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Zirconium ignition in exposed fuel channel
Nuclear Engineering and Design 286 (2015) 205–210
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Zirconium ignition in exposed fuel channel E. Elias ∗ , D. Hasan, Y. Nekhamkin Technion – Israel Institute of Technology, 32000 Haifa, Israel
h i g h l i g h t s • We demonstrate the idea of runaway zirconium–steam reactions in severe accidents in today’s LWRs. • We predict the thermal-hydraulics conditions relevant to cladding oxidation in an exposed fuel channel of a partially uncovered core. • The Semenov theory of metal combustion is extended to define a criterion for runaway oxidation reaction in fuel cladding.
a r t i c l e
i n f o
Article history: Received 23 October 2014 Received in revised form 12 January 2015 Accepted 1 March 2015
a b s t r a c t A theoretical model based on simultaneous solution of the heat and mass transfer equations is developed for predicting the rate of thermo-chemical reaction between zirconium cladding and a hot steam environment. Ignition conditions relevant to cladding oxidation in an exposed fuel channel of a partially uncovered core are predicted based on the theory of metal combustion. A range of decay power, convective heat transfer coefficients, and initial temperatures leading to uncontrolled runaway cladding oxidation is identified. The model could be readily integrated as part of a fuel channel analysis code for predicting possible outcomes of different accident mitigation procedures in light water nuclear reactors under LOCA conditions. © 2015 Elsevier B.V. All rights reserved.
1. Introduction The recent events at the Fukushima plants have underscored the importance to the overall reactor safety of secondary heat generation sources, such as thermo-chemical reactions at the cladding surface. Realistic analysis of metal oxidation in a partially uncovered reactor core is a complicated numerical computation task, requiring the simultaneous solution of a set of energy balance equations in the fuel elements and their metal claddings along with the mass, momentum and energy equations in the corresponding coolant channels (cf., severe accident codes; MAAP, 1992; MELCOR, 2000). Conservative analysis of cladding integrity is typically accounted for by setting a predetermined critical minimum ignition temperature, which is based on out-of-pile oxidation experiments using small specimens of cladding material (Baker and Just, 1962; Cathcart et al., 1977). Accurate modeling of metal oxidation during severe accident must account for the time dependent balance between the thermochemical reaction heat release at the cladding surface, the decay power in the fuel and the rate of cooling by the surrounding steam.
∗ Corresponding author. Tel.: +972 4 8293263; fax: +972 4 8295711. E-mail address: [email protected] (E. Elias). http://dx.doi.org/10.1016/j.nucengdes.2015.03.002 0029-5493/© 2015 Elsevier B.V. All rights reserved.
A step in that direction has been previously described by using a lumped parameters model (Nekhamkin et al., 1998; Hasan et al., 1999; Pickard, 2002) that relies on the Semenov type methodology (Semenov, 1928; Yarin and Hetsroni, 2004; Khaikin et al., 1970; Boddington et al., 1982) to identify critical conditions for runaway clad–steam oxidation reactions. A more realistic cladding oxidation model based on solid combustion theories is described in this paper. The theory of metal oxidation and combustion has been extensively developed in the last century as part of the technological effort to develop efficient solid propellants and later in conjunction with nuclear reactor safety. Conditions for a runaway thermochemical reaction were analytically derived by Chambré (1952) and Frank-Kamenetskii (1969) by solving the one-dimensional steadystate balance equation in a system of finite size. Based on the theory of thermal explosion, the combination of size, conductivity and ambient temperature at which such solution becomes mathematically impossible was defined as the condition of ignition or inflammation. An extensive review of the basic theoretical and engineering literature on metal combustion by Yarin and Hetsroni (2004) demonstrates that the rate of metal oxidation in an oxidizing gas environment depends on the thermal-hydraulic conditions in both the solid and the surrounding gas. The idea of runaway metal–steam reactions was introduced by Khaikin et al. (1970) who
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used the Semenov (1928) analysis to derive a conjugate set of heat and mass balance equations for the time evolution of temperature and oxide layer thickness on a small metal particle placed in an oxidizing gas environment of uniform temperature. In the following the basic combustion theories for metal ignition in steam are extended to account for the special characteristic conditions at a partially exposed nuclear fuel rod. In particular the effect of temperature non-uniformity along the oxidizing specimen as well as the effect of stored and decay heats at the inner surface of a cylindrical cladding shell are considered. Heat and mass balance equations are developed to define runaway timing and the thermal conditions leading to uncontrolled oxidation during severe accident. The paper outlines the possibility for integrating the proposed model in a full-scale fuel channel transient analysis.
temperature, T* , depends on the physical problem solved. In this first case the characteristic temperature is conveniently taken as a constant that equals the ambient steam temperature, i.e., T* = T0 . In Eq. (1) it is assumed that the metal mass remains constant. This holds for the initial stage of the process when the thickness of the oxide film is insignificant. For a planar zirconium slab of unit width, height L and initial thickness a0 , Eq. (1) can be further manipulated by expressing the slab volume and surface area in terms of its initial dimensions,
2. Ignition of metal particles
dı kr B = n exp − T dt ı
Zr CZr a0
dTC dı = Zr Q0 − h(TC − T∗ ). dt dt
(2)
The thickness of the oxide film, ı(t), has been theoretically investigated (Baker and Just, 1962; Khaikin et al., 1970). The simplest diffusion model takes the form,
(3)
Thermo-chemical reaction between steam and zirconium in the fuel cladding could become an important heat source under accident conditions. During normal operation of power reactors, the cladding temperature is about 10◦ above the coolant temperature, i.e., about 330–350 ◦ C, with low oxidation rate and minimal heat release. However, if cladding is partially exposed, because of core coolant loss, the clad temperature could increase by several hundreds degrees. At cladding temperatures above about 900 ◦ C, clad oxidation rate begins to significantly increase, leading to release of heat and formation of hydrogen, according to the following chemical reaction for Zr (Baker and Just, 1962; Whitmarsh, 1982):
where B is the activation energy (B = 22,900 K) and kr = 3.968 × 10−5 m2 /s is a pre-exponent reaction rate, which accounts for the oxidizer concentration at the surface (Baker and Just, 1962). A parabolic law (n = 1) will be used here as it is broadly used for estimating zirconium oxidation rate in nuclear reactors (MAAP, 1992; MELCOR, 2000 codes). Thus, in order to derive an ignition criterion for metal oxidation it is necessary to solve Eqs. (2) and (3) jointly subject to the following initial conditions, ı(t = 0) = ıi and TC (t = 0) = Ti . Eqs. (2) and (3) are first reduced into the following dimensionless form,
Zr + 2H2 O → ZrO2 + 2H2 + Heat
d 1 = exp d
The heat of reaction, Q0 , is 140 kcal/mole (6500 kJ/kg of Zr reacted or 1540 kcal/kg) and the volume of hydrogen generated is 0.5 m3 (at STP conditions) per 1 kg of reacting Zr. Generally, the rate of oxidation depends on the metal temperature, T, and on the rate of oxygen diffusion through the oxidized layer, which is formed on the outer surface of the metal. Thus, predicting the necessary conditions for ignition calls for simultaneous solution of the heat and mass balance equations at the cladding surface. At certain conditions, heat generation by the oxidation reaction may exceed the cooling rate, which leads to an accelerated or runaway oxidation. In the following, a threshold criterion for runaway oxidation is derived for three representative cases: uniform temperature approximation; decay power effect and a general one-dimensional system. 2.1. Uniform temperature approximation To gain a better physical insight into the diffusion and thermal processes taking place during metal oxidation, we consider first a simplified case of a thin walled planar zirconium specimen with initial temperature TC,i instantaneously introduced into an infinite steam environment with a constant and uniform temperature, T0 . As proposed by Khaikin et al. (1970) ignition conditions in this case are determined by considering a balance between the rate of heat release by the oxidation reaction and the rate of heat transfer from the metal to the ambient steam. For a thin walled zirconium metal (low Biot number), a lumped parameters heat balance equation is given by, mZr CZr
dı dTC = Zr Q0 S − hS(TC − T∗ ) dt dt
(1)
where mZr and S are the mass and surface area of the zirconium metal; Zr and CZr are its density and specific heat, h the heat transfer coefficient, ı is the thickness of the oxide film at the surface, Q0 the heat release per unit mass of oxidized metal and TC is the zirconium temperature. The exact definition of the characteristic
d 1 = exp d
1 + ˇ 1 + ˇ
−
ω
(4a)
(4b)
with the following initial conditions, I.C. : at = 0 = i , = i . The following dimensionless variables and parameters are adopted in Eqs. (4a) and (4b), = ˇ=
ı B ; = T∗ ı∗
T
C
T∗
−1
=
1 ˇ
T
T∗ ı2 e1/ˇ 1 ; ω= ; t∗ = ∗ B kr h
C
T∗
−1
; =
t CZr a0 ˇ T∗ ; ı∗ = ; t∗ Q0
Q 2 k e−1/ˇ Zr r 0 ˇT∗
a0 CZr
=
1 a0 CZr Zr . t∗ h (5)
The right hand side of (4a) expresses a balance between the oxidation exponential heat generation in the specimen and the rate of convective cooling by the steam (/ω). The parameter, ω, specifies the ratio between the average rate of heat accumulation in the metal (over a characteristic time, t* ), and that transferred to the ambient steam. The initial value of the dimensionless oxide layer in Eq. (4b) is typically small (i ≈ 0). In the following, the effect of the initial temperature difference, i , will be studied parametrically. In order to demonstrate the utility of the model for predicting zirconium ignition conditions, Eqs. (4a) and (4b) were integrated numerically for a wide range of characteristic temperatures using an initial thickness of a0 = 0.57 mm, which corresponds to typical PWR fuel cladding. As an example, in Figs. 1 and 2 the characteristic temperature, T* , and the initial cladding temperature, TC , were arbitrarily taken as 800 K, yielding i = 0 and ˇ = 0.035. Also, following Khaikin et al. (1970) we use ıi = 10 nm resulting in i = 1.46 × 10−3 . Fig. 1 is a typical example showing the evolution of the dimensionless temperature, vs. dimensionless time, for two values of the parameter ω. It is noticed that there exists a critical minimum value, ω = ωcr , above which the temperature escalates dramatically.
E. Elias et al. / Nuclear Engineering and Design 286 (2015) 205–210
Fig. 1. Dimensionless temperature vs. dimensionless time for I = 0.
Fig. 2. Dimensionless oxide layer thickness vs. dimensionless time for two values of ω and I = 0.
Referring to Eq. (4a), it can be readily shown that at ω > ωcr the oxidation heat generation term exceeds the rate of convection cooling leading to rapid metal temperature increase (runaway). In Fig. 1, runaway oxidation, is experienced at ω ≥ 1.6. The rate of oxide layer buildup also varies significantly during the heating process, as shown in Fig. 2. The critical ignition parameter, ωcr , depends on other parameters that affect the solution of Eqs. (4a) and (4b), namely the characteristic temperature, T* (or ˇ) and the initial temperature difference, i . The dependence of ωcr on ˇ, T* , and i is plotted in Figs. 3 and 4. Negative values of i indicate initial cladding temperature lower than the characteristic ambient steam temperature. Each ωcr line in Figs. 3 and 4 basically divides the parametric space into regions of controlled oxidation below the line, and an uncontrolled runaway oxidation above it. A wide range of characteristic temperatures from 400 to 1400 K was selected in order
Fig. 3. Critical parameter, ωcr vs. characteristic temperature, ˇ, T* , and initial temperature difference, i .
207
Fig. 4. Critical parameter, ωcr of zirconium vs. initial temperature difference, i at various characteristic temperatures, T* .
to cover anticipated core conditions during a hypothetical severe accident. Note that the initial cladding temperature, TC,i , can be readily derived from Eq. (5) for each pair of i and T* (or ˇ) as TC,i = T∗ ˇ i + 1 . In Eq. (5) the parameter, ω, is defined in terms of the characteristic temperature, T* (or ˇ), and the convective heat transfer coefficient, h. The dimensional critical conditions can thus be readily expressed by a physical parametric plot as demonstrated, for example, in Fig. 5 that shows the parameter ω vs. T* for heat transfer coefficients in the range of 0.1–300 W/m2 K. Here again, the critical parameter, ωcr line (for i = 0), shown in Fig. 5 as a dashed line, divides the space into runaway conditions above the line and controlled oxidation conditions below it. As expected the critical characteristic temperature leading to a runaway ignition decreases with decreasing convective coefficient. For instance, Fig. 5 shows that if free convection conditions are assumed in the fuel channel (h ≈ 10 W/m2 K), then ω reaches a critical value (h line intersects the ωcr line) at a characteristic temperature of T∗ ≈ 1060 K, whereas when h ≈ 100 W/m2 K the critical value is reached at a higher temperature of about 1250 K. For characteristic temperatures below about 800 K and i = 0, critical runaway conditions may only be reached if the convective coefficient is unrealistically small (close to adiabatic system). This is clearly demonstrated in Fig. 6 showing the locus of all h/ωcr intersection points, yielding a parametric map of characteristic ignition temperatures as a function of the local heat transfer coefficient. Runaway conditions prevail at the lower right side of the curve (low heat transfer coefficient and high characteristic steam temperature). The effect of the initial temperature difference, i , is relatively small. The preceding model and analysis, while providing physical insight into the oxidation problem, is oversimplified. It relies on the
Fig. 5. The parameter ω as a function of characteristic temperature and heat transfer coefficient.
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E. Elias et al. / Nuclear Engineering and Design 286 (2015) 205–210
θ θ
Fig. 6. Heat transfer coefficient vs. characteristic ignition temperature and initial clad–steam temperature difference.
assumption that oxidation is the sole source of heat in the system and thus, the characteristic coolant temperature could be assumed equal to the constant and uniform ambient steam temperature. A more realistic physical representation of the cladding behavior in a partially uncovered reactor core is considered in the next two sections that account for fuel decay power and steam heating along the fuel channel. First a simplified system with a specified steam mass flow rate and a spatially uniform decay heat is analyzed in Section 2.2. This is followed by a more realistic case of a one-dimensional (axial) analysis of a cladding section subjected to a sine distributed decay heat source and cooled by flowing steam. It will be shown that the effect of both decay power and variable steam temperature could be well accounted for by proper definition of ω and the characteristic temperature, T* . 2.2. Specified steam flow, uniform decay heat Here the threshold condition for a runaway oxidation reaction is developed in a cylindrical shell of zirconium with initial inner radius, R0 , and thickness, a0 , subject to steam flowing axially on its outer surface. A uniform and constant decay power is considered along the inner surface of the shell. For a low Biot number, a lumped parameters’ heat balance over the cylindrical metal shell yields, (6)
where q is the linear decay power, L is the shell length and T is the logarithmic mean temperature difference between the solid surface (TC ) and the steam defined as, TS (L) − TS,0
(7)
T −T
ln T C−T S,0 (L) C
S
Here TS (L) and TS,0 are the steam exit and inlet temperatures, respectively. A heat balance over the steam yields,
˙ S CS TS (L) − TS,0 hST = m
(8)
˙ S and CS are the steam mass flow rate and specific heat, where m respectively. Substituting Eq. (7) into Eq. (8) and solving for the steam temperature yields, TS (L) = TS,0 + (TC − TS,0 )(1 − e− )
with =
2 R0 Lh ˙ S CS m
(9)
For a thin walled cylindrical zirconium shell a0 /2R0 << 1 Eq. (6) can be further manipulated by expressing the shell mass and surface area in terms of its initial dimensions, Zr CZr a0
The governing Eqs. (3) and (10) can now be reduced to a dimensionless form identical to Eqs. (4a) and (4b), with the characteristic temperature defined as, T∗ = TS,0 +
˙ S CS q m dTC dı = − + Zr Q0 [TS (L) − TS,0 ]. 2 R0 2 R0 L dt dt
(10)
q L ˙ S CS (1 − e− ) m
(11)
and the dimensionless group, ω, defined as, ω=
2 R0 a0 LZr CZr ˙ S (1 − e− )t∗ mC
(12)
Similar to the previous case of uniform temperature, the parameter ω in Eq. (12) specifies the ratio between the average rates of heat accumulation in the metal and in the steam. The critical value, ωcr , defines the ignition conditions, averaged over the steam zone, as a function of the system parameters such as the metal mass, steam flow rate, characteristic temperature, heat transfer coefficient, and the thermal properties of the steam and metal. The ignition criteria presented in Figs. 3 and 4 are viable for the present case when the characteristic temperature in the system is defined by Eq. (11). During a LOCA event; the core convective coefficient may deteriorate leading to low values of the parameter . If << 1 the characteristic temperature, T* (Eq. (11)) can then be simplified as, T∗ − TS,0 =
dı dTC mCZr = q L + Zr Q0 S − hST dt dt
T =
Fig. 7. Characteristic temperature, T* , vs. average decay heat flux, q˙ , and convective coefficient, h.
q L q L q˙ ≈ = ˙ S CS h m ˙ S CS (1 − e− ) m
(13)
where q˙ is the decay heat flux averaged over the exposed zirconium zone. Thus, T* is directly proportional to the average decay heat flux and inversely proportional to the convective heat transfer coefficient, h. Note that if decay heat is neglected, the characteristic temperature becomes equal to the inlet steam temperature, TS,0 , and the ignition criterion, ω, in Eq. (12), reduces to the uniform temperature ignition criterion defined in Eq. (5). The characteristic temperature, T* , is conveniently plotted in Fig. 7 over a range of q˙ and h, relevant for LOCA analysis, Winterton (1981). Figs. 3–7 may be readily integrated, in a thermal-hydraulic numerical fuel channel analysis for predicting whether uncontrolled thermo-chemical reaction prevails along the fuel rod during a hypothetical LOCA condition. To this end, q˙ and h are first calculated numerically by solving the nuclear and thermal-hydraulic balance equations in the fuel channel. These are then used to determine, from Fig. 7 (or Eq. (13)), the local characteristic temperature. Figs. 3 and 4 are then used to find the critical parameter ωcr . Uncontrolled thermo-chemical reaction prevails if the value of ω calculated from Eq. (12) exceeds critical value, i.e., if ω ≥ ωcr . We note that the above model is based on uniform decay heat flux and heat transfer coefficient over the entire exposed cladding specimen, which could sometimes extend over a significant portion of the channel. A refined one-dimensional (axial) model based on
E. Elias et al. / Nuclear Engineering and Design 286 (2015) 205–210
dividing the steam-cooling zone into small axial nodes is presented in the next section.
Eqs. (3), (17), and (19) are non-dimensionalized using the dimensionless parameters defined in Eq. (5) and the following definition of the characteristic temperature, T* (z),
2.3. Refined one-dimensional model T∗ (z) = TS (z) + In a LOCA event the thermal-hydraulic conditions may vary along the steam cooled exposed cladding zone. Hence, the ignition criterion derived in Section 2.2, which was based on average thermal conditions along the dry-steam zone, is modified here in order to account for the axially varying steam and surface temperatures. This model can be more readily utilized in a numerical code for fuel channel evaluation. As a practical example we consider here a cylindrical shell of zirconium with initial inner radius, R0 , and thickness, a0 , subject to steam flowing axially on its outer surface and distributed heat flux along its inner surface. Assuming, for instance, a sine shaped linear decay power along the inner surface and a thin walled cylindrical shell, a lumped parameters model may be used to calculate the local clad temperature, TC as,
˙ S CS m
dTS = 2 R0 h · TCS dz
(15)
˙ S is its mass flow-rate. where CS is the steam specific heat and m In a discrete computational cell with a finite length, z and constant cladding temperature, TC , the average temperature difference between steam and cladding, TCS , could be expressed as, T CS =
TS (z + z) − TS (z) TC −TS (z) ln T −T C S (z+z)
=
TS TC −TS (z) ln T −T (z+z) C S
.
(16)
Substituting Eq. (16) in (15) with dTS /dz ≈ TS / z yields, TS = TS (z + z) − TS (z) = [TC − TS (z)](1 − e− )
2 hR0 z ˙ S CS m
2 R0
+ Zr Q0
(18)
˙ S TS (z + z) − TS (z) dı dTC mC − = a0 Zr CZr . 2 R0 dt z dt (19)
It is noticed that for a small z Eq. (17) converges into the following expression, TS = TS (z + z) − TS (z) = [TC − TS (z)] = [TC − TS (z)]
L
z
(21)
˙ S (1 − e− ) mC
q0 sin
z L
(22)
2 R0 h
The characteristic oxide thickness and time are defined as previously expressed in Eq. (5). With the characteristic temperature defined in Eqs. (21) and (22), the resulting dimensionless equations are identical to Eqs. (4a) and (4b) with the dimensionless parameter defining the local thermal ignition conditions becoming a function of z as, 2 R0 a0 zCZr Zr ˙ S (1 − e− )t∗ mC
(23)
Eq. (23) accounts for non-uniform decay heat and non-uniform steam temperature along the heated surface. Eq. (23) could be further simplified by assuming a thin walled shell and << 1 yielding, ω≈
2 R0 za0 CZr Zr Zr = ˙ CS t∗ a0 CZr h m
Q 2 0 ˇT∗ (z)
kr e−1/ˇ
(24)
Note that in this case the ignition conditions depend on the local characteristic steam temperature and not on the clad temperature. Eq. (24) is similar to the criterion developed in Section 2.2 (c.f., Eq. (12)), but here the characteristic temperature varies along the channel. Therefore, once T* (z) is predicted along the channel, Figs. 3–6 may again be used for estimating the local criterion, ωcr , for a given heat transfer coefficient. Since the characteristic temperature depends on the local decay heat flux and local heat transfer coefficient as indicated in Eq. (22), the ignition criterion also varies along the channel. The criterion (24) can be readily utilized in a numerical channel code for estimating whether runaway conditions are reached at any point along the channel. 3. Time-to-ignition
Substituting Eq. (15) in Eq. (14) yields, q˙ 0 sin z L
z
(17)
where is defined as, =
T∗ (z) = TS (z) +
(14)
Here both TC and the average temperature difference between the cladding and steam temperatures, TCS , depend on the axial coordinate z along the cladding. The local steam temperature along the channel, TS , is determined using a one dimensional (axial) sensible heat balance over the steam,
q˙ 0 sin
Note that the characteristic temperature becomes a function of elevation along the dry steam region. Using Eqs. (15) and (17) and assuming a thin walled shell, << 1 yields,
ω=
z q˙ dı dTC − h TCS = a0 Zr CZr . sin + Q0 Zr L 2 R0 dt dt
209
2 hR0 z ˙ S CS m (20)
The governing Eqs. (3), (17), and (19) turn into a complete system of differential equations, which may be solved simultaneously and TS (z) at each computational cell, z, along the for ı, TC , dry cladding. Note that the lumped parameters’ model of Section 2.2 is obtained by substituting, z = L in Eq. (18).
The above oxidation models define runaway conditions in zirconium without addressing the issue of time for reaching the critical ignition conditions (termed here time-to-ignition). Examination of Eq. (4b) reveals that the oxidation process has no strict mathematical steady-state solution since the time derivative of the oxide layer, , is always positive. Eqs. (4a) and (4b) also show that the time evolution of the heating and oxidation processes depends on the characteristic and initial conditions of the system, i.e., ˇ and i , as well as on the initial thickness of the oxidation layer, which in this study is assumed constant (ıi = 10 nm). This section discusses the effect of ˇ and i on the time for reaching runaway conditions, given that ω = ωcr . For simplicity we concentrate on a representative case with i = 0, i.e., having equal characteristic and initial metal temperature. As shown in Figs. 1 and 2, when ω = ωcr the temperature and oxidation layer undergo rapid escalation at a certain dimensionless critical time, cr . The value of cr depends weakly on ˇ and i . In the studied range of ˇ and i , value of cr varies in the range of 5–6.5. For a given cr the dimensional time-to-ignition is readily calculated from Eq. (5) as the product of cr and the characteristic time, t* . For i = 0 Fig. 8 depicts the dimensional time-to-ignition as a function of the characteristic temperature in the range of 400–1400 K. The
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temperature. A general ignition criterion, which accounts for the local convective heat transfer coefficient, the local steam temperature and the residual decay power, is defined. The present solution may be readily integrated as part of a channel code for predicting possible outcomes of different accident mitigation procedures in light water nuclear reactors under LOCA conditions. Acknowledgments
Fig. 8. Time for reaching critical ignition conditions vs. initial characteristic temperature.
strong effect of the characteristic temperature on the ignition time is noticeable. For example, Fig. 8 shows that at ω = ωcr and a characteristic temperature of 600 K runaway conditions are reached in about 108 s, compared to about 100 s at 1200 K. Therefore, one may practically neglect the possibility for a runaway thermo-chemical reaction for characteristic temperature below about 700 K as long as ω ≤ ωcr . Note, however, that ignition may happen almost instantaneously if the parameter ω exceeds its critical value, due to, e.g., a deteriorating heat transfer coefficient at the surface. Fig. 8 shows sample data calculated for i = 0. Longer times-toignition were predicted for i < 0, i.e., initial metal temperatures lower than the characteristic temperature of the system. Fig. 8 may, thus, be considered as a lowest conservative estimate for the timeto-ignition when ω = ωcr . 4. Conclusions The idea of runaway zirconium–steam reactions during severe accidents in light water reactors (LWR) is demonstrated. A theoretical model based on the Semenov (1928) approach is derived for simultaneously solving the heat and mass balance equations along an exposed section of a fuel cladding undergoing heating and thermo-chemical oxidation reaction in a hot steam environment. Critical ignition conditions are estimated for a wide range of thermal parameters. It is shown that the critical conditions leading to runaway thermo-chemical reaction at the cladding surface depend on the local characteristic temperature and not solely on the clad
This work is supported by the Israel Atomic Energy Commission (IAEC) and the Ministry of Energy and Water Resources. Research has been motivated by several invigorating discussions with Professor P.L. Chambré of the Department of Nuclear Engineering, UC Berkeley. Professor Chambré passed away in April, 2013. This work is dedicated to his memory. References Baker Jr., Louis, Just, Louis C., 1962. Studies of Metal–Water Reactions at High Temperatures. Argonne National Lab, ANL 6548. Boddington, T., Gray, P.F.R.S., Scott, S.K., 1982. Modeling thermal runaway and criticality in systems with diminishing reaction rates: the uniform temperature (Semenov) approximation. Proc. R. Soc. Lond. A 380, 29–47. Chambré, P.L., 1952. On the solution of the Poisson–Boltzmann equation with application to the theory of thermal explosions. J. Chem. Phys. 20 (11), 1795–1797. Cathcart, J.V., Pawel, R.E., McKee, R.A., Druschel, R.E., Yurek, G.J., Campbell, J.J., Jury, S.H., 1977. Zirconium Metal–Water Oxidation Kinetics IV. Reaction Rate Studies. ORNL/NUREG-17. Frank-Kamenetskii, D.A., 1969. Diffusion and Heat Transfer in Chemical Kinetics. Nauka, 1967, 2nd edition. Plenum Press, New York. Hasan, D., Nekhamkin, Y., Rosenband, V., Elias, E., Wacholder, E., Gany, A., 1999. Nuclear fuel cores’ thermal hazards’ criteria , Proc. INS-20, The Dead Sea, Israel. Khaikin, B.I., Bloshenko, V.N., Merzhanov, A.G., 1970. On the Ignition of metal particles. UDC 536.46-662.215.1. Translated from FizikaGoreniya i Vzryva, No. 4, pp.474–488, October–December, 1970, also in Combustion, Explosion, and Shock Waves, 6., pp. 4. MAAP BWR, 1992. Application Guidelines. EPRI, Palo Alto, CA, TR-100742. MELCOR, 2000. Computer Code Manuals. NUREG/CR-6119. Nekhamkin, Y., Rosenband, V., Hasan, D., Elias, E., Wacholder, E., Gany, A., 1998. Oxidation and heat transfer considerations in degraded core accident , TOPSAFE98, Valencia. Pickard, James M., 2002. Critical ignition temperature. Thermochim. Acta 392–393, 37–40. Semenov, N.N., 1928. ZhRFKhO Ch. Fiz. 60, 241. Whitmarsh, C.L., 1982. Review of zircaloy-2 and zircaloy-4 properties relevant to N.S. SAVANNAH reactor design, ORNL-3281. Winterton, R.H.S., 1981. Thermal Design of Nuclear Reactors. Pergamon Press, Oxford. Yarin, L.P., Hetsroni, G., 2004. Combustion of Two-Phase Reactive Media. Springer–Verlag, Berlin.
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Zirconium ignition in exposed fuel channel E. Elias ∗ , D. Hasan, Y. Nekhamkin Technion – Israel Institute of Technology, 32000 Haifa, Israel
h i g h l i g h t s • We demonstrate the idea of runaway zirconium–steam reactions in severe accidents in today’s LWRs. • We predict the thermal-hydraulics conditions relevant to cladding oxidation in an exposed fuel channel of a partially uncovered core. • The Semenov theory of metal combustion is extended to define a criterion for runaway oxidation reaction in fuel cladding.
a r t i c l e
i n f o
Article history: Received 23 October 2014 Received in revised form 12 January 2015 Accepted 1 March 2015
a b s t r a c t A theoretical model based on simultaneous solution of the heat and mass transfer equations is developed for predicting the rate of thermo-chemical reaction between zirconium cladding and a hot steam environment. Ignition conditions relevant to cladding oxidation in an exposed fuel channel of a partially uncovered core are predicted based on the theory of metal combustion. A range of decay power, convective heat transfer coefficients, and initial temperatures leading to uncontrolled runaway cladding oxidation is identified. The model could be readily integrated as part of a fuel channel analysis code for predicting possible outcomes of different accident mitigation procedures in light water nuclear reactors under LOCA conditions. © 2015 Elsevier B.V. All rights reserved.
1. Introduction The recent events at the Fukushima plants have underscored the importance to the overall reactor safety of secondary heat generation sources, such as thermo-chemical reactions at the cladding surface. Realistic analysis of metal oxidation in a partially uncovered reactor core is a complicated numerical computation task, requiring the simultaneous solution of a set of energy balance equations in the fuel elements and their metal claddings along with the mass, momentum and energy equations in the corresponding coolant channels (cf., severe accident codes; MAAP, 1992; MELCOR, 2000). Conservative analysis of cladding integrity is typically accounted for by setting a predetermined critical minimum ignition temperature, which is based on out-of-pile oxidation experiments using small specimens of cladding material (Baker and Just, 1962; Cathcart et al., 1977). Accurate modeling of metal oxidation during severe accident must account for the time dependent balance between the thermochemical reaction heat release at the cladding surface, the decay power in the fuel and the rate of cooling by the surrounding steam.
∗ Corresponding author. Tel.: +972 4 8293263; fax: +972 4 8295711. E-mail address: [email protected] (E. Elias). http://dx.doi.org/10.1016/j.nucengdes.2015.03.002 0029-5493/© 2015 Elsevier B.V. All rights reserved.
A step in that direction has been previously described by using a lumped parameters model (Nekhamkin et al., 1998; Hasan et al., 1999; Pickard, 2002) that relies on the Semenov type methodology (Semenov, 1928; Yarin and Hetsroni, 2004; Khaikin et al., 1970; Boddington et al., 1982) to identify critical conditions for runaway clad–steam oxidation reactions. A more realistic cladding oxidation model based on solid combustion theories is described in this paper. The theory of metal oxidation and combustion has been extensively developed in the last century as part of the technological effort to develop efficient solid propellants and later in conjunction with nuclear reactor safety. Conditions for a runaway thermochemical reaction were analytically derived by Chambré (1952) and Frank-Kamenetskii (1969) by solving the one-dimensional steadystate balance equation in a system of finite size. Based on the theory of thermal explosion, the combination of size, conductivity and ambient temperature at which such solution becomes mathematically impossible was defined as the condition of ignition or inflammation. An extensive review of the basic theoretical and engineering literature on metal combustion by Yarin and Hetsroni (2004) demonstrates that the rate of metal oxidation in an oxidizing gas environment depends on the thermal-hydraulic conditions in both the solid and the surrounding gas. The idea of runaway metal–steam reactions was introduced by Khaikin et al. (1970) who
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used the Semenov (1928) analysis to derive a conjugate set of heat and mass balance equations for the time evolution of temperature and oxide layer thickness on a small metal particle placed in an oxidizing gas environment of uniform temperature. In the following the basic combustion theories for metal ignition in steam are extended to account for the special characteristic conditions at a partially exposed nuclear fuel rod. In particular the effect of temperature non-uniformity along the oxidizing specimen as well as the effect of stored and decay heats at the inner surface of a cylindrical cladding shell are considered. Heat and mass balance equations are developed to define runaway timing and the thermal conditions leading to uncontrolled oxidation during severe accident. The paper outlines the possibility for integrating the proposed model in a full-scale fuel channel transient analysis.
temperature, T* , depends on the physical problem solved. In this first case the characteristic temperature is conveniently taken as a constant that equals the ambient steam temperature, i.e., T* = T0 . In Eq. (1) it is assumed that the metal mass remains constant. This holds for the initial stage of the process when the thickness of the oxide film is insignificant. For a planar zirconium slab of unit width, height L and initial thickness a0 , Eq. (1) can be further manipulated by expressing the slab volume and surface area in terms of its initial dimensions,
2. Ignition of metal particles
dı kr B = n exp − T dt ı
Zr CZr a0
dTC dı = Zr Q0 − h(TC − T∗ ). dt dt
(2)
The thickness of the oxide film, ı(t), has been theoretically investigated (Baker and Just, 1962; Khaikin et al., 1970). The simplest diffusion model takes the form,
(3)
Thermo-chemical reaction between steam and zirconium in the fuel cladding could become an important heat source under accident conditions. During normal operation of power reactors, the cladding temperature is about 10◦ above the coolant temperature, i.e., about 330–350 ◦ C, with low oxidation rate and minimal heat release. However, if cladding is partially exposed, because of core coolant loss, the clad temperature could increase by several hundreds degrees. At cladding temperatures above about 900 ◦ C, clad oxidation rate begins to significantly increase, leading to release of heat and formation of hydrogen, according to the following chemical reaction for Zr (Baker and Just, 1962; Whitmarsh, 1982):
where B is the activation energy (B = 22,900 K) and kr = 3.968 × 10−5 m2 /s is a pre-exponent reaction rate, which accounts for the oxidizer concentration at the surface (Baker and Just, 1962). A parabolic law (n = 1) will be used here as it is broadly used for estimating zirconium oxidation rate in nuclear reactors (MAAP, 1992; MELCOR, 2000 codes). Thus, in order to derive an ignition criterion for metal oxidation it is necessary to solve Eqs. (2) and (3) jointly subject to the following initial conditions, ı(t = 0) = ıi and TC (t = 0) = Ti . Eqs. (2) and (3) are first reduced into the following dimensionless form,
Zr + 2H2 O → ZrO2 + 2H2 + Heat
d 1 = exp d
The heat of reaction, Q0 , is 140 kcal/mole (6500 kJ/kg of Zr reacted or 1540 kcal/kg) and the volume of hydrogen generated is 0.5 m3 (at STP conditions) per 1 kg of reacting Zr. Generally, the rate of oxidation depends on the metal temperature, T, and on the rate of oxygen diffusion through the oxidized layer, which is formed on the outer surface of the metal. Thus, predicting the necessary conditions for ignition calls for simultaneous solution of the heat and mass balance equations at the cladding surface. At certain conditions, heat generation by the oxidation reaction may exceed the cooling rate, which leads to an accelerated or runaway oxidation. In the following, a threshold criterion for runaway oxidation is derived for three representative cases: uniform temperature approximation; decay power effect and a general one-dimensional system. 2.1. Uniform temperature approximation To gain a better physical insight into the diffusion and thermal processes taking place during metal oxidation, we consider first a simplified case of a thin walled planar zirconium specimen with initial temperature TC,i instantaneously introduced into an infinite steam environment with a constant and uniform temperature, T0 . As proposed by Khaikin et al. (1970) ignition conditions in this case are determined by considering a balance between the rate of heat release by the oxidation reaction and the rate of heat transfer from the metal to the ambient steam. For a thin walled zirconium metal (low Biot number), a lumped parameters heat balance equation is given by, mZr CZr
dı dTC = Zr Q0 S − hS(TC − T∗ ) dt dt
(1)
where mZr and S are the mass and surface area of the zirconium metal; Zr and CZr are its density and specific heat, h the heat transfer coefficient, ı is the thickness of the oxide film at the surface, Q0 the heat release per unit mass of oxidized metal and TC is the zirconium temperature. The exact definition of the characteristic
d 1 = exp d
1 + ˇ 1 + ˇ
−
ω
(4a)
(4b)
with the following initial conditions, I.C. : at = 0 = i , = i . The following dimensionless variables and parameters are adopted in Eqs. (4a) and (4b), = ˇ=
ı B ; = T∗ ı∗
T
C
T∗
−1
=
1 ˇ
T
T∗ ı2 e1/ˇ 1 ; ω= ; t∗ = ∗ B kr h
C
T∗
−1
; =
t CZr a0 ˇ T∗ ; ı∗ = ; t∗ Q0
Q 2 k e−1/ˇ Zr r 0 ˇT∗
a0 CZr
=
1 a0 CZr Zr . t∗ h (5)
The right hand side of (4a) expresses a balance between the oxidation exponential heat generation in the specimen and the rate of convective cooling by the steam (/ω). The parameter, ω, specifies the ratio between the average rate of heat accumulation in the metal (over a characteristic time, t* ), and that transferred to the ambient steam. The initial value of the dimensionless oxide layer in Eq. (4b) is typically small (i ≈ 0). In the following, the effect of the initial temperature difference, i , will be studied parametrically. In order to demonstrate the utility of the model for predicting zirconium ignition conditions, Eqs. (4a) and (4b) were integrated numerically for a wide range of characteristic temperatures using an initial thickness of a0 = 0.57 mm, which corresponds to typical PWR fuel cladding. As an example, in Figs. 1 and 2 the characteristic temperature, T* , and the initial cladding temperature, TC , were arbitrarily taken as 800 K, yielding i = 0 and ˇ = 0.035. Also, following Khaikin et al. (1970) we use ıi = 10 nm resulting in i = 1.46 × 10−3 . Fig. 1 is a typical example showing the evolution of the dimensionless temperature, vs. dimensionless time, for two values of the parameter ω. It is noticed that there exists a critical minimum value, ω = ωcr , above which the temperature escalates dramatically.
E. Elias et al. / Nuclear Engineering and Design 286 (2015) 205–210
Fig. 1. Dimensionless temperature vs. dimensionless time for I = 0.
Fig. 2. Dimensionless oxide layer thickness vs. dimensionless time for two values of ω and I = 0.
Referring to Eq. (4a), it can be readily shown that at ω > ωcr the oxidation heat generation term exceeds the rate of convection cooling leading to rapid metal temperature increase (runaway). In Fig. 1, runaway oxidation, is experienced at ω ≥ 1.6. The rate of oxide layer buildup also varies significantly during the heating process, as shown in Fig. 2. The critical ignition parameter, ωcr , depends on other parameters that affect the solution of Eqs. (4a) and (4b), namely the characteristic temperature, T* (or ˇ) and the initial temperature difference, i . The dependence of ωcr on ˇ, T* , and i is plotted in Figs. 3 and 4. Negative values of i indicate initial cladding temperature lower than the characteristic ambient steam temperature. Each ωcr line in Figs. 3 and 4 basically divides the parametric space into regions of controlled oxidation below the line, and an uncontrolled runaway oxidation above it. A wide range of characteristic temperatures from 400 to 1400 K was selected in order
Fig. 3. Critical parameter, ωcr vs. characteristic temperature, ˇ, T* , and initial temperature difference, i .
207
Fig. 4. Critical parameter, ωcr of zirconium vs. initial temperature difference, i at various characteristic temperatures, T* .
to cover anticipated core conditions during a hypothetical severe accident. Note that the initial cladding temperature, TC,i , can be readily derived from Eq. (5) for each pair of i and T* (or ˇ) as TC,i = T∗ ˇ i + 1 . In Eq. (5) the parameter, ω, is defined in terms of the characteristic temperature, T* (or ˇ), and the convective heat transfer coefficient, h. The dimensional critical conditions can thus be readily expressed by a physical parametric plot as demonstrated, for example, in Fig. 5 that shows the parameter ω vs. T* for heat transfer coefficients in the range of 0.1–300 W/m2 K. Here again, the critical parameter, ωcr line (for i = 0), shown in Fig. 5 as a dashed line, divides the space into runaway conditions above the line and controlled oxidation conditions below it. As expected the critical characteristic temperature leading to a runaway ignition decreases with decreasing convective coefficient. For instance, Fig. 5 shows that if free convection conditions are assumed in the fuel channel (h ≈ 10 W/m2 K), then ω reaches a critical value (h line intersects the ωcr line) at a characteristic temperature of T∗ ≈ 1060 K, whereas when h ≈ 100 W/m2 K the critical value is reached at a higher temperature of about 1250 K. For characteristic temperatures below about 800 K and i = 0, critical runaway conditions may only be reached if the convective coefficient is unrealistically small (close to adiabatic system). This is clearly demonstrated in Fig. 6 showing the locus of all h/ωcr intersection points, yielding a parametric map of characteristic ignition temperatures as a function of the local heat transfer coefficient. Runaway conditions prevail at the lower right side of the curve (low heat transfer coefficient and high characteristic steam temperature). The effect of the initial temperature difference, i , is relatively small. The preceding model and analysis, while providing physical insight into the oxidation problem, is oversimplified. It relies on the
Fig. 5. The parameter ω as a function of characteristic temperature and heat transfer coefficient.
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θ θ
Fig. 6. Heat transfer coefficient vs. characteristic ignition temperature and initial clad–steam temperature difference.
assumption that oxidation is the sole source of heat in the system and thus, the characteristic coolant temperature could be assumed equal to the constant and uniform ambient steam temperature. A more realistic physical representation of the cladding behavior in a partially uncovered reactor core is considered in the next two sections that account for fuel decay power and steam heating along the fuel channel. First a simplified system with a specified steam mass flow rate and a spatially uniform decay heat is analyzed in Section 2.2. This is followed by a more realistic case of a one-dimensional (axial) analysis of a cladding section subjected to a sine distributed decay heat source and cooled by flowing steam. It will be shown that the effect of both decay power and variable steam temperature could be well accounted for by proper definition of ω and the characteristic temperature, T* . 2.2. Specified steam flow, uniform decay heat Here the threshold condition for a runaway oxidation reaction is developed in a cylindrical shell of zirconium with initial inner radius, R0 , and thickness, a0 , subject to steam flowing axially on its outer surface. A uniform and constant decay power is considered along the inner surface of the shell. For a low Biot number, a lumped parameters’ heat balance over the cylindrical metal shell yields, (6)
where q is the linear decay power, L is the shell length and T is the logarithmic mean temperature difference between the solid surface (TC ) and the steam defined as, TS (L) − TS,0
(7)
T −T
ln T C−T S,0 (L) C
S
Here TS (L) and TS,0 are the steam exit and inlet temperatures, respectively. A heat balance over the steam yields,
˙ S CS TS (L) − TS,0 hST = m
(8)
˙ S and CS are the steam mass flow rate and specific heat, where m respectively. Substituting Eq. (7) into Eq. (8) and solving for the steam temperature yields, TS (L) = TS,0 + (TC − TS,0 )(1 − e− )
with =
2 R0 Lh ˙ S CS m
(9)
For a thin walled cylindrical zirconium shell a0 /2R0 << 1 Eq. (6) can be further manipulated by expressing the shell mass and surface area in terms of its initial dimensions, Zr CZr a0
The governing Eqs. (3) and (10) can now be reduced to a dimensionless form identical to Eqs. (4a) and (4b), with the characteristic temperature defined as, T∗ = TS,0 +
˙ S CS q m dTC dı = − + Zr Q0 [TS (L) − TS,0 ]. 2 R0 2 R0 L dt dt
(10)
q L ˙ S CS (1 − e− ) m
(11)
and the dimensionless group, ω, defined as, ω=
2 R0 a0 LZr CZr ˙ S (1 − e− )t∗ mC
(12)
Similar to the previous case of uniform temperature, the parameter ω in Eq. (12) specifies the ratio between the average rates of heat accumulation in the metal and in the steam. The critical value, ωcr , defines the ignition conditions, averaged over the steam zone, as a function of the system parameters such as the metal mass, steam flow rate, characteristic temperature, heat transfer coefficient, and the thermal properties of the steam and metal. The ignition criteria presented in Figs. 3 and 4 are viable for the present case when the characteristic temperature in the system is defined by Eq. (11). During a LOCA event; the core convective coefficient may deteriorate leading to low values of the parameter . If << 1 the characteristic temperature, T* (Eq. (11)) can then be simplified as, T∗ − TS,0 =
dı dTC mCZr = q L + Zr Q0 S − hST dt dt
T =
Fig. 7. Characteristic temperature, T* , vs. average decay heat flux, q˙ , and convective coefficient, h.
q L q L q˙ ≈ = ˙ S CS h m ˙ S CS (1 − e− ) m
(13)
where q˙ is the decay heat flux averaged over the exposed zirconium zone. Thus, T* is directly proportional to the average decay heat flux and inversely proportional to the convective heat transfer coefficient, h. Note that if decay heat is neglected, the characteristic temperature becomes equal to the inlet steam temperature, TS,0 , and the ignition criterion, ω, in Eq. (12), reduces to the uniform temperature ignition criterion defined in Eq. (5). The characteristic temperature, T* , is conveniently plotted in Fig. 7 over a range of q˙ and h, relevant for LOCA analysis, Winterton (1981). Figs. 3–7 may be readily integrated, in a thermal-hydraulic numerical fuel channel analysis for predicting whether uncontrolled thermo-chemical reaction prevails along the fuel rod during a hypothetical LOCA condition. To this end, q˙ and h are first calculated numerically by solving the nuclear and thermal-hydraulic balance equations in the fuel channel. These are then used to determine, from Fig. 7 (or Eq. (13)), the local characteristic temperature. Figs. 3 and 4 are then used to find the critical parameter ωcr . Uncontrolled thermo-chemical reaction prevails if the value of ω calculated from Eq. (12) exceeds critical value, i.e., if ω ≥ ωcr . We note that the above model is based on uniform decay heat flux and heat transfer coefficient over the entire exposed cladding specimen, which could sometimes extend over a significant portion of the channel. A refined one-dimensional (axial) model based on
E. Elias et al. / Nuclear Engineering and Design 286 (2015) 205–210
dividing the steam-cooling zone into small axial nodes is presented in the next section.
Eqs. (3), (17), and (19) are non-dimensionalized using the dimensionless parameters defined in Eq. (5) and the following definition of the characteristic temperature, T* (z),
2.3. Refined one-dimensional model T∗ (z) = TS (z) + In a LOCA event the thermal-hydraulic conditions may vary along the steam cooled exposed cladding zone. Hence, the ignition criterion derived in Section 2.2, which was based on average thermal conditions along the dry-steam zone, is modified here in order to account for the axially varying steam and surface temperatures. This model can be more readily utilized in a numerical code for fuel channel evaluation. As a practical example we consider here a cylindrical shell of zirconium with initial inner radius, R0 , and thickness, a0 , subject to steam flowing axially on its outer surface and distributed heat flux along its inner surface. Assuming, for instance, a sine shaped linear decay power along the inner surface and a thin walled cylindrical shell, a lumped parameters model may be used to calculate the local clad temperature, TC as,
˙ S CS m
dTS = 2 R0 h · TCS dz
(15)
˙ S is its mass flow-rate. where CS is the steam specific heat and m In a discrete computational cell with a finite length, z and constant cladding temperature, TC , the average temperature difference between steam and cladding, TCS , could be expressed as, T CS =
TS (z + z) − TS (z) TC −TS (z) ln T −T C S (z+z)
=
TS TC −TS (z) ln T −T (z+z) C S
.
(16)
Substituting Eq. (16) in (15) with dTS /dz ≈ TS / z yields, TS = TS (z + z) − TS (z) = [TC − TS (z)](1 − e− )
2 hR0 z ˙ S CS m
2 R0
+ Zr Q0
(18)
˙ S TS (z + z) − TS (z) dı dTC mC − = a0 Zr CZr . 2 R0 dt z dt (19)
It is noticed that for a small z Eq. (17) converges into the following expression, TS = TS (z + z) − TS (z) = [TC − TS (z)] = [TC − TS (z)]
L
z
(21)
˙ S (1 − e− ) mC
q0 sin
z L
(22)
2 R0 h
The characteristic oxide thickness and time are defined as previously expressed in Eq. (5). With the characteristic temperature defined in Eqs. (21) and (22), the resulting dimensionless equations are identical to Eqs. (4a) and (4b) with the dimensionless parameter defining the local thermal ignition conditions becoming a function of z as, 2 R0 a0 zCZr Zr ˙ S (1 − e− )t∗ mC
(23)
Eq. (23) accounts for non-uniform decay heat and non-uniform steam temperature along the heated surface. Eq. (23) could be further simplified by assuming a thin walled shell and << 1 yielding, ω≈
2 R0 za0 CZr Zr Zr = ˙ CS t∗ a0 CZr h m
Q 2 0 ˇT∗ (z)
kr e−1/ˇ
(24)
Note that in this case the ignition conditions depend on the local characteristic steam temperature and not on the clad temperature. Eq. (24) is similar to the criterion developed in Section 2.2 (c.f., Eq. (12)), but here the characteristic temperature varies along the channel. Therefore, once T* (z) is predicted along the channel, Figs. 3–6 may again be used for estimating the local criterion, ωcr , for a given heat transfer coefficient. Since the characteristic temperature depends on the local decay heat flux and local heat transfer coefficient as indicated in Eq. (22), the ignition criterion also varies along the channel. The criterion (24) can be readily utilized in a numerical channel code for estimating whether runaway conditions are reached at any point along the channel. 3. Time-to-ignition
Substituting Eq. (15) in Eq. (14) yields, q˙ 0 sin z L
z
(17)
where is defined as, =
T∗ (z) = TS (z) +
(14)
Here both TC and the average temperature difference between the cladding and steam temperatures, TCS , depend on the axial coordinate z along the cladding. The local steam temperature along the channel, TS , is determined using a one dimensional (axial) sensible heat balance over the steam,
q˙ 0 sin
Note that the characteristic temperature becomes a function of elevation along the dry steam region. Using Eqs. (15) and (17) and assuming a thin walled shell, << 1 yields,
ω=
z q˙ dı dTC − h TCS = a0 Zr CZr . sin + Q0 Zr L 2 R0 dt dt
209
2 hR0 z ˙ S CS m (20)
The governing Eqs. (3), (17), and (19) turn into a complete system of differential equations, which may be solved simultaneously and TS (z) at each computational cell, z, along the for ı, TC , dry cladding. Note that the lumped parameters’ model of Section 2.2 is obtained by substituting, z = L in Eq. (18).
The above oxidation models define runaway conditions in zirconium without addressing the issue of time for reaching the critical ignition conditions (termed here time-to-ignition). Examination of Eq. (4b) reveals that the oxidation process has no strict mathematical steady-state solution since the time derivative of the oxide layer, , is always positive. Eqs. (4a) and (4b) also show that the time evolution of the heating and oxidation processes depends on the characteristic and initial conditions of the system, i.e., ˇ and i , as well as on the initial thickness of the oxidation layer, which in this study is assumed constant (ıi = 10 nm). This section discusses the effect of ˇ and i on the time for reaching runaway conditions, given that ω = ωcr . For simplicity we concentrate on a representative case with i = 0, i.e., having equal characteristic and initial metal temperature. As shown in Figs. 1 and 2, when ω = ωcr the temperature and oxidation layer undergo rapid escalation at a certain dimensionless critical time, cr . The value of cr depends weakly on ˇ and i . In the studied range of ˇ and i , value of cr varies in the range of 5–6.5. For a given cr the dimensional time-to-ignition is readily calculated from Eq. (5) as the product of cr and the characteristic time, t* . For i = 0 Fig. 8 depicts the dimensional time-to-ignition as a function of the characteristic temperature in the range of 400–1400 K. The
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temperature. A general ignition criterion, which accounts for the local convective heat transfer coefficient, the local steam temperature and the residual decay power, is defined. The present solution may be readily integrated as part of a channel code for predicting possible outcomes of different accident mitigation procedures in light water nuclear reactors under LOCA conditions. Acknowledgments
Fig. 8. Time for reaching critical ignition conditions vs. initial characteristic temperature.
strong effect of the characteristic temperature on the ignition time is noticeable. For example, Fig. 8 shows that at ω = ωcr and a characteristic temperature of 600 K runaway conditions are reached in about 108 s, compared to about 100 s at 1200 K. Therefore, one may practically neglect the possibility for a runaway thermo-chemical reaction for characteristic temperature below about 700 K as long as ω ≤ ωcr . Note, however, that ignition may happen almost instantaneously if the parameter ω exceeds its critical value, due to, e.g., a deteriorating heat transfer coefficient at the surface. Fig. 8 shows sample data calculated for i = 0. Longer times-toignition were predicted for i < 0, i.e., initial metal temperatures lower than the characteristic temperature of the system. Fig. 8 may, thus, be considered as a lowest conservative estimate for the timeto-ignition when ω = ωcr . 4. Conclusions The idea of runaway zirconium–steam reactions during severe accidents in light water reactors (LWR) is demonstrated. A theoretical model based on the Semenov (1928) approach is derived for simultaneously solving the heat and mass balance equations along an exposed section of a fuel cladding undergoing heating and thermo-chemical oxidation reaction in a hot steam environment. Critical ignition conditions are estimated for a wide range of thermal parameters. It is shown that the critical conditions leading to runaway thermo-chemical reaction at the cladding surface depend on the local characteristic temperature and not solely on the clad
This work is supported by the Israel Atomic Energy Commission (IAEC) and the Ministry of Energy and Water Resources. Research has been motivated by several invigorating discussions with Professor P.L. Chambré of the Department of Nuclear Engineering, UC Berkeley. Professor Chambré passed away in April, 2013. This work is dedicated to his memory. References Baker Jr., Louis, Just, Louis C., 1962. Studies of Metal–Water Reactions at High Temperatures. Argonne National Lab, ANL 6548. Boddington, T., Gray, P.F.R.S., Scott, S.K., 1982. Modeling thermal runaway and criticality in systems with diminishing reaction rates: the uniform temperature (Semenov) approximation. Proc. R. Soc. Lond. A 380, 29–47. Chambré, P.L., 1952. On the solution of the Poisson–Boltzmann equation with application to the theory of thermal explosions. J. Chem. Phys. 20 (11), 1795–1797. Cathcart, J.V., Pawel, R.E., McKee, R.A., Druschel, R.E., Yurek, G.J., Campbell, J.J., Jury, S.H., 1977. Zirconium Metal–Water Oxidation Kinetics IV. Reaction Rate Studies. ORNL/NUREG-17. Frank-Kamenetskii, D.A., 1969. Diffusion and Heat Transfer in Chemical Kinetics. Nauka, 1967, 2nd edition. Plenum Press, New York. Hasan, D., Nekhamkin, Y., Rosenband, V., Elias, E., Wacholder, E., Gany, A., 1999. Nuclear fuel cores’ thermal hazards’ criteria , Proc. INS-20, The Dead Sea, Israel. Khaikin, B.I., Bloshenko, V.N., Merzhanov, A.G., 1970. On the Ignition of metal particles. UDC 536.46-662.215.1. Translated from FizikaGoreniya i Vzryva, No. 4, pp.474–488, October–December, 1970, also in Combustion, Explosion, and Shock Waves, 6., pp. 4. MAAP BWR, 1992. Application Guidelines. EPRI, Palo Alto, CA, TR-100742. MELCOR, 2000. Computer Code Manuals. NUREG/CR-6119. Nekhamkin, Y., Rosenband, V., Hasan, D., Elias, E., Wacholder, E., Gany, A., 1998. Oxidation and heat transfer considerations in degraded core accident , TOPSAFE98, Valencia. Pickard, James M., 2002. Critical ignition temperature. Thermochim. Acta 392–393, 37–40. Semenov, N.N., 1928. ZhRFKhO Ch. Fiz. 60, 241. Whitmarsh, C.L., 1982. Review of zircaloy-2 and zircaloy-4 properties relevant to N.S. SAVANNAH reactor design, ORNL-3281. Winterton, R.H.S., 1981. Thermal Design of Nuclear Reactors. Pergamon Press, Oxford. Yarin, L.P., Hetsroni, G., 2004. Combustion of Two-Phase Reactive Media. Springer–Verlag, Berlin.