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.
Application:
 Research reactors (TRIGA)  pulse mode operation  normal operation
 Power reactors  reactivity transient  severe accident
Detailed explanation is found in the following
references:
 BellGlasstone: 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, 5258, 1994.
 M. Ravnik, Experimental Verification of Adiabatic
FuchsHansen Pulse Model, 4^{th} Regional Meeting Nuclear Energy
in Central Europe, September 710, 1997, Bled, Slovenia, 450456.
 M. Ravnik et al., PULSTRI1, A computer program for TRIGA reactor pulse calculations, IJSDP5756, January 1990.
Physical description
of pulsing in TRIGA reactor
Schematically:
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:
 fast
withdrawal of transient rod by means of pneumatic system (pulse experiment)
 rod ejection in
PWR (hypothetical transient in power reactor)
Consequences:
 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
increasing
 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 Other rods withdrawn (except for the part compensating
excess reactivity).
 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
reactivity
> reactor becomes subcritical after the
pulse, but would return to power after short time due to cooling (reason
for SCRAM)
Time dependence of reactivity, power and energy are
presented in Fig. 1 and Fig. 2
Physical model of the pulse: Fuchs  Hansen adiabatic
model
Assumption 1: Point kinetics approximation
Where:
 P  total reactor power (MW)
 li  delayed neutron precursors'
decay constant (in average approx. 0.1s^{1}, 1/l_{i}=
10s)
 l  prompt neutron generation time ( 40m sec << 1/l i
)
 b  effective delayed neutron
fraction (importance factor times nuclear delayed neutron fraction for
U235, in TRIGA: from 0.0070 to 0.0073)
 r  reactivity, convenient
unit: b = $ = 0.007=
700pcm
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
reactivity.
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
Evidently
Where:
 a_{f}  fuel
temperature reactivity coefficient
 c_{p}  specific thermal capacity of fuel (multiplying
material only)
 m  total mass of fuel (multiplying material) in reactor
It is assumed that a_{f} and c_{p} do not depend on temperature. Inserting r(t) into equation for P we obtain Fuchs  Hansen model
Initial condition: P(0) = P_{o}
For t = 0, we see immediately:
Exponential power is increasing: a_{o} is initial inverse
period, 0.007/40 ms = 175 s^{1} = 1/ (0.005 s).
Formal solution of FH equation
Kinetics equation:
can be solved analytically by introducing a new variable
y(t).
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
physical meaning.
Equation:
is solved by introducing a new variable u(t):
.
After insertion we obtain:
.
Multiplying with u^{2} (u not 0) yields a
simple nonhomogeneous 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:
.
where:
Total released energy E_{t} equals:
.
Maximum power in pulse P_{max}:
.
Note: E_{t} and P_{max} 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:
P_{max} is proportional to (r')^{2} and m (mass of reactor fuel).
P_{max} is inversely proportional to l, g and a_{f}.
E_{t} is proportional to r' and m .
E_{t} is inversely proportional to g and a_{f}.
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 burnup due to long term steady state operation
between pulsing, xenon effect.
 modifications in core configuration.
Conclusion
FuchsHansen adiabatic model and pulse experiment provides
good insight in reactor physics of reactivity transient important in power
reactor safety analysis.
Figures:
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
P_{max} in dependence of r'^{2}.
Figure 4: Measured and calculated total energy E_{t} in dependence of r'.
Figure 5: Measured maximum fuel temperature in dependence
of r'.
Figure 6: Measured fuel temperature in dependence of
time after pulse.
