How small can a flame get?

Question

Last time I was watching a candle die. After its wick was finished, there remained just a drop of molten wax that was still slowly burning, with a flame that became smaller and smaller, down to a millimeter high, until it disappeared.

It made me wonder how small can a flame get. With controlled conditions, is it possible to make a micrometric flame for instance ? Is there a lower limit ?

Answer 1

Combustion is... complicated. Essentially what is going on in flame is that you have molecules of fuel and oxidizer that mix and start to bounce off each other. If the molecules are moving fast enough (meaning they have enough energy, which we measure as temperature), then when they collide with each other, they start to make the fuel and oxidizer fall apart into other molecules.

Depending on which molecules collide and the energies involved, when things start to fall apart they are moving to lower energy states and the energy that was stored in the chemical bonds gets released as heat (and radiation in the form of light, which may be invisible). If it is happening often enough, the heat raises the temperature (adds energy) of the molecules around it and the process starts to run away. This is how you get a stable flame.

So this means there's at least a fundamental limit to the thickness of a flame -- you couldn't have a flame at lengths smaller than the distance molecules travel before they collide. This distance is called the mean free path, but frankly it's not a useful limit because flames cannot exist on the scale of the mean free path for other reasons.

For a flame to exist and be stable (i.e. not just a spark or something that goes away quickly), the rate of heat release has to be in balance with the rate of heat losses. If heat release exceeds heat losses, the flame will get bigger. If heat release is less than heat losses, the flame will run out of energy.

All of this means it is difficult, if not impossible, to put a general limit on the smallest possible flame. It will depend on the fuel source and how much oxidizer is present (different fuel+oxidizer combinations need different energies to start releasing heat), what the flow around the flame is like (how fast heat is carried away), how much energy the mixture has (higher temperature means more collisions that can break things apart), and how far the molecules need to move before they collide (how dense the mixture is).

The only definitive thing we can say is that the flame needs to be thicker than the mean free path, but anything more precise would require getting specific about the setup.

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For a candle flame, we're looking at what is called a diffusion flame. The fuel (wax) is on one side and it has to vaporize and diffuse/mix with the oxygen in the air before it can properly burn. This is pretty hard. An overview lecture on diffusion flames is available, but it's actually not that easy to define a thickness for diffusion flames.

Suffice to say that the flames can be arbitrarily small, at least for sizes greater than several times the mean free path, provided the heat release is in balance with the heat losses. To be more specific would require a lot more details of the setup.

Answer 2

Fire has to be hot enough to burn. The smaller the flame, the cooler the flame, until it gets so cool that it doesn't burn.

Smaller flames are cooler because they have more surface area in proportion to their volume, so they radiate away heat faster.

Edit: When a fire goes out when it still has fuel and oxygen (assuming it wasn't smothered), it's because it was too small to be hot enough. For instance, the OP's candle flame that went down to a millimeter and then out, did so because the minimum size of a wax fire that's hot enough to sustain the reaction, is larger than the fire he had.

The minimum self-sustaining fire size will depend on the amount of oxygen and the geometry and type of fuel, because those factors affect the rate of heat production of the fire. If you are familiar with starting a campfire, the minimum size of a tinder fire is quite small because tinder has a large surface area for combustion, giving it a high rate of heat production, capable of keeping the tinder above the ignition point despite the high radiative heat loss for a small fire. The minimum size of a log fire is much larger because of the lower surface area, so the log fire has less heat production in proportion to its size. This is why you can't just put a match to the log, and it's also why using a small amount of tinder to ignite one end of the log is likely to fail as soon as you're out of tinder. You need to first use a tinder fire to heat a large section of the log above the ignition point, or several logs close together, before the log fire will be large enough to be hot enough to be self sustaining.

Answer 3

The paper by Sunderland et al. (2011) discusses the characteristics and measurements of candle flame shapes, including an analysis of flame height and width for various wick diameters and lengths. It also touches upon the conditions necessary for sustaining a flame on a candle wick.

Sunderland, P. B., Quintiere, J. G., Tabaka, G. A., Lian, D., & Chiu, C.-W. (2011). Analysis and measurement of candle flame shapes. Proceedings of the Combustion Institute, 33(2), 2489–2496. doi:10.1016/j.proci.2010.06.095

Here’s a summary of the relevant findings:

Critical Size for Combustion

1. Wick Diameter and Length:

   • The study found that the diameter and length of the wick are critical parameters that influence the flame size and stability. Wicks with diameters ranging from about 1 to 9 mm and lengths from about 2 to 10 mm were tested.
   • The aspect ratio (diameter to length) of the wick plays a significant role in flame attachment and stability. Wicks with aspect ratios between 0.1 and 2 were examined.

2. Minimum Wick Size:

   • The paper notes that flames might not survive on wicks shorter than about 1 mm. Specifically, it suggests that dry-out, where the flame extinguishes due to insufficient fuel supply, tends to occur for aspect ratios ($D/L$) of less than 0.15.
   • A more precise threshold is given by the equation indicating no flame for $Ra^{1/4} (L/D) < 2$, where $Ra$ is the Rayleigh number based on the wick diameter. This corresponds to a minimum wick diameter of approximately 0.88 mm for sustaining a flame, depending on the aspect ratio.

Modeling and Practical Implications

3. Flame Height and Width:

   • The height and width of the candle flame were found to be dependent on the wick size and properties of the fuel. The flame height is modeled to be linearly related to the fuel flow rate in laminar flow conditions.
   • The flame width at the top of the wick is related to the heat transfer and burning characteristics of the fuel.

4. Combustion Characteristics:

   • The study concludes with equations that predict the flame height and width based on wick dimensions and fuel properties. For instance, the normalized flame height is given by: $$ \frac{L_f}{D} = C Ra^{1/4} \left(\frac{L}{D} - 2\right)^n $$ with specific constants for different wax types.

In summary, there is a critical minimum size for the wick to sustain combustion, which is approximately around a diameter of 0.88 mm for typical wick lengths. Wicks below this size may not support a stable flame due to insufficient fuel flow and combustion dynamics. This critical mass for combustion ensures that the wick can maintain the necessary heat and fuel supply to sustain the flame.

The paper by Sunderland et al. (2011) develops a model for predicting the flame height and width of a candle flame based on the properties of the wick and the wax. This model can help determine if there is a minimum size for a flame by examining the parameters involved in sustaining a flame.

Model Overview

The model in the paper is based on a combination of empirical observations and theoretical predictions, particularly focusing on the following aspects:

1. Flame Height: $$ \frac{L_f}{D} = C \left( Ra^{1/4} \left(\frac{L}{D} - 2\right) \right)^n $$ where:

   • $ L_f $ is the flame height.
   • $ D $ is the wick diameter.
   • $ L $ is the wick length.
   • $ Ra $ is the Rayleigh number, defined as $ Ra = \frac{g \beta (T_f - T_\infty) D^3}{\nu \alpha} $.
   • $ C $ and $ n $ are constants derived from empirical data (e.g., $ C $ is 0.526 for tetracosane and 0.470 for paraffin, $ n $ is 0.75 for tetracosane and 0.60 for paraffin).
   • $ g $ is gravitational acceleration.
   • $ \beta $ is the coefficient of volumetric expansion.
   • $ T_f $ is the flame temperature.
   • $ T_\infty $ is the ambient temperature.
   • $ \nu $ is kinematic viscosity.
   • $ \alpha $ is thermal diffusivity.

2. Flame Width: $$ \frac{W_f}{D} = 2.71 \left(\frac{0.70}{k} \right) $$ where $ k $ is a parameter that depends on the local heat transfer and burning rate per unit length at the top of the wick.

3. Burning Rate and Heat Transfer Coefficient: The burning rate per unit area for the wick can be expressed using the Spalding B number: $$ \dot{m}'' = \frac{h_c}{c_p} \ln(1 + B) $$ where:

   • $ h_c $ is the convective heat transfer coefficient.
   • $ c_p $ is the specific heat.
   • $ B $ is the Spalding B number, which represents the ratio of the fuel's enthalpy change to the heat needed for vaporization.

Exploring Minimum Flame Size

To determine if there is a minimum size for a flame using this model, we need to examine the behavior of the flame parameters as the wick diameter and length decrease.

1. Flame Height and Minimum Wick Size: According to the model, the normalized flame height $ \frac{L_f}{D} $ becomes undefined or non-physical if the term $ Ra^{1/4} \left(\frac{L}{D} - 2\right) $ becomes too small. This implies there is a critical value below which the flame cannot be sustained.

2. Critical Rayleigh Number: The paper suggests that no flame can be sustained for $ Ra^{1/4} \left(\frac{L}{D} - 2\right) < 2 $. Substituting the expression for $ Ra $: $$ Ra = \frac{g \beta (T_f - T_\infty) D^3}{\nu \alpha} $$ This can be rewritten to find the critical diameter $ D $ and length $ L $: $$ \frac{g \beta (T_f - T_\infty) D^3}{\nu \alpha} \left(\frac{L}{D} - 2\right)^{1/4} < 2 $$ Simplifying, we get: $$ D < \left( \frac{2 \nu \alpha}{g \beta (T_f - T_\infty)} \right)^{1/3} $$

3. Specific Values: Let's plug in some typical values:

   • $ g = 9.81 \, \text{m/s}^2 $
   • $ \beta \approx 1/T_\infty \approx 1/300 \, \text{K}^{-1} $
   • $ T_f - T_\infty \approx 1400 \, \text{K} $
   • $ \nu \approx 15.11 \times 10^{-6} \, \text{m}^2/\text{s} $ (for air at room temperature)
   • $ \alpha \approx 22.5 \times 10^{-6} \, \text{m}^2/\text{s} $ (for air at room temperature)

   Substituting these values: $$ D < \left( \frac{2 \times 15.11 \times 10^{-6} \times 22.5 \times 10^{-6}}{9.81 \times \frac{1}{300} \times 1400} \right)^{1/3} $$ $$ D < \left( \frac{6.80 \times 10^{-10}}{4.58 \times 10^{-2}} \right)^{1/3} $$ $$ D < \left( 1.48 \times 10^{-8} \right)^{1/3} $$ $$ D < 2.48 \times 10^{-3} \, \text{m} $$ $$ D < 2.48 \, \text{mm} $$

Therefore, the model predicts that for a typical candle flame, the wick diameter must be greater than approximately 2.48 mm for a flame to be sustained. Below this size, the flame cannot maintain the necessary combustion conditions due to insufficient fuel flow and heat transfer dynamics.

Conclusion

The model constructed in the paper indicates that there is indeed a minimum size for the flame of a candle, specifically related to the diameter of the wick. The critical wick diameter for sustaining a flame is approximately 2.48 mm, below which the flame is unlikely to survive. This critical size ensures that the necessary conditions for combustion, such as adequate fuel supply and heat transfer, are met.