**3. Problem formulation**

*ux xi; yj* 

*uy xi; yj* 

*uxx xi; yj* 

*uyy xi; yj* 

*<sup>h</sup>*<sup>2</sup> <sup>þ</sup>

values of the function at each boundary can be directly replaced.

So*;* we approximate *<sup>∂</sup><sup>u</sup>*

about the boundary of the region.

method is same as that of Gauss-Seidel method.

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

approximated by using SOR method, which may take the form

**2.2 Linear system solver**

**42**

*ui*þ1*,j* � 2*ui,j*þ*ui*�1*,j*

*Numerical Modeling and Computer Simulation*

quotients:

<sup>¼</sup> *ui*þ1*,j* � *ui*�1*,j*

<sup>¼</sup> *ui,j*þ<sup>1</sup> � *ui,j*�<sup>1</sup>

<sup>¼</sup> *ui*þ1*,j* � <sup>2</sup>*ui,j*þ*ui*�1*,j*

<sup>¼</sup> *ui,j*þ<sup>1</sup> � <sup>2</sup>*ui,j*þ*ui,j*�<sup>1</sup>

*ui,j*þ<sup>1</sup> � 2*ui,j*þ*ui,j*�<sup>1</sup>

*<sup>h</sup>*<sup>2</sup> <sup>¼</sup> <sup>f</sup> *xi; yj*

*ui*þ1*,j* � *ui*�1*,j*

which implies *ui*þ1*,j* ¼ *ui*�1*,j* (8)

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

The boundary conditions of the boundary value problem need to be applied in

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

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.

¼ 0 as

accordance with the difference approximation of PDEs. The simplest of the

*∂n* p

The vanishing of normal derivative means that function *u x*ð Þ *; y* is symmetric

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

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

<sup>2</sup>*<sup>h</sup>* (2)

<sup>2</sup>*<sup>h</sup>* (3)

*<sup>h</sup>*<sup>2</sup> (4)

*<sup>h</sup>*<sup>2</sup> (5)

<sup>2</sup>*<sup>h</sup>* <sup>¼</sup> 0 (7)

(6)

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

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) is shown in **Figure 1**.

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

#### **3.1 Momentum, heat and energy equations**

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 Syed [19].

This problem is constrained in the region where *r*<sup>i</sup> ≤ *r*≤*r*<sup>o</sup> and 0≤*θ* ≤*α* þ *β* because of its geometrical symmetry as shown in **Figure 2**:

$$\frac{\partial^2 u^\*}{\partial R^2} + \frac{1}{R} \frac{\partial u^\*}{\partial R} + \frac{1}{R^2} \frac{\partial^2 u^\*}{\partial \theta^2} = \frac{4}{c} \tag{11}$$

$$\text{where } \mathcal{C} = -\left(\mathbf{1} - \mathbf{R}\_{\mathbf{m}}^2 + 2\mathbf{R}\_{\mathbf{m}}^2 \ln \mathbf{R}\_{\mathbf{m}}\right).$$

transformed into dimensionless form:

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

*∂*2 *τf <sup>∂</sup>R*<sup>2</sup> <sup>þ</sup>

where *u*<sup>∗</sup> is the dimensionless mean velocity.

a. No slip conditions at the solid boundaries:

ð Þ <sup>V</sup> *<sup>∂</sup>u*<sup>∗</sup>

*∂τ*

*∂τs*

*∂τf*

*∂τs*

*∂τf*

*Boundary conditions at the lines of symmetry*

ð Þ VI *<sup>∂</sup>u*<sup>∗</sup>

dimensionless form may be written as:

b.Symmetry conditions:

faces. These may be defined as

**45**

*∂*2 *τs <sup>∂</sup>R*<sup>2</sup> <sup>þ</sup>

1 *R ∂τf ∂R* þ

> 1 *R ∂τs ∂R* þ

1 *R*2 *∂*2 *τf <sup>∂</sup>θ*<sup>2</sup> <sup>¼</sup> *<sup>u</sup>*<sup>∗</sup> *A*<sup>∗</sup>

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

1 *R*2 *∂*2 *τs*

The boundary conditions are applied due to the viscosity of the fluid and sym-

ð Þ<sup>I</sup> *<sup>u</sup>*<sup>∗</sup> <sup>¼</sup> 0 at *<sup>R</sup>* <sup>¼</sup> *R,* ^ <sup>0</sup>≤*<sup>θ</sup>* <sup>≤</sup> *<sup>α</sup>* ð Þ II *<sup>u</sup>*<sup>∗</sup> <sup>¼</sup> 0 at *<sup>θ</sup>* <sup>¼</sup> *<sup>α</sup>, <sup>R</sup>*^ <sup>≤</sup>*R*≤*R*<sup>1</sup> ð Þ III *<sup>u</sup>*<sup>∗</sup> <sup>¼</sup> 0 at *<sup>R</sup>* <sup>¼</sup> *<sup>R</sup>*1*, <sup>α</sup>*<sup>≤</sup> *<sup>θ</sup>* <sup>≤</sup>*<sup>α</sup>* <sup>þ</sup> *<sup>β</sup>* ð Þ IV *<sup>u</sup>*<sup>∗</sup> <sup>¼</sup> 0 at *<sup>R</sup>* <sup>¼</sup> <sup>1</sup>*,* <sup>0</sup>≤*<sup>θ</sup>* <sup>≤</sup>*<sup>α</sup>* <sup>þ</sup> *<sup>β</sup>*

*<sup>=</sup><sup>∂</sup><sup>θ</sup>* <sup>¼</sup> 0 at *<sup>θ</sup>* <sup>¼</sup> <sup>0</sup>*, <sup>R</sup>*^ <sup>≤</sup>*R*≤*R*<sup>1</sup>

*=<sup>∂</sup><sup>θ</sup>* ¼ 0 at *θ* ¼ *α* þ *β, R*<sup>1</sup> ≤*R* ≤1

*τ* ¼ 0 at R ¼ *Rw,* 0≤ *θ* ≤*α* (14)

*<sup>∂</sup><sup>θ</sup>* <sup>¼</sup> 0 at R <sup>¼</sup> <sup>1</sup>*,* <sup>0</sup>≤*<sup>θ</sup>* <sup>≤</sup>*<sup>α</sup>* <sup>þ</sup> *<sup>β</sup>* (15)

*<sup>∂</sup><sup>θ</sup>* <sup>¼</sup> 0 at <sup>θ</sup> <sup>¼</sup> <sup>0</sup>*, Rw* <sup>≤</sup>*<sup>R</sup>* <sup>≤</sup>*Ro* (16)

*<sup>∂</sup><sup>θ</sup>* <sup>¼</sup> 0 at <sup>θ</sup> <sup>¼</sup> <sup>0</sup>*, <sup>R</sup>*^ <sup>≤</sup>*R*≤<sup>1</sup> (17)

*<sup>∂</sup><sup>θ</sup>* <sup>¼</sup> 0 at <sup>θ</sup> <sup>¼</sup> <sup>α</sup> <sup>þ</sup> <sup>β</sup>*, Rw* <sup>≤</sup>*R*≤*R*<sup>1</sup> (18)

*<sup>∂</sup><sup>θ</sup>* <sup>¼</sup> 0 at <sup>θ</sup> <sup>¼</sup> <sup>α</sup> <sup>þ</sup> <sup>β</sup>*, <sup>R</sup>*^ <sup>≤</sup>*R*≤<sup>1</sup> (19)

Inner pipe interface is at *r* ¼ *ri*&0≤*θ* ≤ *α:* (20) Fin lateral surface interface is at *θ* ¼ *α*&*ri* ≤ *r*≤ *r*<sup>1</sup> (21) Fin tip interface is at *r* ¼ *r*<sup>1</sup> and *α* ≤*θ* ≤ *α* þ *β:* (22)

*Constant flux boundary condition at the inner surface of the inner pipe*

*An adiabatic wall temperature condition at the inner surface of the outer pipe*

There are three interfaces where the solid and fluid mediums are contact. These three interfaces are termed here as inner pipe, fin lateral surface and fin tip inter-

metry offered by the domain shown in **Figure 2**. The boundary conditions in

<sup>c</sup> *<sup>u</sup>*<sup>∗</sup> (12)

*<sup>∂</sup>θ*<sup>2</sup> <sup>¼</sup> <sup>0</sup> (13)

**Figure 1.** *Cross section of the finned double pipe.*

**Figure 2.** *Computational domain.*

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

transformed into dimensionless form:

$$\frac{\partial^2 \overline{\sigma}^f}{\partial R^2} + \frac{1}{R} \frac{\partial \overline{\sigma}^f}{\partial R} + \frac{1}{R^2} \frac{\partial^2 \overline{\sigma}^f}{\partial \theta^2} = \frac{u^\*}{A\_c^\* \overline{u}^\*} \tag{12}$$

$$\frac{\partial^2 \tau^\epsilon}{\partial R^2} + \frac{1}{R} \frac{\partial \tau^\epsilon}{\partial R} + \frac{1}{R^2} \frac{\partial^2 \tau^\epsilon}{\partial \theta^2} = \mathbf{0} \tag{13}$$

where *u*<sup>∗</sup> is the dimensionless mean velocity.

The boundary conditions are applied due to the viscosity of the fluid and symmetry offered by the domain shown in **Figure 2**. The boundary conditions in dimensionless form may be written as:

a. No slip conditions at the solid boundaries:

$$\begin{aligned} (\text{I) } u^\* &= \mathbf{0} \text{ at } R = \hat{R}, \mathbf{0} \le \theta \le a\\ (\text{II) } u^\* &= \mathbf{0} \text{ at } \theta = a, \hat{R} \le R \le R\_1 \end{aligned}$$

$$\begin{aligned} (\text{III) } u^\* &= \mathbf{0} \text{ at } R = R\_1, a \le \theta \le a + \beta\\ (\text{IV) } u^\* &= \mathbf{0} \text{ at } R = \mathbf{1}, \mathbf{0} \le \theta \le a + \beta \end{aligned}$$

b.Symmetry conditions:

**Figure 1.**

**Figure 2.**

**44**

*Computational domain.*

*Cross section of the finned double pipe.*

*Numerical Modeling and Computer Simulation*

$$(\mathbf{V})^{\circ \boldsymbol{\mu}^\*/\_{\partial \boldsymbol{\theta}}}/\_{\partial \boldsymbol{\theta}} = \mathbf{0} \text{ at } \boldsymbol{\theta} = \mathbf{0}, \hat{R} \le R \le R\_1$$

$$(\mathbf{VI})^{\circ \boldsymbol{\mu}^\*/\_{\partial \boldsymbol{\theta}}}/\_{\partial \boldsymbol{\theta}} = \mathbf{0} \text{ at } \boldsymbol{\theta} = \boldsymbol{a} + \boldsymbol{\beta}, R\_1 \le R \le 1$$

*Constant flux boundary condition at the inner surface of the inner pipe*

$$
\pi = \mathbf{0} \text{ at } \mathbf{R} = R\_{w\nu} \ 0 \le \theta \le a \tag{14}
$$

*An adiabatic wall temperature condition at the inner surface of the outer pipe*

$$\frac{\partial \pi}{\partial \theta} = 0 \text{ at } \mathbf{R} = \mathbf{1}, \ 0 \le \theta \le a + \beta \tag{15}$$

*Boundary conditions at the lines of symmetry*

$$\frac{\partial \tau^{\ell}}{\partial \theta} = 0 \text{ at } \theta = 0 \text{, } R\_w \le R \le R\_o \tag{16}$$

$$\frac{\partial \mathfrak{v}^f}{\partial \theta} = \mathbf{0} \text{ at } \theta = \mathbf{0}, \hat{R} \le R \le \mathbf{1} \tag{17}$$

$$\frac{\partial \pi'}{\partial \theta} = 0 \text{ at } \theta = \alpha + \beta, R\_w \le R \le R\_1 \tag{18}$$

$$\frac{\partial \mathfrak{v}^f}{\partial \theta} = \mathbf{0} \text{ at } \theta = \mathfrak{a} + \mathfrak{z}, \hat{R} \le R \le \mathbf{1} \tag{19}$$

There are three interfaces where the solid and fluid mediums are contact. These three interfaces are termed here as inner pipe, fin lateral surface and fin tip interfaces. These may be defined as


$$\text{Fin tip interface is at } r = r\_1 \text{ and } a \le \theta \le a + \beta. \tag{22}$$

On these interfaces, we impose the conditions of continuity of temperature and that of heat flux in order to maintain the energy balance. These interface conditions are used to couple conduction (Eq. (13)) in the solid with the energy (Eq. (12)) in the fluid. This forms the so-called conjugate problem.

The interface conditions can be expressed mathematically as given below by using same dimensionless transformations.

*Continuity of fluxes at the solid-fluid interfaces*

$$\frac{\partial \pi'}{\partial R} = \frac{1}{\Omega} \frac{\partial \pi'}{\partial R} \text{ at } R = \hat{R} \text{ and } 0 \le \theta \le a \tag{23}$$

procedure need to be validated. The error criterion given as *ε* ¼ 0*:*00001 was used to terminate the iterative procedures. The present results of the friction factor and Nusselt number for copper have been compared with literature results for *β* = 3°. They were verified by comparing them with results present in literature with same geometrical parameters of heat exchanger. The present results differ with the literature results by less than 0.5%. However, in two exceptional cases, the results are not up to the same level of accuracy. The improved results obtained at all grid levels are of comparable accuracy. The comparison given in **Table 1** confirms the validity of numerical algorithm. This difference between friction factor and Nusselt number

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

gives the overview; the percentage difference greater than '0' depicts the overestimation in present results, and difference less than '0'shows the

**4. Result and discussion**

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

**4.1 Local results**

*4.1.1 Flow behaviour*

*4.1.1.1 Velocity contours*

towards the outer pipe.

**Geometrical parameters**

**Table 1.**

**47**

underestimated values of the present study. Since the difference in the values provided in **Table 1** is negligible, this gives the validity of results in the present study.

In this section, velocity contours are given with respect to different geometrical variations. **Figure 3a** and **<sup>b</sup>** show the velocity contours for *<sup>R</sup>*^ <sup>¼</sup> <sup>0</sup>*:*5*, <sup>β</sup>* <sup>¼</sup> 2o, *H\** = 0.6. The effect of the number of fins is observed by firstly taking *M=6* and then *M = 18.* While observing the contours, it is clear that between two consecutive fins a

**Figure 3a** shows the velocity contours, For *M=6,* the annulus region is filled with

**Figure 3b** shows the velocity contours for *M = 18*, for *H\** = 0.6. For the increased number of fins, the middle of region is surrounded by annular loops, which depict

**M** *H\* f***Re** *f***Re Age change (%) Nu(Old) Nu(Cu) Age change (%)** 6 0.2 21.117 21.1351 0.0857 4.5189 4.5212 0.0508

12 0.2 19.125 19.1580 0.1725 3.8433 3.4873 �9.262

*Validity of present results of momentum equation and energy equation for <sup>β</sup>* <sup>¼</sup> *<sup>3</sup>°, <sup>R</sup>*^ <sup>¼</sup> *0.5.*

0.4 20.401 20.4185 0.0857 4.1846 4.1881 0.0836 0.6 18.848 18.8592 0.0594 3.9827 3.9873 0.1154 0.8 15.783 15.7868 0.0240 3.766 3.3789 �10.278

0.4 19.358 19.3991 0.2123 3.1644 3.1705 0.1928 0.6 20.027 20.0653 0.1912 3.9256 3.9421 0.4203 0.8 17.145 17.1592 0.0828 4.2311 4.2401 0.2127

**Comparison of friction factor Comparison of Nusselt number**

closed loops in the middle of the region, while near the inner wall of outer pipe circular loops are formed. For this fin height, two dimensional effects are more

region of high velocity exists in the middle of annulus.

$$\frac{\partial \boldsymbol{\pi}^{\epsilon}}{\partial \boldsymbol{R}} = \frac{1}{\boldsymbol{\Omega}} \frac{\partial \boldsymbol{\overline{\boldsymbol{\sigma}}^{\epsilon}}}{\partial \boldsymbol{R}} \text{ at } \boldsymbol{R} = \boldsymbol{R}\_1 \text{ and } \boldsymbol{a} \le \boldsymbol{\theta} \le \boldsymbol{a} + \boldsymbol{\beta} \tag{24}$$

$$\frac{\partial \pi'}{\partial \theta} = \frac{1}{\Omega} \frac{\partial \pi'}{\partial \theta} \text{ at } \theta = a \text{ and } \hat{R} \le R \le R\_1 \tag{25}$$

where <sup>Ω</sup> <sup>¼</sup> *<sup>λ</sup><sup>f</sup> <sup>λ</sup><sup>s</sup>* is the ratio of conductivities of fluid by solid. Continuity of temperature at the solid-fluid interfaces

$$\mathfrak{r}' = \mathfrak{r}' \mathfrak{a} \mathfrak{r} R = \hat{R} \text{ and } 0 \le \theta \le a \tag{26}$$

$$\mathfrak{r}' = \mathfrak{r}' \text{ at} \\ \mathsf{R} = \mathsf{R}\_1 \text{ and } a \le \theta \le a + \beta \tag{27}$$

$$T^{\circ} = T^{\circ} \text{ at } \theta = a \text{ and } \hat{R} \le R \le R\_1 \tag{28}$$

#### **3.2 Numerical solutions**

For numerical domain, Poisson equation given in Eq. (1) takes the form

$$\frac{U\_{i+1,j} - 2U\_{i,j} + U\_{i-1,j}}{h^2} + \frac{1}{R\_i} \frac{U\_{i+1,j} - U\_{i-1,j}}{2h} + \frac{1}{R\_i^2} \frac{U\_{i+1,j} - 2U\_{i,j} + U\_{i-1,j}}{k^2} = \frac{4}{C} \tag{29}$$

After combining the coefficient, we get

$$\left(\frac{1}{h^2} + \frac{1}{2R\_i h}\right) U\_{i+1,j} + \left(\frac{-2}{h^2} - \frac{2}{R\_i^2 k^2}\right) U\_{i,j} + \left(\frac{1}{h^2} - \frac{1}{2R\_i h}\right) U\_{i-1,j} + \frac{1}{R\_i^2 k^2} U\_{i,j+1} + \frac{1}{R\_i^2 k^2} U\_{i,j-1} = \frac{4}{C} \tag{30}$$

For our simplicity, writing the coefficient in some standardized form

$$\mathbf{e}\mathbf{U}\_{i+1,j} + \mathbf{p}\mathbf{U}\_{i,j} + \mathbf{w}\mathbf{U}\_{i-1,j} + \mathbf{n}\mathbf{U}\_{i,j+1} + \mathbf{s}\mathbf{U}\_{i,j-1} = \text{rhs}$$

$$\text{where } \mathbf{e} = \left(\frac{1}{h^2} + \frac{1}{2R\_ih}\right), \mathbf{p} = \left(\frac{-2}{h^2} - \frac{2}{R\_i^2k^2}\right),$$

$$\mathbf{w} = \left(\frac{1}{\mathbf{h}^2} - \frac{1}{2\mathbf{R}\_ih}\right), \mathbf{n} = \frac{1}{\mathbf{R}\_i^2\mathbf{k}^2}, \mathbf{s} = \frac{1}{\mathbf{R}\_i^2\mathbf{k}^2}, \text{rhs} = \frac{4}{\mathbf{C}}$$

For energy and heat equations, similar scheme is developed.

#### **3.3 Error analysis and validity of results**

The numerical algorithm described in Chapter 2 has been used to determine the numerical results in the present study. The iterative convergence and interpolation

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

procedure need to be validated. The error criterion given as *ε* ¼ 0*:*00001 was used to terminate the iterative procedures. The present results of the friction factor and Nusselt number for copper have been compared with literature results for *β* = 3°. They were verified by comparing them with results present in literature with same geometrical parameters of heat exchanger. The present results differ with the literature results by less than 0.5%. However, in two exceptional cases, the results are not up to the same level of accuracy. The improved results obtained at all grid levels are of comparable accuracy. The comparison given in **Table 1** confirms the validity of numerical algorithm. This difference between friction factor and Nusselt number gives the overview; the percentage difference greater than '0' depicts the overestimation in present results, and difference less than '0'shows the underestimated values of the present study. Since the difference in the values provided in **Table 1** is negligible, this gives the validity of results in the present study.
