Numerical Analysis of Performance of Annular Fin

Text Only Version

Numerical Analysis of Performance of Annular Fin

College of Mechanical Engineering Technology

Benghazi-Libya

Naser S. Sanoussi

College of Mechanical Engineering Technology

Benghazi-Libya

Gamal G. S. Hashem

Mechanical Engineering dept.

Faculty of Engineering, University of Benghazi Benghazi-Libya

Abstract – The governing equation that represents the temperature distribution in annular fins is solved numerically. The finite difference technique is utilized in two dimensions under transient conditions. No angular variation of temperature is considered. The solution was conducted under the boundary conditions of insulated fin tip. The model was solved under practical ranges of different normalized parameters of radii ratio, R1, aspect ratio, ASR and Biot number, Bi.

The results are generated and presented at different values of the above mentioned parameters. Generally the fin efficiency is shown to decrease with Bi, and the ASR. However, the efficiency was shown to increase with R1.

Keywords–Heat transfer; Annular fine; Fin performance; Temperature distribution

1. INTRODUCTION

Heat conducted through a solid substance is often transferred by convection to the surrounding fluid. Since the convection rate is proportional to the surface area, the heat dissipated can be increased by attaching strips of metal called "fins" to the surface, in the spaces where the large convection resistance occurs.

Fins are generally used on surfaces where the heat transfer coefficient is low. For any application, the problem is to choose a fin which will give a maximum cooling efficiency, minimum material for cost, weight, and space consideration, minimum resistance to the flow of the ambient medium, adequate strength and ease of manufacture. For this purpose detailed information on the performance of fins under different operating conditions and for different geometries is needed.

Fins are used to enhance convective heat transfer in a wide range of engineering applications, and offer a practical means for achieving a large total heat transfer surface area without the use of an excessive amount of primary surface area. Fins are commonly applied for heat management in electrical appliances such as computer power supplies or

substation transformers. Other applications include IC engine cooling, such as fins in a car radiator. It is important to predict the temperature distribution within the fin in order to choose the configuration that offers maximum effectiveness.

Although there are many different situations that involve combined conduction-convection effects, the most frequent application is one in which an extended surface is used specifically to enhance the heat transfer rate between a solid and an adjoining fluid.

To illustrate the function of fins, consider the plane wall of Fig. 1a. If the fin surface temperature, Ts, is fixed, there are two ways by which the heat transfer rate may be increased. Either by increasing the convection coefficient, h, by increasing the fluid velocity, and/or reducing the fluid temperature, T. However; many situations would be encountered in which increasing h to the maximum possible value is either insufficient to obtain the desired heat transfer rate or the associated costs are prohibitive. Such costs are related to the blower or pump power requirements needed to increase h through increased fluid motion. Moreover, the second option of reducing T is often impractical. Examining Fig. 1b, however, we see that there exists a third option. That is, the heat transfer rate may be increased by increasing the surface area across which the convection occurs. This may be done by employing fins that extends from the wall into the surrounding fluid .

The thermal conductivity of the fin material has a strong effect on the temperature distribution along the fin and therefore influences the degree to which the heat transfer rate is enhanced. Ideally, the fin material should have a large thermal conductivity to minimize temperature variations from its base to its tip. In the limit of infinite thermal conductivity, the entire fin would be at the temperature of the base surface, thereby providing the maximum possible heat transfer enhancement, i.e. ideal fin.

A straight fin is any extended surface that is attached to a plane wall. It may be of uniform cross-sectional area, or its cross-sectional area may vary with the distance x from the wall. An annular fin is one that is circumferentially attached to a cylinder, and its cross section varies with radius from the centerline of the cylinder.

Fig.1 Illustration of fin performance

The foregoing fin types have rectangular cross sections, whose area may be expressed as a product of the fin thickness t and the width w for straight fins or the circumference 2r for annular fins. In contrast a pin fin, or spine, is an extended surface of circular cross section. Pin fins may also be of uniform or non-uniform cross section. In any application, selection of a particular fin configuration may depend on space, weight, manufacturing, and cost considerations, as well as on the extent to which the fins reduce the surface convection coefficient and increase the pressure drop associated with flow over the fins.

However, Fins may be classified as straight surface area such as the trapezoidal fin and fins with curved surface area such as the parabolic and the cylindrical fins.

The aim of this study is to numerically study the performance of the annular fins. This study assumes two dimensional variations of temperature thus covering wider range of fin sizes with good accuracy. Our study also studies the performance variations with time.

Nomenclature

(aE, aN, aS, aW, aP) Algebraic equation coefficient ASR Aspect ratio

b Wall coefficient

Bi Biot number

e East control volume interface

ETAT Fin efficiency

Fo Fourier number

h Average convection heat transfer coefficient

k Thermal conductivity

1. Fin length

2. Number of grid points along (R)

n North control volume interface

3. Number of grid points along ()

P Central control volume node

1. Heat transfer rate

qact. Actual heat transfer

qideal Ideal heat transfer

2. Dimensional radial coordinate

R Dimensionaless radial coordinate

R1 Ratio of radius

r1 Inner radius of fin

r2 Outer radius of fin

3. South control volume interface

SS Fo at steady state

4. Time

T Dimensional temperature

Tb Fin base temperature

T Ambient fluid temperature w

x West control volume interface

Greek symbols

a Thermal diffusivity

Dimensional axial coordinate

Fo Incremental Fourier number

R Incremental radial distance

Incremental axial distance

E, W, S, N The interior nodes

P Dimensionless temperature previous at central node

P` Dimensionless temperature present at central node

Dimensionless axial coordinate

2. STATEMENT OF THE PROBLEM

1. Physical Model

The geometry under consideration is an annular fin; the fin is characteized by its total length, L, its thickness, 2, its inside radius r1, and its outside radius, r2. The fin material is assumed to be homogeneous with constant thermal conductivity equals to, k. The fin skin is assumed to be totally surrounded by a fluid at a uniform temperature equals to, T, with an average convective heat transfer coefficient equals to, h. The fin tip is assumed to be perfectly insulated, i.e. all heat dissipated from the fin is from the circular two sides of the fin. The temperature within the fin is assumed to be two dimensional in the radial and transverse directions and changes with time, i.e. transient.

The variations in temperature in the angular direction, are neglected and assumed constant at each instant of time. Fig. 2 shows the annular fin under study with all involved parameters.

Fig. 2 Schematic diagram of annular fin

2. Solution Domain

Due to symmetry around an axis that extends radially and divides the fin into equal halves each with thickness equals to , and since there are no temperature variations in the angular direction, the solution domain is reduced to a rectangular area. This rectangular area extends axially from x = 0 at the symmetry line to x = d at the fin surface, and it extends radially from r = r1 at the fin base to r = r2 at the fin tip, as can be seen in Fig. 3.

F t

o 2

(11)

The governing equation in dimensionless form is found by substituting equations (8) to (11) into equation (2) which

L gives:

r x d

r1

r2

2

R 2

Where;

1

R R

ASR2

2

2

ASR2

F0

(12)

Fig. 3 Solution domain and coordinate system

3. Mathematical Model

ASR r2

(13)

The boundary conditions and the

initial condition in the dimensionless form become:

The general conduction equation in cylindrical coordinate

At R R1 , 1.0

(14)

is given by the equation:

At R 1 ,

0

R

(15)

2T 1 T 2T

2T

q“`

1 T

2 2

G

2

(1)

At 0

,

r r r

x k t

0

(16)

This equation is derived by applying the conservation principle to the differential element defined in cylindrical coordinates. Consider the two dimensional and no heat

At 1

,

Bi ( )

(17)

generation, homogeneous differential equation of heat conduction in the cylindrical system, the governing equation is reduced to:

At Fo 0

Where:

, ( , R , 0) 0

(18)

2T

1 T

2T

1 T

(2)

the Biot Number is

h

r 2

r r

x2

t

Bi

k

(19)

4. Boundary Conditions and Initial Condition

3. METHOD OF SOLUTION

At r r1 ,

At r r2 ,

At x 0 ,

T Tb

T 0

r

T 0

x

(3)

(4)

(5)

The problem of determination of heat transfer through a fin requires the knowledge of temperature distribution in the fin, which in turn requires the solution of mathematical model and the applicable boundary conditions and initial condition.

There are four available methods for the evaluation of the

At x ,

• k T

x

h[T ( ) T ]

(6)

temperature distribution; analytical, graphical, numerical and experimental Method

At t 0

, T (x, r,0) T

(7)

1. Proposed Method of Solution

Our proposal is to use the numerical method which is

To normalize the mathematical model mentioned above with all boundary conditions and initial condition, the following parameters are used:

r

based on finite differences, and therefore it is an approximation computational technique. This method is preferred because it is straightforward, flexible, and it will frequently give a good approximate solution to a problem which its solution is complicated by formal analytical

R

r2

Dimensionless axial coordinate

x

Dimensionless temperature

T( r , x.t ) T

Tb T

Dimensionless Time, (Fourier number),

(8)

(9)

(10)

means.

2. Control-volume formulation

This method includes the tasks of providing a set of algebraic equations for these unknowns and of prescribing an algorithm for solving the equations. This control-volume formulation can be regarded as a special version of the method of weighted residuals. The calculation domain is divided into a number of non-overlapping control volumes such that there is one control volume surrounding each grid point as in the Fig.4. The differential equation is integrated over each control volume.

The result is the discretization equation containing the Integrating the first term of equation (12) as follows:

values of for a group of grid points. The discretization

2

2

o R

E

W

equation obtained in this manner expresses the conservation principle for for the finite control volume, just as the

R F

2 dFo ddR ( R

P )F0

differential equation expresses it for an infinitesimal control

Similarly, the second term becomes:

volume.

Deriving the control volume discretization equation by integrating the differential equation over a finite control volume is a rather roundabout process; the control volume

R F

1

dFo dd R (

E W

)F0

2R

o R R

equation would have been our only way of stating the conservation principle.

And the third term becomes:

R F

2 2

r

2

2 2

dFo dd R

o

r 2 N

2 (

S

2 p

)F0 R

2

And the fourth term becomes:

R F

2

r

2

2

dFo ddR

2

r

2 ( \

p

)R

p

2

o Fo

Fig.4 Three-dimensional cluster

3. Solution Grid

Due to symmetry between upper surface and lower surface in x direction of the annular fin, and since there is no

Substituting by the four equations into the main equation (12), we obtain:

variation in the temperature in the angular direction, the`

a a

solution domain becomes a two dimensional grid. Fig.5P

E E W W

(20)

shows the solution domain and the coordinates x and rin aNNAt symmaetSryliSne: aPP

dimensional form, and and R in dimensionless form.

In the r direction the domain covers from r = r1 (the fin base) to r = r2 (the fin tip), while in the x direction it covers from x = 0 (the fin symmetry line) to x = (the fin surface).

' (a a

P

E

E

W W

At the fin surface:

• aN N

• aP P

) /(1 aS )

(21)

In dimensionless form, the solution domain covers from

' (a a a a ) /(1 b * a )

(22)

R R1 to R 1 in the radial direction and from

P E E W W S S p P N

= 0 to = 1 in the axial direction. At the fin tip:

'

P (aW W

• aN N

• aSS

• aP

P) /(1 aE )

(23)

At the corner of the symmetry line and tip:

'

P (aWW

• aNN

• aPP

) /(1 aE

• aS )

(24)

At the corner of the surface and the tip:

'

P (aWW

• aSS

• aPP

) /(1 aE

• b * aN )

• /ul>

(25)

Where:

a ( 1 1 ) 1

F0

, a ( 1 1 ) 1

F0

Fig. 5 Solution Grid

E R

2R ASR2 R

W R

2R ASR2 R

• Discretization of the Governing Equation

To change the applicable mathematical model, boundary

aN

Fo

2

, S

Fo

2

a

conditions, and initial condition into algebraic form, the

1 2F

2F ,

2 Bi *

approach of the control volume presented by Patankar  is

aP 1 0 0 b

used. Each term of the differential equation (12) is integrated over the limits of the boundary of the control volume for a single node as shown in the cluster of Fig.4. The integration is performed as follows:

ASR2

R2

2

2 Bi *

• Solution Procedure

ETAT 2 R

R 1

R

(30)

Equations (20) to (25) are applied each at its appropriate location and the solution is attained by a point by point Gauss-Seidel iteration method. The iteration continues until a convergence is reached. This convergence is measured by a criterion of the form | (- p)/ |10-5 is achieved at each nodal point. Where is the temperature at the present iteration and p is the temperature of the previous iteration at the same node.

Once a convergence is reached, the solution is progressed to the next time step and continues until steady state conditions are reached.

• Fin Performance Calculation

There are several ways used for measuring the relative performance of fin. The first of these is effectiveness, which is the heat conducted through the fin base to that which would be transferred from the same base area if the fin were not present, and if the base temperature (Tb) remains the same, but this criterion is not a true indication of fin performance, since (Tb) cannot be expected to remain constant if the fin is removed. A second and more realistic measure of fin performance is the ratio of the actual heat dissipated by the fin to that which would be dissipated if the entire fin temperatures were at (Tb), (i.e. ideal fin). This ratio, if calculated under steady state conditions, is known as the fin efficiency.

In this work we use the second measure as an index of performance and refer to it as the fin efficiency.

(1 R2 ) R R1

1

Where: ETAT is the fin efficiency.

H. Selection of Mesh Size

The mesh size was selected on the basis of many trials. Due to the nature of the problem, a huge memory is needed to store the temperature and the finer the mesh the greater the memory needed. The execution time was also another factor that played a major role in selecting a reasonable mesh size.

Our aim is to select a mesh size that gives reasonably accurate solution with optimized execution time and memory size. Our final selection of mesh size was as follows:

40 equally spaced mesh points in the radial direction.

20 equally spaced mesh points in the axial direction.

The step in the time Fo is selected as a very fine step at the fin base equals to 0.0005. This value then increased gradually as we march in the time space by multiplying it by the factor 1.01 until Fo reached a value of 0.001 then kept constant until we reach the steady state conditions.

The steady state conditions were assumed when the value of the total heat transfer from the fin, ETAT, as defined by equation (30), becomes reasonably constant with Fourier number, Fo. The criterion used to assume a steady state condition is to compare the value of ETAT with its value at the previous time step, ETATp as:

1. Calculation of Heat Transfer Parameters

Since the tip is insulated, the heat loss from the fin tip

ETAT ETAT p

ETAT

10 6 , when this criterion is

does not exist and all heat loss is that from the lateral surface.

The actual heat transfer distribution qact from the fin surface area is calculated locally at each radial location, r covering a surface area in the form of a ring equals to (2prr).

This actual heat transfer through the fin surface is given by:

satisfied the solution is stopped and assumed complete.

The value of Fo at which the steady state conditions are reached is called Foss, and the value of ETAT at which the steady state conditions are reached is the steady state fin efficiency.

q k (2 r r)(T

T )

(26)

1. Validation of Accuracy

act.

b

To validate the accuracy of our solution, a comparison in

the form of ETAT with Bi is presented against analytical

And the ideal heat transfer through the entire fin surface area assuming that the whole fin is isothermal at the base temperature Tb is given by:

one-dimensional solution as presented in Ref. . This comparison is shown in Fig. 6.

The comparisons were made at two extreme geometrical

ideal

2

1

b

q h (r 2 r 2 ) (T

T )

(27)

cases, one with too thick and short fin (Minimum ASR and

From Equation (26) and (27) we get:

maximum R1), and the other with too thin and long fin (Maximum ASR and minimum R1). It can be seen clearly

1

i

q qact. 1

2R R

(28)

that the agreement with the thin and long fin is excellent,

qideal

B (1 R 2 )

whereas the comparison with the thick and short fin has

From boundary condition at

1

we have

some deviations. These findings are expected since the

Bi ( )

Then equation (28) can be written in the form:

assumption of one-dimensional solution of Ref.  is not expected to give accurate results when the fins are thick and short. Thus this validates the accuracy of our model, method

q qact.

qideal

2R R

1

(1 R 2 )

( )

(29)

of solution and selected mesh size.

To get the fin efficiency, we need to calculate the ratio of the total actual heat transfer from the fin to the ideal heat loss from the same fin geometry. This can be achieved by numerically integrating equation (29) over the whole fin surface

1

0.9

0.8

0.7

ETAT

0.6

0.5

0.4

0.3

0.2

0.1

0

Present solution at R1

= 0.2 & ASR = 16

0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 3

BI1

Fig. 6 Comparison of present solution with one dimension solution of 

1. RESULTS AND DISCUSSION

The result in the form of temperature distribution in the fin body and the heat transfer rate distribution over the fin surface in dimensionless form are present at different geometrical parameters of ASR and R1 and different values of Bi and Fo". The fin efficiency which is an important parameter is presented at different parameters to show the fin performance at different geometrical and operational parameters.

The range values of ASR covered in this study are 10, 12, 14, and 16, while the range of values of R1 covered are 0.2, 0.4, 0.6, and 0.8. The Biot number, Bi range covered in this study covers the values of 0, 0.04, 0.08, and 0.12. These ranges of parameters cover the practical and meaningful values that are generally encountered in practical life.

1. Fin Efficiency

The fin efficiency is an indication of how much energy a specific fin can dissipate when compared to an ideal fin of similar size. An ideal fin is a fin with zero conductive resistance within the fin material; the ideal fin dissipates the maximum amount of heat that can possibly be dissipated by a fin of that size. Mathematically when Bi equals to zero, the fin becomes ideal. The ideal fin is characterized by its isothermal temperature throughout the fin body and equals to the temperature of the fin base, Tb.

In Fig.7, the fin efficiency, ETAT, is presened as a function of Biot number Bi, at different values of ASR and R1. It can be seen from the figure that as Bi increases the fin efficiency decreases. The rate of decrease is sharp at lower values of Bi. However; as Bi increases to higher values the rate of decrease in efficiency becomes smaller. The values of the efficiency at any Bi are shown to be higher at higher values of R1 and consistently lower at higher values of ASR. This means that R1 has the effect of increasing the efficiency while the ASR has the opposite effect of lowering the values of the efficiency at all values of Bi.

Fig. 7 Variation of fin efficiency with Bi at different values of R1 and ASR

2. Heat Transfer Rate Distribution

The dimensionless heat transfer rate distribution, q along the radial direction, R is presented in eight figures. Fig. 8 to Fig. 11 represent the distribution of the heat transfer rate, q with R at different values of Bi ranging from 0.04 to 0.12. The first two figures, Fig. 8 and Fig. 9, are for minimum ASR of 10 while the other two figures of, Fig. 10 and Fig. 11, are for the maximum value of ASR of 16. The other four figures, from Fig. 12 to Fig. 15, show the heat transfer distribution, q with R at different values of ASR ranging from 10 to 16. The first two figures, Fig. 12 and Fig. 13, are for the minimum value of Bi of 0.04 and the other two figures, Fig. 14 and Fig. 15, are for a maximum value of Bi = 0.12.

These eight figures in all show the effect of ASR, R1, and Bi on the heat transfer distribution along the fin surface. It can be said that the heat transfer rate decreases as Bi increases. The effect of R1 is shown to increase the value of the transfer rate at all values of Bi and ASR. However; the effect of ASR is shown to have the opposite effect of decreasing the heat transfer rate at all values of Bi and R1.

Fig. 8 Distribution of heat transfer rate at ASR = 10, R1 = 0.2 and different values of Bi

Fig. 9 Distribution of heat transfer rate at ASR = 16, R1 = 0.2 and different values of Bi

Fig. 10 Distribution of heat transfer rate at ASR = 10, R1 = 0.8 and different values of Bi

Fig. 11 Distribution of heat transfer rate at ASR = 16, R1 = 0.8 and different values of Bi

Fig. 12 Distribution of heat transfer rate at Bi = 0.04, R1 = 0.2 and different values of ASR

Fig.13 Distribution of heat transfer rate at Bi = 0.04, R1 = 0.8 and different values of ASR

Fig. 14 Distribution of heat transfer rate at Bi = 0.12, R1 = 0.2 and different values of ASR

Fig. 15 Distribution of heat transfer rate at Bi = 0.12, R1 = 0.8 and different values of ASR

Figures, from Fig. 16 to Fig. 19, show the heat transfer rate distribution with R at different values of R1 and covering the limiting values of ASR and Bi by taking the minimum and maximum of each, as shown in the figures. The figures, Fig. 16 and Fig. 17, are for minimum value of Bi = 0.04. Comparing these two figures we can conclude that the R1 consistently increases the heat transfer rate while the ASR decreases the heat transfer rate. Similar findings can be drawn from the next two figures, Fig. 18 and Fig. 19, which are at the maximum value of Bi = 0.12.

The general trend of the heat transfer rate with R has the tendency to decrease at early sections of R and attempt to increase at higher values of R. This is attributed to the fact that as we move towards the fin tip the fin temperature decreases thus the heat transfer rate decreases; however, as we proceed towards the fin tip the surface area of the fin which is a function of R increases thus increases the heat transfer rate. With these two counter acting effects depending on which of the two effects dominates the rate of heat transfer decreases or increases.

Fig. 16 Distribution of heat transfer rate at Bi = 0.04, ASR

= 10 and different values of R1

Fig. 17 Distribution of heat transfer rate at Bi = 0.04, ASR = 16 and different values of R1

Fig. 18 Distribution of heat transfer rate at Bi = 0.12, ASR = 10 and different values of R1

Fig. 19 Distribution of heat transfer rate at Bi = 0.12, ASR = 16 and different values of R1

3. Effect of Fo on the Heat Transfer Rate Distribution

To study the effect of Fo on the distribution of heat transfer from the fins at different parameters, eight figures are presented. Figures, from Fig. 20 to Fig. 27, present the heat transfer rate distribution with R at different values of Fo, starting from small value of Fo to the maximum value at the steady state. All figures show that at early stages of time, (low Fo) the heat transfer rate is very small and concentrates at areas close to the fin base. As Fo increases, the heat transfer rate spreads further towards the fin tip. This is due to the fact that at smaller values of Fo, the fin is storing heat and only small portion is transferred to the surroundings. At the steady state value of Fo, all heat conducted into the fin through the base is dissipated by convection from the fin surface. Thus, the higher the value of Fo the higher the heat transfer rate from the fin.

Comparing these figures, it can be seen that higher values of R1 increases the heat transfer rate distribution. However, both Bi and ASR have an opposite effect on the heat transfer rate distribution for all values of Fo.

Fig. 20 Distribution of heat transfer rate at R1 = 0.2, ASR = 10, and Bi =

0.04 at different values of Fo

Fig. 21 Distribution of heat transfer rate at R1 = 0.2, ASR = 16, and Bi = 0.04 at different values of Fo

Fig. 22 Distribution of heat transfer rate at R1 = 0.8, ASR = 10, and Bi =

0.04 at different values of Fo

Fig. 23 Distribution of heat transfer rate at R1 = 0.8, ASR = 16, and Bi =

0.04 at different values of Fo

Fig. 24 Distribution of heat transfer rate at R1 = 0.2, ASR = 10, and Bi =

0.12 at different values of Fo

Fig. 25 Distribution of heat transfer rate at R1 = 0.2, ASR = 16, and Bi =

0.12 at different values of Fo

0 . 9

0 . 8

0 . 7

0 . 6

X

0 . 5

0 . 4

0 . 3

0 . 2

0 . 1

0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7

R

Fig. 28 Temperature contour lines for fin with R1=0.2, ASR=10, and Bi=0.04

Fig. 26 Distribution of heat transfer rate at R1 = 0.8, ASR = 10, and Bi =

0.12 at different values of Fo

0 . 9

0 . 8

0 . 7

0 . 6

X

0 . 5

0 . 4

0 . 3

0 . 2

0 . 1

0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7

R

Fig. 29 Temperature contour lines for fin with R1=0.2, ASR=16, and Bi=0.04

0 . 9

0 . 8

Fig. 27 Distribution of heat transfer rate at R1 = 0.8, ASR = 16, and Bi =

0.12 at different values of Fo

4. Temperature Contour Lines

In this section, the mechanism of heat conduction through the fin material is presented in qualitative figures under steady state conditions. These figures in general show the path that the heat is selecting through the fin material before it reaches the surface and therefore convected by the

0 . 7

0 . 6

X

0 . 5

0 . 4

0 . 3

0 . 2

0 . 1

0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7

R

surrounding fluid. For this purpose, eight geometries representing the limiting cases of Bi = 0.04 and 0.12, ASR = 10 and 16, and R1 = 0.2 and 0.8. The contour lines of the temperature for all cases start at a value of unity at the fin base and decreases as we move away towards the fin tip. In general, the temperature contour lines show the direction of heat flow and its magnitude. The flow of heat takes paths that are orthogonal to the temperature contour lines. The rate of heat transfer increases as the temperature gradient increases. This can be read by the degree of change in the value of the temperature from one contour line to the next ones. As the changes are larger, the heat transfer rate is also larger and vice versa.

.

Fig. 30 Temperature contour lines for fin with R1=0.2, ASR=10, and Bi=0.12

0 . 9

0 . 8

0 . 7

0 . 6

X 0 . 5

0 . 4

0 . 3

0 . 2

0 . 1

0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7

R

Fig. 31 Temperature contour lines for fin with R1=0.2, ASR=16, and

Bi=0.12

0 . 9

0 . 8

0 . 7

0 . 6

X 0 . 5

0 . 4

0 . 3

0 . 2

0 . 9

0 . 8

0 . 7

0 . 6

X 0 . 5

0 . 4

0 . 3

0 . 2

0 . 1

0 . 1

0 0 . 0 5 0 . 1 0 . 1 5

R

0 0 . 0 5 0 . 1 0 . 1 5

R

Fig. 35 Temperature contour lines for fin with R1=0.8, ASR=16, and

Bi=0.12

Fig. 32 Temperature contour lines for fin with R1=0.8, ASR=10, and Bi=0.04

0 . 9

0 . 8

2. CONCLUSIONS

A numerical analysis of the annular fin was carried out for different geometrical and operational parameters. The effect of varying of ASR, R1, and Bi were presented and discussed. The results presented in the previous chapter can lead to the following conclusions:

0 . 7

0 . 6

X 0 . 5

0 . 4

0 . 3

0 . 2

0 . 1

0 0 . 0 5 0 . 1 0 . 1 5

R

1. The efficiency of the fin decreases as Bi increases, and the rate of decrease is sharp at lower values of Bi while it becomes smaller by increasing of Bi.

2. The values of the efficiency at any Bi are shown to be higher at higher values of R1 and consistently lower at higher values of ASR.

3. The heat transfer rate decreases as Bi increases. Also, it increases by increasing R1 at all values of Bi and ASR, however the effect of ASR is shown

Fig. 33 Temperature contour lines for fin with R1=0.8, ASR=16, and Bi=0.04

0 . 9

0 . 8

0 . 7

0 . 6

X 0 . 5

0 . 4

0 . 3

0 . 2

0 . 1

to have the opposite effect of decreasing the heat transfer rate at all values of Bi and R1.

4. Thick and short fins (with minimum ASR and maximum R1) are found to have better performance than thin and long ones (with maximum ASR and minimum R1).

REFERENCES

1. J P Holman, Heat Transfer, 10th ed. NewYork, McGraw-Hill, 2010.

2. Patankar, s. v., Numerical Heat Transfer and Fluid Flow,

Hemisphere Publishing Corporation, 1980

3. Sucer, J., Heat Transfer, WM. C. Brown, 1st Edition, 1985.

0 0 . 0 5 0 . 1 0 . 1 5

R

Fig. 34 Temperature contour lines for fin with R1=0.8, ASR=10, and Bi=0.12