Giuseppe Sdanghi

Nicolas Vincent

Kevin Mozet

Sébastien Schaefer

Gaël Maranzana

Alain Celzard

Vanessa Fierro
email: vanessa.fierro@univ-lorraine.fr

V Nicolas

Modelling of a hydrogen thermally driven compressor based on cyclic adsorption-desorption on activated carbon

Keywords:

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Hydrogen offers innovative and environment-friendly solutions for the use of energy in the next decades as it can be produced from renewable resources, like biomass and water [START_REF] Hosseini | Hydrogen production from renewable and sustainable energy resources: Promising green energy carrier for clean development[END_REF], and, moreover, its use as fuel does not release pollutants. Nevertheless, in order to make hydrogen a real alternative to fossil fuels, safe and cost-effective transportation and storage solutions need to be found. It has been shown that the cheapest hydrogen storage-recovery mode is obtained by compression and truck delivery, especially for low demands [START_REF] Yang | Determining the lowest-cost hydrogen delivery mode[END_REF]. Moreover, pressurised composite tanks are commercially available nowadays, and ensure storing hydrogen up to 70 MPa, which is the level required by on-board storage systems used in fuel cell-powered vehicles and by hydrogen refuelling stations [START_REF] Kikukawa | Risk assessment of Hydrogen fueling stations for 70 MPa FCVs[END_REF].

Compression is the most widely used method to store hydrogen at present. Traditional hydrogen mechanical compressors, like those based on pistons or diaphragms, rely on mature technologies and are particularly used when moderate flows and high pressures are demanded. These systems allow producing compressed hydrogen, up to 100 MPa, with a flow rate of about 10 -2 kg s -1 [START_REF]High-Pressure Hydrogen Compressors (100MPa-class hydrogen compressors for hydrogen refueling stations)[END_REF]. However, these technologies are affected by several drawbacks: (i) they consist of several mechanical components in movement, with consequent problems related to vibrations and noise; (ii) most of hydrogen mechanical compressors use oil for lubricating the moving parts, increasing the risk of hydrogen pollution; (iii) maintenance of mechanical compressors is very difficult; (iv) effective heat exchange during operation is challenging [START_REF] Tuhovcak | Comparison of heat transfer models for reciprocating compressor[END_REF]; and (v) hydrogen can cause metal embrittlement [START_REF] Dwivedi | Design Considerations for High-Pressure Reciprocating Compressors for Refinery Services[END_REF]. Ionic liquid compressors partly solve the aforementioned drawbacks, but are also considered as mechanical compressors. An ionic liquid compressor is able to compress hydrogen from atmospheric pressure to 90 MPa in only five steps, with flows rate of about 10 -2 kg s -1 , similarly to piston compressors [START_REF] Linde | The Ionic Compressor 90 MPa -IC90[END_REF]. In addition, this technology allows reaching very high efficiency, around 70%, along with reduced size and weight, making them the most suitable out of all mechanical compressors.

Attention is now focused on non-mechanical compressors, e.g. metal hydride [START_REF] Lototskyy | Metal hydride hydrogen compressors: A review[END_REF] and electrochemical compressors [START_REF] Grigoriev | Description and characterization of an electrochemical hydrogen compressor/concentrator based on solid polymer electrolyte technology[END_REF]. Hydrogen compression up to 87.5 MPa was achieved by using a metal hydride compressor [START_REF] Corgnale | Techno-Economic Analysis of High-Pressure Metal Hydride Compression Systems[END_REF], along with a significant reduction of dimensions, cost and weight with respect to a mechanical compressor reaching the same pressure level [START_REF] Stamatakis | Metal hydride hydrogen compressors: Current developments & early markets[END_REF]. In the same way, electrochemical compressors ensure reaching 100 MPa in a compact and silent system [START_REF]HyET Hydrogen[END_REF], with overall efficiency higher than 70% [START_REF] Suermann | Electrochemical Hydrogen Compression: Efficient Pressurization Concept Derived from an Energetic Evaluation[END_REF]. In addition, both metal hydride and electrochemical compressors have no moving parts, preventing all the drawbacks listed above and related to mechanical compressors. Moreover, both kinds of systems indeed proved to be safe and highly reliable, although they are particularly more relevant when a low hydrogen flow is required, e.g. 10 -3 kg s -1 . A comparison between all hydrogen compression technologies in terms of cost and efficiency has been thoroughly reviewed elsewhere [START_REF] Sdanghi | Review of the current technologies and performances of hydrogen compression for stationary and automotive applications[END_REF].

The adsorption-desorption compressor is a non-mechanical compressor which might be called thermally driven compressor as it is based on heat exchanges in a compression tank filled with high surface area-materials. Such a system relies on the capacity of porous materials to adsorb gases such as hydrogen. Carbonaceous materials seem to be particularly relevant for this purpose, and especially activated carbons (ACs) since they present, on average, the highest hydrogen adsorption capacity out of all common carbon adsorbents like carbon nanotubes, carbon nanofibers, zeolite-and MOFs-derived carbons [START_REF] Sdanghi | Hydrogen Adsorption on Nanotextured Carbon Materials[END_REF]. Such high H 2 storage capacity is due to the high surface area and pore volume exhibited by ACs, which can be as high as 3000 m 2 g -1 and 1.5 cm 3 g -1 , respectively [START_REF] Fierro | Adsorption and compression contributions to hydrogen storage in activated anthracites[END_REF]. Furthermore, nitrogen adsorptiondesorption isotherms obtained for the textural characterisation of the ACs commonly used for hydrogen adsorption are of type I, according to the IUPAC classification [START_REF] Brunauer | The Adsorption Of Gases And Vapors[END_REF]. These isotherms exhibit both a high N 2 uptake at very low relative pressures and an extremely narrow knee, indicating the presence of ultra-micropores, i.e., with a width lower than 0.7 nm [START_REF] Fierro | Experimental evidence of an upper limit for hydrogen storage at 77 K on activated carbons[END_REF]. Hydrogen storage capacity is particularly enhanced in carbon materials having both high microporosity and very narrow micropores. Indeed, only one single layer of adsorbed hydrogen can be formed in such narrow pores, thus a high adsorption potential is achieved deriving from the overlap of the Van der Waals forces between hydrogen molecules and both walls of the slit-shape pores confining them [START_REF] Rzepka | Physisorption of Hydrogen on Microporous Carbon and Carbon Nanotubes[END_REF]. Thus, hydrogen adsorption on ACs is a physical phenomenon, since no chemical bonds between hydrogen molecules and the AC surface are involved. Furthermore, the adsorption energies are of the order of 6-8 kJ mol -1 , i.e., considerably lower than in the case of chemical adsorption, which can be as high as 130 kJ mol -1 [START_REF] Kyriakopoulos | Adsorption of pesticides on porous polymeric adsorbents[END_REF]. The basic advantages of physical adsorption are both the reversible and the fast kinetics of adsorption compared to chemical adsorption. Nevertheless, high hydrogen storage capacity can be only obtained at very low temperature, because of the low adsorption enthalpy. To date, the highest hydrogen uptake was reached by using activated carbons at 77 K, and it was equal to 7 wt. % [START_REF] Wang | High Hydrogen Storage Capacity of Porous Carbons Prepared by Using Activated Carbon[END_REF].

In a thermally driven system based on cyclic adsorption-desorption, hydrogen compression derives from system heating that produces hydrogen desorption, i.e., hydrogen moves from the (denser) adsorbed phase to the (lighter) bulk phase. As a result, the density of the hydrogen bulk phase increases, producing high-pressure hydrogen. Richard et al. [START_REF] Richard | Preliminary evaluation of the performance of an adsorption-based hydrogen storage system[END_REF] obtained preliminary results on a similar hydrogen compression system based on adsorption on AC. These authors introduced hydrogen at room temperature into a tank filled with AC initially at 80 K and 0.14 MPa. The consequent increase of the system temperature made the hydrogen pressure increase, reaching a value of about 5 MPa.

As far as we know, hydrogen compression based on adsorption is rather an emerging research topic nowadays, and hence the open literature still lacks both experimental and numerical studies. However, several authors investigated the behaviour of cryo-adsorptive systems for storing hydrogen as an alternative method to compression. Petitpas et al. [START_REF] Petitpas | A comparative analysis of the cryo-compression and cryo-adsorption hydrogen storage methods[END_REF] thus carried out an in-depth comparative analysis of cryo-adsorption and cryo-compression hydrogen storage methods, concluding that it is not possible to assess which one is better than the other as the choice of a hydrogen storage system strictly depends on the end application.

Ahluwalia and Peng [START_REF] Ahluwalia | Automotive hydrogen storage system using cryo-adsorption on activated carbon[END_REF] developed a numerical model to investigate the potential use of a cryo-adsorptive system for on-board applications. They found that the amount of hydrogen recoverable from a particular sorbent strictly depends on the pressure and temperature swings.

In particular, they concluded that a temperature swing of around 50 K was needed to recover more than 80% of the sorption capacity of a super-activated carbon sorbent at 100 K and 10 MPa when the tank was depressurised to 0.3-0.8 MPa. Moreno-Blanco et al. [START_REF] Moreno-Blanco | The fill density of automotive cryo-compressed hydrogen vessels[END_REF] investigated the performances of hydrogen cryo-adsorptive systems for automotive applications, finding that it is possible to reach a filling density of about 76 kg m -3 for a 70 MPa vessel and to ensure a driving distance of 150 km day -1 . Several authors focused also on the thermal effects in a cryo-adsorptive storage system through computational fluid dynamics (CFD), stating that significant temperature variations take place especially during the charging process [START_REF] Xiao | Heat and mass transfer and fluid flow in cryo-adsorptive hydrogen storage system[END_REF][START_REF] Xiao | Simulation of heat and mass transfer in activated carbon tank for hydrogen storage[END_REF][START_REF] Xiao | CFD model for charge and discharge cycle of adsorptive hydrogen storage on activated carbon[END_REF][START_REF] Xiao | CFD simulation for chargedischarge cycle of cryo-adsorptive hydrogen storage on activated carbon[END_REF][START_REF] Xiao | Adsorption-desorption cycle thermodynamics for adsorptive hydrogen storage system[END_REF][START_REF] Xiao | Charge-discharge cycle thermodynamics for compression hydrogen storage system[END_REF].

Hardy et al. [START_REF] Hardy | Hydrogen storage adsorbent systems acceptability envelope[END_REF] found that a material bulk density of about 700 kg m -3 must be achieved in order to make a cryo-adsorptive hydrogen system reaching both the gravimetric and the volumetric capacities foreseen by the 2025 DOE target [START_REF]DOE Technical Targets for Onboard Hydrogen Storage for Light[END_REF].

The main objective of the present study was thus to investigate an adsorption-based system to compress hydrogen in a non-mechanical way. A mathematical model based on mass and energy balances occurring in the system is presented and the corresponding results are compared to the data available in the open literature. We show that a pressure as high as 70

MPa is achievable by using this innovative hydrogen thermally driven compression system.

Description of the hydrogen adsorption compressor

Figure 1 shows a scheme of a hydrogen thermally driven compression system. In the latter, the hydrogen compression tank must be able to withstand both the high pressures that are generated and the embrittlement induced by the contact of the materials with high-pressure hydrogen. The tank, filled with AC, is initially immersed in a Dewar filled with liquid nitrogen (77 K) in order to enhance hydrogen adsorption. Once thermal equilibrium between the tank and the Dewar is reached, medium-pressure hydrogen, around 4 MPa, is introduced in the tank and hydrogen adsorption onto the AC takes place. The charging process lasts as long as a pressure gradient between the inlet manifold and the tank exists, then the inlet valve is closed once the pressure gradient is zero. At this point, the tank is removed from the Dewar and thus heated up by natural convection; alternatively, an electric heating system can be used to speed up the process. Increasing the tank temperature produces hydrogen desorption, and therefore the density of the hydrogen bulk phase increases and high-pressure hydrogen is obtained. A pressure regulator is placed downline the tank to allow high-pressure hydrogen to flow at the desired pressure. As shown below, the potentially achievable pressure level strictly depends on the final temperature, on the free volume inside the tank, and on the mass of AC.

Figure 1 -Scheme of an adsorption compressor system (from [START_REF] Sdanghi | Review of the current technologies and performances of hydrogen compression for stationary and automotive applications[END_REF] with permission of Elsevier).

Description of the numerical model

In order to investigate the mass and heat transfer phenomena occurring in a hydrogen thermally driven compressor, we developed a zero-dimensional mathematical model simulating:

-the temperature (T) and pressure (P) dynamics inside the tank; -the changes of hydrogen amounts both in the adsorbed phase (n a ) and in the bulk phase (n g ); -the volume of hydrogen in the bulk phase (V g ).

For that purpose, several simplifying assumptions were used: (i) the pressure and the temperature inside the tank are uniform; (ii) the gas inside the tank is a homogeneous fluid;

(iii) the hydrogen adsorption kinetics is fast enough to consider that the hydrogen in the bulk phase is in equilibrium with that adsorbed on the AC.

The numerical model consists of several, coupled, ordinary differential equations (ODEs) in which the aforementioned physical properties are the unknowns of the problem. By solving the system of ODEs, the temporal evolution of each property can be obtained. We introduced in the model the hydrogen equation of state, an expression describing hydrogen adsorption on porous material, and the equations of mass and energy conservation. From each of these expressions, an explicit ODE in the form y'= f (t,y) was obtained, wherein the dependent variable y is one of the physical characteristic of the problem. Thus, the 4 th order Runge-Kutta iterative method (RK4) was used to solve the ODEs system. For each ODE, an initial value of the dependent variable y at time t = 0, y 0 , was fixed. The time t = 0 was considered to be the beginning of either the charging or the discharging stage, depending on the process investigated. In the following, each single ODE was derived and discussed.

Temperature

Beyond the heat exchanges required for cooling down the system and hence for favouring adsorption on the one hand, and then for heating it up for desorbing hydrogen on the other hand, several other thermal effects take place in a thermally driven compressor. Adsorption itself is indeed an exothermic process that cannot be neglected. Hermosilla-Lara et al. [START_REF] Hermosilla-Lara | Hydrogen storage by adsorption on activated carbon: Investigation of the thermal effects during the charging process[END_REF] found that the heat of adsorption can contribute up to 22% for all the thermal effects involved in a cryo-adsorptive hydrogen storage system. On the other hand, significant variations of the system temperature can be produced during the charging process itself, when hydrogen at room temperature is fed to the reservoir at 77 K. All these heat transfer phenomena affect the temperature dynamics inside the tank, and all of them are taken into account in the energy conservation equation:

= - + (1)
where m i [kg] is the mass and U i [J kg -1 ] is the specific internal energy of either hydrogen, or AC, or tank walls, H in and H out [J kg -1 ] are the hydrogen enthalpies related to inlet and outlet flows, respectively, and [J s -1 ] is the sum of the heat of adsorption and the heat exchanged with the external environment. Eq. ( 1) can be further developed by taking into account the following simplifying assumptions: (i) U i can be written as the specific heat capacity at constant volume c v,i multiplied by the temperature T; (ii) the total enthalpy can be written as the hydrogen mass flow towards the tank, either during the charging stage, , or during the discharging stage, , multiplied by the specific heat capacity at constant pressure c p,i and by the temperature T; (iii) the heat transfer with the external environment can be modelled by using a global heat transfer coefficient, h [W K -1 m -2 ]. According to the aforementioned assumptions, Eq. ( 1) can be rewritten as follows:

, + , + , + , ! = ", - ", + Δ $% &' & + ℎ)* -+ (2)
In Eq. ( 2), the subscript g stands for hydrogen in the bulk phase, a stands for hydrogen in the adsorbed phase, AC is for activated carbon and w is for the tank walls. T in [K] is the temperature of the low-pressure hydrogen flow introduced into the compression tank, and is generally equal to the room temperature. ∆H ads [J mol -1 ] is the heat of adsorption, considered as constant in the present numerical model. The variation of the total amount of heat released during adsorption is evaluated as the product of the adsorption heat by the differential variation of hydrogen mass in the adsorbed phase. The heat transfer between the compression tank and its environment is calculated as the global heat transfer coefficient h

[W K -1 m -2 ]
multiplied by the surface area where the heat transfer takes place, S [m 2 ], and by the temperature difference T a -T [K]. T a is the temperature of the external environment, and it is equal to 77 K during the charging stage and to 298 K (or more, depending on the heating system used) during the discharging stage. The temperature dependency of both c p,g and c v,g [J kg -1 K -1 ] was calculated from the data of the National Institute of Standards and Technology (NIST):

", = -2 • 10 012 3 + 10 04 5 -0.0024 8 + 3.0534 + 13332

, = -2 • 10 012 3 + 10 04 5 -0.0024 8 + 3.0534 + 9207.6

After introducing the hydrogen molar mass M [g mol -1 ] and developing and rearranging Eq.

(2), the following ODE with the temperature as dependent variable was obtained:

= >' ", ->' ", + *Δ $% -> + &' & -> &' & + ℎ)* -+ > , ' + > , ' + , + , (5)
We further assumed that c v,g ~ c v,a . In all the simulations, the temperature T 0 was fixed at 77 K during the charging stage and when starting the discharging process.

Amount of hydrogen in the adsorbed phase

Dubinin's model [START_REF] Dubinin | Adsorption in micropores[END_REF] was proved to describe quite well gas adsorption on microporous materials, such as ACs, by volume filling of the available adsorption space and not by gradual formation of adsorption layers on the surface of the micropores. This approach leads to the conclusion that adsorption on microporous materials is similar to the subcritical adsorption of vapours. However, in the experimental conditions used herein, hydrogen remains in the supercritical state. In order to deal with adsorption of supercritical fluids in micropores, Dubinin introduced a hypothetical saturated vapour pressure and proposed an empirical equation for it, assuming that the adsorption mechanism of supercritical fluids is the same pore filling mechanism as for subcritical ones [START_REF] Dubinin | Fundamentals of the theory of adsorption in micropores of carbon adsorbents: Characteristics of their adsorption properties and microporous structures[END_REF]. In fact, the interpretation of such approach has been one of the shortcomings of Dubinin's model for supercritical fluids [START_REF] Amankwah | A modified approach for estimating pseudo-vapor pressures in the application of the Dubinin-Astakhov equation[END_REF]. To support the concept of a quasi-saturated vapour pressure, Do and Do [START_REF] Do | Adsorption of supercritical fluids in non-porous and porous carbons: analysis of adsorbed phase volume and density[END_REF] argued that supercritical molecules may form clusters while confined in micropores, and these clusters could be large enough to exert a quasi-saturated vapour pressure in the same way as subcritical fluids exhibit their vapour pressure. Several versions and improvements of Dubinin's model were proposed in order to give a physical interpretation of all the equation parameters. The modified Dubinin-Astakhov equation proposed by Richard et al. [START_REF] Richard | Gas adsorption process in activated carbon over a wide temperature range above the critical point. Part 1: modified Dubinin-Astakhov model[END_REF] proved to be a good analytical tool for representing experimental adsorption data of different gases over a wide range of pressures and temperatures in supercritical conditions. In this model, the quasi-saturation pressure was considered as a fitting parameter, along with the maximum amount of gas adsorbed and the characteristic energy. Hence the modified Dubinin-Astakhov model was used in the present work for describing hydrogen adsorption on AC in supercritical conditions:

? = ? @ A B?C D-E F G + H I 8 J' 8 E K % K IL (6)
Eq. ( 6) describes the evolution of the number of moles of adsorbed hydrogen as a function of pressure and temperature, higher amounts of hydrogen being adsorbed at higher pressures and lower temperatures [START_REF] Bénard | Determination of the Adsorption Isotherms of Hydrogen on Activated Carbons above the Critical Temperature of the Adsorbate over Wide Temperature and Pressure Ranges[END_REF]. x a [mol kg -1 ] is the absolute amount of hydrogen adsorbed onto the AC normalised with respect to the total mass of AC contained in the compression tank, and

x max [mol kg -1 ] is the amount of adsorbed hydrogen corresponding to the saturation of the entire available porous volume. α [J mol -1 ] is an enthalpy factor, β [mol J -1 K -1 ] is an entropy factor, and P s [MPa] is the pseudo-saturation pressure introduced above [START_REF] Do | Adsorption of supercritical fluids in non-porous and porous carbons: analysis of adsorbed phase volume and density[END_REF]. By differentiating Eq. ( 6) with respect to time, the following expression is obtained:

' = % &? &K &K & + % &?
where m s [kg] is the amount of AC contained in the compression tank. The partial derivatives of x a with respect to the two unknowns P and T can be also obtained by differentiating Eq. ( 6):

&? &K = 2? F N-J' ? ? @ A K*G + H + = M 1 (8)
&? & = 2G? J' ? ? @ A *G + H + = M 8 (9)
The ODE having the number of moles of adsorbed hydrogen as dependent variable was thus obtained. The initial condition n a0 was found by evaluating Eq. ( 7) with respect to the compression tank conditions at time t 0 , i.e., n a (t 0 ) = n a (T 0 , P 0 ).

Amount of hydrogen in the bulk phase

The evolution of the amount of hydrogen in the bulk phase can be described by using an appropriate equation of state. In order to take into account the deviation from ideal conditions due to the high pressures reached during compression, the Van der Waals equation was chosen out of all other possible equations of state [START_REF] Nasrifar | Comparative study of eleven equations of state in predicting the thermodynamic properties of hydrogen[END_REF]:

OK + P' 8 Q 8 R Q -' S = ' F (10)
where a and b are the constant parameters of the Van der Waals equation (0.02476 J m 3 mol -2 and 2.661 × 10 -5 m 3 mol -1 , respectively) and R is the universal gas constant, 8.314 m 3 Pa K -1 mol -1 . By differentiating Eq. ( 10) with respect to time and by rearranging the as-obtained expression, the following ODE was obtained:

' = -Q -S' &K & -OK -P E ' Q I 8 + 2PS E ' Q I 5 R &Q & + ' F & & O-SK + 2P' Q -3PS E ' Q I 8 -F R = - T 1 T 8 &K & - T 3 T 8 &Q & + T 5 T 8 & & (11)
The coefficients γ 1 , γ 2 , γ 3 and γ 4 were introduced to clearly show the direct dependence of n g on the other state functions T and P, and on the volume of the bulk phase, V g . Since the Van der Waals equation is nonlinear, it is not possible to calculate the amount of hydrogen in the bulk phase at given physical conditions. For this reason, the initial condition n g0 was evaluated by using the ideal gas law, which is a valid assumption, since the compression tank is still at low pressure at t 0 :

' 2 = K 2 Q 2 F 2 (12)
P 0 and T 0 have been introduced in the previous subsection. The volume V g0 is the volume of hydrogen in the bulk phase at time t 0 and is estimated by the volume balance in the tank:

Q 2 = Q U -Q 2 -Q % ( 13
)
where V tank is the whole volume of the compression tank, V a0 is the volume of hydrogen in the adsorbed phase at t 0 , and V s is the volume of solid adsorbent. V s can be calculated as m s / ρ s , where ρ s is the skeletal density of the AC. A discussion about the volume of the adsorbed phase V a is presented in subsection 3.5 below.

Pressure

Hydrogen pressure is the main state function to be evaluated in an adsorption compressor; it increases with temperature, which leads to hydrogen desorption. Hence, hydrogen moves from the adsorbed to the bulk phase and hence increases its density as long as it remains in a confined volume. Pressure evolution can be obtained from the mass conservation equation:

' + ' = ' -' (14)
where ' [kg s -1 ] is the hydrogen inlet flow during the charging stage and ' [kg s -1 ] is the high-pressure hydrogen outlet flow. By replacing Eq. ( 7) and Eq. ( 11) into Eq. ( 14), the following equation is obtained:

&K & = ' -' + T 3 T 8 &Q & - T 5 T 8 + % M 8 & & % M 1 - T 1 T 8 (15)

Volume of hydrogen in the bulk phase and in the adsorbed phase

Eq. ( 13) shows the volume balance in a hydrogen adsorption compressor; both V tank and V s are definitely constant, as they represent the volume of the tank and of the AC, respectively.

The volume of hydrogen in the adsorbed phase, V a , is defined as a volume in the immediate proximity of carbon surfaces, i.e., wherein the density of the adsorbate is much higher than that of the bulk phase because of the potential field exerted by the adsorbent.

The density of adsorbed hydrogen, ρ a , is often assumed to be equal or even higher than the density of liquid hydrogen [START_REF] Durette | Investigation of the hydrogen adsorbed density inside the pores of MOF-5 from path integral grand canonical Monte Carlo at supercritical and subcritical temperature[END_REF]. Based on this assumption, ρ a can be considered as a constant, thus the volume of the adsorbed phase varies with time. Consequently, by differentiating Eq. ( 13) with respect to time, and by considering that V a can be written as the ratio mass/density of adsorbed hydrogen, the variation with time of the hydrogen volume in the bulk phase, V g , reads:

&Q & = - > V &' & (16)
where M [mol kg -1 ] is the molecular weight of hydrogen, and ρ a [kg m -3 ] is the density of the adsorbed hydrogen. In this study, ρ a was supposed to be equal to the density of liquid hydrogen, i.e., 70.8 kg m -3 . According to this assumption, both V g and V a vary with time, the increase of one corresponding to the decrease of the other.

On the other hand, several authors [START_REF] Do | Adsorption of supercritical fluids in non-porous and porous carbons: analysis of adsorbed phase volume and density[END_REF][START_REF] Richard | Gas adsorption process in activated carbon over a wide temperature range above the critical point. Part 1: modified Dubinin-Astakhov model[END_REF][START_REF] Czerny | Adsorption of Nitrogen on Granular Activated Carbon: Experiment and Modeling[END_REF] considered V a as constant, meaning that all the adsorbed hydrogen occupies a defined volume very close to the carbon surface where the adsorption field exists, thus its density gradually increases up to an asymptotical value.

According to this second definition, both V g and V a are constant, hence it is the density of adsorbed hydrogen that changes with time and they are not dependent variables in the numerical model. Based on this assumption, the model consists of only 4 ODEs in explicit form instead of 5, thus reducing the computational time.

Herein, the two aforementioned ways of considering V a were examined, and we used the Matlab/Simulink software to solve the ODEs system introduced above.

Results and discussion

Model validation

The mathematical model developed herein was checked against the experimental data obtained by Richard et al. [START_REF] Richard | Preliminary evaluation of the performance of an adsorption-based hydrogen storage system[END_REF]. In the latter study, the authors investigated the performances of an adsorption-based hydrogen storage system consisting of a 2.5 L tank filled with 0.67 kg of commercial AC, Maxsorb MSC-30 TM . The tank, initially at 80 K and 0.14 MPa, was soaked into liquid nitrogen. A hydrogen flow of 1.44 g min -1 at 295 K was then introduced into the tank for 27 minutes. As a consequence of such charging process, the heat of adsorption that was released increased the average temperature inside the tank. An increase of hydrogen pressure was thus observed, somehow similar to what occurs in a thermally driven hydrogen compression. [22] and the predicted temperature and pressure dynamics, respectively, by considering V a as a time-dependent variable. On the other hand, Figure 2 c) and d) show the fitting of the same experimental data when a constant V a was used. Although Figure 2 a) and b) show similar trends to those obtained by Richard et al. [START_REF] Richard | Preliminary evaluation of the performance of an adsorption-based hydrogen storage system[END_REF], the fitting was not satisfactory, as a large difference between the predicted and the experimental results was observed. On the other hand, Figure 3 c) and d) show that a much better fit of the experimental data from [START_REF] Richard | Preliminary evaluation of the performance of an adsorption-based hydrogen storage system[END_REF] was obtained when a constant V a was assumed, in agreement with other studies showing that V a approaches the value of the micropore volume in microporous adsorbents [START_REF] Do | Adsorption of supercritical fluids in non-porous and porous carbons: analysis of adsorbed phase volume and density[END_REF]. Thus, it can be assumed that V a is constant and specific for a given AC.

Furthermore, the results of the model validation may lead to another conclusion. According to Eq. ( 16), the time evolution of V g and V a strictly depends on the density of the adsorbed phase, which was initially supposed to be equal to that of liquid hydrogen. The time derivative of V g vanishes when the density of the adsorbed phase is too high, hence both V g and V a can be considered as constant; this finding is in good agreement with studies suggesting that the density of hydrogen in the adsorbed phase can be higher than that of liquid hydrogen, up to 100 kg m -3 [START_REF] Gadiou | The influence of textural properties on the adsorption of hydrogen on ordered nanostructured carbons[END_REF]. From their simulations, Richard et al. [START_REF] Richard | Gas adsorption process in activated carbon over a wide temperature range above the critical point. Part 1: modified Dubinin-Astakhov model[END_REF] obtained an upper limit for the density of adsorbed hydrogen close to the density of solid hydrogen, around 87 kg m -3 . They concluded that this result was not completely surprising, as the adsorption limit corresponds to a state in which the adsorbed phase density is at its maximum, and hydrogen in the adsorbed phase becomes incompressible. Thus, the hypothesis of considering that the volume of the hydrogen adsorbed phase is constant is valid.

The volumes V g and V a obtained in the case where V a was assumed to be a variable were quite different from those calculated by considering V a as a constant, and this explains the large discrepancy observed in Figure 2. Figure 3 shows the evolution of the volumetric ratios of bulk phase volume to tank volume, adsorbed phase volume to tank volume, and AC volume to tank volume, by using the two aforementioned assumptions for V a . Figure 3a shows that V g decreases during both the charging and the cooling stages by assuming variables volumes, and this behaviour is more significant during the charging stage.

On the other hand, V a increases in agreement with Eq. ( 13). This behaviour may be explained by the fact that hydrogen introduced into the tank at time t = 0 is mainly adsorbed onto AC instead of remaining in the bulk phase because of the low temperature. Therefore, V a must increase in order to keep constant the density of the adsorbed phase. Figure 3b shows that the volumes evaluated by assuming V a as constant and equal to the value reported in Table 1 are higher than those in Figure 3a. Indeed, the latter hypothesis implies that the density of the adsorbed phase varies with time while V a remains constant. In addition to the evolution of the volumetric ratios, Figure 3 also shows that in the case of a constant V a , the density of the adsorbed phase increases with time to reach an almost constant value around 66 kg m -3 , somewhat lower than the density of liquid hydrogen, 70.8 kg m -3 . Nevertheless, keeping V a constant definitely gives a better fit of the experimental data.

Table 1 shows the fixed parameters used to validate the model. Information regarding the geometry of the system, i.e., the area where the heat transfer takes place and the mass of the tank, were estimated from literature data [START_REF] Richard | Preliminary evaluation of the performance of an adsorption-based hydrogen storage system[END_REF] related to a prototype container of 150 L and 69 kg. On the other hand, parameters that could not be reliably estimated and that are difficult to measure directly were obtained by fitting the model to the experimental literature data [START_REF] Richard | Preliminary evaluation of the performance of an adsorption-based hydrogen storage system[END_REF]. This is the case of the heat transfer coefficient, h, for which a value of 40 W K -1 m -2 was obtained for the charging process and 15 W K -1 m -2 was obtained for the following cooling stage. These results are in good agreement with the h values normally used in thermal simulations; h is higher in forced convection conditions, such as those of tank filling, than in natural convection ones, such as the following cooling stage.

Table 1 -Parameters used for the validation of the numerical model.

Parameter Value / Equation Reference

ρ S [kg m -3 ] -skeletal density of ACs 2 200 [START_REF] Richard | Preliminary evaluation of the performance of an adsorption-based hydrogen storage system[END_REF] m w [kg] -mass of compression tank 1.15 estimated from [START_REF] Richard | Preliminary evaluation of the performance of an adsorption-based hydrogen storage system[END_REF] A [m 2 ] -heat transfer area 0.12 estimated from [START_REF] Richard | Preliminary evaluation of the performance of an adsorption-based hydrogen storage system[END_REF] ∆H ads [J mol -1 ] -heat of adsorption -6 000 [START_REF] Schaefer | Assessment of hydrogen storage in activated carbons produced from hydrothermally treated organic materials[END_REF] α [J mol -1 ] -parameter of modified DA equation 3 080 [START_REF] Richard | Gas adsorption process in activated carbon over a wide temperature range above the critical point. Part 1: modified Dubinin-Astakhov model[END_REF] β [J mol -1 K -1 ] -parameter of modified DA equation 18.9 [START_REF] Richard | Gas adsorption process in activated carbon over a wide temperature range above the critical point. Part 1: modified Dubinin-Astakhov model[END_REF] x max [mol kg -1 ] -parameter of modified DA equation 71.6 [START_REF] Richard | Gas adsorption process in activated carbon over a wide temperature range above the critical point. Part 1: modified Dubinin-Astakhov model[END_REF] P sat [MPa] -parameter of modified DA equation 1.47 × 10 3 [START_REF] Richard | Gas adsorption process in activated carbon over a wide temperature range above the critical point. Part 1: modified Dubinin-Astakhov model[END_REF] V a [m 3 kg -1 ] -volume of adsorbed phase 0.00143 [START_REF] Richard | Gas adsorption process in activated carbon over a wide temperature range above the critical point. Part 1: modified Dubinin-Astakhov model[END_REF] h [W K -1 m -2 ] -heat transfer coefficient 40 during the charging stage 15 during cooling stage from fits to data [START_REF] Richard | Preliminary evaluation of the performance of an adsorption-based hydrogen storage system[END_REF] c s [J kg -1 K -1 ] -AC specific heat 825 [START_REF] Xiao | CFD model for charge and discharge cycle of adsorptive hydrogen storage on activated carbon[END_REF] c w [J kg -1 K -1 ] -specific heat of tank walls 38 + 3T [START_REF] Richard | Preliminary evaluation of the performance of an adsorption-based hydrogen storage system[END_REF]

Simulation of a hydrogen adsorption compressor

The developed zero-dimensional numerical model presented in Section 3 was used to simulate the performances of a thermally driven hydrogen compressor. The case of an adsorption compressor consisting of a tank of 0.5 L filled up with 0.15 kg of AC was considered. The AC chosen for our simulations was the MAXSORB 30, having a bulk density of around 270 kg m -3 measured with the tapped density analyser AUTOTAP (Quantachrome Instruments)

and following the ASTM D8176-18 standard test method. The parameters of the model used for the numerical investigation were taken from Table 1, except the mass of the compression tank and the heat transfer area, which were estimated from the study of Richard et al. [START_REF] Richard | Preliminary evaluation of the performance of an adsorption-based hydrogen storage system[END_REF] by considering a reservoir volume of 0.5 L. In the beginning of the compression process, both the compression tank and the hydrogen are at 77 K. Hydrogen is initially fed to the tank at 4 MPa, which is the minimum pressure in order to reach the discharge pressure of 70 MPa (Figure 4) required for the development of hydrogen refuelling stations and fuel cell-powered vehicles. The main purpose of the present numerical investigation was to evaluate the time necessary for compressing hydrogen while the compression tank is heated up. The following Figure 5 shows the time evolution of: (i) temperature; (ii) pressure; (iii) hydrogen density in the bulk phase; (iv) mass of hydrogen in the adsorbed phase; (v) high-pressure hydrogen discharge flow; and (vi) low-pressure hydrogen inwards the compressor.

Figure 5a shows the temperature evolution occurring in the thermally driven compressor once the system is heated up. In this stage, the compression tank is taken out of the liquid nitrogen, is kept at room temperature and then exposed to natural convection in ambient air.

For this reason, a heat transfer coefficient h equal to 15 W K -1 cm -2 was chosen. The numerical simulations show that around 400 minutes are necessary to reach the thermal equilibrium between the compression tank and its external environment at room temperature.

During this time, the hydrogen pressure increases up to 70 MPa, since hydrogen desorption takes place by warming up the compression tank (Figure 5b).

According to the thermodynamics of adsorption expressed by Eq. ( 6), 17.85 × 10 -3 kg of hydrogen are adsorbed on the surface of the AC at 77 K and 4 MPa (Figure 5d). On the other hand, the calculated amount of hydrogen in the bulk phase was 3 × 10 -4 kg, corresponding to a hydrogen density of around 12.5 kg m -3 (Figure 5c), which is significantly lower than that of the adsorbed phase when the compression stage starts. This large discrepancy is due to the amount of AC used in the compressor. For 0.25 kg of AC, it was shown that the volume of solid adsorbent V s corresponds to approximately 23% of the total volume of the compression tank, whereas more than 71% of it corresponds to V a and around 6% is left to the bulk phase V g . These amounts were obtained by using the value of V a reported in Table 1. Figure 4d shows that the amount of hydrogen in the adsorbed phase decreases down to 17.08 × 10 -3 kg as a consequence of warming up. This means that 5% of the hydrogen initially adsorbed onto the AC is released towards the bulk phase, whereas 95% of it remains in the adsorbed phase when the system is heated up to room temperature. As a consequence of desorption, hydrogen density in the bulk phase increases, reaching a value as high as 35 kg m -3 .

Figure 5b shows that the maximum pressure of 70 MPa is reached within 50 minutes. The simulation describes the production of a high-pressure hydrogen discharge flow once the desired maximum pressure is reached. Once the pressure inside the compression tank is equal to or higher than the fixed discharge pressure, the discharge valve opens, allowing the highpressure hydrogen to flow outwards the compression tank. The hydrogen discharge flow is modelled by the following expression:

' = -W*K @ A -K+ (17)
where ' [kg s -1 ] has the same meaning as in Eq. ( 14), P max [MPa] is the maximum fixed pressure, P [MPa] is the pressure inside the tank according to Eq. ( 15), and K [kg s -1 MPa -1 ] is a constant relating the discharge flow to the pressure difference. According to Eq. ( 17), the higher is the pressure difference, the higher is the discharge flow rate. The value of the asdetermined constant K strictly depends on the order of magnitude of the amount of hydrogen participating to the mass balance. Physically acceptable flows were obtained for K = 10 -13 kg s -1 MPa -1 . Figure 5e shows that a high-pressure hydrogen discharge flow is produced once the hydrogen pressure has reached the maximum value of 70 MPa. At the exact moment at which the discharge valve is opened, a maximum discharge flow of around 1.3 × 10 -7 kg s -1 is obtained. The discharge flows vanishes within around 100 minutes, as the pressure difference becomes less important as soon as the discharge valve is opened. Once high-pressure hydrogen is depleted, the system is plunged again in liquid nitrogen and thus cooled down to 77 K. As a consequence of temperature decrease and adsorption on AC, the hydrogen pressure inside the tank decreases as shown in Figure 5b. Afterwards, hydrogen at 4 MPa and room temperature is introduced into the compression tank to begin a new hydrogen adsorption cycle. Figure 5f shows the evolution of the low-pressure hydrogen flow introduced into the compression tank, evaluated with the similar principles used for the discharge pressure:

' = W*K @ A -K+ (18)
Low-pressure hydrogen is introduced into the tank once the hydrogen pressure inside the tank becomes lower than the minimum pressure of the system, i.e., 4 MPa in the present work. Figure 5a shows that no significant thermal effects take place during the charging process. As a consequence of both the inlet flow and the cooling process, the amount of hydrogen in the adsorbed phase increases. According to the results from the numerical simulation, around 1000 minutes (i.e., 16 hours), are necessary for the amount of hydrogen in the adsorbed phase to reach the initial value again. Afterwards, more hydrogen can be compressed by heating up the compression tank, giving a cyclic operation.

Discussion

The numerical investigation presented in the previous section was applied to a hydrogen thermally driven compressor consisting of a tank of volume 0.5 L. The results from this study revealed that a high-pressure hydrogen flow of around 10 -4 kg h -1 could be obtained with such designed system. These values are far from those commonly needed by a hydrogen refuelling station, which are at least 4 orders of magnitude higher [START_REF] Reddi | Hydrogen refueling station compression and storage optimization with tube-trailer deliveries[END_REF]. In fact, a thermally driven hydrogen compressor seems to be specially adapted for small-and average-size facilities. Due to the novelty of this kind of compressor, it is not possible to give detailed information about performances and cost. However, it is expected that the capital expenditure for installing it is lower than that demanded for a hydrogen mechanical compressor. A thermally driven hydrogen compressor indeed has no moving parts and hence the maintenance cost should be much lower than that of a mechanical compressor. Moreover, the volume of a thermally driven compressor can be significantly lower, around 22% less than that, much larger, commonly used for a mechanical compressor [START_REF] Sdanghi | Hydrogen Adsorption on Nanotextured Carbon Materials[END_REF]. Nevertheless, the operating costs might be quite high due to the broad range of working temperatures and the use of liquid nitrogen.

Taking again the example of the compression tank of 0.5 L, around 14 L of liquid nitrogen would be needed to reach a stable thermal equilibrium between the compression tank and its external environment. Such a volume can be very difficult to handle, and introduces several issues related to the safety of human operators. Therefore, the volume of liquid nitrogen to be used in a hydrogen adsorption compressor is one of the most relevant drawbacks hindering the development of this technology.

Nevertheless, several aspects must be taken into account for evaluating the potentially achievable high-pressure hydrogen flows by using a thermally driven compressor. One of the severest restrictions of a thermally driven compressor is the too low thermal conductivity of ACs, which is generally around 0.25 W m -1 K -1 [START_REF] Sdanghi | Review of the current technologies and performances of hydrogen compression for stationary and automotive applications[END_REF] and which represents the limiting factor of this kind of system. Heat exchange inside the compression tank is hindered when ACs with low thermal conductivity are used, thus the desorption phenomena following the heating phase and the consequent increase of hydrogen pressure may last longer than the value reported above. Higher thermal conductivities around 2 W m -1 K -1 can be achieved if composite adsorbents based on AC and flexible graphite are used [START_REF] Hardy | Hydrogen storage adsorbent systems acceptability envelope[END_REF], or by imposing a better heat exchange inside the tank through flow coating realised inside the tank in contact with the AC. For this reason, the estimation of the high-pressure hydrogen flow obtained by a thermally driven compressor is quite optimistic when a zero-dimensional model is used. A one-dimensional model taking into account the thermal conductivity may give more realistic results, and this kind of numerical investigation is currently underway.

In addition, the hydrogen discharge flow of a thermally driven compressor also depends on the heat exchange with the external environment, which represents the most important contribution to the thermal balance as shown in Eq. ( 1). Improvements are possible by increasing the temperature of the external environment T a during the compression stage, e.g. by using an electric heating system instead of keeping the compression tank at room temperature. It was shown that 14% of hydrogen may be desorbed when the system is heated up to 353 K, compared to the 5% obtained by heating at room temperature, thereby significantly increasing the amount of high-pressure hydrogen produced by the compressor.

Therefore, a more efficient heat exchange can be obtained by using a higher global heat transfer coefficient h. Figure 6 shows that the compression stage is faster when imposing forced convection (h = 40 W K -1 cm -2 ) instead of natural convection (h = 15 W K -1 cm -2 ).

Whereas the system requires approximately 83 minutes to reach room temperature in natural convection, it takes only 20 minutes in forced convection (i.e., 75% less), making the thermally driven compressor work faster. This feature is beneficial in the perspective of using this technology in hydrogen refuelling stations, as more vehicles might be refuelled in a given time.

However, the aforementioned solutions are not advantageous in terms of efficiency, defined as the ratio of compression work to heat input, and in terms of energy consumption. In fact, 500 W must be supplied to the system when heat exchange by forced convection is imposed, while natural convection is completely free. Nevertheless, methods for further reducing energy consumption, increasing the efficiency and reducing the duration of the compression stage, may consist in using cheap waste heat. The low bulk density of the ACs is another strict limitation for a hydrogen adsorption compressor. ACs are indeed porous adsorbents exhibiting large empty spaces, which can only partly host hydrogen molecules by adsorption. At constant specific surface area, the filling of the compression tank directly depends on the bulk density of the solid adsorbent used inside: the higher the bulk density of the adsorbent material, the higher the mass of adsorbent that can be introduced in the tank. Indeed, the higher the bulk density of the adsorbent, the higher the amount of hydrogen that can be adsorbed onto its surface, and hence the higher the amount of high-pressure hydrogen produced by the adsorption compressor. Hence, the use of porous materials having both high bulk density and high specific surface area is highly recommended. The AC chosen for the present numerical investigation, i.e., the MAXSORB 30, has a bulk density of 270 kg m -3 . Such rather low value somewhat restricts the amount of solid adsorbent that can be used to fill up the compression tank. In the present study, an amount of 0.15 kg of AC was used to fill a 0.5 L tank, leading to a high-pressure hydrogen discharge flow of approximately 1.3•10 -7 kg s -1 , as shown in Figure 5e.

Unlike skeletal density, bulk density is not an intrinsic property of a porous solid thus it can be increased by means of densification processes, or by packing the AC particles closer to each other by tapping and vibrating the stack. Bénard and Chahine [START_REF] Bénard | Modeling of adsorption storage of hydrogen on activated carbons[END_REF] found that bulk densities up to 700 kg m -3 can be achieved by optimising the densification, but the latter lowered the specific surface area from 2778 to 2000 m 2 g -1 . Celzard et al. [START_REF] Celzard | Preparing a Suitable Material Designed for Methane Storage: A Comprehensive Report[END_REF][START_REF] Celzard | Methane storage capacities and pore textures of active carbons undergoing mechanical densification[END_REF] reported that a mixture of two granulometric fractions of a KOH-activated anthracite together with moderate compaction (100 kg cm -2 ) increased by 30% the amounts of stored and delivered methane for an operational vehicle system using adsorbed natural gas and working at MPa and 298 K. A similar strategy might be straightforwardly applied to a hydrogen storage tank, with the promising results expected. In fact, 0.35 kg might be used to fill a volume of 0.5 L if the bulk density of the ACs was increased up to 700 kg m -3 . Thus, the high-pressure hydrogen flow produced in this conditions would be of about 2.2•10 -6 kg s -1 , i.e., one order of magnitude higher than in the case of a bulk density of 270 kg m -3 .

Conclusion

The design and the modelling of a hydrogen compressor based on cyclic adsorptiondesorption on activated carbon (AC) was described herein. A numerical model based on mass and energy balances occurring in a thermally driven compressor was presented and allowed confirming that hydrogen pressures as high as 70 MPa can be obtained by heating the compression tank from 77 K up to room temperature. Moreover, it was shown that hydrogen discharge flows of about 10 -4 kg h -1 can be obtained using a tank volume of 0.5 L filled with 0.25 kg of AC.

However, only 5% of hydrogen in the adsorbed phase was desorbed during the heating/discharging stage, and high volumes of liquid nitrogen were required for the charging stage. These facts restrict the practical applications of a thermally driven hydrogen compressor, and so does the low thermal conductivity of the ACs, which hinders the heat exchange and thus limits the system efficiency. This study aimed at investigating the performances and the feasibility of a hydrogen adsorption compressor through numerical simulations. A prototype tank is presently under construction and the model validation with our own experimental data will be the subject of a forthcoming publication.

Figure 2 a

 2 Figure 2 a) and b) show the comparison between the experimental data from Richard et al.

Figure 2 -

 2 Figure 2 -Fitting of data from Ref. [22]. a) Predicted temperature and b) predicted pressure

Figure 3 -

 3 Figure 3 -Evolution of the volumetric ratios of bulk phase volume to tank volume, adsorbed

Figure 4 -

 4 Figure 4 -Maximum hydrogen pressure reached by a hydrogen thermally driven compressor

Figure 5 -

 5 Figure 5 -Time evolution of: a) temperature; b) pressure; c) hydrogen density in the bulk phase; d) mass of hydrogen in the adsorbed phase; e) high-pressure hydrogen discharge flow; and f) low-pressure hydrogen inlet flow in a thermally driven compressor of inner volume 0.5 L during the compression stage and the following charging stage.

Figure 6 -

 6 Figure 6 -Time evolution of: a) Temperature and b) Pressure in a thermally driven

& & & = % M 1

& + % M 8 & &(7)

Acknowledgements

This study was partly supported by the French PIA project "Lorraine Université d'Excellence", reference ANR-15-IDEX-04-LUE.


domain: info shs

text:
Revisiting the roots of "programs" and why you should care.

Liesbeth De Mol Maarten Bullynck

Please do not circulate. This is a draft. Today, it is still a widely accepted thesis amongst historians and computer scientists that the modern notion of computer programs has its roots in the work of John von Neumann. This is symptomatic for a general tendency amongst academic computer scientists to view the foundations of their field as logico-mathematical and, by consequence, also its historical roots. This, despite clear evidence that, at best, the modern computer was driven by concerns of applied mathematics and developed by a collective of people, mathematicians but also engineers, physicists, (human) computers, etc. Hence, claiming that the most important developments in computer science have their origins, first of all, in the work of mathematicians and logicians, is a distorted view on history. It is not our aim here to repeat some of the arguments why, in computing, one is inclined to reshape history in function of disciplinary identity [START_REF] Tedre | The science of computing:Shaping a Discipline[END_REF][START_REF] Daylight | Why did computer science make a hero out of Turing?[END_REF]. Instead, we would like to revisit the origins of "program" and argue for the need for a deeper historical understanding, not just for the sake of academic history but for the sake of the field itself.

The notion of "program" is a fundamental one in computing. In the flux of historical time and space, "program" underwent significant changes and so "program" in the 1950s would have different connotations from "program" today. Indeed, today one often even uses instead: "software", "apps" or "algorithms" (as in "ethics of algorithms"). Moreover, "program" means different things to different people: a logically-minded computer scientist will have a different understanding of "program" than a software engineer. Nonetheless, as soon as one starts to speak about the historical origins of the term, this plurality of meanings disappears to be replaced by only one, viz. the "stored program". This is anchored in another historical narrative: the modern computer originates in the "stored-program" computer. While this latter notion has been historically scrutinized to some extent,1 the origins of "program" have not been looked at independently of that notion. So what is the classical story here?

A narrative

In the mid 1940s a group of engineers of the Moore School of Electrical Engineering designed and constructed the ENIAC, a large-scale and high-speed machine that would become one of the first computers. Originally, it was a highly parallel and electronic machine with loops and conditionals, and could, essentially, compute any problem provided that its memory would have been unlimited. However, unlike some other large-scale calculators at the time, like the relay-based ASCC/Harvard Mark I or the Bell Lab machines, problems were not set-up on the machine via code but were directly wired on the machine. In a sense, one had to reconfigure the whole machine every time it had to compute another problem. By consequence, setting-up a problem was a timeconsuming and error-prone process. Moreover, the "size" of a problem that could be computed was, in a certain sense, determined by the "size" of the machine. In order to deal with these issues, ENIAC was soon converted in a kind of "stored-program" machine. It is here that John von Neumann enters the story. A few months after he got involved with ENIAC, in the spring of 1945, he wrote the famous "The first draft of a report on the EDVAC" which is considered the blueprint of the modern computer. It is then often assumed that the first "modern" programs must be those that ran on EDVAC-like machines, that is, machines like the converted ENIAC [START_REF] Haigh | Where code comes from: Automatic Control from Babbage to Algol[END_REF]. This goes hand-in-hand with the idea that the roots of our modern notion of program should be sought with von Neumann.

But of course, just as "code" was already in use before it was introduced in the computing context, also "program" was an already existing word. No new term was invented at the time. In fact, as is quite well-known, the word "program" became our modern word because it was extensively used around ENIAC before and after von Neumann entered the scene. Von Neumann himself, however, never really used it -he preferred the common terminology of preparing, planning, setting-up and coding a problem for the EDVAC.2 Before ENIAC it is assumed that the term "program" was not used in a computational setting. Its earlier meanings, according to the entries of the Oxford English Dictionary (OED), referred to such things as: an advance notice, an itinerary, something that is written before some activity happens, orders it and, so, pre-scribes it. A typical example is a theater program or a training schedule. This notion was then picked up in the context of radio broadcasting to refer to radio programs. 3In ENIAC, the notion was introduced by John Mauchly, together with Presper J. Eckert, one of the main ENIAC engineers. It is then assumed that he transposed this common term to a more specific engineering discourse of the ENIAC.

Deconstructing the narrative

We have found clear evidence that, even before ENIAC, there was already an extensive engineering discourse around "program" that goes beyond that mentioned in the OED. First of all, in the context of radio engineering, the growing complexity of the radio broadcasting network increased the need for automation, especially when it concerned the scheduling of radio programs in different networks for different stations and which had to be handled at so-called ``switching points''. This resulted in a discourse in which "program" steadily transposed from radio programs to the technology itself and so one sees the emergence of terms like ``program circuits'', ``program trunks'', ``program switching'', ``program line'', ``program switches'', ``program loop'', ``program transfer'', etc. It are exactly these kind of terms one also finds in the ENIAC context from the start.

More importantly, we have found that there is another engineering discourse using "program" terminology and has its roots in so-called ``programme clocks', a device first developed in the 19th century and used to "furnish a convenient and practical clock, that may be set to strike according to any required programme" [START_REF] Estell | Improvements in programme-clocks[END_REF]). Thus, ``program clocks'' were devices that could be set according to a given schedule or program so that it can ring at preset times. This was very handy, for instance, for a factory work floor, railway stations or a school. From the first clocks onwards one sees the steady development of a more general technology of so-called "program devices" or ``program machines'' to be used in a variety of applications such as a paper cutting machine, a washing machine, an automatic telephone and responding machine or a calculating machine. In that context, ``program'' comes to stand for the automatic carrying out of a sequence of operations or as an automated scheduler.

These different meanings of "program" circulated amongst engineers. Thus, it is highly plausible that Mauchly relied on this existing discourse on the automatic sequencing and scheduling of operations when he introduced the term around ENIAC. This is further substantiated that, as we found, he was not the only one to do so. In an earlier design stage, "program" was also used for the IBM ASCC/Harvard Mark I machine, an important electromechanical large-scale digital calculator. The machine is mostly associated with the name of Howard Aiken, a Harvard physicist, but was designed and built by IBM engineers. The operations of that machine were controlled by the "control tape" in which the sequences of operations were coded with punched wholes. But while "control tape" was the standard term used once the machine was put into operation at Harvard, the original IBM patents show traces of another terminology where the control tape was also called a "program tape" and where the sequences of operations, at some points, were called "programs" (See Fig. 1) instead of "sequences".4 Fig. 1: Graphical representation of the program tape from the ASCC/Mark I patent (Lake et al 1945) This terminology is due to the IBM engineers involved with the design of the machine, amongst others, James W. Bryce and Claire D. Lake. In fact, Bryce already had a patent, predating his first meetings with Howard Aiken, and in which one main technology is a "program device" which is used, amongst others, to do an automatic transfer of control. 5Some have claimed that earlier uses of "program" around ENIAC, before von Neumann became involved, were much more restricted and referred only to "a single operation set up on one of its units" or "the [programming of] operations of the internal circuitry of that unit" [START_REF] Haigh | Where code comes from: Automatic Control from Babbage to Algol[END_REF] This does not take into account this more general discourse which, by that time, had clearly become widespread among engineers working on automatic control. This is clear in the ASCC/Mark I example we discovered but also, for instance, in an Army report written by Eckert, Mauchly and Brainerd to describe the main ideas of the ENIAC. They state that the "purpose […] is to indicate one of many possible ways of setting up a program for the step-by-step solution of a pair of equations commonly used in external ballistics". A device called the "master programmer" was used for the sequencing and scheduling of loops and conditionals6 and was an integral part of the original design of the ENIAC. This goes to show how "program", from the start, referred not just to individual (control) units (as in "program cables") or smaller pieces of an entire program (as in "dummy programs"), but also to the complete problem as a "set-up" on the machine. 7These different uses of "program" circulated around ENIAC and there are different reports in which one can find this terminology. Even though these reports were often written after the EDVAC report they referred to programs for non-EDVAC-like machine but which do have loops, subroutines (closed and open), conditionals, sequences of operations, etc. 8 The addition of using code rather than the technique of wiring a program is, from that conceptual perspective, nonessential. It matters only from the efficiency perspective sketched before. Or, to put it differently, the general understanding of "program" was much more related to the existing discourse on program devices than to specific techniques for implementing them.

Also later this meaning of program persists. Douglas R. Hartree, who was involved with ENIAC and several other computing devices, gives an explicit definition of "program" in his popularizing book on computing machines: "a sequence of operation [s] for a particular calculation" [START_REF] Hartree | Calculating instruments and Machines[END_REF]. Later, Wilkes, Wheeler and Gill in their influential Programs for an electronic digital computer define "program" as: "A sequence of orders for performing some particular calculation". Of course, in this latter context, programs are intended for EDSAC and thus for an EDVAC-like machine but even then, it is clear that this is not part of the definition of program per se. They point to the aspect of program-controlled computing, which has to be distinguished from stored-program control. The former is automatic control through a program, whatever its materiality, the latter is a program stored in the computer's memory, controlling the operations of the machine in the same medium.

Another understanding of "program" which starts to appear in the late 1940s is a definition of program as a "plan". Indeed, in the ACM Glossary of 1951, which was compiled by Grace Hopper, programs are defined as: "a plan for the solution of a problem". A similar understanding can be found in another paper by Maurice Wilkes titled Programme Design for a High-Speed Automatic Calculating Machine. It was also picked up in several glossaries. Note that such definitions are historically connected to the earlier practices of human computation where one spoke of a computation plan.

Thus "program" as used in the discourse around ENIAC and afterwards was not so much anchored in the EDVAC report but in a more general understanding of "program" as a sequencing and scheduling of operations. The EDVAC report then described a technology for achieving this more efficiently. Also in later years, "program", both on EDVAC-like machines and non-EDVAClike machines, was defined in a more general manner, often with reference to the older discourse.

This, in a sense, should not be surprising: while "programs" are very much determined and dependent on the computational technology on which they are ultimately implemented and ran, that need not mean that understandings of "program" should be reduced to properties of and possibilities offered by those technologies. If we would have done that, we would have never had, say, concurrent programs, virtual machines or Docker containers.

Why this matters

Since the 2012 Turing centenary it has become clearer that the academic computing field has a tendency to construct for itself a storyline whereby the presumed theoretical foundations of the field coincide with its historical foundations (the "first" computers and the "first" programs). This is not without effects but only strengthens an academic computing field where one often cares more about formalism than about actual programming (methods) [START_REF] Noble | Notes on postmodern programming[END_REF] and thus contributes to a growing "communication gap" between different communities. Moreover, if one assumes that one approach prevails in the computing field (be it logic or something else), this also has its effects, not just on research and education policies but also on how we understand this field we call computing.

As we showed in this viewpoint, the notion of "program" did not originate with von Neumann or with the stored-program concept, rather it naturally evolved from an engineering context. Program devices for automatic control of operations were developing first for scheduling activities or communications, but were then applied to computing machines as well. In this context, the transfer of meaning happened, preparing the ground for our modern notions of program. Should we derive from this that, actually, computing should be understood first of all as an engineering discipline? No. The point is that as soon as one confines oneself to the perspectives offered by one discipline only, one misses out on the richness of the field of computing as a whole and so lacks a basic understanding: computing is not mathematics, it is not engineering, it is not logic, it is not science but a field on its own and one which should, perhaps, not be reduced to the confines of disciplinary thinking (which is, itself, a construction of the 19 th century). History can and has been used to reinforce such confines but it can also be used against them. We thus would like to make an appeal to you, reader, to not just dismiss this piece as just another nice historical anecdote, but to pick it up as an opportunity to question your own disciplinary confines and assumptions. We hope that by doing so you will not only render your own history more transparent but, perhaps, also make an effort to bridge some communication gaps.

One interesting outcome of that work(Haigh et al

2014) is the observation that the term "stored-program" was not even used until the late 1940s and that it has received different meanings in different times. For those reasons we will gloss the tern "stored-program" here.

"Coding" and "code" were extensively used in the context of the Harvard Mark I which von Neumann knew very well; the idea of "setting-up" a problem on a calculating machine was used in the context of the differential analyzer which he also knew very well.

In(Grier 1997) it is claimed that "program" before ENIAC also referred to "any electrical signal" but we could not find evidence for this in the sources provided.

There were actually two patents, one by the IBM engineers, and one for the Harvard machine.

Note that Bryce was, however, not one of the inventors listed on the ASCC patent. He was, however, very much involved with its design and had the first contacts with Aiken.

Remember that in its original constellation ENIAC was a parallel machine and so scheduling was one basic problem to be counted with.

Note that also the term "set-up" was frequently used instead of "program".

Apart from the original (parallel and decentralized) ENIAC, also the Bell model V machine or IBM calculators were considered.