## Thermoelectric Battery Consist Of 1000 Identical Thermocouples

5.1 The Russians are quite advanced in thermoelectrics and have recently developed a secret material for such use. Data are naturally, hard do obtain, but the CIA discovered that a thermoelectric cooler is being built capable of depressing the temperature by 100 K when the hot junction is at 300 K. What is the figure of merit of the thermocouple?

5.2 The Seebeck coefficient of a junction is a = 100 + T - 10-3T2 /V/K.

1. At what temperature is the Peltier coefficient maximum?

2. What is the coefficient at this temperature?

3. At what temperature are the Thomson coefficients of the two arms equal?

5.3 A thermoelectric cell has a figure of merit of 0.002 K-1 and an internal resistance of 100/l. The average Seebeck coefficient is 200/V/K. Heat flow meters are used to measure the flow of heat, PH, from the hot source to the cell and the flow, PC, from the cell to the cold sink. There is no other heat exchange between the cell and the environment.

The source is at 600 K and the sink at 300 K.

1. With no electric current flowing in the cell, what is the value of PH and PC ?

2. A current, I, is now made to circulate. This modifies PH. What currents cause PH to go to zero?

3. What are the corresponding values of PC?

4. Over what range of currents does the cell act as a power generator?

5.4 A block of metal, maintained at 850 K, is mounted on a pedestal above a platform at 350 K.

The experiment being conducted requires that there be absolutely no heat transfer from the metal block through the pedestal. To achieve this, the pedestal was made into a thermocouple and the appropriate current was driven through it. The thermocouple has a heat conductance of 1 W/K and a resistance of 1 milliohm. Its figure of merit is 0.001 K-1. All these data are average values over the temperature range of interest.

What current(s) must be driven through the thermocouple? What is the voltage across its terminals? What is the electric power required?

5.5 We want to pump 100 W of heat from 210K to 300K. Two stages of thermocouples must be used. The second stage pumps 100 W from 210 K to TH2. The first stage pumps the necessary energy from TCl = TH2 to 300 K.

All thermocouples are made of the same materials whose combined characteristics are:

The geometry of each thermocouple is optimized. The current through the thermoelectric pair is adjusted for maximum cooling. The current through the first stage is not necessarily the same as that through the second.

Within each stage, all elements are connected electrically in series.

The total electric power input to the system depends critically on the choice of TH2. What value of TH2 minimizes the electric power consumption, PE ? What is the value of PE ? What is the voltage applied to each stage?

5.6 A battery of thermocouples delivers 5kW at 24.0 V to a load. The thermocouples were designed to operate at maximum efficiency under the above conditions.

The hot side of the battery is maintained at 1100 K and the cold side, at 400 K.

The figure of merit of each individual thermocouple cell is 0.0015 K-1.

What is the efficiency of the system?

5.7 What is the heat conductance of a metal bar at uniform temperature (T = 400 K) if the bar has a resistance of 4 Q?

5.8 To drive a current of 100 A through a thermocouple at uniform temperature (300 K), a power of 50 W is required.

The same thermocouple, when open-circuited and under a temperature differential from 800 K to 300 K, has 0.5 V between the open terminals, on the cold side.

### Assume that the resistance is independent of T.

How many watts are necessary to drive a current of 100 A through the thermocouple when the differential above is maintained? Is there a unique answer to this question?

5.9 A thermocouple works between 500 K and 300 K. Its resistance is 0.0005 Q, and its heat conductance is 0.2 W/K. The mean Seebeck coefficient (between 500 and 300K) is 0.001 V/K.

What is the open-circuit voltage generated by the thermocouple?

When there is no current, heat flows, of course, from the hot side to the cold side. Is it possible to make the heat that flows from the hot source to the thermocouple equal to zero? If so, what is the heat flow from the cold sink to the thermocouple? What is the electric power involved? What is the voltage across the couple? Does the electric power flow into the thermocouple or out of it (i.e., does the couple act as a generator or a load)?

5.10 A thermocouple operates between 900 K and 300 K. When short-circuited, it delivers 212 A, and when open-circuited, 0.237V. The dimensions of the arm have been optimized for the mean temperature of 600 K. The material in the arms are semiconductors that have negligible lattice heat conductivity.

1. Calculate the heat taken from the hot source when open-circuited?

2. What is the power delivered to a 500 ^0 load?

3. What is the efficiency of the device in Question 2?

5.11 A small thermoelectric generator (single pair) is equipped with two thermal sensors, one measuring TH (in the hotter side) and one measuring TC (in the colder side). An electric heater warms up the hotter side, and all the heat thus generated is delivered to the thermocouple. A feedback system assures that TH is kept at exactly 1000 K. The amount of electric power delivered to the heater can be measured.

On the colder side, a corresponding feedback system assures that TC is kept at a constant 500 K.

The current, I, forced through the thermocouple can be adjusted to a desired value.

When I = 0, it takes 10 W of electric power to operate the heater at the hotter side of the thermocouple. Under such conditions, the thermocouple develops 0.50 V at its open-circuited terminals.

Now, a given current I is forced through the thermocouple so that the electric heater can be disconnected, while TH still remains at 1000 K.

Next, the current I, above, is reversed, and it is observed that it takes 18.3 W to maintain TH = 1000K.

2. What voltages are necessary to drive the currents I and -I.

5.12 A thermocouple is connected to an adjustable current source. Disregard the resistance of the wires connecting the current source to the thermocouple. These wires and the cold side of the thermocouple are at 300 K. Under all circumstances, the current delivered has an absolute value of 100 A. Assume that the resistance of the thermocouple is temperature independent.

1. When TH = TC = 300 K, the absolute value of the voltage across the current source is 0.20 V. What is the voltage if the direction of the current is inverted?

2. When TH = 600 K and TC = 300 K, the absolute value of the voltage is 0.59 V. What are the voltages if the direction of the current is inverted?

5.13 Assume that the materials in the arms of a thermocouple obey strictly the Wiedemann-Franz-Lorenz law. The Seebeck coefficient, a, of the thermocouple is 150 ^V/K independently of temperature, and the electric conductivities, a (the same for the two arms), are also temperature independent. Such unrealistic behavior has been specified to make the problem more tractable; both a and a usually do vary with temperature.

The device is to operate between TH and TC. For the calculation of any temperature dependent quantity, use the arithmetic mean, Tm, of these two temperatures.

The geometry of the thermocouple has been optimized, and the load connected to it always has the value that maximizes the efficiency of the system.

Show that, under the above circumstances, the electric power delivered to the load is independent of the choice of TH provided that AT = Th — TC is always the same.

At 800 K, what is the value of the AR product for this thermocouple?

5.14 A thermoelectric device, consisting of 100 thermocouples electrically in series and thermally in parallel, is being tested as a heat pump. One side is placed in contact with a cold surface so that it cools down to —3C; the other side is maintained at 27 C.

The open-circuit voltage is measured by means of a high-impedance voltmeter and is found to be 900 mV.

Next, the electric output is shorted out, and it is observed that a current of 9 A flows through the short.

The device is now removed from the cold surface, and its cold end is insulated thermally so that absolutely no heat can flow in. A current of 50 A generated by an external source is forced through the device in such a direction that heat is pumped from the cold end to the warm end. A thermometer monitors the final temperature of the cold side. After steady state is reached, the temperature is 260 K. The hot side is still at 27 C.

Is this the lowest temperature that can be achieved? If not, what is the lowest temperature, and what is the necessary current?

5.15 A thermoelectric device is being tested in a laboratory. It consists of a single thermocouple and has its hot side in intimate thermal contact with an electric heater whose total heat output is transferred to the thermocouple. In other words, the power delivered to the heater is equal to the heat power, PH, that flows into the thermocouple. Under all circumstances, the hot side is maintained at 1000 K and the cold side, at 300 K.

The two temperatures, TH and TC, are monitored by thermometers.

The first step in the test reveals that when the device is open-circuited, it draws 14 watts of heat from the hot source and delivers a voltage of 0.28 V. When short-circuited, it delivers a current of 35 A.

1. What is the heat power input when the device is short-circuited?

2. What is the heat power input when 0.4 V are applied in opposition to the open-circuit voltage?

5.16 A Radioisotope Thermal Generator is used as an electric power source aboard a spacecraft. It delivers 500 W at 30 V to a load optimally matched to the thermoelectric generator. Under such circumstances, this generator operates at a 12.6% efficiency.

The hot side of the generator (TH) is at 1300 K, and the cold side (TC) is at 400 K.

Assume that all the characteristics of the thermocouple (a, R, and A) are temperature independent.

1. What would be the efficiency of the generator if the load resistance were altered so that the power delivered fell to 250 W?

Assume now that the heat power source is a radionuclide that delivers a constant heat power independent of the demands of the load. Thus, the temperature becomes a function of the load power. If the radioactive material were inside an adiabatic container, the temperature would rise until the container was destroyed—a steady heat leak must be provided to limit the temperature.

The radionuclides in this problem release heat at a rate of 4984 W. The container, by itself, radiates enough energy to keep its outer skin at constant TL = 1000 K under all circumstances, including when the thermoelectric generator is not installed, and, consequently, only the leakage path removes heat from the source. The cold side of the thermoelectric generator is always at TC = 400 K.

When the thermoelectric generator is attached to the heat source and generates 500 W to a matched electric load, the temperature, TS, falls to 1300 K (as in Question 1, this is TH of the thermoelectric generator). In other words, the heat source delivers heat to two parallel paths: the leakage path and the thermoelectric generator.

2. Calculate the temperature of the heat source when no electric energy is being drawn from the thermoelectric generator.

3. When 250 W dc are drawn from the generator, what is the source temperature? There is more than one answer. Use the current with the smallest absolute value. Set up your equations and use a trial-and-error method.

5.17 Demonstrate that the voltage required to drive a thermoelectric heat pump is independent of the amount of power pumped and of the cold temperature, provided the current has been adjusted for maximum pumping.

5.18 Tungsten has an electric resistivity that (between 1000 and 3600 K) is given with acceptable precision by where p is in 0 m.

Give me your best estimate for the thermal conductivity of tungsten at 1100K and 1600K.

5.19 A perfectly heat-insulated box is equipped with an electric heater, which allows the introduction of heat at an accurately measured rate. The only way heat can be removed is through a Peltier heat pump whose hot side is maintained at a constant 300 K.

The current, I, through the heat pump is controlled by a computer that senses the temperature inside the box and is set to keep it, if possible, at 280 K.

The experimenter chooses the amount of heat dissipated by the electric heater and tabulates the current the computer delivers to the heat pump. Here are the results:

For each case in the table, what is the coefficient of performance (COP) of the heat pump?

5.20 At 300 K, the electric resistivity of a sample is 0.002 Ocm, and its heat conductivity is 0.03 WK"1 cm"1. From these data, determine if the material is a metal or a semiconductor. Explain.

5.21 A thermoelectric battery consists of 1000 identical thermocouples electrically in series and thermally in parallel. It works between 1000 K and 500 K.

Each thermocouple has the following characteristics: Heat conductance: 3W/K. Electric resistance: 200 Seebeck coefficient: 0.0007 V/K.

How much power does this battery deliver to a 0.3 0 load?

5.22 A thermoelectric generator consists of a number of series-connected thermocouples. It operates between 1000 and 400 K. ## Guide to Alternative Fuels

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