Heat transfer engineering an international journal, tập 32, số 6, 2011

New correlations for mass flow rate, radiative heat flux, and dimensionless maximum wall temperature are proposed in the emissivity range from 0.10 to 0.90, convergence angle ranging fro

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C Taylor and Francis Group, LLC

ISSN: 0145-7632 print / 1521-0537 online

DOI: 10.1080/01457632.2010.506166

Correlations for Natural Convection

in Vertical Convergent Channels With Conductive Walls and Radiative

Effects

LUIGI LANGELLOTTO1and ORONZIO MANCA2

1Centro Sviluppo Materiali S.p.A., Rome, Italy

2Dipartimento di Ingegneria Aerospaziale e Meccanica Seconda Universit`a degli studi di Napoli, Naples, Italy

Natural convection in air, in vertical convergent channels, is analyzed to carry out thermal design and optimization criteria.

A scale analysis is developed to estimate the optimal geometrical configuration in terms of total volume and average wall

temperature The best geometrical configuration obtained by this analysis is the parallel-plates channel New correlations

for mass flow rate, radiative heat flux, and dimensionless maximum wall temperature are proposed in the emissivity range

from 0.10 to 0.90, convergence angle ranging from 0◦to 10◦, ratio between minimum and maximum channel spacing in the

range from 0.048 to 1.0, aspect ratio, the ratio between wall length and minimum channel spacing, in the range from 10 to

58, and average channel Rayleigh number in the range from 5.0 to 2.3 × 10 5 For the same convergence angle and ratio

between minimum and maximum channel spacing ranges, new average Nusselt number correlations are also given These

correlations are evaluated for emissivity value equal to 0.90, for aspect ratio, referred to the minimum channel spacing,

ranging from 10 to 80 and average channel Rayleigh number ranging from 2.5 × 10 −2 to 2.3 × 10 5

INTRODUCTION

Cooling technology for electronic equipments and

compo-nents requires a deep knowledge of heat transfer phenomena

The main aim is to maintain a relatively constant component

temperature equal to or lower than the manufacturer’s

max-imum specified service temperature, in order to ensure

sys-tem performance and reliability [1–3] The design of natural

convection thermal control systems using simple relations is

certainly appealing Particular interest has been devoted to the

channel configuration and several contributions have dealt with

this geometry [4] An interesting problem is the heat transfer

in a convergent channel with two uniformly heated flat plates

[5–11]

The first numerical and experimental study of natural

con-vection in water, in a convergent vertical channel, was carried

This work was supported by MIUR with Articolo 12 D M 593/2000 Grandi

Laboratori “EliosLab.” A special acknowledgment is given to the reviewers; their

suggestions have improved the article.

Address corrrespondence to Professor Oronzio Manca, Dipartimento di

Ingegneria Aerospaziale e Meccanica Seconda Universit`a degli studi di Napoli,

Via Roma 29–81031, Aversa (CE), Italy E-mail: oronzio.manca@unina2.it

out in [5] The converging walls were maintained at the sameuniform temperature Natural convection in air in a uniform-wall-temperature convergent channel was investigated experi-mentally in [6], numerically in [7], and both numerically andexperimentally in [8] A numerical study on natural convection,

in vertical convergent channels, with uniform wall ture, for different convergence angles was carried out in [9] ANusselt number composite correlation was proposed for conver-

simula-tion and optimizasimula-tion for a vertically diverging and convergingchannel with laminar natural convection was accomplished in[10] For convergent channels, the results showed that the opti-mal angle between the two walls was approximately zero whenthe Rayleigh number was large The configuration of a verticalconvergent channel was numerically studied in [11] The twoprincipal flat plates, at uniform heat flux, were considered withfinite thickness and thermal conductivity An experimental in-vestigation on natural convection in air, in vertical convergentchannels, with uniform wall heat flux was presented in [12].For the lowest spacing, maximum wall temperature decreasedsignificantly, passing from the configurations of the parallel ver-

439

440 L LANGELLOTTO AND O MANCA

L LANGELLOTTO AND O MANCA 441Radiative effects are particularly interesting in convergent chan-

nels, due to the large view factor toward the ambient [13, 14] In

the first study, a numerical analysis was carried out in laminar,

two-dimensional steady-state conditions, with the two principal

flat plates at uniform heat flux and taking into account wall

con-ductivity and emissivity Average Nusselt numbers were

eval-uated and simple monomial correlations for average Nusselt

numbers, in terms of channel Rayleigh numbers, were

pro-posed In the second study, an experimental investigation on

natural convection in air, in a convergent channel, with uniform

heat flux at the walls, was carried out Average Nusselt numbers

were evaluated and simple monomial correlations for

dimen-sionless maximum wall temperatures and average Nusselt

num-bers were proposed in terms of channel Rayleigh numnum-bers in the

same ranges given in [13] Numerical results, obtained in [13],

were in very good agreement with experimental results given in

[14]

Design charts for the evaluation of thermal and geometrical

parameters, for natural convection in air, were proposed for

nat-ural convection in vertical convergent channels in [15] Thermal

design and optimization of a channel in stack of convergent

channels were obtained employing the correlations among the

more significant dimensionless thermal and geometrical

param-eters

Proposed correlations for natural convection in

conver-gent channels, given in the already-mentioned papers, are

reported in Table 1 In the present paper, a scale analysis

is carried out following the procedure given in references

[16–20] New correlations for convective heat transfer

con-tribution in terms of Reynolds numbers, dimensionless wall

temperature, and global Nusselt numbers are proposed More

accurate new correlations for the ratio between radiative and

global heat flux (radiative and convective heat fluxes) are

evaluated

The new correlations extend the analysis presented in

refer-ences [15] and [20–23] They are obtained by enlarging the

re-sults given in [13] to large values of channel aspect ratio and low

Rayleigh numbers This also allows evaluation of the thermal

behavior of the convergent channels in a possible fully

devel-oped flow The analysis is proposed to evaluate the previously

mentioned variable for vertical convergent channel, with

sur-face emissivity ranging from 0.10 to 0.90, for a single assigned

wall thickness and thermal conductivity, for convergence angle

From a different point of view, the present study may be

conceived as an effort to estimate the right balance between

the control of the maximum wall temperature and an

ap-plied symmetrical wall heat flux Moreover, this attempt can

also be viewed as the maximization of heat transfer for an

assigned available total volume that is constrained by space

Figure 1 Sketch of the configuration: (a) physical domain; (b) computational domain.

limitations This goal has been studied in references [10],[16], [19], and [24], and reviewed lately in [25] and morerecently in [26] The present geometry is important in elec-tronic cooling [9, 10, 27] and in solar energy components[28, 29]

MODEL DESCRIPTION AND NUMERICAL PROCEDURE

Model Description

The physical domain under investigation is shown inFigure 1a It consists of two nonparallel plates that form a ver-tical convergent channel Both plates are thermally conductive,gray, and heated at uniform heat flux The imbalance between

heated plates draws air into the channel The flow in the channel

is assumed to be steady-state, two-dimensional, laminar, pressible, with negligible viscous dissipation All thermophys-ical properties of the fluid are assumed to be constant, exceptfor the dependence of density on the temperature (Boussinesqapproximation), which gives rise to the buoyancy forces Thethermophysical properties of the fluid are evaluated at the am-

442 L LANGELLOTTO AND O MANCA

A two-dimensional conduction model is employed The heat

conduction equations in the steady-state regime with constant

thermophysical properties is:

∂2Ts

∂x2 +∂2Ts

The characteristic variables, for the investigated

configura-tion in this paper, are the dimensionless maximum wall

temper-ature, the channel Rayleigh number, the Reynolds number, and

the channel Nusselt number, defined as follows:

toward the ambient through the lower and upper edges of the

in Figure 1b, by following the approach given in [11–13] Thiscomputational domain allows taking into account the diffusiveeffects peculiar to the elliptic model The imposed boundaryconditions are reported in Table 2 for the fluid domain and inTable 3 for the solid domain The pressure defect is equal tozero at the inlet and outlet boundaries The net radiative heatflux from the surface is computed as a sum of the reflectedfraction of the incident and emitted radiative heat fluxes:

∗The letters in the column are in reference to Figure 1b.

L LANGELLOTTO AND O MANCA 443The computational fluid dynamics code FLUENT [30] was

employed to solve the present problem The segregated method

was chosen to solve the governing equations, which were

lin-earized implicitly with respect to the equation’s dependent

vari-able The second-order upwind scheme was chosen for the

energy and momentum equations The Semi Implicit Method

for Pressure-Linked Equations (SIMPLE) scheme was chosen to

couple pressure and velocity Similar considerations were made

for choice of the discrete transfer radiation model (DTRM),

which assumes all surfaces to be diffuse and grey The

A grid dependence test is accomplished to realize the more

convenient grid size and radiative subdivisions by monitoring

the induced dimensionless mass flow rate and the average

Nus-selt number, referred to the minimum channel spacing for a

detailed description on the numerical model is reported in [13]

A comparison between numerical and experimental [31]

re-sults is reported in Figure 2 In Figure 2a wall temperature

bmin =

37, are shown The comparison between the numerical and

ex-perimental data showed a good agreement with a maximum

percentage discrepancy of about 8% In Figure 2b the

compar-ison, in terms of average Nusselt number, is accomplished A

very good accord between the numerical and experimental data

is observed

Since the numerical results and experimental data are in good

agreement, the assumptions of steady-state, two-dimensional,

laminar, incompressible, with negligible viscous dissipation are

confirmed, as well as the Boussinesq approximation

SCALE ANALYSIS

For the convergent channel, the total volume (channel total

volume) is:

and it is greater than the channel volume as shown in Figure

1a The geometrical optimization of the convergent channel, in

terms of maximum or average wall temperature, should take

into account the channel total volume The heat transfer rate in

the channel is:

Experimental Numerical

Present numerical data Experimental data [14,31]

for an assigned heat transfer rate The optimal channel

that maximizes the heat transfer as a function of the channel

In laminar, fully developed and two-dimensional ral convection between parallel plates, heated at uniformheat flux, the maximum wall temperature is obtained at thechannel outlet section and the minimum Nusselt number is[20]:

k

48

444 L LANGELLOTTO AND O MANCA

Nusselt number is estimated by [20]:

k

12

(21)

The first term on the right-hand side of the Eqs (20) and

(21) is negligible with respect to the square root as given in

In vertical channels, at uniform heat flux, with small

con-vergence angle, the minimum and average Nusselt numbers,

referred to the minimum channel spacing, can be evaluated as

in [20] It is obtained as:

Nub min xw=Lw

Rab min48

For fully developed flow, the comparison between the

parallel-plate channel, Eqs (20) and (21), and the convergent

channel has a higher Nusselt number value; i.e., the

conver-gent channel, with the minimum channel spacing, equal to the

parallel-plate channel spacing, presents lower maximum and

average wall temperature values

For the Nusselt number referred to the average channel

As shown in Eqs (24), the Nusselt number for the

parallel-plate channel, i.e., the wall temperature in the

conver-gent channel is greater than the one in the parallel-plate channel

Further, the convergence angle limit is:

maxand

max

(26)

Also in this case, the Nusselt number for the convergent

parallel-plate channel The convergence angle limit is equal to:

(27)

In laminar, developing, and two-dimensional natural tion along an inclined single plate, heated at uniform heat flux,the average Nusselt number is [32]:

(28)

For developing flow in convergent channels as limit

as a first approximation, the correlation exponent, p, is set equal

to 2

(VtotTw)b min∼ Qbmaxbmin

L LANGELLOTTO AND O MANCA 445

Figure 3 Wall temperature for channel total volume as a function of channel

spacing and convergence angle with reference channel spacing equal to: (a)

b min ; (b) b av (c) Optimal geometrical configurations, in terms of b min , b av and

b max values, as a function of convergence angle.

function of the convergence angle and the minimum and

av-erage channel spacing, are reported in Figures 3a and 3b The

considered convective heat flux the optimal channel spacing is

δlimit, exists and a vertical asymptotic plane is detected forδ →

δlimit, according to Eq (25) The optimal configuration in terms

the results are not reported here

In Figure 3c, the optimal configuration, in terms of channelspacing, is given as a function of the channel convergence an-

represents the optimal configuration for the parallel-plate

convergence angle, the minimum channel spacing decreases

decreases and the Nusselt number increases, Eq (31), and then

total volume increase as the reference channel spacing increases

ANALYSIS AND PROCEDURES FOR CORRELATIONS

The results are obtained by the numerical procedure reported

in [13] In this work, the analysis is focused on the radiativeeffects on natural convection in air, in a convergent channel,uniformly heated at the two principal walls The wall thickness,

Its thermal conductivity is 0.198 W/m-K, with a solid-to-fluid

0.1 to 0.9

for the geometry here considered, are given in Table 4

Mass Flow Rate

Mass flow rate, involved in the heat transfer, is an tant parameter in design and control of electronic equipmentand solar energy in building The following correlations formass flow rate, in a convergent uniformly heated vertical chan-nel, as a function of thermal and geometrical parameters areproposed The mass flow rate for unit of width is defined asfollows:

impor-˙

m= ρuav,bminbmin (34)

section From Eqs (8) and (34):

˙

in Figure 4 The figure shows that, when the Rayleigh ber increases, for fixed aspect ratio and convergence angle, theReynolds number also increases Decreasing the aspect ratio(increasing the spacing), the Reynolds number increases signif-icantly, whereas there is slight change in the mass flow rate inthe emissivity range 0.10–0.90 The maximum percent variation

446 L LANGELLOTTO AND O MANCA

Table 4 Conductive heat losses

Ra∗bmax ≥ 30 L w /b min ≤ 58 0.048 ≤ b min /b max ≤ 1.0 q k /q≈ 3%; q k /q≤ 5%

1.0 ≤ Ra∗bmax ≤ 30 58 ≤ L w /b min ≤ 80 0.048 ≤ b min /b max ≤ 1.0 q k /q≈ 10%; q k /q≤ 15%

L w /b min = 80 0.048 ≤ b min /b max ≤ 0.7

Ra∗bmax ≤ 1.0 L w /b min ≥ 80 0.048 ≤ b min /b max ≤ 1.0 q k /q≥ 15%

between emissivity value equal to 0.90 and 0.10 is about 10%

Figure 4 shows that the Reynolds number is highly dependent

on channel aspect ratio and convergence angle

In order to reduce the Reynolds number scattering, the

bav Lw

−5/2 The plot of Eq (36) is reported in Figure 5a In this

figure, a greater dispersion is observed for high values of

of the aspect ratio in this zone To obtain a monomial correlation

for the mass flow rate, in terms of geometrical and thermal

variables, the following relation is proposed in the form [15]:

Figure 4 Reynolds number versus Rayleigh number for various convergence

angles and wall emissivity values.

700

(b)

Figure 5 (a) Rebmin

bmax as a function of Ra bmax(b av

L LANGELLOTTO AND O MANCA 447

They present a better regression coefficients than the

In Figure 5b, a comparison between the Reynolds number

value from numerical data and the Reynolds number values

calculated by the correlation in Eq (38) are reported together

Radiative Heat Flux

Correlations to evaluate the ratio between radiative heat flux,

Figure 6 Radiative heat flux ratio ( q r

q c +q r), as a function of Ra∗bav , for three different wall emissivity values and different convergence angles.

estimation of radiative heat losses

The ratios between radiative and total heat fluxes, as a

ratio The percentage reduction of heat flux ratio decreases withincreasing convergence angle Furthermore, the ratio values in-crease with decreasing aspect ratio value Increase in the conver-gence angle produces a significant increase in heat flux ratio For

of the total heat flux Figure 6 shows that, for fixed Rayleighnumber, the heat flux ratio decreases with decrease in the wall

bav

For assigned wall length, Figure 6 allows to observe the pendence on the convergence angle and channel spacing whenthe wall heat flux is fixed Furthermore, the figure shows a datascattering In order to reduce the heat flux ratio scattering, the

448 L LANGELLOTTO AND O MANCA

Table 5 Coefficients and exponent of the Eq (43)

ε m n p r 2

In Eq (42), the value of reference heat flux is used to obtain

a dimensionless equation A good accord is observed between

Eq (42) and the numerical data set

where the reference heat flux is equal to the values given in

Eq (42) The coefficient m and exponents n and p, as well as

this case, simpler equations are proposed with respect to the one

given in [15] Moreover, a new global correlation is evaluated

using all available data:

numer-ically and from Eq (44), taking into account Eq (46), are

is observed that the best agreement among the data and

The main advantage of Eq (44), with respect to the

corre-lations in Eqs (42) and (43) and ones in [15], is that it is a

single equation for all emissivity values Furthermore, Eq (44)

Dimensionless Maximum Wall Temperature

Composite correlations between the dimensionless mum wall temperatures and Rayleigh numbers, referred to the

and 0.9 The equations are obtained by means of the asymptoticrelations for the single tilted plates (large Rayleigh number,

Ra > 104) and for the fully developed limit (small Rayleigh

0 1 0

0 1

0

L LANGELLOTTO AND O MANCA 449

Considering the variable

A good agreement is observed in the comparison between the

numerical data and Eq (49) The comparison shows that greater

its validity in the fully developed region with respect to the

equation given in [15]

Nusselt Number Correlation

The average convective Nusselt numbers, defined in the Eq

is given in the following for two asymptotic conditions: fully

bmin >

800 The correlation for fully developed flow depends on the

convergence angle The following correlations are obtained by

means of regression analysis:

Numerical data and the two composite correlations are ported in Figure 8a A good accord between the numerical dataand the correlations is observed A slightly better agreement

re-is observed for higher Nusselt numbers corresponding to thehigher Rayleigh numbers

In order to obtain a single composite correlation that takesinto account all convergence angles, a different monomial cor-

employed The following monomial correlation is proposed in

scale analysis results These functions are reported in Figure 8band a horizontal asymptotic value is observed in both functions.Moreover, the critical values correspond to the zone where thefunctions change from vertical to horizontal asymptote

A similar analysis is given for average total Nusselt number,defined in Eq (9), which takes into account both radiation andconvective heat fluxes Average total Nusselt number, as func-

in the following for the two asymptotic conditions: fully

bmin>

the convergence angle In fact, by means of numerical data, thefollowing correlations are obtained employing the regression

450 L LANGELLOTTO AND O MANCA

Figure 8 (a) Nusselt number versus Rayleigh numbers and correlations given

by Eqs (53) and (54) (b) Coefficient m 0 and exponent n 0 in the Eq (55) (c)

Total Nusselt number vs total Rayleigh numbers and correlations given by Eqs.

bmin< 100 with r2= 997; and

Nu∗0,bmin = 0.492Ra∗bmin 0.392

(58)

bmin< 100 with r2= 992

For the single plate limit, the monomial correlation is:

Nu∗∞,bmin = 0.725Ra∗bmin 0.210

(59)

bmin> 1.0×103with r2= 975

Eqs (57) and (59) resulting in:

re-In the same way employed to obtain Eqs (55), the following

into account all convergence angles:

In Figure 9a, all proposed correlations are reported and they

L LANGELLOTTO AND O MANCA 451

9a, the correlations for convective average Nusselt number, as

a function of convective channel Rayleigh number, referred to

the minimum channel spacing, are compared The figure shows

that the greatest differences among the correlations are detected

In Figure 9b, the correlation for total average Nusselt number,

as a function of total channel Rayleigh number, referred to the

minimum channel spacing, for several wall emissivities, are

reported In this case as well, the greatest differences among the

⎤

⎦

−4+0.660Ra 0.200

reported In this figure the contours in the (b,δ) plane are alsogiven It is observed that the minimum value is obtained for

configuration for convergent channel These values are almostequal to the ones estimated by the scale analysis Moreover, thetwo diagrams in Figures 10a and 3a seem similar This con-firms that the scale analysis provides a good estimation for thisconfiguration

452 L LANGELLOTTO AND O MANCA

In Figure 10b, the optimal channel spacing, evaluated by

means of Eqs (64), (31), (32), and (33), is given as a function of

the convergence angle The figure shows that the curves present

asymptotic values, corresponding to the optimal configuration,

the parallel-plate channel For all curves, the asymptotic values

are almost equal For the dimensionless quantities referred to

optimal configurations obtained by the numerical solution, Eq

(64), and scale analysis, Eq (31), have a similar trend In Figure

10b, it is observed, for Eq (64), that the minimum value of the

By increasing the heat flux the optimal minimum channel

spacing decreases as shown in Figure 10c This result is in

agreement with the results reported in [15] and [33]

CONCLUSIONS

Natural convection in air, in convergent channels,

symmetri-cally heated at uniform heat flux, in a steady-state regime, was

studied A scale analysis allowed estimation of optimal

geo-metrical configurations from Nusselt number correlations, for

single plate and fully developed flow, in terms of the channel

an-gle A new optimization procedure was obtained in terms of the

minimum value of the product of average wall temperature and

minimum, average, and maximum wall spacing It was observed

are different from the ones given in previous papers [10, 12, 13,

15]

Monomial and composite correlations were estimated by the

numerical results obtained by the numerical model proposed in

[13] The correlation equations were accomplished for mass flow

rate, radiative heat flux, and dimensionless maximum wall

tem-perature in the emissivity range from 0.10 to 0.90, convergence

number correlation was proposed for the emissivity value of

in very good accord with the numerical data

The analysis of Nusselt number, in fully developed flow,

all values were along a single asymptotic curve, Eqs (56) and

considered the border line between the fully developed flow and

the developing flow for low Ra values

The new proposed optimization procedure was applied

em-ploying the evaluated Nusselt number correlation The optimal

obtained by means of the scale analysis The optimal value ofminimum channel spacing decreases with increase in the wallheat flux as shown in [15] and [33]

Greek Symbols

L LANGELLOTTO AND O MANCA 453

Superscripts

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Luigi Langellotto is a researcher at Centro Sviluppo

Materiali S.p.A (CSM), Rome Italy He received his Ph.D in mechanical engineering from Seconda Universit`a degli Studi di Napoli (SUN) He is CSM project leader in an RFCS project and several in- dustrial projects with TenarisDalmine S.p.A or ABS S.p.A His main scientific activities are on natural convection in an open-ended cavity; thermal control

of electronic equipment; solar systems; analytical and numerical solutions in material processing such as seamless pipe rolling, strip rolling, and ingot casting; and numerical analysis

of austenite deformation and decomposition He has co-authored more than 10 refereed journal and conference publications.

Oronzio Manca is a professor of mechanical

engi-neering at Facolt`a di Ingegneria della Seconda versit`a degli Studi di Napoli (SUN), Naples, Italy He has been coordinator of the Industrial Engineering Area at SUN since January 2005 His main scien- tific activities are on active solar systems; passive solar systems; refrigerant fluids; natural and mixed convection in an open-ended cavity with and without porous media; conduction in solids irradiated by mov- ing heat sources; combined radiative and conductive fields in multilayer thin films; analytical and numerical solutions in material processing; thermal control of electronic equipment and solar systems; and heat transfer augmentation by nanofluids He is a member of the ATA Campania Committee He is a member of the American Society of Mechanical Engi- neering, and Unione Italiana di Termofluidodinamica UIT He has co-authored more than 270 refereed journal and conference publications He is currently a

Uni-member of the editorial advisory boards for The Open Thermodynamics

Jour-nal, The Open Fuels & Energy Science JourJour-nal, and Advances in Mechanical Engineering.

Copyright
C Taylor and Francis Group, LLC

ISSN: 0145-7632 print / 1521-0537 online

DOI: 10.1080/01457632.2010.506167

An Inverse Analysis for Parameter

Estimation Applied to a Non-Fourier Conduction–Radiation Problem

RANJAN DAS,1 SUBHASH C MISHRA,1 T B PAVAN KUMAR,1

and RAMGOPAL UPPALURI2

1Department of Mechanical Engineering, Indian Institute of Technology Guwahati, Guwahati, India

2Department of Chemical Engineering, Indian Institute of Technology Guwahati, Guwahati, India

Retrieval of parameters in a non-Fourier conduction and radiation heat transfer problem is reported The direct problem is

formulated using the lattice Boltzmann method (LBM) and the finite-volume method (FVM) The divergence of radiative heat

flux is computed using the FVM, and the LBM formulation is employed to obtain the temperature field In the inverse method,

this temperature field is taken as exact Simultaneous estimation of parameters, namely, the extinction coefficient and the

conduction–radiation parameter, is done by minimizing the objective function The genetic algorithm (GA) is used for this

purpose The accuracies of the estimated parameters are studied for the effects of measurement errors and genetic parameters

such as the crossover and mutation probabilities, the population size, and the number of generations The LBM-FVM in

combination with GA has been found to provide a correct estimate of parameters.

INTRODUCTION

Any differential equation governing a transient phenomenon

is subjected to initial and boundary conditions, and such a

prob-lem is mathematically well posed In the area of heat transfer,

with medium and boundary properties known, the objective of

such a problem remains the determination of temperature and/or

heat flux distributions Problems of this type belong to direct

problems However, there are many situations where properties

and/or initial and/or boundary conditions remain unknown and

the temperature/heat flux histories are known from experiments

The estimation of unknown quantities for this class of problems

falls under the purview of inverse problems They are

mathe-matically ill posed and accuracy of their solution depends on the

measured data Inverse problems find applications in many

ar-eas of engineering such as material science [1], circuit analysis

[2], turbomachinery [3], manufacturing science [4], and design

of radiant enclosures [5]

In the area of heat transfer, inverse problems have been

inves-tigated by many researchers [5–16] Erturk et al [5] estimated

Address correspondence to Professor Subhash C Mishra, Department of

Mechanical Engineering, Indian Institute of Technology Guwahati, Guwahati

781039, India E-mail: scm iitg@yahoo.com

boundary conditions in a transient radiative enclosure Reinhardtand Hao [6], Alifanov and Nenarokomov [7], and Chen et al [8]have studied inverse heat conduction problems Reinhardt andHao [6] proposed a method for solving noncharacteristic Cauchyproblems for parabolic equations They used a conjugate gra-dient method in their analysis Alifanov and Nenarokomov [7]solved an inverse boundary heat conduction problem used toinvestigate thermal processes between solids and environment.They used an iterative regularization method based on mini-mizing the residual functional by means of gradient methods ofthe first kind and spline approximation of unknown functions.From the knowledge of temperature measurements within theconducting solid, Chen et al [8] retrieved temperature and heatflux on a surface A Kalman filter scheme was used in theirinverse analysis

With known transient temperature data of a plate-finned tubeheat exchanger, Huang et al [9] estimated thermal contact con-ductance They used the conjugate gradient method for mini-mization Franca and Howell [10] performed inverse design ofradiative enclosures for thermal processing of materials Theirmethodology was based on a truncated singular value decompo-sition method They estimated heat input to a heater located atthe top of a three-dimensional enclosure that can satisfy a pre-scribed time-dependent temperature curve on a surface located

at the base of the enclosure

455

456 R DAS ET AL.

An inverse analysis to simultaneously estimate the

effec-tive thermal conductivity, the effeceffec-tive volumetric heat

capac-ity, and the heat transfer coefficient between a porous medium

and a hot wire was performed by Znaidia et al [11] They used

the Levenberg–Marquardt method to solve the inverse problem

Frankel and Arimilli [12] determined the convective and

radia-tive loads from temperature measurements They used Gauss

filter methods in their analysis Most recently, Pereverzyev et al

[13] carried out an inverse analysis for temperature

reconstruc-tion in a glass cooling process They used the fast derivative-free

iterative method Kim and Baek [14] performed an inverse

anal-ysis in a cylindrical enclosure involving conduction and

radia-tion They estimated heat flux distribution on the design surface

The Levenberg–Marquardt method was used in their analysis

They used the finite-volume method (FVM) for the energy

equa-tion Das et al [15, 16] recently did simultaneous estimation of

parameters for one-dimensional (1-D) [15] and two-dimensional

(2-D) [16] conduction–radiation problems They used the

lat-tice Boltzmann method (LBM) and the FVM [17] in conjunction

with genetic algorithm (GA) [2, 18] All the past studies dealing

with inverse heat transfer problems involved consideration of

Fourier’s law of heat conduction

In Fourier heat conduction, effects of any thermal

distur-bances in the system, in the form of a sudden rise in boundary

temperatures or a sudden appearance of a heat source at any

location in the medium, propagate with infinite speed and hence

establish instantaneously This assumption is not universally

ap-plicable in all situations For example, the validity of Fourier’s

law of heat conduction breaks down when we consider heat

transport through a processed meat/skin [19–22] Further, at

a very low temperature, heat transport by conduction is not

governed by Fourier’s law [23–25] When any material is

sub-jected to a pulse radiation, at short time levels, a discontinuity in

the temperature profile is observed, which cannot be explained

through Fourier’s law of heat conduction [26] In the area of

non-Fourier conduction and/or radiation heat transfer problems,

a few studies have been reported in the past [27–32]

Although literature reports dealing with parameter

estima-tions in heat transfer problems involving Fourier’s law of heat

conduction are available to a good extent, no work has been

reported in the field of inverse non-Fourier heat transfer

in-volving transient conduction–radiation The present work is,

therefore, aimed at estimation of parameters in a non-Fourier

conduction–radiation problem

In the present work, we simultaneously estimate parameters,

namely, the extinction coefficient and the conduction–radiation

parameter, in a non-Fourier conduction–radiation heat transfer

in a planar geometry The LBM and FVM are used to solve

the direct problem This is due to the fact that the LBM is an

efficient numerical method and, being mesoscopic in approach,

it presents a clear physical meaning Most recently the usage of

the LBM has been explored for a wide variety of fluid flow and

heat transfer problems [33–37] The FVM is also an efficient

method to compute the radiative information and is less prone

to the ray effect With transient temperature distributions known

from the direct method, in the inverse method, we use the LBMand the FVM in conjunction with the GA to simultaneouslyretrieve the extinction coefficient and the conduction–radiationparameter The GA is used as an optimization tool because theprobability of obtaining the solutions near to the global optimaldomain is expected to be higher because of its evolutionaryapproach [38] In both the direct and the inverse methods, theFVM is employed to compute the radiative information and theLBM is used to solve the energy equation In the inverse method,optimization is achieved using the GA

FORMULATION

Let us consider a planar participating medium (Figure 1) Itsthermophysical and optical properties are constant The initial

the east boundaries of the medium are maintained at

of the medium are considered black In the absence of tion and heat generation, the governing energy equation for theproblem under consideration is given by

takes some finite time to establish itself in the medium, andthus the assumption of the infinite propagation speed of theconduction wave in Fourier heat conduction does not hold true

and thus for the time scale considered in the present study,radiative transfer is an instantaneous process

With finite propagation speed of the conduction wavefront,the non-Fourier heat conduction equation is given by [23, 24]

 ∂qC

∂t + qC= −k ∂T

diffusivity, and C is the speed of the conduction wavefront.

From Eqs (1) and (2), we obtain the following expression:

R DAS ET AL 457

Figure 1 Schematic of the 1-D planar geometry under consideration along with D1Q2 lattice of the LBM and control volume of the FVM, (b) Intensity I jin the direction  j in the center of the elemental sub-solid angle m.

conduction–radiation parameter N , and incident radiation G

in the following way:

 R= q R

σT4 ref N = kC

4ασT 3 ref G= G∗

ref π

[31], in the present work, the value of A has been taken as 2.0.

Equation (4) in dimensionless form is written as

33], and the LBM is employed to compute the temperature field

in both the direct and the inverse methods In the inverse method,

the optimization is achieved using the GA Next in this article

we provide a brief formulation of the FVM, the LBM, and the

is the dimensionless source term, which for a linear

anisotropic scattering phase function is given by

In Eq (9), in case of a planar medium, incident radiation G

computed from [33]:

π

δ=0

where M is the number of rays considered over the complete

Writing Eq (8) for a discrete direction having index m and

458 R DAS ET AL.

Table 1 Effect of the number of lattices in the LBM and control volumes in

the FVM and number of rays M on temperature distribution at three different

locations: ξ = 0.60, β = 0.5, ω = 0.8, N = 0.01, and θE = 0.5 and

θ W = 1.0., for the non-Fourier conduction–radiation problem

From Eqs (15) and (16), the unknown cell-center intensity

computed from previous time level is calculated from the

and in this case, for any control volume, the cell surface intensity

E isknown

M

/2

LATTICE BOLTZMANN METHOD

In the LBM, the discrete Boltzmann equation withBhatanagar–Gross–Krook (BGK) approximation in dimension-less form is written as [34, 35]

i

is the equilibrium distribution function For a planar medium

Lines: Present work.

Markers: Chu et al

Markers: Chu et al

Figure 2 Comparison of temperature θ distributions in the direct method with

that of Chu et al [28].

In the present case with D1Q2 lattice, temperature and heatflux  are computed from the following [35, 36]:

θ =2

i=1

 =

2

direction, and they are chosen in such a way that the followinghold true:

2

i=1

f i (eq)=

2

i=1

2

i=1

f i eq e i =

2

is analogous to biological evolutions of any species in whichsuccessive generations are conceived, born, and raised until

Figure 3 Effect of different crossover probability and mutation probability on the best fitness: (a) E = 0.0, (b) E = 0.5, (c) E = 1.0, and (d) E = 2.0.

they themselves become ready to reproduce Reproduction,

crossover, and mutation are the three main steps involved in

the GA After generation of an initial population and evaluation

of its fitness, the process of reproduction starts The generations

having good fitness values are replicated in the next

popula-tion Next, the crossover operation starts In this process, pairs

from new strings mate to produce new offspring The crossover

rep-resents the number of individuals in the population undergoing

the crossover operation The parents are replaced by the newly

produced offspring Finally, through an assigned probability

randomly changes the genes in the string The process

contin-ues until a satisfactory fitness value of the objective function is

attained

In the present work, in the inverse analysis, the objective

function is defined as the summation of the squares of the

To account for the effect of measurement errors, biased errors

to the exact temperature field are added Thus, the temperature

θmeasuredwhen an error is included is expressed as [15]

estimation of unknown parameters, the minimization of the jective function [Eq (28)] is required

R DAS ET AL 461

RESULTS AND DISCUSSION

In the following, we present results of the inverse analysis

The boundaries of the planar medium (Figure 1) are assumed to

ξ > 0.0, the east and the west boundaries are at temperatures

for the analysis

In Table 1, we present results of grid and ray independency

1 that beyond 100 control volumes and 12 rays, there is no

the present work we have provided the results considering 100control volumes and 12 rays

In order to verify the results of the direct problem

dimensionless temperatures of the west and the east boundaries

this comparison has been carried out at two different time

0.025, have been investigated It is observed from Figure 2

method (LBM-FVM) compare very well with those given

by Chu et al [28], who have solved the same problem using

(d)(c)

Figure 4 Effect of different population sizes on the best fitness: (a) E = 0.0, (b) E = 0.5, (c) E = 1.0, and (d) E = 2.0.

In the following pages we present results for the inverse

anal-ysis using the LBM-FVM in conjunction with the GA For this

available from the direct method (LBM-FVM), in the inverse

method (LBM-FVM-GA) two parameters, namely, the

have been simultaneously estimated The ranges for the

To demonstrate the workability of the LBM-FVM-GA in the

inverse method, for a population size of 100, in Table 2, effects

ξ=0.3, 0.6 ξ=0.3, 0.6

Figure 5 Comparison of exact and measured temperature profiles for different

measurement errors on temperature distribution E= 2.0.

(0.3, 0.3) The exact values of the parameters that were

observed that for all measurement errors, a crossover

minimum error in the estimated value of the parameters.For parameters considered in Table 2, in order to study the

effects on the number of generations required for the gence, we present a comparison in Figure 3 It can be observed

as compared to other combinations of the crossover and the

ξ = 0.3, 0.6

LBM-FVMLBM-FVM-GA

Figure 6 Comparison of exact and estimated temperature profiles on

temper-ature distribution E= 2.0.

R DAS ET AL 463

Table 4 Comparison of the CPU(s) times required in the LBM-FVM and the LBM-FVM-GA for ξ = 0.60; ω = 0.8, θE = 0.5, and

θ W = 1.0 for the non-Fourier conduction–radiation problem, with number of generations 100, and population size 100

CPU time (s)

(N , β) E LBM-FVM LBM-FVM-GA Ratio of CPU times 

LBM −FVM−GA LBM−FVM

mutation probabilities Further, it can be noticed that in case of

observed that there is no significant change in the variation of the

best fitness beyond 100 generations Thus, in the present work,

the analysis has been done for a maximum of 100 generations

(0.8, 0.03) provides the estimated values with least errors and

also requires a lower number of generations for the convergence

the effect of population size on the accuracy of the estimated

values of the parameters For this comparison, three different

population sizes, e.g., 25, 50, and 100, have been investigated

For all values of measurement errors, it is seen that the estimation

accuracy deteriorates for small population numbers, and as the

population size increases, the accuracy also improves

In Figure 4, we compare the effect of population size on the

variation of best fitness with the number of generations required

is taken It is observed from Figure 4 that for all measurement

errors, the attainment of convergence is faster for a higher

pop-ulation size It is also observed that the converged values are the

least for a population size of 100 Thus, in the present study, a

higher value of population size provides a better result for the

estimated parameters This can be explained in the following

manner A small population number means that in the

popu-lation fewer individuals are present Hence the probability of

obtaining an individual of desired fitness at a particular

gener-ation is also less So, the number of genergener-ations required for

the convergence is more and the presence of fitter individuals

is also less and hence the estimation accuracy reduces On the

other hand, the presence of more individuals in the population

contains more individuals with better fitness and hence require

less generations to attain the convergence

Figure 5 shows the comparison between the exact

that a significant amount of deviation at all time levels, namely,

ξ = 0.3, 0.6, has been applied and the square of error is

mini-mized using the optimization tool in the inverse method In thepresent work, we have used the GA for the accomplishment ofthe same

In order to demonstrate the accuracy of the estimated rameters obtained in the inverse method (LBM-FVM-GA), in

com-puted using the direct method and the inverse method This

compared to other sets of E It is observed that the temperature

θ distributions computed using the direct method and the inversemethod are in excellent agreement with each other

In order to study the CPU times involved in the direct method(LBM-FVM) as well as the inverse method (LBM-FVM-GA),

we present a comparison in Table 4 This comparison

number of generations and the population size is 100 All runswere carried out on a 2.8-GHz CPU (Pentium (R) 4 with 248 MBRAM) It is seen that the inverse method (LBM-FVM-GA) re-quires CPU time approximately 1300–1800 times that required

by the direct method (LBM-FVM) This is due to the fact that

in the inverse method, the GA starts with a random generation

of the initial population The values of the estimated ters corresponding to this initial population deviate greatly with

parame-Table 5 Study of the effect of measurement points on the accuracy of the estimated parameters for the non-Fourier conduction–radiation

problem, with number of generations 100, and population size 100

Exact value

Number of measurement points Estimated value Percent error (N , β)

464 R DAS ET AL.

respect to the actual ones Thus, to attain the converged

solu-tion, the algorithm has to undergo a series of generations and

for a particular generation, in the GA loop there are a number of

processes involved Therefore, in the inverse method the CPU

time is considerably larger than in the direct method

In order to study the effect of number of measurement points

on the accuracy of the estimated results, we present a

sets of measurement points, namely, 10, 25, 50, and 100, are

taken for the present study It is observed from the table that

the estimation error gradually increases with decrease in the

number of measurement points It is also noticed that the

prop-agation of estimated error is more pronounced in the value of

the conduction–radiation parameter, N, thus indicating that it is

more sensitive to the number of measured points as compared

CONCLUSIONS

An inverse method was used for simultaneous estimation

of parameters in a non-Fourier conduction–radiation problem

involving the LBM-FVM-GA Parameters, namely, the

extinc-tion coefficient and the conducextinc-tion–radiaextinc-tion parameter, were

simultaneously estimated and they are compared with their

exact value Effects of different genetic parameters such as

the crossover and the mutation probabilities, the population

size, and number of generations were studied A comparison

of CPU times involved in the direct and the inverse method

was also done The accuracy of the estimated parameters was

checked by comparing the temperature distributions obtained

using the direct and the inverse method The LBM-FVM in

conjunction with the GA has been found to provide reasonably

good estimations for the unknown parameters in a non-Fourier

conduction–radiation problem

NOMENCLATURE

to the characteristic thermal wave time of non-Fourier

conduction (X/C)

 dimensionless total heat flux

R DAS ET AL 465

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Ranjan Das is a Ph.D student in the Department

of Mechanical Engineering at the Indian Institute of Technology (IIT) Guwahati From September 2006

to December 2008, concurrently with his Ph.D search, he has worked in a sponsored research project

re-of the Defense Research and Development tory, Hyderabad He did his master’s thesis in the area of thermal engineering and received his M.Tech.

Labora-in 2006 from the National Institute of Technology Silchar He obtained his bachelor’s degree in 2003 from Bhavnagar University, Gujarat The area of his Ph.D research includes parameter retrieval in heat transfer problems involving thermal radiation, op- timization using genetic algorithms, and lattice Boltzmann-based heat transfer analysis.

Subhash C Mishra holds a B.Sc (engineering) from

BIT Sindri Dhanbad (1989) and an M.Tech (1992) and Ph.D (1997) in mechanical engineering from IIT Kanpur He joined IIT Guwahati as a senior lec- turer in December 1996 and became a full professor

in October 2004 He is the recipient of a research fellowship of the Alexander von Humboldt Founda- tion, Germany and a fellowship by invitation of the Japan Society for Promotion of Science (JSPS) He

is a reviewer of many journals and conferences in the area of heat transfer and has about 150 research papers to his credit.

T B Pavan Kumar holds an M.Tech

(mechani-cal engineering) from IIT Guwahati (2008) During his M.Tech work he extended the usage of the lat- tice Boltzmann method for non-Fourier conduction and radiation He obtained his bachelor’s degree in

2006 from Visvodaya Institute of Technology and Science, Kavali, Andhra Pradesh He did his bach- elor’s project in HAL (Hindustan Aeronautics Ltd.).

He completed his diploma in mechanical engineering (2003) from Govt Polytechnic College, Proddatur, Andhra Pradesh Currently, he is working as a CFD analyst for John Deere, Pune.

Ramgopal Uppaluri holds a B.Tech (chemical

en-gineering) from Andhra University, Visakhapatnam (1997), and an M.Tech (chemical engineering) from IIT Kanpur (1999) He obtained his Ph.D (process in- tegration) from the University of Manchester (2002) Prior to joining IIT Guwahati, he worked as a post- doctoral fellow (2002–2004) at the Robert Gordon University, Scotland, in the field of membrane tech- nology Presently, he is an associate professor in the Chemical Engineering Department, IIT Guwahati His research interests are variegated and he works exclusively in process sys- tems engineering, heat exchanger networks, refinery process design, surfactant enhanced oil recovery, membrane technology, and electrodeless plating.

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ISSN: 0145-7632 print / 1521-0537 online

DOI: 10.1080/01457632.2010.506169

The Effect of Tube Flattening on Flow Boiling Heat Transfer Enhancement

M NASR, M A AKHAVAN-BEHABADI, and S E MARASHI

School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, Iran

The present study investigated the effect of smooth tube flattening on heat transfer enhancement in an evaporator The tubes

with internal diameter of 8.7 mm were flattened into an oblong shape with different inside heights The test setup was

basically a vapor compression refrigeration system equipped with all necessary measuring instruments Refrigerant R-134a

flowing inside the tube was heated by an electrical coil heater wrapped around it The ranges of mass velocities were from

74 to 106 kg/m 2 -s and vapor quality varied from 25% to 95% Analysis of the collected data indicated that the heat transfer

coefficient elevates by increasing the mass velocity and vapor quality in flattened tubes just like the round tube The flow

boiling heat transfer coefficient increases when the flattened tube is used instead of the round tube The highest heat transfer

coefficient enhancement of 172% was achieved for the tube with the lowest inside height at mass velocity of 106 kg/m 2 -s and

vapor quality of 85% Finally, based on the present experimental results, a correlation was developed to predict the heat

transfer coefficient in flattened tubes.

INTRODUCTION

Heat transfer equipment like evaporators and condensers has

an important role in different industries, such as refrigeration,

air conditioning, power plant, and chemical To save limited

available energy resources, design and manufacturing of

effi-cient heat transfer equipment are essential Different methods

have been used by numerous investigators to increase the heat

transfer rate in the heat transfer equipment [1–5] Although these

methods usually increase the heat transfer coefficients, they also

increase the pressure drop in the equipment

Behabadi et al [1] investigated the effect of spiral spring

inserts on augmentation of heat transfer coefficients during

forced convection condensation of R-134a inside horizontal

tubes They reported a maximum heat transfer enhancement

of 80% above the plain tube values on a nominal area basis

Bandarra Filho and Jabardo [2] performed a study on

con-vective boiling of R-134a in herringbone and microfin copper

tubes The ranges of vapor quality were from 5% to 90% and

indicated that the herringbone tube has a distinct heat transfer

The authors express their thanks to the School of Mechanical Engineering,

College of Engineering, University of Tehran, for financial support of the present

experimental work.

Address correspondence to Professor M A Akhavan-Behabadi, School of

Mechanical Engineering, College of Engineering, University of Tehran, Tehran,

Iran E-mail: akhavan@ut.ac.ir

performance for the mass velocity ranges considered in theirstudy Thermal performance of the herringbone tube has beenfound better than that of the standard microfin tube at high mass

-s), for vapor qualities higher than 50%, the performance of theherringbone tube was worse than that of the standard microfintube

Wongwises and Polsongkram [3] investigated on tion heat transfer and pressure drop characteristics of R-134a

evapora-in helically coiled concentric tube-evapora-in-tube heat exchangers, perimentally They conducted their experiments for high mass

effect of changing average vapor quality, mass velocity, tion temperature, and heat flux Their observation showed that

satura-by increasing average vapor quality and mass velocity, bothpressure drop and heat transfer increase They also developedtwo correlations for estimation of refrigerant Nusselt numberand pressure drop

Akhavan-Behabadi et al [4] studied the effect of twistedtape inserts on heat transfer enhancement and pressure drop in

a horizontal tube during swirl flow boiling of R-134a Twistedtapes with different twist ratios of 6, 9, 12, and 15 were madeand used in full length of the test evaporator The use of twistedtape inserts was found to increase the heat transfer coefficients

by as much as 68% above the plain tube values on a nominal areabasis However, the pressure drop also increased substantially,

by as much as 180%

467

468 M NASR ET AL.

Figure 1 Schematic diagram of the experimental setup.

Methods of increasing heat transfer coefficients are divided

into two categories: active techniques and passive techniques

In passive methods, enhanced tubes with specific geometry

or added fluids are used for increasing the heat transfer

rates On the other hand, in active methods, external forces

like electrical or acoustic fields or surface vibrations are

used

One of the passive techniques to enhance the heat transfer

coefficient is the use of flattened tubes in condensers and

evaporators A study conducted by Wilson et al [5] investigated

pressure drop, heat transfer coefficient, and void fraction in

flattened copper passageways in a condenser They investigated

the pressure drop and heat transfer coefficient of R-134a and

and vapor quality ranging from approximately 10% to 80%

The plane tubes used were round tubes of 8.91 mm inner

diameter (I.D.) flattened into an oblong shape with inside

heights of 5.74, 4.15, 2.57, and 0.974 mm They used the

hydraulic diameter to consider the effect of tube-flattened

profile for pressure drop in flattened tubes For calculation of

condensation pressure drop in oblong tubes, they suggested the

use of existing correlations for round tubes by replacing the

tube diameter with the tube hydraulic diameter They reported

the enhancement of condensation heat transfer coefficient as the

tube was flattened According to their study, increasing both the

mass velocity and the vapor quality tended to increase the heat

transfer coefficients during convective condensation in flattened

tubes

In the present research, an experimental investigation has

been carried out to study the enhancement of heat transfer

co-efficient in an evaporator with flattened tube

EXPERIMENTAL FACILITY AND PROCEDURE

The test setup was a well-instrumented vapor compressionrefrigeration system The schematic diagram of the experimentalsetup is shown in Figure 1 The setup included a pre-evaporator,

a test evaporator, and an after-evaporator Refrigerant R-134aflowing inside these three evaporator tubes was heated by elec-

uni-formly wrapped around the tubes The input power for eachheater was controlled by a 2-kW dimmer The evaporators withthe heaters on them were completely insulated to prevent anyheat leakage

The pre-evaporator was used to achieve desired vapor

quali-M NASR ET AL 469

Figure 2 Dimensions of flattened tube cross sections.

To measure the outside tube wall temperatures, wall-mounted

type T (copper–constantan) thermocouples with a calibrated

evaporator at 22-cm intervals At each station, four

thermocou-ples were mounted on the top, bottom, and both sides of the

tube The thermocouples were carefully soldered on the outer

surface of the tubes

The refrigerant pressures at the inlet and outlet of the test

evaporator were also measured by a pressure gauge, with a

calibrated accuracy of 2 kPa, which was welded carefully on

the outer surface of the tube by means of a three-way joint The

refrigerant volumetric flow rate was measured by a flow meter

installed downstream of the condenser The provision was also

made to measure other necessary parameters

In this study, refrigerant R-134a was used as the

work-ing fluid Mass velocities of the refrigerant varied from 74 to

test evaporator varied from approximately 25% to 95% During

each test run, the difference between the inlet and outlet vapor

qualities of test evaporator was about 10% The mean vapor

quality of test section was taken as the average of its inlet and

outlet vapor qualities

after-evaporator were completely insulated by glass wool to prevent

any heat loss, there was a little heat leakage to the surrounding

In order to consider the effect of heat loss in the calculations,

insulation efficiency was defined as

deter-of the inlet and outlet deter-of test evaporator Saturation temperatureand superheated and saturation enthalpies of R-134a were cal-culated from [6], and other properties such as viscosity from[7]

For each test run, the refrigerant-side average heat transfer

coefficient, ¯h, was calculated by:

The average static pressure in the test evaporator was taken

to be the mean of the inlet and outlet pressures The

the saturation temperature corresponding to this average staticpressure

The average outside tube wall temperature of the test

Copyright
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ISSN: 0145-7632 print / 1521-0537 online

DOI: 10.1080/01457632.2010.506170

Superposition Relations for Forced

Convective Local Nusselt Numbers

for Flow Through Asymmetrically

Heated Parallel-Plate Channels

V V SATYAMURTY1and R REPAKA2

1Department of Mechanical Engineering, Indian Institute of Technology, Kharagpur, India

2Department of Mechanical Engineering, National Institute of Technology, Rourkela, India

In this article, explicit relations to calculate the local Nusselt number from the values corresponding to the boundary

condition of the first kind have been developed for forced convective flow through asymmetrically heated parallel-plate

channels The asymmetric thermal boundary condition is characterized by a parameter, A, defined as the ratio of wall

temperatures in excess of the entry fluid temperature The boundary condition of the first kind corresponds to A = 0 or A →

∞ The validity of the superposition relations has been established from the numerical results obtained for the simple model

of hydrodynamically developed and thermally developing flow However, the superposition relations can be expected to be

valid, as long as geometric and flow symmetry are preserved.

INTRODUCTION

Interest in studying heat transfer in asymmetrically heated

ducts in general has been renewed owing to newer applications,

such as cooling of electronic equipment, using materials

in-volving hyperporous media or microchannels, and fuel cells

The configurations of parallel-plate channels and annuli are

be-ing investigated on a regular basis Early pioneerbe-ing studies by

Graetz dealing with forced convective heat transfer in pipes

as-sumed hydrodynamically and thermally fully developed

condi-tions An excellent review on the studies involving laminar heat

transfer in ducts prior to 1978 is available in Shah and London

[1] Heat transfer in parallel-plate channels subjected to

con-stant but unequal temperatures was first studied by Hatton and

Turton [2] Recently, Mitrovic et al [3] referred to this problem

as the asymmetric Graetz problem Hatton and Turton [2]

as-sumed fully developed flow and developing thermal field but

ne-glected axial conduction and obtained a series solution Mitrovic

et al [3] obtained numerical solutions within the same

frame-work and additionally identified the points where the Nusselt

Address correspondence to Dr Ramjee Repaka, Department of Mechanical

Engineering, National Institute of Technology, Rourkela 769008, India E-mail:

ramjee.repaka@gmail.com

number displays unbounded swing and the channel wall turnsadiabatic Mitrovic and Maletic [4, 5] also investigated laminarforced convection heat transfer in asymmetrically heated annuliand channels filled with porous material In all these studies,asymmetric heating (walls of the channel or inner and outerpipes, subjected to unequal temperatures) has been character-ized essentially by considering the difference or the ratio ofthe wall temperatures or a variant In connection with mixedconvection studies in vertical channels, Barletta [6] describedasymmetry in a binary form: unity when the temperatures areunequal, and zero when the temperatures are equal Repaka [7]

characterized the asymmetry by a parameter, A, defined as the

ratio of wall temperatures in excess of the fluid inlet temperature

numbers at the two walls get interchanged

There has been mention of the applicability of superposition

in the literature in obtaining the solutions of energy equation andderived quantities (such as Nusselt number) for different bound-ary conditions In studying the effect of adiabatically preparingthe fluid with viscous dissipation, Barletta and Magyari [8] ar-gued, if viscous dissipation effects are included and a fullydeveloped flow is assumed in duct flows, the entry temperatureneeds to be considered as that would be obtained in an adiabaticduct while the flow becomes developed Barletta and Magyari476

V V SATYAMURTY AND R REPAKA 477[8] termed this the adiabatic preparatory zone Subsequently

they obtained a solution to the energy equation as comprised

of a particular solution accounting for viscous dissipation and

a solution to simple conservation of thermal energy equation,

neglecting axial conduction and assuming fully developed flow

field Such approaches are not valid in general when the flow

field is developing or when axial conduction is included or even

478 V V SATYAMURTY AND R REPAKA

T i

T i

T w1

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ISSN: 0145-7632 print / 1521-0537 online

DOI: 10.1080/01457632.2010.506358

Experimental Studies on Heat

Transfer and Friction Factor

Nanofluid in a Circular Pipe Under

Transition Flow With Wire Coil

Inserts

M CHANDRASEKAR,1 S SURESH,1and A CHANDRA BOSE2

1Department of Mechanical Engineering, National Institute of Technology, Tiruchirappalli, India

2Department of Physics, National Institute of Technology, Tiruchirappalli, India

Heat transfer and friction factor characteristics in a circular tube fitted with wire coil inserts are investigated experimentally

using Al 2 O 3 /water nanofluid as the working fluid The effects of the pitch ratio and nanofluid on the Nusselt number and the

friction factor are determined in a circular tube with fully developed transition flow for Reynolds number in the range of 2500

to 5000 The experiments were performed using wire coil inserts having different pitch ratios with Al 2 O 3 /water nanofluid

having 0.1% volume concentration of nanoparticles as the working fluid Experiments using the plain tube and with wire

coil inserts were also carried out with distilled water as the working fluid for experimental setup validation and comparison.

The experimental results reveal that the use of nanofluids increases the heat transfer rate with negligible increase in friction

factor in the plain tube and the tube fitted with wire coil inserts In addition, empirical correlations are proposed based on

the experimental results of the present study, which are found to be sufficiently accurate for prediction of the heat transfer

and friction factor characteristics.

INTRODUCTION

The thermal load control technologies with extended surface,

such as fins and microchannels, have already reached their

lim-its Hence, the management of high thermal loads in

high-heat-flux applications offers challenges and the thermal conductivity

of heat transfer fluid has become vital Traditional heat transfer

fluids such as water, engine oil, and ethylene glycol are

inher-ently poor heat transfer fluids and thus major improvements

in cooling capabilities have been constrained To overcome the

limited heat transfer capabilities of these traditional heat

trans-fer fluids, micro-/millimeter-sized particles with high thermal

conductivity suspended in them were considered by Ahuja [1]

Address correspondence to Dr S Suresh, Department of Mechanical

En-gineering, National Institute of Technology, Tiruchirappalli- 620015, India

E-mail: ssuresh@nitt.edu

Heat transfer fluids containing suspended particles of /millimeter sizes suffered from numerous drawbacks, like ero-sion of the components by abrasive action, clogging in small pas-sages, settling of particles, and increased pressure drop Hence,they were not accepted as suitable candidate for heat transfer en-hancement and the search for new heat transfer fluids continued.Nanotechnology has come to the rescue by providing opportu-nities to process and produce materials of sizes in the nanometerrange that can be suspended in traditional heat transfer fluids

micro-to produce a new class of engineered fluids with high thermalconductivity and elimination of the mentioned problems asso-ciated with heat transfer fluids containing suspended particles

of micro-/millimeter size This new class of heat transfer fluidswith nanoparticles in suspension is called nanofluids [2].Nanofluids have emerged as an exciting new class ofnanotechnology-based heat transfer fluids and have grown enor-mously in the past few years Scientists and engineers are beingchallenged to discover the many unexpected thermophysical

485

486 M CHANDRSEKAR ET AL.

properties of these fluids, and to propose new mechanisms and

unconventional models to explain their behavior A

comprehen-sive summary of the previous research works on the formulation,

application, thermophysical properties, and heat transfer

char-acteristics of nanofluids is well documented in recent review

papers [3, 4]

Pak and Cho [5] experimentally investigated the convective

number (Nu) was found to increase with the particle volume

concentration and the Reynolds number (Re), the heat transfer

coefficient (h) actually decreased by 3–12% at constant velocity

Xuan and Li [6] investigated experimentally the convective heat

transfer and flow characteristics for Cu/water nanofluid flowing

through a straight tube with a constant heat flux under laminar

and turbulent flow conditions Cu nanoparticles with diameters

below 100 nm were used in their study The results of the

ex-periment showed that the suspended nanoparticles remarkably

enhanced the heat transfer performance of the conventional base

fluid, and their friction factor coincided well with that of the

water According to Xuan and Li, the convective heat transfer

coefficient of Cu/water nanofluid is increased by about 60%

for the nanofluid at 2.0% volume concentration In addition,

they also proposed new convective heat transfer correlations for

the prediction of heat transfer coefficients of the nanofluid for

both laminar and turbulent flow conditions As it is necessary

to study the pressure drop of nanofluids besides the heat

trans-fer enhancement in order to apply nanofluid to practical cases,

they also conducted pressure drop studies for both the laminar

and turbulent flow, which revealed no significant

augmenta-tion in pressure drop for the nanofluid, which indicates that the

nanofluids will not cause an extra penalty in pump power

Wen and Ding [7] experimentally probed the convective heat

signifi-cantly enhance the convective heat transfer in the laminar flow

regime, and the enhancement increases with Reynolds number

and particle volume concentration They also showed that (i) the

enhancement is significant in the entrance region and decreases

with axial distance and (ii) the thermal developing length of

nanofluids is greater than that of pure base liquid They

at-tributed the enhancement of the convective heat transfer to

par-ticle migration, which may result in a nonuniform distribution

of thermal conductivity and viscosity field, which will reduce the

thermal boundary layer thickness Using the same experimental

setup, Ding et al [8] reported a maximum enhancement of

con-vective heat transfer of carbon nanotube (CNT) nanofluids of

more than 350% at a Reynolds number of 800, and the maximum

enhancement may occur at an axial distance of approximately

110 times the tube diameter The observed large enhancement of

the convective heat transfer was attributed to the enhancement

of thermal conductivity, particle rearrangement, shear-induced

thermal conduction enhancement, reduction of thermal

bound-ary layer thickness due to the presence of nanoparticles, and the

very high aspect ratio of CNTs The convection heat transfer

performance of the graphite nanofluids was studied tally by Yang et al [9] in laminar flow through a circular tube,showing that the nanoparticles increase the heat transfer co-efficient of the fluid system in laminar flow, but the increase

experimen-is much less than that predicted by the correlation based onstatic thermal conductivity measurements Hence, Yang et al.concluded that further investigation is needed to develop an ap-propriate heat transfer correlation for non-spherical nanoparticledispersions

He et al [10] carried out experimental study on the flow

through a straight vertical pipe under both laminar and lent flow conditions They observed that for a given Reynoldsnumber and particle size, the convective heat transfer coeffi-cient increased with volume concentration in both the laminarand turbulent flow regimes, and the convective heat transfer co-efficient was insensitive to the changes in particle size Theirmeasured pressure drop of nanofluids was very close to that ofthe base liquid for a given Reynolds number With the sameexperimental setup, recently, Chen et al [11] with nanofluidshaving titanate nanotubes concluded that, compared with ther-mal conduction, the enhancement of the convective heat transferwas much higher and the enhancement depended on nanotubeconcentration, Reynolds number, and the axial position Giventhe nanotube concentration and Reynolds number, the highestenhancement was observed at the entrance region; the enhance-ment decreased with increasing axial distance and approached aconstant value in the fully developed region Nanoparticle shapewas shown to have a significant effect on the observed enhance-ment of the convective heat transfer coefficient, which was con-sistent with the reported results of nanofluids containing carbon

∼0.02) They suggested that the possible mechanisms of vective heat transfer coefficient enhancement include enhancedconduction under both static and dynamic conditions, particlerearrangement under shear, enhanced wettability, particle shapeeffect, and aggregation (structuring) This, however, requiresfurther experimental validation More recently, Duangthongsukand Wongwises [12] presented for the first time the heat transferand flow characteristics of nanofluid consisting of water and

double-tube heat exchanger The results showed that the convective heattransfer coefficient of nanofluid was only slightly higher thanthat of the base liquid by about 6–11% and has a small penalty

in pressure drop The same group [13] (with same experimentalsetup and nanofluid) studied the effect of various thermophys-ical properties models on predicting the forced convective heattransfer performance of nanofluid They reported that the var-ious thermophysical models have no significant effect on thepredicted values of the heat transfer coefficient of the nanofluid.They also concluded that the reliability and accuracy of theexperimental heat transfer coefficient may depend on the ex-perimental system calibration rather than on the thermophysicalproperties models of nanofluid