12.7. Voiding Due to Gas Release from Failed Fuel Pins

The boiling model in SASSYS-1 and SAS4A has been modified to account for voiding due to release of plenum gas from failed fuel pins. The main purpose of this modification is to address the question of whether pin failure in a transient that would otherwise not lead to boiling recovery would be prevented. Another case that can be addressed with this model is one in which pin failure occurs after the onset of boiling and increases the severity of the voiding. DEFORM-5 predicts the time and location of pin failure, or the user can specify the time and location of the pin failure. The gas release then provides a fictitious source of noncondensing vapor for the boiling model. An adjustable rip area and orifice coefficient determine the rate at which gas is released from the pins. Reduction in the plenum pressure is calculated as the gas flows out the rip. The gas fraction in each bubble in the coolant channel is calculated. The presence of gas in a bubble has two effects: it reduces the condensation coefficient and it increases the vapor friction factor. As the gas bubbles out the top of the subassembly a smooth transition is made to normal boiling. Three pin failure groups are used to account for incoherence. Each group represents a fraction of the pins. A separate life fraction is used by DEFORM-5 to predict failure for each pin group. Each pin group has a separate plenum gas pressure and temperature and a separate flow rate out the rupture.

For this model the rate of gas flow from the plenums to the coolant channel is calculated as

(12.7‑1)

\[w_{g} = \sum_{m}^{}{f_{m}A^{*}\sqrt{\frac{2\left( p_{pm} - p_{c} \right){\overline{\rho}}_{gm}}{K_{g}}}\ }\]

where

\(m =\) pin group number

\(p_{pm} =\) plenum pressure for pin group \(m\)

\(p_{c}\) = pressure in the coolant channel

\(A^{*} =\) rip area

\(K_{g} =\) rip orifice coefficient

(12.7‑2)

\[{\overline{\rho}}_{gm} = \frac{\rho_{gm} + \rho_{gc}}{2}\]

\(f_{m} =\) fraction of the pins in group \(m\)

\(\rho_{pm} =\) gas density in the plenum

\(\rho_{gc} =\) gas density in the coolant channel

Note that for \(p_{pm} \gg p_{c}\) and \(K_{g} \text{ ~ } 2\), Eq. 12.7-1 reduces to the expression for choked flow through as orifice, whereas for small pressure differences the equation reduces to the ordinary orifice flow expression.

Also, for a perfect gas:

(12.7‑3)

\[p_{pm}V_{p} = m_{pm}R_{g}T_{pm}\]

or

(12.7‑4)

\[p_{pm} = \rho_{pm}R_{g}T_{pm}\]

and

(12.7‑5)

\[\rho_{gc} = \frac{p_{c}}{R_{g}T_{gc}}\]

where

\(p_{c} =\) pressure in the coolant channel near the rip

\(T_{gc} =\) gas temperature as it enters the coolant channel

\(p_{pm} =\) plenum gas pressure

\(R_{g} =\) gas constant

\(T_{pm} =\) gas temperature in the plenum

The gas flow at the end of the time step is calculated as

(12.7‑6)

\[w_{g}\left( t + \Delta t \right) = \sum_{m}^{}{f_{m}(w_{gm1} + \Delta p_{v}w_{gm3})}\]

where

\(t =\) time at beginning of step

\(\Delta t =\) step size

\(\Delta p_{v} =\) change in coolant channel pressure during the step

(12.7‑7)

\[w_{gm1} = A^{*}\sqrt{\frac{\left[ p_{pm}\left( t \right) - p_{c}(t)\right] 2\overline{\rho}_{gm}(t)}{K_{g}}}\]

(12.7‑8)

\[w_{gm3} = \frac{A^{*}{\overline{\rho}}_{gm}}{K_{g}\sqrt{\frac{\left\lbrack p_{pm}\left( t \right) - p_{c}\left( t \right) \right\rbrack 2{\overline{\rho}}_{gm}(t)}{K_{g}}}}\]

The gas that comes into the coolant channel is converted into enough vapor to have the same pressure-volume product. Thus

(12.7‑9)

\[p_{g} \frac{w_{g}}{\rho_{gc}} = p_{v}\frac{w_{v}}{\rho_{v}}\]

where

\(w_{v} =\) vapor source

\(p_{v} =\) vapor pressure

\(\rho_{v} =\) vapor density

A conversion factor, \(\gamma_{g}\) between gas and vapor is defined as

(12.7‑10)

\[w_{g} = \gamma_{g}w_{v}\]

Then

(12.7‑11)

\[\gamma_{g} = \frac{p_{v}{/\rho}_{v}}{p_{g}/\rho_{g}} = \frac{p_{v}/\rho_{v}}{R_{g}/T_{gc}}\]

The temperature used for \(T_{gc}\), the temperature of the gas entering the coolant channel, is the fuel surface temperature at the rupture node.

The treatment of the gas after it enters the coolant channel depends on whether it is entering a small, nonpressure gradient bubble on a larger, pressure gradient bubble. In a small, nonpressure gradient bubble the gas adds to the heat flow, \(Q_{es}\), from cladding and structure to the vapor:

(12.7‑12)

\[Q_{\text{es}}\left( t + \Delta t \right) = I_{e1} + \delta I_{e1g} + \Delta T_{v}\left( I_{e2} + \delta I_{e2g} \right)\]

\[+ I_{e3}\Delta Z'\left( K,2 \right) + I_{e4}\Delta Z'(K,1)\]

where

\(\Delta T_{v} =\) change in vapor temperature for the time

\(\Delta Z' \left(K, 2 \right) =\) change in lower bubble interface position due to change in bubble pressure

\(\Delta Z' \left(K, 1 \right) =\) change in lower bubble interface position due to change in bubble pressure

\(K =\) bubble number

\(I_{e1}, I_{e2}, I_{e3}, I_{e4} =\) contribution from normal vapor sources

\(\delta I_{e1g}, \delta_I{e2g} =\) gas source contributions

(12.7‑13)

\[\delta I_{e1g} = \frac{\lambda_{v}}{\gamma_{g}}A^{*}\sum_{m}^{}f_{m}\sqrt{\frac{\left\lbrack p_{pm}\left( t \right) - p_{c}\left( t \right) \right\rbrack 2{\overline{\rho}}_{gm}(t)}{K_{g}}}\]

(12.7‑14)

\[\delta I_{e2g} = \frac{\lambda_{v}}{\delta_{g}}A^{*}\frac{\partial p_{v}}{\partial T_{v}}\sum_{m}^{}{f_{m}\frac{{\overline{\rho}}_{gm}(t)}{K_{g}}}\sqrt{\frac{K_{g}}{2{\overline{\rho}}_{gm}\left( t \right)\left\lbrack p_{pm}\left( t \right) - p_{c}\left( t \right) \right\rbrack}}\]

\(\lambda_v =\) heat of vaporization

For a larger, pressure gradient bubble Eq. 12.6-45 gives a mass equation of the form

(12.7‑15)

\[C_{1,J}\Delta p(J) + C_{2,J}\Delta W(J) + C_{3,J}\Delta p(J + 1) + C_{4,J}\Delta W(J + 1) = h_{J}\]

For the mode \(J\) at which the pin rupture occurs the gas adds contributions to \(C_{1,J}, C_{3,J}\) and \(h_{J}\):

(12.7‑16)

\[h_{J} \rightarrow h_{J} + \sum_{m}^{}{\frac{\Delta tf_{m}}{\lambda_{g}}w_{gm1}}\]

(12.7‑17)

\[C_{1,J} \rightarrow C_{1,J} + d_{2}\]

(12.7‑18)

\[C_{3,J} \rightarrow C_{3,J} + d_{2}\]

(12.7‑19)

\[d_{2} = \sum_{m}^{}{\frac{\Delta t}{2\gamma_{g}}f_{m}w_{gm3}}\]

The gas plenum calculations for a time step are done in two parts. First the effects of gas release are calculated. Then the heat flow from the cladding to the gas is accounted for as before.

The gas mass lost from the plenum in pin group \(m\) during a time step is calculated using

(12.7‑20)

\[m_{pgm}\left( t + \Delta t \right) = m_{pgm}\left( t \right) - \Delta t\left( w_{gm1} + \frac{\Delta p_{v}}{2}w_{gm3} \right)\]

where

\(m_{pgm} =\) gas mass in plenum for group \(m\)

Then the remaining gas in the plenum is expanded adiabatically:

(12.7‑21)

\[p_{pm}\left( t + \Delta t \right) = p_{pm}\left( t \right)\left( \frac{m_{pgm}\left( t + \Delta t \right)}{m_{pgm}\left( t \right)} \right)^{\gamma}\]

where

\(\gamma =\) gas specific heat at constant pressure/specific heat at constant volume

The gas temperature, \(T_{pgm}\), for the first part of the step is calculated as:

(12.7‑22)

\[T_{pgm}\left( t + \Delta t \right) = \frac{p_{pm}\left( t + \Delta t \right)V_{p}}{m_{pgm}\left( t + \Delta t \right)R_{g}}\]

where

\(V_{p} =\) gas plenum volume

Then heat flow from the cladding is used to recalculate \(T_{pgm} \left(t + \Delta t \right)\) and \(p_{pm} \left(\ t + \Delta t \right)\).

The amount of gas in each bubble is kept track of, and the gas fraction in a bubble affects the condensation heat transfer coefficient and the friction factor. The mass of gas, \(m_{gbk}\), in bubble \(k\) is calculated using

(12.7‑23)

\[m_{gbk}\left( t + \Delta t \right) = m_{gbk}\left( t \right) + \frac{\Delta t}{\gamma_{g}}\sum_{m}^{}{(w_{gm1} + \frac{\Delta p_{v}}{2}w_{gm3})}\]

The total mass, \(m_{tk}\), in each bubble is calculated as

(12.7‑24)

\[m_{tk} = \int\rho_{v}\left( z \right)A_{c}\left( z \right)dz\]

where

\(A_{c} =\) coolant flow area

The gas fraction, \(f_{gk}\), in bubble \(k\) is then defined as

(12.7‑25)

\[f_{gk} = \frac{m_{gbk}}{m_{tk}}\]

Gas stays in bubble until it blows out the top of the subassembly. When a vapor bubble extends far enough out the top of the subassembly, part of the bubble is assumed to break away and go up into the outlet plenum. If the bubble upper position, \(z_{iuk}\), exceeds the top of subassembly, \(z_{tsa}\), by an amount \(\Delta z_{bu}\) (e.g. \(z_{iuk} \geq z_{tsa} + \Delta z_{bu}\)) then the part of the bubble above \(z_{tsa} + \Delta z_{bl}\) is broken off from the bubble. When this happens \(m_{gbk}\) is decreased by the amount of gas in the part broken off:

(12.7‑26)

\[m_{gbk} \rightarrow m_{gbk} - \Delta m_{g}\]

where

(12.7‑27)

\[\Delta m_{g} = f_{gk}A_{c}\rho_{v}\left( z_{iuk} - z_{tsa} - \Delta z_{bl} \right)\]

Then the upper interface of the bubble is moved down to \(z_{tsa} + \Delta z_{bl}\):

(12.7‑28)

\[z_{iuk} \rightarrow z_{tsa} + \Delta z_{bl}\]

Also, each time that the top part of a bubble is broken off, the velocity of the upper interface is cut in half. The condensation coefficient, \(h_{c}\), in a bubble containing gas is calculated as

(12.7‑29)

\[h_{c} = \frac{h_{co}}{1 + 100,000f_{gk}}\]

where

\(h_{co} =\) normal vapor condensation coefficient.

Gas streaming in a bubble will have a different friction pressure drop than sodium vapor moving at the same velocity, so in this model the vapor friction factor is adjusted when gas is present in a bubble. The friction pressure drop, \(\Delta p_{f}\), due to vapor streaming in the coolant channel is

(12.7‑30)

\[\Delta p_{f} = \frac{L}{D_{h}}f\rho_{v}v^{2}\]

where

\(f =\) friction factor

\(v =\) vapor velocity

\(L =\) length

\(D_{h} =\) hydraulic diameter

When gas is turned into fictitious vapor in the coolant channel, its pressure-volume product is conserved; so its velocity should be approximately correct. On the other hand, fission product gas has a higher density than sodium vapor, so the friction pressure drop will not be correct unless the density ratio is accounted for. In this model, the friction factor is adjusted to account for the density ratio:

(12.7‑31)

\[f = f_{0}\left(\frac{M_{\text{wg}}}{M_{\text{wNa}}}f_{\text{gk}} + 1 - f_{\text{gk}}\right)\]

(12.7‑31)

where

\(f_{0} =\) normal friction factor

\(M_{wg} =\) molecular weight of the gas

\(M_{wNa} =\) molecular weight of sodium, taken to be 46