ANALYSIS OF UNSTEADY HEAT CONDUCTION THROUGH SHORT · PDF file269 Int. J. Mech. Eng. & Rob....

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Int. J. Mech. Eng. & Rob. Res. 2013 Tejpratap Singh et al., 2013

ANALYSIS OF UNSTEADY HEAT CONDUCTIONTHROUGH SHORT FIN WITH APPLICABILITY OF

QUASI THEORY

Tejpratap Singh1*, Sanjeev Shrivastava1 and Harbans Singh Ber2

*Corresponding Author: Tejpratap Singh, [email protected]

The paper is based on the analysis of unsteady heat conduction through short fin with applicabilityof quasi theory. “The area exposed to the surrounding is frequently increased by the attachmentof protrusions to the surfaces, and the arrangement provides a means by which heat transferrate can be substantially improved. The protrusions are called fins”. The fins are commonlyused on small power developing machine as engine used for motor cycle as well as smallcapacity compressor. Earlier, Work under steady state conduction had been carried outextensively. Unsteady heat conduction analysis for the fins is being done for calculation of heattransfer. Unsteady Closed form solutions had been derived earlier by various researchers. Exactsolutions are given for the unsteady temperature in flux-base fins with the method of Green’sFunctions (GF) in the form of infinite series for three different tip conditions. The time ofconvergence is improved by replacing the series part by closed form solution. The presentstudy supplies a new approach to calculate the thermal performance of the short fin. For theshort fin case, exact fin solution and a quasi-steady solution is presented. Numerical values arepresented and the conditions under which the quasi-steady solution is accurate are determined.Dimensionless temperature distribution is presented for both quasi steady theory and exact fintheory.

Keywords: Unsteady heat, Green function, Protrusion, Heat transfer

INTRODUCTIONThe laws which are governing heattransmission are very important to theengineers in the design, construction, testing

ISSN 2278 – 0149 www.ijmerr.comVol. 2, No. 1, January 2013

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Int. J. Mech. Eng. & Rob. Res. 2013

1 Department of Mechanical Engineering (Thermal Engineering), Shri Shankracharya College of Engineering and Technology (SSCET,Bhilai), Chhattisgarh, India.

2 Department of Mechanical Engineering (Thermal Engineering), National Institute of Technology (NIT, Raipur), Chhattisgarh, India.

and operation of heat exchange apparatus.Heat transfer is the study of the rate at whichenergy is transferred across a surface ofinterest due to temperature gradients at the

Research Paper

270


surface, and temperature difference betweenthe different surfaces. This variation intemperature is governed by the principle ofenergy conservation which when applied to acontrol volume or a control mass, states thatthe sum of the flow of energy and heat acrossthe system, the work done on the system andthe energy stored and converted within thesystem is zero. The mechanical engineer dealwith problems of heat transfer in the field ofinternal combustion engines, steamgeneration, refrigeration and heating andventilation. To estimate the cost, the feasibilityand size of the equipment necessary to transfera specified amount of heat in a given time, adetailed heat transfer analysis must be made.The dimensions of boilers, heaters,refrigerators and heat exchangers depend notonly on the amount of heat to be transmittedbut rather on the rate at which heat is to betransferred under given conditions. Thermalsystem contains matter or substance and thissubstance may change by transformation orby exchange of mass with the surroundings.To perform a thermal analysis of a system, weneed to use thermodynamics, which allows forquantitative description of the substance. Thisis done by defining the boundaries of thesystem, applying the conservation principlesand examining how the system participates inthermal energy exchange and conversion.

The unsteady response of fins is importantin a wide range of engineering devicesincluding heat exchangers, clutches, motorsand so on. Heat conduction is increasinglyimportant in various areas, namely in the earthsciences, and in many other evolving areas ofthermal analysis. A common example of heatconduction is heating an object in an oven or

furnace. The material remains stationarythroughout, neglecting thermal expansion asthe heat diffuses inward to increase itstemperature. The importance of suchconditions leads to analyze the temperaturefield by employing sophisticated mathematicaland advanced numerical tools. The sectionconsiders the various solution methodologiesused to obtain the temperature field. Theobjective of conduction analysis is todetermine the temperature field in a body andhow the temperature varies within the portionof the body. The temperature field usuallydepends on boundary conditions, initialcondition, material properties and geometryof the body. Why one need to knowtemperature field. To compute the heat flux atany location, compute thermal stress,expansion deflection, design insulationthickness, heat treatment method, these allanalysis leads to know the temperature field.The solution of conduction problem involvesthe functional dependence of temperature onspace and time coordinate. Obtaining asolution means determining a temperaturedistribution which is consistent with theconditions on the boundaries and alsoconsistent with any specified constraintsinternal to the region.

Many researchers have contributed inunsteady heat conduction through fins.

Donaldson and Shouman (1972) studiedthe transient temperature distribution in aconvecting straight fin of constant area for twodistinct cases, namely, a step change in basetemperature, and a step change in base heatflow rate. The tip of the fin is insulated. Theauthors developed the equations for thetransient temperature distribution and the heat

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flow rate for the two aforementioned cases,and present their results graphically. Alsoincluded is a summary of their experimentalwork to verify their results for the case of a stepfunction in heat flow rate.

Chapman (1959) who studied the transientbehaviour of an annular fin of uniform thicknesssubjected to a sudden step change in the basetemperature. His interest in circular annular finstemmed from the numerous applications ofthese type of fins, especially on cylinders ofair cooled internal combustion engines.Chapman (1959) developed equations thatgive the temperature distribution within the fin,the heat removed from the source and the heatdissipated to the surroundings, all as functionsof time. These equations in graphical form arevery useful for the design engineers.

Suryanarayana (1975 and 1976) alsostudied the transient response of straight finsof constant cross-sectional area. However,rather than using the separation of variablestechnique followed by Donaldson andShouman, he utilized the Laplace transformsin order to develop the solutions for small andlarge values of time, when the base of the finis subjected to a step change in temperatureor heat flux. The tip of the fin is insulated inaddition, the use of the Laplace transformsmade it easier for Suryanarayana to developsolutions for the case of a fin subjected to asinusoidal temperature or heat flux at its base.

Suryanarayana (1976) has provided ananalysis of the heat transfer that takes placefrom one fluid to another separated by a solidboundary with fins on one side.

Aziz and Na (1980) considered the transientresponse of a semi-infinite fin of uniform

thickness, initially at the ambient temperature,subjected to a step change in temperature atits base, with fin cooling governed by a power-law type dependence on temperaturedifference. The choice of a semi-infinitegeometry enabled the transformation of thegoverning nonlinear partial differentialequations into a sequence of similarity typelinear perturbation equations. Aziz and Na alsodiscussed the applicability of the results tofinite fins.

Mao and Rooke (1994) also used theLaplace transform method to study straightfins with three different transients: a stepchange in base temperature; a step changein base heat flux and a step change in fluidtemperature. Transient fins of constant cross-section have also been studied with themethod of Green’s functions (Beck et al.,1992, pp. 60-64), a flexible and powerfulapproach that are applicable to anycombination of end conditions on the fin.

Aziz and Kraus (1995) present a variety ofanalytical results for transient fins, developedby separation of variable and Laplacetransform techniques. Results discussedinclude rectangular fins with three differentbase conditions, rectangular fins with powerlaw convective heat loss and radial fins, alongwith several specific examples. Aziz and Krausalso present a comprehensive literaturereview. The material on transient fins ofconstant cross-section is also included in abook by Kraus et al. (2001, Chap.16).

Kim (1976) developed an approximatesolution to the transient heat transfer instraight fins of constant cross-sectional areaand constant physical and thermal properties.

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The author utilized the Kantorovich methodin the variation formulation to provide a simpleexpression of the exact form of the solution.In some fin applications, Newton’s law ofcooling is not applicable, and a power-lawtype dependence of convective heat flux ontemperature better describes the coolingprocess. Such cases include cooling of finsdue to film boiling, natural convection,nucleate boiling, and radiation to space atabsolute zero.

The work discussed so far has focused onthe transient response of fins of simplegeometry such as circular annular fins andstraight fins. In addition, several simplifyingassumptions were utilized such as uniformthickness, constant cross-sectional area,semi-infinite length, insulated tip and small finthickness-to-length ratio to ensure onedimensional heat conduction. Recently, workhas included fins of various shapes and cross-sections, two and three dimensional heattransfer, and practical applications of finnedheat exchangers.

Campo and Salazar (1996) explored theanalogy between the transient conduction in aplanar slab for short times and the steady stateconduction in a straight fin of uniform cross-section. They made use of a hybridcomputational method, known as theTransversal Method Of Lines (TMOL), to arriveat approximate analytical solutions of theunsteady-state heat conduction equation forshort times in a plane having a uniform initialtemperature and subjected to a Drichletboundary condition. The resulting solutions aresuitable for obtaining quality short-timetemperature distributions within the slab whenit is subjected to a Dirichlet boundary condition,

or a Robin boundary condition for which theconvective heat transfer coefficient is verylarge and/or the thermal conductivity of the slabmaterial is very small.

In an application type study, Saha andAcharya (2003) conducted a detailedparametric analysis of the unsteady three-dimensional flow and heat transfer in a pin-finheat exchanger. The work was motivated bythe desire to enhance the perform-of compactheat exchangers, which are designed toprovide high heat transfer surface area per unitvolume and to alter the fluid dynamics toenhance mixing. There have been severalnumerical studies of transient fins combinedwith complicating factors, such as naturalconvection (Hsu and Chen, 1991; andBenmadda and Lacroix, 1996), spatial arraysof fins (Tafti et al., 1999; and Saha andAcharya, 2004) and phase change materials(Tutar and Akkoca, 2004).

There are few publications on transientexperiments for determining heat transfercoefficients in fins. Mutlu and Al-Shemmeri(1993) studied a longitudinal array of straightfins suddenly heated at the base. Theinstantaneous heat transfer coefficient wasfound at one point on the fin as a ratio of themeasured temperature to the measure heatflux. There are several papers on inversetechnique for determination of heat transfercoefficients from temperatures measured incompact bodies suddenly placed in aConvection environment (Stolz, 1960; andOsman and Beck, 1990). In these studies, theheat transfer coefficient is found from asystematic comparison between the transientdata and a mathematical model of the heatconduction in the body of interest.

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EXACT UNSTEADYSOLUTIONS OF FINS FORTHREE DIFFERENT TIPCONDITIONSConsider a straight fin initially in equilibriumwith the surrounding fluid environment attemperature T

e. The fin has a constant cross-

sectional area, but may be of any shape (pin,rectangular, etc.). For time t > 0 a steady heatflux is applied to the base of the fin. Thetemperature in the fin satisfies the followingequations.

x

TTTm

x

Te

12

2

2

; 0<x<L ...(1)

At t = 0, T(x, 0) – Te = 0 ...(2)

At x = 0, 0qx

Tk

...(3)

At x = L, 022

eTThx

Tk ...(4)

Quantity m is the fin parameter given by

kV

hAm h . The boundary condition at x = L is a

general condition that represents one of threedifferent tip conditions for the fin. For a tipcondition of the first kind, setting k

2 = 0 and h

2

= 1 represents a specified end temperature(at T = T

e). For a tip condition of the second

kind, setting k2 = k and h

2 = 0 represents an

insulated end condition. For a tip condition ofthe third kind, setting k

2 = k represents

convection at x = L. Usually the convectivecoefficient at the end of the fin is taken as sameas that along the sides of the fin (i.e., h

2 = h in

general). Here, the results will be written outfor three tip conditions. For the temperature-end condition (first kind),

1 2220 /cos

2,n

n

ne

Lm

Lx

k

LqTtxT

2222 /exp1 LtLm n ...(5)

Where n = (n – 1/2);

For the insulated end condition (secondkind),

1 222

022

0 /cos2

1,

2

nn

n

tm

eLm

Lx

k

Lq

Lm

e

k

LqTtxT

2222 /exp1 LtLm n ...(6)

Where n = n and for convective end

condition.

1 222

222

2

22

20 /cos

2,n

n

n

n

ne

Lm

Lx

BB

B

k

LqTtxT

2222 /exp1 LtLm n ...(7)

Where satisfies ntan

n = B

2

And where B2 = h

2L/k.

Each solution contains a series that shouldbe considered in two parts: a transient partwith an exponential factor and a steady partwith no exponential factor. Each of the transientseries contains an exponential factor with

argument 222nLm t/L2, which defines the

rate of decay of the unsteady. The decay ratedepends on fin effects (through m2L2) and also

on the tip condition (through 2n ).

The series solution for the unsteadytemperature in a flux-base fin is developed bythe method of Green’s functions.

First, a transformation (Ozisik, 1993) is usedto remove the fin term from the heat conductionequation. Let

tme WeTT 2 ...(8)

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and then transform Equations (1)-(4) to give:

t

w

x

w

1

2

2

; 0 < x < L ...(9)

At t = 0, W(x, 0) = 0 ...(10)

At x = 0, tmeqx

Wk 2

0

...(11)

At x = L, 022

Whx

wk ...(12)

This transformed problem may be solvedby the method of Green’s functions in the form(Beck et al., 1992, p. 165).

tdtxtxGeqk

txWt

t

tm

,0|,,0

0

2...(13)

The Green’s function associated withfunction W is that for a plane wall, given by Cole(2008).

0

00 00,|,

N

xxtxtxG

1

/ 22

n

Ltt

nn

nn neN

xxxx

...(14)

The first term (for n = 0) is needed only for atype 2 (insulated) boundary at x = L. Eigenfunctions X

n, Eigen values

n, and norms N

n

are determined by the boundary conditions onthe fin. For the flux-base fins of interest here,the Eigen functions are:

L

xX n

nn cos ...(15)

And the eigen values and norms are givenin Table 1. The number system in Table 1 forthe three cases listed is X2J where J = 1, 2, or3 to represent tip conditions of the first kind(temperature), second kind (insulated) or thirdkind (convection), respectively. After the timeintegral in Equation (13) is evaluated, the

transformation in Equation (8) can be reversedto find temperature T in the form:

1 222

022

0

0 /cos1,

2

nn

n

tm

eLm

Lx

k

Lq

Lm

e

N

L

k

LqTtxT

2222 /exp1 LtLm n ...(16)

Again, the first term is only used when thefin tip is insulated (see Kraus et al., 2001,p. 765) for an independent derivation of theinsulated-tip case). The above expression,with the eigen values and norms are given inTable 1, is limited to fins with a specified heatflux at the base (x = 0). However, the sameapproach could be used for fins with otherbase conditions with the appropriate planewall Green’s function. The plane-wall Green’sfunctions for the temperature-base fin (type 1boundary at x = 0) and the fin with the basetemperature applied through a contactconductance (type 3 boundary at x = 0) areavailable elsewhere (see Beck et al., 1992).

Table 1: Eigen Values for Three DifferentTip Conditions

CasenNL

n or Eigen Condition

X21 2 21n

X22 2; n 0 n

1; n = 0

X23

222

2

22

22

BB

B

n

n

ntan(

n) = B

2

Improvement of SeriesConvergence

It has long been known that classic solutionsfor the temperature in a body heated on aboundary contain a slowly converging steady-state series (Ozisik, 1993). In this section, the

275


convergence of the transient solution isimproved by replacing the steady series by afully summed form. Although the steady-finsolutions are well known, a unified solution ispresented with the method of Green’sfunctions.

The steady temperature satisfies thefollowing equations.

02

2

eh TT

kV

hA

x

T; 0 < x < L ...(17)

At x = 0, 0qx

Tk

...(18)

At x = L, 022

eTThx

Tk ...(19)

Again, the boundary condition at x = Lrepresents three kinds of tip conditions. Usingthe method of Green’s functions, the steady-fin temperature has the form Cole (2004).

0,20 xxG

k

qTxT JXe ...(20)

The symbol for Green’s function GX2J

denotes a Cartesian coordinate systemsymbol X, boundary of the second kind at x =0 (symbol 2), and boundary of type J at x = L(symbol J) for J = 1, 2, or 3. This numberingsystem is used to catalog the many GFavailable on the Library web site(www.greensfunction.unl.edu). Table belowshows the eigen values for three different tipconditions.

Green’s function GX2J

for the steady-fin isgiven by Cole (2008).

D

eeRxxG

xxLmxxLm

JX

2||2

2 ,

Dee xxmxxm || ...(21)

Where D = 2m (1 – R.e–2mL)

Coefficient R is determined by the tipcondition:

LxattypeBmL

BmLLxattype

Lxattype

R

3

21

11

2

2

Where B2 = h

2L/k.

The above GF may be evaluated at x’ = 0and substituted into

The above temperature expression,Equation (20), to give

mL

mxxLm

eeRmL

eeR

k

LqTxT

2

20

2

...(22)

Where coefficient R is given in the previouspage.

Alternately, steady-fin solutions may beobtained from computer program TFINdescribed previously (Cole, 2004) thatproduce analytical expressions for the steadytemperature in fins under a variety of boundaryconditions. Program TFIN is also available fordownload at the Green’s Function Library(Cole, 2008).

Next the closed-form steady solutions givenabove are used in the transient-fin solutionsgiven earlier to replace the slowly convergingseries by replacing the series steady term withnon series steady term as obtained in theEquation (22). The improved-convergenceform of the transient temperature in flux-basefins are given by:

For the temperature tip condition (firstkind),

mL

xLmmx

eemL

ee

k

LqTtxT

2

20

1,

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1 2220 /cos

2n

n

n

Lm

Lx

k

Lq

2222 /exp LtLm n ...(23)

Where n = (n – 1/2)

For the insulated end condition (secondkind),

22

0

2

20

2

1,

Lm

e

k

Lq

emL

ee

k

LqTtxT

tm

mL

mxxLm

e

1 2220 /cos

2n

n

n

Lm

Lx

k

Lq

2222 /exp LtLm n ...(24)

Where n = n and for convective tip

condition (third kind)

mL

mxxLm

e

eBmLBmL

mL

eeBmLBmL

k

LqTtxT

2

2

2

2

2

2

0

1

,

222

2

22

2

1222

0 /cos2

BB

B

Lm

Lx

k

Lq

n

n

n n

n

2222 /exp LtLm n ...(25)

Where satisfies ntan

n = B

2 and where

B2 = h

2L/k

It is instructive to examine these threetemperature solutions as a group. Eachcontains a steady term and each contains anadditional term, a non-series transient.However, the insulated-tip solution uniquelycontains another term, a non-series transient.

Quasi Steady Solution for ShortFins

The short fin is of interest for our particularapplication. The exact temperature expression

for this case contains two terms: a seriessteady term and a series transient term. Theseries contains an exponential factor withargument m2L2 +

n2. By comparing these

arguments, it is clear that as time increasesthe series term will decay more rapidly. Thissuggests that a quasi-steady solution may beconstructed of the form

Tq(x, t) = Ts(x) + TL(t)

Here Ts is the steady solution term and TL

is the transient solution term. Both the termcontain series term. In quasi steady approach,series steady term is being transformed in toa non series term which transforms theexpression in to an easily computed algebraicexpression. Based on the above discussionof exponential arguments, the quasi-steadysolution should be accurate for later time. Thenumerical results given in the next section arepresented with the following dimensionlessvariables:

2

0// LtdtLxX

kLqTT e

k

Avh

Bi

AvL

BiM h

h

where = Dimensionless temperature

= Dimensionless location

= Dimensionless time

M = Fin parameter

Bi = Biot number

With these parameters, the dimensionlessquasi-steady temperature for short fin is givenby putting the above values in the Equation(25):

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conditions under which the quasi-steadysolution is accurate. Different dimensionlessparameter were considered (M, andlocation) for obtaining the temperaturedistribution curves. While variation of oneparameter was considered the other variableswere kept constant as indicated in the graphs.Based on the analysis, quasi-steady solutionhas been proposed as an accurate solution atlarge dimensionless times, which isindependent of geometry of the problem.

Accuracy of Quasi Steady Solutionfor M = 1 at DimensionlessLocations 0.0, 0.5 and 1.0

Figures 1, 2 and 3 shows the (dimensionless)temperature versus time at three differentpositions on the fin, all for M = 1.0. For allvalues of dimensionless time the quasi-steadytheory estimates the exact values at x/L = 0and x/L = 1.0. For all locations the agreementimproves as time increases.

M

MMM

eBMBMM

BMeeBMe2

22

222

2

22

2

1 22

22 cos

tantan1

tan12

nn

nnn

nn

M

22exp nM ...(26)

And the dimensionless exact f intemperature for short fin is given by:

1 22

22

tantan1

tan12

nnnn

nn

22exp1 nM ...(27)

Where n satisfies

ntan

n = B

2 with

dimension less parameter here n = n

RESULTS AND DISCUSSIONAccuracy of Quasi Steady Solution

The quasi-steady solution is compared withthe exact transient solution to determine the

Figure 1: Temperature History in a Fin of Constant Cross Section for Both Quasi SteadyTheory and Exact Theory for M = 1 at Location x/L = X = 0.0

278



• Figures 1, 2 and 3 shows the(dimensionless) temperature versus time

at three different positions on the fin, all forM = 1.


279


• The quasi-steady theory estimates the exactvalues at x/L = 0.0, 0.5 and 1.0.

• For all locations the agreement improvesas time increases.

Accuracy of Quasi Steady SolutionFor = X = 0 at Fin ParameterM = 0.2, 1.0 and 5.0

Figures 4, 5 and 6 shows temperature versustime at X = 0 for M = 0.2, 1.0 and 5.0. At M =5.0 the fin transient ends quickly so that this finreaches steady-state at about = 0.1. As Mdecreases the temperature distribution takeslonger and longer to reach steady-state. Finparameter M may be interpreted as a ratio ofthermal resistances. Specifically, M2 is thethermal resistance along the fin length dividedby the convective thermal resistance from thesurface of the fin. Thus when M is small, theconvective thermal resistance from the surfaceof the fin is large compared to the thermal

resistance along the fin, producing a long, slowtransient.

Specific values of the dimensionlesstemperature in the quasi-steady theory forseveral values of dimensionless time andseveral values of fin parameter M, all atdifferent values of x/L (dimensionless location).Dimensionless temperature calculated is thenincorporated and analyzed with help of graphsfor both the theories. Followings are the resultsincorporated from the graphs.

• Figures 4, 5 and 6 shows temperatureversus time at X = 0 for M = 0.2, 1.0 and5.0.

• At M = 5 the fin transient ends quickly sothat this fin reaches steady-state at about = 0.1.

• As M decreases the temperaturedistribution takes longer and longer to reachsteady-state.

Figure 4: Temperature History in a Fin of Constant Cross-Section for Both Quasi SteadyTheory and Exact Fin Theory at Location x/L = X = 0 for M = 0.2

280




• Fin parameter M may be interpreted as aratio of thermal resistances.

• Thus when M is small, the convectivethermal resistance from the surface of the

281


fin is large compared to the thermalresistance along the fin, producing a long,slow unsteady state.

CONCLUSIONA unified theory has been presented forunsteady heat transfer in flux-base fins for threetip conditions. The method may be easilyextended to fins with other base conditions. Aquasi steady theory has been applied to a caseof straight fin with short length tip in the form ofa simple, non-series expression for steadyterm.

• The quasi-steady theory is simple andefficient for computing numerical valuescompared to the exact series solution.

• A comparison with an exact series solutionfor the unsteady condition fin shows that thequasi-steady theory is accurate withindimensionless times for all values of the finparameter M.

• The results show that the quasi-steady finmodel is a simple way to find heat transfercoefficients for larger dimensionless times.

• Complicated exact unsteady solutions canbe simplified for temperature distributionanalysis through Green’s function method.

• A unified theory is obtained for unsteadyheat transfer in flux-base fins for three tipconditions.

Based on the analysis, the followingconclusions have been obtained.

• For M > 1 the accurate range extends to alldimensionless times except = 0.

• The accuracy increases for largedimensionless times, where the sensitivityto heat transfer coefficient is largest.

• For M = 0.2, 1.0 and 5.0. For = 0 the quasi-steady theory overestimates the exactvalues at = X = 0 and estimates the exactvalues at = X = 1.0. For all locations theagreement improves as time increases.

• At M = 5 and = X = 0 the fin transient endsquickly so that this fin reaches steady-stateat about = 0.2. As M decreases, thetemperature distribution takes longer andlonger to reach steady-state.

• When M is small, the convective thermalresistance of the fin surface is largecompared to the thermal resistance alongthe fin, producing a long, slow transient.

• For M 1 and M > 1. There is no error fordimensionless time > 0. the region of smallerror extends to earlier time.

• The quasi-steady and exact temperaturesagrees closely except at early time ( = 0).

SCOPE FOR FUTURE WORK• It is suggested that the quasi-steady

approach could be successfully applied toother fin geometries with different tipconditions or other fins for which exactsolutions are difficult to be obtained.

• The results show that the quasi-steady finapproach can be a simple way to find heattransfer coefficient associated with heatloss.

• The heat transfer coefficients obtained bythis method are intended for future use asan external boundary condition for moreelaborate thermal models.

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Greek

Ah

Surface area of fin for convection (m2) Thermal diffusivity (m2s–1)

Bi Biot number, hi (V/A

h)/K

nEigen value [Equation (14)]

B2

Biot number, hL/k Dimensionless temperature

G Green’s function , X Dimensionless x-coordinate

h Heat transfer coefficient (W m–2K–1) , dt Dimensionless time

k Thermal conductivity (W m–1K–1) Superscripts

L Length of fin (m) q Quasi-steady

Nn

Norm [Equation (14)] (m) s Steady state

m Fin parameter, (m–1)

M Dimensionless fin parameter = mL

q0

Heat flux (W m–2)

Q Input heat (W)

T Temperature (K)

t Time (s)

V Fin volume (m3)

W Transformed temperature [Equation (13)]

APPENDIX

Nomenclatures

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