**2. Iterative methods**

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.

*Numerical Modeling and Computer Simulation*

The transfer of heat between fluid and solid while flowing in any heat exchanger

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 pro-

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

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,

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

portional to the ratio of conductivities of solid to fluid.

other is transfer of heat by using Navier-Stokes equations.

Reynolds number varied from 200 to 1400.

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.

**40**

In this chapter, a numerical algorithm for solving elliptic PDEs involving fluid 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 accuracy of the solution.

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 of linear equations.

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 shown in "Results" section.

#### **2.1 Iterative scheme for Poisson equation**

A general Poisson equation in two dimensions ∇<sup>2</sup> *u* ¼ *f x*ð Þ *; y* (1)

can be approximated by the pictorial relation ∇<sup>2</sup> *<sup>u</sup>* <sup>¼</sup> <sup>1</sup> *h*2 1 1 �4 1 1*:* 8 >< >: 9 >= >;

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 helpful. The value of the exact solution function *u x*ð Þ *; y* at a point P(*xi, yj* Þ is denoted by *u xi; yj* � �, 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 this purpose central difference quotients are used as follows:

$$u\_{\mathbf{x}}\left(\mathbf{x}\_{i},\boldsymbol{y}\_{j}\right) = \frac{u\_{i+1,j} - u\_{i-1,j}}{2h} \tag{2}$$

*un*þ<sup>1</sup> *i,j* <sup>¼</sup> *un*

*ε* ¼ 0*:*00001.

**3. Problem formulation**

is shown in **Figure 1**.

where *<sup>C</sup>* ¼ � <sup>1</sup> � *<sup>R</sup>*<sup>2</sup>

Syed [19].

**43**

*a*≤*r*≤*b* and 0 ≤*θ* ≤*α* þ *β* shown in **Figure 2**.

**3.1 Momentum, heat and energy equations**

because of its geometrical symmetry as shown in **Figure 2**:

*∂*2 *u*∗ *<sup>∂</sup>R*<sup>2</sup> <sup>þ</sup>

<sup>m</sup> <sup>þ</sup> <sup>2</sup>*R*<sup>2</sup>

� �

*i,j* þ *ω=*4 *f xi; yj*

order to have faster rate of convergence.

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

� � � <sup>4</sup>*un*

*i,j* � *ai*þ<sup>1</sup>*,jun*

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

where the value of relaxation factor '*ω*' is obtained by hit and trial method in

The SOR iteration procedure can be terminated by using the following convergence criteria depending upon the convergence nature of the system being solved. In order to have computational results with a minimum number of iterations, absolute norm is used to decide the convergence criteria, which is given below:

� �

where *ε* is the order of convergence for each iteration and its value is taken as

In convective forced flow, the assumptions of negligible wall thickness and infinite

In order to describe the conjugate problem mathematically, we need to describe the momentum, energy and heat equations that are to be solved simultaneously. The momentum and energy equations will be described in the prescribed sections and will be transformed into their dimensionless forms by the means of dimensionless variables. Similarly, heat equation is treated and solved for the transfer of heat in the solid part of the domain. A cross-sectional view of finned double pipe (FDP)

The velocity field is then independent of the temperature field. Geometrical symmetries shown in **Figure 1** permit the equations to be only solved in the region

The governing momentum, heat and energy equations in dimensionless form are given in the Eqs. (11)–(13), as below by using the transformations defined in

This problem is constrained in the region where *r*<sup>i</sup> ≤ *r*≤*r*<sup>o</sup> and 0≤*θ* ≤*α* þ *β*

1 *R*2 *∂*2 *u*∗ *<sup>∂</sup>θ*<sup>2</sup> <sup>¼</sup> <sup>4</sup> *c*

1 *R ∂u*<sup>∗</sup> *∂R* þ

mln *R*<sup>m</sup>

conductivity of the fin and wall may cause unrealistic predictions of heat transfer characteristics, because of such ideal assumptions. In the present study, we take into account finite wall thickness of the inner pipe wall for realistic results. Also, we consider finite conductivity of the fin and wall in order to avoid the overestimates of heat transfer characteristics. This require coupling of the heat conduction problem in the wall-fin assembly and convective heat transfer problem in the fluid, and this coupled system is called conjugate heat transfer problem. The constant heat flux boundary condition is applied to the inner side of the inner pipe, and adiabatic thermal condition is applied at the inner wall of the outer pipe. At each solid-fluid interface, the heat flux is continuous. Moreover, temperature is assumed to be continuous.

*e i*ð Þ¼ *; j ui,j* � *ui,j* �

*<sup>i</sup>*þ1*,j* � *ai*�<sup>1</sup>*,ju<sup>n</sup>*þ<sup>1</sup>

h� i

*<sup>i</sup>*�1*,j* � *ai,j*þ1*u<sup>n</sup>*

� < *ε* (10)

*i,j*þ<sup>1</sup> � *ai,j*�1*un*þ<sup>1</sup>

*i,j*�1

(9)

(11)

$$u\_{\mathcal{V}}\left(\mathbf{x}\_{i},\boldsymbol{y}\_{j}\right) = \frac{u\_{i,j+1} - u\_{i,j-1}}{2h} \tag{3}$$

$$u\_{\infty}(\mathbf{x}\_i, \mathbf{y}\_j) = \frac{u\_{i+1,j} - 2u\_{i,j+1}u\_{i-1,j}}{h^2} \tag{4}$$

$$\left(u\_{\mathcal{Y}}\left(\mathbf{x}\_{i},\mathbf{y}\_{j}\right) = \frac{u\_{i,j+1} - 2u\_{i,j+1}u\_{i,j-1}}{h^{2}}\tag{5}$$

The Poisson equation can be approximated by using the above central difference quotients:

$$\frac{u\_{i+1,j} - 2u\_{i,j} + u\_{i-1,j}}{h^2} + \frac{u\_{i,j+1} - 2u\_{i,j} + u\_{i,j-1}}{h^2} = \mathbf{f}\left(\mathbf{x}\_i, \mathbf{y}\_j\right) \tag{6}$$

The boundary conditions of the boundary value problem need to be applied in accordance with the difference approximation of PDEs. The simplest of the boundary conditions is the Dirichlet boundary condition. In this case the difference equation can be applied to all interior points with unknown functions, and known values of the function at each boundary can be directly replaced.

Here, a simple type of Neumann boundary condition is taken into consideration. It is considered that boundary is taken at grid parallel to any of the axes here, e.g., x-axis. This boundary condition requires the normal derivative to be disappeared. This derivative in normal direction can be approximated by the central difference quotients.

$$\left. \text{So, we approximate } \left. \frac{\partial u}{\partial n} \right|\_{\text{p}} = 0 \text{ as } \frac{u\_{i+1,j} - u\_{i-1,j}}{2h} = 0 \tag{7}$$

$$\text{which implies } u\_{i+1,j} = u\_{i-1,j} \tag{8}$$

The vanishing of normal derivative means that function *u x*ð Þ *; y* is symmetric about the boundary of the region.

#### **2.2 Linear system solver**

SOR is an iterative method, which is an important solver for the class of large linear system arising from the finite difference approximation of PDEs. It is not only an efficient solver but also a smoother. The method of relaxation is an iterative scheme which permits one to select the best equation to be used for the faster rate of convergence. Although Gauss Jacobi and Gauss Seidel are taken as one of the good iterative methods, relaxation method is more advantageous because of its faster rate of convergence depending upon the relaxation parameter '*ω:*' When 0 < *ω* < 1, the procedures are called under relaxation methods. When *ω* = 1, then relaxation method is same as that of Gauss-Seidel method.

For 1 < *ω* < 2, the procedure is called over-relaxation.

The general Laplace equation is discretized by using finite differences, and the boundary conditions are approximated by second-order central differences. Fictitious points lying outside the domain, arising from the discretized form of derivative boundary conditions, are expressed in terms of the points and then incorporated into the governing finite difference equations. In order to have faster rate of convergence, the discretized form of Laplace equation in Eq. (9) can be approximated by using SOR method, which may take the form

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

$$u\_{i,j}^{n+1} = u\_{i,j}^n + \omega/4 \left[ \left( f\left(\mathbf{x}\_i, y\_j\right) - 4u\_{i,j}^n - a\_{i+1,j}u\_{i+1,j}^n - a\_{i-1,j}u\_{i-1,j}^{n+1} - a\_{i,j+1}u\_{i,j+1}^n - a\_{i,j-1}u\_{i,j-1}^{n+1} \right) \right] \tag{9}$$

where the value of relaxation factor '*ω*' is obtained by hit and trial method in order to have faster rate of convergence.

The SOR iteration procedure can be terminated by using the following convergence criteria depending upon the convergence nature of the system being solved. In order to have computational results with a minimum number of iterations, absolute norm is used to decide the convergence criteria, which is given below:

$$e(i,j) = \left| \overline{u\_{i,j}} - u\_{i,j} \right| < \varepsilon \tag{10}$$

where *ε* is the order of convergence for each iteration and its value is taken as *ε* ¼ 0*:*00001.
