Keywords

1 Introduction

Thermoelectric devices are pollution-free devices. A thermoelectric generator (TEG) can be used as heat engine (using Seebeck effect), and a thermoelectric refrigerator (TER) can be used as a refrigerator or heat pump (using Peltier effect). They possess encouraging potential in comparison with existing engines and refrigerators. As there is no moving component which increases durability, reduces maintenance cost, and improves span of system life. With change in direction of current, only the same device can be used as refrigerator as well as heat pump. The non-existence of refrigerant removes the hazards of leakages to the environment which are serious issues in vapor compression refrigeration systems. As there is no compressor used, it reduces the noise in the system. Nowadays, lots of new methods have been developed to make better-performing semiconductor materials; therefore, more and more applications and investigations to improve operating of thermoelectric devices are gaining importance. TERs are more appropriate in areas such as small electronic circuits, laser diodes, communication gadgets, medical, pharmaceutical, and defense operations where precise control of temperature is more important than other things. Thermoelectric refrigerators are better choice for the green environment as the ozone depleting working substances do not exist in the system. Due to these advantages, thermoelectric devices are being used in wide areas. The fundamental concepts of the thermoelectric technology and the bright likelihood of their applications have been studied by number of researchers and engineers [1]. As the gap between the energy consumption and finding energy sources in the world is widening day by day. The entire world is trying to face the challenges of discovering new energy sources to satisfy the spurt in energy consumption in day-to-day life along with degradation of the environment with pollution.

Thermoelectric devices are being viewed as alternatives systems which can utilize the waste heat for power generation. The low efficiency of nano-engineering thermoelectric device can be improved by decreasing the thermal conductivity of the materials, and they become a promising alternative for large-scale use because of their excellent performances [2]. A review of the research work to enhance the working of thermoelectric cooling systems was compiled. This work embraces the review of work related to enlist the new materials suitable for designing, making, and analyzing thermoelectric modules [3]. Thermoelectric refrigerators are found more appropriate for the applications where a regulation of temperature in limited space is required. A model was developed and studied to modify vapor content of air using TEC channels. It was observed in this particular work that the model speculated the variation in the temperature of air along the channel with minor error. Numerous experimental and simulation-based analysis of the thermoelectric devices have been done to evaluate their performance in various applications related to power generation and refrigeration or air-conditioning [4, 5].

A thermo-economic optimization based on exergy analysis of a vapor compression and vapor absorption refrigeration systems has been carried out. Thermo-economic analysis in these systems plays a very important role to obtain feasible life-cycle cost. Many researchers have established thermodynamic and thermo-economic objective functions based on second law and thermo-economic principles [6,7,8,9]. Comparison of single-stage and double-stage thermoelectric cooler has been done for specific designs [10]. Exergo-economic analysis has also been done for a multistage thermoelectric cooler [11, 12]. A three-dimensional numerical simulations’ investigation has been done to optimize the design of a thermoelectric cooler using finite element method [13]. A study to evaluate the effect of input parameters on exergy flow in a thermoelectric system was done to establish the optimum value of input parameters [14].

2 Two-Stage Thermoelectric Refrigerator

A double-stage thermoelectric refrigerator (TER) is an arrangement of n and p type elements in which two single-stage TERs are connected thermally in series. That means the total heat absorbed from first stage will be transferred to second stage. There may be equal or different number of elements in the two stages. Figure 1 shows a double-stage TER. The energy balance equation for a two-stage TER can be represented as:

Fig. 1
figure 1

Double-stage TER

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$$Q_{\text{cc}} = \left[ {\alpha I_{\text{c}} T_{\text{cc}} - \frac{{I_{\text{c}}^{2} R}}{2} - k\left( {T_{\text{ch}} - T_{\text{cc}} } \right)} \right]N_{\text{c}}$$
(1)
$$Q_{\text{ch}} = \left[ {\alpha I_{\text{c}} T_{\text{ch}} + \frac{{I_{\text{c}}^{2} R}}{2} - k\left( {T_{\text{ch}} - T_{\text{cc}} } \right)} \right]N_{\text{c}}$$
(2)
$$Q_{\text{hc}} = \left[ {\alpha I_{\text{h}} T_{\text{hc}} - \frac{{I_{\text{h}}^{2} R}}{2} - k\left( {T_{\text{hh}} - T_{\text{hc}} } \right)} \right]N_{\text{h}}$$
(3)
$$Q_{\text{hh}} = \left[ {\alpha I_{\text{h}} T_{\text{hh}} + \frac{{I_{\text{h}}^{2} R}}{2} - k\left( {T_{\text{hh}} - T_{\text{hc}} } \right)} \right]N_{\text{h}}$$
(4)

where Qcc is the heat absorbed at the cold side of the colder stage; Qch is the rate of heat rejected at the hot side of the colder stage; Qhc is the heat absorbed at the colder side of the hotter stage; and Qhh is the rate of rejected at the hotter side of the hotter stage. Ic is the current flow in colder side, and Ih is the current flow in hotter side of TER. In this analysis, the two stages are electrically is series, so the current Ic and Ih are equal. Nh and Nc are number of elements in colder and hotter sides of TER, respectively. For this analysis Nh + Nc = 50 has been considered. Tcc and Tch represent the temperatures of the cold side and hot side of colder stage, and Thc and Thh represent the cold side and hot side temperature of hotter side. It can be assumed that there exists a junction temperature Tm which can be calculated by equating Qch and Qhc. Since the two stages are thermally in series so the heat rejected from first stage will be transferred completely to second stage.

Hence \(Q_{\text{ch}} = Q_{\text{hc}}\)

$$T_{\text{m}} = \frac{{k\left( {T_{\text{cc}} + xT_{\text{hh}} } \right) + \left( {xj^{2} + 1} \right)\frac{1}{2}I_{\text{c}}^{2} R}}{{k\left( {x + 1} \right) + I_{\text{c}} \alpha \left( {xj - 1} \right)}}$$
(5)

where x = Nh/Nc and j = Ih/Ic.

The energy balance equations can be written as:

$$Q_{\text{cc}} = \left[ {\alpha I_{\text{c}} T_{\text{cc}} - \frac{{I_{\text{c}}^{2} R}}{2} - k\left( {T_{\text{m}} - T_{\text{cc}} } \right)} \right]N_{\text{c}}$$
(6)
$$Q_{\text{ch}} = \left[ {\alpha I_{\text{c}} T_{\text{m}} + \frac{{I_{\text{c}}^{2} R}}{2} - k\left( {T_{\text{m}} - T_{\text{cc}} } \right)} \right]N_{\text{c}}$$
(7)
$$Q_{\text{hc}} = \left[ {\alpha I_{\text{h}} T_{\text{m}} - \frac{{I_{\text{h}}^{2} R}}{2} - k\left( {T_{\text{hh}} - T_{\text{m}} } \right)} \right]N_{\text{h}}$$
(8)
$$Q_{\text{hh}} = \left[ {\alpha I_{\text{h}} T_{\text{hh}} + \frac{{I_{\text{h}}^{2} R}}{2} - k\left( {T_{\text{hh}} - T_{\text{m}} } \right)} \right]N_{\text{h}}$$
(9)

COP of double-stage TER:

$$\left( {\text{COP}} \right)_{\text{act}} = \frac{{Q_{\text{cc}} }}{{W_{\text{hh}} + W_{\text{cc}} }} = \frac{{Q_{\text{cc}} }}{{\left( {Q_{\text{hh}} - Q_{\text{hc}} } \right) + \left( {Q_{\text{ch}} - Q_{\text{cc}} } \right)}} = \frac{{Q_{\text{cc}} }}{{\left( {Q_{\text{hh}} - Q_{\text{cc}} } \right)}}$$
(10)

Whh and Wcc are the work inputs in hot and cold sides, respectively. Since the two-stage TER is a combination of two single-stage TER’s which are thermally in series, so the total work input would be equal to the sum of the work input to individual stages. But Qcc would be the heat absorbed at the cold side plate only.

Nh and Nc are number of elements in hot and cold sides, respectively, and Nh + Nc = 50. There may be three possibilities to select Nh and Nc such that Nh > Nc, Nh < Nc, and Nh = Nc. A non-dimensional parameter ‘x’ has been used which is the ratio of number of elements in hot side and cold side in a double-stage TER.

The value of ‘x’ may vary from 0.0204 (Nh = 1 and Nc = 49) to 49 (Nh = 49 and Nc = 1). If the value of x is considered to be the maximum, i.e., 49, the performance of TER is as in Table 1.

Table 1 Variation in performance parameters of double-stage TER with I (x = 49)
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Figure 2 shows the variation in ROR for a double-stage TER with x = 49 with current. The peak value of ROR is 1.49 at I = 31 Å.

Fig. 2
figure 2

ROR of double-stage TER with x = 49

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Figure 3 shows the variation in COP for a double-stage TER with x = 49 with current. The peak value of COP is 0.042 at I = 7 Å.

Fig. 3
figure 3

COP of double-stage TER with x = 49

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The results represented in Table 2 show that COP is maximum with value 0.042 at current input as 7 Å with ROR as 0.354. The comparison of this result with the results at 31 Å shows that the maximum ROR is obtained at 31 Å, i.e., 1.49 but with a loss in COP. There is a significant improvement in ROR at 31 Å. So these results show that the same device should be used at these different values of current to obtain the best performance according to the application and desired effect.

Table 2 Double-stage TER with x = 49 for COP and ROR
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3 Conclusion

A two-stage thermoelectric refrigerator, with 49 elements on hotter side and only one element on colder side, has been analyzed. Rate of refrigeration and coefficient of performance have been calculated to measure the performance. As rate of refrigeration and coefficient of performance both are the desirable performance measuring parameters of a refrigerating device. This work shows the values of current input at which both the desirable parameters, rate of refrigeration and coefficient of performance have the maximum values, respectively. The results may be useful for a user to obtain the best performance according to the application and desired effect. Thermoelectric refrigerators are more useful in the applications where meticulous control of the temperature is needed. So this work may be extended to design a thermoelectric refrigerator for such applications.