feasibility study of electromagnetic driven dream pipe

International Journal of Heat and Mass Transfer 83 (2015) 212–221

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International Journal of Heat and Mass Transfer

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

Feasibility study of electromagnetic driven dream pipe

http://dx.doi.org/10.1016/j.ijheatmasstransfer.2014.11.0720017-9310/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author. Tel.: +81 3 3342 1211; fax: +81 3 3342 5304.E-mail address: [email protected] (M. Furukawa).

Masao Furukawa ⇑, Mimpei Morishita, Shuichi YokoyamaDepartment of Electrical Systems Engineering, Kogakuin University, 1-24-2, Nishi-Shinjuku, Shinjuku-ku, Tokyo 163-8677, Japan

a r t i c l e i n f o

Article history:Received 22 April 2013Received in revised form 24 November 2014Accepted 24 November 2014

Keywords:Dream pipeForced oscillationElectromagnetic driveEnhanced heat transfer

a b s t r a c t

Dream pipes, a kind of forced oscillatory heat pipes, necessarily require some driving mechanisms foroscillations of enclosed working fluids. Commonly fitted up are mechanical shakers but not suited forpractical use because of becoming quite large in volume. Proposed in this study is an innovative typeof dream pipe with an electromagnetically actuated oscillating disk. The driving principle basically fol-lows Lorentz force generated upon electric wires set on the disk, in the radial direction of which a peri-odically varying magnetic field is formed by applying the three-phase alternating current. Feasibilities ofthis new device are theoretically examined by analyses from both thermal and electrical points of view.Heat transfer analysis is first made to determine the required driving force, from which the tidal displace-ment of the fluids is derived to show a resulted possible oscillation amplitude. Joule heat minimizationanalysis is then made to specify a suitable couple of the applied direct and alternating current voltages.Such specified voltages may go down to a lowest level by selecting the driving frequency to become anintrinsic one. The specific power, defined as the power to heat ratio, is introduced as a performance indexof that device. Numerical results show that less specific power than 0.10 is possible in most of supposeddesign cases and that the required magnetic flux density is far smaller than 0.5 T. It is thus concluded thatthe electromagnetic driven dream pipe is realizable. A 400 W m class dream pipe of electromagneticdrive is then design-specified as a demonstrative example.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Availability of oscillatory pipe flows was first mentioned byChatwin [1] and then by Watson [2]. They mathematically demon-strated that reciprocating flows make a remarkable contribution tolongitudinal mass dispersion. Kurzweg recognized that heat diffu-sion might also be enhanced by induced oscillatory flows sincetheir exists an analogy between mass transfer and heat transfer.Kurzweg [3–6] and his coworkers [3,4] thereby made a series ofexperimental/ theoretical studies on enhanced heat conductionby sinusoidal oscillations. A novel type of heat transfer device,named dream pipe, was thus invented by Kurzweg [7]. Thisattracted much attention of researchers. Kaviany [8] and Kavianyand Reckker [9] investigated possibilities of dream-pipe- basedheat exchangers. Zhang and Kurzweg [10,11] made numericalstudies to appropriate dream pipes to enhanced thermal pumping.Katsuta et al. [12] experimentally demonstrated the workability ofdream pipes with a model almost identical with Kurzweg’s one.Expecting much higher thermal conductivities, Nishio et al. [13]proposed a phase-shifted dream pipe. Rocha and Bejan [14]

composed a model applicable to geometric optimization of paralleltubes forming a dream pipe. It therefore seems that dream pipe hasalready arrived at a technology readiness state.

It is however noted that most of studies mentioned above[3–14] were done in late 1980s to early 2000s and that no dreampipes have been put to practical use in the past. The reasons whyno remarkable progress has recently been made in the dream pipetechnology are:

(1a) Mathematical expressions of Watson’s formulas [2] andthose transformed by Kurzweg [4–6] and then recomposed byFurukawa [15] are too sophisticated to actually calculate.(1b) Computational modeling by Ozawa and Kawamoto [16]and analytical modeling by Takahashi [17] are also unsuitablefor design calculations.(2) Mechanical shakers for liquid oscillations usually consume aconsiderable amount of electrical power and become so bulkyto set in a limited space, but no means taking the place of themhave not been presented so far.(3) Self-excited oscillatory heat pipes invented by Akachi[18,19] in early 1990s, now simply called OHP/PHPs (oscillat-ing/pulsating heat pipes), won favor of researchers [20,21],

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Nomenclature

A one-sided or cross-sectional area, m2

a tube inside radius, m; or coefficient paired with b in Eq.(9), s

B magnetic flux density, Tb coefficient paired with a in Eq. (9), sc multiplier noted as ct in Eq. (9) or as cm in Eq. (10),

dimensionlessD tube inside diameter, mE required electric power, WF driving force, NFM magneto-motive force, Af driving frequency, HzH intensity of magnetic field, A/mI electric current, Ak thermal conductivity, W/m KL tube length, m‘ oscillating disk size, mm exponent specifying test function in Eq. (9), dimension-

lessN number of tubes, wires, or windings, dimensionlessP pressure gradient, Pa/mp pressure, PaQ heat load, WQJ Joule heat, WR relative increase of thermal diffusivity, dimensionlessRE electric resistance, Xr radial distance, m; or normalized resistance, dimension-

lessT temperature, Kt time, sV voltage, Vw velocity, m/sz axial distance, m

Greek letters

C temperature gradient, K/mDT temperature difference, KDV tidal volume, m3

Dz tidal displacement, mg specific power, dimensionlessh angular position, rad; or temperature field, m

j thermal diffusivity, m2/sl0 magnetic permeability of air, 4p � 10�7 H/mm kinematic viscosity, m2/sq mass density, kg/m3

r electric conductivity, 1/Xms oscillation period, s/ cross-sectional area ratio, dimensionlessx angular frequency, rad/s

SubscriptsA coil A, appended to FM

AC alternating current, appended to VB coil B, appended to FM; or tube bundle, appended to AC coil C, appended to FM

CR cold reservoir, appended to TD oscillating disk, appended to A or /DC direct current, appended to Ve effective, appended to k or jHR hot reservoir, appended to T‘ leading wire, appended to A; I;N;RE; r, or rm coil, appended to I;N;RE, or rmin minimum, appended to E; F;QJ ;V , or gn viscosity-based, appended to xopt optimal, appended to ft diffusivity-based, appended to x3/ three-phase, appended to B or I1 mainstream, appended to T? tube cross-sectional, appended to A

Superscripts(–) characteristic, appended to w; peak value, appended to

B; F; FM ;H, or I; direct current, appended to V; or stan-dard, appended to RE or Dz

ð�Þ alternating current, appended to Vð^Þ possible, appended to w or hðÞ� reference, appended to RE or Vð�Þ� determined by Galerkin method, appended to a or b

AbbreviationsAC alternating currentDC direct current

M. Furukawa et al. / International Journal of Heat and Mass Transfer 83 (2015) 212–221 213

and thereby their technical interest turned to OHP/PHPs ratherthan dream pipes.

Nevertheless, if some solutions should be found out, dreampipes would be serviceable for industrial use. As for Point 1, designformulas, readily calculable and highly accurate, have recentlybeen presented by Furukawa [22]; who solved momentum-energyequations of the same form as Watson’s [2] by using Galerkinmethod [23,24], a kind of variational technique. This greatly facil-itates design calculations. As for Point 2, Furukawa [22] alsonumerically demonstrated the effectiveness of piezoelectric drivesemployed as a non-mechanical driving way with an expectation ofless power and small volume. Regarding Point 3, OHP/PHPs cansurely serve electronics cooling as most effective heat sink devicesbut are generally not fit for long-distance heat transport frequentlyencountered in various scenes. Dream pipes thus still meet ourfinal object.

Jaegar and Kurzweg [3] and Kurzweg [4] mainly investigatedhigh-frequency oscillations but, according to Furukawa [22], of sig-nificance are rather low- than high-frequency ones. A dream pipe

designed by Kaviany and Reckker [9] ran for 0.5 Hz to 10 Hz andanother one by Katsuta et al. [12] for 1, 4, 8, and 10 Hz. Hishidaet al. [25] set a new device similar to dream pipe in motion for0.5 Hz or 1.0 Hz. Operations under much lower frequencies,0.025 Hz to 1.0 Hz, were then practiced by Hassami and Zulkifli[26]. A technical issue is now if piezoelectric drives may cause suchlow-frequency oscillations as well as mechanical ones [9,12,25,26].Since early 2000s, many attempts have been made to apply piezo-electric actuators to pumps [27–29], fans [30,31], manipulators[32], agitators [33], and so on. This is along a recent trend of min-iaturization of machines but all are of high frequency-oscillations.As mentioned by Park et al. [27], there exist no commercially avail-able piezoelectric cells for low-frequency oscillations. In addition,we need those causing oscillations of larger amplitude. This isbecause we aim at developing a heat-pump-less heat recovery sys-tem, in which vapor compressors [34] would be replaced withdream pipes. We then noticed that Lorentz force may generatelow-frequency high-amplitude oscillations without technical diffi-culty and recognized that electromagnetic drives are popularlyused as linear motors for cryocoolers [35]. A notion of electromag-

214 M. Furukawa et al. / International Journal of Heat and Mass Transfer 83 (2015) 212–221

netic driven dream pipes has thus emerged among us to apply apatent [36]. As claimed there [36] and demonstrated in our lateststudy [37], actuators well adapted for dream pipes are linearmotors of Halbach type due to expected high efficiencies. Never-theless, another type of linear motor is considered in this studybecause the feasibility of electromagnetic driven dream pipe doesnot essentially depend on the type of actuators. That type of thedream pipe will then be easily fabricated, thereby suited for exper-imental investigations prior to development for industrial use.

2. Operations principle and system description

Unlike commonly used heat pipes, the dream pipe is not a sin-gle tube but a capillary tube bundle with reservoirs at both ends. Inaddition to the difference of construction, the two are dissimilar inoperational mechanism. Heat pipes passively go with capillaryforces generated by grooves or wicks, which are not required inoperating dream pipes. By contrast, dream pipes actively workwith oscillatory pipe flows caused by periodically imposed exter-nal forces. In any of traditionally configured dream pipes, the driv-ing force is generated by a mechanical shaker laid on the base ofthe cold reservoir. Fig. 1 conceptually schematically shows a newtype of dream pipes, in which the reciprocating mechanical shakeris replaced with an electromagnetic actuator. That actuator con-sists of two right circular cylinders, named Z and X, intersectingat right angles. As cross-sectionally drawn in four ways on theright-hand part of Fig. 1, the X-cylinder structurally includes theright half of the Z-cylinder serving as the cold reservoir. Visiblefrom the outside is thereby its left half. To clearly describe the sys-tem, three-dimensional Cartesian coordinates are introduced. Thez- axis runs along the tube bundle in a direction from the hot res-ervoir to the cold one while the x-axis perpendicularly penetratesthis page from the back to the face. The y-axis thereby goes in adownward direction on this page. The x- and y-axes are then

HeatHot in

Hot out

Capillary Tube Bundle

z

y

θ

HR

F

B

I

HR: Hot Reservoir, CR: Cold Reservoir, Z: Z-cylinder, X:

A: coil A, B: coil B, C: coil C, EW: Electric Wires, MD: M

xyz: coordinates, (xy): xy-section, (yz): yz-section

Thermal insulations and AC/DC power supply are neces

x yz

Fig. 1. Electromagnetic

placed on the disk-like bottom of the cold reservoir. The origin ofthe xyz-coordinates is naturally the center of that disk.

Both reservoirs are cylindrically shaped and are coaxially con-nected with the circular tube bundle. As depicted in the figure,the bottom of each reservoir is made movable with metallic bel-lows, permitting volumetric expansion and contraction resultedfrom induced liquid oscillations. The cold reservoir is concentri-cally enclosed by half with the Z-cylinder and longitudinallyextends as far as the vertically-cut radial cross-section of theX-cylinder. On that square-formed section, electric wires for directcurrent (DC) are laid along the x-axis. On the external surface of theX-cylinder, three coils named A, B, and C are spaced at equal angu-lar intervals of 120�(=2p=3). Composing a rectangularly shaped cir-cuit on a plane including the x-axis, each coil equally turns inwindings. Coils A, B, and C then differently look in the directionof 0, 120, and, 240�(=0;2p=3;4p=3), where the angle h is takenclockwise from the ð�yÞ-axis on the yz-plane. When the coils arecharged with three-phase alternating currents (3/ACs) of 0,2p=3, and 4p=3 in phase difference, a rotating alternating magneticfield is generated in the yz-plane. This rotating field, however, uni-formly points to the y-direction in the xy-plane. Consequently,when the wires mentioned above are simultaneously charged witha constant DC current, there occurs Lorentz force in the ð�zÞ-direc-tion. That force also sinusoidally varies with time by the samefrequency as the applied 3/ACs. Liquid oscillations are thus elec-tromagnetically induced in the capillary tubes.

3. Theoretical considerations

3.1. Required driving force

Since any of N tubes, circularly forming a tube bundle, is equallysized as D in diameter and L in length, we may consider one tubealone in describing momentum-energy equations. They arethen expressed in reduced cylindrical coordinates, z and r, which

Cold out

Cold in

CR

A

B C

B(C)

A(A)

C(B)

ZX(yz) ZX(xy)

Z(yz) Z(xy)

MD

EW

X-cylinder <Components>

ovable Disk <Subcomponents>

sary but not depicted here.

driven dream pipe.


respectively show the axial and radial distances of a specified pointof the tube. When the movable disk is oscillated with the angular fre-quency x, the liquid pressure p also synchronously changes with itin a way as dp=dz ¼ �P cosxt where P is the induced pressure gra-dient and t the time. The momentum equation for the liquid velocityw then becomes:

@w@t¼ P

qcos xt þ m

@2w@r2 þ

1r@w@r

!ð1Þ

where q is the liquid density and m the liquid kinematic viscosity.Furukawa [22] solved Eq. (1) by using Galerkin method to give apossible velocity field:

w ¼ �w½ðxn=xÞ cos xt þ sin xt�ð1� r2=a2Þ ð2Þ

where a is the inside tube radius expressed as a ¼ D=2. The viscos-ity-based natural angular frequency xn and the characteristic oscil-lation velocity �w are defined in:

xn ¼ 6m=a2 ð3Þ

�w ¼ 32

Px=qðx2 þx2nÞ ð4Þ

An extent of liquid oscillations occurred for a half periods=2ð¼ p=xÞ is generally expressed in terms of the tidal displace-ment Dz, which is mathematically derived from the definition thatpa2Dz ¼

R s=20

R a0 w � 2prdrdt. This definite integral shows the time-

averaged radially-leveled tidal volume DV and yields:

�w ¼ xDz ð5Þ

Equating Eq. (5) to Eq. (4) results in:

Dz ¼ 32

P=qðx2 þx2nÞ ð6Þ

The energy equation for the liquid temperature T, coupled withEq. (1), is:

@T@tþ w

@T@z¼ j

@2T@r2 þ

1r@T@r

!ð7Þ

where j is the liquid thermal diffusivity. Furukawa [22] also solvedEq. (7) by using Galerkin method to give a possible temperaturefield:

T ¼ T1 þC½�zþ hðr; tÞ� ð8Þ

where T1 is the mainstream temperature and C the axial tempera-ture gradient resulted from the temperature difference DT betweenthe hot and cold reservoirs. That gradient is thereby expressed asDT=L. The third term hðr; tÞ in Eq. (8) represents a radially inducedtemperature change expressed in not K but m and takes a vari-ables-separated form:

h ¼ ct �wð~a� cos xt þ ~b� sinxtÞðrm=am � 1Þ ð9Þ

where m is the exponent to be optimally specified and defines thefirst coefficient ct so as to be ðmþ 1Þðmþ 6Þ=2mðmþ 4Þ. The secondand third coefficients of Eq. (9) are not dimensionless but expressedin s. Defining expressions of the two are such that~a� ¼ ðx2 �xnxtÞ=xðx2 þx2

t Þ and ~b� ¼ �ðxn þxtÞ=ðx2 þx2t Þ,

where xt ¼ jðmþ 1Þðmþ 2Þ=ma2. Watson [2] mathematicallyshowed that the multiplicative interaction of the velocity and tem-perature fields increases the axial heat conduction. That interactioncan now be wh, having the same dimension as the thermal diffusiv-ity, m2/s (=(m/s) m). The heat diffusion thereby takes place at therate as jþ wh. Since this rate varies with the time t and the radialdistance r, Watson [2] introduced the effective thermal diffusivitydefined as je ¼

R s0

R a0 ðjþ whÞ � 2prdrdt=spa2 for convenience.

Owing to Eqs. (2) and (9), that integral becomes calculable andresults in:

R � je=j� 1 ¼ ðcm=8ÞðDz=aÞ2 ð10Þ

where R is the relative increase of heat diffusion. The coefficient cm

depends on m in the same way as ct but is defined so as to beðmþ 1Þ2ðmþ 6Þ2=mðmþ 4Þ2, which amounts to a minimum whenm ¼ 1:31253. Eq. (10) then becomes:

R ¼ 0:965ðDz=aÞ2 ð11Þ

For sustainable operations, the dream pipe constantly calls forsome energy corresponding to the work done by the oscillatingliquid. The work rate is expressed as wð�dp=dzÞDz per ring-shapeddifferential cross-section of the tube. That energy is thereby esti-mated as high as ðE=NÞs ¼

R s0

R a0 wð�dp=dzÞDz � 2prdrdt where

E=N shows the required electric power per tube. Substituting Eq.(2) for w and Eq. (5) for �w and again using a relation thatdp=dz ¼ �P cosxt, one has:

E=N ¼ PxnðDzÞ2ðpa2=4Þ ð12Þ

Following Fourier’s law of heat conduction, one also has:

Q=N ¼ keðpa2=LÞDT ð13Þ

where ke is the effective thermal conductivity. Eq. (12) divided byEq. (13) makes the specific power, which is expressed asE=Q ¼ PxnðDzÞ2L=4keDT and then approximated asE=Q � PxnðDzÞ2L=4kRDT because R � je=j � ke=k. As seen fromEq. (6), P and Dz are linearly interrelated. To express E=Q in Dzalone, a reversed form of Eq. (6), P ¼ ð2=3Þqðx2 þx2

nÞDz, is used.The required electric power thus becomes:

E ¼ qxnðx2 þx2nÞðDzÞ3QL=6kRDT ð14Þ

In accordance with the energy conservation law, the energy sup-plied for a half period turns into the tidal work in the tube. This rela-tion is simply written as:

Eðs=2Þ ¼ pDV ð15Þ

When the force F is applied to the disk sized as AD in surface area,such a pressure as p ¼ F=AD arises there. The relation between DVand Dz is expressed as DV ¼ ABDz where AB is the tube-bundlecross-sectional area sized in a way that /D ¼ AD=AB > 1. It followsthat pDV ¼ FDz=/D and it is also noted that s ¼ 2p=x. One conse-quently finds:

E ¼ xFDz=p/D ð16Þ

Equating Eq. (16) to Eq. (14) results in:

F ¼ p/Dqðxn=xÞðx2 þx2nÞðDzÞ2QL=6kRDT ð17Þ

Substituting Eq. (11) for R in Eq. (17), one obtains:

F ¼ 0:173/Dqðxn=xÞðx2 þx2nÞA?QL=kDT ð18Þ

where A? is the tube cross-sectional area expressed as A? ¼ pa2.Since P ¼ p=L; p ¼ F=AD;AD ¼ /DAB, and AB ¼ NA?, one has a relationas P ¼ F=/DNA?L. Due to Eq. (18), this relation becomes:

P ¼ 0:173qðxn=xÞðx2 þx2nÞQ=NkDT ð19Þ

Substitution of Eq. (19) into Eq. (6) makes:

Dz ¼ 0:2595ðxn=xÞQ=NkDT ð20Þ

3.2. Joule heat minimization

We now investigate an ordinary case where any of coils A, B,and C is Nm in number of turns and is equally charged with Im inpeak electric current. Magneto-motive forces FMA; FMB, and FMC


are then generated in the yz-plane of the X-cylinder. Thosethree are, respectively, NmIm cos h cos xt;NmIm cosðh� 2p=3Þcosðxt � 2p=3Þ, and NmIm cosðh� 4p=3Þ cosðxt � 4p=3Þ and addi-tively yield:

FM ¼ FMA þ FMB þ FMC ¼ FM cosðh�xtÞ ð21Þ

where FM is the peak magneto-motive force defined in:

FM ¼32

NmIm ð22Þ

In our case specified as h ¼ 0, Eq. (21) becomes such thatFM ¼ FM cos xt. This force perpendicularly forms a uniform mag-netic field alternating as H cosxt in the X-cylinder. There is how-ever no need of considering cos xt in determining the peakintensity H. Applying Ampere’s contour integration law to asquare-shaped closed path of ‘ in side length, drawn on the yz-cross-section of X-cylinder, yields a relation thatðH þ 0þ 0þ 0Þ‘ ¼ ð3NmIm=p‘2Þ‘2 under no leak of magnetic flux,where 3NmIm=p‘2 is the electric current density on the X-cylindersurface and ‘2 is the square-path- formed area. That relation conse-quently becomes so that H‘ ¼ 3NmIm=p ¼ ð2=pÞFm. Hence,

H ¼ ð2=pÞFM=‘ ð23Þ

where ‘ is the size of the X-cylinder and also of the Z-one. From twodifferently made definitions that AD ¼ /DAB ¼ /DNA? ¼ /DNpðD=2Þ2

and AD ¼ pð‘=2Þ2, one has:

‘ ¼ffiffiffiffiffiffiffiffiffiffi/DN

pD ð24Þ

Due to Eq. (23), the peak magnetic flux density becomes:

B ¼ l0H ¼ l0ð2=pÞFM=‘ ð25Þ

where l0 ¼ 4p � 10�7 H/m, representing the magnetic permeabilityof air. Substituting Eq. (22) for FM in Eq. (25), one finds:

B‘ ¼ ð3=pÞl0NmIm ¼ 12 � 10�7NmIm ð26Þ

When the wires, set up on the moving disk in a way as ‘ in lengthand N‘ in number, are evenly charged with I‘ in intensity, the elec-tric current transmitted there amounts to:

I ¼ N‘I‘ ð27Þ

Complying with B and I in strength, Lorentz force F arises fromthere to multiplicatively become:

F ¼ IB‘ ð28Þ

Substitution of Eqs. (26) and (27) into Eq. (28) results in:

F ¼ 12 � 10�7N‘NmI‘Im ð29Þ

Since F should be equal to F in strength, F is hereafter simply writ-ten as F.

When DC and AC voltages, V and eV , are respectively applied tothe wires and coils, the electric currents, I‘ and Im, follow Ohm’slaw to become:

I‘ ¼ V=RE‘ ð30aÞIm ¼ eV=REm ð30bÞ

where RE‘ and REm are the electric resistances. Substituting Eqs.(30a) and (30b) for I‘ and Im in a mutually transposed expressionof Eq. (29) makes:

V eV ¼ ð107=12ÞFRE‘REm=N‘Nm � V�2 ð31Þ

A reciprocal relation thus holds between V and eV and serves asdefining expression of the reference voltage V�. The resistances,RE‘ and REm, are now expressed in:

RE‘ ¼ r‘R�E ð32aÞ

REm ¼ NmrmR�E ð32bÞ

where r‘ and rm are the normalized dimensionless resistances andR�E is the reference resistance. Eq. (31) then turns into:

V�2 ¼ ð107=12ÞFr‘rmR�2E =N‘ ð33Þ

Since Joule heat generated per wire is V2=RE‘ and that per coil iseV 2=2REm, all the Joule heat QJ generated in the X- and Z-cylindersamounts to:

QJ ¼ N‘ðV2=RE‘Þ þ 3ðeV 2=2REmÞ ð34Þ

Expressing Eq. (34) in terms of V alone by using Eq. (31) and substi-tuting Eqs. (32a) and (32b) for RE‘ and REm, one obtains:

QJ ¼ ð2N‘V2=r‘ þ 3V�4=NmrmV2Þ=2R�E ð35Þ

Eq. (35) has a minimum, which occurs under the condition that:

2N‘V2=r‘ ¼ 3V�4=NmrmV2 ð36Þ

The DC and AC voltages, V and eV , satisfying Eqs. (31) and (36) are:

VDC ¼ ð3r‘=2N‘NmrmÞ1=4V� ð37aÞVAC ¼ ð2N‘Nmrm=3r‘Þ1=4V� ð37bÞ

Substituting Eq. (37a) for V in Eq. (35) and replacing V�2 with Eq.(33), one finally has:

QJmin ¼ ð107=12Þð6r‘rm=N‘NmÞ1=2FR�E ð38Þ

Owing to the unavoidably generated Joule heat, the specific powerg, originally defined in E=Q , should be redefined as:

g ¼ ðEþ Q JminÞ=Q ð39Þ

Upon a definition that x ¼ 2pf , Eq. (16) becomes:

E ¼ 2fFDz=/D ð40Þ

Because usually mentioned is not x but the driving frequency f,more practical is Eq. (40) rather than Eq. (16).

As clearly seen from Eq. (18), F depends on ðxn=xÞðx2 þx2nÞ,

that is, xnðxþx2n=xÞ. Eq. (18) therefore becomes minimum when

x ¼ xn and reduces to:

Fmin ¼ 0:346/Dqx2nA?QL=kDT ð41Þ

The optimal driving frequency f opt is then:

f opt ¼ xn=2p ð42Þ

Under that driving frequency, Eq. (20) reduces to:

Dz ¼ 0:2595Q=NkDT ð43Þ

Eqs. (33), (37a), (37b), and (38)–(40) correspondingly change into:

V�2min ¼ ð107=12ÞFminr‘rmR�2E =N‘ ð44Þ

VDCmin ¼ ð3r‘=2N‘NmrmÞ1=4V�min ð45aÞVACmin ¼ ð2N‘Nmrm=3r‘Þ1=4V�min ð45bÞ

QJminmin ¼ ð107=12Þð6r‘rm=N‘NmÞ1=2FminR�E ð46Þgmin ¼ ðEmin þ Q JminminÞ=Q ð47ÞEmin ¼ 2f optFminDz=/D ð48Þ

The least specific power gmin defined in Eq. (47) serves as a good cri-terion in rating availability of the proposed way of electromagneticdrive. More realistic criteria are the resulted magnetic flux densityB3/ and also the charged alternating current intensity I3/ required

0

1.5

3

4.5

6

0 1 2 3 4

, N

f, Hz

Q =400W

#n:(TCR , ΔT )#1:(300K,20K), #2:(300K,30K), #3:(300K,40K), #4:(300K,50K)#5:(320K,20K), #6:(320K,30K), #7:(320K,40K), #8:(320K.50K)

#1

#5#2

#6/#3

#7/#4

#8

FFig. 2b. Required driving force versus driving frequency.

0.05

0.075

0.1

Δz, m

f =2.0Hz#1

#5

#2

#6/#3


for B3/ at minimum. Eq. (26) connecting B with Im also holds for B3/

and I3/ and thereby becomes:

B3/ ¼ 12 � 10�7NmI3/=‘ ð49Þ

Replacing Im with I3/ and eV with VACmin in Eq. (30b), one readily has:

I3/ ¼ VACmin=REm ð50Þ

Since the X- and Z-cylinders are equally sized as ‘, the contour ofeach one is roughly estimated at 4‘. The reference resistance R�E con-sequently grows such that R�E ¼ 4RE. This modified reference resis-tance is herein defined as RE ¼ ‘=r‘A‘ for a wire of ‘ in length, r‘

in electric conductivity, and A‘ in cross-sectional area. Consideringa 1.0 mm-diameter copper wire specified as r‘ ¼ 5:8 � 107=Xm,one gets a practical relation that RE ¼ 0:022‘.

4. Numerical results and discussion

Eqs. (18), (20), (37a), (37b), (39), (42), (45a), (45b), (47), (49),and (50) are consecutively computed to make an estimate ofF;Dz;VDC ;VAC ;g, f opt ;VDCmin;VACmin;gmin;B3/, and I3/. Main variablesare then Q ; f , or T. Unless otherwise specified, such design param-eter values as D ¼ 2:5 mm, L ¼ 1:0 m, N ¼ 43, /D ¼ 3:0, N‘ ¼ 4,Nm ¼ 400, r‘ ¼ 1:0, and rm ¼ 1:0 are used in computations. Thenumerical results are graphically shown in Figs. 2a, 2b, 3a, 3b, 4a,4b, 5, 6, 7a, 7b, 8, 9a, and 9b corresponding to the equations men-tioned above. In all the figures but Figs. 2b, 3b, and 6, Q is taken asthe abscissa and ranges from 100 W to 900 W. The working fluid iswater in all the cases of computations because we take much inter-est in heat recovery through a process of high-temperature wastewater disposition. Eight curves drawn in the figures other thanFig. 6 respectively correspond to cases 1–8 identified by TCR andDT. It is again noted that TCR is the cold reservoir temperatureand DT the temperature difference between hot- and cold-reser-voirs. Considered are such as TCR ¼ 300 K or 320 K and DT ¼ 20 K,30 K, 40 K, or 50 K.

Figs. 2a and 2b show the required driving force F while Figs. 3aand 3b the resulted tidal displacement Dz. Taken as the abscissaare the heat load Q or the driving frequency f. Graphs are formed

0

1.5

3

4.5

6

100 300 500 700 900

, N

Q, W

f =2.0Hz

#n:(TCR , ΔT )#1:(300K,20K), #2:(300K,30K), #3:(300K,40K), #4:(300K,50K)#5:(320K,20K), #6:(320K,30K), #7:(320K,40K), #8:(320K,50K)

#1

#5#2

#6/#3

#7/#4

#8

F

Fig. 2a. Required driving force versus heat load.

0

0.025

100 300 500 700 900Q, W


#4/#7#8

Fig. 3a. Resulted tidal displacement versus heat load.

against Q in Figs. 2a and 3a while against f in Figs. 2b and 3b. Asseen in Figs. 2a and 3a, F and Dz increase together as Q gains. Thisupward trend is natural. Then, the smaller DT becomes, the larger Fgrows. Possible F values extend over a range from 0.15 N to 6.3 N,corresponding to which possible Dz values are in a range from0.002 m to 0.092 m. A remarkable feature of Fig. 2b is that any ofcurves 1–8 has a minimum within the limits of 0 Hz to 4 Hz. Thisgraphically indicates that there exists an optimal driving frequencyminimizing F. Also seen in Fig. 2b is that Dz decreases as f increases.This downward trend suggests that operationally preferred arelower frequencies. A characteristic common to those four figuresis that curves 2–7 are found between curves 1 and 8. This instructs

0

0.025

0.05

0.075

0.1

0 1 2 3 4

Δz, m

f, Hz

Q =400W


#1#5/#2#6/#3#4/#7#8

Fig. 3b. Resulted tidal displacement versus driving frequency.

0

0.15

0.3

0.45

0.6

100 300 500 700 900

V DC

, V

Q,W

#4

#3

#2#1

#8#7

#6#5

#n: (TCR , ΔT , f )#1:(300K, 20K, 0.5Hz), #2:(300K, 20K, 1.0Hz), #3:(300K, 20K, 1.5Hz)#4:(300K, 20K, 2.5Hz), #5:(320K, 50K, 0.5Hz), #6:(320K, 50K, 1.0Hz)#7:(320K, 50K, 1.5Hz), #8:(320K, 50K, 2.0Hz)

Fig. 4a. Specified direct current voltage versus heat load.

0

5

10

15

20

100 300 500 700 900

V AC

, V

Q, W

#4

#3

#2

#1#8#7

#6

#5

#n:(TCR , ΔT , f )#1:(300K, 20K, 0.5Hz), #2(300K, 20K, 1.0Hz), #3:(300K, 20K, 1.5Hz)#4:(300K, 20K, 2.5Hz), #5(320K, 50K, 0.5Hz), #6:(320K, 50K, 1.0Hz)#7:(320K, 50K, 1.5Hz), #8(320K, 50K, 2.0Hz)

Fig. 4b. Specified alternating current voltage versus heat load.

0

0.25

0.5

0.75

1

100 300 500 700 900

η , -

--

Q, W

#3

#2

#1#8

#7

#6#5

#4: out of scale

#n:(TCR , ΔT , f )#1:(300K, 20K, 0.5Hz), #2:(300K, 20K, 1.0Hz), #3:(300K, 20K, 1.5Hz)#4:(300K, 20K, 2.0Hz), #5:(320K, 50K, 0.5Hz), #6:(320K, 50K, 2.0Hz)#7:(320K, 50K, 1.5Hz), #8:(320K, 50K, 2.0Hz)

Fig. 5. Specific power versus heat load.


us that case 1 (TCR ¼ 300 K and DT ¼ 20 K) and case 8 (TCR ¼ 320 Kand DT ¼ 50 K) are of significance for discussion.

Figs. 4a and 4b respectively display the DC and AC voltages, VDC

and VAC , specified so as to minimize Joule heat. Fig. 5 then showsthe specific power g when Joule heat is minimized. Curves 1–4and 5–8 in Figs. 4a, 4b, and 5, taking Q as a variable, respectivelycorrespond to cases 1 and 8. The driving frequency also serves asa curve identifier in a way as f ¼ 0:5 Hz in curves 1 or 5, 1.0 Hzin curves 2 or 6, 2.0 Hz in curves 3 or 7, and 3.0 Hz in curves 4 or8. Specified values of VDC are in a lower range from 0.042 V to0.608 V while those of VAC extend over a higher range from 1.4 Vto 19.9 V. As expected, VDC and VAC increase as Q goes up but g is

almost fixed. Expected g values read in Fig. 8 are 1.32 (scale-out),0.90, 0.51, 0.37, 0.28, 0.19, 0.10, and 0.06 in from-top-to-bottomorder. It is noticeable that curves 1–8 keep their ordinate distancesin order of 4, 3, 2, 1, 8, 7, 6, and 5 in number. This order is the samein all the three figures of concern. The possible least values ofVDC ;VAC , and g are thereby respectively found from the threecurves numbered 5 in Figs. 4a, 4b, and 5, where curve 5 is identi-fied as TCR ¼ 320 K, DT ¼ 50 K, and f ¼ 0:5 Hz.

Fig. 6 shows the optimal driving frequency f opt which causesfurther minimization of the once minimized Joule heat. As seenfrom Eqs. (3) and (42), that frequency depends on the operatingtemperature T and the tube inside diameter D. Graphs of f opt are

0

0.5

1

1.5

2

290 310 330 350 370

f opt,H

z

T, K

T =(TCR +THR )/2#1:D =1.0mm, #2:D =1.5mm, #3:D =2.0mm, #4:D =2.5mm#5:D =3.0mm, #6:D =3.5mm, #7:D =4.0mm

#1

#2

#3#4#5/#6#7

Fig. 6. Optimal driving frequency versus operating temperature.

0.05

0.1

0.15

0.2

0.25

100 300 500 700 900

V DC

min

, V

Q, W

#1

#2

#5

#3

#6#4

#7

#8

#n:(TCR , ΔT )#1:(300K, 20K), #2:(300K, 30K), #3:(300K, 40K), #4:(300K, 50K)#5:(320K, 20K), #6:(320K, 30K), #7:(320K, 40K), #8:(320K, 50K)

Fig. 7a. Optimally specified direct current voltage versus heat load.

1

3

5

7

9

100 300 500 700 900

V AC

min

, V

Q, W

#1

#2#5

#3#6#4

#7

#8


Fig. 7b. Optimally specified alternating current voltage versus heat load.

0

0.1

0.2

0.3

0.4

100 300 500 700 900

η m

in, -

--

Q, W

#1

#2#5

#3#6#4#7#8


Fig. 8. Minimized specific power versus heat load.


therefore made against T over a range from 290 K to 370 K, result-ing in seven curves identified as D ¼ 1:0 mm, 1.5 mm, 2.0 mm,2.5 mm, 3.0 mm, 3.5 mm, and 4.0 mm. As clearly seen in Fig. 6,f opt goes down as T increases and D grows. Curves 1–7 thereby takeeach position in due order in the figure. Also seen in Fig. 6 is thatf opt becomes the highest when T ¼ 290 K and D ¼ 1:0 mm whilethe lowest when T ¼ 370 K and D ¼ 4:0 mm. Obtained f opt valuesthus extend over a range from 4.2 Hz (scale-out in curve 1) to0.072 Hz (read in curve 7). As previously stated, cases 1–8 aredefined in TCR and DT. Because T ¼ ðTCR þ THRÞ=2 ¼ TCR þ DT=2,one finds that T ¼ 310 K, 315 K, 320 K, 325 K, 330 K, 335 K, 340 K,

and 345 K corresponding to cases 1–8. Attention is now paid tocurve 4 identified as D ¼ 2:5 mm. That curve of interest indicatesthat f opt ¼ 0:422 Hz for case 1, 0.384 Hz for case 2, 0.351 Hz forcase 3, 0.323 Hz for case 4, 0.299 Hz for case 5, 0.278 Hz for case6, 0.260 Hz for case 7, and 0.244 Hz for case 8.

The following five figures show demonstrative examples ofcomputations done over a range from 100 W to 900 W in Q withthe above-mentioned f opt values. As regards Figs. 7a, 7b, 8, 9a,and 9b, it is noted that curves 1–8, corresponding to cases 1–8,show themselves in order of 1, 2, 5, 3, 6, 4, 7, and 8 in number.Figs. 7a and 7b respectively display the optimally specified DC

0

0.05

0.1

0.15

0.2

100 300 500 700 900

B 3Φ

, T

Q, W

#1

#2#5#3#6#4#7#8


Fig. 9a. Minimized magnetic flux density versus heat load.

1

3

5

7

9

100 300 500 700 900

I 3Φ

, A

Q, W

#1

#2#5

#3#6#4

#7

#8


Fig. 9b. Minimized alternating current intensity versus heat load.


and AC voltages, VDCmin and VACmin, finally minimizing Joule heat.Obtained VDCmin values are 0.038 V in case 8 to 0.106 V in case 1when Q ¼ 100 W and are 0.113 V in case 8 to 0.319 V in case 1when Q ¼ 900 W. Then, obtained VACmin values are 1.23 V in case8 to 3.47 V in case 1 when Q ¼ 100 W and are 3.68 V in case 8 to10.4 V in case 1 when Q ¼ 900 W. Comparing curves in Figs. 7aand 7b with those in Figs. 4a and 4b, one finds a remarkable factthat VDCmin and VACmin respectively reduce to roughly half of VDC

and VAC . Fig. 8 then indicates the minimized specific power gmin,representing the attainable least g value. Expected gmin values readin Fig. 8 are 0.36, 0.20, 0.12, 0.083, 0.175, 0.10, 0.065, and 0.045,

respectively, for cases 1–8. Comparisons between Figs. 8 and 5are also made to show that gmin reduces to about 1/4 of g. It is note-worthy that g < 0:10 in only two cases while gmin < 0:10 in fourcases. Driving the dream pipe in f opt is thus so effective.

Figs. 9a and 9b respectively give the smallest values of theresulted magnetic flux density B3/ and the required alternatingcurrent intensity I3/. It is seen in Fig. 9aa that B3/ extends over arange from 0.021 T when Q ¼ 100 W in case 8 to 0.176 T whenQ ¼ 900 W in case 1. It is then seen in Fig. 9b that I3/ extends overa range from 1.23 A when Q ¼ 100 W in case 8 to 10.4 A whenQ ¼ 900 W in case 1. All such values of B3/ and I3/ are acceptablebecause usually employed allowable upper limits are less than0.5 T and 10 A. This numerically proves that there is almost notechnical issue in realizing the electromagnetic driven dream pipe.A dream pipe depicted in Fig. 1 has finally been specified asD ¼ 2:5 mm, L ¼ 1:0 m, and N ¼ 43 under such driving conditionsthat Q ¼ 400 W, TCR ¼ 320 K, DT ¼ 50 K, f ¼ 2:0 Hz, VDC ¼ 0:15 V,and VAC ¼ 5:0 V. The expected performance is then becomes so asg <0.19.

5. Conclusion

The electromagnetic driven dream pipe, devised by the authorsand introduced here, is a new type of forced oscillatory heat pipes.Theoretical analyses, concerned with required driving force andJoule heat minimization, have first been made to present practicalexpressions serviceable for discussion on the feasibility. Computa-tions upon such expressions have extensively been done to safelysay that it is feasible. Proposed design specifications for a400 W m class new-style dream pipe are also technically accept-able. That novel heat transfer device therefore possibly may takeplace of existing mechanical driven dream pipes.

Conflict of interest

None declared.

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