Optimal Design and Control o CPU eat Sink Processes Chyi-Tsong Chen *, Ching-Ko W and Chyi wang Department o Chemical Engineering Feng Chia University Taichng 407, Taiwan Department o Chemical Engineering I-Sho University, Kaoshing 840, Taiwan * E-mail: ctchen@c.ed.tw Astract: This paper considers the optimal plate in design and the control o CPU heat sink processes. First, we apply a inite element method to investigate the heat transer phenomena o a heat sink process. To have a etter heat dispersion perormance, a realcoded genetic algorithm is tilized to search or an optimal set o plate-in shape parameters. The ojective nction to e imized is the entropy generation rate which can take simltaneosly the two major actors, heat transer rate and air resistance, into consideration or design. The comparison with existing method shows that the present optimization scheme is ale to achieve a etter design or heat dispersion. To attenate environmental and time-varying distrances, a direct adaptive control scheme is then developed or the CPU heat sink process. It is ased on sing a onded single neron controller (SNC) along with a parameter tning algorithm to reglate the temperatre o a selected control point. Extensive comparisons o the SNC-ased control perormance with the on-o control as well as a PI controller show that the proposed scheme provides excellent control perormance despite the existence o nexpected process ncertainties. Keywords: eat sink process, Optimal design and control, Real-coded genetic algorithm, Single neron controller, Parameter tning algorithm, Partial dierential eqation system. INTRODUCTION De to the increment o clock speed, switching speed and transistor density, the size o the central processing nit (CPU) ecomes smaller t the eiciency is getting higher. This inevitaly leads to the increase o heat generation rate per volme o the CPU. I the heat cannot e removed appropriately, the CPU lie can e greatly shorten and the overheat sitation can also damage the normal operation o the process (Bar-Cohen, 999). A simplest and eective way to improve the CPU heat dispersion is the se o the air cooling ins. In recent years the plate-in design or the CPU heat dispersion has received considerale attention and many design techniqes have een proposed (Bejan and Morega, 99; Clham and Mzychka, 000; Kreger and Bar- Cohen, 00; Shih and Li, 004). This paper consists o two parts. In the irst part, a real-coded genetic algorithm is applied to the optimal in shape design. Dierent rom conventional genetic algorithms, the real-coded genetic algorithm is easier to se since it does not need the coding and decoding process and is more eicient to allocate the optimal soltion. The ojective nction to e imized is the entropy generation rate, which has een shown to e eective in the compromise etween the heat resistance and transer rate. The optimal design o plate-ins with horizontal air cooling system is considered in this paper. Extensive comparisons o the proposed optimization scheme with existing methods are carried ot. With the optimally designed plate-in air cooling systems, the second part implements a control system in the CPU heat sink process in order to attenate the negative eects o the environmental pertrations on the CPU operation. A onded single neron controller (SNC) is applied along with a devised parameter tning algorithm to reglate the an speed adaptively. The control perormance o the SNC-ased scheme is compared with an on-o control and a PI controller.. DYNAMIC SIMULATION AND ANALYSIS OF TE CPU EAT SINK PROCESS. Governing eqations and ondary conditions The schematic diagram o the CPU heat sink process to e considered is shown in Figre. Fig.. CPU heat sink model.
The geometric symmetry o the conigration allows one to se the system shown in Figre or the sseqent nmerical stdy, in order to take the advantages o comptational eiciency and to maintain the soltion accracy. Fig.. Cross section o the CPU heat sink model. For sch a CPU heat sink process, heat "travels" rom areas o high temperatre to areas o lower temperatre is governed y the heat eqation: ρ C p ( k T ) Q () t ρ is the density, C p the heat capacity, k the coeicient o heat condction, which is material dependent, and Q the volme heat sorce. In () the irst term represents the energy accmlation, the middle term the dierence in the heat lx over the volme, and the last term the volme sink (or sorce). The ondary conditions are o Nemann type with the otward heat lx, n (k T), eing given as n k T he T T () ( ) ( ) ere n is the normal vector o the heat lx. In (), the term h e ( Tin T ) is known as Newton's law o cooling, which speciies the heat lx rom the srrondings de to orced and/or natral convections, T in the external or srronding temperatre. It is noted that the radiative heat transer is neglected here since the temperatres are assmed to e moderate. Also noted is that the heat sink cools down the processor y directing the dissipated heat rom processor to the srrondings throgh heat condction. The an removes the heat rom the heat sink y lowing air across its srace. This convection heat transer throgh is proportional to the dierence in temperatres etween the heat sink and the srrondings p to a actor known as the heat transer coeicient h e. The vale o h e depends on the degree o convection across the srace. That is the higher the convection, the higher the vale. Figre simply depicts a D view o the model eqations and ondary conditions. The otlined area shows a D symmetry approximation.. Dynamic simlation and analysis eat dispersion with natre heat convection. This ssection analyses the heat transer phenomena o the CPU heat sink process withot a an. In this case there is no orced convection and only the condction and natre convection are considered. The simlation stdies or this case are nder the sitations o sing alim ins and silicon CPU. The physical properties o the materials and the operation conditions are listed in Tales and, respectively. in The simlations are done sing the sotware - COMSOL Mltiphysics. (COMSOL Inc.). The simple and rost graphical ser interaces makes it easy to set p a partial dierential eqation (PDE) model, rn a simlation and visalize the reslts throgh inite element methods. Figre 4 depicts the temperatre contor in the plate-ins at time t00, while the steady-state temperatre distrition is plotted in Figre 5. Besides, Figre 6 shows the time evoltion o the temperatre at the control point or varios CPU heat sorces. Notice that the sensor (control point) is located at the ottom o the middle in, which is near to the CPU. As can e seen rom this igre that the temperatre can e over 90 when the an is switched o, which can ths case the atal damage o the CPU or long time operation. Tale Physical properties o materials Material properties Alim Silicon eat capacity, C p ( J / kg K ) 905 705.5 Density, ρ ( kg / m ) 707 0 Condctivity, k ( W / m K ) 00 6 Tale Operation conditions o the CPU heat sink process Parameter Vale Description 400 eat transer coeicient vale ( h ) when an is operated in ll e, speed ( W / m K) h e 0 eat transer coeicient vale ( h ) when an is switched o e, ( W / m K) T in 5 Srrondings temperatre ) Volme 5 6 0 m CPU volme ( ) Power 0 CPU heat sorce ( W ) n ( k T ) he ( Tin T ) n ( k T ) 0 ρ C p ( k T) Q t Figre Governing eqation and ondary conditions eat dispersion with orced heat convection. The simlation reslts shown in the pre-ssection indicate that the CPU temperatre can e too high when the an is switched o. The overheat sitation can e simply avoided y trning on the an to
enorce the air convection. Now, we exae the eectiveness o the heat dispersion or varios an speeds. ere, it is assmed that with a sitale an speed the temperatre at the control point is 40 initially. Figre 7 shows that the temperatre responses or changing the an speed. From this igre it is seen that the higher speed, the lower temperatre at the control point. Besides, Figre 8 indicates that the CPU temperatre can e pertred as the srronding temperatre sddenly changes. T ( ) 55 50 45 40 5 0 5 000 500 000 500 000 T in 5 C T in 0 C T in 0 C T in 5 C Fig. 8. Temperatre changes in response to srrondings temperatre pertrations.. OPTIMAL DESIGN OF CPU EAT SINK PROCESSES Fig. 4. Temperatre contor plot at t00.. Thermal Analysis o eat Sink Processes Beore attempting to perorm the optimal plate-in design, we irst condct the thermal analysis o heat sink processes. The horizontal air inlet system shown in Figre 9 is considered in this paper. t Fig. 5. Steady state temperatre distritions over the ins. T ( ) 0 0 0 00 90 80 70 60 50 40 0 0 000 000 000 4000 5000 6000 Fig. 6. Temperatre evoltions at the control point or dierent power sorces. T( ) 55 50 45 40 5 0 5 800 000 00 400 600 800 000 00 400 600 800 000 0W 5W 40W 5% 50% 00% Fig. 7. Temperatre responses or changing varios an speed (initial an speed is at % open). Fig. 9. A plate-in heat sink process with horizontal inlet low. The model is sject to the ollowing assmptions (Clham and Mzychka, 00): niorm approach low velocity, niorm heat transer coeicient, constant thermal properties, no ypassing low eect, and the adiaatic in tips. De to a in can generate the entropy associated with the external low and can also generate entropy internally ecase o the nonisothermal condition, the entropy generation rate can e given y (Bejan, 98; 996) Q R F V sin S k d gen + () T T in Q represents the heat transer o the in ase, R sin k the overall heat sink thermal resistance, T Q R ) the temperatre dierence etween ( sin k the heat sorce (CPU) and V the air low rate. Flid riction itsel is in the orm o the drag orce, F, d along the direction o air low. This entropy generation rate is contrited y oth the heat sink resistance and viscos dissipation. The overall thermal resistance o the inned srace is given y (Kay and London, 984) t R k ( n R ) + h n L + in (4) sin in e ( ) klw n is the nmer o the ins, and R in thermal resistance o each in, is represented y, the
R in (5) h PkA tanh( m ) o which the parameter m is given y e c h P m kac The perimeter P is the srace area per nit length o ins, and A represents the cross-sectional area or c heat condction o each in. The total drag orce Fd o the heat sink can e otained y the orce alance on the heat sink (Kay and London, 984) Fd n(l L) K ( w) K ( w) app + + sc + (7) se (/) ρv e (6) ch is the apparent riction actor or a hydrodynamically developing low. The channel velocity app V o horizontal inlet low is given y ch tw Vch V ( + ) (8) and the actor or rectanglar channel is given app y (Mzychka and Yovanovich, 998): and h.44 + ( ) app Re Re (9) L L D h L Re (0) D is the hydralic diameter o the channel. Notice that the Reynolds nmer grop Re in Eq. (9), representing the riction actor o lly developed low, is given y (Mzychka and Yovanovich, 998): Re 4.57 4 + 46.7 5 40.89 +.954 6.089 () Besides, the sdden contraction and expansion coeicients K and K in Eq. (7) can e, sc se respectively, calclated sing the ollowing relations (White, 987) K 0.4( σ ) () sc K ( σ ) () se n t σ w Moreover, the convective heat transer coeicient can e compted rom the ollowing relation (Teertstra et al., 00) w (4) Re Pr.65 N + 0.664 Re Pr + (5) Re N h e, Re k L Vch Re, Re, υ and P and υ represent, respectively, the Prandtl r nmer and the kinematics viscosity o the low.. Optimal design: an illstrative example Consider a plate-in conigration as shown in Figre 9, which is eqipped with horizontal inlet cooling low. Both the width and ase length o the in are 5cm. The thermal condctivity o the alim in is 00 W mk. The total CPU heat dissipation o 0 W is niormly applied over the ase plate o the in. The amient air temperatre T in is 5. The air density ρ, kinematics viscosity υ, and the air condctivity k are.77kg / m, 5.6 0 m / s, and 0.067 W / mk, respectively. The ojective nction to e imized is the entropy generation rate Q R F V sin S k d gen +, which Tin Tin provides the trade-o etween the heat resistance and transer rate. The design parameters are the nmer o o ins, n, the height o the in,, the thickness o the in,, and the air low rate V. To solve this optimal design prolem, a real-coded genetic algorithm (Tstsi and Golderg, 00) is tilized since it does not need the coding and decoding process and is more eicient to allocate the optimal soltion as compared with conventional genetic algorithms. In sing the algorithm, we set the chromosome as Θ θ, θ, θ, ] n,,, V ]. The [ θ 4 [ parameter space Ω is given y Θ n 40, 0.05m 0.4m, 0.5m / s V m/ s, and 4 0 m.5 0 m. Besides, the parameters sed in the real-coded genetic algorithm are set as N 00 (nmer o the chromosomes), p r 0. (parameter or the nction o reprodction), p 0. (parameter or the nction o crossover), c p 0.4 (parameter or the nction o mtation), m s 0.5 (positive real nmer), G 80 (imm nmer o generations), and Φ [ 0.5, 0.5] (a pertration vector or mtation). Tale Comparison o optimal shapes o the platein sing dierent design methods methods n (m) (m) V S gen (m/s) (W/K) T ) R sin k proposed 8.4 0. 0.006. 0.008805 7.68 0.89 proposed 8.0 0. 0.0068.4 0.008809 7.805 0.94 Clham and 9.07 0. 0.0060. 0.000607 7.885 0.7 Mzychka (00) Shih and Li (004) 0.0 0.4 0.006.05 0.009670 7.7 0.4 With these parameter settings, we otained the design reslts as listed in Tale, in which the reslts y the methods o Clham and Mzychka (00) and Shin and Li (004) are also shown or comparison. From the tale, it is clearly shown that the real-coded genetic algorithm is ale to get a etter design soltion with less entropy generation rate and ths etter heat dispersion. Notice that in the
second row o the tale, the entropy generation rate in system with 8 ins is slightly higher than that o irst row, in which the nmer o ins is not integer. 4. DIRECT ADAPTIVE SNC CONTROL OF CPU EAT SINK PROCESSES It is noted here that the design is ased on the assmption o constant operation conditions. When the load o the CPU and/or the amient temperatre change, the designed in-plate is no longer optimal as the CPU operation temperatre is not the same as that sed or the design. To maintain the perormance o the heat sink in acing with the environmental distrance, it is reqired to adjst the an speed. To meet this reqirement, a direct adaptive control strategy sing a ond SNC is proposed in this section or the CPU heat sink processes. 4. Control conigration In this ssection, a onded SNC is implemented in the CPU heat sink process to keep the operation temperatre o the CPU at a constant vale. Since the temperatre is a nction o time and position, the heat transer process is descried y partial dierential eqations. As revealed y the simlation reslts in Section and or simplicity, we consider the temperatre control or a single point as shown in Figre 0, in which the shape o the plate-in is optimally designed. Notice that the control point is selected to locate as close to the CPU as possile, which has the highest temperatre in the plate-ins. Fig. 0. The optimal plate-in or the horizontal inlet low and the location o temperatre sensor. T e e ( h e ) d e t d s T d Fig.. A model-ased SNC control system. A model-ased SNC control conigration shown in Figre is sggested, the process model o td s Gm ( s) G( s) e is sed. The onded SNC and the associated parameter tning algorithm are introdced riely in the ollowing ssection. 4. The SNC and parameter tning algorithm The onded SNC is o the orm (Chen and Peng, 999): s y (6) [ ( + ) + ( ) ] exp ( ) [ ( e θ ) ] t + exp [ ( e θ ) ] In Eq. (7), e Td T (7) is the error etween the setpoint and the actal temperatre at the control point. is the controller slope parameter and θ the ias parameter to e tned adaptively. It can e seen rom Eq. (7) that the nction vale o lies within [-, ], which ensres that the controller otpt vales will e conined in the range o ( ) t. For an speed control, one can set h (lly open) and e, h (lly e, closed). To ensre control system perormance, Chen and Peng (999) sggested the ollowing algorithm or tning the ias parameter. θ ( t ) η e ( ) ( + ) sign (8) In this algorithm, η denotes the learning rate representing the tradeo etween the response speed and the level o oscillation. The system response direction sign can e set to + or -, according the response direction o the process. For this heat sink process, the response direction is set - since when an speed is increased the temperatre is lowered. For the rigoros theoretical proo and the derivation o the parameter tning algorithm, see (Chen and Peng, 999). 4. Control simlation and analysis In the ollowing simlation stdies, we assme that the wind speed in etween the in is niorm and constant. This condition is as the same as that sed in Section. Under sch an assmption, the temperatre along the horizontal axis is the same and the temperatre distrition is varied only along the vertical axis. Using a simple open loop test, we otained a irst order pls dead-time model, 0.00 0.5s e. This model indicates clearly that the 0.4s + system response direction is -, i.e., sign. The initial SNC parameter vector and the learning rate are set as (, θ ) (,0 ) and η 0. 5 respectively. Also, according to Tale we set the controller otpt onds as 400 and 0. For comparison, a PI controller with parameters Kc 5 and T i 5.096, which is tned with Ciancone Tning method (Ciancone and Marlin, 99), is sed. Besides, de to the setpoint at the control point is 5 the conventional on-o control with the two-action o 0, T < 4 h e is 400, T > 6 implemented to the control o CPU heat sink process. The control simlation is implemented with the coation o COMSOL Mltiphysics and Matla
Simlink. Figre shows the closed-loop system response y the proposed scheme, therein the reslts y the on-o and PI control strategies are also shown. As can e seen that the commonly sed on-o control reslts in highly oscillatory system response, which is ndesirale. The PI controller can ring the process to the setpoint, t there exists a large overshoot and the settling time is long. Using the SNC controller, the system response is qite satisactory, mainly de to its adaptive aility. To exae the control capaility in handling with distrance, we assme that there is a sdden change in srrondings temperatre rom 5 to occrring at t 600 sec. The control perormance o the SNC controller is still very excellent and rost, there is no signiicant pertration on the system otpt. On the contrary, the on-o controller is reaching to its pper ond while casing a slight oset. The PI controller, thogh can ring the process otpt ack to the setpoint, gives a considerale larger deviation and takes longer time to settle down. It is also oserved that the SNC works well even thogh the process model is not sed (model ree case). The tning progress o the ias parameter shows the eectiveness o the parameter tning algorithm in handling with this PDE system. T ( ) θ h e (W/m K) 8 6 4 0 0 00 400 600 800 000 00 400 00 00 00 0 0 00 400 600 800 000 00 0.5 0-0.5 - Model-ased SNC SNC (model ree) PI On-O -.5 0 00 400 600 800 000 00 Fig.. Control system perormance or the case o horizontal inlet low. 5. CONCLUSIONS This paper has presented an optimization scheme and a direct adaptive control strategy or CPU heat sink processes. The heat transer phenomena o the platein are investigated sing the sotware - COMSOL Mltiphysics. A real-coded genetic algorithm is developed or the optimal design o the plate-in with horizontal inlet air lows. The ojective nction to e imized is the entropy generation rate, which allows one to take into accont o oth the eects o heat sink resistance and viscos dissipation. The proposed real-coded genetic algorithm can give etter designs than existing methods. As indicated y simlation experiments that the CPU heat sink process is sensitive to environmental distrances even thogh the design is optimal with respect to certain operating conditions. To reject these distrances, a direct adaptive SNC control scheme has een sggested. With a parameter tning algorithm, the SNC controller can adjst the an speed adaptively sch that the CPU is operated smoothly at a pre-speciied condition despite the inlence o extra loads and the change o srrondings temperatre. Extensive comparisons revealed that the proposed SNC-ased control scheme is sperior in the control perormance to conventional PI controller and the commonly sed on-o control or CPU heat sink processes. Acknowledgement This work was spported y the National Science Concil o Taiwan (ROC) nder grant NSC 94-4-E-05-00. REFERENCES Bar-Cohen, A. (999). Thermal packaging or the st centry: Challenges and options. Proc. 5 th Theric-International Workshop Thermal Investigations o IC s and Systems, Rome, Italy. Bejan, A. and A. M. Morega (99) Optimal arrays o pin ins and plate ins in laal orced convection. J. eat Transer, Vol. 5, pp. 75 8. Bejan, A. (98) Entropy Generation Throgh eat and Flid Flow, New York: Wiley. Bejan, A. (996) Entropy Generation Minimization, Orlando, FL: CRC. Chen, C. T. and S. T. Peng (999) Learning control o process systems with hard inpt constraints. J. Process Control, Vol. 9, pp. 5-60. Ciancone, R. and T. Marlin (99) Tne controllers to meet plant ojectives. Control, Vol. 5, pp. 50-57. Clham, J. R. and Y. S. Mzychka (00) Optimization o plate in heat sinks sing entropy generation imization. IEEE Trans. Comp. and Packag. Tech., Vol. 4, pp. 59-65. Kay, W. M. and A. L. London (984) Compact eat Exchangers, New York: McGraw-ill. Kreger, W. B. and A. Bar-Cohen (00) Optimal nmerical design o orced convection heat sinks IEEE Trans. Comp. and Packag. Tech., Vol. 7, pp. 47-45. Mzychka, Y. S. and M. M. Yovanovich (998) Modeling riction actors in non-circlar dcts or developing laar low. Proc. nd AIAA Theoretical Flid Mech. Meeting, Algerqe, NM, pp.5-8. Shih, C. J. and G. C. Li (004) Optimal design methodology o plate-in heat sinks or electronic cooling sing entropy generation strategy. IEEE Trans. Comp. and Packag. Tech., Vol. 7, No., pp. 55-559. Teertstra, P., M. M. Yovanovich and J. R. Clham, (00) Analytical orced convection modeling o plate in heat sinks. J. Electron Man., Vol. 0, No. 4, pp. 5-6. Tstsi, S. and D. E. Golderg (00) Search space ondary extension method in real-coded genetic algorithm. Inormation Sciences, pp.9-47. White, F. M. (987) Flid Mechanics, New York: McGraw-ill.