On automodelling processes in chemically active media

102 ON

AUTOMODELLING

PROCESSES

ACTIVE

IN

CHEMICALLY

MEDIA

A. S. P R E D V O D I T E L E V INTRODUCTION A great number of physical phenomena can be divided into two categories, sharply differing from each other from the viewpoint of their mathematical description. One category of phenomena is described by solving one or a set of simultaneous differential equations, with the given initial state of matter and pre-determined conditions on the boundaries of the volume it occupies. To this category also belong all the phenomena which are described by solving boundary problems. There is an equally large group of physical phenomena which by their nature permit one or a set of simultaneous differential equations but for which no boundary conditions can be set, even if the initial state of matter is known. The peculiarities of these phenomena permit their mathematical description on the basis of specific features of differential equations themselves, not on the basis of specific forms of integrals of differential equations. Such phenomena include all automodelling processes, i.e. all processes described by functions which do not contain time in some obvious form. The automodelling factor is especially manifest in coincident processes. In such phenomena these processes are self-coordinated by the time coordinate and we, therefore, cannot use any of the coincident processes as a time-measuring unit. It is common knowledge that any isolated process occurring within a certain period of time may be employed as the basic principle of design for any time-measuring instrument. For example, the motion of an isothermal surface in space is characteristic of thermal propagation in some medium. This motion may constitute the basic design principle of a thermal clock. Needless to say the above statement implies general possibility and not practicability. Due to their basic features a large group of phenomena comprising coincident processes cannot be used as a basis of the clock's design as these phenomena are described by functions which are obviously independent of time. In our opinion a special method of description of physical phenomena can be based, in some cases, on the above-

mer/tioned characteristic of coincident processes. Tile examples analysed below show that this method quickly leads to the desired end and discloses the logical sequence of physical features making up the picture of the phenomenon. Thus the value of this method in this sense alone should attract the attention of scientists and the method itself should be used wherever necessary. LAWS OF T H E R M A L P R O P A G A T I O N AND R A D I A T I O N As was pointed out above coincident processes always occur in a self-consistent, automodelling manner. Consequently, during thermal propagation in a chemically active medium there must be some isothermal surface travelling in space in accordance with the laws of thermal propagation and the laws of thermal radiation in chemical transformations. Let formula f(xl, xa, x3)= S represent this surface for each particular moment of time. As the surface travels, the right part of the equation will undergo changes. The automodelling processes always require a special coordinate # which will be determined as follows : ~ - S t Let us write the equation of thermal conductivity in the form of a linear equation: 0T

C,,p -~f = KAnT

(1)

Let us transform the equation relative to the variable ~. With this aim in view we then perform the following operations: ST St

~ d r . O 2 T _ d 2 T 1 {OS~'+ dT02S 1 t d~: ' ex~ d~ 2 t ~kOxJ d~ Ox~ t

1 dT A S

A 2 T = ~-~-

,

H2d2T

q- ~-d~e~~-

where H expresses the first differential parameter of the surfacer = S.

773

D E T O N A T I O N AND ITS I N I T I A T I O N E q u a t i o n 1 can now be written as follows:

- t ~d T (KA2S + C~,p~) ~ KH 2 d 2 T d~

These functions have the following derivatives: dy

T h e equation thus o b t a i n e d will now certainly include time; thus the described p h e n o m e n o n can be used in the design of the clock. We assumed t h a t , in accordance with the law given above, h e a t propagates in a chemically active medium. L e t us observe the kinetics of the process by a c o n c e n t r a t i o n of some substance. L e t y represent the concentration of this substance; then the rate of reaction m a y be expressed in the following way:

~Y = f(y)

d 2T

t d~

-- f ( y )

CveS=K +$~2] We now h a v e the following equation stabilized coincident processes:

(3)

(CvP)ISI-= K2- + (~t) ~ g2 which will be true only w h e n the second surface limiting the zone of coincident processes is following the first surface with the same velocity. By s u b t r a c t i n g the first of these equations from the second we shall h a v e :

(4)

thus we obtain a differential e q u a t i o n which can be applied to coincident processes of substance transformation a n d to t h e r m a l p r o p a g a t i o n in some medium. Let us place ourselves at some point in the m e d i u m with the instruments to instantaneously measure the t e m p e r a t u r e a n d c o n c e n t r a t i o n of some chemically active substance. Assuming t h a t as a result of observations we have found: y = ~o(t);

T=

for

Here index (0) means t h a t we should take initial state values. Since the process u n d e r consideration is stabilized the coordinate ~ must be constant a n d should coincide with the surface travel velocity limiting the beginning of the process. For the final state of the medium, i.e. w h e n the chemical transformation is completed, we h a v e the following e q u a t i o n :

Let us consider the process characterized by p l a n e symmetry, t h o u g h it is not necessarily so. I n this instance the first differential p a r a m e t e r of the s u r f a c e f = S will t u r n into a unity a n d the second into zero. Bearing this in m i n d we can write e q u a t i o n 3 as follows:

Kd~T ~

(5)

(Cvp)oSo = K2 + (~d~ SoK g o g~

d~: d~ f ( y )

C ~2dy d T t vPr ~ ' f ~ ) ~

Sz

Now we can write e q u a t i o n 4 as follows:

(2a)

1 (KA~S + C~p~) = KH 2 d~T

2S

de

Since the processes are self-consistent, time t m a y be subtracted from equation l a with the aid of equation 2a. H e n c e :

~dy d T

S

Since ~bt coincides with the function f(y) we have : d2T dy 1 S d# ~ (~ ~S]

We have not set a task of analysing one particular case; w h a t we are trying to do is to show, in the clearest m a n n e r possible, a general idea which m a y prove to be quite helpful in m a n y instances. Let us transform e q u a t i o n 2 relative to the variable ~. As a result we h a v e :

~dy

dT

d ~ - 4, g + ~

(2)

6t

S

(la)

Cvp6S + S6(Cvp) = ~

-~6S +

$6 ~

(6)

I n this case symbol 6 denotes finite differences, a n d the b a r means t h a t the arithmetical m e a n is taken from the initial a n d final values of a certain quantity. E q u a t i o n 6 thus obtained can be written as follows:

r

Substituting time t in these functions by obtain :

SlY, we but this e q u a t i o n must contain neither variable t nor v a r i a b l e S in obvious form. Therefore the terms within the brackets t u r n to zero, a n d this 774

A U T O M O D E L L I N G PROCESSES IN ACTIVE MEDIA shows what purpose should achieve:

the

following equations

q~t/o

~kCvp ~t/1

T h e expression thus obtained proves that we cannot always count on describing the normal front velocity by nominal formulae. So far we have considered a quiescent medium, but it is not difficult to make the necessary calculations for a moving medium. In this case partial time derivative in the equation of thermal conductivity should be substituted by complete time d~rivative. This generalization will bring about a substitution of the variable ~ in equality 3a by the following value:

(1)

As soon as the ignition starts the temperature of the medium cannot be considered independent of the concentration of the chemically active substance. Let us express the relationship by: T = q~(y) hence:

aT

oq~

Oy

: --;By 02T

a2~

O~ =

2

,.,

,

"~

(~

yJtyJ c~

-

wn) = h)

The rate of reaction cannot be expressed by partial derivative either and it should be substituted by a full derivative. This generalization will cause changes in equality 2a; instead of variable # it will contain value h . In the final formula, 4, factor 2 2 will be substituted by h 2. Thus, with the front stabilized its propagation velocity will be expressed by the following ratio:

(8)

8t 2 -- a y i f (Y) + ~ y f ( Y ) To fix the initial point we will assume that the concentration y of the chemically active agent is equal to zero. T h e first of equations 7 can be written as follows :

113 = [a ~'r 0--~--~ f(O ) +

Index 0 shows that the square bracket refers to the initial point.

This is a general formula expressing the velocity of the normal flame front. As can be seen, to calculate the formula it is necessary to know the conditions of ignition and the law of chemical reaction. It stands to reason that the normal flame front velocity can also be deduced from the second formula 7. For this purpose we will assume that the concentration of chemically active agent in formulae 8 is maximum, this, however, involves considerable difficulties because frontal burning is attended by incomplete combustion. Let us write formula 9 as follows: g2=

FACTORS CONTROLLING THE PROCESS OF COMBUSTION We shall now try to establish factors controlling the process of combustion of particles in heterogeneous medium. Let us place the origin of coordinates in the centre of a spherical particle. T h e equation of thermal conductivity will have to be written in terms of spherical coordinates. I f we transform this equation relative to variable ~, then in terms of spherical coordinates we shall have : dT ie~_S 2 -- t d~ K ~ ar2 + r ~-r + CvP~ = K d ~ - ~

(K ] ~u(O)Yc0' [1 ' fy(O)~(O)-]

By extracting a square root from this expression we shall find: t ~(0)

g:ia~f(O>l

~ t . f I , f~(o)4~(O)/t / •

After subtracting time with the aid of the equation for chemical reaction rate we get the following equation :

(lOa)

where a stands for temperature conductivity of the medium. I f the second addend in the expression (under the second square root) is considered a small value as compared with unity, then formula 10 can be simplified by substituting the second root with its approximate value. We then have: g

[a

/

~,,(0) _ ~t a~f(O)(~u(O))~

(I I)

(9)

~ f(y)k

~~r~ + r Or] +

dS 2 (12)

In the case of spherical symmetry we shall have for the surface f - - S: (x~ + x~ + x~)~ = r

Consequently the following ratios are true:

02S at:

(10b)

775

O" 8S Or

1

D E T O N A T I O N AND ITS I N I T I A T I O N Variations in the quantity of substance of the new product may be written in the following way:

hence, formula 12 can be written as: ~d~

+ ~

= a

(12a)

d M " _ 4rrr~h ~y

The first addend in the square bracket represents the velocity of travel of an isothermal surface; the second addend expresses the flame front velocity which is constant for the stabilized state in self-consistent processes. T o prevent the front against breaking off from the particle it is necessary to have equality of these two values in absolute value and difference in signs. I n other words the terms in the square bracket must turn to zero, and it means that the second temperature derivative by variable ~ is equal to zero, i.e. we have: dZT --0 d~ 2 We have already seen that during the process of combustion temperature T cannot be considered independent of concentration; therefore we have: d 2 T _ d2cb [dy'l ~+ dq~ d ~ (13) d~ 2 dy2 t~-~) dyd~ 2 Function # ( y ) , connecting temperature with concentration, must possess the following properties. Temperature does not depend on concentration when the latter is varied between: 0 ~y
dy 0 ; ~-~ = const = -- B

(14)

The expression thus obtained with the aid of equation 2a may be written as follows:

ey _ bt

~2 dy ~ B~__2 r d~ r

(14a)

de

7

where r 1 represents the radius of combustion zone beginning, and h combustion zone thickness. Both variations in mass must be equal to each other; hence, we have : 2 dr = 4~rr~h B~e2 4,rr O ~ r

or denoting the surfaces of the spheres by S and Sa respectively, we have: dS= dt

8~rS~.hB~ 2 = 8 , S o

1 + ~ AS~ - | ] ~ Bh ~

2

(15)

The formula thus obtained presents considerable interest. Indeed if we consider value AS. = (S1 -- S) small as compared with the surface S and the combustion zone thickness constant, then the stabilized combustion process will go on with the rate of variation of the particle surface remaining constant. The characteristic feature of our line of argument is that we do not analyse in detail the process of changes in the mass of a particle; therefore the above holds good for the combustion process of a liquid drop and its evaporation; for combustion of a metallic part whose surface is liquid during burning; and, finally, for the combustion of a coal particle which is always enveloped with a cloud of after-burning carbon monoxide. The above law was experimentally found in 1882 by a Russian scientist, Srednevskey, when he was investigating the rate of evaporation of liquid drops. Later it was confirmed by V. A. Fedoseev and D. I. Polestchuk when they were carrying out research work in the field of combustion and evaporation of liquid drops. This law is also applicable to the combustion of sufficiently large coal particles as was proved by research made by V. I. Blinov. Formula 15 is interesting from another point of view: it shows in what conditions Srednevskey's law becomes invalid. I f the evaporating particle is located in the flow then in equation 12 variable ~ should be substituted by:

~- Wh=h Srednevskey's law will hold true in this case under certain circumstances, but first of all the following condition must be kept: aA2S + h = 0 ;

Variations in the quantity of substance of the burning particle per time unit may be expressed as follows: d M _ 4~r2 O __dr dt dt

4m~hB ~2

~

aA2S + ~ -- W n

(16)

and secondly, we should have an equality of the following kind : dy _ h B dt S 776

(16a)

A U T O M O D E L L I N G PROCESSES IN ACTIVE MEDIA I n the case u n d e r consideration value S must not be identified with the sphere radius; thus, the Laplace operator a n d value S in formulae 16 a n d 16a are left unexpanded. C o m b u s t i o n will take place in the m a n n e r illustrated in Figure la, the d a r k p a r t of the illustration being the c o m b u s t i o n zone. As c a n be seen, it h a s the shape of a p a r a chute, a n d this type of b u r n i n g we t e r m e d p a r a c h u t e combustion.

relationship between the angle of blow-out a n d the radius of the streamlined ball for the flow velocity of 10 m/sec. This relationship is illustrated in the following table.

~ D O~D

9

(a)

(b)

Let us a p p r o x i m a t e the volume of the d a r k e n e d area in Figure la by a n equivalent v o l u m e in Figure lb for the sake of simplicity of calculations. Let R 0 represent line OC in Figure lb, a n d R 1 line OD. Bearing this in m i n d we will solve the following integral: =

f.lfoo i,B JRo J0

~

(m)

(deg.)

0-038 0.060 0.084 0.120

144 151 156 158

Velocity (m/see)

0o

10

If, using the table, we d r a w a c h a r t showing the relationship of the function (1 - - c o s 00) to the ball radius, we shall o b t a i n a curve pictured in Figure 2 which shows t h a t the a b o v e relationship is a p p r o x i m a t e d reasonably well b y a direct line not passing t h r o u g h the origin of coordinates. T h e constants of a straight line are connected with flow velocity. T h e true relationship of the function ( 1 cos 00) is shown in the same illustration by a dotted line. It stands to reason t h a t in the area of small values of the radius of the streamlined ball in accordance with the curve there is a rectilinear section passing t h r o u g h the origin of coordinates; a n d it means t h a t the ratio (1 -- cos Oo)/r in this area should b e r e g a r d e d as constant. Since all the values constituting the right p a r t of equation 17 are constant we are justified in d r a w i n g the conclusion t h a t the law of Srednevskey to a

Figure I

dM" dt

Ball radius

r2sin0d0dr2~r

I n this, the integral value S in the first approxim a t i o n m a y be ide.ntified w i t h radius r. This condition allows us to express the mass of substance b u r n t per time u n i t in the d a r k e n e d spherical depression by the following: dM" dt = ~ h B ~ ( R ~ -- R0a)(1 -- cos 0o)

1-cosO

/

= ~I-rBzr(R 1 -- R0)(R 1 + R0)(1 -- cos 00) which can be written a p p r o x i m a t e l y as: dM' d t -- ~:hB27rR~

f

-- cos 00)

where h represents the average thickness of the spherical depression. O n the o t h e r h a n d the mass of b u r n t substance p e r time u n i t can be written as follows: d dt

~rap

_

/

0.05

Figure 2

_ 2 dt

dt

T h e comparison of the last two equations gives :

d___S= 4 ~ h B ~ _ h 1 -- cos 00 dt p r

(17)

A p p a r e n t l y we should take a n angle equal to (180 -- 00) for the angle of blow-out. Some considerable time ago, in the l a b o r a t o r y n a m e d after Zhukovskey of the Moscow U n i v e r sity, G. I. L u k y a n o v experimentally proved the

certain degree of precision holds good for the combustion of particles in the flow. Proceeding from the facts described above we m a y present the following description of combustion of double-phase mixtures. As long as the particle is not washed by the flow of oxidant (air) a zone of combustion is formed a r o u n d it. T h e i n n e r surface of the zone coincides with the isothermal surface corresponding to ignition point.

777 *~7A

0.1 r

D E T O N A T I O N AND ITS I N I T I A T I O N The distance at which this surface is located from the evaporation surface of the drop is determined by the equality of the rate of movement of isothermal surface and diffusive isoconcentration surface. Here we are dealing with a specific principle of Guy-Michelson. I f a gas flow is superimposed on this process, the isothermal surface of combustion zone will be distorted according to the laws of thermal propagation in gas flows. With the increase of velocity the combustion zone will break away and will hang over the drop like a parachute. Further increase of velocity will result in incombustibility of the drop. Now if we turn to the system of drops of such density that the drops are small as compared with average distances between them, which is always the case in practice, we can make use of the principle of 'superposition'. We can assert that the combustibility of such a mixture is determined by the limiting values of drop relative velocities at which their parachute combustion is possible.

778

The turbulization of the flow does not change the description essentially, on the contrary it contributes to the formation of parachute combustion. T h e presence of pulsating velocities m a y bring about rather high relative velocities of burning particles. Careful scrutiny of photographs showing burning turbulent flows may prove that in reality this phenomenon occurs in exactly the same way. T h e problem of turbulent combustion of double phase-mixtures in spite of its great practical significance is still far from b e i n g thoroughly investigated and understood. It is for these reasons that we deemed it necessary to put forward our views on the problem, particularly as the questions raised in this article have not been discussed in respective scientific publications. Apart from this, the views present a certain interest because on their basis one may come to a simple method of calculation of combustion of liquid fuel in modern jet engines.