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Emission instability of thermionic converters
Energy Conversion. Vol. 13, pp. 49-51. Pergamon Press, 1973. Printed in Great Britain
Emission Instability of Thermionic Converters Y, Y, ABRAMOV,A, A, VEOENOVand G, G, GLADUSH (Received 5 July 1972)
Introduction The original instability may appear in thermionic converters, if the operating point of the converter is on the negative slope of the emission S-curve. In this case, the total heat flux from the emitter may be decreased by increasing the temperature, due to the considerable electron cooling. This will destroy the thermal balance on the emitter. A. Schock began to investigate such an instability [1]. The linear approach was developed by Loshkarev [2]. The final stationary states of thermionic converters in which this instability may occur were investigated recently by the authors of the present article [3]. It was found that very inhomogeneous temperature profiles may exist along the emitter. The stability of these final states was also investigated. In all previous papers, the thermal flux from the heat source to the emitter was assumed to be homogeneous. However, if the input thermal flux is not homogeneous, only a certain part of the emitter may be in stable region, while the other part is not. This case, which is of practical importance, is investigated in the present article.
Basic Equations The ignited mode is assumed. The emitter, which is the thin stripe, has the length L. The width of the stripe is assumed to be unity. The load resistance of the converter is R, and the e.m.f, of the additional battery connected in series is E. The current-voltage curves of the converter are assumed to be straight lines with a constant slope: V = RoL[js(r) -- j]
where Cv = specific heat capacity for unit length, q = heat flux density to the emitter from heat source, ~T 4 = interelectrode thermal radiation density, Cj ---- electron cooling flux density, K = xd, where X = thermal conductivity and d = emitter mlcKness. Boundary conditions for (3) are OT_ _ OT -= O. Ox x=0 Ox x=z,
The system (1)-(3) may be reduced to the following integro-differential equation: OT Cv ~[ + aT 4 + Cjs[T(x, t)] L
¢ R + Ro L
E¢ + ~c aZT L(R + Ro) ~"
(5)
For convenience, the following abbreviations are introduced: W ( T ) ---- ~ r 4 + ~js(T), WR(T)
(6)
= crT4 @ 6 R ~ R°RoJS(r), --
e¢
Ro)'
C(x) = q (x)
L(R +
b(x) = C(x) +
R~_~ ( j s ) , R+Ro
(1)
(7/
(8) (9)
with
,f
( j s ) = ~,
jsd~.
In the stationary case, Equation (5) may be rewritten as
Ohm's law for the circuit is
KT" = W ( T ) -- b(x)
L
V = RI,
t)]
= q (x)
converter output voltage, converter internal resistance, current density, short circuit current density, temperature distribution along the emitter.
E+
js[T(~, o
where V= R0 = j = js = T(x) =
(4)
I = f j ( x ) dx
For q ( x ) = constant, and consequently, b ( x ) = const, c(x) = const, Equation (10) may have the homogeneous solution T(x) = const. Tis found from the simple algebraic equation WR = C. This equation commonly has three roots (see Fig. 1). It is evident that the middle root of the equation is unstable with respect to perturbations not depending on
o
and the energy balance equation for the emitter is OT 02T Cv ~ = K ~ + q
-- aT 4 --j¢,
(10)
(2)
(3)
t I. V. K u r c h a t o v Institute of Atomic Energy, Moscow, USSR. 49
50
Y . Y . ABRAMOV, A. A. VEDENOV and G. G. GLADUSH
X, i.e. the temperature region where OWR/OT < 0 is unstable. If the perturbations are periodic and have a form such that the external circuit current remains constant, the stability criterion is not dependent upon R; it is determined by internal converter parameters.
Suppose that L = co; then, for x < 0 and x > 0, Equation (10) is again the equation for particle motion in a potential well. This means that we can use a mechanical analogy again. B*~o~
W
W (7)
B*
B--
BI *I
/i 4
I
1
I
J
i
I
I
I
i
1
I
°
Fig. 2. Two solutions of F_Zluation (10): b < b*, b* < b + b i < bgaax. 'P T
Fig. 1. Components of emitter energy flux dem/ty vs emitter temperature: w, flux by R = 0; wR, flux by R ~ 0; b, 0, input power to emitter.
In particular, if the emitter length is more than a certain critical value (11) the region where OW/OT < 0 is unstable with respect to the perturbation mentioned. This region is greater than OWR/OT < 0 (see Fig. 1). As to inhomogeneous solutions of Equation (10) it is noticed that for q ( x ) = const and b(x)= const, (10) becomes a motion equation for particle moving in a static potential field. Indeed, by substituting T ~ X, x ~ ~- in (10), one can obtain:
dr~d2X--
OXO f [b -- W(X)] dX --
OXO U(X)
Thus, the term U(x) = f (b -- w (x)) dx has the meaning of potential. The boundary conditions become dr
~o
dr
r=L
0
This means that particle velocity is zero at the initial moment and at moment r = L. According to the extent with which L exceeds 11/~, the particle will oscillate one or more times in the potential field; this corresponds to a stationary temperature profile with one or more minima [3]. Discussion and Results
For q (x) :~ const, let us write q and b in following forms
q(x) = q + qtO(x) t" (X) = b + biO(x)
ql = bl
f l, O(x) = ~ 0 ,
x >0 x <0
The accepted form for q means that the particle will be exposed to additional force ql from the moment t = 0. As this force is limited, the velocity of the particle cannot change sharply. The boundary conditions mean that the particle begins to move from one top of a potential curve to another (see Fig. 2). Figure 2 represents the function W(T); three particular values of b are marked. The straight line b* is drawn so that the corresponding potential curve forms the potential well with top values equal to each other. Let b < b* and b* < b + b t < b * a x .. In this case there are two solutions, which conform to the boundary conditions at 4-oo and have the form of 'smooth step'. The low temperature level and high temperature level are determined by the conditions W ( T ) = b, and W(T) = b + ha, respectively. Mechanically it means that the particle at the moment t = oo began to run down from the left top of the potential curve; at the moment t ---- 0, the form of potential curve changes immediately and the particle will climb to the fight top, reaching it at t ~- oo. Using the mechanical analogy, the other cases may also be examined. For instance, if b * < b < bmax. , b + bt > bmax; there are not any solutions when one part of the emitter has the temperature corresponding to the left wing of function W(T), and another part has a temperature corresponding to the right one. In this case, the solution has the form of a comparatively low 'step'. The lower temperature level is the root of equation W(T) = b; the upper level is the root of equation W(T) = b + hi. Both roots are placed on the right wing W(T) (Fig. 3a). I f L =~ oo, let us take q(x) in the form
q(x) = q --b qiO(x -- 1), where 0 < 1 < L. In this case the particle moves from t = 0 (V0 = O) to t----- 1 in the potential field of one form, and from t = 1 to t = L in another (VL = 0). Let us consider again the case b * < b < bmax and b + bi > b*m~. The one solution corresponding to the lower emitter temperature (as compared to the case
51
Emission Instability of Thermionic Converters
where L = CO)is illustrated in Fig. 3b. There are some limitations on 1 and L; in particular, 1 and L may not be too small. As shown in Fig. 3b, for small values of bl a considerable part of emitter length may lie in the unstable region, but the emitter is not turned into a high temperature regime. If bl is not small, the high temperature r&ime is avoided when only a small part of emitter lies in the unstable region. The stability section included as an appendix.
ln the linear approach, we have
By introducing ST(x) =
‘4 = E, ;;!i
$(x>,
T=T (2) = 0
u(x),
We transform Equation (12) into the form - $” + U(X)#(x) = E+(X)
(13)
with boundary conditions 4(x) -+ 0,
X-+&-CC
(14)
It is evident that (13) and (14) are the eigenvalue problem. For stability, it is necessary to have a positive Spector of E. Equation (13) looks like the Shroedinger equation. Substituting the stationary solution T = T(x) in Equation (IO), (with q(x) = q + 410(x), and differentiating it by x, we shall find that + = T&x) has to satisfy the equation Fig. 3. The Mueaee of emitter leagtb on average temperature of emitter: b, finite lagth; a, inlinite length.
-_ i;t
+ u(x) d(x) =
(41/d Kd.
(15)
Equation (15) can be rewritten as Conclusion The performance of the system was studied in the case of practical importance, i.e. then to some part of emitter a very high heat flux from the heat source is applied as compared with the flux to the rest part of emitter. In this case the output heat flux is not sufficient for compensation and temperature of this part of emitter begin to increase. If the length of this part is large, the overheating spread of whole length and emitter is ‘jumped’ into dangerously high temperature mode of operation. In opposite case (if the length of overheated part is small or the redundant heat flux is not very high) the average temperature of the whole emitter remains normal. In this paper only the general approach to the problem is given. For particular determination of the temperature profile the exact values of output and input heat fluxes are necessary. It is shown also that the stability of emitter temperature distribution in the case of nonuniform heat flux input is improved. Acknowledgement-The authors wish to Karetnikov D.V. for a fruitful discussion.
thank
e-rt,
where TO(X)= stationary solution tested, 6T = small perturbation.
dx2
+
[
q&x) 44 - ___ q3(x)= VW 1
0.
(16)
Equation (16) also looks like the Shroedinger equation with the potential u(x) -
c71~(XY4(0)*
(17)
+(x) is the eigenfunction of (16) for zero eigenvalue. As it is positive anywhere @To/ax in our mechanical analogy is the particle velocity, which is positive), it is a function of the ground state. It is well known that the variational derivative 6E/6u for Equations (13) and (14) is positive. This means that the eigenvalues of Equation (13) are greater than the ones of Equation (16). All eigenvalues of (13) are positive, as the addition to the potential cnm1 i-1‘Cd@> is always positive and the ground state of equation with the potential
Dr.
Appendix Some remarks should be made about the stability of stationary solutions obtained. For L = CO, we look for the solution of (5) in the form T(x, t) = TO(X) + ST(x)
-_ d2+
has a zero energy. Thus, all stationary solutions found are stable.
References [l] A. Schock. Proc. Second Int. Conf. Thermionic Power Generation, Stress, Italy (1968). [2] A. I. Loshkarev, High Temperature Physics, 8, [3] Y. Y. Abramov, A. A. Vedenov, G. G. Gladusch, XIV Alimion Emission Electronics Conference. (USSR), 1970.
Electrical 2 (1970).
Theses of Tashkent
Emission Instability of Thermionic Converters Y, Y, ABRAMOV,A, A, VEOENOVand G, G, GLADUSH (Received 5 July 1972)
Introduction The original instability may appear in thermionic converters, if the operating point of the converter is on the negative slope of the emission S-curve. In this case, the total heat flux from the emitter may be decreased by increasing the temperature, due to the considerable electron cooling. This will destroy the thermal balance on the emitter. A. Schock began to investigate such an instability [1]. The linear approach was developed by Loshkarev [2]. The final stationary states of thermionic converters in which this instability may occur were investigated recently by the authors of the present article [3]. It was found that very inhomogeneous temperature profiles may exist along the emitter. The stability of these final states was also investigated. In all previous papers, the thermal flux from the heat source to the emitter was assumed to be homogeneous. However, if the input thermal flux is not homogeneous, only a certain part of the emitter may be in stable region, while the other part is not. This case, which is of practical importance, is investigated in the present article.
Basic Equations The ignited mode is assumed. The emitter, which is the thin stripe, has the length L. The width of the stripe is assumed to be unity. The load resistance of the converter is R, and the e.m.f, of the additional battery connected in series is E. The current-voltage curves of the converter are assumed to be straight lines with a constant slope: V = RoL[js(r) -- j]
where Cv = specific heat capacity for unit length, q = heat flux density to the emitter from heat source, ~T 4 = interelectrode thermal radiation density, Cj ---- electron cooling flux density, K = xd, where X = thermal conductivity and d = emitter mlcKness. Boundary conditions for (3) are OT_ _ OT -= O. Ox x=0 Ox x=z,
The system (1)-(3) may be reduced to the following integro-differential equation: OT Cv ~[ + aT 4 + Cjs[T(x, t)] L
¢ R + Ro L
E¢ + ~c aZT L(R + Ro) ~"
(5)
For convenience, the following abbreviations are introduced: W ( T ) ---- ~ r 4 + ~js(T), WR(T)
(6)
= crT4 @ 6 R ~ R°RoJS(r), --
e¢
Ro)'
C(x) = q (x)
L(R +
b(x) = C(x) +
R~_~ ( j s ) , R+Ro
(1)
(7/
(8) (9)
with
,f
( j s ) = ~,
jsd~.
In the stationary case, Equation (5) may be rewritten as
Ohm's law for the circuit is
KT" = W ( T ) -- b(x)
L
V = RI,
t)]
= q (x)
converter output voltage, converter internal resistance, current density, short circuit current density, temperature distribution along the emitter.
E+
js[T(~, o
where V= R0 = j = js = T(x) =
(4)
I = f j ( x ) dx
For q ( x ) = constant, and consequently, b ( x ) = const, c(x) = const, Equation (10) may have the homogeneous solution T(x) = const. Tis found from the simple algebraic equation WR = C. This equation commonly has three roots (see Fig. 1). It is evident that the middle root of the equation is unstable with respect to perturbations not depending on
o
and the energy balance equation for the emitter is OT 02T Cv ~ = K ~ + q
-- aT 4 --j¢,
(10)
(2)
(3)
t I. V. K u r c h a t o v Institute of Atomic Energy, Moscow, USSR. 49
50
Y . Y . ABRAMOV, A. A. VEDENOV and G. G. GLADUSH
X, i.e. the temperature region where OWR/OT < 0 is unstable. If the perturbations are periodic and have a form such that the external circuit current remains constant, the stability criterion is not dependent upon R; it is determined by internal converter parameters.
Suppose that L = co; then, for x < 0 and x > 0, Equation (10) is again the equation for particle motion in a potential well. This means that we can use a mechanical analogy again. B*~o~
W
W (7)
B*
B--
BI *I
/i 4
I
1
I
J
i
I
I
I
i
1
I
°
Fig. 2. Two solutions of F_Zluation (10): b < b*, b* < b + b i < bgaax. 'P T
Fig. 1. Components of emitter energy flux dem/ty vs emitter temperature: w, flux by R = 0; wR, flux by R ~ 0; b, 0, input power to emitter.
In particular, if the emitter length is more than a certain critical value (11) the region where OW/OT < 0 is unstable with respect to the perturbation mentioned. This region is greater than OWR/OT < 0 (see Fig. 1). As to inhomogeneous solutions of Equation (10) it is noticed that for q ( x ) = const and b(x)= const, (10) becomes a motion equation for particle moving in a static potential field. Indeed, by substituting T ~ X, x ~ ~- in (10), one can obtain:
dr~d2X--
OXO f [b -- W(X)] dX --
OXO U(X)
Thus, the term U(x) = f (b -- w (x)) dx has the meaning of potential. The boundary conditions become dr
~o
dr
r=L
0
This means that particle velocity is zero at the initial moment and at moment r = L. According to the extent with which L exceeds 11/~, the particle will oscillate one or more times in the potential field; this corresponds to a stationary temperature profile with one or more minima [3]. Discussion and Results
For q (x) :~ const, let us write q and b in following forms
q(x) = q + qtO(x) t" (X) = b + biO(x)
ql = bl
f l, O(x) = ~ 0 ,
x >0 x <0
The accepted form for q means that the particle will be exposed to additional force ql from the moment t = 0. As this force is limited, the velocity of the particle cannot change sharply. The boundary conditions mean that the particle begins to move from one top of a potential curve to another (see Fig. 2). Figure 2 represents the function W(T); three particular values of b are marked. The straight line b* is drawn so that the corresponding potential curve forms the potential well with top values equal to each other. Let b < b* and b* < b + b t < b * a x .. In this case there are two solutions, which conform to the boundary conditions at 4-oo and have the form of 'smooth step'. The low temperature level and high temperature level are determined by the conditions W ( T ) = b, and W(T) = b + ha, respectively. Mechanically it means that the particle at the moment t = oo began to run down from the left top of the potential curve; at the moment t ---- 0, the form of potential curve changes immediately and the particle will climb to the fight top, reaching it at t ~- oo. Using the mechanical analogy, the other cases may also be examined. For instance, if b * < b < bmax. , b + bt > bmax; there are not any solutions when one part of the emitter has the temperature corresponding to the left wing of function W(T), and another part has a temperature corresponding to the right one. In this case, the solution has the form of a comparatively low 'step'. The lower temperature level is the root of equation W(T) = b; the upper level is the root of equation W(T) = b + hi. Both roots are placed on the right wing W(T) (Fig. 3a). I f L =~ oo, let us take q(x) in the form
q(x) = q --b qiO(x -- 1), where 0 < 1 < L. In this case the particle moves from t = 0 (V0 = O) to t----- 1 in the potential field of one form, and from t = 1 to t = L in another (VL = 0). Let us consider again the case b * < b < bmax and b + bi > b*m~. The one solution corresponding to the lower emitter temperature (as compared to the case
51
Emission Instability of Thermionic Converters
where L = CO)is illustrated in Fig. 3b. There are some limitations on 1 and L; in particular, 1 and L may not be too small. As shown in Fig. 3b, for small values of bl a considerable part of emitter length may lie in the unstable region, but the emitter is not turned into a high temperature regime. If bl is not small, the high temperature r&ime is avoided when only a small part of emitter lies in the unstable region. The stability section included as an appendix.
ln the linear approach, we have
By introducing ST(x) =
‘4 = E, ;;!i
$(x>,
T=T (2) = 0
u(x),
We transform Equation (12) into the form - $” + U(X)#(x) = E+(X)
(13)
with boundary conditions 4(x) -+ 0,
X-+&-CC
(14)
It is evident that (13) and (14) are the eigenvalue problem. For stability, it is necessary to have a positive Spector of E. Equation (13) looks like the Shroedinger equation. Substituting the stationary solution T = T(x) in Equation (IO), (with q(x) = q + 410(x), and differentiating it by x, we shall find that + = T&x) has to satisfy the equation Fig. 3. The Mueaee of emitter leagtb on average temperature of emitter: b, finite lagth; a, inlinite length.
-_ i;t
+ u(x) d(x) =
(41/d Kd.
(15)
Equation (15) can be rewritten as Conclusion The performance of the system was studied in the case of practical importance, i.e. then to some part of emitter a very high heat flux from the heat source is applied as compared with the flux to the rest part of emitter. In this case the output heat flux is not sufficient for compensation and temperature of this part of emitter begin to increase. If the length of this part is large, the overheating spread of whole length and emitter is ‘jumped’ into dangerously high temperature mode of operation. In opposite case (if the length of overheated part is small or the redundant heat flux is not very high) the average temperature of the whole emitter remains normal. In this paper only the general approach to the problem is given. For particular determination of the temperature profile the exact values of output and input heat fluxes are necessary. It is shown also that the stability of emitter temperature distribution in the case of nonuniform heat flux input is improved. Acknowledgement-The authors wish to Karetnikov D.V. for a fruitful discussion.
thank
e-rt,
where TO(X)= stationary solution tested, 6T = small perturbation.
dx2
+
[
q&x) 44 - ___ q3(x)= VW 1
0.
(16)
Equation (16) also looks like the Shroedinger equation with the potential u(x) -
c71~(XY4(0)*
(17)
+(x) is the eigenfunction of (16) for zero eigenvalue. As it is positive anywhere @To/ax in our mechanical analogy is the particle velocity, which is positive), it is a function of the ground state. It is well known that the variational derivative 6E/6u for Equations (13) and (14) is positive. This means that the eigenvalues of Equation (13) are greater than the ones of Equation (16). All eigenvalues of (13) are positive, as the addition to the potential cnm1 i-1‘Cd@> is always positive and the ground state of equation with the potential
Dr.
Appendix Some remarks should be made about the stability of stationary solutions obtained. For L = CO, we look for the solution of (5) in the form T(x, t) = TO(X) + ST(x)
-_ d2+
has a zero energy. Thus, all stationary solutions found are stable.
References [l] A. Schock. Proc. Second Int. Conf. Thermionic Power Generation, Stress, Italy (1968). [2] A. I. Loshkarev, High Temperature Physics, 8, [3] Y. Y. Abramov, A. A. Vedenov, G. G. Gladusch, XIV Alimion Emission Electronics Conference. (USSR), 1970.
Electrical 2 (1970).
Theses of Tashkent