ReleaseBirthRatio

Release rate to birth rate ratios for Kr and Xe isotopes.

Description

The release rate to birth rate ratio (R/B) is calculated for short-lived krypton and xenon isotopes.

Short-lived krypton and xenon isotopes are produced inside and released from the TRISO fuel, and their release is quantified by a release rate to birth rate ratio.

The source of these isotopes is either from uranium contamination, i.e., dispersed uranium present in the coating layers of the TRISO particles or in the matrix of the fuel element, or from exposed kernels, i.e., TRISO particles with leaking coating layers resulting from manufacturing defects or in-service failures.

Particles with at least one intact PyC layer are assumed to not release any fission gas.

Validity Range

Temperature: assumed valid at all temperatures.

Limits

The functional relationship is calculated for all temperature values.

Release/Birth Ratios

The total release/birth ratio, $var element = document.getElementById("moose-equation-3b0b9c02-a457-481b-a855-297bcb01fdc8");katex.render("RB_{total}", element, {displayMode:false,throwOnError:false});$ (dimensionless), is the sum of the contribution from uranium contamination, $var element = document.getElementById("moose-equation-9ae79a4b-61fd-402c-b7f6-a9e2999c0c52");katex.render("RB_{cont}", element, {displayMode:false,throwOnError:false});$ (dimensionless), and from exposed kernels, $var element = document.getElementById("moose-equation-54387631-d11a-4d94-9987-3f68b6449d1d");katex.render("RB_{exp}", element, {displayMode:false,throwOnError:false});$ (dimensionless). It is given by:

$var element = document.getElementById("moose-equation-2fa2ba40-f6c3-42b7-87ab-2007d3db6641");katex.render(" RB_{total} = RB_{cont} + RB_{exp}", element, {displayMode:true,throwOnError:false});$

ReleaseBirthRatio supports three models: Petti et al. (2004) (FRG, default), Pham et al. (2019) (AGR), and Richards (2019) (Richards).

FRG Model

Release/Birth from Contamination

The correlation for release/birth from uranium contamination comes from the historical German model Petti et al. (2004):

$var element = document.getElementById("moose-equation-2c8e1596-6edf-4b54-9536-f67a23768b3b");katex.render(" RB_{cont} = \\alpha_{cont} \\frac{3}{X_1} \\left[ \\text{coth}(X_1) - \\frac{1}{X_1} \\right]", element, {displayMode:true,throwOnError:false});$

$var element = document.getElementById("moose-equation-1bd7a35d-c46c-43e0-8985-e144f1281901");katex.render(" X_1 = \\left( \\frac{\\lambda}{D^\\prime_1} \\right)^{0.5}", element, {displayMode:true,throwOnError:false});$

where $var element = document.getElementById("moose-equation-5bf0e0a3-d06a-4bc8-8df5-77646624aea4");katex.render("\\alpha_{cont}", element, {displayMode:false,throwOnError:false});$ (dimensionless) is the uranium contamination fraction, $var element = document.getElementById("moose-equation-dc101beb-60c7-4690-ba2d-a39fab2b9bb9");katex.render("\\lambda", element, {displayMode:false,throwOnError:false});$ (s $var element = document.getElementById("moose-equation-d32987e1-24fe-47bb-a8bb-3c9009e15cc9");katex.render("^{-1}", element, {displayMode:false,throwOnError:false});$ ) is the decay constant of the isotope being released (see below), and $var element = document.getElementById("moose-equation-a877dd32-075f-4b48-ba54-9e993da104f7");katex.render("D^\\prime_1", element, {displayMode:false,throwOnError:false});$ (s $var element = document.getElementById("moose-equation-36d69131-1797-4f1e-b05b-16bd75975d81");katex.render("^{-1}", element, {displayMode:false,throwOnError:false});$ ) is a reduced diffusion coefficient. The diffusion coefficient depends on the chemical element:

$var element = document.getElementById("moose-equation-87600a7d-5297-45d4-a188-b32190fb2080");katex.render(" D^\\prime_1 = \\begin{cases} 3 \\times 10^{-5} \\exp{\\left( \\dfrac{-106000}{R T_K} \\right)} & \\text{for Kr}\\\\ 1.7 \\times 10^{-7} \\exp{\\left( \\dfrac{-78600}{R T_K} \\right)} & \\text{for Xe} \\end{cases}", element, {displayMode:true,throwOnError:false});$

where $var element = document.getElementById("moose-equation-a4771a54-c77f-403a-b75a-a93a49f43ece");katex.render("R", element, {displayMode:false,throwOnError:false});$ (8.3145 J/mol-K) is the universal gas constant and $var element = document.getElementById("moose-equation-4fc1654d-28e8-4f38-838b-0ed8e8ee76a3");katex.render("T_K", element, {displayMode:false,throwOnError:false});$ (K) is the average temperature of the matrix.

Release/Birth from Exposed Kernels

The correlation for R/B from exposed kernels comes from the historical German model Petti et al. (2004):

$var element = document.getElementById("moose-equation-018f7cdd-c981-4639-a895-f5d286d3dd97");katex.render(" RB_{exp} = \\alpha_{exp} \\frac{3}{X_2} \\left[ \\text{coth}(X_2) - \\frac{1}{X_2} \\right]", element, {displayMode:true,throwOnError:false});$

$var element = document.getElementById("moose-equation-fb6635cd-3d44-40b3-af4a-22f4be3359b0");katex.render(" X_2 = \\left( \\frac{\\lambda}{D^\\prime_2} \\right)^{0.5}", element, {displayMode:true,throwOnError:false});$

where $var element = document.getElementById("moose-equation-fa71f8c0-1fe2-43ba-863f-d5038e071f5b");katex.render("\\alpha_{exp}", element, {displayMode:false,throwOnError:false});$ (dimensionless) is the fraction of exposed kernels, $var element = document.getElementById("moose-equation-0b2bbecf-5200-41e4-9564-5e9f0aacbc6a");katex.render("\\lambda", element, {displayMode:false,throwOnError:false});$ (s $var element = document.getElementById("moose-equation-1c4f5107-2cc9-4247-9934-99bceb18bb98");katex.render("^{-1}", element, {displayMode:false,throwOnError:false});$ ) is the decay constant of the isotope being released (see below), and $var element = document.getElementById("moose-equation-0a331732-ed6b-4031-a027-e6a8cabc1f91");katex.render("D^\\prime_2", element, {displayMode:false,throwOnError:false});$ (s $var element = document.getElementById("moose-equation-512a2184-5a70-4a76-9684-3725254631b1");katex.render("^{-1}", element, {displayMode:false,throwOnError:false});$ ) is a reduced diffusion coefficient.

The diffusion coefficient is given by:

$var element = document.getElementById("moose-equation-2bcdb68b-1bb2-48d9-8d58-f661bc74bcf7");katex.render(" D^\\prime_2 = \\begin{cases} \\left( \\frac{500}{d_{k}} \\right)^{2} \\times 10^{-\\left(7.97 + \\frac{1920}{T_K}\\right)} & \\text{for } T_K < 1173 \\text{K} \\\\ \\left( \\frac{500}{d_{k}} \\right)^{2} \\times 10^{-\\left(2.6 + \\frac{8220}{T_K}\\right) } & \\text{for } T_K \\ge 1173 \\text{K} \\end{cases}", element, {displayMode:true,throwOnError:false});$

where $var element = document.getElementById("moose-equation-dd282888-2877-4155-927a-1d61a27939cb");katex.render("d_{k}", element, {displayMode:false,throwOnError:false});$ ( $var element = document.getElementById("moose-equation-b3e4374b-1bcc-435c-91fe-cebcfcf7a160");katex.render("\\mu", element, {displayMode:false,throwOnError:false});$ m) is the diameter of the kernel and $var element = document.getElementById("moose-equation-16d19f79-118f-4db9-bb3a-935f3b442b53");katex.render("T_K", element, {displayMode:false,throwOnError:false});$ (K) is the temperature of the kernel.

AGR Model

Release/Birth from Exposed Kernels

$var element = document.getElementById("moose-equation-faf829c4-eafb-40b8-9b90-c1bac28b62d1");katex.render(" RB_p = \\exp{\\left(n \\ln{\\left[\\frac{1}{\\lambda}\\right]}+\\frac{B}{T_K}+C\\right)}", element, {displayMode:true,throwOnError:false});$

where $var element = document.getElementById("moose-equation-f0569ca7-2245-47cd-8f7b-3cbe70f5a005");katex.render("RB_p", element, {displayMode:false,throwOnError:false});$ is the release to birth ratio per failed particle, $var element = document.getElementById("moose-equation-f5b262f8-6958-483c-b1ed-24563a6ff81c");katex.render("\\lambda", element, {displayMode:false,throwOnError:false});$ (s $var element = document.getElementById("moose-equation-ff3268b2-b3de-4690-8822-50b0a8a6405c");katex.render("^{-1}", element, {displayMode:false,throwOnError:false});$ ) is the decay constant of the isotope being released (see below), $var element = document.getElementById("moose-equation-fce15c3d-8ec5-478c-a545-bf8e966ea250");katex.render("T_K", element, {displayMode:false,throwOnError:false});$ (K) is the capsule-average temperature of the capsule, and $var element = document.getElementById("moose-equation-cb6f4c22-a281-400a-b7f4-8f130cc8c94e");katex.render("n", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-4525c848-70c3-4fe5-8624-c428bc19276c");katex.render("B", element, {displayMode:false,throwOnError:false});$ , and $var element = document.getElementById("moose-equation-edde3641-0842-485d-af21-a72a12bf1088");katex.render("C", element, {displayMode:false,throwOnError:false});$ , are given in Table 1.

Table 1: Constants for the AGR model for Kr and Xe isotopes.

Isotope	n	B	C
Kr-85m, Kr-87, Kr-88	0.325	-8572	-1.41
Xe-135, Xe-137, Xe-138	0.302	-7793	-2.73

Release/Birth from Contamination

$var element = document.getElementById("moose-equation-b42e5822-f8e3-47ca-afd6-041a07b87966");katex.render(" RB_{DU} = DU_{factor} RB_p", element, {displayMode:true,throwOnError:false});$

where $var element = document.getElementById("moose-equation-5ba47cd4-295a-49cb-9565-d86cbaff9d42");katex.render("DU_{factor}", element, {displayMode:false,throwOnError:false});$ is given in Table 2.

Table 2: Constants for the AGR model for computing R/B from contamination for Kr and Xe isotopes.

Isotope	$var element = document.getElementById("moose-equation-33323369-aa21-43e2-9f33-ab503578960f");katex.render("DU_{factor}", element, {displayMode:false,throwOnError:false});$	Isotope	$var element = document.getElementById("moose-equation-d7e0760a-e26e-4bef-9246-5df58d48e6d2");katex.render("DU_{factor}", element, {displayMode:false,throwOnError:false});$
Kr-85m	6.4	Xe-135	2.3
Kr-87	6.3	Xe-137	5.2
Kr-88	5.5	Xe-138	4.6

Finally, $var element = document.getElementById("moose-equation-833aa4f2-c754-4d54-acf6-0df2a8096560");katex.render(" RB_{total} = \\alpha_{cont} RB_{DU} + \\alpha_{exp} RB_{p}", element, {displayMode:true,throwOnError:false});$

Richards Model

$var element = document.getElementById("moose-equation-7660048f-8270-4ead-be13-7958de8fddee");katex.render(" RB = f_{gas} \\left(f_{rec}\\eta_{rec}+\\left[1-f_{rec}\\right]RB_{DR}\\right)", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-589d2639-c161-4415-8f9d-9c5347c0be85");katex.render("f_{gas}", element, {displayMode:false,throwOnError:false});$ is a parameter that accounts for decay after leaving the fuel, $var element = document.getElementById("moose-equation-001fd717-43b4-45c4-b617-37a11f49757a");katex.render("f_{rec}", element, {displayMode:false,throwOnError:false});$ is the recoil fraction (dimensionless), and $var element = document.getElementById("moose-equation-9864add2-54fe-41ac-9e8b-e834e4323b78");katex.render("\\eta_{rec}", element, {displayMode:false,throwOnError:false});$ is the fraction of recoiled atoms that do not embed in the surrounding material (0.0156).

$var element = document.getElementById("moose-equation-db2d2164-e895-4d71-9586-ca9a50a8edde");katex.render(" f_{rec} = 0.25 a_g \\mu_{ff}", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-5a8566db-6f25-49be-9455-8782b4850096");katex.render("a_g", element, {displayMode:false,throwOnError:false});$ is $var element = document.getElementById("moose-equation-c350cc05-0226-46b2-83e7-5cc0915553e5");katex.render("3/r_k", element, {displayMode:false,throwOnError:false});$ for spherical kernels (m $var element = document.getElementById("moose-equation-ae9c3141-ba15-4ef1-b308-33a7e819c74e");katex.render("^{-1}", element, {displayMode:false,throwOnError:false});$ ), $var element = document.getElementById("moose-equation-30c90b89-48a5-461f-b04d-e28474ab4191");katex.render("\\mu_{ff}", element, {displayMode:false,throwOnError:false});$ is the fission product range (m), and $var element = document.getElementById("moose-equation-9d4a159d-cbbd-4fb3-ba1c-bc36e2734c72");katex.render("r_k", element, {displayMode:false,throwOnError:false});$ is the radius of the kernel (m).

$var element = document.getElementById("moose-equation-b8d6ddca-4b38-4a8b-9a7a-effcc2c7ed45");katex.render(" RB_{DR} = RB_D + H\\left(1-RB_D\\right)", element, {displayMode:true,throwOnError:false});$ with $var element = document.getElementById("moose-equation-c0a52993-c31d-4904-86f2-9b62525bdead");katex.render("RB_D = 3\\left(\\Gamma G\\right)^{0.5} \\left[\\coth{\\left(\\frac{1}{(\\Gamma G)^{0.5}}\\right)-(\\Gamma G)^{0.5}}\\right]", element, {displayMode:true,throwOnError:false});$ $var element = document.getElementById("moose-equation-112f688c-fae5-4861-81d5-98d93aa0dc3e");katex.render(" H = \\frac{g b_T / \\lambda}{\\lambda + b_F + b_T + g + g b_T / \\lambda}", element, {displayMode:true,throwOnError:false});$ $var element = document.getElementById("moose-equation-9d8ba4b8-3b1b-4795-b1b4-d9e24b16dd64");katex.render(" G = \\frac{\\lambda + b_F + b_T}{\\lambda + b_F + b_T + g + g b_T/\\lambda}", element, {displayMode:true,throwOnError:false});$ $var element = document.getElementById("moose-equation-96964f3a-b173-490c-ba74-e57cd0485097");katex.render("\\Gamma = \\frac{D_0}{\\lambda},\\quad g = c_1 \\dot{F},\\quad b_F = c_2 \\dot{F},\\quad b_T = A_{0,T}\\exp{\\left(-\\frac{T_{A,T}}{T}\\right)},\\quad D_0 = A_0\\exp{\\left(-\\frac{T_A}{T}\\right)}", element, {displayMode:true,throwOnError:false});$ $var element = document.getElementById("moose-equation-36753f93-300f-4168-b389-6232dcf328ec");katex.render("\\lambda", element, {displayMode:false,throwOnError:false});$ is the decay constant (s $var element = document.getElementById("moose-equation-1deb8d14-a9e1-422f-8bab-38e9d2e858f8");katex.render("^{-1}", element, {displayMode:false,throwOnError:false});$ ) with $var element = document.getElementById("moose-equation-469ef9db-6945-42ee-8935-086400cd5582");katex.render("\\dot{F}", element, {displayMode:false,throwOnError:false});$ being the fission rate ( $var element = document.getElementById("moose-equation-bdbdd3a7-80d9-402a-96ef-785b6ca4f548");katex.render("10^{14}", element, {displayMode:false,throwOnError:false});$ fissions/cm $var element = document.getElementById("moose-equation-8b6005d2-609d-4245-9b73-f45bf09fd50f");katex.render("^3", element, {displayMode:false,throwOnError:false});$ /s).

$var element = document.getElementById("moose-equation-7b7beb53-9dab-42d0-96e1-2a333ab31805");katex.render("A_0 = 2.047\\times 10^{-5}", element, {displayMode:false,throwOnError:false});$ for kypton, and $var element = document.getElementById("moose-equation-250fd84f-c41d-4dcc-8480-0738f9cbf3d2");katex.render("A_0 = 3.133\\times 10^{-6}", element, {displayMode:false,throwOnError:false});$ for xenon (s $var element = document.getElementById("moose-equation-9907347b-52b7-4fd0-8c83-c0bac6bed554");katex.render("^{-1}", element, {displayMode:false,throwOnError:false});$ ). $var element = document.getElementById("moose-equation-eb37446b-c3fa-4dc2-8525-ad21abcfedcf");katex.render("T_A = 12722", element, {displayMode:false,throwOnError:false});$ (K), $var element = document.getElementById("moose-equation-2bc45dc8-c180-4112-9480-946a9524b626");katex.render("c_1 = 2.09\\times 10^{-4}", element, {displayMode:false,throwOnError:false});$ ( $var element = document.getElementById("moose-equation-ea9ac1b8-a246-42fb-b29c-7e6a6990acc3");katex.render("10^{-14}", element, {displayMode:false,throwOnError:false});$ cm $var element = document.getElementById("moose-equation-52406813-33c2-43b9-972f-75b5d7261ce3");katex.render("^3", element, {displayMode:false,throwOnError:false});$ /fission), $var element = document.getElementById("moose-equation-da548dfd-0a79-4cd7-aacf-7ab4d1407abb");katex.render("c_2 = 1.976\\times 10^{-5}", element, {displayMode:false,throwOnError:false});$ ( $var element = document.getElementById("moose-equation-61cbd6b5-3e20-42bc-bd3f-c51aaec108dd");katex.render("10^{-14}", element, {displayMode:false,throwOnError:false});$ cm $var element = document.getElementById("moose-equation-f8356607-f6cd-4412-bcb3-894b1dc8d58e");katex.render("^3", element, {displayMode:false,throwOnError:false});$ /fission), $var element = document.getElementById("moose-equation-0856cc72-9324-469a-80d6-75ef0e5b9b24");katex.render("A_{0,T} = 3.988\\times 10^{13}", element, {displayMode:false,throwOnError:false});$ (s $var element = document.getElementById("moose-equation-0d20c6a6-f13f-4425-ab50-0f17b37c5974");katex.render("^{-1}", element, {displayMode:false,throwOnError:false});$ ), and $var element = document.getElementById("moose-equation-a164718d-a5d8-4c89-ad63-26738d169748");katex.render("T_{A,T} = 65000", element, {displayMode:false,throwOnError:false});$ (K).

Finally, $var element = document.getElementById("moose-equation-4e63f6e2-7cc2-470c-b4bb-0dd8c17d8a50");katex.render("RB", element, {displayMode:false,throwOnError:false});$ is multiplied by $var element = document.getElementById("moose-equation-adc9cbe2-02b4-495c-9b8b-3940e9a94aab");katex.render("\\alpha_{exp}", element, {displayMode:false,throwOnError:false});$ .

Decay Constants

The decay constants of the short-lived krypton and xenon isotopes are given in Table 3. These values are from NuDat 3.0.

Table 3: Decay constants of short-lived Kr and Xe isotopes.

Isotope	Decay Constant (s $var element = document.getElementById("moose-equation-9d077fc0-d05b-47c2-a619-193d36375949");katex.render("^{-1}", element, {displayMode:false,throwOnError:false});$ )	Isotope	Decay Constant (s $var element = document.getElementById("moose-equation-064e4919-da51-4134-967b-18e0ea4f67af");katex.render("^{-1}", element, {displayMode:false,throwOnError:false});$ )
Kr-83m	$var element = document.getElementById("moose-equation-e3a7975a-0776-4290-aa81-2c871f758e4c");katex.render("1.052 \\times 10^{-4}", element, {displayMode:false,throwOnError:false});$	Xe-131m	$var element = document.getElementById("moose-equation-31c32e16-255e-4daa-881e-a5b74da70eba");katex.render("6.776 \\times 10^{-7}", element, {displayMode:false,throwOnError:false});$
Kr-85m	$var element = document.getElementById("moose-equation-f5d145db-6b0c-4466-8d00-7e5ebc7b8a25");katex.render("4.298 \\times 10^{-5}", element, {displayMode:false,throwOnError:false});$	Xe-133	$var element = document.getElementById("moose-equation-a8a7da81-a4ea-4735-84f8-d2f776a07453");katex.render("1.529 \\times 10^{-6}", element, {displayMode:false,throwOnError:false});$
Kr-85	$var element = document.getElementById("moose-equation-9a9fb0e8-aa7e-4cf9-ab38-0a277ae09290");katex.render("2.047 \\times 10^{-9}", element, {displayMode:false,throwOnError:false});$	Xe-135m	$var element = document.getElementById("moose-equation-7a4fb05d-c39c-402c-9f0b-86a5f2f181fc");katex.render("7.556 \\times 10^{-4}", element, {displayMode:false,throwOnError:false});$
Kr-87	$var element = document.getElementById("moose-equation-50310149-4d4e-462a-ad69-046c54f1b4d3");katex.render("1.514 \\times 10^{-4}", element, {displayMode:false,throwOnError:false});$	Xe-135	$var element = document.getElementById("moose-equation-03d29625-4d52-4109-958c-96df30be0e75");katex.render("2.107 \\times 10^{-5}", element, {displayMode:false,throwOnError:false});$
Kr-88	$var element = document.getElementById("moose-equation-90f72d92-8e47-4998-8cbc-56f685230d0e");katex.render("6.816 \\times 10^{-5}", element, {displayMode:false,throwOnError:false});$	Xe-137	$var element = document.getElementById("moose-equation-0464d90f-5dab-400c-b66b-e97ab3d34611");katex.render("3.026 \\times 10^{-3}", element, {displayMode:false,throwOnError:false});$
Kr-89	$var element = document.getElementById("moose-equation-ffd9f835-b943-489f-bff9-07c8edb99d9f");katex.render("3.667 \\times 10^{-3}", element, {displayMode:false,throwOnError:false});$	Xe-138	$var element = document.getElementById("moose-equation-3114b14e-0aa1-4a56-8b24-4f559e2fc666");katex.render("8.170 \\times 10^{-4}", element, {displayMode:false,throwOnError:false});$
Kr-90	$var element = document.getElementById("moose-equation-9e75590d-c78e-4c5c-afbc-59e5ff3173f1");katex.render("2.145 \\times 10^{-2}", element, {displayMode:false,throwOnError:false});$	Xe-139	$var element = document.getElementById("moose-equation-6b1ce5ae-d115-40ea-91d7-564ee9224614");katex.render("1.747 \\times 10^{-2}", element, {displayMode:false,throwOnError:false});$

Example Input Syntax

[Postprocessors<<<{"href": "../../syntax/Postprocessors/index.html"}>>>]
  [Kr85m_RB_ratio]
    type = ReleaseBirthRatio<<<{"description": "Release rate to birth rate ratios for Kr and Xe isotopes.", "href": "ReleaseBirthRatio.html"}>>>
    isotope<<<{"description": "The radionuclide isotope: Kr83m Kr85m Kr85 Kr87 Kr88 Kr89 Kr90 Xe131m Xe133 Xe135m Xe135 Xe137 Xe138 Xe139"}>>> = Kr85m
    average_kernel_temperature<<<{"description": "The postprocessor for average kernel temperature"}>>> = average_kernel_temperature
    average_matrix_temperature<<<{"description": "The postprocessor for average matrix temperature"}>>> = average_matrix_temperature
    triso_geometry<<<{"description": "TRISOGeometry user object name"}>>> = particle_geometry
    execute_on<<<{"description": "The list of flag(s) indicating when this object should be executed. For a description of each flag, see https://mooseframework.inl.gov/source/interfaces/SetupInterface.html."}>>> = 'initial timestep_end'
  []
[]

Input Parameters

U_contamination_fractionRatio of uranium in the matrix over the total inventory after fabrication
C++ Type:PostprocessorName
Unit:(no unit assumed)
Controllable:No
Description:Ratio of uranium in the matrix over the total inventory after fabrication
average_kernel_temperatureThe postprocessor for average kernel temperature
C++ Type:PostprocessorName
Unit:(no unit assumed)
Controllable:No
Description:The postprocessor for average kernel temperature
average_matrix_temperatureThe postprocessor for average matrix temperature
C++ Type:PostprocessorName
Unit:(no unit assumed)
Controllable:No
Description:The postprocessor for average matrix temperature
exposed_kernel_fractionFraction of particles with exposed kernels
C++ Type:PostprocessorName
Unit:(no unit assumed)
Controllable:No
Description:Fraction of particles with exposed kernels
isotopeThe radionuclide isotope: Kr83m Kr85m Kr85 Kr87 Kr88 Kr89 Kr90 Xe131m Xe133 Xe135m Xe135 Xe137 Xe138 Xe139
C++ Type:MooseEnum
Options:Kr83m, Kr85m, Kr85, Kr87, Kr88, Kr89, Kr90, Xe131m, Xe133, Xe135m, Xe135, Xe137, Xe138, Xe139
Controllable:No
Description:The radionuclide isotope: Kr83m Kr85m Kr85 Kr87 Kr88 Kr89 Kr90 Xe131m Xe133 Xe135m Xe135 Xe137 Xe138 Xe139
triso_geometryTRISOGeometry user object name
C++ Type:UserObjectName
Controllable:No
Description:TRISOGeometry user object name

Required Parameters

C_U0.4Carbon to uranium ratio. Required for the Richards model.
Default:0.4
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Carbon to uranium ratio. Required for the Richards model.
O_U1.5Oxygen to uranium ratio. Required for the Richards model.
Default:1.5
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Oxygen to uranium ratio. Required for the Richards model.
R8.31446Universal gas constant.
Default:8.31446
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Universal gas constant.
fission_rate0Fission rate. Required for the Richards model.
Default:0
C++ Type:PostprocessorName
Unit:(no unit assumed)
Controllable:No
Description:Fission rate. Required for the Richards model.
initial_density1Initial density. Required for the Richards model.
Default:1
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Initial density. Required for the Richards model.
modelFRGThe model type: FRG AGR Richards
Default:FRG
C++ Type:MooseEnum
Options:FRG, AGR, Richards
Controllable:No
Description:The model type: FRG AGR Richards
mol_weight_gas_medium12Molecular weight of the gaseous medium. Required for the Richards model.
Default:12
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Molecular weight of the gaseous medium. Required for the Richards model.
outside_temperature1250Outside temperature of the fuel element. Required for the Richards model.
Default:1250
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:Outside temperature of the fuel element. Required for the Richards model.
void_fraction0.29Void fraction. Required by Richards model.
Default:0.29
C++ Type:PostprocessorName
Unit:(no unit assumed)
Controllable:No
Description:Void fraction. Required by Richards model.

Optional Parameters

allow_duplicate_execution_on_initialFalseIn the case where this UserObject is depended upon by an initial condition, allow it to be executed twice during the initial setup (once before the IC and again after mesh adaptivity (if applicable).
Default:False
C++ Type:bool
Controllable:No
Description:In the case where this UserObject is depended upon by an initial condition, allow it to be executed twice during the initial setup (once before the IC and again after mesh adaptivity (if applicable).
execute_onTIMESTEP_ENDThe list of flag(s) indicating when this object should be executed. For a description of each flag, see https://mooseframework.inl.gov/source/interfaces/SetupInterface.html.
Default:TIMESTEP_END
C++ Type:ExecFlagEnum
Options:XFEM_MARK, NONE, INITIAL, LINEAR, NONLINEAR_CONVERGENCE, NONLINEAR, POSTCHECK, TIMESTEP_END, TIMESTEP_BEGIN, MULTIAPP_FIXED_POINT_END, MULTIAPP_FIXED_POINT_BEGIN, FINAL, CUSTOM, TRANSFER
Controllable:No
Description:The list of flag(s) indicating when this object should be executed. For a description of each flag, see https://mooseframework.inl.gov/source/interfaces/SetupInterface.html.
execution_order_group0Execution order groups are executed in increasing order (e.g., the lowest number is executed first). Note that negative group numbers may be used to execute groups before the default (0) group. Please refer to the user object documentation for ordering of user object execution within a group.
Default:0
C++ Type:int
Controllable:No
Description:Execution order groups are executed in increasing order (e.g., the lowest number is executed first). Note that negative group numbers may be used to execute groups before the default (0) group. Please refer to the user object documentation for ordering of user object execution within a group.
force_postauxFalseForces the UserObject to be executed in POSTAUX
Default:False
C++ Type:bool
Controllable:No
Description:Forces the UserObject to be executed in POSTAUX
force_preauxFalseForces the UserObject to be executed in PREAUX
Default:False
C++ Type:bool
Controllable:No
Description:Forces the UserObject to be executed in PREAUX
force_preicFalseForces the UserObject to be executed in PREIC during initial setup
Default:False
C++ Type:bool
Controllable:No
Description:Forces the UserObject to be executed in PREIC during initial setup

Execution Scheduling Parameters

control_tagsAdds user-defined labels for accessing object parameters via control logic.
C++ Type:std::vector<std::string>
Controllable:No
Description:Adds user-defined labels for accessing object parameters via control logic.
enableTrueSet the enabled status of the MooseObject.
Default:True
C++ Type:bool
Controllable:Yes
Description:Set the enabled status of the MooseObject.
outputsVector of output names where you would like to restrict the output of variables(s) associated with this object
C++ Type:std::vector<OutputName>
Controllable:No
Description:Vector of output names where you would like to restrict the output of variables(s) associated with this object
use_displaced_meshFalseWhether or not this object should use the displaced mesh for computation. Note that in the case this is true but no displacements are provided in the Mesh block the undisplaced mesh will still be used.
Default:False
C++ Type:bool
Controllable:No
Description:Whether or not this object should use the displaced mesh for computation. Note that in the case this is true but no displacements are provided in the Mesh block the undisplaced mesh will still be used.

Advanced Parameters

prop_getter_suffixAn optional suffix parameter that can be appended to any attempt to retrieve/get material properties. The suffix will be prepended with a '_' character.
C++ Type:MaterialPropertyName
Unit:(no unit assumed)
Controllable:No
Description:An optional suffix parameter that can be appended to any attempt to retrieve/get material properties. The suffix will be prepended with a '_' character.
use_interpolated_stateFalseFor the old and older state use projected material properties interpolated at the quadrature points. To set up projection use the ProjectedStatefulMaterialStorageAction.
Default:False
C++ Type:bool
Controllable:No
Description:For the old and older state use projected material properties interpolated at the quadrature points. To set up projection use the ProjectedStatefulMaterialStorageAction.

Material Property Retrieval Parameters

Input Files

(test/tests/triso/release_birth_ratio/agr.i)
(test/tests/triso/release_birth_ratio/richards.i)
(test/tests/triso/release_birth_ratio/exposed_kernel.i)
(test/tests/triso/release_birth_ratio/uranium_contamination.i)

References

David Petti, Phillipe Martin, Mayeul Phelip, and Ronald Ballinger. Development Of Improved Models And Designs For Coated-Particle Gas Reactor Fuels. Report INEEL/EXT-05-02615, Idaho National Laboratory for the International Nuclear Energy Research Initiative, December 2004.

@techreport{INERI2004,
    author = "Petti, David and Martin, Phillipe and Phelip, Mayeul and Ballinger, Ronald",
    title = "{Development Of Improved Models And Designs For Coated-Particle Gas Reactor Fuels}",
    institution = "Idaho National Laboratory for the International Nuclear Energy Research Initiative",
    year = "2004",
    month = "December",
    number = "INEEL/EXT-05-02615",
    type = "Report"
}

Binh T. Pham, Jeffrey J. Einerson, Dawn M. Scates, John T. Maki, and David A. Petti. AGR-2 and AGR-3/4 release-to-birth ratio data analysis. Report INL/EXT-14-32970, Rev. 2, Idaho National Laboratory, June 2019.

@techreport{Pham2019,
    author = "Pham, Binh T. and Einerson, Jeffrey J. and Scates, Dawn M. and Maki, John T. and Petti, David A.",
    title = "{AGR}-2 and {AGR}-3/4 Release-to-Birth Ratio Data Analysis",
    institution = "Idaho National Laboratory",
    year = "2019",
    month = "June",
    number = "INL/EXT-14-32970, Rev. 2",
    type = "Report"
}

Matt Richards. A theoretical model for fission gas release from UCO TRISO fuel. In Proceedings of ICONE-27. Ibaraki, Japan, May 19–24 2019.

@inproceedings{richards2019model,
    author = "Richards, Matt",
    title = "A theoretical model for fission gas release from {UCO} {TRISO} fuel",
    booktitle = "Proceedings of {ICONE}-27",
    Month = "May 19--24",
    Address = "Ibaraki, Japan",
    year = "2019"
}

(test/tests/triso/release_birth_ratio/exposed_kernel.i)

# Release rate to birth rate ratio (R/B) of short-lived Kr and Xe isotopes from exposed kernels.
# The geometry is a sphere with a diameter of 425 micrometers.
# The temperature is ramped from 673.15 to 2073.15 K.
#
# Bison results are checked against results computed in the *_exact Functions.
#

[GlobalParams]
  U_contamination_fraction = 0.0
  exposed_kernel_fraction = 1e-4
[]

[Mesh]
  coord_type = RSPHERICAL
  [gen]
    type = TRISO1DMeshGenerator
    elem_type = EDGE3
    coordinates = '0 2.125e-4'
    mesh_density = '6'
    block_names = 'particle'
  []
[]

[UserObjects]
  [particle_geometry]
    type = TRISOGeometry
    outer_OPyC = '2'
    outer_SiC = '2'
    outer_IPyC = '2'
    inner_IPyC = '2'
    outer_buffer = '2'
    outer_kernel = '2'
    include_particle = true
    include_pebble = false
  []
[]

[Variables]
  [temperature]
    initial_condition = 673.15
  []
[]

[Functions]
  [temp_function]
    type = PiecewiseLinear
    x = '0          3e6'
    y = '673.15 2073.15'
  []
  [D_1_Kr]
    type = ParsedFunction
    symbol_names = 'R      temp'
    symbol_values = '8.3145 temp_function'
    expression = '3e-5*exp(-106e3/(R*temp))'
  []
  [X_1_Kr85m]
    type = ParsedFunction
    symbol_names = 'decay    D_1'
    symbol_values = '4.298e-5 D_1_Kr'
    expression = 'sqrt(decay / D_1)'
  []
  [D_2]
    type = ParsedFunction
    symbol_names = 'radius     temp'
    symbol_values = '2.125e-4 temp_function'
    expression = 'pow(500*1e-6/radius/2,2)*pow(10,if(temp<1173,-7.97-1920/temp,-2.6-8220/temp))'
  []
  [X_2_Kr85m]
    type = ParsedFunction
    symbol_names = 'decay    D_2'
    symbol_values = '4.298e-5 D_2'
    expression = 'sqrt(decay/D_2)'
  []
  [Kr85m_exact]
    type = ParsedFunction
    symbol_names = 'expo X_2'
    symbol_values = '1e-4 X_2_Kr85m'
    expression = 'expo*3/X_2*(1/tanh(X_2)-1/X_2)'
  []
  [X_2_Kr87]
    type = ParsedFunction
    symbol_names = 'decay    D_2'
    symbol_values = '1.514e-4 D_2'
    expression = 'sqrt(decay/D_2)'
  []
  [Kr87_exact]
    type = ParsedFunction
    symbol_names = 'expo X_2'
    symbol_values = '1e-4 X_2_Kr87'
    expression = 'expo*3/X_2*(1/tanh(X_2)-1/X_2)'
  []
  [X_2_Kr88]
    type = ParsedFunction
    symbol_names = 'decay    D_2'
    symbol_values = '6.816e-5 D_2'
    expression = 'sqrt(decay/D_2)'
  []
  [Kr88_exact]
    type = ParsedFunction
    symbol_names = 'expo X_2'
    symbol_values = '1e-4 X_2_Kr88'
    expression = 'expo*3/X_2*(1/tanh(X_2)-1/X_2)'
  []
  [X_2_Kr89]
    type = ParsedFunction
    symbol_names = 'decay    D_2'
    symbol_values = '3.667e-3 D_2'
    expression = 'sqrt(decay/D_2)'
  []
  [Kr89_exact]
    type = ParsedFunction
    symbol_names = 'expo X_2'
    symbol_values = '1e-4 X_2_Kr89'
    expression = 'expo*3/X_2*(1/tanh(X_2)-1/X_2)'
  []
  [X_2_Kr90]
    type = ParsedFunction
    symbol_names = 'decay    D_2'
    symbol_values = '2.145e-2 D_2'
    expression = 'sqrt(decay/D_2)'
  []
  [Kr90_exact]
    type = ParsedFunction
    symbol_names = 'expo X_2'
    symbol_values = '1e-4 X_2_Kr90'
    expression = 'expo*3/X_2*(1/tanh(X_2)-1/X_2)'
  []
  [X_2_Xe131m]
    type = ParsedFunction
    symbol_names = 'decay    D_2'
    symbol_values = '6.776e-7 D_2'
    expression = 'sqrt(decay/D_2)'
  []
  [Xe131m_exact]
    type = ParsedFunction
    symbol_names = 'expo X_2'
    symbol_values = '1e-4 X_2_Xe131m'
    expression = 'expo*3/X_2*(1/tanh(X_2)-1/X_2)'
  []
  [X_2_Xe133]
    type = ParsedFunction
    symbol_names = 'decay    D_2'
    symbol_values = '1.529e-6 D_2'
    expression = 'sqrt(decay/D_2)'
  []
  [Xe133_exact]
    type = ParsedFunction
    symbol_names = 'expo X_2'
    symbol_values = '1e-4 X_2_Xe133'
    expression = 'expo*3/X_2*(1/tanh(X_2)-1/X_2)'
  []
  [X_2_Xe135]
    type = ParsedFunction
    symbol_names = 'decay    D_2'
    symbol_values = '2.107e-5 D_2'
    expression = 'sqrt(decay/D_2)'
  []
  [Xe135_exact]
    type = ParsedFunction
    symbol_names = 'expo X_2'
    symbol_values = '1e-4 X_2_Xe135'
    expression = 'expo*3/X_2*(1/tanh(X_2)-1/X_2)'
  []
  [X_2_Xe135m]
    type = ParsedFunction
    symbol_names = 'decay    D_2'
    symbol_values = '7.556e-4 D_2'
    expression = 'sqrt(decay/D_2)'
  []
  [Xe135m_exact]
    type = ParsedFunction
    symbol_names = 'expo X_2'
    symbol_values = '1e-4 X_2_Xe135m'
    expression = 'expo*3/X_2*(1/tanh(X_2)-1/X_2)'
  []
  [X_2_Xe137]
    type = ParsedFunction
    symbol_names = 'decay    D_2'
    symbol_values = '3.026e-3 D_2'
    expression = 'sqrt(decay/D_2)'
  []
  [Xe137_exact]
    type = ParsedFunction
    symbol_names = 'expo X_2'
    symbol_values = '1e-4 X_2_Xe137'
    expression = 'expo*3/X_2*(1/tanh(X_2)-1/X_2)'
  []
  [X_2_Xe138]
    type = ParsedFunction
    symbol_names = 'decay    D_2'
    symbol_values = '8.170e-4 D_2'
    expression = 'sqrt(decay/D_2)'
  []
  [Xe138_exact]
    type = ParsedFunction
    symbol_names = 'expo X_2'
    symbol_values = '1e-4 X_2_Xe138'
    expression = 'expo*3/X_2*(1/tanh(X_2)-1/X_2)'
  []
  [X_2_Xe139]
    type = ParsedFunction
    symbol_names = 'decay    D_2'
    symbol_values = '1.747e-2 D_2'
    expression = 'sqrt(decay/D_2)'
  []
  [Xe139_exact]
    type = ParsedFunction
    symbol_names = 'expo X_2'
    symbol_values = '1e-4 X_2_Xe139'
    expression = 'expo*3/X_2*(1/tanh(X_2)-1/X_2)'
  []
  # [Kr85m_exact]
  #   type = ParsedFunction
  #   symbol_names = 'cont X_1'
  #   symbol_values = '1e-6 X_1_Kr85m'
  #   expression = 'cont*3/X_1*(1/tanh(X_1)-1/X_1)'
  # []
[]

[Kernels]
  [heat] # gradient term in heat conduction equation
    type = HeatConduction
    variable = temperature
  []

  [heat_ie] # time term in heat conduction equation
    type = HeatConductionTimeDerivative
    variable = temperature
  []
[]

[BCs]
  [heat_removal]
    type = FunctionDirichletBC
    variable = temperature
    boundary = '2'
    function = temp_function
  []
[]

[Materials]

  [thermal]
    type = HeatConductionMaterial
    block = '1'
    thermal_conductivity = 1.0
    specific_heat = 1.0
  []

  [density]
    type = ParsedMaterial
    block = '1'
    property_name = density
    expression = 2570.0
  []

[]

[Executioner]

  type = Transient

  solve_type = 'PJFNK'

  petsc_options_iname = '-pc_type -pc_factor_mat_solver_package -ksp_gmres_restart'
  petsc_options_value = 'lu       superlu_dist                  51'
  line_search = 'none'

  l_max_its = 50
  l_tol = 1e-2

  nl_max_its = 150
  nl_rel_tol = 1e-8

  start_time = 0.0
  end_time = 3e6
  num_steps = 140

  dtmax = 3e6
  dtmin = 1.0

  [TimeStepper]
    type = IterationAdaptiveDT
    dt = 1e5
    time_t = '1e5'
    time_dt = '1e5'
  []
[]

[Postprocessors]

  [temperature]
    type = ElementExtremeValue
    block = '1'
    value_type = 'max'
    variable = temperature
    execute_on = 'initial timestep_end'
  []

  [average_kernel_temperature]
    type = ElementAverageValue
    block = '1'
    variable = temperature
    execute_on = 'initial timestep_end'
  []

  [average_matrix_temperature]
    type = ElementAverageValue
    block = '1'
    variable = temperature
    execute_on = 'initial timestep_end'
  []

  [_dt]
    type = TimestepSize
  []

  [Kr85m_RB_ratio]
    type = ReleaseBirthRatio
    isotope = Kr85m
    average_kernel_temperature = average_kernel_temperature
    average_matrix_temperature = average_matrix_temperature
    triso_geometry = particle_geometry
    execute_on = 'initial timestep_end'
  []
  [Kr85m_exact]
    type = FunctionValuePostprocessor
    function = Kr85m_exact
    execute_on = 'initial timestep_end'
  []
  [Kr85m_check]
    type = RelativeDifferencePostprocessor
    value1 = Kr85m_RB_ratio
    value2 = Kr85m_exact
  []

  [Kr87_RB_ratio]
    type = ReleaseBirthRatio
    isotope = Kr87
    average_kernel_temperature = average_kernel_temperature
    average_matrix_temperature = average_matrix_temperature
    triso_geometry = particle_geometry
    execute_on = 'initial timestep_end'
  []
  [Kr87_exact]
    type = FunctionValuePostprocessor
    function = Kr87_exact
    execute_on = 'initial timestep_end'
  []
  [Kr87_check]
    type = RelativeDifferencePostprocessor
    value1 = Kr87_RB_ratio
    value2 = Kr87_exact
  []

  [Kr88_RB_ratio]
    type = ReleaseBirthRatio
    isotope = Kr88
    average_kernel_temperature = average_kernel_temperature
    average_matrix_temperature = average_matrix_temperature
    triso_geometry = particle_geometry
    execute_on = 'initial timestep_end'
  []
  [Kr88_exact]
    type = FunctionValuePostprocessor
    function = Kr88_exact
    execute_on = 'initial timestep_end'
  []
  [Kr88_check]
    type = RelativeDifferencePostprocessor
    value1 = Kr88_RB_ratio
    value2 = Kr88_exact
  []

  [Kr89_RB_ratio]
    type = ReleaseBirthRatio
    isotope = Kr89
    average_kernel_temperature = average_kernel_temperature
    average_matrix_temperature = average_matrix_temperature
    triso_geometry = particle_geometry
    execute_on = 'initial timestep_end'
  []
  [Kr89_exact]
    type = FunctionValuePostprocessor
    function = Kr89_exact
    execute_on = 'initial timestep_end'
  []
  [Kr89_check]
    type = RelativeDifferencePostprocessor
    value1 = Kr89_RB_ratio
    value2 = Kr89_exact
  []

  [Kr90_RB_ratio]
    type = ReleaseBirthRatio
    isotope = Kr90
    average_kernel_temperature = average_kernel_temperature
    average_matrix_temperature = average_matrix_temperature
    triso_geometry = particle_geometry
    execute_on = 'initial timestep_end'
  []
  [Kr90_exact]
    type = FunctionValuePostprocessor
    function = Kr90_exact
    execute_on = 'initial timestep_end'
  []
  [Kr90_check]
    type = RelativeDifferencePostprocessor
    value1 = Kr90_RB_ratio
    value2 = Kr90_exact
  []

  [Xe131m_RB_ratio]
    type = ReleaseBirthRatio
    isotope = Xe131m
    average_kernel_temperature = average_kernel_temperature
    average_matrix_temperature = average_matrix_temperature
    triso_geometry = particle_geometry
    execute_on = 'initial timestep_end'
  []
  [Xe131m_exact]
    type = FunctionValuePostprocessor
    function = Xe131m_exact
    execute_on = 'initial timestep_end'
  []
  [Xe131m_check]
    type = RelativeDifferencePostprocessor
    value1 = Xe131m_RB_ratio
    value2 = Xe131m_exact
  []

  [Xe133_RB_ratio]
    type = ReleaseBirthRatio
    isotope = Xe133
    average_kernel_temperature = average_kernel_temperature
    average_matrix_temperature = average_matrix_temperature
    triso_geometry = particle_geometry
    execute_on = 'initial timestep_end'
  []
  [Xe133_exact]
    type = FunctionValuePostprocessor
    function = Xe133_exact
    execute_on = 'initial timestep_end'
  []
  [Xe133_check]
    type = RelativeDifferencePostprocessor
    value1 = Xe133_RB_ratio
    value2 = Xe133_exact
  []

  [Xe135_RB_ratio]
    type = ReleaseBirthRatio
    isotope = Xe135
    average_kernel_temperature = average_kernel_temperature
    average_matrix_temperature = average_matrix_temperature
    triso_geometry = particle_geometry
    execute_on = 'initial timestep_end'
  []
  [Xe135_exact]
    type = FunctionValuePostprocessor
    function = Xe135_exact
    execute_on = 'initial timestep_end'
  []
  [Xe135_check]
    type = RelativeDifferencePostprocessor
    value1 = Xe135_RB_ratio
    value2 = Xe135_exact
  []

  [Xe135m_RB_ratio]
    type = ReleaseBirthRatio
    isotope = Xe135m
    average_kernel_temperature = average_kernel_temperature
    average_matrix_temperature = average_matrix_temperature
    triso_geometry = particle_geometry
    execute_on = 'initial timestep_end'
  []
  [Xe135m_exact]
    type = FunctionValuePostprocessor
    function = Xe135m_exact
    execute_on = 'initial timestep_end'
  []
  [Xe135m_check]
    type = RelativeDifferencePostprocessor
    value1 = Xe135m_RB_ratio
    value2 = Xe135m_exact
  []

  [Xe137_RB_ratio]
    type = ReleaseBirthRatio
    isotope = Xe137
    average_kernel_temperature = average_kernel_temperature
    average_matrix_temperature = average_matrix_temperature
    triso_geometry = particle_geometry
    execute_on = 'initial timestep_end'
  []
  [Xe137_exact]
    type = FunctionValuePostprocessor
    function = Xe137_exact
    execute_on = 'initial timestep_end'
  []
  [Xe137_check]
    type = RelativeDifferencePostprocessor
    value1 = Xe137_RB_ratio
    value2 = Xe137_exact
  []

  [Xe138_RB_ratio]
    type = ReleaseBirthRatio
    isotope = Xe138
    average_kernel_temperature = average_kernel_temperature
    average_matrix_temperature = average_matrix_temperature
    triso_geometry = particle_geometry
    execute_on = 'initial timestep_end'
  []
  [Xe138_exact]
    type = FunctionValuePostprocessor
    function = Xe138_exact
    execute_on = 'initial timestep_end'
  []
  [Xe138_check]
    type = RelativeDifferencePostprocessor
    value1 = Xe138_RB_ratio
    value2 = Xe138_exact
  []

  [Xe139_RB_ratio]
    type = ReleaseBirthRatio
    isotope = Xe139
    average_kernel_temperature = average_kernel_temperature
    average_matrix_temperature = average_matrix_temperature
    triso_geometry = particle_geometry
    execute_on = 'initial timestep_end'
  []
  [Xe139_exact]
    type = FunctionValuePostprocessor
    function = Xe139_exact
    execute_on = 'initial timestep_end'
  []
  [Xe139_check]
    type = RelativeDifferencePostprocessor
    value1 = Xe139_RB_ratio
    value2 = Xe139_exact
  []
[]

[Outputs]
  csv = true
  exodus = false
  [console]
    type = Console
  []
[]

(test/tests/triso/release_birth_ratio/agr.i)

# Release rate to birth rate ratio (R/B) of short-lived Kr and Xe isotopes from exposed kernels.
# The geometry is a sphere with a diameter of 425 micrometers.
# The temperature is ramped from 673.15 to 2073.15 K.
#
# Bison results are checked against results computed in the *_exact Functions.
#

[GlobalParams]
  U_contamination_fraction = 1e-6
  exposed_kernel_fraction = 1e-4
[]

[Mesh]
  coord_type = RSPHERICAL
  [gen]
    type = TRISO1DMeshGenerator
    elem_type = EDGE3
    coordinates = '0 2.125e-4'
    mesh_density = '6'
    block_names = 'particle'
  []
[]

[UserObjects]
  [particle_geometry]
    type = TRISOGeometry
    outer_OPyC = '2'
    outer_SiC = '2'
    outer_IPyC = '2'
    inner_IPyC = '2'
    outer_buffer = '2'
    outer_kernel = '2'
    include_particle = true
    include_pebble = false
  []
[]

[Variables]
  [temperature]
    initial_condition = 673.15
  []
[]

[Functions]
  [temp_function]
    type = PiecewiseLinear
    x = '0          3e6'
    y = '673.15 2073.15'
  []
  [Kr85m_exact]
    type = ParsedFunction
    symbol_names =  'expo cont n     B     C     decay    factor'
    symbol_values = '1e-4 1e-6 0.325 -8572 -1.41 4.298e-5 6.4'
    expression = '(expo + cont * factor)*exp(n*log(1/decay)+B/(673.15 + 1400/3e6*t)+C)'
  []
  [Kr87_exact]
    type = ParsedFunction
    symbol_names =  'expo cont n     B     C     decay    factor'
    symbol_values = '1e-4 1e-6 0.325 -8572 -1.41 1.514e-4 6.3'
    expression = '(expo + cont * factor)*exp(n*log(1/decay)+B/(673.15 + 1400/3e6*t)+C)'
  []
  [Kr88_exact]
    type = ParsedFunction
    symbol_names =  'expo cont n     B     C     decay    factor'
    symbol_values = '1e-4 1e-6 0.325 -8572 -1.41 6.816e-5 5.5'
    expression = '(expo + cont * factor)*exp(n*log(1/decay)+B/(673.15 + 1400/3e6*t)+C)'
  []
  [Xe135_exact]
    type = ParsedFunction
    symbol_names =  'expo cont n     B     C     decay    factor'
    symbol_values = '1e-4 1e-6 0.302 -7793 -2.73 2.107e-5 2.3'
    expression = '(expo + cont * factor)*exp(n*log(1/decay)+B/(673.15 + 1400/3e6*t)+C)'
  []
  [Xe137_exact]
    type = ParsedFunction
    symbol_names =  'expo cont n     B     C     decay    factor'
    symbol_values = '1e-4 1e-6 0.302 -7793 -2.73 3.026e-3 5.2'
    expression = '(expo + cont * factor)*exp(n*log(1/decay)+B/(673.15 + 1400/3e6*t)+C)'
  []
  [Xe138_exact]
    type = ParsedFunction
    symbol_names =  'expo cont n     B     C     decay    factor'
    symbol_values = '1e-4 1e-6 0.302 -7793 -2.73 8.170e-4 4.6'
    expression = '(expo + cont * factor)*exp(n*log(1/decay)+B/(673.15 + 1400/3e6*t)+C)'
  []
[]

[Kernels]
  [heat] # gradient term in heat conduction equation
    type = HeatConduction
    variable = temperature
  []

  [heat_ie] # time term in heat conduction equation
    type = HeatConductionTimeDerivative
    variable = temperature
  []
[]

[BCs]
  [heat_removal]
    type = FunctionDirichletBC
    variable = temperature
    boundary = '2'
    function = temp_function
  []
[]

[Materials]

  [thermal]
    type = HeatConductionMaterial
    block = '1'
    thermal_conductivity = 1.0
    specific_heat = 1.0
  []

  [density]
    type = ParsedMaterial
    block = '1'
    property_name = density
    expression = 2570.0
  []

[]

[Executioner]

  type = Transient

  solve_type = 'PJFNK'

  petsc_options_iname = '-pc_type -pc_factor_mat_solver_package -ksp_gmres_restart'
  petsc_options_value = 'lu       superlu_dist                  51'
  line_search = 'none'

  l_max_its = 50
  l_tol = 1e-2

  nl_max_its = 150
  nl_rel_tol = 1e-8

  start_time = 0.0
  end_time = 3e6
  num_steps = 140

  dtmax = 3e6
  dtmin = 1.0

  [TimeStepper]
    type = IterationAdaptiveDT
    dt = 1e5
    time_t = '1e5'
    time_dt = '1e5'
  []
[]

[Postprocessors]

  [temperature]
    type = ElementExtremeValue
    block = '1'
    value_type = 'max'
    variable = temperature
    execute_on = 'initial timestep_end'
  []

  [average_kernel_temperature]
    type = ElementAverageValue
    block = '1'
    variable = temperature
    execute_on = 'initial timestep_end'
  []

  [average_matrix_temperature]
    type = ElementAverageValue
    block = '1'
    variable = temperature
    execute_on = 'initial timestep_end'
  []

  [_dt]
    type = TimestepSize
  []

  [Kr85m_RB_ratio]
    type = ReleaseBirthRatio
    model = AGR
    isotope = Kr85m
    average_kernel_temperature = average_kernel_temperature
    average_matrix_temperature = average_matrix_temperature
    triso_geometry = particle_geometry
    execute_on = 'initial timestep_end'
  []
  [Kr85m_exact]
    type = FunctionValuePostprocessor
    function = Kr85m_exact
    execute_on = 'initial timestep_end'
  []
  [Kr85m_check]
    type = RelativeDifferencePostprocessor
    value1 = Kr85m_RB_ratio
    value2 = Kr85m_exact
  []

  [Kr87_RB_ratio]
    type = ReleaseBirthRatio
    model = AGR
    isotope = Kr87
    average_kernel_temperature = average_kernel_temperature
    average_matrix_temperature = average_matrix_temperature
    triso_geometry = particle_geometry
    execute_on = 'initial timestep_end'
  []
  [Kr87_exact]
    type = FunctionValuePostprocessor
    function = Kr87_exact
    execute_on = 'initial timestep_end'
  []
  [Kr87_check]
    type = RelativeDifferencePostprocessor
    value1 = Kr87_RB_ratio
    value2 = Kr87_exact
  []

  [Kr88_RB_ratio]
    type = ReleaseBirthRatio
    model = AGR
    isotope = Kr88
    average_kernel_temperature = average_kernel_temperature
    average_matrix_temperature = average_matrix_temperature
    triso_geometry = particle_geometry
    execute_on = 'initial timestep_end'
  []
  [Kr88_exact]
    type = FunctionValuePostprocessor
    function = Kr88_exact
    execute_on = 'initial timestep_end'
  []
  [Kr88_check]
    type = RelativeDifferencePostprocessor
    value1 = Kr88_RB_ratio
    value2 = Kr88_exact
  []

  [Xe135_RB_ratio]
    type = ReleaseBirthRatio
    model = AGR
    isotope = Xe135
    average_kernel_temperature = average_kernel_temperature
    average_matrix_temperature = average_matrix_temperature
    triso_geometry = particle_geometry
    execute_on = 'initial timestep_end'
  []
  [Xe135_exact]
    type = FunctionValuePostprocessor
    function = Xe135_exact
    execute_on = 'initial timestep_end'
  []
  [Xe135_check]
    type = RelativeDifferencePostprocessor
    value1 = Xe135_RB_ratio
    value2 = Xe135_exact
  []

  [Xe137_RB_ratio]
    type = ReleaseBirthRatio
    model = AGR
    isotope = Xe137
    average_kernel_temperature = average_kernel_temperature
    average_matrix_temperature = average_matrix_temperature
    triso_geometry = particle_geometry
    execute_on = 'initial timestep_end'
  []
  [Xe137_exact]
    type = FunctionValuePostprocessor
    function = Xe137_exact
    execute_on = 'initial timestep_end'
  []
  [Xe137_check]
    type = RelativeDifferencePostprocessor
    value1 = Xe137_RB_ratio
    value2 = Xe137_exact
  []

  [Xe138_RB_ratio]
    type = ReleaseBirthRatio
    model = AGR
    isotope = Xe138
    average_kernel_temperature = average_kernel_temperature
    average_matrix_temperature = average_matrix_temperature
    triso_geometry = particle_geometry
    execute_on = 'initial timestep_end'
  []
  [Xe138_exact]
    type = FunctionValuePostprocessor
    function = Xe138_exact
    execute_on = 'initial timestep_end'
  []
  [Xe138_check]
    type = RelativeDifferencePostprocessor
    value1 = Xe138_RB_ratio
    value2 = Xe138_exact
  []

[]

[Outputs]
  csv = true
  exodus = false
  [console]
    type = Console
  []
[]

(test/tests/triso/release_birth_ratio/richards.i)

# Release rate to birth rate ratio (R/B) of short-lived Kr and Xe isotopes from exposed kernels.
# The geometry is a sphere with a diameter of 700 micrometers.
# The temperature is ramped from 773 to 2073 K.
# This test uses the Richards model.
#

[GlobalParams]
  U_contamination_fraction = 0.0
  exposed_kernel_fraction = 1.0
  outside_temperature = 1250
  fission_rate = 1.872e18 # fissions/m^3/s
  O_U = 1.5
  C_U = 0.4
[]

[Mesh]
  coord_type = RSPHERICAL
  [gen]
    type = TRISO1DMeshGenerator
    elem_type = EDGE3
    coordinates = '0 350e-6'
    mesh_density = '6'
    block_names = 'particle'
  []
[]

[UserObjects]
  [particle_geometry]
    type = TRISOGeometry
    outer_OPyC = '2'
    outer_SiC = '2'
    outer_IPyC = '2'
    inner_IPyC = '2'
    outer_buffer = '2'
    outer_kernel = '2'
    include_particle = true
    include_pebble = false
  []
[]

[Variables]
  [temperature]
    initial_condition = 773
  []
[]

[Functions]
  [temp_function]
    type = PiecewiseLinear
    x = '0   13'
    y = '773 2073'
  []
[]

[Kernels]
  [heat]
    type = HeatConduction
    variable = temperature
  []

  [heat_ie]
    type = HeatConductionTimeDerivative
    variable = temperature
  []
[]

[BCs]
  [heat_removal]
    type = FunctionDirichletBC
    variable = temperature
    boundary = '2'
    function = temp_function
  []
[]

[Materials]

  [thermal]
    type = HeatConductionMaterial
    block = '1'
    thermal_conductivity = 1.0
    specific_heat = 1.0
  []

  [density]
    type = ParsedMaterial
    block = '1'
    property_name = density
    expression = 11250.0
  []

[]

[Executioner]
  type = Transient

  solve_type = 'PJFNK'

  petsc_options_iname = '-pc_type -pc_factor_mat_solver_package -ksp_gmres_restart'
  petsc_options_value = 'lu       superlu_dist                  51'
  line_search = 'none'

  l_max_its = 50
  l_tol = 1e-2

  nl_max_its = 150
  nl_rel_tol = 1e-8

  start_time = 0
  end_time = 13
  num_steps = 13

  dtmax = 1
  dtmin = 1
[]

[Postprocessors]
  [temperature]
    type = ElementExtremeValue
    block = '1'
    value_type = 'max'
    variable = temperature
    execute_on = 'initial timestep_end'
  []

  [average_kernel_temperature]
    type = ElementAverageValue
    block = '1'
    variable = temperature
    execute_on = 'initial timestep_end'
  []

  [average_matrix_temperature]
    type = ElementAverageValue
    block = '1'
    variable = temperature
    execute_on = 'initial timestep_end'
  []

  [_dt]
    type = TimestepSize
  []

  [Kr83m_RB_ratio]
    type = ReleaseBirthRatio
    model = Richards
    isotope = Kr83m
    average_kernel_temperature = average_kernel_temperature
    average_matrix_temperature = average_matrix_temperature
    mol_weight_gas_medium = 12
    initial_density = 11250.0
    void_fraction = 0.29
    triso_geometry = particle_geometry
    execute_on = 'initial timestep_end'
  []

  [Kr85m_RB_ratio]
    type = ReleaseBirthRatio
    model = Richards
    isotope = Kr85m
    average_kernel_temperature = average_kernel_temperature
    average_matrix_temperature = average_matrix_temperature
    mol_weight_gas_medium = 12
    initial_density = 11250.0
    void_fraction = 0.29
    triso_geometry = particle_geometry
    execute_on = 'initial timestep_end'
  []

  [Kr85_RB_ratio]
    type = ReleaseBirthRatio
    model = Richards
    isotope = Kr85
    average_kernel_temperature = average_kernel_temperature
    average_matrix_temperature = average_matrix_temperature
    mol_weight_gas_medium = 12
    initial_density = 11250.0
    void_fraction = 0.29
    triso_geometry = particle_geometry
    execute_on = 'initial timestep_end'
  []

  [Kr87_RB_ratio]
    type = ReleaseBirthRatio
    model = Richards
    isotope = Kr87
    average_kernel_temperature = average_kernel_temperature
    average_matrix_temperature = average_matrix_temperature
    mol_weight_gas_medium = 12
    initial_density = 11250.0
    void_fraction = 0.29
    triso_geometry = particle_geometry
    execute_on = 'initial timestep_end'
  []

  [Kr88_RB_ratio]
    type = ReleaseBirthRatio
    model = Richards
    isotope = Kr88
    average_kernel_temperature = average_kernel_temperature
    average_matrix_temperature = average_matrix_temperature
    mol_weight_gas_medium = 12
    initial_density = 11250.0
    void_fraction = 0.29
    triso_geometry = particle_geometry
    execute_on = 'initial timestep_end'
  []

  [Kr89_RB_ratio]
    type = ReleaseBirthRatio
    model = Richards
    isotope = Kr89
    average_kernel_temperature = average_kernel_temperature
    average_matrix_temperature = average_matrix_temperature
    mol_weight_gas_medium = 12
    initial_density = 11250.0
    void_fraction = 0.29
    triso_geometry = particle_geometry
    execute_on = 'initial timestep_end'
  []

  [Kr90_RB_ratio]
    type = ReleaseBirthRatio
    model = Richards
    isotope = Kr90
    average_kernel_temperature = average_kernel_temperature
    average_matrix_temperature = average_matrix_temperature
    mol_weight_gas_medium = 12
    initial_density = 11250.0
    void_fraction = 0.29
    triso_geometry = particle_geometry
    execute_on = 'initial timestep_end'
  []

  [Xe131m_RB_ratio]
    type = ReleaseBirthRatio
    model = Richards
    isotope = Xe131m
    average_kernel_temperature = average_kernel_temperature
    average_matrix_temperature = average_matrix_temperature
    mol_weight_gas_medium = 12
    initial_density = 11250.0
    void_fraction = 0.29
    triso_geometry = particle_geometry
    execute_on = 'initial timestep_end'
  []

  [Xe133_RB_ratio]
    type = ReleaseBirthRatio
    model = Richards
    isotope = Xe133
    average_kernel_temperature = average_kernel_temperature
    average_matrix_temperature = average_matrix_temperature
    mol_weight_gas_medium = 12
    initial_density = 11250.0
    void_fraction = 0.29
    triso_geometry = particle_geometry
    execute_on = 'initial timestep_end'
  []

  [Xe135_RB_ratio]
    type = ReleaseBirthRatio
    model = Richards
    isotope = Xe135
    average_kernel_temperature = average_kernel_temperature
    average_matrix_temperature = average_matrix_temperature
    mol_weight_gas_medium = 12
    initial_density = 11250.0
    void_fraction = 0.29
    triso_geometry = particle_geometry
    execute_on = 'initial timestep_end'
  []

  [Xe135m_RB_ratio]
    type = ReleaseBirthRatio
    model = Richards
    isotope = Xe135m
    average_kernel_temperature = average_kernel_temperature
    average_matrix_temperature = average_matrix_temperature
    mol_weight_gas_medium = 12
    initial_density = 11250.0
    void_fraction = 0.29
    triso_geometry = particle_geometry
    execute_on = 'initial timestep_end'
  []

  [Xe137_RB_ratio]
    type = ReleaseBirthRatio
    model = Richards
    isotope = Xe137
    average_kernel_temperature = average_kernel_temperature
    average_matrix_temperature = average_matrix_temperature
    mol_weight_gas_medium = 12
    initial_density = 11250.0
    void_fraction = 0.29
    triso_geometry = particle_geometry
    execute_on = 'initial timestep_end'
  []

  [Xe138_RB_ratio]
    type = ReleaseBirthRatio
    model = Richards
    isotope = Xe138
    average_kernel_temperature = average_kernel_temperature
    average_matrix_temperature = average_matrix_temperature
    mol_weight_gas_medium = 12
    initial_density = 11250.0
    void_fraction = 0.29
    triso_geometry = particle_geometry
    execute_on = 'initial timestep_end'
  []

  [Xe139_RB_ratio]
    type = ReleaseBirthRatio
    model = Richards
    isotope = Xe139
    average_kernel_temperature = average_kernel_temperature
    average_matrix_temperature = average_matrix_temperature
    mol_weight_gas_medium = 12
    initial_density = 11250.0
    void_fraction = 0.29
    triso_geometry = particle_geometry
    execute_on = 'initial timestep_end'
  []
[]

[Outputs]
  csv = true
  exodus = false
  [console]
    type = Console
  []
[]

(test/tests/triso/release_birth_ratio/exposed_kernel.i)

# Release rate to birth rate ratio (R/B) of short-lived Kr and Xe isotopes from exposed kernels.
# The geometry is a sphere with a diameter of 425 micrometers.
# The temperature is ramped from 673.15 to 2073.15 K.
#
# Bison results are checked against results computed in the *_exact Functions.
#

[GlobalParams]
  U_contamination_fraction = 0.0
  exposed_kernel_fraction = 1e-4
[]

[Mesh]
  coord_type = RSPHERICAL
  [gen]
    type = TRISO1DMeshGenerator
    elem_type = EDGE3
    coordinates = '0 2.125e-4'
    mesh_density = '6'
    block_names = 'particle'
  []
[]

[UserObjects]
  [particle_geometry]
    type = TRISOGeometry
    outer_OPyC = '2'
    outer_SiC = '2'
    outer_IPyC = '2'
    inner_IPyC = '2'
    outer_buffer = '2'
    outer_kernel = '2'
    include_particle = true
    include_pebble = false
  []
[]

[Variables]
  [temperature]
    initial_condition = 673.15
  []
[]

[Functions]
  [temp_function]
    type = PiecewiseLinear
    x = '0          3e6'
    y = '673.15 2073.15'
  []
  [D_1_Kr]
    type = ParsedFunction
    symbol_names = 'R      temp'
    symbol_values = '8.3145 temp_function'
    expression = '3e-5*exp(-106e3/(R*temp))'
  []
  [X_1_Kr85m]
    type = ParsedFunction
    symbol_names = 'decay    D_1'
    symbol_values = '4.298e-5 D_1_Kr'
    expression = 'sqrt(decay / D_1)'
  []
  [D_2]
    type = ParsedFunction
    symbol_names = 'radius     temp'
    symbol_values = '2.125e-4 temp_function'
    expression = 'pow(500*1e-6/radius/2,2)*pow(10,if(temp<1173,-7.97-1920/temp,-2.6-8220/temp))'
  []
  [X_2_Kr85m]
    type = ParsedFunction
    symbol_names = 'decay    D_2'
    symbol_values = '4.298e-5 D_2'
    expression = 'sqrt(decay/D_2)'
  []
  [Kr85m_exact]
    type = ParsedFunction
    symbol_names = 'expo X_2'
    symbol_values = '1e-4 X_2_Kr85m'
    expression = 'expo*3/X_2*(1/tanh(X_2)-1/X_2)'
  []
  [X_2_Kr87]
    type = ParsedFunction
    symbol_names = 'decay    D_2'
    symbol_values = '1.514e-4 D_2'
    expression = 'sqrt(decay/D_2)'
  []
  [Kr87_exact]
    type = ParsedFunction
    symbol_names = 'expo X_2'
    symbol_values = '1e-4 X_2_Kr87'
    expression = 'expo*3/X_2*(1/tanh(X_2)-1/X_2)'
  []
  [X_2_Kr88]
    type = ParsedFunction
    symbol_names = 'decay    D_2'
    symbol_values = '6.816e-5 D_2'
    expression = 'sqrt(decay/D_2)'
  []
  [Kr88_exact]
    type = ParsedFunction
    symbol_names = 'expo X_2'
    symbol_values = '1e-4 X_2_Kr88'
    expression = 'expo*3/X_2*(1/tanh(X_2)-1/X_2)'
  []
  [X_2_Kr89]
    type = ParsedFunction
    symbol_names = 'decay    D_2'
    symbol_values = '3.667e-3 D_2'
    expression = 'sqrt(decay/D_2)'
  []
  [Kr89_exact]
    type = ParsedFunction
    symbol_names = 'expo X_2'
    symbol_values = '1e-4 X_2_Kr89'
    expression = 'expo*3/X_2*(1/tanh(X_2)-1/X_2)'
  []
  [X_2_Kr90]
    type = ParsedFunction
    symbol_names = 'decay    D_2'
    symbol_values = '2.145e-2 D_2'
    expression = 'sqrt(decay/D_2)'
  []
  [Kr90_exact]
    type = ParsedFunction
    symbol_names = 'expo X_2'
    symbol_values = '1e-4 X_2_Kr90'
    expression = 'expo*3/X_2*(1/tanh(X_2)-1/X_2)'
  []
  [X_2_Xe131m]
    type = ParsedFunction
    symbol_names = 'decay    D_2'
    symbol_values = '6.776e-7 D_2'
    expression = 'sqrt(decay/D_2)'
  []
  [Xe131m_exact]
    type = ParsedFunction
    symbol_names = 'expo X_2'
    symbol_values = '1e-4 X_2_Xe131m'
    expression = 'expo*3/X_2*(1/tanh(X_2)-1/X_2)'
  []
  [X_2_Xe133]
    type = ParsedFunction
    symbol_names = 'decay    D_2'
    symbol_values = '1.529e-6 D_2'
    expression = 'sqrt(decay/D_2)'
  []
  [Xe133_exact]
    type = ParsedFunction
    symbol_names = 'expo X_2'
    symbol_values = '1e-4 X_2_Xe133'
    expression = 'expo*3/X_2*(1/tanh(X_2)-1/X_2)'
  []
  [X_2_Xe135]
    type = ParsedFunction
    symbol_names = 'decay    D_2'
    symbol_values = '2.107e-5 D_2'
    expression = 'sqrt(decay/D_2)'
  []
  [Xe135_exact]
    type = ParsedFunction
    symbol_names = 'expo X_2'
    symbol_values = '1e-4 X_2_Xe135'
    expression = 'expo*3/X_2*(1/tanh(X_2)-1/X_2)'
  []
  [X_2_Xe135m]
    type = ParsedFunction
    symbol_names = 'decay    D_2'
    symbol_values = '7.556e-4 D_2'
    expression = 'sqrt(decay/D_2)'
  []
  [Xe135m_exact]
    type = ParsedFunction
    symbol_names = 'expo X_2'
    symbol_values = '1e-4 X_2_Xe135m'
    expression = 'expo*3/X_2*(1/tanh(X_2)-1/X_2)'
  []
  [X_2_Xe137]
    type = ParsedFunction
    symbol_names = 'decay    D_2'
    symbol_values = '3.026e-3 D_2'
    expression = 'sqrt(decay/D_2)'
  []
  [Xe137_exact]
    type = ParsedFunction
    symbol_names = 'expo X_2'
    symbol_values = '1e-4 X_2_Xe137'
    expression = 'expo*3/X_2*(1/tanh(X_2)-1/X_2)'
  []
  [X_2_Xe138]
    type = ParsedFunction
    symbol_names = 'decay    D_2'
    symbol_values = '8.170e-4 D_2'
    expression = 'sqrt(decay/D_2)'
  []
  [Xe138_exact]
    type = ParsedFunction
    symbol_names = 'expo X_2'
    symbol_values = '1e-4 X_2_Xe138'
    expression = 'expo*3/X_2*(1/tanh(X_2)-1/X_2)'
  []
  [X_2_Xe139]
    type = ParsedFunction
    symbol_names = 'decay    D_2'
    symbol_values = '1.747e-2 D_2'
    expression = 'sqrt(decay/D_2)'
  []
  [Xe139_exact]
    type = ParsedFunction
    symbol_names = 'expo X_2'
    symbol_values = '1e-4 X_2_Xe139'
    expression = 'expo*3/X_2*(1/tanh(X_2)-1/X_2)'
  []
  # [Kr85m_exact]
  #   type = ParsedFunction
  #   symbol_names = 'cont X_1'
  #   symbol_values = '1e-6 X_1_Kr85m'
  #   expression = 'cont*3/X_1*(1/tanh(X_1)-1/X_1)'
  # []
[]

[Kernels]
  [heat] # gradient term in heat conduction equation
    type = HeatConduction
    variable = temperature
  []

  [heat_ie] # time term in heat conduction equation
    type = HeatConductionTimeDerivative
    variable = temperature
  []
[]

[BCs]
  [heat_removal]
    type = FunctionDirichletBC
    variable = temperature
    boundary = '2'
    function = temp_function
  []
[]

[Materials]

  [thermal]
    type = HeatConductionMaterial
    block = '1'
    thermal_conductivity = 1.0
    specific_heat = 1.0
  []

  [density]
    type = ParsedMaterial
    block = '1'
    property_name = density
    expression = 2570.0
  []

[]

[Executioner]

  type = Transient

  solve_type = 'PJFNK'

  petsc_options_iname = '-pc_type -pc_factor_mat_solver_package -ksp_gmres_restart'
  petsc_options_value = 'lu       superlu_dist                  51'
  line_search = 'none'

  l_max_its = 50
  l_tol = 1e-2

  nl_max_its = 150
  nl_rel_tol = 1e-8

  start_time = 0.0
  end_time = 3e6
  num_steps = 140

  dtmax = 3e6
  dtmin = 1.0

  [TimeStepper]
    type = IterationAdaptiveDT
    dt = 1e5
    time_t = '1e5'
    time_dt = '1e5'
  []
[]

[Postprocessors]

  [temperature]
    type = ElementExtremeValue
    block = '1'
    value_type = 'max'
    variable = temperature
    execute_on = 'initial timestep_end'
  []

  [average_kernel_temperature]
    type = ElementAverageValue
    block = '1'
    variable = temperature
    execute_on = 'initial timestep_end'
  []

  [average_matrix_temperature]
    type = ElementAverageValue
    block = '1'
    variable = temperature
    execute_on = 'initial timestep_end'
  []

  [_dt]
    type = TimestepSize
  []

  [Kr85m_RB_ratio]
    type = ReleaseBirthRatio
    isotope = Kr85m
    average_kernel_temperature = average_kernel_temperature
    average_matrix_temperature = average_matrix_temperature
    triso_geometry = particle_geometry
    execute_on = 'initial timestep_end'
  []
  [Kr85m_exact]
    type = FunctionValuePostprocessor
    function = Kr85m_exact
    execute_on = 'initial timestep_end'
  []
  [Kr85m_check]
    type = RelativeDifferencePostprocessor
    value1 = Kr85m_RB_ratio
    value2 = Kr85m_exact
  []

  [Kr87_RB_ratio]
    type = ReleaseBirthRatio
    isotope = Kr87
    average_kernel_temperature = average_kernel_temperature
    average_matrix_temperature = average_matrix_temperature
    triso_geometry = particle_geometry
    execute_on = 'initial timestep_end'
  []
  [Kr87_exact]
    type = FunctionValuePostprocessor
    function = Kr87_exact
    execute_on = 'initial timestep_end'
  []
  [Kr87_check]
    type = RelativeDifferencePostprocessor
    value1 = Kr87_RB_ratio
    value2 = Kr87_exact
  []

  [Kr88_RB_ratio]
    type = ReleaseBirthRatio
    isotope = Kr88
    average_kernel_temperature = average_kernel_temperature
    average_matrix_temperature = average_matrix_temperature
    triso_geometry = particle_geometry
    execute_on = 'initial timestep_end'
  []
  [Kr88_exact]
    type = FunctionValuePostprocessor
    function = Kr88_exact
    execute_on = 'initial timestep_end'
  []
  [Kr88_check]
    type = RelativeDifferencePostprocessor
    value1 = Kr88_RB_ratio
    value2 = Kr88_exact
  []

  [Kr89_RB_ratio]
    type = ReleaseBirthRatio
    isotope = Kr89
    average_kernel_temperature = average_kernel_temperature
    average_matrix_temperature = average_matrix_temperature
    triso_geometry = particle_geometry
    execute_on = 'initial timestep_end'
  []
  [Kr89_exact]
    type = FunctionValuePostprocessor
    function = Kr89_exact
    execute_on = 'initial timestep_end'
  []
  [Kr89_check]
    type = RelativeDifferencePostprocessor
    value1 = Kr89_RB_ratio
    value2 = Kr89_exact
  []

  [Kr90_RB_ratio]
    type = ReleaseBirthRatio
    isotope = Kr90
    average_kernel_temperature = average_kernel_temperature
    average_matrix_temperature = average_matrix_temperature
    triso_geometry = particle_geometry
    execute_on = 'initial timestep_end'
  []
  [Kr90_exact]
    type = FunctionValuePostprocessor
    function = Kr90_exact
    execute_on = 'initial timestep_end'
  []
  [Kr90_check]
    type = RelativeDifferencePostprocessor
    value1 = Kr90_RB_ratio
    value2 = Kr90_exact
  []

  [Xe131m_RB_ratio]
    type = ReleaseBirthRatio
    isotope = Xe131m
    average_kernel_temperature = average_kernel_temperature
    average_matrix_temperature = average_matrix_temperature
    triso_geometry = particle_geometry
    execute_on = 'initial timestep_end'
  []
  [Xe131m_exact]
    type = FunctionValuePostprocessor
    function = Xe131m_exact
    execute_on = 'initial timestep_end'
  []
  [Xe131m_check]
    type = RelativeDifferencePostprocessor
    value1 = Xe131m_RB_ratio
    value2 = Xe131m_exact
  []

  [Xe133_RB_ratio]
    type = ReleaseBirthRatio
    isotope = Xe133
    average_kernel_temperature = average_kernel_temperature
    average_matrix_temperature = average_matrix_temperature
    triso_geometry = particle_geometry
    execute_on = 'initial timestep_end'
  []
  [Xe133_exact]
    type = FunctionValuePostprocessor
    function = Xe133_exact
    execute_on = 'initial timestep_end'
  []
  [Xe133_check]
    type = RelativeDifferencePostprocessor
    value1 = Xe133_RB_ratio
    value2 = Xe133_exact
  []

  [Xe135_RB_ratio]
    type = ReleaseBirthRatio
    isotope = Xe135
    average_kernel_temperature = average_kernel_temperature
    average_matrix_temperature = average_matrix_temperature
    triso_geometry = particle_geometry
    execute_on = 'initial timestep_end'
  []
  [Xe135_exact]
    type = FunctionValuePostprocessor
    function = Xe135_exact
    execute_on = 'initial timestep_end'
  []
  [Xe135_check]
    type = RelativeDifferencePostprocessor
    value1 = Xe135_RB_ratio
    value2 = Xe135_exact
  []

  [Xe135m_RB_ratio]
    type = ReleaseBirthRatio
    isotope = Xe135m
    average_kernel_temperature = average_kernel_temperature
    average_matrix_temperature = average_matrix_temperature
    triso_geometry = particle_geometry
    execute_on = 'initial timestep_end'
  []
  [Xe135m_exact]
    type = FunctionValuePostprocessor
    function = Xe135m_exact
    execute_on = 'initial timestep_end'
  []
  [Xe135m_check]
    type = RelativeDifferencePostprocessor
    value1 = Xe135m_RB_ratio
    value2 = Xe135m_exact
  []

  [Xe137_RB_ratio]
    type = ReleaseBirthRatio
    isotope = Xe137
    average_kernel_temperature = average_kernel_temperature
    average_matrix_temperature = average_matrix_temperature
    triso_geometry = particle_geometry
    execute_on = 'initial timestep_end'
  []
  [Xe137_exact]
    type = FunctionValuePostprocessor
    function = Xe137_exact
    execute_on = 'initial timestep_end'
  []
  [Xe137_check]
    type = RelativeDifferencePostprocessor
    value1 = Xe137_RB_ratio
    value2 = Xe137_exact
  []

  [Xe138_RB_ratio]
    type = ReleaseBirthRatio
    isotope = Xe138
    average_kernel_temperature = average_kernel_temperature
    average_matrix_temperature = average_matrix_temperature
    triso_geometry = particle_geometry
    execute_on = 'initial timestep_end'
  []
  [Xe138_exact]
    type = FunctionValuePostprocessor
    function = Xe138_exact
    execute_on = 'initial timestep_end'
  []
  [Xe138_check]
    type = RelativeDifferencePostprocessor
    value1 = Xe138_RB_ratio
    value2 = Xe138_exact
  []

  [Xe139_RB_ratio]
    type = ReleaseBirthRatio
    isotope = Xe139
    average_kernel_temperature = average_kernel_temperature
    average_matrix_temperature = average_matrix_temperature
    triso_geometry = particle_geometry
    execute_on = 'initial timestep_end'
  []
  [Xe139_exact]
    type = FunctionValuePostprocessor
    function = Xe139_exact
    execute_on = 'initial timestep_end'
  []
  [Xe139_check]
    type = RelativeDifferencePostprocessor
    value1 = Xe139_RB_ratio
    value2 = Xe139_exact
  []
[]

[Outputs]
  csv = true
  exodus = false
  [console]
    type = Console
  []
[]

(test/tests/triso/release_birth_ratio/uranium_contamination.i)

# Release rate to birth rate ratio (R/B) of short-lived Kr and Xe isotopes from uranium contamination.
# The geometry is a sphere with a diameter of 425 micrometers.
# The temperature is ramped from 673.15 to 2073.15 K.
#
# Bison results are checked against results computed in the *_exact Functions.
#

[GlobalParams]
  U_contamination_fraction = 3e-7
  exposed_kernel_fraction = 0.0
[]

[Mesh]
  coord_type = RSPHERICAL
  [gen]
    type = TRISO1DMeshGenerator
    elem_type = EDGE3
    coordinates = '0 2.125e-4'
    mesh_density = '6'
    block_names = 'particle'
  []
[]

[UserObjects]
  [particle_geometry]
    type = TRISOGeometry
    outer_OPyC = '2'
    outer_SiC = '2'
    outer_IPyC = '2'
    inner_IPyC = '2'
    outer_buffer = '2'
    outer_kernel = '2'
    include_particle = true
    include_pebble = false
  []
[]

[Variables]
  [temperature]
    initial_condition = 673.15
  []
[]

[Functions]
  [temp_function]
    type = PiecewiseLinear
    x = '0          3e6'
    y = '673.15 2073.15'
  []
  [D_1_Kr]
    type = ParsedFunction
    symbol_names = 'R      temp'
    symbol_values = '8.31446261815324 temp_function'
    expression = '3e-5*exp(-106e3/(R*temp))'
  []
  [D_1_Xe]
    type = ParsedFunction
    symbol_names = 'R      temp'
    symbol_values = '8.31446261815324 temp_function'
    expression = '1.7e-7*exp(-78.6e3/(R*temp))'
  []
  [X_1_Kr85m]
    type = ParsedFunction
    symbol_names = 'decay    D_1'
    symbol_values = '4.298e-5 D_1_Kr'
    expression = 'sqrt(decay/D_1)'
  []
  [Kr85m_exact]
    type = ParsedFunction
    symbol_names = 'cont X_1'
    symbol_values = '3e-7 X_1_Kr85m'
    expression = 'cont*3/X_1*(1/tanh(X_1)-1/X_1)'
  []
  [X_1_Kr87]
    type = ParsedFunction
    symbol_names = 'decay    D_1'
    symbol_values = '1.514e-4 D_1_Kr'
    expression = 'sqrt(decay/D_1)'
  []
  [Kr87_exact]
    type = ParsedFunction
    symbol_names = 'cont X_1'
    symbol_values = '3e-7 X_1_Kr87'
    expression = 'cont*3/X_1*(1/tanh(X_1)-1/X_1)'
  []
  [X_1_Kr88]
    type = ParsedFunction
    symbol_names = 'decay    D_1'
    symbol_values = '6.816e-5 D_1_Kr'
    expression = 'sqrt(decay/D_1)'
  []
  [Kr88_exact]
    type = ParsedFunction
    symbol_names = 'cont X_1'
    symbol_values = '3e-7 X_1_Kr88'
    expression = 'cont*3/X_1*(1/tanh(X_1)-1/X_1)'
  []
  [X_1_Kr89]
    type = ParsedFunction
    symbol_names = 'decay    D_1'
    symbol_values = '3.667e-3 D_1_Kr'
    expression = 'sqrt(decay/D_1)'
  []
  [Kr89_exact]
    type = ParsedFunction
    symbol_names = 'cont X_1'
    symbol_values = '3e-7 X_1_Kr89'
    expression = 'cont*3/X_1*(1/tanh(X_1)-1/X_1)'
  []
  [X_1_Kr90]
    type = ParsedFunction
    symbol_names = 'decay    D_1'
    symbol_values = '2.145e-2 D_1_Kr'
    expression = 'sqrt(decay/D_1)'
  []
  [Kr90_exact]
    type = ParsedFunction
    symbol_names = 'cont X_1'
    symbol_values = '3e-7 X_1_Kr90'
    expression = 'cont*3/X_1*(1/tanh(X_1)-1/X_1)'
  []
  [X_1_Xe131m]
    type = ParsedFunction
    symbol_names = 'decay    D_1'
    symbol_values = '6.776e-7 D_1_Xe'
    expression = 'sqrt(decay/D_1)'
  []
  [Xe131m_exact]
    type = ParsedFunction
    symbol_names = 'cont X_1'
    symbol_values = '3e-7 X_1_Xe131m'
    expression = 'cont*3/X_1*(1/tanh(X_1)-1/X_1)'
  []
  [X_1_Xe133]
    type = ParsedFunction
    symbol_names = 'decay    D_1'
    symbol_values = '1.529e-6 D_1_Xe'
    expression = 'sqrt(decay/D_1)'
  []
  [Xe133_exact]
    type = ParsedFunction
    symbol_names = 'cont X_1'
    symbol_values = '3e-7 X_1_Xe133'
    expression = 'cont*3/X_1*(1/tanh(X_1)-1/X_1)'
  []
  [X_1_Xe135]
    type = ParsedFunction
    symbol_names = 'decay    D_1'
    symbol_values = '2.107e-5 D_1_Xe'
    expression = 'sqrt(decay/D_1)'
  []
  [Xe135_exact]
    type = ParsedFunction
    symbol_names = 'cont X_1'
    symbol_values = '3e-7 X_1_Xe135'
    expression = 'cont*3/X_1*(1/tanh(X_1)-1/X_1)'
  []
  [X_1_Xe135m]
    type = ParsedFunction
    symbol_names = 'decay    D_1'
    symbol_values = '7.556e-4 D_1_Xe'
    expression = 'sqrt(decay/D_1)'
  []
  [Xe135m_exact]
    type = ParsedFunction
    symbol_names = 'cont X_1'
    symbol_values = '3e-7 X_1_Xe135m'
    expression = 'cont*3/X_1*(1/tanh(X_1)-1/X_1)'
  []
  [X_1_Xe137]
    type = ParsedFunction
    symbol_names = 'decay    D_1'
    symbol_values = '3.026e-3 D_1_Xe'
    expression = 'sqrt(decay/D_1)'
  []
  [Xe137_exact]
    type = ParsedFunction
    symbol_names = 'cont X_1'
    symbol_values = '3e-7 X_1_Xe137'
    expression = 'cont*3/X_1*(1/tanh(X_1)-1/X_1)'
  []
  [X_1_Xe138]
    type = ParsedFunction
    symbol_names = 'decay    D_1'
    symbol_values = '8.170e-4 D_1_Xe'
    expression = 'sqrt(decay/D_1)'
  []
  [Xe138_exact]
    type = ParsedFunction
    symbol_names = 'cont X_1'
    symbol_values = '3e-7 X_1_Xe138'
    expression = 'cont*3/X_1*(1/tanh(X_1)-1/X_1)'
  []
  [X_1_Xe139]
    type = ParsedFunction
    symbol_names = 'decay    D_1'
    symbol_values = '1.747e-2 D_1_Xe'
    expression = 'sqrt(decay/D_1)'
  []
  [Xe139_exact]
    type = ParsedFunction
    symbol_names = 'cont X_1'
    symbol_values = '3e-7 X_1_Xe139'
    expression = 'cont*3/X_1*(1/tanh(X_1)-1/X_1)'
  []
[]

[Kernels]
  [heat] # gradient term in heat conduction equation
    type = HeatConduction
    variable = temperature
  []

  [heat_ie] # time term in heat conduction equation
    type = HeatConductionTimeDerivative
    variable = temperature
  []
[]

[BCs]
  [heat_removal]
    type = FunctionDirichletBC
    variable = temperature
    boundary = '2'
    function = temp_function
  []
[]

[Materials]

  [thermal]
    type = HeatConductionMaterial
    block = '1'
    thermal_conductivity = 1.0
    specific_heat = 1.0
  []

  [density]
    type = ParsedMaterial
    block = '1'
    property_name = density
    expression = 2570.0
  []

[]

[Executioner]

  type = Transient

  solve_type = 'PJFNK'

  petsc_options_iname = '-pc_type -pc_factor_mat_solver_package -ksp_gmres_restart'
  petsc_options_value = 'lu       superlu_dist                  51'
  line_search = 'none'

  l_max_its = 50
  l_tol = 1e-2

  nl_max_its = 150
  nl_rel_tol = 1e-8

  start_time = 0.0
  end_time = 3e6
  num_steps = 140

  dtmax = 3e6
  dtmin = 1.0

  [TimeStepper]
    type = IterationAdaptiveDT
    dt = 1e5
    time_t = '1e5'
    time_dt = '1e5'
  []
[]

[Postprocessors]

  [temperature]
    type = ElementExtremeValue
    block = '1'
    value_type = 'max'
    variable = temperature
    execute_on = 'initial timestep_end'
  []

  [average_kernel_temperature]
    type = ElementAverageValue
    block = '1'
    variable = temperature
    execute_on = 'initial timestep_end'
  []

  [average_matrix_temperature]
    type = ElementAverageValue
    block = '1'
    variable = temperature
    execute_on = 'initial timestep_end'
  []

  [_dt]
    type = TimestepSize
  []

  [Kr85m_RB_ratio]
    type = ReleaseBirthRatio
    isotope = Kr85m
    average_kernel_temperature = average_kernel_temperature
    average_matrix_temperature = average_matrix_temperature
    triso_geometry = particle_geometry
    execute_on = 'initial timestep_end'
  []
  [Kr85m_exact]
    type = FunctionValuePostprocessor
    function = Kr85m_exact
  []
  [Kr85m_check]
    type = RelativeDifferencePostprocessor
    value1 = Kr85m_RB_ratio
    value2 = Kr85m_exact
  []

  [Kr87_RB_ratio]
    type = ReleaseBirthRatio
    isotope = Kr87
    average_kernel_temperature = average_kernel_temperature
    average_matrix_temperature = average_matrix_temperature
    triso_geometry = particle_geometry
    execute_on = 'initial timestep_end'
  []
  [Kr87_exact]
    type = FunctionValuePostprocessor
    function = Kr87_exact
  []
  [Kr87_check]
    type = RelativeDifferencePostprocessor
    value1 = Kr87_RB_ratio
    value2 = Kr87_exact
  []

  [Kr88_RB_ratio]
    type = ReleaseBirthRatio
    isotope = Kr88
    average_kernel_temperature = average_kernel_temperature
    average_matrix_temperature = average_matrix_temperature
    triso_geometry = particle_geometry
    execute_on = 'initial timestep_end'
  []
  [Kr88_exact]
    type = FunctionValuePostprocessor
    function = Kr88_exact
  []
  [Kr88_check]
    type = RelativeDifferencePostprocessor
    value1 = Kr88_RB_ratio
    value2 = Kr88_exact
  []

  [Kr89_RB_ratio]
    type = ReleaseBirthRatio
    isotope = Kr89
    average_kernel_temperature = average_kernel_temperature
    average_matrix_temperature = average_matrix_temperature
    triso_geometry = particle_geometry
    execute_on = 'initial timestep_end'
  []
  [Kr89_exact]
    type = FunctionValuePostprocessor
    function = Kr89_exact
  []
  [Kr89_check]
    type = RelativeDifferencePostprocessor
    value1 = Kr89_RB_ratio
    value2 = Kr89_exact
  []

  [Kr90_RB_ratio]
    type = ReleaseBirthRatio
    isotope = Kr90
    average_kernel_temperature = average_kernel_temperature
    average_matrix_temperature = average_matrix_temperature
    triso_geometry = particle_geometry
    execute_on = 'initial timestep_end'
  []
  [Kr90_exact]
    type = FunctionValuePostprocessor
    function = Kr90_exact
  []
  [Kr90_check]
    type = RelativeDifferencePostprocessor
    value1 = Kr90_RB_ratio
    value2 = Kr90_exact
  []

  [Xe131m_RB_ratio]
    type = ReleaseBirthRatio
    isotope = Xe131m
    average_kernel_temperature = average_kernel_temperature
    average_matrix_temperature = average_matrix_temperature
    triso_geometry = particle_geometry
    execute_on = 'initial timestep_end'
  []
  [Xe131m_exact]
    type = FunctionValuePostprocessor
    function = Xe131m_exact
  []
  [Xe131m_check]
    type = RelativeDifferencePostprocessor
    value1 = Xe131m_RB_ratio
    value2 = Xe131m_exact
  []

  [Xe133_RB_ratio]
    type = ReleaseBirthRatio
    isotope = Xe133
    average_kernel_temperature = average_kernel_temperature
    average_matrix_temperature = average_matrix_temperature
    triso_geometry = particle_geometry
    execute_on = 'initial timestep_end'
  []
  [Xe133_exact]
    type = FunctionValuePostprocessor
    function = Xe133_exact
  []
  [Xe133_check]
    type = RelativeDifferencePostprocessor
    value1 = Xe133_RB_ratio
    value2 = Xe133_exact
  []

  [Xe135_RB_ratio]
    type = ReleaseBirthRatio
    isotope = Xe135
    average_kernel_temperature = average_kernel_temperature
    average_matrix_temperature = average_matrix_temperature
    triso_geometry = particle_geometry
    execute_on = 'initial timestep_end'
  []
  [Xe135_exact]
    type = FunctionValuePostprocessor
    function = Xe135_exact
  []
  [Xe135_check]
    type = RelativeDifferencePostprocessor
    value1 = Xe135_RB_ratio
    value2 = Xe135_exact
  []

  [Xe135m_RB_ratio]
    type = ReleaseBirthRatio
    isotope = Xe135m
    average_kernel_temperature = average_kernel_temperature
    average_matrix_temperature = average_matrix_temperature
    triso_geometry = particle_geometry
    execute_on = 'initial timestep_end'
  []
  [Xe135m_exact]
    type = FunctionValuePostprocessor
    function = Xe135m_exact
  []
  [Xe135m_check]
    type = RelativeDifferencePostprocessor
    value1 = Xe135m_RB_ratio
    value2 = Xe135m_exact
  []

  [Xe137_RB_ratio]
    type = ReleaseBirthRatio
    isotope = Xe137
    average_kernel_temperature = average_kernel_temperature
    average_matrix_temperature = average_matrix_temperature
    triso_geometry = particle_geometry
    execute_on = 'initial timestep_end'
  []
  [Xe137_exact]
    type = FunctionValuePostprocessor
    function = Xe137_exact
  []
  [Xe137_check]
    type = RelativeDifferencePostprocessor
    value1 = Xe137_RB_ratio
    value2 = Xe137_exact
  []

  [Xe138_RB_ratio]
    type = ReleaseBirthRatio
    isotope = Xe138
    average_kernel_temperature = average_kernel_temperature
    average_matrix_temperature = average_matrix_temperature
    triso_geometry = particle_geometry
    execute_on = 'initial timestep_end'
  []
  [Xe138_exact]
    type = FunctionValuePostprocessor
    function = Xe138_exact
  []
  [Xe138_check]
    type = RelativeDifferencePostprocessor
    value1 = Xe138_RB_ratio
    value2 = Xe138_exact
  []

  [Xe139_RB_ratio]
    type = ReleaseBirthRatio
    isotope = Xe139
    average_kernel_temperature = average_kernel_temperature
    average_matrix_temperature = average_matrix_temperature
    triso_geometry = particle_geometry
    execute_on = 'initial timestep_end'
  []
  [Xe139_exact]
    type = FunctionValuePostprocessor
    function = Xe139_exact
  []
  [Xe139_check]
    type = RelativeDifferencePostprocessor
    value1 = Xe139_RB_ratio
    value2 = Xe139_exact
  []
[]

[Outputs]
  csv = true
  exodus = false
  [console]
    type = Console
  []
[]

Description
Validity Range
Limits
Release / Birth Ratios
Example Input Syntax
Input Parameters
Input Files
References