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.

■ INTRODUCTION
Meeting the cooling requirements of exothermic processes and
limited by heat transfer in the melt (particularly when the size
of the PCM structure is large compared to the thermal mass
systems is one of the fundamental operational needs in a variety whose temperature it is meant to regulate). Once the melting
of industrial sectors, from chemicals and petrochemicals to process begins, a melt film is formed at the interface between
commercial buildings and data centers.1−3 While the absolute the PCM and thermal mass, as shown in Figure 1. Owing to the
values of such cooling duties can span many orders of lower thermal conductivity of the melted PCM, this film acts as
magnitude, ranging from a few watts for, e.g., a microprocessor, an insulator, preventing heat transfer to the remaining PCM
to megawatts for, e.g., a power plant, the operation of active solid. It is thus possible that using such heat sinks have a
cooling systems frequently entails significant specif ic energy deleterious, rather than beneficial effect. These findings have
consumption (in terms of energy expenditure per unit of heat led to the use of composite devices, comprised of a container
dissipation rate), contributing in no small measure to operating section (e.g., internally finned enclosure, porous matrix) built of
costs. Energy consumption could, in principle, be reduced by a material with high thermal conductivity, and a PCM filler
maximizing the use of passive (e.g., natural convection) cooling. which, due to the container confinement, assumes a high
Increased reliance on passive cooling requires, however, process surface-to-volume ratio.5−8 Several applications have been
equipment of increased dimensions (with, e.g., larger heat reported, ranging from temperature control in electronics to
transfer areas) and, consequently, larger capital costs. improving the energy efficiency of commercial buildings.9−14
Furthermore, there are evident inherent physical limitations However, the design of composite PCM heat sinks has
to this approach. frequently been carried out in view of meeting a static cooling
An intuitive solution for this predicament, in particular in duty,15,16 rather than focusing of performing a thermal
applications where the rate of heat generation and the regulation function in an optimal fashion under transient
associated cooling requirements fluctuate in time, consists of operating conditions (i.e., acting as or supplementing a
using both active and passive cooling in conjunction with a temperature controller).
thermal energy storage system. In this case, a portion of the In this paper, we present a novel, systems-based approach to
heat generated during periods of intense operation is stored PCM-enhanced temperature control and energy management.
(reducing the need for active cooling) and dissipated via passive We begin by analyzing the dynamics of systems with PCM
cooling when the rate of heat generation drops. Phase-change elements in a generic context, demonstrating that the
materials (PCMs) constitute a natural choice of energy storage fundamental effect of latent heat storage is to increase the
medium for implementing this strategy. Phase transitions occur time constant of the response of the system temperature to
with latent heat exchange, i.e., during melting/solidification the changes in heat input without affecting the steady-state gain. As
material stores or releases heat at a constant temperature (the
melting point), having the potential to act as a temperature Special Issue: Process Engineering of Energy Systems
regulator to an adjacent thermal mass. High latent heats of
melting ensure that such devices have a high energy density Received: November 8, 2012
and, consequently, a relatively compact size. Revised: January 9, 2013
Early trials (which can be traced back to the US Space Accepted: January 10, 2013
program4) indicated that the practical use of this concept is Published: January 10, 2013

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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.

a consequence, we argue that PCM-based energy storage in Intuitively, the dynamics of the system depend on the
conjunction with passive cooling is to be relied upon as a sole geometries of the thermal mass and PCM heat sink and on the
cooling system only in limited circumstances. Subsequently, we physical properties (thermal conductivity, heat capacity, latent
focus on composite PCM-based heat sinks consisting of heat of melting) of their respective construction materials, as
spherical PCM elements encapsulated in a conductive matrix well as on the control algorithm implemented in the controller.
material (see, e.g., refs 7 and 17) for which we develop a This makes it impractical to carry out a generic analysis using
rigorous first-principles model. We draw on ideas from first principles arguments. Rather, we will rely on a series of
nonlinear system identification and dynamic optimization to simplifying assumptions to derive transfer function models
formulate a novel stochastic optimization framework for relating the device temperature to the heat generation and
“tuning” the dynamic response of the PCM elements, i.e., for dissipation rates, and to elucidate the effect of the presence of
selecting their size and size distribution such that the energy the PCM-based heat sink on the system dynamics. This analysis
consumption of the active cooling system is minimized. Finally, will serve as the basis for demonstrating the dynamic principles
we illustrate our results with a case study concerning the of PCM-based thermal management; a rigorous, first-principles
cooling of a computer microprocessor, demonstrating potential modeling and an optimal design framework are developed later
for real energy savings. in the paper.

■ PRINCIPLE OF PCM-BASED THERMAL

MANAGEMENT
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
System Description. Let us consider a prototype system the geometry of the thermal mass is such that it can be modeled
consisting of a thermal mass subject to intermittent heating at a as a lumped parameter system, it can be shown that first-order
rate Hhs(t) (Figure 2). In order to maintain the temperature of transfer functions constitute a good approximative way to relate
the thermal mass at a desired value, heat is removed and the device temperature to the rates of heat generation and heat
dissipated at a rate Hd via a combination of active (e.g., forced- dissipation.18 Thus, we can write the corresponding transfer
convective) and passive (e.g., natural convection) cooling functions as
system. A PCM-based sink is present and absorbs heat at a rate K1
Ha. The operation of the active cooling system is managed by a G1 =
τs + 1 (1)
controller.
K2
G2 =
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
Figure 2. Block diagram for PCM-enhanced cooling system. latent heat and occur at constant temperature (Figure 3). The
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Industrial & Engineering Chemistry Research Article

T=
K1(τ3s + 1)
Hhs
(
(τ3s + 1)(τs + 1) − K 2K a τ3 + (t 2 − t1) 1 − ( Tm
M ))s
(6)
It can be verified that the dominant time constant for (6),

1 −B − τ − τ3 + (B + τ + τ3)2 − 4ττ3
− =
τdom 2ττ3 (7)
where
Figure 3. Response of the PCM temperature TPCM to a step change in
the thermal mass temperature T. ⎛ ⎛ T ⎞⎞
B = −K aK 2⎜τ3 + (t 2 − t1)⎜1 − m ⎟⎟
⎝ ⎝ M ⎠⎠ (8)
is larger than the original time constant, τ. On the other hand,
dynamic impact of the ideal PCM heat sink is to maintain the the steady-state gain of (6) is K1. Consequently, the presence of
temperature of the adjacent thermal mass at the PCM melting the PCM heat sink alters the time constant of the response of a
point. Evidently, this temperature regulation is in effect only system to an increase in heat input, without altering its steady-
until the phase transformation is complete, after which the state gain. From a physical perspective, this observation
temperature of the thermal mass will continue to rise. indicates that the temperature-regulation effect of the latent-
The step response of the PCM can be described using a heat cooling system is limited in time.
transfer function of the form Furthermore, these results suggest that a PCM heat sink can
⎛ Tm − 1 ⎞ ⎛ 1 − Tm ⎞ be used as a stand-alone cooling solution only when the PCM
1
G3 = + ⎜⎜ M ⎟e−t1s + ⎜ M ⎟ −t 2s
⎜ τ s + 1 ⎟e
system can be designed such that its bandwidth (i.e., τdom) is
τ3s + 1 ⎝ τ3s + 1 ⎟⎠ ⎝ 3 ⎠ sufficiently narrow to filter disturbances in Hgen. To this end, eq
(3)
7 suggests that τdom can be increased by increasing τ3, the time
where τ3 is the time constant that corresponds to the response constant of the PCM response which, in turn, can be increased
of a material with no phase change but having similar density, by raising the PCM mass. Clearly, this strategy is met with
heat capacity, and thermal conductivity as the PCM, Tm is the physical limitations as outlined in the previous section. As a
PCM melting point, and t1 and t2 are the time instants when consequence, in most cases, PCM heat sinks must be used in
melting starts and, respectively, ends (which depend on the rate conjunction with an active cooling system, with the latter
of heat transfer to the material). Note that this representation is addressing low(er) frequency disturbances for improved
only valid if the melting temperature of the PCM, Tm, is temperature control.
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
■ OPTIMAL DESIGN OF COMPOSITE PCM THERMAL
MANAGEMENT SYSTEMS UNDER FLUCTUATING
the response of the melted material may be different from the OPERATING CONDITIONS
time constant of the solid PCM. This approximation is valid As we have highlighted in the previous sections, the use of
since the thermal conductivity of the PCM is much lower than phase-change materials for temperature regulation is hindered
that of the matrix material (and, respectively, its time constant by heat transfer limitations in the melt film that forms at the
is much higher). Then, the rate at which heat is absorbed by the interface between the PCM and the thermal mass whose
composite heat sink is proportional to the temperature temperature must be controlled. Intuitively, this shortcoming
difference between the PCM and the thermal mass. Specifically, can be mitigated by increasing the contact area between the
Ha = K a(T − G3T ) (4) PCM and the thermal mass, which can be accomplished by
using a composite heat sink, whereby the PCM is embedded in a
Cooling System. The heat dissipation rate by the combined thermally conductive support such that the area/volume ratio
active and passive cooling systems Hd depends on the power of of the PCM is significantly increased. Several such designs have
the active cooling system (e.g., fan), the thermal mass been proposed in the literature, including e.g., graphite matrices
temperature, ambient temperature, etc. In this section, we are impregnated with PCM7 and internally finned enclosures.19
primarily concerned with studying the effect of the PCM on the Other approaches to composite heat sink construction can be
temperature dynamics of the thermal mass and will assume that envisioned, such as the use of structures based on block-
there are no changes in the active cooling system or variations copolymers. In this section, we develop a first-principles
in the ambient temperature and, hence, Hd = 0. mathematical description for composite heat sinks consisting of
On the basis of the above, the transfer function PCM elements encapsulated in a conductive matrix. We
G1 subsequently use this model system to introduce a novel
T= Hhs general framework for optimal design of PCM composite heat
1 − K aG2(1 − G3) (5)
sinks under fluctuating operating conditions.
relates the thermal mass temperature to the heat generation System Description and Model. We consider a
rate. Substituting the expressions of the transfer functions in eq composite heat sink consisting of a thermally conductive
5 and using a first-order Taylor series approximation for the matrix material and a set of encapsulated PCM elements as
time-delay terms, we obtain depicted in Figure 4. For simplicity, we assume that the PCM
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⎧ TP
⎪ ∫ ρP c P,s dT TP < Tm
⎪ Tm
⎪
HP = ⎨ ρP fL TP = Tm
⎪
⎪ TP
∫
⎪ ρP fL + T ρP c P, l dT TP > Tm
⎩ m (12)

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
Figure 4. Composite heat sink system structure. of the enthalpy, and L is the heat of fusion. The solid fraction, 0
< f(r,t) < 1 is defined as

⎧1 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 f = ⎨1 − Hp/(ρP L) TP = Tm
⎪
⎪0
thermal mass which is subject to a time-varying heat flux from ⎩ TP > Tm (13)
below. The exterior of the ensemble is cooled by forced
convection, whose intensity can be modulated by a controller. Boundary Conditions. The rate of heat input to the
Thermal Mass Modeling. Assuming that heat transfer in the thermal mass is specified as
thermal mass is purely conductive, and that the physical
properties of the material are not temperature-dependent, the −k hs∇T |Ω = Hgen(t ) (14)
temperature distribution in the thermal mass can be described
by the heat equation: where Ω describes the boundary between the heat source and
∂Ths k the thermal mass, and Hgen(t) is the heat generation rate as a
= hs ∇2 Ths function of time. The rate of heat generation is typically time
∂t ρhs chs (9) varying, and we assume knowledge of the distribution of the
where Ths is the temperature, khs is the thermal conductivity, ρhs values of Hgen.
is the density, and chs is the heat capacity. At the interface between the thermal mass and the matrix
Matrix Material. We assume that heat transfer in the matrix material, temperatures are equal and the heat fluxes are
is also governed entirely by conduction and the temperature balanced. To reflect this, we use boundary conditions of both
distribution is described by the heat equation: the first and second kind:

∂Tmtx k mtx Ths|g(X ) = 0 = Tmtx (15)

= ∇2 Tmtx
∂t ρmtx cmtx (10)
Ths|g(X ) = 0 = TP, i(r = R i) (16)
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 N
∂TP, i
high (and, intuitively, much higher than the thermal ∇Ths|g(X ) = 0 Ahs = qcoolA mtx + ∑ kP,i AP, i
conductivity of the thermal mass). i=1
∂ri (17)
PCM Elements. Heat transfer in the PCM elements can also
be captured via the heat equation. However, the melting of the where g(X) = 0 defines the boundary of the thermal mass that
material creates a moving melt front, and it is more convenient is in contact with PCM matrix, Ahs is the area of the boundary
to use the material enthalpy, rather than the temperature, to surface of the heat source, which can also be calculated as the
write the energy balance equation: 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
∂HP k ∂ ⎛ 2 ∂TP ⎞ surface area of the ith PCM sphere, and qcool is the heat flux
= P2 ⎜r ⎟
∂t r ∂r ⎝ ∂r ⎠ (11) corresponding to heat dissipation from the system due to
cooling.
where HP is the enthalpy of the PCM, kP is the thermal The effect of active cooling is captured by a boundary
conductivity, r is the radius of the PCM sphere, and TP(r,t) is condition at the interface between the composite heat sink and
the temperature of the PCM. the environment. Assuming convective heat transfer, we can
The models of such systems typically use a “mushy region”20 write the rate of heat dissipation as
to approximate the relationship between temperature and
enthalpy, as well as the temperature dependence of heat qcool = h(Tmtx − Tamb) (18)
capacity, thermal conductivity, and density, across the melt
boundary. The approximation assumes that the phase change where h is the heat transfer coefficient. Note that h is directly
does not occur instantaneously, but through an intermediate related to the energy used by the active cooling system and
stage where the material can assume a state between solid and typically increases as the energy expenditure increases (e.g., if a
melted. In the case of, e.g., enthalpy, this assumption is fan is used for cooling, more energy is needed to drive the fan
captured quantitatively via a piecewise continuous function: in order to increase the flow rate of cooling air).
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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.

■ OPTIMIZATION OF THE DYNAMIC PERFORMANCE

OF PCM-ENHANCED COOLING SYSTEMS
(b) the time integral of the deviations of the temperature at
location Ω̅ on the interface between the device and the
Problem Formulation. The temperature regulation heat sink from a desired set point
performance of the composite heat sink depends on the tf
number of PCM elements in the matrix material and on their J2 = ∫t 0
/(Ths|Ω̅ − Tref )[Ths|Ω̅ − Tref ] dt
(20)
size, both of which determine the total volume of the PCM
material. These parameters can be construed as “tuning where / (x) is the Heaviside function:
parameters” for the PCM-based temperature controller: the ⎧1 x > 0
volume of PCM directly affects the thermal storage capacity of /(x) = ⎨
the heat sink, while the size of the PCM spheres influences the ⎩0 x ≤ 0 (21)
dynamic response (spheres with large radii have a larger storage The the design optimization problem for the composite heat
capacity, while spheres with small radii will melt faster and thus sink can thus be formulated as a single-stage dynamic
have a faster dynamic response). However, as shown above, the optimization problem over a fixed horizon tf:
temperature regulation effect of the PCM sink is limited in
time, i.e., composite heat sinks are not suited as an exclusive min J = C1J1 + C2J2
N ,R i
means for thermal regulation in the presence of persistent heat
loads. Rather, they should be designed and used concurrently N
4
with an active cooling system, in a hierarchical control s.t. ∑ πR i 3 = V
3
structure. From this perspective, the composite heat sink acts i=1
as the fast control component, which rejects fast disturbances, N ∈ Notation for set of natural numbers,
while slow, persistent disturbances are rejected by the active
cooling system. R min < R i
In light of the above, the optimal design of composite heat N
4
sinks (and associated active cooling systems) entails the V= ∑ πR i 3
minimization of an aggregate objective function, that accounts i=1
3
for (i) the power requirements of the cooling system, p, a V < Vmax
crucial term for reducing energy consumption and (ii) a control
performance term that accounts for the deviation of the device model equations 9−18 (22)
temperature (measured, e.g., at the interface between the device
and the heat sink) from a desired value (set point). A further where N is the number of PCM spheres, Ri is the radius of the
complication arises from the fact that the operation of the ith PCM sphere, and C1 and C2 are weighting coefficients.
system is subject to fluctuations in the rate at which heat is The lower bound of the radius of the spheres is a technology
input to the device. The optimization calculations should limit on the smallest radius of the PCM spheres that can be
therefore be stochastic and aimed at minimizing the likelihood manufactured. The upper bound for the total volume, Vmax, is
of the peak temperature exceeding the temperature target, set to 74% of the total volume of the matrix material block,
rather than considering the worst-case scenario of a significant, which corresponds to the maximum packing occupancy of
momentary disturbance (which, intuitively, would result in a spheres.21
very large heat sink size). The objective function to be Solution Strategy. In order to capture the variability of the
minimized is thus comprised of two parts: heat generation rate, Hgen(t), we propose a novel use of
concepts from nonlinear system identification. Specifically, we
(a) the energy use of the active cooling system represent the rate of heat input a pseudorandom multilevel
tf sequence, PRMS,22,23 which is imposed on the system during
J1 = ∫t p dt
(19)
the dynamic optimization iterations. PRMS is a function of time
0 associated with a set of admissible function values (levels); the
value of the function changes in a step fashion at each switching
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Table 1. Optimization Algorithm

time and remains constant until the next switching time is cooling system (consisting of a composite heat sink and active,
reached. The number of levels and probability of reaching any fan-based cooling) for a microprocessor.
given level is defined by the statistical properties of Hgen, and The system is similar to the prototype represented in Figure
over sufficiently long time intervals, the PRMS preserves the 4; the microprocessor and the heat sink are assumed to be
statistical properties (e.g., mean and variance) of the original rectangular, and the active cooling system is assumed to control
variable. In this sense, using a PRMS to describe a random the air flow at the upper boundary of the heat sink. The
variable can be regarded as a “time-unfolding” of that variable’s operation of the microprocessor is subject to fluctuations
distribution (Figure 5). between high duty cycles and idle periods (Figure 6). The
On the basis of physical considerations, the composite heat control objective is to maintain the temperature at the
sink must reject disturbances with frequencies within a microprocessor surface at or below 330 K.
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 time-
integral 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).

■
Figure 6. Histogram for the distribution of the heat generation rates
used in the case study.
CASE STUDY
Problem Formulation. Battery-powered mobile devices, Intuitively, it is beneficial to choose a PCM having the same
ranging from cellular phones and computers to (hybrid) melting point as the temperature set point; this requirement is
electric vehicles have witnessed an explosive growth in the past fulfilled by a 26-carbon-atom paraffin, which has a melting point
decade. In intensive use, such devices generate significant of 330 K.24 We also consider three separate control strategies
amounts of heat (e.g., the thermal design power for a for the active cooling system:
consumer-grade laptop CPU is as high as 55 W), which is 1. On−Off Fan Control. This is the simplest approach to
oftentimes transferred to the environment with the aid of an modulating the operation of the fan (Figure 7a). The control
active cooling system (i.e., a fan) which operates intermittently. law stipulates that if the surface temperature of the processor is
Active cooling places additional demands on the battery and greater than 340 K, the fan switches on, and if the surface
reduces the time the devices can operate before recharging is temperature of the processor is less than 330 K, the fan
required (and, by consequence, their usefulness). switches off. Note that hysteresis switching is used to prevent
This predicament can, in principle, be addressed by rapid shifting between the on and off states. The rate of heat
increasing the capacity or storage density of the battery. The removed by convection can be calculated as
former typically entails increasing the battery size (with a
⎧
⎪ hmax (Tmtx − Tamb) if fan is on
corresponding weight penalty), while the latter involves using a
different (and likely more expensive) battery chemistry. A qfan = ⎨
⎪
⎩ hoff (Tmtx − Tamb) if fan is off (23)
reduction of the parasitic load associated with active cooling
can also be attained by minimizing the amount of time that the where hmax and hoff are, respectively, the heat transfer
active cooling system is operated by adding a composite PCM coefficients for the fan operating at full power and for the
heat sink which is exposed to the environment. In this case case when the fan is completely off (with the latter case
study, we consider the optimal design of such a combined amounting to natural convection).25
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Figure 7. Switching strategies for fan control.

2. Three-Speed Fan. This situation is frequently encoun- Table 2. Nominal Values of the System Parameters
tered in practice (especially in mobile computing applications).
parameters value parameters value
In this case, the fan has an additional, intermediate, operating
level compared to the previous case, with a convective heat ρc (kg/m3) 2330 cc (J/kg) 710
transfer coefficient hmid. Two operation schemes are considered. kc (W/mK) 149 Tm (K) 330
cp,l (J/(kg K)) 2500 cp,s (J/(kg K)) 2500
• Type I: From the off state, the fan is switched to the ρp (kg/m3) 750 L (J/kg) 206000
intermediate operating level if the temperature is greater kp,l (W/(m K)) 0.2 kp,s (W/(m K)) 1.0
than 330K, then switched to the full-on state if the Tamb (K) 293 hmaxAmtx (W/K) 3
temperature is greater than 340 K. When the temper- hoffAmtx (W/K) 1 hmidAmtx (W/K) 1.5
ature is decreasing, the temperature thresholds are 330
and 325 K (Figure 7b). Note that in this case the fan is
consider the problem in a single dimension.Furthermore, heat
switched on before the PCM reaches its melting point.
transfer in the processor is described by the one-dimensional
• Type II: In this case, the switching point from the off
heat equation:
state to the intermediate level is increased by 5 K, and
the switching occurs af ter the PCM has melted. Note ∂Tc k ∂ 2Tc
that this is in effect a modification of the on−off strategy, = c
∂t ρc cc ∂z 2 (25)
whereby the heat transfer coefficient is increased from
hoff to hmid for temperatures ranging from 335 to 340 K where Tc is the temperature of the processor as a function of
(as indicated by the shaded area in Figure 7c). both time and z, kc is the thermal conductivity of the processor,
3. Proportional−Integral (PI) Control. This scheme ρc is the density of the processor, and cc is the heat capacity of
provides a more elaborate temperature control strategy. The the processor. The PCM spheres are modeled using eqs 11 and
transfer function for the controller is 12.
The boundary condition at the heat source side, i.e., z = 0 is
1
Gc(s) = K p + ∂Tc hgen
τIs (24) =−
∂z z=0 kc (26)
where (Kp = 0.15 W/(m2 K2)) and (τI = 8000 s m2 K2/W)
were chosen by trial and error. Two PI controllers with At the interface between the microprocessor and the matrix
different set points are considered. The set point for the first PI material, the following equations are obtained according to eqs
controller is 330 K, and it is 331 K for the second one. A partial 15−17:
antiwindup strategy is implemented, whereby the integral Tc|z = l = TPCM, i|r = R i (27)
component is active only when the system temperature is above
the set point; operating below the set point is considered to be Tc|z = l = Tmtx (28)
safe, and the accumulation of positive errors is not beneficial for
N
control purposes. ∂Tc ∂TP, i
The design objective is to select the PCM particle size and kc Ac = qfanA mtx + ∑ kPCM, i AP, i
∂z z=l i=1
∂ri (29)
density (in terms of PCM volume per unit volume of matrix)
for a composite heat sink, which complements the active where Ac is the contact area between the microprocessor and
cooling system and leads to minimizing the energy expended in the proposed unit and qfan is the heat flux removed by the fan.
its operation. The optimization problem (22) was solved using
The physical properties of the microprocessor are assumed gPROMS;26 the spatial derivatives were discretized using
to be the same as those of silicon. The system is initially centered finite differences, with a grid of 10 equally spaced
assumed to be at ambient temperature Tc(z,t = 0) = Tmtx(t = 0) discretization nodes for the axial domain (z coordinate) in the
= TP,i(r,t = 0) = 293 K. The values of the system parameters are microprocessor and 15 nonuniformly distributed discretization
summarized in Table 2. nodes (with an increased node density at the matrix−PCM
Solution Approach. For simplicity, we assume that the boundary) for the radial domain (r coordinate) in the PCM
thermal conductivity of the matrix material is significantly elements.
higher than that of the processor or the PCM contained within A time horizon of 2 h was used for the dynamic optimization.
the matrix, and, consequently, that the temperature of the The weighting coefficients C1 = 1 and C2 = 60 for the objective
matrix responds very fast and is spatially uniform. Together function were chosen so that both terms in the objective
with the geometry of the system, this assumption allows us to function in (22) are of similar magnitude. The radii of the PCM
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Industrial & Engineering Chemistry Research Article

spheres were assumed to be equal, with a lower bound set to 1 Table 3. Optimization Results
mm. This assumption is justified by the observation that, given
total PCM volume number of sphere radii
a number of spheres and a total possible volume, using spheres variables (mm3) spheres (mm)
of identical radii will result in the maximum surface area, and a
on−off control 2009 41 2.27
larger surface area increases the total amount of heat that can
type I three- 1471 33 2.20
be transferred from the microprocessor to the PCM spheres. speed
Note, however, that this is connected to our assumption that type II three- 3261 40 2.69
the matrix material is highly conductive and has a uniform speed
temperature; it is to be expected that a spatial temperature type I PI control 3341 47 2.57
gradient in the matrix will require using a nonuniform size type II PI control 3069 53 2.40
distribution of the spheres.
The upper bound of the total volume was assumed to be energy storage to operate the fan at an intermediate, low-energy
3600 mm3, corresponding to 40% occupancy of a 30 mm × 30 use level. The case of the type I three-speed fan is somewhat
mm × 10 mm block of the matrix material. The power anomalous and will be explained later in this section.
consumption of the fan was 2.64 W when in the full-on state In order to validate the above results, we considered a set of
and 1.32 W when operating at the middle level, which are simulation scenarios. We compared the energy consumption of
values representative for a realistic computer cooling system. the active cooling system under the five different control
A ten-level PRMS with a switching time of 100 s was used to strategies, with and without the proposed composite heat sink.
simulate Hgen of the system, which is in the order of magnitude A heat generation rate profile with the same statistical
of the time constant of the system. A first-order filter with a properties as the one used in the design optimization
time constant of 5 s was applied to the sequence in order to calculations was considered.
improve the stability of the numerical optimization algorithms. Figure 9 presents a comparative snapshot of the operation of
The levels are from 10 to 50 W with a 5 W increment, systems with and without PCM under on−off fan control. The
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.

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,
RESULTS AND DISCUSSION activating the fan, while the temperature remains under the
The results of the optimization calculations are shown in Table activation threshold with the heat sink present. The same
3. In order to reduce the likelihood of reaching local minima, situation can also be observed at t = 1250 s. These results
three different initial guesses were used for each case; the indicate that over a period of 6 h, using the proposed composite
results were identical each time. heat sink yields a 3.2% energy savings due to reduced use of
The results indicate that more sophisticated control active cooling.
strategies favor the use of larger energy buffers, and, conversely, A comparative snapshot of the operation of systems with and
the simple on−off strategy requires less energy storage. This without PCM under type I three-speed fan control is shown in
result can be interpreted in view of the fact that the energy Figure 10. The temperature profiles and fan usage schedules are
savings achieved by switching the state of the simple on−off similar for both systems as indicated in the figure. Energy is
controller is much larger than the savings achieved by state saved at t = 900 s by the same reason as shown in the on−off
switching in a more sophisticated controller (where the power case. However, in this case, the fan operates for a longer period
consumption difference between states is lower). Thus, energy of time at the intermediate power level, which can be seen at
savings with on−off control result mainly from switching the time t = 1300 s. This is a consequence of the thermal-buffer
fan off, while more sophisticated control strategies rely on effect of the PCM. When the PCM spheres are discharging, i.e.
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Industrial & Engineering Chemistry Research Article

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 Figure 12. Temperature profiles for the original system with active
temperature can remain, for an extended period of time, higher cooling and the system with the proposed combined active cooling−
than the temperature of the original system. As a result of this latent storage strategy for type I PI fan.
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), are closer to the 330 K set point than the previous cases. The
increasing energy use. Consequently, while energy storage does second figure in Figure 12 shows the corresponding heat
reduce energy use for the high operation level, the total energy transfer coefficient as a reflection of the power usage of the PI
consumption during the 6 h operation is increased by 0.7% due fan (see eq 18). As in the previous case, energy is saved when
to the extended operation of the fan at the intermediate power the PCM is melting, but more power is consumed when the
level. PCM is discharging (freezing), as can be seen at time t = 300 s
Figure 11 shows a similar comparison using type II three- and, respectively, t = 1300 s. Note that while, as illustrated
speed fan control. In this case, the energy-buffer effect prevents 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
Figure 11. Temperature profiles for the original system with active
cooling and the system with the proposed combined active cooling−
PCM, the amount of energy expended on the solidification of
latent storage strategy for a type II three-speed fan. 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
the fan from switching to the high level at time t = 900 s. cooling (or, equivalently, increasing reliance of natural
Observe that in this case the onset of active cooling is delayed convection) for removing the latent heat accumulated in the
compared to the type I three-speed fan, and the use of the PCM.
intermediate operation level when the temperature is Table 4 shows the energy saving performance for the
decreasing is shorter; thus, the total energy savings during the different active cooling systems. The integral square errors
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Industrial & Engineering Chemistry Research Article

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.

■ CONCLUSIONS
In this paper, we provided systems-oriented perspective on the
use of latent-energy-storage based on phase-change materials
Figure 13. Temperature profiles for the original system with active
(PCM) as a means for improving temperature control and
cooling and the system with the proposed combined active cooling− energy management in mobile systems. We demonstrated that
latent storage strategy for type II PI fan. 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
(ISE) for temperature above the melting point of PCM are also rigorous first-principles model for composite heat sinks
calculated for each scenario. consisting of PCM elements confined in a thermally conductive
Comparing the above results, it is evident that different matrix material. Using concepts from nonlinear system
switching or continuous control strategies can have a significant identification and dynamic optimization, we formulated a
impact on the performance and energy consumption of the stochastic optimization framework for selecting the optimal size
PCM-enhanced cooling system. Most notably, our results distribution of the PCM elements, which effectively amounts to
suggest that, in designing PCM-based thermal management “tuning” the dynamic response of the PCM system. We
systems, one should avoid using active cooling to remove latent presented a case study concerning the temperature control of a
heat. This tenet is in agreement with our initial assertion that microprocessor, demonstrating that energy savings are possible
energy storage should be used to absorb part of the heat from combining active cooling with the proposed latent-energy
generated during peak operating periods and dissipate it via storage-based thermal management strategy. The case study
passive cooling during periods of nonpeak operation. As an also helped delineate and explain several tenets concerning
example, the type I three-speed fan uses the middle operation storage-enhanced cooling systems, including a potential
level to dissipate the stored heat, resulting in an increase in increase in energy consumption when active cooling is
energy consumption. improperly configured and is used for removing latent heat
The results in Tables 3 and 4 also indicate that there is a (i.e., for “freezing” the PCM). Furthermore, we have
direct (and intuitive) correlation between the amount of energy demonstrated that, with appropriate controller tuning (i.e.,
used by the active cooling system and control performance (as such that the use of active cooling to remove latent heat is
measured by the ISE). Better temperature control performance avoided), the use of latent energy storage can improve the
(lower error) entails an increase in energy consumption. The energy efficiency of active cooling systems (regardless of the
presence of the energy storage buffer also leads to a slight control algorithm implemented) and result in significant energy
control performance improvement (a decrease in ISE) for each savings.

■
control strategy. While seemingly counterintuitive (in view of
our linear analysis, which indicates that the presence of the AUTHOR INFORMATION
PCM increases the time constant of the system, as well as in
view of the increased nonlinearity of the temperature response Corresponding Author
due to the presence of the PCM), this result can be understood *E-mail: mbaldea@che.utexas.edu.
in light of the fact that control performance is explicitly Notes
accounted for in the design objective function. Furthermore, The authors declare no competing financial interest.

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

ISE
control consumption consumption with energy switches switches with ISE for the ISE for the change
algorithm without PCM (J) PCM (J) savings (%) without PCM PCM original system modified system (%)
on−off 5316 5148 3.2 188 178 373000 358000 −4.0
type I three- 9657 9723 −0.7 122 120 172000 153000 −11.0
speed
type II three- 8170 7940 2.8 138 120 470000 419000 −10.8
speed
type I PI 10680 10443 2.2 74100 71800 −3.1
type II PI 9959 9658 3.0 71200 69600 −2.2

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Industrial & Engineering Chemistry Research

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Article

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