COMPARISON OF THE ANALYTICAL AND NUMERICAL SOLUTION OF A ONE-DIMENSIONAL NON-STATIONARY COOLING PROBLEM. László Könözsy 1, Mátyás Benke 2

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1 COMPARISON OF THE ANALYTICAL AND NUMERICAL SOLUTION OF A ONE-DIMENSIONAL NON-STATIONARY COOLING PROBLEM László Könözsy 1, Mátyás Benke Ph.D. Student 1, Unversty Student Unversty of Mskolc, Department of Flud and Heat Engneerng 1. INTRODUCION Modellng non-statonary heat conducton problems s very complcated [1,,3,4]. For nstance, when modellng weldng phenomena, heat sources are mostly travellng, metallurgcal reactons, concentraton-changes and phase-changes are occured durng the physchal process. Thermal and physcal propertes of materals are hghly dependng on parameters of states, but thermal parameters durng the soluton of heat conducton problems are mostly defned as constant values. Authors compared the analythcal and numercal soluton of a non-statonary coolng problem wth Drchlet boundary condtons on both sdes of the plane wall based on Crank-Ncholson numercal scheme [1,5,6]. The am of ths comparson was the accuracy test of the numercal method. The temperature values are computed n the same node-ponts of the doman wth constant temperature conductvty. In order to ncrease the effcency of the heat conducton model, authors determne temperature conductvty by lnear nterpolaton, whch procedure wll be processed n the future.. ANALYTHICAL SOLUTION The temperature dstrbuton and Drchlet boundary condtons on both sdes of the plane wall s shown n Fgure 1. [1]. The basc equaton of the coolng problem: the ntal condton: κ = t x the boundary condtons: x l ( ; n ( 0 ) ( x,0 ) = f( x) ; n ( 0 < x < l) Fgure 1. Temperature dstrbuton and boundary condtons n the plane wall ( 0,t ) = 0 ( lt, ) = 0. 99

2 The well-known analythcal soluton of the analyzed non-statonary coolng problem wth f() x = 0 + mx ntal functon s as follows: n [ ] 1 π ml nπ x n, sn exp κ π n n l l = 1 ( xt) = 0 ( 0 + )( t () where 0 s the ntal temperature value at x = 0 pont, m s the angular coeffcent of the gven ntal functon, l s length of the doman, and κ s the temperature conductvty, whch s consdered to be constant durng the analyzed coolng process. 3. NUMERICAL SOLUTION The applcated numercal dscretzaton s the Crank-Ncolson scheme, because t s a very popular method for solvng parabolc partal dfferental equatons [5]. Ths method computes temperature values between two tme-levels. Intal temperature values are determned n each node-pont wth f() x = 0 + mx ntal functon, as t was defned n case of analythcal soluton. Let us defne the next second-order dfference-operator: L xx = 1 ; ; 1. ( x) ( x) ( x) We obtan the ( dscretzed basc equaton of ths heat conducton problem based on Crank-Ncolson scheme: t ( xx ) ( Lxx ) ( ) L 1ω κ ω κ = 0 (3) ( + n where = s the temperature-dfference between two tme-levels. The ω relaxaton parameter can be choosed from the next form: ( 0 ω. At frst, we obtan the Forward Tme Centered Space (FTCS) scheme n case of ω = 0. If ω = 05. the Crank-Ncolson scheme s obtaned and f ω = 10. the fully mplct scheme s obtaned. A von Neumann stablty analyzs of (3) equaton ndcates [5] that a stable soluton s possble for t 05. x κ( 1 ω ), f ( 0 < 0 5) ω., no restrcton, f ( 05. ω. 100

3 The (3) dscretzed basc equaton after algebrcal transformatons s as follows: ( 1 ω tκ L ) = tκ L xx xx (4) whch can be wrtten n the next extended form: ( ) ( ) ( ) ωκ ( ) ( ) ( ) + ωκ + ωκ = κ n n n t n 1 t n 1 t n t (5) where the coeffcents of the trdagonal L.E.S. whch has to be solved are the next: a b t = ωκ =,..., and ( ), ( N) t = 1+ ωκ, ( N) = 1,...,, =, a n 1 0 c t = ωκ, ( = 1 N,..., and c N = 0, and the rght hand sde of the (5) equaton s defned by the next term: RHS + = t κ + 1 1, ( N) = 1,...,. The trdagonal L.E.S. of ths numercal coolng problem s as follows: b1 c a b c a b c an bn cn an bn N 1 N RHS RHS = RHS RHS RHS 1 N 1 N (6) whch can be solved by the well-known and very effectve Thomas-algorthm or other trdagonal L.E.S. solvers. The temperature values n the followng tme-step are computed by the help of numercal soluton of (6) trdagonal L.E.S. n the next way: 101

4 = +. (7) Accordng to the authors, f κ temperature conductvty s not consdered to be a constant, t s sutable to applcate lnear nterpolaton n each tme-step: ( ) κ κ κ = ( ) + ( + 1 ) κ + 1 ( ) ( ). (8) For nstance, ths lnear nterpolaton can be effectve, f κ temperature conductvty values are determned from measurements. Ths method wll be processed by the authors n the future. 4. COMPARISON OF THE ANALYTHICAL AND NUMERICAL SOLUTION The analythcal and numercal soluton s compared at the same node-ponts of the doman. Absolute and relatve dfferences between the two computatons are computed. The data-system of the comparsons s determned by the parameters of the problem and the f() x = + mx 0 0 ntal functon, where [ ] = K, m = 1.5. Current parameters are the κ = 0.06 [m /s] temperature conductvty, the l = 10 [m] length of the doman, the t = 60 [s] analyzed tme-moment, the n = accuracy of analythcal soluton, the ω = 0.5 relaxaton parameter of the numercal method. The tables of comparsed values are as follows: Index of nodes Analythcal soluton Numercal soluton Relatve error Absolute error Table 1. Comparson of temperature values n case of 1 mesh ponts Index of nodes Analythcal soluton Numercal soluton Relatve error Absolute error

5 Table. Comparson of temperature values n case of 15 mesh ponts Index of nodes Analythcal soluton Numercal soluton Relatve error Absolute error Table 3. Comparson of temperature values n case of 50 mesh ponts Index of nodes Analythcal soluton Numercal soluton Relatve error Absolute error

6 Table 4. Comparson of temperature values n case of 100 mesh ponts Consder the magntude of relatve and absolute dfferences between the analythcal and numercal soluton - n case of ncreasng the number of node-ponts wth the current nput parameters - an average relatve error and absolute error can be detected. In general, ncreasng the number of node-ponts, the numercal errors are decreasng. The analyzed numercal model s obtaned to be effectve and relable to compute the temperature-dstrbuton n case of non-statonary heat conducton problems. Development of ths numercal model s stll on the topc. 5. REFERENCES [1] T. Czbere: Vezetéses hőátvtel, Mskolc Egyetem Kadó, p [] P. Bárczy - Gy. Czél - M. Meer - B. Tolvaj: The MIZA Materal Technologcal Drop Module (MTDM) Drop Tower Days 94. Bremen, Germany, p [3] J. Szöke, A. Roósz, B. Tolvaj: Modelng and testng the Thermal Behavor of UMC, Thrd Internatonal Conference on Soldfcaton and Gravty, Mskolc, p [4] J. Szöke, A. Roósz, B. Tolvaj: Az unverzáls sok-zónás krstályosító hőtechnka modellje, GÉP, XLVIII. évfolyam, szám, p [5] C.A.J. Fletcher: Computatonal Technques for Flud Dynamcs, Vol. I. (Fundamental and General Technques), Sprnger-Verlag, Berln, p [6] Charles Hrsch: Numercal Computaton of Internal and External Flows, Vol. 1. (Fundamentals of Numercal Dscretzaton), John Wley & Sons, New York, 199. p