Author’s Translation


The Cause and the effects of Fukushima Nuclear Accident discovered from Theoretical Physical Analysis

Hajime Nishimura  Prof.
Emeritus Tokyo University

Fumito Kotari       Godfoot Research





1.     The daily release of the radioactive matter to the atmosphere 

2.     The daily release of the radioactive matter to the marine sphere

3.     Comparison with the cases of Chernobyl and Three Mile Island

4.     The complete sequence of events leading to the hydrogen explosion

5.     The capital cause of the disaster concluded from the theoretical analysis




Estimation of the Daily Release of Radioactive Matter into the Atmosphere

Purpose  The estimation of the daily discharge of the radioactive matter from the Fukushima Atomic Power Plant No.1 to the atmosphere.


Method  Our method is the reverse application of the atmospheric pollution model which is normally to predict the pollutant concentration of any given place under the given discharge rate of the pollutant from a stack. By the reverse application, we mean to estimate the rate of the pollutant discharge from the given concentration of some place. Here we apply the method for the radio active matter taking it as the pollutant. In actual application, the intensity of the effect of the radioactive matter to human body measured at monitoring stations and expressed by μSv/h was used as the input data to the whole model by converting it to the radioactive matter concentration Bq/m3 before applying it to the atmospheric pollution model.

Environmental data as input   Fig. 1 shows the intensities of the bodily effect of the radio active matter expressed inμSv/h measured at the seven monitoring stations placed at about 40 kms from the plant site on 22 and 23 March. The figure does not show any uniform distribution but characterized by the appearance of a singular point at Iidate Mura. The non-uniform distribution reveals that the radioactive matter does not disperse by the simple diffusion from the source but is mainly carried away leeward by the wind with some lateral diffusion which expand the lateral distribution of the pollutant along the path of the wind. The resultant form of the distribution is called a plume. Iidate Mura seems to located exactly under the plume or in the plume. As the bodily effect intensities of the neighboring stations at Minami Soma and Tamura separated less then 10 kms from the line connecting the plant site with Iidate Mura are drastically lower than that of Iidate Mura, the plume width seems to be considerably smaller than 10 km at around 40 km from the source, the power plant.


Plume Model  Two types of diffusion model are usually applied to the atmospheric dispersion of the pollutant: the puff model and the plume model. Judging from the distribution of the radioactive matter shown in Fig 1, we have applied the plume model  In the plume Model, the concentration in the plume is assumed to vary primarily with the distance from the source and secondarily from the vertical and horizontal distances from the center of the plume. The concentration distribution in the plume at a given leeward distance from the source is assumed to obey the normal Gaussian distribution described by   exp-(y/σ)  exp-(z/σz). Therefore the average concentration of the plume at the section of the given point x  appears at y=0 and z=σz.  If we approximate the plume as having a rectangular cross section with 2σas its width and 2σz. as its height, the average concentration appears where the rectangular plume touches on the ground.


Application to the present case  In order to apply the plume model to the present case, the primary importance lies on the determination of the height of the origin of the plume. We determined it from the photograph of the white smoke of vapor rising from the site, which is estimated 5 or 6 times as high as the height of the power plant building of 50 m high. Therefore the height of the plume source was found as high as 300 m.

    The next task is the determination of the atmosphere stability from the weather on Mar 24 and 25. As it was cloudy with drizzles of rain under the wind of 2~3 m/s,  the stability was found as D.

     The consequent task is to find the leeward distance for whichσz reaches 200~300m from the D curve in Fig. 2. We know the distance is around 30~40 km from the plant site. From Fig. 1 we know Iidate Mura satisfies the condition and it means so the concentration at the site represent the average concentration  in the plume at this cross section.

     The width 2σ and the height 2σz of the plume at Iidate Mura is respectively estimated as 2000m and 200m. Therefore when the wind velocity was 2 m/s, the total daily volume of the air Q  passing over Iidate Mura as a plume is


           Q = U 2σ2σz ­= 4 x 2 x 105 m/d x 103 x 102 = 8 x 1010


Estimation of atmospheric radioactive emission   Once the air volume Q is known, the daily discharge of the radioactive matter to the atmosphere G can be estimated as

                         G = Q Cav

when the average concentration of the plume Cav at the point is known. 

  We have determined Cav , the concentration of radioactive matter in the air Bq/m3 from the amount of the radioactive matter in bodily tissue expressed by μSv/h.  In the case of the constant daily uptake of the radioactive matter Bq/d,  the amount of the internal effective radioactive matter expressed by Sv/h depends on the mean residence time of the radio active matter.  This principle applies not only for the bodily intake of the radioactively tainted milk and water  but also for the inhalation of the tainted air, which means we can estimate the concentration of the radioactive matter in the air Bq/d  from the internal amount of the radioactive matter Sv/h once we know the amount of air inhaled in day.

 The relation between Bq and Sv/h has been established for Iodine 131 exp


experimentally as 1Bq = 7.4 x 10--3μSv/h  or  1μSv/h130 Bq 

The radioactive matter concentration Bq/ m3 at the site of internal concentration of 1μSv/h can be determined by knowing the amount the air inhaled by an adult man . As it is about 10 m3 the radioactive  concentration in the air corresponding to 1μSv/h  is  13 Bq/ m3

As the internal concentration at Iidate Mura was 10 μSv/h , the atmospheric concentration is supposed to be  130 Bq/ m3 

  Therefore the daily emission of the radioactive matter from the Fukushima plant is estimated as


        130 Bq/ m3 x 8 x 10 10 = 1013 Bq / d = 10 T Bq / d


The daily discharge to the atmosphere was found as

10 Tera Becquerel per day





Estimation of the Daily Discharge of Radioactive Matter to the Coastal Area


Purpose   Estimation of the daily discharge of the radioactive material from Fukushima nuclear power plant No.1 to the neighboring marine sphere


Method   Two methods are available for the purpose. One is the estimation using the data on the overflow water from the unit, the amount of overflow for a day G kg /day and its concentration C Bq/kg.

     The other method is to utilize the diffusion equation for the coastal area together with the radioactive concentration data Bq/kg of the offshore seawater measured by the monitoring boat.


Daily discharge from direct method   The radioactive material concentration in the effluent water from the plant site measured on March 21 at 800 m downstream of the end of the effluent duct was  3 x 104 Bq/kg while the total amount of the water sprayed on the plant on Mar. 21 and 22  is reported as about 200 tons for each day. 

    Therefore the total amount of radio active material Q discharged on March 21 is estimated as


           Q = 3 x 104 Bq/kg x 200 x 103  kg/ d  = 6 GBq/d


The daily discharge of the radioactive material to the coastal area is estimated as about   6 GBq/d


Empirical coastal diffusion equation   Any material continuously discharged from a sewage to the coastal area is dispersed by the turbulent diffusion, which actually consist of the horizontal as well as vertical diffusion. Both diffusions depends on the tidal current and wind and therefore the theoretical prediction is totally impossible. Only the empirical approach based on the numerous measurements of pollutant concentrations in the coastal area is the way for practical solution.

     In 1970s, Nishimura took such approach in the Seto Inland Sea, which is not an inland sea but a channel of 150 km x 40 km with the both ends  connected to the Pacific Ocean. He assembled a vast amount of data in the Seto Inland Sea by taking the fresh water of  rivers as a pollutant to the -ea of saline water.

An empirical diffusion equation derived from such effort is


C = ( 10-3  ~ 10-4 ) ( L / S )


where C and L are the concentration and the daily discharge (kg/day), while S is the surface area of the sea encircled with the contour that connected the the same concentration C.  The coefficient 10-3 ~ 10-4 corresponds to the reciprocal of the diffusion coefficient that include both the horizontal as well as the vertical diffusion. In the outer ocean the coefficient approaches to 10-4.


Daily discharge estimated from diffusion model   A boat is daily monitoring the concentration of the sea water at 30km off coast of the power plant, the result of which is shown in Fig. 3  If we assume a rectangle of 30km x 60 km as the area encircled by the contour of the same concentration, the corresponding concentration was about 44 Bq/ l in Iodine 131 and 16 Bq/ l  Cesium 137 and 60 Bq / l in total.

     Therefore the daily discharge of the radioactive material estimated from the offshore data using the empirical diffusion equation is


L = 104 C S = 104 x 60 x 30 x 60 =  10 Bq/ d  =  1 GBq/ d


The daily discharge of the radioactive material from the Fukushima plant to its coastal area is estimated as about 1 GBq from the empirical diffusion equation.




Comparison with Chernobyl and Three Mile Island Cases


A worthwhile comparison with the present case would be the comparison with the cases at Chernobyl, Three Mile Island and Tomsk in the former Soviet Union. Among the three cases, the quantitative data on the amount of emission is available only for the accident of a fuel reprocessing plant in Tomsk. According to the official disclosure of IAEA, the total amount of Cs 137 released to the atmosphere is estimated as 505 GBq.

    On the Three Mile Island accident, no quantitative official data is                            available. The only data available on Internet is a disclosure by a journal called Walker, which says the total emission of Iodine 131 was 505 GBq to the atmosphere. The amount is absurdly small and cannot be credited at all.

     For the case at Chernobyl, as it was an total explosion of the atomic pile rather than an accident, the estimation is easy when we assume about a half of the radioactive material contained in the reactor was released to the environment. According to a report of OECD, that assumes 50 % of Iodine 131is released, the total amount of discharge is estimated as 1760 PBq, where PBq is a million GBq.





Theoretical Realization of the First Stage Process induced by

the First Impact that finally lead to the Hydrogen Explosion



Any disaster resembles avalanche  Even if the first impact may have caused

the first accident, it cannot cause a disaster unless it started the first stage of

process which lead to the second accident which, by starting the second stage

of process, triggered the final disaster.


Hydrogen explosion in every reactor in operation        As shown in Fig. 4,

hydrogen explosions occurred in reactor N0.1,No.2 and No 3, each of which is

accompanied by a burst of the radioactivity level 8 hours after the explosion.

Each explosion was a huge one blowing off the most part of building. Most

experts of explosion I interviewed by telephone agreed to liken the explosion

as powerful as 100 kg of TNT. The explosion did not only destroyed the

building but would have severely damaged the machines and appliances in

the building which would have been useful in evacuation and restoration.


An accurate disclosure of the fist stage of process is the aim of the paper

The most important lesson everyone can learn from this disaster would be

the reproduction of the first stage that ultimately lead to a disaster in this

case but would have not been inevitable otherwise managed. For useful

lesson, a very accurate knowledge is necessary. Inaccurate supposition might

cause a disadvantage to pupil. Japanese proverb goes akago wo nagasu drain

a baby with bathwater and atsumono nj korjte namasu wo fuku a very hot

pudding makes one very cautious of cold pudding.


Rumored cause of the accident   Rumor goes as

"Tsunami washed away everything including important machines and equipments of the Fukushima nuclear power plant"

"The nuclear reactor could not stand the earthquake and severely damaged"

"Fukushima plant is not equipped with the emergency electric power supply

that should have avoided the cut of the coolant supply to the reactor"

"The used fuel rods cooled in the fuel rod pool the placed at the upper floor

of the reactor produced hydrogen when dried up because of the absence of

cooling water"


How to get an accurate knowledge of the first stage process

As the sophisticated measurement systems temporally failed immediately

after the earthquake and were not restored a while, only measurement data

available is the water level in Unit 1, though the data for the first 10 hours is

still unavailable.

Under this condition, only method to infer the actual process of the first

stage is a genuine theoretical reproduction of the whole process fully

utilizing whole laws of physics. Such inference leaves a limited freedom to

the initial condition and the circumstantial condition. Therefore the

theoretical reproduction is justified only when the estimated situation made

a complete agreement with the limited available data. When the agreement

is complete by tuning the initial and circumstantial conditions, the assumed

conditions are true and actual.

That was the reason for us to have started with the theoretical reproduction of the water level.



Theoretical Reproduction of the Water Level in the Reactor


For the water in the reactor No.1, the change of temperature before boiling and the change of the liquid level after boiling were rigorously estimated using the heat balance equations.


For the change of the temperature T before boiling

              ρC S h0  dT/ dt  = 

For the change of the water level h after boiling 

                  HρS dh/ dt = Q      


where S is the cross sectional area of the reactor, h0 the initial water level and H is the heat of vaporization of water.

     All parameters used in the calculation is shown in Table 2  and the result of estimation is shown in Fig. 5 together with the measured data of the observed water level. The agreement seems to be excellent. It means that the initial condition and the circumstantial condition assumed in the calculation are justified. Its meaning is unexpectedly crucial and vital and will be discussed in the conclusion.



Theoretical Estimate of the Temperature Rise of a Fuel Rod when dried up


When the upper part of the fuel rod is dried up, residual heat generation will cause a harsh temperature rise until the radiation heat release balances with the heat generation. The time course of the temperature rise can be estimated by


              ρC v dT/ dt =  q -sσT  


where  v, s and q respectively stand for the volume, surface area and residual heat generation per unit length of a fuel rod. All parameters used in calculation is shown in Fig. 6.  The graph shows the temperature rises very quickly and within 2 minutes, it almost approaches to the equilibrium temperature that is 779 degree Centigrade.


    At high temperature, the casing of the fuel rod made of zircaloy reacts with water vapor and produces hydrogen. The rate of this reaction steeply depends on temperature as shown in Fig. 7.  An appreciable reaction occurs in the temperature range between 750 C and 800 C. The found temperature exactly falls in this range.




Important Conclusions drawn from the Analysis


1)  The emissions of the radioactive materials into the atmosphere is about a thousand times greater than the emission to the coastal area.


2) When compared with the case of the Chernobyl accident , the daily amount of the atmospheric emission is about a hundred thousandth 1/100,000 of the total emission from the Chernobyl explosion. So it means the total emission from the Fukushima site would not exceed the one thousandth 1/1,000 of the Chernobyl case even if the present emission of March 22,  lasted 100 days


3)  From the exact agreement of the purely theoretical estimation and measured data on the water level, we can infer that the earthquake gave no damage to the reactor itself but gave a leaking failure to the pipe line to or from the steam turbine.


4)  The whole disaster is a sequence of accidents triggered by the foregoing one. The accident was the stop of the coolant water to the reactor which resulted in the dry up of the reactor and the complete exposure of the fuel rods followed by an instant temperature rise to 800 degree centigrade, which produced a large amount of hydrogen that eventually explode and destroyed the building and the important appliances.


5)  The primary cause of the whole series of the accidents is obviously the stoppage of the coolant water supply to the reactor. The reactor was equipped with the emergency electricity supply driven by diesel engine which was designed to start 10 seconds after the stoppage of the whole electricity supply. Therefore the capital cause of the disaster is the failure of functioning of the emergency electricity supply system.


6)  From a detailed analysis of the theoretical calculation, it was discovered the initial water temperature of the reactor was 40 degree centigrade while the temperature under the normal operation is about 140 degree centigrade. This means emergency electric supply system including the Diesel engine worked properly after the earthquake for some duration  enough to cool down the reactor. Its stoppage after a few minutes is a mystery.



All the Figures and Tables are attached.