1

1

28.1 Introduction As discussed in Chapter 9, under normal operating conditions in light-water reactors (LWRs), cladding and core structural materials operate around 350ºC while fuel pellets experience peak centerline temperatures below 1500ºC . At these temperatures, zirconium alloys remain in the alpha phase, the vapor pressures of major fuel constituents remain low and most fission products are retained within the fuel pellets. During operation the fuel and the changes undergo changes due to irradiation, composition changes, corrosion, etc. The following illustrates the changes in fuel condition under normal steady-state conditions. 28.1.1 Example #1: Stresses in Cladding A fuel rod with 1.2-cm-diameter UO2 pellets is loaded at 25oC into 1.224 cm ID Zircaloy cladding. The thickness of the cladding wall is 0.8 mm. (a) What is the initial gap thickness? m120cm012.0)2.1224.1(t 2

1ogap μ==−=

(b) When inserted into the reactor, the normal fuel centerline temperature is 1500 K and the fuel surface temperature is 700 K. The average cladding temperature is 350oC. Find the reduction in gap size upon ascent to power. The coefficients of thermal expansion of the fuel and cladding are: 1.35×10-5 K-1 and 4.4×10-6 K-1, respectively. The change of gap size is equal to the expansion of the cladding less the increase of fuel radius

fuel: [ ] 011.0)4001200)(1035.1()298T()298T(R

52

1sof2

1

f

gap −=+×−=−+−α−=δΔ −

cladding: 0014.0)298273350(104.4)298T(R

6CC

C

gap =−+×=−α=δΔ −

total: m57)0014.0)(1061.0()011.0)(106.0()( 44thermalgap μ−=×+−×=δΔ

(the factor of 104 converts cm to μm) (c) At what burnup (in MWd/kgU) does the gap close? The fission-product swelling rate of the fuel is 0.07% per MWd/kgU. With burnup, the remaining gap (120 - 57 = 63 μm) is consumed by fuel swelling:

( ) 0105.0106.0

63BU107RR

44

31

swellingfpf

f =×

=××=⎟⎟⎠

⎞⎜⎜⎝

⎛ Δ −

Solving for the burnup:

kgUMWd45

1070105.03BU 4 =

××

= −

(d) At what burnup does the azimuthal strain of the cladding reach 1% due to fuel swelling? What are the stresses in the cladding at this burnup?

The fuel continues to swell at the same rate as before gap closure. Fuel swelling is inexorable. An azimuthal strain of the cladding of εθ gives a radius increase of

)avg(CC RR ×ε=Δ θ . The mean radius of the cladding is RC(avg) = 12.24/2+0.8/2=6.5 mm, so for εθ = 0.01, the cladding radius increases by ΔRC = 0.01×6.5 = 0.065 mm, or 65 μm.

2

2

Fuel swelling at fuel-cladding contact is 63 μm (see part c above), so in order to impart a 1% strain to the cladding, the total fuel swelling generates a radius of 63 + 65 = 128 μm. The burnup at which this radius is reached is:

kgUMWd91

106.0128

1073BU 44 =

××= −

This burnup is much larger than what can be achieved in current PWRs (limit of ~60 MWd/kgU). However, if it were achievable, the stresses in the cladding can be determined from the azimuthal strain of 1 % and the equations of elasticity.

The equilibrium equation: ( ) θσ=σ rrdrd

the stress-strain relation: ( )rE1

νσ−σ=ε θθ

where E and ν are Young's modulus and Poisson's ratio, respectively. Eliminating σθ from the above equations and setting εθ = 0.01 yields:

E01.0r

)r(dr

)r(d rr =σ

ν−σ

With the condition σr = 0 at r = RC(OD) (the cladding OD), and η = r/RC(OD), the solution is:

⎟⎟⎠

⎞⎜⎜⎝

⎛−

ην−−=σ ν− 11

1E01.0

1r

note that σr < 0 (compression)

The hoop stress is: ⎟⎟⎠

⎞⎜⎜⎝

⎛ην

−ν−

=σηησ

=σ ν−θθ 1r 1

1E01.0or

d)(d

which is tensile (σθ > 0) For Zircaloy at 350oC: ν = 0.35; E = 17 MPa; σY = 0.064 MPa The ID is RCI = 12.24/2 = 6.1 mm and the OD is RCO = 12.24/2 + 0.8 = 6.9 mm In dimensionless terms, the stress equations cover the radial range 0.88 ≤ η ≤ 1. Figure 28.1 shows plots of the two stress components (as (σ/E) as functions of radial location in the cladding. The azimuthal (hoop) stress is σθ = 0.01×17 = 0.17 MPa, which is nearly 3 times the yield stress.

Fig. 28.1 Stresses in cladding with a hoop strain of 1%

3

3

28.1.2 Cladding Degradation during Reactor Operation As mentioned in Chapters 17, 22, 25, zirconium alloy cladding accumulates radiation damage, corrosion and hydrides during exposure to the reactor environment. Both total hydrogen content and hydride distribution are crucial in determining cladding behavior in the event of a transient that increases the stresses beyond those shown in Fig. 28.1. As illustrated at the bottom of Fig 28.2, most of the hydrogen produced by the corrosion reaction dissolves in the coolant, but a significant fraction (15 - 20%) is absorbed in the cladding. At burnups around 60 MWd/kgU, typical cladding contains 100-800 wt. ppm hydrogen, only about 50 wt.ppm of which is in solution at normal operating temperatures (Eq 22.32) [14]. Hydride precipitates harden the cladding and reduce its ductility, thereby increasing the chances that a transient that expands the fuel against the cladding will result in its failure. This type of failure generally starts with crack initiation in the brittle hydride rim near the outside diameter and proceeds by a mixture of brittle and ductile failure modes through the rest of the cladding wall.

Fig. 28.2 Characteristics of Zircaloy corrosion, especially hydride precipitation (bottom) The mechanical response of the cladding to a particular loading is governed by its constitutive properties and by the failure mechanism(s). The yield stress increases, ductility decreases and strain hardening sets in. Hydrides redistribute in response to temperature and stress gradients (Chap. 22) and form rims or blisters, both of which reduce cladding ductility beyond embrittlement for a uniform hydrogen content[11, 12]. The most likely result is localized-failure initiation at a weak spot in the cladding where hydride has accumulated. Spalling (i.e., flaking) of the oxide corrosion layer creates local cold spots because bare metal is exposed to the coolant. Thermal diffusion (Chap. 5, Sect. 5.8) drives hydrogen to these regions, where it precipitates as hydride blisters. The resulting ductility reduction creates sites for cladding failure early in an RIA transient. Although the failure shown in Fig. 28.3 appears ductile because of the 45° shear angle, the deformation is localized and the overall cladding

4

4

strain is small. These are characteristic of brittle failure. For a massive cladding breach to develop during a transient, the initial cladding failure has to propagate axially. Axial variation of all properties (which properties??) is important in this process.

Figure 28.3 Zircaloy-4 cladding (light gray) with a rim of hydrides; tested to failure at 300°C [13].

28.1.3 Postulated accidents in Light-Water Reactors Two abnormal conditions are of special interest in addressing the safety of nuclear power reactors. Much research has been devoted to the behavior of fuel materials under such postulated accident conditions. Accidents are separated into two groups. The first are called design-basis accidents. These accidents can be terminated by safety systems before the core is damaged. The second group are termed severe accidents. These accidents cause core damage and release radioactivity to the containment vessel. Both the Three-mile Island (TMI) and Chernobyl reactor accidents were in this class. The two most challenging LWR design-basis accidents are reactivity-initiated accidents (RIAs) and loss-of-coolant accidents (LOCAs). The RIAs are the most credible severe overpower events and the LOCAs are the most severe undercooling events. These accidents involve the limiting conditions that fuel materials might be expected to experience. The RIA This postulated accident is initiated by a large (nuclear) reactivity insertion while the reactor is operating normally. This generates a power excursion which can burst the cladding and expel of hot fuel particles into the coolant. The thermal energy of the fuel particles is converted into mechanical energy and a steam explosion is the result. Such a reactivity excursion initiated the severe accident at the Chernobyl plant in April 1986.The postulated RIA accidents are initiated differently in BWRs and PWRs, but the progression of subsequent events is similar in these two reactor types. In a BWR, the control blades are normally operated through the lower head of the reactor pressure vessel (RPV) by drives that move inside the instrument nozzles in Fig. 1.8. An RIA accident can occur if a control blade hangs up as the drive assembly is withdrawn and later

hydrided layer next to coolant

5

5

rapidly drops out of the core. This is termed a rod-drop accident. It produces the largest reactivity excursion when the reactor is initially at cold, zero-power conditions. It has never occurred. In a PWR, the RIA is postulated to occur after a crack develops around an upper head penetration (nozzle) that houses a control-rod drive (see Fig. 1.7). The drive unit and its attached control-rod cluster are ejected under pressure when the housing fails. This event, called a rod-ejection accident, produces the largest reactivity excursion if it occurs starting from the reactor at hot zero-power conditions1. No rod-ejection accidents have occurred in PWRs, but a precursor occurred in 2002 in the Davis Besse plant when a control-rod-drive nozzle remained attached only by its stainless steel liner following nozzle cracking and severe boric-acid corrosion of the upper reactor head (see Sect. 15.9.1). The LOCA A LOCA originates with a failure in the primary coolant circuit, usually a break in a valve or in one of the large pipes through which coolant moves between the reactor core and either the steam generator (PWR) or the turbine (BWR). As the name implies, water cooling of the fuel elements is lost in a LOCA. Unless another water supply, called the emergency-core-cooling system (ECCS), is activated to resume cooling the core, the fuel elements will overheat and rupture and then be destroyed. A LOCA that is terminated by the ECCS before the fuel is damaged to the extent of releasing fission products is considered as a design-basis accident. However, when the accident progresses to the stage of core damage (as in the TMI case), it becomes a severe accident. Severe accidents Another category of accidents are called severe accidents. These accidents do not appear to be credible because of diverse and redundant safety systems that are incorporated into plant designs. Nevertheless they provide bounding estimates of accident consequences. 28.2 Reactivity-Initiated Accidents with Fresh Fuel 28.2.1 Evolution of an RIA Regardless of the speed of a falling BWR control blade or an ejecting PWR control rod, the nuclear reaction soon becomes prompt-critical. From then on, the rate of power increase depends only on nuclear properties of the core, in particular Doppler broadening of the neutron energy spectrum. In general, the resulting neutron/energy pulse has a full-width at half-maximum of a few tens of milliseconds, as illustrated in Fig.28.4. During this time, essentially no heat is transferred from the fuel pellet to the cladding and the event can be considered adiabatic. Consequently, a power pulse in an RIA is characterized by

the maximum fuel enthalpy addition in J/g (We must keep it metric!) and by the pulse half-width in ms. The enthalpy increase in the fuel pellet during the transient can be calculated from the integral of the power curve. 1 This accident is not possible in a PWR under cold conditions because the vessel must be pressurized (and therefore hot) to cause rod ejection.

6

6

Fig. 28.4 Fuel rod power during an RIA with an energy deposition of 700 J/g.

Approximating the pulse in Fig. 28.4 as a triangle with a base b = 0.024 s and height h = 3x104 kW/m. The area under the pulse is 1/2 b×h = 360 kJ/m. Assuming that the pellets are 1 cm in height and weigh 8 g each, the pulse energy is:

gJ450

kJJ10

g8pellet1

pellets100m1

mkJ360 3 =××× fellows - what's wrong here?

Figure 28.5 shows the pulse widths for typical enthalpy additions in a PWR Such pulses are somewhat broader in a BWR because of the looser nuclear coupling (to what?) [2].

change abcissa units to J/g!

Fig. 28.5 Dependence of pulse width on pulse energy (fuel enthalpy change) for beginning-of-cycle (BOC) and end-of-cycle (EOC) conditions in PWRs (after [2]).

What is the purpose of giving the formula for the thermal expansion coefficient of UO2 when the single value given in the example #1 (from MATPRO) will do. Besides, what does it have to do with the fuel temperature rise?

Delete this plot (Enthalpy of UO2 Vs T) Although the cladding temperature does not rise significantly during the 10-ms period of the power pulse, the energy Caution: heat is energy in motion, not energy proper!

7

7

in the pellets eventually passes through the cladding into the coolant. During this process the cladding temperature approaches the fuel temperature. In the following example, the cladding temperature is calculated and compared to the that calculated by the FRAPTRAN [4] code 28.2.2 Example # 2 cladding temperature due to an RIA in a cold, zero-power BWR A power pulse deposits 725 J/g in the UO2 fuel pellets in a Zircaloy-clad fuel rod, which is at zero power and at room temperature. Assuming that the enthalpy rises uniformly in the UO2 pellets and the heat capacity is 85 J/mole-K, the temperature of the pellets increases from 300 K to 2600 K. Since the power pulse in an RIA is very fast, the initial (uniform) temperature that it generates in the fuel can be said to occur at t = 0. Assuming temperature-independent properties, the decay of the initial temperature is analyzed by the transient heat conduction equation in the fuel:

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

=∂∂

ρrTr

rr1k

tTCP

The initial condition is T(r,0) = To and the boundary condition at r = 0 is ∂T/∂r = 0. At the fuel surface (r = RF), the boundary condition is: -kf(∂T/∂r)r=Rf = h[T(RF,t) - Tcool] where Tcool = 300 K is the coolant temperature and h is the overall heat transfer coefficient between the fuel surface and the coolant. Because of fuel expansion due to the high initial temperature, there is no fuel-cladding gap. The series resistances are the cladding and the hydraulic boundary layer in the coolant at the cladding OD:

coolC

C

h1

kh1

+δ

=

This problem is best approached in dimensionless terms, which are:

o

cool

TTT −

=θ f

2fP

f kRC*twhere

*tt

Rr ρ

==τ=η

properties are: ρf = 10.5 g/cm3; Rf = 0.6 cm; kf = 0.03 W/cm-K; hcool = 3 W/cm2-K; δC = 0.8 mm; kC = 0.17 W/cm-K

88.02600

3002600)0,(Kcm

W2.131

17.008.0hs40

03.02706.0855.10*t 2

12

=−

=ηθ−

≅⎟⎠⎞

⎜⎝⎛ +==

×××

=−

The system is:

2403.0

6.02.1kRhHwhere),1(H

0T

TT)0,(with1

f

f

1

0o

coolo

=×

==τθ=⎟⎟⎠

⎞⎜⎜⎝

⎛η∂θ∂

−

=⎟⎟⎠

⎞⎜⎜⎝

⎛η∂θ∂−

=ηθ⎟⎟⎠

⎞⎜⎜⎝

⎛η∂θ∂

ηη∂∂

η=

τ∂θ∂

=η

=η

Carslaw & Jaeger (p. 202) give the solution to this system as:

)(J)A()(J

)exp(H2)0,( n0

22n

n0

1n

2n β+β

ηβτβ−=

ηθθ ∑

∞

=

where J0 is the Bessel function of order zero and βn are the roots of: βJ1(β) = AJ0(β) for H = 24, the roots are:

β1 = 2.30; : β2 = 5.30; : β3 = 8.31; : β4 = 11.34; : β5 = 14.39; : β6 17.44 We first calculate the cladding ID temperature, TC(ID), as a function of time. Because the gap is closed, TC(ID)(t) = T(RF,t), the fuel surface temperature. From the PDE solution, at η = 1:

8

8

∑∞

= +βτβ−

=−

−

1n22

n

2n

coolo

cool)ID(C

H)exp(H2

TTTT

=2H×sum(τ)

After sum(τ) has been computed as a function of time, TC(ID) follows from: TC(ID) = Tcool + 2H×sum(τ)×(To - Tcool) = 300 + 2×24×sum(τ)×2300

Determination of the average cladding temperature from TC(ID) starts with the heat flux: q = (kC/δC)(TC(ID) - TC(OD)) = hcool(TC(OD) - Tcool)

with: 4.108.0/17.0

3/k

h

CC

cool ==δ

, solving for TC(OD) gives:

TC(OD) = (TC(ID) + 1.4Tcool)/2.4

with Tcool = 300 K, the average cladding temperature is:

TC(avg) = 0.5(TC(ID) + TC(OD)) = 0.71TID + 87. With TC(ID) from sum(τ), the average cladding temperature following the power transient is shown in Fig. 28.6 for the analytical method described above and for the output of the FRAPTRAN code. The reasons for the discrepancy between the two curves include: (a) neglecting the heat capacity of the cladding in the present method. Including this factor is the reason for the slow increase of the FRAPTRAN curve in the plot. (b) neglecting the temperature dependence of physical properties (c) neglecting thermal resistance between fuel and cladding (d) unknown (but guessed) heat-transfer coefficient in the water contacting the cladding Because these processes are complicated and difficult to model (even using codes such as FRAPTRAN), the progression of fuel damage during RIAs is customarily determined empirically by testing.

Fig. 28.6 Cladding average-temperature engendered by a energy input of 725 J/g UO2

calculated in Example #2 and by the FRAPTRAN code. (in ordinate label, change oC to K)

9

9

28.2.3 Characteristics of the post-RIA fuel The state of the fuel and the cladding after a test on a fresh fuel rod in the Power-Burst Facility (PBF) at the Idaho National Laboratory (INL) is shown in Fig.28.7. This test generated a fuel enthalpy pulse of 1200 J/g with a width of 13 ms [5]. The cladding can fail in the early part of the transient in a brittle manner if insufficient ductility remains after reactor exposure to survive the loading imposed by pellet-cladding mechanical interaction resulting from fuel expansion. If the cladding survives the initial part of the transient, the high temperatures achieved later can cause failure by oxidation and hydrogen pickup. These impurities embrittle the cladding and can cause failure upon quenching

From the condition of the fuel and cladding seen in Fig.28.7, it is clear that very high cladding temperatures were achieved during the transient.(What makes it clear?) Evidence of the beta phase is seen in the center of the cladding, indicating temperatures above the phase transformation at about 850ºC. Also, thick oxide layers (.......μm) cover the inside and outside surfaces of the cladding; these could not have developed during the transient unless temperatures were well above 1000ºC. The eventual fracture (during quenching) is brittle (as deduced from the fracture surface, not shown). Cladding embrittlement was caused by oxygen diffusing into the bulk metal from the oxides on the ID and the OD. This extent of embrittlement in such a short time indicates temperatures well above 1200ºC. The condition of this section of the fuel and cladding is similar to that which occurs in a loss-of-coolant accident (LOCA). Finally, it is clear that the failure occurred at a low temperature during cool-down because no oxide is present on the fracture surface.

lost the picture! Fig.28.7. Post-Irradiation examination showing damage in cladding of fresh fuel after exposure to a large RIA power pulse (1200 J/g) in the Power Burst Facility test reactor (after [5]and.[6]). Give scale. simplify explanation in lower-left-hand corner. Identify the outer layer 28.2.4 RIA limits A traditional value of 700 J/g total enthalpy has been used by the nuclear industry to indicate the failure level. With fresh fuel, the cladding fails by swelling and rupture rather than by a brittle mechanical mechanism [2]. During an RIA, release of fission products is small, and the major consequence of failure is a steam explosion due to expulsion of hot fuel particles into the coolant from a crack in the cladding . Fuel enthalpies greater than 1150 J/g result in some molten UO2. At a higher enthalpy , thermal expansion during the zirconium phase change causes molten or hot solid particles of UO2 to be expelled from the rod into the coolant. The classic work by MacDonald et al. found damage levels for fresh and low-burnup fuel as shown in Fig. 28.8 [7]. The fuel-enthalpy rise resulting from an RIA is related to the worth of the ejected control rod as shown in Fig.28.9 [2]. Because control-rod worths2 are generally less than $2, it is seen from this figure that the fuel-enthalpy change is usually less than about 300 J/g. If a 100 J/g RIA (this sentence is not clear) pulse in a PWR starts with the fuel at about 300ºC, the total (give number) 2 explain control-rod worth.

It's not clear where the lines end. Put an arrow showing where the end is

10

10

is still less than the 700 J/g enthalpy at which cladding failure occurs and much less than the 1000 J/g at which hot fuel particles are expelled from a fuel rod. Therefore, no cladding failure, fuel dispersal, or steam explosion is expected for an RIA involving fresh LWR fuel of burnups less than 30 MWd/kgU. These limits may not be valid at higher burnups [8, 9].

Fig.28.8. Photographs of fresh fuel rods after RIA testing in the SPERT reactor (after [7]).

The cladding suffers embrittlement during exposure to hot coolant (see chapter 22) causing failure at enthalpy depositions below the limit set for fresh fuel. These changes, which could also include the effects of radiation damage, are discussed in the next section.

What is Beta?

Fig.28.9. Maximum fuel enthalpy change for an RIA in a PWR from hot zero power for two values of the delayed-neutron percentage, 70% at end-of-C (what does C mean??) and 100% at beginning of C?? (after.[2]).

Show only two curves - as it is there is an overload of information that is not necessary.

Peak Fuel enthalpy (J/ 1270 1150 1000 860 610

change to J/g

11

11

28.3 Reactivity-Initiated Accidents in High-Burnup Fuel As of 2010, power reactors achieved rod-averaged fuel burnups above 60 GWd/t. Several properties of high-burnup fuel radically alter the evolution of an RIA in both BWRs and PWRs. Specifically: i) the fissile atom distribution within the fuel pellets is changed; , ii) the pellet-cladding gap is closed, often with pellet-cladding bonding; iii) hydrogen is present in greater (than what?) quantities in the cladding iv) fission-gas bubbles are found within the fuel pellets.3 One risk factor for RIA accidents diminishes with burnup: the maximum energy deposition decreases because less fissile material is available. The comments below are for a given energy deposition and the ability of a rod ejection to deposit a sufficient amount of energy to be assessed separately. This sentence doesn''t make sense. All of the other changes degrade the RIA response of high-burnup fuel, as discussed below. 28.3.1 Nonuniform atom distributions(it is not redistribution, it is nonuniform generation) and due to rim formation Fissile atom nonuniformities can be seen in Fig.28.10(b) in a fuel rod with very high burnup [1]. When irradiation starts, the fissile atom density (235U) is homogeneous throughout the pellet. In parallel with the fissioning of 235U , 238U absorbs neutrons to become 239Pu. As the result of neutron shielding, both this absorption reaction and the fission reaction occur preferentially at the periphery relative to the center of the fuel pellet (see Chap. 16). As burnup progresses plutonium builds up near the pellet surface. In high-burnup fuel, the concentration of surface Pu is more than double that at the centerline. The resulting increase in the number of fissions near the surface causes a much greater concentration of fission. This zone, called the rim region or the high-burnup structure (HBS), exhibits a much different microstructure than that of the pellet centerline, showing a high density of high-pressure bubbles as shown in figure 28.10a. The formation of the HBS has three important consequences for RIA response. First, a larger fraction of the total enthalpy will be deposited next to the pellet surface, and thus closest to the cladding, thereby maximizing cladding strain and increasing the probability of cladding failure. Second, the large inventory of fission gases stored in the rim region is available for release during the transient. Third, fragmentation of the tiny grains in the rim can lead to greater fuel dispersal with the concomitant risk of a steam explosion. As a consequence of the increased deposition of energy near the surface of the pellet, fuel temperatures near the pellet surface are very high early in the power transient (see Fig.28.11), and much higher than the peak temperature in a fresh fuel rod. Because of high fuel-surface temperatures, thermal expansion of the pellet is substantially enhanced. Figure 28.12 (left side) illustrates a typical thermal-expansion algorithm (what is this? is the HBS writing code??) that is used in most computer models, especially for normal operating conditions when the peak temperature is at the pellet centerline. In this (need a better word), the centerline node expands more than the other nodes and the expansion of each node is added to the next. Because UO2 has

3 no: See Fig. 2 in Nogita & Une, Nucl. Instr. Methods Phys Res. B91.(1994) 301. Also my paper " Point-defects in irradiated UO2" Journal of Nuclear Materials 399 (2010) 236–239 and references therein

12

12

no significant tensile strength, each nodal ring cracks and the incremental expansions are additive and insensitive to the number of rings.

Fig.28.10. (a) SEM image showing the ultra-high-burnup rim structure,

and (b) elemental distribution as a function of the distance from the fuel-cladding interface at a burnup of about 105 MWd/kgU(after [10]).

Figure 28.12 (right side) shows the thermal-expansion algorithm that is used for an RIA with high-burnup fuel. It is seen that the outer of the three nodes expands more than the others, and the expansion decreases going from the surface to the center. The outer node determines the expansion of the pellet as if the whole pellet were at the peak (peripheral) temperature. There are two practical problems with this algorithm. First, the temperature that is assigned to the outer node is sensitive to its thickness and hence to the number of nodes chosen. Second, a very thin outer node will not have the ability to deform the cladding outwards without some inwards deformation of the node itself. Although this complicates the analysis, a high-burnup fuel pellet expands considerably more than a fresh or low-burnup pellet for a given fuel enthalpy in an RIA. This is crying for a sample calculation, à la examples #1 - #3 above

show scale here

13

13

Radius

Tem

p.

Tem

p.

Radius

28.3.2 RIA tests Because of the difficulties in calculating pellet expansion, gap closure and pellet-cladding mechanical interaction (PCMI), empirical data is used to obtain cladding strain. Figure 28.13 shows data from RIA tests with high-burnup fuel in the Cabri test reactor in France and the Nuclear Safety Research Reactor (NSRR) in Japan[2]. This figure shows measured plastic strain (no elastic strain) on the ordinate. Despite considerable scatter in the data, it is clear that at 1000

Fig.28.11. Edge-peaked pellet temperatures early in an RIA transient

and center-peaked temperatures after significant heat transfer

(after [8]).

Fig.28.12. Thermal expansion algorithms (these are not algorithms - they are diagrams) for normal operation (highest temperature at pellet centerline) and for a zero-power RIA (highest temperature at the edge) in high-burnup fuel.

14

14

J/g, the total strain is greater than 3%. Although fresh cladding can easily accommodate 1% strain during this power pulse, it is not clear that the same is true for irradiated cladding. . The mechanical response of cladding on high-burnup fuel to an RIA is governed by the loading characteristics (stress state, strain rate) and by cladding temperature, both of which vary during the transient. One of the crucial aspects of cladding failure during an RIA is the temperature at which the maximum load occurs. If this is near the beginning of the transient, the cold cladding may not have enough ductility to survive the accident; if it happens later, the greater ductility afforded by the higher temperature may prevent cladding failure.

Fig.28.13. Plastic strain measured in non-failed cladding as a function of maximum fuel

enthalpy change for tests in Cabri and NSRR (after.[2]). What is HBO? - Home Box Office? This graph has too many points. Use data from only the 3 reactors mentioned in

text. 28.3.3 Cladding loading by PCMI Energy deposition from an RIA causes a sudden rise of the fuel temperature, and the fuel expands rapidly against the cladding. The resulting pellet-cladding mechanical interaction (PCMI) results in compressive radial stresses and tensile hoop stresses on the cladding (see example #1 above) . This loading is somewhere between plane-strain tension (pellet can slide with no friction), and equal biaxial stress (no pellet-cladding bonding). Hydrides generated during normal operation (Fig. 28.2) do not dissolve during the fraction of a second duration of an RIA. They remain and harden the metal, thereby reducing its ductility and causing failure at lower pulse energies than an RIA in fresh fuel (Fig. 28.14). Failure starts with crack initiation in the brittle hydrided rim on the cladding OD and propagates inward by a mixture of brittle and ductile cracking. Two comments regarding Fig. 28.14 are germane. First, such data are usually evaluated as a function of maximum fuel enthalpy addition (this is what the energy pulse is termed) (where is this on the figure?) which drives cladding expansion

15

15

during the transient but does not include expansion associated with the temperature increase on startup. (this comment needs further clarification). Second, at very low hydrogen concentrations the cladding has sufficient ductility to survive the power pulse and ultimately fails around 700 J/g by the high-temperature mechanism described above for fresh fuel. Does this mean that the cladding doesn't fail until 700 J/g? What does (total) mean? Where is the high-temperature mechanism described?.. How is this text related to Fig. 28.14?

0

50

100

150

200

250

300

350

0 50 100 150 200 250 300 350 400 450 500 550 600

Estimated Hydrogen Content (ppm)

Fuel

Ent

halp

y R

ise

(cal

/g)

BIGRCABRIIGRNSRRPBFSPERT

Fig.28.14. Effect of hydrogen content on maximum fuel enthalpy for failure during an RIA

on fuel irradiated to ....MWd/kgU. Solid symbols indicate cladding failure (after [2]). Finally, the mechanism by which fuel dispersal into the coolant from high-burnup fuel leads to a steam explosion is totally different from that in fresh fuel. In the latter, fuel melting is necessary to produce a sudden expansion that expels hot fuel material. In high-burnup fuel on the other hand, fission gas bubbles which accumulate on grain boundaries (in Fig.28.10(a)) can cause fuel expansion. Rapid heating of fuel material, especially in the gassy high-burnup rim region near the pellet periphery, results in breakup of the fuel pellets and entrainment of particles in gas escaping through the cladding breach. Most tests with high-burnup fuel that failed by PCMI experienced fuel dispersal accompanied by a pressure pulses (signifying a steam explosion). According to Fig.28.9, most fuel rods undergo less than 300 J/g fuel-enthalpy change during an RIA. However, PCMI failures at fuel-enthalpy pulses of this order are seen in Fig.28.14. Therefore, irradiated fuel with high-hydrogen-content cladding that is located near high-worth control rods might fail, with the accompanying fuel dispersal and the potential for a steam explosion. 28.3.4 Summary The consequences of a reactivity insertion accident are influenced by a combination of states of the fuel , the cladding and the coolant. Each of these pose limits that cannot be exceeded if cladding rupture is to be avoided. Figure 28.15 shows these limits. The origin indicated by the

16

16

circle is the point of least-likely cladding damage. The distances from the origin and other locations in the cube represent increased likelihood of damage - on the vertical axis is the state of cladding ductility (or brittleness). The more ductile the metal, the less likely is damage. The upper plane of the cube is the least desirable outcome of ductility loss, namely fracture. - the horizontal axis represents burnup of the fuel, with the accompanying radiation damage and

corrosion. Due to the strain energy concentrated in the microstructure of high-burnup fuel, the liklihood of CSED (what is this?) and of cladding damage increases. - The remaining axis is a measure of coolant temperature prior to the RIA event. For a PWR, hot coolant increases the likelihood of damage. In addition, starting from a hot system increases the chances of departure from nucleate boiling (DNB), a situation that reduces heat transfer to the coolant and impedes removal of the energy introduced by the pulse. 28.4 High-temperature Properties of Zr and ZrO2 Analysis of accidents in LWRs that lead to exposure of Zircaloy cladding to steam rely heavily on the properties of the metal and its oxide. 28.4.1 The Zr/ZrO2 phase diagram The phase diagram of the zirconium- zirconium oxide system is shown in Fig. 28.16. Upon heating zirconium metal, a phase change from the alpha phase (hexagonal-close-packed structure) to the beta phase (body-centered-cubic structure) begins at about 850ºC and is complete at about 975ºC. Oxygen is highly soluble in α-Zr; the metal can accommodate ~30 at% oxygen before transforming to the oxide. However, α-Zr is very brittle, so mechanical properties of the cladding are compromised at low temperature. The phase diagram shows that α-Zr is stable at temperatures approaching 2000oC if the oxygen content is sufficiently high. This form

Fig. 28.15 Factors influencing an RIA and the potential effects on the cladding

17

17

of the metal is called oxygen-stabilized alpha zirconium, often designated by α-Zr(O). The pure metal liquefies at ~1850oC. To strengthen the metal, the alloy Zircaloy (denoted by Zry) containing 1.5 wt % Sn and 0.5 wt % Fe + Ni + Cr is often used for nuclear fuel-element cladding. The alloying additions lower the melting point by about 100oC compared to pure Zr.. According to Fig. 28.16, at low temperature the stable oxide has a monoclinic crystal structure. Over ~ 1000oC, the oxide transforms to the tetragonal form. This is a martensitic transformation that is accompanied by a significant volume change that can crack or even powder the oxide. Above 1500oC the stable phase is cubic ZrO2. These crystal structures are illustrated in Sect. 3.7.4. They are all fundamentally fluorite lattices but with distortions in the monoclinic and tetragonal forms. Unlike UO2+x, hyperstoichiometric zirconia does not exist because the metal has no valence greater than 4+. However, hypostoichiometric zirconia (ZrO2-x) is represented by the narrow hatched region on the right of the phase diagram in Fig. 28.16. The other two shaded regions in Fig. 28.16 are the two crystal structures of the metal: hcp α-Zr and bcc β-Zr. The oxygen concentrations in the two limits of α-Zr are denoted by Cα1 and Cα2, where in single-phase α-Zr, 28.4.2 Zirconium (Zircaloy) Oxidation The reaction:

Zr + 2H2O(g) → ZrO2 + 2 H2(g) (28.1) releases hydrogen gas that is swept away with the flowing steam. If the hydrogen escapes from the reactor pressure vessel it can cause an explosion when it mixes with the air in the containment vessel. Such an explosion occurred in the TMI accident. Zirconium exposed to steam oxidizes at a more rapid rate than during normal operation in water at lower temperatures (corrosion). As the temperature increases above 1000ºC, the cladding surface oxidizes rapidly enough that more than 10% of the cladding wall thickness can be consumed during the period of a LOCA before the core is cooled by emergency systems. At high temperature, the ZrO2 product of Zr oxidation in either steam or oxygen forms a dense, layer, or scale on the metal. The high-temperature tetragonal oxide is adherent and protects the cladding surface. Contrary to the oxide produced by corrosion in liquid water at low temperature (~ 400oC ), the high-temperature scale does not crack because of the compressive stress generated by the ~ 50% volume increase when the metal is converted to oxide. At high temperature, the oxide is sufficiently plastic to deform rather than fracture. Local charge neutrality must be maintained in ionic crystal structures of oxides such as ZrO2, in which Zr cations have a valence of +4. Therefore, if an aliovalent impurity such as Nb5+ is incorporated into the oxide, the local lattice defect concentration is modified and the diffusion rate is altered. Note that Sn has a valence of +4 just like Zr, so Sn impurities do not significantly affect diffusion in ZrO2. Consequently alloys like M5 (Zr-1%Nb) would be expected to oxidize at different rates than Zircaloy (Zr-1.4%Sn).

18

18

Fig. 28. 16 ZrO2 - Zr phase diagram - Single-phase regions are shaded the dotted

horizontal line at 1300oC is not part of the phase diagram 28.4.3 Simple parabolic scaling Zircaloy oxidation in steam is described by the weight gain of small specimens exposed to steam in laboratory experiments. In the temperature range of 1000-1500ºC, the weight gain rate obeys parabolic kinetics of the same form as Eq (22.8): tKW = (28.2) where W is weight gain due to oxygen forming ZrO2 in grams per square centimeter of surface area and t is time in seconds. K is a temperature-dependent rate constant. From Eq (28.1) the weight-gain rate is:

2/121

tK

dtdW

= (28.2a)

Its temperature dependence is given by: 'exp( / )K K Q RT= − (28.3)

300 μm

19

19

T is the temperature in Kelvins and Q is the activation energy of the reaction. R is the gas constant (8.314 J/mole-K). The zirconium-oxidation rate constant derived from an early study by Baker and Just [18] overestimates reaction rates by as much as 30% . A more recent and more accurate study was performed by Cathcart, Pawel, et al.[19]. The Cathcart-Pawel correlation is

)T/050,10exp(60.0K −= g/cm2-s1/2 (28.4)

28.4.4 Heat release from oxidation Reaction (28.1) is highly exothermic (ΔHo = -585 kJ/mole [3]) and thermodynamically favored at all temperatures (ΔGo is large and negative). Consequently, Zircaloy oxidation is partly responsible for the generation of high temperatures in the accident sequence and can result in liquefaction of the fuel rods, as in the TMI accident. Fission-product decay heat elevates the fuel rod's temperature, but reaction (28.1) is kinetically restricted until about 1100oC. Then, the oxidation process is autocatalytic: the heat released by the oxidation reaction aids in increasing the temperature, which further accelerates the kinetics of the reaction (see Sect. 28.10). At high temperatures (>~ 1100oC) the enthalpy released by reaction (28.1) adds to the decay heat from fission products. The oxidation heating rate is related to the weight-gain rate by:

heat rate = cmkW

tddW)R2(18

cmcm)R2(

ZrmolekJ585

Omole2Zrmole1

Og16Omole1

scmOg

tddW

C

2

C2 ⎟⎟⎠

⎞⎜⎜⎝

⎛π=π×××

−

(28.5)

The 2πRC term converts the heat generation from per unit area to per unit length of cladding. An example of linear power due to oxidation as a function of temperature is shown in Fig.28.18. The individual curves represent different thicknesses of oxide on the metal, which are related to exposure time by:

2oxox

Kt ⎟

⎠⎞

⎜⎝⎛ ρδ

=

ρox is the density of ZrO2 (g/cm3) and δox is the scale thickness (in cm). From Eq (28.4), at 1300oC (1573 K), the rate constant is K = 1.0×10-3 g/cm2-s1/2. Using these numbers, the time to grow a scale of thickness δox = 10 μm (10-3 cm) is 32 seconds4. At this temperature and scale thickness, Fig. 28.18 gives a heat release rate of ~2 kW/m. The weight-gain rate from Eq (28.2a) is 0.5(1×10-3)/√32 = 9×10-5 g O/cm2-s. Substituting this and the 1.25- cm diameter of the cladding used for Fig. 28.18 into Eq (28.5) yields:

4 This can be compared to growth in reactor coolant water of a 10 μm corrosion scale on Zircaloy cladding at 350oC, which, from Eq (22.9b), requires 5x1012 days. Arthur: something is amiss!

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20

cm/kW0063.0)109)(63.02(18 oxidation todue releaseenergy of rate 5 =××π= −

This is a factor of 3 lower than the value obtained from Fig. 28.18. 28.4.5 Growth of the α-Zr(O) layer An important feature of the oxidation process that has been omitted from the above analysis is the growth of another layer beneath the oxide scale, which clearly shows in Fig. 28.17. This observation and the phase diagram of Fig. 28.16 are consistent in identifying this layer as α-Zr(O).

Fig.28.17. Unirradiated Zircaloy-2 after oxidation in steam at 1200ºC for 600 seconds. The scale reads 300 μm (after [17])

During oxidation, oxygen from the oxide layer diffuses into the cladding metal. Zirconium alloys in the beta phase cannot hold much oxygen, however. Therefore, as oxygen pours into the metal, near-surface regions with high oxygen concentrations change phase again – this time back to the alpha phase, which is stabilize by the oxygen (α-Zr(O), the oxygen-stabilized alpha phase). As shown in Fig.28.16, the oxygen solubility limit in the alpha and beta phases is temperature-dependent. At 1300ºC, for example, the beta phase can hold only about 5 at% oxygen. A sketch of the oxygen concentration profiles in Fig. 28.19 provides the basic parameters for the subsequent analysis. In addition to the presence of a second reaction layer, the movement of the two interfaces must be taken into account; the first interface lies between the oxide and the α-Zr(O) and the second separates α-Zr(O) and β-Zr phases.

21

21

Fig.28.18. Linear power generation from metal-steam reaction for cladding with a

diameter of 1.25 cm and various oxide thicknesses (after [3]). This is a complex multi-phase diffusion problem with moving boundaries, but is simplified by the following5: 1. Oxygen penetration through the α/β interface is neglected; the β-Zr phase does not absorb oxygen 2. The volume change upon conversion of metal to oxide is neglected. This is not a trivial simplification; per atom of Zr, the oxide is 50% larger than the metal. 3. The initial state is pure β-Zr; neither the oxide nor the α-Zr phase is present 4. The β-Zr phase is infinite in the direction of diffusion in the other two phases 5. Both interfaces move in a parabolic fashion; their velocities are proportional to t-1/2. This implies that the O2- diffusion coefficients Dox in the oxide and Dα in the metal are independent of the oxygen concentration, C. The governing kinetic equations are:

in the oxide layer (0 ≤ x ≤ δox): 2

2

ox xCD

tC

∂∂

=∂∂ (28.6)

with boundary conditions: at the steam/oxide interface: C(0,t) = Co (28.7) at the ox/α interface: C(δox,t) = Ca (28.8) Concentrations (Co and Ca) are read off the phase diagram of Fig. 28.16 as atomic % oxygen. 5 Pawel (J. Electrochem. Soc., 126 (1979) 1111) has analyzed this situation without simplifications 1 - 4 above.

22

22

Fig. 28.19 Oxygen concentration profiles in scaling of Zircaloy with an intermediate layer

of α-Zr(O). vα and vβ are interface speeds with respect to the surface of the oxide.

In the α-Zr(O) layer, the origin at the ox/α interface is moving to the right with velocity vα, which imparts a convective term into the oxygen diffusion equation for this phase. The solid appears to move to the left in the diagram at a velocity vα.

in the α-Zr(O) layer (0 ≤ y ≤ δα): y

Cv

yC

Dt

C2

2

∂∂

+∂∂

=∂∂ α

αα

αα (28.9)

in which the Dα is the diffusivity of oxygen in α-Zr(O). Boundary conditions are: at the ox/α interface: Cα(0,t) = Cα1 (28.10a) at the α/β interface: Cα(δα,t) = Cα2 (28.10b) A though the diffusion equations are time-dependent, there are no initial conditions because the layer thicknesses are zero at t = 0. The remaining equations are oxygen balances at the two interfaces:

at the ox/α interface: 10

aox Cvy

CDCv

xCD

ox

ααα

ααδ

+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=+⎟⎠⎞

⎜⎝⎛∂∂ (28.11)

at the α/β interface: 0C)vv(y

CD 2 =−+⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

ααβδ

αα

α

(28.12)

23

23

Equation (28.12) assumes that the solubility and/or diffusivity of O in β-Zr are/is small enough to neglect oxygen penetration into this phase. To facilitate solution of these equations, they are recast in dimensionless form using:

tD2xX

ox

= tD2

yYα

= (28.13)

and:

ao

a

CCCC−−

=φ 21

2

CCCC

αα

αα

−−

=θ (28.14)

Equations (28.6) through (28.12) become:

0dXdX2

Xdd

2

2

=φ

+φ (28.15)

φ(0) = 1 φ(A) = 0 (28.16)

The parameter A represents the oxide side of the ox/α interface and is defined below.

0dYd)BY(

dYd

2

2

=θ

++θ (28.17)

θ(0) = 1 θ(H - B) = 0 (28.18)

The parameters H and B are defined below. The dimensionless forms of the flux balances at the interfaces are:

⎟⎟⎠

⎞⎜⎜⎝

⎛−−

+⎟⎠⎞

⎜⎝⎛ θ⎟⎟⎠

⎞⎜⎜⎝

⎛−−

=⎟⎠⎞

⎜⎝⎛ φ⎟⎟⎠

⎞⎜⎜⎝

⎛−−

−α

α

=α

αα

=α 2o

1a

0Y2o

21

AX2o

ao

CCCC

A2dYd

CCCC

SdXd

CCCC

(28.19)

and

⎟⎟⎠

⎞⎜⎜⎝

⎛−

−=⎟⎠⎞

⎜⎝⎛ θ⎟⎟⎠

⎞⎜⎜⎝

⎛−−

−α

α

−=α

αα

2o

2

BHY2o

21

CCC

)BH(2dYd

CCCC

(28.20)

The above forms of the defining equations depend on the crucial assumption that the two interfaces grow in parabolic fashion:

dt/dt/kv oxδ== αα which leads to: tk2ox α=δ (28.21)

t/kv ββ = = dδox/dt + dδα/dt which yields: t)kk(2 αβα −=δ (28.22) The dimensionless parameters in the equations are:

24

24

αβαα ==== D/kH D/D S A/SB D/kA oxox (28.23)

The objective of the analysis is to determine the rate constants kα and kβ. The solution procedure is as follows: 1. Eqs (28.15) and (28.17).

)A(erf)X(erf1−=φ (28.24)

)B(erf)H(erf)B(erf)BY(erf1

−−+

−=θ (28.25)

Here erf(z) is the error function of z (illustrative variable):

dwe2)z(erfz

0

w 2

∫ −

π= (28.26)

2. Eqs(28.24) and (28.25) are substituted into Eqs (28.19) and (20).

⎟⎟⎠

⎞⎜⎜⎝

⎛−−

π+−⎟⎟

⎠

⎞⎜⎜⎝

⎛−−

=⎟⎟⎠

⎞⎜⎜⎝

⎛−−

α

α−

α

αα−

α 2o

1aB

2o

21A

2o

ao

CCCC

A)B(erf)H(erf

eCCCC

S)A(erf

eCCCC

22

(28.27)

⎟⎟⎠

⎞⎜⎜⎝

⎛−

−π=−⎟⎟

⎠

⎞⎜⎜⎝

⎛−−

αα

α−

α

αα

21

2H

2o

21

CCC

)BH()B(erf)H(erf

eCCCC

2

(28.28)

3. The above equations are solved by trial-and-error for the parameters A and H 4. Input data Dox and Dα have been measured by a number of investigators, and their results are summarized in Table 28.1. The diffusivities in the oxide and α-Zr from various sources are in very good agreement

Table 28.1 Diffusion coefficients (in cm2/s) in the Zr/ZrO2 system at 1300oC Dox Dα Dβ source

2×10-6 3×10-7 - Pawela 2×10-5 3×10-7 3×10-6 Berdyshevb

- 3×10-7 3×10-6 Iglesiasc

a R. E. Pawel, J. Electrochem. S oc. 126 (1979) 1111 b A. Berdshev et al, Russian Academy of Sciences Preprint IBRAE-97-05 (1997); re-interpretation of data of Pawel et al 126 (1979) 1105 and S. Leistikow, Report KFK 2587 (1978), c F. C. Iglesias et al, J. Nucl. Mater. 130 (1985) 36

25

25

The concentrations appearing the above equations are obtained from the 1300oC isotherm in the phase diagram of Fig. 28.16. In units of atom fraction O, they are:

Co = 0.667 Ca = 0.66 Cα1 = 0.31 Cα2 = 0.13

The ratios in parentheses in Eqs (28.27) and (28.28) are calculated from these values. The diffusivity ratio in Eq (28.23) is S = 0.4. Eqs(28.27) and (28.28) are now functions only of the unknowns A and H, for which the trial-and-error solution gives:

A = 0.08 B = 0.2 H = 0.86

From these values and the diffusivities Dox = 2×10-6 cm2/s and Dα = 3×10-7 cm2/s, the parabolic rate constants given by Eq (28.23) are:

kα = 0.08√2×10-6 = 1.1×10-4 cm/√s and kβ = 0.86√3×10-7 = 4.7×10-4 cm/√s

Armed with these results, several calculations can be made. 1. Comparison of the layer thicknesses in Fig. 28.17 with those calculated from Eqs (28.21) and (28.22) for t = 600 s. The latter are: (δox)calc = 2×1.1×10-4√600 = 5.4×10-3 cm = 54 μm; (δα)calc = 2×(4.7 - 1.1)×10-4 √600 = 1.8×10-2 cm = 180 μm for comparison, the layer thicknesses from the photomicrograph in Fig. 28.17 are: (δox)exp = 110 μm (δα)exp = 150 μm (δox)calc is approximately one half of the measured value. (δα)calc is 20% larger than the measured value. Possible reasons are: oxide layer Pawel's diffusion coefficient Dox in Table 28.1 is an order of magnitude smaller that Dox obtained by Berdyshev et al from the same set of data. Were the latter's result correct, the calculated oxide layer thickness would have been 170 μm, which is as far off of the measured value in the overshoot direction as the value of 54 μm obtained by Pawel's equation is in the undershoot direction. α-Zr layer The discrepancy between the experimental and calculated Dα values is too small to seek an explanation on this basis. However, diffusion of oxygen into the β-Zr phase was neglected in the analysis that produced Eqs (28.24) and (28.25). Undoubtedly some oxygen was absorbed in the β-Zr phase, which would have slowed down the growth of the α-Zr layer. Accounting for this neglected process would have reduced (δα)calc.

26

26

2. Concentration profiles Using the calculated values of A and H in Eqs (28.24) and (28.25) yields the concentration variations within the layers. However, if the dimensionless distances X and Y are to be converted to x and y using Eqs (28.13), the time needs to be specified. For t = 600 s, the conversions are:

Y027.0Y6001032y and X069.0X6001022x 76 =××==××= −− From here, the steps below are followed: (a) calculate φ Vs X from Eq (28.24) using A = 0.08 as determined above (b) convert φ to C by Eq (28.13) and X to x by the above equation (c) calculate θ vs Y from Eq (28.25) using the B and H values determined above (d) convert θ to Cα by Eq (28.13) and Y to y by the above equation The result is shown as Fig 28.20. Although the oxygen distribution in the β-Zr phase was not calculated in this example, a likely distribution starting at the oxygen fraction obtained from Fig.28.16 is shown as the dashed curve. Note that the profile in the oxide is effectively linear, which provides a useful approximation for subsequent calculations.

Fig. 28.20 Distribution of oxygen in Zry exposed to steam at 1300oC for 600 s

3. Comparison of weight-gain rates with previous measurements Figure 28.21 collects the results of the most commonly-cited experiments on the temperature effects on the rate constant K expressed in the Arrhenius form of Eq (28.3). To compare these data with the present analysis, the first step is conversion of the oxygen atom fraction (C) to the O/Zr atom ratio. This connection is:

C1

CZrO

−=

In the oxide layer, the absorbed O is given by:

2oxavg

ox2

32

avgox cm

OgZrO74.0cm

OmoleOg16

ZrOg123Zrmole1

cmZrOg68.5

ZrmoleOmole

ZrOW δ⎟

⎠⎞

⎜⎝⎛=δ××××⎟

⎠⎞

⎜⎝⎛=

27

27

Fig. 28.21 Arrhenius plots of weight-gain rate constants for Zircaloy oxidation in steam Because the oxygen profile in the oxide layer is essentially linear, the average O/Zr ratio in this phase is one-half of the sum of the ratios at the two extremes. At the surface exposed to steam, O/Zr = 2. At the ox/α interface, C = 0.66, or O/Zr = 1.94. The average is 1.97. The calculated oxide layer is 54 μm thick, or 5.4×10-3 cm. From the above equation, Wox = 7.9×10-3 g O/cm2. For the α-Zr layer, the oxygen uptake is

2avg

3avg cm

OgZrO1.1cm

OmoleOg16

Zrg91Zrmole1

cmZrg5.6

ZrmoleOmole

ZrOW ααα δ⎟

⎠⎞

⎜⎝⎛=δ××××⎟

⎠⎞

⎜⎝⎛=

With δα = 180 μm and if the O/Zr ratio in the α-Zr layer decreased linearly between the bounding values of 0.45 and 0.15, the above equation gives Wα = 5.9×10-3 g O/cm2. The sum of the contributions from the two layers is 0.014 g O/cm2, and from Eq (28.2) at t = 600 s, K = 0.57×10-3 g O/cm2-s1/2 . This rate-constant value is marked by x on Fig. 28.21. It is ~ 40% below the lowest group of data lines. This suggests again that the diffusivities used in the present analysis may have been too low. 28.5 Loss-of-Coolant Accident (LOCA) A LOCA can be initiated by a pipe break or a leak in a component. Such an accident occurred at Three Mile Island PWR in 1979 when coolant was lost through a relief valve that did not close after a series of unplanned events. This evolved into a severe accident when an emergency core cooling pump, which had automatically started, was turned off because the operators mistakenly thought that the coolant was going “solid.” (i.e. contained no gas). Many lessons were learned

28

28

from the Three Mile Island (TMI) accident, and improved instrumentation and procedures should significantly increase the likelihood that a future LOCA could be successfully terminated before evolving into a severe accident. However the fact that it did occur illustrates the need to consider the consequences of such accidents in reactor licensing in order that any effects are mitigated. Subsequent sections address LOCAs i) up to the point that they are successfully terminated by an emergency-core-cooling system and ii) subsequent effects on the fuel integrity. A LOCA is postulated to begin with a blowdown of steam through the pipe break or other leak as water rapidly evaporates. The voids in the coolant shut down the nuclear reaction, but the enthalpy stored in the fuel and decay heat from radionuclides must be removed to prevent runaway core temperatures. When the break occurs, the reactor shuts down and the fuel rapidly cools and the containment vessel fills with steam. As the pressure drops, there is extensive bulk boiling as the water in the core seeks its equilibrium (saturation) temperature. The sequence of events that characterized the TMI accident are best described by the introduction of a report prepared for EPRI in 1986*: "The uncovering of the TMI reactor core was the result of a slow boil-off of core coolant due to decay heat occurring about 100 min after scram of the reactor. During the initial period of fuel uncovering and heat-up, the reactor core had no forced coolant flow and very little additional coolant was added through the combined effects of makeup and letdown. The period of fuel uncovering is critical in that fuel-pin failure and initial fission-product release occurred during this interval." "Following core uncovering, and until forced flow was re-established around 174 min, the fuel pins failed under the influence of decay heat augmented by the absence of forced coolant flow. At that time, the fission-product heating rate decayed to less than 1.5% of the full reactor power. The rate of steam generation was slow and decreased with the falling water level in the uncovering core. As the fuel rods were uncovered, notwithstanding the small power level of decay heating, they quickly became overheated." "It is important to understand the condition of the fuel pins at the time of quenching in order to assess the possibility of quench-induced fuel rearrangement and the states of the redistributed fuel. At elevated temperatures, Zircaloy reacts with steam to form an oxide along with the simultaneous generation of hydrogen and exothermic release of heat." "This scenario of events emphasizes the need to be able to predict fuel-rod temperatures and the rate of oxidation of the Zircaloy cladding under a slow boil-off condition simulating the uncovering of a reactor core" *R. Viskanta and A. Mohanty, School of Mechanical Engineering, Purdue University and J. H. Kim, EPRI project manager, "The effects of Zircaloy oxidation and steam dissociation on PWR core heat-up under conditions simulating uncovered fuel rods", Interim Report NP-4529 (1986)

29

29

Figure 28.22 depicts the sequence of events that follow a break in the primary coolant circuit piping of a PWR before the emergency core-cooling system (ECCS) is activated. The core is represented by a single fuel rod and the coolant (blue) fills the entire reactor pressure vessel (RPV). Phase A depicts the situation within a few tens of seconds following reactor shutdown in response to the pipe failure. Because of the reduction of energy production in the fuel from normal operation to fission-product decay heating, a rapid decrease of fuel temperature occurs without a substantial change in coolant temperature. Phase B describes the fuel-rod behavior as the RPV starts to empty but before the water level reaches the fuel rods. Nucleate boiling of the now-stagnant water removes some of the decay heat; the remaining energy serves to either increase the fuel-rod temperature or delay its cooling. Phase C illustrates the state of the fuel as evaporation of coolant uncovers the fuel rods. Lacking an efficient heat removal mechanism(only slowly-flowing steam cools the rods), the temperature of the bare portion of the fuel rods begin to increase.

Fig. 28.22 Sequence of events following a pipe break

30

30

28.6 Cooldown from operating temperature (Phase A in Fig. 28.22) Prior to the accident, high fuel temperatures are sustained by fission power. The centerline temperature is usually in the range 900 - 1100oC and the fuel surface is at ~400oC. Between these two limits, the temperature distribution is parabolic. The coolant in the core during normal operation of a PWR is ~ 300oC. In the following analysis of initial fuel cooling, several simplifications are made: 1. All conditions refer to the elevation in the fuel rod of the peak fuel temperature (i.e., near the midplane). 2. axial heat conduction in the fuel is neglected 3. The peak volumetric power generation in the operating fuel is 315 W/cm3, which corresponds to a peak linear heat rate of 200 W/cm and a fuel centerline temperature of 950oC. 4. The decay volumetric heat generation is 6% of the at-power value, or 19 W/cm3; it is constant during transient cooling and uniform over the radius of the fuel pellet. 5. The boundary condition at the fuel surface includes the series resistances of the gap (hgap = 10) the cladding (hc = kc/δc = 1.5) and of nucleate boiling (hboil = 1.1). The units are W/cm2-oC. 6. The coolant temperature remains at its pre-accident value (300oC) during the cooling transient. This implies that the coolant pumps are still operating during this phase of the accident. 28.6.1 Basic equations The heat conduction equation in the fuel is:

QrTr

rr1k

tTCP +⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

=∂∂

ρ (28.29)

Q is the volumetric heat generation due to fission-product decay, which is fixed at 19 W/cm3. The initial condition is that of normal operation:

2f

So

S )R/r(1TT

T)0,r(T−=

−−

(28.30)

To is the centerline temperature, which is 950oC for the pre-accident conditions chosen for this example. For the same conditions, the surface temperature is TS = 420oC. The fuel radius is Rf = 0.45 cm. The boundary condition at the pellet surface (r = Rf) is:

( )coolfR

f T)t,R(ThrTk

f

−=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

− (28.31)

where kf is the thermal conductivity of UO2. Tcool = 300oC is the coolant temperature, which is assumed to remain constant during the transient. The heat transfer coefficient is a combination of the three series resistances:

boilcladgap h1

h1

h1

h1

++= (28.32)

For the values of the individual conductances given above, h = 0.6 W/cm2-oC.

setting: s

cmCk

Rr

Rt

TTTT 2

pff

f

f2fcoolo

cool

ρ=κ=η

κ=τ

−−

=θ (28.33)

31

31

where κ ~ 0.01 cm2/s is the thermal diffusivity of UO2, Eq (28.31) becomes:

Γ+⎟⎟⎠

⎞⎜⎜⎝

⎛η∂θ∂

ηη∂∂

η=

τ∂θ∂ 1 (28.34)

the dimensionless decay-heat generation rate is:

)TT(kQR

coolof

2f

−=Γ (28.35)

For the thermal conductivity of UO2, a constant value of kf = 0.03 W/cm-K is used. This and the values of the other terms yield Γ = 0.20. The initial condition, Eq (28.32), becomes:

θ(η,0) = 1 - θSη2 (28.36) where

82.0300950420950

TTTT

coolo

SoS =

−−

=−−

=θ

The boundary condition of Eq (28.31) is nondimensionalized to:

f

f

1 kRhH here w),1(H =τθ=⎟⎟

⎠

⎞⎜⎜⎝

⎛η∂θ∂

−=η

Eq (28.37)

is the dimensionless heat transfer coefficient at the fuel surface. With the above parameter values, H = 9 28.6.2 Solution to governing equations Following the method used by Carslaw and Jaeger, the solution is sought as the sum of two functions,

θ = U(η) + W(η,τ) (28.38) where U satisfies:

Γ−=⎟⎟⎠

⎞⎜⎜⎝

⎛η

ηηη d

Uddd1 (28.39)

with the boundary condition

)1(UHddU

1

=⎟⎟⎠

⎞⎜⎜⎝

⎛η

−=η

(28.40)

and W satisfies:

⎟⎟⎠

⎞⎜⎜⎝

⎛η∂

∂η

η∂∂

η=

τ∂∂ W1W (28.41)

with the initial condition: W(η,0) = 1 - θSη2 - U(η) (28.42)

and the boundary condition: W(1,τ) = 0 (28.43)

32

32

In Eqs (28.39) and (28.41) , the implicit boundary condition at η = 0 is zero gradient. Adding the differential equations for U and W, the initial condition for W and the boundary condition for U, gives the original equations, (28.34), (28.36) and Eq (28.37). The only twist is the appearance of U(η) in Eq (28.42). This is needed because otherwise the solution of the ODE for U would appear in the sum of U and W. The solution of Eq (28.39) with Eq (28.40) is:

U(η) = )1(H

24

12

1 η−Γ+Γ (28.44)

The solution of Eq (28.41) is sought in the form of a series:

∑∞

=

τα−ηα=τη1n

2nn0n )exp()(JA),(W (28.45)

where J0(x) is the Bessel function of zero order and αn are the roots of Jo(α) = 0. The initial condition of Eq(28.42) is:

∑∞

=

ηα=η−=η1n

n0n2 )(JABC)0,(W (28.46)

The last form of this equation is Eq (28.45) at τ = 0. The constants C and B are:

77.0B and 94.0/1C 41

S41

21 =Γ−θ==Γ−ΗΓ−= (28.47)

The coefficients An are obtained from:

⎥⎦

⎤⎢⎣

⎡ηηαη+ηηαη

α=ηηαηη

α= ∫ ∫∫

1

0

1

0n0

3n0

n21

1

0n0

n21

n d)(JBd)(JC)(J

2d)(J)0,(W)(J

2A

The integrals in this equation are available, so:

⎭⎬⎫

⎩⎨⎧

αα

−α+αα

=n

n2n1

n21n

n)(J

B2)(J)BC()(J

2A (28.48)

The first 4 roots of J0(α) = 0 and the values of An from the above equation are shown below.

n αn An 1 2.405 1.89 2 5.520 -1.72 3 8.654 1.43 4 11.791 -1.23

With this table, W(η,τ) is readily computed from Eq (28.45). Because the roots increase rapidly with n, four terms in the sum suffice. The result of this analysis is shown in Fig. 28.23 as the temperature distribution decreases with time. Only 15 s are required to attain the new steady-state temperature distribution that is

33

33

supported by the decay heat release. This distribution is U(η) of Eq (28.44). It is not changing with time (unless the temperature of the water which surrounds the fuel rod decreases from its initial value of 300oC towards its equilibrium value at the prevailing system total pressure). The flatness of the steady-state distribution implies that subsequent analysis of the LOCA can be based on a lumped capacity treatment of the fuel. This is explained in the following sections.

Fig. 28.23 Collapse of initial fuel temperature distribution (denoted as -1 s) following shutdown from a linear heat rate of 200 W/cm and a coolant temperature of 300oC

The temperature profiles in Fig. 28.23 apply only to the elevation where the fuel centerline temperature is 950oC and the coolant temperature is 300oC. Both of the values change with height in the fuel rod. The steady-state fuel temperatures at other heights analogous to the profile at 15 s in Fig. 28.23 are attained on the same time scale, but the radial averages represented by the horizontal line in the plot differ from the ~ 340oC that characterizes the 950oC/300oC combination. The time scale of the decrease of fuel temperature shown in Fig 28.23 is consistent with the rule-of-thumb that is employed for diffusion problems: the time to cover a distance Δx is Dt . For a thermal diffusivity κ = 0.01 cm2/s over a distance equal to the fuel pellet radius, Rf = 0.45 cm, the characteristic time is κ/R 2

f = 0.452/0.01 = 20 s, which is close to the ~ 15 s required to reach steady-state in Fig. 28.23. 28.7 Cooldown prior to fuel uncovery (Phase B in Fig. 28.21) Even if the ECCS does not kick in, water does not leave the core immediately following the relaxation of the operating temperature distribution, as calculated for Phase A. Time is required for the water in the core to vaporize through the leak before the level in the RPV reaches the top of the core. This time is strongly dependent on the size of the leak, where it occurs, and the rate at which steam escapes from it. During this time interval the fuel rods are always immersed in water.

34

34

Phase B is treated with a broad brush, without delving into the details of processes taking place in the core or the primary components. The system analyzed shown in Fig. 28.24 consists of the main parts of the primary system: reactor core, fuel assemblies, primary piping, reactor pressure vessel (RPV) and pressurizer. The water in the primary system, shown in dark blue in Fig. 28.25, is characterized by its mass and its temperature.

Fig. 28.24 Main components inside the containment building of a PWR. The water in the primary system is dark blue. There are four steam generators, but only one is shown Table 28.2 lists the masses and temperatures of the primary-system components at the start of Phase B. The factor of 4 preceding the numbers for the piping and the steam-generator tubes accounts for the four steam generators attached to the reactor pressure vessel (Fig. 28.24 shows only one steam generator). For simplicity, the pressurizer is filled with water at the start. The fuel/cladding temperatures are the axial averages of the temperatures calculated in phase A (e.g., based on 340oC for the rod midplane shown for 15 s in Fig. 28.23) Table 28.2 Characteristics of the primary system of a 1000 MWe PWR following shutdown

and after the initial fuel-temperature profile has disappeared.

Component Mass, g* Mass water, g* Temp., oC

fuel 9×107 - 330#

cladding 1.8×107 - 330#

Fuel assemblies 1.2×108 - 300 Steam gen. tubes 4×9×107 4×5×107 300

RPV 5×108 1.3×108 300 piping 4×7×106 4×5×106 300

pressurizer 1.1×108 1.5×107 300 * From J. Rust, "Nuclear Power-Plant Engineering", Chap. 1, Holland Co. (1979) # from Phase A calculation, axially-averaged

35

35

The accident can be initiated in various ways. When the water exit is large, conditions change very rapidly. This is called a large-break LOCA. The leak rate from a small opening such as a stuck-open relief valve is much slower, and is in a category called a small-break LOCA. For this example, the TMI-type failure of the relief valve at the top of the pressurizer (where No. 5 in Fig. 28.24 is pointing) is chosen as the source of the leak into the containment building. No liquid water leaves the system. In order to simplify this complex situation, the following restrictions are imposed: - all decay heat from the fuel rods is transferred to the water in the core by nucleate boiling. - water in the system (in dark blue in Fig. 28.24) is characterized by a single temperature. - water boiling from the core collects as steam in the pressurizer - the core is treated as a single unit that acts solely as a heat source - only the water in the primary system absorbs heat; the solid components are not involved - from Table 28.2, the mass of water initially in the primary system is 4×108 g - no make-up water is added to the system - the steam and water are in thermodynamic equilibrium. Their properties are obtained as functions of pressure from the saturated steam tables. - the open relief valve is represented as a length of piping through which the water flow rate is equal to k'Δp, where Δp is the pressure difference between the steam in the pressurizer and the gas in the containment building. k' is a constant which is selected to yield the correct order of magnitude of the time for the system pressure change from saturation at 300oC to the containment-building pressure pcont. Figure 28.25 shows a simplified version of Fig.28.24 that is used in the analysis of phase B. It is simply a pot of water with an internal heat source. The time-dependent variables are: - system pressure, p - the mass of water mW - the mass of steam, mSt These are determined as functions of time by the following conservation equations:

36

36

Fig. 28.25 Simplified version of the primary system during phase B

Mass: td

mdtd

md StW + = - k(p - pcont) (28.49)

Volume: 0W

0W

W

W

St

St mmmρ

=ρ

+ρ

(28.50)

where ρ density in g/cm3 and subscript 0 means initial value. The pot is filled with water at the start of boiling.

Energy: Stcontth h)pp('ktdHd)t(gP −+= (28.51)

Pthg(t) is the thermal power of the heat source. Relating this back to the accident analysis, Pth is the reactor thermal power prior to shutdown (3000 MW) and g(t) is the fraction that appears as decay heat at t seconds after shutdown6: g(t) = 0.095(t + to) -0.26 (28.52) to is the time following reactor shutdown when phase A ends and phase B begins. From Fig. 28.23, to = 15 s. t is the time starting from the beginning of phase B. H in Eq (28.51) is the enthalpy of the contents of the pot:

H = mwhw + mSthSt (28.53) hw and hSt are the specific enthalpies of water and steam in the pot in J/g. The governing equations are recast in dimensionless form with the definitions in the table below:

6 M. El-Wakil, "Nuclear Heat Transport", Internat. Textbook Co. (1971)

37

37

variable water steam

mass MW = mw/mw0 MSt = mSt/mw0 density ΛW = ρw/ρW0 ΛSt = ρSt/ρW0

enthalpy ΩW = hw/hvap,300 ΩSt = hSt/hvap,300 pressure Γ = p/pcont Γ= p/pcont

where hvap,300 = 1405 J/g is the enthalpy of vaporization of water at 300oC. ΛW, ΛSt, ΩW and ΩSt and obtained as functions of Γ from the saturated steam tables once pcont is specified. The remaining dimensionless quantities are:

300,vap0w

th

hm*tP

=Π *tt

=τ where cont

0w

p'km

*t = (28.54)

The dimensionless forms of the three conservation equations are:

1. Mass )1(d

dMd

dM StW −Γ−=τ

+τ

(28.49a)

2. Volume 1MM

W

W

St

St =Λ

+Λ

(28.50a)

3. Energy ( ) )1(MMdd)(g StStStWW −ΓΩ+Ω+Ωτ

=τΠ (28.51a)

The derivatives of the dependent variables are rendered in calculable forms by the following manipulations:

The time-derivative of Eq (28.50a) is: τ

−τΓ

=τ d

dMc

ddb

ddM WSt (28.50b)

where

τ−

ΓΛ

=d

cdMd

db W

st W

StcΛΛ

= (28.55)

Solving Eqs (28.49a) and (28.50b) gives the two mass derivatives in terms of the pressure derivative:

)1(ddb

ddM

)c1( w −Γ−τΓ

−=τ

− (28.49b)

and

)1(cddb

ddM

)c1( St −Γ+τΓ

=τ

− (28.50c)

38

38

Substituting the above two equations into Eq (28.51a) gives:

( )ba

)1)(gdd

vap

vap

Ω+

−ΓΩ−τΠ=

τΓ (28.51b)

where:

τΩ

+τ

Ω=

dd

Md

dMa St

StW

W c1

WStvap −

Ω−Ω=Ω (28.56)

The coefficients a, b and c are functions of the pressure variable Γ (via the densities and enthalpies) and of MW and MSt, but not of their derivatives. Consequently, Eqs (28.49b), (28.50c) and (28.51b) can be solved (using Euler's method) for the three dependent variables as functions of τ given the initial conditions:

Γ = 86 MW = 1 MSt = 0 at τ = 0 (28.57) The first of these assumes that the initial temperature of the system is 300oC and pcont = 1 atm. All constants in Π defined in Eq (28.54) are known excepting t*, which also controls the conversion of real time to dimensionless time. Consequently, only t* need be specified in order to solve the system of equations. Reasonable depressurization kinetics are obtained with t* = 500 min. Figure 28.26 shows the time dependence of the pressure and the water/steam masses.

time, sec

0 500 1000 1500 2000 2500 3000

pres

sure

, atm

0

20

40

60

80

100

m/m

wo

0.0

0.2

0.4

0.6

0.8

1.0

water

steam x 100

Fig. 28.26 Depressurization of the primary system of a PWR following a small-break LOCA with a time constant of 500 min

39

39

The pressure drops rapidly in the first few minutes but then levels off and very slowly approaches the containment pressure. The steam mass rises quickly, but then decreases and reaches a constant value of ~ 0.1% of the initial water mass. The latter decreases rapidly at first, but with depressurization, expulsion of steam drops, and the water-loss rate is much slower, although not zero. At 2700 s, 50% of the initial water has been lost through the open relief valve. It is assumed that this is the point at which core uncovery (phase C) begins. 28.8 Fuel-rod uncovery (Phase C) When the water level drops below the top of the core, the bare portions of the fuel rods heat up because the only heat sink is slowly-flowing steam. The temperatures of the water-covered portion of the fuel elements are kept low because essentially all of the decay heat is consumed in boiling the adjacent water. Figure 28.27 is a representative group of fuel elements in a core partially-filled with water. On the left is the shape of the decay heat. The dashed ends represent the extrapolation distance of the neutron flux in the operating reactor. Table 28.3 gives the thermal properties needed for the analysis and Table 28.4 gives the characteristics of the reactor core used in the calculation. Table 28.3 Properties of the components

Component Density, g/cm3

Heat capacity J/g-K

Fuel (UO2) 10.5 0.31 Zircaloy 6.5 0.28

water 0.95 4.2 steam ∝ p/Tst 2.0

Table 28.4 Fuel-element dimensions

Component symbol value Fuel-pellet radius Rf 0.45 cm

Cladding OD RC 0.55 cm Cladding wall thickness δC 0.1 cm

Number of fuel rods N 6.4×104 Flow area per rod Aflow 0.8 cm2

Fueled length 2Zo 300 cm Phase A dealt with a single axial location on a single fuel rod. Phase B considered the behavior of the entire primary system. Phase C analyzes the thermal performance of a single fuel rod which is partially immersed in water and partially in steam. Phase B ends when the water level reaches the top of the core, which in this example occurs when 50% of the water has escaped into the containment building. 2700 s after initiation of the accident, the system pressure reaches ~ 2 atm (Fig. 28.26) with the corresponding water

Flow area per rod

40

40

Zex

decayheat

upper grid plate

lower grid platez = 0

z = s

z = Zo

z = 2Zo

Fig. 28.27 Partially-uncovered fuel elements

temperature of ~ 120oC. These conditions are assumed to persist throughout the fuel-rod uncovery period analyzed in Phase C. As the water level drops, the upper portions of the fuel rods become exposed to steam emanating from boiling in the lower, water-covered zone. Because of the poor heat transfer rate from the bare fuel rod to the steam, the fuel temperatures increase well above those in the water-covered lower region. The upper portion exposed to steam occupies heights from the water/steam interface level (z = s) to the top of the core (z = 2Zo). The region containing water begins at the core inlet (z = 0) and terminates at the water/steam interface (z = s). 28.8.1 Convective heat transfer to steam (s ≤ z ≤ 2Zo) The increase of the temperature of the upflowing steam is given by:

)TT(hR2dz

dTCGA StFC

StPstflow −π= (28.58)

Here G is the mass flow rate of steam (units of g/cm2-s) generated by boiling in the water- covered region. Aflow is the cross-sectional area available for coolant flow past a single rod (see Table 28.4) and CPst is the heat capacity of steam. TSt is the steam temperature in the uncovered length of the fuel rod and TF is the local fuel temperature. Both are functions of time and

41

41

elevation. h is the heat transfer coefficient from the cladding OD to the bulk steam averaged over the length of the exposed metal. Axial heat conduction and natural convection in the steam are neglected in Eq (28.58).

The Reynolds number is Re = G×Dh/μSt, where Dh = hydraulic diameter = 4Aflow/2πRC and μSt = viscosity of saturated steam at 120oC. For the dimensions in Table 28.4, Dh = 0.9 cm. Figure 28.28b shows that initially, the steam velocity at the interface is high - nearly 3 m/s. At this location, Re ~ 2200 but decreases as the interface recedes. Consequently, the steam flow is laminar.

The single most critical parameter for the analysis of the upper section of the fuel rod is the length-averaged convective heat-transfer coefficient h . Engineering correlations for this quantity are available for laminar or turbulent flow inside horizontal tubes or exterior cross flow across horizontal tubes, but none for exterior flow along vertical tubes, which best fits the geometry of the flow channels in the fuel assembly. For our purposes, the correlation for laminar flow inside a horizontal tube with constant wall temperature suffices for an estimate for h in the fuel-element flow channel.

The correlation7 provides the average Nusselt number, uN = h Dh/kSt, where kSt is the thermal conductivity of steam. For saturated steam, kSt varies from 2.5×10-4 W/cm-oC at 100oC to 6.6×10-4 W/cm-oC at 600oC. The Nusselt number is a function of the Graetz number, Gr = Re×Pr×(Dh/L), where Pr = CPStμSt/kSt is the Prandtl number of steam. For very long tubes (Dh/L << 1, or Gr < ~15) the correlation reduces to uN ~ 5. The long-tube approximation applies to the fuel assemblies in the core, so the length-averaged heat-transfer coefficient is:

2St

9St

64hSt T101T103108.7D/kuNh −−− ×+×+×==

where StT is the average steam temperature (oC) in s ≤ z ≤ 2Zo. The heat transfer coefficient is taken as that at 120oC, or, from the above equation, 3102.1h −×= W/cm2-oC. 28.8.2 Fuel heatup in the uncovered zone (s ≤ z ≤ 2Zo) Assuming that the neutron flux is independent of radius in the core, energy is released from a fuel rod at a rate (Pth/N) g(t), where Pth is the thermal power of the reactor prior to shutdown and N is the number of fuel rods in the core. The fraction of the power that is produced in a unit relative height at elevation z/Zo is:

( ) ( )[ ]( )[ ] ( )∫ −

−=

γπ

γπ

2

0Zz

Zz

2

Zz

2Zz

oo

o

o

d1cos

1cosF (28.59)

From the diagram of Fig. 28.27, γ = (Zex+Zo) /Zo represents the ratio of the axial extrapolation distance of the neutron flux to the fuel-rod half-height. For the present case γ = 1.3 is chosen, and the axial power distribution follows from Eq (28.59):

7 F. Kreith, "Principles of heat trarnsfer", 3rd Ed., p. 438 Internat. Textbook Co. (1965)

42

42

( )[ ]12.1cos645.0)(F

oo Zz

Zz −= (28.59a)

The energy balance on a unit relative length of fuel at height z/Zo fuel is:

)TT(hR2dtTd)R()C()(F)t(g

ZNP

StFCF2

FFPZz

o

tho

−π+πρ= (28.60)

28.8.3 Boiling in the water-cooled zone (0 ≤ z ≤ s) In the lower portion of the fuel rod, heat is transferred to the adjacent water by nucleate boiling. This mode of heat transfer is so efficient that essentially all of the decay heat generated by the fuel is transferred to the water; the fuel does not increase in temperature. The vaporized water creates a mass flux of steam that feeds the upper zone of the fuel rod (i.e., above the water/steam interface). The enthalpy balance in this zone is:

( ) ( ) ( )[ ]{ }12.1sin)2.1sin()t(gNP

2.1645.0dF)t(g

NP

hGAo

o

Zsth

Z/s

0Zoz

Zozth

120,vapflow −+== ∫ (28.61)

The enthalpy of vaporization at 2 atm pressure is hvap,120 = 2206 J/g and the temperature of the water and the immersed fuel rod is 120oC. The integral in this equation was evaluated by substituting Eq (28.59a). Angles are expressed in radians.

28.8.4 Water/steam interface motion (at z = s) Water boiloff from the N fuel assemblies each of flow area Aflow in the core removes water at a rate of NAflowG. The water in the rest of the RPV is not boiling but nevertheless communicates with the water in the fuel assemblies and retards the rate at which the common surface decreases. This rate is given by:

W

Gftdsd

ρ

×−= (28.62)

where ρW is the density of water and:

RPV

flow

ANA

f = (28.63)

is the fraction of the cross-sectional area of the RPV open to coolant flow.

ARPV is the inside area of the RPV, less the room occupied by structures such as the core barrel and the thermal shield. For the present calculation, we take f = 1/2.

The a pair of each of the above equations apply to the two regions of the fuel rod; Eqs (28.58) and (28.60) for the elevations exposed to steam (s ≤ z ≤ 2Zo) and Eqs (28.61) and (28.62) for the submerged zone (0 ≤ z ≤ s).

The time dependence of the decay-heat production has been expressed as:

43

43

g(t) = 0.095(t + to)-0.26 = 0.095(t*)-0.26(τ +τo)-0.26 (28.52a)

Eliminating G between Eqs (28.61) and (28.62) yields:

( )[ ]{ }12.1sin)2.1sin()tt(hA

)N/P(f2.1

645.0095.0dtds

oZs26.0

o120,vapflowW

th −++ρ

×−= − (28.64)

Integration of Eq (28.64) with the initial condition s(0) = 2Zo yields s(t), independent of the processes occurring in the upper uncovered region. Solution of Eq (28.64) is deferred until the appropriate dimensionless variables and parameters are introduced. Time t starts at the beginning of core uncovery, which occurs at to = 2700 s.

28.8.5 Nondimensionalization The four governing equations apply to four dependent variables. They depend on two independent variables. Each of these is converted to dimensionless form by dividing by characteristic parameters which are termed "divisors". The process is summarized in Table 28.5.

Table 28.5 Nondimensionalization of the governing formulas in Phase C

Quantity symbol Units Governing equation

Divisor Dimensionless symbol

Time t s - t* τ

elevation z cm - Zo ξ

Fuel temperature TF oC (28.60) T* Θ

Steam temperature TSt oC (28.58) T* Ψ

Steam flow G g/cm2-s (28.61) G* Ω

Interface s cm (28.64) Zo S

The dimensionless forms in the last column of the table represent the symbol in the second column divided by the divisor in the fifth column (e.g., τ = t/t*). These are defined below.

hR2)R()C(*t

C

2FFP

ππρ

= (28.65)

)hR2(*)t()NZ/P(645.0095.0

*TC

26.0oth

π×

= (28.66)

PStflow

oC

CAZhR2

*Gπ

= (28.67)

44

44

In dimensionless form, the governing equations are:

=ξΨ

Ωdd

Θ - Ψ (28.58a)

+τΘ

=−ξτ+τ −

dd)]1(2.1cos[)( 26.0

o (Θ - Ψ) (28.60a)

M { })]1S(2.1sin[)2.1sin()( 26.0o −+τ+τ=Ω − (28.61a)

where M PSt

120,vap

oth

C26.0

Ch

)NZ/P(hR2

645.0095.0*)t(2.1 π

×= (28.68)

Equation (28.64) is recast in dimensionless form and integrated to yield:

∫ =−+

=S

2 )]1'S(2.1sin[)2.1sin('dS)S(U W [ ]74.0

o74.0

o )( τ−τ+τ (28.64a)

The coefficient in this equation is:

WM

*t*G1065.4M

*t*GZ74.0

f 3

oW

−×=⎟⎟⎠

⎞⎜⎜⎝

⎛ρ

= (28.69)

Tables 28.3 and 28.4 provide most of the values needed to evaluate the parameters for calculating the four dependent variables that together give the complete description of the core uncovery process. The data not in the tables have been cited in the preceding text.

Numerical values of the scaling parameters from Eqs (28.65) - (28.67) are:

t* = 545 s T* = 980 oC G* = 0.35 g/cm2-s (28.70a)

and the two dimensionless constants are:

M = 1.32 W = 0.67 (28.70b)

The dimensionless starting time for Phase C is τo = to/t* = 2700/545 = 5.0.

The explicit relationship between the dimensionless interface elevation S and the dimensionless time τ obtained from Eq (28.64a) is:

( ) o35.174.0

o W/)S(U τ−−τ=τ (28.64b)

In terms of dimensional height and time, this equation is shown in Fig. 28.28a.

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45

With S known as a function of time from Eq (28.64b), division of Eq (28.61a) by M/G* gives the time dependence of G, the mass flux of steam. Division of G by the steam density at 120oC (the temperature of the remaining water covering the fuel rods) gives the velocity of steam leaving the steam/water interface. The variation of the velocity of the interface with time is shown in Fig. 28.28b.

Fig. 28.28a Elevation of the steam/water interface during phase C as a function of time starting when the water level reaches the top of the core

Both the steam/water interface and the velocity leaving the surface decrease precipitously following core uncovery. The reasons are twofold. First, with increasing time, the fission-product decay heat decreases, but only by a factor of 1.5 between the first exposure of the fuel rods to their complete dryout. The more important factor is the volume of water available for boiling, and therefore for generating steam. As the water level drops, this volume decreases until it reaches zero when the interfaces reaches the bottom of the core. 28.8.6 Heatup of the uncovered core During uncovery of the core, the temperature of the fuel rod varies with height as does the temperature of the rising steam. These two temperatures are governed by Eqs (28.60a) and (28.58a), respectively. Solving the two coupled ordinary differential equations would be simple were it not for the complex boundary/initial condition. The lower boundary is the steam-water interface where the temperature of the fuel and the steam are both the same (120oC for this example). However, this boundary moves with time according to Eq (28.64b), which has the shape shown in Fig. 28.28a. The pair of equations can be solved using Euler's method, the details of which are given in Appendix A.

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46

Fig. 28.28b Velocity of steam leaving the interface during phase C. Time is measured from initial uncovery of the core

Figure 28.29 shows two aspects of the thermal behavior of the fuel rod (solid lines) and the steam (dashed curves). In (a), the axial temperature profiles are plotted at two times. The horizontal dotted lines represent the heights reached by the interface at the times indicated. Early in the uncovery period (t = 430 s), the steam-water interface has decreased only to ~ 80% of the core height and both temperatures are low. Much later (t = 2700 s), the interface has dropped to ~ 24% of the core height and both the steam and fuel-rod temperatures have increased dramatically. The steam exits at ~ 700oC and the fuel rod reaches a maximum of 950oC at 3/4 of the height of the core. The reason for this maximum is the combined effects of time and position on the left-hand side of Eq (28.60a). At the long time represented by the right-hand curve of Fig. 28.29a, [(τ + τo)-0.26] is reduced; at the upper elevations, the position factor cos[1.2(ξ-1)] is also decreased compared to the center of the rod. Together, these effects reduce the decay heat rate sufficiently that the left-hand side of Eq (28.60a) to become less that the heat removal term on the right, (Θ-Ψ). Consequently, dΘ/dτ becomes negative and Θ decreases with increasing height for ξ > 1.5. Figure 28.29b shows similar plots as functions of time at fixed elevations. The vertical dotted lines represent the times at which the interface reaches the two heights treated. In this depiction, both the steam temperature and the fuel-rod temperature pass through maxima with respect to time. This behavior is a direct result of the fall off of the decay heat rate with time. The temperature maxima shown in Fig. 28.29 are due to omission of important physical processes involving Zircaloy. The first is ballooning and rupture, which take place at ~ 800oC, and the second is the onset of steam oxidation. This reaction becomes important when the cladding temperature reaches ~ 1000oC, which occurs at about two hours after the water level in the RPV reaches the top of the core. These processes occur simultaneously with rod heatup of phase C, but they are treated separately in the following sections.

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47

(a) height variation at fixed times

(b) time dependence at two elevations

Fig. 28.29 Thermal responses of fuel rod and steam during core uncovery (Phase C)

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Appendix A Fuel/cladding heatup during core uncovery

The purpose of this appendix is to provide the details of the thermal behavior of the fuel rod while the water level in the core drops from full coverage to exposure of the entire fuel rod. To this end, Eqs (28.60a) and (28.62a) must be solved simultaneously with the water/steam interface decreasing with time according to Eq (28.66b) (see Fig 28.28a). In the covered region 0 ≤ z ≤ s (Fig. 28.27), nucleate boiling efficiently removes the decay heat, so that both water, fuel and cladding are maintained at the saturation temperature. For the present case, the final pressure when the contents of the RPV and the containment building are in mechanical equilibrium is ~ 2 atm. The corresponding saturation temperature is 120oC, or in dimensionless terms (see Eq (28.72a), Θ = Ψ = 120/980 = 0.122 in 0 ≤ ξ ≤ S(τ) (refer to Table 28.5 for the nondimensionalization). These constitute boundary conditions on the fuel temperature Θ and the steam temperature Ψ at the moving interface S(τ). Equation (28.60a):

=ξΨ

Ωdd

Θ - Ψ (A1)

and Eq (28.62a) are solved by Euler's method subject to the above interface condition. Because of the large range of the dimensionless time τ (Fig. 28.28a) this variable is changed to:

u = lnτ (A2)

and Eq (28.62a) becomes:

⎭⎬⎫

⎩⎨⎧

Ψ−Θ−τ+

−ξ=

Θ )()e(

)]1(2.1cos[edud

26.0o

uu (A3)

where τo = 2700/545 = 5.0 is the dimensionless time at which core uncovery begins. τ = 0 represents the start of exposure of the fuel rods, but because u = -∞ is not an appropriate initial condition, the start of core uncovery is taken to begin when the water level has dropped from the core top at s = 300 cm to s = 292.5 cm. In dimensionless terms, the latter corresponds to a water/steam interface height of S = 1.95 and from Fig 28.28a, u = -2.0. The fuel-rod temperature equation (A3) is solved for Θ as a function of u at discrete elevations in the range 0.05 ≤ ξ ≤ 1.95. At each elevation, the starting value of u is along the curve of Fig. 28.28a, for which a reasonably accurate approximation is:

⎟⎠⎞

⎜⎝⎛ −+= 1

S2ln767.0784.0u int (A4a)

where (uint,S) is a point on the uncovery curve. uint stands for the value of u corresponding to the water/steam interface at elevation S. In order to integrate Eq (A3), an initial guess of the function Ψ(ξ,u) is required.

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49

The following example is for the height S = 1.6. At this elevation, the starting time from Eq (A4a) is uint = -0.28. The procedure based on Euler's method is as follows: 1. At the interface, Ψ0 = Ψ(1.6,-0.28) = T/T* = 0.122, where from Eq (28.72a), T* = 980oC and T = 120oC is the saturation temperature at the total system pressure of 2 atm. 2. To start the integration Θ0 = Θ(1.6,-0.28) = 0.128 is assumed 3. The starting time is u0 = uint = -0.28 4. The time derivative of Θ is given by Eq (A3):

oueZZYYXX

dud

⎟⎠⎞

⎜⎝⎛ −=Θ∇=

Θ (A5)

where XX = cos[1.2(S - 1)] = cos[1.2(1.6 - 1)] = 0.75 YY = [exp(u0) + τo]0.26 = (e-0.28 + 5)0.26 = 1.58 ZZ = Θ0 - Ψ0 = 0.128 - 0.122 = 0.006 which yields: ∇Θ = 0.35 5. For a time-step size Δu = 0.01, the values of Θ and u at the end of the first time increment are: Θ1 = Θo + ∇Θ×Δu = 0.128 + 0.35×0.01 = 0.132 u1 = u0 + Δu = -0.28 + 0.01 = -0.27 For the next time interval, the starting values are: u0 = u1 Θ0 = Θ1 and Ψ0 = Ψ(S,u1), as given by the initial guess of the function Ψ(ξ,u). 6. Return to step 4. 7. Integration ceases when the desired final time is reached. 8. The process is repeated for the next height S. Equation (A1) is solved for Ψ in a similar fashion for uniform intervals of u. The starting ξ is the interface value S corresponding to the chosen value of u. Inverting Eq (A4), this is:

)02.1u30.1exp(12Sintint −+

==ξ (A4b)

The values of Θ needed for Eq (A1) are those obtained from the preceding steps Nos. 1 - 8. Ω = G/G* is obtained from the curve in Fig. 28.28b and G* = 0.35 g/cm2-s. With the newly-generated function Ψ(ξ,u), the computation returns to step 4 above. The above steps are repeated until the functions Θ(ξ,u) and Ψ(ξ,u) do not change with subsequent iterations.

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