The 9th International Topical Meeting on Nuclear Thermal-Hydraulics, Operation and Safety (NUTHOS-9) Kaohsiung, Taiwan, September 9-13, 2012
N9P0146
Implementation of Flame Acceleration Criterion in CFD Code GASFLOW for Hydrogen Safety Analysis of Nuclear Power Plants J. Xiao, M.S. Kuznetsov Institute for Nuclear and Energy Technologies, Karlsruhe Institute of Technology, Kaiserstraße 12, 76131 Karlsruhe, Germany Email Address:
[email protected]
ABSTRACT The purpose of this paper is to present the algorithm for the criterion of hydrogen flame acceleration which was implemented in CFD code GASFLOW. The algorithm is used to calculate the sigma value, σ , for the mixtures of H2-O2-N2-H2O at various initial temperatures. The algorithm is efficient, robust and has been proved to be conservative for the hydrogen safety analysis. The σ -criterion implemented in CFD code GASFLOW can be used as the necessary criterion to fast evaluate the potential of hydrogen flame acceleration under severe accidents. It provides a quick way for the evaluation and optimization of the hydrogen mitigation measures in the nuclear containment, such as passive catalytic recombiners (PARs) and ignitors. KEYWORDS hydrogen safety, flame acceleration, severe accident, GASFLOW 1.
INTRODUCTION
Hydrogen combustion or explosion could threaten the integrity of the containment under core meltdown accidents, and lead to large scale release of radiation into the environment. The consequences of hydrogen explosion have been shown in the TMI (1979) and Fukushima (2011) accidents and many other safety studies. Uncontrolled hydrogen explosions generally start by the ignition of a flame from either an electrical spark, or the auto ignition of a mixture in contact with a hot surface. After ignition the flame starts initially as a slow quasi-laminar premixed H2-air-steam deflagration. It will preferentially propagate into the direction of high burning rates which is generally along the hydrogen/steam concentration gradient towards the richer and/or dryer mixtures and into regions with high turbulence intensity. This effect and also the self-induced turbulence from the expansion flow of the burned mixture behind the flame can induce a transition from slow laminar to fast turbulent deflagration, and may further undergo a transition to detonation. It will take a great computational effort for a CFD code to simulate the spontaneous flame acceleration from a slow laminar to a fast turbulent deflagration in nuclear power plant containment. To give a fast and conservative estimation of the flame acceleration potential, a flame acceleration criterion, also called σ -criterion, was developed in Karlsruher Institute of Technology (KIT) based on experiments with various scales performed in KIT and Kurchatov 1 / 11
The 9th International Topical Meeting on Nuclear Thermal-Hydraulics, Operation and Safety (NUTHOS-9) Kaohsiung, Taiwan, September 9-13, 2012
N9P0146
Institute Moscow (KI) [1-2]. The criterion of flame acceleration has been implemented in CFD code GASFLOW for conservative hydrogen safety analysis of light water reactors under severe accidents. GASFLOW is a three-dimensional finite-volume Arbitrary-Lagrangian-Eulerian (ALE) hydrodynamics code for solving the time-dependent compressible Navier-Stokes equations for multiple gas species [3-5]. The code can model condensation, heat transfer to walls and internal structures, multiple compartments, chemical kinetics, and fluid turbulence. GASFLOW code is currently developed in KIT, Germany, and has been used widely for the hydrogen safety analysis in the nuclear containments [6-10]. 2. IMPLEMENTATION OF FLAME ACCELERATION CRITERION IN GASFLOW The possibility of development of fast flames is defined mainly by the value of the mixture expansion ratio σ , as shown in the experiments by Kuznetsov [11]. It was also suggested that initial thermodynamic conditions should be expected to influence the critical value of σ critical mainly through changes of the Lewis, Le, and Zeldovich number, β . In GASFLOW code, the sigma index is defined as
σ index =
σ ( H ,O ,H O,T ) σ critical ( H ,O ,T ) 2
2
2
2
(1)
2
where σ ( H 2 ,O 2 ,H 2 O,T ) is the expansion ratio and σ critical ( H 2 ,O 2 ,T ) is the critical value of sigma. σ is dependent on the volume fraction of H2, O2, H2O and the temperature of the mixture, and σ critical is dependent upon the equivalence ratio and the temperature. If σ index is greater than 1, the risk of flame acceleration cannot be excluded, and the potential deflagration to detonation transition (DDT) must be considered in the risk analysis. 2.1. Calculation of critical expansion ratio The critical expansion ratio, σ critical, can be expressed as the function of Ea/RTu, where Ea is the effective activation energy and Tu is the initial flame temperature. Cases of stable and unstable flames can be generalized in one plot within the range of uncertainty, which can be estimated as ±8% in values of σ critical as shown in Figure 1 [1].
The critical expansion ratio which is the borderline in Figure 1 can be obtained by:
σ critical = 0.9 × 10−5 q 3 − 0.0019q 2 + 0.1807 q + 0.2314
(2)
where q= Ea/RTu is calculated by: q = ( 7725 − 406θ + 89.6θ 2 − 0.432θ 4 ) ( RTu )
The equivalence ratio θ of H2–O2 mixture is defined as:
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(3)
The 9th International Topicall Meeting on Nucclear Thermal-Hyydraulics, Operation and Safety (N NUTHOS-9) mber 9-13, 2012 Kaohsiung, Taiwan, Septem
θ=
X H22 2i X OO2
N9P0146 N
(4 4)
where XH2 and XO2 are the volu ume fractionn of H2 and O2. The equ uivalence raatio indicatees which of the reeactive com mponent dom minants (H2 or O2) and what the geeneral sensittivity of thee mixture is. The m mixture is most m reactiv ve when θ equals to 1.
Figgure 1. Ressulting com mbustion reggime as a function f of expansion ratio, σ , and a d dimensionleess effective activation n energy, Ea/RTu, for mixtures oof hydrogen n. n ratio 2.2. Callculation off expansion The exppansion ratioo σ is defi fined as the ddensity ratio of the reactants and pproducts,
σ=
ρ0 ρf
(5 5)
med, σ cann be calculatted as, If a connstant pressuure is assum
σ=
nf Tf
(6 6)
n0T0
nsity, quantiity and tem mperature off the reactannts, ρ f , n f , T f are where ρ 0 , n0 , T0 are the den the denssity, quantitty and adiab batic flame ttemperaturee of the disssociated prooducts. t e, Tq, is thee temperatu ure of the products when w the The adiiabatic frozzen-flame temperature combusstion processs takes placce adiabaticcally withou ut work or changes c in kkinetic enerrgy [12]. Howeveer, at high adiabatic frozen-flam f me temperattures (> 30 000 K) the dissociatio on effect becomees significannt which thus t introduuces additional compllications. T The temperrature of non-disssociating products p is the adiabattic frozen-fflame temperature, Tq . For exam mple, the stoichioometric hyddrogen and pure p oxygenn mixtures (300 K) bu urning will llead to an adiabatic a frozen-fflame temperature Tq at a about 50000 K. At this t high temperature, H2O is disssociated
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The 9th International Topical Meeting on Nuclear Thermal-Hydraulics, Operation and Safety (NUTHOS-9) Kaohsiung, Taiwan, September 9-13, 2012
N9P0146
which absorbs some of the reaction heat, and a lower adiabatic flame temperature, Tf, is therefore reached at the equilibrium state. The simplified chemical reaction is considered: 1 1 + ΔH reac tan t H 2( g ),T0 + O 2( g ),T0 ⎯⎯⎯⎯ → H 2( g ),Tq + O 2( g ),Tq → H 2O( g ),Tq + ΔH r ,Tq 2 2
(7)
where ΔH reactants + ΔH r ,Tq = ΔH r ,T0 + ΔH products Tq ⎛ N Tq ⎛ N ⎞ ⎞ ΔH reactants = ∫ ⎜ ∑ mf i ,reactantsC p ,i ⎟ dT ; ΔH products = ∫ ⎜ ∑ mf i ,productsC p ,i ⎟ dT T0 T0 ⎝ i =1 ⎠ ⎝ i =1 ⎠
Tq ⎛ N Tq ⎛ N ⎞ ⎞ ΔH r ,Tq − ΔH r ,T0 = ∫ ⎜ ∑ mfi ,productsC p ,i ⎟ dT − ∫ ⎜ ∑ mfi ,reactantsC p ,i ⎟ dT T0 T0 ⎝ i =1 ⎠ ⎝ i =1 ⎠
(8)
(9)
mf is the mass fraction of the species. Considering the steam dissociation at T, 1 2
+ ΔQ τ H 2 O ( g ),T ⎯⎯⎯⎯⎯ →τ H 2( g ),T + τ O 2( g ),T dissociation
(10)
where τ is the mole fraction of the dissociated steam at the temperature of T, the heat for the steam dissociation is ΔQdissociation . ΔQdissociation = ΔH r ,Tq − ΔH r ,T f = ∫
Tq N
Tf
∑ ( mf i =1
i ,dissociated
C p ,T ,i )dT
(11)
It should be noted that the degree of the dissociation τ is the function of T. When the adiabatic frozen-flame temperature, Tq, is sufficiently high, the steam dissociation becomes important. The adiabatic flame temperature, Tf, cannot be obtained directly with the specific heat capacity, Cp,Tq, and the heat needed for the dissociation ΔQdissociation,Tq. Numerical iteration should be used to obtain the adiabatic flame temperature, as implemented in the GASFLOW code. 2.2.1. Coefficients for thermochemical properties
The constant-pressure specific heat capacity Cp, the enthalpy difference ΔH and the dissociation degree τ are the functions of the temperature T, and are usually represented by the following power series: Ψ (a0 ,
, an T ) = a0 + a1T + a2T 2 +
+ anT n
(12)
To save the computational cost and be implemented in GASFLOW, the equivalent horner scheme of the polynomial is used, Ψ (a0 ,
, an T ) = a0 + T (a1 + T (a2 (
+ T (an −1 + Tan ) )
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(13)
The 9th International Topical Meeting on Nuclear Thermal-Hydraulics, Operation and Safety (NUTHOS-9) Kaohsiung, Taiwan, September 9-13, 2012
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2.2.2. Specific heat capacity and enthalpy difference
Considering the relation between the specific heat capacity and the enthalpy at the constant pressure:
⎛ ∂H ⎞ Cp = ⎜ ⎟ ⎝ ∂T ⎠ p
(14)
the same set of coefficients CM(1:n) is used for the specific heat capacity and the enthalpy, + CM(n)T n −1
DH (CM T ) = CM(1) + CM(2)T + CM(3)T 2 +
= CM(1) + T (CM(2) + T (CM(3)( CP (CM T ) = CM(2) + 2CM(3)T +
+ T (CM(n-1) + TCM(n)) )
+ (n − 1)CM(n)T n − 2
= CM(2) + T (2CM(3) + T (3CM(4)(
+ T ((n − 2)CM(n-1) + T (n − 1)CM(n)) )
(15)
(16)
2.2.3. Degree of dissociation
At high temperature, we have to consider the steam dissociation. The degree of dissociation τ is defined as the mole fraction of the dissociated steam. It can be derived from the equilibrium constant Kp, which is the function of T. n nH 2 + O 2 → nH 2 O 2 nτ nτ H 2 O → nτ H 2 + O 2 2
(17)
The equilibrium constant Kp(T) can be expressed as the function of dissociation degree τ (T ) ,
K p (T ) =
pH 2 O 1/ 2 pH2 pO2
(1 − τ )
=
⎛ 1 ⎞ ⎜1 + τ ⎟ ⎝ 2 ⎠ 1
⎡ ⎛ 1 ⎞ ⎤ ⎡⎛ 1 ⎞ ⎛ 1 ⎞ ⎤ 2 ⎢(τ ) ⎜1 + 2 τ ⎟ ⎥ ⎢⎜ 2 τ ⎟ ⎜ 1 + 2 τ ⎟ ⎥ ⎝ ⎠ ⎦ ⎣⎝ ⎠ ⎝ ⎠⎦ ⎣
(18)
Rearrange the Eq.18, we can obtain
(1 − ( K (T )) )τ 2
p
3
− 3τ + 2 = 0
(19)
By Solving Eq.19 and selecting the roots in the range of (0,1), we can obtain the degree of dissociation τ as function of T. The polynomial Eq. 20 is used to calculate the dissociation degree τ in GASFLOW code.
τ (g T ) = g(1) + g(2)T + g(3)T 2 +
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+ g(n)T n −1
(20)
The 9th International Topical Meeting on Nuclear Thermal-Hydraulics, Operation and Safety (NUTHOS-9) Kaohsiung, Taiwan, September 9-13, 2012
N9P0146
2.2.4. Algorithm to calculate expansion ratio in GASFLOW code
This algorithm is to calculate the sigma value σ of the adiabatic H2-O2 reaction with the initial mass fraction, mf, of H2, O2, N2, H2O, and the temperature T0. For convenience further sets of coefficients, v and D, are needed, as presented in Eq. 21. ⎛ − 0.002016 ⎞ ⎜ ⎟ − 0.0106 ⎟ D(4,1) = ⎜ (kg/mol) ; v(4,1) = ⎜ ⎟ 0.0 ⎜ ⎟ ⎝ 0.018016 ⎠
⎛ 496.0317460 ⎞ ⎜ ⎟ ⎜ 31.25000000 ⎟ (mol/kg) ⎜ 35.69771178 ⎟ ⎜ ⎟ ⎝ 55.50621670 ⎠
(21)
D is the mass change per unit mole in the reaction of H2 and O2. D(1) and D(2) are the mass consumption of H2 and O2 which are negative. D(4) is the production of steam, and D(3,1) is the mass change of N2 in the reaction which is zero. v is the mole amount per unit mass for H2, O2, N2, H2O respectively. Generally the algorithm includes five steps: 1) calculate the initial enthalpy difference ΔH 0 of the mixture at the temperature of T0; 2) compute the adiabatic frozen-flame temperature Tq after hydrogen combustion without steam dissociation; 3) obtain the degree of steam dissociation τ ; 4) calculate the adiabatic flame temperature, Tf, and the quantities, nf , after the steam dissociation; 5) obtain the sigma value with Eq. 6. More details about the algorithm are given in the flow chart in Fig. 2. 3. COMPARISONS OF GASFLOW AND STANJAN RESULTS
STANJAN [13], which uses the JANAF data tables [14], is an interactive program for chemical equilibrium analysis by the method of element potentials. It handles a gas phase and multiple condensed phases in an extremely robust and efficient manner. The STANJAN calculation results are compared with the sigma values calculated by the GASFLOW code. Two H2 mixtures were considered. One is the H2-O2-N2-H2O mixture, and the other is the H2-air-H2O mixture in which the ratio of oxygen and nitrogen volume fractions is 21/79, as shown in Tables 1 and 2. 3.1. H2-O2-N2-H2O mixture
The cases with various equivalence ratios and temperatures in the H2-O2-N2-H2O mixtures are presented in the Table 1.
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The 9th International Topical Meeting on Nuclear Thermal-Hydraulics, Operation and Safety (NUTHOS-9) Kaohsiung, Taiwan, September 9-13, 2012
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Table 1. H2-O2-N2-H2O mixtures with various equivalence ratios and temperatures cases
T0 (K)
N2 vol%
H2O vol%
1
300
0%
0%
2
400
0%
0%
3
500
0%
0%
4
600
0%
0%
5
700
0%
0%
6
800
0%
0%
7
400
0%
10%
8
400
20%
10%
9
400
40%
10%
10
400
60%
10%
The sigma values in cases 1-10 are presented in Figs. 3 and 4. The GASFLOW results are compared with the STANJAN calculations. When the equivalence ratio is one, the mixture should be the most reactive and the sigma should reach the maximum value. The GASFLOW results obtained a similar tendency as the STANJAN’s results. However, when the equivalence ratio is close to one the sigma values are over predicted compared to the STANJAN’s results, for instance cases 1-8. The reason is that the hydrogen and oxygen dissociations were neglected in GASFLOW, and only the steam dissociation was considered. For the stoichiometric H2-O2 mixtures in cases 1-8, the adiabatic frozen flame temperature is approximately at 5000 K. For such a high temperature, the H2 and O2 dissociations should be considered. However, the reaction becomes more complicated by introducing the species of H, O, HO, HO2 and H2O2. Moreover, it takes more computational effort to calculate the sigma value. To simplify the problem and save computational resource, only the reaction in Eq. 17 was considered. Higher adiabatic flame temperature and bigger sigma value will be therefore predicted without considering the hydrogen and oxygen dissociations. Over-prediction of sigma value is supposed to be conservative and acceptable for the hydrogen safety analysis. In case 9 and 10, due to the lower adiabatic frozen flame temperature generated by the H2-O2 mixtures, the GASFLOW results agree well with the results by the STANJAN.
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N9P0146
Input: Initial mass fraction: mf(1,4)=[H2 ,O2 ,N2 ,H2 O] Initial temperature: T0
Initial total quantity: n0 = mf(1,4)*v(4,1) Initial H2 volume fraction: vf_H2 = mf(1)v(1)/n0 Initial O2 volume fraction: vf_O2 = mf(2)v(2)/n0
vf_H2 > 4% & vf_O2 > 4%
NO
Output: σ=0
YES Enthalpy difference at T0: ΔH0
Enthalpy difference at Tq : ΔHq
Adiabatic frozen-flame temperature: Tq
NO Tq >1800 K YES Steam dissociation degree at Tq : τq=τ (g|Tq )
Enthalpy difference: ΔHq Specific heat capacity: Cp,q Adiabatic flame temperature: Tf
τ = 0 Tf = Tq nf = n0
Tq = Tf
Steam dissociation degree at Tf: τ f =τ (g|Tf )
(τ
q
−τ f
τf
)
