EP3644692A1 - Method for determining a quality factor of an accelerating cavity of a particle accelerator - Google Patents

Abstract

Method for determining a quality factor (QO) of a superconductive cavity (4) accelerating a particle accelerator (1), in particular a linear type particle accelerator, the method comprising the following steps: - a determination of a thermal load (Qdynam) to which a cryomodule (3) is subjected comprising the accelerating cavity (4) and a bath of cryogenic fluid (5), then a determination of a quality factor (QO) based on the determination of the thermal load (Qdynam) during the operation of the particle accelerator.

Description

Technical field of the invention

The invention relates to a method for determining a quality factor of an accelerating cavity of a particle accelerator, in particular a linear particle accelerator. The invention also relates to a method of operating a particle accelerator. The invention further relates to a device for determining a quality factor and to a particle accelerator comprising such a device.

State of the art

Within particle accelerators, in particular linear particle accelerators, superconducting accelerating cavities, called RF cavities, achieve particle acceleration. These cavities are made from a superconductive material at very low temperatures, such as niobium, and are immersed in a volume of cryogenic fluid, such as in particular helium. The performance of an accelerating cavity, and in particular the maximum admissible power of the cavity is a function of its quality factor. This factor is directly related to the surface condition and the geometry of the cavity. Under certain conditions, the cavity can lose its superconductive state, which triggers a phenomenon called "quench" leading to a complete stop of the beam in the particle accelerator. The degradation of the quality factor also degrades the accelerating capacity of the cavity concerned.

Currently, the quality factor is measured using a probe measuring an electromagnetic field in the cavity. This measurement is carried out when the accelerator is stopped, generally before the start of the installation or during maintenance periods. In particular, the document "Improvement of the Q-factor measurement in RF cavities" by Wencan Xu, S. Belomestnykh, and H. Hahn (BNL Technical Note, BNL-98894- (Ipac 13): 2489-2491, 2013 ) describes such a method. However, it is impossible to use this measurement method when the particle accelerator is running.

Subject of the invention

The object of the invention is to provide a method for determining a quality factor of an accelerating cavity of a particle accelerator overcomes the above drawbacks and improving the devices and methods known from the state of the art prior. In particular, the invention makes it possible to carry out a continuous measurement of the quality factor. The measurement obtained has the advantage of being feasible when the accelerator is running. Furthermore, the determination of the quality factor according to the invention can allow detection of the deterioration of the cavity.

• une détermination d'une charge thermique à laquelle est soumise un cryomodule comprenant la cavité accélératrice et un bain de fluide cryogénique, puis
• une détermination d'un facteur qualité basée sur la détermination de la charge thermique au cours du fonctionnement de l'accélérateur de particules.

The invention relates to a method for determining a quality factor of an accelerating superconductive cavity of a particle accelerator, in particular of a linear type particle accelerator, the method comprising the following steps:

• determining a thermal load to which a cryomodule comprising the accelerating cavity and a bath of cryogenic fluid is subjected, then
• a determination of a quality factor based on the determination of the thermal load during the operation of the particle accelerator.

The steps of determining the thermal load and determining the quality factor can be carried out simultaneously and in real time. The step of determining the thermal load can comprise the use of a state observer.

dans laquelle :

• ṁ_comp désigne un débit massique de fluide cryogénique sous forme compressible au travers de la vanne,
• ṁ_incomp désigne le débit massique de fluide cryogénique sous forme incompressible au travers de la vanne, et
• β_T désigne un coefficient de compressibilité isotherme du fluide cryogénique.

The condition observer can include an estimate of a mass flow of cryogenic fluid passing through a valve of the cryomodule in the form $$\dot{m}={\beta }_T\mathrm{.}{\dot{m}}_{\mathit{comp}}+\left(1-{\beta }_T\right)\mathrm{.}{\dot{m}}_{\mathit{incomp}}$$

in which :

• ṁ _comp designates a mass flow of cryogenic fluid in compressible form through the valve,
• ṁ _incomp designates the mass flow rate of cryogenic fluid in incompressible form through the valve, and
• β _T denotes an isothermal compressibility coefficient of the cryogenic fluid.

The state observer can include an estimate of a density and of a specific internal energy of the cryogenic fluid bath.

• d'un volume de fluide cryogénique à l'état liquide calculé à partir d'une mesure d'une hauteur du fluide cryogénique à l'état liquide et/ou calculé à partir d'une mesure de la quantité de fluide cryogénique entrant et sortant du bain de fluide cryogénique ; et
• d'une charge thermique statique et d'une charge thermique dynamique reçues par le bain de fluide cryogénique ; et
• d'une enthalpie spécifique d'entrée et de sortie du bain cryogénique basée sur une mesure de la pression du bain de fluide cryogénique, ou
  d'une température de sortie du bain de fluide cryogénique basée sur une mesure de la pression du bain de fluide cryogénique et du titre massique d'entrée du bain de fluide cryogénique.

Said estimate can be made from:

• a volume of cryogenic fluid in the liquid state calculated from a measurement of a height of the cryogenic fluid in the liquid state and / or calculated from a measurement of the amount of cryogenic fluid entering and leaving cryogenic fluid bath; and
• a static thermal load and a dynamic thermal load received by the cryogenic fluid bath; and
• a specific enthalpy of entry and exit of the cryogenic bath based on a measurement of the pressure of the cryogenic fluid bath, or
  an outlet temperature of the cryogenic fluid bath based on a measurement of the pressure of the fluid bath cryogenic and the mass entry of the cryogenic fluid bath.

The invention also relates to a method of operating a particle accelerator, in particular a linear type particle accelerator, comprising at least one accelerating cavity, the operating method comprising the implementation of the method for determining a quality factor of at least one accelerating cavity as defined above and a step of modifying at least one operating parameter of said accelerating cavity as a function of its quality factor.

Said operating parameter may be a power control value of a radiofrequency wave emitted in the accelerating cavity, and the modification step may include a reduction in the value of the power control if the quality factor of the at least an accelerating cavity crosses a predetermined threshold, the other cavities of the particle accelerator, when they exist, can continue to operate.

The invention also relates to a device for determining a quality factor of at least one accelerating cavity of a particle accelerator, the determination device comprising hardware and / or software elements implementing the method as defined previously, in particular hardware and / or software elements designed to implement the method as defined above.

The invention also relates to a particle accelerator, in particular a linear type particle accelerator, comprising at least one determination device as defined above. The particle accelerator can comprise at least one cryomodule comprising an accelerating cavity or several accelerating cavities and a bath of a cryogenic fluid.

The invention also relates to a computer program product comprising program code instructions recorded on a medium readable by a computer for implementing the steps of the method as defined above when said program operates on a computer or program product for calculator downloadable from a communication network and / or recorded on a data medium readable by a calculator and / or executable by a calculator comprising instructions which, when the program is executed by a calculator, lead the latter to implement the method as defined above.

The invention also relates to a data recording medium, readable by a computer, on which is recorded a program for a computer comprising program code instructions for implementing the method as defined above or recording medium. readable by a computer comprising instructions which, when executed by a computer, lead the latter to implement the method as defined above.

Brief description of the drawings

• The figure 1 is a schematic view of a particle accelerator according to an embodiment of the invention.
• The figure 2 is a schematic view of a cryogenic system equipped with a cryomodule.
• The figure 3 is a schematic view of a connection between a man-machine interface, an industrial programmable controller and sensors of the cryogenic system.
• The Figures 4A, 4B, 4C and 4D are schematic views of different alternative configurations of a cryogenic system.
• The figure 5 is a schematic view of a cryogenic system equipped with a regulation means.
• The figure 6 is a schematic view of different steps of a method for determining a quality factor according to an embodiment of the invention.
• The figure 7 is a schematic view of a thermodynamic modeling of the cryomodule.
• The Figures 8A, 8B and 8C are graphs illustrating the precision of a thermodynamic model according to an embodiment of the invention.
• The figure 9 is a schematic view of a thermal load observer of the cryomodule.
• The figure 10 is a schematic view of a modeling of a flow through a valve of the cryogenic system.
• The Figures 11A and 11B are graphs illustrating the precision of an estimation of a dynamic load applied to the cryomodule.
• The figures 12A , 12B and 12C are helium property tables.

Description of an embodiment Description of the device

The figure 1 schematically illustrates a linear particle accelerator 1 comprising a longitudinal tube 2 capable of transporting particles, and two cryomodules 3, 3 'arranged in series along the longitudinal tube 2. The particle accelerator 1 could include even more cryomodules. Each cryomodule 3, 3 ′ comprises at least one accelerating cavity 4 and a bath of cryogenic fluid 5. The bath of cryogenic fluid 5 is contained in an enclosure 6 enveloping the accelerating cavity 4. The cryogenic fluid bath 5, maintained at a temperature of the order of 4K, is intended to maintain the temperature of the cavity below its critical temperature, in particular below 9.2K. As visible on the figure 1 , a first type of cryomodule 3 comprises a single accelerating cavity 4 and a second type of cryomodule 3 'comprises two accelerating cavities 4. Alternatively, other types of cryomodules could include any number of cavities and a particle accelerator could include any arrangement of cryomodules. The cryomodule can be equipped with a peripheral cooling system (not shown), called a thermal shield, making it possible to maintain its external envelope at a given temperature, for example at a temperature of 70K.

With reference to the figure 2 , we specify the design of a cryomodule 3 equipped with a single cavity 4. An accelerating cavity 4 includes walls 7 made for example of niobium, niobium alloy with titanium, or any other material suitable for manufacturing walls of superconductive accelerating cavities. The walls 7 have a given thickness e. Niobium is a superconductive material when kept at a temperature below 9.2K. The cavity 4 also comprises a radiofrequency antenna 8 capable of emitting electromagnetic waves to accelerate the particles passing through the cavity 4. A perfect or almost perfect vacuum prevails inside of the cavity 4. The cryogenic fluid 5 in which the cavity 4 is immersed is advantageously boiling helium which is partly in the liquid state and partly in the gaseous state. Similarly, other chemical compositions could be envisaged to constitute the cryogenic fluid 5.

Helium in the liquid state is denser than helium in the gaseous state. By gravity helium in the liquid state therefore occupies a lower volume of the enclosure 6 of the cryomodule 3 while helium in the gaseous state occupies a greater volume of the enclosure. The helium bath therefore behaves like a phase separator, that is to say a bath in which an equilibrium occurs between the gaseous state and the liquid state of the same fluid depending on the pressure conditions. and temperature. In the rest of the document, the term “phase separator” will therefore be used interchangeably to designate the helium bath contained in the cryomodule 3. The cryomodule is equipped with an LT level sensor, capable of measuring the height of helium under liquid form within the cryomodule enclosure.

The phase separator is subject to a thermal load which can be broken down into two parts. On the one hand, the phase separator is subjected to a static _static measurable thermal load Q, due to the heat exchanges by conduction, convection and radiation between the external environment of the cryomodule at an ambient temperature (i.e. around 300K) and the cryogenic fluid. at a temperature of 4K. On the other hand, the phase separator is subjected to a dynamic thermal load Q _dynam due to the power of the electromagnetic field in the cavity and / or by the passage of the particles in the cavity. This dynamic load will be determined (in other words estimated, simulated or calculated) in accordance with the description which will be given later. From a thermodynamic point of view, the cavity 4 has no other effect than an additional supply of heat on the helium bath.

The thermal load can be the image of the radiofrequency power injected into the cavity but not only. It can be the image of the degradation of the insulation vacuum, of low-energy electronic emission at the level of a radio frequency coupler or in the cavity, of the emission of fields or of line loss of the beam.

The thermodynamic model of a cryomodule 3 'equipped with two accelerating cavities 4 is equivalent to the thermodynamic model of a cryomodule 3 equipped with a single accelerating cavity 4. Only three parameters of these models differ: the volume of the enclosure Vol containing the cryogenic fluid, the static thermal load Q _static and the dynamic thermal load Q _dynam acting on the cryogenic fluid 5. The invention will be detailed on example d '' a cryomodule equipped with a single accelerating cavity. A person skilled in the art can transpose these lessons to a cryomodule comprising two or more accelerating cavities.

A cryogenic system 10 comprises the cryomodule 3 as well as three valves CV001, CV002, CV005, making it possible to connect the cryomodule 3 to a helium distribution circuit 11. A first valve CV001 is a helium inlet valve and is connected to a lower part of the helium bath, at a point where the helium is in liquid form (once the temperature of the helium is lowered to its temperature Operating). A second valve CV002 is also a helium inlet valve and is connected to an upper part of the helium bath, at a point where the helium is in gaseous form. A third valve CV005 is a helium outlet valve and is connected to the upper part of the helium bath, at a point where the helium is in gaseous form. The first valve CV001 can be used during helium filling of the cryomodule enclosure. The cryogenic system 10 may not understand this first valve CV001 if the helium filling can be obtained in a different way. The first valve CV001 is not used for a regulatory function. The second valve CV002 can be used to regulate the helium level in the cryomodule enclosure. The third valve CV005 can be used to regulate the pressure in the cryomodule enclosure. The first and second valves CV001, CV002, called supply valves, are connected to a two-phase helium supply line. The third valve CV005, called the exhaust valve, is connected to a return line. These lines are not shown on the figure 2 but are replaced by the boundary conditions at BC _in input and the boundary conditions at BC _out output. The boundary conditions at the BC _in input are given by the pressure P _in and the enthalpy H _in at the input of the supply valves CV001 and CV002. The boundary conditions at the BC _out outlet are given by the pressure P _out at the outlet of the CV005 exhaust valve. The opening of each valve can be adjusted to gradually vary the flow of helium passing through it. The position of each of these valves, that is to say its percentage of opening, can be read manually or automatically.

A triplet of variables is associated with each input or output of all the components: the internal pressure P (expressed in absolute bars), the specific enthalpy H (expressed in J / kg) and the mass flow rate ṁ (expressed in kg / s) represented by "M" on the figure 2 . The physical characteristics of helium are thus defined locally. These variables represent the data exchanged between the different elements constituting the model. The index associated with each of the variables indicates whether it is the input (“in”) or the output (“out”) of the model. The exponent defines whether the variable is calculated ("calc") or imposed by a neighboring component ("imp").

The cryogenic system also includes a PT helium pressure sensor (illustrated on Figures 4A to 4D ). This sensor can for example be positioned upstream of the third valve CV005, that is to say between the third valve and the helium bath. The pressure sensor PT and the level sensor LT of liquid helium are capable of carrying out continuous measurements, that is to say that they supply a signal fluctuating as a function of the evolution of the pressure in the bath helium and the height of liquid helium. With reference to the figure 3 , they are connected to an API industrial programmable controller. This automaton can advantageously be itself connected to an HMI man-machine interface, such as a computer or any other display means intended for a user. If the automaton implements the determination method according to the invention, the human-machine interface HMI is only used to display the result. The pressure sensors PT and level LT, the industrial programmable controller API and the man-machine interface IHM (when there is one) are part of a device 9 for determining the quality factor Q ₀ .

There are two ways of implementing the quality factor estimation method: either the calculations are carried out on the PLC industrial programmable controller and the result is communicated to the HMI man-machine interface, or the controller transmits the data of the PT, LT sensors at the HMI man-machine interface and it is the latter that performs the calculations. The API PLC and / or the HMI man-machine interface are computers and include means for implementing the method for estimating the quality factor, in particular a memory and a calculation unit. It is recommended to favor the first option since it avoids possible bugs that may occur on a man-machine interface (a programmable controller being designed to minimize the risk of bug). In the rest of the description, the method implemented by the API PLC is therefore assumed.

The various valves and sensors of the cryogenic system are connected to one or more CTRL controllers capable of regulating the helium pressure and the level of liquid helium inside the cryomodule by acting on the CV002 and CV005 valves. With reference to Figures 4A, 4B, 4C and 4D , four possible control structures are proposed.

The figure 4A illustrates a centralized control: a single CTRL controller is able to act on the two valves CV002 and CV005.

The figure 4B illustrates a decentralized control of the opening of the second valve CV002 and of the third valve CV005: it consists in using two separate CTRL controllers and in fully decoupling the two level and pressure regulations.

The figure 4C illustrates a distributed control: it works on the same principle as the decentralized control but with an interaction between the two CTRL level and pressure controllers.

The figure 4D illustrates a hierarchical control: a Coord coordinator pilots two separate CTRL controllers and ensures the stability of the cryogenic system.

The CTRL and Coord control structures use the estimation of the thermal load acting on the helium bath to improve the overall stability of the system, which we will detail below.

The figure 5 illustrates a cryogenic system 10 equipped with a means 12 for regulating the power emitted by the radiofrequency antenna as a function of the estimation of the quality factor Q0.

Description of the process

We will now describe an embodiment of the method for determining a quality factor Q0 of a cavity through six steps E1 to E6 carried out successively. As shown on the figure 6 , steps E1 to E5 lead to estimating a thermal load Q _dynam acting on the cryomodule. This estimation is carried out using a state observer based on a thermodynamic and thermohydraulic model of the cryomodule. Then, step E6 leads to estimating the value of the quality factor Q ₀ on the basis of the thermal load Q _dynam .

The process is carried out in real time, that is to say that the estimates of the dynamic load Q _dynam and of the quality factor Q ₀ are calculated instantaneously and renewed constantly. By “real time”, it is understood that the determination steps are carried out at a speed adapted to the evolution of the dynamic load Q _dynam and of the quality factor Q ₀ . For example, new values of dynamic load Q _dynam and of quality factor Q ₀ can be calculated at a frequency greater than or equal to 1 Hz, or even greater than or equal to 10 Hz, or even even greater than or equal to 1 kHz. In addition, the method can be carried out while the cavity is in operation, that is to say when the particle accelerator is precisely used to accelerate particles, in particular for experimental purposes. The method is not necessarily implemented during a manipulation dedicated to the measurement of the quality factor. The method can therefore be implemented in parallel with an experiment during which all of the accelerator systems are in operation. The estimate can also be renewed when a sensor in the cryogenic system records a significant variation.

When the description does not specify it, the physical unit associated with a given physical quantity is a unit of the international system.

• le volume Vol de l'enceinte contenant le fluide cryogénique,
• la charge thermique statique Qstatic supportée par le fluide cryogénique, c'est-à-dire l'énergie transmise au cryomodule par les transferts thermiques avec l'extérieur du cryomodule,
• la fonction f1 définissant la hauteur d'hélium liquide h en fonction du volume d'hélium liquide Vliq dans l'enceinte : h = f1 (Vliq). Cette fonction dépend de la géométrie de l'enceinte contenant le fluide cryogénique. Elle peut être calculée au moyen d'un modèle numérique de cette enceinte ou être définie empiriquement.

In a first step E1, characteristics of the phase separator and of the control valves are determined. These characteristics depend directly on the design of the cryomodule and the valves. They can be measured or calculated. These characteristics are:

• the volume Vol of the enclosure containing the cryogenic fluid,
• the static Qstatic thermal load supported by the cryogenic fluid, that is to say the energy transmitted to the cryomodule by heat transfers with the outside of the cryomodule,
• the function f1 defining the height of liquid helium h as a function of the volume of liquid helium Vliq in the enclosure: h = f1 (Vliq). This function depends on the geometry of the enclosure containing the cryogenic fluid. It can be calculated using a numerical model of this enclosure or be defined empirically.

• Le coefficient de débit CV de chaque vanne, c'est-à-dire le coefficient exprimant le débit d'un fluide qui traverse une vanne, à une température donnée, et qui provoque une chute de pression donnée.
• La rangeabilité Rv de chaque vanne, c'est-à-dire le rapport des débits maximal et minimal entre lesquels la caractéristique d'une vanne est maintenue dans certaines limites de précision.

For the second valve CV002 and the third valve CV005, these characteristics are:

• The flow coefficient CV of each valve, that is to say the coefficient expressing the flow of a fluid which passes through a valve, at a given temperature, and which causes a given pressure drop.
• The rangeability Rv of each valve, that is to say the ratio of the maximum and minimum flow rates between which the characteristic of a valve is maintained within certain precision limits.

In a second step E2, a thermodynamic modeling of the cryomodule is carried out. Such modeling is illustrated macroscopically on the figure 7 . The thermodynamic modeling of the cryomodule makes it possible to link by equations of the boundary conditions BC _in , BC _out , the position of the three valves POS _{CV 001} , POS _{CV 002} , POS _{CV 005} , the static load Q _static , the dynamic load Q _dynam , the height h of liquid helium in the enclosure, and the internal pressure P in the enclosure 6. The modeling can be broken down into three sub-steps E21, E22, E23.

In a first substep E21, a modeling of the valves is established. This first substep E21 makes it possible to define the quantity of helium entering ṁ _in into the enclosure and the quantity of helium leaving ṁ _out of the enclosure according to the boundary conditions BC _in , BC _out , and the position of the three valves POS _{CV 001} , POS _{CV 002} , and POS _{CV 005} , as well as the pressure in the cryomodule.

In a second substep E22, an energy model of a phase separator is established. This second sub-step makes it possible to define the density of the helium ρ (expressed in kg / m3) and the specific internal energy u (expressed in J / kg) of the helium contained in the cryomodule as a function of the quantity of helium entering the enclosure ṁ _in and the quantity of helium leaving the enclosure ṁ _out , of the static charge Q _static , and of the dynamic charge Q _dynam , as well as of the specific enthalpy of entry and of the cryomodule outlet H _in and H _out or of the cryomodule outlet temperature and of the cryomodule inlet mass titer.

In a third substep E23, a modeling of the physical properties of the helium bath is established. This third sub-step makes it possible to define the height h of liquid helium in the enclosure, and the internal pressure P in the enclosure as a function of the density of the helium ρ and of the specific energy u of the helium.

We will now detail each of the models established during these three substeps.

Dans laquelle : $$X=\min \left(\frac{{\mathit{P}}_{\mathit{in}}-P_{\mathit{out}}}{P_{\mathit{in}}},X_C\right)$$

$$X_C=\frac{\gamma }{1.4}\mathrm{.}{X}_t$$

$$X_t=\frac{P_{\mathit{in}}-P_{\mathit{out}}}{P_{\mathit{in}}}$$

Avec :

• K un coefficient de conversion entre le système d'unité anglo-saxon et le système d'unité international (K = 7.59.10^-3).
• ρ_in la masse volumique de l'hélium (exprimée en kg/m3) en amont de la vanne. Cette masse volumique pouvant être interpolée à partir d'une table de propriété de l'hélium en connaissant la pression et l'enthalpie en amont de la vanne.
• P_in la pression en amont de la vanne (exprimée en Pa).
• P_out la pression en aval de la vanne (exprimée en Pa).
• γ, le rapport des chaleurs spécifiques (sans unité), défini comme le rapport de la chaleur spécifique à pression constante Cp (exprimée en J/(kg.K)) sur la chaleur spécifique à volume constant Cv (exprimée en J/(kg.K)) de l'hélium, soit : $\gamma =\frac{C_p}{C_v}\mathrm{.}$
  
  Ce rapport peut être interpolé à partir d'une table de propriété de l'hélium en connaissant la pression et l'enthalpie en amont de la vanne.
• CV désignant le coefficient de débit de la vanne (sans unité) et pouvant être calculé par la formule : $$\mathit{CV}=\frac{{\mathit{CV}}_{\mathit{max}}}{R_v}\mathrm{.}\left(\exp \left(\frac{\mathit{open}}{100}\mathrm{.}{R}_v\right)-\left(1-\frac{\mathit{open}}{100}\right)\right)$$
  

dans laquelle :

• CV_max est une constante de dimensionnement de la vanne choisie de sorte à ce que l'ouverture de la vanne soit dans une gamme de fonctionnement convenable (c'est-à-dire que pour que la vanne ne s'ouvre pas ou ne se ferme pas à 100% lors de son utilisation.
• R_v est la rangeabilité de la vanne (sans unité).
• « open » est un niveau d'ouverture de la vanne évoluant de 0, quand la vanne est complètement fermée, à 100, quand la vanne est complètement ouverte. Le niveau d'ouverture « open » représente la position respective d'ouverture de chaque vanne POS _CV001, POS _CV002, et POS _CV005 en pourcentage.

The first substep E21 makes it possible to establish a modeling of the valves. The method is detailed on the example of any particular valve among the three valves CV001, CV002, CV005. First of all, it is considered that the expansion occurring in the valve is isenthalpic, that is to say without any external energy supply. Thus, the enthalpy of helium upstream of the valve is identical to the enthalpy downstream of the valve, namely: H _out = H _in . It is also considered that the valve does not accumulate fluid. Thus, we can also write the equation ṁ _out = ṁ _in . According to ANSI / ISA-75.01.01 standard, the flow rate of a compressible fluid through a valve is written according to the following formula F2: $${\dot{m}}_{\mathit{comp}}=K\mathrm{.}\mathit{CV}\mathrm{.}\left(1-\frac{\mathrm{<figure-callout\; id="3"\; label="V"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">V</figure-callout>}}{3.X_{\mathrm{VS}}}\right)\sqrt{{\rho }_{\mathit{in}}\mathrm{.}{P}_{\mathit{in}}\mathrm{.}X}$$

In which : $$X=\min \left(\frac{{\mathit{P}}_{\mathit{in}}-P_{\mathit{out}}}{P_{\mathit{in}}},X_{\mathrm{VS}}\right)$$

$$X_{\mathrm{VS}}=\frac{\gamma }{1.4}\mathrm{.}{X}_t$$

$$X_t=\frac{P_{\mathit{in}}-P_{\mathit{out}}}{P_{\mathit{in}}}$$

With:

• K a conversion coefficient between the Anglo-Saxon unit system and the international unit system ( K = 7.59.10 ^-3 ).
• ρ _in the density of helium (expressed in kg / m3) upstream of the valve. This density can be interpolated from a helium property table by knowing the pressure and the enthalpy upstream of the valve.
• P _in the pressure upstream of the valve (expressed in Pa).
• P _out the pressure downstream of the valve (expressed in Pa).
• γ , the ratio of specific heats (without unit), defined as the ratio of the specific heat at constant pressure Cp (expressed in J / (kg.K)) to the specific heat at constant volume Cv (expressed in J / (kg .K)) helium, either: $\gamma =\frac{{\mathrm{VS}}_p}{{\mathrm{VS}}_v}\mathrm{.}$
  
  This report can be interpolated from a helium property table by knowing the pressure and the enthalpy upstream of the valve.
• CV designating the flow coefficient of the valve (without unit) and can be calculated by the formula: $$\mathit{CV}=\frac{{\mathit{CV}}_{\mathit{max}}}{R_v}\mathrm{.}\left(\exp \left(\frac{\mathit{open}}{100}\mathrm{.}{R}_v\right)-\left(1-\frac{\mathit{open}}{100}\right)\right)$$
  

in which :

• CV _max is a dimensioning constant of the valve chosen so that the opening of the valve is in a suitable operating range (i.e. so that the valve does not open or close does not close 100% when in use.
• R _v is the rangeability of the valve (without unit).
• "Open" is a level of opening of the valve evolving from 0, when the valve is completely closed, to 100, when the valve is fully open. The “open” opening level represents the respective opening position of each POS _{CV 001} , POS _{CV 002} , and POS _{CV 005} valve in percentage.

Furthermore, the flow rate of an incompressible fluid through a valve is written according to the following formula F3: $${\dot{m}}_{\mathit{incomp}}=K\mathrm{.}\mathit{CV}\mathrm{.}\sqrt{{\rho }_{\mathit{in}}\mathrm{.}\left(P_{\mathit{in}}-P_{\mathit{out}}\right)}$$

In which the variables have the same meaning as in the formula F2.

Isothermal compressibility β _T is defined as a factor of variation in the volume of a system when the pressure in the system varies while its temperature remains constant. This factor indicates how compressible a fluid is. Thus, β _T = 0 when the fluid is incompressible and β _T = 1 when the fluid is compressible. At a given temperature, the factor β _T can be calculated by the following formula: $${\beta }_T={\frac{1}{\rho }\left(\frac{\mathit{d\rho }}{\mathit{dP}}\right)}_T\mathrm{.}$$

This factor is used to weight the flow rate of the fluid passing through the valve according to the formula F2 or F3 presented above. Thus the calculation of the flow rate through the valve can be written with the following formula: $$\dot{m}={\beta }_T\mathrm{.}{\dot{m}}_{\mathit{comp}}+\left(1-{\beta }_T\right)\mathrm{.}{\dot{m}}_{\mathit{incomp}}$$

We will now detail the second substep E22 in which we establish the energy model of the phase separator. It is assumed that the helium bath is in liquid-gas equilibrium. Consequently, the density ρ of helium and the specific energy u of helium (in other words its mass density of energy) are uniformly distributed in the enclosure.

First, we establish a physical relationship between the total mass of helium m _tot in the cryomodule bath, its density ρ and the volume of the enclosure Vol containing helium with the following physical equation: $$m_{\mathit{early}}=\rho \mathrm{.}\mathit{Flight}$$

This formula can be derived so as to be written: $${\dot{m}}_{\mathit{early}}=\dot{\rho }\mathrm{.}\mathit{Flight}$$

Then, a mass balance of the cryomodule makes it possible to link the total mass m _tot of helium inside the enclosure, the incoming mass m _in of helium and the outgoing mass m _out of helium by the following equation: $${\dot{m}}_{\mathit{early}}={\dot{m}}_{\mathit{in}}-{\dot{m}}_{\mathit{out}}$$

The relation linking the total energy U stored by helium, the specific energy u of helium and the total mass of helium m _{tot is} written: $$U=m_{\mathit{early}}\mathrm{.}u$$

This formula can be derived so as to be written according to the following formula: $$\dot{U}={\dot{m}}_{\mathit{early}}\mathrm{.}u+m_{\mathit{early}}\mathrm{.}\dot{u}$$

Finally, an energy balance applied to the helium bath is written with the following formula: $$\dot{U}={\dot{m}}_{\mathit{early}}\mathrm{.}u+H_{\mathit{in}}\mathrm{.}{\dot{m}}_{\mathit{in}}-H_{\mathit{out}}\mathrm{.}{\dot{m}}_{\mathit{out}}+\sum _i{Q}_i$$

• ∑ _i Q_i désigne l'ensemble des charges thermiques agissant sur le bain d'hélium, c'est-à-dire : ∑ _i Q_i = Q_static + Q_dynam .
• H_in désigne l'enthalpie de l'hélium entrant dans l'enceinte du séparateur de phase.
• H_out désigne l'enthalpie de l'hélium sortant de l'enceinte du séparateur de phase.

Equation in which:

• ∑ _i Q _i denotes the set of thermal loads acting on the helium bath, that is to say: ∑ _i Q _i = Q _static + Q _dynam .
• H _in denotes the enthalpy of the helium entering the enclosure of the phase separator.
• H _out designates the enthalpy of the helium leaving the phase separator enclosure.

By combining the above equations, we obtain an equation of the thermodynamic model of the cryomodule: $$\dot{u}=\frac{H_{\mathit{in}}\mathrm{.}{\dot{m}}_{\mathit{in}}-H_{\mathit{out}}\mathrm{.}{\dot{m}}_{\mathit{out}}+\sum _i{Q}_i}{\rho \mathrm{.}\mathit{Flight}}-u\mathrm{.}\frac{\dot{\rho }}{\rho }$$

Finally, we now detail the third substep E23 making it possible to define the height h of liquid helium in the enclosure, and the internal pressure P in the enclosure as a function of the density ρ of the helium and of l specific energy u of helium.

The internal pressure P in the cryomodule enclosure can be directly determined as a function of the density ρ of helium and its specific energy u by exploiting the physical properties of helium. To this end, an interpolation function integrated into a calculation software such as Hepak © and / or a C ++ library such as "CoolProp" can advantageously be used. As an example, a first helium property table is illustrated in figure 12A . In this figure, the density ρ of helium is represented on the ordinate and expressed in kg / m3. The energy specific energy u is represented on the abscissa and expressed in 10 ⁵ .J / kg. The ten curves X1 to X10 are obtained for ten internal pressure levels P from 50 mPa to 5000 mPa.

Thanks to a second helium property table, we can also determine the mass title X of helium as a function of the density ρ of helium and its specific energy u. This second table is shown as an example on the figure 12B . The mass titer X is represented on the vertical axis Z. The density ρ of helium represented on a first horizontal axis Xh1 and expressed in kg / m3. The energy specific energy u is represented on a second horizontal axis Xh2 and expressed in 10 ⁴ .J / kg. By definition, the mass title X of helium satisfies the following formula: $$m_{\mathit{liq}}=m_{\mathit{early}}\left(1-X\right)$$

Formula in which m _liq denotes the mass of liquid helium in the enclosure and m _tot the total mass of helium.

dans laquelle ρ_liq désigne la masse volumique de l'hélium liquide.The volume of helium in liquid form V _liq is defined by the equation: $$V_{\mathit{liq}}=\frac{m_{\mathit{liq}}}{{\rho }_{\mathit{liq}}}$$

in which ρ _liq denotes the density of liquid helium.

Finally, thanks to a third helium property table, illustrated by way of example on the figure 12C , and considering that the helium is in the saturated liquid state, one can also determine the density of the liquid helium ρ _liq (on the ordinate) as a function of internal pressure P (on the abscissa). In this figure, the density of liquid helium ρ _liq is expressed in kg / m3 and the internal pressure P is expressed in 10 ⁵ .Pa. Note, we call saturated liquid, a liquid whose temperature and pressure are such that at constant temperature, any loss of pressure even the smallest would cause the boiling of the liquid. Finally, the volume of helium in liquid form V _{liq is determined} , then the height h of liquid helium is calculated using the function f ₁ defined above.

Finally, thanks to the three sub-steps E21, E22, E23 previously described, a thermodynamic modeling of the cryomodule is obtained. linking by equations the boundary conditions BC _in , BC _out , the position of the three valves POS _{CV 001} , POS _{CV 002} , POS _{CV 005} , the static charge Q _static , the dynamic charge Q _dynam , the height of liquid helium in the enclosure h, and the pressure in the enclosure P.

The accuracy of the thermodynamic modeling of the cryomodule can be verified by comparing the height h _mes of liquid helium measured with the height h _calc of liquid helium estimated by the modeling, and likewise by comparing the pressure P _mes in the enclosure measured with the pressure P _calc estimated by the modeling, when the opening of the valves CV002 and CV005 is varied. For the purposes of this verification, the dynamic thermal load Q _dynam may be kept at zero and the valve CV001 kept closed. The figure 8A illustrates a graph of the opening of valves CV002 and CV005 as a function of time (the value of 100% designates a valve fully open). The figure 8B illustrates the evolution over time of the height of liquid helium measured h _mes and estimated h _calc . The two dotted curves represent the calculation of an uncertainty. More precisely, the two dotted curves represent the height of liquid helium estimated with an opening of the valves increased by 2%, and respectively reduced by 2%. It can be seen that the height h _calc of liquid helium calculated does not depart by more than a few percent from the height h _mes of liquid helium measured. Likewise, the figure 8C illustrates the evolution over time of the helium pressure measured P _mes and estimated P _calc . The two dashed curves also represent the calculation of an uncertainty obtained by simulating an opening of the valves increased by 2%, and respectively reduced by 2%. It can be seen that the simulated helium pressure P _calc does not deviate by more than a few millibars from the measured helium pressure P _mes . This verification therefore makes it possible to validate the accuracy of the established thermodynamic model.

• la pression P_in d'hélium en amont des vannes d'admission CV001, CV002,
• la pression P_out d'hélium en aval de la vanne de sortie CV005.
• L'enthalpie spécifique d'entrée de la vanne d'alimentation : H_in.

In a third step E3, various parameters of the cryogenic system are recorded in the memory of the industrial programmable controller API. In particular, the internal pressure P supplied by the PT pressure sensor. The height h of liquid helium is recorded in the enclosure supplied by the level sensor LT. The boundary conditions are also recorded, namely:

• the helium pressure P _in upstream of the intake valves CV001, CV002,
• the pressure P _out of helium downstream of the outlet valve CV005.
• The specific inlet valve enthalpy: H _in .

The boundary conditions are dependent on the helium distribution circuit 11 and can be measured and / or calculated by means of suitable sensors positioned in the distribution circuit 11. As a variant, and to simplify the calculations, the boundary conditions could be considered to be constants. Such a simplification would nevertheless lead to a less precise estimation of the dynamic load.

The thermodynamic modeling of the cryomodule obtained at the end of the second step E2 includes complex equations to be solved. In order to facilitate the resolution of these equations, the invention provides in a fourth step E4 a linearization of the thermodynamic model, that is to say an approximation of the thermodynamic model by a set of linear differential equations around a point of predetermined operation.

In a first substep E41, an operating point is defined around which the model will be linearized. This operating point can be determined according to the constraints which the cryomodule must respect. For example, the operating point can be defined by an internal pressure P of the helium bath equal to 1200 mBar and a height of liquid helium equal to 90% of the total height of the enclosure.

In a second substep E42, the boundary conditions of the system BC _in and BC _out are defined and the opening values of the valves POS _{CV 001} , POS _{CV 002} , POS _{CV 005} are sought which allow the model to stabilize at the previously defined operating point. The opening value of the first POS _{CV 001} valve can be set to 0% (that is to say fully closed) because this valve is only useful for filling the enclosure. Two regulators of the PID type, that is to say of the “Proportional - Integral - Derivative” action type, can be used to determine the opening values of the two other valves POS _{CV 002} , POS _{CV 005} . During this step, the radio frequency antenna may or may not be activated depending on the operating point around which it is desired to linearize the thermodynamic model.

$$y=C\mathrm{.}x+D\mathrm{.}\left(\begin{array}{c}v\\ w\end{array}\right)$$

Avec :

• A, B, C et D des matrices d'état du système,
• v désignant un vecteur d'entrées commandables et défini par : $$v=\left(\begin{array}{c}{\dot{m}}_{\mathit{in}}\\ {\dot{m}}_{\mathit{out}}\end{array}\right)$$
  
  • ṁ_in désignant le débit massique entrant dans le cryomodule.
  • ṁ_out désignant le débit massique sortant du cryomodule.
• w désignant un vecteur d'entrées non commandables et défini par : $$w=\left(\begin{array}{c}{Q}_{\mathit{static}}\\ Q_{\mathit{dynam}}\\ H_{\mathit{in}}\end{array}\right)$$
  

Q_static désignant la charge statique, Q_dynam désignant la charge dynamique et H_in désignant l'enthalpie au sein du cryomodule.

• x désignant un vecteur d'état, celui-ci étant égal au vecteur de sortie y et étant défini par : $$x=y=\left(\begin{array}{c}u\\ \rho \end{array}\right)$$
  

où p désigne la masse volumique de l'hélium et u son énergie interne spécifique.In a third substep E43, the thermodynamic model is represented as a linear dynamic system. The linear system is defined by the following state representation: $$\dot{x}=\mathrm{AT}\mathrm{.}x+B\mathrm{.}\left(\begin{array}{c}v\\ w\end{array}\right)$$

$$y=\mathrm{VS}\mathrm{.}x+D\mathrm{.}\left(\begin{array}{c}v\\ w\end{array}\right)$$

With:

• A, B, C and D of the system state matrices,
• v designating a vector of controllable inputs and defined by: $$v=\left(\begin{array}{c}{\dot{m}}_{\mathit{in}}\\ {\dot{m}}_{\mathit{out}}\end{array}\right)$$
  
  • ṁ _in designating the mass flow entering the cryomodule.
  • ṁ _out designating the mass flow leaving the cryomodule.
• w designating a vector of non-controllable inputs and defined by: $$w=\left(\begin{array}{c}{Q}_{\mathit{static}}\\ Q_{\mathit{dynam}}\\ H_{\mathit{in}}\end{array}\right)$$
  

Q _static designating the static load, Q _dynam designating the dynamic load and H _in designating the enthalpy within the cryomodule.

• x designating a state vector, this being equal to the output vector y and being defined by: $$x=y=\left(\begin{array}{c}u\\ \rho \end{array}\right)$$
  

where p denotes the density of helium and u its specific internal energy.

$$w_0=\left(\begin{array}{c}{Q}_{\mathit{static}}\left(\mathit{nom}\right)\\ Q_{\mathit{dynam}}\left(\mathit{nom}\right)\\ H_{\mathit{in}}\left(\mathit{nom}\right)\end{array}\right)$$

$$x_0=y_0=\left(\begin{array}{c}u\left(\mathit{nom}\right)\\ \rho \left(\mathit{nom}\right)\end{array}\right)$$

This linear system describes the dynamics of the process around the operating point defined in sub-step E41 and defined by: $${\mathrm{<figure-callout\; id="0"\; label="v"\; filenames="imgf0008.png,imgf0011.png"\; state="\{\{state\}\}">v</figure-callout>}}_0=\left(\begin{array}{c}{\dot{m}}_{\mathit{in}}\left(\mathit{last\; name}\right)\\ {\dot{m}}_{\mathit{out}}\left(\mathit{<figure-callout\; id="0"\; label="last\; name\; w"\; filenames="imgf0008.png,imgf0011.png"\; state="\{\{state\}\}">last\; name\; </figure-callout>},\right)& \\ \end{array},\right)$$

$$w_{}$$$${}_0=\left(\begin{array}{c}{Q}_{\mathit{static}}\left(\mathit{last\; name}\right)\\ Q_{\mathit{dynam}}\left(\mathit{last\; name}\right)\\ H_{\mathit{in}}\left(\mathit{last\; name}\right)\end{array}\right)$$

$$x_0={\mathrm{<figure-callout\; id="0"\; label="y"\; filenames="imgf0008.png,imgf0011.png"\; state="\{\{state\}\}">y</figure-callout>}}_0=\left(\begin{array}{c}u\left(\mathit{last\; name}\right)\\ \rho \left(\mathit{last\; name}\right)\end{array}\right)$$

Thanks to the thermodynamic model previously established during the second step E2, we can determine the values of the state matrices of the system A, B, C and D. For this purpose we can use a linearization function of a calculation tool such as the "linearize" function of Matlab®. In practice, like x = y, the state matrix C is equal to a unitary matrix and the state matrix D is equal to a null matrix. The linear system thus obtained describes the thermodynamic behavior of the cryomodule around the operating point defined in substep E41.

In a fifth step E5, an OBS thermal load observer is set up as illustrated on the figure 8 . The model inputs are the height h of liquid helium, the internal pressure P of the helium bath, the helium pressure P _out downstream of the outlet valve CV005, the helium pressure P _in upstream of the valves CV001 and CV002, the position of the three valves POS _{CV 001} , POS _{CV 002} , POS _{CV 005} , the enthalpy of helium in upstream of the inlet valves H _in . All of these inputs are measured with the exception of H _in , a value estimated from two sensors located upstream of the cryomodule.

In a first substep E51, the signals supplied by the sensors are filtered so as to reduce their noise. First order filtering of the following form can be used: $$H\left(p\right)=\frac{1}{1+{\tau }_{\mathit{filtered}}\mathrm{.}p}$$

In which τ _filter is a time constant of the filter, chosen as a function of a time constant τ _process of the process, such that: $${\tau }_{\mathit{filtered}}\ll 5.{\tau }_{\mathit{proc}édé}$$

In a second substep E52, the density p and the internal energy u of the helium bath are calculated from the level of liquid helium h and the internal pressure P in the enclosure. First, the volume of liquid helium V _liq is calculated from the level measurement according to the formula V _liq = f ₁ (h) in which f ₁ is the function giving the volume of liquid helium V _liq in depending on the height indicated by the LT level sensor. From the volume of liquid helium it is possible to deduce the mass titre X of the fluid contained in the cryomodule using the formula: $$X=\frac{m_{\mathit{gas}}}{m_{\mathit{gas}}+m_{\mathit{liq}}}=\frac{{\rho }_{\mathit{gas}}\mathrm{.}{V}_{\mathit{gas}}}{{\rho }_{\mathit{gas}}\mathrm{.}{V}_{\mathit{gas}}+{\rho }_{\mathit{liq}}\mathrm{.}{V}_{\mathit{liq}}}$$

In which m _gas , ρ _gas , V _gas respectively denote the mass, density and volume of helium in gaseous form and m _liq , ρ _liq , V _liq respectively denote the mass, density and volume of helium in liquid form.

As explained above, the density of liquid helium ρ _liq can be determined using the third helium property table, as a function of internal pressure P and knowing that it is a saturated liquid.

Similarly, the density of helium gas ρ _gas can be determined using the fourth helium property table, depending on the internal pressure P and knowing that it is saturated vapor.

We can thus calculate the mass title X, then from, from a fifth and sixth helium property table, deduce respectively the density ρ and the internal energy u of the helium bath.

In a third substep E53, the flow ṁ through each of the valves CV001, CV002 and CV005 is calculated in accordance with the logic diagram illustrated in the figure 9 . This calculation includes a first substep E531 for adjusting the measurement of the position of a valve and a second substep E532 for calculating the mass flow through a valve by means of the modeling established during the substep. step E21.

où :

• $\mathit{CV}00{i_{\mathit{pos}}}_{\mathit{mes}}^{\mathit{nom}}$
  
  désigne la valeur nominale mesurée de la position de la vanne i au point de fonctionnement défini à la sous-étape E41.
• $\mathit{CV}00{i_{\mathit{pos}}}_{\mathit{sim}}^{\mathit{nom}}$
  
  désigne la valeur nominale simulée de la position de la vanne i au point de fonctionnement défini à la sous-étape E41.

Sub-step E531 aims to compensate for deviations and drifts observed between the simulated mass flow and the mass flow observed through a valve CV00i (i being equal to 1, 2 or 5 depending on the valve considered). In order to compensate for a difference between the simulated mass flow and the observed mass flow, a static offset CV 00 i _offset is defined. _stat applied to the measurement of the position of valve i by the following formula: $$\mathit{CV}00i_{{\mathit{offset}}_{\mathit{stat}}}=\mathit{CV}00{i_{\mathit{pos}}}_{\mathit{my}}^{\mathit{last\; name}}-\mathit{CV}00{i_{\mathit{pos}}}_{\mathit{sim}}^{\mathit{last\; name}}$$

or :

• $\mathit{CV}00{i_{\mathit{pos}}}_{\mathit{my}}^{\mathit{last\; name}}$
  
  designates the measured nominal value of the position of valve i at the operating point defined in substep E41.
• $\mathit{CV}00{i_{\mathit{pos}}}_{\mathit{sim}}^{\mathit{last\; name}}$
  
  designates the simulated nominal value of the position of valve i at the operating point defined in substep E41.

où :

• CV00i_posmes désigne la valeur courante, mesurée de la position de la vanne i.
• $\mathit{CV}00{i_{\mathit{pos}}}_{\mathit{mes}}^{\mathit{nom}}$
  
  désigne la valeur nominale mesurée de la position de la vanne i au point de fonctionnement défini à la sous-étape E41.
• gain désigne un coefficient de proportionnalité à régler à partir des mesures faites sur le système.

In order to correct a drift in the mass flow through the valve, we can also define a dynamic offset by the following formula: $$\mathit{CV}00i_{{\mathit{offset}}_{\mathit{dyn}}}=\left(\mathit{CV}00i_{{\mathit{pos}}_{\mathit{my}}}-\mathit{CV}00{i_{\mathit{pos}}}_{\mathit{my}}^{\mathit{last\; name}}\right)\mathrm{.}\mathit{gain}$$

or :

• CV 00 i _{pos my} designates the current value, measured from the position of valve i.
• $\mathit{CV}00{i_{\mathit{pos}}}_{\mathit{my}}^{\mathit{last\; name}}$
  
  designates the measured nominal value of the position of valve i at the operating point defined in substep E41.
• gain designates a proportionality coefficient to be adjusted from the measurements made on the system.

The corrected position CV 00 i _{pos corr} CV00i valve can then be obtained by adding the measured position, the static offset and the dynamic offset, or by the following equation: $$\mathit{CV}00i_{{\mathit{pos}}_{\mathit{corr}}}=\mathit{CV}00i_{{\mathit{pos}}_{\mathit{my}}}+\mathit{CV}00i_{{\mathit{offset}}_{\mathit{stat}}}+\mathit{CV}00i_{{\mathit{offset}}_{\mathit{dyn}}}$$

The block identified by E531 on the figure 10 illustrates a logic diagram for implementing the CV 00 i _pos calculation formula _corr defined above.

During the second substep E532, the modeling established during the substep E21 is used. This modeling makes it possible to calculate the mass flow through a valve CV00i as a function of the corrected position CV 00 i _{pos corr} of the CV00i valve calculated previously, of the pressure upstream of the valve P _in , of the pressure downstream of the valve P _out and of the enthalpy upstream of the valve H _in (assumed to be identical to the enthalpy downstream of the valve). Sub-step E53 is then repeated for each of the valves CV001, CV002 and CV005 of the system so as to determine the mass flow passing through each of these valves as a function of the current value, measured CV 00 i _{pos my} the valve position i.

In a sub-step E54, a state observer, known as a Kalman observer, is implemented, in accordance with the diagram defined in the figure 8 . The state observer includes the state matrices A, B, C and D defined during sub-step E43. L denotes the gain of the observer calculated for the system. The block made up of the symbol ∫ represents an integrator.

The resulting system is an invariant linear system, for which there is a Kalman estimator obtained by solving a Riccati equation with differences, for example using the "Iqr" function of Matlab® with L = Iqr (A, C, Q, R) in which Q and R are weighting matrices. In other words, it is a question of finding the gain L which minimizes the following criterion: $$J=\sum _{k=1}^{\infty }{x}^T\mathit{Qx}+u^T\mathit{Ru},\mathrm{with}u=-\mathit{Lx}$$

$$R=\left[\begin{array}{cccc}1e3& 0& 0& 0\\ 0& 1e3& 0& 0\\ 0& 0& 1e2& 0\\ 0& 0& 0& 1e2\end{array}\right]$$

For example, the matrices Q and R can be written in the following form: $$Q=\left[\begin{array}{cc}1\mathrm{<figure-callout\; id="2"\; label="e"\; filenames="imgf0001.png,imgf0012.png"\; state="\{\{state\}\}">e</figure-callout>}2& 0\\ 0& 1e3\end{array}\right]$$

$$R=\left[\begin{array}{cccc}1\mathrm{<figure-callout\; id="3"\; label="e"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">e</figure-callout>}3& 0& 0& 0\\ 0& 1\mathrm{<figure-callout\; id="3"\; label="e"\; filenames="imgf0001.png,imgf0002.png"\; state="\{\{state\}\}">e</figure-callout>}3& 0& 0\\ 0& 0& 1\mathrm{<figure-callout\; id="2"\; label="e"\; filenames="imgf0001.png,imgf0012.png"\; state="\{\{state\}\}">e</figure-callout>}2& 0\\ 0& 0& 0& 1\mathrm{<figure-callout\; id="2"\; label="e"\; filenames="imgf0001.png,imgf0012.png"\; state="\{\{state\}\}">e</figure-callout>}2\end{array}\right]$$

The state observer thus implemented makes it possible to determine and observe in real time the dynamic load Q _dynam .

Advantageously, the calculation method thus developed can be validated by means of an experiment on a cryogenic system when the latter comprises a cavity equipped with a heat generation device such a heating resistor with variable power supply, also known as a “heater”. Such a resistance simulates a heat supply identical to that which would be produced by the operation of the cavity in a particle accelerator. The heating resistance delivers heat equivalent to the dynamic thermal load Q _dynam . The figure 11A is a graph representing as a function of time, the power Q _ref delivered by the heating resistor, the dynamic thermal load Q1 calculated without applying any treatment to the nonlinearities (that is to say using the complete linear model of the cryomodule, integrating the modeling of the valves and the calculation of the liquid level of the bath), and the dynamic thermal load Q2 calculated by the state observer described previously. It is observed that the power Q _ref provided by the heating resistor is stable at a value of 47W. The calculated value of the dynamic thermal load Q1 oscillates around a power of around 40W. The calculated value of the dynamic thermal load Q2 oscillates around a power of approximately 47W and converges more quickly towards this value when the heating resistance is activated. The transient regime of the dynamic thermal load Q1 is reduced from 20 to 30 seconds compared to the dynamic thermal load Q2. The figure 11B is a graph representing as a function of time the error of estimation of the dynamic thermal load Q1 and the dynamic thermal load Q2. The error is greater for the calculation of the dynamic thermal load Q1 than for the calculation of the dynamic thermal load Q2. This experience therefore shows that the determination of the dynamic thermal load carried out by the state observer according to the invention is precise and reliable.

In a sixth step E6, the quality factor Q ₀ of the cavity 4 is calculated. The quality factor is a measure of the damping rate of an oscillating system. The quality factor depends on the temperature T of the internal wall of the cavity, assumed to be uniform, of the material of the cavity and of its geometric shape. It is defined by the ratio of the energy U stored in the cavity on the energy dissipated P _loss in the walls of the cavity, per period of oscillation. The quality factor can therefore be expressed by the following formula: $$Q_0\left(T\right)=\frac{\omega \mathrm{.}U}{P_{\mathit{loss}}}$$

In which ω denotes the resonance pulsation of the cavity.

$$P_{\mathit{loss}}=\frac{1}{2}{R}_s{\oint }_s{\left|H\right|}^2\mathit{ds}$$

To calculate the energy U stored in the cavity and the energy dissipated P _loss in the walls of the cavity, we make the hypotheses that a perfect vacuum prevails in the cavity and that the resistivity of the walls of the cavity is uniform over l all of their surfaces. Knowing that the energy stored in the electric field is equal to the energy stored in the magnetic field, that the internal energy of the cavity is calculated on the volume and that the losses are concentrated on the surface of the cavity, we can express U and P _loss by integrals respectively on the volume of the cavity and on the surface of the walls of the cavity. U and P _loss can therefore be expressed according to the following formulas: $$U=\frac{1}{2}{\mathrm{<figure-callout\; id="0"\; label="\mu "\; filenames="imgf0008.png,imgf0011.png"\; state="\{\{state\}\}">\mu </figure-callout>}}_0{\int }_{\mathrm{<figure-callout\; id="2"\; label="v\; H"\; filenames="imgf0001.png,imgf0012.png"\; state="\{\{state\}\}">v\; </figure-callout>}}^{}{\left|H\right|}^{}{\left|\right|}^2\mathit{dv}$$

$$P_{\mathit{loss}}=\frac{1}{2}{R}_s{\oint }_s{\left|\mathrm{<figure-callout\; id="2"\; label="H"\; filenames="imgf0001.png,imgf0012.png"\; state="\{\{state\}\}">H</figure-callout>}\right|}^2\mathit{ds}$$

• µ ₀ désigne la perméabilité magnétique du vide,
• R_s désigne la résistance de surface de la cavité et sa valeur dépend de la température de la cavité,
• H désigne le champ magnétique à l'intérieur de la cavité.

Formulas in which:

• µ ₀ designates the magnetic permeability of the vacuum,
• R _s denotes the surface resistance of the cavity and its value depends on the temperature of the cavity,
• H designates the magnetic field inside the cavity.

Thus, the quality factor Q ₀ can be expressed in the following form: $$Q_0\left(T\right)=\frac{G}{R_s\left(T\right)}$$

Where G is called the geometric factor of the cavity, and is defined by: $$G=\omega \mathrm{.}{\mu }_0\frac{{\int }_{\mathrm{<figure-callout\; id="2"\; label="v\; H"\; filenames="imgf0001.png,imgf0012.png"\; state="\{\{state\}\}">v\; </figure-callout>}}^{}{\left|H\right|}^{}{\left|\right|}^2\mathit{dv}}{{\oint }_s{\left|\mathrm{<figure-callout\; id="2"\; label="H"\; filenames="imgf0001.png,imgf0012.png"\; state="\{\{state\}\}">H</figure-callout>}\right|}^2\mathit{ds}}$$

The geometric factor G is a known and invariant datum which can be directly calculated from a radiofrequency model of the cryomodule. Considering the known geometric factor G, we still have to find the expression of the surface resistance R _s (T) to deduce the quality factor. The surface resistance R _s (T) satisfies the following equation: $$R_S\left(T\right)=R_{\mathit{BCS}}\left(T\right)+R_{\mathit{res}}$$

• R_res désigne la résistance résiduelle du niobium,
• R_BCS(T) est une résistance variable définie par l'équation suivante : $$R_{\mathit{BCS}}\left(T\right)=\frac{A}{T}\mathrm{.}{\mathrm{<figure-callout\; id="2"\; label="f"\; filenames="imgf0001.png,imgf0012.png"\; state="\{\{state\}\}">f</figure-callout>}}^2\mathrm{.}\exp \left(\frac{-\mathrm{\Delta }}{k_B\mathrm{.}T}\right)$$
  

Equation in which:

• R _res denotes the residual resistance of niobium,
• R _BCS (T) is a variable resistance defined by the following equation: $$R_{\mathit{BCS}}\left(T\right)=\frac{\mathrm{AT}}{T}\mathrm{.}{\mathrm{<figure-callout\; id="2"\; label="f"\; filenames="imgf0001.png,imgf0012.png"\; state="\{\{state\}\}">f</figure-callout>}}^2\mathrm{.}\exp \left(\frac{-\mathrm{\Delta }}{k_B\mathrm{.}T}\right)$$
  

• T désigne la température de la paroi interne de la cavité,
• A désigne une constante dépendant des propriétés du matériau utilisé pour la fabrication des parois de la cavité, notamment du niobium,
• f désigne la fréquence de résonance de la cavité,
• Δ désigne l' « energy gap » du matériau utilisé pour la fabrication des parois de la cavité,
• k_B désigne la constante de Boltzmann.

Equation in which:

• T designates the temperature of the internal wall of the cavity,
• A denotes a constant depending on the properties of the material used for the manufacture of the walls of the cavity, in particular niobium,
• f denotes the resonance frequency of the cavity,
• Δ designates the “energy gap” of the material used for the manufacture of the walls of the cavity,
• k _B denotes the Boltzmann constant.

Among all these parameters, only the temperature T is unknown and subject to variations. All the other parameters A, f, Δ, k _B are known or measurable fixed values.

The temperature T of the internal wall of the cavity can precisely be estimated from the power dissipated in the cavity and the temperature of the helium bath. Indeed, the heat dissipated on the interior surface of the cavity is transmitted to the helium bath by conduction through the walls in niobium of the cavity. By considering that the cavity gives up all its heat to the helium bath we deduce that the energy dissipated P _loss in the walls of the cavity is equal to the dynamic thermal load Q _dynam .

In addition, a thermal conduction equation applied to the walls of the cavity is expressed as follows: $$P_{\mathit{loss}}=\frac{\lambda \left(T\right)\mathrm{.}S}{e}\mathrm{.}\left(T_{\mathit{cavity}}-T_{\mathit{bath}}\right)=Q_{\mathit{dynam}}$$

• λ(T) désigne la conductivité thermique du niobium,
• S désigne la surface d'échange entre la cavité et le bain d'hélium,
• e désigne l'épaisseur de la paroi de la cavité,
• T_cavité désigne la température de la paroi interne de la cavité,
• T_bain désigne la température du bain d'hélium.

Equation in which:

• λ (T) denotes the thermal conductivity of niobium,
• S denotes the exchange surface between the cavity and the helium bath,
• e denotes the thickness of the wall of the cavity,
• T _cavity designates the temperature of the internal wall of the cavity,
• T _bath designates the temperature of the helium bath.

The temperature of the helium _bath T _bath can be interpolated from a helium property table by knowing the internal pressure P of the helium bath (regulated around a value of 1200 mBar) and knowing that it it is saturated liquid.

In remark, the internal pressure of the helium bath is supposed in this modelization like uniform. The modeling could be refined by considering the pressure as a function of the height of the point considered in the helium bath. It would then be possible to define a temperature gradient in the helium bath rather than considering a uniform temperature.

Finally, like the thermal conductivity of niobium λ (T) , the surface S of exchange between the cavity and the helium bath, the thickness e of the wall of the cavity are known quantities, and, as we have d '' an estimate of the dynamic thermal load Qdynam provided by the state observer, we can calculate the temperature of the internal wall of the cavity T _cavity . Once this temperature has been determined, it is possible to calculate the value of the variable resistance R _BCS (T), then the surface resistance R _S (T), and finally the quality factor Q ₀ .

Finally, the quality factor Q0 can be expressed by the following formula: $${\mathrm{<figure-callout\; id="0"\; label="Q"\; filenames="imgf0008.png,imgf0011.png"\; state="\{\{state\}\}">Q</figure-callout>}}_0=G/\left(\frac{\mathrm{AT}}{Q_{\mathit{dynam}}\mathrm{.}\frac{e}{\lambda \left(T\right)\mathrm{.}S}+T_{\mathit{bath}}}\mathrm{.}{\mathrm{<figure-callout\; id="2"\; label="f"\; filenames="imgf0001.png,imgf0012.png"\; state="\{\{state\}\}">f</figure-callout>}}^2\mathrm{.}\exp \left(\frac{-\mathrm{\Delta }}{k_B\mathrm{.}\left(Q_{\mathit{dynam}}\mathrm{.}\frac{e}{\lambda \left(T\right)\mathrm{.}S}+T_{\mathit{bath}}\right)}\right)+R_{\mathit{res}}\right)$$

The invention also relates to a method of operating a particle accelerator comprising the implementation of the method for determining the quality factor Q0 as described above, in particular steps E4 to E6 and a step E7 of modifying at least one operating parameter of the accelerating cavity as a function of the quality factor Q0. For example, the operating parameter can be a power control value of a radiofrequency wave emitted in the cavity by the antenna 8. The modification can consist of a reduction in the value of the power control until the stopping the emission of the radiofrequency wave if the quality factor of at least one accelerating cavity crosses a predetermined threshold, the other accelerator cavities, when they exist, can continue to operate. The reduction in the value of the command can possibly be continued until the emission of the radiofrequency wave stops. The invention can also be implemented during a ramp-up of the particle accelerator, for example by gradually increasing the value of the order as a function of the determined quality factor. The modification can also consist of any other modification to the configuration and / or adjustment of the particle accelerator.

Concretely, the control means 12, integrated into the industrial programming PLC API, can for example compare the estimate of the quality factor with a threshold value. If the quality factor Q0 is greater than a predetermined threshold, then the regulating means 12 can send a command order to the radiofrequency system to reduce the power of the waves emitted by the radiofrequency antenna 8 integrated into the cavity. The regulating means 12 can possibly comprise several thresholds beyond which the power of the waves emitted by the radiofrequency antenna will be successively reduced until reaching zero power. Thus, each cavity can be used at optimum power taking into account its quality factor and without impacting the operation of the other cavities of the particle accelerator. Preferably, the operating method comprises several iterations of steps E4 to E7.

During operation, the CTRL and Coord control structures illustrated on the Figures 4A, 4B, 4C and 4D use the estimation of the thermal load acting on the helium bath to regulate the opening of the CV002 and CV005 valves and maintain the cryomodule around an optimal operating point.

Conclusion

The measurement principle illustrated through this invention is fundamentally different from conventional measurement because it is based on the thermal state of the cryogenic fluid bath in which the cavity is immersed and not on a direct measurement of the radiofrequency field in the cavity.

The thermal load is estimated using sensors specific to the cryogenic system, so determining the quality factor Q ₀ does not require a pick-up probe, a network analyzer or any other means dedicated to determining the quality factor. In addition an estimate of the quality factor during operation (and not static) is carried out. This determination can be made deferred or in real time but always during the operation of the particle accelerator, unlike the prior art. In other words, the invention makes it possible to estimate the accelerating potential of an accelerating cavity (the maximum admissible power) at any time and to adapt the power emitted by the radiofrequency antenna accordingly. This estimate does not require any physical modification of the system in place, that is to say that it does not require the addition of a sensor or other measuring device and neither does it require knowledge of a voltage applied to the cavity. Knowledge of the quality factor, in particular in real time, makes it possible to balance both the cryogenic behavior and the behavior of the radiofrequency system acting on the cavity to allow reliable operation of the accelerator. The device produced is sufficiently economical in computing power to be installed on a programmable controller.

The method according to the invention is executable from the injection of radiofrequency power into the superconductive cavities and even before there is a beam. It is also executable with the beam and allows to diagnose certain possible anomalies which would generate an abnormal thermal load in the cavities.

Claims (13)

Method for determining a quality factor (Q0) of a superconductive cavity (4) accelerating a particle accelerator (1), in particular a linear type particle accelerator, the method comprising the following steps: - A determination of a thermal load (Q _dynam ) to which a cryomodule (3) is subjected comprising the accelerating cavity (4) and a bath of cryogenic fluid (5), then - a determination of a quality factor (Q0) based on the determination of the thermal load (Q _dynam ) during the operation of the particle accelerator. Determination method according to claim 1, characterized in that the steps of determining the thermal load (Q _dynam ) and determining the quality factor (Q0) are carried out simultaneously and in real time. Determination method according to claim 1 or 2, characterized in that the step of determining the thermal load comprises the use of a state observer (OBS). Determination method according to claim 3, characterized in that the use of a state observer (OBS) comprises an estimation of a mass flow ( ṁ ) of cryogenic fluid passing through a valve of the cryomodule in the form $$\dot{m}={\beta }_T\mathrm{.}{\dot{m}}_{\mathit{comp}}+\left(1-{\beta }_T\right)\mathrm{.}{\dot{m}}_{\mathit{incomp}}$$

in which : - ṁ _comp designates a mass flow of cryogenic fluid in compressible form through the valve, - ṁ _incomp designates the mass flow rate of cryogenic fluid in incompressible form through the valve, and - β _T denotes an isothermal compressibility coefficient of the cryogenic fluid. Determination method according to claim 3 or 4, characterized in that the state observer comprises an estimate of a density ( ρ ) and of a specific internal energy (u) of the cryogenic fluid bath (5). Determination method according to claim 5 characterized in that said estimation is carried out from: - a volume (V _liq ) of cryogenic fluid in the liquid state calculated from a measurement of a height (h) of the cryogenic fluid in the liquid state and / or calculated from a measurement of the quantity of cryogenic fluid entering and leaving the cryogenic fluid bath (5); and - a static thermal load (Q _static ) and a dynamic thermal load (Q _dynam ) received by the cryogenic fluid bath (5); and - a specific enthalpy of inlet (H _in ) and outlet (H _out ) of the cryogenic bath based on a measurement of the pressure (P) of the bath of cryogenic fluid (5), or
an outlet temperature of the cryogenic fluid bath (5) based on a measurement of the pressure (P) of the cryogenic fluid bath (5) and of the mass titer (X) of the inlet of the cryogenic fluid bath (5) . Method for operating a particle accelerator (1), in particular a linear type particle accelerator, comprising at least one accelerating cavity (4), the operating method comprising implementing the method for determining a quality factor (Q0) of at least one accelerating cavity (4) according to one of the preceding claims and a step of modifying at least one operating parameter of said accelerating cavity as a function of its quality factor (Q0). Operating method according to the preceding claim, characterized in that said operating parameter is a power control value of a radiofrequency wave emitted in the accelerating cavity (4), and in that the modification step comprises a reduction of the value of the power command if the quality factor (Q0) of the at least one accelerating cavity (4) crosses a predetermined threshold, the other cavities of the particle accelerator (1), when they exist, can continue to operate. Device for determining (9) a quality factor (Q0) of at least one accelerating cavity (4) of a particle accelerator (1), the device (9) comprising elements (LT, PT, API , HMI) hardware and / or software implementing the method according to one of claims 1 to 8, in particular hardware elements (LT, PT, API, HMI) and / or software designed to implement the method according to one of claims 1 to 8. Particle accelerator (1), in particular linear type particle accelerator, comprising at least one determination device (9) according to the preceding claim. Particle accelerator (1) according to the preceding claim, characterized in that it comprises at least one cryomodule (3) comprising an accelerating cavity (4) or several accelerating cavities (4) and a bath of a cryogenic fluid (5) . Program product for computer comprising program code instructions recorded on a medium readable by a computer to implement the steps of the method according to one any one of claims 1 to 8 when said program runs on a computer or computer program product downloadable from a communication network and / or recorded on a data medium readable by a computer and / or executable by a computer, characterized in that that it comprises instructions which, when the program is executed by a computer, lead the latter to implement the method according to any one of claims 1 to 8. Data recording medium, readable by a computer (HMI, API), on which is recorded a program for a computer comprising program code instructions for implementing the method according to one of claims 1 to 8 or medium recording readable by a computer comprising instructions which, when executed by a computer, lead the latter to implement the method according to any one of claims 1 to 8.

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