MSR Depletion Method and Results

Contact: Samuel Walker, [email protected]

Governing Equations

The main equations for this problem are the Bateman equations. The methodology for solving the Bateman equations is given in (Betzler et al., 2019), and the equations themselves are given by

$var element = document.getElementById("moose-equation-c1971208-6304-4b8e-8cf9-9b62767e06f8");katex.render(" \\frac{d \\mathbf{n_{pl}}}{dt} = \\mathbf{A_{pl}}(t) \\mathbf{n_{pl}}(t)", element, {displayMode:true,throwOnError:false});$

$var element = document.getElementById("moose-equation-017650ae-598c-46b6-9ba4-bc76561171f0");katex.render(" \\frac{d \\mathbf{n_{og}}}{dt} = \\mathbf{A_{og}}(t) \\mathbf{n_{og}}(t)", element, {displayMode:true,throwOnError:false});$

Here, $var element = document.getElementById("moose-equation-c7fc9ec0-d677-4c9b-9e7a-6194d55154dd");katex.render("\\mathbf{n_{pl}}(t)", element, {displayMode:false,throwOnError:false});$ denotes the time dependent isotope concentration vector for species in the primary loop, $var element = document.getElementById("moose-equation-e1721241-269b-406f-b3db-225bf0addd49");katex.render("\\mathbf{n_{og}}(t)", element, {displayMode:false,throwOnError:false});$ denotes the time dependent isotope concentration vector for species in the off-gas, and $var element = document.getElementById("moose-equation-219a757e-6e0b-4f63-8cea-d62e0e3b655c");katex.render("\\mathbf{A_{pl}}(t)", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-4bf9fdb2-1c83-46b2-af49-86f2b7e98822");katex.render("\\mathbf{A_{og}}(t)", element, {displayMode:false,throwOnError:false});$ denote the time dependent transition matrices for the primary and off-gas loops, respectively.

Griffin builds the complete depletion matrix from the two-region depletion matrices from the above equations and solves the method explicitly with the Chebyshev Rational Approximation Method (CRAM) (Pusa, 2016). The complete depletion matrix is given by

$var element = document.getElementById("moose-equation-e958e025-b34b-405f-b88f-d22e808e5a60");katex.render("\\mathbf{A_2} = \\begin{bmatrix} \\mathbf{A_{og}} & 0 \\\\ 0 & \\mathbf{A_{pl}} \\end{bmatrix}", element, {displayMode:true,throwOnError:false});$

$var element = document.getElementById("moose-equation-2779ed38-c102-40dc-aeb4-b2745377fa9b");katex.render("\\mathbf{n_2} = \\begin{bmatrix} \\mathbf{n_{og}} \\\\ \\mathbf{n_{pl}} \\end{bmatrix}", element, {displayMode:true,throwOnError:false});$

where $var element = document.getElementById("moose-equation-ce9aa984-583c-4fb1-983c-65a40e85e786");katex.render("\\mathbf{n_{pl}}", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-0697a2aa-61d5-48aa-b17a-f359153c1011");katex.render("\\mathbf{n_{og}}", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-5b0b5459-5c04-48e6-ab04-cbe35e1f454d");katex.render("\\mathbf{A_{pl}}", element, {displayMode:false,throwOnError:false});$ , and $var element = document.getElementById("moose-equation-07e29460-12a3-43fd-9190-dd46cb0afba8");katex.render("\\mathbf{A_{og}}", element, {displayMode:false,throwOnError:false});$ are the same as above.

The loss matrix $var element = document.getElementById("moose-equation-414ef071-466c-49c3-8945-0b72d096ffc3");katex.render("\\mathbf{L}", element, {displayMode:false,throwOnError:false});$ is introduced to couple the two regions within the CRAM solver where the $var element = document.getElementById("moose-equation-0f1a902e-8912-41a6-91ab-a3a4804ff227");katex.render("L_n", element, {displayMode:false,throwOnError:false});$ terms on the diagonal are the removal terms for each isotope in the primary loop region. Here the $var element = document.getElementById("moose-equation-3606b66f-99f2-4430-b4ef-ae9e30c891a7");katex.render("L_n", element, {displayMode:false,throwOnError:false});$ terms are reduced order terms that capture the combined physics of chemical volatilization and the mass transfer speed of isotopes from the primary fuel-salt loop to the off-gas system without allowing for the possibility of species to return to the reactor primary loop region. The loss matrix is given as

$var element = document.getElementById("moose-equation-04259dbf-b53a-4906-bc16-3fb710fcbaf6");katex.render("\\mathbf{L} = \\begin{bmatrix} L_1 & \\dots & 0 \\\\ \\vdots & \\ddots & \\vdots \\\\ 0 & \\dots & L_n \\end{bmatrix}", element, {displayMode:true,throwOnError:false});$

Incorporating the loss matrix into the depletion matrix gives the two-region depletion matrix with one way removal which is then solved in Griffin. Since the removal is only one-directional without allowing for species in the offgas system to reenter the primary loop, there is only one loss matrix $var element = document.getElementById("moose-equation-20a4d883-a439-4db9-a8b8-7cacb31663c4");katex.render("\\mathbf{L}", element, {displayMode:false,throwOnError:false});$ in the upper right-hand corner. The two-region depletion matrix is given as

$var element = document.getElementById("moose-equation-53744e6f-35ee-499b-87c8-370c50f5734f");katex.render("\\mathbf{A_2} = \\begin{bmatrix} \\mathbf{A_{OC}} & \\mathbf{L} \\\\ 0 & \\mathbf{A_{IC}} \\end{bmatrix}", element, {displayMode:true,throwOnError:false});$

where the subscripts IC and OC denote in-core and out-of-core quantities, respectively. By only considering flow from one region to another and no return flow, it is possible to make the 2-region depletion matrix comply with the expected sparsity pattern of CRAM. This yielded the performant solves for the one-direction removal system. However, it should be noted that in the event of two-way flow, the sparsity pattern expected by CRAM would be violated, potentially necessitating the use of another matrix exponential solver, which is not in the current scope (Walker et al., 2022).

Results

The Griffin results for both depletion cases are taken from (Walker et al., 2022), and are compared to results obtained from the Monte Carlo SERPENT code. The process for each study starts with a depletion calculation in SERPENT. The spectrum weighted microscopic one-group cross sections and the resulting one group flux are taken from SERPENT and are then exported and converted to ISOXML format to be used in Griffin. Griffin does not perform a neutronics solve, but rather uses this information from SERPENT to only perform a depletion solve. Currently the microscopic cross sections generated by SERPENT are not tabulated by burnup, but instead one set from mid-life burnup are chosen and used in Griffin for the entire burnup calculation.

Depletion results without removal

The first case was designed to ensure that Griffin could match SERPENT results for a depletion problem, within error bounds. The results of the first case are shown in Table 1.

Table 1: Isotope depletion without removal

	SERPENT	Griffin	Absolute difference
Fission reaction rate (fission/(cm $var element = document.getElementById("moose-equation-05a7e0cb-cbd8-4c3c-b722-3665c97e7b86");katex.render("^3", element, {displayMode:false,throwOnError:false});$ $var element = document.getElementById("moose-equation-60ac5edc-93dc-44d2-a15f-e622851e694e");katex.render("\\cdot", element, {displayMode:false,throwOnError:false});$ s))	$var element = document.getElementById("moose-equation-178c87a6-13d0-4224-ade5-0fa3ae9153ad");katex.render("9.89 \\times 10^{12}", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-2a172ee3-b1ba-41e0-9baa-ed5abc4a07a5");katex.render("9.94 \\times 10^{12}", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-a2ff5e79-53b8-4344-a87c-0eb5371b50b4");katex.render("0.5", element, {displayMode:false,throwOnError:false});$ %
I $var element = document.getElementById("moose-equation-e255f8fe-e9d3-412d-90a4-fe91a5c1d80b");katex.render("^{127}", element, {displayMode:false,throwOnError:false});$ in primary loop (atom/(b $var element = document.getElementById("moose-equation-a7b34840-9604-4575-bca6-855f66a105df");katex.render("\\cdot", element, {displayMode:false,throwOnError:false});$ cm))	$var element = document.getElementById("moose-equation-6596a81a-b4bc-4b35-84c4-00df6d111c9e");katex.render("1.04 \\times 10^{-7}", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-d29847dc-50a9-416d-b9ad-7d575d6abaed");katex.render("1.05 \\times 10^{-7}", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-0938ad81-c845-4d8e-adf2-f7c05c5d3edf");katex.render("0.6", element, {displayMode:false,throwOnError:false});$ %
I $var element = document.getElementById("moose-equation-eafddc77-7158-4d81-8bf6-d45fa2b7a1ad");katex.render("^{129}", element, {displayMode:false,throwOnError:false});$ in primary loop (atom/(b $var element = document.getElementById("moose-equation-cff2d3ae-ada2-4f66-9dc5-fd809c5ca58c");katex.render("\\cdot", element, {displayMode:false,throwOnError:false});$ cm))	$var element = document.getElementById("moose-equation-fd078b9e-68c6-4c62-a57b-d7dffee5b0b8");katex.render("4.08 \\times 10^{-7}", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-18a2ea1c-84ee-4554-a139-bd28849beb80");katex.render("4.09 \\times 10^{-7}", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-6e6fc243-29b3-463b-9d9f-1352be19ad37");katex.render("0.2", element, {displayMode:false,throwOnError:false});$ %

Table 1 shows that the fission reaction rate density calculated by SERPENT and Griffin for midlife burnup are in good agreement despite Griffin using a constant flux to perform the burnup. The atomic densities of two iodine isotopes are compared at 70 days burnup and show good agreement. Discrepancies are most likely related to the fact that Griffin is using a one-group flux and one-group cross sections collapsed from the continuous energy cross sections generated by SERPENT for midlife burnup.

Depletion results with removal

The second case included the removal of iodine from the primary region to the off-gas system. The results of the second case are shown in Table 2.

Table 2: Isotope depletion with removal

	SERPENT	Griffin	Absolute difference
I $var element = document.getElementById("moose-equation-cc43c350-c641-4463-baf0-5682db381fa2");katex.render("^{127}", element, {displayMode:false,throwOnError:false});$ in primary loop (atom/(b $var element = document.getElementById("moose-equation-9fe58ff6-509f-4d13-a3f8-0aed0767be3c");katex.render("\\cdot", element, {displayMode:false,throwOnError:false});$ cm))	$var element = document.getElementById("moose-equation-68034fb3-9a5f-4fae-9116-c8c8843246bd");katex.render("1.06 \\times 10^{-7}", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-de00a083-808f-47a3-9173-d649580ecff8");katex.render("1.06 \\times 10^{-7}", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-0de53bc7-3a93-4e6c-b50e-4ac6dc00fe49");katex.render("0.5", element, {displayMode:false,throwOnError:false});$ %
I $var element = document.getElementById("moose-equation-009b3915-db57-45ff-b9a4-19eecdc6ce55");katex.render("^{129}", element, {displayMode:false,throwOnError:false});$ in primary loop (atom/(b $var element = document.getElementById("moose-equation-8eda661d-29c7-4ad8-9bba-499ff2a34d4a");katex.render("\\cdot", element, {displayMode:false,throwOnError:false});$ cm))	$var element = document.getElementById("moose-equation-e2a7c034-3fae-4664-ac16-9e4fd555d44a");katex.render("4.10 \\times 10^{-7}", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-b88e9fea-4de3-419a-b8fc-a5bc83653458");katex.render("4.09 \\times 10^{-7}", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-c27be65c-b12d-4436-be15-75e95a3a2425");katex.render("0.3", element, {displayMode:false,throwOnError:false});$ %

Table 2 shows good agreement between the atomic densities of the same two iodine isotopes that were removed to the off-gas system over a 70-day burnup period. The removal rate for iodine in the system is set at r = 3.85E-5 $var element = document.getElementById("moose-equation-1be7f9a0-4584-41b6-9c36-7ed287297415");katex.render("\\frac{1}{s}", element, {displayMode:false,throwOnError:false});$ , which is very fast when compared with the generation from fission, transmutation, and decay terms. The fast removal rate means nearly all the iodine generated in the primary loop is quickly extracted to the off-gas system, and a slightly greater amount of iodine is allowed to build up in the off-gas system compared with Table 1 since the iodine in the off-gas system does not experience any transmutation loss from the flux in the core.

Iodine solubility effect on iodine removal

Finally, the effect of chemical solubility on the isotopic removal rate used in a depletion analysis was investigated. As before, these results are reported over a 70-day burnup, and are given in Table 3.

Table 3: Effect of Iodine Solubility on Iodine Removal to Off-Gas System

	Decreasing solubility $var element = document.getElementById("moose-equation-d4935e85-55dd-4125-b892-85fd2af93275");katex.render("\\rightarrow", element, {displayMode:false,throwOnError:false});$
Removal rate (1/s)	$var element = document.getElementById("moose-equation-a5b3fe79-7a70-476a-b2ad-18930925f174");katex.render("3.85 \\times 10^{-9}", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-673c9a3a-61af-4cf2-a487-baf31c214f36");katex.render("3.85 \\times 10^{-8}", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-810fda83-f6d6-4368-a637-e02399727ec3");katex.render("3.85 \\times 10^{-7}", element, {displayMode:false,throwOnError:false});$	$var element = document.getElementById("moose-equation-4195d99a-3f05-4a65-863d-0f56bf2a6668");katex.render("3.85 \\times 10^{-6}", element, {displayMode:false,throwOnError:false});$
Percentage I $var element = document.getElementById("moose-equation-029cba9f-86a4-4359-8470-64dd68bade6c");katex.render("^{129}", element, {displayMode:false,throwOnError:false});$ in off-gas	$var element = document.getElementById("moose-equation-0c510464-3c62-4ae6-8a17-c02135293c73");katex.render("1.13", element, {displayMode:false,throwOnError:false});$ %	$var element = document.getElementById("moose-equation-88fc922d-4bd6-454e-b905-8e777509aaea");katex.render("10.5", element, {displayMode:false,throwOnError:false});$ %	$var element = document.getElementById("moose-equation-588cd4dc-61f3-4198-9263-2a6b3f584df7");katex.render("60.1", element, {displayMode:false,throwOnError:false});$ %	$var element = document.getElementById("moose-equation-efd74634-d1be-4609-9377-efdc515acd45");katex.render("95.4", element, {displayMode:false,throwOnError:false});$ %

Table 3 shows the relationship between removal rates for chemical species and the solubility of the species in the fuel-salt and why it is critical to determine these rates accurately for predicting amounts of material accumulating in the off-gas system. It is important to note that performing depletion and predicting species removal to the off-gas system in this way is considered a reduced order approach. Most notably, the spatial dependence of isotope concentrations is not included here which will be significant for isotopes that have a decay half-life on the order of the circulation time of the fuel in the reactor loop. Chemical solubility, mass transfer/reaction kinetics, and flow effects from the velocity and temperature field are all wrapped into a single removal rate term within the Bateman equations. This may be suitable for certain situations, but the removal rate term must be correctly informed by higher fidelity models operating on smaller spatial/time domains.

References

Benjamin R. Betzler, Kursat B. Bekar, William Wieselquist, Shane W. Hart, and Shane G. Stimpson. Molten salt reactor fuel depletion tools in scale. Technical Report 1566988, Oak Ridge National Lab, Oak Ridge, TN, sep 2019.

@techreport{betzler2019,
    author = "Betzler, Benjamin R. and Bekar, Kursat B. and Wieselquist, William and Hart, Shane W. and Stimpson, Shane G.",
    title = "Molten Salt Reactor Fuel Depletion Tools in SCALE",
    institution = "Oak Ridge National Lab",
    address = "Oak Ridge, TN",
    number = "1566988",
    year = "2019",
    month = "sep"
}

Maria Pusa. Higher-order chebyshev rational approximation method and application to burnup equations. Nuclear Science and Engineering, 182(3):297–318, 2016.

@article{mp2016,
    author = "Pusa, Maria",
    title = "Higher-Order Chebyshev Rational Approximation Method and Application to Burnup Equations",
    journal = "Nuclear Science and Engineering",
    volume = "182",
    number = "3",
    pages = "297--318",
    year = "2016"
}

Samuel Walker, Olin Calvin, Mauricio E. Tano, and Abdalla Abou Jaoude. Implementation of isotopic removal capability in griffin for multi-region msr depletion analysis. ANS conference paper, 2022.

@unpublished{walker2022,
    author = "Walker, Samuel and Calvin, Olin and Tano, Mauricio E. and Abou Jaoude, Abdalla",
    title = "Implementation of Isotopic Removal Capability in Griffin for Multi-Region MSR Depletion Analysis",
    year = "2022",
    note = "ANS conference paper"
}

Governing Equations
Results
References