WO2024008316A1

WO2024008316A1 - Method for controlling humidity of a membrane of a fuel cell

Info

Publication number: WO2024008316A1
Application number: PCT/EP2022/069193
Authority: WO
Inventors: Fredrik Blomgren; Staffan LUONG; Johan Lindberg; Vijai Shankar BHAVANI SHANKAR
Original assignee: Volvo Truck Corporation
Priority date: 2022-07-08
Filing date: 2022-07-08
Publication date: 2024-01-11

Abstract

A method of operating a controller for controlling a humidifier of a fuel cell system comprising a fuel cell, the humidifier comprising a humidifier membrane. The method comprises obtaining at least one of a predetermined value of humidity and a predetermined value of temperature of an inlet air stream of the fuel cell that are required for humidifying a membrane of the fuel cell; providing the at least one of the predetermined values to a humidifier control model that repeatedly provides an analytically-derived, numerically stable solution comprising an amount of water vapor to be transferred across the humidifier membrane from the humid side to the dry side of the humidifier membrane, to humidify the membrane of the fuel cell in accordance with the at least one of the predetermined values; and controlling the amount of water vapor to be transferred across the humidifier membrane of the humidifier.

Description

METHOD FOR CONTROLLING HUMIDITY OF A MEMBRANE OF A FUEL CELL

TECHNICAL FIELD

Embodiments herein relate to controlling a water vapor transfer across a humidifier membrane of a humidifier of a fuel cell system, to thereby control a water vapor content in an air intake of the fuel cell system.

The disclosure may be applied to heavy-duty vehicles, such as trucks, busses, and construction equipment.

BACKGROUND

A fuel cell is a power generation device that converts fuels and oxidizing agents electrochemically into electrical energy. This takes place without combustion, and it may be possible to continuously generate electrical energy as long as as fuel and oxidizing agent are supplied. Because fuel cells only emit water vapor and heat, they are gaining increasing interest in various applications due to having lower impact on the environment and assisting in reducing or eliminating reliance on fossil fuels.

In proton-exchange membrane fuel cells (PEMFC), also known as polymer electrolyte membrane (PEM) fuel cells, a proton-conducting polymer electrolyte membrane is used, and the chemical energy in a fuel, such as hydrogen gas, and an oxidizing agent, such as air/oxygen, is converted directly into electrical energy. Among various types of fuel cells, PEM fuel cells are considered to be most suitable for transport applications, as well as for stationary fuel-cell applications and portable fuel-cell applications.

In PEM fuel cells, hydrogen is supplied to an active surface of the anode (anode side) and is broken down into hydrogen ions (protons) and electrons. The electrons are conducted via the anode to an external electric circuit, and hydrogen ions are transported through the electrolyte/the membrane to the cathode. At the active surface of the cathode (cathode side), the oxidizing agent, such as air, is supplied and reacts with the hydrogen ions, forming heat and water. The external electric circuit can be used for, for example, driving a vehicle, charging batteries, or driving peripheral equipment in vehicles or other applications. A number of fuel cells are usually assembled into what is known as a fuel cell stack in order for it to be possible to deliver sufficiently high power and/or voltage for the application concerned.

One of the most important factors in proper performance, reliability, and durability of a PEM fuel cell is the ability to supply an appropriate amount of moisture to a polymer electrolyte membrane or a proton exchange membrane (collectively referred to as a PEM) of a membrane electrode assembly of the fuel cell, in order to retain moisture content. This is because, if the PEM is dried or is insufficiently moisturized, power generation efficiency is abruptly reduced. Potential issues also stem from accumulation of excess water which may cause water flooding (so called cathode flooding) which hinders the transport of oxygen through the PEM by blocking the pores of the cathode and anode catalysts, which has detrimental effects on the performance, durability, and life of the fuel cell.

Several methods exist for humidification of a PEM, among which there is a membrane humidification method of supplying moisture to an intake air of the fuel cell. The PEM humidification may depend on a proper control of transfer of water vapor across the humidifier membrane. Although techniques exist for controlling moisture at the PEM, there are shortcomings related to the lack of accuracy and predictability.

Accordingly, there is a need for methods for controlling moisture at the PEM of a fuel cell.

SUMMARY

Accordingly, the object of the present disclosure is to provide a method of controlling a humidifier of a fuel cell system, to control moisture at a PEM of a fuel cell.

The membrane humidification method involves the use of a humidifier which provides water vapor and heat to the intake air that is supplied to the PEM. The humidifier includes a humidifier membrane that selectively transmits water vapor and heat extracted from an exhaust stream of the fuel cell, in order to deliver an air/oxygen to the PEM having a moisture level that permits proper humidification of PEM. The membrane humidification method is advantageous because it is possible to reduce a weight and size of a humidifier. At the same time, the PEM humidification depends on an accurate control of transfer of water vapor and heat across the humidifier membrane.

Accordingly, the object is achieved by providing a method for controlling a flow of water vapor and heat through a humidifier membrane of a fuel cell system, to maintain proper moisture levels of a fuel cell membrane and thus reduce a risk of degradation of the fuel cell. A numerically more stable and less computationally demanding method is provided that involves calculating mass and heat transfer across the humidifier membrane using a mass and heat transfer equation that is solved analytically.

According to aspects of the present disclosure, the object is achieved by providing a method of operating a controller for controlling a humidifier of a fuel cell system comprising a fuel cell, the humidifier comprising a humidifier membrane separating a humid side from a dry side of the humidifier and being selectively permeable to water vapor such that the water vapor can diffuse through the humidifier membrane from the humid side to the dry side. The method comprises obtaining at least one of a predetermined value of humidity and a predetermined value of temperature of an inlet air stream of the fuel cell that are required for humidifying a membrane of the fuel cell; providing the at least one of predetermined value of humidity and the predetermined value of temperature to a humidifier control model that repeatedly provides an analytically-derived, numerically stable solution comprising an amount of water vapor to be transferred across the humidifier membrane from the humid side to the dry side of the humidifier membrane, wherein the amount of water vapor to be transferred across the humidifier membrane of the humidifier results in humidifying the membrane of the fuel cell in accordance with the at least one of the predetermined value of humidity and the predetermined value of temperature; and controlling the amount of water vapor to be transferred across the humidifier membrane of the humidifier as the fuel cell operates. Furthermore, according to an aspect, the object is achieved, according to embodiments herein, by providing a fuel cell system configured to perform the methods herein. The fuel cell system may be used, for example, in a vehicle. The vehicle may be a truck.

In certain examples, the humidifier extracts, at the humid side of the humidifier, water vapor and heat from an exhaust stream of the fuel cell, and transfers, at the dry side of the humidifier, the extracted water vapor and heat to the inlet air stream of the fuel cell.

In certain examples, controlling the amount of water vapor to be transferred across the humidifier membrane as the fuel cell operates comprises controlling a bypass valve of the fuel cell system.

In certain examples, the humidifier control model is a thermodynamic model describing the first law of thermodynamics of an open system, the thermodynamic model being used for controlling a heat and water mass transfer across the humidifier membrane.

In certain examples, the humidifier control model repeatedly provides the analytically-derived, numerically stable solution so that no unphysical solution is provided.

In certain examples, the humidifier control model provides the analytically- derived, numerically stable solution at certain time intervals.

In certain examples, the humidifier control model includes an analytical expression for heat and water transfer across the humidifier membrane, and wherein the analytical expression is for evaluating a mass and heat change of the humid and dry side of the humidifier, wherein the analytical expression is derived de from solutions to one or both of a diffusion equation

— = DV²c, and a heat

equation

= c V²T, or a variation thereof.

In certain examples, the humidifier is a counter flow heat and mass exchanger. According to aspects of the present disclosure, the object is achieved by providing a controller for controlling a fuel cell system, the controller being configured to perform the method according to embodiments of the present disclosure.

According to aspects of the present disclosure, the object is achieved by providing a fuel cell system comprising the controller according to embodiments of the present disclosure.

According to aspects of the present disclosure, the object is achieved by providing a vehicle comprising the fuel cell system and/or being in communication with the controller according to embodiments of the present disclosure.

According to aspects of the present disclosure, the object is achieved by providing a computer program product comprising instructions, which, when executed on at least one processor, cause the at least one processor to carry out the method according to embodiments of the present disclosure.

According to aspects of the present disclosure, the object is achieved by providing a computer-readable storage medium, having stored thereon a computer program product comprising instructions which, when executed on at least one processor, cause the at least one processor to carry out the method according to embodiments of the present disclosure.

The present disclosure relates to control of the humidifier of the fuel cell system comprising the fuel cell by using a humidifier control model that provides an analytically-derived, numerically stable solution comprising an amount of water vapor to be transferred across a humidifier membrane of the humidifier from a humid side to a dry side of the humidifier membrane. The approach in accordance with aspects of the present disclosure may result in analytically computed values of an amount of water vapor to be transferred across a humidifier membrane that are numerically stable and correspond to values that may occur in an actual physical system. BRIEF DESCRIPTION OF THE DRAWINGS

Embodiments will now be described in more detail in relation to the enclosed drawings, in which:

Fig. 1 A is a side view of an example of a vehicle comprising a fuel cell system in which a method in accordance with aspects of the present disclosure may be implemented.

Fig. 1 B is a diagram illustrating an example of a fuel cell system in accordance with aspects of the present disclosure.

Fig. 2 is a diagram illustrating a fuel cell and a humidifier of the fuel cell system of Fig. 1 B.

Fig. 3 is a diagram illustrating a model of a humidifier and parameters for heat and water transfer that are determined in accordance with aspects of the present disclosure.

Fig. 4 is a graph illustrating a water vapor concentration as a function of time, as determined using existing approaches and a method in accordance with aspects of the present disclosure.

Fig. 5 is a flowchart illustrating a method of operating a humidifier of a fuel cell, in accordance with aspects of the present disclosure.

Figs. 6A and 6B are schematic block diagrams illustrating an example of a controller, in accordance with aspects of the present disclosure.

DETAILED DESCRIPTION

A low membrane moisture content of a membrane of a fuel cell in a fuel cell system may both damage the membrane long term, which results in the fuel cell degradation, and may also entail a short-term loss of power. A high membrane content can cause flooding of the cathode, also resulting in short-term a loss of power. Accordingly, a proper control of a fuel cell membrane moisture is needed for reduction of degradation and power losses in the fuel cell system. The fuel cell system may include a humidifier that humidifies a water vapor delivered to the membrane, such as a PEM, of the fuel cell. The humidifier may include a humidifier membrane. In the fuel cell system, for the humidifier to be able to humidify the water vapor, that is transferred to the PEM, to a proper level, it is required to control a water vapor and heat transfer, e.g., a mass of the water vapor and heat, across the humidifier membrane of the humidifier.

Accordingly, there is a need in improved techniques for controlling a moisture level of a fuel cell membrane, particularly using a humidifier, to prevent a fuel cell degradation and power losses.

Aspects of the present disclosure address the above need by providing a method of operating a controller for controlling a humidifier of a fuel cell system comprising a fuel cell, the humidifier comprising a humidifier membrane separating a humid side from a dry side of the humidifier and being selectively permeable to water vapor such that the water vapor can diffuse through the humidifier membrane from the humid side to the dry side. The method comprises obtaining at least one of a predetermined value of humidity and a predetermined value of temperature of an inlet air stream of the fuel cell that are required for humidifying a membrane of the fuel cell. The method also comprises providing the at least one of the predetermined value of humidity and the predetermined value of temperature to a humidifier control model that repeatedly provides an analytically- derived, numerically stable solution comprising an amount of water vapor to be transferred across the humidifier membrane from the humid side to the dry side of the humidifier membrane, wherein the amount of water vapor to be transferred across the humidifier membrane of the humidifier results in humidifying the membrane of the fuel cell in accordance with the at least one of the predetermined value of humidity and the predetermined value of temperature; and controlling the amount of water vapor to be transferred across the humidifier membrane of the humidifier as the fuel cell operates.

The humidifier control model may provide the analytically-derived, numerically stable solution so that no unphysical solution is provided.

FIG. 1A depicts a side view of a vehicle 10 according to an example embodiment of the present disclosure. The vehicle 10 is shown as a truck, such as a heavy-duty truck for towing one or more trailers (not shown). It should however be appreciated that the present disclosure is not limited to this, or any other specific type of vehicle, but may be used in any other type of vehicle, such as a bus, construction equipment, e.g. a wheel loader and an excavator, a passenger car, an aircraft, and a marine vessel. The present disclosure is also applicable for other applications not relating to vehicles as long as a fuel cell system and a controller are utilized.

As shown schematically in FIG. 1A, the vehicle 10 comprises a fuel cell system 100 which may be used for powering one or more electric motors (not shown) which are used for creating a propulsion force to the vehicle 10. The fuel cell system 100 may additionally or alternatively be used for powering other electric power consumers (not shown) of the vehicle 10, such as an electric motor for a crane, an electric motor for a refrigerator system, an electric motor for an air conditioning system, or any other electric power consuming function of the vehicle 10.

The fuel cell unit or system 100 comprises one or more, typically multiple, fuel cells, which together form a fuel cell stack. The fuel cell system 100 may include one or more fuel cell stacks. Also, the fuel cell system 100 may comprise one or more fuel cell systems, such that the vehicle 10 may have multiple fuel cell systems.

The fuel cell system 100 is arranged to provide the fuel cells with necessary supply of hydrogen fuel (H2) and air, cooling, etc., and the fuel cell system 100 may include various components, some of which are shown in FIG. 1 B.

The vehicle 10 further comprises a control unit or controller 116 according to an example embodiment of the present disclosure. The fuel cell system 100 may be communicatively coupled to the control unit or controller 116. In implementations in which the fuel cell system 100 comprises multiple fuel cell systems, each fuel cell system may comprise its own control system, which may be communicatively coupled to the controller 116. The controller 116 may be used for controlling the fuel cell system 100, and the controller 116 may be configured to control the humidifier of the fuel cell system.

Even though an on-board controller 116 is shown, it shall be understood that the controller 116 may also be a remote controller 116, i.e. an off-board control unit, or a combination of an on-board and off-board control unit or units. The controller 116 may be configured to control the fuel cell system 100 by issuing control signals and by receiving status information relating to the fuel cell system 100 and its components. The controller 116 may also be configured to receive information from various sensors, including one or more of temperature sensors, moisture sensors, and other sensors included in or associated with the vehicle 10. For example, a temperature sensor may be positioned such that it can measure a temperature of inlet air in the inlet air stream that is acquired from the environment.

The controller 116 may be an electronic control unit and may comprise processing circuitry which is adapted to execute a computer program code or computer-executable instructions as disclosed herein. The controller 116 may comprise hardware, firmware, and/or software for performing the method according to embodiments of the present disclosure. The controller 116 may be denoted a computer. The controller 116 may be constituted by one or more separate sub-control units. In addition, the controller 116 may communicate by use of wired and/or wireless communication means.

Further, additionally or alternatively to what is mentioned above, the controller 116 may comprise various other components.

Although the present disclosure is described with respect to a vehicle such as a truck, aspects of the present disclosure are not limited to this particular vehicle, but may also be used in other vehicles such as passenger cars, off-road vehicles, aircrafts and marine vehicles. The present disclosure may also be applied in vessels and in stationary applications, such as in grid-connected supplemental power generators or in grid-independent power generators.

Fig. 1B shows the fuel cell system 100 that comprises a fuel cell stack or fuel cell 102. It should be appreciated that, in the fuel cell system 100, two or more fuel cells form a fuel cell stack. In the fuel cell 102, an electrolyte, such as a fuel cell membrane 104, is sandwiched between two electrodes or catalyst layers - an anode 103 and a cathode 105. The fuel cell membrane 104 may be a solid electrolyte such as a PEM, and it is referred to herein as PEM 104, additionally to distinguish it from a humidifier membrane (which however may also be a PEM). The fuel cell system 100 also comprises a hydrogen tank 106 storing fuel such as hydrogen gas, a humidifier 108 comprising a humidifier membrane 110, an air compressor 112, and a controller 116. The fuel cell system 100 may further comprise a turbine 114 as shown in Fig. 1 B. The fuel cell system 100 may be a fuel cell system of an electric vehicle.

The hydrogen gas is supplied from the hydrogen tank 106 to the anode 103, also referred to as the anode side, where the hydrogen is separated into protons and electrons. The protons are allowed to pass through the PEM 104 to the cathode 105, also referred to as the cathode side, while the electrons are forced to follow an external circuit (not shown) to the cathode and the flow of electrons passing through the external circuit generates electricity which can be used to power a vehicle or another system. The protons and electrons arriving at the cathode side enter into a reaction with oxygen molecules provided in an inlet air stream to the cathode, thereby forming water.

The inlet air, also referred to herein as an inlet air stream, is acquired from an outside environment and may pass through the air compressor 112 where it is compressed. The compressed air is fed to the humidifier 108 where it is humidified, using the recirculated exhaust stream, such that the humidified air is transferred to the PEM 104. The compressed air may also be warmed up at the humidifier 108, such that the warm humidified air is transferred to the PEM 104.

As shown in Fig. 1 B, water and heat produced at the exhaust of the fuel cell 102 are expelled and may be recirculated in the humidifier 108 that uses the moisture and heat in the fuel cell exhaust to humidify the (compressed) air from the inlet air stream. A bypass valve 109, or a bypass control valve, positioned in a path between an outlet of the fuel cell 102 where the fuel cell exhaust is expelled and an inlet of the humidifier 108, may be used to control an amount of the fuel cell exhaust transferred to the humidifier 108. It should be noted however that the amount of the fuel cell exhaust fed to the humidifier 108 may be controlled in any other way.

As shown schematically in Fig. 2, the humidifier membrane 110 of the humidifier 108 separates a humid side from a dry side of the humidifier 108, and the humidifier membrane 110 is selectively permeable to water vapor such that the water vapor can diffuse through the humidifier membrane 110 from the humid side to the dry side. The humidifier membrane 110 may be made from the same material as the fuel cell membrane referred to herein as PEM 104.

At the dry side, the humidifier membrane 110 receives the inlet air stream, which is dry and may be cold or cool. As an exhaust of the humidifier 108, cold and dry exhaust stream may be expelled, at the dry side of the humidifier membrane 110, as shown in Fig. 2.

At the humid side, humidifier membrane 110 receives the fuel cell exhaust stream which may be humid and hot. The air from the inlet air stream is humidified and warmed up using the fuel cell exhaust stream, and the humidified air, which may be hot or warm, is output from the humidifier membrane 110 at its humid side.

The hydrogen gas, e.g., as a fuel inlet stream, is passed from the hydrogen storage tank 106 to the anode 103 of the fuel cell 102. A fuel exhaust stream (i.e., an anode exhaust stream) is provided from the fuel cell 102. At least a portion of the hydrogen included in the fuel exhaust stream may be separated and provided back into the fuel inlet stream, as shown via a dashed line in Fig. 1 B.

The turbine, or turbine generator 114, which may be optional, may be configured to extract energy from the fuel cell exhaust to power the compressor 112. More than one turbine generators 114 may be present, depending on a specific configuration of the fuel cell system 100. It should be appreciated that the fuel cell system 100 may comprise various other components that are not shown in Fig. 1 B.

As further shown in Fig. 1 B, the fuel cell system 100 also comprises or is associated with a controller 116 comprising a humidifier control model 118 in accordance with aspects of the present disclosure. The humidifier control model 118 may be a thermodynamic model, which is discussed in more detail below. As shown in the example of Fig. 1 B, the controller 116 may comprise a memory 120 and a processing circuitry 122, e.g., at least one processor. The processing circuitry 122 may be configured to execute program code stored in memory 120, which may include one or several types of memory such as read-only memory (ROM), random-access memory (RAM), cache memory, flash memory devices, optical storage devices, etc.

The humidifier control model 118 may be stored in the memory 120, e.g., in the form of program code or computer-executable instructions that, when executed by the processing circuitry 122, carry out the method in accordance with embodiments of the present disclosure. The controller 116 may be configured to control extraction of moisture and heat from the exhaust flow based on one or both of a target level of inlet air humidity, i.e. humidity of the inlet air stream of the fuel cell that is required for humidifying the membrane of the fuel cell, and a target temperature of the inlet air stream. In some embodiments, the controller 116 may not use a target temperature of the inlet air stream.

It should be appreciated that the memory 120 and processing circuitry 122 are shown by way of example only, as the controller 116 may also be considered to be included in the processing circuitry 122, or itself may be a processing circuitry. The controller 116 may be part of the fuel cell system 100 or it may be a separate component capable of communicating with the fuel cell system 100. The controller 116 may be a single unit or it may include sub-units. In addition, the controller 116 may be an on-board controller, an off-board controller, or a combination of an on-board and off-board controller(s). The controller 116 may receive data, e.g., sensor data acquired by sensor(s) included in or otherwise associated with the fuel cell system 100 and the controller 116 provides control signals to one or more components of the fuel cell system 100.

The humidifier 108 may be a counter flow heat and mass exchanger. The humidifier 108 may comprise various other components that are not shown in Fig. 1 B. For example, the humidifier 108 may comprise or may otherwise be associated with at least one sensor, for example, a humidity or moisture sensor that acquires, e.g., dynamically, moisture content of the inlet air stream, a temperature sensor for measuring a temperature of an inlet air stream, and other sensors. The moisture sensor may be a heat-resistant sensor. Any other sensors of any suitable type may be used in or in conjunction with the humidifier 108, to monitor parameters that are used for control of the humidifier 108 and the fuel cell’s 102 overall operation.

The humidifier 108 may be controlled to determine a water vapor content in the air intake of the fuel cell system 100. The humidifier 108 operates as a mass and heat exchanger configured to control moisture of the PEM 104.

Humidity of an inlet air stream is not affected by operation of the fuel cell, and is considered to be a suitable feature for controlling the humidify of the fuel cell membrane. Models have been developed to control steady-state operation of the fuel cell, and efforts have been made to develop dynamic models to predict fuel-cell membrane humidity. However, existing models have drawbacks in that they may be not sufficiently reliable and are CPU power intensive. Moreover, unphysical solutions may be generated which, in use, may cause both damage to the fuel cell and loss of power.

Accordingly, the inventors of the present disclosure have recognized that an improved model is required to control an amount of water vapor, transferred across the humidifier membrane from the humid side to the dry side of the humidifier membrane, with improved accuracy and reproducibility. Thus, in aspects of the present disclosure, a humidifier control model is provided comprising a thermodynamic model describing the first law of thermodynamics of an open system as shown in Fig. 3, wherein the thermodynamic model is used for controlling a heat and water mass transfer across the humidifier membrane. In most models, the dry and humid side flow paths are divided into an integer number of slices from inlet to outlet, each slice being described by the first law of thermodynamics of an open system. The thermodynamic model describes the flow of heat and mass in and out of the humidifier.

In a method in accordance with embodiments of the present disclosure, the transfer of heat and mass from humid to dry side (a process occurring within each slice) is described by analytical solutions to the heat and mass transfer equation, such that the humidifier control model is more stable and reliable than existing models. An analytical solution involves framing a problem in a well-understood form and calculating the exact solution according to physics. The analytical solutions allow accurately describing experimental data and modeling a transport of water vapor and heat (mass flow) across a humidifier membrane, to control how much water vapor is to be provided to the humidifier membrane to thereby maintain a desired moisture at a membrane, such as a PEM, of the fuel cell. However, the most advantageous feature obtained by using the analytical solutions to the heat and mass transfer equations is numerical stability. The exactness of the solutions guarantees avoidance of numerical difficulties inherent using gradient based methods. This is exemplified in Fig. 4, which shows negative concentrations obtained using a too large time step in conjunction with a too low starting concentration in the calculation of mass transport.

The method described here adds numerical stability, which reduces both computational power (which may be limited in the controller’s processor) and guarantees that unphysical solutions are avoided as described above.

In embodiments in accordance with aspects of the present disclosure, a humidity of the inlet air stream is controlled by controlling an amount of water vapor to be transferred across the humidifier membrane 110 from the humid side to the dry side of the humidifier membrane 110. A predetermined, target value of humidity of the inlet air stream may be used for humidifying the membrane of the fuel cell 102 such as the PEM 104. Additionally or alternatively, a predetermined, target value of temperature of the inlet air stream may be used for heating the membrane of the fuel cell 102 such as the PEM 104. One or both of the predetermined humidity and temperature values may be fed to the controller 116 comprising the humidifier control model 118 that repeatedly provides an analytically-derived, numerically stable solution comprising an amount of water vapor to be transferred across the humidifier membrane 110 from the humid side to the dry side of the humidifier membrane 110. The controller 116 may provide the analytically-derived, numerically stable solution based on one or both of the predetermined humidity and temperature values. The controller 116 may also take into consideration a state of an inlet air stream, e.g., humidity and temperature, as that air stream is acquired from the environment, as well as any other information, e.g., sensor data acquired by one or more sensors associated with or included in the fuel cell system 100. The determined amount of water vapor to be transferred across the humidifier membrane 110 results in humidifying the membrane 104, such as the PEM, of the fuel cell 102 in accordance with one or both the predetermined value of humidity and the predetermined value temperature of the inlet air stream. In this way, the controller 116 operates to control the amount of water vapor to be transferred across the humidifier membrane 110 as the fuel cell system 100 operates. As used herein, the amount of water vapor may also include an amount of heat that is transferred across the humidifier membrane 110 and provided to humidify, and in some cases heat, the inlet air stream.

Existing approaches to determining an amount of water vapor to be transferred across a membrane of a humidifier include solutions that are not numerically stable. For example, Chen and Peng (2005) describe a model for controlling a vapor transfer across a humidifier membrane that (model) includes equations that is not analytically solved and may provide solutions that are not numerically stable. See Chen, D. & Peng, H. “A Thermodynamic Model of Membrane Humidifiers for PEM Fuel Cell Humidification Control.” ASME. J. Dyn. Sys., Meas., Control. September 2005; 127(3): 424-432. Afshari and Houreh also describe an equation that is also not analytically solved and may provide solutions that are not numerically stable. See Afshari, E. & Houreh, N.B. “An analytic model of membrane humidifier for proton exchange membrane fuel cell.” Environmental Engineering Science, 2014, vol. 2: 83-94. Both these references use numerical gradient methods to describe and calculate mass transport over the humidifier membrane. These gradient based methods may be intrinsically unstable due to the linearization of a nonlinear process: the decrease/increase of concentration inherent with mass transport. If a situation arises in which the starting concentration is low compared to the time step (see Fig. 4 herein), the linearization can induce negative concentrations. Such issue may likewise arise in calculation of energy transport. In both cases, an unphysical situation may arise and the continued calculation can come to catastrophically erroneous results.

Embodiments in accordance with aspects of the present disclosure provide a humidifier control model that includes an analytical expression for heat and water transfer across the humidifier membrane, and wherein the analytical expression is for evaluating a mass and heat change of the humid and dry side of the humidifier. The analytical expression is derived from solutions to one or both of

the diffusion equation

= DV²c, and the heat equation

= aV²T, or a variation thereof.

The humidifier control model may provide a numerically stable solution.

A humidifier used in a fuel cell system is typically a counter flow heat and mass exchanger. Fig. 3 illustrates schematically a humidifier such as the humidifier 108 of Fig. 1 B, comprising the humidifier membrane 110 which is shown in this example as a tube, such as a Nation™ tube. An arrow 303 denotes an input stream, such as a fuel cell exhaust stream, at the humid side of the humidifier membrane. An arrow 305 denotes an input stream, such as a compressed air from the inlet air stream, at the dry side of the humidifier membrane. When making a humidifier model, in this example, a tube as shown in Fig. 3 may be discretized in the flow axial direction. The counter flow geometry considerably complicates these discretized equations which are solved iteratively. The iterative solution is undesirable for control of a fuel cell system in a vehicle where there may be no processor room (i.e. , limited processor, such as a controller processor, capability) for numerical problems such as, e.g., infinite loops and bad convergence. Thus, due to no known ways to address this particular problem, except adding numerical robustness at the cost of computational power, use of existing solutions for control of a fuel cell system in a vehicle such as, e.g., a truck, is limited.

The first law of thermodynamics of an open system may be represented as:

wherein m is mass (kg), u is internal energy (J), Q denotes heat transfer flow (i.e. per a unit of time), m_t denotes mass transfer, h denotes enthalpies, t represents time, m_inh_in is a product of input mass and enthalpy in an inflow, m_outh_out is a product of output mass and enthalpy in an outflow, and m_th_t is a product of transferred mass and enthalpy in the flow across the humidifier membrane. In embodiments in accordance with aspects of the present disclosure, the method is provided that adds numerical stability to the heat and transport equations. In particular, equations for mass and heat transfer over the membrane are solved for m_t and Q shown in Fig. 3. The transport equations may be solved by discretizing the membrane or solving the equations analytically. The first option is numerically more sensitive and more CPU-power intense than the second. Thus, a numerically stable choice is using analytical solutions as described herein, which can be achieved by solving the heat and mass transfer across the membrane, which both may be taken as the same partial differential equation of the type:

wherein it is a variable of interest (water vapor concentration for mass transfer, temperature for heat transfer), t represents time, and a is a positive coefficient called the thermal diffusivity of a medium.

The Laplacian, A, is determined by the choice of coordinates. For a membrane modelled as a tube as shown in Fig. 3, which may be a suitable way to model the humidifier membrane 110 as shown in Fig. 1 B, a cylindrical coordinate system may be used.

Satterfield & Benziger (2008) showed the accuracy of analytical solutions for solving heat and mass transfer equations for water mass transport across membranes. See Satterfield, M.B. & Benziger, J.B. “Non-Fickian water vapor sorption dynamics by Nation membranes.” J Phys Chem B. 2008 Mar 27; 112(12):3693-704. For example, Figure 3 in Satterfield & Benziger shows experimentally measured and calculated (using analytical solutions to the above equation) water transport over a series of Nation™ membranes, which are the standard type of materials used to make both fuel cell membranes and humidifier membranes.

In aspects of the present disclosure, analytically derived expressions for Q denoting heat transfer flow, and m_t, denoting mass transfer, provide numerically stable solution comprising an amount of water vapor to be transferred across the humidifier membrane from the humid side to the dry side of the humidifier membrane. Determining the amount of water vapor to be transferred across the humidifier membrane allows controlling the humidifier.

As described in Satterfield & Benziger (2008), assuming that the ratelimiting step for water absorption is Fickian diffusion with a constant diffusivity, the diffusion equation (Eq. 3) can be solved for the mass gain as a function of time.

wherein D is the diffusion constant or diffusivity coefficient, and V² is the Laplacian of a coordinate system.

The normalized mass change as a function of time is found by solving Eq. (1) with constant diffusivity, and the solution is given by Eq. (4).

wherein m is the index of the eigenvalue solution obtained when solving Eq. (3) using standard methods.

In Eq. (4), M_t is the mass transferred during a chosen time step, i.e. the time integral of m_t in Eq. (1), above. Hence, M_t describes the mass transferred over the membrane and can be plugged into the integrated form of Eq. (1 ) to model the flow of the humid and dry side of the humidifier, given certain humidifier inlet conditions.

The analytical solution provided in accordance with aspects of the present disclosure provides the description of the heat and water transfer that is numerically robust and does not result in an unphysical solution, i.e. a solution that does not actually happen in a physical system.

In embodiments in accordance with the present disclosure, the humidifier control model includes an analytical expression for heat and water transfer across the humidifier membrane, wherein the analytical expression is for evaluating a mass and heat change of the humid and dry side of the humidifier. The analytical expression may be derived from solutions to the diffusion equation (Fick’s second law), defined as follows;

wherein c is a variable of interest (water vapor concentration for mass transfer, temperature for heat transfer), D is the diffusion constant, V² is the de

Laplacian of the coordinate system, and — is a rate of change of the variable of interests at a point over time.

The analytical expression may alternatively or additionally be derived from solutions to the heat equation, defined as follows:

— = aV²T, (6) at ^v '

wherein a is the thermal diffusivity, is a rate of change of temperature at

a point over time, and V² is the Laplacian of the coordinate system, same as in the case above with the diffusion equation.

Furthermore, the analytical expression may be derived from solutions to an equation that is a variation of any of the diffusion equation and the heat equation. Examples of variations include be adding convection terms (dependent on flow velocity), source or radiative terms, or a variation in boundary conditions.

Fig. 4 illustrates a water vapor concentration on a wet side of a membrane as water, modeled as mass and heat flux over a membrane, is transferred across the membrane, such as a humidifier membrane, as a function of time. The water vapor concentration is shown as determined using a conventional approach, e.g., a gradient-based equation (line A) and using a humidifier control model in accordance with the present disclosure (line B).

As shown in Fig. 4, solutions obtained using gradient-based equations to calculate mass and heat flux over the membrane are inherently numerically unstable. In particular, when solving these equations over time using finite time steps, it is possible to obtain solutions in which conservation of mass and energy is not obeyed, as represented by line A. When that happens, calculations provided by the model may drift to physically incorrect solutions and the model needs to be restarted. This results in longer periods of having no information from the model during fuel cell operation, which may result in significant damage to the fuel cell. For example, at time 0, a constant transfer rate is assumed (see, e.g., equation 4 in Chen and Peng (2005)). However, as the time progresses, an initial amount of material (water, in this case) may change. For example, if an assumption of a transfer rate was 20 molecules per second, if the amount of material reduces, e.g., to 10 molecules per second or less, a computation using a gradient-based equation may result in a solution that violates the laws of physics. For example, Fig. 4 illustrates, by line A, that, after a certain time (time step), marked by a dashed line C, computed values of the water vapor concentration may become negative, which is an unphysical solution.

As shown in Fig. 4, when solutions obtained by solving the diffusion and heat equation are used for controlling an amount of water vapor to be transferred across a membrane, as shown by line B, this guarantees that a solution obeys the laws of physics. Thus, such solutions guarantee avoidance of numerical problems. Also, computing these solutions requires low computational costs as compared to iterative gradient-based equations.

Fig. 5 illustrates a method 500 of operating a humidifier of a fuel cell system comprising a fuel cell which may be multiple fuel cells forming a fuel cell stack, or more than one fuel cell stack. The method 500 may be implemented, for example, for controlling operation of the humidifier 108 of the fuel cell system 100 comprising fuel cells such as the fuel cell 102, shown in Fig. 1 B. The method 500 may however be implemented in any other fuel cell system including a humidifier comprising a humidifier membrane and operating, similar to the humidifier 108 of the fuel cell system 100, to humidify an air stream to thereby humidify a fuel cell membrane.

The method 500 may be performed by a controller such as, e.g., a controller 116 of Figs. 1A and 1 B. The controller 116 may be part of the fuel cell system 100, or it may be a separate component operatively coupled to at least one component of the fuel cell system 100, or a part of the controller 116 may be included in the fuel cell system 100. Regardless of its specific implementation, the controller 116 includes and/or executes on its processor, as computer-executable instructions, a humidifier control model that repeatedly provides an analytically- derived, numerically stable solution comprising an amount of water vapor to be transferred across the humidifier membrane, as discussed in more detail below.

The method or process 500 may begin at any suitable time. For example, the method 500 may be executed continuously as the fuel cell system 100 is operating to generate energy, and as the humidifier is thus operating to humidify air obtained from the inlet air stream before the air is supplied to the cathode and thus to the fuel cell membrane. The humidifier comprises a humidifier membrane separating a humid side from a dry side of the humidifier and being selectively permeable to water vapor such that the water vapor can diffuse through the humidifier membrane from the humid side to the dry side.

Action 502. The method 500 comprises obtaining at least one of a predetermined value of humidity and a predetermined value of temperature of an inlet air stream of the fuel cell that are required for humidifying a membrane of the fuel cell. The predetermined value of humidity may represent a desired humidity of the inlet air stream that is delivered to the fuel cell membrane, e.g., the PEM, of the fuel cell, and that is required to maintain water balance in the fuel cell (fuel cell stack) to achieve a proper fuel cell membrane humidity. The predetermined value of temperature may represent a desired temperature of the inlet air stream. Thus, the inlet air stream, acquired from the outside (and which may be compressed), is humidified and heated to achieve the predetermined value of humidity and/or the predetermined value of temperature The predetermined values of humidity and temperature may be set by an operator (e.g., an operator of a system such as a vehicle including the fuel cell system, or by a remote operator), determined and set automatically, or via a combination thereof - e.g., the operator may adjust a set value.

In embodiments in accordance with the present disclosure, the humidifier extracts, at the humid side of the humidifier, water vapor and heat from an exhaust stream of the fuel cell, and transfers, at the dry side of the humidifier, the extracted water vapor and heat to the inlet air stream of the fuel cell. The inlet air stream may be compressed, e.g., by the compressor 112 of Fig. 1 B. The humidifier may be a counter flow heat and mass exchanger. The fuel cell system may be a fuel cell system of an electric vehicle.

Action 504. The method 500 comprises providing the at least one of the predetermined value of humidity and the predetermined value of temperature to the humidifier control model that repeatedly provides an analytically-derived, numerically stable solution comprising an amount of water vapor to be transferred across the humidifier membrane from the humid side to the dry side of the humidifier membrane, wherein the amount of water vapor be transferred across the humidifier membrane results in humidifying the membrane of the fuel cell in accordance with the at least one of predetermined value of humidity and the predetermined value of temperature. The humidifier control model repeatedly provides an analytically-derived, numerically stable solution using the at least one of the predetermined value of humidity and the predetermined value of temperature. The humidifier control model repeatedly provides an analytically- derived, numerically stable solution also using at least one of a current value of humidity and a current value of temperature of an inlet air stream as the inlet air stream is acquired from the outside environment.

In embodiments in accordance with the present disclosure, the humidifier control model is a thermodynamic model describing the first law of thermodynamics of an open system, the thermodynamic model being used for controlling a heat and water mass transfer across the humidifier membrane. The first law of thermodynamics of an open system may be represented as shown, e.g., in Eq. (1), above. The humidifier control model may repeatedly provide the analytically-derived, numerically stable solution so that no unphysical solution is provided.

The humidifier control model may provide the analytically-derived, numerically stable solution at certain time intervals. Thus, the analytically-derived, numerically stable solution may compute the amount of water vapor to be transferred across the humidifier membrane at time intervals such as, e.g., every 1 second, or 1/2 of a second, or 1/5 of a second, or 1/10 of a second, etc.

The humidifier control model includes an analytical expression for heat and water transfer across the humidifier membrane, and wherein the analytical expression is for evaluating a mass and heat change of the humid and dry side of the humidifier. The analytical expression may be derived from solutions to one or de both of the diffusion equation (Fick’s second law),

— = DV²c, and the heat

equation

= c V²T. Furthermore, the analytical expression may be derived from solutions to any one or more equations that are variations of the above equations. The equations for mass and heat transfer over the membrane may be solved analytically for m_t and Q (see, e.g., Fig. 3) as discussed above.

Action 506. The method 500 comprises controlling the amount of water vapor to be transferred across the humidifier membrane as the fuel cell operates. The amount of water vapor is controlled so that the humidifier membrane has a level of humidity that allows delivering target humidity and/or heat amounts to an inlet air stream. Thus, the inlet air stream, acquired from the environment, may be humidified and/or heated using the moisture and heat from the fuel cell exhaust stream. The humidified and/or heated inlet air stream is provided to a cathode side of the fuel cell system, such as the cathode 105 of the fuel cell system 100 of Fig. 1 B.

In embodiments in accordance with the present disclosure, controlling the amount of water vapor to be transferred across the humidifier membrane as the fuel cell operates may comprise controlling a bypass valve of the fuel cell system. For example, the bypass valve 109 of Fig. 1 B may be controlled to supply a determined amount of water vapor to the humidifier membrane 110. Other control device may be used additionally or alternatively, as embodiments in accordance with the present disclosure are not limited in this respect.

The process 500 may be performed while the fuel cell operates.

To perform the method steps described herein, the controller 116 may be configured to perform the processing described in connection with Fig. 5 above, and/or any other examples or embodiments herein. Although the controller 116 is shown in Figs. 1 A and 1 B, Figs. 6A and 6B additionally illustrate an example of an arrangement of a controller, such as the controller 116, in accordance with embodiments of the present disclosure.

As shown in Fig. 6A, the controller 116 may comprise an input and output interface 600 configured to communicate with any necessary components and/or entities of embodiments herein. The input and output interface 600 may comprise a wireless and/or wired receiver (not shown) and a wireless and/or wired transmitter (not shown). The input and output interface 600 may comprise a wireless and/or wired transceiver. The controller 116 may be positioned in any suitable location of the vehicle 10. In implementations in which the controller 116 is an off-board controller, it may be positioned, entirely or in part, outside of the vehicle 10. The controller 116 may use the input and output interface 600 to control and communicate with sensors, actuators, subsystems, and interfaces in the vehicle 10 by using any one or more out of a Controller Area Network (CAN), ethernet cables, Wi-Fi, Bluetooth, and other network interfaces.

The method described herein may be implemented through processing circuitry, e.g., one or more processors, such as the processing circuitry 122 of the controller 116, together with computer program code for performing the functions and actions of the embodiments herein. The processing circuitry 122, as well as memory 120, are also shown in Fig. 1 B. The computer program code mentioned above may also be provided as a computer program medium, e.g., in the form of a computer-readable storage medium carrying computer program code for performing the method in accordance with embodiments herein.

The memory 120 may comprise one or more memory units. The memory 120 comprises computer instructions executable by the processing circuitry 122 of the controller 116. The memory 120 is configured to store, e.g., information, data, etc., to perform the methods in accordance with embodiments herein, when being executed by the processing circuitry 122. The controller 116 may additionally obtain information from an external memory. The methods according to the embodiments described herein may be implemented by means of e.g. a computer program product 680 or a computer program, comprising instructions, i.e. , software code portions, which, when executed on at least one processor, e.g., the processing circuitry 122, cause the at least one processor to carry out the actions described herein, as performed by the controller 116.

In some embodiments, a computer-readable storage medium 690 stores the computer program product 680. The computer-readable storage medium 690 may be, e.g., a disc, a universal serial bus (USB) stick, or similar. The computer- readable storage medium 690, having stored thereon the computer program product, may comprise the instructions which, when executed on at least one processor, e.g., the processing circuitry 122, cause the at least one processor to carry out the actions of the method described herein, as performed by the controller 116.

As shown in Fig. 6A, the controller 116 may comprise an obtaining unit 602. The controller 116, the processing circuitry 122, and/or the obtaining unit 602 may be configured to obtain at least one of a predetermined value of humidity and a predetermined value of temperature of an inlet air stream of the fuel cell that are required for humidifying a membrane of the fuel cell. In some embodiments, the controller 116, the processing circuitry 122, and/or the obtaining unit 602 may be configured to obtain the predetermined value of humidity of the inlet air stream of the fuel cell that is required for humidifying a membrane of the fuel cell. In some embodiments, the controller 116, the processing circuitry 122, and/or the obtaining unit 602 may be configured to obtain both the predetermined value of humidity and the predetermined value of temperature of an inlet air stream of the fuel cell that are required for humidifying a membrane of the fuel cell.

The controller 116 may comprise a providing unit 604. The controller 116, the processing circuitry 122, and/or the providing unit 604 may be configured to provide the at least one of the predetermined value of humidity and the predetermined value of temperature to a humidifier control model that repeatedly provides an analytically-derived, numerically stable solution comprising an amount of water vapor to be transferred across the humidifier membrane from the humid side to the dry side of the humidifier membrane. The amount of water vapor to be transferred across the humidifier membrane of the humidifier results in humidifying the membrane of the fuel cell in accordance with the at least one of the predetermined value of humidity and the predetermined value of temperature.

The controller 116 may comprise a controlling unit 606. The controller 116, the processing circuitry 122, and/or the controlling unit 606 may be configured to control the amount of water vapor to be transferred across the humidifier membrane of the humidifier as the fuel cell operates.

Those skilled in the art will appreciate that the units in the controller 116 described above may refer to a combination of analogue and digital circuits, and/or one or more processors configured with software and/or firmware, e.g., stored in the controller 116, that, when executed by the respective one or more processors such as the processors described above, may carry out the actions or steps of the method(s) in accordance with the present disclosure. One or more of these processors, as well as the other digital hardware, may be included in a single Application-Specific Integrated Circuitry (ASIC), or several processors and various digital hardware may be distributed among several separate components, whether individually packaged or assembled into a system-on-a-chip.

The operational steps described in any of the exemplary aspects herein are described to provide examples and discussion. The steps may be performed by hardware components, may be embodied in machine-executable instructions to cause a processor to perform the steps, or may be performed by a combination of hardware and software. Although a specific order of method steps may be shown or described, the order of the steps may differ. In addition, two or more steps may be performed concurrently or with partial concurrence.

The terminology used herein is for the purpose of describing particular aspects only and is not intended to be limiting of the disclosure. As used herein, the singular forms “a,” “an,” and “the” are intended to include the plural forms as well, unless the context clearly indicates otherwise. As used herein, the term “and/or” includes any and all combinations of one or more of the associated listed items. It will be further understood that the terms “comprises,” “comprising,” “includes,” and/or “including” when used herein specify the presence of stated features, integers, steps, operations, elements, and/or components, but do not preclude the presence or addition of one or more other features, integers, steps, operations, elements, components, and/or groups thereof.

It will be understood that, although the terms first, second, etc., may be used herein to describe various elements, these elements should not be limited by these terms. These terms are only used to distinguish one element from another. For example, a first element could be termed a second element, and, similarly, a second element could be termed a first element without departing from the scope of the present disclosure.

Relative terms such as “below” or “above” or “upper” or “lower” or “horizontal” or “vertical” may be used herein to describe a relationship of one element to another element as illustrated in the Figures. It will be understood that these terms and those discussed above are intended to encompass different orientations of the device in addition to the orientation depicted in the Figures. It will be understood that when an element is referred to as being “connected” or “coupled” to another element, it can be directly connected or coupled to the other element, or intervening elements may be present. In contrast, when an element is referred to as being “directly connected” or “directly coupled” to another element, there are no intervening elements present.

Unless otherwise defined, all terms (including technical and scientific terms) used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this disclosure belongs. It will be further understood that terms used herein should be interpreted as having a meaning consistent with their meaning in the context of this specification and the relevant art and will not be interpreted in an idealized or overly formal sense unless expressly so defined herein.

It is to be understood that the present disclosure is not limited to the aspects described above and illustrated in the drawings; rather, the skilled person will recognize that many changes and modifications may be made within the scope of the present disclosure and appended claims. In the drawings and specification, there have been disclosed aspects for purposes of illustration only and not for purposes of limitation, the scope of the inventive concepts being set forth in the following claims.

Claims

CLAIMS What is claimed is:

1 . A method of operating a controller (116) for controlling a humidifier (108) of a fuel cell system (100) comprising a fuel cell (102), the humidifier (108) comprising a humidifier membrane (110) separating a humid side from a dry side of the humidifier (108) and being selectively permeable to water vapor such that the water vapor can diffuse through the humidifier membrane (110) from the humid side to the dry side, the method comprising: obtaining (502) at least one of a predetermined value of humidity and a predetermined value of temperature of an inlet air stream of the fuel cell that are required for humidifying a membrane (104) of the fuel cell (102); providing (504) the at least one of the predetermined value of humidity and the predetermined value of temperature to a humidifier control model (118) that repeatedly provides an analytically-derived, numerically stable solution comprising an amount of water vapor to be transferred across the humidifier membrane (110) from the humid side to the dry side of the humidifier membrane (110), wherein the amount of water vapor to be transferred across the humidifier membrane (110) of the humidifier (108) results in humidifying the membrane (104) of the fuel cell (102) in accordance with the at least one of the predetermined value of humidity and the predetermined value of temperature; and controlling (506) the amount of water vapor to be transferred across the humidifier membrane of the humidifier as the fuel cell operates.

2. The method of claim 1 , wherein the humidifier extracts, at the humid side of the humidifier, water vapor and heat from an exhaust stream of the fuel cell, and transfers, at the dry side of the humidifier, the extracted water vapor and heat to the inlet air stream of the fuel cell.

3. The method of claim 1 or 2, wherein controlling the amount of water vapor to be transferred across the humidifier membrane as the fuel cell operates comprises controlling a bypass valve (109) of the fuel cell system (100). The method of any one of claims 1 to 3, wherein the humidifier control model is a thermodynamic model describing the first law of thermodynamics of an open system, the thermodynamic model being used for controlling a heat and water mass transfer across the humidifier membrane. The method of any one of claims 1 to 4, wherein the humidifier control model repeatedly provides the analytically-derived, numerically stable solution so that no unphysical solution is provided. The method of any one of claims 1 to 5, wherein the humidifier control model provides the analytically-derived, numerically stable solution at certain time intervals. The method of any one of claims 1 to 6, wherein the humidifier control model includes an analytical expression for heat and water transfer across the humidifier membrane, and wherein the analytical expression is for evaluating a mass and heat change of the humid and dry side of the humidifier, wherein the analytical expression is derived from solutions to one or both of a diffusion

equation

= £)V²c, and a heat equation

= aV²T, or a variation thereof. The method of any one of claims 1 to 7, wherein the humidifier is a counter flow heat and mass exchanger. A controller (116) for controlling a fuel cell system (100), the controller (116) being configured to perform the method of any one of claims 1 to 8. A fuel cell system (100) comprising the controller (116) of claim 9. A vehicle (10) comprising the fuel cell system (100) of claim 10, and/or being in communication with the controller (116) claim 9. A computer program product comprising instructions, which, when executed on at least one processor, cause the at least one processor to carry out the method of any one of claims 1 to 8. A computer-readable storage medium, having stored thereon a computer program product comprising instructions which, when executed on at least one processor, cause the at least one processor to carry out the method of any one of claims 1 to 8.