Theorems and operations performed in

## 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, 3, 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, 3, 4, 5, 9, 10, 11, 12, 13, 14, 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.

## 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

The initial condition is given by

here,

Boundary conditions are given by

In non-dimensionalized variables and parameters we have,

subject to the initial condition

and boundary conditions

where the non-dimensional quantities

where

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])

subject to the initial condition

and boundary conditions

### 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.

### 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])

subject to initial and boundary conditions

#### 2.3.2 Solid moving fins

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

subject to initial and boundary conditions

#### 2.3.3 Porous stationary fins

The model was considered in [27].

subject to initial and boundary conditions

#### 2.3.4 Porous moving fins

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

subject to initial and boundary conditions

### 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

subject to initial and boundary conditions

#### 2.4.2 Solid moving fins

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

subject to initial and boundary conditions

#### 2.4.3 Porous stationary fins

For solid stationary radial fins the model is given by

subject to initial and boundary conditions

#### 2.4.4 Porous moving fins

For porous moving radial fins the model is given by

subject to initial and boundary conditions

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

Considering heat transfer in a one dimensional longitudinal fin of cross area

subject to

In case of a moving radial fin the term

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

subject to

In case of a moving radial fin the term

## 3. Methods of solutions

### 3.1 Brief account on lie symmetry methods

In this subsection we provide a brief theory of Lie point symmetries. This discussion and further account can be found in the book of Bluman and Anco [31].

#### 3.1.1 m dependent and n independent variables

* Total derivative.*The total differentiation operator wrt

where

We seek the one-parameter Lie group of transformations

which leave the system of equation in question invariant. These transformations are generated by the base vector

The

Theorem 1.1 The extended infinitesimals satisfy the recursion relations

Introducing the * Lie Characteristic function*defined by

then

The corresponding (

Theorem 1.2 A differential function

Theorem 1.3 A differential function

where

### 3.2 Approximate methods

#### 3.2.1 p-dimensional differential transform methods

For an analytic multivariable function

The upper and lower case letters are for the transformed and the original functions respectively. The transformed function is also referred to as the T-function, the differential inverse transform is given by

It can easily be deduced that the substitution of (18) into (19) gives the Taylor series expansion of the function

For real world applications the function

We now give some important operations and theorems performed in the

Original function | T-function |
---|---|

In the table

We now provide one result of the

** Theorem**. Proof in [32].

If

then

#### 3.2.2 Variational iteration methods

where

where

where

Taking the variation on both sides of Eq. (25) with respect to the independent variable

or equivalently

which gives

obtained upon using

or equivalently

The extreme condition of

This in turn gives

In general, for the

Having determined

The iteration formula Eq. (36), without restricted variation, should be used for the determination of the successive approximations

## 4. Survey of some solutions

In this section we demonstrate the challenge in the construction of exact solution for heat transfer in pin fin. Also, we consider the work in [5].

### 4.1 Some exact solutions

#### 4.1.1 Example 1

Given the power law thermal conductivity in heat transfer through pin fins, that is in Eq. (3)

Notice that

and hence we start the double reduction first with

This symmetry generator leads to the first order ODE

In terms of original variables one obtains the general exact solution

The difficulty for group-invariant solutions is the satisfaction of the imposed or prescribed boundary conditions. This has been seen in two dimensional steady state problems [7, 16], and 1 + 1 D transient problems [25]. Perhaps the most successful attempt in in [26]. For nonlinear steady state problems, some transformation such as Kirchoff [7, 16], may linearise the two dimensional problems which then becomes easier to solve using standard methods. Linearisation of nonlinear steady state one dimensional problems is possible when thermal conductivity is a differential consequence of heat transfer coefficient [5].

#### 4.1.2 Example 2

In [5], preliminary group classification is invoked to determine the thermal conductivity which lead to exact solutions. It turned out that given a power law heat transfer coefficient, thermal conductivity also takes the power law form. Given Eq. (12) with both

The expressions for fin efficiency and effectiveness can be explicit in this case. Furthermore, this solution led to the benchmarking of the approximate analytical solutions [35]. With established confidence in approximate methods, then one may solve other problems that are challenging to solve exactly.

### 4.2 Some approximate solutions

#### 4.2.1 Three dimensional DTM

In this subsection we consider heat transfer in a cylindrical pin fin. We consider thermal conductivity given as a linear function of temperature

One may determine the value of

To plot a three dimensional figure for this solution one may fix temperature, say at

#### 4.2.2 Two dimensional DTM

The two dimensional DTM solution for a steady heat transfer through the cylindrical fin is given by

and

This solution is plotted in Figure 3

#### 4.2.3 Comparison of one dimensional exact, DTM and VIM solutions

Here the solutions for the one dimensional heat transfer problems are compared, namely the exact solution given in Eq. (38). The VIM solutions is given by

The constant

Likewise, the constant

These solutions are depicted in Figure 4.

## 5. Outlook and some concluding remarks

The interest in heat transfer through fins will continue unabated. This is brought about by applications of fins in engineering appliances. The solutions to the problems give insight into effectiveness and efficiency of different fins. In this chapter we provided a summary of some of the work in recent times. In particular, we reviewed the exact and approximate analytical solutions. We demonstrated that although the models describing hear transfer seem to be simple, they are in fact challenging to solve exactly. When constructed, the exact solutions are used as benchmarks for the approximate solutions. It appears that some models including contracting or expanding have attracted some attention. The analysis of these problems provide insight into heat transfer phenomena and assist in the design of fins. The challenge is the construction of exact solutions, however one may construct approximate analytical solutions. The problems discussed here are not exhaustive.

## Nomenclature

Cross-sectional area

Biot number

Aspect ratio

dimensionless heat transfer coefficient

Heat transfer coefficient

Heat transfer coefficient at the base of the fin

Dimensionless thermal conductivity

Thermal conductivity of the fluid

Thermal conductivity of the fin

Length of the fin

Radius

Radius

time

Base temperature

Fluid temperature

Dimensionless fin length

Fin length

Dimensionless fin length

Fin length

Length of a cylindrical pin fin. Greek letters

Dimentionless time

dimensionless temperature