ANALYTICAL & COMPUTATIVE THERMAL ANALYSIS ON STRAIGHT FINS

Mr. KUNDAN SINGH DESHWAL, Mr. SACHIN GUPTAKUNDAN SINGH DESHWAL- kundan.deshwal88@gmail.com

SACHIN GUPTA–Staywidsachin@gmail.comDEPARTMENT OF MECHANICAL ENGINEERING

SEEMANT INSTITUTE OF TECHNOLOGY PITHORAGARH (UK)A CONSTITUTEINSTITUTE OF UTTARAKHAND TECHNICAL UNIVERSITY

ABSTRACT

Thermal behavior of Rectangular and Triangular fin is

analytically investigated using first principle. Two different

methods of solution are implemented in this study. First, an

analytical closed form solution of the problem under-

consideration is obtained manually since the governing

equations and their boundary conditions are linear. Second,

the solution is developed in modeling form using

HYPERWORKS Software and compared with developed

closed form analytical solution. The results of this study

show that MATLAB can be used effectively and efficiently

to solve challenging heat transfer problems. The length,

base thickness, and end thickness of the fin is specified.

Coarse, medium, and fine mesh types are available.

Thermal conductivity of the fin material is specified. A

constant temperature condition is applied at the base of the

fin. Fully insulated and convective boundary conditions can

be applied at the tip of the fin. The results report base wall

temperature, total area for heat convection, heat dissipation

rate, fin efficiency, fin effectiveness and Contours of

temperature distribution & heat flux.

Keywords: MATLAB, Analytical, Thermal behavior, HYPERWORKS,

INTRODUCTION

Profile under convective conditions was first proposed by

Schmidt (1926) [1] based on a physical reasoning. Later on

Duffin (1959) [2] proved Schmidt criteria using calculus of

variation. . LAI was used for optimizing fin shapes under

convective, radiating, convective-radiating condition, Liu

[3](1962), for fins with heat generation and for variable

thermal conductivity.

Maday (1974) [4] in his pioneering analysis proposed the

correct formulation for the optimization of longitudinal fin

with the elimination of LAI and obtained a profile much

different from Duffin (1959). Maday (1974) [4] further

extended this analysis for radial fins. However, fin shapes

determined by the above procedure are complex and

difficult to manufacture. Sonn [5] (1981) considered the

effect of profile curvature on the optimum dimensions of

longitudinal fins of triangular, concave and convex

parabolic profile. Chung and Kan (1987) [6] determined the

optimum dimensions of spines having different profiles

(cylindrical, conical, concave and convex parabolic) from a

generalized formulation using a numerical procedure. 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. This exercise serves as a visualization tool

for evaluating the effect of shape on fin effectiveness,

efficiency, and temperature distribution. In many heat

transfer applications, it is desirable to increase the surface

area that is available for the heat transfer process; this is

particularly true when one desires to dissipate heat to a low

conductivity medium such as air. On the other hand Ulman

and Kalman (1989) [7] solved the conduction equation for

radial fins of different profiles (straight, hyperbolic,

triangular and parabolic) numerically to find out the rate of

heat dissipation.

HyperWorks:

Altair HyperWorks is the engineering frame work for

product design for maximizing product performance,

automating design process and improving profitability with

an open and flexible environment. Built from the core up to

be programmable, automated and interoperable with all

major commercial CAD system and CAE solvers.

Hyperworks enable innovation by providing high end,

value–based product design solution within one common

engineering environment.HyperWorks comprises a suite of

application that includes Hyper Mesh, optistruct, Radioss,

Hyper Crash, Hyper View. Hyper Graph 3D, Motion View,

media view, Text View, FM Model, Hyper Study,

Assembler process manager, Providing Modeling and

assembly, robust design and Optimization.

OBJECTIVES OF THE PRESENT WORK:

As is evident from the diversity of application areas, the

study of heat transfer in Straight Rectangular and

Triangular fins is very important for the technology of

today and the near future, as developments are following

the trend of miniaturization in all fields. Analysis shows

that the Rectangular and Triangular heat sinks were studied

extensively, but there is limited research related to the

performance study of fins using ALTAIR HYPERWORKS

models. Following from the investigation, this work studies

based on the derivation the equation from the first

principles and heat transfer, which couples in a Rectangular

and Triangular fin and heat conduction in the solids.

The present work is undertaken to study the following aspects of:

ALTAIR HYPER WORKS modeling to understand its

thermal behavior.

Validation of the models by comparing the present

Computational results with the data available in the

Analytical study. Parameter sensitivity study of Straight

fin. Comparison between Straight Rectangular and

Triangular fin through computationally and analytical value

in all aspect.

SPECIFICATION OF PROBLEM:

Consider the two types of the fin, first one is Straight

Rectangular and second one is Straight Triangular fin.

Whose physical geometry is same, that means the length L

is.025 meter, width w is .06 meter and thickness t is .004

meter .These fin are made of two different materials which

is Steel and Aluminium, have a thermal conductivity is

17,237 w/m-k respectively. The convective heat transfer

coefficient is 40w/m2-k. Ambient temperature is 300c and

fin wall temperature is 4000c is constant over the cross-

section.

Figure 1: Specification of Straight Rectangular fin and Triangular fin

BOUNDARY CONDITION:

In fin problems it is always assumed that the base of the fin is maintained at a known temperature [8] [9] [10]

(1)

There are three common possibilities for the other end.

(2)

Insulated tip, or symmetry

(3)

Infinitely long fin

(4)

ANALYTICAL SOLUTION OF STRAIGHT

RECTANGULAR FIN:The rate of heat conduction into the element = the rata of heat conduction out the element +the rate of heat Convection from the element surface

Q(x) = Q(x) + ddx

[Q(x)] d x + h d A s[T(x)-T ∞]

ddx

( ACdtdx

) – h d A s

k dx [T(x)-T ∞] = 0

Using element surface area d A s= pdx, then

d2t (x)

dx2 – h p

k AC [T(x)-T ∞] =0

(5)

Where the cross section area AC , perimeter p, heat transfer coefficient h, thermal conductivity of fin material k are treated constant,

Let θ (x) = [T(x)-T ∞] (6)

And m 2 = h pk AC

(7)

Substituting, in eqn. (5) we get,

d2t (x)

dx2 - m2θ = 0

It is linear homogeneous, second order differential equation, it can be solved by using operator D

(D2-m2) θ =0, d θdx

= - mθ, on integration,

(x)=c1e-mx+c2emx

(8)Where the constant c1 and c2 of integration are evaluated

from the boundary condition specified for the fin. A fin is

usually very thin and long enough so that the heat loss from

the fin tip may be assumed negligible .The boundary

condition for such infinite long fin, at the fin tip.

[ d θdx

] x=L =0

Substituting, in eqn. (5) we get,

C1=C2 eml

e−ml

At x=0, θo = C1 + C2,or, it give C2=θ o e−ml

(eml+e−ml)

And, C1 =θo - C2 It gives C1= θ o eml

(eml+e−ml)

Using the value C1and C2of and in eqn. (5), we get

θ(x)θ o

= T ( x )−T ∞

T o−T ∞ =

em(l−X )+e−m(l−X )

(eml+e−ml)In terms of hyperbolic function

θ(x)θ o

= T ( x )−T ∞

T o−T ∞ =

cosh {m(L−x )}coshmL

(9)

ANALYTICAL SOLUTION OF STRAIGHT TRIANGULAR FIN:

Assuming the sufficient thin fin (L>>t), so the one dimension heat conduction can be considered.

ddx

( ACdtdx

) – h d A s

k dx [T(x) -T ∞] = 0

(10)

For a Straight Triangular fin, the cross-section area varies with fin length. Its value at any x position in terms of length ratio x/L is given by

AC= A(x) = w t xL

The perimeter, P = 2wSurface area, dAs = Pdx = 2wdxSubstituting AC and dAs in energy equation

ddx

(w txL

dTdx

) – h2 w dx

k dx [T(x) -T ∞] = 0

d2

dx2 (T ) + 1x

dTdx

– 2hLk tx

(T -T ∞) =0

Introducing θ = (T -T ∞)

d2

dx2 (θ) + 1x

d θdx

– 2h Lk tx

(θ¿ =0

Introducing, β 2 = 2hLk t

(11)And multiplying throughout by x2, we get

x2 d2

dx2 (θ) + xd θdx

– β 2 x (θ ¿ =0

(12)

The eqn. (9) is a modified Bessel equation and its solution is

(θ ¿ = C1Io (2β √x) + C2ko (2β √x) (15)

Where Io and ko are modified zero order Bessel function of

the first and second kind, respectively and C1and C2 are

constant of integration and are evaluated from boundary

condition. At x=0,θ =finite, dθdx

= 0, It gives

C2 =0,At x=L (at root) θ=θ0, using

θ0= C1Io (2β √L)

RESULTS AND DISCUSSIONS

Figure 2: Temperature distribution along the different position of fin

Figure 3: Variation in temperature distribution for along the different position of fin with thickness t=14 (mm)

Figure 4: Variation in Heat Transfer Rate Q (W) at different position of fin

Figure 5: Variation in fin efficiency at different position of fin.

Figure 6: Variation in fin effectiveness at different position of fin.

(a)

(b)

Figure 7: Contours of temperature distribution for Steel ma- terial (a) Straight Rectangular fin (b) Straight Triangular fin

(a)

(b)

Figure 8: Contours of Heat Flux (w/m2) for Alumin -iummaterial (a) Straight Rectangular fin (b) Straight Triangular fin

CONCLUSIONS & FUTURE SCOPE OF THE WORK

The computational results is successfully validated the analytical data for a Straight Rectangular and Triangular fin. Grid temperature and heat flux contours represent successfully the thermal behavior of the system .Fin wall temperature is decreasing along the length direction of Straight Rectangular and Triangular fin and Steel material, which has lower value of thermal conductivity (k) drop down more temperature. As we are using the same material of fin with larger surface area then we will achieve high heat transfer rate Q (W) and heat flux (W/m2).Higher the heat transfer rate Q (W), higher value of fin effectiveness and fin efficiency. Aluminium material obtains larger value of fin efficiency and fin effectiveness then Steel. Triangular fin contains less amount of material and high amount of

heat flow rate per unit mass in comparison of Rectangular fin. For this reason we should be use Triangular fin. Aluminium material obtains larger value of heat flow rate per unit mass then Steel. Thus, Triangular fin is more suitable for space application. Analysis of the thermal characteristics of Straight Rectangular and parabolic fin using computationally as Altair Hyper Works software.

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Ver. Dtsch. Inc. 70 (1926) 885–951.

[2] R. Duffin, A variational problem relating to cooling

fins, J. Math. Mech. 8 (1959) 47–56.

[3] C.Y. Liu, A variational problem with applications to

cooling fins, J. Soc. Ind. Appl. Math. 10 (1962) 19–27.

[4] C.J. Maday, The minimum weight one dimensional

straight fin, ASME J. Eng. Ind. 96 (1974) 161–165.

[5] A. Sonn, A. Bar-cohen, Optimum cylindrical pin fin,

ASME J. Heat Transfer 103 (1981)814–815.

[6] B.T.F. Chung, R.W. Kan, The optimum dimensions of

convective straight fins with profile curvature effect,

AIChE J. 83 (1987)77–83.

[7] A. Ullmann, H. Kalman, Efficiency and optimized

dimensions of annular fins of different cross-section

shapes, Int. J. Heat Mass Transfer 32 (1989) 1105–

1110

[8] Er. R.K. Rajput, (2010) Heat and Mass Transfer SI

UNITS, Fourth Edition,

[9] Incropera, F. P., and DeWitt, D. P., “Fundamentals of

Heat and Mass Transfer", 5th Ed,

[10] M.M.Rathore, (2012) “Engineering Heat and mass

transfer”, 2nd Ed,