δxh¯+gδx+σk2ρδx=1ρh¯(τi′−τb′). If τ¯b′ is small, the deviation from the time-averaged value τ¯b=μ(du¯/dy)|y=0=3μ(u¯/h¯),
which may be written as (8)τb′=τb−τ¯b≒3μ(uh−u¯h¯)≒3μu′h¯. This implies that no heat is stored in the
subsurface layer. All the heat is transferred at the gas-1iquid interface. If
the flow is laminar, T¯i=T¯b−α(qi/λl)h¯, where T¯b is the time-averaged temperature in the
liquid, λl is the thermal conductivity of the liquid, and α is the heat release factor
(a positive sign indicates that heat is released from the surface to the gas
phase, negative indicates that heat is absorbed at the surface). If the
perturbation of the surface temperature is very small, then (9)Ti=T¯b−αqiλlh=T¯b−αqiλl(h¯+δ). Differentiating Ti with respect to x, and substituting into (3), we obtain (10)τi′=−αqiλlσTδx. By
substituting (8) and (10) into (7) and differentiating with respect to x and
using (6) to eliminate u, we can obtain the following wave equation: (11)(∂∂t+C+∂∂x)(∂∂t+C−∂∂x)δ+3μρh¯2(∂∂t+C0∂∂x)δ=0, where (12)C±=β¯u¯±(β¯−1)β¯u¯2+gh¯+σk2h¯ρ+ασTqiρλl,C0=u¯. The
first term on the LHS of (11) represents the dynamic wave, where C+ and C−, respectively, represent the
forward and reverse propagation velocities. The second term on the LHS of (11)
represents the kinematic wave, where C0 is the propagation
velocity in the direction of flow. Equation (11) represents the surface wave as
a linear combination of the dynamic wave and the kinematic wave. The Marangoni
effect appears in the C± term in the dynamic
wave expression.Figure 3
Subsurface layer flow model and symbols.If the surface perturbation
is expressed asδ=aexpi(kx−ωt), we
obtain the following equation from (11): (13)(ω−kC−)(ω−kC+)+i3μρh¯2(ω−kC0)=0. Hence, ω may be approximated as (14)ω≅{kC+−i(3μ/ρh¯2)(C+−C0)(C+−C−)+kC−−i(3μ/ρh¯2)(C0−C−)(C+−C−)}. If Im(ω)>0,  the dynamic wave is unstable, resulting in C0≧C+. This
then becomes a neutrally stable condition, C0=C+. The following equation describes
the generation of the surface wave: (15)(1−β¯)u¯2=gh¯+σk2h¯ρ+ασTqiρλl. If the
local liquid velocity is defined as U=(US/h¯2)y2(0≤y≤h¯), the liquid
volumetric flow rate per unit width of the tray is Γ=∫0h¯Udy=(1/3)USh¯, the
average liquid velocity is u¯=∫0h¯Udy/h¯=(1/3)US, the
momentum displacement thickness is β=∫0h¯U2dy/u2h¯=9/5, and the heat release factor is α=1. Equation (15) can be rewritten as (16)−31/3415Γcr4/3−31/3(g+σk2ρ)Γcr1/3=σTρλlqi, where Γcr is the critical flow rate required for surface
wave to occur.


### 2.2. Conditions for Generation of a Surface Wave

The neutral stable line obtained from (16) is shown in
Figure 4 for three different wavelengths into a parameter. The following values
are used for calculation: σ=24.17×10−3N/m,σT=−8.16×10−5N/m⋅K,ρ=809kg/m3,λi=149×10−3W/m⋅K, and α=1. As
the hot liquid flows ahead of the flame leading edge due to Marangoni force,
the liquid surface temperature is higher than the gas phase temperature. The qi in horizontal in Figure 4 is positive for heat release from the liquid
surface to the gas phase.Figure 4
The neutral stable line.The area above the neutral stable
line in Figure4 shows an unstable region, which corresponds to the region,
where the surface wave is generated. Theoretical considerations show the
following: (1) shorter wavelength perturbations can easily develop into a
surface wave, (2) the liquid surface is more stable as the heat flux from the
liquid surface to the gas phase increases.In experiments, uniform flame spread changes to
pulsation at a certain temperature below the flash point. The Marangoni force
is proportional to the temperature difference between the hot zone underneath
the flame leading edge and the bulk liquid temperature. When the liquid bulk temperature
is lower, the Marangoni force is greater. This may lead to a subsurface flow
rate increase and the generation of a surface wave.


### 2.3. The Nondimensional Parameter

An
overall energy balance and an overall momentum balance for the subsurface layer,
respectively, produce the following:(17)qi=λlΔTαhT,μUShF=σTΔTL+βρgΔTLhT2, where L and hT are the thermal characteristic length and
depth and hF is the flow
characteristic depth of the subsurface layer circulation defined in Figure 2. The characteristic depth scale ratio is
expressed as (18)γ=hFhT. The difference between the thermal characteristic depth
and flow depth is discussed in Section 4.2. Using (17) and (18), the
following nondimensional parameters are introduced: (19)Fr≡γσTΔT2μg1hThTL,(20)We≡γ2ρσT2ΔT24μ2σλ(hTL)2,(21)Ma≡γ2ρσTΔT4μ2hT(hTL)2,(22)Gr≡γ2β2ρ3g2ΔT4μ2σThT5(hTL)2.
Using (19), (20), (21), and (22),  (15) can be rewritten in this nondimensional form: (23)1−β¯=1Fr2+4π2h¯λ1We−{1Ma+1Gr}2. Equations (19), (20), (21), and (22), all having the
characteristic length scale ratio in their expressions, describe the generation
of a liquid surface wave and its propagation. Therefore, these four equations
may predict the onset of flame pulsation and subsequent pulsating flame spread
over liquids. These predictions will be validated by other researchers’
experimental data as well as our own past and future experiments.


## 2.1. Basic Equation

Our
objective is to study the instability of laminar subsurface layer flow ahead of
a spreading flame. The temperature coefficient of surface tension and other
physical properties are treated as constants. Our instability analysis is based
on the previous study [31–34]. A schematic of subsurface layer flow is
shown in Figure 3. The depth of subsurface layer [18] that we conduct our
analysis is an order of 2 mm (very thin); we use the fully developed flow
assumption. All symbols used are defined in the nomenclature section. The
liquid temperature just ahead of the flame leading edge is higher than the bulk
liquid temperature because the high-temperature liquid generated by the flame
moves in the upstream direction due to surface tension. The continuity and
momentum equations for the subsurface layer can be, respectively, expressed as (1)ht+(hu)x=0,(2)(hu)t+(∫0hU2dy+12gh2)x+(σk2δhρ)x=1ρ(τi−τb). The
third term on the LHS in (2) is due to surface tension, where δ=Aexpik(x−ct) is the
expression for the surface wave, k(≡2π/λ) is the wave number, λ is the wavelength, c is the
wave velocity, and all other symbols are defined in the nomenclature section.
In (2), τi is
the surface shear stress due to the Marangoni effect and is expressed as (3)τi=∂σ∂x=σT∂T∂x,σT≡dσdT. If the
momentum displacement thickness: β(≡∫0hU2dy/u2h) and the continuity equation, (1), are substituted into (2), we can
obtain the following equation: (4)hut+(2β−1)huux+(β−1)u2hx+βxu2h+ghhx+σk2ρ(δxh+δhx)=1ρ(τi−τb). The
perturbation at the gas-liquid interface is very small and the independent
variables (u,h,τ) are expressed as (5)u=u¯+u′,τb=τ¯b+τb′,h=h¯+δ,τi=τ¯i+τi′, where
the superscripts bar and prime indicate time-averaged and perturbation
quantities, respectively. If the velocity remains similar on perturbation, then β=β¯. With the substitution of (5) and (1) into (4),
and if the velocity is fully developed, u¯x=h¯x=0,τ¯i=0,τ¯b=0,     and the higher-order perturbation terms are
negligible; the continuity and momentum equations can then, respectively, be
written as (6)δt+u¯δx+h¯ux′=0,(7)ut′+(2β¯−1)u¯ux′+(β¯−1)u¯2δxh¯+gδx+σk2ρδx=1ρh¯(τi′−τb′). If τ¯b′ is small, the deviation from the time-averaged value τ¯b=μ(du¯/dy)|y=0=3μ(u¯/h¯),
which may be written as (8)τb′=τb−τ¯b≒3μ(uh−u¯h¯)≒3μu′h¯. This implies that no heat is stored in the
subsurface layer. All the heat is transferred at the gas-1iquid interface. If
the flow is laminar, T¯i=T¯b−α(qi/λl)h¯, where T¯b is the time-averaged temperature in the
liquid, λl is the thermal conductivity of the liquid, and α is the heat release factor
(a positive sign indicates that heat is released from the surface to the gas
phase, negative indicates that heat is absorbed at the surface). If the
perturbation of the surface temperature is very small, then (9)Ti=T¯b−αqiλlh=T¯b−αqiλl(h¯+δ). Differentiating Ti with respect to x, and substituting into (3), we obtain (10)τi′=−αqiλlσTδx. By
substituting (8) and (10) into (7) and differentiating with respect to x and
using (6) to eliminate u, we can obtain the following wave equation: (11)(∂∂t+C+∂∂x)(∂∂t+C−∂∂x)δ+3μρh¯2(∂∂t+C0∂∂x)δ=0, where (12)C±=β¯u¯±(β¯−1)β¯u¯2+gh¯+σk2h¯ρ+ασTqiρλl,C0=u¯. The
first term on the LHS of (11) represents the dynamic wave, where C+ and C−, respectively, represent the
forward and reverse propagation velocities. The second term on the LHS of (11)
represents the kinematic wave, where C0 is the propagation
velocity in the direction of flow. Equation (11) represents the surface wave as
a linear combination of the dynamic wave and the kinematic wave. The Marangoni
effect appears in the C± term in the dynamic
wave expression.Figure 3
Subsurface layer flow model and symbols.If the surface perturbation
is expressed asδ=aexpi(kx−ωt), we
obtain the following equation from (11): (13)(ω−kC−)(ω−kC+)+i3μρh¯2(ω−kC0)=0. Hence, ω may be approximated as (14)ω≅{kC+−i(3μ/ρh¯2)(C+−C0)(C+−C−)+kC−−i(3μ/ρh¯2)(C0−C−)(C+−C−)}. If Im(ω)>0,  the dynamic wave is unstable, resulting in C0≧C+. This
then becomes a neutrally stable condition, C0=C+. The following equation describes
the generation of the surface wave: (15)(1−β¯)u¯2=gh¯+σk2h¯ρ+ασTqiρλl. If the
local liquid velocity is defined as U=(US/h¯2)y2(0≤y≤h¯), the liquid
volumetric flow rate per unit width of the tray is Γ=∫0h¯Udy=(1/3)USh¯, the
average liquid velocity is u¯=∫0h¯Udy/h¯=(1/3)US, the
momentum displacement thickness is β=∫0h¯U2dy/u2h¯=9/5, and the heat release factor is α=1. Equation (15) can be rewritten as (16)−31/3415Γcr4/3−31/3(g+σk2ρ)Γcr1/3=σTρλlqi, where Γcr is the critical flow rate required for surface
wave to occur.


## 2.2. Conditions for Generation of a Surface Wave

The neutral stable line obtained from (16) is shown in
Figure 4 for three different wavelengths into a parameter. The following values
are used for calculation: σ=24.17×10−3N/m,σT=−8.16×10−5N/m⋅K,ρ=809kg/m3,λi=149×10−3W/m⋅K, and α=1. As
the hot liquid flows ahead of the flame leading edge due to Marangoni force,
the liquid surface temperature is higher than the gas phase temperature. The qi in horizontal in Figure 4 is positive for heat release from the liquid
surface to the gas phase.Figure 4
The neutral stable line.The area above the neutral stable
line in Figure4 shows an unstable region, which corresponds to the region,
where the surface wave is generated. Theoretical considerations show the
following: (1) shorter wavelength perturbations can easily develop into a
surface wave, (2) the liquid surface is more stable as the heat flux from the
liquid surface to the gas phase increases.In experiments, uniform flame spread changes to
pulsation at a certain temperature below the flash point. The Marangoni force
is proportional to the temperature difference between the hot zone underneath
the flame leading edge and the bulk liquid temperature. When the liquid bulk temperature
is lower, the Marangoni force is greater. This may lead to a subsurface flow
rate increase and the generation of a surface wave.


## 2.3. The Nondimensional Parameter

An
overall energy balance and an overall momentum balance for the subsurface layer,
respectively, produce the following:(17)qi=λlΔTαhT,μUShF=σTΔTL+βρgΔTLhT2, where L and hT are the thermal characteristic length and
depth and hF is the flow
characteristic depth of the subsurface layer circulation defined in Figure 2. The characteristic depth scale ratio is
expressed as (18)γ=hFhT. The difference between the thermal characteristic depth
and flow depth is discussed in Section 4.2. Using (17) and (18), the
following nondimensional parameters are introduced: (19)Fr≡γσTΔT2μg1hThTL,(20)We≡γ2ρσT2ΔT24μ2σλ(hTL)2,(21)Ma≡γ2ρσTΔT4μ2hT(hTL)2,(22)Gr≡γ2β2ρ3g2ΔT4μ2σThT5(hTL)2.
Using (19), (20), (21), and (22),  (15) can be rewritten in this nondimensional form: (23)1−β¯=1Fr2+4π2h¯λ1We−{1Ma+1Gr}2. Equations (19), (20), (21), and (22), all having the
characteristic length scale ratio in their expressions, describe the generation
of a liquid surface wave and its propagation. Therefore, these four equations
may predict the onset of flame pulsation and subsequent pulsating flame spread
over liquids. These predictions will be validated by other researchers’
experimental data as well as our own past and future experiments.


## 3. Experimental Apparatus and Method

### 3.1. Normal Gravity Experiments

A schematic of the
experimental apparatus is shown in Figure5, including a high sensitivity
shadowgraph (HSSG) and a particle-track laser-sheet (PTLS) flow visualization
apparatus.Figure 5
Experimental apparatus.The fuel container is 480 mm long×20mm wide × seven different
heights (2,5,7,10,15,20, and 50 mm). Its bottom is brass and its sides
Pyrex. Both n-propanol and n-butanol were used as test fuels. The container was
filled to its limit in order to eliminate any meniscus. A small pilot flame at
one end of the tray initiated the flame that propagated to the other end. The
flame spread was recorded by a digital video camera. The flame spread rate was
measured from the recorded digital image.Temperature structures created in the liquid were visualized by
our specially designed high sensitivity shadowgraph (HSSG) [34], which measures
the refraction of a beam when it passes across the liquid. The 50 mW He-Ne
laser beam passes through a tracing paper and is changed into scattered light
in order to remove any distortion from the shadowgraph. For flow visualization
and velocity measurement in the liquid phase, a particle-track laser-sheet
(PTLS) technique was used [28], with a high-speed video camera (125 frames/sec,
20 mm ×20mm field of view). PTLS can measure profiles of both streamlines and
the two-dimensional velocity with significantly fewer particles and nearly
instantaneously. Using a 4 W argon-ion laser beam and a cylindrical lens, we
established a thin laser sheet with an approximately 35 degree opening angle.
The fuel was seeded in advance with hollow glass particles (5 μm average diameter, 1.02g/cm3 density) for
visualization of the flow field in the liquid fuel.


### 3.2. Microgravity Experiments

The microgravity
experiments were conducted using the 4.5-second drop tower facility at Microgravity
Laboratory of Japan (MGLAB). A 4.5-second period of microgravity is obtained by
allowing the experiment to free-fall in an evacuated tube through a distance of
about 150 m. A schematic of the test rack for microgravity tests is shown in
Figure6(a).Figure 6
Experimental apparatus for a shallow liquid pool in
microgravity.The
package housed the test rack, containing a wind tunnel, a fuel tray, a fuel
delivery syringe, three video cameras, and a gas control system. A fuel tray (250×20mm, 3 mm deep) was located inside the wind tunnel (7×7cm, 30 cm long, Figure 6(b)). The wind tunnel can provide a steady, laminar, forced air flow.
The averaged airflow rate is 5 cm/s. A hot-wire igniter was set up at the end
of the tray and 1 mm above the fuel surface. A thermocouple was inserted in the
fuel tray to measure the initial fuel temperature. A top view camera and two
side-view cameras recorded the spreading flame. One side-view camera recorded
the visible flame and another recorded an ultraviolet image to record the blue
flame under microgravity. The test rack was contained in a sealed test package
filled with dry air at 1 atm. Once the tray was filled with 24C° n-butanol fuel from a fuel delivery system, a 5 cm/s airflow opposing the direction
of the flame spread was started, and ignition occurred just before the package dropped.
The igniter was engaged until a flame appeared.


## 3.1. Normal Gravity Experiments

A schematic of the
experimental apparatus is shown in Figure5, including a high sensitivity
shadowgraph (HSSG) and a particle-track laser-sheet (PTLS) flow visualization
apparatus.Figure 5
Experimental apparatus.The fuel container is 480 mm long×20mm wide × seven different
heights (2,5,7,10,15,20, and 50 mm). Its bottom is brass and its sides
Pyrex. Both n-propanol and n-butanol were used as test fuels. The container was
filled to its limit in order to eliminate any meniscus. A small pilot flame at
one end of the tray initiated the flame that propagated to the other end. The
flame spread was recorded by a digital video camera. The flame spread rate was
measured from the recorded digital image.Temperature structures created in the liquid were visualized by
our specially designed high sensitivity shadowgraph (HSSG) [34], which measures
the refraction of a beam when it passes across the liquid. The 50 mW He-Ne
laser beam passes through a tracing paper and is changed into scattered light
in order to remove any distortion from the shadowgraph. For flow visualization
and velocity measurement in the liquid phase, a particle-track laser-sheet
(PTLS) technique was used [28], with a high-speed video camera (125 frames/sec,
20 mm ×20mm field of view). PTLS can measure profiles of both streamlines and
the two-dimensional velocity with significantly fewer particles and nearly
instantaneously. Using a 4 W argon-ion laser beam and a cylindrical lens, we
established a thin laser sheet with an approximately 35 degree opening angle.
The fuel was seeded in advance with hollow glass particles (5 μm average diameter, 1.02g/cm3 density) for
visualization of the flow field in the liquid fuel.


## 3.2. Microgravity Experiments

The microgravity
experiments were conducted using the 4.5-second drop tower facility at Microgravity
Laboratory of Japan (MGLAB). A 4.5-second period of microgravity is obtained by
allowing the experiment to free-fall in an evacuated tube through a distance of
about 150 m. A schematic of the test rack for microgravity tests is shown in
Figure6(a).Figure 6
Experimental apparatus for a shallow liquid pool in
microgravity.The
package housed the test rack, containing a wind tunnel, a fuel tray, a fuel
delivery syringe, three video cameras, and a gas control system. A fuel tray (250×20mm, 3 mm deep) was located inside the wind tunnel (7×7cm, 30 cm long, Figure 6(b)). The wind tunnel can provide a steady, laminar, forced air flow.
The averaged airflow rate is 5 cm/s. A hot-wire igniter was set up at the end
of the tray and 1 mm above the fuel surface. A thermocouple was inserted in the
fuel tray to measure the initial fuel temperature. A top view camera and two
side-view cameras recorded the spreading flame. One side-view camera recorded
the visible flame and another recorded an ultraviolet image to record the blue
flame under microgravity. The test rack was contained in a sealed test package
filled with dry air at 1 atm. Once the tray was filled with 24C° n-butanol fuel from a fuel delivery system, a 5 cm/s airflow opposing the direction
of the flame spread was started, and ignition occurred just before the package dropped.
The igniter was engaged until a flame appeared.


## 4. Results and Discussion

### 4.1. Analytical Results

Figure7 shows the critical Ma number versus the thermal subsurface layer depth (defined in Figure
2) with two constant We number cases
plotted: We=1.31 for normal
gravity, and We=0.89 for
microgravity. The We number is
calculated with (22) using the thermal characteristic depth and length of the
subsurface layer circulation of [34].Figure 7
Marangoni number
versus characteristic depth of subsurface layer circulation.The
critical Ma number under microgravity
is much larger than that under normal gravity, suggesting that when the liquid surface
becomes unstable and a surface wave is formed under microgravity, the valuehT will be larger (which means the diameter of subsurface liquid convection
will be larger). This possibility, if validated, could explain NASA’s
microgravity experimental results [16], which showed a large diameter
subsurface circulation for n-butanol compared with its normal gravity behavior
at the same temperature. The large subsurface circulation under
microgravity takes longer to reach the fully-developed state because gravity is
a major contributor to the forces that keep each element of a liquid in
communication with its neighbors. This means that the microgravity experiments
require a longer tray length to achieve a steady state condition than
experiments in normal gravity (which usually employ a 30 to 40 cm long tray).


### 4.2. Thermal and Fluid Scale

Figure8 shows the characteristic length scale ratio hT/L as a function of the
initial fuel depth (H) for n-propanol and n-butanol, measured by
the HSSG technique [34].Figure 8
Ratio of characteristic length and characteristic depth
of subsurface layer circulation versus initial fuel layer depth (fuel is
n-propanol and tray width = 20 mm).The quotienthT/L increased sharply for both n-propanol and
n-butanol with increasing initial fuel layer depth in the range H=2 to 5 mm and reached a very slow
increase for n-propanol with H=5 to
25 mm and for n-butanol with H=5 to
50 mm. This indicates that
there are two different regimes with a division at H=5mm: shallow liquid pools for H<5 mm, and deep liquid pools for H>5 mm. When the fuel
depth is less than 5 mm, the restricted subsurface liquid circulation
reduces hT and increases L. For an initial fuel depth
of H>5 mm, both the characteristic length L and the circulation depth hT increase with
increasing initial fuel depth and reach their constant values, when the deep liquid
pool condition is satisfied.The thermal field is targeted in the experiment, while the
theoretical analysis targets the flow field. Therefore, the shadowgraph and
PTLS methods were used to measure the thermal scale and the flow scale at the
same time. Figure9 shows hT (thermal
characteristic depth) and hF (flow characteristic depth) as functions of the initial fuel layer depth.Figure 9
Thermal and flow characteristic depth of subsurface
layer circulation versus initial fuel layer depth.The characteristic depth in n-butanol was larger than the
depth in n-propanol by a factor of about 1.4.  This value is close to the 1.3
ratio of the viscosity coefficients of n-propanol and n-butanol. As a result, when
a dimensionless number is used, it corrects it by the Prandtl (Pr)
number. The Pr number is defined as
follows:(24)Pr⁡=va, where, ν is the
kinematic viscosity, a is the thermal
diffusivity.


### 4.3. Scale Analysis of Flame Spread

The nondimensional flame spread rate may be
influenced by the quenching distanceδ, associated with the dark region formed between
the flame leading edge and the liquid surface, and the diffusivity, D, of fuel vapor. A nondimensional form
of the flame spread rate, V/VD, can therefore be written as follows: (25)VVD=VδD. This nondimensional flame
spread V/VD is the
same to the vapor Peclet number. We measured the quenching distance at δ=0.8mm in normal gravity
and δ=1.2mm in microgravity. The diffusivity of
fuel vapor, D, was calculated from
the empirical formula [22] for four different types of alcohols, D=1.12cm2/s for ethanol, D=0.93cm2/s for methanol, D=0.75cm2/s for n-propanol,
and D=0.63cm2/s for
n-butanol.In a previous scaling analysis [34], the flame spread
rate (dimensional) was expressed as a function of 1/MaPr. (26)1MaPr⁡=4μ2aγ2ρσTΔTv1hT(LhT)2. However, 1/ Ma Pr was induced in experimental data, hT and hT/L. To improve the scaling analysis, it is thought
that the hT and hT/L can be related by
combining two nondimensional numbers, Gr and Ma. The Gr number is related to hT, and the Ma number is related to L. The experimental data hT(hT/L)2 is expressed as (27)hT(hTL)2=C1(GrMaPr⁡)c2=C1(β2ρ2g2aσTvhT4)C2.By submitting (10) into (9), we can obtain the following
equation: (28)Gr−C2/(1−C2)MaPr⁡=(4C1μ2aγ2ρσTΔTv)1/(1−C2),C1 and C2 are fitted to the
experimental data for deep liquid pool: C1=420,C2=−0.25.The time averaged flame spread rates under subflash conditions
measured in this study (for four different fuels: methanol, ethanol,
n-propanol, and n-butanol) and experimental data from [12, 16] are plotted in Figure 10 as a function of Gr0.2/(Ma⋅Pr). The initial fuel depth that produced these data varies from 1
to 50 mm. A clear difference between deep
liquid pools (Gr0.2/(Ma⋅Pr)<6×10−4) and shallow liquid pools (Gr0.2/(Ma⋅Pr)>6×10−4) can be seen. For deep liquid pools, the flame spread
rate correlates with {Gr0.2/(Ma⋅Pr)}1.5, while for shallow liquid fuel poolsit correlates with {Gr0.2/(Ma⋅Pr)}1.0 in the pulsating flame
spread condition. Both the deep and shallow liquid pool regions include
microgravity data (H=3 and 25 mm).Figure 10
Flame spread rate as a function
of for four different alcohols (◊ methanol: H=1–10 mm; □ ethanol: H=1–5 mm; △ n-pripanol: H=2–20 mm; ○ n-butanol: H=1–50 mm).


## 4.1. Analytical Results

Figure7 shows the critical Ma number versus the thermal subsurface layer depth (defined in Figure
2) with two constant We number cases
plotted: We=1.31 for normal
gravity, and We=0.89 for
microgravity. The We number is
calculated with (22) using the thermal characteristic depth and length of the
subsurface layer circulation of [34].Figure 7
Marangoni number
versus characteristic depth of subsurface layer circulation.The
critical Ma number under microgravity
is much larger than that under normal gravity, suggesting that when the liquid surface
becomes unstable and a surface wave is formed under microgravity, the valuehT will be larger (which means the diameter of subsurface liquid convection
will be larger). This possibility, if validated, could explain NASA’s
microgravity experimental results [16], which showed a large diameter
subsurface circulation for n-butanol compared with its normal gravity behavior
at the same temperature. The large subsurface circulation under
microgravity takes longer to reach the fully-developed state because gravity is
a major contributor to the forces that keep each element of a liquid in
communication with its neighbors. This means that the microgravity experiments
require a longer tray length to achieve a steady state condition than
experiments in normal gravity (which usually employ a 30 to 40 cm long tray).


## 4.2. Thermal and Fluid Scale

Figure8 shows the characteristic length scale ratio hT/L as a function of the
initial fuel depth (H) for n-propanol and n-butanol, measured by
the HSSG technique [34].Figure 8
Ratio of characteristic length and characteristic depth
of subsurface layer circulation versus initial fuel layer depth (fuel is
n-propanol and tray width = 20 mm).The quotienthT/L increased sharply for both n-propanol and
n-butanol with increasing initial fuel layer depth in the range H=2 to 5 mm and reached a very slow
increase for n-propanol with H=5 to
25 mm and for n-butanol with H=5 to
50 mm. This indicates that
there are two different regimes with a division at H=5mm: shallow liquid pools for H<5 mm, and deep liquid pools for H>5 mm. When the fuel
depth is less than 5 mm, the restricted subsurface liquid circulation
reduces hT and increases L. For an initial fuel depth
of H>5 mm, both the characteristic length L and the circulation depth hT increase with
increasing initial fuel depth and reach their constant values, when the deep liquid
pool condition is satisfied.The thermal field is targeted in the experiment, while the
theoretical analysis targets the flow field. Therefore, the shadowgraph and
PTLS methods were used to measure the thermal scale and the flow scale at the
same time. Figure9 shows hT (thermal
characteristic depth) and hF (flow characteristic depth) as functions of the initial fuel layer depth.Figure 9
Thermal and flow characteristic depth of subsurface
layer circulation versus initial fuel layer depth.The characteristic depth in n-butanol was larger than the
depth in n-propanol by a factor of about 1.4.  This value is close to the 1.3
ratio of the viscosity coefficients of n-propanol and n-butanol. As a result, when
a dimensionless number is used, it corrects it by the Prandtl (Pr)
number. The Pr number is defined as
follows:(24)Pr⁡=va, where, ν is the
kinematic viscosity, a is the thermal
diffusivity.


## 4.3. Scale Analysis of Flame Spread

The nondimensional flame spread rate may be
influenced by the quenching distanceδ, associated with the dark region formed between
the flame leading edge and the liquid surface, and the diffusivity, D, of fuel vapor. A nondimensional form
of the flame spread rate, V/VD, can therefore be written as follows: (25)VVD=VδD. This nondimensional flame
spread V/VD is the
same to the vapor Peclet number. We measured the quenching distance at δ=0.8mm in normal gravity
and δ=1.2mm in microgravity. The diffusivity of
fuel vapor, D, was calculated from
the empirical formula [22] for four different types of alcohols, D=1.12cm2/s for ethanol, D=0.93cm2/s for methanol, D=0.75cm2/s for n-propanol,
and D=0.63cm2/s for
n-butanol.In a previous scaling analysis [34], the flame spread
rate (dimensional) was expressed as a function of 1/MaPr. (26)1MaPr⁡=4μ2aγ2ρσTΔTv1hT(LhT)2. However, 1/ Ma Pr was induced in experimental data, hT and hT/L. To improve the scaling analysis, it is thought
that the hT and hT/L can be related by
combining two nondimensional numbers, Gr and Ma. The Gr number is related to hT, and the Ma number is related to L. The experimental data hT(hT/L)2 is expressed as (27)hT(hTL)2=C1(GrMaPr⁡)c2=C1(β2ρ2g2aσTvhT4)C2.By submitting (10) into (9), we can obtain the following
equation: (28)Gr−C2/(1−C2)MaPr⁡=(4C1μ2aγ2ρσTΔTv)1/(1−C2),C1 and C2 are fitted to the
experimental data for deep liquid pool: C1=420,C2=−0.25.The time averaged flame spread rates under subflash conditions
measured in this study (for four different fuels: methanol, ethanol,
n-propanol, and n-butanol) and experimental data from [12, 16] are plotted in Figure 10 as a function of Gr0.2/(Ma⋅Pr). The initial fuel depth that produced these data varies from 1
to 50 mm. A clear difference between deep
liquid pools (Gr0.2/(Ma⋅Pr)<6×10−4) and shallow liquid pools (Gr0.2/(Ma⋅Pr)>6×10−4) can be seen. For deep liquid pools, the flame spread
rate correlates with {Gr0.2/(Ma⋅Pr)}1.5, while for shallow liquid fuel poolsit correlates with {Gr0.2/(Ma⋅Pr)}1.0 in the pulsating flame
spread condition. Both the deep and shallow liquid pool regions include
microgravity data (H=3 and 25 mm).Figure 10
Flame spread rate as a function
of for four different alcohols (◊ methanol: H=1–10 mm; □ ethanol: H=1–5 mm; △ n-pripanol: H=2–20 mm; ○ n-butanol: H=1–50 mm).


## 5. Conclusions

Instability and scaling analyses were conducted on
experimental data quantifying flame spread over liquid fuels obtained both
under normal and microgravity by other researchers and by our group. We arrived
at the following conclusions.(1)
We found
four important (Ma, Gr, Fr, We) pi-numbers and a characteristic length
scale ratio,hT/L, and used them in a general formula to
describe flame spread over liquids. Using this formula, flame spread over
liquid was divided into two separate regimes: the shallow-liquid pool and the
deep-liquid pool.(2)
The flow and
thermal characteristic depths of subsurface circulation were measured by our
specially designed high sensitivity shadowgraph (HSSG) combined with a
particle-track laser-sheet (PTLS) technique. The flow characteristic depth for
n-butanol was twice as deep as the thermal characteristic depth. Based on these
experimental results, the characteristic depth scale ratio:γ was introduced to compensate for the difference between thermal and fluid
subsurface layer depth.(3)
We
correlated the nondimensional flame spread rate,V/VD, over
four different alcohol fuels, the initial fuel depth, and gravity, which ranged
from 1 to 50 mm, with Gr0.2/(Ma⋅Pr). The flame spread rate for shallow liquid pools correlates with {Gr0.2/(Ma⋅Pr)}1.0, while the spread rate for deep liquid pools
correlates with {Gr0.2/(Ma⋅Pr)}1.5.



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*Source: 178292-2008-06-15.xml*