Boiling and Drying Accident of High-Level Liquid Waste in a Reprocessing Plant: Examination of the Time-Dependent Temperature Increase of the Waste and of the Generation Rates of the Individual Components Released into the Gas Phase

Abstract Using the amount, composition, and decay power density of high-level liquid waste in a storage tank, the temperature change of the waste up to 600°C and the corresponding vapor and gas release rates of H2O, HNO3, NO2, NO, and O2 as a function of time after the loss of cooling function were obtained by the following method. The heat balance equations in and around the tank were derived, and the solution of the waste temperature change was numerically obtained using the vaporization rates of H2O and HNO3 and the generation rate of NOx, which were both obtained from the experiments using the simulated liquid waste. Utilizing the temperature versus time curve obtained from the equation, the release rates of the components described above were obtained as a function of time. This information on the progress of the accident can be used to study the Leak Path Factor of radioactive materials, especially of volatilized Ru, and further, it becomes basic information when considering accident management and suppressing the impact of a disaster.


I. INTRODUCTION
The boiling and drying accident of high-level liquid waste (HLLW) has drawn considerable attention among stakeholders because of high potential radiation exposure to the general public.3][4][5][6][7][8][9][10] In order to assess the radiation exposure, the amount of radioactive aerosols and volatilized 106 Ru released into the environment is necessary to be known.To do this, it is necessary to understand the behavior of H 2 O and HNO 3 vaporization and of NO 2 , NO, and O 2 generation, which not only carry the radioactive materials but also affect the Leak Path Factor (LPF), [11] especially of volatilized 106 Ru. [9] Previously, we reported [10] the NO 2 , NO, and O 2 generation rates from the simulated high-level liquid waste (SHLLW) up to 600°C under constant temperature increasing rates of 0.2 and 1°C•min −1 .In this study, using the amount, composition, and decay power density of HLLW in a storage tank, we predict how the waste a temperature up to 600°C b and also the individual generation rates of H 2 O and HNO 3 vapors and NO 2 , NO, and O 2 gases will change with time, based on the heat balance equation related to the inside and outside of the storage tank and experimental results on the H 2 O and HNO 3 vaporization and NOx generation using SHLLW.
]7] However, we consider that the contents of these reports are not sufficient as shown below.
Ishikawa et al. reported that the temperature increase of the waste up to 140°C could be predicted by extrapolating the molar boiling point elevation model. [4]However, HLLW is a nitric acid solution in which various types of nitrates are dissolved at high concentrations and precipitate with increasing temperature.Therefore, we consider that it is difficult to construct and solve the theoretical model up to 140°C.Indeed, the agreement between the predicted values and the experimental ones is not always good as the temperature approaches 140°C.For the analysis from 100°C to 800°C, Yoshida and Ishikawa [3] used the MELCOR code. [12]But, the code cannot handle nitric acid, so they introduced control functions and multiple state input volumes into the code to model the key phenomena of boiling events such as boiling above 100°C, nitric acid vaporization, and NOx gas generation. [3]However, in a subsequent paper [6] from the same group, the results are said to be qualitative.
To obtain the temperature versus time curve up to 600°C, it is necessary to solve the heat balance equation around the storage tank.The H 2 O and HNO 3 vaporization, nitrate decomposition, and heat transfer to the cell air and cell wall proceed simultaneously.This paper describes an integrated model for simulating time-dependent changes in the waste temperature and the released gas composition, as well as the solution method and the results obtained.There is no other report of such contents.
Note that the values used in the following analysis may not reflect the actual plant values.

II. OUTLINE OF THE MODEL TO BE ANALYZED
One tank is installed in one cell.The tank is filled with HLLW.Cell ventilation air is flowing into the cell but stops due to the accident.The corresponding numerical values are as follows: 1.The amount of HLLW is 120 m 3 .
2. The properties of HLLW are as follows.The amount of 0.4 m 3 of HLLW is produced by reprocessing 1 ton of spent fuel (pressurized water reactor fuel, specific power of 38 MW•ton −1 , burnup of 45 000 MWd•ton −1 , and cooling period of 6 years), and its composition is obtained using the ORIGEN-2 code.The decay power of 120 m 3 of HLLW becomes 578 kW.
3. Cell ventilation air is introduced with a flow rate of 57 mol•s −1 and at 25°C.4. All six outer cell walls are surrounded by concrete building walls, and the air in the space and the building walls both have a temperature of 25°C.
The temperature of HLLW rises with time after the cooling function loss and boiling begins.As time passes, the liquid waste dries and solidifies.Therefore, in the following analysis, the model was divided into two stages: the initial stage and the late stage.In the initial stage, the waste remains in the liquid phase, and so, the temperature of the tank wall and inside the tank is considered to be uniform due to a large amount of steam generated by the decay heat.On the other hand, in the late stage, the temperature distribution occurs inside the tank.Based on the experimental results (see Sec. IV.A.2), we assumed that the waste was in a liquid phase up to 130°C and above that was in a solid phase.c

III.A. Heat Balance Equation
The various heat transfer processes and related parameters affecting the waste temperature θ(t) (in degrees Celsius) are shown in Fig. 1.Although most of the decay energy is dissipated in the tank, part of the gamma rays is released from the tank contributing to the temperature rise of the cell wall (gamma heating).The decay energy dissipated in the tank is used not only for the temperature rise of the waste and the tank but also for vaporization of H 2 O and HNO 3 , decomposition of nitrates, and heat loss out of the tank (convection and heat radiation), and as a result, the waste temperature is determined.
The amount of heat transfer associated with the individual heat transfer processes shown in Fig. 1 depends on the following parameters.Variable t (in seconds) means the time after the cooling loss.The value θ (in degrees Celsius) is a function of t, that is, θ(t), and sometimes is described as simply θ.Also, note the following parameters 1 through 8: 1. Initial storage amount of the waste M 0 (m 3 ).
2. Energy release rate due to the radioactive decay R h (kW) and the leak rate of gamma-ray energy out of the tank (gamma heating) R γ (t) (kW); the power of [R h -R γ (t)] is used to heat the waste and the tank.

Amount of H 2 O in the waste
, and molar specific heat c W (kJ•mol −1 •°C −1 ); c W was assumed to be independent of temperature. [13] Amount of HNO 3 in the waste M N (t) (mol), its vaporization rate Y N (t) (mol•s −1 ), latent heat d λ N (θ) (kJ•mol −1 ), and molar specific heat c N (kJ•mol −1 •°C −1 ); c N was assumed to be independent of temperature.[13] 5. Heat absorption rate due to the thermal decomposition of nitrates (salts) R S (t) (kW) and the total specific heat c S (θ) (kJ•kg −1 •°C −1 ); the specific heat of each nitrate was assumed to be equal to that of the oxide, and the oxides were assumed to have a constant total mass M S (kg) irrespective of temperature (adequacy of the assumptions is explained in Sec.VI.D).
6. Mass of the tank including the internal structural material M T (kg) and its specific heat c T (θ) (kJ•kg −1 •°C −1 ).
7. The temperature of the tank is assumed to be the same as that of the waste.8. Heat transfer rate from the tank to the cell air by natural convection R loss1 (θ) (kW) and that from the tank to the cell wall by heat radiation R loss2 (θ) (kW), and their sum R loss (θ) = R loss1 (θ) + R loss2 (θ).The cell ventilation function is assumed to be lost due to the accident.
Using the foregoing parameters, the waste temperature θ(t) is described by the following heat balance equation e : d Since H 2 O and HNO 3 are vaporized from nitric acid solution, the interaction between H 2 O and HNO 3 needs to be taken into account.The influence of this effect is explained in Sec.IV.C.3.
e The tank is made of SUS with a thickness of 0.03 m.The thermal conductivity of SUS is 17 W•m −1 •K −1 , and the heat transfer coefficient through the tank wall is 17/0.03= 570 W•m −2 •K −1 , which is about 100 times larger than the natural convection heat transfer coefficient (2 to 5 W•m −2 •K −1 ), and therefore, the heat transfer resistance from the tank to the cell air is controlled by natural convection.
where Γ(t) = sum of the heat capacities of the waste (H 2 O, HNO 3 , and the nitrates = the oxides f ) (kJ•°C −1 ) and the tank and is given by the following equation: The right side of Eq. ( 1) shows the net power to increase the temperature of the waste and the tank.

III.B. Solution Method
In order to obtain the solution of Eq. ( 1) up to 130°C, it is necessary to know the time change of the individual parameters 1 through 8 shown in Sec.III.A.In the present study, the relationship between the individual vaporization rates of H 2 O and HNO 3 and the waste temperature was obtained from the experimental results as shown in Sec.IV.The relationship between the heat absorption rate accompanying the decomposition of nitrates and the waste temperature is examined in Sec.V.In this examination, the experimental results of the previous report concerning the NOx generation [10] were used.The heat capacity Γ(t) in Eq. ( 1) is treated in Sec.VI, and the heat transfer from the tank including gamma heating is dealt with in Sec.VII.These results are integrated to solve Eq. ( 1) by the difference method in Sec.VIII.Then, using the result of Eq. ( 1), the generation rates of gaseous components are obtained as a function of time.

IV. HEAT ABSORPTION RATE OF H 2 O AND HNO 3 VAPORIZATION
The content in this section can be used for both stages.The heat absorption rate due to liquid vaporization is the sum of the corresponding individual values of H 2 O and HNO 3 .Each heat absorption rate is obtained by multiplying the vaporization rate and the latent heat.The vaporization rates were obtained from the results of experiments using SHLLW as described below.Since H 2 O and HNO 3 are vaporized from nitric acid solution, the interaction between H 2 O and HNO 3 needs to be taken into account to obtain the individual latent heats.This effect is described in Secs.IV.B.3.and IV.C.3.

IV.A.1. Experimental
Two types of the simulated wastes, SHLLW-J and SHLLW-K, were prepared.The individual compositions are shown in Table I.SHLLW-J simulates HLLW, and SHLLW-K contains only the main 12 compounds.The f The meaning of this equal sign is explained in Sec.III.A, No. 5. details are given in the previous report. [9]SHLLW-J was used to examine the vaporization rates of H 2 O and HNO 3 as a function of time, and SHLLW-K was used to examine the effect of the temperature increasing rate on their vaporization ratios versus temperature curves (see Sec. IV.A.2.b).Two runs were carried out using SHLLW-J and five runs using SHLLW-K.Individual experimental conditions are given in Table II.
The apparatus to measure the vaporization rate is shown schematically in Fig. 2. SHLLW-J of 120 mL was charged into a separable flask with an inner diameter of 85 mm and a height of 110 mm (nominal volume of 500 mL).For SHLLW-K, 120 to 200 mL was charged.Two thermocouples were inserted to measure the waste temperature near the bottom and the vapor temperature at the top of the flask.The flask was heated on a hot plate.A 2-mm-thick aluminum plate was placed between the flask and the hot plate so that the heating from the bottom was uniform.In addition, flexible rubber heaters were wound around the side of the separable flask and the lid of the flask.The pipe connecting the steam outlet to the inlet of a Liebig condenser was heated by a ribbon heater.The waste temperature was controlled by the hot plate and the temperatures of the side wall; the lid and the pipe were controlled to be the same as the waste temperature.
Steam generated in the flask flowed into the condenser, and the condensate was recovered periodically.The temperature of the cooling water was 15°C.The recovery was performed at appropriate intervals in a batch manner from the boiling point to about 280°C (the sampling interval can be seen from the data points in Fig. 3 shown in the next section).At temperatures above 120°C where the amount of condensate became small, 10 mL of H 2 O was injected into the condenser before recovering the condensate through the pipe shown in Fig. 2 to wash and collect the condensate attached on the inner wall.
The mass and density of the recovered liquid including the condensate and the washing water were measured for each batch, and then, the liquid was diluted to 1000 to 10 000 times to measure the pH.These results were used to obtain the net amounts of H 2 O and HNO 3 recovered in the condenser.It should be noted that the measured amount of HNO 3 is different from the amount vaporized in the flask as described in the next section.
In order to observe the state of the waste, a vertical slit was made in the rubber heater that was wound around the side of the flask.As boiling began and concentration proceeded, it became unclear whether the waste was liquid or solid.At 130°C, no bubbling was observed, and the waste became like a paste.From the previous experiment, the following is known. [10]When the temperature reached 180°C, the waste was more like a solid than a paste, but the waste taken out from the flask was somewhat moist and gave off a strong HNO 3 odor.The waste heated to 280°C to 300°C was dry.It was solid with many voids.We examined the amount of vaporized H 2 O and HNO 3 using SHLLW-J.The initial amount of H 2 O in the liquid waste obtained was as follows:     The initial amount of H 2 O obtained in this way, 48 200 mol•m −3 , became the same as the final vaporized amount shown in Fig. 3, which shows the time-dependent changes of the measured cumulative amounts of H 2 O (and HNO 3 ) obtained from Runs J1 and J2.In the figure, the cumulative amount is shown as molar quantity per 1 m 3 of initial liquid waste, and the boiling start point corresponds to t = 0.

IV.A.2. Results and Discussion
On the other hand, the measured cumulative amount of HNO 3 at 280°C is about 2600 mol•m −3 for both runs as shown in Fig. 3, which is different from the initial amount of 2000 mol•m −3 .This difference is explained as follows.Previously, we reported [10] that most of the Zr, Ru, and Pd nitrates in SHLLW-J do not produce NOx but produce HNO 3 by 180°C, while HNO 3 left in the drying waste produces Nox mainly above 300°C, and that the difference in the amount between them is 100 mol•m −3 .Therefore, the final vaporized amount of HNO 3 should be equal to 2100 mol•m −3 .The difference between 2100 mol•m −3 and the measured 2600 mol•m −3 should be due to the amount produced from NOx, mainly NO 2 , in the condenser by the following reactions [14] : where H 2 O comes from the condensate and the washing water injected into the condenser.Formulas (3) and (4) show that H 2 O is consumed in absorption of NO 2 and that the amount of H 2 O in mol•m −3 in the initial waste is about 24 times larger than that of HNO 3 , and so, the consumption due to the NO 2 absorption is negligible.The cumulative amount curve of vaporized HNO 3 shown in Fig. 3 was obtained by using the final value of 2100 mol•m −3 and correcting the amount of HNO 3 produced by NO 2 according to its rate of generation.In five runs using SHLLW-K, the heating time from 104°C (boiling start temperature) to 300°C was changed variously as shown in Table II, and the effect of the temperature increasing rate on the vaporization ratios of H 2 O and HNO 3 was examined.Individual vaporization ratios were obtained using the corresponding final vaporization amounts as the denominator and the cumulative vaporized amounts at that temperature as the numerator.The final amounts were obtained in the same way as for SHLLW-J described above.The vaporization ratio versus temperature curves for H

IV.B. Heat Absorption Rate of H 2 O Vaporization
The heat absorption rate due to the vaporization is given by the product of the vaporization rate and the latent heat of vaporization.The vaporization rate can be obtained from the above experimental results, and the latent heat can be obtained from the literature. [15]

IV.B.1. Relationship Between the H 2 O Vaporization Ratio and Temperature
The H 2 O vaporization ratio ξ W (θ), that is, the ratio of the cumulative amount at temperature θ to its final amount, is shown in Fig. 6, which was obtained from the experimental results of Runs J1 and J2.When θ < 104°C (before boiling), ξ W (θ) = 0. Based on the results of the figure, the curve for dξ W (θ)/dθ was obtained as shown in Fig. 7. Using the figure, the following approximate expressions were obtained.The solid lines in the figure correspond to these expressions:  Although the data points scatter above 160°C, it is attributed to a small amount of vaporization and is not important since most of the H 2 O has already been vaporized.

IV.B.2. Vaporization Rate
The H 2 O vaporization rate Y W (t), which is equal to the time derivative of the cumulative amount of H 2 O vaporization M W (0)ξ W (θ), is given as follows: The final cumulative amount of vaporized H 2 O is equal to the initial value of M W (0) as explained in Sec.IV.A.2.a.The value dξ W (θ)/dθ in Eq. ( 6) is given by Eq. ( 5).

IV.B.3. Latent Heat of Vaporization
For the latent heat of H 2 O vaporization λ W (θ) (in kJ•mol −1 ), the following approximate equations created from data from 50°C to 373.95°C [16] were used: Since H 2 O is vaporized from nitric acid solution in the present case, the interaction between H 2 O and HNO 3 needs to be taken into account.This effect was approximated by the heat of dissolution of HNO 3 in water and included in the latent heat of HNO 3 vaporization (see Sec. IV.C.3).

IV.B.4. Heat Absorption Rate
The product of the vaporization rate Y W (t) and its latent heat λ W (θ) obtained above is the heat absorption rate.

IV.C.1. Relationship Between the HNO 3 Vaporization Ratio and Temperature
The HNO 3 vaporization ratio ξ N (θ), that is, the ratio of the cumulative amount at temperature θ to its final amount obtained from Runs J1 and J2, is also plotted in Fig. 6.When θ < 104°C (before boiling), ξ N (θ) = 0. Based on the results of the figure, the curve for dξ N (θ)/dθ was obtained as shown in Fig. 8. Using the figure, the

IV.C.2. Vaporization Rate
The HNO 3 vaporization rate Y N (t), which is equal to the time derivative of the cumulative amount of HNO 3 vaporization M N (0)ξ N (θ), is given as follows: where dξ N θ ð Þ=dθ is given by Eq. ( 7).

IV.C.3. Latent Heat of Vaporization
The latent heat of HNO 3 vaporization from aqueous nitric acid λ N (in kJ•mol −1 ) was approximated as the sum of the latent heat of HNO 3 vaporization from pure HNO 3 and the dissolution heat of HNO 3 in water: The first term on the right side is the latent heat of HNO 3 vaporization, [4] and the second term H sol (in kJ•mol −1 ) is the dissolution heat of HNO 3 in water.The latter was calculated by the following approximate Eq. ( 9) created from the data at 25°C [15] : where ω = molar fraction of HNO 3 .H sol does not depend on temperature as is explained below.
The derivative of the dissolution heat with respect to temperature is equal to the difference between the heat capacity of the solution and the sum of the heat capacities of H 2 O and HNO 3 .Since the heat capacity of nitric acid solution could be assumed to be equal to the sum of the heat capacities of H 2 O and HNO 3 as is explained in Sec.VI.A, the dissolution heat does not depend on temperature.Note that the values of ω and H sol change with time as a result of vaporization.

IV.C.4. Heat Absorption Rate
The product of the HNO 3 vaporization rate Y N (t) and the latent heat of vaporization λ N (θ) obtained above is the heat absorption rate accompanying HNO 3 vaporization.

V. HEAT ABSORPTION RATE OF THERMAL DECOMPOSITION OF NITRATES
The content in this section can be used for both stages.The total heat absorption rate R S (t) associated with the thermal decomposition of nitrates is equal to the sum of products obtained by multiplying the decomposition heats required for the generation of unit amounts of NO 2 and NO by the respective generation rates (see Sec. V.C).In the following, the decomposition heat and the generation rate are examined.

V.A. Heats of Decomposition Generating NO 2 and NO
In the waste, various nitrates are included.The decomposition heat is estimated as follows.Increasing the temperature of the waste, the nitrates generate not only NO 2 and NO but also HNO 3 . [10]However, the decomposition heat generating HNO 3 is small as is explained in Appendix A, and so, it is not taken into consideration.
First, the decomposition heat generating NO 2 is examined taking Nd(NO 3 ) 3 as an example: The enthalpy of this reaction Δ d1 H (in kJ•mol −1 ) is obtained as where Δ f H(Nd 2 O 3 ), Δ f H(NO 2 ), and Δ f H(Nd(NO 3 ) 3 ) are their formation enthalpies.
The decomposition reaction of Nd(NO 3 ) 3 to generate NO and its enthalpy Δ d2 H are given as follows: Therefore, if the formation enthalpies of Nd(NO 3 ) 3 , Nd 2 O 3 , NO, and NO 2 are given, the enthalpy of the nitrate decomposition is obtained.The temperature dependence of the enthalpy is several percent within the scope of this study, [13] so it was not considered, and the standard enthalpy values were used.In Table III, the standard enthalpies related to the various nitrates and oxides in the waste are shown.In the table, the heat absorption accompanying the individual nitrate decomposition is also shown.The heat absorption shown in units of kJ•L −1 is obtained by multiplying Δ d1 H (or Δ d2 H) by the number of moles of nitrate per 1 L of waste (see Table I).Standard formation enthalpies of all oxides and part of the nitrates were obtained from the literature. [13,17]The enthalpies of the nitrates that are not given in the literature were estimated as follows.
Many standard formation enthalpies of chlorides are given in the literature. [13]For Mn, Ni, Sr, Ag, Cd, Cs, Ba, and Ce elements, standard formation enthalpies of chlorides and nitrates are given.There is a good correlation between the two, which can be approximated by the following equation: If the standard formation enthalpy for nitrate was not found in the literature, it was estimated from the standard formation enthalpy of chloride using Eq.(10).The values obtained in this way are listed as superscript a ( a ) in Table III.
The decomposition enthalpies of the individual nitrates generating NO 2 or NO were estimated using the standard enthalpies obtained above and are shown in Table III as Δ d1 H and Δ d2 H.The values of Δ d1 H/ NO 2 and Δ d2 H/NO show the decomposition heat generating 1 mol of NO 2 and NO, respectively.
However, since the standard formation enthalpies of not only nitrate but also chloride of Zr and Ru are not found in the literature, the following estimation was used.Values of Δ d1 H/NO 2 for nitrates excluding alkali and alkaline earth metal nitrates go into the range of 61 to 153 kJ•mol −1 , and so, the average of these values of 120 kJ•mol −1 was adopted for ZrO (NO 3 ) 2 and RuNO(NO 3 ) 3 .Similarly, 176 kJ•mol −1 was adopted for Δ d2 H/NO of Zr and Ru nitrates.
In the following calculation, the generation ratio of NO 2 , i.e., NO = 80:20, reported in the previous paper [10] was used.The decomposition heat of each nitrate in 1 L of initial waste (in kJ•L −1 ) generating NO 2 was calculated as the product of the concentration of the nitrate in the waste (in mol•L −1 ), the amount of NO 2 generated from 1 mol of nitrate (in mol-NO 2 •mol −1 ), and the NO 2 generation heat (in kJ mol-NO 2 −1 ).However, we reported previously [10] that Zr, Ru, and Pd nitrates mostly decomposed by 180°C but that only 17% of N in   The value of Δ f H° was estimated from Δ f H° of chloride using Eq.(10).
these nitrates was released as NOx, and so, 17% of the concentrations of these nitrates given in Table I were used to calculate the above decomposition heat.
In Table III, the decomposition heat of each nitrate obtained above is shown as "Heat," and the amount of NO 2 generated from each nitrate is also shown.In addition, we reported that with rising temperature, 0.66 mol of NOx was generated per 1 L of initial waste by the decomposition of HNO 3 . [10]In Table III, NOx generation from the decomposition of HNO 3 is also included.The amount of NO 2 generated from 1 L of initial waste and the corresponding decomposition heat are 1.86 mol•L −1 and 228 kJ•L −1 , respectively, as shown as "Sum" in the last row in Table III.Therefore, the heat absorption per 1 mol of NO 2 generation is obtained as Similarly,

V.B. NOx Generation Rate
The generation rates of NO 2 and NO, expressed as R NO2 (t) and R NO (t) (in mol•s −1 ), are given by the following Arrhenius-type equations, respectively, which were obtained for a temperature increasing rate of 0.2°C•min −1 previously. [10]The amount of NOx in the equations is the amount generated from the initial waste of 120 m 3 : The value M i (t) (in moles) shows an amount of NO 3 in group i that generates NOx.It is assumed that the decomposition of 1 mol M i (t) generates 1 mol of NO 2 (i = 1 to 5) or 1 mol of NO (i = 6 and 7).The terms A i (in inverse seconds), E i (in kJ•mol −1 ), and R (in kJ•mol −1 •°C −1 ) are the frequency factor, activation energy, and gas constant, respectively.Equation (11)  approximates NO 2 generation by five groups and Eq. ( 12) NO generation by two groups.The values for these parameters previously obtained [10] are reprinted in Table IV after converting the waste volume 1 to 120 m 3 and time minutes to seconds.
Since 0.25 mol of O 2 is generated per 1 mol of NO 2 generation and 0.75 mol of O 2 per 1 mol of NO, the O 2 generation rate R O2 (t) is obtained from R NO2 (t) and R NO (t) as follows [10] :

V.C. Heat Absorption Rate
The heat absorption rate R S (t) accompanying the decomposition of nitrate is obtained as the product of the NOx generation rate described in Sec.V.B and the amount of the heat absorption per 1 mol of NOx generation described in Sec.V.A. Therefore, R S (t) is given as follows:

VI. HEAT CAPACITY
The content in this section can be used for both stages.
In addition to the vaporization of liquid and the decomposition of nitrates, the heat capacities of the waste and the tank are required as the heat absorption terms.These heat capacities are included in Γ in Eq. ( 1).The total heat capacity Γ is examined below.

VI.A. H 2 O and HNO 3
As shown in Eq. ( 2), the heat capacity of the waste is the sum of the products of the individual amounts of H 2 O, HNO 3 , and in the waste their corresponding specific heats.The following values were used for the molar specific heats of H 2 O and HNO 3 : The value of H 2 O at 100°C and that of HNO 3 at 25°C are shown in the literature. [16]Since the temperature dependence on these heat capacities is insignificant, [18] it was not considered.

VI.B. Nitrates
Specific heats of the individual nitrates in the waste were not found in the literature.However, since the dissolved nitrates precipitate and further decompose into oxides with increasing temperature, we assumed that the heat capacity of the nitrate is approximated by that of the oxide even before boiling.Under this assumption, the total amount of the oxides in the initial waste becomes 153 kg•m −3 .The specific heat of 18 oxides lies between 0.27 to 0.79 kJ•°C −1 •kg −1 (at 25°C), [13,15] and the average value weighted by their concentrations in the initial waste is 0.37 kJ•°C −1 •kg −1 .Using the specific heats of Ba, Cr, Fe, Sr, and Zr oxides at 400 and 800 K, [13] the increase rates per 1°C of these oxides were found to be 0.0028°C −1 to 0.0033°C −1 , averaging 0.0029°C −1 .Therefore, the average specific heat of the nitrates c S (θ) (in kJ•kg −1 •°C −1 ) multiplied by the total mass of the nitrates (= oxides) is approximated as follows:

VI.C. Tank
The material of the tank (including the internal structural material) was assumed to be Type 316 stainless steel (SUS-316).Its heat capacity C T (θ) (in kJ•°C −1 ) was obtained by the following formula: where c T (θ) = specific heat of SUS-316 at temperatures between 27°C and 727°C [18] ; 70 000 kg = mass of the tank.

VI.D. Examination of the Nitrate Heat Capacity Estimation Error
Since some assumptions are included in the derivation of the heat capacity of the nitrates, the effect of the error was examined.Using the initial amount of the waste of M 0 = 120 m 3 , the heat capacities of H 2 O, HNO 3 , nitrates, and tank are given as follows: 1. where t b and 104°C are the time and the temperature at the beginning of boiling, respectively.Since the contribution of the nitrates is only 1.6% before boiling, the effect of the error assuming that the heat capacity of the nitrate is equal to that of the oxide is negligible.At 300°C, at which the waste is almost dry and most of the nitrates convert to oxides, the contribution of the oxides to the total heat capacity becomes 19%, and so, the effect of the heat capacity estimation error of the nitrates (= oxides) is small.

VII. HEAT TRANSFER RATE FROM THE TANK IN THE INITIAL STAGE
The way of handling this part in the late stage is described in Sec.IX.

VII.A. Used Parameters and Values
There are three energy loss paths from the tank: convective heat transfer to the cell air, heat radiation to the cell wall, and gamma-ray release.The following parameters and values were used to determine the heat transfer rate of each path: Parameter 1.The temperature of the tank is equal to that of the waste, and the initial value is θ (0) = 50°C.
Parameter 2. In the calculation of the convective heat transfer, the following assumptions were used: a.The tank size is 7-m outside diameter × 4.0-m height, and its surface area S T = 165 m 2 .The thickness of the tank wall is 0.03 m.The heat transfer coefficients at the top and bottom of the tank are given by the same equation [Eq.(B.1)] as that of its vertical side [17] (see Sec. B.II).
b.The inside cell size is an 8.1-m cube, and its surface area S C = 394 m 2 .The thickness of the concrete cell wall is L = 2.0 m. c.The cell air is heated by natural convective heat transfer on the tank outer wall and cooled on the inner cell wall.Since the heat capacity of air is small and the air is well mixed by convection, this heating and cooling can be regarded as balanced.The air temperature inside the cell, θ ai (t) (in degrees Celsius), is assumed to be uniform before and after the accident.The inner surface temperature of the cell θ C (t,0) is determined by the balance of the heat transfer from the gas, the heat radiation from the tank, the heat generation inside the cell by gamma rays, and the heat conduction to the inside of the cell wall.Therefore, it is necessary to find the temperature distribution inside the cell by solving the heat equation.The temperature distribution inside the concrete cell wall denoted by θ C (t,x) is expressed by the following equation: where a C = thermal diffusivity (m 2 •s −1 ); ρ C = density (kg•m −3 ); C C = specific heat of the cell concrete (kJ•kg −1 •°C −1 ) (see Appendix B); q γ (t,x) = heat generation density due to the gamma heating (kW•m −3 ) given by Eq. (C.1) in Appendix C.

The boundary conditions are
where κ C = thermal conductivity of the cell concrete (kW•m −1 •°C −1 ); θ ao = air temperature outside the cell; α co = convective heat transfer coefficient of the outer cell wall surface (kW•m −2 •°C −1 ) given by the same equation as Eq.(B.1); F OW = configuration factor between the outer cell surfaces and the building wall surfaces facing the outer cell surfaces.The second term on the right side of Eq. ( 21) is the net radiant heat transfer rate from the outer cell wall surfaces to their surrounding concrete building surfaces at T OW = 25 + 273.15 K.The outside air temperature θ ao was assumed to be constant at 25°C.F OW was set to 0.2 assuming a distance of 8 m between the two surfaces. [19]hese boundary conditions were rearranged as follows in order to apply the numerical calculation algorithm shown in the literature. [19]Substituting Eq. ( 18) into Eq.( 20) to eliminate θ ai (t) and substituting Eq. ( 16) for R loss2 , the boundary condition at x = 0 can be rewritten as where Λ 1 = overall heat transfer coefficient including the effect of radiant heat transfer.
The boundary condition at x = L, Eq. ( 21), can also be rewritten as Equation ( 19) was numerically solved under the boundary conditions ( 22) and ( 23) and the initial condition, which will be explained later.The 2-m-thick wall was divided into 150 increments (Δx = 0.0133 m), and the time step Δt was set to be 109.7 s, which was derived from the condition that α C Δt/Δx 2 = ½, which enables simplicity and stability of numerical integration. [17]he initial temperature distribution inside the cell wall is the distribution before the accident happens.It was determined as follows.Assuming that the cell ventilation air is continuously fed at 25°C (= θ in a in the following equation) and the temperature of the tank is maintained at 50°C, the temperature of the cell air θ a is given by where c a = molar specific heat of air (kJ•mol −1 •°C −1 ); F Ca = flow rate of cell ventilation air before the accident (mol•s −1 ).
The boundary condition at x = 0 for the time before the accident is where Λ 0 = overall heat transfer coefficient for the time before the accident.Using Eqs. ( 21), (22), and (24), Eq. ( 19) was solved numerically.Starting with an initial uniform distribution at 35°C, a steady-state distribution before the accident was obtained after a sufficient number of time steps (about 400 h).The temperatures of the cell air, the inside surface of the cell wall, and the outside surface of the cell wall are 34°C, 39°C, and 32°C, respectively.This temperature distribution is initial condition for the waste calculation after the accident.

VIII. CALCULATION OF THE HEAT BALANCE EQUATION IN THE INITIAL STAGE
The time change of the waste temperature is obtained by integrating Eq. ( 1) over time.The amounts of H 2 O and HNO 3 , the heat absorption rates due to the liquid vaporization and the nitrate decomposition, and the heat transfer rates are already described in the previous sections as a function of time.
Substituting Eqs. ( 6) and ( 8) into Eq.( 1), the following equation is obtained: where R γ = leak rate of gamma-ray energy out of the tank given by Eq. (C.1) shown in Sec.C.II. Rearranging Eq. ( 25), we get Here, variables whose values change with time are expressed as a function of time.Denoting a time step by Δt, nΔt by t n , and θ(t n ) by θ n and approximating dθ/dt = (θ n+1 − θ n )/Δt and the value of the right side of Eq. ( 26) to be equal to the value at t = t n , we get By repeating this operation, θ can be determined as a function of time.

IX.A.1. Description of the Model
Figure 9 shows the modeled temperature distribution inside and outside the tank.The tank is modeled as a cylinder with a diameter of 7 m, a height of 4 m, and a wall thickness of 0.03 m.The thickness of the debris is 0.32 m (see Sec. C.II).The tank wall is separated into the following two parts.One contacts the debris (bottom and a small portion of the side wall of the tank), and the other does not contact the debris (top and most of the side wall), which are named as the tank bottom and the tank top/side, respectively.The debris is evenly divided into three parts: the upper, the middle, and the bottom.The debris volume was assumed to be a constant 12 m 3 above 130°C.
The temperature of each part is displayed as follows.The temperatures of the tank bottom and the tank top/side are θ Tb (t) and θ Tt (t), respectively.The temperatures of each piece of the debris are denoted as θ du (t), θ dm (t), and θ db (t) assuming that the temperature of each part is uniform.We considered θ db (t) = θ Tb (t) because the heat transfer from the bottom debris to the tank is much larger than that from the tank to the cell.Θ g (t) denotes the gas temperature in the tank, and θ ai (t) denotes the inner cell air temperature.For simplicity, the cell wall temperature θc(t,x) is assumed to be the same regardless of the cell's top, bottom, left, and right positions.
The surface area of each part is displayed as follows and shown in Fig. 9. S 1 is the surface area of each piece of the debris, which is equal to S 2 of the tank top.S 3 is the surface area of the tank side without contacting the debris.S 4 is the area of the side surface of the debris (S 4 /S 3 is the area of the side surface of each piece of the debris).Thus, S 1 = S 2 = 38.5 m 2 , S 3 = 2 , and S 4 7.0 2 .In deriving S 3 and S 4 , the debris thickness of 0.32 m was used.S C is the area of the inner cell surface of 394 m 2 .
The mass of the tank is 39 200 kg, and that of the structural material inside the tank is 30 800 (= 70 000 -39 200) kg.Assuming that the structural material was distributed uniformly according to the longitudinal length of the tank, the mass of the tank bottom is 13 300 (= 10 800 + 2500) kg, that of the side part is 47 500 (19 200 + 28 300) kg, and that of the top is 9140 (9140 + 0) kg (the latter figure in parentheses is the contribution of the structural materials).

IX.A.2. Heat Balance Equation for Each Part Inside and
Outside the Tank

Bottom Debris
It was assumed that the temperature of the tank bottom wall is equal to that of the bottom debris θ db (in degrees Celsius).Therefore, the heat balance equation that combines the bottom debris and the tank bottom is given as follows: where where K d = heat transfer coefficient between two pieces of the debris in contact (W•m −2 •K −1 ); S ss = sum of the cross-sectional area of the tank side wall and that of the internal structure (m 2 ); K ss = transfer coefficient between the tank bottom part contacting with the debris to the tank upper part not contacting with the debris (kW•m −2 •°C −1 ).The equation for R lossb is given as follows: where α 1 = convective heat transfer coefficient between the tank bottom and the inner cell air; σ = Stefan-Boltzmann constant; ε To = emissivity of the tank outer surface.It is assumed that the proportion of the heat radiation from the cell wall reaching the outer surface of the tank is equal to the surface area ratio.Although S 4 is not the bottom surface, it has the same temperature as the bottom surface, so it is included in the equation.
Using the evaporation rates Y Wb and Y Nb given in Eqs. ( 6) and ( 8), Eq. ( 27) can be transformed as follows: In this section, suffixes u, m, and b denote the upper, middle, and bottom

Middle Debris
Similar to Eq. the temperature change of the middle debris θ dm is derived as follows: where Q m = heat input to the middle debris from its surroundings (kW).

Upper Debris
Similarly, the temperature change of the upper debris θ du is derived as follows: where Q u = heat input to the upper debris from its surroundings (kW); R rad = net radiation heat transfer rate from the upper debris surface to the tank inner surface and is expressed as follows: In the equation, the emissivity of the upper surface of the debris and the inner surface of the tank are both treated as 1 (see Sec. B.II).F 12 , F 13 , F 21 , and F 31 are the configuration factors (suffixes 1, 2, and 3 correspond to the upper debris surface, the upper tank surface, and the side tank surface), and their values are 0.365, 0.635, 0.365, and 0.302, respectively.

IX.A.2.b. Equation for the Gas in the Tank
The gaseous components are H 2 O vapor, HNO 3 vapor, NO 2 , NO, and O 2 .The gas temperature θ g is described by the following heat balance equation: where Q g = sum of the gas generation rates from three pieces of the debris (mol•s −1 ), each rate being given by equations similar to Eqs. ( 6), ( 8), (11), and (12); c g = its molar heat capacity; α dg = convective heat transfer coefficient between the upper debris surface and the tank gas; α g2 = convective heat transfer coefficient between the tank top/side surface and the tank gas; c a = specific heat of the air.The first term on the left side of the above equation is the carry-on enthalpy of the gas generated from the debris, and the temperature of the generated gas leaving the debris is assumed to be equal to the temperature of the upper debris.The second term is the heat transfer rate from the debris to the gas in the tank.The first term on the right side is the heat transfer rate from the gas to the top/side wall of the tank, and the second term is the take-out enthalpy of the gas flowing out of the tank.
From the above equation, the gas temperature is described as follows:

IX.A.2.c. Equation for the Tank Top/Side Wall
The heat source of the top/side wall is the convective heat transfer from the gas in the tank (the first term on the right side of the following equation), the heat radiation R rad between the upper debris and the top/side wall, and the conductive heat transfer from the bottom wall to the side wall (the third term on the right side of the following equation).The heat sink R losst is the heat release rate from the tank to the cell.Therefore, the following equation is obtained: where C Tt = heat capacity of the tank top/side

IX.A.2.d. Equation for the Cell Air
The following equation is obtained from the conditions described in parameter 2 in Sec.VII.A: where α 2 = convective heat transfer coefficient between the tank upper/side and the inner cell air; α ci = inner cell wall and the inner cell air.From this, we obtain the following equation, which gives the cell air temperature:

IX.A.3.a. Heat Generation Rate R h of Each Piece of the Debris
The equation where the second suffixes b, m, and u of R h correspond to the bottom, middle, and upper debris, respectively.In the equation, 10 kW is the gamma-ray release out of the tank [see Eq. (C.1)].

IX.A.3.b. Heat Transfer Coefficient Between Two Pieces of the Debris in Contact
The coefficient K d (in kW•m −2 •°C −1 ) is given as follows: where κ d = effective thermal conductivity of debris given as 0.00037 at 150°C to 270°C and 0.00040 at 400°C. [20]e adopted 0.00040 over the entire temperature range.

IX.A.3.c. Heat Transfer Coefficient Between the Tank Bottom to the Tank Top/Side
The coefficient K ss is obtained as follows.If the thermal diffusivity of the material is a (in m 2 •s −1 ) and if the distance is d, the timescale τ (in seconds) required for heat conduction over the distance d (in meters) is given by τ � d 2 =a.The duration of the accident targeted in this study is 100 to 200 h, so the timescale of the temperature change would be about 10 h.Using the thermal diffusivity a = 4.1 × 10 −6 m 2 •s −1 of stainless steel (SUS), it is considered appropriate to set the distance scale for heat conduction in SUS to ffi ffi ffi ffi ffi aτ p = 0.4 m.Therefore, using the thermal conductivity of SUS, which is 0.0165 kW•m −1 •K −1 , [18] the heat transfer coefficient is calculated as K ss = 0.0165/0.4= 0.040 kW•m −2 •°C −1 .The sum of the cross-sectional area of the tank side wall and that of the internal structure is 0.66 + 0.98 = 1.64 m 2 .When the temperature difference between θ db and θ Tt is 200 K, the heat transfer rate from the bottom surface to the side surface is 1.64 × 0.040 × 200 = 13 kW, which is a small value in the overall heat balance of 568 kW (heat generation inside the debris).Therefore, the value of K ss considered in this paper has little effect.

IX.A.3.e. Heat Capacity
The heat capacity of debris C d is given by Eq. (13).The heat capacity of each piece of the debris is The specific heat c T of SUS-316, which makes up the tank and the internal structural material, is given in the literature. [18]Therefore, the heat capacities of the tank bottom C Tb and its top/side C Tt (J•K −1 ) are expressed as follows: The temperature-dependent molar heat capacity of each gas (NO 2 , NO, O 2 , and H 2 O) composing the gas in the tank was obtained from Ref. 15.The molar heat capacity of HNO 3 vapor was assumed to be the same as that of H 2 O.The heat capacities of the nitrates (= oxides) and tank (including the internal structural material) are given by Eqs. ( 13) and (14), respectively.

IX.B.1. Temperature Inside the Concrete Cell Wall
To obtain a numerical solution from Eq. ( 19) in the late stage, it is necessary to know θ C (t,x) at t = t d when the waste temperature reaches 130°C.According to the results of the initial stage, t d = 4.64 × 10 5 s (129 h).Therefore, the various parameters in Eq. ( 19) at t = t d are obtained from the results at the initial stage (see Sec. X.A).
The boundary condition at x = 0 for t > t d can be obtained using the fact that the heat flux into the wall due to convection and radiation on the inner wall surface is equal to the conductive heat flux inside the wall: The temperature of the cell air θ Ai is given by Eq. (32).When obtaining the numerical solution of Eq. (33), it was transformed into the following equation: In the above equation, Λ 2 and Θ are given as follows: The boundary condition at x = L for t > t d is the same as that in the initial stage given by Eq. (21).

IX.B.2. Calculation of the Heat Balance Equation
The temperatures that need to be obtained in the late stage are the temperatures of the bottom, middle, and upper debris, the temperatures of the tank top/side wall and the bottom wall (the bottom temperature is assumed to be equal to the bottom debris), the gas temperature in the tank, the air temperature in the cell, and the temperature distribution inside the cell wall.Time changes in the debris temperatures and the tank temperatures are obtained by solving numerically the heat balance equations, Eqs. ( 28) through (31), with the values at time t d obtained in the initial stage as the initial conditions.The gas temperature and the air temperature can be calculated using the above results and Eqs. ( 31) and (32).The surface temperature of the cell wall is required for radiant heat calculation.This is obtained by numerically solving Eq. ( 19) under the boundary conditions Eqs. ( 33) and ( 21) at each time step and calculating the temperature distribution of the cell wall.The numerical calculation formulas are the same as those of the initial stage model except for the boundary condition at x = 0, and the time step value is also the same (see Sec. VII.C).

X. CALCULATED RESULTS AND DISCUSSION OVER BOTH STAGES X.A. Transition of the Accident with Time
Figure 10 shows the temperatures of the waste, tank walls (tank top/side and tank bottom, both including the internal structure), tank gas, cell air, and inner and outer cell surfaces as a function of time, obtained by solving the heat balance equations.The temperature of the tank bottom is the same as that of the bottom debris.Elapsed time 0 h corresponds to the time when the cooling function loss occurred.Boiling begins after 14 h.Because Ru volatilization is reported [1] to begin when the waste temperature reaches 119°C, it starts after 111 h.The waste reaches 130°C after 129 h (= t d ), when the model has changed from the initial stage to the late stage.The temperatures of the upper and bottom debris rise at almost the same rate, but that of the middle debris increases quickly.When those of the upper and bottom debris reach 600°C, the middle debris exceeds 1200°C.This quick temperature increase is explained by the low effective thermal conductivity of the debris (see Sec. IX.A.3.b) and the sandwich effect between the upper and bottom debris.The heat of the upper debris is transferred to the upper part of the tank, and the heat of the lower debris is transferred to the tank bottom.At time t d , it can be seen that the temperature of the tank side temporarily drops.This is explained by the abrupt model change, i.e., from the homogeneous liquid state to the heterogeneous dry state.
When the bottom debris reaches 180°C after 142 h, the inner cell surface reaches 81°C, and when it reaches 600°C after 456 h, the surface reaches 482°C.The outer cell surface is assumed to be in contact with air at 25°C, and the outer surface temperature increases slowly and reaches 62°C when the bottom debris reaches 600°C.
Figure 11 shows the NO 2 and NO generation rates as a function of time, which was calculated by the Arrhenius-type model explained in Sec.V.B.The bottom Fig. 10.Temperatures of the waste, tank walls (tank top/ side and the tank bottom), tank gas, cell air, and cell inside and outside wall surfaces versus time curves (temperature of the tank bottom is the same as that of the bottom debris).At time t d = 129 h, HLLW temperature reached 130°C, and the stage was changed from the initial stage to the late stage.debris temperature is shown for reference.A small amount of NO 2 is generated when boiling begins, but most of it is generated at temperatures of the debris between 150°C and 350°C.The NO generation curve shows several peaks.This is mainly because the temperature of the middle debris is much higher than those of the upper and bottom debris.
Figure 12 shows the heat absorption rates due to the liquid vaporization and the nitrate decomposition, the heat transfer rate from the tank to the cell, and the sensible heat accumulation rate inside the tank, all as a function of time.The sum of these heat absorption terms should be equal to the value obtained by the energy release rate of leaked gamma rays from 578 kW due to the decay of radioactive materials.Abrupt discontinuous changes of vaporization and heat accumulation curves at around 129 h are due to the model change from the initial stage to the late stage.The latent heat of vaporization is the main heat absorption source until 142 h (180°C bottom debris temperature) when liquid evaporation from the waste is almost gone, and then, the heat absorption source of nitrate decomposition becomes main until 167 h (280°C bottom debris temperature).After that, the heat transfer to the cell wall becomes the main heat absorption source.The sensible heat accumulation rate curve shows two peaks at 130 to 150 h and 155 to 190 h, and a trough around 150 h.This phenomenon is explained by the fact that much of the decay heat is spent on nitrate decomposition at around 150 h, as can be seen in the figure .Figure 13 shows the temperature increasing rate of each debris as a function of time.For reference, the temperature of the bottom debris is indicated by a dotted line.Before boiling, the rate of the liquid waste is 0.06°C•min −1 .After boiling, it decreases to 0.0012°C•min −1 and then increases gradually as the concentration progresses.When the time exceeds 129 h (130°C), the increasing rates of the upper and bottom debris come within the range of 0.02 to 0.2°C•min −1 and that of the middle debris within 0.02 to 1.0°C•min −1 .Each rate curve shows two peaks.The rise of the first peak is explained by the substantial end of the vaporization of H 2 O and HNO 3 .Then, since the nitrate decomposition generating NO 2 begins, the peak decreases forming the first peak.The second peak is formed by the completion of NO 2 generation and the increased heat loss mainly due to heat radiation.Humps seen around 200°C are caused by NO generation.After that, the debris temperatures increase under the balance of the internal heat generation and the heat loss through heat radiation, conduction, convection, and sensible heat accumulation.
Figure 14 shows the changes in the released gas composition and the total gas flow rate corresponding to the bottom debris temperature.At the initial stage between 104°C and 130°C, the major component of the gas is H 2 O vapor.Though it decreases with the increasing temperature, its percentage is still 70% at 180°C, when the waste seems almost solid with a slight nitric acid smell. [9]The figure shows that even at 300°C, the HNO 3 vapor is contained in the gas by 17% although the H 2 O vapor is zero.It should be noted that in this  temperature region, these errors in vapor amount can be large.However, the errors do not affect the waste temperature rise because of their small heat absorption rates (see Fig. 12).Gas flow above 400°C consists of NO and O 2 produced by the decomposition of nitrates.
In order to examine the validity of the present model, we attempted to apply it to the experimental results of other papers.However, no comparable results could be found.

X.B. Application of the Results to a Postulated Accident of a Plant
The above results were obtained using the initial waste volume of 120 m 3 , the decay power density of 4.82 kW•m −3 , and the composition of the initial waste given in Table I.These values will change from time to time in a reprocessing plant.However, information necessary to solve Eq. ( 1) is the decay power density; the initial waste volume; and the initial amounts of H 2 O, HNO 3 , and nitrates.Such information will be obtained from the daily HLLW management.Accordingly, the waste temperature and the release rates of the components transferred to the gas phase are predictable as a function of time after the cooling loss using the HLLW conditions in a plant.
The ultimate goal of our study is to predict the release behavior of radioactive aerosol and volatile Ru into the outside atmosphere under accidental conditions.The results obtained above can be used to study the LPF [11] of radioactive materials, especially volatilized Ru, because the Ru and NO 2 are absorbed into the condensate of vaporized H 2 O and HNO 3 formed in the release path.Since aerosol release will cease by 130°C [8] and volatile Ru by 300°C, [6] the experimental results up to 300°C shown in this section are important.
In the present study, since Excel calculation was used to obtain the solution, some assumptions and approximations were used for simplicity.For example, although the waste changes gradually from liquid phase to solid phase, it was assumed to change suddenly at 130°C.Although the solid phase was continuous, it was divided into three pieces, and the temperature of each piece was made uniform.The solid-phase thermal conductivity will vary with temperature due to the presence of some residual liquid, nitrate decomposition, and volume change, but it was approximated to be constant.In the future, the use of more a sophisticated modeling and numerical solution will bring these assumptions and approximations closer to reality, enabling more accurate predictions.

XI. CONCLUSIONS
We proposed a method for systematically predicting temperatures of the waste, tank, and cell concrete up to 600°C and individual gas release rates as a function of time under accidental conditions.This method would be applicable to HLLW with different decay power densities, volumes, and compositions.
The results obtained in this study are useful not only for examining the LPF but also for developing measures to reduce it.Further, they will become the basic information to examine the accident management measures and to provide the disaster prevention system since the release of radioactive materials and their carrier gases can be quantitatively predicted as a function of time.parameters are also affected by temperature.Using the temperature dependence of ν a and a a obtained from the literature, [18] the following equation (in s 2 •°C −1 •m −4 ) was obtained: where θ Ta = average temperature of the tank and the cell air.
Similarly, the heat transfer between the cell air and the cell wall surface is dominated by turbulent natural convection, and the heat transfer coefficient can be handled in the same manner as described above.Therefore, Eq. (B.1) is applicable to α ci and α co .

B.III. EMISSIVITIES USED FOR THE CALCULATION OF HEAT RADIATION
The following emissivities were assumed from the literature values: 1. 0.92 for the tank inner surface ε Ti (assumed from SUS oxidized black surface values). [19] 0.4 for the tank outer surface ε To (SUS brown surface value due to heating).[19] 3. 0.98 for the upper debris surface ε d (assumed to be equal to that of black leather).[18] 4. 0.94 for the cell concrete surface ε c .[19] Emissivities above 0.9 were approximated as in the calculations. Therre, except for the reflection from the outer surface of the tank, the reflection from the other three surfaces was not considered.

APPENDIX C CALCULATION OF GAMMA-RAY LEAKAGE RATE OUT OF THE TANK AND CORRESPONDING GAMMA HEATING OF THE CELL
The content in this section can be used for both stages.
Of the radiation generated by the decay of radionuclides, all alpha rays and beta rays are absorbed in the tank, but some of the gamma rays leak out of the tank.The gamma-ray leakage rate to the total decay power and the corresponding gamma heating of the concrete cell were examined.

C.I. CALCULATION CONDITIONS
The tank and the cell sizes are given in Sec.VII.A.The amount and the composition of the initial waste are given in Sec.III.According to the progress of the accident, the waste volume changes from 120 to 12 m 3 .12 m 3 corresponds to the debris (dried waste) volume i assuming its density is 1.5 g•cm −3 , which was obtained from the experiment shown in Sec.IV.A. Therefore, the liquid volume in the waste changes from 120 to 0 m 3 .
In order to calculate the gamma-ray leakage rate and the gamma heating of the cell, the Monte Carlo code MCNP5-1.60 and library MCPLIB84 were used. [23]The energy and generated number of gamma rays + X-rays per unit time were obtained from ORIGEN-2.Bremsstrahlung X-ray was taken into account assuming the surrounding medium was UO 2 as the maximum condition, but its influence on the results was less than 3%.Therefore, hereinafter, we do not consider X-rays.
In calculating the gamma heating of the cell wall, we did not distinguish a ceiling, a floor, and four side walls.Therefore, the gamma heating in the depth direction of the cell walls was the same regardless of the position of each wall.

C.II. GAMMA-RAY LEAKAGE RATE FROM THE TANK
Before the start of boiling, the gamma-ray leakage rate was 2.18 kW out of 578 kW.When the waste became an oxide, it was 10.3 kW.For the intermediate time from the boiling start and the dryup, the gamma-ray leakage rate (gamma heating) R γ (t) (in kilowatts) was approximated by the following expressions: where ξ γ (t) = mass vaporization ratio of liquid from the waste.

C.III. GAMMA HEATING OF THE CELL
.1 shows the heating rate per unit depth within the cell concrete as a function of depth from the cell surface.It is the average value in all directions.Using this result and Eq.(C.2), the heating density q γ (t,x) (in kW•m −3 ) corresponding to the progress of the boiling accident is obtained by the following approximate equations:

Fig. 1 .
Fig. 1.Heat transfer processes and parameters affecting the waste temperature at the initial stage.
IV.A.2.a.Time Change of Vaporized Amounts of H 2 O and HNO 3

Fig. 3 .
Fig. 3. Time changes of vaporized amounts of H 2 O and HNO 3 .
2 O and HNO 3 thus obtained are shown in Figs. 4 and 5, respectively.These curves can be approximated by the individual single curves for H 2 O and HNO 3 irrespective of the temperature increasing rate.As shown in Secs.IV.B.1 and IV.C.1, these individual curves are consistent with those obtained from SHLLW-J.By 155°C, almost all H 2 O and 80% HNO 3 are vaporized.The temperature increasing rates up to 155°C used in Table II cover the realistic rates caused by the HLLW's decay power density of 4.82 kW•m −3 as seen in Fig. 13 of Sec.X.A.

Parameter 3 .
Physical property values used for the calculation of the convective heat transfer are given in Appendix B.

972
KODAMA et al. • BOILING AND DRYING ACCIDENT OF HLLW IN A REPROCESSING PLANT NUCLEAR TECHNOLOGY • VOLUME 210 • JUNE 2024 BOILING AND DRYING ACCIDENT OF HLLW IN A REPROCESSING PLANT • KODAMA et al. 973 NUCLEAR TECHNOLOGY • VOLUME 210 • JUNE 2024 db and C Tb = heat capacities of the bottom debris and the tank bottom (kJ•K −1 ); R hb = heat generation rate due to the decay heat in the bottom debris (kW); R Sb = heat absorption rate due to the salt (nitrate) decomposition (kW); λ Wb and λ Nb = latent heats of vaporization of H 2 O and HNO 3 (kJ•kg 1 ); Y Wb and Y Nb = their vaporization rates (kg•s −1 ); Q b = heat input to the bottom debris from its surroundings (kW); R lossb = sum of the convective heat transfer from the tank bottom to the cell air and the heat radiation to the cell wall (kW).The equation for Q b is given as follows:

Fig. 9 .
Fig. 9. Parameters on temperature θ and on surface area S around the tank in the late stage.
BOILING AND DRYING ACCIDENT OF HLLW IN A REPROCESSING PLANT • KODAMA et al. 977

Fig. 11 .
Fig. 11.NO 2 and NO generation rates and the bottom debris temperature (dotted line for reference) versus time curves.

Fig. 13 .
Fig. 13.Temperature increasing rates of the individual debris and the temperature of the bottom debris (dotted line for reference) versus time curves (up to 129 h, the waste remains in the liquid state).

Fig. 12 .
Fig. 12. Vaporization heat absorption rate, nitrate decomposition heat absorption rate, heat transfer rate from the tank, sensible heat accumulation rate in the tank versus time curves (up to 129 h, the waste remains in the liquid state).
i A debris thickness becomes 0.32 m.982KODAMA et al. • BOILING AND DRYING ACCIDENT OF HLLW IN A REPROCESSING PLANT TECHNOLOGY • VOLUME 210 • JUNE 2024

Fig. C. 1 .
Fig. C.1.Total heating rate per unit depth inside the cell concrete as a function of depth from the cell surface.

TABLE I Compositions
a Concentrations were analyzed by inductively coupled plasmaatomic emission spectroscopy or inductively coupled plasmamass spectrometry except oxides and HNO 3 .Oxide concentrations were calculated from the amounts added.b NUCLEAR TECHNOLOGY • VOLUME 210 • JUNE 2024 BOILING AND DRYING ACCIDENT OF HLLW IN A REPROCESSING PLANT • KODAMA et al. 967 NUCLEAR TECHNOLOGY • VOLUME 210 • JUNE 2024 BOILING AND DRYING ACCIDENT OF HLLW IN A REPROCESSING PLANT • KODAMA et al. 969