# Exergy and Exergoeconomic Analyses of a Combined Power Producing System including a Proton Exchange Membrane Fuel Cell and an Organic Rankine Cycle

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}O) absorption cycle. They only investigated the system from an energy viewpoint, and reported a maximum fuel saving ratio of about 35%. Ebrahimi et al. [20] carried out energy and exergy analyses for a micro-CCHP system that is based on a LT-PEMFC. A portion of the produced electric power is utilized to drive a thermoelectric cooler. Energy and exergy efficiencies of up to 76.9% and 53.9%, respectively, are calculated for the proposed system, but the economic aspects were not addressed. Arsalis [21] reviewed micro-CHP systems that are based on fuel cells, and concluded that PEMFCs are the most promising technology for cogeneration configurations.

## 2. System Description

## 3. PEMFC Mathematical Modeling

#### 3.1. LT-PEMFC Electrochemical Modeling

- System operation is at steady state.
- The inlet air contains 79% N
_{2}and 21% O_{2}by volume. - Constant and equal pressure is assumed for the gas flow channels in the fuel cell.
- Hydrogen at a relative humidity of 50% and humidified air at a relative humidity of 100% are supplied to the anode and cathode, respectively.
- The fuel cell operates at a temperature of 85 °C and a pressure of 300 kPa.
- The stream temperature at the channel exits is equal to that of the stack operating temperature.
- The heat losses from the system components to the surrounding environment are negligible.
- Pressure drops inside the fuel cell are neglected.

^{+}ions (protons) are produced:

^{=}ions from the electrolyte:

#### 3.2. Mass Balances

## 4. Thermodynamic Analysis

#### 4.1. Thermal Model of the System (First Law of Thermodynamics)

#### 4.1.1. PEM Fuel Cell

#### 4.1.2. Organic Rankine Cycle

#### 4.2. Exergy Analysis

_{2}, and air entering the system, respectively.

#### 4.3. Exergoeconomic Analysis

^{3}, respectively. In addition, the fuel cell capital cost is taken to be 5000 $/kW and the humidifier capital cost is assumed to be 4% of the fuel cell cost [34]. Table 4 presents the capital cost functions for the other components.

_{k}, which indicates the relative increase between ${\mathrm{c}}_{\mathrm{F},\mathrm{k}}$ and ${\mathrm{c}}_{\mathrm{p},\mathrm{k}}$, is defined as:

## 5. Results and Discussion

_{Nernst}) and the reductions in the activation losses at the electrodes, especially at the cathode.

^{2}for an operating pressure of 300 kPa. With this value of power density, an efficiency of 33.5% is obtained from Figure 5. This efficiency is much lower than the maximum theoretical value, which is around 80%. Referring to Figure 5, much higher efficiencies may be obtained at significantly lower power densities. This indicates that, for a given value output power, choosing an appropriate point on the polarization curve (Figure 4a) could result in a larger fuel cell (with a larger active area) with higher efficiency or a compact fuel cell with less efficiency. In the literature, it has been reported that the maximum power density is not recommended in fuel cell sizing [44]. An operating point corresponding to a cell potential of around 0.65–0.7 V is common practice. Therefore, this results in a power density of 0.4 W cm

^{−2}and an efficiency of around 44% for the proposed system at the operating condition.

^{2}. At higher current densities, the addition of the ORC to the fuel cell results in a lower value of the product unit cost. The difference between the two product unit costs is more prominent at higher values of the fuel cell operating pressure. A comparison between the results in Figure 7 and Figure 10 shows that the operating pressure of 300 kPa is better than the other two pressure values that are indicated in Figure 11. The reasoning is as follows: comparing the results for the operating pressures of 300 kPa and 400 kPa it is observed that ${\mathsf{\epsilon}}_{\mathrm{Overall}}$ does not change much, but ${\mathrm{c}}_{\mathrm{w},\mathrm{overall}}$ is higher for 400 kPa. In addition, although a lower value of ${\mathrm{c}}_{\mathrm{w},\mathrm{overall}}$ is obtained for P = 200 kPa, ${\mathsf{\epsilon}}_{\mathrm{Overall}}$ for this operating pressure is much less than that for P = 300 kPa.

## 6. Conclusions

- An increase in the operating pressure results in an increase in the overall exergy efficiency. However, the fuel cell exergy efficiency is maximized at a specific value of operating pressure.
- The exergy efficiency for the hybrid power system can be higher than the corresponding value for the PEM fuel cell stack, as a standalone system, by up to 4.16%.
- Among the studied fuel cell operating pressures for the proposed system, it is observed that better thermodynamic and economic results are achieved with a pressure of 300 kPa.
- The combination of a PEMFC with an ORC is only economically justified at a specific value of current density, which depends on the fuel cell operating pressure.
- Among all the cycle components, the compressor and ORC condenser have the lowest exergoeconomic factors (less than 10%), which indicates that these components exhibit the worst exergoeconomic performances.
- Increasing the operating pressure of the PEMFC reduces its exergy destruction and also the cost associated with it.
- The overall exergoeconomic factor for the proposed system is observed to be 40.8%. Therefore, more expensive components are expected to enhance the exergoeconomic performance of the proposed system.

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

${\mathrm{A}}_{\mathrm{cell}}$ | Active surface area (cm^{2}) |

$\mathrm{a}$ | Water activity |

$\mathrm{c}$ | Concentration at membrane surface (mol/cm^{3}) |

$\mathrm{c}$ | Cost per unit exergy ($/kJ or $/GJ) |

${\overline{\mathrm{c}}}_{\mathrm{p}}$ | Specific heat capacity (kJ/kmolK) |

$\dot{\mathrm{C}}$ | Cost rate ($/s) |

CRF | Capital recovery factor |

d | Channel height (cm) |

D_{0} | Intradiffusion coefficient of water in membrane (cm^{2}/s) |

${\mathrm{D}}^{\ast}$ | Diffusion coefficient of water in membrane (cm^{2}/s) |

$\mathrm{e}$ | Specific exergy (kJ/kg) |

$\dot{\mathrm{E}}$ | Exergy flow rate (kW) |

$\mathrm{f}$ | Exergoeconomic factor |

$\mathrm{F}$ | Faraday constant (C/mol) |

${\Delta \mathrm{G}}^{0}$ | Change of Gibbs free energy ($\mathrm{kJ}/\mathrm{kmol}$) |

$\mathrm{h}$ | Specific enthalpy (kJ/kg) |

$\mathrm{HHV}$ | Higher heating value (kJ/kmol) |

${\mathrm{h}}_{\mathrm{fg}}^{0}$ | heat of vaporization of water (kJ/kmol) |

$\mathrm{i}$ | Current density (${\mathrm{A}/\mathrm{cm}}^{2}$) |

$\mathrm{I}$ | Stack operating current (A) |

${\mathrm{i}}_{\mathrm{L}}$ | Limiting current density (${\mathrm{A}/\mathrm{cm}}^{2}$) |

${\mathrm{i}}_{\mathrm{n}}$ | Interest rate |

${\mathrm{k}}_{\mathrm{c}}$ | Condensation rate constant (1/s) |

${\mathrm{k}}_{\mathrm{p}}$ | Water permeability (cm^{2}) |

$\dot{\mathrm{m}}$ | Mass flow rate (kg/s) |

$\dot{\mathrm{n}}$ | Molar flow rate (mol/s) |

${\mathrm{N}}_{\mathrm{cell}}$ | Number of cells in stack |

${\mathrm{n}}_{\mathrm{d}}$ | Electro-osmotic drag coefficient (molecules/protons) |

${\mathrm{n}}_{\mathrm{e}}$ | Number of electrons |

$\mathrm{P}$ | Pressure (kPa) |

$\dot{\mathrm{Q}}$ | Heat transfer rate (kW) |

$\mathrm{r}$ | Relative cost difference |

$\mathrm{R}$ | Gas constant (kJ/(kg K)) |

$\overline{\mathrm{R}}$ | Universal gas constant (kJ/(kmol K)) |

${\mathrm{R}}_{\mathrm{int}}$ | Total internal resistance (Ω) |

${\mathrm{r}}_{\mathrm{mem}}$ | Membrane resistivity (Ω cm) |

$\mathrm{s}$ | specific entropy (kJ/(kg K)) |

$\mathrm{t}$ | Temperature (°C) |

$\mathrm{T}$ | Temperature (K) |

t_{m} | Membrane thickness (cm) |

$\mathrm{V}$ | Voltage (V) |

$\dot{\mathrm{W}}$ | Power (kW) |

$\mathrm{x}$ | Mole fraction |

${\mathrm{x}}_{\mathrm{v}}$ | Mole fraction of water vapor |

$\mathrm{Z}$ | Capital cost of a component ($) |

$\dot{\mathrm{Z}}$ | Capital cost rate ($/s) |

Greek letters | |

$\mathsf{\alpha}$ | Ratio of water molecules per proton flux (molecules/protons) |

$\mathsf{\beta}$ | Amplification constant (V(cm^{2}/A)k) |

$\mathsf{\epsilon}$ | Exergy efficiency |

$\mathsf{\eta}$ | Efficiency |

${\mathsf{\eta}}_{\mathrm{s}}$ | Isentropic efficiency |

$\mathsf{\lambda}$ | Stoichiometric rate |

$\mathsf{\mu}$ | Water viscosity (g/cm.s) |

${\mathsf{\rho}}_{\mathrm{m},\mathrm{dry}}$ | Dry membrane density (g/cm^{3}) |

$\mathsf{\omega}$ | Humidity ratio (kg_{v}/kg_{a}) |

$\mathsf{\xi}$ | Empirical coefficient of activation overvoltage |

$\mathsf{\psi}$ | Membrane hydration |

Subscripts | |

0 | Dead (environmental) state |

a | Anode, air |

act | Activation, actual |

C | Cathode |

ch | Chemical |

Cond | Condenser |

Comp | Compressor |

cv | Control volume |

D | Destruction |

elec | Electrical |

F | Fuel |

fc | Fuel cell |

H_{2} | Hydrogen |

H_{2}O | Water |

i | ith stream |

in | Input |

k | kth component |

N_{2} | Nitrogen |

O_{2} | Oxygen |

Out | Output |

P | Pump, Product |

ph | Physical exergy |

q | Heat |

S,l | Sensible and latent heat |

T | Turbine |

th | Thermal |

w | Water |

v | Vapour |

Superscripts | |

l | Liquid |

sat | Saturated |

v | Vapour |

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**Figure 3.**Validation of simulation results for PEMFC performance using the theoretical and experimental data reported by Miansari et al. [8].

**Figure 4.**Characteristic curves of the PEMFC with respect to current density at various operating pressures: (

**a**) Polarization curve; (

**b**) Power density.

**Figure 6.**The variation of heat rejected from the fuel cell and the mole fraction of liquid water at the cathode outlet with current density.

**Figure 7.**Variation of overall exergy efficiency and system net power with current density at P = 300 kPa.

**Figure 8.**Variations in overall and fuel cell exergy efficiency with the fuel cell operating pressure.

**Figure 10.**Variation of the turbine power and cost per unit of turbine power with turbine back pressure.

**Figure 11.**Variation of unit cost of fuel cell net power output (dashed lines) and unit cost of overall system net output power (solid lines) with current density at various operating pressures.

**Figure 12.**Variation of exergy destruction cost rate and total capital cost rate vs. current density for the proposed system.

Variable | Value |
---|---|

Physical Parameters | |

T_{amb} (°C) | 25 |

P_{amb} (kPa) | 101.325 |

n_{e} | 2 |

F (C/mol) | 96,485 |

R (J/molK) | 8.314 |

A_{cell} (cm^{2}) | 232 ^{a} |

N_{cell} | 13,000 ^{a} |

d (cm) | 0.1 ^{b} |

t_{m} (cm) | 0.00254 ^{a} |

k | 1.1 ^{c} |

β | 0.085 ^{c} |

D_{0} (cm^{2}/s) | 5.5 × 10^{−7 b} |

${\mathsf{\rho}}_{\mathrm{m},\mathrm{dry}}$ (g/cm^{3}) | 2 ^{b} |

M_{m,dry} (kg/kmol) | 1100 ^{b} |

k_{c} (s^{−1}) | 100 ^{b} |

HHV (kJ/kmol) | 2.855 × 10^{5 a} |

Fuel cell operating values | |

i (A/cm^{2}) | 0.6 ^{a} |

I_{L} (A/cm^{2}) | 1.5 ^{a} |

T_{fc} (°C) | 85 ^{a} |

P (kPa) | 300 ^{a} |

${\mathsf{\lambda}}_{\mathrm{air}}$ | 2 ^{a} |

${\mathsf{\lambda}}_{{\mathrm{H}}_{2}}$ | 1.2 ^{a} |

${\dot{\mathrm{n}}}_{\mathrm{w},\mathrm{a},\mathrm{in}}^{\mathrm{l}}$ (mol/s) | 0 |

${\dot{\mathrm{n}}}_{\mathrm{w},\mathrm{a},\mathrm{in}}^{\mathrm{v}}$ | Saturated |

${\dot{\mathrm{n}}}_{\mathrm{w},\mathrm{c},\mathrm{in}}^{\mathrm{l}}$ (mol/s) | 0 |

${\mathsf{\eta}}_{\mathrm{s},\mathrm{comp}}$ (%) | 85 ^{a} |

ORC | |

P_{Low} (kPa) | 97.8 ^{a} |

P_{high} (kPa) | 489.08 ^{a} |

${\mathsf{\eta}}_{\mathrm{s},\mathrm{T}}$ (%) | 85 ^{a} |

${\mathsf{\eta}}_{\mathrm{s},\mathrm{P}}$ (%) | 70 ^{a} |

Component | Energy Balance or Expression |
---|---|

Turbine | ${\dot{\mathrm{W}}}_{\mathrm{T}}={\dot{\mathrm{m}}}_{\mathrm{ORC}}({\mathrm{h}}_{13}-{\mathrm{h}}_{14})$ |

Condenser | ${\dot{\mathrm{Q}}}_{\mathrm{cond}}={\dot{\mathrm{m}}}_{\mathrm{ORC}}({\mathrm{h}}_{14}-{\mathrm{h}}_{15})$ |

Pump | ${\dot{\mathrm{W}}}_{\mathrm{p}}={\dot{\mathrm{m}}}_{\mathrm{ORC}}({\mathrm{h}}_{12}-{\mathrm{h}}_{15})$ |

Input heat rate | ${\dot{\mathrm{Q}}}_{\mathrm{fc}}={\dot{\mathrm{Q}}}_{\mathrm{in}.\mathrm{ORC}}={\dot{\mathrm{m}}}_{\mathrm{ORC}}({\mathrm{h}}_{13}-{\mathrm{h}}_{12})$ |

ORC thermal efficiency | ${\mathsf{\eta}}_{\mathrm{th},\mathrm{orc}}=\frac{{\dot{\mathrm{W}}}_{\mathrm{T}}-{\dot{\mathrm{W}}}_{\mathrm{P}}}{{\dot{\mathrm{Q}}}_{\mathrm{in},\mathrm{ORC}}}$ |

**Table 3.**Mole fractions of the components at dead state for a relative humidity of 60% [39].

Component i | N_{2} | O_{2} | CO_{2} | H_{2}O | Other |
---|---|---|---|---|---|

Mole fraction $({\mathrm{x}}_{0,\mathrm{i}})$ | 0.7662 | 0.2055 | 0.0003 | 0.0188 | 0.0092 |

Component | Capital Cost |
---|---|

Compressor | ${\mathrm{Z}}_{\mathrm{c}}=(\frac{{75\dot{\mathrm{m}}}_{\mathrm{air}}}{0.9-{\mathsf{\eta}}_{\mathrm{s},\mathrm{comp}}})({\mathrm{P}}_{\mathrm{out}}/{\mathrm{P}}_{\mathrm{in}})\mathrm{ln}({\mathrm{P}}_{\mathrm{out}}/{\mathrm{P}}_{\mathrm{in}})$ |

Pump | ${\mathrm{Z}}_{\mathrm{p}}=3540{\dot{\mathrm{W}}}_{p}^{0.71}$ |

Steam turbine | ${\mathrm{Z}}_{\mathrm{T}}=6000{\dot{\mathrm{W}}}_{\mathrm{T}}^{0.7}$ |

Condenser | ${\mathrm{Z}}_{\mathrm{cond}}=1773{\dot{\mathrm{m}}}_{\mathrm{ORC}}$ |

**Table 5.**Cost rate balances and auxiliary equations for components of the proton exchange membrane fuel cell-organic Rankine cycle (PEMFC-ORC) system.

Component | Exergy Cost Rate Balance Equation | Auxiliary Equations |
---|---|---|

Anode side humidifier | ${\dot{\mathrm{C}}}_{2}+{\dot{\mathrm{C}}}_{11}+{\dot{\mathrm{Z}}}_{\mathrm{Humidifier}}={\dot{\mathrm{C}}}_{3}$ | ${\mathrm{c}}_{11}={\mathrm{c}}_{\mathrm{water}}$ |

Compressor | ${\dot{\mathrm{C}}}_{5}+{\dot{\mathrm{C}}}_{\mathrm{W},\mathrm{Comp}}+{\dot{\mathrm{Z}}}_{\mathrm{comp}}={\dot{\mathrm{C}}}_{6}$ | $\frac{{\dot{\mathrm{C}}}_{\mathrm{W},\mathrm{Comp}}}{{\dot{\mathrm{W}}}_{\mathrm{comp}}}=\frac{{\dot{\mathrm{C}}}_{\mathrm{W},\mathrm{FC}}}{{\dot{\mathrm{W}}}_{\mathrm{FC}}}$ |

Cathode side humidifier | ${\dot{\mathrm{C}}}_{6}+{\dot{\mathrm{C}}}_{10}+{\dot{\mathrm{Z}}}_{\mathrm{Humidifier}}={\dot{\mathrm{C}}}_{7}$ | ${\mathrm{c}}_{10}={\mathrm{c}}_{\mathrm{water}}$ |

Fuel cell | ${\dot{\mathrm{C}}}_{3}+{\dot{\mathrm{C}}}_{7}+{\dot{\mathrm{C}}}_{12}+{\dot{\mathrm{Z}}}_{\mathrm{FC}}={\dot{\mathrm{C}}}_{4}+{\dot{\mathrm{C}}}_{8}+{\dot{\mathrm{C}}}_{13}+{\dot{\mathrm{C}}}_{\mathrm{W},\mathrm{FC}}+{\dot{\mathrm{C}}}_{9}$ | $\begin{array}{l}\frac{{\dot{\mathrm{C}}}_{13}-{\dot{\mathrm{C}}}_{12}}{{\dot{\mathrm{E}}}_{13}-{\dot{\mathrm{E}}}_{12}}=\frac{{\dot{\mathrm{C}}}_{\mathrm{w},\mathrm{FC}}}{{\dot{\mathrm{W}}}_{\mathrm{FC}}}\\ \frac{{\dot{\mathrm{C}}}_{4}}{{\dot{\mathrm{E}}}_{4}}=\frac{{\dot{\mathrm{C}}}_{3}}{{\dot{\mathrm{E}}}_{3}}\\ \frac{{\dot{\mathrm{C}}}_{8}}{{\dot{\mathrm{E}}}_{8}}=\frac{{\dot{\mathrm{C}}}_{7}}{{\dot{\mathrm{E}}}_{7}}\end{array}$ |

Pump | ${\dot{\mathrm{C}}}_{15}+{\dot{\mathrm{C}}}_{\mathrm{W},\mathrm{p}}+{\dot{\mathrm{Z}}}_{\mathrm{p}}={\dot{\mathrm{C}}}_{12}$ | $\frac{{\dot{\mathrm{C}}}_{\mathrm{w},\mathrm{p}}}{{\dot{\mathrm{W}}}_{\mathrm{p}}}=\frac{{\dot{\mathrm{C}}}_{\mathrm{w},\mathrm{T}}}{{\dot{\mathrm{W}}}_{\mathrm{T}}}$ |

Turbine | ${\dot{\mathrm{C}}}_{13}+{\dot{\mathrm{Z}}}_{\mathrm{T}}={\dot{\mathrm{C}}}_{14}+{\dot{\mathrm{C}}}_{\mathrm{W},\mathrm{T}}$ | $\frac{{\dot{\mathrm{C}}}_{14}}{{\dot{\mathrm{E}}}_{14}}=\frac{{\dot{\mathrm{C}}}_{13}}{{\dot{\mathrm{E}}}_{13}}$ |

Condenser | ${\dot{\mathrm{C}}}_{14}+{\dot{\mathrm{C}}}_{16}+{\dot{\mathrm{Z}}}_{\mathrm{cond}}={\dot{\mathrm{C}}}_{17}+{\dot{\mathrm{C}}}_{15}$ | $\begin{array}{l}\frac{{\dot{\mathrm{C}}}_{14}}{{\dot{\mathrm{E}}}_{14}}=\frac{{\dot{\mathrm{C}}}_{15}}{{\dot{\mathrm{E}}}_{15}}\\ {\mathrm{c}}_{16}=0\end{array}$ |

Component | ${\dot{\mathbf{E}}}_{\mathbf{F}}$ (kW) | ${\dot{\mathbf{E}}}_{\mathbf{P}}$ (kW) | ${\dot{\mathbf{E}}}_{\mathbf{D}}$ (kW) | $\mathsf{\epsilon}$ (%) | ${\dot{\mathbf{Z}}}_{\mathit{k}}$ ($/h) | ${\dot{\mathbf{C}}}_{\mathbf{D}\mathbf{,}\mathbf{k}}$ ($/h) | ${\dot{\mathbf{C}}}_{\mathbf{D}\mathbf{,}\mathbf{k}}\mathbf{+}{\dot{\mathbf{Z}}}_{\mathbf{D}\mathbf{,}\mathbf{k}}$ ($/h) | $\mathbf{r}$ (%) | $\mathbf{f}$ (%) |
---|---|---|---|---|---|---|---|---|---|

Compressor | 162.280 | 145.125 | 17.155 | 89.4 | 0.083 | 1.294 | 1.377 | 12.6 | 6 |

Air humidifier | 153.123 | 131.548 | 21.575 | 85.9 | 0.797 | 1.746 | 2.543 | 23.9 | 31.4 |

H_{2} humidifier | 2707 | 2702.501 | 4.499 | 99.8 | 0.797 | 0.012 | 0.809 | 0.83 | 98.5 |

PEMFC | 2836.804 | 1830.366 | 850.078 | 41.3 | 23.145 | 33.358 | 56.503 | 23 | 41 |

Turbine | 133.128 | 114.445 | 18.683 | 86.0 | 2.201 | 2.024 | 4.225 | 41.4 | 52.1 |

Condenser | 22.737 | 15.108 | 7.629 | 66.5 | 0.116 | 0.938 | 1.054 | 60.5 | 11 |

Pump | 1.881 | 1.334 | 0.547 | 70.9 | 0.074 | 0.114 | 0.188 | 92 | 39.3 |

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**MDPI and ACS Style**

Seyed Mahmoudi, S.M.; Sarabchi, N.; Yari, M.; Rosen, M.A.
Exergy and Exergoeconomic Analyses of a Combined Power Producing System including a Proton Exchange Membrane Fuel Cell and an Organic Rankine Cycle. *Sustainability* **2019**, *11*, 3264.
https://doi.org/10.3390/su11123264

**AMA Style**

Seyed Mahmoudi SM, Sarabchi N, Yari M, Rosen MA.
Exergy and Exergoeconomic Analyses of a Combined Power Producing System including a Proton Exchange Membrane Fuel Cell and an Organic Rankine Cycle. *Sustainability*. 2019; 11(12):3264.
https://doi.org/10.3390/su11123264

**Chicago/Turabian Style**

Seyed Mahmoudi, S. M., Niloufar Sarabchi, Mortaza Yari, and Marc A. Rosen.
2019. "Exergy and Exergoeconomic Analyses of a Combined Power Producing System including a Proton Exchange Membrane Fuel Cell and an Organic Rankine Cycle" *Sustainability* 11, no. 12: 3264.
https://doi.org/10.3390/su11123264