The derivation of the sodium droplet size distribution model based on the maximum entropy principle requires the establishment of appropriate physical constraints. Jaynes (1957) applied the maximum entropy principle to physics to obtain the droplet distribution law, shown as Eq. 1. S = − T ∑ k P k ⁡ ln ⁡ P k (1) where S is the entropy, T is a constant, and P k is the probability of the number of droplets under a certain volume of the leaked sodium droplet.

When the entropy takes a maximum value, the most probable event occurs, that is, the sodium droplet size distribution function with the largest probability of occurrence is obtained (Cao et al., 2008), shown as Eq 2 ∑ k P k ⋅ ln ⁡ P k = M a x . (2)

From the mass conservation of sodium droplets in the system (Cheng et al., 2002), Eq. 3 can be obtained ∑ k P k m k N 0 ( 1 + ε ) = m ˙ (3) where m k is the ejection mass under the action of the pressure difference when the loop system is broken, N 0 is the total number of sodium droplets produced in unit time, ε is the loss of sodium droplets due to chemical reaction or physical actions like pressure difference, and m ˙ is the mass flow rate of sodium droplets ejected from the break.

Considering that the sodium droplet is an incompressible fluid, Eq. 3 can be transformed into Eq. 4 ∑ k P k v k N 0 ( 1 + ε ) = V ˙ (4) where v k is the volume of a single sodium droplet and V ˙ is the total volume flow of sodium droplets ejected from the break.

In various cases, the sum of the number probabilities is 1, shown as Eq. 5 ∑ k P k = 1 (5)

For the above three basic constraints of Eqs 2, 4, and 5, the Lagrange method is used to obtain the function of G m a as Eqs 6, 7 (Mojaver et al., 2020) G m a = ∑ k P k ⋅ ln P k + ( ϕ − 1 ) ( ∑ k P k − 1 ) + ω ( ∑ k P k v k N 0 ( 1 + ε ) − V ˙ ) (6) d G ma d P k = 0 (7)

Then, Eq. 8 can be obtained P k = exp [ − ϕ − ω v k N 0 ( 1 + ε ) ] (8) where ϕ and ω are the undetermined Lagrange multipliers.

Eqs 9, 10 can be obtained through inserting Eq. 8 into Eqs 4, 5 ∑ k exp [ − ϕ − ω v k N 0 ( 1 + ε ) ] ⋅ v k N 0 ( 1 + ε ) = V ˙ (9) ϕ = ln ∑ k exp [ − ω v k N 0 ( 1 + ε ) ] (10)

Thus, Eq. 11 is obtained V ˙ = ∑ k v k N 0 ( 1 + ε ) ⋅ exp [ − ω v k N 0 ( 1 + ε ) ] ∑ k exp [ − ω v k N 0 ( 1 + ε ) ] (11)

Assuming that the droplet diameter of the particles of atomization sodium droplets ejected from the break of the loop system, denoted as D, conforms to the continuous distribution, Eqs 12, 13 can be obtained ϕ = ln ∫ D min D max exp [ − ω π 6 D 3 ⋅ N 0 ( 1 + ε ) ] π 2 D 2 d D (12) V ˙ = π 2 12 N 0 ( 1 + ε ) ∫ D min D max D 5 exp [ − ϕ − π 6 ω N 0 ( 1 + ε ) D 3 ] d D (13)

Eq. 14 can be obtained by solving Eq. 12. Insert Eq. 14 into Eq. 13 and integrate it, and Eq. 15 is obtained ϕ = ln { 1 ω N 0 ( 1 + ε ) { exp [ − ω N 0 ( 1 + ε ) π 6 D min 3 ] − exp [ − ω N 0 ( 1 + ε ) π 6 D max 3 ] } } (14) V ˙ = 1 ω exp [ − π 6 ω N 0 ( 1 + ε ) D max 3 ] [ − π ω N 0 ( 1 + ε ) D max 3 6 - 1 ] { exp [ − π 6 ω N 0 ( 1 + ε ) D min 3 ] − exp [ − π 6 ω N 0 ( 1 + ε ) D max 3 ] }        - 1 ω - exp [ − π 6 ω N 0 ( 1 + ε ) D min 3 ] [ − π ω N 0 ( 1 + ε ) D min 3 6 - 1 ] { exp [ − π 6 ω N 0 ( 1 + ε ) D min 3 ] − exp [ − π 6 ω N 0 ( 1 + ε ) D max 3 ] } (15)

Finally, Lagrange multipliers ϕ and ω can be uniquely solved by Eqs 14, 15.

Define P n i as the probability of droplet size distribution, which can be expressed with the integration of P k in the range of volume intervals [ v i − Δ v i 2 , v i + Δ v i 2 ] . Eq. 16 can be obtained as follows P n i = ∫ v i − Δ v i 2 v i + Δ v i 2 P k d v = ∫ v i − Δ v i 2 v i + Δ v i 2 exp [ − ϕ − w v k N 0 ( 1 + ε ) ] d v = ∫ D i − Δ D i 2 D i + Δ D i 2 exp [ − ϕ − w π D 3 6 N 0 ( 1 + ε ) ] π D 2 2 d D (16)

When the droplet diameter size is in the range of [ D i − Δ D i 2 , D i + Δ D i 2 ] , the probability density function of a sodium droplet of diameter D is defined as dN/dD, and N means the number percentage of sodium droplet diameter. Assuming that the droplet size of the particles of atomization sodium droplets ejected from the break of the loop system conforms to the continuous distribution, the probability of droplet diameter distribution of P n i can be expressed as Eq. 17 P n i = ∫ D i − Δ D i 2 D i + Δ D i 2 d N d D d D (17)

The probability density function of a sodium droplet of diameter D can be obtained as Eq. 18 d N d D = π 2 D 2 ⁡ exp [ − ϕ − ω π 6 D 3 N 0 ( 1 + ε ) ] (18)

Consider that there is a pressure differential drive between the sodium circuit system and the external environment where the pipe breaks. Assume that the minimum diameter D _min of the sodium droplet after ejection is 0 and the maximum diameter is D _max, which can also be denoted as D _cr and determined by We _cr (Hinze, 1955) D c r = σ W e c r ρ A u ¯ 2 (19) where ρ A is the density of the ambient medium; σ is the surface tension of the sodium droplet; We _cr is the critical value of We; and u ¯ 2 is the average value across the whole flow field of the squares of velocity differences over a distance equal to D _cr, which is related to the kinetic energy per unit mass. Therefore, Eq. 20 can be obtained by simplifying Eq. 14 ϕ = ln { 1 − exp [ − ω N 0 ( 1 + ε ) π 6 D c r 3 ] } ω N 0 ( 1 + ε ) (20)

Insert Eq. 20 into Eq. 18 d N d D = π 2 D 2 { ω N 0 ( 1 + ε ) 1 − exp [ − ω N 0 ( 1 + ε ) π 6 D c r 3 ] } ⋅ exp [ − ω N 0 ( 1 + ε ) π 6 D 3 ] (21) where ω can be given by the implicit expression of (1–15). However, for the calculation of the safety analysis of the sodium spray fire accident, it is inconvenient to use the mathematical model of the implicit expression. The semi-empirical relationship based on the constraint of the mass equation will be determined by the method of fitting the experimental data. Eq. 21 can be simplified as Eq. 22 d N d D = a D 2 ⋅ exp ( b D 3 ) (22)

The mathematical expression form with the maximum entropy principle with the mass equation as the constraint condition can be obtained as Eq. 23 ln ( d N d D ) = a + 2 ⁡ ln ⁡ D + b D 3 (23) where a and b are the coefficients determined by fitting the experimental data.

The momentum equation considering the viscous resistance term of the sodium droplet is as follows ∑ i P i ν i N 0 [ u 0 2 + 4 μ l π 2 D i 3 ρ A ( σ D i ρ l ) 1 / 2 ] = A U 0 3 (24) where P i represents the probability of the number of droplets with atomization volume ν i , u 0 is the volume-weighted average velocity, μ l is the integral of the kinetic viscosity of the sodium droplet for the droplet surface, D i is the diameter of the sodium droplet, ρ l is the density of the sodium droplet, A is the cross-sectional area of the break, and U 0 is the average flow velocity of sodium droplets at the break.

Using the Lagrange method, Eq. 25 is obtained between G _mo and the probability of the number of sodium droplets G m o = ∑ i P i ⁡ ln ( P i ) + λ { ∑ i P i ν i N 0 [ u 0 2 + 4 μ l π 2 D i 3 ρ A ( σ D i ρ l ) 1 / 2 ] }        - λ A U 0 3 + ( β - 1 ) ( ∑ i P i - 1 ) (25)

The derivation of G _mo with respect to the number probability is 0. Thus, Eqs 26, 27 can be obtained as follows ln ⁡ P i + 1 + λ ν i N 0 [ u 0 2 + 4 μ l π 2 D i 3 ρ A ( σ D i ρ l ) 1 / 2 ] + β − 1 = 0 (26) P i = exp { − β − λ ν i N 0 [ u 0 2 + 4 μ l π 2 D i 3 ρ A ( σ D i ρ l ) 1 / 2 ] } (27) where λ and β are the Lagrange multipliers.

The probability of droplet size distribution P n i can be expressed with the integration of P i in the range of volume intervals [ v i − Δ v i 2 , v i + Δ v i 2 ] , and Eq. 28 can be obtained P n i = ∫ v i − Δ v i 2 v i + Δ v i 2 P i d v = ∫ v i − Δ v i 2 v i + Δ v i 2 exp { − β − λ ν i N 0 [ u 0 2 + 4 μ l π 2 D i 3 ρ A ( σ D i ρ l ) 1 / 2 ] } d v = ∫ D i − Δ D i 2 D i + Δ D i 2 exp { − β − λ π D 3 6 N 0 [ u 0 2 + 4 μ l π 2 D 3 ρ A ( σ D ρ l ) 1 / 2 ] } π D 2 2 d D = ∫ D i − Δ D i 2 D i + Δ D i 2 exp { − β − λ π D 3 6 N 0 u 0 2 − 2 3 λ N 0 μ l π ρ A ( σ D ρ l ) 1 / 2 } π D 2 2 d D (28)

Set Eqs 29, 30 as follows. Also, assuming that the droplet size of the sodium droplets ejected from the main system break follows a continuous distribution, Eq. 31 can be obtained from Eq. 28 α = π 6 λ N 0 (29) K = 2 3 N 0 μ l π ρ A ( σ ρ l ) 1 / 2 (30) d N d D = exp { − β − α u 0 2 D 3 − λ K D − 1 / 2 } π D 2 2 (31)

According to Eq. 12, Eq. 32 can be obtained ϕ = ln ∫ D min D max exp { − α u 0 2 D 3 − λ K D − 1 / 2 } π D 2 2 d D (32)

Insert Eq. 32 into Eq. 31 d N d D = D 2 ⁡ exp ( − α D 3 u 0 2 − λ K D − 1 / 2 ) ∫ D min D max D 2 ⁡ exp ( − α D 3 u 0 2 − λ K D − 1 / 2 ) d D (33)

Define the coefficient W ₁ as follows W 1 = ∫ D min D max D 2 ⁡ exp ( − α D 3 u 0 2 − λ K D − 1 / 2 ) d D (34)

The mathematical expression form with the maximum entropy principle with the momentum equation as the constraint condition can be obtained as Eq. 35 ln d N d D = − ln ( W 1 ) + W 2 ⁡ ln ⁡ D − W 3 D 3 − W 4 D − 1 / 2 (35) where W ₁, W ₂, W ₃, and W ₄ are the coefficients determined by fitting the experimental data.

The HAN-based liquid spray experiment (Wang, 2017) is selected, in which the density and viscosity are 1300 kg/m³ and 1.2 × 10⁻³ Pas, respectively, for the simulated working fluid of the HAN-based liquid. The droplet size distribution experiments of spray droplets under the pressure differences of 1.8, 2.2, and 2.6 MPa were carried out, where the droplet size distribution of spray droplets was measured with phase Doppler analysis (PDA). The semi-empirical model with the mass conservation equation as the constraint condition and the momentum conservation equation as the constraint condition was used to simulate the experimental data, and the result is shown in Figures 1–3. Under different pressure differences, the two models can both reflect the spray size distribution of HAN-based liquids. In addition, both models can reflect the trend that the median diameter gradually decreases with the increase of the nozzle pressure difference, which is consistent with the experimental conclusion. The result of the semi-empirical model with the momentum equation as the constraint is in good agreement with the experimental data, with little error, especially for the smaller spray diameter.

The model is also verified by the spray droplet distribution experiment of injecting diesel into the air combustion chamber (Hiroyasu, 1985) with the pressure differences of 8.8 and 9.8 MPa, respectively. In addition, the sampling time of the experimental data is 4.5 ms after injection. At this time, the atomization state and droplet size are basically stable. The experimental data under the pressure difference of 8.8 MPa show a normal distribution, that is, the distribution of data points near the peak of probability density is nearly symmetrical, shown as Figure 4. However, under the experimental condition of the pressure difference of 9.8 MPa, the overall distribution of the experimental data is asymmetric and is barbed, shown as Figure 5. The results calculated with the both models can reflect the normal distribution very well. However, when the pressure difference reaches 9.8 MPa, the model with the mass equation as the constraint cannot correctly reflect the spray size distribution. The spray size distribution simulated by the semi-empirical model with the momentum equation as the constraint is in good agreement with the experimental data under a higher pressure difference. It is shown that when the peak of the probability density appears at a small diameter under a high pressure difference, the model constrained by the momentum equation also has better applicability due to the influence of viscous force.

The model is verified by the Italian LISOF sodium spray fire experiment (Torsello et al., 2012) and the sodium spray fire test of JNC (Nagai et al., 1999). The LISOF experiment is carried out for the droplet leakage phenomenon under the small break of the sodium-cooled fast reactor, and it uses sodium with a temperature of 550°C and a 0.2 mm nozzle to simulate the phenomenon of sodium spray under microdefects in a cooling circuit, which is similar to those in SFR. The pressure difference is 0.6 MPa, and the initial mass flow rate is 0.9 g/s. The JNC sodium droplet size distribution experiment is carried out for large-scale sodium spray fire under the pressure nozzle with a pressure difference of 0.2 MPa, an initial mass flow rate of 49.6 g/s, and a sodium temperature of 140°C.

The comparison between the calculation results of the two models and the experimental data of two sodium sprays is shown in Figures 6, 7, which shows similar conclusions with the experimental verification of the liquid fuel spray. In other words, the semi-empirical model with the momentum equation as the constraint can reflect the particle size distribution of sodium spray, whether it is normal distribution or asymmetric distribution, such as barb distribution.

The data from the liquid fuel spray experiment and the sodium spray fire experiment are summarized and compared with the both semi-empirical prediction models. The results are shown as Figures 8, 9. The horizontal axis represents the probability density value measured by the four experiments, and the vertical axis represents the predicted value of the model corresponding to the experiments. The error of the semi-empirical model with momentum as a constraint is in the range of 20%. In addition, the error of the semi-empirical model with the mass equation as a constraint is in the range of 30%, except for some data with large deviation.

The semi-empirical model with the mass equation as a constraint does not seem to be able to correctly reflect the slow reduction of the number of large particles with the increase of particle size. This is because after the droplet is ejected, the momentum will decrease due to viscous force and surface tension; as a result, the reduction is not conducive to breaking the droplets into small particles, leading to an increase in the proportion of large droplets. There exists the same problem in the HAN-based liquid experiment, only due to the small amount of data; the fitting results under the two constraints have little difference.