**4.1 Local results**

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 interface conditions can be expressed mathematically as given below by

*<sup>λ</sup><sup>s</sup>* is the ratio of conductivities of fluid by solid.

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

2*h* þ

1 *<sup>h</sup>*<sup>2</sup> � <sup>1</sup> 2*Rih* � �

eUiþ1*,*<sup>j</sup> þ pUi*,*<sup>j</sup> þ wUi�1*,*<sup>j</sup> þ nUi*,*jþ<sup>1</sup> þ sUi*,*j�<sup>1</sup> ¼ rhs

*, p* <sup>¼</sup> �<sup>2</sup>

<sup>i</sup> k2 *,*<sup>s</sup> <sup>¼</sup> <sup>1</sup> R2

1 2*Rih* � �

> *,* <sup>n</sup> <sup>¼</sup> <sup>1</sup> R2

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

1 *R*2 *i*

*Ui*þ1*,j* � *Ui*�1*,j*

*Ui,j* þ

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

For energy and heat equations, similar scheme is developed.

For our simplicity, writing the coefficient in some standardized form

*<sup>∂</sup><sup>R</sup>* at *<sup>R</sup>* <sup>¼</sup> *<sup>R</sup>*^ and 0<sup>≤</sup> *<sup>θ</sup>* <sup>≤</sup>*<sup>α</sup>* (23)

*<sup>∂</sup><sup>R</sup>* at *<sup>R</sup>* <sup>¼</sup> *<sup>R</sup>*<sup>1</sup> and *<sup>α</sup>*<sup>≤</sup> *<sup>θ</sup>* <sup>≤</sup>*<sup>α</sup>* <sup>þ</sup> *<sup>β</sup>* (24)

*<sup>∂</sup><sup>θ</sup>* at *<sup>θ</sup>* <sup>¼</sup> *<sup>α</sup>* and *<sup>R</sup>*^ <sup>≤</sup>*R*≤*R*<sup>1</sup> (25)

at *<sup>R</sup>* <sup>¼</sup> *<sup>R</sup>*^ and 0 <sup>≤</sup>*<sup>θ</sup>* <sup>≤</sup>*<sup>α</sup>* (26)

*Ui*þ1*,j* � 2*Ui,j* þ *Ui*�1*,j*

1 *R*2

*<sup>i</sup> <sup>k</sup>*<sup>2</sup> *Ui,j*þ<sup>1</sup> <sup>þ</sup>

*,*

C

<sup>i</sup> k2 *,*rhs <sup>¼</sup> <sup>4</sup>

*<sup>k</sup>*<sup>2</sup> <sup>¼</sup> <sup>4</sup>

1 *R*2

*<sup>C</sup>* (29)

*<sup>i</sup> <sup>k</sup>*<sup>2</sup> *Ui,j*�<sup>1</sup> <sup>¼</sup> <sup>4</sup>

(30)

*C*

*<sup>τ</sup><sup>s</sup>* <sup>¼</sup> *<sup>τ</sup><sup>f</sup>* at*<sup>R</sup>* <sup>¼</sup> *<sup>R</sup>*<sup>1</sup> and *<sup>α</sup>* <sup>≤</sup>*<sup>θ</sup>* <sup>≤</sup>*<sup>α</sup>* <sup>þ</sup> *<sup>β</sup>* (27) *<sup>T</sup><sup>s</sup>* <sup>¼</sup> *<sup>T</sup><sup>f</sup>* at *<sup>θ</sup>* <sup>¼</sup> *<sup>α</sup>* and *<sup>R</sup>*^ <sup>≤</sup>*R*<sup>≤</sup> *<sup>R</sup>*<sup>1</sup> (28)

*Ui*�1*,j* þ

*<sup>h</sup>*<sup>2</sup> � <sup>2</sup> *R*2 *i k*2

!

the fluid. This forms the so-called conjugate problem.

*Continuity of fluxes at the solid-fluid interfaces ∂τs <sup>∂</sup><sup>R</sup>* <sup>¼</sup> <sup>1</sup> Ω *∂τf*

> *∂τs <sup>∂</sup><sup>R</sup>* <sup>¼</sup> <sup>1</sup> Ω *∂τf*

where <sup>Ω</sup> <sup>¼</sup> *<sup>λ</sup><sup>f</sup>*

**3.2 Numerical solutions**

*Ui*þ1*,j* � 2*Ui,j* þ *Ui*�1*,j*

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

**46**

1 2*Rih* � �

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

*Ui*þ1*,j* <sup>þ</sup> �<sup>2</sup>

After combining the coefficient, we get

*<sup>h</sup>*<sup>2</sup> � <sup>2</sup> *R*2 *i k*2

!

where *<sup>e</sup>* <sup>¼</sup> <sup>1</sup>

<sup>h</sup><sup>2</sup> � <sup>1</sup> 2Rih � �

<sup>w</sup> <sup>¼</sup> <sup>1</sup>

**3.3 Error analysis and validity of results**

*∂τs <sup>∂</sup><sup>θ</sup>* <sup>¼</sup> <sup>1</sup> Ω *∂τf*

Continuity of temperature at the solid-fluid interfaces

*<sup>τ</sup><sup>s</sup>* <sup>¼</sup> *<sup>τ</sup><sup>f</sup>*

1 *Ri*

using same dimensionless transformations.

*Numerical Modeling and Computer Simulation*

#### *4.1.1 Flow behaviour*

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 region of high velocity exists in the middle of annulus.

#### *4.1.1.1 Velocity contours*

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

**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


**Table 1.**

*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.*

The same geometrical parameters and grid points are taken as that for momen-

tum equation. A comparative study on behaviour of heat transfer is shown in **Figure 4a** and **b**. While using copper for fin and wall geometry, it should be noted that the isotherms corresponding to the zero level represent the geometry. Because of the definition of the dimensionless temperature used, the higher the isotherm level depicts, the lower the value of local temperature. The isotherms corresponding to value 1 have local fluid temperature equal to mean fluid temperature. Those with values lower than 1 have the local temperature greater than the mean temperature, and while greater than 1 indicate that the local temperature is lesser than the meant temperature. From these figures, it is clear that the region of high-temperature gradient near the inner pipe wall and near fin surface shows the high rate of convection. In order to comprise with interfaces and transfer of heat between fluid

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

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

*(a) Isotherms (copper) for R*^ *= 0.5, H\* = 0.6, M = 6. (b) Isotherms (copper) for R*^ *= 0.5, H\* = 0.6, M = 18.*

and solid, continuity of fluxes and of temperature is considered.

**Figure 4.**

**49**

#### **Figure 3.**

*(a) Velocity contours for R*^ *= 0.5, H\* = 0.8, M = 6. (b) Velocity contours for R*^ *= 0.5, H\* = 0.8, M = 6.*

that the region of high velocity starts to develop at the middle of annulus. Also, when a number of fins are increased, then closed loop will break into annular loops.

We must have to notice that increase in the number of fins will also give rise to the value of friction factor, which may slow down the fluid motion in the pipe.

#### *4.1.2 Heat transfer analysis*

The results of energy and heat equations are combined together for analytical study of conjugate heat transfer, while fin and wall are offering finite heat conduction because of material used in them.

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

The same geometrical parameters and grid points are taken as that for momentum equation. A comparative study on behaviour of heat transfer is shown in **Figure 4a** and **b**. While using copper for fin and wall geometry, it should be noted that the isotherms corresponding to the zero level represent the geometry. Because of the definition of the dimensionless temperature used, the higher the isotherm level depicts, the lower the value of local temperature. The isotherms corresponding to value 1 have local fluid temperature equal to mean fluid temperature. Those with values lower than 1 have the local temperature greater than the mean temperature, and while greater than 1 indicate that the local temperature is lesser than the meant temperature. From these figures, it is clear that the region of high-temperature gradient near the inner pipe wall and near fin surface shows the high rate of convection. In order to comprise with interfaces and transfer of heat between fluid and solid, continuity of fluxes and of temperature is considered.

**Figure 4.** *(a) Isotherms (copper) for R*^ *= 0.5, H\* = 0.6, M = 6. (b) Isotherms (copper) for R*^ *= 0.5, H\* = 0.6, M = 18.*

that the region of high velocity starts to develop at the middle of annulus. Also, when a number of fins are increased, then closed loop will break into annular loops. We must have to notice that increase in the number of fins will also give rise to the value of friction factor, which may slow down the fluid motion in the pipe.

*(a) Velocity contours for R*^ *= 0.5, H\* = 0.8, M = 6. (b) Velocity contours for R*^ *= 0.5, H\* = 0.8, M = 6.*

The results of energy and heat equations are combined together for analytical study of conjugate heat transfer, while fin and wall are offering finite heat conduc-

*4.1.2 Heat transfer analysis*

**Figure 3.**

**48**

tion because of material used in them.

*Numerical Modeling and Computer Simulation*

#### *4.1.2.1 Isotherms*

**Figure 4a** shows the isotherms for *M = 6, R*^ = 0.5, *H\** = 0.6, which shows that a region of higher temperature starts to develop in the middle of annulus and significant change occurs in temperature gradient. Whereas, **Figure 4b** shows the isotherms, when *M = 18*, the temperature gradient is higher in the entire cross section, and higher temperature region appears near the outer wall of the inner pipe, while lower near the inner wall of the outer pipe.

**Figure 5b** shows the plots of different values of Nusslet number of copper

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

For *R*^ **= 0.7**, *M=6*, heat transfer coefficient shows monotonic decreasing behaviour as we increase fin height. For *H\** = 0.2, the value is Nusselt number is

While for the increased number of fins, *M = 18*, Nusselt number shows alternating behaviour for different values of fin height. It firstly decreases and then it starts

From the above two comparisons, we can deduce that for the less number of fins, if we increase the fin height, it reduces the increase in heat transfer. While for

The results presented in previous sections can be concluded as follows:

• A comparison of present results with the literature results gives the validity

• The number of fin and fin heights are most effective geometrical parameters.

• The influences of geometrical parameters are dependent on each other.

• The location of the regions of high velocities is dependent on geometrical

• Large velocity gradients exist near the fin tip and outer pipe inner surfaces.

• The value of *R*^ affects the heat transfer rate significantly, for larger fins and for

• The value of Nusselt number at *M = 18*, *<sup>R</sup>*^ <sup>¼</sup> <sup>0</sup>*:*5, gives the optimal value for

• For *<sup>R</sup>*^ <sup>¼</sup> <sup>0</sup>*:*7*,* the value of Nu(Cu) goes on increasing with higher values of M,

∈f0.2, 0.4, 0.6, 0.8} taking fixed value of ratio of radii

against number of fins *H\**

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

increasing. For *H\** = 0.6, it gives its largest value.

proof of the numerical study.

more number of fins, longer fins give more optimal results.

*R*^ = 0.7.

highest.

**5. Conclusion**

parameters.

*H\** = 0.8.

at *H\** = 0.8.

**51**

higher number of fins.

Apparently, in these isotherms conduction in wall and fin assembly is not being shown, but if we increase the number of contours, the temperature gradient is shown in the solid part of the domain too. This is because of the very hightemperature gradient in fluid and because of convection and conduction.

The trend of isotherms is more understandable corresponding to the velocity contours. From these figures, it is observed that the region of high-velocity gradients near the heated surfaces of the inner pipe and fin also has high-temperature gradients indicating the high rate of convection.

There is an equivalent transfer of heat in fin and fluid on the interfaces; interfaces get smoother because of the continuity of fluxes. But in case of temperature difference, it rises more rapidly in fluid than the solid (wall and fin).

#### **4.2 Overall results**

**Figure 5a** shows the plots of different values of Nusselt number of copper against the number of fins *H\** ∈f0.2, 0.4, 0.6, 0.8} taking fixed value of ratio of radii *R*^ = 0.5. For fin height *M=6*, the value of Nusselt number decreases as we increase fin height. The plot of Nusselt number against these values shows monotonic decreasing behaviour. However, the value of Nu(Cu) remains higher for the *H\** = 0.2 than the fin height *H\** = 0.4 and others.

The plots of Nusselt numbers for *M=6* and *M = 18* clearly depict that the behaviour of Nusselt number significantly differs in these two graphs. An optimal value of Nusselt number is observed for some particular value of *H\** = 0.2 used. Thus, for *M=6*, shorter fin gives better rate of heat conduction.

While taking the higher number of fins into account, for *M = 18*, Nusselt number gradually decreases when fin height is increased from *H\** = 0.2 to *H\** = 0.4. Its shows a parabolic behaviour. After reaching its lowest value, it starts increasing and attains its optimal value at the longest fin.

It can be concluded that the value of Nusselt number for *R*^ = 0.5 at *H\** = 0.6 and *H\** = 0.8 is the best choice when *M=9* and *M = 18* are taken, respectively.

**Figure 5.** *(a)* H\* *-Nu(Cu), for R*^ *= 0.5. (b) M-Nu(Cu), for R*^ *= 0.7.*

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

**Figure 5b** shows the plots of different values of Nusslet number of copper against number of fins *H\** ∈f0.2, 0.4, 0.6, 0.8} taking fixed value of ratio of radii *R*^ = 0.7.

For *R*^ **= 0.7**, *M=6*, heat transfer coefficient shows monotonic decreasing behaviour as we increase fin height. For *H\** = 0.2, the value is Nusselt number is highest.

While for the increased number of fins, *M = 18*, Nusselt number shows alternating behaviour for different values of fin height. It firstly decreases and then it starts increasing. For *H\** = 0.6, it gives its largest value.

From the above two comparisons, we can deduce that for the less number of fins, if we increase the fin height, it reduces the increase in heat transfer. While for more number of fins, longer fins give more optimal results.
