For purposes of modeling the potential leakage of aerosols from the canister interior through a CISCC path in the canister wall to the environment, three thermodynamic volumes are defined: everything interior to the inner surfaces of the dry storage canister, the CISCC through wall crack, and everything external to the canister.

Inside the canister, flow is driven by natural convection generated by the temperature gradients imposed by the decay heat of the stored fuel. Aerosol particles being circulated by the natural convection within the canister can adhere to particles/surfaces, settle by gravity, and be resuspended. The magnitude of aerosol deposition on surfaces within a canister is contingent upon several factors, including the concentration and size of the aerosols, as well as the size, roughness, orientation, and shape of the surfaces.

Coagulation and deposition are inherent properties of aerosols (Hinds WC, 1982). Coagulation is the process of aerosol particles colliding and sticking with one another, to form larger particles. Deposition happens through the boundary layer at the surface walls and is only possible when the particles penetrate the stagnant boundary layer in contact with the walls. Both coagulation and deposition can be described using the principle of mass conservation when applied to aerosol particles inside a canister (Chatzidakis, 2018): d C 1 d t = − K 1 c C 1 2 − Q t C 1 V 1 − K 1 g + K 1 d C 1 (1) where K _1c is the coagulation decay rate (hr⁻¹), C ₁ is the particle concentration (kgm⁻³), t is time (s), Q is the volumetric flow rate (m³s⁻¹), V ₁ is the canister’s free volume (m³), K _1g is the particle decay rate due to gravitational settling, and K _1d is the particle decay rate due to diffusion to the surfaces. The first and second terms on the right side represent the rate at which particles are removed from the canister by coagulation and airflow, respectively. The third term represents the rate at which particles are removed by deposition mechanisms on surfaces other than microchannels.

Within a canister, a homogeneously distributed monodisperse aerosol will decay exponentially due to gravitational settling with a decay constant: β s = v s S f V (2) where v _s is the Stokes settling velocity (m/s), S _f the projected area of the canister (m²), and V is the canister volume (m³). For a typical vertical canister with v _s = 3 × 10⁻⁵ m/s, S _f = 1 m² and V = 6 m³, the decay constant for particles of 1 μm aerodynamic diameter is 0.02 h⁻¹, which translates to a half-life of 34.6 h. For 0.1 μm aerodynamic diameter the decay constant is 0.002 h⁻¹ with a half-life of 346 h. Similarly, for 10 μm aerodynamic diameter, the decay constant is 0.2 h⁻¹ with a half-life of 3.46 h. Coagulation and diffusive deposition are expected to decrease these times even further. This shows that aerosol suspension time within a canister decreases with increasing aerodynamic diameter with half of the suspended particles having less than 10 μm depositing within the canister is less than 4 h and after 40 h (or ten half-lives) a 1000-fold reduction of the initial aerosol concentration is expected.

A temperature difference of 0.01°C in a canister with an effective wall length of 1 m can keep an aerosol consisting of particles smaller than 10 μm homogenously distributed inside the canister (Chatzidakis, 2020). When the particle size is so large that the deposition velocity approaches the velocity of the free convection there will arise inhomogeneous aerosol distribution. The particle diameter at which this occurs ( d max can be obtained by taking the Stokes’ settling velocity equal to the velocity of the free convection given by the Prandtl relation: ρ g 18 μ d max 2 F d max = 0.5 g α ∆ T l (3) where µ is the viscosity of the gas (Pa.s), α is the coefficient of thermal expansion of the gas (K⁻¹), ΔT is the average temperature difference causing the convection, ρ is the particle density (kgm⁻³), and l is the wall length (m). F (d_max) may be taken unity for d_max larger than 1 and then the following relation holds: d max = 3 μ ρ 1 / 2 α ∆ T l g 1 / 4 (4)

For the usual range of canister volumes and ΔΤ = 0.01^οC, d_max ranges from 18–24 μm (aerodynamic diameter). Clearly, respirable aerosol particles of a few μm or less would remain homogeneously distributed within a canister. Another way to interpret this result is to compare the convective flow velocity of the order of 10–20 cm/s or higher to the particle’s settling velocity, which is several orders of magnitude lower ∼10^–3 cm/s.

Leak path parameters through which a particle must travel to escape from the canister mainly include crack characteristics such as location, quantity, branching, roughness, area. The principle of mass conservation (Eq. 1) is applied to describe aerosol transport in an arbitrary leak path. An aerosol transport equation can be applied to any cross-sectional shape for which the hydraulic diameter and mass flow rate are known (Williams, 1994). The mass flow rate Q _m for continuum flow can be written as a function of the pressure drop along the flow direction: p u 2 ‐ p d 2 = R g T Q m 2 ∫ 0 L C f Re χ x A 3 x dx (6) where x is the axial distance from the inlet of the crack (or capillary) (m), p _u and p _d are the pressures upstream and downstream from the crack (Pa), respectively, L is the length of the duct (m), χ is the perimeter of the duct (m), and A is the cross-sectional area (m²). This equation can be solved numerically to determine the mass flow rate Q _m (m³sec⁻¹). When this is known, the velocity and volumetric flow rate can be calculated using mass continuity.

Using the approach by Williams (1994) (16), the mass flow rate from Eq. 6 for a constant cross section can be expressed as: Q m 2 = A 3 p u 2 − p d 2 χ C f L R g T (7)

Where, C_f is the friction factor, R_g is the gas constant (Jkg⁻¹K⁻¹), p_u and p_d are the pressure at the upstream and at the downstream of the crack, respectively, T is the fluid temperature (K), L is the length of the duct (m), χ is the perimeter of the duct (m), and A is the cross-sectional area (m²).

The transition from laminar to turbulent flow for pipes supposedly takes place at Re = 2,300. However, microcracks or capillaries with microscale diameters show the transition to occur much earlier, at Re numbers as low as 5–10 for rectangular geometries and Re = 400–600 for cylindrical geometries (Chatzidakis and Scaglione, 2019b). At these low Re numbers, any error in the friction factor has a large influence and this resulted in the need for new friction factor correlations for transient flows. A detailed derivation of the friction factors and flow equations can be found elsewhere (Chatzidakis, 2018).

The current model is based on the aerosol general dynamic equation (Crowe et al., 2005). For the case of aerosol penetration through a microchannel, assuming only external processes, the general dynamic equation (GDE) is reduced to a transport equation, which can be written as follows in one-dimensional form (Mitrakos et al., 2008; Chatzidakis and Scaglione, 2019b): dC x , t dt + 1 A x , t d d x A x , t ⋅ u x , t ⋅ C x , t = ‐ V d x , t χ x , t A x , t C x , t (8)

Where, C is the aerosol mass concentration (kgm⁻³), V _d is the deposition velocity (msec⁻¹), A is the cross-sectional area (m²), χ is the wetted perimeter of the cross section (m), and u is the gas velocity (msec⁻¹). The deposition velocity is calculated as the sum of the deposition velocities corresponding to each individual deposition mechanism. The deposition mechanisms transport aerosols to the walls of the leak path due to gas flow, gradients, or external forces. Five primary deposition mechanisms are included in the model:

1. Gravitational settling: The deposition velocity due to gravitational settling is written as follows (Drossinos et al., 2016):

Angle θ is the angle between the airway direction and the force of gravity (the so-called gravity angle), ρ_p is the particle’s density (kgm⁻³), d_p is the particle’s diameter (μm), Cc is the Cunningham slip correction factor that depends on the particle’s size, g is the acceleration due to gravity (ms⁻²), and μ_g is the dynamic viscosity of the gas (Pa.s).

2. Brownian diffusion: The deposition velocity due to Brownian diffusion is determined using mass transfer theory. It is expressed in terms of the concentration boundary layer thickness, where according to the heat-mass transfer theory analogy, the Nusselt number is replaced by the Sherwood number Shah and London (1978). Specifically,

In practice, any temperature change through the crack flow path would only have a small effect on gas flow, but it could provoke thermophoretic velocities that could affect aerosol deposition. This is, however, not included in the present model. Other second order deposition mechanisms such as electrophoresis and diffusiophoresis are neglected as rough calculations indicate that the contribution of these mechanisms is relatively lesser in the removal of particles as compared to the first order mechanisms considered in the model. The total deposition velocity is given as the algebraic sum of the deposition velocities corresponding to each individual mechanism, namely: V d = V d s e d + V d d i f f + V d i n e r t − i m p   l a m i n a r   f l o w V d s e d + V d e d d y − i m p + V d t u r b − d i f f + V d i n e r t − i m p   t u r b u l e n t   f l o w (16)

As can be seen from Eq. 16, the model is flexible and additional deposition mechanisms can be added if needed. As mentioned in the introduction, studies have shown evidence of partial or complete plugging of aerosol leak paths. Plugging occurs when particulate matter deposits on the surface of the flow channel, changing the internal geometry of the flow area. To model this, the mass of the deposit up to any position S in the crack path, can be obtained in terms of the deposition velocity V_d (ms⁻¹) and the particle concentration C (kgm⁻³) as follows (Mitrakos et al., 2008): M d e p = ∫ 0 S ∫ 0 t 2 π R C V d d x d t (17)

The deposition of particles is assumed to occur uniformly on the path’s circumference. This assumption is valid for mechanisms such as Brownian or turbulent diffusion or for eddy impaction, but it is approximate for directional mechanisms such as gravitational settling. Under this assumption, the change in radius due to plugging is related to the deposit volume, as follows: d R = 1 2 π R d V d e p d x (18)

Where, V d e p = M d e p ρ p , assuming that the deposit material is homogeneous with a density equal to the density of the particles.

Figure 2 shows a flowchart of the code and the different underlying steps that are involved in predicting the penetration, retention, and modifications to the crack, as a result of depressurization. The code currently covers rectangular and cylindrical geometries that are perceived as ideal stress corrosion crack geometries. The numerical solution first calculates the fluid velocity in each time step. Then the particle transport equation is solved using an implicit finite difference scheme. As mentioned previously, the deposition of particles is assumed to occur uniformly along the path’s circumference in the current model. The duct radius is then updated by calculating the amount of the deposited mass (Eq. 18). All the numerical integrations required in this calculation are performed using the trapezoidal rule (Chatzidakis, 2020). The new cross section is then used for the aerosol calculations in the next step.

Durbin et al.‘s experiments aimed at understanding the flow rates and aerosol retention in stress corrosion cracks using an engineered microchannel/slot with characteristic dimensions resembling that of real cracks observed on canisters. The experimental setup (seen in Figure 4) consisted of a 0.908 m³ (240 gal) pressure tank that was used to simulate a canister. A test section was connected to the tank with a mass flow meter to precisely measure the flow from the tank to the test section. An engineered microchannel simulating a crack was mounted in the middle of the test section. The dimensions of the microchannel were 12.7 mm (0.500 in.) wide, 8.86 mm (0.349 in.) long and an average of 28.9 μm (0.0011 in.) deep. A schematic of the microchannel assembly is shown in Figure 5. Cerium oxide (CeO₂) was chosen as the surrogate for spent nuclear fuel because of its relatively high density (7.22 g/cm³) and its commercial availability. The experimental approach was similar to previous studies (Wells and Chamberlain, 1967; Lewis, 1995; Liu and Nazaroff, 2003; Gelain and Vendel, 2008) in that aerosol analyzers are used to characterize the particle size distribution and concentration present in the gas before and after flowing through a simulated crack. A detailed layout and description of the experiment can be found in Durbin et al. (2018).

The engineered microchannel geometry was added to the model and it was initially run for blowdown conditions dealing with transient states. The blowdown results were then compared to the gas flow measurements from Durbin et al. (2018). It is noted that the measurements were taken before particles were released into the tank, so there was no aerosol influence in the flow (pre-aerosol measurements). The canister (tank) was assumed to have a starting initial pressure of 800 kPa (116 psia). Deposition mechanisms depend on flow rate, so the ability to correctly predict flow rate is critical to both the experiment and the predictions using the model. The results of the present model vs. Durbin et al.‘s experimental measurements are shown in Figure 6 for two Reynolds number co-relations. A model with friction factor correlation of Gelain and Vendel, (2008) is also presented for comparison.

The following observations can be made from Figure 6:

• A flow rate based on laminar friction factor significantly overestimates the flow rate. It can be stated that the flow is not laminar, despite Reynolds number being less than 2,300.

To develop a representative friction factor correlation to capture the fluid flow phenomena in a microchannel, the friction factor vs. Reynolds number as derived from measured data was plotted in Figure 7. It is observed that the friction factor is significantly different than the laminar friction factor indicating a flow regime that is not laminar (either transition or turbulent). A slope change is also evident around Re = 300. Thus, parameter tuning was applied on the friction factor Cf within the model to obtain the best fit because of the complexities of determining flow regimes in microchannels. An empirical correlation was developed that matches the measured friction factor: C f = 15.161 R e − 0.823 (19)

The updated flow rate estimates based on the new friction factor are in good agreement with the experimental pre-aerosol measurements.

Aerosol tests in the experimental setup by Durbin et al. involved pressurizing the tank and loading it with a measured amount of aerosols. For modeling the mass flow rate in this scenario, a friction factor was plotted as a function of Reynolds number and an empirical correlation was developed to capture the correct flow regime. Figure 8 shows the friction factor vs. Reynolds number. It can be observed that the slope change occurs at Re = 70 and that the flow cannot be represented using a laminar flow friction factor. The calculated friction factor is: C f = 201.68 R e − 1.348 , R e < 70 (20) C f = 21.154 R e − 0.842 , R e > 70 (21)

The results using the updated friction factor are shown in Figure 9. Good agreement is observed with experimental measurements.

Using the empirical friction factor correlations mentioned above, the depressurization equation (Eq. 5) was used to predict the mass flow rate over time (Figure 10). Given the simplicity of the depressurization equation, the results are in good agreement with the experimental measurements although the difference appears to be larger as time progresses. This can be partially attributed to the simplifying assumptions that were made during the derivation of the depressurization equation. The deposited mass due to coagulation was modelled with Eq. 1, using a coagulation constant of 2 × 10⁻⁷ cm³/s. The present aerosol model was in good agreement with the experimental measurements and correctly predicted the aerosol concentration within the source container (upstream) due to coagulation as shown in Figure 11.

The model was also validated against experiment results from Tian et al. who studied penetration efficiencies of fine aerosol particles (sized <0.3 μm) through capillaries under pressure differences ranging from 60 to 450 kPa (Tian et al., 2017). The capillary bore sizes ranged from 5 to 20 μm and its lengths ranged from 10 mm tο 80 mm respectively. The results showed the aerosol penetration efficiency to decrease significantly with increased capillary length although it was identical for capillaries of different bore sizes. Further, it was also observed that the penetration efficiency correlates strongly with average flow velocity than with the air leakage rate. The present aerosol model was in good agreement with the experimental measurements and correctly predicted the reduction in aerosol concentration as a function of time, within the source container, due to coagulation at a coagulation constant: 2 × 10⁻⁹ cm³/s (Figure 12). It was also found that coagulation was the most dominant mechanism in the early stages of the experiment which can be attributed to the second order nature of coagulation in Eq. 1. After the initial coagulation period, the other two deposition processes, Brownian and gravitational, start to dominate more and more as a result of their first order kinetics nature. However, in this work and given the short-term period of the experimental data only coagulation was used. This approach is confirmed by the good agreement with the experimental measurements and supported by theoretical analysis explained by van de Vate (Mosley et al., 2001). The magnitude of the coagulation constant depends on the particle size and is in agreement with earlier work (Mosley et al., 2001).

In comparison to the coagulation constant used in the previous section, the difference in magnitude is attributed to the much larger particle size used in the experiments by Durbin et al. Further, the model results accurately capture the behavior of aerosol penetration efficiency for various bore sizes as a function of flow velocities obtained by Tian et al. (Tian et al., 2017) (Figure 13).