**Multi-Level Mathematical Modeling of Solid Oxide Fuel Cells**

for Better Environment

*Cliffs, NJ: Prentice-Hall*.

velopment". Elsevier, Oxford, UK.

and Sons, West Sussex, UK.

*ware, 2010 www.fChart.com.*

lication, United Kingdom.

86-92.

*Malabar, FL.*

[1] Moran, M. J. (1982). Availability analysis: A guide to efficiency energy use. *Englewood*

[2] Kotas, T. J. (1995). The Exergy Method of Thermal Plant Analysis. *Reprint ed. Krieger,*

[3] Szargut, J., Morris, D. R., & Steward, F. R. (1988). Exergy Analysis of Thermal, Chem‐

[4] Arcaklioglu, E., Çavuşoglu, A., & Erisen, A. (2005). An algorithmic approach towards finding better refrigerant substitutes of CFCs in terms of the second law of thermo‐

[5] Yumrutas, R., Kunduz, M., & Kanoglu, M. (2002). Exergy analysis of vapor compres‐

[6] Dincer, I., & Rosen, M. A. (2007). Exergy: Energy, Environment and Sustainable De‐

[7] Dincer, I., & Kanoglu, M. (2010). Refrigeration Systems and Applications. John Wiley

[8] Klein & S.A. Engineering Equation Solver (EES). *Academic Commercial, F-Chart Soft‐*

[9] Ataer, O. E., & Gogus, Y. (1991). Comparative study of irreversibilities in an aquaammonia absorption refrigeration system. *International Journal of Refrigeration*, 14,

[10] Dincer, I., & Dost, S. (1996). A simple model for heat and mass transfer in absorption cooling systems (ACSs). *International Journal of Energy Research*, 20, 237-43.

[11] Dincer, I., & Kanoglu, M. (2010). Refrigeration Systems and Applications. Wiley Pub‐

ical and Metallurgical Processes. Hemisphere, New York.

dynamics". *Energy Conversion and Management*, 46, 1595-1611.

sion refrigeration systems. *Exergy, an International Journal*, 2, 266-272.

**References**

52 Clean Energy

Jakub Kupecki, Janusz Jewulski and Jarosław Milewski

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/50724

## **1. Introduction**

In recent years, numbers of questions concerning energy generation have arisen. Emission levels, delivery security, and diversification of the portfolio of technologies have been exten‐ sively discussed. Well-established generation based on fossil fuels in large-scale power sta‐ tions is criticized for big environmental impacts, and limited sustainability due to high fraction of process losses. Not only emissions, but also extraction of resources, alternation of the landscape, transmission and distribution inefficiencies are often pointed as the main downside. As a solution for rapidly increasing energy consumption, and emerging threat of current resources depletion, distributed generation based on highly efficient micro- and small-system was proposed. Moreover, combined heat and power (CHP) units with high achievable efficiency are seen as possible substitutes for stand-alone electricity generators. Most of technologies from that group are currently under development, however selected systems are already reaching market availability. In 2004 European Commission indicated selected systems, with guidelines for promotion and development of highly efficient co-gen‐ erative units [1]. List of technologies, which can provide high electrical and overall efficiency with limited environmental impacts, includes the following:


© 2012 Kupecki et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


**2. Selected systems with Solid Oxide Fuel Cells**

2 it becomes clear that SOFC can be indeed competitive.

**cell**

**Table 1.** Comparison of selected micro-CHP systems with fuel cells.

**Vendor Type of fuel**

system operation.

Over the years fuel cell technology proved to be feasible in a number of applications, includ‐ ing portable energy generation, transportation, stationary back-up systems and energy gen‐ erators in space missions among others Currently, selected fuel cells such as SOFCs are considered as suitable conversion systems for the clean and sustainable energy generation. Development of such systems requires proper modeling approaches, construction of high fi‐ delity numerical simulators and tools able to provide clear insight into various aspects of the

Selected units with SOFCs have already reached proof-of-the-concept stage of development, and in some cases units are already available in for sale. Comparison of market-available systems is presented in Table 1, based on available data [7,8,9,10,11,12,13]. It should be em‐ phasized, that using fuel cells for electricity only generation in micro- and small-units is not economically feasible, therefore it was not considered. SOFC-based electricity-only genera‐ tion is economically feasible only for the capacity range over 100 kW*el*. For such systems ex‐ pected electrical efficiency ranges between 40 and 85%, while capital and peration and maintenance (O&M) costs were estimated for 1500-3000 \$kW and 0.0019 – 0.0153 \$/kW, re‐ spectively [14]. By comparing these numbers with data for other systems presented in Table

> **Power: electrical/ thermal [kW]**

**Hexis** SOFC 1.0/2.0 30-35/>90 42 **CFCL** SOFC 1.5/0.6 60/85 >10 **Vaillant** SOFC 4.6/6.5 30/88 >60 **JX Nippon** SOFC 0.7/1.25 45/87 800 **Baxi** PEFC 1.0/1.7 32/85 >20 **Viessmann** PEFC 2.0/5.0 28/80 10 **Bosch** PEFC 4.6/6.5 29/80 10

Different concepts of large SOFC-based systems were developed and studied [15,16,17]. Among those, various plants proposed by Siemens-Westinghouse with nominal power ranging from single up to hundreds MW*elel* were investigated [18,19], including pressurized systems. Despite the fact that high efficiency and near-zero emissions in such plants were

envisioned, attention has been focused on smaller scales – single and tens of kW*el*.

**Efficiency: η***el***/η***tot* **[%]** **Number of units**

Multi-Level Mathematical Modeling of Solid Oxide Fuel Cells

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55

Further studies were devoted to finding the optimal technology for micro- and small scalesystems suitable for CHP applications. It should be noted that, to distinguish between select‐ ed systems scales, terms micro and small were introduced. In the EU Combined Heat and Power directive [1] the earlier term refers to units with nominal power output 50 kW*el*, while in literature it usually covers systems with nominal power output of single kW*el* [2,3,4]. The later usually refers to system with output of tens of kW*el*.

It was found that three groups of technologies are especially interesting from the technical and economical point of view for systems with single kilowatts power output, namely:


While different energy generating systems with internal combustion and Stirling engines are a well-established technology, fuel cells in stationary generation have been known for not more than two decades. Even though technology is not yet mature, numerous demonstra‐ tion systems have already been operated allowing to gain operating experience. The main reason to consider fuel cells as an alternative to other generation systems is high electrical performance due to the direct conversion of chemical energy of a fuel into electricity.

Evaluation of PEFC and SOFC for micro-CHP application was recently presented [3]. Au‐ thors underlined high efficiency and mulifuel capabilities of the SOFC. Additionally, in case of low-temperature cells, such as PEFC, partial internal reforming cannot be done, hence ef‐ ficiency penalty due to external reforming is observed next to limited fuel flexibility. More‐ over, SOFCs offer utilization of high temperature heat in co-generative systems [4]. Substantial part of the high-grade heat can be recovered from the anodic and cathodic gas streams leaving the SOFC stack at elevated temperature [5] for hot tap water supply or heat‐ ing purposes [6]. Taking into account these advantages, SOFC technology has been selected for futher analysis with different modeling techniques.

## **2. Selected systems with Solid Oxide Fuel Cells**

for Better Environment

**•** Internal combustion engines

**•** Microturbines

**•** Stirling engines

**•** Steam engines

the directive

**•** Organic Rankine cycles

**•** Internal combustion engines

**•** Fuel Cells (PEFC and SOFC)

**•** Stirling engines

**•** Any other type of technology of combination of thereof falling under the definition laid in

Further studies were devoted to finding the optimal technology for micro- and small scalesystems suitable for CHP applications. It should be noted that, to distinguish between select‐ ed systems scales, terms micro and small were introduced. In the EU Combined Heat and Power directive [1] the earlier term refers to units with nominal power output 50 kW*el*, while in literature it usually covers systems with nominal power output of single kW*el* [2,3,4]. The

It was found that three groups of technologies are especially interesting from the technical and economical point of view for systems with single kilowatts power output, namely:

While different energy generating systems with internal combustion and Stirling engines are a well-established technology, fuel cells in stationary generation have been known for not more than two decades. Even though technology is not yet mature, numerous demonstra‐ tion systems have already been operated allowing to gain operating experience. The main reason to consider fuel cells as an alternative to other generation systems is high electrical

Evaluation of PEFC and SOFC for micro-CHP application was recently presented [3]. Au‐ thors underlined high efficiency and mulifuel capabilities of the SOFC. Additionally, in case of low-temperature cells, such as PEFC, partial internal reforming cannot be done, hence ef‐ ficiency penalty due to external reforming is observed next to limited fuel flexibility. More‐ over, SOFCs offer utilization of high temperature heat in co-generative systems [4]. Substantial part of the high-grade heat can be recovered from the anodic and cathodic gas streams leaving the SOFC stack at elevated temperature [5] for hot tap water supply or heat‐ ing purposes [6]. Taking into account these advantages, SOFC technology has been selected

performance due to the direct conversion of chemical energy of a fuel into electricity.

later usually refers to system with output of tens of kW*el*.

for futher analysis with different modeling techniques.

**•** Fuel cells

54 Clean Energy

Over the years fuel cell technology proved to be feasible in a number of applications, includ‐ ing portable energy generation, transportation, stationary back-up systems and energy gen‐ erators in space missions among others Currently, selected fuel cells such as SOFCs are considered as suitable conversion systems for the clean and sustainable energy generation. Development of such systems requires proper modeling approaches, construction of high fi‐ delity numerical simulators and tools able to provide clear insight into various aspects of the system operation.

Selected units with SOFCs have already reached proof-of-the-concept stage of development, and in some cases units are already available in for sale. Comparison of market-available systems is presented in Table 1, based on available data [7,8,9,10,11,12,13]. It should be em‐ phasized, that using fuel cells for electricity only generation in micro- and small-units is not economically feasible, therefore it was not considered. SOFC-based electricity-only genera‐ tion is economically feasible only for the capacity range over 100 kW*el*. For such systems ex‐ pected electrical efficiency ranges between 40 and 85%, while capital and peration and maintenance (O&M) costs were estimated for 1500-3000 \$kW and 0.0019 – 0.0153 \$/kW, re‐ spectively [14]. By comparing these numbers with data for other systems presented in Table 2 it becomes clear that SOFC can be indeed competitive.


**Table 1.** Comparison of selected micro-CHP systems with fuel cells.

Different concepts of large SOFC-based systems were developed and studied [15,16,17]. Among those, various plants proposed by Siemens-Westinghouse with nominal power ranging from single up to hundreds MW*elel* were investigated [18,19], including pressurized systems. Despite the fact that high efficiency and near-zero emissions in such plants were envisioned, attention has been focused on smaller scales – single and tens of kW*el*.


**Scale [m] Structure Phenomena**

Triple phase boundary – electrode, electrolyte and oxidant contact point

**Table 3.** Selected processes taking place during SOFC-system operation and their corresponding length scales.

(Table 3) and characteristic time has been proposed (Fig. 1).

Application-specific criteria and various designs require dedicated methodology for de‐ tailed investigation of processes listed in Table 3. In general, models are used to help under‐ stand and predict behavior of a particular system, to optimize control strategy, thermal balancing, and other aspects. Additionally, optimization tools can provide information on the optimal operational parameters. Moreover, models can be used as predictive tools for performance evaluation under off-design conditions. Modeling can provide crucial informa‐ tion for the system configuration improvements. Work on a prototype design is usually an iterative procedure where modeling is coupled with design definition. In each case, determi‐ nation of criteria is an important step and must correspond to particular requirements. De‐ pending on type of modeling, desired complexity and level of details, sufficient data have to be supplied to model. This section will briefly review different modeling techniques, includ‐ ing 0, 1-2 and 3D models. In a recent and valuable summary of modeling and simulation techniques [20] pictorial illustration of different issues and their corresponding length scales

Models can be divided into macro- and micro-scale, depending on the length scales that are covered by particular approach. In general case, analysis of SOFC at the stack level focuses

Porous media Knudsen diffusion

Flow field Diffusion

Electrochemistry Diffusion through the surface Chemical reaction

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57

Flow through porous media

Chemical reaction

Thermal balancing

Thermal balancing

Processes in the electrical

Integration of the entire

Mass flow Heat exchange

Single cell Transport of oxidant and fuel

Stack Electrical circuits of the cell

System level Control, automatics, safety systems

system

system

10-8–10-7 Electrode material

10-7–10-5

10-5–10-3

10-3–10-2

10-2–100

10-0–10?

**Table 2.** Comparison of selected systems for stand-alone electricity generation.

## **3. Modeling: transition between different length and time scales**

During SOFC-based system operation numbers of different processes are taking place at dif‐ ferent length- and time-scales. Summary of typically considered phenomena is presented in Table 3.

**Figure 1.** Different processes and their corresponding length scales and time frames (after [20])


for Better Environment

**Disel engines**

**Photovoltaic system**

> **Wind turbines**

Table 3.

56 Clean Energy

**Technology Capacity <sup>η</sup>***el* **[%] Capital cost**

**Table 2.** Comparison of selected systems for stand-alone electricity generation.

**3. Modeling: transition between different length and time scales**

**Figure 1.** Different processes and their corresponding length scales and time frames (after [20])

During SOFC-based system operation numbers of different processes are taking place at dif‐ ferent length- and time-scales. Summary of typically considered phenomena is presented in

**[\$/kW]**

500 kW - 50MW 35 200-350 0.005 - 0.015

1 kW - 1MW 6 - 19 6600 0.001 - 0.004

10 kW - 2MW 25 1000 0.01

**Gas turbines** 500 - 5 MW 29 - 42 450 - 870 0.005 - 0.0065

**O&M costs [\$/kW]**

**Table 3.** Selected processes taking place during SOFC-system operation and their corresponding length scales.

Application-specific criteria and various designs require dedicated methodology for de‐ tailed investigation of processes listed in Table 3. In general, models are used to help under‐ stand and predict behavior of a particular system, to optimize control strategy, thermal balancing, and other aspects. Additionally, optimization tools can provide information on the optimal operational parameters. Moreover, models can be used as predictive tools for performance evaluation under off-design conditions. Modeling can provide crucial informa‐ tion for the system configuration improvements. Work on a prototype design is usually an iterative procedure where modeling is coupled with design definition. In each case, determi‐ nation of criteria is an important step and must correspond to particular requirements. De‐ pending on type of modeling, desired complexity and level of details, sufficient data have to be supplied to model. This section will briefly review different modeling techniques, includ‐ ing 0, 1-2 and 3D models. In a recent and valuable summary of modeling and simulation techniques [20] pictorial illustration of different issues and their corresponding length scales (Table 3) and characteristic time has been proposed (Fig. 1).

Models can be divided into macro- and micro-scale, depending on the length scales that are covered by particular approach. In general case, analysis of SOFC at the stack level focuses on development of models for electrochemical processes, chemical reaction, transport phe‐ nomena, and geometry influence. Investigation of the entire system includes studies on the integration, heat and mass exchange, electrical circuits, and equipment.

no mass and heat accumulation occurs. The main disadvantage is the significant limitation in modeling influence of geometry and sizing, especially when those are of a high impor‐

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59

tance, for instance in chemistry modeling.

**Figure 2.** Single-pass system with CPOX reactor

**Figure 3.** Single-pass system with steam reformer

#### **3.1. 0D modeling**

Zero-dimensional methodology allows studying processes that can be analyzed without tak‐ ing into account spatial configuration and geometry. Such approach is justified for systemlevel studies, however might also be used for estimation of certain parameters. Depending on the required precision, 0D models can be used to solve governing equations for planar SOFC, written for each of the cell components: electrodes, electrolyte, interconnects and flow channels [21,22,23,24,25]. Required assumptions include constant fluid properties, air as an incompressible gas and no chemical reactions occurring in the fuel and air channels. Set of governing equations is later solved with desired accuracy by different algorithms. System-level studies can be performed with commercially available software such as Aspen Plus or Aspen Hysys. The later was recently used by Kupecki and Badyda [5] for evaluation of different fuel processing technologies for micro-CHP unit with SOFC. Different designs, presented in Fig. 2-6 were studied for evaluation of heat and mass balances. In the study, characteristics of market-available SOFCs were implemented, and auxiliary equipment was selected for off-the-shelf products. Considering different fuels, including natural gas, diesel, and LPG it was possible to define the optimal processing technology for micro-CHP unit equipped with afterburner. Steam reforming allowed achieving the highest overall system efficiency, even tough it required substantial amount of heat. Generally, 0D method proved to be sufficient for system-level studied, including thermal processes (i.e. heat exchange, heat losses, combustion in the afterburner), electrochemical reactions in the SOFC stack and chemical reactions occurring in the fuel processor.

With introduction of heat capacity, dynamics can be studied to some extent using 0D model‐ ing techniques, as it will be presented in the dynamic modeling section. In certain cases chemistry can also be investigated. The main limitation is the difficulty to explicitly incorpo‐ rate geometry of chemical reactors, although semi-empirical correlations are sometimes ap‐ plicable. Bove and Ubertini [26] suggested using black-box 0D models to investigate impact of fuel composition, oxidant or fuel utilization and overpotentials on the macroscopic per‐ formance of SOFC in terms of efficiency and current-voltage characteristics. Such models should be used when system-level approach is required, without main focus on the SOFC stack itself [27]. In cell-leveling modeling, zero dimensional approach can be efficiently used for solving elementary balance equations for fluids: continuity, momentum, energy and spe‐ cies transport. Since solid oxide fuel cells consists of two porous electrodes separated by an electrolyte, porosity of these materials should be explicitly considered in the governing equations. Once the set of equations is developed, it can easily be transferred from discrete 0D to 1D model to be solved using proper CFD method [28].

Summarizing, the main advantage of zero-dimensional approach is low computational costs, simple formulation of the model. Such models can be freely used for systems where no mass and heat accumulation occurs. The main disadvantage is the significant limitation in modeling influence of geometry and sizing, especially when those are of a high impor‐ tance, for instance in chemistry modeling.

**Figure 2.** Single-pass system with CPOX reactor

for Better Environment

chemical reactions occurring in the fuel processor.

0D to 1D model to be solved using proper CFD method [28].

**3.1. 0D modeling**

58 Clean Energy

on development of models for electrochemical processes, chemical reaction, transport phe‐ nomena, and geometry influence. Investigation of the entire system includes studies on the

Zero-dimensional methodology allows studying processes that can be analyzed without tak‐ ing into account spatial configuration and geometry. Such approach is justified for systemlevel studies, however might also be used for estimation of certain parameters. Depending on the required precision, 0D models can be used to solve governing equations for planar SOFC, written for each of the cell components: electrodes, electrolyte, interconnects and flow channels [21,22,23,24,25]. Required assumptions include constant fluid properties, air as an incompressible gas and no chemical reactions occurring in the fuel and air channels. Set of governing equations is later solved with desired accuracy by different algorithms. System-level studies can be performed with commercially available software such as Aspen Plus or Aspen Hysys. The later was recently used by Kupecki and Badyda [5] for evaluation of different fuel processing technologies for micro-CHP unit with SOFC. Different designs, presented in Fig. 2-6 were studied for evaluation of heat and mass balances. In the study, characteristics of market-available SOFCs were implemented, and auxiliary equipment was selected for off-the-shelf products. Considering different fuels, including natural gas, diesel, and LPG it was possible to define the optimal processing technology for micro-CHP unit equipped with afterburner. Steam reforming allowed achieving the highest overall system efficiency, even tough it required substantial amount of heat. Generally, 0D method proved to be sufficient for system-level studied, including thermal processes (i.e. heat exchange, heat losses, combustion in the afterburner), electrochemical reactions in the SOFC stack and

With introduction of heat capacity, dynamics can be studied to some extent using 0D model‐ ing techniques, as it will be presented in the dynamic modeling section. In certain cases chemistry can also be investigated. The main limitation is the difficulty to explicitly incorpo‐ rate geometry of chemical reactors, although semi-empirical correlations are sometimes ap‐ plicable. Bove and Ubertini [26] suggested using black-box 0D models to investigate impact of fuel composition, oxidant or fuel utilization and overpotentials on the macroscopic per‐ formance of SOFC in terms of efficiency and current-voltage characteristics. Such models should be used when system-level approach is required, without main focus on the SOFC stack itself [27]. In cell-leveling modeling, zero dimensional approach can be efficiently used for solving elementary balance equations for fluids: continuity, momentum, energy and spe‐ cies transport. Since solid oxide fuel cells consists of two porous electrodes separated by an electrolyte, porosity of these materials should be explicitly considered in the governing equations. Once the set of equations is developed, it can easily be transferred from discrete

Summarizing, the main advantage of zero-dimensional approach is low computational costs, simple formulation of the model. Such models can be freely used for systems where

integration, heat and mass exchange, electrical circuits, and equipment.

**Figure 3.** Single-pass system with steam reformer

**Figure 4.** System with recirculation based on an ejector

**Figure 6.** System with recirculation based on a low-temperature fan

ble only to no-load, isothermal stack conditions.

**3.2. 2- and 3D modeling and computational fluid dynamics codes**

In 1-,2- and 3D models, space-dependent governing equations are being solved. In case of three-dimensional approach, mathematical formulas are usually written in form of partial differential equations. Different methods can be used for solving the resulting set of equa‐ tions. In 1D approach, ordinary differential equations may be encountered, and solution can be easily found with simple codes or even analytically. Complex 3D models of SOFC stack are useful for heat and mass exchange modeling [29]. With high fidelity models, different heat exchange means can be studies, and cell voltage under inhomogeneous temperature distribution can be found. Space-continuous models can be applied for material studies and evaluation of process losses. Time-dependent thermal processes can be studied in similar way to proposed nearly twenty years ago by Achenbach [30]. In his work, numerical tool was used to investigate heat conductivity of stack made of ceramic and metallic plates. With the proposed methodology it was possible to find the overall heat conductivities of the com‐ bined SOFC assembly. Additionally, the model was applied to evaluate influence of thermal radiation and the total heat losses from the stack. Such studies are crucial for evaluation of overall system performance, and can indicate dangerous operational modes, which should be avoided. Several analytical models of pressure and flow distribution in the stack have been presented [59,60]. The results have been compared to 3D CFD model, showing accura‐ cy sufficient for engineering calculations. However, analytical models are typically applica‐

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Significant computational power is required to implement fully-3D CFD combined models of the SOFC stack and auxiliary system components. Computational time requirements limits complex optimization of such cases. In particular, when optimization of 3D models of sys‐ tem sections is necessary to approximate integration of SOFC stack with pre-heaters or reform‐

**Figure 5.** System with recirculation based on a high-temperature fan

**Figure 6.** System with recirculation based on a low-temperature fan

for Better Environment

60 Clean Energy

**Figure 4.** System with recirculation based on an ejector

**Figure 5.** System with recirculation based on a high-temperature fan

#### **3.2. 2- and 3D modeling and computational fluid dynamics codes**

In 1-,2- and 3D models, space-dependent governing equations are being solved. In case of three-dimensional approach, mathematical formulas are usually written in form of partial differential equations. Different methods can be used for solving the resulting set of equa‐ tions. In 1D approach, ordinary differential equations may be encountered, and solution can be easily found with simple codes or even analytically. Complex 3D models of SOFC stack are useful for heat and mass exchange modeling [29]. With high fidelity models, different heat exchange means can be studies, and cell voltage under inhomogeneous temperature distribution can be found. Space-continuous models can be applied for material studies and evaluation of process losses. Time-dependent thermal processes can be studied in similar way to proposed nearly twenty years ago by Achenbach [30]. In his work, numerical tool was used to investigate heat conductivity of stack made of ceramic and metallic plates. With the proposed methodology it was possible to find the overall heat conductivities of the com‐ bined SOFC assembly. Additionally, the model was applied to evaluate influence of thermal radiation and the total heat losses from the stack. Such studies are crucial for evaluation of overall system performance, and can indicate dangerous operational modes, which should be avoided. Several analytical models of pressure and flow distribution in the stack have been presented [59,60]. The results have been compared to 3D CFD model, showing accura‐ cy sufficient for engineering calculations. However, analytical models are typically applica‐ ble only to no-load, isothermal stack conditions.

Significant computational power is required to implement fully-3D CFD combined models of the SOFC stack and auxiliary system components. Computational time requirements limits complex optimization of such cases. In particular, when optimization of 3D models of sys‐ tem sections is necessary to approximate integration of SOFC stack with pre-heaters or reform‐ ers, engineering accuracy approximations are often implemented. 3D non-CFD numerical model of SOFC stack has been applied to improve the thermal management of SOFC system through radiant heat transfer from the stack walls to adjacent air preheater panels [61].

In this study, options for minimizing axial and in-plane temperature gradients in the stack have also been identified. The results of subsequent tests, verifying modeling results, sug‐ gested that the use of radiation-based approach significantly improves the management of stack-generated heat [62].

Since porous body, representing electrochemically active part of the SOFC stack, is imper‐ meable in directions other than flow direction in gas channels, simplifications of the 3D CFD stack model is possible, including 2D CFD model with the porous body approach (see Fig. 7). Periodical and ordered geometry of reactant channels in the stack, allows treatment of stack geometry as a porous body, with porosity defined as a ratio of channels cross-sectional area to stack cross-sectional area [63]. In the study, 2-D and 3D CFD SOFC stack models with internal manifolds have been implemented to simulate flow distribution under electric load conditions for the selected fuels. The semi-empirical model of electrochemical kinetics has been implemented. Typical flow arrangements of the inlet and outlet gas supply mani‐ folds (U-flow, Z-flow) have been evaluated, including effects associated water-shift reaction and finite-rate of internal reforming of methane in the stack.

**Figure 8.** Stack representation in the hydraulic network approach

where:

*ΔP<sup>i</sup>*

*ρi*

*Vi*

*fD* Darcy's friction factor

gas density [kg cm-1]

 gas velocity [m s-1] *D* hydraulic diameter [m]

*Re* Reynolds number

In yet another approach to SOFC stack modeling, so-called hydraulic network approach [64], pressure drop is calculated separately for each manifold section and reactant channel section, as shown in Fig. 8. The pressure drop is calculated based on the Darcy''s friction fac‐ tor, incorporating local geometry and stream characteristics. The hydraulic model approach has been implemented for the planar, rectangular geometry of the fuel cells. In the model,

pressure drop is calculated separately for each manifold section and cell section:

*L i Di ρi Vi* 2

<sup>2</sup> (4.1)

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63

Δ*Pi* = *f <sup>D</sup>*

*fD* = *K*/*Re* for the laminar flow and *fD* = *ε/D* for the turbulent flow

pressure drop in the manifold section or cell section [Pa]

*L* length of the manifold section or cell section [m]

*K* constant (64 for the circular channels) *ε/D* relative roughness of the channel

**Figure 7.** Stack representation in the 2D/3D CFD porous body approach

**Figure 8.** Stack representation in the hydraulic network approach

In yet another approach to SOFC stack modeling, so-called hydraulic network approach [64], pressure drop is calculated separately for each manifold section and reactant channel section, as shown in Fig. 8. The pressure drop is calculated based on the Darcy''s friction fac‐ tor, incorporating local geometry and stream characteristics. The hydraulic model approach has been implemented for the planar, rectangular geometry of the fuel cells. In the model, pressure drop is calculated separately for each manifold section and cell section:

$$
\Delta P\_i = f\_D \frac{L\_i}{D\_i} \frac{\rho\_i V\_i^2}{2} \tag{4.1}
$$

where:

for Better Environment

stack-generated heat [62].

62 Clean Energy

ers, engineering accuracy approximations are often implemented. 3D non-CFD numerical model of SOFC stack has been applied to improve the thermal management of SOFC system through radiant heat transfer from the stack walls to adjacent air preheater panels [61].

In this study, options for minimizing axial and in-plane temperature gradients in the stack have also been identified. The results of subsequent tests, verifying modeling results, sug‐ gested that the use of radiation-based approach significantly improves the management of

Since porous body, representing electrochemically active part of the SOFC stack, is imper‐ meable in directions other than flow direction in gas channels, simplifications of the 3D CFD stack model is possible, including 2D CFD model with the porous body approach (see Fig. 7). Periodical and ordered geometry of reactant channels in the stack, allows treatment of stack geometry as a porous body, with porosity defined as a ratio of channels cross-sectional area to stack cross-sectional area [63]. In the study, 2-D and 3D CFD SOFC stack models with internal manifolds have been implemented to simulate flow distribution under electric load conditions for the selected fuels. The semi-empirical model of electrochemical kinetics has been implemented. Typical flow arrangements of the inlet and outlet gas supply mani‐ folds (U-flow, Z-flow) have been evaluated, including effects associated water-shift reaction

and finite-rate of internal reforming of methane in the stack.

**Figure 7.** Stack representation in the 2D/3D CFD porous body approach


Additional pressure losses are calculated for the flow obstacles, such as dividing/combining flows at the manifold/reactant channel junctions, as:

$$
\Delta P\_i = K\_{tot} \frac{\rho\_i V\_i^2}{2} \tag{4.2}
$$

such system. However, it should be noted that fuel cells are generally believed to operate

Multi-Level Mathematical Modeling of Solid Oxide Fuel Cells

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It is well know that the main advantage of solid oxide fuel cells is the ability to operate with number of different fuels including alcohols [32], hydrocarbons [31], pure hydrogen [33,21,31], biofuels [34] and energy carries which can be converted to hydrogen-rich gas, in‐ cluding ammonia [35] and dimethyl ether [36]. Nonetheless, in order to assure high per‐ formance operation of a fuel cell stack, by limiting cells degradation, proper fuel processing has to be selected. As recently reported by Leone et al. [40] different fuel processing technol‐ ogies may be used for fuel cell-based system, however reforming technique can influence cell operating conditions and selection should be made taking into account different factors. Generally, three different technologies can be distinguished for converting fuel before it en‐ ters the SOFC stack: catalytic partial oxidation (CPOX), steam reforming (SR) and autother‐ mal reforming (AT). In certain cases these processes can be accompanied by fuel clean-up

The important part of SOFC system operation is direct thermal integration of stack and fuel reforming. Different implementations have been proposed and corresponding modeling

**1.** Intermediate indirect reforming plates (IIR) can be directly integrated with the SOFC stack [45,46]. In this approach, fuel reforming plates are integrated with the stack struc‐ ture and separated with one or more fuel cells. Reformed fuel from the reforming plates

**2.** Direct internal reforming (DIR) is often implemented, taking advantage of catalytic properties of the SOFC anode material [47]. In this approach fuel is directly reformed on the anode side of the fuel cell. Pre-reforming of the fuel might be necessary in some cases to avoid overcooling of the fuel inlet stack region, particularly for the fuel with

**3.** Thermal integration of SOFC stack and fuel reformer can also be implemented with thermal radiation/convection/conduction conjugate heat transfer between SOFC stack(s) and reformer. In this approach, fuel reformer is placed in a direct vicinity of the stack(s).

In this subsection, theory of different fuel processing technologies will be briefly discussed. Partial oxidation reaction proceeds with the presence of catalyst can be written in a general

2 2 2 2 2 2 ( 3.76 ) (2 2 ) (2 2 0.5 ) 3.76 *C H O x O N n x p H O nCO n x p m H xN n mp* + + + - - ® + - -- + (4.4)

where *x* is the oxygen to fuel molar ratio. This ratio defines the required amount of water for carbon to carbon monoxide conversion, amount of generated hydrogen, and molar concen‐

quite stable when compared to other energy conversion techniques [31].

stage as it is usually done for fuels with significant H2S content.

is redirected to fuel inlet of the adjacent cell(s).

**3.3. Fuel processing technologies**

studies performed:

high methane content.

form for any hydrocarbon [37]

The resulting system of nonlinear equations is solved numerically for each of the flow loops:

$$
\Delta P\_{MI,i} + \Delta P\_{CELL\\_i \star 1} - \Delta P\_{CELL\\_i \star 1} + Q \cdot \Delta P\_{MO,i} = 0 \text{ for } i = 1...N-1\tag{4.3}
$$

Numerical results show good convergence with analytical models (Fig. 9). The hydraulic networks approach is also applicable to SOFC stack modeling under electric load conditions.

**Figure 9.** Comparison of pressure drop across the cathode side of the SOFC cell for a range of flows corresponding to a range of oxidant utilizations (▲– hydraulic model results; ♦ – measurements, 295 K; ■– 3D CFD simulation results)

Computational fluid dynamic system can be also used for system integration [2]. The cou‐ pling different models in one simulator can provide insight into operation of system compo‐ nents such as BoP devices, fuel processor, tail gas combustor, and SOFC stack. Time scale selection for the modeling should be done with caution. As has been noted by Tanaka et al. [29], certain fluctuations during small-scale co-generative SOFC-based unit operation occur. Authors performed detailed uncertainty estimation for 10 kW class units fed with town gas. Research was focused on evaluation of possible fluctuations, including changes in fuel qual‐ ity over time (i.e. deviation of HHV from the nominal value), flow variations, precision of measurement equipment and other factors. The results indicate that the electrical efficiency of system can be determined with 1.0% relative uncertainty at 95% level of confidence for such system. However, it should be noted that fuel cells are generally believed to operate quite stable when compared to other energy conversion techniques [31].

#### **3.3. Fuel processing technologies**

for Better Environment

64 Clean Energy

flows at the manifold/reactant channel junctions, as:

Additional pressure losses are calculated for the flow obstacles, such as dividing/combining

*ρi Vi* 2

The resulting system of nonlinear equations is solved numerically for each of the flow loops:

Numerical results show good convergence with analytical models (Fig. 9). The hydraulic networks approach is also applicable to SOFC stack modeling under electric load conditions.

**Figure 9.** Comparison of pressure drop across the cathode side of the SOFC cell for a range of flows corresponding to a range of oxidant utilizations (▲– hydraulic model results; ♦ – measurements, 295 K; ■– 3D CFD simulation results)

Computational fluid dynamic system can be also used for system integration [2]. The cou‐ pling different models in one simulator can provide insight into operation of system compo‐ nents such as BoP devices, fuel processor, tail gas combustor, and SOFC stack. Time scale selection for the modeling should be done with caution. As has been noted by Tanaka et al. [29], certain fluctuations during small-scale co-generative SOFC-based unit operation occur. Authors performed detailed uncertainty estimation for 10 kW class units fed with town gas. Research was focused on evaluation of possible fluctuations, including changes in fuel qual‐ ity over time (i.e. deviation of HHV from the nominal value), flow variations, precision of measurement equipment and other factors. The results indicate that the electrical efficiency of system can be determined with 1.0% relative uncertainty at 95% level of confidence for

Δ*PMI* ,*<sup>i</sup>* + Δ*PCELL* ,*i*+1 −Δ*PCELL* ,*i*+1 + *Q* ⋅Δ*PMO*,*<sup>i</sup>* =0 *for i* =1...*N* −1 (4.3)

<sup>2</sup> (4.2)

Δ*Pi* = *Ktot*

It is well know that the main advantage of solid oxide fuel cells is the ability to operate with number of different fuels including alcohols [32], hydrocarbons [31], pure hydrogen [33,21,31], biofuels [34] and energy carries which can be converted to hydrogen-rich gas, in‐ cluding ammonia [35] and dimethyl ether [36]. Nonetheless, in order to assure high per‐ formance operation of a fuel cell stack, by limiting cells degradation, proper fuel processing has to be selected. As recently reported by Leone et al. [40] different fuel processing technol‐ ogies may be used for fuel cell-based system, however reforming technique can influence cell operating conditions and selection should be made taking into account different factors.

Generally, three different technologies can be distinguished for converting fuel before it en‐ ters the SOFC stack: catalytic partial oxidation (CPOX), steam reforming (SR) and autother‐ mal reforming (AT). In certain cases these processes can be accompanied by fuel clean-up stage as it is usually done for fuels with significant H2S content.

The important part of SOFC system operation is direct thermal integration of stack and fuel reforming. Different implementations have been proposed and corresponding modeling studies performed:


In this subsection, theory of different fuel processing technologies will be briefly discussed.

Partial oxidation reaction proceeds with the presence of catalyst can be written in a general form for any hydrocarbon [37]

$$\rm{C}\_{n}H\_{n}O\_{p} + x(O\_{2} + 3.76N\_{2}) + (2n - 2\text{x} - p)H\_{2}O \rightarrow nCO\_{2} + (2n - 2\text{x} - p - 0.5\text{m})H\_{2} + 3.76xN\_{2} \tag{4.4}$$

where *x* is the oxygen to fuel molar ratio. This ratio defines the required amount of water for carbon to carbon monoxide conversion, amount of generated hydrogen, and molar concen‐ tration of hydrogen in the reaction products. For *x* = 0 the reaction becomes an endothermic steam reforming, and for *x* = 12.5 it corresponds to a combustion process. Partial oxidation reaction should be controlled in such way, that overall thermal balance would be exother‐ mic. Simple calculations lead to conclusion, that for a low value of x coefficient, higher amounts (or concentrations) of hydrogen should be expected. The main reason for using cat‐ alyst is the reduction of the process temperature. Reaction described by equation (1) to pro‐ ceed without catalyst, however temperature o about 1000° C is required in such case. Because of that fact in most commercial applications, including SOFC- based systems, cata‐ lyst is used.

Second method for turning different fuels into hydrogen-rich gas is the steam reforming. In most fuel cell applications, reaction proceeds at high temperature with addition of water va‐ por. Typical products of steam reforming include hydrogen and carbon dioxide. Ideal reac‐ tion can be written for any hydrocarbon fuel fed in the following form:

$$\rm C\_{n}H\_{2n} + 0.5(n-1)H\_{2}O \rightarrow 0.25(3n+1)CH\_{4} + 0.25(n-1)CO\_{2} \tag{4.5}$$

In most technological processes, steam reforming comprises two stages which can be written for the simplest hydrocarbon in a form:

$$\rm{CH}\_4\star\rm{H}\_2\rm{O}\rightarrow\rm{CO}\_2\star\rm{3H}\_2\tag{4.6}$$

material issues (endothermic reforming reaction) and possibility of carbon deposition to oc‐ cur. Recent study presented in 0D modeling section and available literature [42,43] clearly indicates that steam reforming is the most efficient technology for bioethanol, methane and other hydrocarbons conversion into hydrogen rich-gas. Arteaga et al. [44] performed thor‐ ough evaluation of different fuel processing technologies particularly for SOFC application also finding steam reforming the most suitable. Comparison of different fuel processing technologies for biogas and methane was previously done and reported [53]. It was clearly indicated that steam reforming is the optimal selection for micro-CHP units with SOFCs.

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As it was discussed in previous section, systems with SOFCs require fuel processing. In most cases steam reforming would be selected, and in such system catalysts would be employed.

Since micro-CHP unit of discussed type generated both electricity and heat, two control strategies are possible. Device can operate following electricity or heat/hot water demand. Generally it is believed, that the most optimal strategy is to control electricity generation, considering heat as a by-product which can possibly be stored in sufficiently large water tank. In available literature, different tank sizes were considered. In design of micro-CHP system with power output of about 2 kW by Kupecki and Badyda [5], tank with volume of 600 liters was considered. At average storage temperature of 55° C, the total of about 28 kW*th* can be stored in the tank. This volume was selected based on availability of off-the-shelf products, its reasonable price and sufficient heat capacity. Surprisingly, some authors [65] suggest selection of much larger size like 1000 or even 3000 liters. From the product devel‐ opment point of view, customer expectations and required compact size, this is an a way too large volume. Moreover, according to authors'' own calculations, selection of such a big ves‐ sel has negligible economical gain, and in all cases can lead to increase of capital cost of the system. Price of hot water storage tanks increases exponentially with the capacity increase, therefore considerations of volumes above 600 liters should not have place. Additional as‐ pect of control strategy selection depend on current conditions, including generation price

In a recent study [66] control strategy for the highest energy savings for 20 households using 0.7 kW micro-CHP systems with SOFC was evaluated. Mixed-integer linear programming was used to optimized control for a case, and when each of households has a different consump‐ tion. They clearly found, that electrical load-following is the best strategy, allowing the high‐ est energy-saving effect. Authors also pointed out that wastage of surplus hot water is possible in the summer season, but this can be avoided by selection of slightly larger water tank.

**4. Control strategy for micro-CHP unit**

and cost or resources.

and

$$\rm{CO}\_2 + \rm{H}\_2\rm{O} \rightarrow \rm{CO}\_2 + \rm{H}\_2\tag{4.7}$$

Where equation (4.6) is strongly endothermic and (4.7) is slightly exothermic, therefore the overall reaction requires heat delivery. Typically, steam reforming of gases can also be done as a catalyst-supported process. Usually a metallic nickel catalyst [38,39] either Ni/Al2O3 or Ni on refractory material, containing 5-30% of Ni are used. Lifetime of a catalyst strongly de‐ pends on quality of gases converted in the steam reformer, so-called poisoning is usually the main process leading to rapid performance deterioration. In order to ensure long lasting operation of the catalyst, poisonous impurities should be removed prior the reforming process.

Next to CPOX and SR, internal reforming is also mentioned as a interesting processing tech‐ nology and the most economical way to convert hydrocarbon fuels for tubular and planar SOFCs. Despite the fact that process has number of advantages, it may lead to high tempera‐ ture variations in the fuel cell and stack [41]. Highly endothermic character of the reaction is responsible for local cooling of the cell material leading to cracking and rupture. In a similar way, CPOX reaction along the cell is often claimed to be responsible for cell overheating which can compromise the ceramic material stability in a similar way as internal reforming. Even though, internal reforming is allowed to a certain extent, it is believed that thermal de‐ composition of higher hydrocarbons may lead to carbon formation on the anode compart‐ ments [30]. Usually, limitation on the fraction of higher hydrocarbons is imposed by material issues (endothermic reforming reaction) and possibility of carbon deposition to oc‐ cur. Recent study presented in 0D modeling section and available literature [42,43] clearly indicates that steam reforming is the most efficient technology for bioethanol, methane and other hydrocarbons conversion into hydrogen rich-gas. Arteaga et al. [44] performed thor‐ ough evaluation of different fuel processing technologies particularly for SOFC application also finding steam reforming the most suitable. Comparison of different fuel processing technologies for biogas and methane was previously done and reported [53]. It was clearly indicated that steam reforming is the optimal selection for micro-CHP units with SOFCs.

As it was discussed in previous section, systems with SOFCs require fuel processing. In most cases steam reforming would be selected, and in such system catalysts would be employed.

## **4. Control strategy for micro-CHP unit**

for Better Environment

for the simplest hydrocarbon in a form:

lyst is used.

66 Clean Energy

and

tration of hydrogen in the reaction products. For *x* = 0 the reaction becomes an endothermic steam reforming, and for *x* = 12.5 it corresponds to a combustion process. Partial oxidation reaction should be controlled in such way, that overall thermal balance would be exother‐ mic. Simple calculations lead to conclusion, that for a low value of x coefficient, higher amounts (or concentrations) of hydrogen should be expected. The main reason for using cat‐ alyst is the reduction of the process temperature. Reaction described by equation (1) to pro‐ ceed without catalyst, however temperature o about 1000° C is required in such case. Because of that fact in most commercial applications, including SOFC- based systems, cata‐

Second method for turning different fuels into hydrogen-rich gas is the steam reforming. In most fuel cell applications, reaction proceeds at high temperature with addition of water va‐ por. Typical products of steam reforming include hydrogen and carbon dioxide. Ideal reac‐

In most technological processes, steam reforming comprises two stages which can be written

Where equation (4.6) is strongly endothermic and (4.7) is slightly exothermic, therefore the overall reaction requires heat delivery. Typically, steam reforming of gases can also be done as a catalyst-supported process. Usually a metallic nickel catalyst [38,39] either Ni/Al2O3 or Ni on refractory material, containing 5-30% of Ni are used. Lifetime of a catalyst strongly de‐ pends on quality of gases converted in the steam reformer, so-called poisoning is usually the main process leading to rapid performance deterioration. In order to ensure long lasting operation of the catalyst, poisonous impurities should be removed prior the reforming process. Next to CPOX and SR, internal reforming is also mentioned as a interesting processing tech‐ nology and the most economical way to convert hydrocarbon fuels for tubular and planar SOFCs. Despite the fact that process has number of advantages, it may lead to high tempera‐ ture variations in the fuel cell and stack [41]. Highly endothermic character of the reaction is responsible for local cooling of the cell material leading to cracking and rupture. In a similar way, CPOX reaction along the cell is often claimed to be responsible for cell overheating which can compromise the ceramic material stability in a similar way as internal reforming. Even though, internal reforming is allowed to a certain extent, it is believed that thermal de‐ composition of higher hydrocarbons may lead to carbon formation on the anode compart‐ ments [30]. Usually, limitation on the fraction of higher hydrocarbons is imposed by

*CnH*2*<sup>n</sup>* + 0.5(*n* −1)*H*2*O* →0.25(3*n* + 1)*CH*<sup>4</sup> + 0.25(*n* −1)*CO*<sup>2</sup> (4.5)

*CH*<sup>4</sup> + *H*2*O* →*CO*<sup>2</sup> + 3*H*<sup>2</sup> (4.6)

*CO*<sup>2</sup> + *H*2*O* →*CO*<sup>2</sup> + *H*<sup>2</sup> (4.7)

tion can be written for any hydrocarbon fuel fed in the following form:

Since micro-CHP unit of discussed type generated both electricity and heat, two control strategies are possible. Device can operate following electricity or heat/hot water demand. Generally it is believed, that the most optimal strategy is to control electricity generation, considering heat as a by-product which can possibly be stored in sufficiently large water tank. In available literature, different tank sizes were considered. In design of micro-CHP system with power output of about 2 kW by Kupecki and Badyda [5], tank with volume of 600 liters was considered. At average storage temperature of 55° C, the total of about 28 kW*th* can be stored in the tank. This volume was selected based on availability of off-the-shelf products, its reasonable price and sufficient heat capacity. Surprisingly, some authors [65] suggest selection of much larger size like 1000 or even 3000 liters. From the product devel‐ opment point of view, customer expectations and required compact size, this is an a way too large volume. Moreover, according to authors'' own calculations, selection of such a big ves‐ sel has negligible economical gain, and in all cases can lead to increase of capital cost of the system. Price of hot water storage tanks increases exponentially with the capacity increase, therefore considerations of volumes above 600 liters should not have place. Additional as‐ pect of control strategy selection depend on current conditions, including generation price and cost or resources.

In a recent study [66] control strategy for the highest energy savings for 20 households using 0.7 kW micro-CHP systems with SOFC was evaluated. Mixed-integer linear programming was used to optimized control for a case, and when each of households has a different consump‐ tion. They clearly found, that electrical load-following is the best strategy, allowing the high‐ est energy-saving effect. Authors also pointed out that wastage of surplus hot water is possible in the summer season, but this can be avoided by selection of slightly larger water tank.

## **5. Modeling the dynamic behavior of a singular Solid Oxide Fuel Cell**

## **5.1 Dynamic oriented model of SOFC**

The mathematical model of SOFC for steady state calculations was presented in a few previ‐ ous papers [73,74,75,76,77,78,79,80]. In this section only dynamic oriented relationships are included and commented on.

**Figure 11.** A concept of a model of Solid Oxide Fuel Cell

*dTCell dt* <sup>=</sup>

*d pCathode*,*Out dt* <sup>=</sup>

*d pAnode*,*Out dt* <sup>=</sup>

{

where:

The fuel cell presented in Fig. 11 can be reduced to an 0D model. This is the simplest ap‐ proach, but generates a model of the same class as models of other equipment (compressors,

> *<sup>Q</sup>*˙ *Oxidant* <sup>+</sup> *<sup>Q</sup>*˙ *Fuel* <sup>−</sup>2⋅*Q*˙ *Surrounding* <sup>−</sup>*PSOFC* <sup>+</sup> *<sup>Q</sup>*˙ *Fuel* 2⋅*CManifold* + *CFuel* + *COxidant* + *CCell*

*<sup>Q</sup>*˙ *Oxidant* <sup>=</sup>*m*˙ *Oxidant* <sup>⋅</sup> *cp*,*Oxidant*(*TCathode*,*In* <sup>−</sup>*TCell*) (6.2)

*<sup>Q</sup>*˙ *Surrounding* <sup>=</sup>*kSurrounding* <sup>⋅</sup> *ACell*(*TCell* <sup>−</sup>*TSurrounding*) (6.4)

*<sup>Q</sup>*˙ *Fuel* <sup>=</sup>*m*˙ *Fuel* <sup>⋅</sup> *cp*,*Fuel*(*<sup>T</sup> Anode*,*In* <sup>−</sup>*TCell*) (6.3)

*PSOFC* =*ESOFC* ⋅ *ISOFC* (6.5)

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*<sup>Q</sup>*˙ *Fuel* <sup>=</sup>*m*˙ *Fuel* <sup>⋅</sup>*HH VFuel* <sup>⋅</sup>*<sup>η</sup> <sup>f</sup>* (6.6)

*CManifold* =*mManifold* ⋅ *cp*,*Manifold* (6.7)

(6.1)

pumps, heat exchangers). The set of equations for the 0D model is as follows:

*m*˙ *Cathode*,*In* −*m*˙ *Cathode*,*Out VCathode ROxidant* ⋅*TCell*

*m*˙ *Anode*,*In* −*m*˙ *Anode*,*Out V Anode RFuel* ⋅*TCell*

**Figure 10.** Dimensions of fuel cell plate and the manifolds

As an object for modeling, a singular fuel cell is chosen with dimensions of 5 cm × 5 cm and thickness of 1 mm (see Fig. 10). It was assumed that the manifolds (for fuel and oxi‐ dant) are identical.

Some processes which occur during fuel cell operation are very rapid, thus they can be as‐ sumed to be time independent compared to others. The following processes are assumed to be time independent:


For those processes only the static equations were utilized.

Fig. 11 presents a concept of a model of Solid Oxide Fuel Cell, the fuel cell is equiped with two inlet streams and two outlet streams. Processes which occur during fuel cell operation can be divided into three steps: capture of oxygen atoms from the delivered oxidant (air), oxygen ions passing through the electrolyte layer, and the ions escaping and reacting with the delivered fuel. Material aspects play a crucial role here [68].

**Figure 11.** A concept of a model of Solid Oxide Fuel Cell

The fuel cell presented in Fig. 11 can be reduced to an 0D model. This is the simplest ap‐ proach, but generates a model of the same class as models of other equipment (compressors, pumps, heat exchangers). The set of equations for the 0D model is as follows:

$$\begin{aligned} \frac{d\,T\_{Cell}}{dt} &= \frac{\dot{\mathbf{Q}}\_{Oxidant} + \dot{\mathbf{Q}}\_{Fuel} - \mathbf{2} \cdot \dot{\mathbf{Q}}\_{Surrounding} - \mathbf{P}\_{SOFC} + \dot{\mathbf{Q}}\_{Fuel}}{2 \cdot \mathbf{C}\_{Anrigid} + \mathbf{C}\_{Fuel} + \mathbf{C}\_{Oxidant} + \mathbf{C}\_{Cell}} \\ \frac{d\,p\_{Cathode,Out}}{dt} &= \frac{\dot{m}\_{Cathode,In} - \dot{m}\_{Cathode,Out}}{V\_{Calcide}} \\ &= \frac{d\,p\_{Anode,Out}}{R\_{Oxidant} + T\_{Cell}} \\ \frac{d\,p\_{Anode,Out}}{dt} &= \frac{\dot{m}\_{Anode,In} - \dot{m}\_{Anode,Out}}{R\_{Anode}} \end{aligned} \tag{6.1}$$

where:

for Better Environment

68 Clean Energy

**5.1 Dynamic oriented model of SOFC**

**Figure 10.** Dimensions of fuel cell plate and the manifolds

For those processes only the static equations were utilized.

the delivered fuel. Material aspects play a crucial role here [68].

dant) are identical.

be time independent:

**1.** Electrical processes

**3.** Pressure changes

**2.** Electrochemical processes

included and commented on.

**5. Modeling the dynamic behavior of a singular Solid Oxide Fuel Cell**

The mathematical model of SOFC for steady state calculations was presented in a few previ‐ ous papers [73,74,75,76,77,78,79,80]. In this section only dynamic oriented relationships are

As an object for modeling, a singular fuel cell is chosen with dimensions of 5 cm × 5 cm and thickness of 1 mm (see Fig. 10). It was assumed that the manifolds (for fuel and oxi‐

Some processes which occur during fuel cell operation are very rapid, thus they can be as‐ sumed to be time independent compared to others. The following processes are assumed to

Fig. 11 presents a concept of a model of Solid Oxide Fuel Cell, the fuel cell is equiped with two inlet streams and two outlet streams. Processes which occur during fuel cell operation can be divided into three steps: capture of oxygen atoms from the delivered oxidant (air), oxygen ions passing through the electrolyte layer, and the ions escaping and reacting with

$$\mathbf{Q}\_{\text{Oxidant}} = \dot{m}\_{\text{Oxidant}} \cdot c\_{p,\text{Oxidant}} \{ T\_{\text{Cathode},In} - T\_{\text{Cell}} \} \tag{6.2}$$

$$
\dot{Q}\_{Fuel} = \dot{m}\_{Fuel} \cdot c\_{p,Fuel} \{ T\_{Anode,In} - T\_{Cell} \} \tag{6.3}
$$

$$
\dot{Q}\_{Surrounding} = k\_{Surrounding} \cdot A\_{Cell} \{T\_{Cell} - T\_{Surrounding}\} \tag{6.4}
$$

$$P\_{\text{SOFC}} = E\_{\text{SOFC}} \cdot I\_{\text{SOFC}} \tag{6.5}$$

$$
\dot{\mathbf{Q}}\_{\text{Fuel}} = \dot{m}\_{\text{Fuel}} \cdot \text{HHV} \, V\_{\text{Fuel}} \cdot \eta\_f \tag{6.6}
$$

$$\mathbf{C}\_{\text{Manijfold}} = \mathfrak{m}\_{\text{Manijfold}} \cdot \mathbf{c}\_{p, \text{Manijfold}} \tag{6.7}$$

$$\mathbf{C}\_{\rm Cell} = m\_{\rm Cell} \cdot \mathbf{c}\_{p,\rm Cell} \tag{6.8}$$


**Table 4.** Selected factors of a dynamic oriented mathematical model of SOFC.

The factors used in the above equations are presented in Table 4.

Typical interconnects have a thickness of 3 mm 82, which gives 7.5 cm3 of material for fuel cell dimensions of 5 cm × 5cm × 3 mm. Assuming that the interconnect is made from La‐ CrO3, the interconnect weight is 50 g per fuel cell. The additional weight relates to the mani‐ folds which deliver the working fluids—depending on the current architecture solution of the stack. In this study, it was assumed that the interconnect weight in relation to fuel cell area is 2.03 g cm-2.

The typical channel within which working fluids are delivered has dimensions of 0.5 mm × 1.5 mm, and its length depends on the total fuel cell dimensions (5–8 cm). Usually, the dis‐ tance between the channels are the same as the channels themselves. Assuming a planar fuel cell of dimensions of 5 cm × 5 cm, the channel volume is 0.5 mm × 5 cm × 5 cm–(17 × 17 × 1.5 mm × 1.5 mm × 0.5 mm) = 0.925 cm3 per each fuel cell side and in total 1.85 cm3 for the fuel cell. Relating the volume to the fuel cell area gives a value of 0.074 cm3 /cm2 of the channel volume in relation to fuel cell area.


Coefficient of thermal expansion

Specific interconnect weight [g

**Table 6.** Main material factors of the fuel cell related to fuel cell area.

Density [71] [g cm-2] 3–6.77

Working fluids velocities inside the channels depend on the channel dimensions and quanti‐ ty of flows delivered. To provide an adequate time for reaction as well as mixing of re‐ agents, the velocities of working fluids should be relatively low. Based on the authors'' own calculations, the nominal velocities of working fluids are below 5 m s-1, being on average 1.6 m s-1. Due to such low velocities, the pressure drops along the channels can be omitted [69].

> **Parameter Value** Specific fuel cell weight [g cm-2] 0.6

> Specific volume [cm3 cm-2] 0.074

During fuel cell operation there is a series of processes that affect its performance. The oper‐

The amounts of air and fuel supplied to the fuel cell should enable its proper operation, es‐ pecially the behavior of the quantities of both fuel utilization and oxidant utilization. In ad‐ dition, changes in certain parameters interact in a similar way: maintaining the desired temperature of fuel cells can be achieved by either reducing or increasing the amount of air and its temperature. Both of these parameters are related to each other (you cannot cool the cell with overly hot air, regardless of the amount). Selection of the optimal control strategy

ator affects only some of them; the parameters subject to direct regulation are:

(2–8)⋅10-6

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2.03

[72], ΔL/L/K

**Table 5.** Main material parameters of interconnector.

cm-2]

**5.2. Dynamic behavior of SOFC**

*5.2.1. The control strategy*

**1.** Temperature of inlet air

**2.** Temperature of inlet fuel

**3.** Quantity of inlet air

**4.** Quantity of inlet fuel

in this case is a key issue.

**5.** Electric current draw from the cell


**Table 5.** Main material parameters of interconnector.

for Better Environment

Specific heat of oxidant, *cp,Oxidant*

Heat transfer coefficient to surrounding, *kSurrounding* [W m-2K-1]

[MJ kg-1]

70 Clean Energy

kg-1K-1]

Higher Heating Value of fuel, *HHVFuel*

Specific heat of interconnector material, *cp,Manifold* [kJ kg-1K-1]

Specific heat of fuel cell, *cp,Cell* [kJ

fuel cell area [kg m-2]

area [kg m-2]

area is 2.03 g cm-2.

mm × 1.5 mm × 0.5 mm) = 0.925 cm3

volume in relation to fuel cell area.

Interconnector weight in relation to

Fuel cell weight in relation to fuel cell

**Table 4.** Selected factors of a dynamic oriented mathematical model of SOFC.

The factors used in the above equations are presented in Table 4.

cell. Relating the volume to the fuel cell area gives a value of 0.074 cm3

*CCell* =*mCell* ⋅ *cp*,*Cell* (6.8)

<sup>144</sup> hydrogen

per each fuel cell side and in total 1.85 cm3 for the fuel

fuel cell is isolated

Thermal balancing

LaCrO3

YSZ

/cm2 of the channel

**Factor Value Comment**

[kJ kg-1 K-1 1.156 Electrochemistry air

Specific heat of fuel, *cp,Fuel* [kJ kg-1 K-1] 15.25 hydrogen

0.1

0.5

0.5

20.3

6

Typical interconnects have a thickness of 3 mm 82, which gives 7.5 cm3 of material for fuel cell dimensions of 5 cm × 5cm × 3 mm. Assuming that the interconnect is made from La‐ CrO3, the interconnect weight is 50 g per fuel cell. The additional weight relates to the mani‐ folds which deliver the working fluids—depending on the current architecture solution of the stack. In this study, it was assumed that the interconnect weight in relation to fuel cell

The typical channel within which working fluids are delivered has dimensions of 0.5 mm × 1.5 mm, and its length depends on the total fuel cell dimensions (5–8 cm). Usually, the dis‐ tance between the channels are the same as the channels themselves. Assuming a planar fuel cell of dimensions of 5 cm × 5 cm, the channel volume is 0.5 mm × 5 cm × 5 cm–(17 × 17 × 1.5

> **Parameter LaCrO3** Heat conductivity [67] [W m-1K-1] 1.7–2.5

Working fluids velocities inside the channels depend on the channel dimensions and quanti‐ ty of flows delivered. To provide an adequate time for reaction as well as mixing of re‐ agents, the velocities of working fluids should be relatively low. Based on the authors'' own calculations, the nominal velocities of working fluids are below 5 m s-1, being on average 1.6 m s-1. Due to such low velocities, the pressure drops along the channels can be omitted [69].


**Table 6.** Main material factors of the fuel cell related to fuel cell area.

#### **5.2. Dynamic behavior of SOFC**

#### *5.2.1. The control strategy*

During fuel cell operation there is a series of processes that affect its performance. The oper‐ ator affects only some of them; the parameters subject to direct regulation are:


The amounts of air and fuel supplied to the fuel cell should enable its proper operation, es‐ pecially the behavior of the quantities of both fuel utilization and oxidant utilization. In ad‐ dition, changes in certain parameters interact in a similar way: maintaining the desired temperature of fuel cells can be achieved by either reducing or increasing the amount of air and its temperature. Both of these parameters are related to each other (you cannot cool the cell with overly hot air, regardless of the amount). Selection of the optimal control strategy in this case is a key issue.

In this study, it was assumed that the fuel utilization factor is kept constant at the point of maximum efficiency (in fact at the laboratory scale it means only 4.5% for a fuel utilization factor of 12%). This means that inlet fuel mass flow is correlated with fuel cell current.

fed to the cathode reaches 1000 ml min-1cm-2. Cell voltage drops to the value of 0.75 V and

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An external source of heat is required to support the start-up of a fuel cell. The simplest sol‐ ution is to use the burner boot to warm the cell to a temperature which enables it to work independently. During fuel cell start-up acceptable temperature differences should be pre‐

An active start-up system is proposed, comprising regulating the temperature of gases sup‐

**Figure 13.** Correlation of air temperature used for heating the cell with cell temperature was applied during the simu‐

The amount of air used to heat the cells was determined as the nominal point. The tempera‐ ture of the air is correlated with the cell temperature according to the relation with which the air temperature decreases from the value of 700°C in proportion to the increase in cell

remains without significant change.

served, with the assumed values being:

lation starting from cold state

temperature (see Fig. 13).

plied to the cells depending on cell temperature.

**•** 45°C between inlet and outlet temperatures of working fluids

**•** 90°C between working fluid temperature and fuel cell temperature

*5.2.2. Start-up*

The most important parameter is cell temperature, which must be kept constant. The tem‐ perature is controlled by an inlet air mass-flow, which is regulated by a valve equipped with a PID regulator.


**Table 7.** Optimal parameters of PID controller [70].

The singular PID controller is chosen to keep the fuel cell temperature at set point (800°C). The PID controls inlet air mass flow. Optimal PID parameters are listed in Table 7. For these parameters the optimal parameter settings for the PID controller of the fuel cell are determined:


**Figure 12.** System response to a step change in charging current density (0–1.3 A/cm2) using PID

Fig. 12 shows the cell parameters change with a stepped increase in current density using the PID controller. It can be seen that the quality of control is very good (almost no distor‐ tion: 15°C), and the system reaches a steady state after about 4 minutes. The amount of air fed to the cathode reaches 1000 ml min-1cm-2. Cell voltage drops to the value of 0.75 V and remains without significant change.

#### *5.2.2. Start-up*

for Better Environment

a PID regulator.

72 Clean Energy

**•** *KP* = 3.264

**•** *KD* = 0.5.

= 1.6

**•** *KI*

PI 1

PID 1 ⋅ 36

*K* ⋅ *T*<sup>0</sup>

*K* ⋅ *T*<sup>0</sup>

**Table 7.** Optimal parameters of PID controller [70].

In this study, it was assumed that the fuel utilization factor is kept constant at the point of maximum efficiency (in fact at the laboratory scale it means only 4.5% for a fuel utilization factor of 12%). This means that inlet fuel mass flow is correlated with fuel cell current.

The most important parameter is cell temperature, which must be kept constant. The tem‐ perature is controlled by an inlet air mass-flow, which is regulated by a valve equipped with

**Kp KI KD**

The singular PID controller is chosen to keep the fuel cell temperature at set point (800°C). The PID controls inlet air mass flow. Optimal PID parameters are listed in Table 7. For these parameters the optimal parameter settings for the PID controller of the fuel cell are determined:

**Figure 12.** System response to a step change in charging current density (0–1.3 A/cm2) using PID

Fig. 12 shows the cell parameters change with a stepped increase in current density using the PID controller. It can be seen that the quality of control is very good (almost no distor‐ tion: 15°C), and the system reaches a steady state after about 4 minutes. The amount of air

4.3⋅T<sup>0</sup>

1.6⋅T<sup>0</sup> 0.5⋅T<sup>0</sup>

An external source of heat is required to support the start-up of a fuel cell. The simplest sol‐ ution is to use the burner boot to warm the cell to a temperature which enables it to work independently. During fuel cell start-up acceptable temperature differences should be pre‐ served, with the assumed values being:


An active start-up system is proposed, comprising regulating the temperature of gases sup‐ plied to the cells depending on cell temperature.

**Figure 13.** Correlation of air temperature used for heating the cell with cell temperature was applied during the simu‐ lation starting from cold state

The amount of air used to heat the cells was determined as the nominal point. The tempera‐ ture of the air is correlated with the cell temperature according to the relation with which the air temperature decreases from the value of 700°C in proportion to the increase in cell temperature (see Fig. 13).

The results of simulated rapid increase in power by 10% are shown in Fig. 15. The control system keeps all key parameters in acceptable ranges. Larger changes are noted only for cur‐ rent density (which increases from 2.43–2.68 A cm-2), and the air flow rate, which is the re‐ sult of regulation and oscillates between 1600–2200 ml min-1cm-2 finally stays at 2000 ml

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**Figure 16.** Changes of the fuel cell operating parameters during load decreases by 10%

The results of simulated rapid decrease in power by 10% are shown in Fig. 16. The control system keeps all crucial parameters in the acceptable ranges, so that most of the parameters practically do not change themselves. Larger changes are noted only for the current density (which increases from 2.43–2.19 A cm-2), and the air flow rate which is the result of regula‐

The most likely emergency scenario is a sudden loss of load resulting from load shut down

Fig. 17 presents the simulated behavior of a fuel cell reacting to a sudden loss of load. The fuel cell parameters stabilize after about 4 minutes and the cell goes into idle mode. The cell

tion and oscillates between 1300–1700 ml min-1cm-2 finally stays at 1400 ml min-1cm-2.

(e.g. activation of the safety switch). In this case the fuel cell should be left to idle.

temperature reaches 807°C, which seems to be a safe value.

min-1cm-2.

*5.2.4. Loss of load*

**Figure 14.** Comparison of the simulated start-up procedure with the results obtained from experiments [80]

Fig. 14 presents a comparison of the simulated cell voltage during start-up against the real values (data [80]). The two cells differ in structure as well as in the procedures used during start-up but, qualitatively speaking, the modeled start-up is a very close approximation to the reality.

#### *5.2.3. Continuous operation and changes in power*

**Figure 15.** Changes of the fuel cell operating parameters during load increases by 10%

The results of simulated rapid increase in power by 10% are shown in Fig. 15. The control system keeps all key parameters in acceptable ranges. Larger changes are noted only for cur‐ rent density (which increases from 2.43–2.68 A cm-2), and the air flow rate, which is the re‐ sult of regulation and oscillates between 1600–2200 ml min-1cm-2 finally stays at 2000 ml min-1cm-2.

**Figure 16.** Changes of the fuel cell operating parameters during load decreases by 10%

The results of simulated rapid decrease in power by 10% are shown in Fig. 16. The control system keeps all crucial parameters in the acceptable ranges, so that most of the parameters practically do not change themselves. Larger changes are noted only for the current density (which increases from 2.43–2.19 A cm-2), and the air flow rate which is the result of regula‐ tion and oscillates between 1300–1700 ml min-1cm-2 finally stays at 1400 ml min-1cm-2.

#### *5.2.4. Loss of load*

for Better Environment

*5.2.3. Continuous operation and changes in power*

**Figure 15.** Changes of the fuel cell operating parameters during load increases by 10%

the reality.

74 Clean Energy

**Figure 14.** Comparison of the simulated start-up procedure with the results obtained from experiments [80]

Fig. 14 presents a comparison of the simulated cell voltage during start-up against the real values (data [80]). The two cells differ in structure as well as in the procedures used during start-up but, qualitatively speaking, the modeled start-up is a very close approximation to

> The most likely emergency scenario is a sudden loss of load resulting from load shut down (e.g. activation of the safety switch). In this case the fuel cell should be left to idle.

> Fig. 17 presents the simulated behavior of a fuel cell reacting to a sudden loss of load. The fuel cell parameters stabilize after about 4 minutes and the cell goes into idle mode. The cell temperature reaches 807°C, which seems to be a safe value.

**Figure 17.** Simulated behavior of a fuel cell reacting to a sudden loss of load

On the basis of the simulation it can be concluded that the fuel cell is relatively resistant to a sudden loss of load in the presence of a proper control system.

**Figure 18.** Fuel cell parameters during shut down procedure based on maximizing cathode flow (air side)

unit is relatively low, as is the fuel utilization factor.

pared with the experimental data, with satisfactory results.

The control strategy for a singular solid oxide fuel cell is proposed. The strategy is based on a singular PID controller which controls the amount of air delivered to the cathode side of the fuel cell. Additionally, fuel mass flow is correlated with current density to achieve a fixed fuel utilization factor. In fact, the efficiency of the singular laboratory scale fuel cell

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77

The start-up procedure of the fuel cell must be supported by an external source of heat. The‐ oretically speaking, it is possible to heat up the cell until the point at which it starts to gener‐ ate some voltage (practically, above 0.4 V) and then the fuel cell should be able to heat itself up to working temperature by the applied external load. The simulations performed do not confirm this theoretical speculation. After the load is applied, the voltage drops and no cur‐ rent can be drawn. Thus, adequate correlation of air inlet temperature with cell temperature is proposed in order to reach the nominal temperature. The simulated start-up was com‐

During normal operation, the proposed control system keeps all fuel cell parameters with‐ in acceptable ranges—there are no consequences following a rapid increase/decrease of

The one conceivable emergency scenario was analyzed: rapid loss of external load. The con‐ trol system keeps the key parameters at acceptable levels (e.g. cell temperature reaches 807°C).

**5.3. Discussion**

load by 10%.

#### *5.2.5. Shut down*

During normal operation the fuel cell is able to resist a sudden loss of load. Therefore no special procedures are required to shut down the fuel cell unit. Additional procedures should be used to cool down the fuel cell to ambient temperature. The best option seems to be using a PID controller with variable temperature settings.

Fig. 18 shows the fuel cell characteristics with stepped increase in the quantity of air sup‐ plied to the cathode to its maximum value (6000 ml min-1cm-2). The cathode part of the fuel cell loses heat relatively quickly, reaching ambient temperature after about 10 minutes. By contrast, at the anode side, there is no fuel flow and cooling takes far longer (after 30 mi‐ nutes the temperature drops by only a few degrees). In total, this leads to very large temper‐ ature differences between the anode and cathode side (almost 600°C).

In order to shut down the fuel cell, the flows on both the anode and cathode sides need to be maintained. The simplest solution is to maintain the fuel stream on the anode side, which otherwise experiences a loss of fuel. Another solution is to provide another gas, but it should be an inert gas (the use of air may result in oxidation of the anode surface).

**Figure 18.** Fuel cell parameters during shut down procedure based on maximizing cathode flow (air side)

#### **5.3. Discussion**

for Better Environment

76 Clean Energy

**Figure 17.** Simulated behavior of a fuel cell reacting to a sudden loss of load

sudden loss of load in the presence of a proper control system.

be using a PID controller with variable temperature settings.

ature differences between the anode and cathode side (almost 600°C).

*5.2.5. Shut down*

On the basis of the simulation it can be concluded that the fuel cell is relatively resistant to a

During normal operation the fuel cell is able to resist a sudden loss of load. Therefore no special procedures are required to shut down the fuel cell unit. Additional procedures should be used to cool down the fuel cell to ambient temperature. The best option seems to

Fig. 18 shows the fuel cell characteristics with stepped increase in the quantity of air sup‐ plied to the cathode to its maximum value (6000 ml min-1cm-2). The cathode part of the fuel cell loses heat relatively quickly, reaching ambient temperature after about 10 minutes. By contrast, at the anode side, there is no fuel flow and cooling takes far longer (after 30 mi‐ nutes the temperature drops by only a few degrees). In total, this leads to very large temper‐

In order to shut down the fuel cell, the flows on both the anode and cathode sides need to be maintained. The simplest solution is to maintain the fuel stream on the anode side, which otherwise experiences a loss of fuel. Another solution is to provide another gas, but it

should be an inert gas (the use of air may result in oxidation of the anode surface).

The control strategy for a singular solid oxide fuel cell is proposed. The strategy is based on a singular PID controller which controls the amount of air delivered to the cathode side of the fuel cell. Additionally, fuel mass flow is correlated with current density to achieve a fixed fuel utilization factor. In fact, the efficiency of the singular laboratory scale fuel cell unit is relatively low, as is the fuel utilization factor.

The start-up procedure of the fuel cell must be supported by an external source of heat. The‐ oretically speaking, it is possible to heat up the cell until the point at which it starts to gener‐ ate some voltage (practically, above 0.4 V) and then the fuel cell should be able to heat itself up to working temperature by the applied external load. The simulations performed do not confirm this theoretical speculation. After the load is applied, the voltage drops and no cur‐ rent can be drawn. Thus, adequate correlation of air inlet temperature with cell temperature is proposed in order to reach the nominal temperature. The simulated start-up was com‐ pared with the experimental data, with satisfactory results.

During normal operation, the proposed control system keeps all fuel cell parameters with‐ in acceptable ranges—there are no consequences following a rapid increase/decrease of load by 10%.

The one conceivable emergency scenario was analyzed: rapid loss of external load. The con‐ trol system keeps the key parameters at acceptable levels (e.g. cell temperature reaches 807°C). The simulated shut down procedure was unsuccessful: the PID controller was used to cool the fuel cell, but it only influenced air flow, causing an extremely high temperature gradient. Additional procedures must be applied to cool the fuel cell properly.

**Acknowledgements**

fully acknowledged.

**Author details**

**References**

Jakub Kupecki1,2\*, Janusz Jewulski1

rective 92/62/EEC.

36, 11056-11067.

son of fuel cell technologies.

\*Address all correspondence to: jakub.kupecki@ien.com.pl

1 Fuel Cell Department, Institute of Power Engineering, Poland

management. *Journal of Power Sources*, 196, 3790-3802.

at Hexis. Fuel Cell Seminar 2007 proceedings 5-8.11. 12.

2 Institute of Heat Engineering, Warsaw University of Technology, Poland

Support from the European Regional Development Fund and Ministry of Science and Edu‐ cation under the project no. UDA-POIG.01.01.02-00- 016/08-00, from the National Centre for Research and Development under the project Advanced Technologies for Energy Genera‐ tion, and European Social Fund through the "Didactic Development Program of the Faculty of Power and Aeronautical Engineering of the Warsaw University of Technology" are grate‐

Multi-Level Mathematical Modeling of Solid Oxide Fuel Cells

http://dx.doi.org/10.5772/50724

79

and Jarosław Milewski<sup>2</sup>

[1] European Commission . (2004). 2004/8/EC Directive on the promotion of cogenera‐ tion based on a useful heat demand in the internal energy market and amending di‐

[2] Kattke, K. J., Braun, R. J., Colclasure, A. M., & Goldin, G. (2011). High-fidelity stack and system modeling for tubular solid oxide fuel cell system design and thermal

[3] Blum, L., Deja, R., Peters, R., & Stolten, D. (2001). Comparison of efficiencies of low, mean and high temperature fuel cell systems. *International Journal of Hydrogen Energy*,

[4] Mekhilef, S., Saidur, R., & Safari, A. (2012). Comparative study of different fuel cell

[5] Kupecki, J., & Badyda, K. (2011). SOFC-based micro-CHP system as an example of

[6] DOE Energy Efficiency and Renewable Energy Information Center. (2008). Compari‐

[7] Mai, A., Sfeir, J., & Schuler, A. (2007). Status of sofc stack and systems development

technologies. *Renewable and Sustainable Energy Reviews*, 16, 981-989.

efficient power generation unit. *Archives of Thermodynamics*, 32(3), 33-42.

## **6. Conclusions**

Different mathematical models are useful for evaluation and predicition of fuel cells and en‐ tire system performance. In all cases, specific application-related criteria are selected for de‐ velopment of numerical tool.

Development of advanced energy systems, including micro-CHP units, under various oper‐ ating conditions is possible only with high-fidelity numerical simulator. Tool has to be vali‐ dated against available experimental data. In certain cases, numerical modeling is not possible without supporting experimental measurement.

Different time- and length-scales can be covered with dedicated models, ranging from 0D up to complex full-3D tools. Steady state can be evaluated with available engineering soft‐ ware, including computational fluid dynamic tools.

Analysis of chemical and electrochemical reactions taking place during SOFC operation re‐ quires specific knowledge and in most cases detailed models are needed. Spatial configura‐ tion and influence of geometry can only be studied with space-continuous models or number of simplifications is required.

Based on mathematical modeling, an analysis of the dynamic operation of a singular fuel cell is presented. Based on the analysis a few cases relating to the cell were simulated:


In almost all cases, the singular PID controller is able to keep the fuel cell operation within a safe range. Special procedures are required during start up and shut down. During start up, external heat sources are required to warm the cell to operational temperature. It is pro‐ posed that air temperature be correlated to cell temperature. As regards the shut down pro‐ cedure, a change in fuel cell configuration is required—an inert gas instead of fuel must be delivered in order to cool the cell.

The start up procedure was compared against available experimental data with satisfactory results, qualitatively speaking.

## **Acknowledgements**

for Better Environment

**6. Conclusions**

78 Clean Energy

velopment of numerical tool.

The simulated shut down procedure was unsuccessful: the PID controller was used to cool the fuel cell, but it only influenced air flow, causing an extremely high temperature gradient.

Different mathematical models are useful for evaluation and predicition of fuel cells and en‐ tire system performance. In all cases, specific application-related criteria are selected for de‐

Development of advanced energy systems, including micro-CHP units, under various oper‐ ating conditions is possible only with high-fidelity numerical simulator. Tool has to be vali‐ dated against available experimental data. In certain cases, numerical modeling is not

Different time- and length-scales can be covered with dedicated models, ranging from 0D up to complex full-3D tools. Steady state can be evaluated with available engineering soft‐

Analysis of chemical and electrochemical reactions taking place during SOFC operation re‐ quires specific knowledge and in most cases detailed models are needed. Spatial configura‐ tion and influence of geometry can only be studied with space-continuous models or

Based on mathematical modeling, an analysis of the dynamic operation of a singular fuel

In almost all cases, the singular PID controller is able to keep the fuel cell operation within a safe range. Special procedures are required during start up and shut down. During start up, external heat sources are required to warm the cell to operational temperature. It is pro‐ posed that air temperature be correlated to cell temperature. As regards the shut down pro‐ cedure, a change in fuel cell configuration is required—an inert gas instead of fuel must be

The start up procedure was compared against available experimental data with satisfactory

cell is presented. Based on the analysis a few cases relating to the cell were simulated:

**•** Continuous operation (with power changes in the range of +/-10%)

Additional procedures must be applied to cool the fuel cell properly.

possible without supporting experimental measurement.

ware, including computational fluid dynamic tools.

number of simplifications is required.

**•** Emergency scenario (loss of load)

delivered in order to cool the cell.

results, qualitatively speaking.

**•** Start up

**•** Shut down, and

Support from the European Regional Development Fund and Ministry of Science and Edu‐ cation under the project no. UDA-POIG.01.01.02-00- 016/08-00, from the National Centre for Research and Development under the project Advanced Technologies for Energy Genera‐ tion, and European Social Fund through the "Didactic Development Program of the Faculty of Power and Aeronautical Engineering of the Warsaw University of Technology" are grate‐ fully acknowledged.

## **Author details**

Jakub Kupecki1,2\*, Janusz Jewulski1 and Jarosław Milewski<sup>2</sup>

\*Address all correspondence to: jakub.kupecki@ien.com.pl

1 Fuel Cell Department, Institute of Power Engineering, Poland

2 Institute of Heat Engineering, Warsaw University of Technology, Poland

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## *Edited by Cumhur Aydinalp*

The prospect of producing clean, sustainable power from renewable energy sources is becoming a very important subject, stimulated by recent technological developments that have improved the cost-effectiveness of many renewables and by the increasing concern for the environmental impact and sustainability of conventional fossil and nuclear fuels. This book provides a comprehensive overview of the principal renewable energy sources with a wide range of case studies for each source. It explains the underlying physical and technological principles, and examines the environmental impact of renewable sources and their future prospects. The overall approach is interdisciplinary, covering the economic, social, environmental and policy issues from the point of research on renewable energy. It also tackles the physical and engineering aspects. The book will, therefore, strongly appeal to non-specialist readers who wish to improve their understanding of this complex, fascinating and increasingly important subject.

Photo by monticelllo / iStock

Clean Energy for Better Environment

Clean Energy

for Better Environment

*Edited by Cumhur Aydinalp*