Orthotropy in Annular fins: A Collocation based solution

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ScienceDirect Materials Today: Proceedings 5 (2018) 28402–28407

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ICCMMEMS_2018

Orthotropy in Annular fins: A Collocation based solution Ambuj Shrivastava, Vivek Kumar Gaba, Shubhankar Bhowmick* National Institute of Technology Raipur 492010, India

Abstract The effect of orthotropy in 2D temperature distribution in annular fins is reported using collocation method. Point Collocation Method (PCM) gives results for insulated tip boundary condition problem along with contact resistance at the base. The problem is solved assuming the non-dimensional 4th order polynomial temperature distribution and forcing the residue to zero at Gauss quadrature points. The fin material is assumed to be aluminium (Kr =200w/m k). The fin is orthotropic in nature and thermal conductivity effect has been governed by thermal conductivity ratio (K). The analysis has been done by considering different coefficients of convective heat transfer at fin base (hc), surface (hs) and tip (he) and is reported for three different aspect ratios (α) of fin. The results presented demonstrate the viability for design and optimization of fins of different sizes under a range of convection conditions, applied to many practical applications encountered in engineering industry. © 2018 Elsevier Ltd. All rights reserved. Selection and Peer-review under responsibility of International Conference on Composite Materials: Manufacturing, Experimental Techniques, Modeling and Simulation (ICCMMEMS-2018). Keywords: Orthotropy; Annulus; Point Collocation Method; Aspect ratio.

1. Introduction Fins find many applications in industry and are essential requirement to conduct various engineering operations. The studies conducted earlier reflect that one dimensional studies are not accurate enough to predict the performance of fins as every physical system in the world is three dimensional in nature.

 This is an open-access article distributed under the terms of the Creative Commons Attribution-Non Commercial-ShareAlike License, which permits non-commercial use, distribution, and reproduction in any medium, provided the original author and source are credited. * Corresponding author. Tel.: +91-9575 955040. E-mail address: [email protected]

2214-7853 © 2018 Elsevier Ltd. All rights reserved. Selection and Peer-review under responsibility of International Conference on Composite Materials: Manufacturing, Experimental Techniques, Modeling and Simulation (ICCMMEMS-2018).

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Nomenclature hs hc he K Kz Kr Q



R1 R2 0

Bie Bic Bis Ϛ ɳ α

Convection heat transfer coefficient for the top surface (W/m2k) Convection heat transfer coefficient for the contact surface (W/m2k) Convection heat transfer coefficient for the tip surface (W/m2k) Thermal conductivity ratio (Kz/Kr)0.5 Thermal conductivity along the axial direction (W/mk) Thermal conductivity along the radial direction (W/mk) Heat transfer from the fin (W) Temperature distribution in Ϛ- ɳ coordinate (Non-dimensional Temperature) Inner radius of annulus (m) Outer radius of annulus (m) Half-thickness of annulus (m) Temperature difference at the base (K) Temperature difference at any location (K) Biot number at the tip Biot number at the contact face Biot number at the surface Non-dimensional radius Non-dimensional thickness Aspect Ratio

Yovanovich et al. [1] presented the heat transfer for orthotropic annular fins having convective-tip boundary condition and the contact resistance at the fin base. Kalman [2] reported the heat dissipation of annular fins of definite mass for shapes having constant thickness, constant area of heat flow, triangular and parabolic profiles. Aziz et al. [3] investigated the fin performance in an annular fin with periodic temperature variation at the fin base. Results reveal that the performance with a rectangular shape is best for straight and annular fins followed by parabolic profile and the triangular shape being the least favorable which prompted the selection of rectangular geometry for the present analysis. Kang [4] reported solutions for 2D symmetric fins under different set of values for thickness, heat loss and fin length. The results indicate that, of all the shapes considered, rectangular fin yields maximum heat transfer. The optimal dimensions of convective-radiating circular fins of varying thickness, heattransfer coefficient and thermal conductivity in addition to internal heat generation has been reported by Khan and Zubair [5]. Two-dimensional fin with convective base condition has been dealt in detail by Malik and Rafiq [6]. It has been found that a (quadratic) hyperbolic circular fin gives an optimum performance. Abboudi [7] conducted transient thermal analysis for the approximation of two dimensional rectangular fins with two different convective boundary conditions. Based on inverse regularization method, heat transfer coefficient has been estimated and compared with experimentally determined heat transfer coefficient. Exact solutions for heat transfer in a 2D rectangular fin has been described by Moitsheki and Rowjee [8], using Kirchoff’s transformation and assuming temperature dependent thermal conductivity, internal energy generation function, and heat transfer coefficient. Mustafa et al. [9] calculated the temperature distribution and heat transfer rate for orthotropic two-dimensional, annular fins subject to convective-tip boundary condition and contact resistance at the fin base. Aasi et al. [10] presented the two dimensional performance analysis of orthotropic annular fin of rectangular profile. In the present work, excess temperature is presented for orthotropic annular fin having base contact resistance under insulated tip as well as short fin type boundary condition. Assuming 4th order 2D polynomial, the formulation is based on point collocation residual minimization technique. The fin conductivity is governed by thermal conductivity ratio (K) and analysis is reported for three different fin aspect ratios (α) under different coefficients of convective heat transfer at fin base (hc), surface (hs) and tip (he).

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2. Mathematical Formulation The generalized governing equation obtained for two dimensional heat transfer by applying energy balance on elemental area of fin is given by  2 1   2 (1)   K2 0 r 2

r r

z 2

The governing equation is solved by residual minimization using Point Collocation Method (PCM) applied in form of a 4th order 12-term polynomial approximation for normalized temperature distribution in two dimensional Ϛɳ coordinate. The normalization of fin geometry parameters is reported using Eq. 2, K  r z (2)   ,  ,   , K 2  z 0 0 0 Kr The normalized governing equation, thus obtained, is

 2 1   2   K2 2  0 2    

(3)

Fig. 1. Fin geometry and boundary conditions

The boundary conditions implied over the fin to solve the Eq. (3) are shown in Fig.1. The temperature distribution is approximated using the following linear combination of the polynomial functions in Ϛ- ɳ coordinate: (4)    Cii i = 1, 2, 3..., 12 i

Here, i are polynomial or basis functions as reported in Table 1 and Ci are the unknown coefficients, twelve, in this case. Table 1. Polynomial or basis functions. i

i

1 1

2



3

4

5

6

7

8

9

10

11

12



2



2

3

 2

 2

3

 3

 3

Eq. 4, being an approximation, results in residue which is forced to be zero at the collocation points. The collocation points can be chosen arbitrarily. In the present work, four Gauss quadrature collocation points set has been chosen in both Ϛ- ɳ directions. Further each boundary condition is satisfied at two quadrature points along the individual boundary edges, thus resulting into a total of twelve simultaneous equations to determine twelve unknown coefficients using an in-house MATLAB® code. 3. Results and discussion The present work has been studied for three different aspect ratio and thermal conductivity ratios for insulated tip boundary condition i.e. he = 0. The proposed method has been validated Mustafa et.al. [8]; the comparison is shown in Table 2. From the table, it is evident that a good agreement has been found with the benchmark results as the error is below 5%.

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Table 2. Comparison of heat transfer results with numerical results R0(m)

R1(m)

δ(m)

K

0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05

0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06

0.001 0.001 0.001 0.001 0.001 0.001 0.001 0.001

0.96076 0.96076 0.96076 0.96076 2 2 2 2

Aspect ratio(α) 10 10 10 10 10 10 10 10

Q(PCM), W 19.94 37.74 43.08 50.37 19.98 37.86 43.26 50.72

Q [Ref. 8], W 20.99 38.34 43.67 52.17 21.00 38.40 43.76 52.44

% error 4.99 1.55 1.33 3.43 4.85 1.39 1.12 3.26

Table 3. Cases of parameters studied. Bie 0 0 0

Bic 0.1 0.1 0.1

hs 20 20 20

Aspect ratio(α) 5 10 20

K 0.25, 0.5, 1 0.25, 0.5, 1 0.25, 0.5, 1

This is followed by the calculation of excess temperature for the thermal parameters reported in Table 3. The excess temperature distribution for all the cases considered in Table 3 are computed and plotted in Figs. 2-4. Fig. 2 shows temperature distribution over the surface of the fin for K = 0.25. It has been observed that the temperature at the base near the mid-plane is maximum and starts decreasing in both radial and axial direction. It has also been observed that temperature is more in a fin with lower aspect ratio because of larger thickness and thick fins are having more cross sectional area at the base for heat transfer. Fig. 2 depicts that for K = 0.25, the normalized temperature value at the mid-plane decreases from 0.915 to 0.737 as aspect ratio increases from 5 to 20 thus meaning that thicker fins are not effective. It has also been observed that at higher aspect ratios the heat transfer in axial direction becomes significant and effect of two dimensional study starts to become analogous with one dimensional case, hence it can be said that for a thick fin, a two dimensional study is much more important rather than considering it as a one dimensional case.

Fig.2 Temperature distribution at K = 0.25 for α = 5, 10 and 20 respectively

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Fig.3 Temperature distribution at K = 0.50 for α = 5, 10 and 20 respectively

Fig.4 Temperature distribution at K = 1.00 for α = 5, 10 and 20 respectively

Similar contour plots of excess temperature distribution at K = 0.5 and K = 1.0 for different aspect ratio are shown in Fig. 3 and 4 respectively. The similar pattern has been observed for both the cases as found for K = 0.25. However, Figs 3-4 further reveal that increasing the K value for a particular aspect ratio increases heat transfer in axial direction and variation along thickness reduces. This clearly shows the dependency of the fin heat transfer on thickness. Fig. 5(a-c) represents the radial variation of normalized temperature at the fin centerline for different aspect ratios along with different K values. It is observed that temperature at the mid- plane of the fin is higher for smaller values of K although effect of orthotropic behavior is insignificant at larger values of α.

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Fig. 5 (a-c) Centre-line temperature in orthotropic annulus of (a) α = 5, (b) α = 10 and (c) α = 20 for different thermal conductivity ratio (K)

4. Conclusion The present work reports the effect of orthotropic thermal conductivity on two dimensional annular fins using residue minimization technique, i.e. Collocation method. The solution is validated with a benchmark result and found in good agreement. Results of temperature distribution are reported for fins having aspect ratio; 5, 10 and 20 and conductivity ratio; 0.25, 0.50 and 1.00. The contour plots reveal the necessity of two dimensional study as results show the temperature distribution over the surface are the function of aspect ratio, α and thermal conductivity ratio (K). The results also point towards the necessity of two dimensional analysis for lower aspect ratio. A two dimensional study has been observed to be important while considering the fins which are having a substantial thickness. The results can be used for the design and optimization of annular fins in relation to size for different aspect ratios. References [1] M. M. Yovanivich, J. R. Culham, and T. F. Lemczyk, Int. J. Thermophy., 1988, 2(2), 152–157. [2] H. Kalman, Int. J. Heat Mass Tran., 1989, 32(6), 1105–1110. [3] A. Aziz, M. Mujahid, K.Abu-Abdou, J. King Saud University, 1993, 5(1), 105–122. [4] H.S. Kang, KSME Int. Journal, 1997, 11(3), 311–318. [5] J. Khan, S.M. Zubair, Heat Mass Tran., 1999, 35, 469–478. [6] M.Y. Malik, A. Rafiq, Nonlin. Analysis: Real World Appl., 2010, 11(1), 147–154. [7] S. Abboudi, J. Franklin Inst., 2011, 348(7), 1192–1207. [8] R.J. Moitsheki,A. Rowjee, Math. Prob. Engg., 2011, 1–13. [9] M.T.Mustafa, Zubair, M. Syed ,A.F.M.Arif, App. Therm. Engg., 2011, 31(5), 937–945. [10] H. K. Aasi, V.K. Gaba, A.K. Tiwari, S. Bhowmick, Front. Heat Mass Tran., 2017,8,15.