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A hybrid Peltier-Ettingshausen cooler for cryogenic temperatures
Solid-State
Electronics
Pergamon
Press 1964. Vol. 7, pp. 643-654.
Printed in Great Britain
A HYBRID PELTIER-ETTINGSHAUSEN FOR CRYOGENIC
COOLER
TEMPERATURES
J. R. MADIGAN’ Roy C. Ingersoll Research
Center,
Borg-Warner
(Received
23 January
Corporation,
Des Plaines, Illinois
1964)
Abstract-Expressions are obtained for the electrical power consumed in and the temperature drop across an Ettingshausen cooler as a function of the current in the cooler by a constant coefficient-‘hot junction’ analysis. The temperature drop as a function of the current in the cooler is symmetrical about its maximum value in contrast to the behavior of the temperature drop with current in the hot junction analysis of the Peltier cooler. Assuming equal figures of merit for the Peltier and the Ettingshausen coolers, their relative performance is compared. If certain practical difficulties such as thermal shorting effects due to the electrical contacts are neglected, the theoretical performance characteristics of the Ettingshausen cooler are clearly superior to those of the Peltier cooler. The design of a hybrid cooler in which a Peltier cascade acts as a variable temperature heat sink for an Ettingshausen cooler is presented. RBsumb-On obtient des expressions de la puissance electrique consommee et de la chute de temperature le long d’un refroidisseur Ettingshausen en fonction du courant dans le refroidisseur par une analyse de la ‘jonction chaude’ a coefficient constant. La chute de temperature en fonction du courant dans le refroidisseur est symetrique des deux c&es de sa valeur maximum, contrairement au comportement de la chute de temperature en fonction du courant dans l’analyse de la jonction chaude du refroidisseur Peltier. En assumant des facteurs de merite Bgaux dam les cas des refroidisseurs Peltier et Ettingshausen, leur rendement relatif est compare. Si certaines difficult& pratiques, telles que les effets de court-circuits thermiques dfis aux contacts Blectriques, sont negligees, les caracteristiques de rendement theorique du refroidisseur Ettingshausen sont clairement superieures a celles du refroidisseur Peltier. La construction d’un refroidisseur hybride dam lequel une cascade Peltier se comporte en recepteur de chaleur pour un refroidisseur Ettinghausen est presentee. Zusarnmenfassung-Einfiihrung von Gleichungen ftir den Verbrauch an elektrischer Energie und die die Temperaturabname bei Ettingshausen-Kiihlem als Funktion des Kiihlerstroms mittels einer Analyse, die eine Warmlijtstelle mit konstanten KoeBizienten zugrundelegt. Die Temperaturabnahme als Funktion des Ktihlerstroms ist um ihren M aximalwert symmetrisch, im Gegensatz zum Verhalten der Temperaturabnahme mit dem Strom in der Analyse der Warmllitstelle im Peltier-Kiihler. Unter der Annahme gleicher Leistungsziffern fiir beide Kiihlerarten wird ihr relativer Wirkungsgrad verglichen. Vernachhissigt man gewisse praktische Schwierigkeiten, wie thermische Kurzschlusseffekte an den elektrischen Kontakten, so ist der theoretische Wirkungsgrad des EttingshausenKiihlers offensichtlich dem des Peltier-Ktihlers iiberlegen. Der Entwurf eines Kombinationskiihlers, in dem eine Peltier-Kaskade als verimderliche W5rmeabftihrung fiir einen Ettingshausen-Ki.ihler dient, wird beschrieben. 1. INTRODUCTION ONE CAN reach temperatures
of about 180°K from room temperature with cascaded thermoelectric coolers.(l) Unfortunately, in this mode of operation the maximum temperature difference, ATmax, developed across a stage is directly proportional * Now at Zenith Radio Corporation, Avenue, Chicago 39, Illinois. I
6001
Dickens 643
to the square of the cold junction temperature. Thus, successive stages in the cascade make a rapidly decreasing contribution to the over-all AT unless the Peltier figure of merit, Z, increases strongly with decreasing temperature. Experimentally, we have observed(l) that for BisTes alloys the Peltier figure of merit is roughly independent of temperature over the range
644
200”-300°K.
J.
R.
MADIGAN
Since no materials superior to the BisTes alloys have yet been found for this temperature range, the above facts plus unavoidable interstage coupling losses limit practical cascaded Peltier coolers to four or five stages. Recently the prospects for attaining cryogenic temperatures by solid state means have been brightened considerably by applying a magnetic field to the cooler. The magnetic field modifies the phenomenological expressions for the maximum temperature difference and efficiency of the heat pump depending upon the mode of operation (i.e. the relative orientations of the magnetic field, the electric and heat current densities, and the temperature gradient). Two cases with the magnetic field normal to the plane of the electric and heat current densities and the temperature gradient have been studied experimentahy.(2,s) WOLFE and SMITH@) have studied the effect of a transverse magnetic field on a couple in which the heat and electric current densities are parallel. This mode of operation is essentially the same as that of a normal thermocouple and, therefore, directly shows the effect of a transverse magnetic field on a Peltier heat pump. KOOI et aZ.(s) have investigated the mode in which the temperature gradient and electric current density are perpendicular. This case is called an Ettingshausen cooler and for temperature independent coefficients in the expressions for the heat and electric current densities exact solutions may be obtained for its maximum temperature difference and efficiency just as for the normal Peltier cooler. Unfortunately, none of the other modes including that studied by Wolfe and Smith seem to be amenable to an exact analysis. Presently the thermomagnetic effects are largest in the temperature range of lOO”-200°K and in Bi-Sb rather than BiaTes alloys. In this temperature range it has only been possible to make n-type Bi-Sb alloys so that the Ettingshausen cooler which only requires one type of material is preferable at the moment to Wolfe’s and Smith’s cooler which essentially is a Peltier couple in an applied magnetic field. The AT across the Ettingshausen cooler saturates at some critical value of the magnetic field which depends upon the temperature. The condition for saturation is that the product of the charge carrier mobility and the magnetic field be much greater
than unity. Since the mobility falls with increasing temperature, the magnetic fields required to produce saturation become very large at high temperatures. It is, therefore, necessary to provide a heat sink for the Ettingshausen cooler which will limit the saturation field to reasonable values and allow the cooler to operate in a temperature range which will yield the largest AT or the lowest temperature. Cascaded Peltier coolers should be capable of providing such a heat sink and this paper is concerned with the design of a hybrid PeltierEttingshausen cooler. Since the temperature of the heat sink will depend on the thermal load presented to it by the Ettingshausen cooler, the over-all AT of the hybrid cooler may be greater if the Ettingshausen stage is operated at a current less than that which gives the maximum temperature drop for this stage. Consequently, the first part of the paper derives appropriate expressions for the electrical power consumed in and temperature drop across an Ettingshausen cooler as a function of the current in the cooler. These equations are completely equivalent to the hot junction analysis of a Peltier cooler with temperature independent parameters and the results are compared wherever possible. The design and performance of a hybrid cooler based upon these device equations is then discussed. NOTATION -a(&)/&, vrB, ratio of the electric field in the direction of the current to the thermomagnetic power magnetic field electric current electric current density average current density ratio of the electric field for maximum temperature drop to the actual field Righi-Leduc coefficient dimensions (cm) of the Ettingshausen cooler (&- l)/(l + w (1 -W/(1 + 8:) electrical power consumed in the cooler heat current density resistance of the Ettingshausen cooler in the direction of the electric current isothermal Hall coefficient isothermal Seebeck coefficient average temperature heat sink temperature cold surface temperature for the cooler temperature drop across the cooler isothermal Peltier figure of merit
A HYBRID
z; 2r gI E 81 WB JQ Pi
-VWe)
VT
PELTIER-ETTINGSHAUSEN
COOLER
isothermaJ Ettingshausen figure of merit 2/U +ZtT) d/(1 --zIT) efficiency isothermal Nernst coefficient thermomagnetic power isothermal thermal conductivity isothermal electrical resistivity electric field, negative gradient of the electrochemical potential per unit charge temperature gradient
2. THJiRMOELECTFUC AND TI-IERMOMAGNEl’IC COOLERS
A. The energy balance equation Irreversible thermodynamics gives us a linear relationship between the electric and heat current densities and the gradients of the temperature and electrochemical potential. In the absence of an applied magnetic field these relationships may be expressed in the form
FOR CRYOGENIC
yield the energy balance equation qV2T-27gVT
l
(BxJ)+
x J+ptJ2
TvtB.
5 0
X3 4
/
L3
T&J-K~VT
(2)
c” = pJ+RtBxJ+SrVT+r]cBxVT 0e
q = T&J+
Ll
TvtB x J-
K
l
(4
where J and q are the electric and heat current densities, V(pJe) and VT are the gradients of the electrochemical potential per unit charge and the temperature, and pp, St and K$ are the isothermal resistivity, Seebeck coeflicient and thermal conductivity, respectively. In an applied magnetic field& pt, St, and ~1 become functions of the field, and when B is perpendicular to the plane of J, q and VT, the appropriate expressions for V&/e) and q become -V
(6)
If there is no magnetic field (B = 0), this equation and equations (3) and (4) reduce to the familiar equations which describe the behavior of a Peltier couple with the Thompson coefficient set equal to zero. If as in Fig. 1 we set B = i3B and J = ilJ,
= ptJ+&VT
q =
V
= 0.
0
-V
645
TEMPERATURES
tc&?B x
Th
T(XJ
(3) VT
(4)
where Rt, r]r and 9 are the isothermal Hall and Nernst coefficients and the Righi-Leduc coefficient, respectively. These equations are subject to the restriction
FIG. 1. Schematic
representation cooler. (a) Relative orientation of (b) Schematic temperature (c) Schematic temperature
of the Ettingshausen j, VT, and I?. distribution. gradient.
equation (6) becomes under steady state conditions. If none of the coefficients are functions of position although they may be functions of B, equations (3), (4) and (5)
@T-27@Jz
- TqrB; 2
+ptJ2 = 0. 2
(7)
For a constant current density this equation is
646
J. R. MADIGAN
solvable for any relative orientation of J and VT or q.(4) However, J will only be constant when B = 0. The steady state energy balance [equation (S)] simply requires that V J = 0 and since J = il J, this means only that aJ/axl = 0. Thus, J may vary in a plane normal to the direction of J, e.g. in this case be a function of xs and xs. This is indeed unfortunate for it makes all the magnetic field cases except VT and J perpendicular (the Ettingshausen cooler) intractable. This problem was treated by several people under the assumption of constant J, and was finally solved correctly by KOOI et d.(s) who recognised that although J could vary in a plane normal to itself the gradient of the electrochemical potential in the direction of J is constant (i.e. a(Z.+)/axl = const.). We will restrict our discussion to the Ettingshausen and the Peltier coolers both of which can be treated exactly in the case of temperature independent or suitably temperature dependent coefficients in equations (3) and (4). In fact, if one permits the Seebeck coefficient only to vary with position in equation (2), one obtains a heat balance equation for the Peltier cooler which includes the Thompson effect and which can still be solved exactly.(*) Although this result has important implications for possible improvements in Peltier devices it will be omitted here to facilitate the comparison between Peltier and Ettingshausen coolers. l
II. Comparison oj Ettingshausen and Pelt& coolers The solution of equation (7) with aT/axl = 0 gives an expression for the temperature difference developed across the Ettingshausen cooler of Fig. 1. The conditon aTjaxl = 0 is established by placing the hot side, xs = Ls, in contact with an infinite heat sink of temperature Th. It is then possible to show that the gradient of the electrochemical potential in the direction of J (in this case a(,+)/ax,) and not J itself is constant. One can thuseliminate thecurrent densityfromequation (7) by expressing J in terms of a(+)l and aTJax2. When this is done and the equation solved, one finds that AT is a function of the gradient of the electrochemical potential in the direction of the current and may be maximised with respect to this quantity. This process gives
_a(p/e)
aT= -
ax11opt
$T&&Z;T:,
(8)
for the optimum field to give the maximum AT where 2; = (~#)~/K$pi is the isothermal Ettingshausen figure of merit. We choose this notation to distinguish the fields which give maximum AT and maximum efficiency, cmax. The maximum AT for the unloaded cooler then takes the form
where T = (Th+ Tc)/2 is the average temperature in the cooler. With the aid of equation (9) we can express equation (8) in the form
-We> AT= -- VB To ax1 1opt
(8’)
L2
In general one would not operate the Ettingshausen cooler at the field corresponding to AT,,, but at some fraction I of
AT may then be expressed in the normalised form - 1+ 2; Th( 1 - 1) + Z/2(2; Th)2
AT
-= Th
2; Th
+ 2/[1-2Z;Th(l-Z)+(Z;TiJ2(1--)I -,m
(10)
Similarly, AT/Th for the Peltier cooler as a function of the ratio of the actual to the optimum current density can be expressed in the form AT -= Th
2/(1+2&Th)-1 & Th
z{2zzTh-z(2/[1+2zaTh]-l)} 2{1+z(2/[1+2&Th]-1)}
x
where Z now equals J/J:;
(11)
and
Jt; = [-1+2/(1+2&Th)]K(/s~Ll and Zt = Sf/K+J# is the isothermal Peltier figure of merit. Equations (10) and (11) are plotted in Fig. 2 as a function of Z for various values of ZiTh and &Th. It is evident that the normalised temperature differences of both the Peltier and Ettingshausen coolers as functions of Z are unsymmetrical about their maximum values (I = 1).
A HYBRID
PELTIER-ETTINGSHAUSEN
COOLER
Since 1 is the ratio of current densities in the Peltier case and of electrical fields in the Ettingshausen case, a better comparison of the two coolers can be made if one expresses the temperature difference in the Ettingshausen cooler as a
FOR CRYOGENIC
TEMPERATURES
647
where
VB = -T,,
(13)
p&2
is the average current density in the cooler when the current has the value corresponding to maximum AT. The normalised average current density, Jgoopt, is plotted in Fig. 3 as a function of I for various
FIG. 2. AT as a function of: (1) the electric field in the direction of the electric current (Ettingshausen cooler) ; (2) the electric current density (Peltier cooler).
function of the current. Since the current density is not constant in the Ettingshausen cooler one must integrate J over the plane normal to direction of the current to obtain the current corresponding to a given electric field. In our case this becomes La L8
ss
J dxz dxs
I=
0
0
La L3 =PC
FIG. 3. Relationship between the current density and the electric field in the direction of the current.
values of Z;Th. With the aid of Figs. 2 and 3 we can plot AT/Th for the Ettingshausen cooler as a function of the normalised average current density. Since the current density is constant in a Peltier cooler of uniform cross section, the normalised temperature differences for both the Peltier and Ettingshausen coolers can be plotted in the same figure as a function of the normalised average current density. This has been done in Fig. 4 and while there is no difference in the
0
Dividing both sides of this expression by LsL3 and substituting for AT/T* from equation (10) yields the follow_ing expression for the average current density, J
J = Jopt Z;Th-1+~[1-22;T~(l-1)+(Z;T~)2(1-~)] X
0
z;Th
2.0
1.0 I
I 3.0
=SlS.pt
FIG. 4. AT as a function of the current density.
J.
648
R.
MADIGAN
Peltier case since we are merely replotting the Peltier curves of Fig. 2, there is a great change in the form of the Ettingshausen curves. In particular, we see that these curves are now symmetrical about their maximum values and the AT for all curves goes to zero at twice the average optimum current density. Since the curves for AT/Th in the Ettingshausen cooler as a function ofJ/&,t are symmetrical about their maximum value for the two special values of Z;TJ, shown in Fig. 4, one should be able to show that they are a symmetric function of J/&,t for any value of Z;Th. Comparing equations (10) and (12) we see that AT/Th can be written in the form 2& Solving equation and Z;TI, gives
- l(2 - Z; Th).
= 2;
lh
opt
(12) for I as a function
Z(2- Z;Th) =
2+2(1-
of JJopt
limit for Z;Th(3) but Z#Th for the Peltier cooler is unbounded. AT,,/Ta approaches a limiting value of one as ZiTh -+ 00 in the Peltier cooler. However, ZtTh = 1 seems to be a reasonable practical limit with Ti, = 300°K for the BisTe3 alloys used in the presently available Peltier coolers. Since the thermoelectric cascade will represent a relatively ‘soft’ heat sink for the Ettingshausen cooler compared to some liquid gas sink of essentially infinite capacity, it may be desirable to operate the thermomagnetic device at less than AT,,, to reduce the thermal load on the Peltier cooler. This can increase the overall AT for the combined thermoelectric-thermomagnetic cooler. To determine how AT varies with power in the Ettingshausen cooler one must obtain an expression for the electrical power consumed in the cooler. The electrical power, P, is found by integrating the electric power density over the sample. Thus,
Z;T+ opt
and substituting this quantity in the expression for 2AT/Th enables one to write it in the form = ZIiptR(l -f
AT -=__ T?k which is a symmetric
Z;+
(14)
opt
where function
of J/Jopt
1 = (~(cl/e)/~~l)~~/~(~/e)/~~l,
with a
maximum atJ/yopt = 1 and zeros at 0 and 2. One can compare ATmax/Th for the Peltier and Ettingshausen coolers by assuming that Z; Th= Z$Th and plotting ATmax/Th as a function of ZfTh. This is done in Fig. 5 up to ZrTh = 1. For temperature independent parameters this is the upper
Iopt = (?/d3/&2)T&2L3),
and R =
,&/L2L3.
With the aid of equations (10) and (12) P can be related to AT as follows P = lPopt(l-)Z;T/,)
0.6
1- fz;r,+
g
(15)
t 0.5 -
a4 A_r,, Tl 0.3-
0
a5 Z,Th=ZiTh
FIG. 5. AT/T,, as a function of ZtTh when Zg = Zi.
where Popt = Iz,,R is the electrical power consumed in the cooler at AT,,,. PIPopt is plotted as a function of I for various values of ZiTb in Fig. 6. With the aid of Figs. (2) and (6) one can plot PIPopt as a function of ATjTh. This has been done in Fig. 7 and we see that for Z;Th = 1 one needs only one eighth while for Z;Th = ) one needs about one quarter of the power corresponding to ATmax to achieve a temperature difference equal to 75 per cent of AT,,,.
A HYBRID
PELTIER-ETTINGSHAUSEN
COOLER
FOR CRYOGENIC
TEMPERATURES
649
so that c becomes
CE
- T,A-AT(l-
Z;T)/Z;L,-AZ&/2 (18)
A2Lz - AAT where
(19)
A = %&)/axl/@. The value of A which corresponds efficiency is
AT 6; = - Lz 1-s;
A&
to maximum
(20)
where
8; = 2/[1-
FIG. 6. Electric power in the Ettingshausen cooler vs. the electric field in the direction of the current.
Z;T].
(21)
With this value for A in equation (18) the maximum efficiency of an Ettingshausen cooler becomes Tc 1 -S;T,/Tc
iax=
E
l+s;
.
(22)
The maximum efficiency of a Peltier cooler may be expressed in the form
04 t 0.7 -
TC 6a - Th/ Te EmaX = E
0.125
a25 A7/Th
0.375
0.50
FIG. 7. Electric power in an Ettingshausen cooler vs. ATIT,,.
Having obtained an expression for the power consumed in the cooler one can calculate its etliciency. The efficiency Z, is defined by l =
(16)
42(0)/P
where 42(O) = qref is the refrigeration load on the cold side of the cooler. 42(O) can be expressed in the form
42(O) = - &;T,A-
K$l
- Z;T)-
KsZ;A+ (17)
l+St
(23)
where 66 = ~[l+&T]. One can plot both of these expressions as a function of ZiTh if Zf = Z;. It is necessary to assume that the device is operating at some fixed value of ATJTh or equivalently at some fixed T,/Th. Since ZpTh must be greater than zero unless Tc = Th, we must determine the value of Z{Th corresponding to zero efficiency for a given TJTh. For T,/Th = 718 these values are O-250 for the Ettingshausen and O-327 for the Peltier cooler. In Fig. 8 emaX is plotted as a function of ZrTh assuming that T,JTJ, = 718 and Z; = Za. Since Z{Th < 1 for the Ettingshausen cooler, the limiting value of its efficiency for this case is four and while there is no upper limit on Z#Th for the Peltier cooler the practical limit for currently available thermoelectric materials is about one. At ZfTh = 1 the Ettingshausen etliciency is about 5.5 times the efficiency of the Peltier device for this case. However, these efficiencies have been calculated from the heat current and power densities and while these quantities may be uniform in the Peltier case there is a thermal shorting effect at the ends of the Ettingshausen sample due to the electrical contacts. It has been experimentally
650
J.
R.
MADIGAN
determined@) that the expressions we have used for the performance of an Ettingshausen cooler are only valid if the length to width ratio (&/La in our case) is four or more. This effect will improve the relative performance of the Peltier cooler. The temperature drop across a thermoelectric or thermomagnetic cooler can be increased by using multistage cooling. However, it is not possible to
for the Ettingshausen cooler or the Peltier cooler if nt is substituted for $ where n; = (1 - S;)/(l + S;) and nf = (Sg- l)j(l+&) with S; z 1/1--ZiT and Si w 2/l + ZeT. Two cases of special interest are 2~ constant or .ZfT constant. & constant roughly holds for a Peltier cooler while KOOIet d.(5) have shown that ZeT constant is not a bad approximation for an Ettingshausen cooler. For both of these cases
6 max 2.0
OL 0
FIG. 8. Comparison
of the single stage efficiencies of Ettingshausen and Peltier coolers.
attain the AT’s given by equations (10) and (11) which correspond to z = 0, and at the same time reduce the temperature still further using another stage, since the second stage will generate heat which the first one will have to absorb. It can be shown(a) that the efficiency of an 1zstage cooler is
(24) It is a fairly easy matter to evaluate this expression in the limit of an infinite number of stages all operating at their maximum efficiency. For infinite staging equation (24) becomes(‘) -1
(25)
the integral can be evaluated exactly. The result for the Ettingshausen cooler (Z;T = const.) is Q =((;)““-1r.
(26)
For the Peltier cooler (26 = const.) equation (25) “;‘:s,
=((SCV)
x exp2
&(T+St(T~) [Sr(Th)-ll[S(T,)-ll-l
I
-:
(27)
To compare the infinite staging efhciency of a Peltier cooler and an Ettingshausen cooler it is necessary to substitute numerical values in equations (28) and (29). We will assume for both coolers that Th = 200” and T, = 100°K
A HYBRID
PELTIER-ETTINGSHAUSEN
COOLER FOR CRYOGENIC
TEMPERATURES
651
constant over the temperature range of lOO”-200°K. with ZrT = ZiT = O-4. These are reasonable figures for an Ettingshausen cooler and imply an These results indicate that approximations used in deriving equations (26) and (27) for the infinite effective Zi = 2.67x lo-3(“K)-1 for the thermostaging efficiencies of Ettingshausen and Peltier electric cooler. The efficiencies are then & = 0.006 coolers, respectively, are reasonable. and coo = 0*00015 for Ettingshausen and Peltier coolers, respectively. Thus 1 W will pump 6 mW from 100“ to 200°K with the Ettingshausen cooler while it will require 40 W to do the same job with the thermoelectric unit. Of course, the two units 0.20 are not really comparable in the same temperature range. The Peltier unit works best in 200”-300°K range and here, for constant 26, ZZT > ZiT so z’i Th . that the relative performance of the Peltier unit . . will improve. 0.10
I
C. Experimental results We shall now see how well the available experimental results support the simple temperature independent device theory presented in Section 2.B. It should be noted that we actually made use of the temperature dependent properties when we calculated the infinite staging efficiency of an Ettingshausen cooler since we assumed that ZiT and not Z; was constant. However, it can be shown(a) that the maximum efficiency of a single stage has exactly the same form for Za cc l/T as for constant Z;. It was, therefore, permissible to use the temperature independent result, equation (22), in calculating & for constant ZiT. In Fig. 9 the Peltier figure of merit is plotted as a function of Th for an experimental couple and we
01 A.1 0
“2oiJ
I
I
I
250
300
350
Thr
'K
FIG. 9. Z; as a function of Th. see that it is approximately constant over the range 200”-300°K. The quantity ZiTh for a rectangular Ettingshausen cooler in a magnetic field of 10 kG is plotted in Fig. 10 and we see that this is sensibly
t
l
01 -,. 0
(
1
50
I
I 100
150
I 200
Th, OK
FIG. 10. ZiT as a function of Th. The AT developed across a three stage thermoelectric cascade as a function of heat sink temperature, Th, is shown in Fig. 11. An extrapolation of the straight line drawn through these points indicates that the AT for this thermoelectric cooler will vanish at a heat sink temperature of 105°K. The temperature differences developed across both a rectangular and a shaped Ettingshausen cooler are shown in Fig. 12. For the Ettingshausen device approximately exponential shaping of the cooler in the direction of the temperature gradient is equivalent to cascading in the thermoelectric cooler. The shaped Ettingshausen cooler has a AT equal to greater than that of the thermoelectric cascade below 156°K. However, the magnetic field at which the data of Fig. 12 was taken (8000 G) is below the value required to saturate the thermomagnetic effects at the higher heat sink temperatures. At higher magnetic fields the Lockheed group has recently achieved a AT = 6o”C@) with the heat sink at 156°K. It thus appears that the shaped Ettingshausen cooler will be superior to the cascaded Peltier cooler in the range lOO”-200°K. The lowest cold junction temperature we have been able to achieve with a Peltier cooler with the hot junction at room temperature (25°C) is 156°K. This was done with an unloaded seven-stage thermoelectric cascade and
652
J.
R.
MADIGAN
suggests that one could provide a thermoelectric heat sink for an Ettingshausen cooler with a heat sink temperature between 150” and 200°K depending on the load.
thermoelectric cascade acts as the heat sink for an Ettingshausen cooler. To calculate the AT developed across the combined cooler it is sufficient to regard the cascaded Peltier cooler as a variable
FIG. 11. AT for an unloaded three stage Peltier cooler vs. Tn.
D. Design of a combined thermoelectric-thermomagnetic cooler The discussion of the previous sections indicate that the thermomagnetic cooler seems to be optimum for the temperature range lOO”-200°K while
r (G
20
lo 0
0
I
$4 I
50
Bi (97) Sb(3) Ettingshausen cooler at 6 = 8000 G I 150
I 100
I 200
Tt,.'K
FIG. 12. AT’s for shaped and rectangular coolers vs. Th.
Ettingshausen
the thermoelectric cooler is better in theZOO”-300°K temperature range. This suggests that one might be able to go from room temperature to 150°K or less with a combination thermoelectric-thermomagnetic heat pump. In the combined device a
temperature heat sink whose temperature depends upon the thermal load presented to it by the Ettingshausen cooler. We assume that the cold junction temperature of the unloaded thermoelectric cascade is 180°K and its derating factor is l”C/lOO mW. The load presented to the heat sink for the maximum temperature drop across the Ettingshausen cooler will be assumed to be 2 W so that at the maximum temperature drop across the Ettingshausen cooler Th = 200°K and when the current in the Ettingshausen cooler is zero Th = 180°K. The normalized temperature drop and electrical power consumption, and the electrical power consumed at optimum current in a rectangular Ettingshausen cooler with temperature independent coefficients are given by AT _=-Th
z;T, 2
P&2
I IoOpt
A HYBRID
PELTIER-ETTINGSHAUSEN
COOLER FOR CRYOGENIC
L1 = I&w----~ LzL3
At Th = 200°K we assumed that AT = AT,,, and, therefore, (I/L,,t)2OO”K = 1. A value of Z;Th = 0.50 at 200°K is not unreasonable so that Z; = 2.5 x 10-s. Then for a temperature independent Z; one can calculate Z;Ta for 200 2 Th 2 180°K. The value of Th may be determined for any 0 Q P Q 2 W by using the loading curve for the Peltier cooler to calculate Th. We select a reasonable value of Iopt when Th = 200°K and since Iopt =
‘lIB/&(L2L3)Tfa
=
653
The results of such a calculation together with the values assumed or calculated for the thermomagnetic properties of the Ettingshausen device are given in Table 1. The cold junction temperature of the hybrid cooler is plotted in Fig. 13 as a function of the current in the Ettingshausen cooler. We see that the temperature bottoms out at 7.6 A
and Popt = I&R
TEMPERATURES
ATn,
can calculate A (assumed independent of temperature). If values are now assumed for p{, Ll, Ls, and Ls, one can calculate Iopt and Popt for any temperature. Then knowing P, Popt and Th we can calculate I/l,,,t from the expression for PIPopt and substitute I/I,,t in ATJTa to determine the drop across the Ettingshausen device. This result is then subtracted from the Th of the sink to determine the cold junction temperature of the combination. we
Table 1. Design and performance of a hybrid Ettingshausen-Peltier
cooler
Ti,“K
T,“K Hybrid cooler
Ettingshausen heat sink 200 198 196 194 192 191 190 188 186 184 182 180
Ll
AT”K Ettingshausen cooler 50.00 48.94 47.72 46.32 44.67 43.72 42.67 40.19 36.97 32.51 25 a43 0
Th = 180+ lop,
R=pt-
FIG. 13. T, for a hybrid cooler as a function of the current in the Ettinsghausen stage.
Iopt = 5 X
WV on Peltier cooler 2.0 1.8 1.6 1.4 1.2 1.1 1.0 0.8 0.6 0.4 0.2 0
1(k2Th
, pc = 6.67 x 10-d Sz cm
I(A) Ettingshausen cooler lO*OO 9.53 9.03 8.50 7.92 7.60 7.28 6.57 5.76 4.77 3.45 0
150.00 149.06 148.28 147.68 147.33 147.28 147.33 147.81 149.03 151.49 156.57 180+0
Z; = 2.5 x lO-3/“K Popt = Qx lo-4T;
L2LS
L1 = 3 cm, Ls = O-5 cm, L3 = O-15 cm
= I&R
654
so that this is the optimum Ettingshausen cooler.
J.
R. MADIGAN
current to use in the
3. INCREASING THE AT OF THE HYBRID COOLER The calculation of the minimum temperature attained by a combination Peltier-Ettingshausen cooler was based upon equations which were derived under very restrictive assumptions. The design of both the cascaded Peltier cooler and the Ettingshausen cooler were based upon device equations which assumed that parameters such as the Seebeck and Nernst coefficients, and the electrical and thermal conductivities are independent of temperature. The figures of merit which are based on these parameters will, therefore, be constant. While this result is roughly true for the Peltier cooler, Z.;,T = const. is a better approximation for the Ettmgshausen cooler. KOOI et d.(3,5) have shown that by taking KS independent of temperature and pc and + inversely proportional to the temperature one can solve the energy balance equation exactly and satisfy the requirement ZtT = const. An Ettingshausen cooler with a constant value for ZtT would perform better than one with constant 2; and it would not be difficult to obtain the optimum design for such a hybrid cooler. Another restriction that can be relaxed is the rectangular shape of the Ettingshausen cooler. Cascading can be achieved in the Ettingshausen cooler simply by shaping the crystal in the direction of the temperature gradient. KOOI et aZ.@) have given an approximate calculation of the AT developed across an exponentially shaped cooler. The ratio ATexp/ATrect is a function of the ratio of widths of the hot and cold sides of the
cooler and increases with this ratio. Their calculation predicts a smaller AT than is observed probably partly because of the crudeness of the theory and partly because the shaping is not truly exponential. A more exact treatment of the cascade theory for both the Ettingshausen and Peltier coolers would enable one to make more reliable predictions about the ultimate performance of a hybrid cooler. In addition, any improvements in either thermoelectric or thermomagnetic materials will improve the performance of the cooler. In this regard it should be noted that the colder stages of the Peltier cascade will be in the fringing field of the magnet used with the Ettingshausen cooler. This mode of operation is the same as that studied by WOLFE and SMITH@) and one may substitute a Bi-Sb alloy for the n-type legs in one of these Peltier stages and thereby improve the overall AT. An increase of 4°C was obtained in the AT of a two stage Peltier cooler by this means.(s) REFERENCES 1. G. F. BEOSEN, A. D. REICH and C. E. RUFER, 2. 3. 4. 5. 6. 7. 8. 9.
Infrared Cell Electronically Refrigerated, ASD Tech. Rep. 61-369, September (1961). R. WOLFE and G. E. SMITH, Appl. Phys. Lett. 1, 5 (1962). C. F. Koor, R. B. HORST, K. F. CUFF, and S. R. HAWKINS,J. AppE. Phys. 34,1735 (1963). B. VARGA,A. D. REICH, and J. R. MADIGAN,J. Appl. Phys. 34, (1963). C. F. KOOI et al., Solid State Cryogenics, ASD-TDR 62-1100, February (1963). A. F. IOFFE, Semiconducting Thermoelements and Thermoelectric Cooling, Infosearch, London (1957). R. R. HEIKE~ and R. W. URE, JR., Thermoelectricity, Interscience, New York (1961). C. F. KOOI. Private communication. R. WOLFE and G. F. SMITH. Private communication.
Electronics
Pergamon
Press 1964. Vol. 7, pp. 643-654.
Printed in Great Britain
A HYBRID PELTIER-ETTINGSHAUSEN FOR CRYOGENIC
COOLER
TEMPERATURES
J. R. MADIGAN’ Roy C. Ingersoll Research
Center,
Borg-Warner
(Received
23 January
Corporation,
Des Plaines, Illinois
1964)
Abstract-Expressions are obtained for the electrical power consumed in and the temperature drop across an Ettingshausen cooler as a function of the current in the cooler by a constant coefficient-‘hot junction’ analysis. The temperature drop as a function of the current in the cooler is symmetrical about its maximum value in contrast to the behavior of the temperature drop with current in the hot junction analysis of the Peltier cooler. Assuming equal figures of merit for the Peltier and the Ettingshausen coolers, their relative performance is compared. If certain practical difficulties such as thermal shorting effects due to the electrical contacts are neglected, the theoretical performance characteristics of the Ettingshausen cooler are clearly superior to those of the Peltier cooler. The design of a hybrid cooler in which a Peltier cascade acts as a variable temperature heat sink for an Ettingshausen cooler is presented. RBsumb-On obtient des expressions de la puissance electrique consommee et de la chute de temperature le long d’un refroidisseur Ettingshausen en fonction du courant dans le refroidisseur par une analyse de la ‘jonction chaude’ a coefficient constant. La chute de temperature en fonction du courant dans le refroidisseur est symetrique des deux c&es de sa valeur maximum, contrairement au comportement de la chute de temperature en fonction du courant dans l’analyse de la jonction chaude du refroidisseur Peltier. En assumant des facteurs de merite Bgaux dam les cas des refroidisseurs Peltier et Ettingshausen, leur rendement relatif est compare. Si certaines difficult& pratiques, telles que les effets de court-circuits thermiques dfis aux contacts Blectriques, sont negligees, les caracteristiques de rendement theorique du refroidisseur Ettingshausen sont clairement superieures a celles du refroidisseur Peltier. La construction d’un refroidisseur hybride dam lequel une cascade Peltier se comporte en recepteur de chaleur pour un refroidisseur Ettinghausen est presentee. Zusarnmenfassung-Einfiihrung von Gleichungen ftir den Verbrauch an elektrischer Energie und die die Temperaturabname bei Ettingshausen-Kiihlem als Funktion des Kiihlerstroms mittels einer Analyse, die eine Warmlijtstelle mit konstanten KoeBizienten zugrundelegt. Die Temperaturabnahme als Funktion des Ktihlerstroms ist um ihren M aximalwert symmetrisch, im Gegensatz zum Verhalten der Temperaturabnahme mit dem Strom in der Analyse der Warmllitstelle im Peltier-Kiihler. Unter der Annahme gleicher Leistungsziffern fiir beide Kiihlerarten wird ihr relativer Wirkungsgrad verglichen. Vernachhissigt man gewisse praktische Schwierigkeiten, wie thermische Kurzschlusseffekte an den elektrischen Kontakten, so ist der theoretische Wirkungsgrad des EttingshausenKiihlers offensichtlich dem des Peltier-Ktihlers iiberlegen. Der Entwurf eines Kombinationskiihlers, in dem eine Peltier-Kaskade als verimderliche W5rmeabftihrung fiir einen Ettingshausen-Ki.ihler dient, wird beschrieben. 1. INTRODUCTION ONE CAN reach temperatures
of about 180°K from room temperature with cascaded thermoelectric coolers.(l) Unfortunately, in this mode of operation the maximum temperature difference, ATmax, developed across a stage is directly proportional * Now at Zenith Radio Corporation, Avenue, Chicago 39, Illinois. I
6001
Dickens 643
to the square of the cold junction temperature. Thus, successive stages in the cascade make a rapidly decreasing contribution to the over-all AT unless the Peltier figure of merit, Z, increases strongly with decreasing temperature. Experimentally, we have observed(l) that for BisTes alloys the Peltier figure of merit is roughly independent of temperature over the range
644
200”-300°K.
J.
R.
MADIGAN
Since no materials superior to the BisTes alloys have yet been found for this temperature range, the above facts plus unavoidable interstage coupling losses limit practical cascaded Peltier coolers to four or five stages. Recently the prospects for attaining cryogenic temperatures by solid state means have been brightened considerably by applying a magnetic field to the cooler. The magnetic field modifies the phenomenological expressions for the maximum temperature difference and efficiency of the heat pump depending upon the mode of operation (i.e. the relative orientations of the magnetic field, the electric and heat current densities, and the temperature gradient). Two cases with the magnetic field normal to the plane of the electric and heat current densities and the temperature gradient have been studied experimentahy.(2,s) WOLFE and SMITH@) have studied the effect of a transverse magnetic field on a couple in which the heat and electric current densities are parallel. This mode of operation is essentially the same as that of a normal thermocouple and, therefore, directly shows the effect of a transverse magnetic field on a Peltier heat pump. KOOI et aZ.(s) have investigated the mode in which the temperature gradient and electric current density are perpendicular. This case is called an Ettingshausen cooler and for temperature independent coefficients in the expressions for the heat and electric current densities exact solutions may be obtained for its maximum temperature difference and efficiency just as for the normal Peltier cooler. Unfortunately, none of the other modes including that studied by Wolfe and Smith seem to be amenable to an exact analysis. Presently the thermomagnetic effects are largest in the temperature range of lOO”-200°K and in Bi-Sb rather than BiaTes alloys. In this temperature range it has only been possible to make n-type Bi-Sb alloys so that the Ettingshausen cooler which only requires one type of material is preferable at the moment to Wolfe’s and Smith’s cooler which essentially is a Peltier couple in an applied magnetic field. The AT across the Ettingshausen cooler saturates at some critical value of the magnetic field which depends upon the temperature. The condition for saturation is that the product of the charge carrier mobility and the magnetic field be much greater
than unity. Since the mobility falls with increasing temperature, the magnetic fields required to produce saturation become very large at high temperatures. It is, therefore, necessary to provide a heat sink for the Ettingshausen cooler which will limit the saturation field to reasonable values and allow the cooler to operate in a temperature range which will yield the largest AT or the lowest temperature. Cascaded Peltier coolers should be capable of providing such a heat sink and this paper is concerned with the design of a hybrid PeltierEttingshausen cooler. Since the temperature of the heat sink will depend on the thermal load presented to it by the Ettingshausen cooler, the over-all AT of the hybrid cooler may be greater if the Ettingshausen stage is operated at a current less than that which gives the maximum temperature drop for this stage. Consequently, the first part of the paper derives appropriate expressions for the electrical power consumed in and temperature drop across an Ettingshausen cooler as a function of the current in the cooler. These equations are completely equivalent to the hot junction analysis of a Peltier cooler with temperature independent parameters and the results are compared wherever possible. The design and performance of a hybrid cooler based upon these device equations is then discussed. NOTATION -a(&)/&, vrB, ratio of the electric field in the direction of the current to the thermomagnetic power magnetic field electric current electric current density average current density ratio of the electric field for maximum temperature drop to the actual field Righi-Leduc coefficient dimensions (cm) of the Ettingshausen cooler (&- l)/(l + w (1 -W/(1 + 8:) electrical power consumed in the cooler heat current density resistance of the Ettingshausen cooler in the direction of the electric current isothermal Hall coefficient isothermal Seebeck coefficient average temperature heat sink temperature cold surface temperature for the cooler temperature drop across the cooler isothermal Peltier figure of merit
A HYBRID
z; 2r gI E 81 WB JQ Pi
-VWe)
VT
PELTIER-ETTINGSHAUSEN
COOLER
isothermaJ Ettingshausen figure of merit 2/U +ZtT) d/(1 --zIT) efficiency isothermal Nernst coefficient thermomagnetic power isothermal thermal conductivity isothermal electrical resistivity electric field, negative gradient of the electrochemical potential per unit charge temperature gradient
2. THJiRMOELECTFUC AND TI-IERMOMAGNEl’IC COOLERS
A. The energy balance equation Irreversible thermodynamics gives us a linear relationship between the electric and heat current densities and the gradients of the temperature and electrochemical potential. In the absence of an applied magnetic field these relationships may be expressed in the form
FOR CRYOGENIC
yield the energy balance equation qV2T-27gVT
l
(BxJ)+
x J+ptJ2
TvtB.
5 0
X3 4
/
L3
T&J-K~VT
(2)
c” = pJ+RtBxJ+SrVT+r]cBxVT 0e
q = T&J+
Ll
TvtB x J-
K
l
(4
where J and q are the electric and heat current densities, V(pJe) and VT are the gradients of the electrochemical potential per unit charge and the temperature, and pp, St and K$ are the isothermal resistivity, Seebeck coeflicient and thermal conductivity, respectively. In an applied magnetic field& pt, St, and ~1 become functions of the field, and when B is perpendicular to the plane of J, q and VT, the appropriate expressions for V&/e) and q become -V
(6)
If there is no magnetic field (B = 0), this equation and equations (3) and (4) reduce to the familiar equations which describe the behavior of a Peltier couple with the Thompson coefficient set equal to zero. If as in Fig. 1 we set B = i3B and J = ilJ,
= ptJ+&VT
q =
V
= 0.
0
-V
645
TEMPERATURES
tc&?B x
Th
T(XJ
(3) VT
(4)
where Rt, r]r and 9 are the isothermal Hall and Nernst coefficients and the Righi-Leduc coefficient, respectively. These equations are subject to the restriction
FIG. 1. Schematic
representation cooler. (a) Relative orientation of (b) Schematic temperature (c) Schematic temperature
of the Ettingshausen j, VT, and I?. distribution. gradient.
equation (6) becomes under steady state conditions. If none of the coefficients are functions of position although they may be functions of B, equations (3), (4) and (5)
@T-27@Jz
- TqrB; 2
+ptJ2 = 0. 2
(7)
For a constant current density this equation is
646
J. R. MADIGAN
solvable for any relative orientation of J and VT or q.(4) However, J will only be constant when B = 0. The steady state energy balance [equation (S)] simply requires that V J = 0 and since J = il J, this means only that aJ/axl = 0. Thus, J may vary in a plane normal to the direction of J, e.g. in this case be a function of xs and xs. This is indeed unfortunate for it makes all the magnetic field cases except VT and J perpendicular (the Ettingshausen cooler) intractable. This problem was treated by several people under the assumption of constant J, and was finally solved correctly by KOOI et d.(s) who recognised that although J could vary in a plane normal to itself the gradient of the electrochemical potential in the direction of J is constant (i.e. a(Z.+)/axl = const.). We will restrict our discussion to the Ettingshausen and the Peltier coolers both of which can be treated exactly in the case of temperature independent or suitably temperature dependent coefficients in equations (3) and (4). In fact, if one permits the Seebeck coefficient only to vary with position in equation (2), one obtains a heat balance equation for the Peltier cooler which includes the Thompson effect and which can still be solved exactly.(*) Although this result has important implications for possible improvements in Peltier devices it will be omitted here to facilitate the comparison between Peltier and Ettingshausen coolers. l
II. Comparison oj Ettingshausen and Pelt& coolers The solution of equation (7) with aT/axl = 0 gives an expression for the temperature difference developed across the Ettingshausen cooler of Fig. 1. The conditon aTjaxl = 0 is established by placing the hot side, xs = Ls, in contact with an infinite heat sink of temperature Th. It is then possible to show that the gradient of the electrochemical potential in the direction of J (in this case a(,+)/ax,) and not J itself is constant. One can thuseliminate thecurrent densityfromequation (7) by expressing J in terms of a(+)l and aTJax2. When this is done and the equation solved, one finds that AT is a function of the gradient of the electrochemical potential in the direction of the current and may be maximised with respect to this quantity. This process gives
_a(p/e)
aT= -
ax11opt
$T&&Z;T:,
(8)
for the optimum field to give the maximum AT where 2; = (~#)~/K$pi is the isothermal Ettingshausen figure of merit. We choose this notation to distinguish the fields which give maximum AT and maximum efficiency, cmax. The maximum AT for the unloaded cooler then takes the form
where T = (Th+ Tc)/2 is the average temperature in the cooler. With the aid of equation (9) we can express equation (8) in the form
-We> AT= -- VB To ax1 1opt
(8’)
L2
In general one would not operate the Ettingshausen cooler at the field corresponding to AT,,, but at some fraction I of
AT may then be expressed in the normalised form - 1+ 2; Th( 1 - 1) + Z/2(2; Th)2
AT
-= Th
2; Th
+ 2/[1-2Z;Th(l-Z)+(Z;TiJ2(1--)I -,m
(10)
Similarly, AT/Th for the Peltier cooler as a function of the ratio of the actual to the optimum current density can be expressed in the form AT -= Th
2/(1+2&Th)-1 & Th
z{2zzTh-z(2/[1+2zaTh]-l)} 2{1+z(2/[1+2&Th]-1)}
x
where Z now equals J/J:;
(11)
and
Jt; = [-1+2/(1+2&Th)]K(/s~Ll and Zt = Sf/K+J# is the isothermal Peltier figure of merit. Equations (10) and (11) are plotted in Fig. 2 as a function of Z for various values of ZiTh and &Th. It is evident that the normalised temperature differences of both the Peltier and Ettingshausen coolers as functions of Z are unsymmetrical about their maximum values (I = 1).
A HYBRID
PELTIER-ETTINGSHAUSEN
COOLER
Since 1 is the ratio of current densities in the Peltier case and of electrical fields in the Ettingshausen case, a better comparison of the two coolers can be made if one expresses the temperature difference in the Ettingshausen cooler as a
FOR CRYOGENIC
TEMPERATURES
647
where
VB = -T,,
(13)
p&2
is the average current density in the cooler when the current has the value corresponding to maximum AT. The normalised average current density, Jgoopt, is plotted in Fig. 3 as a function of I for various
FIG. 2. AT as a function of: (1) the electric field in the direction of the electric current (Ettingshausen cooler) ; (2) the electric current density (Peltier cooler).
function of the current. Since the current density is not constant in the Ettingshausen cooler one must integrate J over the plane normal to direction of the current to obtain the current corresponding to a given electric field. In our case this becomes La L8
ss
J dxz dxs
I=
0
0
La L3 =PC
FIG. 3. Relationship between the current density and the electric field in the direction of the current.
values of Z;Th. With the aid of Figs. 2 and 3 we can plot AT/Th for the Ettingshausen cooler as a function of the normalised average current density. Since the current density is constant in a Peltier cooler of uniform cross section, the normalised temperature differences for both the Peltier and Ettingshausen coolers can be plotted in the same figure as a function of the normalised average current density. This has been done in Fig. 4 and while there is no difference in the
0
Dividing both sides of this expression by LsL3 and substituting for AT/T* from equation (10) yields the follow_ing expression for the average current density, J
J = Jopt Z;Th-1+~[1-22;T~(l-1)+(Z;T~)2(1-~)] X
0
z;Th
2.0
1.0 I
I 3.0
=SlS.pt
FIG. 4. AT as a function of the current density.
J.
648
R.
MADIGAN
Peltier case since we are merely replotting the Peltier curves of Fig. 2, there is a great change in the form of the Ettingshausen curves. In particular, we see that these curves are now symmetrical about their maximum values and the AT for all curves goes to zero at twice the average optimum current density. Since the curves for AT/Th in the Ettingshausen cooler as a function ofJ/&,t are symmetrical about their maximum value for the two special values of Z;TJ, shown in Fig. 4, one should be able to show that they are a symmetric function of J/&,t for any value of Z;Th. Comparing equations (10) and (12) we see that AT/Th can be written in the form 2& Solving equation and Z;TI, gives
- l(2 - Z; Th).
= 2;
lh
opt
(12) for I as a function
Z(2- Z;Th) =
2+2(1-
of JJopt
limit for Z;Th(3) but Z#Th for the Peltier cooler is unbounded. AT,,/Ta approaches a limiting value of one as ZiTh -+ 00 in the Peltier cooler. However, ZtTh = 1 seems to be a reasonable practical limit with Ti, = 300°K for the BisTe3 alloys used in the presently available Peltier coolers. Since the thermoelectric cascade will represent a relatively ‘soft’ heat sink for the Ettingshausen cooler compared to some liquid gas sink of essentially infinite capacity, it may be desirable to operate the thermomagnetic device at less than AT,,, to reduce the thermal load on the Peltier cooler. This can increase the overall AT for the combined thermoelectric-thermomagnetic cooler. To determine how AT varies with power in the Ettingshausen cooler one must obtain an expression for the electrical power consumed in the cooler. The electrical power, P, is found by integrating the electric power density over the sample. Thus,
Z;T+ opt
and substituting this quantity in the expression for 2AT/Th enables one to write it in the form = ZIiptR(l -f
AT -=__ T?k which is a symmetric
Z;+
(14)
opt
where function
of J/Jopt
1 = (~(cl/e)/~~l)~~/~(~/e)/~~l,
with a
maximum atJ/yopt = 1 and zeros at 0 and 2. One can compare ATmax/Th for the Peltier and Ettingshausen coolers by assuming that Z; Th= Z$Th and plotting ATmax/Th as a function of ZfTh. This is done in Fig. 5 up to ZrTh = 1. For temperature independent parameters this is the upper
Iopt = (?/d3/&2)T&2L3),
and R =
,&/L2L3.
With the aid of equations (10) and (12) P can be related to AT as follows P = lPopt(l-)Z;T/,)
0.6
1- fz;r,+
g
(15)
t 0.5 -
a4 A_r,, Tl 0.3-
0
a5 Z,Th=ZiTh
FIG. 5. AT/T,, as a function of ZtTh when Zg = Zi.
where Popt = Iz,,R is the electrical power consumed in the cooler at AT,,,. PIPopt is plotted as a function of I for various values of ZiTb in Fig. 6. With the aid of Figs. (2) and (6) one can plot PIPopt as a function of ATjTh. This has been done in Fig. 7 and we see that for Z;Th = 1 one needs only one eighth while for Z;Th = ) one needs about one quarter of the power corresponding to ATmax to achieve a temperature difference equal to 75 per cent of AT,,,.
A HYBRID
PELTIER-ETTINGSHAUSEN
COOLER
FOR CRYOGENIC
TEMPERATURES
649
so that c becomes
CE
- T,A-AT(l-
Z;T)/Z;L,-AZ&/2 (18)
A2Lz - AAT where
(19)
A = %&)/axl/@. The value of A which corresponds efficiency is
AT 6; = - Lz 1-s;
A&
to maximum
(20)
where
8; = 2/[1-
FIG. 6. Electric power in the Ettingshausen cooler vs. the electric field in the direction of the current.
Z;T].
(21)
With this value for A in equation (18) the maximum efficiency of an Ettingshausen cooler becomes Tc 1 -S;T,/Tc
iax=
E
l+s;
.
(22)
The maximum efficiency of a Peltier cooler may be expressed in the form
04 t 0.7 -
TC 6a - Th/ Te EmaX = E
0.125
a25 A7/Th
0.375
0.50
FIG. 7. Electric power in an Ettingshausen cooler vs. ATIT,,.
Having obtained an expression for the power consumed in the cooler one can calculate its etliciency. The efficiency Z, is defined by l =
(16)
42(0)/P
where 42(O) = qref is the refrigeration load on the cold side of the cooler. 42(O) can be expressed in the form
42(O) = - &;T,A-
K$l
- Z;T)-
KsZ;A+ (17)
l+St
(23)
where 66 = ~[l+&T]. One can plot both of these expressions as a function of ZiTh if Zf = Z;. It is necessary to assume that the device is operating at some fixed value of ATJTh or equivalently at some fixed T,/Th. Since ZpTh must be greater than zero unless Tc = Th, we must determine the value of Z{Th corresponding to zero efficiency for a given TJTh. For T,/Th = 718 these values are O-250 for the Ettingshausen and O-327 for the Peltier cooler. In Fig. 8 emaX is plotted as a function of ZrTh assuming that T,JTJ, = 718 and Z; = Za. Since Z{Th < 1 for the Ettingshausen cooler, the limiting value of its efficiency for this case is four and while there is no upper limit on Z#Th for the Peltier cooler the practical limit for currently available thermoelectric materials is about one. At ZfTh = 1 the Ettingshausen etliciency is about 5.5 times the efficiency of the Peltier device for this case. However, these efficiencies have been calculated from the heat current and power densities and while these quantities may be uniform in the Peltier case there is a thermal shorting effect at the ends of the Ettingshausen sample due to the electrical contacts. It has been experimentally
650
J.
R.
MADIGAN
determined@) that the expressions we have used for the performance of an Ettingshausen cooler are only valid if the length to width ratio (&/La in our case) is four or more. This effect will improve the relative performance of the Peltier cooler. The temperature drop across a thermoelectric or thermomagnetic cooler can be increased by using multistage cooling. However, it is not possible to
for the Ettingshausen cooler or the Peltier cooler if nt is substituted for $ where n; = (1 - S;)/(l + S;) and nf = (Sg- l)j(l+&) with S; z 1/1--ZiT and Si w 2/l + ZeT. Two cases of special interest are 2~ constant or .ZfT constant. & constant roughly holds for a Peltier cooler while KOOIet d.(5) have shown that ZeT constant is not a bad approximation for an Ettingshausen cooler. For both of these cases
6 max 2.0
OL 0
FIG. 8. Comparison
of the single stage efficiencies of Ettingshausen and Peltier coolers.
attain the AT’s given by equations (10) and (11) which correspond to z = 0, and at the same time reduce the temperature still further using another stage, since the second stage will generate heat which the first one will have to absorb. It can be shown(a) that the efficiency of an 1zstage cooler is
(24) It is a fairly easy matter to evaluate this expression in the limit of an infinite number of stages all operating at their maximum efficiency. For infinite staging equation (24) becomes(‘) -1
(25)
the integral can be evaluated exactly. The result for the Ettingshausen cooler (Z;T = const.) is Q =((;)““-1r.
(26)
For the Peltier cooler (26 = const.) equation (25) “;‘:s,
=((SCV)
x exp2
&(T+St(T~) [Sr(Th)-ll[S(T,)-ll-l
I
-:
(27)
To compare the infinite staging efhciency of a Peltier cooler and an Ettingshausen cooler it is necessary to substitute numerical values in equations (28) and (29). We will assume for both coolers that Th = 200” and T, = 100°K
A HYBRID
PELTIER-ETTINGSHAUSEN
COOLER FOR CRYOGENIC
TEMPERATURES
651
constant over the temperature range of lOO”-200°K. with ZrT = ZiT = O-4. These are reasonable figures for an Ettingshausen cooler and imply an These results indicate that approximations used in deriving equations (26) and (27) for the infinite effective Zi = 2.67x lo-3(“K)-1 for the thermostaging efficiencies of Ettingshausen and Peltier electric cooler. The efficiencies are then & = 0.006 coolers, respectively, are reasonable. and coo = 0*00015 for Ettingshausen and Peltier coolers, respectively. Thus 1 W will pump 6 mW from 100“ to 200°K with the Ettingshausen cooler while it will require 40 W to do the same job with the thermoelectric unit. Of course, the two units 0.20 are not really comparable in the same temperature range. The Peltier unit works best in 200”-300°K range and here, for constant 26, ZZT > ZiT so z’i Th . that the relative performance of the Peltier unit . . will improve. 0.10
I
C. Experimental results We shall now see how well the available experimental results support the simple temperature independent device theory presented in Section 2.B. It should be noted that we actually made use of the temperature dependent properties when we calculated the infinite staging efficiency of an Ettingshausen cooler since we assumed that ZiT and not Z; was constant. However, it can be shown(a) that the maximum efficiency of a single stage has exactly the same form for Za cc l/T as for constant Z;. It was, therefore, permissible to use the temperature independent result, equation (22), in calculating & for constant ZiT. In Fig. 9 the Peltier figure of merit is plotted as a function of Th for an experimental couple and we
01 A.1 0
“2oiJ
I
I
I
250
300
350
Thr
'K
FIG. 9. Z; as a function of Th. see that it is approximately constant over the range 200”-300°K. The quantity ZiTh for a rectangular Ettingshausen cooler in a magnetic field of 10 kG is plotted in Fig. 10 and we see that this is sensibly
t
l
01 -,. 0
(
1
50
I
I 100
150
I 200
Th, OK
FIG. 10. ZiT as a function of Th. The AT developed across a three stage thermoelectric cascade as a function of heat sink temperature, Th, is shown in Fig. 11. An extrapolation of the straight line drawn through these points indicates that the AT for this thermoelectric cooler will vanish at a heat sink temperature of 105°K. The temperature differences developed across both a rectangular and a shaped Ettingshausen cooler are shown in Fig. 12. For the Ettingshausen device approximately exponential shaping of the cooler in the direction of the temperature gradient is equivalent to cascading in the thermoelectric cooler. The shaped Ettingshausen cooler has a AT equal to greater than that of the thermoelectric cascade below 156°K. However, the magnetic field at which the data of Fig. 12 was taken (8000 G) is below the value required to saturate the thermomagnetic effects at the higher heat sink temperatures. At higher magnetic fields the Lockheed group has recently achieved a AT = 6o”C@) with the heat sink at 156°K. It thus appears that the shaped Ettingshausen cooler will be superior to the cascaded Peltier cooler in the range lOO”-200°K. The lowest cold junction temperature we have been able to achieve with a Peltier cooler with the hot junction at room temperature (25°C) is 156°K. This was done with an unloaded seven-stage thermoelectric cascade and
652
J.
R.
MADIGAN
suggests that one could provide a thermoelectric heat sink for an Ettingshausen cooler with a heat sink temperature between 150” and 200°K depending on the load.
thermoelectric cascade acts as the heat sink for an Ettingshausen cooler. To calculate the AT developed across the combined cooler it is sufficient to regard the cascaded Peltier cooler as a variable
FIG. 11. AT for an unloaded three stage Peltier cooler vs. Tn.
D. Design of a combined thermoelectric-thermomagnetic cooler The discussion of the previous sections indicate that the thermomagnetic cooler seems to be optimum for the temperature range lOO”-200°K while
r (G
20
lo 0
0
I
$4 I
50
Bi (97) Sb(3) Ettingshausen cooler at 6 = 8000 G I 150
I 100
I 200
Tt,.'K
FIG. 12. AT’s for shaped and rectangular coolers vs. Th.
Ettingshausen
the thermoelectric cooler is better in theZOO”-300°K temperature range. This suggests that one might be able to go from room temperature to 150°K or less with a combination thermoelectric-thermomagnetic heat pump. In the combined device a
temperature heat sink whose temperature depends upon the thermal load presented to it by the Ettingshausen cooler. We assume that the cold junction temperature of the unloaded thermoelectric cascade is 180°K and its derating factor is l”C/lOO mW. The load presented to the heat sink for the maximum temperature drop across the Ettingshausen cooler will be assumed to be 2 W so that at the maximum temperature drop across the Ettingshausen cooler Th = 200°K and when the current in the Ettingshausen cooler is zero Th = 180°K. The normalized temperature drop and electrical power consumption, and the electrical power consumed at optimum current in a rectangular Ettingshausen cooler with temperature independent coefficients are given by AT _=-Th
z;T, 2
P&2
I IoOpt
A HYBRID
PELTIER-ETTINGSHAUSEN
COOLER FOR CRYOGENIC
L1 = I&w----~ LzL3
At Th = 200°K we assumed that AT = AT,,, and, therefore, (I/L,,t)2OO”K = 1. A value of Z;Th = 0.50 at 200°K is not unreasonable so that Z; = 2.5 x 10-s. Then for a temperature independent Z; one can calculate Z;Ta for 200 2 Th 2 180°K. The value of Th may be determined for any 0 Q P Q 2 W by using the loading curve for the Peltier cooler to calculate Th. We select a reasonable value of Iopt when Th = 200°K and since Iopt =
‘lIB/&(L2L3)Tfa
=
653
The results of such a calculation together with the values assumed or calculated for the thermomagnetic properties of the Ettingshausen device are given in Table 1. The cold junction temperature of the hybrid cooler is plotted in Fig. 13 as a function of the current in the Ettingshausen cooler. We see that the temperature bottoms out at 7.6 A
and Popt = I&R
TEMPERATURES
ATn,
can calculate A (assumed independent of temperature). If values are now assumed for p{, Ll, Ls, and Ls, one can calculate Iopt and Popt for any temperature. Then knowing P, Popt and Th we can calculate I/l,,,t from the expression for PIPopt and substitute I/I,,t in ATJTa to determine the drop across the Ettingshausen device. This result is then subtracted from the Th of the sink to determine the cold junction temperature of the combination. we
Table 1. Design and performance of a hybrid Ettingshausen-Peltier
cooler
Ti,“K
T,“K Hybrid cooler
Ettingshausen heat sink 200 198 196 194 192 191 190 188 186 184 182 180
Ll
AT”K Ettingshausen cooler 50.00 48.94 47.72 46.32 44.67 43.72 42.67 40.19 36.97 32.51 25 a43 0
Th = 180+ lop,
R=pt-
FIG. 13. T, for a hybrid cooler as a function of the current in the Ettinsghausen stage.
Iopt = 5 X
WV on Peltier cooler 2.0 1.8 1.6 1.4 1.2 1.1 1.0 0.8 0.6 0.4 0.2 0
1(k2Th
, pc = 6.67 x 10-d Sz cm
I(A) Ettingshausen cooler lO*OO 9.53 9.03 8.50 7.92 7.60 7.28 6.57 5.76 4.77 3.45 0
150.00 149.06 148.28 147.68 147.33 147.28 147.33 147.81 149.03 151.49 156.57 180+0
Z; = 2.5 x lO-3/“K Popt = Qx lo-4T;
L2LS
L1 = 3 cm, Ls = O-5 cm, L3 = O-15 cm
= I&R
654
so that this is the optimum Ettingshausen cooler.
J.
R. MADIGAN
current to use in the
3. INCREASING THE AT OF THE HYBRID COOLER The calculation of the minimum temperature attained by a combination Peltier-Ettingshausen cooler was based upon equations which were derived under very restrictive assumptions. The design of both the cascaded Peltier cooler and the Ettingshausen cooler were based upon device equations which assumed that parameters such as the Seebeck and Nernst coefficients, and the electrical and thermal conductivities are independent of temperature. The figures of merit which are based on these parameters will, therefore, be constant. While this result is roughly true for the Peltier cooler, Z.;,T = const. is a better approximation for the Ettmgshausen cooler. KOOI et d.(3,5) have shown that by taking KS independent of temperature and pc and + inversely proportional to the temperature one can solve the energy balance equation exactly and satisfy the requirement ZtT = const. An Ettingshausen cooler with a constant value for ZtT would perform better than one with constant 2; and it would not be difficult to obtain the optimum design for such a hybrid cooler. Another restriction that can be relaxed is the rectangular shape of the Ettingshausen cooler. Cascading can be achieved in the Ettingshausen cooler simply by shaping the crystal in the direction of the temperature gradient. KOOI et aZ.@) have given an approximate calculation of the AT developed across an exponentially shaped cooler. The ratio ATexp/ATrect is a function of the ratio of widths of the hot and cold sides of the
cooler and increases with this ratio. Their calculation predicts a smaller AT than is observed probably partly because of the crudeness of the theory and partly because the shaping is not truly exponential. A more exact treatment of the cascade theory for both the Ettingshausen and Peltier coolers would enable one to make more reliable predictions about the ultimate performance of a hybrid cooler. In addition, any improvements in either thermoelectric or thermomagnetic materials will improve the performance of the cooler. In this regard it should be noted that the colder stages of the Peltier cascade will be in the fringing field of the magnet used with the Ettingshausen cooler. This mode of operation is the same as that studied by WOLFE and SMITH@) and one may substitute a Bi-Sb alloy for the n-type legs in one of these Peltier stages and thereby improve the overall AT. An increase of 4°C was obtained in the AT of a two stage Peltier cooler by this means.(s) REFERENCES 1. G. F. BEOSEN, A. D. REICH and C. E. RUFER, 2. 3. 4. 5. 6. 7. 8. 9.
Infrared Cell Electronically Refrigerated, ASD Tech. Rep. 61-369, September (1961). R. WOLFE and G. E. SMITH, Appl. Phys. Lett. 1, 5 (1962). C. F. Koor, R. B. HORST, K. F. CUFF, and S. R. HAWKINS,J. AppE. Phys. 34,1735 (1963). B. VARGA,A. D. REICH, and J. R. MADIGAN,J. Appl. Phys. 34, (1963). C. F. KOOI et al., Solid State Cryogenics, ASD-TDR 62-1100, February (1963). A. F. IOFFE, Semiconducting Thermoelements and Thermoelectric Cooling, Infosearch, London (1957). R. R. HEIKE~ and R. W. URE, JR., Thermoelectricity, Interscience, New York (1961). C. F. KOOI. Private communication. R. WOLFE and G. F. SMITH. Private communication.