International Journal of Chemical Engineering

Volume 2016, Article ID 4109204, 10 pages

http://dx.doi.org/10.1155/2016/4109204

## An Improved Empirical Fuel Cell Polarization Curve Model Based on Review Analysis

^{1}Lab of Clean Energy Automotive Engineering Center, Tongji University, Shanghai 201804, China^{2}School of Automotive Studies, Tongji University, Shanghai 201804, China^{3}Shanghai Motor Vehicle Inspection Center, Shanghai 201805, China^{4}China National Institute of Standardization, Beijing 100088, China

Received 20 November 2015; Revised 9 March 2016; Accepted 18 April 2016

Academic Editor: Iftekhar A. Karimi

Copyright © 2016 Dong Hao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Based on a review analysis of empirical fuel cell polarization curve models in the literature, an improved model that can predict fuel cell performance with only measured current-voltage data is developed. The fitting characteristics of this new model are validated by fitting bench test data and road test data. In the case of bench test data, a comparison of the new model and two representative models is conducted, and the results show that the new model presents the best fitting effects over a whole range of current densities. Moreover, the fitted ohmic resistances derived from the new model show good agreement with the measured values obtained through a current interruption test. In the case of using road test data, the new model also presents excellent fitting characteristics and convenience for application. It is the author’s belief that the new model is beneficial for the application-oriented research of fuel cells due to its prominent features, such as conciseness, flexibility, and high accuracy.

#### 1. Introduction

A polymer electrolyte membrane fuel cell (PEMFC) is an electrochemical device that converts chemical energy stored in hydrogen directly to electricity and heat with water as the only byproduct of the reaction. With the prominent features of zero emissions, low operating temperature, quick startup, and high efficiency, PEMFCs have been broadly considered the best electrical energy sources for automotive, stationary, and portable power devices [1]. In the area of PEMFCs research, mathematical fuel cell models, including mechanism and empirical models, are widely applied in the design, control, and optimization of PEMFCs [2–7]. Compared with mechanism models, empirical PEMFC polarization curve models are generally used in the application-oriented investigations and industrial research with advantages such as flexibility, simplicity, and acceptable accuracy.

For PEMFC, the polarization curve, which describes the relationship between output voltage and current density, is the most important characteristic of performance. Thus, a variety of empirical fuel cell polarization curve models have been developed in the past decades to reproduce the measured polarization curves containing a series of current-voltage data points. Furthermore, the values of fitting parameters derived from the empirical models are also valuable references for the investigation of PEMFCs.

This work began with a review on existing empirical fuel cell polarization curve models. Based on this, a new empirical model for the entire range of current densities was then presented. Validation of the new model was executed through fitting bench test data and road test data. Regarding bench tests, a polarization curve test and current interruption test were carried out on a fuel cell stack consisting of 57 cells. Taking the measured current-voltage data as reference values, the fitting accuracy of the new model was investigated and compared with two representative models. Finally, the validity of the new model was discussed by comparing the fitted ohmic resistances with the experimental values. Regarding road tests, some discrete current-voltage data of a 90-cell stack, which were sampled from a demonstrating fuel cell sightseeing vehicle, were used to analyze the application effects of different models.

#### 2. Review Analysis of Existing Polarization Curve Models

With the aim of improving fitting accuracy of fuel cell performance throughout the range of operation, a number of researchers have developed numerous empirical and semiempirical fuel cell polarization curve models since the early 1990s. Ten polarization curve models are reviewed in this section. To distinguish fitting parameters from other parameters or constants, fitting parameters are written in boldface throughout this work.

The first empirical fuel cell polarization curve model with five fitting parameters was presented by Kim et al. [8] (denoted by model K):

The parameter can be described aswhere represents activation loss, dedicates ohmic loss, and is an empirical term that approximates mass transfer loss. It is noteworthy that is simply a fitting parameter rather than the open circuit voltage (OCV) or the reversible voltage of the fuel cell. Fitting (1) has been cited by many studies and publications because of its advantages; that is, it fits the current-voltage data over the entire range of current densities under different temperatures, pressures, gas compositions, and so on. However, the measured value of OCV, the corresponding voltage of 0 A cm^{−2}, cannot be used during the fitting process because of the term. In addition, although this equation provides excellent fitting characteristics with both medium and high current densities, there is a minor deficiency in fitting accuracy with small current densities.

Based on model K, several improved models aiming at modifying the term of mass transfer loss have been proposed. The model suggested by Lee et al. [9] is expressed aswhere the last term, containing the pressure ratio logarithm, is a form of the Nernst equation, which is introduced primarily to describe potential changes in the cathode. This term is an empirical constant parameter that has no relationship with current density. In addition, the pressure of oxygen must be measured before using this model.

The model of Squadrito et al. [10] iswhere accounts for an “amplification term” of the logarithmic term; both and are fitting parameters without any physical meaning; is a fitting parameter representing the inverse of the limiting current density. This model improves the fitting characteristic of the mass transfer loss term by increasing a fitting parameter.

The model of Chu et al. [11] can be described aswhere is the smallest current density that causes the voltage to deviate from linearity. When the current density is less than , the term of mass transfer loss in (5) is equal to 0. That is, mass transfer loss occurs only in the high current density region; nevertheless, in fact, it occurs over the entire range of current densities. With the foregoing considerations, an improved model was proposed by Xia and Chan [12]:

In this model, first in (5) is substituted by parameter . As a result, when the current density is less than , the mass transfer loss in (8) is equal to , which means that the relationship between mass transfer loss and current density is linear in the low and moderate current density range. The coefficient indicates that the mass transfer phenomenon occurs over the entire range of current densities. The nonlinear term indicates that the phenomenon occurs at current densities that are higher than .

Similarly, some modifications on the mass transfer loss term were executed by Pisani et al. [13]:where is a fitting parameter describing the flooding phenomenon and is an empirical constant. The derivation of this model is based on the observation that the strongest nonlinear contributions to the cell voltage drop at high current densities arise from interface phenomenon happening in the cathode reactive region. One of the limitations of this model is the fact that there are three fitting parameters and one empirical constant to be included in the mass transfer loss term.

In summary, (3)–(5), (8), and (11) are obtained by modifying the term of mass transfer loss in model K. Each of these models consists of a constant voltage fitting parameter, a logarithmic term approximating the activation loss, a linear term representing the ohmic loss, and one or two terms describing the mass transfer loss. As with model K, the parameters of in these models are merely fitting parameters without any theoretical meanings. Moreover, owing to the existence of the term , which tends towards infinity when the current density decreases to zero, it is difficult to precisely predict the OCV and the performance at small current densities.

Considering that the models mentioned above cannot accurately fit no-load operation and small current densities, Fraser and Hacker [14] executed some modifications to (1) and presented a model (denoted by model F) as follows:

The reversible voltage, which is a constant term, can be estimated with known fuel cell operation temperature and partial pressures of oxygen and hydrogen. Additional fitting parameters and are introduced into the logarithmic term. Consequently, this equation provides accurate fitting characteristics with small current densities and OCV. It is worth noting that the reversible voltage here is merely an approximate value because it is difficult to measure the actual values of the operation temperature and partial pressures of the reactants.

In the case of a hydrogen/air fuel cell, the thermodynamic reversible voltage, , is calculated from the modified Nernst equation, which considers both temperature and pressure changes [9]:where is the standard-state reversible voltage (1.229 V) and is the entropy of the reaction (assuming that it is independent of temperature), which is −163.28 J mol^{−1} K^{−1} at temperature . is the activity of . For an ideal gas, , where is the partial pressure of gas and is the standard-state pressure (1 atm). For a hydrogen/air fuel cell operating at a low temperature with liquid water as a product, the activity of water is 1.

By substituting the known parameters mentioned above into (13), the reversible voltage can be described as a function of temperature and pressure as follows:

As shown by (14), the operation temperature of the fuel cell and partial pressures of oxygen and hydrogen are indispensable for calculating the value of the reversible voltage. To reduce the number of indispensable measured parameters in the model, a fitting term of reversible voltage was used in (15) by Weydahl et al. [15] and in (16) by Poh et al. [16]:

Either (15) or (16) contains the maximum number of fitting parameters among these ten existing models. Compared with model K, two additional fitting parameters are introduced. To regard the reversible voltage as a fitting parameter, it is difficult to obtain a unique, reasonable solution of . As a consequence, there are many solutions for other fitting parameters.

Equation (17), developed by Haji [17], also regards the reversible voltage as a constant term and presents outstanding fitting accuracy over the entire region of current densities. Consider

Although there are only four fitting parameters in this model, the limiting current density and current loss are obtained through measurements. As a result, the range of application of this model is limited.

In addition, an analysis technique to evaluate six sources of polarization losses in hydrogen/air proton exchange membrane fuel cells was developed by Williams et al. [18]. The overall polarization curve was described by two separate equations: one for current densities smaller than and one for current densities greater than . At

At

The approach in their work is very clear and effective for evaluating different sources of polarization losses under laboratory conditions. Based on determinations of some parameters, including reversible voltage, nonelectrode ohmic resistance, cathode electrode ohmic resistance, limited current density, Tafel slope, and exchange current density, the output voltage of fuel cells is calculated. The development of this model is aimed at distinguishing different polarization losses in fuel cells rather than reproducing the whole polarization curve based on voltage/current data. However, it provided useful insights for analyzing the polarization curve to future works on developing a model without any parameters that lack physical significance.

In conclusion, all of the models reviewed above have their merits and drawbacks, and their numbers of fitting parameters range from 4 to 7. The shortages and inconveniences of these models are listed in Table 1.