Heat transfer enhancement by using pulsating flowsEfrén Moreno Benavides Citation: Journal of Applied Physics 105, 094907 (2009); doi: 10.1063/1.3116732 View online: http://dx.doi.org/10.1063/1.3116732 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/105/9?ver=pdfcov Published by the AIP Publishing

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Heat transfer enhancement by using pulsating flowsEfrén Moreno Benavidesa�

Dpto. Motopropulsión y Termofluidodinámica, ETS Ingenieros Aeronáuticos, Universidad Politécnicade Madrid, Pza. Cardenal Cisneros, 3, 28040 Madrid, Spain

�Received 30 April 2008; accepted 15 March 2009; published online 8 May 2009�

The paper presents a theoretical model of convective heat exchangers working with internalpulsating flows. It aims for a better physical understanding of the processes leading to a heat transferenhancement inside these devices. When the frequency of the pulsation is increased, somegeometries exhibit a maximum response, measured by its temperature rise, similar to those obtainedin some dynamical resonant systems. The work explains the nature of this characteristic behaviorand produces a simplified theoretical model that isolates the main physical features of the fluiddynamics involved. Two characteristic frequencies, measured by its Strouhal numbers, aretheoretically found. The first one is associated with the spatial-averaged thermal response of thefluid near the wall and the second with the response of the velocity field. It is found, for a generaldevice, that both Strouhal numbers and the maximum enhancement are mainly defined by thegeometry of the device. Finally, the heat transfer enhancement of a straight channel, a backwardfacing step channel, and a two heated blocks inside an adiabatic channel are used to validate themodel. Enhancements calculated with the present model are compared with the results reported inthe scientific literature showing a good agreement for the tested cases. © 2009 American Instituteof Physics. �DOI: 10.1063/1.3116732�

I. INTRODUCTION

The specific power of thermoacoustic devices is in-creased by using in the heat exchangers solid walls separatedonly by a few thermal penetration depths of gas and by in-creasing the surface to volume ratio.1–4 This feature needs tobe implemented in the design for practical applications andhence several actual designs include flat plates, wires,grooves, sharp bends, stepped walls, and stepped or large-aspect ratio channels, among others. Besides, nowadays,there are many other engineering systems that present similarconfigurations such as microelectromechanical systems, mi-crocombustors, or microcoolers. The state of the art in thetheoretical characterization of these heat transfer processesappears when the pulsating flow is considered, either becauseit appears in a natural way such in some kind of thermoa-coustic devices1–5 or well because it is forced by pulsatingthe flow with vibrating or moving parts placed far enough inthe upstream or downstream path.6,7 Since the refrigerationpower depends on the heat fluxes, an interesting way tomodify these fluxes could be to change the mean velocity ofthe fluid just as it has been experimentally corroborated inbaffled pipes.8 Other interesting applications of the oscillat-ing baffled tubes are the oscillatory flow reactors9,10 wherethe transport phenomena, both mass and heat transfer, are ofthe highest interest.

Due to the interest, the theme is being discussed in thecurrent scientific literature where it remains controversial. Abrief review11 of this situation is best summarized by classi-fying previous work into four categories according to theconclusion being reached: �a� pulsation enhances heattransfer,8,12 �b� pulsation deteriorates heat transfer,13 �c� pul-

sation does not affect heat transfer,11 and �d� heat transferenhancement or deterioration may occur depending on theflow parameters.14

After the review,11 where the main conclusion was thatpulsation neither enhances nor deteriorates heat flow, it hasbeen reported that pulsation has no effect on the time-averaged heat transfer along straight channels15 and thatforced flow pulsation enhances convective mixing and af-fects Nusselt number:16 it reaches its maximum for a specificpulsating frequency and decreases for both higher and lowervalues of frequency.

Since this maximum looks like the one obtained in thecase of many damped resonating dynamical systems, it wassuggested16 that this behavior was due to the existence of anonlinear coupling of resonant nature between thermal ef-fects and fluid dynamics parameters, i.e., it was concludedthat the Nusselt number enhancement appeared to be of aresonant nature. However, since the energy equation is de-coupled from the momentum one for an incompressible flow,the velocity field is not determined by the temperature fieldexcept for the change in properties due to the temperaturevariations. In addition, the decoupled energy equation doesnot have any temporal derivative of order greater than 1, sothat there is no resonant frequency associated with the tem-perature field. A resonant frequency could appear if variableproperties were considered. However, it was verified, bymeans of numerical calculations with a two-dimensional�2D� backward facing step at low Reynolds numbers, that themaximum is obtained even when the properties are consid-ered constant16 and hence the explanation based on the reso-nant frequency is not satisfactory.

Therefore, this study is motivated by the need to find acomprehensive explanation that fixes the basic mechanismsleading to a heat transfer enhancement when a pulsating flowa�Electronic mail: efren.moreno@upm.es.

JOURNAL OF APPLIED PHYSICS 105, 094907 �2009�

0021-8979/2009/105�9�/094907/13/$25.00 © 2009 American Institute of Physics105, 094907-1

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is present. Although the theory is highly simplified, it pro-poses an alternative method to deal with this complex phe-nomenon. Since the formulation of the presented model issupported by integral equations, the main results are generaland, hence, useful for a broad class of geometries. Designerscould use these ideas to devise means to enhance heat trans-fer processes or to improve the current devices.

II. PROBLEM FORMULATION

In order to establish which ones are the main physicalphenomena that explain the above exposed disparity of re-sults, a general geometrical configuration will be studied andthe physics involved will be simplified to retain only therelevant terms, so that, we focus on incompressible flowwithout any source of mechanical work. This approach willretain only the most important aspects for the dynamical re-sponse.

Consider the unsteady temperature field T�x , t�, where xis the three-dimensional vector and t represents the time, andthe three-dimensional unsteady velocity field, u�x , t�, definedin every point of a volume V limited by the surface S�V�. TheReynolds’ transport theorem applied to the energy balanceleads to

dQ

dt= �

V

�

�t��cT + �

u2

2�d3x

+ �S�V�

�cT +P

�+

u2

2���u · dA� . �1�

The term on the left represents the heat flux, � is thedensity, c the specific heat capacity of the fluid, P the pres-sure, and dA a surface element vector. Since the fluid isliquid, P /�cT�1 and u2 /2cT�1 hold, and the energy bal-ance can be reduced to

dQ

dt= �

V

�

�t��cT�d3x + Gc�TH�TH − Gc�TL�TL. �2�

Here, G is the mass flow rate and ��TL�c�TL�TL and��TH�c�TH�TH are the spatial-averaged specific internal ener-gies over the inlet and exit ports, respectively.

We assume a stationary inlet temperature profile. How-ever, since the problem is not stationary, the temperatureprofile at the exit port must exhibit a periodical response.This temporal oscillation is the result of a spatial wave trav-eling toward the exit port. Consequently, the spatial-averagedinternal energy over a region with a characteristic lengthmuch greater than the wavelength must be stationary. Let usdefine V-Vp as the volume where transient effects are negli-gible over the spatial-averaged properties and Vp as the vol-ume where the spatial-averaged internal energy is time de-pendent. By this definition, the internal geometry and theunsteady fluid field have been divided into two characteristicregions with physical relevance. Remembering that the tem-poral oscillation of the internal energy is negligible outsidethe volume Vp, the following equation holds:

dQ

dt= �

Vp

�

�t��cT�d3x + Gc�TH�TH − Gc�TL�TL. �3�

By defining ��Tp�c�Tp�Tp as the spatial-averaged specificinternal energy over the volume Vp, the energy balance equa-tion leads to

dQ

dt= VP

d

dt��TP�t��c�TP�t��TP�t�

+ G�c�TH�TH − c�TL�TL� . �4�

As usual, the heat flow amounts to

dQ

dt= hSW�TW − TP� . �5�

Here h is the spatial-averaged heat transfer coefficientand SW is the wall area at the temperature TW.

III. EFFECT OF THE PULSATION ON THETEMPERATURE FIELD

For stationary flows, it is well known that the heat trans-fer coefficient is obtained, with enough precision, by relatingNusselt, Reynolds, and Prandtl numbers, Nu= f�Re,Pr�. Weassume that this relationship preserves all the importantphysics for our purpose �its validity, for the pulsating flowsconsidered here, is shown in Sec. VII�, so that it is used asthe instantaneous heat transfer model. Here, the Nusseltnumber is defined as usual, Nu=hL /k�TP�, where L is a char-acteristic length associated with the device geometry andk�TP� is the fluid thermal conductivity at the transient repre-sentative temperature in that region; the Reynolds number ischosen in this context as Re=GL / ���TP�A� with A the inletcross-sectional area and ��TP� the viscosity at temperatureTP; and the Prandtl number as Pr=��TP�c�TP� /k�TP�.

In order to obtain an analytical solution by means of aperturbation method around the stationary solution, which islabeled with the subindex 0, the nonstationary flow is mod-eled as G /G0=1+�a�t� with the initial prescription ��1�note that � does not have any physical meaning, it is usedonly with the purpose of obtaining an analytical solution andmust be set to 1 once the solution is found�. Therefore, thesolution is expressed as TP /T0=1+�b�t�+0��2� so hence, theheat transfer coefficient is reduced to h /h0=1+ ���a+�b���+0��2� with �=�-��+��+-��, �= �Re dLnf /d Re�G0,T0

,= �Pr dLnf /d Pr�G0,T0

, �= ��T /k��dk /dT��T0, �= ��T /���d�

/dT��T0, = ��T /c��dc /dT��T0

, and �= ��T /���d� /dT��T0de-

fining the dependency of the fluid properties and the heattransfer coefficient with the temperature and the velocity.

The order-�0 energy equation is

h0SW�TW − T0� = G0�c�TH�TH − c�TL�TL� . �6�

By using this information, the next order in the energyequation leads to

db

d�= �� − 1�za + ��z − 1�b , �7�

t

�= tc =

VP��T0�c�T0��1 + � + �h0SW

, �8�

094907-2 Efrén Moreno Benavides J. Appl. Phys. 105, 094907 �2009�

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z =TW

T0− 1. �9�

The pulsating flow is obtained when a is set as a= �a+ei �+a−e−i �� /2 �subindexes + and � indicate that a+

and a− are conjugate complex constants, and =�tc relatesnondimensional and dimensional pulsations�. Then b is cal-culated from Eq. �7� to be b= �b+ei �+b−e−i �� /2 with

b+ = a+�� − 1�z

1 − �z + i . �10�

The modulus of this amplitude, see Eq. �11� below,shows that the effects of the mass flow pulsation on the tem-perature field are damped if the period of the pulsation ismuch greater than the characteristic time tc,

�b+�2 = b+b− =�� − 1�2z2

�1 − �z�2 + 2 �a+�2. �11�

Note that the denominator does not vanish in any fre-quency, so that there is no resonant frequency in the tempera-ture field. Since the maximum perturbation in the tempera-ture field is obtained for zero frequency, the relativeimportance of �, �, and z determines the maximum response.Coefficients �, �, , and � depend on the properties of thefluid, whereas � and depend on the geometrical configu-ration of the device and on the topology of the fluid fieldinside it. Coefficient � mixes fluid properties and flow topol-ogy and, finally, coefficient z measures the thermal rise in-side the fluid.

Typical values of these coefficients for liquid water17

�which implies z�0.15� are �=0.60, �=−6.37, ���0.17,and �=−0.14. �The specific heat for water has a minimumnear 313 K, so that the coefficient can be positive or nega-tive depending on the temperature.� With respect to the to-pology, if the dominant solid structure inside the channel iscomposed of wires in crossflow surrounded by a liquid atReynolds numbers lower than 1000 and greater than 40,a good approach18 could be to use f�Re,Pr�=0.52 Re0.50

Pr0.62 /Pr�TW�0.25, so that �=0.50 and =0.62. However, fora turbulent movement inside an empty pipe �Re�104� typi-cal values could be �=0.80 and =0.33. Another interestingcase is the incompressible laminar flow in circular tubes thatassumes all the properties to be constant and both hydrody-namic and thermally fully developed, where it is known18

that the Nusselt number in this case does not depend on theReynolds and Prandtl numbers, i.e., �==0. As a result,coefficient � spans from �0.7 to 3.4 �the lower value holdsfor the low-Reynolds-number crossflow case�. By consider-ing other fluids and other topologies, the coefficients couldbe very different from the ones presented here.

Since �z moves from �0.1 to 0.5, the expected error dueto neglecting the variation of the properties with the tempera-ture can become significant. For example, a system charac-terized by a low Reynolds number over a flat plate ��=0.50, =0.33, �=1.5, and z=0.12� would have an error ofthe order of 18% ��z=0.18�. This result compares well withprevious numerical calculations,16 which in similar condi-tions of low Reynolds numbers reported a maximum differ-ence of 15%. However, although just as we have demon-

strated, exact calculations oblige to retain the dependencywith the temperature, this dependency is not necessary forexplaining the nature of the heat transfer enhancement.

Therefore, the properties are fixed as constant and hencethe temperature field response is given by

b+ = a+�� − 1�z1 + i

. �12�

Additionally, the heat transfer coefficient by

h

h0= 1 + �a� + 0��2� . �13�

This result shows that there is no variation in the instan-taneous heat transfer coefficient if �=0 just as someresearches15 pointed out for a range of pulsating frequenciescovering two orders of magnitude. They have shown that acircular tube in the laminar regime under pulsating flow con-ditions does not have any oscillation of the local Nusseltnumber if the flow is both thermally and hydrodynamicallydeveloped. This can be explained with the use of the station-ary solution for a constant wall temperature,18 Nu=3.66,which shows that in this case � is zero and hence, avoids anyfluctuation of the heat transfer coefficient.

However, they also found15 that the Nusselt number var-ies in time in the near-entry region of the pipe. The explana-tion comes again from Eq. �13� by taking into account thatthe thermal entry flow with a fully developed velocity profilehas a behavior given by the Léveque solution18 Nu�Re0.33,i.e., �=0.33, and hence the expansion given by Eq. �13�retains the pulsation effects.

On the other hand, when �=1, the oscillation of theconvection coefficient would be maximum while the oscilla-tion of the temperature would be minimum. This is nearlythe situation in turbulent flows where �=0.8.

IV. EFFECT OF THE TEMPERATURE DEFINITION

Since the heat transfer coefficient replaces the unknownheat flux by the introduction of a temperature difference, itsbehavior is conditioned by the definition of thisdifference.12,18,19 This problem is illustrated in this sectionand solved in the next one.

A quantity related with the heat flow appears if the heattransfer coefficient, Eq. �13�, is multiplied by the temperaturefluctuation, Eq. �12�. This quantity is an initial approach tothe heat flow,

dQ/dt

h0SW�TW − T0�= 1 + ���a − z−1b� − �2�z−1ab + 0��2� .

�14�

Since the time-averaged heat flux is obtained by integrat-ing the above expression along a complete period, the termsof order � vanish, and the nonlinear terms produce a heattransfer given by

�dQ/dt�averaged

h0SW�TW − T0�= 1 +

�2�a+�2

2

��1 − ��1 + v2 + 0��2� . �15�

094907-3 Efrén Moreno Benavides J. Appl. Phys. 105, 094907 �2009�

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This heat flux does not exhibit any resonant behaviorsince all the terms in the denominator are always positivequantities. Indeed, it always decreases with the frequencyand hence does not have any maximum �it is always positive,i.e., it predicts a heat transfer enhancement�. This kind ofsolutions can be found in the literature,8,11 when a periodicalperturbation without any constant component is added to thestationary temperature field to give the periodical response.

For example �note that this solution is maximum for �=1 /2, laminar flow, and negligible frequency, and that themaximum pulsation without reversing flow is given by �a+�=1�, the maximum heat transfer enhancement predicted bythis approach is 12.5% �6% for �a+�=1 /2�. Calculations for asmall-frequency periodic motion imposed on a fully devel-oped steady laminar flow inside a cylindrical pipe show thesame result.11 It is significant that they conclude that theinteraction between the velocity and temperature oscillationsintroduces an extra term in the energy equation which re-flects the effect of the pulsations in producing higher heattransfer rates, which increases approximately 6% comparedto its nonpulsating value when the amplitude is half of themean velocity. Besides, they conclude that the solutionwould depend on the square of the frequency if terms ofhigher order were kept in their solution just as the aboveexpression shows.

However, the solution given by Eq. �15� is not alwayscorrect since the average of the temperature perturbationover one period is implicitly assumed to be zero. Indeed, thegreater the heat transfer established by Eq. �15�, the greaterthe averaged local temperature over the wall, so that theexpansion of the temperature as a series of powers of �should include a constant term of order �2. The new termsmodify the solution and, hence, depending on the model usedfor this feature, the calculated Nusselt number could changesignificantly. As a consequence, it is possible to define dif-ferent Nusselt numbers. It has been shown19 that, for thesame spatial and temporal temperature distribution, differentdefinitions of the average Nusselt number for pulsating flowlead to contradictory results. The difference depends on howthe time-averaged temperature is constructed.

Although a definition of the Nusselt number for pulsat-ing flow, based on the local bulk temperature, can be definedin a rational way,12 the next sections will use the time-averaged heat flux because it is the entity with physicalmeaning. Besides, this quantity can be obtained from theenergy equation as the time-averaged increment of enthalpyalong the device. As it will be shown, only the steady Nusseltnumber, which is well defined in the literature, will beneeded in this article. Of course, since the steady Nusseltnumber depends on the Reynolds number, it must follow themass flux variations. This approach to the problem is devel-oped in Sec. V by using the minimum number of terms in theTaylor expansions that leads to a consistent solution.

V. HEAT TRANSFER FOR AN ARBITRARY GEOMETRY

Consider a general device with the limitations describedin Sec. II where the exit temperature TH is limited by the

wall temperature TW. Then, it is convenient to define a di-mensionless measure of the heating efficiency as

� =TH − TL

TW − TL. �16�

Of course, the fluid temperature between the initial andthe final part of the hot wall changes significantly. This isconsidered, in the calculation of the heat transferred from thewall, by using the logarithmic mean temperature,18

TP = TW −TH − TL

lnTW − TL

TW − TH

. �17�

The energy balance comes from Eqs. �4� and �5�,

hSW�TW − TP� = �VPcdTP

dt+ Gc�TH − TL� . �18�

The definitions of two dimensionless ratios, given byEqs. �19� and �20�, show the energy balance as

�P =TW − TP

TW − TL, �19�

�0 =TW − T0

TW − TL, �20�

h

h0�P +

d�P

d�=

G0c

h0SW

G

G0� . �21�

By using �0 as the stationary heating efficiency, the en-ergy balance is reduced to

� =h0SW

G0c=

�0

�0, �22�

h

h0

�P

�0+

d�P/�0

d�=

G

G0

�

�0. �23�

In addition, the dimensionless form of the equation de-fining the logarithmic temperature �Eq. �17�� is

ln�1 − �� + �/�P = 0. �24�

This equation relates the ratio �P /�0 and the exit tem-perature measured by � /�0. Its zeroth and first order termslead to

�0 = 1 − e−�, �25�

�

�0− 1

�P

�0− 1

= − Z , �26�

Z =1 − �0

�0 + �0 − 1. �27�

Coefficient Z measures the temperature amplification in-side the device. Normally, actual devices have small valuesof �, typically ��0.1, and hence the following holds: �0

094907-4 Efrén Moreno Benavides J. Appl. Phys. 105, 094907 �2009�

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���1, �0�1, and Z�2�−1�1. Since Z is much greaterthan 1, this states that the outlet fluctuations are driven byvery small fluctuations in the spatial-averaged specific inter-nal energy over the volume VP. Although in this case theterms of order 0�1 /Z� could be set aside, it is convenient toretain them for further discussions.

Therefore, when the fluctuation is taken into account,these ratios are expressed for any value of � as

�

�0− 1 = − Zm�t�� + s�2 + 0��3� , �28�

�P

�0− 1 = m�t�� − Ym�t�2�2 − s�2/Z + 0��3� , �29�

where Y is calculated from the logarithmic temperature defi-nition by retaining the second order terms,

Y = Z�0�Z/2 + 1��0 − 1

�1 − �0�2 . �30�

In order to solve the energy balance equation for m�t�and s, it is necessary to know h /h0. This heat transfer coef-ficient is of a convective nature, and hence it must depend ona characteristic velocity over the wall, which, in general, isdifferent from the spatial-averaged one obtained from themass flow rate. The quasisteady behavior suggests to use aTaylor expansion h /h0=1+�d�+��d2�2+0��3�, with � and�� constants and d a function that depends on the wall ve-locity. With this in mind, Eq. �23� evolves to

�1 + Z�m +dm

d�= a − �d , �31�

s =Z

Z + 1 ��d2 + m�Za + �d − Y�m + 2

dm

d���� . �32�

This system of differential equations determines m and sby knowing a and d. Since d�t� represents the oscillation ofthe dimensionless velocity over the wall, a pure-harmonicapproach imposes d= �d+ei �+d−e−i �� /2 with d+ and d− twoconjugate complex constants. �A simplified estimation of d+

is given in Sec. IX�. Then, the solution is obtained from Eqs.�31� and �32� as

m = �m+ei � + m−e−i ��/2, �33�

m+ =a+ − �d+

1 + Z + i , �34�

s =Z

2�Z + 1�

�0 + �1

Z + 1+ �2�

Z + 1�2

1 + �

Z + 1�2 , �35�

�0 =Z�Z + 1� − Y

�Z + 1�2 �a+�2 + ��� − �2Z + 1 + Y

�Z + 1�2 ��d+�2

− �Z2 − 1 − 2Y

�Z + 1�2 �a+��d+�cos��a+− �d+

� , �36�

�1 = ��a+��d+�sin��a+− �d+

� , �37�

�2 = ���d+�2. �38�

Note that the time-averaged outlet temperature is the onethat a thermocouple would measure in the exit port of anexperimental arrangement and hence the constant part of s isa measure of the heat transfer enhancement. Note also thatthe effect of fluid oscillation amplitude on heat transfer en-hancement appears in Eqs. �35�–�38�. In Eq. �35� the heattransfer enhancement s depends on the parameters �0, �1,and �2, given by Eqs. �36�–�38�. In this scenario the param-eters a+ and a−, which are the complex numbers related tothe phase and to the amplitude of the mass flow rate fluctua-tion, and d+ and d−, which are related to the phase and am-plitude of the velocity over the wall, explicitly appear. Thus,this heat transfer modeling takes into account the effectspresent in oscillatory baffled tubes where the pulsating flowcreates a periodic mass flow rate that contributes to an in-crease in the cycle-average velocity near the walls. Further-more, the model predicts that the dependence is quadratic inthe parameters describing the mass flow rate and the velocitynear the wall. Besides the peak superficial fluid velocity ismeasured by the modulus of d+ that appears in �1 and �2 aslinear and quadratic, respectively. Therefore, both the peaksuperficial fluid velocity and the averaged flux are used tocharacterize the flow, just as it is normally done for describ-ing the mixing in oscillatory baffled tubes.20 However, Eq.�35� also depends on the frequency and on the relative phase.In fact, this is a proof that shows that more than two param-eters must be considered for a full description.

For very low frequencies, this solution �Eq. �35�� canproduce the enhancement or deterioration of the heat transferprocess depending on the sign of the coefficient �0 whichchanges, for example, with the relative phase between thevelocity over the wall and the mass flow rate. The same canhappen at very high frequencies depending on the sign of �2

which changes with the sign of ��, which normally is notpositive and hence it will not give an enhancement of theheat transfer process. Besides, the solution given by Eq. �35�presents an extreme at the frequency,

Z + 1=

�2 − �0

�1����2 − �0

�1�2

+ 1. �39�

This frequency gives the maximum response and hence es-tablishes a critical Strouhal number. Since this extreme dis-appears for those devices where �1 vanishes, the above ex-pression is able to reproduce all the results found in thecurrent scientific literature. In the next sections, this solutionis validated for several practical devices.

VI. HEAT TRANSFER IN STRAIGHT CHANNELS

As shown before, for very high frequencies, the solutiongiven by Eq. �35� can be approached by

s =��Z

2�Z + 1��d+�2. �40�

094907-5 Efrén Moreno Benavides J. Appl. Phys. 105, 094907 �2009�

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This states that the final response depends on the parameter Z�which depends on the efficiency of the device�, on thesquare of the amplitude of the pulsation, and on the param-eter �� �which depends on the behavior of the stationaryconvective coefficient�. Assuming that the stationary convec-tive coefficient follows a potential law of the form h�Re�,its Taylor expansion gives ��=���-1� /2 which is negativefor those applications with ��1. For example, a turbulentflow ��=0.8� suffers a discouragement of the heat transferprocess. From the expression, we obtain that the maximumdiscouragement is produced for �=0.5 �laminar flow� and Ztending to infinity �very low efficiency�. Without reverseflow on the wall, the maximum value for �d+� is 1, and hencethe maximum attainable discouragement is near 6%.

Another interesting limit is the one obtained for verylong pipes, where the ratio length/radius �l /R� tends to infin-ity. For a pipe, �=h0SW / �G0c� can be expressed as �=2�l /R�Nu0 / �Pr Re0�, with the Nusselt and Reynolds num-bers based on the radius of the pipe. This means that � canbe much greater than 1 if the length increases sufficiently:l /R�Pr Re0 /Nu0. In this limit, ��1, �0��−1, Z��e−�

�1, and Y �� /2�1 hold, and hence, the solution given byEq. �35� tends to zero exponentially independently of theflow parameters. This means that the difference between pul-sating and nonpulsating flows decreases along the tubelength just as it has been found in a previous work.21

In addition, for fully developed stationary flows in cir-cular pipes, the value of � is 1 �Ref. 18� and thus ��=���-1� /2=0 holds: the discouragement is near zero forhigh frequencies independently of the length of the pipe andof the amplitude of the oscillation. It has been theoreticallydemonstrated12 that the difference between the pulsating andnonpulsating flows is very small �less than 1% of the steadyone� for very low frequencies and zero for high frequencies.

For turbulent flows ���0.8, ���−0.08� or pipe entries���0.33, ���−0.11�, the behavior depends on the heatflow imposed by the stationary mass flow rate, on the opera-tional temperatures, and on the propagation of the pulsationtoward the wall against the viscosity. This last fact obliges usto seek a model, as general as possible, that relates the aver-aged velocity over the wall and the heat transfer coefficient.Section VII aims to attain this, while Sec. IX describes amodel that relates the velocity over the wall with the massflow rate pulsation.

VII. ESTIMATION OF THE HEAT TRANSFERCOEFFICIENT FOR AN ARBITRARY GEOMETRY

This estimation lies in the dimensionless equations gov-erning the heat transfer for an incompressible flow under theassumption that the properties of the fluid are temperatureindependent18 �Einstein’s notation is applied with i=1,2 ,3�

�i�i = 0, �41�

St0 �0� j + �i�i� j = − � jp +1

Re0�i�i� j , �42�

St0 �0� + �i�i� =1

Pe0�i�i� −

Ec0

Re0� j�i��i� j + � j�i� . �43�

Dimensional and nondimensional variables are related toeach other by means of

�

�t= ��0,

�

�xi= L−1�i, xi = L�i, ui = �i

G0

�A,

P = pG0

2

�A2 , T = TW − ��TW − TL� ,

St0 =�AL�

G0, Re0 =

LG0

�A, Pe0 =

LG0c

kA,

Ec0 =G0

2

�2A2c�TW − TL�. �44�

Note that the dimensionless numbers St0, Re0, Pe0, andEc0 are evaluated at the mean mass flow rate, and hence donot change with the pulsation. This fact is indicated by thesubindex 0. Eckert number Ec0 only affects the temperaturefield and only has to be taken into account when frictiongives rise to a noticeable warming of the fluid which is notthe case because the fluid velocity is considered to be negli-gible when compared with the speed of sound and there is noany large velocity gradient �fluid is being modeled as incom-pressible�. In this situation, the equation governing the maxi-mum attainable heat transfer enhancement is

Pe0 St0 �0� + Pe0 �i�i� = �i�i� . �45�

The Strouhal number St0 typically is near to one in theseapplications, and hence the complete device cannot betreated as stationary. However, the Péclet number Pe0 is nor-mally much greater than 1, and hence the dominant effects inthe heat transfer forced by the wall temperature can be re-stricted to those that are important in a region near the wall.Indeed, it fixes the thickness of the thermal convectiveboundary layer. The thickness of this thermal boundary layercan be obtained by calculating the characteristic length thatleads to a Péclet number of order of 1, i.e., ��kA / �G0c��L. The energy equation can be formally rewritten for thislayer by changing the characteristic length L for ��L, sothat, near the wall, St0����0�� /L��1 and Pe0����1 hold.�Here, the nomenclature Pe0��� and St0��� indicate that thosenumbers have been calculated by using � instead of L.� Thisfact allows the removal of the temporal derivative from theenergy equation near the wall. Therefore, although the com-plete device is not stationary, the thermal boundary layer canbe modeled as quasisteady, and Eq. �45� can be reduced to

Pe0 �i�i� = �i�i� . �46�

Note that we have retained the convective term in the energyequation near the wall because it has to be as important asthe dominant one. Since it is proportional to the velocity, themaximum heat transfer enhancement is obtained when thevelocity near the wall is maximum. This allows us to find anupper bound to the heat transfer enhancement by consideringan inviscid flow, i.e., by taking the Reynolds number to in-finity. Thus, the vanishing viscosity allows to approach thevelocity near the wall by a parallel flux �i=1� which does notdepend on the transversal coordinate �i=2�. In this situation,

094907-6 Efrén Moreno Benavides J. Appl. Phys. 105, 094907 �2009�

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the thermal conduction in the direction of the flow can beneglected as well as all the derivatives in the third compo-nent �i=3�, and hence the equation governing the maximumattainable heat transfer enhancement results

Pe0 �1�1� = �2�2� . �47�

In this equation �1 is assumed to be maximum and indepen-dent of the coordinates i=1,2 ,3. Although we have pre-sumed an infinite Reynolds number, the Péclet number Pe0

has been maintained finite because in a convective heattransfer problem, it is always as important as the conductiveheat transfer. Since liquids have Prandtl numbers �Pr=Pe /Re� above 1, and then a thermal boundary layer thinnerthan the velocity boundary layer �removed in the presentcase because of the large Reynolds number assumed�, thelast assumption could seem so drastic. However, it is usefulbecause it emphasizes the effects derived from the velocityfluctuation while retaining a finite heat transfer rate.

A solution which satisfies the boundary conditions���1 ,0�=0 and ��0,� / �2���=�P is

� = �Pe−��2/�1 Pe0��1 sin���2� . �48�

In this expression, ��0 is a constant related to the upperlimit of the convective boundary layer, � �note that the fluxof heat is zero at �2=� / �2��=� /L�. The heat transferredover the entire wall amounts to

�0

SW

− k� �T

�x2�

x2=0dSW =

kSW�TW − TL�L

�0

1 � ��

��2�

�2=0d�1

=kSW�TW − TP�

L

�1 Pe0

�

��1 − e−��2/�1 Pe0�� �49�

This heat has to be equal to the one calculated from theconvective coefficient by means of the expression hSW�TW

−TP� and hence, by using the averaged thickness of the ther-mal boundary layer, �, the convective coefficient is con-verted to

h =2k

�L�1 Pe0����1 − e−��2L/�/4�1 Pe0����� . �50�

Remembering that Pe0����1 and that � /L�1, theabove expression can be approached by

h =2k Pe0���

�L�1. �51�

This solution is formally similar to that calculated for athermal entry in a laminar tubular flow where the velocityprofile is described by the Hagen–Poiseuille´s law, Nu=3.66+0.05 /X, with X=L / �2R Pe� and R the pipe radius,18

if the heat transfer at zero velocity is discounted. It is inter-esting to note that this solution assumes a finite low Rey-nolds number, whereas the one derived here assumes theReynolds number to be infinite with the purpose of obtainingan upper bound to the actual heat in a given configuration.

Since the Péclet number is the product of the Reynoldsand Prandtl numbers, the model obtained for the heat transferprocess is

Nu =2�

�LRe0 Pr �1, � = 1, �� = 0. �52�

The advantage of this approach is that it explicitly re-tains the effect of the local velocity near the wall. This ex-pression states that the mean Nusselt number is proportionalto the dimensionless velocity over the wall. Since this veloc-ity is induced by the pulsating flow, it has to exhibit aninduced oscillation which also makes the instantaneous meanNusselt number to oscillate. The effect of the pulsation onthe velocity near the wall can be described by �1 /�10=1+�d�t�, and hence the instantaneous spatial-averaged heattransfer coefficient as h /h0=1+��d�t�+�2��d�t�2=1+�d�t�.�It does not have any term of order greater than � because ofthe linear model obtained.�

In this situation, the energy balance �Eqs. �35� and �38��provides the following equations for the heat transfer:

s =Z

2�Z + 1�

�0 + �1

Z + 1

1 + �

Z + 1�2 , �53�

�0 =Z�Z + 1� − Y

�Z + 1�2 �a+�2 −Z + 1 + Y

�Z + 1�2 �d+�2 −Z2 − 1 − 2Y

�Z + 1�2 �a+�

��d+�cos��a+− �d+

� , �54�

�1 = �a+��d+�sin��a+− �d+

� . �55�

The time-averaged energy transfer is measured by thetime-averaged exit temperature and hence by s. In additionthe Strouhal number and the dimensionless pulsation are re-lated to each other by means of a characteristic Strouhalnumber Stc defined by

St0 =�LA�

G0=

�LA

G0tc = Stc . �56�

Since the mean thickness of the thermal boundary layeris given by �, the volume VP can be approached by VP

=SW�, and hence the characteristic time by tc=�c� /h0 so thatthe characteristic Strouhal number leads to

Stc =L

�

A

SW� . �57�

This solution shows that the heat transfer enhancement isdominated by the relative phase between the pulsating massflow rate and the displacement induced over the wall. Inparticular, the expression has an extreme at the frequencygiven by Eq. �39� with �2=0.

If the solution given for small frequencies coincides withthe one obtained for nonpulsating flows, �0 should be zeroand, hence, the averaged phase between the velocity near theinlet port and the wall should accomplish

094907-7 Efrén Moreno Benavides J. Appl. Phys. 105, 094907 �2009�

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cos��a+− �d+

� =Z�Z + 1� − Y

Z2 − 1 + 2Y

�a+��d+�

−Z + 1 + Y

Z2 − 1 + 2Y

�d+��a+�

.

�58�

It is convenient to note that the condition �0=0 is notattainable when �0 approaches to 1. In fact, near this point ofmaximum efficiency, ��1, �0��−1, Z��e−��1, and Y�� /2�1 hold and hence, the last expression does not pro-duce any real solution for the cosine except when �a+�= �d+�.This establishes that the heat transfer enhancement that willbe obtained in Sec. VIII does not exist for high efficiencydevices.

For those cases where the pulsation on the wall velocitycoincides with the mass flow rate pulsation, i.e., �1=0 and�a+�= �d+�, there is no any effect in the heat transfer process atany frequency, just as occurs for straight channels with com-pletely developed flows. This result reproduces the one ob-tained in Sec. VI.

It is also interesting to note that the result can be anenhancement of the heat transfer flow if �1�0 and a dis-couragement if �1�0. The first case is obliged by the cau-sality if the hot wall is placed near the inlet port �the phasedifference is positive and less than �� and the second one ifthe distance transversal to the flow from the hot wall to theinlet port is adequately increased. The maximum enhance-ment will be produced if the pulsation over the wall had aphase delay near � /2.

Note that the dominant parameters retained in this modelfor the heat transfer process near the wall are the velocityover the wall and the thickness of the thermal boundarylayer. Since the nonslip condition imposed by the viscosityproduces also a viscous boundary layer, the exact value ofthe averaged velocity near the wall is not adequately deter-mined. This crucial problem has been solved in this sectionby substituting the exact solution of the problem by an upperbound of it. Since the maximum enhancement will comefrom the maximum heat transfer coefficient, the partial de-rivative equations of the fluid mechanics have been simpli-fied in order to obtain the maximum attainable heat transfercoefficient near the wall. As shown, this solution comes fromretaining the thermal boundary layer and from removing thenonslip condition. Hence, the velocity over the wall is justrelated with the velocity in the nonviscid core. This approachis feasible because the thermal and momentum equations aredecoupled in the incompressible problem presented.

VIII. DISCUSSION AND VERIFICATION FOR LOWVALUES OF �

When ��1, the heating efficiency of the device is verylow, just as occurs in some devices of reduced size. Thisapplication can be modeled by the following practical limit:�0���1, �0�1, and Z�2 /��Y �1. In this limit, theabove expressions lead to

s =1

2

�0 + �1St0

Z Stc

1 + � St0Z Stc

�2 , �59�

�0 = �a+�2 − �a+��d+�cos��a+− �d+

� , �60�

�1 = �a+��d+�sin��a+− �d+

� . �61�

In the case of having no effect at zero frequency �Eq.�58� holds�, Eq. �59� can be simplified to

s =�a+���d+�2 − �a+�2

2

St0/�Z Stc�1 + �St0/�Z Stc��2 . �62�

From Eq. �57�, the Strouhal number for maximum en-hancement can be expressed as

Z Stc = 2L

�

A

SW. �63�

This is the ratio of two volumes: the first one is related tothe size of the device, and the second one, with the volumeof fluid affected by the convective heating near the hot walls.

This formula states that the Strouhal number for maxi-mum enhancement is given by St0=Z Stc, which is of orderof 1 in those cases where LA / ��SW� is of order of 1 and leadsto a maximum enhancement given by

smax =�a+�4

��d+�2 − �a+�2. �64�

In addition, it states that this kind of enhancement re-quires that the relative importance of the pulsation over themean velocity must be greater in the wall than in the inletport, i.e., �a+�� �d+�, something that can be implemented bysuddenly increasing the cross-sectional area of the channel.

For example, a backward facing step like the one pre-sented in Fig. 1 exhibits this feature.16 Since the flow channelduplicates its area, the mean velocity is divided by a factor of2 �this will be derived in Sec. IX�, and hence, the amplitudeof the pulsation can be estimated by considering that itdoubles the mean velocity over the wall: a good approach tothe maximum enhancement can be obtained by considering�a+�=1 and �d+�=2. This produces a maximum value for theheat transfer enhancement of 43% which compares well withthe previous literature16 which states that the maximum Nus-selt number is 42% higher than in the steady case for idealwater and 44% greater if the properties of the fluid are con-sidered to depend on the temperature.

The device �establishing the characteristic length L asthe outlet height� is described16 by L=450 �m and SW /A=10 and works with ideal water �k=0.598 W K−1 m−1, �=10−3 Pa s, c=4180 J kg−1 K−1, and �=998 kg m−3� in anoperational point identified by Nu0=5.18 and Re0=100.These values give the following nondimensional character-

FIG. 1. Dimensionless definition of the 2D backward facing step. Not atscale.

094907-8 Efrén Moreno Benavides J. Appl. Phys. 105, 094907 �2009�

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ization of the stationary heat flow: �=0.074, �0=0.071, �0

=0.96, Z=26, and Y =0.34. By looking at the configurationof the device, it is plausible to assume that the thermalboundary layer extends over the 5% of the channel height�� /L�0.05�. This leads to consider Stc=0.037, and hence,the maximum enhancement is obtained for a Strouhal num-ber equal to Z Stc=0.96. These values have been employedto feed Eq. �62� whose results are represented in Fig. 2 to-gether with data proceeding from previous numericalcalculations16 for different frequencies.

Discrepancies observed in Fig. 2 between the numericalcalculations and the approach given in this section can bedue to the neglected nonlinear terms, which can introduceharmonics of order greater than 1 and to the fact of havingneglected the influence of the Strouhal number on the ampli-tude and phase of the wall velocity. Additionally, the lack ofexperimental data produces an extra error whose amplitudedepends on the assumptions of the 2D numerical model,mainly on the boundary conditions. However, presented re-sults show that the main physical phenomena governing thepulsating heat transfer process, which have been taken intoaccount by the present work, are �i� the existence of an in-ternal characteristic transient temperature, given by themodel that is summarized by Eq. �24�, �ii� the transient re-sponse of an internal volume of fluid, given by the modelleading to Eq. �35�, and �iii� the convective heat transferdriving the transient response of that internal volume, givenby the characteristic limit that leads to Eq. �52�.

Besides, it can be anticipated that the results obtained byconsidering �d+� constant in Eq. �62� will be incorrect if thedevice geometry exhibits a hot wall that is not flat in thedirection of the flow. These kinds of geometries need to pre-dict �d+� as a function of the Strouhal number, in the mostgeneral way possible.

IX. ESTIMATION OF THE VELOCITY OVER THE WALL

In a general device there are different phenomena con-ducting to a pulsation on the wall velocity. One of them isthe direct effect that modifies the intensity of the entire ve-locity field with a delay depending on the device geometry.

Another is the indirect effect given by the displacement ofsome fluid characteristic structures such as stagnation pointsor contracting or expanding recirculation regions. These phe-nomena appear only if the geometry of the device is ad-equate, and add an extra velocity in the wall that, in general,will have a phase different from the forcing pulsation:G /G0=1+�a�t� with a= �a+ei �+a−e−i �� /2. In Sec. IX A,we will characterize the longitudinal problem in the corewhere the effects due to the viscosity can be neglected, andafterwards, the viscous problem near the wall.

A. Longitudinal distribution of pressure imposed bythe nonviscid core

Providing that the Reynolds number is much greater than1, the mass flow rate along the channel is related with thepressure by means of the momentum equation applied to thenonviscid core. The integral equation of momentum in thelongitudinal direction is then expressed as

�V

���u1��t

d3x + �S�V�

�u1uidAi = F1. �65�

Since the fluid has been assumed to be incompressible,the longitudinal velocity can be obtained from the mass fluxas

u1 =G

�A�x1�. �66�

With this change, the integral of volume leads to thelength of the channel, l, and the surface integral, to the dy-namic pressures in the inlet and exit ports whose areas are,respectively, A and AE,

ldG

dt+

G2

�A� A

AE− 1� = F1. �67�

The force over the volume, which includes the one dueto the pressure in the inlet and outlet ports and the one overthe walls, can be modeled by a force linked to the pressuredrop and a drag force. The drag is modeled as a frictioncoefficient cf multiplied by the inlet dynamic pressure andthe dimensionless length of the channel,

F1 = �PI − PE�A −cf

2

G2

�A

l

L. �68�

After some algebra, the momentum equation can bewritten in a nondimensional form,

�pI − pE�L

l= St0�0

G

G0+ � G

G0�2� cf

2+

L

l� A

AE− 1�� . �69�

Normally, these devices have an aspect ratio muchgreater than 1, i.e., l /L�1, while A�AE and cf �1 due tothe pulsation,22 and hence the second term in the squarebracket can be neglected. When the pulsation is taken intoaccount, the mass flow rate is given by G /G0=1+�a, and thelast equation can be expanded up to terms of first order in �.During this expansion, the friction coefficient in the channelcan be considered as constant. In addition, the first term is

0.3

0.4

0.5

sheattransfer

cement

0

0.1

0.2

0 0.5 1 1.5 2 2.5 3

Dimensionles

enhan c

Strouhal number (St0)

FIG. 2. Comparison between the theoretical dimensionless heat transferenhancement �solid line� and data obtained from direct numerical simulation�Ref. 16� �diamond points� for different values of the frequency at Re0

=100.

094907-9 Efrén Moreno Benavides J. Appl. Phys. 105, 094907 �2009�

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related to the longitudinal gradient of pressure and, hence,the pressure in the nonviscid core is governed by the follow-ing equation:

− �1p =cf

2+ �St0�0a + cfa�� + 0��2� . �70�

B. Longitudinal distribution of velocity

The solution over the walls that are parallel to the flowcomes from considering the external gradient of pressuregiven by the last equation and by retaining the viscosity, suchas the following momentum equation �derived from Eq. �42��expresses:

St0 �0�1 + �1�1�1 = − �1p +1

Re0�2�2�1. �71�

By considering the velocity over the wall as

�1/�10 = 1 + ��d+ei�0 + d−e−i�0�/2, �72�

and the pressure gradient imposed by the inviscid core �Eq.�70��, the order-� contribution of Eq. �71� results

�10St0 �0d+ + �102 �1d+ −

�10

Re0�2�2d+ = St0 �0a+ + cfa+.

�73�

The spatial derivatives can be removed by using thedominant component of its spatial Fourier expansions,

�1d+ = − i�1d+, �74�

�2d+ = − i�2d+. �75�

With this in mind, the pulsating solution evolves to

d+ =a+

�10

St0 i + cf

�St0 − �10�1�i +�2

2

Re0

. �76�

We are interested in the modulus, which is

�d+� =�a+��10� St0

2 + cf2

�St0 − �10�1�2 + � �22

Re0�2 . �77�

This expression shows that the velocity over the wall canbe amplified if the Strouhal number coincides with the eigen-value describing the dominant spatial distribution of thegeometry.

C. Comparison with previous 2D numericalsimulations

This expression takes into account the behavior of thevelocity near the wall and can be used to explain the resultsfor the device described in the Ref. 22. This is a 2D channelwith two heated blocks working with a pulsating flow of air�see Fig. 3�.

Numerical calculations22 of the heat transfer enhance-ment are plotted in Fig. 4 for two different Reynolds num-bers and a fixed amplitude of the pulsation as a function of

the Strouhal number. The global Nusselt number that is rep-resented in Fig. 4 for the full device can be obtained bytaking into account that the Nusselt numbers for the steadynonpulsating flow are 10.3 and 11.2, at respective Reynoldsnumbers of 500 and 700, for the first block and 7.50 and 8.96for the second block.22

From these results, we can calculate that the Strouhalnumber for maximum enhancement is near 5. However, theisothermal lines22 indicate that the dimensionless thicknessof the region affected by the transient field of temperature isnear 0.25, and therefore � /L�0.25, in addition, the hot sur-face is Sw /A=3. This allows to calculate Z Stc�2.7 which isroughly the half of the one observed in Fig. 4. The discrep-ancies are solved by using Eq. �77�.

The value of �1 in Eq. �77� arises from the longitudinaldistribution of the blocks, so that the main longitudinalwavelength introduced by this configuration is obtained byconsidering that one block induces a complete period�2� rad� over its dimensionless length �1�, i.e, by taking�1�2�.

The coefficient �2 is related to the reciprocal of thethickness of the viscous boundary layer, so that it can beapproached by a nth-power law of the Reynolds number:�2=k Re0

n. For stationary, laminar, and flat boundary layers,it is known18 that the value n=0.5 holds. Although, the prob-lem considered here is not flat and neither stationary, theboundary layer can be modeled as quasisteady since itsthickness is expected to be very low and hence n is fixed tobe 0.5. Considering that the dimensionless boundary layerthickness is near 5% of the channel height, k can be obtainedfrom 5001/20.05k�� /2 which leads to k�1.4.

The averaged velocity over the wall inside the thermallayer, �10, is governed by the mean flow over the wall of the

FIG. 3. Dimensionless definition of the 2D two-heated-block channel.

0.15

0.20

0.25

sheattransfer

cement

0.00

0.05

0.10

0 2 4 6 8 10 12 14

Dimensionles

enhanc

Strouhal number (St0)

Re0=500

Re0=700

FIG. 4. The effect of the Strouhal number on the heat transfer enhancementwith an amplitude of the pulsation of 0.2 for the device described in Fig. 3.Points are numerical calculations carried out with a direct simulation at twodifferent Reynolds numbers �Ref. 22� and solid lines are the theoreticalcalculation carried out by using Eqs. �62� and �77�.

094907-10 Efrén Moreno Benavides J. Appl. Phys. 105, 094907 �2009�

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channel and by the defect of momentum due to the viscouslayer. Since the thermal boundary layer and the obstacleshave a similar dimensionless thickness �0.25�, which is muchgreater than the viscous layer �0.05�, it is plausible to expectthat the dominant effect was due to the continuity equation,which takes into account the averaged area of the channel,

E�10 = 1, �78�

E =1

l�

0

l A�x1�A

dx1. �79�

Note that E is 2 �this result was used in Sec. VII toestablish �d+�=2� for the device of Fig. 1 and 0.75 over theblocks in the device of Fig. 3. However, the second devicehas four vertical hot walls that are not present in the firstdevice. Since these vertical walls are not contemplated bythe averaged velocity given by Eq. �79�, it must be correctedin order to consider their contribution to the averaged veloc-ity over the wall. Every vertical wall generates a stagnationpoint in its lower corner that reduces the velocity to zero inits surrounding. Since the velocity field near a stagnationpoint changes linearly with the coordinate, the averaged ve-locity over the vertical walls is roughly half of the one at theupper corner. In addition the vertical component of the ve-locity on the upper corner is reduced considerably by thedeflection imposed by the main stream and the viscous layerreduces even more this velocity. Hence, we neglect the ve-locity on the vertical walls: this gives a dimensionlessspatial-averaged velocity of �10�0.9.

Since the pulsation modifies the velocity field, the forceover the blocks is typically amplified by the pulsating flow.For example, the amplitude of the instantaneous friction fac-tor increases dramatically, easily reaching 50 times the sta-tionary value:22 for �a+�=0.2, Re0=500, and St0�6.5, thefriction factor under pulsating conditions is near 35 times thestationary one, which is typically of order one of 1. We haveused values of cf =29 for Re0=500 and cf =48 for Re0=700.The plots in Fig. 4 compare the numerical results22 with thetheoretical ones obtained by using Eq. �62� with �d+� given byEq. �77�.

Figure 5 shows the effect of the Strouhal number on thevelocity over the wall. As St0 increases, the gain in the pul-

sating wall velocity gradually increases to a maximumaround St0�5.6 and decreases afterward. This peak is due tothe dominant spatial harmonic induced by the blocks.

Therefore, in this case, the heat transfer enhancement isgiven by the effect of the pulsating mass flow rate on thevelocity field. The effect on the temperature field does notproduces a significant enhancement, although it is slightlyappreciated in the change in the slope in the region nearSt0�2.7 for the lower Reynolds number �see Fig. 4�.

The physics implemented in this scheme is the follow-ing: the mean flow pulsation induces a velocity pulsationover the wall, which depends on the changes in the cross-sectional area available to the flow, in the augmentation ofthe pressure forces, in the longitudinal velocity distributionalong the channel, and in the viscous boundary layer thick-ness. The gas dynamic, described mainly by these param-eters, substantially increases the pulsating wall velocity for agiven Strouhal number �measured by Eq. �77�� and results inthe augmentation of convective thermal transport �measuredby Eq. �62��.

X. CONCLUSION

A theoretical study was carried out for determining theeffect of pulsating flows in channels with different solidstructures on the inside. The work shows that, although thetemperature field is substantially affected by such pulsation,the heat transfer is not changed for straight channels withfully development profiles of velocity. The study also showsthat there are changes in the heat transfer where the velocityprofile is not developed, for example, near the pipe entry orwhere there are abrupt changes in the internal geometry ofthe channels.

Two different geometries of this last case have beenstudied in representation of the typical topologies that ap-pears in these types of devices. The first one is a 2D back-ward facing step with the hot surface parallel to the flow inan adiabatic channel. In this case, a maximum enhancementappears for a characteristic Strouhal number defined by theratio of two volumes: one related with the size of the deviceand the other to the volume of the fluid region supportingtransient effects on the spatial-averaged temperature near thehot wall. This number results to be quite stable for a givendevice. The second one is a 2D channel with two heatedblocks at uniform temperature in an adiabatic channel. In thiscase, the maximum enhancement is mainly controlled by theresponse of the wall velocity to the pulsating excitation. Thisnew effect introduces another characteristic Strouhal numberfor the device which is mainly defined by the periodical lon-gitudinal distribution of the blocks along the channel. Resultsshow that this number is also quite stable.

The relative importance of these two physical phenom-ena establishes the final response for a given device. In par-ticular, the maximum attainable enhancement should be ex-pected when both Strouhal numbers coincide. Besides, it hasbeen proven that, due to the first effect, the enhancementexists even when there is no resonant frequency in the sys-tem.

Another interesting conclusion is that, while the steady

3

4

5

6

d +| Re0=700

0

1

2

3

0 2 4 6 8 10 12 14

|d

Strouhal number (St0)

Re0=500

0

FIG. 5. Dimensionless amplitude of the pulsation of the velocity over thewall estimated as a function of the Strouhal number for the device describedin Fig. 3.

094907-11 Efrén Moreno Benavides J. Appl. Phys. 105, 094907 �2009�

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heat transfer depends on the variations in fluid propertieswith the temperature, the heat transfer enhancement does notdepend significantly on them. As a result, the performance ofthese devices is summarized in an equation which shows thatthe enhancement is easily obtained in those devices whoseefficiency is low if the pulsation over the wall has the correctamplitude and phase.

Although the validation has been made in straight andconstricted 2D channels operating with a pulsating flow su-perimposed on a continuous and constant mass flow rate, theconclusions of the theory are quite general because of theintegral formulation of the physical laws. This allows the useof the main results for complex three-dimensional geom-etries.

Additionally, by using an upper bound of the convectiveheat transfer in the boundary layer, it has been concluded thatthe main information relating to the details of the device isthe closure model for the characteristic distribution of thevelocity over the wall, as a function of the inlet flow pulsa-tion, and the averaged thickness of the thermal boundarylayer. This result reveals that this model is very sensitive tothe geometry and to the flow patterns in the channel, espe-cially when stagnant or recirculation regions are involved. Acomparison with numerical calculations obtained from theopened literature shows that the results agree well, at leastfor the cases presented, if these models are correctly imple-mented. It also shows that, as expected, the results are verydependent on the pressure over the blocks under pulsatingconditions because the pressure field determines the spatialdistribution of the velocity. Therefore, the use of the modelwith complex geometries requires the correct estimation ofthe velocity near the wall, that requires a good approach forthe drag force and a good knowledge of the spatial distribu-tion of the velocity; both could be the object of future studiesfor particular configurations.

NOMENCLATURE

a dimensionless measure of the flow rateoscillation

A inlet cross-sectional area of VAE cross-sectional area of the exit port

b dimensionless measure of the temperatureoscillation

c specific heat capacitycf friction coefficientd dimensionless measure of the oscillation of the

velocity near the wallEc Eckert numberF force exerted over the fluidG dimensional mass flow rateh convective heat transfer coefficientk thermal conductivityL characteristic length of the devicel longitudinal length

m dimensionless measure of the first-order term ofthe heat efficiency oscillation

Nu Nusselt number

P dimensional pressure fieldp dimensionless pressure field

Pe Péclet numberPr Prandtl numberQ heatR radius

Re Reynolds numberS boundary of the control volumes dimensionless measurement of the second-order

heat efficiency oscillationSt Strouhal number

Stc characteristic Strouhal numberSW area of the solid surface on the boundary of Vp

T dimensional temperature fieldt time

T0 spatial-averaged stationary temperature of Vp

tc characteristic time that measures the delay ofthe device response

TL inlet low temperatureTH outlet high temperatureTp spatial-averaged transient temperature of Vp

u dimensional velocity fieldV control volume

Vp volume inside V where the spatial-averaged in-ternal energy is time dependent

x spatial vectorY dimensionless coefficient affecting the second-

order term in the transient temperaturez dimensionless coefficient measuring the station-

ary jump of temperature imposed by the wallZ dimensionless coefficient comparing pulsation

effect on the heating efficiency with the pulsa-tion effect on the wall temperature jump.

� dimensionless coefficient that measures the in-fluence of Re in Nu

�� dimensionless coefficient measuring the influ-ence of the velocity oscillation in the second-order term of h

dimensionless coefficient that measures the in-fluence of Pr in Nu

� dimensionless coefficient that measures the in-fluence of T in h

� characteristic thickness of the thermal boundarylayer

� positive dimensionless number used for themathematical treatment

dimensionless coefficient that measures the in-fluence of T in c

� dimensionless measurement of the heatingefficiency

� dimensionless coefficient related to the tem-perature field

�0 dimensionless measure of the temperature jumpnear the wall at stationary conditions

�p dimensionless measure of the temperature jumpnear the wall at oscillating conditions

� dynamic viscosity dimensionless pulsation

094907-12 Efrén Moreno Benavides J. Appl. Phys. 105, 094907 �2009�

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� dimensionless coefficient that measures the in-fluence of T in k; also is the dimensionless spa-tial coordinate

� density� dimensionless coefficient that measures the in-

fluence of T in �� dimensionless time� dimensionless velocity field� phase of a complex number �the number is in-

dicated by a subindex�� ratio of the stationary efficiency to the station-

ary dimensionless jump of the temperature nearthe wall

� dimensionless coefficient that measures the in-fluence of T in �

� dimensional angular pulsation

Subindexes+, � indicate complex conjugated numbers

i , j indicate the spatial coordinate0 indicates stationary conditions; also indicates

temporal coordinate

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