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Thermal analysis of straight rectangular fin using homotopy perturbation method
Alexandria Engineering Journal (2016) 55, 2269–2277
H O S T E D BY
Alexandria University
Alexandria Engineering Journal www.elsevier.com/locate/aej www.sciencedirect.com
ORIGINAL ARTICLE
Thermal analysis of straight rectangular fin using homotopy perturbation method Pranab Kanti Roy *, Ashis Mallick Department of Mechanical Engineering, Indian School of Mines, Dhanbad 826004, Jharkand, India Received 11 November 2015; revised 12 May 2016; accepted 18 May 2016 Available online 10 June 2016
KEYWORDS Sink temperature; Homotopy perturbation method; Efficiency; Variable surface emissivity
Abstract In this study a simple and accurate semi-analytical method called Homotopy Perturbation Method (HPM) is used for solving nonlinear energy equation in a straight rectangular fin. The thermal conductivity and surface emissivity are considered as temperature dependent with some constant source of internal heat generation. The problem is solved for two main cases of thermo geometric fin parameter Nc = 1 and for Nc = 0.25. The results are presented for the temperature distribution, efficiency and optimum dimensionless parameters are effective and convenient for practical fins. Also it is found that this method can achieve more suitable results as compared to the other methods available in the literature. Ó 2016 Faculty of Engineering, Alexandria University. 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/).
1. Introduction The fins are extended surface that are used to dissipate heat from the primary surface to the surrounding environment [1]. Heat transfer through the straight is common because of their low manufacturing cost and simplicity. The fins are sometimes subjected to both convective and radiative environments [2]. Under the circumstances of both convective and radiative modes of heat transfer the fin may generate heat internally due to passage of electric current as in electric filament or due to an atomic or chemical reaction as in an atomic reactor [3–6]. The internal heat generation may be assumed to be at constant rate with respect to the volume of fin. But to utilize the fin materials effectively the model includes the thermal conductivity varies * Corresponding author. Tel.: +91 9609551679. E-mail addresses: [email protected] (P.K. Roy), [email protected] (A. Mallick). Peer review under responsibility of Faculty of Engineering, Alexandria University.
linearly with temperature and in most cases the real surface of the fin material varies linearly with temperature. The energy equation for a convective–radiative fin along with heat generation with two variable thermal properties results in highly nonlinear terms. The resulting equation does not admit the exact solution. Consequently the energy equation with nonlinear terms is solved either numerically or using a variety of approximate analytical methods. Many different methods have been introduced to solve nonlinear problems, the Homotopy analysis method (HAM), Galerkin method (GM), Spectral collocation method (SCM), Adomian decomposition method (ADM). Homotopy perturbation method (HPM) is one of the semi analytical methods for solving nonlinear boundary value problem introduced by He [7]. An HPM possesses all the advantages of perturbation method and also it is independent of assumption of small parameter. As compared to the Adomian decomposition method, Homotopy perturbation method does not require the calculation of Adomian polynomials but it requires only the initial approximation [8–10]. Also the method does not require determination of ‘h-curves’ which
http://dx.doi.org/10.1016/j.aej.2016.05.020 1110-0168 Ó 2016 Faculty of Engineering, Alexandria University. 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/).
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P.K. Roy, A. Mallick
Nomenclature Nr C k ka es T P Tb Ta Ts b x
radiation–conduction parameter constant which represents the temperature temperature dependent thermal conductivity (W/ (mK)) thermal conductivity corresponding to ambient condition (W/(mK)) the surface emissivity corresponding to radiation sinks temperature, Ts temperature (K) fin perimeter (m) fin’s base temperature (K) convection sink temperature (K) sink temperature for radiation (K) length of the fin (m) axial coordinate of the entire fin (m)
makes it different from the Homotopy Analysis method. Bhowmik et al. [11] predicted the dimensions of rectangular and hyperbolic fins with variable thermal properties. Hazarika et al. [12] established an analytical method to predict the performance and design parameters of T-shape fin with simultaneous heat and mass transfer. Several researchers applied Collocation method (CM) to obtain the analytical solution of unsteady motion of fluid particles in conjunction with other numerical technique [13,14]. Rahimi-Gorji et al. [15] used Galerkin Method to obtain the heat transfer characteristics of micro channel heat sink cooled by different nanofluids in porous media. Pourmehran et al. [16] studied the thermal analysis of a fin shaped micro channel heat sink cooled by different nanofluids using least square method. Spectral collocation method has been used to study the performance parameter of simple and complex cross section of moving rod in thermal processing of continuous casting and rolling [17,18]. Sun et al. [19] used the Lagrange interpolation polynomials to approximate the temperature distribution at the spectral collocation points of nonlinear heat transfer problems. Ma et al. [20] presented the spectral collocation method to predict the thermal performance of porous fin with temperature dependent heat transfer coefficient, surface emissivity and internal heat generation. From the above discussion it is clear that in fin design application the selection of choice of conductive–convective parameter is very important for practicing engineer. The practical fins work at low values of conductive convective fin parameter. HPM is used to study the variation of surface emissivity, thermal conductivity parameter, heat generation number on the temperature distribution and efficiency of fin working under practical range of operating condition.
cross-sectional area of the entire fin (m2 ) dimensionless axial coordinate thermal conductivity parameters the surface emissivity parameters
Ac X A B
Greek symbols a slope of the thermal conductivity-temperature curve (K1 ) b slope of the surface emissivity-temperature curve (K1 ) h dimensionless temperature of the fin ha dimensionless convection sinks temperature hs dimensionless radiation sinks temperature r Stefan–Boltzmann constant (W=ðm4 K4 Þ) e emissivity
end side and the effect of heat transfer in vertical direction is neglected. The steady state one dimensional energy equation is given by d dT hPðT Ta Þ eðTÞrPðT4 T4s Þ kðTÞ þq¼0 ð1Þ dx dx Ac Ac With the insulated boundary condition dT ¼ 0 at dX
x¼0
ð2aÞ
T ¼ Tb
x¼b
ð2bÞ
at
For the better utilization of the fin materials the thermal conductivity of fin material is assumed to be linear function of temperature. Also the emissivity of all real surfaces is not constant, and it sometimes varies linearly with temperature. Therefore both of the parameters can written as below kðTÞ ¼ ka ½1 þ aðT Ta Þ
ð3Þ
eðTÞ ¼ es ½1 þ bðT Ts Þ
ð4Þ
h,
T
Conducon
dx
x
2. Mathematical formulations Let us consider rectangular fin geometry of various details as shown in Fig. 1. The thermal conductivity k and surface emissivity e of the fin materials are temperature dependent. The heat generated inside the fin material is assumed to be at the constant rate q. The fin is assumed to be insulated at the free
Convection
Radiation
Tb
b
X
Radiation sink temperature, Ts Convection sink temperature, Ta Figure 1
The geometry of straight rectangular fin.
Thermal analysis of straight rectangular fin
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In order to express Eq. (1) in non-dimensional forms, the following dimensionless parameters are defined as T h¼ Tb A ¼ aTb
Ta ha ¼ Tb
Ts hs ¼ Tb
B ¼ bTb
x X¼ b 2 hPb es rT3b Pb2 N2c ¼ Nr ¼ ka Ac ka Ac
Q¼
b2 q ka Tb ð5Þ
The formulation of the fin problem reduces to the following equations: 2 d2 h d2 h dh d2 h þ Ah 2 þ A Aha 2 N2c ðh ha Þ 2 dX dX dX dX 4 4 ð6Þ Nr ½1 þ Bðh hs Þ h hs þ Q ¼ 0 With the following boundary conditions:
For p ¼ 1 we obtain Hðh; 1Þ ¼ ½LðhÞ þ NðhÞ fðrÞ
ð14Þ
Using the perturbation technique by taking by taking into account of small values of p, then the solution of Eq. (10) can be written as h ¼ h0 þ ph þ p2 h2 þ p3 h3 þ p4 h4 þ
ð15Þ
The series converges for p ¼ 1 the solution for h can be given by h ¼ h0 þ h þ h2 þ h3 þ h4 þ
ð16Þ
4. HPM formulation Using Eq. (12), Eq. (6), can be written as in HPM form
dh ¼ 0 at X ¼ 0 dX
ð7aÞ
h ¼ 1 at X ¼ 1
ð7bÞ
Hðh; pÞ ¼ ð1 pÞLðh h0 Þ " 2 # dh 2 2 d h d2 h d2 h Aha dX 2 þ Ah 2 þ A dX 2 Nc ðh ha Þ þ p dX dX ¼0 Nr h4 h4s þ Bh5 Bh4s h Bhs h4 þ Bh4s ð17Þ
3. Homotopy Perturbation Method (HPM)
This can be arranged as
Homotopy perturbation method (HPM) is a semi-numerical method for solving linear or nonlinear, homogeneous or inhomogeneous boundary value problem first introduced by He [7]. As compared to the Adomian decomposition method this method does not require the calculation of Adomian polynomial and leads to convergent solution rapidly. Moreover this method requires only initial condition as input for its solution. To illustrate the basic idea of HPM according to He [7], consider the following nonlinear differential equation, AðhÞ fðrÞ ¼ 0;
r 2 X:
With the boundary conditions @h B h; ¼ 0 r 2 C; @X
ð8Þ
ð9Þ
where A is a general differential operator, B is a boundary operator, fðrÞ is a known analytic function, and C is the boundary of the domain X. The operator A can be generally divided into linear and nonlinear parts say LðhÞ and NðhÞ. Therefore Eq. (8) can be written as LðhÞ þ NðhÞ fðrÞ ¼ 0
ð10Þ
Define the Homotopy mðr; pÞ : X ½0; 1 ! R that satisfies Hðh; pÞ ¼ ð1 pÞLðh h0 Þ þ p½LðhÞ þ NðhÞ fðrÞ ¼ 0:
ð11Þ
Hðh; pÞ ¼ LðhÞ Lðh0 Þ þ pðh0 Þ " # dh 2 2 d2 h d2 h Aha dX Ah dX 2 þ A dX 2 Nc ðh ha Þ þp ¼0 Nr h4 h4s þ Bh5 Bh4s h Bhs h4 þ Bh4s ð18Þ Substituting h, from Eqs. (15) to (18), and separating the variables of identical power of p. p0 : h ¼ h0
ð19Þ
From the boundary condition (7) it is clear that the solution is becoming meaningless. Therefore in order to predict the solution physically meaningful the hð0Þ must be an arbitrary constant. This h0 ¼ C is taken as initial input for HPM. And p1 :
" # 0 2 2 2 Ah0 ddXh20 þ A dh Aha ddXh20 N2c ðh ha Þ d2 h1 dX þ ¼0 dX2 Nr h40 h4s þ Bh50 Bh4s h0 Bhs h4 þ Bh5s
ð20Þ
dh1 ¼ 0 at X ¼ 0; h1 ¼ 0 at X ¼ 0 dX
ð21Þ
And p : 2
This can be rearranged as Hðh; pÞ ¼ LðhÞ Lðh0 Þ þ pLðh0 Þ þ p½NðhÞ fðrÞ ¼ 0:
ð12Þ
2
d where L ¼ dX 2 and h0 is the initial approximation. Here h ¼ fðXÞ and p is the embedding parameter such that p 2 ½0; 1. It is clear that
Hðh; 0Þ ¼ ½LðhÞ Lðh0 Þ
ð13Þ
2 2 3 dh dh d2 h d h d2 h 2 d2 h2 4 A h0 dX21 þ h1 dX20 þ A 2 dX0 dX1 Aha dX21 Nc ðh ha Þ 5 þ ¼0 dX2 Nr 4h3 h1 þ 5Bh4 h1 Bh4 h1 Bhs 4h3 h1 0
0
s
0
ð22Þ
dh2 ¼ 0 at X ¼ 0; h2 ¼ 0 at X ¼ 0 dX
ð23Þ
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P.K. Roy, A. Mallick
p3 :
3 2 2 2 2 2 0 dh2 1 A h0 ddXh22 þ h1 ddXh21 þ h2 ddXh20 þ A 2 dh þ ðdh Þ dX dX dX 7 d2 h3 6 6 3 4 !7 2 2 3 2 þ6 7¼0 þ B 5h 4h h þ 6h h h þ 10h h 2 2 2 2 0 0 1 0 0 1 5 4 Ah d h2 N2 ðh h Þ N dX a dX2 a r c Bh4s h2 Bhs 4h30 h1 þ 6h20 h21
ð24Þ
dh3 ¼ 0 at X ¼ 0; h3 ¼ 0 at X ¼ 0 dX
ð25Þ
p4 :
2 2 3 2 2 2 A h0 ddXh23 þ h1 ddXh22 þ h2 ddXh21 þ h3 ddXh20 6 7 6 7 dh0 dh3 d2 h3 dh1 dh2 2 7 d2 h4 6 þ 2ð Þð Þ Ah N ðh h Þ þA 2 a dX2 a c 6 7 dX dX dX dX þ 3 4 !7 ¼ 0 2 3 3 3 2 dX2 6 6 7 4h h þ 12h h h þ 4h h h þ 20h h h þ 10h h þ B 5h 0 1 0 3 0 1 2 0 3 0 1 2 1 0 4 N 5 3 r 4 2 3 Bhs h3 Bhs 4h0 h3 þ 12h0 h1 h2 þ 4h0 h1
dh4 ¼ 0 at X ¼ 0; h4 ¼ 0 at X ¼ 0 dX
ð27Þ
By increasing number of terms in the solution higher accuracy will be obtained. Solving Eqs. (20)–(27) results h1 ; h2 ; h3 ; h4 ; . . . can be obtained as below
h2 ¼
2 C7 X4 þ 4Nr24
7 X4
þ 8Nr Bh24s C
2
8Nr Bh24s C 2 2
þ 5Nr B24C
9 X4
2
aC 5Nr Nc Bh 24
5
2
2 2 4
2 4 C3 X4 s 4Nr h24
3 X4
6Nr B 24hs C 4 X4
þ 5Nr N24c C
9Nr B 24hs C
þ 10Nr B24hs C
3 X4
2
5 X4
2
4Nr Nc24ha C 2 2
8 X4
4
5 X4
þ 6Nr Nc24BC
2
4 X4
2 2 5
4
8
2
sX þ Nr Bh24s QX þ Nr Bh 24
3 X6
2
c QC 80Nr N720
10
3 8
2 X6
2 4
2 2 9 4 Nr B24hs X
4 2 4 s ha X þ Nr Nc Bh 24
3 X6
7 X6
3 2 9
3 X6
3 2 2
10 X6
280Nr B720hs C
3 X6
2 2 4
QC3 X4
s þ 4Nr Bh24
2
sC 5Nr Nc Bh 24
4
þ
4 X4
4
4N2r B2 h2s C7 X4 24
24
2 4
þ 5
4Nr N2c Bhs ha C3 X4 24
QX ha X c hs X Nc24 Nr N24 þ Nr Nc24Bhs X Nc 24 2
4
2
CX þ Nc24 Aha2QX þ Nr Ah2a C 4
4N2r B2 h6s C3 X4
Nr ABh2a hs CX Nr ABha2hs C 2
4 X2
4 X2
4
2
4
2
5
2
4
4
Nr Ah2a hs X þ Nr ABh2a C 2
þ Nr ABh2a hs X Nc Ah2 a X 2
2 þ AQCX 2
5 2 ANr C2 X
4 2 þ ANr h2s CX
6 X2 ANr BC 2
þ
2
þ ANr Bh2s C
5 X2
ANr Bh5s CX2 2
AN2c C2 X2 2
5
2
4
4 X6
4 X6
hs QC þ 232Nr B720
7
2 2
3 3
13
þ 85Nr B720C
X6
X6
5
c Bhs C þ 160Nr N720 2
3 X6
2
3 X6
4 2
c ha C þ 36Nr N720 8
2 2
hs QC 200Nr B720 3 3 4
3 X6
155Nr B720hs C
9 X6
2
c Bhs C 192Nr N720
145Nr B720QC
2 2 5
4 X6
4 X6
6 X6
2
X6
3 2 8
4
2
C þ 60Nr BQ 720
11
þ 175Nr B720hs C
8 X6
2
2
2 X6
3 2
Bhs ha C þ 176Nr Nc720 2 X6
7 X6
376Nr B720hs C
c Bhs C 220Nr N720
Bhs ha C 144Nr Nc720
Nc C þ 45Nr 720
2
2
2
2 4
2
þ 250Nr N720c BC
3 X6
6 X6
2
2
8 X6
2
7 X6
7 X6
2 X6
X6
3 X6
2
Bha QC þ 120Nr Nc720
221Nr B720hs C 3 3
12
X6
4 3 3 5
8 X6
3 3 9
4 X6
3 3 6
7 X6
þ 395Nr B720hs C 175Nr B720hs C 328Nr B720hs C
2 2 5
c B ha C 145Nr N720 2
.. .
8 X6
2 2 4
2
2 2 C9 X6
cB þ 155Nr N720
2 2 4
3 3 8
þ 71Nr B720hs C
4 X6
c B hs C 142Nr N720
B hs ha C þ 232Nr Nc 720
7 X6
c B hs C 250Nr N720
2
2
3 X6
4 X6
c Bha C 130Nr N720
2 2
B hs ha C þ 130Nr Nc 720
200Nr Nc B720hs ha C 4
ANr Bh4s C2 X2
3 2 6
c BQC 130Nr N720
5 X2
2
4
264Nr B720hs C
4 X6
2
7 X6
4
3
Bhs C 328Nr 720
hs ha C þ 72Nr Nc720
2 X6
3 2 4
11 X6
Nc C þ 96Nr 720
390Nr B720hs C
X6
2
2
6 X6
9
3
2
2
3
2
2 X6
2
BC þ 188Nr720
Bhs C 108Nr 720
Bhs ha C þ 200Nr Nc720
hs QC þ 130Nr B720
2
2 2 þ 2Nc Ah2a CX
2
3 2 10 C2 X6
c ha C 80Nr N720
12
2
c Bha C 232Nr N720
þ 108Nr B720hs
2 Bh4 CX4 s 2Nr Nc24
3 X6
3 2
þ 156Nr B720hs C
4
8
B C þ 221Nr720
3 2 5
4
4 2 2 8 þ Nr B 24hs CX
3
ha QC þ 72Nr Nc720
2 X6
c ha C 88Nr N720
6 X6
Bhs C þ 140Nr 720
5
2
6 X6
7 X6
2
BQC 232Nr720
s QC 144Nr Bh 720 3 4
5
3
2 X6
hs C 88Nr720
10 X6
Bhs C þ 264Nr 720
þ 656Nr B720hs C 4 X4
6 X6
2
3
4
2
sC þ 72Nr Qh 720
s QC þ 704Nr Bh 720
rC þ 52N720
X6
c hs C 80Nr N720 2
6 X6
2
QC 88Nr720
3 X6
hs C þ 36Nr720
5Nr BQC 24
4 X4
4
2
sC 156Nr Bh 720
4 4 4 2 sC X 10Nr Bh 24
8 X4 2 þ 9Nr BC 24
2 X6
s QC þ 200Nr Bh 720
3
QX2 Nr C4 X2 Nr h4s X2 Nr BC5 X2 Nr Bh4s CX2 þ þ 2 2 2 2 2 Nr Bhs C4 X2 Nr Bh5s X2 N2c ha X2 N2c CX2 þ þ 2 2 2 2
h1 ¼
3 4 4Nr C24QX
2
Q C h3 ¼ þ 36Nr 720
ð26Þ
2 2
2 2 5
c B hs C þ 220Nr N720 2
4
c BC þ 71Nr N720
5 X6
2
2
4 X6
2 2
4
2
þ 188Nr B720hs C
8 X6
þ 140Nr B720hs
11 X6
3 3 10 C3 X6
3 X6
3 2 12 X6
hs QX hs Nr B720 Nr B720 6
3 3 2
5 X6
Bha C þ 60Nr Nc720
2 2 8
5 X6
3 3 12
hs CX Nr B 720
6
Thermal analysis of straight rectangular fin 2
2
5 X8
3
rQ C h4 ¼ þ 1944N40320 4
3
r Bhs QC þ 16968N40320 2
2
r Nc QC 4368N40320
r Bhs C 15004N40320 3
2
r Nc C þ 3136N40320
6 X8
6 X8
4
4
2 4
r B hs C 19132N40320
3
4
r Bhs C 4792N40320 4 8
13 X8
5 X8
4 2 5
4 2 8
7 X8
r B hs C 31284N40320
13 X8
r B hs C 16512N40320
4 2 2
B hs C þ 7368Nr40320
4 2 10 C5 X8
r B hs þ 11664N40320
r Nc ha C 4368N40320 2
6 X8
4
6 X8
r B hs C þ 43572N40320
14 X8
r B hs C þ 20448N40320
5
2
4
7 X8
r Nc C þ 2436N40320
10 X8
Nc B ha C 15420Nr 40320
4 3 9
7 X8
c B hs ha C þ 22872Nr N40320
4 3 2
14 X8
4 3 10 C6 X8 2
4
2 2
11 X8
2 2 4
3
4 3 6
r B hs C 42492N40320
7 X8
3
2
3
2
10 X8
3
2 2 5
2
4
8 X8
r Nc BC þ 7599N40320
6 X8
9 X8
6 X8
6 X8
2
12 X8
2 2
3
B hs QC 7444Nr40320
3 X8
15 X8
4 3 8
r B hs C þ 17389N40320
10 X8
2 2 5
6 X8
8 X8
8 X8
Nc B hs C þ 44688Nr40320 3 2 8
2
Nc Bha QC þ 12600Nr 40320
2 2
2 2 4
5 X8
11 X8
4 3
c B hs ha C þ 25536Nr N40320 3
4 2
2
7 X8
10 X8
r Nc ha C þ 1944N40320
r B hs C 24048N40320
r Nc B hs C 24988N403200 3
2
12 X8
r Nc B C þ 16330N40320
6 X8
3
Nc Bhs C 9408Nr 40320
3 2
3
4
r Nc Bhs C 25944N40320
r B QC 15420N40320
3 2 5
11 X8
r Nc BC þ 12828N40320
6 X8
r B hs QC 34464N40320
7 X8
c B hs ha C 33216Nr N40320
r Nc Bha C 13872N40320
2
4 3 4
11 X8
r B hs þ 30564N40320
10 X8
5
2
r B hs C 22730N40320
16 X8
4 3 5
3
3
3 X8 11 X8
Nc Bhs C þ 9864Nr40320 2
3 2
4 3
5 X8
2
9 X8
4 2 4
4
2
r BQ C þ 6300N40320
rB C þ 10505N40320
r B hs C þ 19596N40320
2
3 X8
r B hs C 28228N40320
15 X8
c Bhs ha C þ 8616Nr N40320
3
9
4
Nc Bha C 13668Nr40320 3
2
r Bhs C 5832N40320
9 X8
7 X8
r B hs C 34436N40320
3
2
14 X8 3
4 2 6
2
r B hs C þ 56476N40320
6 X8
3
2
Nc ha C 88Nr 40320
Nc Bhs ha C þ 88248Nr 40320
B hs QC þ 26376Nr40320
2
8
9 X8
6 X8
7 X8
r Nc BQC 13872N40320
5
4 2 9
3 2 4
r B hs QC þ 22872N40320
4
10 X8
c ha QC þ 72Nr N40320
4
4 2
10 X8
5 X8
r BC þ 6068N40320
rB C þ 12024N40320
Nc Bhs ha C 11656Nr 40320 2
9 X8
r Bhs C þ 10768N40320 4
4 2
3
r BQC 11868N40320 5
3
r Bhs C þ 10668N40320
2 4
3
5 X8
r Bhs QC 7776N40320 4 4
r Nc hs C 4368N40320
5 X8
9 X8
r hs C 2872N40320
13 X8
3
4
3
r Qhs C þ 3880N40320
r Bhs QC þ 10728N40320
r hs C þ 1944N40320
10 X8
9 X8
rC þ 1288N 40320
10 X8
Nc hs ha C þ 3880Nr 40320 3
r QC 28720N 40320
2273
Nc B hs C 25656Nr 40320
7 X8
3
2 2
2
4
2
Nc Bha C þ 3100Nr40320 3 2 12
B hs C 1444Nr40320
11 X8
6 X8
3 X8
.. . Substituting the values of h0 ; h1 ; h2 ; h3 and h4 ; . . ., in Eq. (13) the formulation of non-dimensional temperature can be obtained as a X h¼ hm ¼ h0 þ h1 þ h2 þ h3 þ h4 þ
ð28Þ
0
Now the temperature field, h can be evaluated if the fin tip temperature C is known whose value lies in the interval (0, 1). Using an arbitrary initial guess value for C, for the temperature field h computed from the above Eq. (26) and applying Newton–Raphson method satisfying the boundary conditions (6) the actual temperature field can be obtained. 4.1. Fin efficiency ðgÞ The fin efficiency is defined as the ratio of actual heat transfer rate to the ideal heat transfer rate. The ideal heat transfer rate takes place in the fin if the whole fin is subjected to base temperature. The actual rate of heat transfer can be obtained by computing the heat losses from the base of the fin dT Qactual ¼ KðTÞAc ð29Þ dx x¼b The ideal heat transfer rate Qideal obtained if the entire fin is kept at the base temperature and can be expressed as
Qideal ¼ hPbðT Ta Þ þ es rPb½1 þ bðT Ts Þ T4 T4s ð30Þ
The fin efficiency can be expressed using non-dimensional terms by Eq. (5) g¼
Qactual Qideal
k a Ac T b dh ½1 þ Ac ðh ha ÞdX b X¼1
¼ hPbðT Ta Þ þ es rPb½1 þ bðT Ts Þ T4 T4s dh ½1 þ Aðh ha ÞdX X¼1 ¼ 2 Nc ð1 ha Þ þ Nr ½1 þ Bð1 hs Þ 1 h4s
ð31Þ
5. Optimization The optimization of the fin can be made by either minimizing the volume for any required heat dissipation or maximizing the heat dissipation for any given fin volume [22,23]. Assuming that the volume of the fin is constant the fin volume is defined as V ¼ Ac b The heat dissipated per unit volume is q ka Ta dh ¼ 2 f1 þ Aðh ha Þg V dX X¼1 b
ð32Þ
The dimensionless form of the above equation [21] is q Ac ka Ta dh ¼ 2 f1 þ Aðh ha Þg ð33Þ Ac N¼ ka Ta V dX b
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P.K. Roy, A. Mallick
The maximum heat dissipation value expressed the optimum fin characteristics and the dimensions are the optimum fin configuration. The optimization procedure is also done to fix the profile area Ac to express N as a function of b and then search the optimum value of b. 6. Results and discussion The validation of results is done by considering the both sink temperature, hs ; ha and surface emissivity parameter, B and heat generation number, equating to zero Eq. (6) is converted into the previous work found in the literature [21]. The present works consider five terms and the accuracy results are compared with the previous work as available in the literature as shown in Tables 1 and 2. In fin design application the selection of the values of Nc is very important. The fins are categorized as practical fins depending on the value of conductive–convective fin parameter Nc whose value lies below 0.5. Depending on the surface condition of the fin materials the surface emissivity can increase with temperature ðB > 0Þ or decrease with temperature ðB < 0Þ. In practice values of B range from 0.6 to +0.6 [8]. In order to study the effect of surface emissivity on the temperature distribution and efficiency of the fin, highest, lowest and constant values of surface emissivity are taken i.e. B = 0.6, 0, +0.6 respectively in the present analysis. The idea of maintaining nonzero environmental temperature ðhs and ha Þ is that in actual practice it is not possible for a
fin to maintain one end at high temperature and dissipating heat to the absolute zero. The study of effect of the environmental temperature is also found effective on the thermal performance. The range values of environmental temperature and heat generation number are found in the literature [6]. Fig. 2 illustrates the variation of tip temperature hð0Þ, with respect to heat generation number, Q for three different values 1.00
Fixed parameters A=0,Nr=1, a=0, s=0
0.95
NC=0.25
0.90 0.85 0.80 0.75 0.70
NC=1
0.65 0.60
B=+0.6 B=0 B=-0.6
0.55 0.50 0.0
0.1
0.2
0.3
0.4
Heat generation number,Q
Figure 2 number.
The tip temperatures function of heat generation
Table 1 Comparison of HPM solutions with Adomian Decomposition Method (ADM), Galerkin Method (GM) and Boundary Value Problem (NM). Nc ¼ 1; Nr ¼ 0:2; A ¼ 0:2; Q ¼ 0; hs ¼ 0; ha ¼ 0; B ¼ 0 X
HPM (Present)
ADM [6]
GM [21]
NM(BVP) [21]
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.666858541 0.669984372 0.679386188 0.695136900 0.717357999 0.746220026 0.781944267 0.824806704 0.875145547 0.933237796 0.9999999999
0.666858541 0.669984372 0.679386188 0.695136900 0.717357999 0.746220026 0.781944267 0.824806704 0.875145547 0.933237796 0.999999999
0.667013018 0.670132740 0.679514048 0.695228804 0.717399183 0.746198567 0.781855307 0.824659376 0.874971905 0.933237591 0.999999999
0.667013597 0.670132745 0.679514005 0.695229254 0.717399776 0.746198734 0.781855061 0.824655933 0.874972380 0.933237796 0.999999999
Table 2
Effect of number of terms in HPM on the temperature distribution of rectangular fin.
Nc ¼ 1; Nr ¼ 0:2; A ¼ 0:2; Q ¼ 0; hs ¼ 0; ha ¼ 0; B ¼ 0 Number of terms (m)
Distance, X 0.0000
0.1000
0.2000
0.300
0.400
0.5000
0.6000
0.7000
0.8000
0.9000
1.0000
2 3 Absolute error difference With m = 2 and m=3
0.6608 0.6687 0.0079
0.6638 0.6719 0.0081
0.6730 0.6812 0.0082
0.6884 0.6969 0.0085
0.7102 0.7194 0.0092
0.7389 0.7479 0.0090
0.7746 0.7835 0.0089
0.8180 0.8260 0.0080
0.8696 0.8760 00064
0.9300 0.9338 0.0038
0.9999 0.9999 0.0000
3 4 Absolute error difference With m = 3 and m=4
0.6687 0.6668 0.0019
0.6718 0.6700 0.0018
0.6812 0.6794 0.0018
0.6970 0.6951 0.0019
0.7192 0.7173 0.0019
0.7479 0.7462 0.0017
0.7835 0.7819 0.0016
0.8260 0.8284 0.0012
0.8760 0.8751 00009
0.9338 0.9334 0.0004
0.9999 0.9999 0.0000
Thermal analysis of straight rectangular fin
2275
of surface emissivity parameters, i.e. B ¼ 0:6; 0; þ0:6 while other parameter maintained at zero except Nr ¼ 1. The surface emissivity lines increase linearly with increase in internal heat generation number Q ¼ 0 to Q ¼ 0:4. The internal heat source affects the surface of the fin materials. The relative spacing between the emissivity lines is higher when conduction–convection parameter, Nc is maintained at low values. This is because at lower value conduction–convection parameter, Nc, the convective heat loss is higher and hence temperature gradient is higher. Figs. 3 and 4 demonstrates the variation of temperature of the fin materials for different values of thermal emissivity parameters. The thermal conductivity of the fin materials is maintained at A ¼ 0:6 and sinks temperature and heat generation maintained at some fixed value. It is evident from the both figures that the surface emissivity lines of fin materials are more affected when the values of Nc are maintained below the range of 0.5. The surface of the fin materials is more affected at higher values of heat generation number which is evident from Fig. 4.
Fig. 5 illustrates the variation of fin efficiency of the fin materials g with respect to heat generation number Q. It is clear from the figure that the efficiency of the fin materials is very low without internal heat generation number Q ¼ 0. Fin efficiency increases with the increase in internal heat generation number Q. The efficiency corresponding to lower values of conduction–convection parameter, Nc is steeper as compared to other. That may be because of lower values of conduction–convection parameter, the simultaneous internal heat generation and surface heat loss enhances the fin efficiency. Fig. 6 shows the variation of fin efficiency with respect to the radiation sink temperature hs . The efficiency of surface emissivity curves corresponding to negative values of surface emissivity is higher. This observation has been found to be true in case of space radiative fin [9]. The efficiency is higher when fin maintained at zero sink temperature, and decreases beyond the radiation sink temperature 4.5.
0.90
Fixed parameters A=-0.2, a= s=0.5,Nr=1
0.85 1.00
Nc=0.25
0.80 0.95
B=+0.6 B=-0.6 B=0
0.75 0.70
Nc=0.25
0.85
Effeciency,
Temperature distribution,
0.90
0.80 0.75
0.65 0.60 0.55 0.50
0.70 0.65
0.45
Nc=1
Nc=1
0.40
0.60
0.35
Fixed parameters =1, a= s=0.1,Q=0.1 r
0.55 0.50 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.0
0.1
0.2
0.3
0.4
Heat generation number,Q
1.0
Figure 5 The effect of fin efficiency g on the heat generation number, Q.
Distance,X
Figure 3 The effect of surface emissivity on the temperature distribution for Q = 0.1. 0.8
1.00
Fixed parameters A=-0.2, a=0.5,Nr=1,Q=0.1
NC=0.25
0.95
Nc=1 0.6
0.85
Efficiency,
Temperature distribution,
0.90
0.80 0.75
B=+0.6 B=0 B=-0.6
NC=1
0.70
0.4
0.65
Fixed parameters A=-0.6,Nr=1, a= s=0.1,Q=0.4
0.60 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Distance,X
Figure 4 The effect of surface emissivity on the temperature distribution for Q = 0.4
Nc=0.25
B=-0.6 B=+0.6 B=0
0.2
0.0
0.1
0.2
0.3
0.4
0.5
Radiation sink temperature,
0.6
0.7
0.8
s
Figure 6 The variation of fin efficiency g with respect to the radiation sinks temperature hs .
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P.K. Roy, A. Mallick
0.60
B=-0.6,Nc=0.25 B=+0.6,Nc=0.25 B=0, Nc=0.25 B=-0.6, Nc=1 B=+0.6,Nc=1 B=0, Nc=1
0.55 0.50
Effeciency,
0.45 0.40 0.35 0.30 0.25
Fig. 7 shows the variation of fin efficiency with respect to radiation conduction parameter Nr . As the radiation conduction parameter increases the efficiency corresponding to different surface emissivity lines decreases gradually. At higher values of radiation conduction parameter the radiation heat losses are higher but at negative values of surface emissivity parameter the efficiency is higher. Fig. 8 shows the optimum dimensionless parameter N as function surface emissivity parameter, B for different values of conduction–convection parameter. Fig. 9 presents optimum dimensionless parameter N is a function of fin length. The best fin length can be obtained under a given volume V, profile area Ac and thermal parameter.
A=-0.2, a= s=0.5,Q=0.1
0.20
0.8
1.2
1.6
2.0
2.4
2.8
3.2
3.6
4.0
7. Conclusion
Radiation conduction parameter,N r
Figure 7 The variation of fin efficiency g with respect to radiation conduction parameter Nr .
50 45
Nc=0.25
40
Nc=1
Nx100
35 30
References 25 20 15 10 -0.6
A=-0.2, a= s=0.5,Nr=0.2,Q=0.1,b=0.05 -0.4
-0.2
0.0
0.2
0.4
0.6
Surface emissivity parameter,B
Figure 8 The optimum dimensionless parameters N function of surface emissivity parameter B. 50
A=-0.5, a= s=0.5,Nr=0.2,Q=0.1
40
30
NX100
In this study Homotopy Perturbation Method (HPM) is used for solving nonlinear energy equation in a straight rectangular fin with variable thermal conductivity and surface emissivity with some constant source of heat generation. The problem is solved for two main real and industrial use of thermo geometric fin parameter Nc ¼ 0:25 and Nc ¼ 1. The results are presented for the temperature distribution, efficiency and optimum dimensionless parameters are effective and convenient. Also it is found that this method cans achieve more suitable results as compared to the other methods available in the literature.
Nc=0.25 Nc=1
20
10
0 0.05
0.06
0.07
0.08
0.09
0.10
0.11
0.12
0.13
0.14
Fin length,b
Figure 9 The optimum dimensionless parameters N function of fin length b.
[1] D.Q. Kem, D.A. Kraus, Extended Surface Heat Transfer, McGraw-Hill, New York, 1972. [2] Balaram Kundu, Samchai Wongwises, A decomposition analysis on convective-radiating rectangular plate fins for variable thermal conductivity and heat transfer coefficient, J. Frankling Inst. 349 (2012) 966–984. [3] A. Aziz, M.N. Bouaziz, A least square method for a longitudinal fin with temperature dependent internal heat generation and thermal conductivity, Energy Convers. Manage. 52 (2011) 2876– 2882. [4] Balaram Kundu, Kwan-Soo Lee, Analytical tools for calculating the maximum heat transfer of annular stepped fins with internal heat generation and radiation effects, Energy 76 (2014) 733–748. [5] R.K. Singla, R. Das, Application of Adomian decomposition method and inverse solution for a fin with variable thermal conductivity and heat generation, Int. J. Heat Mass Transf. 66 (2013) 496–506. [6] Pranab K. Roy, Hiranmoy Mondal, Ashis Mallick, A decomposition method for convective-radiative fin with heat generation, Ain Shams Eng. J. 6 (2015) 307–313. [7] J.H. He, Homotopy perturbation method: a new non linear analytical technique, Appl. Math. Comput. 135 (2003) 73–79. [8] Arslanturk Cihat, Performance analysis and optimization of radiating fins with a step change in thickness and variable thermal conductivity by homotopy perturbation method, Heat Mass Transf. 47 (2011) 131–138. [9] Pranab K. Roy, Apurba Das, Hiranmoy Mondal, Ashis Mallick, Application of homotopy perturbation method for a conductive-radiative fin with temperature dependent thermal conductivity and surface emissivity, Ain Shams Eng. J. 6 (2015) 1001–1008. [10] Ashis Mallick, Sujeet Ghosal, Prabir Kr. Sarkar, Rajiv Ranjan, Homotopy perturbation method for thermal stresses in an
Thermal analysis of straight rectangular fin
[11]
[12]
[13]
[14]
[15]
[16]
annular fin with variable thermal conductivity, J. Therm. Stress 38 (2015) 110–132. A. Bhowmik, R.K. Singla, P.K. Roy, D.K. Prasad, R. Das, R. Repaka, Predicting geometry of rectangular and hyperbolic fin profiles with temperature-dependent thermal properties using decomposition and evolutionary methods, Energy Convers. Manage. 74 (2013) 535–547. Saheera Azmi Hazarika, Dipankar Bhanja, Sujit Nath, Balaram Kundu, Analytical solution to predict performance and optimum design parameters of a constructal T-shaped fin with simultaneous heat and mass transfer, Energy 84 (2015) 303–316. M. Rahimi-Gorji, O. Pourmehran, M. Gorji-Bandpy, D.D. Ganji, An analytical investigation on unsteady motion of vertically falling spherical particles in non-Newtonian fluid by collocation method, Ain Shams Eng. J. 6 (2015) 531–540. O. Pourmehran, M. Rahimi-Gorji, M. Gorji-Bandpy, D.D. Ganji, Analytical investigation of squeezing unsteady nanofluid flow between parallel plates by LSM and CM, Alexandria Eng. J. 54 (2015) 17–26. M. Rahimi-Gorji, O. Pourmehran, M. Hatami, D.D. Ganji, Statistical optimization of micro channel heat sink (MCHS) geometry cooled by different nanofluids using RSM analysis, Eur. Phys. J. Plus 130 (22) (2015) 1–21. O. Pourmehran, M. Rahimi-Gorji, M. Hatami, S.A.R. Sahebi, G. Domairry, Numerical optimization of microchannel heat sink (MCHS) performance cooled by KKL based nanofluids in saturated porous medium, J. Taiwan Inst. Chem. Eng. 55 (2015), http://dx.doi.org/10.1016/j.jtice.2015.04.016.
2277 [17] Ya-Song Sun, Jing Ma, Ben-Wen Li, Spectral collocation method for convective–radiative transfer of a moving rod with variable thermal conductivity, Int. J. Therm. Sci. 90 (2015) 187– 196. [18] Ya-Song Sun, Jin-Liang Xu, Thermal performance of continuously moving radiative–convective fin of complex cross-section with multiple nonlinearities, Int. Commun. Heat Mass Transf. 63 (2015) 23–34. [19] Yasong Sun, Jing Ma, Benwen Li, Zhixiong Guo, Predication of nonlinear heat transfer in a convective-radiative fin with temperature-dependent properties by the collocation spectral method, Numer. Heat Transf. Part B: Fund. Int. J. Comput. Methodol. 69 (1) (2016). [20] Jing Ma, Ya-Song Sun, Ben-Wen Li, Hao Chen, Spectral collocation method for radiative-conductive porous fin with temperature dependent properties, Energy Convers. Manage. 111 (2016) 279–288. [21] D.D. Ganji, M. Rahgoshay, M. Rahimi, M. Jafari, Numerical investigation of fin efficiency and temperature distribution of conductive, convective and radiative straight fins, Int. J. Res. Rev. Appl. Sci. (August) (2010). [22] Lien-Tsai Yu, Cha’-O-Kuang Chen, Application of Taylor transformation to optimize rectangular fins with variable thermal parameters, Appl. Math. Model. 22 (1998) 11–21. [23] Balaram Kundu, Kwan-Soo Lee, Antonio Campo, An ease of analysis for optimum design of an annular step fin, Int. J. Heat Mass Transf. 85 (2015) 221–227.
H O S T E D BY
Alexandria University
Alexandria Engineering Journal www.elsevier.com/locate/aej www.sciencedirect.com
ORIGINAL ARTICLE
Thermal analysis of straight rectangular fin using homotopy perturbation method Pranab Kanti Roy *, Ashis Mallick Department of Mechanical Engineering, Indian School of Mines, Dhanbad 826004, Jharkand, India Received 11 November 2015; revised 12 May 2016; accepted 18 May 2016 Available online 10 June 2016
KEYWORDS Sink temperature; Homotopy perturbation method; Efficiency; Variable surface emissivity
Abstract In this study a simple and accurate semi-analytical method called Homotopy Perturbation Method (HPM) is used for solving nonlinear energy equation in a straight rectangular fin. The thermal conductivity and surface emissivity are considered as temperature dependent with some constant source of internal heat generation. The problem is solved for two main cases of thermo geometric fin parameter Nc = 1 and for Nc = 0.25. The results are presented for the temperature distribution, efficiency and optimum dimensionless parameters are effective and convenient for practical fins. Also it is found that this method can achieve more suitable results as compared to the other methods available in the literature. Ó 2016 Faculty of Engineering, Alexandria University. 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/).
1. Introduction The fins are extended surface that are used to dissipate heat from the primary surface to the surrounding environment [1]. Heat transfer through the straight is common because of their low manufacturing cost and simplicity. The fins are sometimes subjected to both convective and radiative environments [2]. Under the circumstances of both convective and radiative modes of heat transfer the fin may generate heat internally due to passage of electric current as in electric filament or due to an atomic or chemical reaction as in an atomic reactor [3–6]. The internal heat generation may be assumed to be at constant rate with respect to the volume of fin. But to utilize the fin materials effectively the model includes the thermal conductivity varies * Corresponding author. Tel.: +91 9609551679. E-mail addresses: [email protected] (P.K. Roy), [email protected] (A. Mallick). Peer review under responsibility of Faculty of Engineering, Alexandria University.
linearly with temperature and in most cases the real surface of the fin material varies linearly with temperature. The energy equation for a convective–radiative fin along with heat generation with two variable thermal properties results in highly nonlinear terms. The resulting equation does not admit the exact solution. Consequently the energy equation with nonlinear terms is solved either numerically or using a variety of approximate analytical methods. Many different methods have been introduced to solve nonlinear problems, the Homotopy analysis method (HAM), Galerkin method (GM), Spectral collocation method (SCM), Adomian decomposition method (ADM). Homotopy perturbation method (HPM) is one of the semi analytical methods for solving nonlinear boundary value problem introduced by He [7]. An HPM possesses all the advantages of perturbation method and also it is independent of assumption of small parameter. As compared to the Adomian decomposition method, Homotopy perturbation method does not require the calculation of Adomian polynomials but it requires only the initial approximation [8–10]. Also the method does not require determination of ‘h-curves’ which
http://dx.doi.org/10.1016/j.aej.2016.05.020 1110-0168 Ó 2016 Faculty of Engineering, Alexandria University. 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/).
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P.K. Roy, A. Mallick
Nomenclature Nr C k ka es T P Tb Ta Ts b x
radiation–conduction parameter constant which represents the temperature temperature dependent thermal conductivity (W/ (mK)) thermal conductivity corresponding to ambient condition (W/(mK)) the surface emissivity corresponding to radiation sinks temperature, Ts temperature (K) fin perimeter (m) fin’s base temperature (K) convection sink temperature (K) sink temperature for radiation (K) length of the fin (m) axial coordinate of the entire fin (m)
makes it different from the Homotopy Analysis method. Bhowmik et al. [11] predicted the dimensions of rectangular and hyperbolic fins with variable thermal properties. Hazarika et al. [12] established an analytical method to predict the performance and design parameters of T-shape fin with simultaneous heat and mass transfer. Several researchers applied Collocation method (CM) to obtain the analytical solution of unsteady motion of fluid particles in conjunction with other numerical technique [13,14]. Rahimi-Gorji et al. [15] used Galerkin Method to obtain the heat transfer characteristics of micro channel heat sink cooled by different nanofluids in porous media. Pourmehran et al. [16] studied the thermal analysis of a fin shaped micro channel heat sink cooled by different nanofluids using least square method. Spectral collocation method has been used to study the performance parameter of simple and complex cross section of moving rod in thermal processing of continuous casting and rolling [17,18]. Sun et al. [19] used the Lagrange interpolation polynomials to approximate the temperature distribution at the spectral collocation points of nonlinear heat transfer problems. Ma et al. [20] presented the spectral collocation method to predict the thermal performance of porous fin with temperature dependent heat transfer coefficient, surface emissivity and internal heat generation. From the above discussion it is clear that in fin design application the selection of choice of conductive–convective parameter is very important for practicing engineer. The practical fins work at low values of conductive convective fin parameter. HPM is used to study the variation of surface emissivity, thermal conductivity parameter, heat generation number on the temperature distribution and efficiency of fin working under practical range of operating condition.
cross-sectional area of the entire fin (m2 ) dimensionless axial coordinate thermal conductivity parameters the surface emissivity parameters
Ac X A B
Greek symbols a slope of the thermal conductivity-temperature curve (K1 ) b slope of the surface emissivity-temperature curve (K1 ) h dimensionless temperature of the fin ha dimensionless convection sinks temperature hs dimensionless radiation sinks temperature r Stefan–Boltzmann constant (W=ðm4 K4 Þ) e emissivity
end side and the effect of heat transfer in vertical direction is neglected. The steady state one dimensional energy equation is given by d dT hPðT Ta Þ eðTÞrPðT4 T4s Þ kðTÞ þq¼0 ð1Þ dx dx Ac Ac With the insulated boundary condition dT ¼ 0 at dX
x¼0
ð2aÞ
T ¼ Tb
x¼b
ð2bÞ
at
For the better utilization of the fin materials the thermal conductivity of fin material is assumed to be linear function of temperature. Also the emissivity of all real surfaces is not constant, and it sometimes varies linearly with temperature. Therefore both of the parameters can written as below kðTÞ ¼ ka ½1 þ aðT Ta Þ
ð3Þ
eðTÞ ¼ es ½1 þ bðT Ts Þ
ð4Þ
h,
T
Conducon
dx
x
2. Mathematical formulations Let us consider rectangular fin geometry of various details as shown in Fig. 1. The thermal conductivity k and surface emissivity e of the fin materials are temperature dependent. The heat generated inside the fin material is assumed to be at the constant rate q. The fin is assumed to be insulated at the free
Convection
Radiation
Tb
b
X
Radiation sink temperature, Ts Convection sink temperature, Ta Figure 1
The geometry of straight rectangular fin.
Thermal analysis of straight rectangular fin
2271
In order to express Eq. (1) in non-dimensional forms, the following dimensionless parameters are defined as T h¼ Tb A ¼ aTb
Ta ha ¼ Tb
Ts hs ¼ Tb
B ¼ bTb
x X¼ b 2 hPb es rT3b Pb2 N2c ¼ Nr ¼ ka Ac ka Ac
Q¼
b2 q ka Tb ð5Þ
The formulation of the fin problem reduces to the following equations: 2 d2 h d2 h dh d2 h þ Ah 2 þ A Aha 2 N2c ðh ha Þ 2 dX dX dX dX 4 4 ð6Þ Nr ½1 þ Bðh hs Þ h hs þ Q ¼ 0 With the following boundary conditions:
For p ¼ 1 we obtain Hðh; 1Þ ¼ ½LðhÞ þ NðhÞ fðrÞ
ð14Þ
Using the perturbation technique by taking by taking into account of small values of p, then the solution of Eq. (10) can be written as h ¼ h0 þ ph þ p2 h2 þ p3 h3 þ p4 h4 þ
ð15Þ
The series converges for p ¼ 1 the solution for h can be given by h ¼ h0 þ h þ h2 þ h3 þ h4 þ
ð16Þ
4. HPM formulation Using Eq. (12), Eq. (6), can be written as in HPM form
dh ¼ 0 at X ¼ 0 dX
ð7aÞ
h ¼ 1 at X ¼ 1
ð7bÞ
Hðh; pÞ ¼ ð1 pÞLðh h0 Þ " 2 # dh 2 2 d h d2 h d2 h Aha dX 2 þ Ah 2 þ A dX 2 Nc ðh ha Þ þ p dX dX ¼0 Nr h4 h4s þ Bh5 Bh4s h Bhs h4 þ Bh4s ð17Þ
3. Homotopy Perturbation Method (HPM)
This can be arranged as
Homotopy perturbation method (HPM) is a semi-numerical method for solving linear or nonlinear, homogeneous or inhomogeneous boundary value problem first introduced by He [7]. As compared to the Adomian decomposition method this method does not require the calculation of Adomian polynomial and leads to convergent solution rapidly. Moreover this method requires only initial condition as input for its solution. To illustrate the basic idea of HPM according to He [7], consider the following nonlinear differential equation, AðhÞ fðrÞ ¼ 0;
r 2 X:
With the boundary conditions @h B h; ¼ 0 r 2 C; @X
ð8Þ
ð9Þ
where A is a general differential operator, B is a boundary operator, fðrÞ is a known analytic function, and C is the boundary of the domain X. The operator A can be generally divided into linear and nonlinear parts say LðhÞ and NðhÞ. Therefore Eq. (8) can be written as LðhÞ þ NðhÞ fðrÞ ¼ 0
ð10Þ
Define the Homotopy mðr; pÞ : X ½0; 1 ! R that satisfies Hðh; pÞ ¼ ð1 pÞLðh h0 Þ þ p½LðhÞ þ NðhÞ fðrÞ ¼ 0:
ð11Þ
Hðh; pÞ ¼ LðhÞ Lðh0 Þ þ pðh0 Þ " # dh 2 2 d2 h d2 h Aha dX Ah dX 2 þ A dX 2 Nc ðh ha Þ þp ¼0 Nr h4 h4s þ Bh5 Bh4s h Bhs h4 þ Bh4s ð18Þ Substituting h, from Eqs. (15) to (18), and separating the variables of identical power of p. p0 : h ¼ h0
ð19Þ
From the boundary condition (7) it is clear that the solution is becoming meaningless. Therefore in order to predict the solution physically meaningful the hð0Þ must be an arbitrary constant. This h0 ¼ C is taken as initial input for HPM. And p1 :
" # 0 2 2 2 Ah0 ddXh20 þ A dh Aha ddXh20 N2c ðh ha Þ d2 h1 dX þ ¼0 dX2 Nr h40 h4s þ Bh50 Bh4s h0 Bhs h4 þ Bh5s
ð20Þ
dh1 ¼ 0 at X ¼ 0; h1 ¼ 0 at X ¼ 0 dX
ð21Þ
And p : 2
This can be rearranged as Hðh; pÞ ¼ LðhÞ Lðh0 Þ þ pLðh0 Þ þ p½NðhÞ fðrÞ ¼ 0:
ð12Þ
2
d where L ¼ dX 2 and h0 is the initial approximation. Here h ¼ fðXÞ and p is the embedding parameter such that p 2 ½0; 1. It is clear that
Hðh; 0Þ ¼ ½LðhÞ Lðh0 Þ
ð13Þ
2 2 3 dh dh d2 h d h d2 h 2 d2 h2 4 A h0 dX21 þ h1 dX20 þ A 2 dX0 dX1 Aha dX21 Nc ðh ha Þ 5 þ ¼0 dX2 Nr 4h3 h1 þ 5Bh4 h1 Bh4 h1 Bhs 4h3 h1 0
0
s
0
ð22Þ
dh2 ¼ 0 at X ¼ 0; h2 ¼ 0 at X ¼ 0 dX
ð23Þ
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P.K. Roy, A. Mallick
p3 :
3 2 2 2 2 2 0 dh2 1 A h0 ddXh22 þ h1 ddXh21 þ h2 ddXh20 þ A 2 dh þ ðdh Þ dX dX dX 7 d2 h3 6 6 3 4 !7 2 2 3 2 þ6 7¼0 þ B 5h 4h h þ 6h h h þ 10h h 2 2 2 2 0 0 1 0 0 1 5 4 Ah d h2 N2 ðh h Þ N dX a dX2 a r c Bh4s h2 Bhs 4h30 h1 þ 6h20 h21
ð24Þ
dh3 ¼ 0 at X ¼ 0; h3 ¼ 0 at X ¼ 0 dX
ð25Þ
p4 :
2 2 3 2 2 2 A h0 ddXh23 þ h1 ddXh22 þ h2 ddXh21 þ h3 ddXh20 6 7 6 7 dh0 dh3 d2 h3 dh1 dh2 2 7 d2 h4 6 þ 2ð Þð Þ Ah N ðh h Þ þA 2 a dX2 a c 6 7 dX dX dX dX þ 3 4 !7 ¼ 0 2 3 3 3 2 dX2 6 6 7 4h h þ 12h h h þ 4h h h þ 20h h h þ 10h h þ B 5h 0 1 0 3 0 1 2 0 3 0 1 2 1 0 4 N 5 3 r 4 2 3 Bhs h3 Bhs 4h0 h3 þ 12h0 h1 h2 þ 4h0 h1
dh4 ¼ 0 at X ¼ 0; h4 ¼ 0 at X ¼ 0 dX
ð27Þ
By increasing number of terms in the solution higher accuracy will be obtained. Solving Eqs. (20)–(27) results h1 ; h2 ; h3 ; h4 ; . . . can be obtained as below
h2 ¼
2 C7 X4 þ 4Nr24
7 X4
þ 8Nr Bh24s C
2
8Nr Bh24s C 2 2
þ 5Nr B24C
9 X4
2
aC 5Nr Nc Bh 24
5
2
2 2 4
2 4 C3 X4 s 4Nr h24
3 X4
6Nr B 24hs C 4 X4
þ 5Nr N24c C
9Nr B 24hs C
þ 10Nr B24hs C
3 X4
2
5 X4
2
4Nr Nc24ha C 2 2
8 X4
4
5 X4
þ 6Nr Nc24BC
2
4 X4
2 2 5
4
8
2
sX þ Nr Bh24s QX þ Nr Bh 24
3 X6
2
c QC 80Nr N720
10
3 8
2 X6
2 4
2 2 9 4 Nr B24hs X
4 2 4 s ha X þ Nr Nc Bh 24
3 X6
7 X6
3 2 9
3 X6
3 2 2
10 X6
280Nr B720hs C
3 X6
2 2 4
QC3 X4
s þ 4Nr Bh24
2
sC 5Nr Nc Bh 24
4
þ
4 X4
4
4N2r B2 h2s C7 X4 24
24
2 4
þ 5
4Nr N2c Bhs ha C3 X4 24
QX ha X c hs X Nc24 Nr N24 þ Nr Nc24Bhs X Nc 24 2
4
2
CX þ Nc24 Aha2QX þ Nr Ah2a C 4
4N2r B2 h6s C3 X4
Nr ABh2a hs CX Nr ABha2hs C 2
4 X2
4 X2
4
2
4
2
5
2
4
4
Nr Ah2a hs X þ Nr ABh2a C 2
þ Nr ABh2a hs X Nc Ah2 a X 2
2 þ AQCX 2
5 2 ANr C2 X
4 2 þ ANr h2s CX
6 X2 ANr BC 2
þ
2
þ ANr Bh2s C
5 X2
ANr Bh5s CX2 2
AN2c C2 X2 2
5
2
4
4 X6
4 X6
hs QC þ 232Nr B720
7
2 2
3 3
13
þ 85Nr B720C
X6
X6
5
c Bhs C þ 160Nr N720 2
3 X6
2
3 X6
4 2
c ha C þ 36Nr N720 8
2 2
hs QC 200Nr B720 3 3 4
3 X6
155Nr B720hs C
9 X6
2
c Bhs C 192Nr N720
145Nr B720QC
2 2 5
4 X6
4 X6
6 X6
2
X6
3 2 8
4
2
C þ 60Nr BQ 720
11
þ 175Nr B720hs C
8 X6
2
2
2 X6
3 2
Bhs ha C þ 176Nr Nc720 2 X6
7 X6
376Nr B720hs C
c Bhs C 220Nr N720
Bhs ha C 144Nr Nc720
Nc C þ 45Nr 720
2
2
2
2 4
2
þ 250Nr N720c BC
3 X6
6 X6
2
2
8 X6
2
7 X6
7 X6
2 X6
X6
3 X6
2
Bha QC þ 120Nr Nc720
221Nr B720hs C 3 3
12
X6
4 3 3 5
8 X6
3 3 9
4 X6
3 3 6
7 X6
þ 395Nr B720hs C 175Nr B720hs C 328Nr B720hs C
2 2 5
c B ha C 145Nr N720 2
.. .
8 X6
2 2 4
2
2 2 C9 X6
cB þ 155Nr N720
2 2 4
3 3 8
þ 71Nr B720hs C
4 X6
c B hs C 142Nr N720
B hs ha C þ 232Nr Nc 720
7 X6
c B hs C 250Nr N720
2
2
3 X6
4 X6
c Bha C 130Nr N720
2 2
B hs ha C þ 130Nr Nc 720
200Nr Nc B720hs ha C 4
ANr Bh4s C2 X2
3 2 6
c BQC 130Nr N720
5 X2
2
4
264Nr B720hs C
4 X6
2
7 X6
4
3
Bhs C 328Nr 720
hs ha C þ 72Nr Nc720
2 X6
3 2 4
11 X6
Nc C þ 96Nr 720
390Nr B720hs C
X6
2
2
6 X6
9
3
2
2
3
2
2 X6
2
BC þ 188Nr720
Bhs C 108Nr 720
Bhs ha C þ 200Nr Nc720
hs QC þ 130Nr B720
2
2 2 þ 2Nc Ah2a CX
2
3 2 10 C2 X6
c ha C 80Nr N720
12
2
c Bha C 232Nr N720
þ 108Nr B720hs
2 Bh4 CX4 s 2Nr Nc24
3 X6
3 2
þ 156Nr B720hs C
4
8
B C þ 221Nr720
3 2 5
4
4 2 2 8 þ Nr B 24hs CX
3
ha QC þ 72Nr Nc720
2 X6
c ha C 88Nr N720
6 X6
Bhs C þ 140Nr 720
5
2
6 X6
7 X6
2
BQC 232Nr720
s QC 144Nr Bh 720 3 4
5
3
2 X6
hs C 88Nr720
10 X6
Bhs C þ 264Nr 720
þ 656Nr B720hs C 4 X4
6 X6
2
3
4
2
sC þ 72Nr Qh 720
s QC þ 704Nr Bh 720
rC þ 52N720
X6
c hs C 80Nr N720 2
6 X6
2
QC 88Nr720
3 X6
hs C þ 36Nr720
5Nr BQC 24
4 X4
4
2
sC 156Nr Bh 720
4 4 4 2 sC X 10Nr Bh 24
8 X4 2 þ 9Nr BC 24
2 X6
s QC þ 200Nr Bh 720
3
QX2 Nr C4 X2 Nr h4s X2 Nr BC5 X2 Nr Bh4s CX2 þ þ 2 2 2 2 2 Nr Bhs C4 X2 Nr Bh5s X2 N2c ha X2 N2c CX2 þ þ 2 2 2 2
h1 ¼
3 4 4Nr C24QX
2
Q C h3 ¼ þ 36Nr 720
ð26Þ
2 2
2 2 5
c B hs C þ 220Nr N720 2
4
c BC þ 71Nr N720
5 X6
2
2
4 X6
2 2
4
2
þ 188Nr B720hs C
8 X6
þ 140Nr B720hs
11 X6
3 3 10 C3 X6
3 X6
3 2 12 X6
hs QX hs Nr B720 Nr B720 6
3 3 2
5 X6
Bha C þ 60Nr Nc720
2 2 8
5 X6
3 3 12
hs CX Nr B 720
6
Thermal analysis of straight rectangular fin 2
2
5 X8
3
rQ C h4 ¼ þ 1944N40320 4
3
r Bhs QC þ 16968N40320 2
2
r Nc QC 4368N40320
r Bhs C 15004N40320 3
2
r Nc C þ 3136N40320
6 X8
6 X8
4
4
2 4
r B hs C 19132N40320
3
4
r Bhs C 4792N40320 4 8
13 X8
5 X8
4 2 5
4 2 8
7 X8
r B hs C 31284N40320
13 X8
r B hs C 16512N40320
4 2 2
B hs C þ 7368Nr40320
4 2 10 C5 X8
r B hs þ 11664N40320
r Nc ha C 4368N40320 2
6 X8
4
6 X8
r B hs C þ 43572N40320
14 X8
r B hs C þ 20448N40320
5
2
4
7 X8
r Nc C þ 2436N40320
10 X8
Nc B ha C 15420Nr 40320
4 3 9
7 X8
c B hs ha C þ 22872Nr N40320
4 3 2
14 X8
4 3 10 C6 X8 2
4
2 2
11 X8
2 2 4
3
4 3 6
r B hs C 42492N40320
7 X8
3
2
3
2
10 X8
3
2 2 5
2
4
8 X8
r Nc BC þ 7599N40320
6 X8
9 X8
6 X8
6 X8
2
12 X8
2 2
3
B hs QC 7444Nr40320
3 X8
15 X8
4 3 8
r B hs C þ 17389N40320
10 X8
2 2 5
6 X8
8 X8
8 X8
Nc B hs C þ 44688Nr40320 3 2 8
2
Nc Bha QC þ 12600Nr 40320
2 2
2 2 4
5 X8
11 X8
4 3
c B hs ha C þ 25536Nr N40320 3
4 2
2
7 X8
10 X8
r Nc ha C þ 1944N40320
r B hs C 24048N40320
r Nc B hs C 24988N403200 3
2
12 X8
r Nc B C þ 16330N40320
6 X8
3
Nc Bhs C 9408Nr 40320
3 2
3
4
r Nc Bhs C 25944N40320
r B QC 15420N40320
3 2 5
11 X8
r Nc BC þ 12828N40320
6 X8
r B hs QC 34464N40320
7 X8
c B hs ha C 33216Nr N40320
r Nc Bha C 13872N40320
2
4 3 4
11 X8
r B hs þ 30564N40320
10 X8
5
2
r B hs C 22730N40320
16 X8
4 3 5
3
3
3 X8 11 X8
Nc Bhs C þ 9864Nr40320 2
3 2
4 3
5 X8
2
9 X8
4 2 4
4
2
r BQ C þ 6300N40320
rB C þ 10505N40320
r B hs C þ 19596N40320
2
3 X8
r B hs C 28228N40320
15 X8
c Bhs ha C þ 8616Nr N40320
3
9
4
Nc Bha C 13668Nr40320 3
2
r Bhs C 5832N40320
9 X8
7 X8
r B hs C 34436N40320
3
2
14 X8 3
4 2 6
2
r B hs C þ 56476N40320
6 X8
3
2
Nc ha C 88Nr 40320
Nc Bhs ha C þ 88248Nr 40320
B hs QC þ 26376Nr40320
2
8
9 X8
6 X8
7 X8
r Nc BQC 13872N40320
5
4 2 9
3 2 4
r B hs QC þ 22872N40320
4
10 X8
c ha QC þ 72Nr N40320
4
4 2
10 X8
5 X8
r BC þ 6068N40320
rB C þ 12024N40320
Nc Bhs ha C 11656Nr 40320 2
9 X8
r Bhs C þ 10768N40320 4
4 2
3
r BQC 11868N40320 5
3
r Bhs C þ 10668N40320
2 4
3
5 X8
r Bhs QC 7776N40320 4 4
r Nc hs C 4368N40320
5 X8
9 X8
r hs C 2872N40320
13 X8
3
4
3
r Qhs C þ 3880N40320
r Bhs QC þ 10728N40320
r hs C þ 1944N40320
10 X8
9 X8
rC þ 1288N 40320
10 X8
Nc hs ha C þ 3880Nr 40320 3
r QC 28720N 40320
2273
Nc B hs C 25656Nr 40320
7 X8
3
2 2
2
4
2
Nc Bha C þ 3100Nr40320 3 2 12
B hs C 1444Nr40320
11 X8
6 X8
3 X8
.. . Substituting the values of h0 ; h1 ; h2 ; h3 and h4 ; . . ., in Eq. (13) the formulation of non-dimensional temperature can be obtained as a X h¼ hm ¼ h0 þ h1 þ h2 þ h3 þ h4 þ
ð28Þ
0
Now the temperature field, h can be evaluated if the fin tip temperature C is known whose value lies in the interval (0, 1). Using an arbitrary initial guess value for C, for the temperature field h computed from the above Eq. (26) and applying Newton–Raphson method satisfying the boundary conditions (6) the actual temperature field can be obtained. 4.1. Fin efficiency ðgÞ The fin efficiency is defined as the ratio of actual heat transfer rate to the ideal heat transfer rate. The ideal heat transfer rate takes place in the fin if the whole fin is subjected to base temperature. The actual rate of heat transfer can be obtained by computing the heat losses from the base of the fin dT Qactual ¼ KðTÞAc ð29Þ dx x¼b The ideal heat transfer rate Qideal obtained if the entire fin is kept at the base temperature and can be expressed as
Qideal ¼ hPbðT Ta Þ þ es rPb½1 þ bðT Ts Þ T4 T4s ð30Þ
The fin efficiency can be expressed using non-dimensional terms by Eq. (5) g¼
Qactual Qideal
k a Ac T b dh ½1 þ Ac ðh ha ÞdX b X¼1
¼ hPbðT Ta Þ þ es rPb½1 þ bðT Ts Þ T4 T4s dh ½1 þ Aðh ha ÞdX X¼1 ¼ 2 Nc ð1 ha Þ þ Nr ½1 þ Bð1 hs Þ 1 h4s
ð31Þ
5. Optimization The optimization of the fin can be made by either minimizing the volume for any required heat dissipation or maximizing the heat dissipation for any given fin volume [22,23]. Assuming that the volume of the fin is constant the fin volume is defined as V ¼ Ac b The heat dissipated per unit volume is q ka Ta dh ¼ 2 f1 þ Aðh ha Þg V dX X¼1 b
ð32Þ
The dimensionless form of the above equation [21] is q Ac ka Ta dh ¼ 2 f1 þ Aðh ha Þg ð33Þ Ac N¼ ka Ta V dX b
2274
P.K. Roy, A. Mallick
The maximum heat dissipation value expressed the optimum fin characteristics and the dimensions are the optimum fin configuration. The optimization procedure is also done to fix the profile area Ac to express N as a function of b and then search the optimum value of b. 6. Results and discussion The validation of results is done by considering the both sink temperature, hs ; ha and surface emissivity parameter, B and heat generation number, equating to zero Eq. (6) is converted into the previous work found in the literature [21]. The present works consider five terms and the accuracy results are compared with the previous work as available in the literature as shown in Tables 1 and 2. In fin design application the selection of the values of Nc is very important. The fins are categorized as practical fins depending on the value of conductive–convective fin parameter Nc whose value lies below 0.5. Depending on the surface condition of the fin materials the surface emissivity can increase with temperature ðB > 0Þ or decrease with temperature ðB < 0Þ. In practice values of B range from 0.6 to +0.6 [8]. In order to study the effect of surface emissivity on the temperature distribution and efficiency of the fin, highest, lowest and constant values of surface emissivity are taken i.e. B = 0.6, 0, +0.6 respectively in the present analysis. The idea of maintaining nonzero environmental temperature ðhs and ha Þ is that in actual practice it is not possible for a
fin to maintain one end at high temperature and dissipating heat to the absolute zero. The study of effect of the environmental temperature is also found effective on the thermal performance. The range values of environmental temperature and heat generation number are found in the literature [6]. Fig. 2 illustrates the variation of tip temperature hð0Þ, with respect to heat generation number, Q for three different values 1.00
Fixed parameters A=0,Nr=1, a=0, s=0
0.95
NC=0.25
0.90 0.85 0.80 0.75 0.70
NC=1
0.65 0.60
B=+0.6 B=0 B=-0.6
0.55 0.50 0.0
0.1
0.2
0.3
0.4
Heat generation number,Q
Figure 2 number.
The tip temperatures function of heat generation
Table 1 Comparison of HPM solutions with Adomian Decomposition Method (ADM), Galerkin Method (GM) and Boundary Value Problem (NM). Nc ¼ 1; Nr ¼ 0:2; A ¼ 0:2; Q ¼ 0; hs ¼ 0; ha ¼ 0; B ¼ 0 X
HPM (Present)
ADM [6]
GM [21]
NM(BVP) [21]
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.666858541 0.669984372 0.679386188 0.695136900 0.717357999 0.746220026 0.781944267 0.824806704 0.875145547 0.933237796 0.9999999999
0.666858541 0.669984372 0.679386188 0.695136900 0.717357999 0.746220026 0.781944267 0.824806704 0.875145547 0.933237796 0.999999999
0.667013018 0.670132740 0.679514048 0.695228804 0.717399183 0.746198567 0.781855307 0.824659376 0.874971905 0.933237591 0.999999999
0.667013597 0.670132745 0.679514005 0.695229254 0.717399776 0.746198734 0.781855061 0.824655933 0.874972380 0.933237796 0.999999999
Table 2
Effect of number of terms in HPM on the temperature distribution of rectangular fin.
Nc ¼ 1; Nr ¼ 0:2; A ¼ 0:2; Q ¼ 0; hs ¼ 0; ha ¼ 0; B ¼ 0 Number of terms (m)
Distance, X 0.0000
0.1000
0.2000
0.300
0.400
0.5000
0.6000
0.7000
0.8000
0.9000
1.0000
2 3 Absolute error difference With m = 2 and m=3
0.6608 0.6687 0.0079
0.6638 0.6719 0.0081
0.6730 0.6812 0.0082
0.6884 0.6969 0.0085
0.7102 0.7194 0.0092
0.7389 0.7479 0.0090
0.7746 0.7835 0.0089
0.8180 0.8260 0.0080
0.8696 0.8760 00064
0.9300 0.9338 0.0038
0.9999 0.9999 0.0000
3 4 Absolute error difference With m = 3 and m=4
0.6687 0.6668 0.0019
0.6718 0.6700 0.0018
0.6812 0.6794 0.0018
0.6970 0.6951 0.0019
0.7192 0.7173 0.0019
0.7479 0.7462 0.0017
0.7835 0.7819 0.0016
0.8260 0.8284 0.0012
0.8760 0.8751 00009
0.9338 0.9334 0.0004
0.9999 0.9999 0.0000
Thermal analysis of straight rectangular fin
2275
of surface emissivity parameters, i.e. B ¼ 0:6; 0; þ0:6 while other parameter maintained at zero except Nr ¼ 1. The surface emissivity lines increase linearly with increase in internal heat generation number Q ¼ 0 to Q ¼ 0:4. The internal heat source affects the surface of the fin materials. The relative spacing between the emissivity lines is higher when conduction–convection parameter, Nc is maintained at low values. This is because at lower value conduction–convection parameter, Nc, the convective heat loss is higher and hence temperature gradient is higher. Figs. 3 and 4 demonstrates the variation of temperature of the fin materials for different values of thermal emissivity parameters. The thermal conductivity of the fin materials is maintained at A ¼ 0:6 and sinks temperature and heat generation maintained at some fixed value. It is evident from the both figures that the surface emissivity lines of fin materials are more affected when the values of Nc are maintained below the range of 0.5. The surface of the fin materials is more affected at higher values of heat generation number which is evident from Fig. 4.
Fig. 5 illustrates the variation of fin efficiency of the fin materials g with respect to heat generation number Q. It is clear from the figure that the efficiency of the fin materials is very low without internal heat generation number Q ¼ 0. Fin efficiency increases with the increase in internal heat generation number Q. The efficiency corresponding to lower values of conduction–convection parameter, Nc is steeper as compared to other. That may be because of lower values of conduction–convection parameter, the simultaneous internal heat generation and surface heat loss enhances the fin efficiency. Fig. 6 shows the variation of fin efficiency with respect to the radiation sink temperature hs . The efficiency of surface emissivity curves corresponding to negative values of surface emissivity is higher. This observation has been found to be true in case of space radiative fin [9]. The efficiency is higher when fin maintained at zero sink temperature, and decreases beyond the radiation sink temperature 4.5.
0.90
Fixed parameters A=-0.2, a= s=0.5,Nr=1
0.85 1.00
Nc=0.25
0.80 0.95
B=+0.6 B=-0.6 B=0
0.75 0.70
Nc=0.25
0.85
Effeciency,
Temperature distribution,
0.90
0.80 0.75
0.65 0.60 0.55 0.50
0.70 0.65
0.45
Nc=1
Nc=1
0.40
0.60
0.35
Fixed parameters =1, a= s=0.1,Q=0.1 r
0.55 0.50 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.0
0.1
0.2
0.3
0.4
Heat generation number,Q
1.0
Figure 5 The effect of fin efficiency g on the heat generation number, Q.
Distance,X
Figure 3 The effect of surface emissivity on the temperature distribution for Q = 0.1. 0.8
1.00
Fixed parameters A=-0.2, a=0.5,Nr=1,Q=0.1
NC=0.25
0.95
Nc=1 0.6
0.85
Efficiency,
Temperature distribution,
0.90
0.80 0.75
B=+0.6 B=0 B=-0.6
NC=1
0.70
0.4
0.65
Fixed parameters A=-0.6,Nr=1, a= s=0.1,Q=0.4
0.60 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Distance,X
Figure 4 The effect of surface emissivity on the temperature distribution for Q = 0.4
Nc=0.25
B=-0.6 B=+0.6 B=0
0.2
0.0
0.1
0.2
0.3
0.4
0.5
Radiation sink temperature,
0.6
0.7
0.8
s
Figure 6 The variation of fin efficiency g with respect to the radiation sinks temperature hs .
2276
P.K. Roy, A. Mallick
0.60
B=-0.6,Nc=0.25 B=+0.6,Nc=0.25 B=0, Nc=0.25 B=-0.6, Nc=1 B=+0.6,Nc=1 B=0, Nc=1
0.55 0.50
Effeciency,
0.45 0.40 0.35 0.30 0.25
Fig. 7 shows the variation of fin efficiency with respect to radiation conduction parameter Nr . As the radiation conduction parameter increases the efficiency corresponding to different surface emissivity lines decreases gradually. At higher values of radiation conduction parameter the radiation heat losses are higher but at negative values of surface emissivity parameter the efficiency is higher. Fig. 8 shows the optimum dimensionless parameter N as function surface emissivity parameter, B for different values of conduction–convection parameter. Fig. 9 presents optimum dimensionless parameter N is a function of fin length. The best fin length can be obtained under a given volume V, profile area Ac and thermal parameter.
A=-0.2, a= s=0.5,Q=0.1
0.20
0.8
1.2
1.6
2.0
2.4
2.8
3.2
3.6
4.0
7. Conclusion
Radiation conduction parameter,N r
Figure 7 The variation of fin efficiency g with respect to radiation conduction parameter Nr .
50 45
Nc=0.25
40
Nc=1
Nx100
35 30
References 25 20 15 10 -0.6
A=-0.2, a= s=0.5,Nr=0.2,Q=0.1,b=0.05 -0.4
-0.2
0.0
0.2
0.4
0.6
Surface emissivity parameter,B
Figure 8 The optimum dimensionless parameters N function of surface emissivity parameter B. 50
A=-0.5, a= s=0.5,Nr=0.2,Q=0.1
40
30
NX100
In this study Homotopy Perturbation Method (HPM) is used for solving nonlinear energy equation in a straight rectangular fin with variable thermal conductivity and surface emissivity with some constant source of heat generation. The problem is solved for two main real and industrial use of thermo geometric fin parameter Nc ¼ 0:25 and Nc ¼ 1. The results are presented for the temperature distribution, efficiency and optimum dimensionless parameters are effective and convenient. Also it is found that this method cans achieve more suitable results as compared to the other methods available in the literature.
Nc=0.25 Nc=1
20
10
0 0.05
0.06
0.07
0.08
0.09
0.10
0.11
0.12
0.13
0.14
Fin length,b
Figure 9 The optimum dimensionless parameters N function of fin length b.
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