LA-6949-MS, RevisedClC-l 4 REPORT CC UECT!ON

FE!TKNWCT!O?Jc, 3 cow

LOS Alamos National Laboratory la operated by the Unlverslty of California for the United States Department of Enerav under contract w-7405 -ENG.36.

LDSA[ao mm Los Alamos National LaboratoryLos Alamos,New Mexico 87545

At Affkmativa Actiorr/Eqrraf Oppatutttty ESSSphy’SZ. .

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This report was issued ori@nally in January 1978.

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DISCLAIMER

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This reprt was prepared as an account of work sponsored by-% agency of the United States Cotmrrnesrt.Neither the United Statss Covamment nor any agency thereof, nor any of tftcii employees, makes arsywarranty, express or implied, or asrrrrnesany legat Iiabitity or resporraibilltyfor the accuracy, completeness,or usefulness of any information, apparatus, product, or process disclosed, or represents that its w wouldnot infringe privately owned rights. References herein to any specitlc ccmrrnerdalproduct, prrscus, orsaMce by trade name, trademark, maturfacturer,or otherwise, does not newsrarily constitute or imply itsendorsement, recommendation, or favoring by the United States Covemment or any agency thereof. The

~ews andopinionsof authortexprettedhereindo not necesatily ttateor reflectthow of theUnitedStates Government or any agency thereof.

LA-6949-MS, Revised

UC-45Issued: September 1981

EXPLO: Explosives Thermal Analysis

Computer Code

Dwight L. Jaeger

~~~~k)~~~ LosAlamos,NewMexico87545LosAlamos National Laboratory

CONTENTS

ABSTRACT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

I. INTRODUCTION . . . . .’. . . . . . . . . . . . . . . . . . . . . . . . 1

II. THERMAL ANALYSIS THEORY. . . . . . . . . . . . . . . . . . . . . . . . 3

A. Problem Description. . . . . . . . . . . . . . . . . . . . . . . . 3

B. External Convection Analysis . . . . . . . . . . . . . . . . . . . 3

c. Internal Convection Analysis . . . . . . . . . . . . . . . . . . . 4

D. Conduction Analysis. . . . . . . . . . . . . . . . . . . . . . . . 6

III. COMPUTER CODE ORGANIZATION . . . . . . . . . . . . . . . . . . . ...13

IV. MATERIAL LIBRARY . .

v. EXAMPLE PROBLEMS . .

A. PBX 9502 Sphere .

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B. Plutonium-Beryllium Sandwich

c. Skid Model . . . . . . . . .

REFERENCES. . . . . . . . . . . . . .

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APPENDIX: EXPI.O INPUTS. . . . . . . . . . . . . . . . . . . . . . . . . . .33

NOMENCLATURE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .36

EXPLO: EXPLOSIVES THERMAL ANALYSIS COMPUTER CODE

by

Dwight L. Jaeger

ABSTRACT

The finite difference method is used to calcu-late temperature fields and times to initiation forexplosive materials. The code is one-dimensionaland is programmed for Cartesian, cylindrical, andspherical coordinates. Temperature-dependent prop-erties, phase changes, and multiple heat sourceterms are allowed. Multiple source terms use Nth-order Arrhenius kinetics for each material compo-nent. Temperature, flux, convection, or radiationboundary conditions may be employed. Internalconvection is considered for materials that undergoa solid-liquid phase change. The Crank-Nicholsonimplicit solution method, which allows large timesteps and short running time, is used.

I. INTRODUCTION

WX Division of the Los Alamos National Laboratory has been concerned with

thermal initiation of explosives for many years. Stringent weapons safety

requirements, aerodynamic heating in low-altitude high-speed aircraft, and

explosives that are in contact with radioactive materials are examples of the

thermal initiation problems of current interest. The code EXPLO is a derivative

of the TEPLO code developed by Anderson.’ EXPLO deviates from TEPLO in the

following major areas: (1) variable conductivity between each node, (2) Nth-

order rather than zero-order Arrhenius kinetics, (3) multiple heat generation

terms, and (4) internal convection for melt problems.

I

EXPLO solves the conduction equation with N source terms:z

r{

()-Ej/RT

pCp ~= v(kvT) + ~ P Sj - Wj N QjZj e + QJ + pQI

j=l

+ PFQM(T - Tf) s

where W must obey

5=(s,..dT

W<)N e-

and where

T=

T =

c=

kp =

w=

N=

Q.

z =

E/R=QJ =

QI =Tf .

F=

Qm =

J J

E/RT

temperature,

time,

density,

heat capacity,

thermal conductivity,

original fraction of species j,

current burned fraction of species j,

reaction order,

heat of reaction,

collision number,

activation temperature,

heat input at layer interfaces,

heat generation,

transition (melt) temperature,

constant, and

phase transition energy.

(1)

(2)

For problems involving melt and internal convection a free convection analysis

is performed, which uses Prandtl and Grashof numbers to determine the Nusselt

number. This is then used to determine an effective thermal boundary layer

2

thickness. Finally, the thermal conductivity of the liquid region outside the

thermal boundary layer is increased artificially to simulate the heat flow

caused by the free movement of the liquid region. The conduction equation

[Eq. (l)] can then be used to simulate the conduction-convection problem.

II. THERMAL ANALYSIS THEORY

A. Problem Description . .

EXPLO was developed as a heat transfer tool for a variety of explosives

thermal initiation problems. These include aerodynamic heating of external

weapons on high-speed aircraft, irradiated materials in contact with explosives,

the possibility of exposure of weapons to fire, and many others. The high

explosives can have nonlinear material properties, internal heat generation, and

phase changes, which most heat transfer codes can not handle. A one-dimensional,

multilayer, finite difference approach was selected for this computer code.

Temperature, flux, and convective external boundary conditions were required.

The capability for either temperature or energy initialization was required with

an exponential form for the energy initialization. Phase change and internal

convection were required for materials, such as TNT, that melt long before ig-

nition temperatures are reached. Nth-order Arrhenius kinetics with multiple

species were also required to simulate mixed explosives. The code uses either

steady-state or transient boundary c~nditions, with either fixed or variable

time steps. The classical convection analysis using Prandtl, Reynolds, Grashof,

and Nusselt numbers is incorporated into the code.

B. External Convection Analysis

Routines are available within EXPLO to perform either free or forced con-

vection on the external boundary. The free convection analysis assumes a ver-

tical plate theory and uses the correlation

‘NU= C(NGRNPR)M

with the following coefficients and exponents.

3

‘GRNPR c m

O < NGRNPR < 2713 1.585 0.125

727’3 < ‘GRNPR < ‘“378X ‘o

0.59 0.25

78“378 x ‘0 < ‘GRNPR< M

0.129 0.33333

The exact location of the transition between coefficients and exponents, based

on the product of Grashof and Prandtl numbers, was selected to give a piecewise

smooth correlation with the Nusse.lt number. The analysis is reasonably accurate

for vertical plates and vertical cylinders. EXPLO uses this analysis for all

problems, including spherical problems, and thus caution is in order.

the

The

The forced convection analysis uses three continuous correlations among

Nusselt, Prandtl, and Reynolds numbers from a flat plate theory.3

0 < ‘RE < 3305 (3)

1/3 ${2‘NU = 0“332 ‘PR

3305 < ‘RE< 2,846,758

‘NU= 0.0292

2’846’758 < ‘RE

.185 N;;3‘RE

‘NU =(lo9 NRE)

2.584– “

flat plate theory was used to develop

ther code development may be required for

problems.

c. Internal Convection Analysis

the programmed correlations, and fur-

certain cylindrical and spherical

TNT thermal initiation experiments conducted at Los Alamos4 indicated

that a substantial amount of internal convection was taking place in the liquid

TNT . To simulate the convection phenomena, a model was devised that calculated

I the effective thermal boundary layers and then artificially increased the

4

conductivity of the material that was above the melting point and outside the

boundary layer. In this way a conduction code can simulate the heat conduction

plus the heat transport. To illustrate this point, consider a vertical heated

plate (Fig. 1 ). The low-density hot air near the wall tends to rise, estab-

lishing a velocity profile and a thermal boundary layer. The classical defini-

tion of a boundary layer is

Tw - TBL

Tw - TAMB = 0“99 s (4)

where

Tw = wall temperature,

‘BL= temperature at maximum thickness, and

‘AMB= ambient temperature.

Therefore, there is sufficient mixing in the fluid outside the boundary layer

to maintain a temperature gradient near zero. The internal convection problem

can be simulated by defining an effective boundary layer with a resistance to

heat flow equal to the real boundary layer’s resistance and then setting the

conductivity of the free moving fluid sufficiently high to restrain the tempera-

ture variations in this region to <l%. The interface resistance can be calcu-

lated by using the free convection correlations, defined in Sec. II. B., to

define the Nusselt number, and hence h, the convection coefficient. Then, using

the relation for equal thermal resistance, it follows that

w

z /-1n FLOW BOUNDARY LAYEI?nuzur /

/?.

EFFECTIVE THERMALBOUNDARY LAYER

VELOCITY

TEMPERATURE

TEMPERATURE

Fig. 1.Vertical heated plate.

5

&h(T .TAMB)+ ,w (5)

where

A=

Tw =

‘AMB =k =

AT =

Ax =

This can be

xeff =

heat flow,

area,

convection coefficient,

wall temperature,

ambient temperature,

conductivity,

temperature drop in boundary layer, and

boundary layer thickness.

rewritten to define the effective thermal boundary layer as

(6)

The internal convection problem is thereby reduced to a conduction problem that

uses both the real conductivity of the fluid in the boundary layer and the con-

duction through the central portion of the fluid pool with its conductivity made

high enough to ensure a <1% deviation of the temperature through the boundary

layer.

D. Conduction Analysis

The previous sections defined the boundary conditions and translated the

internal convection problem into a conduction problem. The solution to this con-

duction problem is different from Anderson’s classical solutionl in that the

thermal conductivity variations at the fluid-boundary layer intersection are

now large, and the intersection moves with time.

The conduction equation [Eq. (l)] can be expanded, using a Taylor series

expansion, and formulated according to the Crank-Nicholson method5 to give

6

T’m -

[

(l+)h ~(jmT,m-+ (1-6) T’m+l +T’m_lcP

-~Tm+~2~lm+,-T,m-]1

a-

-[

J.

( )]6ATFQT+~8Tm+1+Tm1- 2T+#T -T=Tm+~ mmm+l -

‘~ ~ @j - ‘mj) ‘jzj:Ej~Tm ‘~ ~J + ‘Q1 - ‘FQmT~ ‘ ●

j=l

where

T’ = temperature at a new time step;

m = node number;

K = O, 1, or 2 for Cartesian, cylindrical, or spherical geometry,

respectively;

a = thermal diffusity;

T = time;

r = spatial coordinate;

s = initial species fraction;

w = species at T;

Q = heat of decomposition;

z = collision number;

E/R = activation temperature; and

13 = O-backward difference, .5-Crank-Nicholson, l-forward difference.

Equation (7) can be simplified to give

[-C1(l-@)T’m_l + CC 1-f& FQm(l-6) + 2(1-f3) T’m - C2(l-B)Tm+l

= “lTm-l+(CC+* FQm- 26) Tm + BC2TIV+1

$( )

-Ej/RT+ pAr2

k Sj - Wmj QjZje ‘m +$(QJ+PQI-

)P FQmTf

(7)

.where

C,=l-g$ , and

C2 =1+% .

Equation (8) is valid over a region of equally spaced intervals with constant

conductivity. The normal finite difference scheme applies Eq. (8) over a de-

fined layer and adds a dummy node at each end of the layer for the boundary

conditions or layer intersections.l However, if a melt transition is moving

through a layer in which the conductivity of the liquid differs from the solid,

the temperature distribution is not calculated correctly. The solution is to

consider all layers to be one element thick and to use two dummy nodes to define

the interface with the next element. The dummy nodes may then be eliminated

algebraically to give a matrix equal in size to that of a one-layer problem and

smaller than that in a multilayer problem solved by the conventional procedure.

The nodal pattern (Fig. 2) consists of one real element and two boundary

elements for each node. Equation (8) is valid for node m with the boundary

conditions in which the temperature and flux must be continuous. Thus, for the

boundary between nodes 1 and m, the boundary conditions are

‘z - ‘2+1 = ~ ‘m-1 - ‘m:+( Axtt m AXm 3

n-1 n+lr --, +1 ~--,

L --J UL-–d.

(9)

Fig. 2.Nodal Pattern

8

and

‘~ + ‘2+1. ‘m-l + ‘m2 2 3 (lo)

where

‘1= conductivity of element 1,

km = conductivity of element m,

Axl = size of element l?,and

Axm= size of element m.

Solving Eq. (10) for Tt+l, substituting the result into Eq. (9), and solving

for Tm-1 ‘ives

1

T - ‘lm ‘nlmm-1 ‘l +ntm Tm+l+nZmT~ ‘

where

A similar express

.

(11)

on can be derived for the interface between m and n giving

I -n 2nT ‘m Tm+~ Tn ,m+l ‘l+nnm

nm

where

knAXmn

kmAXn “nm=—

(12)

Substitution of Eqs. (11) and (12) into Eq. (8) gives the final form of the

conduction equation:

2nRm ~,

-(1-10cl ~ 2[ ()

+ CC- (1-B) ~ FQM + 2(1-6) - c1 (1-8 #g: (13)

[

=BC1>Tl+CC+~FQm -

(3+ ’c&lTm‘B + ‘cl l+nlm

Em

2nnm

2( )

-Ej/RTm+fic — T +~

2 l+nnm nS. - Wm j QjZjeJ 9.=

Equation (13) is an implicit equation capable of large time steps and it can be

solved by Gaussian elimination. It is valid for all interior nodes and is un-

affected by layer intersections. Note that if AXM = AXn and km = kn (on the

interior of a given layer), then ~nm = 1, qgm = 1, and Eq. (13) reduces to a

more conventional equation.1,3,4

Temperature, flux, and convection boundary conditions are of interest in

thermal initiation problems. If node ~ (Fig. 2) is on the boundary and a tem-

perature is imposed on the boundary between it and the dummy node L-1, the

average temperature of the two nodes must equal the boundary temperature (TO);

thus, the temperature boundary condition can be stated as

T,+T2=2T0 .

The flux boundary condit

states (defined for heat

(14)

on is obtained from Fourier’s conduction law, which

flow into the system as positive).

10

(15)

A Taylor series expansion of Eq. (15) provides the final form for the flux

boundary condition,

‘1 -T2+ . (16)

The convection-radiation boundary condition is obtained by equating Newton’s

law of cooling to Fourier’s conduction law:

(17)

A Taylor series expansion of Eq. (17) provides the final form of the convection-

radiation boundary condition,

[

l+p+~

1[ 1(Tl +T2)3 T1 + -1 +#&+#(T1 +T2)3 T2

=~TO+~ TR4 . (18)

Equations (14), (16), and (18) are the boundary

fers to ambient temperature, subscript 1 refers

equations where subscript O re-

to the dummy boundary node,

subscript 2 refers to the first real node, and subscript R refers to the radia-

tion ambient temperature. The boundary conditions can be combined with the con-

duction equation [Eq. (13)] to generate a Crank-Nicholson tridiagonal matrix

(Dij) of the form

[

‘11 ’12 0

’21 ’22 ’23

0. . . . . . . . . . . . . . . . . . . .0

0. . . . . . . . . . . . . . . . . . ..0

0’32 ’33 ’34” “ “ “ “ ‘ “ “ “ ““ “ “ “ “ “ “ “ “0. . . . . . .

. . . . . . .

. . . . . . .

0 0 Di,i-1

Dii .

D.l,i+l

o. . . . . . . . . . . .0. . . . . . .

I. . . . . . .. . . . . . .0.. . . . . . . . . . . ()

Lo..... . . . . . . . 0

0

0

‘k-l,k-2 ‘k-l,k-l

o’‘k,k-l

‘k-1,

‘kk

11

where the conduction equation has the form

IDijl ~i[=lvil’where

’11 = ‘kk=1 .O, or-l .O, orl.0+~ +#(T1+T2)3

(19)

(20)

and

OSAT (Tl + T2)3’12 = ‘k,k-l

=1.0, or-1.0, or-l.O+~+~ (21)

for temperature, flux, or convection-radiation boundary conditions, respec-

tively; and where

D.“lnLm

l,i-1 ‘~ (1-6) ,~m(22)

D2c2nnm (1-6)

i, i+l ‘~ snm

and also

hAXT + as,Ax~R4or9#,0r~‘1

= Vn = 2T0, k

(24)

(25)

for temperature, flux, or convection boundary conditions, respectively; and

finally, where

.

12

The above analysis has been programmed for the CDC 7600 and VAX/VMS compu-

ters at Los Alamos and is used to simulate a variety of tests. Temperature dis-

tribution errors and energy check errors were prevalent in all problems before

the ~Lm and ~nm were included in the analysis. There was an early attempt to

fix the error problems by including the nonlinear term 6k6T/dXdX in the analysis

and defining a linear conductivity variation, k. + BT, across the melt inter-

face. However, this proved unsuitable. The solution to the conduction equation

given here is the only solution that handles properly the moving interface

between the low-conductivity boundary layer and the high-conductivity liquid

pool between the boundary layers.

III. COMPUTER CODE ORGANIZATION

EXPLO is written in FORTRAN IV. The main program is only a controller; as

many input, output, and execution steps as possible are organized into subrou-

tines. Data transmission between the subroutines is through four common blocks.

The first contains the major arrays, such as temperature, which are used by most

of the subroutines. The second contains layer-independent variables such as

geometry definition and boundary conditions. The third contains layer-dependent

variables such as dimensions, kinetics constants, and material properties. The

fourth contains parameters that are generated by the code such as total heat,

flux, and internal energy. A list of the inputs and their functions is in the

Appendix.

Special features of some inputs and their effect on program execution need

further discussion. The code calculates an average temperature for each time

step, which, in turn, is used to calculate temperature-dependent material

properties. Temperatures from a previous time step are stored in the T array,

and new calculations are stored in the TT, or T’ array. At the beginning of

13

each new time step, the T and TT arrays contain the same temperatures. If the

number of variable property iterations (IKOUNT) is set to zero, the material

properties are determined from the previous time step. However, if IKOUNT is

set to one, the code first uses the previous temperatures to determine material

properties and calculates the TT array. Then, the code determines material pro-

perties from the average of the T and TT arrays and calculates a new TT array.

This process continues IKOUNT times. Normally, IKOUNT need not be set greater

than one. However, for problems involving melt with internal convection, prob-

lems that burn beyond initiation (IDR or IDM = 2), or if the code output con-

tains energy warning errors, IKOUNT may have to be set to three or greater.

Some problems involve a fixed flux at one boundary and a time-dependent

temperature at the other boundary. Often the system should be allowed to equi-

librate to a temperature distribution defined by the flux and the initial bound-

ary temperature. If the 1SS input is set to 1 (Card 2, see the Appendix), the

code calculates an initial temperature distribution before it runs the transient

analysis.

The code has a variable time step feature. If the maximum temperature

change for all nodes is >4°C, the time step is decreased by half. If the maxi-

mum temperature change for all nodes is <10% of the maximum change allowable,

the time step is increased by a factor of 2. If the initial time step is set

to O or is not input, the code will calculate the initial time step.

The third card contains an input called GLOAD. GLOAD is a gravity-loading

coefficient; it is used only by the internal convection routine to calculate

the Grashof number. It is intended for use on studies involving the aerodynamic

heating of an explosive beyond its melting point in aircraft or missile systems

subject to aerodynamic forces other than gravity.

A point should be made about the code’s operation in reacting boundary

layer calculations. If a liquid explosive is heated through a boundary layer,

thermal initiation occurs within the boundary layer. The”code applies Nth-order

kinetics to each element. In a real case, part of the explosive reacts, vapor-

izes, and floats away leaving a void to be filled. To represent this within the

code, only for cases with internal convection (IDM = 2), the array that contains

the burned fraction is reinitialized to zero before each calculation. Thus,

each time step generates only the total energy available in an element, but that

energy is replaced for the next time step. This use of the code gives better

14

results for the initiation of liquids; however, it invalidates the code’s use

for liquid burn problems.

If the energy initialization option is used, the code requires input of

the average initial energy deposit (ENGA) and the front face energy deposit

(ENGF) for each layer of materfal. The code then fits an exponential curve of

the form

-clxEi = (ENGF)e , (27)

where Cl is selected such that the average energy criteria are met, Ei is the

nodal initial energy, and X is the spatial distance.

The specific heat and thermal conductivity are defined as functions of

temperature. The user may define the value of specific heat and conductivity

for up to five temperatures. If a phase transition (such as melting) exists,

the user can input the phase transition energy at any or all of the five tem-

peratures. The computer code will linearly interpolate the defined values to

determine the value at any temperature during the transient.

The use of Nth-order kinetics hints that the code may be valid for burn

calculations. The reaction indicator (IDR) can be set equal to two, to allow

the code to continue calculating beyond the initiation point. The code assumes

that the heat is generated in the solid or liquid phase and is available for

conduction from the generation point. In a real case the high explosive vapor-

izes and releases heat as gas. Any heat that is not lost must be convected

and conducted back into the solid. Therefore, the code calculates reaction

rates that may be considerably different than reality.

The user can simulate test configurations with the code and plot thermo-

couple data on the space plots generated by the code. If this option is used,

inputs for the number of thermocouples (NTH) and the number of data points per

thermocouple (NDATA) must be specified on Card 2. The code uses the thermo-

couple data to interpolate data points linearly in time and then places these

points on the space plots.

IV. MATERIALLIBRARY

A material library has been constructed to be compatible with the EXPLO

computer code. This library (MATLIB) greatly reduces the time required to

15

prepare an input file. A library search is initiated if the user defines

the reaction indicator (IDR) to be a number greater than 99. A list of the

materials that are available for use with EXPLO is printed on the output file

during each execution of the computer code. The data format used in the mate-

rial library is the same as that used for the normal EXPLO input.

v. EXAMPLE PROBLEMS

EXPLO assumes that the user will have the figures output on the color FR80

system at Los Alamos. To reduce the amount of data that appears on each figure,

EXPLO makes extensive use of color coding according to layer. Therefore, al-

though some of the figures in this document may look confusing to the reader,

the normal color film output will be clear.

From the many thermal analyses performed at Los Alamos, three were selected

for simulation with EXPLO. The first is an analysis of a 2.54-cm-diam PBX 9502

sphere that was tested in the Los Alamos unconfined critical temperature test

chamber.6

This example shows the type of problems the code was designed to

solve. The second example shows how the code can be used to calculate the tem-

perature distribution of a plutonium-beryllium sandwich subjected to pulsed

heating. The third example shows how a user may add a FORTRAN subroutine that

can add energy (as a function of time) between layers of a composite. This par-

ticular example is a theoretical model that describes a potential mechanism by

which HE is ignited during a skid impact.

A. PBX 9502 Sphere

We first prepared an input deck (Fig. 3) that consisted of a title card

(line 1), two cards of control information (lines 2 and 3), three cards

.6

pbx-WQi!pbx-S5a!?

;2-%::

Fig. 3.Input deck for PBX 9502 sphere.

16

●

describing the transient temperature on the external surface (lines 4, 5, and

6), and four lines describing the material properties and geometry (lines 7

through 10). In this case, the material properties for PBX 9502 were found

from material 102 in the material library. The sphere was divided into four

layers to facilitate comparison of the analysis with test results. The terminal

output (Fig. 4) shows the maximum temperature for each layer at each time. This

particular analysis shows that we would have a thermal ignition at the center

of the sphere after 2395 seconds.

l%:223.289.354 ●

419.465.~5i,616.68i? *?47.813.878.944.

1909.1075.l14@.1206.li!71.1337.:4&:

1534.1688*1s65.1738.1796.186s01827.1992.g:.

2123.21890mi4.e320.2382.

2394.0312396,3982395.4372395,440

Iamat.r 9502 s h-r-itlfWl~ll T H~ERATUfjE FOR

& 293. 293. i?93.293. 293. i293.

293: 293. 296. ;;&296. 297. 305.302. 395. 314. 314:310. 313. 323. 323.319. 322. 332. 332*328. ;~;. ;:;, ;~;.337 ●

346. 349: 359: 359:354. 3S8. 368. 369.363. 367. 377. 37a.3?2. 376. 386. 387.381. 385. 38S. 39S.yO: 394. 4@6. 40S.

403. 414. 414.408. 41%. 423. 423.4;7. 4ai, 43a. 43a.486. 438, 441. 441.436. 438. 460. 4s0.445. 448, 4s9. M.454. 467. 488. 4690463. 48S. 4779 478.472. 475. 486, 487.481. 48s0 4m. 486.4989 4949 SOS. 68S.498. 6@3. 514. S14,5989 gi: y: 8239618. ma.687. 631. 541, 541.638. S48. 543. 643.S46. 646. 64S. S43e56* . 650. 648. 543.5630 563. 6s99 543.

so#m#mTxOn6 CCWL21. 562. 643.

S5~: 659e 6S3- 643.668. 662, 6s4. 543,6S6. 666. 6S8. S43.589. 68*. 5S8, 643.616. 61S, WE. 643.642, 64ee 688. 643.y;: y;, 662. 643.

. 662. 543.

Icn

AT TIME =

IGNITION BEGINS aT DEPTH ● 0.90E-03 CR IN IAVER i2395.4;O#3* 670. S~~o S43.

PLOT PACES ● . UORD9 wyAPHICS CL ~ U

58374

0.888E+04

Fig. 4.Terminal output for PBX 9502 sphere.

17

The first frame on the film file contains a description of the geometry and

a summary of the room-temperature material properties (Fig. 5). The subsequent

plots that are printed on the film are dependent on the output option selected

on card 2. For option 1, the code will generate a temperature versus space plot

for the initial temperature distribution (Fig. 6) and the final temperature dis-

tribution (Fig. 7). If EXPLO had detected more than 0.1% decomposition of a

high explosive (HE) component, it will also generate a burned fraction versus

space plot (Fig. 8). EXPLO then generates plots of the boundary flux and in-

ternal heat generation versus time (Fig. 9) and temperature (Fig. 10), which

are helpful when comparing EXPLO results with differential thermal analysis

results. EXPLO then generates temperature versus time plots for each layer

(Fig. 11) followed by a composite (Fig. 12) of all layers (the film output is

color coded). Finally, EXPLO will generate plots of specific heat, internal

energy, and conductivity versus temperature for each layer.

B. Plutonium-Beryllium Sandwich

This example demonstrates EXPLO’S ability to deposit energy a$ a function

of both space and time. The first three cards of the input deck (Fig. 13) are

similar to those discussed in the first example. Lines 4 through 29 define a

time-dependent multiplication factor (QFI - fourth column) for the internal heat

generation. The geometry section (lines 30 through 41) is similar to the first

example except for the energy generation term (QI) in the right-hand column of

2.!i4-cM-oinHcrcR ?502 W@7r

aRl 0. m m- 1.s0- O.UDZ r- 293.

I nx-m C.ml !,m O.mlno 0.229 718. 5SUS5.o.3t8c. ai2 MZ-9YM 0.6X I.m 0.00Iuo 0.239 716. =wu3. o.lla.a3 mxqw O.#l* !.s9 0.00Izx 0.229 718. 5s3s3. o.31a.m3 mx+.oa O.ml I.EQ O WIZ?I (1.2M ?K SW%. C. JIK.JO

Fig. 5.Geometry description and summary ofroom-temperature material properties(PBX 9502 sphere).

18

.

.

O.mx 0.125 0.Z5.I o. 17s 0. m 0.625 0.750 0.05,

I. L-m 1.1X !.210 1.S71 l.m01 STf3NCC lCfil

Fig. 6.Temperature distribution for PBX 9502 sphere.

,, 2.54-CM-DIFIMCTER 9502 SPHERE

Tlnc

0.240 C.04S

i 1

.

z?!.

M-

ii

is

9-*. 1

ii=j”!

-.

il-? !

5,O.ml 0.13 0.2!! o.3m 0.502 0.U5 0.7=4 0.C75 ,,m 1.12s I.m 1.$7s I.kll

Olsifwcc (ml

Fig. 7.Final temperature distribution (PBX 9502 sphere).

19

2.54-CM-DIFIMETZR 9’502 SPHERE

‘0!

- o.cm0.1250.2s4 9,375 0. Sm 0.625 0. 7S41 0.675 l.C@ 1.1?3 1.2YI 1.$? I.w

D! STRNCE (Ml

Fig. 8.Burned fraction versus space plot (PBX 9502 sphere).

FI.iJX GRPOICNTtmr GRII,III:wT

2.5’l-CM-DIRMErER 9502 SPHLRE-..,,1‘1

‘;~ a ~ ~ ~0.0 m. 0 Sm.o Two Ima. o 12s’3.0 1503.0 !m.o Xm. o Wm. o 2503.02750.

i!’lC [S1

Fia. 9.

.

.

Boundary flux and internal heat gen~ration versus time (PBX 9502 sphere).

20

.2.54-cl -D RMETER 9502 SPHERE

0.0 Uc.o Isa. 0 Ku, 0 +40.0 w.2.0 \m.o——~

564.0 EOO.O 640.0 W.O . 7M. OTf31PUi13TURE !KJ

Fig. 10.Boundary flux and internal heat generation versus temperature(PBX 9502 sphere).

2.54-CM-DIFIMETER 95 LRYiZR 2PBX-9502;

iiI-.E

---~F’k?.SiIiJ;-7

E5

Elk“0.0 2W. O Sm. o 793.0 Ioco.o lm. a !W.o 1.=4.0 Zxa. o 22W0 Z!ix. o 2753.0 Ica). o

TI$lc [5)

Fig. 11.Temperature versus time (PBX 9502 sphere) (individual layer).

21

Te

2.54-CM-DIf7ME”l_CR 9502 SPHERE;1 I

iiJ___ —-l0.0 m.o SCO.3 ?’50.0 laO.9 1250.0 !,,02.0 1750.0 xul.o 22W.O 25m.0 27s0.0 3c01.o

TIME [S1

Fig. 12.lrature versus time (PBX 9502 sphere) (composite of all layers).

1 pU-~E S~;B ST~DV~ Ii) 199 1

.00020. 00:: 00::

. ‘oai6

.09(!0

. 1(W6

.1100. ~,?oe

. 13@fa

.14e0

. tsoe

. 160e

. 17ee

.Iaee● 1900wee.21ee.23Qe.2s00. i?9c@● 330%.3708.4ie0.4549.59e9.s509.6%3

imoecw53ss e633S O53’a5 ●5305 06305 05335 e5335 e539s e63a5 O5365 a6383 es3e3 o

606.000.$30e.eoe.090.000.eee.eaa.@w.we.eee.eee.eea.em.em.000,088.e2e.eee.003.000.800.em.O?o.0?0.9.02O.eaO.eee.ei!:.;:

eIe2O.CN?e.ea●.@2.:;:

@o@.eOe.me.efae.eoe .Ooe.eee.eee.eee.009.eeo.@e@●

eeo.eee.me.ea..080.900.em.eee.eee.eae.cu.me.we.293.f?93.293.293.293.293.293.293.293.293.293.E93.

212223e42s

i%2829303132333435363738394e41

8

1 26

.e~l, 2G3.6?2

1.7124.1208.86.??

:;.;;;

1;:;;;

4:81S2.8151.8161.3951.1s11.022

.93a

.8e3

.604

.369

.197; g;

● 044

:8$

003

6.56.56.56.56.s6.s6.S6.66.7

0:::fm.o

,

Fig. 13.Input deck (plutonium-beryllium sandwich).

22

data. The magnitude of

product of QFI and QI.

arbitrary but the other

(Fig. 14) shows themax

The first plot

the internal heat generation for each time step is the

Therefore, the magnitude of either QFI or QI may be

must be selected appropriately. The terminal output

.

mum temp(

on the f

rature for each layer as a function of time.

lm file (Fig. 15) shows the geometry and pro-

of the material properties. EXPLOthen gener-

n the previous example, which include transient

(Fig. 16) and a composite transient temperature

vides a room-temperature summary

ates several plots as described

temperature plots for each layer

plot (Fig. 17). As previously described, EXPLO will follow these plots with

material property plots for each layer.

:::,

3<?5:31$.3.3“?;?.

A.W4.317.33?.3!>0.3S8.

a::.293.2’>7.3ot.3’3s .3d8.311.313.31s.316.317.318.319.329.322.325.g;.

331:333.335.3.36.337.338.339*34$.341.34%.343.343.344.344.34s .34s .

i.?293.293.i?93 .

i?l)l,294.t!!J5 .i?’J5 .~<J7.290.2s9 .3Q1.3?!2 .3W .33s.~~le315.318.321.~tll&:2“?D.331.33’2.334.335.337.338+339.343*341.341.342.343.344.

o.i4b-l{w 189. 31{9 .0.1s44$0 411. 411.t).ii-2403 426. 426.0.170400 436. 436.o.17R4tio 442. 442.0.1SG400 445. 445.0.194400 448. 448.

0.204 4s0. 449.0.220 452. 451.0.236 452. 452.0.260 451?. ::::9.292 450.e.3i!4 447. ::;.0.356 443.0.388 438. 437:0.429 432. 431.0.4%? 426. 42S...484 42e. 419.0.516 41S. 414.0.S48 489. 409.e.sso 44s. 404.8.612 46e. 399.0.644 396, 39S.0.676 392. 391.9.708 388. ;;:.0.749 3E5.0.772 322. m:

a.

31!U.419.425.433.438.44e .441.441.439,438.4;s.432.429.42s.421B.:;:.

40s :409.396.392.389.3ss.3s2.379.37? .374.372.37e .366.

‘LETED363.361.359.3S6 .3s3.3s1,3s1 .

;= 1

388.4@9.423.430.434.:::,

433:431.428.42s ●

421.418.414.41*.4@6.401.397.~90

.2S6 .383.3n*377.376.37a .370.

3%:364.

nM:l

359,357 ●

355.

F&:

leik%

31Jti .4afi.4111.424.426.42s.424 ●

422.418.41s.4s2.468.48s .4%? .393.394 ●

391.387.334.381.3?8 .376.3?3 .371.369.xi?.

3U;!.400.4e9.412.412.411.498.495.;:;.

395:392.389.387.3!34 ●

3$31.378.3?6 .;;;.

369:353.36C.364.363.362.361.368.3s9.3s7.

.y&.

3ss.y;:

3s1.3se.350.

396.392.393.391.388.386.333.379.376.3-?4 .372.371.370.368.xx.;:; .

362:361.3E8.3s9.SS8.3s7.356.3S6.Xr5.355.3s4 .ZS3.

‘aa353.2’52.351.~s@●

3:9.349.349.

361.

we .359.3s? .3ss .3s3.3s1.3s0 .350.35e .356.350.3W.359.358.35Q .350 ●

259.3s@ .3s9 .350.;:; .

349:349.349.349.349.

4j5.441.445,447.448.449.449.

Ik‘3.4.

448,446.442.438.433.428.422.416.411.406.491.397.393.389.386.392.3ri3.377.374.370.

l?y;m

364:361.3s? .3s4 .3s1.3s1.

66.

4(3’3(

366.364.363.369.

a3;g*.

3s?.35s .353.3S2.3s0.3se.

_——.o:aai 379. 5?;..0.842 376. 37S.0.926 3

4

346 ●

348.347.347 ●

348.349.34s.

iS.;3.;8 .;7 .;4.il.

349 ●

349.349.349.349 ●

349.34s.

345.345.346.347.343.348.349.

3

:3u

;1.IRDSPLOT

1s ‘-”

Fig. 14.Terminal output (plutonium-beryllium sandwich).

23

no - l.%?.”o.lil%l 0- 0.00XI

mm mm:m. Pllcx WI x mot z

0.020 1s.7s0.020 IS. Z5O.oao 1s.750.020 15.750.020 15.750.020 15.750.02C 15.710.220 15.7s0.020 1s.750.020 15.750.29C 1.840$10 1.84

0.022O.ou0.032O.olaO.oIa0.012O.o’Ja0. onO.ou0.0320.4190.419

0.0.0.0.0.0.0.0.0.0.0.a,

Fiq. 15.Geometry description and room-temperature summaryof material properties (Pu-Be sandwich).

Fia. 16.Transient temperature-plot for individual layer(Pu-Be sandwich).

24

.?~-

‘XLT13-PLUT

UEIITR-PLIIT

DELT!3-PLUT

i-OELTfl PLUT

OELTR-PLUT

-?

~ ~-

OELT17-PLU1

OELTfi-PLUT

~?0ELTf3-PLUT

&5’-nELTt3-PLUT

BERYLLIL#l

BERYLLILNI

?

5“

?E“

L_____–~–0.0 0.2 0.4 0.6 0.8 1.0 ,.2 1.4 !.6 . 2.4

TltE (S1

Fig. 17.Composite transient tempe~ature plot (Pu-Be sandwich).

c. Skid Model

EXPLO has also been modified, on several occasions, to accept user-defined

routines that calculate energy input as a function of other variables. As an

example, we developed one subroutine that calculated the energy generated by

a sliding surface as a function of several variables and time. We are using

this subroutine, with EXPLO, to try to understand better the initiation of HE

caused by a skid impact. The subroutine requires one additional line of input

data (line 9, Fig. 18) in the input file. The first input (the number 3) indi-

cates the layer to the left of which the energy will be added. The second input

is the drop height (cm), and the third is the weight (g). The fourth and fifth

inputs are temperatures that correspond to shear strengths, the sixth and

seventh inputs. The last input is the skid angle.

The first frame on the film output (Fig. 19) shows the geometry used for

the calculation. We simulated the moving HE with a thick and a thin layer. We

used the thin layer to obtain accurate calculations at the skid interface. The

target was a gold layer, backed by a copper layer, which, in turn, was backed

by an aluminum layer. The energy from skid was input between layers 2 and 3.

The TTY output (Fig. 20) shows that

It also shows that, even though the

the specimen rebounded 2.1 ms after impact.

HE was heated to 648 K, the specimen did

25

1- —. ... . . . ..— —.— ..—.

I

10.0. e2e●.@@l

.6875

.0175

. 2MIB. 8674.

1

293.

4e.e.e.40e.00

931 .5e. .s8:. 08.

e6 .pbx 9S431

e: 00.Fbx ‘%01

o. w.qo (dCoppor

ea. slkd2.2” 2.6008 4s ●

1- ..— —— -.. . —

Fig. 18.Input deck (skid model).

..,J=M ‘Kin HOOU. 9501 -GOIJI

RI - I.ffi m - I.Zo -G. oxo a - O.CCW

LW27 WKCfW. mllx mm K C?oc z

1 Pot 550! O.m 1.84 o.cmzm 0.234 ,ms. 527m. IJ.SJX.2U2 ma %0) 0.001 !.89 D.WILWI 0.238 505. 527xI 0.5bX@23I w-n 0.W2 /9.33 1.0512+2 0.021 0. 0. o.Nauu+ mf-cn 0.0I7 0.94 9.9472X2 Q.093 O. 0. mcca-m5 RJJI; w n.2m 2.M 0.-294M3 O.m c. 0. O.axr. m

Fia. 19.Geometry used for skid model calculation.

.-

1...J

not ignite. The temperature distribution after rebound (Fig. 21) shows that

this target was able to conduct heat away from the HE and thus prevent ignition.

The burned fraction plot (Fig. 22) shows that 3% of the explosive at the inter-

face has reacted. The flux gradient plot (Fig. 23) shows that the energy input

was very high for approximately the first 10 PS while the HE was still cold and

strong. By 25 us, the HE has thermally softened at the interface, and the

heat input is constant until rebound. The temperature of the HE at the inter-

face increases rapidly for approximately 10 us (Fig. 24). After 10 us the HE

is able to transfer most of the generated heat to the gold layer. The HE tem-

perature does, however, continue to increase, but at a much lower rate. The

26

aw3.

27

T:W

o. Z30E G,%

,, S1.f?u 5<[0 r-?ooll.S5(-JI-GOLCI

I.’in 1.’ka

Fia. 21.Temperatl~re distribution ”~fter rebound (skid model)

.

.

Fig. 22.Burned fraction plot (skid model).

.

.

F

5[. (1[1 ‘SKlll M-IIJI;I.. ‘J!,t J1-[;Ill .13-I, C:Y %1

?s J

Fig. 23.ux gradient plot (skid model).

“ SLfli2SKID MODEL 9L0 LfiYER 2PBX-!3501

ii ~—.-——r ,.,-.—— ---r —0.0 . 5.0 7.s 12.5 :5.0 17.-3 X.o 22.5 25.0 V.G A3

TItlC ,s; ●10’

Fig. 24.Transient temperature plot for individual layer(skid model).

I

temperature-time curve is not smooth, partly because of the phase transitions

within the HE. The composite transient plot (Fig. 25) shows the complexity of

the heat transfer that takes place. Finally, Figs. 26, 27, and 28 show the

material property plots for PBX 9501.

SLflB W.[i.lMO12LL 9!][J1GCll,OG

~

Pflx .9’,01

Pox 5s01‘i GOLO

COPPER

flLLll IMJfl

-?

~il-

~:

c.

I!ii

i

?

R-

;. ~ ~. —- —r—— T——0.0 25 .

.—--f-~---l7.5 12.5 15.9 17.5 X.o 22.s . n.% 30.0

:IY [s1 9110*

Fig. 25.[Composite transient temperature plot (skid model).

SLRB SKID MOCICL.95[1 LflYtIR 1P13X 9501;!.;1

$0

i

R6L- L—--r---F---.-yin-r-- —

,. ——,—..

am xao 3m.o 4m.0--T--T—-T--7

*43.@ 520.0 w. o km 0 7m.0 7m.o

TW?ERWUW [K)

Fig. 26.Heat capacity versus temperature (skid model)

.

.

30

.

.

‘.) L,IIU :~k; 1[.1 110[11 “L :1!)(J L.1I iL’1{ 1I ‘IJX !I!)LI 1

:/ ————---r --- -T——-r--r. ~-7200,0 UO.o 360.0 ~m.o WO.C 4M.O 520. C SiO. o 6M.O SQO.O bml. o 7W,0

1f fR’LRtiTduL : K I

Fig. 27.Internal energy versus temperature (skid model).

—

Cl MWI..1. !J’.,(l LIIYI.’I( 11’ljX 9W1

+--— --.—~.—__.T 3

ao. o Sa3.o 360.0 U9.o +40.0 Wn.0 Sm.o .=.3 a.m.a 6u1.O 687.0 7a.9 7%0.0T~MPfRfITLIF/C [ < ]

Fig. 28Conductivity versus temperature (skid model).

31

—

REFERENCES

1.

2.

3.

4.

5.

6.

C. A. Anderson, “TEPLO: A Heat Conduction Code for Studying Thermal Ex-plosions in Laminar Composites,” Los Alamos Scientific Laboratory reportLA-4511 (November 1970).

J. Zinn and R. N. Rogers, “Thermal Initiation of Explosives,” J. Phys.Chem. 6&, 2646 (1962).

A. J. Chapman, Heat Transfer, 3rd edition (MacMillan Publishing Co., Inc.,NewYork, 1974).

A. Popolato, J. J. Ruminer, A. S. Vigil, N. K. Kernodle,.andl,D. Lo Jawr~“Thermal Response of Explosives Subject to External Heating, Los AlamosScientific Laboratory report LA-7667-MS (February 1979).

J.Crankand P. Nicholson, “A Practical Method for Numerical Evaluation ofSolutions of Partial Differential Equations of the Heat-Conduction Type,”Proc. Carob.Phil. Sot. ~, 50 (1947).

D. L. Jaeger, “Thermal Response of Spherical Explosive Charges Subjectedto External Heating,” Los Alamos Scientific Laboratory report LA-8332(August 1980) .

.

.

32

IAPPENDIX

EXPLOINPUTS

Card 1 5A1O.

.

1-60 - Title

Card 2 1615

1- 5 - IGEOM -6-10 - K

11-15 - INDOUT:

16-20 - NCYC -21-25 - NCYP -26-30 - IIBC -

31-35 - IOBC -

36-40 - NOTIM -41-45 - IDT -46-50 - IZZY -51-55 - IKOUNT-56-60 - ISKID -61-65 - NTli -66-70 - NDATA -71-75 - 1SS -76-80 - BETA -

Card 3 8E1O

O-Slab, l-Cylindrical, 2-SphericalNumber of layers (10 max)Output indicator- O-Time-temperature summary- l-Detailed time-temperature- 2-Above plus plots- 3-Above plus temperature of each nodeCycles per printNumber of cycles printedInner B.C.- O-Temperature- l-Flux- 2-Convection- 3-Conv-radiationOuter B.C.- O-Temperature- I-Flux- 2-Convection- 3-Conv-radiationNumber of time-dependent boundary conditionsVariable time? O-Yes, l-NoInitialization, O-Temperature, l-EnergyVariable property iterationsO-Normal, l-For skid tests, 2-For hot wire initiationNumber of thermocouple locationsNumber of data pointsInitial steady-state indicatorO-Euler forward, .5-Crank-Nicholson, l-Euler backward

1-1o -DT -11-20 - TIMEMX -21-30 - RI -31-40 - TQIB -41-50 - TQOB -51-60 - SSFR -61-70 - VOLR -71-80 - PRESMX -

Initial time increment(s)Maximumcalculation timeInner radius (cm)Temp or flux on inner boundary (K, or cal/cm2/s)Temp or flux on outer boundary (K, or cal/cm2/s)Solid state reaction functionInitial HE volume ratio (<1.0)Maximum container containment pressure (atmos)

Next Card (If External Convection or Radiation) 8E1O.3

1-1o - DIM - Characteristic dimension (cm)11-20 - ALT - Altitude - (meters)

33

21-30 - VELOC - Velocity (m/s)31-40 - DELTAM - Ambient temperature deviation from standard41-50 - EMIS - Emissivity51-60 - TOR - Radiation temperature base

New Card(s) (If Time-Dependent Boundary) 8E1O.3

1-1o - TIMB - Time(s)11-20 - TQIB - Temperature or flux on inner boundary (K, cal/cm2/s)21-30 - TQOB - Temperature or flux on outer boundary (K, cal/cm2/s)31-40- QFI - Time-dependent multiplication factor for QI41-50- QFJ - Time-dependent multiplication factor for QJ51-60 - ALT - Time-dependent altitude, transient convection61-70 - VELOC - Time-dependent velocity (m/s), transient convection71-80- QRAD - Radiation flux

Next Card(s) (l-Each Layer) 14, 213, 6F1O.O, AlO

1-4-NNL -5-7-IDR -

8-10 - IDM11-20 - THICK21-30 - TAVI31-40 - ENGA41-50 - ENGF51-60- QI;]-~: - QJ

- MTL

Number of elements per layerReaction, O-No, l-Yes, 2-Continue analysis beyond initia-tion, if >99, find all material data in material libraryMelt, O-no, l-yes, 2-interna~ convectionThickness of layer (cm)Initial average temperature (K)Initial average energy (cal/gm3)Initial front face energy (cal/gm3)Internal heat generation (cal/g/s)Heat generation at interface (cal/cmp/s)Material label

Next Card(s) (If IDM=2) 8E1O.3

1-1o - VISC(l)- Viscosity coefficient11-20 - VISC(2)- Viscosity coefficient21-30 - VISC(3)- Viscosity coefficient31-40 - BET - Coefficient of thermal expansion

Next Card(s) (5 maximum) 10F1O.O

1-1o - TMLT -11-20 - RHO -21-30 - CP -31-40 - EK -41-50 - HTFUS -51-60 - E -61-70 -Q -71-80 - Z -81-90 - ORDR -91-100- SPFR -

Transition temperatureDensity (g/cm3)Specific heat (liquid phase, cal/g/K)Conductivity (liquid phase, cal/cm/s/K)Heat of fusion (cal/g)Activation energy (cal/mole)Heatrelease(cal/g)Frequency factor (1/s)Reaction orderSpecies fraction (can be O for last)

For Each Above Card (5 maximum) 10F1O.O

51-60 - E - Activation energy (cal/mole)

.

.

34

61-70 -Q - Heat release (cal/g)71-80 - Z - Frequency factor (1/s)81-90 - ORDR - Reaction order

, 91-100- SPFR - Species fraction (can be O for last)End with a blank card

. Next Card(s) (If Test Data) 20E6.O (There should be NTH points)

1-100- RTST - Location of each thermocouple

Next Card(s) (If Test Data) 20E6.O (There should be NDATA cards)

1-1o - TIMTST - Time of thermocouple reading11-100- TEMTST - Thermocouple temperatures

35

NOMENCLATURE

Symbol

c

Cc=g

kAr.—c1 = 1 2r

C2=1+%

Dij

E.J

h

K

i,j,k,l,m,n,

N

‘GR

‘NU

‘PR

‘RE

Qj

QI

QJ

R

Sj

T

‘AMB

‘BL

Description

Constant

Coefficient of finite difference equation

Geometry coefficient

Geometry coefficient

Matrix coefficient

Activation energy of jth component

Convection coefficient

Thermal conduction coefficieflt

Exponent or subscript

Number of explosive speties present

Grashof number

Nusselt number

Prandtl number

Reynolds number

Energy of decomposition

Internal energy generation

Interface energy generation

Gas content

Initial species concentration

Temperature

Ambient temperature

Boundary layer temperature

36

Svmbol

Tw

‘R

v

w.J

AX

AXeff

z

a

E

A,6

.&i‘i ji

P

T

Description

Wall temperature

Ambient temperature for radiation

Force vector

Species present at any given time

Incremental distance

Effective boundary layer thickness

Collision number

Thermal diffusivity

Emissivity

Denotes difference

Coefficient in difference equation

Density

Stefan-Boltzmann Constant

Time

37

Domestic NTlsPageRange Price Rice Code

00142s s S.oo A02

026450 6.00 A03

0514375 7.00 A04

076-100 8.00 A05

101-125 9.00 A06

126-1S0 10.00 A07

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Domestic NTIS DOmemic NTlsPage Range Rice Rice Code page Range Ptice price Code

151-17s S1l.oo A08 301-32S $17.00 A14176-200 12.00 A09 326-3S0 18.W AIS201.225 13.00 AIO 3s1-37s 19.00 A16226-2S0 14.00 All 37640 20.00 A172S 1.275 1s.00 A12 40142S 21.00 A18276.3oo 16.00 A13 426450 22.00 A19

Domestic NTlsPage Range price Price Cade

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