Application of differential transformation method (DTM) for heat and mass transfer in a porous channel

Application of differential transformation method (DTM) for heat and mass transfer in a porous channel Q2 Q1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 3[.]

10:0:1465=WUnicodeDec222011ị 6ỵ model JPPR : 119 Prod:Type:FTP pp:028col:fig::NILị ED: PAGN: SCAN: Propulsion and Power Research ]]]];](]):]]]–]]] 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 HOSTED BY http://ppr.buaa.edu.cn/ Propulsion and Power Research www.sciencedirect.com ORIGINAL ARTICLE Q2 Application of differential transformation method (DTM) for heat and mass transfer in a porous channel Q1 S Sepasgozara,n, M Farajib, P Valipourc a Department of Civil Engineering, Babol University of Technology, Babol, Iran Department of Mechanical Engineering, Babol University of Technology, Babol, Iran c Department of Textile and Apparel, Qaemshahr Branch, Islamic Azad University, Qaemshahr, Iran b Received 10 April 2015; accepted March 2016 KEYWORDS Differential transformation method (DTM); Axisymmetric channel; Rotating disk; Porous media; Non-Newtonian fluid Abstract In the present paper a differential transformation method (DTM) is used to obtain the solution of momentum and heat transfer equations of non-Newtonian fluid flow in an axisymmetric channel with porous wall The comparison between the results from the differential transformation method and numerical method are in well agreement which proofs the capability of this method for solving such problems After this validity, results are investigated for the velocity and temperature for various values of Reynolds number, Prandtl number and power law index & 2017 National Laboratory for Aeronautics and Astronautics Production and hosting by Elsevier B.V This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Introduction The problem of non-Newtonian fluid flow has been under a lot of attention in recent years Various applications in different fields of engineering specially the interest in heat transfer n Corresponding author E-mail address: s.sepasgozar@yahoo.com (S Sepasgozar) Peer review under responsibility of National Laboratory for Aeronautics and Astronautics, China problems of non-Newtonian fluid flow, such as hot rolling, lubrication, cooling problems and drag reduction was the main reason for this considerable attention Debruge and Han [1] studied a problem concerning heat transfer in channel flow, which can be considered as an application of the previous works reported by Yuan [2], White [3] and Treill [4] Increasing the resistance of the blades against the hot stream around the blades for cooling was interested However the cooling process gives rise to excess energy consumption which leads to largely decrease of turbine efficiency http://dx.doi.org/10.1016/j.jppr.2017.01.001 2212-540X & 2017 National Laboratory for Aeronautics and Astronautics Production and hosting by Elsevier B.V This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Please cite this article as: S Sepasgozar, et al., Application of differential transformation method (DTM) for heat and mass transfer in a porous channel, Propulsion and Power Research (2017), http://dx.doi.org/10.1016/j.jppr.2017.01.001 Q3 Q4 57 58 59 60 61 62 63 64 65 66 67 68 69 70 2 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 S Sepasgozar et al Nomenclature A, B C Cn f F xk Kr k n p qn ðηÞ Pr Re T symmetric kinematic matrices specific heat blade-wall temperature coefficients velocity function transformation of ƒ general coordinates rotation parameter fluid thermal conductivity power law index in temperature distribution fluid pressure temperature function Prandtl number injection Reynolds number temperature Most of phenomena in our world are essentially nonlinear and are described by nonlinear equations Nonlinear differential equations usually arise from mathematical modeling of many physical systems Some of them are solved using numerical methods and some are solved using analytic methods such as perturbation Perturbation techniques are based on the existence of small or large parameters, this is called perturbation quantity Unfortunately, many nonlinear problems in science and engineering not contain such kind of perturbation quantities Therefore, same as the HAM [5,6], HPM [7–10], ADM [11,12] and OHAM [13,14] can overcome the foregoing restrictions and limitations of perturbation methods One of the semi-exact methods which does not need small parameters is the differential transformation method This method constructs an analytical solution in the form of a polynomial It is different from the traditional higher-order Taylor series method The Taylor series method is computationally expensive for large orders The differential transform method is an alternative procedure for obtaining an analytic Taylor series solution of differential equations The main advantage of this method is that it can be applied directly to nonlinear differential equations without requiring linearization, discretization and therefore, it is not affected by errors associated to discretization The concept of DTM was first introduced by Zhou [15], who solved linear and nonlinear problems in electrical circuits Chen and Ho [16] developed this method for partial differential equations and Ayaz [17] applied it to the system of differential equations Jang et al [18] applied the two-dimensional differential transform method to the solution of partial differential equations Sheikholeslami et al [19] used this method to solve the problem of nanofluid flow between parallel plates considering Thermophoresis and Brownian effects Sheikholeslami and Ganji [20] applied DTM to solve the problem of nanofluid flow and heat transfer between parallel plates considering Brownian motion They concluded that Nusselt number increases with augment of nanoparticle volume fraction, Hartmann number while it decreases with increase of the squeeze number Natural convection of a nonNewtonian copper–water nanofluid between two infinite parallel ur ; uz V r; θ; z δvm δxn δam δxn velocity components in r, z directions, respectively injection velocity cylindrical coordinate symbols velocity gradients acceleration gradients Greek symbols ρ τij ϕk φ η ψ fluid density stress tensor component viscosity coefficients dissipation function dimension less coordinates in z direction stream function vertical flat plates is investigated by Domairry et al [21] They conclude that as the nanoparticle volume fraction increases, the momentum boundary layer thickness increases, whereas the thermal boundary layer thickness decreases New analytical and numerical method has been developed in recent year in different field of science [22–56] In this study, the purpose is to solve nonlinear equations via DTM It can be seen that this method is strongly capable for solving a large class of coupled and nonlinear differential equations without tangible restriction of sensitivity to the degree of the nonlinear term Mathematical formulation 2.1 Flow analysis This study is concerned with simultaneous development of flow and heat transfer for non-Newtonian viscoelastic fluid flow on the turbine disc for cooling purposes The problem to be considered is depicted schematically in Figure The r-axis is parallel to the surface of disk and the z-axis is normal to it Figure Schematic diagram of the physical system Please cite this article as: S Sepasgozar, et al., Application of differential transformation method (DTM) for heat and mass transfer in a porous channel, Propulsion and Power Research (2017), http://dx.doi.org/10.1016/j.jppr.2017.01.001 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 Application of differential transformation method (DTM) for heat and mass transfer in a porous channel 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 The porous disc of the channel is at zẳ ỵL The wall that coincides with the r-axis is heated externally and from the other perforated wall non-Newtonian fluid is injected uniformly in order to cool the heated wall As can observed in Figure the cooling problem of the disk can be considered as a stagnation point flow with injection For a steady, ax symmetric, non-newtonian fluid flow the following equations can be written in cylindrical coordinates The continuity equation: rur ị ruz ị ỵ ẳ0 ∂r ∂z ð1Þ And the momentum equations: ∂ður Þ ∂ður ị ỵ uz r z
 P rr rz ỵ ẳ ỵ rr  ị þ ρ ∂r ρ ∂r r ∂z ur ð2Þ τzz ẳ Azz ỵ Azz ỵ Bzz 10ị rz ẳ Arz ỵ Arz ỵ ϕ3 Brz ð11Þ For the solution of the problem depicted in Figure in the case of axially symmetric flow it is convenient to define a stream function so that the continuity equation is satisfied: ψ ¼ Vr f ị Where ẳ and the velocity components can be derived as: Vr f ị ur ẳ 13ị L uz ẳ  2Vf ị L2 P 000 ỵ f ∂r ρVL ρV r ϕ ϕ 00 000 00 ỵ 22 f  2f f ị ỵ 32 f  2f f iv Þ ρL ρL 00 f  2f f ¼  ð3Þ Here τrr ; τrz ; τzr ; τzz are the components of stress matrix The analytical model under consideration leads to the following boundary conditions: @z ¼ ur ẳ uz ẳ 4ị @z ẳ L ur ẳ 0; 5ị Here ur ; uz are the velocity components in the r and z directions and V is the injection velocity; ρ; P are the density, and pressure For particular class of viscoelastic and viscoinelastic fluids Rivlin [36] showed that if the stress components τij at a point xk k ẳ 1; 2; 3ị and time t are assumed to be m polynomials in the velocity gradient v xn m; n ẳ 1; 2; 3ị and am the acceleration gradients xn m; n ẳ 1; 2; 3ị, and if in addition the medium is assumed to be isotropic the stress matrix can be expressed in the form ij ẳ I ỵ A ỵ B ỵ A ỵ ::: 6ị Here I is the unit matrices, A and B are symmetric kinematic matrixes defined by:   δvi δvj  ; A ẳ  ỵ x xi   j  ai aj vm vm    B ẳ  ỵ 7ị ỵ2 xj xi xi xj  And k k ¼ 0; 1; 2; 3Þ are polynomials in the invariants of A, B, A2 This study is restricted to second order fluids for which ϕk ðk ¼ 0; 1; 2; 3ị are constant and k k ẳ 4; 5; ị are zero So that the stress components are as follows: rr ẳ Arr ỵ Arr ỵ Brr ð14Þ Using Eqs (12)–(14) the equations of motion reduce to: 15ị Q5 12ị z L ur uz ẳ  V 9ị ẳ A ỵ A ỵ B uz ị uz ị ỵ uz ∂r ∂z
 ∂P ∂τzr zz ỵ ẳ ỵ rz ỵ z r r ∂z ð8Þ L2 ∂P ϕ 00 2 f ρVL ρV ∂z   ϕ r 00 000 00 ỵ 22 14f f ỵ f f L ρL   ϕ3 r 00 000 00 000 ỵ 11f f ỵ f f ỵ f f L L 4f f ẳ  ð16Þ The pressure term can be eliminated by differentiating Eq (15) with respect to z and Eq (16) with respect to r and subtracting the resulting equations This gives the following equations: 000  2f f ¼ 00 f iv Re 00 000  K ð4f f þ 2f f iv Þ 000 ð17Þ  K 4f f ỵ 2f f iv ỵ 2f f v ị Where K ẳ L is the injection Reynolds ; K2 ¼ ρL2 number For K ¼ 0, the equation turned to: 000 00 000 f iv ỵ Re f f  K Re 4f f ỵ 2f f iv ị ¼ ð18Þ The boundary conditions are: f ð0Þ ¼ 0; f ð0Þ ¼ 0; f ð1Þ ¼ 1; f 1ị ẳ 0: 19ị 2.2 Heat transfer analysis The energy equation for the present problem with viscous dissipation in non-dimensional form is given by:   ∂T ∂T ỵ uz C ur 20ị ẳ k2 T ỵ r z   ur ur uz ur uz ỵ ỵ zz ỵ rz ỵ ẳ rr r r ∂z ∂z ∂r ð21Þ Please cite this article as: S Sepasgozar, et al., Application of differential transformation method (DTM) for heat and mass transfer in a porous channel, Propulsion and Power Research (2017), http://dx.doi.org/10.1016/j.jppr.2017.01.001 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 4 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 S Sepasgozar et al Here ur ; uz are the velocity components in the r and z directions and V is the injection velocity; ρ; P; T; C; k are the density, pressure, temperature, specific heat, and heat conduction coefficient of fluid, respectively φ is the dissipation function Letting the blade wall (z¼ 0) temperature distribution be  n P Tw ẳ T0 ỵ Cn Lr nẳ0 And assuming the uid temperature to have the form of Ref [1] T ¼ T0 ỵ X Cn r n nẳ0 L qn ðηÞ ð22Þ Where T is the temperature of the incoming coolant (z¼ L) and neglecting dissipation effect the following equations and boundary conditions are obtained: 00 qn  Pr Re ð f qn  2f qn Þ ¼ 0; ð23Þ ðn ¼ 0; 2; 3; 4; :::Þ qn 0ị ẳ 1; qn 1ị ẳ 24ị Fundamentals of differential transform method [20] In Eq (6), f ðηÞ is the original function and F ðk Þ is the transformed function which is called the T-function (it is also called the spectrum of the f ðηÞ at η ¼ η0 , in the k domain) The differential inverse transformation of F ðkÞ is defined as: X F kị  ịk 26ị kẳ0 Table method Some of the basic operations of differential transformation Original function Transformed function f ị ẳ gị7hị n f ị ẳ d dg ị n f ị ẳ gịhị f ị ẳ sin ỵ ị F ẵk ẳ Gẵk7H ẵk ! F ẵk ẳ kỵnị k! Gẵk ỵ n Pk F ½k ¼ m  F ½mH ½k m F ẵk ẳ f ị ẳ cos ỵ ị F ẵk ẳ f ị ẳ e F ị ẳ ỵ ịm F ẵk ẳ f ị ẳ m F ẵk ẳ k k k! sin ỵ ị k k k! cos ỵ ị k k! m m  1ị :::m  kỵ1ị k! ( F ẵk ẳ k  mị ẳ Eq (27) implies that the concept of the differential transformation is derived from Taylor's series expansion, but the method does not evaluate the derivatives symbolically However, relative derivatives are calculated by an iterative procedure that is described by the transformed equations of the original functions From the definitions of Eqs (25) and (26), it is easily proven that the transformed functions comply with the basic mathematical operations shown in below In real applications, the function f ðηÞ in Eq (27) is expressed by a nite series and can be written as: f ị ẳ N X kẳm ; k am 28ị kẳ0   P Eq (9) implies that f ị ẳ 1k ẳ Nỵ1 F kị  ịk is negligibly small, where N is series size Theorems to be used in the transformation procedure, which can be evaluated from Eqs (25) and (26), are given below (Table 1) Now differential transformation method into governing equations has been applied Taking the differential transforms of Eqs (18) and (23) with respect to χ and considering H ẳ gives: k ỵ 1ịk ỵ 2ịk ỵ 3ịk ỵ 4ịF ẵk ỵ 4 ! k X F ẵk  mm ỵ 1ị ỵ Re m ỵ 2ịm ỵ 3ịF ẵm ỵ 3 mẳ0 k  m ỵ 1ịk  m ỵ 2ị k XB C  Re K @ F ẵk  m ỵ 2m ỵ 1ị A mẳ0 m ỵ 2ịm ỵ 3ịF ẵm ỵ 3 k  m ỵ 1ịF ẵk  m ỵ 1 k X B m ỵ 1ịm ỵ 2ịm ỵ 3ị C  Re K @ Aẳ0 mẳ0 m ỵ 4ịF ẵm ỵ 4 29ị F ẵ0 ẳ 0; F ẵ1 ẳ 0; 30ị F ẵ2 ẳ ; F ẵ3 ẳ k ỵ 1ịk ỵ 2ịQn ẵk ỵ 2 k X  nPrRe m ỵ 1ịF ẵm ỵ 1Qn ẵk  m ị mẳ0 ỵ 2PrRe k X 31ị m ỵ 1ị Qn ẵm ỵ 1F ẵk  m ị ẳ mẳ0 Qn ẵ0 ẳ 1; ; F ðkÞðη  η0 Þk Solution with DTM Basic definitions and operations of differential transformation are introduced as follows Differential transformation of the function f ðηÞ is defined as follows:
 dk f ðηÞ F k ị ẳ 25ị k! dk ẳ f ị ẳ by combining Eq (25) and Eq (26), f ðηÞ can be obtained: 
k X d f ðηÞ ð η  η0 Þk ð27Þ f ðηÞ ¼ dηk η ¼ η0 k! k¼0 Qn ½1 ¼ λ ð32Þ where F ðkÞ; Qn ðkÞ are the differential transforms of f ðηÞ; qn ðηÞ and α; β; λ are constants which can be obtained through boundary condition, Eqs (19) and (24) This problem can be solved as followed: Please cite this article as: S Sepasgozar, et al., Application of differential transformation method (DTM) for heat and mass transfer in a porous channel, Propulsion and Power Research (2017), http://dx.doi.org/10.1016/j.jppr.2017.01.001 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 Application of differential transformation method (DTM) for heat and mass transfer in a porous channel 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 F ẵ0 ẳ F ẵ1 ẳ F ị ẳ ỵ ỵ ỵ Re K ị   2 ỵ 65 Re K ỵ 24 Re K ! 2 Re ỵ 44  30 Re K ỵ þ ::: 3 þ 64 Re K F ẵ2 ẳ F ẵ3 ẳ F ẵ4 ẳ Re K F ½5 ¼ F ½6 ¼ 2  30 Re ỵ 44 Re K 3 ỵ 64 Re K α β 22  70 Re β2  105 α Re2 K β 3 2 1632 ỵ 132 35 Re K ỵ 35 Re K 4 ỵ 265 Re K 2 34  11 35 Re K αβ  35 Re K α β 3 7632 ỵ 1752 35 Re K ỵ 35 Re K 5 ỵ 768 Re K F ẵ7 ẳ F ẵ8 ẳ 2 Re K ỵ 24 Re K 33ị ỵ 2 2 45 Pr Re n ỵ 15 Pr þ 15 Pr Re2 n K β2 Re2 nK Pr Re2 K ỵ 45 Pr Re3 nK α2 β  15  15 Pr Re2 n ! ỵ::: Qn ẵ1 ẳ 35ị   Qn ị ẳ þ λη þ 13 Pr Re nα η3   þ 16 Pr Re n αλ þ 14 Pr Re n β  16 Pr Re αλ η4 ! 20 Pr Re n ỵ Pr Re nK ỵ  10 Pr Re Qn ẵ0 ẳ 36ị by substituting the boundary condition from Eqs (19) and (24) into Eqs (35) and (36) in point η ¼ 1, it can be obtained the values of α; β; λ Qn ½2 ¼ Qn ½3 ¼ 13 PrRenα Qn ½4 ¼ 16 PrRen ỵ 14 PrRen  16 PrRe Qn ẵ5 ẳ 20 PrRen ỵ 25 PrRe2 nK F 1ị ẳ ỵ ỵ þ ð2 Re K αβÞ   2 ỵ 65 Re K ỵ 24 Re K α β   2 3 64 ỵ  30 Re þ 44 Re K αβ þ Re K α β  10 PrReβλ Qn ẵ6 ẳ 45 Pr Re2 n2 ỵ 15 PrRe2 nK ỵ 15 PrRe2 n K ỵ 45 PrRe3 nK 2  15 PrRe2 K αβλ  15 Pr2 Re2 n 1 Qn ẵ7 ẳ 126 Pr Re2 n2 ỵ 28 Pr Re2 n2 βα  126 Pr2 Re2 nα2 λ þ 17 PrRe2 nλK β2 þ 47 PrRe3 nλK α2 β  210 PrRe2 nαβ 2 3 64 ỵ 44 35 PrRe nK ỵ 35 PrRe nK ỵ ::: ẳ 37ị F 1ị ẳ ỵ ỵ ỵ Re K ị   2 ỵ 65 Re K ỵ 24 Re K α β ð34Þ   2 2 35 PrRe λK β  35 PrRe λK α β 2 2 2 21 Pr Re n ỵ 63 Pr Re 13 Qn ẵ8 ẳ 70 Pr Re2 n2 ỵ 210 Pr Re3 n2 K α2 β 53  840 Pr2 Re2 nαβλ þ 224 Pr2 Re2 n2 β2 2  280 PrRe2 n ỵ 33 35 PrRe nK 3 ỵ 48 35 PrRe nλK α β  560 PrRe nβ 11  420 PrRe3 nα2 K β þ 33 70 PrRe nK β 2 4 32 ỵ 204 35 PrRe nK ỵ PrRe nK 2 ỵ 840 PrRe2  11 35 PrRe λK αβ 3  16 35 PrRe λK α β  Pr Re K α βn 1  28 Pr2 Re2 n2 ỵ 24 Pr2 Re2 The above process is continuous By substituting Eqs (33) and (34) into the main equation based on DTM, it can be obtained that the closed form of the solutions is: þ 2 Re αβ þ 44  30 Re K αβ ! 3 ỵ 64 Re K ỵ ::: ẳ 38ị   Qn 1ị ẳ þ λ þ 13 Pr Re nα   þ 16 Pr Re n ỵ 14 Pr Re n  16 Pr Re 3  ỵ 20 Pr Re n ỵ Pr Re2 nK αβ  10 Pr Re βλ  2 25 ỵ 45 Pr Re n ỵ 15 Pr Re2 nK αβλ Pr Re2 n K ỵ Pr Re3 nK α2 β 5  2 Pr Re K αβλ Pr2 Re2 α2 n  15 15 ỵ ::: ẳ ỵ 39ị By solving Eqs (37)(39) gives the values of α; β; λ By substituting obtained α; β; λ into Eqs (35) and (36), it can be obtained the expression of F ðηÞ; Qn ðηÞ Please cite this article as: S Sepasgozar, et al., Application of differential transformation method (DTM) for heat and mass transfer in a porous channel, Propulsion and Power Research (2017), http://dx.doi.org/10.1016/j.jppr.2017.01.001 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 6 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 S Sepasgozar et al Table f ðηÞ values in different steps of differential transformation method at: Re ¼ 0:5; K ¼ 0:01; Pr ¼ 1; n ¼ η N¼ Error N ¼6 Error 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.031767 0.116853 0.239782 0.384867 0.536216 0.677728 0.793091 0.865789 0.879096 0.00214 0.007411 0.013885 0.01927 0.020762 0.014878 0.002704 0.037278 0.095201 0 0.031767 0.116874 0.239975 0.385834 0.539633 0.68739 0.816482 0.916281 0.978892 0.002141 0.007432 0.014078 0.020237 0.024179 0.024541 0.020687 0.013214 0.004596 η N¼ Error N ¼10 Error 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.029687 0.109737 0.226588 0.366737 0.516936 0.664418 0.797125 0.903891 0.974563 0.000059 0.000295 0.000692 0.00114 0.001481 0.001569 0.00133 0.000823 0.000266 0 0.02955 0.109271 0.225724 0.365524 0.515532 0.66306 0.796061 0.903288 0.974391 0.000076 0.000171 0.000173 0.000073 0.000077 0.000211 0.000267 0.00022 0.000094 Figure Velocity component profile ðf Þ for variable Re at K ¼ 0:01 Figure Temperature K ¼ 0:01; Re ¼ profile   qn for variable n Figure Velocity component profile ðf Þ for variable Re at K ¼ 0:01 Results and discussion The objective of the present study was to apply the differential transformation method to obtain an explicit analytic solution of heat transfer equation of a non-Newtonian fluid flow in an axisymmetric channel with a porous wall for turbine cooling applications (Figure 1) The results that obtained by the differential transformation method were well matched with the results carried out by the numerical solution obtained by a four-   Figure Temperature profile qn for at K ¼ 0:01 Please cite this article as: S Sepasgozar, et al., Application of differential transformation method (DTM) for heat and mass transfer in a porous channel, Propulsion and Power Research (2017), http://dx.doi.org/10.1016/j.jppr.2017.01.001 at 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 Application of differential transformation method (DTM) for heat and mass transfer in a porous channel 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 order Runge–Kutta method as shown in Table In these tables, the error is introduced as followed:    Error ¼  f ðηÞ  f ðηÞ ð40Þ NM DTM As shown in Figures and for constant value of K , velocity values increase as a result of Reynolds number increase At low Reynolds numbers the velocity profile exhibit center line symmetry indicating a Poiseuille flow for nonNewtonian fluids At higher Reynolds numbers the maximum velocity point is shifted to the solid wall where shear stress 00 becomes larger as the Reynolds number grows Since f ð0Þ is measure of friction force, it is advisable to use viscoinelastic fluids as a coolant fluid for industrial gas turbine engines In Figure temperature profiles for different values of power law index (n) are shown It shows that for constant value of η temperature increase if power law index decrease Increasing Reynolds number leads to increasing the curve of temperature profile and decreasing of qn ðηÞ values as shown in Figure Conclusion In this paper, non-Newtonian fluid flow in an axisymmetric channel with a porous wall for turbine cooling applications problem (Figure 1) has been solved via a sort of analytical method, differential transformation method (DTM), and this problem has been also solved by a numerical method (the Runge–Kutta method of order 4) The observed good agreement between the present method and numerical results shows that differential transformation method is a powerful approach for solving nonlinear differential equation such as this problem Some facts were observed through the results Friction force increased as a result of Reynolds numbers increase Increasing power law index leads to decreasing temperature value between two plates Nusselt number increases as a result of Reynolds number increase Prandtl number increase leads to Nusselt number increase as well Also Nusselt number increase as an effect made by power law increase References [1] L.L Debruge, L.S Han, Heat transfer in a channel with a porous 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porous channel, Propulsion and Power Research (2017), http://dx.doi.org/10.1016/j.jppr.2017.01.001 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 ... Schematic diagram of the physical system Please cite this article as: S Sepasgozar, et al., Application of differential transformation method (DTM) for heat and mass transfer in a porous channel, ... ð21Þ Please cite this article as: S Sepasgozar, et al., Application of differential transformation method (DTM) for heat and mass transfer in a porous channel, Propulsion and Power Research (2017),... http://dx.doi.org/10.1177/1740349911433468 Please cite this article as: S Sepasgozar, et al., Application of differential transformation method (DTM) for heat and mass transfer in a porous channel, Propulsion and Power Research (2017),