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International Journal of Heat and Mass Transfer 53 (2010) 2022–2034

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International Journal of Heat and Mass Transfer

journal homepage: www.elsevier .com/locate / i jhmt

Convective heat transfer enhancement in low Reynolds number flows withwavy walls

Fernando V. Castellões a, João N.N. Quaresma b,c, Renato M. Cotta c,*

a Petrobras R&D Center – CENPES, Rio de Janeiro, RJ, Brazilb School of Chemical Engineering, Universidade Federal do Pará – FEQ/UFPA, Belém, PA, Brazilc Laboratory of Transmission and Technology of Heat – LTTC, Mechanical Engineering Department – POLI & COPPE/UFRJ, Universidade Federal do Rio de Janeiro, Cx. Postal 68503,Rio de Janeiro, RJ 21945-970, Brazil

a r t i c l e i n f o

Article history:Received 14 April 2009Received in revised form 14 December 2009Accepted 14 December 2009Available online 3 February 2010

Keywords:Low Reynolds number flowsMicro-channelsWavy wallsHeat transfer enhancementForced convectionIntegral transforms

0017-9310/$ - see front matter � 2010 Elsevier Ltd. Adoi:10.1016/j.ijheatmasstransfer.2009.12.054

* Corresponding author. +55 21 5562 5566; fax: +5E-mail address: [email protected] (R.M. Cotta

a b s t r a c t

The present work reports the analysis of combining low Reynolds number flows and channels with wallcorrugation and the corresponding thermal exchange intensification achieved. The proposed modelinvolves axial heat diffusion along the fluid and adiabatic regions both upstream and downstream tothe corrugated heat transfer section, in light of the lower values of Reynolds numbers (and consequentlyPeclet numbers) that can be encountered in the present class of problems. Aimed at developing a fast andreliable methodology for optimization purposes, the related laminar velocity field is obtained by anapproximate analytical solution valid for smooth corrugations and low Reynolds numbers, typical forinstance of micro-channel configurations, locally satisfying the continuity equation. A hybrid numeri-cal-analytical solution methodology for the energy equation is proposed, based on the Generalized Inte-gral Transform Technique (GITT) in partial transformation mode for a transient formulation. The hybridapproach is first demonstrated for the case of a smooth parallel-plates channel situation, and the impor-tance of axial heat conduction along the fluid is then illustrated. Heat transfer enhancement is analyzed interms of the local Nusselt number and dimensionless bulk temperature along the heat transfer section.An illustrative sinusoidal corrugation shape is adopted and the influence of Reynolds number and corru-gation geometric parameters is then discussed.

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1. Introduction

Within the scope of thermal engineering, energy conservationand sustainable development demands have been driving researchefforts towards more energy efficient equipments and processes.The petroleum and process industries have been quite active inprogressively incorporating heat transfer enhancement solutionsto the efficiency increase requirements along the years [1]. Morerecently, heat exchangers employing micro-channels with charac-teristic dimensions below 500 microns have been calling the atten-tion of researchers and practitioners, towards applications thatrequire high heat removal demands and/or space and weight lim-itations [2]. Motivated by the search for optimal solutions in heatexchange rates, Steinke and Kandlikar [3] critically analyzed vari-ous heat transfer enhancement techniques as applied to the mi-cro-channels scale. Among the passive enhancement techniquesthen discussed, the authors emphasize the utilization of treatedsurfaces, rough or corrugated walls and additives for working flu-ids. Several other approaches were disregarded in light of the dif-

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5 21 5562 8383.).

ficulties in manufacturing or mechanically modifying the thermalsystem at the micro-scale. In parallel, a few previous works haveaddressed the interest in investigating channel corrugations atthe micro-scale, either for liquid or gaseous flows [4,5].

The research here reported addresses the convective heat trans-fer in low Reynolds number flows within channels possibly en-hanced by the presence of axial heat conduction along the fluidand wall corrugations. First, the typical low Reynolds numbers inapplications such as micro-systems may lead to low values of thePeclet number that bring up some relevance to the axial heat dif-fusion along the fluid stream, especially for regions close to the in-let. Then, both the upstream and downstream sections of themicro-channel that are not actually part of the heat transfer sec-tion, may participate in the overall heat transfer process, and final-ly yield different predictions than those reached by making use ofconventional macro-scale relations for ordinary liquids or gases.Therefore, our first objective is to inspect such effects of the axialheat diffusion within the fluid. For this reason, it was initially nec-essary to identify the range of governing parameters to be ana-lyzed, in order to allow for an appropriate modeling of therelevant physical phenomena that may appear at this dimensionalscale. Second, either due to the inherent difficulties in achieving

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Nomenclature

Aij; Bni; Cmn coefficients defined by Eqs. (24a), (24b) and (24c),respectively

d half the distance between the flat wallsDmn; Emn coefficients defined by Eqs. (24d) and (24e), respectivelyfy�0; fy0 function that describes lower wall boundary, dimen-

sional and dimensionless, respectivelyfy�1; fy1 function that describes upper wall boundary, dimen-

sional and dimensionless, respectivelyFmn coefficient defined by Eq. (24f)g source function defined by Eq. (11g)�gm coefficient defined by Eq. (24h)Gmn coefficient defined by Eq. (24g)L�, L total length of the channel, dimensional and dimension-

less, respectivelyL�1; L1 length of region 1, dimensional and dimensionless,

respectivelyL�2; L2 total length of the regions 1 and 2, dimensional and

dimensionless, respectivelyMS sub-domain discretization parameter (maximum al-

lowed step size)ny0

;ny1unit normal vector associated with the lower and upperwall boundaries, respectively

N1; N2; N3 truncation orders for the temperature expansions inregions 1, 2 and 3, respectively

Nu0; Nu1 local Nusselt numbers associated with the lower andupper wall boundaries, respectively

Nwi;NCm norms defined by Eqs. (16e) and (17e), respectivelyPe Peclet numberRe Reynolds numbert�, t time variable, dimensional and dimensionless, respec-

tivelyT�; TA temperature field, dimensional and dimensionless,

respectivelyTm bulk temperature

T1;i; T2;m; T3;i transformed potentials for the regions 1, 2, and 3,respectively

T�w;0; T�w;1 temperatures of the lower and upper wall boundaries,

respectivelyTA;w;1 dimensionless temperature of the upper wall boundaryu�, u longitudinal velocity component, dimensional and

dimensionless, respectivelyum average velocityv�, v transversal velocity component, dimensional and

dimensionless, respectivelyx� longitudinal coordinatex1; x2; x3 normalized longitudinal coordinatesy�, y transversal coordinate, dimensional and dimensionless,

respectively

Greek lettersa�, a channel amplitude, dimensional and dimensionless,

respectivelyaf fluid thermal diffusivitybm eigenvalues defined by Eq. (17f)Cm eigenfunctions defined by Eq. (17d)DTc reference temperature differenceli eigenvalues defined by Eq. (16f)n dimensionless longitudinal coordinate/ filtered potential�/m coefficient defined by Eq. (24i)wi eigenfunctions defined by Eq. (16d)

Subscripts and superscripts_ integral transformed quantitiesi, j, m, n expansions indices1, 2, 3 relative to the regions 1, 2, and 3, respectively

F.V. Castellões et al. / International Journal of Heat and Mass Transfer 53 (2010) 2022–2034 2023

smooth surfaces during micro-fabrication processes or to the ac-tual purpose of improving mixing and/or heat transfer, micro-channels with irregularly shaped walls started gaining some focusin the heat and mass transfer literature, as pointed out in the briefreview above. Thus, the analysis of laminar forced convectionwithin micro-channels with corrugated walls, and the possibleheat transfer enhancement effect achieved, is another goal of thepresent study.

The steady two-dimensional flow problem was handled byadopting an approximate analytical solution that essentiallyadapts the fully developed velocity profile to the wall geometricvariations, satisfying the fully developed momentum equationand the continuity equation [6]. Such simplified approach wasintroduced in [6] aimed at the solution of incompressible flowproblems in low Reynolds number situations and gradual geomet-ric variations in converging or diverging walls. The solution meth-odology for the energy equation first introduces a domaindecomposition strategy, redefining the coordinates systems forthe three heat transfer regions, so as to rewrite the problem inthe form of a system of equations within the same mathematicaldomain, coupled at the interfaces of the three regions. Then, a hy-brid numerical-analytical solution based on the Generalized Inte-gral Transform Technique (GITT) is proposed [7–9], whichconsists on the elimination of the transversal coordinate byintegral transformation, and results in a coupled system of one-dimensional partial differential equations for the transformed tem-peratures. This system is then handled numerically with local error

control by making use of the Method of Lines implemented in theMathematica system [10]. This so-called partial transformationmode of the GITT allows for the accurate and flexible solution ofmultidimensional partial differential systems [11] and has beenpreviously employed in the solution of transient and periodicforced convection within smooth parallel-plates micro-channels[12–14]. This alternative hybrid solution strategy to the more usualfull integral transformation mode is of particular interest in thetreatment of transient convection-diffusion problems with a pref-erential convective direction. In such cases, the partial integraltransformation in all but one space coordinate, may offer an inter-esting combination of relative advantages between the eigenfunc-tion expansion approach and the selected numerical method forhandling the coupled system of one-dimensional partial differen-tial equations that results from the transformation procedure. Itshould be recalled that the GITT approach has already been previ-ously applied to the analysis of heat transfer within smooth rectan-gular micro-channels [15,16] for steady state situations.

The situation of a smooth parallel-plates channel is here firstanalyzed, for typical values of the governing parameters, so as toprovide validations of the hybrid numerical-analytical solutionfor the energy equation, while also illustrating the importance ofthe axial heat diffusion along the fluid, especially in the transitionfrom the first adiabatic region and the heat transfer section. Then,the approximate analytical solution of the flow problem is demon-strated, as compared to benchmark results of the two-dimensionalNavier-Stokes equations, as obtained from the GITT itself in previ-

2024 F.V. Castellões et al. / International Journal of Heat and Mass Transfer 53 (2010) 2022–2034

ous works [17]. The illustrative situation of sinusoidal symmetriccorrugated walls is considered more closely, allowing for paramet-ric variations on the corrugation geometry. Finally, the heat trans-fer enhancement is inspected for a few different combinations offlow, thermal and geometric parameters, in terms of both thedimensionless bulk temperature and local Nusselt number alongthe heat transfer section. The present contribution is thus relatedto the fundamental analysis of forced convection in low Reynoldsnumber flows, here illustrated by flow within micro-channels asrequired for the design of micro-heat exchangers, including the ef-fects of axial heat conduction and wall corrugation or roughness onheat transfer enhancement.

2. Analysis

We consider transient laminar forced convection within chan-nels formed by smooth or corrugated plates. Three regions alongthe channel are considered in the problem formulation, as de-scribed in Fig. 1 below, which is a general representation of a chan-nel with asymmetric irregular walls. First, an adiabatic region withsmooth walls, followed by the heat transfer section with pre-scribed temperatures at the corrugated walls, and the third one,following the corrugated region, is again made of smooth adiabaticwalls.

The two-dimensional steady flow is assumed to be laminar andincompressible, with temperature independent thermophysicalproperties, while viscous dissipation and natural convection effectsare neglected. Also, the continuum hypothesis is considered to bevalid throughout the flow region, without slip or temperature jumpat the boundaries, such as in the case of liquids flowing in micro-channels. Due to the possible low values of Peclet number, in lightof the lower range of Reynolds numbers, axial diffusion along thefluid is not disregarded. Also, the flow is assumed to be fully devel-oped at the first section entrance, but varies along the axial coordi-nate once the corrugated section is reached.

In obtaining the velocity field along the flow, the full Navier-Stokes equations should be employed, yielding the variable veloc-ity components and pressure field along the transversal and longi-tudinal directions, as recently demonstrated in [17], where theGeneralized Integral Transform Technique (GITT) has been em-ployed in the hybrid numerical-analytical solution of this laminarflow. However, for sufficiently low Reynolds numbers and smoothvariations on the wall corrugations, an approximate solution hasbeen previously proposed [6], essentially by accounting for the var-iable cross section within the local mass balance equation in a qua-

Fig. 1. Geometry and coordinates system fo

si-fully developed formulation, thus neglecting the inertia termsinfluence on the velocity components modification. These explicitsolutions for the velocity components are particularly handy, espe-cially in pre-design and optimization tasks, and shall be hereexploited to demonstrate the transient thermal problem solution.Thus, following the approach in [6], and for sufficiently smoothwall geometries defined by the x* functions fy�0 and fy�1, as depictedin Fig. 1, the velocity components are analytically derived as:

u�ðx�; y�Þ ¼ �ðy�0 � y�1Þ½fy�0ðx�Þ � fy�1ðx�Þ�

3 f�y�2 þ y�½fy�0ðx�Þ þ fy�1ðx�Þ�

� fy�0ðx�Þfy�1ðx�Þg ð1aÞ

v�ðx�; y�Þ ¼ �ðy�0 � y�1Þ½fy�0ðx�Þ � fy�1ðx�Þ�

4 f6½y� fy�0ðx�Þ�½y� fy�1ðx�Þ�

� ffy�01 ðx�Þ½y� fy�0ðx�Þ� � fy�00 ðx�Þ½y� fy�1ðx�Þ�gg ð1bÞ

It can be noticed that according to the approximate solution inEqs. (1a) and (1b), at the entrance and exit of the corrugated sec-tion, as well as for a smooth heat transfer section, the flow be-comes the classical parabolic fully developed velocity profile forparallel-plates, given in the present coordinates system by:

u�ðy�Þ ¼ 6

ðy�0 � y�1Þ2 ½�y�2 þ y�ðy�0 þ y�1Þ � y�0y�1� ð2Þ

while the transversal velocity component vanishes in these regions.Once the velocity field is available, the energy equation in the

whole region is given as:

@T�

@t�þ u�

@T�

@x�þ v� @T�

@y�¼ af

@2T�

@x�2þ @

2T�

@y�2

!;

0 < x� < L�; fy�0ðx�Þ < y� < fy�1ðx�Þ; t� > 0 ð3aÞ

T�ðx�; y�;0Þ ¼ T�0ðx�; y�Þ;

T�ð0; y�; t�Þ ¼ T�eðy�; t�Þ;@T�

@x�

��x�¼L�

¼ 0 ð3b—dÞ

@T�

@y�

��y�¼fy�0ðx�Þ

¼ 0

@T�

@y�


¼ 0

8>>><>>>:

; for 0 < x� < L�1;

T�ðx�; y� ¼ fy�0ðx�Þ; t�Þ ¼ T�w;0T�ðx�; y� ¼ fy�1ðx�Þ; t�Þ ¼ T�w;1

(; for L�1 6 x� 6 L�2 ð3e — hÞ

r heat transfer in corrugated channel.


@T�

@y�


¼ 0

@T�

@y�


¼ 0

8>>><>>>:

; for L�2 < x� < L� ð3i; jÞ

The following dimensionless groups are defined:

n ¼ af

umd2 x� ¼ 1Pe

x�

d; L ¼ af

umd2 L� ¼ 1Pe

L�

d;

L1 ¼af

umd2 L�1 ¼1Pe

L�1d; L2 ¼

af

umd2 L�2 ¼1Pe

L�2d;

y ¼ y�

d; uðn; yÞ ¼ u�ðx�; y�Þ

um; vðn; yÞ ¼ v�ðx�; y�Þ

um;

t ¼ af

d2 t�; TAðn; y; tÞ ¼T�ðx�; y�; t�Þ � T�w;0

DTc;

Pe ¼ umdaf

; d ¼ y�1 � y�02

ð4a—kÞ

Then, the dimensionless form of the temperature problem iswritten as:

@TA

@tþ uðn; yÞ @TA

@nþ Pevðn; yÞ @TA

@y¼ 1

Pe2

@2TA

@n2 þ@2TA

@y2 ;

0 < n < L; fy0ðnÞ < y < fy1ðnÞ; t > 0 ð5aÞ

TAðn;y;0Þ¼ TA;0ðn;yÞ; TAð0;y;tÞ¼ TA;eðy;tÞ;@TA

@n

��n¼L

¼0 ð5b—dÞ

@TA

@y

��y¼fy0ðnÞ

¼ 0

@TA

@y

��y¼fy1ðnÞ

¼ 0

8>>><>>>:

; for 0 < n < L1;

TAðn; y ¼ fy0ðnÞ; tÞ ¼ 0TAðn; y ¼ fy1ðnÞ; tÞ ¼ TA;w;1

�; for L1 6 n 6 L2 ð5e — hÞ

@TA

@y

��y¼fy0ðnÞ

¼ 0

@TA

@y

��y¼fy1ðnÞ

¼ 0

8>>><>>>:

; for L2 < n < L ð5i; jÞ

It can be observed from Eqs. (5g) and (5h) that the formulationdoes not impose symmetry to the thermal problem boundary con-ditions in the heat transfer section, but if symmetry prevails, Eq.(5h) shall be an homogeneous one.

In light of the discontinuity on the boundary conditions at thewall junction of the three regions, it is more convenient to proposea domain decomposition to handle the three mathematical prob-lems coupled at the cross sections between each pair of regions.The three problems formulation should therefore include the con-tinuity conditions of temperature and heat flux at the fluid inter-faces between the regions. Thus, the problem formulation for thefirst adiabatic region becomes:

@T1

@tþ uðn; yÞ @T1

@nþ Pevðn; yÞ @T1

@y¼ 1

Pe2

@2T1

@n2 þ@2T1

@y2 ;

0 < n < L1; fy0ðnÞ < y < fy1ðnÞ; t > 0 ð6aÞ

T1ðn; y;0Þ ¼ TA;0ðn; yÞ ð6bÞ

T1ð0; y; tÞ ¼ TA;eðy; tÞ;@T1

@n

��n¼L1

¼ @TA;2

@n

��n¼L1

;

@T1

@y

��y¼fy0ðnÞ

¼ 0;@T1

@y

��y¼fy1ðnÞ

¼ 0 ð6c—fÞ

For the heat exchanging section, we have:

@TA;2

@tþ uðn; yÞ @TA;2

@nþ Pevðn; yÞ @TA;2

@y

¼ 1Pe2

@2TA;2

@n2 þ@2TA;2

@y2 ; L1 < n < L2; fy0ðnÞ < y < fy1ðnÞ;

t > 0 ð7aÞ

TA;2ðn; y;0Þ ¼ TA;0ðn; yÞ ð7bÞ

TA;2ðn ¼ L1; y; tÞ ¼ T1ðn ¼ L1; y; tÞ;TA;2ðn ¼ L2; y; tÞ ¼ T3ðn ¼ L2; y; tÞ ð7c;dÞ

TA;2ðn; y ¼ fy0ðnÞ; tÞ ¼ 0; TA;2ðn; y ¼ fy1ðnÞ; tÞ ¼ TA;w;1 ð7e; fÞ

and finally for the exiting section, also adiabatic, we may write:

@T3

@tþ uðn; yÞ @T3


@y¼ 1

Pe2

@2T3

@n2 þ@2T3

@y2 ;

L2 < n < L; fy0ðnÞ < y < fy1ðnÞ; t > 0 ð8aÞ

T3ðn; y; 0Þ ¼ TA;0ðn; yÞ ð8bÞ

@T3

@n

��n¼L2

¼ @TA;2

@n

��n¼L2

;@T3

@n

��n¼L

¼ 0;@T3

@y

��y¼fy0ðnÞ

¼ 0;

@T3

@y

��y¼fy1ðnÞ

¼ 0 ð8c—fÞ

The temperature problem for the heated section remains non-homogeneous in Eq. (7f). In order to homogenize the problem inthe transversal direction y, a filtering solution, /ðn; yÞ, is employedfor the potential TA;2ðn; y; tÞ, in the form:

TA;2ðn; y; tÞ ¼ T2ðn; y; tÞ þ /ðn; yÞ ð9Þ

A simple and sufficiently general form for the filter /ðn; yÞ is ob-tained by satisfying the diffusion operator in the transversal direc-tion, which yields:

/ðn; yÞ ¼ TA;w;1fy0ðnÞ � y

fy0ðnÞ � fy1ðnÞð10Þ

Then, the homogeneous problem for the corrugated section isgiven by:

@T2

@tþ uðn; yÞ @T2


@y

¼ 1Pe2

@2T2

@n2 þ@2T2

@y2 þ gðn; yÞ; L1 < n < L2;

fy0ðnÞ < y < fy1ðnÞ; t > 0 ð11aÞ

T2ðn; y; t ¼ 0Þ ¼ TA;0ðn; yÞ � /ðn; yÞ ð11bÞ

T2ðn ¼ L1; y; tÞ ¼ T1ðn ¼ L1; y; tÞ � /ðn ¼ L1; yÞ;T2ðn ¼ L2; y; tÞ ¼ T3ðn ¼ L2; y; tÞ � /ðn ¼ L2; yÞ ð11c;dÞ

T2ðn; y ¼ fy0ðnÞ; tÞ ¼ 0; T2ðn; y ¼ fy1ðnÞ; tÞ ¼ 0 ð11e; fÞ

where

gðn; yÞ ¼ 1Pe2

@2/

@n2 � Pevðn; yÞ @/@y� uðn; yÞ @/

@nð11gÞ

Next, the decomposed domain is described with three differentcoordinates systems, which match at the two interfaces, as shownin Fig. 2 below. The normalized longitudinal coordinates are thencomputed in terms of the original dimensionless coordinate as:

Fig. 2. Coordinates systems for the decomposed domain.


x1 ¼1L1

n; x2 ¼1

ðL2 � L1ÞðL2 � nÞ; x3 ¼

1ðL� L2Þ

ðn� L2Þ ð12Þ

Thus, in terms of the redefined coordinates systems the dimen-sionless problem formulation is given, for each of the three regions,as:

@T1

@tþ uðx1; yÞ

@T1

@x1

1L1þ Pevðx1; yÞ

@T1

@y

¼ 1Pe2

1L2

1

@2T1

@x21

þ @2T1

@y2 ; 0 < x1 < 1; y0 < y < y1; t > 0 ð13aÞ

T1ðx1; y; t ¼ 0Þ ¼ T1;0ðx1; yÞ ð13bÞ

T1ðx1 ¼ 0; y; tÞ ¼ T1;eðy; tÞ;@T1

@x1

��x1¼1¼ � L1

L2 � L1

@T2

@x2

��x2¼1

ð13c;dÞ

@T1

@y

��y¼y0

¼ 0;@T1

@y

��y¼y1

¼ 0 ð13e; fÞ

@T2

@t� 1ðL2 � L1Þ

uðx2; yÞ@T2

@x2þ Pevðx2; yÞ

@T2

@y

¼ 1Pe2

1

ðL2 � L1Þ2@2T2

@x22

þ @2T2

@y2 þ gðx2; yÞ;

0 < x2 < 1; fy0ðx2Þ < y < fy1ðx2Þ; t > 0 ð14aÞ

T2ðx2; y; t ¼ 0Þ ¼ T2;0ðx2; yÞ � /ðx2; yÞ ð14bÞ

T2ðx2 ¼ 0; y; tÞ ¼ T3ðx3 ¼ 0; y; tÞ � /ðx2 ¼ 0; yÞ;T2ðx2 ¼ 1; y; tÞ ¼ T1ðx1 ¼ 1; y; tÞ � /ðx2 ¼ 1; yÞ ð14c;dÞ

T2ðx2; y ¼ fy0ðx2Þ; tÞ ¼ 0; T2ðx2; y ¼ fy1ðx2Þ; tÞ ¼ 0 ð14e; fÞ

/ðx2; yÞ ¼ TA;w;1fy0ðx2Þ � y

fy0ðx2Þ � fy1ðx2Þ;

gðx2; yÞ ¼1

Pe2ðL2 � L1Þ2@2/

@x22

� Pevðx2; yÞ@/@y

þ 1L2 � L1

uðx2; yÞ@/@x2

ð14g;hÞ

@T3

@tþ 1ðL� L2Þ

uðx3; yÞ@T3

@x3þ Pevðx3; yÞ

@T3

@y

¼ 1Pe2

1

ðL� L2Þ2@2T3

@x23

þ @2T3

@y2 ; 0 < x3 < 1;

y0 < y < y1; t > 0 ð15aÞ

T3ðx3; y; t ¼ 0Þ ¼ T3;0ðx3; yÞ ð15bÞ

@T3

@x3

��x3¼0¼ � L� L2

L2 � L1

@T2

@x2

��x2¼0

;@T3

@x3

��x3¼1¼ 0 ð15c;dÞ

@T3

@y

��y¼y0

¼ 0;@T3

@y

��y¼y1

¼ 0 ð15e; fÞ

One may observe that after the coordinates systems redefini-tion, the problems are coupled only at the interfaces and all the do-main limits are the same, and therefore the dependent variablesmay be algebraically interpreted as applied to one single domainin the longitudinal coordinate (0 < x < 1).

Following the formalism in the GITT [7–9], the auxiliaryproblems are now defined to construct the eigenfunction expan-sions in each region. For regions 1 and 3 we adopt the sameeigenvalue problem, with second kind boundary conditions, gi-ven by:

d2wi

dy2 þ l2i wiðyÞ ¼ 0; y0 < y < y1; ð16aÞ

dwi

dy

��y¼y0

¼ 0;dwi

dy

��y¼y1

¼ 0 ð16b; cÞ

whose solution in terms of eigenfunctions, norms and eigenvaluesis readily found as

wiðyÞ ¼1; i ¼ 0cos½liðy� y0Þ�; i ¼ 1;2;3 . . .

�;

Nwi ¼Z y1

y0

w2i ðyÞdy ¼

y1 � y0; i ¼ 0y1�y0

2 ; i ¼ 1;2;3 . . .

(;

li ¼0; i ¼ 0

ipy1�y0

; i ¼ 1;2;3 . . .

(ð16d—fÞ

For region 2, the auxiliary problem has to account for the irreg-ular walls, which is incorporated into the eigenfunctions andeigenvalues via the functions of x that describe the transversal do-mains, as shown below:

@2Cm

@y2 þ b2mðxÞCmðx; yÞ ¼ 0; fy0ðxÞ < y < fy1ðxÞ ð17aÞ

Cm½y ¼ fy0ðxÞ� ¼ 0; Cm½y ¼ fy1ðxÞ� ¼ 0 ð17b; cÞ


Thus, the x-dependent eigenfunctions, norms and eigenvaluesare given by:

Cmðx; yÞ ¼ sinfbmðxÞ½y� fy0ðxÞ�g;

NCmðxÞ ¼Z fy1ðxÞ

fy0ðxÞC2

mðx; yÞdy ¼ fy1ðxÞ � fy0ðxÞ2

;

bmðxÞ ¼mp

fy1ðxÞ � fy0ðxÞ; m ¼ 1;2;3 . . . ð17d — fÞ

Once the eigenvalue problems have been defined and solved,the integral transform pairs (transform-inverse) are constructedas:

T1;iðx; tÞ ¼Z y1

y0

T1ðx; y; tÞwiðyÞdy;

T1ðx; y; tÞ ¼X1i¼0

1Nwi

T1;iðx; tÞwiðyÞ ð18a;bÞ

T2;mðx; tÞ ¼Z fy1ðxÞ

fy0ðxÞT2ðx; y; tÞCmðx; yÞdy;

T2ðx; y; tÞ ¼X1m¼1

1NCmðxÞ

T2;mðx; tÞCmðx; yÞ ð19a;bÞ

T3;iðx; tÞ ¼Z y1

y0

T3ðx; y; tÞwiðyÞdy;

T3ðx; y; tÞ ¼X1i¼0

1Nwi

T3;iðx; tÞwiðyÞ ð20a;bÞ

One may start with the integral transformation of Eqs. (13) forregion 1, operating with

R y1y0�wiðyÞdy, and after substitution of the

inverse formula (18b) and some manipulation, the transformedsystem becomes:

@T1;i

@tþ 1

L1

X1j¼0

Aij@T1;j

@x¼ 1

Pe2L21

@2T1;i

@x2 � l2i T1;iðx; tÞ;

0 < x < 1; t > 0 ð21aÞ

T1;iðx; t ¼ 0Þ ¼ T1;0;iðxÞ ð21bÞ

T1;iðx ¼ 0; tÞ ¼ Te;iðtÞ;

@T1;i

@x

��x¼1

¼ � L1

L2 � L1

X1n¼1

1NCnðx ¼ 1ÞBinðx ¼ 1Þ@T2;n

@x

��x¼1

ð21c;dÞ

Similarly, Eqs. (14) are operated on withR fy1ðxÞ

fy0ðxÞ�Cmðx; yÞdy to

yield the transformed version for the temperature problem in re-gion 2:

@T2;m

@t¼ 1

Pe2ðL2 � L1Þ2@2T2;m

@x2 þX1n¼1

@T2;n

@x

� 1

Pe2ðL2 � L1Þ22

ddx

1NCnðxÞ

� �NCnðxÞdmn þ

2NCnðxÞ

FmnðxÞ� �"

þ 1L2 � L1

1NCnðxÞ

DmnðxÞ� ��

þX1n¼1

T2;nðx; tÞ

� 1

Pe2ðL2 � L1Þ2d2

dx2

1NCnðxÞ

� �NCnðxÞdmn þ 2

ddx

1NCnðxÞ

� � "

�FmnðxÞ þ1

NCnðxÞGmnðxÞ

�þ 1

L2 � L1

� 1NCnðxÞ

CmnðxÞ þddx

1NCnðxÞ

� �DmnðxÞ

� �

�Pe1

NCnðxÞEmnðxÞ

� �� b2

mðxÞdmn

�þ �gmðxÞ; 0 < x < 1; t > 0

ð22aÞ

T2;mðx; t ¼ 0Þ ¼ T2;0;mðxÞ � �/mðxÞ ð22bÞ

T2;mðx ¼ 0; tÞ ¼X1j¼0

1Nwj

Bmjðx ¼ 0ÞT3;jðx ¼ 0; tÞ !

� �/mðx ¼ 0Þ

ð22cÞ

T2;mðx ¼ 1; tÞ ¼X1j¼0

1Nwj

Bmjðx ¼ 1ÞT1;jðx ¼ 1; tÞ !

� �/mðx ¼ 1Þ

ð22dÞ

And finally, for region 3, Eqs. (15) are operated on withR y1y0�wiðyÞdy, to furnish:

@T3;i

@tþ 1

L� L2

X1j¼0

Aij@T3;j

@x¼ 1

Pe2ðL� L2Þ2@2T3;i

@x2

� l2i T3;iðx; tÞ; 0 < x < 1; t > 0 ð23aÞ

T3;iðx; t ¼ 0Þ ¼ T3;0;iðxÞ ð23bÞ

@T3;i

@x

��x¼0

¼ � L� L2

L2 � L1

X1n¼1

1NCnðx ¼ 0ÞBinðx ¼ 0Þ@T2;n

@x

��x¼0

;

@T3;i

@x

��x¼1

¼ 0 ð23c;dÞ

The coefficients that appear on the transformed system Eqs.(21)–(23) are analytically obtained from the followingintegrations:

Aij ¼1

Nwj

Z y1

y0

uðyÞwiðyÞwjðyÞdy;

BniðxÞ ¼Z y1

y0

wiðyÞCnðx; yÞdy ð24a;bÞ

CmnðxÞ ¼Z fy1ðxÞ

fy0ðxÞuðx; yÞCmðx; yÞ

@Cn

@xdy;

DmnðxÞ ¼Z fy1ðxÞ

fy0ðxÞuðx; yÞCmðx; yÞCnðx; yÞdy ð24c;dÞ

EmnðxÞ ¼Z fy1ðxÞ

fy0ðxÞvðx; yÞCmðx; yÞ

@Cn

@ydy;

FmnðxÞ ¼Z fy1ðxÞ

fy0ðxÞCmðx; yÞ

@Cn

@xdy ð24e; fÞ

GmnðxÞ ¼Z fy1ðxÞ

fy0ðxÞCmðx; yÞ

@2Cn

@x2 dy;

�gmðxÞ ¼Z fy1ðxÞ

fy0ðxÞgðx; yÞCmðx; yÞdy;

�/mðxÞ ¼Z fy1ðxÞ

fy0ðxÞ/ðx; yÞCmðx; yÞdy ð24g—iÞ

Eqs. (21)–(23) form an infinite system of partial differentialequations having as independent variables the unified dimension-less longitudinal coordinate, x, and dimensionless time, t, and asdependent variables the transformed temperatures in each region.For computational purposes, the system is truncated to a finite or-der, truncating the eigenfunction expansions for each field in a suf-ficiently large number of terms for each region (N1; N2 and N3).Due to the x-variable nature of the system coefficients, Eqs. (24),the PDE system has to be numerically solved, for instance employ-

Table 2Convergence behavior of the centerline temperature as function of the dimensionlesstime for Pe = 30 along the three sub-regions of a smooth channel with differenttruncation orders N1 and N2 ¼ 8, N3 ¼ 4 and MS = 50.

t n N1 ¼ 2 N1 ¼ 4 N1 ¼ 6 N1 ¼ 8

Centerline temperature0.1 0.05 1.00186 0.99669 0.99815 0.99771

0.09 0.99858 0.98834 0.99064 0.989990.13 0.98290 0.96775 0.97071 0.969880.17 0.94467 0.92650 0.92975 0.928840.21 0.87445 0.85637 0.85943 0.858570.25 0.76784 0.75289 0.75534 0.754620.29 0.62938 0.61934 0.62100 0.620460.33 0.47374 0.46877 0.46967 0.469300.37 0.32224 0.32120 0.32158 0.32134


ing the Method of Lines with local error control as implemented onthe function NDSolve of the Mathematica symbolic-numerical com-putation platform [10].

For the thermal problem, results are reported in terms of thebulk temperature and the local Nusselt numbers at the two chan-nel walls (yielding the same results for the symmetric situation), asdefined below:

Tmðn; tÞ ¼R fy1ðnÞ

fy0ðnÞuðn; yÞTAðn; y; tÞdyR fy1ðnÞfy0ðnÞ

uðn; yÞdyð25aÞ

Nu1ðn; tÞ ¼�4

Tmðn; tÞðrTA � ny1

Þ;

Nu0ðn; tÞ ¼4

Tmðn; tÞðrTA � ny0

Þ ð25b; cÞ

In the chosen example discussed below, the problem formula-tion adopts the same geometry and boundary conditions as pre-sented by Wang and Chen [18]. The walls boundaries are thendescribed by the following functions along the longitudinal coordi-nate which describe a symmetric sinusoidal wavy profile:

fy�0ðx�Þ ¼

y�0; 0 < x� < L�1

y�0 þ a�sin pðx� � L�1Þ 12L�2

� �; L�1 < x� < L�2

y�0; L�2 < x� < L�

8>><>>: ð26aÞ

fy�1ðx�Þ ¼

y�1; 0 < x� < L�1

y�1 � a�sin pðx� � L�1Þ 12L�2

� �; L�1 < x� < L�2

y�1; L�2 < x� < L�

8>><>>: ð26bÞ

0.5 0.05 1.00002 0.99962 0.99974 0.999700.14 0.99881 0.99672 0.99715 0.997020.23 0.98762 0.98218 0.98316 0.982870.32 0.94645 0.93767 0.93925 0.938800.41 0.85043 0.84088 0.84280 0.842260.50 0.68889 0.68140 0.68316 0.682660.59 0.48152 0.47710 0.47828 0.477920.68 0.27631 0.27454 0.27511 0.274900.77 0.12341 0.12326 0.12347 0.123390.86 0.04082 0.04118 0.04124 0.04123

1.0 0.10 0.99985 0.99984 0.99984 0.999840.26 0.99964 0.99943 0.99946 0.999450.42 0.99664 0.99547 0.99562 0.995570.58 0.97813 0.97480 0.97526 0.975100.74 0.91262 0.90717 0.90809 0.907800.90 0.76446 0.75879 0.76010 0.759721.06 0.52500 0.52262 0.52223 0.523081.22 0.26621 0.26700 0.26704 0.267141.38 0.10090 0.10103 0.10104 0.101071.54 0.02708 0.02702 0.02703 0.02704

1.5 0.30 0.99985 0.99984 0.99984 0.999840.52 0.99962 0.99949 0.99950 0.999500.74 0.99654 0.99573 0.99581 0.995770.96 0.97918 0.96987 0.97210 0.971641.18 0.75643 0.75519 0.75517 0.755621.40 0.45739 0.45761 0.45764 0.457881.62 0.24305 0.24358 0.24361 0.24372

3. Results and discussion

A computational code was developed and implemented on theMathematica system [10] and the transformed partial differentialEqs. (21)–(23) were solved by employing the subroutine NDSolve,which uses the numerical Method of Lines to handle a wide rangeof coupled PDEs. Numerical results were produced for bulk tem-perature and local Nusselt numbers in both situations of smoothand wavy channels. First, a brief convergence analysis of the tem-perature field is presented and a few comparisons with literatureresults for simpler models are provided for validation purposes.Second, a physical interpretation of the axial conduction effectsin smooth ducts is offered, in light of the possible low values ofthe Peclet number, followed by the relevant parametric analysisin terms of the Reynolds number and the wavy wall geometry ef-fects on heat transfer enhancement.

The convergence behavior of the proposed eigenfunctionexpansion for the temperature fields is analyzed in terms of the fol-lowing parameters: truncation orders of the expansions for eachregion ðN1; N2; N3Þ; maximum step size along the spatial coordi-nate during the numerical solution of the transformed system(MaxStepSize parameter of the NDSolve subroutine [10]). Such

Table 1Dimensionless lengths of the sub-domains for each Peclet number used in thesimulations.

Pe Dimensionless lengths

L1 L2 L

1 200 230 28010 5 10 1530 1 5 6100 0.15 4.15 4.45

parameter is indirectly controlled via the minimum number ofsteps in the domain discretization allowed for in the algorithm(parameter MS, which stands for Max Steps).

The values of Peclet number adopted in the present analysiswere Pe = 1, 10, 30 and 100. The dimensionless longitudinal lengthsused in the simulations for each Peclet number are listed in Table 1below, which have proven to be compatible with the physicalproblem here formulated, i.e., they are sufficiently large so as towarrant satisfaction of the prescribed boundary conditions at theinlet and outlet of the channel.

The constructed computer code was first verified against thethermal problem results for transient convection within smoothparallel-plates channels [13,14] with one single region, i.e., justthe heat transfer section. A thorough convergence analysis wasthen undertaken on the bulk and centerline temperature results

1.84 0.10587 0.10622 0.10623 0.106272.06 0.03380 0.03389 0.03389 0.03390

1.9 0.20 0.99985 0.99985 0.99985 0.999850.48 0.99985 0.99984 0.99984 0.999840.76 0.99962 0.99951 0.99952 0.999521.04 0.96859 0.95545 0.95366 0.957291.32 0.63269 0.63222 0.63220 0.632641.60 0.36454 0.36431 0.36430 0.364531.88 0.19564 0.19567 0.19568 0.195792.16 0.09006 0.09023 0.09024 0.090282.44 0.03111 0.03120 0.03121 0.031222.72 0.00679 0.00680 0.00680 0.00680

Table 3Convergence behavior of the bulk temperature as function of the dimensionless timefor Pe = 30 along the three sub-regions of a smooth channel with different truncationorders N1 and N2 ¼ 8; N3 ¼ 4 and MS = 50.

t n N1 ¼ 2 N1 ¼ 4 N1 ¼ 6 N1 ¼ 8

Bulk temperature0.1 0.05 0.98772 0.98774 0.98779 0.98779

0.09 0.96227 0.96278 0.96289 0.962910.13 0.91567 0.91735 0.91755 0.917580.17 0.84351 0.84663 0.84689 0.846940.21 0.74526 0.74919 0.74947 0.749520.25 0.62488 0.62838 0.62860 0.628650.29 0.49100 0.49303 0.49314 0.493170.33 0.35628 0.35667 0.35668 0.356680.37 0.23501 0.23438 0.23434 0.23433

0.5 0.05 0.99902 0.99892 0.99892 0.998920.14 0.99139 0.99096 0.99094 0.990940.23 0.96276 0.96240 0.96242 0.962420.32 0.89243 0.89335 0.89345 0.893470.41 0.76603 0.76870 0.76889 0.768920.50 0.58916 0.59222 0.59242 0.592450.59 0.39093 0.39261 0.39273 0.392740.68 0.21385 0.21397 0.21402 0.214010.77 0.09167 0.09130 0.09132 0.091330.86 0.02933 0.02913 0.02915 0.02916

1.0 0.10 0.99982 0.99981 0.99981 0.999810.26 0.99903 0.99889 0.99888 0.998880.42 0.99243 0.99193 0.99190 0.991900.58 0.96205 0.96136 0.96135 0.961340.74 0.87438 0.87460 0.87465 0.874650.90 0.70359 0.70537 0.70550 0.705551.06 0.41799 0.41997 0.42005 0.420211.22 0.19611 0.19680 0.19683 0.196901.38 0.07252 0.07260 0.07261 0.072631.54 0.01904 0.01899 0.01900 0.01901

1.5 0.30 0.99983 0.99982 0.99982 0.999820.52 0.99929 0.99919 0.99918 0.999180.74 0.99381 0.99338 0.99336 0.993350.96 0.95015 0.94894 0.94903 0.949611.18 0.58119 0.58104 0.58104 0.581391.40 0.34425 0.34446 0.34448 0.344661.62 0.18146 0.18187 0.18189 0.181981.84 0.07810 0.07836 0.07837 0.078402.06 0.02452 0.02458 0.02458 0.02459

1.9 0.20 0.99985 0.99985 0.99985 0.999850.48 0.99984 0.99983 0.99983 0.999830.76 0.99936 0.99927 0.99927 0.999271.04 0.83701 0.83763 0.83759 0.838191.32 0.48041 0.48011 0.48010 0.480431.60 0.27577 0.27560 0.27560 0.275771.88 0.14738 0.14741 0.14742 0.147502.16 0.06729 0.06742 0.06743 0.067462.44 0.02293 0.02300 0.02300 0.023012.72 0.00491 0.00491 0.00491 0.00492

Table 4Convergence behavior of the centerline temperature as function of the dimensionlesstime for Pe = 30 along the three sub-regions of a corrugated channel ða ¼ 0:1Þ withdifferent values of the sub-domains discretization parameter MS and N1 ¼ 4; N2 ¼ 8and N3 ¼ 4.

t n MS = 100 MS = 150 MS = 200

Centerline temperature0.1 0.05 0.96101 0.96103 0.96102

0.08 0.88657 0.88662 0.886610.11 0.74938 0.74940 0.749390.14 0.55846 0.55842 0.558420.17 0.35510 0.35501 0.355020.20 0.18778 0.18778 0.18780

0.5 0.05 0.99962 0.99962 0.999620.14 0.99672 0.99672 0.996720.23 0.98218 0.98218 0.982180.32 0.93768 0.93767 0.937670.41 0.84089 0.84088 0.840880.50 0.68140 0.68139 0.681390.59 0.47707 0.47708 0.477080.68 0.27451 0.27454 0.274550.77 0.12328 0.12330 0.123300.86 0.04121 0.04121 0.04120

1.0 0.10 0.99984 0.99984 0.999840.26 0.99943 0.99943 0.999430.42 0.99547 0.99547 0.995470.58 0.97480 0.97480 0.974800.74 0.90717 0.90716 0.907160.90 0.75880 0.75879 0.758781.06 0.56627 0.56578 0.565671.22 0.33640 0.33612 0.336041.38 0.09140 0.09139 0.091371.54 0.01859 0.01857 0.01857

1.5 0.30 0.99984 0.99984 0.999840.52 0.99949 0.99949 0.999490.74 0.99573 0.99573 0.995730.96 0.97002 0.96971 0.969601.18 0.90198 0.90066 0.900341.40 0.33787 0.33746 0.337351.62 0.15549 0.15529 0.155241.84 0.10100 0.10089 0.100852.06 0.02252 0.02250 0.02249

1.9 0.20 0.99985 0.99985 0.999850.48 0.99984 0.99984 0.999840.76 0.99951 0.99951 0.999511.04 0.99679 0.99667 0.996391.32 0.59252 0.59161 0.591371.60 0.21974 0.21939 0.219301.88 0.17429 0.17401 0.173952.16 0.04220 0.04214 0.042132.44 0.02116 0.02114 0.021132.72 0.00370 0.00369 0.00369


so as to provide confidence on the numerical results to be reported.For instance, Tables 2 and 3 bring the convergence behavior of thecenterline and bulk temperatures, respectively, for Pe = 30 in thecase of a smooth walls duct. Clearly, the expansions convergencerates improve with both increasing dimensionless time and longi-tudinal coordinate, as expected, within the range of these indepen-dent variables as covered in Tables 2 and 3. In synthesis, theadoption of the truncation orders N1 ¼ 8; N2 ¼ 8 and N3 ¼ 4 wassufficient to provide three to four converged significant digits inthe temperature field throughout these tabulated results, in a fairlywide range of dimensionless time and longitudinal coordinate andfor the proposed values of Peclet number. It was also observed,though not evident from the single Pe number tables here pre-sented, that the truncation order of the expansions is required togradually increase as Pe increases.

In the sequence, Tables 4 and 5 illustrate the convergencebehavior of the centerline and bulk temperatures, respectively,for the wavy channel case, with Pe = 30 and a ¼ 0:1, by varyingthe sub-domains discretization parameter MS (maximum numberof steps) and with N1 ¼ 4; N2 ¼ 8 and N3 ¼ 4. Here the objective isto illustrate the effect of demanding more refined grids from theMethod of Lines numerical solution, and it has been verified thatthe results are apparently converged to at least four significant dig-its with MS = 150. Thus, this particular combination of parametersvalues was utilized for all subsequent simulations.

Next, an attempt was made of validating the present analysis,for the smooth parallel-plates case, against a previously reportedanalytical solution [19] that accounts for the upstream adiabaticregion, under steady laminar forced convection for low Pecletnumbers. Fig. 3 thus presents such comparison in terms of thedimensionless bulk temperature, where only the behavior along

Table 5Convergence behavior of the bulk temperature as function of the dimensionless timefor Pe = 30 along the three sub-regions of a corrugated channel ða ¼ 0:1Þ withdifferent values of the sub-domains discretization parameter MS and N1 ¼ 4; N2 ¼ 8and N3 ¼ 4.

t n MS = 100 MS = 150 MS = 200

Bulk temperature0.1 0.05 0.90490 0.90492 0.90491

0.08 0.78181 0.78184 0.781830.11 0.61260 0.61260 0.612600.14 0.42326 0.42322 0.423220.17 0.25115 0.25109 0.251110.20 0.12509 0.12509 0.12510

0.5 0.05 0.99892 0.99892 0.998920.14 0.99096 0.99096 0.990960.23 0.96241 0.96240 0.962400.32 0.89335 0.89335 0.893350.41 0.76870 0.76870 0.768700.50 0.59221 0.59221 0.592210.59 0.39259 0.39260 0.392600.68 0.21396 0.21397 0.213980.77 0.09132 0.09132 0.091330.86 0.02916 0.02915 0.02915

1.0 0.10 0.99981 0.99981 0.999810.26 0.99889 0.99889 0.998890.42 0.99193 0.99193 0.991930.58 0.96137 0.96136 0.961360.74 0.87460 0.87460 0.874600.90 0.70543 0.70539 0.705371.06 0.49158 0.49103 0.490861.22 0.24372 0.24354 0.243481.38 0.05385 0.05383 0.053831.54 0.01254 0.01253 0.01252

1.5 0.30 0.99982 0.99982 0.999820.52 0.99919 0.99919 0.999190.74 0.99338 0.99338 0.993380.96 0.95029 0.94923 0.948821.18 0.71454 0.71342 0.713141.40 0.20636 0.20609 0.206021.62 0.12681 0.12666 0.126611.84 0.07906 0.07897 0.078942.06 0.01336 0.01335 0.01335

1.9 0.20 0.99985 0.99985 0.999850.48 0.99983 0.99983 0.999830.76 0.99927 0.99927 0.999271.04 0.91993 0.91767 0.917081.32 0.37605 0.37549 0.375341.60 0.17770 0.17741 0.177341.88 0.13025 0.13004 0.129992.16 0.02830 0.02826 0.028252.44 0.01768 0.01766 0.017652.72 0.00219 0.00219 0.00218

0.001 0.01 0.1 1 10

ξ-L1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Bul

k Te

mpe

ratu

re

Present WorkTan and Normandia [19]

PeT=50

PeT=10

PeT=1

PeT=0.5

Fig. 3. Comparison of dimensionless bulk temperatures from present GITT solutionand from analytical steady state solution in [19] for a smooth parallel-plateschannel with an upstream adiabatic region and low Peclet number.


the heat transfer section is presented. Different values of a modi-fied Peclet number are employed, here called PeT , according tothe definition in [19]. A fairly good adherence between the twoindependent solutions can be observed providing further valida-tion to the proposed hybrid methodology. Also, one may alreadyobserve the marked influence of the axial diffusion along the fluidon the temperature behavior within the actual heat transfersection.

The influence of the axial heat diffusion on the channel tran-sient thermal response is now more closely examined and Fig. 4aand b show a comparison of the transient behavior of the bulk tem-perature in a smooth channel for the cases of Pe = 1 and 30 and atdifferent dimensionless times. One must observe that the longitu-dinal coordinate was multiplied by the Peclet number in thesegraphs, so that the resulting variable is essentially the channellength normalized by the transversal dimension. Clearly, the lowerPe number case (Pe = 1) involves a much more significant partici-

pation of the upstream region in the channel thermal behavior,within a longer transient process as well. For the higher value ofPe = 30, the participating upstream region is quite reduced in com-parison with the previous case, while in terms of dimensionlesstime the steady state is achieved much faster.

Fig. 5 then illustrates the influence of the axial heat diffusionwithin the fluid for the steady state dimensionless bulk tempera-tures along the channel for Pe = 1, 10, and 30, and especially acrossthe interface between the upstream adiabatic region and the heattransfer section. The pre-cooling (or pre-heating) effect providedby the presence of the upstream region is quite noticeable forthe lower values of Peclet number (Pe = 1 and 10), which is quiterelevant along the transient state as well, as shown in Fig. 4. As aresult, a heat transfer enhancement effect is in fact observed,resulting in lower temperature values within the same physical re-gion, for decreasing Peclet number. As can be deduced, significanterrors may occur if experimental results are employed to estimateNusselt numbers that assume the inlet temperature at the heattransfer section inlet as having the same value as the uniform tem-perature at the upstream region inlet, as usually considered for or-dinary liquids at the macro-scale.

The next step in the present analysis is the illustration of theadequacy of the approximate velocity field, Eqs. (1), in describingthe flow behavior within the wavy channel, as shown below inFig. 6. As previously mentioned, the simplified approach is ex-pected to provide reasonable results for lower values of the Rey-nolds number and for smoother corrugations. Thus, numericalresults for the full Navier-Stokes formulation were employed inthe verification of the present approximate analytical solutions,as obtained from the GITT hybrid numerical-analytical solutionavailable in [17]. For instance, Fig. 6a presents the longitudinalvelocity component at different axial positions along the corru-gated section, for the chosen values of Re = 10 and a ¼ 0:1, whileFig. 6b and c present the longitudinal velocity component for pro-gressively less smooth corrugations, with a ¼ 0:2 and a ¼ 0:3,respectively. It can be noticed that the increase in wall corrugationamplitude leads to a less accurate velocity component as obtainedfrom the approximate relations of Eqs. (1), especially in regionscloser to the inlet of the wavy region. Also, increasing the Reynolds

0 10 20 30 40 50 60

ξPe

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Bul

k Te

mpe

ratu

re

Pe=1t=50t=95t=140t=180t→∞

0 2.5 5 7.5 10 12.5 15 17.5 20 22.5 25 27.5 30

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Bul

k Te

mpe

ratu

re

Pe=30t=0.5t=1.0t=1.5t→∞

ξPe

a

b

Fig. 4. Transient behavior of the dimensionless bulk temperature in smoothparallel-plates channel along both the upstream adiabatic region and the heattransfer section (dashed line is the interface between the two regions): (a) Pe = 1;(b) Pe = 30.

-25 -20 -15 -10 -5 0 5 10 15 20

(ξ-L1)Pe

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Bul

k Te

mpe

ratu

re

Pe=1Pe=10Pe=30

Fig. 5. Comparison of dimensionless bulk temperatures in smooth parallel-plateschannel along both the upstream adiabatic region and heat transfer section atsteady state for different values of Peclet number (dashed line is the interfacebetween the two regions).


number leads to some loss of accuracy in the simplified solution,but one can see from Fig. 6d below that the results are still reason-ably accurate for the case of Re = 100 and a ¼ 0:1, and apparentlythe increase in corrugation amplitude of Fig. 6b and c was moresignificant in deviating the approximate solution from the con-verged GITT results for the full Navier-Stokes formulation thanthe increase in Reynolds number of Fig. 6d. The comparisons ofthe longitudinal velocity component are emphasized since the re-lated convective term is the most important one in the heat trans-fer problem to be addressed.

Fig. 7 illustrates the transient behavior of the bulk temperaturein the wavy channel for Pe = 10 and a ¼ 0:1, where the dashed linedenotes the interface between the adiabatic region 1 and the cor-rugated section, region 2. Clearly, we may see that due to the

low Peclet number, the axial diffusion along the fluid promotes asensible effect on the bulk temperature evolution within the accessregion (region 1) along the transient period. The steady state re-sults are however closer to the situation of an unheated inlet sec-tion, which would be obtained by the model that neglects axialdiffusion of heat within the fluid. Therefore, for micro-channelapplications that involve low Peclet numbers, the behavior of thethermal wave front can be markedly affected by the presence ofan adiabatic inlet section. It can also be observed that the bulktemperature behavior downstream already presents a slight fluctu-ating shape due to the presence of the wall corrugations, thoughthe lower values of Peclet tend to smear our the oscillations.

Next, Fig. 8 illustrates the effect of the Peclet number on thebulk temperatures temperature in the wavy channel with a ¼ 0:1for the steady state situation, by taking the two values Pe = 10and 30. Similarly to the smooth channel situation, one may clearlyobserve the more significant participation of region 1 in the ther-mal exchange due to the lower value of Pe, but also the more pro-nounced effects on the bulk temperature fluctuations due to thewall corrugations in the case of a smaller axial diffusion of heat(Pe = 30), when the transversal effects start playing a major rolewith respect to the longitudinal conduction.

For the heat transfer enhancement analysis it is of interest toevaluate the behavior of the Nusselt number under different corru-gation conditions. Fig. 9, for instance, illustrates the local Nusseltnumber results for Pe = 10 and a ¼ 0:1 and 0.2. The smooth paral-lel-plates case is also plotted for reference purposes, as the solidblack line. One may see that even with the lower corrugationamplitude value some noticeable heat transfer enhancement is al-ready evident, and marked increases in the local heat transfer coef-ficient are achieved for the higher corrugation amplitude value forthis value of Pe.

4. Conclusions

Laminar forced convection in low Reynolds number flows with-in channels with smooth and corrugated walls was analyzed and

0 2 4 6 8 10 12 14 16 18 201 3 5 7 9 11 13 15 17 19

ξ

-1.5

-1

-0.5

0

0.5

1

1.5

y

Present WorkSilva et al. [17]Re=10; α=0.1

0 2 4 6 8 10 12 14 16 18 201 3 5 7 9 11 13 15 17 19

ξ

-1.5

-1

-0.5

0

0.5

1

1.5

y


0 2 4 6 8 10 12 14 16 18 201 3 5 7 9 11 13 15 17 19

ξ

-1.5

-1

-0.5

0

0.5

1

1.5

y


0 2 4 6 8 10 12 14 16 18 201 3 5 7 9 11 13 15 17 19

ξ

-1.5

-1

-0.5

0

0.5

1

1.5

y


a

b

c

d

Fig. 6. Comparison of longitudinal velocity component for corrugated channel from approximate solution [6] and GITT solution [17] for: (a) Re = 10 and a ¼ 0:1; (b) Re = 10and a ¼ 0:2; (c) Re = 10 and a ¼ 0:3; (d) Re = 100 and a ¼ 0:1.


0 2.5 5 7.5 10 12.5 15 17.5 20 22.5 25

ξPe

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Bul

k Te

mpe

ratu

re

Pe=10; α=0.1t=2t=3t=5t→∞

Fig. 7. Transient behavior of bulk temperature for corrugated channel on regions 1and 2, Pe = 10, a ¼ 0:1.

-10 -5 0 5 10 15 20

(ξ-L1)Pe

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Bul

k Te

mpe

ratu

re

α=0.1Pe=10Pe=30

Fig. 8. Influence of Peclet number on the bulk temperature behavior for corrugatedchannel along regions 1 and 2 in steady state, Pe = 10 and 30 and a ¼ 0:1.

12 13 14 15 16 17 18 19 20 21 22 23 24 25

ξPe

0

4

8

12

16

20

24

28

32

36

40

Loc

al N

usse

lt N

umbe

r

Pe=10Smooth Wall - α=0.0Corrugated Wall - α=0.1Corrugated Wall - α=0.2

Fig. 9. Local Nusselt numbers at steady state for smooth and corrugated channelswith Pe = 10 and a ¼ 0 (parallel-plates), 0.1 and 0.2.


discussed to investigate possible heat transfer enhancement ef-fects. The physical modeling for the thermal problem includes dis-continuities in the boundary conditions at the channel walls,considering the interfaces between entrance and exit adiabatic re-gions and an intermediate heat transfer section. The thermal prob-lem was then solved with the Generalized Integral TransformTechnique (GITT) applied in partial transformation mode and witha transient formulation. The presented results exploited the bulktemperature field and the local Nusselt number within the heattransfer section of the channel. The axial diffusion effect alongthe fluid was first examined for a smooth parallel-plates channel,

in light of the lower values of Peclet number achievable in suchapplications due to the expectable low values of Reynolds number.The importance of accounting for an upstream adiabatic regionwas discussed, illustrating the resulting marked changes on thetemperature distribution behavior within the actual heat transfersection. Then, for the corrugated heat transfer section case, thevelocity field was obtained by making use of an approximate solu-tion methodology, shown to be appropriate for small scale corruga-tions and low Reynolds numbers. From such results, one maynotice the combined influence on the heat transfer enhancementalong the heat transfer section due to both the low values of Pecletnumber and the presence of corrugated walls. The analysis maynow proceed towards the utilization of optimization schemesaimed at the identification of wall profiles for optimally enhancedheat transfer, which can then be reconfirmed by the more accuratesolution of the flow problem employing the GITT itself, as previ-ously accomplished in [17].

Acknowledgements

The authors would like to acknowledge the partial financialsupport provided by CNPq and CENPES/Petrobras S.A., both in Bra-zil, along the developments here reported.

References

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