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Mathematical solutions for explosions in spherical vessels
C O M B U S T I O N A N D F L A M E 26, 201-217 (1976)
201
Mathematical Solutions for Explosions in Spherical Vessels DEREK BRADLEY and ALAN MITCHESON Mechanical Engineering Department, University of Leeds, Leeds, LS2 9JT.
Equations, assumptions and previous solutions for central ignition of premixed gases in closed spherical vessels are reviewed. Three new categories of solution are presented: the Approximate Computer Solution (Case 1), the Dimensionless Universal Expression (Case 2), and the Complete Computer Solution (Case 3). Case 3 is the most accurate and shows the importance of accurate values of burning velocity as the unburnt gas is compressed in the course of the explosion. Recommendations are made concerning such values for methane-air mixtures. Case 2 solutions do not require a computer and they appear to be of good accuracy. During the explosions studied dimensionless pressure, Pr, is proportional to dimensioness time, t, raised to the power 4.5.
1. Introduction The experimental measurement of pressure in closed vessel explosions was one of the earliest areas of combustion research. Possibly the Rev. W. Cecil in 1820 [1] was the first to o b t a i n a c r u d e i n d i c a t i o n o f the maximum explosion pressure of a hydrogen-mr mixture, by making explosions rupture strings of known breaking strength. L a t e r d e v e l o p m e n t s were related to explosion h a z a r d s and c o m b u s t i o n in i n t e r n a l combustion engines and much experimental data have been accumulated. Theoretical analyses were developed which related burning velocity, flame radius and pressure. T h e m o s t c o n v e n i e n t starting point for theoretical analyses is laminar combustion of a quiescent, uniform, premixed gas with central ignition in a closed sphere, to the walls o f w h i c h t h e r e is no e n e r g y loss. In the course of much work by many different researchers, further assumptions have been inv o k e d to enable practical solutions o f the basic equations to be obtained, and some of these are discussed in the present study. A simple model is presented which is amenable to computer simulation and this is used to derive pressure-time curves in dimensionless form which are universally applicable for all explosions,
From these curves it is clearly seen that corresponding dimensional curves are v e r y sensitive to the value of burning velocity. Both t e m p e r a t u r e and pressure o f the unburnt gas are changing during the explosion and d a t a are r e q u i r e d for the a s s o c i a t e d changes in burning velocity. Some data of this type are available from the experimental work of Babkin and K o z a c h e n k o [2]. Their data are recalculated in the present work, making allowance for the effects of flame thickness, and are presented in the form of an equation which relates the burning velocity of a methane-air mixture to both temperature and pressure. With the availability o f such data it has b e e n worthwhile to d e v e l o p a m u c h more accurate computer model based upon finite difference techniques and less restrictive assumptions than formerly used. Some results o f s u c h c o m p u t a t i o n s are p r e s e n t e d and c o m p a r e d with t h o s e f r o m the s i m p l e r model. T h e d e v e l o p m e n t o f the a c c u r a t e model is seen also as a first step in the simulation of vented explosions. 2. The Basic Equations It is assumed that central ignition occurs in a rigid sphere and that a laminar flame that is smooth and spherical, propagates outwards
Copyright © 1976 by The Combustion Institute Published by American Elsevier PublishingCompany, Inc.
202
D. BRADLEY and A. MITCHESON
without any significant movement due to natural c o n v e c t i o n . The p r e s s u r e , P, is equalised throughout the vessel [3,4] and the unburnt gas is isotropic. Mass conservation gives: mo = m u + m b , dmu_-dm dt
b
(I)
(2)
dt
(8)
d (Pbr3b) = 3r2p, S ,
dt
b , ,
where Pb is the mean density of the gas within the radius rb. Its accurate evaluation presents severe problems. The further d e v e l o p m e n t of these equations has led to the postulation of several simplifying assumptions, which are accompanied by limitations in the accuracy of derived results.
in which m is the mass of gas and subscripts u, b and o indicate the unburnt and burnt gas
states at time t, and the initial reference state at time t equal to zero, respectively. Define n, the burnt gas fraction as:
3. Simplified Analyses It might be assumed that the unburnt gas is compressed isentropically according to the law:
mb
n=
(3) mo
.Tu Pou
=
a constant
(9)
It follows from the definition of burning velocity, Su, that: dm dt
'
u _ -47rr~PuSuu
(4)
where rb is the radius of the inner boundary of the unburnt gas of density Ou. If V represents volume, then volume conservation gives: Vo =
+ Vb
(5)
The equation of state is: P V = mRgT,
in which 3' is the ratio of specific heats. Equations (7) and (9) yield:
drb Su - - d ~
where R is the vessel radius. Burnt gas conservation yields:
3r2b TuP
dt
(10) '
(6)
in which R g is the mass basis gas constant and T is the absolute temperature. Equation (4) yields: dt
dP
a result derived by Fiock and Marvin [5]. Lewis and yon Elbe [6] made several other assumptions and deduced that the fractional pressure rise is proportional to the fractional mass burnt: P-P
d [ ( R a _ r g ) ~ ] =_3r2bPuS
(R3 - r~ )
(7)
_m b _
Pe-Po /720
H.
(11)
Subscript " e " indicates the condition when the explosion is completed. This equality simplifies the equations considerably and is of great utility. The assumptions upon which it rests cannot be regarded as entirely valid, but the complete computer solutions pre-
EXPLOSIONS IN SPHERICAL VESSELS
203
sented in Section 5.2 enable the validity of this useful expression to be assessed. 3.1 Case 1: The Approximate Computer Solution
Pressure-time curves were obtained by increasing " n " in elemental steps from 0 to 1 on the I.C.L 1906A Computer at Leeds University. The computation procedure will be appreciated from the following equations. From Eq. (11) e=n(Pe-Po)+P o •
(12)
From Eqs. (6) and (9)
A pressure-time solution derived from these equations is presented in Section 6, where it is compared with the corresponding results from the complete computer solution.
3.2 Case 2: The Dimensionless Universal Expression This attempts to obtain the maximum generality in the pressure-time relationship. Previous assumptions are utilised and dimensionless relationships developed. Equations (1), (3), (4), (9), (ll) and (16) yield:
dt
Tu - 1 7u
m u = mo(1 - n ) .
(14)
From Eqs. (5) and (6)
4 rrr~ = 4 r r R a - m u R g u Tu 3 3 P
RPo
(13)
From Eqs. (1) and (3)
(18)
This equation was derived by Benson and Burgoyne [7] and also Wilson [8], by a slightly different method. The former workers obtained solutions for cetane-air and Wilson assumed a burning velocity relationship for pentane-air which gave rise to solutions in poor agreement with experiment. Dimensionless pressures and times are defined in terms of conditions at the end of the explosion:
(15) P=P/P
3m u t012 --
lieu Re _ e_] 2/3
de _ 3supu (e e _eo)
,
(19)
(20)
(16)
47r(R a - r~) " t = t/t.
From Eq. (4)
(21)
A mean flame speed, Sin, is defined by: dt -
-dm
u 4 7rr2bPuS u
(17) s m = Rit e .
(22)
D. BRADLEY and A. MITCHESON
204 In non dimensional form Eq. (18) becomes
Where S, = drddt, the flame speed. Eqs. (1), (9) and (11) give:
1/~u drd/~-
)3Su S (1 m-fi°)("o~~-
1 -
1 -
(23)
The ratio R/rb can be eliminated from Eq. (26) using Eq. (27) and it is seen that, for given values of 3'= and Po, the ratio S,/S= is a function of P. That is: dr b at - Ss = Su f(ff)
This equation has been solved using finite difference techniques, for different values of SJS,, and Po. Such solutions are presented in Figs. la, b and c, in the form of curves relating reduced pressure, i~r ( = (P - Po) (Pe .p,)-l) to dimensionless time, T. The generality in these curves is further extended by that also exhibited in the ratio SdS=, as now is demonstrated. dP/dt in Eq. (10) can be evaluated using Eq. (11). Differentiation of the latter gives:
d P _ dn (Pc" Po)" dt dt
(24)
(27)
(28)
and te
fo
dr b R - dt = fo S u f d ) "
(29)
If S= is assumed to be constant throughout the explosion, then:
teSu _ S U _ f o 1 d(rb/R __ ) R Sm f(p )
(30)
Equations (1), (4), (9) and (14) yield:
dt
-
rb
n-l+ (V)
(25)
and substitution of Eqs. (19), (20), (24) and (25) into Eq. (10) gives:
7uP
rb~ - 1
(26)
Here Sm is a mean flame speed, Rite, and the equation can be evaluated with the aid of Eqs. (26) and (27). It is clear that SJSm has a unique value that depends only upon the values of y= and Po and that this is otherwise independent of mixture composition. Computerised numerical integration has yielded the following values of SJS=: 0.244 for (po)-i = 8.99; 0.290 for (Po)-I = 7.00 and 0.365 for (Po)-' = 5.00. In each case a value of 1.36 was taken as a mean value for 3'= for a stoichiometric methane-air mixture during isentropic compression from ambient conditions. Values of S=/S= for other conditions can be obtained very readily. These values explain the ranges of values used in the c o m p u t e d dimensionless p r e s s u r e - t i m e curves shown in Figs. la, b and c.
EXPLOSIONS IN SPHERICAL VESSELS
205
1.0
1-0
t
0.8
0.8
Pr
0.6
0.6
0.4 _
_o°:I
0.25\
0.4
0.2
0.2
0.2
0-4
0.6
0.8
1.0
0
0.2
0.4
(a)
/ ////
0.8 0.25,, 0.3.,," 035\
0-6
0.2
0
0.4
I s
I
04
REE 10\
/
R ,
j #
t /
0.z;
0.6 T
1.0
//
0.2
/
0.2
1.0
O.E
0.4,,,\ \
0.4
0.8 T---~
(b)
1.0
I
0.6
0.8
1.0
~
0
iIII/ .I 0.2
~ E q . I(11
I --
0.4
i
I i I J 0.6
0.8
1.0
n ~
(c) . (d) Fig. 1 (a)-(c). Dimensionless pressure-time curves for Universal Expression with y= = 1.36. (a) (Po)-1 = 8.99 (marked points for Case 3 solution); (b) (Po)-I = 7; (c) (Po)-1= 5; (d) Dimensionless pressure-fractional mass burnt curve. Circled points are complete computer solution, Po = 1.01325 × 10~N m-~, To = 298 °K, vessel radius = 0.1 m. Such c u r v e s c a n be o f p r a c t i c a l utility. F r o m them, there is no difficulty in deriving actual pressure-time curves for particular c o n d i t i o n s . First, the m a x i m u m e x p l o s i o n pressure for the mixture should be found [6] and Po evaluated. The value of SJSm should then be determined and from a knowledge of S=, Sra, (=R/te), is derived. F o r a given vessel radius this gives te, which gives the time scale for the explosion. The p r o b l e m presented by the fact that S= is not c o n s t a n t but varies during the c o m pression process a c c o m p a n y i n g combustion,
might be c i r c u m v e n t e d by the selection of an appropriate mean value of Su for the whole e x p l o s i o n . G u i d a n c e as to w h i c h value to select is obtained f r o m a c o m p a r i s o n o f the p r e s e n t Case 2 solutions with the C a s e 3, c o m p l e t e c o m p u t e r solutions, p r e s e n t e d in Section 6. 3.3 Other Approximate Solutions
Before proceeding to the complete c o m p u t e r solution it is o p p o r t u n e at this stage to summarise some of the other solutions and their u n d e r l y i n g a s s u m p t i o n s . T h e simplicity o f
206
D. BRADLEY and A. MITCHESON
the Case 1 and Case 2 solutions rests upon the assumed equality of the fractional pressure rise and the fractional mass burnt, Eq. (11). An alternative route involves a knowledge of the changes in the mean density of the burnt gas, Pb. This is not easy because of the non uniformity of the burnt gas temperature; the gas burnt first attains a t e m p e r a t u r e higher than that burnt later, a fact demonstrated in 1906 by H o p k i n s o n [9]. Some workers have nevertheless made an assumption of a global isentropic compression law, namely:
PPb -7b = constant.
(31)
Perlee, Fuller and Saul [10] utilised Eqs. (31) and (9), with values of Yb and y , calculated from polynomials in T and P. The temperature of the burnt gas was assumed to be that at the centre of the vessel. This overestimation of temperature was compensated by additionally using a value of pb appropriate to the true burnt gas condition immediately behind the flame front [11]. Their values of Pb are given as a polynomial in T and P. The equations of mass and volume conservation, the perfect gas equation of state, Eq. (4), and an a s s u m e d equality of the burnt and unburnt gas constants give rise to:
with P, T~ rb, Ss and n plotted against t in Ref. [ 10]. In order to examine the validity of the assumptions made by these workers, their results are presented as the reduced pressure, Pr, plotted against the fractional mass burnt, n, in Fig. 1.d. Also plotted are the values obtained from the complete computer solution for the same mixture. There is significant difference between the two sets of results. The straight line on the graph represents Eq. (11) and this is seen to be closer to the complete computer solution. It would seem that procedures based on an " a priori" assumption of this equation are preferable to those based on assumptions about the value Ofpb. Earlier Nagy et al. [12] had assumed Yb = y~ = y, a constant, to derive:
drb _ dt
SuO
"lfPe~/7
f-~ -(P~)1/~~t 3-27+37(34) dP _ 37SUP P:/3~/(Pel/'[_ dt R p 1/7
Pol/"/)1/3
o
(35) and
dP _ 37u Su r2b P(Pu/Pb - 1) dt
R a [1 + (rb/R)a(Tu/7 b - 1)]
(32) po/\1/7 ' 1-
and
Ss = P__~u Pb Su
rb dP 31[bP dt
(33)
Solutions of these equations were obtained numerically and presented in graphical form
(36)
Te)
which were evaluated numerically. Suo is the burning velocity at the initial temperature and pressure and /3 is an exponent in the pressure variation of S,,. A further simplification of these workers was the assumption that the compression
EXPLOSIONS IN SPHERICAL VESSELS
207
was isothermal, with T~ = To and yb = 7~= 1. They obtained the following equations, the solutions of which are presented in Ref. [ 12], for a number of gaseous-air and dust-air mixtures. (~) 3-1 -P/P
(37) '
,
- -
=S
u -
I
1 -
(38)
-
and
dp-3Su Pe2/3 pP~ (Pe-Po )1/3 (I _p/p)2~3. dt R 0 (39) The analyses later were modified to take turbulence into account by the introduction of a constant, a , such that the turbulent burning velocity was a times the laminar burning velocity. The analyses also were extended to axi-symmetric flame propagation in a non-spherical vessel, on the assumption, based on the observations of Ellis and Wheeler [ 13], that the flame front adopted the shape of the vessel. This leads to 4rrr~, the surface area of the flame front, being replaced by As(VdVo)2Is, where A, is the surface area of the vessel in Eq. (4). Solutions are p r e s e n t e d in Ref. [14] for several gaseous-air and dust-air mixtures. It is useful to have the pressure rise algebraically expressed in terms of time. Perlee, Fuller and Saul [ 10] assumed Sa, pu/pb and 3'u = 3'b, all to be constant and showed that for the early stages of the explosion Eq. (32) could be modified to give: 3 3
P-Po _KSut 190
R3
(40)
in which K is a constant. This equation was also proposed by Zabetakis [15] and Maisey [ 16]. It has been partially verified experimentally by Knapton, Stobie and Krier [ 17]. 4. Influence of Temperature and Pressure Changes on Burning Velocity All solutions indicate their sensitivity to the numerical value of burning velocity. It is therefore necessary to know with some accuracy not only the initial value of burning velocity, but also the changes in it that occur during an explosion, as a result of the compression of the unburnt gas. Unfortunately, there is little data available for such conditions. Some data exist for the separate variation of temperature and pressure, but what are required for the p r e s e n t theoretical studies are the changing values of Su associated with isentropic compression. Rallis, Garforth and Steinz [18] have presented a three-dimensional S,,-P - Tgraph for acetylene-air but give no equation. Wilson [8] used the burning velocity data of Gerstein, Levine and Wong [ 19] for pentane and combined this with the temperature dependence of Dugger [20] for propane-air and the pressure dependence of Manson [21] to make an estimate of the combined temperature and pressure influence on the burning velocity of pentane-air. Hiroyasu and Kadota [22] assumed that the burning velocity of octane was dependent only upon temperature and derived a relationship from the works of Sachsse and Bartholom6 [23] and Dugger and Heimel [24]. Perlee et al. [10] considered the burning velocity of stoichiometric methane-air and combined the pressure dependence of Agnew and Graiff [25] with their own temperature dependence to give:
Su= Tu
2 (32.9 - 6.78 In ( ÷ o
)) cm sec'l. (41)
208
D. BRADLEY and A. MITCHESON
Better agreement with experiment was obtained with the form:
/ T\~
S u = (Suo - {3, ln(-~ -P ) ) l ~ l
ro
S u = (10 + 0.000371 T2) "--0.0052 T 1"5 log10P
cm sec"1 ,
(42)
Vo/
where Suo, the burning velocity at the initial conditions, /31 and /32 were constants chosen to give the best agreement with experiment. They did not give the values of/31 and/3~, but took Suo -- 44.29 cm sec-1. Nagy et al. [ 12, 14] used:
Su
u0
(43)
for a variety of mixtures, in which /3 was zero for the isothermal and 0.25 for the adiabatic model. S,,o was chosen to give good agreement with experiment. The complete computer solution presented in Sec. 5.2 is for a methane-air mixture. It was therefore important to have accurate burning velocity data for this mixture. As far as the present authors are aware, Babkin and Kozachenko [2] have carded out the most comprehensive experimental investigation of the influence of temperature and pressure during the period of c o n s t a n t pressure methane-air burning in a bomb explosion. They used the well-known expression, derived from Eq. (33):
Su = P__b_bS Pu
relationship was found to give reasonable agreement with the corrected data:
(44)
to obtain the burning velocity. The effect of the finite thickness of the flame is to reduce the value of Pb and this has been allowed for in a recalculation of Su values from the original data [26, 27]. It was assumed that the maximum temperature of the burnt gas was the ideal temperature. For a 9.5% CH4 -air mixture the following
cm sec"1
(45)
in which T is in QK and P in atmospheres. The equation is valid in the range: 298 < T < 473 1
cm sec"1
(46)
EXPLOSIONS IN SPHERICAL VESSELS
209
D VALUE AT ID 19.4 ATMOS.
Vl
2
3
4
5
6
7 B 9 10 P (ats)
15
20
Fig. 2. Burning velocity of 9.5% CH4--air mixture at different temperatures and pressures.
5. Accurate Analysis In recent years computer based analyses have been d e v e l o p e d with emphasis on chemical kinetic effects relevant to exhaust emissions. Hiroyasu and K a d o t a [22] and Blumberg[30] have described models in which allowance is made for the anisotropy of the burnt gas, but both groups of workers have assumed PW laws for compression of burnt and unburnt gas and have used average values of yu and yb. The former group assumed each element of mass burnt at constant volume and made allowances for heat loss to the vessel walls and the effect of turbulence on burning velocity, whilst Blumberg [30] assumed combustion at constant pressure and used an experimental burnt gas mass fraction-crank angle relationship to introduce time into the analysis. The present work is less concerned with kinetics and more c o n c e r n e d with deriving a c c u r a t e pressure-time solutions.
5.1 Case 3: The Complete Computer Solution
The assumptions and equations are those of Section 2. Flame propagation is seen as the consumption of unburnt gas in small mass decrements, dm,,. This mass does not become burnt gas instantaneously, but first passes through a reaction zone of finite thickness. The basic statement is that of conservation of total volume, which is made up of the three volumes of unburnt, reacting, and burnt gas. Consider the consumption of the " n " t h elemental unburnt mass decrement, dmu.n. This moves into the reaction zone with a temperature Tu,n-1, burns at the constant pressure Pn-1, and a proportion of the gas attains the ideal equilibrium temperature T~,n, for these conditions. This reaction must, in practice, increase the pressure and in the model this is assumed to follow the constant pressure combustion and to be isentropic. The pressure throughout the vessel becomes
210
D. BRADLEY and A. MITCHESON
P . and the uniform unburnt gas temperature becomes T,,,.. At this stage the radius of the inner boundary of the unburnt gas is rb,.. Equations (1) and (6) give:
raises the freshly burnt gas from a temperature T,.n to Tb.n. Thus the volume of burnt gas produced during the decremental change is: Vbn = !dmu, n - (mr, n
- mf, n.
1 )] Rgb,n Tb,n . (52)
e
i=H
mu'n = m° - 1Z'=Idmu'i
(47) The total volume of burnt gas up to the " n " t h decrement is:
and Vu n = mu'nRgu Tu'n
i =£ n lfn i= l
_1
(48)
P,
Vb,n
In Eq. (47) " i " refers to the generalized ele[dmu, i - (mr, i - mf, i. 1) ] Rgbi, nTbi, n
i=n
ment and E dmua is the sum of all the mass i=l decrements up to the "n"th. The value of Rgu is constant throughout the explosion. The flame thickness, 8., between the temperatures T~..,-x and T:,. can be calculated and the volume, Vt.., associated with this flame thickness is found from: VL n = -~-47r [r~, n _(rb,n_Sn)a]
,
(49)
(53)
where " i " refers to a generalised element. Rg~,n and Tbt,, are the values of gas constant and temperature, respectively, for the gases burnt in the " i " t h decrement, that obtain at the " n " t h decrement. It should be noted that mt, o = O. Volume conservation gives: Vo = Vu,n + Vf, n + Vb, n .
(54)
where 3 V )1/3 rb'n = (Ra - 4--~ u,n
(50)
With regard to the volume of the equilibrated burnt gas, a l l o w a n c e must be made for the changing mass within the flame thickness, 8.. Associated with the movement of the " n " t h mass decrement, dm,,.n, from the unburnt region there is a mass increase, d m b , . , of the burnt gas. Because the gas must be within the flame thickness before it is burnt, dmb,. is seen to be given by:
Equation (54) can be solved using Eqs. (48), (49) and (53). This was done numerically using the I C L 1906A c o m p u t e r at Leeds University and the CDC 7600 computer at the University of Manchester Regional Computing Centre. Solutions took the form of mass-flame radius-pressure relationships. Time is introduced into the solutions by means of Eq. (4) and the appropriate value of S~: dt =
- dm u
(55)
4 7rr2bPuS u " dmb, n =dmu, n -(mr, n - m f , n . 1) ,
(51)
where m,., and m1.n-1 are masses within the flame thickness. The isentropic compression immediately following c o n s t a n t p r e s s u r e c o m b u s t i o n
The computational method is summarised on the flow chart, Fig. 3. Very small mass decrements, of the order of mo x 10 -~, were used for the early stages of the explosion. A check was incorporated to ensure
EXPLOSIONS IN SPHERICAL VESSELS
211
that dm~ > m:. To save computing time, larger mass decrements were used for the later stages of the explosion, resulting in no appreciable loss of accuracy. Equilibrium temperatures were computed from the JANAF [31] thermochemical data in the usual way. The presence of eleven chemical species was a s s u m e d in the methane-air explosions investigated, namely, CO, CO2, H, H2, H20, OH, N2, N, NO, O and 02, and from their concentrations could be obtained the burnt gas constant, Rgb. Equilibrium constants were evaluated from values of Gibbs function. Enthalpies and entropies for each species were conveniently obtained from sets of polynomials derived from the JANAF data. The equations took the form:
h'/= A/T" + B/T 3 + CjT 2 + I~T + E/
(56)
where " n ~ " is the n u m b e r of moles of species '~j" in a total number of " n " moles, /~g is the mole basis gas constant and ~ j represents the sum of the parameters over all species. The unburnt gas was assumed isotropic but each burnt gas element, arising from each mass decrement in mu, must be treated separately in order to evaluate Tbt,n after isentropic compression. Any energy exchange between gas elements, either by conduction or convection, was neglected. The thickness of the flame may be regarded as comprised of two zones: a preheat zone extending from the unburnt gas at temperature, T~, to gas iat the ignition temperature, Tig, in which negligible chemical reaction occurs, and a reaction zone in which the gas temperature increases from T~g to T:, the ideal equilibrium temperature. For a one dimensional laminar flame the temperature distribution in the preheat zone is given by [32]:
and T = a exp
sq=
+ B/T2 + C/T + D/ ln T + F/
in which h °. and s °. are the specific enthalpies anJd entropnes, .1 respectively, of species '~/" at 1 atmosphere pressure, and temperature T °K, on a mole basis. As, B~, C/, D/, E~ and F~ are coefficients for the species '~"', and these were arranged in two groups, one for the temperature range 2001400 °K, the other for the range 1300-3500 OK. Values of entropy for unburnt and burnt gas were evaluated and maintained constant during the compression processes. Allowance must be made for the entropy of mixing and pressure changes from one atmosphere, so that the entropy of the mixture at a pressure of P atmospheres is:
S =~,n.s°-R]11
g~(nj'lnP)-Rg~(nJ
+ b,
(59)
(57)
-In n ) ) '
(58)
where Cp is the mass basis specific heat at constant pressure, h is the thermal conductivity, x is the distance measured in the direction of the gas flow and a and b are constants. This solution assumes that Cp and X are constant values. Because, in practice, they are not, the preheat thickness was divided into a number of elemental thicknesses for which this solution was employed, with mean values of Cp and k, appropriate to the strip. The lowest temperature was arbitrarily taken as (Tu + l) °K and the preheat zone thickness, 8p, evaluated from the sum of all the elemental thicknesses:
Xn,T t
212
D. BRADLEY and A. MITCHESON
ISTEP"n"thMASSDECREMENT,dmo,.I [CALCULATEIDEALEOUILIBRIUMTEMPERATURE,TF.] [CALCULATE FLAME THICKNESS a~w I
t
ICALCULATE BURNING VELOCITYfSu I ICALCULATE UNBURNT MASS, mul
-
T
IESTIMATE PRESSURE ,Pn I
t
CALCULATE ISENTROPIC TEMPERATURE R SE N UNBURNT GAS , Tu
t
ICALCULATE UNBURNT GAS
VOLUME ,Vu..I
t
ICALCULATE FLAME VOLUME.V,.n I
T
C~CULATE MASS WITHIN FLAME~rnf n I
t
CALCULATE COMPOSITION~ISENTROPIC TEMPERATURE RISE.Tb.n AND GAS CONSTANT, RobpOF EACH PREVIOUSLYBURNT GAS ELEMENT
t
ICALCULATE BURNT GAS VOLUME ?Vb,n I
[CALCULATE TIME, dt I
YES
NO
Fig. 3. Flow Chart for Complete Computer Solution.
It was assumed that h/ho = (T/To) °'s [33], where ho = 0.025 Jm -1 sec-1 (°K)-~ for 10% methane-air at the reference temperature, To = 273 °K [34] and values of Cp were interpolated from tables [35, 36]. To obtain the thickness of the reaction zone, 8,, the mean temperature gradient across it, (T I - T~g)/Sr, was equated to the temperature gradient at T~g which was obtained from the preheat zone analysis. The flame thickness was found from the addition of 8p and 8r. Such values of thickness were found to be lower than those measured experimentally [28, 37]. In Reference [28] a flame thickness of 1.2
stoichiometric methane-air flame at atmospheric t e m p e r a t u r e and pressure. The t h e o r e t i c a l l y derived value of t h i c k n e s s therefore was multiplied by a factor to make it equal to the experimental value. The same value of this factor was used for all determinations of flame thickness at the different pressures and unburnt gas temperatures.
5.2 Results from the Complete Computer Solution Results are presented for a stoichiometric methane-air mixture at an initial temperature and pressure of 292.1 °K and 1.014 x 105N m-z, respectively with central ignition in a sphere of 160.2 mm diameter. Figure 4 shows the radial progression of the flame front through the mixture. The broken curves show the trajectories of imaginery particles initially situated at the quarter, half and three-quarter radius positions. The gas flow reversal that occurs at the flame front is apparent and the slope of the curves gives the gas velocity at that radius and instant. Close to the centre unburnt gas velocities of more than 2 m sec -1 are observed. Close to the walls inward burnt gas velocities are larger than outward unburnt gas velocities. The radial distribution of burnt gas temperature at the instant of peak pressure is shown in Fig. 5. The first burnt is at a temperature more than 500 °K greater than the last burnf gas. This distribution affects the theoretical value of maximum pressure of an explosion and this is shown in Fig. 6, where values of P J P o are plotted against equivalence ratio, @. The results are for an initial temperature and pressure of 298 °K and 1.01325 x l0 s N m -2, respectively. The full line curve was obtained using the computational method described in the previous section. The broken curve was obtained on the assumption of instantaneous constant volume combustion of the initial mixture. It is seen that the latter assumption leads to a slight underestimation of the maximum pressure, the difference, however, being very small. Both methods neglect energy transfer to the walls and experimental values of maximum pressure are lower than those of the full line curve.
213
EXPLOSIONS IN SPHERICAL V E S S E L S 0.06
0-06
UNBURNT
0.05
/
GAS
~" /
0-0~
~....\..
/
0-02 . . . . . . . . .
~ ~
~ ~ " Z ~"
" BURNT
0.0
~
"V~
GAS
I
I
0.01
I
0-02
I
0,03
TIME (sec)
I
0-04
0.05
Fig. 4. Flame propagation and particle trajectories. Stoichiometric methane-air. 30OC
290
t 280
'~ 27O(
o. :E 0
260(
250(
240
230
0
0.~
0!0~
0.'03
070,
0.$5
0.00 '
0,0~ '
RADIUS(metres)
0108
0.09
0!1
Fig. 5. Burnt gas temperatures at instant of maximum pressure. Stoichiometric methane-air, Po = 1.01325 × 105N m-2, To = 298 °K.
tained for a range o f initial t e m p e r a t u r e s and p r e s s u r e s o f s t o i c h i o m e t r i c methane-air. T h e v e s s e l r a d i u s w a s t a k e n as 0.1 m. It w a s
f o u n d that the f o l l o w i n g e x p r e s s i o n g a v e a g o o d c o r r e l a t i o n for the o b s e r v e d p r e s s u r e rises:
214
D. 1.025 Nm-2 ,
C? -%
(61)
for lo5
and 290 < To < 450 “K.
The pressure-time relationship for a single stoichiometric methane-air explosion is shown in Fig. 7. The conditions correspond to the experimental ones of Garforth [381 whose measured points are indicated. The complete computer solution, Case 3, gave the full line curve. Agreement between theory and experiment is good until the later stages of the explosion, when measured pressures and rates of pressure increase fall below theoretical values. The measured peak pressure is approximately 1% below the theoretical value. These discrepancies are not surprising in view of the neglect of energy transfer to the vessel walls and between burnt gas elements in the computer programme. SO-
BRADLEY and A. MITCHESON
6. Discussion Also shown, by the chain dotted curves, in Fig. 7 is the complete computer solution with a constant burning velocity value of 0.45 m set-‘. This deviates significantly from the complete computer solution, which makes allowance for changes in burning velocity as a result of isentropic compression, and shows the importance of making this allowance. Indeed, it is seen from Fig. 7 that the Case 1, approximate computer solution, shown by the dotted curve, in which the variation in burning velocity is given by Eq. (45), gives a more accurate pressure-time relationship than does Case 3 in which S, is constant. The validity of Eq. (1 I), upon which both Case 1 and Case 2 solutions rest, was examined using the accurate Case 3 solutions. The points shown on the graph of fractional pressure rise, P,, against fractional mass burnt, n, in Fig. 1.d were obtained from the Case 3 solution for a stoichiometric methane-air mixture. Equation (11) gives the straight line. The closeness of the points to the straight line suggests a sound basis for Eq. (11) and its application in Case 1 and Case 2 so-
COMPLETE COMPUTER SOLUTION
&5-
CONSTANT
VOLUME
SOLUTION 8.0 -
t
7.5 -
a”lB 7.
-
6.5-
6.0 -
5.5-
5.0’ 0.4
0.5
0.6
I
I
o.7
0.8
1 0.9
ECGLENCE
I ZATIO
1.2 $I -
Fig. 6. Maximum pressures attained for different methane-air equivalence = 0.t m.
1
I
I
1.3
1.4
1.5
ratios. Vessel radius
EXPLOSIONS IN SPHERICAL VESSELS
2]5
~0 X
EXPERIMENTAL POINTS
(38)
//
//1
8.0
CASE 3 BUT WITH Su =0-/45 m SeE"t (CONSTANT)
/,,
7.0
o
//I
6o
K 7E
z
5.(3
4.0
3.0
2.G
1.0
()
I 0-01
I 0-02
I 0.03
I 0.0/*
I 0.05
, I 0-6
TIME (s¢cs)
Fig. 7. P r e s s u r e - t i m e c u r v e s for s t o i c h i o m e t r i c m e t h a n e - a i r . P o = 1.014 x V e s s e l r a d i u s = 8.01 c m .
lutions. The Case 3 solution values are in good agreement with those calculated by Rallis and co workers [39, 40], but not with those of Perlee et al. [10], which are also shown in Fig. 1.d. In view of this satisfactory finding, the question arises as to whether the Case 2, Dimensionless Universal Expression solutions, such as those given in Fig. la-c might be both valid and of wide applicability. They have the advantage of not entailing the use of computers. The Case 2 solutions, however, are based on the value of S= remaining constant throughout the explosion. For a stoichiometric methane-air mixture with (Po)-I = 8.99 and 3' = 1.36, the Case 2 solutions gives S=/Sm = 0.244. The more accurate Case 3 solution gives Sm = 1.613 m sec-~ in a sphere of 160.2 mm diameter and an arithmetic mean value of burning velocity of 0.48 m sec -~. With these two values, S,,/Sm = 0.298. The dimensionless pressure-time points
l(P N m -2, To = 292.1 ° K ,
given by the Case 3 solution for (po)-i = 8.99, are shown in Fig. la. These points lie very close indeed to the Case 2 solution for SdSm = 0.3. This suggests that if the arithmetic mean value of burning velocity for the explosion is taken for S=, then the solution curves for the Dimensionless Universal Expression are valid. In Section 3.3 Eq. (40) suggests that the pressure rise is proportional to ta during the early stages of an explosion. The complete c o m p u t e r solutions for stoichiometric methane-air give Pr = t 4.5
(62)
for o . o l "<
This is in close agreement with the exo perimental results of Perlee et al. [ 10] which
216
yield a time exponent of 4.3, but is much higher than the value of 3.5 computed from Garforth's data [38]. 7. Conclusions (1) The basic equations and associated assumptions for flame propagation in a spherical vessel have been reviewed. Different solutions have been compared~with Complete Computer Solutions, which involve minimal assumptions. (2) Support has been found for those solutions which rest upon the simplifying assumption that the fractional pressure rise is proportional to the fractional mass burnt. (3) The simplest solution to apply is that of the Dimensionless Universal Expression in which the burning velocity value is the arithmetic mean of initial and final values. (4) The Complete Computer Solutions show the importance of accurate values of burning velocity. The concept of isentropic variation of burning velocity is introduced and values suggested for methane-air mixtures. (5) The Complete Computer Solution is in good agreement with accurate experimental values of the pressure-time relationships. Theoretical pressure rises are predicted from this model. (6) For a significant period of an explosion it has been demonstrated that/3 ~ 74.5. We are indebted to our colleagues, Mr. Don Appleyard, Dr. Ian Swift, and the late Mr. Steve Arbury for their contributions towards the d e v e l o p m e n t o f the Complete Computer Solution. We thank the British Gas Corporation for a Research Scholarship to Alan Mitcheson.
References 1. Clerk, D., The Gas, Petrol and Oil Engine, Longmans, London, 1910. 2. Babkin, V. S. and Kozachenko, L. S., Fizika Goreniya i Vzryva, 2, 77-86 (1966). 3. Eschenbaeh, R. C., Use of the Constant Volume Bomb Technique for Measuring Burning Velocities, Ph.D. Thesis, Purdue University, Indiana (1957). 4. Graiff, L. B., The pressure Dependence of the
D. B R A D L E Y and A. M I T C H E S O N Laminar Burning Velocity, Ph.D. Thesis, Purdue University, Indiana (1959). 5. Flock, E. F. and Marvin, C. F., Chem. Revue 21, 367 (1937). 6. Lewis, B. and Von EIbe, G., Combustion, Flames and Explosion o f Gases, 2nd ed., Academic Press Inc., New York, 1961. 7. Benson, R. S. and Burgoyne, J. H., British Shipbuilding Research Assoc., Rep. No. 76 (1951). 8. Wilson, M. J. G., Relief of Explosions in Closed Vessels, Ph.D. Thesis, University of London (1954). 9. Hopkinson, B., Proc. Roy. Soc., Ser. A 77, 387 (1906). 10. Perlee, H. E., Fuller, F. N. and Saul, C. H., U.S. Department of Interior, Bureau of Mines, Rep. of Investigation RI 7839 (1974). 11. Gordon, S. and Zelezulk, F. J., NASA Tech. Note, TN D-1737 (1963). 12. Nagy, J., Conn. J. W. and Verakis, H. C., U.S. Department of Interior, Bureau of Mines, Rep. of Investigation RI 7279 (1969). 13. Ellis, O. C. de C. and Wheeler, R. V., Fuel 7, 169 (1928). 14. Nagy, J., Seller, E. C., Conn, J. W. and Verakis, H. C., U.S. Dept. of Interior, Bureau of Mines, Rep. of Investigation RI 7507 (1971). 15. Zabetakis, M. G., U.S. Dept. of Interior, Bureau of Mines, Bull. 627, (1965). 16. Maisey, H. R., Chem. and Proc. Eng. 527 (1965). 17. Knapton, J. D., Stobie, I. C. and Krier, H., Cornbust. Flame 21,211 (1973). 18. Rallis, C. J., Garforth, A. M. and Steinz, J. A., Combust. Flame 9, 345 (1965). 19. Gerstein, M., Levine, O. and Wang, E. L., J. Am. Chem. Soc., 73, 418 (1951). 20. Dugger, G. L., Effect of Initial Temperature on Flame Speed of Methane-air, Propane-air and Ethylene-air Mixtures, NACA Rep. 1061 (1952). 21. Manson, N., Fuel 32, 186 (1953). 22. Hiroyasu, H. and Kadota, T., Fifteenth Symposium (International) on Combustion, Combustion Institute, Pittsburgh, Pa., 1975, p. 1213. 23. Sachsse, H. and Bartholomt, E., Z. Elektrochem 53, 183 (1949). 24. Dugger, G. L. and Heimel, S., Flame Speeds of Methane-air, Propane-air and Ethylene-air at Low Initial Temperatures, N A C A Tech. Note 2624 (1952). 25. Agnew, J. T. and Graiff, L. B., Combust. Flame 5, 209 (1961). 26. Andrews, G. E. and Bradley, D., Combust. Flame 18, 133 (1972). 27. Andrews, G. E. and Bradley, D., C o m b u s t . Flame,20, 77 (1973). 28. Andrews, G. E. and Bradley, D., Combust. Flame 19, 275 (1972). 29. Halstead, M. P., Pye, D. B. and Qulnn, C. P., Cornbust. Flame 22, 89 (1974). 30. Blumberg, P. N., Combust. Sci. Tech. 8, 5 (1973).
E X P L O S I O N S IN S P H E R I C A L V E S S E L S 31. J A N A F Thermochemical Tables, U.S. Dept. of Commerce, Clearinghouse for Federal, Scientific and Technical Information, PB 168370, 1965. 32. Bradley, J. N., Flame and Combustion Phenomena, 1st ed., Methuen and Co. Ltd., London, 1969. 33. Andrews, G. E., Bradley, D. and Hundy, G. F., Int. J. Heat and Mass Transfer 15, 1765 (1972). 34. Mason, E. A. and Saxena, S. C., Phys. Fluids 1,361 (1958). 35. Haywood, R. W., Thermodynamics Tables in SI Units, Cambridge Univ. Press, 1968. 36. Mayhew, Y. R. and Rogers, G. F. C., Thermodynamic and Transport Properties of Fluids, Blackwell, Oxford, 1971. 37. Janlsch, G., Chemi. Ing. Tech. 43, 561 (1971). 38. Garforth, A. M., Measurement of Rapidly Varying
217 Density, and Hence Temperature, by Laser Interferometry in the Unburnt Gas Region of a Spherical Constant Volume Combustion Vessel During Flame Propagation, Univ. of the Witwatersrand, Sch. Mech. Eng., Rep. No. 57 (1974). 39. Rallis, C. J. and Tremcer, G. E. B., Equations for the Determination of Burning Velocity in a Spherical Constant Volume Vessel, Univ. of the Witwatersrand, Sch. Mech. Eng., Rep. No. 14 (1963). 40. Rallis, C. J., Garforth, A. M. and Steinz, J. M., The Determination of Laminar Burning Velocity with Particular Reference to the Constant Volume Method, Pt. 3--Experimental Procedure and Results, Univ. of the Witwatersrand, Sch. Mech. Eng., Rep. No. 26 (1965). Received 7 November 1975
201
Mathematical Solutions for Explosions in Spherical Vessels DEREK BRADLEY and ALAN MITCHESON Mechanical Engineering Department, University of Leeds, Leeds, LS2 9JT.
Equations, assumptions and previous solutions for central ignition of premixed gases in closed spherical vessels are reviewed. Three new categories of solution are presented: the Approximate Computer Solution (Case 1), the Dimensionless Universal Expression (Case 2), and the Complete Computer Solution (Case 3). Case 3 is the most accurate and shows the importance of accurate values of burning velocity as the unburnt gas is compressed in the course of the explosion. Recommendations are made concerning such values for methane-air mixtures. Case 2 solutions do not require a computer and they appear to be of good accuracy. During the explosions studied dimensionless pressure, Pr, is proportional to dimensioness time, t, raised to the power 4.5.
1. Introduction The experimental measurement of pressure in closed vessel explosions was one of the earliest areas of combustion research. Possibly the Rev. W. Cecil in 1820 [1] was the first to o b t a i n a c r u d e i n d i c a t i o n o f the maximum explosion pressure of a hydrogen-mr mixture, by making explosions rupture strings of known breaking strength. L a t e r d e v e l o p m e n t s were related to explosion h a z a r d s and c o m b u s t i o n in i n t e r n a l combustion engines and much experimental data have been accumulated. Theoretical analyses were developed which related burning velocity, flame radius and pressure. T h e m o s t c o n v e n i e n t starting point for theoretical analyses is laminar combustion of a quiescent, uniform, premixed gas with central ignition in a closed sphere, to the walls o f w h i c h t h e r e is no e n e r g y loss. In the course of much work by many different researchers, further assumptions have been inv o k e d to enable practical solutions o f the basic equations to be obtained, and some of these are discussed in the present study. A simple model is presented which is amenable to computer simulation and this is used to derive pressure-time curves in dimensionless form which are universally applicable for all explosions,
From these curves it is clearly seen that corresponding dimensional curves are v e r y sensitive to the value of burning velocity. Both t e m p e r a t u r e and pressure o f the unburnt gas are changing during the explosion and d a t a are r e q u i r e d for the a s s o c i a t e d changes in burning velocity. Some data of this type are available from the experimental work of Babkin and K o z a c h e n k o [2]. Their data are recalculated in the present work, making allowance for the effects of flame thickness, and are presented in the form of an equation which relates the burning velocity of a methane-air mixture to both temperature and pressure. With the availability o f such data it has b e e n worthwhile to d e v e l o p a m u c h more accurate computer model based upon finite difference techniques and less restrictive assumptions than formerly used. Some results o f s u c h c o m p u t a t i o n s are p r e s e n t e d and c o m p a r e d with t h o s e f r o m the s i m p l e r model. T h e d e v e l o p m e n t o f the a c c u r a t e model is seen also as a first step in the simulation of vented explosions. 2. The Basic Equations It is assumed that central ignition occurs in a rigid sphere and that a laminar flame that is smooth and spherical, propagates outwards
Copyright © 1976 by The Combustion Institute Published by American Elsevier PublishingCompany, Inc.
202
D. BRADLEY and A. MITCHESON
without any significant movement due to natural c o n v e c t i o n . The p r e s s u r e , P, is equalised throughout the vessel [3,4] and the unburnt gas is isotropic. Mass conservation gives: mo = m u + m b , dmu_-dm dt
b
(I)
(2)
dt
(8)
d (Pbr3b) = 3r2p, S ,
dt
b , ,
where Pb is the mean density of the gas within the radius rb. Its accurate evaluation presents severe problems. The further d e v e l o p m e n t of these equations has led to the postulation of several simplifying assumptions, which are accompanied by limitations in the accuracy of derived results.
in which m is the mass of gas and subscripts u, b and o indicate the unburnt and burnt gas
states at time t, and the initial reference state at time t equal to zero, respectively. Define n, the burnt gas fraction as:
3. Simplified Analyses It might be assumed that the unburnt gas is compressed isentropically according to the law:
mb
n=
(3) mo
.Tu Pou
=
a constant
(9)
It follows from the definition of burning velocity, Su, that: dm dt
'
u _ -47rr~PuSuu
(4)
where rb is the radius of the inner boundary of the unburnt gas of density Ou. If V represents volume, then volume conservation gives: Vo =
+ Vb
(5)
The equation of state is: P V = mRgT,
in which 3' is the ratio of specific heats. Equations (7) and (9) yield:
drb Su - - d ~
where R is the vessel radius. Burnt gas conservation yields:
3r2b TuP
dt
(10) '
(6)
in which R g is the mass basis gas constant and T is the absolute temperature. Equation (4) yields: dt
dP
a result derived by Fiock and Marvin [5]. Lewis and yon Elbe [6] made several other assumptions and deduced that the fractional pressure rise is proportional to the fractional mass burnt: P-P
d [ ( R a _ r g ) ~ ] =_3r2bPuS
(R3 - r~ )
(7)
_m b _
Pe-Po /720
H.
(11)
Subscript " e " indicates the condition when the explosion is completed. This equality simplifies the equations considerably and is of great utility. The assumptions upon which it rests cannot be regarded as entirely valid, but the complete computer solutions pre-
EXPLOSIONS IN SPHERICAL VESSELS
203
sented in Section 5.2 enable the validity of this useful expression to be assessed. 3.1 Case 1: The Approximate Computer Solution
Pressure-time curves were obtained by increasing " n " in elemental steps from 0 to 1 on the I.C.L 1906A Computer at Leeds University. The computation procedure will be appreciated from the following equations. From Eq. (11) e=n(Pe-Po)+P o •
(12)
From Eqs. (6) and (9)
A pressure-time solution derived from these equations is presented in Section 6, where it is compared with the corresponding results from the complete computer solution.
3.2 Case 2: The Dimensionless Universal Expression This attempts to obtain the maximum generality in the pressure-time relationship. Previous assumptions are utilised and dimensionless relationships developed. Equations (1), (3), (4), (9), (ll) and (16) yield:
dt
Tu - 1 7u
m u = mo(1 - n ) .
(14)
From Eqs. (5) and (6)
4 rrr~ = 4 r r R a - m u R g u Tu 3 3 P
RPo
(13)
From Eqs. (1) and (3)
(18)
This equation was derived by Benson and Burgoyne [7] and also Wilson [8], by a slightly different method. The former workers obtained solutions for cetane-air and Wilson assumed a burning velocity relationship for pentane-air which gave rise to solutions in poor agreement with experiment. Dimensionless pressures and times are defined in terms of conditions at the end of the explosion:
(15) P=P/P
3m u t012 --
lieu Re _ e_] 2/3
de _ 3supu (e e _eo)
,
(19)
(20)
(16)
47r(R a - r~) " t = t/t.
From Eq. (4)
(21)
A mean flame speed, Sin, is defined by: dt -
-dm
u 4 7rr2bPuS u
(17) s m = Rit e .
(22)
D. BRADLEY and A. MITCHESON
204 In non dimensional form Eq. (18) becomes
Where S, = drddt, the flame speed. Eqs. (1), (9) and (11) give:
1/~u drd/~-
)3Su S (1 m-fi°)("o~~-
1 -
1 -
(23)
The ratio R/rb can be eliminated from Eq. (26) using Eq. (27) and it is seen that, for given values of 3'= and Po, the ratio S,/S= is a function of P. That is: dr b at - Ss = Su f(ff)
This equation has been solved using finite difference techniques, for different values of SJS,, and Po. Such solutions are presented in Figs. la, b and c, in the form of curves relating reduced pressure, i~r ( = (P - Po) (Pe .p,)-l) to dimensionless time, T. The generality in these curves is further extended by that also exhibited in the ratio SdS=, as now is demonstrated. dP/dt in Eq. (10) can be evaluated using Eq. (11). Differentiation of the latter gives:
d P _ dn (Pc" Po)" dt dt
(24)
(27)
(28)
and te
fo
dr b R - dt = fo S u f d ) "
(29)
If S= is assumed to be constant throughout the explosion, then:
teSu _ S U _ f o 1 d(rb/R __ ) R Sm f(p )
(30)
Equations (1), (4), (9) and (14) yield:
dt
-
rb
n-l+ (V)
(25)
and substitution of Eqs. (19), (20), (24) and (25) into Eq. (10) gives:
7uP
rb~ - 1
(26)
Here Sm is a mean flame speed, Rite, and the equation can be evaluated with the aid of Eqs. (26) and (27). It is clear that SJSm has a unique value that depends only upon the values of y= and Po and that this is otherwise independent of mixture composition. Computerised numerical integration has yielded the following values of SJS=: 0.244 for (po)-i = 8.99; 0.290 for (Po)-I = 7.00 and 0.365 for (Po)-' = 5.00. In each case a value of 1.36 was taken as a mean value for 3'= for a stoichiometric methane-air mixture during isentropic compression from ambient conditions. Values of S=/S= for other conditions can be obtained very readily. These values explain the ranges of values used in the c o m p u t e d dimensionless p r e s s u r e - t i m e curves shown in Figs. la, b and c.
EXPLOSIONS IN SPHERICAL VESSELS
205
1.0
1-0
t
0.8
0.8
Pr
0.6
0.6
0.4 _
_o°:I
0.25\
0.4
0.2
0.2
0.2
0-4
0.6
0.8
1.0
0
0.2
0.4
(a)
/ ////
0.8 0.25,, 0.3.,," 035\
0-6
0.2
0
0.4
I s
I
04
REE 10\
/
R ,
j #
t /
0.z;
0.6 T
1.0
//
0.2
/
0.2
1.0
O.E
0.4,,,\ \
0.4
0.8 T---~
(b)
1.0
I
0.6
0.8
1.0
~
0
iIII/ .I 0.2
~ E q . I(11
I --
0.4
i
I i I J 0.6
0.8
1.0
n ~
(c) . (d) Fig. 1 (a)-(c). Dimensionless pressure-time curves for Universal Expression with y= = 1.36. (a) (Po)-1 = 8.99 (marked points for Case 3 solution); (b) (Po)-I = 7; (c) (Po)-1= 5; (d) Dimensionless pressure-fractional mass burnt curve. Circled points are complete computer solution, Po = 1.01325 × 10~N m-~, To = 298 °K, vessel radius = 0.1 m. Such c u r v e s c a n be o f p r a c t i c a l utility. F r o m them, there is no difficulty in deriving actual pressure-time curves for particular c o n d i t i o n s . First, the m a x i m u m e x p l o s i o n pressure for the mixture should be found [6] and Po evaluated. The value of SJSm should then be determined and from a knowledge of S=, Sra, (=R/te), is derived. F o r a given vessel radius this gives te, which gives the time scale for the explosion. The p r o b l e m presented by the fact that S= is not c o n s t a n t but varies during the c o m pression process a c c o m p a n y i n g combustion,
might be c i r c u m v e n t e d by the selection of an appropriate mean value of Su for the whole e x p l o s i o n . G u i d a n c e as to w h i c h value to select is obtained f r o m a c o m p a r i s o n o f the p r e s e n t Case 2 solutions with the C a s e 3, c o m p l e t e c o m p u t e r solutions, p r e s e n t e d in Section 6. 3.3 Other Approximate Solutions
Before proceeding to the complete c o m p u t e r solution it is o p p o r t u n e at this stage to summarise some of the other solutions and their u n d e r l y i n g a s s u m p t i o n s . T h e simplicity o f
206
D. BRADLEY and A. MITCHESON
the Case 1 and Case 2 solutions rests upon the assumed equality of the fractional pressure rise and the fractional mass burnt, Eq. (11). An alternative route involves a knowledge of the changes in the mean density of the burnt gas, Pb. This is not easy because of the non uniformity of the burnt gas temperature; the gas burnt first attains a t e m p e r a t u r e higher than that burnt later, a fact demonstrated in 1906 by H o p k i n s o n [9]. Some workers have nevertheless made an assumption of a global isentropic compression law, namely:
PPb -7b = constant.
(31)
Perlee, Fuller and Saul [10] utilised Eqs. (31) and (9), with values of Yb and y , calculated from polynomials in T and P. The temperature of the burnt gas was assumed to be that at the centre of the vessel. This overestimation of temperature was compensated by additionally using a value of pb appropriate to the true burnt gas condition immediately behind the flame front [11]. Their values of Pb are given as a polynomial in T and P. The equations of mass and volume conservation, the perfect gas equation of state, Eq. (4), and an a s s u m e d equality of the burnt and unburnt gas constants give rise to:
with P, T~ rb, Ss and n plotted against t in Ref. [ 10]. In order to examine the validity of the assumptions made by these workers, their results are presented as the reduced pressure, Pr, plotted against the fractional mass burnt, n, in Fig. 1.d. Also plotted are the values obtained from the complete computer solution for the same mixture. There is significant difference between the two sets of results. The straight line on the graph represents Eq. (11) and this is seen to be closer to the complete computer solution. It would seem that procedures based on an " a priori" assumption of this equation are preferable to those based on assumptions about the value Ofpb. Earlier Nagy et al. [12] had assumed Yb = y~ = y, a constant, to derive:
drb _ dt
SuO
"lfPe~/7
f-~ -(P~)1/~~t 3-27+37(34) dP _ 37SUP P:/3~/(Pel/'[_ dt R p 1/7
Pol/"/)1/3
o
(35) and
dP _ 37u Su r2b P(Pu/Pb - 1) dt
R a [1 + (rb/R)a(Tu/7 b - 1)]
(32) po/\1/7 ' 1-
and
Ss = P__~u Pb Su
rb dP 31[bP dt
(33)
Solutions of these equations were obtained numerically and presented in graphical form
(36)
Te)
which were evaluated numerically. Suo is the burning velocity at the initial temperature and pressure and /3 is an exponent in the pressure variation of S,,. A further simplification of these workers was the assumption that the compression
EXPLOSIONS IN SPHERICAL VESSELS
207
was isothermal, with T~ = To and yb = 7~= 1. They obtained the following equations, the solutions of which are presented in Ref. [ 12], for a number of gaseous-air and dust-air mixtures. (~) 3-1 -P/P
(37) '
,
- -
=S
u -
I
1 -
(38)
-
and
dp-3Su Pe2/3 pP~ (Pe-Po )1/3 (I _p/p)2~3. dt R 0 (39) The analyses later were modified to take turbulence into account by the introduction of a constant, a , such that the turbulent burning velocity was a times the laminar burning velocity. The analyses also were extended to axi-symmetric flame propagation in a non-spherical vessel, on the assumption, based on the observations of Ellis and Wheeler [ 13], that the flame front adopted the shape of the vessel. This leads to 4rrr~, the surface area of the flame front, being replaced by As(VdVo)2Is, where A, is the surface area of the vessel in Eq. (4). Solutions are p r e s e n t e d in Ref. [14] for several gaseous-air and dust-air mixtures. It is useful to have the pressure rise algebraically expressed in terms of time. Perlee, Fuller and Saul [ 10] assumed Sa, pu/pb and 3'u = 3'b, all to be constant and showed that for the early stages of the explosion Eq. (32) could be modified to give: 3 3
P-Po _KSut 190
R3
(40)
in which K is a constant. This equation was also proposed by Zabetakis [15] and Maisey [ 16]. It has been partially verified experimentally by Knapton, Stobie and Krier [ 17]. 4. Influence of Temperature and Pressure Changes on Burning Velocity All solutions indicate their sensitivity to the numerical value of burning velocity. It is therefore necessary to know with some accuracy not only the initial value of burning velocity, but also the changes in it that occur during an explosion, as a result of the compression of the unburnt gas. Unfortunately, there is little data available for such conditions. Some data exist for the separate variation of temperature and pressure, but what are required for the p r e s e n t theoretical studies are the changing values of Su associated with isentropic compression. Rallis, Garforth and Steinz [18] have presented a three-dimensional S,,-P - Tgraph for acetylene-air but give no equation. Wilson [8] used the burning velocity data of Gerstein, Levine and Wong [ 19] for pentane and combined this with the temperature dependence of Dugger [20] for propane-air and the pressure dependence of Manson [21] to make an estimate of the combined temperature and pressure influence on the burning velocity of pentane-air. Hiroyasu and Kadota [22] assumed that the burning velocity of octane was dependent only upon temperature and derived a relationship from the works of Sachsse and Bartholom6 [23] and Dugger and Heimel [24]. Perlee et al. [10] considered the burning velocity of stoichiometric methane-air and combined the pressure dependence of Agnew and Graiff [25] with their own temperature dependence to give:
Su= Tu
2 (32.9 - 6.78 In ( ÷ o
)) cm sec'l. (41)
208
D. BRADLEY and A. MITCHESON
Better agreement with experiment was obtained with the form:
/ T\~
S u = (Suo - {3, ln(-~ -P ) ) l ~ l
ro
S u = (10 + 0.000371 T2) "--0.0052 T 1"5 log10P
cm sec"1 ,
(42)
Vo/
where Suo, the burning velocity at the initial conditions, /31 and /32 were constants chosen to give the best agreement with experiment. They did not give the values of/31 and/3~, but took Suo -- 44.29 cm sec-1. Nagy et al. [ 12, 14] used:
Su
u0
(43)
for a variety of mixtures, in which /3 was zero for the isothermal and 0.25 for the adiabatic model. S,,o was chosen to give good agreement with experiment. The complete computer solution presented in Sec. 5.2 is for a methane-air mixture. It was therefore important to have accurate burning velocity data for this mixture. As far as the present authors are aware, Babkin and Kozachenko [2] have carded out the most comprehensive experimental investigation of the influence of temperature and pressure during the period of c o n s t a n t pressure methane-air burning in a bomb explosion. They used the well-known expression, derived from Eq. (33):
Su = P__b_bS Pu
relationship was found to give reasonable agreement with the corrected data:
(44)
to obtain the burning velocity. The effect of the finite thickness of the flame is to reduce the value of Pb and this has been allowed for in a recalculation of Su values from the original data [26, 27]. It was assumed that the maximum temperature of the burnt gas was the ideal temperature. For a 9.5% CH4 -air mixture the following
cm sec"1
(45)
in which T is in QK and P in atmospheres. The equation is valid in the range: 298 < T < 473 1
cm sec"1
(46)
EXPLOSIONS IN SPHERICAL VESSELS
209
D VALUE AT ID 19.4 ATMOS.
Vl
2
3
4
5
6
7 B 9 10 P (ats)
15
20
Fig. 2. Burning velocity of 9.5% CH4--air mixture at different temperatures and pressures.
5. Accurate Analysis In recent years computer based analyses have been d e v e l o p e d with emphasis on chemical kinetic effects relevant to exhaust emissions. Hiroyasu and K a d o t a [22] and Blumberg[30] have described models in which allowance is made for the anisotropy of the burnt gas, but both groups of workers have assumed PW laws for compression of burnt and unburnt gas and have used average values of yu and yb. The former group assumed each element of mass burnt at constant volume and made allowances for heat loss to the vessel walls and the effect of turbulence on burning velocity, whilst Blumberg [30] assumed combustion at constant pressure and used an experimental burnt gas mass fraction-crank angle relationship to introduce time into the analysis. The present work is less concerned with kinetics and more c o n c e r n e d with deriving a c c u r a t e pressure-time solutions.
5.1 Case 3: The Complete Computer Solution
The assumptions and equations are those of Section 2. Flame propagation is seen as the consumption of unburnt gas in small mass decrements, dm,,. This mass does not become burnt gas instantaneously, but first passes through a reaction zone of finite thickness. The basic statement is that of conservation of total volume, which is made up of the three volumes of unburnt, reacting, and burnt gas. Consider the consumption of the " n " t h elemental unburnt mass decrement, dmu.n. This moves into the reaction zone with a temperature Tu,n-1, burns at the constant pressure Pn-1, and a proportion of the gas attains the ideal equilibrium temperature T~,n, for these conditions. This reaction must, in practice, increase the pressure and in the model this is assumed to follow the constant pressure combustion and to be isentropic. The pressure throughout the vessel becomes
210
D. BRADLEY and A. MITCHESON
P . and the uniform unburnt gas temperature becomes T,,,.. At this stage the radius of the inner boundary of the unburnt gas is rb,.. Equations (1) and (6) give:
raises the freshly burnt gas from a temperature T,.n to Tb.n. Thus the volume of burnt gas produced during the decremental change is: Vbn = !dmu, n - (mr, n
- mf, n.
1 )] Rgb,n Tb,n . (52)
e
i=H
mu'n = m° - 1Z'=Idmu'i
(47) The total volume of burnt gas up to the " n " t h decrement is:
and Vu n = mu'nRgu Tu'n
i =£ n lfn i= l
_1
(48)
P,
Vb,n
In Eq. (47) " i " refers to the generalized ele[dmu, i - (mr, i - mf, i. 1) ] Rgbi, nTbi, n
i=n
ment and E dmua is the sum of all the mass i=l decrements up to the "n"th. The value of Rgu is constant throughout the explosion. The flame thickness, 8., between the temperatures T~..,-x and T:,. can be calculated and the volume, Vt.., associated with this flame thickness is found from: VL n = -~-47r [r~, n _(rb,n_Sn)a]
,
(49)
(53)
where " i " refers to a generalised element. Rg~,n and Tbt,, are the values of gas constant and temperature, respectively, for the gases burnt in the " i " t h decrement, that obtain at the " n " t h decrement. It should be noted that mt, o = O. Volume conservation gives: Vo = Vu,n + Vf, n + Vb, n .
(54)
where 3 V )1/3 rb'n = (Ra - 4--~ u,n
(50)
With regard to the volume of the equilibrated burnt gas, a l l o w a n c e must be made for the changing mass within the flame thickness, 8.. Associated with the movement of the " n " t h mass decrement, dm,,.n, from the unburnt region there is a mass increase, d m b , . , of the burnt gas. Because the gas must be within the flame thickness before it is burnt, dmb,. is seen to be given by:
Equation (54) can be solved using Eqs. (48), (49) and (53). This was done numerically using the I C L 1906A c o m p u t e r at Leeds University and the CDC 7600 computer at the University of Manchester Regional Computing Centre. Solutions took the form of mass-flame radius-pressure relationships. Time is introduced into the solutions by means of Eq. (4) and the appropriate value of S~: dt =
- dm u
(55)
4 7rr2bPuS u " dmb, n =dmu, n -(mr, n - m f , n . 1) ,
(51)
where m,., and m1.n-1 are masses within the flame thickness. The isentropic compression immediately following c o n s t a n t p r e s s u r e c o m b u s t i o n
The computational method is summarised on the flow chart, Fig. 3. Very small mass decrements, of the order of mo x 10 -~, were used for the early stages of the explosion. A check was incorporated to ensure
EXPLOSIONS IN SPHERICAL VESSELS
211
that dm~ > m:. To save computing time, larger mass decrements were used for the later stages of the explosion, resulting in no appreciable loss of accuracy. Equilibrium temperatures were computed from the JANAF [31] thermochemical data in the usual way. The presence of eleven chemical species was a s s u m e d in the methane-air explosions investigated, namely, CO, CO2, H, H2, H20, OH, N2, N, NO, O and 02, and from their concentrations could be obtained the burnt gas constant, Rgb. Equilibrium constants were evaluated from values of Gibbs function. Enthalpies and entropies for each species were conveniently obtained from sets of polynomials derived from the JANAF data. The equations took the form:
h'/= A/T" + B/T 3 + CjT 2 + I~T + E/
(56)
where " n ~ " is the n u m b e r of moles of species '~j" in a total number of " n " moles, /~g is the mole basis gas constant and ~ j represents the sum of the parameters over all species. The unburnt gas was assumed isotropic but each burnt gas element, arising from each mass decrement in mu, must be treated separately in order to evaluate Tbt,n after isentropic compression. Any energy exchange between gas elements, either by conduction or convection, was neglected. The thickness of the flame may be regarded as comprised of two zones: a preheat zone extending from the unburnt gas at temperature, T~, to gas iat the ignition temperature, Tig, in which negligible chemical reaction occurs, and a reaction zone in which the gas temperature increases from T~g to T:, the ideal equilibrium temperature. For a one dimensional laminar flame the temperature distribution in the preheat zone is given by [32]:
and T = a exp
sq=
+ B/T2 + C/T + D/ ln T + F/
in which h °. and s °. are the specific enthalpies anJd entropnes, .1 respectively, of species '~/" at 1 atmosphere pressure, and temperature T °K, on a mole basis. As, B~, C/, D/, E~ and F~ are coefficients for the species '~"', and these were arranged in two groups, one for the temperature range 2001400 °K, the other for the range 1300-3500 OK. Values of entropy for unburnt and burnt gas were evaluated and maintained constant during the compression processes. Allowance must be made for the entropy of mixing and pressure changes from one atmosphere, so that the entropy of the mixture at a pressure of P atmospheres is:
S =~,n.s°-R]11
g~(nj'lnP)-Rg~(nJ
+ b,
(59)
(57)
-In n ) ) '
(58)
where Cp is the mass basis specific heat at constant pressure, h is the thermal conductivity, x is the distance measured in the direction of the gas flow and a and b are constants. This solution assumes that Cp and X are constant values. Because, in practice, they are not, the preheat thickness was divided into a number of elemental thicknesses for which this solution was employed, with mean values of Cp and k, appropriate to the strip. The lowest temperature was arbitrarily taken as (Tu + l) °K and the preheat zone thickness, 8p, evaluated from the sum of all the elemental thicknesses:
Xn,T t
212
D. BRADLEY and A. MITCHESON
ISTEP"n"thMASSDECREMENT,dmo,.I [CALCULATEIDEALEOUILIBRIUMTEMPERATURE,TF.] [CALCULATE FLAME THICKNESS a~w I
t
ICALCULATE BURNING VELOCITYfSu I ICALCULATE UNBURNT MASS, mul
-
T
IESTIMATE PRESSURE ,Pn I
t
CALCULATE ISENTROPIC TEMPERATURE R SE N UNBURNT GAS , Tu
t
ICALCULATE UNBURNT GAS
VOLUME ,Vu..I
t
ICALCULATE FLAME VOLUME.V,.n I
T
C~CULATE MASS WITHIN FLAME~rnf n I
t
CALCULATE COMPOSITION~ISENTROPIC TEMPERATURE RISE.Tb.n AND GAS CONSTANT, RobpOF EACH PREVIOUSLYBURNT GAS ELEMENT
t
ICALCULATE BURNT GAS VOLUME ?Vb,n I
[CALCULATE TIME, dt I
YES
NO
Fig. 3. Flow Chart for Complete Computer Solution.
It was assumed that h/ho = (T/To) °'s [33], where ho = 0.025 Jm -1 sec-1 (°K)-~ for 10% methane-air at the reference temperature, To = 273 °K [34] and values of Cp were interpolated from tables [35, 36]. To obtain the thickness of the reaction zone, 8,, the mean temperature gradient across it, (T I - T~g)/Sr, was equated to the temperature gradient at T~g which was obtained from the preheat zone analysis. The flame thickness was found from the addition of 8p and 8r. Such values of thickness were found to be lower than those measured experimentally [28, 37]. In Reference [28] a flame thickness of 1.2
stoichiometric methane-air flame at atmospheric t e m p e r a t u r e and pressure. The t h e o r e t i c a l l y derived value of t h i c k n e s s therefore was multiplied by a factor to make it equal to the experimental value. The same value of this factor was used for all determinations of flame thickness at the different pressures and unburnt gas temperatures.
5.2 Results from the Complete Computer Solution Results are presented for a stoichiometric methane-air mixture at an initial temperature and pressure of 292.1 °K and 1.014 x 105N m-z, respectively with central ignition in a sphere of 160.2 mm diameter. Figure 4 shows the radial progression of the flame front through the mixture. The broken curves show the trajectories of imaginery particles initially situated at the quarter, half and three-quarter radius positions. The gas flow reversal that occurs at the flame front is apparent and the slope of the curves gives the gas velocity at that radius and instant. Close to the centre unburnt gas velocities of more than 2 m sec -1 are observed. Close to the walls inward burnt gas velocities are larger than outward unburnt gas velocities. The radial distribution of burnt gas temperature at the instant of peak pressure is shown in Fig. 5. The first burnt is at a temperature more than 500 °K greater than the last burnf gas. This distribution affects the theoretical value of maximum pressure of an explosion and this is shown in Fig. 6, where values of P J P o are plotted against equivalence ratio, @. The results are for an initial temperature and pressure of 298 °K and 1.01325 x l0 s N m -2, respectively. The full line curve was obtained using the computational method described in the previous section. The broken curve was obtained on the assumption of instantaneous constant volume combustion of the initial mixture. It is seen that the latter assumption leads to a slight underestimation of the maximum pressure, the difference, however, being very small. Both methods neglect energy transfer to the walls and experimental values of maximum pressure are lower than those of the full line curve.
213
EXPLOSIONS IN SPHERICAL V E S S E L S 0.06
0-06
UNBURNT
0.05
/
GAS
~" /
0-0~
~....\..
/
0-02 . . . . . . . . .
~ ~
~ ~ " Z ~"
" BURNT
0.0
~
"V~
GAS
I
I
0.01
I
0-02
I
0,03
TIME (sec)
I
0-04
0.05
Fig. 4. Flame propagation and particle trajectories. Stoichiometric methane-air. 30OC
290
t 280
'~ 27O(
o. :E 0
260(
250(
240
230
0
0.~
0!0~
0.'03
070,
0.$5
0.00 '
0,0~ '
RADIUS(metres)
0108
0.09
0!1
Fig. 5. Burnt gas temperatures at instant of maximum pressure. Stoichiometric methane-air, Po = 1.01325 × 105N m-2, To = 298 °K.
tained for a range o f initial t e m p e r a t u r e s and p r e s s u r e s o f s t o i c h i o m e t r i c methane-air. T h e v e s s e l r a d i u s w a s t a k e n as 0.1 m. It w a s
f o u n d that the f o l l o w i n g e x p r e s s i o n g a v e a g o o d c o r r e l a t i o n for the o b s e r v e d p r e s s u r e rises:
214
D. 1.025 Nm-2 ,
C? -%
(61)
for lo5
and 290 < To < 450 “K.
The pressure-time relationship for a single stoichiometric methane-air explosion is shown in Fig. 7. The conditions correspond to the experimental ones of Garforth [381 whose measured points are indicated. The complete computer solution, Case 3, gave the full line curve. Agreement between theory and experiment is good until the later stages of the explosion, when measured pressures and rates of pressure increase fall below theoretical values. The measured peak pressure is approximately 1% below the theoretical value. These discrepancies are not surprising in view of the neglect of energy transfer to the vessel walls and between burnt gas elements in the computer programme. SO-
BRADLEY and A. MITCHESON
6. Discussion Also shown, by the chain dotted curves, in Fig. 7 is the complete computer solution with a constant burning velocity value of 0.45 m set-‘. This deviates significantly from the complete computer solution, which makes allowance for changes in burning velocity as a result of isentropic compression, and shows the importance of making this allowance. Indeed, it is seen from Fig. 7 that the Case 1, approximate computer solution, shown by the dotted curve, in which the variation in burning velocity is given by Eq. (45), gives a more accurate pressure-time relationship than does Case 3 in which S, is constant. The validity of Eq. (1 I), upon which both Case 1 and Case 2 solutions rest, was examined using the accurate Case 3 solutions. The points shown on the graph of fractional pressure rise, P,, against fractional mass burnt, n, in Fig. 1.d were obtained from the Case 3 solution for a stoichiometric methane-air mixture. Equation (11) gives the straight line. The closeness of the points to the straight line suggests a sound basis for Eq. (11) and its application in Case 1 and Case 2 so-
COMPLETE COMPUTER SOLUTION
&5-
CONSTANT
VOLUME
SOLUTION 8.0 -
t
7.5 -
a”lB 7.
-
6.5-
6.0 -
5.5-
5.0’ 0.4
0.5
0.6
I
I
o.7
0.8
1 0.9
ECGLENCE
I ZATIO
1.2 $I -
Fig. 6. Maximum pressures attained for different methane-air equivalence = 0.t m.
1
I
I
1.3
1.4
1.5
ratios. Vessel radius
EXPLOSIONS IN SPHERICAL VESSELS
2]5
~0 X
EXPERIMENTAL POINTS
(38)
//
//1
8.0
CASE 3 BUT WITH Su =0-/45 m SeE"t (CONSTANT)
/,,
7.0
o
//I
6o
K 7E
z
5.(3
4.0
3.0
2.G
1.0
()
I 0-01
I 0-02
I 0.03
I 0.0/*
I 0.05
, I 0-6
TIME (s¢cs)
Fig. 7. P r e s s u r e - t i m e c u r v e s for s t o i c h i o m e t r i c m e t h a n e - a i r . P o = 1.014 x V e s s e l r a d i u s = 8.01 c m .
lutions. The Case 3 solution values are in good agreement with those calculated by Rallis and co workers [39, 40], but not with those of Perlee et al. [10], which are also shown in Fig. 1.d. In view of this satisfactory finding, the question arises as to whether the Case 2, Dimensionless Universal Expression solutions, such as those given in Fig. la-c might be both valid and of wide applicability. They have the advantage of not entailing the use of computers. The Case 2 solutions, however, are based on the value of S= remaining constant throughout the explosion. For a stoichiometric methane-air mixture with (Po)-I = 8.99 and 3' = 1.36, the Case 2 solutions gives S=/Sm = 0.244. The more accurate Case 3 solution gives Sm = 1.613 m sec-~ in a sphere of 160.2 mm diameter and an arithmetic mean value of burning velocity of 0.48 m sec -~. With these two values, S,,/Sm = 0.298. The dimensionless pressure-time points
l(P N m -2, To = 292.1 ° K ,
given by the Case 3 solution for (po)-i = 8.99, are shown in Fig. la. These points lie very close indeed to the Case 2 solution for SdSm = 0.3. This suggests that if the arithmetic mean value of burning velocity for the explosion is taken for S=, then the solution curves for the Dimensionless Universal Expression are valid. In Section 3.3 Eq. (40) suggests that the pressure rise is proportional to ta during the early stages of an explosion. The complete c o m p u t e r solutions for stoichiometric methane-air give Pr = t 4.5
(62)
for o . o l "<
This is in close agreement with the exo perimental results of Perlee et al. [ 10] which
216
yield a time exponent of 4.3, but is much higher than the value of 3.5 computed from Garforth's data [38]. 7. Conclusions (1) The basic equations and associated assumptions for flame propagation in a spherical vessel have been reviewed. Different solutions have been compared~with Complete Computer Solutions, which involve minimal assumptions. (2) Support has been found for those solutions which rest upon the simplifying assumption that the fractional pressure rise is proportional to the fractional mass burnt. (3) The simplest solution to apply is that of the Dimensionless Universal Expression in which the burning velocity value is the arithmetic mean of initial and final values. (4) The Complete Computer Solutions show the importance of accurate values of burning velocity. The concept of isentropic variation of burning velocity is introduced and values suggested for methane-air mixtures. (5) The Complete Computer Solution is in good agreement with accurate experimental values of the pressure-time relationships. Theoretical pressure rises are predicted from this model. (6) For a significant period of an explosion it has been demonstrated that/3 ~ 74.5. We are indebted to our colleagues, Mr. Don Appleyard, Dr. Ian Swift, and the late Mr. Steve Arbury for their contributions towards the d e v e l o p m e n t o f the Complete Computer Solution. We thank the British Gas Corporation for a Research Scholarship to Alan Mitcheson.
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