Temperature Control and Optimal Energy Management using Latent

Article pubs.acs.org/IECR

Temperature Control and Optimal Energy Management using Latent Energy Storage Siyun Wang and Michael Baldea* McKetta Department of Chemical Engineering, The University of Texas at Austin, 200 East Dean Keeton Street, Stop C0400, Austin, Texas 78712, United States ABSTRACT: Cooling is a fundamental need as well as a significant energy consumer in a plethora of practically important applications. In this paper, we analyze latent-heat storage using phase-change materials (PCM) as a means for improving temperature control and energy management in cooling systems. We propose a novel, systems-centric approach to PCM-based thermal management and establish a connection between the quantity and geometric properties of the PCM, the dynamics of the integrated system, and potential energy savings. We show that the melting/solidification cycles of PCM provide a thermal buffer effect which can be relied upon to balance the use of passive and active cooling, reducing energy consumption. Subsequently, we focus on composite heat sinks consisting of PCM elements encapsulated in a conductive matrix material as a practical implementation of PCM-enhanced thermal management. Relying on concepts from nonlinear system identification and dynamic optimization, we formulate a novel stochastic optimization framework for selecting the optimal size and size distribution of the PCM elements for minimizing energy consumption under fluctuating loads. Finally, we illustrate our results with a case study.

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INTRODUCTION Meeting the cooling requirements of exothermic processes and systems is one of the fundamental operational needs in a variety of industrial sectors, from chemicals and petrochemicals to commercial buildings and data centers.1−3 While the absolute values of such cooling duties can span many orders of magnitude, ranging from a few watts for, e.g., a microprocessor, to megawatts for, e.g., a power plant, the operation of active cooling systems frequently entails significant specif ic energy consumption (in terms of energy expenditure per unit of heat dissipation rate), contributing in no small measure to operating costs. Energy consumption could, in principle, be reduced by maximizing the use of passive (e.g., natural convection) cooling. Increased reliance on passive cooling requires, however, process equipment of increased dimensions (with, e.g., larger heat transfer areas) and, consequently, larger capital costs. Furthermore, there are evident inherent physical limitations to this approach. An intuitive solution for this predicament, in particular in applications where the rate of heat generation and the associated cooling requirements fluctuate in time, consists of using both active and passive cooling in conjunction with a thermal energy storage system. In this case, a portion of the heat generated during periods of intense operation is stored (reducing the need for active cooling) and dissipated via passive cooling when the rate of heat generation drops. Phase-change materials (PCMs) constitute a natural choice of energy storage medium for implementing this strategy. Phase transitions occur with latent heat exchange, i.e., during melting/solidification the material stores or releases heat at a constant temperature (the melting point), having the potential to act as a temperature regulator to an adjacent thermal mass. High latent heats of melting ensure that such devices have a high energy density and, consequently, a relatively compact size. Early trials (which can be traced back to the US Space program4) indicated that the practical use of this concept is © 2013 American Chemical Society

limited by heat transfer in the melt (particularly when the size of the PCM structure is large compared to the thermal mass whose temperature it is meant to regulate). Once the melting process begins, a melt film is formed at the interface between the PCM and thermal mass, as shown in Figure 1. Owing to the lower thermal conductivity of the melted PCM, this film acts as an insulator, preventing heat transfer to the remaining PCM solid. It is thus possible that using such heat sinks have a deleterious, rather than beneficial effect. These findings have led to the use of composite devices, comprised of a container section (e.g., internally finned enclosure, porous matrix) built of a material with high thermal conductivity, and a PCM filler which, due to the container confinement, assumes a high surface-to-volume ratio.5−8 Several applications have been reported, ranging from temperature control in electronics to improving the energy efficiency of commercial buildings.9−14 However, the design of composite PCM heat sinks has frequently been carried out in view of meeting a static cooling duty,15,16 rather than focusing of performing a thermal regulation function in an optimal fashion under transient operating conditions (i.e., acting as or supplementing a temperature controller). In this paper, we present a novel, systems-based approach to PCM-enhanced temperature control and energy management. We begin by analyzing the dynamics of systems with PCM elements in a generic context, demonstrating that the fundamental effect of latent heat storage is to increase the time constant of the response of the system temperature to changes in heat input without affecting the steady-state gain. As Special Issue: Process Engineering of Energy Systems Received: Revised: Accepted: Published: 3247

November 8, 2012 January 9, 2013 January 10, 2013 January 10, 2013 dx.doi.org/10.1021/ie303073n | Ind. Eng. Chem. Res. 2013, 52, 3247−3257

Industrial & Engineering Chemistry Research

Article

Figure 1. PCM melting: (a) the temperature of the thermal mass increases. (b) In the ideal case, the temperature of the thermal mass will remain at the value of the melting point of the PCM, Tm, as long as the material continues to melt. (c) In practical cases, the low heat conductivity of the melt film hinders heat transfer from the thermal mass to the PCM, and the temperature of the thermal mass will continue to rise.

Intuitively, the dynamics of the system depend on the geometries of the thermal mass and PCM heat sink and on the physical properties (thermal conductivity, heat capacity, latent heat of melting) of their respective construction materials, as well as on the control algorithm implemented in the controller. This makes it impractical to carry out a generic analysis using first principles arguments. Rather, we will rely on a series of simplifying assumptions to derive transfer function models relating the device temperature to the heat generation and dissipation rates, and to elucidate the effect of the presence of the PCM-based heat sink on the system dynamics. This analysis will serve as the basis for demonstrating the dynamic principles of PCM-based thermal management; a rigorous, first-principles modeling and an optimal design framework are developed later in the paper. Component Transfer Functions. Thermal Mass. Heat transfer inside the thermal mass is assumed to be dominated by conduction. At relatively small Biot numbers and assuming that the geometry of the thermal mass is such that it can be modeled as a lumped parameter system, it can be shown that first-order transfer functions constitute a good approximative way to relate the device temperature to the rates of heat generation and heat dissipation.18 Thus, we can write the corresponding transfer functions as

a consequence, we argue that PCM-based energy storage in conjunction with passive cooling is to be relied upon as a sole cooling system only in limited circumstances. Subsequently, we focus on composite PCM-based heat sinks consisting of spherical PCM elements encapsulated in a conductive matrix material (see, e.g., refs 7 and 17) for which we develop a rigorous first-principles model. We draw on ideas from nonlinear system identification and dynamic optimization to formulate a novel stochastic optimization framework for “tuning” the dynamic response of the PCM elements, i.e., for selecting their size and size distribution such that the energy consumption of the active cooling system is minimized. Finally, we illustrate our results with a case study concerning the cooling of a computer microprocessor, demonstrating potential for real energy savings.

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PRINCIPLE OF PCM-BASED THERMAL MANAGEMENT System Description. Let us consider a prototype system consisting of a thermal mass subject to intermittent heating at a rate Hhs(t) (Figure 2). In order to maintain the temperature of the thermal mass at a desired value, heat is removed and dissipated at a rate Hd via a combination of active (e.g., forcedconvective) and passive (e.g., natural convection) cooling system. A PCM-based sink is present and absorbs heat at a rate Ha. The operation of the active cooling system is managed by a controller.

G1 =

K1 τs + 1

(1)

G2 =

K2 τs + 1

(2)

Note that the above transfer functions have different gains to account for the fact that heat generation and heat removal occur at different locations in the device. Furthermore, K2 < 0 because heat dissipation reduces temperature. However, based on the lumped-parameter approximation, the two transfer functions have the same time constant. PCM Heat Sink. The dynamic behavior of the PCM heat sink is strongly influenced by the melting and solidification cycles of the material. These phase transformations entail the absorption (and, respectively, the release) of a large amount of latent heat and occur at constant temperature (Figure 3). The

Figure 2. Block diagram for PCM-enhanced cooling system. 3248

dx.doi.org/10.1021/ie303073n | Ind. Eng. Chem. Res. 2013, 52, 3247−3257


Article

T= K1(τ3s + 1)

(

(

(τ3s + 1)(τs + 1) − K 2K a τ3 + (t 2 − t1) 1 −

Tm M

))s

Hhs (6)

It can be verified that the dominant time constant for (6), −

(3)

G1 Hhs 1 − K aG2(1 − G3)

2ττ3

(7)

(8)

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OPTIMAL DESIGN OF COMPOSITE PCM THERMAL MANAGEMENT SYSTEMS UNDER FLUCTUATING OPERATING CONDITIONS As we have highlighted in the previous sections, the use of phase-change materials for temperature regulation is hindered by heat transfer limitations in the melt film that forms at the interface between the PCM and the thermal mass whose temperature must be controlled. Intuitively, this shortcoming can be mitigated by increasing the contact area between the PCM and the thermal mass, which can be accomplished by using a composite heat sink, whereby the PCM is embedded in a thermally conductive support such that the area/volume ratio of the PCM is significantly increased. Several such designs have been proposed in the literature, including e.g., graphite matrices impregnated with PCM7 and internally finned enclosures.19 Other approaches to composite heat sink construction can be envisioned, such as the use of structures based on blockcopolymers. In this section, we develop a first-principles mathematical description for composite heat sinks consisting of PCM elements encapsulated in a conductive matrix. We subsequently use this model system to introduce a novel general framework for optimal design of PCM composite heat sinks under fluctuating operating conditions. System Description and Model. We consider a composite heat sink consisting of a thermally conductive matrix material and a set of encapsulated PCM elements as depicted in Figure 4. For simplicity, we assume that the PCM

(4)

Cooling System. The heat dissipation rate by the combined active and passive cooling systems Hd depends on the power of the active cooling system (e.g., fan), the thermal mass temperature, ambient temperature, etc. In this section, we are primarily concerned with studying the effect of the PCM on the temperature dynamics of the thermal mass and will assume that there are no changes in the active cooling system or variations in the ambient temperature and, hence, Hd = 0. On the basis of the above, the transfer function T=

(B + τ + τ3)2 − 4ττ3

is larger than the original time constant, τ. On the other hand, the steady-state gain of (6) is K1. Consequently, the presence of the PCM heat sink alters the time constant of the response of a system to an increase in heat input, without altering its steadystate gain. From a physical perspective, this observation indicates that the temperature-regulation effect of the latentheat cooling system is limited in time. Furthermore, these results suggest that a PCM heat sink can be used as a stand-alone cooling solution only when the PCM system can be designed such that its bandwidth (i.e., τdom) is sufficiently narrow to filter disturbances in Hgen. To this end, eq 7 suggests that τdom can be increased by increasing τ3, the time constant of the PCM response which, in turn, can be increased by raising the PCM mass. Clearly, this strategy is met with physical limitations as outlined in the previous section. As a consequence, in most cases, PCM heat sinks must be used in conjunction with an active cooling system, with the latter addressing low(er) frequency disturbances for improved temperature control.

where τ3 is the time constant that corresponds to the response of a material with no phase change but having similar density, heat capacity, and thermal conductivity as the PCM, Tm is the PCM melting point, and t1 and t2 are the time instants when melting starts and, respectively, ends (which depend on the rate of heat transfer to the material). Note that this representation is only valid if the melting temperature of the PCM, Tm, is contained between the lower and upper values of the input T. For simplicity, the time constants of the three terms in (3) are assumed to be the same (i.e., τ3), although the time constant of the response of the melted material may be different from the time constant of the solid PCM. This approximation is valid since the thermal conductivity of the PCM is much lower than that of the matrix material (and, respectively, its time constant is much higher). Then, the rate at which heat is absorbed by the composite heat sink is proportional to the temperature difference between the PCM and the thermal mass. Specifically, Ha = K a(T − G3T )

−B − τ − τ3 +

⎛ ⎛ T ⎞⎞ B = −K aK 2⎜τ3 + (t 2 − t1)⎜1 − m ⎟⎟ ⎝ ⎝ M ⎠⎠

dynamic impact of the ideal PCM heat sink is to maintain the temperature of the adjacent thermal mass at the PCM melting point. Evidently, this temperature regulation is in effect only until the phase transformation is complete, after which the temperature of the thermal mass will continue to rise. The step response of the PCM can be described using a transfer function of the form ⎛ Tm − 1 ⎞ ⎛ 1 − Tm ⎞ 1 M ⎟ −t 2s ⎟e−t1s + ⎜ + ⎜⎜ M ⎜ τ s + 1 ⎟e τ3s + 1 ⎝ τ3s + 1 ⎟⎠ ⎝ 3 ⎠

τdom

=

where

Figure 3. Response of the PCM temperature TPCM to a step change in the thermal mass temperature T.

G3 =

1

(5)

relates the thermal mass temperature to the heat generation rate. Substituting the expressions of the transfer functions in eq 5 and using a first-order Taylor series approximation for the time-delay terms, we obtain 3249



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⎧ TP ρP c P,s dT TP < Tm ⎪ ⎪ Tm ⎪ TP = Tm HP = ⎨ ρP fL ⎪ TP ⎪ ⎪ ρP fL + T ρP c P, l dT TP > Tm ⎩ m

∫

∫

where ρP is the density of the PCM, cP,s and cP,l are the heat capacity of solid and liquid PCM, respectively, Tm is the melting point of the PCM and it is also set as the reference temperature of the enthalpy, and L is the heat of fusion. The solid fraction, 0 < f(r,t) < 1 is defined as

Figure 4. Composite heat sink system structure.

⎧1 TP < Tm ⎪ ⎪ f = ⎨1 − Hp/(ρP L) TP = Tm ⎪ ⎪0 TP > Tm ⎩

elements are spherical and that they are randomly distributed within the matrix. On one side, the heat sink is in contact with a thermal mass which is subject to a time-varying heat flux from below. The exterior of the ensemble is cooled by forced convection, whose intensity can be modulated by a controller. Thermal Mass Modeling. Assuming that heat transfer in the thermal mass is purely conductive, and that the physical properties of the material are not temperature-dependent, the temperature distribution in the thermal mass can be described by the heat equation: ∂Ths k = hs ∇2 Ths ∂t ρhs chs

−k hs∇T |Ω = Hgen(t )

(14)

where Ω describes the boundary between the heat source and the thermal mass, and Hgen(t) is the heat generation rate as a function of time. The rate of heat generation is typically time varying, and we assume knowledge of the distribution of the values of Hgen. At the interface between the thermal mass and the matrix material, temperatures are equal and the heat fluxes are balanced. To reflect this, we use boundary conditions of both the first and second kind:

(9)

Ths|g(X ) = 0 = Tmtx

(15)

Ths|g(X ) = 0 = TP, i(r = R i)

(16)

(10)

where the subscript mtx denotes the matrix material. Following the basic premise of the composite heat sink design, the thermal conductivity of the matrix material is considered to be high (and, intuitively, much higher than the thermal conductivity of the thermal mass). PCM Elements. Heat transfer in the PCM elements can also be captured via the heat equation. However, the melting of the material creates a moving melt front, and it is more convenient to use the material enthalpy, rather than the temperature, to write the energy balance equation: ∂HP k ∂ ⎛ 2 ∂TP ⎞ ⎟ ⎜r = P2 ∂t r ∂r ⎝ ∂r ⎠

(13)

Boundary Conditions. The rate of heat input to the thermal mass is specified as

where Ths is the temperature, khs is the thermal conductivity, ρhs is the density, and chs is the heat capacity. Matrix Material. We assume that heat transfer in the matrix is also governed entirely by conduction and the temperature distribution is described by the heat equation: ∂Tmtx k mtx = ∇2 Tmtx ∂t ρmtx cmtx

(12)

N

∇Ths|g(X ) = 0 Ahs = qcoolA mtx +

∑ kP,i i=1

∂TP, i ∂ri

AP, i

(17)

where g(X) = 0 defines the boundary of the thermal mass that is in contact with PCM matrix, Ahs is the area of the boundary surface of the heat source, which can also be calculated as the area of the curve g(X) = 0, Amtx is the outside area of the PCM matrix that is exposed to the active cooling system, AP,i is the surface area of the ith PCM sphere, and qcool is the heat flux corresponding to heat dissipation from the system due to cooling. The effect of active cooling is captured by a boundary condition at the interface between the composite heat sink and the environment. Assuming convective heat transfer, we can write the rate of heat dissipation as

(11)

where HP is the enthalpy of the PCM, kP is the thermal conductivity, r is the radius of the PCM sphere, and TP(r,t) is the temperature of the PCM. The models of such systems typically use a “mushy region”20 to approximate the relationship between temperature and enthalpy, as well as the temperature dependence of heat capacity, thermal conductivity, and density, across the melt boundary. The approximation assumes that the phase change does not occur instantaneously, but through an intermediate stage where the material can assume a state between solid and melted. In the case of, e.g., enthalpy, this assumption is captured quantitatively via a piecewise continuous function:

qcool = h(Tmtx − Tamb)

(18)

where h is the heat transfer coefficient. Note that h is directly related to the energy used by the active cooling system and typically increases as the energy expenditure increases (e.g., if a fan is used for cooling, more energy is needed to drive the fan in order to increase the flow rate of cooling air). 3250



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Figure 5. Pseudo random multilevel sequence for simulating fluctuations in heat generation rate. In this example, the data are based on the assumption that Hgen is a normally distributed deviation variable, and thus, H̅ gen = 0.

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OPTIMIZATION OF THE DYNAMIC PERFORMANCE OF PCM-ENHANCED COOLING SYSTEMS Problem Formulation. The temperature regulation performance of the composite heat sink depends on the number of PCM elements in the matrix material and on their size, both of which determine the total volume of the PCM material. These parameters can be construed as “tuning parameters” for the PCM-based temperature controller: the volume of PCM directly affects the thermal storage capacity of the heat sink, while the size of the PCM spheres influences the dynamic response (spheres with large radii have a larger storage capacity, while spheres with small radii will melt faster and thus have a faster dynamic response). However, as shown above, the temperature regulation effect of the PCM sink is limited in time, i.e., composite heat sinks are not suited as an exclusive means for thermal regulation in the presence of persistent heat loads. Rather, they should be designed and used concurrently with an active cooling system, in a hierarchical control structure. From this perspective, the composite heat sink acts as the fast control component, which rejects fast disturbances, while slow, persistent disturbances are rejected by the active cooling system. In light of the above, the optimal design of composite heat sinks (and associated active cooling systems) entails the minimization of an aggregate objective function, that accounts for (i) the power requirements of the cooling system, p, a crucial term for reducing energy consumption and (ii) a control performance term that accounts for the deviation of the device temperature (measured, e.g., at the interface between the device and the heat sink) from a desired value (set point). A further complication arises from the fact that the operation of the system is subject to fluctuations in the rate at which heat is input to the device. The optimization calculations should therefore be stochastic and aimed at minimizing the likelihood of the peak temperature exceeding the temperature target, rather than considering the worst-case scenario of a significant, momentary disturbance (which, intuitively, would result in a very large heat sink size). The objective function to be minimized is thus comprised of two parts: (a) the energy use of the active cooling system J1 =

∫t

tf 0

p dt

(b) the time integral of the deviations of the temperature at location Ω̅ on the interface between the device and the heat sink from a desired set point J2 =

∫t

tf

/(Ths|Ω̅ − Tref )[Ths|Ω̅ − Tref ] dt

0

(20)

where / (x) is the Heaviside function: ⎧1 x > 0 /(x) = ⎨ ⎩0 x ≤ 0

(21)

The the design optimization problem for the composite heat sink can thus be formulated as a single-stage dynamic optimization problem over a fixed horizon tf: min J = C1J1 + C2J2 N ,R i

N

s.t.

∑ i=1

4 πR i 3 = V 3

N ∈ Notation for set of natural numbers, R min < R i N

V=

∑ i=1

4 πR i 3 3

V < Vmax model equations 9−18

(22)

where N is the number of PCM spheres, Ri is the radius of the ith PCM sphere, and C1 and C2 are weighting coefficients. The lower bound of the radius of the spheres is a technology limit on the smallest radius of the PCM spheres that can be manufactured. The upper bound for the total volume, Vmax, is set to 74% of the total volume of the matrix material block, which corresponds to the maximum packing occupancy of spheres.21 Solution Strategy. In order to capture the variability of the heat generation rate, Hgen(t), we propose a novel use of concepts from nonlinear system identification. Specifically, we represent the rate of heat input a pseudorandom multilevel sequence, PRMS,22,23 which is imposed on the system during the dynamic optimization iterations. PRMS is a function of time associated with a set of admissible function values (levels); the value of the function changes in a step fashion at each switching

(19)

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Table 1. Optimization Algorithm

time and remains constant until the next switching time is reached. The number of levels and probability of reaching any given level is defined by the statistical properties of Hgen, and over sufficiently long time intervals, the PRMS preserves the statistical properties (e.g., mean and variance) of the original variable. In this sense, using a PRMS to describe a random variable can be regarded as a “time-unfolding” of that variable’s distribution (Figure 5). On the basis of physical considerations, the composite heat sink must reject disturbances with frequencies within a bounded range. Due to thermal inertia, the device will naturally filter high frequency disturbances (see the previous section). Conversely, low-frequency disturbances are addressed by the active cooling system. Consequently, the switching frequency of the PRMS is determined based on the time constant of the device, e.g., νswitch = (kτ)−1, with 0.9 < k < 1.1, where the time constant, τ, can be determined from a step test. Figure 5 shows an example of a five-level PRMS that is sampled from a normal distribution. The optimization calculations then proceeds according to Algorithm 1 (see Table 1). Imposing the PRMS disturbance in the course of the dynamic simulation, along with the timeintegral objective function and a sufficiently long time horizon, allows the system to efficiently sample and account for its possible states in a Monte Carlo fashion. Note that in the present case, we are interested in minimizing a time integral objective function that inherently accounts for path constraints on temperature and do not explicitly introduce path constraints (such constraints could, however, be easily imposed).

cooling system (consisting of a composite heat sink and active, fan-based cooling) for a microprocessor. The system is similar to the prototype represented in Figure 4; the microprocessor and the heat sink are assumed to be rectangular, and the active cooling system is assumed to control the air flow at the upper boundary of the heat sink. The operation of the microprocessor is subject to fluctuations between high duty cycles and idle periods (Figure 6). The control objective is to maintain the temperature at the microprocessor surface at or below 330 K.

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Figure 6. Histogram for the distribution of the heat generation rates used in the case study.

CASE STUDY Problem Formulation. Battery-powered mobile devices, ranging from cellular phones and computers to (hybrid) electric vehicles have witnessed an explosive growth in the past decade. In intensive use, such devices generate significant amounts of heat (e.g., the thermal design power for a consumer-grade laptop CPU is as high as 55 W), which is oftentimes transferred to the environment with the aid of an active cooling system (i.e., a fan) which operates intermittently. Active cooling places additional demands on the battery and reduces the time the devices can operate before recharging is required (and, by consequence, their usefulness). This predicament can, in principle, be addressed by increasing the capacity or storage density of the battery. The former typically entails increasing the battery size (with a corresponding weight penalty), while the latter involves using a different (and likely more expensive) battery chemistry. A reduction of the parasitic load associated with active cooling can also be attained by minimizing the amount of time that the active cooling system is operated by adding a composite PCM heat sink which is exposed to the environment. In this case study, we consider the optimal design of such a combined

Intuitively, it is beneficial to choose a PCM having the same melting point as the temperature set point; this requirement is fulfilled by a 26-carbon-atom paraffin, which has a melting point of 330 K.24 We also consider three separate control strategies for the active cooling system: 1. On−Off Fan Control. This is the simplest approach to modulating the operation of the fan (Figure 7a). The control law stipulates that if the surface temperature of the processor is greater than 340 K, the fan switches on, and if the surface temperature of the processor is less than 330 K, the fan switches off. Note that hysteresis switching is used to prevent rapid shifting between the on and off states. The rate of heat removed by convection can be calculated as ⎧ if fan is on ⎪ hmax (Tmtx − Tamb) qfan = ⎨ ⎪ ⎩ hoff (Tmtx − Tamb) if fan is off

(23)

where hmax and hoff are, respectively, the heat transfer coefficients for the fan operating at full power and for the case when the fan is completely off (with the latter case amounting to natural convection).25 3252



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Figure 7. Switching strategies for fan control.

2. Three-Speed Fan. This situation is frequently encountered in practice (especially in mobile computing applications). In this case, the fan has an additional, intermediate, operating level compared to the previous case, with a convective heat transfer coefficient hmid. Two operation schemes are considered. • Type I: From the off state, the fan is switched to the intermediate operating level if the temperature is greater than 330K, then switched to the full-on state if the temperature is greater than 340 K. When the temperature is decreasing, the temperature thresholds are 330 and 325 K (Figure 7b). Note that in this case the fan is switched on before the PCM reaches its melting point. • Type II: In this case, the switching point from the off state to the intermediate level is increased by 5 K, and the switching occurs af ter the PCM has melted. Note that this is in effect a modification of the on−off strategy, whereby the heat transfer coefficient is increased from hoff to hmid for temperatures ranging from 335 to 340 K (as indicated by the shaded area in Figure 7c). 3. Proportional−Integral (PI) Control. This scheme provides a more elaborate temperature control strategy. The transfer function for the controller is Gc(s) = K p +

1 τIs

Table 2. Nominal Values of the System Parameters parameters

value

parameters

value

ρc (kg/m3) kc (W/mK) cp,l (J/(kg K)) ρp (kg/m3) kp,l (W/(m K)) Tamb (K) hoffAmtx (W/K)

2330 149 2500 750 0.2 293 1

cc (J/kg) Tm (K) cp,s (J/(kg K)) L (J/kg) kp,s (W/(m K)) hmaxAmtx (W/K) hmidAmtx (W/K)

710 330 2500 206000 1.0 3 1.5

consider the problem in a single dimension.Furthermore, heat transfer in the processor is described by the one-dimensional heat equation: ∂Tc k ∂ 2Tc = c ∂t ρc cc ∂z 2

(25)

where Tc is the temperature of the processor as a function of both time and z, kc is the thermal conductivity of the processor, ρc is the density of the processor, and cc is the heat capacity of the processor. The PCM spheres are modeled using eqs 11 and 12. The boundary condition at the heat source side, i.e., z = 0 is ∂Tc ∂z

(24)

where (Kp = 0.15 W/(m2 K2)) and (τI = 8000 s m2 K2/W) were chosen by trial and error. Two PI controllers with different set points are considered. The set point for the first PI controller is 330 K, and it is 331 K for the second one. A partial antiwindup strategy is implemented, whereby the integral component is active only when the system temperature is above the set point; operating below the set point is considered to be safe, and the accumulation of positive errors is not beneficial for control purposes. The design objective is to select the PCM particle size and density (in terms of PCM volume per unit volume of matrix) for a composite heat sink, which complements the active cooling system and leads to minimizing the energy expended in its operation. The physical properties of the microprocessor are assumed to be the same as those of silicon. The system is initially assumed to be at ambient temperature Tc(z,t = 0) = Tmtx(t = 0) = TP,i(r,t = 0) = 293 K. The values of the system parameters are summarized in Table 2. Solution Approach. For simplicity, we assume that the thermal conductivity of the matrix material is significantly higher than that of the processor or the PCM contained within the matrix, and, consequently, that the temperature of the matrix responds very fast and is spatially uniform. Together with the geometry of the system, this assumption allows us to

=− z=0

hgen kc

(26)

At the interface between the microprocessor and the matrix material, the following equations are obtained according to eqs 15−17: Tc|z = l = TPCM, i|r = R i

(27)

Tc|z = l = Tmtx

(28)

kc

∂Tc ∂z

N

Ac = qfanA mtx + z=l

∑ kPCM, i i=1

∂TP, i ∂ri

AP, i

(29)

where Ac is the contact area between the microprocessor and the proposed unit and qfan is the heat flux removed by the fan. The optimization problem (22) was solved using gPROMS;26 the spatial derivatives were discretized using centered finite differences, with a grid of 10 equally spaced discretization nodes for the axial domain (z coordinate) in the microprocessor and 15 nonuniformly distributed discretization nodes (with an increased node density at the matrix−PCM boundary) for the radial domain (r coordinate) in the PCM elements. A time horizon of 2 h was used for the dynamic optimization. The weighting coefficients C1 = 1 and C2 = 60 for the objective function were chosen so that both terms in the objective function in (22) are of similar magnitude. The radii of the PCM 3253



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spheres were assumed to be equal, with a lower bound set to 1 mm. This assumption is justified by the observation that, given a number of spheres and a total possible volume, using spheres of identical radii will result in the maximum surface area, and a larger surface area increases the total amount of heat that can be transferred from the microprocessor to the PCM spheres. Note, however, that this is connected to our assumption that the matrix material is highly conductive and has a uniform temperature; it is to be expected that a spatial temperature gradient in the matrix will require using a nonuniform size distribution of the spheres. The upper bound of the total volume was assumed to be 3600 mm3, corresponding to 40% occupancy of a 30 mm × 30 mm × 10 mm block of the matrix material. The power consumption of the fan was 2.64 W when in the full-on state and 1.32 W when operating at the middle level, which are values representative for a realistic computer cooling system. A ten-level PRMS with a switching time of 100 s was used to simulate Hgen of the system, which is in the order of magnitude of the time constant of the system. A first-order filter with a time constant of 5 s was applied to the sequence in order to improve the stability of the numerical optimization algorithms. The levels are from 10 to 50 W with a 5 W increment, mimicking the intermittent use of a computer. Figure 8 shows the heat generation pattern for the first 2 h operation. The ten levels obey a bimodal distribution that is shown in Figure 6.

Table 3. Optimization Results variables on−off control type I threespeed type II threespeed type I PI control type II PI control

total PCM volume (mm3)

number of spheres

sphere radii (mm)

2009 1471

41 33

2.27 2.20

3261

40

2.69

3341 3069

47 53

2.57 2.40

energy storage to operate the fan at an intermediate, low-energy use level. The case of the type I three-speed fan is somewhat anomalous and will be explained later in this section. In order to validate the above results, we considered a set of simulation scenarios. We compared the energy consumption of the active cooling system under the five different control strategies, with and without the proposed composite heat sink. A heat generation rate profile with the same statistical properties as the one used in the design optimization calculations was considered. Figure 9 presents a comparative snapshot of the operation of systems with and without PCM under on−off fan control. The

Figure 9. Temperature profiles for the original system with active cooling and the system with the proposed combined active cooling− latent storage strategy for on−off temperature control. Figure 8. Heat generation rate profile for the first 2 h.

most notable effect of the PCM is to prevent or delay the activation of the fan; for example, at t = 600 s, the temperature of the system without the composite heat sink reaches 340 K, activating the fan, while the temperature remains under the activation threshold with the heat sink present. The same situation can also be observed at t = 1250 s. These results indicate that over a period of 6 h, using the proposed composite heat sink yields a 3.2% energy savings due to reduced use of active cooling. A comparative snapshot of the operation of systems with and without PCM under type I three-speed fan control is shown in Figure 10. The temperature profiles and fan usage schedules are similar for both systems as indicated in the figure. Energy is saved at t = 900 s by the same reason as shown in the on−off case. However, in this case, the fan operates for a longer period of time at the intermediate power level, which can be seen at time t = 1300 s. This is a consequence of the thermal-buffer effect of the PCM. When the PCM spheres are discharging, i.e.

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RESULTS AND DISCUSSION The results of the optimization calculations are shown in Table 3. In order to reduce the likelihood of reaching local minima, three different initial guesses were used for each case; the results were identical each time. The results indicate that more sophisticated control strategies favor the use of larger energy buffers, and, conversely, the simple on−off strategy requires less energy storage. This result can be interpreted in view of the fact that the energy savings achieved by switching the state of the simple on−off controller is much larger than the savings achieved by state switching in a more sophisticated controller (where the power consumption difference between states is lower). Thus, energy savings with on−off control result mainly from switching the fan off, while more sophisticated control strategies rely on 3254



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operating period under consideration are significantly larger than in the case of the type I strategy, i.e., around 2.8%. The comparison for systems using type I PI control is shown in Figure 12. Temperatures for the system with energy storage

Figure 10. Temperature profiles for the original system with active cooling and the system with the proposed combined active cooling− latent storage strategy for a type I three-speed fan.

solidifying, the PCM prevents the system temperature from dropping. Therefore, during cooling, the PCM-based system temperature can remain, for an extended period of time, higher than the temperature of the original system. As a result of this undesirable effect, active cooling is fact used to “freeze” the PCM (note that the fan remains on when the temperature of the processor is equal to the melting temperature of the PCM), increasing energy use. Consequently, while energy storage does reduce energy use for the high operation level, the total energy consumption during the 6 h operation is increased by 0.7% due to the extended operation of the fan at the intermediate power level. Figure 11 shows a similar comparison using type II threespeed fan control. In this case, the energy-buffer effect prevents

Figure 12. Temperature profiles for the original system with active cooling and the system with the proposed combined active cooling− latent storage strategy for type I PI fan.

are closer to the 330 K set point than the previous cases. The second figure in Figure 12 shows the corresponding heat transfer coefficient as a reflection of the power usage of the PI fan (see eq 18). As in the previous case, energy is saved when the PCM is melting, but more power is consumed when the PCM is discharging (freezing), as can be seen at time t = 300 s and, respectively, t = 1300 s. Note that while, as illustrated above, a cooling control strategy that involves switching between (multiple) discrete levels can be designed to avoid using active cooling to “freeze” the PCM, this phenomenon is inevitable when using a PI controller, since the proportional part of the controller will always determine the power usage of the fan based on the difference between the current system temperature and the set point. When the system is cooling, the PCM spheres solidify at constant temperature while the temperature of the system without the composite heat sink would keep decreasing. As a result, the fan will remain on for a longer period of time than when the heat sink is not present. This phenomenon notwithstanding, using the heat sink results in a net energy savings of 2.2% over a 6-h period of operation. The comparison for systems using type II PI control is shown in Figure 13. The type II PI controller differs from type I by having a higher temperature set point (spefically, the set point is 1 K above the melting point of the PCM). As expected the performance of the two controllers (as shown in Figures 12 and 13) is apparently similar. However, because the set point for type II PI controller is greater than the melting point of PCM, the amount of energy expended on the solidification of PCM is reduced. Consequently, the total energy savings in this case is 3.0%, a 0.8% increase compared to the type I controller. This benefit is obtained by reducing the amount of active cooling (or, equivalently, increasing reliance of natural convection) for removing the latent heat accumulated in the PCM. Table 4 shows the energy saving performance for the different active cooling systems. The integral square errors

Figure 11. Temperature profiles for the original system with active cooling and the system with the proposed combined active cooling− latent storage strategy for a type II three-speed fan.

the fan from switching to the high level at time t = 900 s. Observe that in this case the onset of active cooling is delayed compared to the type I three-speed fan, and the use of the intermediate operation level when the temperature is decreasing is shorter; thus, the total energy savings during the 3255



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the nonlinear response is, in fact, beneficial to temperature control by maintaining the chip−matrix interface at the PCM melting point (which, in turn, is identical or close to the temperature set point) for extended periods of timea phenomenon that results in lower ISE values. Finally, we note that the energy savings that stem from using energy storage−enhanced cooling in battery-powered mobile devices are significant. For example, considering the base-case of a 6-h battery life (which is reasonable for a laptop computer), reducing the energy used for cooling by 3.2% could, in principle, prolong the operation of the device by about a quarter of an hour per charge, with a very small increase in physical size and a likely minimal change in cost.

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CONCLUSIONS In this paper, we provided systems-oriented perspective on the use of latent-energy-storage based on phase-change materials (PCM) as a means for improving temperature control and energy management in mobile systems. We demonstrated that the fundamental effect of latent-heat storage is to increase the time constant of the response of the system to changes in heat input without affecting the steady-state gain. We developed a rigorous first-principles model for composite heat sinks consisting of PCM elements confined in a thermally conductive matrix material. Using concepts from nonlinear system identification and dynamic optimization, we formulated a stochastic optimization framework for selecting the optimal size distribution of the PCM elements, which effectively amounts to “tuning” the dynamic response of the PCM system. We presented a case study concerning the temperature control of a microprocessor, demonstrating that energy savings are possible from combining active cooling with the proposed latent-energy storage-based thermal management strategy. The case study also helped delineate and explain several tenets concerning storage-enhanced cooling systems, including a potential increase in energy consumption when active cooling is improperly configured and is used for removing latent heat (i.e., for “freezing” the PCM). Furthermore, we have demonstrated that, with appropriate controller tuning (i.e., such that the use of active cooling to remove latent heat is avoided), the use of latent energy storage can improve the energy efficiency of active cooling systems (regardless of the control algorithm implemented) and result in significant energy savings.

Figure 13. Temperature profiles for the original system with active cooling and the system with the proposed combined active cooling− latent storage strategy for type II PI fan.

(ISE) for temperature above the melting point of PCM are also calculated for each scenario. Comparing the above results, it is evident that different switching or continuous control strategies can have a significant impact on the performance and energy consumption of the PCM-enhanced cooling system. Most notably, our results suggest that, in designing PCM-based thermal management systems, one should avoid using active cooling to remove latent heat. This tenet is in agreement with our initial assertion that energy storage should be used to absorb part of the heat generated during peak operating periods and dissipate it via passive cooling during periods of nonpeak operation. As an example, the type I three-speed fan uses the middle operation level to dissipate the stored heat, resulting in an increase in energy consumption. The results in Tables 3 and 4 also indicate that there is a direct (and intuitive) correlation between the amount of energy used by the active cooling system and control performance (as measured by the ISE). Better temperature control performance (lower error) entails an increase in energy consumption. The presence of the energy storage buffer also leads to a slight control performance improvement (a decrease in ISE) for each control strategy. While seemingly counterintuitive (in view of our linear analysis, which indicates that the presence of the PCM increases the time constant of the system, as well as in view of the increased nonlinearity of the temperature response due to the presence of the PCM), this result can be understood in light of the fact that control performance is explicitly accounted for in the design objective function. Furthermore,

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

Table 4. Energy Saving Performance for the Different Control Algorithms Considered control algorithm

consumption without PCM (J)

consumption with PCM (J)

energy savings (%)

switches without PCM

switches with PCM

ISE for the original system

ISE for the modified system

ISE change (%)

on−off type I threespeed type II threespeed type I PI type II PI

5316 9657

5148 9723

3.2 −0.7

188 122

178 120

373000 172000

358000 153000

−4.0 −11.0

8170

7940

2.8

138

120

470000

419000

−10.8

10680 9959

10443 9658

2.2 3.0

74100 71200

71800 69600

−3.1 −2.2

3256



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Temperature Control and Optimal Energy Management using Latent

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