Optimization of linear inductive magnetohydrodynamic machine considering longitudinal end effec

Valdmanis, Ya. YA.

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OPTIMIZATION OF A LINEAR INDUCTIVE MHD MACHINE CONSIDERING THE
LON61TUDINAL END EFFECT
OnTV1MV13Au !-, JIViHEOHOO V1H,,]YKuMHHOV1 Mf ,^ MAL ViHbI C YHETOM n p o4OJIb-
HOPO KPAEBOPO 3^(:^EKTA
Optimizatsiva Lineynoy Induktsionnov MGD Mashiny s uchetom Pro-
dol'noco Krayev000 Effekta
by
Ya.Ya. Valdmanis
Izvestiya of the ncdemy of Sciences of the Latvian SSR
V13BECTA^ AKAAEMVV1 HAYK JIATBNOCKON CCP (Izvesti y a Aka-
demii Nauk Latviiskoy SSR)
Series on Physics and Enaineerinq Science
1968, No. 4. ff/,^o-/^6
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Translated by the Center for Forei gn Technology,
Pasadena, CalifornJ_a, on 11 Aug. 1969.
Preoared for and issued by the Jet Propulsion Laboratory,
California Institute o= Technology, Pasadena, Calif.,
under NASA contract NTAS 7-100.
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Izvestiya of the Academy of Sciences of the Latvian SSR
PhysicaZ and Engintering Sciences Series
1968 3 No. 4
UDK 621.313.39:538.4
Ya. Ya. Valdmanis
OPTIMIZATION OF A LINEAR INDUCTIVE MHD MACHINE CONSIDERING TNT
LONGITUDINAL END EFFECT
By :Weans of utilizing the variation method, a distri-
bution of the current load in a linear induction MHD-
machine has been found in which the efficiency has a
maximum value. Presented are-the dependencies of the
optimal efficiency on the relative length of the MHD-
machine. The investigatior. was conducted both ftr
the pumping and the generator regimes (operating con-
ditions).
The theory of the lona 4.tudinal end effect allows for t`-.: pos.-
sibility of creating this or that for,n of a magnetic field in the:
work (operating) space of the MHD stator by varying the current
loads (Refs. 1 and 2). Usually, preference is given to the field
in the form of a free-playing wave which is created by utilizing
an analogous current load with various compensating devices. But,
there has not been enough exact research verifying either one or
another field distribution for operating conditions. It _J s well
knoc.,n that additional loses appear under such conditions that are
related to current diffusion beycnd the limits of the active part
of the stator. These loses are especially essential for generator
conditions of operation where, in agreement with Ref. 3, short MHD
devices are more advantageous (in terms of polar division). For
such generators it would be desireable to reduce the additional
loses, conditioned by the longitudinal end effect, to a minimum.
;1i.th this goal in mind in Ref. 3, we propose to utilize the addi-
tional compensating poles and to divide the operating channel, out-
side of the active zone, by isolating partitions in the longitudi-
nal direction. In Ref. 3 we also indicate some experimental re-
sults that corroborate the effectiveness of the improvements. How-
ever, there are no sufficient theoretical bases for showing that
these methods are the most effective. That is why this question
needs to be further explained.
Let us look at a model of the MHD machine shown in Fig. 1
where ?.L is the length of the active part of the device, V is the
velocity of movement of the conductive body. Here are the assumed
simplifications: the model is limitless laterally; the operating
body is a reliable conductor and fills up the entire space; the
magnetic field forms a comlete closed circuit within the limit of
the stator's active part (TZI<L); the Field is horizontally paral-
lel (riot depending on x). We presuppose the density of the longi-
1.
a
2tudinal current load (jo=1 0 16) in the area IZI<L, whose distri-
bution we shall determine from maximum conditions of efficiency.
The latter is determined for the pumping conditions by the ex-
pression
L
Re { J jB*vdz }
,s,
Ti„ - -
II
L
Re	 jE-dz
For generator operating condition effeciency we obtain from
I r = 1 /qH	 01)
after the formal calculation (1). All values in (1) are scalar in-
so-far as they differ from zero by the components j=
'
, Bx=B, EU=E,
ant vz=v. The quantities in the numerator and deno3^inatc,r U.) are
temporarily neutralized (dependency is on time j, B and F-eiwt).
Since j, B, and E depend on the longitudinal current load jo, the
dependency n H and nr from jo is functional. In order to determine
jo, leading to maximum efficiency, it is necessary to find variant
r,H and r,r and to approximate zero. But a strong dependency of J.,
B, and E on j o must be ;mown to do this. We assume that 1 j ; << 1 j o ,
then we get the Following by integrating Maxwell's equation
H= f jo(zi)dzi +Ch, E=i(oµo f H(z i )dz,+CE,	 .2)
—L	 —L
j=a(E+vB)
L
CH is determined by condition ftldz=0:
—L
L	 to
CH-- 4') L f f jo(z2)dz2dzj . 	 ( )
—L —L`
In the future we will differentiate two cases: a) with
z > L the channel is separated by isolating partitions in the
.ongitudinal direction; b) the partitions are missing. In the
--first instance
L	 L	 t,
f jdz =0, CE=- 
iwEt°
.ZL	 H(zz)dzzdz, ,	 ( 4a )
—L	 —L —L
in the second	 CF-0.
	
(46)
We can rewrii those elements entering the actual parts
from the product of complex function in the following way:
L	 L
Re { f jB'adz } _	 f u(jB-+Bj-)dz	 (5)
—L	 —L
,.,nd we can variate jo & i'o independently. When this happens
the equations for ; -o & jo' become complexl y linked. We get an
equation for '/o , by varying i-0:
2v = f + vE (1 + 9„) + icy f ;2q H E (z,) + vB (z,) (l + Ti„) ] dz, - const.	 (6)
—L
3The forms of equation (6) for (4a) & (4b) coincide.
However, they contain quantities B & E; the latter is ex-
presses through (2) which contains the constant CE, differ-
ing for (4a) & (4b). From (6) it is easier to go to the
differential equation for %o' . Utilizing H'=jo & E'=i(,)B
we must differentiate equation (6) according to z three
times.
2v2B'+iwv (1 4
 r)„) B +ic,, [2q,.E+v (1 +rl„) B] =0,	 (7)
2v2i 'o + iWv ( I + 11„) jo+ ia ► [2i ^ . ^9,, H +v(1+T1„)H']=0,	 (8)
lG)
	 2j^^o+ v (I +rl „) j'o— U` ^^jo=0 . 	(9)
The primes in (7) , (8) , & (9) refer to the differen-
tial process according to z.. jo is determined from (9) while
(6) , ( 7) , & (8)  are needed to determine constant integration,
- they act as limiting factors. Since E is differen-
tiated only by a constant for (4a) & (4b) & (8) & (9) will
serve as common in both cEses.
The general solution (9)
jo = C,e -iw'r z +  C2e-ib),'V n,z
	
(10)
combines two flat waves, moving in the direction of trans-
ference of the conducting body. The velocity of the move-
ment of the first waNe equals the velocity of the conductor's
movement, while the velocity of the second one exceeds it by
( 1)N- I )	 -
To determine C,, C_ 2 &'T), let us utilize condition (7) ,
(8) , & (1) . For (4a) we get the following from (7) & (8)
rl„C, sin Q+C2
 sin S2q,,=0,
(11)
r1 M2C, sin n+C2
 sin Qvjm=O,
where
Q==w1vL.
	 (12)
System (11) has solutions distinguished from zero given
Timm
nn
 s^	 L = 0, (13)
inn:
io n = C2 ne L	 ( 14)
where n - is a whole positive quantity (11*0). In each solu-
tion (14) conditions (1) are executed identically.
The dependency ►1„ from parameter SP_ is illustrated in
Fig. 2 by a set of curves (1). Those parts of the curve,
which correspond to the absolute optimum, are represented
by a solid line. The other parts of the curves (1) correspond
only to relative optimums & are indicated by a broken curve.
We can observe a certain periodicity in the dependency of the
t
t
IfV
4optimum Q from g1 , , whose amplitude diminishes with the
growth of Q. When.	 reaches its maximum value.
In area? <apumping conditions are impossible. In Fig. 2
there is a definite solution that corresponds to each part
of curve 1.	 (14) i.4Z there is a definite winding coefficient
with a whole quantity of waves for length 2L.
Taking the formal correlation we may investi-
gate the generator conditions of operation by an analogous
method. In this way we get a set of straight linos rather
than a set of hyperboles (13).
S?
nn
	 v,
It is obvious that the generator conditions of opera
tion are possible with all ?; the same winding coefficient
with a whole quantity of polar paris lead to the optimum.
For coefficients C„ & C 2 we get the following equation
for case (4b)
rt „2C, sin Q +C2 sin ffil;,=0,
,1„2 C, cos Q+C2
 cos f? q ,, 	 (I6)
For %o we have a set of solutions (17) where
%on= C^(e -iw!rI_ (_ I)nq,,2e- i0'r".nz]. (17)
nil
TiHn=l--	 (1R)
The condition (1) for (17) becomes an identity because
of the orthogpnal character of the 1st and 2nd term in the
interval !z -L.
	
The optimal T1N i.s attributed to the (:urve
with the least
	
rI. Z.C.,when a= I (curve 3 Fig. 2) . In the
entire area Q the optimum )I. is less than in case (4a)
(only at points Q =2n, 3n. ,	 the curves 1 & 3 cross each
other) .
For generator conditions of operation
Q
'1 r ” 	 ( l9)S2+.nn
while the optimum il, is expressed by (19) when n= I (curve 4
fig. 2)
When clearly running Fields manifest as B.& E.in figure
2, n is greater both for generator and pumping conditions of
operation. This is only possible when there are isolating
dividers present (4a) , when z >L then Ei =0 & Q- 0• and both B
& E card take on any value at the end of the stator. Condi-
tions jBdz =0 lead to the necessity of a whole quantity of
waves-Llong length 2L. In case (4b) it is necessary to sat-
isfy conditions E=0 when ;z,-L ; otherwise, in the reverse situ-
ation the current will spill out in area z>L at that the
leakage will be considerable. Such conditions cannot be
achieved by means of a freely running current. An altered
Scurrent (the superimposition if two waves) presents a sol-
ution and leads to a reduction of efficiency. When SO_ are
great then for both situations we have an efficiency that
approximates one.
In the above instance we assume	 !ji^;loJ	 ; however, it
is not difficult to generalize for various instances. For
inference (2) we must not use
rot H- jo,	 (20)
but
rot H= j -f- jo	 (20')
or in (2) /o r^;us t 1` P considered a summary density of current
io 
=j+ jo and jo taken as the current density of the magnetic
field.
_,et us take the following to examine case (4a)
B= B oe
 
-
 
iwi'r „- .
	 (21)
From equation rot E _ - i(uB
	
we find
Eo = - LBO-
	 (22)
and^ y^
j= —aLµo (1- IN) Ho 	 (23)
YIN
Now, (20) leads to , (
	
w	 (1-^1„>
jo= L-i v TIN +ovµo	 ^ F	Ho.	 (24)
Analogous to (4b) we obtain
jo == —i v Ho e-	 + [ ( i - ^/S2) +	 , e-	 } ,	 (23)
where	 wµoQT^
n-
	 (2G)
out, T is the polar division determined by the first member
(25) ;i.e. , w/V=: n / T ) .	 If for case (4a) the dam. tribution of
the charge remained the same (only the amplitude and phase
changed), for case (4b) the relationship between the 1st and
2nd time in (25) and (17) changes and with it the distribu-
tion of 7
In fig. 3a we hove the instant values lo (curve-l') and jo
(curve 2) when Q 2^ , e== I & toi = 0.; fi5. 3 b shows the same given
a,t =z/2.
	 Fig. 31 also indicates the distribution of the ef-
fective value of jo along the stators length (curve 3). Evi-
dently, the distribution of a linear current load must possess
a certain assymmetry with great current concentration at the
inlet. This assymmetry is conditioned by the fact that the
field is carried along by the moving conductor, while the sx=-
mary current must be symmetrical.
To clarify the
cient of type (17) ,
rate of comparison.
absent arid e = 1 ( for
but p =2 ---curve 6 -
dicated by the dott
expediency in utilizing a winding coeffi-
the following claculatien was made for the
We assumed that the 2nd time in (17) was
the quantity of polar pairs p-1---curve 5,
Fig. 2). The corresponding curves are fin-
ed line wnen a <<1 , or when ;j l<t- jo
li%.
6
The last case was first explored in Ref. 5, and, evidently,
the maximal onE! does not exceed 18% independent from the quan-
tity of polar pairs efficiency. If e-1 we can achieve greater
efficiency, but they would be far from the optimal given ^!.
Besides, with utilization of (17), we investigated the depend-
ence i1 (e) for fixated Q (^^. ^5 was not considered) . Fig. 4
shows dependence >> (e) when S? =2a ( curve 1) and Q = 3a ( curve 2) .
The dotted line shows the optimum value, which is given by Y1
wi:iding coefficients (25) and does not depend on e. As we can
see from these curves, the calculation (25) improves the effi-
ciency significantly when the end effect is e.
Even though above we only considered joule-effect loses
in the working body and calculation of other loses (loses in
the sheath, wall, etc.) could change the obtained corYes-
pondences, nevertheless, the results can be applied in the
construction c various types of MHD devices.
References
1 .	 Ya. VI-"" cL.anis and Ya.Ya. Liyelp:^6cr, Yak netic I:ydrodyna: cs (I:amitraya
o:;inw-ika), 1 967, 1 , 115.
A. I. V ol,dek, I'MVSh, E1ect, roTichh,—'.n-cs and Automiatics (Elektromckhanika i
V I L .ica), 1959, 2, 1-
3. D.j. Cleriri c.nd D.G. Elliott, Eighth Sympcsi.un on Engineering Aspects of
...,g :etohydro(,yn&-.tics, Stanford, 1967s 1 1 .
µ. -A.;. Vol'dek, :-.v. Vuzov. Elektro;rekhan^ka (Izvestiya of Education in Electro—
mechanics), 1960, 3, 3.
5. L.:. Make, Proceedin&s !EEL, pt. A, 1957, 1 04, 1 3, 49.
of Physics of the	 Suh.-nitted on 6 ;-:ay 196E
Acade::. - of Sciences of the Latvian SS.
."uc. 36.
tip
Fuc. 3a.
0,5
Puc. I.
a---
/	 s'
Puc. 1.
o f 	 f.
o,s	 1.0
Puc. 4.


Optimization of a linear inductive MHD machine considering the longitudinal end effect

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