The modelling aims to determine the electrochemical and thermal performance of a methane fed internally reforming SOFC operating at 800 °C. The model is programmed using Engineering Equation Solver (EES).

In a fuel cell, it is the change in the Gibbs free energy of formation, AG/, that gives the energy released during the electrochemical reaction in. This change is the difference between the Gibbs free energy of the products and the Gibbs free energy of the reactants, as shown in Eq. 4.1 (Larmine and Dicks 2003).

The quantities of the products and reactants are usually considered in their per mole form gf (J mol^{-1}). The overall reaction occurring in an SOFC operating on methane is shown in Eq. 4.2 (Pilatowsky et al. 2011).

The product is two moles of H2O and one mole of CO2, the reactants are one mole of CH4 and two moles of O2. The change in Gibbs free energy per mole of substance is shown in Eq. 4.3.

The Gibbs free energy of formation for the reaction shown in Eq. 4.3 is not constant; it changes with temperature and state. At a temperature of 800 °C, where the water product is in the steam phase, Agj = -818,400 J mol^{-1} (Blomen 1993). The negative value denotes energy released. If the electrochemical process was reversible i.e. it were ideal with no losses to the environment, then all this Gibbs free energy would be converted to electricity, however in reality some energy is converted to heat.

The Gibbs free energy of formation (Agy) can be used to find the reversible open circuit voltage (E_{0}) of the methane fed SOFC. This is shown in Eq. 4.4 (Tuyen and Fujita 2012).

F is the Faraday’s constant 96,472.44 °C mol^{-1} and z denotes the number of electrons that pass through the external circuit for each mole of reactant. As shown in Eq. 4.5, for every mole of methane fed to the SOFC, eight electrons are released and circulate in the external circuit, thus z = 8 (Pilatowsky et al. 2011).

Using Eq. 4.4, an SOFC operating at 800 °C will have a reversible open circuit voltage (E_{0}) of 1.06 V. However, E_{0} assumes no irreversibilities. In practice the actual operating voltage (Vc_{ell}) is lower due to cell voltage drops. Some of the irreversibilities even apply when no current is drawn. The actual operating voltage of the SOFC (V_{cell}) is given in Eq. 4.6.

EOCV is the non-reversible open circuit voltage of the cell, and considers the impact of fuel composition and ratio of fuel to oxidant and products. EOCV is determined using the Nemst equation, shown in Eq. 4.7, and is a function of the partial pressures of the gases in the cell (Pilatowsky et al. 2011).

E_{0} is the reversible open circuit voltage. R is the universal gas constant (8.314 J mol^{-1} K). P denotes the concentrations of the gas mixtures within the cell, given as partial pressures relative to atmospheric pressure (101,325 Pa). For the SOFC CHP system simulation, the partial pressures have been assumed constant, with P_{CH4} = 25,330 and P_{H2O} = 75,990 Pa, values taken from experimental data (Peters et al. 2002). Using these constants equates to a steam to carbon ratio (S/C) of 3:1. A minimum S/C ratio of two is required to achieve complete internal reforming of the methane fuel in the SOFC stack (Blum et al. 2011). As detailed below, the recycling of anode off-gas into the inlet fuel stream provides enough water for the internal reforming reaction to occur. Because the SOFC is fed by air, the partial pressure of the oxygen in the cell has been assumed constant at P_{O2} = 21,270 Pa, as this is the partial pressure of oxygen in air. The partial pressure of the carbon dioxide product, P_{CO2} = 31,900, is taken from the experimental work of Laosiripojana and Assabumrungrat (2007).

A description and quantification of the operating voltage losses are provided below.

Ohmic losses (AV_{o}h_{m}) are caused due to resistance to the flow of electrons through the anode, cathode, electrolyte and various inter-connections. Equation 4.8 quantifies the Ohmic losses (Tuyen and Fujita 2012).

i is the current density in mA cm^{-2} of the cell, R_{ohm} is the electrical resistance of the cell in m^ cm^{-2}. Tuyen and Fujita (2012) state that the largest contributor to Ohmic losses is from the transport resistance of O^{2-} in the electrolyte, and is strongly dependent upon temperature. The resistances of the other components can be considered constant due to their weak dependence on temperature and their small contribution to the total resistance. For a planar type SOFC with a YSZ electrolyte, Ni-YSZ anode and LSM cathode, the total electrical resistance can be calculated using Eq. 4.9 (Tuyen and Fujita 2012).

T is the operating temperature of the cell, and has been set at 800 °C.

Activation losses (AV_{act}) are caused by the slowness of the reactions taking place on the surface of the electrodes, and are the sum of those occurring at the anode and cathode, shown in Eq. 4.10.

The activation losses occurring at the electrodes, V_{act},_{e} for both the anode and cathode, are determined using Eq. 4.11 (Al-Sulaiman et al. 2010).

io_{e} is the exchange current density of the electrodes. In this model, 0.65 A cm^{-2 }for the anode and 0.25 A cm^{-2} for the cathode have been selected, a reasonable assumption for a planar SOFC with Ni-YSZ anodes and LSM cathodes (Al-Sulaiman et al. 2010).

Transmission losses (AE_{trans}) are caused due to the change in concentration of the reactants at the surface of the electrodes as the fuel is used. Equation 4.12 quantifies the transmission losses.

Equation 4.12 is an empirical correlation. Tuyen and Fujita (2012) state that it provides a robust and accurate prediction of the transmission losses in a SOFC. m and n are constants, 3.1 x 10^{-5} V and 8.1 x 10^{-3} mA cm^{-2} have been selected respectively (Tuyen and Fujita 2012).

Once the cell operating voltage is known, the DC power output in Watts, from the SOFC stack is calculated using Eq. 4.13.

I is the current in Amps of one cell and is determined by multiplying the cell area by the operational current density. ncell is the number of cells in the stack, and is selected based on the power output requirements. If the cells are connected in series the voltage of the stack is the sum of all the cell voltages. The operating cell efficiency is calculated using Eq. 4.14.

u_{f} is the fuel utilisation factor of the cell, and has been assumed as 80 % for the current model, a reasonable assumption for a planar type SOFC (Al-Sulaiman et al. 2010; Blum et al. 2011). To calculate the cell efficiency with respect to the SOFC operating voltage, the heating value of the methane fuel needs to be given in eV. For a SOFC operating at 800 °C, the H_{2}O product will be in the vapour phase, thus the lower heating value (LHV_{eV}) is used, for methane this is 1.039 eV (Blomen et al. 1993). Once the SOFC DC power output and cell efficiency is known, the total energy input in Watts to the SOFC Q_{CH4} is calculated using Eq. 4.15.

The mass flow rate of the methane fuel into the SOFC stack is determined using Eq. 4.16. The LHV of CH_{4} is given in J kg^{-1} = 5 x 10^{7}

The net stack AC power output in Watts is determined using Eq. 4.17.

r|mve_{r}te_{r} is the inverter efficiency, and has been assumed to be 90 %, a reasonable assumption for AC power production in small scale fuel cell systems (Blum et al. 2011). WW is the electrical consumption in Watts of the balance of plant (BoP) for components such as fans, pumps and control devices. Blum et al. (2011) states that the BoP power consumption is highly dependent upon the system, but based on empirical experience, the BoP will consume 5-10 % of the gross DC output from the SOFC stack. In this model W_{BoP} has been assumed as 10 %.

Heat up of an SOFC stack to operating temperature is usually achieved through the combustion of fuel (methane) in the afterburner. In this modelling work, only instantaneous SOFC CHP system performance is investigated and thus it is assumed that the SOFC system is already up to temperature. Heat-up energy input is not considered. Because of the high operating temperatures of SOFCs and the exothermic nature of the electrochemical reactions, a large quantity of heat is produced whilst generating electricity. In order to improve the overall efficiency of the SOFC system, this heat should be recovered. The thermal energy output and resulting waste heat recovery (<2_{WHR} from the SOFC system is derived from the hot gases leaving the SOFC stack (temperature assumed to be equal to the SOFC operating temperature) plus the heat of combustion of the unconsumed hydrogen fuel leaving the anode and flowing to the afterburner (Blum et al. 2011; Tuyen and Fujita 2012).

Anode gas recycling means that a portion of the unconsumed fuel leaving the SOFC stack, which is a function of the fuel utilisation factor, is re-circulated back to the inlet gas stream, and the rest is passed on to the afterburner. The anode gas recycle ratio has been set at 65 % as this provides enough water in the inlet fuel stream for internal reforming to occur (Blum et al. 2011). Although anode gas recycling reduces the amount of fuel entering the afterburner and thus the resulting heat output of the SOFC system, it does simplify the entire system due to the omission of steam generation components. For the modelled SOFC system, the mass ratio of air to fuel (X) entering the fuel cell stack is 3.8. In such a design, this ratio provides enough air for the reactions and sufficient stack cooling.

The hot gases leaving the afterburner are used to pre-heat the inlet fuel and air streams to the required cell temperature, and finally transfer heat to the waste heat recovery O^HR) circuit. The recoverable thermal energy from the SOFC system (6_{W}aste,s_{OF}c) in Watts is shown in Eq. 4.18.

This recoverable thermal energy is used to heat water in the WHR circuit, which is then either used for direct heating applications or to regenerate desiccant solution depending if the system is operating in a CHP or tri-generation configuration. All heat exchanger processes have been evaluated using the e method, with ep_{X }assumed as 80 %. The net recoverable heat from the SOFC CHP system (<2_{WHR} is determined using Eq. 4.19.

m_{water} is the mass flow rate of water in the WHR circuit, and has been assumed constant at 0.042 kg s^{-1}. c_{p} is the specific heat capacity of the water in J kg^{-1} K, and is a function of temperature. T_{WHR},_{flow} and r_{WHR},_{re}t_{urn} are the respective flow and return temperatures of the water in the WHR circuit, given in °C. During simulation r_{WHR},_{re}t_{urn} = 45 °C. The SOFC CHP system sub-component schematic in Fig. 4.1 shows the WHR circuit and anode gas recycling concepts described above. The net electrical efficiency of the SOFC CHP system is related to the net AC electrical output, and is calculated using Eq. 4.20.

The combined heat and power or co-generation heating efficiency is the sum of the electrical and thermal outputs, and is shown in Eq. 4.21.

An important parameter in CHP systems is the heat to power ratio (HP) which indicates the proportion of generated heat to electrical power. Equation 4.22 is used to determine the SOFC CHP systems HP. For a planar type SOFC CHP system of less than 10 kW_{e}, the HP will be between 0.5 and 1 (Elmer et al. 2015b).

Liu and Barnett (2003) provide experimental electrochemical performance data for an internally reforming SOFC with thin YSZ electrolytes on porous Ni-YSZ anodes, operating on methane at a cell temperature of 800 °C. Validation of the developed SOFC model with the experimental data is provided in Fig. 4.3. Over a current density range of 0-1000 mA cm^{-2} the developed SOFC model shows good agreement with the published experimental data.

The SOFC model has been validated, to a suitable degree, with published experimental data. The SOFC model can therefore be used with confidence in the SOFC parametric and tri-generation system analysis.

Next, Sect. 4.2.1.2 presents the SOFC CHP system parametric analysis.