## Point Dynamics Equations

A simple **point dynamics model** is based on point kinetics equations, but here we should take into account the influence of the fuel and the moderator temperature on the reactivity. We assume, there are no other feedbacks and therefore this simple model can be applied only on PWRs. For example, the void coefficient is here neglected. In systems with **boiling conditions**, such as boiling water reactors (BWR), the void coefficient is of prime importance during reactor operation.

Thus, the simplest point dynamics model of PWR should take into consideration the time variations of the fuel and coolant temperature. The point dynamics model consist of the following equations:

- The first equation is the
**equation for neutrons.**The first term on the right hand side is the production of prompt neutrons in the present generation, minus the total number of neutrons in the preceding generation. The second term is the production of delayed neutrons in the present generation. - The second equation is the
**equation for precursors**. There is the balance between the production of the precursors of i-th group and their decay after the decay constant λ_{i}. As can be seen, the rate of the decay of precursors is the radioactivity rate (λ_{i}C_{i}) and the rate of production is proportional to the number of neutrons times**β**_{i}**, which**is defined as the fraction of the neutrons which appear as**delayed neutrons in the i**.*th*group - The third equation expresses the dependence of the reactivity on various parameters. But in this case, there is a dependence on the coolant and the fuel temperature only. ρ
_{0}is the initial reactivity, whereas ρ_{C}(t) is time dependent reactivity inserted by reactor control system (e.g. by control rods or by boron dilution). This is the feedback equation. - The equations of the heat balance for fuel and coolant are interconnected via
*h(T*_{F}*– T*_{C}, which represents the heat transfer from the fuel into the coolant. In these equations,*)**m*and_{F}*m*are the mass of fuel and coolant in the core, respectively,_{C}*c*and_{pF}*c*are specific heat capacities of fuel and coolant,_{pC }is the heat transfer coefficient between fuel and coolant,*h**m*dotted is the coolant mass flow rate [kg/s] and T_{C}_{C}and T_{C,in}are the average and inlet coolant temperature, respectively.

To solve the point dynamics equations it is necessary to specify the initial conditions like in the case of point kinetics.

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