Accuracy of the Quasistatic Treatment of Spatial Reactor Kinetics - 1969
Authors:
- K. O. Ott (Perdue)
- D. A. Meneley (Argonne)
Codes:
QX-1: quasistatic treatment of excursions
Notes:
General problem: solution of the time-dependent multigroup diffusion equation
Modal/Nodal Method:
- Modal-method: “expansion of the time-dependent neutron flux into a series of time-independent modes or nodes with time-dependent coefficients.”
- Sometimes coefficients are also dependent on energy.
- Nodal: Functions are only applied over nonoverlapping regions, index means the region, uses “coupling coefficients” for interaction between regions.
Factorization
- Factorization of total flux into amplitude and shape
- Another separation condition is required for t>0.
- Many can be used, only real requirement is that the shape function is positive and bounded for all points in space, energy, and time.
- Henry’s condition fulfills those requires and allows transition to point kinetics formulation.
- Any weighting function can be used.
- Want strong coupling in only one direction for rapid convergence.
- Chosen splitting: space-energy shape of shape function depends only weakly on the amplitude function.
- Once reduced to the point kinetics equation form, the integral quantities (reactivity, beta, etc) must be averaged with the time-dependent shape function.
- Makes things more complicated, until you assume that the time-dependence of the shape function is of “lesser importance” than the time-dependence of the amplitude function.
Improved Quasistatic Approximation
- Replaces the derivative of the shape function by a backwards difference of the first order.
- Shape function is slowly varying: can apply first-order difference over a much larger deltaT than would be possible for the amplitude function
- Delayed source term is calculated directly from the flux history
Quasistatic Approximation
The time derivative is neglected entirely.
Adiabatic Approximation
- Does not distinguish the shape of the delayed neutron source from prompt.
- Ignores both time derivatives, uses eigenvalue to get stationary expression for the shape function.
- Ignores a lot of the coupling.
- Removal and source still depend on the total flux.
Adiabatic Approx. with Precomputed Shape Functions
- Ignores coupling of removal and source with total flux.
- Diff. eq. for amplitude and shape are completely independent, can pre-calculate time-dependent shape functions.
- Calculate shape functions for the time-dependent situation, calculate integral quantities, used as inputs to point kinetics to get amplitude function.
- “Point kinetics method”
Comparison of Quasistatic with others
- Quasistatic diverges from exact solution due to neglecting the shape function derivative, this can be improved via the IQM
- Adiabatic: significant errors without any real reduction in complexity.
Comparison methodology
- Accuracy of QSM is investigated by comparison with direct numerical
solutions calculated by
WIGLE - Used two-energy-group slab systems
- Simulated excursions in a large water-cooled thermal rx and a fast reactor, three regions (I,II,III), center region has higher buckling to obtain a relatively flat initial flux”
- Transients modeled as linear increase in region I
- One-group delayed neutrons (beta 0.0084, lambda 0.08/sec)
Quasistatic Errors
- By ignoring the time derivative term, it is ignoring the finite transition time between the two different neutron population shapes.
- Implies you can estimate the error expressing it in terms of this “retardation”
- Error is only dependent on inserted reactivity and the degree of coupling.
- Errors can only be compared over the same flux rise.
Conclusion/Summary
- Asymmetric insertions were applied to very loosely coupled systems.
- Errors in fast reactors excursions are <5%
- Errors in thermal reactors <80%
- QSM is sufficient for fast reactors, but not thermal reactors due to the slow flux-shape adaptation.