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Volumetric power density dependence on volume

  1. Oct 26, 2013 #1
    I've been reading about nuclear reactors lately, and I wondered how scientists in the 50s, without modern computing equipment, or maybe even now could predict the thermal power outputs of reactors (I'm guessing primarily determined by the number of fissions per second after the short lived decay heat generating fission products build up to an equilibrium) under various conditions. And then I thought that maybe with a lot of moderated reactors, as long as the reactors being compared are built with similar geometry, materials, fuel enrichment, etc., the thermal power density maybe won't vary as much with the reactor volume (for volumes much larger than what is needed to maintain criticality with similar reactor design), so they could make predictions based on experiments with smaller reactors before building something with 1 GW or more thermal power output. Also, I've read that its impossible or nearly impossible to produce a uranium bomb with less than ~90% U-235. So, I'm wondering if there is a theoretical limit to the number of fissions occurring per time per unit volume that could be imposed solely by the volumetric fissile isotope density w/o any non-fissile neutron absorbing isotopes, moderator, or coolant present (below a certain maximum fissile isotope density that may vary for different isotopes), or, if this is not the case, if such a theoretical limit (on volumetric fission rate density) imposed solely by volumetric fissile isotope and non-fissile neutron absorbing isotope density could be reached practically and safely in a reactor. Also, I was wondering how long a neutron might last on average (mean) after a fission event before it gets absorbed, causes another fission, or decays in both fast and moderated reactors (I know with only the latter, spontaneous decay, they last about 10 minutes, but I'm not sure how long they do with the previous 2). I know moderated neutrons' "targets" have higher absorption and fission cross-sections, but I suspect that thermal neutrons, since they travel much slower, last longer before getting absorbed or inducing fission, but don't cover as much distance before such an event happens as fast neutrons.
     
  2. jcsd
  3. Oct 26, 2013 #2

    Astronuc

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    The earliest nuclear reactors were small experimental reactors, and it was pretty easy to design, construct and operate. The AEC and military did in fact have computational facilities with which to do designs, otherwise they use slide rules, calculators and pencil/pen and paper like so many others.

    Slide rule - http://en.wikipedia.org/wiki/Slide_rule

    Commercial reactors are pretty much an extension of the reactor built for the naval program. The companies that manufactured naval reactors, fuel or components (Westinghouse, GE, Combustion Engineering, B&W) adapted the technology to commercial reactors. They had access to computers, e.g., IBM, CDC or other mainframes.

    For a nuclear reactor, it is a matter of calibrating the instrumentation and getting the thermal (heat) balance right, whether it is a fossil fired system or a nuclear reactor. The above companies had experience in supplying conventional power systems.

    I'm not sure why nuclear weapons were introduced into the discussion. The physics is quite different from a controlled reactor system.

    Most neutrons survive for less than milli-seconds, but a small fraction of neutrons will be 'delayed' since they come not from fission or moderated fast neutrons, but they are emitted from particular fission products. The 'delayed' neutrons allow the reactor to be controlled in conjunction with soluble boron (in PWRs) and control rods/blades in BWRs.

    The power distribution in a reactor is controlled by variations in enrichment (for fresh cores), and use of 'burnable' aborbers such as gadolinia (mixed with the fuel), ZrB2 (coated on fuel pellets) or B-bearing ceramics in special absorber assemblies that sit in guide tube positions of PWR fuel assemblies.

    As the reactor operates, the enrichment is depleted through fissions. In fresh cores, the lowest enrichment fuel is discharged after one cycle, but the higher enrichment fuel continues for an additional cycle or two. Fresh fuel is added to the core to replace discharged fuel. The 'burnable' absorbers 'hold down' the power in the fresh fuel, while the depleted enrichment holds down the power in the older fuel. While enrichment is depleted, so is the burnable absorber material, and the power actually increases in the fuel containing burnup absorbers.

    A lot of research (calculation and experimental work) went into understanding the physics of nuclear reactors and how the fuel enrichment and burnable poisons behaved with irradiation. The result was the development of nuclear reactor core simulation codes (neutronics or depletion codes) with which the reactor is modeled. Such codes have been fine tuned over the years with experience.

    Nuclear reactors have instrumentation. Some instrumentation is located outside the core, but still inside the reactor vessel, and some reactors are designed for instrumentation within the reactor. Neutron detectors and thermocouples can be inserted through special tubing into the core. Some thermocouple systems sit in special instrument tubes above the core, just near the outlet of the core where they measure the exit temperature. It is assumed that the core inlet temperature is pretty uniform from the mixing of the coolant in the bottom plenum of the reactor vessel. Knowing the coolant exit and inlet temperatures, one can determine the enthalpy rise in the core. With a set of thermocouples, one can monitor the enthalpy rise at different locations. Combining the thermal data with the neutronics data, one can develop a map of the power distribution in the core. In a commercial reactor, various process computers can monitor coolant flow rates and temperatures, or steam rates, and power output of electrical generators, so it is possible to know with reasonable accuracy the power balance in the nuclear plant and the core.
     
  4. Oct 26, 2013 #3

    Astronuc

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    The power density is determined by fuel rod and assembly lattice design, and by the maximum heat flux that the cladding can accommodate under given operating conditions without experiencing excessive corrosion or failure during a postulated transient, or if there is failure, the number of failures is limited.

    Currently, commercial LWR fuel is limited to 5% enrichment. The number of assemblies loaded at the beginning of the reactor operating cycle will depend on the enrichment, cycle length and capacity factor, which will produce a given burnup (burnup = energy produced per unit mass of fuel, e.g., GWd/tU, MWd/kgU, . . . ).
     
  5. Oct 27, 2013 #4
    I'm also wondering about limits to power density, if we ignore practical issues like corrosion or melting of the fuel (or reactor vessel for liquid fuel), and only consider the following constraints: isotopic composition and density of the fuel are what one might use for a fast molten salt fueled reactor, we pretend the initial density of the fuel is as high as it can be in a liquid state at ~10 bar, and the critical region of the reactor has infinite volume and homogenous composition and density. I want to know if any of the constraints imposed on the system I'm describing limit its power density, and very roughly what that limit might be. I read what I think was a German educated packaging engineer's description of a molten salt fast reactor w/ an ellipsoid core and a thermal power density of about 600 MW/m^3 in the fissile zone of the core, and the following fissile fuel salt composition: 43% NaCl, 23% KCl, 25.5% UCl3, 4.5% PuCl3, 4% fission product chlorides, assume the uranium is natural (if the fuel salt can't go critical w/ that composition, then assume the uranium has 1.5 times minimum U-235 proportion needed for the fuel salt, w/ infinite homogenous volume, to go critical). I'm wondering how the maximum power density of such a fissile mass would compare to the hypothetical operating power density of the reactor the engineer described.
     
  6. Oct 27, 2013 #5
    The fuel composition was given in mole fractions.
     
  7. Oct 27, 2013 #6

    Astronuc

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    As power density increases, the temperature of the fuel would increase, the moderation would decrease (in a water cooled reactor, the moderator density decreases with temperature and with phase change (from liquid to vapor)), and the Doppler resonance absorption increases. At some point, the spectrum shifts from thermal to epithermal and perhaps to fast.

    Perhaps the practical maximum power density is the achieved when the molten fuel starts voiding. The enrichment is certainly a consideration, but perhaps the most significant consideration is reactivity control.

    Voiding and melting temperatures of structural materials is a critical concern regarding structural integrity of the core, and the ability to maintain control of the system. Maintaining control implies coolability and reactivity control. Reactivity control means the ability so control the power level and shutdown the reactor upon demand.
     
  8. Oct 28, 2013 #7

    QuantumPion

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    There is no limit to fission power density as neutron flux can be arbitrarily large. A reactor that becomes prompt critical will increase in power until it is disassembled.
     
  9. Oct 28, 2013 #8

    Astronuc

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    Nevertheless, Thou shall not allow a reactor to become prompt critical! That's a cardinal rule in commercial nuclear energy.

    Certainly a critical system is limited by expansion and subsequent disassembly.

    Prompt criticality is never allowed.

    Some historic criticality accidents.
    http://www.cddc.vt.edu/host/atomic/accident/critical.html [Broken]

    Wikipedia article on prompt criticality has some claims about prompt supercriticality (or near prompt) accidents.
    http://en.wikipedia.org/wiki/Prompt_critical

    Criticality implies k = 1, and power is constant
    Supercriticality implies k > 1, and power increases
    k = 1+ρ, where ρ = Δk. The greater Δk, the faster power increases.

    There was a test, Kiwi-TNT.
    http://www2.gwu.edu/~nsarchiv/radiation/dir/mstreet/commeet/meet6/brief6/tab_l/br6l1k.txt
    http://www.cddc.vt.edu/host/atomic/images/ktntb.jpg [Broken]

    This is why we don't allow prompt-criticality.
     
    Last edited by a moderator: May 6, 2017
  10. Oct 29, 2013 #9

    QuantumPion

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    Neat!
     
  11. Oct 29, 2013 #10

    Astronuc

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    A colleague has some video of a nuclear rocket motor being tested. It was a graphite/carbide core and hydrogen (I think it was hydrogen) was pumped through. They used a sapphire mirror over the nozzle throat so the video camera looks down into the core.

    One could see the oscillations of the fuel assemblies, which eventually started failing, and pieces start flying out of the reactor. From another camera angle, it looked like tracers coming out of the rocket nozzle.

    Nuclear ceramics are fired at > 1600°C, and often > 1800°C during manufacturing. One cannot see anything in the furnace since it is bright white (or maybe a yellowy creamy white) incandescent.
     
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