Natural Circulation and Decay Heat

[mudweez0009]

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)

 

[jim hardy]

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

 

[FermiAged]

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.

 

[FermiAged]

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.

 

[mudweez0009]

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!

 

[FermiAged]

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)