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Thermally Developing Forced Convection of Non-Newtonian Fluids Inside Elliptical DuctsCASSIO ROBERTO MACEDO MAIA a , JOÃO BATISTA APARECIDO a & LUIZ FERNANDO MILANEZ ba Department of Mechanical Engineering, University of São Paulo State, Ilha Solteira, Brazilb Department of Mechanical Engineering, State University of Campinas, BrazilPublished online: 17 Aug 2010.

To cite this article: CASSIO ROBERTO MACEDO MAIA , JOÃO BATISTA APARECIDO & LUIZ FERNANDO MILANEZ (2004): ThermallyDeveloping Forced Convection of Non-Newtonian Fluids Inside Elliptical Ducts, Heat Transfer Engineering, 25:7, 13-22

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Heat Transfer Engineering, 25(7):13–22, 2004Copyright C©© Taylor & Francis Inc.ISSN: 0145-7632 print / 1521-0537 onlineDOI: 10.1080/01457630490495805

Thermally DevelopingForced Convection ofNon-Newtonian FluidsInside Elliptical Ducts

CASSIO ROBERTO MACEDO MAIA and JOAO BATISTA APARECIDODepartment of Mechanical Engineering, University of Sao Paulo State, Ilha Solteira, Brazil

LUIZ FERNANDO MILANEZDepartment of Mechanical Engineering, State University of Campinas, Brazil

Laminar-forced convection inside tubes of various cross-section shapes is of interest in the design ofa low Reynolds number heat exchanger apparatus. Heat transfer to thermally developing,hydrodynamically developed forced convection inside tubes of simple geometries such as a circulartube, parallel plate, or annular duct has been well studied in the literature and documented invarious books, but for elliptical duct there are not much work done. The main assumptions used inthis work are a non-Newtonian fluid, laminar flow, constant physical properties, and negligible axialheat diffusion (high Peclet number). Most of the previous research in elliptical ducts deal mainlywith aspects of fully developed laminar flow forced convection, such as velocity profile, maximumvelocity, pressure drop, and heat transfer quantities. In this work, we examine heat transfer in ahydrodynamically developed, thermally developing laminar forced convection flow of fluid inside anelliptical tube under a second kind of a boundary condition. To solve the thermally developingproblem, we use the generalized integral transform technique (GITT), also known asSturm-Liouville transform. Actually, such an integral transform is a generalization of the finiteFourier transform, where the sine and cosine functions are replaced by more general sets oforthogonal functions. The axes are algebraically transformed from the Cartesian coordinate systemto the elliptical coordinate system in order to avoid the irregular shape of the elliptical duct wall.The GITT is then applied to transform and solve the problem and to obtain the once unknowntemperature field. Afterward, it is possible to compute and present the quantities of practicalinterest, such as the bulk fluid temperature, the local Nusselt number, and the average Nusseltnumber for various cross-section aspect ratios.

INTRODUCTION

Problems involving non-Newtonian fluids constitutea field of interest in mechanical engineering of great eco-

Address correspondence to Dr. Joao Batista Aparecido, Department ofMechanical Engineering, University of Sao Paulo State, Avenida Brasil56, Ilha Solteira—SP—15.385-000, Brazil. E-mail: jbaparecido@dem.feis.unesp.br

nomic importance. Researches of rheologic nature arenecessary for the accurate knowledge of deformationrelations as well as for the understanding of the wholedynamics related to this class of fluids. Therefore, thedevelopment of techniques for the solution of problemsin this area is relevant, especially those that permit toobtain parameters of interest in the design of thermohy-draulic equipment. The contribution of the present work

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relates to the calculation of heat transfer coefficients forthe flow of one class of these fluids inside ducts of anelliptical section.

The several methods and techniques of solution re-lated to convection problems are mostly concerned withthe flow of Newtonian fluids and are usually more ac-curate and powerful when applied to the simplest prob-lems, where the results are already known. On the otherhand, analytical solutions, and even solutions of a hybridnature (analytical-numerical), are scarcer for complexcases involving the flow in ducts of irregular geome-tries submitted to variable boundary conditions or forsituations where the fluid properties exhibit some de-pendence with, for example, temperature [1]. Solutionsfor problems that deal with the flow of non-Newtonianfluids are even scarcer.

It is worth mentioning, however, that the GeneralizedIntegral Transform Technique (GITT [2]) has been con-sistently developed for the solution of complex diffusiveand diffusive-convective problems that cannot be solvedby the techniques of the integral transform or separationof variables. Among the problems solved successfullyby means of the GITT are the flow in ducts of irregularshape [3–5] with time varying coefficients and boundaryconditions [6,7]; problems involving space dependencefor the heat transfer coefficients and boundary condi-tions [8]; problems with thermally and hydrodynami-cally developing flow [9]; diffusive problems with mov-ing boundaries [10]; and problems of non-Newtonianfluid flows [11–13].

Continuing the development of this line of research,the calculation of heat transfer parameters for the flowof non-Newtonian fluids are presented here, in problemsof thermally developing flow inside ducts of ellipticalsection, subjected to the second kind of boundary con-dition. In the present work, only fluids that follow thepower law are considered because they constitute thegreat majority of the known non-Newtonian fluids. Inparticular, a convenient change of variables for the solu-tion of this problem was used to determine the velocityprofile inside the elliptical duct and also to transform theelliptical profile into a new geometry in order to sim-plify the application of the boundary conditions. Even-tually, the generalized integral transform technique wasapplied to the energy equation corresponding to thisproblem, thus enabling to obtain the temperature fieldin the flow and, consequently, the computation of theheat transfer parameter of interest.

ANALYSIS

For the formulation of the present problem, hydrody-namically developed and thermally developing flow isassumed, with a uniform temperature profile at the en-

Figure 1 Geometry of the problem.

try and in steady state. The viscous dissipation and theaxial conduction are neglected, and the fluid propertiesare assumed to be constant in the whole domain. Thus,the energy equation, according to the coordinate systempresented in Figure 1, is given by

ρcpV (x, y)∂T (x, y, z)

∂z= k

[∂2T (x, y, z)

∂x2

+ ∂2T (x, y, z)

∂y2

]; {(x, y) ∈ �; z > 0}, (1)

where ρ is the fluid density, cp is the specific heat, k is thethermal conductivity, and T (x, y, z) is the temperaturefield. The velocity profile V (x, y) for fully developedflows of fluids that follow the power law in ducts ofelliptical section is given by [14]

V (x, y) =[

3n + 1

n + 1

]{1 −

[x2

a2+ y2

b2

] n+12n

}Vav, (2)

where n is the fluid behavior index and Vav is the flowaverage velocity.

In the present work, it is considered constant heat fluxq ′′

o at the contour �,

−k∂T (x, y, z)

∂η= q ′′

o ; {(x, y) ∈ �; z > 0; η ⊥ �}. (3)

The others boundary conditions and the condition atthe entrance of the problem can be written as:

∂T (x, y, z)

∂x= 0; {x = 0, 0 < y < b, z > 0}, (4)

∂T (x, y, z)

∂y= 0; {0 < x < a, y = 0, z > 0}, (5)

T (x, y, 0) = To; {(x, y) ∈ �}. (6)

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Transformation of Coordinates

The temperature potential and other physical andgeometrical parameters were written in dimensionlessform as follows:

θ(X, Y, Z ) = To − T (x, y, z)

q ′′o Dh/k

, (7)

X = x/Dh ; Y = y/Dh ; Z = z/(DhPe);

η∗ = η/Dh, (8)

α = a/Dh ; β = b/Dh ; Dh = 4Asc/P;

r = b/a, (9)

Asc = πab;

P = 4a∫ π/2

0

√1 − (1 − r2)sen2 θ dθ;

A∗sc = Asc/D2

h ; P∗ = P/Dh, (10)

U (X, Y ) = V (x, y)/Vav, (11)

where Pe is the Peclet number, Asc is the area of theelliptical section, P is the perimeter of the ellipticalcontour, and r is the aspect ratio of the ellipsis. Withthese new variables, the energy equation is rewritten as

U (X, Y )∂θ(X, Y, Z )

∂Z= ∂2θ(X, Y, Z )

∂X2+ ∂2θ(X, Y, Z )

∂Y 2.

(12)

The dimensionless velocity profile defined byEq. (11) can be written as

U (X, Y ) = 3n + 1

n + 1

{1 −

[X2

α2+ Y 2

β2

] n+12n

}, (13)

and the entry and boundary conditions

θ(X, Y, 0) = 0; {(X, Y ) ∈ �}, (14)

∂θ(X, Y, Z )

∂η∗ = 1; {(X, Y ) ∈ �, Z > 0}, (15)

∂θ(X, Y, Z )

∂X= 0; {X = 0, 0 < Y < β, Z > 0}, (16)

∂θ(X, Y, Z )

∂Y= 0; {0 < X < α, Y = 0, Z > 0}. (17)

The orthogonal elliptical coordinate system (u, v) isutilized to transform the original domain, with an ellip-tical contour in the plane (X, Y ) in one domain and arectangular contour in the transformed plane (u, v):

X = α∗ cosh(u) cos(v); Y = α∗ sinh(u) sin(v), (18)

α∗ = α/ cosh(uo);

uo = arctanh(β/α); {0 < β < α}. (19)

The metric coefficients hu(u, v) and hv(u, v) and theJacobian J (u, v) of the transformation of the system ofcoordinates (X, Y ) to the system (u, v) are given by:

hu(u, v) = hv(u, v) = h(u, v)

= α∗√

sinh2(u) + sin2(v), (20)

J (u, v) = ∂(X, Y )

∂(u, v)= α∗2[ sinh2(u) + sin2(v)]. (21)

With these new variables, the energy equation can bewritten as:

H (u, v)∂θ(u, v, Z )

∂Z

= ∂2θ(u, v, Z )

∂u2+ ∂2θ(u, v, Z )

∂v2, (22)

H (u, v) = J (u, v)U (u, v), (23)

U (u, v) = 3n + 1

n + 1

{1 −

[α∗2

α2cosh2(u) cos2(v)

+ α∗2

β2 sinh2(u) sin2(v)

] n+12n

}(24)

The entry and boundary conditions in the new coor-dinate system are

θ(u, v, 0) = 0; {(u, v) ∈ �}, (25)

∂θ(u, v, Z )

∂u= h(u, v);

{u = uo, 0 < v < π/2, Z > 0}, (26)

∂θ(u, v, Z )

∂u= 0; {u = 0, 0 < v < π/2, Z > 0}, (27)

∂θ(u, v, Z )

∂v= 0; {0 < u < uo, v = 0, π/2; Z > 0}.

(28)

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In order to apply the GITT, it is convenient to carryout the homogenization of the boundary conditions.To accomplish this, consider the following change ofvariables

θ(u, v, Z ) = θ∗(u, v, Z ) + u2

2uoh(uo, v). (29)

In this new variable, the energy equation transformsinto

H (u, v)∂θ∗(u, v, Z )

∂Z

= ∂2θ∗(u, v, Z )

∂u2+ ∂2θ∗(u, v, Z )

∂v2+ G(u, v), (30)

G(u, v) = h(uo, v)

uo+ α∗u2

2h(uo, v)uo

{cos(2v)

−[

α∗

2h(uo, v)sin2(2v)

]2}. (31)

The entry and boundary conditions are redefined as:

θ∗(u, v, 0) = − u2

2uoh(uo, v); {(u, v) ∈ �}, (32)

∂θ∗(u, v, Z )

∂u= 0; {u = 0, 0 < v < π/2, Z > 0}, (33)

∂θ∗(u, v, Z )

∂u= 0; {u = uo, 0 < v < π/2, Z > 0}, (34)

∂θ∗(u, v, Z )

∂v= 0; {0 < u < uo; v = 0, π/2; Z > 0}.

(35)

Application of the GITT

To solve the energy equation, Eq. (30), submitted tothe conditions given by Eqs. (33–35), the GeneralizedIntegral Transformed Technique is applied. In order toaccomplish this, consider the following auxiliary eigen-value problem for the variable v [15],

d2ψ(v)

dv2+ µ2ψ(v) = 0; {0 ≤ v ≤ π/2}, (36)

dψ(v)

dv= 0; {v = 0}, (37)

dψ(v)

dv= 0; {v = π/2}. (38)

The eigenvalues and eigenfunctions associated to thisproblem are

µi = 2(i − 1); i = 1, 2, 3 . . . , (39)

ψi (v) = cos(µiv). (40)

The above eigenfunctions exhibit the property of or-thogonality, which permit the development of the fol-lowing transform-inverse pair:

θ∗i (u, Z ) =

∫ π/2

0Ki (v)θ∗ (u, v, Z ) dv, (41)

θ∗(u, v, Z ) =∞∑

i=1

Ki (v) θ∗i (u, Z ), (42)

where Ki (v) are the normalized eigenfunctions givenby

Ki (v) = ψi (v)

N 1/2i

, (43)

Ni =∫ π/2

0ψ2

i (v) dv ={

π/2, i = 1π/4, i > 1

. (44)

From the internal product of Ki (v) and θ∗(u, v, Z )with Eqs. (30) and (36), respectively, and by making useof the boundary conditions given by Eqs. (35), (37), and(38), the following coupled system of partial differentialequations of second order can be obtained:

∞∑j=1

Aij(u)∂θ

∗j (u, Z )

∂Z+ µ2

i θ∗i (u, Z )

= ∂2θ∗i (u, Z )

∂u2+ Ci (u), (45)

Ai j (u) =∫ π/2

0Ki (v)K j (v)H (u, v)dv, (46)

Ci (u) =∫ π/2

0Ki (v)G(u, v)dv. (47)

Let us now consider the following eigenvalue prob-lem relative to variable u:

d2φ(u)

du2+ λ2φ(u) = 0; {0 ≤ u ≤ uo} , (48)

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dφ(u)

du= 0; {u = 0}, (49)

dφ(u)

du= 0; {u = uo}. (50)

The eigenvalues and the eigenfunctions for this newproblem are:

λm = (m − 1)π/uo, (51)

φm(u) = cos(λmu). (52)

The above eigenfunctions are orthogonal and permitthe development of the transform-inverse pair

¯θ∗im(Z ) = t

∫ uo

0

∫ π/2

0Ki (v) Zm(u) θ∗(u, v, Z ) dv du,

(53)

θ∗(u, v, Z ) =∞∑

i=1

∞∑m=1

Ki (u)Zm(u) ¯θ∗im(Z ), (54)

where Zm(u) are the normalized eigenfunctions,

Zm(u) = φm(u)

M1/2m

, (55)

Mm =∫ uo

0φ2

m(u) du ={

uo, m = 1uo2 , m > 1

. (56)

For the computation of the transformed potential¯θ∗im(Z ), the procedure is similar to that used to transform

the energy equation relatively to the axis v. By applyingthe internal product of Zm(u) and also of θ

∗i (u, Z ) to

Eqs. (45) and (48), respectively, and using the boundaryconditions, Eqs. (33), (34), (49), and (50), the follow-ing coupled system of ordinary differential equations offirst order is obtained:

∞∑n=1

∞∑j=1

Bijnm

d ¯θ∗jn(Z )

d Z+ (

µ2i + λ2

m

)

× ¯θ∗im(Z ) + Dim = 0, (57)

Bijnm =∫ uo

0Zm(u)Zn(u)A(u)du

=∫ uo

0

∫ π/2

0Ki (v)K j (v)Zm(u)Zn(u)H (u, v)dvdu,

(58)

Dim = −∫ uo

0Zm(u)Ci (u) du

= −∫ uo

0

∫ π/2

0Ki (v)Zm(u)G(u, v)dvdu. (59)

Parameters Bijmn and Dim can be integrated and there-fore known. The solution of this system of equations,when submitted to the transformed entry condition,

¯θ∗im(0) =

∫ uo

0

∫ π/2

0Ki (v) Zm(u) θ∗(u, v, 0) dvdu

= −∫ uo

0

∫ π/2

0Ki (v) Zm (u)

u2

2uoh(uo, v) dvdu,

(60)

allows the transformed potential ¯θ∗im(Z ) to be obtained.

Therefore, the temperature potential θ(u, v, Z ) can becalculated numerically by Eq. (57), truncating the ex-pansion for a given order i = N e m = M :

θ(u, v, Z ) =N∑

i=1

M∑m=1

Ki (v) Zm(u) ¯θ∗im(Z )

+ u2

2uoh(uo, v). (61)

N∑n=1

M∑j=1

Bijnm

d ¯θ∗jn(Z )

d Z+ (µ2

i + λ2m)

× ¯θ∗im(Z ) + Dim = 0, (62)

Obviously, the higher N and M , the greater the ac-curacy of the results.

Calculation of the Bulk Temperature and of theNusselt Number

The average temperature Tav of the fluid at a givenduct section can be determined by means of an energybalance:

q ′′o Pz = ρVav Acp(Tav − To). (63)

Using the dimensionless variables defined by Eqs.(7–11), the average temperature in its dimensionlessform is determined by

θav(Z ) = 4Z . (64)

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Table 1 Average wall temperature for aspect ratio b/a = 0.5

Z n = 0.1 n = 0.15 n = 0.2 n = 0.5 n = 1.0 n = 1.5 n = 2.0 n = 5.0

0.00010 0.02550 0.02790 0.02958 0.03430 0.03683 0.03788 0.03845 0.039580.00020 0.03275 0.03573 0.03783 0.04374 0.04693 0.04825 0.04897 0.050400.00050 0.04591 0.04987 0.05267 0.06063 0.06496 0.06676 0.06774 0.069690.00100 0.05973 0.06461 0.06809 0.07805 0.08351 0.08578 0.08703 0.089490.00200 0.07844 0.08441 0.08870 0.10116 0.10805 0.11092 0.11250 0.115630.00500 0.11471 0.12237 0.12799 0.14465 0.15404 0.15798 0.16016 0.164480.01000 0.15612 0.16524 0.17206 0.19273 0.20461 0.20964 0.21242 0.217980.02000 0.21801 0.22872 0.23689 0.26235 0.27736 0.28377 0.28733 0.294470.05000 0.36141 0.37419 0.38417 0.41648 0.43624 0.44483 0.44964 0.459330.10000 0.57223 0.58607 0.59698 0.63290 0.65535 0.66521 0.67075 0.681990.20000 0.97675 0.99088 1.00205 1.03904 1.06233 1.07260 1.07840 1.090170.50000 2.17733 2.19147 2.20265 2.23968 2.26301 2.27331 2.27912 2.290941.00000 4.17733 4.19147 4.20265 4.23968 4.26301 4.27331 4.27912 4.29094

Table 2 Local Nusselt number for aspect ratio b/a = 0.5

Z n = 0.1 n = 0.15 n = 0.2 n = 0.5 n = 1.0 n = 1.5 n = 2.0 n = 5.0

0.00010 39.833 36.357 34.265 29.500 27.447 26.681 26.279 25.5210.00020 31.299 28.625 27.005 23.288 21.677 21.074 20.758 20.1610.00050 22.776 20.891 19.736 17.056 15.882 15.442 15.211 14.7740.00100 17.943 16.499 15.604 13.504 12.577 12.228 12.045 11.6970.00200 14.196 13.087 12.391 10.735 9.9953 9.7163 9.5694 9.29090.00500 10.559 9.7687 9.2603 8.0226 7.4606 7.2474 7.1349 6.92120.01000 8.6119 7.9846 7.5725 6.5475 6.0749 5.8948 5.7996 5.61870.02000 7.2456 6.7238 6.3739 5.4840 5.0670 4.9075 4.8232 4.66270.05000 6.1955 5.7408 5.4296 4.6194 4.2329 4.0844 4.0058 3.85610.10000 5.8060 5.3742 5.0766 4.2936 3.9163 3.7707 3.6934 3.54620.20000 5.6578 5.2388 4.9491 4.1833 3.8120 3.6683 3.5920 3.44620.50000 5.6391 5.2227 4.9347 4.1723 3.8022 3.6588 3.5827 3.43711.00000 5.6390 5.2227 4.9347 4.1723 3.8022 3.6588 3.5827 3.4371

Table 3 Thermal entry length as a function of the aspect ratio and of the behavior index

Z n = 0.1 n = 0.15 n = 0.2 n = 0.5 n = 1.0 n = 1.5 n = 2.0 n = 5.0

0.125 1.35 1.32 1.30 1.24 1.21 1.20 1.19 1.180.250 0.31 0.30 0.29 0.29 0.29 0.29 0.29 0.290.500 0.079 0.079 0.080 0.081 0.081 0.084 0.084 0.0850.800 0.042 0.042 0.043 0.044 0.047 0.047 0.048 0.0490.900 0.040 0.040 0.040 0.042 0.044 0.045 0.046 0.0470.990 0.039 0.039 0.039 0.041 0.043 0.044 0.045 0.046

Table 4 Limit Nusselt number as a function of the aspect ratio and of the behavior index

Z n = 0.1 n = 0.15 n = 0.2 n = 0.5 n = 1.0 n = 1.5 n = 2.0 n = 5.0 Ref. [18]

0.125 1.8188 1.5894 1.4416 1.0921 0.9432 0.8891 0.8611 0.8091 0.94330.250 3.9066 3.5263 3.2709 2.6274 2.3331 2.2225 2.1645 2.0553 2.3330.500 5.6390 5.2227 4.9347 4.1723 3.8022 3.6588 3.5827 3.4371 3.8020.800 6.1570 5.7451 5.4578 4.6865 4.3052 4.1562 4.0766 3.9239 —0.900 6.2026 5.7913 5.5042 4.7326 4.3507 4.2012 4.1214 3.9682 —0.990 6.2173 5.8056 5.5183 4.7462 4.3639 4.2143 4.1344 3.9810 4.364

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However, the dimensionless average temperature canalso be determined as

θav(Z ) = D2h

Asc

∫�

θ(X, Y, Z ) U (X, Y ) d�. (65)

Thus, in the plane (u, v), θav(Z ) is given by

θav(Z ) = 1

A∗sc

∫ π/2

0

∫ uo

0

{[N∑

i=1

M∑m=1

Ki (v)Zm(u) ¯θ∗im(Z )

]

+ u2

2uoh(uo, v) H (u, v)

}du dv. (66)

Eqs. (64) and (66) are appropriate to verify the ac-curacy of the numerical results when the expansion istruncated in orders i = N and m = M because theyshould provide the same results when utilizing an infi-nite number of terms in the series expansion.

The average wall temperature is obtained by the in-tegration of the temperature profile at the wall

θw,av(Z ) = 4

P∗

∫ π/2

0

N∑i=1

M∑m=1

Ki (v) Zm (uo) ¯θ∗im

× (Z )h(uo, v) dv. (67)

The Nusselt number is defined by Nu(z) = h(z)Dhk ,

with h(z) written in the form

h(z) = q ′′o (z)

Tw,av(z) − Tav(z). (68)

Using the dimensionless variables

Nu(Z ) = 1

θw,av − θav

(69)

The average Nusselt number is obtained integratingnumerically Eq. (69),

Nuav(Z ) = 1

Z

∫ Z

0Nu(Z ′)d Z ′. (70)

The thermal entry length Lth is defined as the positionwhere the local Nusselt number is 5% higher than theNusselt number in the region where the flow is fullydeveloped [16]. Thus,

Lth ≡ positive root of {1.05 Nu(∞) − Nu(Z) = 0}.

(71)

RESULTS AND DISCUSSION

For the calculation of the transformed potential¯θ im(Z ), the expansion, given by Eq. 61, was truncatedfor several orders N and M . It was observed that theconvergence is relatively slower when the aspect ratiob/a → 1 and when the behavior index n → 0. Fortruncations after orders of M = 30, and N = 30, itwas verified that the values of the Nusselt numbers cal-culated for Z > 0.0001 converged around four digitsor more for all cases analyzed. Thus, the results in thiswork were obtained with the truncation of the expansionin M = 30 and N = 30.

The integration involved in the calculations of theperimeter, parameters Bijmn and Dim, and the aver-age temperature was performed numerically by theGauss Method (36 points). The system of differen-tial equations for the transformed potential, Eq. (62),was solved using the routine DIVPAG of the Li-brary IMSL [17]. The heat transfer parameters wereobtained for several ellipsis eccentricities (b/a =1/8, 1/4, 1/2, 8/10, 9/10, and 99/100) and behaviorindices (n = 0.1, 0.15, 0.2, 0.3, 0.5, 0.7, 1.0, 1.5, 2.0,0.3, and 5.0). For b = a, the cross-section geometry isthe circle, and the problem can be solved more easilyusing the cylindrical coordinate system as in Cotta andOzisik [18].

The difference between the average temperature ob-tained by Eq. (64) and the average temperature calcu-lated by Eq. (66) is less than 10−5 for Z > 0.0001, forall the cases analyzed. Tables 1 and 2 present resultsobtained for the average wall temperature and for theNusselt number as a function of the behavior index n,for flow in ducts with aspect ratio b/a = 0.5. The be-havior of the local Nusselt number in the flow is shownin Figure 2 as a function of the behavior index n for

Figure 2 Local Nusselt number as a function of the behavior in-dex for aspect ratio b/a = 0.5.

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Figure 3 Local Nusselt number as a function of the aspect ratiofor behavior index n = 0.5.

aspect ratio b/a = 0.5, and in Figure 3 as a function ofthe aspect ratio b/a for behavior index n = 0.5.

Results obtained for the Nusselt number in the regionwhere the flow is thermally fully developed can be seenin Table 3. It may be observed in Figure 4 that the limitNusselt number is more sensitive to the variation ofthe aspect ratio when b/a < 0.5, and to the variationof the behavior index n for the pseudoplastic fluids,n < 1. It can also be observed that the Nusselt numberis smaller for dilatant fluids, n > 1, when comparedto pseudoplastic fluids due to the fact that the apparentviscosity increases with n for the fluids that obey thepower law. Results were found in the literature only forthe particular case of flow of Newtonian fluids [19]. InTable 4, there is excellent agreement between the resultsfor this case.

Finally, Table 3 exhibits the results obtained for thethermal entry length Lth . It may be observed that thethermal development is less sensitive to the variationof the flow behavior index and strongly dependent on

Figure 4 Limit Nusselt number as a function of the aspect ratioand of the behavior index n.

the aspect ratio b/a. The thermal development dimin-ishes with the aspect ratio, and the thermal entry lengthincreases substantially when b/a → 0.

CONCLUSIONS

In the present work, the heat transfer parameters forlaminar flow of non-Newtonian fluids inside ducts ofsection elliptical section, subjected to a second kindof boundary condition, were determined. The problemconsidered here was the thermally developing flow offluids that follow the power law rheological model. Inorder to enable the application of the boundary con-ditions, a suitable change of variables was performedby transforming the real elliptical contour of the ductsection into a rectangular contour in the new coordinatesystem. As the energy conservation equation in this newsystem has inseparable variables, the GITT was appliedto obtain the temperature distribution in the flow. Theexpansion in eigenfunctions for the temperature poten-tial resulted in a relatively slow convergence, leading toa series with at least thirty terms for variables u and v,to obtain a satisfactory convergence. Parameters of in-terest, such as the average wall temperature; local, aver-age, and limit Nusselt number; and thermal entry length,were obtained for several aspect ratios b/a of the ellip-tical section and for several indices of behavior n of theflow. The results presented here exhibit a strong depen-dence of the heat transfer parameters with the aspectratio. The flow behavior index n has a marked influencein pseudo-plastic fluids in the region where n → 0. Theresults obtained showed an excellent agreement withthose found in the literature, for a Newtonian fluid onlyn = 1.

NOMENCLATURE

a, b major and minor ellipsis semi-axis, mAsc, A∗

sc cross section area, Eq. (10)Bijmn, Dim equation coefficients, Eqs. (58) and (59)cp specific heat, J/kg KDh hydraulic diameter, mh averaged heat transfer coefficient, W/m2 KH, G equation coefficients, Eqs. (23) and (31)h, hu, hv metric coefficients, Eq. (20)J Jacobian determinant, Eq. (21)k fluid conductivity, W/m KKi , Zm normalized eigenfunctions, Eqs. (43) and

(55)Lth thermal entry length, Eq. (71)n flow behavior indexNi , Mm normalization integrals, Eqs. (44) and

(56)

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Nu, Nuav local and average Nusselt numbers,Eqs. (68) and (69)

P, P∗ ellipsis perimeter, Eq. (10)Pe Peclet numberq ′′

o heat flux at the contour, W/m2

r aspect ratio, Eq. (9)T, Tav temperature and average temperature, KTo constant inlet temperature, KV velocity field, m/sVav flow average velocity, m/su, v dimensionless elliptical coordinates, Eq.

(18)U dimensionless velocity field, Eq. (11)uo coordinate contour, Eq. (19)x, y, z coordinate axes, mX, Y, Z dimensionless coordinates, Eq. (8)

Greek Symbols

α, β dimensionless ellipsis semi-axes, Eq. (9)α∗ dimensionless focal distance, Eq. (19)η normal coordinate, mη∗ dimensionless normal coordinate, Eq. (8)ρ fluid density, kg/m3

θ, θ∗ dimensionless temperatures, Eqs. (7) and(29)

θ∗i ,

¯θ∗im transformed potentials, Eqs. (41) and (53)

θav dimensionless average temperature,Eq. (64)

θw,av dimensionless average wall temperature,Eq. (67)

φm eigenfunction related to the coordinate u,Eq. (52)

λm eigenvalue associated to the eigenfunctionφm(u)

ψi eigenfunction related to the coordinate v,Eq. (40)

µi eigenvalue associated to the eigenfunctionψi(v)

�, � cross-section domain and cross-sectioncontour

Subscripts

i, j, m, n integer indices

Superscripts

∼ transform operator related to the coordi-nate v

- transform operator related to the coordi-nate u

REFERENCES

[1] Shah, R. K., and London, A. L., Laminar Flow Forced Con-vection in Ducts, pp. 1–4, Academic Press Inc., New York,1978.

[2] Cotta, R. M., Integral Transforms Method in Thermal and Flu-ids Science and Engineering, pp. 27–53, Begell House Inc.,New York, 1998.

[3] Aparecido, J. B., Cotta, R. M., and Ozisik, M. N., AnalyticalSolutions to Two-Dimensional Diffusion Type Problems in Ir-regular Geometries, Journal of the Franklin Institute, vol. 326,pp. 421–434, 1989.

[4] Maia, C. R. M., Aparecido, J. B., and Milanez, L. F., HeatTransfer in Laminar Forced Convection inside Elliptical Tubewith Boundary Condition of First Kind, Proc. 3rd EuropeanThermal Sciences, Heidelberg, Germany, vol. 1, p. 347, 2000.

[5] Maia, C. R. M., Aparecido, J. B., and Milanez, L. F., HeatTransfer in Laminar Forced Convection inside Elliptical CrossSection Ducts with Boundary Condition of Second Kind, Proc.22nd Congresso Ibero-Latino Americano de Metodos Com-putacionais em Engenharia—CILAMCE, vol. 1, p. 198, Camp-inas, Brazil, 2001.

[6] Cotta, R. M., and Ozisik, M. N., Laminar Forced Convectionin Ducts with Periodic Variation of Inlet Temperature, Interna-tional Journal of Heat and Mass Transfer, vol. 29, pp. 1495–1501, 1986.

[7] Cotta, R. M., and Ozisik, M. N., Diffusion Problems with Gen-eral Time-Dependent Coefficients, Brazilian Journal of Me-chanical Sciences, vol. 9, no. 4, pp. 269–292, 1987.

[8] Vick, B., and Wells, R. G., Laminar Flow with an AxiallyVarying Heat Transfer Coefficient, Int. Journal of Heat andMass Transfer, vol. 29, pp. 1881–1889, 1986.

[9] Silva, J. B. C., Aparecido, J. B., and Cotta, R. M., Solutionsto Simultaneously Developing Laminar Flow Inside ParallelPlate Channels, Int. Journal of Heat and Mass Transfer, vol.35, pp. 887–895, 1992.

[10] Diniz, A. J., Aparecido, J. B., and Cotta, R. M., Heat Con-duction with Ablation in a Finite Slab, Heat and Technology,vol. 8, pp. 30–43, 1990.

[11] Nascimento, U. C. S., Macedo, E. N., and Quaresma, J. N.N., Thermal Entry Region Analysis through the Finite Inte-gral Transform Technique in Laminar Flow of Bingham Flu-ids within Concentric Annular Ducts, International Journal ofHeat and Mass Transfer, vol. 45, pp. 923–929, 2001.

[12] Quaresma, J. N. N., and Macedo, E. N., Integral Transform So-lution for the Convection of Herschel-Bulkley Fluids in Cir-cular Tubes and Parallel-Plates Ducts, Brazilian Journal ofChemical Engineering, vol. 15, pp. 77–89, 1998.

[13] Macedo, E. N., and Quaresma, J. N. N., Transient LaminarForced Convection Heat Transfer in Circular and Parallel-Plates Ducts of Herschel-Bulkley Fluids, Proc. 14th BrazilianCongress of Mechanical Engineering, Bauru, Brazil, 1997 (onCD-ROM).

[14] Maia, C. R. M., Aparecido, J. B., and Milanez, L. F., PressureDrop for Flow of Power Law Fluids Inside Elliptical Ducts,Proc. 17th International Congress of Mechanical Engineering,Sao Paulo, Brazil, Paper 1705, 2003 (on CD-ROM).

[15] Aparecido, J. B., How to Choose Eigenvalue Problemswhen Using Generalized Integral Transforms to Solve Ther-mal Convection-Diffusion Problems, Proc. 14th BrazilianCongress of Mechanical Engineering, Bauru, Brazil, 1997 (onCD-ROM).

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[16] Shah, R. K., and London, A. L., Laminar Flow Forced Con-vection in Ducts, Academic Press Inc., New York, p. 50, 1978.

[17] IMSL Math/Library, Visual Numerics, 10th ed., Houston, TX,1994.

[18] Cotta, R. M., and Ozisik, M. N., Laminar Forced Convection ofPower Law Non-Newtonian Fluids inside Ducts, Warme-undStoffubertragung, vol. 20, pp. 211–218, 1986.

[19] Iqbal, M., Khatry, A. K., and Aggarwala, B. D., On the SecondFundamental Problem of Combined Free and Forced Convec-tion through Vertical Non-Circular Ducts, Applied ScientificResearch., vol. 26, pp. 183–208, 1972.

Cassio R. M. Maia is a teacher in the mechanicalengineering department at the University of SaoPaulo State at Ilha Solteira-Brazil. He obtainedhis undergraduate degree in physics and his grad-uate degree in nuclear and mechanical engineer-ing from the University of Campinas and Univer-sity of Sao Paulo. His research interests includeanalytical-numerical methods in heat transfer andthe development of the heat transfer exchanger.

Joao Batista Aparecido obtained his undergrad-uate degree in mechanical engineering from theFederal University of Uberlandia (1981), his Mas-ters degree from University of Sao Paulo Stateat Guaratingueta (1986), and his Doctorate De-gree from the Technology Institute of Aeronau-tics (1988). He has been employed at Universityof Sao Paulo State at Ilha Solteira since 1982. Heis currently an associate professor of MechanicalEngineering. He has been teaching courses in thethermal field for about twenty years. His current

research interests include numerical algorithms, computer programming, an-alytical methods, numerical methods, and computational heat transfer andfluid dynamics.

Luiz F. Milanez is a professor of mechanical en-gineering at Universidade Estadual de Campinasin Brazil. He received his B.Sc. degree in mechan-ical engineering from Universidade Estadual deCampinas in 1973. He obtained an M.Sc. in chem-ical engineering from the University of Salford,England, in 1976. Dr. Milanez received his Ph.D.in mechanical engineering from Universidade Es-tadual de Campinas, Brazil, in 1982. His researchinterests include thermodynamics and conductionand convection heat transfer.

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