Fluid momentum and Natural circulation in Nuke reactors

[Mordukon]

How is the net driving pressure difference within a PWR a function of T and H

delta P = delta T(Core) * g * partial Rho(T, P(RCS))/partial T * delta H

I get how this correlates to the pressure differential and is found with bernoulli's equation but how does the temperature dependence get added? Do you need to derive from the conservation of energy?

Sorry first post so I am not sure if this is as coherent as I am hoping it is.

 

[Astronuc]

The net driving force/pressure involves the difference in head between hot and cold vertical chambers/piping, if the system relies on natural convection. The head is a function of height and density (per unit length).

The cold leg from the exit of the steam generator (primary heat exchanger) to the core inlet would be considered to have the same temperature and density. The core has a density gradient due to the ΔT (T_exit - T_inlet) across the core, and the hot leg has the lowest density at T_hot = T_exit.

Current LWRs use forced convection.

 

[jim hardy]

simplify your thinking. Some people i believe actually do think in equations but i have to resort to pictures..

One of our guys said : "Being as natural circulation is natural, you got to work really hard to foul it up."

try this simplified thought experiment :

remember how a manometer works - in a u shaped tube half filled with liquid , liquid in both sides comes to same level. Try it in your bathtub with a piece of clear plastic tube..

what if the two legs of the manometer are filled with different liquids?
If left tube were filled to one inch above bottom with mercury, it would take 13.65 inches of water above bottom of right tube to balance it.

That's because rho X g X h for water is 1/13.65th of mercury's. For same g & h mercury makes 13.65X more pressure.

If instead the two tubes contained water of slightly differing densities, difference in heights(pressures) would be way less of course but still there.
you could use water and cooking oil for a visual demonstation...

Now continue thought experiment:
completely fill your manometer and tie its tubes together at top;
warm the right tube (makes its water less dense) and cool left tube(makes its water more dense)
right tube's level wants to rise to balance the heavier water in left tube
but the warm water can't go up, the connecting tube is horizontal so it can only only flow over into left side of manometer - and there it gets cooled back off...
SO- you have built a simple machine that moves water from one tube to another as a result of adding heat one place and removing that heat at another. It has no moving parts and requires only a heat source and a heat sink to work. Well, gravity too.

Observe the motive force for the water is [rho X g X h cool side] - [rho X g X h warm side]

which can be simplified! to the partial differential equation you gave.
{okay, i admit you'll get more accurate result by express rho & t as f(h) and integrate - i love math but math only loves to humiliate me}
Bernoulli would let you figure velocities i suppose. But your question was more basic than Bernoulli.

Natural circulation requires only that pipes remain filled and heat sink be above source. That's why in PWR's steam generator is placed well above reactor.

hope this helps.
old jim

 

[Mordukon]

Beautiful thank you both so much, can't believe I forgot about expanding fluids

 

[Astronuc]

Also, the flow rate determines the temperature/enthalpy rise across the core.

BWRs allow for boiling, so soluble boron is not used for reactivity control. Reactivity control must be accomplished by control rods/blades. The benefit of boiling is the reduced density as compared to single phase.

In PWR using natural convection, one does not have the benefit of bulk boiling, although there could be limited nuclear boiling.

 

[jim hardy]

yep, gravity pulls down harder on the dense side.
but you already knew that.

a Westinghouse PWR could, in theory, get to around 15% power on natural circulation and thereby bootstrap itself up into operation.

But that'd be quite a maneuver. In our plant such "cowboying" would go strictly against procedures, legal regulations and good sense.
Might be a handy feature in a submarine though - i don't know if they can do that.

old jim

 

[gmax137]

The 'practical' application of natural circulation in PWRs is during loss of offsite power events, where the reactor is tripped and the normal reactor coolant pumps are de-energized. The N/C flow is sufficient to transfer the decay heat from the core to the steam generator without the core temperature rise exceeding the normal full power value. This works because (1) the SG is high enough above the core (as old jim mentioned), (2) because the decay heat is much lower than the full power, and (3) because the coolant pipes etc. are large enough that the resistance to flow is small (at the low flow rates seen during N/C conditions).