The Iowtemperature heat pump is a simple and reliable device for attaining cryogenic temperatures. Heat is pumped from low temperature to ambient. The pressure on both sides of the displacer is the same except for a small differential pressure caused by the pressure drop produced when gas is flowing through the regenerator. Thus, no appreciable work is required to move the displacer and no net work is done by the displacer. This paper presents an exergy analysis of the device in order to estimate the various losses. The exergy balance reveals that the exergetic efficiency of the device is only 9.4% hence this approach may enable the designer to minimise the losses and make the unit highly reliable.

Exergy analysis of a low temperature heat pump

H.A. Rangrej and K.G. Narayankhedkar

Key words: heat pumps, exergy, cryogenic

Nomenclature

C constant

Cp specific heat at constant pressure

Cv specific heat at constant volume

e exergy

Ae change in exergy, exergy utilised

e loss of exergy

m mass

PH high pressure

PL low pressure

Qr refrigerating effect

Qnet net refrigerating effect

R gas constant

T temperature

Tc cold end temperature

To environment temperature, dead state temperature

U internal energy

V Volume of expansion space

VR regenerator void volume

r/c, ts isothermal compression efficiency

3' ratio of specific heats

Gifford and McMahon reported a low temperature heat pump 1 as a simple and reliable cryogenic refrigerator. The device consists of a thin-walled stainless steel cylinder into which a displacer is fitted in the form of a fairly closely fitting piston which is free to move a limited distance lengthwise. Fig. 1 shows a schematic diagram of this device. Both the cylinder and the displacer are poor conductors of heat and they are subjected to large longitudinal tem- perature gradient. The variable volumes at the two ends of the cylinder are connected through a regenerator. The volume available at one end is warm and the other cold they will be referred to as Volume 1 and Volume 2 respectively. The cycle of operation consists of four processes, namely pressure build up, intake, pressure release and exhaust.

Pressure build up. With the displacer at the bottom of the cylinder and the exhaust valve dosed, the intake valve is opened, building pressure from PL to PH in volume 1 and the regenerator. Volume 2 is negligibly small at this time.

Intake. With the intake valve open, the displacer is moved from the bottom to the top of the cylinder, displacing

the high pressure gas from Volume 1 to Volume 2. Due to lowering the gas temperature, additional gas enters through the intake valve to maintain the pressure during this part of the stroke.

x Intake

Volume I --Gas seal i ~ PH1 ~Cylinder

Displacer PL

- - Volume 2

valve Exhaust ,~ valve |

t Regeneratr

Load

Fig. 1 Gifford McMahon low temperature heat pump 0011-2275/83/030148-03 $03.00 1983 Butterworth & Co (Publishers) Ltd

Vs

Volume 2

148 CRYOGENICS. MARCH 1983

Pressure release. With the displacer at the top of the cylinder, the intake valve is closed and the exhaust valve opened. The pressure drops from PH to PL.

Exhaust. After the pressure has dropped to PL, the displacer is moved to the bottom of the cylinder displacing the gas from Volume 2 to Volume 1.

Fig. 2 shows the temperature history during the cycle. Due to pressure build-up, the gas temperature increases to T~" in Volume 1 which then mixes with additional gas from the intake valve at T~' before it flows to the top of regene- rator. The mixed gas has an intermediate temperature between that of the feed gas from the intake valve and that from Volume 1 (T~). The final exhaust temperature from the regenerator is T 4 which is very close to the temperature of the mixture that entered. Thus the gas leaving the system is hotter than the intake feed gas. Heat removed in this way is equal to the total net refrigeration achieved. The result is that heat is pumped out at a low temperature and fed into the exhaust gas stream above room temperature. Fig. 2 shows corresponding states on the T-s plane.

Exergy

Every body possesses energy, however, we utilise the quality of energy, exergy, and not the quantity. The exergy of a system is the maximum work that can be obtained from it in reducing its state to the dead state (state 0). The physical exergy per unit mass of the system in a flow process depends on the states of both the system and the dead state (state 0). The expression for exergy is as follows:

e = (h -Tos) - (h 0nToso)

= h - Tos + C (1)

For the purpose of calculations, one is concerned with the exergy differences. Thus for a constant dead state, the exergy difference between states 1 and 2 is given by

e = (h i -Tos l ) - (h2 -Tos2)

= e l - e2 (2 )

Analysis

For the purpose of analysis, temperatures at various states as shown in Figs 2 and 3 should be known. The process of pressure build-up is similar to that of filling the tank. The gas temperature at the end of pressure build-up

77" r,

Fig. 2

~o Regenerator _ _ _ ~ AT

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

~ ~ Net --J -~ "~ I X I S -- ~1 / refrigeration E "~ ~ o

Z > >~ ~ So

-,-o ~ g -- Jn - - - ~- ~ 'er~- tor -~ ~,~E__ in - - - e. ~ ~ "B g[ . . . . . . I regenerator E ~ ~ ~ I I

Temperature history of low temperature heat pump cycle

2

/ i4 ,, //O

Fig. 3 T- s diagram

can be determined by applyingfirst law of thermodynamics. The following assumptions are made:

1. The process of pressure build-up is adiabatic 2. Kinetic and potential energy terms are negligibly

small 3. The gas used is an ideal gas with constant values of

Cp and Cv 4. Mass of gas in the regenerator void volume is

evaluated at mean temperature

An increase in the internal energy (AU) of the system consisting of Vol. 1 can be expressed as

Am i h i = AU (3)

where ~g/ i is the differential mass entering Vol. 1 and h i the enthalpy of the mass entering Vol. 1. If T 1 " and TI ' represent final and initial temperatures of gas in Vol. 1 and T1 temperature of gas after mixing, (3) can be expressed as:

TI" = PH [[ PH -PL)/'~TI'I + eL/T1] -1 (4)

With knowledge of TI', TI" can be evaluated by trial and error assuming a certain value of T1 and checking back for this temperature after mixing.

Similarly, application of first law of thermodynamics to Vol. 2 yields an expression for the refrigerating effect, Qr as follows:

Qr = (PH -eL ) V (5)

where V is the volume of expansion space. Due to regenerator ineffectiveness, part of the refrigerating effect is consumed in cooling the gas from T2' to 7"2 (Fig. 3).

CRYOGENICS . MARCH 1983 149

Thus the net refrigerating effect available can be expressed as:

Qnet = (PH -PL ) V-mCp (T2 ' - T2) (6)

where m is the mass of gas admitted to the expansion space per cycle. Equation (6) accounts for the exergy utilised. The difference between exergy supplied and exergy utilised is to be accounted for in the losses in various components such as compressor, regenerator, expansion space, mixing, low pressure line etc.

The exergy analysis has been carried out with the data of Table 2. With the help of (4), TI" and T1 have been calculated by trial and error procedure. These values are 406.15 K and 330 K respectively. Table 1 represents the exergy values at various states. Using (6), the net refrigerating effect is 0.015636 kJ cycle -1 . Thus, exergy utilised is equal to

Ae = Qnet (To - Tc)lTc

= 0.0208 kJ cycle -1 (7)

Exergy supplied is equal to m (e'l - eo). The mass com- pressed by the compressor per cycle, m can be expressed as:

m = PH VR R 2

+ I-PH V2 /RTc - P2 Vi /R T41 (8)

Thus, the exergy supplied is equal to 0.13309 kJ cycle -~ . Actual work supplied after accounting for isothermal com- pression efficiency works at 0.2218 kJ cycle -1 .

The loss of exergy in various components can be computed as follows:

1 .V ecomp = Actual work - Ideal work = 0.0887 kJ cyclC 1

2.V ereg = m 1 e 1 + m 3 e 3 - m 4 e 4 - m2'e2 ~

2(Pr i -eL ) VR PH V2 where , m 1 = m4 = +

TH+Tc RT2

m3 = m2' - PH V2 RT~

Table 1, Exergy values at various states where To = 300 K

Exergy Temperature Entropy Enthalpy e,(17-Tos)

State 7-, K s, k Jkg- lK -1 h, kJkg -1 kJkg -1

1" 406.15 3.7082 1306.15 193.68 1 330 3.5013 1230.00 179.60 2' 133 2.5918 1033.00 255.44 2 130 2.5~1 1030.00 258.57 3 130 3.1637 1030.00 80.89 4 327 4.088 1227.00 00.63

Table 2. Exergy balance

Percentage of Item kJ cycle -1 exergy supplied

Exergy loss in

a - compressor 0.08870 40.00 b - regenerator 0.03038 13.70 c - cold end volume 0.07150 32.20 d - mixing 0.00170 00.76 e - low pressure line 0.00042 00.18

Exergy utilised 0.02080 9.40

Unaccounted loss 0.00830 3.75

Data: PH = 8.16 bar, PL = 1.02 bar, V 1 = V 2 = 25 cm 3, V R = 0.25 V1, T c = 130 K, T R = 133 K, T O = T 1' = 300 K, rtc, is = 0.6. Exergy supplied = 0.2218 kJ cycle -1

Thus, Vereg = 0.03038 kJ cyclC 1

3. Exergy loss in cold end volume, ev2 can be determined from the balance,

Vev2 = m2 e2 + &e - m3 ea = 0.0715 kJ cycle -I

4. Exergy loss in mixing

Vemix = ml 'e i " + ml 'e l ' - mz el =

0.0017 kJ cycle -1

5. Exergy loss in low pressure line

VeLe = m (e4 -eo) = 0.00042 kJ cyclC 1

Table 2 summarises the exergy balance.

Conclusion

From Table 2, it is seen that the major losses occur in the compressor, regenerator and cold end volume. The losses in the compressor and regenerator depend on the efficiency achieved in practice, while the losses in the cold end volume are due to the isothermal process. The exergetic efficiency is only 9.4%. Although, for miniature units reliability is much more important than efficiency, this analysis gives an estimate of various losses and thus enables the designer to attempt to minimise these losses in addition to making the unit highly reliable.

Authors

The authors are at the Department of Mechanical Engineering, Indian Institute of Technology, Bombay, India. Paper received 9 July 1982.

References

Gifford, W.E., McMahon, H.O. Low temperature heat pump, Proc Tenth Intern Congress of Refrigeration, Copenhagen, 1 (1960) 100-105

150 CRYOGENICS. MARCH 1983