The following experimental impact facilities at AEEW were typical of those involved in collaborative agreements with French and German companies. As explained in Section 6.2 the provision of micro-concrete which accurately replicates the bonding between actual concrete, aggregate and steel armatures is of paramount importance. Because commercial suppliers were considered unable to meet the required standards of consistency, a small manufacturing laboratory was constructed with an associated suite of measuring devices to test the starting materials of cured concrete and reinforcing steel [106]. Considerable care was also taken to ensure the precise locations of reinforcements. Some 70 test specimens were taken in the form of cubes, cylinders and beams from each concrete mix, and some 10 to 20 pull-out discs were included in each target to assay the quality of the cured material. These tests generally confirmed a ratio ofcompressive to tensile strength of 10:1, which is typical of a prototype. Material data was

obtained using a 3 MN hydraulic press which could induce constant, ramp, triangular, sinusoidal or step loadings on a variety of test geometries.

To satisfy the velocities for replica scaling shown in Table 6.4, the two compressed-air guns shown in Figure 6.7 were constructed. The earlier and smaller Missile Launcher had the performance specifications:

Maximum projectable energy Interchangeable barrel IDs Projectile velocity range Target abutment

Compressed air within a reservoir provided the variable energy source and thereby a variable projectile velocity. A tubular barrel was separated from the reservoir by a thin diaphragm of metal or melamine according to the required operating pressure. Firing was activated as appropriate by an explosively fired metal dart or by electrically fusing an overlaid matrix of thin copper wires. By initiating a penetration longer than the critical length [96], a diaphragm collapsed virtually instantaneously and the missile itself or in a wooden sabot was accelerated by compressed air along the barrel. Missile velocities up to impact were measured by three independent systems: light beams, fine transverse wires and high­speed cine at 10,000 fps. In the event of target penetration, the subse­quent missile velocity was recorded by a similar fine-wire system, high speed cine and two induction loops. As well as visual records of the target’s response, analog tape recorders (3dB at 80kHz) monitored as many as 120 transients from linear displacement transducers and resistance-type strain gauges. Typical diamond sawn cross-sections of damaged reinforced concrete targets are shown in Figure 6.8 where a smooth missile entry occurs because concrete is some 10 times stronger in compression than in tension.

The Horizontal Impact Facility was constructed later to investigate primarily the regulatory compliance of irradiated fuel transport flasks. It had the performance specifications

Maximum projectile energy Maximum projectile mass

Maximum projectile velocity Interchangeable barrel IDs Post-stressed concrete abutment

Projectiles were fired from the 0.5 and 1.0 m barrels in the same way as with the Missile Launcher, but firings with the 2 m barrel generally required special arrangements like that shown in Figure 6.9. Here a driver plate guided by four rails first propelled the cradled missile along

six supporting and aligning rails in the breech. After a short distance the plate and trolley were rapidly arrested by buckling sacrificial lengths of aluminium tubing to release the missile along the barrel. Up to five high-speed cine cameras monitored an impact and an infrared system measured missile velocity. As can be seen by the wires atop of the driver plate in Figure 6.9, transducers were sometimes carried by the missile itself. Any induced pitch, roll or yaw on a projectile evidently reduces the direct energy of an impact, so careful engineering was necessary to restrict these to just ±1°. Drop height during flight was also constrained to within 120mm. Quarter-scale replicas of existing steel transport flasks and proposed reinforced concrete designs were successfully tested.

Though valuable in themselves, the principal benefit of these impact experiments lies in underwriting the development of computer simulations. These can now be confidently applied to a wide range of geometries and situations as outlined next in Section 6.4. In passing, the Missile Launcher and Horizontal Impact Facility found commercial applications such as bird strikes on helicopter blades, the survival of

air-transport containers in extreme accidents and the effects of various projectiles on glass windscreens.

With material power production, temperatures and other phenomena change the effective multiplication constant K. Higher temperatures increase the vibrations of component nuclei and decrease their densities. One effect is to widen the effective resonance absorption bands of U-238. Because neutrons are slowed down by scattering in energy steps many times larger than these resonance widths, their non­fissile capture rates by U-238 increase with fuel temperature [58]. This negative reactivity feedback mechanism is called the Doppler Effect,[33] and it is an important re-stabilizing influence on the neutron populations of both fast and thermal reactors. Indeed fast reactor designs in particular have progressively evolved with greater U-238 content and less energetic neutrons so as to exploit the effect. Denoting the mean absolute temperature of the fuel by T, then in terms of the effective multiplication factor it is found that under normal operation

—T = d with d > 0 (2.30)

dT

where the Doppler Constant d is specific to a reactor design. Equation (2.30) shows that due to the Doppler Effect the effective multiplication factor is inversely proportional to the logarithm of the mean fuel-
temperature ratio. Other temperature feedback effects on reactor dynamics are generally associated with

— variations in coolant-density either by thermal expansion or by vaporization that alter the absorption or moderation of neutrons

— thermal expansion of the fuel and control rods similarly alters their macroscopic cross-sections

— thermal expansion of a moderator leading to faster neutrons (i. e., a “harder spectrum”) with more resonance absorptions by U-238 nuclei

— thermal expansion of the fuel cladding (zircalloy or stainless steel tubing) whose “bowing” excludes coolant.

These interactions and their associated time delays are significant features of nuclear reactor dynamics.

Neutron absorption is also significantly affected by the in-pile dwell time of the fuel, and its preceding 7 to 47 h power history due to developing concentrations of Xe-135 and Sm-149. Glasstone and Edlund [58] quantify the former as the more dominant “neutron poison,” and it is the daughter of the fission product I-135 whose half-life is 9.17 h [76]. Whilst a reactor is at power, Xe-135 is transmuted (“burned up”) faster by neutron capture than by its natural decay rate. However, in the event of a complete operational shutdown (trip), its concentration increases progressively even for as long as 12 h because of the relatively faster disintegration of I-135. As a result, it could be impossible to restart power production in this period unless sufficient latent reactivity has been held in reserve (i. e., by pre-trip control rod insertions).

The direct cycle RMBK reactor at Chernobyl was moderated by both graphite and a light water coolant, which was partially converted to steam for electric power production. Unlike heavy water, light water is both an effective moderator and absorber,12 so its conversion to less dense steam reduces both neutron moderation and absorption. In the unauthorized fateful incident, the night-shift operators cancelled trip settings and withdrew all 211 control rods [12]. Progressive reductions in inlet-water flow then resulted in a growing volume of steam in the reactor core, and a net reduction in neutron absorptions eventually

See Section 1.8.

occurred because moderation by its graphite was still sufficient. In this way a progressive positive feedback process was initiated that produced an exponentially increasing reactor power with the prompt time con­stant [58,80,117] of around 1ms, and some hundred times [12] full­rated power resulted. Subsequently by

i. the contact of molten fuel with liquid water [59,212],

ii. the accretion of then more mobile fission products into sizeable bubbles, whose external pressures increase as the surface ten­sion effect is reduced [210], and

iii. hydrogen production [12] from the chemical reduction of steam by graphite,

there was the recorded explosive destruction of the site. A nuclear explosion was not involved.

Due to astute design of core-lattice geometry [61] and fuel enrich­ment, light water reactors outside Russia have always been designed to become under-moderated with increasing steam production in order to stifle such potentially explosive events. Indeed a negative power- reactivity coefficient is a necessary prerequisite for licensing by European Regulatory Authorities.

5.1 A HISTORY AND A MIXING ANALYSIS

A potential explosion when molten reactor fuel mixes with its vaporizable coolant is an example of a more general phenomena. Outside the nuclear industry other highly destructive thermal detonations involve iron + water [186,187]; aluminum + water [188]; liquefied natural gas + seawater [189], soda ash + water [190]; coal tar + water [191]; and glass + water [192]. However, the sparseness of reports indicates that such explosions are far from frequent events. Indeed, if they were other than rare, the corresponding industrial processes would now have been discontinued. Moreover, many technical recordings show molten lava from on-land or subsea volcanoes mixing passively with seawater, and the last documented detonation was at Krakatau in August 1883. This rarity in usually passive industrial and natural processes clearly indicates that special conditions are necessary for a detonation.

Though true scientific investigation began in the nineteenth century [186], the physical processes underlying an explosive interaction only became partially understood after 1960 [193]. Even now, however, knowledge of some aspects remains incomplete. From pioneering

Nuclear Electric Power: Safety, Operation, and Control Aspects, First Edition. J. Brian Knowles.

© 2014 John Wiley & Sons, Inc. Published 2014 by John Wiley & Sons, Inc.

101

experiments with aluminum and water in 1970 Laker and Lennon [194] suggest that the formation of a “macro-mixture” is a necessary precur­sor for an explosive interaction. Then a localized disturbance, such as the partial collapse of a coolant vapor film around the molten metal, triggers a propagating detonation.1 They (correctly) conjecture that the shock wave would escalate in intensity by finely fragmenting the melt to achieve the rapid heat transfer rate[72] [73] [74] for the detonation timescale of around 1 ms. Finally, though hydrogen production is recorded, later experiments by Robinson and Fry [192] prove the oxidation of alumi­num to have been irrelevant. The necessity of first creating a particular coarse-mixture morphology of the two liquids is confirmed by subse­quent experiments with kilogram quantities of molten metals or ceramics. Available evidence establishes that a potentially explosive coarse mixture has a sponge-like structure with a cell size of around 10 mm. Figure 5.1 depicts part of a typical untriggered coarse mixture

recovered from an aluminum-water experiment at AEEW. The follow­ing simplified fluid dynamics analysis confirms the necessity for first creating an appropriate coarse mixture over a timescale significantly longer than the 1 ms of a detonation.

If a one-step division process over t seconds transforms a contigu­ous mass M into identical spheres each separated by their own diameter d, then the work done against the surrounding liquid drag forces is of order [195]

with

p—coolant density.

The thermal energy change of the initially molten corium is

Ec ‘ MCPDT with Cp ‘ 500 J/kg (5.2)

where AT is the temperature difference in the corium prior to and then after a postulated interaction. Computer simulations of Severe Accidents suggest that a fast reactor provides the extreme molten corium tempera­ture of 5000 K, so it is assumed here that

25009 AT <4000 tC (5.3)

with:

Psodium ‘ 800 kg/m3 and Pwater ‘ 1000 kg/m3 (5.4)

As described in Section 5.7 the diameter of a fragmented corium particle must be below about 250 mm for its energy to be transferred over the 1 ms timescale of a detonation. Over this restricted size-range experiments provide the approximate mass mean diameter

d ‘ 100 mm

Granted the total elapsed time of an MFCI event as 1 ms, the above equations give

3.75M<Ef1/Ec <6.0M water coolant

(5.6)

4.70M<Ef1/Ec<7.50M sodium coolant

Hypothetically, if the entire 100 tonne fuel inventory of a PWR were to be involved, it follows that

3.8 x 105<Ef1 /Ec <6.0 x 105 in a PWR (5.7)

and similarly for a 20 tonne fast reactor fuel load

0.94 x 105<EFi/Ec <1.5 x 105 in a fast reactor (5.8)

On the other hand, consider a two-stage process in which a contiguous mass M is first mixed into spheres of diameter D in 1s, and then finely fragmented into spheres of diameter d in t second. The approximate number N of larger spheres is evidently

and their mixing energy requirement is obtained from equation (5.1) as EM ‘ ^4^^)M2 with D ‘ 10 mm (5.10)

Similarly the energy E for finely fragmenting just one sphere of diameter D in t second is

Equations (5.9)—(5.11) yield the two-stage fragmentation energy as

and from equation (5.2)

For the range of temperature differences in equation (5.3) and a total PWR fuel inventory of 100 tonne, equation (5.13) evaluates for a 1 ms detonation as

3.8 x 10_3<Ef2/Ec< 6.0 x 10“3 in a PWR (5.14)

and for a 20 tonne fast reactor fuel inventory

0.93 x 10_2<Ef2/Ec<1.5 x 10“3 in a fast reactor (5.15)

Though the above are merely “scoping calculations,” the orders of magnitude involved indicate that an explosive MFCI with a one-step fine fragmentation process for a sizeable portion of a reactor fuel load is impossible. However, an experimentally consistent two-stage process could involve just a plausibly small fraction of the fuel’s thermal energy, so MFCIs cause concern in water and fast reactor safety assessment [59]. Significantly, the coarse-mixing energy term in equation (5.13) is totally dominant which possibly explains the natural rarity of detonatable morphologies.

Some 715 GW of hydroelectric power are already installed worldwide, and in 2006, it supplied 20% of the global electricity demand and 88% of that from all renewable sources [4]. Large schemes of more than about 30 MW involve the construction of a convex dam across a deep river gorge whose sides and bottom must be geologically sound. In addition, a sufficiently large upstream area must exist for water storage (i. e., availability) and sufficient precipitation or glacial melt must be available to maintain this reservoir level. Viable large hydroelectric sites thus necessitate a special topography and geology, but are never­theless more numerous and powerful than geothermal ones as indicated by Table 1.1. Both renewable sources, however, are reliable and can accommodate the variations in power demanded by an industrialized economy. Water below a dam is drawn-off in large pipes (penstocks) to

drive vertically mounted turbines whose blades are protected from cavitation by a slightly rising outfall to downstream [10].

Formal legislation on carbon emissions [1] and the increasing costs of fossil fuels have been driving global construction programs for hydro­electricity. Suitable large-scale sites in the United Kingdom were fully developed during 1940-1950, and future opportunities will focus on small or microscale plants (< 20 MW) whose total potential is estimated at 3% of national consumption [5]. Redundant factories from the UK’s industrial revolution provide opportunities for microgeneration like the 50kW rated plant at Settle [6], but even after a copious rainfall the claim to supply 50 homes is optimistic. It is to be concluded that no large-scale hydro-sources are available now to compensate materially for the impending demise of the UK’s aging fossil and nuclear power stations. The situation [21] in the United States is that large and small-scale hydro-generation have remained largely unchanged over the past 10 years and that future renewable energy development will center on wind turbines [7].

Dams are sometimes breached by river spates or earthquakes despite the inclusion of such statistics in their design. For example environmental damage and a serious loss of life ensued from the failure of the Banqiao Dam [11] (China). Here there were 26,000 immediate fatalities and a further 145,000 from subsequent infections. No worse nuclear accident could be envisaged than that in 1986 of the RMBK reactor at Chernobyl which is designated 7 on the IAEA scale of 1 to 7. The 186 exposed settlements with a total population of some 116,000 were evacuated within 12-13 days. In the specific context of health issues, the International Chernobyl Project [13] of the IAEA reported

i. “Adverse health effects attributed to radiation have not been substantiated.”

ii. “There were many psychological problems of related anxiety and stress.”

iii. “No abnormalities in either thyroid stimulating hormone (TSH) or thyroid hormone (TH) were found in the children examined.”

The earlier Three Mile Island accident (1979) did not directly cause any on or off-site fatalities, though some occurred from remote road accidents due to the absence of an organized evacuation plan. Historic

catastrophic failures of large hydroelectric dams have thus caused far greater fatalities than the worst nuclear power plant accident, but their relative probabilities require of course quantification,[1] which must now account for the lessons learnt and practiced. Though all large dams are potential terrorist targets, the Ruhr-dam bombing raids in World War II demonstrate that success necessitates a scientifically sophisticated attack.

Several correlations of experimental data are available for predicting the damage to simple reinforced concrete panels from the impact of hard flat-nosed cylindrical steel billets. Scabbing damage corresponds to dislodgement of a portion of the target’s rear face, and perforation has its usual connotation. More than 150 experimental results are available for comparison against these formulae. Neilson [286] estab­lishes that the revised NRDC correlation affords the most accurate predictions for missile penetration depth, and for the scabbing to perforation transition. Though the missile velocity for the onset of

scabbing is accurate in 71% of cases, predictions of the perforation velocity are only 57% successful due to the lack of a reinforcement term in the correlation. During an attempted perforation steel reinforcement bars absorb energy by bending and shear, as well as by providing a net to retain broken concrete within the panel. Experimental results from Winfrith [286] in Figure 6.10 confirm a progressive increase in perforation velocity Vp with the amount of reinforcement, and they are correlated by

where

r — amount of square mesh reinforcement (% EWEF)

% EWEF — per cent of cross-sectional area occupied by the same square (EW) steel reinforcement just below the surface of each face (EF)

EW — “each way” (horizontal + vertical)

Other Winfrith experiments show that the perforation velocity for steel-faced concrete panels can be predicted by equation (6.5) if the thickness of a rear plate is converted to an equivalent reinforcement percentage. For example, a 1 mm thick rear plate on a 100 mm thick concrete panel corresponds to 1% EW. In the case of a front steel panel, the perforation energy of the composite is the sum of their individual perforation energies. The Ballistics Research Laboratory and Winfrith data are best correlated [286] especially for thicker steel panels by

E ‘ 1.44 x 109(hd)15 within ± 15% (6.6)

where

E — perforation energy (J); h — steel panel thickness (m) d — missile diameter (m); hd<3.4 x 10“3

Ohte et al. [287] confirm that conically-nosed hard missiles perforate targets more readily than flat-nosed ones. For specific missile-target combinations the perforation energy for a hard missile having a 45° half-angle nose is consistently about half that predicted by the BRL formula. With modification of panel thickness and missile diameter as a function of nose-angle in the BRL correlation satisfactory predictions of perforation energy are obtained.

Impacts of irregularly shaped fragments from a disintegrating steam turbine on reinforced concrete, metal panels or major pipe work are important safety issues especially for nuclear plants. Also of real concern is the perforation of a reinforced-concrete containment by aircraft whose geometries and crushing strengths are axially non­uniform. Though correlations are often sufficient for their specific physical situations having regular simple geometries, applications outside the experimental databases can lead to erroneous predictions. For example a linear extrapolation of the Canfield-Clator correlation [288] for steel projectiles impacting reinforced concrete wrongly suggests that no perforation would occur [289] at velocities less than 150 m/s. Empirical correlations for simple missile and target geometries are therefore inadequate for nuclear plants. Furthermore, the construction of complex replica models would not be cost — effective.

Accordingly the preferred option is the development of comprehensive physics-based computer simulations that are systematically validated by means of replica experiments with simple, but representative portions of the pertinent structures. Finite element techniques [290-293] were developed at Winfrith to pursue this strategy from about 1980.

In essence a finite element calculation divides the region for integrating a partial differential equation into sub-regions (finite elements) that are usually triangular or rectangular, though curved boundaries can be readily accommodated [293]. Values of the required solution at the elements corners constitute the “unknowns,” and for small enough elements the solution varies linearly across each element. However, higher-order numerical approximations can be formulated to allow larger mesh sizes [292]. Elliptic equations and the biharmonic equations of structural dynamics are both well suited to finite element techniques because their solutions are equivalent to the minimization of a quadratic functional (e. g., energy). In this form the required values become the unique solution of a linear equation having a positive definite matrix [115]. Galerkin’s method is equivalent, if applicable, to a variational approach but the resulting matrix is less sparse and has weaker numerical conditioning [115,292]. Early implementations of the finite element method involved a manual construction and labelling of the mesh that is evidently tedious, error prone and therefore a sizeable portion of the overall cost. Specific computer codes now automate data preparation and the interactive creation of a suitably graded mesh. However, considerable skill and experience are still required to achieve a successful outcome. Numerical solutions with finite elements are presently available for structures involving crushable materials, strain-rate dependent elasto-plastic materials and reinforced concrete with interfacial friction [296]. Dedicated software also assists the interpretation and presentation of computed

solutions. The following example illustrates the overall methodology for an aircraft crashing into a reinforced reactor containment.

Prior to unification the large number of military overflights led the Federal German Government to legislate that a nuclear reactor containment must withstand the impact of a Phantom RF-4E aircraft at 215 m/s. However, no such prescriptive requirements apply in the United Kingdom where each site must be separately assessed. Two principal safety concerns are

i. whether a crashing aircraft can perforate the secondary con­tainment or cause unacceptable damage

ii. the nature of vibrations transmitted to the rest of the structure.

Because the crushing strength of an impacting aircraft is so much less than the collapse loading of a reinforced concrete containment, it is therefore a soft missile and deformation of the concrete structure can be neglected in calculating the imposed transient loading. Riera [67] first quantified the situation as part of a safety assessment for the Three Mile Island installation near Harrisburg Airport. His principal assumptions are that an impact produces a region of compacted stationary debris and that the undistorted portion of an airframe continues on towards the target. By Newton’s second and third laws of motion the total reaction force R(t) on the building is

R(t) = — (mV) = mV + m V (6.7)

where m and V denote respectively the instantaneous mass and velocity of the undistorted portion of the aircraft. If a structure is slowly crushed in a hydraulic press, the involved mass remains constant throughout so the term mV is termed the crushing force Fc of the intact portion. With this nomenclature, equation (6.7) becomes

R(t) = Fc(t) + mV2 (6.8)

where Fc(f)now contains an allowance for strain-rate enhancement and m denotes the instantaneous mass per unit length of the residual airframe. By further assuming that impact velocities remain constant at the approach velocity, the aircraft manufacturer’s drawings then

provide data for a conservative evaluation of the transient reaction loading on a building. A refinement of the above analysis represents each structurally different longitudinal section of an aircraft, and the loss of mass by dispersed fragmentation should its ultimate compressive

strength be exceeded [294,295]. These reaction loadings [289] for the impacts of a Boeing 707, Phantom RF-4E, and a Tornado on a rigid structure are graphed in Figure 6.11.

Replica 1/25th scale experiments were performed at Winfrith for an aircraft impacting a proposed reactor containment at around 220 m/s. The aircraft model is shown in Figure 6.12, and the curved reinforced containment was simulated by a flat panel with a massive circumferen­tial ring-beam to provide an edge restraint representative of the remain­ing structure. Axial mass and stiffness distribution of the model aircraft were engineered to reasonably match the design reaction loading like that in Figure 6.11 The scaled reaction loading in Figure 6.13 formed

the input to a DYNA-3D finite element code as a transient pressure on the impact zone of the experimental reinforced concrete barrier. A computer simulation then predicted its crack formation etc allowing for interaction between the concrete and its reinforcement. Code validation was as usual sought by comparison with the replica experi­ments. Though cracking was generally over-predicted [289], good agreement was achieved with regard to transient deflections [296]. Moreover, though model experiments indicated the perforation of a 1 m thick prototype barrier and severe damage at 1.5 m thickness, a 2 m thick containment appeared to be essentially undamaged.

To conclude, the range of Winfrith replica experiments have extensively validated the DYNA-3D code modules for crushable mate­rials, strain-rate-dependent elasto-plastic materials and reinforced con­crete with interfacial friction. Nuclear safety assessments can now be confidently implemented with regard to the impact of disintegrating plant items or crashing aircraft on concrete buildings, steel panels or pipe work.

Section 2.4 describes how power production changes the effective multiplication factor of a nuclear reactor so that

K = K (N) (2.31)

As a result equation (2.26) is non-linear. However, for control analysis it is sufficient to examine this equation for small perturbations about an operating point for which the effective multiplication factor is a constant derived from neutron diffusion and possibly thermal-hydraulic calculations. Effecting the Laplace transformation of equation (2.26) and involving the definition

j

b = b

j=1

(2.33)

For analytical purposes, equation (2.33) is more conveniently expressed in terms of the core reactivity

r, (K — 1)/k

to give

The algebraic roots of

^ bi’Tj’S

F(s) = Is — r(1s + 1)^^ = 0 (2.36)

j=f rjs + 1

define the poles of the neutron population’s kinetics, and the real root closest to the origin is termed dominant whose reciprocal T* defines the reactor period. As reactivity is increased from zero, the dominant pole moves into the right half s-plane causing the neutron population to diverge exponentially in the form Noexp (tT*). The location of the dominant pole s* can be determined iteratively using the Newton — Raphson algorithm

where

s—an estimated location of the dominant pole and

For small enough reactivities, the reactor period is estimated from the above equations as

which from Table 2.2 further approximates to

so that the radioactive decay periods of the precursor groups govern the growth of a neutron population. On the other hand for large reactivities the dominant root of equation (2.36) is far removed from the {1/tj}, and under these conditions it behaves as

1s — p(1s + 1)+ b = 0 (2.41)

The corresponding reactor period is then asymptotic to

1(1 — p)/(p — b) (2.42)

which is dominated by the lifetime of prompt neutrons, because as seen from Figure 2.4 these alone constitute a self-sustaining subpopulation. Typical data in Table 2.2 and the above analysis enable a simple digital computation of the stable reactor period as a function of normalized reactivity (p/b). Reactivity are often quantified in this normalized form for operational purposes, and the so-called prompt critical value of unity is ascribed a magnitude of one dollar ($1) with corresponding subdivisions of cents. Once a reactor enters the super-prompt critical regime, the reactor period is seen from Figure 2.5 to decrease dramati­cally: especially for fast reactors.[34] Bearing in mind the typical rate

constraints associated with induced thermal stresses in a power plant, and the interval necessary for emergency intervention, then reactivity changes in normal operation must be restricted to a few cents (1/100th of 1$) to achieve a stable reactor period of no less than about 30 seconds [80,117]. After circumspect increases in reactor power by restricting withdrawal of control rods, the negative reactivity feedbacks described in Section 2.4 restablize a neutron population. In terms of equation (2.40) such increases in reactor power or neutron population correspond to an infinite reactor period T* with K = 1 or p = 0. An alternative viewpoint from equation (2.36) is that a neutron population (reactor power) in equilibrium corresponds to the dominant pole at the origin with p = 0.

A reliable upper bound for the fraction of detonatable mixture[75] is patently an important parameter in reactor safety assessments. In this respect the explosive power of MFCIs restricts experiments to no more than 100 kg of corium stimulants, whereas Severe Accidents could involve tonne-quantities of reactor materials. Such a wide extrapolation
is obviously a moot point. The total heat content of a melt-mass is broadly conserved as

_/

Surface area of melt/thermal capacity « (mass) 3 (5.16)

which implies that with increasing mass its average temperature remains higher for longer. It does not imply that the fraction of coarse mixture asymptotically approaches 1 as the total melt mass increases. Figure 5.1 and volcanic lava flows clearly reveal that a coarse mixture could not form where the melt’s surface is close to or below its freezing point. In fact the formation of a coarse mixture involves many physical processes such as heat diffusion in melt and coolant, heat transfer from it by principally radiation,[76] melt viscosity and contact mode of melt and coolant. The two-dimensional CHYMES code [197] was developed to consolidate the opinion that partial freezing would impose an upper bound on the fraction of coarse mixture if large quantities of corium were to be involved. Though the computed vapor flow rates match experiments [184], thereby suggesting a reasonable simulation of an evolving potentially pre-detonatable mass, the required upper bound did not become available. Nevertheless, present knowledge enables the convincing safety assessment in Section 5.7.

The analysis in Section 5.1 and later in Section 5.7 reveal that debris sizes must be less than about 250 mm for an effective contribution to the explosive energy of an MFCI, and that these fine particles largely stem from the fine fragmentation of a coarse mixture across a shock front. On this basis the CORECT2 and THINA experiments with a sodium coolant provide a 40% upper bound on the detonatable fraction of a coarse mixture [82,206]. However, urania-sodium experiments at AEEW show that hydrogen generated by the prerequisite cleansing of the solidified debris with methanol, and then its vacuum distillation, augments the fine debris fraction.[77] Moreover, unlike a urania-water interaction, a sodium coolant experiment very often produces a number of incoherent weaker detonations [86], which plausibly augment the recovered fine debris. Accordingly, urania-water experimental data
from Rig A and the Molten Fuel Test Facility at AEEW give the arguably more reliable upper bound of 20%. Figure 5.2 schematically illustrates the MFTF in which up to 20 kg of urania from a depleted uranium-molybdenum thermite mix is injected into water so as to replicate in some ways corium slumping into the residual water in a PWR pressure vessel. While the experimental and reactor contact modes are not too dissimilar, the pertinent volumetric ratios of corium to water differ by orders of magnitude. Specifically, experimental ratios

Release mechanism

Coo ant eve

Debris tray

vesse

Figure 5.2 The Molten Fuel Test Facility for the SUS Experiments

are at least 1:1000 but in a reactor situation[78] the ratio can be as low as 1:30. Experimental results show that explosive energies are markedly reduced when “fuel-rich” mixtures are involved because

i. A shortage of coolant restricts the formation of a coarse mixture.

ii. A reduced inertial constraint (a tamp) allows less durable heat transfer between melt particles and coolant.

Expert opinion [59,65] identifies Severe Accidents in PWRs that could probably result in molten corium slumping into residual water in the lower head at pressures in the range 1 to 155 bar. Data on MFCI at higher than atmospheric pressure is sparse, but available evidence [98,198] indicates that with increasing pressure more violent triggers are required and that the energy release is greater. An actual reactor experiment at EG and G-Idaho observed an MFCI at the highest recorded ambient pressure of 64 bar.

Section 4.5 describes the inception of a Severe Accident in a fast reactor subassembly, in which potential MFCI might occur in a radically different geometry from that of a PWR or the MFTF in Figure 5.2. SCARABAEE [200] and TRAN [201] tests establish that the sodium content of an affected subassembly would first vaporize before any melting of the steel-clad fuel pins. At this juncture MFCI are patently impossible. Due to decay heat the steel cladding subsequently melts at ‘1200 °C to be followed by the mixed oxide at ‘3000 °C. Because the boiling point of steel is around 3000 °C, its vapor condenses at the cooler subassembly inlet and outlet to form strong blockages. Molten corium is then considered to perforate the sub­assembly wrapper and thereby allow pressurized injections of molten corium into the inter-wrapper gap or a neighboring subassembly. So far significant quantities of molten fuel would not be involved, and therefore there would be little chance of damaging escalations. How­ever, due to fluid inertias, corium ejection would end by the develop­ment of a negative differential pressure around 2 bar which would encourage re-entry of liquid sodium. As part of a European research program [86] also involving the CORECT2 and THINA experiments,
the MFTF was modified as in Figure 5.3 to investigate this different contact mode.

These independent tests each elicited an erratic series of relatively small MFCIs. Furthermore, with the original unrestricted contact mode in Figure 5.2 just one of these SUS tests had any semblance of coherence with a principal interaction followed by a series of very much smaller ones.

Release mechanism

Upper nozzle

Charge container

To ballast vessel preset I to 10 bar

Charge injection

tube

Flow meter Containment vessel

capacity 1.7 m

Shroud tube

Argon gas

Wrapper

Sodium pool

Punch

Flow meter actuator

Sodium storage tank

Figure 5.3 General Arrangements for B-Series Experiments

Interactions with sodium therefore appear markedly different from and far less damaging than with a water coolant. This can reasonably be attributed to the two orders greater thermal diffusivity of sodium compared to water.[79] As described in Section 5.4, this enables a much higher heat transfer rate from the melt and by promoting localized freezing obstructs an essentially coherent propagation. The most probable outcome for this particular Severe Accident appears to be a less rapid and damaging pressurization of the reactor vessel by an erratic series of small MFCIs or by slower conventional[80] heat transfer from larger corium particles (a so-called Q*-event [202]). Finally, fast — reactor safety is enhanced by the 2000 tonne or so of sodium in its primary circuit. If 50 tonne of molten corium passively equilibrated with this coolant, the predicted temperature rise is only about 42 ° C, which emphasizes that the hazard of an MFCI resides in a localized almost coherent heat transfer to a vaporizable coolant.

Photoelectricity was discovered by Hallwachs [16] in 1888, and its quantum mechanical analysis was provided by Einstein in 1905. However, the necessary research toward viable electrical power units actually began in 1954 with transistor development by Bell System Laboratories NJ. Solar cells for this purpose are now [17] series — connected arrays of p-n junctions in ribbon polycrystalline silicon which have a quoted life expectancy of 30 years.[2] Though mono­crystalline devices offer a somewhat greater conversion efficiency of sunlight into electrical energy, ribbon technology is cheaper with a theoretical maximum conversion efficiency [17] of 29%. By manu­facturing ever-thinner devices charge carrier recombination during diffusion has been reduced so as to achieve efficiencies of around 18%. Conversion losses also occur as a result of atmospheric or bird deposits and in the thyristor inverters between domestic and Grid networks. Because solar radiation has no cost, a low conversion efficiency principally aggravates capital investment and environmen­tal impact.

During the four winter months Table 1.2 and Figure 1.1 show that the average of 1-2 sunshine hours around mid-day are well outside the UK’s national peak demands between 1600 and 2100h. Though solar cells provide some twilight output the 17% capacity factor for UK solar arrays from Table 1.2 suggests an inadequate annual return on capital for commercial plants. However, Spain and the United States lead the

world in the use of solar energy [19]. Spain has a currently installed capacity of 432 MW with plans for a total 900 MW, and the United States has presently 457 MW with a large 968 MW unit under con­struction in Riverside County, California. As well as Capacity Factors twice that of a UK plant, their solar output conveniently peaks with that of summer noon-time electricity demand for air conditioning. By avoiding the synchronization of low merit order[3] fossil-fired stations, Spanish and US solar units further enhance their economics and reduce carbon emissions. Moreover, these countries have large arid or other­wise unusable areas of land whose purchase offers no impediment to commercially sized developments. For example in the United States, a Boston plant [1] of 1.3 MW was recently built on the contaminated land of a derelict gas works with a utilization of 1.9 hactares per installed MW. On the other hand high land prices, climate and incompatibility with the national electricity demand further militate against solar generation in the United Kingdom. Indeed the United Kingdom reduced its subsidy for commercial generation [30] (>50kW) in February 2011, and later in February 2012 attempted to cut the feed-in tariff for domestic roof-top units of a few kW.

7.1 NATURAL CONVECTION IN NUCLEAR PLANTS

Following the immersion of a heated surface in a fluid, molecular heat conduction raises local temperatures and thereby reduces fluid densities [208,209]. Buoyancy forces then lift these localized lighter fluid packets to induce a continuous laminar or turbulent flow called natural circulation or natural convection. Industrial research on natural con­vection began in the 1930s but largely stalled in the mid 1950s because the enhanced heat transfer rates with forced convection (i. e., pumps) enable more compact and therefore more cost-effective plant designs.1 After the Three Mile Island incident in 1979, natural circulation research restarted to develop passive safety systems [108] that would function even after a total loss of emergency power. An IAEA Conference [298] in 1991 noted that passive safety systems based on natural circulation are a desirable method of enhancing the

1 Even domestic central-heating boilers now have forced circulation on both flue and water sides. [105] simplification and reliability of essential safety functions. Though some new innovative designs for small integrated reactors (< 100 MWe) now propose natural circulation as a means of core cooling during normal operation, these have yet to gain formal compliance with European Utility Requirements [59,109].

The principal advantage of natural circulation systems lies in the elimination of active power supplies and pumps, so simplifying plant architecture, maintenance and operation. It also eliminates certain accident scenarios in nuclear power plants, and an increase in channel power intrinsically increases its mass flow rate. On the other hand with a stable forced convective channel an active control system is required to increase its mass flow rate with input power. Specifically, momentum conservation in a vertical channel gives the steady-state pressure gradient [117]

— dp = PS + D (GG/2p) + dZ (G[106]/p) (7.1)

where

p — density; g — gravitational acceleration; G — mass flux (kg/m2s) D — hydraulic diameter; f — Fanning friction factor

Local density and dynamic viscosity m for a two-phase flow may be calculated as a function of steam quality x from the saturated values as

1/p = x/pG + (1 — x)/pL and 1/m = x/mG + (1 — x)/pL (7.2)

and for clean2 reactor or boiler channels with single or two-phase flow

1/f = -4log1o(1.25/Re f (7.3)

where

Re = GD/m — Reynolds Number

The driving pressure from modern pumps is only weakly dependent on flow rate, so for present purposes it can be assumed constant. An increase in channel power clearly decreases local density which accord­ing to equation (7.1) then increases local pressure gradients. Thus to preserve a constant overall driving pressure, a channel’s mass flux must suffer an inappropriate decrease. For this reason boiler feed water flow in conventional forced convective stations is controlled by cascaded throttle valves whose opening increases with demanded output[107] power.

The principal disadvantage of heat removal by natural circulation is that the driving forces from density disparities and gravity are relatively small. Their increase necessitates increases in loop heights and decreases in loop resistances to achieve prescribed heat transfer rates. Because 60 to 70% of the capital costs for nuclear stations reside in civil engineering, the economic viability of plants of order 1 GWe appears questionable. Also larger reactor cores with their smaller bucklings [58] are more susceptible to a spatially unstable neutron flux, and the thermal-hydraulic problems associated with boiling channels are also exacerbated under natural circulation. Specifically, flow stability at economic output powers (exit steam quality) requires the insertion of inlet ferrules to increase the liquid-phase pressure drop as described in Section 3.2. However, this artifice patently reduces the circulating mass flow rate unless loop heights are increased with the penalty of higher construction costs. Moreover, lower mass fluxes and higher steam qualities in two-phase flows encourage a radical reduction in heat transfer that occurs when the liquid phase can no longer “wet” fuel pin surfaces. Under these burn-out[108] or critical heat flux conditions[109] the pressurized fission product gases can locally puncture the fuel cladding whose creep strength [96] progressively weakens with increas­ing temperature. It is concluded here that heat removal by natural circulation is most cost-effective and reliable for single-phase liquid flows in safety systems removing just the decay heat in Grid-sized reactors (>100MWe). To meet the United Kingdom’s present peak demand of circa 60 GW, economic and environmental considerations favor large stations of around 1 GWe due to the limited availability of suitable sites (Section 4.4) and the existing form of transmission network.

A pioneering paper by Lorenz in 1881 analyzed natural convection heat transfer from a uniformly heated vertical plate in terms of molecular conduction to the neighboring fluid whose reduced density results in its buoyant upward laminar flow [219]. His formula for the corresponding heat transfer coefficient has the form

Nu = Function of Gr(Pr)1 (7.4)

where

Nu — Nusselt Number; Gr — Grasshof Number; Pr — Prandtl Number

l — a constant

Later in 1902, Bourinesq showed that when viscosity is negligible (as in very many turbulent flows) the above index l equals 2. Further investigations [219,303,305] during the twentieth century still generally relate to flat plates or the outsides of horizontal tubes. The earlier experiments reveal that natural circulation can become turbulent with an enhanced heat transfer rate, and this is encouraged in reactor fuel pin bundles by diagrids [268,304]. Heat transfer and pressure drop in tube bundles also depend on their pitch-diameter ratio so the relevant correlations are very much design-specific. Because the gas-side heat — transfer resistance in AGR steam generators is a significant part of the overall primary to secondary-side thermal resistance, specific correla­tions are necessary for sufficiently accurate predictions of steady-state and dynamic performances [117]. Likewise by virtue of the lower mass fluxes in natural circulation, pertinent experimental heat transfer and pressure drop data are necessary to underwrite proposed safety system designs: especially if the stability of a two-phase channel flow is in question. Thermo-hydraulic transitions in natural circulation are a tractable numerical problem and they can be negotiated like those found under forced convection [64,117]. However, stratified cold sections can be potentially created prior to ECCS operation and these would block decay heat removal by natural circulation, as would an accumulation of nitrogen or hydrogen in the upper U-bend of a steam generator (a loop seal) [306]. Thus the operation of heat removal systems by natural convection must be unequivocally confirmed by experimental rigs. Because full-scale whole plant replicas are impractical, one adopted approach is to adjudge representative cross-sectional sizes and to replicate exactly the differential heights between components so as to preserve gravitational driving forces. A superbly engineered rig relying on this principle has been constructed for example at CEN Grenoble, but the APEX and MASLWR rigs in the United States also scale differential heights by 1/4 or 1/3, respec­tively [109].

Though natural circulation is proposed for the on-load cooling of Generation IV water reactors [109,298,302], it is installed in presently operational plants only for decay heat removal in potential Severe Accidents. In this context, the separate steam generators of existing PWRs have an important safety role [59,65]. If the primary recircu­lation pumps in Figure 1.3 were to fail and the system were depres — surized,[110] then sufficient coolant injected into the secondary-side boiler inlets is seen to encourage natural circulation in the original flow direction of the reactor circuit. Deaerator or emergency tanks are presently available for this purpose. Moreover, the intrinsic water inventory and that of the ECCS ensure that fuel degradation is a progressive process [66,93,213] so time is available to secure civilian fire engines [65], or to reconfigure the plant to use its steam turbine condensers as heat sinks [59].

An alternative scenario to MFCIs during a Severe Accident in PWRs is a progressively increasing mass of corium on its lower core support structures and at the bottom of its pressure vessel, whose creep strengths [96] are diminished by heat transfer. At high pressures the vessel itself fails first,[111] whereas at lower pressures the supports fail some computed 15 min before the vessel [65]. Steam, fission products and hydrogen from oxidation of the fuel cladding would then be discharged into the reinforced concrete containment building, whose hydrogen content would further increase from a rapidly developing corium-concrete interaction [65]. A major concern is then a potential rupture of the building from a hydrogen explosion with the consequential atmospheric release of radioactive isotopes. Ejected steam, air and gaseous fission products diminish the likelihood of the initial 7 to 16% by volume of hydrogen [307] to detonate: as do igniters and catalytic recombiners which innocuously dissipate locally concentrated pockets. However, the ejected hot material also threatens the normally well-sealed building by over-pressurization. To restrict the excess pressure to well below its ultimate strength of 5 bar, elevated rings of chemically doped[112] sprays are automatically activated at around 2 bar [66,307]. These produce fine droplets with Sauter diameters of 448 to 544 mm [308], and their large surface area to aggregate volume effects a rapid condensation of steam and ambient cooling, whilst evidently increasing the volumetric concen­trations of hydrogen. Herein lies a trade-off problem between spray cooling to counter over-pressurization and the increased risk of a sig­nificant hydrogen explosion. It evidently demands serious investigation.

Spray cooling within a reactor containment building creates multi­dimensional natural circulation flows which to a greater or lesser extent are complicated by

i. Interactive heat transfer between all solid, liquid and gaseous components.

ii. Permanent gases which inhibit steam condensation [219,311]. Also they increase the fugacity [3] of the contents which alters thermodynamic state equations from those of the pure sub­stance [210].

iii. Multicomponent two-phase turbulent flows.

iv. Thermal disequilibrium at liquid-vapor interfaces; see Section 5.5.

v. Dissolved fission products decreasing the vapor pressure of water (boiling point increases)—a fugacity effect.

vi. The efficiency of sump screens in removing core debris from water to be recirculated [310].

Due to the daunting physical and mathematical complexity of a containment’s spray cooling process, a combined step-by-step exper­imental and computer simulation investigation is first necessary to identify the most significant phenomena and then their pertinent interactions. The TOSQAN program is a European initiative along these lines. However, unlike the replica scaling described in Chapter 6, Zuber’s time-preserving Hierarchical Two Tiered Scaling Method [307,309] is adopted for this far more complex problem. Though the TOSQAN experiments are not necessarily dynamically similar to reactor-scale processes, the above step-by-step creation of validated computer models eventually results in a reliable simulation for safety assessments at reactor-scale [312,313].

3.1 STEAM DRUM WATER-LEVEL CONTROL

Grid-connected power stations are operated in either decoupled or coupled mode [80,117], which describes the form of their response to grid frequency variations1 from changes in network demand. These alternatives are illustrated in Figures 3.1 and 3.2 where C1 (s) and C2 (s) represent controller transfer functions. By creating strict limits on steam plant temperature and pressure variations, decoupled control promotes steam turbine efficiency and boiler longevity (e. g., by reducing the exfoliation of tube magnetite deposits). Consequently, decoupled con­trol is generally the economic choice for large base-load stations having a high thermal efficiency and capital investment. With a coupled control scheme, a fossil or nuclear heat source is modulated to sustain boiler pressure when changes in Grid frequency operate the turbine control valve. Because Grid frequency fluctuations widely outpace heat-source dynamics, boiler-pressure changes can be offset only by exploiting the thermal energy stored in the rest of the plant. Such spontaneous changes of stored energy in some plant components are readily accommodated and their responses help to relieve thermally induced stresses in other less robust items [141].

1 See Section 3.3.

Nuclear Electric Power: Safety, Operation, and Control Aspects, First Edition. J. Brian Knowles.

© 2014 John Wiley & Sons, Inc. Published 2014 by John Wiley & Sons, Inc.

The evaporator section of the La Mont boiler system [117] in Figure 3.3 produces a low steam quality (< 10%) flow into a large steam drum which separates the saturated steam for superheating. Feed water to match steam generation is usually injected downwards from a sparge pipe near the base of a drum. A Richardson number analysis [117] establishes that its mixing with less dense saturated liquid from the evaporator is thereby largely suppressed,[35] so the drum’s liquid content remains markedly stratified.

When the turbine control valve opens under coupled control, the increased mass flow rate of steam to the turbine reduces boiler pressure, so some of the upper layer of saturated water in the drum flashes rapidly into steam to preserve thermodynamic equilibrium. This extra steam partially supports system pressure, and there is a corresponding fall in drum water level. Because a large enough reduction would cause the drawdown of steam into the recirculation pump [117], drum water-level control is necessary to prevent cavitation damage to its impeller. On the other hand, too high a water level would impair steam separation leading to damaging thermal shocks to the superheater tubing.

Figure 3.3 depicts the drum water-level control scheme for a 250MW(e) fast reactor nuclear power plant. The intermediate heat exchangers (IHXs) provide an additional safety barrier to obstruct an explosive ingress of water into the sodium-cooled reactor circuit. Because improvements in station efficiency of as little as 0.1% are financially material, the feed pump was driven from the same high-pressure steam supply as the main turbine to exploit this opportunity. Under coupled control a reduction in Grid frequency opens the turbine control valve via ytg and the drum pressure falls as a result of the increased steam flow. Its saturated surface-water then flashes very rapidly into steam, so a

correspondingly fast increase in feed-water flow is necessary to prevent the drawdown of steam. However, because the driving pressure to the feedpump turbine has reduced, the feed-flow is reduced just when it’s needed. Hence, the main turbine and feedpump turbine control-valve settings ytg and yyv are strongly interactive making the plant’s transfer function matrix far from diagonally dominant. It can be readily appre­ciated therefore that SISO control system design techniques proved unsatisfactory. By consummate skill, Hughes [81] devised a broadly satisfactory MIMO control system design using a 3 x 3 matrix controller with proportional plus integral diagonal elements. Nevertheless the incipient deployment of an independent electrically driven feedpump would have circumvented the problem to allow largely independent SISO control schemes and a more transparent design of ad hoc accident management strategies. This example highlights the importance of engineering insight and awareness of plant operating conditions: thereby demonstrating the industry specific nature of control engineering. Finally, it should be noted that the relatively high initial capital and low fuel costs of nuclear stations usually favours their top merit order placings with operation at maximum power under decoupled control.