Effect of interface thermal resistance on ignition of reactive material by a hot particle

International Journal of Heat and Mass Transfer 97 (2016) 146–156

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Effect of interface thermal resistance on ignition of reactive material by a hot particle Jiuling Yang, Supan Wang, Haixiang Chen ⇑ State Key Laboratory of Fire Science, University of Science and Technology of China, Hefei, Anhui 230026, PR China

a r t i c l e

i n f o

Article history: Received 7 July 2015 Received in revised form 17 January 2016 Accepted 29 January 2016 Available online 18 February 2016 Keywords: Reactive material Hot particle Ignition Interface thermal resistance Reactant consumption

a b s t r a c t An ignition model of reactive material (RM) by a hot particle, taking into account the interface thermal resistance (ITR) and reactant consumption is established. The heat transfer process between RM and the particle was described by a radiation boundary condition at the interface. Five regimes were identified from the ignition model through numerical simulations. The criticality of reaction heat on the ignition regime was determined by the sensitivity analysis technique. It was found that a larger ITR would result in a longer ignition delay time, a smaller burnout area and a higher critical reaction heat of RM, and a higher critical radius and initial temperature of the particle. The effects of particle size, initial particle temperature and particle shape on the ignition delay time were more significant at a larger ITR. This work shows that both ITR and reactant consumption have unusual and significant influences on the ignition process of RM by a hot particle, and may have an instructive significance for the hot-spot ignition theory and RM management. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction Investigation of ignition of reactive materials (RMs) initiated by hot particles is of great practical importance for avoiding fires and explosion hazards [1–9]. In some fire accidents, the hot particles, generated from the firecracker [8], welding process [10–12] or processing equipment [13], are casually contacted with or completely embedded into RMs. The heat transfer between the hot particles and RMs can initiate chemical reactions in RMs and thus lead to fire hazards under appropriate conditions. In order to reveal the ignition mechanism and the critical condition for ignition, some theoretical models and numerical calculations were proposed in the literature. Most previous models adopted the assumption that the particle and RM were satisfied the ideal contact condition at the interface, i.e. the temperature and heat flux across the boundary are continuous at any moment [2,4,5,14–16]. However, it is known that, when the hot particle contacts with the ‘‘cold” material, the surface temperatures of the particle and material are different and they need time to reach the same. In order to avoid temperature incontinuity, a theoretical analysis assumed that at time zero, the temperatures at the hotspot surface rise instantaneously to an intermediate ‘‘jump” temperature, which is related the initial temperature and the ⇑ Corresponding author. Tel.: +86 551 63600275; fax: +86 551 63601669. E-mail address: [email protected] (H. Chen). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.01.070 0017-9310/Ó 2016 Elsevier Ltd. All rights reserved.

thermal inertia of the hotspot and RMs [17]. Therefore, the transient heat transfer process after contacting, which is controlled by the interface thermal resistance (ITR) [18–21], should be considered for practical applications. ITR appears in the contact region when two solid surfaces are met together. As we know, solid surfaces are comprised of peaks and valleys of varying size and shape and this unevenness is the basic cause of ITR. When two surfaces are pressed together, they actually touch at only a limited number of discrete points. As a consequence, the heat flow between the two surfaces is primarily confined to the limited areas of the true solid-to-solid spot-like contacts. An ITR appears in the contact region, characterized as the temperature discontinuity at the interface. In contrast, the temperature is continuous at the interface when the contact is assumed ideally. ITR is a function of the temperature difference and heat flux across the interface and is dependent of material properties [22]. ITR has received special interest and attention in literature. For example, Liu [20] analyzed the dual-phase-lag heat transfer in a layered cylinder using a radiation boundary condition model and discussed how ITR affected the transmission–reflection phenomenon at the interface. Wu [23] employed an acoustic mismatch model with a radiation boundary condition to model ITR between thin-film and substrate. However, to the best of the authors’ knowledge, there is no work related to ITR in the hot-spot ignition research.

J. Yang et al. / International Journal of Heat and Mass Transfer 97 (2016) 146–156

147

Nomenclature Dimensional parameters T temperature k thermal conductivity c specific heat q density Q reaction heat a mass fraction of RM r radial distance t time R0 particle radius A pre-exponential factor R universal gas constant E activation energy T⁄ reference temperature j constant; see Eq. (5a)

For RMs with relatively low or a moderately exothermicity, the stable ignition may occur far away from the hotspot surface with a substantial degree of reactant consumption [1,7,24,25]. With a constant power of the particle, a counterintuitive behavior, that is, two distinct critical power values for small-sized power sources to ignite RM with low exothermicity were observed [1]. With a fixed initial temperature of the particle, the ignition mechanism is more complicated [2]. There exists only a narrow band of fuel exothermicity where the traveling wave is actually established. When the wave stalls, the stalling mechanism is the same for the constant power hotspot, involving rapid burnout of the fuel resulting in insufficient reactivity in the fuel bed to support propagation [16]. A physical explanation for those counterintuitive ignition behavior [1,16,24] was given as the required equilibrium between the fast depletion of the reactant near the particle surface and the maintenance of a high temperature away from the surface [6]. More interestingly, four regimes of the ignition process initiated by a high-temperature hotspot were found, including nonignition, ignition with one flash, unstable ignition, stable ignition with double-flash [2]. This work will consider ITR with a radiation boundary condition in the hotspot ignition model. RM is a combustible material with relatively low exothermicity and thus the reaction consumption is also considered. In addition, the particle property is analogous to that of steel sphere [11,12]. The purpose of this study is to analyze the effects of ITR and reaction consumption of RMs on the ignition regimes numerically by establishing a one-dimensional mathematical model. Based on this model, the ignition position, ignition delay time and critical conditions for ignition are discussed. This is the first time to study the effect of ITR on the ignition of RMs by hot particles. The results could be employed to provide the safe guidelines for avoiding industrial fires caused by a localized hot spot in processing equipment such as blenders, mills and screw feeders.

Dimensionless parameters h dimensionless temperature s dimensionless time n dimensionless space b volumetric heat capacity ratio d dimensionless FK number g dimensionless conversion fraction Td Todes parameter r⁄ reference radius t⁄ reference time Subscripts 0, h initial values of RM and particle 1 hot particle 2 RM

as Refs. [1,2,7,24]. Thus, a one-dimensional spherically symmetric model is built for the fundamental research nature of hotspot ignition, which represents the closest approximation to the practical situation [24,26]. The assumptions used in this model are as follows. (1) For simplicity, a widely used reaction scheme as Eq. (1) in thermal ignition theory [1,2,6,11,16,24,26,27], ignoring the consumption of oxygen and the diffusion of gas products, is adopted in this work.

RM ! Solid residue

ð1Þ

(2) The reactant consumption is characterized by the density decrease rather than the volume deformation of RM. This exothermic reaction is assumed to obey a first-order reaction as Refs. [1,2,7,24]. (3) The phase transition and the diffusion of gas products are absent during reactant consumption. Therefore, the solid residue remains condensed with the same constant thermal properties as RM. c (4) The Biot number Bi ¼ hL of the particle is very small. Here k1 h ¼ jðT 21 þ T 22 ÞðT 1 þ T 2 Þ is an equivalent radiation heat transfer coefficient in the radiation boundary condition [20], approximately equals to 74.12 W/m2 K when j = 5.67  108 W/m2 K4, T1 = 973 K and T2 = 293 K. The qualitative length of sphere particle is Lc = V/A = R0/3. So Bi ¼ 0:0015 << 0:1 for the particle with R0 = 15 mm (large enough). It is also assumed that the thermal conductivity k1 of the inert particle is very high and its radial size R0 is small and thus R20 c1 q1 << k1 t ch is satisfied, which can

2. Formulation of the problem Equipment such as blenders, mills and screw feeders can develop a localized hot spot in combustible solids during the handling process and thus develop a potential fire hazard through initiating combustion of the materials and, under certain circumstances, the combustion can propagate throughout the entire bulk materials [1]. Generally, a non-reactive spherical metallic particle with an initial high temperature are assumed to be embedded completely at the center of spherical RM (see Fig. 1)

Fig. 1. Schematic diagram of the spherical axisymmetric ‘‘particle-RM” system.

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guarantee the inner thermal diffusion time of particle is much less than the characteristic time of the initiation of chemical process tch [2]. Therefore, the internal temperature gradient inside the hot particle can be neglected in accordance with the lumped capacitance assumption. The particle and RM are in a non-ideal thermal contact at the interface, where ITR is considered.

Some dimensionless parameters are defined as follows:

b¼

h¼

RT  E E

RT 

3. Mathematical model

t ¼ Taking into account the above simplifications, a onedimensional numerical model for the ignition of RM by a hot inert particle is proposed in the spherical axisymmetric coordinate.

c 1 q1


R0 @T 1 @T 1

¼ k1 3 @t @r
r¼R

0 6 r 6 R0

ð2aÞ

0

c 2 q2

  @T 2 k2 @ @T 2 r2 ¼ 2 þ QAq2 a expðE=RT 2 Þ r P R0 @t r @r @r

@ðq2 aÞ ¼ Aq2 a expðE=RT 2 Þ r P R0 @t

ð3aÞ ð4aÞ

Here subscripts 1 and 2 represent the particle and RM respectively, and a is the mass fraction of RM. The energy conversation equation (Eq. (2a)) of the hot particle indicates that the particle enthalpy change is equal to the heat flux of its outer surface due to the negligible inner temperature gradient. Combining the energy equation (Eq. 3(a)) and the depletion equation (Eq. 4(a)) of RM, it can be seen that the heat release rate is proportional to the reactant density. The boundary conditions are

k1

  @T 1 R0 @T 1 r ¼ R0 ¼ c1 q1 ¼ j T 41  T 42 @r 3 @t

  @T 1 @T 2 k1 ¼ k2 ¼ j T 41  T 42 @r @r

@a ¼ 0 r ¼ Rþ0 @r

ð5aÞ ð6aÞ

which consider the discontinuous thermal boundary condition caused by ITR. Eq. (5a) is used to calculate the particle temperature and Eq. (6a) is the boundary condition for energy equation (Eq. (3a)) þ of RM. Here R 0 means the outer surface of hot particle while R0 means the interface between the particle and RM. This discontinuous thermal boundary condition is firstly used in the hotspot ignition theory, at which ITR is specified with a radiation boundary condition model [18–21,23]. It is noted that the constant j is a function of the properties of the two contacted solid surfaces and higher j represents lesser ITR [23]. At the center of the particle (Eq. (7a)), the temperature gradient is zero. At the outer edge of RM (Eq. (8a)), the infinite Biot number condition (ha = 1) is employed [7,24,28]. This boundary condition is indeed equivalent to the ambient temperature condition (T 2 jr¼R1 ¼ T a ) [7], where Ta is the ambient temperature. A zero gradient boundary condition is applied to the mass fraction of RM.

@T 1 ¼0 r¼0 @r k2

ð7aÞ

@T 2 ¼ ha ðT 2  T a Þ @r

@a ¼ 0 r ¼ R1 @r

ð8aÞ

The initial conditions are presented in Eq. (9a), where the initial temperatures of the particle and RM are uniform and equal to a high-temperature Th and the ambient temperature Ta, respectively.

T1 ¼ Th

T2 ¼ Ta

a ¼ a0 ¼ 1 t ¼ 0

ð9aÞ

ðT  T  Þ ¼ 2

  1 T  T b T

c2 q2 RT 2 expðE=RT  Þ EQAq2 a0

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2 k2 RT 2 t ¼ expðE=RT  Þ r ¼ c 2 q2 EQAq2 a0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi QAq2 a0 E expðE=RT  Þ d ¼ R0 =r  ¼ R0 k2 RT 2

s ¼ t=t ; n ¼ r=r ; g ¼ ða0  aÞ=a0 Kk ¼ hh ¼

k1 ; k2 E RT 2

b¼

c 2 q2 ; c 1 q1

ðT h  T  Þ;

Td ¼ h0 ¼

c2 RT 2 QEa0

E RT 2

ðT 0  T  Þ

ð10Þ

Here b is the dimensionless adiabatic temperature excess [1] and the reference temperature T⁄ is selected as the initial particle temperature Th [11]. Other dimensionless parameters are listed in the nomenclature. Introducing the dimensionless parameters (Eq. (10)) into Eq. ((2a)–(9a)) leads to the following dimensionless equations

@h1 3jT 2 R0 ¼ b½ð1 þ bh1 Þ4  ð1 þ bh2 jn¼d Þ4  0 6 n 6 d @s bk2 d2

ð2bÞ

    @h2 @ @h2 h2 n2 nPd ¼ 2 þ ð1  gÞ exp @s @n 1 þ bh2 n @n

ð3bÞ

  @g h2 nPd ¼ T d ð1  gÞ exp @s 1 þ bh2

ð4bÞ

@h1 3jT 2 R0 ¼ b½ð1 þ bh1 Þ4  ð1 þ bh2 jn¼d Þ4  n ¼ d @s bk2 d2

ð5bÞ

i @h2 jT 3 R0 h ¼  ð1 þ bh1 Þ4  ð1 þ bh2 jn¼dþ Þ4 ; @n bk2 d

@g ¼ 0; @n

n ¼ dþ ð6bÞ

@h1 ¼0 n¼0 @s

ð7bÞ

@h2 ¼ 0; @n

ð8bÞ

h1 ¼ hh

@g ¼0 n¼1 @n h2 ¼ h0

g¼0 s¼0

ð9bÞ

In order to overcome the mathematical difficulties induced by the nonlinear boundary conditions in Eq. (5b) and (6b), the nonlinear term is linearized as

ð1 þ bh1 Þ4  ð1 þ bh2 Þ4 ¼ ½ð1 þ bh1 Þ2 þ ð1 þ bh2 Þ2 ½ð1 þ bh1 Þ þ ð1 þ bh2 Þ½ð1 þ bh1 Þ  ð1 þ bh2 Þ

ð11Þ

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J. Yang et al. / International Journal of Heat and Mass Transfer 97 (2016) 146–156

Then Eqs. (5b) and (6b) are expressed as

@h1 ¼ Bðh1  h2 Þ n ¼ d @s

ð5cÞ

@h2 ¼ Aðh2  h1 Þ n ¼ dþ @n

ð6cÞ

Here

3

A ¼ jkT2

R0 d

h i ð1 þ bh1 Þ2 þ ð1 þ bh2 Þ2 ½ð1 þ bh1 Þ þ ð1 þ bh2 ,

B ¼  3b Ad

and stable numerical results. In order to eliminate the constraints between the time step and space step, the implicit difference scheme was adopted except Eq. (5c), which was discretized by an explicit difference method. The dimensional space Dr ranges from 0.0001 m to 0.0005 m, and the dimensional time step Dt varies from 0.01 s to 0.1 s. The total spatial nodes (>500) and time nodes (>1000) are sufficient enough to guarantee the computational stability, i.e. Dr and Dt in each simulation run also satisfy the convergence condition in Eq. (12).

Dt 1 6 Dx2 2

ð12Þ

4. Numerical method

d

The radiant boundary conditions (Eqs. (5b), (6b)) at the interface bring mathematical difficulty of strong nonlinearity. In this work, the finite difference method is used to obtain the accurate

Here d ¼ k2 =ðq2 c2 Þ is the thermal diffusion coefficient of reactant. The grid independence examination of all the cases show that reducing the dimensional space (Dr) and time step (Dt) by a factor of 5 results in less than 10 K variation of the peak RM temperature. The partial equations and boundary conditions are discretized as follows:

Table 1 Thermal property parameters of RM and hot particle. Parameter

Value

Unit

Description

Q E A c1

1.73  106 1  105 1  105 465 8000 25 1300 300 0.04

J/kg J/mol 1/s J/(kg K) kg/m3 W/(m K) J/(kg K) kg/m3 W/(m K)

Present Present [1] [11] [11] Present Present Present Present

q1 k1 c2

q2 k2

study study

hnþ1  hn1 1 ¼ Bðhn1  hn2 jn¼dþ Þ 0 6 n 6 d k

gnþ1  gn k study study study study

0

 ¼ T d ð1  gÞ exp

hn2 1 þ bhn2

ð13Þ

 d6n

ð14Þ

n nþ1 nþ1 nþ1 nþ1 nþ1 hnþ1 Z  1 h2;mþ1  h2;m1 h2;mþ1  2h2;m þ hm1 2;m  h2;m þ Da d < n ¼ þ 2 Xm k 2h h ð15Þ

0

-2 -4

interface particle

-6 -8 0

particle

-1

-2 200

400

600

800 1000 0

4

' 200

400

600

800 1000

2

0 1 interface particle

-4

0 particle

-8 0

200

400

600

-1 800 1000 0

4

600

800 1000

2

0

1 interface particle

-4

0

400

3

8

-8

200

0

C 200

400

600

800 1000 0

particle

C

200

400

600

800 1000

Fig. 2. Temperature histories of material interface and particle with the radius of (A) R0 = 4 mm, (B) 5 mm and (C) 6.6 mm. (T0 = 293 K, Th = 1000 K, Q = 1.73  106 J/kg, j = 5.67  108 W/m2 K4).

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J. Yang et al. / International Journal of Heat and Mass Transfer 97 (2016) 146–156

n  hnþ1 Z  1  nþ1 2;1  h2;1 A h2;1  hn1 ¼ d k   n nþ1 nþ1 A hnþ1 h2;2  h2;1 2;1  h1 þ2 þ Da n ¼ d 2 2 h h

absolute coefficient on the diagonal line is greater than the summation of other absolute coefficients in each equation, which guarantees the computational convergence due to round-off error

n nþ1 hnþ1 hnþ1 2;M  h2;M 2;M  h2;M1; þ Da n ¼ d ¼ 2 2 k h

ð16Þ

(Oð105 Þ). This matrix equation was solved by a computationally efficient tridiagonal matrix algorithm (TDMA) [29]. The values of the parameters in the simulation are listed in Table 1.

ð17Þ

2

where Z = 1, 2, 3 corresponds to the slab, cylinder and sphere coordinate system, respectively. X m ¼ d þ ðm  1Þh,Da ¼ ð1  gn Þ hn

nþ1 2 expð1þbh n Þ. k and h are the time and space steps, respectively. h2;m 2

is the material temperature at time step n + 1, and the subscript m represents the mth space step. M means the last space step, i.e. the outer boundary of RM. Since the hot particle is fully embedded into the reactive material, a sufficiently large computational domain must be selected to ensure that the outer boundary effect of RMs can be neglected. In each simulation case, the size of space domain is 50 times larger than the particle radius, which is proven to be adequate for the simulation. Please refer to the nomenclature table for other symbols. Rearrangement of Eqs. (13)–(17) yields a matrix equation: [C] {h} = f (see Eq. (18)). The coefficient matrix [C] is a sparse tridiagonal matrix and satisfies that jbi j > jai j þ jci j ði ¼ 1; 2; . . . ; nÞ i.e. the

0

b

32 : 3 2 3 : 7 6 76 : 7 6 : 7 76 7 6 76 nþ1 7 7 6 76 hm1 7 6 : 7 7 76 7 6 n 76 hnþ1 7 7 6 76 m 7 6 hm þ d 7 7 76 nþ1 7 ¼ 6 7 76 hmþ1 7 6 : 7 76 7 6 7 76 6 : 7 76 : 7 7 6 7 7 7 7 6 c 56 4 : 5 4 : 5

c

6a b c 6 6 6 a b c 6 6 : : 6 6 6 : 6 6 6 6 4

: :

:

a b

c

a b

:

a b

ð18Þ

:

5. Results and discussion This section presents the simulation results for a series of cases. Firstly the effects of particle radius, initial particle temperature and chemical reaction heat on ignition regimes are discussed, followed by the effects ITR on the heat transfer process. Finally the factors affecting the ignition delay time are investigated.

0.0

interface particle

-3

particle

-0.2 -0.4

-6 -0.6 -9 0

40

80

3

120

160

interface particle

-0.8 0

40

80

120

0.0

160

particle

0 -0.2 -3 -0.4 -6 -0.6 -9 0

50 100 150 200 250 300 350 0

9

50 100 150 200 250 300 350

interface 1.5 particle 1.0

6 3

particle

0.5

0 -3

0.0

-6 -9 0

-0.5

C 100

200

300

400

500

0

C 100

200

300

400

500

Fig. 3. Temperature histories of material interface and particle with the initial temperature of (A) Th = 850 K, (B) 900 K and (C) 940 K. (T0 = 293 K, R0 = 9 mm, Q = 17.3  105 J/kg, j = 5.67  108 W/m2 K4).

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J. Yang et al. / International Journal of Heat and Mass Transfer 97 (2016) 146–156

For R0 = 6.6 mm (i.e. d = 63.733), as presented in Fig. 2C and C0 , the particle temperature undergoes an initial insignificant drop but quickly turns to a small increase amplitude, due to the first rapid flash of the material (the sharp peak temperature in Fig. 2C). After the first flash, the material temperature keeps nearly constant until s  320 when the heated layer is thick enough to assist the combustion wave propagating stably and the particle temperature increases once again. The comparison of Fig. 2B and Fig. 2C indicates that stable ignition can occur with one flash or without flash, which depends on the formation speed of the heated layer. All the results shown in Fig. 2 agree well with the numerical results in literatures [2,15], but the stable ignition without flash is not fully addressed in literature [2], which is determined potentially by ITR .

5.1. Effect of particle radius on ignition Thermal energy reserved in the hot particle is the function of the volume and initial temperature of particle. For a constant initial temperature, the particle radius determines thermal energy. Fig. 2 shows that as the particle radius increases, RM is heated by the hot particle and transits from the non-ignition region to the ignition region. For R0 = 4 mm (i.e. d = 38.626) (Fig. 2A and A0 ), the temperature of material interface increases firstly due to heat transfer from the hot particle and then decreases gradually due to the failure of initiation of chemical reactions in RM. The particle temperature decreases monotonously due to cooling. This phenomenon is in accordance with the non-ignition regime in literature [2]. For R0 = 5 mm (i.e. d = 48.283), as presented in Fig. 2B and B0 , the material is ignited by the hot particle with an adequate thermal energy. According to Burkina’s idea [2] that a heated layer must be formed to support the ignition of RM, the heated layer in this case is formed so slowly (i.e. the chemical reaction rate is very low) that the ignition without flash occurs after a long time, which is not reported in literature without considering ITR. During the period of the heated layer formed, the interface temperature varies insignificantly, whereas the particle temperature decreases as the result of the conductive heat loss (Fig. 2B0 ). Subsequently, a substantial increase of the particle temperature at s  400 implies that the material has been ignited and the combustion process heats the particle inversely.

5.2. Effect of initial particle temperature on ignition The ignition process is also sensitive to the initial particle temperature according to the experimental observations [11] and the numerical predictions [16] in literature. It is mentioned that the time and space steps are affected by the initial hotspot temperature, according to the definition of dimensionless parameters. Thereby, the time steps are variable in Fig. 3. The non-ignition regime is found for Th = 850 K in Fig. 3A, because of insufficient thermal energy reserved in the particle. For Th = 900 K in Fig. 3B and B0 , the interface temperature sharply increases (i.e. the first flash) at s  50 due to the instantaneous

0

2 0

-2

-2 -4

interface particle

interface particle

-4

-6

-6 -8

-8

(1) 0

150

300

(2) 450

600

-10

0

150

300

450

600

750

6 6 3

3

0

0

-3

interface particle

-6

interface particle

-3 -6

(3)

-9 0

200

(4)

-9 400

600

800 1000

0

500

1000

1500

2000

Fig. 4. Temperature histories of particle and material interface with the chemical reaction heat of (1) Q = 1.5  106 J/kg (2) 1.75  106 J/kg (3) 2.0  106 J/kg (4) 2.5  106 J/kg. (T0 = 293 K, Th = 973 K, R0 = 6 mm, j = 5.67  108 W/m2 K4).

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Q = 2.0  106 J/kg, as shown in Fig. 4(3), there is a steep rise of material interface temperature at s  120 before it decays gradually to a plateau at s  200. The difference between Fig. 4(2) and (3) may due to the burnout of material at the interface in the case of higher reaction heat. This region is regarded as stable ignition with one flash, which is also observed in literature without considering ITR [2]. A special ignition regime captured in Fig. 4(4) for Q = 2.5  106 J/kg shows two steep peaks for material interface temperature. The first peak implies the first flash due to the burnout of the combustible component of RMs near the particle surface but the heated layer has not been formed in this stage. Subsequently, the combustion wave quenches for a short moment but the material continues obtaining heat from the hot particle and shortly the second flash occurs away from the interface followed by the stable combustion wave traveling through the material, which is similar with the stable ignition with double flashes regime in literature [2].

burnout of the combustible component of RM near the particle surface. Meanwhile, an insignificant increase of particle temperature for a short period appears due to the fleeting flash of RM. Soon afterwards both the temperatures of the particle and material interface descend monotonically because that the thermal energy is not enough to form the heated layer in RM. So the combustion wave creases after the first flash, which corresponds to the unstable ignition regime. Another ignition regime is found in Fig. 3C and C0 where Th is 940 K and the temperature histories are similar to those in Fig. 2C and C0 . The comparison of Fig. 3B with C indicates that, for a lower initial particle temperature, the first flash occurs but the lack of sufficient heated layer leads the combustion wave to extinguish instantly; however, for a higher initial temperature, the first flash appears due to a faster reaction rate of RM and then the combustion wave can spread stably due to the accumulation of reaction heat. 5.3. Effect of chemical reaction heat on ignition

5.4. Effect of ITR on ignition The reaction source term in the energy conservation equation of RM (Eq. (3a)) is the function of chemical reaction heat Q, so the value of Q has a direct influence on the ignition regimes. Fig. 4(1) illustrates the non-ignition regime for a lower value Q = 1.5  106 J/kg due to insufficient reaction heat released from RM. For Q = 1.75  106 J/kg in Fig. 4(2), the stable ignition without flash is captured at s  200 when the required heated layer of material for ignition is formed. For a much higher value

The ideal thermal contact between RM and the particle is assumed in literatures [2,4,14–16] and thus the temperatures of particle surface and material interface are the same throughout the process. In real scenarios, this assumption is unreasonable due to the existence of ITR, which is because that the actual contact area accounts for only a tiny proportion of the macroscopically contact area and the imperfect contact causes the thermal

-2

c

-4

b

-6

a

-8 0

200

400

600

800

1000

600

800

1000

0

a -2

b -4

c -6 0

200

400

-4

-6

c

-8

b a 0

200

400

600

800

1000

1200

Fig. 5. Temperature histories of (1) material interface and (2) particle and (3) temperature profile of material interface at s = 100 for (a) j = 5.67  1010, (b) 5.67  109 and (c) 5.67  108 W/m2 K4. (T0 = 293 K, Th = 1000 K, Q = 17.3  105 J/kg, R0 = 4 mm).

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properties of RMs and the radius, initial temperature and shape of the particle. For j = 1.5  108 W/m2 K4 in Fig. 6(1)-A, the ignition does not occur but for j = 2.0  108 W/m2 K4, stable ignition occurs at s  650 and the maximum dimensionless interface temperature h2 is about 6.0. Similarly, for the cases of j = 3.0  108 W/m2 K4 and j = 5.67  108 W/m2 K4 in Fig. 6(3)–(4), the ignition delay time is s  300 and s  120, respectively, but slight difference exists between their maximum dimensionless interface temperatures. It is concluded that ITR has a significant influence on the ignition delay time but a slightly effect on peak interface temperature, that is, the higher ITR existed at the interface is, the higher the ignition delay time is. This work also considers material consumption and ITR together and this section analyzes the effect of ITR on material consumption, which is not found in literatures. The initial value of conversion is zero, meaning no material consumption. The conversion increases when RM is reacted and the conversion reaches one if RM is fully combusted. In Fig. 6(1) for j = 1.5  108 W/m2 K4, the higher ITR limits heat transfer from the hot particle to RMs and thus the reaction rate of RMs is very low. The conversion region is just a tiny area near the particle surface (g is far less than 1). For j = 2.0  108 W/m2 K4 in Fig. 6(2), the conversion rate g is less than 1 before s = 627(a–d), which implies the incomplete depletion of RM near the hot particle surface, however, when s = 836(e), the conversion rate g = 1 in the region of n 6 200 and

contraction. Considering the difficulty of determining the specific value of the resistance, a parameter j in Eqs. (5a), (6a) is used to characterize ITR as Refs. [20,22,23]. The empirical constant j decreases inversely with ITR, analogous to the interfacial conductance [21]. 5.4.1. Effect of ITR on heat transfer process Fig. 5 presents the temperature of material interface and particle with different values of j in the non-ignition region. It is shown that a higher value of j (i.e. lower ITR) leads to a quicker and higher increase of material interface temperature, because the hot particle transfers heat to RM more quickly due to a weaker thermal resistance. Meanwhile, the particle temperature in Fig. 5 (2) decreases more quickly for a higher value of j. Besides the temperature of material interface, the inside temperature of material also increases more quickly for a higher value of j, as seen in Fig. 5(3). 5.4.2. Effect of ITR on the ignition delay time The ignition moment is determined using the condition of zero time derivative of the surface particle temperature [2]. The ignition delay time, which is usually determined from the moment when the hot particle contacts RMs to the ignition moment, is a key indicator of material flammability. A shorter ignition delay time means a greater potential fire risk of material. The ignition delay time is affected by many factors, such as the contact conditions, thermal

0

0.3

-2 interface particle

-4

a-f

0.1

-6 -8 0

0.2

0.0 200

400

600

800

1000

0

20

40

60

80

100

600

800

1000

e

f

600

800

8 4

0.8

0

f

a-d

-8 0

e

0.4

-4

200

400

600

800

0.0 1000 0

200

400

c

d

200

400

8 0.8

4 0

0.4 -4

a-b

-8 0

200

400

600

800

0.0 1000 0

1000

f

4

0.8

0

-8 0

b

0.4

-4

200

400

600

800

0.0 1000 0

c

e

d

a 200

400

600

800

1000

Fig. 6. Temperature histories of material interface and particle (A) and conversion depth of RM (B) for (1) j = 1.5  108, (2) 2.0  108, (3) 3.0  108 and (4) 5.67  108 W/m2 K4 at s = 105(a), 209(b), 418(c), 627(d), 836(e), 1046(f), (T0 = 293 K, Th = 973 K, Q = 2.5  106 J/kg, R0 = 6 mm).

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g = 0 in the region of n > 200 show that n = 200 is a turning point, before which RM has been consumed completely while after which RM still keeps its origin properties. Since the combustion wave has been formed, the burnout region expands gradually and arrives at n  460 (s = 1046) (f). Therefore, a dimensionless traveling speed of the combustion wave m, is calculated as the ratio of the space interval of the adjacent conversion turning points and the time interval, i.e. m ¼ DDsn. The dimensional speed is expressed as u ¼ DDsn rt ¼ m rt (1.28 mm/s) and the burnout percentage of RM is about 46% at s = 1046 in this case. For j = 3.0  108 W/m2 K4, the depth of transformation increases a lot in Fig. 6(3)-B. The speed of combustion wave is 1.04 mm/s as well as the burnout percentage of RM is about 75% at s = 1046. For a more lower ITR (j = 5.67  108 W/m2 K4), RM has been consumed completely at s = 1046 (Fig. 6(4)-B(f)). The conversion rate is 100% yet the propagation speed of combustion wave is 1.14 mm/s. Comparisons of the four groups of figures in Fig. 6 indicate that ITR not only significantly affects the ignition delay time but also the depth of RM transformation (i.e. the burnout region of the material). A lower ITR leads to a larger burnout area of RM and a smaller ignition delay time. The ignition delay time is also affected by the radius and initial temperature of the particle. As seen from Fig. 7, the ignition delay time decreases with j, i.e. increases with a larger ITR. The ignition delay time at Th = 973 K decreases from 59.90 s (R0 = 6 mm) to 41.08 s (R0 = 8 mm) at j = 2.0  108 W/m2 K4, while from 11.50 s

(1)

6 mm 7 mm 8 mm

Ignition delay time (s)

20 10 -8

3.0x10

-8

4.0x10

-8

5.0x10

-8

6.0x10

(2)

Ignition delay time (s)

60 50

973 K 1000 K 1023 K

40

For the case in Table 2 (R0 = 6 mm, Th = 973 K, T0 = 293 K, Q = 2.5  106 J/kg), the effect of particle shape on the ignition delay time (s) are more significant under a larger ITR (j = 2.0  108 W/m2 K4) compared to that under the ideal contact (j = 5.67  108 W/m2 K4). The ignition delay time is the smallest with the slab particle but the largest with the sphere particle under the same conditions.



30

-8

ð19Þ

g absolute sensitivity is defined by sðg ; Q Þ ¼ j ddQ j, where g⁄ is the

40

2.0x10

@h @ 2 h Z  1 @h þ Da ¼ þ @ s @X 2 X @X

5.4.3. Effect of ITR on the critical reaction heat It is well known that there is a critical reaction heat of RM to sustain ignition and combustion, below which RM cannot be ignited. Here the sensitivity analysis technique [30] is used to calculate the reactant consumption and thus determine the critical reaction heat for j = 5.67  108 and 2.0  108 W/m2 K4. The

60 50

(R0 = 6 mm) to 10.65 s (R0 = 8 mm) at j = 5.67  108 W/m2 K4 (see Fig. 7(1)). Analogously, the ignition delay time for the R0 = 6 mm particle decreases from 59.90 s (Th = 973 K) to 29.92 s (Th = 1023 K) at j = 2.0  108 W/m2 K4, while from 11.50 s (Th = 973 K) to 7.01 s (Th = 1023 K) at j = 5.67  108 W/m2 K4 (see Fig. 7(2)). So the effects of particle size and initial particle temperature on the ignition delay time are more significant at a larger ITR (a smaller j). The particle shape also affects the ignition delay time. The equations for the slab-, cylinder- and sphere-shaped particle are often described by the Cartesian, cylindrical and spherical coordinate system, respectively. Generally, the dimensionless form of energy equation is expressed by Eq. (19), where Z = 1, 2, 3 corresponds to the Cartesian, cylindrical and spherical coordinate system, respectively.

30

normalized conversion of RM. The criticality is defined as the value of Q at which the absolute sensitivity takes a maximum value. i.e. g⁄ is most sensitive to the change of Q, which is explained in detail in literature [30]. For j = 5.67  108 W/m2 K4 in Fig. 8(1), the curve of normalized conversion g⁄ divides the space into two regions. The division line is about at Q = 2.10  106 J/kg, in the left of which the normalized conversion is almost zero yet in the right part, the value is 1 (RM is fully consumed). In other words, the left demarcation (Q < 2.1  106 J/kg) is the non-ignition region while the right side is the stable ignition region. Correspondingly, the absolute sensitivity s experiences a sudden sharp rise at about Q = 2.10  106 J/kg, which confirms that the critical reaction heat is 2.10  106 J/kg for j = 5.67  108 W/m2 K4. In order to investigate the effect of ITR on the critical reaction heat, a smaller value of j = 2.0  108 W/m2 K4 is chosen and the results are depicted in Fig. 8(2). The watershed between the non-ignition and stable ignition regimes is at about Q = 3.68  106 J/kg. It is obvious that the critical reaction heat to sustain RM ignition by hot particles becomes larger for a higher ITR.

20 Table 2 Ignition delay time (s) against particle shape at different j.

10 -8

2.0x10

-8

3.0x10

-8

4.0x10

-8

5.0x10

-8

6.0x10

Fig. 7. Ignition delay time (s) vs. j (inversely proportional to ITR) at (1) different particle radius with Th = 973 K, (2) different initial particle temperature with R0 = 6 mm (T0 = 293 K, Q = 2.5  106 J/kg).

Z 1 (slab) 2 (cylinder) 3 (sphere)

j = 2.0  108

j = 3.0  108

j = 5.67  108

W/m2 K4

W/m2 K4

W/m2 K4

27.92(s) 38.22(s) 59.90(s)

16.67(s) 21.06(s) 29.70(s)

8.64(s) 10.38(s) 11.50(s)

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(1)

8

0.005 stable ignition region

non-ignition region

0.8

|d */dQ|

|d /dQ| 0.003

0.6

0.002

0.4

0.001

0.2

Particle radius R0(mm)

0.004

7

1.0

theoretical prediction of Gol'shleger's Model

6 5 4 3 2 1

0.000

(2)

0.0 6

6

1x10

2x10

6

Q [J/kg]

3x10

0 900

1000

6

4x10

1100

1200

1300

1400

Initial particle temperature T h (K)

0.006 1.0 |d */dQ| stable ignition region

*

|d */dQ|

0.004 0.003

non-ignition region

0.002

0.8 0.6

16

0.2

0.000

0.0 6

6

6

2x10

3x10

theoretical prediction of Gol'shleger's Model

0.4

0.001

1x10

Fig. 9. Critical relationship between particle radius and initial temperature to ignite RM with the reaction heat of Q = 1.73  106 J/kg. The results for j = 5.67  108 W/m2 K4 (solid circles), j = 2.0  108 W/m2 K4 (solid triangles) and the predictions of Gol’dshleger’s model (solid squares) are presented. (T0 = 293 K).

6

4x10

Q [J/kg] Fig. 8. Sensitivity analysis of reactant consumption and reaction heat for (1) j = 5.67  108 and (2) 2.0  108 W/m2 K4. (T0 = 293 K, Th = 973 K, R0 = 5 mm).

14 Particle Radius R0(mm)

0.005

5.67 10

12

2.0 10

-8

-8

2

W /m K 2

W /m K

4

4

10 8 6 4 2

5.4.4. Effect of ITR on the critical radius and initial temperature of particle For a given RM, such as polystyrene foam [9], the occurrence of ignition by a hot particle is dependent on the radius and initial temperature of the particle. There is a critical relationship between the radius and the initial temperature. The literature has investigated the critical values of particle radius and initial temperature without considering ITR and the depletion of RM, and the following equation, proposed by Gol’dshleger et al [5], is wildly used

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3 dcr  0:4 b þ 0:25nðn þ 1Þðb þ 0:1b Þð2:25ðn  1Þ  h0 Þ2  ð1  0:5bh0 Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi k RT 2h E r cr ¼ dcr exp RT h qAQ E

ð20Þ

ð21Þ

This work also determines the critical particle radius and initial temperature to ignite RM with a pre-known reaction heat by the sensitive analysis method. The results in Fig. 9 show that the present model with considering RM burnout and ITR has a higher critical radius than Gol’dshleger’s prediction under the same conditions. The critical radius decreases monotonically versus the initial particle temperature, which is qualitatively consistent with the trend of the theoretical prediction curve of Gol’dshleger’s model and experimental results of Wang et.al [9]. A comparison between the critical curves for j = 5.67  108 W/m2 K4 and j = 2.0  108 W/m2 K4 indicates that a greater critical radius is needed for a higher ITR. For a fixed initial temperature value, the

5.0x10

5

1.0x10

6

1.5x10

6

2.0x10

6

2.5x10

6

3.0x10

6

Q[J/kg] Fig. 10. Critical relationship between particle radius (Th = 973 K) and the reaction heat of RM. The results for j = 5.67  108 (solid circles), j = 2.0  108 (solid triangles) and the predictions of Gol’dshleger’s model (solid squares) are presented. (T0 = 293 K).

critical radius is the largest for j = 2.0  108 W/m2 K4 (i.e. lowest ITR) and the smallest for the Gol’dshleger’s model (without ITR). It is obvious that Gol’dshleger’s model underestimates the critical particle radius, which is mainly due to the negligible influence of material depletion and TIR in their work. The critical condition of RM ignited by a hot particle is also related to RM properties (for example, reaction heat). This work also investigates the critical radius of particle with a given high temperature to ignite RM with different reaction heat, as shown in Fig. 10. The results also show that the critical lines of this work are higher than that predicted by Gol’dshleger’s model. It is concluded that ITR plays a significant role in the critical particle radius and initial temperature as well as the reaction heat of RM. If the thermal resistance increases, the critical curves in Fig. 8 and Fig. 10 are expected to move upward. 6. Conclusions This work investigated the ignition of RM by a hot particle with an initial high temperature, by taking into account the interface thermal resistance (ITR) with a radiant boundary model. Five

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ignition regimes were identified in the present model: non-ignition, unstable ignition, stable ignition without flash, stable ignition with one flash and stable ignition with double flashes. The stable ignition without flash is not reported in literature, which is potentially affected by ITR. It was found that a higher ITR would result in a longer ignition delay time, a smaller burnout area and a higher critical reaction heat of RM, and a higher critical radius and initial temperature of the particle. RM was ignited after 59.90 s by a hot particle with the radius of 6 mm and the initial temperature of 973 K and was almost completely consumed under j = 5.67  108 W/m2 K4, while ignited after 29.70 s with only 75% burnout of combustibles under j = 3  108 W/m2 K4. The effects of particle size, initial particle temperature and particle shape on the ignition delay time were more significant at larger ITR. The critical particle radius and initial temperature predicted by the present model through sensitivity analysis were larger than that of Gol’dshleger’s model, indicating that the critical conditions for ignition were underestimated if ITR and reactant consumption were not considered. This work is an extension of the hot-spot ignition theory and has an instructive significance for RM management. Acknowledgements This work was sponsored by National Natural Science Foundation of China (Nos. 51576184 and 51176179) and National Basic Research Program of China (973 Program, No. 2012CB719702). HX Chen was supported by Fundamental Research Funds for the Central University (WK2320000034). References [1] J. Brindley, J. Griffiths, A. McIntosh, Ignition phenomenology and criteria associated with hotspots embedded in a reactive material, Chem. Eng. Sci. 56 (6) (2001) 2037–2046. [2] R.S. Burkina, E.A. Mikova, High-temperature ignition of a reactive material by a hot inert particle with a finite heat reserve, Combust. Explos. Shock Waves 45 (2) (2009) 144–150. [3] R. Burkina, V. Vilyunov, Initiation of chemical reaction at a ‘‘hot spot”, Combust. Explos. Shock Waves 16 (4) (1980) 423–426. [4] D. Glushkov, P. Strizhak, Heat and mass transfer at ignition of solid condensed substance with relatively low calorific power by a local energy source, J. Eng. Thermophys. 21 (1) (2012) 69–77. [5] U. Gol’dshleger, K. Pribytkova, V. Barzykin, Ignition of a condensed explosive by a hot object of finite dimensions, Combust. Explos. Shock Waves 9 (1) (1973) 99–102. [6] A.C. McIntosh, J. Brindley, J.F. Griffiths, An approximate model for the ignition of reactive materials by a hot spot with reactant depletion, Math. Comput. Model. 36 (3) (2002) 293–306. [7] A. Shah, J. Brindley, A. McIntosh, J. Griffiths, Ignition and combustion of lowexothermicity porous materials by a local hotspot, Proc. R. Soc. A: Math. Phys. Eng. Sci. 463 (2081) (2007) 1287–1305.

[8] S. Wang, H. Chen, L. Zhang, Thermal decomposition kinetics of rigid polyurethane foam and ignition risk by a hot particle, J. Appl. Polym. Sci. 131 (4) (2014) 39359. [9] S. Wang, H. Chen, N. Liu, Ignition of expandable polystyrene foam by a hot particle: an experimental and numerical study, J. Hazard. Mater. 283 (2015) 536–543. [10] J.C. Jones, Thermal calculations on the ignition of a cotton bale by accidental contact with a hot particle, J. Chem. Technol. Biotechnol. 65 (2) (1996) 176– 178. [11] R.M. Hadden, S. Scott, C. Lautenberger, A.C. Fernandez-Pello, Ignition of combustible fuel beds by hot particles: an experimental and theoretical study, Fire Technol. 47 (2) (2011) 341–355. [12] J.L. Urban, C.D. Zak, C. Fernandez-Pello, Cellulose spot fire ignition by hot metal particles, Proc. Combust. Inst. 35 (3) (2015) 2707–2714. [13] P.J. Caine, S. Puttick, J. Brindley, A. McIntosh, J. Griffiths, R. Mullins, Ignition of Bulk Solid Materials by a Localised Hotspot, University of Leeds, 2010. [14] D. Glushkov, G. Kuznetsov, P. Strizhak, Numerical simulation of solid-phase ignition of metallized condensed matter by a particle heated to a high temperature, Russ. J. Phys. Chem. B 5 (6) (2011) 1000–1006. [15] G. Kuznetsov, G.Y. Mamontov, G. Taratushkina, Numerical simulation of ignition of a condensed substance by a particle heated to high temperatures, Combust. Explos. Shock Waves 40 (1) (2004) 70–76. [16] J.E. Staggs, A theoretical investigation of initiation of travelling combustion fronts in fuel beds from embedded hotspots, J. Fire Sci. 30 (5) (2012) 437–456. [17] A. Linan, M. Kindelan, Ignition of a reactive solid by an inert hot spot, Combust. React. Syst. (1981) 412–426. [18] W.-B. Lor, H.-S. Chu, Effect of interface thermal resistance on heat transfer in a composite medium using the thermal wave model, Int. J. Heat Mass Transfer 43 (5) (2000) 653–663. [19] E.T. Swartz, R.O. Pohl, Thermal boundary resistance, Rev. Mod. Phys. 61 (3) (1989) 605. [20] K.C. Liu, Numerical analysis of dual-phase-lag heat transfer in a layered cylinder with nonlinear interface boundary conditions, Comput. Phys. Commun. 177 (3) (2007) 307–314. [21] S. Shenogin, L. Xue, R. Ozisik, P. Keblinski, D.G. Cahill, Role of thermal boundary resistance on the heat flow in carbon-nanotube composites, J. Appl. Phys. 95 (12) (2004) 8136–8144. [22] M. Kelkar, P.E. Phelan, B. Gu, Thermal boundary resistance for thin-film highTc superconductors at varying interfacial temperature drops, Int. J. Heat Mass Transfer 40 (11) (1997) 2637–2645. [23] S.-K. Wu, H.-S. Chu, Inverse determination of surface temperature in thinfilm/substrate systems with interface thermal resistance, Int. J. Heat Mass Transfer 47 (14) (2004) 3507–3515. [24] A. Shah, J. Brindley, J. Griffiths, A. Mcintosh, M. Pourkashanian, The ignition of low-exothermicity solids by local heating, Process Saf. Environ. Prot. 82 (2) (2004) 156–169. [25] A. Shah, J. Brindley, A. Mcintosh, J. Griffiths, Gas-phase and heat-exchange effects on the ignition of high-and low-exothermicity porous solids subject to constant heating, J. Eng. Math. 56 (2) (2006) 161–177. [26] J. Brindley, J. Griffiths, A. Mcintosh, J. Zhang, Initiation of combustion waves in solids, and the effects of geometry, ANZIAM J. 43 (01) (2001) 149–163. [27] J. Brindley, J. Griffiths, A. Mcintosh, Potential hazards from spherical hotspots in reactive solids, in: Proc. of the Third Int. Seminar on Fire and Explosion Hazards, April 2001, 2001. [28] A. Shah, A. Mcintosh, J. Brindley, J. Griffiths, M. Pourkashanian, The effect of oxygen starvation on ignition phenomena in a reactive solid containing a hotspot, Combust. Theor. Model. 7 (3) (2003) 509–523. [29] L.H. Thomas, Elliptic problems in linear difference equations over a network, Watson Sci. Comput. Lab. Rept., Columbia University, New York, 1949. [30] R. Weber, H. Sidhu, D. Sawade, G. Mercer, M. Nelson, Using the method of lines to determine critical conditions for thermal ignition, J. Eng. Math. 56 (2) (2006) 185–200.