# Full text of "DTIC ADA072483: An Approximate Representation of Neumann's Solidification Solution."

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rr, 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,