https://orcid.org/0000-0003-4228-8762
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Abstract
Temperature distribution plays a crucial role in the safety performance assessment and thermal dimensioning design in a deep geological repository for disposing high-level waste. In this study, a two-dimensional axisymmetric model of a single container for heat transfer was created. The fully analytical solution to temperature distribution in the repository was derived by utilizing the methods of separation of variables, impulse theorem, and Fourier transform.
The fully analytical solution was validated by comparing with the existing semi-analytical solution and line heat source solution. The temperature change in the near field around the container was analyzed using the present solution, and the influences of different parameters on the container surface temperature were investigated. Furthermore, the proposed fully analytical solution was used to predict the results of the in situ test.
The findings indicate that the temperature in the buffer layer rapidly increases and reaches its peak value within the first 2 years, then gradually decreases thereafter with time. The thickness of the bentonite pellet layer had a greater effect on the container surface temperature than that of the bentonite block layer. A comparison between the fully analytical solution and the results of the in situ heating test demonstrated that the proposed fully analytical solution can accurately predict the temperature variations in the in situ heating test.
Nomenclature
| A = | = | side surface area of the container (m2) |
| A1m, B1m = | = | constant coefficients |
| a = | = | radius of the container (m) |
| ai = | = | coefficients determined by the cooling time |
| c = | = | infinite radius (m) |
| d = | = | outer radius of bentonite block layer (m) |
| e = | = | outer radius of bentonite pellet layer (m) |
| F = | = | |
| g(t) = | = | auxiliary function, g(t) = |
| H = | = | height of the container (m) |
| Ji(·) = | = | i-order Bessel function of the first type |
| L = | = | rock thickness for the model area (m) |
| m = | = | temperature gradient in the vertical direction |
| P(0) = | = | initial container power (W) |
| P(r) = | = | spatial function |
| P(t) = | = | |
| Q(t) = | = | time function |
| r = | = | radius distance from the container axis (m) |
| T = | = | temperature (°C) |
| Tcontainer(t) = | = | container surface temperature (°C) |
| t = | = | time (year) |
| = | ||
| t0 = | = | initial uniform temperature on the ground (◦C) |
| t1 = | = | temperature at z=l (◦C) |
| ti = | = | time constants (years) |
| u(·) = | = | unit step function |
| w = | = | Fourier transform variable, w = nπ/l, n = 1, 2, 3, … ∞ |
| Yi(·) = | = | i-order Bessel function of the second type |
| z = | = | vertical distance from the ground (m) |
| z1 = | = | top of the canister (m) |
| z2 = | = | bottom of the canister (m) |
Greek
| βm = | = | eigenvalues |
| β = | = | separation constant |
| γ = | = | specific heat [J/(K·kg)] |
| ΔTB(t) = | = | temperature increments of bentonite block layer (°C) |
| ΔTP(t) = | = | temperature increments of bentonite pellet layer (°C) |
| λ = | = | thermal conductivity [W/(K·m)] |
| θ = | = | temperature after homogenized (°C) |
| ν = | = | temperature after applying impulse theorem (°C) |
| ρ = | = | density (kg/m3) |
| τ = | = | time integral variable |
| ϕ = | = | ratio of the side surface area of the container to the total surface area |
| χ = | = | thermal diffusivity (m2/s) |
Subscripts
| 1 = | = | homogeneous equation |
| 2 = | = | inhomogeneous equation |
| B = | = | bentonite block layer |
| P = | = | bentonite pellet layer |
| R = | = | rock layer |
Disclosure Statement
No potential conflict of interest was reported by the authors.