Difference between revisions of "SuperFINIX theory manual"
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== Field transfers from SuperFINIX meshes to FINIX meshes == | == Field transfers from SuperFINIX meshes to FINIX meshes == | ||
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=== Power from SuperFINIX to FINIX === | === Power from SuperFINIX to FINIX === | ||
− | SuperFINIX receives power densities in blocks that have to be discretized in the radial nodes of FINIX. This discretization is based on assumption that the number of axial elements remains unchanged before and after the discretization. Therefore the discretization is only performed in the radial dimension. The main solvers in FINIX assume linear scaling of power density (and most other variables) between two nodal points. Because of this assumption, the power densities in nodes have to adjusted when integrating over a volume element that is spanned by two radial nodes in different power density blocks. This conserves the total power over the volumes of single elements as well as the whole fuel rod volume. As such, the following condition has to hold true: | + | SuperFINIX receives power densities in blocks that have to be discretized in the radial nodes of FINIX. This discretization is based on assumption that the number of axial elements remains unchanged before and after the discretization. Therefore the discretization is only performed in the radial dimension. The main solvers in FINIX assume linear scaling of power density (and most other variables) between two nodal points. Because of this assumption, the power densities in nodes have to adjusted when integrating over a volume element that is spanned by two radial nodes in different power density blocks. This conserves the total power over the volumes of single elements as well as the whole fuel rod volume. As such, the following condition has to hold true for the scenario of <xr id="fig:SuperFINIXDiscretizationGraph_3"/>: |
+ | |||
+ | <figure id="fig:SuperFINIXDiscretizationGraph_3"> | ||
+ | [[File:SuperFINIXDiscretizationGraph_3.png|thumb|800px|<xr id="fig:SuperFINIXDiscretizationGraph_3"/>: Discretization of a power field in SuperFINIX-FINIX coupling.]] | ||
+ | </figure> | ||
<equation id="eqn:PowerConservation1SuperFINIX"> | <equation id="eqn:PowerConservation1SuperFINIX"> | ||
− | {{NumBlk|:|<math> P = \int\limits_{r_i}^{ | + | {{NumBlk|:|<math> P = \int\limits_{r_i}^{r'_{i+1}} \rho'_j \cdot 2\pi r h \text{d}r + \int\limits_{r'_{i+1}}^{r_{i+1}} \rho'_{i+1} \cdot 2\pi r h \text{d}r</math>|<xr id="eqn:PowerConservation1SuperFINIX" nolink />}} |
</equation> | </equation> | ||
− | + | and | |
<equation id="eqn:PowerConservation2SuperFINIX"> | <equation id="eqn:PowerConservation2SuperFINIX"> | ||
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|- | |- | ||
|<math> r </math> | |<math> r </math> | ||
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− | |||
|Radial coordinate of a radial node in FINIX, | |Radial coordinate of a radial node in FINIX, | ||
|- | |- | ||
− | |<math> | + | |<math> r' </math> |
|Radial coordinate at the beginning of a power density block in SuperFINIX, | |Radial coordinate at the beginning of a power density block in SuperFINIX, | ||
|- | |- | ||
|<math> \rho </math> | |<math> \rho </math> | ||
− | |||
− | |||
− | |||
|Power density of a radial node in FINIX, | |Power density of a radial node in FINIX, | ||
|- | |- | ||
− | |<math> \ | + | |<math> \rho' </math> |
|Power density of a block in SuperFINIX, | |Power density of a block in SuperFINIX, | ||
|- | |- | ||
|<math> h </math> | |<math> h </math> | ||
− | |Axial length of the volume element | + | |Axial length of the volume element. |
|} | |} | ||
− | <xr id="eqn: | + | <xr id="eqn:PowerConservation2SuperFINIX"/> can be solved for <math> \rho_{i+1} </math> to obtain the power densities at all radial nodes as: |
<equation id="eqn:DiscretizePowerDensitySuperFINIX"> | <equation id="eqn:DiscretizePowerDensitySuperFINIX"> | ||
− | {{NumBlk|:|<math> \rho_{i+1} = \left(\ 1 + \frac{( | + | {{NumBlk|:|<math> \rho_{i+1} = \left(\ 1 + \frac{(r_{i+1} - r_i)(r_{i+1}^2 - r_i^2 - \frac{P}{\rho_i \pi h})}{r_i(r_{i+1}^2 - r_i^2)-\frac{2}{3}(r_{i+1}^3 - r_i^3)} \right)\ </math>|<xr id="eqn:DiscretizePowerDensitySuperFINIX" nolink />}} |
</equation> | </equation> | ||
− | + | here P is the total power of the element which can be solved from <xr id="eqn:PowerConservation1SuperFINIX"/> and the power density of the first node <math> \rho_1 </math> is always equal to the power density of the first block. For situations in which adjacent FINIX radial nodes are separated by several SuperFINIX blocks, the calculation of total power P can be generalized as: | |
+ | |||
+ | <equation id="eqn:PowerConservation3SuperFINIX"> | ||
+ | {{NumBlk|:|<math> P = \int\limits_{r_i}^{r'_{i+1}} \rho'_i \cdot 2\pi r h \text{d}r + \int\limits_{r'_{i+1}}^{r'_{i+2}} \rho'_{i+1} \cdot 2\pi r h \text{d}r + ... + \int\limits_{r'_{i+N}}^{r_{i+1}} \rho'_{i+N} \cdot 2\pi r h \text{d}r</math>|<xr id="eqn:PowerConservation3SuperFINIX" nolink />}} | ||
+ | </equation> | ||
=== Temperature from FINIX to SuperFINIX === | === Temperature from FINIX to SuperFINIX === | ||
+ | |||
+ | FINIX calculates the fuel temperatures at nodal points, from which temperatures of the SuperFINIX blocks have to be calculated. First the temperatures have to be linearly interpolated to calculate the temperatures at the beginning of each block: | ||
+ | |||
+ | <equation id="eqn:TemperatureInterpolationSuperFINIX"> | ||
+ | {{NumBlk|:|<math> T'_{j+1} = T_i + (T_{i+1} - T_i) \frac{r_{j+1} - r_{i}}{r_{i+1}-r_i}</math>|<xr id="eqn:TemperatureInterpolationSuperFINIX" nolink />}} | ||
+ | </equation> | ||
+ | |||
+ | Once the interpolated temperatures <math> T' </math> are known, the average temperature of each block <math> T_avg </math> can be calculated by integrating the linearly scaling temperature over the volume of the block and then dividing it with the total volume. For a block of scenario <xr id="fig:FINIX-SuperFINIXTemperature"/> containing two radial nodes this is done as: | ||
+ | |||
+ | <equation id="eqn:TemperatureIntegralSuperFINIX"> | ||
+ | {{NumBlk|:|<math> T_{avg} = \frac{\int\limits_{r'_i}^{r_{i}} \left(T'_i + (r - r'_i) \cdot \frac{T_{i} - T'_i}{r_{i} - r'_{i}} \right)\, \cdot 2\pi r h \text{d}r + \int\limits_{r_i}^{r'_{i+1}} \left(T_i + (r - r_i) \cdot \frac{T'_{i+1} - T_i}{r'_{i+1} - r_{i}} \right)\, \cdot 2\pi r h \text{d}r}{V_j} = \frac{\int\limits_{r'_{i}}^{r_{i}} \left(T'_i + (r - r'_i) \cdot \frac{T_{i} - T'_i}{r_{i} - r'_{i}} \right)\, \cdot 2\pi r h \text{d}r + \int\limits_{r_i}^{r'_{i+1}} \left(T_i + (r - r_i) \cdot \frac{T'_{i+1} - T_i}{r'_{i+1} - r_{i}} \right)\, \cdot 2\pi r h \text{d}r}{\pi ({r'}_{i+1}^2 - {r'}_{i}^2) h} </math>|<xr id="eqn:TemperatureIntegralSuperFINIX" nolink />}} | ||
+ | </equation> | ||
+ | |||
+ | <math> = \frac{\left( T'_{i} - (T_{i} - T'_{i}) \frac{r'_i}{r_i - r'_{i}}\right)\ (r_{i}^2 - {r'}_{i}^2) + \frac{2}{3}\frac{T_{i} - T'_{i}}{r_{i} - r'_{i}}(r_{i}^3 - {r'}_{i}^3) + \left( T_{i} - (T'_{i+1} - T_{i}) \frac{r_{i}}{r'_{i+1} - r_{i}}\right)\ | ||
+ | ({r'}_{i+1}^2 - r_{i}^2) + \frac{2}{3}\frac{T'_{i+1} - T_{i}}{r'_{i+1} - r_{i}}({r'}_{i+1}^3 - r_{i}^3)}{{r'}_{i+1}^2 - {r'}_{i}^2} </math> | ||
+ | |||
+ | For scenarios in which a SuperFINIX block contains several FINIX radial nodes, the formula can be generalized as: | ||
+ | |||
+ | <equation id="eqn:TemperatureIntegralSuperFINIX"> | ||
+ | {{NumBlk|:|<math> T_{avg} = \frac{\int\limits_{r'_i}^{r_{i}} \left(T'_i + (r - r'_i) \cdot \frac{T_{i} - T'_i}{r_{i} - r'_{i}} \right)\, \cdot 2 r\text{d}r + \int\limits_{r_i}^{r_{i+1}} \left(T_i + (r - r_i) \cdot \frac{T_{i+1} - T_i}{r_{i+1} - r_{i}} \right)\, \cdot 2 r \text{d}r + ... + \int\limits_{r_i+N}^{r'_{i+1}} \left(T_{i+N} + (r - r_{i+N}) \cdot \frac{T'_{i+1} - T_{i+N}}{r'_{i+1} - r_{i+N}} \right)\, \cdot 2 r \text{d}r}{{r'}_{i+1}^2 - {r'}_{i}^2} </math>|<xr id="eqn:TemperatureIntegralSuperFINIX" nolink />}} </equation> | ||
+ | |||
+ | <figure id="fig:FINIX-SuperFINIXTemperature"> | ||
+ | [[File:FINIX-SuperFINIXTemperature2.PNG|thumb|800px|<xr id="fig:FINIX-SuperFINIXTemperature"/>: Calculation of a temperature field in FINIX-SuperFINIX coupling.]] | ||
+ | </figure> | ||
+ | |||
+ | [[Category:SuperFINIX]] | ||
+ | [[Category:Theory manuals]] |
Latest revision as of 08:25, 10 June 2020
Field transfers from SuperFINIX meshes to FINIX meshes
Power from SuperFINIX to FINIX
SuperFINIX receives power densities in blocks that have to be discretized in the radial nodes of FINIX. This discretization is based on assumption that the number of axial elements remains unchanged before and after the discretization. Therefore the discretization is only performed in the radial dimension. The main solvers in FINIX assume linear scaling of power density (and most other variables) between two nodal points. Because of this assumption, the power densities in nodes have to adjusted when integrating over a volume element that is spanned by two radial nodes in different power density blocks. This conserves the total power over the volumes of single elements as well as the whole fuel rod volume. As such, the following condition has to hold true for the scenario of Figure 1:
![](/kraken/images/thumb/9/9e/SuperFINIXDiscretizationGraph_3.png/800px-SuperFINIXDiscretizationGraph_3.png)
-
(1)
and
-
(2)
where:
The total power in a volume element, Radial coordinate of a radial node in FINIX, Radial coordinate at the beginning of a power density block in SuperFINIX, Power density of a radial node in FINIX, Power density of a block in SuperFINIX, Axial length of the volume element.
Equation 2 can be solved for to obtain the power densities at all radial nodes as:
-
(3)
here P is the total power of the element which can be solved from Equation 1 and the power density of the first node is always equal to the power density of the first block. For situations in which adjacent FINIX radial nodes are separated by several SuperFINIX blocks, the calculation of total power P can be generalized as:
-
(4)
Temperature from FINIX to SuperFINIX
FINIX calculates the fuel temperatures at nodal points, from which temperatures of the SuperFINIX blocks have to be calculated. First the temperatures have to be linearly interpolated to calculate the temperatures at the beginning of each block:
-
(5)
Once the interpolated temperatures are known, the average temperature of each block
can be calculated by integrating the linearly scaling temperature over the volume of the block and then dividing it with the total volume. For a block of scenario Figure 2 containing two radial nodes this is done as:
-
(7)
For scenarios in which a SuperFINIX block contains several FINIX radial nodes, the formula can be generalized as:
-
(7)