ABSTRACT
In order to solve a system of partial differential equations, a numerical solution was
examined as well as the analytical solution. Steady state heat transfer in a clad nuclear fuel rod
is presented. A numerical solution was performed in COMSOL, and results verified with the
analytical solution. The required steps to model, solve and validate the solution of the systems
are shown.
INTRODUCTION
Finite element analysis (FEA) software is an engineering tool that can be used to obtain
solutions to a system in question. However, results from FEA software should be verified to
ensure the tool is working properly. Today’s discussion will examine steady state heat transfer in
a clad nuclear fuel rod, a real world application. When designing nuclear power plants, the
calculation of the power produced in the reactor core and its removal by the coolant are very
important. Coolant is circulated through the core and heat flows from the fuel rods to the
coolant. Because the heat is generated inside the fuel rods, temperature gradients are established
inside the rods that enable the heat to flow outwards from the rods to the coolant, which results
in a temperature profile within the rods. The rod consists of cylindrical pellets, surrounded by a
cladding. The heat transfer geometry is shown in the Figure 1 below:
TC
r1
r2
Figure 1: 2D Cylindrical System where Temperature varies Radially
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Heat is generated in the fuel pellet of radius r1 and flows radially through the fuel pellet and the
cladding to reach the coolant. When verifying results from FEA software, it is helpful to have a
comparison. In this instance, the system of equations is known, and the exact solution can be
solved for and compared to the results obtained from COMSOL.
FINITE ELEMENT MODELING
When one first opens up the COMSOL Multiphysics FEA tool, the Model Wizard is what
is first seen. Here a space dimension must be selected in which the user prefers to work in. As is
the case for the problem at hand, 2D Axisymmetric is selected. Since we are searching for results
to the steady state heat transfer in a clad fuel rod, Heat Transfer in Solids will next be selected
from the Add Physics menu. Finally, a Study Type must be chosen. Once again, steady state heat
transfer is wanted, therefore the Stationary Study Type will be selected.
A Geometry must then be added. Because 2D Axisymmetric was chosen for the model,
the fuel cell and cladding will be looked at from the side. Therefore, two rectangles are added as
the geometry. Once the system is solved for at the end, COMSOL will revolve the two coincided
rectangles and a cylinder will be seen. The problem at hand has the fuel rod with a radius of
0.015 m and a cladding of 0.003 m thickness surrounding and a height of 0.01 m and all can be
seen built in COMSOL in Figure 2 below.
Figure 2: Two Coinciding Rectangles Built in COMSOL Geometry
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Next, material properties must be added to the geometry. The larger rectangle on the left
hand side represents Uranium Dioxide (UO2), a popular fuel cell material. The smaller rectangle
on the right hand side represents Zircaloy-2, whose main use is in nuclear technology as cladding
of fuel rods in nuclear reactors. These two materials aren’t built in to the COMSOL software
therefore, they must be entered manually. Density (ρ), heat capacity (Cp), and thermal
conductivity (k) values are the only ones needed as these are required to solve for heat transfer.
Density (ρ), heat capacity (Cp), and thermal conductivity (k) values are readily available for both
the materials listed above, by doing a quick search on the internet and reliable sources were
found. 1 Both uranium dioxide and zircaloy-2 heat transfer coefficients vary with temperature
therefore, all values were plotted versus temperature and a best fit line was found using Excel.
The best fit line can be seen in Appendix A. These best fit lines were then entered into the
COMSOL model for the uranium dioxide and zircaloy materials.
After the geometry is built and material properties set, the Heat Transfer physics needs to
be set. This is where boundary conditions need to be applied. A Heat Source is added to the
larger rectangle on the left hand side that is representing the fuel rod. A General source was
𝑊
selected as the Heat source and a value of 4 x 106 𝑚3 entered. Finally, a Heat Flux is added to the
far right hand edge of the geometry. A heat flux is the rate of heat energy transfer through a
given surface. The fuel rod is producing heat, transferring it to the surrounding cladding material,
and then that material is transferring it to the coolant in which the clad fuel rod is surrounded. An
𝑊
inward heat flux is selected from the options, a heat transfer coefficient of 100 𝑚2 ∙𝐾 is entered,
and a Text value of 573.15 K is also entered. Text stands for the external temperature at which the
coolant is kept at.
Finally, a Mesh can be added to the COMSOL model. A mathematical model must be
discretized into finite elements. This is commonly called discretization process meshing, or
meshing for short. A meshed model is easy to illustrate, as can be seen in Figure 3 below. But a
meshed model may be confusing as it implies a mesh is just imposed on model geometry. Nodes
connected by lines show finite elements, but nothing of the original geometry remains.
Continuous geometry is replaced by nodes and interactions between nodes are defined by
elements connecting the nodes.
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Figure 3: Geometry in COMSOL after Mesh has been added
Finally in COMSOL, the solution can be computed, though the results still need to be verified.
After computing, the graphics window of the geometry will look like Figure 4 below.
Figure 4: COMSOL Numerical Solution of Fuel Rod Problem
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One can see that the inside of the fuel cell is at a higher temperature than the zircaloy perimeter
material, and that the temperature varies radially, as was expected.
FORMULATION OF ANALYTICAL SOLUTION
The removal of heat from the cylindrical fuel element occurs in the radial direction,
through a series of heat resistances by conduction. The heat balance equation is:
1 𝜕
𝑟 𝜕𝑟
𝜕𝑇
𝜕𝑇
(𝑘𝑟 𝜕𝑟 ) + 𝑞 = 𝜌𝐶𝑝 𝜕𝑡
(1)
where ρ is the density, Cp is the heat capacity, k is the thermal conductivity, and q is the heat
generated per unit volume in the fuel pellet.
Equation 1 above is the equation for the transient case of the system. Equations 2 and 3
below are used for the steady state case of the system. One can see the right-side of Equations 2
and 3 below are not time dependent. Also Equation 3 has no q value, as the cladding material is
not generating heat.
1 𝑑
(𝑘𝐹𝐶 𝑟
𝑟 𝑑𝑟
𝑑𝑇𝐹𝐶
𝑑𝑟
)+𝑞 =0
𝑑
(𝑟
𝑑𝑟
(2)
𝑑𝑇𝐶𝐿
𝑑𝑟
) = 0 (3)
Using Equations 2 and 3 above and the four boundary conditions listed below, one can solve the
steady-state case analytically.
𝑑𝑇𝐹𝐶
Boundary Condition 1
Boundary Condition 2
Boundary Condition 3
Boundary Condition 4
𝑑𝑟
𝑘𝐹𝐶
−𝑘𝐶𝐿
𝑑𝑇𝐹𝐶
𝑑𝑟
𝑑𝑇𝐶𝐿
𝑑𝑟
|
𝑟=0
|
𝑟=𝑟1
|
𝑟=𝑟2
=0
= 𝑘𝐶𝐿
𝑑𝑇𝐶𝐿
𝑑𝑟
|
𝑟=𝑟1
= ℎ[𝑇𝐶𝐿 (𝑟2 ) − 𝑇𝐶 ]
𝑇𝐹𝐶 (𝑟1 ) = 𝑇𝐶𝐿 (𝑟1 )
SOLUTION
In order to solve the numerical steady state case, a COMSOL model was created in 2D
Axisymmetric, using the Heat Transfer in Solids function. Geometry was defined, similar to
Figure 1 above. The correct materials were chosen for each region and data for the thermal
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properties of uranium dioxide (the fuel rod material) and zircaloy (the cladding material) were
found as this data is not contained in COMSOL.1 Density (ρ), heat capacity (Cp), and thermal
conductivity (k) values were all plotted versus temperature and a best fit line was found using
Excel and can be seen in Appendix A. These best fit lines were then entered into the COMSOL
model for the uranium dioxide and zircaloy materials. A Heat Source was added to the uranium
dioxide material and a Heat Flux was also added to the model. Originally a mesh size of Fine was
selected and the results computed.
In order to analytically solve for the steady state case, Equations 2, 3 and the four
boundary conditions above are used. Some steps are excluded, but all can be seen in Appendix
B.
Equation 2 is differentiated on both sides to begin with:
𝑑
∫ 𝑑𝑟 (𝑘𝐹𝐶 𝑟
𝑑𝑇𝐹𝐶
𝑑𝑇𝐹𝐶
𝑑𝑟
) = − ∫ 𝑞𝑟
−𝑞𝑟
= 2𝑘
𝑑𝑟
𝐹𝐶
+𝑘
𝐶1
𝐹𝐶
𝑟
(4)
(5)
Now Boundary Condition 1 can be used and substituted into Equation 5 from above and C1 is
solved for and substituted back in to Equation 5 above.
𝑑𝑇𝐹𝐶
𝑑𝑟
−𝑞𝑟
= 2𝑘
𝐹𝐶
(6)
Next, Equation 3 is differentiated on both sides.
𝑑
∫ 𝑑𝑟 (𝑟
𝑑𝑇𝐶𝐿
𝑑𝑟
𝑑𝑇𝐶𝐿
𝑑𝑟
=
) = ∫0
𝐶3
(7)
(8)
𝑟
Then Boundary Condition 2 is used with Equations 6 and 8 and C3 is solved for and substituted
back in to Equation 8 above.
𝑑𝑇𝐶𝐿
𝑑𝑟
−𝑞𝑟1 2
= 2𝑘
𝐶𝐿
𝑟
(9)
Next, Equation 9 is differentiated on both sides.
∫
𝑑𝑇𝐶𝐿
𝑑𝑟
−𝑞𝑟1 2
= ∫ 2𝑘
𝐶𝐿 𝑟
(10)
6
𝑇𝐶𝐿 (𝑟) =
−𝑞𝑟1 2
2𝑘𝐶𝐿
ln(𝑟) + 𝐶4
(11)
Then Boundary Condition 3 is used with Equations 9 and 11 and C4 is solved for and substituted
back in to Equation 11 above.
𝑇𝐶𝐿 (𝑟) =
−𝑞𝑟1 2
2𝑘𝐶𝐿
𝑞𝑟 2
𝑞𝑟 2
2
𝐶𝐿
ln(𝑟) + 2ℎ𝑟1 + 2𝑘1 ln(𝑟2 ) + 𝑇𝐶
(12)
Next, Equation 6 is differentiated on both sides.
∫
𝑑𝑇𝐹𝐶
𝑑𝑟
−𝑞𝑟
= ∫ 2𝑘
𝑇𝐹𝐶 (𝑟) =
−𝑞𝑟 2
4𝑘𝐹𝐶
(13)
𝐹𝐶
+ 𝐶2
(14)
Then Boundary Condition 4 is used with Equations 12 and 14 and C2 is solved for and substituted
back in to Equation 14 above.
𝑇𝐹𝐶 (𝑟) =
−𝑞𝑟 2
4𝑘𝐹𝐶
+
−𝑞𝑟1 2
2𝑘𝐶𝐿
𝑞𝑟 2
𝑞𝑟 2
𝑞𝑟 2
2
𝐶𝐿
𝐹𝐶
ln(𝑟1 ) + 2ℎ𝑟1 + 2𝑘1 ln(𝑟2 ) + 𝑇𝐶 + 4𝑘1
(15)
The analytical solutions for the temperature as a function of the radius are listed above in
Equations 12 and 15. Equation 12 works for the range from r = 0.015 m to r = 0.018 m. Equation
15 works for the range r = 0 m to r = 0.015 m.
RESULTS
The numerical solution can be seen in Figure 5 below. One can see the inside of the fuel
cell is at a higher temperature than the perimeter, which is the zircaloy material, and the
temperature varies radially, as was expected.
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Figure 5: COMSOL Numerical Solution of Fuel Rod Problem
The analytical solution was performed in Excel and compared to the COMSOL numerical
solution as can be seen in Figure 6 below.
Temperature vs. Radius with Different
Mesh Sizes
Temperature (K)
870
865
860
855
850
845
840
835
830
825
820
-0.002
T_Analytical
T_Extremely Coarse
T_Extremely Fine
T_Normal
0.003
0.008
0.013
0.018
Radius (m)
Figure 6: Temperature versus Radius at Steady-State for both Numerical and Analytical
Solutions with Different Mesh Sizes
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It can be seen that all values lie closely on top of each other. Though the values don’t vary greatly
between different mesh sizes chosen, it is ensured that the solution is not diverging towards an
unknown solution. There is a converged solution present here.
Now that the COMSOL model and physics applied has been verified, a solution can be
examined where it is more difficult to find the analytical solution by hand. The same clad fuel rod
is examined in COMSOL, and the Heat Transfer in Solids tab is closely looked at once again.
Where a Heat Flux was only added to the far right hand edge of the original geometry, we can
now change it and understand more about how steady state heat flux works. When only having
the heat flux on the far right hand edge, it essentially means the system in question is an infinitely
long cylinder, which makes the analytical solution easier to solve. However, this isn’t necessarily
true in real world applications. The heat flux is added to the top and bottom edges of the entire
geometry and the computed results looked upon, as can be seen in Figure 7 below.
Figure 7: COMSOL Numerical Solution of Short Fuel Rod Problem
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The infinitely long cylinder was first chosen as the problem in order to be able to
accurately verify if the COMSOL model was working properly. Once the COMSOL model is
verified, a different system that is much more applicable to real world systems can be computed
and solved for in COMSOL. The short clad fuel rod is solved for and the solution is compared to
that of the long clad fuel rod. The difference between the two can be seen in Figure 8 below.
Temperature vs. Radius with Different
Boundary Conditions
900
Temperature (K)
850
800
750
T_Long Rod
T_Short Rod
700
650
600
-0.002
0.003
0.008
0.013
0.018
Radius (m)
Figure 8: Temperature versus Radius at Steady-State for Two Different Physics
DISCUSSION
When reviewing the results, one can see that although the mesh size matters, it doesn’t
need to be so small that computer resources are wasted. A small enough mesh size to be sure that
the values converge is plenty, as is proven with the less than 1% percentage differences. It is
interesting to note that the percentage differences between different mesh sizes at critical
locations went from the largest in the center to the smallest out at the surface of the system. The
steady state model in COMSOL took approximately 15 seconds to solve as this problem isn’t
hugely computer intensive.
When solving the analytical solution for the long clad fuel rod and comparing it to the
numerical solution of the same system, the final analytical solution matches very closely with that
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of the numerical solution in COMSOL. The biggest percentage difference for all those data points
is 0.0006%.
After COMSOL results have been verified, problems that are more difficult to do by hand
can be examined and the results computed. As can be seen from Figure 8 above, adding the heat
flux to all the outside edges of the geometry, which is for a short clad fuel rod, the temperature
profile varies the same along the radius, however the overall temperature values are lower, as can
be expected. Now that a heat flux is added to all the edges, heat is escaping through all those
surfaces, as a heat flux is the rate of heat energy transfer through any given surface.
CONCLUSIONS
In conclusion, a system of partial differential equations was closely examined and solved
for given initial conditions. It is known that one of the most important factors in designing power
plants is the calculation of the power produced in the reactor core and its removal by the coolant.
Furthermore, it was proven that COMSOL is a successful FEA modeling program for problems
similar to the one used here. Once results are verified for a simpler problem, harder problems can
be solved for with the confidence of having a correct answer.
REFERENCES
[1] IAEA (2006). Thermophysical properties database of materials for light water reactors and
heavy water reactors. http://www-pub.iaea.org. Retrieved on December 1, 2013, from
http://www-pub.iaea.org/MTCD/publications/PDF/te_1496_web.pdf
Cengel, Y. (2007). Heat and Mass Transfer. New York, NY: McGraw-Hill.
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