This repo 1; I.js been prepared

for inform:’' -0 record

purposes and is net to be referenced

in any publication.

NATIONAL BUREAU OF STANDARDS REPORT

3625

A PROBLEM IN SELF-HEATING

OF A

SPHERICAL BODY

by

Samuel M. Genensky

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
U. S. DEPARTMENT OF COMMERCE

Sinclair Weeks, Secretary NATIONAL BUREAU OF STANDARDS

A. V. Astin, Director

THE NATIONAL BUREAU OF STANDARDS

The scope of activities of the National Bureau of Standards is suggested in the following listing of

the divisions and sections engaged in technical work. In general, each section is engaged in specialized research, development, and engineering in the field indicated by its title. A brief description

of the activities, and of the resultant reports and publications, appears on the inside of the back

cover of this report.

Electricity. Resistance and Reactance Measurements. Electrical Instruments. Magnetic

Measurements. Electrochemistry.

Optics and Metrology. Photometry and Colorimetry. Optical Instruments. Photographic

Technology. Length. Engineering Metrology.

Heat and Power. Temperature Measurements. Thermodynamics. Cryogenic Physics. Engines

and Lubrication, Engine Fuels. Cryogenic Engineering.

Atomic and Radiation Physics. Spectroscopy. Radiometry. Mass Spectrometry. Solid State Physics. Electron Physics. Atomic Physics. Neutron Measurements. Infrared Spectros- copy. Nuclear Physics. Radioactivity. X-Ray. Betatron. Nucleonic Instrumentation. Radiological Equipment. Atomic Energy Commission Radiation Instruments Branch.

Chemistry. Organic Coatings. Surface Chemistry. Organic Chemistry. Analytical Chemistry.

Inorganic Chemistry. Electrodeposition. Gas Chemistry. Physical Chemistry. Thermochemistry.

Spectrochemistry. Pure Substances.

Mechanics. Sound. Mechanical Instruments. Fluid Mechanics. Engineering Mechanics. Mass

and Scale. Capacity, Density, and Fluid Meters. Combustion Control.

Organic apd Fibrous Materials. Rubber. Textiles. Paper. Leather. Testing and Specifications. Polymer Structure. Organic Plastics. Dental Research.

Metallurgy. Thermal Metallurgy. Chemical Metallurgy. Mechanical Metallurgy. Corrosion.

Mineral Products. Porcelain and Pottery. Glass. Refractories. Enameled Metals. Concreting

Materials. Constitution and Microstructure.

Building Technology. Structural Engineering. Fire Protection. Heating and Air Conditioning. Floor, Roof, and Wall Coverings. Codes and Specifications.

Applied Mathematics. Numerical Analysis. Computation. Statistical Engineering.

Electronics. Engineering Electronics. Electron Tubes. Electronic Computers. Electronic

Instrumentation. Process Technology.

Radio Propagation. Upper Atmosphere Research. Ionospheric Research. Regular Propagation

Services. Frequency Utilization Research. Tropospheric Propagation Research. High Frequency

Standards. Microwave Standards.

^Office of Basic Instrumentation ^Office of Weights and Measures.
NATIONAL BUREAU OF STANDARDS REPORT

NBS PROJECT NBS REPORT

1002 - 10-1029 August l8, 195^ 3625

A PROBLEM IN SELF-HEATING

OF A

SPHERICAL BODY

by

Samuel M. Genensky

Fire Protection Section

Building Technology Division

NBS

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

The publication,

unless permisslo

25, D.C. Such

cally prepared I

or in part, is prohibited

P

FStandards, Washington Approved for public release by the

Director of the National Institute of

Standards and Technology (NIST) ^ on October 9, 201 5.

1 report has been specifi-

A Problem in Self-Heating of a Spherical Body

by

S. M. Genensky

National Bureau of Standards

Abstract

An analytic steady-state solution is developed for a spherical

body in which heat is generated according to a first order unimolecular reaction law, and lost at the surface in accordance

with Newton's law of cooling. The temperature within the sphere

depends chiefly on the radial distance from the center, but also

on the ambient temperature, the surface heat loss coefficient,

and the material properties which are assumed constant.

Introduction

The analysis described in this paper was pursued in connection

with an investigation into the self-heating of fibrous materials

carried on at the National Bureau of Standards.

Comparison of the results of this analysis with the experimental results indicated that the assumption of a single first

order unimolecular reaction was an over-simplification for the

material investigated experimentally. However, the analysis serves

to indicate the relative importance of the various parameters entering into the problem, and may be helpful in indicating the kind

of data that would be of greatest use in further investigations

in which the assumptions may be applicable.

A homogeneous sphere of radius B is generating heat , under

steady-state conditions, in accordance with a first order unimolecular reaction law Ae“pp and losing heat from its surface
' ' -J

-Y r J, “

....

• -.4 'rtX- i' i " - .\

*u.

Lii

'

^

'

•

-. . •"

r-. -,: M 4

' r; ; t>*0'R

-

. ’'‘r ;• .; £:iO-t J \ A.'..' . . -j, v.'‘

b'jj 3-0sa;.: ir;.«* >#f£

’ ;-r 4rJv

' ' ' '"'(.A:; 5'0

zc,,: ^:-.-l:\:S'.i.' /' 15?^.; ‘:.j XU>^‘ -' '

X 1^, •':• •:'-

'

,

. •• '

*

• y •

t - '.

, • r«

. aiisV/ t-jjrji- ;;.:.A4, *u.X;^4’'

- .-,, ? 1 .. r v> -f's> • . ..

. Vj •. '.• 1 vi.j. ft iiu 4**' ;-«iy. Av.,

'.AA .

' - : ' X f j.«;is:f‘5 -

r,£s- 2 bv

; <-7£) 7^‘ A „'- X

i;^. V-

' '> Vl‘l/

'

:•

«. . , r-s

'i

./.'f iS..-L-n'<r'^y h'S^-

V--

' ; > •>' r^'

, .. . *. ."y ' .,.V u:^ V ’.J;'

'

,'. •^v• \.r ^rcix.r-' >, 5vJt .rVvt 9lt -^ .'

•

'< _ , .

• - '. •..- . :r "..S

o,:ri,^! “‘.i.Di'iX ni Jby'yrxxr .,>t'-r J'x : ..[

. -•'t.4

^

-' '-vf’.t Y.;:;! ni:,:^.; 'U} ^£«X b 'wc-i./

r s! ': ; . ,. J r ’••’f' 'fr.l,

a’;',' 4 . 4c, "iiq.5'

'

'

. .>- >jb5-'

::'!;x 4 li- 'dv'lv hV>j(l , faf' U

•it.

^'>1 ' ‘‘^ ':A noi ' .‘

. .

' \ r,' ri , I I f

’. ;'k.
- 2 -

to an atmosphere at constant temperature according to Newton's

law of cooling. The temperature T(r) of the sphere is independent

of time and depends upon the radial distance r from the sphere's

center. Within the sphere transfer of heat by means other than

conduction is considered negligible. The thermal conductivity

k, gas constant R, rate of reaction-frequency product A and

activation energy E of the material of the sphere are assumed to

dT be known constants as are the temperature Tq and gradient ^

= 0

at the center of the sphere. The problem is to find the surface

temperature of the sphere Tg at r = B and the heat transfer

coefficient h between the surface of the sphere and a

surrounding atmosphere.

Analysis

Mathematically the problem becomes:

( 1 )

T = Tq at r = 0 (2)

( 3 )

= -hZ TB-Ta_/ ( 4 )

Let

Then equations (l)-(4) become:

d^V ^

2

^ 1

+ Ce y

— 0 1 ) ( 5 )

-3-

V = Vo = g

0 ( 6 )

dr^ ^ at

n

= 0

. ^ BhR /-^ 7

E ^ Ta_/

(7)

( 8 )

It is further assumed that (a) the temperature gradient

within the sphere exists and is continuous throughout the interval 0

^1 and (b) that the absolute value of the difference

between 1 (7^) and Tq , is small in comparison to Tq.

Under these restrictions a technique presented by Chambre^ proves

useful to observe that;

1 1 ^ ^

V Vq-^^V - ^ Zl/(i-f )_7 - ^

V. I€ (9)

i=0

where AV = Vn - V

Now since series coverges and for a sufficiently

1 ^i large positive integer N, 7— (7 is a very good

^ Vo o approximation to y.

Therefore ( 5 ) may be written approximately as

lb? + 2 dv

^

d)^2 7

dry

Now

N

42 + a 2 ce-% ^

h:o = 0 (0 < v^< 1 )

N A ^ V

• 21 ( 7

^)^ = Z1 (1-

“ = (N+1) - BqV + B2V2 - i=0 i-0 °

+ (-1)%VN

( 10 )

N n:

% “ (n-k)' ki

n=0 V, k

k = 1,2, : . . ,N

where

- 4 -

and in the special case V = V

(N+l)-BiV + B2V2- ... +(-l) Bp|V =1 N = 0,1,2,

Therefore (10) may be written

^

+ Ce-fy e*7Z

ZB1V-B2V2 + -(-l)»BMvh7 _= 0

Since (11) is analytic and regular except at = 0 and

n

further since 0 at = 0 the solution of (11) may be

written in the form:

V = V(/|) -

3-iliY]^ (o^y|^i)

Recalling (6), equation (12) at yields:

Vo - V(0) =_ TqR _ E

3-0

and further differentiating (12) once and considering (7)

it is found that

aq = 0

Differentiating (12) twice, substituting these first

two derivatives into equation (11) and rearranging terms:

Oca

ZI(l2+i)a -r.l-2 = _De+ - BaVS + ... -(-1)%?” _7

i=2 1 \ o N±l

where D = Ce“ Vq

and the aq are given by

ai

iz.2 q _ ]\T _

1 dZDe-*-v7ZBlV-B2V2 . - (-D'^'BnV”/ 7

(i^+i ) (i-2 ) d/Y"2

i=2,3,4...

( 11 )

(12)

( 13 )

( 14 )

( 15 )

(16)

- 5

-

Using the coefficients obtained from (13), (1^), and

(l6) and substituting them into equation (12), V can be

computed for 0^

In particular for 7|^= 1 equation (12) gives the

dimensionless surface temperature V(l) and thus recalling

that T (/|) = pV (/^) the surface temperature T(l) which equals

Tb is easily found. Differentiating equation (12) once

and evaluating this derivative at

7| = 1, h can then be

found by solving equation (8), for all the other factors

involved in this equation are either given or have been

computed

.

The first six non-zero coefficients have been evaluated

and are

^o

C _ 1

a2 -

- ^

e Vo

^4 - 120 Vq2

8 C _ 2 -r _ 1-

"6 = Iw8'7o3 ^ 15 Vo 3-/ "

c5 ,-122c£ 122C 10 7 - -

as - 51840 Vo^ - 63 Vq2 ' 21 Vq 3

-5®

^ q1 -4032 c3 8428 i486 C 460 7 5 ®10 4435200 Vq 5 ^ 405 Vo3 135 Vo^ 27 Vo ' 9 -^e- Vq

Observe that (i ~ l,2,3,o..) are zero. This

follows from ap = 0 and that spherical symmetry of the problem.

Figure 1 is a plot of V(l), as a function of C and 1 /Vq

,

where C was allowed to vary over the range 10“l<j^C^10l9

and 1 /Vq took on selected values in the range 5^1/V^lOO.

- 6 -

The points were obtained using

10

( 17 )

over its range of applicability for the values of C and I/Vq

within the above limits.

In the foregoing, it has been assumed that the temperature

at the center of the sphere Tq is known, and also that the

physical condition indicated by equation (8) is satisfied.

The more common steady state problem of finding the

temperature at the center of a sphere which generates heat

according to a first order unimolecular reaction law and loses

heat according to Newton's law of cooling may be solved using

the analysis described above. In this case E, R, k. A, h, T^,

Tb and B are assumed known.

If the sphere being considered really satisfies the

assumptions of the problem then its temperature must satisfy

equation (1). Since equation (1) is of the second order only

two boundary conditions can be preassigned . Further since the

temperature at the center of the sphere must remain finite, one

dT of these preassigned boundary conditions must be = 0 at r - 0.

This leaves but one free boundary condition. However at the

surface of the sphere both

( 18 )

and

T = Tb at r = B (19)

-7-

must be satisfied; and since only one of these two boundary

conditions may be formally assigned, it must be assumed that the

other is consistent with the problemo Therefore Figo 1, which

represents equation (17) ? may be used to solve this problemo

Since R, E, k, A and B are known, both C and V(l) (recall that

Tb = ^-*V(1)) can be computed* Thus using Fig* 1, 1 /Vq may be

K

found by interpolation provided that the point (C, V(l)) under

consideration lies within the region of applicability of equation

(17) as indicated on Page 6* Since the center temperature of

E

the sphere is T^ , that is, ^

problem is solvedo

Here no use was made of condition (l8) and as mentioned

earlier this condition must be satisfied independently, if the

assumptions made are fulfilled* Thus the degree of consistency

between conditions (l8) and ( 19 ) serves as an indication of the

applicability of the assumptions made in this analysis*
vrJI ^ '1 7

'

/-.

- =:,«

<•_•-,_• , ^ '-s- -

• . ,

.• *" ^ j~ i

..: J-' ^ -:^;;ra; 7V' ' ^ :s£-'

^ -*' ''^ !,' " ?

'

. '[i^ i--; ,-,^.i!,}^ •6^-' j;t>ri,r v- (^.Q

‘U ,.'^M ^vV

(,' -j»<j' V.-’'” ,./. vj */jj'"-i.,'r <ia mj/'-/} '.

i» '

«

'

'^' -"^5%

' ’y -.-

'. ’-t i:.-

.. .

' '

. ^'> ' ^ !(*V/ 4iid - ; iiali^'-r. Xv

^•'' *»r? •jj;' ', ,V r

< =^./ .i

i'jijS®

••

ir-

' ;*ar4’ • ;•'#•. .. . * ^-S'*'

Ck -'

.. r p

' .

;,. , i,i' -.i; 4^^Z^.n :.-. , L-tHl ' ,i ii' XisvC^/.r

:> ^

'; ' ^-^i.,--:'-- :,• K.r’J r' ..t -iV '.\y

<. , t •

- 1:

.'^.'ii

- t :v ;.^i. •., ?

... ^,;;v- „ , .-,

' "'A ( i.'N’

= ^-;'v

“v.- >v^ •

*. ,.^ ’ih if< #»* j? (•• *’ 'ifjiJte" , v<'

•'^, >•• *..*/ tU’ ..t

Li '<

1 '.

• fJS' .f';S ’*
- 8 -

References

1, Mitchell, No D» "New Light on Self Ignition" National Fire

Protection Association Quarterly October, 195lo

2o Chambre , Po L, "On the Solution on the Poisson-Boltzman

Equation with Application to the Theory of Thermal Explosions"

Jo Chemo Physo Volo 20, Noo 11, pp» 1795-1797, 1952o

\l/

THE NATIONAL BUREAU OF STANDARDS

Functions and Activities

The functions of the National Bureau of Standards are set forth in the Act of Congress, March

3, 1901, as amended hy Congress in Public Law 619, 1950. These include the development and

maintenance of the national standards of measurement and the provision of means and methods

for making measurements consistent with these standards; the determination of physical constants and properties of materials; the development of methods and instruments for testing materials,

devices, and structures; advisory services to Government Agencies on scientific and technical

problems; invention and development of devices to serve special needs of the Government; and the

development of standard practices, codes, and specifications. The work includes basic and applied

research, development, engineering, instrumentation, testing, evaluation, calibration services, and

various consultation and information services. A major portion of the Bureau’s work is performed

for other Government Agencies, particularly the Department of Defense and the Atomic Energy

Commission. The scope of activities is suggested by the listing of divisions and sections on the

inside of the front cover.

Reports and Publications

The results of the Bureau’s work take the form of either actual equipment and devices or published papers and reports. Reports are issued to the sponsoring agency of a particular project

or program. Published papers appear either in the Bureau’s own series of publications or in the

journals of professional and seientific societies. The Bureau itself publishes three monthly periodicals, available from the Government Printing Office: The Journal of Research, which presents

complete papers reporting technical investigations; the Technical News Bulletin, which presents

summary and preliminary reports on work in progress; and Basic Radio Propagation Predictions,

which provides data for determining the best frequencies to use for radio communications throughout

the world. There are also five series of nonperiodical publications: The Applied Mathematics

Series, Circulars, Handbooks, Building Materials and Structures Reports, and Miscellaneous

Publications.

Information on the Bureau’s publications can be found in NBS Circular 460, Publications of

the National Bureau of Standards ($1.25) and its Supplement ($0.75), available from the Superintendent of Documents, Government Printing Office. Inquiries regarding the Bureau’s reports and

publications should be addressed to the Office of Scientific Publications, National Bureau of

Standards, Washington 25, D. C.