The effect of pressure on the characteristics of laminar jet diffusion flame: A similarity analysis and experimental study

Highlights

• Similarity analysis for flow and soot models is developed.

• Flame characteristics for ethanol pool fires under high pressure are presented.

• Flame height, width, oscillation length, frequency and volume were correlated.

• Globle relationship for non-dimensional radiation fraction was verified.

Abstract

Although the effect of pressure on the characteristics of laminar diffusion flame has been extensively studied, there is need of simple similarity correlations for flame shape and radiation fraction, which can be useful for application to turbulent flows and fire safety. In this paper, similarity analysis for flow and soot models is developed, where non-dimensional parameters were defined for the flame geometry and the radiation fraction. For validation, small ethanol pool fires with diameter of 1.5 cm were conducted in a pressure chamber at different pressures ranging from 1 to 5 atm, where the flame shape and radiation fraction were measured. Results show the characteristic lengths of the flame geometry including flame height, flame width, flame oscillation length, flame frequency and flame volume could be well correlated using the non-dimensional similarity parameter. Furthermore, the globle relationship for non-dimensional radiation fraction was verifed based on measured value in current and previous study, where the non-dimension radiation fraction could be well correlated by the −0.30 power of $\frac{\mathrm{S}{\dot{\mathrm{m}}}_{\mathrm{f}}}{\mu {\left(\frac{{\nu }^2}{\mathrm{g}}\right)}^{\frac{1}{3}}}$. Comparison with literature experiments are also presented and discusssed.

Introduction

Pressure and buoyancy effects on the laminar diffusion flame have been extensively explored in different gravity and pressure environments [1], [2], [3], [4], [5]. However, related studies about simple similarity correlations for flame shape and radiation fraction are limited [6], even though they can be useful for application to turbulent flows and fire safety.

Some classical correlations [7], [8] have been developed to describe the flame shape of laminar jet diffusion flame under different pressures, where the flame was considered as stable and constant. Recently, Sunderland et al. [4] conducted laminar jet diffusion flames for methane, ethane, and propane over the pressure ranging from 0.25 to 2 atm, where camera equipped with a CH-line filter was used to obtain stoichiometric flame sheet accurately. They suggested that stoichiometric flame length is proportional to fuel mass flow rate and independent of pressure while flame width decreases with pressure owing to radial diffusion. Yuan et al. [3] created a simple theoretical model for buoyant laminar jet flames, where radial diffusion of species and the buoyancy effects were considered. In the model, the flame radius is proportional to the −1/2 power of pressure, and flame height remains constant under different pressure, which were verified by jet diffusion flames having low Reynolds number (Re) in reduced pressure environment. Based on these similar relationships for flame height and flame width, the flame volume was deduced to be proportional to −1 power of pressure in the study of Fang et al. [9], [10], and was verified by methane flame experiments using a circular nozzle with diameter of 12 mm at 0.45–1 atm in a confined cabin.

Radiation fraction is also important characteristic parameter for diffusion flame. In our previous study, a global soot formation model was proposed [5], where soot formation time and soot oxidation time were defined. From this model, soot formation chemical time is proportional to the smoke height whereas the smoke point height is shown to be inversely proportional to pressure, which is in agreement with experiments. Based on the introduced relationship between soot formation chemical time and pressure, Fang el al. [9] predicted that the radiation fraction was proportional to P^0.5, where the soot characteristic flow time was analyzed to be roughly proportional to the air pressure. The predicted relationship was compared with fitted value for laminar methane diffusion flame at 0.45–1 atm.

As briefly reviewed in the previous paragraph, the flame was assumed to be stable and constant. However, flame would become unstable or flicker with the increase of pressure or fuel flow rate, and consequently results in flame oscillation, which has been observed in recent studies [6], [11]. Sato et al. [11] found the normalized flame length could be well correlated with Reynolds number for the steady flame cases, but the relationship collapsed for flickering flame. Chen et al. [6] also observed that the change of flame height with pressure was determined by the flame structure. The vortices induced by oscillation would influence the flow field of the flame, and the scale of vortex would influence the characteristic lengths of the flame geometry. In addition, the soot concentration is related with the laminar smoke point height and the flow conditions. The change of the characteristic lengths of the flame geometry will influence the flame volume and characteristic soot flow time. So, how the characteristic lengths of the flame, in both normal and reduced pressures, affects the flame radiation fraction also needs to be clarified. These behaviors have not been revealed in the past and are to be quantified in the present work.

The major contributions of the current work are claimed as follows: (1) A flow model of laminar jet flame is presented in this work where the main difference from previous analysis in [4], [7], [9], [10] lies in the consideration of different characteristic lengths of the flame. (2) Base on the flow model above, a soot model similarity [5] is developed, where the soot concentration is expressed in terms of the laminar smoke point height and the flow conditions, and a global relationship for non-dimensional radiation fraction is introduced to provide a good interpretation of the data for soot concentrations in small scale pool flames including effects of variable pressure environments. (3) Flame characteristic of ethanol pool fire under high-pressure were measured and presented including the flame shape and flame radiation fraction.

In this paper, a similarity analysis for flow and soot models was developed in the Section 2. In the Section 3, small scale ethanol pool fires with diameter of 1.5 cm were conducted under pressure of 1–5 atm, where the flame height, flame width, flame oscillation frequency and flame radiation fraction were measured in the experiments. Finally, experimental results in our and previous studies are correlated to verify the flow and soot formation model for laminar diffusion flame under different pressure, which are important for fundamentally understanding turbulent combustion and for promoting fire prevention.

Based on our previous work [5] and the numerical work of the Gülder et al. [2], we have developed the following similarity relations for the flame heights and characteristic flame widths of laminar jet flames issuing from a nozzle of diameter d and having velocity $u_s$. It should be noted that in the following presentation we are using (a) properties (density, dynamic viscosity) at ambient conditions instead of flame conditions, observing that the flame temperature is nearly the same for the fuels we are dealing with and assuming that mixture fraction and absolute normalised temperature are self-similar and (b) thermal and mass diffusion properties equal (dynamic, thermal and specie diffusivity). This limitation can be relaxed in further developments by using direct numerical simulations (DNS).

The following dimensionless parameters are derived to determine the characteristic lengths of the flame geometry and frequency of oscillations:

• For the stable laminar jet flame, a general relation for flame height is expressed as ${\mathrm{L}}_{\mathrm{f}}\propto \frac{{\dot{\mathrm{S}\mathrm{m}}}_{\mathrm{f}}}{\mu }$, where the fuel and oxidizer coming together in a stoichiometric reaction zone. In this equation, S is the mass stoichiometric air-to-fuel mass ratio and$\mu$is dynamic viscosity. Note that $\frac{{\mathrm{L}}_{\mathrm{f}}\mu }{{\dot{\mathrm{S}\mathrm{m}}}_{\mathrm{f}}}$ is defined as dimensionless flame height.

• It was found that the flame would flicker as ambient pressure or the fuel flow rate increases. Flame oscillation is related to the buoyancy instability, where the upward movement of burnt gas driven by buoyancy causes the generation of vortex on the flame surface. The instability develops visually as the flow rate increases. In the study of Yuan et al. [12], the oscillating behaviour away from the fuel source is modelled as a thermal plume boundary layer, and the characteristic oscillating length was obtained as $l^{\ast }\sim {\nu }^{2/3}{g}^{-1/3}$ by numerical solution and similarity analysis. Therefore, $\frac{\mathrm{S}{\dot{\mathrm{m}}}_{\mathrm{f}}}{\mu {\left(\frac{{\nu }^2}{\mathrm{g}}\right)}^{\frac{1}{3}}}$ is introduced to express the relation of oscillation with the flame height.

• We note that the fuel source can also introduce another vortex of size d, and additional oscillations at the rim of the pan may be observed. Therefore, the diameter of burner would also influence the dimensionless flame length and is related to the scale of vortex, so that the dimensionless ratio $\frac{\mathrm{d}}{{\left(\frac{{\nu }^2}{\mathrm{g}}\right)}^{\frac{1}{3}}}$ is defined.

• The Reynolds number at the burner outlet may also influence the flame geometry (especially for momentum dominated jet flames), expressed as$\mathrm{R}\mathrm{e}=\frac{{\mathrm{u}}_{\mathrm{s}}\mathrm{d}}{\nu }$.

Based on above scaling analysis and proposed non-dimensional quantities, the dimensionless flame height could be represented by the following function:

$$\frac{{\mathrm{L}}_{\mathrm{f}}\mu }{{\dot{\mathrm{S}\mathrm{m}}}_{\mathrm{f}}}=\mathrm{f}(\frac{\mathrm{S}{\dot{\mathrm{m}}}_{\mathrm{f}}}{\mu {\left(\frac{{\nu }^2}{\mathrm{g}}\right)}^{\frac{1}{3}}},\frac{\mathrm{d}}{{\left(\frac{{\nu }^2}{\mathrm{g}}\right)}^{\frac{1}{3}}},\frac{{\mathrm{u}}_{\mathrm{s}}\mathrm{d}}{\nu })$$

The flame height ${\mathrm{L}}_{\mathrm{f}}$, can be replaced by flame width ${\mathrm{W}}_{\mathrm{f}}$ or flame oscillation length A_f defined as the difference of maximum flame height and minimum flame height. We emphasize here that the large differences in density should be included in detailed analysis.

The flame volume is the cube of the dimensionless length, so the following relation is expected:

$$\frac{{\mathrm{V}}_{\mathrm{f}}}{{{\dot{(\mathrm{S}\mathrm{m}}}_{\mathrm{f}}/\mu )}^3}=\mathrm{f}(\frac{\mathrm{S}{\dot{\mathrm{m}}}_{\mathrm{f}}}{\mu {\left(\frac{{\nu }^2}{\mathrm{g}}\right)}^{\frac{1}{3}}},\frac{\mathrm{d}}{{\left(\frac{{\nu }^2}{\mathrm{g}}\right)}^{\frac{1}{3}}},\frac{{\mathrm{u}}_{\mathrm{s}}\mathrm{d}}{\nu })$$

We noted that the characteristic velocity derived based on buoyancy conditions is proportional to ${\mathrm{u}}_{\mathrm{f}}\sqrt{\frac{\mathrm{\Delta }{\mathrm{T}}_{\mathrm{f}}}{{\mathrm{T}}_{\infty }}\mathrm{g}{\mathrm{L}}_{\mathrm{f}}}$and the induced frequency of oscillations would be proportional to $\frac{{\mathrm{u}}_{\mathrm{f}}}{{\mathrm{L}}_{\mathrm{f}}}$. Because the flame temperature is assumed to be constant for the fuels in our study, the characteristic velocity could be simplified as ${\mathrm{u}}_{\mathrm{f}}\sqrt{\mathrm{g}{\mathrm{L}}_{\mathrm{f}}}$. Finally, we can conclude that the oscillating frequency should be expressed as:

$$\frac{\mathrm{f}}{\left(\frac{{\mathrm{u}}_{\mathrm{f}}}{{\mathrm{L}}_{\mathrm{f}}}\right)}=\mathrm{f}(\frac{\mathrm{S}{\dot{\mathrm{m}}}_{\mathrm{f}}}{\mu {\left(\frac{{\nu }^2}{\mathrm{g}}\right)}^{\frac{1}{3}}},\frac{\mathrm{d}}{{\left(\frac{{\nu }^2}{\mathrm{g}}\right)}^{\frac{1}{3}}},\frac{{\mathrm{u}}_{\mathrm{s}}\mathrm{d}}{\nu })$$

which is also equivalent to:

$$\frac{\mathrm{f}}{{\left(\frac{g^2}{\nu }\right)}^{1/3}}=\mathrm{f}\left(\frac{\mathrm{S}{\dot{\mathrm{m}}}_{\mathrm{f}}}{\mu {\left(\frac{{\nu }^2}{\mathrm{g}}\right)}^{\frac{1}{3}}},\frac{\mathrm{d}}{{\left(\frac{{\nu }^2}{\mathrm{g}}\right)}^{\frac{1}{3}}},\frac{{\mathrm{u}}_{\mathrm{s}}\mathrm{d}}{\nu }\right)$$

Base on the flow model above, a soot model similarity [5] is developed, where the soot concentration is expressed in terms of the laminar smoke point height and the flow conditions, and a global relationship for non-dimensional radiation fraction is introduced. The flame radiation is assumed to be mainly contributed from the soot in the flame, as evidenced by the yellow flame recorded in the most experimental conditions. It should be noted that the assumption is not exactly correct for the methanol flames at low pressure.

The radiation fraction could be written as:

$${\chi }_R~\frac{\sigma T_r^4{V}_f{\varphi }_s}{\dot{Q}}$$

where $\sigma$ is the Stefan–Boltzmann constant, $T_r$ is the characteristic flame radiation temperature, $V_f$ is the flame volume, ${\varphi }_s$ is the soot volume fraction. $\dot{Q}$ is the heat release rate and is determined as the fuel mass flow rate multiplied by the heat of combustion.

Based on the global soot formation model introduced in our previous study [5], the soot volume fraction is proportional to the characteristic flow time divided by the soot formation time ($\frac{{\tau }_{\mathrm{F}}}{{\tau }_{\mathrm{s}}}$), and could be written as

$${\varphi }_{\mathrm{s}}=\frac{{\rho }_{\mathrm{g}}}{{\rho }_{\mathrm{s}}}{\mathrm{Y}}_{\mathrm{s}}~\frac{{\rho }_{\mathrm{g}}}{{\rho }_{\mathrm{s}}}\ast \frac{{\tau }_{\mathrm{F}}}{{\tau }_{\mathrm{s}}}$$

where the ${\mathrm{Y}}_{\mathrm{s}}$, ${\rho }_{\mathrm{g}}$, ${\rho }_{\mathrm{s}}$ is the soot mass fraction, gas density and soot density, respectively.

The soot formation time, ${\tau }_{\mathrm{s}}$, is proportional to the smoke point heat release rate and varies inversely proportionally to pressure for the same fuel:

$${\tau }_s={\tau }_{\mathit{so}}\frac{p_o}{p}$$

where ${\tau }_{\mathit{so}}$ is soot formation time at reference pressure $p_o$.

By introducing Eq. (6) into Eq. (7), ${\varphi }_{\mathrm{s}}$ the soot volume fraction could be written as

$${\varphi }_{\mathrm{s}}~\frac{{\rho }_{\mathrm{g}}}{{\rho }_{\mathrm{s}}}\ast \frac{{\tau }_{\mathrm{F}}}{{\tau }_{\mathrm{s}\mathrm{o}}{(\mathrm{p}}_{\mathrm{o}}/\mathrm{p})}$$

According to the induced flow model above, the flow time ${\tau }_{\mathrm{F}}$ is expressed as

$${{\tau }_{\mathrm{F}}=\frac{{\mathrm{L}}_{\mathrm{f}}}{{\mathrm{u}}_{\mathrm{f}}}=\left(\frac{\nu }{{\mathrm{g}}^2}\right)}^{\frac{1}{3}}=\mathrm{f}(\frac{\mathrm{S}{\dot{\mathrm{m}}}_{\mathrm{f}}}{\mu {\left(\frac{{\nu }^2}{\mathrm{g}}\right)}^{\frac{1}{3}}},\frac{\mathrm{d}}{{\left(\frac{{\nu }^2}{\mathrm{g}}\right)}^{\frac{1}{3}}},\frac{{\mathrm{u}}_{\mathrm{s}}\mathrm{d}}{\nu })$$

Consider the decrease of flame temperature due to the radiative heat loss, the relationship between temperature and radiation fraction could be assumed as [13]

$$\sigma {\mathrm{T}}_{\mathrm{r}}^4~\frac{1}{{\chi }_R^3}$$

Then, the radiation fraction has the following equation:

$${\chi }_{\mathrm{R}}~({\frac{{\mathrm{V}}_{\mathrm{f}}{\varphi }_{\mathrm{s}}}{\dot{\mathrm{Q}}})}^{1/4}$$

According to Eqs. (7), (8), (9), (10), (11), the equation for radiation fraction could be converted as

$${\chi }_{\mathrm{R}}~{\left(\frac{{(\mathrm{S}{\dot{\mathrm{m}}}_{\mathrm{f}}/\mu )}^3\times \mathrm{f}(\frac{\mathrm{S}{\dot{\mathrm{m}}}_{\mathrm{f}}}{\mu {\left(\frac{{\nu }^2}{\mathrm{g}}\right)}^{\frac{1}{3}}},\frac{\mathrm{d}}{{\left(\frac{{\nu }^2}{\mathrm{g}}\right)}^{\frac{1}{3}}},\frac{{\mathrm{u}}_{\mathrm{s}}\mathrm{d}}{\nu })\times \frac{{\rho }_{\mathrm{g}}}{{\rho }_{\mathrm{s}}}\times \frac{1}{{\tau }_{\mathrm{s}\mathrm{o}}{(\mathrm{p}}_{\mathrm{o}}/\mathrm{p})}\times {\left(\frac{\nu }{{\mathrm{g}}^2}\right)}^{\frac{1}{3}}\mathrm{f}(\frac{\mathrm{S}{\dot{\mathrm{m}}}_{\mathrm{f}}}{\mu {\left(\frac{{\nu }^2}{\mathrm{g}}\right)}^{\frac{1}{3}}},\frac{\mathrm{d}}{{\left(\frac{{\nu }^2}{\mathrm{g}}\right)}^{\frac{1}{3}}},\frac{{\mathrm{u}}_{\mathrm{s}}\mathrm{d}}{\nu })}{{\dot{\mathrm{m}}}_{\mathrm{f}}\mathrm{\Delta }\mathrm{H}}\right)}^{1/4}$$

Then

$$\frac{{\chi }_{\mathrm{R}}\times {\left({\dot{\mathrm{m}}}_{\mathrm{f}}\mathrm{\Delta }\mathrm{H}\right)}^{1/4}}{{\left(\mathrm{S}{\dot{\mathrm{m}}}_{\mathrm{f}}/\mu \right)}^{3/4}\times {\left[\frac{{\rho }_{\mathrm{g}}}{{\rho }_{\mathrm{s}}}\ast \frac{1}{{\tau }_{\mathrm{s}\mathrm{o}}{(\mathrm{p}}_{\mathrm{o}}/\mathrm{p})}\right]}^{1/4}{\left(\frac{\nu }{{\mathrm{g}}^2}\right)}^{\frac{1}{12}}}=\mathrm{F}\mathrm{n}\mathrm{c}\left(\frac{\mathrm{S}{\dot{\mathrm{m}}}_{\mathrm{f}}}{\mu {\left(\frac{{\nu }^2}{\mathrm{g}}\right)}^{\frac{1}{3}}},\frac{\mathrm{d}}{{\left(\frac{{\nu }^2}{\mathrm{g}}\right)}^{\frac{1}{3}}},\frac{{\mathrm{u}}_{\mathrm{s}}\mathrm{d}}{\nu }\right)$$

And if the effect of ${\rho }_{\mathrm{s}}$ and ${\tau }_{\mathrm{s}\mathrm{o}}$ as being constant, are ignored:

$$\frac{{\chi }_{\mathrm{R}}\times {\left({\dot{\mathrm{m}}}_{\mathrm{f}}\mathrm{\Delta }\mathrm{H}\right)}^{1/4}}{{\left(\mathrm{S}{\dot{\mathrm{m}}}_{\mathrm{f}}/\mu \right)}^{3/4}\times {\left[\frac{{\rho }_{\mathrm{g}}}{{\mathrm{p}}_{\mathrm{o}}}\ast \frac{\mathrm{p}}{{\mathrm{p}}_{\mathrm{o}}}\right]}^{1/4}{\left(\frac{\nu }{{\mathrm{g}}^2}\right)}^{\frac{1}{12}}}=\mathrm{F}\mathrm{n}\mathrm{c}\left(\frac{\mathrm{S}{\dot{\mathrm{m}}}_{\mathrm{f}}}{\mu {\left(\frac{{\nu }^2}{\mathrm{g}}\right)}^{\frac{1}{3}}},\frac{\mathrm{d}}{{\left(\frac{{\nu }^2}{\mathrm{g}}\right)}^{\frac{1}{3}}},\frac{{\mathrm{u}}_{\mathrm{s}}\mathrm{d}}{\nu }\right)$$