Mathematical Problems in Engineering

Volume 2018, Article ID 6947565, 18 pages

https://doi.org/10.1155/2018/6947565

## A Conjugate Heat Transfer Analysis of a Triangular Finned Annulus Based on DG-FEM

^{1}Department of Mathematics, COMSATS University Islamabad (CUI), Vehari Campus, Pakistan^{2}Centre for Advanced Studies in Pure and Applied Mathematics (CASPAM), Bahauddin Zakariya University, Multan, Pakistan^{3}Department of Mathematics, Government Emerson College, Multan, Pakistan^{4}Department of Basic Sciences & Humanities, University College of Engineering & Technology, Bahauddin Zakariya University Multan, Pakistan

Correspondence should be addressed to Muhammad Ishaq; moc.liamtoh@151qahsim

Received 28 February 2018; Accepted 14 June 2018; Published 11 July 2018

Academic Editor: Filippo de Monte

Copyright © 2018 Muhammad Ishaq et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A DG-FEM based numerical investigation has been performed to explore the influence of the various geometric configurations on the thermal performance of the conjugate heat transfer analysis in the triangular finned double pipe heat exchanger. The computed results dictate that Nusselt number in general rises with values of the conductivity ratio of solid and fluid, for the specific configuration parameters considered here. However, the performance of these parameters shows strong influence on the conductivity ratio. Consequently, these parameters must be selected in consideration of the thermal resistance, for better design of heat exchanger.

#### 1. Introduction

A review of the literature [1–23] of the heat transfer enhancement shows that this is an emergent area. The investigation of the solid-wall conductivities on the heat transfer enhancement in internally finned tube was done by Soliman [1]. Krishen [2] investigated the wall conduction effects on conjugate heat transfer problem in a pipe with finite thickness. Barrozi and Pagliarini [3] showed the conjugate heat transfer results in a circular pipe based on their numerical simulation. Sakakibara et al. [4] performed their analytical study of conjugate heat transfer in the annulus. Their results dictated that conduction has inverse relation to the ratio of the conductivities. Tao [5] performed his work on conjugate heat problem in internally finned tube. One of the outcomes of his study was that heat capacities have stronger influence on heat transfer enhancement. Pagliarini [6] studied the influence of wall conductance in the conjugate problem in the tube. He suggested various limits of the values of wall thickness. A finned double pipe (FDP) heat exchanger, comprising two circular pipes, uses one of the most efficient augmentation techniques in which fins of various shapes are augmented on the inner pipe. Agrawal and Sengupta [7] investigated the heat transfer problem with periodic circular FDP and reported substantial heat transfer rate improvement. Kettner et al. [8] investigated the study of the internally finned tube and recommended that the fin height may be more than 40% relative to radius of pipe for enhancement of heat with respect to conductivities ratio. Suryanarayana and Apparao [9] reported enhancement of the heat transfer rate in FDP. Fiebig et al. [10, 11] did their investigation of heat reversal occurrence and its avoidance in a finned tube. A review of FDP and multitube was presented by Taborek [12]. Syed [13] numerically investigated the heat transfer enhancement in the rectangular FDP. He investigated the influence of various parameters of the configuration of the FDP on the heat transfer characteristics. While investigating the relative performance of the two heated surfaces, he showed that the fin surface is highly effective compared with the inner pipe surface in promoting the convective heat transfer. Dorfman and Renner [14] presented their review on the conjugate problems on heat transfer. They discussed the analytical solutions of such problems. An investigation based on the finite element method for the study of the fluid and the heat transfer characteristics in the triangular FDP heat exchanger with fins of equal height was presented by Syed et al. [15]. A uniform heat flux boundary condition (H1) was imposed at the inner pipe. The comparison of triangular and rectangular fins shows that former fin is low in weight compared to latter due to thin top and thus results in low cost. Due to the complexity and mathematical convenience, in the heat transfer problem, some assumptions were always made relax so that the problem may be explored easily. The assumptions of infinitesimal inner pipe wall thickness and infinite thermal conductivity were considered in [15]. Their investigation presented that the Nusselt number is enhanced more than four times as a result of augmentation of triangular fins. Iqbal et al. [16] studied the parabolic FDP and suggested many optimal configurations of FDP based on the Nusselt number. Ishaq et al. [17] studied the heat transfer enhancement in FDP with different fin heights of triangular shape. They showed that a group of triangular fins with unequal heights perform better than the equal heights. Iqbal et al. [18] proposed many optimal fin shapes in FDP. Salman et al. [19] numerically presented the heat transfer enhancement in a tube induced by elliptic-cut and classical twist tape (ECT). Their results showed that the performance of ECT is better than classical twist tape. Iqbal et al. [21] reported many optimum FDP configurations using genetic algorithm based on the weight, heat quantity, cost, and structural integrity of FDP. Syed et al. [22] numerically investigated FDP with variable thickness of the tip of the triangular fins. Their results recommended that the tip thickness may be considered in the designing of FDP. Waseem et al. [23] presented the performance of exponential FDP numerically. They found that Nusselt number may have significant effect on the conductivities ratio values of solid and fluid.

In the present investigation, problem [15] is enhanced by taking into account the assumptions of finite thickness of inner pipe wall of FDP and finite thermal conductivity in the inner pipe wall-fin assembly, carried out in chapter 6 of Ph.D. thesis of corresponding author [20]. It may be remembered that in most of the finned annulus fin surface plays a dominant role in heat transfer. Therefore, it is necessary for an accurate solution of the heat transfer problem in FDP to consider fin conduction in wall-fin assembly. Thus, two-dimensional energy equation is considered in the triangular fins and inner pipe wall, in the present investigation. In this way, the present problem takes the form of conjugate heat transfer analysis by considering the conduction in the solid and the convection in the fluid, simultaneously. At the solid-fluid interface of such problems, the temperature and heat flux are considered as continuous functions. In the present work, this condition is implemented successfully. The 2-dimensional energy equation in the inner pipe wall and fin will be solved subject to H1-boundary conditions. The effect of this more realistic situation of wall-fin conductance will be investigated on the triangular FDP design by studying the temperature distribution and the Nusselt number with the influence of the variations in the height of the triangular fin (), the inner pipe wall thickness (), the ratio of radii of both pipes (), and the number of triangular fins ().

The outline of present work consists of 5 sections. Section 2 deals with the present problem and its mathematical model. The DG-FEM based numerical method is described in Section 3. Sections 4 and 5 are, respectively, devoted to results and discussion, as well as the conclusions.

#### 2. Problem Statement and Its Mathematical Model

The cross section of the triangular FDP is drawn in Figure 1(a). FDP consists of two concentric circular pipes with a number of longitudinal triangular fins augmented on the inner pipe. Triangular fins are nonporous, straight, and uniformly distributed around the periphery of the outer surface of inner pipe. Thus, the geometry is described by five parameters: the ratio of radii of both pipes, the number of triangular fins, fin half angle, inner pipe wall thickness, and the triangular fin height. The computational domain is sketched in Figure 1(b).