The purpose of this presentation is:
- physical description
of pulse experiment
- derivation of
Fuchs - Hansen adiabatic model
- comparison of
the theoretical model with the experimental results.
- Research reactors (TRIGA) - pulse mode operation - normal operation
- Power reactors - reactivity transient - severe accident
Detailed explanation is found in the following
- Bell-Glasstone: Nuclear Reactor Theory, Chapter
9.6. Large power excursions.
- I. Mele, M. Ravnik, A. Trkov, "TRIGA Mark II Benchmark
Experiment, Part II: Pulse Operation," Nuclear Technology 105, 52-58, 1994.
- M. Ravnik, Experimental Verification of Adiabatic
Fuchs-Hansen Pulse Model, 4th Regional Meeting Nuclear Energy
in Central Europe, September 7-10, 1997, Bled, Slovenia, 450-456.
- M. Ravnik et al., PULSTRI-1, A computer program for TRIGA reactor pulse calculations, IJS-DP-5756, January 1990.
of pulsing in TRIGA reactor
Large reactivity insertion -> prompt criticality -> power increasing -> fuel temperature increasing ->
-> reactivity decreasing due to negative temperature
reactivity effect ('feedback' reactivity) -> power decreasing,
temperature increasing ->
-> further reactivity decreasing to zero or even
negative value -> power stabilizing at (relatively) low level.
Reactivity transient from r = 0 -> r = b in 0.1sec result of:
withdrawal of transient rod by means of pneumatic system (pulse experiment)
- rod ejection in
PWR (hypothetical transient in power reactor)
- r >> b , reactor prompt critical -> exponential power increase with short period (0.01 s)
- Released thermal
energy accumulates in fuel:
heat generation -> conductive heat removal
- fast fuel temperature
- fast internal
gas pressure increasing
- fast thermal expansion
of fuel meat (but not the cladding) leading to fuel damage and disintegration if
energy released in pulse exceeds design limit.
TRIGA research reactors are designed for pulsing. Pulse energy in TRIGA reactors is limited by negative and prompt fuel temperature reactivity effect.
Negative: due to spectrum hardening and Doppler effect. Typical value:
Prompt: due to homogeneous reactor (uranium homogeneously mixed with hydrogen, fission and moderation in fuel)
Practical pulse procedure:
- Reactor critical with transient rod completely inserted.
- Low power (100 W).
- Transient rod upper position preset by adjusting vertical
position of the piston stopper.
- Other rods withdrawn (except for the part compensating
- Preset SCRAM approx. 5sec after pulse.
- Start pulse (Fire signal)
- valve from air high pressure tank opens
- air pressure moves transient rod drive mechanism piston
- transient rod connected to the piston moves out until
upper stopper is reached (typically in 0.1 s).
- Pulse (from 0.1 s to 1 s after start signal, depending
on inserted reactivity).
- SCRAM after preset time (typically 5 s).
- Cooling (typically 15 min).
- Repeat procedure.
Inserted reactivity is regulated by transient rod upper position - defined by the operator.
"Feedback" reactivity depends on fuel temperature and temperature reactivity coefficient:
If DT small (small pulse) , then feedback reactivity smaller than inserted
reactivity -> reactor remains critical at power defined
by the power defect (reason for SCRAM)
If DT large (large pulse), then feedback reactivity greater than inserted
-> reactor becomes subcritical after the
pulse, but would return to power after short time due to cooling (reason
Time dependence of reactivity, power and energy are
presented in Fig. 1 and Fig. 2
Physical model of the pulse: Fuchs - Hansen adiabatic
Assumption 1: Point kinetics approximation
- P - total reactor power (MW)
- li - delayed neutron precursors'
decay constant (in average approx. 0.1s-1, 1/li=
- l - prompt neutron generation time ( 40m sec << 1/l i
- b - effective delayed neutron
fraction (importance factor times nuclear delayed neutron fraction for
U-235, in TRIGA: from 0.0070 to 0.0073)
- r - reactivity, convenient
unit: b = $ = 0.007=
Assumption 2: Contribution of delayed
neutrons during pulse negligible
Valid only if r >> b.
Assumption 3: Transient rod withdrawal
time is short, rod is withdrawn before temperature feedback effect on reactivity
For convenience we introduce: r' = r(t = 0)
- b, as pulse prompt
Assumption 4: Reactivity decrease during
pulse is proportional to accumulated energy
Assumption implies that:
- there is no energy transfer from multiplying part of
the fuel element during pulse (adiabatic approximation)
- coefficient g isconstant, independent of temperature
- af - fuel
temperature reactivity coefficient
- cp - specific thermal capacity of fuel (multiplying
- m - total mass of fuel (multiplying material) in reactor
It is assumed that af and cp do not depend on temperature. Inserting r(t) into equation for P we obtain Fuchs - Hansen model
Initial condition: P(0) = Po
For t = 0, we see immediately:
Exponential power is increasing: ao is initial inverse
period, 0.007/40 ms = 175 s-1 = 1/ (0.005 s).
Formal solution of FH equation
can be solved analytically by introducing a new variable
If we make second derivative of y(t) we obtain:
Inserting P(t) and dP/dt into initial equation yields
nonlinear second order differential equation
It can be integrated:
Constant c can be determined from the initial condition
at t = 0:
negative solution for c is omitted as it has no
is solved by introducing a new variable u(t):
After insertion we obtain:
Multiplying with u2 (u not 0) yields a
simple non-homogeneous first order differential equation:
Its solution is:
Free coefficient a is calculated from the initial
condition at t = 0:
By reinserting u and y into P(t) we get final solution:
Total released energy Et equals:
Maximum power in pulse Pmax:
Note: Et and Pmax are zero for inserted reactivity up to 1$ - the model is obviously not
valid for reactivity insertions smaller than prompt reactivity. According to Fuchs - Hansen model:
Pmax is proportional to (r')2 and m (mass of reactor fuel).
Pmax is inversely proportional to l, g and af.
Et is proportional to r' and m .
Et is inversely proportional to g and af.
Comparison of model and experiment
Comparison of measured and theoretical results is
presented in Figures 3 - 6. Discrepancies between measured and calculated results
are consequence of deficiencies in the physical model and deficiencies in the measurement.
Main deficiencies of the physical model:
- temperature reactivity coefficient and fuel thermal
capacity independent of temperature.
- delayed neutrons (and their power) neglected, important
at low reactivity insertions and for energy released after pulse.
- point kinetics (more important in big reactors).
Source of systematic experimental errors:
- pulse channel calibration and sensitivity of
the detector on the local flux variations.
- transient rod reactivity worth and influence of
other control rods.
- fuel burn-up due to long term steady state operation
between pulsing, xenon effect.
- modifications in core configuration.
Fuchs-Hansen adiabatic model and pulse experiment provides
good insight in reactor physics of reactivity transient important in power
reactor safety analysis.
Figure 1: Dependence of reactivity, power
and energy on time during pulse (schematically).
Figure 2: Measured P(t) for several pulses with
different inserted reactivity.
Figure 3: Measured and calculated maximum power
Pmax in dependence of r'2.
Figure 4: Measured and calculated total energy Et in dependence of r'.
Figure 5: Measured maximum fuel temperature in dependence
Figure 6: Measured fuel temperature in dependence of
time after pulse.