MHD Generators

04/08/23 NATIONAL INSTITUTE OF TECHNOLOGY, CALICUT

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MHD GENERATORS


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What is Magneto-hydrodynamics?

• Otherwise known as magneto-fluid dynamics , as well

• Deals with the dynamics of electrically-conducting fluids and their interactions with magnetic materials

• Can be used for electric power generation through the DEC route


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Important aspects of MHD

Following are the two basic aspects of MHD:

1. Accelerate fluids through Lorentz force J x B which refers to MHD accelerators

2. Convert the kinetic energy of a fluid and its enthalpy into electric energy

[This refers to the MHD generators]


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MHD GENERATORS

Disk MHD Generator

Disk MHD Generator


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The conversion process in MHD (Basic Principle)

MHD engines show promise as a way of generating electricity on a ‘larger scale’

This is also being studied as rockets for the propulsion of spacecraft in inter-planetary travel

If a perpendicular magnetic field is applied to a charged particle moving at velocity ‘v’ to the right, then the magnetic force acting on the particle is given by,

F q v B


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If q is +ive , the force acting on the particle points upward (and vice versa)

Instead of 2 particles, if we have a mass of ionized gas molecules (moving at an avg. velocity ‘v’ toward the right), the +ive ions are accelerated towards plate P1 (and the –ive ions are accelerated toward plate P2)

Even if ions collide with each other, the avg. drift of the +ive ions is towards P1 (and –ive ions is towards P2 )

If P1 & P2 are connected through an external load, then a +ive electrical current flow from P1 to P2 through the load


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Basic components of a MHD Generator

An MHD generator extracts mechanical energy from the mean flow of ionized gas and converts it to electrical energy

This process is a “Basically Reversible” one.

If instead of a load, an external source (of emf with sufficiently large magnitude) is connected in the direction opposite to the natural direction of current, the latter can be reversed (and energy fed into the mean flow)


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The essential components of a MHD generator, which converts heat into electrical energy in a closed cycle operation include the following:Boiler with super heater (HX)NozzleMagnetic coil & electrodesCondenser & cooling fluidPump

The superheated gas is expanded through a nozzle to obtain a high mean flow velocity


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The ionization of gas is accomplished by the natural process of thermal agitation and is sometimes aided by seeding into the gas by a small amount of easily ionizable material (or) by using an RF field

The ionized gas is passed through the energy conversion chamber in which the generation of electricity takes place


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Afterwards the gas is condensed and pumped back into the boiler

A reversed operation is also possible, like in the case of the MHD ramjet

The inlet gas is accelerated and is passed through the conversion chamber, and then it is ejected at a higher velocity


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12


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Transformation of Electromagnetic Field

Using a moving reference system the MHD process can be analyzed more rigorously

The electro-magnetic force acting on a moving charge is given as,

With a single charged particle moving at a single , the six components of and can’t be determined from the three components of

F q E v B

V

E

B

F


14

However, by measuring the forces on particles moving at different velocities, the vectors and can be completely determined

Hence, in general

with

E

B

F q X v Y

&E X B Y


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Now, let and represent the electro-magnetic fields as seen in a stationary system

However, we need to find the values measured by a laboratory set up which is moving at a velocity,

Since the actual velocity (relative to the stationary system) is the force becomes,

Hence it becomes now that,

E

B

' '&E Bv

'

v v

' 'F q E v v B q E v B v B

' ' & E E v B B B


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Generator & Motor effects of an MHD Engine

To understand the physical implications of , consider a schematic arrangement as shown

An electrical load of resistance R is connected across the plates P1 & P2

A +ive current I flows from plate P1 through the load to plate P2

and a voltage V=IR is developed across the load

The electrical intensity between the plate is,

….Where d is the distance between the plates

' '&E B

z

VE

d


17

Fig5.4&5.5


18


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To determine the effects of electro-magnetic fields & B, on the moving gas, imagine a ‘laboratory moving with the gas’

The electrical & magnetic fields as seen by a person within the laboratory is,

zE

' '1&z zE E Bv Bvd V B B

d


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The following two observations shall be made now:

(i) The physical laws as seen by a person within the laboratory are exactly same as that seen by a person outside

(ii) The motion of various particles in the ionized gas constitute an objective fact (As such, it remains the same; no matter who sees it!)


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The electro-magnetic fields & B, acting on the moving gas produce exactly the same forces on the ions

[as electro- magnetic fields acting on a gas with zero average velocity]

The electric field, can be produced by an emf , Ei in series with the external voltage, V where

Ei =Bvd

As far as any external measurements can determine, Ei is an internal emf of the engine

zE

' &zE B

'zE


22

A volume of ionized gas with zero average velocity behaves like a passive resister to external measurements

i.e., The MHD engine has an internal emf , Ei (=Bvd) and an internal resistance

And, the flow of current, I causes a force, F (=BId) acting on the ions (and is eventually transmitted to the entire gas, through collision)

idR A


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The current, I in the gaseous medium is in the z-direction and F is in the minus x-direction

The force F tends to slow down the mean flow of the gas (or) to cause the pressure to build-up in the minus x-direction

In any case, it extracts from the mean flow, a mechanical

energy equal to,

iFv BIdv E I


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This equation gives the energy balance in the conversion of mechanical to electrical energy

The MHD engine is a “basically reversible” one!

Suppose, an external source of emf, Eo is connected by moving the switch arm to position 2 (instead of load at position 1)

And if Eo>Bvd, then the direction of the current is reversed


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Consequently, the direction of force acting on the ions is also reversed

It tends to speed up the mean flow rather than to retard the latter

The ejected gas is at a much higher speed than the inlet gas

In the meantime, the reactive force acting on the magnet tends to push the MHD engine in the minus x-direction


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Thus the MHD engine acts like a jet engine, with the exception that the input is electrical energy (rather than a HC fuel)

This explains the basic process by which an MHD engine converts mechanical energy into electrical energy (and vice versa)


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The theory elaborated so far doesn’t take into account the following 2 factors:

(i) The pressure, temperature & mean flow velocity vary from point to point and are functions of x

(ii) The conduction of electricity through a stationary gas is not unaffected by the Pressure of a strong magnetic field (under certain conditions)


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THREE VELOCITIES

The problem of gaseous conduction has the following two essential aspects:

The ionization & recombination of electrons & ions

The average motion of charge carriers


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The first aspect deals with the densities of charge carriers and the dependence of the densities on macroscopic conditions such as composition, pressure & temperature of the gas

The second aspect deals with the observable effects which are consequences of the motions of charge carriers, such as conductivity, Hall effect, etc.


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The following three types of motions of any single charge carrier need to be clearly distinguished:

(i) The “mean flow "is the average movement of the gas as a whole (the velocity of mean flow is denoted as )

(ii) The thermal movement is the Brownian motion due to thermal agitation (its velocity is denoted as )

(iii) The drift motion is due to electro-magnetic force acting on a charged particle

(its average velocity is denoted as )

PS: A neutral atom or molecule has only

V

u

du

&V u


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IONIZATION PROCESS

When an orbital electron of a gas molecule acquires sufficient energy through thermal agitation, it escapes from the molecule and becomes a free electron

The remaining molecule becomes a +ively charged ion

Sometimes, a molecule separates into a +ive ion and a –ive ion

At the same time electrons & ions (or +ive & -ive ions) recombine into neutral molecules


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Under equilibrium conditions the density (or number per unit volume) of ionized molecules satisfy the following equation:

Where C - the proportionality constant

φ - the energy deficiency which binds the electron to the rest of the atom (or the two ions together)

- the density of the un-ionized molecules

312 4 exp 2i on Cn T kT

on


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Assuming electrical neutrality (on the average), the densities of +ive & -ive ions are identical & are both represented by

The above eqn. for is only an approximation and it holds only if kT<< φ [in which case is small compared to ]

As T increases, also increases sharply through the exponential dependence

At kT = φ, a substantial number of molecules are ionized, and the above equation for is no longer valid!

on

in

inin

in

in


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DERIVATION FOR ni

At all times, the +ive & -ive ions collide with each other

When such a collision takes place, there is a probability, pr that they recombine into a neutral molecule

The frequency of collision is proportional to the densities of ions and the speed (or √T)

Hence, the rate of recombination is,

1 . . .r i i rR C n n p T


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In the mean time, the neutral molecules are dissociating into ions at a rate,

The ratio, is proportional to

This accounts for the probability of molecules acquiring the necessary energy, φ to ionize through thermal agitation

2d o dR C n p

d

r

pp

2 kTT e


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Under equilibrium condition, and we have

i.e.,

Presence of other types of molecules doesn’t change the above equation!

d tR R

32 2 2 expi on C n T kT

3142 expi on Cn T kT


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Example:

In a He gas seeded with K, no is the density of K atoms

While He is far more difficult to ionize, its presence is completely immaterial (with the same no, the same ni is produced, whether He is present or not!)

This occurs because the number of collisions per second between electrons & K-ions is not changed by the presence (or pressure) of He!!


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GAS Vs PLASMA

The energy, φ which binds a valance electron to an atom has a value of flow eV; whereas the energy which binds an inner-most electron to a nucleus is proportional to the square of the nuclear charge!

The latter varies from a few eV for H2 to about 105eV for the heaviest nuclei

Since k = 8.616×10-5 eV per degree, it takes a temp. of at least “few thousand” degrees to cause substantial ionization of valance electrons from their molecules (or atoms)


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But, much higher temperature is required to cause complete ionization of an atom

A gas with a substantial number of ionized particles in it is one kind of “plasma”

There is no clear demarcation line of what plasma is & what is not! (Since, a gas without ions are yet to be found!!)

Hence, the word “plasma” in the practical sense means a gas which has a fairly large electrical conductivity value


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As this is essential for the operation of MHD engines, they are also called plasma engines, as well!

Ionized gases most commonly used in MHD engines are evaporated alkali metal (or) an ordinary gas seeded with a small trace of an alkali metal

Eg: Conductivity of about 10mho per meter at 2000K can be

obtained by seeding air with about 1% of K vapor!


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GASEOUS CONDUCTION

The valance electrons of some gas molecules or atoms (of a high temperature gas) are dissociated from the rest of atoms

The latter carries +ive charge and is called +ive ions

Thus, both types of charge carriers are usually present, and the gas as a whole is neutral!

The charge carriers & neutral atoms (or molecules) more at high speeds with the average K.E. of each particle in each degree of freedom equal to ½ kT

[As obtained below (next page!)]


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Here m is the mass of the particles & are the velocity components

As per this equation the lighter particles moves at higher velocity!

The motion of the particle is deflected every time it collides with another particle (as shown)

2 2 21 1 1 1

2 2 2 2x y zmu mu mu kT

, &x y zu u u


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In the absence of any external field, the particle starting at 1 end at 2 after some time; not much away from the initial location

However, in the presence of an electric field, E (pointing towards right), the charge carriers (electrons) are accelerated towards left (as shown)

This acceleration is given by eE/me, causing greater average drift towards left (wiping out the earlier accumulated drift velocity)


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The average drift velocity is simply the average accumulated drift velocity between collisions

…Where τ is the average interval between

collisions

The above equation shows that, the drift velocity is proportional to E and the mobility of the charge carrier is defined as,

2de

eEu

m

du

E


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Hence, for electrons, it follows from the above equations,

The current density due to the drift of electrons is,

Where - density of electrons

- electrons crossing unit area

e - charge of electrons

2ee

e

m

2

2e

e e de

n e EJ en u

M

en

e dn u


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Therefore the conductivity due to electrons is,

This equation also holds for +ive ions with replaced by which are the

density, mass and average interval between collisions of +ive ions, respectively

2

2e e

ee

J n e

E M

, &e en m , &i i in m


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As mi for H2 ions is 1800 times of me and mi for Ce ions is 2,00,000 times of me, the conductivities of +ive ions are entirely negligible, compared to that of electrons

Hence σe can be regarded as the conductivity of the gas and subscript ‘e’ may be dropped!


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The average length, l traveled by an electron between two collisions is called the mean free path of the electrons, given by,

Here A0 and Ai denote the collision cross section between an electron & a neutral particle and between an electron & a positive ion, respectively

And n0 and ni are the densities of neutral particles and +ive ions, respectively

0 0

1

i i

lA nA


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If ue denote the average speed of an electron, then τ = l/ue

Substituting these in the equation for σ , we get

The following conclusions can be obtained from this eqn.:

1. For a gas, in which electrons are the predominant form of -ive ions, ne = ni and both densities increases rapidly with temperature

2

0 0

1

2

e

e e i i

n e

m u A nA


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2. At a relatively low temperature niAi is negligible compared

to n0A0, and the conductivity σ rises with n0

3. As the collision cross section Ai is ‘few 1000 times’ as large as A0; however, it takes only an ion density of about 0.1% for the term niAi to dominate!

4. Hence, both the numerator and denominator of this equation increases approximately in proportion to ni as the temperature rises

5. And the conductivity σ saturates at a value of

2

2 e e i

e

m u A


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Also it can be proved that, σ varies with gas pressure

The partial pr. of neutral particles, ions and electrons are

Also

Then the equation for σ becomes

0 0p n kT i ip nkT e ep n kT2 2 2 2

3e e e x e y e zm u m u m u m u kT

2 2

0 0 0 0

3 1

6( ) ( ) 12

e e

e

i i e i i e

P e P e

P A PA m kT P A PA m kT


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A closer (and more correct) calculation for σe gives

Conductivity of gas seeded with 1% K vapor in the temperature range 1500-4000K indicates a sharp increase in σ

2

0 0

2 3

3 ( )

o

e

i i e

P e

P A PA m kT


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THE HALL EFFECT In the presence of magnetic field, the electrons no longer follow a

straight line path in the flow field

There is no force in the direction of magnetic field, and the motion is still a linear function of time

By balancing the magnetic force with centrifugal force, the radius of the projection trajectory can be obtained as

2e e e e

e

m u m uu B r

r B


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In the presence of a magnetic field the electrical conductivity is a vector (rather than a scalar)

If we connect a voltage across plates 3 & 4 (as shown), so that an electric field E is established, as shown

In this case, the plate 5 will be -ively charged and the plate 6 will be +ively charged (see the figure)

If we connect these plates together, electrically, then a ‘positive’ current will flow from the bottom plate (6) to the top plate (5)


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This effect is called the “Hall effect”

The ‘Hall effect’ can be used to determine the nature of the charge carrier

In order to have an MHD engine operate properly, the conductivity of ionized gas must be as high as possible

The Hall Effect can be rendered insignificant by requiring e < r (or) σ ≤ Pe/(2BVT)

Where Pe is the partial pressure of electrons and VT=kT/e


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DIMENSIONAL ANALYSIS

The relation between the size and performance of MHD engines

The mean velocity (v), current density (J) and flux density (B) are assumed to be independent of size

And the MHD engine of various sizes are assumed to be proportional

L is the characteristic length of the engine (separation between electrodes or cubic root of the total volume)


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1. Converted power, P=EI = Bvd and JA= JBvK1L3, …where K1 is a constant independent of L

2. i2R loss inside the plasma,

3. Friction loss at the walls,

….where f is the fraction of ρv2 per unit area and Aw

is the wall area

4. Heat loss at the walls, H=K4 L2

5. i2R loss of the magnetizing coil is

2

32 2

1p

JQ d J K L

2 23 3

wF f v A v K v L

2 2( )m

lQ i R ni

Ac


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2

2 2 5

2 2

0 0

/ ,

m

c

m o m

Also

ni Bl l

ll B B K LQm

A

…where lm is the length of the magnetic path

As R=ρln2/Ac

….where ρ - resistivity of the wire material

Ac - total copper cross section

N - number of turns

l - mean length per turn


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The significant conclusions of the foregoing analysis include:

1. The converted power is proportional to the volume of the engine

2. For a significantly large MHD engine, all the losses are small (compared to the energy converted)

3. The i2R loss in the plasma is proportional to the energy converted

Hence, the MHD engine is found to be efficient at high power levels

[but rather inefficient at low power levels!]

Therefore, it is unlikely to become commercially feasible for any application below the 10-100MW power levels!!


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CONSTANT AREA MHD ENGINE

A. Assumptions:

1. The cross section of the flow (A) between plates P1 and P2 is constant

2. The pressure (p), average velocity (v) and temperature (T) of the ionized gas are functions of x only

3. For steady flow, these are independent of t, as well

4. The friction and heat loss at the walls are negligible

5. The flux density (B) is a constant

6. The ionized gas follow the ideal gas flow => p/ρ=R1T

where R1 is the gas constant


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In the analysis, the conversion chamber is divided into slices along the x-direction

The variables in each slice are treated to be constant and the basic MHD relations derived are made applicable

B. Basic Equations: Consider the volume of gas bounded by two imaginary

planes, P’ and P’’ at x1 and x2 respectively, which move with the mean flow

At a small internal δt second later, the two planes P’ and P’’ move to x1+δx1 and x2+δx2, where δx1 = ν(x1)δt and δx2= ν(x2)δt


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Let I12 denote the total current between x1 and x2

All the three conservation (matter, momentum and energy) principles need to be satisfied in the flow field

1) Conservation of matter:

Since the planes P’ and P’’ move with the mean motion, the total amount of matter bound between them don’t change with time

Thus the mass of gas bound between x1 and x2 is the same as the mass of gas bound between x1+δx1 and x2+δx2:


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i.e.,

Substracting from both sides, the integral , we get

As δx1 and δx2 can be made arbitrarily small by making δt small the above equation becomes,

2 x2+ x2

1 x1+ x1

( ) ( )x

x

A x dx A x dx

2

1 1

( )x

x x

A x dx

x1+ x1 x2+ x2

1 x2

( ) ( )x

A x dx A x dx

1 1 2 2( ) ( )A x x A x x


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Substituting for δx1 and δx2 in this equation gives,

This equation indicates that ρv is constant and independent of x; otherwise q=Aρv, where q is the mass flow rate

2) Conservation of Momentum:

For the mass of gas bound between P’ and P’’, we have the following quantities in the momentum equation

Ap(x1)-Ap(x2) ….Forces due to static pressure in x-direction

-BI12d ....Magnetic force in the x-direction

1 1 2 2( ) ( ) ( ) ( )A x v x A x v x


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The rate of change of momentum becomes,

x2+ x2 x2

x1 x1+ x1

x2+ x2 x1+ x1

2 x1

2 2 2 1 1 1

2 22 2 1 1

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

x

Ax v x dx x v x dx

t

Ax v x dx x v x dx

t

A x v x x A x v x x

t t

A x v x x v x


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By the application of Newton’s 2nd law gives,[Rate of change of momentum = Total force]

Alternatively we can write this as ,

…..where w is the width

PS: The expression (p+ρv2) is called the stagnation pressure and is denoted as ps [That is, ps= p+ρv2]

122 22 2 1 1 1 2( ) ( ) ( ) ( ) ( ) ( )

BI dx v x x v x p x p x

A

122 2

2 1( ) ( )

at x at x

BIp v p v

w


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3) Conservation of Energy:

Assuming conduction of heat negligible, we have ΔQ=0

For a slice of gas with thickness dx, the total internal energy is

The increase in internal energy between P’ and P’’ is,

2

1

1( )

2l

tic

ThermalMassKinetic

dU Adx Cv T v

1

2 2 2 2 2 1 1

1 1 1 2 1x

x x x x x x x

x x x x x

U dU dU dU dU

2

2 2 1 2 2

1( ) ( ) ( )

2A x x Cv T x v x

21 1 1 1 1

1( ) ( ) ( )

2A x x Cv T x v x


The electrical work output is => VI12δt,

….where V is the voltage between P1 and P2

The mechanical work done by the gas is

Therefore =>

By the application of 1st law of Thermodynamics we get,

2 2 1 1( ) ( )Ap x x Ap x x

12 2 2 1 1( ) ( )W VI t Ap x x Ap x x

2 21 2 2 12

2

( ) 1( ) ( )

( ) 2

p xq Cv T x v x VI

x

1 21 1 1

1

( ) 1( ) ( )

( ) 2

p xq Cv T x v x

x


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The physical explanation of this equation is ,

…where hT is the enthalpy of the mass of gas q, which flows across the cross section at x every second (enthalpy flux)

The internal energy of the gas is given by,

21

1

2T

pq Cv T v h

22 2

1

1 1 1

2 2 2

v

v v

qC TU mC T qv qv qC T qv

M


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And the volume occupied by the gas, V=q/ρ

Using these two equations, the energy conservation equation my be written as HT=U+pV

Physically, the enthalpy flux is the rate of flow of total energy across a given surface

The enthalpy flux, hT can be expressed in many other ways, as well (next page)


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2

1

2

1 1

1

1

1

1

1( )

21

( )1 2

/

T p

T

P V

P

V

h q C T v

ph q v

C R C

C

C

P R T

As

and


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Using the concept of enthalpy flux, we can write

This equation is the demonstration of conservation of energy in the MHD engine

Another equation gives the current I12 as a function of gas variables

i.e.,

1 2 12( ) ( )T Th x h x vI

1

2 2

12

1 1

( ) ( ) ' ( ) ( ) ( )

( ) ( ) ( )x

z

x x

x x

J x x E x x Bv xd

I wJ x dx w x Bv x dxd


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C. Conversion Effectiveness: The conversion effectiveness of an MHD engine is the ratio

of electric power output to the enthalpy flux at the input

i.e., …where Iol is the total conduction current between the two plates P1 and P2

Denoting the input plane as x=0 and the output plane as

x= l we get,

The conversion effectiveness is quite similar to the thermal efficiency of heat engines

( )

ol

c

T

VI

h o

( ) ( ) ( )1

( ) ( )

T T T

c

T T

h o h l h l

h o h o


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D. Normalized Equations:

Continuity equation may be used to eliminate 8 from momentum and energy equations, as follows

Momentum

Energy

A natural limit for v is v0=V/dB

At a mean flow speed v=v0, the generated emf Bvd is just sufficient to balance out the terminal voltage, V and hence, no current will flow

s

qvP p

A

2

1

1 2Th Apv qv


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Multiplying the above equations by A/qvo and 1/qv02

From the above two equations, the natural limit for p seems to be qv0/A

[Rate of momentum flow per unit area at a mean flow speed, vo]

0

0 0 0

2

20 0 0 0

1

1 2

T

AP AP v

qv qv v

h AP v v

qv qv v v


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The governing equations for the analysis of MHD engine become,

20 0 0 0

; ; ;s T

s

Ap Ap h vP P H Q

qv qv qv v

2

20 0

20

1

1 2

( ) ( )

( ) ( )

( )1

( )

ox ox

s s

ox

c

sP P Q

H PQ Q

ABI VIP o P x

qv x qv

VIH o H x

qv

H l

H o

Defining the four non-dimensional variables Ps, H, P and Q as,


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E. Stagnation Pressure Vs Enthalpy:

The GEs of the MHD engines can be readily solved with the aid of the Ps-H diagram (see the figure)

The following points become apparent in this figure:

I. For each Q, the locus (H, Ps) traces out a straight line as P varies

II. The envelope of the of straight lines with different values of Q is a parabola with equation

2

2

22( 1) sH P


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III. This parabola (envelope of constant Q lines) is the sound barrier where the mean flow velocity v is equal to the velocity of

sound, as

….where a is the velocity of sound and M is the Mach number

2 20 2

20

1Q qv v v vM

p AP v p a

IV. Each constant Q line is divided by its tangential point on the sound barrier into 2 segments with lower segment respectively supersonic speeds (upper segment represents subsonic speeds)


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V. The sound barrier divides the Ps-H plane into 2 regions and the region below sound barrier is physically inaccessible

VI. The locus of H(x), Ps(x) is called the trajectory of the MHD engine (Eg. AA’ in the figure). This is a straight line with unity slope towards the lower left corner on the Ps-H diagram

VII. From this trajectory we can read the values of Q as x increases, representing the progress of the plasma down the engine

VIII. The length of the trajectory depends on the length of the engine and the conductivity of the gas. The two natural boundaries which the trajectory can’t cross are,

Sound barrier corresponding to M=1 (flow is unstable)

Current density (J) barrier corresponding to Q=1 (J drops to zero as electric and magnetic fields balance out)


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IX. The conductivity of the gas affect only the correspondence between x and the length of the trajectory (it can’t change the position of the trajectory) and the performance of an MHD engine with constant cross section can be calculated from the initial and final points on the trajectory without knowing σ value

X. The conversion effectiveness can be calculated from the H values at the starting and end points of the trajectory


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The other variables are determined as follows :

Ps and Q are read from the Ps-H diagram

P is calculated as Ps-Q

The actual variables are calculated from P & Q as,

0

0

20

0

; ;

1;

1

qv Pp v v Q

Aq v QP

TAv Q R


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XI. Since a trajectory is either supersonic or subsonic, (but is never mixed); the above diagram can be separated into two diagrams

XII. For operation as a jet, Q<1, and the representative points move on a straight line inclined at 45o from lower left to the upper right!


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