Several examples are presented to clarify certain sections of the foregoing text

a. Routh-Hurwitz criterion (refer to Section 2.5.3.1). Figure 2.30 shows a simple feedback loop for an imaginary fluid-fueled system in which the fuel produces a power P while in the critical configuration of the core but gives up a proportion of its heat a while on a path through an IHX. The circuit time is в. The equations are:

Fuel

dT/dt = P— h(T— Ті)

Neutron kinetics

dPjdt = (6k PIl*) — (aTjl*) + (aTjl*)

Circuit

Ti = aT with a delay of 6

No delayed neutrons are assumed. Taking the Laplace transform,

ST — T0 = P — hT + hT{ (2.27)

sP-P0= (6k PJl*) + (6k0 P/i*) — (aTjl*) (2.28)

= «77(1 + 6s) (2.29)

and in the steady state it is assumed that

Ti0 = aT0 and 6k0 == 0

P0 = h(T0 — Ti0) and take P0 = 1 (2.30)

From these relations the frequency response of each pair of variables Tj6k, P/6k, and TJdk may be calculated. Defining G(s) = P/6k" and H(s) = 6k’ jP, where

6k" = 6k—6k’ and 6k’ = (aT—aT0) (2.31)

the characteristic equation is given by

so 1 + G{s)H{s) — 0 gives a cubic equation:

s3l*6 + s4*( 1 + 6h) + s(/*A(l + a) + «0) + a = 0 (2.34)

The Routh-Hurwitz criterion for stability gives the following array:

1*6 l*h( + a) + ad

/*(1 + Bh) a

had2 _ (2.35)

(1 + 6h)

0

from which the criterion of no sign changes in the first column gives:

Both of these statements are satisfied by a positive a, so the system is stable for all negative Doppler coefficients (—a).

b. Nyquist criterion. (1) A system in which G(s) = —K(s + 2)_1 is shown in Fig. 2.31. To draw the locus of G{im) notice that for

(o = 0, G(ico) = —K

(o = oo, G(im) = 0

Thus the locus starts from the origin and has a diameter of K (Fig. 2.32). The number of poles P of G(s) in the right half-plane is zero (only s = — 2 exists in the left half-plane) so, for stability (Z = 0), there should be no encirclements of the origin by 1 + G(s). To avoid any encirclements by G(s) of the point — 1 and to ensure stability for this system, the criterion is that К must be less than 2. Any adjustment to the system to make К smaller to obtain this condition will be a gain adjustment.

Fig. 2.32. Nyquist plot (7) for G(/co).

(2) If the system is now represented by

G(s) = Ks-^s + a)-1^ + b)-1 (2.38)

with a feedback loop shown in Fig. 2.33, the locus in the G(ico) plane is shown in Fig. 2.34a as for

a> = 0, G(ico) = oo

со = go, G(ico) -= 0

The plot in the s plane is shown in Fig. 2.34b. Here P = 0 for a, b > 0 so there should be no encirclements. The stability criterion is now that K/ab(a + b) must be less than unity to avoid encirclements of the point — 1. Thus the maximum gain К is ab(a + b). This is called the gain margin when related to the actual value of K.

Note that in this example an wth order pole at the zero point in the s plane is mapped into a counterclockwise rotation of rm of infinite mag­nitude in the G{s) plane. H(s) is, of course, unity in these examples.

(3) The effect of a time lag is that it tends in general to make the closed — loop system more unstable. The lag exp(—sT) in association with G(s)=K/s

Fig. 2.34a. The s plane contour (7).

Fig. 2.34b. Nyquist diagram (7) for G(iw) = Kl[io)(ia) + a)(iw + b)].

gives the Nyquist diagram shown in Fig. 2.35, where the time lag is a linearly increasing, frequency-dependent function. For this system, the stability criterion is that К should be less than nT.

c. Bode plot. For a system characterized by

G(s) = K/(l + sT)» (2.40)

the gain is

20 log] G{i<o) I = 20 log К — n20 log] (1 + іТш) | (2.41)

The separate parts of this expression can be plotted and added as shown in Fig. 2.36.

If now a feedback H(s) = F(1 + s9) is required, then the gain of G(s)H(s) is given by the sum

total gain = 20 log AT — 20nlog|(l + гТсо)| + 201ogF + 20 log] (1 + івш)|

(2.42)

for all frequencies со. In order to establish the stability margin, this gain is plotted against the phase after eliminating the frequency parameter. The criterion of Section 2.5.3.2 is used.

Following local damage it is important to contain the existing damage as locally as possible and to prevent the spread of damage. One must consider a plant protective system, multiple path inlets to assemblies to avoid blockage, strong assembly canning to contain any local failure, and emergency core cooling system (and pony motors on main pumps) to avoid failure after loss of cooling.

3.4.3.2 Prevention of Criticality Following Fuel Melt-down (Consequence Limiters)

The core fuel after fuel melt-down would need to be prevented from accumulating into a critical mass (see also Section 5.6.1). Preventatives could include large heat sinks at subassembly inlets to freeze molten fuel from single subassembly, dispersion cones in and out of vessel, melt catchers (assembly or core size), terminal cooling systems, or ground disposal to permanently avoid criticality.

As the fuel slumps against the subassembly duct wall, heat fluxes reach several million Btu/hr-ft2. The exact configuration of fuel against the wall does not matter because, even with small amounts, the fuel farthest away from the wall is soon at its vaporization temperature of 6500°F if no credit is taken for heat removal in the failed assembly.

A heat transfer calculation for this slumped fuel and the two adjacent subassembly walls and the neighboring subassembly may be calculated for a simple slab model to determine whether boiling (and thus an effective further failure) can be induced in the second subassembly. Figure 4.18 shows some typical results in which the adjacent subassembly is analyzed for its temperature distribution, a distribution due to an incoming heat flux from its neighbor. The COBRA code (29, 30) was used to compute the cross flows and the temperatures in each subchannel. In the particular case shown, the highest coolant temperatures occur in the shut-down case in which it is assumed that the reactivity feedback from the failed assembly has given a high flux signal and tripped the reactor. In this case a maximum tempera­ture of 1660°F is reached, which shows that boiling may occur only if the system is at atmospheric pressure. However, primary systems are usually slightly pressurized and boiling would not occur. Not only does the slumped

fuel result in high heat fluxes to the adjacent can wall, but it also melts a certain amount of the can wall itself, attaining some sort of steady state in about 10-20 sec.

The final consequence of the subassembly failure depends on whether the reactor has been tripped or not due to reactivity feedbacks, whether or not the subassembly duct has been ruptured due to high pressure pulses, whether or not high heat fluxes have caused adjacent channel boiling, and what the final configuration of fuel in the failed subassembly is.

These answers depend on some experimental information that is still required: How does the film remaining after voiding grow and dry out, how does fragmented fuel behave in a voiding environment, how does fuel fragment, how do molten fuel pins slump, what is the response of the duct to pressure pulses within the assembly and locally against the duct wall, etc. It is expected that when these answers are forthcoming, surety will be obtained that failure cannot propagate beyond a single subassembly.

Whereas there is only slight evidence for a genetic damage threshold, there is good evidence for a threshold for somatic effects. It is just over 20 rem (Fig. 5.2). Below this level no somatic effects or bodily effects to the individual himself occur and above this the level of damage rises until for acute exposures, it is possible to talk of the percentage of fatalities at certain whole body doses above 100-200 rem.

A scale of values for damage in terms of doses for an average individual is the following: less than 20 rem causes no observable reactions; 30-50 rem bring detectable changes in the blood; 50-100 rem produces nausea and vomiting; 400-500 rem gives an individual a 50-50 chance of survival without medical care; and 600-1000 rem may be lethal.

Work at the Naval Ordnance Laboratory produced a correlation to relate the strain in a vessel wall to the chemical explosive charge (22, 23).

The correlation of Eq. (5.10) expresses a relation between the shock energy W and the strain є in terms of the dimensions of the vessel and material properties of the vessel wall

1.407o-t££°-85(3.41 + 0.117 Rilh0)(Re2 — Rf)1*5
10s(l.47 + 0.0373 /?i/A0)°-15 R°-u

where

0*t (Ту — f — (£/£ц)[(Тц(1 £u) (Ту] (5.11)

and (Ту is the yield stress (psi), є is the strain, £u is the ultimate strain, eru is the ultimate stress (psi), q is the density (lb/ft3), 7?e is the external radius (ft), Rt is the internal radius (ft), h0 equals Re — Ru IF is the blast energy (lb TNT). Use of the correlation assumes that the vessel absorbs the strain energy radially and uniformly in the cylindrical wall, that the internal structure in the vessel has no effect, and that the classic stress-strain curve applies. The correlation applies only to radial sections of the vessel and another method must be used to assess damage to the bottom head. The top head is assumed not to see shock damage due to the large attenuation of shock pressures before arrival.

The Proctor correlation has been applied to an analysis of the SL-1 accident vessel damage to determine which observed radial damage strains could be attributed to shock and which could be attributed to blast and water hammer effects. The application provided a good representation of the observed facts and provides therefore some confirmation for the use of the correlation (23).

Figure 5.9 provides a plot of the SL-1 circumferential residual strain as a

function of the vessel length showing the shadowing effect of the thermal shield.

From this measured strain, the strain energy of each of the 24 rings, into which the vessel was divided, was determined by integrating the classical stress up to the strain value for each ring according to

SE — inRmth (ode (5.12)

J о

where SE is the strain energy (ft-lb), Rm is the initial radius to the center of the ring (in.), t is the thickness of ring (in.), h is the height of ring (in.), a is the classical stress in ring (psi), and e is the strain in the ring (in./in.). This strain energy was then summed for different regions of the vessel on the assumption that different mechanisms for damage existed in each region.

It was assumed that the water coolant was cold at 300°K and that the energy release was 50 MW-sec. This energy release was equivalent to that part of the energy produced in the inner fuel elements that could be rapidly transferred to the water. The 50 MW-sec was assumed to be equivalent to

26.2 lb of pentolite producing about 459 moles of detonation gases with a pressure of 21,230 psi, calculated from the ideal gas law and a charge volume of 0.276 ft3.

Now calculating the reduction of this pressure due to the expansion of the vessel, the figure of 1780 psia was obtained and a value for the work expansion was calculated from a P dv integration. The study assumed that the gas bubble accelerated the water above the core until it hit the vessel head, and calculated the work done in this expansion by another P dv integration for the increase in gas volume due to the fact that it has moved the water upward. This expansion then left a residual pressure of 133 psia, and produced a hammer velocity of 147 ft/sec.

Table 5.11 shows a comparison of the damage estimates for the SL-1 and the Proctor calculations. The correlation of Eq. (5.10) was used to predict a maximum strain from the initial charge weight, on the assumption that the damage expected for a slow energy release would be similar to that from a pentolite explosion of one-eighth of the prompt nuclear energy release. The correlation predicted a strain of 0.028 in./in. against the meas­ured maximum of 0.023 in./in.

It is recognized that using the Proctor correlation for the prediction of shock damage to the vessel wall is pessimistic in view of the different char­acteristics of nuclear and chemical explosions. However, Proctor (22, Ta­ble 3) does give recommended reductions in effective charge weight for slower releases.

The principal fossil fuels, bituminous coal, natural gases, and oil, give off a variety of pollutants that include carbon monoxide, oxides of sulfur and nitrogen, hydrocarbons, and particulate matter. Of these, sulfur dioxide is the most critical.

Oil and gas also give rise to large quantities of nitrogen dioxide, while coal and oil contribute fly ash particulates. The exposure to these pollutants is measured in an individual by the action on his lungs, and in some cases, standards apply which limit the concentration of the pollutants in the air.

To place the emission of fossil-fuel pollutants in perspective, it is worth noting (13) that the discharge quantities of sulfur dioxide amount to be­tween 27 lb/MWe-yr for gas and 306,000 lb/MWe-yr for coal, with oil coming in between. These figures, when compared to the concentration standard of 0.3 parts per million, imply that a dilution of the pollutant to the required standard may need up to nearly 2-10nm3 of air/MWe-yr (see Table 6.4).

In contrast, a PWR plant may discharge 5-10-103 //Ci/MWe-yr. In order to dilute this to the concentration standard required by the AEC 10CFR20 regulations, 10-7//Ci/cm3, approximately 5-10-104m3 of air/ MWe-yr is needed. In other words, the required dilution is something greater than a million times easier for the nuclear plants.

The RBE for radium and thorium from fossil-fuel plants is much higher than for the nuclear plants’ krypton and iodine, therefore only a little radium release can have a relatively large effect. This is exhibited in Table 6.4 and also by the dose rates quoted for three power plants: Con­necticut Yankee PWR (1968) gave 1.2 • 10~e [irem/hr-MWe, Dresden I BWR(1968) gave 8.7 • 10-2 [xrem/hr-MWe, while Widows Creek fossil plant gave 3.5 • 10-5 [xrem/hr-MWe. Thus the fossil plant gave just as much radioactive emission as nuclear plants even though it relied on an 800 ft stack to dilute effluent by a factor of 100 {9b).

Fast reactors do not differ very significantly from the LWR plants al­though they do discharge less and will have no difficulty in bettering the already excellent PWR standard.

The discharge of pollutants from fossil-fueled plants is being reduced. Effective methods for the control of particulates already exist and are in use. A Dolomite system for the removal of sulfur dioxide involves using finely divided limestone in a combustion chamber {13). Following a wet scrubber treatment, using an aqueous suspension of limestone or lime particles, it is hoped that up to 80% of the sulfur may be removed. This system is now being tested on some operating power plants. Significant reductions in the nitrogen oxides will require new combustion processes; in the meantime plants are being built with taller stacks to increase the dilution factor for the remaining discharged pollutants.

In sodium, boiling exhibits quite different characteristics than in water and stable boiling does not appear possible at reactor flow rates. The specific volume of the vapor at low pressure is high and it requires very high

f See Tilbrook and Macrae (10).

velocities for stability; higher than can be provided by available pump designs.

So in a sodium cooled system, inadvertent boiling is presumed to lead rapidly to a bulk-boiling region in which an annular flow representation is the best model. In this model the sodium vapor expands against the inertia of sodium liquid slugs above and below it. The value of Tc is constant at the saturation value because all the heat goes into converting more liquid into vapor. The bubble can extend outside the core into a region of signifi­cant condensation (Fig. 1.10).

The mathematical representation uses the conservation equations of mass and energy for the saturated region to define the size of the bubble, the velocity of the interface between vapor and liquid, and the pressure within the bubble. This model has no need of special boundary representations between boiling and nonboiling regions, but there are considerable un­certainties. The uncertainties include:

(a) The superheat value at which boiling occurs, which may range as high as 500°F but is expected to be as low as 30-50°F in a reactor environ­ment.

(b) Heat transfer data and the conditions under which a liquid film is retained on the fuel-pin cladding adjacent to the bubble.

(c) Condensation effects within a real system.

The condensation that the bubble experiences outside the core is most important since this defines the mode of sodium-vapor bubble collapse, which allows the sodium liquid slugs to reenter the core. This gives rise to
presumed chugging motion (see also Section 4.4.2) in which the vapor bubble alternately grows and collapses until fuel failure results from the reduced heat transfer.

However boiling within a sodium-cooled breeder is an accident condition and may give rise to rapid fuel failure. Thus, in a design model of the core, sodium boiling would not be included and the design would almost certainly have considerable margins before boiling might be initiated. For a 35 psi cover gas system the sodium might boil at 1850°F whereas the highest coolant temperature would be approximately 1200°F, leaving a temperature margin of over 600°F.

It is clear that these three systems will have very different responses to flow perturbations and because of the high pressures involved, flow per­turbations in the gas — and steam-cooled systems due to system rupture are classified as depressurizations (Section 2.2.4).

2.2.1 System Modeling

The modeling of the coolant flow has been treated in the first chapter. The flow enters the core at a low temperature and is raised in temperature through the core. It then transfers the removed heat to a heat sink in the form of a heat exchanger or a turbine.

While in the core, the heat balance in the coolant is represented by an equation of the form

mccc dTJdt = hf(Tf— Tc) — mccct>c dTJdz (2.1)

where the heat-transfer coefficients are flow dependent, according to some correlation between the Nusselt and the Reynolds and Prandtl numbers:

К = a + 0.025(Re)°-8(Pr)0-8 = hfdJkA (2.2)

Thus a reduction of flow reduces the heat removal term in Eq. (2.1) directly through the velocity vc and so Tc increases. The flow reduction also decreases the heat transfer ht and the fuel temperature T{ increases by more than just the increase in T0. It is noted that the coolant temperature in­creases first and is a primary indication of flow reduction.

Flow perturbations arise from: (a) system malfunctions such as pump failures, a loss of system pressure, or blockages which might arise from isolation valve malfunction; and (b) local blockages within the subassembly or at its inlet. The latter cases of local blockages are treated in Section 4.4; this section is concerned with overall system malfunctions.

In a gas-bonded fuel pin, failure might be defined as being coincident with cladding rupture, which, if the pin were originally unvented, would allow the release of fission gases.

The rupture will depend on conditions within the fuel pin that depend in
turn on the power history of the pin and its burn-up. The structural changes may be summarized as follows:

a. At start-of-life. Figure 3.1 shows a cross section of a gas-bonded, stainless-steel-clad oxide pellet with a fuel density of from 85 to 95% theo­retical density. The design includes a 5 mil cold gap between the fuel and the cladding.

Fig. 3.1. Structure of fuel, cold and at the start-of-life.

b. At low bum-up. Fuel above the sintering temperature (3272°F) sinters and lenticular voids migrate up the temperature gradient and a central void is formed. The fuel above the sintering temperature attains the theoretical density and columnar grains mark the migration routes of the lenticular voids. Fuel at less than the sintering temperature remains at its original density. The void migration rates fall off rapidly over a small radius in­crement.

c. At higher burn-up. Continued irradiation (to about 25,000 MWD/tonne) causes fuel swelling and fission gases are generated. The rate of fuel swelling is higher than that of the cladding and so the fuel meets the cladding and then there will be a fuel-cladding contact in the higher rated regions. Fission gases are released from the fuel and they diffuse to the fuel pin gas plenum (Fig. 3.2).

d. Later in life. At about 70,000 MWD/tonne the central void starts to decrease and could close eventually. The cladding at the end-of-life is weaker due to thermal cycling, erosion, and corrosion. Reductions in cladding tensile strength have been reported of between 15 and 30% over 90,000 hr irradiation {2a), and the strain on the cladding might be as much

as % (2b). At this stage the fuel pin is most sensitive to adverse conditions as in addition, the pressure of fission gases within the fuel pin plenum may be between 800 and 3500 psia depending on the size of plenum (2a).

It is assumed that the amount of fission gas released from the fuel is a function of the temperature of the fuel (3a, b): 100% is released at tempera­tures over 3272°F; 50% is released at temperatures between 2912°F and 3272°F; and 4% is released at temperatures less than 2912°F. Thus the total gas released can be calculated by integrating over the fuel temperature distribution. Values of about 65-70% can be expected. During a transient when the temperatures increase little additional gas will emerge.

In addition to this normal operational data, the TREAT facility has provided additional information on abnormal conditions in a transient (4):

(a) Cladding deformations of greater than 1% result in gross cracking on the inner cladding surface due to heavy grain precipitation.

(b) No foaming of irradiated oxide fuel occurs even when 70-80 vol % of the fuel is melted.

(c) Agreement with the calculated temperatures is fairly good. This indi­cates that the fuel condition can be successfully modeled.

(d) In irradiated fuel, the failure mechanism appears to be cladding melting due to contact with molten fuel.

With this information the following failure criteria may be derived.

The presence of bubbles has been noted as providing valuable nucleation sites (6) at which coolant boiling might start, if the heat removal from the system were inadequate.

In pure sodium, superheats of more than 500°F have been observed before boiling commenced due to overheating (7). If such superheats were possible in reactor accident conditions, then Table 4.2 shows that the time

needed to void a channel completely would be much shorter and the excess channel pressures would be much greater. Thus, although accident condi­tions are to be avoided, if they can not be avoided then it would be prefer­able to have sodium boiling occur relatively quietly at little or no super­heating.

The radiation present in the reactor core is expected to limit practical superheats to less than 200°C, but it has been estimated (6) that one bubble of radius 10-3 cm in every 100 cm3 of sodium would prevent superheats of more than 10 or 20°C. Absorbed gas appears to have little or no effect, but some entrained small bubbles are likely to maintain conditions at the lower end of Table 4.2. Fortunately, in a reactor system there will always be one or two small bubbles.