Energy Conversion & Storage Technologies
Dr. Farschad Torabi Homework # 3
1. Assume the thermocouple shown in the picture works as a generator. The gen- erated current,I, passes through an ex- ternal load,RLwhich is proportional to the thermocouple load, i.e., RL =mR.
Show that:
Arm A Arm B
Heat conducting electrically insulating sheets
HeatSource(T)H HeatSink(T)C
P
HP
C I+VL
RL
(a) The generated current is:
I = α(TH −TC) R+RL
(b) Calculate the output power of the generator.
(c) Calculate the efficiency of the thermocouple, with respect to Carnot cycle efficiency, i.e.
η=ηCarnot×η2
(d) The second law efficiency, η2, can be expressed as a function of figure of merit, Z = α2/ΛR. It can be seen that the efficiency of the thermocouple is maximized when Z is maximized. Show that with a constant α, the maximum Z is achieved when:
lAAB lBAA =
√λAσA
λBσB
in which, l, A, λ and σ are length, cross section area, thermal conductivity and electrical conductivity of each arm, respectively. Then calculate ΛR.
(e) Show that for the maximum efficiency, the proportionality facor of the external load can be found as
m=√
1+< T > Z in which
< T >= TH +TC 2
2. A thermocouple works between 500 K and 300 K. Its resistance is 0.0005Ω and its heat conductance is 0.2 W K−1. The mean Seebeck coefficient (between 500 and 300 K) is 0.001 V K−1. 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
Due: 1392/12/20 1 of 2
Energy Conversion & Storage Technologies
Dr. Farschad Torabi Homework # 3
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)?
3. A thermoelectric device, consisting of 100 thermocouples electrically in series and ther- mally 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 −3◦C and 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 ow 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?
4. 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. Tungsten has an electric resistivity that (between 1000 and 3600 K) is given with accept- able precision by
ρ=−1.23×10−7+ 3.49×10−10T
where ρ is in Ω m. Give your best estimate for the thermal conductivity of tungsten at 1100 K and 1600 K.
Due: 1392/12/20 2 of 2
