TRIGA mark II is a research nuclear reactor from General Atomics, at this page you can find more informations about it. Here, what you need to know is a just a little about it’s geometry: it is a water pool type reactor, core have a radius of 22.85 cm and an 36 cm tall (more or less). Each fuel rod have a radius of 1.9 cm. At this page you can find some more informations.
Index:
Neutronic model
For neutronics well known 6 gropus precursor model is adopted.
\frac{dP}{dt} = \frac{\rho -\beta}{\Lambda}P + \sum{\lambda_i C_i}
\frac{dC_i}{dt} = \frac{\beta_i}{\Lambda}P -\lambda_i C_i
In the table below are reported neutronic parameters, you can find them in: “A zero dimensional model for simulation of TRIGA Mark II dynamic response” paper.
| \beta | 730e-5 |
| \lambda_1 [1/s] – \beta_1/\beta | 0.0124 – 0.033 |
| \lambda_1 [1/s] – \beta_2 /\beta | 0.0305 – 0.219 |
| \lambda_3 [1/s] – \beta_3 /\beta | 0.111 – 0.196 |
| \lambda_4 [1/s] – \beta_4 /\beta | 0.301 – 0.395 |
| \lambda_5 [1/s] – \beta_5 /\beta | 1.14 – 0.115 |
| \lambda_6 [1/s] – \beta_6 /\beta | 3.01 – 0.042 |
| \Lambda [s] | 60e-6 |
Finally reactivity is function of fuel and coolant feedback coefficients:
\rho = \rho_0 + \alpha_f(T_f - T_f^0) + \alpha_c(T_c - T_c^0)Thermal Hydraulic model
For TH model we have at least two choice regarding coolant mass flow rate. In the experiment here reported coolant undergoes natural convection, mass flow rate is around 6 kg/s. However coolant temperature variation is quite small, so small that we can simply impose \frac{dT_c}{dt} = 0 without loosing too much accuracy. However in the evaluation of global heat transfer coefficient we have to consider the effect of fluid motion, this is a crucial aspect.
So, below you see the complete model, which consider natural circulation mass flow rate (A zero dimensional model for simulation of TRIGA Mark II dynamic response) which model is developed considering the equilibrium between buoyancy forces and friction.
c_fM_f\frac{dT_f}{dt} =P - K(T_f - T_c)
c_cM_c\frac{dT_c}{dt} =K(T_f - T_c) + 2\Gamma C_c(T_c-T_{in})
Of course mass flow rate \Gamma is function of coolant temperature, the expession is:
\Gamma = \sqrt{2\frac{\delta_{in} g L \nu }{\alpha_2}(T_c - T_{in})}
Even if the experiment is performed without external recirculation, T_{in} was considered constant, equal to ambient temperature. This is true for short transient.
The other parameters are:
| g | Gravity acceleration |
| \delta_{in} | water density at the inlet |
| L | Core heigth |
| \nu | water thermal expansion coefficient |
| \alpha_2 | Friction factor |
For the numerical value you should look at:
A zero dimensional model for simulation of TRIGA Mark II dynamic response, A. Cammi et al.
For what concerns fuel heat capacity, GA provide a formula for the single fuel element, which is:
C_f^1 = 750 +1.55(T_f-25)So that the total heat capacity is M_fc_f = n_{fe}C_f^1
Heat transfer coefficient K is obtained from steady state formulation: K_0 = \frac{P_0}{T_f^0-T_c^0}, if you know initial temperatures it’s easy, if not you can manipulate this formula in terms of sum of thermal resistances.
K = \frac{n_{fe}V_f}{\pi r_{fo}^2\left( \frac{1}{8\pi r_{fo}^2} + \frac{1}{2\pi k_c}ln\left(\frac{r_{co}}{r_{ci}}\right) + \frac{1}{2\pi r_{gap} h_{gap}} + \frac{1}{2\pi r_{co}h_{coolant}} \right)}
Where n_{fe} and V_f are the number of fuel elements and the volume of single fuel element. The term at the denominator is the sum of thermal resistances. note that fuel temperature in lumped model is an average temperature, in order to obtain this value just hlaf of thermal resistance is considered. You can find this relation in “Nuclear systems” by Todreas and Kazimi and on the paper cited before.
I considered constant heat conductivity for fuel and cladding, but for gap and coolant it is necessary to consider a power or temperature dependent relation. For coolant htc i used the classic Dittus Boelter relation, even if Reynolds number is below the range of applicability (moreover this do not consider subculled boiling). For the gap i used an empirical formulation (power in kW):
h_{gap} [kW/m^2K]= 0.0239 P^2 - 1.472P + 1593.1
This latter relation is obtained from: “Experimental heat transfer analysis of the IPR-R1 TRIGA reactor, Amir Zacaris Mesquita, Hugo Cesar Rezende”.
Comparison with exerimental data
70 kW 
95 kW
Final comments
Aim of this work was to evaluate fuel feedback coefficient from experimental data, to do that a series of value was tested in an iterative way and just the one providing best fit (lower mean absolute error) was considered. If this is the goal, a better procedure could be to consider also a variable value for K instead of using a physical model. Look for example at the following plot, where both \alpha_f and K are randomly varying simultaneously.
The fact that some parameters are temperature dependent is quite important, in my analysis i considered both heat capacity and thermal conductivity as function of temperature and so these values can change during the transient.
Some values of material properties have an huge impact on the results, it’s the case, for example, of water heat conductivity, up to now i considered a value of k = 1, which is slightly different than the actual value k = 0.6 but provide better results.