**Abstract**

In this chapter we provide the review and a narrative of some obtained results for steady and transient heat transfer though extended surfaces (fins). A particular attention is given to exact and approximate analytical solutions of models describing heat transfer under various conditions, for example, when thermal conductivity and heat transfer are temperature dependent. We also consider fins of different profiles and shapes. The dependence of thermal properties render the considered models nonlinear, and this adds a complication and difficulty to solve these model exactly. However, the nonlinear problems are more realistic and physically sound. The approximate analytical solutions give insight into heat transfer in fins and as such assist in the designs for better efficiencies and effectiveness.

**Keywords:** exact solutions, approximate solutions, lie symmetry methods, approximate methods, heat transfer, fins

## **1. Introduction**

In the study of heat transfer, a fin may be a solid or porous and stationary or moving that extends from an attached body to rapidly cool off heat of that surface. Cooling fins find application in a large real world phenomena particularly in engineering devices. Fins increase the surface area of heat transfer particularly for cooling of hot bodies. These come in different shapes, geometries and profiles. These differences provide variety of effectiveness and efficiencies. The literature with regard to the study of heat transfer in fins is well documented (see e.g. [1]). The solutions either exact, numerical or approximate analytical continue to be of immerse interest and this is due to continued use of fins in engineering devices.

Much attention has been given to linear one dimensional models [2–4] whereby Homotopy Analysis Method (HAM) was used to determine series solutions for heat transfer in straight fins of trapezoidal and rectangular profiles given temperature dependent thermal properties; nonlinear one dimensional models [5] wherein preliminary group classification methods were utilised to contract invariant (symmetry) solutions; heat transfer in linear two dimensional trapezoidal fins [6]; heat transfer in two dimensional straight nonlinear fins were considered [7] wherein Lie point symmetries and other standard methods were invoked and recently nonlinear three dimensional models [8] were considered wherein three dimensional Differential Transform Methods (DTM) were employed to construct approximate analytical solutions. The dependence of thermal properties on the temperature renders the equations highly nonlinear. The non-linearity brings an added complication or difficulty in the construction of solutions and particularly exact solutions.

Few exact solutions are recorded in the literature, for example for one dimensional problems [2–5, 9–15], two dimensions [6, 7, 16, 17]. An attempt to construct exact solutions for the three dimensional problems is found in [8], however these were general solutions. For this reason, either approximate analytical or numerical solutions are sought. However, the accuracy of numerical schemes is obtained by comparison with he exact solutions.

This chapter summaries the work of Moitsheki and collaborators in the area of heat transfer through fin. In their work, they employed Lie symmetry methods to construct exact solutions. These methods include, the preliminary group classification, the Lie point symmetries, conservation laws and associate Lie point symmetries, non-classical symmetry methods and recently non classical potential symmetries. It appeared that most of the constructed exact solutions do not satisfy the prescribed boundary conditions. The idea then becomes, start with the simple model that satisfy the boundary conditions and compare it with the approximate solutions to establish confidence in the approximate methods, then extend analysis to problems that are difficult to solve exactly.

We acknowledge that some scholars employed many other approximate methods to solve boundary value problems (BVPs); for example the Homotopy Analysis Method [18], Collocation Methods (CM) [19], Homotopy Perturbation Methods (HPM) [20], Haar Wavelet Collation Methods (HWCM) [21], Collocation Spectral Methods (CSM) [22], modified Homotopy Analysis Method (mHAM) [23], Spectral Homotopy Analysis Methods (SHAM) and the Optimal Homotopy Analysis Methods [24]. In this chapter we restrict discussions to Lie symmetry methods for exact solutions, and DTM and VIM for approximate analytical methods.

fin is nonuniform and temperature dependent and that the internal heat source or sink is neglected. Furthermore, the temperature-dependent thermal conductivity is assumed to be the same in both radial and axial directions. The model describing the

*Survey of Some Exact and Approximate Analytical Solutions for Heat Transfer in Extended…*

þ 1 *R ∂ ∂R*

*T*ð Þ¼ 0, *R*, *Z Ts*, 0≤*R*≤ *Ra*, 0≤*Z* ≤ *L*,

*T t*ð Þ¼ , *R*, *L Tb*, 0≤*R*≤*Ra*, *t*> 0,

*<sup>∂</sup><sup>Z</sup>* <sup>¼</sup> 0, *<sup>Z</sup>* <sup>¼</sup> 0, 0 <sup>≤</sup>*R*≤*Ra*, *<sup>t</sup>*<sup>&</sup>gt; <sup>0</sup>*:*

*<sup>∂</sup><sup>R</sup>* <sup>¼</sup> 0, *<sup>R</sup>* <sup>¼</sup> *Ra*, 0≤*<sup>Z</sup>* <sup>≤</sup>*L*, *<sup>t</sup>*>0,

<sup>þ</sup> *<sup>E</sup>*<sup>2</sup> <sup>1</sup> *r ∂ ∂r*

*θ*ð Þ¼ 0,*r*, *z* 0, 0 ≤*z*≤1, 0 ≤*r*≤1,

*k*ð Þ*θ r ∂θ ∂r*

*<sup>∂</sup><sup>R</sup>* ¼ �*H T*ð Þ *<sup>T</sup>* � *Ts* ½ �, *<sup>R</sup>* <sup>¼</sup> 0, 0<sup>≤</sup> *<sup>Z</sup>* <sup>≤</sup> *<sup>L</sup>*, *<sup>t</sup>*<sup>&</sup>gt; 0,

*K T*ð Þ*<sup>R</sup> <sup>∂</sup><sup>T</sup> ∂R*

*:* (1)

, (2)

*K T*ð Þ *<sup>∂</sup><sup>T</sup> ∂Z* 

heat transfer in pin fins is given by the BVP (see e.g. [17])

here, *Ts* is the temperature of the surrounding fluid.

In non-dimensionalized variables and parameters we have,

*<sup>k</sup>*ð Þ*<sup>θ</sup> <sup>∂</sup><sup>θ</sup> ∂z* 

*ρcp ∂T <sup>∂</sup><sup>t</sup>* <sup>¼</sup> *<sup>∂</sup> ∂Z*

*DOI: http://dx.doi.org/10.5772/intechopen.95490*

The initial condition is given by

*Schematic representation of a pin fin.*

**Figure 1.**

Boundary conditions are given by

*K T*ð Þ *<sup>∂</sup><sup>T</sup>*

subject to the initial condition

**87**

*∂T*

*∂T*

*∂θ <sup>∂</sup><sup>τ</sup>* <sup>¼</sup> *<sup>∂</sup> ∂z*

#### **2. Mathematical descriptions**

Mathematical descriptions represent some physical phenomena in terms of deterministic models given in terms of partial differential equations (PDEs). These differential equations become non-linear when heat transfer coefficient and thermal conductivity depend on the temperature (see e.g. [5]). This non-linearity was introduced as a significant modifications of the usually assumed models see e.g. [2].

In this chapter we present a few models for various heat transfer phenomena.

#### **2.1 2 + 1 dimensional transient state models**

Mathematical modelling for heat transfer in fins may be three dimensional models.

#### *2.1.1 Cylindrical pin fins*

We consider a two-dimensional pin fin with length *L* and radius *R*. The fin is attached to a base surface of temperature *Tb* and extended into the fluid of temperature *Ts*. The tip of the fin is insulated (i.e., heat transfer at the tip is negligibly small). The fin is measured from the tip to the base. A schematic representation of a pin fin is given in **Figure 1**. We assume that the heat transfer coefficient along the

*Survey of Some Exact and Approximate Analytical Solutions for Heat Transfer in Extended… DOI: http://dx.doi.org/10.5772/intechopen.95490*

**Figure 1.** *Schematic representation of a pin fin.*

point symmetries and other standard methods were invoked and recently nonlinear three dimensional models [8] were considered wherein three dimensional Differential Transform Methods (DTM) were employed to construct approximate analytical solutions. The dependence of thermal properties on the temperature renders the equations highly nonlinear. The non-linearity brings an added complication or

Few exact solutions are recorded in the literature, for example for one dimensional problems [2–5, 9–15], two dimensions [6, 7, 16, 17]. An attempt to construct exact solutions for the three dimensional problems is found in [8], however these were general solutions. For this reason, either approximate analytical or numerical solutions are sought. However, the accuracy of numerical schemes is obtained by

This chapter summaries the work of Moitsheki and collaborators in the area of heat transfer through fin. In their work, they employed Lie symmetry methods to construct exact solutions. These methods include, the preliminary group classification, the Lie point symmetries, conservation laws and associate Lie point symmetries, non-classical symmetry methods and recently non classical potential

symmetries. It appeared that most of the constructed exact solutions do not satisfy the prescribed boundary conditions. The idea then becomes, start with the simple model that satisfy the boundary conditions and compare it with the approximate solutions to establish confidence in the approximate methods, then extend analysis

We acknowledge that some scholars employed many other approximate methods to solve boundary value problems (BVPs); for example the Homotopy Analysis Method [18], Collocation Methods (CM) [19], Homotopy Perturbation Methods (HPM) [20], Haar Wavelet Collation Methods (HWCM) [21], Collocation Spectral Methods (CSM) [22], modified Homotopy Analysis Method (mHAM) [23], Spectral Homotopy Analysis Methods (SHAM) and the Optimal Homotopy Analysis Methods [24]. In this chapter we restrict discussions to Lie symmetry methods for exact solutions, and DTM and VIM for approximate analytical methods.

Mathematical descriptions represent some physical phenomena in terms of deterministic models given in terms of partial differential equations (PDEs). These differential equations become non-linear when heat transfer coefficient and thermal conductivity depend on the temperature (see e.g. [5]). This non-linearity was introduced as a significant modifications of the usually assumed models see e.g. [2]. In this chapter we present a few models for various heat transfer phenomena.

Mathematical modelling for heat transfer in fins may be three dimensional

We consider a two-dimensional pin fin with length *L* and radius *R*. The fin is attached to a base surface of temperature *Tb* and extended into the fluid of temperature *Ts*. The tip of the fin is insulated (i.e., heat transfer at the tip is negligibly small). The fin is measured from the tip to the base. A schematic representation of a pin fin is given in **Figure 1**. We assume that the heat transfer coefficient along the

difficulty in the construction of solutions and particularly exact solutions.

comparison with he exact solutions.

*Heat Transfer - Design, Experimentation and Applications*

to problems that are difficult to solve exactly.

**2. Mathematical descriptions**

models.

**86**

*2.1.1 Cylindrical pin fins*

**2.1 2 + 1 dimensional transient state models**

fin is nonuniform and temperature dependent and that the internal heat source or sink is neglected. Furthermore, the temperature-dependent thermal conductivity is assumed to be the same in both radial and axial directions. The model describing the heat transfer in pin fins is given by the BVP (see e.g. [17])

$$
\rho c\_p \frac{\partial T}{\partial t} = \frac{\partial}{\partial Z} \left[ K(T) \frac{\partial T}{\partial Z} \right] + \frac{1}{R} \frac{\partial}{\partial R} \left[ K(T) R \frac{\partial T}{\partial R} \right]. \tag{1}
$$

The initial condition is given by

$$T(0, R, Z) = T\_s, \quad 0 \le R \le R\_s, \quad 0 \le Z \le L\_s$$

here, *Ts* is the temperature of the surrounding fluid. Boundary conditions are given by

$$T(t, R, L) = T\_b, \quad 0 \le R \le R\_d, \quad t > 0,$$

$$\frac{\partial T}{\partial Z} = 0, \quad Z = 0, \quad 0 \le R \le R\_d, \quad t > 0.$$

$$K(T) \frac{\partial T}{\partial R} = -H(T)[T - T\_s], \quad R = 0, \quad 0 \le Z \le L, \quad t > 0,$$

$$\frac{\partial T}{\partial R} = 0, \quad R = R\_d, \quad 0 \le Z \le L, \quad t > 0,$$

In non-dimensionalized variables and parameters we have,

$$\frac{\partial \theta}{\partial \sigma} = \frac{\partial}{\partial \mathbf{z}} \left[ k(\theta) \frac{\partial \theta}{\partial \mathbf{z}} \right] + E^2 \frac{1}{r} \frac{\partial}{\partial r} \left[ k(\theta) r \frac{\partial \theta}{\partial r} \right],\tag{2}$$

subject to the initial condition

$$
\theta(0, r, z) = 0, \quad 0 \le z \le 1, \quad 0 \le r \le 1,
$$

and boundary conditions

$$\theta(\tau, \mathbf{1}, r) = \mathbf{1}, \quad 0, \le r \le 1, \quad \tau > 0,$$

$$\frac{\partial \theta}{\partial z} = 0, \quad z = 0, \quad 0 \le r \le 1, \quad \tau > 0,$$

$$k(\theta) \frac{\partial \theta}{\partial z} = -Bih(\theta)\theta, \quad z = 0, \quad 0 \le z \le 1, \quad \tau > 0,$$

$$\frac{\partial \theta}{\partial r} = 0, \quad r = 1, \quad 0 \le z \le 1, \quad \tau > 0,$$

*2.2.1 Cylindrical pin fins*

*DOI: http://dx.doi.org/10.5772/intechopen.95490*

*2.2.2 Rectangular straight fins*

*2.3.1 Solid stationary fins*

*2.3.2 Solid moving fins*

*∂θ <sup>∂</sup><sup>τ</sup>* <sup>¼</sup> *<sup>∂</sup>*

*2.3.3 Porous stationary fins*

*2.3.4 Porous moving fins*

and is given by

**89**

*∂θ <sup>∂</sup><sup>τ</sup>* <sup>¼</sup> *<sup>∂</sup>*

For steady state problem, the heat transfer is independent of the time variable.

*Survey of Some Exact and Approximate Analytical Solutions for Heat Transfer in Extended…*

For steady state problem, the heat transfer is independent of the time variable.

For solid stationary straight fins the model is given by (see e.g. [25, 26])

*∂x*

It appear, as far as we know, this is still an open problem and in preparation.

� *Nc*ð Þ *<sup>θ</sup>* � *<sup>θ</sup><sup>a</sup> <sup>n</sup>*þ<sup>1</sup> � *Nr <sup>θ</sup>*<sup>4</sup> � *<sup>θ</sup>*<sup>4</sup>

The model describing heat transfer in porous moving fin is considered in [28]

� *<sup>M</sup>*<sup>2</sup>

� *<sup>M</sup>*<sup>2</sup>

*∂θ ∂x x*¼0

*<sup>θ</sup>h*ð Þ� *<sup>θ</sup> Pef x*ð Þ *<sup>∂</sup><sup>θ</sup>*

*∂x x*¼0

*∂x x*¼0

*∂x*

*a*

*θh*ð Þ*θ* , 0 ≤*x*≤ 1*:* (4)

¼ 0, *τ* ≥ 0*:*

¼ 0, *τ* ≥ 0*:*

, 0≤ *x*≤1*:* (6)

¼ 0, *τ* ≥ 0*:*

, 0≤ *x*≤ 1*:* (5)

For example, the time derivative in Eq. (2) vanish (see e.g. [16]).

For example, the time derivative in Eq. (3) is zero (see e.g. [7]).

*<sup>∂</sup><sup>x</sup> f x*ð Þ*k*ð Þ*<sup>θ</sup> <sup>∂</sup><sup>θ</sup>*

*θ*ð Þ¼ 0, *x* 0, 0 ≤*x*≤ 1, *θ τ*ð Þ¼ , 1 1;

*∂x*

*<sup>θ</sup>*ð Þ¼ 0, *<sup>x</sup>* 0, 0 <sup>≤</sup>*x*<sup>≤</sup> 1, *θ τ*ð Þ¼ , 1 1; *<sup>∂</sup><sup>θ</sup>*

*<sup>θ</sup>*ð Þ¼ 0, *<sup>x</sup>* 0, 0 <sup>≤</sup>*x*<sup>≤</sup> 1, *θ τ*ð Þ¼ , 1 1; *<sup>∂</sup><sup>θ</sup>*

**2.3 1 + 1 dimensional transient model for straight fins**

*∂θ <sup>∂</sup><sup>τ</sup>* <sup>¼</sup> *<sup>∂</sup>*

subject to initial and boundary conditions

*<sup>∂</sup><sup>x</sup> f x*ð Þ*k*ð Þ*<sup>θ</sup> <sup>∂</sup><sup>θ</sup>*

subject to initial and boundary conditions

The model was considered in [27].

*<sup>∂</sup><sup>x</sup> f x*ð Þ*k*ð Þ*<sup>θ</sup> <sup>∂</sup><sup>θ</sup>*

*∂x*

subject to initial and boundary conditions

where the non-dimensional quantities *<sup>E</sup>* <sup>¼</sup> *<sup>L</sup> <sup>δ</sup>*, and *Bi* <sup>¼</sup> *Hb<sup>δ</sup> Ka* , are the fin extension factor and the Biot number respectively. Also,

$$\begin{aligned} t &= \frac{L^2 \rho c\_p}{K\_d} \tau, \quad Z = L\tau, \quad R = R\_d r, \\ K &= K\_d k, \quad H = H\_b h, \quad T = (T\_b - T\_s)\theta + T\_s. \end{aligned}$$

where *τ*, *z*, *r*, *k*, *h* and *θ* are all dimensionless variables. *Ka* and *Hb* are the ambient thermal conductivity and the fin base heat transfer coefficient respectively.

Notice that other terms may be added, for example internal heat generation (source term) and fin profile.

#### *2.1.2 Rectangular straight fins*

Following the similar pattern, in dimensionless variables we have (see e.g. [8])

$$\frac{\partial \theta}{\partial \pi} = \frac{\partial}{\partial \mathbf{x}} \left[ k(\theta) \frac{\partial \theta}{\partial \mathbf{x}} \right] + E^2 \frac{\partial}{\partial \mathbf{y}} \left[ k(\theta) \frac{\partial \theta}{\partial \mathbf{y}} \right],\tag{3}$$

subject to the initial condition

$$
\theta(0, \varkappa, \underline{\chi}) = 0, \quad 0 \le \varkappa \le 1, \quad 0 \le \underline{\chi} \le 1,
$$

and boundary conditions

$$\theta(\tau, \mathbf{1}, \mathbf{y}) = \mathbf{1}, \quad \mathbf{0}, \le \mathbf{y} \le \mathbf{1}, \quad \tau > \mathbf{0},$$

$$\frac{\partial \theta}{\partial \mathbf{x}} = \mathbf{0}, \quad \mathbf{x} = \mathbf{0}, \quad \mathbf{0} \le \mathbf{y} \le \mathbf{1}, \quad \tau > \mathbf{0},$$

$$k(\theta) \frac{\partial \theta}{\partial \mathbf{y}} = -Bih(\theta)\theta, \quad \mathbf{y} = \mathbf{0}, \quad \mathbf{0} \le \mathbf{x} \le \mathbf{1}, \quad \tau > \mathbf{0},$$

$$\frac{\partial \theta}{\partial \mathbf{y}} = \mathbf{0}, \quad \mathbf{y} = \mathbf{1}, \quad \mathbf{0} \le \mathbf{x} \le \mathbf{1}, \quad \tau > \mathbf{0},$$

#### **2.2 Two-dimensional steady state models**

In this section we consider the two dimensional steady state models. The symmetry analysis of these models have proven to be challenging. In some cases standard method such as separations of variables have been employed to determine exact solutions.

*Survey of Some Exact and Approximate Analytical Solutions for Heat Transfer in Extended… DOI: http://dx.doi.org/10.5772/intechopen.95490*

#### *2.2.1 Cylindrical pin fins*

and boundary conditions

*∂θ*

*Heat Transfer - Design, Experimentation and Applications*

*∂θ*

where the non-dimensional quantities *<sup>E</sup>* <sup>¼</sup> *<sup>L</sup>*

*ρcp Ka*

*∂θ <sup>∂</sup><sup>τ</sup>* <sup>¼</sup> *<sup>∂</sup> ∂x*

*∂θ*

*∂θ*

*<sup>k</sup>*ð Þ*<sup>θ</sup> <sup>∂</sup><sup>θ</sup>*

**2.2 Two-dimensional steady state models**

exact solutions.

**88**

factor and the Biot number respectively. Also,

*<sup>t</sup>* <sup>¼</sup> *<sup>L</sup>*<sup>2</sup>

(source term) and fin profile.

*2.1.2 Rectangular straight fins*

subject to the initial condition

and boundary conditions

*<sup>k</sup>*ð Þ*<sup>θ</sup> <sup>∂</sup><sup>θ</sup>*

*θ τ*ð Þ¼ , 1,*r* 1, 0, ≤*r*≤1, *τ* >0,

*<sup>∂</sup><sup>z</sup>* <sup>¼</sup> 0, *<sup>z</sup>* <sup>¼</sup> 0, 0<sup>≤</sup> *<sup>r</sup>*<sup>≤</sup> 1, *<sup>τ</sup>* <sup>&</sup>gt;0,

*<sup>∂</sup><sup>r</sup>* <sup>¼</sup> 0, *<sup>r</sup>* <sup>¼</sup> 1, 0≤*z*<sup>≤</sup> 1, *<sup>τ</sup>* <sup>&</sup>gt;0,

*τ*, *Z* ¼ *Lz*, *R* ¼ *Rar*,

where *τ*, *z*, *r*, *k*, *h* and *θ* are all dimensionless variables. *Ka* and *Hb* are the ambient thermal conductivity and the fin base heat transfer coefficient respectively. Notice that other terms may be added, for example internal heat generation

*K* ¼ *Kak*, *H* ¼ *Hbh*, *T* ¼ ð Þ *Tb* � *Ts θ* þ *Ts:*

Following the similar pattern, in dimensionless variables we have (see e.g. [8])

*θ*ð Þ¼ 0, *x*, *y* 0, 0 ≤*x*≤ 1, 0≤*y* ≤1,

*θ τ*ð Þ¼ , 1, *y* 1, 0, ≤*y* ≤1, *τ* >0,

*<sup>∂</sup><sup>x</sup>* <sup>¼</sup> 0, *<sup>x</sup>* <sup>¼</sup> 0, 0 <sup>≤</sup>*y*≤1, *<sup>τ</sup>* <sup>&</sup>gt;0,

*<sup>∂</sup><sup>y</sup>* <sup>¼</sup> 0, *<sup>y</sup>* <sup>¼</sup> 1, 0 <sup>≤</sup>*x*<sup>≤</sup> 1, *<sup>τ</sup>* <sup>&</sup>gt;0,

In this section we consider the two dimensional steady state models. The symmetry analysis of these models have proven to be challenging. In some cases standard method such as separations of variables have been employed to determine

*<sup>∂</sup><sup>y</sup>* ¼ �*Bih*ð Þ*<sup>θ</sup> <sup>θ</sup>*, *<sup>y</sup>* <sup>¼</sup> 0, 0≤*x*≤1, *<sup>τ</sup>* <sup>&</sup>gt;0,

<sup>þ</sup> *<sup>E</sup>*<sup>2</sup> *<sup>∂</sup> ∂y*

*<sup>k</sup>*ð Þ*<sup>θ</sup> <sup>∂</sup><sup>θ</sup> ∂y* 

*<sup>k</sup>*ð Þ*<sup>θ</sup> <sup>∂</sup><sup>θ</sup> ∂x* 

*<sup>∂</sup><sup>z</sup>* ¼ �*Bih*ð Þ*<sup>θ</sup> <sup>θ</sup>*, *<sup>z</sup>* <sup>¼</sup> 0, 0<sup>≤</sup> *<sup>z</sup>*≤1, *<sup>τ</sup>* <sup>&</sup>gt;0,

*<sup>δ</sup>*, and *Bi* <sup>¼</sup> *Hb<sup>δ</sup>*

*Ka* , are the fin extension

, (3)

For steady state problem, the heat transfer is independent of the time variable. For example, the time derivative in Eq. (2) vanish (see e.g. [16]).

#### *2.2.2 Rectangular straight fins*

For steady state problem, the heat transfer is independent of the time variable. For example, the time derivative in Eq. (3) is zero (see e.g. [7]).

#### **2.3 1 + 1 dimensional transient model for straight fins**

#### *2.3.1 Solid stationary fins*

For solid stationary straight fins the model is given by (see e.g. [25, 26])

$$\frac{\partial \theta}{\partial \mathbf{r}} = \frac{\partial}{\partial \mathbf{x}} \left[ f(\mathbf{x}) k(\theta) \frac{\partial \theta}{\partial \mathbf{x}} \right] - M^2 \theta h(\theta), \ 0 \le \mathbf{x} \le \mathbf{1}. \tag{4}$$

subject to initial and boundary conditions

$$\theta(0,\infty) = 0, \quad 0 \le \varkappa \le 1, \quad \theta(\tau, \mathbb{1}) = 1; \quad \left. \frac{\partial \theta}{\partial \mathfrak{x}} \right|\_{\varkappa = 0} = 0, \quad \tau \ge 0.$$

#### *2.3.2 Solid moving fins*

It appear, as far as we know, this is still an open problem and in preparation.

$$\frac{\partial \theta}{\partial \mathbf{r}} = \frac{\partial}{\partial \mathbf{x}} \left[ f(\mathbf{x}) k(\theta) \frac{\partial \theta}{\partial \mathbf{x}} \right] - M^2 \theta h(\theta) - P \mathbf{f} f(\mathbf{x}) \frac{\partial \theta}{\partial \mathbf{x}}, \ \mathbf{0} \le \mathbf{x} \le \mathbf{1}. \tag{5}$$

subject to initial and boundary conditions

$$\theta(0,\infty) = 0, \quad 0 \le \varkappa \le 1, \quad \theta(\tau, 1) = 1; \quad \left. \frac{\partial \theta}{\partial \mathbf{x}} \right|\_{\mathbf{x}=0} = \mathbf{0}, \quad \tau \ge \mathbf{0}.$$

#### *2.3.3 Porous stationary fins*

The model was considered in [27].

$$\frac{\partial \theta}{\partial \tau} = \frac{\partial}{\partial \mathbf{x}} \left[ f(\mathbf{x}) k(\theta) \frac{\partial \theta}{\partial \mathbf{x}} \right] - N\_c (\theta - \theta\_d)^{n+1} - N\_r \left( \theta^4 - \theta\_a^4 \right), \ 0 \le \mathbf{x} \le \mathbf{1}. \tag{6}$$

subject to initial and boundary conditions

$$\theta(0,\infty) = 0, \quad 0 \le \varkappa \le 1, \quad \theta(\tau, \mathbb{1}) = 1; \quad \left. \frac{\partial \theta}{\partial \mathbb{x}} \right|\_{\varkappa = 0} = 0, \quad \tau \ge 0.$$

#### *2.3.4 Porous moving fins*

The model describing heat transfer in porous moving fin is considered in [28] and is given by

*Heat Transfer - Design, Experimentation and Applications*

$$\begin{split} \frac{\partial \theta}{\partial \tau} &= \frac{\partial}{\partial \mathbf{x}} \left[ f(\mathbf{x}) k(\theta) \frac{\partial \theta}{\partial r} \right] - N\_c (\theta - \theta\_a)^{n+1} - N\_p (\theta - \theta\_a)^2 - N\_r \left( \theta^4 - \theta\_a^4 \right) \\ &- P \mathbf{e} f(\mathbf{x}) \frac{\partial \theta}{\partial \mathbf{x}}, \quad \mathbf{0} \le \mathbf{x} \le \mathbf{1}. \end{split} \tag{7}$$

subject to initial and boundary conditions

*DOI: http://dx.doi.org/10.5772/intechopen.95490*

*d*

In case of a moving radial fin the term

non-dimensional variables, one obtains

1 *r d*

In case of a moving radial fin the term

is added to Eq. (13) (see e.g. [30]).

one obtains

subject to

is added to Eq. (12).

subject to

**91**

*<sup>θ</sup>*ð Þ¼ 0,*<sup>r</sup>* 0, 0 <sup>≤</sup>*r*≤1, *θ τ*ð Þ¼ , 1 1; *<sup>∂</sup><sup>θ</sup>*

Considering heat transfer in a one dimensional longitudinal fin of cross area *Ac* with various profiles. The perimeter of the fin is denoted by *P* and length by *L:* The fin is attached to a fixed prime surface of temperature *Tb* and extends to the fluid of temperature *T*∞*:* in non-dimensional variables,

*Survey of Some Exact and Approximate Analytical Solutions for Heat Transfer in Extended…*

**2.5 One-dimensional steady state model for straight fins**

*dx f x*ð Þ*k*ð Þ*<sup>θ</sup> <sup>d</sup><sup>θ</sup>*

**2.6 One-dimensional steady state model for radial fins**

*dr rf r*ð Þ*k*ð Þ*<sup>θ</sup> <sup>d</sup><sup>θ</sup>*

*dx* � *<sup>M</sup>*<sup>2</sup>

*dx* � *<sup>M</sup>*<sup>2</sup>

*θ*ð Þ¼ 1 1,

*dθ*ð Þ 0 *dx* <sup>¼</sup> <sup>0</sup>*:*

*f x*ð Þ*Pe <sup>d</sup><sup>θ</sup> dx*

Considering heat transfer in a one dimensional stationary radial fin of cross area

*Ac* with various profiles. The perimeter of the fin is denoted by *P* and length by *Lrb* � *rt* The fin is attached to a fixed prime surface of temperature *Tb* and extends to the fluid of temperature *T*∞*:* One may assume that at the tip of the fin *rt* ¼ 0*:* In

*<sup>θ</sup>*ð Þ¼ <sup>1</sup> 1, *<sup>d</sup>θ*ð Þ <sup>0</sup>

*f r*ð Þ*Pe <sup>d</sup><sup>θ</sup> dr*

*dr* <sup>¼</sup> <sup>0</sup>*:*

*∂r r*¼0

¼ 0, *τ* ≥ 0*:*

*θh*ð Þ¼ *θ* 0, 0 ≤*x*≤1*:* (12)

*θh*ð Þ¼ *θ* 0, 0 ≤*x*≤1*:* (13)

subject to initial and boundary conditions

$$
\theta(\mathbf{0}, \mathbf{x}) = \mathbf{0}, \quad \mathbf{0} \le \mathbf{x} \le \mathbf{1}, \quad \theta(\boldsymbol{\tau}, \mathbf{1}) = \mathbf{1}; \quad \frac{\partial \theta}{\partial \mathbf{x}}\Big|\_{\mathbf{x} = \mathbf{0}} = \mathbf{0}, \quad \boldsymbol{\tau} \ge \mathbf{0}.
$$

#### **2.4 1 + 1 dimensional transient model for radial fins**

#### *2.4.1 Solid stationary fins*

For solid stationary radial fins thge model is given by

$$\frac{\partial \theta}{\partial \tau} = \frac{1}{r} \frac{\partial}{\partial r} \left[ \eta f(r) k(\theta) \frac{\partial \theta}{\partial r} \right] - M^2 \theta h(\theta), \ 0 \le r \le 1. \tag{8}$$

subject to initial and boundary conditions

$$\theta(0, r) = 0, \quad 0 \le r \le 1, \quad \theta(\tau, 1) = 1; \quad \left. \frac{\partial \theta}{\partial r} \right|\_{r=0} = 0, \quad \tau \ge 0.$$

#### *2.4.2 Solid moving fins*

For solid moving radial fins the model is given by (see e.g. [29]),

$$\frac{\partial \theta}{\partial \tau} = \frac{1}{r} \frac{\partial}{\partial r} \left[ r f(r) k(\theta) \frac{\partial \theta}{\partial r} \right] - M^2 \theta h(\theta) - f(r) \text{Pe} \frac{\partial \theta}{\partial r}, \quad 0 \le r \le 1. \tag{9}$$

subject to initial and boundary conditions

$$\theta(0, r) = 0, \quad 0 \le r \le 1, \quad \theta(\tau, 1) = 1; \quad \frac{\partial \theta}{\partial r}\Big|\_{r=0} = 0, \quad \tau \ge 0.$$

#### *2.4.3 Porous stationary fins*

For solid stationary radial fins the model is given by

$$\frac{\partial \theta}{\partial \tau} = \frac{1}{r} \frac{\partial}{\partial r} \left[ \eta^{\xi}(r) k(\theta) \frac{\partial \theta}{\partial r} \right] - N\_p (\theta - \theta\_a)^2 - N\_r (\theta^4 - \theta\_a^4), \ 0 \le r \le 1. \tag{10}$$

subject to initial and boundary conditions

$$\theta(0, r) = 0, \quad 0 \le r \le 1, \quad \theta(\tau, 1) = 1; \quad \frac{\partial \theta}{\partial r}\Big|\_{r=0} = 0, \quad \tau \ge 0.$$

#### *2.4.4 Porous moving fins*

For porous moving radial fins the model is given by

$$\frac{\partial \theta}{\partial \tau} = \frac{1}{r} \frac{\partial}{\partial r} \left[ \eta^{\xi}(r) k(\theta) \frac{\partial \theta}{\partial r} \right] - N\_p \left( \theta - \theta\_d \right)^2 - N\_r \left( \theta^4 - \theta\_a^4 \right) - f(r) Pe \frac{\partial \theta}{\partial r}, \ 0 \le r \le 1. \tag{11}$$

*Survey of Some Exact and Approximate Analytical Solutions for Heat Transfer in Extended… DOI: http://dx.doi.org/10.5772/intechopen.95490*

subject to initial and boundary conditions

$$
\theta(0, r) = 0, \quad 0 \le r \le 1, \quad \theta(\tau, 1) = 1; \quad \frac{\partial \theta}{\partial r}\Big|\_{r=0} = 0, \quad \tau \ge 0.
$$

#### **2.5 One-dimensional steady state model for straight fins**

Considering heat transfer in a one dimensional longitudinal fin of cross area *Ac* with various profiles. The perimeter of the fin is denoted by *P* and length by *L:* The fin is attached to a fixed prime surface of temperature *Tb* and extends to the fluid of temperature *T*∞*:* in non-dimensional variables, one obtains

$$\frac{d}{d\mathfrak{x}}\left[f(\mathfrak{x})k(\theta)\frac{d\theta}{d\mathfrak{x}}\right] - M^2\theta h(\theta) = 0, \ 0 \le \mathfrak{x} \le 1. \tag{12}$$

subject to

*∂θ <sup>∂</sup><sup>τ</sup>* <sup>¼</sup> *<sup>∂</sup>*

*<sup>∂</sup><sup>x</sup> f x*ð Þ*k*ð Þ*<sup>θ</sup> <sup>∂</sup><sup>θ</sup>*

� *Pef x*ð Þ *<sup>∂</sup><sup>θ</sup>*

*2.4.1 Solid stationary fins*

*2.4.2 Solid moving fins*

*∂θ <sup>∂</sup><sup>τ</sup>* <sup>¼</sup> <sup>1</sup> *r ∂*

*2.4.3 Porous stationary fins*

*2.4.4 Porous moving fins*

*<sup>∂</sup><sup>r</sup> rf r*ð Þ*k*ð Þ*<sup>θ</sup> <sup>∂</sup><sup>θ</sup>*

*∂r*

*∂θ <sup>∂</sup><sup>τ</sup>* <sup>¼</sup> <sup>1</sup> *r ∂*

**90**

*∂θ <sup>∂</sup><sup>τ</sup>* <sup>¼</sup> <sup>1</sup> *r ∂*

*∂x*

*∂θ <sup>∂</sup><sup>τ</sup>* <sup>¼</sup> <sup>1</sup> *r ∂*

subject to initial and boundary conditions

*∂r*

*Heat Transfer - Design, Experimentation and Applications*

subject to initial and boundary conditions

**2.4 1 + 1 dimensional transient model for radial fins**

For solid stationary radial fins thge model is given by

*θ*ð Þ¼ 0,*r* 0, 0 ≤*r*≤1, *θ τ*ð Þ¼ , 1 1;

*<sup>∂</sup><sup>r</sup> rf r*ð Þ*k*ð Þ*<sup>θ</sup> <sup>∂</sup><sup>θ</sup>*

subject to initial and boundary conditions

*<sup>∂</sup><sup>r</sup> rf r*ð Þ*k*ð Þ*<sup>θ</sup> <sup>∂</sup><sup>θ</sup>*

subject to initial and boundary conditions

*θ*ð Þ¼ 0,*r* 0, 0 ≤*r*≤1, *θ τ*ð Þ¼ , 1 1;

For solid stationary radial fins the model is given by

*θ*ð Þ¼ 0,*r* 0, 0 ≤*r*≤1, *θ τ*ð Þ¼ , 1 1;

For porous moving radial fins the model is given by

� *Np*ð Þ *<sup>θ</sup>* � *<sup>θ</sup><sup>a</sup>* <sup>2</sup> � *Nr <sup>θ</sup>*<sup>4</sup> � *<sup>θ</sup>*<sup>4</sup>

*∂r*

*<sup>∂</sup><sup>r</sup> rf r*ð Þ*k*ð Þ*<sup>θ</sup> <sup>∂</sup><sup>θ</sup>*

For solid moving radial fins the model is given by (see e.g. [29]),

� *<sup>M</sup>*<sup>2</sup>

� *Np*ð Þ *<sup>θ</sup>* � *<sup>θ</sup><sup>a</sup>* <sup>2</sup> � *Nr <sup>θ</sup>*<sup>4</sup> � *<sup>θ</sup>*<sup>4</sup>

*∂r*

*∂r*

� *<sup>M</sup>*<sup>2</sup>

*<sup>θ</sup>*ð Þ¼ 0, *<sup>x</sup>* 0, 0 <sup>≤</sup>*x*<sup>≤</sup> 1, *θ τ*ð Þ¼ , 1 1; *<sup>∂</sup><sup>θ</sup>*

� *Nc*ð Þ *<sup>θ</sup>* � *<sup>θ</sup><sup>a</sup> <sup>n</sup>*þ<sup>1</sup> � *Np*ð Þ *<sup>θ</sup>* � *<sup>θ</sup><sup>a</sup>* <sup>2</sup> � *Nr <sup>θ</sup>*<sup>4</sup> � *<sup>θ</sup>*<sup>4</sup>

, 0≤*x*≤1*:* (7)

*∂x x*¼0

*∂θ ∂r r*¼0

*<sup>θ</sup>h*ð Þ� *<sup>θ</sup> f r*ð Þ*Pe <sup>∂</sup><sup>θ</sup>*

*∂θ ∂r r*¼0

*∂θ ∂r r*¼0

*a* � *f r*ð Þ*Pe <sup>∂</sup><sup>θ</sup>*

*∂r*

*a*

*a* 

¼ 0, *τ* ≥ 0*:*

*θh*ð Þ*θ* , 0≤*r*≤1*:* (8)

¼ 0, *τ* ≥ 0*:*

¼ 0, *τ* ≥ 0*:*

, 0≤*r*≤ 1*:* (10)

¼ 0, *τ* ≥ 0*:*

*∂r*

, 0≤*r*≤ 1*:* (11)

, 0≤ *r*≤ 1*:* (9)

$$
\theta(\mathbf{1}) = \mathbf{1}, \quad \frac{d\theta(\mathbf{0})}{dx} = \mathbf{0}.
$$

In case of a moving radial fin the term

$$f(\mathbf{x})Pe\frac{d\theta}{d\mathbf{x}}$$

is added to Eq. (12).

#### **2.6 One-dimensional steady state model for radial fins**

Considering heat transfer in a one dimensional stationary radial fin of cross area *Ac* with various profiles. The perimeter of the fin is denoted by *P* and length by *Lrb* � *rt* The fin is attached to a fixed prime surface of temperature *Tb* and extends to the fluid of temperature *T*∞*:* One may assume that at the tip of the fin *rt* ¼ 0*:* In non-dimensional variables, one obtains

$$\frac{1}{r}\frac{d}{dr}\left[\eta^{\prime}(r)k(\theta)\frac{d\theta}{d\mathfrak{x}}\right] - M^{2}\theta h(\theta) = 0, \ 0 \le \mathfrak{x} \le 1. \tag{13}$$

subject to

$$
\theta(\mathbf{1}) = \mathbf{1}, \quad \frac{d\theta(\mathbf{0})}{dr} = \mathbf{0}.
$$

In case of a moving radial fin the term

$$f(r) \\ Pe\frac{d\theta}{dr}$$

is added to Eq. (13) (see e.g. [30]).
