Dr. Pletser’s experience includes 30 years of working with the European Space Agency as a Senior Physicist/Engineer and coordinating their parabolic flight campaigns, and he is the Guinness World Record holder for the most number of aircraft flown (12) in parabolas, personally logging more than 7,300 parabolas.
\\n\\n
Seeing the 5,000th book published makes us at the same time proud, happy, humble, and grateful. This is a great opportunity to stop and celebrate what we have done so far, but is also an opportunity to engage even more, grow, and succeed. It wouldn't be possible to get here without the synergy of team members’ hard work and authors and editors who devote time and their expertise into Open Access book publishing with us.
\\n\\n
Over these years, we have gone from pioneering the scientific Open Access book publishing field to being the world’s largest Open Access book publisher. Nonetheless, our vision has remained the same: to meet the challenges of making relevant knowledge available to the worldwide community under the Open Access model.
\\n\\n
We are excited about the present, and we look forward to sharing many more successes in the future.
\\n\\n
Thank you all for being part of the journey. 5,000 times thank you!
\\n\\n
Now with 5,000 titles available Open Access, which one will you read next?
Preparation of Space Experiments edited by international leading expert Dr. Vladimir Pletser, Director of Space Training Operations at Blue Abyss is the 5,000th Open Access book published by IntechOpen and our milestone publication!
\n\n
"This book presents some of the current trends in space microgravity research. The eleven chapters introduce various facets of space research in physical sciences, human physiology and technology developed using the microgravity environment not only to improve our fundamental understanding in these domains but also to adapt this new knowledge for application on earth." says the editor. Listen what else Dr. Pletser has to say...
\n\n\n\n
Dr. Pletser’s experience includes 30 years of working with the European Space Agency as a Senior Physicist/Engineer and coordinating their parabolic flight campaigns, and he is the Guinness World Record holder for the most number of aircraft flown (12) in parabolas, personally logging more than 7,300 parabolas.
\n\n
Seeing the 5,000th book published makes us at the same time proud, happy, humble, and grateful. This is a great opportunity to stop and celebrate what we have done so far, but is also an opportunity to engage even more, grow, and succeed. It wouldn't be possible to get here without the synergy of team members’ hard work and authors and editors who devote time and their expertise into Open Access book publishing with us.
\n\n
Over these years, we have gone from pioneering the scientific Open Access book publishing field to being the world’s largest Open Access book publisher. Nonetheless, our vision has remained the same: to meet the challenges of making relevant knowledge available to the worldwide community under the Open Access model.
\n\n
We are excited about the present, and we look forward to sharing many more successes in the future.
\n\n
Thank you all for being part of the journey. 5,000 times thank you!
\n\n
Now with 5,000 titles available Open Access, which one will you read next?
\n'}],latestNews:[{slug:"stanford-university-identifies-top-2-scientists-over-1-000-are-intechopen-authors-and-editors-20210122",title:"Stanford University Identifies Top 2% Scientists, Over 1,000 are IntechOpen Authors and Editors"},{slug:"intechopen-authors-included-in-the-highly-cited-researchers-list-for-2020-20210121",title:"IntechOpen Authors Included in the Highly Cited Researchers List for 2020"},{slug:"intechopen-maintains-position-as-the-world-s-largest-oa-book-publisher-20201218",title:"IntechOpen Maintains Position as the World’s Largest OA Book Publisher"},{slug:"all-intechopen-books-available-on-perlego-20201215",title:"All IntechOpen Books Available on Perlego"},{slug:"oiv-awards-recognizes-intechopen-s-editors-20201127",title:"OIV Awards Recognizes IntechOpen's Editors"},{slug:"intechopen-joins-crossref-s-initiative-for-open-abstracts-i4oa-to-boost-the-discovery-of-research-20201005",title:"IntechOpen joins Crossref's Initiative for Open Abstracts (I4OA) to Boost the Discovery of Research"},{slug:"intechopen-hits-milestone-5-000-open-access-books-published-20200908",title:"IntechOpen hits milestone: 5,000 Open Access books published!"},{slug:"intechopen-books-hosted-on-the-mathworks-book-program-20200819",title:"IntechOpen Books Hosted on the MathWorks Book Program"}]},book:{item:{type:"book",id:"6346",leadTitle:null,fullTitle:"Machine Learning - Advanced Techniques and Emerging Applications",title:"Machine Learning",subtitle:"Advanced Techniques and Emerging Applications",reviewType:"peer-reviewed",abstract:"The volume of data that is generated, stored, and communicated across different industrial sections, business units, and scientific research communities has been rapidly expanding. 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\n
1. Preamble
\n
Uncertainty is beyond awareness our indisputable decision-maker. A meeting announced to start at 12:00 may implicitly be understood to start in the time interval 12:00–12:01. Hence, we should have arrived at 12:01, at the latest. Alternatively, the interval could be 12:00–12:05. The communicated uncertainty of the start of the meeting is clearly ambiguous: accustomed to analog clocks discretized in 5-minute intervals, the latter is plausible, but used to digital clocks the former makes more sense. A meeting scheduled at 12, however, means something quite different to most of us. In that case, it can start as late as 12:30. The invisible practice in everyday life is to communicate uncertainty through a vaguely perceived precision, suggesting random variability. It is more often than not confused with accuracy, or systematic deviation (see Figure 1).
\n
Figure 1.
Illustrations [1] of precision (left) and accuracy (middle) of four samples (●), and corresponding schematic probability density for the population of all possible outcomes (right), often utilized in uncertainty quantification.
\n
Results repeated within ±1% variation tell nothing about the range of possible errors or uncertainty. An entirely deterministic algorithm has perfect precision. This is normally the situation of scientific modeling, before uncertainties are considered. The precision usually thought of as random variability for any given set up is often re-interpreted as the total variability between known different situations. That is a dubious strategy to assign numbers of uncertainty. Without extensive consideration, it is generally impossible to assess whether or not the considered history is representative for the current problem. For instance, errors in modeling of fluid flow velocities and electromagnetic fields at nearly singular points in space or time, such as sharp corners, or deficiencies in describing collective phenomena like resonances, are usually far too complex to be understood by studying examples only. An extensive analysis based on a large or even infinite set of hypothetical variations is required. The widely practiced intuitive assessment of uncertainty exemplified above, based on experience and communicated with precision, jeopardizes decision-making: uncertainties of this kind are subjective and encourage different interpretations. Invalid uncertainty assessment is also a major cause of false rejection of modeling as a general tool, depriving us all means for making educated guesses through scientific model prediction of important matters, like future weather conditions and risk of major nuclear power accidents.
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1.1. The goal
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Uncertainty quantification targets objective association of quantitative traceable numbers representing uncertainty to modeling, simulation, and calculation results. By applying a well-documented and widely accepted method with known performance, for the last 20–30 years of so, such a methodology has been established and widely recognized for measurement models, to the extent a quantitative assessment of uncertainty now almost always is required for measurement apparatus. It is not yet so for scientific modeling, as the advanced computations in modern science and technology generally are far more difficult to analyze than measurement models. The uncertainty should predict the range of possible modeling errors, but without exaggeration. If so, modeling results and observations are consistent, which means no more than they are not contradictory. Expressed in terms of conventional mathematical statistics developed by Fisher [2] and Popper [3], the hypothesis that the model accurately reproduces observations cannot be falsified. These perspectives, outlined in the early 20th century while studying, e.g., crop growth in agriculture and demography, still hold well for modern uncertainty quantification addressing complex applications, such as nuclear power generation, fatigue testing, etc. Mathematical statistics is indeed the genesis of most uncertainty quantification approaches and techniques utilized today.
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The mere evaluation of uncertainty is, however, not automatically of any value. Unwarranted assumptions of uncertainties entering the evaluation are deceiving. Respecting what is not known is usually far more important than accurately describing what is known. Lack of knowledge tends to increase the uncertainty and often leads to ambiguity, an important ingredient in qualitative science. In quantitative science addressed here though, any lack of well-defined information is normally defied by bold simplifying assumptions, simply because current methodologies require complete knowledge. Closing the gap of ambiguity in this way reflects willful ignorance [4]. Therefore, it is important to consider alternative hypotheses of uncertainty. For instance, parameter correlations are very rarely known, but nevertheless have a major influence on the evaluated uncertainty. In this respect, it is important to view the model with all of its parameters as one composite unit. The hypothesis touched upon above, stating that the model reproduces observations, implies that propagated parameter errors combine coherently, according to the behavior of the deterministic model equations. Correlations are thus essential components of uncertainty, as they may attenuate or amplify contributions from different uncertain parameters by means of destructive or constructive interference. If such effects are not taken into account, uncertainty quantification may evolve into con artistry.
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1.2. The preparation
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In many respects and for good reasons, methods of uncertainty quantification (UQ) [5] are in their infancy. The need of viable and credible UQ methods is rapidly increasing, with higher utilization of advanced computations. The excess computational power at disposal for UQ is unfortunately not increasing nearly as rapidly as the total resources. The reason is simple. Most computational models are discretized in space and time, truncated, or simplified by neglecting minor but complicated contributions. Such approximations cannot be traced to lack of knowledge or ability, but are often required to enable computation. As soon as the resources increase, eliminating these model reductions as much as possible is most logical and desirable. Weather forecasting [6] illustrates the principle. Proper propagation of disturbances requires comparable resolution in space and time. Reducing the unit cell of analysis from 10 km × 10 km down to 5 km × 5 km to render more detailed forecasts increases the computational load no less than 24 = 16 times. Even so, the unit cell will still be larger than desired. Additional resources will therefore mainly be spent on improvements of the deterministic model formulation in the future, leaving a relatively small fraction to be spent on improved UQ. However, with model samples that can be evaluated independently in different computer kernels, the challenge of improved UQ by additional sampling translates into an economical issue. Then it does not compete with the advancement of computer architecture required to solve the dependent deterministic equations.
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UQ combines several advanced mathematical disciplines and can be applied to a plethora of disparate applications not only in technology and science, but also in econometrics and for risk assessment. This makes the subject exceedingly difficult to master, but also hard to understand and learn by studying examples. Physical modeling usually provides the basis for setting up the underlying deterministic model. Major simplifications as well as coarse assumptions are common. For instance, Navier-Stokes equations of fluid flow may require both physical and mathematical idealizations like continuous media and differentiability, as well as neglect of higher-order turbulence contributions. Already at this first stage, contributions to uncertainty are building up. Finite element methods (FEMs) discretize physical fields in space and time caused by fixed (solids) or moving (fluids) matter. Signal processing techniques such as temporal sampling, digital filtering, and state space formulations for Kalman filtering and model prediction control convert infinite-dimensional continuous physical differential models to finite systems of difference equations, suitable for computers. Numerical methods then provide the means for solving these equations, with maximum efficiency and minimum error. Preferably with known error estimates, which may be re-phrased in terms of uncertainty in the proceeding UQ. Knowledge of computer science is needed for efficient programming and maintaining numerical precision throughout the calculation, but also for managing large complex software modules. The studied system may also exhibit critical properties. The chaotic nature of weather forecast models is one example. More than 50 years ago, Lorenz assessed an absolute upper prediction horizon of about two weeks [6]. Explained by “the butterfly effect” [6, p. 206], this limit is still believed to be accurate: Even the slightest possible change in initial conditions may render a monumental change in the forecast after some time, which clearly is a major complication for credible UQ. Understanding these preparatory stages is crucial, as they accommodate many sources of uncertainty.
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1.3. Overview
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Uncertainty quantification can now be addressed. Statistics of all kinds of uncertain quantities are then propagated in two possible directions, as explained in Figure 2 (adapted from Ref. [7]).
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Figure 2.
Uncertainty quantification (UQ) and model calibration, or inverse UQ. Identifying or matching the model against identification data often requires simplified surrogate models. The model should be checked or validated before it is utilized for prediction comprising a best estimate and its uncertainty.
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Fundamentally, statistics of populations rather than finite samples drawn from them are propagated, which avoids sampling variance, the principal complication addressed in mathematical statistics with statistical inference [2]. There are thus two generic types of uncertainty1 to some extent corresponding to accuracy and precision, respectively:
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Epistemic uncertainty, i.e., unknown and unpredictable systematic but repeatable errors due to lack of knowledge and imperfect simplifications.
Aleatoric uncertainty, i.e., non-repeatable errors of a statistical nature. Typically, the variable outcome of finite random draws (sampling variance).
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Applications of UQ are typically concerned with epistemic uncertainty due to imperfect modeling, calculation and signal processing, finite discretization (FEM) as well as inaccurate boundary and initial conditions, etc. Mathematical statistics, on the other hand, focuses on aleatoric uncertainty due to finite statistical sampling. In the latter case, modeling has an entirely different character. The quantities of interest are usually not a result of a complex model implemented in a large computer program but rather directly observable, like mean and variance of some measure of performance, frequency, length, or response time. In that case, the uncertainty due to the variability of small observation sets presumably dominates over model errors.
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1.4. Some common tools
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Bayesian approaches [8] make the difference between epistemic and aleatoric uncertainties almost invisible. Generalizing observed frequencies of observation to also include other kinds of knowledge requires a shift of perspective from experimental testing, to the observer and his/her degree of belief. Since our belief rarely is complete or totally absent, this still has the appearance of probability, but is conceptually different. Nevertheless, belief is the enabler for unifying epistemic and aleatoric uncertainty consistently within the same framework of UQ. Our belief often refers to a model’s track record, or how it has performed in different situations over a long period of time. That may be difficult to assess quantitatively, but could in principle be made with multimodel calibration. Only independent data sets/model results must be included, as dependencies will underestimate the uncertainty severely. Worth emphasizing is also that any piece of prior information available before the uncertainty is quantified must reflect some kind of knowledge or experience. Any reduction of uncertainty due to a guessed prior is purely hypothetical and deceptive.
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Random sampling reduces the difference between the practices of UQ and mathematical statistics even further by introducing sampling variance of finite random ensembles, making it a primary target to control in both fields. The basic motivation for random sampling is its simplicity, while a severe drawback is the added sampling variance. Much larger ensembles than the computational power allows for may be required. The obvious work-around is to substitute the full model with a much less demanding approximate surrogate model, which allows for excessive sampling. The surrogate is often affine, i.e., linear in uncertain parameters and obtained with traditional linear regression. Aleatoric sampling errors are then exchanged with presumably smaller epistemic ones. Alternatively, the sampling variance may be reduced by imposing deterministic components in the random sampling methodology, like stratified sampling, perhaps combined with latin-hypercube [9] or orthogonal sampling exclusion rules. It is indeed possible to extend these amendments of determinism into entirely deterministic sampling, as in the unscented Kalman filter [10]. The sampling variance is then completely(!) exchanged with sampling errors due to imperfections of the reproducible sampling rule [11]. Just knowing the modeling error is entirely reproducible is of great value when differential changes are of primary interest, as in product development.
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Model calibration or inverse UQ is an inverse problem usually requiring an implicit solution. The high complexity of the full model normally prohibits ubiquitous trial-and-error search and steepest descent methods like the Newton-Raphson method [12], to minimize the model prediction error. Just as for excessive random sampling, surrogate models are often utilized. In this case though, the iterative character of many inverse solutions requires even higher computational efficiency. The maximum likelihood method is perhaps the most common approach to inverse propagation of uncertainty. Virtually all methods require complete statistical information. That is a major issue since available information normally is incomplete. Just like Bayesian estimation can be invalidated by faulty prior distributions, inappropriate assumptions of unknown calibration data statistics may invest far too much credibility in the calibrated model, making it likely to fail any validation test. What is particularly detrimental is the ubiquitous assumptions of uncorrelated calibration errors. Allowance of incomplete statistical information in model calibration is therefore one of the most urgent tasks to address in future development of model calibration, to remedy overconfident faulty model predictions.
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\n
1. Introduction
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In the study of heat transfer, a fin may be a solid or porous and stationary or moving that extends from an attached body to rapidly cool off heat of that surface. Cooling fins find application in a large real world phenomena particularly in engineering devices. Fins increase the surface area of heat transfer particularly for cooling of hot bodies. These come in different shapes, geometries and profiles. These differences provide variety of effectiveness and efficiencies. The literature with regard to the study of heat transfer in fins is well documented (see e.g. [1]). The solutions either exact, numerical or approximate analytical continue to be of immerse interest and this is due to continued use of fins in engineering devices.
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Much attention has been given to linear one dimensional models [2, 3, 4] whereby Homotopy Analysis Method (HAM) was used to determine series solutions for heat transfer in straight fins of trapezoidal and rectangular profiles given temperature dependent thermal properties; nonlinear one dimensional models [5] wherein preliminary group classification methods were utilised to contract invariant (symmetry) solutions; heat transfer in linear two dimensional trapezoidal fins [6]; heat transfer in two dimensional straight nonlinear fins were considered [7] wherein Lie point symmetries and other standard methods were invoked and recently nonlinear three dimensional models [8] were considered wherein three dimensional Differential Transform Methods (DTM) were employed to construct approximate analytical solutions. The dependence of thermal properties on the temperature renders the equations highly nonlinear. The non-linearity brings an added complication or difficulty in the construction of solutions and particularly exact solutions.
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Few exact solutions are recorded in the literature, for example for one dimensional problems [2, 3, 4, 5, 9, 10, 11, 12, 13, 14, 15], two dimensions [6, 7, 16, 17]. An attempt to construct exact solutions for the three dimensional problems is found in [8], however these were general solutions. For this reason, either approximate analytical or numerical solutions are sought. However, the accuracy of numerical schemes is obtained by comparison with he exact solutions.
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This chapter summaries the work of Moitsheki and collaborators in the area of heat transfer through fin. In their work, they employed Lie symmetry methods to construct exact solutions. These methods include, the preliminary group classification, the Lie point symmetries, conservation laws and associate Lie point symmetries, non-classical symmetry methods and recently non classical potential symmetries. It appeared that most of the constructed exact solutions do not satisfy the prescribed boundary conditions. The idea then becomes, start with the simple model that satisfy the boundary conditions and compare it with the approximate solutions to establish confidence in the approximate methods, then extend analysis to problems that are difficult to solve exactly.
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We acknowledge that some scholars employed many other approximate methods to solve boundary value problems (BVPs); for example the Homotopy Analysis Method [18], Collocation Methods (CM) [19], Homotopy Perturbation Methods (HPM) [20], Haar Wavelet Collation Methods (HWCM) [21], Collocation Spectral Methods (CSM) [22], modified Homotopy Analysis Method (mHAM) [23], Spectral Homotopy Analysis Methods (SHAM) and the Optimal Homotopy Analysis Methods [24]. In this chapter we restrict discussions to Lie symmetry methods for exact solutions, and DTM and VIM for approximate analytical methods.
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2. Mathematical descriptions
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Mathematical descriptions represent some physical phenomena in terms of deterministic models given in terms of partial differential equations (PDEs). These differential equations become non-linear when heat transfer coefficient and thermal conductivity depend on the temperature (see e.g. [5]). This non-linearity was introduced as a significant modifications of the usually assumed models see e.g. [2].
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In this chapter we present a few models for various heat transfer phenomena.
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2.1 2 + 1 dimensional transient state models
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Mathematical modelling for heat transfer in fins may be three dimensional models.
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2.1.1 Cylindrical pin fins
\n
We consider a two-dimensional pin fin with length \n\nL\n\n and radius \n\nR\n\n. The fin is attached to a base surface of temperature \n\n\nT\nb\n\n\n and extended into the fluid of temperature \n\n\nT\ns\n\n\n. The tip of the fin is insulated (i.e., heat transfer at the tip is negligibly small). The fin is measured from the tip to the base. A schematic representation of a pin fin is given in Figure 1. We assume that the heat transfer coefficient along the fin is nonuniform and temperature dependent and that the internal heat source or sink is neglected. Furthermore, the temperature-dependent thermal conductivity is assumed to be the same in both radial and axial directions. The model describing the heat transfer in pin fins is given by the BVP (see e.g. [17])
where the non-dimensional quantities \n\nE\n=\n\nL\nδ\n\n\n, and \n\nBi\n=\n\n\n\nH\nb\n\nδ\n\n\nK\na\n\n\n\n, are the fin extension factor and the Biot number respectively. Also,
where \n\nτ\n\n, \n\nz\n\n, \n\nr\n\n, \n\nk\n\n, \n\nh\n\n and \n\nθ\n\n are all dimensionless variables. \n\n\nK\na\n\n\n and \n\n\nH\nb\n\n\n are the ambient thermal conductivity and the fin base heat transfer coefficient respectively.
\n
Notice that other terms may be added, for example internal heat generation (source term) and fin profile.
\n
\n
\n
2.1.2 Rectangular straight fins
\n
Following the similar pattern, in dimensionless variables we have (see e.g. [8])
In this section we consider the two dimensional steady state models. The symmetry analysis of these models have proven to be challenging. In some cases standard method such as separations of variables have been employed to determine exact solutions.
\n
\n
2.2.1 Cylindrical pin fins
\n
For steady state problem, the heat transfer is independent of the time variable. For example, the time derivative in Eq. (2) vanish (see e.g. [16]).
\n
\n
\n
2.2.2 Rectangular straight fins
\n
For steady state problem, the heat transfer is independent of the time variable. For example, the time derivative in Eq. (3) is zero (see e.g. [7]).
\n
\n
\n
\n
2.3 1 + 1 dimensional transient model for straight fins
\n
\n
2.3.1 Solid stationary fins
\n
For solid stationary straight fins the model is given by (see e.g. [25, 26])
2.5 One-dimensional steady state model for straight fins
\n
Considering heat transfer in a one dimensional longitudinal fin of cross area \n\n\nA\nc\n\n\n with various profiles. The perimeter of the fin is denoted by \n\nP\n\n and length by \n\nL\n.\n\n The fin is attached to a fixed prime surface of temperature \n\n\nT\nb\n\n\n and extends to the fluid of temperature \n\n\nT\n∞\n\n.\n\n in non-dimensional variables, one obtains
2.6 One-dimensional steady state model for radial fins
\n
Considering heat transfer in a one dimensional stationary radial fin of cross area \n\n\nA\nc\n\n\n with various profiles. The perimeter of the fin is denoted by \n\nP\n\n and length by \n\n\nLr\nb\n\n−\n\nr\nt\n\n\n The fin is attached to a fixed prime surface of temperature \n\n\nT\nb\n\n\n and extends to the fluid of temperature \n\n\nT\n∞\n\n.\n\n One may assume that at the tip of the fin \n\n\nr\nt\n\n=\n0\n.\n\n In non-dimensional variables, one obtains
In this subsection we provide a brief theory of Lie point symmetries. This discussion and further account can be found in the book of Bluman and Anco [31].
\n
\n
3.1.1 \n\nm\n\n dependent and \n\nn\n\n independent variables
\n
\n\n\nm\n\n dependent variables \n\nu\n=\n\n\nu\n1\n\n\nu\n2\n\n…\n\nu\nm\n\n\n\n and \n\nn\n\n independent variables \n\nx\n=\n\n\nx\n1\n\n\nx\n2\n\n…\n\nx\nn\n\n\n\n, \n\nu\n=\nu\n\nx\n\n\n with \n\nm\n≥\n2\n,\n\n arise in studying systems of differential equations. We consider extended transformations from \n\n\nx\nu\n\n−\n\nspace to \n\n\nx\nu\n\nu\n\n1\n\n\n\nu\n\n2\n\n\n…\n\nu\n\nk\n\n\n\n−\n\n space. Here \n\n\nu\n\nk\n\n\n\n denotes the components of all \n\nk\n\nth-order partial derivatives of \n\nu\n\n wrt \n\nx\n.\n\n.
\n
definitionTotal derivative. The total differentiation operator wrt \n\n\nx\ni\n\n\n is defined by
Theorem 1.2 A differential function \n\nF\n\nx\nu\n\nu\n1\n\n…\n\nu\n\np\n\n\n\n\n\n\n\np\n≥\n0\n,\n\n is a \n\np\n\nth-order differential invariant of a group \n\nG\n\n if
Theorem 1.3 A differential function \n\nF\n\nx\nu\n\nu\n1\n\n…\n\nu\n\np\n\n\n\n\n\n\n\np\n≥\n0\n,\n\n is a \n\np\n\nth-order differential invariant of a group \n\nG\n\n if
\n
\n\n\nX\n\np\n\n\nF\n=\n0\n,\n\n
\n
where \n\n\nX\n\np\n\n\n\n is the \n\np\n\nth prolongation of \n\nX\n.\n\n.
The upper and lower case letters are for the transformed and the original functions respectively. The transformed function is also referred to as the T-function, the differential inverse transform is given by
It can easily be deduced that the substitution of (18) into (19) gives the Taylor series expansion of the function \n\nf\n\n\nx\n1\n\n\nx\n2\n\n…\n\nx\np\n\n\n\n about the point \n\n\n\nx\n1\n\n\nx\n2\n\n…\n\nx\np\n\n\n=\n\n.
For real world applications the function \n\nf\n\n\nx\n1\n\n\nx\n2\n\n…\n\nx\np\n\n\n\n is given in terms of a finite series for some \n\nq\n,\nr\n,\ns\n∈\nℤ\n\n. Then (19)becomes
We now give some important operations and theorems performed in the \n\np\n\n-dimensional DTM in Table 1. Those have been derived using the definition in (18) together with previously obtained results [32].
\n
\n
\n
\n\n
\n
Original function \n\nf\n\n\nx\n1\n\n\nx\n2\n\n…\n\nx\np\n\n\n\n\n
where \n\nL\n\n and \n\nN\n\n are linear and nonlinear operators, respectively, and \n\ng\n\nx\n\n\n is the source inhomogeneous term. He [33], proposed the VIM where a correctional functional for Eq. (22) can be written as
where \n\nλ\n\n is the general Lagrange multiplier, which can be be identified optimally via the variation theory, and \n\n\n\nθ\n˜\n\nn\n\n\n is a restricted variation, which means \n\nδ\n\n\nθ\n˜\n\nn\n\n=\n0\n\n [34]. The Lagrange multiplier can be a constant or a function depending on the order of the deferential equation under consideration. The VIM should be employed by following two essential steps. First we determine the Lagrange multiplier by considering the following second order differential equation,
The extreme condition of \n\n\nθ\n\nj\n+\n1\n\n\n\n requires that \n\n\nδθ\n\nj\n+\n1\n\n\n=\n0\n\n. Equating both sides of Eq. (30) to \n\n0\n\n, yields the following stationary conditions
The iteration formula Eq. (36), without restricted variation, should be used for the determination of the successive approximations \n\n\nθ\n\nj\n+\n1\n\n\n\nx\n\n,\nj\n⩾\n0\n\n, of the solution \n\nθ\n\nx\n\n\n. Consequently, the solution is given by
In this section we demonstrate the challenge in the construction of exact solution for heat transfer in pin fin. Also, we consider the work in [5].
\n
\n
4.1 Some exact solutions
\n
\n
4.1.1 Example 1
\n
Given the power law thermal conductivity in heat transfer through pin fins, that is in Eq. (3)\n\n\nk\n\nθ\n\n=\n\nθ\nn\n\n\n. The model admits four finite symmetry generators. Amongst the others, the two dimensional Lie subalgebra is given by
and hence we start the double reduction first with \n\n\nX\n1\n\n\n which implies \n\n\nτ\nr\nθ\n\n\n are invariants and leads to a steady state problem. Hence writing \n\nθ\n=\nF\n\nτ\nr\n\n\n and substitute in the original equation, one obtains
The difficulty for group-invariant solutions is the satisfaction of the imposed or prescribed boundary conditions. This has been seen in two dimensional steady state problems [7, 16], and 1 + 1 D transient problems [25]. Perhaps the most successful attempt in in [26]. For nonlinear steady state problems, some transformation such as Kirchoff [7, 16], may linearise the two dimensional problems which then becomes easier to solve using standard methods. Linearisation of nonlinear steady state one dimensional problems is possible when thermal conductivity is a differential consequence of heat transfer coefficient [5].
\n
\n
\n
4.1.2 Example 2
\n
In [5], preliminary group classification is invoked to determine the thermal conductivity which lead to exact solutions. It turned out that given a power law heat transfer coefficient, thermal conductivity also takes the power law form. Given Eq. (12) with both \n\nk\n\nθ\n\n\n and \n\nh\n\nθ\n\n\n given by \n\n\nθ\nn\n\n\n then one obtains the solution
The expressions for fin efficiency and effectiveness can be explicit in this case. Furthermore, this solution led to the benchmarking of the approximate analytical solutions [35]. With established confidence in approximate methods, then one may solve other problems that are challenging to solve exactly.
\n
\n
\n
\n
4.2 Some approximate solutions
\n
\n
4.2.1 Three dimensional DTM
\n
In this subsection we consider heat transfer in a cylindrical pin fin. We consider thermal conductivity given as a linear function of temperature \n\n1\n+\nβθ\n\n and a power law heat transfer coefficient. The three dimensional DTM solution of Eq. (2) is given by
To plot a three dimensional figure for this solution one may fix temperature, say at \n\nτ\n=\n0.4\n\n The results are shown in Figure 2.
\n
Figure 2.
Approximate analytical solutions for a two-dimensional cylindrical spine fin with a constant thermal conductivity (\n\nβ\n=\n0\n\n) for \n\nτ\n=\n0.4\n\n. The parameters are set such that \n\nE\n=\n2\n\n, \n\nBi\n=\n0.2\n\n, and \n\nm\n=\n3\n\n. (see also, [8]).
\n
\n
\n
4.2.2 Two dimensional DTM
\n
The two dimensional DTM solution for a steady heat transfer through the cylindrical fin is given by
Approximate analytical solutions for a two-dimensional cylindrical spine fin with a constant thermal conductivity (\n\nβ\n=\n0\n\n) for \n\nτ\n=\n0.4\n\n. The parameters are set such that \n\nE\n=\n2\n\n, \n\nBi\n=\n0.2\n\n, and \n\nm\n=\n3\n\n. (see also, [8]).
\n
\n
\n
4.2.3 Comparison of one dimensional exact, DTM and VIM solutions
\n
Here the solutions for the one dimensional heat transfer problems are compared, namely the exact solution given in Eq. (38). The VIM solutions is given by
A temperature distribution in a rectangular fin for varying values of \n\nn\n\n, \n\nM\n=\n1.7\n\n.(see also [36]).
\n
\n
\n
\n
\n
5. Outlook and some concluding remarks
\n
The interest in heat transfer through fins will continue unabated. This is brought about by applications of fins in engineering appliances. The solutions to the problems give insight into effectiveness and efficiency of different fins. In this chapter we provided a summary of some of the work in recent times. In particular, we reviewed the exact and approximate analytical solutions. We demonstrated that although the models describing hear transfer seem to be simple, they are in fact challenging to solve exactly. When constructed, the exact solutions are used as benchmarks for the approximate solutions. It appears that some models including contracting or expanding have attracted some attention. The analysis of these problems provide insight into heat transfer phenomena and assist in the design of fins. The challenge is the construction of exact solutions, however one may construct approximate analytical solutions. The problems discussed here are not exhaustive.
\n
\n
Conflict of interest
The authors declare no conflict of interest.
\n
Nomenclature
\n\n\n\n\n\n\nA\nc\n\n\n\n\n\n
Cross-sectional area
\n\n\n\n\n\n\nBi\n\n\n\n\n
Biot number
\n\n\n\n\n\n\nE\n\n\n\n\n
Aspect ratio
\n\n\n\n\n\n\nh\n\n\n\n\n
dimensionless heat transfer coefficient
\n\n\n\n\n\n\nH\n\n\n\n\n
Heat transfer coefficient
\n\n\n\n\n\n\n\nH\nb\n\n\n\n\n\n
Heat transfer coefficient at the base of the fin
\n\n\n\n\n\n\nh\n\n\n\n\n
Dimensionless thermal conductivity
\n\n\n\n\n\n\n\nK\na\n\n\n\n\n\n
Thermal conductivity of the fluid
\n\n\n\n\n\n\nK\n\n\n\n\n
Thermal conductivity of the fin
\n\n\n\n\n\n\nL\n\n\n\n\n
Length of the fin
\n\n\n\n\n\n\nR\n\n\n\n\n
Radius
\n\n\n\n\n\n\n\nR\na\n\n\n\n\n\n
Radius
\n\n\n\n\n\n\nt\n\n\n\n\n
time
\n\n\n\n\n\n\n\nT\nb\n\n\n\n\n\n
Base temperature
\n\n\n\n\n\n\n\nT\ns\n\n\n\n\n\n
Fluid temperature
\n\n\n\n\n\n\nx\n\n\n\n\n
Dimensionless fin length
\n\n\n\n\n\n\nX\n\n\n\n\n
Fin length
\n\n\n\n\n\n\ny\n\n\n\n\n
Dimensionless fin length
\n\n\n\n\n\n\nY\n\n\n\n\n
Fin length
\n\n\n\n\n\n\nZ\n\n\n\n\n
Length of a cylindrical pin fin. Greek letters
\n\n\n\n\n\n\nτ\n\n\n\n\n
Dimentionless time
\n\n\n\n\n\n\nθ\n\n\n\n\n
dimensionless temperature
\n\n\n\n
\n',keywords:"exact solutions, approximate solutions, lie symmetry methods, approximate methods, heat transfer, fins",chapterPDFUrl:"https://cdn.intechopen.com/pdfs/74742.pdf",chapterXML:"https://mts.intechopen.com/source/xml/74742.xml",downloadPdfUrl:"/chapter/pdf-download/74742",previewPdfUrl:"/chapter/pdf-preview/74742",totalDownloads:20,totalViews:0,totalCrossrefCites:0,dateSubmitted:"July 22nd 2020",dateReviewed:"December 15th 2020",datePrePublished:"January 22nd 2021",datePublished:null,dateFinished:"January 11th 2021",readingETA:"0",abstract:"In this chapter we provide the review and a narrative of some obtained results for steady and transient heat transfer though extended surfaces (fins). A particular attention is given to exact and approximate analytical solutions of models describing heat transfer under various conditions, for example, when thermal conductivity and heat transfer are temperature dependent. We also consider fins of different profiles and shapes. The dependence of thermal properties render the considered models nonlinear, and this adds a complication and difficulty to solve these model exactly. However, the nonlinear problems are more realistic and physically sound. The approximate analytical solutions give insight into heat transfer in fins and as such assist in the designs for better efficiencies and effectiveness.",reviewType:"peer-reviewed",bibtexUrl:"/chapter/bibtex/74742",risUrl:"/chapter/ris/74742",signatures:"Raseelo Joel Moitsheki, Partner Luyanda Ndlovu and Basetsana Pauline Ntsime",book:{id:"10299",title:"Heat Transfer - Design, Experimentation and Applications",subtitle:null,fullTitle:"Heat Transfer - Design, Experimentation and Applications",slug:null,publishedDate:null,bookSignature:"Dr. Miguel Araiz",coverURL:"https://cdn.intechopen.com/books/images_new/10299.jpg",licenceType:"CC BY 3.0",editedByType:null,editors:[{id:"230662",title:"Dr.",name:"Miguel",middleName:null,surname:"Araiz",slug:"miguel-araiz",fullName:"Miguel Araiz"}],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. Mathematical descriptions",level:"1"},{id:"sec_2_2",title:"2.1 2 + 1 dimensional transient state models",level:"2"},{id:"sec_2_3",title:"2.1.1 Cylindrical pin fins",level:"3"},{id:"sec_3_3",title:"2.1.2 Rectangular straight fins",level:"3"},{id:"sec_5_2",title:"2.2 Two-dimensional steady state models",level:"2"},{id:"sec_5_3",title:"2.2.1 Cylindrical pin fins",level:"3"},{id:"sec_6_3",title:"2.2.2 Rectangular straight fins",level:"3"},{id:"sec_8_2",title:"2.3 1 + 1 dimensional transient model for straight fins",level:"2"},{id:"sec_8_3",title:"2.3.1 Solid stationary fins",level:"3"},{id:"sec_9_3",title:"2.3.2 Solid moving fins",level:"3"},{id:"sec_10_3",title:"2.3.3 Porous stationary fins",level:"3"},{id:"sec_11_3",title:"2.3.4 Porous moving fins",level:"3"},{id:"sec_13_2",title:"2.4 1 + 1 dimensional transient model for radial fins",level:"2"},{id:"sec_13_3",title:"2.4.1 Solid stationary fins",level:"3"},{id:"sec_14_3",title:"2.4.2 Solid moving fins",level:"3"},{id:"sec_15_3",title:"2.4.3 Porous stationary fins",level:"3"},{id:"sec_16_3",title:"2.4.4 Porous moving fins",level:"3"},{id:"sec_18_2",title:"2.5 One-dimensional steady state model for straight fins",level:"2"},{id:"sec_19_2",title:"2.6 One-dimensional steady state model for radial fins",level:"2"},{id:"sec_21",title:"3. Methods of solutions",level:"1"},{id:"sec_21_2",title:"3.1 Brief account on lie symmetry methods",level:"2"},{id:"sec_21_3",title:"3.1.1 \n\nm\n\n dependent and \n\nn\n\n independent variables",level:"3"},{id:"sec_23_2",title:"3.2 Approximate methods",level:"2"},{id:"sec_23_3",title:"Table 1.",level:"3"},{id:"sec_24_3",title:"3.2.2 Variational iteration methods",level:"3"},{id:"sec_27",title:"4. Survey of some solutions",level:"1"},{id:"sec_27_2",title:"4.1 Some exact solutions",level:"2"},{id:"sec_27_3",title:"4.1.1 Example 1",level:"3"},{id:"sec_28_3",title:"4.1.2 Example 2",level:"3"},{id:"sec_30_2",title:"4.2 Some approximate solutions",level:"2"},{id:"sec_30_3",title:"4.2.1 Three dimensional DTM",level:"3"},{id:"sec_31_3",title:"4.2.2 Two dimensional DTM",level:"3"},{id:"sec_32_3",title:"4.2.3 Comparison of one dimensional exact, DTM and VIM solutions",level:"3"},{id:"sec_35",title:"5. Outlook and some concluding remarks",level:"1"},{id:"sec_39",title:"Conflict of interest",level:"1"},{id:"sec_36",title:"Nomenclature",level:"1"}],chapterReferences:[{id:"B1",body:'\nA.D. Kraus, A. Aziz and J. Welte, Extended Surface Heat Transfer, Wiley, New York, 2001. https://onlinelibrary.wiley.com/doi/pdf/10.1002/9780470172582.fmatter [Accessed: 15 September 2020]\n'},{id:"B2",body:'\nF. Khani and A. Aziz, Thermal Analysis of a Longitudinal trapezoidal fin with temperature-dependent thermal conductivity and heat transfer coefficient, Communications in Nonlinear Science and Numerical Simulation. 2010;15(3): 590–601. https://www.sciencedirect.com/science/article/abs/pii/S1007570409002202 [Accessed: 15 September 2020]\n'},{id:"B3",body:'\nF. Khani, M. A. Raji, and H. H. 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Sobamowo, Analysis of heat transfer in porous fin with temperature-dependent thermal conductivity and internal heat generation using Chebychev Spectral Collocation Method, Journal of Computational Applied Mechanics. 2017;48(2): 271–284. https://jcamech.ut.ac.ir/article_63384.html [Accessed: 15 September 2020]\n'},{id:"B20",body:'\nH.A. Hoshyar, I. Rahimipetroudi, D.D. Ganji, A.R. Majidian, Thermal performance of porous fins with temperature-dependent heat generation via the homotopy perturbation method and collocation method. Journal of Applied Mathematics and Computational Mechanics. 2015;14(4):53–65 http://amcm.pcz.pl/?id=view volume=14 issue=4 article=6 [Accessed: 15 September 2020]\n'},{id:"B21",body:'\nG. Oguntala and R. Abd-Alhameed, Haar Wavelet Collocation Method for thermal analysis of porous fin with temperature-dependent thermal conductivity and internal heat generation, Journal of Applied and Computational Mechanics. 2017;3(3): 185–191. https://bradscholars.brad.ac.uk/handle/10454/12045 [Accessed: 15 September 2020]\n'},{id:"B22",body:'\nY. Sun, J. Ma, B. Li and Z. Guo, Prediction of nonlinear heat transfer in a convective-radiative fin with temperature dependent properties by the collocation spectral method. Numerical Heat Transfer Part B Fundamentals. 2015;69(1):68–83 https://www.tandfonline.com/doi/abs/10.1080/10407782.2015.1081043?journalCode=unhb20 [Accessed: 15 September 2020]\n'},{id:"B23",body:'\nS.S. Motsa, P. Sibanda and S. Shateyi, A new spectral-homotopy analysis method for solving a nonlinear second order BVP, Communications in Nonlinear Science and Numerical Simulation. 2010; 15(9): 2293–2302 https://www.sciencedirect.com/science/article/abs/pii/S1007570409004754 [Accessed: 07 October 2020]\n'},{id:"B24",body:'\nM. Esmaeilpour and D.D. Ganji, Solution of the Jeffery–Hamel flow problem by optimal homotopy asymptotic method. Computers and Mathematics with Applications. 2010;59(11), 3405–3411. https://www.sciencedirect.com/science/article/pii/S0898122110002002 [Accessed:07 October 2020]\n'},{id:"B25",body:'\nR.J. Moitsheki and C. Harley, Transient heat transfer in longitudinal fins of various profiles with temperature-dependent thermal conductivity and heat transfer coefficient, Pramana Journal of Physics. 2011;77: 519–532. https://link.springer.com/article/10.1007/s12043-011-0172-6[Accessed: 31 September 2020]\n'},{id:"B26",body:'\nM.D. Mhlongo, R.J. Moitsheki and O.D. 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International Journal of Computer Mathematics. 2005;82(3) 369–380. https://www.tandfonline.com/doi/abs/10.1080/0020716042000301725 [Accessed: 5 September 2020]\n'},{id:"B33",body:'\nJ.H. He, Variational iteration method — a kind of non-linear analytical technique: some examples, International Journal of Non-Linear Mechanic. 1999;(34) 699—708. https://tarjomefa.com/wp-content/uploads/2018/06/9165-English-TarjomeFa.pdf [Accessed: 20 September 2020]\n'},{id:"B34",body:'\nA.M. Wazwaz, The variational iteration method for solving linear and nonlinear ODEs and scientific models with variable coefficients. Central European Journal of Engineering, 2014;4: 64–71. https://link.springer.com/article/10.2478/s13531-013-0141-6 [Accessed: 15 September 2020]\n'},{id:"B35",body:'\nP.L. Ndlovu and R.J. Moitsheki, Analytical solutions for steady heat transfer in longitudinal fins with temperature-dependent properties. 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