Introduction
Many multi-chip modules such as portable electronic devices have variable power dissipation rates. As a consequence, conventional electronics module coolers must be designed for peak power operation. Incorporation of a Thermal Energy Storage ͑TES͒ mechanism into the module cooler will allow for a smaller and quieter cooling system that is sized for some intermediate heat load. Then, heat is stored in the TES-system during periods of high power operation, and it is subsequently released from the system during periods of reduced power operation. Phase Change Materials ͑PCM͒ formulated to undergo phase transition at key temperatures can provide this load-leveling capability via the latent heat effect.
TES units have been employed in a variety of temperature stabilization applications including thermal control of electronics. Leoni and Amon ͓1͔ reported a transient thermal design of wearable computers with embedded electronics using PCMs. Fossett et al. ͓2͔ investigated the avionics passive cooling with microencapsulated PCMs. They demonstrated that microencapsulated PCM technology can be used to provide an effective passive heat sink for remotely located, intermittently operated avionics and suitable analysis may be used with confidence to design passive heat sinks. Wirtz et al. ͓3͔ developed a simple mathematical model that simulates the performance of a cooler/heater storage unit. Alawadhi and Amon ͓4͔ numerically studied the effectiveness of the PCM thermal storage unit under various operating conditions. They found that for varying power conditions, the thermal balance is a function of the varying power magnitude, period and PCM quantity. Gurrum et al. ͓5͔ reported the work on thermal management of high temperature pulsed electronics using solid-liquid PCMs. Based on numerical modeling, they investigated the feasibility of using solid-liquid PCMs for microwave power transistors, which are periodic power dissipating devices. By experiments, Hodes et al. ͓6͔ quantified the effectiveness of transient thermal management of a handset as a function of PCM material, power dissipated, orientation and wind in terms of increased talk-time and recovery time. Zheng and Wirtz ͓7͔ developed a thermal model for optimizing the design of the hybrid TES-unit.
Thermal analysis of a TES unit will give design guidelines. However, these analyses involve complicated numerical calculations or experiments to determine the design. For engineering application purpose, a simple algorithm is needed for a preliminary assessment of the feasibility of a hybrid TES-unit.
The objective of this paper is to develop figures of merit that can be used to quickly evaluate various design alternatives of plate-type TES units. A modification of a previously benchmarked semi-empirical finite volume model in Part 1 is used to study the characteristics of various TES-unit designs. A series of optimal designs, subject to various design constraints, is obtained by linking the model to an optimization algorithm. This database of optimal designs is used to develop two figures of merit that can be used to characterize the physical attributes of TES-units proposed for specific applications. Figure 1 shows the general layout of these energy storage devices. One side of the storage volume is in thermal contact with the heat source ͑the electronics͒. The other side connects to the system's heat exchanger, which is sized for some nominal heat load. The storage volume contains the PCM and a metal conducting path that is in thermal contact with the PCM. The metallization is necessary since the PCM that may be employed for this application generally have low thermal conductivity. The metallization is designed so that it can convey a nominal heat load through the storage volume to the hybrid cooler's heat exchanger while at the same time it facilitates heat transfer to the PCM. The conducting path might consist of discrete elements such as parallel pins or plates, or it might be aluminum foam or honeycomb material. Or, the ''metallization'' could involve impregnation of the PCM mass with a conductivity enhancer. The thermal performance of the TES-unit is characterized by the metallization thermal resistance, R metal and the TES-unit thermal capacity. These quantities are functions of the physical attributes of the TES-unit.
TES-Unit Configuration
A plate-type TES-unit, shown in Fig. 2 , is studied in the current work. Thin, conductive plates span the gap between heat spreader plates that are in thermal contact with the heat source and the system heat exchanger, respectively. The conducting plates, of thickness, t pl and spacing, s, are the ''metallization'' of this design. The PCM fills the spaces between the plates, forming the thermal storage volume, V storage . The lower base plate ͑heat spreader plate͒ is assumed to be in contact with the heat source such that a time dependent heat load, Q(t) is imposed on the lower boundary. The upper heat spreader plate is in contact with the system heat exchanger, which is characterized by its overall thermal conductance, (UA). (UA) is sized for the nominal heat load of the application such that at steady state operation
where T o,ss and T a are the steady state operating temperature of the TES-unit's base and the heat sink temperature, respectively. Eq. ͑1͒ neglects the thermal resistance of the heat spreader plates.
Solid solutions of the polyalcohols pentaglycerine ͓PG͔ (C 5 H 12 O 3 ) and neopentylglycol ͓NPG͔ (C 5 H 12 O 2 ) are typical of PCM's employed in the present application. These materials show promise as a candidate PCM for the electronics application due to their high latent heat over the appropriate temperature range, solid-state phase transition resulting in a ''dry'' process, and low cost. Furthermore, these materials can be formulated to transition at any temperature over the range of 24°C to 83°C. This provides increased design flexibility. Figure 3 shows the phase diagram of the PG/NPG binary system. We are interested in the ␣-␥ solid-state phase transition that occurs over the range 0 to 60% ͑mole͒ NPG. The latent heat of the PG/NPG system will range from about 81 J/gm at the eutectic point up to about 176 J/gm ͑pure PG͒. This is comparable to paraffin compounds currently in use. The melt temperature of this system ͑partly shown in Fig. 3͒ ranges from about 138°C at the eutectic up to 200°C ͑0% NPG͒; so inadvertent melting of the PCM is unlikely when applied to the electronics temperature control application.
Thermal Performance Model
The semi-empirical finite volume model in Part 1 is used to model the thermal response characteristics of the hybrid TES-unit. The previous model incorporated an effective-specific-heat to model the latent heat effect. We have found that this approach is inaccurate when applied to systems subject to rapid heating/ cooling. The model has been upgraded to incorporate the latent heat effect via a heat source/sink.
From the PG/NPG phase diagram, shown in Fig. 3 , it is seen that at fixed PCM composition ͑for example 25 mol % NPG in PG͒, transition occurs along a vertical line with a phase change occurring at temperature T L or T H , depending on whether the mass of material is being heated or cooled. The time for phase transition, ␦t s ͑storage time͒, depends on the input energy, q, the latent heat, h tr and mass, m of the PCM; given as
The PCM starts to store or release the energy at transition temperature, The thermal response model is linked to VisualDOC ͓8͔, a same design optimization algorithm used in Part 1, such that the best ͑optimal͒ design, subject to various design constraints, can be obtained. Table 1 lists parameters that characterize the thermal performance of a plate-type TES unit. Design requirements include: the steady operating temperature relative to the ambient, T ss (0)-T a ; initial heat loading, Q ss ; heat loading excursion, ⌬Q; and, the storage time, ␦t s . Important TES-unit characteristics include the thermal conductance of the unit, as characterized by the thermal resistance of the metallization, R metal ; and, the PCM thermodynamic properties ͑the specific latent heat, h tr ; and, transition temperature, T tr ). Design outcomes are: the unit size, as characterized by the storage volume, V storage ; and, the temperature control sensitivity of the unit, as characterized by the maximum tem-1 This model neglects the PCM composition change process during the phase transition ͑Chandra ͓9͔͒. perature excursion of the base of the unit relative to the transition temperature of the PCM, T 0 max ϪT tr . These parameters are summarized in Table 1 . Using ''Power Product method'' ͓10͔, the dimensional analysis gives the characteristic equations for plate-type TES-units if considering two design outcomes is the function of design inputs:
Dimensional Analysis
where
with
In the above equations, V storage is the actual storage volume of the PCM. The thermodynamic volume of a TES unit, V thermo reflects the maximum PCM volume needed for given design requirements and is related to the thermodynamic properties of the PCM employed. The actual storage volume, V storage which is related to the physical characteristics of the design will generally be smaller than V thermo since the TES unit, with finite thermal resistance, R metal will operate at an elevated temperature during increased heat loading. As a consequence, a portion of ⌬Q will route to the heat exchanger so that the rate of heat storage is reduced. Ṽ is a function of the metallization thermal resistance across the TES unit, R metal , and thus a function of the physical design of the unit. R total includes the thermal resistance of the TES-unit in addition to the resistance of the heat exchanger/cooler that connects the TES-unit to the ambient.
The temperature control sensitivity of the TES-unit can be characterized by the maximum temperature excursion, ͓T 0 max ϪT tr ͔. It is the unit's base temperature-to-PCM transition temperature difference during the heating process. Its dimensionless form is given by written as ⌬T .
Parametric Study of a Hybrid TES Unit
In order to investigate the effect of various design requirements on the optimal design, six case studies are completed. In each, one design requirement is systematically varied while all others are held constant. Then the overall result is used to develop correlations of Ṽ and ⌬T in terms of R metal and ⌬Q .
Baseline Optimal Design. The system is assumed to be fabricated of aluminum (kϭ168 W/m•K). The PCM employed has transition temperature, T tr (ϭT L )ϭ60°C and latent heat, h tr ϭ150 Joule/gm. This corresponds to a 75 mol % PG/25 mol % NPG solid solution. The initial power across the unit is set at Q ss ϭ100 W, with an excursion to 130 W (⌬Qϭ30 W). The storage time, ␦t s is set to at least 120 min and the maximum base temperature-to-transition temperature excursion, ͓T 0 max ϪT tr ͔ is constrained to no more than 5°C, so that V thermo ϭ1920 cm 3 (117 in 3 ) and ⌬T ϭ0.125. The objective is to minimize the footprint area of the unit, A b while the height, H is limited to 50.8 mm ͑2 inches͒. Case I: Vary ''Metallization'' Thermal Conductivity. We consider TES-units fabricated from materials having thermal conductivity ranging from 168 W/mK to 2300 W/mK, as shown in Table  3 . The material thermal capacity, c is essentially constant, and in any case, it represents only a small fraction of the TES-units overall thermal capacity. Since the PCM characteristics, temperature control sensitivity, storage time interval and heat increment are fixed, V thermo ϭ1920 cm 3 (117 in 3 ), ⌬T ϭ0.125, and ⌬Qϭ0.3. The results of the optimization show that with both V thermo and ⌬T fixed, both R metal and Ṽ are essentially constant. In retrospect, Transactions of the ASME this is not surprising. With the PCM characteristics and ⌬T fixed, the R metal requirement is established, and this will give rise to optimal designs having the same Ṽ .
Case II: Fix V thermo and Vary ⌬T . The unit is fabricated from aluminum and the heat load increment, storage time and PCM characteristics are fixed at the baseline values, so that V thermo ϭ1920 cm 3 (117 in 3 ) is fixed. The maximum temperature excursion is varied from 10°C down to 3°C so that 0.075р⌬T р0.250. Table 4 lists the optimal designs for increasingly stringent temperature control sensitivity. The optimization gives designs having metallization thermal resistances that are proportional to the increasingly stringent control sensitivity requirement. This gives rise to TES-units having increasingly larger volume requirements due to increased ''metallization,'' so that Ṽ ranges from 0.39 to 0.81. 3 ) . ⌬T ϭ0.125 is fixed. With the storage time increasing, the required storage volume will increase while the thermodynamic volume, V thermo will increase too. The results of the optimization listed in Table 6 show that both R metal and Ṽ are essentially the same for varying storage time.
Case V: Vary Heat Loading Excursion, ⌬Q. The unit is fabricated from aluminum and the heat loading excursion, ⌬Q is varied from 30 W to 100 W. Q ss ϭ100 W is fixed. This corresponds V thermo varying from 1920 cm 3 (117 in 3 ) to 6400 cm 3 (391 in 3 ). Table 7 lists the resulting optimal designs for increasing heat loading excursion ⌬Q. It shows that Ṽ is increased with ⌬Q increasing and at the same time R metal is decreased. This means that for higher ⌬Q, more storage volume and smaller metallization thermal resistance are needed to keep all other design requirements satisfied.
Case VI: Vary Initial Heat Loading Q ss . The initial heat loading, Q ss is varied from 100 W down to 20 W while ⌬Qϭ30 W is fixed. In this case, V thermo ϭ1920 cm 3 (117 in 3 ) and ⌬T ϭ0.125 are fixed. Table 8 gives the optimal designs due to Q ss changing. In this case, both Ṽ and R metal are decreased with the Q ss increasing. From two figures, we can see that the relations between Ṽ and R metal /⌬Q; ⌬T /⌬Q and R metal /⌬Q are nearly linear. Note also that the limiting case, R metal /⌬Q→0, there would be no metallization thermal resistance across the storage unit. However, there remains a small thermal resistance between the conducting plates of the storage volume and the PCM. This gives rise to ⌬T Ϸ0.08⌬Q and Ṽ Ϸ0.93.
A linear regression analysis gives the correlation for these two relations ͑Eqs. ͑3͒ and ͑4͒͒:
The maximum relative errors ͑with respect to plotted points͒ for these two equations are 5.3% and 13% respectively. These two simple correlations can be used to quickly assess design alternatives for plate-type TES units, as described in examples that follow.
Characteristic Time, s . Wirtz et al. ͓3͔ introduced a characteristic time, s that is the temperature storage time, ␦t s occurring when a TES-unit experiences a doubling in input power level. They showed that for simple devices described by a simple thermal performance model, s characterizes the heat-up/cool-down process of a TES-unit, and the storage time, ␦t s is a function of the incremental increase or decrease in power level,
The characteristic time, s is related to the volumetric figure of merit, Ṽ via the storage volume, V storage :
Using the above-described baseline of optimal design, we exercised the model to numerically experiment with unit responses for different heat loading ratios. Figure 6 plots the storage time, ␦t s ͑min.͒ versus heat loading ratio, ⌬Q/Q ss for the baseline design. The correlations, in conjunction with Eqs. ͑10͒ and ͑11͒ appear to hold for this relatively complex shape.
Application of Ṽ and s to TES-Unit Design Assessment
The above can be used to estimate the time response characteristics of a given TES-unit once the physical attributes have been determined. Given requirements and candidate PCM, V thermo and s are established. Eq. ͑9͒ can give the characteristics of the physical design, R metal . Then combining Eq. ͑8͒ and Eq. ͑5͒ gives the actual storage volume, V storage . This is described in the following examples.
Example I: Reproduce the Baseline Optimal Design. The baseline design has V thermo ϭ1920 cm 3 (117.17 in 3 ), ⌬T ϭ0.125, and ⌬Qϭ0.3. The metallization thermal resistance across the unit can be obtained from Eq. ͑9͒, giving R metal ϭ0.037°C/W. From Eq. ͑8͒, Ṽ ϭ0.7, and from the definition of Ṽ ͑Eq. 5͒ V storage ϭṼ
•V thermo ϭ1347 cm 3 (82.18 in 3 ). For the plate-type configuration of the TES unit, V storage can be written as
where f m is the metal fraction in the TES unit and is defined as N•t pl /L. R metal can be approximately written as Transactions of the ASME Eq. ͑11͒, the characteristic time, s ϭ36 min. Examination of Table 2 shows that we have reproduced the baseline design.
Example II: Reproduce the Low-k Optimal Design of Zheng and Wirtz [7] . Table 9 lists the details of the given conditions, design requirement, and results comparison using the same method described in example I.
From the results in table 9, the volumetric figure of merit, Ṽ and its correlation are very useful tools to assess the design alternatives of the plate-type TES-units.
Conclusion
A previously benchmarked semi-empirical finite volume model in part I is used to study the characteristics of TES unit designs. The old effective specific heat model is replaced with a new latent heat effect model to correctly describe the phase transition process. The finite volume model is integrated into a commercial design optimization software package to obtain the best design of the TES unit subject to certain constraints.
Two figures of merit, Ṽ and ⌬T are proposed. They characterize the TES-unit in terms of the unit's global parameters R metal and ⌬Q. Six different cases are investigated to find out the relation between Ṽ and ⌬T with R metal and ⌬Q. The study shows that
• Ṽ is a linear function of the R metal /⌬Q.
• ⌬T is a linear function of R metal and ⌬Q.
• No metallization results in V storage →V thermo and ⌬T only a linear function of the ⌬Q.
Ṽ can be related to the transient response characteristic time, s of a TES-unit. The figures of merit and correlations developed here can be easily used to evaluate design alternatives of platetype TES units.
The applicable range of the two correlations is limited to the current study. It is valuable to expand this range in the further study.
It is good idea that the future work for this study maybe includes the CFD simulations under same conditions and the comparison with the present works. 

{'paper_id': '16196677', 'title': 'A Hybrid Thermal Energy Storage Device, Part 2: Thermal Performance Figures of Merit', 's2_url': 'https://api.semanticscholar.org/CorpusID:16196677', 'source': 'S2ORC'}