Hi there! I realized that I wasn't currently happy with the methods used to calculate the energy of Sukuna's furnace released in Shibuya and Shinjuku, so I wanted to propose an alternative approach and ask what other people thought.

Current Model of Fuga

Currently, Furnace (or Fuga) is modelled as a single thermobaric explosion, where the most commonly used method of calculating it's energy using the energy required to vaporize the surrounding area. However, this model holds a flaw within it.

When unleashed in Shibuya, Sukuna notes that he reduces his domain's range from it's maximum of 200 meters down to 140 meter in order to allow Megumi (who lies unconcious outside the domain's barrier) to survive. The resulting Fuga unleased on Mahoraga completely annihilates him, with the assumption that all pulverized matter was turned into fuel to allow the thermobaric explosion.

This calculation would produce a resulting explosion size much larger than the set domain radius of 140m, which would convincingly kill Megumi nearby. The solution I propose is the manipulation of domain barrier, where Sukuna in Shibuya manipulated the conditions of his domain's barrier in order to prevent any energy from escaping the radius.

When extending this assumption to the explosion present in Shinjuku (seen in the attached image), there is some limitations in the validity of the idea that Malevolent Shrine contains the entire energy of Fuga. It's clear some forms of energy is escaping past the 200m radius. However, the explosion area of Fuga is somewhat unnatural (in my eyes), with a much larger vertical profile than you would expect.

These two aspects lead to to the hypothesis that the energy Fuga can be contained by a variable value dependent on the domain conditions. I will use this in my calculations.

Furnace Mechanism

To quickly discuss Furnace, we know that it is a thermobaric explosion. When used in a domain, Furnace works through first allowing the constant dismantles to pulverize all matter in the surrounding area. This creates an enormous mass of dust, which by the JJK wiki is imbued 'with the same explosive cursed energy as Divine Flame'. This energy is prevented from escaping the domain's radius through altering the barrier conditions, essentially turning Malevolent Shrine into a large pressure cooker.

Once the dust particles of Malevolent Shrine are at their limit of energy, Sukuna releases his Devine Fire Arrow, which triggers a 'deflagration-to-detonation transition across these particles, generating instantaneous high temperatures, shockwaves, decompression and intense pressurization'.

The Equation for Furnace's Energy

As seen above, furnace is proportional to the mass of explosive material and the Specific Energy of dust imbued with 'explosive CE'. My goal in this section is to create a function for Energy that could be applied to every possible combination of variables.

Mass of Particles

The mass of explosive material can be set as a function of time and location. For Shibuya and Shinjuku (which takes place in high density urban sprawl in Tokyo), we will characterize the maximum amount of mass enclosed by Malevolent Shrine as C_max, the radius of the domain as R, the average height of buildings as H, the percentage of ground covered by buildings as f_area, and the average density of physical material within each building as p_build.

C_max = (πR************^(2)) × H × f_area × p_build.

Then, to find the mass of actually pulverized matter, we will create a half-life function of how long it is required to pulverize the entirety of C_max. This function will use the variables of time the domain was open (t), and time required to destroy half of the physical matter within the domain, or the Pulverization Half-Life (k). Therfore, the final mass of pulverized particles in the domain can be described as:

C(t) = C_max × (1-2^(-(t/k)))

For this equation, k is not set in stone. In Shibuya, the time required to destroy the building is a lot larger than the time required in Shinjuku. Therefore, we will define our value for k whenever calculating a feat.

Specific energy

The Cursed Energy imbued into the dust particles is difficult to quantify, so we will set an equation to explore different values of specific energy.

We know that the dust particles are charged by Sukuna's cursed energy. We will assume that this energy is the maximum amount of energy the particle can hold before it combusts. Therfore, each particle holds the energy:

E = m × H_c

This equation holds H_c, or the specific energy of the particle. We will define this variable as a set of possible values, which will allow us to find lowball and highball calculations.

H_c =

• 25 (equivalent to the specific energy of coal)
• 31 (equivalent to the specific energy of Aluminum Powder)
• 50 (equivalent to the specific energy of Gasoline/Aviation Fuel)

The Barrier Function

This part of the equation will calculate the effect of Malevolent Shrine's barrier conditions. We know that the domain is capable of containing partial or all of the explosion's energy. In shibuya, the explosion has no escapeage of energy, causing the energy to be constrained to a smaller space. We will quantify this through considering the effects of Blast Wave Reflection (W), where an explosion can increase the power of the explosion up to a factor of 3.

W = (1+2(1-β))

In this equation, β represents an arbitrary value, and can be thought of as the percent of energy that can escape the domain. When β = 0, no energy can escape the domain, and the resulting blast has an increased AP of x3. When β = 1, all energy can escape the domain, and there is no increase in AP.

TLDR: the Final Equation

E = (C_max × (1-2^(-(t/k)))) × H_c × (1+2(1-β))

Note I've converted the following calcs into tones of TNT equivalent.

Calculating Shibuya

C_max =

• R = 140m
• H = 30m
• f_area = 50%
• p_building = 300 kg / m^(3)

t,k = unknown, but given the rate of slashes and time of domain, we can assume that all matter in the domain was pulverized.

H_c = {25, 31, 50}

β = 0 (megumi remained unharmed, therefore no energy escaped)

Resulting E = 4.97 - 9.93 Megatons of TNT (Small City to City Level)

Calculating Shinjuku

C_max =

• R = 200m
• H = 30m
• f_area = 50%
• p_building = 300 kg / m^(3)

t = 99 seconds (this aspect is incredibly important, as it sets a limit to how long the domain has to pulverize matter. This may be related to the binding vow Sukuna made to open the domain, as he lost a large amount firepower in exchange for opening the domain)

k = unknow, however it was shown that the rate of slashes was much lower when compared to Shijuku. It will be assumed that 60% of the matter in the domain was pulverized, k will be set to 70

H_c = {25, 31, 50}

β = 0.25 (it is shown some form of energy is escaping. However, the main component of the explosion appears to be contained within Malevolent Shrine. I will assume 25% of energy escapes the barrier)

Resulting E = 5.07 - 10.1 Megatons of TNT (Small City to City Level)

Analysis and Conclusions

This method can be used to calculate Sukuna's Furnace attack given any possibility of combinations. This suggests that this attack is most likely a City Level attack. The additional factor of considering barrier conditions can scale up the attack to as much as a factor of 3.

I suggest it's reasonable to assume the higher values of specific energy for this calculation. Cursed energy is imbued into the matter, which our model treats as the replacement for chemical bonds. It allows the city to undergo combustion, where normally city materials would be too unreactive to do so.

It's important to note that the value of specific energy can be held higher. Depending on the energy contained within sukuna's cursed energy, the upper bound of specific energy would be 142 MJ/kg, equivalent to hydrogen gas. Higher values would be unrealistic for what we view in the series, where lower values would prevent the deflagration-to-detonation transition.

I hope you enjoyed reading this yap fest! If anyone has any notes or questions, I'd love to answer them :D