Optimizing the cooling efficiency of a convex spine fin with wetted characteristics beneficial in automotive components: an execution of Charlier polynomial collocation method

doi:10.1007/s10483-025-3278-9

Abstract

Abstract:

Fins are extensively utilized in heat exchangers and various industrial applications as they are lightweight and can benefit in various systems, including electronic cooling devices and automotive components, owing to their adaptable design. Furthermore, spine fins are introduced to improve performance in applications such as automotive radiators. They can be shaped in different ways and constructed from a collection of materials. Inspired by this, the present model examines the effects of internal heat generation and radiation-convection on the thermal distribution in a wetted convex-shaped spine fin. Using dimensionless terms, the proposed fin model involving a governing nonlinear ordinary differential equation (ODE) is transformed into a dimensionless form. The study uses the operational matrix with the Charlier polynomial collocation method (OMCCM) to ensure precise and computationally efficient numerical solutions for the dimensionless equation. In order to aid in the analysis of thermal performance, the importance of major parameters on the temperature profile is graphically illustrated. The main outcome of the study reveals that as the radiation-conductive, wet, and convective-conductive parameters increase, the heat transfer rate progressively improves. Conversely, the ambient temperature and internal heat generation parameters show an inverse relationship.

Key words: fin, wet fin, spine, internal heat generation, Charlier polynomial collocation method

2010 MSC Number:

O242.21

A. N. MALLIKARJUNA, S. K. ABHILASHA, R. S. VARUN KUMAR, F. GAMAOUN, B. C. PRASANNAKUMARA. Optimizing the cooling efficiency of a convex spine fin with wetted characteristics beneficial in automotive components: an execution of Charlier polynomial collocation method. Applied Mathematics and Mechanics (English Edition), 2025, 46(8): 1609-1630.

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Figures/Tables 20

Fig. 1

Fig. 2

Table 1

Comparative analysis of the heat transfer rate QS‐OMCCM and the efficiency ηS‐OMCCM of the spine using the OMCCM for various values of λ with an absolute error evaluation"

$λ$	Exact solution		OMCCM result		Absolute error
$λ$	$Q ˜ S$	$η ˜ S$	$Q S ‐ OMCCM$	$η S ‐ OMCCM$	$Q S AE$	$η S AE$
0	2.072 319 58	0.414 463 92	2.073 294 71	0.414 658 94	$9.75 × 10 − 4$	$1.95 × 10 − 4$
0.5	2.185 353 35	0.397 336 97	2.186 083 86	0.397 469 79	$7.31 × 10 − 4$	$1.33 × 10 − 4$
2.0	2.493 965 47	0.356 280 78	2.494 110 87	0.356 301 55	$1.45 × 10 − 4$	$2.08 × 10 − 5$
3.0	2.680 126 54	0.335 015 82	2.680 044 23	0.335 005 53	$8.23 × 10 − 5$	$1.03 × 10 − 5$
10	3.734 629 91	0.248 975 33	3.737 369 39	0.249 157 96	$2.74 × 10 − 3$	$1.83 × 10 − 4$

Table 1

Table 2

Comparative analysis of the heat transfer rate QS‐OMCCM and the efficiency ηS‐OMCCM of the spine using the OMCCM for various values of Nc with an absolute error evaluation"

$N c$	Exact solution		OMCCM result		Absolute error
$N c$	$Q ˜ S$	$η ˜ S$	$Q S ‐ OMCCM$	$η S ‐ OMCCM$	$Q S AE$	$η S AE$
1	1.193 575 98	0.596 787 99	1.195 588 54	0.597 794 27	$2.01 × 10 − 3$	$1.01 × 10 − 3$
2	1.540 269 93	0.513 423 31	1.542 192 61	0.514 064 20	$1.92 × 10 − 3$	$6.41 × 10 − 4$
6	2.493 965 47	0.356 280 78	2.494 110 87	0.356 301 55	$1.45 × 10 − 4$	$2.08 × 10 − 5$
8	2.854 280 86	0.317 142 32	2.854 117 83	0.317 124 20	$1.63 × 10 − 4$	$1.81 × 10 − 5$
12	3.465 617 64	0.266 585 97	3.466 712 77	0.266 670 21	$1.10 × 10 − 3$	$8.42 × 10 − 5$

Table 2

Table 3

Maximum and minimum absolute error results for various N cases"

$N$	3	4	5	6
$Θ max ‐ AE$	$1.811 × 10 − 2$	$1.114 × 10 − 2$	$7.768 × 10 − 3$	$5.548 × 10 − 3$
$Θ min ‐ AE$	$6.994 × 10 − 15$	$1.299 × 10 − 13$	$2.300 × 10 − 13$	$2.113 × 10 − 11$

Table 3

Fig. 3

Fig. 4

Fig. 5

Table 4

Validation of the OMCCM results by evaluation with those in Ref. [47]"

$X$	$Θ (X)$
$X$	ADM^[47]	OMCCM	Error
0	0.648 055	0.648 037	$1.08 × 10 − 5$
0.2	0.661 059	0.661 047	$1.20 × 10 − 5$
0.4	0.700 594	0.700 585	$9.00 × 10 − 6$
0.6	0.768 246	0.768 238	$8.00 × 10 − 6$
0.8	0.866 731	0.866 722	$9.00 × 10 − 6$
1	1	0.999 983	$1.70 × 10 − 5$

Table 4

Fig. 6

Fig. 7

Fig. 8

Fig. 9

Fig. 10

Fig. 11

Fig. 12

Fig. 13

Fig. 14

Fig. 15

Fig. 16

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