An analytical analysis for the prediction of nonlinear inverse temperature profiles in solid explosives

INT. COMM. HEAT MASS TRANSFER Vol. 20, pp. 811-820, 1993 Printed in the USA

0735-1933/93 $6.00 + .00 Copyright°1993 Pergamon Press Ltd.

AN ANALYTICAL ANALYSIS FOR THE PREDICTION OF NONLINEAR INVERSE TEMPERATURE PROFILES IN SOLID EXPLOSIVES

Awad R. Mansour Department of Chemical Engineering S. Taqieddin and Y. Abdel Jawad Department of Civil Engineering Jordan University of Science and Technology Irbid, Jordan

(Communicated by J.P. Hartnett and W.J. Minkowycz)

ABSTRACT A novel analytical method developed by Mansour and Hussein (11) has been used for the evaluation of critical thermal conditions for a non-linear inverse temperature distribution model of an explosive. The analytical results presented in this work have been compared with previously published numerical results and shown to be accurate. The advantage of this new method is that it can be easily applied to other inverse nonlinear models arising in boundary layer, kinetics, electronics, vibration, combustion ... etc.

Introduction In recent years, increasingattention has been devoted to the applicationof approximate and closed-form analyticaltechniquesto non-linearheat transferproblems. Analyticaltechniquesprovide researchers and engineers with major advantagesover numericalmethodssuch as:l2.

Providinghelpful physicalinsight into the nature of the problem solution. Analyticaltechniquescan also be used in conjunctionwith a partitioningscheme for the thermal analysis of individualcomponentsof practical structures.

3.

Numericaldiscretizationtechniqueshave the major drawback that the calculationmust be performed at each time or space step until the final time or length is reached. Therefore, the numerical solution is discrete and stepwise while the analyticalsolutionis continuousalong all axes

811

812

4.

A.R. Mansour, S. Taqieddin and Y. Abdel Jawad

Vol. 20, No. 6

The output of the numerical technique is a set of numbers of doubtable correctness, while the analytical solution is expressed in terms of functions that can be easily used for subsequent differentiation and integration when expressions of heat transfer rate or thermal efficiency for systems, or applying different operating or boundary conditions are needed.

5.

Stability and convergence problems are usually encountered when numerical solutions are chosen, and these problems become more complicated when non-linear thermal problems are considered. Owing to no general mathematical theory to analytically solve non-linear partial and ordinary differential

equation models, various approximate methods [1-13] have been published to fred analytical solutions. In some technical applications, the heat production in a solid satisfies a non-linear relation between heat and temperature. Similar behavior is encountered in some chemical reactions of higher order. In these cases it has been found [9] experimentally that the non-linear heat generation follows an exponential law such that f(0) = ~e0. The problem in such equation lies in that it is non-linear and its non-linearity is in the dependent variable. Moreover, these problems have singularities at some end points of the interval and Nenmann and Din'chlet boundary conditions have to be satisfied. The objective of this paper is to present a straight forward approximate hybrid method which can be easily used for different inverse differential equations describing temperature prof'fles within the explosive body and for similar applications in chemical reactions within porous catalyst particles.

S c h e m e o f the Pronosed H v b i r d M e t h o d Consider a general non-linear thermal problem described by the following differential equation: U(O,x) =0

in

¢2

(1)

R

(la)

with the boundary conditions

B(0,x) = 0

on

where D and B are non-linear differential operators. W is the domain of the problem and R its boundary. The application of the present hybrid analytical - numerical technique can be described by the following steps: (1) Linearization of the non-linear terms inside Equation (1) which will become of the linear form:

~O,x) = o

(2)

Where. [) is a linear differential operator. (2)

If any 0 terms remain non-llneur after the linearization step, they are assumed temporarily constant to permit linear solution of Eq. (2)

(3)

Equation (2) is solved by one of the standard methods used for linear differential equations, and the obtained solution is put in the form 0 i+l = f ( 0 i ' x )

(3)

Vol. 20, No. 6

TEMPERATURE IN SOLID EXPLOSIVES

813

where 0i is the temperature profile at the ith iteration level which has been assumed constant in step(2), and 0i+l is the temperature profile at the (i+l)th level iteration which represents the dependent variable solution of Equation (3) (4)

An alternative procedure is performed till convergence between qi+l and qi is oblained at each x value in the interval of interest. An iterative solution is said to be convergent if

i+l

, where e is the

tolerance error.

Illustrative

Examnles

In the following examples, three cases are studied to illustrate the applicability of the present approximate hybrid method to non-linear thermal problems. Example 1:

Heat

Conduction

in Solid Explosive

Cylinder

with Non-Liner

Exponential

Heat Generation The model describing this process is given by the following differential equation.

0" + (l/r)0' = - 8 exp(0)

(4)

subject to the boundary conditions q'(O) = 0

(5)

q(1) = 0

(6)

where d is a dimensionless thermal conductivity Method of Approximate Solution Let exp (q) : 1 + q

(7)

Substitute Eq. (7) into Eq. (4) to yield Eq. (8)

0"+ ( 1 / r ) 0 ' + ~0 = - ~

(8)

The analytical solution of this differential equation can be written in terms of Bessel functions as follows: 0 ( r ) = c 1 J 0 ( ' v / - g r) + c 2 Y o ( ' X ~ r ) - I

(9)

By applying the boundary conditions given in Eqs. (5) and (6), we obtain the following solution

0(r) -

J 0(-V/-~ r )

-1

J°("M/-~') Another Approximate

(10) Solution

If we assume that the fwst three terms of exp (0) series axe enough to represent it as follows:

02 exp(0) = 1+ 0 + - 2

(11)

By Substituting Eq. (11) into Eq. (4), the following differential equation is obtained: 01,

+ (l/r)0' + 5 0

~ --

~ - (~/2)02

(12)

814

A.R. Mansour, S. Taqieddin and Y. Abdel Jawad

Vol. 20, No. 6

Now assume 02 = ct2 = constant in the right-hand side of Eq. (12) to get the following: 0" + (l/r)0" + 50 = - 5 - (5/2)a 2

(13)

The complete solution of Eq. (13) is: 0(r)=

J0('~/~'r) j0(,V¢.~. )

a2 +--~--1 (14)

Or in an iterative form as:

j0(-V/-~ r) 0 i + l ( r ) -- j 0 ( . V / - ~ )

0 2" + --~--- 1 (15)

Where Oi+l and Oi are successive tempoatme values which can be iteratively computed at each r,adiul value till convergence is obtained. The two analytical solutions obtained in this section have been compared with the closed-form exact solution of Eq. (4) which can be given as follows:

0 exact(r)= ln(

8---B-/5- 2 ~

~,(Br 2 + 1) )

(16)

Where B is the integration constant in Eq. (16), which is given by the following equation: 8B/5

= (B + I) 2

(17)

For 0 < $ < 2, Eq. (17) has two distinct real roots and thus two solutions of Eq. (4) occur. For 8 = 2, Eq. (17) has only one root when B=I, and only one solution of Eq. (4) occurs. For 5 > 2, no solutions exist. From Figs.(l-3) it can be noticed that the two approximate solutions obtained in the present study are very accurate compared to the exact solution. 2.& 2.4 .

2.2

i' ~

J.,

FIG. 1 T~mre Dislaibution Inside Cylindrical Solid Explosive for 5=0.1

t2

.6

.2 0

.1

2

.3

.4

.$ I~ldlUJ, r

J

.7

.ll

.9

Vol. 20, No. 6

T E M P E R A T U R E IN S O L I D E X P L O S I V E S

-

815

Exact Seiufim ([q,16)

m

~e

7.5

J s

l

FIG. 2

22;

Temperature Distribution Inside Cyinderical Solid Explosive for 5 = 0.5. 0

.1

.2

.3

.4

5

.6

.?

.l

.9

1

Rldl~,r

3s

4

-"~

"~,-- A,~x.cw s , t ~ .

25 2t

FIG. 3

1s

Temperature Distribution Inside Cyinderical Solid Explosive Explosive for Various Values of Dimensionless Thermal Conductivity, 8.

lo 5 e I

.1

.2

.3

.t

.5

.1

Ridimm. r

.7

.I

.9

816

A.R. Mansour, S. Taqieddin and Y. Abdel Jawad

Example 2:

Vol. 20, No. 6

Steady State Heat Conduction With Zeroth Order Arrhenius Heat Generation In

Solid Spherical Explosives

The model of this process is described by the following second order non-linear differential equation: 0" + ( a / x ) 0 ' = - 8 e x p (0/(1 + 0 / T ) )

(18)

with the following implicit boundary conditions: x=0:0'=0

(19)

x=l

(20)

:v0+0'=0

where a: is a geometric factor and it takes the values of 0, 1 and 2 for the slab, cylindrical and spherical geometries respectively. ~: is dimensionless thermal conductivity y and v: are thermal characteristic constants of the system. Analytical

Solution:

Assuume that exp(0/(l+0/y)) = 1+ 0/(l+0/y)

(21)

and substitute Eq. (21) into (18) to obtain Eq. (22) 0"+(aJx) 0 ' + 50/(1+0/~,)=-5

(22)

Equation (22) is still non-linear, hence the following assumption is made: Let 0 = constant = a temporarily in the denominator of the third term, therefore, Eq. (22) becomes linear as follows,: 0"+(a/x)+[~20=-5

(23)

wheae

(24)

2=5/(l+a/.D The general solution of Eq. (23) is: 0(x) = x - l / 2 ( A Jl/2(l~x)+B J.1/2([ix))-5/[~ 2

(25)

or .

sinl~ + BCOS[lX x

e(x) = A ~

5

[~2 (26)

If we apply the boundary conditions to Eq. (26), it becomes as follows: 0(x) =

v~osl~x

5

x~2((V - 1)cos[~ - ~ s i n ~ )

I~2

(27)

Vol. 20, No. 6

TEMPERATURE

IN S O L I D E X P L O S I V E S

817

or in an iterative form:

O i+ l(X)

v~c°sPix

8

X [ ~ ( ( V -- I)COS[~ i -- ~ i s i n [~ i)

2 [~i

(28)

where 2 ~i --~/(1 + 0/T)

(29)

The values of 0i+l can be successively computed based on the assumed values of 0i at each x value in the interval of interest, till convergence condition. 10i + 1 - Oi I< E is statisfied at each point The approximate hybrid solution obtained in Eq. (28) for different values of v (v = 2, 5, 10 and 100 respectively), y = 10, a=2 and 8 = 2 has been compared with the accurate numerical solution obtained by applying Rungc-Kutta-Gill method with tolerance error of 10-5 o~der, and shown (see Fig. 4) to have good agtccmenL

2.6 "-., 2.Z~

\

2.2

,,

2

a • o

Numerical Solution V = 100 DELTA =3.352 V = 10 DELTA = 2.819 V = 5 DELTA =2.359

•

v: 2 .DELTA: ;:52

1.8 .

1.6 1.4 1.2 1

.,8 .6 .4 .2

0 0

.1

.2

.3

.4

.5

.6

.7

.8

Radial Distance, X FIG. 4 Temperature Distribution Inside Cyindetical Solid Explosive for Various Values of 7 and 8.

818

A.R. Mansour, S. Taqieddin and Y. Abdel Jawad

Example 3:

Vol. 20, No. 6

Heat and Mass Transfer within a Porous Catalyst

This model is described by the following ordinary non-linear differential equation: 0" + ( a / x ) 0 ' = 0 2 0 e x p (1 TI](1 - 0)

¥ ~-(i-~)/

(30)

with the boundary conditions: x=0 : 0'=0

(31)

x=l : 0=0

(32)

Here ~2, a, Vand 15are parameters of the system. This equation can not be integrated analytically and only numerical solution is possible. Analytical Approximate Solution:

"~i(1 - 0 ) ~) ) - - 1 + exp (~.-1+ ~-(i-

I+T~(1-~I-0)0)

(33)

and substitute Eq. (33) into Eq. (30) to obtain: O,,+(a/x)O,_020=02<

T~(1-O)

)

1 + ~ 1 - 0)

(34)

Now, let 0 = ct = constant in the right-hand side of Eq. (34) to become

0"+ (a/x)0'-020=0 2 1 T ~ ] - 0 )

(35)

"lhe complete solution of Eq. (35) is

o=

l+'~,l+~l-a))/

co-~

.ll

.

(Vfl(1-a)

]c°sh0 x- ~[i ~-~1 2 ~) )

(36)

or in an iterative form as

1+ ill+-ff(i-L.~)

0i+l =

I

cosh 0

I

"~B(1-Oi)

cosh 0x - 0i 1 ~1-:

Oi)

/

(37)

The analytical solution obtained in Eq. (37) has been compared with the Runge-Kutta-Gill numerical solution and shown to be accurate as it can be noticed in Figs. (5) and (6).

Vol. 20, No. 6

1.2

TEMPERATURE

IN SOLID EXPLOSIVES

819

A AnalyticalSol~lm

1.¼

•

I~m'lcal $oluflca

1.12 1.18 o

!,

1.Or •

•

•

.

A

,

.

•

,

_

--

0.~2 FIG. 5

0.111

Temperature Distribution In a Slab for a = O. 7 = 2 0 , TI]= 14,0 = 16.

0.Ilk 0.8

O

0.1

0.2

0.3

O.k

03

0.6

0.? I.I

0.9

1

Distance.X

0.~ 0.8

0.7

i

0.1 05 0£ 0..t

FIG. 6

0.2

Temperature Distribution for a = 0, y = 20, T~ = 2, 0 =16.

11.1 0 II

t.!

0.2

0.'I

0.4 t.5 O.i 0.1 Obhece, X

O.ll

U

1

820

A.R. Mansour, S. Taqieddin and Y. A b d e l Jawad

Vol. 20, No. 6

Conclusions The method of hybrid analytical numerical techniques presented in this paper has been easily and efficiently applied to a number of important inverse engineering equations and shown to be of good accuracy. The proposed method can be used to other engineering applications as well.

Acknowled2ement The computational efforts of Hasan Atteih during the work of this study is acknowledged.

References 1.

T.Y., Na and S. C. Tang, "A Method for the Solution of Conducting Heat Transfer with Nonlinear Heat Generation, "Z. angew. Math. Mech. Vol 49, pp. 45-52 (1969).

2.

B. Vujanovic, "Application of the optimal linearization method to the heat transfer problem," Int. J. Heat Mass Transfer 16, 111-1117 (1973).

3.

M. Kubicek and V. Hlavacek, "Direct Evaluation of Branching Points for Equations Arising in the Theory of Explosions of Solid Explosives," J. Computational Physics, 17, pp. 79-86 (1975).

4.

A. Aziz and J. Y. Benzies, "Application of perturbation techniques to heat transfer problems with variable thermal properties," Int. J. Heat Mass Transfer 19, 271-276 (1976).

5.

A.K. Noor and C. D. Balch, "Hybrid perturbation Bobnov-Galerkin-technique for nonlinear thermal analysis," AIAA J. 22, 287-297 (1984).

6.

A. Lippke, "Multiple Solutions for Nonlinear One and Two Dimensional Thermal Conduction with Exponential Heat Generation," Z. angew. Math. Mech. Vol. 68, pp. 252-255 (1988).

7.

D. Wacker, "A Contribution for Nonlinear, One Dimensional Thermal Conduction with Exponential Heat Generation, "Z. angew. Math. Mech. Vol. 66, pp. 378-379 (1989).

8.

A. Lippke, "An Efficient Continuation Algorithm," Preprint 194, Technische Universitat, Berlin, Federal Germany, (1989).

9.

A. Lippke, "Analytical Solutions and Sinc Function Approximations in Thermal Conduction with NonLinear Heat Generation, J. Heat Transfer, Vol. 113, p. 5-11 (1991).

10.

H.T. Chert and J. Y. Lin, "Hybrid Laplace transform technique for non-linear transient thermal problems," Int. J. Heat Mass Transfer 34, 1301-1308 (1991).

11.

Awad R, Mansour and A. Hussein, "An Approximate Solution for the Nonlinear Inverse Axial Dispersion Model," Int. Comm. Heat, Mass Transfer 17, 823-830 (1991).

12.

B. Jubran, A. R. Mansour, M. Hamdan and B. Tashtoush, "A New Approximate Analytical Solution for the Prediction of the Thermal and Hydrodynamic Characteristics of the Trombe wall, "accepted by J. Heat Transfer (1992).

13.

Awad R. Mansour and R. Jumah, "An Approximate Analytical Solution to Non-Darcy Heat and Mass Transfer by Natural Convection from a Flat Plate Embedded in a Fluid Saturated Porous Meditma, Submitted for publication (1992).

Received June 22, 1993