Analog computing

[gary36]

I was looking for a all analog computation of the reactor point kinetic equations(studied in under grad course) using opamps

The equation is described as

I referred to the article https://analogparadigm.com/downloads/alpaca_30.pdf. But could not quite get , as to how the circuit represents the differential equation.

 

[D.A.(Tony)Stewart]

Revised

kinetics simulation is a central part of every nuclear reactor simulation.1 Following [Fenech et al. 1973, p. 128] neutron kinetics can be described by
n ̇ = n (δK − β) + X λici + s

Where/how is s defined? s represents an external neutron source (which is often used to “start” a nuclear reactor)

...
I now see s is defined in the 1971 article of your last link. I guess it gets ignored in the 2021 link as it is just a lower level trigger condition and if it falls out of equation for the dCi/dt .

If dn/dt is the flux velocity of neutrons and n is population then perhaps the simulation is for the integral which makes sense to me ( for n= output) summing junction for C and sum(RiCi) feedback with Rload from input.

Since reactors must always operate at 100% power then the RL load must be constant. So there must be a modulator to limit the integration not shown.

--- Updated Oct 6, 2024 ---

GPT4 seems to agree with me.


"
In nuclear reactor kinetics, the equation you referenced is indeed fundamental for describing neutron population changes. The term s, as you noted, typically represents an external neutron source, which can help initiate fission reactions during startup or during certain operational conditions.


Understanding s and the Kinetics Equation


The equation you provided can be summarized as follows:

n˙=n(δK−β)+∑λi ci +s

n: Neutron population
n˙: Rate of change of neutron population
δK: Neutron multiplication factor
β: Delayed neutron fraction
λi : Decay constants for the neutron precursors
ci : Concentrations of the neutron precursors
s: External neutron source

Role of The external source s is critical during the reactor startup phase. Once the reactor reaches a steady state, or during normal operation,

s may be negligible. However, during transients or startups, its contribution becomes significant for maintaining neutron population until the fission chain reaction is self-sustaining.


Modulation and Integration


Regarding the modulator you mentioned, you're correct that a control mechanism is essential to maintain reactor power levels.

1. Feedback Mechanisms: In a nuclear reactor, feedback loops often include reactivity feedback mechanisms, where changes in temperature, power output, and neutron population influence control rod positions or other control systems to adjust the neutron population.
2. Integration and Output: When you refer to the integration for n as output, you're highlighting the importance of continuously monitoring and adjusting the neutron population based on both internal (reactivity) and external (source s) factors. The system is designed to keep the neutron population stable, ideally at a constant level, to achieve and maintain 100% power.
3. RL Load: The concept of a constant resistive-inductive (RL) load implies that the power output must remain steady. This requires dynamic adjustments to prevent oscillations in power, often facilitated by control systems that modulate the external source or the control rods.

Conclusion


In summary, the external source s plays a crucial role in reactor dynamics, particularly during startups. The integration of neutron population dynamics with feedback mechanisms is essential for maintaining operational stability and achieving the desired power levels. The control strategies ensure that any fluctuations are managed effectively to maintain the reactor within safe operational limits."

Don't trust this 100%. I had to delete the duplicates that GPT injects on a paste of formatted characters weirdly. I hope I did it write

--- Updated Oct 6, 2024 ---

I don't think the hand drawn circuit performs the equation.
Equation has Vi as "output", circuit has Vi as input. Maybe
if you put the whole thing inside a feedback loop, that
would have an output at Vi which truly did what the text
indicates.

The equation looks correct for a dyslexic . Murphy's Law again(?) but if you solve for Vo= you get the transfer function of the circuit while simultaneously solving KCL produces the solution of the derivatives for Vi=

 

[BradtheRad]

I did a matlab simulation and observed that due to absence of feedback the output from integrator keeps increasing without control. Can we perhaps reset the integrator so that cycle starts all over again?

When I think of controlled nuclear fission I think of the necessity to prevent runaway chain reaction. It resembles a chaotic oscillator more than anything I can think of. The Chua circuit is built around an op amp. Gain is adjusted just right so the waveform gets amplified a bit more during each cycle.

When it's about to go out of bounds (saturate, pin to a supply rail), it transitions to the opposite polarity where it continues to oscillate at small amplitude. (Resembling a reset and resuming a growth phase). Again it grows stronger with each cycle. Eventually it transits back to the other polarity.

In this simulation gain is just enough to amplify oscillations gradually til they reach the supply rails at left or right (almost reaching critical mass). Then circuit behavior like control rods prevents a chain reaction. If the waveform had remained pinned to the supply rail then that would resemble runaway fission, wouldn't it?

Link below is my schematic above, running in Falstad's simulator.
By clicking it you can navigate to the website and run it on your own computer.

tinyurl.com/yw8qs4qp

 

[danadakk]

Revised

kinetics simulation is a central part of every nuclear reactor simulation.1 Following [Fenech et al. 1973, p. 128] neutron kinetics can be described by
n ̇ = n (δK − β) + X λici + s

Where/how is s defined? s represents an external neutron source (which is often used to “start” a nuclear reactor)

...
I now see s is defined in the 1971 article of your last link. I guess it gets ignored in the 2021 link as it is just a lower level trigger condition and if it falls out of equation for the dCi/dt .

View attachment 194437

If dn/dt is the flux velocity of neutrons and n is population then perhaps the simulation is for the integral which makes sense to me ( for n= output) summing junction for C and sum(RiCi) feedback with Rload from input.

Since reactors must always operate at 100% power then the RL load must be constant. So there must be a modulator to limit the integration not shown.

--- Updated Oct 6, 2024 ---

GPT4 seems to agree with me.


"
In nuclear reactor kinetics, the equation you referenced is indeed fundamental for describing neutron population changes. The term s, as you noted, typically represents an external neutron source, which can help initiate fission reactions during startup or during certain operational conditions.


Understanding s and the Kinetics Equation


The equation you provided can be summarized as follows:

n˙=n(δK−β)+∑λi ci +s

n: Neutron population
n˙: Rate of change of neutron population
δK: Neutron multiplication factor
β: Delayed neutron fraction
λi : Decay constants for the neutron precursors
ci : Concentrations of the neutron precursors
s: External neutron source

Role of The external source s is critical during the reactor startup phase. Once the reactor reaches a steady state, or during normal operation,

s may be negligible. However, during transients or startups, its contribution becomes significant for maintaining neutron population until the fission chain reaction is self-sustaining.


Modulation and Integration


Regarding the modulator you mentioned, you're correct that a control mechanism is essential to maintain reactor power levels.

1. Feedback Mechanisms: In a nuclear reactor, feedback loops often include reactivity feedback mechanisms, where changes in temperature, power output, and neutron population influence control rod positions or other control systems to adjust the neutron population.
2. Integration and Output: When you refer to the integration for n as output, you're highlighting the importance of continuously monitoring and adjusting the neutron population based on both internal (reactivity) and external (source s) factors. The system is designed to keep the neutron population stable, ideally at a constant level, to achieve and maintain 100% power.
3. RL Load: The concept of a constant resistive-inductive (RL) load implies that the power output must remain steady. This requires dynamic adjustments to prevent oscillations in power, often facilitated by control systems that modulate the external source or the control rods.

Conclusion


In summary, the external source s plays a crucial role in reactor dynamics, particularly during startups. The integration of neutron population dynamics with feedback mechanisms is essential for maintaining operational stability and achieving the desired power levels. The control strategies ensure that any fluctuations are managed effectively to maintain the reactor within safe operational limits."

Don't trust this 100%. I had to delete the duplicates that GPT injects on a paste of formatted characters weirdly. I hope I did it write

--- Updated Oct 6, 2024 ---

The equation looks correct for a dyslexic . Murphy's Law again(?) but if you solve for Vo= you get the transfer function of the circuit while simultaneously solving KCL produces the solution of the derivatives for Vi=

Since reactors must always operate at 100% power

Thats not correct, often during testing, maintenance, docking (case of subs, carriers) reactors "idle"
at very low power levels. Fission flux is managed by control rods, and uncovering fuel brings up the
power level the reactor outputs. Transient power, in the case of the 5 MW I operated, as much as
150%, very short duration. Basically a back emergency bell to stop the boat.

Regards, Dana.

 

[gary36]

I have referred several patents on analog reactimeter, which solve this very equation, with assumption of S=0. The issue is with the precursor term. But if we evolve the transfer function of this term, then Ci(s)/N(s)= Beta(i)/L/s+Lamda(i), with k(t)=1, resembles a filter. I wonder why the output integrates infinitely.