Laminar fully developed flow through square and equilateral triangular ducts with rounded corners subjected to H1 and H2 boundary conditions

International Journal of Thermal Sciences 49 (2010) 1763e1775

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Laminar fully developed flow through square and equilateral triangular ducts with rounded corners subjected to H1 and H2 boundary conditions S. Ray a, *, D. Misra b a b

Institute of Thermal Engineering, Technische Universität Bergakademie, Freiberg, Gustav-Zeuner-Strasse 7, D 09596 Freiberg, Germany Department of Mechanical Engineering, Jadavpur University, Kolkata 700 032, India

a r t i c l e i n f o

a b s t r a c t

Article history: Received 25 January 2008 Received in revised form 22 March 2010 Accepted 23 March 2010 Available online 26 May 2010

The present paper deals with the evaluation of pressure drop and heat transfer characteristics of laminar fully-developed flow through ducts of square and equilateral triangular cross sections with rounded corners, for both H1 and H2 boundary conditions. The dimensionless radius of curvaturep(R ffiffiffi c) of both type of ducts is varied from zero to the maximum possible value (1 for square duct and 1= 3 for triangular duct). The solutions for velocity and temperature are considered in the form of a harmonic series. The constants of the series are evaluated by ‘least square technique’. From the velocity and temperature solutions, fRe and Nu are calculated. The results show that for square duct, at lower values of Rc, both fRe and Nu increase rapidly with Rc and for higher values of Rc, both fRe and Nu asymptotically assume their values corresponding to that for the circular duct. For triangular ducts, Nu shows a similar behaviour. fRe, on the other hand, shows a similar behaviour only for lower values of Rc. At moderate Rc, fRe attains its maximum value around Rc z 0.35 and with further increase in Rc, fRe drops slightly and finally tends to its value corresponding that of a circular ducts in an asymptotic manner. For both type of ducts, Nu for H1 boundary condition is always higher than that for H2 boundary condition. It is also observed that the straight portion of the duct is always more effective than the circular duct, where as, the effectiveness of the rounded portion, which is always less than the circular duct, increases with the increase in Rc. Correlations, in two different forms, are obtained for all the cases and they show excellent agreement with the computed data. Ó 2010 Elsevier Masson SAS. All rights reserved.

Keywords: Laminar Fully developed Square Triangular Rounded corner

1. Introduction Study of thermal hydraulic characteristics for laminar fullydeveloped flow through ducts of different cross section assumes importance in design of heat exchangers of existing geometric configuration, as well as, for development of newer types of flow passages in them in order to achieve better thermal hydraulic performance. The primary objective of the heat exchanger designer is to work with duct geometries that yield (i) a high value of heat transfer area to volume ratio, (ii) a high value of heat transfer coefficient and (iii) a corresponding low value of friction factor. As a result, substantial research work is focused towards development of compact and efficient duct geometries, those satisfy the above criteria. In compact heat exchangers, due to smaller system dimensions, the hydraulic diameter is low, which, for most of the practical cases, yields essentially laminar flow. It is therefore essential to investigate

* Corresponding author. Fax: þ49 (0) 3731 393942. E-mail address: [email protected] (S. Ray). 1290-0729/$ e see front matter Ó 2010 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.ijthermalsci.2010.03.012

the thermal hydraulic behaviour of ducted flows, particularly in the laminar regime. These studies find its relevance in different areas, such as, aerospace, nuclear, chemical and process industries, biomedical, electronics and instrumentation. In most of the heat exchangers in service, especially in shell and tube type, generally circular duct is used. However, it may be noted here that a polygonal duct (number of sides ¼ n), offers higher surface area to volume ratio as compared to a circular duct. This ratio increases with the decrease in n. As a result, triangular ducts offer largest surface area to volume ratio and for the circular duct, for which n is infinite, it is the least. Hence, from the view point of compactness, the triangular duct should be the most preferred geometry and the square ducts should be the next choice. Circular ducts, on the other hand, should have the least priority. Offering maximum compactness, i.e., highest surface area to volume ratio, however, is not the sole criterion for selection of duct geometry. A designer should also look into the overall thermal hydraulic behaviour of the flow through the ducts. In this regard, it may be mentioned here that the sharp corners of the polygonal ducts offer least effective heat transfer surface as the bulk flow tends to by-pass them. The corners, in fact, act as hot spots, where

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Nomenclature A Ac aj, bj C Cp cj, dj dh, Dh f h k m Nu n p Ph, Pw q_ 00 ; q_ 00p

Half side of the duct, m Cross-sectional area, m2 j-th. constants for velocity solution Constant in momentum equation, (ms)1 Specific heat at constant pressure, J/kg K j-th. constants for temperature solution Hydraulic diameter (4Ac/P), m Friction factor Heat transfer coefficient, W/m2 K Thermal conductivity, W/m K Number of boundary points Nusselt number Number of terms in the series Pressure, N/m2 Heated and wetted perimeter, m Average and local heat flux for H2 boundary condition, W/m2

the temperature is significantly higher than the rest of the duct surface. As a result the overall heat transfer performance of the sharp cornered ducts reduces to a certain extent. Since the sharp corners of the polygonal ducts exhibit ineffective heat transfer characteristics and as a result affect the overall thermal hydraulic performance, such ducts with rounded corners may offer a possible remedy. The present study is devoted to address this issue. Extensive research work has been carried out by numerous researchers to study the thermal hydraulic behaviour of laminar flow through ducts of various geometries as the problem is of practical importance and relevant to heat exchanger industries. Upto 1978, exhaustive literature review has been presented by Shah and London [1]. Subsequently, Hartnett and Kostic [2] have compiled the existing experimental and numerical data for Newtonian and non-Newtonian fluid flow through rectangular ducts in both laminar and turbulent regimes. As more and more articles are still being published in this area, it is apparent that further monographs are essential on this topic. A detailed review of all the literature covering various thermal hydraulic aspects of flow through ducts of different geometries, subjected to various possible boundary conditions, is left out of the scope of the present work. However, in the present context, literatures are reviewed only on works covering generalized techniques for flow and heat transfer through arbitrary shaped ducts. As reported by Shah and London [1], there are various methods available for solution of relevant momentum and energy equations applicable for laminar fully-developed ducted flows, ranging from analytical treatments to computational solutions by finite difference/element techniques. Some of these methods are applicable only for simple geometries. For example, ‘exact solution’ is possible, only when the concerned geometry and the boundary conditions are relatively straightforward. On the other hand, some of the methods, like numerical solutions, are applicable to almost all kind of duct geometries and boundary conditions, although they are quite costly. As far as the analytical or semi-analytical methods are concerned, in the past, various researchers have used the ‘Conformal Mapping’ technique [3], the ‘Generalised Integral Transform’ technique [4], the ‘Variational Method’ [5] and the ‘Series Solution Method’ [6e13]. Among these various approximate semi-analytical methods, solutions by ‘Point-Matching-Methods’ and ‘Least Square Methods’, those fall into the general class of series solution method, require special mention, as these methods are very general in nature and are applicable to fully-developed flow through ducts of very

0 Q_ Re rc, Rc r s t tb, tw T w wm w* z

a Dp m n r q G

Heat input per unit axial length, W/m Reynolds number Radius of curvature, m Radial coordinate, m Distance along the duct wall, m Temperature,  C Bulk and wall temperatures,  C Transformed temperature Axial velocity, m/s Average axial velocity, m/s Transformed axial velocity, m2 Axial coordinate, m Thermal diffusivity, m2/s Pressure drop along axial length L, N/m2 Dynamic viscosity, N-s/m2 Kinematic viscosity, m2/s Density, kg/m3 Angular coordinate, rad The boundary of the duct

complex shape, as long as a constant wall heat flux type of boundary condition is used. In general, for fully-developed ducted flow, if the dependent variables are suitably transformed, a Laplace and a Poisson equation can represent the momentum and the energy equations respectively. The energy equation, after decomposing the dependent variable (temperature) into complementary and particular solutions, can be further transformed to a Laplace equation. The general solution of Laplace equation is obtained by a linear combination of harmonic functions in the form of a truncated infinite series, consisting of N (starting from zero) terms. Therefore, the method involves solution of n ¼ 2N þ 1 unknown coefficients. The solutions exactly satisfy the governing equations, although the implementation of boundary conditions requires special treatment. In order to solve the coefficients, when n number points are chosen on the boundary, and same number of unknown coefficients is solved, the method is termed as ‘point-matching method’. The final solution (the series) satisfies the boundary conditions at the chosen points, although at other points on the boundary, errors are expected. Sparrow et al. [6,7] have applied the point matching method, using algebraicetrigonometric polynomials, for longitudinal flow over array of cylinders. Cheng and Jamil [8] have adopted the point matching technique to study flow and heat transfer in cylindrical ducts with diametrically opposite flat sides. In the ‘least square method’ more than n points (say, m, usually 2n to 3n points) along the boundary are employed to determine n unknown coefficients in the truncated infinite series. The coefficients of the series are evaluated by solving m linear algebraic equations by minimising the squared error of the boundary conditions at m (m > n) chosen points. Thus, in the least square method, the exact fit at the discrete boundary points (as in ‘point-matching method’) is sacrificed to obtain a better fit to the boundary as a whole. The least square method has been employed Ratkowsky and Epstein [9] and Hagen and Ratkowsky [10] to study laminar flow in regular polygonal ducts with circular centered cores and in cylindrical ducts with regular polygonal cores respectively. The least square approximation was first adopted by Sparrow and HajiSheikh [11] employing Gram-Schmidt orthonormalisation to study flow and heat transfer in arbitrary shaped ducts. However, they furnished results only for circular ducts and sectors. Subsequently, Shah [12] obtained various averaged parameters like friction factor in the form of fRe and Nusselt number for flow through ducts of various shapes, for example, rectangular, isosceles triangular,

S. Ray, D. Misra / International Journal of Thermal Sciences 49 (2010) 1763e1775

sinusoidal and equilateral triangle with rounded corners employing Golub’s method (using Householder reflections). It may be mentioned here that although the methods of Sparrow and Haji-Sheikh [11] and Shah [12] are similar to each other, in a sense that both these papers presented the least square method to obtain the unknown coefficients, the former considered the governing equations in Cartesian coordinates, where as, the latter adopted the polar form of the governing equations. As a result, Sparrow and HajiSheikh [11] considered the real and imaginary part of the complex variable ðx þ iyÞj and Shah [12] adopted the algebraic-trigonometric polynomial, in the form r j ½aj cosðjqÞ þ bj sinðjqÞ as individual solutions for the problem.1 It is obvious that adding extra terms in the latter method requires much less effort and hence, the least square method, presented by Shah [12], has been adopted for the present study. It is worthwhile to mention at this point that the present problem could also be solved using commercial or self developed (in-house) CFD codes. However, the advantages of the adopted semi-analytical method, with least square minimisation technique, are as follows (as will be shortly apparent): (i) the present method is much faster compared to CFD solutions as it boils down to solving two least square minimisation problems e one for the velocity and the other for the temperature, where as, the CFD solution would require solution on either unstructured or on block structured curvilinear grids, (ii) the grid generation requires only generation of the boundary points and not the interior points and (iii) the velocity and temperature fields are known in terms of series solution and hence, using the coefficients, these fields can be evaluated at any point in the domain, whereas the CFD solutions are actually obtained only on specified nodes. In the present study, two specific geometries, square and equilateral triangular ducts with rounded corners, are considered for the analysis. The investigations are carried out with the radius of curvature (rc) of these corners varying from zero to the maximum possible. For both the cases, the basic geometries are represented when rc is zero. Corresponding to the respective maximum values of rc, both of the basic geometries are transformed to that of a circular duct. Although various types of thermal boundary conditions could be imposed at the duct walls [1], in the present study only H1 and H2 boundary conditions are explored. It may be noted here that the other commonly used boundary condition, namely the constant wall temperature boundary condition, could not solved using the present method owing to the requirement of iterative solution involving the axial convection, which does not assume a constant value under the assumption of laminar fullydeveloped flow through ducts with constant wall temperature. Section 2 of this article presents the brief mathematical formulation along with the method of solution and evaluation of important quantities, like friction factor and heat transfer. The results of the present study are reported in section 3. The correlations obtained for fRe and Nu for all the cases are also presented in this section. The major conclusions of this article are summarised in Section 4. 2. Mathematical formulation The present analysis is performed for hydrodynamically, as well as, thermally fully-developed laminar flow through square and triangular ducts with rounded corners. The fluid is assumed to be incompressible and with constant properties, which also obeys Newton’s law of viscosity and Fourier law of heat conduction. For all the problems the effect of viscous dissipation and pressure work are neglected. The solutions are obtained for both H1 (i.e., for constant

1

Both for each j (j ¼ 0 to N).

1765

axial heat input and uniform peripheral wall temperature) and H2 (i.e., for uniform axial, as well as, peripheral heat input) boundary conditions, on the duct surface. 2.1. Problem geometry The geometries of the square and triangular ducts with rounded corners are shown in Fig. 1. All the lengths for both the geometries are normalised with respect to half of the sides (a) of the corresponding basic duct. It may be easily recognised that the dimensionless radius of curvature, Rc(¼rc/a), is the only variable geometric parameter, which identifies the shape of the duct. For square duct, Rc can vary from 0 to 1. When Rc is zero, a sharp cornered square duct is obtained, whereas, for Rc ¼ 1, the geometry reduces pffiffiffito a circular duct. For triangular duct, Rc can vary from 0 to 1= 3. Here also, a sharp cornered pffiffiffitriangular duct and a circular duct are obtained for Rc ¼ 0 and 1= 3 respectively. Thus, with the variation of Rc, from its minimum value to the maximum, a family of rounded cornered ducts is obtained for each of the basic geometries, where, the minimum value corresponds to the basic duct and the maximum value corresponds to the circular duct. The geometrical quantities of interest for these ducts are the cross-sectional area (Ac), the wetted perimeter (Pw) and the hydraulic diameter (dh). For rounded corner square duct, they are obtained as follows,

Acs ¼ 4a2  ð4  pÞrc2

(1)

Pws ¼ 8ða  rc Þ þ 2prc

(2)

The hydraulic diameter is defined as,

dh ¼ 4Ac =Pw

(3)

Hence, using Eqs. (1) and (2), the dimensionless hydraulic diameter for rounded corner square duct is written as,

Dhs ¼

dh 4  ð4  pÞR2c ¼ a 2ð1  Rc Þ þ pRc =2

(4)

It may be noted here that in both the limits (Rc ¼ 0 and 1), Dh has the same value and is equal to 2. Similarly, for triangular ducts with rounded corners, Ac, Pw and Dh are obtained as,

  pffiffiffi  a  Act ¼ 3 rc þ pffiffiffi a  3rc þ prc2 3  pffiffiffi  Pwt ¼ 6 a  3rc þ 2prc

Dht

 pffiffiffi pffiffiffi  2 Rc þ 1= 3 1  3Rc þ 2pR2c =3 dh  ¼ ¼ pffiffiffi  a 1  3Rc þ pRc =3

(5)

(6)

(7)

pffiffiffi Here pffiffiffi also, in both the limits of Rc ð0 and 1= 3Þ, Dh is obtained as, 2= 3. Fig. 2 shows the variation of Dh with normalised radius of curvature, Rc/Rc,max for square and triangular ducts, where Rc,max is the maximum allowable radius of curvature that generates the circular duct. 2.2. Velocity problem 2.2.1. Governing momentum equation For hydrodynamically fully-developed flow (no transverse velocities), the governing momentum equation for the axial direction can be simplified and written as,

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a

b

B rc α=0

r

n

n α = π/6

r

y

s

θ

x

y

A

θ

x rc A a

a

a

Square Duct with Rounded Corner

B

s a

Triangular Duct with Rounded Corner

Fig. 1. Geometry of the ducts of square and triangular cross sections with rounded corners.

  1 v vw 1 v2 w 1 dp r þ 2 ¼ ¼ C m dz r vr vr r vq2

V2 w ¼

(8)

where, C is a constant (since, under the assumption of fully-developed flow, the pressure gradient in the axial direction is a constant. It may also be recognised that Eq. (8) is a Poission equation. In order to convert the same to a Laplace equation, the following transformation is used.

  r2 w ¼ C w*  4

(9)

The corresponding obtained as,

V2 w* ¼

1 v vw* r r vr vr

transformed

! þ

momentum

equation

1 v2 w * ¼ 0 r 2 vq2

is

(10)

2.2.2. Velocity boundary conditions There are two types of boundary conditions, encountered for velocity solutions in case of a ducted flow problem. They are (i) wall boundary condition and (ii) symmetry boundary condition. The no-slip boundary condition at the walls is given by w ¼ 0. The corresponding transformed boundary condition is given by,

2.15

1.35

w* ¼ r 2 =4

(11)

On the line of symmetry, the boundary condition for a general variable, f is given by,

vf=vn ¼ 0

(12)

where ‘n’ denotes the outward normal on the boundary of the solution domain. Projecting the terms in the r and q directions, the expression for normal derivative is obtained as,

vf vf 1 vf ¼ cosða  qÞ þ sinða  qÞ vn vr r vq

(13)

where, a is the angle made by the outward normal with the x-axis. Thus, vw=vn ¼ 0 on the line of symmetry results in to the following condition for w*,

vw* r ¼ cosða  qÞ 2 vn

(14)

where, vw* =vn is evaluated from Eq. (13). 2.2.3. Series solution for velocity The algebraic-trigonometric polynomials in the form of r j cos jq and r j sinjq for every j ¼ 0, 1, 2, ., N, satisfy Laplace equation.2 Since, the equation is linear, the general solution for velocity can be obtained by superimposition as,

w* ¼ ao þ

N X

r j aj cosðjqÞ þ bj sinðjqÞ

(15)

j¼1

Hydraulic Diameter, Dh

1.30 2.10 1.25

Eq.(7), Triangular Duct

2.05

1.20

Eq.(4), Square Duct 2.00

1.15

1.95 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Normalised Radius of Curvature, R c /R c,max Fig. 2. Variation of hydraulic diameter with radius of curvature.

where, a0, aj and bj are arbitrary constants. It may be recognised that the total number of constants associated with the Eq. (15) is n ¼ 2N þ 1. The above solution exactly satisfies the momentum equation, although it does not satisfy the boundary conditions. One way of solving the constants is to select exactly n points on the boundary, which generates n equations from the boundary conditions and hence, the constants are solved from these equations. The method is termed as ‘Point-Matching-Method’ in the literature. In this method, the velocity solution satisfies the boundary conditions only at the chosen points, although, some error is encountered at

1.10 1.0 2 Similar analysis had already been presented by Shah [1], except for the symmetry boundary condition. As a result, the analysis is briefly presented here with appropriate modifications for the sake of completeness.

S. Ray, D. Misra / International Journal of Thermal Sciences 49 (2010) 1763e1775

the other left out points on the boundary. In the present analysis, however, m points are chosen on the boundary, where, m > n, which leads to an over-posed problem. The constants are evaluated by ‘Least-Square-Matching-Method’ (see Shah [12] for details). In this method, the solution given by Eq. (15) may not satisfy the boundary conditions at any of the chosen points. However, as reported in the literature, the total error associated with this method is estimated to be much less than that of ‘Point-MatchingMethod’. The details of the solution procedure by least square minimisation technique are discussed by Laha [13] and hence, they are not presented here. When the ith point on the boundary (at ri, qi) is located on the wall, the transformed boundary condition in Eq. (11) is represented as,

ao þ

N X

ri j aj cosðjqi Þ þ bj sinðjqi Þ ¼

j¼1

ri2

(16)

4

On the line of symmetry, the expression for normal derivative, is obtained using Eqs. (13) and (15) as,

vw* =vn,

N X

vw* jr j1 aj cosfa þ ðj  1Þqg þ bj sinfa þ ðj  1Þqg ¼ vn j¼1

(17)

Therefore, when the ith point on the boundary is on the line of symmetry, the transformed boundary condition in Eq. (14) is written as, N X j¼1

Dp Lz

¼

2f rw2m Dh

1767

(21)

where, Dp is the pressure drop over an axial length Lz . Using the definition of C ¼ Dp/mLz from Eq. (8) and the above equation, the average axial velocity (wm/C) can also be expressed as,

D2h wm  ¼ w*m ¼ C 2fRe

(22)

where, Re is the Reynolds number based on the hydraulic diameter and is defined as,

Re ¼ wm Dh =n

(23)

Thus, using Eq. (22), the product of friction factor and Reynolds number (fRe) is obtained as,3

fRe ¼

D2h D2h ¼ 2ðwm =CÞ 2w*m

(24)

The dimensionless axial velocity can be obtained from Eqs. (19) and (22) as,

2 3 N X r2  w 2fRe4 j ¼ r aj cosðjqÞ þ bj sinðjqÞ  5 ao þ wm 4 D2h j¼1

(25)

2.3. Temperature problem j1 jri aj cosfai

¼

þ ðj  1Þqi g þ bj sinfai þ ðj  1Þqi g

ri cosðai  qi Þ 2

(18)

Thus, by choosing m points on the boundary of the solution domain and applying the appropriate boundary conditions, as described in Eqs. (16) and (18), the unknowns, a0, aj and bj are solved in a least square sense as described earlier. 2.2.4. Calculation of average axial velocity and friction factor Once the constants in Eq. (15) are obtained, the solution for axial velocity is completely known in terms of C (which is, however, still not known). Using the definition of w* from Eq. (9) and Eq. (15), the axial velocity is written as, N X

w  ¼ ao þ r j aj cosðjqÞ þ bj sinðjqÞ  4 C j¼1

r2

(19)

The cross section averaged velocity, wm can be evaluated as,

wm 1  ¼ Ac C

Z  w  dAc C Ac

1 ¼ Ac

Z Ac

¼

w*m

3 N X r2  j 4ao þ r aj cosðjqÞ þ bj sinðjqÞ  5dAc 4 j¼1 2

2.3.1. Governing energy equation The solutions for temperature are obtained for either H1 boundary condition, or, for H2 boundary condition. For hydrodynamically fully-developed flow, the governing energy equation is written as,

V2 t ¼

  1 v vt 1 v2 t w vt r þ 2 2 ¼ aT vz r vr vr r vq

When constant wall heat flux is applied on the walls of the duct, the temperature gradient in the axial direction is given by,

vt dtw dt ¼ b ¼ dz dz vz

The area integral in the right hand side of Eq. (20) is generally obtained in a problem specific manner. In the present problem, this is evaluated by choosing a curvilinear coordinate to fit the duct. The entire area is divided into a number of small curvilinear control volumes. The local Jacobian of the coordinate transformation provides the area of each small control volume. It is important to note that a pure numerical value is obtained after evaluation of the integral in Eq. (20), which is denoted by w*m. The overall friction factor, f, through the duct is evaluated from the following definition,

(27)

At this point, let us denote that Q_ 0 is the heat input to the duct per unit axial length (for both H1 and H2 boundary conditions). For H2 boundary condition, Q_ 0 is obtained as,4

Q_ 0 ¼ q_ 00 Ph ¼ q_ 00 Pw

(28)

q_ 00

where, is the applied wall heat flux for H2 boundary condition. The rate of increase in bulk temperature in the axial direction can be obtained from a simple energy balance as follows,

dtb Q_ 0 ¼ rCp wm Ac dz ð20Þ

(26)

(29)

Therefore, using the above equation, the energy equation can be written as,

V2 t ¼

   0 1 v vt 1 v2 t w Q_ r þ 2 2 ¼ kAc wm r vr vr r vq

(30)

Substituting w/wm from Eq. (25), the above equation is written as,

3 The analysis clearly shows that for laminar fully-developed flow through ducts, f w 1/Re 4 Here, Ph is the heated perimeter, which, in principle, could be different from the wetted perimeter, Pw, when adiabatic surfaces are present.

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S. Ray, D. Misra / International Journal of Thermal Sciences 49 (2010) 1763e1775

2

3 N 2 _ 0 f Re X  2 Q r 4ao þ V2 t ¼ r j aj cosðjqÞ þ bj sinðjqÞ  5 4 kAc D2h j¼1

(31)

In order to convert the above equation to a solvable form, a new modified temperature, T, for H1 boundary condition is defined as,

T ¼

ðt  tw ÞkPw D3h 8Q_ 0 f Re

(32)

This definition is useful for H1 boundary condition, since, this allows one to set T ¼ 0 (as, t ¼ tw) on the duct walls. However, when H2 boundary condition is applied on the walls of the duct, a slightly modified form of Eq. (32) is used to define the modified temperature, which is given by,

T ¼

tkPw D3h tkD3h ¼ 8q_ 00 f Re 8Q_ 0 f Re

(33)

Using the transformation, given in either Eq. (32), or, Eq.(33), the energy equation (31) is written as,

V2 T ¼ ao þ

N X

r2 r j aj cosðjqÞ þ bj sinðjqÞ  4 j¼1

(34)

vT * ¼ vn

 _ 00  3 q p Dh vTp  q_ 00 8fRe vn

where, vTp =vn may be obtained using Eqs. (13) and (36) as,

 3  vTp r r ¼ cosða  qÞ  a0 vn 16 2 N jþ1 X r aj fðj þ 2Þcosða  qÞcosðjqÞ  4ðj þ 1Þ j¼1  jsinða  qÞsinðjqÞg þ bj fðj þ 2Þcosða  qÞsinðjqÞ

 jsinða  qÞcosðjqÞg

T ¼ T * þ Tp

(35)

The symmetry boundary condition is obtained simply by setting q_ 00 p ¼ 0 in Eq. (41). 2.3.3. Series solution for temperature Since T * satisfies the Laplace equation, the solution of Eq. (37) is also obtained in terms of algebraic-trigonometric polynomial functions as,

T * ¼ co þ

Tp ¼ ao

4

þ

r jþ2

aj cosðjqÞ þ bj sinðjqÞ 



4ðj þ 1Þ j¼1

r4 64

T ¼ co þ (36)

T * is the complementary solution, which satisfies the following Laplace equation,

V2 T * ¼

vT *

1 v r vr r vr

! þ

N X

r j cj cosðjqÞ þ dj sinðjqÞ

(43)

where, c0, cj and dj are arbitrary constants. Here also, like velocity solution, the constants are evaluated by least square minimisation technique. Using Eqs. (36) and (43), the complete temperature solution in Eq. (35) can be written as,

where, Tp is the particular solution and is obtained as, N X

ð42Þ

j¼1

In order to solve the above equation, the solution for T is sought in the following form,

r2

(41)

v2 T *

1 ¼ 0 r 2 vq2

(37)

þ

N X

r2 r j cj cosðjqÞ þ dj sinðjqÞ þ ao 4 j¼1 N X

r4 r jþ2 aj cosðjqÞ þ bj sinðjqÞ  4ðj þ 1Þ 64 j¼1

ð44Þ

When the ith point on the boundary is located on the duct wall and H1 boundary condition is applied, the following equation is obtained,

co þ

N X

ri j cj cosðjqi Þ þ dj sinðjqi Þ

j¼1

2.3.2. Temperature boundary conditions There are three types of boundary conditions, encountered for the temperature solutions in case of a ducted flow problem. They are (i) H1 boundary condition at the wall, (ii) H2 boundary condition at the wall and (iii) symmetry boundary condition. The H1 boundary condition is rather simple and is obtained by setting

t ¼ tw ; i:e:; T ¼ 0

(38)

For H2 boundary condition, the wall heat flux is uniform along the periphery, as well as, along the axial direction. Thus, the condition is written as (here, q_ 00p is the locat heat input, which could be zero for adiabatic surfaces),

vt k ¼ q_ 00 p vn

(39)

Using the definition of T from Eq. (33), the above condition is written as,

vT ¼ vn



q_ 00

p

q_ 00



D3h 8fRe

Using Eq. (35) the above equation can also be written as,

(40)

¼

jþ2 N

ri ri4 r2 X  ao i  a cosðjqÞ þ bj sinðjqÞ 64 4 j ¼ 1 4ðj þ 1Þ j

ð45Þ

It may be noted here that since the constants, a0, aj and bj are already known from the velocity solution, the right hand side of Eq. (45) is also known. For H2 boundary condition on the duct periphery, the expression for vT * =vn is required, which may be written using Eqs. (13) and (43) as, N X

vT * jr j1 cj cosfa þ ðj  1Þqg þ dj sinfa þ ðj  1Þqg ¼ vn

(46)

j¼1

Therefore, when the ith point on the boundary lies on the duct surface with H2 boundary condition, Eq. (41), with the help of Eq. (46), may be written as, N X j¼1

jrij1 cj cosfai þ ðj  1Þqi g þ dj sinfai þ ðj  1Þqi g

¼

 _ 00  3 q p Dh vTp  ðr ; q Þ q_ 00 8f Re vn i i

(47)

S. Ray, D. Misra / International Journal of Thermal Sciences 49 (2010) 1763e1775

where, vTp =vnðri ; qi Þ is obtained from Eq. (42) by substituting appropriate values of (ri, qi). The symmetry boundary condition is also obtained from Eq. (47), by setting q_ 00 p ¼ 0. It may be recognised here that when H2 boundary condition is applied on the duct walls, the temperature solution is not unique.5 Hence, the value of c0 can not be fixed (solved) form any of the equations, describing the boundary conditions. In such cases, the solutions are obtained only for cj and dj, where as, c0 is set to zero (could be set to any arbitrary value). It would be shortly apparent that the value of c0 is actually not required to obtain the heat transfer coefficients and hence, the Nusselt numbers. Once, the constants are solved by least square minimisation technique, the solution for temperature is completely known from Eq. (44). 2.3.4. Calculation of local and average Nusselt number The quantities of further interest would be the local and average Nusselt numbers. For this purpose, the local and average wall temperatures and the bulk temperature of the fluid is required. For H1 boundary condition, the average wall temperature is simply equal to tw (since the temperature is uniform throughout the periphery of the duct) and hence, T w ¼ 0. When H2 boundary condition is applied, the local temperature on the duct periphery is obtained from Eq. (44), by substituting appropriate values of (ri, qi). The average wall temperature for such cases is obtained from the following relationship,

t w kD3h 1 ¼ Pw 8q_ 00 f Re

Tw ¼

Z Tw dPw

(48)

Pw

The modified bulk temperature, Tb, is defined as,

Tb ¼

1 Ac

Z 

 w TdAc wm

(49)

Ac

For H1 boundary condition, from the definition of the modified temperature, given in Eq. (32), following relationship is obtained,

ðtw  tb ÞkPw D3h ¼ Tb 8Q_ 0 f Re

(50)

Similarly, for H2 boundary condition, from Eq.(33), Tb may be expressed as,

tb kD3h ¼ Tb 8q_ 00 fRe

(51)



vTp

vT

vT *

¼ þ vn L vn L vn L

1769

(54)

The values of ðvT * =vnÞL and ðvTp =vnÞL are obtained from Eqs. (46) and (42) respectively by substituting the appropriate values of (ri, qi). For H2 boundary condition, the right hand side of Eq. (52) can be replaced by q_ 00 and hence, using Eqs. (33) and (51), the local Nusselt number is obtained as,

NuL ¼

D4h hL Dh ¼ k 8fReðTw  Tb Þ

(55)

The average heat transfer coefficient is defined as,

hPw ðt w  tb Þ ¼ Q_ 0 ¼ q_ 00 Pw

(56)

Using the expressions for average wall temperature and the bulk temperature, the average Nusselt number for both H1 and H2 boundary conditions may be obtained as,

Nu ¼

D4h hDh ¼ k 8fReðT w  Tb Þ

(57)

It may be noted here that T w is zero for H1 boundary condition, where as, for H2 boundary condition, this is given by Eq. (48). For both the cases, Tb is obtained from Eq. (49). 3. Results and discussion In the present study, analysis is carried out for two basic duct geometries. They are (i) Square duct with rounded corner and (ii) Triangular duct with rounded corner. Results are obtained for varying radius of curvature, Rc. For square duct, Rc. varies pffiffiffi from 0 to 1, where as, for triangular duct, Rc. ranges from 0 to 1= 3. Most of the results, presented in this paper, are generated with N ¼ 25 (total number of coefficients being n ¼ 51) and m ¼ 400. The results are, however, tested for their accuracy by increasing the values of both N and m. They are accepted when they are found to be independent of these values. In this section, results obtained by the present method are first compared with the available data. The results of fRe, local and average Nusselt numbers are subsequently presented, first for the square duct, followed by those for the triangular duct. Correlations are also developed from the present study to predict these important thermalehydraulic parameters. In the end, the non-uniformity in the wall temperature distribution for H2 boundary condition and the area goodness factor for both H1 and H2 boundary conditions are presented for square and triangular ducts with rounded corners.

The local heat transfer coefficient is defined as,

vt hL ðtw  tb Þ ¼ k

vn L

3.1. Comparison with the available results

(52)

where, the suffix ‘L’ stands for ‘local’ quantities. From this definition, the local Nusselt number for H1 boundary condition is obtained as,

NuL ¼

hL Dh D ðvT=vnÞL ¼  h k Tb

(53)

where, the local normal modified temperature gradient, ðvT=vnÞL , for H1 boundary condition is obtained from the following relationship,

5 This condition states that if T is a solution for temperature then T þ c0 is also a solution.

Shah and London [1] have reported results for laminar forced convection through ducts of various shapes. However, only four cases are found to be similar to those for the present study. These are (i) Perfectly square duct (ii) Circular duct (iii) Equilateral triangular duct and (iv) Equilateral triangular duct with all the three rounded corners (Rc ¼ 1/3). For all the cases, the comparison is made for H1 boundary condition. The comparison is shown in Table 1. The table clearly shows that for simple geometries the results of the present study are in excellent agreement with those of the previous works. 3.2. Results for square duct 3.2.1. Variation of fRe with rc The variation of fRe with Rc for square ducts with rounded corners is presented in Fig. 3. As expected, at the two limits of Rc, fRe is bounded by its values for perfectly square and circular ducts. It is observed from

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S. Ray, D. Misra / International Journal of Thermal Sciences 49 (2010) 1763e1775

Table 1 Comparison with Previous Works. Duct Geometry

fRe

Perfect Square Circle Equilateral Triangle Equilateral Triangle (Rc ¼ 1/3)

Present Study

Shah and Lond0on [1]

Present Study

Shah and London [1]

Present Study

14.227 16.000 13.333 15.993

14.226 15.998 13.333 16.028

3.608 4.364 3.111 4.205

3.608 4.364 3.111 4.209

3.091 4.364 1.892 3.780

3.102 4.364 1.915 3.783

the figure that for lower value of Rc, fRe increases rapidly. However, as Rc approaches unity, fRe asymptotically attains its value for circular duct. Two different correlations are obtained for fRe as a function of Rc. One, in the form of a polynomial series, and is given as,

" f Re ¼ 14:226 1 þ

5 X

# Ak Rkc

(58)

k¼1

where, the constants, Ak, in the above equation are presented in Table 2. Maximum error associated with this correlation is only 0.006%. The other correlation is obtained as,

"

(

f Re ¼ f Rec 1 

1



NuH2

NuH1

Shah and London [1]

f Res f Rec

#0:1204 1=0:1204 ) ð1  Rc Þ2:4467 (59)

where, fRes(¼14.226) and fRec(¼16) are the corresponding fRe values for square and circular ducts respectively. It may be noted here that Eq. (59) satisfies the fRe values for both the limits of Rc. This correlation predicts the computational data within a maximum error of 0.15%. Fig. 3 also shows the correlations presented in Eqs. (58) and (59). It is observed from the figure that these correlations almost coincide with each other. Thus, it may be recommended that any one of these correlations may be used to obtain fRe for rounded corner square duct. 3.2.2. Presentation of local Nusselt number The variation of local Nusselt number (NuL) along the periphery of the rounded corner square duct is presented in Fig. 4 for different values of Rc, where (a) is presented for H1 boundary condition and (b) shows the variation for H2 boundary condition. In this figure, the distance along the wall (s) is normalised with respect to Pw/4 where, the measurement of s starts from the midpoint of the straight portion (point A in Fig. 1). The figure also shows NuL for

16.5

circular duct, which, as expected, is uniform and is given by Nuc ¼ 48/11. From the figure it is clear that for both the cases, the heat transfer coefficient is maximum at the middle of the straight portion (location ‘A’ in Fig. 1) of the duct (which is even higher than Nuc), where as, it is minimum at the middle of rounded part of the duct (location ‘B’ in Fig. 1). This is expected since the rounded portion acts as heat pocket due to lower axial velocity of the fluid in this region. The higher values of NuL, particularly near the straight portion of the duct, clearly indicate that these parts of the duct are more effective than the circular duct. The difference in the maximum and the minimum values of NuL, however, reduces with the increase in Rc, since, with the increase in Rc, the effectiveness of the rounded portion also increases. Comparison of Fig. 4(a) and (b) shows that for a fixed value of Rc, the non-uniformity in NuL is more for H1 boundary condition. The same comparison also shows that the part of the duct, where the values of NuL are higher than Nuc, is significantly more for H1 boundary condition for same value of Rc. The variations of NuL, in the form of absolute difference between NuL and Nuc, with Rc at locations ‘A’ (midpoint of the straight portion) and ‘B’ (midpoint of the curved portion) are shown in Fig. 5, for both H1 and H2 boundary conditions. The figure clearly shows that for both the cases, NuL at ‘B’, which is consistently lower than Nuc, continuously increases (and hence, jNuL  Nuc j continuously decreases) with the increase in Rc. This indicates that the effectiveness of the rounded portion is always less than the circular duct and with the increase in Rc, the effectiveness of this part continuously increases. It is also obvious from the figure that jNuL  Nuc j for H1 boundary condition is always higher than that for H2 boundary condition, irrespective of the location and the value of Rc, which also clearly indicates that the nonuniformity in NuL is higher for the former case. From the figure, it is observed that NuL at ‘A’ is always higher than Nuc irrespective of the value of Rc, indicating that the effectiveness of the straight portion is always more than that of the circular duct. Irrespective of the type of boundary condition, the value of NuL at this location initially increases with the increase in Rc (thereby, increasing the effectiveness further), then attains a maximum around Rc z 0.4 and finally decreases with the further increase in Rc. 3.2.3. Presentation of average Nusselt number The variations of average Nusselt number, Nu, with Rc for square duct with both the boundary conditions are shown in Fig. 6. The overall observations are similar to that observed for fRe. The figure

16.0

f.Re

15.5 Table 2 Constants of the Correlations (Polynomial form).

15.0

Constants, Ak

Present Computation

A1

Eq.(58)

14.5

Eq.(59) 14.0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Radius of Curvature, R c Fig. 3. Variation of fRe with Rc for square duct.

0.8

0.9

1.0

A2

A3

A4

A5

A6

fRe Square 0.4316 0.5549 0.2067 0.1451 0.1040 — Triangular 1.3754 2.2964 2.5717 9.8460 6.6662 — Nu Square (H1 B.C.) 0.4258 0.00903 0.7139 0.7976 0.2909 — Square (H2 B.C.) 0.5552 0.8649 3.1650 4.3328 3.0036 0.8184 Triangular 1.2725 2.2012 15.0072 23.3487 12.0328 — (H1 B.C.) Triangular 2.5044 7.2173 27.7104 35.9759 19.2608 — (H2 B.C.)

S. Ray, D. Misra / International Journal of Thermal Sciences 49 (2010) 1763e1775

4.50

6

Data for H1 B.C.

Nuc

Local Nusselt Number, NuL

5

Average Nusselt Number, Nu

a

4

3

Rc= 0.05 2

Rc= 0.25 Rc= 0.5

Eq.(60)

4.25

Eq.(61) 4.00

3.75

3.50

Data for H2 B.C. Eq.(62)

3.25

Rc= 0.75

1

0.1

Eq.(63)

Rc= 0.95

H1 B.C. 0 0.0

1771

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

3.00 0.0

0.1

0.2

0.3

1.0

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Radius of Curvature, Rc

Distance Along the Wall, 4s/Pw Fig. 6. Variation of average Nusselt number with radius of curvature for square duct.

b

6

Nuc

Local Nusselt Number, NuL

5

of Rc. This difference reduces considerably with the increase in Rc as both the curves asymptotically tend to Nuc for Rc/1. Here also, for both the boundary conditions, two alternative correlations are developed. For H1 boundary condition, the first correlation is given by,

4

3

"

Rc= 0.05 2

5 X

NuH1 ¼ 3:608 1 þ

Rc= 0.25 Rc= 0.75 Rc= 0.95

H2 B.C. 0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Distance Along the Wall, 4s/Pw

The constants in the above equation are listed in Table 2. The maximum error associated with this correlation is 0.007%. The alternative correlation for NuH1 is as follows,

"

(

NuH1 ¼ Nuc 1 

1

Fig. 4. Variation of local Nusselt number along the duct periphery for square duct. (a) H1 boundary condition, (b) H2 boundary condition.

shows that as expected, at the two limits of Rc, Nu is bounded by its values for square and circular ducts. It is also observed from the figure that Nu for H1 boundary condition is substantially higher than that for H2 boundary condition, particularly at lower values

5

At Location 'A'

H2 B.C.

|NuL- Nuc|

where, Nus,H1 ¼ 3.608, is the average Nusselt number for perfectly square duct for H1 boundary condition. At the two bounds of Rc (i.e., at 0 and 1), the values of Nus,H1 predicted by Eq. (61) correspond to those for a sharp cornered square duct and a circular duct. The maximum error associated with this correlation is around 0.126%. Both the correlations are shown in Fig. 6. It is clear from the figure that although these correlations significantly differ in their mathematical form, they are hardly distinguishable. For H2 boundary condition, the correlations are given as,

" NuH2 ¼ 3:102 1 þ

H1 B.C.

6 X

# Ak Rkc

(62)

k¼1

3

where, the constants of the above equation are reported in Table 2.

" 2

H1 B.C.

NuH2 ¼ Nuc

(



Nus;H2 1 1 Nuc

#40:1005 1=40:1005 ) 1:9270 ð1  Rc Þ (63)

1

H2 B.C. 0 0.0

) #0:5393   Nus;H1 1=0:5393 ð1  Rc Þ2:0353 Nuc (61)

At Location 'B'

4

(60)

k¼1

Rc= 0.5 1

# Ak Rkc

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Radius of Curvature, R c Fig. 5. Variation of local Nusselt number with radius of curvature at midpoints of straight and curved portions of square duct.

where, Nus,H2 is the average Nusselt number for perfectly square duct with H2 boundary condition and its value is given by 3.102. The maximum error associated with the first correlation in Eq. (62) is around 0.006%, where as, the same for the second correlation in Eq. (63) is about 0.375%. All of these correlations are also presented in Fig. 6 and they show excellent agreement with the computed data.

1772

S. Ray, D. Misra / International Journal of Thermal Sciences 49 (2010) 1763e1775

a

3.3.1. Variation of fRe with rc The variation of fRe with Rc for triangular ducts with rounded corners is presented in Fig. 7. Here also p it ffiffiffiis observed that at the two limits of Rc (i.e., at Rc ¼ 0 and Rc ¼ 1= 3), the values of fRe correspond to those for equilateral triangular and circular ducts. It is observed from the figure that fRe increases rapidly with the increase in Rc at its lower range. The rate of increase is also observed to be more than that observed for square duct. However, it reaches a maximum at an intermediate Rc (around 0.35) and finally reaches to the value of fRe for the circular duct in an asymptotic manner. This feature is quite distinct for triangular ducts and is not observed for their square counterpart. Here also, like square duct, two different correlations are tried to fit the computed data. The first correlation is given as,

fRe ¼ 13:333 1 þ

5 X

fRe ¼ fRec 1 

Ak Rkc

(64)



fRet 1 fRec

1=5:4430 )

pffiffiffi 1  3Rc

4:8812

#5:4430

(65) where, fRet ¼ 13.333 is the corresponding value for triangular duct. Since the nature of the variation of fRe for triangular duct shows a maximum and also an asymptote, the second correlation in Eq. (65) does not produce good result. As a result, the maximum error associated with this correlation is around 0.501%. Both the correlations are also shown in Fig. 7, which clearly shows this fact. 3.3.2. Presentation of local Nusselt number In Fig. 8(a) and (b), the variations of local Nusselt number for different values of Rc along the periphery of the rounded corner triangular duct are presented for H1 and H2 boundary conditions respectively. In these figures, the distance along the wall is normalised with respect to Pw/3, starting from point A in Fig. 1, the middle point of the straight portion. Here also, like square duct, the figure shows NuL for circular duct, which is uniform at Nuc ¼ 48/11.

16.5 16.0 15.5

f.Re

15.0 14.5 14.0

Present Computation Eq(64)

13.5 13.0 0.0

Eq(65)

0.1

4

Rc= 0.05

3

Rc= 0.15 2

0 0.0

Rc= 0.3 Rc= 0.45 H1 B.C. 0.1

Rc= 0.55 0.2

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.8

0.9

1.0

Distance Along the Wall, 3s/Pw

The constants in the above equation are presented in Table 2. The maximum error associated with Eq. (64) is about 0.014%. The second correlation is as follows,

(

Nuc

5

#

k¼1

"

6

1

0.3

0.4

Radius of Curvature, R c Fig. 7. Variation of fRe with Rc for triangular duct.

0.5

0.6

b

8

Rc= 0.05

7

Local Nusselt Number, NuL

"

8 7

Local Nusselt Number, NuL

3.3. Results for triangular duct

Rc= 0.15 Rc= 0.3

6

Rc= 0.45 5

Nuc

Rc= 0.55

4 3 2 1 0 0.0

H2 B.C. 0.1

0.2

0.3

0.4

0.5

0.6

0.7

Distance Along the Wall, 3s/Pw Fig. 8. Variation of local Nusselt number along the duct periphery for triangular duct. (a) H1 boundary condition, (b) H2 boundary condition.

The overall observation and the general trend appear to be similar to that observed for square duct (refer to Fig. 4 for clarification) and hence, they are not discussed here in detail. The figure clearly shows that for both the cases, NuL is the maximum at the middle of the straight portion of the duct and it is the minimum at the middle of rounded part of the duct. As expected, here also the nonuniformity in NuL reduces with the increase in Rc. The variations of the absolute difference between NuL and Nuc, with Rc at locations ‘A’ and ‘B’ are shown in Fig. 9, for both H1 and H2 boundary conditions. The figure shows that NuL at ‘A’ is higher for H2 boundary condition for lower values of Rc (upto Rc z 0.325), an observation, which is distinctly different from that observed for square duct. At this location, NuL attains maxima at Rc ¼ 0.25 and Rc ¼ 0.075 for H1 and H2 boundary conditions respectively. From the figure it is also observed that NuL at ‘B’ for H2 boundary condition continuously increases with the increase in Rc (this observation is similar to that observed for case of square duct). However, for H1 boundary condition, NuL at the same location initially remains almost constant for very low values of Rc and with further increase in Rc, NuL increases and finally attains its value for the circular duct. It is also evident from the figure that jNuL  Nuc j at ‘B’ is always higher for H1 boundary condition (i.e., NuL is always lower), which indicates that the effectiveness of the rounded portion is more for H2 boundary condition.

S. Ray, D. Misra / International Journal of Thermal Sciences 49 (2010) 1763e1775

"

5

H2 B.C.

At Location 'B'

4

|NuL- Nuc|

NuH1 ¼ Nuc 1 

At Location 'A'

H1 B.C.

H1 B.C. 1

"

H2 B.C.

0.2

0.3

0.4

0.5

Fig. 9. Variation of local Nusselt number with radius of curvature at midpoints of straight and curved portions of triangular duct.

3.3.3. Presentation of average Nusselt number The variations of average Nusselt number for triangular duct with Rc for both the boundary conditions are shown in Fig. 10. Although the variation of fRe for triangular ducts shows an interesting trend by exhibiting a maximum, the variation of Nu is quite straightforward. The general trend of variation is similar to that observed for square ducts. In this case also, Nu increases quite rapidly from its value for perfectly triangular duct at lower values of Rc and finally reaches its value for circular duct in an asymptotic manner. It is also observed from the figure that Nu for H1 boundary condition is substantially higher than that for H2 boundary condition, particularly at lower values of Rc. Here also, two different correlations are obtained for both H1 and H2 boundary conditions to predict the variation of Nu with Rc. The first correlation for H1 boundary condition is as follows,

NuH1 ¼ 3:111 1 þ

! Ak Rkc

(66)

k¼1

where, the constants are shown in Table 2. This correlation yields a maximum error of 0.078%. The other correlation is obtained as,

"

4.00

3.50

3.00

Data for H1 B.C. Eq.(66)

2.50

Eq.(67) Data for H2 B.C. Eq.(68)

2.00

Eq.(69) 0.1

0.2

0.3

0.4

0.5

0.6

Radius of Curvature, Rc Fig. 10. Variation of average Nusselt number with radius of curvature for triangular duct.

( 1

(68)

)   Nut;H2 1=11:9822 Nuc #11:9822

 pffiffiffi 1:9722  1  3Rc

(69)

where, Nut,H2 ¼ 1.915, is the average Nusselt number for perfect triangular duct for H2 boundary condition. The maximum error associated with this correlation is around 0.767%. Both the correlations are shown in Fig. 10. Although these correlations differ significantly in their mathematical form, they are hardly distinguishable. 3.4. Variation in wall temperature and area goodness factor It may be noted here that the temperature distribution along the duct periphery is strongly non-uniform for H2 boundary condition. Under this condition, the maximum wall temperature occurs at the corner (location B, in Fig. 1) of the duct where a hot temperature spot is localised. On the other hand, the minimum wall temperature occurs at the midpoint of the side of the ducts (location A in Fig. 1). In order to account for the temperature variation along the duct periphery, a dimensionless wall temperature may be defined in the following manner;

qw ¼ 4.50

# Ak Rkc

where, the constants in the above equation are listed in Table 2. The maximum error associated with this correlation is 0.1%. The alternative correlation for NuH2 is as follows,

NuH2 ¼ Nuc 1 

5 X

(67)

k¼1

0.6

Radius of Curvature, Rc

Average Nusselt Number, Nu

5 X

NuH2 ¼ 1:915 1 þ 0.1

  Nut;H1 1=4:8286 Nuc #4:8286

In the above equation, Nut,H1 is the average Nusselt number for triangular duct with H1 boundary condition and its value is 3.111. The correlation fits the computed data with a maximum error of 0.432%. Both these correlations are also presented in Fig. 10 and they show excellent agreement with the computed data. For H2 boundary condition, these correlations are obtained as follows,

2

1.50 0.0

1

1773

)

 pffiffiffi 2:5180  1  3Rc

3

0 0.0

(

tw  tc Tw  Tc ¼ t w  tc T w  Tc

(70)

where, tw is the local wall temperature, tc is the centreline temperature and t w is the average wall temperature. For circular duct, however, qw is unity, irrespective of the location on the periphery of the duct. The variations in the maximum wall temperature, in the form of qw,max1, or equivalently ðtw;max  t w Þ=ðt w  tc Þ and the minimum wall temperature, in the form of 1qw,min, or equivalently ðt w  tw;min Þ=ðt w  tc Þ are presented as functions of the normalised radius of curvature in Fig. 11 for both square and triangular ducts with rounded corners. In this figure, the normalised radius of curvature is pffiffiffi defined as Rc/Rc,max (i.e., ¼ Rc for square duct and ¼ 3Rc for triangular duct). The figure clearly shows that the temperature nonuniformity at the wall is higher for triangular duct, irrespective of the value of radius of curvature. It is also interesting to note that although at lower values of radius of curvature the temperature non-uniformity is higher at the corner, beyond a certain value of normalised radius of curvature (approximately 0.46 for square duct and approximately 0.48 for triangular duct) more non-uniformity is observed at the midpoint of the straight portion of the ducts.

1774

S. Ray, D. Misra / International Journal of Thermal Sciences 49 (2010) 1763e1775

maintained at the same value. At this point, it may be mentioned here that although the area goodness factor provides some idea about the effectiveness of the duct, it is not the only criterion based on which the performance of a duct should be judged. Rather, one requires setting different constraints, as suggested by Webb and Eckert [14], Bergles et al. [15,16], Bergles [17], Webb [18] and Webb and Bergles [19] and subject the Performance Evaluation Criterion (i.e., the objective functions) to these constraints in order to determine the optimal shape of the duct geometry. Such a study is, however, beyond the scope of the present paper.

M a x i m u m , M i n i m u m W a l l T em p e r a t u r e

0.8 0.7

Square Duct Triangular Duct

θ w,max - 1

0.6

1 - θ w,min 0.5 0.4 0.3 0.2

4. Conclusions

0.1

θ w,max - 1

0.0 0.0

0.1

0.2

1 - θ w,min 0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Normalised Radius of Curvature, Rc/Rc,max Fig. 11. Variations of normalised maximum and minimum wall temperature with normalised radius of curvature for square and triangular ducts for H2 boundary condition.

The area goodness factor, which is the ratio of Colburn factor, j and the Fanning friction factor, f, is given as;

G ¼

j NuPr1=3 ¼ fRe f

(71)

However, for hydrodynamically and thermally fully-developed flow through ducts, having constant cross sections in the axial direction, both the Nusselt number and the friction factor (in the form of fRe) are independent of Prandtl number. Therefore, for the present purpose, one may consider only the variations of GPr1/3 ¼ (j/f)Pr1/3 ¼ Nu/fRe as functions of the normalised radius of curvature. In Fig. 12, the variations of area goodness factor, in the form of Nu/ fRe, are shown for both square and triangular ducts with rounded corners. From the figure it is obvious that the area goodness factor is higher for the square duct as compared to that for the triangular ducts for same normalised radius of curvature, irrespective of the boundary condition and attains its maximum value for the circular duct in an asymptotic manner. Further, the figure also shows that the area goodness factor is higher for the H1 boundary condition, as compared to that for the H2 boundary condition, irrespective of the shape of the duct, as long as the normalised radius of curvature is

0.30

H1 B.C.

(j/f)Pr1/3 = Nu/fRe

0.25

0.20

H2 B.C.

0.15

Square Duct Triangular Duct 0.10 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Normalised Radius of Curvature, R c /R c,max Fig. 12. Variation of area goodness factor with normalised radius of curvature.

In the present work, analysis is carried out to study the thermal hydraulic characteristics of laminar fully-developed flow through ducts of square and triangular cross sections with rounded corners, employing both H1 and H2 boundary conditions. For square duct, varied from 0 to 1 and the dimensionless radius of curvature, Rc, is p ffiffiffi for triangular ducts, Rc is varied from 0 to 1= 3. For both the cases, at lower limit of Rc, respective perfect polygonal duct shape is obtained, where as, the upper limit of Rc represents a circular duct. The fluid flow results are presented for friction factor, in the form of fRe for both types of ducts, where as, the heat transfer results are obtained in the form of local and average Nusselt numbers for all the cases. Comparisons of fRe and Nu, obtained from the present analysis, with the existing results show excellent agreement. The major conclusions of the present study are as follows, 1. The friction factor (fRe) for both square and triangular ducts increases rapidly with the increase in Rc, particularly at lower values of Rc. For triangular duct, the fRe data shows maxima at Rc z 0.35. However, for both the cases, fRe tend to their values corresponding to that of the circular duct (fRec ¼ 16) in an asymptotic manner as Rc tends to the respective maximum limit. For both types of duct, two alternative correlations are obtained. For square duct, both of these correlations show excellent agreement with the computed data. However, for triangular duct, the correlation obtained in the form of a polynomial works better than the other correlation and hence, it is recommended. 2. Variation of the local Nusselt number along the periphery of the duct shows the presence of non-uniformity in local heat transfer coefficient. This non-uniformity, however, decreases with the increase in Rc, as the duct shape tends to that of a circular duct (for which the local Nusselt number is uniform). 3. For square ducts, the non-uniformity in NuL is always higher for H1 boundary condition. It is also observed that the effectiveness of the rounded portion (which acts as ‘hot-spots’ and hence, is less effective than the circular duct) continuously increases with the increase in Rc and the effectiveness of this part is always higher for H2 boundary condition. For the straight portion, the effectiveness, which is always more for H1 boundary condition, first increases with the increase in Rc and attains a maximum value around Rc z 0.4. The local Nusselt number at this location finally drops to the value corresponding to that of the circular duct. 4. For triangular ducts, the effectiveness of the rounded portion (which is always less for H1 boundary condition) increases continuously with the increase in Rc for H2 boundary condition. However, for H1 boundary condition and for very low values of Rc, NuL initially remains almost constant and increases continuously for larger values of Rc. It is also observed that unlike square duct, the effectiveness of the straight portion (which is always more than that of the circular duct) at lower values of Rc is more for H2 boundary condition and it continues to be more upto Rc z 0.325. For both the boundary conditions,

S. Ray, D. Misra / International Journal of Thermal Sciences 49 (2010) 1763e1775

however, the local Nusselt number initially increases, attains maxima at Rc ¼ 0.25 and Rc ¼ 0.075 for H1 and H2 boundary conditions respectively and subsequently drops to the value corresponding to that of the circular duct at the maximum limit of Rc. 5. Although the local Nusselt numbers show different behaviour for square and triangular ducts, the variations of average Nusselt numbers with Rc show almost similar general trend. For all the cases, correlations are obtained in two different forms. Although these forms vary significantly in their appearance, they almost coincide with each other and predict the computed data with reasonable accuracy. 6. Presentation of area goodness factor shows that the best performance is obtained with the circular duct. However, this issue should be taken up later, by considering the optimisation based on performance evaluation criterion. References [1] R.K. Shah, A.L. London, Laminar flow forced convection in ducts. Adv. Heat Transfer(Suppl. 1) (1978). [2] J.P. Hartnett, M. Kostic, Heat transfer to Newtonian and non-Newtonian fluids in rectangular ducts. Adv. Heat Transfer vol. 19 (1989). [3] U.A. Sastry, Heat transfer by laminar forced convection in a pipe of curvilinear polygonal section. J. Sci. Eng. Res. vol. 7 (1963) 281e292. [4] R.M. Cotta, Integral Transform in Computational Heat Transfer and Fluid Flow. CRC Press, Boca Raton, Florida, USA, 1993. [5] E.M. Sparrow, R. Siegel, A variational method for fully developed laminar heat transfer in ducts. Trans. ASME, J. Heat Transfer vol. 81 (1959) 157e167. [6] E.M. Sparrow, A.L. Loeffler Jr., Longitudinal laminar flow between cylinders arranged in regular array. AICHE J. vol. 5 (1959) 325e330.

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[7] E.M. Sparrow, A.L. Loeffler Jr., H.A. Hubbard, Heat transfer to longitudinal laminar flow between cylinders. Trans. ASME, J. Heat Transfer vol. 83 (1961) 415e422. [8] K.C. Cheng, M. Jamil, Laminar flow and heat transfer in circular ducts with diametrically opposite flat sides and ducts of multiply connected cross sections. Can. J. Chem. Eng. vol. 48 (1970) 333e334. [9] D.A. Ratkowsky, N. Epstein, Laminar flow in regular polygonal ducts with circular centered cores. Can. J. Chem. Eng. vol. 46 (1968) 22e26. [10] S.L. Hagen, D.A. Ratkowsky, Laminar flow in cylindrical ducts having regular polygonal shaped cores. Can. J. Chem. Eng. vol. 46 (1968) 387e388. [11] E.M. Sparrow, A. Haji-Sheikh, Flow and heat transfer in ducts of arbitrary shapes with arbitrary thermal boundary conditions. Trans. ASME, J. Heat Transfer vol. 88c (1966) 351e358. [12] R.K. Shah, Laminar flow friction and forced convection heat transfer in ducts of arbitrary geometry. Int. J. Heat Mass Transfer vol 18 (1975) 849e862. [13] G. Laha, Pressure Drop and Heat Transfer Characteristics of Laminar Fully Developed Flow Through Square and Equilateral Triangular Ducts with Rounded Corners, M.E. Thesis, Jadavpur University, India, 2001. [14] R.L. Webb, E.R.G. Eckert, Application of rough surfaces to heat exchanger design. Int. J. Heat Mass Transfer vol. 15 (1972) 1647e1658. [15] A.E. Bergles, A.R. Blumenkrantz, J. Taborek, Performance evaluation criterion for enhanced heat transfer surfaces, 5th International Heat Transfer Conference, Tokyo, vol. II, 1974, pp. 239e243. [16] A.E. Bergles, R.L. Bunn, G.H. Junkhan, Extended performance evaluation criterion for enhanced heat transfer surfaces. Lett. Heat Mass Transfer vol. 1 (1974) 113e120. [17] A.E. Bergles, Application of heat transfer augmentation. in: S. Kakac, A. E. Bergles, F. Mayinger (Eds.), Heat Exchangers: Thermal-Hydraulic Fundamentals and Design. Hemisphere Publishing Corporation, Washington, D.C, 1981, pp. 883e911. [18] R.L. Webb, Performance evaluation criteria for use of enhanced heat transfer surfaces in heat exchanger design. Int. J. Heat Mass Transfer vol. 24 (1981) 715e726. [19] R.L. Webb, A.E. Bergles, Performance evaluation criterion for selection of heat transfer surface geometries used in low Reynolds number heat exchangers. in: S. Kakac, R.K. Shah, A.E. Bergles (Eds.), Low Reynolds Number Flow Heat Exchangers. Hemisphere Publishing Corporation, Washington, D.C, 1983, pp. 735e742.