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NAVORD REPORT 4 267

HE SOLUTION OF THE STEADY-STATE HEAT CONDUCTION EQUATION WITH

CHEMICAL REACTION FOR THE HOLLOW CYLINDER

.'• 18 APRIL 1956

ü. S. NAVAL ORDNANCE LABORATORY WHITS OAK, MARYLAND

1 NAVOHD Report ^.26?

THE SOLUTION 0^ THE STh-ADY-STAlt HEA? CONDUCTION EQUATION WITH CHEMICAL REACTION FOR THE H0Li>UW CYLINDER

by

J. W. ENIG

Approved by: E. C. Noonan Chief, Fuels and Propellants Division

ABSTRACT: The non-linear steady-state heat conduction equation which arises in the theory of thermal explosions, was solved by Frank-Kamenetzlcy for the case of tae semi-infinite slab, and by Chambre for tne solid cylinder and sphere geometries. Solutions for the case of the :ollow cylinder are presented and it is shown t:iat from these, the slab and solid cylinder solutions can be deduced as special cases. Trie design of large rocket grains, where seif-heating may result in "spontaneous" ignition during manufacture or storage, is considered.

U.S. NAVAL ORDNANCE LABORATORY White Oak, Silver Spring, Maryland

NAVORD Report ^267 l8 April 1956

These corr.putatlon^ were, performed under Task N0L-S6-2d-2-l-56J "Fundamentals of Solid Propellant Ignition and Burning". While present rocket grain webs are restricted to thicknesses which render spontaneous combustion of conventional propellants unlikely, the possibility must not be ignored. In particular, propellants with a low activation energy, low thermal con- ductivity, high density and high rate of decomposition are suspect. Within the assumptions made, the results of this study are believed to be accurate and are the responsibility of the author and the Fuels and Propellants Division of the Naval Ordnance Laboratory.

W. W. WILBOURNE Commander Captain, USN

E. ABLARD y direction

11

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•. NAVOKD Report U2t>7

TABU. OK CONTENTS

Page

INTRODUCTION 1

MATHEMATICAL MODEL 1

SEJ1I-INFINITE SLAB LIMIT 5

SOLID CYLINDER LIMIT 7

ENGINEERING APPLICATIONS 8

REFERENCES 8

li

Hi

• r ^ NAVORD Report [4.267

THL SOLUTION OF THE STEADY-STATE HEAT CONDUCTION EQUATION WITH CHEMICAL REACTION FOR THE HOLLOW CYLINDER

INTRODUCTION

1

Let a combust with heat loss to loss Is assumed to the combustible vo is that the energy greater than that when the heat loss the combustible is

ible undergo an exothermic chemical reaction the walls of the containing vessel where this take place via a conductive process inside

lume. The criterion for tiiermal explosion liberated during the reaction must be

lost through the surfaces. The condition exactly compensates energy production in described by the steady temperature state«

as

MATHEMATICAL MODEL

The steady state heat conduction equation can be written

XVlT=-.PaZe.f(-^) , (1) where "T is the combustible temperature, A the thermal con- ductivity, fi the density, GL the heat of reaction, Z, the frequency factor, E the activation energy, R the gas con- stant, and ^2 the Laplacian operator. It is assumed that the reaction is unimolecular. The solution of Eq. (1) for the semi- infinite slab immersea in an isothermal bath at temperaturei

Tt , under the assumption* T-Tk« Tfc was first ( Frank-Kamenetzky (D, With the same assumption, Cl showed that the solution of Eq. (1) for solid cylindrical and spherical geometries can be obtained in terms of known functions.

given by Chambrl(^)

Consider a hollow combustible cylinder of inner radius a. and outer radius b whose isothermal surfaces are at

temperatures JT^ and Tb respectively. To solve Eq. (1) assume ihat T-Tb «Tw Then to first ord^r in T-T. /\

» The unpublished numerical integration of Eq. (1) for the slab and solid cylinder performed at the U. S. Naval Ordnance Laboratory has shown that this assumption is very good for the range E/RT ^ b .

.

I NAVORD Report ^26?

Setting ■ö-2^!^1!») )?=Y'

Equation (1) becomes In cylindrical coordinates,

where r is the radial space coordinate and

(2)

(3)

(U)

Let 2sm whenrra • Equation (3) Is to be solved subject to the boundary conditions

0=0, 2=1 at the outer surface (5)

(6)

and at the inner surface,

0 = -e'm)2=m

r r NAVORD Report [(.267

which when compared with Equation (3), yields

The solution of this equation is

' 4

(io)

dl)

where

Boundary condition (5) gives

so that

Boundary condition (6) is used to evaluate A , yielding

(12)

(13)

(lil)

I- w -z* (15)

where *" = «xj>(-J^"n,) o Equation (12) still contains the parameter •ft whose relationship to T will be found. Sub- stituting Equation (Ik) into Equation (3) gives

S= 8i A(l-A). Eliminating A between Equations (1?) and (16) yields

3

(16)

(17)

I

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NAVORD Report ^267

which is the desired relationship between a and "R for given m and Jf • From Equation (17) it can be shown that for given m and Y , £ has a maximum critical value, Se , for which the

steady-state solution is still possible. When the value of S as defined by Equation (i|) exceeds tc , then thermal explosion will occur.

Introduce the new parameter

OC = W* (18)

so that Equation (17) becomes

\J2Avm/ rv^O-**)* a9)

Letting **r/jt/= 0 , It is foimd that the root, «c , of the

equation

oia-1

I ~)

NAVORD Report ^26?

If -f and h represent the energy flux and energy transfer rate per unit axial length, then

dT dT d r o ^ (25)

Equation (25) *

*„ = rx-yo^. 2.V

(28)

Substituting Equation (28) into Equation (21) gives

V"=-zX"-[;^| »*• l + ^c Cy-w\Kc)o(t i-6e

m(i-of*) (29)

Consider the case when the inner and outer surfaces are at the same temperature, i.e., X~\ , Table 1 gives the values of «c^c» F'ltm » F»»« »rt»a»v,»H»a» » 2. » and •ö1 _.- » for various values of rv» .

SEMI-IN FINITE SLAB LIMIT

From Table 1 it is seen that as »«-» 1 , o

r-

r NAVORD Report I4.267

JU, !S^ s-l j JU^^sJL^llflii.^ |.»«*4 (30)

where 0-a} . There is a symmetric temperature distribution in the slab whose isothermal faces have equal temperature« Hence the fluxes at the surfaces are of equal magnitude but of opposite sign, showing that energy flows out from both faces. The diraensionless temperature at the center of the slab* is a maximum and equal to 1,18614..

5 For a slab of thickness 2. b , Equation (3) is ^Tä ~~$*Xf(fi) , i-rank-Kamenetzky found the critical temperature distribution to be

where £=0.87846 , the maximum of s

s = 2 [c.st-' e.Kiem„)]%*(.(-*„,„), '*' occurs at & ? 1.1844- which is the root of

1

j l^fP^MJ [c<

O" is the slab center

1

["i>^»)-i]i[c.ir'..f(i^,x)]"r(-i*w.«)=i ■ (33:

temperature•

_J

— - — - — ■

■

NAVORD Report 1^.267

n

From Equations (20) and (3°)» Equation (33) may be obtained» Hence the temperature defined by Equation (30) does indeed correspond :o the slab center temperature. The critical value of S , S8 « for the slab of thickness x (b-a) , can be obtained i'rom Equation (I4.). This gives

«» _ k »• / _ (l-ry>)

»31 i

i i Therefore one finds

.

I

which has previously been found using Equation (32),

(3^)

SOLID CYLINDER LIMIT

The solution for the solid cylinder can be shown to be a special case of the more general hollow cylinder solution. This is obtained by setting D=0 in Equation (10) since the temperature gradient must vanish on the axis because of symmetry considerations, Tlierefore ■& ^ o at *vt= o • From Equation (16)

^c = 2 ; A = Y , (36)

so that from Equation (li+)»

•0-. = i*t4 -Zjkv(zz +0; %mM-JU4 (37)

These results for *, , "0\ , and -Q- mmmm , are indentical with Chambre's.

_J

.

NAVORD Report 426?

ENGINEERING APPLICATIONS

Of late there has been a trend toward the design of large rocket grains. This trend toward larger types has made it necessary to determine how large a grain can be made before it presents a spontaneous ignition hazard. It is unfortunate that all the necessary physical constants needed for this determination are available for only a very few propellants or explosives.

A sample calculation of the critical size of cordite propellant for surface temperature at 300oK and 3390K (1500F) is given. The numerical constants for corditev37 follows r

are as

£= 50,000 cal mole":L,-Z= ID21'8 ua c i -.. IO"^- cal cm-1 """^ -»-~"i

so"1, Q,= 770 cal cm"3, and A = 5*3 x 10"^ cal om~Jm sec~J deg"x. Table 2 gives the critical radius b (in meters) which corresponds to these two temperatures for various values of m (inner/outer diameter).

ACKNOWLEDGEMENT

The author wishes to acknowledge the help of Mr» L. D, Krider in programming the numerical work for the IBM Type 650 Magnetic Drum Data Processing Machine so that Table 1 could be obtained«

REFERENCES

(1) D. A. Frank-Kamenetzky, Acta Physicochimica U.R.S.S, 10,365 (1939).

(2) P. L. Chambre, J. Chem. Phys. 20, 1795 (1952).

(3) D. M. Clemmow and J. D. Huffington, Royal Aircraft Establishment, Farnborough, Technical Note Not R.P.D, 12Ö, Sept. 1955.

8

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NAVORD REPORT 4267

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NAVORD REPORT 4267

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