Released this past November, the list is based on data collected from the Web of Science and highlights some of the world’s most influential scientific minds by naming the researchers whose publications over the previous decade have included a high number of Highly Cited Papers placing them among the top 1% most-cited.
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We wish to congratulate all of the researchers named and especially our authors on this amazing accomplishment! We are happy and proud to share in their success!
IntechOpen is proud to announce that 179 of our authors have made the Clarivate™ Highly Cited Researchers List for 2020, ranking them among the top 1% most-cited.
\n\n
Throughout the years, the list has named a total of 252 IntechOpen authors as Highly Cited. Of those researchers, 69 have been featured on the list multiple times.
\n\n\n\n
Released this past November, the list is based on data collected from the Web of Science and highlights some of the world’s most influential scientific minds by naming the researchers whose publications over the previous decade have included a high number of Highly Cited Papers placing them among the top 1% most-cited.
\n\n
We wish to congratulate all of the researchers named and especially our authors on this amazing accomplishment! We are happy and proud to share in their success!
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She is a member of the National Scientific and Technological Research Council (CONICET) of Argentina in the genetic toxicology field, the Latin American Association of Environmental Mutagenesis, Teratogenesis, and Carcinogenesis (ALAMCTA), the Argentinean Society of Toxicology (ATA), the Argentinean Society of Genetics (SAG), the Argentinean Society of Biology (SAB), and the Society of Environmental Toxicology and Chemistry (SETAC). She has authored more than 380 contributions in the field, including scientific publications in peer-reviewed journals and research communications. She has served as a review member for more than 30 scientific international journals. She has been a plenary speaker in scientific conferences and a member of scientific committees. She is a specialist in issues related to genetic toxicology, mutagenesis, and ecotoxicology.",institutionString:"National University of La Plata",position:null,outsideEditionCount:0,totalCites:0,totalAuthoredChapters:"2",totalChapterViews:"0",totalEditedBooks:"6",institution:{name:"National University of La Plata",institutionURL:null,country:{name:"Argentina"}}},coeditorTwo:null,coeditorThree:null,coeditorFour:null,coeditorFive:null,topics:[{id:"677",title:"Soil Fertility",slug:"soil-fertility"}],chapters:[{id:"68355",title:"Opening History: Gaining Perspectives",slug:"opening-history-gaining-perspectives",totalDownloads:347,totalCrossrefCites:0,authors:[null]},{id:"67534",title:"The State of the Soil Organic Matter and Nutrients in the Long-Term Field Experiments with Application of Organic and Mineral Fertilizers in Different Soil-Climate Conditions in the View of Expecting Climate Change",slug:"the-state-of-the-soil-organic-matter-and-nutrients-in-the-long-term-field-experiments-with-applicati",totalDownloads:507,totalCrossrefCites:1,authors:[null]},{id:"68604",title:"Composting",slug:"composting",totalDownloads:484,totalCrossrefCites:0,authors:[null]},{id:"68090",title:"Plant Growth Biostimulants from By-Products of Anaerobic Digestion of Organic Substances",slug:"plant-growth-biostimulants-from-by-products-of-anaerobic-digestion-of-organic-substances",totalDownloads:415,totalCrossrefCites:1,authors:[null]},{id:"68380",title:"Compost Tea Quality and Fertility",slug:"compost-tea-quality-and-fertility",totalDownloads:666,totalCrossrefCites:0,authors:[null]},{id:"67957",title:"Efficacy of Different Substrates on Vermicompost Production: A Biochemical Analysis",slug:"efficacy-of-different-substrates-on-vermicompost-production-a-biochemical-analysis",totalDownloads:453,totalCrossrefCites:0,authors:[null]},{id:"67917",title:"Organic Fertilizer Production and Application in Vietnam",slug:"organic-fertilizer-production-and-application-in-vietnam",totalDownloads:529,totalCrossrefCites:0,authors:[null]}],productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"},personalPublishingAssistant:{id:"287827",firstName:"Gordan",lastName:"Tot",middleName:null,title:"Mr.",imageUrl:"https://mts.intechopen.com/storage/users/287827/images/8493_n.png",email:"gordan@intechopen.com",biography:"As an Author Service Manager my responsibilities include monitoring and facilitating all publishing activities for authors and editors. From chapter submission and review, to approval and revision, copyediting and design, until final publication, I work closely with authors and editors to ensure a simple and easy publishing process. I maintain constant and effective communication with authors, editors and reviewers, which allows for a level of personal support that enables contributors to fully commit and concentrate on the chapters they are writing, editing, or reviewing. I assist authors in the preparation of their full chapter submissions and track important deadlines and ensure they are met. I help to coordinate internal processes such as linguistic review, and monitor the technical aspects of the process. As an ASM I am also involved in the acquisition of editors. 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Larramendy",coverURL:"https://cdn.intechopen.com/books/images_new/4616.jpg",editedByType:"Edited by",editors:[{id:"14764",title:"Dr.",name:"Marcelo L.",surname:"Larramendy",slug:"marcelo-l.-larramendy",fullName:"Marcelo L. Larramendy"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"5358",title:"Soil Contamination",subtitle:"Current Consequences and Further Solutions",isOpenForSubmission:!1,hash:"e4d136df9f1658ae17f3ba7b3c992460",slug:"soil-contamination-current-consequences-and-further-solutions",bookSignature:"Marcelo L. Larramendy and Sonia Soloneski",coverURL:"https://cdn.intechopen.com/books/images_new/5358.jpg",editedByType:"Edited by",editors:[{id:"14764",title:"Dr.",name:"Marcelo L.",surname:"Larramendy",slug:"marcelo-l.-larramendy",fullName:"Marcelo L. Larramendy"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}}]},chapter:{item:{type:"chapter",id:"66617",title:"A Comparative Study of Maximum Likelihood Estimation and Bayesian Estimation for Erlang Distribution and Its Applications",doi:"10.5772/intechopen.85627",slug:"a-comparative-study-of-maximum-likelihood-estimation-and-bayesian-estimation-for-erlang-distribution",body:'\n
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1. Introduction
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Erlang distribution is a continuous probability distribution with wide applicability, primarily due to its relation to the exponential and gamma distributions. The Erlang distribution was developed by Erlang [1] to examine the number of telephone calls that could be made at the same time to switching station operators. This distribution can be expressed as waiting time and message length in telephone traffic. If the duration of individual calls are exponentially distributed then the duration of succession of calls is the Erlang distribution. The Erlang variate becomes gamma variate when its shape parameter is an integer (for details see Evans et al. [2]). Bhattacharyya and Singh [3] obtained Bayes estimator for the Erlangian queue under two prior densities. Haq and Dey [4] addressed the problem of Bayesian estimation of parameters for the Erlang distribution assuming different independent informative priors. Suri et al. [5] used Erlang distribution to design a simulator for time estimation of project management process. Damodaran et al. [6] obtained the expected time between failure measures. Further, they showed that the predicted failure times are closer to the actual failure times. Jodra [7] showed the procedure of computing the asymptotic expansion of the median of Erlang distribution.
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The probability density function of an Erlang variate is given by
where \n\nλ\n\n and \n\nk\n\n are the rate and the shape parameters, respectively, such that is k an integer number.
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1.1 Graphical representation of pdf for Erlang distribution
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In this chapter, Erlang distribution is considered. Some structural properties of Erlang distribution have been obtained. The parameter estimation of Erlang distribution is obtained by employing the maximum likelihood method of estimation, method of moments and Bayesian method of estimation in different sections of this chapter. In Bayesian approach, the parameters are estimated by using Jeffrey’s and Quasi priors under different loss functions (Figure 1).
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Figure 1.
Pdf’s of Erlang distribution for different values of lambda and k.
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1.2 Relationship of Erlang distribution with other distributions
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The gamma distribution is a generalized form of the Erlang distribution.
If the shape parameter k is 1, then Erlang distribution reduces to exponential distribution.
If the scale parameter is 2, then Erlang distribution reduces to Chi-square distribution with 2 degrees of freedom.
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Thus from the above descriptions, we can say that exponential distribution and Chi-square distribution are the sub-models of Erlang distribution.
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2. Methods used for parameter estimation
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In this chapter, we have used different approaches for parameter estimation. The first two methods come under the classical approach which was founded by Fisher in a series of fundamental papers round about 1930.
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The alternative approach is the Bayesian approach which was first discovered by Reverend Thomas Bayes. In this chapter, we have used two different priors for the parameter estimation. Also three loss functions are used which are discussed in their respective sections. A number of symmetric and asymmetric loss functions used by various researchers; see Zellner [8], Ahmad and Ahmad [9], Ahmad et al. [10], etc.
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These methods of estimations are elaborated in their respective sections accordingly.
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2.1 Maximum likelihood (MLH) estimation
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The most general method of estimation is known as maximum likelihood (MLH) estimators, which was initially formulated by Gauss. Fisher in the early 1920 firstly introduced MLH as general method of estimation and later on developed by him in a series of papers. He revealed the advantages of this method by showing that it yields sufficient estimators, which are asymptotically MVUES. Thus the important feature of this method is that we look at the value of the random sample and then select our estimate of the unknown population parameter, the value of which the probability of getting the observed data is maximum.
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Suppose the observed data sample values are \n\n\n\nx\n1\n\n\nx\n2\n\n…\n\nx\nn\n\n\n\n. When X is a discrete random variable, we can write \n\nP\n\n\n\nX\n1\n\n=\n\nx\n1\n\n\n\n\nX\n2\n\n=\n\nx\n2\n\n\n…\n\n\nX\nn\n\n=\n\nx\nn\n\n\n\n=\nf\n\n\nx\n1\n\n\nx\n2\n\n…\n\nx\nn\n\n\n\n, which is the value of joint probability distribution at the sample point \n\n\n\nx\n1\n\n\nx\n2\n\n…\n\nx\nn\n\n\n\n. Since the sample values has been observed and are therefore fixed numbers, we consider \n\nf\n\n\nx\n1\n\n\nx\n\n2\n,\n\n\n…\n\nx\nn\n\nλ\n\n\n as the value of a function of the parameter \n\nλ\n\n, referred to as the likelihood function.
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Similarly the definition applies when the random sample comes from a continuous population but in that case \n\nf\n\n\nx\n1\n\n\nx\n\n2\n,\n\n\n…\n\nx\nn\n\nλ\n\n\n is the value of joint pdf at the sample point \n\n\n\nx\n1\n\n\nx\n2\n\n…\n\nx\nn\n\n\n\n. That is, the likelihood function at the sample value \n\n\n\nx\n1\n\n\nx\n2\n\n…\n\nx\nn\n\n\n\n which is given by
Since the principle of maximum likelihood consists in finding an estimator of the parameter which maximizes the likelihood function for variation in the parameter. Thus if there exists a function \n\n\nλ\n̂\n\n=\n\nλ\n̂\n\n\n\nx\n1\n\n\nx\n2\n\n…\n\nx\nn\n\n\n\n of the sample values which maximizes \n\nL\n\nx\nλ\n\n\n for variation in \n\nλ\n\n, then \n\n\nλ\n̂\n\n\n is to be taken as the estimator of \n\nλ\n\n. Usually we call \n\n\nλ\n̂\n\n\n as ML estimators. Thus \n\n\nλ\n̂\n\n\n is the solution
Since \n\nL\n\nx\nλ\n\n>\n0\n\n, so \n\nlog\nL\n\nx\nλ\n\n\n which shows that \n\nL\n\nx\nλ\n\n\n and \n\nlog\nL\n\nx\nλ\n\n\n attains their extreme values at \n\n\nλ\n̂\n\n\n. Therefore, the equation becomes
a form which is more convenient from practical point of view.
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The MLH estimation of the rate parameter of Erlang distribution is obtained in the following theorem:
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Theorem 2.1: Let \n\n\n\nx\n1\n\n\nx\n2\n\n…\n\nx\nn\n\n\n\n be a random sample of size n from Erlang density function Eq. (1), then the maximum likelihood estimator of \n\nλ\n\n is given by
One of the simplest and oldest methods of estimation is the method of moments. The method of moments was discovered by Karl Pearson in 1894. It is a method of estimation of population parameters such as mean, variance, etc. (which need not be moments), by equating sample moments with unobservable population moments and then solving those equations for the quantities to be estimated. The method of moments is special case when we need to estimate some known function of finite number of unknown moments.
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Suppose \n\nf\n\nx\n\nλ\n1\n\n\nλ\n2\n\n…\n\nλ\np\n\n\n\n be the density function of the parent population with p parameters \n\n\n\nλ\n1\n\n\nλ\n2\n\n…\n\nλ\np\n\n\n\n. Let \n\n\nμ\ns\n′\n\n\n be the sth moment of a random variable about origin and is given by
In general \n\n\n\nμ\n1\n′\n\n\nμ\n2\n′\n\n…\n\nμ\np\n′\n\n\n\n will be the functions of parameters \n\n\n\nλ\n1\n\n\nλ\n2\n\n…\n\nλ\np\n\n\n\n. Let \n\n\n\nx\ni\n\n\ni\n=\n1\n\n2\n…\nn\n\n\n be a random sample of size n from the given population. The method of moments consists in solving the p-equations (i) for \n\n\n\nλ\n1\n\n\nλ\n2\n\n…\n\nλ\np\n\n\n\n in terms of \n\n\n\nμ\n1\n′\n\n\nμ\n2\n′\n\n…\n\nμ\np\n′\n\n\n\n. Then replacing these moments \n\n\n\nμ\ns\n′\n\n\ns\n=\n1\n\n2\n3\n…\np\n\n\n by the sample moments
where \n\n\nm\ni\n\n\n is the ith moment about origin in the sample.
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Then by the method of moments \n\n\n\n\nλ\n̂\n\n1\n\n\n\nλ\n̂\n\n2\n\n…\n\n\nλ\n̂\n\np\n\n\n\n are the estimators of respectively.
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The MM estimation of the rate parameter of Erlang distribution is obtained in the following theorem:
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Theorem 2.2: Let \n\n\n\nx\n1\n\n\nx\n2\n\n…\n\nx\nn\n\n\n\n be a random sample of size n from Erlang density function Eq. (1), then the moment estimator of \n\nλ\n\n is given by
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\n\n\nλ\n̂\n\n=\n\n\nk\n\n\nx\n¯\n\n\n\n\n.\n\n
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Proof: If the numbers \n\n\n\nx\n1\n\n\nx\n2\n\n…\n\nx\nn\n\n\n\n represents a set of data, then an unbiased estimator for the rth moment about origin is
\n\n\nx\n¯\n\n=\n\nk\nλ\n\n\n, where \n\n\nx\n¯\n\n\n is the mean of the data and
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\n\n\nλ\n̂\n\n=\n\n\nk\n\n\nx\n¯\n\n\n\n\n.\n\n
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3. Bayesian method of estimation
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Nowadays, the Bayesian school of thought is garnering more attention and at an increasing rate. This thought of statistics was given by Reverend Thomas Bayes. He first discovered the theorem that now bears his name. It was written up in a paper “An Essay Towards Solving a Problem in the Doctrine of Chances.” This paper was found after his death by his friend Richard Price, who had it published posthumously in the Philosophical Transactions of the Royal Society in 1763. Bayes showed how inverse probability could be used to calculate probability of antecedent events from the occurrence of the consequent event. His methods were adopted by Laplace and other scientists in the nineteenth century. By mid twentieth century interest in Bayesian methods was renewed by De Finetti, Jeffreys and Lindley, among others. They developed a complete method of statistical inference based on Bayes’ theorem. Bayesian analysis is to be used by practitioners for situations where scientists have a priori information about the values of the parameters to be estimated. In everyday life, uncertainty often permeates our choices, and when choices need to be made, past experience frequently proves a helpful aid. Bayesian theory provides a general and consistent framework for dealing with uncertainty.
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Within Bayesian inference, there are also different interpretations of probability, and different approaches based on those interpretations. Early efforts to make Bayesian methods accessible for data analysis were made by Raiffa and Schlaifer [11], DeGroot [12], Zellner [13], and Box and Tiao [14]. The most popular interpretations and approaches are objective Bayesian inference and subjective Bayesian inference. Excellent expositions of these approaches are with Bayes and Price [15], Laplace [16], Jeffrey’s [17], Anscombe and Aumann [18], Berger [19, 20], Gelman et al. [21], Leonard and Hsu [22], De-Finetti [23]. Modern Bayesian data analysis and methods based on Markov chain Monte Carlo methods are presented in Bernardo and Smith [24], Robert [25], Gelman et al. [26], Marin and Robert [27], Carlin and Louis [28]. Good elementary introductions to the subject are Ibrahim et al. [29], Ghosh [30], Bansal [31], Koch [32], Hoff [33].
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In Bayesian statistics probability is not defined as a frequency of occurrence but as the plausibility that a proposition is true, given the available information. The parameters are treated as random variables. The rules of probability are used directly to make inferences about the parameters. Probability statements about parameters must be interpreted as “degree of belief.” We revise our beliefs about parameters after getting the data by using Bayes’ theorem. This gives our posterior distribution which gives the relative weights we give to each parameter value after analyzing the data. The posterior distribution comes from two sources: the prior distribution and the observed data. This means that the inference is based on the actual occurring data, not all possible data sets that might have occurred.
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In this section, posterior distribution of Erlang distribution is obtained by using Jeffrey’s prior and Quasi prior. The rate parameter of Erlang distribution is estimated with the help of different loss functions. For parameter estimation we have used the approach as is used by Ahmad et al. [34], Ahmad et al. [35], etc. Some important prior distributions and loss functions which we have used in this article are given as below:
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3.1 Prior distributions used
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Prior distribution is the basic part of Bayesian analysis which represents all that is known or assumed about the parameter. Usually the prior information is subjective and is based on a person’s own experience and judgment, a statement of one’s degree of belief regarding the parameter.
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Another important feature of the Bayesian analysis is the choice of the prior distribution. If the data have sufficient signal, even a bad prior will still not greatly influence the posterior. We can examine the impact of prior by observing the stability of posterior distribution related to different choices of priors. If the posterior distribution is highly dependent on the prior, then the data (the likelihood function) may not contain sufficient information. However, if the posterior is relatively stable over a choice of priors, then the data indeed contains significant information.
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Prior distribution may be categorical in different ways. One common classification is a dichotomy that separated “proper” and “improper” priors.
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A prior distribution is proper if it does not depend on the data and the value of integral \n\n\n∫\n\n−\n∞\n\n∞\n\ng\n\nλ\n\ndλ\n\n or summation \n\n∑\ng\n\nλ\n\n\n is one. If the prior does not depend on the data and the distribution does not integrate or sum to one then we say that the prior is improper.
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In this chapter, we have used two different priors Jeffrey’s prior and Quasi prior which are given below:
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3.1.1 Jeffrey’s prior
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An invariant form for the prior probability in estimation problems is given by Jeffery’s [36].
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The general formula of the Jeffreys prior, which is defined by
where \n\nI\n\nλ\n\n\n is the Fisher information for the parameter \n\nλ\n\n.When there are multiple parameters I is the Fisher information matrix, the matrix of the expected second partials
Jeffrey suggested a thumb rule for specifying non-informative prior for parameter \n\nλ\n\n as
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Rule 1: if \n\nλ\n∈\n\n\n−\n∞\n\n∞\n\n\n take \n\ng\n\nλ\n\n\n to be constant, i.e., \n\nλ\n\n to be uniformly distributed.
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Rule 2: if \n\nλ\n∈\n\n0\n∞\n\n\n take \n\ng\n\nλ\n\n∝\n\n1\nλ\n\n\n, i.e., \n\nlog\nλ\n\n to be uniformly distributed.
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Under linear transformation, rule 1 is invariant and under any power transformation of \n\nλ\n\n, rule 2 is invariant.
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3.1.2 Quasi prior
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When there is no more information about the distribution parameter, one may use the Quasi density as given by \n\ng\n\nλ\n\n=\n\n1\n\nλ\nd\n\n\n\n;\nλ\n>\n0\n\nand\n\nd\n>\n0\n.\n\n
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3.2 Loss functions used
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The concept of loss function is old as Laplace and was reintroduced in statistics by Abraham Wald [37]. In statistics, typically a loss function is used for parameter estimation, and the event in question is some function of the difference between estimated and true values for an occurrence of data. In the context of economics, loss function is usually economic cost. In optimal control, the loss is the penalty for failing to achieve a desired value.
\n
The word “loss” is used in place of “error” and the loss function is used as a measure of the error or loss. Loss function is a measure of the error and presumably would be greater for large error than for small error. We would want the loss to be small or we want the estimate to be close to what it is estimating. Loss depends on sample and we cannot hope to make the loss small for every possible sample but can try to make the loss small on the average. Our objective is to select an estimator that makes this error or loss small, also which makes the average loss (risk) small and ideally select an estimator that has the small risk.
\n
In this chapter, we have used three different Loss Functions which are as under:
\n
\n
3.2.1 Precautionary loss function (PLF)
\n
The concept of precautionary loss function (PLF) was introduced by Norstrom [38]. He introduced an alternative asymmetric loss function and also presented a general class of precautionary loss functions as a special case. These loss functions approach infinitely near the origin to prevent underestimation, thus giving conservative estimators, especially when low failure rates are being estimated. These estimators are very useful when underestimation may lead to serious consequences. A very useful and simple asymmetric precautionary loss function (PLF) is
where \n\nλ\n\n and \n\n\nλ\n̂\n\n\n represents the true and estimated values of the parameter. This loss function is frequently used because of its analytical tractability in Bayesian analysis.
\n
\n
\n
3.2.3 LINEX loss function (LLF)
\n
The idea of LINEX loss function (LLF) was founded by Klebanov [40] and used by Varian [41] in the context of real estate assessment. The formula of LLF is given by
where \n\nλ\n\n and \n\n\nλ\n̂\n\n\n represents the true and estimated values of the parameter and the constant c determines the shape of the loss function.
\n
\n
\n
\n
3.3 Posterior density under Jeffrey’s prior
\n
Let \n\n\n\nx\n1\n\n\nx\n2\n\n…\n\nx\nn\n\n\n\n be a random sample of size n having the Erlang density function Eq. (1) which is given by
where \n\nI\n\nλ\n\n=\n−\nnE\n\n\n\n\n∂\n2\n\nlog\nf\n\nx\nλ\nk\n\n\n\n∂\n\nλ\n2\n\n\n\n\n\n is the Fisher’s information matrix for the probability density function Eq. (1).
Let \n\n\n\nx\n1\n\n\nx\n2\n\n…\n\nx\nn\n\n\n\n be a random sample of size n having the Erlang density function Eq. (2) and the likelihood function Eq. (2).
In this section, parameter estimation of Erlang distribution is done by using Jeffreys’ prior under different loss functions. The procedure of calculating the Bayesian estimate is already defined in Section 3. The estimates are obtained in the following theorems:
\n
Theorem 4.1: Assuming the loss function \n\n\nL\np\n\n\n\nλ\n̂\n\nλ\n\n\n, the Bayesian estimator of the rate parameter \n\nλ\n\n, if the shape parameter \n\nk\n\n is known, is of the form
Theorem 4.2: Assuming the loss function \n\n\nl\nA\n\n\n\nλ\n̂\n\nλ\n\n\n, the Bayesian estimator of the rate parameter \n\nλ\n\n, if the shape parameter \n\nk\n\n is known, is of the form
Theorem 4.3: Assuming the loss function \n\n\nL\nl\n\n\n\nλ\n̂\n\nλ\n\n\n, the Bayesian estimator of the rate parameter \n\nλ\n\n, if the shape parameter \n\nk\n\n is known, is of the form
In this section, parameter estimation of Erlang distribution is done by using QUASI prior under different loss functions. The procedure for obtaining the Bayesian estimate is available in Section 3. The estimates of parameter are obtained in the following theorems.
\n
Theorem 5.1: Assuming the loss function \n\n\nL\np\n\n\n\nλ\n̂\n\nλ\n\n\n, the Bayesian estimator of the rate parameter \n\nλ\n\n, if the shape parameter \n\nk\n\n is known, is of the form
Remark: Replacing d = 1 in Eq. (25), the same Bayes estimator is obtained as in Eq. (19).
\n
Theorem 5.2: Assuming the loss function \n\n\nl\nA\n\n\n\nλ\n̂\n\nλ\n\n\n, the Bayesian estimator of the rate parameter \n\nλ\n\n, if the shape parameter \n\nk\n\n is known, is of the form
Minimization of the risk with respect to \n\n\nλ\n̂\n\n\n gives us the optimal estimator, i.e., \n\n\n∂\n\n∂\n\nλ\n̂\n\n\n\n\n\nR\n\n\nλ\n̂\n\n\n\n\n=\n0\n\n
Remark: Replacing d = 1 in Eq. (27), the same Bayes estimator is obtained as in Eq. (21).
\n
Theorem 5.3: Assuming the loss function \n\n\nL\nl\n\n\n\nλ\n̂\n\nλ\n\n\n, the Bayesian estimator of the rate parameter \n\nλ\n\n, if the shape parameter \n\nk\n\n is known, is of the form
Remark: Replacing d = 1 in Eq. (29), the same Bayes estimator is obtained as in Eq. (23).
\n
\n
\n
6. Entropy estimation of Erlang distribution
\n
The concept of entropy was introduced by Claude. Shannon [42] in the paper “A Mathematical theory of Communication.” This concept of Shannon’s entropy is the central role of information theory, sometimes referred as measure of uncertainty. Shannon entropy provides an absolute limit on the best possible lossless encoding or compression of any communication, assuming that the communication may be represented as a sequence of independent and identical distributed random variables. Entropy is typically measured in bits, when the log is to the base 2, and nats, when the log is to the base n.
\n
Shannon’s definition of entropy, when applied to an information source, can determine the minimum channel capacity required to reliably transmit the source as encoded binary digit. The entropy of a random variable is defined in terms of its probability distribution and can be shown to be a good measure of randomness or uncertainty. For deriving the entropy of probability distributions, we need the following two definitions that are more discussed in Cover et al. [43].
\n
Definition (i): The entropy of the discrete random variable defined on the probability space is given by
In this section, entropy estimation of two parameter Erlang distribution is discussed which given as below.
\n
Theorem 6.1: Let \n\n\n\nx\n1\n\n\nx\n2\n\n…\n\nx\nn\n\n\n\n be n positive independent and identically distributed random samples drawn from a population having Erlang density function Eq. (2), then the Shannon’s entropy of two parameter Erlang distribution is given by
For model selection the approach of Akaike information criterion (AIC) and Bayesian information criterion (BIC) based on entropy estimation are used. The Akaike information criterion (AIC) was introduced by Hirotsugu Akaike [44] and proposed it as a measure of goodness of fit of an estimated statistical model. It is a measure of the relative quality of a statistical model for a given set of data. It has been found in information theory that it offers a relative estimate of the information lost when a given model is used to represent the process that generates the data.
\n
The AIC is not a test of the model in the sense of hypothesis testing; rather it is a test between models—a tool for model selection. Given a data set, several competing models may be ranked according to their AIC, with the one having the lowest AIC being the best.
where K is the number of parameters and \n\nL\n\n\nλ\n̂\n\n\n\n is the maximized value of the likelihood function for the estimated model.
\n
AICC was first introduced by Hurvich and Tsai [45] and its different derivations were proposed by Burnham and Anderson [46]. AICC is AIC with a correction for finite sample sizes and is given by
Burnham and Anderson [47] strongly recommended that we should use AICC instead of AIC when the sample size is small or if K is large. Since AICC converges to AIC as the sample size is getting large. Using AIC, instead of AICC, when the sample size is not many times larger than K2, increases the probability of selecting models that have too many parameters, i.e., of over fitting. The probability of AIC over fitting can be substantial, in some cases.
\n
The Bayesian information criterion (BIC) also known as Schwarz Criterion is used as a substitute for full calculation of the Bayes’ factor since it can be calculated without specifying prior distribution. In BIC, the penalty for additional parameters is stronger than that of the AIC.
We have generated the data for Erlang distribution of different sample sizes (15, 30 and 60) in R Software for each pairs of \n\n\nλ\nk\n\n\n, where \n\n\n\nk\n=\n1\n\n2\n\n\n and \n\n\n\nλ\n=\n0.5\n\n1.0\n\n\n. The value for the loss parameter (C1 = −1, 1) and (a = 0.5, 1.0). The values of extension are (C = 0.5, 1.0). The estimates of rate parameter for each method are calculated. The results are presented in the following tables.
\n
\n
\n
9. Comparison of Erlang distribution (ED) with its sub-models
\n
The flexibility and potentiality of the Erlang distribution is compared with its sub models, which is examined by using different criterions like AIC, BIC and AICC with the help of the following illustration.
\n
Illustration I:
\n
We provide the compatibility of the Erlang distribution (ED) with their sub-models; Chi square and exponential distributions. For this purpose, we generated the data set for Erlang distribution of large sample size (i.e., 200) in R Software for each pairs of \n\n\nλ\nk\n\n\n, where \n\n\n\nk\n=\n2\n\n\n\n and \n\n\n\nλ\n=\n2.5\n\n\n\n. The data analysis is given in the following table:
\n
\n\n\n
\n
Illustration II:
\n
The data set is taken from Lawless [48]. The observations involves the number of million revolutions between failures for each of 23 ball bearings, the individual bearings were inspected periodically to determine whether “failure” had occurred. Treating the failure times as continuous, the 23 failure times are:
We primarily studied the maximum likelihood (MLH) estimation and Bayesian estimation to estimate the rate parameter of Erlang distribution. In Bayesian method, we use Jeffreys’ prior and Quasi prior under three different loss functions. These methods are compared through simulation technique and the results are presented in the Tables 1 and 2 respectively.
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n\n
\n
N
\n
k
\n
\n\n\nλ\n\n\n
\n
\n\n\n\n\nλ\n̂\n\nML\n\n\n\n
\n
\n\n\n\n\nλ\n̂\n\np\n\n\n\n
\n
\n\n\n\n\nλ\n̂\n\nA\n\n\n\n
\n
\n\n\n\n\nλ\n̂\n\nl\n\n\n\n
\n
\n
\n
c = −1
\n
c = 1
\n
a = 0.5
\n
a = 1.0
\n
\n\n\n
\n
15
\n
1
\n
0.5
\n
0.15350
\n
0.16179
\n
0.10530
\n
0.18428
\n
0.13350
\n
0.12597
\n
\n
\n
2
\n
1.0
\n
0.51869
\n
0.54673
\n
0.43100
\n
0.58896
\n
0.47443
\n
0.44430
\n
\n
\n
30
\n
1
\n
0.5
\n
0.03480
\n
0.03238
\n
0.03238
\n
0.03266
\n
0.04191
\n
0.03259
\n
\n
\n
2
\n
1.0
\n
0.06850
\n
0.04258
\n
0.07049
\n
0.02085
\n
0.03077
\n
0.02171
\n
\n
\n
60
\n
1
\n
0.5
\n
0.03910
\n
0.03320
\n
0.02856
\n
0.03488
\n
0.03110
\n
0.03060
\n
\n
\n
\n
\n
2
\n
1.0
\n
0.10349
\n
0.09918
\n
0.08981
\n
0.10244
\n
0.09398
\n
0.09200
\n
\n\n
Table 1.
Mean squared error for \n\n\nλ\n̂\n\n\n under Jeffrey’s prior.
ML = maximum likelihood estimate, p = precautionary LF, A = Al-Bayyati’s LF, l = LINEX LF.
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n\n
\n
N
\n
k
\n
\n\n\nλ\n\n\n
\n
d
\n
\n\n\n\n\nλ\n̂\n\nML\n\n\n\n
\n
\n\n\n\n\nλ\n̂\n\np\n\n\n\n
\n
\n\n\n\n\nλ\n̂\n\nA\n\n\n\n
\n
\n\n\n\n\nλ\n̂\n\nl\n\n\n\n
\n
\n
\n
c = −1
\n
c = 1
\n
a = 0.5
\n
a = 1.0
\n
\n\n\n
\n
15
\n
1
\n
0.5
\n
0.5
\n
0.15350
\n
0.14710
\n
0.11081
\n
0.18979
\n
0.13901
\n
0.13148
\n
\n
\n
1.0
\n
0.15350
\n
0.16179
\n
0.10530
\n
0.18428
\n
0.13350
\n
0.12597
\n
\n
\n
2
\n
1.0
\n
0.5
\n
0.51869
\n
0.51528
\n
0.43951
\n
0.59747
\n
0.48294
\n
0.45281
\n
\n
\n
1.0
\n
0.51869
\n
0.54673
\n
0.43100
\n
0.58896
\n
0.47443
\n
0.44430
\n
\n
\n
30
\n
1
\n
0.5
\n
0.5
\n
0.03480
\n
0.03286
\n
0.02179
\n
0.02206
\n
0.21630
\n
0.02200
\n
\n
\n
1.0
\n
0.03480
\n
0.03238
\n
0.03238
\n
0.03266
\n
0.04191
\n
0.03259
\n
\n
\n
2
\n
1.0
\n
0.5
\n
0.06850
\n
0.05828
\n
0.06361
\n
0.01397
\n
0.02389
\n
0.01483
\n
\n
\n
1.0
\n
0.06850
\n
0.04258
\n
0.07049
\n
0.02085
\n
0.03077
\n
0.02171
\n
\n
\n
60
\n
1
\n
0.5
\n
0.5
\n
0.03910
\n
0.03206
\n
0.03206
\n
0.03534
\n
0.03156
\n
0.03106
\n
\n
\n
1.0
\n
0.03910
\n
0.03320
\n
0.02856
\n
0.03488
\n
0.03110
\n
0.03060
\n
\n
\n
2
\n
1.0
\n
0.5
\n
0.10349
\n
0.09641
\n
0.09021
\n
0.10284
\n
0.09439
\n
0.09241
\n
\n
\n
\n
\n
\n
\n
1.0
\n
0.10349
\n
0.09918
\n
0.08981
\n
0.10244
\n
0.09398
\n
0.09200
\n
\n\n
Table 2.
Mean squared error for \n\n\nλ\n̂\n\n\n under Quasi prior.
ML = maximum likelihood estimate, p = precautionary LF, A = Al-Bayyati’s LF, l = LINEX LF.
\n
From the results obtained in Tables 1 and 2, we observe that in most of the cases, Bayesian estimator under Al-Bayyati’s loss function has the smallest mean squared error (MSE) values for Jeffrey’s prior and Quasi prior as compared to other loss functions and the maximum likelihood estimator. Thus we can conclude that Bayes estimator under Al-Bayyati’s loss function is efficient when the loss parameter C is −1.
\n
Also we estimated the unknown parameters of sub-models of Erlang distribution. The Akaike information criterion (AIC), Bayesian information criterion (BIC), and the corrected Akaike information criterion (AICC) are used to compare the candidate distributions. The best distribution corresponds to lower −logL, AIC, BIC, AICC statistics value.
\n
From the results obtained in Tables 3 and 4, we observe that the Erlang distribution is a competitive distribution as compared to its sub-models (i.e., exponential distribution and Chi-square distribution). In fact, based on the values of the AIC, BIC and AICC criteria, it shows clear picture that the Erlang distribution provides the best fit for these data among all the models considered.
\n
\n
\n
\n
\n
\n
\n
\n
\n\n
\n
Model
\n
Parameter estimate
\n
Standard error
\n
Measures
\n
\n
\n
\n\n\n−\nlog\nl\n\n\n
\n
AIC
\n
BIC
\n
AICC
\n
\n\n\n
\n
Exponential
\n
\n\n\n\nλ\n̂\n\n\n = 0.5945
\n
0.04203
\n
315.6853
\n
633.3705
\n
636.6689
\n
581.9245
\n
\n
\n
Chi-square
\n
k = 1.090
\n
0.0586
\n
312.3096
\n
625.6346
\n
628.9329
\n
633.3902
\n
\n
\n
Erlang
\n
k = 2
\n
289.9327
\n
581.8654
\n
585.1637
\n
625.6543
\n
\n
\n
\n
\n
\n\n\n\nλ\n̂\n\n\n = 1.1890
\n
0.0594
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n\n
Table 3.
AIC, BIC and AICC criterion for different sub-models of ED.
\n
\n
\n
\n
\n
\n
\n
\n
\n\n
\n
Model
\n
Parameter estimate
\n
Standard error
\n
Measures
\n
\n
\n
\n\n\n−\nlog\nl\n\n\n
\n
AIC
\n
BIC
\n
AICC
\n
\n\n\n
\n
Exponential
\n
\n\n\n\nλ\n̂\n\n\n = 0.01387
\n
0.002877
\n
121.4324
\n
244.8648
\n
246.0003
\n
245.0248
\n
\n
\n
Chi-square
\n
k = 2.9420
\n
1.1745
\n
177.5835
\n
357.1670
\n
358.3025
\n
357.327
\n
\n
\n
Erlang
\n
k = 1
\n
113.0298
\n
230.0597
\n
232.3307
\n
230.5212
\n
\n
\n
\n
\n
\n\n\n\nλ\n̂\n\n\n = 0.05577
\n
0.01683
\n
\n
\n
\n
\n
\n
\n
\n
\n
\n\n
Table 4.
AIC, BIC and AICC criterion for different sub-models of ED.
\n
\n
\n
11. Conclusions
\n
In this paper we have generated three types of data sets with varying sample sizes for Erlang distribution. These data sets were simulated and behavior of the data was checked in case of parameter estimation for Erlang distribution in R Software. By the virtue of the data analysis we are able predict the estimate of rate parameter for Erlang distribution under three different functions by using two different prior distributions. With the help of these results we can also do comparison between loss functions and the priors.
\n
Also the comparison of Erlang distribution with its sub-models was carried out. The results acquired in Tables 3 and 4, it shows the clear picture that Erlang distribution performs better as compared to its sub-models. Thus we can say that Erlang distribution is efficient as compared to its sub-models (i.e., exponential distribution and Chi-square distribution) on the basis of the above procedures.
\n
\n
Acknowledgments
\n
The authors acknowledge editor of this journal for encouragement to finalize the chapter. Further, the authors acknowledge profound thanks to anonymous referee for giving critical comments which have immensely improved the presentation of the chapter. Also I extend my sincere thanks to all the authors whose papers I have consulted for this work.
\n
Conflict of interest
The authors declare that there is no conflict of interests regarding the publication of this chapter.
\n',keywords:"Erlang distribution, prior distributions, loss functions, simulation study, applications",chapterPDFUrl:"https://cdn.intechopen.com/pdfs/66617.pdf",chapterXML:"https://mts.intechopen.com/source/xml/66617.xml",downloadPdfUrl:"/chapter/pdf-download/66617",previewPdfUrl:"/chapter/pdf-preview/66617",totalDownloads:356,totalViews:0,totalCrossrefCites:1,totalDimensionsCites:1,hasAltmetrics:0,dateSubmitted:"September 13th 2018",dateReviewed:"March 4th 2019",datePrePublished:"September 27th 2019",datePublished:"February 26th 2020",dateFinished:null,readingETA:"0",abstract:"In this chapter, Erlang distribution is considered. For parameter estimation, maximum likelihood method of estimation, method of moments and Bayesian method of estimation are applied. In Bayesian methodology, different prior distributions are employed under various loss functions to estimate the rate parameter of Erlang distribution. At the end the simulation study is conducted in R-Software to compare these methods by using mean square error with varying sample sizes. Also the real life applications are examined in order to compare the behavior of the data sets in the parametric estimation. The comparison is also done among the different loss functions.",reviewType:"peer-reviewed",bibtexUrl:"/chapter/bibtex/66617",risUrl:"/chapter/ris/66617",book:{slug:"statistical-methodologies"},signatures:"Kaisar Ahmad and Sheikh Parvaiz Ahmad",authors:[{id:"274770",title:"Dr.",name:"Kaisar",middleName:null,surname:"Ahmad",fullName:"Kaisar Ahmad",slug:"kaisar-ahmad",email:"ahmadkaisar31@gmail.com",position:null,institution:null},{id:"287089",title:"Dr.",name:"Sheikhv Parvaiz",middleName:null,surname:"Ahmad",fullName:"Sheikhv Parvaiz Ahmad",slug:"sheikhv-parvaiz-ahmad",email:"sprvz@yahoo.com",position:null,institution:null}],sections:[{id:"sec_1",title:"1. Introduction",level:"1"},{id:"sec_1_2",title:"1.1 Graphical representation of pdf for Erlang distribution",level:"2"},{id:"sec_2_2",title:"1.2 Relationship of Erlang distribution with other distributions",level:"2"},{id:"sec_4",title:"2. Methods used for parameter estimation",level:"1"},{id:"sec_4_2",title:"2.1 Maximum likelihood (MLH) estimation",level:"2"},{id:"sec_5_2",title:"2.2 Method of moments (MM)",level:"2"},{id:"sec_7",title:"3. Bayesian method of estimation",level:"1"},{id:"sec_7_2",title:"3.1 Prior distributions used",level:"2"},{id:"sec_7_3",title:"3.1.1 Jeffrey’s prior",level:"3"},{id:"sec_8_3",title:"3.1.2 Quasi prior",level:"3"},{id:"sec_10_2",title:"3.2 Loss functions used",level:"2"},{id:"sec_10_3",title:"3.2.1 Precautionary loss function (PLF)",level:"3"},{id:"sec_11_3",title:"3.2.2 Al-Bayyati’s loss function (ALF)",level:"3"},{id:"sec_12_3",title:"3.2.3 LINEX loss function (LLF)",level:"3"},{id:"sec_14_2",title:"3.3 Posterior density under Jeffrey’s prior",level:"2"},{id:"sec_15_2",title:"3.4 Posterior density under Quasi prior",level:"2"},{id:"sec_17",title:"4. Estimation of parameters under Jeffrey’s prior",level:"1"},{id:"sec_18",title:"5. Estimation of parameters under Quasi prior",level:"1"},{id:"sec_19",title:"6. Entropy estimation of Erlang distribution",level:"1"},{id:"sec_20",title:"7. AIC and BIC criterion for Erlang distribution",level:"1"},{id:"sec_21",title:"8. Simulation study of Erlang distribution",level:"1"},{id:"sec_22",title:"9. Comparison of Erlang distribution (ED) with its sub-models",level:"1"},{id:"sec_23",title:"10. Results and discussion",level:"1"},{id:"sec_24",title:"11. Conclusions",level:"1"},{id:"sec_25",title:"Acknowledgments",level:"1"},{id:"sec_28",title:"Conflict of interest",level:"1"}],chapterReferences:[{id:"B1",body:'Erlang AK. The theory of probabilities and telephone conversations. Nyt Tidsskrift for Matematik B. 1909;20(6):87-98\n'},{id:"B2",body:'Evans M, Hastings N, Peacock B. Statistical Distributions. 3rd ed. New York: John Wiley and Sons, Inc.; 2000\n'},{id:"B3",body:'Bhattacharyya SK, Singh NK. Bayesian estimation of the traffic intensity in M/Ek/1 queue Far. East. Journal of Mathematical Sciences. 1994;2:57-62\n'},{id:"B4",body:'Haq A, Dey S. Bayesian estimation of Erlang distribution under different prior distributions. Journal of Reliability and Statistical Studies. 2001;4(1):1-30\n'},{id:"B5",body:'Suri PK, Bhushan B, Jolly A. Time estimation for project management life cycles: A simulation approach. International Journal of Computer Science and Network Security. 2009;9(5):211-215\n'},{id:"B6",body:'Damodaran D, Gopal G, Kapur PK. A Bayesian Erlang software reliability model. Communication in Dependability and Quality Management. 2010;13(4):82-90\n'},{id:"B7",body:'Jodra P. Computing the asymptotic expansion of the median of the Erlang distribution. Mathematical Modelling and Analysis. 2012;17(2):281-292\n'},{id:"B8",body:'Zellner A. Bayesian estimation and prediction using asymmetric loss function [PhD thesis]. Journal of American Statistical Association exponential distribution using simulation. Iraq: Baghdad University; 1986. vol. 81, pp. 446-451\n'},{id:"B9",body:'Ahmad SP, Ahmad K. Bayesian analysis of Weibull distribution using R Software. 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Philosophical Transactions of the Royal Society A. 1763;53:370-418 Reprinted in Biometrika, 48, 296-315\n'},{id:"B16",body:'Laplace PS. Eassi Philosophiquesur Les Probabilities, Paris. This Book Went Through Five Editions (The Fifth was in 1825) Revised By Laplace. The Sixth Edition Appeared In English Translation By Dover Publications, New York, In 1951. While This Philosophical Essay Appeared Separately In 1814, It Also Appeared As A Preface To His Earlier Work. Theoric Analytique Des Probabilites; 1774. pp. 621-656\n'},{id:"B17",body:'Jeffrey H. Theory of Probability. III edition (I edition 1939, II edition 1994) ed. Oxford University Press: Clarendon Press; 1961\n'},{id:"B18",body:'Anscombe F, Alhnann R. A definition of subjective Probability. The Annals of Mathematical Statistics. 1963;34(1):199-205\n'},{id:"B19",body:'Berger JO. Statistical Decision Theory and Bayesian Analysis. 2nd ed. New York: Springer-Verlag; 1985\n'},{id:"B20",body:'Berger JO, Bernardo JM. Estimating a product of means: Bayesian analysis with reference priors. American Journal of Mathematical Statistical Association. 1989;84:200-207\n'},{id:"B21",body:'Gelman A, Carlin JB, Stern HS, Rubin DB. Bayesian Data Analysis. London: Chapman and Hall; 1995\n'},{id:"B22",body:'Leonardo T, Hsu JSJ. Bayesian Methods. Cambridge: Cambridge University Press; 1999\n'},{id:"B23",body:'De Finetti B. A critical essay on the theory of probability and on the value of science. “probabilismo”. Erkenntnis. 1931;31:169-223 English translation as “Probabilism”\n'},{id:"B24",body:'Bernardo JM, Smith AFM. Bayesian Theory. Chichester, West Sussex: John Wiley and Sons; 2000\n'},{id:"B25",body:'Robert CP. The Bayesian Choice. 2nd ed. New York: Springer-Verlag; 2001\n'},{id:"B26",body:'Gelman A, Carlin JB, Stern HS, Rubin DB. Bayesian Data Analysis. 2nd ed. Boca Raton, FL: Chapman and Hall/CRC; 2004\n'},{id:"B27",body:'Marin JM, Robert CP. Bayesian Core: A Practical Approach to Computational Bayesian Statistics. Springer-Verlag; 2007\n'},{id:"B28",body:'Carlin BP, Louis TA. Bayesian Methods for Data Analysis. IIIrd ed. Boca Raton, FL: Chapman and Hall/CRC; 2008\n'},{id:"B29",body:'Ibrahim JG, Chen M-H, Sinha D. Bayesian Survival Analysis. New York: Springer Verlag; 2001\n'},{id:"B30",body:'Ghosh JK, Delampady M, Samanta T. An Introduction to Bayesian Analysis Theory and Methods. Springer-Verlag; 2006\n'},{id:"B31",body:'Bansal AK. Bayesian Parametric Inference. New Dehli: Narosa Publishing House; 2007\n'},{id:"B32",body:'Koch KR. Introduction to Bayesian Statistics. New York: Springer-Verlag; 2007\n'},{id:"B33",body:'Hoff PD. A First Course in Bayesian Statistical Methods. Springer-Verlag; 2009\n'},{id:"B34",body:'Ahmad K, Ahmad SP, Ahmed A. Classical and Bayesian approach in estimation of scale parameter of inverse Weibull distribution. Mathematical Theory and Modeling. 2015;5. ISSN 2224-5804\n'},{id:"B35",body:'Ahmad K, Ahmad SP, Ahmed A. Classical and Bayesian approach in estimation of scale parameter of Nakagami distribution. Journal of Probability and Statistics. 2016;2016 Article ID 7581918, 1-8\n'},{id:"B36",body:'Jefferys H. An invariant form for the prior probability in estimation problems. Proceedings of The Royal Society Of London, Series. A. 1946;186:453-461\n'},{id:"B37",body:'Wald A. Statistical Decision Functions. Wiley; 1950\n'},{id:"B38",body:'Norstrom JG. The use of precautionary loss functions in risk analysis. IEEE Transactions on Reliability. 1996;3:400-403\n'},{id:"B39",body:'Al-Bayyati. Comparing methods of estimating Weibull failure models using simulation [PhD thesis]. Iraq: College of Administration and Economics, Baghdad University; 2002\n'},{id:"B40",body:'Klebnov LB. Universal loss function and unbiased estimation. Dokl. Akad. Nank SSSR Soviet-Maths. Dokl. T. 1972;203:1249-1251\n'},{id:"B41",body:'Varian HR. A Bayesian approach to real estate assessment. In: Fienberg SE, Zellner A, editors. Studies in Bayesian Econometrics and Statistics in Honor of Leonard J. Savage. Amsterdam: North-Holland; 1975. pp. 195-208\n'},{id:"B42",body:'Shannon CE. A mathematical theory of communication. Bell System Technical Journal. 1948;27:623-659\n'},{id:"B43",body:'Cover TM, Thomas JA. Elements of Information Theory. New York: Wiley; 1991\n'},{id:"B44",body:'Akaike H. Information theory and an extension of the maximum likelihood principle. In: Petrov BN, Csaki F, editors. 2nd International Symposium on Information Theory. Budapest: Akademia Kiado; 1973. pp. 267-281\n'},{id:"B45",body:'Hurvich CM, Tsai CL. Regression and time series model selection in small samples. Biometrika. 1989;76:297-307\n'},{id:"B46",body:'Burnham KP, Anderson DR. Multimodel inference: Understanding AIC and BIC in model selection. Sociological Methods & Research. 2004;33:261-304\n'},{id:"B47",body:'Burnham KP, Anderson DR. Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach. 2nd ed. Springer-Verlag; 2002\n'},{id:"B48",body:'Lawless JF. Statistical Models and Methods for Lifetime Data. 2nd ed. Wiley-InterScience; 2002\n'}],footnotes:[],contributors:[{corresp:"yes",contributorFullName:"Kaisar Ahmad",address:"ahmadkaisar31@gmail.com",affiliation:'
Department of Statistics, Sheikh-ul-Alam Memorial Degree College, India
Department of Statistics, University of Kashmir, India
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1. Introduction
Fuel cells are electrochemical devices that convert chemical energy into work in the form of electric energy and heat. Any system producing energy obeys the laws of thermodynamics. The amount of work/heat produced depends on thermodynamic values for reversible reactions, while for irreversible reactions overpotential is required to complete the work. Hydrogen and oxygen are used to illustrate the simplest case. A general thermodynamic analysis of hydrogen fuel cells of the reversible work for the reversible reaction is performed. The concepts enthalpy, specific heat, entropy and Gibbs free energy are related to the reacting systems in fuel cells. Gibbs free energy is the thermodynamic potential that measures the reversible work by a thermodynamic system at constant pressure and temperature. Change in enthalpy and change in entropy are significant in particular to fuel cells; they indicate spontaneity of the adsorption process and increased randomness of adsorbate molecules on the solid surface, respectively. Specific heat is a measure of the amount of heat energy required to increase the temperature of a substance by 1°C. The fuel cell performance is examined through the reversible voltage, and the actual output voltage is after overpotential. The efficiency of a fuel cell is the useful energy output which is the electrical energy produced, and the energy input is the enthalpy of hydrogen.
2. The thermodynamics of a fuel cell
A fuel cell, also known as a galvanic or voltaic cell, is a well-known example of a device that works by changing chemical energy into electrical energy, which is exhibited in terms of cell potential and electrical current output. The maximum possible electrical energy output and the corresponding electrical potential difference between the cathode and anode are achieved when the fuel cell is operated under thermodynamically reversible conditions, as shown in Figure 1, a fuel cell system to which fuel and oxidant streams enter and product stream exits. Unfortunately, it is inevitable that some of the energy will be dissipated as heat.
Figure 1.
Simple H2/O2 fuel cell diagram.
The overall electrochemical reactions occurring inside the fuel cell system boundary are described as follows:
H2fuel+O2oxidant→W+Q+H2OproductE1
where W is the rate of electrical work done by the system and Q is the rate of heat transferred into the system from the surroundings at constant pressure and temperature.
Electrical work is, in general, described by the relation:
W=EI∆tE2
where E is the cell voltage and I is the current. In a fuel cell reaction, electrons are transferred from the anode to the cathode, generating a current.
The amount of electricity (IΔt) transferred when the reaction occurs is given by nF, where n is the number of electrons transferred and F is Faraday’s constant = 96,493 coulombs. The electrical work can hence be calculated as:
W=−nFEcellE3
The Gibbs free energy is the maximum amount of work done on the system:
Wel=∆GE4
∆G=−nFEcellE5
Hence the maximum cell potential or the reversible cell potential becomes:
Erev=−∆GnFE6
where E° is also called the reversible voltage, because it is the maximum possible voltage without any irreversible losses. This is the maximum possible voltage of an electrochemical cell, since it is attained assuming a reversible process. If we are looking at the redox reaction on a per-mole-of-fuel basis, the absolute Gibbs function is equivalent to the molar specific value. All fuel cell losses are associated with deviation from this maximum. Since F and n are constants for a particular global redox reaction, the functional dependence of the maximum possible voltage of an electrochemical cell is related strictly to the dependencies of the Gibbs free energy, namely, temperature and pressure of the reactants and products. If all the potential chemical energy for a reaction went into electrical work and there was no heat transfer, there would be no entropy change; dG = dH. In this case, we can show that:
Erev=−∆HnFE7
For a generic reaction:
For a general reaction or process of A and B giving products C and D:
aA+bB→cC+dDE8
∆Gf=∆Gof+RTlnaCcaDdaAaaBbE9
where a’s are the thermodynamic activity coefficients for the reacting species. To convert to voltage, we can divide it by nF:
ETP=Eo⏟I−RTnFlnaCcaDdaAaaBb⏟IIE10
where I is the standard voltage evaluated at 1 atm pressure for all components and II accounts for the thermodynamic activity dependence on the Nernst voltage.
For an ideal gas, a = Pi/P°, where Pi is the partial pressure of the species of interest and P° is the reference pressure, 1 atm.
For water vapour, the partial pressure of the vapour cannot exceed the saturation pressure, Psat, which is a function of temperature. Thus, the reference pressure is set to Psat, and a = Pv/Psat, which is the relative humidity, RH. This can normally be considered to be 1.0 in the immediate molecular region of the water-generating electrode. This is a reasonable assumption because water generation is always at the catalyst surface and the activity of water here is 1.0. Also, the reaction itself is not limited by the product water concentration at this surface:
ETP=Eo−RTnFlnPCPocPDPodPAPoaPBPobE11
where the partial pressures are evaluated at the particular electrode where the reaction involving the species occurs. Using this expression, we can solve for the expected maximum (Nernst) voltage for a given fuel cell reaction. Two important points are as follows:
The Nernst equation is a result of the equilibrium established at the electrode surfaces. A particular gradient can exist between the concentration of a species in the channel of a fuel cell and the electrode, especially under high-current-density conditions, which cannot be considered a true thermodynamic equilibrium situation anyway.
Only species directly involved in the electrochemical reaction of Eq. (8) are represented directly in the activity terms of Eq. (10). Species not participating in the electrochemical charge transfer reaction only indirectly alter the voltage through the species mole fractions of the participating species.
For the H2/O2 fuel cell potential, the open-circuit voltage is the maximum operating voltage (when no current is flowing) and is determined by the chemical thermodynamics of the overall cell reaction. The Nernst equation provides a relationship between the standard potential (Eo) for the cell reaction and the open-circuit voltage, where it can be determined at the partial pressures of reactants and products at temperature (T):
To understand how the reversible voltage varies with temperature and pressure, respectively, we have previously shown that the Gibbs free energy is related to the reversible cell voltage by Eq. (5):
∆G=−nFEcell
At constant pressure, the above relationship produces a Maxwell relation that links the change in open cell voltage with temperature T (a measurable quantity) to the change in entropy S [1]:
∂E∂TQ=−∂S∂QTE13
∂E∂TP=∆SnFE14
∆S=nF∂E∂TPE15
Erev=Eorev+∆SnFT−ToE16
At constant temperature, Eq. (5) produces an equation that links voltage with pressure, p, to the change in volume:
∂E∂PT=−∂vnFE17
If the volume change of the reaction is negative (if fewer moles of gas are generated by the reaction than consumed, for instance), then the cell voltage will increase with increasing pressure.
Usually, only gas species produce an appreciable volume change. Assuming that the ideal gas law applies, we can write Eq. (17) as:
∂E∂PT=−∆ngRTnFpE18
where Δng represents the change in the total number of moles of gas upon reaction. Pressure, like temperature, turns out to have a minimal effect on reversible voltage:
Erev=Eorev−∆nRTnFlnPE19
2.1 Enthalpy of reaction for a fuel cell
Enthalpy is the thermodynamic quantity that states the total heat content of the system, which is the sum of all internal process in a closed system [2]. For homogeneous systems, enthalpy is solely based on the size of the system as it is an extensive property. Enthalpy (H) is the sum of the internal energy of the system (U) and the product of pressure (P) and volume (V) of the system. The change in enthalpy in a system is equal to the heat gained or lost in the system:
H=U+PVE20
The enthalpy change (ΔH) for a reaction in a fuel cell indicates the full amount of heat released by the reaction at a constant pressure; hence, enthalpy is simply equal to the heat released:
H=QpE21
At constant pressure and volume, the thermodynamic properties of the cell are related to the behaviour of its potential and are defined from the Gibbs-Helmholtz equation [3]:
∆H=∆G−T∂∆G∂TpE22
In accordance with Eq. (5), in terms of electrochemical processes, change in enthalpy can be written as:
∆H=nFT∂EcellTp−EcellE23
The overall reaction in Eq. (1) is the same as the reaction of hydrogen combustion. Combustion is an exothermic process, which means that there is energy released in the process [4, 5]:
H2+½O2→H2O∆H<0E24
From the table of enthalpies of formation (hof) and absolute entropies of formation (sof) obtained from the basic thermodynamic data (see Table 1), the heat of formation of both liquid and vapour water can be calculated using the equation above to form:
hof (kJ/mol)
sof (kJ/mol.K)
∆Gof (kJ/mol)
Hydrogen, H2
0
0.131
0
Oxygen, O2
0
0.205
0
Water (liquid), H2O (l)
−285.8
0.070
−237.2
Water (vapour), H2O (g)
−241.8
0.189
−228.6
Table 1.
Enthalpies of formation and absolute entropies of formation of fuel cell reactants and products (at 25°C and 1 atm).
The heat (or enthalpy) of a chemical reaction is the difference between the heat of formation of products and reactants. This means [6]:
The enthalpy of the hydrogen combustion reaction (Eq. (25)) is also called hydrogen’s heating value. The 285.83 kJ/mol is known as hydrogen’s higher heating value (HHV), which means that 1 mol of hydrogen is fully combusted with ½ mol of oxygen and cooled down to 25°C. If hydrogen is combusted with sufficient excess oxygen and cooled down to 25°C, the value will become 241.82 kJ/mol, which is known as hydrogen’s lower heating value (LHV) [7]. The difference between the LHV and HHV of 44.01 kJ/mol is equal to the molar latent heat of water vaporisation at 25°C.
In the heating value for reactions involving water as a product, there is a choice in the calculation of thermodynamic voltages between a high heating value (HHV) and a low heating value (LHV), defined as follows for a given reaction:
High heating value: It is assumed all the product water is in the liquid phase.
Low heating value: It is assumed all the product water is in the gas phase.
Note that all calculations are based on HHV or LHV and do not necessarily correspond to the actual physical state of the product water at the fuel cell electrode. The terms HHV and LHV are used in combustion calculations as well, where the product water is nearly always in the gas phase. The difference between the two values is proportional to the latent heat of vaporisation of the liquid. The use of the LHV (gas-phase vapour product) will result in a lower calculated thermal voltage, since some energy is used for the latent heat of vaporisation of the liquid. In practice, the LHV is completely appropriate for high-temperature fuel cells, but the HHV is also commonly used. An important point regarding low-temperature fuel cells that is often confusing is that the choice of HHV or LHV is arbitrary and 100°C is not a point of demarcation between the two. Often 100°C is thought of as a natural boundary between the HHV and LHV because it is the phase change temperature of water at 1 atm pressure. The delineation between liquid and gas, however, is more complex and is related to the local vapour pressure and total pressure.
2.2 Entropy (S)
The concept of entropy is one of the thermodynamic parameters that are important to the science of fuel cells to understand. Entropy is defined as the measure of the unavailable energy in a closed thermodynamic system that is usually considered to be a measure of the system’s disorder, known as the second law of thermodynamics. The total entropy of a system increases over time, as the molecular disorder increases. Therefore, if the system is in equilibrium, the change between the initial state and the final state, the system is going through a reversible change. Since entropy represents the unavailable energy used in the system, a system of zero entropy optimises the work output of the system. The fuel cell generates the amount of electricity and rejects an amount of thermal energy Q to its environment. As there is heat transfer, and it is a real system, there must be an increase in entropy. The amount of heat rejected and the maximum amount of electrical power that a fuel cell will generate can be determined by formulating the entropy changes occurring in the cell:
H2+O2→H2O+QE26
The entropy of H2 and O2 will disappear, but the new entropy of H2O and heat formation will appear. As long as the process is reversible, which is the assumption for the fuel cell, the entropy appearing in the rejected heat can be written as [8]:
∆S=∆QrevTE27
The equation for the change of entropy (ΔS) is equal to the change in enthalpy (∆Q) divided by the temperature (T) of the system.
As there is no heat transfer in electrical work, the entropy is zero. The entropy in an H2/O2 fuel cell can be calculated using the absolute entropy values given in Table 1. The second law of thermodynamics requires that in a fuel cell, there will be a net increase in entropy. Therefore, the entropy that shows up in the rejected heat and the product water (liquid water) must be greater than the entropy contained in the reactants (H2 and O2) [9, 10]:
Entropy gain ≥ entropy loss
QT+∑products≥∑reactantsE28
Q≥Tproducts−reactantsE29
To calculate the amount of heat rejected per mole of H2:
As heat capacity effects are generally minor, ∆H and ∆S values are usually assumed to be independent of temperature. A simplified entropy calculation can be with entropy values obtained from Table 1:
From the second law of thermodynamics, the change in free energy, or maximum useful work, can be obtained when a ‘perfect’ fuel cell operating irreversibly is dependent upon temperature. Therefore, Wel, the electrical power output, is [11]:
ΔG=ΔH−TΔSE31
where H is the total energy of the system, S is the ‘unavailable’ energy and G is the ‘free’ energy, or the energy available to do useful work.
The change in Gibbs free energy varies with both temperature and pressure. It can be shown that for a H2/O2 fuel cell:
∆Gf=∆Gof+RTlnpH2OpH2p√O2E32
where ∆Gof is the change in Gibbs free energy at standard pressure, which varies with the temperature T of the fuel cell, in Kelvin; pH2, pO2 and PH2O are the partial pressure of the hydrogen, oxygen and vapour, respectively; and R is the universal gas constant (8.314 J/(kg.K)). The fact that the value of ∆Gof is negative means that the energy is released from the reaction [12, 13, 14].
For the H2/O2 fuel cell reaction, the change in Gibbs free energy is:
The maximum possible electrical energy output and the corresponding electrical potential difference between the cathode and anode are achieved when the fuel cell is operated under the thermodynamically reversible condition. This maximum possible cell potential is called ‘reversible cell potential’, one of the significantly important parameters for FC.
From Eq. (23) we calculated ∆H to be 285.830 kJ/mol for hydrogen’s HHV and 241.98 kJ/mol for hydrogen’s LHV, while for ∆S in Eq. (28), the entropy of reaction is 0.163 kJ/mol for HHV and for LHV 0.044 kJ/mol.
To calculate Gibbs free energy for H2/O2 fuel cell reaction:
The fact that the value of ∆Gof is negative means that the energy is released from the reaction. From Eq. (6), the potential or reversible open cell voltage Eo of any electrochemical device is defined as:
Eo=−∆GnF
where n is the amount of exchanged electrons and F is Faraday’s constant. For the hydrogen oxidation or water formation, n = 2. The free enthalpies ∆G of water formation are either:
∆Gf,H2Ol=−237.13kJ/molor∆Gf,H2Og=−228.57kJ/mol
The corresponding potential is therefore (Table 2):
∆H (kJ/mol)
∆G (kJ/mol)
∆S (kJ/mol.K)
E (V)
H2 + 1/2O2 → H2O (l)
−285.8
−237.1
−0.163
1.23
H2 + 1/2O2 → H2O (g)
−241.8
−228.6
−0.045
1.18
Table 2.
Enthalpies, entropies and Gibbs free energy of H2/O2 fuel cell reaction in (kJ/mol.K) and the resulting theoretical cell potential at 25°C.
Another property that is important in thermodynamics and the study of fuel cells is the specific heat. The specific heat of a solid or liquid is usually defined as the heat required to raise unit mass through 1 degree temperature rise. For a gas there are an infinite number of ways in which heat may be added between any two temperatures, and hence a gas could have an infinite number of specific heats. However, only two specific heats for gases are defined, as specific heat at constant volume, cv, and specific heat at constant pressure, cp.
For a reversible ideal gas, a non-flow process at constant pressure and at constant volume is given as, respectively:
dQ=CpdTE34
and
dQ=CvdTE35
For a substance that is modelled as incompressible, the specific heats are assumed to be equal, Cp = Cv [15]:
Cp=Cv=dQdTE36
For an ideal gas in which a fuel cell is assumed to be, the specific enthalpy depends only on temperature:
hf=dQ=CpdTE37
hT=h298.15+∫298.15TCPdTE38
where h298.15 is the enthalpy at a reference temperature.
The specific heat can also be related to specific entropy at temperature T:
dS=dQT=CpdTTE39
sT=s298.15+∫298.15T1TCPdTE40
The specific enthalpy and entropy for H2/O2 fuel cell are given by [16]:
The values of molar entropy and enthalpy of formation at 298.15 K are given in Table 1 [17]. Eqs. (38) and (40) can be used to determine the specific heat capacity, Cp, at constant pressure. Specific heat relationships are generally modelled with a high-order polynomial, such as those listed below for hydrogen fuel cell gases, valid in the range of 300–1000 K [18]. It is reported that over a range of temperatures, Cp is not constant, while over the range of 300–350 K, the obtained Cp values are 0.6% accurate [4].
The heat capacity values for hydrogen, oxygen and water as a function of temperature are shown in Figure 2. The data were obtained from Kabza [4].
Figure 2.
Specific heat values for hydrogen, oxygen and water as a function of temperature.
3. Fuel cell thermodynamic efficiency
In the process of energy conversion in a fuel cell, the initial chemical energy between the enthalpy of the products and reactants is converted into electrical energy and thermal energy, as stated in the first law of thermodynamics. The efficiency of any energy conversion device is defined as the ratio between useful energy output and energy input [19, 20, 21, 22]:
η=actual electrical workmaximum available work
η=∆G∆H=∆H−T∆S∆HE43
The maximum possible thermodynamic efficiency of a fuel cell can be written as [9]:
η==1−T∆S∆HE44
In the case of a fuel cell, the useful energy output is the electrical energy produced, and the energy input is the enthalpy of hydrogen, that is, hydrogen’s HHV. Assuming that all of the Gibbs free energy can be converted into electrical energy (the reaction is reversible), the maximum theoretical efficiency of a fuel cell is [23, 24] (Figure 3):
Figure 3.
Energy inputs and outputs for a fuel cell as an energy conversion device [25].
η=∆G∆H=237.34286.02x100%=83%
For hydrogen’s LHV, the fuel cell efficiency would be [26]:
η=∆G∆H=228.74241.98×100%=94.5%
The LHV has higher efficiency compared to HHV, because the reversible efficiency of the fuel cell decreases as the operating temperature increases [27].
The expected fuel cell efficiency is not always achieved due to thermodynamic and electrochemical irreversible losses [28].
4. Irreversible losses
Other than calculating energy quantities during the conversion of chemical energy to electrical energy, there is also the matter of electron flow through materials in the fuel cell process. The single fuel cell provides a voltage dependent on operating conditions such as temperature, applied load and fuel/oxidant flow rates [29, 30]. If a fuel cell is supplied with reactant gases, but the electric current is not closed, it will not generate any current, and one would expect the cell potential to be at the theoretical cell potential for the given conditions (temperature, pressure and concentration of reactants). In reversible conditions, the energy loss is the heat lost towards the environment, T∆s, due to negative entropy [12].
However, in practice, the thermodynamic cell potential is decreased from its ideal potential, usually less than 1 V, due to irreversible losses known as overpotential or polarisation [17]. The fuel cell performance overpotential is due to [14, 31]:
Activation overpotential: The activation polarisation is related to the charge transfer processes occurring during the electrochemical reactions on electrode surfaces. The losses are caused by the slowness of the reactions taking place on the surface of the electrodes [32]. Activation polarisation depends on the nature of type of electrode, ionic interactions, ion-solvent interactions and the electrode-electrolyte interface [33].
Ohmic overpotential: In most fuel cells, the most important contribution to this resistance is the electrolyte, due to the ionic nature of its conductivity, resistance to the flow of electrons through the electrodes and the contact resistance at the cell terminals.
Mass transport (concentration) overpotential: Concentration polarisation occurs due to a decrease in the concentration of the reactants at the electrode-electrolyte interface. Due to diffusion or convection problems in the electrolyte, the concentration of the reactants is not maintained at the initial level. Reaction product accumulation can also cause a dilution of reactants. The concentration gradient thereby formed causes a drop in electrode activity, and the terminal voltage is reduced.
Fuel crossover overpotential: ‘Crossover’ is one of the common effects occurring in alcohol fuel cells [22]. Although the electrolyte, a polymer membrane, is not electrically conductive and practically impermeable to reactant gases, some amount of fuel will diffuse from anode to cathode to react with oxygen, resulting in fewer electrons in the generated current of electrons that travel through an external circuit [34]. With this transit the cathode potential decreases, thus reducing the overall efficiency of a fuel cell. It occurs when the intermediates generated by fuel oxidation have higher concentration than oxygen at the cathode. The increase of temperature escalates the crossover effect [35].
The standard measure of performance is the polarisation curve, which represents the cell voltage behaviour against operating current density (Figure 4). From the figure, the voltage loss caused by mixed potential and crossover, activation polarisation, ohmic polarisation and mass transport losses is the most significant in the tail of the I-V curve. The maximum fuel cell is then examined through the reversible voltage of the system, which is calculated using thermodynamics and the actual voltage of the system [36]. The final voltage is lower than the thermodynamic voltage and is usually between 0.5 and 1.0 V. Although polarisations cannot be eliminated, material choice and electrode designs can contribute to their minimisation [37]:
E=Etherm−ηact−ηohm−ηconcE45
Figure 4.
H2/O2 polarisation curve at equilibrium and voltage losses in fuel cell [30].
where E is the cell potential, Etherm is the thermodynamic potential, ηact is the voltage loss due to activation polarisation, ηohm is the voltage loss due to ohmic polarisation and ηconc is the voltage loss due to mass transport polarisation and the entropy generation results [38, 39, 40, 41]:
S=EnFTE46
In addition, the related heat lost for irreversibility can be calculated as (Figure 4):
q=T∆s−nFE=∆H−∆G−nFEE47
5. Conclusion
Thermodynamics is used to understand the process of energy conversion in fuel cells. The determination of a fuel cell’s performance depends on thermodynamic evaluation. The heat potential of a fuel is given by the enthalpy of the reaction. Not all heat potential of a fuel can be used to perform useful work; the reversible work of a fuel is defined by Gibbs free energy, which is the electrical work. The study of the electrical effects shows that the molar flow of the fuel used is proportional to the electric current and the reversible work is proportional to the reversible voltage. The cell voltage varies with temperature, pressure and reactant/product activities. Irreversible losses cause a difference in the efficiency of reversible and real processes, with efficiency of real processes always less than reversible processes. The losses are due to two major reasons, namely, irreversible kinetic losses and fuel utilisation losses.
Acknowledgments
We greatly appreciate the National Research Foundation and the Tertiary Education Support Programme for financial support as well as the Chemistry Department at the University of the Western Cape.
\n',keywords:"enthalpy, entropy, Gibbs free energy, specific heat, fuel cell efficiency, hydrogen fuel cell",chapterPDFUrl:"https://cdn.intechopen.com/pdfs/70166.pdf",chapterXML:"https://mts.intechopen.com/source/xml/70166.xml",downloadPdfUrl:"/chapter/pdf-download/70166",previewPdfUrl:"/chapter/pdf-preview/70166",totalDownloads:872,totalViews:0,totalCrossrefCites:1,dateSubmitted:"June 26th 2019",dateReviewed:"October 14th 2019",datePrePublished:"November 22nd 2019",datePublished:null,dateFinished:null,readingETA:"0",abstract:"Thermodynamics is the study of energy change from one state to another. The predictions that can be made using thermodynamic equations are essential for understanding fuel cell performance, as a fuel cell is an electrochemical device that converts the chemical energy of a fuel and an oxidant gas into electrical energy. When a fuel cell is operating, some of the input is used to create electrical energy, but another portion is converted into thermal energy, depending on the type of fuel cell. Based on the first and second laws of thermodynamics, one can write down thermodynamic potentials to specify how energy can be transferred from one form to another. This chapter examines how electrical energy and thermal energy are transferred in the hydrogen fuel cell system. It also defines how reversible fuel cell voltages, which are the maximum fuel cell performances, are affected by departures from the standard state. Basic thermodynamic concepts allow one to predict states of the fuel cell system, including the potential, temperature, pressure, volume and moles of a fuel cell. The specific topics explored in this chapter include enthalpy, entropy, specific heat, Gibbs free energy, net output voltage irreversible losses in fuel cells and fuel cell efficiency.",reviewType:"peer-reviewed",bibtexUrl:"/chapter/bibtex/70166",risUrl:"/chapter/ris/70166",signatures:"Lindiwe Khotseng",book:{id:"8572",title:"Thermodynamics and Energy Engineering",subtitle:null,fullTitle:"Thermodynamics and Energy Engineering",slug:"thermodynamics-and-energy-engineering",publishedDate:"July 29th 2020",bookSignature:"Petrică Vizureanu",coverURL:"https://cdn.intechopen.com/books/images_new/8572.jpg",licenceType:"CC BY 3.0",editedByType:"Edited by",editors:[{id:"12354",title:"Prof.",name:"Petrică",middleName:null,surname:"Vizureanu",slug:"petrica-vizureanu",fullName:"Petrică Vizureanu"}],productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"}},authors:null,sections:[{id:"sec_1",title:"1. Introduction",level:"1"},{id:"sec_2",title:"2. The thermodynamics of a fuel cell",level:"1"},{id:"sec_2_2",title:"2.1 Enthalpy of reaction for a fuel cell",level:"2"},{id:"sec_3_2",title:"2.2 Entropy (S)",level:"2"},{id:"sec_4_2",title:"2.3 Gibbs free energy",level:"2"},{id:"sec_5_2",title:"2.4 Specific heat capacity",level:"2"},{id:"sec_7",title:"3. Fuel cell thermodynamic efficiency",level:"1"},{id:"sec_8",title:"4. Irreversible losses",level:"1"},{id:"sec_9",title:"5. Conclusion",level:"1"},{id:"sec_10",title:"Acknowledgments",level:"1"}],chapterReferences:[{id:"B1",body:'Yong-Qiang X, Chun-Hui L, Jin-Chuan F. The electrochemical thermodynamics for chemical reactions in dispersed cells. Journal of Colloid and Interface Science. 1999;217:107-110. DOI: 10.1006/jcis.1999.6221'},{id:"B2",body:'Tran D, Tanner G, Yang K, Woodley R. Thermodynamic effects of nanotechnological augmentation of hydrogen fuel cells. PAM Review. 2017:76-86. DOI: 10.5130/pamr.v4i0.1443'},{id:"B3",body:'Spiegel C. 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In: Zhao TS, Kreuer K-D, Van Nguyen T, editors. Advances in Fuel Cells. Oxford: Elsevier; 2007, p. 1-46'},{id:"B21",body:'Haynes C. Clarifying reversible efficiency misconceptions of high temperature fuel cells in relation to reversible heat engines. Journal of Power Sources. 2001;92:199-203. DOI: 10.1016/S0378-7753 (00)00541-3'},{id:"B22",body:'Dos Santos MC, Parreira LS, De Moura Seuza F, Camarge Junior J, Gentil T. Fuel Cells: Hydrogen and Ethanol Technologies. In: Hashmi S, editor. Reference Module in Materials Science and Materials Engineering. Oxford: Elsevier; 2017. p. 1-21. DOI: 10.1016/8978-0-12-803581-8.09263-8'},{id:"B23",body:'Pavelka M, Marski F. Detailed thermodynamic analysis of polymer electrolyte membrane fuel cells efficiency. International Journal of Hydrogen Energy. 2013;38(17):7102-7113. DOI: 10.1016/j.ijhydene.2013.03.149'},{id:"B24",body:'Barbir F. Fuel cell basics. In: Kakaç S, Pramuanjaroenkij A, Vasilie L, editors. 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Engineering Thermodynamics. 3rd ed. Laxmi: New Delhi; 2007. 922 p'},{id:"B36",body:'Srikar VT, Turner KT, Yung TIA, Spearing SM. Structural design considerations for micro-machined solid-oxide fuel cells. Journal of Power Sources. 2004;125(1):62-69. DOI: 10.1016/j.jpowsour.2003.07.002'},{id:"B37",body:'Umberto L. Overview on fuel cells. Renewable and Sustainable Energy Reviews. 2014;30:164-169. DOI: 10.1016/j.rser.2013.09.025'},{id:"B38",body:'Moran MJ, Shapiro HN. Fundamentals of Engineering Thermodynamics. 5th ed. New York, NY: Wiley; 2006. 831 p'},{id:"B39",body:'Wark K Jr. Advanced Thermodynamics for Engineers. New York, NY: McGraw-Hill; 1995 622 p'},{id:"B40",body:'Bejan A. Advanced Engineering Thermodynamics. Hoboken, NJ: Wiley; 2016. 741 p'},{id:"B41",body:'Fuel Cell Technology. Hydrogen Fuel Cell Engines and Related Technologies. 2001. Available from: https://energy.gov/sites/prod/files/2014/03/f9/fcm00r0.pdf [Accessed: 09-09-2019]'}],footnotes:[],contributors:[{corresp:"yes",contributorFullName:"Lindiwe Khotseng",address:"lkhotseng@uwc.ac.za",affiliation:'
Department of Chemistry, University of the Western Cape, Cape Town, South Africa
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This has encouraged the use of phytotherapic agents such as Copaifera multijuga, Apis mellifera (propolis), and Libidibia ferrea.",signatures:"Maria Fulgência Bandeira, Nikeila Conde, Carina Toda,\nAna Regina Casaroto, Geisy Lima, Marne Vasconcellos, Leilane Bentes,\nLarissa Silva, Samuel Lungareze and Joyce Meira",authors:[{id:"264331",title:"Prof.",name:"Maria Fulgência",surname:"Bandeira",fullName:"Maria Fulgência Bandeira",slug:"maria-fulgencia-bandeira",email:"fulgencia@ufam.edu.br"},{id:"272847",title:"Prof.",name:"Nikeila",surname:"Conde",fullName:"Nikeila Conde",slug:"nikeila-conde",email:"nikeilaconde@ufam.edu.br"},{id:"272848",title:"Prof.",name:"Carina",surname:"Toda",fullName:"Carina Toda",slug:"carina-toda",email:"carinatoda@yahoo.com.br"},{id:"272849",title:"Dr.",name:"Ana Regina",surname:"Casaroto",fullName:"Ana Regina Casaroto",slug:"ana-regina-casaroto",email:"ana_casaroto@yahoo.com.br"},{id:"272851",title:"MSc.",name:"Leilane",surname:"Bentes",fullName:"Leilane Bentes",slug:"leilane-bentes",email:"leilane.bentes@gmail.com"},{id:"272852",title:"MSc.",name:"Geisy",surname:"Lima",fullName:"Geisy Lima",slug:"geisy-lima",email:"geisylima@hotmail.com"},{id:"272886",title:"MSc.",name:"Samuel",surname:"Lungareze",fullName:"Samuel Lungareze",slug:"samuel-lungareze",email:"contato@lungareze.com.br"},{id:"273005",title:"MSc.",name:"Larissa",surname:"Silva",fullName:"Larissa Silva",slug:"larissa-silva",email:"acasarotom@gmail.com"},{id:"273006",title:"MSc.",name:"Joyce",surname:"Meira",fullName:"Joyce Meira",slug:"joyce-meira",email:"moreschi.maxilofacial@gmail.com"},{id:"273007",title:"Prof.",name:"Marne",surname:"Vasconcellos",fullName:"Marne Vasconcellos",slug:"marne-vasconcellos",email:"marne@ufam.edu.br"}],book:{title:"Biomaterial-supported Tissue Reconstruction or Regeneration",slug:"biomaterial-supported-tissue-reconstruction-or-regeneration",productType:{id:"1",title:"Edited Volume"}}}],collaborators:[{id:"264331",title:"Prof.",name:"Maria Fulgência",surname:"Bandeira",slug:"maria-fulgencia-bandeira",fullName:"Maria Fulgência Bandeira",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:null},{id:"267962",title:"Dr.",name:"Zeljka",surname:"Peric Kacarevic",slug:"zeljka-peric-kacarevic",fullName:"Zeljka Peric Kacarevic",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:null},{id:"272848",title:"Prof.",name:"Carina",surname:"Toda",slug:"carina-toda",fullName:"Carina Toda",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:null},{id:"272849",title:"Dr.",name:"Ana Regina",surname:"Casaroto",slug:"ana-regina-casaroto",fullName:"Ana Regina Casaroto",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:null},{id:"272851",title:"MSc.",name:"Leilane",surname:"Bentes",slug:"leilane-bentes",fullName:"Leilane Bentes",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:null},{id:"272852",title:"MSc.",name:"Geisy",surname:"Lima",slug:"geisy-lima",fullName:"Geisy Lima",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:null},{id:"272886",title:"MSc.",name:"Samuel",surname:"Lungareze",slug:"samuel-lungareze",fullName:"Samuel Lungareze",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:null},{id:"273005",title:"MSc.",name:"Larissa",surname:"Silva",slug:"larissa-silva",fullName:"Larissa Silva",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:null},{id:"273006",title:"MSc.",name:"Joyce",surname:"Meira",slug:"joyce-meira",fullName:"Joyce Meira",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:null},{id:"273007",title:"Prof.",name:"Marne",surname:"Vasconcellos",slug:"marne-vasconcellos",fullName:"Marne Vasconcellos",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:null}]},generic:{page:{slug:"OA-publishing-fees",title:"Open Access Publishing Fees",intro:"
The Open Access model is applied to all of our publications and is designed to eliminate subscriptions and pay-per-view fees. This approach ensures free, immediate access to full text versions of your research.
As a gold Open Access publisher, an Open Access Publishing Fee is payable on acceptance following peer review of the manuscript. In return, we provide high quality publishing services and exclusive benefits for all contributors. IntechOpen is the trusted publishing partner of over 118,000 international scientists and researchers.
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*These prices do not include Value-Added Tax (VAT). Residents of European Union countries need to add VAT based on the specific rate in their country of residence. Institutions and companies registered as VAT taxable entities in their own EU member state will not pay VAT as long as provision of the VAT registration number is made during the application process. This is made possible by the EU reverse charge method.
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Services included are:
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English language copyediting and proofreading, including the correction of grammatical, spelling, and other common errors
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XML Typesetting and pagination - web (PDF, HTML) and print files preparation
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Discoverability - electronic citation and linking via DOI
\\n\\t
Permanent and unrestricted online access to your work
What isn't covered by the Open Access Publishing Fee?
\\n\\n
If your manuscript:
\\n\\n
\\n\\t
Exceeds 20 pages (for chapters in Edited Volumes), an additional fee of 40 GBP per page will be required
\\n\\t
If a manuscript requires Heavy Editing or Language Polishing, this will incur additional fees.
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\\n\\n
Your Author Service Manager will inform you of any items not covered by the OAPF and provide exact information regarding those additional costs before proceeding.
\\n\\n
Open Access Funding
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To explore funding opportunities and learn more about how you can finance your IntechOpen publication, go to our Open Access Funding page. IntechOpen offers expert assistance to all of its Authors. We can support you in approaching funding bodies and institutions in relation to publishing fees by providing information about compliance with the Open Access policies of your funder or institution. We can also assist with communicating the benefits of Open Access in order to support and strengthen your funding request and provide personal guidance through your application process. You can contact us at oapf@intechopen.com for further details or assistance.
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For Authors who are still unable to obtain funding from their institutions or research funding bodies for individual projects, IntechOpen does offer the possibility of applying for a Waiver to offset some or all processing feed. Details regarding our Waiver Policy can be found here.
\\n\\n
Added Value of Publishing with IntechOpen
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Indexing and listing across major repositories, see details ...
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Long-term archiving
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Visibility on the world's strongest OA platform
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Live Performance Metrics to track readership and the impact of your chapter
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Dissemination and Promotion
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Benefits of Publishing with IntechOpen
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Proven world leader in Open Access book publishing with over 10 years experience
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+146,150 citations in Web of Science databases
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The Open Access Publishing Fee (OAPF) is payable only after your full chapter, monograph or Compacts monograph is accepted for publication.
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OAPF Publishing Options
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1,400 GBP Chapter - Edited Volume
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10,000 GBP Monograph - Long Form
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4,000 GBP Compacts Monograph - Short Form
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*These prices do not include Value-Added Tax (VAT). Residents of European Union countries need to add VAT based on the specific rate in their country of residence. Institutions and companies registered as VAT taxable entities in their own EU member state will not pay VAT as long as provision of the VAT registration number is made during the application process. This is made possible by the EU reverse charge method.
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Services included are:
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An online manuscript tracking system to facilitate your work
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Personal contact and support throughout the publishing process from your dedicated Author Service Manager
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Assurance that your manuscript meets the highest publishing standards
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English language copyediting and proofreading, including the correction of grammatical, spelling, and other common errors
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XML Typesetting and pagination - web (PDF, HTML) and print files preparation
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Permanent and unrestricted online access to your work
What isn't covered by the Open Access Publishing Fee?
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\n\t
Exceeds 20 pages (for chapters in Edited Volumes), an additional fee of 40 GBP per page will be required
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If a manuscript requires Heavy Editing or Language Polishing, this will incur additional fees.
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\n\n
Your Author Service Manager will inform you of any items not covered by the OAPF and provide exact information regarding those additional costs before proceeding.
\n\n
Open Access Funding
\n\n
To explore funding opportunities and learn more about how you can finance your IntechOpen publication, go to our Open Access Funding page. IntechOpen offers expert assistance to all of its Authors. We can support you in approaching funding bodies and institutions in relation to publishing fees by providing information about compliance with the Open Access policies of your funder or institution. We can also assist with communicating the benefits of Open Access in order to support and strengthen your funding request and provide personal guidance through your application process. You can contact us at oapf@intechopen.com for further details or assistance.
\n\n
For Authors who are still unable to obtain funding from their institutions or research funding bodies for individual projects, IntechOpen does offer the possibility of applying for a Waiver to offset some or all processing feed. Details regarding our Waiver Policy can be found here.
\n\n
Added Value of Publishing with IntechOpen
\n\n
Choosing to publish with IntechOpen ensures the following benefits:
\n\n
\n\t
Indexing and listing across major repositories, see details ...
\n\t
Long-term archiving
\n\t
Visibility on the world's strongest OA platform
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Live Performance Metrics to track readership and the impact of your chapter
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Dissemination and Promotion
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Benefits of Publishing with IntechOpen
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Proven world leader in Open Access book publishing with over 10 years experience
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Most competitive prices in the market
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Optimized processes, enabling publication between 8 and 12 months
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Personal support during every step of the publication process
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+146,150 citations in Web of Science databases
\n\t
Currently strongest OA platform with over 130 million downloads
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