Exact solutions and ignition parameters in the Arrhenius conduction theory of gaseous thermal...

Journal of Applied Mathematics and Physics (ZAMP) 0044-2275/82/003392-14 $ 4.30/0 Vol. 33, May t982 �9 Birkh~iuser Verlag Basel, 1982

Exact solutions and ignition parameters in the Arrhenius conduction theory of gaseous thermal explosion*

By K. E. Gustafson, Dept. of Mathematics, and B. E. Eaton, Dept. of Chemical Engineering, University of Colorado, Boulder, Colorado, USA

1. Introduction

Of principal importance in the pure conduction theory of gaseous thermal explosion is the calculation of the critical ignition parameters 2 and e for the equation

AO+2exp[O/(l+eO)]=O in f2, (1) 0 = 0 on 8 0 .

In this equation, which we shall refer to as the true Arrhenius equation, e, 0, and 2 represent (up to scale factors, see the following section) respectively the surface temperature and the interior temperature of a reaction vessel (2, and the exothermicity of the reactant mixture.

All of these quantities are nonnegative and it is known that for e = 0 there is a critical 2c beyond which solutions no longer exist. The latter is taken as the condition of inflamation, since it marks the limit for which the rate of heat produced in the reaction term may be matched by the rate of heat dissipation in the volume. We will wish to refer later to the equations in this approximation, which is usually called the Frank-Kamenetskii approximation [1], that is,

AO+2exp[O]=O in f2, 0 = 0 on 8 ~ . (2)

For the true equation (I), there is no such finite 2c and positive solutions exist for all positive 2 and e. For small e > 0 it is known that there exists a critical 2c(e) for which the smallest positive solution of (1) reaches an infinite change of temperature with respect to 2. This value of 2c is then

*) Partially supported by NSF Grant NSF MCS 80-12220.

Vol. 33, t982 Exact parameters in the Arrhenius theory of thermal explosion 393

taken as the critical ignition parameter for a given e. The largest e for which such ignition occurs is therefore a critical value ec of principal importance and has been the subject of numerous investigations.

For purposes of studying the true equat ion (1) and in particular the critical value ec, Bazley and Wake [2, 3] recently introduced the model equation

A 0 + 2 e x p [ 0 / ( I +e l f011) ]=0 in f2, 0 = 0 on ~f2, (3)

where ]l 0 i[ = max 10(x)[, x ~ O . This equation is analytically solvable for the cases of infinite slab and infinite circular cylinder geometries, respectively. For the case of a sphere geometry, it may be solved in terms of a known (in tabulated form, that is) isotherm function ~r. In this way Bazley and Wake [2, 3] obtain some interesting comparisons and conjecture that equation (3) will serve to closely approximate the qualitative behavior of the true equation (1).

By means of careful numerical calculation and some analysis we will show that the conjecture while true to some degree has a number of important limitations. In particular we obtain the exact solutions and the exact (2, I] 0 [[) bifurcation diagram for the sphere geometry case for the true Arrhenius equation (1). F rom these results we show that, a l though solutions of (1) and (3) are reasonably close along their minimal branches, and although generally (3) better approximates (1) than does (2), equations (1) and (3) exhibit radically different behavior on the higher branches. Also, for the model problem (3) the bifurcation diagrams are monotone in e, whereas for the true problem (1) there can occur crossings.

Solutions to the true equation (1) have, in general, a wider "hot spot" than those for the model equation (3), and a shape significantly different from that for the Frank-Kamenetski i approximat ion (2).

Further conclusions and dicussion will be found in section 4.

2. Background and formulation

The steady state energy balance equat ion for a selfheating chemical whose reaction velocity follows the Arrhenius law and which dissipates energy by conduction only is

{ A T + k - I Q o A e x p ( - E / R T ) = O in f2 T=To on ~s (4)

where T = temperature, k = thermal conductivity, Q = latent heat of reaction (energy/mass), 0 = density (e.g., concentration), A = Arrhenius preexponen- tial factor, E = activation energy, R = gas constant, and To is a nonzero

394 K.E. Gustafson and B. E. Eaton ZAMP

boundary temperature. All coefficients are assumed constant. Introducing dimensionless variables

O=(E/RT~) ( T - To) and e= RTo/E ,

one obtains the dimensionless form equation (1), where

EQ oA 2 = - - exp ( - E/R To). (5)

RT~ok

We note in passing that equation (1) follows from equation (4) immediately from the identity 0/(1 + e 0) = (E/R To) - (E/R T).

Note that the Frank-Kamenetskii approximation (2) for a finite nonzero surface temperature T o corresponds to assuming that the chemical has an infinite activation energy whereas letting e ~ oo corresponds to assuming a finite activation energy in an infinitely hot chemical. Similar implications for 2 written in the form i -1 = O(P E exp (e-l)) are available. Both dimensional and physical analyses indicate that 0 is an appropriate scaling of the temperature increment in the explosion range.

For more background and in particular interpretations of subcritical 2 < 2c and supercritical 2 > 2c, see the book by.Frank-Kamenetskii [4]. For more physical theory see the several journals devoted exclusively to flame and combustion problems. For recent developments emphasizing energy asymptotics and fluid mechanical approaches see the forthcoming book by Buckmaster and Ludford [5].

For a short introduction to the bifurcation diagram, which will be the starting point of our approach, and to the solvable ignition problem in one dimension, see Gustafson [6]. For an extensive review and large bibliography of related mathematical methods exployable for existence proofs see Amann [7]. For some references and results later than [7], see Bebemes and Kassoy [8]. For even later information and as an indication of the wide range of current activity on such problems, see the symposia talks at the 1981 SIAM meeting [9].

An important paper giving further earlier references is that of Joseph and Lundgren [10], where, among others, the equation (2) is rather fully analyzed for all dimensions via a phase plane approach with e ~ obtained as the limit of the nonlinearities (1 +/~-1 0) ~ as /~ increases without bound. Spherical symmetry is assumed in [10] and in many earlier works.

By use of the recent result of Gidas, Ni, and Nirenberg [11], it may be asserted that all solutions of equations (1), (2), and (3) on a spherical geometry s are themselves spherically symmetric. Thus all partial differential equations (1), (2), and (3) may be studied as ordinary differential equations and the currently and previously obtained results for the latter will be complete. We proceed now to the latter.

Vol. 33, 1982 Exact parameters in the Arrhenius theory of thermal explosion 395

Expanding the Laplacian in one dimension under the just mentioned condition of a spherical domain f2 of radius normalized to one, the Arrhenius equation (1) becomes

d20 J - ' d 0 + 2 e x p ( 0 / ( l + e 0 ) ) = 0 0 - < r < l , (6) dr ~ + r dr

0'(0) = 0 (1 ) = 0 ,

where j = 0 for the infinite slab geometry, j = 1 for the infinite cylinder geometry, j = 2 for the sphere in three dimensions. We will emphasize the latter most important physical case in our calculations.

3. Numerical considerations and results

Equation (6) was analyzed by a scheme based upon a shooting method combined with a Newtonian-Raphson technique and certain boundary value expansions. This enabled us to obtain the exact solutions, bifurcation diagrams, and critical ignition parameters for the true Arrhenius equation (1). Minor modifications then permitted the same results for the model equation (3) and the Frank-Kamenetskii equation (2). A full description of the numerical scheme and a complete tabulation of solutions along with their bifurcation curves and critical ignition parameters may be found in a report [12] currently under compilation. The essential details of the method are given here in the Appendix.

To obtain high precision values of the critical ignition parameters ec and 2c we then extended the method used by Kubi6ek and Hlav~ek [13] to calculate 2c given e, by consideration of the curvature characteristics of the bifurcation diagram. See the Appendix. The results (see Table 1 at the end of the paper) are accurate to six significant figures with higher accuracy possi- ble if desired.

Figure 1 through Figure 8 illustrate the results obtained. Figure 1 shows the bifurcation diagrams for values of e = 0, 0.17323, and

0.2138 for the model equation (3) in the case of infinite slab geometry (j = 0). This similar to the bifurcation diagrams of [2, 3, 10] and for e = 0 is the bifurcation diagram for the Frank-Kamenetskii equation (2). Note that 2c=0.88 for the latter, in agreement with Frank-Kamenetskii 's results [1], and that for the model equation (3), we found er to four digit accuracy, in agreement with the value reported in Bazley and Wake [2], and that 2c = 1.303.

Figure 2 shows the bifurcation diagrams for values of e = 0.1732 and 0.2138 for the true Arrhenius equation (1) for the same case of infinite slab geometry. The value of e = 0.17323 is the critical value e~ for the model


A1-6 1.4 ~-0.2138

1.0 0.8 O.E

0.4 ~ ~0 0.2

% 2 4 6 8 10 17 1'4 1'6 18 ilOi] Figure 1 Bifurcation Diagrams for the Frank-Kamenetskii Equation (2) and the Model Equation (3) for Slab Geometry.

equation (3) in the case of infinite cyl inder geomet ry ( j = 1) and, as on Figure 1, leads to a corresponding bi furcat ion curve below criticality for the infinite slab geometry. Note also that a l though the qual i ta t ive features are the same, quant i ta t ive ly the bifurcat ion curves for the true equat ion (1) fall below that for the model equat ion (3) shown in F igure 1. F r o m the relat ion

O 0 -- --< 0 (7)

l + ll01i l+ 0-

for nonnegative e and 0, it follows immediately that a solution to the model equation (3) is a lower solution for the true equation (1) and that a solution to (1) is a lower solution to the Frank-Kamenetskii approximation (2). As pointed out in [2], this provides upper and lower bounds for the solutions to the true equation (1) so long as one remains on the minimal branch. Note that Figures I and 2 give quantitative refinement to the just mentioned qualitative comparison, e,g., see how closely the solution norms for (1), (2), and (3) agree for example at 2 = 0.6 and how the solution 0(0) to the model equation (3) is only slightly less than the solution 0 (0) to the true equation (1) at ,t = 1.0 for the case e = 0.17323. Note also, on the other hand, that on

tl. 4 1.2

1s

0.~

0.6

Figure 2 Bifurcation Diagrams for

e-0.2138

- 0.17323

O.4

O.2

the true Arrhenius Equation (l) for Slab Geometry. 1'4 1'8 llell


6

J

4 6 g 1'0 1'2 14 1'6 1'6 11(9ll Figure 3 Bifurcation Diagrams for the Frank-Kamenetskii Equation (2) and the Model Equation (3) for Sphere Geometry.

the upper branches the just mentioned qualitative comparison is reversed and weakened. This is what we meant when saying the curves for (1) fall below those for (3). Solutions for the model equation are larger, and considerably larger, than those for the true equation. For example at 2 = 1.0 the solution norms are 6.0 and 3.7, respectively. This is a quantitative beginning of a limitation on the qualitative approximation of (1) by (3) hoped for in [3].

Figure 4 2 4 6 8 1'0 1~2 14 16 18 IIOll Bifurcation Diagrams for the true Arrhenius Equation (1) for Sphere Geometry.


Figure 3 shows the bifurcation diagrams for values of e = 0, 0.1, and 0.13 for the model equation (3) for the sphere geometry. These agree with those of [10, Figure 1] for the e = 0 case of equation (2) and for ~ > 0 are not unlike those in [3, Figure 1]. Although not plotted in Figure 3, ec was found to be 0.124095 in agreement with the value reported in [3], and 2c = 4.41184.

Figure 4 now gives for comparison the bifurcation diagrams for e = 0.1, 0.2, 0.3, and 0.005 for the true equation (1). Like Figure 2, these are new bifurcation curves for the exact model (1), and, as pointed out in [3], are of high interest physically in trying to understand phenomena such as double ignition and double extinction. The critical ignition parameters were found to be ec = 0.238797, as compared to the value ec = 0.2389 obtained in 1978 by Kordylewski as reported in [3], and ).c = 5.04111. Also of particular interest is the crossing and failure of monotonici ty of the curves exhibit in Figure 4.

Figure 5 (note the three different temperature scales used) shows that if one goes to high enough temperatures the bifurcation curve for equation (1) for the sphere geometry eventually will turn back up as required by the theory. We were at first somewhat surprised at how far out on the graph we had to go to confirm this. As shown, for e = 0.1, the m i n i m u m 2 ~ 0.12 was not achieved (modulo temperature increments of 10) until [I 0 [[ = 140, 2 was still below 0.2 at !! 01[ = 530, and 2 did not exceed its early m a x i m u m (modulo temperature increments of 0.5) of 3.8 achieved at ]] 01] = 2 until It 0 II = 13 700 (modulo temperature increments of 50). This behavior becomes however, transparent upon solving either of the differential equations (6) or (7) with ( 0 / l + e S ) ~ e -1 for large 0, f rom which it is seen that 2 - ~ 6 e x p ( - I / e ) II 0 ][, an asymptotic behavior for 2 thus monotonically increasing in e.

Figure 6 is a solution plot for the Frank-Kamenetski i approximat ion (2) for the sphere geometry. Note that as Ii 0 [[ increases, the solution of (2) approaches a 6-function behavior and resembles a localized hot spot.

Figure 7 is a solution plot for [[ 0 n = 5, 10, 15, and 20 of the model approximation (3) for the sphere geometry with e = 0.1. Each of these solutions is

Figure 5 Large Temperature behavior for the Bifurcation Diagrams of the true Arrhenius Equation (1) with small e parameter.

A 4

~ !, SO00' SO'O0 ' 10(}00' 12000' 14'000'11ell


16

tlell 14

Figure 6 Frank-Kamenetskii Equation (2) solution behavior as a function of center temperature.

8

4 15

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110r

IIOII

18

121 ~\;t = 4.994 ~,,/ ~A- Z= 4.261

6I)1.: 3.884

Figure 7 Model Equation (3) solution behavior as a function of center temperature. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 tOr


2~

tlOlr 1

Figure 8 True Arrhenius Equation (1) solution behavior as a function of center temperature.

. . . . . . . . . . . . . r

on the minimal branch. Note that the model equation (3) has a wider hot spot than does the solution of (2).

Figure 8 is a solution plot for ]1011=5, 10, 15, and 20 for the true Arrhenius equation (1) for the sphere geometry, also with e=0.1. All of these solutions lie beyond the minimal branch. Note that the true solutions spread the hot spot even more than those for the model equation (3) approximation plotted in Figure 7, and that the latter are more realistic, as hoped for in [3], than those for the Frank-Kamenetskii approximation (2).

4. Conclusions and discussion

1. Exact solutions, bifurcation diagrams, and ignition parameters are obtained for the true Arrhenius equation (1). These are new, and as pointed out in [3], of high importance in understanding the nature of explosive flamability. They reveal behavior at higher temperatures quite distinct from the Frank-Kamenetskii approximation (2) and significantly although less distinct from the Bazley-Wake approximation (3). Full tabulation may be found in [12].


2. Some agreement and important limitations are seen in the qualitative approximation of (2) by (3) conjectured in [3], implying also and more generally the need for better general theory for qualitative comparison on other than the lowest branch of nonlinear eigenvalue problems.

3. The exact Arrhenius equation (I) possesses bifurcation curves which fail in monotonicity in e.

4. One can describe and contrast our approach as follows. By use of Chandrasekhar's tabulated values of the isothermal functions, Bazley and Wake [3] were able to resolve the approximate model equation (3), whereas in the present paper we resolve the true Arrhenius equation (1) directly. This approach possesses two significant advantages. First, it enabled the exact calculation of ec for the true Arrhenius problem (1). Secondly, very accurate solutions themselves are obtained.

In particular, this resolves the difficulty pointed out in [2, end of section 1] as concerns ec and the important question of resolving the exact equation (1) as pointed out in [3, near the end of section 3 (b)].

5. Rather than solving for A = ]1 0 It with 2 specified, which can cause numerical difficulties near branching points, it is generally better to, as we did here, specify A and solve for 2, thereby obtaining all solutions and the full bifurcation diagram. This brings us back to an old folklore truth that 2 should be plotted vertically in the bifurcation diagram rather than horizon- tally "so as to avoid vertical slopes".

6. Of some interest is the case of finite Biot number, wherein the surface temperature need not be constant. See for example Boddington, Gray and Wake [14]. Our scheme (see the Appendix) includes general boundary conditions and hence enables the calculation of critical ignition parameters for this case also.

Table 1 Critical Ignition Parameters for the True Arrhenius Equation (1) and the Model Equation (3).

Geometry

j = O j = l j = 2

Eqn. (1) ec 0.245780 0.242106 0.238797 2c 1.30737 3.00630 5.04111 ]]O[Ic 4.89654 5.94324 7.18494

Eqn. (3) 0.213815 0.173233 0.124095

2c 1.30273 2.88629 4.41184 ilO]ic 4.82835 5.33500 4.69978


Appendix

In this appendix we give the essentials of the numerical schemes we employ. More details and full results may be found in [12].

As mentioned in Section 3, the method is basically that of shooting. This is combined with a Newton-Raphson iteration for 2 from a boundary expansion of 0 about the endpoint r = 1. As ment ioned in Section 4.5, we recommend the advantages of first specifying 0(0) and then solving for 2 when seeking a full bifurcation diagram. Later (see below) we extend the method of Kubi~ek-Hlav~ek so that we may solve for both 2 and e in an exact criticality scheme including general boundary conditions.

Letting Yl = O, Y2 = 0 '= y~, equation (6) is equivalent to the first order system

/ Y~ = Y2 y~ = - ( j / r ) y z - 2 exp (yl/(1 + e yl)) yl (0) = A

y (0) = 0 (8)

where A = II 0 II. Note that by the symmetry and curvature in the problem all solutions are positive and that A = max~ 101 = [0 (0) l- Rather than adjusting A which would correspond to the traditional shooting approach, we will specify A and e and determine 2 by shooting for

y~ (1) = 0. (9)

Assuming that the solution is a continuously differentiable function of 2, we may write the Taylor series expansion of yl evaluated at the endpoint r = 1, truncated after the linear term,

y~(1; 2 i+1) = y~ (1; 2 i) + (2 i+1- 2 ~) dy~ (1) I (10) T 2i"

The desired endpoint is y~(1;2 i+l )=0. We adjust 2 by the Newton- Raphson technique

2i+x = 2i yx (1, 2 ~) yl,~ (1; 2 i) (11)

where we use the notation Y1,x = Yl, z (1; 2 ~) = dyl(1)/d2 evaluated at 2 = 2 ~. Finding a value for y~,z is accomplished by integrating auxiliary

equations obtained by differentiating the equations for y~ and yz with respect to 2. Assuming the solution is twice continuously differentiable in 2, we arrive at

y~,z = - ( j / r ) Y2,z- 1 + 2 exp(yJ(1 + eyD) y~,a (0) = 0 (1 + eye) 21 ]

Y2,a (0) = 0. (12)

Vol. 33, 1982 Exact parameters in the Arrhenius theroy of thermal explosion 403

Changing variables according to ya=yl ,~ , and Y4 =-y2,>., the equations (12) may be written as

y;~ = - ( j / r ) y , - [1 + ), (y3/(1 + ey,) 2] exp (y,/(1 + ayl)) (0) = 0

Y4 (0) = 0 .

The derivatives for the Jacobian matrix for an ODE solver f~ . . . . ,f4 denote the right hand side of the ODE system (8), (13):

= 0 , = I , ~Y, eYa 8y~ ~y~

are,

- 2(1 + eye) -2 expCvJ(1 + ey~)),

= - ( j / r ) , ey3 - y--7 0 ;

= Ofa= ~ f 3 = 0 8 f 3 = 1 . ~Y2 ~Ya ' ~Y4 '

exp (,Yl/(1 + gYl)) [ (1 + eyl) a (1 + eyl) = I

- - = 0, Of 4 = _ 2(1 + eyl) -~ exp(yl/(1 + e y 0 ) , OY3

auxiliary

(13)

where

(14a)

(14b)

(14c)

= -- ( j / r ) . ( 1 4 d ) ay4

The system (8), (13) was solved using the "Gear" package of Lawrence- Livermore Laboratories (multistep, variable stepsize, variable order, Adams method). This provided the needed values yl(1; 2 i) and yl ,a(1,2 i) for the Newton-Raphson iterations�9

For the first t ime step, the limit as r -+ 0 may be seen to be

lim (j/r) dO~dr = 0"(0) . (15) r~0

This follows from 0' (0) = 0 and elementary considerations. F rom this the differential equation (6) becomes

d20 2 [ - d - 7 + - ~ e x p ( 0 / ( l + e 0 ) ) = 0 , r = 0 ,

[ 0 ' ( 0 ) = 0 ( 1 ) = 0 . (16 )

From (16) there results a modificat ion of equations (8), (12), (13), (14), the details omitted here, which was then used for r = 0, followed by the original (8), (12), (13), (14) for r > 0.


The model equation (3) was treated in the same way. We omit the detailing of the modifications needed as these are obvious and may be supplied by the interested reader.

We obtain the full bifurcation curves by starting with A = 0 and incrementing by a user-specified AA. Very accurate values of 2c(e) can be obtained in principle by refinement of AA. However, it is more efficient to employ the method of Kubi~ek and Hlav~trek [13] which includes an expression for d2/dA = 0. The method described above is used to evaluate the quantities in that expression.

For calculation for the inflection point 2c(ec), we need in addition to be able to include an expression for dZZ/dA z= 0, and these considerations require an extension of the method of Kubirek-Hlav~t~ek [13]. Briefly, let

FI (A, 2, e) = e~ y~ (1, A, 2, e) + fl~ y~ (1, A, 2, e) - ?q. (17)

From the (now general) boundary condition

FI(A, 2, e) = 0

the critical condition ~2/8A = 0 corresponds to

84 = ~Fa/~A 1).,~ = O.

For the critical condition ~22/~A2 -- 0 one has

0F1 ~ZF 1 _ ~F 1 ~ZF 1 82 [. ~2F I

822 [ ~A 8A82 82 0A 2 ~A \ 8 2 ~ A

I ~A ~

(18)

(19)

82 ~A 822 / = 0 , ( 8Fl/2

82 / (20)

wherein it is clear which variables are being held constant. The full details of implementation of this method are somewhat extensive and may be found in [12].

References

[1] D.A. Frank-Kamenetskii, Acta Physicochimia URSS 10, 365 (1939) and 13, 738 (1939). [2] N. W. Bazley and G. C. Wake, The disappearance of criticality in the theory of thermal

ignition. J. Appl. Maths. Phys. (ZAMP) 29, 971 (1978). [3] G. C. Wake and N. W. Bazley, Criticality in a model for thermal ignition in three or more

dimensions. J. Appl. Maths. Phys. (ZAMP) 32, 594 (1981). [4] D. A. Frank-Kamenetskii, Diffusion and heat exchange in chemical kinetics. Translated by

N. Thon, Princeton University Press, Princeton, N. J. (1955). [5] J. Buckmaster and G. S. S. Ludford, Theory of laminar flames. Cambridge Monograph

Series in Applied Maths. and Mechanics, to appear. [6] K. E. Gustafson, Introduction to partial differential equations and Hilbert space methods.

John Wiley and Sons, New York (1980).


[7] H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Rev. 18, 620 (1976).

[8] J. Bebernes and D. Kassoy, A mathematical analysis of blow up for thermal reactions - The spatially nonhomogeneous case. SIAM J. Appl. Math., to appear.

[9] Symposium on Combustion, Final announcements of SIAM National Meeting. 8-10 June, 1981, SIAM, Philadelphia, PA.

[10] D. D. Joseph and T. S. Lundgren, Quasilinear dirichlet problems driven by positive sources. Arch. Rat. Mech. Anal 9, 241 (1973).

[11] B. Gidas, W. M. Ni, and L. Nirenberg, Symmetry and related properties via the maximum principle. Comm. Math. Phys. 68, 209 (1979).

[12] K. E. Gustafson and B. Eaton, Exact solutions, bifurcation diagrams, and ignition parameters in explosion theory. Univ. of Coldorado Report, to appear.

[13] M. Kubi~ek and V. Hlav~ek, Direct evaluation of branching points for equations arising in the theory of explosives of solid explosives. J. Comp. Phys. 17, 79 (1975).

[14] T. Boddington, P. Gray, and G. Wake, Criteria for thermal explosions with and without reactant consumption, Proc. Roy. Soc. London A 357, 403 (1977).

Summary

Exact solutions, bifurcation diagrams, and critical ignition parameters are obtained for the true Arrhenius equation for conductive gaseous thermal explosion.

Zusammenfassung

Exakte L6sungen, Verzweigungsdiagramme, und kritische Ztindungsparameter werden ftir die genaue Arrhenius-Gleichung ffir thermische Explosionen in leitenden Gasen erhalten.

(Received: September 9, 1981; revised: February 9, 1982.)

Exact solutions and ignition parameters in the Arrhenius conduction theory of gaseous thermal...

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