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AD-A072 483 


UNCLASSIFIED 


I of I 

AD 

A072A83 


NORTHWESTERN UNIV EVANSTON IL TECHNOLOGICAL INST F/G 20/13 

AN APPROXIMATE REPRESENTATION OF NEUMANN'S SOLIDIFICATION SOLUT— ETC(U) 
NOV 78 B A BOLEY» L ESTENSSORO N00014-75-C-1042 

TR-1978-1 NL 

END 

DAT! 

FILMED 

9 79 


1 








DDG JlLEJOPY AD A 072483 



Luis Estenssoro 

Wiss- Janney-Elstner Associates, 330 Pfingsten, Northbrook, Illinois 60062 
and 

Bruno A. Boley 

Northwestern University, Evanston, Illinois 60201 


Int roduct ion 

The solution of Neumann's change-of-phase problem (i.e., the solidification 
or melting of a slab, whose temperature is initially uniform and is maintained 
constant at the surface [l]) by approximate analytical means has received some 
attention in the recent literature [2j. Most of the solutions available, how- 
ever, present a single approximation, and are not readily adaptable to obtain- 
ing further, and hopefully more accurate, approximations. In the present work 
a method of so doing is presented, in which the multiple penetration-depth 
technique of [3j is applied to the formulation of Neumann's problem in the 
form established in [4-]. Some numerical results presented at the end indicate 
that, at least for the case of constant properties, the proposed approach is 
workable and satisfactory. 


Analysis 

The problem at hand refers to the slab>>0, initially liquid at a uniform 
temperature (for the case of solidification), whose surface temperature 

is maintained at the constant value At any time t >o the 

liquid occupies the region^(t)<> c «, , and the solid the region 0< * < 

The temperatures T and in the liquid and solid respectively must satisfy 
the Fourier heat-conduction equation (with primes and dots indicating differen- 
tiation with respect to ? and t? respectively): 


This work was supported by a grant from the Office of Naval Research. It 
formed part of the first author's M.S. thesis in the department of Civil 
Engineering at Northwestern University (1976). 


I 



NUMptM 


*/ 1 Ac c 



TRACT OR GRANT NUMt)Eh(a) 


(/0pi4-75-C-ld42 


j "1978-1 


— T*-T^e-fwr<rsu®rfffp) I / 

AN APPROXIMATE REPRESENTATION OF NEUMANN'S / 


SOLIDIFICATION SOLUTION ' 


auI«lO(U*j — J 

Bruno A. /Bol ev ^Dea n of Technological Institute! 
Luis /Es tens so ro~l W i s s - Janney-E 1 s t ne r Associates 


I 3. PERFORMING ORGAN! Z AT ION NAME AND ADDRESS 

i Northwestern University, Evanston, IL 60201 
J 330 Pfingsten, Northbrook, IL 60062 


10. PROGRAM ELEMENT. PROJECT. TASK 
AREA A WORK UNIT NUMUERS 


NR-064-401 


//! 


I II, CONTROLLING OFFICE NAME AND ADDRESS 

j OFFICE OF NAVAL RESEARCH 

j Arlington, VA 22217 

1 

| I « MON iTORlNG AGENCY NAME A AODRESSfU dllf«r»ol Rom Coni rolling OH! cm) 


*r: REPORT PATE 

Nov. 1-978 / 




OFFICE OF NAVAL RESEARCH 
Chicago Branch Office 
536 So. Clark St. 


| Chicago, IL 60605 



13. NUMBER OF PAGES 

5 


15. SECURITY CLASS, (o< Ihl, import) 

Unclass i tied 


DECLASSIFICATION - DOWnGHA OiNG 
schedule 


1 

[ 

16 Dt ST RltiU Tl ON STATEMENT (ol It, la Report) 

l 

Qualified requesters may obtain copies of this report from DDC 

• 


»7. DISTRIBUTION STATEMENT (ol the mbatrmet entered In Block 20, II dlllerent Itom Report) 

l 

. 

. 


• a. supplementary notes 



19. KEY WORDS ( Continue on reveree aide II neceaamry end Identlly by block number ) 

Melting, solidification, heat transfer, approximate solutions 


1 

^0 ABSTRACT (Continue on reeerae aide II neceaamry end Identlly by block number) 

An approximate method of solution of Neumann's change-of-phase problem 

Is presented. The solution makes use of a multiple penetration-deplh 
approach. 

« 









2 


XsT- 

pc (T 
J L.sV. L, 

the initial and 

boundary 

conditions 


• T 

y > c? 


T 

i 

t ^ o 

H 

/ Jl> 

Q 

T. 

t > o 


and subject to the solid-liquid interface conditions 

X [<<:), t V T s[^ t ] ’ T» b> ' 


U'-U , ' , f>l* 

S L L J (4) 

with si(p) = 0- We note that the solution is of the form 

A(t)~ 1 (5) 

where "X is an unknown constant and reference diffusivity. 

Let now [4] two auxiliary solut ionsT B, C) be defined as satisfying 

eq. (1) and the following boundary and initial conditions- 

T s («.t ; B,T.> T. T t (»,t;T,.cj=C 

T 1 ( y -°; B -. T «V T . + s T[>.o;T, c). T7 

L '*• / ' » 1 (6^ 

B,T«V T c+B Tj-.t-.T.tV T, 

It is then easily verified that the solution of the desired problem is obtain* 
by choosing the constants ^,C and Xso as to satisfy eqs. (3) and (4). 

An approximate form for the auxiliary temperatures can he taken in terms of 
m penetration depths ^£t)as [3]: 

Tfr fc 8T\ = T+R- B 7 <L .-.Yl- \ 


T J 

where 


B z> 5 -'6- Q h 

'')ZA l x (l- 


3 


[1 

o 


x< X, 


X > X. 




l 


( 8 ) 


' 0s >L, 

and where it is assumed that 




(9) 


The unknowns l and are calculated by satisfying (1) approximately by means 
of the "method of moments" , i. e . , by setting 


O 

5 


[(*T') O, V\* 0,1, 

For the case of constant properties, it is found [3] that 

<L . = A. fy t l> '° 

v'-S i L L !l« 5 ; L > 


( 10 ) 


<1- 


3> : ,, 


L > O 


( 11 ) 


'VL 




where the values of A. and I) are to be obtained numerically under the re- 

V t 

striction of eqs. (9). 

There now remain to satisfy the interface relations (3) and (4). The former 
gives T _ 


B - 


C = 


and the latter reduces to the form 


i - IA S * ^ s') 

v r o 


(12) 


T> - ^ 


(13) 


wnerc 


R. 




l f 


> P 


— * cv 

X-t J 

w o 




(13a) 





The solution of eqs. (13) involves a trial and error procedure, in order to 
obtain values of/i(t)which lie (since for between zero and the 

largest of the^,(t-^. In the cases of one and two penetration-depths , was 
smaller than the smallest penetration-depth, so that the step function in 
these equations could be ignored; but for the case of three penetration-depths 
this was not the case and the complete expressions had to be used. A numerical 
comparison of the result obtained from one, two and three penetration depth 
with the exact values is presented in the accompanying figures, for the case 
in which the solid and the liquid have the same properties («<»|), for various 
values of p . It is clear that improvements result from the use of several 
penetration depths, although for this particular problem even one penetration 
depth leads to quite accurate results. 


REFERENCES 

1. H.S. Carslaw and J.C. Jaeger, Conduction of Heat in Solids , 2nd Ed., 

Oxford University Press (1959). 

2. B.A. Boley, "An Applied Overview of Moving Boundary Problems", in Moving 
Boundary Problems , edited by D.G. Wilson, A.D. Solomon and P.T. Boggs, 
Academic Press (1978), pp. 205-231. 

3. B.A. Boley and Luis Estenssoro , "Improvements on Approximate Solutions in 
Heat Conduction", Mech. Res. Comm., vol. 4, no. 4, pp. 271-279 (1977). 


4. B.A. Boley, "On a Melting Problem with Temperature-Dependent Properties", 
in Trends in Elasticity and Thermoe last icitv , W. Nowacki Anniversary 
Volume, Wolters-Noordhof f Publishing Co., Groningen, The Netherlands,