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?
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\n
1. Introduction
\n
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
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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.
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Notice that other terms may be added, for example internal heat generation (source term) and fin profile.
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2.1.2 Rectangular straight fins
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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.
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2.2.1 Cylindrical pin fins
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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]).
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2.2.2 Rectangular straight fins
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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]).
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2.3 1 + 1 dimensional transient model for straight fins
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2.3.1 Solid stationary fins
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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
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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
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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].
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3.1.1 \n\nm\n\n dependent and \n\nn\n\n independent variables
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\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.
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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
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\n\n\nX\n\np\n\n\nF\n=\n0\n,\n\n
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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].
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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].
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4.1 Some exact solutions
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4.1.1 Example 1
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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].
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4.1.2 Example 2
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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.
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4.2 Some approximate solutions
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4.2.1 Three dimensional DTM
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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.
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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]).
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4.2.2 Two dimensional DTM
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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]).
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4.2.3 Comparison of one dimensional exact, DTM and VIM solutions
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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]).
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5. Outlook and some concluding remarks
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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.
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Conflict of interest
The authors declare no conflict of interest.
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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. Nejad, Analytical solutions and efficiency of the nonlinear fin problem with temperature-dependent thermal conductivity and heat transfer coefficient, Communications in Nonlinear Science and Numerical Simulation. 2009;14(8): 3327–3338. https://www.sciencedirect.com/science/article/abs/pii/S1007570409000355 [Accessed: 16 September 2020]\n'},{id:"B4",body:'\nF. Khani and M.A. Raji, S.H. Nezhad, A series solution of the fin problem with a temperature-dependent thermal conductivity, Communications in Nonlinear Science and Numerical Simulation. 2009;14(7): 3007–3017. https://www.sciencedirect.com/science/article/abs/pii/S1007570408004115 [Accessed: 16 September 2020]\n'},{id:"B5",body:'\nMoitsheki RJ, Hayat T, Malik Y, Some exact solutions of the fin problem with a power law temperature-dependent thermal conductivity, Nonlinear Analysis: Real World Applications 2010;(11)5: 3287–3294 https://www.sciencedirect.com/science/article/pii/S1468121809003459 [Accessed: 16 September 2020]\n'},{id:"B6",body:'\nH.S. Kang and D.C. Look, Two dimensional trapezoidal fins analysis, Computational Mechanics. 1997; 19: 247–250 https://link.springer.com/article/10.1007/s004660050173 [Accessed: 16 September 2020]\n'},{id:"B7",body:'\nR.J. Moitsheki and A. Rowjee, Steady Heat Transfer through a two-dimensional rectangular straight fin http://downloads.hindawi.com/journals/mpe/2011/826819.pdf [Accessed: 16 September 2020]\n'},{id:"B8",body:'\nN. Fallo and R.J. Moitsheki, Approximate Analytical Solutions for Transient Heat Transfer in Two-Dimensional Straight Fins, Nonlinear Dynamics and Systems Theory. 2019;19: 133–140, http://www.e-ndst.kiev.ua/v19n1-SI/3(67).pdf [Accessed: 14 September 2016]\n'},{id:"B9",body:'\nR.J. Moitsheki and M.D. Mhlongo, Classical Lie point symmetry analysis of a steady nonlinear omne-dimensionsl fin problem. Journal of Applied Mathematics. 2012; Article ID 671548, 13 pages. https://www.hindawi.com/journals/jam/2012/671548/ [Accessed: 14 September 2020].\n'},{id:"B10",body:'\nR.J. Moitsheki, Steady heat transfer through a radial fin with rectangular and hyperbolic profiles, Nonlinear Analysis: Real World Applications 2011;12(2):867–874 https://www.sciencedirect.com/science/article/pii/S1468121810001975 [Accessed: 15 September 2020].\n'},{id:"B11",body:'\nR.J. Moitsheki, Steady one dimensional heat flow in a longitudinal triangular and parabolic fin, Communications in Nonlinear Science and Numerical Simulation. 2011;16(10): 3971–3980 https://www.sciencedirect.com/science/article/abs/pii/S1007570411000426 [Accessed: 15 September 2020].\n'},{id:"B12",body:'\nA. Moradi, Analytical solutions for fin with temperature dependant heat transfer coefficient, International Journal of Engineering and Applied Sciences, 2011;3(2): 1–12. https://dergipark.org.tr/en/download/article-file/217660 [Accessed: 20 September 2020].\n'},{id:"B13",body:'\nM. Turkyilmazoglu, An optimum profile of a special pin fin: full solutions, International Journal of Numerical Methods for Heat and Fluid Flow. 2020. Ahead of print. https://www.emerald.com/insight/content/doi/10.1108/HFF-11-2019-0801/full/html [Accessed: 30 September 2020]\n'},{id:"B14",body:'\nM. Turkyilmazoglu, Heat transfer through longitudinal fins, Journal of Thermophysics and Heat Transfer. Journal of Thermophysics and Heat Transfer. 2014;28(4): 806–811. https://arc.aiaa.org/doi/abs/10.2514/1.T4348?journalCode=jtht [Accessed: 15 September 2020]\n'},{id:"B15",body:'\nM. Turkyilmazoglu, Exact heat-transfer solutions to radial fins of general profile. Journal of Thermophysics and Heat Transfer. 2016;30(1): 89–93. https://arc.aiaa.org/doi/abs/10.2514/1.T4555 [Accessed: 15 September 2020]\n'},{id:"B16",body:'\nR.J. Moitsheki and C. Harley, Steady thermal analysis of two-dimensional cylindrical pin fin with a nonconstant base temperature. Mathematical Problems in Engineering. 2011; Article ID 132457. 17 pages. https://www.hindawi.com/journals/mpe/2011/132457/ [Accessed: 15 September 2020]\n'},{id:"B17",body:'\nN. Fallo, R.J. Moitsheki and O.D. Makinde, Analysis of heat transfer in a cylindrical spine fin with variable thermal properties, Defect and Diffusion Forum, 2018;387: 10–22 https://www.scientific.net/DDF.387.10 [Accessed: 15 September 2020]\n'},{id:"B18",body:'\nA. Moradi, A.P.M. Fallah, T. Hayat, O.M. Aldossary. On solution of natural convection and radiation heat transfer problem in a moving porous fin. Arabian J Sci Eng. 2014;39:1303–12. https://link.springer.com/article/10.1007/s13369-013-0708-9 [Accessed: 15 September 2020]\n'},{id:"B19",body:'\nM.G. 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. Makinde, Transient response of longitudinal rectangular fins to step change in base temperature and in base heat flow conditions, International Journal of Heat and Mass Transfer. 2013;57(1), 117–125. https://www.sciencedirect.com/science/article/abs/pii/S0017931012007788 [Accessed: 01 October 2020]\n'},{id:"B27",body:'\nP.L. Ndlovu, The significance of fin profile and convective-radiative fin tip on temperature distribution in a longitudinal fin, Nano Hybrids and Composites. 2019;26: 93–105. https://www.scientific.net/NHC.26.93 [Accessed: 7 September 2020]\n'},{id:"B28",body:'\nP.L. Ndlovu, Analytical study of transient heat transfer in a triangular moving porous fin with temperature dependant thermal properties, Defect and Diffusion Forum. 2019;393:31–46. https://www.scientific.net/DDF.393.31 [Accessed:12 September 2020]\n'},{id:"B29",body:'\nP.L. Ndlovu and R.J. Moitsheki, Analysis of transient heat transfer in radial moving fins with temperature-dependent thermal properties, Journal of Thermal Analysis and Calorimetry. 2019;138: 2913–2921. https://link.springer.com/article/10.1007/s10973-019-08306-5 [Accessed:29 September 2020]\n'},{id:"B30",body:'\nP.L. Ndlovu and R.J. Moitsheki, Analysis of temperature distribution in radial moving fins with temperature dependent thermal conductivity and heat transfer coefficient. International Journal of Thermal Sciences. 2019;145: 106015 https://www.sciencedirect.com/science/article/abs/pii/S1290072919301784 [Accessed: 10 September 2020]\n'},{id:"B31",body:'\nG.W. Bluman and S. Anco, Symmetry and Integration Methods for Differential Equations. Applied Mathematical Sciences, Springer, New York. 2002. https://www.springer.com/gp/book/9780387986548 [Accessed: 7 September 2020]\n'},{id:"B32",body:'\nA. Kurnaz, G. Oturanç and M.E. Kiris, n-Dimensional differential transformation method for solving PDEs. 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. Mathematical Problems in Engineering. 2013, Article ID 273052. https://www.hindawi.com/journals/mpe/2013/273052/ [Accessed: 22 September 2020]\n'},{id:"B36",body:'\nP.L. Ndlovu and R.J. Moitsheki, Predicting the Temperature Distribution in Longitudinal Fins of Various Profiles with Power Law Thermal Properties Using the Variational Iteration Method. Defect and Diffusion Forum, 387: 403–416 https://www.scientific.net/DDF.387.403 [Accessed: 10 December 2020]\n'}],footnotes:[],contributors:[{corresp:"yes",contributorFullName:"Raseelo Joel Moitsheki",address:"raseelo.moitsheki@wits.ac.za",affiliation:'
School of Computer Science and Applied Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, South Africa
Department of Mathematical Science, College of Science, Engineering and Technology, University of South Africa, South Africa
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