Specific heating load of thermoelectric heat pumps

Energy Convers.Mgmt Vol. 35, No. 6, pp. 459-464, 1994

Pergamon

0196-41904(93)E0017-F

SPECIFIC

HEATING

Elsevier Science Ltd. Prinl~d in Great Britain 0196-8904/94 $7.00+ 0.00

LOAD OF THERMOELECTRIC HEAT PUMPS

C. WU and W. SCHULDEN Department of Mechanical Engineering, U.S. Naval Academy, Annapolis, MD 21402-5000, U.S.A.

(Received 8 January 1993; receivedfor publication 30 November 1993) Abstract--A mathematical model for calculating the performance of a thermoelectric heat pump to achieve a specific heating load is presented. The effects of Joulean heating and thermal conduction are considered. An elaborate heat transfer model is employed to relate the thermal resistances between the thermoelectric system and its surroundings. A specific heating load, heating load per unit of total heat exchanger surface area, is adopted as the objective function for the thermoelectric heat pump performance analysis in this paper. Thermoelectrics

Heat pump

Finite-time thermodynamics

INTRODUCTION

The thermoelectric heat pump phenomenon was discovered by Peltier in 1834. He noticed during his experiments that, when a small current was passed through the junction of two dissimilar wires, the junction was cooled. This is called the Peltier effect, and it forms the basis for thermoelectric heat pumps. From the time of Peltier's basic discovery to about 1950, the technology of thermoelectric heat pumps was primitive, mainly because suitable materials were not available. However, since 1950, the rate of discovery of new materials and th.e rate of development of techniques to make excellent semiconductor materials have been extremely rapid. In recent years, numerous papers have described the great strides accomplished in theoretical studies and advances in material technology. Currently, thermoelectric heat pumps have found many interesting applications. They are available in the market and are preferred in some applications, because of their small size, simplicity, quietness and reliability. A practical thermoelectric heat pump, in its simplest form, contains semiconductor materials, a heat source and a heat sink as shown in Fig. 1. The formulation and analysis of the thermoelectric heat pump in textbooks, e.g. by Angrist [1], assumes an externally reversible system, ignoring the heat transfer resistances between the thermoelectric system and its surroundings. Unfortunately, the externally reversible thermoelectric heat pump can only take place in the limit of an infinitely large heat exchanger surface area. No practicating engineer wants to design a heat pump with a huge heat exchanger to add a finite amount of heating load to the heat sink. Therefore, real thermoelectric heat pumps seldom attain the level of performance of externally reversible thermoelectric devices. Industrial heat pump optimization usually takes the form of determining a minimum heat exchanger area per unit heating load or a minimum cost per unit heating load. A specific heating load, heating load per unit total heat exchanger surface area, is, therefore, adopted to be the objective function for heat pump optimization in this paper. The conventional exoreversible and real thermoelectric heat pump systems are described, analyzed and compared in the following sections. ANALYSIS

AND OPTIMIZATION

The analysis of the performance of a thermoelectric heat pump is considered with reference to Figs 1 and 2 and the following assumptions: (1) The Seebeck coefficient, electric resistivity and thermal connductivity of the thermoelectric elements are independent of temperature; 459

460

WU

and

SCHULDEN:

THERMOELECTRIC HEAT PUMPS

(2) The geometrical sizes of the thermoelectric elements are fixed; (3) The thermoelectric elements are of uniform cross-section and perfectly insulated on their sides; (4) The connecting joint straps are perfect conductors of electricity with negligible thermal capacity; (5) The temperature gradients from hot to cold junctions are linear; (6) The overall heat transfer coefficients of the heat exchangers between the heat pump and its surroundings are fixed. These assumptions usually are not significant limitations, particularly under static conditions. Consider the thermoelectric heat pump system to include the cold and hot junctions, two thermoelectric n- and p-semiconductor legs, and the heat source and heat sink reservoirs. Referring to Fig. 1, heat is pumped from the heat source reservoir at TL to the cold junction at Tc, along the two legs to the hot junction at Tw, and then to the heat sink reservoir at 7".. For steady-state conditions, the rates of heat transfer from the source to the cold junction (QL),

sink TH

Heat

QHI

TH,= TW Hot junction

Tw

P "--'-t

N

Tc

Cold junction !'---

Heat source

TL

P (Power input) Fig. I. Externally reversible and internally irreversible (exoreversible)-thermoelectric heat pump.

WU and SCHULDEN: THERMOELECTRIC HEAT PUMPS

461

Heat sink TH

HI''OH'H Hot junction

Tw N

Cold junction To

i

"

.AL

---t

Heat source

TL

P (Power input) Fig. 2. Externally and internally irreversible (real)-thermoeloctric heat pump.

to the thermoelectric heat pump at the cold junction (~L), from the thermoelectric heat pump at the hot junction (Q.) and from the hot junction to the heat sink ((~8) are defined by the following relations ~L---- U L A L ( T L - Tc)

~L

~"

a T c I - 0.5I~R - K ( r w -

0.8 = aTw/

+ 0.5I:R

- K(T w -

~8 = UHAH(Tw-

1".)

(1) Tc)

(2)

rc)

(3) (4)

where UL, U8 = the overall heat transfer coefficients of the heat exchangers between the thermoelectric heat pump and its heat source and heat sink reservoirs, respectively,

462

w u and SCHULDEN: THERMOELECTRIC HEAT PUMPS A L , A H -----the

heat transfer areas of the heat exchangers between the thermoelectric heat pump and its heat source and heat sink reservoirs, respectively, a, R, K = the Seebeck coefficient, internal electrical resistance and thermal conductance of the semiconductor legs, respectively, I2R = the rate of heat generation in the two semiconductor legs caused by the Joule effect of which one-half flows to the cold junction and another half flows to the hot junction, K ( T w - Tc) = the rate of heat transfer from the hot junction along the legs to the cold junction, a T l = the rate of heat absorption at the hot and cold junctions caused by the Peltier effect. The Seebeck coefficient, internal electrical resistance and thermal conductance of the semiconductor legs are given by a = l a . l + [ap l

(5)

R = r . l . / A . + rplp/A

(6)

K = k . A . / l . + kpAp/lp

(7)

where a., ap = the Seebeck coefficients of the n- and p-semiconductor legs, respectively, l., !e = the lengths of the n- and p-semiconductor legs, respectively, A., Ap = the cross-section areas of the n- and p-semiconductor legs, respectively, r., rp = the electrical resistivities of the n- and p-semiconductor legs, respectively, k., kp = the thermal conductivities of the n- and p-semiconductor legs, respectively. The electrical power input (P), voltage (V) and coefficient of performance (B) of the thermoelectrical heat pump are given by the equations P = F R + a I ( T w - Tc)

(8)

V = I R + a ( T w - Tc)

(9)

(lo)

B = O.le.

Increasing efforts are now being made worldwide in industry to apply thermoelectric heat pumps. One critical parameter of the heat pump is the specific heating load. The specific heating load, t)H, is defined as the heating load per unit total heat transfer surface area of the heat exchangers between the heat pump and its surroundings, i.e.

(11)

~I. = Q . / ( A . + AL).

The following discussion considers the performance of two classes of thermoelectric heat pumps. These are: (1) an exoreversible thermoelectric heat pump defined as an externally reversible and internally irreversible system, and (2) a real thermoelectric heat pump defined as an externally and internally irreversible system. Table I. Performancedata of thermoelectricheat pumps Endoreversible 273 Heat sink temperature (K), Ta 288 Heat source temperature (K), TL 273 Hot junction temperature (K), Tw 288 Cold junction temperature (K), Tc 1 Overall heat transfer coefficient(W/cm2(K)), U8 1 Overall heat transfer coefficient(W/cm2(K)), UL 2 Power input (W), P 32.8 Current (As), 1 oo Heat exchanger surface area (cm2),An oo Heat exchanger surface area (cm2), AL 4.83 Heating load (W), Q~ 2.42 Coefficientof performance, B 0 Specific heating load (W/cm2), qa

Real 273 288 278 285 1 I

2 32.1 I 1

4.57 2.28 2.28

WU and SCHULDEN:

THERMOELECTRIC

HEAT PUMPS

463

( A ) Exoreversible thermoelectric heat p u m p (Fig. 1)

It is known that a thermoelectric heat pump operating between two thermal reservoirs at different temperatures (TH and TL) displays its highest coefficient of performance under externally reversible conditions (i.e. Tn = Tw and Tc = TL). The classical approach to analyzing the performance of a thermoelectric heat pump is to assume an exoreversible system. In this approach, equations (2), (3), (8) and (9) become O-L = a T L I -- 0.512R -- K(TH - TL)

(12)

QH = a T H I + 0.512R - K ( T a - TL)

(13)

P = I2R + aI(TH -- TL)

(14)

V = I R + a(TH -- TL).

(15)

In addition, to transfer a finite amount of heat QH from the hot junction at Ta to the heat sink also at Ta, an infinitely large heat exchanger surface area (AH = ~ ) is required. Similarly, to transfer a finite amount of heat OL from the heat source at TL to the cold junction also at TL requires an infinitely large heat exchanger surface area (AL = ~). Hence, applying equation (11), the specific heating load of the exoreversible thermoelectric heat pump is equal to zero. (B) R e a l thermoelectric heat p u m p (Fig. 2)

Although the exoreversible thermoelectric heat pump displays the highest coefficient of performance, it does not produce any specific heating load. Real thermoelectric heat pumps are designed to deliver a specific heating load. Therefore, the coefficient of performance of the exoreversible thermoelectric heat pump is a useless measure of merit to practical engineers. It only gives an upper bound that is too high to reach for any real thermoelectric heat pump. Hence, there is a need to find the coefficient of performance and specific heating load for a real thermoelectric heat pump. Consider the input power (P) to a thermoelectric heat pump to be fixed. Combining equations (1) and (2) gives Tw = Tc + (aTc l - 0.512R - UL AL ( TL -- T c ) ) / K .

(16)

Similarly, combining equations (3) and (4) yields Tc = T w ( a T w I + 0.512R - UHAH(Tw -- TH))/K.

(17)

Solving equations (16), (17) and (8) for the three unknowns Tw, Tc and I and substituting them into equations (3), (10) and (11), one is able to obtain the heating load (Qn), coefficient of performance (B) and specific heating load (0n) of a real thermoelectric heat pump. NUMERICAL EXAMPLE We now consider a set of numerical calculations for the performance characteristics of the exoreversible and real thermoelectric heat pumps. The thermoelectric couples of the heat pumps are made of n-type semiconductor (75% BizTe3 and 25% Bi2Se3) and p-type semiconductor (25% BizTe3, 75% Sb2Te3 and 1.75% excess Se) materials with the following average properties and dimensions per couple:

a r k 1 A

V/K ohm-cm W/cm(K) cm cm ~

n-type

p-type

- 1 6 5 × l0 -6 1.05 x 10 -3 0.013 1.0 1.0

+ 2 1 0 × 10 -6 0.98 x 10 -3 0.012 1.0 1.0

Applying equations (8)-(10) to calculate the Seebeck coefficient (a), the internal electrical resistance (R) and the thermal conductance (K) for the thermoelectric couple, we have a = 375

x 10 -6

V/K, R = 0.00203 ohms, K = 0.025 W/K.

464

WU and SCHULDEN: THERMOELECTRICHEAT PUMPS

The exoreversible and real thermoelectric heat pumps operate between an available heat source temperature of 288 K (TL) and heat sink temperature of 273 K (T.). The power input to the heat pumps is 2 W per couple, The overall heat transfer coefficients UM and UL are both 1 W/cm2(K), and for the real thermoelectric heat pump, the heat source and heat sink heat exchanger surface areas AM and AL, respectively, are both 1 cm 2. The calculations of all relevant performance characteristics of an exoreversible and a real thermoelectric heat pump are carried out, and the results are listed in Table 1 for comparison. CONCLUSION Thermoelectric heat pumps for heating, traditionally, have been analyzed assuming that the hot and cold junction temperatures are at the same temperatures of the sink and source reservoirs. However, it is a relatively poor guide to the coefficient of performance of the heat pump. Also, the externally reversible thermoelectric heat pump does not provide any specific heating load. This paper presents an externally irreversible thermoelectric heat pump model to account for the external irreversibility effect. A specific heating load is, therefore, defined and adopted as an objective function for the performance of the heat pump. This provides a much more realistic measure of performance than does the traditional exoreversible model. REFERENCE 1. S. W. Angrist, Direct Energy Conversion, 4th Edn. Allyn & Bacon, Boston, Mass. (1982).