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# The fluctuations of the outdoor temperature as a source of immaterial entropy production in energy conversion processes

Michel Pons, Lionel Mathelin

► To cite this version:

Michel Pons, Lionel Mathelin. The fluctuations of the outdoor temperature as a source of immaterial entropy production in energy conversion processes. Energy Conversion and Management, 2026, 347, pp.120485. 10.1016/j.enconman.2025.120485. hal-05282663

HAL Id: hal-05282663

https://universite-paris-saclay.hal.science/hal-05282663v1

Submitted on 25 Sep 2025

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Research Paper

# The fluctuations of the outdoor temperature as a source of immaterial entropy production in energy conversion processes

Michel Pons* and Lionel Mathelin

Université Paris-Saclay, CNRS, Laboratoire Interdisciplinaire des Sciences du Numérique, Route du Belvédère, 91400 Orsay, France

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## ARTICLE INFO

### Keywords:

- Outdoor temperature fluctuations
- Heat storage
- Control strategy
- Energy efficiency
- Second Law
- Immaterial entropy production

## ABSTRACT

In fluctuating outdoor conditions, adding a heat storage system between the energy conversion processes and the outdoor air, and employing it appropriately, helps to reduce primary energy consumption and, consequently, total entropy production. This combination is studied numerically for a heat pump heating a house in real winter conditions. Three strategies for using the heat storage are compared. One strategy uses no prediction of the upcoming outdoor temperature. The second assumes that it is at its lowest around dawn. The third uses exact prediction over 24 h. With reasonably sized heat storage, the first strategy reduces energy consumption by 2–4%, whereas the third reduces it by 5–7% thanks to using the heat storage at the coldest time of the day. The second strategy is almost as efficient, yet much more realistic.

The entropy analysis of processes in fluctuating outdoor temperatures is conducted using the absolute upper limit of performance (i.e. the highest possible outdoor temperature) as a reference point. The entropy balance derived in this way reveals an unusual entropy production involving the outdoor temperature and the heat flux exchanged with the outdoor air. This production is not physical but *immaterial*. Its magnitude (up to 30% of the total entropy production) depends on the chosen strategy, which is an immaterial characteristic of the heat storage system. The results and formal derivations demonstrate that the entropy budget would be incomplete without this *immaterial* term.

This robust analysis leads to a systemic interpretation of the role of heat storage.

## 1. Introduction

The temperature of the outdoor air (or ‘ambient’, or ‘atmospheric’) fluctuates, more or less cyclically, between day and night. The amplitude of these circadian temperature fluctuations is typically in the order of 10–20 K. Since all energy conversion processes (power generation, refrigeration, heat pumping ...) involve heat exchange with the outdoor air, their energy efficiency (the ratio of the energy produced by the process and the energy supplied to it) also fluctuates. These fluctuations can be significant for domestic processes (e.g. heat pumping in winter or air conditioning in summer), as the temperature difference between the outdoor and indoor air is comparable to the circadian amplitude. The negative effects of these fluctuations can be counteracted by various means, such as the use of nocturnal ventilation of dwellings during hot summers. Nocturnal ventilation consists in ventilating the house or apartment as much as possible during the night, so that the air conditioning can be switched on later the next day, or even not at all, thanks to the freshness stored in the walls. Ramponi et al. compare the effects of

various European climates on wind-induced ventilation in an isolated office building [1]. Bonetti and Kokogiannakis study the exergy storage performance of seven types of building wall [2]. Solgi et al. state in their review [3] that climates with high amplitudes of outdoor temperature fluctuations are the most favourable for nocturnal ventilation. Piselli et al. examine how nocturnal ventilation performance can be improved by incorporating phase change materials into building envelopes [4]. Ali-zadeh and Sadrameli review the technical and economic aspects of phase change materials [5]. Since a specific heat storage system can replace the thermal inertia of a building’s walls, Faraj et al. review the different types of heat storage material, how they are integrated into walls, and their interaction with ventilation modes [6]. Hirschey et al. describe how such a heat storage can reduce both the heating demand of a building and the power consumption of its heat pump [7]. Ma et al. combine the cold energy storage with the use of a free or inexpensive cold source [8]. Food refrigeration and preservation processes can also benefit from such heat storage systems [9], as can solar-powered absorption chillers. These units use heat-powered three-temperature cycles where the temperature difference between the condenser and the

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: michel.pons@lisn.fr (M. Pons).</footnote>
Nomenclature

Variables

A<sub>α</sub> solar absorption parameter [m<sup>2</sup>]</td>

E Internal energy [J]</td>

GHI Global horizontal insolation [W.m<sup>-2</sup>]</td>

H Heaviside function [-]</td>

MC Heat capacity [J.K<sup>-1</sup>]</td>

P<sub>S</sub> Entropy production [J.K<sup>-1</sup>]</td>

q, Q Heat quantity [J]</td>

Q<sub>d</sub> Cumulative heat demand function [J]</td>

s, S Entropy [J.K<sup>-1</sup>]</td>

t Time [s]</td>

T Temperature [K]</td>

T̃<sub>o</sub> Entropic mean outdoor temperature [K]</td>

UA Heat exchange parameter [W.K<sup>-1</sup>]</td>

w non-dimensional power consumption [-]</td>

W Power consumption [J]</td>

X Size parameter [-]</td>

Symbols - Greek letters

δ<sub>·</sub> elementary amount of quantity · [same unit as ·]</td>

δ<sub>2</sub> indicator function for Mode 2 [-]</td>

η Second Law efficiency of the heat pump [-]</td>

Indexes

a refers to the point a shown in Fig. 3

av refers to the average periodic day

A, B two control configurations to be compared

b, c refer to the points b and c shown in Fig. 3

D demand

h heating

HP supplied by the heat pump

l limit

o outdoor

s heat storage

T total

Superscripts

+ maximum outdoor temperature

* reference storage configuration

0 apparently reversible

hp heat pump

n estimation at the next time step

o outdoor air

rev reversible

s heat storage

t threshold

Mathematical operators

• Time derivative of quantity •

¯ Time averaging of quantity •

Abbreviations

HP heat pump

HS heat storage

evaporator (one of which exchanges heat with the outdoor air) is around 30–40 K (see textbook [10]). The energy efficiency of such units can be improved by adding a heat storage with either latent [11] or sensible [12] heat. Energy conversion processes equipped with heat storage either produce more with the same input, or require less input for the same output.

In process thermodynamics, the Second Law establishes a close relationship between energy efficiency and entropy production. Entropy *s* is a macroscopic function of state whose differential is defined by *ds* = *δq/T*, where *δq* is the elementary heat quantity received by an elementary material volume at temperature *T*. Clausius stated the Second Law of Thermodynamics: the entropy of an isolated system cannot decrease [13]. In the field of mechanics of macroscopic continuous media, there are five causes of entropy production: heat diffusion, viscous friction, species diffusion, chemical reactions, and the Joule effect [14]. At least one of these five physical phenomena is present in every real (or simulated) energy conversion process: real processes are irreversible. However, each of them has a reversible counterpart, with the same purpose, the same energy sources, but without any entropy production. If all heat source temperatures are constant (this is the thermostatic context), three properties are verified:

1. The energy efficiency of the reversible process (called here the *reversible efficiency*) is the upper limit of the efficiency of the process under consideration.

2. This upper limit depends only on the temperatures of the heat sources (typically, the *Carnot efficiency*), meaning it is the same for any process (real or simulated) designed for the same purpose and with the same energy sources.

3. The gap between the *reversible* and the real efficiencies is fully explained by the entropy productions occurring in the process. Moreover, all the entropy productions are equivalent: each unit of entropy production has the same effect on the energy efficiency, regardless of its causes or circumstances.

These three properties are important for process optimisation. Optimisation requires comparing the energy efficiency of processes designed for the same purpose and using the same energy sources but with different designs. By virtue of the three properties above, such a comparison is exactly equivalent to comparing the entropy productions of these processes. Entropy production thus aggregates all possible changes in energy efficiency, regardless of their nature and causes, into a single physical quantity. Entropy production also establishes a direct relationship between the energy efficiency, a global property of the process, and local irreversible phenomena occurring within the process. If the reduction in entropy production resulting from a change in the design of a component is known, then the increase in overall energy efficiency is also known. The benefits of various changes can therefore be compared. Second Law analysis is therefore a powerful tool not only for understanding the causes of the efficiency gap mentioned in property #3, but also for providing valuable information on how to reduce the gap. This is described and explained in many textbooks which are based either on entropy analysis [15–18] or on exergy analysis [19–21]. Second Law analysis is useful thanks to the equivalence between low entropy production and high energy efficiency.

In the *thermostatic* context, storing heat between the process and the outdoor air would be pointless. Conversely, in the real context of fluctuating outdoor temperatures (*non-thermostatic*), this enables energy savings and improves energy efficiency, as reported by all the references [1–9,11,12], despite the variety of processes considered. Furthermore, all of these references describe their operating *strategy* in detail, i.e. how their heat storage is operated in relation to outdoor temperature fluctuations. If this transient operation matches the transient behaviour of the outdoor temperature well, the storage is *used appropriately*. This improves energy efficiency. Consequently, this also reduces entropy production. In a *non-thermostatic* context, the fluctuating outdoor temperature, the heat storage, and its operating *strategy* affect the entropy balance. It is uncertain whether Second Law analyses developed in the usual thermostatic context can account for this effect. The wide variety
of definitions of exergy and expressions of the reversible efficiency in the relevant literature [2,22–26] illustrates this uncertainty well. To rigorously analyse the effect of strategy on entropy production, a new entropy analysis is developed here that is adapted to the non-thermostatic context while maintaining the aforementioned three properties. This entropy analysis is then applied to numerical simulations of a heat pump and heat storage (HP + HS) system, which heats a house as if it were exposed to real winter weather conditions. Three strategies are compared (when the references cited above considered only one). These strategies correspond to three levels of prediction of the upcoming outdoor temperature, ranging from ‘no prediction at all’ (strategy S1) to ‘exact knowledge of the next 24 h’ (strategy S3). The aim is then to apply the new entropy analysis to the numerical data obtained in this way, and to study the effect of strategy on the various causes of entropy production, including one that is specific to non-thermostatic contexts.

First, the article introduces the case in question: the house and its location, as well as the meteorological data measured in winter a few years ago. It then describes the numerical models used to mimic the operation of the heat pump and heat storage. A preliminary study, carried out on an average periodic day, indicates how the heat storage can be used appropriately. The three control strategies are then elaborated and applied to the process. The numerical results are discussed in terms of both thermal behaviour and energy savings.

Next, the new entropy analysis is developed; it involves an extremum of the outdoor temperature (here, the maximum) in the definition of a reversible efficiency independent of the process. This new approach identifies an immaterial entropy production, which is generated by the interaction between the process and the outdoor air. The numerical results of the three strategies are then analysed within this new framework. The analysis shows that the entropy balance would be incomplete without considering immaterial entropy production and ultimately leads to a comprehensive, systemic analysis of the role of heat storage.

## 2. The case under consideration

Residential applications globally represent a significant consumption of energy, with large greenhouse gas emission. The process considered here is a heat pump installed in a house in the northern half of France (Paris Île-de-France region), where heating in winter is the main concern in terms of both comfort and energy consumption. For a given heating demand, the heat pump consumes less electricity when the outdoor air temperature is relatively warm (daytime) than when it is relatively cold (at night). If the heat pump (HP) is complemented with a heat storage system (HS) consisting of a tank plus two heat exchangers, one with the heat pump and one with the outdoor air (see Fig. 1, left side), it consumes less power (electricity) than without heat storage, if the heat storage is appropriately managed [7,26–28]. The heat pump HP almost continuously supplies heat ($\dot{Q}_h$) to the indoor air at $T_h$ while consuming electricity ($\dot{W}$). It extracts heat either from the outdoor air at $T_o$, or from the heat storage tank HS at $T_s$. The heat storage tank can then replace the

outdoor air as the heat pump’s cold source when it is warm enough for the heat pump to consume less electricity for the same heating demand. Meanwhile, the temperature in the storage tank drops. Later, when the outdoor air eventually warms up, the tank is thermally charged via the second heat exchanger with the warm outdoor air. Such arrangements are reported in the literature [7]. Once the HP + HS system is designed, a specific control procedure must determine when the storage should be either used (instead of the outdoor air), charged or isolated.

### 2.1. Location, meteorological data

The present study employs actual meteorological data: the outdoor air temperature ($T_o$, as measured by a CS215 temperature probe) and global irradiance on a horizontal plane (GHI, as measured by a Kipp and Zonen CMP22 pyranometer). The data were kindly provided by the SIRTA (Site Instrumental de Recherche par Télédétection Atmosphérique), which is situated in Palaiseau, France (48.71°N, 2.21°E, at approximately 20 km south of Paris) [29]. The dataset covers two heating seasons (2016–2017 and 2017–2018) from October to March, which encompasses the period when heating systems are turned on in the Île-de-France region (from late October to mid-March). Here, the data time step is of 5 min. As an example, Fig. 2 shows the outdoor temperature and GHI measured over three days. The first and third days are rather sunny: the temperature is minimal around sunrise, rising to reach its maximum between 13:00 and 14:00. The second day is rather cloudy, probably rainy (almost no solar radiation), and the temperature behaves completely differently.

### 2.2. The house, the heat pump and the heat storage system

The house considered here is maintained at 19 °C ($T_h$ = 292.15 K) in winter by the HP + HS process. The house, the heat pump, and the heat storage are modelled using first-order equations with as few parameters as possible, to focus attention on the relationship between three phenomena: the operating strategy for the HP + HS process in actual weather conditions, the resulting energy savings, and the different entropy productions.

#### 2.2.1. Numerical model of the house

The house is assumed to receive from the heat pump exactly the heat flux $\dot{Q}_h$ required to maintain the indoor temperature at $T_h$. Here, a linear model relates the heat flux $\dot{Q}_h$ to the outdoor temperature $T_o$ and the solar radiation GHI, both time-dependent:

$$ \dot{Q}_h = \max[UA_h \cdot (T_h - T_o) - A_\alpha \cdot GHI, 0] \quad (1) $$

Here, the global heat transfer parameter $UA_h$ corresponds to a two-story house, moderately insulated according to the figures given by Lartigue et al. [30], and with a total surface area of 100 m²: $UA_h = 160 \text{ W.K}^{-1}$. The parameter $A_\alpha$ is reasonably chosen so that the total solar input, once integrated over time, represents 10 % of the total heat demand. The

<figure><img src="image_5.png" /><figcaption>**Fig. 1.** Left: Schematic representation of the combination of a heat pump HP and a heat storage system HS (tank and heat exchangers) for residential application. Right: The three operating modes of the process HP + HS. Mode 0: The heat pump extracts heat from the outdoor air and the storage tank is isolated. Mode 1: The heat pump extracts heat from the storage tank; no heat transfer from the outdoor air. Mode 2: The outdoor air transfers heat to both the heat pump and the storage tank.</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>Fig. 2. Meteorological data for three consecutive days (2017, November 26 to 28). Black solid line: outdoor temperature. Blue thin lines: global horizontal irradiance. The segments and numbers at the top of the graph are referred to in Section 4.4 and Appendix B.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 3. Optimal operation of the (HP + HS) system under periodic conditions. Graph A shows the averaged outdoor temperature (thin, dotted, solid line O) and the heat storage temperature in three cases: (i) 'reference' (solid line R), (ii) 'tiny' (curve T with short dashes), and (iii) 'huge' storage (curve H with long dashes). Graph B shows the corresponding non-dimensional power consumption, either without heat storage (thin dotted curve O) or with heat storage (curves T, R and H as in graph A). The graph shows the periods during which modes 0, 1 and 2 are active (the numbers at the top) and the transition times between these modes (the vertical dash-dotted lines between the numbers and the points a, b, and c on the curves). For clarity, these are only shown for the 'reference' case R; the corresponding times and points on the curves T and H can be easily deduced.</figcaption></figure>

corresponding value is $A_α = 3.2 \text{ m}^2$. With only two parameters, this model makes the heating demand primarily dependent on the outdoor temperature, and to a lesser extent on the global insolation. The latter plays a minor role in the total energy balance, but, as solar energy is only available for a few hours around noon, it often temporarily compensates

for the dissipated flux $UA_h \cdot (T_h - T_o)$ and, then, can play a significant role on instantaneous fluxes (cf. Section 5). The max function in Eq. (1) insures that $\dot{Q}_h$ is never negative: the heat pump is designed to provide heating, not cooling.

### 2.2.2. The heat pump plus heat storage process (HP + HS process)

For heat pumps, the difference between the two heat fluxes is the power (electricity) consumed. Here, the warm source is the house at $T_h$, and the cold source is the outdoor air at $T_o$ (time-dependent). Fig. 1, right side, describes the operation of the HP + HS process: the heat storage system is in one of three modes. In Mode 0, it is thermally isolated and the heat pump operates as if it were alone. In Mode 1, the storage system is in use and supplies heat to the heat pump. In Mode 2, the outdoor air supplies heat to both the tank and the heat pump.

### 2.2.3. Numerical model of the heat pump

Similarly to the house model, the heat pump model uses as few parameters as possible. It assumes that the Second Law efficiency (or exergy efficiency) of the heat pump, $\eta$, remains constant in each mode: $\eta_s$ in Mode 1 and $\eta_o$ in Modes 0 and 2. Although this model is probably too simplistic with respect to the details of heat pump operation, it provides a reasonable estimation of the heat pump's performance on average. The heat pump's power consumption is then given by:

$$ \dot{W} = \dot{Q}_h \cdot \frac{T_h - T_\bullet}{\eta_\bullet \cdot T_h} \qquad (2) $$

where the symbol $\bullet$ stands for either o (outdoor air - Modes 0 and 2) or s (storage tank - Mode 1). In the numerical simulations, $\eta_s$ and $\eta_o$ are equal to 0.4, a value that corresponds to fairly efficient heat pumps. However, to formally account for the possibility of inequality, the criterion for deciding between Mode 0 and Mode 1 compares the tank temperature to a threshold value, called $T_s^e$, where both modes would have the same power consumption. This threshold tank temperature also fluctuates, it is given by:

$$ T_s^e(t) = T_h - (T_h - T_o(t)) \cdot \eta_s / \eta_o \qquad (3) $$

### 2.2.4. Numerical model of the heat storage

The heat storage simulated here is based on sensible heat only, its temperature is uniform and its thermal insulation is perfect. The storage system is then characterized by the global heat capacity of the storage tank, $MC_s$, and the global heat transfer parameter between the tank and the outdoor air, $UA_s$; both are temperature-independent. The values of $MC_s$ and $UA_s$ are explored below. Both are proportional to the reference values $MC_s^e$ and $UA_s^e$, which characterise the reference configuration (see Section 3.2):

$$ \frac{MC_s}{MC_s^e} = \frac{UA_s}{UA_s^e} = X_s \qquad (4) $$
$MC_s$ and $UA_s$ then only depend on the non-dimensional proportionality factor $X_s$ which is hereafter referred to as 'the heat storage size'. When the storage tank delivers heat to the heat pump (Mode 1), the tank temperature $T_s$ decreases according to:

$$ \dot{Q}_s = \dot{W} - \dot{Q}_h = \dot{Q}_h \cdot \left( \frac{T_h - T_s}{\eta_s \cdot T_h} - 1 \right) = MC_s \cdot \frac{dT_s}{dt} < 0 \quad (5) $$

The first equality is the First Law applied to a heat pump with zero inertia. The second equality is obtained by replacing $\dot{W}$ by its expression (2). The third equality is the differential equation governing the temperature $T_s$ during operation in Mode 1.

When the condition for Mode 1 is not met, the storage tank is either warmer or colder than the outdoor air. In the first case, Mode 0 is operated and the temperature $T_s$ of the storage tank, which is isolated, remains constant. In the second case, Mode 2 is activated (the tank receives heat from the outdoor) and the tank temperature $T_s$ is governed by the differential equation:

$$ \dot{Q}_s = UA_s \cdot (T_o - T_s) = MC_s \cdot \frac{dT_s}{dt} > 0 \quad (6) $$

### 3. Simulations on a periodic average day

This section examines optimal cycles operated on a periodic average day. The construction of this day is described in the Appendix A. Analysing these optimal periodic cycles yields a simple rule that can later be applied to one of the strategies using real weather data.

#### 3.1. The periodic average day and optimal operation

The periodic average day is defined by two functions: $T_{o,av}(t)$ for the average outdoor temperature (curve O in Fig. 3A) and $\dot{Q}_{h,av}(t)$ for the average heating demand. Applying Eq. (2) to these functions gives the corresponding power consumption without heat storage $\dot{W}_{o,av}(t)$, the average value of which over the whole day is $\overline{\dot{W}_{o,av}}$. The non-dimensional rate of power consumption $\dot{w}_{o,av}(t) = \overline{\dot{W}_{o,av}(t)/\dot{W}_{o,av}}$ is shown in Fig. 3B (curve O, thin, dotted solid line).

Each graph, 3A or 3B, also shows three other curves (denoted as T, R and H). Graph A shows tank temperatures, and graph B shows non-dimensional rates of power consumption with heat storage, $w_{s,av}(t) = \overline{\dot{W}_{s,av}(t)/\dot{W}_{s,av}}$, where the power consumption $\dot{W}_{s,av}$ is also referred to $\overline{\dot{W}_{s,av}}$. The curves result from numerical simulations performed as described below, using three sizes of the storage system, i.e. three values of $X_s$. The points a, b, and c (at times $t_a$, $t_b$, $t_c$) on each curve R (reference heat storage) indicate the times at which the system switches from one mode to the next (from Mode 0 to 1 at $t_a$, etc.). They can be commented as follows:

The cycle consists of three periods. The central period is Mode 1 operation (path a-b), during which the tank temperature must exceed the threshold: $T_s > T_s^t$. The storage tank delivers heat to the heat pump, causing its temperature $T_s$ to decrease according to Eq. (5) (see graph A). Mode 1 is the only period during which power consumption is lower than without heat storage (see graph B).

As soon as $T_s \le T_s^t$ (point b, when the outdoor temperature rises after sunrise), the system switches to Mode 2. As the outdoor air is warmer than the tank ($T_s \le T_o$), the outdoor air reheats the storage tank according to Eq. (6) (path b-c). [Note: in the simulations, it can occur that $T_s > T_o$ at point b; Mode 0 is then selected; however, the system may return to Mode 1 if the condition $T_s > T_s^t$ is met again.].

In the afternoon, the outdoor temperature drops again. So does $T_s^t$, which eventually becomes colder than the storage tank $T_s$ (point c). In this situation, the system switches to Mode 0 (the tank is isolated and its temperature is constant) because operating in Mode 1 will be more efficient later on, at time $t_a$.

The same pattern applies to curves T and H, just with different positions for points a, b, and c.

A heat storage size $X_s$ and a value of the time $t_a$ define a periodic cycle. Periodicity is achieved by iterating the simulation on the periodic weather data until the tank temperatures $T_{sa}$, $T_{sb}$, and $T_{sc}$ at the times $t_a$, $t_b$, and $t_c$ eventually converge to their final values within less than 0.01 K (the times $t_b$ and $t_c$ also converge). Periodicity usually occurs after five or six iterations. The periodic cycles shown in Fig. 3 are optimised by finding the time $t_a$ (with a resolution of 5mn) that minimises the power consumption (it is assumed that the energy cost of switching between modes is negligible).

#### 3.2. Three heat storage systems

The reference system results from a compromise between three constraints that significantly reduce the arbitrariness of the choice. (i) The heat storage must play a non-negligible role in the entire process without taking over completely. (ii) Both the tank volume (and its heat capacity) and the outdoor air heat exchanger must not be too large in relation to the size of the house. (iii) The resulting operation must also be well balanced over time between the three modes. The reference heat storage system is characterized by $MC_s^* = 10.75 \text{ MJ.K}^{-1}$ and $UA_s^* = 920 \text{ W.K}^{-1}$. If the heat storage medium were an aqueous solution containing 25 % of mono-propylene glycol [which, with a freezing point of 263.15 K (-10 °C), can face measured outdoor temperatures as low as -6°C], the corresponding mass and volume would be 2700 kg and 2.65 m³. For the heat exchanger, assuming a heat transfer coefficient of 40 W.m⁻².K⁻¹ (turbulent convection with air), the heat transfer area would be 23 m². This storage volume and this exchanger area are not negligible, but still reasonable (the tank volume is about 1 % of the house volume). Fig. 3A shows that this reference heat storage system (code R and solid line) operates for 5 h in Mode 1 (path a-b, with a temperature drop of 3.8 K), 8 h in Mode 2 (path b-c, with a return to the initial temperature), and 11 h in Mode 0. In Fig. 3B, the gap between the paths a-b on the line O (the uppermost one) and on the solid line R shows the reduction in power consumption, here 4 % of the total consumption without storage.

The second system, called 'tiny' is defined by $X_s = 0.055$ ($MC_s = 0.59 \text{ MJ.K}^{-1}$, $UA_s = 50.6 \text{ W.K}^{-1}$). With this system (code T and short dashes), Mode 1 is very short, only 20 min (four time steps) and the tank temperature drop is greater than in the reference case (4.7 K). Mode 2 is longer than in the reference case (10 h), and the system is in Mode 0 for more than half of the cycle. Due to the brevity of Mode 1, power reduction is only 0.16 %.

The third system, called 'huge', is defined by $X_s = 19$ ($MC_s = 203.9 \text{ MJ.K}^{-1}$, $UA_s = 17 \text{ 480 W.K}^{-1}$). With this system (code H and long dashes), operation in Mode 1 lasts almost 19 h, with a small temperature drop (0.65 K). Then, the storage is charged very efficiently (in 4 h only) and the system spends slightly less than one hour in Mode 0. The resulting power reduction is very significant: 20 %.

The tiny and huge systems are both extreme: the former because the storage would be used up too quickly, and the latter because the tank would occupy 19 % of the house itself. Although they are not realistic, studying them provides valuable information about what makes the cycles optimal. In addition, the huge case provides information about the limit of infinite storage systems, a key consideration in the entropy analysis. The ratio of the two storage size parameters, $MC_s$ and $UA_s$, is kept constant here; the effect of changing only one of them on the optimal periodic cycles was studied in a previous article [26].

#### 3.3. What makes the cycles optimal?

With the tiny storage (curves T), the optimal operation in Mode 1 is practically near the minimum outdoor temperature. With the reference storage (curves R), the optimal Mode 1 is almost centred on the same moment. The coincidence of Mode 1 with low outdoor temperatures
(and thus low threshold temperatures $T_s^t$) makes sense for three reasons:

(i) For a given tank temperature $T_s$ (above $T_s^t$), the lower the outdoor temperature, the greater the efficiency difference between Modes 1 and 0, and then, the larger the power saving in Mode 1.

(ii) In addition, the greater the heat demand too: consequently, the better efficiency just mentioned applies to a high power rate.

(iii) Lastly, the greater the possible drop in $T_s$ during operation in Mode 1, i.e., the greater the amount of heat supplied to the heat pump.

Therefore, operating Mode 1 when the outdoor temperature is at its lowest is expected to be the most efficient way to use the heat storage. Here, the most efficient way is the *appropriate* way.

## 4. Strategies with the actual weather data

As real weather conditions change daily, it is not possible to iterate the operational scheme as done in Section 3 and determine the optimal distribution between modes 1, 2 and 0. A *strategy* is then a set of rules for switching the HP + HS system between modes, with the aim of reducing power consumption. For Mode 2, the operating conditions are well defined in Section 3: the outdoor air temperature ($T_o$) must be higher than the tank temperature ($T_s$) in order to reheat the tank, and $T_s$ must be lower than the threshold temperature $T_s^t$. However, between Modes 0 and 1, deciding which mode to operate depends on the predicted weather. This decision is therefore uncertain. Three *strategies* are considered here, that base their decisions on the respective positions of the three time-dependent temperatures $T_s$, $T_s^t$ and $T_o$ and their own level of prediction for the upcoming outdoor temperature.

### 4.1. Features common to all simulations and all strategies

Two heating seasons are simulated: 2016–2017 and 2017–2018, from October 25 to March 10. Heating demand is usually low around these two dates, so changing these limits by one or two days has negligible impact on the global energy budget. Before each heating season, the system is prepared as any homeowner could do. From 15 to 19 October, the storage tank takes the average outdoor temperature for those five days. Then, from 19 to 24 October, the tank is charged: as soon as the outdoor air is warmer than the tank, it transfers heat to the latter, whose temperature increases according to Eq. (6). On October 25, the heating season begins and lasts until March 10, that is, for 137 days. It is assumed that the controller knows with good accuracy (as if they were measured) (i) the total heat capacity $MC_s$ and Second Law efficiencies $\eta_o$ and $\eta_s$ (all assumed constant); (ii) the storage tank temperature $T_s^t$; (iii) the heat demand $\dot{Q}_h$ and the current outdoor temperature $T_o$ from which the threshold temperature $T_s^t$ is determined. The procedure described in Section 3 is applied (Mode 2 always starts and ends accordingly). As in the periodic case, the time resolution for switching between modes is 5 min. All the energy budgets will be done over the two heating seasons, assuming that switching from one mode to another has no energy cost.

### 4.2. Description of strategy S1

Strategy S1 does not use any predictions about the future weather; instead it reduces power consumption by operating in Mode 1 as soon as $T_s > T_s^t$. At each time step, the ordinary differential equation (5) is solved over one time step using the current values of $T_s$, $T_s^t$, $\dot{Q}_h$. The result is an estimate $T_s^\eta$ of the tank temperature at the next time step. If the condition $(2 \cdot T_s^\eta + T_s \ge 3 \cdot T_s^t)$ is satisfied, then the tank temperature should be above the threshold for more than half of the time step, Mode 1 should be more advantageous than Mode 0 and will therefore be selected by the controller. If the condition is not met, the controller will select Mode 0.

### 4.3. Description of strategy S2

Rather than using predictions about future weather, strategy S2 uses a simple general rule to roughly anticipate when the outdoor temperature should be at its next minimum: the coldest hour is just before dawn. The principle of this strategy S2 is therefore to centre operation in Mode 1 symmetrically around sunrise. To calculate the time of sunrise at a given location on a given day, all you need are simple astronomical formulae, completely independent of the weather. This can be done with a simple local calculator. Mode 1 is then operated around this time, half before, half after. At the end of Mode 2 (in general around 14:00), and as long as the system is in Mode 0, the controller regularly evaluates the time $\Delta t$ after which the storage tank is expected to reach the current threshold temperature $T_s^t$ if it were operated in Mode 1. This is done by solving the differential part of Eq. (5),

$$ \frac{dT_s}{dt} = \frac{-(\dot{Q}_h - \dot{W})}{MC_s} = \frac{\dot{Q}_h}{MC_s} \cdot \left( \frac{T_h - T_s}{\eta_s \cdot T_h} - 1 \right) \quad (7) $$

where $\dot{Q}_h$ is set to a constant equal to its current value: once the equation is linearised, its solution is an exponential function. The initial value of $T_s$ is the current tank temperature and the final value is set to the current threshold $T_s^t$. The resulting $\Delta t$ is the expected duration of operation in Mode 1. As long as the current time is less than $\Delta t/2$ before sunrise, the system is operated in Mode 0; it is operated in Mode 1 otherwise (points a in Fig. 3). Once in Mode 1, the controller follows the procedures described above for entering and, later, exiting Mode 2 (points b and c in Fig. 3). And so on.

### 4.4. Description of strategy S3

This strategy is intended to approach the best possible performance by operating in Mode 1 when the outdoor temperature is at its lowest during the next 24 h. In practice, this strategy would require extremely accurate weather prediction for the next 24 h, which is unrealistic given that no weather forecast can provide such a high level of accuracy and detail. Nevertheless, this strategy is relatively straightforward to simulate, since data for the following day is always available for computation. Therefore, it is worthwhile to compare the performance of the simple, easily achievable strategy S2, and that of the theoretical limit case strategy S3.

The principle is to calculate a threshold outdoor temperature value, $T_b$, that will separate operation in Mode 1 (when $T_o(t) < T_b$) and in Mode 0 (otherwise) for the next 24 h, such that two heat quantities are equal. The first quantity is the amount of heat required to heat the house during periods within the next 24 h when the outdoor temperature is lower than $T_b$, according to the heat demand function. The second is the amount of heat delivered by the heat pump in Mode 1 when extracting heat from the storage tank while its temperature $T_s$ decreases from the initial current value, $T_s^t$, to the threshold temperature determined by $T_l$ [$T_s = T_s^t(T_l)$]. For the sake of readability, the method is detailed in Appendix B. Once this value of $T_l$ is calculated, the control strategy S3 simply consists of switching the system to Mode 1 when the outdoor temperature is below $T_l$ ($T_o \le T_b$), and switching it back to Mode 0 otherwise. There may be several alternations between Modes 0 and 1 in the same day. This strategy ensures that Mode 1 is activated during the periods when the outdoor temperature is the lowest.

## 5. Details of operation during three successive days

The next three figures show how the simulated HP + HS system behaves during the three days of Fig. 2, depending on the storage size $X_s$ and for the three strategies. The codes used in the following graphs are the same as in Fig. 3: O (red thin lines) indicate the time evolution of the outdoor temperature and the heat pump's power consumption without
heat storage; the curves labelled *T*, *R* and *H* refer to the *tiny*, *reference*, and *huge* storage systems described in Section 3. Graphs A show the storage tank temperature and graphs B show the power consumption. Here the threshold tank temperature $T_s^*$ equals the outdoor temperature.

There are some periods during the first and the third day when the power consumption vanishes. This is due to the fact that the heating demand vanishes during these periods because of the combination of a high outdoor temperature with a large enough solar input. Such situations were mentioned when commenting on Eq. (1).

The tank temperature curves in Mode 2 (storage charge) of the *tiny* case and the *reference* case are very similar, especially when their initial values are close. This is because the solution of the differential equation (6) depends only on the initial condition and the ratio $MC_s/UA_s$, which is here a constant ($= MC_s^*/UA_s^*$).

### 5.1. Operation with strategy S1: Fig. 4

Similarly to Fig. 3, the tank temperature of the *tiny* storage (curve *T* in graph A) drops very rapidly during operation in Mode 1. Since the strategy switches to Mode 1 as soon as possible, the tank temperature never really deviates from the outdoor (threshold) temperature. As a result, the saving in power consumption compared to the case without storage is barely noticeable (see graph B).

With the *reference* storage (curves *R*), the temperature drop in Mode 1 is much slower. For the displayed days, it is even quite similar to that of the outdoor (threshold) temperature itself: see the periods around the times marked as 18, 52, and 69 h. However, when the outdoor temperature drops suddenly, such as around $t = 63$hrs, then the tank temperature deviates significantly from the former, resulting in a visible reduction in power consumption (graph B between 63 and 69 h).

With the *huge* heat storage, the system remains in Mode 1 over the three days and the heat pump uses only the heat storage. The tank temperature (curve *H* in graph A) decreases very slowly (roughly from 13 to 11 °C in three days), and remains above the outdoor temperature throughout the three days. This is because, three days before the days under consideration, on the 23rd of November, the outdoor temperature reached a maximum of 16 °C (289.15 K) and could heat the tank to 14.9 °C (288.05 K). The outdoor temperature then dropped, but the *huge* amount of heat stored that day was enough to keep the system in Mode 1 for several days. The power consumption is therefore very significantly reduced, see curve *H* in graph B. These features of the *huge* case are reproduced by the other two strategies and will not be commented again.

### 5.2. Operation with strategy S2: Fig. 5

As intended by this strategy, the operation in Mode 1 with the *tiny* and *reference* storages is almost periodic in time (around sunrise). With the *tiny* storage (dashed curves *T*), it lasts less than 20 min, but now the decrease in power consumption can be seen (see the central inset) and it resembles the one observed on the periodic day shown in see Fig. 3. With the *reference* storage (solid lines *R*), Mode 1 can last for a few hours, and the drop in power consumption, which is also similar to that in Fig. 3, often occurs when the power consumption is at, or close to its maximum (first and third day), but not always (see the second day, at $t \approx 32$hrs).

### 5.3. Operation with strategy S3: Fig. 6

Fig. 6 shows that, as intended, strategy S3 activates Mode 1 at times when the outdoor temperature is at its lowest. Strategy S3 has two main differences from strategy S2. First, the operation in Mode 1 often deviates from the dawn, e.g., by a few hours on the first day and by up to 10 h on the second day. Second, the operation in Mode 1 may be divided into two or more sub-periods separated by local maxima of outdoor temperature (as for the *reference* case on the first day, see the inset).

Strategy S3 thus fine-tunes the activation of Mode 1 and adapts it to the detailed fluctuations in the outdoor temperature.

Thanks to the preset limit outdoor temperature $T_b$, the control procedure looks simple: the controller just has to compare the current outdoor temperature with $T_b$. In fact, strategy S3 combines extremely accurate weather predictions with the following rule: 'Use Mode 1 when the outdoor air is at its coldest'. Thanks to the Second Law analysis developed in Section 7, it can be proved that this rule is the best possible. Therefore, strategy S3 is the most efficient once the size $X_s$ of the HP + HS system and the weather conditions are known.

## 6. Comparison of global performance

The primary figure of merit here compares power consumption with and without storage, $\dot{W}_s(t)$ and $\dot{W}_o(t)$. These two quantities are calculated with Eq. (2), where the symbol $\cdot$ is either ‘o’ alone for $\dot{W}_o(t)$, or alternatively ‘o’ and ‘s’ depending on the activated modes for $\dot{W}_s(t)$. The latter then depends on the size parameter $X_s$ and on the control strategy. $\dot{W}_s(t)$ and $\dot{W}_o(t)$ are integrated over the entire simulation period (the periodic average day, or from October 25 to March 10 for the two heating seasons), and averaged, as done for the periodic day. The non-dimensional power reduction, defined as $(\overline{\dot{W}_o} - \overline{\dot{W}_s})/\overline{\dot{W}_o}$, is shown in Fig. 7 as a function of the size parameter $X_s$, for the periodic day (dashed red curve ‘Per’) and for the three strategies S1 to S3 (solid black curves).

Fig. 7 mainly considers values of $X_s$ less than a few units where the influence of strategy is significant; the inset describes the asymptotic behaviour for large values of $X_s$, which is also of theoretical interest. As expected, the larger the storage size $X_s$, the less power the heat pump consumes. Meanwhile, the marginal benefit of a given increase $\Delta X_s$ (i.e., the slope of each curve) tends to decrease, except on the first part of the curve S1 ($X_s < 1.5$). The trend of decreasing marginal benefit is common in the field of energy efficiency.

### 6.1. Small values of $X_s$

Close to the origin $X_s \approx 0$, the slope of the curve S1 is significantly

<figure><img src="image_8.png" /><figcaption>Fig. 4. Results of the strategy S1 during the three days of Fig. 2: temperatures (graph A) and corresponding power consumptions (graph B). The curves *O* (red, thin, dotted, solid lines) refer to the case without heat storage; the curves *T* (short dashes) refer to the *tiny* heat storage, the curves *R* (solid lines) to the *reference* storage, and the curves *H* (long dashes) to the *huge* one. The temperature curve *T* and the power curve *H* are underlined with grey thin solid lines for better readability.</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>Fig. 5. Results of strategy S2 during the three days of Fig. 2: same notations and legends as in Fig. 4. The central inset zooms in on the detail of the operation in Mode 1 with the tiny storage on the first day ($t \approx 8:20$).</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 6. Results of strategy S3 during the three days of Fig. 2: same notations and legends as in Fig. 4. The central inset zooms in on the detail of the operation in Mode 1 with the tiny and reference storages during the first hours of the first day (0:00 < t < 8:00).</figcaption></figure>

smaller than that of the other curves; its tangent at the origin even looks horizontal. Strategy S1 saves power, but not by much. Strategies S2 and S3 are more efficient because they delay the start of Mode 1 until either the outdoor temperature is low on average (strategy S2) or it falls below the calculated threshold $T_l$ (strategy S3).

Remarkably, the difference in performance between strategies S2 and S3 is less than 1 %. This is a very small difference compared to the contrast in the accuracy of weather prediction used in the two strategies: only the time of the next sunrise in the former (S2), the values of both the outdoor temperature and the heating demand for the next 24 h in the latter (S3). It is beyond the scope of this article to develop an analysis of concepts such as ‘complexity’ or ‘amount’ of information which are more intuitive than clearly defined. Suffice it to say that both strategies S2 and S3 use a relevant rule based on a thermodynamics: “Use Mode 1 when the outdoor temperature is the least favourable”, here at its lowest. Strategy S2

<figure><img src="image_11.png" /><figcaption>Fig. 7. Power reduction as a function of the storage size $X_s$ (both non-dimensional) for optimal periodic cycles (red dashed curve Per) and for the strategies S1, S2, and S3. The inset shows the asymptotic behaviour.</figcaption></figure>

also uses another rule: “The outdoor temperature is generally lowest just before sunrise”. This rule is simple but it is based on correct statistical observations and it complements the first rule very well. The combination of these two rules is very relevant to the process under consideration and can rival exact predictions about the next 24 h.

## 6.2. Large values of $X_s$ and the asymptotic limits

In Fig. 7, the inset extends the range of the axes to large values of $X_s$. For $X_s = 400$, the non-dimensional power reduction for the optimal periodic cycles (dashed red curve) is 0.2515. The corresponding asymptotic limit would be with an infinite size $X_s$ of the heat storage, so that the tank supplies all the heat required by the heat pump, while its temperature remains constant and equal to the maximum of the outdoor temperature, $T_o^+$. As the heat exchange parameter $UA_s$ is also infinite, the storage tank can be fully charged when the outdoor temperature is at its maximum. This behaviour, described in more details in ref. [26], fits well with the extrapolation of the curve H in Fig. 3A: the tank temperature curve becomes more and more horizontal, and closer and closer to $T_o^+$. For the periodic day, the maximum outdoor temperature $T_o^+$ is 8.21 °C (281.36 K). At this asymptotic limit, the ratio on the right-hand side of the expression of the power consumption $\dot{W}_s(t)$, Eq. (2), becomes constant and equal to $(T_h - T_o^+)/(\eta_s \cdot T_h)$. The numerical value is 0.261; this is a likely asymptotic limit of the curve shown in the inset.

For the real weather data and strategies S1 to S3 (solid black lines in the inset), the maximum of the outdoor temperature is 16.3 °C ($T_o^+ = 289.45$ K); the corresponding limit of the non-dimensional power reduction is 0.821. Therefore, the latter curves lie above the previous one.

The asymptotic limit discussed above is important, because it leads to minimum power consumption and therefore maximum efficiency. In the thermostatic context, maximum efficiency is equivalent to reversibility (typically the Carnot cycle), and this maximum does not depend on the process (it is universal: no cycle can be more efficient than the Carnot cycle) but only on the heat source temperatures. Here, one of the two heat sources is not thermostatic, and the energy efficiency of the process depends on both the size of the heat storage and the way it is
used. The maximum efficiency must then be that of a reversible process equipped with the best possible heat storage, i.e., the infinite one. This statement is a cornerstone of the entropy analysis of processes immersed in a fluctuating outdoor temperature developed below.

# 7. Entropy analyses in fluctuating outdoor conditions

## 7.1. The usual combination of the energy and entropy balances

### 7.1.1. Stationary equations and the framework for entropy analysis and process optimisation

When considered globally, the HP + HS process extracts a heat flux ($\dot{Q}_o$) from the outdoor air at $T_o$ and delivers the flux $\dot{Q}_h$ to the indoor air at $T_h$, while consuming mechanical energy ($\dot{W}$), mostly in the form of electricity (power). The non-stationary energy and entropy balances are:

$$ \dot{E}_T = (-\dot{Q}_h) + \dot{Q}_o + \dot{W} \qquad (8) $$

$$ \dot{S}_T = -\frac{\dot{Q}_h}{T_h} + \frac{\dot{Q}_o}{T_o} + \dot{P}_S \qquad (9) $$

where $E_T$ and $S_T$ are the total internal energy and total entropy contained in the HP + HS system, and $\dot{P}_S$ is the rate of entropy produced by irreversibility. $\dot{P}_S$ involves two contributions: one ($\dot{P}_S^{hp}$) is generated within the heat pump itself due to Second Law efficiencies $\eta_\bullet$ less than one (see Eq. (2)), the other ($\dot{P}_S^{os}$) is generated in the heat exchanger between the outdoor air and the storage tank, see Fig. 1. One has $\dot{P}_S = \dot{P}_S^{hp} + \dot{P}_S^{os}$; the expressions of $\dot{P}_S^{hp}$ and $\dot{P}_S^{os}$ are given in the next sections.

For a system in steady state, all the terms in the equations (8) and (9) are constant and the time derivatives $\dot{E}_T$ and $\dot{S}_T$ vanish. The combination [(8)–$T_o$·(9)] eliminates $\dot{Q}_o$ and leads to a simple expression of the power consumption $\dot{W}$ for a given heat demand $\dot{Q}_h$:

$$ \dot{W} = \frac{\dot{Q}_h \cdot (T_h - T_o)}{T_h} + T_o \cdot \dot{P}_S \qquad (10) $$

The three properties mentioned in the introduction are verified in this equation. (i) The power consumption $\dot{W}$ has a lower bound, equivalently the ‘real’ energy efficiency $\dot{Q}_h/\dot{W}$ has an upper bound, $T_h/(T_h-T_o)$, that depends only on the temperatures of the heat sources (and not at all on the process); (ii) this limit is the efficiency of the reversible process (with $\dot{P}_S=0$); and (iii) the gap between the real and reversible efficiencies is fully explained by the total entropy production $\dot{P}_S$, where all the individual entropy productions are equivalent.

Thanks to these three properties, Eq. (10) provides an efficient framework for process optimisation. Now consider two possible design changes, say A and B, that would improve energy efficiency (or in this case, reduce power consumption) and have the same marginal ‘investment cost’ $\Delta C$. This ‘investment cost’ can refer to money, greenhouse gas emissions or the amount of material used to manufacture the components of the process. Such a local change could be increasing the area of a heat exchanger or the diameter of a pipe, or reducing the rotation speed of a turbine or compressor; it reduces the rate of entropy production in the modified component, but also in the neighbouring components. The resulting changes in the total entropy production, $\Delta \dot{P}_S^A$ and $\Delta \dot{P}_S^B$, both negative, can therefore be evaluated and compared. If $\Delta \dot{P}_S^A < \Delta \dot{P}_S^B$ (or $|\Delta \dot{P}_S^A| > |\Delta \dot{P}_S^B|$), then modification A is preferred. Furthermore, the double modification that associates A with the opposite of B will increase the overall energy efficiency (since $\Delta \dot{P}_S^A - \Delta \dot{P}_S^B < 0$) at constant ‘cost’. This brief development shows (i) how Second Law analysis helps to optimise a process, and (ii) that the changes in entropy productions are

more important than their absolute values.

### 7.1.2. Non-stationary equations in the non-thermostatic context

Like any energy conversion process, the HP + HS process here is in contact with the outdoor air and exchanges heat with it. As the outdoor temperature $T_o$ fluctuates, so does the temperature field in the heat exchanger involved in this heat exchange, and so do the internal energy $E_T$ and the entropy $S_T$ of the system: the left-hand sides of Eqs. (8) and (9) do not vanish, nor that of the combination [(8)–$T_o$·(9)]: $\dot{E}_T - T_o \cdot \dot{S}_T$. The important figure of merit is the average power consumption. Averaging (indicated by the bar symbol) consists of first integrating the power rate over the entire period of operation considered (either the periodic average day or the two consecutive heating seasons: $2 \times 137$ days), and then dividing the latter integral quantities by the period considered. The average of the left-hand side of [(8)–$T_o$·(9)] is: $\overline{\dot{E}_T - T_o \cdot \dot{S}_T}$. In the periodic case, by definition, the initial and final states of the system are the same, as are the corresponding values of $E_T$: $\overline{\dot{E}_T}$ vanishes. In the ‘real data’ case, these initial and final values are close enough that the difference plays a negligible role in the global balance compared to the other terms, which are integrated over 137 days. $\overline{\dot{E}_T}$ can therefore be neglected in the overall balance. However, this is not the case for $-T_o \cdot \dot{S}_T$, since both $T_o$ and $\dot{S}_T$ are time-dependent. The time derivative of the total entropy, $\dot{S}_T$, is mainly due to the variation of the entropy of the storage tank (the thermal state of the heat pump, i.e., the temperature field in its various components, varies much less than that of the storage tank, whose heat capacity is, by design and in addition, much more significant). In addition, the storage tank is designed to be charged (i.e., to receive heat, which means $\dot{S}_T > 0$) when $T_o$ is high, and to be used (i.e., to deliver heat: $\dot{S}_T < 0$) when $T_o$ is low. As a result, the average value of $-T_o \cdot \dot{S}_T$, which is very likely to be negative, cannot be neglected. Consequently, in a fluctuating environment, the average power consumption is:

$$ \overline{\dot{W}} = \frac{\overline{\dot{Q}_h \cdot (T_h - T_o)}}{T_h} + \overline{\dot{T}_o \cdot \dot{P}_S} - \overline{\dot{T}_o \cdot \dot{S}_T} \qquad (11) $$

This equation looks similar to Eq. (10), but with three terms on the right-hand side instead of two.

(i) The first term on the right-hand side, $\overline{\dot{Q}_h \cdot (T_h - T_o)} / T_h$, looks like the reversible power consumption in Eq. (10) (with $\dot{P}_S = 0$): $\dot{Q}_h \cdot (T_h - T_o) / T_h$. However, the time average $\overline{\dot{Q}_h \cdot (T_h - T_o)}$ depends on how the time dependent heat demand $\dot{Q}_h(t)$ is phased with respect to the time dependent outdoor temperature $T_o(t)$, and therefore not only on the heat source temperatures.

(ii) The second term on the right, $\overline{\dot{T}_o \cdot \dot{P}_S}$, looks like the term with the entropy productions $T_o \cdot \dot{P}_S$ in Eq. (10). However, as $T_o$ is now time dependent, the entropy productions that occur when the outdoor temperature $T_o$ is high (typically in the afternoon) weigh more than those that occur when $T_o$ is low (typically around dawn). The equivalence between all entropy productions regardless of their circumstances, previously ensured in Eq. (10), is broken in Eq. (11): one would have to compare not the entropy productions alone, but their product with the outdoor temperature (in K) at the time when the entropy productions occur.

(iii) The status of the third term on the right-hand side, $-\overline{\dot{T}_o \cdot \dot{S}_T}$, is uncertain, except that it is certainly not entropy production. This product (time derivative of the total entropy times outdoor temperature) is unknown in thermodynamics. Its sign itself depends on how the respective variations are phased with respect to each other and is therefore uncertain. In the present case, $-\overline{\dot{T}_o \cdot \dot{S}_T}$ is very likely to be negative (see above), but this property is specific and cannot yet be generalised to other processes. If the
sign is negative, then the sum $\overline{\dot{Q}_h \cdot (T_h - T_o)}/T_h - \overline{T_o \cdot \dot{S}_T}$ is less than its first term (which then loses its status of “minimum power consumption” as stated in point (i) above). It might therefore seem reasonable to associate this third term to the first one in the definition of the reversible ($\dot{P}_S \equiv 0$) and minimum power consumption, denoted as $\overline{\dot{W}}$:

$$ \overline{\dot{W}} = \frac{\dot{Q}_h \cdot (T_h - T_o)}{T_h} - \overline{T_o \cdot \dot{S}_T} \quad (12) $$

However, this definition is only likely and specific to the present process. It also makes the minimum power consumption dependent on what happens within the process and when. On the contrary, in a development to come (Section 7.5), it seems reasonable to associate $-\overline{T_o \cdot \dot{S}_T}$ with the entropy production $\overline{T_o \cdot \dot{P}_S}$. Thus, depending on the question addressed, the term $-\overline{T_o \cdot \dot{S}_T}$ plays a role either in the minimum power consumption or as if it were an entropy production, which it formally is not. Finally, it should also be noted that while the entropy productions are local, this term $-\overline{T_o \cdot \dot{S}_T}$ involves (i) a global quantity, the total entropy $S_T$ of the whole system, and (ii) how its variations are correlated with those of the outdoor temperature. The presence of this third term breaks the direct relationship between local phenomena and the global performance through the entropy productions as in Eq. (10).

Equation (11) is correct, but the framework that it provides for analysing the performance and power consumption of processes raises many difficulties. Probably the most important is that it leads to a definition of maximum efficiency (or minimum power consumption) that does not necessarily correspond to reversibility and depends on internal phenomena within the process. The apparent upper limit of energy efficiency in this framework could be exceeded in another process equipped with a larger heat storage. Furthermore, the equivalence between all entropy productions is lost.

Compared to the thermostatic framework of Section 7.1.1, the framework of Eq. (11) unnecessarily complicates and confuses Second Law analyses of processes. However, these issues can be avoided by taking a different approach.

## 7.2. A new combination of the energy and entropy balances

For all these reasons, the approach developed here takes as a cornerstone that the upper limit of energy efficiency is universal and applies to any possible process designed for the same purpose and with the same heat sources. A reversible heat pump [$\dot{P}_S \equiv 0$ in Eq. (10)] is one obvious condition; an infinite size $X_s$ of the heat storage system is the second condition, see section 6.2 and Fig. 7. Increasing the heat capacity $MC_s$ of the storage tank reduces the temperature drop experienced by the heat storage tank during operation in Mode 1, while ensuring the heat pump receives the required amount of heat to meet the heating demand. At the infinite limit, the tank temperature remains constant during operation in Mode 1. Increasing the $UA_s$ parameter of the heat exchanger between the outdoor air and the storage tank reduces the temperature difference between the outdoor air and the storage tank during operation in Mode 2. At the infinite limit, this temperature difference vanishes and the tank temperature reaches the maximum $T_o^+$ of the outdoor temperature, which is then the tank temperature during total operation (Modes 1 and 2). The reversible heat pump would then operate as if its cold heat source were a thermostat at $T_o^+$; its power consumption would then be:

$$ \overline{\dot{W}}^{\text{rev}} = \overline{\dot{Q}_h \cdot (T_h - T_o^+)} / T_h \quad (13) $$

$\overline{\dot{W}}^{\text{rev}}$ is not only similar to the first term on the right hand side of Eqs. (11) and (12), it is certainly its lower bound. In addition, no other heat pump operating with the same heat sources can consume less power

than $\overline{\dot{W}}^{\text{rev}}$: this minimum is absolute and unique, it is universal. Thirdly, the minimum power consumption per unit of heat demand, $\overline{\dot{W}}^{\text{rev}}/\overline{\dot{Q}_h}$, only depends on the heat source temperatures. This expression (13) is also obtained by combining Eqs. (8) and (9) according to [(8) $-T_o^+ \cdot (9)$]. In this combination, $T_o^+$ is a constant. Using Eq. (9) multiplied by the constant factor $T_o^+$, time averaging is straightforward and the average power consumption is given by:

$$ \overline{\dot{W}} = \overline{\dot{Q}_h \cdot (T_h - T_o^+)} / T_h + \left( \overline{\dot{E}_T} - T_o^+ \cdot \overline{\dot{S}_T} \right) + T_o^+ \cdot \overline{\dot{P}_S} - \overline{\dot{Q}_o \cdot (1 - T_o^+ / T_o)} \quad (14) $$

On the right-hand side, the first term is $\overline{\dot{W}}^{\text{rev}}$; the second bracket either vanishes (in periodic conditions) or can be neglected (it is integrated over a very long time and the difference between the initial and final states is small); the third term multiplies the entropy productions by a constant; and the last term, which never appears in the steady state framework, can be rewritten slightly differently. Equation (14) then becomes:

$$ \overline{\dot{W}} = \overline{\dot{W}}^{\text{rev}} + T_o^+ \cdot \left[ \dot{P}_S + \dot{Q}_o \cdot \left( \frac{1}{T_o} - \frac{1}{T_o^+} \right) \right] \quad (15) $$

### 7.2.1. Reversible power consumption and efficiency

Reversibility here consists of a zero rate of entropy production $\dot{P}_S$ plus an infinite size $X_s$ of the heat storage system. The so defined reversible HP + HS process exchanges heat with the outdoor air only at the exact time when $T_o(t)$ is at its maximum $T_o^+$. The rest of the time, the heat pump receives heat from the storage tank, the temperature of which remains constant (thanks to the infinite size) and equal to $T_o^+$. This reversible process, like the Carnot cycle, is only theoretical. In the Carnot cycle, there is no temperature difference between the working fluid and the heat sources, nor any pressure drop in the pipes, turbine or compressor. Physically and materially, these two assumptions correspond to infinitely large heat exchangers and infinitely large diameters. Similarly, the present reversible process assumes an infinitely large heat storage system. In any finite configuration, the storage tank would cool below $T_o^+$ (finite $MC_s$) or charge at times when $T_o(t)$ is below $T_o^+$ (finite $UA_s$). As a result, the heat pump would consume more than $\overline{\dot{W}}^{\text{rev}}$, and then have a lower efficiency than the reversible limit. Table 1 shows the values of the reversible efficiency in the two weather configurations considered here (periodic average day and real weather data) where $T_o^+$ takes two different values.

### 7.2.2. The physical entropy production $\dot{P}_S$

The entropy production $\dot{P}_S$ in the equations above is produced by physical phenomena, with two contributions: $\dot{P}_S^{\text{hp}}$, within the heat pump, and $\dot{P}_S^{\text{os}}$, in the heat exchanger between the outdoor air and the storage tank. One has: $\dot{P}_S = \dot{P}_S^{\text{hp}} + \dot{P}_S^{\text{os}}$.

The first component $\dot{P}_S^{\text{hp}}$ is obtained by combining the entropy balance such as in Eq. (9) with the First Law balance without inertia, $\dot{Q}_h = \dot{Q}_o + \dot{W}$, where the index $\bullet$ is either $o$ or $s$, depending on the operating mode. For a given heat demand $\dot{Q}_h$, the power consumption $\dot{W}$ depends

**Table 1**

Maximum outdoor temperature $T_o^+$, and reversible power consumption $\overline{\dot{W}}^{\text{rev}}/\overline{\dot{Q}_h}$, and energy efficiency $\overline{\dot{Q}_h}/\overline{\dot{W}}^{\text{rev}}$ in the two weather conditions considered here.

<table><thead><tr><th>Cases:</th><th>Periodic average day<br>Per</th><th>Real weather data, with strategies S1, S2, or<br>S3</th></tr></thead><tbody><tr><td>$T_o^+$:</td><td>281.36 K (8.21 °C)</td><td>289.45 K (16.3 °C)</td></tr><tr><td>$\overline{\dot{W}}^{\text{rev}}/\overline{\dot{Q}_h}$:</td><td>36.9 × 10<sup>-3</sup></td><td>9.2 × 10<sup>-3</sup></td></tr><tr><td>$\overline{\dot{Q}_h}/\overline{\dot{W}}^{\text{rev}}$</td><td>27.1</td><td>109</td></tr></tbody></table>
on the current operating mode. In Modes 0 and 2, the heat pump receives the heat flux $\dot{Q}_o$ from the outdoor air, and its Second Law efficiency is $\eta_o$. In Mode 1, the heat pump receives $\dot{Q}_s$ from the storage tank, and its Second Law efficiency is $\eta_s$. After substitution of $\dot{W}$ with its expression (2), the result is:

$$ \dot{P}_S^{hp} = \dot{Q}_h \cdot \frac{1 - \eta_\bullet}{\eta_\bullet} \cdot \left( \frac{1}{T_\bullet} - \frac{1}{T_h} \right) = \frac{1 - \eta_\bullet}{T_\bullet} \cdot \dot{W} \quad (16) $$

where $\eta_\bullet$ and $T_\bullet$ are either $\eta_s$ and $T_s$ (operation in Mode 1) or $\eta_o$ and $T_o$ (operation in Mode 0 or 2); $\dot{Q}_h$, $\dot{W}$, $T_o$, and $T_s$ are time-dependent.

For $\dot{P}_S^{os}$, the second component of $\dot{P}_S$, one has:

$$ \dot{P}_S^{os} = \delta_2(t) \cdot \dot{Q}_s \cdot \left( \frac{1}{T_s} - \frac{1}{T_o} \right) \quad (17) $$

where $\delta_2(t) = 1$ during operation in Mode 2 and $\delta_2(t) = 0$ otherwise, and where the heat flux $\dot{Q}_s$ transferred from the outdoor air to the storage tank is given by Eq. (6). The two components $\dot{P}_S^{hp}$ and $\dot{P}_S^{os}$ are integrated over time and then averaged as described in Section 7.1.2.

### 7.2.3. An immaterial entropy production

According to Eq. (15), the difference between $\overline{\dot{W}}$ and $\overline{\dot{W}}^{rev}$ is not fully explained by the two components of $\dot{P}_S$ described in the previous subsection; another term contributes to the difference, denoted by $\dot{P}_S^{+o}$:

$$ \dot{P}_S^{+o} = \dot{Q}_o \cdot (1/T_o - 1/T_o^+) \quad (18) $$

As $\dot{Q}_o$ is non-negative (the heat pump and heat storage never transfer heat to the outdoor air) and $T_o$ is by definition smaller than or equal to $T_o^+$, then $\dot{P}_S^{+o}$ cannot be negative. It does not contradict the Second Law to state that $\dot{P}_S^{+o}$ is an entropy production. As $\dot{P}_S^{+o}$ vanishes if the outdoor air is a thermostat [$T_o(t) \equiv \text{constant} = T_o^+$], this term never appears in Second Law analyses developed in the thermostatic context. This novel term has never been mentioned in literature before and therefore deserves a deeper analysis.

According to its expression, $\dot{P}_S^{+o}$ would be the entropy production generated by the transfer of the heat flux $\dot{Q}_o$ from the heat source at the constant temperature $T_o^+$ to the outdoor air at $T_o(t)$. The paradox is that this heat transfer is not real: in fact, the heat flux $\dot{Q}_o$ is not received by the outdoor air at $T_o(t)$, but is delivered by the latter to the HP + HS system. In a first step, the paradox is partially resolved by considering that the reversible power consumption $\overline{\dot{W}}^{rev}$ refers to a single value of the outdoor temperature, $T_o^+$, just as if the outdoor air were a thermostat at $T_o^+$. Adopting $\overline{\dot{W}}^{rev}$ as the reversible limit recreates a two-temperature ($T_h$ and $T_o^+$) thermostatic context in which the heat flux $\dot{Q}_o$, transferred from the outdoor air to the system, can only be seen as initially extracted from the source at $T_o^+$ (there is no heat source below $T_o^+$ in this context). In this context, the temperature drop from $T_o^+$ to $T_o(t)$ generates the entropy production $\dot{P}_S^{+o}$. Since the heat flux $\dot{Q}_o$ is not really transferred between these two temperatures, this entropy production is said here *immaterial* (it has no physical support). Nevertheless, it is meaningful in the present context because it gives the control strategy (how the process is operated and managed in relation to outdoor temperature fluctuations) a thermodynamic meaning. This provides valuable insight into the energy efficiency of the process. This topic is covered in more detail in Section 8.

The entropy production $\overline{\dot{P}}_S^{+o}$ is maximal when there is no heat storage at all ($X_s = 0$), it decreases when the storage size parameter $X_s$ increases, and it vanishes when $X_s$ is infinite (heat is only transferred when $T_o(t) = T_o^+$). As shown in Figs. 3 to 6, all the cases above demonstrate that Mode 1 (in which the HP + HS system is isolated from the cold outdoor air) is

operated for longer as $X_s$ increases. Consequently, the average outdoor temperature during the heat transfer from the outdoor air to the HP + HS system (Modes 0 and 2) increases, and finally the entropy production $\overline{\dot{P}}_S^{+o}$ decreases. Figs. 3 to 6 also show that an appropriate control strategy has a similar effect: the evolution from strategy S1 to strategy S3 causes the system to increasingly avoid extracting heat from the outdoor air when its temperature is cold. If this interpretation is correct, then the numerical results should show a relevant dependence of $\dot{P}_S^{+o}$ on $X_s$ and strategy.

### 7.2.4. The global entropy balance

Finally, the average power consumption per unit of average heat demand is given by:

$$ \frac{\overline{\dot{W}}}{\overline{Q}_h} = \frac{T_h - T_o^+}{T_h} + T_o^+ \cdot \frac{\overline{\dot{P}}_S^{hp} + \overline{\dot{P}}_S^{os} + \overline{\dot{P}}_S^{+o}}{\overline{Q}_h} \quad (19) $$

where the three entropy productions (given by Eqs. (16), (17), and (18) respectively) are all equally weighted, regardless of their causes and circumstances: the three properties mentioned in the introduction are preserved by the present approach.

## 7.3. Entropy budget of optimal periodic cycles

Fig. 8 shows how the terms of Eq. (19) depend on the parameter $X_s$ for the optimal periodic cycles. The symbols + alone represent the ratio $\overline{\dot{W}} / \overline{Q}_h$ on the left-hand side of Eq. (19). The height of the reversible zone below the horizontal line represents the reversible first term on the right. The heights of the three zones labelled hp, os and +o indicate the respective magnitudes of the next three terms, $T_o^+ \overline{\dot{P}}_S / \overline{Q}_h$, which are successively added to the reversible value. The upper curve thus represents the total sum of the right-hand side; this curve is obtained independently of the + symbols. The complete correspondence between the curve and the symbols in Fig. 8 proves the correctness of the present entropy analysis.

<figure><img src="image_12.png" /><figcaption>Fig. 8. Non-dimensional entropy budget of the optimal periodic cycles as a function of the storage size $X_s$. Horizontal line: Reversible power consumption; the three zones hp, os and +o represent, respectively, the entropy productions $\overline{\dot{P}}_S^{hp}$, $\overline{\dot{P}}_S^{os}$, and $\overline{\dot{P}}_S^{+o}$, all three multiplied by $T_o^+ / \overline{Q}_h$. The sum of these four components (upper curve) perfectly agrees with the average power consumption $\overline{\dot{W}} / \overline{Q}_h$ (symbols +).</figcaption></figure>
The decrease (as $X_s$ increases) in the height of the hp zone and the entropy production $\vec{P}_S^{hp}$ is easily explained: the greater the size of the heat storage, the longer it is possible to operate in Mode 1, the purpose of which is to reduce the power consumption $\dot{W}$ of the heat pump; the decrease in the irreversibility $\vec{P}_S^{hp}$ directly results from the latter decrease, see Eq. (16). The height of the os zone (entropy production $\vec{P}_S^{os}$ occurring in the storage heat exchanger) is obviously zero at the origin $X_s = 0$ (there is no heat exchanger); it increases with $X_s$ because the amount of heat exchanged through it increases. The last zone, $+o$, represents the immaterial entropy production $\vec{P}_S^{+o}$. First, its height is not negligible. Moreover, close to the origin $X_s = 0$, the slope of the upper curve is steeper than that at the top of the hp zone ($\vec{P}_S^{hp}$). In fact, the immaterial entropy production is the component that changes the most with the storage size: the changes in the power consumption could not be explained without taking it into account.

In Appendix C, Table C1 provides a more detailed comparison of cases without heat storage ($X_s = 0$, column 2) and cases with reference heat storage ($X_s = 1$, column 3). The budget resulting from the entropy analysis agrees with power consumption within the margin of rounding. Sixty-five percent (65 %) of the calculated energy saving thanks to the heat storage ($4.9 \times 10^{-3}$) is due to the reduction in immaterial entropy production ($3.2 \times 10^{-3}$); the reduction in physical entropy production ($1.7 \times 10^{-3}$) only accounts for 35 % of the saving.

## 7.4. Entropy budget with strategies S1, S2, and S3

### 7.4.1. Entropy budget in absolute values

Fig. 9 shows the same quantities as in Fig. 8, but for the numerical simulations carried out with the real weather data and the strategies S1 (solid lines), S2 (long dashes) and S3 (short dashes). Compared to Fig. 8, the reversible zone is much smaller (in good agreement with the data in Table 1), while the $+o$ zone is much larger. Both differences have the same origin: the maximum outdoor temperature $T_o^+$ is higher in the real data (289.45 K, 16.3 °C) than in the periodic case (281.35 K, 8.2 °C). This increase in $T_o^+$ has two opposite consequences.

<figure><img src="image_13.png" /><figcaption>Fig. 9. Non-dimensional entropy budget with the strategies S1 (solid lines), S2 (long dashes), and S3 (short dashes); same notations as in Fig. 8.</figcaption></figure>

(i) The reversible power consumption $\vec{W}^{rev} / \vec{Q}_h$ decreases. Since this ratio is linear with $(T_h - T_o^+)$ (see Eq. (13)), and since $T_h$ ($=292.15$ K, $19$ °C) is not much higher than $T_o^+$, increasing $T_o^+$ by 8 K reduces the difference $(T_h - T_o^+)$ by a factor of 4; the same factor applies to the reversible zones from Fig. 8 to Fig. 9. In absolute terms, this zone is reduced by $28 \times 10^{-3}$.

(ii) The immaterial entropy production $\vec{P}_S^{+o}$, which is linear with $(T_o^+ - T_o)$ in the first order (see Eq. (18)), also increases. If $T_o^+$ increases by $\Delta T_o^+$, the non-dimensional average entropy production $T_o^+ \vec{P}_S^{+o} / \vec{Q}_h$, equal to $\vec{Q}_o \cdot (T_o^+ - T_o) / T_o / \vec{Q}_h$, increases by $\Delta T_o^+ \vec{Q}_o / (\vec{Q}_h \cdot \vec{T}_o)$ in the first order. Invoking the First Law balance $\vec{Q}_o = \vec{Q}_h - \vec{W}$ and the average value of $\vec{W} / \vec{Q}_h$ ($\approx 0.12$), this expression becomes: $0.88 \times \Delta T_o^+ / \vec{T}_o$. With $\Delta T_o^+ = +8$K, the increase is of the order of $25 \times 10^{-3}$.

The two consequences largely cancel each other out, but not completely.

As with Fig. 8, the $+$ symbols ($\vec{W} / \vec{Q}_h$) and the upper curves (right-hand side of Eq. (19)) in Fig. 9 are in perfect agreement. Furthermore, as in Fig. 7, the curves for the strategy S1 differ from those for strategies S2 and S3, which are very close to each other. Their slopes at the origin $X_s = 0$ are also similar: quasi-horizontal for the curve S1 and finite (here negative) for S2 and S3. These correspondences all indicate that the entropy analysis developed here is consistent.

### 7.4.2. Changes in entropy productions

The comment above on the effect of the value of $T_o^+$ shows that, as already mentioned in section 7.1.1, the absolute values of the entropy productions are less important than their respective variations. Therefore, attention is now focused on how the entropy productions are reduced with respect to their values without heat storage (at $X_s = 0$), i.e., on the difference ($\vec{P}_S^{(0)} - \vec{P}_S^{(X_s)}$), where the symbol $\bullet$ represents a source of entropy production: hp, os, or $+o$. In Fig. 10, the 'immaterial' entropy production ($\vec{P}_S^{+o}$) is considered separately while the two 'physical' ones ($\vec{P}_S^{hp}$ and $\vec{P}_S^{os}$) are summed up. It can be seen that the immaterial component changes more than the physical one: more than half of the change in power consumption is explained by the changes in the immaterial entropy production alone. Moreover, the physical curve for the strategy S1 (solid line with open circles) shows a paradoxical behaviour: for values of $X_s$ below 1.5, the reduction of $\vec{P}_S^{hp} + \vec{P}_S^{os}$ is negative, meaning that the physical entropy production is greater than without storage. There are two explanations for this unexpected behaviour.

(i) As Fig. 7 shows, operating a small heat storage system using strategy S1 results in only a slight reduction in power consumption $\dot{W}$. As $\vec{P}_S^{hp}$ is almost linear with $\dot{W}$ (see Eq. (16)), the reduction in $\vec{P}_S^{hp}$ is also limited.

(ii) On the other hand, the entropy production $\vec{P}_S^{os}$ occurring in the heat exchanger between the outdoor air and the storage tank increases with $X_s$ (see the expansion of the 'os' zone in Fig. 9). As this increase is greater than the decrease in $\vec{P}_S^{hp}$, in total, the physical entropy production with heat storage is larger than without, until $X_s \approx 1.5$.

Meanwhile, the immaterial entropy production $\vec{P}_S^{+o}$ decreases as the storage size $X_s$ increases (see the top three curves in Fig. 10) and this
<figure><img src="image_14.png" /><figcaption>**Fig. 10.** Reduction of the non-dimensional entropy productions, either physical (hp + os components: the lower three curves, with open circles and grouped by an ellipse) or immaterial (+o component: the upper three curves, grouped by an ellipse), all non-dimensionalised by $T_o^o/Q_h$, as a function of the storage size $X_s$ and for the strategies S1 (solid lines with symbol '1'), S2 (long dashes), and S3 (short dashes).</figcaption></figure>

decrease overcompensates for the total physical increase. If only physical entropy production were considered, analysing configurations with $X_s < 1.5$ would lead to an apparent contradiction: as $X_s$ increases, power consumption decreases, but the physical entropy production increases. This effect is clearly illustrated in Table C1 (Appendix C) for cases in real weather conditions, without heat storage (column 4), and with reference heat storage ($X_s = 1 < 1.5$) and strategy S1 (column 5). Although the calculated power consumption (row 3) decreases by $2.3 \times 10^{-3}$, thanks to the heat storage, this does not correspond to a decrease in physical entropy production. In fact, the physical entropy production increases by such an amount that the power consumption would actually increase by $0.6 \times 10^{-3}$ (row 5). However, this is only an apparent contradiction, because the simultaneous decrease in immaterial entropy production reduces power consumption by $2.9 \times 10^{-3}$ (row 6). If entropy (Second Law) analysis only considered physical entropy production (excluding the immaterial part from the analysis), it would lead to conclusions that contradict energy (First Law) analysis. This development shows the importance of taking immaterial entropy production into account.

### 7.4.3. Comparison between strategies

Contrary to strategy S1, strategies S2 and S3 significantly reduce the physical entropy production $\overline{P_s^{hp}} + \overline{P_s^{os}}$. In fact, these two strategies are designed to reduce the power consumption $\dot{W}$ by operating Mode 1 at times when the outdoor temperature is low, either on average (S2) or almost exactly (S3). Reducing $\dot{W}$ also reduces $\overline{P_s^{hp}}$ (see Eq. (16)), and the latter reduction is significant enough to overcompensate for the increase in $\overline{P_s^{os}}$ mentioned in the previous subsection [item (ii)].

Strategies S2 and S3 generate practically the same physical entropy production, but their immaterial entropy productions $\overline{P_s^{+o}}$ are different. Strategy S3 is the one that reduces $\overline{P_s^{+o}}$ the most, and this feature explains why it is ultimately the most efficient (see Fig. 7).

It can also be noted that, when $X_s$ is greater than 1.25, strategy S1

reduces the immaterial entropy production $\overline{P_s^{+o}}$ more than strategy S2 does. This unexpected result can be explained in the following way. First, the average rate of decrease in the outdoor temperature ($\dot{T}_o$) between 15:00 and 01:00 is of -0.45 K.hr$^{-1}$. Second, with an average heat flux for heating the house of 2 kW, the average heat flux delivered to the heat pump by the heat storage is 1.75 kW. With $X_s = 1.25$ (i.e., $MC_s = 13.5$ MJ.K$^{-1}$), this gives an average rate of decrease in the storage tank temperature ($\dot{T}_s$) of -0.47 K.hr$^{-1}$, i.e., practically the same value as for the outdoor temperature. The performance of strategy S1 depends on whether the storage tank cools down faster ($X_s < 1.25$) or slower ($X_s > 1.25$): in the first case, it is better to wait until the outdoor temperature is close to its minimum; in the second case, there is no reason to wait.

As shown in Table C1, reference heat storage with strategy S3 reduces power consumption by $6.2 \times 10^{-3}$, compared to the case without storage. Sixty percent (60 %) of this saving is due to the reduction in immaterial entropy production ($3.7 \times 10^{-3}$), while only 40 % is due to the reduction in physical entropy production ($2.4 \times 10^{-3}$).

### 7.4.4. Short summary of the analysis of Fig. 10

This analysis can be summarised as follows:

- Physical entropy production alone cannot explain the reduction in power consumption achieved by each strategy, nor the differences between strategies;

- The variations in immaterial entropy production are at least as significant as those in physical entropy production;

- The entropy analysis would not be complete without considering the immaterial entropy production.

## 7.5. A practical application: Strategy S3 is the most efficient

Appendix D compares the difference in power consumption of the HP + HS system depending on the value of the outdoor temperature when Mode 1 is in operation (this value can be more or less cold). Three approaches are considered: the First Law approach, based on Eq. (2); the Second Law approach presented here and based on Eq. (15); and the Second Law approach based on Eq. (11). The mathematical derivations (somewhat tedious) demonstrate that operation in Mode 1 when the outdoor temperature is at its coldest is always the most efficient. Therefore, strategy S3, which is based on this rule, is the most efficient possible. The derivations also show that the results of the analysis based on Eq. (11) are difficult to interpret. In contrast, the current Second Law approach based on Eq. (15) is entirely consistent and aligns well with the three properties outlined in the introduction.

# 8. General discussion: Towards a systemic approach to heat storage

## 8.1. Practical engineering significance

Beyond the appropriate verification of findings presented in Appendix E, and before discussing the thermodynamic consequences of the immaterial entropy production, the above results are considered from a practical engineering point of view.

Fig. 7 shows that, with reasonable heat storage ($X_s = 1$), around 4 % of power consumption can be saved, and up to 10 % with a heat storage system three times larger. Whether the corresponding investment would be worthwhile is far beyond the scope of this article. Nevertheless, it is clear that the greater the average temperature difference between day and night, the greater the saving [3]. This day-night temperature difference depends on the climate: desert climates are favourable, while tropical ones are not. Savings also depend on to the average temperature difference between the outdoor air and the process's other heat source. In domestic applications, the 'other' source is indoor air, and the latter temperature difference is similar to the range of outdoor temperature
fluctuations. Therefore, applications such as heat pumping in winter and air conditioning in summer can benefit greatly from such a heat storage. The same applies to food refrigeration above 0 °C. However, in applications involving much larger temperature differences between heat sources, such as in power generation, the efficiency gains would probably be small. To answer this question definitively, a comprehensive optimisation analysis needs to consider and compare all the possible means of improving energy efficiency. The heat storage examined here is one of such means.

## 8.2. The possibility of heat storage should always be considered

In the thermostatic context, there is no need for heat storage between the processes and their heat sources. Outdoor temperature fluctuations, which are a reality, change the context and such a heat storage system can be used to reduce power consumption. Fig. 7 shows that the magnitude of this positive effect depends on the size $X_s$ of the heat storage system (i.e., on $MC_s$ and $UA_s$). These two physical design parameters are like any other design parameter (e.g. heat exchange area, insulation thickness, or compressor size): they affect performance. Any optimisation procedure should consider their effect and compare it with that of other parameters, even if the system under consideration is initially designed without heat storage. Remembering that non-existent heat storage is the least efficient, the possibility of adding heat storage to a process should always be considered when looking for a more efficient design.

## 8.3. Restoring the thermostatic framework reveals a new kind of irreversibility

The outdoor temperature fluctuations make the outdoor air act as an infinite number of elementary thermostatic heat sources within a temperature range determined by the local climate, but with one limitation: only one of these heat sources is available at any given time. In the absence of heat storage between the process and the outdoor air, the process must passively follow the outdoor temperature fluctuations, without any degree of freedom. The presence of heat storage provides a degree of freedom, as it is possible to choose (at least to some extent) which heat sources to use. The larger the heat storage size $X_s$, the greater the degree of freedom. At the infinite limit of its size, a heat storage allows the process to exchange heat with the outdoor air only when the outdoor temperature is the most favourable. The infinite heat storage is an ideal and theoretical limit (just as the Carnot cycle is an ideal theoretical limit of engine cycles) that reduces power consumption the most. For heat pumping, as here, the most favourable outdoor temperature is the highest. For air conditioning, it would be the lowest. (Actually, both conditions can be described as 'the closest to the indoor temperature'.) With an infinitely large heat storage, the whole range of thermostatic elementary heat sources can be reduced to a single one. This restores the classical thermostatic context and makes the three properties of the Introduction valid again.

Restoring the thermostatic context for the analysis does not stop the outdoor temperature from fluctuating. With a real, finite heat storage, some heat exchange with the fluctuating outdoor air occurs when the outdoor temperature is not at its most favourable level. Compared to the infinite case, this necessarily results in a loss of energy efficiency and, equivalently, an increased entropy production. In this restored thermostatic context, the formal derivation of power consumption (or energy efficiency) reveals a previously unknown entropy production related solely to heat exchanges with the outdoor air when the outdoor temperature differs from the most favourable. This immaterial entropy production, which is not due to irreversible physical phenomena, only exists in processes that exchange heat with a fluctuating outdoor source.

## 8.4. The strategy as an action to reduce immaterial entropy production

Fig. 7 shows that the energy conversion efficiency depends on the way in which the heat storage is used, that is called here the strategy. The strategy is not a physical design parameter, but it affects energy efficiency, and also entropy production. Beyond its physical support (the electronic controller that applies it, and the heat storage system itself, especially its size), a strategy consists, at least initially, of a simple rule: "Use the storage as soon as possible" for strategy S1, "Use the storage around the next dawn" for strategy S2, and "Use the storage at the minimum of the upcoming outdoor temperature" for strategy S3. These rules eventually become algorithms. Rules and algorithms are not physical; they are immaterial. The strategy is an immaterial design parameter that has physical consequences for power consumption and entropy production, particularly immaterial entropy production.

Figs. 8 and 9 show that the physical part of entropy production, $\dot{P}_S^{hp} + \dot{P}_S^{os}$, is affected by the strategy. However, its changes alone do not explain the reduction in average power consumption, see Fig. 10. This reduction can only be explained if the immaterial entropy production $\dot{P}_S^{+o} = \dot{Q}_o \cdot (1/T_o - 1/T_o^*)$ is included in the analysis. This immaterial entropy production vanishes when the outdoor air temperature is constant ($T_o(t) \equiv \text{Constant} = T_o^+$): it cannot appear in the usual thermostatic framework, where it is unknown. Its average $\dot{P}_S^{+o}$ depends on the phasing between the outdoor heat flux $\dot{Q}_o$ and the outdoor temperature $T_o$ (both time-dependent). It therefore represents how the process uses the range of elementary thermostatic heat sources offered by the outdoor temperature fluctuations: if part of the heat flux received from the outdoor air, $\dot{Q}_o$, is shifted from periods of low outdoor temperature $T_o$ to periods of higher outdoor temperature, $\dot{P}_S^{+o}$ is reduced. The heat storage enables this shift to be achieved, and the larger the size $X_s$ of the heat storage, the more significant the shift can be. By choosing the times when the system operates in Mode 1, a strategy removes elementary outdoor sources at low temperatures from the entropy production $\dot{P}_S^{+o}$ and replaces them with those used to charge the storage tank (system in Mode 2), whose temperatures are obviously higher than the previous ones. The progression from strategy S1 to strategy S3 amplifies this shift, which reduces immaterial entropy production and, consequently, power consumption.

## 8.5. A systemic interpretation of the role of the heat storage

To illustrate this effect on a graph, the concept of entropic mean temperature is applied to the heat fluxes received from the outdoor air. The entropic mean outdoor temperature is the temperature of a thermostatic heat source that, when transferring the average heat flux $\dot{Q}_o$, would also transfer the average flux of entropy actually transferred by the outdoor air: $\dot{Q}_o/T_o$ [31–33]:

$$ \tilde{T}_o = \overline{\dot{Q}_o} / \left( \overline{\dot{Q}_o} / T_o \right) \qquad (20) $$

where $\dot{Q}_o(t) = 0$ during operation in Mode 1 and temperatures $T$ are in Kelvin.

Zühlsdorf et al. also apply the notion of entropic mean temperature to outdoor air [34], but for a heat pump in a steady-state in the thermostatic context. They focus on the temperature profiles of the air flows when exchanging heat with the heat pump. Each entropic mean temperature (indoor or outdoor) is therefore the spatial ‘entropic mean’ of the air temperature between the inlet and outlet of each heat exchanger. In the current non-thermostatic context, however, the heat pump does not remain in a steady-state, and equation (20) relates to the fluctuating outdoor heat source itself, calculating the temporal ‘entropic mean’ of its temperature.
Fig. 11 shows how $\tilde{T}_o$ (converted to Celsius) depends on $X_s$ and on strategy. As the important quantity is $(T_o^+ - \tilde{T}_o)$ with $T_o^+ = 16.3$ °C, a variation of $\tilde{T}_o$ by 1 K is rather significant. Since one has: $\dot{P}_S^{\cdot o} = \overline{\dot{Q}}_o \cdot (1/\tilde{T}_o - 1/T_o^+)$, the three curves are almost the same as the *immaterial* ones in Fig. 10. And strategy S3 really appears as the one that shifts $\overline{\dot{Q}}_o$ the most toward high temperatures. This temperature shift is the physical change by which the strategy, an *immaterial* parameter, alters the *immaterial* component of the entropy production, and finally helps to reduce power consumption.

The above analysis broadens the perspective on the role of the heat storage. When considering energy alone, the heat storage is described as a substitute for the outdoor air, helping to reduce power consumption. This 'First Law' description only considers operation in Mode 1, regarding the other periods. The Second Law analysis developed above does not contradict this description, but complements it by giving the charge phase (operation in Mode 2) an equally important role as that of Mode 1. Globally, the role of the heat storage is to shift as much of the heat exchange with the outdoor air as possible from periods when the outdoor temperature is the least favourable for the process, towards periods when it is more favourable, potentially the most favourable. The latter description takes into account the entire operation of the HP + HS system, including the periods when the storage tank is isolated (Mode 0); they are important for operating Mode 1 at the most efficient times. In this systemic interpretation, the strategy plays its full role.

Finally, it should be noted that the notion of immaterial entropy production and the latter interpretation of the role of the heat storage do not depend on the simplifying assumptions employed in modelling the house and the HP + HS system. More detailed models would produce different numerical results, but would still lead to the same qualitative conclusion.

## 9. Perspectives

Firstly, the numerical models of the heating demand, heat pump and storage tank can obviously be improved and refined by including (i) the

house's thermal inertia, (ii) temperature gradients in the storage tank or the use of phase change materials, and (iii) the on-off regulation of the heat pump, possibly coupled with daily variations in the indoor temperature setpoint (e.g. according to the presence or absence of inhabitants).

Secondly, this approach can easily be applied to air conditioning in summer. The issue is symmetrical. In this case, the most favourable outdoor temperature is the lowest (typically at night) and the heat storage tank will preferably be used in the afternoon. This shifts the release of heat to the outdoor air from periods of high temperature to cooler periods. In the immaterial entropy production, this released heat flux is now multiplied by the difference between the current outdoor temperature and the minimum temperature over the period under consideration.

Thirdly, this approach can also be applied to seasonal heat storage, e. g. in the underground. There is a fundamental thermodynamic difference between the transfer of heat between seasons and the issue studied above. During the heat pumping period in winter, the range of outdoor temperature fluctuations does not encompass the indoor air temperature $T_h$. The same applies to air conditioning in summer. Conversely, when considering the whole year, the range of outdoor temperatures includes the indoor temperature. Therefore, an infinitely large heat storage system could remain at the same temperature $T_h$ all year round, providing the house with both heating in winter and cooling in summer without consuming any power, except for that necessarily consumed by auxiliaries such as circulators and fans. In such a context, the indoor temperature $T_h$ can used for defining the exergy state function, as proposed by Bonetti and Kokogiannakis [2]. This infinitely large system would exchange heat reversibly with either the indoor air (both are at $T_h$) or the outdoor air (when the latter is exactly at $T_h$). This would eliminate the need for a heat pump in winter and an air conditioner in summer. When considering a reversible heat pump that provides an elementary heat quantity $\delta q$ to the house at a time when the outdoor temperature is $T_o$, it simultaneously extracts $\delta q \cdot T_o / T_h$ from the outdoor air and consumes $\delta q \cdot (T_h - T_o) / T_h$ as power. The latter expression is precisely the same as that of the additional power consumption due to the *immaterial* entropy production $\delta p_S^{\cdot o}$ caused by extracting $\delta q \cdot T_o / T_h$ from the outdoor air at $T_o$, see Eqs. (15) and (18) where the reference heat storage temperature $T_o^+$ equates to $T_h$. The same short derivation applied to a reversible air conditioner in summer yields the same conclusion. Consequently, the three processes (heat pumping in winter, air conditioning in summer and possibly a part of seasonal heat storage in between) can be analysed, possibly together, within the proposed framework. In this framework, even the reversible power consumption of the heat pump or air conditioner appears avoidable, because it is related to an immaterial entropy production that could be eliminated by using an infinitely large heat storage system. With infinite storage, the reference power consumption is zero, meaning an infinite coefficient of performance COP. A Second Law analysis based on efficiency (i.e. real efficiency compared to Carnot efficiency) cannot handle an infinite reference for COP. However, since the present analysis framework is based on power consumption rather than COP, it can easily use zero power consumption as a reference and compare it to real-world power consumption via all causes of entropy production, the physical ones and the immaterial. As well as being robust, the analysis is flexible enough to account for the fact that comfort is achieved within a range of indoor temperatures rather than at a single specific temperature. For example, people generally feel comfortable between 19 and 26 °C. Exchanging heat with outdoor air within this temperature range can contribute to comfort without consuming any power. A good example of this is nocturnal ventilation in summer [3–5]. Within the present entropy analysis framework, it can be easily stated that these heat exchanges do not cause any immaterial entropy production. Only releasing heat towards outdoor air above 26 °C, and extracting heat from outdoor air below 19 °C would then cause immaterial entropy production. This extension of the analysis

<figure><img src="image_15.png" /><figcaption>Fig. 11. Mean entropic outdoor temperature $\tilde{T}_o$ (converted to °C) as a function of storage size $X_s$ and for strategies S1 (solid line), S2 (long dashes), and S3 (short dashes).</figcaption></figure>
framework is currently unexplored, but it would be much closer to real-
life practice in dwellings than the thermostatic approach. Additionally,
since indoor comfort conditions lie within a temperature range (here,
19-26 °C) rather than at a single temperature, comfort could be ach-
ieved using a very large, yet finite, heat storage system. The thermo-
dynamic features of such systems, which involve energy conversion and
large heat storage systems operating in a varying environment, cannot
be analysed in the thermostatic context. However, they, can be analysed
using the current entropy analysis framework.

10. Conclusion

As the outdoor temperatures fluctuate, adding a heat storage system between an energy conversion process and the outdoor air can help reduce the process’s power consumption. This is used here for a heat pump that heats a house in winter. The amount of power saved depends on the size of the heat storage system and on the strategy used to employ it. Every strategy uses some form of prediction of the upcoming outdoor temperature, which can be more or less accurate.

The results show that power consumption can be reduced by 2–4 % with a reasonably sized heat storage system and ‘no prediction at all’. Using the heat storage system only when the outdoor temperature is at its lowest, thanks to ‘precise prediction of the next 24 h’, achieves a reduction of 5–7 %. However, this perfect strategy is unrealistic. A much more realistic strategy based on the general knowledge that ‘the coldest hour usually occurs around dawn’ is almost as efficient.

The context of fluctuating outdoor temperatures requires a new,
rigorous entropy analysis that is also consistent with the properties of
the familiar thermostatic framework. The mathematical derivation of
the new entropy balance yields an *immaterial* entropy production, which
is unknown in the thermostatic framework. This *immaterial* entropy
production is related to the phasing of heat exchange with outdoor air in
relation to outdoor temperature fluctuations. The strategy employed
largely determines this phasing. It affects power consumption, primarily
through the immaterial entropy production. Thus, the strategy is an
immaterial characteristic of the heat storage system.

Immaterial entropy production can account for up to 30 % of the
total entropy production. In addition, its presence explains how
concomitant increases in energy efficiency and physical entropy pro-
duction can occur. Therefore, without accounting for this immaterial

entropy production, the entropy budget would be incomplete in both
quantitative and qualitative terms.

This analysis provides a systemic interpretation of the role of the heat
storage, which is to shift the heat exchange with the outdoor air as much
as possible towards periods when the outdoor temperature is most
favourable (i.e. when it is closest to indoor comfort temperatures).

This new thermodynamic approach opens wide perspectives,
particularly in the analysis of systems with seasonal heat storage.

Declaration of generative AI and AI-assisted technologies in the
writing process

The authors used the DeepL-Write application for English corrections
during the preparation of this work. After using this tool/service, the
authors reviewed and edited the content as needed and take full re-
sponsibility for the content of the published article.

CRediT authorship contribution statement

**Michel Pons:** Writing – review & editing, Writing – original draft, Supervision, Software, Methodology, Investigation, Formal analysis, Conceptualization. **Lionel Mathelin:** Writing – review & editing, Validation, Software, Investigation.

Declaration of competing interest

The authors declare that they have no known competing financial
interests or personal relationships that could have appeared to influence
the work reported in this paper.

Acknowledgments

This research did not receive any specific grant from funding
agencies in the public, commercial, or not-profit sectors.

The authors gratefully thank Vincent Bourdin (GEEPS, Université Paris-Saclay, Gif-sur-Yvette, France) and SIRTA (Site Instrumental de Recherche par Télédétection Atmosphérique, Palaiseau, France) for having supplied the meteorological data. Without these data, this work could not have been done.

Appendix A. Construction of the average periodic day used in section 3 and Fig. 3

Days are different. On sunny days with clear skies, the magnitude of the fluctuations in outdoor temperature, between its minimum (early morning) and its maximum (early afternoon), is on average of the order of 5–6 Kelvin. This is significant compared to the average temperature difference between the indoor and outdoor air (about 15 K), and the presence of a heat storage can then improve the energy performance of the heat pump. However, on very cloudy or rainy days, the outdoor temperature fluctuates little (if at all) and the presence of the heat storage has little or no effect. Simply averaging the meteorological data would take into account too many days where the presence of a heat storage plays no role. The data is then weighted in order to construct an average periodic day on which the storage tank could have a performance close to its average performance on real days. As the temperature difference ΔT between the warmest and coldest time on the same day scales the effect of the heat storage on performance, the days with a large ΔT should weigh more than those with a more stable outdoor temperature. The procedure for averaging the meteorological data is as follows. Firstly, only the days with a ΔT greater than 2 K are kept. Second, for the retained days, the outdoor temperature is weighted by the ΔT of the day, but the global horizontal insolation is not. And third, the heating demand dot Qh(t) is recalculated by applying the formula (1) to the modified outdoor temperature and the measured insolation. The very close proximity in Fig. 7 of the dashed red curve Per (calculated with this average periodic day) to the solid black curves S2 and S3 (calculated with the actual data and strategies S2 and S3) shows that this averaging method makes sense in this context: the results of Fig. 7 can then lead to relevant rules for the elaboration of strategies.

Appendix B. Calculation of the threshold temperature T<sub>l</sub> for strategy S3

The model first extracts the values of the upcoming outdoor temperature ($T_o$) and heat demand ($\dot{Q}_h$) for the next 24 h from the dataset, as if they were perfectly predicted. This extraction is done when the computation time is 13:00 (Mode 2 very often ends around this time). Then, and in order to operate in Mode 1 when the outdoor temperature is effectively at its lowest for the next 24 h, a threshold value of the outdoor temperature, $T_b$, to be applied for the next 24 h is determined as follows.

First, from the two extracted dataset $T_o(t)$ and $\dot{Q}_h(t)$, a cumulative heat demand function $Q_d(T_t)$ is constructed according to the following integral
over the next 24 h:

$$Q_d(T_l) = \int H(T_l - T_o(t)) \cdot \dot{Q}_h(t) \cdot dt \qquad (B.1)$$

where the Heaviside function $H(T_l - T_o(t))$ is 1 when $T_o(t) < T_l$ and 0 otherwise. $Q_d(T_l)$ is the amount of heat required to heat the house during periods (within the next 24 h) when the outdoor temperature is lower than $T_l$. There may be several periods, separated by local maxima of $T_o$. Fig. B1 shows the $Q_d(T_l)$ curves for the three ‘days’ numbered at the top of Fig. 2.

<figure><img src="image_16.png" /><figcaption>Fig. B1. Cumulative amount of heat $Q_d$ to be delivered to the house when the outdoor temperature is below $T_l$ [cf. Eq. (B.1)].</figcaption></figure>

Then, at the end of Mode 2 (points c in Fig. 3: the tank temperature is $T_s^c$), the controller must determine a temperature $T_l$ such that, until the next Mode 2 begins, the system will be in Mode 1 if the outdoor temperature is below $T_b$, and in Mode 0 if it is not. Ideally, the next Mode 2 will begin when Mode 0 and Mode 1 are equally efficient, i.e., when $T_s(t) = T_s^c(t)$. This relation is rewritten as $T_s = T_s^c(T_l)$. The unknown temperature $T_l$ is the one that equates two amounts of heat. The first, $Q_D$, is the heat to be supplied to the house while the outdoor temperature is below $T_l$ for the next 24 h; this is given by the function $Q_d$ defined above: $Q_D = Q_d(T_l)$. The second one, $Q_{HP}$, is the energy that the heat pump in Mode 1 can supply to the house when extracting heat from the storage tank, whose temperature thus decreases from $T_s^c$ to $T_s = T_s^c(T_l)$. $Q_{HP}$ is determined by the amount of heat $MC_s \cdot (T_s^c - T_s^t(T_l))$ received from the heat storage tank when it cools from $T_s^c$ to the threshold temperature $T_s^t(T_l)$. The first two equations of Eq. (5), with an average tank temperature during operation in Mode 1 approximated by $(T_s^c + T_s^t(T_l))/2$, then lead to the expression of $Q_{HP}$:

$$Q_{HP} \approx \frac{\eta_s \cdot T_h}{(\eta_s - 1) \cdot T_h + (T_s^c + T_s^t(T_l))/2} \cdot MC_s \cdot (T_s^c - T_s^t(T_l)) \qquad (B.2)$$

When $T_l$ is low, $Q_D$ is small, possibly zero, and $Q_{HP}$ is large. When $T_l$ is high, the former is large and the latter is small, possibly zero. Equality occurs for a value of $T_l$.

## Appendix C. Quantitative comparison of power consumptions and entropy productions

Table C1 provides a quantitative comparison of the calculated power consumptions and entropy productions calculated for various cases. The reference heat storage ($X_s = 1$) is compared to a system without heat storage ($X_s = 0$), both operated either on the average periodic day or under the measured weather conditions. In the latter case, the results compare the three strategies.

The reported quantities are those presented in Figs. 8 and 9. The ‘calculated power consumption’ (row #3) is $\overline{W} / \overline{Q}_h$, where $\overline{W}$ is directly calculated in the numerical simulation. The ‘reversible’ power consumption (row #4) is $\overline{W}^{rev} / \overline{Q}_h$, where $\overline{W}^{rev}$ is given by Eq. (13). The term resulting from the ‘physical’ entropy production (row #5) is $(\overline{P}_S^{hp} + \overline{P}_S^{os}) \cdot T_o^+ / \overline{Q}_h$, see Eqs. (16) and (17). The term resulting from the ‘immaterial’ entropy production (row #6) is $\overline{P}_S^{+o} \cdot T_o^+ / \overline{Q}_h$, see Eq. (18). The ‘budget’ (row #7) is the sum of the latter three terms according to the right hand side of Eq. (19).
Table C1

Quantitative comparison of power consumptions and entropy productions between $X_s = 0$ (no heat storage) and $X_s = 1$ (reference heat storage) either on the average periodic day (columns #2 and #3) or with the real weather data and strategies S1 to S3 (columns #4 to #7). All numerical values refer to the unit of heat demand and are multiplied by 1000.

<table><thead><tr><th rowspan="2">Operation:</th><th colspan="2">0</th><th colspan="1">1</th><th colspan="1">0</th><th colspan="3">1</th></tr><tr><th colspan="2">Average periodic day</th><th colspan="1">No storage</th><th colspan="1">Strat. S1</th><th colspan="1">Strat. S2</th><th colspan="1">Strat. S3</th></tr></thead><tbody><tr><td>Calculated power consumption:</td><td>124.9</td><td>120.0</td><td>129.8</td><td>127.5</td><td>124.5</td><td>123.6</td></tr><tr><td>Reversible:</td><td></td><td>36.9</td><td></td><td></td><td>9.2</td><td></td></tr><tr><td>Physical:</td><td>76.1</td><td>74.4</td><td>81.7</td><td>82.3</td><td>79.4</td><td>79.3</td></tr><tr><td>Immaterial:</td><td>11.9</td><td>8.7</td><td>38.8</td><td>35.9</td><td>35.8</td><td>35.1</td></tr><tr><td>Budget</td><td>124.9</td><td>120.0</td><td>129.7</td><td>127.4</td><td>124.4</td><td>123.6</td></tr></tbody></table>

## Appendix D. A practical application: Proof that strategy S3 is the most efficient

This appendix deals with the question of the optimal strategy. The approach is to consider an elementary choice between two equivalent options in terms of heat demand, and formally derive their difference in power consumption. The elementary choice is between (i) operating in mode 1 at one of two different times and in mode 0 at the other time, and (ii) vice-versa. Consider an elementary amount of heat demand, $\delta q_h$, which is delivered twice by the heat pump, at $t_1$ and at $t_2$, with the respective outdoor temperatures $T_{o1}$ and $T_{o2}$. At one of these two times, the heat pump is operated in Mode 1: it receives the elementary heat quantity $\delta q_s$ from the storage tank at $T_s$. At the other time, it is operated in Mode 0: it receives the elementary heat quantity $\delta q_o$ from the outdoor air at $T_o$. Between $t_1$ and $t_2$, the process is in Mode 0, so that swapping the two modes would not change the rest of the process operation, except for the power consumption during the two elementary time steps $\delta t_1$ and $\delta t_2$. Thanks to the latter condition, it can be assumed that the storage tank will be charged later, at a time $t_3$ with the outdoor temperature $T_{o3}$ independent of when mode 1 is activated.

### D.1 First Law approach.

Table D1 describes the two configurations A and B (Mode 1 is operated at $t_1$ in configuration A, and at $t_2$ in configuration B); the last row shows the formal expression of the difference in power consumption ($\delta w_A - \delta w_B$) / $\delta q_h$, as derived from Eq. (2). This expression is obtained by subtracting the two cells in row 3. The terms that are not cancelled out by another term in this subtraction are highlighted in **bold**. Table D1 shows that configuration A is more efficient ($\delta w_A < \delta w_B$) if and only if $T_{o1} < T_{o2}$. So, when considering two times $t_1$ and $t_2$ with $T_{o1} < T_{o2}$, and a storage tank temperature warm enough for delivering the required amount of heat, then it is always more efficient to operate Mode 1 at $t_1$ rather than at $t_2$. By extension, it is optimal to operate Mode 1 at times when the outdoor temperature is at its minimum. Strategy S3 is based on this rule.

### D.2 Second Law approach.

Due to the equivalence between lower power consumption and lower entropy production, see Eq. (10), the formal relationship between ($\delta w_A - \delta w_B$) and ($T_{o1} - T_{o2}$) obtained above should also result from a Second Law analysis. Table D2 solves the same problem as Table D1, but using the entropy analysis based on the maximum outdoor temperature $T_o^+$ proposed here, see Eq. (15). The elementary heat demand $\delta q_h$ corresponds to elementary entropy productions $\delta p_s$, either physical (superscripts hp and os) or immaterial (superscript +o). The expressions for the internal entropy production, $\delta p_{s\bullet}^{hp}$, shown in row 3, are derived from Eq. (16), where the time derivatives are replaced by elementary amounts. The second part of physical entropy production, $\delta p_{s\bullet}^{os}$, shown in the first line of row 4, concerns the elementary amount of heat $\delta q_s$ that the storage tank receives from the outdoor air at time $t_3$; $\delta q_s$ is such that:

$$ \delta q_s / \delta q_h = (T_s / T_h - (1 - \eta_s)) / \eta_s \qquad (D.1) $$

The immaterial entropy production $\delta p_{s\bullet}^{+o}$ is divided into two components. The first one (index s and temperature drop from $T_o^+$ to $T_{o3}$) is related to $\delta q_s$ and can easily be added to $\delta p_{s\bullet}^{os}$ (see the second line of row 4). The second (index o, as shown in row 5) is related to the elementary heat quantity transferred from the outdoor air to the heat pump in Mode 0, $\delta q_o$; the expression of $\delta q_o$ has the same pattern as Eq. (D.1), except that the index s is replaced by o, o2 or o1 depending on the time when the operation in Mode 0 takes place. The temperature drop is from $T_o^+$ to $T_{o2}$ or $T_{o1}$. All these elementary entropy productions are summed (rows 3–5) and the sum is multiplied by $T_o^+$ according to the right hand side of Eq. (15): the result in row 6 is exactly the expression obtained in Table D1. In this summation, many terms cancel each other out, except for two. It is worth noting that these two non-cancelling terms, highlighted in bold characters, only appear in the immaterial entropy productions. This immaterial entropy production is formally essential for the consistency between the First Law and Second Law analyses. Addressing the same question in the framework provided by Eq. (11) (entropy analysis based on the current outdoor temperature $T_o(t)$) leads to Table D3. The individual entropy productions (superscripts hp and os alone) are now multiplied by the outdoor temperature at the time when they occur: $T_{o1}$, $T_{o2}$ or $T_{o3}$ (see rows 3 and 4). However, in Eq. (11), the entropy budget also includes the term $-T_o \cdot \dot{S}_T$, in which the rate of change in total entropy, $\dot{S}_T$, is multiplied by the current outdoor temperature. Here, the change in entropy is the elementary change $\delta S_T = \delta q_s / T_s$, where $\delta q_s$ is the heat received by the storage tank as given by Eq. (D.1) (temperature is assumed to be constant in the other components); $\delta q_s$ and $\delta S_T$ are positive at $t_3$ and negative at $t_1$ or $t_2$. To obtain the same final formal expression in row 6 as in the previous two tables, the term $-T_o \cdot \delta S_T / \delta q_h$ (in row 5) must be accounted for and added to those due to entropy production (in rows 3 and 4), even though it is negative. This term acts as if it were an entropy production, but its sign shows that it is not. Similarly, since the term $-T_o \cdot \dot{S}_T$ in Eq. (11) can be positive or negative, it cannot be considered as entropy production.
**Table D1**

Difference between two power consumptions $\delta w_*$ of configurations A and B with a given elementary heat demand $\delta q_h$ and where Modes 1 and 0 are operated at one of two times, $t_1$ and $t_2$, with different outdoor temperatures, $T_{o1}$ and $T_{o2}$, respectively. The terms that are not cancelled by another term are highlighted in bold.

<table><thead><tr><th>Configuration:</th><th>A</th><th>B</th></tr></thead><tbody><tr><td>Mode 1 ...</td><td>... operated at t<sub>1</sub> (outdoor at T<sub>o1</sub>)</td><td>... operated at t<sub>2</sub> (outdoor at T<sub>o2</sub>)</td></tr><tr><td>and Mode 0 ...</td><td>... operated at t<sub>2</sub> (outdoor at T<sub>o2</sub>)</td><td>... operated at t<sub>1</sub> (outdoor at T<sub>o1</sub>)</td></tr><tr><td>$\delta p_{SA}^{\text{out}}/\delta q_h$</td><td>... operated at t<sub>2</sub> (outdoor at T<sub>o2</sub>)</td><td>... operated at t<sub>1</sub> (outdoor at T<sub>o1</sub>)</td></tr><tr><td>$\delta p_{SA}^{\text{out}}/\delta q_h + (\delta p_{SA}^{\text{out}}/\delta q_h)_s$</td><td>$(1 - \eta_s) \cdot (1/T_s - 1/T_h) / \eta_s + (1 - \eta_o) \cdot (1/T_{o2} - 1/T_h) / \eta_o$</td><td>$(1 - \eta_s) \cdot (1/T_s - 1/T_h) / \eta_s + (1 - \eta_o) \cdot (1/T_{o1} - 1/T_h) / \eta_o$</td></tr><tr><td>$\delta p_{SA}^{\text{out}}/\delta q_h)_o$</td><td>$[(T_s/T_h - (1 - \eta_s)) / \eta_s] \cdot (1/T_s - 1/T_{o3}) + [(T_s/T_h - (1 - \eta_s)) / \eta_s] \cdot (1/T_{o3} - 1/T_o)$</td><td>$[(T_s/T_h - (1 - \eta_s)) / \eta_s] \cdot (1/T_{o3} - 1/T_o)$</td></tr><tr><td>$(\delta w_A - \delta w_B)/\delta q_h = T_o^+ \cdot \sum (\delta p_{SA}^{\text{out}} - \delta p_{SB}^{\text{out}})/\delta q_h = (T_{o1} - T_{o2}) / (\eta_o \cdot T_h)$</td><td>$\frac{T_{o2}/T_h - (1 - \eta_o)}{\eta_o} \cdot \left(\frac{1}{T_{o2}} - \frac{1}{T_o}\right)$</td><td>$\frac{T_{o1}/T_h - (1 - \eta_o)}{\eta_o} \cdot \left(\frac{1}{T_{o1}} - \frac{1}{T_o}\right)$</td></tr></tbody></table>

**Table D2**

Use of the entropy analysis based on Eq. (15) to derive the same difference between power consumptions $\delta w_*$ at prescribed elementary heat demand $\delta q_h$ from the expressions of the elementary entropy productions $\delta p_{SA}^{\text{out}}$ and for the same two configurations as in Table D1. The terms that are not cancelled are highlighted in bold.

<table><thead><tr><th>Configuration:</th><th>A</th><th>B</th></tr></thead><tbody><tr><td>Mode 1 ...</td><td>... operated at t<sub>1</sub> (outdoor at T<sub>o1</sub>)</td><td>... operated at t<sub>2</sub> (outdoor at T<sub>o2</sub>)</td></tr><tr><td>and Mode 0 ...</td><td>... operated at t<sub>2</sub> (outdoor at T<sub>o2</sub>)</td><td>... operated at t<sub>1</sub> (outdoor at T<sub>o1</sub>)</td></tr><tr><td>$\delta p_{SA}^{\text{out}}/\delta q_h$</td><td>$(1 - \eta_s) \cdot (1/T_s - 1/T_h) / \eta_s + (1 - \eta_o) \cdot (1/T_{o2} - 1/T_h) / \eta_o$</td><td>$(1 - \eta_s) \cdot (1/T_s - 1/T_h) / \eta_s + (1 - \eta_o) \cdot (1/T_{o1} - 1/T_h) / \eta_o$</td></tr><tr><td>$\delta p_{SA}^{\text{out}}/\delta q_h + (\delta p_{SA}^{\text{out}}/\delta q_h)_s$</td><td>$[(T_s/T_h - (1 - \eta_s)) / \eta_s] \cdot (1/T_s - 1/T_{o3}) + [(T_s/T_h - (1 - \eta_s)) / \eta_s] \cdot (1/T_{o3} - 1/T_o)$</td><td>$[(T_s/T_h - (1 - \eta_s)) / \eta_s] \cdot (1/T_{o3} - 1/T_o)$</td></tr><tr><td>$\delta p_{SA}^{\text{out}}/\delta q_h)_o$</td><td>$\frac{T_{o2}/T_h - (1 - \eta_o)}{\eta_o} \cdot \left(\frac{1}{T_{o2}} - \frac{1}{T_o}\right)$</td><td>$\frac{T_{o1}/T_h - (1 - \eta_o)}{\eta_o} \cdot \left(\frac{1}{T_{o1}} - \frac{1}{T_o}\right)$</td></tr><tr><td>$(\delta w_A - \delta w_B)/\delta q_h = T_o^+ \cdot \sum (\delta p_{SA}^{\text{out}} - \delta p_{SB}^{\text{out}})/\delta q_h = (T_{o1} - T_{o2}) / (\eta_o \cdot T_h)$</td><td></td><td></td></tr></tbody></table>

**Table D3**

Attempt to use the entropy analysis based on Eq. (11) to derive the same difference between power consumptions $\delta w_*$ at prescribed elementary heat demand $\delta q_h$ of the same two configurations as in Table D1. The terms that are not cancelled are highlighted in bold.

<table><thead><tr><th>Configuration:</th><th>A</th><th>B</th></tr></thead><tbody><tr><td>Mode 1 ...</td><td>... operated at t<sub>1</sub> (outdoor at T<sub>o1</sub>)</td><td>... operated at t<sub>2</sub> (outdoor at T<sub>o2</sub>)</td></tr><tr><td>and Mode 0 ...</td><td>... operated at t<sub>2</sub> (outdoor at T<sub>o2</sub>)</td><td>... operated at t<sub>1</sub> (outdoor at T<sub>o1</sub>)</td></tr><tr><td>$T_o \cdot \delta p_{SA}^{\text{out}}/\delta q_h$</td><td>$T_{o1} \cdot [(1 - \eta_o) \cdot (1/T_s - 1/T_h) / \eta_s] + T_{o2} \cdot [(1 - \eta_o) \cdot (1/T_{o2} - 1/T_h) / \eta_o]$</td><td>$T_{o2} \cdot [(1 - \eta_s) \cdot (1/T_s - 1/T_h) / \eta_s] + T_{o1} \cdot [(1 - \eta_o) \cdot (1/T_{o1} - 1/T_h) / \eta_o]$</td></tr><tr><td>$T_o \cdot \delta p_{SA}^{\text{out}}/\delta q_h$</td><td>$[(T_s/T_h - (1 - \eta_s)) / \eta_s] \cdot (T_{o3}/T_s - 1)$</td><td>$[(T_s/T_h - (1 - \eta_s)) / \eta_s] \cdot (T_{o3}/T_s - 1)$</td></tr><tr><td>$-T_o \cdot \delta S_{TA}/\delta q_h$</td><td>$\frac{T_s/T_h - (1 - \eta_s)}{\eta_s \cdot T_s} \cdot (T_{o3} - T_{o1})$</td><td>$\frac{T_s/T_h - (1 - \eta_s)}{\eta_s \cdot T_s} \cdot (T_{o3} - T_{o2})$</td></tr><tr><td>$(\delta w_A - \delta w_B)/\delta q_h = \frac{\sum (T_o \cdot \delta p_{SA}^{\text{out}} - T_o \cdot \delta p_{SB}^{\text{out}}) + (T_o \cdot \delta S_{TA} - T_o \cdot \delta S_{TB})}{\delta q_h} = \frac{T_{o1} - T_{o2}}{\eta_o \cdot T_h}$</td><td></td><td></td></tr></tbody></table>

**Appendix E. Appropriate verification of findings**

To the best of the authors' knowledge, the impact of installing a heat storage system between the heat pump and the outdoor air and controlling it appropriately on the performance of the system operating under real weather conditions has not been investigated before. There is no other data, either experimental or numerical, with which the present results can be compared. Only the results of the Second Law analysis can be verified. The Second Law is known to result in an inequality: entropy production cannot be negative. However, when applied to energy conversion processes, the Second Law results in an equality, which can be expressed as equations (15) or (18) [other forms of these equations relate the real efficiency (or COP) to reversible efficiency (or COP) and entropy production]. In the literature, it is not often verified that this equality is actually satisfied by the numerical data. Here, this verification can be found in various places: in Figs. 8 and 9, where there is excellent agreement between the '+' symbols and the upper solid line; in Table C1, where the quantitative data in the third and final rows agree within rounding margin; and in Tables D1 and D2, where the final rows of each table show the same formal expression.

**Data availability**

The authors do not have permission to share data.

**References**

[1] Ramponi R, Angelotti A, Blocken B. Energy saving potential of night ventilation: sensitivity to pressure coefficients for different European climates. Appl Energy 2014;123:185-95. https://doi.org/10.1016/j.apenergy.2014.02.041.

[2] Bonetti V, Kokogiannakis G. Dynamic exergy analysis for the thermal storage optimization of the building envelope. Energies 2017;10:1. https://doi.org/10.3390/en10010095.

[3] Solgi E, Hamedani Z, Fernando R, Skates H, Orji NE. A literature review of night ventilation strategies in buildings. Energ Buildings 2018;173:337-52. https://doi.org/10.1016/j.enbuild.2018.05.052.

[4] Piselli C, Prabhakar M, de Gracia A, Saffari M, Pisello AL, Cabeza LF. Optimal control of natural ventilation as passive cooling strategy for improving the energy performance of building envelope with PCM integration. Renew Energy 2020;162: 171-81. https://doi.org/10.1016/j.renene.2020.07.043.

[5] Alizadeh M, Sadrameli SM. Development of free cooling based ventilation technology for buildings: thermal energy storage (TES) unit, performance enhancement techniques and design considerations - a review. Renew Sust Energ Rev 2016;58:619-45. https://doi.org/10.1016/j.rser.2015.12.168.

[6] Faraj K, Khaled M, Faraj J, Hachem F, Castelain C. Phase change material thermal energy storage systems for cooling applications in buildings: a review. Renew Sust Energ Rev 2020;119:18. https://doi.org/10.1016/j.rser.2019.109579.
[7] Hirshey J, Li ZN, Gluesenkamp KR, LaClair TJ, Graham S. Demand reduction and energy saving potential of thermal energy storage integrated heat pumps. Int J Refrig 2023;148:179-92. https://doi.org/10.1016/j.ijrefrig.2023.01.026.

[8] Ma Y, Tao Y, Deng D, Wang Y, Tu JY. Experimental and numerical investigation on the charging and discharging process of a cold energy storage for space cooling of buildings. Renew Energy 2024;222:12. https://doi.org/10.1016/j. renene.2024.119974.

[9] Compingt A, Blanc P, Quidort A. Slurry for refrigeration industrial kitchen application. In: Kauffeld M, editor. 8th IIR Conference on Phase Change Materials and Slurries for Refrigeration and Air Conditioning. Karlsruhe, Germany: Int. Inst. Refrigeration IIR-IIF; 2009. p. 135-44.

[10] Herold KE, Radermacher R, Klein SA. Absorption chillers and heat pumps. Bocca Raton, Florida: CRC Press; 1996.

[11] Helm M, Keil C, Hiebler S, Mehling H, Schweigler C. Solar heating and cooling system with absorption chiller and low temperature latent heat storage: energetic performance and operational experience. Int J Refrig 2009;32(4):596-606. https:// doi.org/10.1016/j.ijrefrig.2009.02.010.

[12] Boubakri A. A new conception of an adsorptive solar-powered ice maker. Renew Energy 2003;28(5):831-42.

[13] Clausius R. Théorie Mécanique de la Chaleur. 1868 ed. Paris (France): Soc. Ingén. Civils; 1868.

[14] Jou D., Casas-Vazquez J., Lebon G. Extended Irreversible Thermodynamics. Berlin, Germany: Springer Verlag; 1996. 3-540-60789-7.

[15] Bejan A. Entropy generation through heat and fluid flow. New-York: John Wiley & Sons, Inc. (Wiley-Interscience Publication); 1982. 0-471-09438-2.

[16] Bejan A. Advanced engineering thermodynamics. New York: Wiley; 1988. 0-471- 83043-7.

[17] Bejan A, Tsatsaronis G, Moran MJ. Thermal Design and Optimization. Hoboken. USA: Wiley-Interscience; 1995. p. 978-0471584674.

[18] Bejan A. Entropy generation minimization. Boca Raton, Florida: CRC Press; 1996. 0-8493-9651-4.

[19] Borel L. Thermodynamique et énergétique. Lausanne, Switzerland: Presses Polytechniques et Universitaires Romandes; 2005. 2-88074-545-4.

[20] Dincer I, Rosen MA. Exergy: Energy, Environment and Sustainable Development. 2nd ed. Amsterdam, Netherlands: Elsevier; 2012. p. 978-0080970899.

[21] Szargut J, Morris DR, Steward FR. Exergy analysis of thermal, chemical and metallurgical processes. New-York: Hemisphere Pub 1988:0-89116-574-6.

[22] Reini M, Casisi M. The Gouy-Stodola Theorem and the derivation of exergy revised. Energy 2020;210. https://doi.org/10.1016/j.energy.2020.118486.

[23] Martinaitis V, Streckienė G, Biekša D, Šiupšinskas G, Bielskus J. Functional exergy efficiency of an air heat recovery exchanger under varying environmental temperature. Int J Exergy 2018;25(2):93-116.

[24] Acar C, Dincer I. Energy and exergy analyses of a residential cold thermal energy storage system. Int J Exergy 2016;19(4):441-58. https://doi.org/10.1504/ ijex.2016.075879.

[25] Michaelides EE, Michaelides DN. The effect of ambient temperature fluctuation on the performance of geothermal power plants. Int J Exergy 2011;8(1):86-98. https://doi.org/10.1504/ijex.2011.037216.

[26] Pons M. Exergy analysis and process optimization with variable environment temperature. Energies 2019;12(24):19. https://doi.org/10.3390/en12244655.

[27] Pons M. On the reference state for exergy when ambient temperature fluctuates. Int J Thermodynamics 2009;12(3):113-21.

[28] Ma Y, Luo YL, Xu HX, Du RQ, Wang Y. Review on air and water thermal energy storage of buildings with phase change materials. Exp Comput Multiphase Flow 2021;3(2):77-99. https://doi.org/10.1007/s42757-020-0064-4.

[29] Pavlov M, Migan A, Bourdin V, Pons M, Haeffelin M, Badosa J, et al. Experimental and Numerical Study of the Influence of String Mismatch on the Yield of PV Modules Augmented By Static Planar Reflectors. IEEE 42nd Photovoltaic Specialist Conference. New-Orleans, USA: IEEE; 2015.

[30] Lartigue B, Biewesch L, Marion F, Cochran E, Thellier F. Energy performance certificates in the USA and in France-a case study of multifamily housing. Energ Effi 2022;15(5):21. https://doi.org/10.1007/s12053-022-10036-x.

[31] Neveu P, Castaing-Lasvignottes J. Analyse thermodynamique des transformateurs thermochimiques solide-gaz. In: Nikanpour D, Hosatte S, editors. Ab-Sorption '96, Int Absorption Heat Pump Conf. Montreal, Canada: CANMET; 1996. p. 261-8.

[32] Pons M. Analysis of the adsorption cycles with thermal regeneration based on the entropic mean temperatures. Appl Therm Eng 1997;17(7):615-27.

[33] Meunier F, Neveu P, Castaing-Lasvignottes J. Equivalent Carnot cycles for sorption refrigeration. Int J Refrig 1998;21(6):472-89.

[34] Zühlsdorf B, Jensen JK, Elmegaard B. Heat pump working fluid selection - economic and thermodynamic comparison of criteria and boundary conditions. Int J Refrig 2019;98:500-13.