Analytical Study of Magnetohydrodynamic Propulsion Stability

Chabahar Maritime University, Chabahar 99717-56499, Iran

Analytical Study of Magnetohydrodynamic Propulsion Stability

M.Y. Abdollahzadeh Jamalabadi

Chabahar Maritime University, Chabahar 99717-56499, Iran

In this paper an analytical solution for the stability of the fully developed flow drive in a magneto-hydro-dynamic pump with pulsating transverse Eletro-magnetic fields is presented. To do this, a theoretical model of the flow is developed and the analytical results are obtained for both the cylindrical and Cartesian configurations that are proper to use in the propulsion of marine vessels. The governing parabolic momentum PDEs are transformed into an ordinary differential equation using approximate velocity distribution. The numerical results are obtained and asymptotic analyses are built to discover the mathematical behavior of the solutions. The maximum velocity in a magneto-hydro-dynamic pump versus time for various values of the Stuart number, electro-magnetic interaction number, Reynolds number, aspect ratio, as well as the magnetic and electrical angular frequency and the shift of the phase angle is presented. Results show that for a high Stuart number there is a frequency limit for stability of the fluid flow in a certain direction of the flow. This stability frequency is dependent on the geometric parameters of a channel.

stability frequency; Stuart number; transient flow; Eletro-magnetic interaction number; duct flow; MHD propulsion

1 Introduction

Electromagnetic propulsion (EMP) is the theory of speeding up a fluid by the use of electrical and magnetic fields. When a current flows through a conductor in a magnetic field, a force known as the Lorentz force pushes the conductor in a direction perpendicular to the conductor and the magnetic field. In spite of electric motors, the electrical energy used for the EMP is not used to produce rotational energy for motion. The laws were known in the nineteenth century from the work by Hartmann on electromagnetic pumps in 1918. The EMP and its applications for seagoing ships and submarines (without the aid of either propellers or paddles) have been investigated since at least 1958 when Warren Rice filed a patent explaining the technology US 2997013 (Rice, 1961). The collection consists of a water channel open at both ends extending longitudinally through or attached to the ship, a magnetic field throughout the water channel, electrodes at each side of the channel and a source of power to send direct current through the channel at rightangles to the magnetic flux in accordance with the Lorentz force (see Fig. 1).

Fig. 1 Magneto-hydrodynamic propulsion principle (http://www.skewsme.com/mhd.html#ren4)

The Yamato 1, an experimental MHD propulsion craft, is propelled by two MHD thrusters (without any moving parts), a liquid helium-cooled superconductor (cooled in order to maintain its zero-resistance property); the seawater is used as the electrically conducting fluid, and the craft can travel at 15 km/h (Sasakawa, 1997). This ship used an AC/DC converter in order to obtain stationary electromagnetic fields. On the other hand, one very promising MHD thruster idea is the AC internal duct propulsion mechanism. In this arrangement, the electric field and the magnetic field alternate directions in phase, so that the Lorentz force is always directed towards the rear of the vessel. The benefits of this approach are considerable. If the frequency is in the order of 1 to 4 kHz, the polarizing effects of the solution can be avoided. However, if the frequency can be increased to over 5 kHz, the molar conductivity of an electrolytic solution will increase due to the vanishing of the time-of-relaxation effect. The alternating current should also prevent buildup of reactant products on the electrode surfaces. Therefore, the choice of electrode materials may be limited due to oxidation and reduction reactions occurring on both electrode surfaces.

Many prototype cases for MHD propulsion in seawater use AC or DC power sources. The pulsating fields generated by AC power sources can cause internal fluctuations and instabilities even if the conditions of the sea are calm. The fluid flow under constant fields generated by DC power sources could be unstable because of the externaloscillations of violent circumstances in the sea. Then, the stability criterion could be obtained in both cases.

Besides vessel propulsion, the MHD viscous flows have appeared as an essential field of study in recent times owing to its use in numerous different fields of engineering and technology along with astrophysics, geophysics and nuclear sciences. The flow of polymer extrusion processes can be finely controlled by external magnetic fields in order to increase the quality of the final product. Thus, the phenomenon is under extensive study by the research community due to its diverse applications. The MHD pump has been a topic of many researches for simulation (Wanget al., 2004), fabrication (Zhonget al., 2002) and experimental study (Penget al., 2008) in recent years. The application of the MHD covers a very wide range of physical areas of liquid metals to cosmic plasmas, for example, the intensely heated and ionized fluids in an electromagnetic field in astrophysics, geophysics, high-speed aerodynamics and plasma physics. In addition, it has many industrial applications for nuclear magnetic resonances (Homsyet al., 2007), micro-fluidics and micro-systems (Qian and Bau, 2009), actuators and stirrers (Qian and Bau, 2005), controlling fluid flow on an electronic chip (Weston and Fritsch, 2012), micro-pumps for chromatography (Eijkelet al., 2003), mixers (Kang and Choi 2011), induction pumps (Kirillov and Obukhov,2003), the Micro-channel fabrication process in microelectronics (Malecha and Golonka, 2008), and nano-wears (Wanget al., 2013).

In the literature, the effects of the gusts on the performance of the propulsion system are not addressed. For example, the effect of unsteady forces on a foil (a flat plate in a potential flow) caused by chance upon a transverse gust (sudden step-like change) is related to the Küssner function. The important problems with the gust special effects on the ship include the FSI problems and gust modeling, such as the response of a typical flexible section at both the thin and thick airfoils (Arasekiet al., 2004), the dynamic response (deflection) of an elastic wing and pressure distributions (Arasekiet al., 2000), and gust modeling (Arasekiet al., 2006).

One of the most interesting matters concerning the MHD is its stability. This theme followed with annular linear induction pumps (Arasekiet al., 2000), double-supply-frequency pressure pulsations (Arasekiet al., 2006), sodium flow rate measurements (Arasekiet al., 2012), open-cycle power generation systems (Hayanoseet al., 1998; Ishikawaet al., 1996; Matsuoet al., 1998), advanced Tokamak (Yiet al., 2010), dumping resistors (Hayanoseet al., 2001), subsonic disk generators (Inoueet al., 2003; Li and Cai, 2013; Aibaet al., 2009), liquid metal jet flows (Kanget al., 2006), perforated and parallel walls (Ospinaet al., 2008), compressible and radiative flows (Qinet al., 2012), Jeffery–Hamel flows (Makinde and Mhone, 2007), toroidal devices (Gonget al., 2008), gyro-kinetics (Lauberet al., 2007), torus (Lugovtsov and Kotelnikova, 2010), free-surfaces (Giannakiset al., 2009), supersonic generators and diffusers (Matsuoet al., 1994), turbulence and nonlinear dynamics (Moffatt, 1989), anisotropic MHD (Maiellaro and Labianca, 2002), design of medical diagnostic devices which make use of the interaction of magnetic fields with tissue fluids (Ikbalet al., 2009), and the electromagnetic control of the boundary layer flow on a ship’s hull (Bakhtiari and Ghassemi, 2014).

The experimental findings on liquid metal which were conducted at the UCLA MTOR facility shows that the magnetic field changes the turbulent flow from 2D column-type disturbances to ordinary hydrodynamic turbulence wave structures (Yingaet al., 2004). The Orr–Sommerfeld and induction equations govern the linear stability of temporal normal modes in incompressible, parallel MHD. These equations have mainly been applied to study the stability of flows with fixed domain boundaries in the presence of an external magnetic field. In MHD, numerical investigations of the stability of the modified plane Poiseuille flow subjected to a transverse magnetic field, also known as the Hartmann flow, began in 1973 with the work of Potter and Kutchey (1973) for small Hartmann numbers (Ha＜6). A major challenge of hydrodynamic-stability problems at high Reynolds numbers is the existence of thin boundary layers. A Chebyshev method for plane Poiseuille and plane Couette flows in the presence of a transverse magnetic field was later developed by Takashima (Takashima, 1996) for high Reynolds (107) and Hartmann numbers (103). The temporal development of small disturbances in the MHD Jeffery-Hamel flows was investigated in Ref. (Makinde, 2003). Makinde concluded from her results that the magnetic field across the gap has a strong stabilizing effect on the flow (Makinde, 2003).

Many methods have been developed to study the turbulence MHD. One of the most recent is the shell model. This model describing the statistics of homogeneous and isotropic MHD turbulence in spectral space is presented in (Plunianet al., 2013) with their advantages and weaknesses. Other conditions of turbulence such as the Hall effect on unsteady MHD Couette flows, which causes the transverse velocity components was studied by Attia (2003). A theoretical analysis of the steady state solutions and critical values (pitchfork type bifurcations) of the MHD equations in the incompressible case of suction is given in (Ospinaet al., 2008) as a function of the Reynolds number, magnetic Reynolds number, and Alfvenic Mach number, for some of the asymptotic limits.

As seen from the literature review, the stability of MHD for ship propulsion applications has not been studied. In this study, the effects of the Stuart number, electro-magnetic interaction number, Reynolds number, aspect ratio, as well as the magnetic and electrical angular frequency and the shift of the phase angle on the stability of the seawater flow through the propulsion systems in Cartesian, cylindrical, and annular geometries are considered.

2 Geometry and mathematical model

Up to the present time, four basic procedures for MHDpropulsion have been proposed. These include internal flow direct current (DC), internal flow induction (AC), external flow (DC), and external flow induction (AC). With the induction devices, a magnetic field is varied alongside the length of the MHD channel with the strength proportional to a sine function. The purpose of this study is to analyze the feasibility of the stability of the MHD thruster as a driving force device for the naval vehicles. It is praiseworthy to note that the MHD thruster concept offers potentially easier maintenance because of no moving parts and as a result makes the vessel undetectable by sonar radars. In this study, a physical dimensional model is developed to address the overall stability of an MHD thruster based on the conservation of mass and momentum from AC power sources in Cartesian and circular geometries. Fig. 2 shows the submersible with rectangular MHD Channels respectively.

Fig. 2 Submersible with rectangular MHD channels (Gilbert and Lin, 1991)

2.1 Governing equations in cartesian geometry

Consider transient, hydro-dynamically and thermally fully-developed laminar flows of an incompressible fluid between two parallel plates. Furthermore, magnetic and electric properties are constant. Both plates are assumed to be stationary. The momentum equation in thex-direction is described as:

in which the Lorentz force can be written as:

whereEis the electric field intensity inz-direction (E=Emaxsin(ωt+φ)),Bis the magnetic density iny-direction (B=Bmaxsin(ωt+φ)) ,σis the electrical conductivity of the fluid,pis pressure,μis kinematic viscosity,ρis density of the fluid (1 000 kg?m-3is a reference density). Seawater conducts electricity on a modest scale by electrolytic ion exchange. While its conductivity (4 ?-1?m-1) is several orders of magnitude lower than that of metals, it is significantly higher than that of fresh water. The boundary conditions of Eq. (1) are:

whereVAis the ship velocity,LandWare half of the channel length and width in theyandzdirections respectively,VWis the gust velocity,cwis equal to zero for the fully electrically insulated walls andcwtends to infinity at the perfect conducting boundary condition. Here it is expected that thecwapproaches to infinity and the boundary condition at the wall is supposed to be perfectly conducting. So from the induction equation, the magnetic field strength through the entire section is constant.

2.1.1 General solution of Cartesian geometry

To clarify the relationship of this mathematical analysis with the practical MHD propulsion (or pumping) in a pipe, it is worthwhile to remember that the domain of this study is the laminar flow or low turbulence flow regime. Because the purpose is to find the stability criteria. In the usual analyses of standard flows, like the Poiseuille flow, the Orr-Summerfeld stability equation is solved and the critical Reynolds number, the wave length of the 2D Tolimin-Schilichting waves, and the maximum stable frequency of the system is obtained. In spite of the usual condition, here the driving force is not constant and is varied by the frequency of the electro-magnetic fields. By simplifications of the current study, it is assumed that the fluid layers slip on each other and they do not mix like fully turbulent flows. In other words, fluid layers are accelerated together in a way which conserves a similar shape between various times but differs in amplitude. Correspondingly,there is not a momentum change in the system originated from circulation or mixing of fluid layers.

The solution of the previous ordinary differential equation (Eq. (3)) is in the form of:

Regrettably, the explicit formula of the integral in Eq. (4) could not be found because the integral of the esin(t)in the q is the integrating factor and is unknown. By the initial condition of(t=0)=1, the equation (4) can be simplified as:

or at the zero shift of the phase angle ( 0=φ) and by ignoring the viscous effects

The maximum velocity starts fromVAand approaches to 9VAS/8.

2.1.2 Limiting case: low interaction number, low Reynolds number

For low interaction numbers (high Stuart number and low electro-magnetic parameter) and a low Reynolds number, in the limit 0S→ and 0Re→ , the equation (3) is:

The solution of this equation,(S=M/N→ 0,Re→ 0), satisfying the initial condition,(t= 0)=1, is

or by ignoring the viscous attenuation effects

Accordingly with high interaction numbers and a low Reynolds number, because of the decreasing outcome of the magnetic field, the maximum velocity, which starts fromVA, after some time approaches to zero.

2.1.3 Limiting case: high interaction number, low Reynolds number

For high interaction numbers (a high electro-magnetic parameter and a low Stuart number) and a low Reynolds number, in the limitM→ ∞ and 0Re→ the Eq. (3) is:

By choosing the integral factor asq=, the solution of this equation,(M/N→ ∞,Re→ 0), satisfying the initial condition,(t= 0)=1, is:

At simultaneous electric and magnetic fields with the same phase angle ( 0φ= ) and steady state solution (t→ ∞ ) the maximum velocity versus time behaves such as:

the ratio of the amplitude of the harmonic termsto the constant term (cos(φ) ) is equal toand so for the stabilized velocity profile, if the maximum value of the velocity fluctuation to the mean value is less than “m” then the frequency of the field should be higher than a constant value

2.2 Governing equations in cylindrical geometry

Considering the transient to be hydro-dynamically and thermally fully-developed, the laminar flow of an incompressible fluid is through the pipe. Furthermore the magnetic and electric properties are constant. The pipe wall is assumed to be stationary. The momentum equation in thex-direction is described as:

with the boundary conditions as:

whereDis the pipe diameter. Also the annular geometry can be used for MHD propulsion. So the equation (10) can be modified for the annular channel as:

whereDiis the inner pipe diameter andDois the outer pipe diameter.

General solution of the cylindrical configuration

In the general form, the velocity is a function of time and position (u=u(x,r,θ,t)). In the pervious section it is presumed that the MHD tube has an adequate length in which the flow touches the fully-developed form. In the fully-developed flow the crosswise velocities are ignored (vr=vθ= 0). Moreover, based on mass conservation, the deviation of the velocity to the axis of the pipe direction is zeroHitherto, the analytical solution of the equation (7) (i.eu=u(r,t)) is a difficult challenge and its solution requires the use of numerical methods. So by choosing the approximate velocity profile as (Schlichting, 1986)

3 Results

To solve the governing nonlinear differential equations, an in-house finite-volume code is developed and utilized. The code is based on a structured grid for time and the Runge-Kutta method (Antia, 1991). The residual for the velocity is set at 106. The program uses the input data and solves the momentum equations to obtain the maximum of the developed velocity field along the MHD pump. For verification purposes, Figure 3 shows the current as it is compared with a computational model known case (flow inside a rectangular duct) by Wanget al. (2004). As shown, the volumetric flow rates for both studies show a similar trend. The dependence of the flow velocity as a function of the channel width is shown in Fig. 3.

Fig. 3 Simulated flow velocity as a function of channel width; the channel depth is 2 mm, the input current is 0.5 A and the magnetic flux density is 18 mT

According to Fig. 3, the flow velocity is drastically reduced due to the resistance of the frictional effect of the channel side walls when the width is less than 10 mm. When the width is larger than 20 mm, the frictional forces on both sidewalls of the channel no longer affect the flow velocity. As for the dependence of the volumetric flow rate on the channel width, the flow rate increases linearly with the channel width. In the Figs. 4-12 the effect of the geometry parameters is hidden in the modified Reynolds number and is refered to as the Reynolds number. So the effect of the aspect ratio on the rectangular geometry is notstudied here separately. With the one-dimensional laminar flow, the flow velocity profile in the channel along the sidewall direction could not be exactly calculated since the Poiseuille type of flow neglects the frictional effects on both sidewalls. Although the predictions using the one-dimensional laminar flow model are not significantly different from those of the two-dimensional model when the aspect ratio of the channel reduces to near unity, it should be noted that the design of the MHD pump can be quite inaccurate if the one-dimensional model is generally applied.

Fig. 4 shows the effect of the Stuart number on the transient dimensionless velocity profile at a Reynolds number equal to 100 and without the effect of the interaction number. As shown, because of the decreasing outcome in the magnetic field without the benefit of the electric field, the moving particles of the fluid are stopped after some time.

Fig. 4 Effect of the Stuart number on the transient dimensionless velocity profile (M=0,Re=100)

Fig. 5 displays the effect of the Stuart number (Nfrom 1 to 10) on the transient dimensionless velocity profile for the electro-magnetic parameter equal to 1 and a Reynolds number equal to 100. Being exposed to the diminishing result of the Stuart number is compensated by the electro-magnetic parameter. The maximum velocity is stabilized near the initial condition for a Stuart number equal to 2 (near theN=9/4).

Fig. 5 Effect of the Stuart number in the presence of the electro-magnetic parameter on the transient dimensionless velocity profile (M=1,Re=100)

Fig. 6 illustrates the effect of the interaction number on the transient dimensionless velocity profile at a Reynolds number equal to 100 and by ignoring the effect of the Stuart number. As shown, because of the augmentation of the interaction number, the fluid is accelerated constantly with time.

Fig. 6. Effect of the electro-magnetic parameter on the transient dimensionless velocity profile (N=0,Re=100)

Fig. 7 shows the effect of the Stuart number (N=1) on the maximum velocity for the condition of Fig. 6. As is visible, the shrinking result of the Stuart number on the interaction number makes the amount of the velocity half at the same time approximately.

Fig. 7 Effect of the electro-magnetic parameter in the presence of the Stuart number on the transient dimensionless velocity profile (N=1,Re=100)

Figs. 8-10 show the effect of the viscous term in the presence of the Stuart number and electro-magnetic parameter on transient velocity. The dashed line represents the frictionless limit. Similar to Figs. 4-7, the Stuart number has a discontinuing effect and the electro-magnetic parameter has an accelerated effect on the fluid motion. Fig. 8 clarifies that the viscous term decreases the effect of the Stuart number. The similar effect is clear in Fig. 9 for the electro-magnetic parameter and in Figure 10 for both. Although the viscous effect changes the final value of the velocity, it does not affect the rate of acceleration of the fluid and can be ignored for Reynolds numbers greater than 10.

Fig. 8 Effect of the Reynolds number in the presence of the electro-magnetic parameter on the transient dimensionless velocity profile (N=1,M=0). The dashed line is the limit ofRe=∞

Fig. 9 Effect of the Reynolds number in the presence of the Stuart number on the transient dimensionless velocity profile (N=0,M=1). The dashed line is the limit ofRe=∞

Fig. 10 Effect of the Reynolds number in the presence of the Stuart number and electro-magnetic parameter on the transient dimensionless velocity profile (N=1,M=1). The dashed line is the limit ofRe=∞

Fig. 11 reveals the effect of angular velocity on transient dimensionless velocity (M=N=Re= 1). As presented by the increase of the dimensionless angular velocity, the amplitude of the fluctuations decreases. By the increase of the dimensionless angular velocity from 0.1 to 1, the velocity fluctuation amplitude is lessened by four times. Furthermore, Fig. 12 discloses the consequence of the difference between the phase angles of the magnetic field and the electric field for the transient dimensionless velocity at the same condition. As demonstrated, the amplitude of the fluctuations decreases near the 0 and 360 degrees of the phase angle and is maximum for the 180 degrees of difference.

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Fig. 11 Effect of angular velocity on the transient dimensionless velocity profile (M=1,N=1,Re=1)

Fig. 12 Effect of the phase angle on the transient dimensionless velocity profile (M=1,N=1,Re=1)

The stability chart representing the velocity profile based on the maximum velocity fluctuation in relation to the mean velocity parameter is illustrated in Fig. 13. This inequality is a balance between the largeness of the harmonic terms and demonstrates that for a specified value “m” there is a minimum frequency of stabilization.

Fig. 13 Stability chart of the velocity profile based on the maximum velocity fluctuation to the mean velocity

Fig. 14 shows a submersible with circular MHD channels.The practical dimensionless velocity profiles for various Reynolds numbers in the circular passages are shown in Fig. 15. The dashed line is the parabolic shape for the laminar flow regime. Another geometry which is discussed in the current study is the annular configuration in which the application is plotted in Fig. 16. All of the figures of the dimensionless velocity can be evaluated for the circular and the annular geometries based on the modified value of the Reynolds number and the electro-magnetic parameter as a function of the Reynolds number which is shown in Fig. 17.

Fig. 14 Submersible with circular MHD channels

Fig. 15 Velocity profiles for various Reynolds numbers in circular channels

Fig. 16 Submersible with annular MHD channels

Fig. 17 Reynolds number and electro-magnetic ratio of the circular section versus the Reynolds number

4 Conclusions

In this study, the effects of the Stuart number as well as magnetic and electrical angular frequency on the velocity distribution in a magneto-hydro-dynamic pump are scrutinized. Also a criterion for the stability of the velocity field has been derived for the laminar, transient problem in a Poiseuille flow between plane parallel plates with the Lorentz force. Results show that, as the Stuart number approaches zero, the velocity profile becomes similar to that of the fully developed flow in a pipe. Furthermore, for a high Stuart number there is a frequency limit for stability of the fluid flow in a certain direction of the flow. This stability frequency is dependent on the geometric parameters of a channel. The maximum velocity and volumetric flow rate increase with the increasing magnetic flux density and electric current. The maximum temperature increases with the increasing electric current but is not varied considerably with magnetic flux density variations. The effects of the Stuart, Reynolds, and interaction numbers on the transient velocity have been discussed in terms of the time.

Nomenclatures

Aaspect ratio,L/w

Bmagnetic field strength iny-direction T

cconductivity constant m

C1coefficient for the annular cross sections,

C2coefficient for the annular cross sections,

C3coefficient for the annular cross sections,

Dpipe diameter m

Eelectric field strength inz-direction V/m

Fforce N

Lhalf of the channel width in thezm direction

mvelocity fluctuation to mean ratio

Melectro-magnetic parameter,

Mcircleelectro-magnetic parameter for circular

cross sections,

Mannularelectro-magnetic parameter for annular

cross sections,

npower of the approximate velocity profilen≈0.75log10(Re) +2

Nstuart number,

ppressure Pa

qintegral factor,

rradial coordinate m

ReReynolds number,Re=VAL/υ

Remmodified Reynolds number for rectangular duct geometry,Rem=2VALw2/3(w2+L2)υ

Remcmodified Reynolds number for circular tube geometry,

Remamodified Reynolds number for annular tube geometry,

ttime s

dimensionless time

u,vvelocity components m/sdimensionless velocity,

VAship velocity m/s

whalf of the channel length in theym direction

x,y,zCartesian coordinate components m

Greek letters

μconstant viscosity of water

ρfluid density kg/m3

σelectrical conductivity of the fluid

φphase angle

ωangular velocity rad/s

dimensionless angular velocity,

Subscripts

0 stationary

aannular

ccircular

iinner

LLorentz

mmodified

max maximum

oouter

Superscript

′ differentiation with respect to t 1/s

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Author biography

M.Y. Abdollahzadeh Jamalabadiis an assistant professor at the Chabahar Maritime University. He received his B. Sc. in fluid mechanical engineering from Iran university of Science and technology in 2006, his Master′s Degree in energy conversion from Sharif University of Technology in 2008, and his Ph.D. degree in Mechanics from Khajeh Nasir Toosi University in 2012. His research interests include fluid flow simulation, CFD, MHD, heat transfer, thermal radiation, and SOFC simulation. He is currently a senior research scientist at the Gyeongsang National University.

1671-9433(2014)03-0281-10

Received date: 2014-02-25.

Accepted date: 2014-06-13.

＊Corresponding author Email: muhammad_yaghoob@yahoo.com

? Harbin Engineering University and Springer-Verlag Berlin Heidelberg 2014