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rtf - K.f.u.p.m. OCW

VIEWS: 7 PAGES: 9

									   Electrohydrodynamic Induction Pumping of Liquid
         Film in Vertical Annular Configuration

                    Salem Ahmed Aldini                                                Jamal Seyed-Yagoobi
          Department of Mechanical Engineering                         Mechanical, Material, and Aerospace Engineering
                 Texas A&M University                                                     Department
         College Station, Texas 77843-3123, USA                                Illinois Institute of Technology
                     salem@tamu.edu                                             Chicago, Illinois 60616, USA
                                                                                       yagoobi@iit.edu


Abstract— Electrohydrodynamic induction pumping of two-                Pr    Prandtl number= c p m k .
phase medium is attractive for terrestrial and outer space
                                                                       R Radius, m.
applications since it is non-mechanical, lightweight, and involves
no moving part. In addition to pure pumping purposes, EHD              r     Radial space coordinate, m.
induction pumps are also used for the enhancement of heat              Re Real component of a complex number.
transfer, as an increase in mass transport often translates to an      S     Slip coefficient = [o (-Kw)/].
augmentation of the heat transfer. Applications include two-phase      T Temperature, oC.
heat exchangers (evaporators and condensers), heat pipes and           u     Velocity in r direction, m/s.
capillary pumped loops. A theoretical model for the EHD
                                                                       w Velocity in z direction, m/s.
induction pumping of an annular liquid/vapor medium where the
charges are induced at the two-phase interface as well as within       z     Space coordinate in streamwise direction, m.
the bulk of the liquid phase in vertical configuration is presented.    Liquid film thickness, m.
The dimensionless numerical results are obtained and the flow           Electric permittivity, F/m.
physics are discussed in conjunction with the effect of the             Electric permittivity of vacuum= 8.854, F/m.
controlling parameters. The controlling parameters include:
                                                                        Wavelength, m.
liquid film thickness, voltage, wavelength, frequency, external
pressure, and gravity. The dimensionless numerical results              Dynamic viscosity, kg/ms.
obtained provide an essential tool for designing and optimizing         e Electric potential at the electrode, V.
                                                                         ˆ
two-phase liquid /vapor EHD induction pumps.
                                                                       r Density, kg/m3.
                                                                          Circumferential angle.
   Keywords-Electrohydrodynamics; interface; pumping; two-
phase flow;
                                                                          Electric conductivity, S/m.
                                                                          Shear stress, N/m2.
                             NOMNUCLUTURE                                  Angular frequency = 2 f , Hz.
   Cp Specific heat at constant pressure, J/kg.K.                             Absolute magnitude of a complex number.
   C Electric potential function, V.                                   Subscripts and Superscripts
     r
    E Electric field strength vector, V/m.
                                                                       b0     Bulk value at pump entrance.
   f    Frequency, Hz.
                                                                       e      Electric.
   g    Gravitational constant= 9.81, m/s2.
                                                                       i      Space coordinate in tensor notation.
   G Gravitational force density= r g , N/m3.
                                                                       j      Space coordinate in tensor notation.
   I0 Modified Bessel function of the first kind order zero.           l      Liquid.
   I1 Modified Bessel function of the first kind order one.            r      In r direction.
   j    Imaginary number = 1 .                                        zr     In z direction acting on a plane with normal r.
   k    Thermal conductivity, W/m.K.                                   T      Total.
   K Wave number = 2  , 1/m.                                         z      In z direction.
   L Pump length, m.                                                   v      Vapor.
   m* Dimensionless mass flux.                                         ^      Peak value.
                                                                       -      Time-averaged value.
   M e r b0 e0jˆ e 2m .
      *          2   2
                     b0
                                                                       '      Conjugate complex.
         m 0 e0 r b0 s b0 d2 .
          b                                                            *      Dimensionless value.
   P     Pressure, Pa.
                       I.    INTRODUCTION                              liquid film thicknesses are carried out to study the effect of the
    Electrohydrodynamic EHD induction pumping utilizes the             controlling parameters: vapor height, liquid film thickness,
charges induced in the liquid film due to the existence of an          applied voltage, wavelength, and frequency. The results of this
electric conductivity gradient in the bulk of the film and/or at       study are pertinent to the pumping or flow management during
the liquid/vapor interface. This gradient exists due to the            internal condensation or evaporation processes in the presence
temperature within the liquid film and/or discontinuity of             and absence of gravity. This study helps in explaining the
electric conductivity at the liquid/vapor interface. Upon the          physics of EHD induction pumping and serves as a necessary
application of an electric field in the form of a traveling wave,      tool for optimum design of two-phase EHD pump.
the induced charges will get attracted or repelled leading to
fluid motion. EHD induction pump is non-mechanical, and                                   II.   THEORETICAL MODEL
lightweight. It involves no moving parts, produces no vibration,            The model presented considers a two-dimensional
and the electric power input is generally negligible. These            developing annular flow of a thin liquid film in an
pumps have a great potential for more than just pure pumping           axisymmetric vertical tube as illustrated in Fig. 1. The liquid
applications. Applications include single-phase and two phase          film is separated from the vapor by a flat interface. Boundary
heat exchangers, heat pipes, or capillary pumped loops.                condition of electric traveling wave is imposed in the liquid
Enhancing heat transfer in phase change processes by                   film adjacent the wall of the tube while symmetric boundary
controlling the liquid film flow is one of the practical               condition for the electric field is imposed at the tube centerline
applications. In addition, mechanical pumping of liquid film, in
                                                                       (i.e. Erv  0 ). In addition to the charges, and therefore electric
both terrestrial and outer space applications, is usually, not
possible. However, EHD induction pumping provides a                    shear stress, at the liquid/vapor interface the model considers
practical technology.                                                  charges throughout the bulk of the liquid film. Due to an
                                                                       electric conductivity gradient in the liquid film, charge
          The first theoretical model for two-phase attraction         induction takes place. This gradient may exist as a result of a
induction pumping of a poorly conducting fluid was developed           temperature gradient, which is a consequence of viscous
in 1966 by Melcher [1]. He considered an analytical domain             dissipation (negligible), Joule heating, or cooling or heating of
constituted by an electrode generating the electric traveling          the boundaries. The following assumptions are made with
wave placed above the liquid film, and a conducting plate as a         regards to the electric field and liquid film flow field:
lower boundary. Only charges induced at the liquid/air
interface were considered. He later improved his work by                  1) the vapor has properties of the vacuum;
expanding it not only to cover both attraction and repulsion              2) the electric field is irrotational due to low electric
pumping, but also the pumping of a liquid/liquid interface [2].              current;
The improved model was later used by Crowley [3] to                       3) all fluid properties in the liquid film are temperature
incorporate the effect of temperature induced charges in the                 dependent;
bulk of the liquid for single phase pumping process. For the
purpose of latter studies, a fluid exhibiting a continuous                4) charge transport at the interface due to ion mobility and
temperature gradient was modeled as having two layers at                     surface conduction, and bulk conduction in z-direction
distinct constant temperature separated by a temperature jump.               are negligible; and
Wawzyniak and Seyed-Yagoobi [4] further developed an                      5) the flow is two-dimensional and in vertical
analytical model for EHD induction pumping of a stratified                   configuration;
liquid/vapor medium. They assumed charges to be present only
                                                                          6) the flow field may be modeled as flow between two
at the interface and analyzed four different electrode
                                                                             concentric cylinders, the outer one being stationary and
geometries. As an extension to their work Wawzyniak and
                                                                             the inner one moving at the constant speed of the
Seyed-Yagoobi [5] considered charges not only present at the
                                                                             liquid/vapor interface in the z-direction;
liquid/vapor interface but throughout the bulk of the liquid for
only one electrode configuration. Brand and Seyed-Yagoobi                 7) the flow is steady state, laminar, incompressible , but
[6] further extended the work done by [5] to analyze and                     not fully developed;
investigate the effect of four different electrode configurations.        8) the flow is rotationally symmetric, i.e., the variables do
    The current study is an extension of the theoretical model               not depend upon  ;
presented by [5] and [6] who studied the EHD induction                     9) the pressure in r-direction is uniform, the pressure
pumping of a stratified liquid/vapor medium in a horizontal                     gradient in z-direction is constant;
channel. This current paper investigates the EHD induction
pumping of liquid film in vertical annular configuration in the            The model assumes a flat interface and neglects the
presence of gravitational force and external load which has not        possibility of surface waves due to forces, which may arise
been studied previously with charges existing not only at the          from the radial electric field, the gravity, or the surface tension.
liquid/vapor interface but also throughout the bulk of the fluid.      It was reported by Benjamin [7] that films falling vertically
In addition, three different temperature profiles are specified at     solely due to gravity are unstable, even at small Reynolds
the entrance of the pipe to investigate their effect on the electric   numbers. However, for the case presented in the presence of
conductivity gradient, thus the electric shear stress. A               gravitational force, one can assume that the flat interface
theoretical model in a dimensionless cylindrical coordinates is        corresponds to long wave length associated with liquid film
presented. A numerical parametric study for two different
                                                                                                                                                 The time-averaged electric shear stress and the Joule heating
                                                           RT                                                                                    expressed in terms of the electric potential are [8]
                                                   Rv
                                                                                                                                                                                                         dC ' 
                                                                                                                                                                                                                        *


                                                                                                                                                                               *
                                                                                                                                                                                           Re  jK *C * * 
                                                                                                                                                                                             *                                                    (5)
                                                                                                                                                                                e ,rz                    dr 
                                                                                                                                                                                                              




                                                                   liquid
                                                       vapor
                                                                                                                                                                                 1  2                                          
                                                                                                                                                                                                                            2
                                                                                                                                                                                              2  dC *
                                                                                                                                                                             E  *  K * C * 
                                                                                                                                                                                 *      *2
                                                                                                                                                                                                                                                 (6)
                                                                                                                                                                                 2              dr *                           
                                                                                                                                                                                                                               




                                                                                             traveling wave
                                                                                electrodes
                                        symmetry


                                                                                                                                                    The non-dimensional slip coefficient is
                                                       g                                                                                                                                        *  K * w*
                                                                                                                                                                                        S*                                                       (7)
                                                                                                                                                                                                   *

                                           z                                                                                                         The corresponding dimensionless boundary conditions for
                                                                                                                                                 the flow field are
                                                               r
                                                                                                                                                                                u *  0 @ r*  RT
                                                                                                                                                                                                *
                                                                                                                                                                                                                                                  (8)

          Figure 1. Schematic of the vertical analytical domain                                                                                                                 w*  0 @ r*  RT
                                                                                                                                                                                               *
                                                                                                                                                                                                                                                  (9)

instability corresponding to relatively flat interface. It is also                                                                                                              u *  0 @ r*  Rv*                                               (10)
important to mention that the film thickness is assumed to be
constant in this study. In the absence of any phase change, the                                                                                                                                              
                                                                                                                                                                                                                   C  
                                                                                                                                                                                                              *
                                                                                                                                                                                                     I1 K * Rv  *C         2                2
                                                                                                                                                                                M e * K *
                                                                                                                                                                                                 2
                                                                                                                                                                                  *                                                     *R
only source of potential heat is the Joule heating. However, the                                                                                                                                                  C
mass evaporation rate generated due to the Joule heating under
                                                                                                                                                      *
                                                                                                                                                           w*
                                                                                                                                                                            
                                                                                                                                                                                                             
                                                                                                                                                                                                     I 0 K * Rv 
                                                                                                                                                                                                              *
                                                                                                                                                                                                                
                                                                                                                                                                                                                                                 (11)
the conditions considered here is three orders of magnitude less                                                                                           r *                                        1      * *2 
                                                                                                                                                                                                        S*  S  
                                                                                                                                                                  r*  Rv
                                                                                                                                                                        *
than the mass flow rate generated due to the electric field. This
                                                                                                                                                                                                                   
suggests that it is acceptable to assume no phase change, thus
constant film thickness.
                                                                                                                                                                                                          z *  0, z  L
                                                                                                                                                                                                                    *   *
                                                                                                                                                                                u*  0 @                                                         (12)
    The flow is governed by conservation equations of mass,
momentum, and energy. The momentum and energy                                                                                                                                   w*
                                                                                                                                                                                     0 @                   z *  0, z  L
                                                                                                                                                                                                                      *   *
                                                                                                                                                                                                                                                 (13)
conservation equations contain terms based on the electric field                                                                                                                z *
solution. The governing equations can be expressed in a
dimensionless form as (see [8] for more details)                                                                                                 Equation (11) states the balance of viscous and electric shear
                                                                                                                                                 stresses at the interface.
                                                   
                                        * .  * v *  0                                                                                  (1)
                                                                                                                                                    For the energy equation, the dimensionless boundary
                                                                                                                                                 conditions are
                       
    *v * . v *  p*  . *  *G*                  
                                                                                                                                                                                T * Prbo  "
                                                        Me  1      * 
                                                                                                                                           (2)                         k                                       @ r*  RT
                                                                                                                                                                                                                         *

                                                                                                                             
                                                           *
                                                                                                                                                                                            q                                                    (14)
                                                        *  * * r* rz                                                                                                        r * bo e
                                                                                                                                                                                         ˆ 2 wall
                                                        N e  r r      
                                                                                                                                                                                T * Prbo  "
                                                                                                                                                                   k                              @ r*  Rv
                                                                                                                                                                                                                *
                                           1 * * * *                                                                                                                                        q                                                    (15)
          c u .T
              * *
                p
                            *   *
                                               k  T  * E *2                                                                           (3)                                  r* boeˆ 2 int erface
                                          Prbo
    To solve the above equations, the electric shear stress in (2)                                                                               The imposed dimensionless temperature profile at the channel
and the Joule heating in (3) must be expressed in terms of the                                                                                   entrance becomes
dielectric potential. The basic electrostatic relations for the                                                                                                                              bo c pbo
                                                                                                                                                                       T * r *                      T r       @ z 0
                                                                                                                                                                                                                                    *
complex amplitude of the electric potential can be reduced in a                                                                                                                                                                                  (16)
dimensionless form to [8]                                                                                                                                                                     bo e2
                                                                                                                                                                                                  ˆ

                                                                                                                                                  The dimensionless boundary conditions for the electric
     *  j* *  w* K *  1 d *  r * dC*   d * dC*
                                            
                                            *       *    *
                                                                                                                                                 potential equation are
                           r dr
                              *
                                    dr  dr dr                                                                                            (4)
                                                                                                                                                                                     C *  1 @ r*  RT
                                                                                                                                                                                                     *

                                                                                                                         
                                                                                                                                                                                                                                                 (17)
                                        j   w K  K C  0
                                                   *                    *               *                     *       *           *2   *
                                                      
         *
where C is the complex amplitude of the electric potential [8]
                                         N e K *C *
                                           *                      
                                                                   *
                                                          I1 K * Rv  *    j 
                                                                         *
                                                                                                  governing equations iteratively, using the penalty method to
                                                                   * 
                                                                                                  ensure the satisfaction of the continuity equation. Each
               C *
                                
                                                                  
                                                          I 0 K * Rv     S 
                                                                                           (18)   iteration starts by updating all the fluid properties based on the
               r *                                       1 
                                                                       2
                                                                                                  temperature of the last step (see [8] for further details).
                      r*  Rv
                                                        * 
                                                          *2
                            *


                                                         S                                      Computations were carried out on a domain with
                                                                                                  dimensionless liquid height and axial length of one and
   The dimensionless fluid properties are given below,                                            twenty, respectively, on a 20x100 grid. The parameters are
                                                                                                  varied over a range corresponding to stable operation of EHD
                               c          k          
   *         , *      , c*  p , k *      , *      ,                                       induction pumping. Instability of EHD induction pumps can
           bo        bo                             bo
                             p
                                c pbo      kbo                                                    manifest itself in a sudden drop/jump in pump output. The
                                                                                           (19)   instability can also result in alternating/bi-directional flow.
                                 2 
     ,
    *
                      where c  bo bo  0                                                        The stability criteria defined in [9] shows that the erratic
      c                          bo   Ne                                                        behavior of the unstable pump can be eliminated by a proper
and the dimensionless length scales are as follows                                                selection of geometric and liquid film parameters as well as the
                                                                                                  traveling electric wave frequency. The convergence criteria of
           z                    r               Rv                 RT                  L          the numerical results presented with respect to dimensionless
    z*        , r*                , Rv 
                                       *
                                                          , RT 
                                                             *
                                                                           , L*           (20)
                                                                                             velocity, potential, and temperature is 10 -5. It is important to
                                                                                                  mention that all the numerical results presented are
   The dimensionless electric parameters are given below
                                                                                                  corresponding to laminar flow with Reynolds number less than
                                                                                                  2500.
                            E              
                                           ˆ
                    E*        , where Ec  e ,
                            Ec                                                                                       IV.   NUMERICAL RESULTS
                            C           C             '                                               The factors controlling the performance of a two-phase
                    C*        , C '*     ,                                                      induction pump can be summarized in the following three
                            
                            ˆe          e
                                        ˆ
                                                                                                  categories:
                                K               1
                    K*            , where K c  ,                                                  1.   Thermal
                                Kc              
                                                                                           (21)          a)   Entrance temperature profile
                                           
                    *        , where c  bo 2 ,
                            c             bo                                                          b) Heat flux at the wall of the pipe and interface;
                                                                                                             ''      ''
                            S                                                                             qwall , qint erface
                    S*         , where N e  bo 2 0 ,
                                          *

                            Ne*
                                             bo  bo                                                   c)   Transport properties     of      the   working   fluid;
                                e ,rz                                 bo bo 
                                                                               ˆ   2                          , , k , c p
                    *,rz                , where e ,rzc                         e

                                e ,rzc                                  2 bo
                     e
                                                                                                    2.   Electrical
and finally, the dimensionless flow parameters are defined as                                            a)   Applied voltage; 
                                                                                                                               ˆ
follows
                                                                                                         b) Frequency; f
                      u              
           u*           , where uc  bo ,                                                                    Wavelength, 
                      uc             bo                                                                c)
                      w                                                                                 d) Electrical properties of the working fluid, , 
            w*          , where wc  bo ,
                      wc             bo                                                           3.   Physical
                p               2
            p *
                   , where pc  bo 2 ,                                                                   a)   External load or pressure drop
                pc             bo                                                        (22)
                                                                                                         b) Liquid film thickness, vapor radius, the length and
               g                2                                                                          the radius of the pumping section.
           G   *
                  , where g c  2 bo 3 ,
               gc              bo                                                                   The effect of the above parameters on the performance of a
                T              2  ˆ    2  M *                                                 vertical induction pump is discussed in this section. Since the
           T *  , where, Tc  bo e  bo bo e                                                     transport and the electrical properties of any working fluid to
                Tc             bo c pbo  o c pbobo
                                                                                                  be considered in this study are function of temperature and
                                                                                                  require temperature dependent relations, a dimensionless
                                                                                                  parametric study is conducted using refrigerant R-123 as
                           III.           NUMERICAL METHODS                                       reference working fluid. The transport and the electrical
   A finite element code was developed using the finite                                           properties of the working fluid (R-123) are function of
element software Fastflo Version 3.0 (developed at CSIRO                                          temperature and are given in [10]. Three different temperature
Mathematical and Information Sciences) to solve the                                               profiles at the entrance corresponding to a uniform profile of 20
o
 C (Profile No. 1), and a two linear temperature profiles             film thickness of 33 % of the total radius of the pipe responded
producing an average entrance temperature of 20 oC (Profile           more positively to the effect of temperature gradient. I t is also
No. 2) and 22.0 oC (Profile No. 3), respectively, are considered.     important to not that for Case 1 the dimensionless radius of 9
On the other hand the heat fluxes at the wall of the tube and at      and 10 correspond to the liquid/vapor interface and the pipe
the interface are taken as zero because no phase change is            wall, respectively. On the other hand, dimensionless radius of 2
taking place as it was justified in Section II.                       and 3 correspond to the liquid/vapor interface and pipe wall,
                                                                      respectively, for Case 2.
    The effect of the parameters such as the applied voltage  ,
                                                              ˆ
electric wave angular velocity wave number K, the vapor                Fig. 2 displays the dimensionless mass flux as a function of
radius Rv, and gravity g are investigated through the following       the dimensionless electric wave number, which is inversely
dimensionless parameters Me*, *, K*, RV*, and G*,                    proportional to the wave length, for Case 1. The dimensionless
respectively. Each of these parameters is studied separately          mass flux is calculated at the pipe exit by integrating *w* over
while the others are maintained constant. Since the working           the dimensionless liquid film thickness. The behavior of all the
fluid is fixed, the electric permittivity through * and the          four curves presented in Fig. 2 is similar. The mass flux rises to
                                                                      its peak by increasing the electric wave number and then starts
electric conductivity * through Ne* will not be varied to
                                                                      to decrease. The reason of such behavior is that at large wave
accommodate for different working fluids, however, they are
                                                                      number (i.e., short wave length), the electric force is limited to
function of temperature. To account for the effect of the liquid
                                                                      a smaller space and the electric field is confined to the vicinity
film thickness, two base cases corresponding to two different
liquid film thicknesses of 10% and 33 % of the total radius of
the pipe are considered. However, for Case 2 only the results               TABLE I.          BASE CASES IN DIMENSIONAL FORM FOR R-123
that show different behavior than Case 1 are presented.
                                                                                        Rv                                    f        e
                                                                                                                                           ˆ
    The dimensional and the dimensionless parameters of the                            (mm)           (mm)         (mm)         (Hz)       (V)
two base cases for Profiles No. 1 and No. 2 are given in Tables         Case 1         12.6            1.4         4.4          19         501
I and II. The effect of the external pressure gain or load on the       Case 2         9.34           4.66         9.8          19         501
pumping section of EHD system or loop is also studied. This
pressure gain or load, which plays a critical role in operating
the EHD pump, is present in the liquid film only, for example,        TABLE II.        BASE CASES IN NON- DIMENSIONAL FORM FOR R-123 AT 20
                                                                                                        O
due to the existence of other EHD (or mechanical) pumping                                                 C
sections or upstream and downstream connecting lines. In                                                                 
general a negative pressure gradient is a favorable load that                      R*v                                      M*e    N*e
assists the pumping, whereas the positive gradient is
unfavorable and is expected to work against the pumping                Case 1      9.0          1.0          2.0          720     7000     6.0e-5
direction. However, in the results presented here the opposite is      Case 2      2.0          1.0          3.0         8000     7000     5.4e-6
true since the pump is operating in repulsion mode (i.e.
negative velocity, corresponding to negative z in Fig. 1).
                                                                      of the electrodes, in effect limiting its penetration depth into the
    In each of the following figures four cases are compared: 1)      liquid film, resulting in a lower mass flux. In addition, slip
numerical solutions with a uniform temperature profile at the         coefficient (7), which is the measure of the lag between the
entrance and the Joule heating is set to zero; 2) numerical           traveling electric wave and the charges induced in the liquid
solutions with a uniform temperature profile at the entrance          film at the interface, is small. This also means that large wave
(Profile No. 1) and the Joule heating present; 3) numerical           numbers correspond to small wave speeds. On the other hand,
solutions with a linear temperature profile at the entrance           at small wave numbers (i.e., long wave length) the distances
(Profile No. 2) with 21.0 oC at the pipe wall and 19.0 oC at the      between the electrodes are long; hence the electric field is not
liquid/vapor interface, and producing an average entrance             weak resulting in small electric shear stress, therefore, small
temperature of 20.0 oC; 4) numerical solutions with a linear          mass flow rate. Furthermore, at a given frequency and very
temperature profile at the entrance (Profile No. 3) with 25.0 oC      small wave numbers the slip coefficient grows too large again
at the pipe wall and 19.0 oC at the liquid/vapor interface, and       producing small electric shear stress (7).
producing an average entrance temperature of 22.0 oC. While
the first two cases are presented to study the effect of the Joule        Fig. 2 also reveals that the inclusion of the Joule heating is
heating on the pump performance, the last two are presented to        insignificant regardless of the value of the wave number. This
illustrate the effect of the temperature gradient coupled with        is due to the proportionality of the Joule heating to the electric
Joule heating on the pump performance.                                conductivity which, under the operating conditions considered,
                                                                      is on the order of 10-8 (S/m). However, the presence of the
     When the film thickness is larger, the electric shear stresses   temperature gradient (Profiles No. 2 and No. 3) becomes
are smaller because the dominant induced charges at the               significant only when the wave number is greater than unity. At
liquid/vapor interface are far away from the electrodes. This is      larger wave numbers, the bulk electric shear stress within the
why the results for Case 1 show higher mass fluxes than those         liquid film is responsible for the increase in the mass flow rate.
of Case 2. It is noteworthy to mention that the effect of the         This is due to the fact that the electric shear stress is
temperature gradient is more noticeable when the generated            proportional to the square of the wave number. In addition, the
velocity falls in the lower range of the Reynolds number              electric shear stress is a function of the electric conductivity
(~<600). For this reason the numerical results produced for the
gradient. A greater temperature gradient gives a greater                            Figure 3. Dimensionless electric shear stress distribution at the entrance, for
conductivity gradient. Therefore, as can be seen from Fig. 3,                                                         Case 1
the electric shear stress is the largest with Profile No. 3, which
in turns gives the highest mass flux. Note that the electric                        increase in the frequency. It is noteworthy to mention that this
shear stress is the largest close to the electrodes where the                       critical value of the angular velocity is lower with higher
electric field is the strongest. Fig. 2 also shows that as the wave                 wavelength (i.e. small wave number). The trend shown in Fig.
number increases, the inclusion of the temperature Profiles No.                     4 depicts the role of the electric shear stress within the liquid
2 and 3 yields higher mass flow rate compared to the case of                        film. Figure 4 indicates an increase in the mass flux compared
Profile No. 1. For example, at wave number of 2.0, the mass                         to the case of Profile No. 1 of about 4.7% and 15% for Profiles
flow flux with temperature Profiles No. 2 and No. 3 increased                       No. 2 and No. 3, respectively. These values are approximately
by 4% and 14.5%, respectively. With the wave number of 4.0,                         constant for the range of frequency considered here.
the mass flux increased by 19% and 88% with Profiles No. 2                              The influence of Me*, which is proportional to the square of
and No. 3, respectively.                                                            the applied voltage, is shown in Fig. 5 for the Case 2. Similar
    The effect of the electric wave angular velocity on the mass                    results can be obtained for Case 1.The mass flux increases with
flux is presented in Fig. 4. In light of the slip coefficient, it is                increasing Me* as expected. For example, at Me* of 7000
obvious that the mass flux has analogous dependency on the                          Profiles No. 2 and No. 3 result in an increase in the mass flux
electric wave angular velocity as it has on the wave number. As                     of about 9% and 32%, respectively. In addition, Fig. 5 reveals
the frequency increases, the mass flow rate increases until it                      an important observation that for high values of Me* (grater
reaches a critical value, and then it falls gradually with further                  than 1000), inclusion of Joule heating alone can enhance the


                                                                                           -1500
           0
                                                                                                                           Profile No. 1, w/o Joule heating
                                                                                                                           Profile No. 1, w/ Joule heating
      -500                                                                                 -1750                           Profile No. 2, w/ Joule heating
     -1000
                                                                                                                           Profile No. 3, w/ Joule heating
                                                                                           -2000
     -1500
                                                         Profile No. 1 w/o
  m* -2000                                                                             m* -2250
                                                         Joule heating

     -2500                                               Profile No. 1, w/
                                                         Joule heating                     -2500
     -3000                                               Profile No. 2, w/
                                                         Joule heating
                                                                                           -2750
     -3500                                               Profile No. 3, w/
                                                         Joule heating
     -4000                                                                                 -3000
                0     1                2          3           4              5
                                                                                                   0           500           1000            1500             2000
                                           K*                                                                                 *


 Figure 2. Dimensionless mass flux as a function of dimensionless electric           Figure 4. Dimensionless mass flux as a function of dimensionless electric
                       wave number for Case 1.                                                         wave angular velocity for Case 1.

          10                                                                        the mass flux.
          9.9                                                                           Fig. 6 presents the mass flux as a function of the
                                                                                    dimensionless vapor radius for Case 1. The liquid film
          9.8
                                                                                    thickness is kept constant. Varying the vapor radius only and
          9.7                                                                       keeping the liquid film thickness fixed is identical to varying
                                                                                    the pipe diameter. For example, for Case 1 with R v* of 9.0, the
          9.6
                                                                                    dimensionless liquid film thickness is 1.0 resulting in the tube
      *
     r 9.5                                                                          dimensionless radius of 10.0 (implying that the film thickness
          9.4
                                                                                    is 10% of the total radius). However, if Rv* is 1.0 and the
                                                                                    dimensionless liquid film thickness is 1.0 the tube
          9.3              Profile No. 1                                            dimensionless radius becomes 2.0 implying that the film
          9.2              Profile No. 2                                            thickness is 50% of the total radius. Figure 6 states that the
                           Profile No. 3
                                                                                    inclusion of the Joule heating only results in negligible effect
          9.1                                                                       on the mass flux under the operating conditions considered.
           9                                                                        According to Fig. 6, the mass flux initially increases with Rv*
           -1000    -950        -900       -850       -800        -750       -700   and then it reaches an asymptotic value.
                                           e*
                                                                                          1).
          0
                                                                                                   0
                                                                                                                                   Profile No. 1, w/o Joule heating
     -1000
                                                                                                 -500                              Profile No. 1, w/ Joule heating
                                                                                                                                   Profile No. 2, w/ Joule heating
     -2000                                                                                      -1000                              Profile No. 3, w/ Joule heating

                                                                                                -1500
 m* -3000
                                                                                            m* -2000

     -4000                                                                                      -2500
                              Profile No. 1, w/o Joule heating
                              Profile No. 1, w/ Joule heating
     -5000                                                                                      -3000
                              Profile No. 2, w/ Joule heating
                              Profile No. 3, w/ Joule heating                                   -3500
     -6000
              0       10000       20000   30000 40000       50000        60000    70000         -4000
                                               Me*                                                  -600      -400       -200       0        200         400          600
                                                                                                                                 (dP/dz)*

    Figure 5. Dimensionless mass flux as a function of Me* for Case 2.
                                                                                           Figure 7. Dimensionless mass flux as a function of dimensionless pressure
                                                                                                                    gradient for Case 1.

         -500
                                               Profile No. 1, w/o Joule heating
                                               Profile No. 1, w/ Joule heating                      0
                                                                                                                                    Profile No. 1, w/o Joule heating
        -1000                                  Profile No. 2, w/ Joule heating
                                                                                                                                    Profile No. 1, w/ Joule heating
                                               Profile No. 3, w/ Joule heating                   -200
                                                                                                                                    Profile No. 2, w/ Joule heating

        -1500                                                                                                                       Profile No. 3, w/ Joule heating
                                                                                                 -400
   m*

        -2000                                                                               m* -600


                                                                                                 -800
        -2500

                                                                                                -1000

        -3000
                  0           5           10          15            20            25            -1200
                                               Rv*                                                   -200   -100     0     100     200    300      400     500        600
                                                                                                                                 (dP/dz)*
  Figure 6. Dimensionless mass flux as a function of dimensionless vapor
                           redius for Case 1.                                              Figure 8. Dimensionless mass flux as a function of dimensionless pressure
                                                                                                                    gradient for Case 2.
As Rv* increases the symmetry boundary condition imposed at
                                                                                          According to Fig. 7 the curves are almost linear (properties are
pipe centerline is moved further away from the liquid/vapor.
                                                                                          temperature dependent) for all three profiles for the two cases.
Therefore, it is influence on the electric field distribution in the
                                                                                          As an example, according to Fig. 7, Profiles No. 2 and No. 3
liquid film and at the liquid/vapor interface becomes less
                                                                                          enhance the mass flux for dimensionless pressure gradient of -
significant until it losses any influence.
                                                                                          500 by about 7% and 25%, respectively, compared to Profile
The dependency of the mass flux on the external pressure                                  No. 1. Figure 8 presents the results for the negative pressure
gradient is shown in Figs. 7 and 8 for the two cases considered,                          gradient up to -150 for Profile No. 1 and -50 for Profiles No. 2
respectively. The external load plays a crucial role in operating                         and No. 3. The reason is that decreasing the external negative
the EHD pump. As stated previously, generally a negative                                  pressure gradient further results in a bi-directional flow or a
pressure is a favorable load which speeds the pumping,                                    flow in the opposite direction (in positive z-direction). This
whereas the positive pressure gradient is unfavorable and tends                           behavior will be explained further in light of the velocity
to lower the velocity. However, in the results presented here the                         distribution. Nonetheless, Profiles No. 2 and No. 3 result in an
opposite is true since the pump is operating in repulsion mode                            increase in the mass flux of about 23% and 62%, respectively,
resulting in the fluid flow in the negative z-direction (see Fig.                         at pressure gradient of -50 compared to Profile No. 1. This
                                                                                          enhancement ratio decreases as the pressure gradient increases.
    To further explain the effect of the external pressure                                            Figure 10. Dimensionless velocity profile at the pipe entrance for
gradient on the pump performance, the actual velocity profiles                                                      (dP/dz)*=-250 and -400, for Case 2.
at the entrance of the pipe for the three temperature profiles and
selected pressure gradients are presented in Figs. 9 and 10 for                                  Bi-directional liquid film flow, in general, is not desirable and
Case 2. Similar results can be obtained for Case 1. In these                                     in designing an EHD pump, the load should not exceed its limit
figures, the velocity profiles are subjected to the effects of the                               in order to prevent this occurrence.
electric shear stress as well as the external pressure gradient.                                     Finally, Fig. 11 presents the effect of the gravitational force
The effect of the electric shear stress alone (i.e. zero pressure                                on the mass flux for Case 1. Note that a dimensionless wave
gradient, Fig. 9) on the velocity profile results in a negative                                  number of 0.5 was chosen instead of 2.0 to allow for wider
velocity distribution, as expected for EHD pump operating in                                     variation of the gravitational force before the results reach
repulsion mode. The positive external pressure gradient is                                       values corresponding to the region of transition-turbulent.
expected to assist the electric shear stress resulting in a higher                               Here, the wave number of 0.5 corresponds to a wave length of
negative velocity (Fig. 9). However, when the external pressure                                  17.6 mm. Figure 11 shows that the gravitational force
gradient is negative, the overall effect depends on the dominant                                 accelerates the flow, thus enhances the mass flow rate. Another
force. If the electric shear stress and the negative pressure                                    observation is that the gravitational force, as it grows larger,
gradient are comparable, then the velocity profile could be bi-                                  dominates the electric shear stress.
directional as illustrated in Fig. 10. When the negative pressure
is dominant, then the velocity profile is expected to be solely in
the positive z-direction (see Fig. 10).                                                               -2000

          3                                                                                           -3000
                                       ,
                          Profile No. 1 (dP/dz)*=0.0
                          Profile No. 2, (dP/dz)*=0.0                                                 -4000
                          Profile No. 3, (dP/dz)*=0.0
        2.8                            ,
                          Profile No. 1 (dP/dz)*=200.0                                                -5000
                          Profile No. 2, (dP/dz)*=200.0
                          Profile No. 3, (dP/dz)*=200.0                                           m* -6000
        2.6
                                                                                                      -7000
   r*
                                                                                                      -8000                                    Profile No. 1, w/ Joule heating
        2.4
                                                                                                                                               Profile No. 2, w/ Joule heating
                                                                                                      -9000
                                                                                                                                               Profile No. 3, w/ Joule heating
        2.2                                                                                          -10000
                                                                                                          -5000      -4000        -3000          -2000        -1000              0
                                                                                                                                          G*
          2
           -250           -200           -150             -100             -50               0
                                                                                                     Figure 11. Dimensionless mass flux as a function of dimensionless
                                                  w*                                                              gravitational force for Case 1 (K*=0.5).

     Figure 9. Dimensionless velocity profile at the pipe entrance for                           Thus, the presence of temperature profiles within the liquid has
                  (dP/dz)*=0.0 and 200, for Case 2.                                              practically no effect on the resultant liquid film mass flow rate.

                                                                                                                             V.    CONCLUSIONS
                                ,
                     Profile # 1 (dP/dz)*=- 250.0                           ,
                                                                 Profile # 1 (dP/dz)*=- 400.0
                     Profile # 2, (dP/dz)*=- 250.0               Profile # 2, (dP/dz)*=- 400.0       A theoretical model was developed for EHD induction
                     Profile # 3, (dP/dz)*=-250.0                Profile # 3, (dP/dz)*=- 400.0   pumping of liquid film in vertical configuration. The model
         3                                                                                       accounted for the charges not only at the interface but also
                                                                                                 throughout the liquid film. Three different temperature profiles
        2.8                                                                                      were considered to investigate the effect of the Joule heating
                                                                                                 and the temperature gradient throughout the bulk of the film on
                                                                                                 the mass flux for two cases corresponding to two different film
        2.6                                                                                      thicknesses. A parametric study including liquid film thickness,
   r*                                                                                            voltage, wavelength, frequency, external pressure, and gravity
        2.4                                                                                      was carried out. The numerical results indicated that for the
                                                                                                 electric conductivity and voltage level of interest, including the
        2.2
                                                                                                 Joule heating in the analysis proved to be insignificant. In
                                                                                                 addition, the results showed that higher mass flow rate can be
                                                                                                 achieved by increasing the voltage, pipe diameter, fluid
         2                                                                                       temperature, and also by increasing the external pressure gain.
              -40   -20          0          20            40        60           80        100   On the other hand, external pressure load, at certain level and
                                                 w*                                              beyond, yielded flow reversal or bi-directional flow, which is
potentially undesirable. The entrance temperature profile
played an important role in the operation of the pump. Finally,
the gravitational force could easily dominate the effect of the
electric shear stress.

                             REFERENCES
[1]  Melcher, J. R., “Traveling - Wave Induced Electroconvection,”
     Physics of Fluids, Vol. 9, No. 8, (1966), pp. 1548-1555.
[2] Melcher, J. R., “Continuum Electromechanics,” Cambridge, MA: MIT
     Press, pp. 5.94-5.54, 1981.
[3] Crowley, J. M., “Stability of EHD Induction Pump,” Conf. Rec. IEEE-
     IAS Annual Meeting, Mexico City, Mexico, (1983), pp. 1149-1153.
[4] Wawzyniak, M. and Seyed-Yagoobi, J., “An Analytical Study of
     Electrohydrodynamics Induction Pumping of a Stratified Liquid/Vapor
     Medium,” IEEE-IAS Transactions, Ind. Applicat., 35, (1999), pp. 231-
     239.
[5] Wawzyniak, M. and Seyed-Yagoobi, J., “Electrohydrodynamic
     Induction Pumping of a Stratified Liquid/Vapor Medium in the Presence
     of Volumetric and Interface Electric Charges,” IEEE-IAS Transactions,
     Ind. Applicat., 37, No. 4 (2001), pp. 950-958.
[6] Brand, K. and Seyed-Yagoobi, J., “Effect of Electrode Configuration on
     Electrohydrodynamics Induction Pumping of a Stratified Liquid/Vapor
     Medium,” IEEE-IAS Transactions, Ind. Applicat., 38, No. 2 (2002), pp.
     389-400.
[7] B. Benjamin, “Wave Formation in Laminar Flow Down an Inclined
     Plane,” J. Fluid Mech., Vol. 2, pp. 554-574, 1957.
[8] Aldini, S., “Electrohydrodynamic Induction and Conduction Pumping of
     Dielectric Liquid Flim: Theoretical and Numerical Studies,” Ph.D.
     dissertation, Dept., Mech. Eng., Texas A&M Univ., College Station, TX,
     2005.
[9] Aldini, S. and Seyed-Yagoobi, J., “Stability of Electrohydrodynamic
     Induction Pumping of Liquid Film in Vertical Annular Configuration,”
     IEEE-IAS Transactions, Ind. Applicat., (in press).
[10] J. E. Bryan, “Fundamental Study of Electrohydrodynamic Enhanced
     Convective and Nucleate Boiling Heat Transfer,” Ph.D. dissertation,
     Dept., Mech. Eng., Texas A&M Univ., College Station, TX, 1998

								
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