GB2228823A - Thermo-electric generators and heat pumps - Google Patents

Abstract

A thermo-couple comprises matching p-type and n-type thermo-elements designed to have a Seebeck co-efficient that varies from a maximum at the hot junction to a low or zero value at the cold junction in the case of a generator, and from a maximum at the cold junction to a low or zero value at the hot junction in the case of a pump. The variation in Seebeck co-efficient is obtained by altering the amount of dopant along the length of the element. The amount of dopant is a prescribed function of the temperature of the element. The elements can be prepared from powders by separately metering the basic material and dopant, mixing, pressing and sintering.

Description

THERMO-ELECTRIC GENERATORS AND HEAT PUMPS This invention relates to the design and construction of thermo-electric devices for generating power and for pumping heat from a source at one temperature to a sink at a higher temperature. Its purpose is to improve the efficiency of generators and the performance of pumps.

Thermo-electric devices have been investigated by a number of workers, notably by Ioffe, A.F. ("Semi-conductor thermo-elements and thermoelectric cooling", Infosearch, London, 1957); Goldsmid, H.J. ("Thermoelectric refrigeration", Heywood, London, 1964); Wright, D.A. ("Direct generation of electricity", Academic Press, New York and London, 1965); and Ure, R.W. and Heikes, R.R. (2science and engineering of thermoelectricity", Interscience Publishers, New York and London, 1961). These and other investigators have examined the characteristics of devices that depend on the Peltier and Seebeck effects but have often ignored the Thomson effect or regarded it as of minor importance. In the present invention the Thomson effect plays a major role.

Introduction In describing the devices the following symbols are used: a = Seebeck co-efficient, in micro-volts per degree Celsius.

k = thermal conductivity, in Watts per degree Celsius, per centimetre.

p = electrical resistivity, in ohm-centimetres.

t = Thomson co-=efficient, in micro-volts per degree Celsius.

L = length of thermo-element, in centimetres.

n = Peltier co-efficient, in micro-volts.

J = current density in the element, in amperes per square centimetre.

ST = interval between two given temperatures, in degrees celsius.

6a = difference between two seebeck co-efficients.

In the following account the term thermo-element or element is used to refer to any arm, bar, dice or pellet of thermo-electric material, and unless otherwise stated the description relates to single elements. The term dopant is used to denote any substance or substances added to the basic material in order to alter its polarity, Seebeck co-efficient or other properties. For simplicity the two thermo-elements forming a thermo-couple are assumed to be identical except for polarity, and of unit cross-section and constant electrical and thermal conductivity: in the examples given below p = 0.001 ohm-cm, k = 0.02 W deg1 cm1.

Methods of allowing for variation in these parameters when they are temperature dependent and when they vary independently of temperature are known to the art and discussed in the literature. Such variations do not affect the basic principles of the invention.

The invention is illustrated by the diagrams shown.

Figure 1 is the circuit diagram of a thermo-electric generator.

Figure 2 is the circuit diagram of a thermo-electric pump.

Figure 3 depicts a section of a bar carrying an electric current in a temperature gradient.

Figure 4 shows the relation between the current and the cold junction heat flow when using two heat sources.

Figure 5 is the circuit diagram for a thermo-couple taking in heat from a single source.

Figure 6 is the circuit diagram for a four-element couple taking in heat from a single source.

Figure 7 is the circuit diagram for a generator with four couples thermally in parallel, electrically in series.

Figure 8 is the circuit diagram for a two-stage cascaded generator.

Figure 9 is the circuit diagram for a two-stage generator with segmented elements.

Figure 10 depicts equipment suitable for producing thermo-elements with a concentration of dopant that varies along the element length.

Figure 11 shows an apparatus for producing a layer of thermo-electric material with a concentration of dopant that varies through the thickness of the layer.

Background In order to consider the nature and scope of my invention it is necessary first to examine the principles on which thermo-electric devices are based. They depend on effects attributed to Seebeck, Peltier and Thomson.

The seebeck effect is the appearance of an electro-motive force in a circuit consisting of two different conductors the junctions of which are maintained at different temperatures. Figure 1 is a schematic diagram of a thermo-electric generator in which A is a heat reservoir at temperature Th and B, B are reservoirs at temperature Tc. The reservoirs are connected to two bars of thermo-electric material, one of which, marked p, has a positive seebeck co-efficient and the other, marked n, has a negative co-efficient. When Th exceeds Te a current flows through the load resistance R in the direction shown by the arrow. The two elements form a couple.

The Peltier effect is the heating or cooling at the junctions of two different conductors forming a couple which arises when a direct current passes through them. Figure 2 is a schematic diagram of a heat pump comprising two thermo-elements, one of which, marked p, has a positive seebeck co-efficient and the other, marked n, has a negative coefficient. The ends of the elements are joined by heat reservoirs A, A at temperature Tb and heat reservoir B at temperature Tc. The reservoirs are maintained at different temperatures by passing a current from source C through the circuit in the direction shown by the arrow. Heat is absorbed from reservoir B and rejected from reservoirs A, A. The Seebeck and Peltier coefficients are related by the expression a = When the seebeck co-efficient varies along the length of an element heat is absorbed or generated within it when a current flows.This is the Thomson effect. The amount of heat absorbed or generated is given by the expression

tJdT, where t is the Thomson co-efficient and Th and Te are the temperatures of the hot and cold junctions respectively, and J the current. The Thomson co-efficient is related to the Seebeck coefficient by the expression t = T(da/dT), where T is the temperature at any point along the element and da/dT is the gradient at that point.

The voltage developed at the hot junction of a thermo-element is ah Th t where as is the Seebeck co-efficient and Tb the temperature at the junction. Similarly, the voltage at the cold junction is aT. The net voltage between the junctions is ab to - acTc. If ab # as a voltage arises from the Thomson effect. This is equal to i

(tb - t)dT.

The total voltage of a thermo-element is the sum of the junction and Thomson voltages.

The thermo-electric generators and pumps described in the literature and currently used for power production and refrigeration are based on the Seebeck and Peltier effects. Their thermodynamic performance is temperature dependent and inferior to that of the corresponding Carnotcycle machine. This has restricted application of the devices. The present invention seeks to surmount these limitations.

Description of the invention.

Thermo-electric generators and pumps according to my invention are designed and constructed with couples comprising matching p-type and ntype thermo-elements, each element having a Seebeck co-efficient varying from a high value at one end of the element to a low or zero value at the other, the seebeck co-efficient at any point in the element being a function of the temperature of the element or of the difference in temperature between the hot and cold junctions of the element. The required functional relationship is obtained by varying the concentration of dopant along the length of the element during manufacture.

For the best performance certain conditions must be met. To establish these it is necessary first to examine the behaviour of a characteristic thermo-electric device. In a thermo-electric generator or heat pump heat flow takes place from the hot to the cold junction of each element even in the absence of a current because of thermal conduction. This is the zero-current heat flow. Joule heat is also produced within the element because of its electrical resistance. If ab a Thomson heat is also present. These effects give rise to a temperature gradient along the element.

The relationships between the properties of a thermo-element carrying an electric current in a temperature gradient can be derived by considering an infinitesimal section of an element of unit cross-section carrying a current J along a temperature gradient in the x direction, where x is the distance from the origin (Figure 3).

The heat entering the section through the left face is -k (dT/dx). The heat leaving the section through the right face is the thermal conductivity times the temperature gradient at the right face. It is -k((dT/dx) + (d/dx) (dT/dx) dxj. The Joule heat generated in the volume is J2 eodx. The Thomson heat generated or absorbed within the volume is i tJ(dT/dx)dx. When this term is negative heat is absorbed, when positive it is generated.

In the steady state the heat out of the section must equal the heat into the section plus the heat generated or absorbed within it. The heat balance condition gives the differential equation k(d2T/dx2) itJ(dT/dx) + J2/o = 0 (1) It can be solved by applying the boundary conditions T = Tc at x = 0 and T = Tb at x = L, where L is the length of the element.

It is convenient to write tJ/x = a and J 2p/k = b; also Th ~ tic = 6T.

The equation is then (d2T/dx2) i a(dT/dx) + b = 0 (2) when the Thomson term is absent the solution is T = Tc + bx (L - x) (3) When the Thomson term is positive the solution is T = Te + [ 6T + (b/a)L] [ 1 - exp(ax) ] / [ 1-exp(aL) - (b/a)x (4) When the Thomson term is negative the solution is T = Tc + [6T - (b/a)L] [1 - exp(ax) ] / [ 1-exp(aL) ] + (b/a)x (5) The equations are reduced by expanding the exponentials and retaining the first two terms.

on substituting for a and b: When the Thomson term is absent T = Tc + (x/L) [ 6T + J2Lp(L - x)/2k ] (6) When the Thomson term is positive T = Tc + (x/L) [ 6T + 6TJt(L - x)/2k - J2Lp(L - x)/2k) ] (7) When the Thomson term is negative T = Tc + (x/L) t6T + 6TJt(L - x)/2k + J2Lo(L - x)/2k)] (8) From these equations the heat flow at any point along the element can be derived, using the expression H = k(dT/dx) where H is the flow.When the Thomson term is absent the heat flows at the junctions are Be = k6T/L + JLp towards the cold junction (9) Bb = k6T/L - FJ Lp towards the hot junction (10) If the Thomson co-efficient at the hot junction is tb and that at the cold junction is tc:: When the Thomson term is positive the heat flows are Bc = k6T/L + J2Lp + FJtc6T towards the cold junction (11) Hh = k6T/L - J2Lp - FJte6T from the hot junction (12) When the Thomson term is negative the heat flows are Hc = k6T/L + J2Lo - Jtc#T towards the cold junction (13) Bc = k6T/L - J2Lp + FJtb6T towards the cold junction (14) These equations enable the conditions for optimum performance to be established.

Since t = T(da/dT) the equations for heat flow may be expressed in terms of the temperature and the seebeck gradient, da/dT. When the Thomson term is absent equations (9) and (10) apply. When the Thomson term is positive the heat flows are Hc = k6T/L + J2Lo + JTc (da/dT)6T towards the cold junction (15) Hh = k6T/L - J2Lp - JTb (da/dT)6T from the hot junction (16) When the Thomson term is negative the heat flows are He = k6T/L + J2Lp - JTc(da/dT)6T towards the cold junction (17) Hh = k6T/L - J2Lp + JTb (da/dT)6T from the hot junction (18) The Thomson co-efficient at any point along the element depends on the functional relationship between the Seebeck co-efficient and the temperature.For example, if a = clnT, where c is a constant, the Thomson co-efficient is also constant, and if a = cosh AT (A a constant) the Thomson co-efficient is Asinh AT. In my invention the Seebeck coefficient is a linear or other function of the temperature along the element or the temperature interval between the hot and cold junctions.

In the embodiment of my invention described herein the difference between the seebeck co-efficients at the hot and cold junctions is chosen to be a linear function of the temperature difference between the junctions; that is, 6a/6T is constant over the region of interest.

The sinale-stace Generator In a single-stage thermo-electric generator designed in accordance with my invention the Seebeck co-efficient is chosen to be a maximum at the hot junction and low or zero at the cold junction. The sign of the Thomson term is negative so as to absorb heat within the thermo-element.

If the seebeck co-efficient is equal to ac at Tc and ah at Th r when #α/αT is constant (αh - αc)/(Th - Tc) = da/dT. From equations (17) and (18): Hc = k6T/L + J2Lo ~ FJTe (αh - ac ) (19) Hh = k6T/L - J2Lo + JTb (ab - ac) (20) In a thermo-electric generator the input to the hot junction is equal to the Peltier heat, ahThJ, plus the heat flow, Hbl from the junction, hence if Qh is the input Qh = αh ThJ +JTh (ab - ac) - J2Lp + k6T/L (21) When the Thomson term is absent Qh = α ;ThJ- J2Lp + k6T/L (22) When ae = 0 and Thomson heat is absorbed within the element Qh = 3 αh ThJ - J2Lo + k6T/L (23) 2 The input to the cold junction, Qc, is equal to the Peltier heat, α=TcJ, plus the heat flow to the junction, Hc, hence Qc = αc TcJ acTcJ +JTc (ab - a) + J2Lo + k6T/L (24) When the Thomson term is absent Qc = aTcJ + J21%o + k6T/L (25) When ac = 0 and Thomson heat is absorbed within the element Qe = -αhTcJ + 2J Lp + k6T/L (26) Equations (21) to (26) enable the conditions for maximum output and efficiency to be established.

The output of a generator is Qb - Qc With the Thomson term absent Qb - Qc = a6T - J2Lp This can be maximised by differentiating with respect to JL and setting to zero, when JLp = ȧ6T and (Qh - Qc) max = (a6T)2/4Ro. When ac = 0 and heat is absorbed within the thermo-element Qb - Qc = ȧ(3Th + T)J - J2Lp (28) on maximising as before JLp = ab (3Tb + Te )/4 and (Qb - Q) max = ab (3Tb + Tc) 2/16Lp (29) The efficiency of a generator is given by (Qb - Qc )/Qb In the literature the efficiency of a device having elements with a constant Seebeck co-efficient is stated in terms of the figure of merit, z, where z = a2 /ko, and it is shown that the efficiency, e, is e = (6T/Tb) ([1 + hZ(Th + Tc)]@ - 1) / ([1 + hZ(Th + Tc)@ + Tc/Th) (30) This is always less than the carnot-cycle value. When as = 0 and Thomson heat is absorbed the condition for maximum efficiency can be found by differentiating equation (26) with respect to JL and setting to zero, when JLp = habTc, and then substituting for Jlp in equation (26) and equating to zero, when 6T = (αTc)/8ko. The conditions for maximum efficiency are therefore JIP = hahTe and 6T = (abTc) 2/8ko.

The power at maximum efficiency is found from equation (28), giving Qb - Qc = 3αh 2ThTc/4Lo.

As with elements having a constant Seebeck co-efficient, the efficiency at maximum power is (Qb - Qe )max/Qb and depends on the temperature interval, 6T.

In the following example the conditions chosen are ah = 200#V, ac = O, Th = 300K and L = 1cm. At maximum efficiency Tc = 280.5x and J = 28A, the open circuit voltage is 0.118v and the heat input 2.52W. Since the input to the hot junction is equal to the output, J2R, where R is the external resistance, R = 0.0032Ohm. At maximum power J becomes 59A, Qb = 3.96W, (Qb - QG )max = 3.48W and Qc = 0.48W, giving an efficiency of 0.88. The external is equal to the internal resistance of 0.001 ohm. A couple consisting of p-type and n-type elements in the circuit of Figure 1 produces twice the power and has twice the internal and external resistance.

Sinale and double source devices By changing appropriate variables a couple as described above may be used to extract heat from one source or from two. In a thermo-element according to my invention the effects of altering one or more variables are as follows.

When the Thomson term is negative and as is zero equation (26) can be written kαT + (JL) 2P - αhTcJL = 0 (27) If JL is changed by i6(JL) and the other quantities remain the same the expression for cold junction power becomes k6T + h [ JL+6(JL) ] 2 p - FahTo [JLf+α(JL)] = QcL (28) When 6(JL) is finite QcL is finite and positive, hence when JL is increased or decreased heat is evolved at the cold junction and the efficiency falls. If 6T is increased or ah To diminished heat is rejected at the cold junction, and again the efficiency falls.

If 6T is diminished or αhTc increased heat is taken in at the junction.

Thus, if the element is designed with values of ab, Tb, Tc and JL to give zero cold end flux heat is taken in at both junctions if Tb is reduced or Te increased. When two heat sources are used in this way a change in J results in a corresponding change in the heat extracted from the sources.

Figure 4 shows the relation between the heat flow to or from the cold junction and the current drawn from the element for Th = 300K, Te = 290K, ab = 2OOoV, ac = 0 and L = 1cm. At J = 29A the heat taken in at the cold junction is a maximum of 0.22W and at J = 8A and 50A no heat is taken in.

when J = 8A the total power output is 0.89W and when J = 29A it is 2.61W; when J = 50A the power is 3.45W. Thus if 6T is less than the value derived from equation (27) Qc is zero for two values of current, the higher value yielding both maximum efficiency and close to maximum output. In a similar way, if the element is designed with values of ah, Th, Te and JL to give zero heat at the cold junction, when ah is greater than the original design figure heat is taken in at the hot and cold junctions. Thus if ab = 400oV and Ts = 290K, then J = 3.5 or 112.5A at Q = 0, and Qc (max) = -1.48W at J = 58A. If ab = 400 ; and Tc = 280.5K, then J = 47.5 or 104.5A at Qc = 0, and Qc (max) is equal to -1.18W at J = 56A.

No heat is let in at the cold junction if it is insulated. In a thermocouple with thermally insulated cold junctions the temperature they reach depends on the current taken from the thermo-couple and the heat extracted from the hot junctions. When the latter are maintained at a fixed temperature, as more or less current is withdrawn the cold junction temperature falls or rises. It floats between limits and stabilises at a value that depends on the hot junction temperature, the current and the parameters of the couple. From equation (27) the expression for cold junction temperature is Tc = [ 2kTb + (JL) / [ ah JL + 2k]. A device designed to extract energy from the surroundings can be started by warming the hot junctions or cooling the cold so as to establish a temperature gradient and heat flow from the hot junction to the cold.

Figure 5 shows the circuit diagram for a thermo-couple designed to take in heat from a single source. Heat reservoirs A, A at temperature Th deliver heat to thermo-elements p and n, of opposite polarity, the cold ends of which are joined electrically by a thermally insulated bridge B.

Current produced when heat is drawn from reservoirs A, A passes through load resistance R in the direction shown by the arrows, and the temperature of the cold junctions falls to Tc. An alternative arrangement, using four thermo-elements, is presented in the circuit diagram of Figure 6. Here the thermo-elements p, p and n,n are placed in opposition, with the two positive cold junctions together at (a) and the two negative cold junctions together at (b). The hot junctions draw heat from reservoirs A, A at common temperature Th. The cold junctions at temperature Tc are joined by load resistance R. current passes through the circuit in the direction of the arrows.

By way of example the behaviour of a couple comprising two thermoelements in the circuit of Figure 5 is described. With a start up temperature of 301.5K and load resistance of 0.24Ohm the current is 1A and the output 0.24W when ah = 200V, ac = 0 and L = 1cam. When a load resistance of 0.022Ohm is introduced and Tb = 300K the current drawn becomes 10A, Te falls to 288K and the power output is 2.2W. if the load is reduced to 0.0027Ohm the current rises to 50A, the power increases to 6.9W and the cold junction temperature to 290K. With an external resistance of 0.OO2olm the current is 60A and at a cold junction temperature of 300K the maximum theoretical output of 7.2W is reached.

The behaviour of a couple consisting of four elements connected as in the circuit of Figure 6 is similar but the output is doubled.

A single-source device according to my invention may act as a refrigerator if the source is finite so that extraction of heat causes its temperature to fall. The temperature of the source depends on the amount of current withdrawn. For example, if heat is extracted from an enclosure initially at 300K so that the temperature falls to 280K it can be maintained at the latter temperature by adjustment of the load resistance so that the current ranges between 0 and 56A and the heat withdrawn from the enclosure equals the in-leak. The minimum cold junction temperature of 262K is reached at a current of 28A.

Multi-stage generators In order to increase the voltage and temperature span couples can be connected thermally and electrically in series or in parallel.

Generators of this type having thermo-elements with a constant Seebeck co-efficient are discussed in the literature. Multi-stage generators according to my invention make use of the Thomson effect.

The couples can be arranged thermally in parallel and electrically in series, as depicted in Figure 7, where A is a heat reservoir at temperature Tb, B a reservoir at temperature Tc and c, c are layers of electrically insulating material of high thermal conductivity. R is the load resistance. The p-type and n-type elements are joined in series by straps of high electrical conductivity. If the number of couples is N the voltage, output and load resistance are N times that of a single couple but the current and temperature span are the same as for one couple.Thus a unit operating between 300K and 280.50K and consisting of four couples, each with elements having the values L = 1cam, ab = 200or, ac = 0, has an internal resistance of 0.008Ohm, an external resistance of 0.0258Ohm and produces a current of 28A, a total voltage of 0.944V and output of 20.16W.

In order to increase the temperature span couples can be arranged in cascade, with the heat output from one couple (or group of couples thermally in parallel and electrically in series) forming the input to the next couple or group of couples. The electrical outputs can be separate or series-connected. Two-stage generators are described here but the same principles can be used for devices with many stages.

Subscripts 1 and 2 are used to denote the first and second stages.

The first stage operates between the maximum temperature, Th, and the intermediate junction temperature, Ti, the second stage between Ti and the cold junction temperature, T. The Seebeck co-efficients of the first stage are ab at the hot junction and (αi)1 at the intermediate junction. Those of the second stage are (ai)2 at the intermediate junction and ac at the cold junction.

The thermal input to the hot junction of the first stage, Qb, is Qh = αThJ1 + kαT1/L1 - J1 L1# i F (αh - (ai)l )TbJ (2R) The heat given out at the intermediate junction, Qi, is Qi = (ai)lTiJ1 + kαT1/L1 (αh - (αi)1)TiJ2 + J1 l1# In the steady state this is equal to the heat absorbed at the hot junction of the second stage, which is Qi = (ai)2TiJ2 + k6T2/L2 i ( - ( i)2)TiJ2 - J2 2L2,o (30) The heat given out at the cold junction of the second stage, Qc, is Q@ = ae Te αc Tc J2 + k8T2/L2 i ((αi) - α ;c) Tc J2 + J22 L2 p (31) These equations can be solved for given conditions and parameters.

The simplest case is one in which the seebeck co-efficient of the first, or high-temperature stage, element is constant, so (ai)i = ab, whilst that of the second stage varies from a maximum of ab at the intermediate junction to zero at the cold junction. The stages are in cascade thermally but separate electrically.

The heat input to the first stage, Qb is Qh = αh Th J1 + k#T1/L1 - J1 2L1@ (32) The heat given out at the intermediate junction, Qi, is Qi = αh Ti J1 + k#T1 /L1 + J1 2L1@ (33) The heat absorbed by the first stage, Qb - Qi, is therefore Qb - Qi = ab6TiCl -J1 Llp (34) The maximum value of this expression is obtained by setting to zero its derivative with respect to J1l1 , when J1 L1@ = hab6T, and (Qb - Qi)max = (αh #T1) /4L1@. The heat absorbed is equal to J1 R1 where R1 is the external resistance.

When the Thomson term is negative the heat taken in at the intermediate junction of the second stage, Qi, is Qi = 3αb Ti J2 + k6T2/L2 - FJ2 L2o (35) 2 In the steady state this is equal to the heat given out at the cold junction of the first stage, so 3 αb Ti J2 - fJ2 L2@ + k#T2 /L2 = αb Ti J + J1 L2p + k#T1 /L1 (36) 2 The heat given out at the cold junction of the second stage, Qc, is Qe = -αh Tc J2 + k6T2/L2 + hJ2 L2n (37) This is zero when J2L2,O = αb Tc and 6T = (αh Tc) /8k# When the parameters are known these equations enable the other quantities to be calculated.

If Tb = 800R, Te = 300K, L1 = 1cm and ab = 200@V, the first stage open circuit voltage is 0.0955v, J1 = 47.75A, R1 = 0.001ohm and the output is 2.28W. For the second stage, Ti = 322.5K, L2 = 0.218cm, J2 = 137.7A, R2 = 0.000725Ohm, the open circuit voltage is 0.0968V and the output 13.77W.

The total output is 16w. The circuit diagram for a two-stage cascaded generator with one couple in each stage is shown in Figure 8. Here A is a heat reservoir at temperature Tb and B, B are heat reservoirs at temperature Tc. The first stage couple comprises elements pl and ni and the second stage elements p2 and n2. The first stage couple passes current J1 through load resistance Ri and the second stage couple passes current J2 through load resistance R2. Strap C consists of electrically insulating material of high thermal conductivity so as to maintain both intermediate junctions at temperature Ti.

The first and second stage elements can be series connected so that the same current passes through both, as shown in Figure 9. Here A is a heat reservoir at temperature Th and B, B are reservoirs at temperature Tc.

The external resistance is R and the current J. The first stage elements are pl and ni and the second stage p2 and n2 . The interfaces between pi and p2 and between nl and n2 are assumed to be at common temperature Ti. If the seebeck co-efficient of the first stage, ab, is constant and that of the second stage varies from ah at temperature Ti to zero at temperature Te the previous equations (32) to (37) apply with J1 = J2 = J.

The output is equal to JZR.

If ab = 200yv, Te = 273K and L2 = 1cam, then Th = 315s, Ti = 291.5K, L1 = 1.08cm and Qh = J2R = 1.73W. The open circuit voltage is 0.1218v and the external resistance 0.0024Ohm.

A greater total temperature span can be achieved if the elements of each stage have a maximum seebeck co-efficient at the higher temperature junction and zero at the other. For simplicity it is assumed here that the maximum co-efficient is the same for each stage. Using the previous notation and procedure the results obtained are:: At the hot junction of the first stage Qh = 3 ahTbJ + k#T1 L1 - JL1@# (38) 2 At the intermediate junction Q1 = -αh Ti J + k#T@L1 + J2L1# = 3αh Ti J (39) 2 For the second stage, JL2# = αh Tc and 6T2 = (α;h Tc) /8k# From these expressions 2ahTiJLl - JL12# - k6Tl = 0 (40) on differentiating with respect to JLi and setting to zero JL1# = 2abTi and 6T1 = 2(ahTi) /ko When Tc = 273K, Ti = 291.5K and Th = 631.5s, JLl = 116.6A-cm and JL2 = 27.3A-cm. If a current of 100A is withdrawn L1 = 1.17cm and L2 = 0.273cm; the corresponding internal resistance is 0.00144Ohm, the external resistance 0.00189Ohm and the power 18.94W.Further stages can be added provided the operating temperature of the thermo-element material is not exceeded. As with single-stage devices, two couples may be placed in opposition so as to withdraw energy from the hot junction only, or couples may be designed as above and operated with values of 6T which enable heat to be taken in from hot and cold sources.

The single-stage heat cup A thermo-couple may act as a pump if an electric current is passed through it, absorbing heat from a source and pumping it up to a higher temperature sink. The heat rejected is equal to the heat absorbed together with the pumping energy. The quantities of interest are the heat absorbed from the source, the co-efficient of performance and the maximum temperature difference the device can produce.For a single thermo-element the expression for the heat absorbed from the source, Qc, is Qc = acTcJ - JL# - k6T/L #(αc - ab)TcJ (41) When the Thomson term is absent and αc = αh = α the heat absorbed from the cold junction, Q'c, becomes Q'c = aTJ - JL# - k6T/L (42) The heat delivered to the hot junction, Q'h, is Q'h = αTh J + J L# - k6T/L (43) The heat absorbed can be maximised by differentiating equation (42) with respect to JL and setting to zero, when JLp = aTc and Q'@ (max) = α ; Tc /2L# - k6T/L (44) The pumping power, Q', is Q'p = Q'b - Q'c,= a6TJ + J Lp (45) The co-efficient of performance, , is therefore 9 = [ aTeJ - J2Lo - k6T/L ] /!a6TJ + JL#] (46) The maximum temperature span is found by setting Q'c to zero and solving for 6T, when 6T(max) = (αTc) /2k# When the Thomson term is absent and a = 200 V, L = lem, Te = 280K and Tb = 320K, Q'c and are zero at J = 16 or 96A, 6T(max) is 78.4deg and Q@ (max) is 0.76W at J = 56A. The maximum co-efficient of performance is 0.386 at J = 30.3A.

In a heat pump according to my invention the thermo-element is designed to have a maximum seebeck co-efficient at the cold junction, a minimum (preferably zero) co-efficient at the hot junction and a positive Thomson term. When ab is zero the heat absorbed from the cold junction, Qe, is Q@ = 3 αc Tc J - FJ No - k6T/L (48) 2 The heat delivered to the hot junction, @@ is Qh = -ȧcTbJ + J L# - k6T/L (49) The voltage between the junctions is -ȧc (3Tc + Th ) or -ȧc (4Tc + 6T) and the resistance is Lo.The pumping power, Qp , is Qp = Qb - Qc = J2Lp -αc (4Tc + 6T)J (50) When Qp is negative excess pumping energy is available and heat is dissipated at the hot junction; when it is positive an external voltage must be applied, as shown in Figure 2. The pump may be started by inserting a resistance into the circuit and cooling ther cold junction or heating the hot.With L and 6T fixed the condition for maximum heat absorption, Qc (max), is found by differentiating Qc with respect to JL and setting to zero, when 2JLp = 3ac Tc and Qe (max) = (9/8)(acTc) 2/Lo - k6T/L (51) The maximum temperature difference is obtained by setting Qe to zero and solving for 6T, when 6T(max) = 9(acTc)2/8ko (52) The co-efficient of performance, , is Qc/Qp, hence = = [ 3acTcJL - h(JL)lo - k6T ] / [ (JL)2,o - αc(4Tc+#T)JL] (53) 2 This is zero when Qe is zero, and infinite when Qc is finite but Qp is zero.With Th fixed removal of heat from a finite source at a temperature slightly less than Tb causes To and Qe to fall until a balance is reached at the value of current that makes the pumping power zero. The temperature of a finite source can be controlled by introducing into the circuit an external resistance R to make J(R + Lo) = ha(4Tc + 6T).

By way of example a thermo-element is chosen with ac = 200pV, ah = 0, Tb = 320R, Tc = 280K, L = 1 cm. The heat absorption rate is zero at J = 10 and 158A and a maximum of 2.73W at J = 84A. The maximum temperature difference the device can produce when Tc = 280K is 176.4deg. Qp is zero and the co-efficient of performance infinite when J = 116A. The heat absorption rate is then 2.2W and equal to the heat rejected at the hot junction. If the current exceeds 116A energy must be supplied from an outside source. An external voltage of 0.042v produces a current of 158A and zero Qc and , the energy being rejected at the hot junction.

Multi-stae pumas Multi-stage pumps may be thermally in parallel and electrically in series or separate, thus multiplying the rate of heat absorption but not the temperature span, or they may be thermally in cascade and electrically in series. In the latter case the thermal output from the hot junction of the first or low-temperature stage forms the input to the cold junction of the next, and so on. The absorption rate remains the same as for a single stage but the temperature span is increased. Pumps in parallel do not require separate treatment, and the following description is of a two-stage thermally cascaded pump.

In a series-connected device, using subscript 1 for the first or lowtemperature stage and subscript 2 for the second stage, the heat absorption rate, Qc, is Qe = 3acTcJ - FJ L1 n - k6Tl/Ll (54) 2 As before, the maximum heat absorption rate, Qc(max), is Qc(max) = (9/8)(acTc)2/Lln - k6Tl/Ll (55) In the steady state the heat delivered to the intermediate junction, Qi, is equal to the heat absorbed from it, hence when a low-temperature stage element with maximum co-efficient at the cold and zero co-efficient at the intermediate junction is matched with a second-stage element having a maximum seebeck co-efficient at the intermediate junction and zero coefficient at the hot junction Qi = -ȧcTiJ + fJ2L1n - k6Tl/Ll = 3acTiJ - FJ22L2io - k6T2/L2 (56) 2 The heat delivered to the hot junction, Qb, is Qh = -αc Th J + J2L2p - k6T2/L2 (57) The pumping power, Qp is Qp = Qb - Qc ac -ȧ0(3T + Th) -ȧc(3Tc+Ti) + J L1 # + J L # (58) assuming that the maximum seebeck co-efficient is the same for each stage.

The voltage produced by the combined stages, v, is equal to -ȧc [ (3Tc + Ti) + (3Ti + Th )]. The resistance is (L1 + L2)eO. The condition for maximum co-efficient of performance, #(max), is J(L1 + L2)##αc(8Tc + 56T1 + #T2) (59) By solving equation (59) L1 or 8T2 can be found if the other quantities are known. By way of example, if αc = 200 V, ab = 0, Tc = 280K, Ti = 320K, J = 84A and L1 = 1cm, then 6T2 is positive when L2 exceeds 1.9cm; Qc = 2.73W and # is infinite. As with the single-stage device, the heat absorption rate can be controlled by introducing a variable resistance into the circuit. To increase the temperature span of the pump further stages may be added.

In a multi-stage pump according to my invention one or more elements with varying Seebeck co-efficient may be combined with one or more elements having a constant Seebeck co-efficient. If the stages are thermally cascaded and electrically separate those with constant Seebeck coefficient require an external source of pumping power, as indicated in equation (45). They do not need separate consideration. If the stages are thermally and electrically in series (that is, with segmented elements) they can be treated by the previous methods. The present description refers to two-stage devices, but the same principles can be applied to pumps with more stages.

An element segment with constant Seebeck co-efficient may succeed or precede one with a co-efficient that varies from a maximum at the lowtemperature junction to zero at the high-temperature junction. When the segment with constant Seebeck co-efficient forms the second, or hightemperature, stage the heat absorption is that given by equation (42).

The second-stage voltage is ae6T2 and the first-stage voltage is -ȧc(3Tc + Ti), giving a total of -ȧc (4To + 6T1 - 26T2). This is less than the voltage in equation (59). When the segment with constant Seebeck co-efficient forms the first or low-temperature stage the heat absorption rate is given by equation (42) and the total voltage is -ȧc (4Te + 6T2 - 6T1). In both cases the overall performance is inferior to that of a two-stage device with series-connected couples having a varying Seebeck co-efficient.

Design of thermo-elements When the maximum seebeck co-efficient and the parameters k and o are known, if Th or Te is specified JL can be found from the foregoing equations, and in the case of a fully efficient generator 6T is also determined. When the required temperature span exceeds the temperature interval obtainable from a single stage a multi-stage device may be used and JL and 6T found for each stage. In the case of a pump Te and 6T are usually fixed, and this settles JL and Qc. To obtain a given temperature span and to maximise both Q and it may be necessary to use two or more stages and calculate JL for each stage. For a given value of JL either J or L is at choice.In a fully efficient generator the heat input is equal to the electrical output, or Q = J R, where R is the resistance, hence when the resistive load is known J can be found; this fixes L.

With L known the elements can be designed to have the required dopant distribution. In a pump, if JL is found for (max) the corresponding value of Qe can be calculated and hence L.

In the present invention the amount of dopant is varied along the element so as to confer a maximum seebeck co-efficient at one end and a minimum (preferably zero) co-efficient at the other, with a predetermined gradient along it. The required gradient is obtained by varying the distribution of dopant along the element during its manufacture.

The dopant concentration may be expressed a a function of the Seebeck coefficient; that is, c = f2 (a), where c is the concentration. The gradient at any point is da/dc. The relation between the amount of dopant and the distance along the element may be described as a function of the distance; that is, c = f,(x). At any point along the element the gradient is dc/dx. The Thomson co-efficient, t, is equal to (da/dT)T, or t = fi(a,T). By substituting for t in equations (7) and (8) the temperature can be expressed in terms of the distance along the element, x.

When the Thomson term is positive T = [ Te + (X/L)6T + x(L - x) (J2co/2k ] / [ l + J6a(L - x) (x/L)(2k ] (60) When the Thomson term is negative T = [ Te + (x/L)6T - x(L - x) (JPp/2k ] / [ l - J6a(L - x) (x/Li2k ] (61) The Thomson gradient may be written da/dT = (da/dc)(dc/dx)(dx/dT) If within the region of interest the relation between the Seebeck coefficient and the temperature interval is substantially linear da/dT can be replaced by a constant, C.On re-arranging: dc/dx = c(dT/dx)(dc/da) f3 (x) = CTf2 (a) (62) The procedure is : (1) Find the expression for T in terms of the distance from the cold junction and the parameters of the thermo-element, as in equations (60) and (61). (2) Establish the relation between the Seebeck co-efficient experimentally to give f2 (a). (3) Calculate the required distribution of dopant from the expression for f (x).

If da/dT is not constant but can be expressed as an empirical or analytic function an additional term is introduced into the formula for f (x). An alternative approach is to plot the functions against the corresponding variables and from the curves obtain a graphical solution.

Production of thermo-elements Thermo-elements according to my invention may be prepared in many ways.

When the materials are available in powder form the most convenient method is that based on powder metallurgy techniques known to the art and described in the literature; that is, by metering the materials in powder form into a die, cold pressing and sintering, or hot pressing, using dies, plungers and furnaces and controlled atmospheres. The starting materials are: (1) undoped or lightly doped powder, and (2) highly doped powder. It may sometimes be convenient to use more than two powders; when, for example, p-type and n-type elements are formed with a common junction, or when segmented elements are required. The proportions of the materials are varied along the length of the element by differential metering of the separate powders into the die or onto a platen in such a way as to give the desired gradient of dopant in the final product.

By way of example, a method of manufacturing a p-type element of lead telluride is described. Two basic materials are used: (1) lead telluride powder made by melting lead and tellurium together, casting into ingots and then grinding to 5 to 200-micron size; (2) similarly prepared doped powder made by melting lead telluride and sodium telluride to give a powder containing at least 2.5% sodium. The powders are held in separate vibrated hoppers closed at the base by conical bells. The latter are raised or lowered by electrically actuated solenoids to allow the powders to discharge, mix, and flow into a die. The rate at which each powder is delivered is controlled by varying the current through the solenoid windings. Alternatively, the bells may be operated mechanically by rods which are raised or lowered by rotary cams.In a subsequent operation the powder is hot-pressed or cold-pressed and sintered in an inert atmosphere for a time and at a temperature determined by experiment.

In another method of making the powder compacts the basic material and doped material are delivered to a mixing chute by worms rotating at the base of the hoppers, programmed to discharge the materials at the required rate.

Figure 10 is a diagram of an apparatus for producing thermo-elements such as lead telluride with a concentration of dopant varying from a maximum of about 2.5% sodium at one end of the element to zero at the other. At station 1 hopper A containing undoped lead telluride powder is closed by conical bell B which is raised or lowered by solenoid C when a current is passed through winding D. When bell B is lowered the powder falls into chute E. Hopper At containing doped lead telluride powder is closed by bell B' which is raised or lowered by solenoid Ct when a current passes through winding Dt. As bell B' is lowered the doped powder discharges into E'.The movement of the bells is arranged so that when undoped powder is delivered into chute E at the maximum rate no doped powder is discharged into chute Ett and when doped powder is delivered at the maximum rate into E' no undoped powder is delivered into chute E.

Between these limits the rates of discharge of the powders are progressive so as to give the desired proportions of doped and undoped lead telluride in the product. The powders mix in common chute F and fall into die G, which is vibrated to consolidate the mixture H. When the required quantity of powder has been delivered die G is moved to station 2. At this station the powder compact H is compressed in die G by graphite or ceramic punches J,J and sintered for about 1 hour at 700 to 900K. The die assembly is contained in a housing K filled with heat insulating material L and surrounded by induction coil M through which high-frequency alternating current is passed. Alternatively, the powder compact may be cold pressed, transferred to a controlled atmosphere furnace and then sintered.

Thermo-elements according to my invention may be prepared in the form of a layer by discharging the mixed powder on to a moving platen so that highly doped powder appears at the top of the layer and undoped powder at the bottom, or the converse. It is cut into sections and sintered, or sintered and subsequently cut up. A method of preparing the layer is shown in Figure 11.

The two (or more) powders are mixed as described previously and illustrated in Figure 10. The mixture from chute F is distributed along the platen N by flexible pipe Ft which reciprocates above the platen whilst the latter is moved longitudinally, as shown by the arrow. The motions of the pipe and platen are synchronised so as to produce a layer of the requisite thickness and gradation in composition. The layer of powder P is confined laterally by cheeks R, consolidated by roller S and transferred to the sintering furnace. several layers of powder may be built up, one on top of the other, by using two or more chutes and distributing pipes and their associated hoppers.

other methods of preparing the thermo-elements may be used, such as electrodeposition, thermal decomposition of vapour-phase compounds, sputtering and metal spraying. The components may be deposited simultaneously or consecutively, followed by thermal diffusion. Elements in sheet form may be made by pressing together thin layers of basic and doped material and diffusing under heat and pressure to give the required composition gradient.

Claims (28)

1. A thermo-couple comprising a p-type thermo-element and a matching n-type thermo-element, each element being designed to have a high Seebeck co-efficient at one end of its length and a low or zero co-efficient at the other and a pre-determined linear or non-linear gradient of seebeck co-efficient in between the ends, the required gradient being obtained by varying the concentration of dopant along the length of the element during manufacture.

2. A thermocouple as in claim 1 wherein the gradient of the Seebeck co-efficient is a function of the temperature of the element.

3. A thermocouple as in claims 1 and 2 wherein the gradient of the seebeck co-efficient is a linear function of the difference in temperature between the ends of the element.

4. A thermoelectric generator comprising a thermocouple as in claims 1, 2 and 3 wherein heat is taken in at the hot junction and (a) the maximum Seebeck co-efficient is at the hot junction and a low or zero coefficient at the cold junction; (b' the difference in temperature between the ends of the element, 6T, is equal to (abTc) 2/8ko, where ab is the Seebeck co-efficient at the hot junction, TO the temperature of the cold junction, k the thermal conductivity and the electrical resistivity of the element; and (c) JLo = haTc, where J is the current passing through the element and L its length.

5. A thermo-electric generator as in claims 1 to 4 wherein the temperature of the hot junction is less and/or the temperature of the cold junction greater than that required to give zero heat rejection at the cold junction; that is, dT < (ahT)2/8kp.

6. A thermo-electric generator as in claims 1 to 5 wherein the amount of heat taken in at the cold junction is controlled by a variable resistance in the circuit.

7. A thermo-electric generator as in claims 1 to 6 wherein heat is taken in at the hot junctions, and the cold junctions of the thermocouple are thermally insulated.

8. A method of extracting heat from a finite source with the aid of a thermo-couple as described in claims 1 to 7 wherein the hot junctions of the couple form the finite source.

9. A thermo-electric generator as in claims 1 to 4 wherein the couples are placed in opposition, as shown in Figure 6.

10. A thermo-electric generator comprising thermo-couples as described in claims 1 to 9, wherein the couples are arranged thermally in parallel and electrically in series, as depicted in Figure 7.

11. A thermo-electric generator comprising two or more stages thermally in cascade but electrically separate wherein the lowest temperature stage comprises a thermo-couple as in claims 1 to 4 and the thermo-elements of the higher temperature stages have a constant seebeck co-efficient.

12. A thermo-electric generator comprising two or more stages thermally and electrically in series wherein the stage operating at the lowest temperature consists of thermo-couples as described in claims 1 to 4 and the higher temperature stages have a constant seebeck co-efficient.

13. A thermo-electric generator comprising two or more stages seriesconnected thermally and electrically wherein each stage consists of couples with elements having a Seebeck co-efficient varying from a maximum at the hot junction to a low or zero value at the cold junction of the couple, as described in claims 1 to 4.

14. A thermo-electric generator as in claims 1 to 4 and 10 to 13 wherein the electrical output is controlled by an electrical resistance in the circuit.

15. A heat pump comprising thermocouples as in claims 1 to 3 wherein the Seebeck co-efficient varies from a maximum at the cold junction to a low or zero value at the hot junction.

16. A heat pump as in claims 1 to 3 and 15 wherein 2540 = 3acTc, where J is the current, L the element length, C the electrical resistivity, ae the Seebeck co-efficient at the cold junction and Te the temperature at that junction.

17. A heat pump as in claim 16 wherein the voltage produced by each element is made equal to or greater than the product of the length of the element, its resistivity and the current flowing through it.

18. A heat pump as in claims 1 to 3 and 15 wherein the heat absorption rate and the temperature of a finite source may be altered by introducing a variable resistance into the circuit.

19. A multi-stage heat pump wherein couples as described in claims 15 to 18 are arranged thermally in parallel and electrically separate, in series, or in parallel.

20. A multi-stage heat pump wherein couples as described in claims 15 to 18 are arranged thermally in cascade and electrically separate, in series or in parallel.

21. A multi-stage heat pump wherein couples as described in claims 15 to 18 are series-connected thermally and electrically and the seebeck coefficients of the elements of the lowest temperature stage vary from a maximum at the cold junctions to a minimum or zero at the hot junctions, whilst the seebeck co-efficients of the succeeding stages are constant.

22. A multi-stage heat pump as in claims 15 to 18 wherein the Seebeck co-efficient of each element in each stage varies from a maximum at the cold junctions to a minimum or zero at the hot junctions of each element.

23. A heat pump as in claims 15 to 18 and 21 and 22 wherein the total voltage produced by the elements of each couple is equal to or greater than the product of the current, the length of the element and its resistivity.

24. A heat pump as in claims 15 to 22 wherein the heat absorption rate is controlled by a variable resistance in the pump circuit.

25. A method of manufacturing pellets or bars of material for use in thermo-couples as described in claims 1 to 24 wherein the basic thermoelectric material and dopant or dopants in powder form are mixed in varying proportions by differential metering of the substances, which are then delivered to a die and pressed and sintered for a time and at a temperature depending on the nature of the thermo-electric materials so as to produce elements with the desired gradation of dopant along the element length.

26. A method of preparing thermo-elements as in claim 25 wherein the basic material and dopant are contained in hoppers closed by bells at the lower ends, as illustrated in Figure 10, the bells being raised or lowered by mechanically operated rods and cams or by electrically actuated solenoids programmed to deliver the prescribed proportions of the materials into a die at the prescribed rate, followed by subsequent pressing and sintering.

27. A method of preparing thermo-elements as in claim 26 wherein the basic material and dopant are contained in hoppers fitted with worms at the base which are programmed to deliver, on rotation, the prescribed amounts of material at the prescribed rate, for subsequent pressing and sintering.

28. A method of manufacturing a layer of thermo-electric material in which the basic material and dopant as described in claims 25 to 27 are distributed over a moving platen by a reciprocating flexible pipe or similar means, the movement of the pipe and the platen being synchronised so as to produce a layer of the required thickness and composition, as illustrated in Figure 11.