AN INVERSE STEFAN PROBLEM RELEVANT TO BOILOVER: Heat ...€¦ · Key words: fire, boilover, Stefan problems, heat balance integral, traveling wave-like solution, Koseki’s thermal

AN INVERSE STEFAN PROBLEM RELEVANT TO BOILOVER:

Heat Balance Integral Solutions and Analysis

by

Jordan HRISTOV

Original scientific paperUDC: 519.876.5:662.982

BIBLID: 0354-9836, 11 (2007), 2, 141-160

Stefan prob lems rel e vant to burn ing oil-wa ter sys tems are for mu lated. Twomov ing bound ary sub-prob lems are de fined: burn ing liq uid sur face andfor ma tion of a dis til la tion (“hot zone”) layer be neath it. The ba sic modelcon sid ers a heat trans fer equa tion with in ter nal neat gen er a tion due to ra di -a tion flux ab sorbed in the fuel depth. In verse Stefan prob lem cor re spond ingto the first case solved by the heat bal ance in te gral method anddimensionless scal ing of semi-an a lyt i cal so lu tions are at is sue.

Key words: fire, boilover, Stefan problems, heat balance integral,traveling wave-like solution, Koseki’s thermal wave, criticalfuel depth, time to boilover

Introduction

The “boil over” oc curs when the burn ing fuel is ex pelled vi o lently from the tanks due to the va por iza tion of the un der ly ing wa ter, usu ally col lected due to con den sa tion ef -fects [1, 2]. Num ber of sys tem atic stud ies have ad dressed the fuel layer-water bed pa ram -e ters [3-7] pre de fin ing the boil over ap pear ance and in ten sity. The term, com monly re -ferred as “thin-layer boil over”, has been ap plied also to the burn ing of thin slicks of oilspills spread af ter ac ci den tal re leases [5, 7] – see fig.1.

Boil over prob lem have been stud iedin ten sively in ex per i ments [8-14] and bymod els [1, 12-17, 18 (pres ent is sue)] to -wards deep un der stand ing of the con trol -ling phe nom ena. Ex per i men tal re sultsde vel oped be fore 1993 were re viewed by Koseki [19], while scale of anal y sis ofmod els was per formed re cently in [20,21]. The heat ing mod els of fuel-wa terstrat i fied lay ers are based on a gen eral as -sump tion of heat con duc tion [1, 9, 14-16, 22] and ig nored fuel con vec tion. Thepres ent work ad dress heat con duc tion

DOI:10.2298/TSCI0702141H 141

Figure 1

equa tions de fined in [21] as in-depth ab sorp tion model (DAM) with a vol u met ric heat ing sources and sur face ab sorp tion model (SAM) in cases with both lin ear and non-lin earbound ary con di tions (with a phase-change at the flam ing sur face).

The Stefan bound ary con di tions at the flam ing sur face [21] and vol u met ric heat -ing (DAM) make the prob lem com plex and needs ap prox i mate or nu mer i cal meth ods [7]. The pres ent work es pe cially ap plies an a lyt i cal meth ods al low ing scal ing, gen er a tion ofdimensionless group, and in ves ti ga tion of as ymp totic be hav iour of the so lu tions. An In -verse Stefan Prob lem [24] is solved here by the heat bal ance in te gral (HBI) method[22-24] with a front-fix ing tech nique (FFT) in ab la tion ap prox i ma tion [23] in ab sence ofhot zone be neath the sur face. Spe cial at ten tion is paid on the cor rect in ter pre ta tion of theequa tions and math e mat i cal ar ti facts [23-25] re sult ing from the FFT ap proach.

The burn ing liq uid layer (see fig. 2) (de creases in depth) with a re gres sion rater(t):

V y ty t

t

ma

s

F

= = =¢¢

&( )( ) &¶

¶ r(1)

that is a function of the fuel properties and the vessel diameter [5-7, 26, 27] (see also [18]) but it is independent of the fuel layer thickness. The net heat feedback from the flame (per unit area) to the surface of the burning liquid is a fraction c of the total heat released [27,28]:

& ( )¢¢ =æ

èç

ö

ø÷ -q C T T T Ds p

ff

fg4c

rp

4 4 4 (2)

where c µ [(1 – e–KD)/D1/2]0.61 [26] and K is extinction coefficient.Here af ter, we will as sume that &¢¢qs is con stant for a given fuel and pool di am e ter

and for seek of sim plic ity it is de noted as &¢¢ =q Fs too. Re fer ring to fig. 2a we may de finetwo mov ing bound ary prob lems rel e vant to the fuel-wa ter layer heat ing, namely:– in verse 1-D Stefan prob lem con sid ers the fuel-wa ter heat ing his tory with known and

sharp phase-change line (fuel sur face) (fig. 2b) de fined as ab la tion ap prox i ma tion [23,24]. The cor rect term is in -verse Stefan prob lem [25]since it ad dresses the tem -per a ture pro file across thefuel layer. In the op po sitecase when the ve loc ityshould be de fined we haveto solve the for ward (di -rect) Stefan prob lem [25].This sim pli fied sit u a tioncor re sponds to com bus tionof fuel that do not form iso -ther mal layer be neath thesur face (hot zone).

142

THERMAL SCIENCE: Vol. 11 (2007), No. 2, pp. 141-160

Figure 2

– compound Stefan problem with a second moving boundary due to the formation of ahot zone (HZ) (or convection mixed layer (CML), or isothermal layer, distillationzone) beneath the flaming surface (see fig. 2c) .The hot zone depth increases in timewith a velocity that really defines a direct Stefan problem since both the front line ofCML and the temperature profile are unknown. This compound problem is beyondthe subject of the present discussion.

Hristov et al. [20] re viewed the ex ist ing 1-D heat trans fer equa tion and per -formed large scal ing stud ies. The pres ent work ad dresses the same prob lem with the HBIand FFT tech niques. The one-di men sional in-depth ab sorp tion model, ig nor ing fuel con -vec tion, dis cussed in [20], is:

¶

¶

¶

¶

¶

¶

T

ta

T

y

q

yq q y= +

¢¢¢¢ = ¢¢ -F

rr s e

2

2

&, & & m (3a)

witht = 0, T = T4 and T = Ts, y = ys(t) (3b)

and Stefan boundary condition (SBC) at the liquid surface:

& ( ) & ,¢¢ = + ¢¢ ¢¢ = -=

q H r t q qT

y y yss V F c cr

¶

¶(4)

As a first ap prox i ma tion we as sume [5-7] no dif fer ences in ther mal prop er ties offuel and wa ter. The wa ter heat ing poses a se ri ous ques tion about the limit of super heatsince the ex per i men tal mea sure ments [5, 18, 20] and the phys i cal ex per i ments [28, 29]pro vide dif fer ent data.

Inverse Stefan problem and front-fixing techniques (FFT)

A mov ing bound ary prob lem can be easy trans formed into a fixed-bound ary one by a sim ple mo tion of the in ter face with at tached ref er ence frame (see fig. 3a) to wards the flame at the ve loc ity [25, 30]. For in stance, the FFT con trib ute the SAM equa tions with aterm (Va rFCpF)dT/dy [30]:

r r lF pF a F pF Fd

dC

T

tV C

T

y

T

y

¶

¶

¶

¶- =

2

2(5)

with y = 0, T = Ts, and x = 4, T = T4, and dT/dy = 0 (6)For con stant prop er ties lF, rF, and CpF eq. (5) has a closed form so lu tion [30],

namely:

T T

T T

V y

a-

-=

-4

4s

ea

F (7)

Sim i larly the in-depth ab sorp tion model (eq. 3a) with FFT be comes:

143

Hristov, J.: An Inverse Stefan Problem Relevant to Boilover: Heat Balance ...

¶

¶

¶

¶

¶

¶

¶

¶

T

tV

T

ya

T

y C

q

y- = +

¢¢a F

F pF

r2

2

1

r

&(8)

The added enthalpy fluxterm (Va rF CpF)¶T/¶y in eq.(8) is as so ci ated with the neg -a tive Va (op po site to the ori -en ta tion of the y-axis) and is a re sult of the change of the ref -er ence frame. It can not be as -so ci ated with any con vec tivemo tions in the liq uids as itwas done in [1, 17]. More to the point, Va can con trib ute to dimensionless num ber NDHS = y0r(t)/aF = y0Va/aF

[20] de fined only by the bound ary con di tions but never to the Reynolds num ber that isde pend ent on the bulk per for mance of the liq uid). The model (eqs. 3-4) con sid ers headcon duc tion as a trans port mech a nism only. The phys i cal der i va tion of SBC (eq. 4) con -sid ers con ser va tion of en ergy in which the two phases are as sumed to be in com press -ible and at rest [25]. With two phases of dif fer ent den si ties r1 and r2, re spec tively, themass con ser va tion can only be sat is fied if the liq uid phase moves with a ve loc ity VFgiven by r1ds/dt = r2[(ds/dt) – VF)], x = s(t), where s(t) de notes the mov ing phase--change line. In the liq uid re gion a con vec tive term con trib utes the heat-flow equa tion[25] and with FFT yields:

¶

¶

¶

¶

¶

¶

¶

¶

¶

¶

T

tV

T

yV

T

ya

T

y

q

x- + = +

¢¢a F F

2

2

&(9)

From eqs. (8) and (9) it is clear that the “con vec tive” terms VF¶T/¶x and–Va¶T/¶x have dif fer ent phys i cal mean ings and or i gins. The for mer rep re sents the con -vec tion con tri bu tion to the heat trans fer while the lat ter is a math e mat i cal ar ti fact fa cil i -tat ing the math e mat i cal so lu tion only. In ab sence of con vec tion (VF = 0 we get ex actly the model used in [17, 31, 32] to sim u late fuel con vec tion (sic!). In ad di tion, this is the modelpro posed by Broekmann and Schecker [1].

HBI solution and analysis

Solution by the HBI method

The ex per i ments known so far in di cate two type of tem per a ture pro filessketched in fig. 2b and c. The pres ent study ad dresses that with out CML and looks for a

144


Figure 3

so lu tion by the HBI method for three prin ci ple rea sons: (1) ex plicit re la tion ships be tween the ba sic model pa ram e ters; (2) scal ing, and (3) as ymp totic anal y sis of the re sults.

As sum ing the fuel layer with a depth y0 is d(t), where d(t) is the ther mal pen e tra -tion depth (see fig. 3b) and with a bound ary con di tion lF¶T/¶y = –F(t) the so lu tion fol -lows Good man [22] the sur face flux F(t) = q" = const. [20] and F = F(y) will be used, thatis the vol u met ric heat source is ¶F/¶x = &¢¢q r . In te gra tion of eq. (8) over d (see fig. 3b) giveseqs. (10) and (12):

¶

¶

¶

¶

¶

¶

T

yV

T

yy a

T

yy=

æ

èçç

ö

ø÷÷ +

æ

èçç

ö

ø÷÷ +ò ò ò

0 0

2

20

d d d

a Fd d1

0r

d

F pF

dC

F

yy

¶

¶

æ

èçç

ö

ø÷÷ò (10)

with boundary conditions:

¶

¶

¶

¶

T

yF H V

T

yT T

x y

x

= =

== - + = =0

10

lr

d

dF

F v a( ), , 4 (11a, b, c)

d

da s

F

FF v a

F pFtT V T T

aF H V

F

C( ) ( ) ( )q d

lr

r+ = - + + -4 4 (12)

where qd

= òT yd0

is the heat balance integral (HBI) [22]. Further:

VT

yy V T T V T Ty ya a a sd

¶

¶

æ

èçç

ö

ø÷÷ = - = -ò = =

0

0

d

d( ) ( )4 (13)

and in accordance with boundary conditions (11a, b, c) and Fy=0 = F, and assuming Fd == 0 we get:

aT

yy a

T

y

T

yy

F Fd¶

¶

¶

¶

¶

¶

2

20

æ

èçç

ö

ø÷÷ =

æ

èçç

ö

ø÷÷ -

æ

èçç

ö

øò

=

d

d

÷÷

é

ë

êê

ù

û

úú

=

= -æ

èçç

ö

ø÷÷

é

ë

êê

ù

û

úú

= +

=

=

y

y

aT

y

aF

0

0

0FF

F

¶

¶ l( rF v aH V ) (14)

1 1 1

0

0r r r

d

dF pF F pF F pF

dC

F

yy

CF F

Cy y

¶

¶

æ

èçç

ö

ø÷÷ = - = -ò = =( ) ( F ) (15)

With a qua dratic tem per a ture pro file T(y, t) = b0 + b1y +b2y2, and bound ary

conditions (11a, b, c) we get bi(t):

bl

bl

blb

r0 1 21

2

1 1

2= + = - = = +T F H V4 F F F F, , , F v a (16)

145


Sub sti tu tion of eq. (16) in eq. (3a) and then in eq. (12) yields an equa tion for d(t), namely:

d

dwitha F

dd d

223 6 1 0 0

tV a

Ft t+ = -

æ

èç

ö

ø÷ = =

F( ) , (17)

In case of fixed boun dary (Va = 0) and no flux at the bound ary (F = 0) and F == const. eq. (17) is the Good man’s so lu tion [22]. The so lu tion to eq. (17) is [33]:

d2 32

1 1( ) ( ),ta B

VB

FV tF= - = -

-F F

a

e aF

(18a, b)

There fore, the ther mal pen e tra tion depth is d(t) = (2aFBF/Va)1/2(1– e a-3V t )1/2

that cor rectly gives at t = 0 ® d = 0. The so lu tion con firms the Good man’s es ti ma tiond º (6aFt)

1/2 since ex press ing the ex po nents 1 – e a-3V t » 1 – (1 – 3Va/1!) we can read d(t) » (2aFBF/Va)

1/2(3Vat)1/2 » (6aFt)

1/2BF1/2 º (6aFt)

1/2.There fore, the vol u met ric heat ing af fects the pen e tra tion depth by a fac tor (BF

1/2 < 1) and the tem per a ture pro file be -comes:

T y t T

T T T T

y

ZZ a t B

( , )

( ),

-

-=

--

æ

èç

ö

ø÷ =4

4 4s F sF F

F

21 6

2

l(19a, b)

Scaling of the thermal penetration depth d(t)

Let us try to check the ini tial as sump tion of the HBI that y0 o d(t). Re ally the time to boil over tB0 is the only time limit we have from ex per i ments. The wa ter heat ing clearlyin di cates that the en tire layer is ther mally pen e trated, so the an swer is clear, but from anac a demic stand point we should es tab lish when with fuel depth de creas ing in time thecon di tion y = d(t) will be sat is fied. It might be rea son able to es tab lish the up per limit of y t

y h=

=d( ) at t = th that with scal ing de vel oped ear lier yields h = d(th) = (6aFth)

1/2BF1/2.

This gives th = (1/6)(h2/aF)1/BF = t0/6BF and the time re quired the pen e tra tion depth to

reach the fuel bot tom h is ap prox i mately 6 time shorter than the ther mal dif fu sion time t hh0

2= /ah in ab sence of vol u met ric heat sources and mov ing sur face. Tak ing into ac -count that Foe ~ O(l) [20], more ex actly Foe is close to 1 at t = tB0, we might sug gest that d(th) » d(t

h0

/6) » h. How ever, we have to test the con tri bu tion of BF to eq. (18a). From eq.(18b) we read:

BF F

F H V

H V

F HF

F v a

F v a

p

= - = -+

= - +æ

èç

ö

ø÷ = -

æ

è

çç

ö-

1 1 1 1 11

1

F r

r

ø

÷÷

-1

(20)

where Hp = F/rFHvVa = &¢¢q /Hv & ¢¢m a dimensionless number defined in [20, 21] through theSBC.

This num ber Hp [21] is a ra tio of three dimensionless num bers char ac ter iz ing the fuel layer heat ing (see tab. 1 and 2), that is Hp = BSASte/NDHS, where NDHS = Vay0/aF, Ste == CP(Ts – T4)/Hv is the Stefan num ber, and BSA = &¢¢q y0/lF(Ts – T4) is dimensionless num -ber an a logue to con vec tive num ber in case of ra di a tion-con duc tion heat ing [21]. In fact,

146


147


Table 1. Data summarized from various experiments and results derived in [21]*

Ref.D

[m]y0

(3)

[mm]tB0

(1)

[s]UT

(1)

[103 mm/s]Foe (1)

Va(average)[102 mm/s]

NDHSNVA[103]

[6, 7]

Heating oil(aF = 0.877·10

–7 m2/s)c

Ste = 1.373

0.15

19 945 20.1 0.22

0.01 (2)

1.9 7.02

17 830 20.48 0.24 1.7 5.75

13 625 20.8 0.31 1.5 3.31

9 450 20.0 0.47 1.3 1.58

7 340 20.58 0.59 0.7 0.99

4 165 24.42 0.88 0.4 0.33

2 90 22.22 1.94 0.2 0.007

0.23

17 710 23.94 0.21

0.011 (2)

1.7 7.01

15 620 24.19 0.23 1.5 4.71

13 530 24.52 0.27 1.3 4.1

9 340 26.47 0.36 0.9 2.02

4 125 32.0 0.66 0.4 0.39

3 75 40 0.71 0.3 0.21

2 30 66.66 0.64 0.2 0.009

0.5

15 345 43.47 0.13

0.017 (2)

1.5 6.95

13 265 49.05 0.13 1.3 6.04

11 190 57.89 0.13 1.1 4.33

7 90 77.77 0.15 0.7 2.89

5 70 71.42 0.24 0.5 0.85

3 15 200 0.24 0.3 0.329

[10]

Arabian light crude-oil(aF = 0.679·10

–7 m2/s)c

Ste = 1.703

0.3 3.5 612 57.18 0.003 no data

0.620 492 40.65 0.083 2.33 6.87 6.87

69 942 73.25 0.013 3.33 33.87 33.87

1

20 681 42.73 0.079 2.91 8.59 8.59

40 978 40.89 0.041 3.66 21.6 21.6

60 1310 45.87 0.024 4.0 35.34 35.34

100 1926 51.92 0.013 3.66 54 54.0

2 20 411 48.66 0.069 3.08 9.08 9.08

3.5 (4) 27 402 67.16 0.037 3.83 15.24 15.24

* These data shows experimental conditions and order of magnitude of dimensionless groups appearing in the *,solutions. No convection mixing layer (CML) formation in the burning fuelNotes: (1) – calculated in [21]; (2) – calculted in [21] with data recovered from various papers of Garo et al.; (3) –

from [6]; (4) – the diameter of the circular pan with the same area, while the real square pan is 2.7 ́ 2.7 m pan;UT and RAV data of Koseki experiments are from the original work [10], but recalculated in mm/s

BSA is a ra tio of two length scales, BSA = y0/Z0, where Z0 = &¢¢q /lF(Ts – T4). The es ti mates in[21] re veal that Hp n 1 that al lows to 1/Hp o 1 and there fore BF » 1. Hence, from eq. (20)we have th » t0/6 and there is no ef fect of the ra di a tion flux. The lat ter im plies that with in -ter nal heat gen er a tion the tem per a ture pro file [22] is T(d,t) = (Q/rFCpF)[1 – (1 – y/d)

3] that re sem bles the pres ent so lu tion (19). In this case the pen e tra tion depth is d = [(24aF Q

t 20ò ·

·dt)Q–2]1/2 where Q = & ( )¢¢ò q t tt d0 with non-time vary ing sur face flux, & ( )¢¢q t = F(t) = const. In

view of Q = Ft we can read d = [24aF(Q2t/Q2)]1/2 = (24aFt)

1/2. Fur ther, we read th = th0

/24that is shorter than th » t

h0

/6. There fore, with a cer tain pro cess ap prox i ma tion we may as -sume that at the be gin ning d = y0 = h that sim plify the so lu tion, i. e. the HBI should be ap -plied over the en tire fuel layer thick ness h. Such an ex am ple is de vel oped next.

Alternative HBI solution in ablation approximation with d > h

Linear boundary conditions

Equa tion (5) un der sta tion ary con di tions with y = 0, T = Ts, and x = 4, T = T4, and dT/dy = 0 at y = d = h has a so lu tion ex pressed as (7). The tran sient heat trans fer so lu tioncan be ex pressed as [34]:

148


Table 2. Treatment of the data of Arai et al. [9] concerning the boilover onset(D = 0.048 m; y0 = 10 mm)

FuelaF

(1)

[107 m2/s]Ti

[K]Tb

(1)

[K]Va(average)

(1)

[105 m/s]Foe (2) Ste NDHS

Toluene 1.03

293

383 1.35

0.43 0.462

1.35

318 0.29 0.332

323 0.42 0.305

325 0.45 0.295

341 0.74 0.214

355 1.01 0.143

Ethylbenzene

0.88

293

409 1.5

0.45 0.558

1.69323 0.24 0.432

330 0.18 0.398

345 0.08 0.322

n-Decane 0.753

293

433 1.19

0.34 1.149

1.58

323 0.22 0.892

348 0.11 0.694

355 0.10 0.636

362 0.07 0.581

Notes: (1) – The paper of Garo et al. [6] was used as a source for some average values of the fuel properties due todeficiences in the original paper [21]; (2) – from [21]. NVA was calculated with approximately constant flametemperature of about 1100 K typical for hydrocarbon pool fire through eq. (2)

T T T T D t

V y

a= + --

æ

èçç

ö

ø÷÷

4 4( ) ( )s ea

F (21)

The time-de pend ent func tion D(t) can be de ter mined by HBI with d(t) > y0 = h,namely:

d

de ds s

aa

FD t

tT T y a T T

V

a

V y

ah

FF

h( )( ) ( )- = -

æ

èçç

ö

ø÷÷

-

ò 4 40 0

ò-

2

e da

F

V y

aD t y( ) (22)

d

dea

F

a

FD t

t

V

aD t D t D t

V

a( ) ( ) ( )=æ

èçç

ö

ø÷÷ Þ =

2

0

2

(23)

The ini tial con di tion D0 = D(0) could be de ter mined from the tem per a ture at theflam ing sur face (y = 0 at t = 0) that rises im me di ately to T = Ts. This con di tion pro vides D0 = = D(0) = 1and we read:

T T

T Tt

V y

a

V

a-

-=

-4

4s

e ea

F

a2

F (24)

Note: The approach to determine D0 = D(0) differs from that suggested by Zubarev [34]where the condition T4 = (1/h) T

h0ò (y,t)dy at y = 0 at t = 0 is used.

Non-linear (Stefan) boundary condition

The gen eral time-de pend ent so lu tion to eq. (5) with y ® 4, T = T4 and y = 0,–¶T/¶x = = (1/lF)(F + rFCpFHv) = F/lF sim i lar to eq. (21) is:

T Ta

VP t

V

ay

= +-

4 FF

a

ea

F ( ) (25)

Ap pli ca tion of HBI about P(t) just like eq. (23) yields:

d

dea

F

a

FP t

t

V

aP t P t P t

V

a( ) ( ) ( )- = Þ =2

00

2

(26)

The Stefan bound ary con di tion –¶T/¶y = F/lF at y = 0 sim ply pro vides P0 == P(0) = 1, that leads to:

T T

T T T T

a

V

V

ay

V

at-

-=

-

-4

4 4s F s

F

a

e ea

F

a

FF

l ( )

2

(27)

Analysis of the time constant Va2 /aF

The ex po nent (Va2/aF)t in eqs. (26) and (27) can be re ar ranged as:

149


V

at

V

h

h

at

V

ht t

V

ht

tha

F

a

F

a a Fo2 2

2

22

0

2

02= =

æ

èç

ö

ø÷ =

æ

èç

ö

ø÷ = 0

0

2

t

æ

èçç

ö

ø÷÷ Fo

h (28)

Here t h0 = h2/aF is the thermal diffusivity time scale and Fo

h = t/t h0 = h2t/aF. With hmax = y0

(at t = 0) as a length scale [21] the Fourier number is Fo = y t02 /aF and t0 = y0

2/aF. Thetheoretical time required for complete combustion of the fuel t0 = h/Va.

Data about tB0 and Foe = tB0/t0 are avail able else where [20, 21]. Thus, the ex po nent

of D(t) at t = tB0 is (Va2/aF)tB0 = (t0/t0)

2 Foe = Va2/U0UT. Here U0 = y0/t0 and UT = y0/tB0

(Koseki’s wave) are two hy po thet i cal ve loc i ties de fined through mac ro scopic vari ables.There fore we can ex press (Va

2/aF)t at t = tB0 as (Va2/aF)tB0 = NDHS(Va/UT). Con se quently,

the pro file (27) reads at:

QF

BB

s F s

F

a

Foe ea

F DHSe

00 2=

-

-=

-

-T T

T T T T

a

V

V y

a N4

4 4l ( )( ) (29)

Boilover onset – dimensionless scaling of thesemi-analytical solution of SAM

With the pro file (29), data about Foe and the lo ca tion of the wa ter y = yF we mayde fine the tem per a ture at the fuel-wa ter (F/W) in ter face T y y= W = TB0. This case for mu -lates a di rect A prob lem. Al ter na tively, def i ni tion of Foe through (29) with known T = TB0for mu lates an in verse B prob lem.

· A problem – Time to boilover tB0. Simple rearrangement of expression (29) at t = tB0yields:

AT T

a

V

V y

a NB B

F s

F

a

Foe ea

F DHSe

0 0

1

2=

-

æ

è

çç

ö

ø

÷÷

=-

-

QF

l ( )( )

4

(30)

Foe B

DHS

BB

DHS F

= Þ =ln lnA

Nt

A

N

y

a0

2 00

2

02

(31)

· B problem – Temperature T = TB0 at F/W interface. From (29) we get:

QB BFoe DHS0 0

2= B N

e[( ) ] (32)

withB

T T

a

VN

H

V y

aB

F s

F

a pFe

a

F0 0 1

1=

-= +

æ

è

çç

ö

ø

÷÷

-F

ld

( )4(33)

Here N0 = Fy0/lF(Ts – T4) is the radiation-conduction parameter [20]; dF = yF/y0 withyF/y0 + yw/y0 = 1. Equations (30-33) call for scaling analyses addressing AB0, BB0, and Foe.

150


Dimensionless scaling of Foe , AB0 , and BB0

Sim ple re ar range ment of the nom i na tor of eqs. (32) or (33) with ap prox i ma tion ofthe ex po nent in a se ries (valid for small NDHSdF) since NDHS » 1 (see [20]) and dF = yF/h < 1pro vides:

Ay

T T NN

y

TFB B

s DHSDHS F B

F s0 0

0

1

001»

-

é

ëê

ù

ûú »

-

QF

QF

ld

l( ) (4 -

é

ëê

ù

ûú

-

T )4dF

1

(34a, b)

Since F = F + rFHvVa (see eq. (20) we read from eq. (34) that Fy0/lF(Ts – T4) = N0 ++ NDHS/Ste and:

A NB N

AN

HB BSA DHS

F BB

FSte0 0 0

1

00

0

1» +æ

èç

ö

ø÷

é

ëê

ù

ûú Þ »

-

QQ

dd

p (35)

since from eq. [21]:

HH

HB N

pp

pSA DHS

Sten1 1

121® + » = [ ] (36)

Now, tak ing into ac count that lnx » x – 1 with 0 < x £ 2, eq. (31a) scales as:

Fo Foe

B p

F

DHS

e B

Fp

DHS

» Þ » -æ

èçç

ö

ø÷÷ln

Q

Q

0

0

2

0

02

11

H

N

N NH

N

d

d(37)

In view of the or ders of mag ni tude of the dimensionless num bers, i. e., QB0 ~ O(1),dF ~ O(1), N0 ~ O(1) for lab o ra tory fire and N0 ~ O(10

2) for large fire [21] and NDHS ~ O(1) weget Foe º Hp/N DHS

2 that con firms the scal ing re la tion ship Foe ~ (QB02 /Bu)H mp es tab lished in

[21] through a phys i cal anal y sis only. Fur ther, in ex plicit (di men sional) form the last scal ingre la tion ship gives tB0 º y0

2Hp/aF(Vay0/aF)–2 º (aF/Va

2)(F/Hv & ¢¢m ) º D1/2 Þ tB0 ~ D

1/2 since F ~~ D1/2 (see eq. 2). The last scal ing tB0 ~ D

1/2, es tab lished in [20] and con firmed here, im pliesthat both heat con duc tion and ra di a tion flux ab sorp tion con trib ute the heat bud get.

From eq. (32), in view that NDHS ~ O(1) [21] (see tab. 1 and 2 too) and with Foe £

£ 1 [21], we get e Fo FoDHS e DHSeN N

21 2» - . This al lows the fuel/wa ter crit i cal tem per a ture

to be ex pressed as QB0 º N0HpdF(1 – N DHS2 Foe). This is a rea son able since from [21] we

know that N0 ~ O(1). Hence we have QB0 ~ O(1) or more pre cisely QB0 < 1. The es ti mateQB0 ~ O(1) need a def i ni tion of the wa ter tem per a ture at t = tB0. For in stance, for pure wa -ter the super heat tem per a ture is about TW–SH = 270 °C in view of ho mog e nous nu cle ationas ex plo sion on set [33, 34]. In fuel stor age tanks the bot tom wa ter is mainly due to con -den sate so im pu ri ties of any kind ex ist and can re duce the wa ter super heat tem per a tureless than TW–SH = 270 °C. Garo et al. [6, 7] re port TB0 = 373 K and T4 = 20 °C that givesQB0 = 0.335 [21]. The ex per i ments of Koseki et al. [10] pro vide QB0 = 0.432 [21]. Thesenu mer i cal val ues will be used in the wave anal y sis per formed next.

151


Wave-like analysis of the semi-analytical solutions

Thermal penetration depth and wave-like solutions

The geo met ri cal terms of eqs. (24) and (27) de fine a heat pen e tra tion depth yp asa dis tance where the sur face tem per a ture is at ten u ated e times that dif fers phys i cally fromthe ther mal pen e tra tion depth d(t) used by HBI method. Hence, from eq. (24) we haveQ = (T – T4)/(Ts – T4) = e a F

-V y a/ that through lnQ = ln(1/e) = –Vay/aF de fines the heat pen e -tra tion depth yp0 = aF/Va. With a time-de pend ent term we get ypt = aF/Va + Vat. Sim i larlyfrom eq. (27) we get:

ya

V

a

V T Tp0-non-linear

F

a

F

a F s

= +-

é

ëê

ù

ûú

ìíî

üýþ

1 ln( )

F

l 4(38)

The term in the square brack ets of eq. (38) is dimensionless, closes unity and fromeq. (38) we have yp0-non-lin ear = (aF/Va){1 + ln[(aF/Va)F/lF(Ts – T4)]} = (aF/Va)(1 + lnY) » »aF/Va. Since Y ~ O(1) be cause Y = [Fy0/lF(Ts – T4)](1/y0) » NVA/y0 º O(1) (see tab. 1 and 2).For most of fu els (tab. 1 and 2) we have aF/V » 8 mm. For fu els with y0 > 8 mm we have NVA »» (3-8)10–3 . Both eqs.(24) and (27) can be rep re sented as at ten u ated heat pulses prop a gat ingdown ward to the tank bot tom with a ve loc ity Va, namely:

Q QF

= =-

- - - -

e and ea

Fa

a

Fa

F s

F

a

V

ay V t

V

ay V t

T T

a

V

( ) ( )

( )l 4(39a, b)

The pulse mag ni tude (see eq. 39b) de pends on the dimensionless num ber Np == [F/lF(Ts – T)4](aF/Va) = Fyp0/lF(Ts – T)4 sim i lar to the ra di a tion con duc tion num ber N0[21]. Fur ther, Np = N0

~yp0 with ~ / /y a V y Np F a DHS0 0 1= = that is Np = N0/NDHS.

Koseki’s heat wave

Koseki [19] de fines ex per i men tally a ther mal wave with ve loc ity UT = y0/tB0.The val ues of QB0 com mented pre vi ously gives lnQB0(Garo) = ln0.355 = –1.093 andlnQB0(Koseki) = ln0.432 = –0.839 that are very close to ln(l/e) = –1 de fin ing the ther malpen e tra tion depth yp. For seek of sim plic ity we as sume that at t = tB0 the left-hand side ofboth eqs. (39) is QB0 = 1/e that yields:

ln ~~

QBp

Fa

BT

a F p0

0

0 00

0

0 0

1= - = -æ

èçç

ö

ø÷÷Þ = -

y

yy

V

yt

U

V

y

y y yÞ ºU yT 0

(40a, b, c)

Here ~yF = yF/y0 is the dimensionless fuel depth and ~y yw w= /y0 = 1 –

~yF where 1/(~yF y0 –

– yp0) > 0. Estimations (40b, c) confirms that UT is linearly proportional to the fuel depth, UT µ Va, which is the higher burning rates, the shorter pre-boilover times. Similarly in caseof SBC we get UT/Va = y0/[~yF y0 – yp0ln(N0/NDHS)] Þ UT º y0. Then, eqs.32 and 33 provide:

152


Fo Foe B

DHSF

p

DHS

e

DHS

F= = --æ

èçç

ö

ø÷÷

Þ ºt

t Ny

N

N N

y

y0

0

1 1 1~ ln

0

(41a, b)

Bear ing in mind the Bouguer num ber Bu = my0 and that Hp µ NDHS (see eq. 36)we get from eqs. (41b, c) a power-law Foe ~ H mp /Bu [21]. For in stance, the data of Koseki[10] scaled in [21] re veal that Foe ~ 1/Hp º 1/NDHS that gives im me di ately eqs. (41a, b).

Simplified estimation of the time to boilover

Heat conduction dominating heating

The pen e tra tion depth con cept and the as sump tion ln(1/e) » –1 al low sug gest ingthat at t = tB0 the tem per a ture at the F/W in ter face be comes Q » e. Thus, since the fueldepth de creases in time, we might sug gest that at t = tB0, we get y t tF B( )= 0 = yp » aF/Va thatgives y t tF B( )= 0 = yp = y0 – VatB0 and tB0 = (y0 – yp)/Va = y0/Va – aF/Va

2 = t0 – aF/Va2. Be sides,

with a sim ple al ge bra we get tB0 = t0(1 – 1/NDHS) that is in fact the es ti ma tion tB0 ~ 1/NDHSwe got ear lier. Ad di tion ally, tak ing into ac count that eq. (36), that is, Hp = BSANDHS/Stewe get:

tH B

tF

t

y

Bp SA

B BSte

0 0 00

011

21

1= -

æ

è

çç

ö

ø

÷÷

Þ º -æ

èç

ö

ø÷ Þ º

º

tt

0 0 0 001 1

Ft y

Dt

y

DDÞ º Þ µB B

(42)

that confirms estimations derived in [20, 21] through scaling of experimental data. Thelast expression of eq. (42) defines the Koseki’s heat wave velocities, for heat conductiondominating regime, as UT = y0/tB0 = Va[NDHS/(NDHS – 1)] upon the condition NDHS > 1.

Critical fuel thickness

The re quire ment NDHS > 1 (i. e. y0 > aF/Va) ap plied to the heat ing oil used by Garo et al. [6] de fines y0 ³ 8·10

–3 m (see data with NDHS ³ 1 in tab. 1). The same test ap plied tothe fu els of Arrai et al. [9] listed in tab. 2 pro vides as fol lows: y0 ³ 7.6 mm for to lu ene;y0 ³ 5.8 mm for ethyl ben zene; and y0 ³ 6.3 mm for n-Dec ane. These ex per i ments are gen -er ally de fined in the lit er a ture as thin-layer boil over on the ba sis of the ini tial fuel layerthick ness. Now, we de fine a prin ci ple thermophysical cri te rion dis tin guish ing thin andthick layer boil over. The same nu mer i cal test ap plied to the data of Koseki et al. [10] with a mean burn ing ve loc ity of about 3·10–5 m/s gen er ally de fine the bulk of his ex per i mentsas thick-layer boil over.

Cal cu la tions of tB0 as tB0 = t0(1 – 1/NDHS) and through eq. (42) are sum ma rized in tab. 3. The cal cu la tions are rough but re veal that the pre dicted tB0 ap proaches the ex per i -men tally de fined val ues when NDHS ³ 1.3 for the fuel of Garo [6]. In the ex per i ment ofKoseki et al. [10] as sum ing 0.7 £ tB0(ex per i men tal)/tB0 (eq. 42) £ 1.2 the range of vari a tions of

153


154


ot emi

T .3 elba

Trevolio

b s

noitaluclac el

bane

noitamrof

ni yrailixua

htiw se

ulav detci

derp

dna lat

nemire

pxe hg

uroht

s

qe,)24( .

)b ,a84(

dna ,)c74(

.feR

D]

m[y 0

]m

m[t 0

]s[V

)egareva(a 01[

2]s/

mm

NS

HD

uB

t0

B)

1(

.p

xe ]s[t

0B

24 .

qe ]s[

t

t

B0

(exp

.)

B0

eq.

()

42

t0

B

)c7

4( .qe ]s[

t

t

B0(

exp

.)

B0

eq.

c(

)47

t0

B

]s[s

qe)

b ,a8

4( .

t

t

B0(

exp

.)

B0

eqs.

a, b

()

48

]7 ,

6[

lio

gnitae

H(a

F0

1·7

78.

0 =

7–

m2

]s/)

3(

37

3.1

= etS

y 0 ³

a VF a

mm

8 =

mm

26

2 =

1– ]1

2 ,7 ,

6[

51.

0

91

00

91

1

0.0

)2(

9.1

79.

45

49

00

95

0.1

71

00

71

7.1

54.

40

38

00

78

1.1

31

00

31

5.1

04.

35

26

33

44

4.1

90

09

3.1

63.

20

54

70

27

1.2

70

07

7.0

38.

10

43

38

90.

49

93

4.3

40

04

4.0

50.

15

61

28

20.

26.

79

7.1

20

02

2.0

25.

00

94

87

0.1

00

19.

0

32.

0

71

54

51

1

10.

0)

2(

7.1

54.

40

17

63

61

1.1

51

36

31

5.1

39.

30

26

00

54

2.1

31

18

11

3.1

04.

30

35

7.2

72

49.

1

98

18

9.0

63.

20

43

57

35.

42.

98

18.

3

43

63

4.0

50.

15

21

57

66.

12.

98

4.1

33

72

3.0

68

7.0

57

65.

57

99.

05.

48

88.

0

22

81

2.0

25.

00

33

7.6

79

3.0

1.1

93

3.0

5.0

51

28

8

7

10.

0)

2(

5.1

39.

35

43

49

27

1.1

31

56

73.

10

4.3

56

24.

67

15.

1

11

74

61.

18

8.2

09

18.

85

32.

3

72

14

7.0

38.

10

93

0.2

75

2.1

38

70.

1

54

92

5.0

13.

10

78.

84

79.

07

0.8

52.

1

36

71

3.0

68

7.0

51

8.8

40

3.0

70.

85

13.

0

]0

1[

th

gil nai

barA

lio-e

durc

(aF

01·

97

6.0

= 7–m

2 )s/

)3(

30

7.1

= etS

y 0 ³

a VF a

m 6

2.2

3.0

5.3

atad

on

98.

52

16

54

1

6.0

02

85

83

3.2

78.

62

94

33

77

16.

0

96

27

02

33.

37

8.3

32

49

01

02

86

4.0

1

02

78

61

9.2

95.

81

86

72.

70

61

1.1

04

29

01

66.

36.

12

87

92

40

18

39.

0

06

00

51

0.4

43.

53

01

31

75

41

99

8.0

00

12

37

26

6.3

45

62

91

18

62

81

7.0

20

29

46

80.

38

0.9

11

46

38.

77

51

17.

0

5.3

)4(

72

50

73

8.3

42.

51

20

43

07.

85

60

16.

0

:seto

N

– )2( ;]

12[

ni detal

uclac – )

1(detl

uclac f

o srepa

p su

oirav

morf

derev

ocer atad

htiw ]

12[

ni ora

G

la te

eht

htiw

nap ral

ucric eht f

o retemai

d eht

– )4( ;]

6[ m

orf – )

3( ;.

7.2 si

nap era

uqs laer e

ht elih

w ,aera emas

´

;na

p m

7.2

UT

d

na R

VA

fo ata

d i

keso

Ks/

mm

ni detal

uclacer tu

b ,]0

1[ kr

ow la

nigir

o eht

morf era st

nemire

pxe

NDHS is 8 £ NDHS £ 54. More strong con di tions of 0.9 £ tB0(ex per i men tal)/tB0 (eq. 42) £ 1.1 de -fines a range of 8 £ NDHS £ 21. The ses dis crep an cies call for at ten tion on the change of the fuel heat ing mech a nism. The pro posed em pir i cal method pro vides rea son able times toboil over for thick fuel lay ers de fined by the con di tion y0 ³ aF/Va. In fact y0 ³ aF/Va de finesthe min i mum fuel thick ness be yond which the 1-D heat con duc tion equa tion is ap pli ca -ble. The lat ter im plies that with y0 £ aF/Va the en tire bed is heated vol u met ri cally by theab sorbed flame ra di a tion with neg li gi ble tem per a ture gra di ents.

Simplified estimation of the time toboilover – dominating radiation heating

Ne glect ing all terms with tem per a ture gra di ents in eq. (9) yields a ki netic equa -tion:

¶

¶

T

t

q

C

N N

Ny=

¢¢Þ = -

é

ëê

ù

ûú

-&s

F pF

DHSeBu

Fo Fom

rm Q 0 3

2

03 (43a, b)

with a length scale L0 = lF(Ts – T4)/ &¢¢qs , time scale t = L02 /aF, Q = (T – T4)/(Ts – T4), and

Fo3 = t/t = taF/L02 and T = T4 at t = 0.

The con di tion to ne glect the pseudo-con vec tive term (Va rFCpF)(¶T/¶y) in eq. (9)[21] with a length scale L0 is VaL0/aF n1ÞaF/Va o L0 Þ Vay0/aF n y0/L0 Þ NDHS n N0. Fur -ther with dom i nat ing source term ¶ ¢¢q r /¶y its dimensionless co ef fi cient has to be of or der ofmag ni tude 1, that is L0 » 1/m Þ lF(Ts – T4)/ &¢¢qsy0 » 1/my0 ÞN0 » Bu. Since Bu can be cal cu -lated be fore the ex per i ments (see tabs. 1-3) we get NDHS < Bu as a fi nal es ti mate. From tab.3 with av er age value of Bu about 1-1.3 we get NDHS < 1 Þ y0 < aF/Va as a fi nal es ti mate.From tab. 3 with av er age value of Bu about 1-1.3 and we get NDHS that gives y0 £ 8 mm.Hence, the ra di a tion ab sorp tion as a heat trans fer mech a nism dom i nates within lay ers withy0 £ 8 mm. Fur ther, with my £ 1 a trun cated se ries ap prox i ma tion of the ex po nent in eq.(43a) yields:

¶

¶

¶

¶

¶

¶

T

t

q

C

T

t

q

Cy

ty=

¢¢Þ »

¢¢- Þ »-

& &( )

*

s

F pF

s

F pF

em

r

m

rmm 1

Q 1

0mLy- * (44)

The dimensionless tem per a ture pro file is Q = (1/mL0 – y*)t* with y* = y/L0 and

Q = 0 at t = 0. At the boil over on set t boilover* = tB0aF/L0

2 and ac cor dance with the com mentsabove Q » 0.335 in the ex per i ments of Garo et al. [6, 9]. The tem per a ture pro file of eq.(44) gives:

tL

L yt

L

L y

L

a*

* *=

-Þ =

-Q QB B B

F0

0

00 0

0

0

02

1 1

m

m

m

m(45a, b)

Fur ther, the con di tion L0 » 1/m Þ N0 » Bu sim pli fies the so lu tion and with y0/L0 == N0 » Bu we read:

ty

y

L

y

y

at

yB B

FB0 0

0

02

02

2

02

00

1

1

0 335

0 4

1=

-

æ

è

çç

ö

ø

÷÷

Þ =Q.

. m

æ

èçç

ö

ø÷÷

2

02y

aF(46a, b)

155


The rel a tive thick ness of the re sid ual fuel layer is about D/y0 = (1 – y)/y0 » 0.4 asan av er age value es ti mated from the ex per i ments of Garo et al. [6, 9] and cal cu lated in[21] that is 1 – y/y0 » 0.6:

ty

at

V y

a

yB

FB

a

FBu Bu0

202

0

20 0084

1084

1=

æ

èç

ö

ø÷ Þ =

æ

èç

ö

ø÷. .

VN

aDHS

BuÞ

æ

èç

ö

ø÷084

12

0. t (47a, b, c)

The nu mer i cal pre-fac tor in eq. (47) 0.84 ® 1 and we may as sume it, by con ven -tion, as equal to unity. Hence, we get ap prox i mately:

ty

at NB

FB DHS

Buor

Bu0

202

0

2

01 1

=æ

èç

ö

ø÷ =

æ

èç

ö

ø÷ t (48a, b)

In ac cor dance with (47c) the time to boil over is com pletely pre dict able bymac ro scopic vari able known be fore the ex per i ments with the only dis crim i nat ing con -di tion NDHS < 1 (i. e. y0 < aF/Va). The pre dicted val ues of a tB0 by eq. (47c) and eqs.(48a, b) are sum ma rized in tab. 3. In fact eqs. (47b, c) de fines the Koseki’s heat waveve loc ity in dom i nat ing ra di a tion re gime, namely: UT = y0/tB0 = 1.2 Bu

2(aF/y0) and UT == y0/tB0 = 1.2Bu

2(Va/NDHS).The cal cu lated val ues strongly re veal that around the crit i cal fuel thick ness de -

fined by y0 = aF/Va there is a zone of large dis crep ancy where both heat con duc tion and ra -di a tion ab sorp tion take place in equal or der of mag ni tude. As the fuel layer grows to wardthicker to depths ap prox i mately (2-2.5)(aF/Va) the ra tio TB0(exp)/tB0(eq. 42) ap proaches 1.Sim i larly in case of ra di a tion dom i nat ing re gime the ra tios TB0(exp)/tB0(eq. 47c) orTB0(exp)/tB0(eq. 48a,b) are about 1 with fuel lay ers of depths about 0.5(aF/Va). These ap prox i -mate lim its fig ure the width of the zone where both heat trans fer mech a nism take placeand for mu lates a new prob lem be yond the scope of the pres ent work.

Conclusions

HBI method and scal ing anal y sis of both the ini tial gov ern ing equa tions and theso lu tions were performed thor oughly through the text de vel op ment but some prin ci ple is -sues can be out lined again.(1) The initial analysis formulated two principle Stefan problems pertinent to pool fire

boilover. The inverse Stefan problem only in absence of CML simplifies the analysisand allows demonstrating the power of the HBI method. The closed form solutionsallow scaling and provide results directly applicable to the time to boilover.

(2) The wave-like forms of the solutions permits to evaluate the Koseki’s heat wavevelocity as a part of the rate controlling group pertinent to evolution of thetemperature profiles across the burning fuel layer. Koseki’s heat wave velocity can beexpressed through dimensionless independent variables known before the experiment that is in fact a theoretical derivation of this parameter pertinent to boilover andintroduced intuitively in [10].

156


(3) The approximate analysis based on the heat penetration depth led to reasonableresults in calculations of the time to boilover in two distinguished heating regimes:dominating heat conduction and dominating heat radiation absorption. The principleresults are the critical fuel depth y0 = aF/Va distinguishing the thin-layer boilover andthick-layer counterparts. It is based on fuel physical properties pertinent to their heat(heat diffusivity) and mass transfer characteristics (burning rate) and avoid anygeometrically based definitions.

Acknowledgments

The work was sup ported by the au thor’s home uni ver sity (UCTM) through a re -search grant NN2/60-2007.

Nomenclature

aF – thermal diffusivity of fuel, [m2s–1]

BF – dimensionless flux defined by eq. (18)Cp – specific heat of fame gases, [Jkg

–1K]CpF – heat capacity of fuel, [Jkg

–1K]D – pool (tank) diameter, [m]F – surface density of the flux, [Wm–2s–1] (a symbol used to replace &¢¢q in the formulae for

– clarity of expression)g – gravity acceleration, [ms–2]Hv – the latent heat of vaporization, [Jkg]h – fuel depth (in coordinates assumed h = ys), [m]& ¢¢m – mass burning rate per unit surface area of the pool ( & ¢¢m = rFVa), [kgm

–2s–1]Q – total heat accumulated by the layer over a limited time interval, [W]&¢¢qc – heat conduction flux from the interface toward the fuel depth, [Wm

2s–1]&¢¢q r – volumetric heat sources (eqs. 4a,b), [Wm

3s–1]&¢¢qs – surface density of the flux, [Wm

2s–1] r(t) – surface regression rate (in the present context, r(t) = Va), [ms

–1]T – temperature, [K]Tf – flame temperature, [K]Ts – vaporization temperature of the fuel, [K]T4 – temperature of the environment undisturbed by the heat release from the flame, [K]T4

f – temperature of the fire environments, i. e. the air, [K]t – time, [s]th – time required the thermal penetration layer d to reach the bottom (t = th at d = h), [s]t0 – time scale (t0 = y0

2/aF), [s]tB0 – time of boilover onset, [s]y – vertical coordinate, [m]yF – fuel depth, [m]yp – heat penetration depth, [m]ys – current position of the flaming interface moving with a velocity Va, [m]yw – water depth, [m]

157


y0 – initial fuel-water layer depth (y0 = h t=0), [m]UT – Koseki’s heat wave velocity expressed as UT = h/tB0 or UT = y0/tB0, [ms

–1]U0 – hypothetic velocity (U0 = y0/t0 = aF/y0), [ms

–1]Va – velocity of the interphase interface motion (fuel regression rate), [ms

–1]VF – convective velocity in the fuel, [ms

–1]Vs – velocity of propagation of the convection mixing layer (hot zone), [ms

–1]

Greek letters

d – thermal penetration depth in accordance with the HBI assumption, [m]Q – dimensionless temperatureQB0 – dimensionless temperature at the boilover onsetq – heat balance integral (HBI)lF – thermal conductivity of the fuel, [Wm

–1K–1]m – effective average radiation absorption (or extinction) coefficient, [m–1] rF – density of fuel, [kgm

–3]rw – density of water, [kgm

–3]r4 – density of fire environments (i. e. the surrounding air), [kgm

–3]t0 – hypothetic time for complete burning of the fuel layer without boilover, (t0 = y0/Va), [s]F – flux released at the surface (eq. 16) due to radiation and phase change, [Wm–2s]c – radiative fraction of the radiation feedback to the pool surface), [–]

Dimensionless groups

Bu – Bouger number, [–]BSA = &¢¢qsy0/lF(Ts – T4) – dimensionless numberFo = y0t/aF – Fourier number, [–]Foe = y0tb0/aF – Fourier number at the boilover onsetHp = &¢¢q / & ¢¢m HV – dimensionless numberN0 = &¢¢qsy0/lF(Ts – T4) – dimensionless group (radiation-conduction number)NDHS = y0r(t)/aF = y0Va/aF – dimensionless group (pseudo Peclet number)Ste = CpF(Ts – T4)/HV – Stefan number, [–]

Subscripts

c – conductivityF – fuelf – flames – surfacev – vapourw – waterW– SH – water superheat4 – ambient conditions

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158


Superscripts

f – flame

Special symbols

µ – proportional toº – order of magnitude~ – about equal

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Author's address:

J. HristovDepartment of Chemical Engineering, University of Chemical Technology and Metallurgy8, Kl. Ohridsky blvd., 1756 Sofia, Bulgaria

E-mail: [email protected], [email protected]., [email protected]

Paper submitted: December 15, 2006Paper revised: April 20, 2007Paper accepted: May 5, 2007

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AN INVERSE STEFAN PROBLEM RELEVANT TO BOILOVER: Heat ...€¦ · Key words: fire, boilover, Stefan problems, heat balance integral, traveling wave-like solution, Koseki’s thermal