**1. Introduction**

The theory of heat transfer is widely used in many fields of engineering industries and also in applied sciences. Heat exchangers are being used in power generation houses and nuclear reactor centres, in order to generate and convert energy for unlimited purposes. The design of the heat exchangers, according to its usage, also is a matter of great importance.

As for boiling, condensing and radiating the fluid and other things, the size of heat transfer equipment is always taken into account. In aerospace, equipment requires the limitation of weights, while in nuclear reactors, a deep study of heat transfer analysis is needed to avoid the unbearable damages [1]. In heating, generally these heat exchangers show the very low rate of heat transfer. The performance of such heat exchanger can be signified by various techniques. Özisik [2] has given a detailed study of augmented fin surfaces which are of great help in enhancement of heat transfer rate. A similar study of heat flow was made by Nasiruddin and Kamran [3], for vortex generation by applying baffles in circular ducts. A study on convective heat transfer with variable fin heights was made by Zeitoun and Hegazy [4], in which a rise in heat transfer rate was observed, with low friction factor. Suryanarayana and Apparao [5] mentioned in his work that one of the criteria for evaluating the performance of a heat exchanger with extended surfaces is the pumping power required for a specified heat duty. He reported that average heat transfer coefficient increases with an increase in the frequency of the number of

fins. Another study for heat transfer was made by using elliptic pin finned tube by Qingling et al. [6]. Adegun et al. [7] proposed a new method to increase the heat transfer rate by using circular pipes making them inclined, with different inclinations. They investigated that heat transfer rate is rapid till 15<sup>o</sup> inclination and fin height H = 0.2 and increase in fin height is just a waste of material and causes more expenses. But before the last few decades, it was not found by any good mathematical approach. Pagliarini [8] indicated that the idealization of infinite conduction may cause unrealistic approach in the analysis of heat transfer characteristics. So, they proposed the idea of finite conductivity offered by the material used in it.

if we increase the conductivity of fin. Mazhar [20] extended the work of Syed [19] by considering laminar conjugate heat transfer in the thermal entrance region of the finned annulus. In his work, finite difference method (FDM) was used for the simulation of the problem of hydrodynamically fully developed and thermally developing and fully developed flow. The investigation made by him was clearly showing that the entrance region is more affected by the ratio of conductivities and rate of heat transfer rate at entrance is higher than that of the fully developed

*Finite Difference Solution of Conjugate Heat Transfer in Double Pipe with Trapezoidal Fins*

The methods in the previous discussion are conventional methods for enhancing heat transfer rate. They have their own limitations such as the need of extended surfaces, enhancement in thermal processing equipment sizes and increase in

In this chapter, a numerical algorithm for solving elliptic PDEs involving fluid

Iterative methods are widely used in order to solve the difference equations, which are obtained from elliptic PDEs. Among these iterative methods, SOR method is widely used for its fast convergence for a class of large linear systems arising from difference equations. SOR method is a quick solver for a large number

The momentum and energy equations are solved by using the algorithm given in the next section, for the behaviour of fully developed laminar flow through a finned double pipe. A comparative study of literature results and the present work are

> *<sup>u</sup>* <sup>¼</sup> <sup>1</sup> *h*2

8 ><

>:

*u* ¼ *f x*ð Þ *; y* (1)

9 >=

>;

Þ is denoted

1 1 �4 1 1*:*

A general Poisson equation in two dimensions ∇<sup>2</sup>

helpful. The value of the exact solution function *u x*ð Þ *; y* at a point P(*xi, yj*

The function *u x*ð Þ *; y* can be replaced by the value at the discrete nodes of the region. In order to discretize the function, a square grid with step size h can be

� �, and its approximated value is denoted as *u i*ð Þ *; <sup>j</sup>* . For the chosen discretization of the function, the partial differential equations are approximated at the grid points by using the discrete value of the function *u i*ð Þ *; j :* The first- and second-order partial derivatives are approximated by the difference quotients. For

flow and of heat transfer analysis is described. This algorithm uses multigrid discretization for nested iterations to accelerate the rate of convergence at higher levels with less computation. The well-known successive over-relaxation (SOR) method is used to solve the problem, by giving a fixed value to the relaxation parameter. Problems of steady-state viscous flow and steady temperature can be brought into the category of elliptic PDEs with appropriate boundary conditions. In order to solve these PDEs numerically, a higher order accuracy rate with less computation is more preferred. Moreover, estimation of error helps to ensure the

pumping power to acquire the desirable efficiency level.

*DOI: http://dx.doi.org/10.5772/intechopen.82555*

region.

**2. Iterative methods**

accuracy of the solution.

of linear equations.

by *u xi; yj*

**41**

shown in "Results" section.

**2.1 Iterative scheme for Poisson equation**

can be approximated by the pictorial relation ∇<sup>2</sup>

this purpose central difference quotients are used as follows:

The transfer of heat between fluid and solid while flowing in any heat exchanger is governed by two different kinds of equations, as transfer of heat in fluid is governed by the elliptical Navier-Stokes equation or by the parabolic boundary layer equation, and the heat transfer inside the body is governed by the elliptical Laplace equation or by the parabolic differential equation [9]. This forms a so-called model of conjugate problem. Conjugate heat transfer problems have been analysed in various geometric configurations. Kumar [10] examined two conjugate problems of heat transfer in the laminar boundary layer at the boundary of a semi-infinite porous medium on the assumption that fluid filters continuously through the porous surface and that the injection velocity varies as *x*1/2. Barozzi and Pagliarini [11] used finite element method to examine a conjugate problem of a laminar flow in a pipe when the outer wall is being heated uniformly to observe the effects of wall conduction.

Mori et al. [12] investigated the conjugate problem in a circular pipe and observed the conduction effects. In this study, it was proposed that conductivities of the wall and of fluid make remarkable affects in heat transfer properties when thickness of the wall is increased, while thin wall effects on the heat transfer properties are negligible. An analysis on conjugate heat transfer by using three types of boundary conditions, as constant heat flux, constant wall temperature and constant heat transfer coefficients, was made by Sakakibara et al. [13]. They used Duhamel's theorem to calculate the interfacial temperature. They reported that conduction in the wall is inversely proportional to the ratio of conductivities of solid to fluid.

Kettner et al. [14] numerically investigated that the ratio of thermal conductivities of the solid to fluid has no noticeable effect when the fins of the small height are considered. However, this conductivity ratio has a significant effect when the fin height relative to pipe radius is taken more than 0.4. A similar conjugate problem was studied in finned tube, and it was reported that fin efficiency has a great influence on heat flux and heat transfer coefficient by Fiebig et al. [15]. The conjugate heat transfer problem was investigated in different geometries by Nguyen et al. [16]. Nordstorm and Berg [17] investigated the Navier-Stokes equations for modified interface conditions. They have computed conjugate problem with two approaches: one is by using heat equation for the transfer of heat in solid, and the other is transfer of heat by using Navier-Stokes equations.

Sohail and Fakhir [18] gave numerical investigation of double-pipe heat exchanger with circumferential fins in longitudinal to study the effect of fin pitchto-height (P/H) ratio on heat transfer and fluid flow characteristics at various Reynolds numbers, using water as the working fluid. Systematic analysis is carried out by changing geometric and flow parameters. Geometric parameters include varying the pitch-to-height ratio from 0.55 to 26.4, while for the flow parameters, Reynolds number varied from 200 to 1400.

Syed et al. [19] made numerical simulation of finned double-pipe heat exchanger, where fins are distributed around the outer wall of the inner pipe. By using H1 (constant heat flux) and T1 (constant wall temperature) boundary conditions and one-dimensional fin equation, he concluded that the fin heat loss increases

### *Finite Difference Solution of Conjugate Heat Transfer in Double Pipe with Trapezoidal Fins DOI: http://dx.doi.org/10.5772/intechopen.82555*

if we increase the conductivity of fin. Mazhar [20] extended the work of Syed [19] by considering laminar conjugate heat transfer in the thermal entrance region of the finned annulus. In his work, finite difference method (FDM) was used for the simulation of the problem of hydrodynamically fully developed and thermally developing and fully developed flow. The investigation made by him was clearly showing that the entrance region is more affected by the ratio of conductivities and rate of heat transfer rate at entrance is higher than that of the fully developed region.

The methods in the previous discussion are conventional methods for enhancing heat transfer rate. They have their own limitations such as the need of extended surfaces, enhancement in thermal processing equipment sizes and increase in pumping power to acquire the desirable efficiency level.
