0D David Taylor Research CenterBethesda, MD 20084-5000

NN4: DTRC-90/009 April 1990

SPropulsion and Auxiliary Systems Department DTIC FILE COPYResearch and Development Report

Propulsive Efficiencies of MagnetohydrodynamicSubmerged Vehicular PropulsorsbyS. H. Brown, J.S. Walker, N.A. Sondergaard,P.J. Reilly, and D.E. Bagley

DTICELECTE,AYO 2 1990D

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12 SHIP SYSTEMS INTEGRATION DEPARTMENT

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27 PROPULSION AND AUXILIARY SYSTEMS DEPARTMENT

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Bethesda, Maryland 20034-5000 62936N 1-2712-131 ZF66412001 50054011. TITLE ({citd• e Seaey C0auhonan)

Propulsive Efficiencies of Magnetohdrodynamic Submerged Vehicular Propulsors1•. PERSONAL AUTHOR(S)

Samuel H. Brown, John S. Walker, Neal A. Sondergaard, Patrick J. Reilly, and David E. Bagley13L. TYPE OF REPORT iab. TiME COVERED f14. DATE OF REPORT (YEARL MtONTH. DAY) 15. PAGE COUNT

Final FROM TO April 1990 (416. SUPPLEMENTARY NOTATON This work was a cooperative effort with the Deparment of Maechanical and Industrial Engineering

at the University of Illinois, Urbana 111. 6180117. COSATI CODES IL SUBJECT TERMS (Con'we on mrae . neoaay and kWerty by back nLt.)

FIELD GROUP SUB-GROUP Magnetohydrodynamic propulsion, marine propulsion, seawater pump,

superconducting magnets

19. ABSTRACT (Coniu. on revae. ineaa y and OwWy by blMock nig"

"Magnetohydrodynamic (MMD) ship propulsion is the process of propelling a vehicular structure by a seawater elec-tromagnetic pump. This propulsion system can be applied to a surface ship or a submerged vehicle; however, in thiswork only submerged vehicles at depths where wave effects can be neglected were considered. Although a number ofdifferent arrangements for a MHD propulsion system are possible, the general characteristics of such systems are mosteasily determined by a simple, ideal MHD rectangular duct of constant cross-sectional area. A mathematical model wasdeveloped at the-Daid Taylcr R:3sa -2-enter (WR• for calculating the propulsive efficiencies of such a rectangularduct propelling a submerged vehicle. Numerical propulsive efficiencies are presented in terms of many different param-eters. Assumptions were generally made in the model that tend to maximize the propulsive efficiency of the MI-ID sys-tem. Thus, the propulsive efficiencies calculated from the model overestimate the efficiencies of the)corresponding realMHD propulsion system. These numerical results can be used for engineering estimates of the propulsive efficiencies ofreal MHlD propulsion systems.

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A comparison of the ideal model MHD rectangular duct propulsive efficiencies with the corresponding effi-ciency of a conventional submerged vehicle propeller (order of 0.85) generally shows the MHD efficiencies to beless under most conditions. The cases where MI-ID propulsive efficiencies are in ranges that approach that of a pro-pza!.er correspond to MI) duct volumes greater Lhen 300 m 3 , with small surface-area-to-volume ratios, or the mag-netic inductions are in the order of 20 tesla or more. Magnetic inductions of about 6 to 10 tesla in large volumes ofspace, such as MHD ducts, are probably about the highest field strengths possible with the use of present-day super-conducting magnet construction technology; to produce even these fields will require significant engineering effortin magnet construction. Therefore an important area of concern for MHD propulsion is the construction of practicalsuperconducting magnets and support structures of reasonable mass.

According to the calculations using the mathematical model, a weakness of MID propulsion, for a viable sub-merged vehicle propulsion system, is the generally low propulsive efficiencies of the system at duct magnetic induc-tions of 10 tesla or less, when compared with conventional propellers. Furthermore, the general trend is for propul-sive efficiencies to decrease as vehicle speed increases. Future work will address the degrading effects on the rectan-gular MHD channel, caused by fringing magnetic induction and current distributions at the duct ends.

A very important and potentially promising aspect of MHD propulsion is the possible reduction of the sub-merged vehicle's acoustic noise signature. Unlike a propeller, the MHD duct has no mechanical moving parts. Thismay make the MHID propulsor intrinsically quieter (acoustically) than a rotating propulsor. Also, the replacement ofthe propeller by a MHD thruster eliminates the requirement for a high pressure rotating shaft seal. Therefore, inspite of the model's lower predicted propulsive efficiencies, the acoustic aspects of MBD propulsive should be in-vestigated in detail, and the possibility of designing and constructing a practical, acoustically quiet submerged ve-hicle, even at the possible expense of propulsive efficiency, should be explored.

Accession For

NTIS GRA&IDTIC TAB ElUnamnounced E-Justification

ByDistribution/ ,..

Availability Codes

!Avail rnd/orDist Spec1.al

CONTENTS

Page

A bstract ............................................................ I

Administrative Information ........................................... 2

Introduction ........................................................ 2

H istory ........................................................... 2

M -iD Propulsion Concept ............................................ 2

M HD Project Goals ................................................. 2

Goals of this Technical Report ......................................... 3

Traditional Propeller Theory and ApproximateExtension to M IlD Ducts .............................................. 4

Background ........................................................ 4

Approximate Relationship of Propeller to MHD Duct ....................... 10Mathematical Model of Magnetohydrodynamic (MHD) ShipPropulsion with Propulsive Efficiency ................................... 13

Fundamental Equations of MMD Propulsion .............................. 13Conservation of Linear Momentum ................................... 14Principle of Action and Beaction ..................................... 16Rate of Change of Kinetic Energy (Energy Theorem) ..................... 17

Electromagnetic Energy Input ....................................... 19Efficiencies for MHD Propulsion ..................................... 20

Numerical Results ................................................. 22

Discussion and Conclusions ............................................ 51

Acknowledgm ents .................................................... 53

Appendix A. Three-Dimensional Effects ................................. 54

References .......................................................... 57

FIGURES

1. Stream tube for propeller system...................................... 5

2. Control volume in space ............................................. 6

3. Control volume for propeller mathematical model ......................... 74. Simple M MIID rectangular duct ........................................ 10

5. M IID rectangular duct ............................................... 13

6. Propulsive magnetohydrodynamic rectangular duct configuration ............ 23

7. Power consumption versus submerged vehicle speed ...................... 24

DTRC--90/009

8. Propulsive efficiency of square MID duct versus vehicle speed

in m eters per second . ............................................... 26

9. Thrust required to propel submerged vehicle at constant speed V. ............ 28

10. Propulsive efficiency of square MH) duct versus magneticinduction in tesla ................................................... 29

11. Propulsive efficiency versus electrical conductivity of seawater .............. 30

12. Propulsive efficiency of square M-D duct versus duct cross-sectional area..... 31

13. Propulsive efficiency of square duct versus duct length ..................... 32

14. Propulsive efficiency T11 versus duct height (duct width w = 1 m) ............ 34

15. Propulsive efficiency 71r versus duct height (duct width w = 2 m) ............ 35

16. Propulsive efficiency Tj versus duct height (duct width w = 4 m) ............ 16

17. Propulsive efficiency 1 I versus duct height (duct width w = 8 m) ............ 37

18. Propulsive efficiency 71, versus duct height (duct width w = 16 m) ........... 38

19. Streamlines due to uniform body force JB distributed overrectangular volume with no duct walls ................................. 55

TABLES

1. Baseline numerical results for square rectangular duct ..................... 25

2. Propulsive efficiency TJhX102 versus submerged vehicle speed and lengthof square duct at 2 tesla .......... .................................. 39

3. Propulsive efficiency 11 Ix 102 versus submerged vehicle speed and lengthof square duct at 6 tesla ............................................. 40

4. Propulsive efficiency T1IX1O 2 versus submerged vehicle speed and lengthof square duct at 10 tesla ............................................ 41

5. Propulsive efficiency 1xI10 2 versus submerged vehicle speed and lengthof square duct at 6 resla ............................................. 42

6. Propulsive efficiency I1IX102 versus submerged vehicle speed and lengthof square duct at 10 tesla ............................................ 43

7. Propulsive efficiency 11IX10 2 versus electrical conductivity andsubmerged vehicle speed ............................................ 44

8. Propulsive efficiency JI1Xl02 versus electrical conductivity andmagnetic induction of square duct ..................................... 45

9. Propulsive efficiency 111xl 02 versus hydrodynamic resistance ofsubmerged vehicle (hydrodynamic resistance R = 1000 V2 newtons) .......... 46

10. Propulsive efficiency TjI x102 versus hydrodynamic resistance ofsubmerged vehicle (hydrodynamic resistance R = 2000 V2 newtons) .......... 47

iv DTRC-90/009

11. Propulsive efficiency 11ixl 10 2 versus hydrodynamic resistance

of submerged vehicle (hydrodynamic resistance R = 3000 V2 newtons) ........ 48

12. Propulsive efficiency 11 I x 102 versus hydrodynamic resistance

of submerged vehicle (hydrodynamic resistance R = 4000 V2 newtons) ........ 49

13. Propulsive efficiency 11 1XI 02 versus hydrodynamic resistance

of submerged vehicle (hydrodynamic resistance R = 5000 V2 newtons) ........ 50

DTRC--90/009 v

ABSTRACT

Magnetohydrodynamic (MHD) ship propulsion is the process of propelling a vehic-ular structure by a seawater electromagnetic pump. This propulsion system can beapplied to a surface ship or a submerged vehicle; however, in this work only submergedvehicles at depths where wave effects can be neglected were considered. Although anumber of different arrangements for a MHD propulsion system are possible, the generalcharacteristics of such systems are most easily determined by a simple, ideal MHD rect-angular duct of constant cross-sectional area. A mathematical model was developed atthe David Taylor Research Center (DTRC) for calculating the propulsive efficiencies ofsuch a rectangular duct propelling a submerged vehicle. Numerical propulsive efficien-cies are presented in terms of many different parameters. Assumptions were generallymade in the model that tend to maximize the propulsive efficiency of the MHD sys-tem. Thus, the propulsive efficiencies calculated from the model overestimate theefficiencies of the corresponding real MHD propulsion system. These numerical resultscan be used for engineering estimates of the propulsive efficiencies of real MHD propul-sion systems.

A comparison of the ideal model MHD rectangular duct propulsive efficiencies withthe corresponding efficiency of a conventional submerged vehicle propeller (order of0.85) generally shows the MHD efficiencies to be less under most conditions. The caseswhere MHD propulsive efficiencies are in ranges that approach that of a propeller corre-spond to MHD duct volumes greater then 300 m3 , with small surface-area-to-volumeratios, or the magnetic inductions are in the order of20 tesla or more. Magnetic induc-tions of about 6 to 10 tesla in large volumes of space, such as MHD ducts, are probablyabout the highest field strengths possible with the use of present-day superconductingmagnet construction technology; to produce even these fields will require significant en-gineering effort in magnet construction. Therefore an important area of concern forMHD propulsion is the construction of practical superconducting magnets and supportstructures of reasonable mass.

According to the calculations using the mathematical model, a weakness of MHDpropulsion, for a viable submerged vehicle propulsion system, is the generally low pro-pulsive efficiencies of the system at duct magnetic inductions of 1O tesla or less, whencompared with conventional propellers. Furthermore, the general trend is for propulsiveefficicncies to decrease as vehicle speed increases. Future work will address the degrad-ing effects on the rectangular MHD channel, caused by fringing magnetic induction andcurrent distributions at the duct ends.

A very important and potentially promising aspect of MHD propulsion is the possi-ble reduction of the submerged vehicle's acoustic noise signature. Unlike a propeller, theMHD duct has no mechanical moving parts. This may make the MHD propulsor intrinsi-cally quieter (acoustically) than a rotating propulsor. Also, the replacement of thepropeller by a MHD thruster eliminates the requirement for a high pressure rotating shaftseal. Therefore, in spite of the model's lower predicted propulsive efficiencies, the acous-tic aspects of MHD propulsive should be investigated in detail, and the possibility ofdesigning and constructing a practical, acoustically quiet submerged vehicle, even at thepossible expense of propulsive efficiency, should be explored.

DTRC-90/009

ADMINISTRATIVE INFORMATION

This work was a cooperative effort between DTRC and the Department of Mechani-cal and Industrial Engineering at the University of Illinois, Urbana, 111. 61801. The workwas supported by the DTRC Independent Research Program, Director of Naval Research(OCNRI0), and administered by the Research Director (DTRC0113) under Program Ele-ment 62936N, Task Area ZF66412001, Work Unit 1-2712-131, project title "TheFundamental Conceptual Design and Analysis of Magnetohydrodynamic Propulsors." Inaddition, this work was partially supported by the DTRC Block Program sponsored by

ONT (Gene Remmers), Work Unit 1-2710-197, Project Title "Electric Propulsion," Sub-Task "MHD Propulsion Assessment," Task Area RB23P1l, Element 62323N.

INTRODUCTION

HISTORY

Because nuclear reactors use circulating liquid metals as a heat transfer medium, asubstantial effort in the past has been devoted to developing liquid metal electromagneticpumps.'- 3 These pumps, besides being quite reliable, are unusual in their design, sincethey have essentially no moving parts but the electrical conducting fluid beingpumped. The high electrical conductivity of the liquid metals permits pump power effi-ciencies of nearly 20% under certain conditions.

The direct current, Faraday-type pump is the most elementary electromagneticpump. The liquid metal is pumped through a duct of rectangular cross section. Two op-posite walls of the duct serve as electrodes which, when maintained at an electricpotential, cause a high current to flow through the fluid. A strong magnetic induction ispassed through the other two walls. Thus, the current direction and the field are mutuallyperpendicular to each other and perpendicular to the fluid flow in the pump caused by theLorentz IV ..y force.

MHD PROPULSION CONCEPT

In the mid 1960's the electromagnetic pump concept was converted into an overallmarine propulsion concept by Stewart Way.4 7 The basic idea was to pass a current

through seawater in the presence of a magnetic induction, in order to accelerate the sea-water and to provide a reaction force on the marine vehicle holding the magnet andelectrodes. To our knowledge this magnetohydrodynamic (MHD) propulsion concept hasnot changed significantly since then. The concept can be applied to surface ships andsubmerged marine vehicles. A number of references are available in the literature onMHD propulsion. 8-17

MlRD PROJECT GOALS

The objectives of the M-D marine propulsion research and development projectinitiated at the David Taylor Research Center (DTRC) were to examine fundamental prin-ciples underlying these MHD propulsion systems and then to produce critical scientificand engineering data and conceptual design information. This program is being coordi-nated with the Defense Advanced Research Projects Agency (DARPA) MHD PropulsionProgram, a cooperative effort with the Naval laboratories, Argonne National Laboratory,

2 DTRC-90/009

industry, and academia. All the parameters that characterize NIHI) propulsion systemsare to be extensively studied. This approach will establish a baseline group of exper-imental data and theoretical numerical results to assess the feasibility of marine NIHDpropulsion for Naval surface ships or submerged vehicles.

It is well known that, in general, MWD propulsion systems have significantly lowerpropulsive efficiencies than conventional propellers under basically the same propulsionconditions. However, it is generally assumed that these systems will generate much lcssacoustic noise than propellers, thus making MHD systems more difficult to detect acous-tically than conventional propulsion systems. Along with the elimination of rotatingseals, gears and shafting, this noise reduction is the major selling point for MvfID propul-sion systems for Naval propulsion application. Therefore, the principal starting point forthis project in fiscal year (FY) 1990 will be to study theoretically the physical principlesinvolved in the generation of acoustic noise in MI-D propulsors on submerged ve-hicles. Little reliable scientific information exists on acoustic signature of these MHDsystems.

GOALS OF THIS TECHNICAL REPORT

The MI-D propulsion program at DTRC was initiated late in FY89 with a smalltheoretical effort to investigate MHD propulsion efficiencies, and to develop plans forpossible MOHD experiments to study acoustic noise in later work. This technical reportdocuments the initial theoretical effort on the MBD program.

A literature search was performed on marine MHD propulsions-•17 Since the nu-merical results in the literature were often too incomplete and sketchy for reliabledecision-making, a fundamental set of equations which describe the propulsive efficien-cies of an ideal rectangular duct propulsion system was developed into a computerprogram. Assumptions in the mathematical model were generally made which tend tomaximize the propulsive efficiencies of the MNHD system. These numerical results can beused for engineering estimates of the propulsive efficiencies of real MHD propulsion sys-tems. The computer program was exercised for a wide variety of cases; the work isdescribed in this report.

This report begins with a background description of fundamental propellertheory. It is shown that the propeller propulsion theory is approximately analogous to aMT4D propulsive rectangular duet. Next, the propulsion theory of an MHD rectangularduct was derived from the basic principles of fluid dynamics. From this work, d n-dimensional control volume mathematical model for an ideal rectangular MHD propul-sive duct was developed, 8', 19 and the numerical results for propulsive efficiency ofdifferent system conditions are presented here in the form of curves and tables.

The MHD propulsion mathematical model will be modified in FY 90 to includechannel end effects. The theory of Hughes and McNab2° was developed for MIHID liquidmetal pumps and will be applied to MHD seawater propulsion channels. It is expectedthat these end effects will degrade the propulsive efficiencies calculated for the idealM]HD propulsion model without end effects. End effects are known to have a degradingeffect on MHD liquid metal pumps. However, channel duct design in MHD propulsionsystems can be used to lessen these end effects.

DTRC-90/009 3

TRADITIONAL PROPELLER THEORY AND APPROXIMATEEXTENSION TO MHD DUCTS

BACKGROUND

We will begin the discussion with the traditional propeller theory, since this topicclosely relates to the understanding of the magnetohydrodynamic (MiHD) propulsionproblem. The ship being considered is driven at a constant velocity V with a single pro-peller. The frame of reference is chosen to be moving with the ship. Thus, the fluidunaffected by the propeller has velocity V and pressure P0 . The propeller'- is simplymodelled as a porous disk with area AD through which water flows. The water flowingthrough the disk experiences a pressure increase, AP, as it passes through the disk repre-senting the propeller. We will designate the pressure on the upstream side of the disk asP,, which is less than the ambient pressure P0 . Because the relationship P,, < P0 holds,the water is accelerated from V to VD as it approaches the propeller. Because of this fluidacceleration, the water passing through the propeller comes from a far upstream streamtube with area, A, which is larger than AD. As the water passes through the disk, its ve-locity remains VD, by conservation of mass, but its pressure increases to (PM + AP), whichis greater than the ambient pressure P,. The high pressure water that leaves the propellermust merge with the surrounding water at PF,, so the water accelerates to the exit velocityV,. which is greater than the velocity VD. There is a corresponding reduction in exit areato A, and a return of the pressure to P,. In the moving reference frame far downstream ofthe propeller, we have a jet with elevated velocity V, and cross-sectional area A•, sur-rounded by water with velocity V and pressure at P,. Figure 1 shows a diagramrepresenting this traditional propeller system.

There are four stations on the diagram. Far upstream at station 1, the ambient pres-sure is P,, the area of the free stream tube is A, and the velocity of the fluid is V. Atstation 2, immediately upstream of the propeller, the pressure P, is less than the ambientpressure Po, P, < P,, the area of the propeller is AD and the velocity into the propeller isVD. At station 3, immediately downstream of the propeller, the pressure jumps from P.immediately upstream to P, +AP immediately downstream. P, +AP is greater than theambient pressure (P. +AP) > P,,, the area of the propeller is AD, and the velocity comingout of the propeller is VD. Far downstream the ambient pressure is P ; the exit area is A•,and the exit velocity is V,.

From conservation of mass the following relationship holds: VA = VDAD = V.A,.The law of conservation of energy can be used to represent the conservation of mechani-cal energy per unit of mass. This water flowing from a to b along a streamline can berepresented by:

l'a Va2 Pb V-S+ 2 + gza = -+- gzb + 1 + Wf+ = Constant along streamline (1)

Q 2 LD 2

where:z = vertical coordinate, with positive direction upward,V2

- = kinetic energy per unit mass,2

4 DTRC--90/009

AXISYMMETRIC FREE STREAM TUBEjl• DISK REPRESENTING PROPELLER

P. (Pu + AP)PO

vD.

0Fig. 1. Stream tube for propeller system.

W, = work done by fluid per unit mass,W!r = energy lost to friction flowing from a to b, per unit mass,gz = potential energy in elevation per unit mass, andP- = potential energy in pressure per unit mass.Q

Like all potential energies, both of these have arbitrary datum, i.e., we can measure allelevations from any horizontal plane and all pressures from any arbitrary constant pres-sure.

Some authors refer to Eq. 1 as a modified Bernoulli equation. The following as-sumptions are made in order to simplify Eq. 1:

"* Flows are essentially horizontal, and we can subtract out hydrostatic pres-

sure variation, so we ignore the g: terms.

"• It is assumed that the propeller is frictionless (not for MI-D duct) so that

Wf = 0 for the propeller case.

Referring to Figure 1 for the diagram for the propeller flow, Eq. 1 from stations 1 to

2 is:

L,+ = P+• VD2 (2)

e 2 p 2

DTRC-90/009 5

Note that only the propeller does work on fluid (negative W,). Therefore, W, = 0 fromstations 1 to 2, and from stations 3 to 4:

Pu + AP VD2 Po Ve2

S= + •(3)e9 2 Q 2

Equation 1 from stations 2 to 3 is:

WS AP(4Wvs = -~ . (4)

Q

We will define T here to represent the forward thrust on the propeller disk which isequal to the afterward negative thrust of propeller on water. The thrust from the propelleris:

T = (AP) AD (5)

The thrust - T on the submerged vehicle is caused by the reaction force and is equal inmagnitude and opposite in direction to the force on the fluid T.

The afterward thrust on the water from the propeller can be calculated from th-: con-servation of linear momentum for steady, incompressible flow through a control volume(that is, a fixed volume in space with control surface). (See Fig. 2.)

T= XF f fJV VdA, 6

contrr 'surface

dA

n= OUTWARD UNITNORMAL TOCONTROL SURFACE

Vn V. n

Fig. 2. Control volume in space.

6 DTRC--901009

where:V, = outward normal component of velocity at a point on control surface,

dA = differential element of control surface,

dQ = V,, dA = volumetric flow across dA per unit time, and

F sum of all external forces acting on water which is instantaneously inside

control surface, including pressure forces from water outside control surface.

Figure 3 shows the control volume for the propeller with the force T acting on thewater.

For the control surface for the propeller we assume that P = P0 over entire sur-face. Therefore, for this problem pressures give zero net force. This is the traditionalassumption, which is not strictly true. With this assumption the only external force forthe system is the propeller thrust T. At station 1, V,, = - V (fluid inflow) and V. = + V,

and the entrance area equals A. At station 4, V, = + V, (outward flow of fluid),

Vy = + Ve, and exit area equals A,. At the stream tube surface, V, = 0. Substituting

these results into Eq. 6 results in the well known expression for the propeller thrust:

T = e(V)(-V)A + O(Ve)(Ve)Ae , (7)

STREAM TUBE

SENTRANCE EI--•._

""-P0

0

Fig. 3. Control volume for propeller mathematical model.

DTRC-90/009 7

But from the incompressibility of the water the following relationships holds:

Q = VA = VDAD =. VeAe , (8)

where Q = volumetric flow rate (m3/s) . Thus Eq. 7 can be conveniently expressed as:

T = eQ(Ve - V) , (9)

where Q = mass flow rate (mass per unit time), and oQ(V, - V) = change in linear mo-

mentum per unit time.

First, we consider a traditional manipulation of Eq. 1-9. Then we separate indepen-dent from dependent variables in these equations. Equating Eqs. 5 and 9, we obtain:

(AP)AD =LOQ(Ve - V , (10)

Since Q = VDAD, this gives

A = vD(ve - V) (

Subtracting Eq. 2 from Eq. 3:

(AlP) = 1 (V 2 V2) 1(ve+ V)VeV) . (12)

Lo 2 2

Equating Eqs. II and 12, we obtain

1VVD = 2 We + V) . (13)

This is a full solution of the energy equation and the momentum equation.

Now consider dependent and independent variables involved in the problem. Takeship velocity V and propeller disk area as the only two independent variablesAD = (x/4)D2 . The force T, required to move the ship at constant velocity V, is uniquelydetermined by V and the hull shape. Consider a streamlined submerged vessel with onlyskin friction drag. Then the following equation holds:

Skin friction drag = T = 10V2CSw , (14)

where:Sw = wetted surface area of vessel;Cf = skin friction coefficient, which is a function of Reynold's number; and

8 DTRC-90/009

ReL = L , where L, equals vessel length, L equals density of water, and U equals vis-

cosity of water.

The Reynold's number can be used to determine the skin friction coefficient Cf. If

1.33the flow is laminar (ReL <5 x 105) then Cf can be represented by C1 = 1. At(ReL)'/ 2 At

the transition point where the Reynold's number has the range 5 x 105 < ReL < 107, f

can be represented by Cf = 0.074 -1740. In the turbulent region (ReL > 107), the(ReL)/ 5 ReL

frictional coefficient Cf can be represented by the Prandtl-Schlichting equation,0.455 1700

(log 10 ReL) 2 -s8 ReL

Therefore, once we specify V, the required T is known from the skin friction drag,Eq. 14. Since AD must be specified, the following relationship holds from Eq. 5:

AP= T (15.1)AD

AP = iev ((15.2)

Using Eqs. 12 and 15.2, we obtain:

Ve = 1 + Cf Sw\1/2 (16)AD J

From Eq. 13, VD is easily obtained:

VD = [1+ AD1 , C (17)

Using the relationship VA = VDAD = VA, from the incompressibility of the fluid, we ob-tain:

= A + (18.1)2-AD9

DTRC-90/009 9

Ie = AD .+ 1 f -/ (18.2)2 1 ADJ ) J

Everything is now determined from V and AD plus CQ and Sw.

APPROXIMATE RELATIONSHIP OF PROPELLER TO MHD DUCT

We will consider a simple MI-D rectangular duct like that of Fig.4. Converging ordiverging nozzles attached to a simple duct in general tend to degrade the propulsive effi-ciency of the duct due to surface friction effects. Therefore, we will discuss a simplerectangular duct in this section, as opposed to a more complicated configuration, becauseat this point we are interested in the upper limits of propulsive efficiency.

The duct problem is approximately analogous to the traditional propeller problem

where the force T is produced by the 7 X B Lorentz body force rather than the propellerforce. For a detailed discussion of this analogy, the reader should consult Appendix A.

z

x

r PoIAI AD AD Ae

I0

SPU (PA + P)

. PO FREE STREAMTUBE

FREE STREAMTUBE

Fig. 4. Simple MHD rectangular duct.

10 DTRC-90/009

Equations 16, 17, 18.1, and 18.2 give the expressions for V,, VD,A and A,. The duct di-mensions are length L = (Ay), distance between electrodes w = (Ax), distance between

insulators h = (Az), and cross sectional area AD = wh. The Cartesian coordinate system

(x,yz) is shown in Fig. 4, where . are the unit vectors. Throughout the duct, we ig-nore end effects. It is well known that duct end effects degrade propulsiveefficiency. The velocity of fluid flow in the duct is a constant V = VD), because the duct

has a constant cross-sectional area. The electric current density is designated as j = A,

the electric field as f = E.! = (Ol/w)V, and the voltage difference between electrodes as

b. The magnetic field is represented by B -^ B, the Lorentz body force as

jxB = JBY', and the "back" electromotive force is VxB = - VDB•.

The conservation of energy equation from the entrance to the exit of the duct isVentrance = Veit, 7enrane = 7.Xit

AP =-Ws-W , (19)

Q

where:

= W,-- = work done on a unit mass by electromagnetic body force, 7xB, asQ

mass travels from entrance to exit, and

= (i) (z_ ) frictional losses per unit mass.

where:

Dh = hydraulic diameter = 4AD =2whPer (w + h)'

Per= perimeter = 2(w + h), and

f = friction factor, which is a function of ReD, - and M.Dh

ReD = QV.A = Reynold's number based on mean duct velocity and hydraulic

diameter, andc= characteristic wall roughness size, which is considered a property of the wall

material. For example, for cast iron the value of e = 0.00085 ft, while for wroughtiron, its value is 0.00015 ft.

e

= dimensionless roughness size.DD

DTRC-90/009 1

B/h(V)

M = Bh 12= Hartmann number based on hydraulic diameter, and

a = electrical conductivity of fluid.

Combining Eqs. 15 and 19, we obtain the electric current density required to over-come the frictional resistance in the duct and provide a pressure rise, AP, needed to drivethe ship at a steady-state condition.

2B ( ) +(20)

skin friction drag of hull + internal friction of duct flow

The only role of ship length is in the relationship of ReL to Cf and in S&. Now wemust determine what voltage and power is needed to produce this J.

From the x-component of Ohm's law 22:

J=U( .- VDB) (21.1)

or O Wj+VB(21.2)

Equation 21 gives the voltage difference between plates required for this thrust. The pro-pulsive efficiency j7 is represented by the relationship:

TVTV= (22)

T is given by Eq. 5 and 4 is given by Eq. 21.2, in terms of J and VD. The total cur-rent across the rectangular channel is I = JhL.

12 DTRC-90/009

MATHEMATICAL MODEL OF MAGNETOHYDRODYNAMIC (MHD)SHIP PROPULSION WITH PROPULSIVE EFFICIENCY

FUNDAMENTAL EQUATIONS OF MMD PROPULSION

This section presents the development of the fundamental equations for an MHDpropulsive rectangular duct on a submerged vehicle. The duct configuration under con-sideration is presented in Fig. 5. The fluid flow into, through, and out of the duct isrepresented by a stream tube of total surface area I = 11+ 12 + 13 + A + A, and total

volume V '= Vj' + V2' + V3'. The duct together with the stream tube is surrounded by an

ambient pressure P,. The duct is of uniform cross-sectional area AD. The entrance areaand exit area of the stream tube are A and A, respectively. The pressure at the entrance ofthe stream tube is PF, at the duct entrance P,, at the duct exit P, + AP, and at the stream

tube exit P,. The following inequalities hold for the pressures: P0 > P,, P, < P, + AP ,

and P, + AP > P,. The velocity of the fluid entering the stream tube is V, entering theduct VD, leaving the duct VD, and leaving the stream tube V,. The unit normal to thestream tube 9 is outward from the surface.

The Cartesian coordinate system (x, y, z) has the origin fixed in the center of the

duct and moves with the duct at constant velocity V. The observer is theretore considered

to be travelling with the duct. The constant, homogeneous magnetic induction B is in thenegative z-direction. The constant, homogeneous current density vector lines are in thepositive x-direction. The fluid flows in the positive y-direction.

~~7-4

12

n2

oAD V2 A, V3' vP"

Pu (P= +,AP) -2 Ve

L PO PO FREE STREAM"nj TUBE

FREE STREAMTUBE

- A + A9 +*-+2 + +Z

V" - VIP + V2 ' + V3V

Fig. 5. MHD rectangular duct.

DTRC-90/009 13

Using the above rectangular duct configuration, we shall develop the general funda-mental equations for deriving the propulsive efficiency of the rectangular MHD duct in asteady state condition. The following fundamental principles from theoretical physics forfluid flow will be used for the derivation.

"* Conversation of linear momentum for a stationary control volume

"* Rate of change of kinetic energy for a stationary control volume (EnergyTheorem)

"* The principle of action and reaction

"* Ohm's law for electrical conductors

This derivation implies the conservation of mass, linear momentum, energy, and angularmomentum for the system.

Also, the approximations to these equations are developed for the mathematicalmodel that was used to produce the numerical computer results presented in the report.This development follows well-known concepts of fluid mechanics for fluid flow througha stationary control volume and is discussed in detail to clarify the concepts involved inan MHD duct.

Conservation of Linear Momentum

Newton's Second Law states that the sum of all external forces acting on an assem-blage of masses equals the rate of change of the total linear momentum of theassemblage. Our assemblage is the mass of fluid instantaneously inside the control vol-ume V' in Fig. 5. Since fluid is continuously entering and leaving our fixed controlvolume, there are two different assemblages of masses at two different times. A change ofthe linear momentum of the assemblage instantaneously inside the control volume ap-pears in two forms. First, the velocity at points inside the control volume may beincreasing or decreasing with time, so that the assemblage of masses inside the controlvolume at time (t + dt) has more or less linear momentum than the assemblage at t. Thesetwo assemblages differ only by the control surface, I = A + A, + I 1+12+ 13, duringthe short interval from t to (t + dr). This first change of linear momentum is given by

a o VjdV', (23.1)a (

where Vi are the components of the velocity V, and dV' is a differential volumeelement. The second realization of a linear momentum change is that the fluid leaving thecontrol volume may have more or less linear momentum that the fluid entering it. Thissecond change of linear momentum is given by

f Q ViVknk dS, (23.2)

14 DTRC-90/009

where nk are the components of the outward unit normal 9" at each differential element dSof the control surface S. The volume of fluid crossing the element dS of the control sur-face per unit time is Vk nk dS, which is positive or negative for fluid leaving or entering,respectively, while the linear momentum of this fluid per unit volume is OV. Therefore,Newton's Second Law for a Control Volume, which is an Open System, is

0v+feoViVknkdFj = t Lo, VdV ' + LdS, (24)

at fVf

where 7 Fj is the sum of all external forces acting on the mass instantaneously inside

the control volume. The sum of forces is given by

I Fj =Jf bi dV'+JfmtjdS (25.1)

where bi is the external body force per unit volume at each point inside the control vol-ume, and t; is the external traction force per unit area at each point on the control surface.For a viscous fluid

t=- Pi + oik' nk , (25.2)

where P is the pressure and aik' is the viscous shear stress. 3 If P represents the deviationof the actual pressure from the hydrostatic pressure, than we can ignore the gravitationalbody force, so that our only body force is the Lorentz force per unit volume,

bi = EQi k JJBk , (25.3)

where eijk is the permutation tensor, Ji is the electric current density, and Bk is the mag-

netic induction. We consider a steady flow, so that the velocity at each point inside thecontrol volume is independent of time, and Newton's Second Law is

J@ViVkflkdS= I k dV'+L aik'nf dS- f Pni dS (26)

The sum of the external forces on the right-hand side act on the seawater which isinside the control surface S. The first term is the total Lorentz body force on the fluid inthe rectangular channel plus end effects. The second term represents the frictional forcesof the fluid along the surfaces I I + 2I + 3. The frictional force along the inside walls

of the duct 12 is more important than the frictional force along 7 1 + 3 3. The third

term includes the pressure forces from the fluid external to the control surface. The fol-lowing integrals can be evaluated in Eq. 26.

DTRC.--0/009 is

L o VjVj- nk dS = LOQ(Ve - V) (2.1J n_(27.1)

where Q = VA = VMD = VA,..

f PfidS 0 (27.2)

f ai'nk dS f= ik'nk dS (27.3)

Substituting these integrals in Eq. 26 results in the following equation:

LOQ(Ve V1X= f(< B9)dV'+ f Cjknk dS (28.1)

For the interior surface of the duct 2,,

aik'nk =-TW ' = -- f VD2 , (28.2)

where %,, is the viscous wall shear stress andf is the Moody friction factor. If we ignore

all electromagnetic end effects, the x-component of Eq. 28.1 is

oQ(Ve - V) = JBV2 ' - 1 fe VD2L (Per)8- (28.3)

T = Lorentz force - Internal Friction force

V2' = LWh

where L, W, h, V2', and (Per) = 2(w+h) are the length, width, height, interior volume,and cross sectional perimeter, respectively. Equation 28.3 is also obtained by substitutingEq. 19 into Eq. 10. The term T is designated as the effective thrust on the fluid down thechannel. Equation 28.3 is the fundamental momentum equation for the operation of theMHD duct.

Principle of Action and Reaction

The reaction force-T to the effective Lorentz body force T propels the submergedvehicle at a steady state velocity V in the fluid. The reaction force-T balances the hydro-dynamic resistance R of the vehicle. The reaction force-T is caused by the magnetic

16 DTRC-90/009

induction B from the current density in the channel interacting with the current in thechannel magnetic coil electrical system. Therefore, the useful power in propelling the

submerged vehicle by the MI-ID duct is expressed as

Useful power in propelling vehicle = TV (29)

The reaction to the force JBV2 ' acts directly on the MHD channel magnets.

For a more detailed discussion of the reaction force see Appendix A.

Rate of Change of Kinetic Energy (Energy Theorem)

The dot product between Newton's Second Law for a single mass and the mass'svelocity vector gives an equality between the rate of change of the mass's kinetic energy

and the work done on it per unit time. When we sum the Newton's Second Law for anassemblage of masses, all internal forces between the masses sum to zero, since the forcesbetween two masses are equal and opposite. When we sum the kinetic energy equationfor an assemblage of masses, the work done by the internal forces does not sum to zero.In a continuum model, the internal forces are represented by the pressure and viscousstresses. The pressure force is conservative for a liquid, so it can only transfer kinetic en-ergy between masses with no net change in the total kinetic energy of the assemblage.The viscous stresses are not conservative and produce an irreversible conversion of kinet-ic energy to heat. The rate of energy conversion per unit volume is called the viscous

dissipation 0) and is proportional to the product of the viscous stress between neighbor-

ing masses and their difference in velocity, as reflected in the velocity gradient,

=Oik'E- = viscous dissipation. (30)aXk

The rate of change of the kinetic energy of the assemblage of masses instantaneously in-side the control volume is equal to the rate of increase of kinetic energy inside the controlvolume, plus the difference between the kinetic energies in the fluid entering and leavingthe control volume. Thus, the rate of change of kinetic energy for a control volume is

at (1/2)LV IdV'+ J•(1/ 2 )QV2Vk nk dS

=J (by1i) dV'+jtiVi dS- akF_~ (31)

where again bi and ti represent the external body force and surface traction.

Alternatively, we can begin with the First Law of Thermodynamics, which statesthat the heat added to an assemblage of masses, plus the work done by the external forces,equals the change in total energy, which is the sum of the kinetic and internal energies.The internal energy represents the internal energy of the molecular states. We can split theFirst Law of Thermodynamics into two separate equations, with the heat and work in theinternal and kinetic energy equations, respectively. In making this split, we must add

DTRC-90/009 17

equal and opposite source terms to each equation to represent the energy transfer betweeninternal and kinetic energies. In a gas, internal energy can be converted to kinetic energythrough volumetric expansion, but in a liquid, the only energy transfer is the viscous con-version of kinetic energy to internal energy. After the split of the First Law, the kineticenergy equation is Eq. 31.

Introducing Expressions 25.2 and 25.3, where again P represents the deviation fromhydrostatic pressure, Eq. 31 becomes

:jr f (1/2)QV2dV'+ [(1/2)eV2+P] Vknk dS

f V, (Vi'EijkJjBk) dV'+ f c~ik'Va nk dS f V uik'-' dV' . (32)"j v 'I aX

Our flow is steady, so that the first term in Eq. 32 is always zero. We can applyEq. 32 separately to the smaller control volume V1' or V3' in Fig. 5. We are neglecting

viscous stresses in these volumes, and j x B = 0 here since we neglect electromagneticend effvcts. Therefore, Eq. 32 becomes

Q[(1/2)oVD2 + PM - (1/2)9V2 - Poj = 0 (33.1)

Q[(1/2)OV,2 + P, - (I/2)oVD2 - P"-_AP] = 0 (33.2)

for volumes V1' and V3', respectively. The Eqs. 33.1 and 33.2 are identical to the con-

servation of energy equations ( Eqs. 2 and 3 ) after dividing through by Q. When weapply Eq. 32 to the other smaller control volume 112' in Fig. 5, we must determine thetwo viscous terms on the right-hand side. We use a one-dimensional flow model with

VD= ViD throughout V2'. Therefore, -ýV- = 0, and the viscous work is the work doneaxk

by the fluid moving at velocity Vox_ against the viscous wall shear stress, - r,, X at 12,

which contributes

- r,,.VDL(Per) --fOVD 3L(Per) (34)8

to the right side of Eq. 32. In reality, the fluid at 12 must be at rest, so that

Oik'VPlk = 0 at I ., and it is the final viscous dissipation term which represents the

work done against viscous forces. Actually, this distinction is moot because the Moodyfriction factor is def'med by equating the Expressic-n 34 tn the power input needed tomaintain a constant kinetic energy in spite of frictional losses. For the control volume for

V2', Eq. 32 becomes

18 DTRC-90/009

1Q(AP) = VDJBV2' - fQVD3L(Per) (35)

If we divide this equation by oQ = pVDAD, it becomes Eq. 19.

Electromagnetic Energy Input

In our frame of reference moving with the duct, there is an electric filed E9, a mag-

netic induction B, and an electric current density j in the fluid with electricalconductivity o inside our control volume. The electromagnetic energy input to the fluidper unit time is2'-

f, d(J (36)

If we introduce Ohm's law without Hall effects,

a (E+,P × ) (37)

in order to eliminate f, then Expression 36 becomes

J (J2/cT)dV'+ J v '(JX )dV' (38)

The second term in Eq. 38 is the work done on the fluid by the Lorentz body force (Eq.25.3), and this term appears in our kinetic energy equation (Eq. 32). The first term in Eq.38 represents a heat input because an electric current flowing in a conductor producesheat. This heat input is called the Joulean heating and appears as a term in the conserva-tion of internal energy equation, along with the viscous dissipation (Eq. 30) as twovolumetric heat sources. Since we cannot get any useful work out of the internal energyof a liquid, the Joulean heating represents an energy loss for our system. Equation 36 rep-resents the total electrical power input to the MHD system and is equal to

JEV2' = 10 = VDJBV2' +1 2Re , (39)

where I = JLh = the total current between the electrodes, 4- = Ew = total voltage differ-

ence between electrodes, h = height between insulating walls, w = width betweenelectrodes, and R, = (w/aLh) is the electrical resistance of the duct.

DTRC-90/009 19

Efficiencies For MHD Propulsion

The basic propulsion equations for the MIHD rectangular duct were derived andpresented previously in the report. We assume that the following quantities are known:

V = velocity of the vessel;G = skin friction coefficient of the vessel;Sý,. = wetted surface area of the vessel;

L, w, h = length, width and height of MHD propulsion duct;B = strength of uniform magnetic induction inside duct.

Other quantities are given by the following equations:T = skin friction drag resistance of vessel which equals thrust developed by propulsion

unit, from Eq. 14;A.P = pressure rise through the duct, from Eq. 15;V, = exit velocity from control volume, from Eq. 16;VD = velocity inside duct, from Eq. 17;A = area at the entrance of the control volume, from Eq. 18.1;A, area at the exit of the control volume, from Eq. 18.2;J - uniform electric current density between electrodes inside duct, from Eq. 20;4) = total voltage difference between electrodes, from Eq. 21.2;t7 = propulsive efficiency, from Eq. 22.

For the purposes of interpretation, we can split the propulsive efficiency into threeseparate efficiencies,

TV1 7 =?)e1 7 rertd . (40.1)

The electrical efficiency

VD JBLwh 2 R 2Rere= = 1 - - (40.2)01 01

is the ratio of the work done by the Lorentz force on fluid per unit time to the total electri-cal power input to duct. Equation 39 has been substituted in order to get the secondexpression, which shows that (1 -in,) is the fraction of the electrical input power which islost to Joulean heating. The thrust efficiency

17 = _T V (1/8)foVD 3L(Per) (40.3)

VD JBLwh VDJBLwh

is the ratio of the energy gained by the fluid per unit time inside the duct to the work doneby the Lorentz force per unit time. Equation 28.3 has been substituted in order to obtainthe second expression, which shows that (1 - '7) is the fraction of the work done by theLorentz force which is lost to the internal friction of the duct. The hydraulic efficiency

20 DTRC-90/009

TV (I /2)QQ(Ve -V) 2 (404)TVD TVD

is the ratio of the power driving the vessel to the energy gained by the fluid inside theduct per unit time. Equations 9 and 13 were substituted to obtain the second expression.In a fixed frame of reference, the water entering our control volume has zero velocity,while that leaving has a velocity (V,-V). The kinetic energy in the stream of the duct rep-

resents a loss of energy per unit time, which is (1 12)eQ(V. - V) 2 .

Up to this point we have implicitly assumed that the MHD propulsion duct is insidethe vessel's hull, so that the only skin friction drag resistance is that of the hull. If Jhe ductis housed in an external pod which is connected by a strut to the hull, then the thrust de-livered to the hull is reduced by the skin friction drag on the pod and strut. Thus, Eq. 28.3must be replaced by

T = JBLwh - (1/8)foVD2L(Per) - (1 /2)LV 2C1 ISP , (41)

where T is the thrust delivered to the hull, which balances the skin friction drag of thehull, as given by Eq. 14, Cfp, is the skin friction coefficient for the pod and strut, and Sp isthe wetted surface area of the pod and strut. Then Eq. 20 is replaced by

j T +[L _VD2+fP SPV242

BLw. 2B I-[ Lwh (

As a conservative estimate of the pod's skin friction drag, we use

Cf = 0.075 0910 - 2 +0.0005 (43.1)

Sp = L(Per) , (43.2)

where Cip is given by a variation of the Schoenherr equation which applies for0.0026 < C1 P :5 0.0037. Of course, Sp must be much larger than L(Per) because the

pod must contain the large magnets needed to produce B. Therefore, Eq. 42 becomes

TVD2 + f4pV2) (44)BLwh 2BDh

DTRC-90/009 21

In addition, Eq. 17 is replaced by

VD= 1+ 1+ )+ (45)S2 ýAD V 2 Dh

Equations 21.2, 44, and 45 now give 0, and Eq. 22 gives the propulsive efficiency.

Rather than apply this analysis to specific submerged vessels, we will replace Eq. 14 by

T=KV2 , (46)

with K = 1000, 2000, 3000, 4000 or 5000 kg/m for some typical submerged vessels. If theM-D propulsion duct is in an external pod, then we use Eqs. 21.2, 43.1,44, 45, and 46 tocompute the propulsive efficiency which we denote by 172. If the duct is inside the ves-

sel's hull, we use the same equations with Cfp = 0 and denote the propulsive efficiency by'7'.

Numerical Results

The objective at this point of the work was to calculate the propulsive efficiencies

;71 and '72 of the one-dimensional control volume MHD system at different steady stateconditions. These numerical curves presented herein will determine the approximate up-per limits of the M-D propulsive vehicular system efficiencies, which can be used forengineering judgments in estimating the propulsive efficiencies under different conditionsof "real world" systems used on submerged vehicles. The efficiency is defined as the use-ful power to propel the vehicle divided by the total electrical power to the duct 10,

where I is the total current across the duct and 0 is the voltage drop across the duct. * For

the efficiency '7I the useful power to propel the duct is defined mathematically as theideal thrust in the duct multiplied by the vehicular speed V minus the frictional force in

the duct (1/8)[fQVD2L(Per)] multiplied by V. The efficiency 172 corresponds to an MHDduct in the form of a pod attached by struts to the vehicle hull. The propulsive efficiency172 is defined analogously to the propulsive efficiency 7 1. In this case, the useful powerto propel the submerged vehicle is the ideal thrust in the duct multiplied by V minus both

the frictional force in the duct due to fluid flow (1/8)[Mf'VD 2L(Per)] and the force d.:e to

the fluid flow outside the pod (1/2)[LV 2L(Per)Cp,], where both frictional terms are mul-

tiplied by V. From these definitions 171 > 172 under the same propulsion conditions.

The MHD propulsive mathematical model using equations from the previous sec-tion were programmed on the computer. All equations are in the SI system of units. TheMIHD rectangular duct system was considered to be propelling a submerged vehicle at asteady-state condition V. Transient conditions were not considered at this point of the

* Note that this number does not account for power generation or distribution efficiencies.

22 DTRC-90/009

work. End effects, bubble generation power losses, and the effects of bubbles in the ductwere also omitted. Computer calculations were performed using various input parametersfor the rectangular duct and various hydrodynamic resistances for the submerged vehicle.

To properly characterize the MHI propulsive system propelling a submerged ve-hicle of hydrodynamic resistance 3,000 V2 newtons, we present the base line numericalnumbers in Table 1. We consider a rectangular duct height h of 4 meters, width w of4 meters, and length L of 10 meters (see Fig. 6). A realistic magnetic induction B of 6 tes-la across the duct was chosen. The electrical conductivity a and density of seawater Qwere chosen to be 4 siemens/m and 1026 kg/M3, respectively. The remainder of the calcu-lated data with dimensional units is shown in Table 1. The Moody friction factorf (i.e.,f= (4Cfd) for inside the MHD duct corresponds approximately to that for smooth pipesfor all numerical results in this report.

Figure 7 presents the total power consumption of the vehicular system versus ve-hicle speed V. The propulsive efficiency 171 was used in this plot. The solid curve

-4z _+ .. -F= J x B (Lwh)

-4

VV

B PO

Fig. 6. Propulsive magnetohydrodynamic rectangular duct configuration.

DTRC-90/009 23

109I I I I I I I I I I

4-

2-

108- 24% --8=L 25%s - -

6 -"28% _-

4---30%/o 3,000 VA W

2-. 33%0/-2/

1 10107-$ •-

8 37%

4

2 -2

106I WIDTH OF DUCT, w = 4 m8= HEIGHT OF DUCT, h = 4 m

6- LENGTH OF DUCT, L = 1ml-4- MAGNETIC INDUCTION, B - 6T

% REPRESENTS PROPULSIVE EFFICIENCY, 1n1 x 102105=

a-

6- I 85 = REPRESENTATIVE PROPULSIVE COEFFICIENTOF SUBMERGED VEHICLE DRIVEN BY

4SINGLE ROTATION PROPELLER-68%

2-

104 - 1 1 1 1 1 1 1 1 1

0.0 0.4 0.8 12.0 16.0 20.0 24.0 28.0

VEHICLE SPEED, V (m/s)

Fig. 7. Power consumption versus submerged vehicle speed.

24 DTRC-90/009

Table 1. Baseline numerical results for square rectangular duct.

Height of square duct, h 4 meters Joulean heat loss in duct,12 R 4.59 x 106 watts

Width of square duct, w 4 meters Electrical current in duct, I 1.36 x 104 amperes

Length of square duct L 10 meters Coefficient of skin frictionoutside in duct, Cd 0.00262

Magnetic induction of duct, B 6 tesla Coefficient of skin frictionoutside of duct, C, 0.002637

Velocity of vehicle, V 10 meters/second Total input power to propul-sive system 8.12 x 106 watts

Velocity of fluid in duct, VD 10.8 meters/second Propulsive efficiency,171 0.37Viscosity of seawater, j7' 103 newton-seconds/ Useful power to propel ve-

meter hide 3.0 x 106 watts

Electrical conductivity of Power to overcome skin fric-seawater 4.0 siemens/meter tion of duct 028 x 106 wattsDensity of seawater, go 1026 kilograms/meter' Power to overcome external

skin friction 0.0000Electrical potential across Power lost to kinetic energyelectrodes 599.0 volts aft 0.25 x 106 watts

Electrical resistance of Hydrodynamic resistance ofduct, P, 0.025 ohms vehicle 3.0 x 105 newtons

Electrical current density, J 338.9 amperes/meter2

Nondimensional parameters of duct based on hydraulic diameter, Dh(Transverse dimension related to fully developed fluid flow)

Reynold's number ReD = QDh = 44.3 X 1 inertial force7)' viscous force

induced magnetic inductionMagnetic Reynold's number Re, = VDDhag = 21.7 x 10-5 = induced magnetic inductionexternal magnetic induction

Interaction number ND = = 5.2 X 10-2 =

QVD inertial force

Hartmann number MD = LB( )= 3.8 x i& ponder1=otive force17 P viscous force

Nondimensional parameters of duct based on length, L(Axial dimension related to boundary growth at entrance of channel)

QVLoB 2L

ReL = __ = 110.8 X 106 NL = V 13X

Re,,t = VDLaU = 54.3 x 10-5 ML =LB (_) = 9.5 x 103

DTRC-90/009 25

presents the power in watts required to propel the submerged vehicle at constant speed.The dimensions of the duct are the same as in Table 1. The magnetic induction B was6 tesla. The dotted line shows the total input power required to propel the vehicle system.The percentages above the dotted line show the propulsive efficiency V 1. It should becarefully noted that a representative propulsive efficiency of a submerged vehicle drivenby a single screw rotation propeller could be considered to be about 85%. These curvesshow clearly that the propulsive efficiency of this MHD system decreases significantlywith vehicle speed V, and that the propulsive efficiency is much less than that of a propel-ler.

Figure 8 presents the propulsive efficiency curves rh and r72 of a square MHDpropulsive duct with a 6 tesla magnetic induction B across the duct, propelling a sub-merged vehicle at a steady state speed V. The duct has the same linear dimensions as theduct in the previous figure. The hydrodynamic resistance of the vehicle was again 3,000V2 newtons and the power required to propel the vehicle is thus 3000 V3 watts. The pro-pulsive efficiency 771 > 172, as would be expected because 172 incorporates the loss of

0.70 " -I - I I

DUCT LENGTH, L - 10mnDUCT HEIGHT, h 4 m

0.60" DUCT WIDTH, w 4 mMAGNETIC INDUCTION IN DUCT, B - 6 THYDRODYNAMIC RESISTANCE OF VEHICLE, R - 3,000 1 N

0.50"

0.40"

0.30" 7/2

0.20"

0.10"

0.00- I I , I I I I I

0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0

VELOCITY, V(m/s)

Fig. 8. Propulsive efficiency of square MHD duct versus vehicle speed inmeters per second.

26 DTRC-90/009

power due to the flow of fluid over the external surface of the duct. This corresponds to

the duct as a pod attached to the vehicle. Both curves of propulsive efficiency T) I and '12

show a significant decrease in propulsive efficiency with speed V.

Figure 9 presents the effective thrust T in newtons versus vehicle speed V in metersper second required to propel a submerged vehicle with a hydrodynamic resistance in sea-water of order 3,000 V2 newtons. The thrust curve is proportional to V2 and thus risessteeply with V. One knot is equal to 0.5144 m/second.

Figure 10 presents the propulsive efficiencies jit and T]2 of the previously de-

scribed propulsive MHD duct versus constant, homogeneous magnetic induction B acrossthe duct at vehicular speed V of 10 m/second. Using state-of-the-art superconductingmagnets, a magnetic induction of 6 to 10 tesla in a large volume is probably the highestmagnetic induction that could be obtained in practice. The 6 and 10 tesla limits are shown

by two vertical dotted lines on the figure. The propulsive efficiencies tit and '12 increase

significantly with the magnitude of the magnetic induction B. The power to propel the

vehicle T is 3000 V3 watts. The magnitude of the separation between j/' and '12,

b? t - '121, increases with increasing B.

Figure 11 presents the propulsive efficiencies of the MHD propulsive duct at a ve-hicular speed of 10 m/second versus electrical fluid conductivity a in a 6 tesla magnetic

induction B. The propulsive efficiencies 'i' and '12 increase significantly with fluid elec-trical conductivity o in siemens per meter. The propulsive efficiencies of the MHDsystem in the seawater at the polar ice caps (i.e. a = 0.4 siemens/m) is well below an effi-

ciency of 0.10 for either ']i or '12. The hydrodynamic resistance of the submergedvehicle was again 3000 V2 newtons.

Figure 12 presents the propulsive efficiencies of a MHD propulsive duct of length Lequal to 10 m and varying square cross-sectional area from 4 m2 to approximately 81 m2.The variation of cross-sectional area is shown in the horizontal axis. The magnetic induc-tion across the duct is 6 tesla, and the power required to propel the submerged vehicle is

3000 V3 watts. The efficiencies 171 and '12 increase significantly with increasing cross-

sectional duct area. The magnitude of separation Ill - '121 between '11 and 712 becomes

significant at large duct cross-sectional areas. It should be realized that construction ofsuperconducting magnets for a duct with width w and height h much greater than 4 mwould probably be impossible with present technology.

Figure 13 presents the propulsive efficiencies of a duct of width w of 4 m and aheight h of 4 m with a varying length L in meters at a vehicular speed of 10 m/second.

The efficiencies rh and '12 increase slowly with duct length L in meters. The magneticinduction across the duct B is 6 tesla. The maximum duct length considered in this figureis 20.0 m.

Figures 14 through 18 show the effect of changing the NIH) rectangular duct widthw through the values 1, 2, 4, 8, and 16 m on the duct propulsive efficiencies j11. On each

figure for a specific duct of width w the value of the duct height h is increased from 4 to

DTRC-90/009 27

I 08.. I.I.I. I I I I I I I8 --

6-

4-

- THRUST REQUIRED TO PROPEL SUBMERGED VEHICLE

2- AT CONSTANT SPEED (THRUST REQUIRED TO BALANCEHYDRODYNAMIC RESISTANCE OF VEHICLE, R)

107- T = THRUST REQUIRED TO PROPEL VEHICLE8

4-

2- 3,000 V2 N

106--\8=

_ 6 T

6-4

2-

10',

6:

4-

2-

0.0 4.0 8.0 12.0 16.0 20.0 24.0 28.0

VEHICLE SPEED, V (m/s)

Fig. 9. Thrust required to propel submerged vehicle at constant speed V.

28 DTRC-901009

0 ,70. I I I

DUCT LENGTH, L - 10 m 171DUCT HEIGHT, h - 4 m

0.60- DUCT WIDTH, w = 4 mVEHICLE SPEED, V - 10 m/sHYDRODYNAMIC RESIS- 172

0.5 TANCE OF VEHICLE, R -

3,000 VAN

0.40-(i) PROBABLE PRACTICAL LIMITS

0.30 I OF HIGH FIELDS FOR LARGE0.30 VOLUMES USING SUPERCON-

DUCTING MAGNETS

0.20-

0.10-

0.00 , I ,

0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0

MAGNETIC INDUCTION IN DUCT B (M)

Fig. 10. Propulsive efficiency of square MHD duct versus magnetic in-duction in tesla.

32 m. The M-D duct length L was maintained at 10 m. On each figure two groups of nu-merical curves for submerged vehicle speeds of 2.5, 5, and 10 m/second are presented formagnetic inductions of 6 and 10 tesla. The vessel hydrodynamic resistance was 3000 V2

newtons. The corresponding propulsive efficiency curves /il on each figure for identical

submerged vehicle speed have higher t/ values at 10 tesla than at 6 tesla on each figure,

as would be expected. On each figure the propulsive efficiency values j/l of each curve

increases with duct height h when h increases from 4 to roughly 10 m and then decreasesslightly to 32 m. The main point that each figure shows is that the corresponding propul-sive efficiency curve values i/1 increase with duct width w.

The propulsive efficiency j/1 can be derived from Eq. 39 and 44 to be

7V (v (1/8) !QVD2V2L l + (hlw)J} ( + I EB1T CBV- )29 + VDB (47)

DTRC--W0009 29

where the current I in the channel is determined from

TL)I = -T + L0 ( + , h ) 2 (48)

Bw 4B /(4

In Eq. 47, the frictional term containing the friction factorf is much smaller thanBV, and in Eq. 48 the friction term containingf is much smaller than T/Bw, since L is only10 m. Increasing L to large values will significantly increase the frictional resistance inthe MHD duct. From Eq. 48 it can be seen that increasing w decreases 1, the total currentin the duct, which increases t, in Eq. 47, as the numerical results show.

The MHD propulsive efficiencies reported here which approach those of a propellercorrespond to duct volumes greater than 160 m3 , with small surface area-to-volume ra-tios. These large-volume magnetic inductions of 6 to 10 tesla may be unattainable inpractice.

0.76II06-MAGNETIC INDUCTION, 6T ?

0.50-0.40-

0.30-

DUCT HEIGHT, h 4 4m0.20- //"DUCT WIDTH, w 4 m

0.2 VEHICLE SPEED, V - 10 m/sHYDRODYNAMIC RESISTANCE OF

0.10- ¢ VEHICLE, R = 3,000 k N

,Aý- 0.4 = CONDUCTIVITY AT POLAR ICE CAPS0.00 I

0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0

ELECTRICAL CONDUCTIVITY OF SEAWATER, a (S/m)

FiRg. 11. Propulsive efficiency versus electrical conductivity of seawater.(Electrical conductivity of normal seawater is approximately 4 S/rn.)

30 DTRC-90/009

0.7 1 1

0.60-

zl

0.5( /

E 0.40-

01l) DUCT LENGTH, L 10 bni 0.30" MAGNETIC INDUCTION, B = 6 T

VEHICLE SPEED, V = 10 rn/sHYDRODYNAMIC RESISTANCE OF

0.20 VEHICLE, R = 3,000 V2 N

0. 10")

0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0 100.0

SQUARE CROSS-SECTIONAL AREA OF DUCT A

Fig. 12. Propulsive efficiency of square MHD duct versus ductcross-sectional area.

Tables 2 to 4 present the propulsive efficiencies /il for different length L square

ducts having height h of 4 m and width w of 4 m versus vehicle speed V in meters persecond. (Note: 1 m/second = 1.944 knots.) The magnetic induction B across the duct is 2tesla in Table 2, 6 tesla in Table 3, and 10 tesla in Table 4. The hydrodynamic resistanceof the submerged vehicle is 3000 V2 newtons and thus requires a power of 3000 V3 wattsto propel the vehicle at a constant speed V. The propulsive efficiency 171 increases signifi-cantly with an increase in magnetic induction B. Bear in mind that a magnetic inductionof 6 to 10 tesla across the duct is a practical upper limit, imposed mainly through structur-al considerations using available superconducting magnet technology. The efficiency Idecreases significantly with decreasing vehicle speed V. For the 2-tesla magnetic induc-tion B the efficiency r/l tends to increase slightly as duct length increases to 40 m. Forthe 6-tesla magnetic induction the efficiency generally tends to increase slowly with ductlength L to 40 m. With the 10 tesla magnetic induction B the efficiency r7j generallytends to increase slowly with duct length L to 40 m. In these numerical calculations theconductivity of seawater C was assumed to be 4 siemens/m.

DTRC-90/009 31

0.70 III

DUCT WIDTH, w 4mr0.6- DUCT HEIGHT, h =4 mn

MAGNETIC INDUCTION, B = 6 TVEHICLE SPEED, V = 10 m/s

0.5or HYDRODYNAMIC RESISTANCE OF VEHICLE,

LA.

(wi)

0.30

2i

0.2(ý

0.10

0.00 IIII

0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0

DUCT LENGTH, L (in)

Fig. 13. Propulsive efficiency of square duct versus duct length.

Tables 5 and 6 present the propulsive efficiencies 'i? for the rectangular duct ofwidth w of 4 mn and height h of 4 mn versus duct length L in meters and vehicle speed V inmeters per second for magnetic inductions B of 6 and 10 tesla, respectively. The tablesshow that extremely long duct lengths L of over 50 m generally degrade the propulsiveefficiency j~ with the efficiency decreasing with increasing length. This would be ex-pected since the propulsive efficiencies decrease because of the increased skin frictiou inthe duct with increased surface area.

Table 7 presents the propulsive efficiency ;7 for the preceding MEHD duct with alength L of 10 mn versus vehicle speed V in meters per second and conductivity of seawa-ter a7 in siemens/meter. The magnetic induction across the duct is 6 tesla. Thehydrodynamiic resistance of the submerged vehicle in the seawater is 3000 Vl2. The pro-pulsive efficiency i~ increased significantly with increases in fluid electricalconductivity a and decreases in vehicle speed V. The conductivity of seawater can beincreased by seeding. However, it does not appear that this is practical for MN])propulsion.

32 DTRC-90/009

Table 8 presents the propulsive efficiency j7 , for the duct versus magnetic induc-

tion B across the duct and conductivity aT of the fluid flowing through the duct. Thehydrodynamic resistance of the submerged vehicle in seawater is 3000 V2 newtons. Thepropulsive efficiency ;/ increases significantly with increasing fluid conductivity Cr

and increasing magnetic induction B. However, the conductivity of seawater is generallyabout 4 siemens per meter and the practical limit to B is about 10 tesla with today's pres-ent superconducting magnet technology.

Tables 9 through 13 present the propulsive efficiency 1i, for the square MHD pro-

pulsive duct versus vehicle speed V in meters per second and magnetic induction B intesla. The hydrodynamic resistance R of the vehicle is 1000 V2 newtons in Table 9, 2000V2 newtons in Table 10, 3000 V2 newtons in Table 11, 4000 V2 newtons in Table 12,and 5000 V2 in Table 13. For each table the propulsive efficiency 17 , increases with

magnetic induction B and decreases with vehicle speed V in meters per second. Thesetable also show that the propulsive efficiency 71, for the MHD system generally increases

as the hydrodynamic resistance of the submerged vehicle decreases.

DTRC-90/009 33

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Table 5. Propulsive efficiencyri , x 102 versus submerged vehicle speed and length of

square duct at 6 tesla. (Duct height h = 4 meters, duct width w = 4 meters, magnetic in-duction in duct B = 6 tesla, hydrodynamic resistance R = 3000 V2 newtons, skin frictioncoefficient in duct Cfd = 0.00262. Velocity of submerged vehicle Vin meters per second;length of duct L in meters.)

VW 5 10 15 20 25 30 35 40 45 50

L =25 59.0 48.2 40.7 35.2 31.0 27.8 25.1 22.9 21.1 19.5

L = 50 55.4 48.4 42.9 38.6 35.0 32.1 29.6 27.4 25.6 24.0

L - 75 50.0 44.8 40.6 37.1 34.2 31.7 29.5 27.6 26.0 24.5

L = 100 45.1 41.0 37.6 34.7 32.2 30.1 28.2 26.5 25.1 23.8

L = 125 40.9 37.5 34.7 32.2 30.1 28.2 26.6 25.1 23.8 22.6

L - 150 37.3 34.5 32.0 29.9 28.0 26.4 24.9 23.6 22.4 21.4

L - 175 34.4 31.9 29.7 27.8 26.2 24.7 23.4 22.2 21.1 20.2

L - 200 31.8 29.6 27.7 26.0 24.5 23.1 22.0 20.9 19.9 19.0

L = 225 29.6 27.6 25/9 24.3 23.0 21.8 20.7 19.7 18.8 18.0

L =250 27.6 25.8 24.3 22.9 21.6 20.5 19.5 18.6 17.8 17.0

L -275 25.9 24.3 22.8 21.6 20.4 19.4 18.4 17.6 16.8 16.1

L 300 24.4 22.9 21.6 20.4 19.3 18.4 17.5 16.7 16.0 15.3

L 325 23.1 21.7 20.4 19.3 18.3 17.4 16.6 15.9 15.2 14.6

L 350 21.9 20.6 19.4 18.4 17.4 16.6 15.8 15.1 14.5 13.9

L 375 20.8 19.6 18.5 17.5 16.6 15.8 15.1 14.5 13.9 13.3

L - 400 19.8 18.7 17.6 16.7 15.9 15.1 14.5 13.8 13.3 12.7

L-425 18.9 17.8 16.9 16.0 15.2 14.5 13.9 13.3 12.7 12.2

L-450 18.1 17.1 16.1 15.3 14.6 13.9 13.3 12.7 12.2 11.7

L -,475 17.3 16.4 15.5 14.7 14.0 13.4 12.8 12.2 11.7 11.3

L - 500 16.6 15.7 14.9 14.1 13.5 12.9 12.3 11.8 11.3 10.9

42 DTRC-90/009

Table 6. Propulsive efficiency 7/1 x 102 versus submerged vehicle speed and length of

square duct at 10 tesla. (Duct height h = 4 meters, duct width w = 4 meters, magnetic induc-tion in duct B = 10 tesla, hydrodynamic resistance R = 3000 V2 newtons, skin frictioncoefficient in duct Cid = 0.00262. Velocity of submerged vehicle V in meters per second,length of duct L in meters.)

VW- 5 10 15 20 25 30 35 40 45 50

L ,25 69.0 63.0 58.0 53.7 50.0 46.8 440 41.5 39.2 37.2

L , 50 61.1 57.8 54.8 52.1 49.6 47.4 45.4 43.5 41.8 40.2

L - 75 54.0 51.7 49.5 47.6 45.8 44.1 42.5 41.1 39.7 38.4

L = 100 48.1 46.4 44.7 43.2 41.7 40.4 39.1 38.0 36.8 35.8

L , 125 43.4 41.9 40.6 39.3 38.1 37.0 36.0 35.0 34.0 33.1

L n 150 39.4 38.2 37.1 36.0 35.0 34.1 33.2 32.3 31.5 30.7

L m 175 36.2 35.1 34.1 33.2 32.3 31.5 30.7 29.9 29.2 28.5

L n 200 33.4 32.5 31.6 30.8 30.0 29.2 28.5 27.9 27.2 26.6

L = 225 31.0 30.2 29.4 28.7 28.0 27.3 26.7 26.0 25.5 24.9

L - 250 28.9 28.2 27.5 26.8 26.2 25.6 25.0 24.4 23.9 23.4

L = 275 27.1 26.4 25.8 25.2 24.6 24.0 23.5 23.0 22.5 22.1

L w 300 25.5 24.9 24.3 23.7 23.2 22.7 22.2 21.7 21.3 20.8

L = 325 24.1 23.5 23.0 22.4 21.9 21.5 21.0 20.6 20.2 19.8

L w 350 22.8 22.3 21.8 21.3 20.8 20.4 19.9 19.5 19.1 18.8

L m 375 21.6 21.2 20.7 20.2 19.8 19.4 19.0 18.6 18.2 17.9

L, -400 20.6 20.1 19.7 19.3 18.9 18.5 18.1 17.7 17.4 17.1

L w 425 19.7 19.2 18.8 18.4 18.0 17.7 17.3 17.0 16.6 16.3

L - 450 18.8 18.4 18.0 17.6 17.3 16.9 16.6 16.3 15.9 15.6

L = 475 18.0 17.6 17.3 16.9 16.6 16.2 15.9 15.6 15.3 15.0

L w 500 17.3 16.9 16.6 16.2 15.9 15.6 15.3 15.0 14.7 14.4

DTRC-90/009 43

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DISCUSSION AND CONCLUSIONS

A simple magnetohydrodynamic (MHD) propulsive configuration can be a straightrectangular duct employed to produce the thrust for a submerged vehicle. The MHD pro-pulsive duct can be within the hull of the submerged vehicle, in the configuration of a"pod" attached by struts to the main vehicle, or in an array around the outside surface ofthe vehicle. In the mathematical model discussed herein, the friction on the duct wallsdue to fluid flow was approximately represented by the ordinary smooth pipe flow fric-tional formulation. The vertical duct walls contained electrodes, the horizontal wallscontained superconducting magnets, and the duct wall thickness was assumed to be zero.Neither the power supplied to the duct's superconducting magnet configuration nor themagnet masses were considered in the mathematical model. End effects, which degradepropulsive efficiency, and bubble generation, were also omitted.

The fundamental rectangular duct M-D propulsion system was mathematicallymodelled to compute the propulsive efficiencies of the system. The propulsive efficiencyof the duct within the hull of the vehicle was designated as j7 ; the propulsive efficiencyof the duct as an external pod was designated as 17 2. The propulsive efficiency

17 2 <17 , because of the increased skin friction of the system due to the external skin fric-

tion of the pod. Numerical solutions were presented for the propulsive efficienciesj7 and i7 2 of the MHD propulsion duct in regard to such variables as duct geometry,duct magnetic induction, electrical conductivity of fluid, submerged vehicle speed, sub-merged vehicle hydrodynamic drag, and skin friction fluid drag inside the duct.

The propulsive efficiencies calculated from the model in general overestimate theefficiencies of the corresponding real MIH-D propulsion system. The numerical propul-sion efficiency curves in this report can be used for engineering estimates of thepropulsive efficiencies of real MI-D propulsion systems.

In this work, a base MHD duct was chosen with width 4 meters, height 4 meters,length 10 meters, and magnetic induction of.6 tesla across the duct propelling asubmerged vehicle of 3,000 V2 newtons at a vehicle speed of 10 meters per second. Thepropulsive efficiencies j7 , and 172 related to the base MED propulsive duct propelling

the submerged vehicle have the following properties:

"* In a 6 tesla field 17 , and 17 2 decrease from 0.67 and 0.61 at 2.0 meters per

second to 0.24 and 0.22 at 20.0 meters per second, respectively, a signifi-cant decrease in efficiency with increasing vehicle speed.

"• At a constant vehicle speed of 10 meters per second . , and .7 2 increase

from 0.07 and 0.06 at 2 tesla to 0.67 and 0.63 at 14 tesla , respectively, asignificant increase in efficiency with magnetic induction.

" 17 , and T7 2 increase from 0.07 and 0.06 at an electrical conductivity of sea-

water of 0.4 siemens per meter to 0.67 and 0.62 at conductivity of 20.0siemens per meter, respectively, a significant increase in efficiency with sea-water electrical conductivity. If possible at all, the conductivity of theseawater in the duct could be increased in practice only by "seeding." (The

DTRC-90/009 51

electrical conductivity of seawater is on the average approximately 4 sie-mens per meter.)

"* At a constant vehicle speed of 10 meters per second and a magnetic induc-tion of 6 tesla, r/] and t , increase from 0.14 and 0.13 at a duct cross

section of 4 meters 2 to 0.62 and 0.53 at a cross section of 64 meters2

"* At 6 tesla Y] , and j7 2 increase from 0.13 and 0.12 at a duct length of 2.0

meters to 0.46 and 0.39 at duct length of 20.0 meters, respectively, a signifi-cant increase in efficiency with moderate duct length at 6 tesla. Increasingthe duct length to 200 meters deceases /i 1to 0.30, as would be expected

because of the increased skin friction inside an extremely long duct.

"* At vehicle speed of 10 meters per second, magnetic induction of 6 tesla, andduct length 10 meters, r/ increases from 0.24 to 0.43 with an increase in

the electrode height (i.e., duct height) from 4 meters to 32 meters, a signifi-cant increase in efficiency with electrode height.

" At vehicle speed of 10 meters per second, magnetic induction of 6 tesla,duct height of 4 meters, and duct length of 10 meters, 17 1 increases from

0.14 at a duct width of 1 meter (i.e., distance between electrodes) to 0.60 ata width of 16 meters, a significant increase in efficiency.

* At a vehicle speed of 10 meters per second, j7 1 generally decreases with

increases in the hydrodynamic resistance of the submerged vehicle from1000 V2 newtons to 5000 V2 newtons.

Thus, in summary, according to the mathematical model presented here and therange of parameters studied, the ideal MtlD propulsive duct efficiencies increase in mag-nitude with increase in the following factors: channel magnetic induction, square ductcross sectional area at moderate duct lengths, fluid electrical conductivity, duct length,distance between electrodes, and electrode height.

The ideal MHD propulsive efficiencies decrease with vehicle speed, and generallydecrease with increasing vehicle hydrodynamic drag. Extremely long ducts significantlydecrease propulsive efficiencies because of the increase in internal duct friction.

A comparison of the ideal MHD propulsive efficiencies ,7 and 7 2 for a sub-

merged vehicle with the corresponding efficiencies of a conventional propeller* (i.e.,0.85), generally shows the MHD efficiencies to be less under most conditions. The MHDefficiencies j7 , and 7 2 which approach that of a propeller correspond to ducts of very

large volumes (order of magnitude of 300 meters3 with small surface to volume ratios), orrequire very high duct field values (20 or 25 tesla). In large volumes of space such asMHD ducts, magnetic inductions of about 6 to 10 tesla are about the highest fieldstrengths possible with the use of present day superconducting magnet construction tech-

"McCarthy, J.N., Head, Naval Hydromechanics Division at DTRC (Code 154), Private communication(Oct 1989).

52 DTRC-90/009

nology,** and producing these fields would require a significant engineering effort inmagnet construction.

MHD propulsive ducts have no mechanical moving parts and consequently mayproduce less acoustic noise than propellers under the same propulsion conditions. Also,not having a propeller removes the problem of propeller hull shaft seals. Using MHD pro-pulsion on a submerged vehicle to reduce the acoustic signature as compared with aconventional propeller-driven vehicle is possibly a very promising feature, and it shouldbe investigated, even though the MHD system generally has a lower propulsive efficiency.Therefore, it should be determined if the MHD propulsion technology is available or canbe developed in the future to build a submerged vehicle that produces less acoustic noisethen conventional submerged vehicles.

ACKNOWLEDGMENTS

The authors would like to thank LJ. Argiro, H.O. Stevens, MJ. Superczynski, R.C.Smith, and D. B. Larrabee, all of DTRC, for many helpful technical discussions andmuch encouragement while this work was in progress. Also, we would like to express ourappreciation to Dr. W. J. Levedahl, of DTRC, for many helpful discussions involving themathematical modelling. The work was also supported by many technical discussionswith Dr. Kenneth Tempelmeyer, Secretary of the Navy Fellow at the U.S. Naval Acade-my during 1989 and Professor at the College of Engineering at Southern IllinoisUniversity.

Superczynski. MJ., Superconductive Project Coordinator at DTRC. Private Communication(Apr 1989).

DTRC-90/009 53

APPENDIX ATHREE-DIMENSIONAL EFFECTS

This report presents a one-dimensional control-volume model for an MHD ductwhich is moving through an unbounded fluid and which is accelerating part of that fluidso as to develop forward thrust. In this model, the velocity has only an axial component,while the velocity and pressure are uniform across each cross section of the flow. Clearly,the flow upstream and downstream of the duct must actually have three nonzero velocitycomponents, not one; neither are the pressure and velocity uniform across each cross sec-tion. The present model is an approximate extension of the one-dimensional model for apropellor in an open sea. The key difference between the two applications is that theopen-sea propellor provides a pressure rise across a disk of essentially zero thickness withno other solid surfaces, while the MHD duct provides a pressure rise over the axial lengthL of the duct. The effect of the finite-length duct walls is very similar to the effect of acylinder or shroud placed around a propellor. This effect is treated in detail byKuchemann and Weber.25 Here we only summarize the key features and their implica-tions for the MHD propulsion duct.

The propellor alone in an open sea creates a certain two-dimensional, axisymmetricflow with a radially inward flow associated with the axial acceleration. If a cylinder offinite length is placed around the propellor, it locally blocks the radial velocity compo-nent. In essence, each section of the cylinder acts as an airfoil in the radially convergingflow created by the propellor alone. To be an effective airfoil, the flow must not separateat the leading edge of the cylinder and this dictates a minimum radius of curvature of thewall here. In other words, the wall of the cylinder cannot be too thin at the front.

The effects of the cylinder as an airfoil can be stated in several equivalent ways.First, we can say that the circulation associated with the airfoil section increases the flowthrough the propellor. For a given thrust, the increase in velocity is smaller with moreflow, so that there is less kinetic energy lost in the jet aft of the propellor. An ideal shroudalways increases the efficiency, but an actual shroud adds skin-friction drag, which tosome degree offsets the gain, depending on the particular case. A second way to describethe effect of the cylinder is to say that the airfoil produces a pressure on the forward, in-side surface of the cylinder which is further below the ambient pressure than the localpressure would be without the cylinder. This lower pressure has two effects. First, itdraws more fluid into the cylinder, resulting in more flow through the propellor. Second,the propellor still provides an increase in pressure, but now the upstream pressure is fur-ther below ambient pressure, so that the exit pressure is not as far above ambientpressure. As a result, the jet aft of the propellor does not accelerate and contract as much,so again there is less kinetic energy lost.

The cylinder cannot be treated as an airfoil in the open-sea propellor flow. The pres-ence of the cylinder changes the flow created by the propellor, so that the airfoil-cylinderand the pressure-jump propellor disk must be treated together. For axisymmetricshrouded propellors, Kuchemann and Weber2 present solutions involving axisymmetricvortex sheets which produce flows satisfying the boundary conditions at the shroud wallsand at the propellor disk. The thrust is now divided between the propellor and shroud orcylinder. The propellor thrust is still equal to the pressure increase times the disk area.The pressures which are below the ambient pressure near the leading edge of the cylinderor shroud produce an additional forward thrust on the cylinder. An airfoil alone would

54 DTRC-90/009

have no net axial force. However, the propellor produces an abrupt pressure jump on theinside surface of the airfoil-cylinder. Thus the pressures on the forward and aftward partsof the airfoil are lower and higher, respectively, than they would be without the propellor,and there is a net force on the cylinder or shroud. If the cylinder wall has zero thicknessand if the leading edge separation is erroneously ignored, then the pressure at the leadingedge is minus infinity. This infinite negative pressure acting over the zero area of theleading edge gives the finite forward thrust on the cylinder.

The MHD equivalent of an open-sea propellor would be a uniform axial body force,JB, acting over the rectangular volume of length L, but with no duct walls. The stream-lines which would result from such an open-sea body force distribution are sketched inFig. 19 The forward thrust on the ship associated with this acceleration would be appliedto the magnet coils producing the magnetic field. When we add the actual duct walls, theylocally block the transverse velocity components and therefore act as airfoils with inwardlift forces. There is now an additional forward thrust on the duct walls, associated primar-ily with the much lower pressures near the entrance associated with the circulationproduced by the airfoil-duct walls. Again the duct walls would require a finite thicknessand minimum radius of curvature to prevent separation at the entrance. Indeed, therewould certainly be fairings fore and aft of the straight duct walls since straight wallsalone would not represent the optimal airfoil cross sections.

The two-dimensional, axisymmetric analysis of Kuchemann and Weber2s for ashrouded propellor cannot be extended to the MI-fD duct in a straightforward fashion.First, the pressure jump across the zero-thickness propellor is replaced by a distributedbody force. Second, the flow is in general three-dimensional, although some designs for

Fig. 19. Streamlines due to uniform body force JB distributedover rectangular volume with no duct walls.

DTRC-901009 55

the MHD propulsion of submerged vehicles might involve axisymmetric flows. Since anew two- or three-dimensional analysis would by needed to treat the effect of the ductwalls, it would be inappropriate to keep a one-dimensional model of the MHD bodyforce. One would want to include the effects of the spatially decaying, fringing magneticfields and electric currents near the entrance and exit. We have not presented such a two-or three-dimensional model here. Our present purpose was to provide the best predic-tions possible with a one-dimensional model.

While we have not determined the increase in the flow through the duct due to theairfoil effect of the duct walls, we can make a few qualitative statements based on the re-sults of Kuchmenn and Weber. They treated a propellor at the middle of a long cylinderand found that the cylinder increases the velocity at the propellor. The effect on the ener-gy balance for a fixed ship velocity and total thrust is to decrease the kinetic energy lostaft of the propellor. For most of the cases considered here, the lost kinetic energy was rel-atively small compared to other losses, particularly Joulean heating. Therefore, even arelatively large reduction on the lost kinetic energy may not significantly improve theoverall efficiency of the propulsion unit. In addition, an increase in the velocity inside theduct increases the frictional losses, which vary as the square of the duct velocity, so thatany gains are offset to a degree which depends on the particular case.

56 DTRC-90/009

REFERENCES

1. Blake, L.R., "Conduction and Induction Pumps for Liquid Metals," The Proceed-ings of the Institution of Electrical Engineers, Part A, pp. 49-63; discussion pp.63--67 (February 1957).

2. Cage, J.F. Jr., "Electromagnetic Pumps for High-Temperature Liquid Metal,"Mechanical Engineering, pp. 467-471 (June 1953).

3. Barnes, A.H. and J.F. Cage, Jr., "Pumps-Electromagnetic, "Liquid Metals Hand-book, Sodium-NaK Supplement, pp. 288-305 (1 July 1955).

4. Way, S., "'Propulsion of Submarines by Lorentz Forces in the Surrounding Sea,"American Society of Mechanical Engineers, ASME Paper 64 WA/ENER7 (Nov1964).

5. Way, S. and C. Devlin, "Prospects for the Electromagnetic Submarine, "AIAA3rd Propulsion Joint Specialist Conference, AIAA Paper 67-432 (July 17-21,1967).

6. Way, S. and C. Devlin, "Construction and Testing of a Model Electromagnetical-ly Propelled Submarine," Proceedings, Eighth Symposium on the EngineeringAspects of Magnetohydrodynamics, Stanford University, California, pp. 153-154(March 28-30, 1967).

7. Way, S., "Electromagnetic Propulsion for Cargo Submarine," Journal of Hydro-nautics, Vol. 2, No. 2, pp. 49-57 (April 1968).

8. Doragh, R.A., "Magnetohydrodynamic Ship Propulsion Using SuperconductingMagnets," Society of Naval Architects and Marine Engineers, SNAME Transac-tions, Vol. 71, pp. 370-386 (1963).

9. Friauf, James B., "Electromagnetic Ship Propulsion," American Society of NavalEngineers, ASNE Journal, pp. 139-142 (February 1961).

10. Phillips, O.M., "The Prospects for Magneto-Hydrodynamic Ship Propulsion,"Journal of Ship Research, Vol. 5, No. 4, pp. 43-51 (March 1962).

11. Resler, E.L. Jr., "Magnetohydrodynamic Propulsion for Sea Vehicles," SeventhSymposium on Naval Hydrodynamics, Rome, Italy, pp. 1437-1445 (1968)

12. Vasil'ev A.P. and I.M. Kirko, "Optimization of a Marine MHD Propeller," Trans-lated from Zhurnal Prikladnoi Mekhanikii Tekhnicheskoi Fiziki, No. 3, pp. 86-94(May-June 1981).

13. Yakovlev, V.I., "Theory of an Induction MHID Propeller With a Free Field,"Translated from Zhurnal Prikladnoi Mekhanikii Tekhnickeskoi Fiziki, No. 3, pp.105-116 (May-June 1980).

14. Mitchell, D.L. and D.U. Gubser, Journal of Superconductivity, Vol. 1 (1988).

15. Saji, Y., M. Kitano, and A. Iwata, Advanced Cryogenic Engineering, Vol. 23(1978).

DTRC-90/009 57

16. Iwata, A., E. Tada, and Y. Saji, "Experimental and Theoretical Study of Super-conducting Electromagnetic Ship Propulsion," 5th LIPS Propeller Symposium,No. 2 (1983).

17. Iwata, A., Y. Takeda, and Y. Saji, Journal of Marine Engineering of Japan, Vol.16(1982).

18. Walker, J.S., S.H. Brown, and N.A Sondergaard, Journal of Applied Physics, Vol.64 No. 1 (July 1988).

19. Walker, J.S., S.H. Brown, and N.A. Sondergaard, Journal of Applied Physics,Vol. 64 No. 4 (15 August 1988).

20. Hughes, W.F. and I.R. McNab, "A Quasi-One-Dimensional Analysis of an Elec-tromagnetic Pump Including End Effects," Liquid-Metal Flows andMagnetohydrodynamics, H. Branover, A. Yakot, and P.S. Lykoudis, eds., Vol. 84

of Progress in Astronautics and Aeronautics (1983).

21. Attwood, E.L. and H.S. Pengelly, Theoretical Naval Architecture, Longmans,Green and Co. Ltd., London, England (1953).

22. Hughes, W.F. and FJ. Young, The Electromagnetodynamics of Fluids, JohnWiley and Sons, Inc., New York (1966).

23. Landau, L.D. and E.M. Liftshitz, Fluid Mechanics, Pergamon Press, Addison-Wesley Publishing Co., Inc., Reading Massachusetts (1959).

24. Shercliff, J.A., A Textbook of Magnetohydrodynamics, Pergamon Press, NewYork (1965).

25. Kuchemann, Dietrich and Johanna Weber, Aerodynamics of Propulsion,McGraw Hill, New York (1953).

58 DTRC-90/009

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Office of Chief of Naval Research 3 154 Justin H. McCarthyOffice of Naval Technology 3 1544 Dr. Frank B. Peterson800 N. Quincy St. 2 1561 Geoffrey CoxArlington, VA 22217-5000 2 1942 Dr. Theodore M.

12 DTIC Farabee

20 Dr. John S. Walker 2 27 Larry J. Argiro

University of Illinois at 2 2752 Robert J. Chomo

Urbana-Champaign 2 2701 James L. CorderDept. Mech. & Ind. Engineering 5 2702 Dr. William Levedahl144 Mechanical Engineering Bldg 2 2704 Dr. Earl Quandt, Jr.1206 West Green St. 2 271 Howard 0. Stevens, Jr.Urbana, IL 61801 10 2711 David E. Bagley

3 Dr. Daniel W. Swallom 2 2711 Robert C. SmithAvco Research Lboratory 10 2712 Dr. Samuel H. Brown2385 Revere Beach Parkway 10 2712 Patrick J. ReillyEverett, MA 02149 10 2712 Dr. Neal A. Sondergaard

3 Dr. Michael Petrick 2 2712 Michael J.Argonne National Laboratory Supercznyski, Jr.9700 South Cass Ave. 2 272 Timothy J. DoyleArgonne, IL 60439 2 2720 Dr. Herman Urbach

3 Dr. Kenneth E. Tempelmeyer 2 2743 David B. Larrabeec/o Office of the DeanCollege of Engineering and 2 275 Dr. Cyril Krolick

Technology 3 2772 John G. StrickerSouthern Illinois University 2 2812 Louis F. ApriglianoCarbondale, IL 62901 1 3411 Margaret L. Knox

3 Dr. Basil Picologlou 1 3421 TIC CarderockEngineering Physics Division 2 3422 TIC AnnapolisArgonne National Laboratory9700 South Cass Ave.Argonne, Illinois 60439

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