rv3 modelling aproaches

7/31/2019 RV3 Modelling Aproaches

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Modelling Approaches for relevant technologies

Index:

1. Fuel cell modelling

a. SOFCb. PEMFC

2. Balance of Plant models

3. TEG modelling

4. Pyroelectric

5. Electric modelsa. Battery

b. Solar Panelc. DC converters

d. Brushless DC motor

6. Summary


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1. Fuel cell Modelling

As with any mathematical model, a number of approaches can be followed toproduce models of different fidelity and complexity. A pure black box model

will require a large amount of training and experimental data, but can be

relatively simple. More physical models describe the individual sub processesto represent the overall system.

In a physical fuel cell model some of the relevant sub processes that need to be

considered may be: Charge balance / Mass balance / Energy balance /Momentum balance / Chemical reaction balances.

A modelling 1-D steady state approach for a SOFC cell is highlighted from

[ref].

The assumptions made are:

- Uniform distribution of fuel gas to each cell

- Fully developed laminar flow in gas channels

- Lumped temperature of solid cell structure

- Adiabatic boundaries at the cell inlet/outlet

- Electrodes/current collector act iso-potentially- 100% current efficiency (no gas crossover)

- Extrapolation of a single cell to whole stack (no cell interferences)- Area of interconnect is 100% electro-active


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a. Electrochemical Processes

The maximum work extracted by this process is given by the change in Gibbs free

energy of the process, which in a way describes the potential of the system to do

work.

G = The change in Gibbs free energy

ne = Number of moles of electrons transferred in the reaction

F Faraday constant

En Nernst potential

Where the Nernst potential is the difference between the chemical potentials of the

reactant and product gases and the product of neF the amount of charge transferred.

The charge/ potential balance then describes the incurred losses of the cell, reducing

the output voltage below the Nernst potential.

Where h refers to the activation or concentration voltage losses, i the current

transferred and Rcell is the cell resistance.

To model these losses and produce the important V I relationship of the cell the

following relationship has been proposed:


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Since it is often appropriate to neglect the last 2 terms (fuel cell operates in the linear

region of the V-I graph) the equation can be simplified to:

b. Thermo chemical Model

To establish the behaviour of the chemical reactions, determined by gas concentration

and pressure rises 2 separate models are dealt with. One focuses on the balance of

gaseous matter (fuel and air) and the second on solid (cell and interconnect)

These models are solved for slices of x providing a varying solution in one

dimension (x-axis)


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a. Material balances

Here the flow rate changes due to chemical reactions and the mass transfer of oxygenfrom the cathode to the anode side.

Assuming steady state of the process the relevant equation is rewritten to:

Where v is the stoichiometric coefficient, r dot the reaction rates and n dot is the

volumetric change in molar flow of hydrogen.This equation can be re written as a summation of the elements of each chemical in

the reaction integrated of the relevant area and the final mass balance equations for

Oxygen, Hydrogen and, Carbon can be established.

b. Energy balances

A representation of the relevant energy processes in an axial direction slice x.

The modelling split into 4 parts, the Fuel gas (anode) compartment, Air gas (cathode)compartment, the solid cell and the cell interconnect.


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(Fuel gas compartment)

(Air gas compartment)

(Solid Cell)

(Interconnect Air side)

(Interconnect Fuel side)


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c. Gas Phase Hydrodynamics

The flow through the small cells is commonly assumed to be laminar and fully

developed. Pressure losses between the entrance 1 and exit 2 are proposed to be

estimated according to:

Exit and entrance effects can be dropped if interior flows are analyzed.

Additionally the flow acceleration terms are negligible compared to the corefriction term, further reducing the equation.

d. Numerical Solution techniques

The summarized governing equations are solved via a finite difference scheme, where

the cell is divided into sub slices x. Each slice has an own equilibrium potential,

temperature and pressure, but since the voltage is assumed to be constant along the

whole cell, the slice current can be solved for.


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The equations themselves are coupled since outputs, such as gaseoustemperatures and partial pressure are inputs to equations, such the solution

procedure will need to incorporate an iterative solver.

2. PEMFC modelling considerations

Although the procedures for the PEM model are similar as described earlier, thecorrect representation of the water content of each stage is essential, since the

membrane hydration affects efficiency.

a. Membrane Conductivity

Highly dependent on water content and difficult to model theoretically, this sub

model is essential for an overall effective PEMFC representation. The lowaccuracy of theoretical models often makes it more appropriate to use

experimental data from the literature.

refers to the water content and rages between 0 and 14 (22) for Nafion

(Common PEM membrane material)

b. Gas transport in the diffusion layer

Where N dot refers to the reactant fluxes and N is the oxygen content fraction.

The 3 processes involved are a migration of the gas to the gas channel (GC) by

convection and diffusion into the Gas diffusion layer (GDL). Oxygen further

diffuses into the membrane and dissolves in the water rich environment.The same type of process can be applied to the anode side with hydrogen.


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c. Water transport

The water management of a FC is critical since the membrane resistivity to ion

transfer is dependent on the water content. High water content in the membrane is

therefore desirable; however on the other hand high content in the GC and GDLcan lead to condensation and blocking of the pores reducing the efficiency of the

cell. Water is the only substance that is free to move through the whole cell in

both directions, and its transport is subject to a number of factors.

A coefficient can be introduced describing the flow of water from the anode to the

cathode.

Substituting average molar fractions of water , yields a new relationship for an

average

are the stoichometric flow rates.

The transfer through the membrane is dependent on:

- Electro osmotic drag

- Back diffusion

- Convection


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The overall water transportation equations can be solved by considering allregions of interest (Gas channel, ADL and CDL with the membrane region). Some

simplifications may be introduced to solve this iteratively.

Approach 1:

- Neglect water transport due to pressure difference

- Electro osmotic drag is independent of membrane water content.

- Water diffusivity is a linear function of water content.

This yields an expression for only dependent on water sorption values at the gas

channels (Anode and Cathode)

Approach 2:

- Assume a linear concentration profile (water content variation with

membrane)Results in representation of as a function of gas channel values and average water

content.

Approach 3:

Here an analytical expression of the water content is sought- Membrane water sorption characteristics are linear

- Water diffusivity in the membrane phase is constant

- No membrane swelling

- Water content in membrane is linear

This results in a cube relationship for alpha:


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d. Overall fuel cell stack model

The water transfer coefficient drives the anode and cathode mass balance, which in

turn affect the losses, reducing the Nernst potential and so the cell output. The water

transport coefficient is also important to output, to monitor the health of the

membrane

3. Balance of Plant Models

The following systems are proposed for the FC plant:

- Air supply system & Humidifier

o

Supply air to the Fuel cell and make sure the membrane is kept atappropriate water content. A pressure release valve is used to

regulate the inlet pressure to the cell

- Hydrogen supply system

o Reduce the high pressure of hydrogen from the storage medium to

the required inlet pressure at the cell and assure required delivery

flow rates.

- Cooling liquid circuit

o Circulate cooling liquid through the cell and remove waste heat

keeping the cell at an appropriate temperature. Additionally anheating element is incorporated to provide temperature control

during starting and low power operations.


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a. Air delivery system

A voltage input to a motor regulates the compressed air mass flow rate in the air

delivery circuit. The rest air is the exhausted in an outlet to ambient pressure.

Such a system has 3 time constants, referring to the electrical, mechanical and

hydraulic part. The Voltage in the air compressor is defined by the motor constant the

resistance and inductance.

The mechanical description focuses on the torque requirements, where the motor load

is proportional to the required pressure jump


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The common assumption of a linear relationship between mass flow rate and

compressor speed results in a 3rd relationship

b. Hydrogen delivery system

Similar to the air delivery system, an input voltage to a motor drives a pump, which

controls the hydrogen flow rate. At the same a pressure regulator is employed

between the high pressure H2 reservoir and the delivery circuit.

The same type of equations as previously stated can be derived and represent

this subsystem.

c. Thermal management system

The waste heat generated in the FC processes has to be dumped in a cold reservoir. A

pump circulates liquid through the cell which picks up the heat and losses it in a heatexchanger. A liquid reservoir is used with a build in heating element to provide extra

energy and liquid if needed.

The heat exchange can be liquid to liquid or liquid to air, depending on the

application. In this particular work it was a l to l exchanger.


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Cooling Pump

There are again 3 parts to this system ( 3 time constants), however since the electricaland mechanical constants are very small, they may be neglected and a steady state

expression can be established

Since heat production by the fuel cell is required, a thermal representation needs to be

established. The assumptions for a simple lumped capacitance model are:

- Fuel cell is considered isothermal

- Condensation not accounted for

- Gasses are supplied at cell temperature

- Conduction is much less than convection (negligible)

With these assumptions the lumped model takes the following form:

Taken the natural convection, fuel cell losses and forced internal convection intoaccount. The Temperature at the exhaust T2 is then:

Where Trv is the temperature at the reservoir and Tfc at the fuel cell


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Reservoir temperature

Similar to the fuel cell temperature, the reservoir can be described by

Where Q-heat is the added heat through the heating element and Krv, the natural heat

convection coefficient includes effects from the reservoir, heat exchange and liquid

tubing.

Liquid Liquid heat exchanger

The ability of a heat exchanger to cool a liquid may be expressed by t he NTUanalogy. The number of transfer units is a characteristic for a given Hx an given as:

With a given NTU value the efficiency of the Hx can be calculated and the

corresponding overall heat transfer with given chill water and inlet temperatures.

The mass flow rate of chill water and cooling liquid are important parameters to

consider since they dictate the validity of the above relationships.

Liquid Air heat exchanger

With this system ambient air is used a cool reservoir. The air main be moved by a fan

giving some kind of control over the dumping procedure. Simple on off design or

variable speed fans can be incorporated.The assumption that the heat transfer is proportional to the temperature difference

between ambient air and cooling liquid leads to a proportionality factor Khx


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4. Thermo Electric Generators Modelling

These devices may be modelled on a modular or junction level. Both will needextensive material parameters to describe the heat flow, current generation and

voltage. A possible electrical and thermal modular modelling scheme is shown below:

Considering a single junction the following equations can be used to estimate the

Voltage and current for a load connected to the circuit.

Where the s are the Seebeck coefficients of the n and p semiconductors used, m is

the number of thermocouples per module and N is the number of modules. Rtem is

the internal resistance of the generator and Rl the load resistance.

There is a difference in temperature between the surface of the element and the

surface of the junction (separated by a ceramic plate). For an open circuit, this

temperature difference can be estimated as:

Where T is the temperature at the junction and T at the surface of the element. With

these temperatures given Vcc and I can be calculated. The values of K and Kh , Kc

refer to thermal conductances of the hot and cold junction and the pair.


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The Heat input for steady state operation to the junction and the whole module can bedescribed by the material properties of the junction and the module.

R refers to the electrical resistance of the thermo element pair and is given by

This depends on the Area of the legs, the length and electrical resistivity.

The thermal model accounts for losses between the applied thermal energy Qj and the

heat input to the module Qin

Qb is the heat leakage through the assembly, Qins is the heat flow through theinsulation matter.

5. Pyroelectric Modelling

6. Electrical systems modelling

Since most current and future systems will involve some form of electrical system, for

a multitude of functions (main propulsion / control/avionic/payload systems) some

relevant modelling approaches for important energy/power related components are

presented.

a. Photo voltaic cells.

A first and simple way to model a cell is with the use of a current source model.


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Where Iph - current source (pv source), Id diode current, I l is the load resistance.

Rsh is the shunt resistance and Rs the series resistance of the material. Since Rsh ismuch greater than Rs its effect on the current can be neglected.

q is the charge of an electron, K the Boltzmann constant, A the ideality factor and T

the temperature of the cell. Io is the reverse saturation current defined by

Where Tr is the reference temperature, Egap the band gap energy and T the

temperature. The charge generation of the PV cell can be described by

Si is the insolation on the panel (mW/cm^2). Of course a separate model is required to

estimate this value accurately, including the many factors which affect it. (global

position, time of day / year, incidence angle , Altitude )

Since Temperature generally affects the cells efficiency, a thermal model is required.

The following approach assumes a constant cell temperature.

Reflected:

With S Area , r reflection factor and insolation


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Converted:

incidence angle cell efficiency and area efficiency.

With the cell efficiency depending on temperature:

Thermal radiation:

Involving emissivity, Boltzmanns constant and the ambient and cell temperatures

Cell conduction:

Lamba is the thermal conductivity and d is the thickness of the cell.

Convection:

Withthe heat transfer coefficient.

b. Battery modelling

The simplest model will represent the battery as an ideal source, with constant opencircuit voltage and internal resistance. This however is not the real case, since the

state of charge of a battery dictates these values and their relation needs to beestimated. Additionally other factors, such as slow discharging procedures during no

load operation and over-charge/discharge resistances.

The Thevenin model includes effects, such as overvoltage resistance and battery plate

capacitance, but still assumes constant values for all battery conditions.


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An improved model was supposed to describe the dynamic behaviour of such a modeland takes the following form:

2 resistors and diodes are used to describe the charge and discharge behaviour. The

Dynamic change in Vp can be described by:

V o is the open circuit voltage, Vb the terminal battery voltage, Vp the internalcapacitor voltage.

The open circuit voltage and SOC had to be estimated by measurements.

A simpler empirical model for Li-Ion batteries was presented, and gives the variation

in Voc and Rint as a function of SOC:


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d. Brushless DC motors

These highly efficient new generation of electric motors have seen a rapid increasedusage over the past years. The main novelty is the commutation, which is now done

by electronic switching of the external stator coils. By following a set sequence,

magnetic field of the permanent magnet rotor interacts with the coils fields and atorque is induced.

Complex control and measurements of the rotor position is required to run such an

engine. A diagram of the components of a 3 phase BLDC motor is shown below,

using Hall sensors in the stator to measure the rotor position.

The microprocessor handles the correct switching of the diodes to run current

through the correct coils of the start in the correct moments.

A simpler model of an ideal DC motor may be of the following form:


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The armature resistances and inductances are included. A mechanical model of the

torque is of the form:

With B, as the viscous friction coefficient and the motor Inertia J. Assuming that

output torque is linear to applied current, and emf Voltage is linear to motor speed, weget the following coupled equations:

With the relevant proportional coefficients Kt and Ke

This ideal model may be quite inaccurate, and can be improved by including the

coulomb resistance:

7. Summary

rv3 modelling aproaches

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