Atakan 2024 - Carnot-Battery Fundamentals

Thermodynamic Analysis of Carnot Batteries

Abstract

• Carnot batteries (Pumped Thermal Energy Storages - PTES) are a potential option for affordable and efficient energy storage.
• These systems charge a thermal energy storage using a heat pump and discharge it with a power cycle.
• Reversible thermodynamics doesn't clearly define the selection of storage temperatures; literature shows tendencies toward both higher and lower temperatures for high roundtrip efficiencies (RTE).
• The study analyzes the sensitivity and robustness of idealized, irreversible cycles concerning temperature levels, heat transfer irreversibilities, and internal cycle efficiencies.
• Global sensitivity analysis is used to assess the importance of various variables.
• Higher mean storage temperatures are generally favorable, but excessively high temperatures don't significantly improve RTE.
• At high temperatures, internal cycle efficiencies are the most sensitive factor for RTE.
• Multi-objective optimizations were performed considering the coupling of power terms to driving temperature differences, heat exchanger areas times heat transfer coefficients (UA), the ratio of discharging to charging time, and roundtrip efficiency.
• Using the same heat exchangers for charging and discharging is beneficial for similar orders of magnitude of discharging and charging power, and shorter discharging to charging periods.
• Separate heat exchangers are favorable for long discharge periods at low power relative to the charging power, which is useful for shifting solar energy to night hours.
• Reducing roundtrip efficiencies slightly near the optimal point can lead to significant reductions in heat exchanger areas and costs.
• The presented methods offer general guidance in Carnot battery design and parameter selection.
• Keywords: Thermal energy storage, Carnot-Battery, Thermodynamics, Global Sensitivity Analysis, Multi-Objective Optimization

1. Introduction

• The demand for energy storage is increasing due to climate change action and the growth of fluctuating renewable sources (wind, solar).
• Various energy storage methods exist, differing in energy density, time scales, maturity, costs, etc.
• There's a need for diverse storage technologies to supplement each other, particularly high energy density storage for longer time scales and cheap storages for small scales (few kWh).
• Thermal energy storage is geographically flexible and can have relatively high energy density using low-cost materials.
• Combining a heat pump and a power cycle is favorable for storing and retrieving electrical energy, known as a Carnot Battery or Pumped Thermal Energy Storage (PTES).
• The term "Carnot battery" will be used consistently.
• From thermodynamics, 100% roundtrip efficiency $\Psi$ is theoretically possible with reversible charging and discharging:
  $\Psi = \frac{E<em>{\text{dis}}}{E</em>{\text{ch}}}$ $(1)$
  $\frac{t<em>{\text{dis}}}{t</em>{\text{ch}}} = \frac{P<em>{\text{ch}}}{P</em>{\text{dis}}}$ $(2)$
• Real devices operating with finite rates experience losses.
• Power-to-power efficiencies are not equivalent to roundtrip efficiency because energy is conserved, while the power ratio is coupled through the storage to the ratio of discharging to charging times.
• In the reversible case, both power values would be zero while both time intervals would be infinity.
• Thess analyzed and optimized the process with internally reversible cycles but irreversible heat transfer, aiming for maximum discharging power output, the roundtrip efficiency should rise slowly with the storage temperature
• Maximum power condition results in high driving temperature differences, leading to large entropy generation and low roundtrip efficiencies.
• Research combining a compression heat pump and an (Organic) Rankine cycle using an inverse engineering approach with abstract optimal fluids.
• Limiting roundtrip efficiencies as a function of temperature for maximum and reduced power output were analyzed.
• Discharging at 80% of maximum power leads to strongly improved efficiencies.
• Including non-ideal compressors and expanders resulted in realistic roundtrip efficiencies below 50% at temperatures below 430 K, with efficiency falling smoothly with temperature due to the saturation curves approaching isothermal Carnot cycles.
• The general temperature dependence of roundtrip efficiency will be analyzed here, neglecting specific fluid properties.
• Numerous publications address specific Carnot battery designs, comparisons, parameter variations, optimizations, and cost estimations.
• Different Carnot battery technologies and configurations are reviewed in various publications, with some concentrating on time-dependent changes in storages for different materials.
• Energy entering a system has positive values.
• Dumont et al. discuss different Carnot battery configurations and energy accumulation in irreversible charging-discharging cycles, but not storage temperature dependence or time scales.
• Recent reviews analyze specific configurations and working fluids, lacking a real ordering theoretical framework and unresolved storage temperature dependence.
• Steinmann et al. compared cycles and configurations with gases, condensable fluids, or fluids passing critical parameters.
• They analyzed processes for fixed temperature differences and the influence of isentropic machine efficiencies (80-90%) on round-trip efficiencies.
• Promising roundtrip efficiencies of 60-70% were predicted, differing between cycles at different temperature levels.
• The general influence of temperature levels and global sensitivity wasn't explicitly addressed.
• Blanquiceth et al. analyzed complex Carnot battery configurations thermodynamically for a restricted temperature range, showing the compromise between roundtrip efficiency, power, and other parameters.
• Higher storage and lower ambient temperatures were reported as favorable for roundtrip efficiency, with the largest effect from improved machine efficiencies, but it's not addressed if this holds for different configurations.
• Local sensitivities for a specific Carnot battery system are briefly mentioned in [16].
• Expansion adiabatic efficiency is most sensitive, followed by storage exergy efficiency.
• A recent paper describes a dynamic model of Carnot batteries using driving temperature differences and temperature levels, focusing on a Brayton cycle with molten salt storage and n-pentane, which helps with specific dynamics but not general parameter selection.
• Dynamics were also investigated for an absorption-based Carnot battery with LiBr/H2O solutions and hot storage temperatures below 100°C, finding optimal concentrations for roundtrip efficiencies above 45% at moderate costs with respect to roundtrip efficiency and energy storage density.
• The off-design performance of a Carnot battery for solar applications was analyzed in [19], emphasizing machine performance along typical days with varying discharging times and pressure levels.
• Different heat exchanger areas are selected for charging and discharging with R1233zd(E) as the working fluid, and several recommendations are given.
• It's not discussed whether this specific configuration with separate heat exchangers is generally needed or leads to further possibilities or restrictions.
• Charging and discharging power at the design point were selected as 2.4 MW and 1.6 MW, respectively, while the relation to the total heat exchanger area and ratios of discharging to charging time aren't discussed in general.
• Wang et al. [20] found that exergetic efficiency rises with storage temperature along a parameter variation for thermally integrated Carnot batteries using R245fa, but it's unclear if this is a general conclusion for much higher temperatures.
• The present article will address this question.
• McTigue et al. [21] discussed configurations with respect to levelized costs of storage (LCOS) and roundtrip efficiencies, including thermodynamic losses, and analyzed charging to discharging time ratios, finding LCOS optima near values of one.
• For specific storage materials, roundtrip efficiencies are highest for materials capable of storing at higher storage temperatures, but without a general derivation.
• Multi-objective optimization shows the trade-off between costs and roundtrip efficiency, finding that higher storage temperatures are beneficial.
• Finite time thermodynamics was applied by Dai et al, finding reduced roundtrip efficiencies with temperature and charging and discharging times, also addressing the effect of one shared heat exchanger and varying driving temperature differences.
• While the findings are valid, it's unclear if they can be generalized.
• Thermodynamic orientation could help guide such work, but the question of favorable storage temperature and the global sensitivity of roundtrip efficiency and power on basic parameters remains unresolved.
• The ratio between discharging power and charging power, the ratio of charging to discharging time, and the relation between heat exchanger sizing and roundtrip efficiency were not analyzed fundamentally.
• The design process includes costs, which are not analyzed directly but scale with the heat exchanger size, while operating costs and revenues are directly related to the roundtrip efficiency.
• Invertible cycles, running as heat pumps and ORC with the same fluid, machines, and heat exchangers, are attractive, but the loss in flexibility will be briefly discussed using thermodynamic fundamentals.
• Fundamental investigations combining internally reversible thermodynamic power cycles (pc) with externally irreversible heat transfer through finite temperature differences allow analysis of aspects like the Pareto front of efficiency and power, and optimum temperature levels for maximum power output.
• The concept is expanded to address the situation where not only heat transfer is irreversible but also the thermodynamic cycles themselves.
• The importance of parameters within realistic regimes is assessed, and if they lead to some preferred temperature regime for thermal energy storage.
• When trying to improve roundtrip efficiency, one may need to consider internal irreversibilities (expansion, compression) or external ones (heat transfer).
• The article wants to address such topics fundamentally, neglecting fluid properties.
• Research questions addressed:
  • I. How should storage temperatures be selected without specific fluids or configurations, and which irreversibilities are crucial for roundtrip efficiency at different temperatures? At what temperatures should electrical heaters be used instead of heat pumps?
  • II. What are the thermodynamically limiting Pareto optimal parameter combinations of roundtrip efficiency, (lumped) total heat exchanger area, and either discharge power or discharging time? What is the consequence of using the same heat exchangers for charging and discharging?
• The work addresses roundtrip efficiency first, before discussing the problem that power, time, heat exchanger size, and roundtrip efficiency have different optima, leading to Pareto fronts.
• A simple model with uniform mean temperature differences between cycle and heat sources/sinks is analyzed, along with internal cycle-efficiencies, before a global sensitivity analysis is performed with multiple parameters bounded in realistic value regimes.
• Exemplary results of a multi-objective optimization are presented and discussed.

Nomenclature

• Abbreviations:
  • PTES: Pumped thermal energy storage.
• Subscripts:
  • eq: equilibrium, h: high (temperature), hp: heat pump, i: parameter number, irr: irreversible, l: low (temperature), pc: power cycle, rev: reversible, s: source or sink, st: storage, tot: total
• Greek symbols:
  • $\varepsilon$: Coefficient of performance, $\eta$: Internal cycle-efficiency, $\Psi$: Roundtrip efficiency
• Symbols:
  • E: Mean value (Expectation), P: power, Q: heat, $\dot{Q}$: heat flow rate, T: Temperature, UA: Overall heat transfer coefficient × area, V: variance, W: work

2. Model

2.1 Thermodynamic Model

• The model simplifies the process with temperatures of the (initial) heat source (Ts) and storage (Tst) being the levels for heat source and sink along charging, and inverse for discharging.
• The Carnot battery has a high-temperature storage and a low-temperature heat source, initially in equilibrium with the surrounding.
• The reasoning holds for a low-temperature storage and a heat source initially at a different temperature, but the formulations are more difficult to read, so that only this case is discussed.
• Heat pump and power cycle deviate from reversible cycles due to:
  • External irreversibilities: Heat transfer through finite temperature differences.
  • Internal cycle irreversibilities: Expansion, compression, throttling, friction, or internal heat transfer, summarized as global internal cycle efficiencies relative to reversible (Carnot) cycles.
• Finite rate charging and discharging of a storage requires temperature differences between the storage and the cycle or the (initial) source and the cycle are needed.
• Along charging, the heat transfer from the low temperature heat source (index l), generally the surrounding, to the cycle requires a slightly lower temperature level in the cycle by $\Delta T_{hp,l}$.
• To reject this heat to the storage the cycle temperature must be at a slightly higher temperature level $\Delta T<em>{hp,h}$ than the storage temperature $T</em>{st}$.
• For discharging, the signs of the temperature differences are inverse.
• Overall heat transfer coefficient times area values (UA) for heat transfer to or from the cycle are first fixed at exemplary levels as well, as the second-law cycle efficiencies.
• All temperatures and temperature differences should be regarded as appropriate mean values.
• If the storage temperature changes linearly from surrounding temperature Tsur to a maximum storage temperature Tst,max, use the thermodynamic mean value Tst = (Tsur + Tst,max) / 2.
• When heat transfer rates are to be calculated the logarithmic mean temperature differences have to be used, as known from heat and mass transfer textbooks.
• Starting with two simple connected cycles, the energy balances are:
  • $\dot{Q}<em>h = \dot{Q}</em>{hp}$ $(3)$
  • $\dot{W} = \dot{Q}<em>{pc} - \dot{Q}</em>l$ $(4)$
• If the cycle and compressor were reversible, the coefficient of performance from the Carnot cycle can be used and the work needed for the heat pump would be
  $\dot{W}<em>{rev} = \dot{Q}</em>h \frac{T<em>h - T</em>l}{T_h}$ $(5)$
• Due to internal irreversibilities, the real net-work is larger than the reversible work and can be quantified by a second-law efficiency $\eta<em>{hp}$, as defined in (6), which can also be used to calculate the coefficient of performance ($\varepsilon$) of the heat pump from the reversible $\varepsilon</em>{hp,rev}$:
  • $\eta<em>{hp} = \frac{\dot{W}</em>{rev}}{\dot{W}}$ $(6)$
  • $\varepsilon = \frac{\dot{Q}<em>h}{\dot{W}} = \eta</em>{hp} \cdot \varepsilon<em>{hp,rev} = \eta</em>{hp} \cdot \frac{T<em>h}{T</em>h - T_l}$ $(7)$
• Combining equations (5) and (6) gives an expression for the total charging work transferred to the heat pump:
  • $\dot{W} = \frac{\dot{Q}<em>h}{\eta</em>{hp}} \cdot \frac{T<em>h - T</em>l}{T_h}$ $(8)$
• To obtain finite heat transfer, the upper temperature of the cycle must be somewhat above the storage temperature $(T<em>{hp,h} = T</em>{st} + \Delta T<em>{hp,h})$ and the lower temperature must be somewhat below the source temperature $(T</em>{hp,l} = T<em>s - \Delta T</em>{hp,l})$. $\Delta T<em>{hp,l} > 0$ , \Delta T{hp,h} > 0
  • $\dot{W} = \frac{\dot{Q}<em>h}{\eta</em>{hp}} \cdot \frac{(T<em>{st} + \Delta T</em>{hp,h}) - (T<em>s - \Delta T</em>{hp,l})}{T<em>{st} + \Delta T</em>{hp,h}} = \frac{\dot{Q}<em>h}{\eta</em>{hp}} \cdot \frac{T<em>{st} - T</em>s + \Delta T<em>{hp,h} + \Delta T</em>{hp,l}}{T<em>{st} + \Delta T</em>{hp,h}}$
• If the work input to the heat pump is larger than the heat which shall be stored (Whp ≥ |Qh|.), the sign of Ql would change and direct electrical heating is favourable and Whp = |Qh|.
• The coefficient of performance must be larger than one throughout, otherwise a part of the work would have to be rejected at low temperature, which would be meaningless.
• Power cycle (subscript: pc):
  $\dot{W}<em>{pc} = \eta</em>{pc} \cdot \dot{Q}<em>h \cdot \frac{T</em>h - T<em>l}{T</em>h} = \eta<em>{pc} \cdot \dot{Q}</em>h \cdot \frac{(T<em>{st} - \Delta T</em>{pc,h}) - (T<em>s + \Delta T</em>{pc,l})}{(T<em>{st} - \Delta T</em>{pc,h})}$
• The roundtrip efficiency can then easily be calculated from the ratio of the last two expressions:
  • $\eta = \frac{\dot{W}}{\dot{W}<em>{hp}} = \eta</em>{hp} \eta<em>{pc} \frac{T</em>{st} - T<em>{pc,h} - T</em>{S} - T<em>{pc,l}}{T</em>{st} - T<em>{pc,h}} \cdot \frac{T</em>{hp,h}}{T<em>{st} - T</em>{sur} + T<em>{hp,h} - T</em>{hp,l}}$
• A quasi-steady state will be analyzed for each cycle, charging and discharging, respectively.
• This also holds for storages and cycles with changing parameters, but then an integration has to be performed or the discussed values must be taken as thermodynamic mean values.
• Including the heat flow calculations with given UA values leads to some additional restrictions:
  • Not all temperature differences can be freely selected, but each cycle has one temperature difference calculated from the energy balance.
  • The temperature difference at low temperature for the heat pump $\Delta T<em>{l}$ and the temperature difference between cycle and storage (subscript h) for the power cycle $\Delta T</em>{h}$ are independent variables, the remaining two are calculated from the energy balance.
  • The UA values are used as an indicator of heat exchanger sizing.
  • The calculation is a rough estimate, the overall heat transfer coefficient U is not constant for different flow situations.
• The charging power is coupled for an assumed steady-state according to:
  • $\dot{Q} = UA \Delta T = \frac{\Delta T}{R}$
  • $\dot{Q}<em>{hp} = (UA)</em>{hp} \Delta T_{hp}$ $(12)$
• Which can be integrated for a time interval of $\Delta t$ to come back to equation (2), but the charging and discharging times differ in general.
  • $\dot{E} = \int<em>{0}^{\Delta t} \dot{Q}(t) dt = UA \int</em>{0}^{\Delta t} \Delta T(t) dt$
• Due to the fixing of the lower temperature difference, it appeared to be more convenient to use the coefficient of performance for refrigeration of a Carnot cycle $\varepsilon<em>{rev}$, although the heat flows vanish for the reversible case with temperature differences of zero: $\varepsilon</em>{c,rev} = \frac{T<em>l}{T</em>h - T_l}$
  • $\dot{Q} = \frac{\dot{W}}{\varepsilon}$ $(15)$
  • $\varepsilon = \frac{T + T<em>{l}}{T</em>h - T<em>L + T</em>{HP, h}}$ $(16)$
• The final equation together with equation (13) has only $\Delta T_{l}$ as unknown, when prescribing the input power and the UA values, and is solved numerically.
• A similar reasoning leads to the equation for the power output of the power cycle, which is solved for $\Delta T_{h}$:
  • $\dot{W}_{pc}$
• With the known temperature differences, the heat flows can be calculated, as well as the discharging power.
• The equations were programmed in Python, and the root finding was carried out with the scipy.optimize.root function
• The function receives the two external efficiencies, the four UA values, the two temperature levels, the power input, and $\Delta T_{h}$ as input and calculates all remaining temperature differences, the discharge power, the roundtrip efficiency, and the ratio of discharging time to charging time and will be used for multi-objective optimization.

2.2 Sensitivity Analysis

• A variance-based sensitivity analysis was performed according to the method of Sobol and Saltelli, as implemented in SALib.
• The first order and the total (Sobol-) sensitivity coefficients for all other parameters were evaluated as a function of storage temperature for ranges of all parameters, within their typically found values using a quasi-random (Monte Carlo) sample of typical size N = 1000, leading to 16000-22000 evaluations (N × 2k) per storage temperature, depending on the number of model parameters k.
• The parameters for the power calculations involve also the heat transfer coefficients times area (UA) values as parameters, for each heat exchanger separately.
• The first order sensitivity coefficients for a function Y = f(X1, X2, …Xk) with mean E and variance V are defined as:
  $\S<em>i = \frac{V</em>X(E(Y/X))}{V(Y)}$ $(19)$
• The subscript $\sim i$ indicates that the mean is taken for all other values, but fixing the value for Xi.
• The aim of the sensitivity analysis is to obtain an overview of the importance of different parameters within this multi-parameter evaluation, in order to find out on which parameters one should concentrate to obtain large improvements in either roundtrip efficiency or in discharge power.
• The results of the Monte Carlo calculation can also be used to visualise the influence of fluctuations within the parameter regimes on the roundtrip efficiency.
• This will be visualized in histograms and gives an impression of the most probable output, when parameters are fluctuating, as due to changes in weather or running in part load, which leads to off-design operation.

2.3 Multi-Objective Optimization

• Several parameters can be selected nearly freely: The four UA-values, the second law efficiencies of the two cycles and at least one temperature difference.
• Assuming that the input power is fixed, as is the charging time, different outputs are of interest, the roundtrip efficiency is:
  • The sum of UA- values, as an approximate measure of the heat exchanger area, which contributes to the investment costs, the output power and the ratio of discharging to charging time.
• The latter two are coupled to the round-trip efficiency via Equation (2).
• Since generally a minimum of the total area (here approximated via $\sum UA$) is aimed while a maximum of the other parameters is desirable, this leads to a multi-objective optimization problem.
• This is solved using the popular NSGA II algorithm as implemented in pymoo
  $\min F(X) = (f<em>1(x), f</em>2(x),…,f_n(x))^T, x \in X$ $(20)$
• All parameters are restricted between minimal and maximal values, as given in Table 1.
• The internal efficiencies were restricted to the interval [0.7,0.75] both, but because they always are optimal at their highest value, they will not further be discussed later.
• And in a final optimization they were no longer optimized, but fixed to a value of 0.75.
• This range also limits the reachable roundtrip efficiency to 56.25%.
• The initial source temperature was fixed to 293.15 K.
• The storage temperature was exemplarily fixed to 363.15 K, having non-pressurized water in mind, when isothermal, but this could also stand for a storage which is heated from 293.15 K to 433.15 K, as could be done with mineral oils.
• The input charging power was fixed to 1000 W, while the value is arbitrary, the discharging power and all heat exchanger UA values scale linearly, thus the reader can scale them easily.
• The minimum discharging power is selected as Pmin = 50 W.
• Different combinations of objectives are optimized, as will be given in the results section.
• The presented calculations were carried out for a population size of 4000 with 1000 offsprings and either 250 or 300 generations, which did not influence the results markedly.
• Further details are found in the Python scripts, which can also be used by the reader, when different parameter regimes shall be analyzed.

3. Results and Discussion

3.1 Roundtrip Efficiency Evaluation

• The storage temperature dependence of the roundtrip efficiency is evaluated for two values of the internal cycle-efficiencies and three temperature differences between the working fluid and the storage or the surrounding, respectively.

• The low (source) temperature level was fixed to 290 K throughout.

• The low temperature and the storage temperature have throughout to be understood as thermodynamic mean temperatures.

• This means they would represent isothermal cases at the given temperature, as well as any temperature change along heat transfer, with the same mean value, e.g. a linear temperature change from 280 to 300 K, would also be represented by T=290 K.

• For these calculations, both internal cycle-efficiencies of the heat pump and the power cycle were identical, either 70% or 90%, as indicated.

• Also, all four temperature differences $\Delta T$ between the working fluid and either source or sink were set to be identical in each calculation and selected between 1 K driving temperature difference and 15 K.

• For the smallest temperature difference, the roundtrip efficiency is dominated by the internal cycle- efficiencies, the limiting roundtrip efficiencies are 0.72 = 49% and 0.92 = 81%, and a small temperature dependency is only found at low storage temperatures.

• The chosen lower second law efficiency is already above typical values which can be expected using piston compressors and expanders (see e.g.: [34]) but realistic for larger systems e.g. with turbomachines.

• If the driving mean temperature difference is larger, as it often cannot be avoided for working fluids with phase change and some additional enthalpy change in the single-phase regime, it is seen that the temperature dependence of the roundtrip efficiency gets large and that higher storage temperatures are favorable.

• The reason is clear: the entropy production in heat transfer through a finite temperature difference depends on the absolute (mean) temperature levels; at low temperatures the heat transfer through the same driving temperature difference leads to a larger entropy production than at high temperatures; this favors higher storage temperatures.

• As long as the heat pump and the power cycle both transfer work to or gain heat from the surrounding with approximately the same temperature, this loss cannot be influenced much, while the losses in heat transfer to the storage vanish with increasing temperature and they can be influenced by selecting higher storage temperatures.

• There is no optimal temperature, but efficiency curves get flatter with storage temperature, so selecting even higher temperatures will hardly be favorable beyond a certain point.

• A curve is included for a combination of electrical heating ($\epsilon_{hp} = 1$) with a power cycle, having an internal second law efficiency of 70% and 15 K driving temperature differences.

• This curve gets identical with the initially heat pump driven cycle between 900 and 1000 K as soon as $\epsilon<em>{hp} \cdot \eta</em>{hp} = 1$, this depends on the driving temperature differences.

• A heat pump generally leads to higher roundtrip efficiencies at temperatures below 900K.

• At higher temperatures, only the power cycle influences the round-trip efficiency, and thus higher temperatures are favorable.

• If the internal second law efficiencies would be at values of 0.9, electrical heating would only reach the heat pump efficiencies at much higher (and unrealistic) storage temperatures.

• The storage temperature plays a crucial role, and if opposing temperature dependencies are found for real or hypothetical fluids, this can be attributed to an increasing mismatch between the temperature profile of the working fluid in the cycle and the heat source or sink, leading to reduced cycle efficiencies.

• The thermodynamic reasoning shows that the improved roundtrip efficiency with storage temperature has a fundamental reason.

• Although the exergy losses get larger and the roundtrip efficiency smaller with larger temperature differences and lower internal cycle-efficiencies, at least the storage size is reduced. (Figure 3)

• The net work is always a difference between compressor and expander work (or the work lost along throttling), see eq. (21).

• For each pressure change, the work added is larger relative to the reversible work and the work transferred by an expander is lower than the reversible work, as expressed in eq. (22).

• The reversible technical work of flow processes is the mean specific volume times the pressure change along compression or expansion, so that the net reversible work can be expressed in these terms (eq. (23)).

• The difference in density between compression and expansion within each charging or discharging cycle should be as large as possible (eq. (24)), and the efficiency of only one machine is important (eq. (25)).

• This is often expressed as work ratio or back-work ratio which is unfavorable for pure gas cycles, where density approximately scales with temperature, and better for Rankine cycles, due to the much larger density difference between liquids and vapors.

  $\dot{W}<em>{net} = \dot{W}</em>{comp} - \dot{W}_{exp}$

  $\dot{W}<em>{comp} > \dot{W}</em>{revers. Comp}$

  \dot{W}_{exp} < \dot{W}_{revers. exp}
  

  $\dot{W}_{rev} = \int V dP$

  $(\rho<em>{high} - \rho</em>{low})$
  $\dot{W}<em>{net} \approx \dot{W}</em>{Comp}$

• These arguments lead to Carnot battery designs with not too high temperatures, to avoid gas cycles.

• For compression heat pumps, throttling is used, which can only lead to small losses, when the mean density along throttling remains much lower, than the mean density at compression.

• For two-phase or transcritical cycles, throttling should take place after subcooling to nearly the temperature of the surrounding, and the evaporation should only lead to low quality saturated states.

3.2 Sensitivity of Roundtrip Efficiency Regarding Stored Work

• Sensitivity on driving temperature differences, internal efficiencies, and variations in surrounding temperature are analyzed.
• Input distributions were uniform (temperature differences: [1, 15] K, surrounding temperature: [274, 300] K).
• For the internal cycle efficiencies, two cases are analyzed. In case 0, both efficiencies were separately within the bounds [0.5, 0.7], while case 1 had [0.7, 0.9] as bounds.
• For each of the 100 temperatures, 16834 conditions were evaluated, and both first order and total sensitivity coefficients were calculated and analyzed.
• The sum was always 1 within 1-2%, showing that higher-order sensitivities are of minor importance, therefore, only the first order sensitivity coefficients S1 are presented in Figure 4.
• The roundtrip efficiency is sensitive to temperature differences and the surrounding temperature below 500 – 600 K only.
• Interestingly, the sensitivity on the low temperature difference in the power cycle is the most sensitive temperature difference, followed by the high temperature power cycle $\Delta T_{pc,h}$, the sensitivities of the driving temperature differences of the heat pump remain below 10%, but cannot be neglected.
• At most temperatures, the sensitivity of the roundtrip efficiency on the internal efficiencies is most important, only when the temperatures are getting above 600 K for case 0, the heat pump sensitivity falls because the cases with a coefficient of performance of one (= electrical heating) dominate.
• The sensitivities depend strongly on the bounds regarded, this shall be shown for one case only, where the internal efficiency bounds differ between the heat pump and the power cycle, and both intervals are halved.
• The bounds are selected as [0.5,0.6] for the heat pump and [0.6, 0.7] for the power cycle in case 0 now, and as [0.7,0.8] and as [0.8,0.9] in case 1.
• The product of the mean values would lead to similar limiting roundtrip efficiencies as in the previous calculation shown in Figure 4.
• The sensitivities on the internal efficiencies differ; the sensitivity of the roundtrip efficiency on the less efficient cycle, here the heat pump, is increased.
• The sensitivity on the $\Delta T$ values are increased, especially at temperatures below 500 K. In such cases, the proper design of the heat exchangers gets especially important.
• The sensitivity of the roundtrip efficiency on $\Delta T_{hp,h}$, the driving temperature difference in the condenser, remains the lowest.
• Performing such a global sensitivity analysis is strongly recommended prior to detailed design, to obtain information about the parts of the system which must be selected most carefully, and to find the less important parts.
• In addition, the influence of fluctuations and off-design conditions on the total output can be assessed.

3.3 Power, Time Ratio, and Multi-Objective Optimization

• The efficiency increases with lower driving temperature differences $\Delta T$ in the heat exchangers, but the heat exchanger area increases with decreasing $\Delta T$.
• The sum $\sum UA$ can be used as a measure of total heat exchanger area.
• Roundtrip efficiency and $\sum UA$ will not have their optimal values for the same condition.
• The driving temperature differences influence the ratio of discharging time to charging time.
• Small $\Delta T$ values lead to long times, low heat flow rates, and low discharging power and vice versa.
• For storages for solar energy, where daytime harvested energy is used throughout the night, the charging power is much higher than the discharging power, while the inverse is true for the periods.
• For private households, a storage of ca. 1 kWh per kW of electrical energy from solar is often installed, and the solar panels are selected such that the battery is loaded within ca. 1 h.
• The discharging time is then often 5 – 15 h (e.g. 6h in [22]) with a discharging power between 50 and 100 W per kW charging power.
• A multi-objective optimization is performed with roundtrip efficiency, UAtotal, the discharging power, and the ratio of discharging time per charging time being the objectives.
• All exemplary optimizations were carried out for a mean storage temperature of 363.15 K.
• Roundtrip efficiencies above 50% can be achieved for discharging time ratios above 9.5. The discharging power per charging power is, thus, restricted to values near 50 W/kW which is the direct result of Equation (2).
• At time ratios of 10, typical optimal UAtotal value is 2280 W/K with roundtrip efficiencies of 50%, while at time ratios of 9, the efficiency drops to 45% and UAtotal to 1150 W/K.
• The internal cycle efficiencies always ended at the highest bound of 75% for both cycles.
• If the mean U values would be on the order of 1000 W/(m2 K), as it would be the case for liquid water on one side and a condensing or evaporating working fluid on the other side, the total heat exchanger area per charging kW of electrical power would be ca. 2 m2 for the first case and 1 m2 for the second one.
• The heat exchanger areas of the optimizations lead to UA values of 800 W/K for both heat exchangers in the heat pump for the first case and ca. 300 W/K for each power cycle heat exchanger.
• In the second case, the orders of magnitude are 450 W/K in the heat pump heat exchangers and 110 W/K in the power cycle heat exchangers.
• The driving temperature differences are throughout larger in the heat pump than in the power cycle.
• Using the same heat exchangers for these time ratios would only be meaningful if the U