Here is why our fires should be built to look the same. Consider the model of a burning pile of fuel as a volume V of height H, base dimension D and absolute temperature T. The volume is illustrated as a cone or pyramid in Fig. 1, but it can have any shape (such as a parallelepiped), provided that the height of the body is H and that the base area has a single length scale D (for example, round or square). The scale analysis reported below does not capture the effect of volume shape, cone versus pyramid, except for the aspect ratio of the volume profile, H/D.

Figure 1 The fire temperature (T) as a function of the shape of the profile of the pile of fuel. The clashing asymptotes define this behavior and the method of intersecting the asymptotes18 pinpoints the architecture, the design. Photographs taken by Adrian Bejan. Full size image

The body of fuel is a porous medium through which fluid flows because of the buoyancy effect due to low density (hot) fluid inside the body and high density (cold) fluid outside the body. For a hot volume of height H, the pressure difference scale that drives fluid through the cone structure of temperature T is18

where ρ is the air density, averaged between the air outside and inside, β is the coefficient of volumetric thermal expansion, ΔT is the temperature difference T − T 0 , where T 0 is the ambient temperature and g is the gravitational acceleration.

The flow of air and combustion gases through the body is modeled as Darcy flow through a saturated porous medium with known permeability K, which scales with the pore diameter squared18. The heat transfer from the surface of the body to the ambient is modeled as black body radiation. We consider this flow in the two extremes, a tall pile (H D) and a shallow pile (H D). The treatment is based on scale analysis, which means that we neglect dimensionless factors of order 1.

Tall limit

The volume averaged velocity of the fluid that permeates through the body is oriented in the vertical direction,

where μ is the fluid viscosity, which is treated as constant. Combining Eqs. (1) and (2) we find that v ~ Kgβ ΔT/ν, where ν is μ/ρ. The scale of the mass flow rate through the tall porous body is

where D2 is the scale of the body cross sectional area. The heat generated by combustion inside the body is proportional to the flow rate of oxidant therefore we represent it as where C accounts for the reaction of combustion and the heating value of the fuel19. The heat generation rate is transferred to the ambient (T 0 ) via radiation,

with the observation that σ is the Stefan-Boltzmann constant and HD is the scale of the external surface of the tall body. Combining Eqs. (3) and (4) and noting that when T > T 0 we can approximate ΔT ~ T and T4 – ~ T4 and obtain the scale of the body temperature

This result shows that the body temperature decreases as the body becomes taller. In practice, needed is a high temperature, therefore a shorter body is of interest. This is why we turn our attention to the opposite extreme:

Shallow limit

In a shallow porous body the volume averaged fluid velocity is oriented horizontally,

The mass flow rate is where HD is the scale of the cross sectional area pierced by u. The radiation heat transfer rate is where D2 is the scale of the external surface of the body. In place of Eq. (4) we write and obtain the scale of the shallow body temperature,

This shows that the shallow design is also inferior, because the body becomes colder as it becomes more shallow.

Intersection of asymptotes

Together, the two extremes covered by the preceding analysis are represented by the two asymptotes sketched in Fig. 1. The actual curve that relates T to the aspect ratio H/D is a bell-shaped curve that fits under the intersection of the two asymptotes. Most useful is the design that offers the hottest fire. This design is easy to identify by finding the H/D location of the intersection of the two asymptotes, Eqs. (5) and (7) which is

The best edifice of fuel must be such that its height is comparable with its diameter at the base. This geometric conclusion is independent of the thermophysical properties that were necessary in constructing the model of the flow phenomenon (g, ρ, μ, ν, K, σ, C, β).

Alternatively, the analysis outlined between Eqs. (2)–(8), can be repeated while regarding the volume of fuel as fixed (V ~ D2H) and searching for the height H, or the base length D, for which T is at its peak. After the intersection of the tall and shallow asymptotes, the conclusion is that H opt ~ V1/3, or D opt ~ V1/3, is the same as in Eq. (8).

This conclusion does not change if we replace β with T–1, which is recommended by the ideal gas model for the fluid19. The scale of the peak temperature achieved when H ~ D is found from either Eq. (5) or Eq. (7), with β = T–1,

This confirms common knowledge: the fire is hotter when it “breathes” better, which happens when the permeability K increases.