# Modelling of a Solid Oxide Fuel Cell CHP System Coupled with a Hot Water Storage Tank for a Single Household

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. General Overview

#### 1.2. Literature Review

_{2}emissions for typical heating, cooling loads, and electricity demand profiles, for different SOFC systems to current standard technologies have been compared [9]. Besides none of these works focus on heat-to-power limitations of SOFC micro-CHP systems.

#### 1.3. Methodology

## 2. Energy System Modeling

- The fuel cell is assumed to be operated under steady-state conditions.
- The fuel cell reactions are assumed to be in equilibrium.
- Syngas consists of the following gas species, j = {H
_{2}, CO, CH_{4}, CO_{2}, H_{2}O, N_{2}}. - Air that enters the fuel cell consists of 79% N
_{2}and 21% O_{2}. - The cathode and anode inlet temperature of the fuel cell are assumed to be equal.
- The cathode and anode exit temperature of the fuel cell are assumed to be equal.
- There is a temperature gradient (ΔT) across the fuel cell. The temperature of the solid structure (T) is homogeneous and midway between the inlet and exit temperatures.
- All gases behave as ideal gases.
- Gas leakage is negligible.
- Heat loss to the environment occurs only in the fuel cell.

## 3. Modelling of the Micro-CHP System

#### 3.1. Modelling the Solid Oxide Fuel Cell Fuelled with Coal Syngas

_{2}and other minority species [11].

_{2}oxidation is considered to contribute to the electrochemical power generation, while CH

_{4}is reformed to CO, which is then converted to CO

_{2}and H

_{2}through water-gas shift reaction [12,13,14]. Consequently, the steam reforming reaction for methane, the water-gas shift reaction and the electrochemical reactions occur simultaneously in the SOFC and are summarized as follows:

_{4}+ H

_{2}O → CO + 3H

_{2}(reforming)

_{2}O → CO

_{2}+ H

_{2}(water gas shift)

#### 3.2. Mass Balance

^{−1}), is related to the current by Faraday’s law:

^{−1}is Faraday’s constant. For a known fuel utilization factor, ${U}_{f}$ , the amount of hydrogen supplied, ${\dot{n}}_{{\mathrm{H}}_{2-\mathrm{s}}}$ (mol·s

^{−1}), is given by:

^{−1}) is thus:

_{2}, CO, CH

_{4}, CO

_{2}, H

_{2}O, N

_{2}}.

^{−1}) is thus:

^{−1}) becomes:

#### 3.3. Electrochemical Descriptions

^{−1}·K

^{−1}is the universal gas constant, $\mathrm{\Delta}g\xb0\left(T\right)=\mathrm{\Delta}h\xb0-T\mathrm{\Delta}s\xb0$ stands for the molar Gibbs free energy change at ${p}_{0}=1\text{atm}$ which also depends on temperature [15,16,17,19], ${p}_{{\mathrm{H}}_{2}}$ , ${p}_{{\mathrm{O}}_{2}}$ and ${p}_{{\mathrm{H}}_{2}\mathrm{O}}$ are the partial pressures of reactants H

_{2}, O

_{2}, and H

_{2}O, respectively.

- (1)
- Activation overpotential depends on the kinetics of the electrochemical reactions occurring at the anode and cathode. According to the general Butler-Volmer equation, the respective activation overpotentials of the anode and cathode can be calculated as:$$\begin{array}{c}{V}_{act,a}=\frac{2RT}{{n}_{e}F}{\text{sinh}}^{-1}(\frac{i}{2{i}_{0,a}})\\ {V}_{act,c}=\frac{2RT}{{n}_{e}F}{\text{sinh}}^{-1}(\frac{i}{2{i}_{0,c}})\end{array}$$
- (2)
- Ohmic overpotential is caused mostly by resistance to conduction of ions and electrons and by contact resistance between the fuel cell components. In the present study, Ohmic losses are simulated as follows assuming a series electrical scheme:$$\begin{array}{c}{V}_{ohm}=I{{\displaystyle \sum}}^{}{R}_{k}=iA{{\displaystyle \sum}}^{}\frac{{L}_{k}}{{\mathrm{\sigma}}_{k}A}=i{{\displaystyle \sum}}^{}\frac{{L}_{k}}{{\mathrm{\sigma}}_{k}}=i(\frac{{L}_{e}}{{\mathrm{\sigma}}_{e}}+\frac{{L}_{a}}{{\mathrm{\sigma}}_{a}}+\frac{{L}_{c}}{{\mathrm{\sigma}}_{c}}+\frac{{L}_{int}}{{\mathrm{\sigma}}_{int}}){V}_{act,c}=\frac{2RT}{{n}_{e}F}{\text{sinh}}^{-1}(\frac{i}{2{i}_{0,c}})\\ {\mathrm{\sigma}}_{e}={C}_{1e}\text{exp}({C}_{2e})\\ {\mathrm{\sigma}}_{a}=\frac{{C}_{1a}}{T}\text{exp}(\frac{{C}_{2a}}{T})\\ {\mathrm{\sigma}}_{c}=\frac{{C}_{1c}}{T}\text{exp}(\frac{{C}_{2c}}{T})\\ {\mathrm{\sigma}}_{int}=\frac{{C}_{int}}{T}\text{exp}(\frac{{C}_{int}}{T})\end{array}$$
**Table 1.**Constants used for the fuel cell model.Parameter Symbol Value Ambient temperature $(K)$ ${T}_{0}$ 298 Operating pressure $(atm)$ ${p}_{0}$ 1 Fuel utilization ${U}_{\text{f}}$ 0.8 Air utilization ${U}_{a}$ 0.2 Number of electrons ${n}_{e}$ 2 Ade exchange current density $(A{m}^{-2})$ ${i}_{0,a}$ 6500 Cathode exchange current density $(A{m}^{-2})$ ${i}_{0,c}$ 2500 Limiting current density $(A{m}^{-2})$ ${i}_{L}$ 9000 Anode thickness $(\mathrm{\mu}m)$ $La$ 500 Anode conductivity constants ${C}_{1a};{C}_{2a}$ 95 × 10 ^{6}; −1150Cathode thickness $(\mathrm{\mu}m)$ $Lc$ 50 Cathode conductivity constants ${C}_{1c};{C}_{2c}$ 42 × 10 ^{6}; −1200Electrolyte thickness $(\mathrm{\mu}m$ $Le$ 10 Electrolyte conductivity constants ${C}_{1e};{C}_{2e}$ 3.34 × 10 ^{4}; −10,300Interconnect thickness (cm) ${L}_{int}$ 0.3 Interconnect conductivity constants ${C}_{1int};{C}_{2int}$ 9.3 × 10 ^{6}; −1100Air blower power consumption factor ${\mathrm{\eta}}_{ab}$ 10% - (3)
- Concentration overpotential is the voltage drop due to mass transfer limitations from the gas phase into and through the electrode. In the present study, the calculation of concentration overvoltage is as follows:$$\begin{array}{c}{V}_{conc,a}=\frac{RT}{{n}_{e}F}\text{ln}(1-\frac{i}{2{i}_{L,a}})\\ {V}_{conc,c}=\frac{RT}{{n}_{e}F}\text{ln}(1-\frac{i}{2{i}_{L,c}})\end{array}$$

#### 3.4. Air Blower

#### 3.5. Combustor

#### 3.6. Energy Balance

^{−1}) and ${\mathrm{c}}_{\mathrm{p},\mathrm{i}}$ denoting the heat capacity of component i (J·mol

^{−1}·K

^{−1}). The total enthalpy change for the SOFC section is therefore determined as:

#### 3.7. Modelling the Heat Exchangers

## 4. Micro-CHP Heat-to Power Ratio Performance

Species | Molar Fraction | |
---|---|---|

Syngas | Natural Gas | |

H_{2} | 0.293 | 0.0 |

CO | 0.287 | 0.0 |

CO_{2} | 0.118 | 0.0024 |

N_{2} | 0.030 | 0.0421 |

H_{2}O | 0.272 | 0 |

CH_{4} | 0.000 | 0.9488 |

^{2}). In fact, at 3000 A/m

^{2}, a system fueled by syngas shows a $H2PRatio$ equal to 1 whereas a system fuelled by natural gas has a $H2PRatio$ around 0.7. We can conclude that a system fueled by syngas will have a $H2PRatio$ range of operation between 0.8 and 1.95.

**Figure 3.**Heat-to-power ratio of a micro-CHP system fueled by syngas and natural gas at fixed temperature (

**a**) and variable temperature (

**b**).

## 5. Stratified Storage Heat Tank Model Design

Symbol | Description | Constant | Unit |
---|---|---|---|

$x$ | Distance traversed by the fluid in the tank | -- | $\mathrm{m}$ |

$T$ | Temperature of the water in the tank | -- | °C |

${T}_{amb}$ | Ambient temperature | 25 | °C |

$\mathrm{\rho}$ | Water density | 990 | kg·m^{−3} |

${C}_{p}$ | Heat capacity of water | 4180 | kj/(kg·K) |

$\mathrm{A}$ | Tank area | -- | m^{2} |

$\mathrm{\psi}$ | Tank perimeter | -- | $\mathrm{m}$ |

$L$ | Tank height | -- | $\mathrm{m}$ |

$\dot{\mathrm{m}}$ | Water mass flow rate | -- | kg·s^{−1} |

$v$ | Average linear velocity of water | -- | m·s^{−1} |

$h$ | Heat transfer coefficient of the walls | 0.02 | W/(m·K) |

$\mathrm{\lambda}$ | Thermal effective conductivity of water | 0.63 | W/(m·K) |

- x a point in the domain;
- t, the current time;
- u(), the current solution (temperature along the tank);
- du/dx(), the current solution spatial derivative.

- c(:), the coefficients of du/dt;
- f(:), the term to which d/dx is to be applied;
- s(:), the source term.

## 6. Tank Sizing

**Figure 5.**(

**a**) Plot of temperature distribution inside the tank in the case of micro-CHP fuelled by natural gas; (

**b**) Plot of temperature distribution inside the tank in the case of micro-CHP fuelled by syngas.

Value | mCHP Fueled by Natural Gas | mCHP Fueled by Syngas | Unit |
---|---|---|---|

mCHP electric output | 1 | 1 | kW |

Lower limit heat to power ratio mCHP | 0.5 | 0.8 | – |

mCHP system heat recovered mCHP | 0.5 | 0.8 | kW |

Tank radius | 0.2 | 0.2 | m |

Tank height | 0.96 | 1.55 | m |

Tank Volume | 120 | 194 | m^{2} |

Heat Accumulation | 4.5 | 7.2 | kW |

Heat flow rate | 5 | 800 | W |

## 7. Conclusions

^{2}). This leads to assume that a system fuelled by syngas will have a $H2PRatio$ range of operation between 0.8 and 1.95. It was concluded that tank should able to store the heat produced by the micro CHP system when operating at the low hand of the $H2PRatio$ for around 8 h (i.e., night period).

## Author Contributions

## Nomenclature

$E$ | Theoretical open-circuit voltage (volts) |

$\dot{n}$ | Molar flow rate (mol·s^{−1}) |

$H2PRatio$ | Heat-to-Power ratio |

$i$ | Current (Amp) |

$p$ | Pressure (atm) |

$T$ | Temperature (K) |

${U}_{f}$ | Fuel utilization |

${U}_{a}$ | Air utilization factor |

$V$ | Voltage (Volts) |

${x}_{j}$ | Species molar fraction (-) |

## Appendix

## Conflicts of Interest

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**MDPI and ACS Style**

Liso, V.; Zhao, Y.; Yang, W.; Nielsen, M.P.
Modelling of a Solid Oxide Fuel Cell CHP System Coupled with a Hot Water Storage Tank for a Single Household. *Energies* **2015**, *8*, 2211-2229.
https://doi.org/10.3390/en8032211

**AMA Style**

Liso V, Zhao Y, Yang W, Nielsen MP.
Modelling of a Solid Oxide Fuel Cell CHP System Coupled with a Hot Water Storage Tank for a Single Household. *Energies*. 2015; 8(3):2211-2229.
https://doi.org/10.3390/en8032211

**Chicago/Turabian Style**

Liso, Vincenzo, Yingru Zhao, Wenyuan Yang, and Mads Pagh Nielsen.
2015. "Modelling of a Solid Oxide Fuel Cell CHP System Coupled with a Hot Water Storage Tank for a Single Household" *Energies* 8, no. 3: 2211-2229.
https://doi.org/10.3390/en8032211