Conceptual Design Analysis of an MHD Power

8/14/2019 Conceptual Design Analysis of an MHD Power

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Conceptual Design Analysis of An MHD PowerConversion System for Droplet-Vapor Core

Reactors 75811- -77Samim AnghaieGirish Saraph

Innovative Nuclear space Power ancInstit uteUniversity of FloridaGainesville, Florida

Propulsion

Final ReportDOE Grant No. DE-FG05-93ER75871

PrOjeCt Title:Droplet-Vapor Core Reactors with MHD Power ConversionUnits

1995


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This report was prepared as an account of work sponsored by an agency of theUnited States Government Neither the United States Government nor any agencythereof, nor any of their employees, m a k e any wuranty , exprcy or implied. orassumes any legal liability or responsibility for the accuracy, cornpkencu. or use-fulness of any infomation, apparatus, product, or procas disclosed. or -nuthat its use would not infringe privatcly owned rights. Reference h d 0 any rpe-a l i c commtrchl product, process, or Xnticc by trade name. trademark manufac-turer, or otherwise does not ncccssariiy constitute or imply iu endorsement. recorn-mendation. or favoring by the United States Government or any agency thereof.T h e views and opinions of authors expressed herein do not a d l y tate orreflect thosc of tbe United States Government or any agency thereof.


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DiSCLAIMERPortions of this docum ent may be illegiblein electronic imag e products. Imag es areproduc ed from th e best available originaldocument.


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CHAPTERS

T.ABLE OF CONTENT

1 HISTORY AND INTRODUCTION TO MHD . . 11.1 Historical Background . . . . , . . . . . . . . 1

Applications . . . . . . . . . . . . . . . . . 31 .2 The Salient Feactures of MHD and its Spe cial1.3 Outline of the Present Stu dy . . . . . . . . . . . 52 THE BASIC OPERATING PRINCI PLES OF THE MHD POWER GENERATOR. . . . . . . . . 6

2.1 The MHD Generator . . . . . . . . . . . . . 62.2 The Governing Equations in Electrodynamics . . . . 72 . 3 The Governing Equatio ns in Fluid Mechanics . . . . 132 . 4 The Geometry of the MHD Generator . . . . . . . . 1 53 THE PROPOSED NUCLEAR DRI VEN MHD SYSTEM ANDFORMULATION OF THE MODELING PROBLEM . .

3 . 1 Space Power System Descripti on . . . . . .3.2 The Design Constraints on the MHD Generator .3 . 3 Formulation of the Problem . . . . . . . .4 SOLUTION METHOD . . . . . . . . .5 RESULTS AND DISCUSSION . . . . . . . . . . . . .

. 2 1

. 2 1. 2 3. 2 6

. 2 8

. 375.1 Designs of 2 5 , 100 and 4 0 0 MW MHD Generators . . . 375 . 2 Variation of Parameters along the Flow Direction . 415.3 Different Designs for the 1000 MW MHD Generator. . 4 55 . 4 Analysis of the Electrodynamic Phenomena . . . . . 495.5 The Effect of Changing the Length of the MHD Channel. . . . . . . . . . 59

6 THREE-DIMENSIONAL COMPUTER MODEL . I . 6 86.1 Extension of the Work to a 3 - D M od el . . 686.2 Electrod:Jnamic M o d o ? . . . . . . . . . . . . . . . 706 . 3 The Ir,tcraction wi th -the F l u i d Mechanics t.!o.:!el . . 77

REFERENCES . 78


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CHAPTER 1HISTORY AND INTRODUCTION TO MHD

1.1 Historical Backeround- -The first idea of magnetohydrodynamic phenomenon was

conceived by Michael Faraday in 18 32 , during his originalinvestigation of electromagnetic induction [l]. Early in the20th century a large number of magnetohydrodynamic (MHD)devices and machines were invented and patented. In 1 9 0 7 thefirst M H D pump was designed by Northrup [ 2 ] . An MHD pump isa device which converts the electrical energy of the currentsupplied into mechanical energy of the pumped fluid. Byinverting the operating principle o f the M H D p u m p , a n M H Dgenerator was invented and patented by Karl ovit z and Halaczi n 1 9 1 0 [ 3 ] .

I n 1 9 3 0 Williams published the result s o f the firstlaboratory studies of M H D flows in pipes and ducts [ 4 ] . Avery comprehensive theoretical study of this subject wascarried o u t by Hartmann and Lazarus in 1 9 3 6 and 1 9 3 7 [ 5 , 6 1 .In the early 1 9 4 0 s a large and otherwise sophisticated Hall-current type MHD generator was built by Westinghouse ElectricCorp. whi ch failed due to insufficie nt knowledge of theproperties of the ionized gases [ 7 ] . Plasma studies in the1 9 5 0 s , created a sufficient databank of these properties.

1


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2In 1959 , an experimental MHD generator was built at

the Avco Everett Research Laboratory that produced 11.5 k w ofpower and obtained a n appreciable pressure drop with theinteraction between the gas and magnetic field [ 8 ] . In theearly sixties many MHD generators operating at higher powerlevels were reported. The subsequent development of the MHDgenerators was pursued o n basically three lines of approach:firstly, rare gas MHD gene raco rs, which used cesium seededand thermally heated rare gas to get sufficient conductivity;secondly, liquid metal MHD generators , which suffered theinherent drawback of low flow r ates ; third ly, combustion gasMHD gene rato rs, which utilized combustion products producedafter burning the conventional type fuel s. The earlierresearch work was concentrated o n using the MHD generators torep lac e the conve tiona l turbines. Although, the MHDgenerators have higher efficiencies ( 8 5 - 9 0 % ) , but as the MHDsystems are more expensive the earlier res ear ch efforts werecurtailed. The MHD systems are not expected, as yet , tocompete with the conventional generators in the commercialpower plants.

But as the oil and natural gas reserves get consumedthe MHD systems will become cost ef fect ive, due to theirhigher efficiency and lowe r fuel consumption. Added to thiswill be the environmental benefits due to considerably lesspollution. The ever increasing applications in the specialfields are giving further boost to the MHD research.


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31.2 The Salient Features of MHD a nd its SD eci al A ~ ~ l i c a t i o n s

A s an energy dev ice, the MHD generator has veryinteresting characteristics whi ch make it suitable for avariety of special applications. In a n MHD generator thefunctions of -b ot h turbine and electric power generator arecombined into ;-single unit with a simple compact geometry.There are no rotating parts which eliminates vibrations andnoise and limits wear and tear. The MHD sys tems operate athigher temperatures than mechanical turbines with consequentsaving i n ar ea , weight and cost of the radiator of the spacepower syste ms and it also le ads to better efficiencies. Highpower turbines operating at high temperature s and speeds areless reliable than MHD systems due to high stress values.The DC type electric power output and the sta rt- up time asl o w as a few milliseconds makes this power generat ion schemevery attractive.

The compactness and l ow wei ght -to -po wer ratio make MHDenergy conv ersi on a very promising option for space powerapplications. It also eliminat es the possibility of anyproblems due to turbine vibrations and gyroscope effect.Efforts are being made to use MHD as a conversion system ina nuclear electric system. The major limitati on on thedesign o f nuclear reactor is to attain temperatures above3000K to get sufficient ionization of the gases.

Th e MHD systems are widely used i n a compact andmobile power source for military applications. Use of MHD on


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4a tactical aircraft for battlefield illumination was reportedby Air Force Aero Propulsion Labor ator y, Wrig ht-P atte rson AirForce Ba se , Ohio [ 9 j . To escape from detection, submarinesneed a low noise and compact power source and clos ed-c ycleliquid-me tal MHD or nucle ar-dr iven MHD are two possiblesch eme s whic h a-re being loo ked int o.

T h e p r i n c i p l e o f MHD pump ca n be utilized toaccelerate a gas stream in a hypersonic wind- tunne l facility.The concept led to constru ctio n of the MHD pilot scalefacility at the Arnold Engineering Development Center (AEDC)in Tul laho ma, Tennessee [ l o ] . The MHD gener ator, known asLOR HO, was designed by AEDC and achieved peak powers of 18 MWfor about 10 seconds. A series of mult i-meg a-wat t MHDgenerators called Mark V , Mark V I , Mark V I 1 were constructedby Avco Laboratory for the military research.

The MHD channel can be used as a flow mete r. Theelectric current output of a MHD gives a direct measurementof the flow rate of the fluid with know n cond ucti vity . TheMHD channel can also cause population inversion in a flowinggas medium which is essential for the lazing action. Theresearch is continuing in the fiel d o f MHD-laser systems.

The MHD systems have become a significant part o fresearch and development in the field of space power andmilitary applications. This research work concentrates o nthe study of proposed nuclear driven MHD system for themulti -mega -watt space power applications.


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1.3 Outline o f the Present Studv

The objective of this rese arch work is to develop ac o m p u t e r m o d e l t o p e r f o r m d e s i g n a n a l ys i s a nd s t u dyelectrodynamics of the linear MHD generator for the proposedspace power -system. It is essential to study the basicoperating principles and governing equations of the MHDgenerator before embarking o n to the present research work.These principles are discussed in Chapter I1 of this report.

The Chapter 111 descri bes the proposed space powersystem and the constraints it imposes on the design of theMHD generator. The Chapter IV discusses the solution methodincorporated in developng the qu asi -on e dimensional computermodel for the linear MHD generator. A detailed parametricstudy is performed using this computer model.

The optimum designs a re determined at 25, 100 and 400MU power l evel s and the resul ts are tabulated in Chapter V.The analysis of the variation of state parameters and basicelectrodynamic phenomena is als o performed. The Chapter Vfurther suggests three design alternatives for operating atoutput power level of 1000 MW.

For more accurate studies it is essential to developa three dimensional model. An innovative scheme for a 3 - Dmodel is formulated and discussed i n details i n Chapter V I .


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CHAPTER 2THE BASIC OPERATING PRINCIPLE S OF TH E MHD POWER GENERATOR4

- .MHD generator converts mechanical energy of the fluid

directly into electric energy. I n a MHD channel a conductingfluid is passed through a transverse magnetic fiel d B . Thefree charge carriers inside the conducting fluid have acomponent of velocity U along the flow dire ctio n same as thatof the neutral particles. The charge particles experience aforce perpendicular to both U and B. This leads to flow ofelectric current in the perpendicular direction.

T h e F i g . 2 . 1 s h o w s a l i n e a r M H D c h a n n e l wi threc ta ng ula r cr oss -se cti on indicating the directions ofmagnetic field B , flow velocity U and current density J . Thetotal current I passing through the load resistor R createsvoltage drop V (and electric field E) between the electrodeplates. This electric field acts in the dire ctio n oppositeto the curren t densit y. The electric power output isobtained i n the direction mutually perpendicular to bothmagnetic field and fluid flow as shown in the figure.

6


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72 . 2 The Governine Equations i n Electrodynamics

2.2.1 Maxwell's eauations

The basic laws of electromagnetism are explained interms o f four vector relations stated bel ow, which are know nas the Maxwell's equations. These relations are expressed in- -the form of gradient and curl o f the electric field E andmagnetic field B vectors.

C J . E = Q

. . . (2.1)

v . B = 0V x E = - mSt

V X = p . ( 3 + 6.m )S twhere, Sa is the free charge density (coul/m 3 ) ,

J is the current densit y (amp/m2),p is the permeability of the medium (Henry/m),E is the permittivity o f the medium (Farad/m).A n o t h e r i m p o r t a n t l a w , w h i c h i s k n o w n as the

conservation o f electric cha rge , can be derived using Eq.2.1and Eq.2.4. This law is given by the following eq uation.

v . J = - U . . . ( 2 . 5 )6 t

. . . (2.2)

. . . ( 2 . 3 )

. . . (2.4)

At steady state the Eq.2 .5 gives, V . J = 0


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82 . 2 . 2 Generalized Ohm's Law and Darametric analysis

The most com mon form of Ohm's Law in terms o f voltageV and current I i s , V = 1 . R (where, R is the resistance)or in terms of current density J and electric field E i s ,J = a . E (where, 0 is the electrical conductivity of the- - .medium). In the Gener alized Ohm's La w there are certai nadditional terms which are significant only in some specificsituations.

In the MHD generator a working fluid with a limitedcon duc tiv ity (a partially ionized gas mixture for thepr op os ed system) passes throu gh the applied transversemagnetic field B with velocity U. Any charge q moving insidethe magnetic field experiences a Lorentz force, F 1 , given byEq. . 6 .

F1 = q.(UxB) . . . (2.6)The motional emf corresponding to the Lorentz force

can be written as UxB ,whi ch adds up vectorially to theThus thelectric field E , applied by the electrodes.

current density J in the MHD channel can now be written a s,

J = o.(E + UxB) . . . ( 2 . 7 )Once the current density distribution is established,

as given by Eq.2.7, the charge particles move inside themedium with equilibrium velocity or the drift velocity W.This drift velocity, W , which is perpendicular to the


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9magnetic field further g i v e s rise to motional e m f , WxB. knownas Hall fiel d. The drift velocity of the electrons W e isusually far greater th an that of the ions W i and byneglecting the later we get,

J = a : ( E + UXB) + a.(WexB)- - . . . (2.8)When the first term in the Eq.8 is the main term we

can use the following relations.a . W e = p . J = (=)..JB . . . (2.9)

where, p is the mobility of the electrons in the medium ,w is the cyclotron freq uenc y,T is the free flight time of the electrons,

(w.7) is known as the Hall parameter f or the medium.By combining Eq.2.8 and Eq.2.9 we g et ,J = 0 . E + UxB) + u.J x B ) . . . (2.10)BThe gas parameters are n o t uniform throughout the

volume of the MHD channel and consequently the electrondensity is also nonuniform in the volume. The negativegradient of the partial pressure o f the e l e c t r o n s , P e , a c tsas a driving force for the electrons further modifying thecurrent density pattern.

When the electric or magnetic fiel d or the gasparameters are changed very rapidly the current density


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10pattern a l s o tends to change accordingly. But as the chargecarriers cannot change their velocities instantaneously theytend to lag behind at the extremely hig h freq uenc ies . Thisphenomenon adds a transient term in the current densityequation.

The Gen-eralized Oh ms Law combin es all the abovephenomena into a single equation a s given below.

J = a . ( E + U X B ) + = . ( J x B ) + 0 . V P e 7 . u . . . (2.11)B n e . Dt

w h e r e , ne is free electron density (m-3),e is the electronic charge (coul).

2.2.3 Parametric Analysis

To determine the relative importance of each term inthe Generalized Ohm s Law parametric analysis is performedtaking into account the typical range of operating conditionsfor the proposed MHD system 1111.

Velocity U = 200 - 300 0 m/s,Magnetic field B = 1 - 8 tesla,Electric field E =: - 0.5 x ( U . B ) ,Conductivity o = 10 - 500 mhos/m,

First term - + a.(E + UxB) = 1000 - 5 x 1 0 7 amp/m2,Pressure P = 2 x 1 0 6 - 8x106 P a ,Particle densityCollision cross-section

n = 4 ~ 1 0 ~ ~m 3 ,Q = 2x10- 9 m 2 ,


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11Electron velocity Ce = 4x105 m / s ,

Free flight t i m e for electron T = 1 = 3 ~ 1 0 - l ~e c ,n.Q.C,

Electronic charge e = 1 . 6 ~ l O - l ~o u l ,Electronic mass me = 9 . 1 ~ 1 0 - ~ g ,

Hall parameter W T = e . B . 7 = 0.05 - 0 . 5 ,- - . me

w.7 = 0.05 (tesla-),B

Second term -- + =.(JxB) = 5 0 - 2 . 5 ~ 1 0 mp/m2B

Third term --, 0 .VPe =. 1 .VTene. 11606

To get the magnitude of the third term comparable tothe first two terms (i. e. above l o 3 amp/m2) the gra die nt ofthe electronic temperature must b e at least l o 7 K/m. Suchextremely high values of gradients are likely only i nboundary layers and in waves. Temperature gradients in theboundary layers of a gaseous f lo w may be expected up to l o 5K/m, but in the non-eq uili briu m state the temperature ofelectrons Te can be a l ot higher t han the gas temperature andthe gradient of Te may have still higher values.

Fourth term --, 7 . u = 3 x 1 0 ~ ~ ~JDt Dt

The four th term will be significant only when the13current density fluctuates with frequencies higher than 10

Hz. If the magnetic field is switched on instantly or theelectric circuit is connected then the current density


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12changes from 0 to lo5 amp/mz, but this process will belimited by this term and the t i m e taken will be of the orderof 10 n s e c .

A s ca n be seen from the above an alys is, only thefirst two terms of the Generalized Oh m s Law are significantfor the normal-Dp-eration o f the MHD generator. Therefore wewill use Eq .10 for determining the current density J for theMHD channel.

2 . 2 . 4 Magne tic Re vnold s Number R,

For an y MHD channel a dimensionless quantity kno wn asMagnetic Reynolds Number is defined as follows:

R, = p.0.U.L . . . ( 2 . 1 2 )where, L is length of the MHD channel (L = 0.5 - 3 . 0 m).

Th e Magnetic Reynolds Number is a measure of anextent to which the gas motion can modify or deflect theapplied magnetic field. When R, = 1.0, then the inducedmagnetic field Bi is approximately equal to the appliedmagnetic field Bo. With the typical operating valuesconsi dered earlier we get the ra ng e of R, from 0 . 0 5 to 2.50.For the design o f a n 100 M W MHD generato r, which is discussedin details late r, the R, = 0.2.


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1 32 . 2 . 5 Induced maenetic f i e l d

The current density J set up inside the MHD channelinduces its ow n magnetic f i e l d B i . The component o f inducedmagnetic field d B 5 due to each current element ( I . d s ) at anygiven point F is gi ven by Eq.13, which known as the Biot-Savart Law. This law can be derived from the Maxwell's--equation (Eq.4).

d B i = u.I d;ixR4 x . . . ( 2 . 1 3 )

w h e r e , R is the displacement vector fr om the current elementto the point P and R is its magnitude (m).

dBi .Ihe net induced field Bi at point P isThe induced magnetic field Bi adds up vectorially to

the applied magnetic field Bo to give the total magneticfield B.

Th e stan dard fluid mechanics equations f o r flowthrough any channel also apply for the MHD channel. At anypoint inside the MHD volume the mas s, momentum and energy o fthe fluid are conserved.

The conservation of mass law applied to a fluidpassing through an infinite simal, fixed control volume yieldsthe equation of continuity, which is given by Eq.2.14.


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14

+ v . ( y u > = 0St . . . (2.14)wher e. is the density o f the fluid (kg/m 3 ) .

New ton 's Second Law applied to a fluid passingthrough the - control volume yields a followingequation: - -

9 . a + p . v u = f +v .x i jSt

momentum

. . . (2.15)

wbere, x i j is the s tress te nsor (N/m2),f is the body force per unit volu me, whic h includesthe gravitational force and the Lorentz force fl

terms, fl = J x B . . . r om E q . 2 . 6Substituting the values for each component of the

stress tensor we get three scalar equations, which are kno wnas Navier -Stokes equations [ 1 2 ] . By assuming incompressibleflow with coefficient of viscosity p , the equation reduces t oa much simpler form give n as follows:

q.m + ? U . ( v . U ) F - - J P + / ~ . v ' 2 US t

. . . ( 2 . 1 6 )

The First Law of Thermodynamics applied to a fluidthrough the control volume yields the energy equation.9 . h + g U . V e + P.(V.U) = 49 + k . q 2 T + F.U + p . . . (2.17)6t St


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15w h e r e , e is the internal energy per unit mass ( e = C,.T),

k is the thermal conductivity o f the fluid (W/'K/m>,cp is the viscous diss ipat ion function (W/m2),

is the rate of heat produced per unit volume byexternal agencies such as Joule dissipation.St

F o r a n -M-HD channel we can substitute,

hQ=E6t u

2 . 4 The Geometry o f MHD Generator

. ( 2 . 1 8 )

MHD generators are designed in two entirely differentgeometries. The first geometry is known as linear MHD, whichhas a simple duct type shape with a mostly rectangular cr oss-section. The typical linear MHD generator is sh own earlierin Fig. 2.1.

T h e o t h e r g e o m e t r y c o n s i s t s of two concentriccircular electrodes with magnetic field applied along theaxis a s shown in F i g . 2.2. The conducting flui d flows inbetween the two electrodes with radial velocity V, andtangential velocity Ve. This generator is called disc typeMHD generator. The dis c-M HD generator has a more compactgeometry and its power density is higher th an the typicalline- MHD generator.

Linear MHD generators differ from each other in termsof the electrode geom etry . The different types o f electrodegeometries are considered below.


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16

2.4.1 Faradav generatorThe Faraday generator is sh own in Fig. 2.1. Since t h e

electrodes are continuous along the axis of the duct theelectric field along the axis E, = 0. Using E q . 2 . 1 0 , we getthe expressions f o r the Faraday current density J y in thenormal direction and the leakage Hall curren t density J, as--f o l l o w s :

J y = 0 . ( U B - E y > j1 + W2?*J, = - w 7 0 . ( U B - EY) i

1 + W2?2. . . ( 2 . 1 9 ). . . ( 2 . 2 0 )

A s the electric field Ey has direction opposite tothe current density J y it appears with the negative sign.The Hall current finds a leak age path through e ach electrodeplate completing the loop. The useful electric power can beobtained only from Faraday current and output power densityis given by ,

- U . ( U B - Er).Ey1 + w272E Yo = J . E = J . . . (2.21)

The approximate value f or the overall power densityfor the MHD channel is given by Eq.2. 22.

P o = 0 . 2 u U2B2 . . . (2.22)2.4 .2 Segmented Faradav nenerator

In this geometry the electrodes are segmented into


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17many separa te pai rs , each pair feeding a separate load a sshown in Fig . 2. 3. The basic advantage o f segmenting is tob l o c k th e Ha ll cur ren t and eliminate the Joule heatdissipation losses due t o this component.

2 . 4 . 3 Hall generator- -When Hall parameter ( W T ) of the medium has high value, i . e . , T 2 1.0, i t is advantageous to extract power from theHall component of the current rather than the Faradaycomponent. This i s achieved by the type of g eome try, calledas the Hall gen erat or, shown in F i g . 2 . 4 . The Hall currentdensity J, and the output power density is given as follows,

J, = -u .(wTUB - Ex) i1 4- W 2 T 2

P o = J . E = J x . E x = uwr .(UB - Ex).Ex1 + w2r2

. . . ( 2 . 2 3 )

. . . ( 2 . 2 4 )

The Hall generator has the advantage of high outputvoltage and the highest efficiency is obtained under heavyload cond itio ns, i. e. , just the opposite of a Faradaygenerator. The pressure gradient produced in the Hallgenerator is in the direction perpendicular to the fluid flowdue to which it suffers the drawback of nonuniform fluidproperties transverse to fluid flow.

2 . 4 . 4 Diae onal ly connected generatorThe electrodes which would lie on the same potential

sur fac e in a normally operating Farad ay generator are


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i aconnected. As a result no Hall current flow s and obtains theefficiency advantage of the Faraday connection plus the Highoutput voltage and reduced number o f output circuits of theHall connecti on. To get maximum efficiency the angle of thediagona l connect ion has to be optim ized for every value o fWT. For the intermediate values of W T , i . e . , W T =: 3 . 0 , thisgenerator is more efficient than both Hall generator andFaraday generat or. The Fig. 2.5 shows the diagonallyconnected generator. The diagonally connected electrodeconfiguration is further modified to obtain the frame typeelectrode configuration.

A l t h o u g h a m o n g t he l i n e a r MHD g e n e r a t o r s t h erectangular cross-s ection geometry is the most common, theMHD ducts with circular and oval cro ss-section s are alsostudied. A circular or an oval cro ss-s ectio n has structuraland aerodynamic advantages over a rectangular o ne , but itcauses distortions i n the electric field and current flowpattern. The output power of a circ ular channel MHD as afunction of an angular width of an electrode is studied by F .J. Fishman and the optimum value lies i n between 3 0 to 4 0 '[ 1 3 ] . The frame type MHD with oval cross-sec tion is studiedby V. A . Kirillin and its comparison with other geometriesis reported [ 1 4 ] .


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5.07 MPa

KF L I2125K5.07 MPAKF (GI2665K

Cycle Efficiency 22% Primarv Radiator Area 910m -KF L )1920 K5.07 MPa

Secondary Radiator Area 54 m290 ka/sP-=4 Okw

TTP I T \ I Secondarv I Il l7 t r _ \ b92oKP=75hrp

P=200MWe.07 MPa1815 K59 kg/s f

=666MWou t

MHD080 K5.07 MPa Nozzle Generator46 kg/


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2 0

F i g u r e 2 . 3 T h e S e g m e n t e d F a r a d a y t y p e M H D g e n e r a t o r 1111

r LOOd 1

F i g u r e 2 . 4 The H a l l t y p e MHD g e n e r a t o r [ l l ]

F i g u r e 2 . 5 The D i a g o n a l l y c o n n e c t e d M H D g e n e r a t o r ill1


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CHAPTER 3THE PROPOSED NUCLEAR DRIVEN MHD SYSTEM ANDFORMULATION OF THE MODELING PROBLEM

3 . L - SDace Power System DescriDtion

The proposed system is an innovative space nuclearpower concept for closed cycle operation for 1 0 0 - 4 0 0 MW,output power leve ls. An Ultrahigh Temperature Vapor CoreNuclear Reactor (UTVR) combined with a linea r MHD generatorto operate on a dir ect , closed Rankine type cycle is beingstudied (Fig. 3.1).

The central vapor core region of the reactor containsa mixture of metal fluoride (e. g. potassium fluoride KF) andhighly enriched (85%) uranium-tetrafluoride UF4 vapor. Themole frac tion of KF is about 0 . 9 0 . The vapor fuel mixtureexiting the reactor passes through the nozzle and enters theMHD generator at a temperature o f about 4000 'K and pressureof about 5 MPa. The mass flo w rate of the closed cyclesystem at 200 MWe power level is estimated to be 286 kg/s.The gas mixture after exiting MHD generator passes through adiffuser and a radiator heat exchanger. The metal fluoridecondenses in the radiator and heat exchanger. Then it ispressurized andinjected back into the reactor, where i t isvaporized and mixed wi th the UF4 gas.

2 1


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2 2

W tl2125 K5.07 W A

F i g u r e 3 . 1 . S c h e m a t i c D i ag r sr n o f U T V R - M H D R a n k i n eP o w e r C y c l e [ 1 5 ]


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2 33 . 2 The Desien Constraints o n the MHD Generator

A s discussed in Section 5 1 . 2 , since the MHD convertoris compact and has a very lo w weight to power ra ti o, it ishighly useful for space power applications. The use of MHDgenerator also eliminates the possibility of any problems dueturbine vibrations and gyroscope effect in the space vehicle.--To attain high power lev els in the MHD channel the operatingflui d, a gaseo us mixture of UF4 and KF, must have highelectrical conduc tivi ty. The gas temperature in the excessof 3 0 0 O o K is necessary to get sufficient conduc tivity, whichis achieved with the help of ultr a-h igh temperature gas corereactor. The high operating temperatures of the proposednuclear dri ven MHD system also ensure high efficiency ofenergy reje ctio n in the radiators. The redu ctio n in theare a, weight and cost of the radiators fu rthe r makes theproposed space power scheme more attr acti ve.

There are two types of MHD generator s, as discussedin Section $ 2 . 4 , a disk-MHD and a l i n e - M H D . A 1 though thedisk-MHD has higher power density, the proposed space powersystem is a closed-loop system and the gas at the exit of theMHD generator must be fully collected and recycled. Thisrecycling design adds excessive volume and weight to thes y s t e m a n d l i m i t s t he u t i l it y of the disk type MHDgenerator. In this report a linear MHD is being looked intoas the possible design option for the proposed syst em.


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2 4The linear M H D can have different types of electrode

configurations as described in Section $ 2 . 4 . The typicalvalue of the Hall parameter for the proposed system liesbetween 0.15 t o 0.8 (refer to Section 2 . 2 ) , whi ch means thatthe Faraday type MHD generator is the most efficient type ofdesign in c ~m pa ri so n with the Hall type and diagonallyconnected MHD generators. To utilize the maximum volume oft h e F a r a d a y t y p e MHD cha nne l a conti nuou s electrodec o n f i g u r a t i o n i s p re f e r r e d o v e r t h e s e g m e n t e d t y p eelectrodes.

The basic requi remen t of compact geometry is met byoperating the MHD in the medium and strong (R, = 0.2 to 1.5)interaction regime, The higher the temperature of the gasmixture at the entrance of the MHD c han nel , the higher willbe the electrical conductivity leading to the larger outputpower density. The average temperature at the entrancecros s-se ctio n is limited by the temperature at the exit ofthe U T V R . The MHD system is being designed f or the powerlevels from 2 5 MWe up to 1000 MWe and typical active Volumeo f the MHD channel is of the order of 1.0 m3. The Eq.2.22gives the approximate form ula for the overall power density.

The proposed workin g fuel/fluid is a vapor mixture ofuranium tetra-fluoride U F4 ( ~ 1 0 % ) nd potassium fluoride KF( ~ 9 0 % ) n the partially ionized state. The gas mixturepassing through the MHD channel also contains fissionproducts. In addition to thermal excitation there also non-


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fission25

equ ili briu m ion izat ion induced because of thefragments.

The typical values of t he working parameters aredetermined by the required power leveland the constraints dueto overall system design. The superheated vapor mixture atthe exit of the-.UTVR has a temperature of 3000 to 4500K anda pressure of 3 to 8 MPa, which sets limits on the values o fthe operating ranges of temperat ure and pressure fo r the MHDgenerator. To achieve the power level s of few hundreds ofmeggawatts the typical operating ranges for the appliedfield, inlet velocity, inlet temperature and inlet pressureare selected to be from 2 to 8 tesla, 800 to 1800 m / s , 2800to 4 5 0 0 ' K and 3 to 8 MP a, respecti vely. The electricconductivity of the gaseous mixture i s a fu nction of gastemperatur e, pressure, composition. The conductivity isfurther enhanced due to the presence of fission frgments andhigh values of cu rrent density and for the proposed system itlies between 10 to 500 mhos/m.

T h e s e d e s i g n c o n s t r a i n t s d e f in e t h e r an ge s ofoperating conditions for the MHD generator. Within theboundaries of these constr aint s, the system parameters areadjusted so a s to optimize the performance of the overallsystem. Certain additional criteria such as keeping velocitymore or less constant along the MHD channel or a constantpower density or a specified temperature or pressure dropacross the channel may also be imposed on the system.


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263 . 3 Formulation of the ?roblem

The aim of this study is t o perform preliminaryanalysis of the MHD channel for the proposed system. Anattempt is made t o develop a simplified computer modelincorporating both the electrodynamics and flu id mechanicsaspects of the- MHD. This computer model is intended t opredict the basic electrodynamic phenomena in the MHD channeland to estimate the variation of te mper atur e, pressure,velocity and current density in the flow direction andvelocity and current density variation across the flow.

The proposed system is being designed to operate atpower lev els ranging from 25 YW up to 1000 MW. The operatingvalues of different parameters a re adjusted to ke ep the axialvelocity more or less constant along the fl ow direction.This computer model is intended to calculate the net changein pres sure , temperature and velocity and also the totaloutput cu rr en t, voltage and electric power. The geometry ofthe channel plays a n important role in the performance of thesystem and the optimum geometry has to be determined at eachpower level.

To dev elop a computer model for the line ar MHD channelfor the prediction of electrodynamic phenomena and parametricstudy, the fuel / working fluid mixture is simulated byhelium gas . This is mainly due t o a lac k of available dataon t her mal , electrical and transport properties of uraniumtetrafluoride and metal fluorides at such high temperatures.


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2 7The electrical conductivity of the fuel / working fluidmixture is assumed to follow the variat ion as that o f seededhelium. Non-ther mal ionization of the MHD gas mixture by thefiss ion fragments has been hypothesized to significantlyenhance the electrical conductivity. The actual functionaldependence of the conductivity for the UF 4 and KF mixture attemperatures above 2 5 0 0 O K is being investigated. F o rdeveloping this computer model it is essential to kno w thevalue of electrical conductivity at eac h point inside the MHDchannel. Fully seeded helium gas at similar temperature andpressure is used to substitute the working fluid mixture fordeterming the electrical properties. How ev er, it should benoted that these assumptions are justified only for thee v a l u a t i o n of the electrodynamic phenomena in the MHDgenerator. The thermal and transport properties of uraniumf l u o r i d e a n d me t a li c f l u o r i d e s a r e e x p e c t e d t o b econsiderably different from inert gases.

-

To perform this preliminary analysis a quasi-onedimensional computer model is dev elo ped , which solves thefluid mechanics equations discussed in Sectio n 0 2 . 3 takinginto account the electro-magnetic f o r c e and energy terms.The deta iled solution method is described i n the nextchapter.


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CHAPTER 4SOLUTION METHOD

A quas i-on e dimensional com puter model is developedto perform phenomenological study and to simulate the-- .interplay between the system state variables and differentdesign parameters. The model takes into acco unt the majorphysical aspec ts of the system involvi ng both electrodynamicsand fluid mechanics.

In t h i s i n c o m p r e s s i b l e q u a s i - o n e d i me n si o na lapp roa ch, the component of the fluid velocity in thetransverse direction (Y, Z directions) is assumed to be verysmall and is neglected. But the axial velocity, U ,temperature, T , and pressure, P, are described as functionsof position (x,y,z). The solution proceeds i n the axialdirection (X direction) from one axial node to the next,using a finite difference technique. The first step in thismodel is to calculate the average values of the stateparam eter s for the node cross-sectio n. The governingequations in fluid mechanics as give n i n Section 2.3 areconverted into one dimensio nal form. The one dimensionalcontinuity equation can be written as follows:

b(?.U.A) = 0d x . . . ( 4 . 1 )2 8


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2 9The equations of conservation of momentum and energy

are further modified t o include the effects o f electric andmagnetic fie lds. The momentum equation includes Lorentzforce per unit volu me, fl and the energy balance equationincludes the energy consumed t o d o work against f l and energyco nv er te d from. ,electrical to thermal because of Joul eheating. These equations can be written as follows:

-{ ( T V Z + P ) A } = - ( F f + f1.A) . . . ( 4 . 2 )d x

where, ,u is the viscosity of the gas (N.sec/m2),A is the c ross-sect ional area o f the channel (m2),Ff is the frictional drag force per uni t len gth (N/m)fl is the Lorentz forc e per unit volume (fl = Jy.B,).For the Reynolds number Re in the range of l o 6 , the

frictional drag force ca n be determined by the followingempirical relation ship, know n as the Blasius re lation:

where, C f is the turbulent frictio n fact or,Dh is the hydraulic diameter of the chan nel, which is

given by Dh = 4 x AreaPerimeterThe Reynolds Number is given by, ReD = 9 . U . L

PThe energy balance equation is give n by Eq.4.4.


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3 0-{UA.(fCVT +c;u:! + P ) ) = d ( k A . a ) +JnA -U(Ff+fl.A) -H,d X 2 dx dx (J . . . ( 4 . 4 )where, k is the coefficient of thermal conduc tivit y,

H, is the rate of heat l o s s t o the sidewalls andele-ctrodes per unit le ng th .

The gas-.flowing thro ugh the MHD channel is at veryhigh temperat ures (above 3 0 0 0 K) and the Ideal Gas Lawrelation is assumed to hold and is used t o substitute for thegas density , as given by E q . 4 . 5 .

P = 7 . R . T or p = P . . . ( 4 . 5 )R.TThe electrical condu ctivity of the gas mixture is a

strong fun ction of its temperature and it a l s o depends on thegas press ure, gas composition, current density and density offission fragme nts and hence on neutr on flu x. The exactvariation of the conductivity of the working fluid in thetemperature range of 2500 to 4500K is not yet known. Theconductivity of the gas mixture is enhanced due the presenceof the fissi on fragments, which induce non-equilibriumionization. A s discussed in Section $ 3 . 3 , the electricalproperties of the gas mixture are simulated by helium seededwith cesium (He + 1.41% C s ) . An approximate function for thegas conductivity, u , is used, which is gi ven as follows [ l l ] :

4 . 0 0 . 5CT = 8 0 . [ T mhos/m . . . ( 4 . 6 )

3000 ] . [+]


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3 1w h e r e , T is the gas temperature in OK,

P is the gas pressure in MPa.T h e R e y n o l d s N u m b e r a t t h e t y p i c a l o p e r a t i n g

conditions is l o 6 or hig her , whic h implies a turbulent typeflow. A t t h e inlet o f the MHD channel a fully-developedvelocity profil-e is assumed w hic h can be gi ve n by the po we r-law approximation as follows:

, n = 7 . . . (4.7)

w h e r e , U is the velocity at a distance Y from the wal l,U is the maximum value o f the gas velocity,W is the total width of the channel.The temperat ure profile in the transverse dir ecti on is

gi ve n by the Reynolds Analogy for the fully-develop edincompressible turbulent fl ow through a pipe. It assumes athermal boundary layer equal to the momentum thickness, 6 andassumes linea r variation o f temperature across the boundarylayer.

w/ 2S = I /Uo . ( 1 - u/Uo) .d y

y=o. . . ( 4 . 8 )

w h e r e , u is the axial velocity at distance y from the wallor the electrode (m/s),

U, is the axial veloc ity at the cen ter (y = W/2)For the velocity profile expressed by Eq. 4.7. S =


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3 2In the core o f the fl ow , the stagnation temperature,

T o , is assumed t o be constant.T o = T + Z

2 c P. . ( 4 . 9 )

The rate of heat loss to the wall per unit area HW',is given by the-.following eq uation :

Hw' = h.(T&-T,) . . . ( 4 . 1 0 )whe re, h is the heat transfer coefficient at the wall ,

h = k/S (W/ma/"K),Tg is the temperature at the edge of the boundary layer ,

Hw = H,'.(wall perimeter)The ma ss , momentum and energy balance equations in the

one dimensional strong conservation form are transformed intomatrix equation as represented by Eq.4.11.

. . , (4.11)

w h e r e , [ A ] - - Coefficient matrix (3x3)[B] - - Value matrix (3x1)[ U ] - - State parameter matrix ( 3 x 1 )


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3 3

The individual elements o f each o f these matrices aregiven below:

U.RT-u2c;T

-9.U.uA d x

-qUz.u C f w J y . B ,A d x 2Dh d t U 1 =d x

-Udx-Pd x-Tdx

The average values of state parameters are updatedand advanced along the flow direction (X direction) using afinite differe nce method. The values of all the elements inthe [ A ] and [B] matrices are calculated at the axial node (I)and the matrix equation is solved t o get the axial gradientof the [U] matrix. The fini te differ ence method incorporatesa predictor-corrector scheme similar to the Explicit Mac-Cormak Method. The predictor step uses forward differencingand a corrector step with central differencing is used toeliminate the one-s ided bias of the forward differencingstep.


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3 4

The steps used for advancing the solution from axialnode (I) to node ( I + 1 ) are described below:S t e p - I Evaluate each element in the ;A] and [ B ] matricesusing the values of the state parameters at the node (I) andform the matrix equatio n,- -

Ste o-I 1 Solve this equation using matrix inversion andevaluate [u(I)]dxSte D-I 11 Estimate the values of the state parameters at thenode (I+1) using their differential values along the axiald i r e c t i o n .differencing.

( T h i s i s a pre dic tor step with forwardThe asteri x mark above the state parameter U

indicates that it is only a first guess at the value.)

u*(I+1) = u(1) + 6 x . d U ( I ) + Sx 2. d2 U(1) . . . ( 4 . 1 2 )d x 2 dx2SteD-IV Calculate the average value of the state parameterfor the interval from node (I) to node (I+1)

SteD-V

U(I+%) = [U(I) + U"(I+l)] / 2.0Redefine [ A ] , [ E ] us in g U(I+%) to gi ve ,

. . . ( 4 . 1 3 )

and substitute i n the matrix equation (Eq.4.11)


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3 5SteD -VI Solve for d (I+k) using matrix inversiondxSteD -VI1 Repeat step -I11 to calculate U(I+l) using the newd i f f e r e n t i a l v a l u e s o f t h e s t a t e p a r a m e t e r s a t t hei n t e r m e d i a t e n o d e (I+%). (This is a corrector stepincorporating- central differencing.)--This method is stable only in the subsonic range offlow velocities as the solu tion advances only i n the forwarddirection. The typical operating velocities are chosen t o befrom 800 to 1800 m/s, whi ch are well below the sonic velocityin helium at the operating temperatures (about 3500 m / s ) .

This method estimates the variatio n o f average valueso f pressure, temperat ure, density and velocity at each axialnode. The local values of v eloc ity, current density andtemperature at each c ross -sec tion are differe nt from theaverage values. By incorporating the local values of stateparameters in the momentum balance equation (Eq. 4 . 2 ) thevariations of axial velocity and current density in thetransverse directions ( z -directi ons) at each axial node arecalculated. The average values o f the thermal and flowparameters wh ich are estimated at earlier ste ps, serve tocalculate normalizing factors.

Another important feature of the medium and stronginteraction M H D generator is the significant modification inthe applied magnetic field due to fluid motion. Each currentelement i n the MHD channel induces its own magnetic field inthe surrounding space which is given by the Biot -Savart law


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36( E q . 2.13). The total induced magnetic field due t o eachcurrent segm ent, adds vectorially t o the applied field.This modification in the magnetic field changes the currentdensity distribution and magnetic force valu es, whic h in turnchange the values of the state parameters i n the chan nel .The changes in-_the urrent densit y pattern fu rthe r change theinduced magnetic field valu es. To incorporate these coupledflow and electrodynamic phen omen a, the computer model usesthe following iterative scheme,

Estimate the variation in stateparameters and calculate thecurrent density (J) pattern

- - T - - - - - - - +Applied magneticfield (Bo) t.II&Determine the net + - - - - - - - - - Calculate the induced

magnetic field ( B ) magnetic field (Bi)Figure 4.1 The Iterative Scheme

The rate of convergence i n this iterative scheme isvery fast for low values of the Mag net ic Re yn ol ds Num ber R,.For a wide range o f medium and strong interaction MHDchannel problems which were mode led, the convergence wasachieved within 4 iter ati ons fo r R, I . 4 and within up to10 iterations for R, 1 0.7 (to give better tha n 1% accuracy).


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CHAPTER 5RESULTS AND DISCUSSION

5.1 De-siqn of 2 5 . 100 and 4 0 0 MW MHD Generators- -Using the quas i-o ne dimensional computer model a

deta iled study was performed s o as to select typicaloperating parameters for the 2 5 , 100 and 400 MW, MHD powergenerator. The operating ranges of the gas temperature,pressure and flow velocity and the s trength of the appliedmagnetic fie ld are chosen to enable achieving the desiredpower le vels of the MHD channel and to suit the des ign of theoverall system.

The variation of the c ross-sec tional area o f the MHDchannel plays a major role in determining the variation inother parameters of the system and the performance of the MHDgenerator. A 25 MWe MHD generator with a constant crosssection area is selected and the operating conditions arespecified in Table 5.1. The variation of the axial velocityalong the le ngt h of the channel is plotted in Fig. 5.1. Thedrop in the pressure in the axial direction is faster thanthe drop in temperature which leads to decrease in the gasdensity along the flo w direction. This results in monotonic

3 7


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3 8

0

C a s eP o w e r = 25 MW) /

0.25 0.5 0.75 1 1.25 1.5

F i g u r e 5.1. V a r i a t i o n o f t h e A x i a l V e l o c i t y o f t h e G a sw i t h t h e D i s t a nc e a l o n g t h e M HD C h a n n e l .


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3 9increase in the flow velocity along the channel length asillustrated by the graph.

The increase in the gas velocity at constant area isextremely fast at higher power levels. The current densityand power density a l s o increase along with the gas velocity.For effective -utilization of the MHD channel volume andavailable operating con dit ions , it is desirable to choose adiverging type MHD channel s o as to maintain the velocityalmost constant.

The two electrode plates apply the electric field Eyacross the MHD channel, which can be given by Ey = V/W,where V is the potential difference between the electrodes involts and W is the widt h (in the Y direction) in meters. Tokee p the current density con sta nt , the electric field has tobe maintained at the same value a long the MHD channel(Eq.2.19). Hence the width W of the MHD channel ismaintained constant and the breadth along the Z direction isincreased to get a diverging type channel geomet ry.

The results were obtained using the computer modelunder various operating c ondi tion s and a comparative studywas performed. The angle of divergence of the cross sectionarea was adjusted to match the inlet and outlet velocities.The reas on for keeping the axial velocity more or lessconstant along the channel is to have a uniform distr ibut ionof current density and he nc e, the power dens ity , ensuring afull utilization of the channel volume. A set of typical


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4 0operatin g values for each parameter are selecte d for theoutput power levels of 2 5 , 100 and 400 MW,. The op er at in gconditio ns and performance resu lts of the MHD system aregiven in Table 5.1.

Table 5 . 1 Design of 2 5 . 100 and 4 0 0 MW MHD Generators

Channel Geomet rv -Length (X-direction) = 1 . 5 m , Width (Y-direction) =i 0 . 7 m ,Workine Fluid - Partially ionized helium

Case +

Flow rate (Q)Inlet Tempera ture (Ti)Inlet Pres sure (Pi)Inlet Velocit y (Vi)Inlet Area (Ai)Ex it Te mp er at ur e (T,)Exit Pressure (P,)Ex it Ve lo ci ty (U,)Exit Area (A,)Magnetic Field (B)Mag. R eyn old s No.(Rm)Variation i n B ( B)Wall Temperat ure (T,)Heat loss to walls (H)Internal Energy (C v T)Work d one (W,)Kinetic Energy (PKE)Voltage (V)Current (I)Output Po wer (Pout)

I2 5 MW2 7 5 kg/sec2 8 0 0 OK4 . 0 0 MPa800 m/sec0 . 5 m22 7 7 5 . 4 OK3 . 7 7 MPa8 4 2 . 7 m/sec0 . 5 m 22 . 2 tesla0.100 . 0 62 2 0 0 OK+ 0 . 3 4 MW- 2 1 . 1 MW- 1 3 . 3 MW+ 9 . 6 M W448 V5 5 . 1 KAmp2 4 . 3 MW

I1100 MW

2 7 5 kg/sec2 8 0 0 OK4 . 0 0 MPa8 0 0 m/sec0 . 5 m22 7 3 0 . 3 OK3 . 3 7 MPa7 9 7 . 8 m/sec0 . 5 8 m24 . 0 tesla0.100 . 1 22 2 0 0 OKt 0 . 3 2 MW- 5 9 . 9 MW- 3 9 . 9 MW- 0 . 5 MW8 9 6 V1 1 0 . 6 KAmp9 9 . 7 MW

I11400 MW

2 7 0 kg/sec3500 O K4 . 0 0 MPa9 8 0 m/sec0 . 5 m23 2 1 6 . 3 O K2 . 3 5 MPa9 8 5 . 7 m/sec0 . 7 7 m24 . 0 tesla0 . 2 40 . 4 12 2 0 0 OK+ 0 . 5 5 MW- 2 4 2 . 2 MW- 1 5 2 . 7 MW+1.5 MW1 0 9 7 . 6 V3 6 4 . 7 KAmp4 0 0 . 3 MW


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41

5. 2 Variation of Parameters along the Flow Direction

Using the computer mo del , the variation of variousparameters along the flo w direction is estimated and thedetailed results for Case I11 (output power = 400 MW,) arepresented here _. The analysis of this variation enhances thebasic understanding of the physics of the problem and c an beused for improving the performance of the MHD system.

The cross-sectional area of the MHD channel increasesalong the f low direction and the angle of divergence is S Oadjusted t o match the inlet and outlet velocities. Thevariation of the axial velocity of the gas along the channelis plotted in Fig .5. 2. A s can be see n from the graph thevelocity does not remain constant but varies along the lengthof the channel. The velocity kee ps decreasing until the mid-section of the channel and then increases back to theoriginal value.

This typical variation of the velocity is the resultof interplay between all the parameter s, but it can bepartially understood by analyzing the variation of pressurealong the length of the channel (Fig .5.3) and the work inputdue to the pressure drop. The linear variation of pressureand area lead to a peaking of their product at the mid -section as c an be seen from the results sho wn in Table 5 .2.


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920

900

4 2

C a s e I 1 1( P o w e r = 4 0 0 MW)

1-

i\\

,?I

i I1 1/ 1,d ,

\

I.

F i g u r e 5 . 2 . V a r i a t i o n o f t h e A x i a l V e l o c i t y U o f t h e G a sw i t h t h e D i s t a n c e X a l o n g t h e M H D channel


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4 3

4

25

C a s e I11

4 1.25 1.5i2 6-25 0.5 0.75 1

F i g u r e 5 . 3 . V a r i a t i o n of t h e P r e s s u r e P w i t h t h eD i s t a n c e X a l o n g t h e MHD C h a n n e l


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44Table 5 . 2 Calculation Table

1 Cross- Area Pres ure P . A Velocity P.A.USection -4 (mz) P (MPa) ( N ) U (m/s) Wo(MW)Inlet 0.50 4 . 0 0 2 . 0 0 980.0 1960.0Middle 0 . 6 6 3.175 2.032 930.0 1874.9Exit 0.78-_ . 2.35 1.832 985.7 1805.8

The net -in put of power due t o the work associated withthe pressure is 15 4. 2 MW , whic h is the difference between theinlet and outlet Wo values. It is noticed f rom the tablethat the product P . A peaks at the center. The lowe r value ofvelocity at the mi d-s ect ion ensures that the value of W odecreases more or less uniformly along the c han nel , whichmeans a continuous input of the work associated with thepressure.

The current density Jy is directly proportional t o theaxial velocity of the conducting gas and hen ce , its variationalong the channel, as sh own in Fig.5.4 , is similar to that ofthe velocity. A s can be seen from the gr aph , the currentdensity inc reases more rapidly near the exit of the M H Dchannel. This rapid increase can be attributed to twofactors. Fi rs t, the percentage drop in the pressure alongthe flo w direction is far higher than the percentage drop int e m p e r a t u r e , w h i c h r e su l ts in i n c r e a s e d e l e c t r i c a lconductivity towards the exit. Sec ond , the value of the netmagnetic field keeps rising along the channel and reaches


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4 5maximum at the exit (reference F i g . 5 . 6 ) . Since the currentdensity is directly proportional to the conductivity andmagnetic field, the rise is very fas t at the exit.

The graph of temperature versus axial dis tanc e, asshown in F i g . 5 . 5 , is more or l e s s linea r, but i t has slightdrooping nature-nea r the exit. More energy is converted intoMHD output towards the exit end due to high values of currentdensity. The kinetic energy also is again built up in thelatter half of the MHD chan nei. This energy is extracted5rom the internal energy of the gas leading to a faster dropin temperature near the exit.

5 . 3 Different Designs for the 1000 MW MHD GeneratorAt higher power levels (of the order of 1000 MW,)

maintaining the volume of the MHD channel almost constantrequires very high power densities . This can be achievedeither by increasing the axial velocity or the temperature orthe strength of magnetic fie ld. These three alternativeswere analyzed using the computer model and the resu lts aretabulated i n Table 5 . 3 . The values of the temperature drop,pressure d rop , flow ra te, heat loss rate and output voltageca n be compared using the tabl e. A comparative study ofthese designs is helpful in selecting a set of operatingconditions best suited for the design of the overall sys tem.

It should be noted that in Case IV the flo w rate ofthe gas is considerably higher a s compared to other cases due


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4 6

260

250

C a s e I 1 1( P o w e r = 4 0 0 MW)

/i

F i g u r e 5 . 4 .

0 0.25 0.5 0.75X ( m > -+ 1.25 1.5V a r i a t i o n o f the C u r r e n t D e n s i t y J y w i t h the

D i s t a n c e X a l o n g t h e MHD C h a n n e l


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47

F i g u r e 5 . 5 . V a r i a t i o n of t h e T e m p e r a t u r e T w i t h t h eD i s t a n c e X a l o n g t h e M HD C h a n n e l


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48to high value of the inlet velocity . The high operatingtemperature in Case V leads to higher electrical conductivityand henc e, ,he output current is larger .

T a b l e - 5 . 3 D e si g ns f o r 1000 MW M H D Generator- -Channel Geometry - Length L = 1 . 5 m , Width W = 0.7 m ,

WorkinP Fluid - Partially ionized Helium

Case -+

F l o w rate (Q)Inlet Temperature (Ti)Inlet Pressure (Pi)Inlet Velocity ( V i )Inlet Area (Ai)Exit Temperature (T,)Exi t Pre ss ur e (P,)Ex it Vel oc it y (U,)Exit Area (A,)Magnetic Field (B)Mag. Reynol ds No.(R,)Variation in B ( E )Wall Temperature (T,)Heat l o s s to walls (H)Internal Energy (Cv T)Wo rk done (W,)Kinetic Energy ( P K E )Voltage (V)Current (I)Output Power (Pout)

IVHieh V e l .7 0 5 kg/ s ec3 5 0 0 OK6 . 0 0 MPa1707 m/sec0.5 m23 2 3 3 . 7 O K3 . 5 4 MPa1 6 9 3 m/sec0 . 7 9 5 m 24 . 0 tesla0 . 4 10 . 8 12 2 0 0 O K+ 0 . 5 6 MW- 5 9 1 . 8 MW- 3 6 9 . 8 MW- 1 4 . 8 MW1 9 1 5 . 2 V524.0 KArnp1 0 0 3 . 6 MW

vHigh Temp.418 kg/sec4 1 5 5 OK6 . 0 0 MPa1200 m/sec0 . 5 m 23 6 8 6 . 5 O K2 . 9 4 MPa1 2 1 2 m/sec0 . 9 0 5 m24 . 0 tesla0 . 4 40 . 9 22 2 0 0 OK+ 0 . 9 3 MW- 6 0 9 . 4 MW- 3 7 7 . 1 M W+ 6 . 1 MW1 3 4 4 . 0 V7 4 5 . 3 KAmp 1L 0 0 1 . 4 MW

V IHigh Mag. B4 9 6 kg/sec3 5 0 0 O K6 . 0 0 MPa1200 m/sec0 . 5 m 23 1 0 5 . 5 OK2 . 9 1 MPa1 2 2 1 m/sec0 . 9 0 m 25.65 tesla0 . 3 10 . 8 52 2 0 0 OKt 0 . 5 4 MW- 6 1 5 . 3 MW- 3 8 1 . 5 MWt 1 2 . 3 MW

1 8 8 3 . 2 Vj 3 1 . 2 KAmpL 0 0 0 . 7 MW


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4 95 . 4 Analysis of the Electrodvnamic Phenomena

Three different types of electrodynamic phenomena areanalyzed here, which are peculiar to the MHD generat or. Theanalysis of these phenomena give us a deeper insi ght into thecharacteristics of the s ystem.- -5.4 .1 The Modification i n A ~ ~ l i e dagnetic Field

A s discussed i n Section $ 2 . 2 . 5 , the current densitypattern in the MHD channel volume induces its own magneticfield at each point. The induced magnetic field addsvectorially to the applied uniform magnetic fi eld , resultingin a nonuniform net magnetic field. This modification in themagnetic field depends upon the electrical conductivity ofthe flui d, its axial vel ocity , the strength of the appliedmagnetic field and the MHD channel length.

The variations o f the net magnetic field along thelength of the MHD channel are plotted for cases 11 , 1 1 1 , I Vand V in F i g . 5.6. Each graph shows that the the inducedmagnetic field opposes the applied magnetic f ield in thesection near the inlet of the MHD channel resulting indecreased values of the net magnetic field. Incontrast , inthe section near the exit both the fields add to give highernet magnetic field values.


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5 0

4.25?B( t e s l a )

5-75

C a s e V

, i I V,/'. P I' /'

I 1

F i g u r e 5 . 6 .

0.5 c.75 1 i25V a r i a t i o n o f t h e N e t M a g n e t i c F i e l d w i t h t h eD i s t a n c e X a l o n g t h e M H D c h a n n e l


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51

45

35

20

15 -

10 -'

5 -

--

c a s e f

I 11 I I 1 I I

0 0.2 0.4 0.6 0.8Rrn +

F i g u r e 5 . 7 . P e r c e n t a g e V a r i a t i o n ( B / Bx 1 00 ) i n t h eM a g n e t i c F i e l d E x p r e s s e d a s a F u n c t i o nof M a g n e t i c R e y n o l d s N u m b e r R,


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5 2The net change in the magnetic field ( B) in the axial

direction can be expressed in terms of the percentage valueof the applied magnetic field. The resulting percentagevalue is plotted as a fun ctio n of the Magnetic Reynoldsnumber (Rm), which is defined earlier in Section 0 2 . 2 . 4 , inFig. 5 . 7 . The graph shows that as the system operates inthe high Magnetic Reynolds number regime . i.e. strongermagnetic interaction with the fluid fl ow , the effect ofinduced magnetic fi eld becomes more signif icant.

-

5 . 4 . 2 Analvsis of the Transverse Profiles of Axial Velocityand Current Densitv

Fig. 5.8 shows the axial velocity profiles in thetransverse direction (Z -direction ) for case I 1 1 ( 4 0 0 MW,) atthe inlet and exit of the MHD channel. The resu lts show thatin the presence of the magnetic fie ld, the original velocityprofile gets modified.

The current density near the sidewalls is lower ascompared to that at the center due to the lower conductivityand the smaller velocity. The current density may have anopposite direction at the two edges. This resu lts in smallervalue of opposing Lorentz for ce near the sidewalls and thevelocity tends to build up here. The axial velocity profilechanges its shape as the fluid progresses i n the flowdirection. The typical velocity profile at the exit shows


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800

700

500

400

- ntrance- id-sectionU Exit- 1 .0 -0.8 -0.6 -0.4 -02 0.0 02 0.4 0.6 0.8 1.0

X


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5 3

I n l e t

C a s e I11

I I II i I i I l l ! ; * i i7L02 0.4 0.6 0.0 1 i.2 1.4. 1.6 1.8

F i g u r e 5 . 8 . T h e A x i al V e l o c i t y U P r o f i l e i n T r a n s v e r s e ZD i r e c t i o n ( a l o n g th e M a g n et i c F i e l d B)N o te : T h e b r e a d t h a l o n g t h e Z - d i r e c t i o n i s n o r m a l i z e d t o 2 m .


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54

340 1320 '300 '280 -260 -'

m 2200 -'180 -160 -

C a s e I:

140 1120 -loo -'8 9 -6 0 -

0 1 \ 1 i \ ~1 1 1 1 1 1 1 i 1 1 \ 1 10 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Figure 5 . 9 . T h e C u r r e n t D e n s i t y P r o f i l e i n T r a n s v e r se ZD i r e c t i o n ( a l o n g t h e M a g n e t i c F i e l d B )

N o t e: T h e b r e a d t h a l o n g t h e Z - d i r e c t i o n is n o r m a l i z e d t o 2 m .


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55thinning of the boundary laye rs near the sidew alls and slightovershoots at the boundary la yers .

I t is important to analyze the physical cause of theseovershoots at the sidewalls. When the value of axialvelocity in th-e boundary layer becomes equal to that at thecent er, the current density is still smaller due to the lowertemperature and hence the lower conductivity. Thu s, the

--

velocity can further increase leading t o overshoots. Theresult ing thinning of the boundary layer increa ses thetemperature restricting the overshoot. The viscous dragforce at the walls also serves t o limit the overshoot. Theeffects are more pronounced at higher power levels.

Fig. 5 . 9 shows the current density profile at theinlet and at the exit of the MHD channel for the Case 1 1 1 .The current density profile is a combination of the axialvelocity profile and the electrical conductivity profile.The curre nt density at the exit is more uniforml y spread outand almost flat .

5 . 4 . 3 The Presence of SDace-charp e in the MHD Channel

For the Faraday type MHD chan nel, the axial currentdensity Jx is given byY . . . Section $ 2 . 4, = - w7.J

At standard operating conditions the Hall parameter(WT) is less than 0 . 2 . When a diverging type channel is used


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5 6the current density J y is maintained nearly constant alongthe axis. Hen ce,

6 J x = 0 . 2 h J y =: 0S X 6 X . . . (5.1)

At steady sta te, the conservation of electric charge- -implies that,.J = 0 ==> L J X + L J y + S J , = 0 . . . (5.2)

SX 6 Y s z

Hence, L J y = 0 , or the Jy is constant in Y - d i r e c t i o n .SYJy(y) = constant = a.(BZ.Ux(y) - Ey(y)) . . . (5.3)

Thus the electric field Ey adjusts itself according tothe variation of axial velocity U, in the Y - d i r e c t i o n , s o asto maintain the value of the current density Jy constant.The electric field Ey has an integral boundary conditiongiven by ,

w/2Ey.dy = V = 1 . RIw/2

where, I is the total current through the external loadresistance R and V is the voltage drop a cross it.

In the boundary layer near the electrodes the partialderivative of the axial velocity with respect to distance inY -dir ecti on is very high. High value of L U X implies that

6 Y

. . . ( 5 . 4 )


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5 7the value of L E y is large. A l s o in the neighbourhood o f

6 Ythe metallic electrode the tangential components of theelectric field E, and E, are zero.

The Maxwells equation (Eq.2.1) reduces to ,L E Y = Q- . . . or E,=E,=OSY E . . . ( 5 . 5 )

where, E is the permittivity of the medium,SI is the fr ee space-cha rge density.

Thus the equations predict the presence of space-charge near the electrodes and its density directly reflectsthe variation of the axial velocity in the Y -direction. Anexperimental m ethod c an be devised to measure the variationin the electric fiel d (Ey) values along the Y -directio n andfrom this variation the profile of the axial velocity in theY -directi on can be calculated.

5 . 4 . 4 The Effect of ChanPinF the Load Factor

The load fac tor, Lf , of a n MHD generator is defined asthe ratio of the value of the transverse electric field (Ey)applied by the electrodes t o the emf generated by thevelocity of the conducting fluid transverse to the magneticfield (U.B). In the present study the load fa ctor is definedusing the inlet velocity and the applied magnetic fiel d.

Lf = E-Y -Ui.B . . . ( 5 . 6 )


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58

380

C a s e I 1 1( P o w e r = 4 00 MW)

.- I{J.

F i g u r e 5.10. V a r i a t i o n o f t h e O u t pu t P o w e r w i t h t h eL o a d F a c t o r Lf


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59The net power output of the MHD generator depends on

the value the load factor. Theoret ically the power output ofthe MHD system is maximum for Lf = 0 . 5 . The variation ofthe power output of the sys tem, estimated using the computermodel , with the load factor is plotted in Fig .S. lO. Thegraph sh ows that the maximum o f the output power is atLf = 0 . 4 . The reason for this descrepancy is due to theaverage value of the axial velocity being lowe r than thevalue at the inlet of the MHD channel. Hen ce, the averagevalue o f the load factor is higher than what is defined here.

-

5 . 5 Th Eff ct of Chan n nP the Leneth of the MHD Chann 1It is important i n the desi gn analysis of the MHD

generator to study the effect of changing the length of theMHD channel. By increasing the length of the ch anne l, theMagnetic Reynolds Number of the system also increases. A s aresu lt, the variation in the magnetic field ( B) is inducedto a greater ex ten t, as discussed in Section $ 5 . 4 . 1 . Fourdifferent cases are studied here by changing the length ofthe channel (L = 1 . 5 , 2.25, 3 , 4 m ) and net magnetic fieldresults are plotted in Fig.5.1 1. In each case in the cross-sectional are a at the inlet is 0 . 2 m2 and the flow rate ismaintained at 8 6 kg/s.

A s the length of the MHD channel is i ncre ased , thecharacteristics of the axial variation of the temperature,pressure, axial velocity and current den sit y, as discussed in


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6 0Section $ 5 . 2 , become more pronounced. These characteristicscan be compared from the graphs in the Figs.5.12-5.15.

The n e t output power verses the length of the channelis plotted in the Fig .S. 16 and the variation is almostlinear. But the average value of the cr oss- sect iona l area ishigher for the channels with l onger leng th, henc e, theaverage power density is not the same for all the cases. Thevariation of the average power density with the leng th of theM HD channel is shown in Fig. 5.17 . For longer l egth channelsthe temperature drop is more which lowers the electricalconductivity towards the ex it; consequently the currentdensity cannot rise back to its entrance level (refer toFig.5.15). The lower values of the average curr ent densitylead to lower average power density for the long er channels.

--


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61

1.11b1 L = 4 . 0 rn

0 0.4 0.8 2.4 28

F i g u r e 5 . 1 1 .

3 . 02 . 2 51 . 5

12 1.6 2X n = X -+L

V a r i a t i o n o f t h e N et M a g n e t i c F i e l d w i t h t h eNorm aliz ed Dist anc e X, alo ng the MH D chan nelf o r D i f f e r e n t V a l u e s o f t h e C h a n n e l L e n g t h L.


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6 2

3.5

3.4

3.3

3

2.9

2.80

ii;;;I i I I 1 I I 1 I I I i I I

0.4 0.8 1.2 2 2.4 2.8

F i g u r e 5.12. V a r i a t i o n o f t h e T e m p e r a t u r e T w i t h t h eN o r m a l i z e d D i s t a n c e X, a l o n g t h e M H D c h a n n e lf o r D i f f e r e n t V a l u e s o f t h e C h a n n e l L e n g t h L .


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6 3

1.4 41.2 ! i I I I I I I 1 I I I i I l

0.4 0.8 1.2 1.6 2 2.4 2.8

F i g u r e 5 . 1 3. V a r i a t i o n o f t h e P r e s s u r e P w i t h t h eN o r m a l i z e d D i s t a n c e X n a l o n g t h e M H D c h a n n e lfo r D i f f e r e n t V a l u e s of t h e C h a n n e l L e n g t h L .


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6 4

820 I

670660I L4.*i54 I I I I I I

1.6 2 2.4 2.8.4 0,0 1.2x ( m > +

F i g u r e 5.14. V a r i a t i o n o f t h e A x i a l V e l o c i t y U with t h eN o r m a l i z e d D i s t a n c e X n a l o n g t h e MHD c h a n n e lf o r D i f f e r e n t V a l u e s o f t h e C h a n n e l L e n g t h L.


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2x

210

2w190

150

1400 0.4 0.8 1.2 1.6 2

x ( m > +2.4 2.8

F i g u r e 5 . 1 4 . V a r i a t i o n o f t h e C u r r e n t D e n s i t y J w i t h t h eN o r m a l i z e d D i s t a n c e Xn a l o n g t h e MHD c h a n n e lf o r D i f f e r e n t V a l u e s o f t h e C h a n n e l L e n g t h L .


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6 6

310xx)

270q

I du100 r[ I i I I I I I t i I I I 4I1.4 1 a 2 2 I I2.6 3 3.4 3.8

F i g u r e 5 . 1 6 . V a r i a t i o n o f t h e P o w e r O u t p u t P o w i t h t heL e n g t h L of t h e M HD Channel.


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6 7

320

310

300

290

280270260250

220

230220

210200

0

- - .

1 3

F i g u r e 5 . 1 7 . V a r i a t i o n o f t h e P o w e r D e n s i ty P D o w i t h t h eL e n g t h L o f t h e MHD C h a n n e l .


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CHAPTER 6THREE-DIMENSIONAL COMPUTER MODEL

6 . 1 - Extension of the Work to a 3 - D M o de l- - -A qua si- one dimensional model reported in the earlier

chapters is very useful for the preliminary studies. Themodel is based o n a simplified phenomenological app roach andthe results obtained give an overall view of the response ofthe system to various types of operating cond itio ns. Formore accurate and reliable results, a three dimensional modelencompassing a l l the fluid mechan ics, electrodynamic andthermal phenomena has to be developed. A detailed 3 - D modelw i l l be an extension o f this work.

A scheme for the development o f the 3 - D model forthe MHD channel is cited here. In this scheme the fluidm e c h a n i c s e q u a t i o n s a n d e l e c t r o d y n a m i c e q u a t i o n s a redecoupled from each other and solved separat ely. The fluidme ch an ic s mo de l a nd electrodynamics model are coupledexternally and the solution is iterated back an d forth tosatisfy both sets of equations. An additional input to thee l e c t r o d y n a m i c s m o d e l i s p r o v i d e d b y t h e e l e c tr i c alproperties package. The electrical conductivity of the gasis a fun ctio n of t emp erat ure, pressure and various other

6 8


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6 9factors. The local value of the conductivity at every pointin the MHD channel volume is essential. The couplingbetween the models is indicated by the diagram (Fig. 6.1) ofthe overall scheme o f the proposed 3 - D model.

ConductivityElectrical Property Package - - - - + - - 0emp. T - - + - - -Pressure P 11I- It

Velocity Field VFluid Mechanic s - - - ~ - - - - - - - - - - - - - ~ - - - Electrodynamics

Model - - - + - - - - - - - - - - - - - c - - - ModelLorentz Force f(MHD Work & Joule Heat)

1

Energy Q iFigure6.1. Overall Scheme of the 3 - D Model

An experimental work has to be done to express theelectrical conductivity of the gas mixture i n an exactfunctional for m. The Lorentz force term f acts as a bodyforce term and energy term Q acts like a externa l source termfor the fluid mechanics model. In both flu id mechanics andelectrodynamics model the solution is advanced and updated infinite time steps. The solution converges to the steadystate solu tion as it progresses i n time. This approach ofsolving the equations is very c ommo n in fluid mechanics and alarge number of advanced techniques such a s Mac Cormak methodor Beam Warming method are available. In Section 0 6 . 2 , theefforts are made to formulate the solut ion scheme for theelectrodynamics model.


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706.2 Electrodvnamics Model

6. 2. 1 Basic EauationsThe proposed model assumes an initial solution and as

it differs from the steady state solution it generatestransient terms whic h update the solution as it advances intime. A s the -time progresses the tr ansien t t erms tend torestore the balance and the so luti on converges to a steadystate solution. A s the steady state sol uti on approaches thetransient terms tend die out and all the variables reach aconstant values in time.

The basic equation s in electrodynamics are the Maxwellequation s, which are discussed earlier in Section 82.2 :

v . E = QE

. . . (2.1)

'i7. B = O

V X E = - =6t

V X B = p . ( J + c . m )S t

. . . (2.2)

. . . ( 2 . 3 )

. (2.4)

The Eq.2.1 ti 2.2 are the scalar equations. The Eq.2 .3& 2 . 4 a r e the vector equations and lead to three scalarequations each. This for ms a set of eight equations. Thefirst two equations do not have differential terms with


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71respect to time and hence cannot generate any transientterms and it is advantageous to satisfy them separately.

Th e first two equations (Eq. 2.1 & 2.2) can beeliminated because of the following rea sons :(i) The Eq.2.3 & 2 . 4 form a set of six scalar equations andsix unknowns (3-_components of E and B each). Wher e as theEq .2 .1 is the seventh equation and introduces an additionalvariable, which is the local free charge density Q(x,y,z,t).Hence it does not help in solving the equations for thedesired variables. This equation ca n be used at the end ofthe solu tion to calculate the local space charge density as afunction of position inside the MHD channel.(ii) The E q . 2 . 2 , in its special case fo rm , can be derivedfrom the Eq.2.3 as follows:

V x E = - & & . . . ( 2 . 3 )6t

v.B = constant (not a f unct ion of time) . . . (6.1)In this model, the so luti on is updated using very

small but finite time ste ps, till the transient terms waneout and the solution reaches a steady stat e. The Eq.6.1implies that as time changes the local value of the V . B


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72remains cons tant . Hence if we ensure the continuity o f themagnetic field ( V . B = 0) all across the volume in thebeginning as the initial condition , it w i l l be maintained asthe solution progresses in time. But any small errors maykeep accumulating and the continuity o f the magnetic fieldwill be eventually lost. To eliminate this possibility arebalancing of the magnetic field has to be performed after afixed numb er of time steps.

- -

The six scalar equations from the remaining twoequations, Eq.2. 3 & 2. 4 are converted into a matrix equationa s follows [ls]:

The subscript to each matrix i n the above equationi n d i c a t e squantity.below:

-BXBYBZE Q I = PEE^W E YP E ,-

the partial derivative with respect to thatThe individual elements o f each matrix are given

- r 1000 0

- EYEX[ S I = - P J x-PJyBY- Bx0 -PJ,- , -. . . ( 6 . 3 )

wh er e, J, = a.(Ex + vBZ - w B y ) + pe.(JyB, - JzBy)Jy = a . ( E y + wB, - uB,) + pe.(J,Bx - JxB,)J, LT.(E, + uBy - vB,) + pe.(J,By - JyBx). . . rom Eq.2.8

whe re, pe is the mobility of the electrons in the medium.


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7 3fie = e.7 = 0.05 (for typical operating conditions)

m eRefer to Section 8 2 . 2 for more detai ls about pe .

6 . 2 . 2 InDut of the ModelTo define the value of current density (J -+J,,Jy,Jz)

we need the velocity field a t every poin t, whic h is obtainedas an input from the fluid mechanics model.

-Velocity Field V = u i + v j + wk

where, u = u(x,y,z,t), v = v(x,y,z,t), w = w ( x , y , z , t )6.2.3 Output o f the Model

The following quantities are sent as an output to thefluid mechanics model:(i) Lore ntz Force f (force per unit volume)f = J xB = f,i + f j + fzkwhere, f, = f x ( x , y , z , t ) = Jy.B, - Jz.By,

Y

fy = f ( x , y , z , t ) = Jz.Bx - J,.B,,Yfz = f z ( X , y , Z , t ) = Jx.By - J y * B x

(ii) Energy term Q (energy per unit volume per unit time)Q = Q ( x , y , z , t ) QJ + QfJoule dissipat ion Q J = J2/oWork done Qf = f . V = f x . u + f y . v + f,.w

. . . ( 6 . 4 )

. . . ( 6 . 5 )

. . . ( 6 . 6 )6 . 2 . 4 Solution Method

Th e wh ol e MH D v olu me is divided into a threedimensional mesh. At the beginning an initial state isassumed by assuming the values of B , E and J at e ach point.


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7 4As the assumed values do not correspond to the steady statevalues the partial derivatives o f these variables withrespect to time are non -ze ro. These partial derivatives canbe calculated by solving the matrix equation (Eq. 6.2) f orthe [Qlt value-s. By assumi ng a finit e time step the valuesof the variables are upda ted. The matrix equation isredefined and solved again for [Q lt . The values of the

- -variables are continuously updated with thi s iterative schemetill the values of the variables assume a n invariant state atevery point in the volume. This invariant state correspondsto a stea dy state solution of the problem. A precaution hasto be taken to be taken to ensure the continuity of themagnetic fiel d. The field rebalancing has to be performedafter a fixed number of time steps each, as discussedearlier

6.2.5 In tial ConditionsThe magnetic field B everywhere at the beginning is

assumed to be equal to the applied magnetic field Bo. Thevelocity field V is provided as an input by the fluidmechanics model. The electric field E and the currentdensity J are considered t o be only in Y -direction. Thevalues E,, and Jy are calculated from the strength of themagnetic field Bo by solving simultaneously the following

/

equations (Eqs. 6.7 & 6 . 8 ) := l- = &.(Jy.A)Ey w W W . . . ( 6 . 7 )


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7 5

. ( 6 . 8 )w h e r e , A is the projection of the area of the electrode in

the XZ-plane ,g o is th-e average value of the elect rical conduc tivit y

over the entire M H D v o l u m e ,- -uo is the average axial component of the velocity.Outside the MHD volume the electric field E y and

current density J are zero everywhere.y6 . 2 . 6 Boundarv Conditions

Th e exact boundary condi tions for the MHD problem canonly be defined at the infinity. But solving Eq.6.6 forinfinitely large volume is impossible and hence a set ofapproximate boundary c ondi tion s, whic h matches closely toreality is defined. The boundary conditions are specifiedon the enclosure as shown in the diagram given below. Thelarger the si ze of the enclosure the more accurate will bethe boundary conditions.

The boundary conditions are s hown by the diagram inFig. 6 . 1 . They are expressed in terms of B o , BI.W h e r e , Bo is the externally applied magnetic field,

B I is the magnetic du e to current I.B I is given by the Biot Savart Law ( E q . 2 . 1 3) .

. . . ( 2 . 1 3 )


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F i g u r e 6 . 2 T h e B o u nd a r y C o n d i t i o n s f o r t h e MHD Channel

T h e t o t a l c u r r e n t I f l o w i n g t h r o u g h t h e e x t er n alc i r c u i t i s g i v e n by E q . 6.9.

I = 1 J.dAA ( e l e c t r o d e )

. . . ( 6 . 9 )

T h e v o l t a g e d r o p V L a t t h e e x t e r n a l l o a d r e s i s t a n c eRL a p p e a r s a c r o s s t h e t w o e l e c t r o d e s . T h i s g i v e s t h ea d d i t i o n a l b o u n d a r y c o n d i t i o n a s f o l l o w s :


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7 7Anode

VL = I.RL = - E.dl = - 1 E y .d yCathodes . . ( 6 . 1 0 )

6 . 3 The Interaction with the Fluid Mechanics ModelInitiall-y a s te ad y state velocity field V is

calculated using the fluid mechanics model assuming themagnetic forc e and energy terms to be zero . This act5 a s afirst input to the electrod ynamics model. After reachingsteady state in the electrodyna mics model a force field f andenergy field Q are sent back to fluid mechanics model as aninput. This introduces new transient fluid mechanics termsand after certain time steps again steady state is reached.Now a diff erent steady state velocity field V appears as aninput to electrodynam ics model and the previous steady statevalues act as the initial conditions. This iterative schemecontinues till both the models re ach a steady state t o givethe final results.


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1.

2 .3.

4.

5.6 .

7.

8.

9 .

10

REFERENCES

Faraday M. , 18 32, Experimental Researches inE l e c t r i c i t y , P h il . T r a n s. R oy a l S O C . , 1 5 , 1 7 5 .Northrup E. F . , 1 9 0 7, P hy s. R e v . , 2 4 , 4 7 4 .Karlovitz B Y , J. Halacz. 1910, History of the K and HGenerator, McGraw Hill , New York.Williams E. J . , 1930, Proc. Phys. S O C . , 4 2 , 4 6 6 .H ar tm an n J . , 1 9 3 7 , H g - Dy n a m ic s I , M a th - F y s . M e d d . , 1 5 , 6 .Hartmann J . , Lazarus F ., 193 7, Hg-Dynamics 1 1 , M a t h - F y s .M e d d . , 1 5 , N o . - 7 .Mather N. W . , Sutton G . W. (eds.), 19 64 , Pro c. 3rd Symp.Eng. Aspects MHD, 187- 204 .Louis J . F . , Lothrop J. and Brogan T. R . , 1964, Phys.F l u i d s , 7 , 3 6 2 - 3 7 4 .Airborn e MHD Generator Deve lope ment , 196 9, Air ForceAero Propusion L ab ., Wright- Patterso n Air Force Base,

O h i o , T ec h . R e p t . , A F A P L - T R - 6 9 - 3 .Ten0 J . . 1969, Proc. 10th Symp. Eng. Aspects MHD ,Cambridg e, Massachusetts.

11. Rosa R. J . , 1 9 8 7 , M a g n e t o h y d r o d y n a m i c Energy Conversion,Hemisphere Publ. Go rp. , New York.12. Anderson D. A . , 198 4, Computational Fluid Mechanics andHeat Transfer, McGraw-Hill, New York.1 3 . F is h ma n F . , 1964, J. Advanced Energy Co nv. , 4 ( 1 ) , 1-14.14. Kirillin V . A. , Sheyndlin A. E., 1983, Progress inAstronautics and Aeronautic s, AIAA , 1 0 1 .

78


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1 5 . E d w a r d D u g a n , G e r a rd W e l c h , 1 9 8 8 , ' U l t r a - H i g hT e m p e r a t u r e V a p o r C o r e N u c l e a r R e a c t o r / MHDG e n e r a t o r S p a c e P o w e r S y s t e m , ' T ec h n i c a l R e p o r t ,U n i v e r s i t y o f F l o r i d a .16. V . S h a n k a r , W. H a l l , 1 9 8 9 , A I A A 9 t h C o m p . F l u i d D y n a mi c sC o n f . , 5 5 1.

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Conceptual Design Analysis of an MHD Power