Spectral collocation method for radiative–conductive porous fin with temperature dependent properties

Energy Conversion and Management 111 (2016) 279–288

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Energy Conversion and Management journal homepage: www.elsevier.com/locate/enconman

Spectral collocation method for radiative–conductive porous fin with temperature dependent properties Jing Ma a,⇑, Yasong Sun b,⇑, Benwen Li c, Hao Chen a a

Key Laboratory of Shaanxi Province for Development and Application of New Transportation Energy, School of Automobile, Chang’an University, Xi’an 710064, China Beijing Key Laboratory of Multiphase Flow and Heat Transfer for Low Grade Energy, North China Electric Power University, Beijing 102206, China c Institute of Thermal Engineering, School of Energy and Power Engineering, Dalian University of Technology, Dalian 116024, China b

a r t i c l e

i n f o

Article history: Received 23 August 2015 Accepted 23 December 2015

Keywords: Spectral collocation method Porous fin Combined convection and radiation Temperature dependent properties Optimization

a b s t r a c t In this work, spectral collocation method is presented to predict the thermal performance of convective– radiative porous fin with temperature dependent convective heat transfer coefficient, fin surface emissivity and internal heat generation. In this approach, the dimensionless fin temperature distribution is approximated by Lagrange interpolation polynomials at spectral collocation points. The differential form of the governing equation is formulated by the Darcy model, and is transformed to a matrix form of algebraic equation. The accuracy of the SCM is verified by compared with numerical results by the homotopy perturbation method and the finite volume method. The node convergence rate of the SCM approximately follows an exponential law, and the computational time of the SCM do not significantly increase with the increasing of collocation points. The effects of various geometric and thermo-physical parameters on the dimensionless fin temperature, fin efficiency and heat transfer rate are comprehensively analyzed. In addition, optimum design analysis is also carried out. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction Fins are generally used as a heat dissipation technique to enhance heat transfer between hot surface and the environment [1]. They are not limited with the traditional thermal engineering applications, such as air conditioning, refrigeration and internal combustion engines. They are also very hopeful in cooling of electronics and heat dissipation systems of space vehicles [2]. Using porous fins is one way for increasing the heat dissipation in a constant volume of equipment [2]. Early in 2001, Kiwan and Al-Nimr [2] adopted the finite element code FIDAP 7.06a to investigate the porous fin to enhance heat transfer from a given surface. The thermal performance of the porous fin was also estimated and compared with that of the conventional solid fin. Kiwan [3] used fourth order Runge–Kutta method to solve porous fin in natural convection environment. Kiwan and Zeitoun [4] developed the finite volume model for predicting the thermal performance of porous fin mounted around the inner cylinder of the annulus between two concentric cylinders. They found that the porous fin enhanced heat transfer rate by 75% compared with the traditional solid fin. Kundu and Bhanjia [5] numerically investigated the thermal performance and optimum ⇑ Corresponding authors. Tel.: +86 29 82334430 (J. Ma). E-mail addresses: [email protected] (J. Ma), [email protected] (Y. Sun). http://dx.doi.org/10.1016/j.enconman.2015.12.054 0196-8904/Ó 2015 Elsevier Ltd. All rights reserved.

dimensions of rectangular porous fin by a domain decomposition method (ADM). Kundu et al. [6] also employed the ADM to solve the porous fin heat transfer in various profiles. They concluded that the heat transfer rate through the porous fin was significantly increased for any geometric fin compared with the solid fin. In the last five years, other numerical methods, such as the differential transformation method (DTM) [7], the least square method (LSM) [8–10], and the homotopy perturbation method (HPM) [11–13], were introduced to solve the nonlinear porous fin heat transfer exposed in convective environment. In the case of high temperature, radiation dissipation from fin surface plays a critical role. The emissivity of a real surface is not a constant, but rather varies with temperature [14]. Kiwan [15] developed a simple model to analyze the effect of radiation losses on the heat transfer from a porous fin attached to a vertical isothermal surface. Gorla and Bakier [16] used the fourth order Runge– Kutta method (RKM) to study the natural convection and radiation in the rectangular profile fin. Their results showed that the radiation transferred more heat than a similar model without radiation. Das [17] extended the RKM to analyze coupled conductive, convective and radiative heat transfer in the cylindrical porous fin. Bhanja et al. [18] employed an ADM model to determine temperature distribution, fin efficiency and optimum design parameters for the moving porous fin losing heat by simultaneous convection and radiation. Hatami and Ganji [19] studied thermal behavior of

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Nomenclature A Ai;j Bi Bi c 1 , c2 , c 3 c0i cp ð1Þ Di;j

cross-sectional area of the fin m2 spectral coefficient matrix defined in Eq. (24) Biot number spectral coefficient matrix defined in Eq. (25) constants of internal heat generation coefficient of Lagrange interpolation polynomials 1 specific heat capacity of fluid (J kg K1 ) the first order derivative matrix

U

vw

V e V wi W x X

dimensionless volume velocity of fluid passing through the porous fin (m s1 ) function defined in Eq. (37) function defined in Eq. (38) integral matrix coefficient width of the fin (m) coordinate in x-direction (m) dimensionless axial coordinate

ð2Þ

Di;j F i;j g Gi h hb

the second order derivative matrix spectral coefficient matrix defined in Eq. (28) gravitational acceleration (m s2 ) spectral coefficient matrix defined in Eq. (29) convective heat transfer coefficient (W m2 K) convective heat transfer coefficient at fin base (W m2 K) Lagrange interpolation polynomials hi K permeability of the porous fin (m2 ) L length of the fin (m) m parameter of heat transfer coefficient N number of grids Ncc convective–conductive parameter Nrc radiative–conductive parameter q conductive heat flux (W  m1 ) q internal heat generation (W m3 ) q0 internal heat generation at ambient temperature (W m3 ) Q dimensionless heat transfer rate through the porous fin Q0 dimensionless internal heat generation at ambient temperature Q ideal ideal fin heat transfer rate R1 dimensionless flow parameter R2 dimensionless convective parameter R3 dimensionless radiative parameter Rbenchmark benchmark solution RSCM numerical solution obtained by SCM si Chebyshev–Gauss–Lobatto collocation points Sh porous parameter source term at control volume center Sp T temperature (K) Tb temperature at the base (K) T1 temperature at ambient fluid (K)

longitudinal convective–radiative porous fins with temperature dependent internal heat generation and variable section profiles by LSM. Moradi et al. [20] adopted the DTM to assess the thermal performance of convective–radiative porous fin of triangular profile with temperature dependent thermal conductivity. Recently, Atouei et al. [21] used two analytical methods, namely the LSM and the collocation method, to analyze the temperature distribution in the semi-spherical fin with temperature dependent heat generation and thermal conductivity. It is well known that the fin heat transfer rate decreases with the increase of fin length; and thus, the entire fin surface may not be equally utilized. So, many fin designers ongoing effort to look for the optimum shape of the fin that will maximum the heat transfer rate for a specified fin or minimize the fin volume for a given heat transfer rate. Jany and Bejian [22] used the generalization of Schmidt’s argument to investigate the optimum shapes for straight fins with temperature dependent conductivity. Copiello and Fabbri [23] investigated the optimization of heat transfer rate from wavy fins cooled by a laminar flow under conditions of forced convection and from a multi-objective point of view. Arslanturk [24] developed simple correlation equations to

Greek symbols a coefficient of surface emissivity b coefficient of thermal expansion (K1 ) cx finite difference weighing factor d semi-thickness of the porous fin (m) dx distance between two adjacent grid points Dx width of the control volume e surface emissivity ea surface emissivity at ambient temperature eSCM integral averaged relative error g fin efficiency H dimensionless temperature H1 dimensionless ambient temperature efficient thermal conductivity (W m1 K1 ) keff kf thermal conductivity of fluid (W m1 K1 ) kr thermal conductivity ratio thermal conductivity of solid (W m1 K1 ) ks qf density of fluid (kg m3 ) r Stefan–Boltzmann constant (W m2 K4 ) u porosity of the porous medium w dimensionless fin length Subscripts e; w east and west boundary of control volume P E; W east and west neighbors of control volume P P control volume i; j solution node indexes Superscript  the last iterative value

optimize annular fins with cross section. Sharqawy and Zubair [25] found optimized thickness for the maximum heat transfer rate in the fully wet and partially wet fins considering with heat and mass transfer. Kundu et al. [26–28] obtained optimum design of fin with different profiles, like step reduction in cross section profile, T-shape profile, trapezoidal profile etc. Furthermore, Kundu et al. [6] obtained performance and optimum design of convective porous fin with various profiles. Aziz and Beers-Green [29] investigated the optimum design of a longitudinal rectangular fin attached to a convectively heated wall of finite thickness. Recently, Mosayebidorcheh [30] determined the optimal dimensions of convective–radiative longitudinal fins with temperature-dependent properties and different section shapes. The heat transfer rate, effectiveness and efficiency for optimum fin design are also obtained for this problem. Different from above mentioned methods, the spectral collocation method (SCM) is a high-order numerical method which is based on Lagrange interpolation polynomials and spectral collocation points [31]. It can provide exponential convergence and high accuracy with relative few grid points [32]. For example, for one dimensional heat transfer, the CSM can achieve ten digits of

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accuracy with 16 collocation points while the finite different method would get three digits of accuracy [33]. Besides, the mathematical implementation of the CSM is relatively easy. Due to the high accuracy and mathematical simplicity, SCM has been taken as an efficient tool in science and engineering computations, such as computational fluid dynamics [34], magneto-hydrodynamics [35], and thermal radiation heat transfer [36]. Recently, Sun et al. [37,38] used SCM to analyze the radiative–convective solid fin heat transfer. To the best of our knowledge, no research to date has aimed at using SCM to solve convective–radiative heat transfer in the porous fin with temperature dependent internal heat generation and properties. The main objective of this work is to evaluate the thermal performance of convective–radiative porous fin with temperature dependent internal heat generation, convective heat transfer coefficient and surface emissivity. In the following, the physical model and mathematical formulations of convective–radiative porous fin are presented in the second section. The accuracy and node convergence rate of SCM are demonstrated in the third section. In addition, dimensionless temperature and fin efficiency are also discussed for various values of dimensionless parameters in the third section. Finally, the conclusions are summarized in the last section.

The present study considers a rectangular porous fin profile with temperature dependent internal heat generation and properties. The dimensions of this porous fin take the length of L, the width of W and the thickness of 2d. The porous nature allows the flow of infiltration through the fin. The hot fin surface loses heat through both convection and radiation. Radiation would play a more remarkable role if the forced convection is weak or absent or when only radiation occurs. Similar as Ref. [8], the following assumptions are made to obtain the governing equations of the prescribed problem: (1) The porous medium is isotropic, homogeneous and saturated with single-phase fluid. (2) The surface of porous fin is gray and diffuse. (3) Darcy formulation is utilized for simulating the interaction between the porous medium and the fluid. (4) The transverse Biot number is assumed to be very small, and the temperature variation in the transverse direction is neglected [39]. Thus, heat conduction is assumed to occur in the longitudinal direction. (5) Radiative heat transfer between the fin and the tip is neglected. (6) Convective heat transfer coefficient, fin surface emissivity and internal heat generation are assumed to be the functions of temperature and can be defined as follows:

T  T1 Tb  T1

m ð1Þ

   T  T1 e ¼ ea 1 þ a Tb  T1 " q ¼ q0 1 þ c1

dq þ q A ¼ qf v w Wcp ðT  T 1 Þ þ 2Whð1  uÞðT  T 1 Þ dx þ 2W erðT 4  T 41 Þ

ð4Þ

where A ¼ dW is the cross section area, and v w is the velocity of fluid passing through the porous fin that can be determined from Darcy model as follows [3]:

vw ¼

gKb

mf

ðT  T 1 Þ

ð5Þ

where g is the gravitational acceleration, K is the permeability of the porous fin, b is the coefficient of thermal expansion, and v f is the kinematic viscosity of the fluid. According to the Fourier’s law, the conductive heat flux can be written as:

q ¼ keff A

dT dx

ð6Þ

where keff is the effective thermal conductivity of the porous fin and can be obtained by the following equation,

ð7Þ

where u is the porosity of the porous fin, kf and ks are the thermal conductivities of fluid and solid, respectively. Substituting Eqs. (5)–(7) into Eq. (4) yields,

2.1. Physical and mathematical models





keff ¼ ukf þ ð1  uÞks

2. Physical model and mathematical formulation

h ¼ hb

Based on the above assumptions, the steady state energy equation of the fin per unit width can be expressed as:

ð2Þ

  2  3 # T  T1 T  T1 T  T1 þ c2 þ c3 Tb  T1 Tb  T1 Tb  T1



ð3Þ

"

2

keff d

d T

þ q0 d 2

dx



1 þ c1

  2  3 # T  T1 T  T1 T  T1 þ c2 þ c3 Tb  T1 Tb  T1 Tb  T1

ðT  T 1 Þmþ1 vf ðT b  T 1 Þm      T  T1 þ 2ea 1 þ a rð1  uÞ T 4  T 41 Tb  T1

¼

qf cp gKb

ðT  T 1 Þ2 þ 2hb ð1  uÞ

ð8Þ

As shown in Fig. 1, the fin base is assumed to be constant temperature T b , and the fin tip is assumed adiabatic. Then, the boundary conditions for Eq. (8) can be written as:

Tjx¼0 ¼ T b

ð9Þ

 dT  ¼0 dx x¼L

ð10Þ

For convenience of analysis, the energy equation and the corresponding boundary conditions can be written in dimensionless form as:

"

2

d H dX

2

þ Q 0 1 þ c1







H  H1 H  H1 þ c2 1  H1 1  H1

2

 þ c3

H  H1 1  H1

3 #

¼ Sh ðH  H1 Þ2 þ Ncc ðH  H1 Þmþ1     H  H1  4 þ Nrc 1 þ a H  H41 1  H1

ð11Þ

HjX¼0 ¼ 1

ð12Þ

 dH ¼0 dX X¼1

ð13Þ

where the dimensionless parameters are defined as follows,

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Fig. 1. Schematic of the convective–radiative porous fin.

H ¼ TTb ; R1 ¼ Sh ¼

H1 ¼ TT1b ;

qf cp gKbT b hb v f

BiR1 kr w2

;

;

X ¼ xL ;

w ¼ dL ; kr ¼

R2 ¼ ð1H1 Þm ;

Ncc ¼

2

BiR2 ð1uÞ ; kr w 2

R3 ¼

Nrc ¼

keff ks

;

Bi ¼ hkbsd ;

2ea T 3b

r

ð14Þ

hb

BiR3 ð1uÞ ; kr w 2

1.0 0.8

h1

L2 q0

Q0 ¼ T

b keff

  q dH  Q¼ ¼ kr w ks T b dX X¼0

h3

0.6

h

According, the heat transfer rate through porous fin can be determined by Fourier’s law of heat conduction at the base,

h5

0.4

h7

0.2

ð15Þ

h9

0.0

The ideal fin heat transfer rate is realized if the entire fin surface is maintained at the base temperature and can be expressed as:

-0.2

  Q ideal ¼ Sh ð1  H1 Þ2 þ Ncc ð1  H1 Þmþ1 þ N rc ð1 þ aÞ 1  H41

-1.0

-0.8

-0.6

-0.4

-0.2

To indicate the effectiveness of the porous fin in transferring a given quantity of heat, the fin efficiency is defined as the ratio of the actual fin heat transfer rate to the ideal heat transfer rate,

g¼

Q

ð17Þ

Q ideal

2.2. Spectral collocation method In the theory of SCM, the dimensionless temperature can be approximated by Lagrange interpolation polynomials [32],

HðsÞ 

N X

Hðsi Þhi ðsÞ

ð18Þ

i¼1

where si are the Chebyshev–Gauss–Lobatto (CGL) collocation points; hi are the Lagrange interpolation polynomials, and can be calculated by the barycentric interpolation formula [32],



si ¼  cos

pði  1Þ N1

 ;

i ¼ 1; 2;    ; N

ð1Þi1 c0i =ðs  si Þ hi ðsÞ ¼ PN 0 j¼1 c j =ðs  sj Þ

ð19Þ

ð20Þ



0.2

0.4

0.6

0.8

1.0

Fig. 2. Lagrange interpolation polynomials of degree N = 9 which are based on CGL collocation points.

hj ðsi Þ ¼

1; i – j 0;

i–j

ð22Þ

which ensures that the expansions coefficients in Eq. (18) coincide with nodal values. Substituting Eq. (18) into Eq. (11), the spectral discretization of energy equation can be expressed in matrix form as follows, N X Ai;j Hj ¼ Bi ;

i ¼ 1; 2;    ; N

ð23Þ

i¼1

where Ai;j and Bi are, ð2Þ

Ai;j ¼ 4Di;j

ð24Þ

2

mþ1 Bi ¼ Sh Hi  H1 þ Ncc Hi  H1     i Hi  H1 h  4 ðHi Þ  H41 þ Nrc 1 þ a 1  H1 "          # Hi  H1 Hi  H1 2 Hi  H1 3  Q 0 1 þ c1 þ c2 þ c3 1  H1 1  H1 1  H1 ð25Þ

where,

c0i

0.0

s

ð16Þ



1=2;

i ¼ 1; N

where H denotes the latest iterative value of dimensionless temð2Þ

ð21Þ

perature, Di;j are entries of the second order derivative coefficient

Fig. 2 shows the distributions of Lagrange interpolation polynomials of degree N ¼ 9. As shown in Fig. 2, the black nodes are the CGL collocation points, and the Lagrange interpolation polynomials at the CGL collocation points meet the following relationship,

matrix. Before solving the matrix form of energy equation, it is required to impose the discretized boundary conditions. After importing the boundary conditions, the following matrix form equation is obtained,

¼

1;

i ¼ 2; 3;    ; N  1

J. Ma et al. / Energy Conversion and Management 111 (2016) 279–288

1 1 Dx ¼ ðdxÞe þ ðdxÞw 2 2

283

ð32Þ

In the control volume, we assume that there is a linear relationship among the two volume-interface temperatures and volumecenter temperature,

ð26Þ

HP ¼ cx HE þ ð1  cx ÞHW

ð33Þ

where cx is the finite difference weighting factor and its value is normally considered to be 0.5 for the central difference scheme. Imposing the Neumann boundary condition, the fin tip temperature can be expressed as [40] ð1Þ

where Di;j is the first order derivative coefficient matrix. Stripping the first row and column of the first matrix, Eq. (26) can be rearranged as, N 1 X F i;j Hjþ1 ¼ Gi ;

i ¼ 1; 2;    ; N  1

Aiþ1;jþ1 ; ð1Þ Di;jþ1 ;

( Gi ¼

i ¼ 1; 2;    ; N  2 i¼N1

Biþ1  Aiþ1;1 H1 ; ð1Þ

DN;1 H1 ;

i ¼ 1; 2;    ; N  2

i¼N1

ð28Þ

2.4. Optimization analysis

ð29Þ

The implementation of SCM for convective–radiative porous fin heat transfer can be carried out according to the following routines: Step 1. Mesh Chebyshev–Gauss–Lobatto collocation points with the solution domain, and compute the first order and the second order derivative coefficient matrices. Step 2. Make an initial guess for dimensionless temperature H . (For example, zero of temperature expect for boundaries.) Step 3. Assemble the spectral coefficient matrix Ai;j and Bi by Eqs. (24) and (25). Step 4. Impose the boundary conditions according to the algorithm described above. Step 5. Solve the matrix form of algebraic Eq. (27) to get new dimensionless temperature on each collocation points. Step 6. Terminate the iteration process if the criterion is satisfied. Otherwise, go back to step 3. Step 7. Calculate the fin heat transfer rate and fin efficiency. In this article, the maximum relative error of dimensionless

It is well known that, when the fin is attached in the primary surface, the weight and the cost of the equipment increase. Hence, to achieve the optimum thermal performance of the fin, it is necessary to know the optimum length. Fin designers are continuously making efforts to determine the optimum shape by two ways, namely, maximization of heat transfer rate for a given fin volume and minimization of fin volume for a given heat transfer rate. In the present study, optimization is performed in order to obtain the highest heat transfer rate for a specified fin volume. The volume of the fin per unit width can be represented in dimensionless form as [18], 2

U¼

hb V k2s

2

¼



@Q @Bi

Integrating Eq. (11) over the control volume and using the concept of FVM, we get,



1 1 1 1 þ HP ¼ HE þ HW  Sp Dx ðdxÞe ðdxÞw ðdxÞe ðdxÞw

ð30Þ

where

2 Sp ¼ Sh Hp  H1 þ Ncc ðHp  H1 Þmþ1     Hp  H1  4 þ Nrc 1 þ a Hp  H41 1  H1 "       # Hp  H1 Hp  H1 2 Hp  H1 3  Q 0 1 þ c1 þ c2 þ c3 1  H1 1  H1 1  H1 ð31Þ

ð35Þ

     @U @Q @U  ¼0 @w @w @Bi

ð36Þ

Eq. (36) can be solve by the constraint equation. Either the heat transfer rate or the fin volume can be considered as a constraint. The constraint can be expressed as follows,

( VðBi; wÞ ¼

2.3. Finite volume method

Bi w

To determine these optimum parameters, the optimality criterion is necessary which can be derived from Euler equation after eliminating the Lagrange multiplier,

temperature, 106 , is taken as the stopping criterion for iteration.



ð34Þ

ð27Þ

where,

(

ðDxÞ2 SN 2

In order to start the solution process of Eqs. (30)–(34), an initiation temperature guess should be made, for example, zero or a suitable value. The convergence criteria are required, and the final results are obtained after satisfying the necessary condition of convergence as in SCM.

i¼1

F i;j ¼

HN ¼ HN1 

 kr w ddXH X¼0 ¼ 0 Bi2 w

U ¼0

ð37Þ

Substituting Eqs. (15) and (35) into Eq. (36) yields,

      @ dH  @ dH  dH  ~ VðBi; wÞ ¼ Bi þ 2w þ 2 ¼0 @Bi dX X¼0 @w dX X¼0 dX X¼0 ð38Þ In order to determine the optimum fin dimensions, Eqs. (37) and (38) can be solved simultaneously by using generalized Newton–Raphson method. The approximate root values for Newton– Raphson method for any iteration can be obtained by using just previously iterative values. For the numerical calculation, it is required to satisfy the necessary and sufficient convergence criteria. To start the solution process, the initial guess values for the roots have been taken cautiously so that the convergence criteria for each iteration have been satisfied.

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J. Ma et al. / Energy Conversion and Management 111 (2016) 279–288

3. Results and discussions

locations are listed in Table 1. It is observed that the maximum relative error between SCM and FVM results is not more than

Based on the above physical and mathematical models, the numerical results, mainly the dimensionless temperature and the fin efficiency, are presented for the convective–radiative porous fin with temperature dependent internal heat generation and properties. Before exhibiting these numerical results by the present study, the accuracy and efficiency of the SCM method should be validated referring to the results in the literature. Unfortunately, it cannot be done directly due to unavailability of similar type in the existing literature. Thus, the porous fin only considering convection heat transfer with available results in the literature is chosen to validate the SCM method. In addition, the FVM is also used to solve the convective–radiative porous fin with temperature dependent internal heat generation and properties for the validation purpose. The numerical solutions of the SCM are performed using a computer with Inter Core I5 2.40 GHz processor and 2.0 GB RAM memory. The integral averaged relative error which is used for quantitatively analyzing the accuracy of the SCM is defined as follows,

1:12  105 . Fig. 4 depicts the node convergence characteristics of the SCM and the FVM against the number of grid points. It can be seen that,

R

eerror ¼

jRSCM ðxÞ  RBenchmark ðxÞjdx R RBenchmark ðxÞdx

ð39Þ

3.1. Accuracy and convergence rate of SCM A special case of convective porous fin heat transfer, which is also taken as a test case for the HPM [11], is adopted to validate the SCM. In this case, internal heat generation and surface radiation heat transfer are neglected; and the convective heat transfer coefficient is assumed to be constant. The validation of the SCM is indicated by the case of N rc ¼ Q 0 ¼ 0, N cc ¼ Sh ¼ 1, m ¼ 0 and H1 ¼ 0, and the accordance is given in Fig. 3. Fig. 3 shows that both SCM and HPM results are in good agreement with each other, and the integral averaged relative error is 1:75%. In order to further prove the accuracy of the SCM model, more comparisons between results of SCM and FVM are made. In this case, the surface of porous fin loses heat through both convection and radiation; internal heat generation, convective heat transfer and surface emissivity are functions of temperature; and the dimensionless parameters are Q 0 ¼ 0:1, c1 ¼ 0:4, c2 ¼ 0:4, c3 ¼ 0:4, N cc ¼ 1:0, N rc ¼ 1:0, Sh ¼ 1:0, a ¼ 0:25 and H1 ¼ 0:5. The dimensionless temperatures obtained by SCM and FVM at different

the integral averaged relative error of the FVM is about 1:88  104 for the value of N ¼ 15; while the integral averaged relative error of the SCM decreases to less than 2:04  109 for the same nodes. When the node number is greater than N ¼ 15, the integral averaged relative error of the SCM does not obviously decrease. Simultaneously, the integral averaged relative error of the FVM decreases linearly with increasing nodes, while the integral averaged relative error of SCM decreases very fast and approximately follows an exponential law trend (from N ¼ 5 to N ¼ 15). In addition, the CPU time of the SCM varies from 0.0765 s to 0.1531 s when the number of nodes increases from 5 to 21. It can be seen that the CPU time of the SCM is very short and the efficiency of the SCM for solving radiative–conductive porous fin is very high. 3.2. Dimensionless temperature In this section, Figs. 5–9 have presented the distribution of dimensionless temperatures obtained by SCM. Fig. 5 is plotted to show the dimensionless temperature variation with internal heat generation parameters Q 0 and ci , respectively. As shown in Fig. 5, the distribution of dimensionless temperature along the length of longitudinal porous fin increases with the increasing of Q 0 . An increase in Q 0 increases the temperature dependent internal heat generation as expressed in Eq. (3). The higher internal heat generation leads to higher dimensionless fin temperature because the fin must dissipate a larger amount of heat to surrounding environment. Similarly, as shown in Fig. 5, the dimensionless temperature of porous fin decreases when the internal heat generation parameter ci decrease. Fig. 6 presents the dimensionless temperature of porous fin with different values of parameter of heat transfer coefficient m and convective–conductive parameter N cc . The dashed curves marked with m ¼ 0 correspond to the dimensionless temperature of porous fin with constant heat transfer coefficient h ¼ hb ; while the solid curves marked with m ¼ 2 imply the dimensionless temperature of porous fin with temperature dependent heat transfer 2

1 coefficient h ¼ hb ðTTT Þ . As expected, temperature dependent T 1 b

heat transfer coefficient (m ¼ 2) can produce higher dimensionless temperature of porous fin compared with constant heat transfer coefficient (m ¼ 0). This is due to the fact that the averaged heat transfer coefficient of m ¼ 2 is lower than that of m ¼ 0. Fig. 6 also shows the dimensionless temperature of porous fin decreases as Table 1 Dimensionless temperatures at different locations obtained by FVM and SCM.

Fig. 3. Validation of SCM results with HPM results.

X

FVM

SCM

Relative error

0.00000000 0.01253604 0.04951557 0.10908426 0.18825510 0.28305813 0.38873953 0.50000000 0.61126047 0.71694187 0.81174490 0.89091574 0.95048443 0.98746396 1.00000000

1.00000000 0.99296920 0.97333126 0.94491590 0.91255093 0.88050208 0.85185397 0.82836032 0.81063382 0.79844596 0.79099424 0.78713394 0.78560024 0.78522722 0.78520003

1.00000000 0.99296818 0.97332759 0.94490895 0.91254110 0.88049028 0.85184139 0.82834746 0.81062076 0.79843297 0.79098143 0.78712124 0.78558757 0.78521455 0.78518736

0.00E+00 1.03E06 3.77E06 7.36E06 1.08E05 1.34E05 1.48E05 1.55E05 1.61E05 1.63E05 1.62E05 1.61E05 1.61E05 1.61E05 1.61E05

Bold values indicate the numerical results of SCM.

J. Ma et al. / Energy Conversion and Management 111 (2016) 279–288

285

Fig. 4. Node convergence rates and computation times of the SCM and the FVM.

Fig. 7. Dimensionless temperature distributions for different values of a and N rc .

Fig. 5. Dimensionless temperature distributions along the length of porous fin for different values of Q 0 and ci . Fig. 8. Dimensionless temperature distributions for different values of Sh .

Fig. 6. Dimensionless temperature distributions for different values of m and N cc .

the value of convective–conductive parameter N cc increases from 1 to 10. N cc is defined to be the ratio of heat dissipation by convection to heat transfer by conduction. As convective–conductive parameter N cc increases, convective heat dissipation from the surface of porous fin increases and thus it maintains low dimensionless temperature at the fin tip.

Fig. 9. Dimensionless temperature distributions for different values of H1 .

The effects of surface emissivity parameter a and radiative–conductive parameter N rc on the dimensionless temperatures of porous fin are illustrated in Fig. 7, while other dimensionless parameters are constant. It is illustrated that the dimensionless

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temperature of porous fin rapidly decreases when the surface emissivity parameter a or the radiative–conductive parameter N rc increases. The surface emissivity parameter a is proportional to the surface emissivity (see Eq. (2)); and N rc is the ratio of heat dissipation by radiation to heat transfer by conduction. The increasing surface emissivity parameter a or radiative–conductive parameter N rc strengthens radiative heat exchange between the surface of porous fin and the ambient environment. Fig. 8 shows the effect of variation of porous parameter Sh on the dimensionless temperature of porous fin. The values of Sh are varied with 0.5, 1.0 and 5.0, and the other dimensionless parameters are constant as Q 0 ¼ 0:1, c1 ¼ 0:4, c2 ¼ 0:4, c3 ¼ 0:4, N cc ¼ 1:0, N rc ¼ 1:0, m ¼ 2, a ¼ 0:25 and H1 ¼ 0:5. As shown in Fig. 8, the distribution of dimensionless temperature along the length of porous fin rapidly decreases as porous parameter Sh increases. The reason is that, as porous parameter Sh increases, the porous fin cools down fast and quickly reaches the ambient temperature. Fig. 9 illustrates how the dimensionless temperature of porous fin is affected by the dimensionless ambient temperature H1 . It is observed that, as the dimensionless ambient temperature H1 increases, convective and radiative heat dissipation from the porous fin surface decrease and thus keeping high dimensionless temperature of porous fin.

The efficiency of porous fin varying with radiative–conductive parameter N rc under different surface emissivity coefficients, a ¼ 0:25 and 0.50, are depicted in Fig. 12. It is noted that the porous fin efficiency considerably drops with the increasing of radiative–conductive parameter N rc ,. Compared with the descending trend in Fig. 10, this trend in Fig. 12 is more obvious. The reason is clear, radiative heat dissipation plays more obvious impact on the fin efficiency. It is due to that the radiative heat dissipation is proportional to the difference of fourth power of temperature between the ambient and the surface. It is also noticed that the porous fin efficiency significantly drops with the surface emissivity parameter a. The effects of porous parameter Sh and dimensionless ambient temperature H1 on the efficiency of porous fin are presented in Fig. 13. As shown in Fig. 13, as porous parameter Sh increases, the porous fin cools down fast and consequently leads to quickly reduction of the porous fin efficiency. On the other hand, for a certain value of dimensionless ambient temperature Sh , porous fin efficiency for the case of H1 ¼ 0:7 is higher than that for the case of H1 ¼ 0:5.

3.4. Optimum fin design

Fig. 10 exhibits the variation of porous fin efficiency with respect to the internal heat generation parameters Q 0 and ci . It is noted that the porous fin efficiency increases with the increasing internal heat generation parameters Q 0 and ci . As explained for Fig. 5, the higher internal heat generation leads to higher dimensionless temperature of porous fin. As expressed in Eq. (17), porous fin efficiency is directly related to dimensionless temperature H. Thus, the porous fin efficiency increases with the internal heat generation. Fig. 11 shows the effect of convective–conductive parameter N cc on porous fin efficiency for different values of heat transfer coefficient parameters m. It can be seen that, as the convective–conductive parameter N cc increases, it attributes more convective heat dissipation from the porous fin surface, and results in a decrease in the fin efficiency. As shown in Fig. 11, compared with the case of constant heat transfer coefficient (m ¼ 0), the porous fin heat transfer with temperature-dependent coefficient (m ¼ 2) is more efficient.

Fig. 14 shows the heat transfer rate in optimum condition for different ambient temperature, H1 ¼ 0:5 and H1 ¼ 0:7. As shown in Fig. 14, the heat transfer rate decreases as the ambient temperature increases. As expressed in Eq. (11), the parameter H1 can be decreased either by decreasing ambient temperature or by increasing base temperature of the fin surface. In both the conditions, the temperature difference between the fin surface and ambient becomes high. Therefore, both the convective and radiative heat transfer rates are augmented. The trend in the variation of loci of maximum heat transfer rate is found to be linear. It is also to be mentioned that the optimum heat transfer rate with low H1 is obtained at high aspect ratio w that indicates a higher fin thickness or lower fin length. Fig. 15 illustrates the variation in actual heat transfer rate with porosity parameter u. It is noticed that the actual heat transfer rate is increased by increasing porosity parameter u. The basic philosophy behind using porous fins is to increase the effective surface area through which heat is transferred to the ambient fluid. Although, effective thermal conductivity of the porous fin decreases, due to removal of solid material but this reduction is compensated with the increase in effective surface area. Thus,

Fig. 10. Efficiencies of porous fin for different values of Q 0 and ci .

Fig. 11. Efficiencies of porous fin for different values of N cc and m.

3.3. Fin efficiency

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Fig. 12. Efficiencies of porous fin for different values of N rc and a.

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Fig. 15. Prediction of the heat transfer rate through the porous fin as a function of w and /.

4. Conclusions A dimensionless mathematical model describing nonlinear heat transfer in a porous fin with temperature-dependent properties has been introduced and solved by the SCM. The SCM results have been compared with FVM solutions and available numerical results in the literature, and excellent agreement was found. The node convergence rate of the SCM approximately follows an exponential law, and the computational time of the SCM do not significantly increase with the increasing of collocation points. A small number of collocation points is needed to obtain accurate results; and thus, the method is very efficient in solving nonlinear porous fin heat transfer. Conclusions can be summarized as follows:

Fig. 13. Efficiencies of porous fin for different values of H1 and Sh .

 Comparison of the HPM results with the numerical outcomes shows that the SCM is convenient and powerful method in nonlinear porous fin heat transfer problems.  Effects of various dimensionless parameters on the distribution of dimensionless temperature are analyzed. Dimensionless temperature increases with Q 0 , ci , m and H1 . In contrast, the opposite trend has been found for parameters, N cc , N rc and Sh .  The efficiency of convective–radiative porous fin is improved with the increasing of Q 0 , ci , m and H1 . Meanwhile, the porous fin efficiency is also improved with the decreasing of a, N cc , N rc and Sh .  The heat transfer rate decreases with increase the ambient temperature H1 . For the entire conditions the heat transfer rate initially increased with reaches to a maximum value and then decreases. Moreover, the variation of loci of the maximum heat transfer rate is linear.

Acknowledgements This work was supported by the Natural Science Foundation of China (Nos. 51406014 and 51176026), and the Fundamental Research Funds for the Central Universities (No. 310822151021). References Fig. 14. Prediction of the heat transfer rate through the porous fin as a function of H1 and w.

the heat transfer rate is augmented remarkably with higher porosities. It is observed that the loci of optimum design for maximization of heat rate with w may be a straight line.

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