International Journal of Chemical Engineering

Volume 2008, Article ID 178292, 10 pages

http://dx.doi.org/10.1155/2008/178292

## Scaling Analysis on Pulsating Flame Spread over Liquids

^{1}Department of Intelligent Machines and System Engineering, Hirosaki University, Hirosaki 036-8561, Japan^{2}Department of Mechanical Engineering, Oita National College of Technology, Oita 879-0152, Japan^{3}Department of Mechanical Engineering, University of Kentucky, Lexington, KY 40506, USA

Received 16 February 2008; Accepted 12 May 2008

Academic Editor: Adrian Schumpe

Copyright © 2008 Kozue Takahashi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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, (= , where is the quenching distance and 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 , while for the deep liquid pool it was correlated with .

#### 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.

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 number, Grashof (Gr) number, Weber (We) number, and Froude (Fr) number, and a characteristic length
scale ratio were introduced for scaling analysis.
The characteristic length scale ratio, 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 and , and thermal characteristic length *L*, determined from a shadowgraph image
and a particle-track laser-sheet image [34].

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 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 The
third term on the LHS in (2) is due to surface tension, where is the
expression for the surface wave, is the wave number, is the wavelength, *c* is the
wave velocity, and all other symbols are defined in the nomenclature section.
In (2), is
the surface shear stress due to the Marangoni effect and is expressed as If the
momentum displacement thickness: and the continuity equation, (1), are substituted into (2), we can
obtain the following equation: The
perturbation at the gas-liquid interface is very small and the independent
variables are expressed as where
the superscripts bar and prime indicate time-averaged and perturbation
quantities, respectively. If the velocity remains similar on perturbation, then . With the substitution of (5) and (1) into (4),
and if the velocity is fully developed, and the higher-order perturbation terms are
negligible; the continuity and momentum equations can then, respectively, be
written as If is small, the deviation from the time-averaged value ,
which may be written as 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, , where is the time-averaged temperature in the
liquid, is the thermal conductivity of the liquid, and 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 Differentiating with respect to *x*, and substituting into (3), we obtain 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: where The
first term on the LHS of (11) represents the dynamic wave, where and , respectively, represent the
forward and reverse propagation velocities. The second term on the LHS of (11)
represents the kinematic wave, where 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 term in the dynamic
wave expression.

If the surface perturbation
is expressed as , we
obtain the following equation from (11): Hence, * ω* may be approximated as If , the dynamic wave is unstable, resulting in . This
then becomes a neutrally stable condition, . The following equation describes
the generation of the surface wave: If the
local liquid velocity is defined as , the liquid
volumetric flow rate per unit width of the tray is , the
average liquid velocity is , the
momentum displacement thickness is and the heat release factor is . Equation (15) can be rewritten as where 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: , and . 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 *q _{i}* in horizontal in Figure 4 is positive for heat release from the liquid
surface to the gas phase.

The area above the neutral stable line in Figure 4 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: where *L* and are the thermal characteristic length and
depth and is the flow
characteristic depth of the subsurface layer circulation defined in Figure 2. The characteristic depth scale ratio is
expressed as 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:
Using (19), (20), (21), and (22), (15) can be rewritten in this nondimensional form: 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 Figure 5, including a high sensitivity shadowgraph (HSSG) and a particle-track laser-sheet (PTLS) flow visualization apparatus.

The fuel container is 480 mm long mm wide × seven different heights ( 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 mm 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 m average diameter, 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 Figure 6(a).

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 (mm, 3 mm deep) was located inside the wind tunnel (cm, 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 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

Figure 7 shows the critical Ma number versus the thermal subsurface layer depth (defined in Figure 2) with two constant We number cases plotted: for normal gravity, and for microgravity. The We number is calculated with (22) using the thermal characteristic depth and length of the subsurface layer circulation of [34].

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 value 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

Figure 8 shows the characteristic length scale ratio as a function of the
initial fuel depth (*H*) for n-propanol and n-butanol, measured by
the HSSG technique [34].

The quotient increased sharply for both n-propanol and
n-butanol with increasing initial fuel layer depth in the range to 5 mm and reached a very slow
increase for n-propanol with to
25 mm and for n-butanol with to
50 mm. This indicates that
there are two different regimes with a division at mm: shallow liquid pools for mm, and deep liquid pools for mm. When the fuel
depth is less than 5 mm, the restricted subsurface liquid circulation
reduces and increases *L*. For an initial fuel depth
of mm, both the characteristic length *L* and the circulation depth 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. Figure 9 shows (thermal characteristic depth) and (flow characteristic depth) as functions of the 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: 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 distance , 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, , can therefore be written as follows: This nondimensional flame
spread is the
same to the vapor Peclet number. We measured the quenching distance at mm in normal gravity
and mm in microgravity. The diffusivity of
fuel vapor, *D*, was calculated from
the empirical formula [22] for four different types of alcohols, for ethanol, for methanol, for n-propanol,
and for
n-butanol.

In a previous scaling analysis [34], the flame spread
rate (dimensional) was expressed as a function of 1/MaPr. However, 1/ Ma Pr was induced in experimental data, and . To improve the scaling analysis, it is thought
that the and can be related by
combining two nondimensional numbers, Gr and Ma. The Gr number is related to , and the Ma number is related to *L*. The experimental data is expressed as

By submitting (10) into (9), we can obtain the following equation: and are fitted to the experimental data for deep liquid pool: .

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 . The initial fuel depth that produced these data varies from 1 to 50 mm. A clear difference between deep liquid pools and shallow liquid pools can be seen. For deep liquid pools, the flame spread rate correlates with , while for shallow liquid fuel poolsit correlates with in the pulsating flame spread condition. Both the deep and shallow liquid pool regions include microgravity data ( and 25 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, , 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: was introduced to compensate for the difference between thermal and fluid subsurface layer depth.(3)We correlated the nondimensional flame spread rate, , over four different alcohol fuels, the initial fuel depth, and gravity, which ranged from 1 to 50 mm, with . The flame spread rate for shallow liquid pools correlates with , while the spread rate for deep liquid pools correlates with .

*Nomenclature*

A: | Amplitude of surface wave, m |

A: | Thermal diffusivity, m^{2}/s |

: | Forward and reverse propagation velocities of dynamic wave, m/s |

: | Propagation velocity of kinematic wave, m/s |

C: | Wave velocity, m/s |

D: | Diffusion coefficient of fuel vapor, m^{2}/s |

Fr: | Froude number |

Gr: | Grashof number |

g: | Acceleration due to
gravity, m/s^{2} |

H: | Initial fuel layer depth, m |

h: | Sub-surface layer depth, m |

: | Flow characteristic depth of sub-surface layer circulation, m |

: | Thermal characteristic depth of the sub-surface layer circulation, m |

: | Characteristic length scale ratio |

k: | Wave number |

L: | Thermal characteristic length of the sub-surface layer circulation, m |

Ma: | Marangoni number |

Pr: | Prandtl number |

: | Heat flux from the liquid surface, W/m^{2} |

T: | Temperature, K |

: | Liquid bulk temperature, K |

: | Flash point temperature, K |

U: | Local liquid velocity m/s |

: | Liquid surface velocity, m/s |

u: | Average liquid velocity , m/s |

V: | Flame spread rate (time averaged), m/s |

: | Vapor diffusion rate, m/s |

We: | Weber number |

x: | Flow direction coordinate, m |

y: | Normal direction to the wall, m |

*Greek symbols*: | Heat release factor |

: | Momentum displacement thickness , m/s |

: | Characteristic depth scale ratio |

: | Temperature difference between flash point, and liquid bulk temperature, , K |

: | Quenching distance, m |

: | Liquid volumetric flow rate per unit width of the tray ,
m^{2}/s |

: | Critical
flow rate required for surface wave to occur, m^{2}/s |

: | Wavelength, m |

: | Thermal conductivity of the liquid, W/(mK) |

: | Kinematic
viscosity, m^{2}/s |

:ρ | Density
of the liquid, kg/m^{3} |

: | Surface tension force, N/m |

: | Temperature derivative of surface tension coefficient, N/mK |

: | Shear stress in the liquid, Pa |

: | Surface shear stress due to Marangoni effect, Pa |

*Subscripts*b: | Liquid phase (bulk) |

i: | Interface between gas and liquid phase |

t: | Partial derivatives in time |

x: | Partial
derivatives in the x-direction |

Superscript bar: | Time-averaged quantity |

Superscript prime: | Perturbation quantity. |

#### Acknowledgments

This study was supported in part by NASA under Grant NAG3-2567 and in part by the Japan Space Forum on “Ground-based Research Program for Space Utilization.” We would like to thank Dr. K. Kuwana and Dr. F. Miller for valuable technical discussions.

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