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Thermal performance and efficiency of convective–radiative T-shaped fins with temperature dependent thermal conductivity, heat transfer coefficient and surface emissivity

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Thermal performance and efficiency of convective–radiative T-shaped fins with temperature dependent thermal conductivity, heat transfer coefficient and surface emissivity
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  Thermal performance and ef  fi ciency of convective – radiative T-shaped  fi ns withtemperature dependent thermal conductivity, heat transfer coef  fi cient andsurface emissivity ☆ Mohsen Torabi  a, ⁎ , A. Aziz  b a Department of Mechanical and Biomedical Engineering, City University of Hong Kong, 83 Tat Chee Avenue, Hong Kong  b Department of Mechanical Engineering, School of Engineering and Applied Science, Gonzaga University, Spokane, WA 99258, USA a b s t r a c ta r t i c l e i n f o Available online 17 July 2012 Keywords: T-shaped  fi nsAnalytical solutionLinear emissivity and thermal conductivity – temperature variationsTemperature dependent heat transfercoef  fi cient This paper analytically investigates the thermal performance of convective – radiative T-shape  fi n with simul-taneous variation of thermal conductivity, heat transfer coef  fi cient and surface emissivity with temperature.Unlike some other  fi n studies, the sink temperatures for convection and radiation are assumed to benon-zero. This model is a more realistic representation of   fi ns in actual engineering practice. The collectionof graphs provided should facilitate design and performance evaluation of T-shaped  fi ns.© 2012 Elsevier Ltd. All rights reserved. 1. Introduction The vast literature on extended surface heat transfer is compre-hensively covered in a treatise by Kraus et. al. [1]. The classical  fi nanalyses assume one dimensional, steady conduction along thelength of the  fi n, constant  fi n thermal conductivity, and a uniformconvection heat transfer coef  fi cient over the exposed surfaces of the fi n. These simplifying assumptions have led to the development of alarge number of analytical solutions for different geometries and dif-ferent thermal conditions [1]. Over the past few decade years, severalre fi nementsof basic fi n analysis have been introduced to take into ac-count temperature or coordinate dependent thermal conductivity,temperature or coordinate dependent heat transfer coef  fi cient, andsimultaneous convection – radiation surface heat loss. Such re fi nedanalyses appearing until the year 2000 are also given in [1]. More re-cent work that is closely linked to the present work will now be brief-ly reviewed.The effect of temperature dependentthermalconductivity and heattransfer coef  fi cient on the performance of   fi ns has continued unabatedduringthelastdecadedespitethefactthatsuchstudieswereconductedin mid-70s. For example, Arslanturk [2] obtained correlation equationsfor optimum design of annular  fi ns with temperature-dependent ther-mal conductivity. Kulkarniand Joglekar[3]proposedand implementeda numerical technique based on residue minimization to solve thenonlinear differential equation, which governs the temperature distri-bution in straight convective  fi ns having temperature-dependent ther-mal conductivity. Many more studies of   fi ns with temperaturedependent thermal conductivity can be found in references [e.g. 4 – 7].If the fi n is made of a functionally graded material, its thermal conduc-tivitydependsonthelocationratherthantemperature.Oneexampleof suchstudyisprovidedbyAzizandRahman[8]whoanalyzedaradial fi nof uniform thickness whose thermal conductivity was a function of theradial coordinate. Another example is the work by Khan and Aziz [9]who studied the transient response of a functionally graded longitudi-nal convecting  fi n.Many convective heat transfer processes, such as natural convec-tion, boiling, etc., occurring on the  fi n's surface can be adequatelymodeled by a power law type dependence of the form  h ∝ ( Δ T  ) m where  m  is a constant. Fouladi et al. [10] utilized the variational iter-ation method (VIM) to generate an approximate analytical solutionfor a straight  fi n of constant thickness. Khani and Aziz [11] used thehomotopy analysis method (HAM) to develop an analytical solutionfor the thermal performance of a straight  fi n of trapezoidal pro fi lewhen both thermal conductivity and heat transfer coef  fi cient aretemperature dependent. When  fi ns are natural convection cooled oroperate at high temperatures, the effect of radiative heat from thesurface of the  fi n becomes signi fi cant and introduces an additionalnonlinearity in the  fi n equation. Bouaziz and Aziz [12] introducedthe novel concept of the double optimal linearization to derive simpleand accurate expressions for predicting the thermal performance of aconvective – radiative  fi n with temperature-dependent thermal con-ductivity. Aziz and Beers-Green [13] have opted in favor of a numer-ical approach to study nonlinear convective – radiative rectangular  fi n International Communications in Heat and Mass Transfer 39 (2012) 1018 – 1029 ☆  Communicated by W.J. Minkowycz. ⁎  Corresponding author. E-mail address:  Torabi_mech@yahoo.com (M. Torabi).0735-1933/$  –  see front matter © 2012 Elsevier Ltd. All rights reserved.doi:10.1016/j.icheatmasstransfer.2012.07.007 Contents lists available at SciVerse ScienceDirect International Communications in Heat and Mass Transfer  journal homepage: www.elsevier.com/locate/ichmt  problems.Torabietal.[14]solvedtheenergyequationfortheconvec-tive – radiative moving  fi ns with variable thermal conductivity usingthe differential transformation method (DTM). They assumednon-zero convection and radiation sink temperature in their analysis.Recently, Aziz and Torabi [15] numerically studied convective – radia-tive straight fi ns with simultaneous variation of thermal conductivity,heat transfer coef  fi cient, and surface emissivity with temperature.As high heat  fl ux electronics systems continue to grow, novel  fi nstructures are being developed to provide effective air cooling of these devices. Cascaded  fi ns made of two materials in series and  fi nsin a tree like structure have been proposed and studied. The cascaded fi n facilitates the savings of expensive materials and reduces bothinvestment and operational costs because of its higher thermal ef  fi -ciency compared with that of a single material  fi n. The combinationsof different geometries such rectangular – rectangular, rectangular – triangular and rectangular – parabolic have been suggested as cascad-ed fi n designs [e.g. 16,17]. The idea of a tree like fi n structure srcinal-ly analyzed in [1] using linear transformation theory has recentlyemerged on a stronger scienti fi c footing as a result of the develop-ment of constructal theory where the constructal structure is deter-mined by minimizing the global resistance between the heat sourceand the sink subject to size constraint [18 – 22] or by minimizing theconstructal entransy dissipation rate [23].A pure convective T-shaped  fi n in isolation and not as a part of atree like structure [18,19] has been recently investigated by Kunduand Bhanja [24]. They established an energy equation for the  fi nwithsimultaneousvariationofthermalconductivityandheattransfercoef  fi cient. The solutions for the temperature distributions in thestem and  fl ange (See Fig. 1) were obtained by using the Adomian de-composition method (ADM), and used subsequently to derive the re-sults for the ef  fi ciency of the  fi n and optimum design parameters. In alater paper [25], they extended the work to include surface radiation. However, the convective heat loss was modeled by a constant heattransfer coef  fi cient.The present work develops a model for the T-shaped  fi n with si-multaneous convective and radiation heat losses. Both the thermalconductivity of the  fi n and surface emissivity of the  fi n are treatedas linear functions of temperature. The convective heat transfercoef  fi cient  h  is allowed to vary as a power law function of the form h ∝ ( Δ T  ) m . The model also allows for non-zero convection and radia-tion sink temperatures. Because such a model has not been investi-gated in the literature, the present results are believed to be novel.The nonlinear model is solved using the differential transformmeth-od (DTM) to obtain the temperature distribution in the  fi n. Becausethe differential approach associated with Fourier's law of heat con-duction is inappropriate for determining the heat transfer rate andthe  fi n ef  fi ciency, a new analytical expression based on the integraltransform methodis usedtocalculatethe fi nef  fi ciency. Theaccuracyof the present procedure was con fi rmed by comparing the presentresults with the results of Kundu and Bhanja [24] which is a specialcase of the present model. Results for the temperature distributionand  fi n ef  fi ciency are presented as functions of ten parameters Fig. 1.  Schematic diagram of a T-shape  fi n. Nomenclature  A  Dimensionless thermal conductivity parameter B  Dimensionless surface emissivity parameter D  Domain H   Constant h  Convection heat transfer coef  fi cient,  W  /  m 2 K h b  Convection heat transfer coef  fi cient at the base of the fi n,  W  /  m 2 K k  Thermal conductivity,  W  /  mK k a  Thermal conductivity at convection sink temperature, W  /  mK L 1  Stem length,  mL 2  Semi- fl ange length,  mL R  Ratio of lengths m  Parameter in variable heat transfer coef  fi cient model Nc   Convection – conduction parameter Nr   Radiation – conduction parameter q  f   Fin heat transfer rate,  W q ideal  Ideal  fi n heat transfer rate,  W t  1  Stem thickness,  mt  2  Flange thickness,  mT  1  Local stem temperature,  K T  2  Local  fl ange temperature,  K T  a  Convection sink temperature,  K T  b  Stem base temperature,  K T  s  Radiation sink temperature,  K  X   Dimensionless axial distance measured from the tip of the  fi n  X  ( k ) Transformed analytical function  x  Axial distance measured from the tip of the  fi n,  m x ( t  ) Original analytical function Greek symbols α   Thickness ratio  β   A measure of surface emissivity variation with tem-perature,  K  − 1 λ  A measure of thermal conductivity variation withtemperature,  K  − 1 ψ  Thickness to length ratio of stem ε   Emissivity of   fi n ε  s  Emissivity of   fi n at the radiation sink temperature  η   Fin ef  fi ciency σ   Stefan – Boltzmann constant,  W  /  m 2 K  4 θ 1  Dimensionless stem temperature θ 2  Dimensionless  fl ange temperature θ a  Dimensionless convection sink temperature θ s  Dimensionless radiation sink temperature 1019 M. Torabi, A. Aziz / International Communications in Heat and Mass Transfer 39 (2012) 1018 – 1029  characterizingthethermalperformanceofT-shaped fi ns,namely thethermal conductivity parameter,  A , surface emissivity parameter,  B ,the exponent  m  associated with convective heat transfer coef  fi cient,convection – conduction parameter,  Nc  , radiation – conduction pa-rameter,  Nr  , convection sink temperature,  θ a , radiation sink temper-ature,  θ s , length ratio,  L R , thickness ratio,  α  , and thickness to lengthratio of stem,  ψ . Because a broad range of governing parametersare investigated, the results should be useful in a number of engi-neering applications. 2. Mathematical formulation We consider a T-shaped  fi n as shown in Fig. 1. The stem with a uniform thickness  t  1  is attached to a primary surface at a constanttemperature  T  b . The axial coordinate  y  for the stem is measuredfrom the interface between the stem and the  fl ange. The  fl ange hasa uniform thickness  t  2 with the coordinate  x  measured from the tipof the  fi n. Both the stem and the  fl ange lose heat by convection andradiation to the same convection sink temperature  T  a and effective ra-diation sink temperature  T  s . Following Kundu and Bhanja [24,25], weassume the  fl ange tip to be adiabatic and temperature continuity atthe interface between the stem and the  fl ange. The heat  fl ow fromthe stem (at  y =0) divides equally between the right and left halvesof the  fl ange. The thermal conductivity  k  of the  fi n, the convectiveheat transfer coef  fi cient  h , and the  fi n surface emissivity  ε   are as-sumed to be functions of temperature as follows: k  ¼  k a  1 þ λ  T  − T  a ð Þ½  ð 1a Þ h  ¼  h b T  − T  a T  b − T  a  m ð 1b Þ ε   ¼  ε  s  1 þ  β   T  − T  s ð Þ½  ð 1c Þ where  k a  is the thermal conductivity of the  fi n at the convection sinktemperature T  a ,  h b  is the convection heat transfer coef  fi cientcorresponding to the temperature difference, T  b − T  a  and  ε  s  is the fi n surface emissivity at the radiation sink temperature T  s . The con-stants  λ  and  β   are measures of variation of thermal conductivityand surface emissivity with temperature, respectively. The analysisassumes negligible radiative interactions between the stem and the fl ange, and between the  fi n (stem and  fl ange) and the primary sur-face. Furthermore, the convecting  fl uid is assumed to be radiativelynonparticipating. Theheatconductioninboththestemandthe fl angeis considered to be one-dimensional (along  y  for the stem and along  x for the  fl ange) and steady.Theenergybalanceequationsfor thestemand fl angeofthe fi nperunit width may be written as follows: k a ddy  1 þ λ  T  1 − T  a ð Þ½  dT  1 dy  − 2 h b t  1 T  1 − T  a T  b − T  a  m T  1 − T  a ð Þ − 2 σε  s t  1 1 þ  β   T  1 − T  s ð Þ½   T   41 − T   4 s   ¼  0 0  b  y  b  L 1 ð 2a Þ k a ddx  1 þ λ  T  2 − T  a ð Þ½  dT  2 dx  − 2 h b t  2 T  2 − T  a T  b − T  a  m T  2 − T  a ð Þ − 2 σε  s t  2 1 þ  β   T  2 − T  s ð Þ½   T   42 − T   4 s   ¼  0 0  b  x  b  L 2 ð 2b Þ where  T  1 (  y )is the local temperature in the stem, and  T  2 (  x )is the localtemperature in the  fl ange.Based on the description in the  fi rst paragraph of this section, wemay write the following boundary conditions: T  1  L 1 ð Þ ¼  T  b  ð 3a Þ T  2  L 2 ð Þ ¼  T  1  0 ð Þ ð 3b Þ 2 t  2 dT  2 dx   x ¼ L 2 ¼  t  1 dT  1 dy   y ¼ 0 ð 3c Þ dT  2 dx   x ¼ 0 ¼  0 :  ð 3d Þ Introducing the following dimensionless parameters θ 1  ¼  T  1 T  b θ 2  ¼  T  2 T  b θ a  ¼  T  a T  b θ s  ¼  T  s T  b  X   ¼  xL 2 Y   ¼  yL 1  A  ¼  λ T  b  B  ¼  β  T  b Nc   ¼  h b t  1 T  mb k a  T  b − T  a ð Þ m  Nr   ¼  σε  s t  1 T  3 b k a L R  ¼  L 2 L 1 α   ¼  t  2 t  1 ψ  ¼  t  1 L 1 ð 4 Þ the energy Eq. (2) and the boundary conditions(3) take the followingforms: ddY   1 þ  A  θ 1 − θ a ð Þ½  d θ 1 dY   − 2 Nc  ψ 2  θ 1 − θ a ð Þ m þ 1 − 2 Nr  ψ 2  1 þ  B  θ 1 − θ s ð Þ½   θ 41 − θ 4 s   ¼  0 0  b  Y   b  1 ð 5a Þ ddX   1 þ  A  θ 2 − θ a ð Þ½  d θ 2 dX   − 2 L 2 R Nc  ψ 2 α θ 2 − θ a ð Þ m þ 1 − 2 L 2 R Nr  ψ 2 α  1 þ  B  θ 2 − θ s ð Þ½   θ 42 − θ 4 s   ¼  0 0  b  X   b  1 ð 5b Þ with the following boundary conditions: θ 1  1 ð Þ ¼  1  ð 6a Þ θ 2  1 ð Þ ¼  θ 1  0 ð Þ ð 6b Þ 2 α  L R d θ 2 dX   X  ¼ 1  ¼  d θ 1 dY   Y  ¼ 0  ð 6c Þ d θ 2 dX    X  ¼ 0 ¼  0 :  ð 6d Þ Eqs.(5a)and(5b)arecoupledthroughtheboundaryconditions(6b) and (6c). The temperature distributions in the stem and the  fl ange,and hence the ef  fi ciency of the  fi n are seen to depend on ten dimen-sionless parameters, namely the thermal conductivity parameter  A ,emissivity parameter  B , the exponent  m  associated with convectiveheat transfer coef  fi cient, convection – conduction parameter  Nc  , radi-ation – conduction parameter  Nr  , the two temperature ratios,  θ a  and θ s , that characterize the temperatures of convection and radiationsinks, length ratio,  L R , thickness ratio,  α  , and thickness to length ratioof stem, ψ . 1020  M. Torabi, A. Aziz / International Communications in Heat and Mass Transfer 39 (2012) 1018 – 1029  3. Fin ef  fi ciency  It is more convenient to determine the heat  fl ow through the base of the stem i.e. primary surface by calculating the total heat loss from thesurface of the  fi n (stem plus  fl ange) by convection and radiation. With this approach, the heat transfer rate  q  f   may be written as q  f   ¼ ∫  x ¼ L 2  x ¼ 0 2 h b T  b − T  a ð Þ m  T  2 − T  a ð Þ m þ 1 þ 2 σε  s  1 þ  β   T  2 − T  s ð Þð Þ  T   42 − T   4 s  dx þ ∫  y ¼ L 1  y ¼ 0 2 h b T  b − T  a ð Þ m  T  1 − T  a ð Þ m þ 1 þ 2 σε  s  1 þ  β   T  1 − T  s ð Þð Þ  T   41 − T   4 s  dy : ð 7 Þ The ideal  fi n heat transfer  q ideal  is realized if the entire  fi n surface was at the base temperature  T  b  and may be determined as q ideal  ¼  2  L 1  þ  L 2 ð Þ h b T  b − T  a ð Þ m  T  b − T  a ð Þ m þ 1 þ 2  L 1  þ  L 2 ð Þ σε  s  1 þ  β   T  b − T  s ð Þ½   T   4 b − T   4 s  :  ð 8 Þ The  fi n ef  fi ciency  η   is de fi ned as the ratio of   q  f   and  q ideal  η   ¼ q  f  q ideal :  ð 9 Þ Employing the dimensionless parameters in Eq. (4), the  fi n ef  fi ciency may be expressed as  η   ¼  12 ∫ Y  ¼ 1 Y  ¼ 0  Nc   θ 1 − θ a ð Þ m þ 1 þ  Nr   1 þ  B  θ 1 − θ s ð Þð Þ  θ 41 − θ 4 s hi dY   þ ∫  X  ¼ 1  X  ¼ 0  Nc   θ 2 − θ a ð Þ m þ 1 þ  Nr   1 þ  B  θ 2 − θ s ð Þð Þ  θ 42 − θ 4 s hi dX Nc T  b − T  a ð Þ m þ 1 þ  Nr   1 þ  β   T  b − T  s ð Þ½   T  4 b − T  4 s   :  ð 10 Þ 4. Differential transformation method (DTM) Let  x ( t  ) be analytic in a domain  D  and let  t  = t  i  represents anypoint in D . The function  x ( t  ) is then represented by a power serieswhose center is located at t  i . A Taylor series expansion function of   x ( t  ) is expressed as follows:  x t  ð Þ ¼ X ∞ k ¼ 0 t  − t  i ð Þ k k ! d k  x t  ð Þ dt  k "# t  ¼ t  i ∀ t  ∈ D :  ð 11 Þ The particular case of Eq. (11) when  t  i =0 is referred to as theMaclaurin series of   x ( t  ) and is expressed as:  x t  ð Þ ¼ X ∞ k ¼ 0 t  ð Þ k k ! d k  x t  ð Þ dt  k "# t  ¼ 0 :  ð 12 Þ The differential transformation of the function  x ( t  ) is de fi ned asfollows [26]:  X k ð Þ ¼  H  ð Þ k k ! d k  x t  ð Þ dt  k "# t  ¼ 0 ð 13 Þ where  x ( t  ) is the srcinal function and  X  ( k ) is the transformed func-tion. The differential spectrum of   X  ( k ) is con fi ned within theinterval t  ∈ [0, H  ], where  H   is a constant. The differential inverse trans-form of   X  ( k ) is de fi ned as follows:  x t  ð Þ ¼ X ∞ k ¼ 0 t H   k  X k ð Þ :  ð 14 Þ It is clear that the concept of differential transformation is basedupon the Taylor series expansion. Values of the function  X  ( k ) at valuesof argument  k  are referred to as discretes, i.e.  X  (0) is known as thezerodiscrete,  X  (1)isthe fi rstdiscrete,andsoon.Thelargerthenumberof discretes, the more precise the recovery of the unknown function.The function  x ( t  ) consists of   T  -function  X  ( k ), and its value is given bythesumofthe T  -functionwith  t H   k asitscoef  fi cient.Inrealapplications,with the right choice of constant H  , for the larger values of argument k ,the discrete of spectrum reduces rapidly. The function  x ( t  ) is expressedby a  fi nite series and Eq. (14) can be written as:  x t  ð Þ ¼ X nk ¼ 0 t H   k  X k ð Þ :  ð 15 Þ Mathematical operations performed by differential transformmethod are listed in Table 1. 5. DTM solution Now we apply the DTM to Eq. (5). Taking the differential trans-form of Eq. (5) with respect to  Y   and  X  , respectively, and considering H  =1 according to Table 1, giveFor  m  ¼  0 1 −  A θ a ð Þ  k  þ 2 ð Þ  k  þ 1 ð Þ Θ 1  k  þ 2 ð Þ þ  A X kv ¼ 0 Θ 1  v ð Þ  k  þ 2 − v ð Þ  k  þ 1 − v ð Þ  Θ 1  k  þ 2 − v ð Þ ! þ  A X kv ¼ 0 v  þ 1 ð Þ  Θ 1  v  þ 1 ð Þ  k  þ 1 − v ð Þ  Θ 1  k  þ 1 − v ð Þ ! − 2 NrB ψ 2 X km ¼ 0 Θ 1  k − m ð Þ X mv ¼ 0 Θ 1  m − v ð Þ X vw ¼ 0 Θ 1  v − w ð Þ X wu ¼ 0 Θ 1  w − u ð Þ  Θ 1  u ð Þ !!!! þ 2  NrB θ s − Nr  ð Þ ψ 2 X km ¼ 0 Θ 1  k − m ð Þ X mv ¼ 0 Θ 1  m − v ð Þ X vw ¼ 0 Θ 1  v − w ð Þ  Θ 1  w ð Þ !!! þ 2  NrB θ 4 s − Nc   ψ 2  Θ 1  k ð Þ þ 2  Nc  θ a  þ Nr  θ 4 s − NrB θ 5 s  ψ 2  δ  k ð Þ ¼  0 ð 16a Þ  Table 1 The fundamental operations of differential transform method.Original function Transformed function  x ( t  )= α   f  ( t  )±  β   g  ( t  )  X  ( k )= α  F  ( k )±  β  G ( k )  x t  ð Þ ¼  df t  ð Þ dt   X  ( k )=( k +1) F  ( k +1)  x t  ð Þ ¼  d 2  f t  ð Þ dt  2  X  ( k )=( k +1)( k +2) F  ( k +2)  x ( t  )= t  m  X k ð Þ ¼  δ  k − m ð Þ ¼  1  k  ¼  m 0  k ≠ m   x ( t  )=exp( λ t  )  X k ð Þ ¼  λ k k !  x ( t  )=  f  ( t  )  g  ( t  )  X k ð Þ ¼ P kl ¼ 0 F l ð Þ G k − l ð Þ 1021 M. Torabi, A. Aziz / International Communications in Heat and Mass Transfer 39 (2012) 1018 – 1029  1 −  A θ a ð Þ  k  þ  2 ð Þ  k  þ 1 ð Þ Θ 2  k  þ  2 ð Þ þ  A X kv ¼ 0 Θ 2  v ð Þ  k  þ 2 − v ð Þ  k  þ  1 − v ð Þ  Θ 2  k  þ  2 − v ð Þ ! þ  A X kv ¼ 0 v  þ 1 ð Þ  Θ 2  v  þ  1 ð Þ  k  þ 1 − v ð Þ  Θ 2  k  þ 1 − v ð Þ ! − 2 L 2 R NrB ψ 2 α  X km ¼ 0 Θ 2  k − m ð Þ X mv ¼ 0 Θ 2  m − v ð Þ X vw ¼ 0 Θ 2  v − w ð Þ X wu ¼ 0 Θ 2  w − u ð Þ  Θ 2  u ð Þ !!!! þ 2 L 2 R  NrB θ s − Nr  ð Þ ψ 2 α  X km ¼ 0 Θ 2  k − m ð Þ X mv ¼ 0 Θ 2  m − v ð Þ X vw ¼ 0 Θ 2  v − w ð Þ  Θ 2  w ð Þ !!! þ 2 L 2 R  NrB θ 4 s − Nc   ψ 2 α Θ 2  k ð Þ þ 2 L 2 R  Nc  θ a  þ  Nr  θ 4 s − NrB θ 5 s  ψ 2 α δ  k ð Þ ¼  0 ð 16b Þ For  m  ¼  2 1 −  A θ a ð Þ  k  þ 2 ð Þ  k  þ 1 ð Þ Θ 1  k  þ 2 ð Þ þ  A X kv ¼ 0 Θ 1  v ð Þ  k  þ 2 − v ð Þ  k  þ 1 − v ð Þ  Θ 1  k  þ 2 − v ð Þ ! þ  A X kv ¼ 0 v  þ 1 ð Þ  Θ 1  v  þ 1 ð Þ  k  þ 1 − v ð Þ  Θ 1  k  þ 1 − v ð Þ ! − 2 NrB ψ 2 X km ¼ 0 Θ 1  k − m ð Þ X mv ¼ 0 Θ 1  m − v ð Þ X vw ¼ 0 Θ 1  v − w ð Þ X wu ¼ 0 Θ 1  w − u ð Þ  Θ 1  u ð Þ !!!! þ 2  NrB θ s − Nr  ð Þ ψ 2 X km ¼ 0 Θ 1  k − m ð Þ X mv ¼ 0 Θ 1  m − v ð Þ X vw ¼ 0 Θ 1  v − w ð Þ  Θ 1  w ð Þ !!! − 2 Nc  ψ 2 X kv ¼ 0 Θ 1  k − v ð Þ X vw ¼ 0 Θ 1  v − w ð Þ  Θ 1  w ð Þ !! þ 6 Nc  θ a ψ 2 X km ¼ 0 Θ 1  k − m ð Þ  Θ 1  m ð Þ ! þ 2  NrB θ 4 s − 3 Nc  θ 2 a  ψ 2  Θ 1  k ð Þ þ 2  Nc  θ 3 a  þ Nr  θ 4 s − NrB θ 5 s  ψ 2  δ  k ð Þ ¼  0 ð 16c Þ 1 −  A θ a ð Þ  k  þ 2 ð Þ  k  þ 1 ð Þ Θ 2  k  þ 2 ð Þ þ  A X kv ¼ 0 Θ 2  v ð Þ  k  þ 2 − v ð Þ  k  þ 1 − v ð Þ  Θ 2  k  þ 2 − v ð Þ ! þ  A X kv ¼ 0 v  þ 1 ð Þ  Θ 2  v  þ 1 ð Þ  k  þ 1 − v ð Þ  Θ 2  k  þ 1 − v ð Þ ! − 2 L 2 R NrB ψ 2 α  X km ¼ 0 Θ 2  k − m ð Þ X mv ¼ 0 Θ 2  m − v ð Þ X vw ¼ 0 Θ 2  v − w ð Þ X wu ¼ 0 Θ 2  w − u ð Þ  Θ 2  u ð Þ !!!! þ 2 L 2 R  NrB θ s − Nr  ð Þ ψ 2 X km ¼ 0 Θ 2  k − m ð Þ X mv ¼ 0 Θ 2  m − v ð Þ X vw ¼ 0 Θ 2  v − w ð Þ  Θ 2  w ð Þ !!! − 2 L 2 R Nc  ψ 2 α  X kv ¼ 0 Θ 2  k − v ð Þ X vw ¼ 0 Θ 2  v − w ð Þ  Θ 2  w ð Þ !! þ 6 L 2 R Nc  θ a ψ 2 α  X km ¼ 0 Θ 2  k − m ð Þ  Θ 2  m ð Þ ! þ 2 L 2 R  NrB θ 4 s − 3 Nc  θ 2 a  ψ 2 α Θ 2  k ð Þ þ 2 L 2 R  Nc  θ 3 a  þ  Nr  θ 4 s − NrB θ 5 s  ψ 2 α δ  k ð Þ ¼  0 ð 16d Þ Letting  θ 1  0 ð Þ ¼  a 1 ;  d θ 1 dY  Y  ¼ 0  ¼  b 1 ;  θ 2  0 ð Þ ¼  a 2 ;  d θ 2 dX  X  ¼ 0  ¼  0 j   andexercising the transformation Θ 1  0 ð Þ ¼  a 1  ð 17a Þ Θ 1  1 ð Þ ¼  b 1  ð 17b Þ Θ 2  0 ð Þ ¼  a 2  ð 17c Þ Θ 2  1 ð Þ ¼  0  ð 17d Þ accordingly from Eq. (16) we can calculate  Θ 1 ( k +2) and  Θ 2 ( k +2)for the stem and the  fl ange parts of the T-shaped  fi n, respectively. Fi-nally, the following solutions by the differential inverse transform of  Θ 1 ( k ) and  Θ 2 ( k ) can be obtained: θ 1  Y  ð Þ ¼ X nk ¼ 0 Y  k Θ 1  k ð Þ ð 18a Þ θ 2  X  ð Þ ¼ X nk ¼ 0  X  k Θ 2  k ð Þ :  ð 18b Þ Using Eq. (18), the values of   a 1 , a 2  and  b 1  can be determined fromboundary Eqs. (6a) to (6c) using the  fsolve  command in Maple 14.The results presented in this paper are based on the evaluation of twenty terms of the series, i.e.  n =20. The 20-term truncated serieswas found to provide a suf  fi ciently accurate when compared withthe direct numerical solutions of Eqs. (5 – 6). 6. Numerical method Eq. (5) along with the boundary conditions (6) was solvednumeri-callyusinga fi nitedifference technique inconjunction withRichardsonextrapolation. The domains of   X   and  Y   were each discretized using 500nodes. The iterative process was continued until a tolerance of 10 − 6 was achieved. 7. Results and discussion For convenience of discussion, we would divide this section intothree subsections. The  fi rst subsection deals with the validation of DTM. The validation is achieved by comparing the DTM results withthe results published by Kundu and Bhanja [24] for a purely convec-tive T-shaped  fi n. As a further validation, the DTM predictions arecomparedwith the fi nite differencesolutions of the problem.The sec-ond subsection details the temperature distribution results for the  fi nand illustrates how the temperature distribution in the  fi n is affectedby the changes in some of the dimensionless parameters identi fi ed inthe previous section. The last subsection shows graphs of   fi n ef  fi cien-cy as a function of the some of the dimensionless parameters. Fig. 2.  Comparison of DTM results with (a) Kundu and Bhanja results [24], circle, (b) and the  fi nite difference solution.1022  M. Torabi, A. Aziz / International Communications in Heat and Mass Transfer 39 (2012) 1018 – 1029
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