Natural Circulation and Decay Heat

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Discussion Overview

The discussion revolves around the theoretical aspects of natural circulation in a nuclear plant during shutdown conditions, specifically focusing on calculating core temperature differences (ΔT) and natural circulation flow rates in relation to decay heat. Participants explore relevant equations and principles that govern these phenomena.

Discussion Character

  • Homework-related
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • One participant expresses difficulty in finding equations related to natural circulation and decay heat, seeking assistance with specific problems involving initial and final ΔT and flow rates.
  • Another participant suggests simplifying the problem to a fluid flow scenario, noting that the driving head is proportional to ΔT and that flow is proportional to the square root of the head.
  • A different participant agrees with the approach and mentions that ΔT and flow are proportional to fractional powers of decay heat, hinting at the involvement of exponents 2 and 3.
  • One participant references licensing submittals from NuScale Power, indicating that their small modular reactor (SMR) design relies on natural circulation rather than forced circulation.
  • Another participant shares equations for relating power and core temperature, as well as natural circulation flow rate and core temperature, concluding that the problems can be solved through ratios.
  • A later reply provides a derivation involving relationships between work, pressure, density, and temperature differences, ultimately leading to a formula for ΔT in terms of power.

Areas of Agreement / Disagreement

Participants generally agree on the approach to solving the problems through ratios and proportional relationships, but there is no consensus on the derivation details or the underlying assumptions of the equations presented.

Contextual Notes

Some participants express curiosity about the derivation of the equations used, indicating a potential gap in understanding the foundational principles behind the ratios. Additionally, the discussion does not resolve how the specific equations are derived or their applicability in different scenarios.

mudweez0009
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Homework Statement



I am having issues with some problems relating to a plant theoretically shut down on natural circulation, and calculating the core ΔT and natural circulation flow rate.. Can anyone provide some equations or theory I could use to assist me? I'm not familiar with this material and have spent hours searching Google and cannot find much.

Problem 1:
Givens:
Initial Decay heat = 2.5% rated thermal power.
Initial Core ΔT = 14 deg F
Final Decay Heat = 1% rated thermal power
Find: Final Core ΔT.

Problem 2:
Givens:
Initial natural circulation flow rate = 3.5% full power flow rate.
Initial Core ΔT = 15 deg F
Final Core ΔT = 8 deg F
Find: Final natural circulation flow rate


Homework Equations



I don't know any equations for this, I have searched for hours and cannot seem to find anything. If someone could reference me to any material they know of I can attempt a solution on my own first.

The Attempt at a Solution



(see 2. above)
 
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thought about this overnight

make it a simple fluid flow problem.

piping configuration does not change.

driving head is in proportion to ΔT

flow is in proportion to √head

heat removed (power) is product of ΔT and flow

should be able to ratio things out

old jim
 
Jim Hardy has outlined the correct approach. You will find that delta T and flow are proportional to fractional powers of decay heat. I will leave the specifics to you as an exercise! (Hint: the exponents involve 2 and 3). At SONGS, I had to do this evaluation to show the NRC that operating at 70% power led to a better natural circulation flow to power ratio. Alas, we never restarted.
 
You might also check any licensing submittals that NuScale Power has made to the NRC for their SMR as their design does not used forced circulation and relies on natural circulation.
 
Okay I got the equations that we are supposed to use...

For power and core temp:
ΔT1/ΔT2=Q1(2/3)/Q2(2/3)
where, Q=power (%)

For natural circulation flow rate and core temp:
ΔT1/ΔT2=m12/m22


From here, the problems are incredibly easy. So yes, I guess it is just a ratio. Not sure how the equations come about, but I guess proofs are not my main concern at the moment. However... It's me, and I'm curious.

Thanks for the responses!
 
Her's a derivation:

W = Sqrt(Delta P)

W=Sqrt(Delta (rho*g*h))

W=Sqrt(Delta T*g*h) = Sqrt(Delta T)

Q=W*Delta T

Delta T = Q/W

W=Sqrt(Q/W)

W^2 = Q/W

Q=W^3

So, W=Q^(1/3)

Q=Q^(1/3)*Delta T

So, Delta T = Q^(2/3)
 
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