Edited by: Nesrin Ozalp, University of Minnesota Duluth, United States
Reviewed by: Gongnan Xie, Northwestern Polytechnical University, China; Xiaofeng Guo, ESIEE Paris, France; Patrick Oosthuizen, Queen's University, Canada
This article was submitted to Thermal and Mass Transport, a section of the journal Frontiers in Mechanical Engineering
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The Ragone relation is a facile approach to assess and compare electro-chemical battery performance in terms of two critical performance parameters: power density and energy density. This power and energy nexus is equally relevant for thermal energy storage materials for thermal management applications that require a balance between energy storage capacity and on-demand cooling or heating rates. Here, thermal energy storage is evaluated for sensible heating and for phase-change materials (PCMs). We propose an analytic expression using a lumped mass model for thermal storage through an analogy with heat diffusion that allows for intuitive mapping of materials and components in power-energy space. In addition, a previously proposed figure-of-merit,
Lumped thermal capacitance is placed in the middle of thermal conduction in order to effectively match the time constant in dynamic thermal response.
A half century ago, Ragone published an overview of electro-chemical and fuel-cell batteries (Ragone,
Thereafter, the Ragone plot has become an essential mapping method to compare different electrochemical energy storage technologies. For example, Christen and Carlen (Christen and Carlen,
Here, we focus on thermal energy storage for potential use in a variety of time-dependent thermal energy and thermal management systems in which heat must be efficiently stored and transferred. Conventional thermal energy storage has been studied as an inexpensive alternative to electro-chemical batteries as a form of energy storage and sometimes as a provider of larger energy capacity, e.g., geothermal energy (Lund and Freeston,
This article first considers the close analogy between thermal and electrical systems, and provides a comparison and discussion of their similarities and differences (Thornton et al.,
The concept of energy storage capacity is common to both thermal and electrical regimes. For either conducting or insulating materials, the analogy described herein is generally applicable for solids or fluids at rest (i.e., no convection or advection), as there is no equivalent to convective processes in the electrical regime. Consequently, some electrical circuit analysis methods are generally applicable for transient and steady-state energy analysis of thermal problems (e.g., Robertson and Gross,
Properties and characteristics in the thermal and electrical analogy based on an energy flow in a rectangular solid with length
Potential | ||
Current | ||
Capacitance | ||
Resistance | ||
Time constant | τ_{th} = |
τ_{el} = |
Diffusivity | α_{th} = |
In thermal management of electronics, for example, the heat flow quantity allowed for conduction path in limited dimensions is an important factor of technology. This applies to thermal storage where the thermal current behaves similar to electrical current in electrochemical batteries. Trade-off between heat power and energy capacities is qualitatively equivalent to that of electrochemical batteries for EVs.
To clarify the nature of dynamic thermal storage, we first analyze a case of a lumped single-phase thermal mass, applying the equivalent circuit approach drawn from the electrical analog. We consider a volume consisting of a rectangular block with a geometry of length
Schematics of the physical representations of the cases of contacting a hot
A simple thermal circuit diagram can be developed in an analog to electrical circuits. In this first-order model, a lumped capacitance exists within the total thermal resistance (
Thermal network diagram of a lumped mass thermal model. Depending on contacting to the hot or cold, the energy flow direction is changed by the switch (SW).
By utilizing the Thermal Quadrupoles method (Maillet et al.,
where,
where the effective capacitance is lumped to the mid point of total thermal resistance. The block of the lumped system is symbolized in
The time constant of the transient thermal response of a single-phase volume is determined as,
The effective resistance contribution to the transient response is assumed to be one half (mean point). This assumption is verified in subsequently with a distributed thermal mass model. The time-dependent temperature
The time-dependent stored specific energy in the solid per unit temperature rise
The specific power (per thermal storage mass and temperature rise)
This function only decreases as time increases. Hence the specific power is maximum at
The specific power consists of a ratio of two elemental material properties
Thermal Ragone plot for single phase materials (
Material properties of selected solids.
Aluminum | 2700 | 896 | 207 | 8.56 × 10^{−5} |
Copper | 8960 | 386 | 380 | 1.10 × 10^{−4} |
Indium | 7310 | 225 | 81.8 | 4.97 × 10^{−5} |
Diamond | 3530 | 516 | 2200 | 1.21 × 10^{−3} |
Graphite | 2260 | 720 | 800 | 4.92 × 10^{−4} |
Silicon | 2329 | 710 | 140 | 8.47 × 10^{−5} |
SiN | 3200 | 700 | 30 | 1.34 × 10^{−5} |
SiO_{2} | 2200 | 700 | 1.4 | 9.09 × 10^{−7} |
GaN | 6150 | 490 | 130 | 4.31 × 10^{−5} |
Polyethylene | 1030 | 1256 | 0.188 | 1.45 x 10^{−7} |
Paraffin wax | 774 | 2160 | 0.15 | 8.97 × 10^{−8} |
A distributed thermal mass model has been demonstrated previously (Jackson and Fisher,
Normalized temperature responses of the discrete model (
Utilization of latent heat enhances the thermal energy capacity per mass [J/kg] for energy storage applications. A lumped mass dynamic model with an effective heat capacity is developed here to include phase-change with a latent heat contribution. Two sets of properties must be considered for the liquid and solid phases, for which suffixes
where
Therefore, the three elements (
Lumped mass model for a phase-change material (PCM). The equivalent analytic thermal circuit
The temperature response is found by a single lumped thermal mass model based on (Equation 5) but with the foregoing expression for
To validate the approximate dynamic models described above, we compare these results against the exact analytical solution for a semi-infinite medium with constant temperature boundary condition. The analytical solution of the two-region Neumann-Stefan problem in 1-D is:
In the ideal limit of a small temperature difference across the phase transition region (
The parameter λ_{2} is much greater than unity when the material is single-phase. In the case of a hot contact for instance, the Stefan number
where
Normalized end wall temperature
Here, the thermal Ragone relation can be extended by utilizing
The previous sections provide a means to analyze relative trade-offs in cooling power and thermal energy storage by analyzing a particular test geometry. However, practical thermal storage problems consist of unique geometries and boundary conditions that may complicate comparisons among different PCMs, and are time-dependent. Following a parallel approach, Shamberger introduced a cooling power figure-of-merit (
where, λ_{2} is the parameter found by solving (Equation 14) as discussed in the previous section, and implicitly requires a working temperature range, Δ
Here, we adopt
Thermal Ragone plot of
Because thermophysical parameters of materials including the effective enthalpy of fusion,
In summary, metallic phases tend to have the highest
The Ragone relationships for thermal storage materials designed for thermal management were explored, based on an electro-thermal analogy. The dynamic thermal response can be derived by temporal energy balance equations in a continuous medium. We demonstrated that a lumped thermal mass model worked well to determine the time constant along with the quick positioning of heat power-energy space, which is Ragone relation from the property information. Latent heat of fusion provides a significant increase in heat capacitance per given physical mass or volume, which drastically extends the energy capacity in the Ragone relation. The analysis for phase-change materials (PCMs) were also conducted by introducing enthalpy method. The lumped model for PCMs observes a discrepancy from the exact model but still the prediction is useful for first order estimate in thermal power and energy space. As a metric specifically for material selection for thermal storage, we utilize
TF conceived the structure of the paper and made substantial contributions to each of many drafts. PS contributed the discussion on modeling and the figure-of-merit of phase change materials. KY contributed to the development of an effective capacitance model and carried out numerical simulations.
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
This material is based on research sponsored by Air Force Research Laboratory under agreement number FA8650-14-2-2419. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes not withstanding any copyright notation thereon.
The authors thank the following individuals for helpful discussions and suggestions: Peter Bermel and Galen R. Jackson. The authors also acknowledge the helpful advice and support from the consortium members of the Center for Integrated Thermal Management of Aerospace Vehicles (CITMAV), including AFRL, Boeing, Honeywell, Lockheed Martin, Northrop Grumman, Raytheon, and Rolls-Royce.
area, m^{2} | |
heat capacity, J/K | |
specific heat, J/kg/K | |
energy, J | |
electron charge, Coulomb | |
latent heat of fusion, J/kg | |
electrical current, A | |
thermal conductivity, W/m/K | |
length, m, heat of fusion, J/kg, or Lorentz number, (-) | |
mass, kg | |
number | |
heat, W | |
thermal resistance, K/W | |
temperature, °C or K | |
time, s | |
voltage, V | |
Greek symbols | |
α | thermal diffusivity, m^{2}/s |
ε | permittivity, F/m |
λ | solution of transcendental equation, (-) |
temperature difference, K | |
ρ | density, kg/m^{3} |
σ | electrical conductivity, 1/Ω |
τ | time constant, s |
Subscripts | |
0 | initial state |
eff | effective value |
el | electrical |
l | liquid phase |
m | melting |
s | solid phase |
th | thermal |
w | wall |
* | specific value, */kg |