Reactor TRIGA

Reactor Physics of Pulsing:

Fuchs - Hansen Adiabatic Model

M. Ravnik

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 references:

  • 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.


Physical description 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:

  • fast 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 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



  • P - total reactor power (MW)
  • li - delayed neutron precursors' decay constant (in average approx. 0.1s-1, 1/li= 10s)
  • 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= 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




  • af - fuel temperature reactivity coefficient
  • cp - specific thermal capacity of fuel (multiplying material only)
  • 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

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.



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 of r'.


Figure 6: Measured fuel temperature in dependence of time after pulse.


Jožef Stefan Institute, Jamova 39, 1000 Ljubljana, Slovenia, Phone: (01) 477 39 00, Fax: (01) 251 93 85