Two special problems concerning the combustion of liquids with a free surface, i.e., flame quenching during the mixing of a burning liquid inside a container and liquid burnout from a porous layer, are analyzed using a quasi-one-dimensional model. The critical parameters corresponding to the quenching of a burning fluid with a free surface are determined. Determinations are also made of the limiting pressure gradients corresponding to the transition from the combustion mode where the liquid evaporates from the surface of a porous layer to the mode where the phase transition surface lies inside the porous layer

Milkov, S. N.

Sukhov, G. S.

Yarin, L. P.

English

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NASA TECHNICAL MEMORANDUM
	
NASA TM-88459
COMBUSTION THEORY FOR LIQUIDS WITH :A FREE FURFACE
III. SPECIAL PROBLEMS
S. N. Mi1'kov, G. S. Sukhov, L. P. Yarin
Translation  of "Teoriya goreniya zhidkostey so svobodnoy
poverkhnost'yu. III. Spets-.al l nyye zadachi," Fizika goreniya i
vzryva, No. 3, 1985, pp. 3-8 (A85-46216).
(NASA-TM-88459) COMBUSTION TELORY FOR	 N86-29011
LIQUIDS WITH A IEEE SURFACE. 3» SPECIAL
PROBLEMS (National Aeronautics and Space
Administ otion) 15 p 	 CSCL 21B	 Unclas
G3/25 43480
NATIONAL AERONAU'.'ICS AND SPACE ADMINISTRATION
..	 WASHINGTON, D.C. 20546 	 JUNE, 1986
r
t
STANDARD TITLE PAGE
1.	 Report No,
NASA TM-88459
2.	 Government Accession No, 3,	 Recipient's Catalog No.
4.	 Title and Subtitle
COMBUSTION THEORY FOR LIQUIDS WITH A
5.	 Report Dote
June 1986
6.	 Performing Organisation CodeFRBE SURFACE III - SPECIA I. PROBL EMS
7.	 Authors) 8.	 Performing Organisation Report No.
S.	 N. Mil' kov; C.	 S.	 Sukhov; L.	 P.	 Yarin 10.	 Work Unit No.
11.	 Contract or Grant No.
9.	 Performing Orgonlsatlon Nome and Address NASW-4006
The Corporate Word
1102 Arrott Building 13,	 T YP• of Report and Period Covered
Pittsburgh, PA	 15222 Translation
12.	 Sponsoring Agency Name and Address
National Aeronautics and Space Administration
Washington, D.C.	 20546
Etfng Agency Code
15.	 Supplementary Notes
Translation of "Teoriya goreniya zhidkostey so svobodnoy poverkhnost'yu
III.	 - Spetsial'nyye zadachi," IN; Fizika goreniya i vzryva, No.	 3,
I
1985, pp,	 3-8.	 (A85-46216)
16. " Abstract
Two special problems concerning the combustion of liquids with a
free surface, i.e., flame quenching during the mixing of a burning
liquid inside a container and liquid burnout from a porous layer, are
analyzed using a quasi-one-dimensional 'model.
	
The critical parameters
corresponding to the quenching of a buriing fluid with a free surface
are determined.	 Determinations are also, made of the limiting pressure
gradients corresponding tc the transition from the combustion mode where
the liquid evaporates from^the surface of a porous layer to the mode
where the phase transition surface lies inside the porous layer.
17.	 Key Words (Selected by Author(%)) 18.	 Distribution Statement
U.S. Government Agencies and
Contractors only
19.	 Security Classif. (of this report) 20.	 Security Classif, (of this page) 21. No. of Pages 22.	 Priaa
Unclassified Unclassified 13
2
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COMBUSTION THEORY FOR LIQUIDS WITH A FREE SURFACE
111. SPECIAL PROBLEMS
S. K. Mil'kov, G. S. Sukhov, L. P. Yarin
The quasi-one-dimensional theory developed in [1,2] can be
	 /3*
used to solve a broad group of problems related to combustion
of liquids with a free surface. We will illustrate this with
examples of approximate calculation of two processes w1lich are
interesting in tera.:i of technical application: extinguishing a
flame when a liquid in a container burns and when a liquid from
a porous layer burns.
A flame above a free surface can be extinguished by rapid
mixing [3). The mechanism of this process involves the
following. As the liquid is mixed, the free surface's
temperature and, therefore, evaporation rate drop. If heat
removal from the flame is nearly constant, the decrr^ase in flow
of vapor to the front reduces the reaction rate and,
u],timatel,y, results in extinction -- critical heat exchange
conditions (4).
To find the values of parameters corresponding to
extinction, we will use a one-dimensional model of a flame
(fig. 1) and a quasi-heterogenous diagram of a flame front
[5]. Let us write the equations for material and thermal
balance at the front
G,y
—o — Ga +o = W01,	 (1)
q(G[b —o "" Gs +o) — Q(pto — Qa —o,	 (2)
where G 0 _ 0 and G^+O are, respectively, diffused flows of
vapors approaching the flame front and departing from it;
*Numbers in the margin indicate pagination of the foreign text.
3
r°^ I
ii
1
0
^s
To
r'0
l2
TM
Qo_ 0 and Q#+0, thermal flows at the front; TV4,—ap^cbbX
Xexp(-E/Rxm) , the rate of effective quasihet erogenous reaction.*
Fig. 1. Temperature
distribution during
combustion of a liquid
with a free surface.
la , b	 Gas medium;
2 - Reacting Liquid;
3 - Substrate.
Flows incorporated into equations (1) and (2) can be found
by solving the system of quasi-one-dimensional transfer
equations followed by calculation. of the derivatives according
to known temperature and concentration profiles fl]. To
simplify the computations, in future approximation equations
for diffusion and thermal flows we will use
dCb l	 C^ --- C^^
00-0--   pap 
	
PiD 
Gcb+o = PiD
 d )41+0 P,D bcb(D	
(4)
Qo+o	 (—a—-0
dr, 1
	,.,	
TID " T o
dT_ T^ — Tl*
QO-O ^a d Ito ^" X1 	 x	 (6)
ii
/4
kX is the nominal thickness of the layer of oxidizer above the
free surface, at whose boundary c ^c .
a apo
Let us introduce the coefficient for vapor combustion
*The remaining values in (1) and (20) and subsequent equations,
as well as subscripts, are taken from [1,2].
efficiency
q, — 1 ~ CbW/GDM,
	 (7)
where cb--^c,o(i-I to/ta) is vapor concentration at section t-4
if there is no combustion.
Substituting equations (3) and (4) into (1) with regard for
(7) yields
r^	
eYla^ r,
	
(8)
T — z/D • to (ta -- tw^ 4a3 e,—,RWE.
Considering to"tai , we obtain. T L ztm/D. . Assuming that
parameter A"% !ff, during steady combustion until flame-out, we
obtain an approximated equation for tS (1) .
D!/r 1 st+ , in (1 + ;
Where P . cbo; i --Qlca *- Using this relationship and Clapeyron-
Clausius' equation	 —S xe^ '{—f o ,^ I/ bEf3^), we express T as an
exponential function of the free surface's temperature
t
T Co[ 1—x exp (~qa /C3o)]-'/
'C, . Z I vac	 1 In (t + ^)r Qa c 4IIRIERb.
Transforming (2) using (3) - (7) , we obtain*
	
^e ^ 60) cPl E.	 (9)
^2 °
	 Cb.gR
a the scalar concentration of vapor e bm is sufficiently
i
deriving (9), it was assumed that i,e=1.
5
' 
tF
,l 1
":..
	
..a.Y`_}.. iRT.	 ^,	 1
close to 0 b0 we can, assuming that
co,„ ^V x oxp (-- WE)o),	
(10)
give (9) the following form:
i	 S
where ? — 9Rx1cpjE-
To determice the free surface's temperature CE O , one must
use the equation for teat balance at the phase interface. Heat
flow incorporated into these equations reflects both heat
release from gas to liquid and transfer of heat by thermal
conductivity and convection. in the liquid. The solution to
	
35
the problem includes calculation of transfer processes in the
gas and liquid if there is :Forced convection (mixing) in the
latter. However, to analyze the effect of mixing on the heat
mode of combustion, there is no need to solve such a
complicated problem. It can be limited merely to considering
the relationship between combustion temperature (reaction rate)
and free surface temperature, since physical considerations
make it clear that mixing the liquid results in a reduction in
100.
t
p	 Oo	 vCP
Fig. 2. - diagram
(opl is heat release,
q, 2 is heat removal) .
..^	
YID	 ^ '.-,
t.I
la^
rip
0 fb'
50 z
Fig. 3. Temperature distribution
during combustion of a liquid from a
porous layex.
1 - Gas phase; 2 -,Porous substance
(lb', region of vapor filtration
through the porous layer).
Therefore, regarding 90 as a parameter, let us consider
the path of curves f 1 (p.+) and T 2 {(W in variables ap-$ (fig.
2) . The function 
c"2 (() f ) is descri bed by an S-curve starting
at origin and, as ®t", having an asymptotegp =l. increasing
® o (acclerating mixing in the gas ;flame) shifts the curve
a l (6^) to origin. The function T2 ( ()0 ) is 7.inear„ A change
in I00 shifts it along the x-axis and abruptly changes its
slope. This reflects two consequences of the change in free
surface temperature: a change in heat removal from the front,
which is proportional to 4-o p , and vapor concentration at
the phase interface, which is an exponential function of 00.
Figure 2 clearly shows that a reduction in 8Q causes
combustion temperature to drop and vapor concentration at the
flame front to rise. At a certain critical A B , curves
°p1 (00) and cp2 (v  j lose their common point at the upper
temperature level, which corresponds to flame-out (point rl) .
Using (8) and (11) , we will find the critical free surface
temperature. Assuming that curves op l (QO) andC'2 (e^) are
tangent, we derive an equation which relates critical
,mbustion efficiency and critical temperature
ze^ (12)
Equating the right sides of (11) and (12), we find critical 	 {
,a
7
'W" -
temperature as a function of free surface temperature and
adjusted calorific value q:
q + 280
 oxp C 
ho !
/T q +2®o exp ^u !	 q eo oxpo id	 _ 	 ..4g
L 4 •}' exp ( in/1660T	 o
9 + exp \ eo
The minus sign before the radical in (13) represents	 L6
combustion; the plus sign, extinction. From (8) and (11), we
have
( o)	 1l9y 	 (14)
Equality (14) is va]id for any steady state, including critical.
Solving the system of equations (13) and (14) , one can find
the critical free surface temperature E)Ocr as a .function of
assigned parameters q and qn.
Let us consider the problem of combustion of a liquid from
the porous layer of a substance. Experiments show that two
different liquid combustion modes can be employed, depending on
the system's hydraulic characteristics [6]. Given low porous
layer hydraulic resistance (high medium porosity, low reacting
liquid viscosity), the filtration supply mechanism meets the
requirements of the reaction. Equilibrium is established in
the system, and the evaporation front is localized at the
layer's surface. If the medium's permeability diminishes, the
equilibrium is disrupted, and the evaporation front shifts to
within the layer. Its outer surface heats to a temperature
above the liquid's boiling point. Therefore, the liquid- and
gas-phase flow field can be divided into four typical zones
(fig. 3). In the first, 2, the reactive liquid, exposed to
1
I
III
(11)
T-Tr,_ 7: 7T, 3777
capillary forces and pressure applied from ouLa I. e_r is filtered
through a layer of porous substance of thickness ►Vhe
liquid warms due to heat supplied to the porous substance from
the flame front, in region lb", vapor formed during
evaporation is filtered through a layer of porous substance of
thickness 46 1 . in zones lb and la, the vapor mixes with the
oxidizer found in the environment, and the components interact
chemically.
in a quasi-one-dimensional approximation, diffusion
combustion of a liquid filtered through a layer of porous
substance is written as follows in a systera related to the
boiling front*
de
	
ePLUI 
'dT .
pjD  _' acc
for the region tai<t<+-,
	
dc	 d C
PLUI -T, PAD TV,
d T,	 d
2 
TI
' PIUICVI	 - I
R,U	 dpf
T'—V I — — 
w , . P, P'I - I;b T•
for the region 0 < t < t6t, 11
	
dT	 d 
277
	
MNMPI^ 	 ^2 '2
dt'
	
42	 dp
	
K02	 W
for the region t6a<t <0, where m and ko are porosity and the
*At Le=l, one diffusion equation is sufficient to describe
heat- and mass-transfer processes in the gas above the porous
layer.
(15)
(16)
(17)
(18)
(19)
(20)
9
MOT 7
	
171.
filtration factor] ^' 1
 and ^ 2 , effective coefficients of
Cher-mal conductivity for the filled porous medium in zones lbI
and 2
Boundary conditions which are taken into account during
integration of system (15) 4 (20) take the form
t -+ + oo: T --k T 1 *, c -j- 0, (2 1 )
t — to .: T • To, c - 6a.,
t— to. T	 T6 1 ' P — Pi *r,
t	 0: ' — Tr a^• car
t
	 t62- T— Ts*, P "' P2*-
Balance equations reflecting the preservation of heat and
substance flows toward typical surfaces should be added to
conditions (21):
dri
	 dTj o-O X= qpD ( 
de6	 (22)
(dT,	 r 
(iT ) &-O
dT' 	 (23)fox, 
^l dj A!+o
	
"
(dT,	 (dT2)
_o
^'^i \d^ /+o ^2 d 	° 4aPaUz ►nr	 ( 2 4)
	
( drb
 
7t ) 0 PIt4I p2u2m"'	 (25)
5x ex C -^ u	 ,P	 To/T .1*
(26)
However, without fully solving the problem, one can identify
certain essential features of this process on the basis of the
simplest possible model.* For example, it is of practical
*Integrating system (15)-(20) does not involve any particular
difficulties.
Z
f•
interest to evaluate limit pressure drop Apap o* ps* --pi*,.
at which the evaporation front shifts inward into the porous
layer. To .find it, assuming	 T*>Toi 10 — To, ,ko,mkoa=ko,
k 4)61+0	 61-0	
61
d7't	 dxi}rat 'a-'—►,—x- \dd,s ^x 	 to y'' tat
(h Is the depth of the porous layer; T..21 the liquid's
boiling point), we obtain a system of three algebraic equations
ro	 o r61 •- rt«	 (27)
°--^ 6 L 	 •— ^.g
', —tot 	
^^p2U,znio.	 (28 )
	
46i/pi +
is 
-x 6,
	 ( 2 9)
which define T , is , and u 2 as a function of thermal,
filtration, and geometric characteristics of the subject
system.. Specifically, solving (27)-(29) relative to 6p, we find
Qp.^. 6^i ` 
rl p'
.	
./%, .- —%a
rH? 	
^.1 VaPIp2k, nE	 T	 !& -- 46i
introducing C,^,=O into OW, we obtain the desired equation,
which relates limit values for parameters representing the
transition of combustion from a rode in which the liquid
evaporates from the porous material's surface to a mode where
the surface of the phase transition lies within the porous
substance layer
epnp .
h
 ^ m^ Xi T4,
_ ^2 T H2 --T )
	
m,^	 h
Assuming that the layer of porous substance is sufficiently
thick and the thermal effect of the chemical reaction is high
(T,lT1# ^fl, a T„2 —~•Tz* ,	 we obtain the following
	
dpnn -^ a,Ja/mkmi* ` A,:%5z • 0/007 l	 (32)
where Pn: p,/p:; 'a,==%j/cp,p,• Equation (32) shows that, all other
(30)
(31)
11
E
sl
conditions being equal, the limit gradient is governed by the
ratio of reaction heat to vapor-formation heat.
Using da ta crom r 6l r we can approximate &PAPp , which
corresponds to the movement of the boiling front to the outer
surface of the porous layer. For example, for kerosene in
sodium chloride (h=120 mm, d-16 mm, rn=0.4), the transition from
filte.ed combustion (boiling front within the layer) to free
combustion (evaporation from the layer's surface) occurs when
the grain cross section changes from 10 to 340 microns. The
values kO6p=0.15'l0_ 2 and 2.16 ` 10 -5 , which correspond
to these conditions, also determine the range of possible 1.4imit
values koAPap * Calculation with (32) and data from [3j over
the length OL the flame yields koAP.-P =0.09*10- 5 m'2,/sec,
which is close to the lower limit of this range.
12
rRBFERRNC4S
14 M l'kov, S. N., Sukhov, G. S., Xarin, L. P., FGV [Pizika
goreniya i vzri.va], 1985, 21, 1.
2. Mil'kov, S. N., Sukhov, G. S., Yarin, L. P., FGV, 1985, 21,
2.
3. Slinov, V. z., Khudyakov, G. N., Diffuzionnoye goreni,ye
zhidkostey [Diffusion Combustion of Liquids]. Moscow:
USSR Academy of Sciences Press, 1961.
4. Vulis, L. A., Teplovoy rezhim goreni.ya (Thermal Conditions
for Combustion]	 Moscow; Gosenergoizdat Press, 1959.
5. Vulis, L. A., Yershin, Sh. A., Yarin, L. P., 4snovy teorii
gazovogo fakela [Gas Flame Theory]. Leningrad: Lnergiya
Press, 1968.
6. Bakhman, N. N., Aldabayev, L. T., Sabinsk y, N. D., in
Goreni.ye i vzryv [Combustion and Explosion]. Moscow:
Nauka Press, 1977.
13


Combustion theory for liquids with a free surface. 3: Special problems

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