Numerical Solution for Non-Stationary Heat Equation in Cooling of Computer Radiator System

Transcription

1 (JZS) Journal of Zankoy Sulaiani, 9, 1(1) Part A (97-1) A119 Nuerical Solution for Non-Stationary Heat Equation in Cooling of Coputer Radiator Syste Aree A. Maad*, Faraidun K. Haa Sal**, and Najadin W. Abdulraan* * Departent of Coputer **Departent of Mateatics College of Science, University of Sulaiani, Kurdistan Region, Iraq. Abstract In te paper a ateatical odel is developed by using nuerical solution to study te non stationary eat equation in one diension. Tis odel leads to incorporate a cooling syste for a coputer radiator. Te result of te odel is copared wit te stationary eat transfer odel. Te siulation results sow te perforance of non stationary odel in ters of variation of teperature. Keywords Cooling Syste Finite Difference Modelling Siulation Non-Stationary Heat Equation Diensionless Linear Partial Differential Equation (LPDE) Introduction For cooling te seiconductor device of a coputer syste, tere are various known tecniques, suc as teral conduction or air-cooling, or te use of a eat pipe, or liquid cooling [1,]. It is typical to use a eat sink wit te central processing unit of a coputer to increase te eat-dissipating surface area of te central processing unit for ore effective cooling. Te odel used in tis work is based on air cooling by non stationary eat equation in one diension. Partial differential equation is a relation involving an unknown function of several independent variables and its partial derivatives wit respect to tose variables [3]. In recent years seeking exact solution of linear partial differential equation is of great significance as it appears tat te LPDEs are ateatical odels of coplex pysics penoena arising in pysics, ecanics and engineering. Te siulation of te developed odel is tested by using finite difference etods [1]. Te advantages of tis odel are low coputation, low coplexity of te ipleented algorit and optiization of odel efficiency. To analyze teoretically and nuerically te relative iportance of te cooling - radiator ecanis, one as to take in account te evolution low of LPDE. Te condition tat rends te proble in one diension is te tinness of te radiator (T depend only on x). Te typical stages of a nuerical siulation [4] are illustrated in figure (1). Pysical penoena (Observation and odelling) Mateatical odel Evolution of LPDE Nuerical analysis (Sapling of LPDE) Prograing Optiization C++ progra Fig (1): General odel diagra Tis paper is organized as follows. Section, we set in equations te proposed E-ail:karita3@yaoo.co 97

2 (JZS) Journal of Zankoy Sulaiani, 9, 1(1) Part A A119 proble, wile in section 3 all te steps wic are necessary to analyze ateatically te equations are discussed. Te results and all relevant discussions appear in section 4. Finally te ain conclusions are suarized in section 5. Non-Stationary Heat Equations Te geoetry of te proposed syste is sowing in te figure (). Te first ter describes te internal flow of eat wile te second ter represents te external flow of eat. Te last ter of te above equation [6] indicates te loss eat by air, were q is te flux of eat by te unit of surface and tie [Watt/²], c is a surface coefficient of teral transfer [W/(² Kelvin)], p is te perieter of te surface, is te density [kg/] and C is te ass of eat [Joule/(Kg Kelvin)]. Dividing equation (1) by S dx for dx, because te radiator layer is sufficiently tin. Te Fourier law expresses te proportionality between eat flux and gradient of teperature [7] as follows: q ( x ) k T x () Were k is teral conductivity [W/(k)]. Finally, te distribution of teperature will be described by te ODE: T t d T a( x ) T T a ( 3) dx (a) Radiator (b) One layer Fig (): Geoetry of te odel Te geoetry is described by te lengt L, te transversal section S, te teperature generated by te processort, and te abient teperaturet a. To find te teperature distribution in te radiatort, it is supposed tat te layer is sufficiently tin to consider our proble one-diensional. Te teral equilibriu of an infinitesial eleent dx is as follows[5]: q( x) S q( x dx) S c ( x)( pdx)( T T ) T C( ( Sdx)) (1) t k Were [ / s] and a(x) = C c ( x) pl ks Equation (3) defined for x L, and ten te boundary conditions ust be added [8]: T ( ) T (Diriclet type condition) dt dx xl (Neuan type condition) Te condition in x=l expresses te fact tat te eat transfers troug te last section q(l) is ignorable wit respect to te initial section q(). 98

3 (JZS) Journal of Zankoy Sulaiani, 9, 1(1) Part A A 119 Mateatical Analysis Te ateatical odel developed in tis paper consists of te following steps: Step1: ake eq (3) in diensionless for. Te diensionless variables are defined as: x T Te t t( ), x and L L Te (, t ) ( T Te ), ( 1, t ), t [, tmax ] x Step: explicit discretization in space and tie: Te first step in sapling consist of decoposition te segent, L in M intervals of lengt L M. We obtain tat te es consists of M 1 points of abscises x,,1,..., M. Te idea of te nuerical resolution is to find an approxiation of te exact solution ( x ). To obtain te discrete equation wic take te values, te derivation operator ust be sapled by using Taylor series expansion about x M1 L [, L] x, x, x, (5) M =1,,, M-1 tat for a linear syste wit M-1 equations and M+1 unknowns. By substituting te above conditions into eq (3), one can get te explicit discretization in tie can be obtained by sapling te tie interval [..tax]. t N N1 Max Max n n t n (6) n [, t ] t, t, t n t, t Te values of solution (, t) two points n ( x, t )will be ( x, t ). Using Taylor n n d developent [8] for two independence a( x ) ( 3) (4) variables ( x, t), we get, t dx w ere x 1, t t 3 M ax 4 1 ( x ) ( x) ( x) ( x) ( x) O ( ) 6 For notation siplicity te * can be reoved 3 4 and for reducing te calculation tie c( x) is 1 ( x ) ( x) ( x) ( x) ( x) O ( ) 6 considered as a constant. Non-stationary equation (4) becoe One can obtain fro [8]: stationary if t Max. ( x, tn t ) ( x, tn) t ( x, tn) ( x 3, ) [ tn, tn 1] t t t (7) and ( x, t ) ( x, t ) ( x, t ) x n n n 1 n n 1 n ( x, t ) O( ) n (8) Substitute (7) and (8) into (4): n1 n n n n 1 1 a (9) t were =1,,, M-1. As n 1 ( x, t ) is known, can be calculated by recurrence using te relation n1 n t n n n ( 1 a t) ( 1 1) r ( 1 r a t ) r (1) n 1 n n n 1 1 were r t. Under vector for: 99

4 (JZS) Journal of Zankoy Sulaiani, 9, 1(1) Part A A 119 n 1 n H H linear operator fro te equation (4). 1 1r a1 r r 1 r a r r 1r a3 r r 1 r a M1 M1 Te ost iportant in te practical point of view is te convergence of te nuerical solution toward te exact solution wen te discretization steps t and tend to zero, siilar convergence studied in [9]. Figure (3) presents te siulation results of te tie evolution in te radiator layer wen te condition ( x,), x 1 is applied. Te non stationary solution is converged to te stationary solution wen te grid points of discretization M=75. Siulation Results Consider te case tat te following constants are taken into account for one layer of Aluiniu, of section 4[] * 5[] and L=4[]. Te teperature of te processor T 46C and te abient teperature is C. For Aluiniu, T a te teral conductivity is 164 W /( k) and te coefficient of eat transfer take te constant value W /( ) c k. Te variation of teperature distribution now can be studied on te Aluiniu layer range, L. Te nuerical solution of te non stationary eat equation needs soe odification copared wit te stationary equation [1]: 1. te reference tie t / is calculated L wit te value of teral diffusion x1 [ / s];. in te initialization loop, it ust be precise ( x, t ); we considered two distinct conditions: ( x,) (,), x 1 and, ( x,), x 1 3. te tie step is given by te stability condition t. Fig (3): Coparison between stationary and non stationary eat transfer for different values of M (3, 75), c=, t. 5. In bot cases te diinution of te teperature is observed witin te tin layer. On te oter and, te nuerical calculations are valid because of te convergence toward te exact solution especially wen te steps of discretization are taken 75. Figure (4) sows te distribution of teperature in te radiator layer for te sae values of given paraeters as it is used for 1

5 (JZS) Journal of Zankoy Sulaiani, 9, 1(1) Part A A figure (3) except te value of [ / c W s]. Figure (4) also explains tat te decreasing of te teperature is very low about1 C. In fact te value of c depends on te air flow and it is directly proportional wit it. Fig (4): Coparison between stationary and non stationary eat transfer for different values of M (75), c=, t Conclusions: In tis paper, te siulation results sow te efficiency of te proposed syste in bot teperature diinution and precision. In te case wen te surface coefficient of te teral transfer and te discretization steps are equal to, 75 respectively, our syste grants te convergence (i.e., optial solution). Finally, te test results sow tat te effect of te oter involved paraeters (like; p, S, ) on te teperature distribution is very sall. Te future work will be te study of non stationary eat equation in te coputer radiator in D (T(x,y)). Anoter perspective of tis researc will be te easure of te produced error by approxiation te continuous operators by discrete operators troug te consistency concept. References 1. Yunus, A., Heat Transfer: A Practical Approac, MacGraw-Hill, nd Edition,,. ISBN -1: Sa,A., Saakia, B., and Sriari, H., A Nuerical Study of te Teral Perforance of an Ipingeent Heat Sink-Fin Sape Optiization, IEEE On Teral Penoena,, (), Butcer, J., Nuerical Metods for Ordinary Differential Equations,, 3.Wiley, Rev edition, ISBN 1: Salle, S., Albert, Y., and Bakar, A., Coputing for Nuerical Metods using Visual C++, 7.Wiley, ISBN -1: , 5. Saini,M., and Weeb,R., Heat Rejection Liits of Air Cooled Plane Fin Heat Sink For Coputer Cooling, IEEE Transactions on Coponents and Packaging Tecnologies, 3, 6 (1), Naba, K., Niekawa, J., and Kiura, Y., Heat-Pipes for Electronic and Evaluation of teir Teral Perforance, IEEE Transactions on Coponents and Packaging Tecnologies,, 3(1), Vladiir, G., and Yury, F., Low Noise Cooling Syste for PC on te base of loop eat pipes, Applied Teral Engineering, 7, 7, Danaila,I. and Hect,F., Siulation Nuérique en C++,3, Paris, ISBN ,. 9.Jon H. Matews and Kurtis D. Fink, Nuerical etod Using Matlab, Tird Edition, 1999, United States of Aerica, ISBN ,. 11

6 (JZS) Journal of Zankoy Sulaiani, 9, 1(1) Part A A 119 التحلیل العددى لتبرید نظام رادیتر الكومبیوتر باستخدام معادلة الحرارة غیر ي ارى على محمد / كلیة العلوم-قسم علوم الحاسبات. المستقرة كلیة العلوم-قسم جامعة السلیمانیة الریاضیات / أقلیم كردستان/العراق الخلاصة فى ھذا البحث تم تطویر نموذج ریاضى لدراسة المعادلة غیر ثابتة للحرارة ذات بعد واحد.و نتاي ج ھذا البحث قورنت بنظام انتقال الحرارى الثابت. Received 15\3\8, Accepted 6\6\